RF and Microwave CoupledLine Circuits Second Edition
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RF and Microwave CoupledLine Circuits Second Edition R. K. Mongia I. J. Bahl P. Bhartia J. Hong
Library of Congress CataloginginPublication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN13: 9781596931565 Cover design by Yekaterina Ratner 2007 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 Sonnet Lite and Sonnet are trademarks of Sonnet Software, Inc., Syracuse, New York. Mathcad is a trademark of Mathsoft, Inc., Needham, Massachusetts. MATLAB is a trademark of The MathWorks, Inc., Natick, Massachusetts. All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. 10 9 8 7 6 5 4 3 2 1
In memory of Dr. K. C. Gupta—a friend, colleague, and mentor
Contents
Foreword to the First Edition
xv
Preface to the Second Edition
xvii
Preface to the First Edition
xxi
CHAPTER 1 Introduction
1
1.1 Coupled Structures 1.1.1 Types of Coupled Structures 1.1.2 Coupling Mechanism 1.2 Components Based on Coupled Structures 1.2.1 Directional Couplers 1.2.2 Filters 1.3 Applications 1.4 Scope of the Book References
1 3 4 6 6 8 11 13 13
CHAPTER 2 Microwave Network Theory
17
2.1 Actual and Equivalent Voltages and Currents 2.1.1 Normalized Voltages and Currents 2.1.2 Unnormalized Voltages and Currents 2.1.3 Reflection Coefficient, VSWR, and Input Impedance 2.1.4 Quantities Required to Describe the State of a Transmission Line 2.2 Impedance and Admittance Matrix Representation of a Network 2.2.1 Impedance Matrix 2.2.2 Admittance Matrix 2.2.3 Properties of Impedance and Admittance Parameters of a Passive Network 2.3 Scattering Matrix 2.3.1 Unitary Property 2.3.2 Transformation with Change in Position of Terminal Planes
17 18 21 22 24 26 26 27 27 28 30 31
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2.4
2.5
2.6 2.7
2.3.3 Reciprocal Networks 2.3.4 Relationship Between Normalized and Unnormalized Matrices Special Properties of Two, Three, and FourPort Passive, Lossless Networks 2.4.1 TwoPort Networks 2.4.2 ThreePort Reciprocal Networks 2.4.3 ThreePort Nonreciprocal Networks 2.4.4 FourPort Reciprocal Networks Special Representation of TwoPort Networks 2.5.1 ABCD Parameters 2.5.2 Reflection and Transmission Coefficients in Terms of ABCD Parameters 2.5.3 Equivalent T and ⌸ Networks of TwoPort Circuits Conversion Relations Scattering Matrix of Interconnected Networks 2.7.1 Scattering Parameters of Reduced Networks 2.7.2 Reduction of a ThreePort Network into a TwoPort Network 2.7.3 Reduction of a TwoPort Network into a OnePort Network 2.7.4 Reduction of a FourPort Network into a TwoPort Network References
CHAPTER 3 Characteristics of Planar Transmission Lines 3.1 General Characteristics of TEM and QuasiTEM Modes 3.1.1 TEM Modes 3.1.2 QuasiTEM Modes 3.1.3 Skin Depth and Surface Impedance of Imperfect Conductors 3.1.4 Conductor Loss of TEM and QuasiTEM Modes 3.2 Representation of Capacitances of Coupled Lines 3.2.1 Even and OddMode Capacitances of Symmetrical Coupled Lines 3.2.2 ParallelPlate and Fringing Capacitances of Single and Coupled Planar Transmission Lines 3.3 Characteristics of Single and Coupled Striplines 3.3.1 Single Stripline 3.3.2 EdgeCoupled Striplines 3.4 Characteristics of Single and Coupled Microstrip Lines 3.4.1 Single Microstrip 3.4.2 Coupled Microstrip Lines 3.5 Single and Coupled Coplanar Waveguides 3.5.1 Coplanar Waveguide 3.5.2 Coplanar Waveguide with Upper Shielding 3.5.3 ConductorBacked Coplanar Waveguide with Upper Shielding 3.5.4 Coupled Coplanar Waveguides 3.6 Suspended and Inverted Microstrip Lines
32 32 32 33 34 35 36 38 38 40 41 43 45 47 48 49 50 51
53 53 57 58 59 60 61 62 66 68 69 73 74 75 81 83 84 86 87 88 88
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3.7 BroadsideCoupled Lines 3.7.1 BroadsideCoupled Striplines 3.7.2 BroadsideCoupled Suspended Microstrip Lines 3.7.3 BroadsideCoupled Offset Striplines 3.8 SlotCoupled Microstrip Lines References
93 94 95 96 99 102
CHAPTER 4 Analysis of Uniformly Coupled Lines
105
4.1 Even and OddMode Analysis of Symmetrical Networks 106 4.1.1 EvenMode Excitation 108 4.1.2 OddMode Excitation 109 4.2 Directional Couplers Using Uniform Coupled Lines 111 4.2.1 ForwardWave (or Codirectional) Directional Couplers 114 4.2.2 BackwardWave Directional Couplers 116 4.3 Uniformly Coupled Asymmetrical Lines 120 4.3.1 Parameters of Asymmetrical Coupled Lines 121 4.3.2 Distributed Equivalent Circuit of Coupled Lines 126 4.3.3 Relation Between Normal Mode (c and ) and Distributed Line Parameters 130 4.3.4 Approximate Distributed Line or NormalMode Parameters of Asymmetrical Coupled Lines 132 4.4 Directional Couplers Using Asymmetrical Coupled Lines 133 4.4.1 ForwardWave Directional Couplers 133 4.4.2 BackwardWave Directional Couplers 136 4.5 Design of Multilayer Couplers 138 4.5.1 Determination of Capacitance and Inductance Parameters Using Sonnet Lite 139 4.5.2 Coupler Design 140 References 147 CHAPTER 5 Broadband ForwardWave Directional Couplers
149
5.1 ForwardWave Directional Couplers 5.1.1 3dB Coupler Using Symmetrical Microstrip Lines 5.1.2 Design and Performance of 3dB Asymmetrical Couplers 5.1.3 UltraBroadband ForwardWave Directional Couplers 5.2 CoupledMode Theory 5.2.1 Nature of Coupling Coefficient K 12 and K 21 5.2.2 Waves on Lines 1 and 2 in the Presence of Coupling 5.2.3 CoupledMode Theory and Even and OddMode Analysis 5.2.4 Coupling Between Asymmetrical Lines 5.3 CoupledMode Theory for Weakly Coupled Resonators References
150 151 153 155 156 158 158 160 161 163 165
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CHAPTER 6 ParallelCoupled TEM Directional Couplers
167
6.1 Coupler Parameters 6.2 SingleSection Directional Coupler 6.2.1 Frequency Response 6.2.2 Design 6.2.3 Compact Couplers 6.2.4 Equivalent Circuit of a QuarterWave Coupler 6.3 Multisection Directional Couplers 6.3.1 Theory and Synthesis 6.3.2 Limitations of Multisection Couplers 6.4 Techniques to Improve Directivity of Microstrip Couplers 6.4.1 Lumped Compensation 6.4.2 Use of Dielectric Overlays 6.4.3 Use of Wiggly Lines 6.4.4 Other Techniques References
167 169 169 171 176 176 177 177 184 186 186 189 189 192 194
CHAPTER 7 Nonuniform Broadband TEM Directional Couplers
197
7.1 Symmetrical Couplers 7.1.1 Coupling in Terms of EvenMode Characteristic Impedance 7.1.2 Synthesis 7.1.3 Technique for Determining Weighting Functions 7.1.4 Electrical and Physical Length of a Coupler 7.1.5 Design Procedure 7.2 Asymmetrical Couplers References
197 199 201 206 209 210 214 217
CHAPTER 8 Tight Couplers
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8.1 Introduction 8.2 BranchLine Couplers 8.2.1 Modified BranchLine Coupler 8.2.2 ReducedSize BranchLine Coupler 8.2.3 LumpedElement BranchLine Coupler 8.2.4 Broadband BranchLine Coupler 8.3 RatRace Coupler 8.3.1 Modified RatRace Coupler 8.3.2 ReducedSize RatRace Coupler 8.3.3 LumpedElement RatRace Coupler 8.4 Multiconductor Directional Couplers 8.4.1 Theory of Interdigital Couplers 8.4.2 Design of Interdigital Couplers 8.5 Tandem Couplers
219 220 223 225 230 233 233 237 239 240 242 243 245 252
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8.6 Multilayer Tight Couplers 8.6.1 Broadside Couplers 8.6.2 ReEntrant Mode Couplers 8.7 Compact Couplers 8.7.1 LumpedElement Couplers 8.7.2 Spiral Directional Couplers 8.7.3 Meander Line Directional Coupler 8.8 Other Tight Couplers References
255 255 258 261 262 262 263 264 265
CHAPTER 9 CoupledLine Filter Fundamentals
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9.1 Introduction 9.1.1 Types of Filters 9.1.2 Applications 9.2 Theory and Design of Filters 9.2.1 Maximally Flat or Butterworth Prototype 9.2.2 Chebyshev Response 9.2.3 Other ResponseType Filters 9.2.4 LC Filter Transformation 9.2.5 Filter Analysis and CAD Methods 9.2.6 Some Practical Considerations 9.3 ParallelCoupled Line Filters 9.3.1 Design Example 9.4 Interdigital Filters 9.4.1 Design Examples 9.5 Combline Filters 9.5.1 Design Example 9.6 The HairpinLine Filter 9.6.1 Design Example 9.7 ParallelCoupled Bandstop Filter 9.7.1 Design Example References
269 270 270 271 272 272 275 275 279 280 283 285 287 287 290 291 295 297 300 301 304
CHAPTER 10 Advanced CoupledLine Filters
307
10.1 Introduction 10.2 CoupledLine Filters with Enhanced Stopband Performance 10.2.1 Design Using UnevenlyCoupled Stages 10.2.2 Design Using Periodically Nonuniform Coupled Lines 10.2.3 Design Using Meandered ParallelCoupled Lines 10.2.4 Design Using Defected Ground Structures 10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics 10.3.1 Filters with CrossCoupled Resonators 10.3.2 Filters with SourceLoad Coupling 10.3.3 Filters with Asymmetric Port Excitations
307 307 307 314 320 324 327 327 335 347
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10.4 Interdigital Filters Using Stepped Impedance Resonators 10.4.1 Narrowband Design 10.4.2 Wideband Design 10.5 DualBand Filters References
352 354 356 359 367
CHAPTER 11 Filters Using Advanced Materials and Technologies
371
11.1 Introduction 11.2 Superconductor CoupledLine Filters 11.2.1 Cascaded Quadruplet and Triplet Filters 11.2.2 HighOrder Selective Filters with GroupDelay Equalization 11.3 Micromachined Filters 11.3.1 Miniature Interdigital Filters on Silicon 11.3.2 Overlay Coupled CPW Filters 11.4 Filters Using Advanced Dielectric Materials 11.4.1 LowTemperature Cofired Ceramic Filters 11.4.2 Liquid Crystal Polymer Filters 11.5 Filters for UltraWideband (UWB) Technology 11.5.1 Optimum Stub Line Filters 11.5.2 Multimode CoupledLine Filters 11.5.3 MicrostripCoplanar Waveguide CoupledLine Filters 11.5.4 UWB Filters with Notch Band 11.6 Metamaterial Filters References
371 371 371 377 385 387 390 391 392 396 400 401 406 410 421 428 438
CHAPTER 12 CoupledLine Circuit Components
443
12.1 DC Blocks 12.1.1 Analysis 12.1.2 Broadband DC Block 12.1.3 Biasing Circuits 12.1.4 MillimeterWave DC Block 12.1.5 HighVoltage DC Block 12.2 CoupledLine Transformers 12.2.1 OpenCircuit CoupledLine Transformers 12.2.2 Transmission Line Transformers 12.3 Interdigital Capacitor 12.3.1 Approximate Analysis 12.3.2 FullWave Analysis 12.4 Spiral Inductors 12.5 Spiral Transformers 12.6 Other CoupledLine Components References
443 443 446 446 449 451 452 452 456 461 462 464 465 472 475 476
Contents
CHAPTER 13 Baluns
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13.1 13.2 13.3 13.4
Introduction MicrostriptoBalanced Stripline Balun Analysis of a CoupledLine Balun Planar Transmission Line Baluns 13.4.1 Analysis 13.4.2 Examples 13.5 Marchand Balun 13.5.1 Coaxial Marchand Balun 13.5.2 Synthesis of Marchand Balun 13.5.3 Examples of Marchand Baluns 13.6 Other Baluns 13.6.1 Coplanar Waveguide Baluns 13.6.2 Triformer Balun 13.6.3 PlanarTransformer Balun References
481 482 486 490 493 496 498 501 505 509 518 518 518 519 524
About the Authors
529
Index
531
Foreword to the First Edition It has been a privilege for me to go through the manuscript of the RF and Microwave CoupledLine Circuits. Sections of coupled transmission structures are critical components in distributed RF and microwave passive circuits. Significance of their role as basic building blocks is second only to the sections of single transmission structures. Their applications in design of directional couplers and filters are well known, but equally important is the role played by coupledline sections in the design of baluns, capacitors, inductors, transformers and dc blocks. Availability of high dielectric constant materials has extended the usage of coupledline sections to lower microwave and RF frequencies. Traditionally, coupled sections consisting of two single lines have been used extensively. However, as the circuit designers understand modeling and characterization of multiple coupled lines, we can look forward to significantly larger applications of multiple coupled transmission line structures. Threeline balun structures reported recently are a step in this direction. Also, as the multilayer RF and microwave circuits become more popular, couplings among the transmission lines at different levels of a multilayer structure become a critical design consideration. In some cases this multilayermulticonductor coupling can be advantageous as a useful circuit component, while in other cases this coupling can become an undesirable effect that should be mitigated. Both of these situations need the modeling and characterization of multiconductor transmission line structures. Recognizing the role of coupled lines, it is hard to comprehend why a comprehensive book on this topic has not been available so far. But then, someone has to be the leader. Bahl and Bhartia have a history of providing to the microwave design community several wellneeded ‘‘firsts,’’ and this book is their most recent contribution. Congratulations Rajesh, Inder, and Prakash on a book which I am confident will be very well received in the RF/microwave community. K. C. Gupta University of Colorado at Boulder November 1998
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Preface to the Second Edition
The first edition of RF and Microwave CoupledLine Circuits was published in 1999. While the fundamentals of coupled line circuits have not changed in the past 8 years, further innovations in coupled line filters and other applications have occurred with changes in technology and use of new fabrication processes, such as the use of low temperature cofired ceramic (LTCC) substrates. In this case for example, it is common to use multilayer structures, with 10–25 layers being quite common. Thus, multilayer coupling needs to be better understood and explained for realizing an optimum threedimensional design and structure that LTCC permits. Over the years the first edition of this book has been well received in the marketplace and has been used extensively in industry by microwave engineers. Practitioners have pointed out some errors that crept into print in the first edition and over the years have suggested topics that should be added for completeness, or deleted in some cases, as they were not very useful in practice. Coupled circuits are fundamental to the realization of a large number of microwave and RF circuits, which, in turn, are essential to the development of electronic warfare, radar, communication, and navigation systems, and hence support the need for comprehensive textbooks in this area. In the past few years, mainly driven by the desire towards miniaturization, many novel configurations of coupledline components, such as directional couples, filters, and baluns, have been reported. Most practicing microwave and RF engineers are not fully aware of the advancements in this area. In view of these concerns, the authors, encouraged by the publishers, felt that a revised version of RF and Microwave CoupledLine Circuits would be useful for the microwave community. To ensure a fresh look, Dr. JiaSheng Hong, an eminent specialist in the microwave filters area, was invited to become a coauthor for the second edition. Dr. Hong has made suggestions on modifications to the book and has contributed fully in the preparation of some chapters that are closer to his area of expertise. With the deletion of some chapters in their entirety and the addition of new chapters and new material in other chapters, overall the book remains the same size as the first edition. The first few chapters reflect only minor changes, as these incorporate the fundamentals of microwave transmission lines, networks, and coupled lines, which have not changed. Some additions and changes have been made to accommodate the multilayer design of coupled lines for the sake of having a selfcontained, complete text. Most of the major changes occur in the ‘‘Applications’’ part of the text (i.e., Chapter 8 onward). Thus, Chapter 9, on filters, includes
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the design of bandstop filters using coupled lines and a discussion of software packages used for filter design, together with their limitations and strengths. Chapters 10 and 11 are new, building on the discussion of filters in Chapter 9. These two chapters discuss advanced filter technology, and the design of filters using new materials and technologies. Chapter 10 concentrates on coupled line filters with many specialized characteristics that are often encountered in practice. This includes designs using unevenly coupled stages, nonuniform coupled lines, meandered parallel lines, and defected ground structures. A number of other currently important topics such as filters with sourceload coupling and asymmetric port excitation are also covered. Chapter 11 takes a different direction, tackling filters using advanced materials and technologies. These include superconductor coupledline filters, micromachined filters, miniature interdigital filters on silicon, LTCC filters which require multilayer coupling, liquid crystal polymer filters, and ultrawideband filters. These topics encompass the new direction and materials in filter technology that will replace the older thick film and organic substrate technology over the coming decade as demands and pressures for smaller, lighterweight, lowercost filters mount. Chapters 12 and 13 are essentially the former Chapters 10 and 11 with revisions as appropriate and the inclusion of new material to update the chapters and make them current. Thus, Chapter 12 discusses the design of common microwave components requiring coupled line technology. This includes dc blocks, transformers, interdigital capacitors, and spiral inductors. Chapter 13 covers baluns. Baluns in different configurations (e.g., microstrip to balanced stripline, planar transmission line, and Marchand type) are discussed in detail. The former Chapters 12 and 13 on highspeed circuit interconnects and multiconductor transmission lines have been deleted in their entirety. The main reason for this deletion is that although the material was relevant and useful, most engineers using the text did not feel that these chapters added much and preferred to see a greater expansion of the coverage on filters, which we have included. Recognizing the current reality that engineers use software packages for their design and no longer do hand calculations, we have included a short discussion in each chapter where possible about current software packages that allow one to design the circuit discussed in that chapter. As an example, in Chapter 9, Section 9.2.5 covers the current software packages that are available for designing the types of filters discussed, together with their strengths, limitations, and shortcomings. We expect this feature to be of significant interest to the design engineer. In all, there is about 35–40% new material in this second edition, though we have endeavored to keep the overall length the same as the first edition. In summary, this second edition includes one thoroughly revised and two new chapters by Dr. Hong, two less important chapters have been deleted, less important sections have been replaced by current topics, and the number of figures and their sizes have been reduced. The authors believe that this edition will be as well accepted by professors, researchers, practicing engineers, and students, as the first edition was. Overall, this edition is more comprehensive in that equations that are not too commonly
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used, as well as the lesserused tables, have been eliminated, and material that renders the text more understandable has been added. The preparation of any text such as this requires significant cooperation and coordination. The authors express their gratitude to colleagues from several organizations for assisting with this work and for providing permission for use of copyright figures and tables. We would specially like to thank Dr. Protap Pramanick for his suggestions and input. The help provided by Dr. James Rautio and other members of staff at Sonnet Software, Inc., is acknowledged. We appreciate the support of our work organizations and especially our families for their understanding, support, and encouragement and putting up with us during the writing phase. Last but not least, we appreciate the input from the reviewers and the support and cooperation that we received from the Artech House staff, in particular, Mark Walsh, Barbara Lovenvirth, and Rebecca Allendorf. R. K. Mongia I. J. Bahl P. Bhartia J. Hong May 2007
Preface to the First Edition There are a number of textbooks on microwave transmission lines. Recent ones include extensive information on the modern planar lines such as microstrip, slotlines, coplanar waveguides, and the like. At the next level of complexity are the various functional circuits such as couplers, hybrids, filters, and baluns, which use the elemental transmission line in different configurations to achieve the desired functionality and meet system performance requirements. Much of this functionality involves coupling between transmission lines, and extensive research has been conducted in the design and analysis of such structures. Initially, much of the literature was oriented to coaxial lines and waveguides. With the evolution and the popularity of planar transmission lines, however, it was felt desirable to put together all aspects of coupled circuits using these lines under one cover. Most current texts, we found, contained perhaps a chapter or two on some specific components, especially couplers and filters. This text attempts to treat the topic in its entirety, starting with the fundamental theory of coupled structures and the application of this to the design and analysis of specific components such as couplers, filters, baluns, and so forth. This treatment emphasizes planar transmission lines, the CAD tools available for the design of these structures, use of fullwave analyses and accurate semiempirical equations for component design, novel structures and configurations, and new applications. This book is primarily intended for design engineers and research and development specialists who are involved in the area of coupledline circuit design, analysis, development, and fabrication. The layout of the book facilitates its use as a text for a graduate course and for short courses on specific component design. The book is divided into 13 chapters. The first chapter introduces the reader to the topic and covers the nature of coupled structures, the importance of these structures in microwave circuits, and some applications. A good introduction to the principal components using coupled lines (i.e., directional couplers and filters) is also given. The second chapter establishes the basic circuit parameters and representation of microwave networks. Fundamental network analysis tools such as impedance and scattering matrix techniques are introduced together with the properties of two, three, and fourport networks. Relationships between the commonly used matrix representation forms such as ABCD, scattering, and impedance are established to permit the researcher or designer to work in the system of his or her preference. The fundamental building blocks for coupledline circuits (i.e., transmission lines) are covered in Chapter 3. In particular, the characteristics of the commonly
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used planar lines, such as microstrip, coplanar, and striplines are covered in detail. In addition, the characteristics of coupled lines in these configurations under different conditions such as broadside coupling, edge coupling, or, in the case of coplanar waveguides, coupling with shields present are discussed. Whereas Chapter 3 concentrates on characteristics of physical transmission lines, Chapter 4 presents the general analysis of uniformly coupled asymmetrical lines, including forward and backward couplers. These fundamentals permit a more indepth coverage of the coupling of uniform lines, which is covered in the next chapter. Even and oddmode analysis is covered together with an analysis of uniformly coupled asymmetrical lines. Forward and backward directional coupler design methods using the aforementioned techniques are also given in the chapter. The next few chapters are devoted to the design of various types of directional couplers. Many directional couplers by their very nature and design have a narrowband performance. In a number of applications, broadband performance is essential. The design and performance of forwarddirectional couplers using asymmetrical coupled lines are the subject of Chapter 5. Coupledmode theory, also discussed in this chapter, is very useful for the analysis of general weakly coupled systems. Parallelcoupled backward TEM directional couplers using a single section or multisections are discussed in Chapter 6, together with limitations and methods for improving the directivity of such couplers. This permits the reader to have a good understanding of how these circuits work and the methods, including lumpedelement compensation and dielectric overlay, that can be used to improve directivity. While we dealt with broadband couplers using multisection couplers in Chapter 6, one can also obtain this type of characteristic of performance using nonuniform lines. Additional flexibilities, and at the same time complexities, are introduced with this line, but in many cases it is essential to resort to this process because of physical or performance constraints imposed by the overall circuit design. In Chapter 7, the design and synthesis procedure for such couplers is outlined, together with some other techniques to obtain broadband performance. Finally, the last type of coupler that requires special treatment is the tight coupler. Tight couplers, as described in Chapter 8, can be designed and fabricated in a number of configurations. Some of the most prevalent forms are the branchline coupler, ratrace coupler, and lumpedelement coupler. These are fully covered in this chapter, together with a large number of other layouts including the multiconductor couplers and tandem couplers. A number of novel designs are also discussed, including the interdigital Lange couplers and compact couplers for wireless applications. The material provided gives the designer a good grasp of the principles and techniques involved in the design of these coupler types, together with the advantages and disadvantages of the specific couplers. This information and understanding is critical to the designer in assisting him or her to choose the appropriate coupler type to meet not only the electrical performance characteristics but also to meet any form, fit, and function requirements imposed. Besides the directional couplers covered in the previous chapters, perhaps the most commonly used form of coupled line circuits is the filter. This is covered extensively in Chapter 9, starting with a definition of filter parameters and leading on to filter synthesis, design, and realization. Modern miniature filters are discussed,
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as they are critical to the wireless communications area, and an assessment of the capabilities of a number of software packages available for filter design is provided. The next two chapters delve into a number of other commonly used coupled circuits. Chapter 10 covers the analysis and design of a number of dc blocks, coupledline transformers, interdigital capacitors, spiral inductors, and transformers, while Chpater 11 treats the design and analysis of baluns. In particular, the Marchand balun is discussed in detail, together with other types of baluns such as the coplanar waveguide balun, triformer balun, and planar transformer balun. Whereas the preceding chapters have used coupling as a means of achieving a specific function and performance, in many cases coupling is not desirable and can create problems. This is typically the electromagnetic compatability/electromagnetic interference problem that is encountered by any circuit designer. To try to cover the topic of coupled circuits in its entirety, we have included highspeed digital interconnections in Chapter 12 to bring about an awareness of the crosstalk problem and provide ways to mitigate this problem. Finally, many of the passive devices covered could use multiconductor lines for their design. The literature on this topic is very dispersed. In Chapter 13, we have provided the essential information for the designer to permit the use of multiconductor lines as the building block for the type of coupledline circuit one wished to design. As with any comprehensive treatment of a topic, one must draw upon the works of a large number of researchers and authors. We have taken special care to reproduce equations and diagrams and believe that this text is a valuable addition to the microwave circuit designer’s library. The preparation of this text has depended on a number of very supportive individuals and organizations. Naturally, the time spent during evenings and weekends comes at the expense of time with our families. For their support and understanding we are eternally grateful. The organizations that we work for have also supported this project in many ways and we wish to express our thanks to them. While always dangerous to mention specific names because some others will feel left out, we have no hesitation in recognizing the contributions and acknowledging with our thanks the assistance of Josie Dunn for typing the manuscript and Bob Gervais of the Defence Research Establishment, Ottawa, who devoted large blocks of time in preparing the illustrations. Part of the manuscript has been handled efficiently by Tanya Morrision of ITT GaAsTEK, Roanoke. The Artech House team did an excellent job on the final book. We would like to thank Mark Walsh, Barbara Lovenvirth, Hilary Sardella, Judi Stone, Steve Cartisano, and Elaine Donnelly for their patience, support, and cooperation. Finally, we want to thank the reviewers for their thoroughness and excellent suggestions for improving the text. R. K. Mongia I. J. Bahl P. Bhartia
CHAPTER 1
Introduction
In microwave circuits, transmission lines are normally used in two ways: (1) to carry information or energy from one point to another; and (2) as circuit elements for passive circuits like impedance transformers, filters, couplers, delay lines, resonators, and baluns. Passive elements in conventional microwave circuits consist mainly of the distributed type and employ transmission line sections and waveguides in different configurations, thereby achieving the desired functionality and meeting performance specifications. This functionality is largely achieved by the use of coupled transmission lines. In this chapter, we briefly describe the various types of coupledline structures, the coupling mechanism, and coupledline components and their applications.
1.1 Coupled Structures When two unshielded transmission lines (as shown in Figure 1.1) are placed in close proximity to each other, a fraction of the power present on the main line is coupled to the secondary line. The power coupled is a function of the physical dimensions of the structure, mode of propagation [TEM (transverse electromagnetic) or nonTEM], the frequency of operation, and the direction of propagation of the primary power. In these stuctures, there is a continuous coupling between the electromagnetic fields of the two lines. These parallel coupling lines are called edgecoupled structures. The structure shown in Figure 1.1(d) is an exception and is called a broadsidecoupled structure. Coupled lines can be of any form, depending on the application and generally consist of two transmission lines, but may contain more than two. The lines can be symmetrical (i.e., both conductors have the same dimensions) or asymmetrical. Both lines are placed in close proximity to each other so that the electromagnetic fields can interact. The separation between the lines may be either constant or variable. The closer the lines are placed together, the stronger the interaction that takes place. When one port is excited with a known signal, a part of this signal appears at other ports. This interaction effect known as desirable coupling is used to advantage in realizing several important microwave circuit functions, such as directional couplers, filters, and baluns, with the coupled line length usually being approximately a quarterwave long. In addition, in closely packed hybrid and monolithic integrated circuits, parasitic coupling can take place between the distributed matching elements or closely
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Figure 1.1
Coupled transmission lines: (a) coaxial lines, (b) striplines, (c) microstrip lines, and (d) broadside striplines.
spaced lumped elements, affecting the electrical performance of the circuit in several ways. It may change the frequency response in terms of frequency range and bandwidth and degrade the gain/insertion loss and its flatness, input and output voltage standing wave ratio (VSWR), and many other characteristics including output power, poweradded efficiency, and noise figure, depending on the type of circuit. This coupling can also result in the instability of an amplifier circuit or create a feedback resulting in a peak or a dip in the measured gain response or a substantial change in a phaseshifter response. In general, this parastitic coupling is undesirable and an impediment to obtaining an optimal solution in a circuit design. However, this coupling effect can be taken into account in the design phase by using empirical equations and by performing electromagnetic (EM) simulations, or it can be reduced to an acceptable level by maintaining a large separation between the matching elements. Multiconductor microstrip lines (Figure 1.2) are used in verylargescale integrated (VLSI) chips for digital circuit applications and threedimensional microwave integrated circuits. Here, numerous closely spaced interconnection lines in different planes are used to integrate the components on a chip. The design of these interconnections is very important to satisfy the size, power consumption, clock frequency, and propagation delay requirements. Signal distortion, increase in background noise, and crosstalk between the lines from coupling are some of the undesirable characteristics. Proper design of these interconnections can reduce the distortion
1.1 Coupled Structures
Figure 1.2
3
Crosssectional view of a multiconductor and multilevel coupledline configuration.
and crosstalk to acceptable levels and has played a significant role in the evolution of highspeed VLSI technology. 1.1.1 Types of Coupled Structures
Coupledline structures are available for all forms and types of transmission lines/ dielectric guides and waveguides. Striplines, microstrip lines, coplanar waveguides, image guides, and insular and inverted stripguides are the most popular planar forms. In Figure 1.3, cross sections of microstrip coupled lines and microstriplike lines are shown. In these structures, practical spacing limitations between the lines limits the tight coupling achievable to about 8 dB over /4 sections. These configurations are edgecoupled structures. On the other hand, broadsidecoupled lines (shown in Figure 1.4) are used extensively to realize tight couplings of the order of 2 to 3 dB. All three structures support TEM modes in the case of a homogeneous dielectric medium and quasiTEM modes in the case of a nonhomogeneous media. In Figure 1.5, we show coupled dielectric waveguides. They support nonTEM modes and forwardwave couplers are realized using these structures. These structures are commonly used at millimeterwave frequencies, and continuous coupling occurs from one guide to another when they are placed in close proximity to each other. The configurations shown in Figure 1.1, 1.3, 1.4, and 1.5 use equal widths for both conductors and guides and constant spacing between the conductors and
Figure 1.3
Coupled microstriplike transmission lines: (a) microstrip lines, (b) inverted microstrip lines, (c) suspended microstrip lines, and (d) coplanar waveguide.
4
Introduction
Figure 1.4
Figure 1.5
Coupled broadside transmission lines: (a) broadsidecoupled striplines, (b) broadsidecoupled inverted microstrip lines, and (c) broadsidecoupled suspended microstrip lines.
Coupled dielectric guides: (a) image, (b) insular, and (c) inverted strip.
guides. These structures are therefore called symmetric and uniformly coupled. Figure 1.6 shows an asymmetrically coupled microstrip line configuration with constant spacing between lines of unequal width. This structure is called a uniformly coupled asymmetric line. Figure 1.7 shows an example of a symmetric coupled line with variable spacing between the microstrip conductors, called a nonuniformly coupled symmetric line. 1.1.2 Coupling Mechanism
The symmetric coupledline structures, as shown in Figure 1.1, support two modes: even and odd. The interaction between these modes induces the coupling between the two transmission lines, and the properties of the symmetric coupled structures may be described in terms of a suitable linear combination of these modes. The field distributions for the even and odd modes on coupled microstrip lines are
Figure 1.6
Coupled microstrip lines with unequal impedances (asymmetric lines).
1.1 Coupled Structures
Figure 1.7
5
Nonuniformly coupled symmetric lines.
shown in Figure 1.8. In evenmode excitation, both microstrip conductors are at same potential while the odd mode delineates equal but of opposite polarity potentials with respect to the ground. The even and odd modes have different characteristic impedances, and their values become equal when the separation between the conductors is very large (lines are uncoupled). The evenmode characteristic impedance (Z 0e ) is the impedance from one line to the ground when both lines are driven inphase from equal sources of equal impedances and voltages. The oddmode characteristic impedance (Z 0o ) is defined as the impedance from one line to the ground when both lines are driven out of phase from equal sources of equal impedances and voltages.
Figure 1.8
Even and oddmode field configurations in coupled microstrip lines.
6
Introduction
The velocities of propagation of the even and odd modes are equal when the lines are embedded in a homogeneous dielectric medium (e.g., stripline). For transmission lines such as coupled microstrip lines, however, the dielectric medium is not homogeneous, and a part of the field extends into the air above the substrate, resulting in different propagation velocities for the two modes. Consequently, the effective dielectric constants (and the phase velocities) are different for the two modes. This nonsynchronous feature deteriorates the performance of circuits using these types of coupled lines. The voltage coupling coefficient of a coupling structure is generally expressed in terms of the even and oddmode characteristic impedances, effective dielectric constants, and coupled structure line length. For a quarterwave coupled section in a homogeneous dielectric medium, the coupling coefficient k is given by k=
Z 0e − Z 0o Z 0e + Z 0o
(1.1)
1.2 Components Based on Coupled Structures There are numerous microwave passive components realized using coupledline sections. They include directional couplers, filters, baluns, impedance transformer networks, resonators, inductors, interdigital capacitors, dc blocks, and others, of which directional couplers and filters are the most popular. A brief history of the latter is presented next.
1.2.1 Directional Couplers
Directional couplers perform numerous functions in microwave circuits and subsystems. They are used to sample power for temperature compensation and amplitude control and in power splitting and combining over an ultrabroadband frequency range. In balanced amplifiers they help obtain good input and output VSWRs. In balanced mixers and microwave instruments (including network analyzers) they help in sampling incident and reflected signals. They have matched characteristics at all four ports, making them ideal for insertion in a circuit or subsystem. A historical account of microwave directional couplers including an extensive reference list is given by Cohn and Levy [1]. The first directional coupler using a quarterwavelong twowire configuration was reported in 1922, and during the 1940s and 1950s, significant progress was made in waveguide couplers using apertures in the common wall. Directional couplers using planar TEM lines such as coupled striplines were developed in the mid1950s. Numerous papers were published in the 1950s and 1960s describing the theory, design, fabrication, and measured data for TEMline edgecoupled directional couplers and significant contributions were made in the development of planar couplers. These couplers can provide coupling in the 8 to 40dB range. Early work on these homogeneous couplers (single and multisections) can be found in [1–5]. These couplers are also known as backwardwave couplers because the coupled wave on the secondary
1.2 Components Based on Coupled Structures
7
line travels in the opposite direction compared with the incident wave on the primary line when excited with a microwave signal. In several applications, a tight coupler such as a 3dB coupler is required and the crosssection shown in Figure 1.1(b) is difficult to realize as the very tight spacings required are limited by current photoetch techniques. This problem is alleviated by using the threedielectriclayer broadsidecoupled striplines (including offset coupled strips [6]) and tandemconnection directional couplers [7] of Figure 1.9 or the vertically installed coupledline configuration of Figure 1.10. Multioctave bandwidth in the abovementioned couplers is realized by using multistages of equallength coupled sections. When these sections are joined, however, abrupt discontinuities in coupling and line widths occur resulting in coupling error and directivity degradation. Continuously tapered TEM couplers [8, 9] yield improved electrical performance including better bandwidth characteristics. After widespread use of microstrip lines in microwave circuits, attention turned to microstrip line couplers [10–19]. One of the driving forces for the development of microstrip couplers was the higher level of integration of microwave circuits on a single substrate, including both active and passive components. Because a
Figure 1.9
Figure 1.10
(a) Offset coupled striplines and (b) tandemconnection of directional couplers.
Vertically installed coupledline coupler.
8
Introduction
microstrip is inhomogeneous, the even and oddmode propagation velocities for a coupled pair of microstrip lines are not equal, resulting in poor directivity, which becomes worse as the coupling is decreased. For example, a 10% difference in phase velocities reduces the directivity of 10, 15, and 20dB couplers to 13, 8, and 2 dB, respectively, from the theoretically infinite value with equalphase velocities. Thus, the deterioration in directivity is higher for loose coupling and becomes worse with higher phase velocity differences. Therefore, couplers fabricated on lowdielectric constant substrates such as plastic (⑀ r = 2.5), have better directivity performance than those on alumina or GaAs. The directivity of stripline and homogeneous broadside directional couplers is much better than that of microstrip couplers. Figure 1.11 depicts several techniques for equalizing or compensating for the difference in the modal phase velocities. Of these ‘‘wiggly lines’’ [10], dielectric overlays [11, 12] and capacitive compensation methods [13, 14, 17] are the most commonly used. Tight microstrip couplers suffer from the same problem as their stripline counterparts; that is, requirements of impractically small spacing between the conductors. This problem was solved by Lange [20] in 1969 with his interdigital coupler (Figure 1.12) using four narrow strips. In this design, alternate strips in pairs are connected with wires or airbridges and the gaps, however small, are realizable. This coupler and its variations [20–25] are widely used in the microwave industry. Figure 1.13 shows other configurations for tight couplers, which include reentrant structures, asymmetric broadsidecoupled microstrip lines, and slotcoupled microstrip lines. 1.2.2 Filters
Next to directional couplers, filters are the most important passive components used in microwave subsystems and instruments. Most microwave systems consist of many active and passive components that are difficult to design and manufacture with precise frequency characteristics. In contrast, microwave passive filters can be designed and manufactured with remarkably predictable performance. Consequently, microwave systems are usually designed so that all of the troublesome components are relatively wide in frequency response with filters being incorporated to obtain the precise system frequency response. Because filters are the narrowest bandwidth components in the system, it is usually the filters that limit such system parameters as gain and group delay flatness over frequency. The first use of filters was reported in 1937. A historical survey of microwave filters including an extensive reference list is given by Levy and Cohn [26]. Pioneering work in coupledline filters using TEM striplines was performed in the 1950s, 1960s, and 1970s. The most popular filter configurations are parallelcoupledline [27, 28], interdigital [29, 30], combline [31, 32] and hairpinline [33, 34]. Most filters are of the bandpass or bandstop type. Coaxial interdigital and combline configurations are shown in Figure 1.14, while stripline coupledline configurations are shown in Figure 1.15. Microstrip coupled line filters are similar to those shown in Figure 1.15. Because microstrip is a nonhomogeneous medium, the different even and oddmode phase
1.2 Components Based on Coupled Structures
9
Figure 1.11
(a) Wiggly twoline coupler, (b) parallel coupled microstrip with overlay compensations, and (c) lumped capacitor compensation of microstrip coupler.
Figure 1.12
Ninetydegree hybrid coupler using interdigital Lange configuration.
10
Introduction
Figure 1.13
Other tight coupler configurations: (a) reentrant structures, (b) asymmetric broadside coupled lines, and (c) slotcoupled microstrip lines.
Figure 1.14
Coaxialline filters: (a) interdigital, and (b) combline.
velocities result in the filter having an asymmetrical passband response, deteriorating the upper stopband performance and moving the second passband (which is at about twice the center frequency) toward the center frequency. To overcome this problem, phase velocity equalization techniques similar to those employed for directional couplers can also be used [35]. Work on coupled line filters can also be found in the literature [2, 36–38]. In microstrip filters, temperature variation of ⑀ r , changes in ⑀ r from lot to lot, and substrate thickness variations usually mean that the bandwidth has to be wider than desired to accommodate manufacturing tolerances.
1.3 Applications
Figure 1.15
11
Typical coupledline filter configurations: (a) parallel coupled, (b) interdigital, (c) combline, and (d) hairpinline.
1.3 Applications In the past decade, five major areas in the development of coupledline components have been emphasized: (1) development of CAD tools, (2) fullwave analysis and accurate semiempirical expressions to enhance component designs, (3) the search for new structures and configurations, (4) advanced materials and technologies, and (5) the search for new applications. Broader bandwidth, ease of fabrication and integration, compact size, and lower cost have been the driving factors. For example, in wireless applications, compact size and lower cost requirements triggered investigation of new configurations and the transformation of existing structures into new layouts such as meander line and spiral geometry to realize compact components.
12
Introduction
Other applications of coupledline sections are in baluns, impedance transformers, dc blocks, interdigital capacitors, and spiral inductors. The spiral inductor is the most popular and is used extensively in hybrid and monolithic microwave integrated circuits (MICs). In particular, the compact size and lowcost circuits used for wireless applications in the L and Sbands are based on inductors as matching elements. Over the past two decades, because of the rapidly growing use of MICs in radar, satellite and mobile communications, electronic warfare (EW) and missiles, couplers and filter technologies have undergone a substantial change in terms of bandwidth, size, and cost. For example, in wirelsss applications, a 90degree hybrid/ coupler (whose output ports have signals of equal magnitude but with 90degree phase difference) is needed to determine the phase error of the transmitter using the /4 quadrature phaseshift keying (QPSK) modulation scheme common to digital cellular radio systems. Basic requirements for this coupler are small size, low cost, and tight amplitude balance and quadrature phase between the output ports. This was met by the coupled microstrip line couplers using the meanderline approach [39–41] and spiral configuration [41] on highdielectric constant substrate compatible with MICs, and meander configuration [39] on a GaAs substrate compatible with monolithic microwave integrated circuits (MMICs). These couplers have the potential to meet the $2 to $5 price goals when housed in plastic packages and produced in large quantities. Satellite, airborne communications, and EW systems require small size, lightweight, lowcost filters. Coupledmicrostrip and stripline filters are very suitable for wideband applications where the demand on selectivity is not severe. Various kinds of filters, shown in Figure 1.15, can be realized using microstriptype structures. For wireless applications, however, miniature versions of these filters are required because of space and cost constraints. Hairpinline and combline filters using resonators on highdielectric constant (⑀ r = 80 and 90) substrate or embedded in dielectric cavities have been developed and can be designed using traditional methods and/or EM simulators. Each filter shown in Figure 1.15 has several other versions to make them compact, either by folding or modifying the layout to fit into a small size. Other applications of filters include dual band communication such as a wireless local area network (WLAN) [42] and ultrawideband (UBW) communication and imaging [43]. Advanced materials/technologies such as high temperature superconductor (HTS) substrates, micromachining, multilayer monolithic, lowtemperature cofired ceramic (LTCC), and liquid crystal polymer (LCP) are commonly used in the development of advanced coupledline components [44, 45]. Coupler and filter technologies are keeping up with the emerging applications. For example, tight couplers are designed using multilayer fabrication processes and electromagnetic metamaterials, and UWB filters are being developed with the help of advanced CAD tools and new modified coupledlines such as defected ground structures (DGSs). Various types of coupled lines, including unevenly coupled [46], periodically nonuniform [47], meandered [48], crosscoupled [49], steppedimpedance resonators [50], and DGS configurations [51], have been studied to suppress the harmonic spurious passbands in advanced coupledline filters. Passive components
1.4 Scope of the Book
13
compatible with CMOS technologies are being developed up to millimeter wave frequencies.
1.4 Scope of the Book Microwave components based on coupledline structures have been in use for over half a century. This text deals exclusively with these components. The purpose of this book is twofold; first, to help the reader understand the theory and working of coupledline components, and second, to provide indepth design information to supplement commercial CAD tools in the design of microwave integrated circuits. As far as possible, enough information has been included to permit the design of passive components for wireless applications covering radio frequency (RF), microwave, and millimeter wave frequencies.
References [1]
[2]
[3] [4] [5] [6]
[7]
[8]
[9]
[10] [11]
[12] [13]
Cohn, S. B., and R. Levy, ‘‘History of Microwave Passive Components With Particular Attention to Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, September 1984, pp. 1046–1054. Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks and Coupling Structures, New York: McGrawHill, 1964 (reprinted by Artech House, Dedham, MA, 1980). Levy, R., ‘‘Directional Couplers,’’ in Advances in Microwaves, Vol. 1, New York: Academic Press, 1966, pp. 184–191. Howe, H., Stripline Circuit Design, Dedham, MA: Artech House, 1974. Malherbe, J. A. G., Microwave Transmission Line Couplers, Norwood, MA: Artech House, 1988. Shelton, J. P., Jr., ‘‘Impedances of Offset ParallelCoupled Strip Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, pp. 7–15, January 1966, and correction IEEE Trans. Microwave Theory Tech., Vol. MTT14, May 1966, p. 249. Shelton, J. P., and J. A. Mosko, ‘‘Synthesis and Design of Wideband EqualRipple TEM Directional Couplers and Fixed Phase Shifters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, October 1966, pp. 462–473. Tresselt, C. P., ‘‘The Design and Construction of Broadband, High Directivity, 90Degree Couplers, Using Nonuniform Line Techniques,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, December 1966, pp. 647–657. Kammler, D. W., ‘‘The Design of Discrete NSection and Continuously Tapered Symmetrical Microwave TEM Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, August 1969, pp. 577–590. Podell, A., ‘‘A High Directivity Microstrip Coupler Technique,’’ IEEE MTTS Int. Microwave Symp. Dig., 1970, pp. 33–36. Sheleg, B., and B. E. Spielman, ‘‘Broadband Directional Couplers Using Microstrip with Dielectric Overlays,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT22, 1974, pp. 1216–1220. Paolino, D. D., ‘‘MIC Overlay Coupler Design Using Spectral Domain Techniques,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT26, 1978, pp. 646–649. Kajfez, D., ‘‘Raise Coupled Directivity with Lumped Components,’’ Microwaves, Vol. 17, No. 3, March 1978, pp. 64–70.
14
Introduction [14] [15] [16]
[17]
[18] [19] [20] [21] [22] [23] [24] [25] [26]
[27] [28] [29] [30]
[31] [32]
[33] [34] [35] [36] [37]
March, S. L., ‘‘Phase Velocity Compensation in ParallelCoupled Microstrip,’’ IEEE MTTS Int. Microwave Symp. Digest, 1982, pp. 410–412. Davis, W. A., Microwave Semiconductor Circuit Design, New York: Van Nostrand, 1983. Horno, M., and F. Medina, ‘‘Multilayer Planar Structures for HighDirectivity Directional Coupler Design,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT34, December 1986, pp. 1442–1449. Dydyk, M., ‘‘Accurate Design of Microstrip Directional Couplers with Capacitive Compensation,’’ IEEE MTTS Int. Microwave Symposium, digest of papers, 1990, pp. 581–584. Uysal, S., Nonuniform Line Microstrip Directional Couplers and Filters, Norwood, MA: Artech House, 1993. Gupta, K. C., et al., Microstrip Lines and Slot Lines, 2nd ed., Norwood, MA: Artech House, 1996. Lange, J., ‘‘Interdigitated Stripline Quadrature Hybrid,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, December 1969, pp. 1150–1151. Waugh, R., and D. LaCombe, ‘‘Unfolding the Lange Coupler,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT20, November 1972, pp. 777–779. Ou, W. P., ‘‘Design Equations for an Interdigitated Directional Coupler,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT23, February 1973, pp. 253–255. Paolino, D., ‘‘Design More Accurate Interdigitated Couplers,’’ Microwaves, Vol. 15, May 1976, pp. 34–38. Presser, A., ‘‘Interdigited Microstrip Coupler Design,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT26, October 1978, pp. 801–805. Bhartia P., and I. J. Bahl, Millimeter Wave Engineering and Applications, New York: Wiley, Ch. 7, 1984. Levy, R., and S. B. Cohn, ‘‘A History of Microwave Filter Research, Design and Development,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, September 1984, pp. 1055–1067. Cohn, S. B., ‘‘ParallelCoupled TransmissionLine Resonator Filters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT6, April 1958, pp. 223–231. Ozaki, H., and J. Ishii, ‘‘Synthesis of a Class of Stripline Filters,’’ IRE Trans. Circuit Theory, Vol. CT5, June 1958, pp. 104–109. Matthaei, G. L., ‘‘Interdigital Bandpass Filters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT10, November 1962, pp. 479–491. Wenzel, R. J., ‘‘Exact Theory of Interdigital Bandpass Filters and Related Coupled Structures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, September 1965, pp. 559–575. Matthaei, G. L., ‘‘CombLine Bandpass Filters of Narrow or Moderate Bandwidth,’’ Microwave J., Vol. 6, August 1963, pp. 82–91. Wenzel, R. J., ‘‘Synthesis of CombLine and Capacitively Loaded Interdigital Bandpass Filters of Arbitrary Bandwidth,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT19, August 1971, pp. 678–686. Cristal, E. G., and S. Frankel, ‘‘Hairpin Line/HalfWave ParallelCoupledLine Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT20, November 1972, pp. 719–728. Gysel, U. H., ‘‘New Theory and Design for HairpinLine Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT22, May 1974, pp. 523–531. Bahl, I. J., ‘‘Capacitively Compensated HighPerformance Parallel Coupled Microstrip Filters,’’ IEEE MTTS Int. Microwave Symp. Dig., 1989, pp. 679–682. Malherbe, J. A. G., Microwave Transmission Line Filters, Dedham, MA: Artech House, 1979. Bahl, I. J., and P. Bhartia, Microwave SolidState Circuit Design, New York: Wiley, 1988, Ch. 6.
1.4 Scope of the Book [38] [39] [40]
[41]
[42]
[43] [44] [45] [46]
[47]
[48]
[49]
[50] [51]
15
Sheinwald, J., ‘‘MMIC Compatible Bandpass Filter Design: A Survey of Applicable Techniques,’’ IEEE MTTS Int. Microwave Symp. Dig., 1989, pp. 679–682. Arai, S., et al., ‘‘A 900MHz Degree Hybrid for QPSK Modulator,’’ IEEE MTTS Int. Microwave Symp. Dig., 1991, pp. 857–860. Tanaka, H., et al., ‘‘2GHz One OctaveBand 90 Degree Hybrid Coupler Using Coupled Meandered Line Optimized by 3D FEM,’’ IEEE MTTS Int. Microwave Symp. Digest, 1994, pp. 906–906. Tanaka, H., et al., ‘‘Miniaturized 90 Degree Hybrid Coupler Using High Dielectric Substrate for QPSK Modulator,’’ IEEE MTTS Int. Microwave Symp. Dig., 1996, pp. 793–796. Tsai, L. C., and C. W. Hsue, ‘‘DualBand Bandpass Filters Using Equal Length CoupledSerialShunted Lines and ZTransform Technique,’’ IEEE Trans. Microwave Theory Tech., Vol. 52, April 2004, pp. 1111–1117. MiniSpecial Issue on UltraWideband, IEEE Trans. Microwave Theory Tech., Vol. 52, September 2004. Lancaster, M. J., Passive Microwave Device Applications of HighTemperature Superconductors, Cambridge, U.K.: Cambridge University Press, 1997. Hong, J.S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, New York: John Wiley & Sons, 2001. Jiang, M., M. H. Wu, and J. T. Kuo, ‘‘ParallelCoupled Microstrip Filters with OverCoupled Stages for Multispurious Suppression,’’ IEEE MTTS Int. Microwave Symp. Dig., 2005, pp. 687–690. Lopetegi, T., et al., ‘‘New Microstrip ‘WigglyLine’ Filters with Spurious Passband Suppression,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT49, September 2001, pp. 1593–1598. Wang, S. M., et al., ‘‘Miniaturized Spurious Suppression Microstrip Filter Using Meandered Parallel Coupled Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT53, 2005, pp. 747–753. Hong, J.S., and M. J. Lancaster, ‘‘Couplings of Microstrip Square OpenLoop Resonators for CrossCoupled Planar Microwave Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT44, November 1996, pp. 2099–2109. Pang, H.K., et al., ‘‘A Compact Microstrip /4SIR Interdigital Bandpass Filter with Extended Stopband,’’ IEEE MTTS Int. Microwave Symp. Dig., 2004, pp. 1621–1625. Vela´zquezAhumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filters with Floating GroundPlane Conductor for SpuriousBand Suppression,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT53, May 2005, pp. 1823–1828.
CHAPTER 2
Microwave Network Theory Microwave coupled lines and components can be classified as Nport networks such as twoport, threeport, fourport, and so on. If the inputoutput parameters of an Nport network are known, its behavior under various conditions of excitation and termination can be determined. For example, if two ports of a fourport network are terminated in open circuit, the inputoutput parameters of the remaining twoport network can be determined from a knowledge of the parameters of the original fourport network. Further, in a microwave system or subsystem, many individual components are cascaded together and the inputoutput parameters of the cascaded network can be determined if those of the individual networks are known. The inputoutput relationship of a linear microwave network can be described in many equivalent ways [1]. In this chapter, we discuss how an Nport network can be characterized by its impedance, admittance, or scattering matrix. Although any form of matrix can be used to describe a network, one form may be more suitable then another. In general, the scattering matrix representation has been the most popular way of describing the inputoutput relationship of a microwave network. Because a network is usually constructed to have specific reflection and transmission properties, one can directly express the desired response in terms of a scattering matrix. These quantities can also be easily measured using vector network analyzers. We discuss scattering matrices in more detail in this chapter together with the conditions imposed by the losslessness and reciprocity on the various representative matrices of a passive network. Some special properties of two, three, and fourport networks are described. The ABCD representation of twoport networks is then discussed. This representation is very useful when a number of twoport networks are cascaded to form a single twoport network. The relationship among various forms of matrices are also given. If P ports of an Nport network are connected to P ports of another Mport network, a network with M + N − 2P ports results. Given the scattering matrices of individual networks, the scattering matrix of the overall network can be determined. We give these relationships and apply them to some specific cases to find the modified scattering parameters of two, three, and fourport networks whose ports are not matchterminated.
2.1 Actual and Equivalent Voltages and Currents For lowfrequency networks, one can define (and measure) unique voltages and currents at various locations in the circuit. Unfortunately, the same is not true at
17
18
Microwave Network Theory
microwave frequencies and beyond, where it is possible to define unique (actual) quantities only for transmission lines carrying power in the TEM mode. Examples of transmission lines supporting the TEM mode of propagation are a coaxial line, stripline, microstrip line,1 and so forth. Many other commonly used transmission lines such as hollow waveguides, dielectric guides, and fin lines do not support the TEM mode of propagation. Therefore, one resorts to the concept of equivalent voltages and currents, and this can be applied to both TEM and nonTEMmode transmission lines. Relationships involving equivalent voltages or currents lead to unique physical quantities such as reflection and transmission coefficients, normalized input impedance, and the like. Equivalent voltages and currents can be defined on a normalized or unnormalized basis. Because the representative matrix of a network may define a relationship between normalized or between unnormalized quantites, it is essential to understand their meaning. 2.1.1 Normalized Voltages and Currents
Figure 2.1(a) shows a twoport network. The power flows into and out of the network by means of transmission lines connected to the network.2 Each transmission line may carry a wave propagating toward the network defined as the incident wave or away from the network defined as the reflected wave. If power is incident in the transmission line connected to port 1, the mode in which the power flows is a characteristic of the type of transmission line. Associated with a mode are unique electromagnetic fields. The transverse components of electric and magnetic fields (transverse to the direction of propagation) have a unique phase associated with them, which is the same for both fields. Further, the zvariation of the incident electromagnetic wave (assuming that the power flow is in the positive zdirection) can be described by a simple factor e −j 1 z, where  1 is a unique quantity and denotes the phase constant of the wave in the transmission line of port 1. To determine the normalized voltage and current waves, we assume that the incident voltage and current waves have the same phase as that of the transverse electric and magnetic field components of the incident electromagnetic wave.3 Further, the zvariation of voltage and current waves is also given by the same factor as that for the field components (e −j 1 z ). Mathematically, the normalized incident voltage and current waves in the transmission line of port 1 can then be expressed as + + + Vˆ1 (z) = Vˆ10 e −j 1 z =  Vˆ10  e j i1 e −j 1 z
(2.1)
+ + + Iˆ1 (z) = Iˆ10 e −j 1 z =  Iˆ10  e j i1 e −j 1 z
(2.2)
and
1. 2. 3.
Microstrip line supports quasiTEM mode. The term transmission line is used in a general sense to denote any physical waveguide structure that can be a microstrip line, coaxial line, rectangular waveguide, optical waveguide, and so on. The phase of an electromagnetic wave is unique and can be measured even if the wave is of a nonTEM type.
2.1 Actual and Equivalent Voltages and Currents
Figure 2.1
19
(a) Normalized and unnormalized voltage and current waves on transmission lines of a twoport network. (b) A twoport network connected to a source and load.
+ + where i1 denotes the phase of the incident wave at z = 0, and Vˆ10 and Iˆ10 are the complex voltage and current, respectively, at the same terminal plane (z = 0). To reemphasize, the value of i1 is the same as that of the transverse components of the electric and magnetic fields of the incident wave. The symbol ‘‘^’’ has been added to denote that the respective quantities are normalized. When the characteristic impedance of a transmission line is real, the voltage and current waves can be expressed in terms of the incident and reflected power.4 At microwave frequencies, the characteristic impedances of practical transmission + + lines are generally real. To compute the values of  Vˆ10  and  Iˆ10  , we force the condition that the average power flow is given by + ˆ+  Vˆ10  I10  = P1+
(2.3)
+ + + where P1 denotes the incident power, and  Vˆ10  and  Iˆ10  denote the rms quantities.
4.
This treatment is, however, not valid if the characteristic impedance is complex [2, 3].
20
Microwave Network Theory + + To determine  Vˆ10  and  Iˆ10  , we need to have another relation between them. To define normalized quantities, we choose +  Vˆ10  + = 1 ˆ  I10 
(2.4)
+ +  Vˆ10  =  Iˆ10  = √P1+
(2.5)
and hence from (2.3) and (2.4):
+ + Substituting the values of  Vˆ10  and  Iˆ10  from (2.5) in (2.1) and (2.2), we obtain + + Vˆ1 (z) = Iˆ1 (z) =
+ j i1 −j 1 z
√P1 e
e
(2.6)
Equation (2.6) is in a form that aids in understanding the physical meaning of normalized voltage and current waves. − When the incident power reaches the network, a part of it (say, P1 ) is reflected back. By analogy with incident voltage and current waves, the reflected waves can be expressed as − − Vˆ1 (z) = Iˆ1 (z) =
− j r1 j 1 z
√P1 e
e
(2.7)
where the superscript ‘‘−’’ is used to denote the reflected waves. r1 is the phase of the transverse components of the electric and magnetic fields of the reflected wave at z = 0 and is a unique quantity. Because the reflected wave propagates in the negative zdirection, its zdependence is described by the factor e j 1 z. The total normalized voltage (at any value of z) in the transmission line of port 1 is thus given by, + − Vˆ1 (z) = Vˆ1 (z) + Vˆ1 (z) + − =  Vˆ10  e j i1 e −j 1 z +  Vˆ10  e j r1 e j 1 z
=
+ j i1 −j 1 z
√P1 e
e
+
(2.8)
− j r1 j 1 z
√P1 e
e
On the other hand, the total current at any value of z is given by + − Iˆ1 (z) = Iˆ1 (z) − Iˆ1 (z) + − = Vˆ1 (z) − Vˆ1 (z)
(2.9)
+ − =  Iˆ10  e j i1 e −j 1 z −  Iˆ10  e j r1 e j 1 z
=
+ j i1 −j 1 z
√P1 e
e
−
− j r1 j 1 z
√P1 e
e
2.1 Actual and Equivalent Voltages and Currents
21
Because the current flows in the axial direction, the net current is given by the difference of the currents flowing in the positive and negative zdirections. We now show that the net power flow (into the network) across any z = constant plane in the transmission line of port 1 is given by the usual lowfrequency relation, that is, P = Re (vi *), where v and i denote the total rms voltage and current, respectively, at the reference plane. In the present case, the term Vˆ1 Iˆ1* can be expressed as + − + − Vˆ1 Iˆ1* = 冠Vˆ1 + Vˆ1 冡冠Vˆ1 − Vˆ1 冡*
(2.10)
+ − After substituting for Vˆ1 and Vˆ1 from (2.6) and (2.7) and noting that the conjugate of a complex number (A + jB) is equal to (A − jB), we obtain + 2 − 2 Vˆ1 Iˆ1* =  Vˆ1  −  Vˆ1  + imaginary term + 2 − 2 =  Vˆ10  −  Vˆ10  + imaginary term +
(2.11)
−
= P1 − P1 + imaginary term or + − Re 冠Vˆ1 Iˆ1* 冡 = P1 − P1
(2.12)
which is the desired result. 2.1.2 Unnormalized Voltages and Currents
If the ratio between the voltage and current of the incident wave (and the reflected wave) is chosen to be different from unity (2.4), the resulting quantities are called unnormalized. For TEM transmission lines, this ratio is generally chosen to be equal to the actual characteristic impedance of the line. In that case, the unnormalized voltages and currents reduce to actual quantities on the line. For nonTEM transmission lines, the characteristic impedance depends on the definition used. Referring to Figure 2.1(a), the unnormalized incident voltage and current waves in the transmission line of port 1 can be expressed as +
V1 (z) = +
I1 (z) =
+ j i1 −j 1 z
√Z 01 P1 e
√
e
(2.13)
+
P1 j i1 −j 1 z e e Z 01
(2.14)
where Z 01 denotes the characteristic impedance of the transmission line of port 1. In the above equations, the symbol ‘‘^’’ has been dropped to denote unnormalized quantities. The unnormalized reflected voltage and current waves in the transmission line of port 1 can be expressed as
22
Microwave Network Theory −
V1 (z) =
− j r1 j 1 z
√Z 01 P1 e
−
I1 (z) =
√
e
(2.15)
−
P1 j r1 j 1 z e e Z 01
(2.16)
Example 2.1
Consider the circuit shown in Figure 2.1(a). Assume that unit power is incident in the transmission line of port 1, of which a quarter is reflected back from the network and that the phase of the incident and reflected waves at z = 0 is equal to 0 and /6 radians, respectively. From (2.6) and (2.7), the normalized voltage and current waves in the input transmission line are + + Vˆ1 (z) = Iˆ1 (z) = − − Vˆ1 (z) = Iˆ1 (z) =
+ j i1 −j 1 z
√P1 e
− j r1 j 1 z
√P1 e
= e −j 1 z
(2.17)
= 0.5e j /6 e j 1 z
(2.18)
e
e
To determine the unnormalized voltage and current waves in the transmission line, we need to specify a value of the characteristic impedance. Let this value be 50 ohms. Using (2.13) and (2.14), the incident unnormalized voltage and current waves are then +
V1 (z) = +
I1 (z) =
+ j i1 −j 1 z
√Z 01 P1 e
√
e
=
√50e
−j 1 z
(2.19)
+
P1 j i1 −j 1 z 1 −j 1 z e e = e Z 01 √50
(2.20)
Using (2.13) and (2.14), the unnormazlied reflected voltage and current waves in the transmission line can be expressed as −
V1 (z) = −
− j r1 j 1 z
√Z 01 P1 e
I1 (z) =
√
e
=
√12.5e
j /6 j 1 z
e
(2.21)
−
P1 j r1 j 1 z 0.5 j /6 j 1 z e e = e e Z 01 √50
(2.22)
2.1.3 Reflection Coefficient, VSWR, and Input Impedance
Referring to Figure 2.1(b), the voltage reflection coefficient ⌫1 in the transmission line of port 1 is defined as ⌫1 (z) =
− − Vˆ1 (z) V1 (z) = + + Vˆ1 (z) V1 (z)
(2.23)
2.1 Actual and Equivalent Voltages and Currents
23
− + Substituting values of Vˆ1 (z) and Vˆ1 (z) from (2.6) and (2.7) in (2.23), we have
⌫1 (z) = =
−  Vˆ10  e j 1r + j 1 z +  Vˆ10  e j 1i − j 1 z −
(2.24)
+ j( 1r − 1i ) 2j 1 z
√P1 /P1 e
e
We easily conclude from (2.24) that the reflection coefficient is a unique quantity, and the square of its modulus gives the fraction of the incident power that is reflected back. The ratio of the reflectedtoincident power is commonly referred to as return loss. The return loss (in decibels), which is a positive quantity, is given by RL (dB) = −10 log  ⌫1 (z)  = −20 log  ⌫1 (z)  2
(2.25)
The voltage standing wave ratio (VSWR) in the transmission line of port 1 is given by VSWR =
1 +  ⌫1 (z)  1 −  ⌫1 (z) 
(2.26)
More often, it is the practice to use the ratio of total voltage and current, which can be termed as input impedance. The ratio of total normalized voltage to current is defined as the normalized input impedance and is given by + − Vˆ (z) Vˆ1 (z) + Vˆ1 (z) 1 + ⌫1 (z) Zˆ in (z) = 1 = = + Iˆ1 (z) Vˆ1 (z) − Vˆ1− (z) 1 − ⌫1 (z)
(2.27)
Because ⌫1 (z) is a unique quantity, the normalized input impedance is also a unique quantity. Similarly, the unnormalized input impedance Z in is given by 1 + ⌫1 (z) Z in (z) = Z 01 1 − ⌫1 (z)
(2.28)
where Z 01 denotes the characteristic impedance of the transmission line connected to port 1. Using (2.27), the reflection coefficient in terms of normalized input impedance is expressed as ⌫1 (z) =
Zˆ in (z) − 1 Zˆ in (z) + 1
and in terms of the unnormalized input impedance:
(2.29)
24
Microwave Network Theory
Z in (z) −1 Z (z) − Z 01 Z 01 ⌫1 (z) = = in Z in (z) + Z 01 Z in (z) +1 Z 01
(2.30)
Transformation of Impedance
With Z in (t 1 ) as the input impedance at the terminal plane t 1 looking into the network, the iput impedance at the terminal plane t 1′ (which is closer to the generator compared with terminal plane t 1 ) is Z (t ) + jZ 01 tan  1 l Z in (t 1′ ) = Z 01 in 1 Z 01 + jZ in (t 1 ) tan  1 l
(2.31)
where l is the distance between terminal planes t 1 and t 1′ . If Z in (t 1 ) denotes the load impedance Z L , the input impedance Z in at distance l away (toward the generator) can be expressed as Z + jZ 01 tan  1 l Z in = Z 01 L Z 01 + jZ L tan  1 l
(2.32)
where  1 = 2 / g denotes the propagation constant of the line, which is assumed to be lossless. 2.1.4 Quantities Required to Describe the State of a Transmission Line
Consider an Nport network as shown in Figure 2.2. The ports are numbered from m = 1 to m = N. The power is carried into and away from the network by means of transmission lines connected to each port. The characteristic impedance of the transmission line of the mth port is denoted by Z 0m . Because the voltages and currents vary along the length of the transmission line, fictitious terminal planes are located in each transmission line. Voltage or current at port m denotes the respective quantity at the specified terminal plane in the transmission line of port m. We use the notation of the sections above for normalized and unnormalized quantities and incident and reflected quantities. + + − − Note that Vˆn and Iˆn (similarly, Vˆn and Iˆn ) are not independent quantities. The normalized quantities satisfy the following relationship: + − Vˆn Vˆn = + − = 1 Iˆn Iˆn
(2.33)
+ + − − If the two quantities Vˆn (or Iˆn ) and Vˆn (or Iˆn ) are known, the total voltage and current can be determined using + − + − Vˆn = Vˆn + Vˆn = Iˆn + Iˆn
and
(2.34)
2.1 Actual and Equivalent Voltages and Currents
Figure 2.2
25
An Nport network.
+ − + − Iˆn = Iˆn − Iˆn = Vˆn − Vˆn
(2.35)
+ + − − Therefore, if any two of the four quantities Vˆn , Iˆn , Vˆn (or Iˆn ) and Vˆn (or Iˆn ) are known, all others can be determined. The same conclusion holds for unnormalized quantities if the characteristic impedances of all the transmission lines are known. For unnormalized quantities: +
−
Vn
Vn
In
In
+ =
−
= Z 0n
(2.36)
Vn = Vn + Vn = Z 0n 冠In + In 冡 +
−
+
+
−
I n = In − In =
+
−
(2.37)
−
Vn − Vn Z 0n
(2.38)
Relationship Between Normalized and Unnormalized Quantities
The normalized and unnormalized quantities are related by ±
Vn ± Vˆn = √Z 0n
(2.39a)
Vn Vˆn = √Z 0n
(2.39b)
26
Microwave Network Theory ± Iˆn =
√Z 0n In
±
(2.39c)
Iˆn =
√Z 0n In
(2.39d)
2.2 Impedance and Admittance Matrix Representation of a Network 2.2.1 Impedance Matrix
Consider the Nport network shown in Figure 2.2. In the impedance matrix representation, the voltage at each port is related to the currents at the different ports as follows: V1 = Z 11 I1 + Z 12 I2 + . . . + Z 1N IN V2 = Z 21 I1 + Z 22 I2 + . . . + Z 2N IN ⯗ ⯗
⯗ ⯗
⯗ ⯗ ⯗ ⯗
(2.40)
⯗
VN = Z N1 I1 + Z N2 I2 + . . . + Z NN IN In matrix notation, this set of equations can be expressed as [V] = [Z][I]
(2.41)
[V] =
V1 ⯗ VN
冤 冥
(2.42)
[I] =
冤 冥
(2.43)
where
I1 ⯗ IN
and
[Z] =
冤
Z 11 Z 21 ⯗ Z N1
Z 12 Z 22 ⯗ Z N2
... ... ⯗ ...
Z 1N Z 2N ⯗ Z NN
冥
(2.44)
The impedance matrix [Z] is unnormalized because it relates unnormalized voltages and currents. The impedance matrix relating normalized voltages and currents is called normalized, and will be denoted as [Zˆ ] with the ‘‘^’’ symbol. The normalized impedance matrix is denoted as
2.2 Impedance and Admittance Matrix Representation of a Network
[Zˆ ] =
冤
Zˆ 11 Zˆ 21 ⯗ Zˆ N1
Zˆ 12 Zˆ 22 ⯗ Zˆ N2
... ... ⯗ ...
27
Zˆ 1N Zˆ 2N ⯗ Zˆ NN
冥
(2.45)
2.2.2 Admittance Matrix
In the admittance matrix representation, the current at each port of the network as shown in Figure 2.2 is related to the voltages at the different ports as follows: [I] = [Y][V]
(2.46)
where [V] and [I] are column vectors as defined by (2.42) and (2.43), respectively, and
[Y] =
冤
Y 11 Y 21 ⯗ YN1
Y 12 Y 22 ⯗ YN2
... ... ⯗ ...
Y 1N Y 2N ⯗ YNN
冥
(2.47)
2.2.3 Properties of Impedance and Admittance Parameters of a Passive Network
For a network not containing any nonreciprocal media (ferrite, plasma, and so forth), Zˆ mn = Zˆ nm
(2.48)
Yˆmn = Yˆnm
(2.49)
and
Similar relationships are also satisfied by the elements of unnormalized impedance and admittance matrices, that is: Z mn = Z nm
(2.50)
Ymn = Ynm
(2.51)
and
Note that networks containing dielectrics and conductors are reciprocal and satisfy the abovementioned properties. For a lossless network, all the elements of an impedance or admittance matrix are imaginary. This is an expected result because any resistive element would imply loss.
28
Microwave Network Theory
The mnth element of the unnormalized impedance matrix is related to the corresponding element of the normalized impedance matrix by Z mn = Zˆ mn √Z 0m Z 0n
(2.52)
where Z 0m and Z 0n denote the characteristic impedances of transmission lines of ports m and n, respectively. Similarly, the mnth element of the unnormalized admittance matrix is related to the corresponding element of the normalized admittance matrix by Ymn = Yˆmn √Y 0m Y 0n
(2.53)
where Y 0m = 1/Z 0m and Y 0n = 1/Z 0n denote the characteristic admittance of the transmission lines of ports m and n, respectively.
2.3 Scattering Matrix A very popular method of representing microwave networks is by the scattering matrix. The scattering matrix is generally represented in a normalized form. In this representation, the normalized reflected voltage at each port of the network as shown in Figure 2.2 is related to the normalized incident voltages at the ports of the network as follows: − + + + Vˆ1 = Sˆ 11 Vˆ1 + Sˆ 12 Vˆ2 + . . . + Sˆ 1N VˆN − + + + Vˆ2 = Sˆ 21 Vˆ1 + Sˆ 22 Vˆ2 + . . . + Sˆ 2N VˆN
⯗ ⯗
⯗
⯗
⯗
⯗ ⯗ ⯗
(2.54)
⯗
− + + + VˆN = Sˆ N1 Vˆ1 + Sˆ N2 Vˆ2 + . . . + Sˆ NN VˆN
In matrix notation, the above set of equations can be expressed as [Vˆ − ] = [Sˆ ][Vˆ + ]
(2.55)
where
[Vˆ − ] =
− Vˆ1 ⯗ − VˆN
冤 冥
(2.56a)
[Vˆ + ] =
+ Vˆ1 ⯗ ˆVN+
(2.56b)
冤 冥
2.3 Scattering Matrix
29
and
[Sˆ ] =
冤
Sˆ 11 Sˆ 21
Sˆ 12 Sˆ 22
⯗
⯗
Sˆ N1
Sˆ N2
Sˆ 1N Sˆ 2N
... ... ⯗ ...
⯗ Sˆ NN
冥
(2.57)
The scattering parameter Sˆ mn is therefore given by ˆ − ˆS mn = Vm + Vˆn

+ Vˆp = 0 where p = 1, . . . , N ; p ≠ n
(2.58)
+
In terms of the incident power Pn in the nth transmission line, the amplitude of the normalized incident voltage wave at the nth port is given by
 Vˆn+  = √Pn+
(2.59)
Similarly, the amplitude of the normalized reflected voltage wave5 at the mth port is given by
 Vˆm−  = √Pm−
(2.60)
−
where Pn denotes the reflected power at port m. + − When the values of  Vˆn  and  Vˆm  from the last two equations are substituted in (2.58), we obtain
 Vˆm−  ˆ S = =  mn   Vˆn+ 
√
−
Pm
(2.61)
+
Pn
From the above equation, we see that  Sˆ mn  denotes the ratio of power coupled + from port n to port m when Vˆp = 0, where p = 1, . . . , N; p ≠ n. The condition + Vˆp = 0, where p = 1, . . . , N; p ≠ n is readily esured by exciting only the nth port and terminating all the ports in matched loads. Similarly: 2
 Sˆ nn  2 =  ⌫n  2 =
 Vˆn−  2
−
P = n+ + 2  Vˆn  Pn
(2.62)
2 where ⌫n denotes the reflection coefficient at port n. Further,  Sˆ nn  denotes the fraction of the incident power that is reflected back at port n. The ports of a typical microwave network are usually matchterminated. Therefore, if some power is incident in one of the ports, the reflected power and the
5.
In the terminology used, any wave traveling toward the network is called the ‘‘incident’’ wave, and any wave traveling away from the network is called the ‘‘reflected’’ wave.
30
Microwave Network Theory
power coupled to the other ports of the network can be easily determined if the normalized scattering matrix is known. 2.3.1 Unitary Property
The elements of a normalized scattering matrix satisfy the following equation, which results from the Law of Conservation of Power:
冤
* Sˆ11 ˆS12 * ⯗ ˆS1N *
* Sˆ21 ˆS22 * ⯗ ˆS2N *
... ... ⯗ ...
* Sˆ N1 ˆS N2 * ⯗ ˆS NN *
冥冤
Sˆ 11 Sˆ 21 ⯗ ˆS N1
Sˆ 12 Sˆ 22 ⯗ ˆS N2
... ... ⯗ ...
Sˆ 1N Sˆ 2N ⯗ ˆS NN
冥冤 =
1 0 ⯗ 0
0 1 ⯗ 0
... ... ⯗ ...
冥
0 0 ⯗ 1 (2.63)
where the symbol * denotes the complex conjugate. Because the [Sˆ ] matrix satisfies the above relationship, it is called a unitary matrix. Power conservation is true for reciprocal as well as for nonreciprocal networks. Therefore, the normalized scattering matrix of any reciprocal or nonreciprocal lossless network is unitary. In a compact form, (2.63) can be expressed as [Sˆ *]t [Sˆ ] = U
(2.64)
where [Sˆ *] denotes the matrix formed by the conjugate of the elements of the [Sˆ ] matrix and [Sˆ *]t denotes the transpose of matrix [Sˆ *]. U is a unit matrix of order N. All the diagonal elements of a unit matrix are 1, while all its nondiagonal elements are 0. From (2.63), it is seen that if the nth row of the [Sˆ *]t matrix is multiplied with the nth column of the [Sˆ ] matrix, the following equation results: * + Sˆ 2n Sˆ 2n * + . . . + Sˆ Nn Sˆ Nn * =1 Sˆ 1n Sˆ 1n or
 Sˆ 1n  2 +  Sˆ 2n  2 + . . . +  Sˆ Nn  2 = 1
(2.65)
where n = 1, 2, . . . , N. On the other hand, if the nth row of the [Sˆ *]t matrix is multiplied with the mth column of the [Sˆ ] matrix with m ≠ n, then the following equation results: ˆ * Sˆ 1m + Sˆ 2n * Sˆ 2m + . . . + Sˆ * Sˆ 1n Nn S Nm = 0
(2.66)
where m = 1, . . . , N; n = 1, . . . , N; and m ≠ n . The unitary properties of the scattering matrix as given by (2.65) and (2.66) lead to very useful predictions about the properties of a lossless network. For example, the unitary property of a scattering matrix leads to the result that it is impossible to match a lossless, threeport
2.3 Scattering Matrix
31
reciprocal network at all its ports simultaneously. We state some special properties of two, three, and fourport lossless networks derived using the unitary property of the scattering matrix later in this chapter. 2.3.2 Transformation with Change in Position of Terminal Planes
Assume that the scattering matrix of the network with the location of terminal planes denoted by t p , p = 1, . . . N as shown in Figure 2.3 is given by (2.57). If the terminal plane in each transmission line is moved to new locations denoted by t p′ where p = 1, . . . N, we define the scattering matrix with the location of ports denoted by t p′ as [Sˆ ′], which is related to the scattering matrix [Sˆ ] as
[Sˆ ′] =
冤 ×
e −j 1 l 1 0 ⯗
e
0
冤
0
...
0
⯗
... ⯗
0 ⯗
0
...
e −j N l N
−j 2 l 2
e −j 1 l 1 0 ⯗ 0
0 e
冥冤 冥
...
0
⯗
... ⯗
0 ⯗
0
...
e −j N l N
−j 2 l 2
Sˆ 11 Sˆ 21 ⯗ Sˆ N1
Sˆ 12 Sˆ 22 ⯗ Sˆ N2
... ... ⯗ ...
Sˆ 1N Sˆ 2N ⯗ Sˆ NN
冥
(2.67)
where  p (p = 1, . . . , N) denotes the phase constant of the wave in the transmission line of the pth port. From (2.67), it follows that the mnth element of the modified
Figure 2.3
An Nport network. The elements of a representative matrix change when the location of ports (terminal planes) is changed.
32
Microwave Network Theory
scattering matrix is related to the corresponding element of the original scattering matrix by Sˆmn ′ = Sˆ mn e
−j m l m − j n l n
(2.68)
2.3.3 Reciprocal Networks
If a network does not contain any nonreciprocal media (e.g., ferrite, plasma), then the following relation holds for the elements of its normalized scattering matrix: Sˆ mn = Sˆ nm
(2.69)
In matrix notation, the condition of reciprocity is stated as [Sˆ ] = [Sˆ ]t
(2.70)
2.3.4 Relationship Between Normalized and Unnormalized Matrices
The mnth elements of the unnormalized and normalized scattering matrices are related by S mn = Sˆ mn
√
Z 0m Z 0n
(2.71)
where Z 0m and Z 0n denote the characteristic impedances of the transmission lines at ports m and n. The above equation leads to an important conclusion that if all the ports of a network have the same characteristic impedance, then the unnormalized scattering matrix of the network is the same as the normalized matrix. In microwave networks, all the ports usually have the same characteristic impedance. Therefore, in these cases it is not necessary to specify whether the scattering matrix is normalized or unnormalized. However, if the characteristic impedances of all the ports of a network are not the same, such as in the case of asymmetrical coupled lines, impedancetransforming couplers, baluns, and so forth it should be specified whether the scattering matrix is normalized or unnormalized.
2.4 Special Properties of Two, Three, and FourPort Passive, Lossless Networks All the elements of a network matrix of a passive lossless network cannot be chosen independently. For example, if a network does not contain any nonreciprocal media (e.g., ferrite, plasma), then Sˆ mn = Sˆ nm . Therefore, one cannot design a network containing a passive reciprocal medium having different values of Sˆ mn and Sˆ nm . As another example, if a network has a plane of symmetry, then Sˆ mm = Sˆ nn where m and n are symmetrical ports. The preceding properties described are true in general for any Nport network. There are, however, some special properties of a
2.4 Special Properties of Two, Three, and FourPort Passive, Lossless Networks
33
network depending on the number of ports it has. In the following, we discuss some special properties of passive lossless, two, three, and fourport networks. 2.4.1 TwoPort Networks
Figure 2.4 shows a twoport passive and lossless network. The normalized scattering matrix of a twoport network can be written as [Sˆ ] =
冋
Sˆ 11 Sˆ 21
册冋
Sˆ 12 Sˆ 22
=
 Sˆ 11  e j 11  Sˆ 21  e j 21
册
 Sˆ 12  e j 12  Sˆ 22  e j 22
(2.72)
The unitary property of the normalized scattering matrix as given by (2.65) and (2.66) leads to the following relations:
 Sˆ 11  2 +  Sˆ 21  2 = 1
(2.73a)
 Sˆ 12  2 +  Sˆ 22  2 = 1
(2.73b)
* Sˆ 22 = 0 * Sˆ 12 + Sˆ 21 Sˆ 11
(2.73c)
The solution of this set of equations leads to the following conclusions:
 Sˆ 11  =  Sˆ 22 
(2.74)
 Sˆ 12  =  Sˆ 21 
(2.75)
11 + 22 = 12 + 21 ⫿ rad
(2.76)
The above relations are true for any twoport network—reciprocal or nonreciprocal. Equations (2.74) and (2.75), along with (2.73), show that for a lossless network
Figure 2.4
A passive, lossless, twoport network.
34
Microwave Network Theory
(reciprocal or nonreciprocal) one needs to know the amplitude of one scattering element to determine the amplitude of all other scattering elements. Equation (2.75) also shows that even if the twoport network is composed of nonreciprocal media,  Sˆ 12  =  Sˆ 21  . In physical terms, this relation states that the ratio of the power coupled from port 1 to port 2 when the power is incident at port 1 is the same as that coupled from port 2 to port 1 when power is incident at port 2. This is an interesting result; one might then wonder about how a twoport isolator works. An isolator is supposed to have a very small attenuation between its ports when the power is incident at one port, and a large attenuation when the power is incident at the other port. The difficulty is resolved by noting that the unitary property is valid only for lossless networks. On the other hand, an isolator employs some kind of lossy elements to achieve different values of  Sˆ 12  and  Sˆ 21  . When the network is reciprocal, Sˆ 12 = Sˆ 21 , which leads to 12 = 21 , or from (2.76):
12 = ( 11 + 22 ± )/2 rad Using the above equation and (2.74) and (2.75), the scattering matrix of a lossless, reciprocal twoport network can be expressed in the form
[Sˆ ] =
冋
 Sˆ 11  e j 11
√1 −  Sˆ 11 
2 j( + ± )/2 √1 −  Sˆ 11  e 11 22
册
2 j( 11 + 22 ± )/2
e
 Sˆ 11  e j 22
(2.77)
Further, noting that  Sˆ 11  ≤ 1, (2.77) reduces to
[Sˆ ] =
冋
册
cos ␣ e j 11
sin ␣ e j( 11 + 22 ± )/2
sin ␣ e j( 11 + 22 ± )/2
cos ␣ e j 22
(2.78)
where cos ␣ =  Sˆ 11  . For a twoport network, any desired values of 11 and 22 can always be chosen by changing the location of the terminal planes. If 11 and 22 are chosen to be zero by appropriately locating the terminal planes, then the scattering matrix of the twoport network becomes [Sˆ ] =
冋
Sˆ 11 Sˆ 21
册冋
Sˆ 12 Sˆ 22
=
cos ␣ ± j sin ␣
± j sin ␣ cos ␣
册
(2.79)
2.4.2 ThreePort Reciprocal Networks
For networks having three ports, the unitary property of the scattering matrix leads to the result that it is impossible to match a passive, lossless reciprocal network at all its ports simultaneously. Therefore, a threeport network enclosing reciprocal media cannot have a scattering matrix of the form
2.4 Special Properties of Two, Three, and FourPort Passive, Lossless Networks
[Sˆ ] =
冤
0 ˆS 12 Sˆ 13
Sˆ 12 0 ˆS 23
Sˆ 13 Sˆ 23 0
冥
35
(2.80)
The above condition holds only for a lossless reciprocal network, but by incorporating lossy elements in the network (such as in a Wilkinson’s power divider), a threeport network can be matched at all its ports simultaneously. For a reciprocal threeport network, the unitary property also leads to another important result. If two of the three ports of the network are completely matched, then the third port is completely isolated from the other two ports. The scattering matrix of this network is then given by
[Sˆ ] =
冤
0 1 0
1 0 0
冥
0 0 1
(2.81)
where it is assumed that the phase of nonzero scattering elements has been adjusted to zero. 2.4.3 ThreePort Nonreciprocal Networks
The unitary property leads to the result that a passive, lossless threeport nonreciprocal network (Figure 2.5) can be matched at all its ports simultaneously. With this, it can behave only as a circulator. The scattering matrix of a circulator is of the following form:
Figure 2.5
A passive, lossless, nonreciprocal threeport network. The network acts as a circulator if it is matched at all its ports.
36
Microwave Network Theory
[Sˆ ] =
冤
0
e j 21 0
e
0
e j 13
0
0
j 32
0
冥
(2.82)
or  Sˆ 13  =  Sˆ 21  =  Sˆ 32  = 1. The lossless circulator has thus an important property that if power is incident at port 1 then all the power is transmitted to port 2. If the power is incident at port 2, all the power is transmitted to port 3, and if the power is incident at port 3, all the power is transmitted to port 1. This is shown schematically in Figure 2.5. 2.4.4 FourPort Reciprocal Networks
Figure 2.6 shows a fourport network. For a passive, lossless reciprocal fourport network, the unitary property of the scattering matrix leads to the result that it is possible to match all the four ports of the network simultaneously. If all four ports are matched, the network behaves like a directional coupler. The scattering matrix of a directional coupler is of the form
冤
0 0 [Sˆ ] = Sˆ 13 Sˆ 14
0 0 Sˆ 23 Sˆ 24
Sˆ 13 Sˆ 23 0 0
Sˆ 14 Sˆ 24 0 0
冥
(2.83)
The directional coupler can be considered to be composed of two pairs of ports, with ports of each pair matched and isolated from each other. As seen from (2.83), ports 1 and 2 of the network are matched and isolated from each other. Similarly, ports 3 and 4 are matched and isolated from each other. The elements of the scattering matrix of a directional coupler as given by (2.83) also satisfy the following relationships:
Figure 2.6
A passive, lossless, reciprocal fourport network. The network behaves like a directional coupler if it is matched at all ports.
2.4 Special Properties of Two, Three, and FourPort Passive, Lossless Networks
37
 Sˆ 13  =  Sˆ 24  ,  Sˆ 14  =  Sˆ 23  Using the results of the above two equations, the elements of the scattering matrix can be expressed as Sˆ 13 = C 1 e j 13, Sˆ 23 = C 2 e j 23, Sˆ 14 = C 2 e j 14, Sˆ 24 = C 1 e j 24 where C 1 =  Sˆ 13  =  Sˆ 24  and C 2 =  Sˆ 14  =  Sˆ 23  . Thus, (2.83) can be expressed as
[Sˆ ] =
0
0
C 1 e j 13
C 2 e j 14
0
0
C 2 e j 23
C 1 e j 24
C 1 e j 13
C 2 e j 23
0
0
C 2 e j 14
C 1 e j 24
0
0
冤
冥
(2.84)
We can easily show that all the phase factors of the various scattering elements 13 , 23 , 14 , and 24 cannot be chosen independently. Assume that desired values of 13 and 14 have been chosen by varying the positions of ports 3 and 4 respectively. The phase factor 23 can be independently chosen by controlling the position of port 2. The remaining phase factor 24 cannot be changed now because both ports 2 and 4 have already been adjusted. From the unitary property of the scattering matrix, one obtains the value of 24 in terms of other phase factors as
24 = 14 + 23 − 13 ± rad
(2.85)
The amplitudes of scattering elements satisfy 2
2
C1 + C2 = 1
(2.86)
The two forms of matrices to which the scattering matrix of a directional coupler can always be reduced by appropriately locating the position of terminal planes are now derived. Let us first choose 13 = 14 = 23 = 0. From (2.85) we obtain 24 = ± rad, or the scattering matrix of a directional coupler takes the form
冤
0 0 [Sˆ ] = C 1 C2
0 0 C2 −C 1
C1 C2 0 0
C2 −C 1 0 0
冥
(2.87)
For deriving the second form, we choose 13 = 0 and 23 = 14 = ± /2 rad. From (2.85), we then obtain 24 = 0, or 24 = ± 2 rad. The scattering matrix thus reduces to
冤
0 0 [Sˆ ] = C 1 ±jC 2
0 0 ±jC 2 C1
C1 ±jC 2 0 0
±jC 2 C1 0 0
冥
(2.88)
38
Microwave Network Theory
We see in later chapters that the scattering matrix of many useful fourport microwave networks can be represented either by (2.87) or (2.88). For example, a ratrace hybrid can be represented by (2.87), whereas quadrature hybrids, Lange couplers, and so on can be represented by (2.88).
2.5 Special Representation of TwoPort Networks A typical microwave subsystem consists of a cascade of twoport networks such that the output of one network is connected to the input of the next and so on. The twoport networks can be represented by their impedance, admittance, or scattering parameters. It is often more useful, however, to represent twoport networks by ABCD parameters because knowing the ABCD parameters, the matrix of the overall cascaded network can be computed by multiplying the matrices of the individual networks. 2.5.1 ABCD Parameters
Figure 2.7 shows a twoport network. In the ABCD matrix representation, the voltage and current flowing into the network at the input of the network are related to the voltage and current flowing away from the network at the output as follows: V1 = AV2 + BI2
(2.89)
I1 = CV2 + DI2 or in matrix form
冋 册 冋 册冋 册 V1 I1
=
A C
B D
V2 I2
(2.90)
Note that in the ABCD matrix representation, the direction of positive current flow at the output as shown in Figure 2.7 is taken in an opposite sense than what is done in the impedance, or admittance matrix representation. To explain the advantage of representation in terms of ABCD parameters, consider the twoport networks cascaded together in Figure 2.8. The ABCD parameters of the individual networks are also shown in the same figure. We are interested
Figure 2.7
A twoport network. In the ABCD matrix representation, the direction of positive current flow at the output is opposite of that used in impedance and admittance matrix representation.
2.5 Special Representation of TwoPort Networks
Figure 2.8
39
Cascade of two twoport networks and their equivalent representation.
in finding the relationship between the input and output of the overall cascaded network. The voltages and currents at the input and output of the first network are related by the following matrix equation:
冋册冋 V1 I1
A1 C1
=
册冋 册
B1 D1
V2 I2
(2.91)
For the second network, V2 represents the input voltage and I2 represents the input current. So we can write
冋册冋 V2 I2
A2 C2
=
Substituting V2 and I2 into (2.91):
冋册冋 V1 I1
=
A1 C1
B1 D1
册冋 册
B2 D2
册冋
A2 C2
V3 I3
B2 D2
(2.92)
册冋 册 V3 I3
(2.93)
or the ABCD matrix of the overall network between ports 1 and 3 can be expressed as
冋
At Ct
册冋
Bt Dt
=
A1 C1
B1 D1
册冋
A2 C2
B2 D2
册
(2.94)
Therefore, the ABCD matrix of the overall network is the product of the ABCD matrices of the individual networks. The same is true for any number of twoport networks connected in cascade. Properties of ABCD Parameters
Consider the twoport network as shown in Figure 2.7. Let Z 01 and Z 02 denote the characteristic impedances of ports 1 and 2, respectively. The normalized and unnormalized ABCD parameters are related as follows:
40
Microwave Network Theory
Aˆ = A Bˆ =
√
Z 02 Z 01
B
√Z 01 Z 02
(2.95)
Cˆ = C √Z 02 Z 01 ˆ =D D
√
Z 01 Z 02
For a passive, lossless, twoport reciprocal network: AD − BC = 1
(2.96)
This relation is also satisfied by normalized ABCD parameters. For a lossless, twoport symmetrical network: ˆ Aˆ = D
(2.97)
2.5.2 Reflection and Transmission Coefficients in Terms of ABCD Parameters
Consider the reciprocal twoport network shown in Figure 2.9. The characteristic impedances of transmission lines of the input and output ports are assumed to be Z 0 and both ports are assumed to be matchterminated. The reflection coefficient at the input port is given by ⌫in =
A + B /Z 0 − CZ 0 − D A + B /Z 0 + CZ 0 + D
(2.98)
Furthermore, the reflection coefficient at output port is given by ⌫o =
−A + B /Z 0 − CZ 0 + D A + B /Z 0 + CZ 0 + D
(2.99)
The return loss (in decibels, which is a positive quantity) is given by (2.25). The transmission coefficient between input and output ports is given by
Figure 2.9
Reflection and transmission coefficients of a twoport network.
2.5 Special Representation of TwoPort Networks
T=
41
− − Vˆ2 V2 2 = = + + A + B /Z 0 + CZ 0 + D Vˆ1 V1
(2.100)
The insertion loss between input and output (in decibels, which is a positive quantity) is given by Insertion loss (dB) = −20 log  T 
(2.101)
For a lossless network.
 ⌫in  2 + T 2 =  ⌫o  2 + T 2 = 1
(2.102)
In the above example, we have assumed that the characteristic impedances of the input and output ports are the same. In case they are not the same, the unnormalized scattering parameters of the network can be obtained from the unnormalized ABCD parameters using the conversion relation given in Table 2.1. The unnormalized scattering parameters can then be normalized using the equations given in Section 2.3.4. The elements Sˆ 11 and Sˆ 21 of the normalized scattering matrix directly give the reflection and transmission coefficients. The unnormalized and normalized ABCD parameters of some elementary networks are given in Table 2.2. More complex networks can be obtained by cascading a number of elementary networks. The ABCD matrix of the overall cascaded network can then be determined by multiplying the ABCD matrices of elementary networks. It is quite simple to determine the ABCD parameters of elementary networks. Consider, for example, the circuit shown in Figure 2.10. Inspecting the circuit, the following equations are obtained: V1 = V2 + ZI2
(2.103)
I1 = I2 By comparing this set of equations with (2.89), we obtain A = 1, B = Z, C = 0, and D = 1 Further, using (2.95), the normalized ABCD parameters can be found as follows: Aˆ =
√
Z 02 ˆ ,B= Z 01
Z
√Z 01 Z 02
ˆ = , C = 0, D
√
Z 01 Z 02
2.5.3 Equivalent T and ⌸ Networks of TwoPort Circuits
If the impedance parameters of a twoport reciprocal network are known, the network can be represented as shown in Figure 2.11(a). Similarly, if the admittance parameters of a twoport reciprocal network are known, the circuit can be represented as shown in Figure 2.11(b). T and ⌸ forms are only two of many possible
42
Microwave Network Theory Table 2.1 Conversion Relationships Between Various Representative Matrices of TwoPort Networks
冋 冋
册 册
冋 冋
Z 12 Y22 1 = Y11 Y22 − Y12 Y21 −Y21 Z 22
−Y12 Y11
Y11 Y21
Y12 Z 22 1 = Z Z − Z Z Y22 11 22 12 21 −Z 21
−Z 12 Z 11
冋
册 冋 冋 册 冋 冋 册 冋 冋 册 冋 Z 11 Z 21
Z 12 1 = C Z 22
Y 11 Y21
A C
B D
A C
冋
S 11 S 21
册 冉
S 12 = Z 11 S 22 +1 Z 01
×
冋
S 11 S 21
册 冉
S 12 = S 22
冤
冋
Y11 Y21
册
冉
冤
Y12 1 = B Y22
=
B D
冊冉
D −1
Y22 1 −Y21 Y11 Y22 − Y12 Y21
冊冉
冊
冊
Z 22 Z Z + 1 − 12 21 Z 02 Z 01 Z 02 Z 2 21 Z 01
冉
1−
册
−(AD − BC) A
册 册
(Z 11 Z 22 − Z 12 Z 21 ) Z 22
Z 11 1 Z 21 1
=
册
AD − BC D
A 1
1 Y11
1 Z 22 Z Z + 1 − 12 21 Z 02 Z 01 Z 02
Z 11 −1 Z 01
Y 1 + 11 Y01
×
册 册
Z 11 Z 21
冊冉
1 1+
Y11 Y01
冊冉
Y22 Y02 1+
冊
−
Y22 Y02
冉
Z 11 +1 Z 01
冊冉
Z 2 12 Z 02
冥
冊
Z 22 Z Z − 1 − 12 21 Z 02 Z 01 Z 02
Y12 Y21 Y01 Y02
冊
+
Y12 Y21 Y01 Y02
Y −2 21 Y02
冉
1+
Y11 Y01
冊冉
Y −2 12 Y01 1−
Y22 Y02
冊
+
冥
Y12 Y21 Y01 Y02
Y12 1 = (1 + S 11 ) (1 + S 22 ) − (S 12 S 21 ) Y22 ×
冋
Y01 [(1 − S 11 ) (1 + S 22 ) + S 12 S 21 ] −2Y02 S 21
−2Y01 S 12 Y02 [(1 + S 11 ) (1 − S 22 ) + S 12 S 21 ]
册
2.6 Conversion Relations
43
Table 2.1 (continued)
冋
A C
冋
S 11 S 21
B D
册
1 = (2S 21 )
册
Z 11 Z 21
Z 02 [(1 + S 11 ) (1 + S 22 ) − S 12 S 21 ] Z 01 [(1 − S 11 ) (1 + S 22 ) + S 12 S 21 ] Z 02
冤
S 12 1 = (B + CZ 01 Z 02 ) + (AZ 02 + DZ 02 ) S 22 ×
冋
[(1 + S 11 ) (1 − S 22 ) + S 12 S 21 ] 1 [(1 − S 11 ) (1 − S 22 ) − S 12 S 21 ] Z 01
冋
册
(B − CZ 01 Z 02 ) + (AZ 02 − DZ 01 ) 2Z 02
2Z 01 (AD − BC) (B − CZ 01 Z 02 ) − (AZ 02 − DZ 01 )
册
Z 12 1 = (1 − S 11 ) (1 − S 22 ) − S 12 S 21 Z 22 ×
冋
Z 01 [(1 + S 11 ) (1 − S 22 ) + S 12 S 21 ] 2Z 01 S 21
冥
2Z 02 S 12 Z 02 [(1 − S 11 ) (1 + S 22 ) + S 12 S 21 ]
册
By substituting Z 01 = Z 02 = 1, the above relations can be used for the conversion of normalized parameters.
ways in which the equivalent circuit of a twoport network can be expressed. The other forms may contain a combination of a length of a transmission line, transformer, reactance and susceptance elements, and the like [1].
2.6 Conversion Relations In the following, conversion relations among admittance, impedance, and scattering matrices are given. The conversion relations between a scattering matrix and impedance and admittance matrices are given by assuming that the respective matrices are normalized. To convert unnormalized matrices, the unnormalized parameters should be first normalized using the equations given earlier in various sections. The normalized matrix can then be converted from one type to the desired type, and later unnormalized if necessary. [Z] = [Y]−1
(2.104)
[Y] = [Z]−1
(2.105)
The above equations are also valid for normalized parameters. [Sˆ ] = ([Zˆ ] − [U]) ([Zˆ ] + [U])−1
(2.106)
Another expression for [Sˆ ] in terms of [Zˆ ] is, [Sˆ ] = ([Zˆ ] + [U])−1 ([Zˆ ] − [U])
(2.107)
44
Microwave Network Theory Table 2.2 ABCD Parameters of Elementary TwoPort Networks
Figure 2.10
An impedance Z in series between two transmission lines.
2.7 Scattering Matrix of Interconnected Networks
Figure 2.11
45
(a) T and (b) ⌸ network representation of a twoport network.
[Zˆ ] = ([U] − [Sˆ ])−1 ([U] + [Sˆ ])
(2.108)
[Yˆ ] = ([U] + [Sˆ ])−1 ([U] − [Sˆ ])
(2.109)
In some cases, the conversion formulas cannot be used. For example, if the determinant of matrix ([U] − [Sˆ ]) is zero, the impedance matrix becomes indeterminant. Similarly, if the determinant of matrix ([U] + [Sˆ ]) is zero, the admittance matrix becomes indeterminant. Frequently, it is required to convert one form of matrix into another. The conversion relations between twoport matrices are given in Table 2.1 [4]. These relations are general and valid for nonreciprocal networks also. By substituting Z 01 = Z 02 = 1 in these equations, conversion between normalized parameters can be obtained.
2.7 Scattering Matrix of Interconnected Networks A typical microwave system or subsystem results after interconnection of many intermediate networks. Consider two networks as shown in Figure 2.12. Networks I and II are assumed to have M + P and P + N ports, respectively. The P ports of each network are directly connected to each other. The overall network has therefore M + N accessible ports, and its scattering matrix is of the order (M + N) × (M + N). Given the scattering matrices of networks I and II, the scattering matrix of the overall network can be determined. Let the ports of network I be so numbered that its M ports (m = 1, . . . , M) represent the accessible ports, and the remaining P ports (m = M + 1, . . . , M + P) represent those connected to network II. All the accessible ports are assumed to be terminated in matched loads. The scattering matrix of network I can be expressed as Sˆ I =
冋
[Sˆ AA ] [Sˆ BA ]
[Sˆ AB ] [Sˆ BB ]
册
(2.110)
46
Microwave Network Theory
Figure 2.12
Interconnection of two multiport networks.
where [Sˆ AA ] and [Sˆ BB ] are matrices of order M × M and P × P, respectively. [Sˆ AB ] and [Sˆ BA ] are matrices of order M × P and P × M, respectively. Further, let the ports of network II be numbered in a manner that its P ports (m = 1, . . . , P) represent those connected to network I and the remaining N ports (m = P + 1, . . . , P + N) represent the free ports. The scattering matrix of network II can be represented as Sˆ II =
冋
[Sˆ CC ] [Sˆ DC ]
[Sˆ CD ] [Sˆ DD ]
册
(2.111)
where [Sˆ CC ] and [Sˆ DD ] denote the matrices of order P × P and N × N, respectively. [Sˆ CD ] and [Sˆ DC ] denote matrices of order P × N and N × P, respectively. The scattering matrix of the overall network of (M + N) ports can be easily derived using matrix algebra [5, 6]. It follows that the scattering matrix of the overall network denoted as Sˆ R can be expressed in concise notation as Sˆ R =
冋
[Sˆ 1 ] [Sˆ 3 ]
册
[Sˆ 2 ] [Sˆ 4 ]
(2.112)
where matrices [Sˆ 1 ], [Sˆ 2 ], [Sˆ 3 ], and [Sˆ 4 ] are given by [Sˆ 1 ] = [Sˆ AA ] + [Sˆ AB ] (U − [Sˆ CC ][Sˆ BB ])−1 [Sˆ CC ][Sˆ BA ] [Sˆ 2 ] = [Sˆ AB ] (U − [Sˆ CC ][Sˆ BB ])−1 [Sˆ CD ]
(2.113)
[Sˆ 3 ] = [Sˆ DC ] (U − [Sˆ BB ][Sˆ CC ])−1 [Sˆ BA ] [Sˆ 4 ] = [Sˆ DD ] + [Sˆ DC ] (U − [Sˆ BB ][Sˆ CC ])−1 [Sˆ BB ][Sˆ CD ] In (2.113) [U ] denotes a unit matrix and ()−1 denotes the inverse of a matrix. A unit matrix is a square matrix. All the diagonal elements of a unit matrix are
2.7 Scattering Matrix of Interconnected Networks
47
unity, while all its nondiagonal elements are zero. The scattering matrix of the overall network [Sˆ R ] is a square matrix of order M + N. 2.7.1 Scattering Parameters of Reduced Networks
The scattering parameters of a network are defined by assuming that all its ports are terminated in matched loads. The ports of a network are usually matchterminated when connected in a system. The elements of the scattering matrix thus directly give the reflection coefficient at each port and coupling between various ports. When one or more ports of a network are not matchterminated, however, reflections take place from these ports, and the scattering parameters of the remaining network are modified. For example, if the ‘‘direct’’ and ‘‘coupled’’ ports of a backward TEM directional coupler are opencircuited, the resulting twoport network behaves as a bandpass filter. Consider a network of M + P ports as shown in Figure 2.13, whose scattering matrix is assumed to be given by (2.110). The ports i = 1, . . . , M of the network are assumed to be terminated in matched loads, while the ports i = M + 1, . . . , M + P are assumed to be connected in loads of reflection coefficient ⌫L1 , ⌫L2 , . . . ⌫LP , respectively. It is required to find the scattering matrix of the reduced network. The scattering matrix of the reduced network will be of the order M × M. We can assume that the given network and the loads shown in Figure 2.13 represent networks I and II of Figure 2.12, respectively, where N = 0. The matrices [Sˆ CD ], [Sˆ CD ], and [Sˆ DD ] are therefore null matrices, and the matrix [Sˆ CC ] is a square diagonal matrix given by
[Sˆ CC ] =
冤
⌫L1 0 ⯗ 0
0 ⌫L2 ⯗ 0
... ... ⯗ ...
0 0 ⯗ ⌫LP
冥
(2.114)
The scattering matrix of the reduced network can be found using (2.113). It is found that the scattering matrix of the reduced Mport network is given by
Figure 2.13
An (M + P) port network whose P ports are terminated in arbitrary loads.
48
Microwave Network Theory
[Sˆ R ] = [Sˆ 1 ]
(2.115)
[Sˆ 1 ] = [Sˆ AA ] + [Sˆ AB ] ([U ] − [Sˆ CC ][Sˆ BB ])−1 [Sˆ CC ][Sˆ BA ]
(2.116)
where
In the above equation [Sˆ AA ], [Sˆ AB ], [Sˆ BB ], and [Sˆ CC ] denote the partitioned matrices of the original M + P port network as defined by (2.110). [Sˆ CC ] is given by (2.114). 2.7.2 Reduction of a ThreePort Network into a TwoPort Network
The use of the above formulas is first demonstrated by finding the modified scattering matrix of a threeport network, one of whose ports is not matchterminated as shown in Figure 2.14. Let the scattering matrix of the threeport network shown in Figure 2.14 be given by
[S ] =
Sˆ 11 Sˆ 21 Sˆ 31
Sˆ 12 Sˆ 22 Sˆ 32
冤
Sˆ 13 Sˆ 23 Sˆ 33
冥
(2.117)
Referring to the notation used earlier in this section, we have [Sˆ AA ] =
冋
Sˆ 11 Sˆ 21
[Sˆ AB ] =
册
Sˆ 12 Sˆ 22
冋 册 Sˆ 13 Sˆ 23
[Sˆ BA ] = [Sˆ 31
Sˆ 32 ]
[Sˆ BB ] = [Sˆ 33 ] and
Figure 2.14
A threeport network with its port three terminated in arbitrary load.
(2.118)
(2.119) (2.120) (2.121)
2.7 Scattering Matrix of Interconnected Networks
49
[Sˆ CC ] = [⌫L3 ]
(2.122)
+ Vˆ3 Z L − Z 03 ⌫L3 = − = Vˆ3 Z L + Z 03
(2.123)
where
The matrices [Sˆ CD ], [Sˆ DC ], and [Sˆ DD ] are null matrices. Substituting the values of the above matrices in (2.115) and (2.116), we find that the scattering matrix of the reduced twoport network is given by [Sˆ R ] =
冋
册冋 册
Sˆ 11 Sˆ 21
Sˆ 12 Sˆ 22
+
Sˆ 13 Sˆ 23
(1 − Sˆ 33 ⌫L3 )−1 ⌫L3 [Sˆ 31
Sˆ 32 ]
(2.124)
which leads to
冤
Sˆ Sˆ ⌫ Sˆ 11 + 13 31 L3 1 − Sˆ 33 ⌫L3 [Sˆ R ] = Sˆ Sˆ ⌫ Sˆ 21 + 23 31 L3 1 − Sˆ 33 ⌫L3
Sˆ Sˆ ⌫ Sˆ 12 + 13 32 L3 1 − Sˆ 33 ⌫L3 Sˆ Sˆ ⌫ Sˆ 22 + 23 32 L3 1 − Sˆ 33 ⌫L3
冥
(2.125)
Equation (2.125) thus describes the scattering matrix of the reduced twoport network. Note that the scattering matrix of the reduced twoport does not satisfy unitary conditions if the load connected to port 3 is lossy. In the following, we give only the final equations describing how the normal scattering parameters of two and fourport networks are modified when some of their ports are not matchterminated.
2.7.3 Reduction of a TwoPort Network into a OnePort Network
Let the scattering matrix of a twoport network as shown in Figure 2.15 be given by [S ] =
Figure 2.15
冋
Sˆ 11 Sˆ 21
册
Sˆ 12 Sˆ 22
A twoport network with its port two terminated in arbitrary load.
(2.126)
50
Microwave Network Theory
If port 2 of the network is terminated in a load of reflection coefficient ⌫L , then the reflection coefficient at port 1 of the network is modified as ˆ ˆ ′ = Sˆ 11 + S 12 ⌫L S 21 Sˆ11 1 − Sˆ 22 ⌫L
(2.127)
′ denotes the reflection coefficient at port 1 of the reduced oneport where Sˆ11 network and ⌫L =
Z L − Z 02 Z L + Z 02
(2.128)
denotes the reflection coefficient at port 2. 2.7.4 Reduction of a FourPort Network into a TwoPort Network
Let the scattering elements of the fourport network as shown in Figure 2.16 be expressed as
冤
Sˆ 11 Sˆ 21 [S ] = Sˆ 31 Sˆ 41
Sˆ 12 Sˆ 22 Sˆ 32 Sˆ 42
Sˆ 13 Sˆ 23 Sˆ 33 Sˆ 43
冥
Sˆ 14 Sˆ 24 Sˆ 34 Sˆ 44
(2.129)
Assume that ports 3 and 4 of the network are terminated in arbitrary loads. The reflection coefficients are given by ⌫L3 =
+ Vˆ3 Z L3 − Z 03 − = Vˆ3 Z L3 + Z 03
(2.130)
⌫L4 =
+ Vˆ4 Z L4 − Z 04 − = Vˆ4 Z L4 + Z 04
(2.131)
and
Figure 2.16
A fourport network with its two ports terminated in arbitrary load.
2.7 Scattering Matrix of Interconnected Networks
51
Using (2.115) and (2.116), the scattering parameters of the resulting twoport network are found as [7] ⌫L3 Sˆ ij′ = Sˆ ij +

Sˆ i3 Sˆ 3j Sˆ 4j
 
−Sˆ i3 Sˆ 34 ⌫L4 Sˆ 3j + ⌫ L4 ˆ ˆ 1 − Sˆ 44 ⌫L4 −S i4 S 4j 1 − Sˆ 33 ⌫L3 −Sˆ 34 ⌫L4 ˆ −S 43 ⌫L3 1 − Sˆ 44 ⌫L4


1 − Sˆ 33 ⌫L3 Sˆ i4 Sˆ 43 ⌫L3
 (2.132)
where i = 1, 2 and j = 1, 2 and Sˆ ij′ denotes the element of the ith row and jth column of the scattering matrix of the remaining network.
References [1] [2] [3] [4]
[5] [6]
[7]
Collin, R. E., Foundations for Microwave Engineering, New York: McGrawHill, 1966. Kurokawa, K., ‘‘PowerWaves and the Scattering Matrix,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, March 1965, pp. 194–202. Marks, R. B., and D. F. Williams, ‘‘A General Waveguide Circuit Theory,’’ J. Res. Natl. Inst. Stand. Technol., Vol. 97, September–October 1992, pp. 533–561. Beatty, R. W., and D. M., Kerns, ‘‘Relationships Between Different Kinds of Network Parameters, Not Assuming Reciprocity or Equality of the Waveguide or Transmission Line Characteristic Impedance,’’ Proc. IEEE, Vol. 52, January 1964, p. 84, Corrections: April 1964, p. 420. Sazanov, D. M., A. N. Gridin, and B. A. Mishustin, Microwave Circuits, Moscow: Mir Publishers, 1982. von Abele, T.A., ‘‘Uber di streumatrix allgemein zusammengeschalteter mehrpole,’’ (‘‘The Scattering Matrix of a General Interconnection of Multipoles’’), Arch. Elek. Ubertragung, Vol. 14, Pt. 6, 1960, pp. 262–268. Otoshi, T. Y., ‘‘On the Scattering Parameters of a Reduced Multiport,’’ IEEE Trans. Microwave Theory Tech., MTT17, September 1969, pp. 722–724.
CHAPTER 3
Characteristics of Planar Transmission Lines Transmission lines used at microwave frequencies can be broadly divided into two categories: those that can support a TEM (or quasiTEM) mode of propagation and those that cannot. For TEM (or quasiTEM) modes, the determination of important electrical characteristics (such as characteristic impedance and phase velocity) of single and coupled lines reduces to finding the capacitances associated with the structure. Furthermore, the conductor loss of TEM (or quasiTEM) mode transmission lines can be determined in terms of variation of the characteristic impedance with respect to the geometrical parameters. This chapter discusses the general characteristics of single and coupled planar TEM and quasiTEM mode transmission lines. Further, design equations of some single and coupled popular planar integrated transmission lines are given. The transmission lines considered are a stripline, microstrip line, coplanar waveguide, and their variants. Of all the planar transmission lines, microstrip is still the most popular for realizing microwave integrated circuits. We discuss characteristics of the microstrip line therefore in greater detail.
3.1 General Characteristics of TEM and QuasiTEM Modes It is an important property of any twoconductor lossless transmission line placed in a homogeneous dielectric medium that it supports a pure TEM mode of propagation. Common examples of these lines are a twinwire line, coaxial line, and shielded stripline as shown in Figure 3.1. If a twoconductor transmission line is enclosed in an inhomogeneous dielectric medium, the mode of propagation is pure TEM only in the limit of zero frequency. The most common example of such a transmission line is a microstrip line as shown in Figure 3.2(a). Some other examples of inhomogeneous transmission lines are a slotline and a coplanar waveguide (CPW) as shown in Figure 3.2(b, c), respectively. If the separation between the conductors of an inhomogeneous transmission line is very small compared to the wavelength, the mode of propagation on the line can be considered to be close to TEM. This mode is called a quasiTEM mode. The characteristic impedance and complex propagation constant of a TEM or a quasiTEM mode transmission line can be described in terms of basic parameters of the line (i.e., its per unit length resistance R, inductance L , capacitance C, and conductance G). The equivalent circuit of a transmission line of length dz is shown
53
54
Characteristics of Planar Transmission Lines
Figure 3.1
Common TEM mode transmission lines: (a) coaxial line, (b) twin wire line, and (c) shielded stripline.
Figure 3.2
Common quasiTEM mode transmission lines: (a) microstrip line, (b) slot line, and (c) coplanar waveguide.
in Figure 3.3. For a transmission line placed in an inhomogeneous medium, the relations given are valid in the quasistatic limit, which means that the operating frequency is assumed to be low enough so that the distance between the conductors of the transmission line is very small compared to the wavelength (≈ g /20 or smaller). For the present discussion, we assume that the conductors of the transmission line have a finite but very high conductivity. It is also assumed that the dielectric loss in the material surrounding the conductors of the transmission line is finite but small. The parameters of interest for a transmission line are its characteristic impedance Z 0 , phase constant  (or phase velocity v p ), and attenuation constant ␣ . In
3.1 General Characteristics of TEM and QuasiTEM Modes
Figure 3.3
55
Equivalent circuit of a TEM or quasiTEM transmission line of length dz.
terms of parameters R, G, L, and C expressed per unit length, the characteristic impedance and the propagation constant ␥ of a transmission line are given by [1] Z0 =
√
R + j L G + j C
(3.1)
␥ = √(R + j L) (G + j C)
(3.2)
At microwave frequencies, lowloss conditions L Ⰷ R and C Ⰷ G are usually satisfied for transmission line conductors fabricated out of normal metals and enclosed in a low dielectric loss medium. Equations (3.1) and (3.2) then reduce to
√
Z0 =
冋
␥ = j √LC 1 +
L C
(3.3)
R G + 2j L 2j C
册
(3.4)
By substituting (3.3) into (3.4), the complex propagation constant ␥ can also be expressed as,
␥ = ␣ + j =
冉
冊
1 R + GZ 0 + j √LC 2 Z0
(3.5)
where = 2 f denotes the angular frequency. From (3.5):
=
= √LC rad/unit length vp
(3.6)
where  and v p denote the phase constant and phase velocity, respectively, along the direction of propagation. The attenuation constant ␣ is given by
␣=
冉
冊
1 R + GZ 0 Np/unit length 2 Z0
(3.7)
56
Characteristics of Planar Transmission Lines
It is common to express the attenuation in decibels (dB) rather than in nepers (Np). The loss in dB is obtained by multiplying the loss in Np by 8.686. The attenuation of the transmission line can therefore also be expressed as
␣ = 4.343
冉
冊
R + GZ 0 dB/unit length Z0
(3.8)
Relation Between Characteristic Impedance Z 0 , Line Capacitance C, and Phase Velocity v p
Eliminating L from (3.3) and (3.6) leads to the following very significant result: Z0 =
1 vp C
(3.9)
Equation (3.9) shows that the characteristic impedance of a transmission line is related to the phase velocity along the transmission line and the capacitance (per unit length) between the conductors of the transmission line. It is also possible to express the phase velocity in terms of the ratio of the actual capacitance of the transmission line to the capacitance of the same transmission line obtained by assuming the dielectric constant of the medium in which it is placed to be unity. Therefore, the problem of determining the characteristic impedance and phase velocity of the structure reduces essentially to the problem of finding the capacitance of the structure. QFactor
Equation (3.7) shows that the total line attenuation is due to two factors: the series resistance R and shunt conductance G. The total attenuation can therefore be expressed as
␣ = ␣c + ␣d
(3.10)
where
␣c =
R Np/unit length 2Z 0
(3.11)
GZ 0 Np/unit length 2
(3.12)
denotes the conductor loss, and
␣d =
denotes the dielectric loss. The attenuation of a transmission line can also be expressed in terms of the Qfactor. The Qfactor of a halfwavelength transmission line resonator is given by
3.1 General Characteristics of TEM and QuasiTEM Modes
Q=
  = 2␣ 2(␣ c + ␣ d )
57
(3.13)
where the attenuation ␣ is expressed in Np/unit length. We can also define the Qfactors for conductor (Q c ) and dielectric loss (Q d ) separately as Qc =
 2␣ c
(3.14)
Qd =
 2␣ d
(3.15)
and
Using (3.13) and (3.14), the overall Qfactor can be expressed as 1 1 1 + = Q Qc Qd For dispersive lines, (3.13) and (3.14) are incorrect and require that the term  in (3.13) and (3.14) be replaced by /v g , where v g denotes the group velocity [1]. For example, (3.13) then becomes Q=
= 2v g ␣ 2v g (␣ c + ␣ d )
(3.16)
Equation (3.16) is more general than (3.13) and is valid for nonTEM modes as well. 3.1.1 TEM Modes
We now specialize some of the above equations for the case when the transmission line is placed in a homogeneous dielectric medium. Some examples of these types of lines are shown in Figure 3.1. For these lines, the velocity of propagation v p along the transmission line is independent of the type of the transmission line and frequency of operation and is given by vp =
c √⑀ r
(3.17)
where c is the velocity of light in freespace and ⑀ r denotes the dielectric constant (relative permittivity) of the medium. The phase constant along the transmission line is therefore given by
=
√⑀ r = = k 0 √⑀ r rad/unit length vp c
(3.18)
58
Characteristics of Planar Transmission Lines
where k 0 = 2 / 0 denotes the freespace propagation constant and 0 denotes the freespace wavelength. Substituting the value of phase velocity from (3.18) in (3.9) leads to the following relation between the characteristic impedance and capacitance of the line: Z0 =
√⑀ r = √⑀ r = 1 cC c⑀ r C 0 c √⑀ r C 0
(3.19)
where C 0 denotes the capacitance between the conductors of the transmission line assuming that the transmission line is placed in a medium of a unity dielectric constant. The determination of dielectric loss ␣ d is also straightforward in this case. It is given by
␣d =
k 0 √⑀ r  tan ␦ = tan ␦ Np/unit length 2 2
(3.20)
or
␣ d = 4.343 tan ␦ = 4.343k 0 √⑀ r tan ␦ = 27.3 √⑀ r
tan ␦ dB/unit length 0 (3.21)
where tan ␦ denotes the loss tangent of the dielectric material. In general, the loss tangent tan ␦ is also a function of frequency. On the other hand, conductor loss ␣ c depends on the type of line, the conductivity of the transmission line, the frequency of operation, and geometrical parameters of the line and is discussed in detail in a later section.
3.1.2 QuasiTEM Modes
Some examples of quasiTEM mode transmission lines are shown in Figure 3.2. For quasiTEM modes, the effective dielectric constant ⑀ re is defined as follows:
⑀ re =
c2 v p2
(3.22)
In qualitative terms, the effective dielectric constant ⑀ re takes into account the relative distribution of electric energy in the various regions of the inhomogeneous medium. The relation between phase constant  , effective dielectric constant ⑀ re , and phase velocity v p is
=
√⑀ re = = √⑀ re k 0 vp c
(3.23)
3.1 General Characteristics of TEM and QuasiTEM Modes
59
⑀ re is a function of frequency and strictly speaking should be evaluated using (3.22) where the phase velocity v p is computed using some rigorous method based on Maxwell’s equations. However, in the quasistatic limit, ⑀ re can be assumed to be ⑀ re =
C C0
(3.24)
where C denotes the capacitance between conductors of the transmission line in the inhomogeneous dielectric medium and C 0 denotes the capacitance between the same conductors in a medium of unity dielectric constant. Using (3.9), (3.23), and (3.24), the characteristic impedance of a quasiTEM mode transmission line can be expressed as Z0 =
√⑀ re = cC
1 Z 1 = = 0a c √CC 0 c √⑀ re C 0 √⑀ re
(3.25)
where Z 0a denotes the characteristic impedance of the same transmission line placed in a medium of unity dielectric constant. The dielectric loss of a quasiTEM mode transmission line is given by
␣ d = 27.3
⑀ r (⑀ re − 1) tan ␦ dB/unit length √⑀ re (⑀ r − 1) 0
(3.26)
The conductance G of a transmission line can be expressed in terms of loss tangent tan ␦ as follows: G=
2 ⑀ r (⑀ re − 1) tan ␦ Z 0 √⑀ re (⑀ r − 1) 0
(3.27)
Furthermore, it is customary to define the effective filling fraction q of a quasiTEM mode transmission line as follows:
⑀ −1 q = re ⑀r − 1
(3.28)
3.1.3 Skin Depth and Surface Impedance of Imperfect Conductors
At high frequencies, the current flowing in a conductor tends to get confined near the outer surface of the conductor. The skin depth of a conductor is defined as the distance in the conductor (along the direction of the normal to the surface) in which the current density drops to 37% of its value at the surface (the current decays to a negligible value in a distance of about 4 to 5 skin depths) and is given by
␦s =
√
2
(3.29)
60
Characteristics of Planar Transmission Lines
where = r 0 , 0 = 4 × 10−7 H/m denotes the permeability of free space, and denotes the conductivity (S/m) of the conductor. Further, r denotes the relative permeability of the material. Its value is almost equal to unity except for magnetic materials. Equation (3.29) shows that the skin depth of a perfect conductor ( = ∞) is zero. The conductivity of normal metals (which are used as conductors) is very high, although finite. For normal metals, the skin depth is therefore very small at microwave frequencies (e.g., the conductivity of copper is 5.8 × 107 S/m and the skin depth at 10 GHz is 0.66 m). The tangential electric field at the surface of a conductor is not zero due to finite conductivity of the conductors of a transmission line.1 The surface impedance (ohms/square) of a conductor (defined as the ratio of tangential electric and magnetic fields at the surface) is given by Z s = R s + j L s =
1+j ␦ s
(3.30)
or 1 Rs = Ls = ␦ s
(3.31)
(e.g., the surface impedance of copper at 10 GHz is 0.026 + j0.026 ohm/square). 3.1.4 Conductor Loss of TEM and QuasiTEM Modes
An ingenious way to determine the conductor loss of TEM or quasiTEM mode transmission lines was given by Wheeler [2]. This method of determining the conductor loss is also known as the incremental inductance rule. The rule is valid only if the thickness of conductors of the transmission lines and the radius of curvature of conductor surfaces are at least five to six times the skin depth. These conditions can usually be satisfied at microwave frequencies except near very sharp edges. According to this rule, the conductor Qfactor of a transmission line is given by [3] Qc =
′ Z0a ′ − Z 0a ) (Z0a
(3.32)
′ denotes the impedance of the same where Z 0a has been defined before and Z0a transmission line placed in a medium of unity dielectric constant, but assuming that the thickness of all conductors is reduced by ␦ s /2 from each surface where the fields are present. This is shown in Figure 3.4, where solid lines show the surfaces of the actual microstrip transmission line of a finite strip thickness and dashed lines show the surfaces of the fictitious microstrip transmission line obtained by removing a depth of ␦ s /2 from each surface of the conductor of the original transmission line. Note that the conductor Qfactor (but not the attenuation per unit length) of a TEM or quasiTEM mode transmission line is independent of the 1.
The tangential electric field at the surface of a perfect conductor is zero.
3.2 Representation of Capacitances of Coupled Lines
Figure 3.4
61
Illustration of incremental inductance rule showing the original and perturbed geometry of a microstrip line.
dielectric constant of the medium in which it is placed. Using (3.14), (3.23), and (3.32), the attenuation factor ␣ c can be expressed as
␣ c = √⑀ re k 0
′ − Z 0a ) √⑀ re f (Z0a ′ − Z 0a ) (Z0a = ′ ′ c 2Z0a Z0a
(3.33)
where f is the frequency of operation.
3.2 Representation of Capacitances of Coupled Lines The coupling between lines can be expressed in terms of self and mutual capacitances. It is therefore useful to discuss the representation of capacitances of coupled transmission lines. Figure 3.5 shows the cross section of two coupled transmission lines having a common ground conductor with the capacitances associated with the coupled structure as shown. If Q 1 and Q 2 denote the charges and V1 and V2 denote the voltages of conductors 1 and 2, respectively, the charges Q 1 and Q 2 can be expressed in terms of voltages and capacitances as Q 1 = C a V1 + C m (V1 − V2 ) = (C a + C m )V1 − C m V2
(3.34)
Q 2 = C m (V2 − V1 ) + C b V2 = −C m V1 + (C b + C m )V2
(3.35)
The capacitance matrix of two coupled transmission lines is represented as [4]
Figure 3.5
Representation of capacitances of coupled lines.
62
Characteristics of Planar Transmission Lines
[C] =
冋
C 11 C 21
册
C 12 C 22
(3.36)
where C 11 and C 22 are defined as selfcapacitances of lines 1 and 2, respectively, in the presence of each other. The capacitance matrix denotes the relation between charges and voltages on the two transmission lines as follows: Q 1 = C 11 V1 + C 12 V2
(3.37)
Q 2 = C 21 V1 + C 22 V2
(3.38)
Hence the capacitance matrix of coupled lines can be expressed as [C] =
冋
C 11 C 21
册冋
C 12 C 22
=
Ca + Cm −C m
−C m Cb + Cm
册
(3.39)
The inductance matrix of a coupled line is given by [L] = 0 ⑀ 0 [C 0 ]−1
(3.40)
where [C 0 ] denotes the capacitance matrix of the transmission lines obtained by assuming that these are placed in a medium of unity dielectric constant. 3.2.1 Even and OddMode Capacitances of Symmetrical Coupled Lines
When the coupled lines are identical (also called symmetrical coupled lines), their capacitance matrix can be expressed in terms of even and oddmode capacitances. EvenMode Excitation
A cross section of uniformly coupled symmetrical lines is shown in Figure 3.6(a). In this case, the capacitance C m shown in Figure 3.5 has been broken into two capacitances of values 2C m each in series. With evenmode excitation, equal and inphase voltages (V1 = V2 = Ve ) are applied to both lines. Because the geometry under consideration is symmetrical, it is clear that if equal voltages of the same polarity are applied to both the lines, the charges on the two lines would also be the same (i.e., Q 1 = Q 2 = Q e ). Denoting the ratio Q e /Ve by C e , (3.34) and (3.35) then reduced to Ca = Cb =
Qe = Ce Ve
(3.41)
OddMode Excitation
In the oddmode excitation, equal but outofphase voltages (V1 = −V2 = Vo ) are applied to the two lines as shown in Figure 3.6(b). It follows from the symmetry
3.2 Representation of Capacitances of Coupled Lines
Figure 3.6
63
(a) Even and (b) oddmode excitation of symmetrical coupled lines.
of the structure that if equal voltages but of opposite polarity are applied to the symmetrical lines, equal charges but of opposite polarity will be induced on the two lines (i.e., Q 1 = −Q 2 = Q o ). With the ratio Q o /Vo denoted by C o , (3.34) and (3.35) reduced to Q o = (C a + 2C m )Vo
(3.42)
Q C a + 2C m = o = C o Vo
(3.43)
or
Substituting the value of C a from (3.41) in (3.43), we obtain Cm =
Co − Ce 2
(3.44)
Therefore, once the even and oddmode capacitance parameters of coupled symmetrical lines are known, C a , C b , and C m can be determined from (3.41) and (3.44). The capacitance matrix of coupled lines can then be determined using (3.39).
64
Characteristics of Planar Transmission Lines
In the evenmode of excitation, the symmetry plane PP ′ as shown in Figure 3.6(a) acts as a magnetic wall (open circuit). The determination of the evenmode capacitance reduces to finding the capacitance of either line with the plane of symmetry PP ′ replaced by a magnetic wall such as shown in Figure 3.7(a). This results in a great simplification of the problem. Similarly, in the oddmode of excitation, the symmetry plane behaves as an electric wall (short circuit). The determination of the oddmode capacitance reduces to finding the capacitance of either line by replacing the plane of symmetry by an electric wall as shown in Figure 3.7(b). The relationships between the even and oddmode capacitances and impedances are given by Z 0e =
1 = v pe C e  e C e
(3.45)
Z 0o =
1 = v po C o  o C o
(3.46)
and
Figure 3.7
Representation of capacitances of (a) even and (b) odd modes of symmetrical coupled lines.
3.2 Representation of Capacitances of Coupled Lines
65
where Z 0e , v pe , and  e denotes the characteristic impedance, phase velocity, and phase constant, respectively, of the even mode of the coupled lines; and Z 0o , v po , and  o denote the same quantities for the odd mode. If the lines are placed in a homogeneous medium of dielectric constant ⑀ r , the even and oddmode phase velocities are equal and are given by v pe = v po =
c √⑀ r
(3.47)
However, if the lines are placed in an inhomogeneous dielectric media (such as coupled microstrip lines), the even and oddmode phase velocities are, in general, different and are given by v pe =
c √⑀ ree
(3.48)
v po =
c √⑀ reo
(3.49)
and
where ⑀ ree and ⑀ reo are defined as the even and oddmode effective dielectric constants, respectively. These can be determined using
⑀ ree =
Ce C 0e
(3.50)
⑀ reo =
Co C 0o
(3.51)
and
where C 0e and C 0o denote, respectively, the even and oddmode capacitance of either line obtained by replacing the relative permittivity of the surrounding dielectric material by unity. C e and C o denote the corresponding capacitances in the presence of the inhomogeneous dielectric medium. Using (3.48) to (3.51), (3.45) and (3.46) reduce to Z 0e =
1 c √C e C 0e
(3.52)
Z 0o =
1 c √C o C 0o
(3.53)
and
66
Characteristics of Planar Transmission Lines
3.2.2 ParallelPlate and Fringing Capacitances of Single and Coupled Planar Transmission Lines
So far we have discussed the capacitances of single and coupled transmission lines in general. In many cases, it is possible to visualize the various components of the total capacitance of the structure. This helps in obtaining a physical understanding of the problem and its analysis. For example, for planar transmission lines, the total capacitance can be broken into its various components such as parallelplate and fringing capacitances. To explain the various components, we consider for simplicity single and coupled microstrip lines.
Single Line
The electric field distribution of a single microstrip line is shown in Figure 3.8(a). Because of the finite width of the microstrip line, fields not only exist directly below the strip conductor, but extend to the surrounding regions as well. The latter are known as fringing fields. The capacitance that results from the electric fields in the region directly below the strip is known as the parallelplate capacitance, while that resulting from the fringing fields is known as the fringing capacitance. The total capacitance associated with a single microstrip line can be represented as shown in Figure 3.8(b) and is given by C = C p + 2C f
(3.54)
⑀ ⑀W Cp = 0 r h
(3.55)
where
Figure 3.8
(a) Electric field distribution of single microstrip line, and (b) equivalent capacitance representation.
3.2 Representation of Capacitances of Coupled Lines
67
denotes the parallelplate capacitance and C f denotes the fringing capacitance from either edge of the microstrip line. Once the value of C f is known, the total capacitance of the line can be determined using (3.54) and (3.55). Conversely, if the characteristic impedance and effective dielectric constant of a microstrip line are known, the capacitance C can be found using (3.25), and using (3.54) and (3.55), the fringing capacitance C f can be determined. Symmetrical Coupled Lines
A crosssection of symmetrical coupled microstrip lines is shown in Figure 3.9(a) with the electric field distribution in Figure 3.9(b) for onehalf of the structure for
Figure 3.9
(a) Cross section of symmetrical coupled lines, (b, c) electric field distribution and capacitance representation of onehalf of the structure for evenmode excitation, and (d, e) electric field distribution and capacitance representation of onehalf of the structure for oddmode excitation.
68
Characteristics of Planar Transmission Lines
the case of evenmode excitation. In this case, the normal component of the electric field at the plane of symmetry PP ′ is zero, because the plane of symmetry behaves like a magnetic wall. The evenmode capacitance of either of the coupled lines, which can be represented as shown in Figure 3.9(c), is given by, C e = C p + C f + C fe
(3.56)
where C p is given by (3.55). If the two lines are not of very narrow width, it can be assumed that the value of C f is the same as that of a single microstrip line having the same width as that of either of the coupled lines. The electric field distribution of onehalf of the coupled structure is shown in Figure 3.9(d) for the case of oddmode excitation. The tangential electric field at the plane of symmetry PP ′ is zero, because in this case, the plane of symmetry behaves like an electric wall. The oddmode capacitance of either of the coupled lines is given by C o = C p + C f + C fo
(3.57)
where C fo denotes the fringing capacitance from the inner edges of the coupled lines. When the spacing between the lines is small (S/2 is small compared to the height of the substrate h), nearly all the fringing fields that start from the inner edge of one of the lines terminate on the mid plane PP ′. In that case, the capacitance of either of the coupled lines, which can be represented as shown in Figure 3.9(e), is given by, C o = C p + C f + C fo = C p + C f + C ga + C gd
(3.58)
where the fringing capacitance C fo is assumed to consist of two capacitances C ga and C gd in parallel; that is, C fo = C ga + C gd
(3.59)
The characteristic impedances and phase velocities of the even and odd modes can be found by determining the parallel plate and fringing capacitances associated with the structure. Conversely, if the characteristic impedances and phase velocities of the even and odd modes are known, the corresponding fringing capacitances can be determined. The breakup of the total capacitance into various components involves certain approximations. For example, (3.55) can only be termed as approximate because the fields under the strip are not exactly vertical, especially near the edges. The breakup of the total capacitance into various components, however, helps in a simple, although approximate design of various coupledline components such as a Lange coupler.
3.3 Characteristics of Single and Coupled Striplines A commonly used stripline is shown in Figure 3.10. The strip conductor is sandwiched between two flat dielectric substrates having the same dielectric constant.
3.3 Characteristics of Single and Coupled Striplines
Figure 3.10
69
Cross section of a stripline.
The outer surfaces of the dielectric substrates are metallized and serve as ground conductors. The signal is applied between the strip conductor and ground. In a commonly used fabrication technique, the strip conductor is etched on one of the dielectric substrates by the process of photolithography. The thickness of both the substrates is generally the same, although it is not essential. A lossless stripline can support a pure TEM mode of propagation at all frequencies. Two striplines can be coupled by placing strip conductors side by side as shown in Figure 3.11 in the edgecoupled configuration. In this configuration, both the strip conductors lie in the same plane, and although it is very convenient for realizing coupled line circuits, it has the disadvantage that tight coupling between the lines cannot be achieved. For tight coupling (about 8 dB or tighter), the width of the strip conductors and the spacing between them becomes quite small, making fabrication difficult. Further, the small dimensions lead to large current densities on the strip conductor leading to higher conductor loss. The edgecoupled configuration is thus suitable for the design of loose couplers having coupling of 8 to 30 dB. Tighter coupling (e.g., 3 dB) can be obtained using broadsidecoupled striplines, as discussed in Section 3.7. 3.3.1 Single Stripline
Based upon the SchwartzChristoffel transformation, the exact expression for the characteristic impedance of a lossless stripline of zero thickness is given by [5, 6] Z 0 √⑀ r = 30
K(k) K(k ′)
(3.60)
where k = sech
冉 冊
W ; k′ = 2b
√1 − k
2
In the above expression, K denotes the complete elliptic integral of the first kind and sech denotes the hyperbolic secant. An approximate expression for K(k)/K(k ′), which is accurate to within 8 ppm, is given by [7]
Figure 3.11
Cross section of edgecoupled striplines.
70
Characteristics of Planar Transmission Lines
冉
1+ K(k) K(k) 1 = = ln 2 K(k ′) K ′(k) 1− =
冊
√k , 0.5 ≤ k 2 ≤ 1 √k
冉
ln 2
√k ′ 1 − √k ′ 1+
冊
(3.61)
, 0 ≤ k 2 ≤ 0.5
Effect of Finite Strip Thickness on Characteristic Impedance
The capacitance between the strip conductor and ground is increased if the thickness of the strip is finite. This happens because of the additional capacitance resulting from the strip conductor edges of finite thickness. The characteristic impedance of a stripline of finite thickness is given by Wheeler [8] as
再
Z 0 √⑀ r = 30 ln 1 +
冋
1 (16h / W ′ ) (16h / W ′ ) + 2
√(16h / W ′ )
2
+ 6.27
册冎 (3.62)
where W ′ denotes the effective width of the stripline. When the thickness of the strip conductor is zero, the effective width is the same as the physical width. For finite strip thickness, the effective width W ′ is given by W ′ = W + ⌬W
(3.63)
where ⌬W 1 = ln t
2.718
√冋
册 冋
册
m 2 1 1/4 + 4h /t + 1 W /t + 1.1
m=
6 3 + t /h
The error in Wheeler’s equation is expected to be less than 1%. Closedform expressions for the synthesis of a stripline have also been given by Wheeler [8]. These are as follows: W = W ′ − ⌬W ′ where W ′ 16 = h
√(e
4 r
(e
− 1) + 1.568 4 r
− 1)
(3.64)
3.3 Characteristics of Single and Coupled Striplines
r= ⌬W ′ 1 = ln t
71
√⑀ r Z 0 376.7 2.718
√冋
册 冋
册
m 2 1 1/4 + 4h /t + 1 W ′/t − 0.26
m=
6 3 + t /h
The characteristic impedance of a stripline as a function of strip width for various values of t is shown in Figure 3.12 together with the characteristic impedance of striplines with strip conductor of square and circular cross section (Figure 3.13). The propagation constant and dielectric loss of a stripline are given by (3.18) and (3.21), respectively. The conductor loss depends on geometrical parameters of the line and can be computed in a rather simple manner using Wheeler’s incremental inductance rule as discussed earlier [2]. For a stripline, the application of the rule leads to the following expression for the conductor loss [9]:
␣c =
Figure 3.12
冋
册
R s √⑀ r ∂Z 0 ∂Z 0 ∂Z 0 Np/m − − 376.7Z 0 ∂b ∂W ∂t
(3.65)
Characteristic impedance of finite thickness stripline versus W/h for various values of t/h. Curves marked 䊐 and 䊊 are valid for stripline having stripconductor of square or circular cross section, respectively, as shown in Figure 3.13. (From: [8]. 1978 IEEE. Reprinted with permission.)
72
Characteristics of Planar Transmission Lines
Figure 3.13
Stripline with stripconductor of square or circular cross section.
where R s is the surface resistance (ohm/square) given by (3.31) and ∂ denotes the partial derivative. The normalized conductor loss (h/Q c ␦ s ) of a stripline is shown in Figure 3.14 where Q c and ␦ s denote the conductor Qfactor and skin depth, respectively. The skin depth ␦ s , is given by (3.29). For given values of W and h, the conductor Qfactor Q c can be determined using Figure 3.14. The conductor loss in Np/m can then be determined using (3.14). It may be noted that the normalized conductor loss (h/Q c ␦ s ) is independent of the dielectric constant of the substrate.
Figure 3.14
Normalized conductor Qfactor (h/Q c ␦ s ) of a stripline versus W/h for various values of t/h (valid for arbitrary value of ⑀ r ). Curves marked 䊐 and 䊊 are valid for stripline having stripconductor of square or circular cross section, respectively, shown in Figure 3.13. (From: [8]. 1978 IEEE. Reprinted with permission.)
3.3 Characteristics of Single and Coupled Striplines
73
For a wide stripline, the cutoff frequency of the first higher order mode is given by [10] 1 15 GHz (W /b + /4) b √⑀ r
fc =
(3.66)
where W and b are in centimeters. 3.3.2 EdgeCoupled Striplines
The even and oddmode charateristic impedances of edgecoupled striplines of zero thickness (Figure 3.11) are given by the following expressions, which are exact [11]: Z 0e √⑀ r = 30
K(ke′ ) K(k e )
(3.67)
where k e = tanh
冉 冊 冋 W 2 b
tanh
册
(W + S) ; ke′ = 2 b
√1 − k e2
and Z 0o √⑀ r = 30
K(ko′ ) K(k o )
(3.68)
where k o = tanh
冉 冊 冋 W 2 b
coth
册
(W + S) ; ko′ = 2 b
√1 − k o2
The functions K(k e )/K(ke′ ) and K(k o )/K(ko′ ) can be evaluated using (3.61). The impedances (3.67) and (3.68) have been plotted in Figure 3.15. It is seen that when S/b → ∞, both Z 0e and Z 0o approach the same value equal to the characteristic impedance of a single stripline of width W. Synthesis equations for the design of zerothickness coupled striplines are also given by Cohn [11]. These are as follows: W 2 = tanh−1 b
√k e k o
(3.69)
冉 √冊
(3.70)
and S 2 = tanh−1 b
1 − ko 1 − ke
ke ko
74
Characteristics of Planar Transmission Lines
Figure 3.15
Characteristic impedances of even and oddmodes of edge coupled striplines. (From: [11]. 1995 IEEE. Reprinted with permission.)
For given values of Z 0e and Z 0o , the values of k e and k o can be determined by numercially solving (3.67) and (3.68), respectively, with the aid of (3.61). For example, if (3.67) is plotted as a function of k e with the aid of (3.61), the value of k e for a given value of Z 0e can be found. The values of W /b and S/b can then be determined by substituting the values of k e and k o in (3.69) and (3.70), respectively. The propagation constants are equal for the even and odd modes and are given by (3.18). Similarly, the dielectric losses are also equal for the even and odd modes and are given by (3.21). The conductor losses, however, are different for the two modes. Expressions for these are given in [12]. The effect of finite thickness of strip conductors on the characteristic impedances of even and odd modes has also been reported in [11].
3.4 Characteristics of Single and Coupled Microstrip Lines A microstrip transmission line is shown in Figure 3.16. Compared with a stripline, a microstrip line uses only a single dielectric substrate. The height of the substrate is chosen to be much smaller than the wavelength in the dielectric. The mode of propagation along a microstrip line is not pure TEM. However, because of the
3.4 Characteristics of Single and Coupled Microstrip Lines
Figure 3.16
75
A microstrip line.
simplicity of the structure and ease of fabrication, the microstrip line is the most popular transmission line for realizing microwave integrated circuits (MIC). Commonly used substrates are alumina (Al2O3 ), teflonbased materials such as RT duroid for hybid MICs, and gallium arsenide (GaAs) for monolithic MICs. Numerous methods have been used to determine the characteristics of single and coupled microstrip lines. For design purposes, many simple closedform empirical expressions have been reported in the literature [13–22]. A microstrip line is dispersive in nature and its characteristic impedance and effective dielectric constant vary with frequency. The accuracy of various dispersion formulas has also been experimentally verified [23, 24]. The characteristics of microstrip lines are generally described by two sets of equations. One gives the characteristics that are valid in the quasistatic limit, while the other set accounts for dispersion. Quasistatic relations are sufficiently accurate as long as the height of the dielectric substrate is very small compared with the wavelength. 3.4.1 Single Microstrip
The quasistatic problem of a single microstrip line was solved analytically by Wheeler [14]. Over the years, this model has generally served as the basis for deriving simple semiempirical relations for the characteristics of a microstrip line. An accurate expression for effective dielectric constant defined according to (3.22) is given by [18]
⑀ re (0) =
冉
冊
⑀r + 1 ⑀r − 1 10 1+ + 2 2 u
−AB
where u = W /h, and A=1+
册
冋
冋 冉 冊册
u 4 + (u/52)2 u 3 1 1 + 1 + ln ln 49 18.7 18.1 u 4 + 0.432 B = 0.564
冉
⑀ r − 0.9 ⑀r + 3
冊
0.053
(3.71)
76
Characteristics of Planar Transmission Lines
The stated accuracy of (3.71) is better than 0.2% at least for ⑀ r ≤ 128 and 0.01 ≤ u ≤ 100. The quasistatic characteristic impedance of a microstrip line of zero thickness can be expressed as Z 0 (0) =
60 ln √⑀ re
冋
f (u) + u
√ 冉 冊册 1+
2 2 u
(3.72)
冊 册
(3.73)
where
冋冉
f (u) = 6 + (2 − 6) exp −
30.666 0.7528 u
The symbol (0) in ⑀ re (0) and Z 0 (0) denotes that the formula is valid in the quasistatic limit. For ⑀ r = 1, the stated accuracy of (3.72) is better than 0.01% for u ≤ 1 and 0.03% for u ≤ 1,000. For other values of ⑀ r , the accuracy of the expression is determined by the accuracy with which the effective dielectric constant is known. FiniteThickness Microstrip
Improved and more general equations for the analysis and synthesis of microstrip lines have also been given by Wheeler [3]. The characteristic impedance of a microstrip line of finite thickness is given by Z0 =
再
冉 冊 冋冉 冊冉 冊
4h 42.4 ln 1 + W ′ √⑀ r + 1 +
√冉
14 + 8/⑀ r 11
冊冉 冊 4h W′
14 + 8/⑀ r 2 4h 2 1 + 1/⑀ r 2 + 11 W′ 2
(3.74)
册冎
where W ′ denotes the effective width of the microstrip. The effect of finite thickness of the strip is taken into account by assuming that the effective width of the microstrip W ′ is greater than its physical width W and is given by W ′ = W + ⌬W
(3.75)
where
冉
1 + 1/⑀ r ⌬W = t 2
冊
ln
10.872
√冉 冊 冉
冊
2 t 2 1/ + h W /t + 1.1
The above equations are supposed to be accurate for arbitrary values of ⑀ r and aspect ratio W /h (within the quasistatic limit). Note that if a general expression
3.4 Characteristics of Single and Coupled Microstrip Lines
77
is available to determine the characteristic impedance of a microstrip line as a function of the dielectric constant of the substrate, it is not necessary to have a separate formula for the effective dielectric constant. The effective dielectric constant can then be determined using (3.25), that is:
⑀ re =
冉 冊
Z 0a 2 Z0
where Z 0a denotes the characteristic impedance of the microstrip line assuming that the dielectric constant of the substrate is unity. The synthesis equations for a microstrip line are as follows: W = W ′ − ⌬W ′
(3.76)
where
W′ =8 h
冉 冋 √ e
Z0 ⑀ +1 42.4 √ r
冊 − 1册 7 + 4/⑀
r
11
冋冉 e
Z0 ⑀ +1 42.4 √ r
+
1 + 1/⑀ r 0.81
冊 − 1册
and
冉
⌬W ′ 1 + 1/⑀ r = t 2
冊
ln
10.872
√冉 冊 冉
冊
2 t 2 1/ + h W ′/t − 0.26
The characteristic impedance of a microstrip line is shown in Figure 3.17(a) as a function of effective width W ′ for various values of ⑀ r [3]. For a microstrip line of given thickness and width, the effective width can be found using (3.75). Figure 3.17(a) can therefore be used to find the characteristics of a microstrip line of finite thickness. The results shown in Figure 3.17(a) can also be used to find the effective dielectric constant of a microstrip line using (3.25). The characteristics of a microstrip line printed on some commonly used dielectric substrates are shown in Figure 3.17(b).
Example 3.1
Determine the dimension ratio W /h and effective dielectric constant of a microstrip line (t = 0) of characteristic impedance of 50⍀ printed on a dielectric substrate of dielectric constant (⑀ r = 2). From the curve labeled as ⑀ r = 2 in Figure 3.17(a), we find that W /h should be chosen as equal to 3.3 to obtain a characteristic impedance (Z 0 ) of 50⍀. Further,
78
Characteristics of Planar Transmission Lines
Figure 3.17
(a) Quasistatic characteristic impedance and effective dielectric constant of a microstrip line for various values of ⑀ r . ⑀ re = (Z 0a /Z 0 )2. (From: [3]. 1977 IEEE. Reprinted with permission.) (b) Quasistatic characteristic impedance and effective dielectric constant of a microstrip line for some commonly used dielectric substrates. (From: [12]. 1988 John Wiley and Sons. Reprinted with permission.)
for the same value of W /h and ⑀ r = 1, we find from the graph that Z 0a = 66⍀. Using (3.25), we then find that
⑀ re =
冉 冊
66 2 = 1.74 50
3.4 Characteristics of Single and Coupled Microstrip Lines
79
Conductor and Dielectric Loss
Once a general expression for the characteristic impedance of a microstrip is available in terms of the structural parameters, the conductor loss can be easily computed using (3.14) and (3.32). The conductor and other losses of a microstrip line on dielectric and ferrite substrates are discussed in [17]. The dielectric loss of a microstrip line is given by (3.26). The normalized conductor loss (h/Q c ␦ s ) of a microstrip line (which is independent of the dielectric constant of the substrate) is shown in Figure 3.18(a) where Q c and ␦ s denote the conductor Qfactor and skin depth, respectively [3]. For given values of W and h, this figure can be used to determine the conductor Qfactor Q c and hence the conductor loss in Np/m from (3.14). The conductor and dielectric loss of a microstrip line on some commonly used dielectric substrates is shown in Figure 3.18(b).
Microstrip Dispersion
If the frequency of operation is high, so that the height of the substrate is not very small compared with the wavelength in the dielectric, the quasistatic expressions presented above are not accurate enough. In some other situations also, these expressions may be inadequate. For example, if a digital signal with short rise time propagates along a microstrip line, the signal can be assumed to contain a number of highfrequency harmonics. An accurate dispersion model for propagation along microstrip lines is therefore required. The frequency dependence of the effective dielectric constant of a microstrip line is well represented by the following relation [19]:
⑀ re ( f ) = ⑀ r −
⑀ r − ⑀ re (0) 1 + ( f /f 50 )m
(3.77)
where f 50 =
冉
f K, TM 0
0.75 + 0.75 −
c tan−1 f K, TM 0 =
冉√ ⑀r
冊
(3.78)
0.332 W ⑀ r1.73 h
⑀ re (0) − 1 ⑀ r − ⑀ re (0)
冊
(3.79)
2 h √⑀ r − ⑀ re (0) m = m0 mc
m0 = 1 +
冉
冊
3 1 1 + 0.32 1 + √W /h 1 + √W /h
80
Characteristics of Planar Transmission Lines
Figure 3.18
mc =
(a) Normalized conductor Qfactor of a microstrip line (valid for arbitrary value of ⑀ r ). Curves marked 䊐 and 䊊 are valid for microstrip line having stripconductor of square or circular cross section respectively. (From: [3]. 1977 IEEE, Reprinted with permission.) (b) Conductor and dielectric loss of microstrip line on some commonly used dielectric substrates. (From: [13]. 1996 Artech House. Reprinted with permission.)
冦
1+
1
冋
冉
−0.45f 1.4 0.15 − 0.235 exp 1 + W /h f 50
冊册
for W /h ≤ 0.7 for W /h > 0.7
Here c is the velocity of light in free space. It is claimed that the above model has an accuracy of better than 2.7% in the range 0.1 ≤ W /h ≤ 10, 1 ≤ ⑀ r ≤ 128 and for any value of h/ 0 .
3.4 Characteristics of Single and Coupled Microstrip Lines
81
The dispersion of a microstrip line fabricated on a dielectric substrate of ⑀ r = 8 is shown in Figure 3.19. The characteristic impedance of a microstrip line is also frequency dependent and shows a small positive increase with an increase in frequency. The following relation describes this dependence quite accurately [18]: Z 0 ( f ) = Z 0 (0)
√
⑀ re (0) [⑀ re ( f ) − 1] ⑀ re ( f ) [⑀ re (0) − 1]
(3.80)
For many practical purposes, the variation of characteristic impedance with frequency can be neglected. 3.4.2 Coupled Microstrip Lines
Coupled microstrip lines are shown in Figure 3.20. The equations for the quasistatic characteristics of coupled microstrip lines have been given by many authors including Hammerstad and Jenson [18], Garg and Bahl [13, 21], and others. For the oddmode case, the Hammerstad and Jenson equations were later modified by Kirschning and Jansen [22] to incorporate the effect of dispersion. Although the latter expressions are lengthy, these are simple to program and believed to be accurate to within 1% in the range of parameters 0.1 ≤ u ≤ 10, 0.1 ≤ g ≤ 10, and 1 ≤ ⑀ r ≤ 18. The quasistatic evenmode effective dielectric constant for coupled microstrip lines for zero conductor thickness is given by
⑀ ree (0) = 0.5(⑀ r + 1) + 0.5(⑀ r − 1) ⭈ (1 + 10/ )−a e ( ) ⭈ b e (⑀ r )
Figure 3.19
(3.81)
Dispersion characteristics of a microstrip line printed on a dielectric substrate of dielectric constant ⑀ r = 8. (From: [19]. 1988 IEEE. Reprinted with permission.)
82
Characteristics of Planar Transmission Lines
Figure 3.20
Edgecoupled microstrip lines.
where
= u(20 + g 2 )/(10 + g 2 ) + g ⭈ exp (−g) a e ( ) = 1 + ln {[ 4 + ( /52)2 ]/( 4 + 0.432)}/49 + ln [1 + ( /18.1)3 ]/18.7 b e (⑀ r ) = 0.564[(⑀ r − 0.9)/(⑀ r + 3)]0.053 and u = W /h, g = S/h. The quasistatic oddmode effective dielectric constant for zero conductor thickness is similarly given by
⑀ reo (0) = [0.5(⑀ r + 1) + a o (u, ⑀ r ) − ⑀ re (0)] ⭈ exp 冠−c o g d o冡 + ⑀ re (0)
(3.82)
where a o (u, ⑀ r ) = 0.7287[⑀ re (0) − 0.5(⑀ r + 1)] ⭈ [1 − exp (−0.179u)] b o (⑀ r ) = 0.747⑀ r /(0.15 + ⑀ r ) c o = b 0 (⑀ r ) − [b 0 (⑀ r ) − 0.207] ⭈ exp (−0.414u) d o = 0.593 + 0.694 ⭈ exp (−0.562u) and ⑀ re (0) denotes the effective dielectric constant of a single microstrip of width W. Quasistatic Even and OddMode Characteristic Impedances
The quasistatic evenmode characteristic impedance of coupled microstrip lines is given by [22] Z 0e (0) = Z 0
√
⑀ re (0) 1 ⑀ ree (0) {1 − [Z 0 (0)/377] [⑀ re (0)]0.5 Q 4 }
where −1 Q 4 = (2Q 1 /Q 2 ) ⭈ 再exp (−g) ⭈ u Q 3 + [2 − exp (−g)] ⭈ u −Q 3 冎
(3.83)
3.5 Single and Coupled Coplanar Waveguides
83
with Q 1 = 0.8695 ⭈ u 0.194 Q 2 = 1 + 0.7519g + 0.189 ⭈ g 2.31 Q 3 = 0.1975 + [16.6 + (8.4/g)6 ]−0.387 + ln (g 10/[1 + (g/3.4)10 ]/241 Similarly, the quasistatic oddmode characteristic impedance of coupled microstrip lines is expressed by Z 0o (0) = Z 0
√
⑀ re (0) 1 ⑀ reo (0) {1 − [Z 0 (0)/377] [⑀ re (0)]0.5 Q 10 }
(3.84)
with Q 5 = 1.794 + 1.14 ⭈ ln [1 + 0.638/(g + 0.517g 2.43 )] Q 6 = 0.2305 + ln (g 10/[1 + (g/5.8)10 ]/281.3 + ln (1 + 0.598g 1.154 )/5.1 Q 7 = (10 + 190g 2 )/(1 + 82.3g 3 ) Q 8 = exp [−6.5 − 0.95 ln (g) − (g/0.15)5 ] Q 9 = ln (Q 7 ) ⭈ (Q 8 + 1/16.5) −1 Q 10 = Q 2 ⭈ {Q 2 Q 4 − Q 5 ⭈ exp [ln (u) ⭈ Q 6 ⭈ u −Q 9 ]}
Equations for the frequency dependence of the even and oddmode effective dielectric constants and characteristic impedances were also given by Kirschning and Jansen [22]. These are quite involved, however, and are not repeated here. The frequencydependent characteristics of coupled microstrip lines printed on some typical dielectric substrates are shown in Figures 3.21 and 3.22 [22]. The results show that the variation of characteristic impedance with frequency is much smaller than the variation of an effective dielectric constant. Further, it is seen that the evenmode parameters show a greater variation with frequency than the oddmode parameters.
3.5 Single and Coupled Coplanar Waveguides A few coplanar waveguide (CPW) configurations are shown in Figures 3.23 to 3.25. In a coplanar waveguide, the signal is applied between the center conductor and two outer conductors that lie in the same plane [25, 26]. The outer conductors (which are of finite width in practice) are at the same (ground) potential. The coplanar waveguide has some inherent advantages, making it suitable for hybrid and monolithic microwave integrated circuits. Because both the center and ground conductors of a CPW lie in the same plane, active devices can be easily placed in
84
Characteristics of Planar Transmission Lines
Figure 3.21
Frequency dependent even and oddmode characteristic impedances and effective dielectric constants of coupled microstrip lines printed on a substrate of dielectric constant ⑀ r = 2.35, h = 0.79 mm. (From: [22]. 1984 IEEE. Reprinted with permission.)
a series or shunt across the transmission line without requiring a via hole geometry. The mode of propagation along a coplanar line is quasiTEM. Both numerical and analytical methods have been used for the analysis in [27–34]. The dispersion in CPW is generally smaller than in a microstrip line [33].
3.5.1 Coplanar Waveguide
Consider the CPW printed on a finitethickness dielectric substrate (Figure 3.23). The width of the dielectric substrate is assumed to be infinite. The quasistatic parameters of this transmission line are given by [30]: Z0 =
⑀ re = 1 + where
30 K(k ′) √⑀ re K(k)
⑀ r − 1 K(k ′) K(k 1 ) 2 K(k) K(k1′ )
(3.85)
(3.86)
3.5 Single and Coupled Coplanar Waveguides
85
Figure 3.22
Frequency dependent even and oddmode characteristic impedances and effective dielectric constants of coupled microstrip lines printed on a substrate of dielectric constant ⑀ r = 9.7, h = 0.64 mm. (From: [22]. 1984 IEEE. Reprinted with permission.)
Figure 3.23
Coplanar waveguide on a finite thickness dielectric substrate.
Figure 3.24
Cross section of a coplanar waveguide with upper shielding.
86
Characteristics of Planar Transmission Lines
Figure 3.25
Cross section of a conductorbacked coplanar waveguide with upper shielding.
k = a /b k′ =
√1 − k 2
k 1 = sinh ( a /2h)/sinh ( b /2h) k1′ =
√1 − k 1
2
Further, K is the complete integral of the first kind. The values of functions K(k)/K(k ′) and K(k 1 )/K(k1′ ) can be found using (3.61). The effective dielectric constant and characteristic impedance of a CPW are shown in Figure 3.26 for a value of ⑀ r = 13.0. Equations for the case of finite conductor thickness are presented by Wadell [35]. 3.5.2 Coplanar Waveguide with Upper Shielding
The cross section of a CPW with upper shielding is shown in Figure 3.24. The quasistatic parameters of the transmission line are given by [31]
Figure 3.26
(a) Effective dielectric constant, and (b) characteristic impedance of CPW shown in Figure 3.23 as a function of aspect ration a/b for various values of h/b, ⑀ r = 13. (From: [13]. 1996 Artech House. Reprinted with permission.)
3.5 Single and Coupled Coplanar Waveguides
87
⑀ re = 1 + q(⑀ r − 1)
(3.87)
where q is the filling fraction given by K(k 1 ) K(k1′ ) q= K(k 2 ) K(k) + K(k2′ ) K(k ′)
(3.88)
In (3.88), parameters k, k 1 and k 2 are given by k = a /b k′ =
√1 − k 2
k 1 = sinh ( a /2h)/sinh ( b /2h)
(3.89)
k 2 = tanh ( a /2h 1 )/tanh ( b /2h 1 ) k i′ =
√1 − k i
2
The values of functions K(k)/K(k ′) and K(k i )/K(k ′i ) can be found using (3.61). The characteristic impedance is then obtained from Z0 =
60 1 √⑀ re K(k 2 ) + K(k) K(k2′ ) K(k ′)
(3.90)
The structure shown in Figure 3.24 reduces to that shown in Figure 3.23 when h 1 → ∞. 3.5.3 ConductorBacked Coplanar Waveguide with Upper Shielding
The cross section of a conductorbacked CPW with upper shielding is shown in Figure 3.25. This structure has a backside ground plane. The lower ground plane adds mechanical strength to the circuit and increases its power handling capability. The quasistatic effective dielectric constant of this transmission line can also be expressed in the form
⑀ re = 1 + q(⑀ r − 1) where K(k 3 ) (k3′ ) q= K(k 3 ) K(k 4 ) + K(k3′ ) K(k4′ )
(3.91)
88
Characteristics of Planar Transmission Lines
In (3.91), parameters k 3 and k 4 are given by [31] k 3 = tanh ( a /2h)/tanh ( b /2h) k 4 = tanh ( a /2h 1 )/tanh ( b /2h 1 ) k i′ =
(3.92)
√1 − k i
2
The expression for the characteristic impedance is Z0 =
60 1 √⑀ re K(k 3 ) + K(k 4 ) K(k3′ ) K(k4′ )
(3.93)
The characteristic impedance and effective dielectric constant of a conductorbacked CPW with upper shielding are shown in Figure 3.27 as a function of a/b for various values of h/b and ⑀ r = 10. 3.5.4 Coupled Coplanar Waveguides
The cross section of coupled CPWs is shown in Figure 3.28. Unfortunately, very few simple formulas are available in the literature for the determination of parameters of coupled CPWs. For a value of ⑀ r = 12.9, the even and oddmode characteristic impedances of coupled CPWs are shown in Figure 3.29 [32]. For the same parameters of the structure, the even and oddmode effective dielectric constants are shown in Figure 3.30.
3.6 Suspended and Inverted Microstrip Lines The cross section of suspended and inverted microstrip lines is shown in Figures 3.31(a) and 3.31(b), respectively. These lines achieve a lower loss (or higer Q) than possible with microstrip lines. Further, these lines have a much lower effective dielectric constant (compared with that of a microstrip line), thus leading to their performance being less sensitive to dimensional tolerances at high frequencies. Further, the wide range of impedance values achievable using these lines makes them particularly suitable in realizing filters. The quasistatic equations for the design of suspended and inverted microstrip lines have been given by Pramanick and Bhartia [36] and Tomar and Bhartia [37]. The characteristic impedances of inverted and supsended microstrip lines (t Ⰶ h) are given by the following expression [36]: Z 0 (0) = where
60 ln √⑀ re
冋 √ f (u) + u
1+
冉 冊册 2 2 u
(3.94)
3.6 Suspended and Inverted Microstrip Lines
Figure 3.27
89
(a) Characteristic impedance, and (b) effective dielectric constant of a conductorbacked CPW with upper shielding shown in inset. (From: [31]. 1987 IEEE. Reprinted with permission.)
90
Characteristics of Planar Transmission Lines
Figure 3.28
Cross section of coupled coplanar lines. (From: [32]. 1996 IEEE. Reprinted with permission.)
冋冉
f (u) = 6 + (2 − 6) exp −
冊 册
30.666 0.7528 u
The parameter is u = W /(a + b) for suspended microstrips and u = W /b for inverted microstrips. The effective dielectric constant ⑀ re of a suspended microstrip is given by
冋 冉
√⑀ re = 1 +
a W a 1 − b 1 ln b b
冊冉
1 −1 √⑀ r
冊册
−1
(3.95)
where
冉 冉
冊 冊
a 1 = 0.8621 − 0.1251 ln
a 4 b
b 1 = 0.4986 − 0.1397 ln
a 4 b
The effective dielectric constant ⑀ re of an inverted microstrip is given by a
冉
W
√⑀ re = 1 + b a 1 − b 1 ln b where
冉 冉
冊 冊
a 1 = 0.5173 − 0.1515 ln
a 2 b
b 1 = 0.3092 − 0.1047 ln
a 2 b
冊 冠√
⑀ r − 1冡
(3.96)
3.6 Suspended and Inverted Microstrip Lines
Figure 3.29
91
Characteristic impedance of (a) even and (b) odd modes of coupled coplanar lines shown in Figure 3.28. (From: [32]. 1996 IEEE. Reprinted with permission.)
The above analysis equations are accurate to within 1% for 1 ≤ W /b ≤ 8, 0.2 ≤ a/b ≤ 1, and ⑀ r ≤ 6. For ⑀ r ≈ 10, the error is less than ± 2%. The characteristic impedance and phase velocity of a suspended microstrip are shown in Figure 3.32(a) for a value of ⑀ r = 3.78. The characteristic impedance and effective dielectric constant of an inverted microstrip line are shown in Figure 3.32(b) for various values of ⑀ r . Closedform formulas are generally not available for coupled suspended and inverted striplines. Electromagnetic simulators, however, can be used to design circuits using such lines.
92
Characteristics of Planar Transmission Lines
Figure 3.30
Effective dielectric constant of even and odd modes of coupled coplanar lines shown in Figure 3.28. (From: [32]. 1996 IEEE. Reprinted with permission.)
Figure 3.31
Cross section of (a) suspended, and (b) inverted microstrip lines.
3.7 BroadsideCoupled Lines
Figure 3.32
93
(a) Characteristic impedance and phase velocity of suspended microstrip line. ⑀ r = 3.78. (b) Characteristic impedance and effective dielectric constant of inverted microstrip line. (From: [36]. 1985 IEEE. Reprinted with permission.)
3.7 BroadsideCoupled Lines In earlier sections, we have considered coupling between lines that lie in the same plane (edgecoupled lines). Although edgecoupled lines are easier to fabricate, they
94
Characteristics of Planar Transmission Lines
are not suitable for tight coupling. Tight coupling can be obtained by placing coupled lines in a broadside manner. Further, in keeping with the trend toward miniaturization, multilayer microwave circuits that use broadside coupling between lines are becoming a reality and are described in Chapter 8. 3.7.1 BroadsideCoupled Striplines
General broadsidecoupled microstrip lines are shown in Figure 3.33. For ⑀ r1 = ⑀ r2 = ⑀ r , the structure reduces to broadsidecoupled striplines. For coupled striplines, the even and oddmode effective dielectric constants are the same and are given by
⑀ ree = ⑀ reo = ⑀ r The characteristic impedance of the odd mode can be found using the following expression [38]: a
a
Z 0o √⑀ r = Z 0∞ − ⌬Z 0∞
(3.97)
where a
Z 0∞ = 60 ln
Figure 3.33
冋 √冉 3S + W
冊
S 2 +1 W
册
(3.98)
(a) Evenmode and (b) oddmode field distribution of general broadside coupled microstrip lines. (From: [38]. 1988 IEEE. Reprinted with permission.)
3.7 BroadsideCoupled Lines
95
and
a
⌬Z 0∞ =
冋
P
for
W ≤ 1/2 S
PQ
for
W ≥ 1/2 S
冦
冉
P = 270 1 − tanh 0.28 + 1.2
Q = 1 − tanh−1
冤冉
0.48
√
1+
√
b−S S
2W −1 S
冊
b−S 2 S
冊册
冥
Furthermore, the evenmode characteristic impedance can be found using Z 0e =
60 K(k ′) √⑀ r K(k)
(3.99)
where k = tanh
冉
293.9S /b Z 0o √⑀ r
冊
(3.100)
and K(k ′)/K(k) can be determined using (3.61). It has been reported that (3.97) and (3.99) offer an accuracy within 1% of spectral domain results [38]. 3.7.2 BroadsideCoupled Suspended Microstrip Lines
The structure shown in Figure 3.33 reduces to broadsidecoupled suspended microstrip lines for ⑀ r1 = ⑀ r ≥ 1, ⑀ r2 = 1. The even and oddmode characteristic impedances of broadsidecoupled suspended microstrip lines are given by a
Z 0e =
Z 0e
√⑀ ree
(3.101)
a
Z 0o = a
a
Z 0o
√⑀ reo
(3.102)
where Z 0e and Z 0o are the even and oddmode characteristic impedances of the corresponding airfilled homogeneous broadsidecoupled striplines (⑀ r = 1). Their values can be found using (3.97) and (3.99), respectively. Furthermore, the evenand oddmode effective dielectric constants are given by [38]
96
Characteristics of Planar Transmission Lines
⑀ reo =
1 (⑀ − 1) (⑀ r + 1) + q r 2 2
(3.103)
where q = q∞ qc
冉
q∞ = 1 +
a(U) = 1 +
冊
5S −a(U) b(⑀ r ) W
冋
U 4 + (U/52)2 1 ln 49 U 4 + 0.432
册
+
冋 冉 冊册
U 3 1 ln 1 + 18.7 18.1
U = 2W /S b(⑀ r ) = 0.564
冉
⑀ r − 0.9 ⑀r + 3
冊
0.053
and
冋
冉 冊 冉 冊册 冠√
q c = tanh 1.043 + 0.121
再 冋
⑀ ree = 1 +
b−S S
W S a − b 1 ln b 1 b
− 1.164
冎
⑀ r − 1冡
冉 冊册 S b−S
(3.104)
2
where a 1 = [0.8145 − 0.05824 ln (S /b)]8 b 1 = [0.7581 − 0.07143 ln (S /b)]8 These equations offer an accuracy of about 1% for ⑀ r ≤ 16, S/b ≤ 0.4, and W /b ≤ 1.2. These conditions are usually met in practice. The even and oddmode characteristic impedances of coupled suspended microstrip lines are shown in Figure 3.34(a) for ⑀ r = 2.32. For the same parameters, the effective dielectric constants are shown in Figure 3.34(b). 3.7.3 BroadsideCoupled Offset Striplines
Broadsidecoupled offset striplines are shown in Figure 3.35. This structure is more general than the broadsidecoupled striplines configuration discussed in Section 3.7.1 or the edgecoupled stripline configuration shown in Figure 3.11. Shelton [39] has given closedform expressions for the analysis and synthesis of broadsidecoupled offset lines. Here, we present the synthesis equations only as they are more
3.7 BroadsideCoupled Lines
97
Figure 3.34
(a) Characteristic impedance and (b) effective dielectric constants of coupled broadside coupled suspended microstrip lines. ⑀ r = 2.32. (From: [38]. 1988 IEEE. Reprinted with permission.)
Figure 3.35
Broadside coupled offset striplines.
98
Characteristics of Planar Transmission Lines
frequently used. Two sets of equations are given, one for tightly coupled lines and the other for loosely coupled lines. The conditions for tight and loose coupling are defined by w′ ≥ 0.35 1 − s′
Tight coupling case:
(3.105)
w′c ≥ 0.7 s′ w′ ≥ 0.35 1 − s′
Loose coupling case:
(3.106)
2w′o ≥ 0.85 1 + s′ In these equations, s′ = S/b, w′ = W /b, w′c = Wc /b, and w′o = Wo /b denote the normalized values. The coupling between TEM lines can be expressed in terms of even and oddmode characteristic impedances. For a TEM coupler that is matched at all its ports: Z 0e Z = 0 Z 0 Z 0o
(3.107)
Defining as Z
Z
0e 0 √ = Z 0 = Z 0o
(3.108)
we obtain the synthesis equation given here: Tight coupling case: A = exp
B=
(B − 1) p=
A−2+
+
r= C fo =
1
再
−
√A
冉
2
冊册
− 4A
2
冉 冊 √ 1 + s′ 2
冋
60 2 1 − s′ √⑀ r Z 0 √
冋
(B − 1)2
冉 冊
1 + s′ 2 + 4s′B 2
2
s′B p
pr 2 1 ln s′ + ln 1 − s′ s′ (p + s′) (1 + p) (r − s′) (1 − r)
册冎
3.8 SlotCoupled Microstrip Lines
99
Co =
w′ = w′o =
1 2
再
120 √
√⑀ r Z 0
s′(1 − s′) (C o − C fo ) 2
(1 + s′) ln
冋
(1 + p) (r − s′) p + (1 − s′) ln r (s′ + p) (1 − r)
Loose coupling case: C o =
⌬C =
K=
a=
√冋
w′c =
120 √
√⑀ r Z 0
(3.109)
120 ( − 1) √⑀ r Z 0 √
1 ⌬C exp 2
冉 冊
−1
册
(s′ − K) 2 (s′ − K) +K − (s′ + 1) (s′ + 1) q=
C fo =
册冎
冋 冋 冋
K a
册 冉 冊册 冉 冊 冉 冊册
1 2 1+a 1 ln − ln q 1 + s′ a(1 − q) 1 − s′ 1−q 1 q (s′ ln + (1 − s′) ln a 1+a
C f (a = ∞) = −
1 1 − s′ 1 + s′ 2 1 + ln ln 1 + s′ 2 1 − s′ 2
w′ =
1 − s′ 2 [C o − C fo − C f (a = ∞)] 4
3.8 SlotCoupled Microstrip Lines Slotcoupled microstrip lines are shown in Figure 3.36. This configuration is useful for realizing coupling in multilayer MICs. Directional couplers realized using this configuration can achieve both tight and loose coupling values. The quasistatic evenmode effective dielectric constant and the characteristic impedance of the structure are given by [40]
100
Characteristics of Planar Transmission Lines
Figure 3.36
Slotcoupled microstrip lines.
K′(k 1 ) K(k 2 ) + K(k 1 ) K′(k 2 ) ⑀ ree = K′(k 1 ) K(k 2 ) + K(k 1 ) K′(k 2 )
⑀r
Z 0e =
60 1 K′(k ) K(k 2 ) 1 √⑀ ree + K(k 1 ) K′(k 2 )
(3.110)
(3.111)
In (3.112), parameters k 1 and k 2 are given by k1 =
√
sinh2 ( G/4h) sinh2 ( G/4h) + cosh2 ( W/4h) k 2 = tanh ( W/4h o )
(3.112) (3.113)
√
2 Furthermore, K′(k i ) = K(k i′ ), where k i′ = 1 − k i . The quasistatic oddmode effective dielectric constant and the characteristic impedance of the structure are given by
K(k 3 ) K(k 4 ) + K′(k 3 ) K′(k 4 ) ⑀ reo = K(k 3 ) K(k 4 ) + K′(k 3 ) K′(k 4 )
⑀r
Z 0e =
60 1 √⑀ reo K(k 3 ) + K(k 4 ) K′(k 3 ) K′(k 4 )
(3.114)
(3.115)
with parameters k 3 and k 4 given by k 3 = tanh ( W/4h)
(3.116)
k 4 = tanh ( W/4h o )
(3.117)
3.8 SlotCoupled Microstrip Lines
Figure 3.37
101
Even and oddmode (a) characteristic impedances, and (b) effective dielectric constants of slotcoupled microstrip lines shown in Figure 3.36. (From: [40]. 1991 IEEE. Reprinted with permission.)
Figure 3.37(a) shows the variation of even and oddmode characteristics impedances of the structure as a function of strip width for a value of ⑀ r = 9.9, while Figure 3.37(b) gives the variation of even and oddmode effective dielectric constants. In this chapter, we have described some commonly used single and coupled strip transmission lines. Because it is not in the scope of this book to cover all
102
Characteristics of Planar Transmission Lines
transmission lines, readers are referred to the Transmission Line Design Handbook by Wadell [35] and Microstrip Lines and Slotlines by Gupta et al. [13], which provide a comprehensive treatment of printed transmission lines.
References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
[20] [21] [22]
Collin, R. E., Field Theory of Guided Waves, 2nd ed., New York: IEEE Press, 1991. Wheeler, H. A., ‘‘Formulas for the Skin Effect,’’ Proc. IRE, Vol. 30, 1942, pp. 412–424. Wheeler, H. A., ‘‘Transmission Line Properties of Strip on a Dielectric Sheet on a Plane,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT25, August 1977, pp. 631–647. Kammler, D. W., ‘‘Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT16, November 1968, pp. 925–937. Howe, H., Jr., Stripline Circuit Design, Dedham, MA: Artech House, 1974. Cohn, S. B., ‘‘Characteristic Impedance of Shielded Strip Transmission Lines,’’ IRE Trans., Vol. MTT2, July 1954, pp. 52–55. Hilberg, W., ‘‘From Approximations to Exact Relations for Characteristic Impedances,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, May 1969, pp. 259–265. Wheeler, H. A., ‘‘Transmission Line Properties of a Stripline Between Parallel Planes,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT26, November 1978, pp. 866–876. Cohn, S. B., ‘‘Problems in Strip Transmission Line,’’ IRE Trans., Vol. MTT3, March 1955, pp. 119–126. Vendelin, G. D., ‘‘Limitations on Stripline Q,’’ Microwave J., Vol. 13, May 1970, pp. 63–69. Cohn, S. B., ‘‘Shielded Coupled Strip Transmission Line,’’ IRE Trans., Vol. MTT3, October 1955, pp. 29–38. Bahl, I. J., and Bhartia, Microwave SolidState Circuit Design, New York: John Wiley & Sons, 1988. Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House, 1996. Wheeler, H. A., ‘‘Transmission Line Properties of Parallel Strips Separated by a Dielectric Sheet,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, 1965, pp. 172–185. Schneider, M. V., ‘‘Microstrip Lines for Microwave Integrated Circuits,’’ Bell System Tech. J., Vol. 48, 1969, pp. 1421–1444. Pucel, R. A., et al., ‘‘Losses in Microstrip,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT16, 1968, pp. 342–350. Corrections: ibid, MTT16, 1968, p. 1064. Denlinger, E. J., ‘‘Losses in Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT28, June 1980, pp. 513–522. Hammerstad, E., and O. Jenson, ‘‘Accurate Models for Microstrip ComputerAided Design,’’ IEEE MTTS Int. Microwave Symp. Dig., 1980, pp. 407–409. Kobayashi, M., ‘‘A Dispersion Formula Satisfying Recent Requirements in Microstrip CAD,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT36, August 1988, pp. 1246–1250. Bianco, B., et al., ‘‘Frequency Dependence of Microstrip Parameters,’’ Alta Frequenza, Vol. 43, 1974, pp. 413–416. Garg, R., and I. J. Bahl, ‘‘Characteristics of Coupled Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT27, July 1979, pp. 700–705. Kirschning, M., and R. H. Jansen, ‘‘Accurate WideRange Design Equations for the FrequencyDependent Characteristics of Parallel Coupled Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, January 1984, pp. 83–90. Corrections: IEEE Trans. Microwave Theory Tech., March 1985, p. 288.
3.8 SlotCoupled Microstrip Lines [23]
[24]
[25]
[26] [27] [28] [29] [30] [31]
[32]
[33] [34]
[35] [36]
[37]
[38]
[39]
[40]
103
Veghte, R. L., and C. A. Balanis, ‘‘Dispersion of Transient Signals in Microstrip Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT34, December 1986, pp. 1427–1436. York, R. A., and R. C. Compton, ‘‘Experimental Evaluation of Existing CAD Models for Microstrip Dispersion,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT38, March 1990, pp. 327–328. Wen, C. P., ‘‘Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonreciprocal Gyromagnetic Device Applications,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, December 1969, pp. 1087–1090. Wen, C. P., ‘‘CoplanarWaveguide Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT18, June 1970, pp. 318–322. Ghione, G., and C. Naldi, ‘‘Parameters of Coplanar Waveguides with Lower Ground Planes,’’ Electronics Letters, Vol. 19, September 1983, pp. 734–735. Rowe, D. A., and B. Y. Lao, ‘‘Numerical Analysis fo Shielded Coplanar Waveguides,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT31, November 1983, pp. 911–925. Leong, M. S., et al., ‘‘Effect of a Conducting Enclosure on the Characteristic Impedance of Coplanar Waveguides,’’ Microwave J., Vol. 29, August 1986, pp. 105–108. Ghione, G., and C. Naldi, ‘‘Analytical Formulas for Coplanar Lines in Hybrid and Monolithic,’’ Electronics Letters, Vol. 20, February 1984, pp. 179–181. Ghione, G., and C. Naldi, ‘‘Coplanar Waveguides for MMIC Applications: Effect of Upper Shielding, Conductor Backing, Finite Extent Ground Planes and LinetoLine Coupling,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, March 1987, pp. 260–267. Cheng, K. K. M., ‘‘Analysis and Synthesis of Coplanar Coupled Lines on Substrates of Finite Thickness,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT44, April 1966, pp. 636–639. Shih, Y. C., and T. Itoh, ‘‘Analysis of ConductorBacked Coplanar Waveguide,’’ Electronics Letters, Vol. 18, June 1982, pp. 538–540. Kitazawa, T., and R. Mittra, ‘‘QuasiStatic Characteristics of Asymmetrical and Coupled CoplanarType Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT33, September 1985, pp. 771–778. Wadell, B. C., Transmission Line Design Handbook, Norwood, MA: Artech House, 1991. Pramanick, P., and P. Bhartia, ‘‘CAD Models for MillimeterWave Finlines and SuspendedSubstrate Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT33, December 1985, pp. 1429–1435. Tomar, R. S., and P. Bhartia, ‘‘New QuasiStatic Models for the ComputerAided Design of Suspended and Inverted Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, April 1987, pp. 453–457. Corrections: IEEE Trans. Microwave Theory Tech., November 1987, p. 1076. Bhartia, P., and P. Pramanick, ‘‘ComputerAided Design Models for BroadsideCoupled Striplines and MillimeterWave Suspended Substrate Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT36, November 1988, pp. 1476–1481. Corrections: IEEE Trans. Microwave Theory Tech., October 1989, p. 1658. Shelton, J. P., ‘‘Impedances of Offset ParallelCoupled Strip Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, January 1966, pp. 7–15. Corrections: IEEE Trans. Microwave Theory Tech., 1996, p. 249. Wong, M. F., et al., ‘‘Analysis and Design of SlotCoupled Directional Couplers Between DoubleSided Substrate Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT39, December 1991, pp. 2123–2128.
CHAPTER 4
Analysis of Uniformly Coupled Lines
Traditionally, two approaches have been used to study coupling between transmission lines (i.e., normalmode and coupledmode). The normalmodes of symmetrical coupled lines are the even and odd modes. The coupling between symmetrical lines can be determined in terms of phase velocities and characteristic impedances of the even and odd modes of coupled lines [1–4]. When the coupled lines are asymmetrical, the even and odd modes are no longer the normal modes of the structure and are now designated as the c and modes. Knowing the c and mode parameters, the coupling between asymmetrical lines can be determined. The normal mode theory provides an exact method of analysis for coupled lines. In some cases however, (e.g., when two transmission lines in nonreciprocal media are coupled), the task of determining normalmode parameters is very complicated. In this case, another approach known as the coupledmode theory may prove to be easier and more intuitive. This theory is discussed in the next chapter. In this chapter, we discuss the normalmode analysis of uniform symmetrical and asymmetrical coupled lines. We first show how the analysis of a symmetrical fourport network is reduced to analyzing two twoport networks using the evenand oddmode analysis. We also examine conditions under which a fourport network composed of symmetrical coupled lines behaves as a forwardwave or backwardwave directional coupler as well as the unique properties of backwardwave and forwardwave directional couplers. Section 4.3 describes the normalmode analysis of asymmetrical coupled lines. Also given are the relations between distributed line parameters (e.g., L, C, L m , and C m ), phase velocities, and characteristic impedances of normal modes along with the Z (impedance) parameters of a fourpart network consisting of asymmetrical coupled lines. Once we know the Zparameters of a linear network, we can determine the response of the network to any arbitrary excitation and termination. Determining normalmode parameters of asymmetrical coupled lines is generally quite complicated. Furthermore, design data are available in the literature for only a few cases. In Section 4.3, we first describe an approximate method from which the parameters of asymmetrical coupled lines are determined from the data of symmetrical coupled lines. We then discuss how these parameters can be determined using a popular EM simulator that is available at no cost. We also discuss the design of asymmetrical backwardwave and forwardwave directional couplers. Finally, we present a simple method for the design of backward multilayer couplers. A few design examples are also discussed.
105
106
Analysis of Uniformly Coupled Lines
4.1 Even and OddMode Analysis of Symmetrical Networks A fourport network (shown in Figure 4.1) is assumed to be symmetrical about the plane PP ′. It is also assumed that the impedances terminating various ports are the same. When one or more ports of the network are connected to a source(s), waves propagating in either direction are generally set up on both the lines. These waves are referred to as incident or reflected. The relationship between incident and reflected voltage waves at different ports of the network as shown in Figure 4.1 can be expressed as −
+
冤冥 冤冥 V1
V1
−
V2
−
V3
+
= [S]
−
V2
(4.1)
+
V3
+
V4
V4
where
冤
S 11 S 21 [S] = S 31 S 41
S 12 S 22 S 32 S 42
S 13 S 23 S 33 S 43
冥
S 14 S 24 S 34 S 44
(4.2)
denotes the scattering matrix of the network. Note that because all the ports of the network are assumed to be terminated in identical loads (denoted by impedance Z 0 ), the unnormalized scattering matrix is identical to the normalized scattering matrix as discussed in Section 2.3.4. Not all the elements of the scattering matrix of the network shown in Figure 4.1 are independent. For example, because of the assumed symmetry and the reciprocal nature of the structure:
Figure 4.1
A fourport symmetrical network. PP ′ is a plane of symmetry.
4.1 Even and OddMode Analysis of Symmetrical Networks
107
S 21 = S 12 , S 31 = S 13 , S 41 = S 14 , S 32 = S 23 S 42 = S 24 , S 43 = S 34 S 33 = S 11 , S 44 = S 22 , S 34 = S 12 , S 23 = S 14 The scattering matrix of (4.2) can therefore be expressed as
冤
S 11 S 21 [S] = S 31 S 41
S 21 S 22 S 41 S 42
冥
S 31 S 41 S 11 S 21
S 41 S 42 S 21 S 22
(4.3)
In a more compact notation, the matrix (4.3) can be expressed as [S] =
冋
[S A ] [S B ]
册
(4.4)
册
(4.5)
册
(4.6)
[S B ] [S A ]
where [S A ] =
冋
S 11 S 21
S 21 S 22
[S B ] =
冋
S 31 S 41
S 41 S 42
and
(4.1) then becomes −
冤冥冋
+
V1
−
V2
−
V3
−
V4
=
[S A ] [S B ]
[S B ] [S A ]
册
冤冥 V1
+
V2
+
V3
(4.7)
+
V4
It may be noted that symmetrical structures show some special electrical behavior. If the symmetrical ports 1 and 3 are connected to equal magnitude and inphase sources, the voltages at ports 1 and 3 will be equal in magnitude and in phase. Similarly, the voltages will be equal in magnitude and inphase at ports 2 and 4. This scheme is shown in Figure 4.2(a) and is called an evenmode excitation. Furthermore, if the symmetrical ports 1 and 3 are connected to equal magnitude but outofphase sources, the voltages at ports 1 and 3 will be equal in magnitude but out of phase. The voltages at ports 2 and 4 will also be equal in magnitude but out of phase. This excitation scheme is shown in Figure 4.2(b) and is called an
108
Analysis of Uniformly Coupled Lines
Figure 4.2
A fourport symmetrical network excited by (a) evenmode and (b) oddmode sources.
odd mode excitation. The algebraic sum of these two excitations is equivalent to the excitation scheme shown in Figure 4.1. 4.1.1 EvenMode Excitation
Figure 4.2(a) shows a symmetrical structure excited by equal magnitude and inphase sources at ports 1 and 3. The incident and reflected voltages set up at the ± ± ± different ports are also shown in the same figure. Let V3 = V1 = V1e and ± ± ± V4 = V2 = V2e , where the suffix e has been used to denote the even mode. Further± ± ± ± more, by the substitution of V1 , V2 , V3 , and V4 in (4.7), the reflected voltages can be expressed in terms of incident voltages as follows: −
冤 冥冋
+
V1e −
V2e −
V1e −
V2e
From (4.8), we obtain
=
[S A ] [S B ]
[S B ] [S A ]
册
冤冥 V1e +
V2e +
V1e +
V2e
(4.8)
4.1 Even and OddMode Analysis of Symmetrical Networks
冋 册
109
−
冋 册 +
V1e
= ([S A ] + [S B ])
−
V2e
V1e +
V2e
(4.9)
4.1.2 OddMode Excitation
Figure 4.2(b) shows the symmetrical structure excited by equal magnitude but outofphase sources at ports 1 and 3. The incident and reflected voltages set up at various ports by the oddmode sources are also shown in the same figure. Let ± ± ± ± ± ± V1 = −V3 = V1o and V2 = −V4 = V2o , where the suffix o has been used to denote that the quantities correspond to the odd mode. Furthermore, by substitution of ± ± ± ± V1 , V2 , V3 , and V4 in (4.7), the reflected voltages at various ports can be expressed in terms of incident voltages as follows: −
冤 冥冋
+
V1o −
V2o
=
−
−V1o
[S A ] [S B ]
[S B ] [S A ]
−
册
−V2o
冤 冥 V1o +
V2o
+
−V1o
(4.10)
+
−V2o
which gives
冋 册 −
V1o −
V2o
冋 册 +
= ([S A ] − [S B ])
V1o +
V2o
(4.11)
Scattering Matrix in Terms of Even and OddMode Parameters
Equation (4.9) can be written as
冋 册 冋 册 −
V1e
+
= [S e ]
V1e +
(4.12)
[S e ] = [S A ] + [S B ]
(4.13)
−
V2e
V2e
where
Similarly, (4.11) can be written as
冋 册 冋 册 −
V1o
+
= [S o ]
V1o +
(4.14)
[S o ] = [S A ] − [S B ]
(4.15)
−
V2o
V2o
where
110
Analysis of Uniformly Coupled Lines
From (4.13) and (4.15) we obtain [S A ] =
[S e ] + [S o ] 2
(4.16)
[S B ] =
[S e ] − [S o ] 2
(4.17)
and
Therefore, if the scattering matrices [S e ] and [S o ], which are matrices of order 2 × 2, are known, then the scattering matrices [S A ] and [S B ] can be determined using (4.16) and (4.17), respectively. Furthermore, if the scattering matricies [S A ] and [S B ] are known, the complete scattering matrix of the fourport network can be determined using (4.4). Let the elements of the scattering matrices [S e ] and [S o ] be given by [S e ] =
冋
S 11e S 21e
S 21e S 22e
册
(4.18)
[S o ] =
冋
S 11o S 21o
S 21o S 22o
册
(4.19)
and
Using (4.16) and (4.17), the 2 × 2 scattering matrices [S A ] and [S B ] are then found to be
冤
S 21e + S 21o 2 S 22e + S 22o 2
冥
(4.20)
冤
S 21e − S 21o 2 S 22e − S 22o 2
冥
(4.21)
S 11e + S 11o 2 [S A ] = S 21e + S 21o 2 and S 11e − S 11o 2 [S B ] = S 21e − S 21o 2
Furthermore, using (4.4), the various elements of the scattering matrix of the fourport network are given as follows:
4.2 Directional Couplers Using Uniform Coupled Lines
111
+ S 11o S S 11 = 11e , S 12 = S 21 , S 13 = S 31 , S 14 = S 41 2 + S 21o + S 22o S S S 21 = 21e , S 22 = 22e , S 23 = S 41 , 2 2 − S 22o − S 11o S S , S 31 = 11e , S 32 = S 41 , S 24 = 22e 2 2
(4.22)
S 33 = S 11 , S 34 = S 21 − S 21o S S 41 = 21e , S 42 = S 24 , S 43 = S 21 , S 44 = S 22 2 In the case of even and oddmode excitations, the plane of symmetry behaves like a magnetic or electric wall, respectively. The reduced circuit for determining scattering parameters for even and odd modes is shown in Figure 4.3.
4.2 Directional Couplers Using Uniform Coupled Lines Having described the basic theory behind even and oddmode analysis, we now discuss the conditions under which a symmetrical fourport network composed of uniformly coupled lines as shown in Figure 4.4 can act as a directional coupler. The scattering parameters of an ideal directional coupler were discussed in Chapter
Figure 4.3
Reduced circuit for determining scattering matrices of the even and oddmodes of the structure shown in Figure 4.1.
Figure 4.4
A fourport network composed of uniform coupled symmetrical lines.
112
Analysis of Uniformly Coupled Lines
2, which showed that if all ports of a fourport network are matched, then the network behaves like a directional coupler. Because the network as shown in Figure 4.4 is assumed to be symmetrical about the midplane PP ′, matching of ports 1 and 2 automatically ensures that the ports 3 and 4 are also matched. Therefore, the condition S 11 = S 22 = 0 leads to the result that the network is a directional coupler. In terms of even and oddmode reflection coefficients, the scattering parameters S 11 and S 22 are given by (4.22) as + S 11o S S 11 = 11e 2
(4.23)
+ S 22o S S 22 = 22e 2
(4.24)
The equivalent circuit for determining the evenmode scattering parameters S 11e and S 22e is shown in Figure 4.5(a), where Z 0e and  e denote the characteristic impedance and propagation constant of the evenmode of symmetrical coupled lines. Similarly, the equivalent circuit for determining the oddmode scattering parameters S 11o and S 22o is shown in Figure 4.5(b), where Z 0o and  o denote the
Figure 4.5
Equivalent circuit for determining scattering matrix of the (a) even mode and (b) odd mode of the structure shown in Figure 4.4.
4.2 Directional Couplers Using Uniform Coupled Lines
113
characteristic impedance and propagation constant of the odd mode of symmetrical coupled lines. To obtain S 11 = S 22 = 0, which are the conditions for realizing a directional coupler, the following possibilities exist:
Case I
S 11e = S 11o = S 22e = S 22o = 0
(4.25)
When the above values are substituted in (4.23) and (4.24), we obtain S 11 = S 22 = 0. Furthermore, using (4.22), we obtain S 13 = S 31 = S 42 = S 24 = 0
(4.26)
Therefore, in this case [when (4.25) is satisfied], no power is coupled to the backward port. For example, if power is incident at port 1, then no power is coupled to port 3 on the coupled line. Similarly, no power is coupled between ports 2 and 4. However, power can be coupled between ports 1 and 4 (or between ports 2 and 3). These types of couplers are called forwardwave or codirectional couplers and are discussed further in this chapter.
Case II
From (4.23) and (4.24), we see that S 11 = 0 and S 22 = 0 can also be obtained if the following conditions are satisfied: S 11e = −S 11o
(4.27)
S 22e = −S 22o
(4.28)
and
where S 11e , S 11o , S 22e , and S 22o are not equal to zero. Using (4.22), we then find that S 31 ≠ 0, S 42 ≠ 0 In this case, power is thus coupled to the backward port. From the properties of a directional coupler, we know that if S 31 ≠ 0, then either S 41 = 0 or S 21 = 0 To ensure that no power is coupled in the forward direction on the coupled line, it is required that S 41 = 0. Using (4.22), we find that this condition is satisfied when
114
Analysis of Uniformly Coupled Lines
S 21e = S 21o
(4.29)
Therefore, when conditions given by (4.27) through (4.29) hold, the structure behaves as a backwardwave directional coupler. 4.2.1 ForwardWave (or Codirectional) Directional Couplers
As discussed earlier, the fourport network as shown in Figure 4.4 behaves like a forwardwave directional coupler when (4.25) is satisfied. Referring to the equivalent circuits shown in Figure 4.5(a, b) for the even and odd modes, respectively, the above condition is satisfied for any arbitrary length l of the coupling section if Z 0e = Z 0o = Z 0 The above condition can be nearly met in practice by keeping a relatively large spacing between the lines. Substituting S 11e = S 22e = 0 in the following equation (this equation follows from the unitary property of the scattering matrix):
 S 11e  2 +  S 21e  2 =  S 22e  2 +  S 21e  2 = 1 we obtain
 S 21e  = 1 or S 21e = e −j e
(4.30)
where e denotes the phase difference between ports 1 and 2 for the evenmode signal. Similarly, by substituting S 11o = S 22o = 0 in the equation
 S 11o  2 +  S 21o  2 =  S 22o  2 +  S 21o  2 = 1 we obtain S 21o = e −j o
(4.31)
4.2 Directional Couplers Using Uniform Coupled Lines
115
where o denotes the phase difference between ports 1 and 2 for the oddmode signal. Because the coupled structure is assumed to be uniform, we can further write
e = el
(4.32)
o = ol
(4.33)
and
where  e and  o denote the propagation constants of the even and oddmode signals, respectively, and l is the length of the coupling section. Furthermore, using (4.22), the scattering parameters of an ideal forwardwave directional coupler are given by S 11 = S 22 = S 33 = S 44 = 0
(4.34)
+ S 21o e −j e l + e −j o l S = S 12 = S 21 = S 34 = S 43 = 21e 2 2
(4.35)
=e
−j(  e +  o )l 2
cos
冋
(  e −  o )l 2
册
− S 21o e −j e l − e −j o l S S 14 = S 41 = S 23 = S 32 = 21e = 2 2 = −je
−j(  e +  o )l 2
sin
冋
(  e −  o )l 2
(4.36)
册
S 13 = S 31 = S 24 = S 42 = 0
(4.37)
The fractional power coupled from port 1 to port 4 is thus given by
冋
P4 (  e −  o )l 2 =  S 41  = sin2 P1 2
册
(4.38)
while the fractional power coupled from port 1 to port 2 is given by
冋
P2 (  e −  o )l 2 =  S 21  = cos2 P1 2
册
(4.39)
Notice that  S 21  +  S 41  = 1, accounting for all the incident power. It should be apparent that forwardwave directional couplers cannot be obtained using TEM mode lines such as coaxial lines because for the TEM mode, the propagation constants of the even and odd modes are equal, and therefore as shown by (4.36), 2
2
116
Analysis of Uniformly Coupled Lines
there is no coupling between ports 1 and 4 or between ports 2 and 3. Forwardwave coupling exists only in nonTEM lines such as metallic waveguides, fin lines, and dielectric waveguides and can also exist in quasiTEMmode transmission lines such as microstrip lines at high frequencies. In these transmission line structures, in general, the phase velocities of the even and odd modes are not equal. Remarks on ForwardWave or Codirectional Couplers
1. From (4.38), we see that complete power can be transferred between lines if the length l of the directional coupler is chosen as l=
0 =   e −  o  2  冠√⑀ ree − √⑀ reo 冡 
(4.40)
This result is significant in the sense that even for arbitrarily small values of difference in the propagation constants of even and odd modes, complete power can be transferred between the lines if the length of the coupler is chosen according to (4.40). We show later that it is not possible to completely transfer power from one line to another in the case of backwardwave directional couplers. 2. By comparing (4.35) and (4.36), we find that the phase difference between S 41 and S 21 is 90 degrees. The wave on the ‘‘coupled’’ line is thus 90 degrees out of phase with the ‘‘direct’’ wave. 3. In deriving equations for the forwardwave coupling, we assumed that the condition given by (4.25) is satisfied, which leads to zero coupling between ports 1 and 3 or between ports 2 and 4. The directivity and isolation of the coupler are thus infinite. In general, however, the above condition cannot be completely satisfied. Therefore, some finite amount of backwardwave coupling, however small, always exists between coupled lines. The exact amount of backwardwave coupling (S 31 and S 42 ) can be determined using (4.22), if the values of S 11e , S 11o , S 22e , and S 22o are known. 4.2.2 BackwardWave Directional Couplers
As discussed earlier, a symmetrical fourport network as shown in Figure 4.4 behaves like a backwardwave directional coupler if the following conditions are satisfied: S 11o = −S 11e S 22o = −S 22e and S 21o = S 21e where S 11o , S 11e , S 22o , and S 22e are not equal to zero.
4.2 Directional Couplers Using Uniform Coupled Lines
117
The above conditions are easily satisfied if the coupled lines are of the TEM type similar to striplines and the even and oddmode characteristic impedances of the lines are properly chosen. The equivalent circuits for the even and odd modes are shown in Figures 4.5(a) and 4.5(b), respectively. Using these equivalent circuits, the ABCD matrices of the coupler can be determined for the even and odd modes. For example, using Table 2.2, the ABCD matrices for the even and odd modes are given, respectively, by Ae Ce
Be De
册冤
cos (  l ) j sin (  l ) Z 0e
jZ 0e sin (  l )
=
冋
Ao Co
Bo Do
册冤
cos (  l ) j sin (  l ) Z 0o
jZ 0o sin (  l )
cos (  l )
冥
(4.41)
and
冋
=
cos (  l )
冥
(4.42)
where we assume that the propagation constants are the same for the even and odd modes and are denoted by  . Because the coupled lines are terminated in an impedance of Z 0 , the even and oddmode reflection coefficients can be shown from (2.98) and (2.99) to be
冉
冊
Z 0e Z − 0 sin  l Z 0 Z 0e S 11e = S 22e = Z Z 2 cos  l + j 0e + 0 sin  l Z 0 Z 0e j
冉
冊
(4.43)
and
冉
冊
Z 0o Z − 0 sin  l Z 0 Z 0o S 11o = S 22o = Z Z 2 cos  l + j 0o + 0 sin  l Z 0 Z 0o j
冉
冊
(4.44)
Comparing (4.43) and (4.44), we find that the conditions S 11e = −S 11o and S 22e = −S 22o are satisfied for any arbitrary value of length l when Z 0e Z = 0 Z 0 Z 0o or 2
Z 0e Z 0o = Z 0
(4.45)
118
Analysis of Uniformly Coupled Lines
The scattering parameters S 21o and S 21e can be computed from (2.100) as follows: S 21e =
2 Z 0e Z 2 cos  l + j + 0 Z 0 Z 0e
冊
sin  l
S 21o =
2 Z 0o Z 2 cos  l + j + 0 Z 0 Z 0o
冊
sin  l
冉
(4.46)
and
冉
(4.47)
We see that when (4.45) holds, S 21e = S 21o as required by (4.29). Therefore, (4.45) gives the necessary condition for TEM backwardwave directional couplers. Once the values of S 11o , S 11e , S 21o , S 21e are known, the scattering parameters of an ideal backwardwave directional coupler can be easily determined using (4.22) as follows: S 11 = S 22 = S 33 = S 44 = 0
(4.48)
S 14 = S 41 = S 23 = S 32 = 0
(4.49)
+ S 21o S = S 21e = S 21o S 12 = S 21 = S 34 = S 43 = 21e 2 =
2 Z 0e Z 0o 2 cos  l + j + Z0 Z0
冉
冊
(4.50)
sin  l
− S 11o S S 13 = S 31 = S 24 = S 42 = 11e = S 11e = −S 11o 2
冉
冊
(4.51)
Z 0e Z 0o − sin  l Z0 Z0 = Z Z 2 cos  l + j 0e + 0o sin  l Z0 Z0 j
冉
冊
From (4.45) and (4.50), we obtain
S 21 =
√1 − k
2
2 √1 − k cos + j sin
Furthermore, from (4.45) and (4.51),
(4.52)
4.2 Directional Couplers Using Uniform Coupled Lines
S 31 =
119
jk sin
(4.53)
2 √1 − k cos + j sin
where =  l, and k=
Z 0e − Z 0o Z 0e + Z 0o
(4.54)
The maximal amount of coupling between ports 1 and 3 (or between ports 2 and 4) occurs when
= l =
rads 2
(4.55)
or l=
g = 2 4
where g denotes the guide wavelength in the medium of the transmission line. The maximum value of coupling is found by substituting = /2 in (4.53), which gives
 S 13  =  S 31  =  S 24  =  S 42  = k
(4.56)
Furthermore, when = /2,
 S 12  =  S 21  =  S 34  =  S 43  = √1 − k 2
(4.57)
Thus, at the frequency where =  l = /2, the scattering matrix of a backwardwave coupler can be represented as follows:
[S] =
冤
0
−j √1 − k 2
k
0
−j √1 − k 2
0
0
k
k
0
0
−j √1 − k 2
k
−j √1 − k
0
2
0
冥
(4.58)
This scattering matrix is valid at the frequency where the length of the coupler is a quarterwave long. We can calculate, however, the frequency response of the backwardwave coupler at any other frequency using (4.48), (4.49), (4.52), and (4.53). The frequency response of ideal backwardwave couplers of various coupling values is given in Chapter 6.
120
Analysis of Uniformly Coupled Lines
From (4.45) and (4.54), the relationships between even and oddmode characteristic impedances and the voltage coupling coefficient k are given by Z 0e = Z 0 Z 0o = Z 0
√ √
1+k 1−k
(4.59)
1−k 1+k
Example 4.1
Compute even and oddmode characteristic impedances of a 20dB quarterwave, 50ohm backwardwave coupler. For a 20dB coupler, k = 10−20/20 = 0.1. Given that the terminal impedance is 50 ohms, then from (4.59), Z 0e = 55.3 ohms and Z 0o = 45.2 ohms. Coupling k in Terms of Capacitance Parameters
Substituting values of Z 0e and Z 0o from (3.45) and (3.46), respectively, along with (3.41) and (3.43) in (4.54), we obtain k=
Cm Ca + Cm
(4.60)
where C a and C m denote the capacitances of coupled lines as shown in Figure 3.5 (C b = C a ). Remarks on BackwardWave Directional Couplers
1. Equation (4.53) shows that there exists a maximum value of backwardwave coupling that can be achieved between two coupled lines. The maximum value of coupling which is given by (4.56) occurs when the length of the coupler is a quarterwave long (or odd multiples thereof). This is unlike the symmetrical forwardwave directional couplers case where arbitrary coupling can be achieved between the lines by properly choosing the length of the coupling section. 2. By comparing (4.52) and (4.53), we find that the wave coupled to the ‘‘backward’’ port (S 31 ) is 90 degrees out of phase with the wave coupled to the ‘‘direct’’ port (S 21 ). This relationship is independent of the electrical length of the coupling section. These types of couplers are thus capable of being used as quadrature phaseshifters over a wide frequency range.
4.3 Uniformly Coupled Asymmetrical Lines Symmetrical coupled lines represent a very useful but restricted class of coupled lines. In many practical cases, it might be more useful or even necessary to design
4.3 Uniformly Coupled Asymmetrical Lines
121
components using asymmetrical coupled lines. For example, the bandwidth of a forwardwave directional coupler using asymmetrical coupled lines is greater than one formed using symmetrical coupled lines. Also, in some situations, the terminal impedances of one of the coupled lines may be different from those of the other. It may then be more useful to choose two coupled lines with different characteristic impedances. In this section, the analysis and design of asymmetrical coupled quasiTEM mode lines is described. The normalmode parameters of asymmetrical coupled lines are first defined. It is shown that six independent parameters are required to characterize asymmetrical coupled lines. The relation between normalmode parameters (i.e., characteristic impedances, phase velocities) and line parameters (i.e., per unit length inductance, capacitance) are derived. Because symmetrical lines are a special case of asymmetrical coupled lines, various expressions given in the following sections can also be used for symmetrical coupled lines. A concise but excellent analysis of asymmmetrical coupled lines is found in [5], which forms the principal basis for the analysis below. 4.3.1 Parameters of Asymmetrical Coupled Lines
A set of two coupled lines can support two fundamental independent modes of propagation (called normal modes). For asymmetrical coupled lines, the two normal modes of propagation are known as the c and modes. Strictly speaking, a structure composed of two coupled lines can support four independent modes of propagation: two traveling in the backward direction and two traveling in the forward direction. The characteristics (phase velocity and characteristic impedance) of a backwardtraveling mode, however, are the same as those of the corresponding forwardwave traveling mode. The cmode is an evenlike mode, while the mode is an oddlike mode. c Mode
Figure 4.6 shows two uniformly coupled quasiTEM mode asymmetrical transmission lines. The assumption of quasiTEM mode is made because it is possible to define unique voltages and currents in this case as compared with nonTEM mode transmission lines. Let the voltage and current waves on asymmetrical coupled
Figure 4.6
Voltage and current waves on uniform asymmetrical coupled lines for the c mode.
122
Analysis of Uniformly Coupled Lines
lines for the c mode be denoted as shown. The forwardtraveling voltage waves + + on lines 1 and 2 are denoted by V1c e −␥ c z and V2c e −␥ c z, respectively; and the + −␥ c z + −␥ c z corresponding current waves by I1c e and I2c e , respectively. Similarly, − − − − V1c e ␥ c z and V2c e ␥ c z, I1c e ␥ c z and I2c e ␥ c z denote the corresponding quantities for the backwardtraveling mode. The characteristic impedance of line 1 (in the presence of line 2) can be defined as +
Z c1 =
−
V1c
V1c
I1c
I1c
= +
(4.61)
−
Similarly, for the same mode the characteristic impedance of line 2 (in the presence of line 1) can be defined as +
Z c2 =
−
V2c
V2c
I2c
I2c
= +
(4.62)
−
Furthermore, let the ratio of voltages on the two lines be defined by a parameter R c as follows: +
Rc =
V2c +
V1c
−
=
V2c −
V1c
A c mode is therefore characterized by four parameters: ␥ c , the propagation constant of the mode; Z c1 and Z c2 , which are, respectively, the characteristic impedances of lines 1 and 2; and R c , the ratio of the voltages on the two lines. Mode
Similar to the c mode, a mode is also characterized by four parameters: ␥ , the propagation constant of the mode; Z 1 and Z 2 , which are, respectively, the characteristic impedances of lines 1 and 2; and R , the ratio of the voltages on the two lines. Of the eight quantities discussed above (four each for c and modes), only six are independent. The currents and voltages of the two modes satisfy the following relationships: +
V2c
= +
V1c
−
V2c
+
−
I 1 I1 =− + =− − − V1c I2 I 2
and +
V 2
= +
V1
−
V 2
+
−
I1c I1c =− + =− − − V 1 I2c I2c
4.3 Uniformly Coupled Asymmetrical Lines
123
Therefore +
Rc =
−
V2c
= +
V1c
V2c
+
−
I1 I 1 = − = − − + − V1c I 2 I2
(4.63)
and +
R =
V 2 +
V1
−
=
V 2 −
V 1
+
I1c
−
I1c
=− + =− − I2c I2c
(4.64)
Using (4.63) and (4.64), the relations between characteristic impedances Z c1 , Z c2 , Z 1 , and Z 2 and ratio parameters R c and R are found to be Z c2 Z 2 = = −R c R Z c1 Z 1
(4.65)
Therefore, a total number of six quantities (i.e., ␥ c , ␥ , Z c1 [or Z c2 ], Z 1 [or Z 2 ], R c , and R ) is required to characterize asymmetrical coupled lines. It is not necessary to specify both Z c1 and Z c2 , as they are related by (4.65). The same holds for Z 1 and Z 2 . On the other hand, symmetrical coupled lines are completely characterized by four parameters: the even and oddmode characteristic impedances of any line (as both lines are identical) and the even and oddmode phase constants. For symmetrical coupled lines, R c = 1 and R = −1. Z and YParameters of a FourPort Network in Terms of NormalMode Parameters
Figure 4.7 shows a fourport network composed of asymmetrical coupled lines. When one or more ports of the structure are excited, voltage and current waves are set up on both the lines. The voltage and current waves can be expressed as a linear sum of forward and backwardtraveling c and mode waves. For example, the voltage and current waves on the two lines can be represented as V1 (z) = A 1 e −␥ c z + A 2 e ␥ c z + A 3 e −␥ z + A 4 e ␥ z
Figure 4.7
A fourport network composed of uniform asymmetrical coupled lines.
(4.66)
124
Analysis of Uniformly Coupled Lines
V2 (z) = A 1 R c e −␥ c z + A 2 R c e ␥ c z + A 3 R e −␥ z + A 4 R e ␥ z
(4.67)
I1 (z) = A 1 Yc1 e −␥ c z − A 2 Yc1 e ␥ c z + A 3 Y 1 e −␥ z − A 4 Y 1 e ␥ z
(4.68)
I2 (z) = A 1 R c Yc2 e −␥ c z − A 2 R c Yc2 e ␥ c z + A 3 R Y 2 e −␥ z − A 4 R Y 2 e ␥ z (4.69) where Yci = 1/Z ci ; Y i = 1/Z i (i = 1, 2) and A 1 , A 2 , A 3 , and A 4 are constants depending on the sources and terminations. By substituting z = 0 and z = l in (4.66) to (4.69), the voltages and currents at all the ports can be found. For example, the voltage at port 2(V2 ) can be found by substituting z = l in (4.66). The voltages and currents at various ports are found to be V1 = A 1 + A 2 + A 3 + A 4
(4.70a)
V2 = A 1 e −␥ c l + A 2 e ␥ c l + A 3 e −␥ l + A 4 e ␥ l
(4.70b)
V3 = A 1 R c + A 2 R c + A 3 R + A 4 R
(4.70c)
V4 = A 1 R c e −␥ c l + A 2 R c e ␥ c l + A 3 R e −␥ l + A 4 R e ␥ l
(4.70d)
I1 = A 1 Yc1 − A 2 Yc1 + A 3 Y 1 − A 4 Y 1
(4.71a)
−I2 = A 1 Yc1 e −␥ c l − A 2 Yc1 e ␥ c l + A 3 Y 1 e −␥ l − A 4 Y 1 e ␥ l
(4.71b)
I3 = A 1 R c Yc2 − A 2 R c Yc2 + A 3 R Y 2 − A 4 R Y 2
(4.71c)
−I4 = A 1 R c Yc2 e −␥ c l − A 2 R c Yc2 e ␥ c l + A 3 R Y 2 e −␥ l − A 4 R Y 2 e ␥ l (4.71d) The set of equations given by (4.71a) to (4.71d) can be solved to obtain the coefficients A i in terms of I1 , I2 , I3 , and I4 . Furthermore, substituting these in (4.70a) to (4.70d), the Zparameters can be determined from the resulting equations by inspection. The Zparameters of the fourport network are found in terms of normalmode parameters to be: Z 11 = Z 22 =
Z c1 coth ␥ c l Z 1 coth ␥ l + (1 − R c /R ) (1 − R /R c )
Z 13 = Z 31 = Z 24 = Z 42 =
Z c1 R c coth ␥ c l Z 1 R coth ␥ l + (1 − R c /R ) (1 − R /R c )
=− Z 14 = Z 41 = Z 32 = Z 23 =
Z coth ␥ l Z c2 coth ␥ c l − 2 R (1 − R c /R ) R c (1 − R /R c )
R c Z c1 R Z 1 + (1 − R c /R ) sinh ␥ c l (1 − R /R c ) sinh ␥ l
4.3 Uniformly Coupled Asymmetrical Lines
125
Z c1 Z 1 Z 12 = Z 21 = + (1 − R c /R ) sinh ␥ c l (1 − R /R c ) sinh ␥ l R Z coth ␥ c l R Z 2 coth ␥ l Z 33 = Z 44 = − c c2 − R (1 − R c /R ) R c (1 − R /R c ) =
R c2 Z c1 coth ␥ c l R 2 Z 1 coth ␥ l + (1 − R c /R ) (1 − R /R c )
R c2 Z c1 R 2 Z 1 + (1 − R c /R ) sinh ␥ c l (1 − R /R c ) sinh ␥ l
Z 34 = Z 43 =
(4.72)
Furthermore, the Yparameters of the fourport network shown in Figure 4.7 are given by Y coth ␥ c l Y 1 coth ␥ l Y11 = Y22 = c1 + (1 − R c /R ) (1 − R /R c ) Y13 = Y31 = Y24 = Y42 = −
Yc1 coth ␥ c l Y coth ␥ l − 1 R (1 − R c /R ) R c (1 − R /R c )
Yc1 Y 1 + Y14 = Y41 = Y23 = Y32 = (R − R c ) sinh ␥ c l (R c − R ) sinh ␥ l Yc1 Y 1 − Y12 = Y21 = − (1 − R c /R ) sinh ␥ c l (1 − R /R c ) sinh ␥ l Y33 = Y44 = −
R c Yc2 coth ␥ c l R Y 2 coth ␥ l − R (1 − R c /R ) R c (1 − R /R c )
R c Yc2 R Y 2 + Y34 = Y43 = R (1 − R c /R ) sinh ␥ c l R c (1 − R /R c ) sinh ␥ l
(4.73)
Z Parameters of Interdigital TwoPort Network
If the two ports (ports 2 and 3) of the fourport network shown in Figure 4.7 are terminated in an open circuit, a twoport network such as the one shown in Figure 4.8 results with the Zparameters given by
冋
Z 11 Z 21
册
Z 21 Z 22
= −j −j
冋 冋
cot c Z c1 (1 − R c /R ) R c csc c cot Z 1 (1 − R /R c ) R csc
R c csc c R c2 cot c
册 册
(4.74)
R csc R 2 cot
where c =  c l and =  l. This subcircuit finds extensive application in planar microwave circuits such as bandpass filters.
126
Analysis of Uniformly Coupled Lines
Figure 4.8
A prototype open circuited section composed of uniform asymmetrical coupled lines, its equivalent circuit and ABCD parameters. (From: [5]. 1975 IEEE. Reprinted with permission.)
4.3.2 Distributed Equivalent Circuit of Coupled Lines
The distributed equivalent circuit of two uniformly coupled lossless asymmetrical transmission lines is shown in Figure 4.9. The voltages and currents on the coupled lines are governed by the following differential equations [6, 7]: ∂V1 (z, t) ∂I (z, t) ∂I (z, t) + L1 1 + Lm 2 =0 ∂z ∂t ∂t
(4.75)
∂I1 (z, t) ∂V (z, t) ∂V (z, t) + C1 1 − Cm 2 =0 ∂z ∂t ∂t
(4.76)
∂I (z, t) ∂I (z, t) ∂V2 (z, t) + L2 2 + Lm 1 =0 ∂z ∂t ∂t
(4.77)
∂V (z, t) ∂V (z, t) ∂I2 (z, t) + C2 2 − Cm 1 =0 ∂z ∂t ∂t
(4.78)
4.3 Uniformly Coupled Asymmetrical Lines
Figure 4.9
127
Lumped equivalent circuit of coupled transmission lines.
where Vi (z, t) and Ii (z, t) denote the voltage and current, respectively, on line i (i = 1, 2) as a function of distance z along the transmission line and time t. L 1 and C 1 denote the (per unit) selfinductance and selfcapacitance of line 1 in the presence of line 2. Similarly, L 2 and C 2 denote the selfinductance and selfcapacitance of line 2 in the presence of line 1. Furthermore, L m and C m denote the mutual inductance and mutual capacitance between the lines, respectively. More specifically, self and mutual inductance and capacitance parameters are the elements of inductance and capacitance matrices [L] and [C], where
冋 冋
[L] =
[C] =
L 11 L 21
C 11 C 21
册冋 册冋
L 12 L 22 C 12 C 22
=
=
L1 Lm
C1 −C m
Lm L2
册
(4.79)
册
−C m C2
(4.80)
For a lossless case, the inductance matrix [L] is given by [8] [L] = ⑀ 0 0 [C]−1
(4.81)
128
Analysis of Uniformly Coupled Lines
where [C 0 ] denotes the freespace capacitance matrix of the coupled lines. Equation (4.81) is general and is valid for any number of coupled lines. In the frequency domain, (4.75) and (4.78) reduce to ∂V1 (z) + j L 1 I1 (z) + j L m I2 (z) = 0 ∂z
(4.82)
∂I1 (z) + j C 1 V1 (z) − j C m V2 (z) = 0 ∂z
(4.83)
∂V2 (z) + j L 2 I2 (z) + j L m I1 (z) = 0 ∂z
(4.84)
∂I2 (z) + j C 2 V2 (z) − j C m V1 (z) = 0 ∂z
(4.85)
By solving the set of coupled equations given by (4.82) to (4.85), the propagation constants and other parameters of asymmetrical coupled lines defined earlier in this section can be determined. The complex propagation constants of the c and modes are found to be
␥ c2 =
a1 + a2 1 + [(a 1 − a 2 )2 + 4b 1 b 2 ]1/2 2 2
(4.86)
␥ 2 =
a1 + a2 1 − [(a 1 − a 2 )2 + 4b 1 b 2 ]1/2 2 2
(4.87)
and
In these equations a1 = y1 z1 + ym zm a2 = y2 z2 + ym zm
(4.88)
b1 = z1 ym + y2 zm b2 = z2 ym + y1 zm where z 1 = j L 1 z 2 = j L 2 z m = j L m and
(4.89)
4.3 Uniformly Coupled Asymmetrical Lines
129
y 1 = j C 1 y 2 = j C 2
(4.90)
y m = −j C m By substituting values of a 1 , a 2 , b 1 , and b 2 from (4.88) in (4.86) and (4.87), we obtain the phase velocities of the c and modes: vc =
冋
(4.91)
L 1 C 1 + L 2 C 2 − 2L m C m + √(L 1 C 1 − L 2 C 2 )2 + 4(L m C 1 − L 2 C m )(L m C 2 − L 1 C m ) 2
册
−1/2
and v =
冋
(4.92)
L 1 C 1 + L 2 C 2 − 2L m C m − √(L 1 C 1 − L 2 C 2 )2 + 4(L m C 1 − L 2 C m )(L m C 2 − L 1 C m ) 2
册
−1/2
where v c, = / c, and  c, = −j␥ c, . Furthermore, the characteristic impedances and admittances of the lines for the c and modes and R c and R parameters are given by Z c1 =
1 ␥c
冉冊
2 z1 z2 − zm 1 = z 2 − z m R c Yc1
(4.93)
Z c2 =
冉 冊 Rc ␥c
2 z1 z2 − zm 1 = z 1 R c − z m Yc2
(4.94)
Z 1 =
1 ␥
冉 冊
2 z1 z2 − zm 1 = z 2 − z m R Y 1
(4.95)
Z 2 =
冉 冊
2 z1 z2 − zm 1 = z 1 R − z m Y 2
(4.96)
R ␥
Rc =
(a 2 − a 1 ) + [(a 2 − a 1 )2 + 4b 1 b 2 ]1/2 2b 1
(4.97)
R =
(a 2 − a 1 ) − [(a 2 − a 1 )2 + 4b 1 b 2 ]1/2 2b 1
(4.98)
and
130
Analysis of Uniformly Coupled Lines
Inductive and Capacitive Coupling Coefficients
The inductive coupling coefficient between the lines is defined by [6] kL =
Lm √L 1 L 2
(4.99)
whereas the capacitive coupling coefficient between the lines is given by kC =
Cm √C 1 C 2
(4.100)
4.3.3 Relation Between Normal Mode (c and ) and Distributed Line Parameters Symmetrical Lines
The relations between c and parameters and distributed line parameters (selfand mutual inductances and capacitances) reduce to simple forms for some special cases. For example, for symmetrical coupled lines, L 1 = L 2 and C 1 = C 2 . In this case, the c and modes reduce to even and odd modes, respectively. From (4.86) and (4.87),
␥ c = j e = j √(L 1 + L m ) (C 1 − C m )
(4.101)
␥ = j o = j √(L 1 − L m ) (C 1 + C m )
(4.102)
and
Furthermore, for symmetrical coupled lines R c = 1 and R = −1 from (4.97) and (4.98). Substituting the values of R c = 1, R = −1 and propagation constants ␥ c and ␥ from (4.101) and (4.102) in (4.93) to (4.96), we obtain Z c1 = Z c2 = Z 0e = Z 1 = Z 2 = Z 0o =
√ √
(L 1 + L m ) (C 1 − C m )
(4.103)
(L 1 − L m ) (C 1 + C m )
(4.104)
Using (4.101) to (4.104), the distributed line parameters are found in terms of characteristic impedances and propagation constants of the even and odd modes to be L1 = L2 =
1 (  Z +  o Z 0o ) 2 e 0e
冉
 1 o C1 = C2 = + e 2 Z 0o Z 0e
冊
(4.105)
(4.106)
4.3 Uniformly Coupled Asymmetrical Lines
131
Lm =
1 (  Z −  o Z 0o ) 2 e 0e
Cm =
冉
 1 o − e 2 Z 0o Z 0e
冊
(4.107)
(4.108)
Note that for lines supporting pure TEM mode of propagation, the even and oddmode phase velocities are the same. (4.101) and (4.102) then lead to Lm Cm = L1 C1
(4.109)
kC = kL
(4.110)
or from (4.99) and (4.100):
The inductive and capacitive coupling coefficients are therefore equal. Equation (4.110) is true for TEM asymmetrical coupled lines as well.
Asymmetrical Coupled Lines
For lossless TEMmode coupled lines, the propagation constants of both the c and modes are the same, and are given by
␥ c = ␥ = j = jk 0 √⑀ r
(4.111)
The following relations are satisfied by line parameters in this case: L1 C1 = L2 C2 Cm = √C 1 C 2
(4.112)
Lm √L 1 L 2
(4.113)
冉 冊
(4.114)
Furthermore, R c and R are given by Z 2 1/2 R c = −R = Z1
where Z 1 = (L 1 /C 1 )1/2 and Z 2 = (L 2 /C 2 )1/2. In this case, the Zparameters of a fourport network as shown in Figure 4.7 are given by
132
Analysis of Uniformly Coupled Lines
冉 冊
j Z 1 1/2 Z 11 = Z 22 = − (Z c + Z ) cot 2 Z2 j Z 13 = Z 31 = Z 42 = Z 24 = − (Z c − Z ) cot 2 j Z 14 = Z 41 = Z 32 = Z 23 = − (Z c − Z ) csc 2
(4.115)
冉 冊 冉 冊 冉 冊
j Z 1 1/2 Z 12 = Z 21 = − (Z c + Z ) csc 2 Z2 j Z 2 1/2 Z 33 = Z 44 = − (Z c + Z ) cot 2 Z1 j Z 2 1/2 Z 34 = Z 43 = − (Z c + Z ) csc 2 Z1 where Z c = (Z 1 Z 2 )1/2
冋
Z = (Z 1 Z 2 )1/2
冋
册
1 + y m /(y 1 y 2 )1/2 1/2 1 − y m /(y 1 y 2 )1/2
and
册
1 − y m /(y 1 y 2 )1/2 1/2 1 + y m /(y 1 y 2 )1/2
4.3.4 Approximate Distributed Line or NormalMode Parameters of Asymmetrical Coupled Lines
The performance of a network consisting of coupled asymmetrical lines can be determined if the distributed line parameters (i.e., L 1 , L 2 , C 1 , C 2 , L m , and C m ) or the normalmode parameters (i.e., Z c1 [or Z c2 ], Z 1 [or Z 2 ], ␥ c , ␥ , R c , and R ) are known. The computation of distributed line or normalmode parameters, however, is quite complicated and can only be carried out using field theoretical methods [8]. Commercially available programs based on this method are also available [9, 10]. Ikalainen and Matthaei [11] have given an approximate technique from which the inductance and capacitance parameters of asymmetrical coupled lines can be determined from the characteristic impedances and effective dielectric constants of the even and odd modes of symmetrical coupled lines. This approach is useful in practice because the even and oddmode parameters of symmetrical coupled lines are generally more readily available. Consider two coupled lines of width W1 and W2 , each with separation S between them as shown in Figure 4.10(a). It is assumed that the mutual inductance and capacitance between the lines is the same as that between symmetrical lines
4.4 Directional Couplers Using Asymmetrical Coupled Lines
Figure 4.10
133
(a) Asymmetrical coupled lines of width W 1 and W 2 , (b) symmetrical coupled lines of width (W 1 + W 2 )/2, (c) symmetrical coupled lines of width W 1 , and (d) symmetrical coupled lines of width W 2 .
of width (W1 + W2 /2, each with separation S as shown in Figure 4.10(b). Using the even and oddmode data of coupled symmetrical lines as shown in Figure 4.10(b), the values of L m and C m can be computed using (4.107) and (4.108). Furthermore, it is assumed that the selfinductance and capacitance of line 1 in the presence of line 2 is the same as if line 2 has the same width as line 1. Therefore, by using the even and oddmode data of coupled symmetrical lines of width W1 each and separated by a distance S as shown in Figure 4.10(c), the selfcapacitance and inductance of line 1 can be computed using (4.105) and (4.106). Similarly, by using even and oddmode data of coupled symmetrical lines of width W2 each and separated by a distance S [Figure 4.10(d)], the selfcapacitance and inductance of line 2 can be computed. Once the distributed line parameters have been found, the normalmode parameters can be found using (4.86), (4.87), and (4.93) through (4.98).
4.4 Directional Couplers Using Asymmetrical Coupled Lines 4.4.1 ForwardWave Directional Couplers
It is known that if the phase velocities of the two normal modes of asymmetrical coupled lines are different, energy is coupled from one line to another in the forward direction. Because a microstrip line is essentially a quasiTEM line, the even and oddmode phase velocities of coupled microstrip lines are not equal. Therefore, coupling occurs both in the forward and backward directions. Usually, backwardwave couplers are realized in microstrip configuration by properly choosing the characteristic impedances of the even and odd modes. The directivity of microstrip backwardwave couplers is generally poor, however, because of the forwardwave coupling that takes place because of unequal even and oddmode phase velocities. The backwardwave coupling can be reduced to negligibly small values by choosing a relatively large separation between the lines. On the other hand, appreciable power can be made to couple in the forward direction, if the length of the coupling
134
Analysis of Uniformly Coupled Lines
section is properly chosen. The bandwidth of an asymmetrical forwardwave coupler is larger than that of a symmetrical forwardwave coupler. This makes them useful in practice [11, 12]. Figure 4.11 shows an asymmetrical microstrip coupler. It is assumed that the backwardwave coupling between the lines is negligible and each line is terminated in a matched load. With unit power incident at port 1, the forwardtraveling voltage waves on the two lines can be expressed as a linear combination of c and mode voltage waves as follows: +
V1 (z) = A 1 e −␥ c z + A 2 e −␥ z +
V2 (z) = A 1 R c e −␥ c z + A 2 R e −␥ z
(4.116) (4.117)
The voltages in these equations denote their actual values. As discussed in Chapter 2, the concept of actual voltages and currents is restrictive and is applicable to TEM and quasiTEM mode transmission lines only. On the other hand, the concept of normalized voltages is more general and can be applied to nonTEM mode transmission lines as well. Using (4.116), (4.117), and the conversion relations between normalized and actual voltages given by (2.39a) to (2.39d), the normalized voltage waves on the two lines can be expressed as A 1 −␥ c z A2 + Vˆ1 (z) = e + e −␥ z √Z c1 √Z 1
(4.118)
A R A R + Vˆ2 (z) = 1 c e −␥ c z + 2 e −␥ z √Z c2 √Z 2
(4.119)
Because unit power is assumed to be incident at the input port, the initial + + conditions are Vˆ1 = 1 and Vˆ2 = 0 at z = 0. Substituting these conditions in (4.118) and (4.119), we obtain A1 =
Figure 4.11
√Z c1 R 1− c R
(4.120)
A forward (codirectional) directional coupler using uniform asymmetrical coupled lines.
4.4 Directional Couplers Using Asymmetrical Coupled Lines
A2 = −
135
√Z 1 R c 1−
(4.121)
Rc R R
Furthermore, substituting the values of A 1 and A 2 in (4.118) and (4.119), the normalized voltage wave on line 1 is + Vˆ1 (z) =
冉
1 1 e −␥ c z − R R 1− c 1− c R R
冊
冉
冊
R c −␥ z e R
After some straightforward algebraic manipulations, this equation reduces to
冋
册
(  −  )z 1−p (  −  )z e −j( c +  )z/2 (4.122) −j sin c Vˆ1 (z) = cos c 2 1+p 2 where p=−
Rc R
(4.123)
Similarly, by substituting the values of A 1 and A 2 from (4.120) and (4.121) in (4.119), the normalized voltage wave on line 2 is given by Vˆ2 (z) = −2j
√p sin (  c −  )z e −j( c +  )z/2 1+p 2
(4.124)
Using (4.122) and (4.124), the scattering parameters between different ports of the coupler shown in Figure 4.11 can be determined. In deriving (4.122) and (4.124), we assumed that unit power was incident at port 1. The scattering parameters Sˆ 21 and Sˆ 41 are therefore given by + Sˆ 21 = Vˆ1 (z)  z=l
(4.125)
+ Sˆ 41 = Vˆ2 (z)  z=l
(4.126)
P2 2 =  Sˆ 21  P1
(4.127)
and
or
and
136
Analysis of Uniformly Coupled Lines
P4 2 =  Sˆ 41  P1
(4.128)
Note that (4.122) and (4.124) are quite general and are valid for both quasiTEM and nonTEM mode asymmetrical coupled transmission lines, or for that matter any two coupled waves. For nonTEM modes or waves, however, it is not possible to determine p using (4.123). This is because for the nonTEM modes, R c and R (which are defined on the basis of actual voltages (4.63) and (4.64), respectively) cannot be determined. In this case, the parameter p should be determined as described in [11].
Example 4.2
The design of a 3dB 冠 Sˆ 21  = Sˆ 41  = √1/2 冡 directional coupler in microstrip form is now discussed. Let the length of the coupler be chosen as lg =
c − 
(4.129)
From (4.124) and (4.126), we obtain 2 √p = 1+p
√
1 2
or −
Rc =p=3± R
√8 = 5.828, or 0.1715
(4.130)
The width of the coupled lines and separation between them should be chosen such that the values of R c and R satisfy (4.130). 4.4.2 BackwardWave Directional Couplers
It may be of interest in certain applications to design backwardwave couplers using asymmetrical coupled lines. For example, if the terminating impedances are different for the two lines, it may be advantageous to choose different characteristic impedances for the two lines. Note, however, that unlike asymmetrical forwardwave couplers, backwardwave asymmetrical couplers do not offer any advantages over symmetrical couplers in terms of bandwidth [13]. Their main advantage is that one does not require an additional impedance transformer to match the impedance of a low or highimpedance device to that of the coupler. Cristal [13] has given equations for the design of backwardwave couplers using asymmetrical coupled lines. Figure 4.12(a) shows such a coupler. Assume that lines 1 and 2 are terminated in conductances G a and G b , respectively. Further, assume that the
4.4 Directional Couplers Using Asymmetrical Coupled Lines
Figure 4.12
137
(a) A backward directional coupler using uniform asymmetrical coupled lines, and (b) capacitance representation of coupled lines.
different capacitances of the lines are as shown in Figure 4.12(b). The capacitance matrix of the coupled lines can be expressed as (see Section 3.2) [C] =
冋
C1 C 21
册冋
C 12 C2
=
Ca + Cm −C m
册
−C m Cb + Cm
(4.131)
If k 2 denotes the power coupling coefficient between the lines, then the values of C a , C b , and C m should be chosen according to C a 376.7 冠G a − k √G a G b 冡 = ⑀ 2 √⑀ r √1 − k
C b 376.7 冠G b − k √G a G b 冡 = ⑀ 2 √⑀ r √1 − k
(4.132)
C m 376.7k √G a G b = ⑀ 2 √⑀ r √1 − k where ⑀ = ⑀ 0 ⑀ r , and ⑀ 0 denotes the permittivity of free space.
Example 4.3
Compute the perunit length capacitances of a 10dB asymmetrical coupler whose lines are terminated in loads of 50 ohms and 75 ohms, respectively.
138
Analysis of Uniformly Coupled Lines
The given quantities are k = 10(−10/20) = 0.316 G a = 1/50 = 0.02 ohms G b = 1/75 = 0.0133 ohms From (4.132), we then obtain Ca = 5.895 ⑀ Cb = 3.244 ⑀ Cm = 2.046 ⑀ Once the capacitance parameters are known, the required physical dimensions of the coupler can be determined using data relevant to the transmission line media in which the coupler is to be realized. Commercially available general programs can be used for this purpose [9, 10]. A general uniform fourport coupler with arbitrary terminations and shunting capacitors is shown in Figure 4.13. Formulas useful for the synthesis of a backwardwave directional coupler in the configuration shown in Figure 4.13 are reported in [14].
4.5 Design of Multilayer Couplers Recent progress in miniaturization of microwave circuits has resulted from advances in manufacturing technology such as LTCC and multilayer board fabrication techniques. The analysis of multilayer structures is more challenging, as there are more physical variables (dielectric layers and their thicknesses) compared to single layer
Figure 4.13
A general fourport uniform coupler with shunting capacitors and arbitrary terminations. (From: [14]. 1990 IEEE. Reprinted with permission.)
4.5 Design of Multilayer Couplers
139
circuits. Furthermore, for coupled lines of the same width, but placed on different layers of a multilayer circuit, electric symmetry does not exist about the mid plane. Therefore, for analysis and design of multilayer circuits, EM simulation tools are generally required. A systematic design procedure of multilayer asymmetric couplers based on the normalmode analysis of c and modes has been reported [15]. Although the method is rigorous, it depends on results of EM tools to determine the rather cumbersome normalmode parameters. Furthermore, the design optimization is based on an iterative procedure. For uniformly coupled lines in a multilayer configuration, the inductance and capacitance parameters can be very useful for the design [16–18]. Investigations on numerical data of coupled transmission lines show some very interesting behavior of the capacitive and inductance parameters of coupled lines. This leads to a very simple method for the design of multilayer backward directional couplers [19]. For example, using this method, the width of coupled lines can be found using data of single transmission lines. Furthermore, the spacing between the lines can be determined using mutual capacitance and inductance data which can be found using Sonnet Lite, as described in the next section or any other simulation tool (e.g., [10]). The design is considerably simplified, as two of the three unknowns (the three unknowns are the widths of two lines and the spacing between them) are easily determined. Before discussing the design, we discuss a technique to determine the capacitance and inductance parameters. 4.5.1 Determination of Capacitance and Inductance Parameters Using Sonnet Lite
With the recent progress in the capabilities of EM simulation tools and the increase in speed and memory of personal computers, these tools are now used regularly for the analysis of microwave circuits. For planar multilayer circuits, many tools are now available (generally referred to as 2D or 2.5D EM simulators). One of the advantages of 2D or 2.5D EM simulators over 3D EM simulators is that it is quite simple to set up the problem for analysis. One such EM simulation tool that is available at no cost is Sonnet Lite. It is a limited feature version of the popular Sonnet. Sonnet Lite can be used to directly determine the inductance and capacitance parameters of coupled transmission lines. The first step in setting up analysis is to define the dielectric layers, conductors, and box size. The geometrical shape of the structure to be analyzed can then be defined. A sonnet geometry for two uniformly coupled lines is shown in Figure 4.14. To determine the inductance and capacitance parameters directly, the input ports of all the lines should be first numbered sequentially. The output ports are then defined in the same order. For example, for two coupled lines, the ports are labeled as shown in Figure 4.14. Before the analysis is run, the frequency range of analysis needs to be defined. This can be done using Analysis → Setup from the main window. In this case, a very low frequency should be chosen for analysis, as we are interested in static line parameters. It is generally sufficient to choose an analysis frequency of about 10 MHz for 10mmlong lines. After the analysis has been run (which may take less than a few seconds on an average machine), one
140
Analysis of Uniformly Coupled Lines
Figure 4.14
Sonnet geometry for determining coupled transmission line parameters.
can look at the response by choosing Project → View Response → New Graph. This opens an emgraph window, where the response of the coupled lines is shown. On the main menu from this window, go to Output → Ncoupled Line Model File to look at the inductance and capacitance parameters. If the metal conductors are defined to be of finite conductivity and there is dielectric loss defined for the dielectric layers, the program also determines the resistance and conductance parameters of the coupled lines. It may be noted that different programs use different formats for the data on inductance and capacitance parameters. For example, for two coupled lines, Sonnet Lite version 10.511 gives three values each for the inductance and capacitance parameters. If the inductance parameters from Sonnet Lite are L aa , L bb , and L cc , then the inductance matrix of the coupled lines as defined by (4.79) is given by
冋
册冋
L 11 L 12
[L] =
L 12 L 22
=
L1 Lm
Lm L2
册冋 =
L aa + L bb L bb
L bb L bb + L cc
册
(4.133)
Similarly, if the capacitance parameters from Sonnet Lite are C aa , C bb , and C cc , then the capacitance matrix of coupled lines as defined by (4.80) is given by [C] =
冋
C 11 C 12
册冋
C 12 C 22
=
C1 −C m
册冋
−C m C2
=
C aa − C bb C bb
C bb −C bb + C cc
册 (4.134)
Sonnet Lite gives the net inductance and capacitance for the length of lines considered. To convert the parameters to per unit length, proper scaling can be used. Once the capacitance and inductance parameters are known, the inductive and capacitive coupling coefficients k L and k C can be determined using (4.99) and (4.100), respectively. 4.5.2 Coupler Design
Figure 4.15 shows symmetric offset coupled strip transmission lines. This structure is very useful for the design of highdirectivity tight couplers. Table 4.1 shows the parameters of the inductance and capacitance matrices for various values of spacing s between the lines. Figure 4.16 shows the structures of line 1 and line 2 in absence 1.
For a different version of the software, the format may be different.
4.5 Design of Multilayer Couplers
Figure 4.15
141
Offset coupled strip transmission lines.
Table 4.1 Inductance and Capacitance Parameters of Structure Shown in Figure 4.15 (⑀ r1 = ⑀ r2 = ⑀ r3 = 2.2, H 1 = H 3 = 254 m, H 2 = 50 m, W 1 = W 2 = 250 m) s (mm)
L 11 ( H/m)
L 12 ( H/m)
L 22 ( H/m)
C 11 (nF/m)
C 12 (nF/m)
C 22 (nF/m)
0.07 0.11 0.15 0.19 0.23 0.25 0.29 0.33 0.39 0.49 0.59 0.79 0.99
0.3418 0.3337 0.3281 0.3265 0.3302 0.3342 0.3421 0.3474 0.3515 0.3540 0.3549 0.3552 0.3553
0.2374 0.2177 0.1959 0.1718 0.1441 0.1292 0.1013 0.0794 0.0557 0.0314 0.0179 0.0058 0.0019
0.3418 0.3337 0.3281 0.3265 0.3302 0.3342 0.3421 0.3474 0.3515 0.3540 0.3549 0.3552 0.3553
0.1384 0.1277 0.1160 0.1037 0.0916 0.0862 0.0785 0.0744 0.0715 0.0697 0.0692 0.0690 0.0690
−0.0961 −0.0833 −0.0692 −0.0546 −0.0400 −0.0333 −0.0232 −0.0170 −0.0113 −0.0062 −0.0035 −0.0011 −0.0004
0.1384 0.1277 0.1160 0.1037 0.0916 0.0862 0.0785 0.0744 0.0715 0.0697 0.0692 0.0690 0.0690
of each other (single lines). The line parameters of single lines are shown in Table 4.2. It is interesting to note from Table 4.1 that the selfinductances L 11 and L 22 of coupled lines are nearly constant (within 10%) for nearly all cases of coupling. Their values are nearly the same as those of the respective single lines of Figure 4.16(a, b) of the same physical parameters. Furthermore, in Table 4.3, we have shown values of coupling coefficients determined using (4.99) and (4.100). 2 2 Also, shown in Table 4.3 are the variables C 11 (1 − kC ) and C 22 (1 − kC ). It is interesting to note that the values of these variables also remain nearly constant and are nearly the same as the capacitance of the respective single lines shown in Table 4.2. In a classic paper, Oliver has shown that for the case of a coupler in an homogeneous medium, the variables L 11 C 11 and L 22 C 22 should vary as (1 − k 2 )−1, where k = k C = k L . Both L ii and C ii can vary, but the essential variation is due to the capacitance alone [4]. Furthermore, Oliver has shown that the wave on the
142
Analysis of Uniformly Coupled Lines
Figure 4.16
Structure of Figure 4.15 with only (a) line 1, and (b) line 2.
Table 4.2 Capacitance, Inductance, and Characteristic Impedance of Single Lines of Structures Shown in Figure 4.16 Line
Line Parameters
Line 1 Line 2
L 10 = 0.3552 H/m, C 10 = 0.0689 nF/m, Z 01 = 71.7 ohms L 20 = 0.3552 H/m, C 20 = 0.0689 nF/m, Z 02 = 71.7 ohms
Table 4.3 Computed Coupling Coefficients and Normalized SelfCapacitances of the Structure Shown in Figure 4.15 (⑀ r1 = ⑀ r2 = ⑀ r3 = 2.2, H 1 = H 3 = 254 m, H 2 = 50 m, W 1 = W 2 = 250 m) 2
s (mm)
kL
kC
C 11 (1 − kC ) (nF/m)
0.07 0.11 0.15 0.19 0.23 0.25 0.29 0.33 0.39 0.49 0.59 0.79 0.99
0.6946 0.6524 0.5971 0.5262 0.4364 0.3866 0.2961 0.2285 0.1585 0.0888 0.0503 0.0163 0.0053
0.6946 0.6524 0.5971 0.5261 0.4364 0.3865 0.2960 0.2284 0.1584 0.0887 0.0502 0.0162 0.0053
0.7163 0.7336 0.7463 0.7501 0.7415 0.7328 0.7158 0.7050 0.6967 0.6916 0.6900 0.6893 0.6896
2
C 22 (1 − kC ) (nF/m) 0.7163 0.7336 0.7463 0.7501 0.7415 0.7328 0.7158 0.7050 0.6967 0.6916 0.6900 0.6893 0.6896
4.5 Design of Multilayer Couplers
143
coupled line appears only on the ‘‘backward’’ port if the two lines are terminated in characteristic impedances defined by Z T1 = Z T2 =
√ √
L 11 C 11
(4.135)
L 22 C 22
(4.136)
It has been demonstrated that backward asymmetric couplers in inhomogeneous medium can be designed with a very high directivity, if the physical parameters are chosen such that k C ≈ k L and the lines are terminated in impedances given by (4.135) and (4.136). Next, we consider the case of coupling between asymmetric microstrip lines on different layers of a multilayer substrate as shown in Figure 4.17. The corresponding results for inductance and capacitance parameters are shown in Tables 4.4 through 4.6. It is seen that the evidence is even more compelling in this case where it is seen that the selfinductances L 11 and L 22 are nearly con
Figure 4.17
Coupling between microstrip lines on different layers of a multilayer structure.
Table 4.4 Inductance and Capacitance Parameters of the Structure Shown in Figure 4.17 (⑀ r1 = 9.8, ⑀ r2 = 3.0, H 1 = 254 m, H 2 = 100 m, W 1 = 350 m, W 2 = 150 m) s (mm)
L 11 ( H/m)
L 12 ( H/m)
L 22 ( H/m)
C 11 (nF/m)
C 12 (nF/m)
C 22 (nF/m)
0.20 0.22 0.24 0.26 0.28 0.32 0.36 0.40 0.50 0.60 0.80 1.00 1.20
0.3587 0.3581 0.3573 0.3566 0.3558 0.3551 0.3554 0.3562 0.3582 0.3591 0.3594 0.3596 0.3593
0.2629 0.2564 0.2487 0.2400 0.2304 0.2093 0.1868 0.1649 0.1202 0.0894 0.0528 0.0334 0.0222
0.5608 0.5577 0.5549 0.5529 0.5518 0.5530 0.5578 0.5639 0.5751 0.5802 0.5838 0.5841 0.5838
0.2682 0.2651 0.2616 0.2577 0.2537 0.2453 0.2376 0.2316 0.2234 0.2205 0.2190 0.2187 0.2187
−0.0599 −0.0576 −0.0549 −0.0517 −0.0483 −0.0408 −0.0334 −0.0269 −0.0159 −0.0098 −0.0040 −0.0018 −0.0009
0.0731 0.0726 0.0718 0.0708 0.0696 0.0671 0.0645 0.0623 0.0593 0.0583 0.0577 0.0577 0.0576
144
Analysis of Uniformly Coupled Lines Table 4.5 Capacitance, Inductance and Characteristic Impedance of Single Lines of Structure of Figure 4.17 Line
Line Parameters
Line 1 Line 2
L 10 = 0.3594 H/m, C 10 = 0.2187 nF/m, Z 01 = 40.5 ohms L 20 = 0.5840 H/m, C 20 = 0.0576 nF/m, Z 02 = 100.6 ohms
Table 4.6 Computed Coupling Coefficients and Normalized SelfCapacitances for the Structure of Figure 4.17 (⑀ r1 = 9.8, ⑀ r2 = 3.0, H 1 = 254 m, H 2 = 100 m, W 1 = 350 m, W 2 = 150 m) 2
s (mm)
kL
kC
C 11 (1 − kC ) (nF/m)
0.20 0.22 0.24 0.26 0.28 0.32 0.36 0.40 0.50 0.60 0.80 1.00 1.20
0.5862 0.5737 0.5585 0.5405 0.5200 0.4723 0.4195 0.3679 0.2648 0.1959 0.1152 0.0730 0.0486
0.4278 0.4155 0.4006 0.3830 0.3631 0.3181 0.2699 0.2242 0.1382 0.0863 0.0358 0.0164 0.0083
0.2191 0.2194 0.2196 0.2199 0.2202 0.2205 0.2203 0.2200 0.2191 0.2188 0.2187 0.2187 0.2187
2
C 22 (1 − kC ) (nF/m) 0.0597 0.0600 0.0602 0.0604 0.0605 0.0603 0.0598 0.0592 0.0582 0.0578 0.0577 0.0576 0.0576
2
stant within about 5% and the same is true of parameters C 11 (1 − kC ) and 2 C 22 (1 − kC ). It is interesting to see this behavior for a structure with high inhomogenity. We now show how the above observations lead to a simple design of backward couplers. From the results of Tables 4.1 through 4.6, we can write, semiempirically, L 11 ≈ L 10
(4.137)
L 22 ≈ L 20
(4.138)
C 11 ≈ C 22 ≈
C 10 2
(4.139)
2
(4.140)
1 − kC C 20 1 − kC
where L 10 , L 20 , C 10 , and C 20 denote the parameters of single lines. Substituting results from (4.137) through (4.140) in (4.135) and (4.136), we obtain: Z 01 ≈
Z T1
√
2
1 − kC
(4.141)
4.5 Design of Multilayer Couplers
145
Z 02 ≈
Z T2
√
2
1 − kC
(4.142)
where Z 01 = √L 10 /C 10 and Z 02 = √L 20 /C 20 denote the characteristic impedances of single lines. To reemphasize, L 10 , C 10 , and Z 01 denote the parameters of line 1 in the absence of line 2. Similarly, L 20 , C 20 , and Z 02 denote the parameters of line 2 in the absence of line 1. In a coupler design, Z T1 and Z T2 should be chosen as equal to the desired port impedances to have a perfect match. Let the port impedances be Z 0 . Equations (4.137) through (4.140) then show that the width of the lines shall be chosen to have single line characteristic impedances of Z 01 ≈
Z 02 ≈
Z0
√
2
1 − kC Z0
√1 − kC 2
(4.143)
(4.144)
For example, for a 6dB coupler k C = 0.5, which leads to Z 01 = Z 02 = 57.7 ohms. The widths of the lines can be chosen corresponding to characteristic impedances of 57.7 ohms. Once the widths of the lines have been determined, the coupling can be determined as a function of spacing, using the procedure outlined in Section 4.5.1. Using this information, spacing for the desired coupling can be obtained.
Example 4.4
We now demonstrate the accuracy of the method with different examples. We first choose the case of a coupler in the broadside offset coupled stripline configuration as shown in Figure 4.15 using duriod substrate. The parameters were chosen according to commercially available thicknesses. The coupler parameters were chosen as ⑀ r1 = ⑀ r2 = ⑀ r3 = 2.2, H 1 = H 3 = 790 m, H 2 = 127 m. For a 3dB coupler, k = 0.707, which leads to Z 01 = Z 02 = 70.7 ohms using (4.143) and (4.144). Using Sonnet Lite for a single transmission line structure, we find that W 1 = W 2 = 780 m (Sonnet Lite directly gives the value of characteristic impedance in the response window. A frequency of about 1 GHz or greater should be chosen to avoid error messages). Once the widths of the lines have been determined, the inductance and capacitance parameters are found for various values of spacing s between lines. The inductive and capacitive coupling coefficients can then be computed using (4.99) and (4.100), respectively. The coupling is plotted in Figure 4.18. It is found that for s = 320 m, k = k C = k L = 0.7, which is close to the desired value. To determine the accuracy of the method, the whole coupler structure was EM simulated using Sonnet Lite. The simulated performance is shown in Figure 4.19. The length of the coupler was assumed to be 10 mm. It is interesting to see from
146
Analysis of Uniformly Coupled Lines
Figure 4.18
Coupling as a function of spacing between offset coupled strip transmission lines shown in Figure 4.15. ⑀ r1 = ⑀ r2 = ⑀ r3 = 2.2. H 1 = H 3 = 0.79 mm, H 2 = 0.127 mm. W 1 = W 2 = 0.78 mm.
Figure 4.19
EM simulated response of coupler in configuration of Figure 4.15. ⑀ r1 = ⑀ r2 = ⑀ r3 = 2.2. H 1 = H 3 = 0.79 mm, H 2 = 0.127 mm. W 1 = W 2 = 0.78 mm, length of coupler = 10 mm.
the figure that the coupling is very close to 3 dB. Furthermore, the return loss and isolation are both better than about 30 dB.
Example 4.5
Example 4.1 considered was for a homogeneous medium. In practice, multilayer circuits are inhomogeneous. To verify the accuracy of the method, a multilayer coupler of the form shown in Figure 4.17 was considered with same substrate parameters as considered in [15]. The substrate parameters were ⑀ r1 = 12.8, ⑀ r2 = 6.8, H 1 = 152.4 m, and H 2 = 1.8 m. Using the procedure as described
4.5 Design of Multilayer Couplers
Figure 4.20
147
EM simulated response of coupler in configuration of Figure 4.17. ⑀ r1 = 12.8, ⑀ r2 = 6.8. H 1 152.4 m, H 2 = 1.8 m, W 1 = 42 m, W 2 = 48 m, length of coupler = 2 mm.
earlier, we obtain W 1 = 42 m, W 2 = 48 m. Furthermore, using Sonnet Lite, one finds that a spacing s = 0 is required. For this case, k C = 0.67 and k L = 0.72, or k ≈ (k C + k L )/2 = 0.695. Since in this case, k C and k L are slightly different, the directivity will not be as good. The EM simulated performance of the coupler of length 2 mm is shown in Figure 4.20. It is seen that the coupling is close to 3 dB. Also, the return loss and directivity are better than 20 dB.
References [1] [2] [3] [4] [5]
[6]
[7]
[8] [9]
Lippmann, B. A., Theory of Directional Couplers, M.I.T. Rad. Lab. Rep., No. 860, December 28, 1945. Reed, J., and G. J. Wheeler, ‘‘A Method of Analysis of Symmetrical FourPort Networks,’’ IRE Trans., Vol. MTT4, October 1956, pp. 246–253. Sazanov, D. M., et al., Microwave Circuits, Moscow: Mir Publishers, 1982. Oliver, B. M., ‘‘Directional Electromagnetic Couplers,’’ Proc. IRE, November 1954, pp. 1686–1692. Tripathi, V. K., ‘‘Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT23, September 1975, pp. 734–739. Krage, M. K., and G. I. Haddad, ‘‘Characteristics of Coupled Microstrip Transmission LinesI: CoupledMode Formulation of Inhomogeneous Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT18, April 1970, pp. 217–222. Krage, M. K., and G. I. Haddad, ‘‘Characteristics of Coupled Microstrip Transmission LinesI: Evaluation of CoupledLine Parameters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT18, April 1970, pp. 222–228. Wei, C., et al., ‘‘Multiconductor Transmission Lines in Multilayered Dielectric Media,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, April 1984, pp. 439–450. Djordjevic, A. R., et al., MULTILIN for Windows: Circuit Analysis Models for Multiconductor Transmission Lines, Software and User’s Manual, Norwood, MA: Artech House, 1996.
148
Analysis of Uniformly Coupled Lines [10] [11]
[12] [13]
[14]
[15]
[16]
[17] [18]
[19]
Djordjevic, A. R., et al., LINPAR for Windows: Matrix Parameters for Multiconductor Transmission Lines, Software and User’s Manual, Norwood, MA: Artech House, 1995. Ikalainen, P. K., and G. L. Matthaei, ‘‘Wideband, ForwardCoupling Microstrip Hybrids with High Directivity,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, August 1987, pp. 719–725. Ikalainen, P. K., and G. L. Matthaei, ‘‘Design of Broadband Dielectric Guide 3dB Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, July 1987, pp. 621–628. Cristal, E. G., ‘‘CoupledTransmissionLine Directional Couplers with Coupled Lines of Unequal Characteristic Impedances,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, July 1966, pp. 337–346. Sellberg, F., ‘‘Formulas Useful for the Synthesis and Optimization of General, Uniform Contradirectional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT38, August 1990, pp. 1000–1010. Tsai, C., and K. C. Gupta, ‘‘A Generalized Model for Coupled Lines and Its Applications to TwoLayer Planar Circuits,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT40, December 1992, pp. 2190–2199. Sachse, K., ‘‘The Scattering Parameters and Directional Coupler Analysis of Characteristically Terminated Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT38, April 1990, pp. 417–425. Emery, T., et al., ‘‘Analysis and Design of Ideal Non Symmetrical Coupled Microstrip Directional Couplers,’’ IEEE MTTS Int. Microwave Symp. Digest, 1989, pp. 329–332. Sachse, K., and A. Sawicki, ‘‘QuasiIdeal Multilayer Two and ThreeStrip Directional Couplers for Monolithic and Hybrid MICs,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT47, September 1999, pp. 1873–1882. Mongia, R. K., ‘‘A SemiEmpirical Method for Design of MultiLayer QuarterWave Directional Couplers,’’ IEEE Microwave and Wireless Components Letters, submitted for publication, 2007.
CHAPTER 5
Broadband ForwardWave Directional Couplers As discussed in the previous chapter, forwardwave coupling exists between uniform symmetrical coupled lines if the even and oddmode phase velocities of the coupled lines are unequal. Furthermore, the backwardwave coupling between these lines can be reduced to a very small value by keeping a relatively large separation between the lines (such that the even and oddmode characteristic impedances of the coupled lines are nearly equal). These types of couplers are known as forwardwave directional couplers and can be realized using nonTEM mode transmission lines such as metallic waveguides, dielectric guides, and the like. The bandwidth of forwardwave directional couplers realized using symmetrical coupled lines is generally small and can be increased by using asymmetrical coupled lines. In this chapter, the design and performance of forwardwave couplers realized using uniform asymmetrical lines is first discussed. In the previous chapter, normalmode analysis of symmetrical and asymmetrical coupled lines was discussed. Although the normalmode analysis is rigorous, its application may prove to be very tedious in certain cases. Another approach that can be used to study coupled structures is known as the coupledmode theory. In its simpler form (which is also its most useful form), the theory is valid for weakly coupled structures [1–3]. The theory has been used extensively in the past in numerous applications for the analysis of both passive and active coupled circuits. In its early development, the theory was used mainly for the analysis of microwave circuits such as mode conversion in multimoded waveguides, parametric amplifiers, beamwave interaction in TWTs, and so forth. In recent years, the theory has also been used for the design and analysis of fiber optics and optoelectronics circuits and components. A good review of the coupledmode theory and its applications has been given by Haus and Huang [3]. The theory is quite general and can be used to study coupling phenomenon between any two waves. The two waves may represent two modes of two different transmission lines or of the same transmission line. For example, if a transmission line is bent along its length, the coupledmode theory can be used to study the conversion of power from one mode to another. The coupledmode theory leads to explicit expressions showing how individual waves are modified in the presence of coupling. The theory also leads to an important result that complete power can be transferred between two lossless lines (or two waves) only if both the lines (waves) have the same phase constants. The equivalence between normal and coupledmode theories is also discussed.
149
150
Broadband ForwardWave Directional Couplers
5.1 ForwardWave Directional Couplers The forwardwave coupling between ports 1 and 4 of symmetrical coupled lines shown in Figure 5.1 is given by
 冉
 S 41  = sin
⌬n eff L f c
冊
(5.1)
where L is the length of the coupler, c is the velocity of light in free space, and f is the operating frequency. ⌬n eff is the difference between the square roots of the effective dielectric constants of the even and odd modes: ⌬n eff =
√⑀ ree − √⑀ reo
(5.2)
The direct coupling between ports 1 and 2 can be expressed as
 冉
 S 21  = cos
⌬n eff L f c
冊
(5.3)
Equation (5.1) shows that the coupling is a function of frequency. Assuming that ⌬n eff is independent of frequency, the forwardwave coupling becomes a sinsusoidal function of frequency. In general, however, ⌬n eff is also a function of frequency. Its variation with frequency depends on the type of transmission line and its parameters. It is also seen from (5.1) that the maximum coupling that can be obtained using symmetrical coupled lines by appropriately choosing the length L is 0 dB (complete power transfer). Equation (5.1) is plotted in Figure 5.2 where it is assumed that ⌬n eff is independent of frequency. It is seen from Figure 5.2 that the coupling versus frequency curve is flat (zero first derivative) at the frequency where the coupling is 0 dB. For any other coupling value, however, the coupling versus frequency curve is not flat at the frequency where the desired coupling is obtained. For example, the coupling versus frequency response is not flat at the frequency where 3dB coupling is obtained. It is therefore expected that a 0dB
Figure 5.1
Forwardwave coupler using symmetrical coupled lines.
5.1 ForwardWave Directional Couplers
Figure 5.2
151
Coupling response of symmetrical forward coupler as function of frequency ratio f / f 0 . f 0 denotes the frequency where maximum coupling is achieved.
symmetrical coupler has a wider bandwidth than does a coupler designed for any other coupling value. It is known that it is possible to achieve complete power transfer only between symmetrical coupled lines. If the lines are symmetrical,1 the maximum coupling that can be achieved between the lines is less than 0 dB. The amount of maximum coupling depends on the difference between the phase constants of the asymmetrical coupled lines. This is described in more detail later in this chapter while discussing the coupledmode theory. The asymmetrical couplers will have a flat coupling versus frequency response at the frequency where maximum coupling is obtained. Therefore, if an asymmetrical coupler is designed such that the maximal coupling which can be obtained is equal to the desired coupling value, then such a coupler will have a wider bandwidth than a symmetrical coupler. This principal has been used to demonstrate wideband 3dB forwardwave couplers [4]. A more comprehensive explanation on the broadband properties of asymmetrical couplers can be found in [5].
5.1.1 3dB Coupler Using Symmetrical Microstrip Lines
Usually, the microstrip configuration is used to realize quarterwave backwardwave couplers by choosing suitable values of the even and oddmode impedances. 1.
By asymmetrical lines, we mean lines that have different phase constants when uncoupled.
152
Broadband ForwardWave Directional Couplers
It becomes, however, difficult because of fabrication tolerances to achieve very tight backwardwave coupling (e.g., 3 dB) in parallelcoupled microstrip lines. Further, the directivity of backwardwave couplers realized using microstrip lines tends to be quite poor. This is because the even and oddmode phase velocities of coupled microstrip lines are unequal, resulting in forwardwave coupling. In general, the directivity becomes poorer as the frequency is increased. By choosing a relatively large separation between coupled microstrip lines (such that the evenand oddmode impedances are nearly equal), the backwardwave coupling can be reduced to a very small value. Further, a desired level of forwardwave coupling can be obtained by appropriately choosing the length of the coupler, which can be determined using (5.1). The strip pattern of a 3dB coupler using symmetrical coupled microstrip lines is shown in Figure 5.3. The spacing between the lines is tapered toward the ends of the coupler. This is done to avoid any abrupt physical discontinuities in the structure, which will lead to reflections causing power to couple to port 3, which is designed to be the isolated port. The theoretical and experimental results of a 3dB symmetrical forwardwave coupler are shown in Figure 5.4. The coupler was designed to operate at 10 GHz. The substrate material has a dielectric constant of 2.2 and a thickness of 0.762 mm. The width of the microstrip line corresponds to an impedance of 50⍀ and is a constant throughout. The length of the straight middle part is 113 mm, which is equivalent to 5.2 guide wavelengths at 10 GHz. The spacing between the lines in the middle section is twice the substrate thickness. The curved sections have a radius of curvature of 102 mm. The coupling of the straight section was determined
Figure 5.3
Strip pattern of symmetrical 3dB forwardwave coupler. (From: [5]. 1987 IEEE. Reprinted with permission.)
Figure 5.4
Theoretical and experimental response of a symmetrical 3dB forwardwave coupler. (From: [5]. 1987 IEEE. Reprinted with permission.)
5.1 ForwardWave Directional Couplers
153
using (5.1). To determine the coupling between curved sections, they were considered to consist of 20 small, straight segments. The coupling of each small segment was determined using (5.1), with the overall coupling determined by summing coupling from various sections. The coupled microstrip lines were analyzed using formulas given in [6]. Figure 5.4 shows that the shapes of the measured coupling curves match well with the theoretically predicted values, except that the measured center frequency is somewhat lower. The measured directivity of the coupler is about 40 dB. It is seen that in the frequency band considered, the coupling increases monotonically with frequency. 5.1.2 Design and Performance of 3dB Asymmetrical Couplers
The design equations of a 3dB directional coupler using asymmetrical coupled microstrip lines were given in Section 4.4.1. The length of the coupler is given by lg =
c − 
(5.4)
where  c and  denote the phase constants of the c and modes, respectively. Furthermore, the ratio of R c and R should be chosen as −
Rc =3± R
√8 = 5.828, or 0.1715
(5.5)
where R c and R denote the voltage ratios on the two lines for the c and modes, respectively. In the design of the coupler, the width of asymmetrical coupled microstrip lines was chosen to correspond to (uncoupled) impedances of 50⍀ and 100⍀. The design is completed by choosing the separation between coupled lines such that the ratio of R c and R satisfies (5.5). For given width of the lines and assumed separation between them, the self and mutual inductance and capacitance parameters can be found using the coupled microstrip data [6] and the technique described in Section 4.3.4. Further, if the self and mutual inductance and capacitance parameters of the coupled lines are known, the values of R c and R can be found using (4.97) and (4.98). The coupler was fabricated on a substrate having the same parameters as that used for the symmetrical coupler discussed in the last section. The top view of the strip pattern of the coupled section and the input and output lines are shown in Figure 5.5. It was found that a 1.81mm separation between the lines is needed to obtain the required value of R c /R . In practice, a 1.65mm separation was used, which gave more than 3dB coupling at the center frequency but gave a wider bandwidth for 1dB amplitude balance. The theoretical and experimental results of the coupling and isolation for this coupler are shown in Figure 5.6. The length of the coupler was found to be 220 mm using (5.4), where  c and  were obtained from (4.91) and (4.92), respectively. Because the feed lines also contribute to some coupling in the fabricated coupler, the length of the uniform coupled section was
154
Broadband ForwardWave Directional Couplers
Figure 5.5
Strip pattern of asymmetrical forwardwave coupler. (From: [5]. 1987 IEEE. Reprinted with permission.)
Figure 5.6
Theoretical and experimental response of an asymmetrical 3dB forwardwave coupler. (From: [5]. 1987 IEEE. Reprinted with permission.)
chosen to be 190 mm and the curved feed lines had a radius of curvature of 102 mm. It is seen from Figure 5.6 that the agreement between theory and experiment is quite good, considering that the conductor and dielectric losses were not accounted for in the theory. The coupler has a bandwidth of about 60% for 1dB amplitude balance. The isolation is better than about 40 dB in the complete frequency range. It is easily verified by comparing Figures 5.4 and 5.6 that a uniform forwardwave asymmetrical coupler offers more bandwidth than a uniform symmetrical coupler. Note, however, that unlike a symmetrical coupler, the phase difference between the output ports of an asymmetrical coupler is not 90 degrees. The theoretically computed and measured phase differences between the coupled and direct port of the asymmetrical coupler are shown in Figure 5.7(a). It is interesting to note that at the frequency where the phase difference between the output ports (ports 2 and 4) is 0 degree when port 1 is the driven port, the phase difference between output ports is 180 degrees when the input is at port 3. In general, an asymmetrical coupler that has endtoend symmetry satisfies the following phasedifference relationship: (∠ S 41 − ∠ S 21 ) + (∠ S 23 − ∠ S 43 ) = 180 degrees
(5.6)
It is possible to achieve approximately the desired phase difference between output ports over a reasonably wide bandwidth in an asymmetrical coupler using an extra length of line as a phase compensating element. For example, if one adds a length of line having a phase shift of 106 degrees at 9.6 GHz to ports 3 and 4, the outputs (S 21 and S 41 ) will be in phase quadrature at that frequency. Further
5.1 ForwardWave Directional Couplers
Figure 5.7
155
(a) Computed and measured phase difference between output ports of an asymmetrical 3dB forwardwave coupler. (b) Theoretically computed phase difference between the coupled and through ports of the asymmetrical coupler with reference planes chosen to approximate quadrature or magicT performance. ⌬ 1 and ⌬ 2 are defined in part (a). (From: [5]. 1987 IEEE. Reprinted with permission.)
more, the outputs are held in phase quadrature within about 12 degrees over a frequency range of 7.0 to 12.2 GHz. By adding another quarterwavelong line section at ports 3 and 4, a phase difference of 180 degrees between the outputs can be obtained. This is shown in Figure 5.7(b), where the two cases are labeled as ‘‘quadraturetype’’ and ‘‘magicTtype,’’ respectively.
5.1.3 UltraBroadband ForwardWave Directional Couplers
The bandwidth of forwardwave couplers realized using asymmetrical coupled lines is greater than those realized using symmetrical coupled lines. It is still not possible, however, to achieve a very broadband coupling (multioctave) using uniform asymmetrical coupled lines. Very broadband coupling can be achieved by continuously varying the phase constants of coupled lines (  1 and  2 ) and the coupling coefficient between them along the length of the structure. This results in a nonuniform structure. This principle of broadband coupling is known as the normalmode warping [7–9]. The cross section of such a structure varies continuously along the
156
Broadband ForwardWave Directional Couplers
length. The essential features of the normalmode warping can be summarized as follows [8]: •
•
Adjust the geometrical parameters at the input of the structure such that the input excitation is identical with one of the normal modes of the coupled structure. Gradually warp the normal mode (the mode in which the power is launched) by continuously varying the structure along the longitudinal direction until the distribution of the normal mode is identical with the desired output. The normal mode at the cross section of the output now contains power in both the coupled modes in the desired ratio.
Note that nonuniform couplers tend to be very long (tens to hundreds of wavelengths). Therefore, they are mainly useful at high millimeterwave and optical frequencies where their physical lengths can be kept reasonably small.
5.2 CoupledMode Theory Consider two lines that are uniformly coupled over a certain length as shown in Figure 5.8. As already discussed, these lines may represent two actual transmission lines, or in a more general case, any two waves. The lines are assumed to be weakly coupled. By the term ‘‘weakly’’ coupled, we mean that the impedances of individual lines (or waves) are affected by a very small amount in the presence of coupling. There is, therefore, negligible coupling in the backward direction, and the predominant coupling takes place in the forward direction only. For example, if power is incident at port 1 as shown in Figure 5.8, the power coupled between the lines appears at port 4 only. The forwardwave coupling between the lines, perunit wavelength of the coupling section, is also assumed to be small. The forwardtraveling waves on the two lines (in the absence of any coupling between them) can then be expressed, respectively, as
Figure 5.8
Uniformly coupled asymmetrical coupled lines.
5.2 CoupledMode Theory
157
Vˆ1 = Ae −j 1 z
(5.7)
Vˆ2 = Be −j 2 z
(5.8)
and
where Vˆ1 and Vˆ2 denote the normalized voltage waves on lines 1 and 2, respectively, and are complex quantities. The power carried by lines 1 and 2 are thus given by  Vˆ1  2 and  Vˆ2  2, respectively.  1 and  2 denote the respective phase constants of lines 1 and 2. The coupledmode theory as given by Miller [1] on which the present discussion is based is valid for complex values of  1 and  2 , but for the sake of simplicity, we assume that these are real quantities. By differentiating (5.7) and (5.8) with respect to z, we obtain, respectively, dVˆ1 = −j 1Vˆ1 dz
(5.9)
dVˆ2 = −j 2Vˆ2 dz
(5.10)
and
In the presence of coupling between the lines as shown in Figure 5.8, the existing voltage waves on both the lines are perturbed. According to coupledmode theory, (5.9) and (5.10) representing voltage waves on the two lines are modified as follows in the presence of coupling: dVˆ1 = −j(  1 + K 11 )Vˆ1 − jK 12Vˆ2 dz
(5.11)
dVˆ2 = −jK 21 Vˆ1 − j(  2 + K 22 )Vˆ2 dz
(5.12)
and
where K 11 and K 22 are the selfcoupling coefficients, and K 12 and K 21 are the mutualcoupling coefficients. Their dimensional unit is perunit length. When the two lines are uncoupled, the propagation constants of the two lines are given by  1 and  2 , respectively. When the two lines are brought closer, the propagation constant of each line changes because of the presence of the other line. The modified propagation constant of line 1 due to the presence of line 2 is denoted by (  1 + K 11 ). Similarly, (  2 + K 22 ) denotes the modified propagation constant of line 2 in the presence of line 1. For weak coupling between lines:
 K 11  Ⰶ  1  K 22  Ⰶ  2  K 12  ,  K 21  Ⰶ  1 ,  2
(5.13)
158
Broadband ForwardWave Directional Couplers
With the above assumptions, (5.11) and (5.12) reduce to dVˆ1 = −j 1Vˆ1 − jK 12Vˆ2 dz
(5.14)
dVˆ2 = −jK 21Vˆ1 − j 2Vˆ2 dz
(5.15)
and
Note that while deriving (5.14) and (5.15), the terms containing K 12 and K 21 in (5.11) and (5.12) have been retained, while those containing K 11 and K 22 have been neglected. The reason for this is that although  K 12  and  K 21  are much smaller than  1 and  2 , it is not necessary that  K 12 Vˆ2  Ⰶ   1 Vˆ1  or  K 21 Vˆ1  Ⰶ   2 Vˆ2  for all values of z. 5.2.1 Nature of Coupling Coefficients K 12 and K 21
The total power on the two lines at any cross section is given by 2 2 W =  Vˆ1  +  Vˆ2 
(5.16)
The principle of conservation of power requires that if the lines are lossless, the total power remains the same at all cross sections. In mathematical terms: d ˆ 2 冠  V1  +  Vˆ2  2 冡 = 0 dz
(5.17)
d ˆ ˆ* ˆ ˆ* 冠V V + V2 V2 ) = 0 dz 1 1
(5.18)
dVˆ1* ˆ * dVˆ1 ˆ dVˆ2* ˆ * dVˆ2 + V1 + V2 + V2 =0 Vˆ1 dz dz dz dz
(5.19)
or
or
Substituting values of first derivatives in (5.19) from (5.14) and (5.15), we obtain K 12 = K 21 = K
(5.20)
where K is purely real and denotes the coupling coefficient between the lines. 5.2.2 Waves on Lines 1 and 2 in the Presence of Coupling
Let us assume that initially there is a wavecarrying unit power on line 1 only; that is:
5.2 CoupledMode Theory
159
Vˆ1 = 1, Vˆ2 = 0 at z = 0
(5.21)
Solution of coupled equations (5.14) and (5.15) with the initial condition of (5.21) gives Vˆ1 =
冋
册
1 ( 1 −  2 ) e −j s z + 2 2 (  −  )2 + 4K 2 √ 1 2
冋
+
(5.22)
册
1 ( 1 −  2 ) e −j f z − 2 2 (  −  )2 + 4K 2 √ 1 2
and Vˆ2 =
K
√(  1 −  2 )2 + 4K 2
e −j s z −
K
√(  1 −  2 )2 + 4K 2
e −j f z
(5.23)
where
s =
( 1 +  2 ) + 2
√(  1 −  2 )2 + 4K 2
(5.24)
f =
( 1 +  2 ) − 2
√(  1 −  2 )2 + 4K 2
(5.25)
2
and
2
The above equations show that in the presence of coupling, the waves on the two lines can be represented as interference between two waves having different phase constants from those of the uncoupled waves. One of these waves (which can be termed as a slow wave) has a phase constant equal to  s , while the other (which can be termed as a fast wave) has a phase constant equal to  f . The phase constants  s and  f depend on the phase constants of individual lines (when the lines are uncoupled) and the coupling coefficient K. These waves with phase constants  s and  f may be considered to represent two normal modes of the coupled structure. Equations (5.22) and (5.23) express the wave on each line in terms of interference between normal modes. The coupling coefficient K depends on the specifics of the structure, and its determination requires the use of field theory methods [10]. Irrespective of the value of the coupling coefficient K, however, (5.22) and (5.23) can be used to draw some significant conclusions on uniform coupling between two lines (or waves). For example, these equations lead to a significant conclusion that if the phase constants of two lines (waves) are different (  1 ≠  2 ), it is not possible to completely transfer power between the two lines (waves).
160
Broadband ForwardWave Directional Couplers
Coupling Between Symmetrical Lines
Let the two coupled lines be symmetrical (  1 =  2 ). Using (5.24) and (5.25), we obtain
s =  0 + K
(5.26)
f =  0 − K
(5.27)
and
where
0 = 1 = 2 Substituting the values of  s and  f from (5.26) and (5.27) in (5.22) and (5.23), we obtain (e −jKz + e jKz ) −j 0 z Vˆ1 = e 2
(5.28)
= cos (Kz) e −j 0 z (e −jKz − e jKz ) − 0 z Vˆ2 = e 2
(5.29)
= −j sin (Kz) e −j 0 z The fraction of power coupled from line 1 to line 2 over a length z of the coupling section is then given by r=
 Vˆ2 (z)  2  Vˆ1 (z = 0)  2
= sin2 (Kz)
(5.30)
where it is assumed that all the power is in line 1 at z = 0. The power coupled to line 2 appears at port 4. 5.2.3 CoupledMode Theory and Even and OddMode Analysis
Using (5.26) and (5.27): K=
s − f 2
(5.31)
0 =
s + f 2
(5.32)
and
5.2 CoupledMode Theory
161
Substituting the values of K and  0 from (5.31) and (5.32) in (5.28) and (5.29), we obtain
冋
册
冋
册
−j (  s −  f )z Vˆ1 = cos e 2
(s + f ) z 2
(5.33)
and −j (  s −  f )z Vˆ2 = −j sin e 2
(s + f ) z 2
(5.34)
It is interesting to find that (5.33) and (5.34) are identical to (4.35) and (4.36), respectively, if  s and  f are assumed to denote the phase constants of even and odd modes, respectively. Equations (4.35) and (4.36) were derived using the evenand oddmode approach, while (5.33) and (5.34) were derived using the coupledmode theory. 5.2.4 Coupling Between Asymmetrical Lines
Let the phase constants of two asymmetrical coupled lines be  1 and  2 , respectively. It is assumed that initially a unit amount of power is incident in line 1 (i.e., Vˆ1 = 1 at z = 0). Using (5.22) and (5.23), the voltage wave on line 1 is given by Vˆ1 = e
−j
( 1 +  2 ) z 2
V1′
(5.35)
where V1′ = cos −j
冋冉√
(  1 −  2 )2 4K 2
( 1 −  2 ) 2K
冋√
冊册
+ 1 Kz 1
(  1 −  2 )2 +1 4K 2
册
sin
冋冉√
(  1 −  2 )2 4K 2
冊册
+ 1 Kz
The voltage wave on line 2 is expressed as Vˆ2 = e
−j
( 1 +  2 ) z 2
V2′
(5.36)
where V2′ = −
j
√
(  1 −  2 )2 4K 2
sin +1
冋冉√
冊册
(  1 −  2 )2 + 1 Kz 4K 2
162
Broadband ForwardWave Directional Couplers
The maximum power transfer from line 1 to line 2 takes place when
冉√
(  1 −  2 )2 4K
2
冊
+ 1 Kz =
2
(5.37)
The maximum fractional power coupled between lines 1 and 2 is given by 1 2 rmax =  Vˆ2  max = (  1 −  2 )2 4K 2
(5.38) +1
It is seen from (5.38) that r max (which represents the maximum power that can be coupled between lines) is less than unity if the values of  1 and  2 are different from each other. The value of r max is plotted as a function of normalized difference in phase velocities of the two lines in Figure 5.9. It is seen that if the difference in phase velocities of the two lines is much greater than the coupling coefficient [(  1 −  2 ) Ⰷ K], only a small amount of power can be coupled between the lines. For example, for a value of (  1 −  2 )/K = 10, the maximal coupling that can be achieved between the lines is only 14.2 dB. After having discussed the normal (Chapter 4) and coupled modes, it is useful to summarize the essential difference between the two. The normal modes of a uniform structure are those that can propagate independent of each other along the structure. Each normal mode is characterized by a unique phase velocity and field distribution. For example, the various TE mn and TM mn modes are the normal
Figure 5.9
Maximum power coupled between asymmetrical coupled lines.
5.3 CoupledMode Theory for Weakly Coupled Resonators
163
modes of a straight, uniform rectangular metal waveguide. Similarly, the even and odd modes are the two normal modes of symmetrical coupled lines. An important property of normal modes is that there is no conversion of energy from one normal mode to another. For example, if an even mode is launched along a symmetrical coupled structure, the energy remains in the even mode all along the length of the structure. On the other hand, energy is continuously exchanged between coupled modes. The individual waves on two coupled lines are an example of two coupled modes. There is a continuous exchange of energy between the waves on the two lines. As another example of coupled modes, consider a rectangular waveguide that is uniformly bent along the direction of propagation. The various TE mn and TM mn modes that are the normal modes of a straight waveguide are no longer the normal modes of the bent waveguide. The various TE mn and TM mn modes are now the coupled modes because energy will continuously be exchanged between these modes along the bend of the waveguide. Usually, it is sufficient to consider coupling between two or three modes only to determine the state of a weakly coupled system.
5.3 CoupledMode Theory for Weakly Coupled Resonators The theory of weakly coupled resonators can be developed in a similar fashion as is for the weakly coupled lines. Consider two isolated resonators having resonant frequencies as 1 and 2 , respectively. The timevarying amplitudes of two uncoupled resonators are given by [3] daˆ 1 = j 1 aˆ 1 dt
(5.39)
daˆ 2 = j 2 aˆ 2 dt
(5.40)
and
where aˆ 1 and aˆ 2 denote the instantaneous normalized amplitudes of resonators 1 and 2, respectively. When two resonators are weakly coupled, the governing equations for the resonator amplitudes are modified as daˆ 1 = j 1 aˆ 1 + jKaˆ 2 dt
(5.41)
daˆ 2 = jKaˆ 1 + j 2 aˆ 2 dt
(5.42)
and
where K denotes the coupling coefficient between the resonators. The dimensional unit of K in this case is perunit time. It may be noted than in case of coupled
164
Broadband ForwardWave Directional Couplers
lines, the dimensional unit of K is perunit length. (5.41) and (5.42) are similar to (5.14) and (5.15), respectively. The solution of the coupled equations (5.41) and (5.42) is therefore similar to the solution of coupled equations (5.14) and (5.15). The coupling between resonators causes the resonant frequencies of the normal modes of the coupled system to be different from 1 and 2 . More specifically, the resonant frequencies of normal modes (denoted by a and b ) are given by
a =
( 1 + 2 ) + 2
√( 1 + 2 )2 + 4K 2
(5.43)
b =
( 1 + 2 ) + 2
√( 1 − 2 )2 + 4K 2
(5.44)
2
and
2
When two resonators having the same resonant frequency ( 0 = 1 = 2 ) are coupled, the resonant frequencies of the coupled resonators become
a = even = 0 + K
(5.45)
b = odd = 0 − K
(5.46)
In microwave and RF circuits, it is common practice to denote coupling between resonators using the equivalent circuit approach [11, 12]. For example, coupling between two identical resonators can be represented by the circuit shown in Figure 5.10 where the resonators are coupled through a mutual inductance M. In the absence of coupling, the resonant frequency of either resonator is given by 1
2
f0 =
(5.47)
2
4 LC
In the presence of coupling, the effect of the mutual inductance is either additive or subtractive to the selfinductance. The resonant frequencies of the coupled resonators are then given by 2
feven =
Figure 5.10
1 4 2(L + M)C
=
1
冉
4 2LC 1 +
M L
冊
=
1 4 2LC (1 + k)
Lumped equivalent circuit of coupled identical resonators.
2
=
f0 (5.48) 1+k
5.3 CoupledMode Theory for Weakly Coupled Resonators
2 fodd
=
1 4 2(L − M)C
=
1
冉
4 2LC 1 −
M L
165
冊
2
f0 (5.49) = = 2 1 −k 4 LC (1 − k) 1
where k = M /L is the voltage coupling coefficient. For k Ⰶ 1, which is usually the case, (5.48) and (5.49) then lead to the following very useful relations: + fodd f f 0 ≈ even 2
(5.50)
f − feven k ≈ odd 2
(5.51)
Equations (5.50) and (5.51) are extensively used in the design of filters as discussed later in Chapters 9 to 11.
References [1] [2] [3] [4] [5]
[6]
[7] [8]
[9] [10] [11] [12]
Miller, S. E., ‘‘Coupled Wave Theory and Waveguide Applications,’’ Bell Syst. Tech. J., Vol. 33, 1954, pp. 661–719. Louisell, W. H., CoupledMode and Parametric Electronics, New York: Wiley, 1960. Haus, H. A., and W. Huang, ‘‘CoupledMode Theory,’’ Proc. IEEE, Vol. 79, October 1991, pp. 1505–1518. Ikalainen, P. K., and G. L. Matthaei, ‘‘Design of Broadband Dielectric Waveguide 3dB Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, July 1987, pp. 621–628. Ikalainen, P. K., and G. L. Matthaei, ‘‘Wideband, ForwardCoupling Microstrip Hybrids with High Directivity,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, August 1987, pp. 719–725. Kirsching, M., and R. H. Jansen, ‘‘Accurate WideRange Design Equations for the FrequencyDependent Characteristics of ParallelCoupled Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, January 1984, pp. 83–90. Corrections, Vol. MTT33, March 1985, p. 288. Cook, J. S., ‘‘Tapered Velocity Couplers,’’ Bell Syst. Tech. J., Vol. 34, July 1955, pp. 807–822. Fox, A. G., ‘‘Wave Coupling by Warped Normal Modes,’’ Bell Syst. Tech. J., Vol. 34, July 1955, pp. 823–852. Also see Fox, A. G., ‘‘Wave Coupling by Warped Normal Modes,’’ IRE Trans., Vol. 3, December 1955, pp. 2–6. Louisell, W. H., ‘‘Analysis of Single TaperedMode Coupler,’’ Bell Syst. Tech. J., Vol. 34, July 1955, pp. 853–870. Yariv, A., ‘‘CoupledMode Theory for Guided Wave Optics,’’ IEEE J. on Quantum Electronics, Vol. QE9, September 1973, pp. 919–933. Cohn, S. B., ‘‘Bandpass Filters Containing High Q Dielectric Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT16, April 1968, pp. 218–227. Van Bladel, J., ‘‘Weakly Coupled Dielectric Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT30, 1982, pp. 1907–1914.
CHAPTER 6
ParallelCoupled TEM Directional Couplers
In Chapter 4 it was demonstrated that if two identical TEM lines are parallelcoupled as shown in Figure 6.1, then by properly choosing the even and oddmode impedances of the coupled lines, a fourport directional coupler can be obtained. The coupler shown in Figure 6.1 is also called a backwardwave directional coupler because the coupling takes place in the backward direction on the coupled line. For example, if power is incident at port 1, power is coupled to port 3. The maximum coupling between ports 1 and 3 (or between ports 2 and 4) takes place at a frequency where the coupler is a quarterwave long (or odd multiples thereof). Because the electrical length of a coupler varies with frequency, the coupling also varies with frequency. The variation of coupling with frequency can be reduced by employing multisection TEM couplers. In a multisection coupler, a number of coupled sections are cascaded. Each coupled section is a quarterwave long at the center frequency and has different even and oddmode impedances compared with those of the adjacent sections. Multisection couplers that have endtoend symmetry are known as symmetrical couplers, while those that do not have endtoend symmetry are known as asymmetrical couplers. This chapter discusses the theory and design of single and multisection parallelcoupled TEM directional couplers. Simple analytical expressions for the design of singlesection couplers exist. Unfortunately, no simple analytical formulas are possible for the design of optimal multisection couplers. Some design tables are available in the literature for equalripple symmetrical and asymmetrical couplers [1–3]. The microstrip line is the most popular transmission line for realizing microwave integrated circuit (MIC) components. When microstrip is used in backwardwave couplers, however, the directivity is generally poor because the even and oddmode phase velocities of coupled microstrip lines are unequal. In this chapter, various techniques that can be used to equalize the even and oddmode phase velocities of coupled microstrip lines are also discussed.
6.1 Coupler Parameters The schematic of a fourport directional coupler is shown in Figure 6.2. The four ports are labeled as ‘‘input,’’ ‘‘direct’’ (through), ‘‘coupled,’’ and ‘‘isolated.’’ Two
167
168
ParallelCoupled TEM Directional Couplers
Figure 6.1
A single section TEM coupler.
Figure 6.2
Schematic of a fourport directional coupler.
important factors that characterize a directional coupler are its coupling and directivity, defined here: P Coupling (dB) = 10 log 1 P3
(6.1)
P Directivity (dB) = 10 log 3 P4
(6.2)
where P1 is the power input at port 1 and P3 and P4 are the power outputs at ports 3 and 4, respectively. Note that all the ports are assumed to be matchterminated. There is no power at port 4 in the ideal case; in practice, a small amount of power is always coupled to this port. If the coupling and directivity are known, the isolation of the coupler can be determined. The isolation is defined as P Isolation (dB) = 10 log 1 P4
(6.3)
or Isolation (dB) = Coupling (dB) + Directivity (dB) All the parameters described above are normally expressed in decibels and are defined here as positive quantities.
6.2 SingleSection Directional Coupler
169
6.2 SingleSection Directional Coupler 6.2.1 Frequency Response
In Section 4.2.2, it was shown that if two identical parallel TEM lines are coupled over a length l as shown in Figure 6.1, then under the condition 2
Z 0e Z 0o = Z 0
(6.4)
the scattering parameters of the network are given by S 11 = S 22 = S 33 = S 44 = 0
(6.5)
S 14 = S 41 = S 23 = S 32 = 0
(6.6)
S 12 = S 21 = S 34 = S 43 = S 21e =
(6.7)
√1 − k
√1 − k
2
2
cos + j sin
S 13 = S 31 = S 24 = S 42 = S 11e =
(6.8) jk sin
√1 − k
2
cos + j sin
where =  l denotes the electrical length of the coupler and k is given by k=
Z 0e − Z 0o Z 0e + Z 0o
(6.9)
Further, S 11e and S 21e denote the reflection and transmission coefficients, respectively, of the coupled lines for the case of evenmode excitation. The maximum amount of coupling between ports 1 and 3 (or between ports 2 and 4) occurs when
= l =
rads 2
(6.10)
or l=
g = 2 4
The properties of an ideal parallelcoupled TEM directional coupler were described in Chapter 4. Frequency Bandwidth Ratio
The frequency bandwidth ratio B of a directional coupler (single or multisection) is defined as
170
ParallelCoupled TEM Directional Couplers
f B= 2 f1
(6.11)
where f 2 and f 1 are the upper and lower frequencies in between which the coupling is within the tolerance amount ␦ compared with its midband value as shown in Figure 6.3. The tolerance amount ␦ can be arbitrarily specified. Fractional Bandwidth
The fractional bandwidth w of a directional coupler is defined as f − f1 w= 2 f0
(6.12)
f + f2 f0 = 1 2
(6.13)
where
is the center frequency of the coupler. The frequency bandwidth ratio B and the fractional bandwidth w are related by B−1 B+1
(6.14)
1 + w /2 1 − w /2
(6.15)
w=2 and B=
Figure 6.3
Typical variation of coupling in a single section TEM coupler.
6.2 SingleSection Directional Coupler
171
Useful Operating Bandwidth
The variation of coupling to the ‘‘direct’’ and ‘‘coupled’’ ports of an ideal 3 ± 0.3dB coupler is shown in Figure 6.4(a) as a function of frequency ratio f / f 0 , where f 0 refers to the frequency where the coupler is a quarterwave long. Because we have assumed a tolerance amount of ± 0.3 dB, this coupler is designed to have a coupling (to the ‘‘coupled’’ port) of 2.7 dB at the midband. The performance of 6 ± 0.3 and 10 ± 0.5dB singlesection couplers is shown in Figure 6.4(b, c). The variation of coupling to the ‘‘coupled’’ port of a 20 ± 0.5dB coupler is shown in Figure 6.4(d). In this case, the power coupled to the direct port is nearly 0 dB. The useful frequency operating range of singlesection couplers can be determined by referring to these plots. For example, it is seen from Figure 6.4(a) that for a tolerance in coupling of ± 0.3 dB, a 3dB coupler can be operated over a frequency bandwidth ratio of about 2 ( f 2 : f 1 ≈ 2 : 1). Singlesection couplers are generally useful for operation over a frequency bandwidth ratio (B) of approximately 2. 6.2.2 Design
From (6.9), we can write Z 0e /Z 0o in terms of a voltage coupling coefficient k as Z 0e 1 + k = Z 0o 1 − k
Figure 6.4
(6.16)
(a) Variation of coupling to the direct and coupled ports of a TEM coupler designed for nominal coupling of 3 dB. (b) Variation of coupling to the direct and coupled ports of a TEM coupler designed for 6dB nominal coupling. (c) Variation of coupling to the direct and coupled ports of a TEM coupler designed for nominal coupling of 10 dB. (d) Variation of coupling to the coupled port of a TEM coupler designed for a nominal coupling of 20 dB.
172
ParallelCoupled TEM Directional Couplers
Figure 6.4
(continued).
Furthermore, the simultaneous solution of (6.4) and (6.16) gives Z 0e = Z 0
√
Z 0o = Z 0
√
1+k 1−k
(6.17)
and 1−k 1+k
(6.18)
In a coupler design, for a given voltage coupling coefficient k and characteristic impedance Z 0 , we first determine the even and oddmode impedances using (6.17) and (6.18), respectively. The dimensions of the coupler are then calculated using the physical data of coupled transmission lines such as those discussed in Chapter 3 for various transmission lines. The physical length l of the coupler is chosen as l=
g 4
(6.19)
where g is the guide wavelength of the TEM wave in the transmission line medium at the design frequency f 0 .
6.2 SingleSection Directional Coupler
Figure 6.4
173
(continued).
If the coupling from port 1 to port 3 is given as C dB (where C is a positive quantity), then k is related to C as k = 10−C/20
(6.20)
Example 6.1
A directional coupler of 10 ± 0.5dB coupling is desired in the configuration as shown in Figure 6.1 at a frequency of 10 GHz. Determine the physical dimensions of the coupler assuming that ports are terminated in an impedance of 50⍀ and the coupler is realized in a stripline medium of ⑀ r = 2.25. The coupler is designed to have a coupling of 9.5 dB at the midband because a tolerance of ± 0.5 dB in the coupling value has been specified. Using (6.20), the voltage coupling coefficient k is found to be k = 10−9.5/20 = 0.335 The even and oddmode impedances of the coupled lines are obtained as
174
ParallelCoupled TEM Directional Couplers
Figure 6.4
(continued).
Z 0e = Z 0
√
Z 0o = Z 0
√
1+k = 50 1−k
√
1 + 0.335 = 70.84⍀ 1 − 0.335
and 1−k = 50 1+k
√
1 − 0.335 = 35.29⍀ 1 + 0.335
Therefore,
√⑀ r Z 0e = 106.26⍀ √⑀ r Z 0o = 52.94⍀ From Figure 3.15, S/b ≈ 0.03 and W/b ≈ 0.65 if we assume that the thickness of the strip conductors is negligible. Thus, if the separation between the ground plane is 1 mm, the gap between conductors, S ≈ 0.03 mm, and conductor width, W ≈ 0.65 mm. A stripline supports a pure TEM mode of propagation. In a medium of ⑀ r = 2.25, the guide wavelength in the medium at a frequency of 10 GHz is given by
6.2 SingleSection Directional Coupler
175
g =
0 30 = = 20 mm √⑀ r √2.25
The physical length l of the coupler is therefore given by l=
g = 0.005m = 5 mm 4
The useful bandwidth of the ideal coupler is found to be 62.5% from Figure 6.4(c).
Example 6.2
Design a 20 ± 0.5dB directional coupler in the microstrip configuration at 5 GHz. Determine physical dimensions of the coupler realized on 0.635mmthick alumina substrate having ⑀ r = 9.7. The coupler is designed to have a 19.5dB midband coupling. From (6.20), the voltage coupling coefficient k is given by k = 10−19.5/20 = 0.106 Furthermore, from (6.17) and (6.18), the even and oddmode impedances of the coupled lines are obtained as Z 0e = 55.6⍀ and Z 0o = 45.0⍀ From Figure 3.22, W/h ≈ 0.95 and S/h ≈ 1.3 (approximately extrapolated). For a 0.635mmthick substrate, W ≈ 0.6 mm and S ≈ 0.83 mm. The physical length of the coupler is calculated using
=
e + o 2 冠√⑀ ree + √⑀ ree 冡 = l = 90 deg 2 0 2
or 360 60
冠√7.2 + √6 冡 2
l = 90 deg
which gives l ≈ 5.85 mm. The useful bandwidth of the ideal coupler is found to be 60% from Figure 6.4(d).
176
ParallelCoupled TEM Directional Couplers
6.2.3 Compact Couplers
When size and cost requirements are stringent, compact directional couplers are mandatory. The coupled line approaches for such couplers are reported in the literature [4–6]. For such couplers, we can use either MIC or MMIC technology. Basically, there are two techniques to design such MIC couplers: one is to use highdielectric constant (⑀ r ≈ 30 − 100) substrates to reduce the size, and second to fold the coupler length in different shapes such as spiral and meander. MMIC couplers that generally use GaAs substrates (⑀ r = 12.9) employ the latter technique to make compact couplers. One of the important applications of these couplers is in wireless communications. The design, fabrication, and test results of these couplers are discussed in Chapter 8.
6.2.4 Equivalent Circuit of a QuarterWave Coupler
The coupling to the direct and backward ports of a parallelcoupled TEM coupler is given in terms of evenmode parameters by (6.7) and (6.8), as follows: S 21 = S 21e S 31 = S 11e where S 11e and S 21e denote the reflection and transmission coefficients, respectively, of the coupled lines for the case of evenmode excitation. The equivalent circuit of an ideal parallelcoupled TEM directional coupler is therefore as shown in Figure 6.5, where Z 0e denotes the evenmode characteristic impedance of the coupled lines [7]. S 11e and S 21e , which are, respectively, the reflection and transmission coefficients of the twoport circuit shown in Figure 6.5 give, respectively, the coupling to the backward (S 31 ) and direct (S 21 ) ports of the fourport coupler shown in Figure 6.1. We thus see that the analysis of a parallelcoupled directional coupler reduces to analyzing a simple circuit consisting of a length of transmission line of characteristic impedance Z 0e terminated by an impedance of Z 0 at either of its ends. This analogy is extremely useful in the design and synthesis of parallelcoupled directional couplers, as it is relatively much simpler to analyze single transmission line circuits.
Figure 6.5
Equivalent circuit of an ideal singlesection TEM directional coupler.
6.3 Multisection Directional Couplers
177
6.3 Multisection Directional Couplers 6.3.1 Theory and Synthesis
To obtain a nearconstant coupling over a wider frequency bandwidth than is possible using a singlesection coupler, a number of coupled sections must be cascaded, as shown in Figure 6.6(a). Each section is a quarterwave long at the center frequency. By properly choosing the even and oddmode impedances of the various sections, the bandwidth of the coupler can be increased. By analogy with the equivalent circuit of a singlesection coupler, the equivalent cascaded transmission line circuit for finding the coupling to the backward and direct ports of the multisection coupler of Figure 6.6(a) is shown in Figure 6.6(b). The reflection and transmission coefficients of the circuit shown in Figure 6.6(b) give, respectively, the coupling to ports 3 and 2 of the coupler shown in Figure 6.6(a). The evenand oddmode impedances of the ith section of the multisection coupler are related by 2
Z 0oi =
Z0 Z 0ei
(6.21)
where Z 0 denotes the impedance terminating the ports of the directional coupler. A multisection coupler can be either symmetrical or asymmetrical. In multisection couplers, the term symmetrical is used to denote a coupler that has endtoend symmetry. A symmetrical coupler employs an odd number of sections. In a
Figure 6.6
(a) An Nsection asymmetrical parallelcoupled multisection directional coupler. (b) Equivalent circuit of directional coupler shown in part (a).
178
ParallelCoupled TEM Directional Couplers
symmetrical coupler, the ith section will be identical to the N + 1 − ith section as shown in Figure 6.7(a). The equivalent circuit for analyzing the symmetrical coupler is shown in Figure 6.7(b). If the coupler does not have endtoend symmetry [Figure 6.6(a)], it is referred to as an asymmetrical coupler. An asymmetrical coupler can employ an even or odd number of sections. A significant property of symmetrical couplers is that in their case, the signal coupled to the direct port is 90 degrees out of phase with the signal coupled to the backward port (∠ S 31 = ∠ S 21 + 90 degrees). This phase relationship is independent of the frequency. Because of this property, 3dB symmetrical directional couplers find extensive use in diplexers, multiplexers, directional filters, balanced mixers, and in other devices where the 90degree phase difference property is required. Asymmetrical couplers do not exhibit the phase property of symmetrical couplers and are generally used where couplers are designed to obtain broadband power division only. The response of an optimal fivesection symmetrical coupler is shown in Figure 6.8. Before discussing the synthesis of multisection TEM couplers, we need to define the power loss ratio of a directional coupler. Power Loss Ratio
The power loss ratio is defined as L=
Figure 6.7
1
 S 21  2
(6.22)
(a) An Nsection symmetrical parallelcoupled multisection directional coupler. (b) Equivalent circuit of directional coupler shown in part (a).
6.3 Multisection Directional Couplers
Figure 6.8
179
Typical response of a symmetrical fivesection parallelcoupled TEM directional coupler.
where S 21 is the transmission coefficient between the input and direct ports. The power loss ratio is a positive quantity and is always greater than or equal to unity. For example, for the directional couplers shown in Figures 6.6 and 6.7, the power loss ratio L can be expressed as L=
1
 S 21 
2
=
1
(6.23)
 S 21e  2
Note that S 21 denotes the scattering parameter between ports 1 and 2 of the fourport directional coupler, whereas S 21e denotes the scattering parameter between ports 1 and 2 of the equivalent twoport network. In decibels: 10 log L = −20 log  S 21  = −20 log  S 21e 
(6.24)
The quantities on the righthand side in the above equation denote the insertion loss in decibels between the input and direct ports. The function L is, therefore, also called the insertion loss function. The relationship between the scattering parameters of an ideal directional coupler (S 11 = S 41 = 0) is given by
 S 21  2 +  S 31  2 = 1
(6.25)
or using (6.22) and (6.25), we obtain 1
 S 31  2 = 1 − L
(6.26)
In terms of ABCD parameters, the power loss ratio L of a network is given by [1, 3] L=1+
冋
冉
冊册
2 B 1 (A − D)2 − − CZ 0 4 Z0
(6.27)
The equivalent circuit of a singlesection coupler is shown in Figure 6.5. Its ABCD parameters (for a lossless case) are given by,
180
ParallelCoupled TEM Directional Couplers
A = cos , B = jZ 0e sin , C =
j sin , D = cos Z 0e
The power loss ratio of a singlesection coupler is therefore given by L=1+
冉
冊
1 Z 0e Z 2 − 0 sin2 4 Z 0 Z 0e
(6.28)
Similarly, the power loss ratio of a symmetrical three section coupler is given by L=1+
冉
1 4
再冋 冉 2
2
冊 冉 冊 冎
Z 0e1 Z0 − Z0 Z 0e1
Z 0e1 Z Z − − 0 2 0e2 Z 0 Z 0e2 Z 0e1
+
sin3
Z 0e2 Z0 − Z0 Z 0e2
冊册
sin cos2
(6.29)
2
where Z 0e1 denotes the evenmode impedance of the first and third sections and Z 0e2 denotes the evenmode impedance of the middle section. Power Loss Ratio of an Ideal Directional Coupler
The characteristics expected of an ideal directional coupler are that over a given frequency band, the values of  S 21  and  S 31  be constant. This requires the function L to be also constant over this frequency band. For example, if it is required to have a 3dB flat coupling over 5:1 frequency bandwidth ratio, the form of the corresponding function L in the frequency band of interest is as shown in Figure 6.9. Outside this frequency range, the function L can take any form.
Figure 6.9
Power loss ratio L of an ideal directional coupler having a frequency bandwidth ratio of B = 5.
6.3 Multisection Directional Couplers
181
It is impossible to realize any arbitrary power loss ratio function using physical networks. For example, the power loss ratio function L of the form shown in Figure 6.9 cannot be realized. Siedel and Rosen [8] have stated the necessary and sufficient conditions for the form of power loss ratio function L that can be realized using homogeneous stepped impedance networks as shown in Figures 6.6 and 6.7. For example, the power loss ratio function L of the form L = PN (sin2 )
(6.30)
can be realized using an asymmetrical network as shown in Figure 6.6(b), where PN is a polynomial of degree N whose value is greater than or equal to unity for all real values of . If the physical network is to be symmetrical as shown in Figure 6.7(b), an extra condition is imposed on the power loss function that can be realized. The necessary and sufficient condition that a power loss ratio function L can be realized using a symmetrical homogeneous stepped impedance network of N equallength sections as shown in Figure 6.7(b) is that it be of the form L = 1 + [PN (sin )]2
(6.31)
where PN is an odd polynomial in sin of degree N. It may be verified that the power loss ratio of a threesection symmetrical coupler as given by (6.29) satisfies the condition of (6.31). Based on the discussion so far, the synthesis of a multisection coupler can be summarized as follows: 1. Find an optimal polynomial L as a function of electrical length that satisfies the conditions imposed by (6.30) if the coupler is asymmetrical or (6.31) if the coupler is to be symmetrical. The polynomial L is to be optimal in the sense that for a given number of sections, response type (equalripple or maximally flat), a given mean coupling, and a given coupling tolerance (ripple level), the polynomial should exhibit maximal bandwidth. 2. Compute the impedance of each section of the network using network synthesis techniques once the optimal polynomial L has been found. The above approach has been used by Levy [1] to design optimal asymmetrical couplers and by Cristal and Young [3], and Toulios and Todd [9] to design symmetrical couplers. Unfortunately, analytical design expressions tend to be very cumbersome even for couplers with a small number of sections [1, 9]. For aiding designers, Levy has generated design tables for equalripple asymmetrical couplers for various values of coupling and bandwidth for up to six sections [2]. For equalripple and maximally flat symmetrical couplers, Cristal and Young have generated similar tables for up to nine sections [3]. For equalripple response symmetrical couplers, these are reproduced in Table 6.1.
182
ParallelCoupled TEM Directional Couplers Table 6.1 Tables of Parameters for Symmetrical TEMMode CoupledTransmissionLine Directional Couplers
␦
Z1
Z2
w
B
(a) Normalized evenmode impedances for equalripple symmetrical 3.01dB couplers of three sections (Z 4 − i = Z i ) 0.10 1.17135 3.25984 1.00760 3.03063 0.20 1.20776 3.41242 1.17199 3.83085 0.40 1.27036 3.66560 1.35225 5.17521 0.60 1.32964 3.90585 1.46353 6.45616 0.80 1.38970 4.15648 1.54440 7.77966 1.00 1.45274 4.43120 1.60798 9.20361 (b) Normalized evenmode impedances for equalripple symmetrical 6dB couplers of three sections (Z 4 − i = Z i ) 0.10 1.10298 2.09445 0.91996 2.70356 0.20 1.12090 2.14693 1.07404 3.31984 0.40 1.15038 2.22865 1.24518 4.29931 0.60 1.17680 2.29968 1.35201 5.17291 0.80 1.20208 2.36724 1.43006 6.01830 1.00 1.22698 2.43431 1.49150 6.86621 (c) Normalized evenmode impedances for equalripple symmetrical 8.34dB couplers of three sections (Z 4 − i = Z i ) 0.10 1.07434 1.71858 0.89286 2.61290 0.20 1.08644 1.74864 1.04355 3.18211 0.40 1.10606 1.79461 1.21159 4.07347 0.60 1.12339 1.83365 1.31688 4.85550 0.80 1.13973 1.86993 1.39403 5.60103 1.00 1.15560 1.90510 1.45488 6.33787 (d) Normalized evenmode impedances for equal ripple symmetrical 10dB couplers of three sections (Z 4 − i = Z i ) 0.20 1.06945 1.57423 1.03140 3.12968 0.40 1.08475 1.60708 1.19816 3.98852 0.60 1.09817 1.63470 1.30282 4.73738 0.80 1.11075 1.66014 1.37959 5.44739 1.00 1.12290 1.68458 1.44020 6.14545 (e) Normalized evenmode impedances for equalripple symmetrical 20dB couplers of three sections (Z 4 − i = Z i ) 0.20 1.02070 1.14914 1.00980 3.03958 0.40 1.02497 1.15617 1.17423 3.84396 0.60 1.02866 1.16197 1.27772 4.53804 0.80 1.03208 1.16720 1.35381 5.19011 1.00 1.03534 1.17213 1.41398 5.82570 Z 0e1 = Z 1 Z 0 , Z 0e2 = Z 2 Z 0 , Z 0e3 = Z 3 Z 0
Example 6.3
Design a symmetrical multisection TEM coupler with the following specifications: Mean coupling = 6 dB Maximal ripple level = ± 0.1 dB Frequency bandwidth ratio B = 8 The schematic of a symmetrical coupler is shown in Figure 6.7(a). To design a symmetrical coupler, we use the design Table 6.1(q) [3]. We find that a 6dB, ninesection coupler designed for a ripple level of ± 0.1 dB exhibits a frequency
6.3 Multisection Directional Couplers
183
Table 6.1 (continued)
␦
Z1
Z2
Z3
w
B
(f) Normalized evenmode impedances for equalripple symmetrical 3.01dB couplers of five sections (Z 6 − i = Z i ) 0.10 1.07851 1.37268 3.97615 1.32559 4.93114 0.20 1.10921 1.44029 4.21023 1.45184 6.29714 0.40 1.16266 1.54541 4.57491 1.58152 8.55845 0.60 1.21370 1.63864 4.90924 1.65791 10.69292 0.80 1.26555 1.73013 5.25363 1.71196 12.88720 1.00 1.31988 1.82466 5.62978 1.75370 15.24047 (g) Normalized evenmode five sections (Z 6 − i = Z i ) 0.10 1.04501 0.20 1.06052 0.40 1.08633 0.60 1.10969 0.80 1.13217 1.00 1.15438
impedances for equalripple symmetrical 6dB couplers of 1.21972 1.25302 1.30203 1.34262 1.37978 1.41542
2.38181 2.46010 2.57332 2.66727 2.75470 2.84048
1.25446 1.37766 1.50548 1.58135 1.63520 1.67673
4.34522 5.45738 7.08866 8.55462 9.96482 11.37370
(h) Normalized evenmode impedances for equalripple symmetrical 8.34dB couplers of five sections (Z 6 − i = Z i ) 0.10 1.03211 1.15690 1.89019 1.23184 4.20727 0.20 1.04271 1.17918 1.93414 1.35395 5.19150 0.40 1.06012 1.21142 1.99635 1.48104 6.70767 0.60 1.07565 1.23760 2.04658 1.55670 8.02323 0.80 1.09039 1.26114 2.09210 1.61050 9.26959 0.95 1.10119 1.27785 2.12484 1.64253 10.18985 (i) Normalized evenmode five sections (Z 6 − i = Z i ) 1.00 1.10476 0.20 1.03418 0.40 1.4784 0.60 1.05996 0.80 1.07140 1.00 1.08249
impedances for equalripple symmetrical 10dB couplers of
(j) Normalized evenmode five sections (Z 6 − i = Z i ) 0.20 1.01016 0.40 1.01406 0.60 1.01747 0.80 1.02066 1.00 1.02371
impedances for equalripple symmetrical 20dB couplers of
1.28331 1.14316 1.16808 1.18815 1.20606 1.22280
1.04183 1.04855 1.05386 1.05851 1.06280
2.13562 1.70922 1.75305 1.78805 1.81943 1.84912
1.17873 1.18767 1.19463 1.20073 1.20638
1.65206 1.34442 1.47118 1.54675 1.60053 1.64210
1.32734 1.45350 1.52888 1.58261 1.62420
10.49635 5.10148 6.56407 7.82513 9.01322 10.17639
4.94656 6.31936 7.49038 8.58338 9.64410
Z 0e1 = Z 1 Z 0 , Z 0e2 = Z 2 Z 0 , Z 0e3 = Z 3 Z 0 Z 0e4 = Z 4 Z 0 , Z 0e5 = Z 5 Z 0
bandwidth ratio B = 7.99. Because this figure of B is very close to the specified value of B = 8, it is sufficient to employ a ninesection coupler to achieve the desired specifications. From the same table, we determine evenmode impedances of the various sections. Once the evenmode impedances are known, we can find the oddmode impedances using (6.21). The even and oddmode impedances, the voltage coupling coefficient k, and the coupling in decibels of various sections of the coupler are given in Table 6.2. We see that the coupling of the center section is 2.17 dB, which is much tighter than the overall coupling for which the coupler has been designed (6 dB). The coupling of other sections is smaller than the overall coupling.
184
ParallelCoupled TEM Directional Couplers Table 6.1 (continued)
␦
Z1
Z2
Z3
Z4
w
B
(k) Normalized evenmode impedances for equalripple symmetrical 3.01dB couplers of seven sections (Z 8 − i = Z i ) 0.10 1.05240 1.18406 1.56753 4.61180 1.49705 6.9531 0.20 1.07950 1.23581 1.65795 4.90662 1.59539 8.8860 0.40 1.12798 1.31754 1.79367 5.35611 1.69388 12.0666 0.60 1.17505 1.39069 1.91172 5.76434 1.75090 15.0578 0.80 1.22323 1.46258 2.02682 6.18437 1.79087 18.1270 1.00 1.27399 1.53668 2.14566 6.64407 1.82155 21.4147 (l) Normalized evenmode (Z 8 − i = Z i ) 0.10 1.02686 0.20 1.04246 0.40 1.06580 0.60 1.08735 0.80 1.10831 1.00 1.12915
impedances for equalripple symmetrical 6dB couplers of seven sections 1.10756 1.13419 1.17408 1.20755 1.23839 1.26805
1.32930 1.37278 1.43416 1.48367 1.52841 1.57104
2.62516 2.72038 2.85438 2.96391 3.06523 3.16446
1.44052 1.53802 1.63645 1.69378 1.73404 1.76487
6.1494 7.6583 10.0026 12.0626 14.0396 16.0119
(m) Normalized evenmode impedances for equalripple symmetrical 8.34dB couplers of seven sections (Z 8 − i = Z i ) 0.10 1.02033 1.07694 1.23301 2.03194 1.42127 5.9117 0.20 1.02963 1.09518 1.26167 2.08436 1.51889 7.3140 0.40 1.04538 1.12204 1.30136 2.15641 1.61749 9.4572 0.60 1.05972 1.14417 1.33267 2.21361 1.67500 11.3077 0.80 1.07350 1.16423 1.36040 2.26506 1.71543 13.0563 1.00 1.08702 1.18319 1.38629 2.31408 1.74643 14.7747 (n) Normalized evenmode impedances for equalripple symmetrical 10dB couplers of seven sections (Z 8 − i = Z i ) 0.20 1.02360 1.07622 1.20802 1.81699 1.51198 7.1965 0.40 1.03597 1.09725 1.23839 1.86715 1.61028 9.2638 0.60 1.04718 1.11444 1.26213 1.90649 1.66773 11.0383 0.80 1.05786 1.12991 1.28298 1.94149 1.70815 12.7059 1.00 1.06834 1.14444 1.30229 1.97446 1.73917 14.3359 (o) Normalized evenmode impedances for equalripple symmetrical 20dB couplers of seven sections (Z 8 − i = Z i ) 0.20 1.00697 1.02256 1.05976 1.20128 1.49853 6.9766 0.40 1.01052 1.02846 1.06767 1.21112 1.59672 8.9188 0.60 1.01369 1.03320 1.07372 1.21863 1.65421 10.5678 0.80 1.01669 1.03740 1.07894 1.22515 1.69472 12.1029 1.00 1.01958 1.04129 1.08368 1.23116 1.72584 13.5903 Z 0ei = Z i Z 0
6.3.2 Limitations of Multisection Couplers
One of the major limitations of a multisection coupler is that the coupling of at least one of the sections is much tighter than the overall coupling as shown in the above example. This can create fabrication problems in microstrip technology where it is difficult to achieve tight coupling. Further, because the even and oddmode impedances of each section of a multisection coupler are different from those of the adjacent ones, the dimensions of the coupler abruptly change at the start and end of each section. Because of practical considerations, it may be necessary to join adjacent sections using small lengths of tapered transmission lines as shown in Figure 6.10. If the operating frequency is high, the extra reactances produced by these abrupt discontinuities or extra lengths of joining transmission lines can
6.3 Multisection Directional Couplers
185
Table 6.1 (continued)
␦
Z1
Z2
Z3
Z4
Z5
w
B
(p) Normalized evenmode impedances for equalripple symmetrical 3.01dB couplers of nine sections (Z 10 − i = Z i ) 0.10 1.04112 1.12024 1.29488 1.74863 5.18240 1.6012 9.030 0.20 1.06598 1.16366 1.36260 1.85696 5.52654 1.6807 11.528 0.40 1.11149 1.23397 1.46548 2.01711 6.04655 1.7594 15.627 0.60 1.15624 1.29789 1.55536 2.15551 6.51769 1.8046 19.475 0.80 1.20234 1.36117 1.64277 2.29038 7.00316 1.8362 23.421 1.00 1.25107 1.42660 1.73250 2.42995 7.53602 1.8604 27.644 (q) Normalized evenmode impedances for equalripple symmetrical 6dB couplers of nine (Z 10 − i = Z i ) 0.10 1.02201 1.06888 1.17282 1.42807 2.83542 1.5550 0.20 1.03437 1.09137 1.20736 1.47877 2.94305 1.6345 0.40 1.05615 1.12599 1.25686 1.54902 3.09269 1.7136 0.60 1.07658 1.15561 1.29716 1.60504 3.21427 1.7594 0.80 1.09661 1.18320 1.33370 1.65546 3.32658 1.7913 1.00 1.11663 1.20989 1.36852 1.70342 3.43663 1.8157
sections 7.989 9.943 12.969 15.622 18.166 20.702
(r) Normalized evenmode impedances for equalripple symmetrical 8.34dB couplers of nine sections (Z 10 − i = Z i ) 0.10 1.01536 1.04904 1.12341 1.30048 2.15200 1.5392 7.681 0.20 1.02379 1.06452 1.14687 1.33341 2.21025 1.6190 9.498 0.40 1.03846 1.08798 1.17989 1.37809 2.28925 1.6985 12.265 0.60 1.05206 1.10771 1.20622 1.41290 2.35152 1.7444 14.650 0.80 1.06523 1.12579 1.22966 1.44356 2.40740 1.7766 16.901 1.00 1.07823 1.14302 1.25158 1.47211 2.46063 1.8011 19.111 (s) Normalized evenmode impedances for equalripple symmetrical 10dB couplers of nine (Z 10 − i = Z i ) 0.20 1.01889 1.05161 1.11743 1.26387 1.90628 1.6133 0.40 1.03041 1.07004 1.14313 1.29777 1.96074 1.6927 0.60 1.04103 1.08543 1.16344 1.32390 2.00313 1.7386 0.80 1.05127 1.09945 1.18139 1.34672 2.04073 1.7708 1.00 1.06133 1.11271 1.19805 1.36779 2.07614 1.7954
sections
(t) Normalized evenmode impedances for equalripple symmetrical 20dB couplers of nine (Z 10 − i = Z i ) 0.20 1.00555 1.01529 1.03447 1.07471 1.21931 1.6024 0.40 1.00886 1.02054 1.04153 1.08328 1.22965 1.6818 0.60 1.01187 1.02485 1.04700 1.08974 1.23748 1.7278 0.80 1.01474 1.02871 1.05175 1.09527 1.24426 1.7601 1.00 1.01753 1.03232 1.05608 1.10028 1.25049 1.7848
sections
Z 0e1 = Z i Z 0
Table 6.2 Parameters of a NineSection Symmetrical Coupler Section
Z 0e /Z 0
Z 0o /Z 0
k
C (dB)
1, 2, 3, 4, 5
1.02201 1.06888 1.17282 1.42807 2.83542
0.97846 0.93556 0.85265 0.70025 0.35268
0.02177 0.06651 0.15807 0.34197 0.77875
33.24 23.54 16.02 9.32 2.17
9 8 7 6
9.345 12.016 14.303 16.450 18.547
9.061 11.571 13.697 15.674 17.588
186
ParallelCoupled TEM Directional Couplers
Figure 6.10
Typical physical layout of a symmetrical multisection coupler. Adjacent quarterwave sections are joined by small length of transmission line sections.
reduce the input match and directivity of the coupler. In this case, a better solution may be to use nonuniform couplers, which are discussed in the next chapter.
6.4 Techniques to Improve Directivity of Microstrip Couplers There are mainly three techniques for improving the directivity of microstrip couplers [10]: 1. By adding lumped capacitances at the ends of the coupled lines; 2. By using a dielectric overlay on top of the coupled lines; 3. By using wiggly lines. 6.4.1 Lumped Compensation
In this technique, lumped capacitances are added at the ends of a coupler as shown in Figure 6.11 [10–12]. The lumped capacitor can be added at only one of the ends or in the middle of coupled section [13]. With the addition of lumped capacitances, the electrical length of the coupler can be made to be equal for the even and odd modes at the design frequency. The addition of lumped capacitances does not affect the evenmode signal, but it affects the oddmode signal. This can be easily explained. In the case of evenmode excitation, the midplane PP ′ as shown in Figure 6.11(b) behaves like an open circuit. This was discussed in Section 4.1. The equivalent circuit of onehalf of the network for the even mode is shown in Figure 6.11(c). We see that the lumped capacitance has no effect on the overall capacitance between the strip and the ground because one of the ends of the capacitor is open circuited. On the other hand, in the case of oddmode excitation, the midplane PP ′ behaves like a short circuit. The equivalent circuit of onehalf of the network for the case of oddmode excitation is shown in Figure 6.11(d). In this case, the lumped capacitance is parallel with the capacitance between the strip and ground conductors. The overall capacitance between the strip and the ground is therefore increased. Because the phase velocity along a line is related to the capacitance as given by (3.6), the phase velocity of the odd mode is reduced because of the additional lumped capacitance. We can show that the electrical length of the coupler can be made equal for the even and oddmode signals by choosing the value of the lumped capacitance C ab as [12]
6.4 Techniques to Improve Directivity of Microstrip Couplers
Figure 6.11
187
(a) Top view of lumped capacitor compensated microstrip coupler, (b) top view of coupled section showing plane of symmetry pp′ and lumped capacitors, (c) equivalent circuit for evenmode excitation, and (d) equivalent circuit for oddmode excitation.
C ab =
1 4 f 0 Z 0o tan 0
(6.32)
where
0 =
2
√
⑀ reo rad ⑀ ree
In the above equation, ⑀ ree and ⑀ reo denote the even and oddmode effective dielectric constants of the coupled microstrip lines, respectively, Z 0o denotes the oddmode impedance of the coupled structure, and f 0 is the design center frequency. The physical length of a capacitor compensated quarterwave coupler is given by [10]
188
ParallelCoupled TEM Directional Couplers
− tan−1 ( f 0 C ab Z 0e ) 2 lc = k 0 √⑀ ree
(6.33)
where k 0 is the free space propagation constant. Experiments have shown that (6.32) and (6.33) lead to a fairly accurate design if the coupling required between the lines is tight (10 dB or tighter) [10]. For weaker coupling, (6.32) and (6.33) may not yield a very accurate design. In that case, it is more useful to optimize the value of the capacitance C ab and the length l c of the coupler using computer simulation programs. The values given by (6.32) and (6.33) can be used as starting values for optimization purposes. The typical improvement in the directivity of a lumped capacitor compensated 15.7dB coupler on an alumina substrate is shown in Figure 6.12 [10].
Example 6.4
Given the parameters of coupled microstrip lines as ⑀ ree = 6.7713, ⑀ reo = 5.5194, Z 0e = 88.83⍀, Z 0o = 28.14⍀, and f 0 = 3 GHz, compute the value of compensating capacitance C ab and length l c of a quarterwave coupler.
Figure 6.12
Directivity improvement of a lumped capacitor compensated 15.7dB microstrip coupler on alumina substrate. (From: [10]. 1982 IEEE. Reprinted with permission.)
6.4 Techniques to Improve Directivity of Microstrip Couplers
189
Using (6.32) and (6.33), we obtain C ab = 0.145 pF and l c = 8.86 mm. On the other hand, the physical length of an uncompensated coupler is 10.09 mm. 6.4.2 Use of Dielectric Overlays
If an additional layer of dielectric is deposited over coupled microstrip lines as shown in Figure 6.13, then by properly choosing the thickness and dielectric constant of the layer, near equalization of even and oddmode phase velocities can be achieved over a reasonably wide frequency band [14–17]. The dielectric overlay can completely cover the bottom dielectric layer or may only cover the region containing the strips. The dielectric overlay also affects quite significantly the backwardwave coupling between the lines. Therefore, the effect of overlay should be considered while computing the dimensions of the strips, the spacing between them, and the length of the coupler. A successful design of the overlay coupler depends on the availability of accurate data on the phase velocities and the characteristic impedances of the even and odd modes of the overlay structure. Broadband directional couplers with high directivity have been demonstrated [14]. Figure 6.14 provides design curves for coupled microstrip lines on alumina substrate using alumina for the overlay [15]. The top alumina layer is assumed to cover the bottom alumina layer completely. Here the overlay thickness is the same as the thickness of the substrate. The strip conductor thickness has been assumed to be zero. Simulated and measured data have shown about 10dB improvement in the directivity of the 8.34dB coupler designed using the curves in Figure 6.14. Similar improvements have been demonstrated [17] using lowdielectric constant substrate have ⑀ r = 2.48. For a 10dB coupler at Sband, the design parameters are ⑀ r = 2.48, W = 3.2 mm, h = d = 1.42 mm, S = 0.4 mm, and coupled length L = 20.5 mm. 6.4.3 Use of Wiggly Lines
Although use of lumped capacitances or dielectric overlay structures lead to an improved directivity of microstrip directional couplers, both techniques complicate fabrication and may undermine the advantages of MICs. Another technique to equalize the phase velocities of even and oddmode signals that is compatible with MIC technology is to use wiggly lines instead of straight lines [18–20]. A top view of wigglycoupled lines is shown in Figure 6.15(b). It is assumed that by wiggling
Figure 6.13
Parallelcoupled microstrip coupler with dielectric overlay compensation.
190
ParallelCoupled TEM Directional Couplers
Figure 6.14
Design curves for coupled microstrip lines covered with dielectric overlay ⑀ r = 10.1, d /h = 1.0. (From: [15]. 1978 IEEE. Reprinted with permission.)
the lines, the oddmode phase velocity is slowed down, whereas the evenmode phase velocity is not affected. Further, wiggling affects only the mutual capacitance between the coupled lines. Although these approximations are not strictly valid, it has been found that these give practically useful results [19]. The geometrical parameters of straightcoupled and wigglycoupled lines are defined in Figure 6.15. To consider the effect of wiggling, let us consider the capacitance of a section of length ⌬L (between reference planes AA and BB) of straight and wigglycoupled sections. For the straightcoupled section, the oddmode capacitance between reference planes AA and BB is given by C o = (C f + C p + C fo ) ⌬L
(6.34)
In these equations, C f , C p , and C fo denote perunit length capacitances as defined in Figure 3.9. On the other hand, the oddmode capacitance of wigglycoupled lines is given by C ow = (C f + C p ) ⌬L + C fo L w
(6.35)
The above relation results because in the case of wigglycoupled lines, the effective length seen by the capacitance C f and C p between the reference planes AA and BB is ⌬L, which is the same as that for the straightcoupled lines. The effective length seen by the oddmode fringing capacitance C fo in this case is L w , however, which is achieved by wiggling the lines. To equalize the odd and evenmode phase velocities, the following relation should be satisfied:
6.4 Techniques to Improve Directivity of Microstrip Couplers
Figure 6.15
191
Top view of (a) parallelcoupled straight lines, (b) wigglycoupled lines, and (c) exploded view of wigglycoupled lines between planes AA and BB.
⑀ C ow = ree C o ⑀ reo
(6.36)
where ⑀ ree and ⑀ reo denote the effective dielectric constants for the even and odd modes, respectively. Using (6.34) to (6.36), we find that the length L w of wigglycoupled lines should be chosen as L w = ⌬L
′ C fo C fo
(6.37)
where
冉
冊
⑀ ⑀ ′ = (C p + C f ) ree − 1 + ree C fo C fo ⑀ reo ⑀ reo
192
ParallelCoupled TEM Directional Couplers
To obtain the value of L w as given by (6.37), the wiggle depth d should be chosen as d=
⌬L 2
√冉 冊
′ 2 C fo −1 C fo
(6.38)
The capacitance parameters C p , C f , and C fo are defined in Section 3.2.2. For microstrip lines, the capacitances C p and C f can be determined as follows:
⑀ ⑀W Cp = 0 r h 2C f =
√⑀ re − C p cZ 0
where c is the velocity of light in free space, and Z 0 and ⑀ re denote the quasistatic characteristic impedance and effective dielectric constant, respectively, of a single microstrip line of width W. Furthermore, the oddmode fringing capacitance C fo can be determined using C fo = C o − C p − C f where Co =
√⑀ reo cZ 0o
In this equation, C o denotes the oddmode capacitance, and Z 0o and ⑀ reo denote, respectively, the characteristic impedance and effective dielectric constant of the odd mode. 6.4.4 Other Techniques
Many other techniques to improve the directivity of microstrip couplers have also been reported in [21–23]. Figure 6.16(a) shows a schematic of a microstrip coupler where a shunt inductive feedback is used between the direct ports [21]. By properly choosing the impedance and length of the feedback element, an isolation zero can be obtained at the desired frequency. The physical layout of a shunt feedback compensated coupler is shown in Figure 6.16(b). The design can be carried out by a microwave circuit simulator using equivalent circuits for various discontinuities and coupled lines. For more accurate designs especially at high frequencies, EM simulators should be used. It has been reported that this scheme leads to about a 5–30dB improvement in directivity. The improvement is obtained over a 15%– 20% bandwidth. Figure 6.17(a) shows another technique. In this technique, coupled spur lines sections are added close to the various ports [22]. The physical layout of a practical
6.4 Techniques to Improve Directivity of Microstrip Couplers
193
Figure 6.16
(a) Schematic and (b) layout of a coupler with shunt inductive feedback.
Figure 6.17
(a) Schematic and (b) layout of a coupler with coupled spur lines attached at all ends.
circuit is shown in Figure 6.17(b). Since the circuit contains many junctions and discontinuities, its design should be carried out using microwave circuit simulators or EM simulations. This technique also adds an isolation zero at a fixed frequency. The directivity improvement is about 10 dB over a bandwidth of less than 20%.
194
ParallelCoupled TEM Directional Couplers
A disadvantage of this technique, however, is that the return loss bandwidth is reduced considerably. The 20dB return loss bandwidth is reduced to less than about 20%. It has been reported that the directivity of microstrip couplers can also be improved by using meandered coupled line sections such as shown in Figure 6.18 [23]. The coupling between the parallel sections in the meandered region (controlled by separation D as shown in Figure 6.18) is responsible for increasing the phase velocity of the even mode, which leads to an improvement in the directivity. Using this technique, a measured directivity of greater than 20 dB was reported over an octave bandwidth, for a 10dB coupler [23].
Figure 6.18
Coupler with meandered section.
References [1] [2] [3]
[4] [5]
[6]
[7] [8]
Levy, R., ‘‘General Synthesis of Asymmetric Multielement CoupledTransmissionLine Directional Couplers,’’ IRE Trans., Vol. MTT11, July 1963, pp. 226–237. Levy, R., ‘‘Tables for Asymmetric Multielement CoupledTransmissionLine Directional Couplers,’’ IRE Trans., Vol. MTT12, May 1964, pp. 275–279. Cristal, E. G., and L. Young, ‘‘Theory and Tables of Optimum Symmetrical TEMMode CoupledTransmissionLine Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, September 1965, pp. 544–558. Arai, S., et al., ‘‘A 900MHZ 90Degree Hybrid for QPSK,’’ IEEE MTTS Int. Microwave Symp. Dig., 1991, pp. 857–860. Tanaka, H., et al., ‘‘2GHz OneOctaveBand 90Degree Hybrid Coupler Using Coupled Meander Line Optimized by 3D FEM,’’ IEEE MTTS Int. Microwave Symp. Dig., 1994, pp. 903–906. Tanaka, H., et al., ‘‘Miniaturized 90Degree Hybrid Coupler Using High Dielectric Substrate for QPSK Modulator,’’ IEEE MTTS Int. Microwave Symp. Dig., 1996, pp. 793–796. Young, L., ‘‘Stepped Impedance Transformers and Filter Prototypes,’’ IRE Trans., Vol. PGMTT10, September 1962, pp. 339–359. Seidel, H., and J. Rosen, ‘‘Multiplicity in Cascade Transmission Line Synthesis—Part I,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, May 1965, pp. 275–283; and Part II, July 1965, pp. 398–407.
6.4 Techniques to Improve Directivity of Microstrip Couplers [9]
[10] [11] [12] [13]
[14]
[15] [16]
[17]
[18] [19]
[20] [21]
[22] [23]
195
Touplios, P. P., and A. C. Todd, ‘‘Synthesis of Symmetrical TEMMode Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, September 1965, pp. 536–544. March, S. L., ‘‘Phase Velocity Compensation in ParallelCoupled Microstrip Line,’’ IEEE MTTS Int. Microwave Symp. Dig., 1982, pp. 410–412. Schaller, G., ‘‘Optimization of Microstrip Directional Couplers with Lumped Capacitors,’’ AEU, Vol. 31, July–August 1977, pp. 301–307. Kajfez, D., ‘‘Raise Coupler Directivity with Lumped Compensation,’’ Microwaves, Vol. 27, March 1978, pp. 64–70. Dydyk, M., ‘‘Microstrip Directional Couplers with Ideal Performance Via SingleElement Compensation,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT47, June 1999, pp. 956–964. Sheleg, B., and B. E. Spielman, ‘‘Broadband Directional Couplers Using Microstrip with Dielectric Overlays,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT22, December 1974, pp. 1216–1220. Paolino, D. D., ‘‘MIC Overlay Coupler Design Using Spectral Domain Techniques,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT26, September 1978, pp. 646–649. Klein, J. L., and K. Chang, ‘‘Optimum Dielectric Overlay Thickness for Equal Even and OddMode Phase Velocities in Coupled Microstrip Circuits,’’ Electronics Letters, Vol. 26, 1990, pp. 274–276. Su, L., T. Itoh, and J. Rivera, ‘‘Design of an Overlay Directional Coupler by a FullWave Analysis,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT31, December 1983, pp. 1017–1022. Podell, A., ‘‘A HighDirectivity Microstrip Coupler Technique,’’ IEEE MTTS Int. Microwave Symp. Dig., 1970, pp. 33–36. Uysal, S., and H. Aghvami, ‘‘Synthesis, Design and Construction of UltraWideband Nonuniform Directional Couplers in Inhomogeneous Media,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT37, June 1989, pp. 969–976. Uysal, S., Nonuniform Line Microstrip Directional Couplers and Filters, Norwood, MA: Artech House, 1993. Chen, J. L., S. F. Chang, and C. T. Wu, ‘‘A High Directivity Directional Coupler with Feedback Compensation,’’ IEEE MTTS Int. Microwave Symp. Digest, 2002, pp. 101–104. Chang, S. F., et al., ‘‘New High Directivity Coupler Design with Coupled Spurlines,’’ IEEE Microwave and Wireless Components Letters, Vol. 14, February 2004, pp. 65–67. Wang, S. M., C. H. Chen, and C. Y. Chang, ‘‘A Study of Meandered Microstrip Coupler with High Directivity,’’ IEEE MTTS Int. Microwave Symp. Digest, 2003, pp. 63–66.
CHAPTER 7
Nonuniform Broadband TEM Directional Couplers
A major disadvantage of using multisection directional couplers to obtain broadband coupling is that there is an abrupt change in the transverse dimensions of the coupler (e.g., width of the lines and spacing between them) at the start and end of each section. The physical discontinuities that occur from a change in the dimensions lead to poor match and directivity of the coupler. These effects increase in severity with increasing frequency. To avoid the abrupt change, a continuous variation in the coupling between the lines along the length of the coupler can be implemented. This requires a continuous variation in the transverse dimensions of the coupler, and these types of couplers are called nonuniform couplers. In this chapter, we discuss the theory and design of symmetrical and asymmetrical nonuniform TEM couplers. Symmmetrical couplers have endtoend symmetry and have a property that the phase difference between the coupled ports is 90 degrees. On the other hand, asymmetrical couplers do not have endtoend symmetry. By proper design, asymmetrical couplers can be designed to have the phase property of a magicT. The design of symmetrical and asymmetrical couplers reported here is based largely on the work of Tresselt [1] and DuHamel and Armstrong [2].
7.1 Symmetrical Couplers Figure 7.1 shows a symmetrical nonuniform TEM coupler. The width of the lines and the spacing between them vary continuously along the length of the structure. This leads to a continuous variation in the coupling along the length of the coupler. The coupler is symmetrical about the planes PP ′ and AA′ which ensures that the signal coupled to the backward port is 90 degrees out of phase with that coupled to the direct port. Because the coupler is assumed to be symmetrical about the plane PP ′, we can analyze it in terms of even and oddmode parameters. The scattering parameters of a symmetrical fourport network are given by (4.22) as follows:
197
198
Nonuniform Broadband TEM Directional Couplers
Figure 7.1
Symmetrical nonuniform TEM coupler. Even and oddmode characteristic impedances vary continuously along the length of the structure.
+ S 11o S S 11 = 11e 2 + S 21o S S 21 = 21e 2
(7.1)
− S 11o S S 31 = 11e 2 − S 21o S S 41 = 21e 2 where S 11e and S 11o denote the reflection coefficients of the even and oddmode signals respectively and S 21e and S 21o denote the transmission coefficients of the even and oddmode signals, respectively. The nonuniform coupling is achieved basically by varying the even and oddmode characteristic impedances of the coupled lines along the length of the structure. If the dimensions of the nonuniform coupler are chosen such that at any cross section, the values of the even and oddmode impedances satisfy the following condition: 2
Z 0e (z) Z 0o (z) = Z 0
(7.2)
then, as in the case of uniform couplers discussed in Section 4.2.2, the scattering parameters of a nonuniform coupler reduce to S 11 = S 41 = 0 S 31 = S 11e S 21 = S 21e
(7.3)
7.1 Symmetrical Couplers
199
The equivalent circuit for obtaining the scattering parameters S 11e and S 21e of a nonuniform coupler is shown in Figure 7.2(a) where Z oe (z) denotes the characteristic impedance of the nonuniform transmission line as a function of longitudinal coordinate z. 7.1.1 Coupling in Terms of EvenMode Characteristic Impedance
We now proceed to find the reflection coefficient S 11e at port 1 of the circuit shown in Figure 7.2(a) in terms of the evenmode characteristic impedance of the coupled lines. Because the evenmode characteristic impedance of the coupled lines varies continuously along the length of the structure, it can be represented as shown in Figure 7.2(b). [The oddmode characteristic impedance also varies along the length of the structure, but because it is always related to the evenmode impedance by (7.2), it is possible to analyze the network in terms of evenmode impedances only.] The incident wave is partially reflected at every step because of a mismatch of the impedance. The total reflection coefficient at port 1 can be found by using the smallsignal reflection [3]. Using this theory, the total reflection coefficient at the input is found by summing differential contributions of the reflection coefficient
Figure 7.2
(a) Equivalent transmission line circuit of nonuniform TEM coupler shown in Figure 7.1; and (b) equivalent circuit for determining coupling.
200
Nonuniform Broadband TEM Directional Couplers
from each step in proper phase. The differential reflection coefficient at the step defined by plane BB ′ is given by d⌫ =
Z 0e + dZ 0e − Z 0e dZ 0e ≈ Z 0e + dZ 0e + Z 0e 2Z 0e
=
1 d(ln Z 0e ) 2
=
1 d (ln Z 0e ) dz 2 dz
(7.4)
where Z 0e is a function of z, and it is assumed that 2Z 0e Ⰷ dZ 0e . The contribution to the total reflection coefficient at port 1 (z = 0) from the differential reflection coefficient d⌫ at plane BB ′ is obtained by multiplying (7.4) by the phase term e −2j z, or dS 11e = d⌫e −2j z =
1 −2j z d e (ln Z 0e ) dz 2 dz
(7.5)
where  is the phase constant of the wave along the direction of propagation. Summing reflections from z = 0 to z = d, we obtain d
1 S 11e = 2
冕
e −2j z
d (ln Z 0e ) dz dz
(7.6)
0
Furthermore, using (7.3), the coupling between ports 1 and 3 of the configuration shown in Figure 7.1 is given by d
1 S 31 = S 11e = 2
冕
e −2j z
d (ln Z 0e ) dz dz
(7.7)
0
This equation is valid if it is assumed that the reflection at any step is quite small compared with unity. The coupler is assumed to be symmetrical about the plane z = d/2. It is useful to define another coordinate u as u=z−
d 2
(7.8)
such that the structure is symmetrical with respect to u = 0. Equation (7.7) then becomes
7.1 Symmetrical Couplers
201 d /2
冕
1 S 31 = e −j d 2
e −2j u
d (ln Z 0e ) du du
(7.9)
−d /2 d /2
=e
冕
−j d
e −2j u p(u) du
−d /2
where 1 d (ln Z 0e ) 2 du
p(u) =
(7.10)
The function p(u) is an odd function of u and hence (7.9) reduces to d /2
S 31 = −je
−j d
冕
sin (2 u) p(u) du
(7.11)
−d /2
Inspection of (7.11) shows that the amplitude of the coupled signal is symmetrical about  = 0; that is,  S 31 (  )  =  S 31 (− )  . Furthermore, the phase of the coupled signal is given by ∠ S 31 =
−  d rad, for  > 0 2
=−
冉
(7.12)
冊
+  d rad, for  < 0 2
Information on the amplitude and phase response for negative values of  are required for the synthesis of the coupler. Using (7.9) or (7.11), we can determine the coupling of a symmetrical nonuniform coupler, if the distribution of the evenmode characteristic impedance is known, and (7.9) and (7.11) can be easily evaluated numerically. 7.1.2 Synthesis
For a coupler of finite length d, the evenmode impedance varies only over the length of the coupler (  u  ≤ d/2). The function p(u) given by (7.10) is therefore zero for  u  ≥ d/2. The lower and upper limits of integration in (7.9) or (7.11) can therefore be changed to u = −∞ and u = ∞, respectively, without affecting the value of the integral. In that case, (7.9) becomes ∞
S 31 = e
−j d
冕
−∞
e −2j u p(u) du
202
Nonuniform Broadband TEM Directional Couplers
or ∞
S 31 e
j d
=
冕
e −2j u p(u) du
(7.13)
−∞
It is interesting to note that the quantity on the righthand side of (7.13) denotes the Fourier transform of the function p(u). The coupling is therefore given by the Fourier transform of the function p(u). In a synthesis problem, however, it is required to find the function p(u) for a given coupling response S 31 , and this can be done by using the inverse Fourier transform. Thus, we obtain 1 p(u) = 2
∞
冕
S 31 e j d e j2 u d(2 )
(7.14)
−∞
where d(2 ) denotes that the integration is with respect to 2 . Equation (7.14) can be used for the synthesis of a nonuniform TEM directional coupler. Ideal Directional Coupler
We now discuss the synthesis of an ideal directional coupler having a flat amplitude response in the range 2 = 0 to 2 = 1 as shown in Figure 7.3. The synthesis can be easily extended to a coupler in a different frequency range by scaling the length of the coupler as discussed later in the chapter. Figure 7.3 shows the amplitude of the coupling of an ideal coupler only in the range 2 > 0. As shown by (7.14), however, we need to know the amplitude and phase response of the coupler in the range −∞ < 2 < ∞ to synthesize a coupler. Because the amplitude response of the coupler is symmetrical about  = 0 as discussed earlier, we have
Figure 7.3
Coupling response of an ideal directional coupler.
7.1 Symmetrical Couplers
203
 S 31  = R, for  2  < 1
(7.15)
= 0, for  2  > 1 If the coupling is specified in decibels, then R = 10−C/20
(7.16)
where C is the coupling in decibels, and is a positive quantity. The phase constant  is related to the frequency f by the relation
=
2 f v
(7.17)
where v is the phase velocity along the direction of propagation. The phase of S 31 is given by (7.12). Using (7.12) and (7.15), we obtain S 31 = Re j( /2 −  d), for 0 < 2 < 1 = Re −j( /2 +  d), for −1 < 2 < 0
(7.18)
= 0 otherwise Substituting S 31 from (7.18) in (7.14), we obtain p(u) = −
R sin2 (u/2) u/2
(7.19)
which is plotted in Figure 7.4 for an arbitrary assumed value of coupling R. Note from (7.19) that the function p(u) extends from u = −∞ to u = ∞. In other words, if it is desired to obtain the coupling response shown in Figure 7.3, then the coupler will have to be infinitely long. However, it is also seen from Figure 7.4 that the
Figure 7.4
The function p(u) required to obtain the coupling response shown in Figure 7.3.
204
Nonuniform Broadband TEM Directional Couplers
function p(u) becomes quite small when  u  becomes large. Therefore, if the length d of the coupler is chosen as equal to 4n (corresponding to n positive and n negative lobes of Figure 7.4), the coupling response is not expected to deviate much from the ideal response, if the value of n is sufficiently high. Figure 7.5 shows the variation in coupling for different lengths of the coupler, for n = 1, . . . , 4; that is, d = 4 , d = 8 , d = 12 , and d = 16 . From these figures we see that as the length of the coupler is increased, the response approaches the ideal response. We also see that for a coupler of length d = 4n , the number of ripples in the frequency band of interest is equal to n. It may be emphasized that the stated length corresponds to a coupler designed to operate in the range  = 0 to  = 1/2 where  is related to frequency by (7.17). Although increasing the length of the coupler brings the response of the coupler closer to the ideal response, it may be noted that there is an overshoot always present near  = 0. The value of the overshoot does not decrease in amplitude with increasing the number of lobes used. This phenomenon is known as Gibb’s phenomenon and occurs when a step function such as shown in Figure 7.3 is approximated by a finite number of Fourier terms. The problem is resolved by employing weighting functions to remove the problem of overshoot and to make the level of ripples equal. In this technique, instead of using function p(u) for the synthesis as given by (7.19), we use the weighted function
Figure 7.5
Coupling response of a nonuniform coupler of length (a) 4 , (b) 8 , (c) 12 , and (d) 16 .
7.1 Symmetrical Couplers
205
p w (u) = w(u) p(u)
(7.20)
where w(u) is the weighting function and p(u) is given by (7.19). A common form of the weighting function is shown in Figure 7.6. The value of the weighting function remains constant over a length equal to 4 of the coupler (e.g., the value of the weighting function is w 2 in the intervals −4 < u < −2 and 2 < u < 4 ). The weighting function can be determined by a termbyterm comparison of a known equal ripplelevel function with a function containing a few weighted Fourier series terms. A technique for determining weighting functions is discussed later in Section 7.1.3. Before that, we discuss the final step for the synthesis of a nonuniform coupler.
Computing Z 0e (u) from p(u) or p w (u)
Once the value of p(u) or p w (u) is known, the corresponding value of Z 0e (u) needs to be determined for the realization of the physical circuit. Integrating (7.10) with respect to u, we obtain

1 1 + ln Z 0e (u) = ln Z 0e 2 2 u = −d /2
u
冕
−d/2
where p(u) is given by (7.19). Furthermore, because Z 0e  u = −d /2 = Z 0 (7.21) can also be expressed as
Figure 7.6
Typical weighting function of a nonuniform coupler.
p(u) du
(7.21)
206
Nonuniform Broadband TEM Directional Couplers u
1 Z (u) = ln 0e 2 Z0
冕
u
R p(u) du = −
−d/2
冕
sin2 (u/2) du u/2
(7.22)
−d/2
Unfortunately, the integral in (7.22) cannot be evaluated in closed form. It is, however, quite simple to evaluate the above integrals numerically using available computer software programs such as MATHCAD or MATLAB. If the weighted function p w (u) given by (7.20) is used for the synthesis of the coupler, the evenmode impedance is then given by u
1 Z (u) R = ln 0e 2 Z0
冕
u
R w(u) p(u) du = −
−d/2
冕
w(u)
sin2 (u/2) du u/2
(7.23)
−d/2
Evaluation of (7.23)
Let us assume that in a particular case, the length d of the coupler is chosen as 20 (i.e., it extends from u = −10 to u = 10 ) and it is desired to find the evenmode impedance at u = − . Assuming that the weighted function is of the form as shown in Figure 7.6, the value of Z 0e at u = − can be found using (7.23) as follows: Z 0e  u = − 1 R =− ln 2 Z0
冤
−8
w5
sin2 (u/2) du u/2
−10
−6
+ w4
冕
冕
sin2 (u/2) du + w 3 u/2
−8 −2
+ w2
冕
−4
−4
冕
sin2 (u/2) du u/2
(7.24)
−6
sin2 (u/2) du + w 1 u/2
−
冕
sin2 (u/2) du u/2
−2
冥
Each integral in (7.24) can now be easily evaluated numerically. 7.1.3 Technique for Determining Weighting Functions
As discussed earlier, (7.7) is valid if the reflection at any step is small compared with unity. The synthesis technique described earlier is thus strictly valid for realization of couplers having coupling of less than about 10 dB. However, (7.23) can also be used to synthesize a ‘‘tight coupler’’ if the weighting function terms w i are chosen properly. A general technique for deriving the weighting terms valid for ‘‘tight’’ as well as ‘‘loose’’ couplers is given by Tresselt [1]. In this technique, a symmetrical multisection coupler as discussed in the previous chapter is first designed for given coupler specifications. Let the multisection coupler employ 2n − 1 sections with
7.1 Symmetrical Couplers
207
impedance levels denoted as shown in Figure 7.7. Such a coupler will have n ripples in the frequency band of interest. Similarly, a nonuniform coupler that has a length equal to 4n 1 will have n ripples in the frequency band. Therefore, the response of a nonuniform coupler can be made identical to the response of a multisection coupler of 2n − 1 sections if the length of the nonuniform coupler is chosen equal to 4n . The nonuniform coupler can then be synthesized using (7.23) where the weighting function terms w i are determined by the following method. Consider the function shown in Figure 7.8. The value of the function for 0 ≤ ≤ represents the value of desired coupling R. The Fourier series representation of the function in the range 0 < < is g( ) = R
1.
4
冉
sin +
sin 3 sin 5 sin 7 sin 9 + + + +... 3 5 7 9
冊
Figure 7.7
Symmetric multisection coupler of 2n − 1 sections.
Figure 7.8
Gotte function whose value is equal to desired coupling R for 0 < < .
(7.25)
This length is for a coupler designed to operate from 2 = 0 to 2 = 1. For a coupler in a different frequency range, the length of the coupler is different.
208
Nonuniform Broadband TEM Directional Couplers
With a few terms from the above series used to describe the function, the response would be similar to that shown in Figure 7.5; that is, there will exist an overshoot near = 0 and the ripples would be unequal. If only n terms of the series (7.25) are used, it is more appropriate to construct a new function f ( ) as follows: f ( ) = R
冋
sin 3 sin (2n − 1) 4 w 1 sin + w 2 + . . . + wn 3 (2n − 1)
册
(7.26)
where w 1 , . . . , w n are the weighting function terms. With a suitable choice of these terms, the function f ( ) can be made an equal ripple. A suitable equal ripple function can be easily determined by using the design tables of symmetrical multisection couplers. If (7.11) is used to find the coupling of a symmetrical coupler of 2n − 1 sections shown in Figure 7.7, the coupling (excluding the phase factor) is given by n
h( ) =
∑
r=1
Z ln 0e (n + 1 − r) sin (2r − 1) Z 0e (n − r)
(7.27)
In (7.27), denotes the electrical length of each section and is given by
=
d 2n − 1
(7.28)
where  is the phase constant, d denotes the total physical length of the coupler, and Z 0e (n − r)  r = n = Z 0
(7.29)
If the various impedances in (7.27) correspond to that of an equal ripple symmetrical coupler, the function h( ) is also equal ripple. Comparing term by term (7.26) and (7.27), we can determine the weighting function terms w i .
Example 7.1
Determine the weighting function terms for the design of a nonuniform coupler having a mean coupling C = 3.01 dB, a frequency bandwidth ratio (B) = 8, and a ripple tolerance ␦ = ± 0.20 dB. From (7.16), we find that R = 10−.15005 = 0.707 Further, from Table 6.1(k) in the last chapter, we find that a symmetrical multisection coupler with the above specifications requires seven sections. This gives
7.1 Symmetrical Couplers
209
N = 2n − 1 = 7 or n=4 Also from the same table, the evenmode impedances of the various sections are Z oe1 = 1.07950 Z0 Z oe2 = 1.23581 Z0
(7.30)
Z oe3 = 1.65795 Z0 Z oe4 = 4.90662 Z0 (7.27) then becomes h( ) = 1.085 sin + 0.29385 sin 3 + 0.13522 sin 5 + 0.076497 sin 7 (7.31) Substituting R = 0.707 and n = 4 in (7.26), we obtain f ( ) = 0.90034w 1 sin + 0.30011w 2 sin 3
(7.32)
+ 0.18006w 3 sin 5 + 0.12862w 4 sin 7 Comparing term by term (7.31) and (7.32), the weighting function terms are found as follows: w 1 = 1.205 w 2 = 0.979
(7.33)
w 3 = 0.751 w 4 = 0.595 7.1.4 Electrical and Physical Length of a Coupler
The synthesis described so far is for a coupler having an equalripple response in the range 2 = 0 to 2 = 1 (the value of  at the center frequency is 1/4). For such a coupler, the length of the coupler corresponds to those of 2n lobes (n positive and n negative) and is therefore equal to 4n . The number of ripples for such a coupler is equal to n in the frequency band of interest. The total electrical length of the nonuniform coupler at the center frequency is therefore
210
Nonuniform Broadband TEM Directional Couplers
c =
1 4n = n 4
(7.34)
If  c denotes the phase constant of the wave in the medium of the coupler corresponding to the center frequency of design f 0 , then
c =  c d = n
(7.35)
where d is the total physical length of the coupler, or d=
n c
(7.36)
In (7.36),  c = 2 f 0 /v, where v is the velocity of propagation in the medium of the coupler. Length of Multisection Coupler
The electrical length of each section of a multisection coupler is /2 rad at the center frequency of design. The electrical length of a multisection coupler of 2n − 1 sections is therefore
 c d = (2n − 1)
2
(7.37)
Comparison of (7.34) and (7.37) shows that a nonuniform coupler is a quarterwave longer than a multisection coupler with the identical response. 7.1.5 Design Procedure
Based on the discussions so far, the design of a nonuniform symmetrical TEM mode coupler can be summarized as follows: •
•
Step 1 1. Specify coupling level C(dB), or voltage coupling factor R. Find one from the other using (7.16). 2. Specify ripple level ␦ in dB. 3. Specify frequency bandwidth ratio B, and center frequency of design f 0 (or frequency range of operation f 1 to f 2 ). If the frequency range of operation of the coupler is specified, find B and f 0 using (6.11) to (6.13), respectively. Step 2 Using Table 6.1 from Chapter 6, find the minimum number of sections required for a multisection symmetrical coupler for the given values of C (coupling), ␦ (coupling ripple), and B (bandwidth ratio). Let the number of sections required be N, then
7.1 Symmetrical Couplers
211
n= •
•
•
•
•
•
N+1 2
Step 3 Using the same tables, find Z 0e1 , Z 0e2 , . . . , Z 0en . Step 4 Using computed values of Z 0e1 , Z 0e2 , . . . , Z 0en , construct the function h( ) using (7.27). Step 5 Construct the function f ( ) using (7.26). Step 6 Comparing term by term the functions h( ) and f ( ) computed using steps 4 and 5, respectively, find the weighting function terms w 1 , . . . , w n . Step 7 Evaluate Z 0e (u) numerically using (7.23) for discrete values of u in the range −2n ≤ u ≤ 2n , and by substituting d/2 = 2n . Step 8 The values of u for which computations are made in step 7 correspond to the design of the coupler having a mean value of  = 1/4. To determine the values of u for  =  c , divide the values of u computed in step 7 by a factor of 4 c , where  c is the phase constant in the medium of the coupler at the center frequency of the design.
Example 7.2
Design a nonuniform coupler in a homogeneous dielectric medium (⑀ r = 2.32) with the following specifications: Mean coupling (C) = 8.34 dB Frequency of operation = 1–11 GHz Ripple level (␦ ) = ± 0.3 dB •
Step 1 From these specifications, we have R = 10−C/20 = 0.3828 f + f 2 1 + 11 f0 = 1 = = 6.0 GHz 2 2 f B = 2 = 11 f1 At 6.0 GHz, the phase constant in free space is given by
212
Nonuniform Broadband TEM Directional Couplers
0 =
2 f 0 = 125.67 rad/m 0.3
or, the phase constant in the medium of the coupler at the center frequency of design is given by
 c = √⑀ r  0 = 1.524  0 = 191.51 rad/m •
Step 2 From Table 6.1(r) in Chapter 6, we find that with a mean coupling level of C = 8.34 dB, ripple level = ± 0.3 dB, a ninesection symmetrical coupler will exhibit a frequency bandwidth ratio of B = 10.96, which is very close to the specified value. We thus have n=
•
N+1 9+1 = =5 2 2
(7.38)
Step 3 Using the same table, the impedances of the different sections of the ninesection coupler is Z oe1 = 1.03134 Z0 Z oe2 = 1.07697 Z0 Z oe3 = 1.16469 Z0
(7.39)
Z oe4 = 1.35771 Z0 Z oe5 = 2.25315 Z0 •
Step 4 Substituting these values in (7.27), we obtain h( ) = 0.5071 sin + 0.1532 sin 3 + 0.0785 sin 5
(7.40)
+ 0.0433 sin 7 + 0.0309 sin 9 •
Step 5 Substituting R = 0.3828 in (7.26): f ( ) = 0.4873w 1 sin + 0.1624w 2 sin 3 + 0.09745w 3 sin 5 + 0.06961w 4 sin 7 + 0.05414w 5 sin 9
(7.41)
7.1 Symmetrical Couplers
•
213
Step 6 Comparing (7.40) and (7.41) term by term, we obtain the following values for the weighting functions: w 1 = 1.040 w 2 = 0.943 w 3 = 0.805
(7.42)
w 4 = 0.623 w 5 = 0.569 •
•
Step 7 The values of Z 0e (u) were computed for values of u in the range −10 ≤ u ≤ 10 using (7.23) by numerical evaluation of the integral using software package Mathcad. The obtained values of Z 0e (u) are shown in Table 7.1. The computed values of u are valid for a coupler having  c = 1/4. Step 8 The values of u for the specified coupler (  c = 191.51) were determined by dividing the values of u derived in step 7 by a factor 4 c . These values are also shown in Table 7.1.
Because the coupler is symmetrical about u = 0, it is sufficient to find the structure parameters for 0 ≤ u ≤ d/2. The overall length of the coupler is d = 2 × 4.1 = 8.2 cm.
Table 7.1 Design Procedure for a Nonuniform TEM Coupler u (m) Normalized Coupler  c = 1/4 rad/m
u (cm) Actual Coupler  c = 191.51 rad/m
Z ln 0e Z0
Z 0e Z0
10 9.5 9 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0
4.10 3.89 3.69 3.48 3.28 3.08 2.87 2.67 2.46 2.25 2.05 1.85 1.64 1.44 1.23 1.03 0.82 0.62 0.41 0.21 0
0.00 0.00268 0.01486 0.0277 0.0309 0.0346 0.0517 0.0702 0.0743 0.0806 0.1113 0.1449 0.1532 0.1647 0.2231 0.2911 0.3087 0.3369 0.5087 0.7852 0.9263
1.00 1.002 1.0149 1.028 1.031 1.035 1.053 1.073 1.077 1.084 1.117 1.156 1.165 1.1790 1.2500 1.3379 1.3616 1.400 1.663 2.1929 2.525
214
Nonuniform Broadband TEM Directional Couplers
Once we determine the evenmode characteristic impedances of the coupler at various cross sections of the coupler, we can find the corresponding oddmode characteristic impedances using (7.2). Furthermore, the physical dimensions of the lines (i.e., their width and the spacing between them) can be determined. The design and performance of nonuniform couplers using microstrip lines is described in [3, 4]. The design procedure described above requires data of symmetrical multisection filters, which is available only for some specific cases as discussed in the last chapter. A design procedure of nonuniform couplers that is based on optimization methods is described in [5].
7.2 Asymmetrical Couplers A nonuniform asymmetric coupler which exhibits a broadband coupling and the phase properties of a magicT is shown in Figure 7.9. The structure does not have an end to end symmetry but has a symmetry with respect to plane PP ′. The coupler consists of a section of coupled lines of length extending from z = −d to z = 0 and a section of uncoupled lines of length . The coupling between lines increases from z = −d to z = 0. However, it reduces abruptly to zero at z = 0. The value of is 90 degrees at midband frequency. The uncoupled section is used to give the structure of the properties of a magicT. The scattering parameters of the coupler can be determined using the evenmode analysis (as discussed earlier in Section 7.1) if the even and oddmode characteristic impedances of the coupled line section at any cross section are chosen to satisfy 2
Z 0e (z) Z 0o (z) = Z 0
(7.43)
The evenmode equivalent circuit of the coupler is shown in Figure 7.10. In terms of evenmode scattering parameters, the scattering parameters of the coupler are given by
Figure 7.9
Asymmetric nonuniform coupler.
7.2 Asymmetrical Couplers
Figure 7.10
215
Equivalent circuit to determine coupling of the asymmetric nonuniform coupler shown in Figure 7.9.
S 11 = S 22 = S 33 = S 44 = 0
(7.44)
S 41 = S 14 = S 23 = S 32 = 0
(7.45)
S 21 = S 12 = S 34 = S 43 = S 21e = S 12e
(7.46)
S 31 = S 13 = S 11e , S 24 = S 42 = S 22e
(7.47)
The evenmode reflection coefficient S 11e at port 1 can be determined by summing reflections in proper phase as discussed in Section 7.1. If the evenmode impedance of the coupled line section from z = −d to z = 0− is so tapered (z = 0− denotes location just to the left of the abrupt discontinuity at z = 0) such that the contribution of the taper section to the overall reflection coefficient S 11e is zero, the reflection coefficient at port 1 results solely from the abrupt impedance discontinuity at z = 0. The reflection coefficient S 11e is given by S 11e =
Z 0 − Z 0 /a −2j a − 1 −2j = e e Z 0 + Z 0 /a 1+a
(7.48)
where it is assumed that the evenmode characteristic impedance of coupled lines is Z 0 /a at z = 0−. Similarly, S 22e =
Z 0 /a − Z 0 −2j 1 − a −2j = e e Z 0 + Z 0 /a 1+a
(7.49)
The properties of lossless twoports were discussed in Chapter 2. Using (2.77), the transmission coefficient between ports 1 and 2 of the network shown in Figure 7.10 is found as S 21e = S 12e =
2 √a −2j e 1+a
(7.50)
216
Nonuniform Broadband TEM Directional Couplers
Furthermore, using (7.44)–(7.47), the scattering matrix of the coupler can be expressed as
冤
0  [S] = −␣ 0
 0 0 ␣
−␣ 0 0 
0 ␣  0
冥
(7.51)
where
␣=
2 √a −2j 1 − a −2j ,= e e 1+a 1+a
(7.52)
For a 3dB coupler, it is found that a = 0.1717 or Z 0e = 5.83Z 0 and Z 0o = 0.1717Z 0 at z = 0−. It is very difficult to obtain coupled lines with such extreme values of even and oddmode characteristic impedances. However, two 8.36dB asymmetric couplers can be connected in tandem as shown in Figure 7.11 to achieve a 3dB coupler. For an 8.36dB coupler, a = 0.446 or Z 0e = 2.24Z 0 and Z 0o = 0.44Z 0 . These values are relatively easier to obtain in practice using planar transmission lines such as broadsidecoupled offset striplines discussed in Section 3.7.3. As discussed earlier, the design of a nonuniform asymmetric coupler requires a reflectionless taper from z = −d to z = 0−. There are many possible designs. One
Figure 7.11
Tandem connection of two asymmetric nonuniform couplers.
7.2 Asymmetrical Couplers
217
frequently employed taper design is due to Klopfenstein [6]. In practice, it is not possible to design a completely reflectionless taper using a finite length. Furthermore, it is not possible to lay out a structure in which the coupling changes abruptly (at z = 0 in Figure 7.9). However, it is still possible to obtain performance which is quite close to the ideal performance [2].
References [1]
[2]
[3]
[4] [5]
[6]
Tresselt, C. P., ‘‘The Design and Construction of Broadband, HighDirectivity, 90Degree Couplers Using Nonuniform Techniques,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT14, December 1966, pp. 647–656. DuHamel, R. H., and M. E. Armstrong, ‘‘A Wideband Monopulse Antenna Utilizing the TaperedLine MagicT,’’ USAF Antenna Research and Development Program 15th Symp., University of Illinois, 1965, pp. 1–30. Uysal, S., and H. Aghvami, ‘‘Synthesis, Design, and Construction of UltraWideBand Nonuniform Quadrature Directional Couplers in Inhomogeneous Media,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT37, June 1989, pp. 969–976. Uysal, S., Nonuniform Line Microstrip Directional Couplers, Norwood, MA: Artech House, 1993. Kammler, D. W., ‘‘The Design of Discrete NSection and Continuously Tapered Symmetrical Microwave TEM Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, August 1969, pp. 577–590. Klopfenstein, R. W., ‘‘A Transmission Line Taper of Improved Design,’’ Proc. IEEE, Vol. 44, January 1954, pp. 31–35.
CHAPTER 8
Tight Couplers 8.1 Introduction In planar quasiTEM transmission line media such as microstrip, it is difficult to obtain tight coupling between lines because it requires a very small spacing between them. Tight couplers, especially 3dB couplers, are used in many practical circuits such as balanced mixers, amplifiers, and so forth. In this chapter, we concentrate on discussing directional couplers specifically suited for obtaining tight coupling values. These include branchline couplers, ratrace couplers, Lange couplers, tandem couplers, and several other structures using new concepts and multilayer dielectric configurations. Although conventional branchline and ratrace couplers do not use coupled lines, many of their recent modifications employ coupled structures to enhance their bandwidths or use folded coupled lines or inductors to make them compact. Branchline and ratrace couplers are easily analyzed using the even and oddmode approach [1–4]. Only the final design equations for branchline and ratrace couplers are given in this chapter. The equations given, however, are valid for arbitrary division of power between the ports of a branchline or a ratrace coupler. These couplers are inherently narrowband (< 20% bandwidth) circuits; some techniques used to enhance their bandwidth are described briefly. Although branchline and ratrace couplers are more suitable for obtaining tight coupling values (such as 3 dB), they can also be useful for obtaining loose coupling values in certain applications. For a branchline coupler designed for loose coupling, the impedance of the shunt branches becomes very high and cannot be easily realized using planar transmission lines. A modified branchline coupler in which the highimpedance shunt branches are replaced by coupled lines is described. The modified branchline coupler can be easily implemented in microstrip configuration. The size of conventional branchline and ratrace couplers becomes quite large at low frequencies (below 2 GHz). Further, their size is ‘‘too large’’ for their implementation in MMICs even at frequencies as high as 10 GHz. The realization of branchline and ratrace couplers using lumped inductive and capacitive elements is also described. The size of lumpedelement couplers is very small compared with that of conventional couplers. Reducedsize branchline and ratrace couplers that use only lumped capacitors and small sections of transmission lines (smaller than g /4) are also described. These couplers are quire suitable for realization in MMICs. Tight coupling between edgecoupled lines can also be achieved by connecting a number of lines in an interdigital manner. This is essentially the basis of the
219
220
Tight Couplers
Lange coupler, which is widely used in microwave circuits. Equations for the design of a general Nconductor interdigital coupler are given. Next, the operation of a tandem coupler is explained. We show that connecting two loose couplers in tandem results in a tight coupler. Finally, compact couplers for wireless applications and tight couplers using multilayer dielectric structures are described.
8.2 BranchLine Couplers Figure 8.1 shows a branchline coupler. It consists of two quarterwavelong transmission line sections of characteristic impedance Z 0s each, connected by two shunt branches. The shunt branches are quarterwavelong transmission line sections of characteristic impedance Z 0p each. By properly choosing the values of Z 0s and Z 0p , the circuit can be made to operate like a directional coupler. At the center frequency the scattering parameters of a branchline coupler are given by S 21 = −j
Z 0s Z0
(8.1)
S 31 = −
Z 0s Z 0p
(8.2)
S 11 = 0
(8.3)
S 41 = 0
(8.4)
where Z 0 denotes the impedance of various ports of a branchline coupler. The scattering parameters of a branchline coupler also satisfy the following condition, which follows from the principle of conservation of energy:
Figure 8.1
Layout of a branchline coupler in planar circuit configuration.
8.2 BranchLine Couplers
221
 S 21  2 +  S 31  2 = 1
(8.5)
where it is assumed that S 11 = S 41 = 0. Substituting values of scattering parameters from (8.1) and (8.2) in (8.5), it is found that Z 0s and Z 0p should satisfy the following condition: 2
2
Z 0s
Z 0s
Z0
Z 0p
+ 2
2
=1
(8.6)
A branchline coupler as shown in Figure 8.1 has two planes of symmetry. The scattering matrix of a branchline coupler can therefore be expressed as
[S ] =
冤
0
Z −j 0s Z0
Z −j 0s Z0
0
0
−
Z 0s Z 0p
0
0
Z −j 0s Z0
Z − 0s Z 0p
Z −j 0s Z0
0
−
0
−
Z 0s Z 0p
0 Z 0s Z 0p
冥
(8.7)
The characteristic impedance of the mainline and shunt branches of a branchline coupler can be computed using (8.1) and (8.2) as Z 0s = Z 0  S 21  = Z 0 √1 −  S 31  2
(8.8)
and Z 0p =
Z 0s =  S 31 
Z 0s
√1 −  S 21 
(8.9) 2
Example 8.1
Compute the characteristic impedances of the mainline and shunt branches of a branchline coupler given Z 0 = 50⍀ and coupling from port 1 to ports 2 and 3 is equal (S 21 = S 31 = −3.01 dB). Given that coupling from port 1 to port 2 is 3.01 dB, therefore
 S 21  = 10−3.01/20 = 0.707 Substituting the value of  S 21  in (8.8), we obtain Z 0s = 0.707Z 0 = 35.4⍀
222
Tight Couplers
Further, substituting Z 0s = 35.4⍀ and  S 21  = 0.707 in (8.9), the value of Z 0p is found as Z 0p = 50⍀ The transmission line sections having characteristic impedances of 35 and 50⍀ can be easily realized using planar transmission lines such as microstrip line. A 3dB branchline coupler is also known as a 90degree hybrid. Properties of the BranchLine Coupler
In a branchline coupler, the following performance is obtained only at the center frequency (i.e., the frequency at which the mainline and shunt branches are each a quarterwave long): S 11 = S 22 = S 33 = S 44 = 0
(8.10)
S 14 = S 41 = S 23 = S 32 = 0
(8.11)
A branchline coupler is thus perfectly matched only at its center frequency, at which there is also complete isolation between ‘‘decoupled’’ ports. In the case of backward parallelcoupled TEM couplers, these properties are satisfied independent of the frequency (as discussed in Section 6.4). In parallelcoupled TEM couplers, the relationship of phase quadrature between the signals at direct and coupled ports is satisfied independent of the operating frequency. In a branchline coupler, the relationship of phase quadrature between the signals at direct and coupled ports is satisfied only at the center frequency. The VSWR, coupling, and isolation of a 3dB branchline coupler as a function of frequency are plotted in Figure 8.2. It is seen that the input VSWR is unity (equivalent to  S 11  = 0) at the center frequency, but increases rapidly away from the center frequency and reaches a value of 2 (equivalent to  S 11  = −9.54 dB) about 20% away from the center frequency. Similarly, the isolation between ‘‘decoupled’’ ports falls to about 15 dB at frequencies about 10% away from the center frequency. Branchline couplers are useful in applications requiring less than about 20% frequency bandwidth only. Physical Implementation of BranchLine Couplers
The layout of a branchline coupler in microstrip form is shown in Figure 8.1. It is seen that there is a physical discontinuity present at each junction. The effect of these discontinuities is to add extra reactances that alter the response of the physical coupler compared with the ideal one, and should be considered in evaluating the performance of the circuit, especially at high frequencies. This can be done using the equivalent circuit models of discontinuities, which are available in the literature [5] and also in many commercially available microwave CAD software. A more accurate analysis of such couplers is generally performed using electromagnetic (EM) simulators.
8.2 BranchLine Couplers
Figure 8.2
223
(a) Variation of VSWR with frequency of a 3dB branchline coupler. (b) Variation of coupling with frequency of a 3dB branchline coupler. (c) Variation of isolation with frequency of a 3dB branchline coupler.
8.2.1 Modified BranchLine Coupler
Branchline couplers are generally used for equal power division. For loose coupling, parallelcoupled TEM couplers are preferred. If parallelcoupled directional couplers are realized in microstrip configuration, the directivity of the coupler may not be very high because of unequal even and oddmode phase velocities. A branchline coupler may be a better choice under this situation [6]. When a branchline coupler is designed for loose coupling ( S 31  < −10 dB), the characteristic impedance of the shunt branches becomes quite high. The impedances of the mainline and shunt branches of a 20dB coupler are found to be nearly 50 and 500⍀, respectively, using (8.8) and (8.9). However, a high impedance line of 500⍀ cannot be realized
224
Tight Couplers
using microstrip. A similar problem is faced in the design of planar multisection branchline couplers. The impedances of the end sections of a multisection branchline coupler tend to be quite high. A highimpedance transmission line section can be realized using coupled lines as shown in Figure 8.3(a). The electrical length of the coupled line is and each line is shorted at one of its ends. The equivalent circuit of the coupled lines between ports 1 and 2 is shown in Figure 8.3(b). When = 90 degrees (which is equivalent to lines being g /4 long), the shorting stubs of characteristic admittance Y 0e appear as open circuit across the main transmission line. Under this condition, the equivalent circuit further reduces to that as shown in Figure 8.3(c) where the characteristic impedance Z c of the line is given by Zc =
2Z 0e Z 0o Z 0e − Z 0o
(8.12)
Note, however, that the length of the equivalent transmission line is 270 degrees compared with 90 degrees for the coupled lines, where an additional 180degree phase is introduced by the shortcircuited ends of the coupledline section. Equation (8.12) shows that very high values of Z c can easily be obtained using coupled lines (by choosing Z 0e ≈ Z 0o ). In practice, it is feasible to obtain impedances above a certain value only. For example, for coupled lines spaced 1 mil on a 25mil substrate, the lowest impedance Z c that can be realized is about 115 to 120⍀ for a dielectric constant of 2. Similarly, on a substrate of dielectric constant 10, the lowest value of impedance Z c that can be realized using coupled lines is about 70⍀. A coupled section as shown in Figure 8.3(a) can therefore be used to replace the shunt branches of highimpedance values in a branchline coupler designed for loose coupling. The modified branchline coupler is shown in Figure 8.4. Although
Figure 8.3
(a) Shorted coupledline pair and (b) equivalent circuit. (c) Equivalent circuit of coupledline pair of part (a) when = 90 degrees.
8.2 BranchLine Couplers
Figure 8.4
225
Modified branchline coupler.
the equivalent length of the line realized using coupled lines is 180 degrees longer than that of the coupled lines, the performance of a modified branchline coupler at the midband frequency is not affected, in comparison to a conventional branchline coupler. The frequency response of a modified branchline coupler will be somewhat different from that of a conventional branchline coupler, however, because the equivalent circuit of the coupled lines shown in Figure 8.3(c) will differ from that shown in Figure 8.3(b) when the value of is different from 90 degrees. The simulated response of a 20dB conventional branchline coupler is shown in Figure 8.5(a). The isolation of an ideal conventional branchline coupler is greater than 40 dB and 35 dB over frequency bandwidths of 12% and 20%, respectively. The directivity of the conventional design is, therefore, greater than 20 dB and 15 dB over a bandwidth of 12% and 20%, respectively. Figure 8.5(b) shows the simulated response of a modified branchline coupler. In the modified design, the shunt branches of 500⍀ impedance have been replaced by coupled sections of even and oddmode impedances of 100⍀ and 71⍀, respectively. When these values are substituted in (8.12), a characteristic impedance of 490⍀ is obtained, which is quite close to the desired value. The directivity of the modified branchline coupler is greater than 20 dB and 15 dB over a bandwidth of 8% and 14%, respectively. Although the bandwidth of a modified branchline coupler is narrower than that of a conventional coupler, the modified design can be realized in microstrip confirguration. For example, the coupled section can be realized by printing 5milwide lines spaced 24 mil apart on a 25mil alumina substrate. It has been reported that a simulated response of a modified branchline coupler matches closely with experimental results [6]. 8.2.2 ReducedSize BranchLine Coupler
In MMICs, lumped capacitors can be easily realized and have become attractive in reducing the size of passive components. Reducedsize branchline hybrids that use only lumped capacitors and small sections of transmission lines (smaller than g /4) have also been reported [7]. The size of these hybrids is about 80% smaller
226
Tight Couplers
Figure 8.5
(a) Frequency response of a conventional 20dB branchline coupler shown in Figure 8.1 [6]. (b) Frequency response of modified 20dB branchline coupler shown in Figure 8.4 [6].
than those for conventional hybrids and is therefore quite suitable for MMICs. Reducedsize branchline couplers are discussed in the following. A transmission line section of impedance Z c and electrical length of 90 degrees is shown in Figure 8.6(a). This section serves as a basic building block of a branchline coupler. Its ABCD matrix is given by (8.13). The quarterwave section can be replaced by a section shown in Figure 8.6(b), which comprises a transmission line of characteristic impedance Z and electrical length and shunt capacitances C at either end. By choosing Z > Z c , the electrical length can be shorter than 90 degrees. The ABCD matrix of the circuit shown in Figure 8.6(a) is given by
冋 册 冤 A C
B = D
0 j Zc
jZ c 0
冥
(8.13)
On the other hand, the ABCD matrix of a lumpedelement circuit as shown in Figure 8.6(b) is given by
8.2 BranchLine Couplers
Figure 8.6
227
(a) Quarterwavelong transmission section. (b) Reducedsize circuit equivalent to quarterwave section.
冋 册 冋 A C
B 1 = D j C
=
冤
j
册冤
0 1
cos sin j Z
jZ sin cos
冥冋
1 j C
册
0 1
(8.14)
cos − CZ sin
jZ sin
sin + 2j C cos − j ( C)2 Z sin Z
cos − CZ sin
冥
It can be easily shown that the above matrix becomes identical to (8.13) if the values of Z and C are chosen as follows: Z=
Zc sin
(8.15)
cos Zc
(8.16)
and
C =
where denotes the electrical length of the shortened transmission line shown in Figure 8.6(b). For example, if the value of Z /Z c is chosen to be equal to 2, then using (8.15), we obtain sin =
1 , or = 30 degrees 2
In other words, the length of the shortened transmission line becomes equal to g /12, which is onethird of the usual quarterwave section. The relationship between , Z /Z c , and C is shown in Figure 8.7.
228
Tight Couplers
Figure 8.7
Relation between elements of Figure 8.6. (From: [7]. 1990 IEEE. Reprinted with permission.)
The characteristic impedances of the mainline and shunt branches of a conventional 3dB branchline coupler are Z 0 /√2 and Z 0 , respectively. If the characteristic impedances of the mainline and shunt branches are replaced by Z, where Z > Z 0 , the lengths of the mainline and shunt branches become less than g /4. More specifically, the lengths of the mainline ( 1 ) and shunt ( 2 ) branches and the value of capacitance C b are given by
1 = sin−1 y y √2
(8.18)
√1 − y 2 + √2 − y 2
(8.19)
2 = sin−1 Cb Z0 =
(8.17)
where y = Z 0 /Z. When y = 1/√2, or Z = √2Z 0 , 1 = 45 degrees and 2 = 30 degrees. Therefore, if the mainline and shunt branch impedances of a 50⍀ branchline coupler are chosen equal to 70.7⍀, the lengths of the mainline and shunt branches become equal to g /8 and g /12, respectively. The value of C b required can be determined using (8.19). The reducedsize branchline hybrid is shown in Figure 8.8. The bandwidth of the reducedsize hybrid is a little wider than that of the purely lumped hybrid but is narrower than that of the conventional quarterwavelength hybrid. The calculated phase difference between the signals at the direct (S 21 ) and coupled (S 31 ) ports is shown in Figure 8.9. Reducedsize hybrids can be easily implemented in MMICs. A photograph of a reducedsize 25GHz branchline hybrid is shown in Figure 8.10(a). The circuit
8.2 BranchLine Couplers
229
Figure 8.8
Reducedsize branchline hybrid.
Figure 8.9
Calculated phase difference between S 21 and S 31 of the reducedsize hybrid, the conventional hybrid, and the purely lumpedelement hybrid. (From: [7]. 1990 IEEE. Reprinted with permission.)
230
Tight Couplers
Figure 8.10
(a) Photomicrograph of the fabricated 25GHz reducedsize branchline hybrid. (From: [7]. 1990 IEEE. Reprinted with permission.) (b) Measured performance of the 25GHz reducedsize branchline hybrid [7].
is fabricated on a GaAs substrate. All the transmission lines are 70⍀ coplanar waveguides with a 10 m center conductor width. The lengths of the mainline and shunt branches are g /8 and g /12, respectively. Metalinsulatormetal (MIM) shunt capacitors are located at the four Tjunctions and between the inner conductors and the ground metal. The size of the overall hybrid is 500 m × 500 m, representing an 80% savings over a conventional branch line hybrid. Its measured performance is shown in Figure 8.10(b). 8.2.3 LumpedElement BranchLine Coupler
The size of a branchline coupler using sections of transmission lines becomes quite large at frequencies below about 2 GHz. At these frequencies, a branchline can be realized using lumped inductor and capacitor elements. The lumpedelement 90degree hybrid can be realized either in a ‘‘pi’’ or a ‘‘tee’’ equivalent network.
8.2 BranchLine Couplers
231
In MMICs a ‘‘pi’’ network is preferred to ‘‘tee’’ because it uses fewer inductive elements which have lower Q and occupy more space. In lumpedelement implementation, basically each transmission line section shown in Figure 8.11(a) is replaced by an equivalent ‘‘pi’’ lumpedelement network as shown in Figure 8.11(b). The values of the lumped elements are obtained by equating the ABCD matrix parameters for both these structures. The ABCD matrix of a lossless transmission line section of characteristic impedance Z c and electrical length is given by
冋 册 冤 A C
B = D
cos sin j Zc
jZ c sin cos
冥
(8.20)
On the other hand, the ABCD matrix of a lumpedelement circuit as shown in Figure 8.11(b) is given by
冋 册 冋 冋 A C
B 1 = D j C =
0 1
册冋
1 0
j L 1
册冋
1 j C
1 − 2LC
j L
2j C − j 3LC 2
1 − 2LC
册
0 1
册
(8.21)
where denotes the radian frequency. By equating the matrix elements in (8.20) and (8.21) and simplifying, we obtain L= C=
Figure 8.11
1 Zc
Z c sin
√
1 − cos 1 + cos
(a) Quarterwave section and (b) its lumped equivalent.
(8.22a)
(8.22b)
232
Tight Couplers
The lumped circuit shown is therefore equivalent to a quarterwavelong transmission line of characteristic impedance Z c , if the elements of the lumped circuit are chosen as follows: L=
Zc
C=
1 Zc
Note that the circuits shown in Figure 8.11 have identical response at the frequency at which the transmission line section is quarterwave long. At other frequencies, the response of both the circuits will be different in general. Therefore, the bandwidth of circuits realized using transmission line sections will be different from those realized using lumped elements. A branchline coupler as shown in Figure 8.1 uses quarterwave transmission line sections of characteristic impedances Z 0s and Z 0p . By replacing these sections with equivalent lumped elements, the circuit as shown in Figure 8.12 is obtained. The values of the lumped inductive and capacitive elements are given by L1 =
Z 0s 1 , C1 = 0 Z 0s 0
(8.23)
and
Figure 8.12
Lumpedelement equivalent circuit model for the 90degree hybrid shown in Figure 8.1.
8.3 RatRace Coupler
233
L2 =
Z 0p 1 , C2 = 0 Z 0p 0
(8.24)
where 0 denotes the radian frequency corresponding to the center frequency of the branchline coupler. More specifically, the element values of a lumped 3dB branchline coupler are given by L1 =
Z0 √2 , C1 = Z0 0 √2 0
(8.25)
Z0 1 , C2 = 0 Z0 0
(8.26)
and L2 =
where Z 0 denotes the terminal impedance of the ports of the branchline coupler. Typical lumpedelement values for a 900MHz coupler designed for 50⍀ terminal impedance are L 1 = 6.3 nH, L 2 = 8.8 nH, and C t = 8.5 pF, where C t = C 1 + C 2 . Over 900 ± 45 MHz the calculated value of amplitude unbalance and the phase difference between the output ports is ± 0.2 dB and 90 ± 2 degrees, respectively. The bandwidth of these couplers can be increased by using more sections of ‘‘pi’’ or ‘‘tee’’ equivalent networks; that is, two sections of 45 degrees or three sections of 30 degrees to realize a 90degree section, and so on, or by properly selecting highpass and lowpass networks [8, 9]. In general, two to three sections are sufficient to realize a broadband 90degree hybrid.
8.2.4 Broadband BranchLine Coupler
As already discussed, the bandwidth of a branchline coupler is quite narrow. To increase its bandwidth, a multisection branchline coupler can be used. The synthesis of multisection branchline couplers has been described by Levy and Lind [10]. The synthesis has been described for both maximally flat and Chebyshev type of response of VSWR and directivity of the coupler. Similar techniques can also be applied to reducedsize and lumpedelement couplers [11].
8.3 RatRace Coupler The strip conductor layout of a ratrace coupler in microstrip form is shown in Figure 8.13. In the case of a branchline coupler as shown in Figure 8.1, the spacing between all adjacent ports is g /4. In the case of a ratrace coupler, however, the spacing between two of the adjacent ports is 3 g /4 and g /4 between all other adjacent ports. By properly choosing the values of Z 01 and Z 02 , the circuit can be made to operate like a directional coupler. At the center frequency the scattering parameters of a ratrace coupler are given by
234
Tight Couplers
Figure 8.13
Layout of a ratrace coupler in planar circuit configuration.
Z0 Z 02
(8.27)
Z0 Z 01
(8.28)
S 21 = −j S 41 = j
S 11 = S 31 = 0
(8.29)
Z0 Z 01
(8.30)
Furthermore, S 32 = −j and S 42 = 0
(8.31)
The impedances Z 01 and Z 02 should satisfy the condition: 2
2
Z0
Z0
Z 01
Z 02
+ 2
2
=1
(8.32)
The above equation follows from the following condition that needs to be satsified because of the principle of conservation of power:
 S 21  2 +  S 41  2 = 1
(8.33)
where it is assumed that S 11 = S 31 = 0. The complete scattering matrix of a ratrace coupler at the midband frequency can be expressed as
8.3 RatRace Coupler
235
冤
0
−j
[S ] =
−j
Z0 Z 02 0
Z j 0 Z 01
Z0 Z 02 −j
0 −j
0
Z0 Z 01 0
j
Z0 Z 01
Z0 Z 01 0
0 −j
Z −j 0 Z 02
冥
Z0 Z 02 0
(8.34)
The scattering matrix of a ratrace coupler makes it useful in many applications. For example, consider a ratrace coupler designed for equal power division between two coupled ports. When an input signal is incident at port 2, the power is equally divided between ports 1 and 3 and no power reaches port 4. The signals arriving at ports 1 and 3 are in phase. When power is incident at port 1, no power is coupled to port 3 and the power is equally divided between ports 2 and 4 as in the previous case. In this case, however, the signals arriving at ports 2 and 4 are out of phase (i.e., 180degree phase difference). This special property makes the ratrace useful in applications, such as in balanced mixers, where the effect of local oscillator AM noise can be canceled by connecting the RF signal to port 3 and local oscillator to port 1. A 3dB ratrace coupler is also known as a 180degree hybrid. Design of a RatRace Coupler
From the scattering matrix of (8.34), we have Z 02 =
Z0 =  S 21 
Z 01 =
Z0 =  S 41 
Z0 2 √1 −  S 41 
(8.35)
and Z0
√1 −  S 21 
2
Example 8.2
Compute the characteristic impedances of the various sections of a ratrace coupler shown in Figure 8.13, given Z 0 = 50⍀ and the coupling coefficient from port 1 to ports 2 and 3 equals (S 21 = S 41 = −3.01 dB). Given
 S 21  =  S 41  = 10−3.01/20 = 0.707 Using (8.35) and (8.36), we then obtain
(8.36)
236
Tight Couplers
Z 01 = Z 0 /0.707 = 70.7⍀ and Z 02 = Z 0 /0.707 = 70.7⍀ Properties of the RatRace Coupler
A ratrace coupler has disadvantages similar to that of a branchline coupler. For example, the ports of a ratrace are matched only at the center frequency. Further, there is complete isolation between decoupled ports (such as between port 1 and port 3 in Figure 8.13) at the center frequency only. Furthermore, the phase difference of 180 degrees between signals arriving at ports 2 and 4 exists only at the center frequency. The scattering parameters of a 3dB ratrace coupler as a function of frequency are shown in Figure 8.14. Here, port 1 is the input, and ports 2 and 4
Figure 8.14
(a) Variation of VSWR with frequency of a 3dB ratrace coupler. (b) Variation of coupling with frequency of a 3dB ratrace coupler. (c) Variation of isolation with frequency of a 3dB ratrace coupler.
8.3 RatRace Coupler
237
are the direct and coupled ports, respectively. The ratrace coupler finds use in applications where the frequency bandwidth requirement is less than about 20%. Because of physical discontinuities at the locations of junctions of a rat race, its performance deviates from that of an ideal rat race. The effect of these discontinuities must be considered in the design, especially at high frequencies.
8.3.1 Modified RatRace Coupler
The bandwidth of a ratrace coupler is only about 20% but can be increased somewhat by using multisections, or significantly increased (to about an octave) by replacing the 3 g /4 section by an equivalent coupledline section such as shown in Figure 8.15. The coupledline section of electrical length 90 degrees is equivalent to a single transmission line of length 270 degrees as described in Section 8.2.1. The threequarterwavelong section of a ratrace coupler (of impedance Z 01 ) can therefore be replaced by a quarterwavelong coupledline section. In the modified coupler, and equivalent impedance of Z 01 can be obtained using coupled lines if the even and oddmode impedances of the coupled section are chosen as follows: Z 0e = 2.414Z 01 Z 0o =
Z 01 2.414
(8.37) (8.38)
Using the above information, a broadband rat race was simulated. The performance of a broadband rat race is compared with a conventional rat race in Figure 8.16. Bandwidth improvement is quite obvious in these plots. Another advantage of the broadband rat race is that its size is smaller compared to that of a conventional rat race.
Figure 8.15
Broadband hybrid ring using a shorted parallelcoupled quarterwave section.
238
Tight Couplers
Figure 8.16
Broadband hybrid response: (a) coupling, (b) isolation, and (c) return loss.
8.3 RatRace Coupler
239
8.3.2 ReducedSize RatRace Coupler
Ratrace hybrids (3dB ratrace couplers) have unique applications in microwave circuits because they can divide an input signal into signals that are either in phase or out of phase. They can also be used to combine two signals (say, A and B) to obtain a ‘‘sum’’ (A + B) signal or a ‘‘difference’’ (A − B) signal. The size of a ratrace hybrid can also be reduced by employing techniques discussed in Section 8.2.2. A conventional 3dB ratrace coupler is shown in Figure 8.13. It uses a transmission line section of length 3 g /4 and three sections of length g /4 each. The characteristic impedance of each section is √2Z 0 . The transmission line section of length 3 g /4 can be replaced by a lumpedequivalent circuit as shown in Figure 8.17(a). Further, the g /4 sections of characteristic impedance √2Z 0 can be replaced by transmission line sections of characteristic impedance 2Z 0 and length g /8 with shunt capacitances at the two ends as shown in Figure 8.17(b). The resulting reducedsize hybrid is shown in Figure 8.18(a). (In this figure, the port labeling is different from that shown in Figure 8.13.) In the reducedsize ratrace hybrid of Figure 8.18(a), note that parallel LC elements (L a and C b ) between port 1 and the ground (and similarly between port 2 and ground) offer a relatively high shunt impedance at the center frequency. These elements can therefore be removed as shown in Figure 8.18(b) without significantly affecting the response of the circuit. Alternatively, to compensate for the effect of removal of these elements, the impedances of the transmission line sections, their length, and the values of other capacitances can be optimized using CAD tools. Another advantage of the reducedsize ratrace hybrid is its flexible port arrangement. The usual port arrangement is shown in Figure 8.19(a). In this arrangement, the right edge of the bottom plate of the metalinsulatormetal (MIM) capacitor is connected to port 1 and the left edge of the top plate is connected to port 2. Port 1 can be connected to the left edge of the bottom plate without affecting the performance because the whole bottom plate is at the same potential. Similarly, port 2 can be connected to the right edge of the top plate. The resulting layout is shown in Figure 8.19(b). This layout is more convenient for mixer applications. Figure 8.20(a) shows a photograph of a 25GHz reducedsize ratrace hybrid [7]. The circuit consists of a 100⍀ coplanar waveguide and MIM capacitors. The center conductor width of the coplanar waveguide is 10 m. The measured results
Figure 8.17
(a) Lumpedelement circuit to replace 3 g /4 section; and (b) reducedsize circuit to replace g /4 section of a ratrace hybrid.
240
Tight Couplers
Figure 8.18
(a, b) Reducedsize ratrace hybrids; 0 is the center angular frequency (= 2 f 0 ).
of this hybrid are shown in Figure 8.20(b). Note that the insertion loss of the ratrace hybrid is much smaller than that of the branchline coupler shown in Figure 8.10(b). A very small, lowloss MMIC ratrace hybrid using elevated coplanar waveguides has been reported by Kamitsuna [12]. A 15GHz rat race was developed on a chip size of 0.5 × 0.55 mm. 8.3.3 LumpedElement RatRace Coupler
The design of lumpedelement ratrace hybrids [Figure 8.21(a)] is similar to that of a lumpedelement 90degree hybrid described in the previous section. A lumped element equivalent circuit model for the 180degree hybrid is shown in Figure 8.21(b). The 90degree sections are replaced with a lowpass ‘‘pi’’ network as shown in Figure 8.11 and the 270degree (or 90degree) section is replaced with an equivalent highpass ‘‘tee’’ network [13] shown in Figure 8.21(a). Following the same procedure as described for the 90degree hybrid, the lumped elements for the ‘‘pi’’ section and the ‘‘tee’’ can be expressed as L1 = C1 =
√2Z 0 sin 1
1 √2Z 0
√
1 − cos 1 1 + cos 1
(8.39a)
(8.39b)
8.3 RatRace Coupler
Figure 8.19
241
Port interchange in the reducedsize ratrace hybrid: (a) usual port layout; and (b) port layout convenient for mixer applications. (From: [7]. 1990 IEEE. Reprinted with permission.)
L2 = −
C2 =
√2Z 0
sin 2
1 √2Z 0
√
1 + cos 2 1 − cos 2
(8.40a)
(8.40b)
When 1 = 90 degrees and 2 = 270 degrees or −90 degrees, element values for a 50⍀ system become L 1 = L 2 = 11.25/ f nH and C 1 = C 2 = 2.25/ f pF, where f is in gigahertz.
242
Tight Couplers
Figure 8.20
(a) Photomicrograph of the fabricated 25GHz reducedsize ratrace hybrid. (b) Measured performance of the 25GHz reducedsize ratrace hybrid. (From: [7]. 1990 IEEE. Reprinted with permission.)
8.4 Multiconductor Directional Couplers The coupling between two TEM lines increases as the mutual capacitance between them is increased. The mutual capacitance between the lines can be increased by decreasing the spacing between them. However, to obtain tight coupling values such as 3 dB between planar transmission lines such as microstrip lines, the spacing between the lines becomes too small to be realized in a convenient manner using commonly used photolithographic techniques. Further, very small transverse dimensions lead to increased conductor loss. Lange [14] described a scheme using multiconductors in interdigital configuration in which the mutual capacitance between
8.4 Multiconductor Directional Couplers
Figure 8.21
243
(a) Equivalent lumped circuit of a transmission line section of length 3 g /4. (b) Lumpedelement 180degree hybrid.
the lines can be increased without the need for a small spacing between them. This section describes the theory and design of interdigital couplers [14–19]. 8.4.1 Theory of Interdigital Couplers
Figure 8.22 shows two coupled lines and their capacitances. C s and C mu denote the capacitances of lines (perunit length) with respect to the ground and the mutual capacitance between them, respectively. To increase the mutual capacitance, the two lines can be divided into two lines of halfwidth each and arranged in an interdigital manner as shown in Figure 8.23(a) with alternate conductors connected together at the ends. The spacing between adjacent conductors remains the same as in the undivided case shown in Figure 8.22(a). The capacitance of each divided line with respect to the ground is approximately C s /2 (because of halfwidth), whereas the mutual capacitance between adjacent lines is somewhat smaller than between the lines in the configuration shown in Figure 8.22(a). The total capacitance of each line with respect to the ground is therefore approximately C s as in the original case. However, the mutual capacitances between neighboring conductors
244
Tight Couplers
Figure 8.22
(a) Top view of coupling between parallel TEM lines, (b) side view, and (c) equivalent capacitance network showing static perunitlength selfcapacitance of the lines and mutual capacitance between them.
Figure 8.23
(a) Top view of an interdigital coupler, (b) side view, and (c) equivalent capacitance network showing static perunitlength selfcapacitance of the lines and mutual capacitance.
add together to give an overall value of mutual capacitance that is much larger than between the lines in the configuration shown in Figure 8.22(a). This is essentially the basis of a Lange coupler. The equivalent capacitances of the interdigital configuration are shown in Figure 8.23(b, c).
8.4 Multiconductor Directional Couplers
245
Figure 8.24 shows how a coupler consisting of two parallelcoupled lines can be rearranged. For the same spacing between adjacent lines, the coupler configurations shown in Figure 8.24(b, c) offer larger coupling than the configuration shown in Figure 8.24(a). The configurations shown in Figure 8.24(b, c) are identical electrically, but the configuration shown in Figure 8.24(c), which is also known as a Lange coupler, offers an extra advantage in that both the direct and coupled ports are on the same side. Figure 8.24(b) is known as an unfolded Lange coupler.
8.4.2 Design of Interdigital Couplers
In this section, the design relations of an Nconductor (Neven) interdigital coupler as shown in Figure 8.25 are presented. It is assumed that the number of conductors (N) is even. The design of a Lange coupler is obtained when N = 4. All the lines
Figure 8.24
TEM couplers using (a) two parallelcoupled lines, (b) interdigital configuration known as unfolded Lange coupler, and (c) interdigital configuration known as a Lange coupler.
246
Tight Couplers
Figure 8.25
(a) Top view of an Nconductor interdigital coupler and its (b) side view; and (c) side view of two parallelcoupled lines having the same width and spacing between them as the Nconductor interdigital coupler.
in the configuration shown in Figure 8.25 are assumed to have the same width. The spacing between all adjacent lines is also assumed to be the same. For given values of N, Z 0 (the impedance of various ports) and coupling, the design is obtained in terms of even and oddmode impedances of a pair of coupled lines as shown in Figure 8.25(c). Once the even and oddmode impedances of one pair of coupled lines are known, the width (W ) of the lines and the spacing (S) between them can be determined. The design is obtained in terms of even and oddmode impedances of one pair of coupled lines because these data are available in the literature for a large class of transmission lines. The design equations of an Nconductor (N even) interdigital coupler are given by [15, 16]
k=
Z=
Z 0o = Z0
(N − 1) (1 − R 2 ) (N − 1) (1 + R 2 ) + 2R
√R[(N − 1) + R] [(N − 1)R + 1] (1 + R)
R=
Z 0o Z 0e
(8.41)
(8.42)
(8.43)
8.4 Multiconductor Directional Couplers
247
where k is the voltage coupling coefficient between the input and coupled ports at the center frequency of the design and N (N is even) is the total number of conductors. Z 0e and Z 0o denote, respectively, the even and oddmode impedances of a pair of coupled lines having the same width and spacing between them as any pair of Nconductor interdigital coupler. It may be worth remarking that the usual relation Z 0 = √Z 0e Z 0o is valid in the case of an interdigital coupler only when N = 2. For other values of N, this relation is not satisfied. The length of the interdigital coupler at the center design frequency is given by ᐉ=
g 4
(8.44)
where g = 0.5( ge + go ) is the wavelength in the medium of the coupler at the center frequency of the design. Here ge and go are the guide wavelengths for the even and odd modes, respectively. For a given value of voltage coupling factor, k, and number of conductors N, (8.41) can be used to find the value of R and (8.42) can be used to determine the oddmode impedance Z 0o . Further, the evenmode impedance Z 0e can be determined using (8.43). Using now the known values of the even and oddmode impedances, the dimensions of the lines and the spacing between them can be determined either by using available nomograms, computer programs [17] or design equations [5]. Equations (8.41) to (8.43) are not exact but are based on the following assumptions: • •
•
The mode of propagation along the structure is TEM. The length of bonding wires is negligible compared with the wavelength at the frequency of operation. The mutual capacitance between any two neighboring conductors of the interdigital coupler is the same as for the two conductor lines shown in Figure 8.25(c).
Results
Using (8.41), the value of R has been plotted as a function of coupling in Figure 8.26 for values of N = 2 and 4 [15, 16]. Similarly, the normalized oddmode impedance has been plotted as a function of coupling in Figure 8.27 for values of N = 2 and 4. In Table 8.1 numerical values are given for an interdigital coupler for a few values of coupling and number of conductors N. The input/output impedances of the coupler are assumed to be 50⍀. It may be verified that for N = 2, the values of even and oddmode impedances given in Table 8.1 are the same as those given by (8.41)–(8.43). Design Data for a Lange Coupler
A Lange coupler is usually realized in microstrip configuration. In Figure 8.28, the normalized dimensions of the coupler (W /h and S/h) are given as a function of
248
Tight Couplers
Figure 8.26
Impedance ratio R (= Z 0o /Z 0e ) as a function of coupling. (From: [16]. 1978 IEEE. Reprinted with permission.)
Figure 8.27
Normalized oddmode impedance (= Z 0o /Z 0 ) as a function of coupling. (From: [16]. 1978 IEEE. Reprinted with permission.)
8.4 Multiconductor Directional Couplers
249
Table 8.1 Impedance Parameters of an NConductor Interdigital Coupler
Figure 8.28
N
Coupling (dB)
Z 0e ⍀
Z 0o ⍀
2
3 6 10
120.70 86.60 69.37
20.71 28.87 36.04
4
3 6 10
176.20 142.50 118.30
52.61 67.96 76.30
6
3 6 10
243.10 204.30 181.11
82.55 105.10 122.10
The dimensional ratios of a Lange coupler printed on a dielectric substrate (⑀ r = 10) and Z 0 = 50⍀ as a function of coupling. (From: [16]. 1978 IEEE. Reprinted with permission.)
coupling for Z 0 = 50⍀ and printed on a dielectric substrate of ⑀ r = 10 [16]. The data are based on the computation of even and oddmode impedance using (8.41) through (8.43). Further, using the computed values of even and oddmode impedances, the dimension ratios W /h and S/h of the coupler have been found using the program for coupled microstrip lines given by Bryant and Weiss [17]. Simple expressions given in reference [5] can also be used to obtain the values of W /h and S/h. A Lange coupler is frequently designed for 3dB coupling. In Figure 8.29, the dimensions of a 3dB, 50⍀ Lange coupler are given as a function of the dielectric constant of the substrate material on which the coupler is printed. The dimensions given in Figures 8.28 and 8.29 are valid for zerothickness conductors. In practice, however, the conductors will always have a finite thickness. Presser [16] has given an empirical formula for the correction factor that was found by performing a large number of experiments. The effect of thickness can be taken
250
Tight Couplers
Figure 8.29
The dimensional ratios of a 3dB Lange coupler as a function of dielectric constant of the substrate. (From: [16]. 1978 IEEE. Reprinted with permission.)
into account by increasing the separation between adjacent lines and decreasing the width of the lines as S S 0 ⌬S = + h h h
(8.45a)
W W 0 ⌬S = − h h h
(8.45b)
and
where W 0 and S 0 denote the width and the separation between lines, respectively, for a coupler employing zerothickness conductors. The value of ⌬S/h in the above equations is given by t /h ⌬S = h √⑀ re
冉
1 + ln
4 W 0 /h t /h
冊
(8.46)
where t denotes the thickness of the metalization and ⑀ re can be assumed to be the effective dielectric constant of a single, uncoupled microstrip line having width W. Furthermore, Presser [16] found from sensitivity analysis that the most sensitive parameters in the design of a Lange coupler are the gap dimensions and the metalization thickness, whereas 10% changes in the width and the dielectric constant cause practically insignificant deviations in the performance. Many computer programs are now available, which give the impedance data for coupled microstrip lines taking the thickness of the conductors into account. With such a program, we can directly compute the dimensions of the interdigital structure using the values of even and oddmode impedances computed using (8.42) and (8.43). Commercial CAD tools can also be used to design such couplers. More accurate design can also be performed using EM simulators.
8.4 Multiconductor Directional Couplers
251
In a Lange coupler, the conductor widths and the spacing between the coupler’s conductors can be produced with standard thin film manufacturing processes on thick lowdielectric constant substrates (thickness > 250 m, ⑀ r < 10). However, on thin GaAs (⑀ r = 12.9) substrates (thickness less than 100 m), tightly coupled structures are difficult to realize because the conductor width and the spacing become prohibitively small. For example, a 3dB coupler on a GaAs substrate requires approximately W /h and S/h values on the order of 0.07. Therefore, a broadband 3dB Lange coupler on a 75 mthick substrate requires approximately 5 m conductor width and gap dimensions. A sixfinger 3dB Lange coupler [20], as shown in Figure 8.30, was designed and tested on a 75 m GaAs substrate using a multilayer MMIC process [21, 22]. In this structure the finger lines were fabricated on a thin polyimide dielectric layer, which is placed on top of the GaAs substrate. This structure has two distinct features: (1) for a given line impedance it increases the line width, and (2) it reduces the microstrip line loss. The design parameters for a Xband coupler are given in Table 8.2. Table 8.3 summarizes the line lengths and capacitor values for several sixfinger Lange couplers working over the 5–37GHz frequency range. The shunt capacitors connected between the input and coupled ports, and between the direct and isolated ports reduce the length and improve the directivity of the coupler. Other parameters for these couplers are the same as given in Table 8.2. The capacitors are of the metal insulator metal (MIM) type, using SiN with 300 pF/mm2
Figure 8.30
Sixfinger microstrip coupler configuration: (a) top view and (b) side crosssectional view.
252
Tight Couplers Table 8.2 XBand SixFinger Lange Coupler Parameters Number of fingers = 6 Line width, W = 9 m Line spacing, S = 7 m Length, L = 2,400 m Capacitor, C = 0.16 pF GaAs substrate, ⑀ r = 12.9 Thickness, h = 75 m Polyimide, ⑀ rd = 3.2 Thickness, d = 10 m Gold conductors, t = 4.5 m
Table 8.3 Line Lengths and Capacitor Values for Several SixFinger Lange Couplers Frequency range (GHz) Physical length, L ( m) Capacitor C (pF)
5–9 4,800 0.22
7–13 3,000 0.16
10–24 1,800 0.12
16–37 1,200 0.07
capacitance density. The measured coupling was 3.3 ± 0.5 dB with a maximum amplitude variation of ± 0.6 dB between the coupled and direct ports, over a 8.5GHz bandwidth. The measured return loss at all ports was greater than 13 dB from 6 to 15.5 GHz. The measured minimum isolation was 15.5 dB across 6 to 16 GHz, and better than 20 dB from 10.5 to 16 GHz. The phase difference was 94 ± 8°.
8.5 Tandem Couplers A tight coupler can also be obtained by connecting two loose couplers in tandem [23] as shown in Figure 8.31. In this arrangement, the direct and coupled ports of
Figure 8.31
Schematic of a tandem coupler.
8.5 Tandem Couplers
253
the first coupler are connected to the isolated and input ports of the second coupler, respectively. Both the couplers are of the TEM type, with scattering parameters of the form given by (8.47) and (8.48). If the voltage coupling factors for the two couplers are k 1 and k 2 , respectively, then the scattering matrix of the first coupler at the center frequency (the center frequency is the one at which the electrical length of the TEM coupler is 90 degrees) is given by, −j cos ␣ 1 0 0 sin ␣ 1
冤
0 −j cos ␣ 1 [S 1 ] = sin ␣ 1 0
sin ␣ 1 0 0 −j cos ␣ 1
冥
0 sin ␣ 1 −j cos ␣ 1 0
(8.47)
where sin ␣ 1 = k 1 is the voltage coupling coefficient for coupler 1. Similarly, the scattering matrix of the second coupler can be expressed as −j cos ␣ 2 0 0 sin ␣ 2
冤
0 −j cos ␣ 2 [S 2 ] = sin ␣ 2 0
sin ␣ 2 0 0 −j cos ␣ 2
冥
0 sin ␣ 2 −j cos ␣ 2 0
(8.48)
where sin ␣ 2 = k 2 denotes the voltage coupling coefficient for the second coupler. + With a wave voltage amplitude of unity (V1 = 1), incident on port 1 of coupler − − − − 1, the reflected voltages V1 , V2 , V3 , and V4 are given by −
冤冥冤 V1
−
V2
−
V3
−
V4
0 −j cos ␣ 1 = sin ␣ 1 0
−j cos ␣ 1 0 0 sin ␣ 1
sin ␣ 1 0 0 −j cos ␣ 1
冥冤 冥
0 sin ␣ 1 −j cos ␣ 1 0
1 0 0 0
(8.49)
which gives −
V1 = 0, −
V2 = −j cos ␣ 1 ,
(8.50)
−
V3 = sin ␣ 1 , and −
V4 = 0 Furthermore, the reflected voltages at ports 2 and 3 of coupler 1 are, respectively, the incident voltages for the ports 4′ and 1′ of the second coupler. Therefore +
−
+
−
V1′ = V3 = sin ␣ 1 V4′ = V2 = −j cos ␣ 1
(8.51a) (8.51b)
254
Tight Couplers
The output at the second coupler can then be found using −
冤冥冤 V1′
−
V2′
−
V3′
− V4′
0 −j cos ␣ 2 = sin ␣ 2 0
−j cos ␣ 2 0 0 sin ␣ 2
sin ␣ 2 0 0 −j cos ␣ 2
0 sin ␣ 2 −j cos ␣ 2 0
sin ␣ 1 0 0 −j cos ␣ 1
冥冤 冥 (8.52)
which gives −
−
V1′ = V4′ = 0
(8.53a)
V2′ = −j cos ␣ 2 sin ␣ 1 − j sin ␣ 2 cos ␣ 1 = −j sin (␣ 1 + ␣ 2 )
(8.53b)
−
and −
V3′ = sin ␣ 2 sin ␣ 1 − cos ␣ 2 cos ␣ 1 = −cos (␣ 1 + ␣ 2 )
(8.53c)
Now let us choose
␣1 = ␣2 =
8
(8.54)
or k 1 = k 2 = sin
冉冊 8
= 0.3827
which is also equivalent to coupling in decibels for the two couplers as C 1 and C 2 , respectively, where C 1 = C 2 = −20 log (0.3827) = 8.34 dB Substituting the values of ␣ 1 and ␣ 2 from (8.54) in (8.53), we obtain −
冉冊 4
=−
j √2
(8.55a)
−
冉冊
=−
1 √2
(8.55b)
V2′ = −j sin and V3′ = −j cos
4
The fractional power coupled from port 1 of the first coupler to port 3′ of the second coupler is therefore given by
8.6 Multilayer Tight Couplers
255 −
P3′
+ P1
=
 V3′− 
2
2  V1+ 
=
1 2
(8.56)
which is equivalent to 3dB coupling. Therefore, by connecting two 8.34dB couplers in tandem, a 3dB coupler is obtained. For physical realization of tandem couplers, a scheme such as shown in Figure 8.32 is used. Note that crossovers are required in tandem couplers to achieve proper interconnections between the couplers as required by the connection scheme symbolically shown in Figure 8.31.
8.6 Multilayer Tight Couplers A recent upsurge in MMICbased system demands and wireless applications has led to new configurations for tight directional couplers. These are broadside, embedded microstrip, reentrant, and compact directional couplers. Compact couplers include lumpedelement, spiral, and meander line structures, which are described briefly in Section 8.7.
8.6.1 Broadside Couplers
The asymmetrical broadsidecoupled microstrip lines configuration is the simplest technique to realize tight coupling. Several different configurations and fabrication technologies to design 3dB couplers have been reported in the literature [24–29]. A basic configuration of a MIC/MMIC asymmetric broadside coupler, which consists of two conductors separated by a thin layer of polyimide dielectric (⑀ r2 = 3.2), is shown in Figure 8.33. This requires multilayer MMIC technology. The 3dB coupler design is usually done using an EM simulator and involves an optimal solution for polyimide dielectric thickness and conductor width for given GaAs
Figure 8.32
Physical configuration of a tandem coupler.
256
Tight Couplers
Figure 8.33
Cross section of an asymmetric broadsidecoupled microstrip line coupler.
substrate thickness. The conductors can be folded or meandered to reduce the overall chip size. A coupledmultilayer microstrip line structure as shown in Figure 8.34, which works similar to symmetrical broadsidecoupled striplines structure, was reported by Okazaki and Hirota [29]. The coupler was fabricated using multilayer thinfilm microstrip line technology and consists of four parallelstrip conductors. The ground plane, which is 1 m thick, for these conductors is placed on the top surface of the GaAs. Diagonal conductors as shown in the figure are connected at the ends, and the structure behaves like symmetrical broadsidecoupled striplines, and the couplers can be designed using broadsidecoupled stripline formulas given in Section 3.7. Various dimensions for the coupler reported in [29] are as follows. Conductors A and C are buried in polyimide layers (⑀ r ≅ 3.3) and have widths (W 1 ) of 3 m. The gap (S 1 ) between them is 5 m and are placed above the ground plane level (h 1 ) at 6.5 m. The other two conductors B and D have widths (W 2 ) of 5 m and the gap between them (S 2 ) is also 5 m. They are placed above the ground plane level (h 2 ) at 9 m. Conductors A and D, and B and D, are
Figure 8.34
Structure of the multilayer symmetric broadside coupler. (From: [29]. 1997 IEEE. Reprinted with permission.)
8.6 Multilayer Tight Couplers
257
connected at both ends. The width of the entire coupled line is only 15 m, which allows layout in meander shape (Figure 8.35) to reduce the chip size, which measures only 1.3 × 0.4 mm for the X/Kuband coupler design. Figure 8.36 shows the measured performance of this coupler; coupling is 4.2 ± 0.4 dB, return loss and isolation better than 20 and 15 dB, respectively, over the 10 to 17.5GHz frequency range. The measured phase difference between the direct and coupled ports was about 91 ± 5 degrees over the 10 to 17.5GHz frequency range. Another 3dB asymmetric broadside coupler (Figure 8.33) was developed using the multilevel plating (MLP) process [22] on a 75 mthick GaAs substrate. Figure 8.37 shows the topsectional view of the broadband coupler, which operated over the 6–16GHz frequency range. The physical length of the coupler is 3,000 m. The bottom and top conductor line widths are 40 and 60 m, respectively, and result in lower dissipated loss than a Lange coupler on GaAs. The dielectric between the broadside conductors is 7 mthick polyimide. As shown in Figure 8.38, the measured coupling to the coupled and direct ports are 3.3 ± 0.5 dB and 3.5 ± 0.5 dB, respectively. Measured return loss was better than 18 dB.
Figure 8.35
Photograph of the MMIC symmetric broadside coupler. (From: [29]. 1997 IEEE. Reprinted with permission.)
Figure 8.36
Amplitude characteristics of the symmetric, broadside microstrip coupler. (From: [29]. 1997 IEEE. Reprinted with permission.)
258
Tight Couplers
Figure 8.37
Physical layout of the 3dB asymmetric broadside coupler.
Figure 8.38
Measured coupling coefficient for the 3dB coupler.
8.6.2 ReEntrant Mode Couplers
Reentrant mode couplers have been designed to obtain tight coupling in coaxial [30], stripline [31], and microstrip [32, 33] media. Design procedures of semireentrant [34, 35] and reentrant [36] couplers have also been discussed. The basic theory of such couplers is very simple and can be described by referring to Figures 8.39 and 8.40(a). In Figure 8.39, the top conductor 1 is floating while in Figure 8.40(a), the underneath conductor is floating. The latter coupler consists of a parallelcoupled microstrip line (conductors A and B) with another conductor C floating underneath. This is again a multilayer configuration, the dielectric constant and the thickness of the dielectric layer between conductors A or B and C are determined to achieve the required coupling coefficient. The even and oddmode impedances, like any other symmetrical edgecoupled lines, can be determined by placing magnetic and electric walls at the plane of symmetry P–P ′ as shown in Figure 8.40. In the case of even mode shown in Figure
8.6 Multilayer Tight Couplers
Figure 8.39
259
(a) A schematic view of a singlesection semireentrant coupler. (b) A cross sectional view of a semireentrant coupled section.
8.40(b), the impedance of floating conductor C (having characteristic impedance of Z 01 ) is in series with the transmission lines A–C and B–C, each having the characteristic impedance of Z 02 . By placing a magnetic wall that bisects the coupler’s cross section, the evenmode characteristic impedance becomes Z 0e = Z 02 + 2Z 01
(8.57)
In the case of the odd mode, Figure 8.40(b), the electric wall passing through the middle of conductor C sets the conductor at ground plane reference, and the oddmode characteristic impedance of the coupled structure is Z 0o = Z 02 Therefore, the coupling coefficient in this case can be expressed as k=
Z 0e − Z 0o Z 01 = Z 0e + Z 0o Z 01 + Z 02
(8.58)
and the termination impedance Z 0 = √Z 0e Z 0o . Equation (8.58) shows that the coupling coefficient, in this case, does not depend upon the spacing between the
260
Tight Couplers
Figure 8.40
(a) Microstrip reentrant mode coupler cross section. (From: [33]. 1990 IEEE. Reprinted with permission.) (b) Even and oddmode excitations.
conductors A and B, but mostly depends upon the Z 02 value, which can easily be controlled by the parameters ⑀ r2 and d. Table 8.4 lists typical design parameters for several 3dB couplers matched approximately to 50⍀. Figure 8.41 shows the measured performance of a reentrant coupler fabricated on 0.381mmthick alumina substrate. The top gold conductors were about 3 to 4 m thick and coupler worked over the 5 to 19GHz range.
Table 8.4 Typical Dimensions for Various ReEntrant Microstrip Couplers Substrate
W 1 (mm)
W 2 (mm)
d (mm)
h (mm)
Z 01 (⍀)
Z 02 (⍀)
Alumina (⑀ r1 = 9.9; ⑀ r2 = 3.7) GaAs (⑀ r1 = 12.9; ⑀ r2 = 6.8)
0.254
0.0685
0.0075
0.381
59.83
18.14
0.075 0.115
0.015 0.015
0.0023 0.0025
0.100 0.150
49.8 49.8
18.00 18.00
GaAs (⑀ r1 = 12.9; ⑀ r2 = 3.7)
0.115
0.015
0.006
0.150
49.8
18.00
8.7 Compact Couplers
Figure 8.41
261
Coupler performance: (a) coupled and direct power, (b) return loss, and (c) phase difference between the coupled and direct ports. (From: [33]. 1990 IEEE. Reprinted with permission.)
8.7 Compact Couplers In cellular wireless microwave applications, quadrature 3dB couplers are required to determine the phase error of a transmitter using the QPSK modulation scheme. The basic requirements for such couplers include small size, low cost, tight amplitude balance, and 90degree phase difference between the coupled and direct ports. At the Lband, the distributed couplers are big in size and also expensive. An equivalent lumpedelement implementation is compact in size and has the potential to be low cost. For example, the size of a monolithic coupler on GaAs substrate has to be of the order of about 1 to 2 mm2 to be cost effective for lowfrequency wireless applications. Several existing coupler configurations have been transformed into new layouts to meet size target values. Some of these new configurations such
262
Tight Couplers
as lumpedelement couplers [37], the spiral directional coupler [38], and meander line couplers [39–41] are described briefly in this section. 8.7.1 LumpedElement Couplers
The coupler shown in Figure 8.42(a) can be modeled as a lumpedelement equivalent circuit as shown in Figure 8.42(b). The values for L, M, C g and C c in terms of Z 0e , Z 0o , and are given as follows [37]: L=
M=
(Z 0e + Z 0o ) sin 4 f o
(Z 0e − Z 0o ) sin 4 f o
Cc =
Cg =
冉
tan ( /2) Z 0e 2 fo
1 1 − Z 0o Z 0e
冊
tan ( /2) 4 f o
(8.59a)
(8.59b)
where fo is the center frequency and = 90 degrees at fo . For a given coupling, using (4.59), the values of Z 0e and Z 0o are determined and then lumpedelement values are calculated using (8.59). The self and mutual inductors are realized using a spiral inductor transformer and the capacitors C g and C c are of MIM type and their partial values are also included in the transformers. 8.7.2 Spiral Directional Couplers
To obtain miniature directional couplers with tight coupling, a coupled structure in a spiral shape (also known as ‘‘spiral coupler’’) is realized. Printing the spiral conductor on high dielectric constant materials further reduces the size of the
Figure 8.42
Twoconductor coupled microstrip line models: (a) distributed elements, and (b) lumped elements.
8.7 Compact Couplers
263
coupler. In this case, tight coupling is achieved by using loosely coupled parallelcoupled microstrip lines placed in a closeproximity spiral configuration. This structure as shown in Figure 8.43 uses two turns and resembles a multiconductor structure. Design details of such couplers and their modifications are given in [38] and are briefly summarized below. An accurate design of such structures, however, is only possible by using EM simulators. As reported in [38], the total length of the coupled line, on the alumina substrate, along its track is 0 /8, where 0 is the freespace wavelength at the center frequency and D ≅ 0 /64 + 4W + 4.5S. Longer lengths result in tighter couplings. Typical line widths and spacings are approximately 500 m and 40 m, respectively, for a 0.635mm alumina (⑀ r = 9.6) substrate. In the spiral configuration, coupling is not a strong function of spacing between the conductors. The conductors were about 5 m thick. Measured coupled power, direct power, return loss, and isolation at 800 MHz for the twoturn spiral coupler, were approximately −3.5 dB, −3.5 dB, 22 dB, and 18 dB, respectively. 8.7.3 Meander Line Directional Coupler
A compact coupler can also be realized by meandered line edgecoupled microstrip lines [39–41] using high dielectric constant substrates. Figure 8.44 shows a meandertype coupler and a spiraltype 90degree coupler for comparison. In these couplers, tight coupling is achieved by placing coupled pair sections in close proximity, which also results in compact size. These couplers are normally designed using EM simulators. The physical dimensions of meander and spiraltype couplers, designed to work over the Sband, are given in Table 8.5, where W, S, and G are the conductor width, separation between the conductors, and gap between the neighboring pair
Figure 8.43
Top conductor layout of a twoturn spiral coupler.
264
Tight Couplers
Figure 8.44
Compact coupledline directional couplers: (a) meander type and (b) spiral type. (From: [41]. 1990 IEEE. Reprinted with permission.)
Table 8.5 Physical Dimensions of Two Compact Couplers Parameters
Meander Type
Spiral Type
W ( m) S ( m) G ( m) Line length (mm) Substrate thickness ( m) Chip size (mm)
30 10 120 6.5 300 1.5 × 1.5
30 20 60 7.5 300 1.5 × 1.5
lines. Measured performance of these couplers is summarized in Table 8.6. The spiraltype configuration provides tighter coupling than the meander type.
8.8 Other Tight Couplers There are several other types of tight couplers described in the literature [42–54]. These include embedded microstrip couplers [42], braided microstrip [43, 44], vertically installed [45, 46], slotcoupled [47, 48], combline [49, 50], finline [51, 52], coplanar waveguide [53], wiggly twoline [54], and dielectric waveguide [55–57] directional couplers. Other couplers include threestrip [58], folded line [59], coupledline hybrid [60], and artificial transmission lines [61]. Monolithic microwave integrated circuit (MMIC), lowtemperature cofired ceramic (LTCC) [62], Sibased monolithic integrated circuit [63], micromachining
8.8 Other Tight Couplers
265
Table 8.6 Measured Electrical Performance of Two Compact Couplers Configuration
Meander Type
Spiral Type
Frequency (GHz) Insertion loss (dB) Amplitude balance (dB) Phase difference (deg) Return loss (dB)
1.8–3.8 4.5–7.5 ± 0.7 93 ± 2 > 15
1.8–3.8 3.8–5.0 ± 0.5 90 ± 2 > 15
techniques [64], and metamaterials [65–67] have stimulated a rapid development of new coupled line directional couplers. Multilayer and multiconductor technologies have resulted in tight and improved directivity couplers. In order to obtain lower loss in Sibased couplers and improve electrical performance at millimeterwave frequencies, micromachining techniques are very suitable. A 20dB micromachined directional coupler with less than 0.5dB insertion loss working over the 10–60GHz frequency range has been reported [64]. Recently, several composite right/lefthanded (CRLH) transmission line structures composed of highpass (lefthanded, LH) and lowpass (righthanded, RH) components cascaded in series have been used to design 3dB couplers [65–67]. Such structures are also known as metamaterials. A major advantage of the CRLHbased couplers is that they can be designed with an arbitrary level of coupling, even up to 0 dB. Because it is not within the scope of this book to go into such detail, readers are referred to the abovementioned references.
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[63] [64] [65]
[66] [67]
Uysal, S., Nonuniform Line Microstrip Directional Couplers and Filters, Norwood, MA: Artech House, 1993. Kim, D. I., ‘‘Directly Connected Image Guide 3dB Couplers with Very Flat Couplings,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT32, June 1984, pp. 621–627. Ikalainen, P. K., and G. L. Matthaei, ‘‘Design of Broadband Dielectric Waveguide 3dB Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT35, July 1987, pp. 621–628. Rodriguez, J., and Prieto, ‘‘WideBand Directional Couplers in Dielectric Waveguide,’’ IEEE Trans. Microwave Theory Tech., Vol, 35, August 1987, pp. 681–686. Sachse, K., and A. Sawicki, ‘‘QuasiIdeal Multilayer Two and ThreeStrip Directional Couplers for Monolithic and Hybrid MICs,’’ IEEE Trans. Microwave Theory Tech., Vol. 47, September 1999, pp. 1873–1882. Sattaluri, R. K., et al., ‘‘Design of Compact Multilevel FoldedLine RF Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. 47, December 1999, pp. 2331–2339. Park, M.J., and B. Lee, ‘‘CoupledLine 180° Hybrid Coupler,’’ Microwave Optical Technology Letts., Vol. 45, April 2005, pp. 173–176. Liu, Z., and R. M. Weikle, ‘‘Hybrid Based on Interdigitally Coupled Asymmetrical Artificial Transmission Lines,’’ IEEE MTTS Int. Microwave Symp. Dig., 2006, pp. 1555–1558. AlTaei, S., P. Lane, and G. Passionpoulos, ‘‘Design of High Directivity Directional Couplers in Multilayer Ceramic Technologies,’’ IEEE MTTS Int. Microwave Symp., 2001, pp. 51–54. Zhu, Y., and H. Wu, ‘‘A 10–40 GHz 7 dB Directional Coupler in Digital CMOS Technology,’’ IEEE MTTS Int. Microwave Symp. Dig., 2006, pp. 1551–1554. Robertson, S. V., et al., ‘‘A 10–60 GHz Micromachined Directional Coupler,’’ IEEE Trans. Mirowave Theory Tech., Vol. 46, November 1998, pp. 1845–1849. Caloz, C., A. Sanada, and T. Itoh, ‘‘A Novel Composite Right/LeftHanded CoupledLine Directional Coupler with Arbitrary Coupling Level and Broad Bandwidth,’’ IEEE Trans. Microwave Theory Tech., Vol. 52, March 2004, pp. 980–992. Caloz, C., and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, New York: WileyIEEE Press, 2005. Nguyen, H. V., and C. Caloz, ‘‘Simple Design and Compact MIM CRLH Microstrip 3dB CoupledLine Coupler,’’ IEEE MTTS Int. Microwave Symp. Dig., 2006, pp. 1733–1736.
CHAPTER 9
CoupledLine Filter Fundamentals 9.1 Introduction Typically, a microwave circuit consists of a number of components, or parts, the functions of which depend on the specific application in mind. Engineering these components for a desired frequency response is often difficult and cost prohibitive, and usually the required frequency response may be obtained by the use of filters. Filters can be fabricated from lumped or distributed elements or a combination of both and can usually be designed for the precise frequency response required, at low cost. Thus, they have been used for a very long time and are popular microwave components, present in virtually every microwave subsystem. The primary parameters of interest in a filter are the frequency range, bandwidth, insertion loss, stopband attenuation and frequencies, input and output impedance, group delay, and transient response. Consider Figure 9.1, where P in is the incident power, P R the power reflected back to the generator, PA the power absorbed by the filter, and P L power transmitted to the load: P in = P R + PA
(9.1)
and if the filter is lossless and there are no reflections, P L = PA and P L = P in . The insertion loss (in decibels) at a particular frequency can be defined as
Figure 9.1
(a) General filter network configuration; and (b) equivalent circuit for powertransfer calculations.
269
270
CoupledLine Filter Fundamentals
IL = −10 log (P L /P in )
(9.2)
while the return loss is given by RL = −10 log (P R /P in ) = −10 log
冋
册
VSWR − 1 2 VSWR + 1
(9.3)
The group delay ( D ), which is a measure of the time taken by a signal to propagate through the filter, is given by 1 d ⌽T 2 df
(9.4a)
T = arg (S 21 )
(9.4b)
D = − where
and S 21 is the transmission coefficient. For no frequency dispersion, the group delay should be constant over the required frequency band. Finally, the transient and steadystate response of a filter may be different. This feature is an important consideration for certain applications. In general, transient effects can be ignored if pulsewidths are no longer than the group delay. 9.1.1 Types of Filters
Filters may be classified in a number of ways. An example of one such classification is reflective versus dissipative. In a reflective filter, signal rejection is achieved by reflecting the incident power, while in a dissipative filter, the rejected signal is dissipated internally in the filter. In practice, reflecting filters are used in most applications. The most conventional description of a filter is by its frequency characteristic such as lowpass, bandpass, bandstop, or highpass. Typical frequency responses for these different types are shown in Figure 9.2. In addition, an ideal filter displays zero insertion loss, constant group delay over the desired passband, and infinite rejection elsewhere. However, in practice, filters deviate from these characteristics and the parameters in the introduction above are a good measure of performance. 9.1.2 Applications
As mentioned above, virtually all microwave receivers, transmitters, and so forth, require filters. Typical commonly used circuits that require filters include mixers, transmitters, multiplexers, and the like. Multiplexers are essential for channelized receivers. System applications of filters include radars, communications, surveillance, ESM receivers, satellite communications (SATCOM), mobile communications, direct broadcast satellite systems, personal communication systems (PCS), and microwave FM multiplexers. In many instances, such as PCS, miniature filters
9.2 Theory and Design of Filters
Figure 9.2
271
Basic filter responses: (a) lowpass, (b) highpass, (c) bandpass, and (d) bandstop.
are a key to realizing the required reduction in size. There is, however, a significant reduction in power handling capacity and an increase in the insertion loss. The former is not a severe limitation in such systems, however, and the latter can be compensated for by subsequent power amplification. In this chapter we constrain ourselves to dealing mostly with coupledline filters. In addition, a small section on computeraided design and synthesis software is included. Finally, because of the importance of filter miniaturization for some applications, we discuss some issues related to this.
9.2 Theory and Design of Filters An ideal bandpass filter with no attenuation or phase shift of the passband frequencies and total attenuation of all outofband frequencies is impractical to realize. In practice, a polynomial transfer function such as Butterworth, Chebyshev, or Bessel is used to model the filter response. A combination of inductors and capacitors, as shown in Figure 9.3, will obviously result in a lowpass filter, and we can develop a prototype normalized to 1⍀ and a 1rad cutoff frequency. From here, it is simply a matter of scaling the g values to obtain the desired frequency response and insertion loss. In addition, other filter types such as highpass, bandpass, and bandstop merely require a transformation in addition to the scaling to obtain the desired characteristics.
272
CoupledLine Filter Fundamentals
Figure 9.3
Lowpass filter prototype.
9.2.1 Maximally Flat or Butterworth Prototype
In the Butterworth lowpass prototype, the insertion loss should be as flat as possible at zero frequency and rise monotonically as fast as possible with increasing frequency. With n as the order of the filter (i.e., the number of reactive elements required to obtain the desired response), f 1 the defined 3dB bandedge point, and f the frequency of interest, the insertion loss is given by IL = 10 log [1 + ( f / f 1 )2n ]
(9.5)
Nomographs, as shown in Figure 9.4, can be used to determine the stopband attenuation versus number of sections for the desired bandwidth. For example, an eightsection filter gives an attenuation of approximately 48 dB at f / f 1 = 2.0 in the stopband, while it results in an attenuation of 0.35 dB at f / f 1 = 0.8 in the passband. The Butterworth prototype values can be calculated from the equations below and are tabulated in Table 9.1 for filters with n = 1 to 10 reactive elements: g0 = 1 g l = 2 sin
冋
册
(2l − 1) , l = 1, 2, . . . , n 2n
(9.6)
g n + 1 = 1 for all n These g values can be scaled for the desired filter input termination resistance R and cutoff frequency 1 = 2 f 1 as L = gR / 1
(9.7)
C = g /( 1 R)
(9.8)
9.2.2 Chebyshev Response
In the Chebyshev response filter, the insertion loss remains less than a specified ripple level A c , up to a specified frequency 1 , and then rises quickly and monotonically with frequency. For an nth order filter, with A c the ripple magnitude (in decibels) and 1 , the bandwidth over which the insertion loss has maximum ripple, the insertion loss is given by
9.2 Theory and Design of Filters
Figure 9.4
273
Nomograph for number of sections of a Butterworth filter for a given insertion loss in the stopband. (From: [1]. 1985 Microwave and RF. Reprinted with permission.)
Table 9.1 Element Values for a Butterworth Filter with g 0 = 1, 1 = 1, and n = 1 to 10 Value of n
g1
g2
g3
g4
g5
g6
g7
g8
g9
g 10
g 11
1 2 3 4 5 6 7 8 9 10
2.000 1.414 1.000 0.7654 0.6180 0.5176 0.4450 0.3902 0.3473 0.3129
1.000 1.414 2.000 1.848 1.618 1.414 1.247 1.111 1.000 0.908
1.000 1.000 1.848 2.000 1.932 1.802 1.663 1.532 1.414
1.000 0.7654 1.618 1.932 2.000 1.962 1.879 1.782
1.000 0.618 1.414 1.802 1.962 2.000 1.975
1.000 0.5176 1.247 1.663 1.879 1.975
1.000 0.445 1.111 1.532 1.782
1.000 0.3902 1.000 1.414
1.000 0.3473 0.908
1.000 0.3129
1.000
IL =
再
10 log [1 + (10A c /10 − 1) cos2 (n cos−1 / 1 )] 10 log [1 + (10
A c /10
2
− 1) cosh (n cosh
−1
/ 1 )]
≤ 1 ≥ 1
(9.9)
As in the case of the Butterworth response, a nomograph as shown in Figure 9.5 can be used to determine the filter characteristics. With the cutoff defined as the ripple value, the lowpass prototype g values are
274
CoupledLine Filter Fundamentals
Figure 9.5
Nomograph for number of sections of a Chebyshev filter for a given ripple and stopband insertion loss. (From: [1]. 1985 Microwave and RF. Reprinted with permission.)
g0 = 1 g1 =
2a 1 ␥
gk =
4a k − 1 a k bk − 1 gk − 1
k = 2, 3, . . . , n
gn + 1 = 1
for n odd
g n + 1 = coth2 (  /4)
for n even
The passband VSWR maximum is related to the ripple level A c by VSWR max = where
1+A 1−A
(9.10)
9.2 Theory and Design of Filters
275
A = 10 [1 − 10−A c /10 ]1/2 and
冋
册 冉 冊 冉 冊 冉 冊 (2k − 1) 2n
k = 1, 2, . . . , n
k b k = ␥ 2 + sin2 n
k = 1, 2, . . . , n
a k = sin
Ac 17.37
 = ln coth ␥ = sinh
(9.11)
 2n
Notice that for n even, the terminating impedances are not equal. The g values are tabulated in Table 9.2 for g 0 = 1, 1 = 1, and n = 1 to 10 for various ripple values. In general, a ripple value in the 0.01 to 0.2dB range is used. 9.2.3 Other ResponseType Filters
Some other responsetype filters are also commonly used, including the elliptic function response, the Bessel response, and the generalized Chebyshev response [1–9]. The elliptic function response is a popular type, and some characteristics of this are discussed next. Here the stopband has a series of peaks and a minimal attenuation level L m . However, no simple equation for the insertion loss is possible. These filters are treated in detail in [2–5], with specific element values for different n values from 3 to 9 given in [7]. These filters can be designed using the coupling matrix approach as discussed in Chapters 10 and 11. This type of filter provides a much steeper stopband skirt for a given n and passband/stopband insertion loss than either the Butterworth or Chebyshev response filters. 9.2.4 LC Filter Transformation
As mentioned before, highpass, bandpass, or bandstop filters require a transformation in addition to scaling, and these transformations are discussed below. For lowpass filters, scaling to the desired frequency band and impedance level is accomplished by using the equations given next for the series inductors and shunt capacitors. In this case
/ 1 = / Lpb L k = g k (Z 0 / Lpb )
(9.12)
C k = g k (1/ Lpb Z 0 )
(9.13)
276
CoupledLine Filter Fundamentals
Table 9.2 Element Values for a Chebyshev Lowpass Prototype with g 0 = 1, 1 = 1, and n = 1 to 10 for Different Ripple Values Value of n g1
g2
1 2 3 4 5 6 7 8 9 10
1.0000 0.4077 0.9702 1.2003 1.3049 1.3600 1.3924 1.4130 1.4270 1.4369
g3
g4
g5
g6
g7
g8
g9
g 10
g 11
1.1007 0.7969 1.5554 1.8043 1.9055
1.0000 0.7333 1.4270 1.6527
1.1007 0.8144 1.5817
1.0000 0.7446
1.1007
1.3554 1.1811 1.9444 2.1345 2.2026
1.000 0.8778 1.4425 1.5821
1.3554 1.1956 1.9628
1.000 0.8853
1.3554
1.5386 1.3722 2.1349 2.3093 2.3720
1.0000 0.8972 1.3938 1.5066
1.5386 1.3860 2.1514
1.0000 0.9034
1.5386
1.9841 1.7372 2.5093 2.6678 2.7231
1.0000 0.8796 1.2690 1.3485
1.9841 1.7504 2.5239
1.0000 0.8842
1.9841
0.01dB ripple 0.0960 0.4488 0.6291 0.7128 0.7563 0.7813 0.7969 0.8072 0.8144 0.8196
1.1007 0.6291 1.3212 1.5773 1.6896 1.7481 1.7824 1.8043 1.8192
1.0000 0.6476 1.3049 1.5350 1.6331 1.6833 1.7125 1.7311
1.1007 0.7563 1.4970 1.7481 1.8529 1.9057 1.9362
1.0000 0.7098 1.3924 1.6193 1.7125 1.7590
0.1dB ripple 1 2 3 4 5 6 7 8 9 10
0.3052 0.8430 1.0315 1.1088 1.1468 1.1681 1.1811 1.1897 1.1956 1.1999
1.0000 0.6220 1.1474 1.3061 1.3712 1.4039 1.4228 1.4346 1.4425 1.4481
1.3554 1.0315 1.7703 1.9750 2.0562 2.0966 2.1199 2.1345 2.1444
1.0000 0.8180 1.3712 1.5170 1.5733 1.6010 1.6167 1.6265
1.3554 1.1468 1.9029 2.0966 2.1699 2.2053 2.2253
1.0000 0.8618 1.4228 1.5640 1.6167 1.6418
0.2dB ripple 1 2 3 4 5 6 7 8 9 10
0.4342 1.0378 1.2275 1.3028 1.3394 1.3598 1.3722 1.3804 1.3860 1.3901
1.0000 0.6745 1.1525 1.2844 1.3370 1.3632 1.3781 1.3875 1.3938 1.3983
1 2 3 4 5 6 7 8 9 10
0.6986 1.4029 1.5963 1.6703 1.7058 1.7254 1.7372 1.7451 1.7504 1.7543
1.0000 0.7071 1.0967 1.1926 1.2296 1.2479 1.2583 1.2647 1.2690 1.2721
1.5386 1.2275 1.9761 2.1660 2.2394 2.2756 2.2963 2.3093 2.3181
1.0000 0.8468 1.3370 1.4555 1.5001 1.5217 1.5340 1.5417
1.5386 1.3394 2.0974 2.2756 2.3413 2.3728 2.3904
1.0000 0.8838 1.3781 1.4925 1.5340 1.5536
0.5dB ripple 1.9841 1.5963 2.3661 2.5408 2.6064 2.6381 2.6564 2.6678 2.6754
1.0000 0.8419 1.2296 1.3137 1.3444 1.3590 1.3673 1.3725
1.9841 1.7058 2.4758 2.6381 2.6964 2.7239 2.7392
1.0000 0.8696 1.2583 1.3389 1.3673 1.3806
where Lpb is the required lowpass cutoff radian frequency and Z 0 is the input and output termination impedance. Highpass Transformation
Transposing the series inductances into series capacitances and the shunt capacitances into shunt inductances transforms the lowpass prototype of Figure 9.3 into a highpass filter (see Figure 9.6). Thus,
9.2 Theory and Design of Filters
Figure 9.6
277
Highpass filter schematic and typical frequency response.
Hpb =− 1
(9.14)
where Hpb is the bandedge frequency. The element values obtained are Ck =
1
(9.15a)
g k Hpb Z 0
Lk =
Z0 g k Hpb
(9.15b)
Bandpass Transformation
Bandpass filters also require transformation and scaling. Series inductors of the lowpass prototype are transformed into a series combination of an inductor and a capacitor, while the shunt capacitors are transformed into a parallel combination of an inductor and a capacitor (see Figure 9.7). Hence the bandpass filter has twice the number of elements. With a lower cutoff frequency f l and upper cutoff frequency fu defined for the bandpass, the center frequency fo , bandwidth BW, and fractional bandwidth FBW are defined by f0 =
√ f ᐉ fu
(9.16)
BW = fu − f ᐉ
(9.17)
FBW = BW / f 0
(9.18)
A typical filter structure that results is shown in Figure 9.7. In this case
f = 0 1 BW and the element values obtained are
冉
f f − 0 f0 f
冊
(9.19)
278
CoupledLine Filter Fundamentals
Figure 9.7
Bandpass filter structure with typical frequency response.
Z0 2 BW L k, SR = g k (Series elements) ; C k, SR = 2 2 BW gk Z0 0 L k, SH =
2 BWZ 0 2 gk 0
gk ; C k, SH = (Shunt elements) 2 BWZ 0
(9.20)
(9.21)
where
02 =
1 1 = L k, SR C k, SR L k, SH C k, SH
(9.22)
Bandstop Transformation
Here, the shunt capacitor of the lowpass prototype is transformed into a series inductor and capacitor in shunt to ground and the series inductor is replaced with a parallel inductorcapacitor in series as shown in Figure 9.8. In this case
9.2 Theory and Design of Filters
Figure 9.8
279
Typical bandstop filter structure and its frequency response.
1 f = 0 BW
冉
f f − 0 f0 f
冊
(9.23)
with paralleltuned circuit element values
0 C k, SR =
0 1 = 0 L k, SR 2 BWZ 0 g k
(9.24a)
and seriestuned circuit element values
0 L k, SH =
0 Z0 1 = 0 C k, SH 2 BWg k
(9.24b)
9.2.5 Filter Analysis and CAD Methods Filter Analysis
While the earlier discussion has covered some analytical aspects of various types of filters, in general, to account for phase characteristics and finite Q of circuit elements, we resort to the use of either ABCD matrices or Kirchhoff’s equations. The
280
CoupledLine Filter Fundamentals
ABCD matrix method is limited to ladder networks, while Kirchhoff’s equations can be applied in general to any network. Both of these techniques are well covered in [10]. Knowing the ABCD matrices for a variety of elements as given in Chapter 2, we can obtain the overall matrix of the circuit by simple matrix multiplication, taking care to perform the multiplication in the right order. This process is considerably simplified by the use of computers. The ABCD matrix method, however, cannot handle reentrant combinations (i.e., nonladder networks) such as are present in highperformance bandpass filters. This limitation is readily overcome through the use of Kirchhoff’s equations. ComputerAssisted Design
Computers can be used effectively to simplify and speed up the design of filters, and in some cases are the only means to render practical the synthesis of filter transfer approximations. Software packages are available to simulate performance before prototype construction, thus permitting fine tuning of parameters for optimization, taking into account practical realities and fabrication constraints. Some of these packages include LINMIC + [11], which uses a fullwave field solution, and planarfield simulators such as Sonnet em, IE3D, and SFPMIC [12–15]. These codes can deal with multiple dielectric layers and multiple conductors and are excellent for design verification. For threedimensional circuits, other packages, such as the HP HighFrequency Structure Simulator (HFSS) [16], which handles multiport structures with unrestricted dielectric and conductor geometry, can be used. In general, we commence the design process by selecting the appropriate circuit construction, modeling the filter response against the physical parameters, and then using accurate models for the structure to determine the response and optimize parameters for the application at hand. Most of these aspects can be performed iteratively on a computer using an optimization routine. One excellent software package for performing the above operation is the Genesys software suite from Eagleware Corporation [17]. The tools are available in this package for the design of a wide set of LC filter topologies including conventional, narrowband, flatdelay, symmetric, elliptic, zigzag, bandpass, lowpass, bandstop, and highpass structures. It also supports a large number of prototype transfer functions including the most commonly used, such as Butterworth, Chebyshev, and Bessel. The advantage of this package is that there are other files included that permit the design of an overall circuit network including the filter, oscillators, matching networks, and equalizers. The filters can be designed in the media of interest here (e.g., stripline, microstrip) in most configurations such as hairpin, interdigital and combline. 9.2.6 Some Practical Considerations
In the previous sections we started with the lumpedelement filters. From this we can move into distributedline filters and from there to coupledline filters. The distributed lines can be realized in any desired form such as waveguide, coaxial lines, or planar transmission lines. Kuroda’s identities [18, 19] allow one to realize
9.2 Theory and Design of Filters
281
lowpass structures using shunt elements with identical response and with Richards transformations [19, 20] one can establish the distributed line parameters. Table 9.3 provides a listing of transmission line lengths with the equivalent RLC network, which can simplify design considerably. However, as the passband for distributed filters can reoccur at frequencies of twice or thrice the initial passband frequency, stopband attenuation is severely compromised. In addition, discontinuities such as open ends, steps in linewidth, and T and crossjunction effects can degrade filter performance. A number of practical considerations and limitations often determine the actual filter construction. For example, for satellite systems applications, naturally, size, weight, and the like are major considerations. In other cases, such as for PCS, while size and weight are again major factors, cost is critical to maintaining overall system costs at acceptable levels. Other aspects, such as finite Q , group delay, temperature effects, powerhandling capacity, and tunability may also be important factors in determining filter configuration and design [10]. For low selectivity, wideband applications, striplines, and microstrip filters are ideal. They do suffer from temperature effects, however, and are difficult to tune. Suspendedsubstrate filters give a higher Q over microstrip, resulting in lower filter loss, sharper band edges, and better temperature stability. Dielectric resonator filters have been developed over a wide range of frequencies, and high dielectric constant materials (e.g., barium tetratitanate) can be used to decrease the size. When one considers the losses in filters (e.g., finite Q of resonators) the passband insertion loss is increased or, as in the case of equalripple response filters, ripples are suppressed. The loaded Q of the resonators normally determines the bandwidth of the filter. The loaded Q in turn depends on the losses and the external circuit. When the Q of the external circuit is much less than the unloaded Q of the filter, the filter bandwidth is almost independent of the unloaded Q but, as both these Q values become comparable, the circuit becomes lossy and the selectivity is degraded, insertion loss is increased, and stop band rejection is reduced. Printed circuit filters can handle a few hundred watts of power. The specific powerhandling capability depends on the filter topology and will depend on the transmission media used. In general, the type of media discussed in this text (i.e., stripline, microstrip) are good for lowpower levels but can handle highpulse power levels (up to a few kW) and approximately 50W average power levels. Environmental temperature changes result in changes to the physical dimensions of the filter structure and therefore changes to the electrical characteristics. These effects are analyzed and discussed in [10]. Finally, the group delay of a filter depends on the selected prototype and number of sections with both band center group delay and group delay variation increasing with an increase in the number of sections. In many applications, a constant group delay over the desired bandwidth is required. The design of these filters is discussed in the literature [21]. There are different types of coupled filters. For example, directcoupled resonator filters tend to have excessive lengths and generally can be reduced significantly in size by the use of parallelcoupled lines. Since parallel coupling results in much tighter coupling than with the use of endcoupled structures, greater bandwidths are possible. The first spurious response occurs at two or three times the center
282
CoupledLine Filter Fundamentals Table 9.3 Equivalent RLC Networks for Transmission Line Lengths
9.3 ParallelCoupled Line Filters
283
frequency, depending upon the media, with a larger gap being permitted between adjacent strips. A broader bandwidth is also obtained for a given gap tolerance. For compactness, resonator sections are placed sidebyside. Some commonly used designs using this concept include the parallelcoupled, interdigital, combline and hairpin line filters, and these configurations are shown in Figure 1.15 (Chapter 1). These parallelcoupled line filters will be discussed in greater detail below. The tapped inputoutput structures shown in the figure have a relatively narrow bandwidth.
9.3 ParallelCoupled Line Filters The design of parallelcoupled line filters was formulated by Cohn [22] and has been refined or modified by others for specific design situations or operating conditions [23–29]. For a typical n section parallelcoupled filter, one starts with computing the g 1 , g 2 , . . . , g n + 1 values of the lowpass prototype for either maximally flat or equalripple response, using (9.6) or (9.10). As shown in Figure 9.9(a), the filter is assembled from n + 1 sections of equal length ( /4) at the center frequency, giving a structure of n resonators. The electrical design is specified by the even and oddmode characteristic impedances Z 0e , Z 0o on the parallel conductors. Using these impedance values, the transmission line dimensions of each section can be determined. A filter section and its equivalent are shown in Figures 9.9(b) and 9.9(c), respectively. The ABCD matrix of the ideal impedance inverter can be obtained by substituting = −90 degrees and Z 0 = K in the ABCD matrix of a transmission line of electrical length and characteristic impedance Z 0 . Therefore, the ABCD matrix of the filter section is
冋 册 冤 A C
B D
=
cos j sin Z0
jZ 0 sin cos
冥冤
0 j − K
−jK 0
冥冤
cos j sin Z0
jZ 0 sin cos
冥 (9.25)
The K inverters for the various sections are defined in terms of the g 0 , g 1 , . . . elements as follows [22]: Z0 = K 01
√
⌬f 2 1 g 0 g 1
Z0 ⌬f = , j = 1 to n − 1 K j, j + 1 2 1 √g j g j + 1 Z0 = K n, n + 1
√
⌬f 2 1 g n g n + 1
(9.26a)
(9.26b)
(9.26c)
where ⌬ f is the fractional bandwidth (FBW), and 1 has been defined previously. After calculating K, the even and oddmode impedances of the coupled lines are calculated from the following relations:
284
CoupledLine Filter Fundamentals
Figure 9.9
(a) Parallelcoupled transmission line resonator filter; (b) parallelcoupled line resonator; (c) K invertertype equivalent circuit of the coupled line resonator; and (d) resonators with end effects.
冉
冊
(9.27)
冉
冊
(9.28)
(Z 0e )j + 1 Z0 2 Z0 =1+ + , j = 0 to n Z0 K j, j + 1 K j, j + 1 (Z 0o )j + 1 Z0 2 Z0 =1− + , j = 0 to n Z0 K j, j + 1 K j, j + 1
The physical dimensions of the filter sections are then calculated [30] for the desired Z 0e and Z 0o . An approximate value for the physical length is obtained from the average value of the even and oddmode velocities, that is:
=
2 2 l= 0
√⑀ ree + √⑀ reo l = 2
2
(9.29)
9.3 ParallelCoupled Line Filters
285
A more accurate value for the length of the resonator is obtained by taking into account the openend discontinuity capacitance [30], which gives rise to an additional length ⌬l as shown in Figure 9.9(d). In this case, the line length calculated from (9.29) is shortened at each end by ⌬l : ⌬l = 0.6C o Z 01 / 冠√⑀ ree +
√⑀ reo 冡
(9.30)
where ⌬l is in millimeters, the openend capacitance C o in picofarads [30], and Z 01 = √Z 0e Z 0o . 9.3.1 Design Example
Consider a design of threepole (n = 3) parallelcoupled microstrip line bandpass filter that has a fractional bandwidth FBW = 0.15 at a center frequency f 0 = 2 GHz. A Chebyshev prototype with a 0.1dB ripple is chosen for the design. From Table 9.2, the element values for the lowpass prototype can be found, which are g 0 = g 4 = 1.0, g 1 = g 3 = 1.0315, and g 2 = 1.1474. Using the design equations (9.26) to (9.28) with terminal impedance Z 0 = 50 ohms, the pairs of even and oddmode impedances of coupled sections for the filter design are calculated: Z 0e1 = 85.3181 ohms Z 0o1 = 37.5243 ohms Z 0e2 = 63.1744 ohms Z 0o2 = 41.5163 ohms Since the filter is symmetrical, only these two pairs of design parameters need to be considered. The next step of the filter design is to find the dimensions of coupled microstrip lines whose even and oddmode impedances match to those above calculated. For example, referring to Figure 9.9(a), W 1 and S 1 are so determined that the resultant even and oddmode impedances match to Z 0e1 and Z 0o1 given above. Assume that the microstrip filter is constructed on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. The final layout of the filter designed on the substrate is illustrated in Figure 9.10(a). The physical length of each parallelcoupled line section was mainly determined based on (9.29) for an electrical length of 90 degrees with some adjustments to take into account the effects of openend and impedancestep discontinuities. The filter performance, obtained by fullwave electromagnetic (EM) simulation, is presented in Figure 9.10(b), showing a desired passband from 1.85 to 2.15 GHz, which is equivalent to a 15% fractional bandwidth centered at 2 GHz. Figure 9.10(c) is the wideband response of the filter, showing a spurious passband at approximately 2f 0 . The issue of suppressing spurious responses of parallelcoupled line filters will be addressed in Chapter 10. In parallelcoupled microstrip filters, the physical lengths of the coupled sections are the same for both the even and odd modes, and we obtain an asymmetrical
286
CoupledLine Filter Fundamentals
Figure 9.10
(a) Threepole parallelcoupled microstrip line filter on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) EMsimulated passband performance. (c) Wideband response.
9.4 Interdigital Filters
287
passband response with deterioration of the upper stopband from differences in phase velocities in these two modes. To improve the stopband performance, the phase velocities of the two modes should be equalized. Various techniques can be used to do this, including overcoupling the resonators, suspending the substrate, using parallelcoupled stepped impedance resonators or using capacitors at the end of the coupled sections [29]. We can also introduce wiggles in the coupled lines to accomplish this [26]. Some of these techniques are described in Chapter 6. Similarly, we can use other methods such as halfwavelength broadsidecoupled microstrip lines to enhance bandwidth [23]. These techniques are useful for MIC and MMIC applications.
9.4 Interdigital Filters The interdigital filter is popular because it is compact and uses the available space efficiently. It can be designed for both narrow and wide (30–70%) bandwidths. A typical interdigital bandpass structure is shown in Figure 1.15(b). Accurate design procedures are available for these filters [31] together with an exact design theory [32]. This filter is not as compact as the combline filter [33] but has higher unloaded Q , thus making it a good choice for narrowbandwidth filter applications. For large bandwidths, the capacitivelyloaded parallelcoupled line filter is ideal. This also reduces the filter size. In addition, the tolerances required in their manufacture can be relaxed, spurious responses are not present, the rates of cutoff and strength of the stopbands can be increased by multiple poles of attenuation at dc and at even multiples of the center frequency of the first passband, and these filters can be fabricated without dielectrics, thereby eliminating dielectric loss. The design equations for parallelcoupled line interdigital filters are given in [31, 32] and will not be repeated here. Essentially these equations yield the various line capacitances per unit length of the line and these can be used to determine the physical dimensions of the line. Alternately, narrowtomoderatebandwidth bandpass filters using resonators can be designed by calculating or measuring the coupling coefficient between resonators and the external quality factor of the input and output resonators [10]. These coupling values are then related to a normalized lowpass prototype value and can be used to realize all possible response shapes. This procedure is the most practical design method when the filter structure is complex or its equivalent circuit model is not readily available. The procedure described below is applicable to all types of coupled resonator filters, whether realized in microstrip or in any other medium. 9.4.1 Design Examples Narrowband Design
The first step in the design procedure is finding the necessary normalized coupling coefficients in terms of lowpass prototype element values and design frequencies, as follows: k n, n + 1 =
BW f 0 √g n g n + 1
(9.31)
288
CoupledLine Filter Fundamentals
The next step is to determine physical dimensions of the coupled resonators, depending on the transmission medium used. The final step is to determine the loaded Q of the first and the last resonators required to connect the coupled resonators to input and output terminals. In a filter design, the singlyloaded Q is calculated from QL =
f0 f g = 0 g BW 1 BW n + 1
(9.32)
To illustrate the abovedescribed design method, the following microstrip example is considered: Center frequency Response Bandwidth 35dB attenuation points Substrate
4 GHz Chebyshev with 0.2dB ripple 0.4 GHz 3.6 and 4.4 GHz ⑀ r = 9.8, h = 1.27 mm
From the nomograph (Figure 9.5), we find that the required number of resonators is 5. The prototype values are g 0 = 1.0, g 1 = g 5 = 1.3394, g 2 = g 4 = 1.337, and g 3 = 2.166. From (9.31) the coupling parameters are determined to be k 12 = k 45 = 0.0747, k 23 = k 34 = 0.0588, and from (9.32), Q L = 13.4. The filter can be realized using coupled microstrip medium on RT/duroid substrate or alumina substrate. Figure 9.11 shows [25] measured coupling coefficients as a function of S/h for ⑀ r = 2.22 and W/h = 1.8, which corresponds to a singlestrip impedance Z 0I of approximately 70⍀, and for alumina substrate (⑀ r = 9.8) and W/h = 0.7 (Z 0I = 58⍀). Similar curves can be obtained for other line widths and dielectric substrates. In these filters, the final step in the design is to determine the tappoint location for a given Q L . The tap point ᐉ /L [Figure 1.15(b)] can be calculated from the following equation [25]: QL = Z 0 /Z 0I 4 sin2 ( ᐉ /2L)
(9.33)
Thus, the filter dimensions in Figure 1.15(b) on a 1.27mmthick alumina substate become h = 1.27 mm W 0 = 1.27 mm (Z 0 = 50⍀) W = 0.889 mm S 12 = 2.16 mm = S 45 S 23 = 2.54 mm = S 34 ᐉ = 2.25 mm L = 7.5 mm
9.4 Interdigital Filters
Figure 9.11
289
Measured coupling coefficients versus S/h for (a) RT/duroid and (b) alumina. (From: [25]. 1979 IEEE. Reprinted with permission.)
where S 12 and S 23 are the spacings between lines 1 and 2 and lines 2 and 3, respectively. Wideband Design
In this example, a microstrip interdigital filter is designed with a center frequency f 0 = 1.45 GHz and a fractional bandwidth of 0.333. The filter is implemented on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. The design example is based on a ninepole Chebyshev lowpass prototype with a return loss of −20 dB (or a ripple of 0.04321 dB). The lowpass element values, which can be obtained from the formulas given in (9.10) and (9.11), are
290
CoupledLine Filter Fundamentals
g 0 = g 10 = 1, g 1 = g 9 = 1.0235, g 2 = g 8 = 1.4619, g 3 = g 7 = 1.9837, g 4 = g 6 = 1.6778, and g 5 = 2.0649. Using the design equations available in [34], the odd and evenmode impedances are determined, which are given in Table 9.4, where the k values are calculated from k i, i + 1 =
Z 0ei, i + 1 − Z 0oi, i + 1 Z 0ei, i + 1 + Z 0oi, i + 1
However, these pairs of even and oddmode impedances generally cannot be matched for fixed line width resonators as required in the case of symmetrical coupled line designs. For this design example, all interdigital line resonators have the same width of 2 mm on the substrate. Therefore, the alternative approach is to determine the coupling coefficients k defined earlier. To this end, the different set of even and odd mode impedance values for the fixed width resonators are determined, which are aimed to match the coupling coefficients or k values given in the Table 9.4. In this way, all the spacings between adjacent coupled resonators can be found. The location of tappedlines at input and output (I/O) resonators are also readily determined based on a procedure documented in [34]. Figure 9.12(a) shows the layout of the designed filter, where all the physical dimensions are in millimeters. Each of the resonators is about a quarterwavelength long at the center frequency f 0 . The I/O resonators are slightly longer to compensate for resonant frequency shift due to the effect of the tapped input and output. The filter was fabricated on an RT/Duroid 6006 substrate and the measured passband performance is plotted in Figure 9.12(b). The measured center frequency is lower than the designed one, which is mainly due to the tolerances in the substrate and manufacturing process. The wideband frequency response of the filter is demonstrated in Figure 9.12(c), which clearly shows that the first spurious response appears at about 3f 0 because of the use of the quarterwavelength resonators.
9.5 Combline Filters A typical combline filter configuration is shown in Figure 1.15(c). Together with the capacitively loaded interdigital filter, the combline filter is one of the most commonly used bandpass structures. This filter is more compact than the interdigital, is generally easier to fabricate, and is an attractive alternative to other filter types, especially for narrow bandwidths. Here the resonators consist of TEM mode transmission line elements that are shortcircuited at one end and have a lumped
Table 9.4 Calculated Odd and EvenMode Impedance and Corresponding k Value i
Z 0ei, i + 1
Z 0oi, i + 1
k i, i + 1
1 2 3 4
60.399 56.154 55.492 55.310
39.303 41.337 41.703 41.806
0.212 0.152 0.142 0.139
9.5 Combline Filters
Figure 9.12
291
(a) The layout of the designed ninepole microstrip interdigital filter on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. (b) Measured passband performance of the ninepole microstrip interdigital filter. (c) Wideband response of ninepole microstrip interdigital filter.
capacitance between the other end of each resonator line element and ground. Without these capacitances, the resonator lines would be exactly /4 at resonance and the structure would have no passband. Hence the lines must be less than 90 degrees long and capacitively loaded to achieve resonance. The capacitances are made relatively large and the lines 45 degrees or less in length, thereby resulting in a very compact and efficient coupling structure. With the line elements /8 at the primary passband, the second passband is far removed and the attenuation is infinite at a frequency where the line length is /4. So beyond the passband, the attenuation is very high and cutoff at the upper side of the passband can be very steep. Further, adequate coupling can be maintained between resonator elements with sizeable spacing, thereby providing more margin in tolerance requirements. Design procedures for combline filters are discussed in [33, 35, 36]. The process is parallel to the design process of the interdigital filter discussed above and therefore is not covered in any detail here. Refer to [33, 35, 36] for the actual design equations to calculate the distributed line capacitances and from there the dimensions of the structure. 9.5.1 Design Example
Coaxial combline filters [Figure 1.14(b)] using ceramic block are commonly used in applications from 200 to 3,000 MHz. These filters have typically a 2dB insertion
292
CoupledLine Filter Fundamentals
Figure 9.12
(continued).
loss and have bandwidth from 1% to 20%. Ceramic filters are temperaturestable, and their temperature range of operation is normally from −30°C to +85°C. They are surfacemountable, and in high volume their cost is $2 to $5. Figure 9.13 shows a schematic of a threeresonator combline ceramicblock filter. The ceramic materials have highdielectric constant (i.e., ⑀ r ≅ 40 − 80). The coupling between pairs of adjacent resonators is realized by a circular or rectangular air hole. The inhomogeneous interface between the highdielectic constant ceramic and air hole gives rise to different phase velocities for the even and oddmodes of the coupled lines. This difference provides the required couplings between the resonators to realize a filter. The design of such filters is straightforward but requires numerical methods, such as EM simulators, to determine coupling between the resonators. Normally, filters are designed empirically and tuned after fabrication
9.5 Combline Filters
Figure 9.13
293
High⑀ r ceramic block combline bandpass filter. All the surfaces are metallized except the top surface. A, B, and C are metallized coaxial resonators. Metallized sidewalls of the ceramic block act as outer conductors.
using ceramic grinders and metal scrapers. Analysis, design, and test results for various ceramicblock filters have been discussed by many authors [37–42]. In this section, we describe briefly the design of such filters. In Figure 9.13, A, B, and C are the metallized center conductors of the coaxial resonators. All resonators are shortcircuited at the bottom and opencircuited at the top and are designed to be /4 long at the operating center frequency [38]. Resonators A, B, and C are coupled to each other for filter action through air holes between them. The first and last resonators are coupled to input and output ports, respectively, by coupling pads P 1 and P 2 located near them. The capacitive coupling between the filter and input and output is usually accomplished by a cutandtry method. Figure 9.14 shows a lumpedelement equivalentcircuit for this filter; /4 resonators are represented by parallel resonant circuits (C i , L i ). Air holes provide magnetic (inductive) coupling (L ij ), and the filter is connected to the input and output (usually 50⍀) through capacitive coupling represented by C in and C out . At the center frequency fo , the length of each resonator is given by L = 0.25 = 0.25
0 0.25c = √⑀ re f 0 √⑀ re
(9.34)
where 0 , c, and ⑀ re are the freespace wavelength, velocity of light, and the effective dielectric constant, ⑀ re , is obtained from
⑀ re =
⑀ ree + ⑀ reo 2
(9.35)
294
CoupledLine Filter Fundamentals
Figure 9.14
Lumpedelement equivalent circuit for the threeresonator filter.
where ⑀ ree and ⑀ reo are, respectively, the even and oddmode effective dielectric constants of the medium in which the coaxial resonators are embedded. The coupling coefficient is given by [39] k=
2冠√⑀ ree −
√⑀ ree
√⑀ reo 冡 + √⑀ reo
(9.36)
Figure 9.15 shows a crosssectional view of an air hole coupledline structure with dimensions, and Figure 9.16 shows the calculated values of the even and oddmode effective dielectric constants. Finite difference method [39] was used to analyze the structure, with ⑀ r = 80, D = 2.4 mm, H = 6 mm, and S = 0.8 mm. Figures 9.17(a) and 9.17(b) show the coupling coefficient versus air hole radius, and separation between resonator and air hole, respectively. More extensive data for the coupling coefficient has been published by Yao et al. [41, 42].
Figure 9.15
Crosssectional view of the threeresonator ceramicblock combline filter.
9.6 The HairpinLine Filter
Figure 9.16
295
Effective dielectric constant for even and odd modes as a function of air hole radius. ⑀ r = 80, D = 2.4 mm, H = 6 mm, and S = 0.8 mm.
To illustrate a design example of a bandpass ceramicblock filter, the following specifications are chosen: Center frequency Response Bandwidth Number of resonators
900 MHz Chebyshev with 0.05dB ripple 30 MHz 3
This filter can be designed by using normalized coupling coefficients in terms of lowpass prototype element values and design frequencies as given in (9.31) and (9.32). The normalized bandwidth, BW/ f 0 = 30/900 = 1/30. From Table 9.2, the lowpass prototype element values are g 0 = g 4 = 1, g 1 = g 3 = 0.8794, and g 2 = 1.1132. From (9.31), the coupling coefficients k 12 = k 23 = 0.0339. For the structure shown in Figure 9.15, and using data in Figure 9.17, the filter design parameters are ⑀ r = 80, H = 6 mm, D = 1.2 mm, R = 1.2 mm, and S = 0.7 mm. Figure 9.18 shows the measured frequency response of the threeresonator ceramicblock filter where the input and output couplings were obtained by experiments. In the 50MHz bandwidth, the measured insertion loss was better than 1.5 dB. The ceramic material’s loss tangent (tan ␦ ) was 2.5 × 10−4 and the temperature coefficient was 3 ppm/°C.
9.6 The HairpinLine Filter The interdigital and combline filters described above require ground connections, which may be difficult to achieve when using microstrip lines on ceramic substrates.
296
CoupledLine Filter Fundamentals
Figure 9.17
(a) Coupling coefficient as a function of air hole radius. ⑀ r = 80, D = 2.4 mm, H = 6 mm, and S = 0.8 mm. (b) Coupling coefficient as a function of separation between the metallized resonator and air hole. ⑀ r = 80, D = 2.4 mm, H = 6 mm, and R = 1.2 mm.
When stripline or microstrip is used, the hairpinline filter is one of the preferred configurations. This is particularly useful when one is interested in MIC or MMIC circuits. The hairpinline filter can be considered basically to be a folded version of a halfwave parallel coupledline filter. It is much more compact, though, and gives approximately the same performance. A typical schematic of this type of filter is shown in Figure 1.15(d). As the frequency increases, the lengthtowidth ratio is smaller for a given substrate thickness, so that folding the resonator becomes
9.6 The HairpinLine Filter
Figure 9.18
297
Measured frequency response of the threeresonator combline filter.
impractical. In general, the hairpinline filter is larger than the combline or interdigital filter. But because no grounding is required, it is amenable to mass production as a large number of filters can be simultaneously printed on a single substrate, thereby lowering production costs. The design of hairpinline filters has been discussed by various researchers [43–47]. Although one could normally use the design expressions for parallelcoupled filters, here, the line between the resonators decreases the length of the coupling sections, so that the coupled sections are less than a quarterwavelength. In addition, the bend discontinuities are difficult to handle. Using the parallelcoupled bandpass design and compensating for the end capacitance, however, dissimilar propagation velocities, shortened coupling elements, and bends through an optimization routine on a computer permits rapid design of such a filter. We can use the methods given above for the design of hairpinline filters, so details are not provided here. The interarm spacings in the hairpinline filter design should be kept large, so that coupling (≥ 15 dB) between the arms can be neglected. However, this leads to a larger filter size. Cristal and Frankel’s [43] unified design method takes the interarm coupling into account, but assumes negligible phase shift over the line joining the arms of a hairpin. Thus, we must exercise caution when highdielectric constant substrates are used, as small physical length variations will result in large phase shifts. Cristal and Frankel’s equations are also not applicable for inhomogeneous media and do not account for the effects of rightangled bends and corners. A good comparison of several types of hairpin filters is given by Matthaei [48]. 9.6.1 Design Example
Hairpinline filters can also be designed by calculating the interresonator coupling as a function of the spacing between resonators for a given set of substrate parameters, frequency, and microstrip width. The microstrip width is selected to obtain maximum Q for the resonators. Figure 9.19 shows measured coupling coefficients for hairpin resonators on alumina and high dielectric constant substrates. In these filters, the tap point (ᐉ /L) is calculated using the following relation [25]:
298
CoupledLine Filter Fundamentals
Figure 9.19
(a) Measured coupling coefficient for hairpinline resonators, ⑀ r = 9.8, h = 1.27, W/h = 0.7. (From: [25]. 1979 IEEE. Reprinted with permission.) (b) Computed and measured coupling for high dielectric constant resonators. ⑀ r = 80, h = 2 mm, f 0 = 905 MHz; and (c) ⑀ r = 90, h = 1 mm, f 0 = 854 MHz. (From: [46]. 1994 International Journal of Microwave and MillimeterWave ComputerAided Engineering. Reprinted with permission.)
9.6 The HairpinLine Filter
299
QL = 2 (Z 0 /Z 0I ) 2 sin ( ᐉ /2L)
(9.37)
The physical layout of a lowfrequency hairpinline filter designed using microstrip on a highdielectric constant material is shown in Figure 9.20, while Figure 9.21 shows the measured data. The filter was designed having fivepole Chebyshev
Figure 9.20
Hairpinline filter layout. (From: [46]. 1994 International Journal of Microwave and MillimeterWave ComputerAided Engineering. Reprinted with permission.)
Figure 9.21
Measured response of a 905MHz hairpinline filter. (From: [46]. 1994 International Journal of Microwave and MillimeterWave ComputerAided Engineering. Reprinted with permission.)
300
CoupledLine Filter Fundamentals
response centered at 905 MHz with 46MHz bandwidth and 20dB return loss. The substrate thickness and the dielectric constant are 2 mm and 80, respectively. The substrate material is a solid mixture of barium titrate and barium zirconate. For regular hairpinline filters, we can use Eagleware’s M/FILTER software [47] for quick results. We start with a filter topology such as edgecoupled, hairpin, interdigital or stepped Z, together with selecting the frequency response, such as Butterworth, Chebyshev, or Bessel. The transmission line format is selected among microstrip, stripline, and so on, together with the performance parameters such as lower cutoff frequency, upper cutoff frequency, and passband ripple. Finally, the substrate parameters are entered, including dielectric constant and thickness. Entering these, we obtain a layout of the filter, while the frequency response can be evaluated using Eagleware’s Superstar software and the output file of M/FILTER. Should further optimization be required, it can be accomplished by an iterative process of reloading the final values into the filter program to arrive at the final dimensions of the circuit. The output plot file can be used to create another complete file to drive a numerically controlled milling machine and thereby achieve prototype fabrication.
9.7 ParallelCoupled Bandstop Filter Parallel coupledline bandstop filters are very popular in MIC and planar circuit applications. They are easy to fabricate and massproduce using printed circuit technology. The basic configuration of a parallel coupledline bandstop filter is shown in Figure 9.22. It consists of a main line attached to a set of Lshaped lines of which the horizontal section is parallel coupled to the main line. Each arm of the Lshaped line has 90 degrees of electrical length. Figure 9.23 shows the equivalent transmission line circuit of each section of the filter. The element values are obtained from Table 9.5 [49]. Each combination of series section and a shunt section can be replaced by a parallel coupled line resulting in the circuit form in Figure 9.22. The procedure is shown in Figure 9.24. However, since the number of series sections is one less than the number shunt sections in Figure 9.23, the designer should add
Figure 9.22
Layout of a parallel coupledline bandstop filter.
9.7 ParallelCoupled Bandstop Filter
Figure 9.23
301
Equivalent circuit of parallel coupledline filter with open circuited shunt loading.
an extra series section of impedance Z A right before the first shunt stub, and then use the conversion procedure in Figure 9.24 [49]. In Table 9.5 g i (i = 0, 1, . . . N) are the lowpass prototype values given in Tables 9.1 and 9.2. Other parameters are defined as a = cot
冉 冊 1 2 0
(9.38)
where 1 is lower cutoff frequency of the bandstop filter. ⌳ = a 1′
(9.39)
where 1′ is the cutoff frequency of the lowpass prototype. 9.7.1 Design Example
A standard commercial software package like WAVECON [50] can be used to implement the above procedure. Table 9.6 shows the WAVECON design file of an Lband stripline bandstop filter. Figure 9.25(a) shows the layout of the filter and Figure 9.25(b) shows the computed frequency response.
302
CoupledLine Filter Fundamentals Table 9.5 Element Values for Parallel CoupledLine Bandstop Filters N = number of stubs Z A , Z B = terminating impedances Z j (j = 1 to N) = impedances of open circuit shunt stubs Z j − 1, j (j = 2 to N) = connecting line impedances g j = lowpass prototype element values ⌳ = 1′a where 1′ is the cutoff frequency of the lowpass prototype and a is the bandwidth parameter defined in (9.38). The terminating impedance Z A is arbitrary. Case of N = 1 Z1 =
ZA ⌳g 0 g 1
ZB =
ZA g2 g0
Case of N = 2
冉
冊
Z 12 = Z A (1 + ⌳g 0 g 1 ),
1 Z1 = ZA 1 + , ⌳g 0 g 1 Z2 =
ZB = ZA g0 g3 .
ZA g0 , ⌳g 2
Case of N = 3. Z 1 , Z 2 , and Z 12 are same as case N = 2. Z3 =
冉
冊
ZA g0 1 1+ , g4 ⌳g 3 g 4
Z 23 = ZB =
ZA g0 (1 + ⌳g 3 g 4 ), g4
ZA g0 . g4
Case of N = 4.
冉 冉
Z1 = ZA 2 + Z2 = ZA
冊
1 , ⌳g 0 g 1
Z 12 = Z A
冊
1 g0 + , 1 + ⌳g 0 g 1 ⌳g (1 + ⌳g g )2 2 0 1
Z3 =
ZA , ⌳g 0 g 3
Z4 =
ZA 1 1+ , g0 g5 ⌳g 4 g 5
冉
冊
冉 冉
冊
1 + 2⌳g 0 g 1 , 1 + ⌳g 0 g 1
冊
Z 23 =
ZA g0 ⌳g 2 + , g0 1 + ⌳g 0 g 1
Z 34 =
ZA (1 + ⌳g 4 g 5 ), g0 g5 ZA . g0 g5
ZB =
Case of N = 5 Z 1 , Z 2 , Z 3 , Z 12 , and Z 23 are same as case N = 4.
冉
冊
1 ZA g6 + , g 0 1 + ⌳g 5 g 6 ⌳g (1 + ⌳g g )2 4 4 5 Z g 1 , Z5 = A 6 2 + g0 ⌳g 5 g 6 Z4 =
冉
冊
冉
冊
Z 34 =
ZA g6 ⌳g 4 + , g0 1 + ⌳g 5 g 6
Z 45 =
Z A g 6 1 + 2⌳g 5 g 6 , g0 1 + ⌳g 5 g 6
ZB =
ZA g6 . g0
冉
冊
9.7 ParallelCoupled Bandstop Filter
Figure 9.24
303
Parallel coupled line and shunt stub line equivalence.
Table 9.6 CAD File for the LBand Bandstop Filter in Figure 9.25(a) Stripline Coupled Line Bandstop Filter 08/25/2006 09:34:36 2.7050 GHz CENTER FREQUENCY 0.1000 dB RIPPLE 0.1000 INCHES GROUNDPLANE SPACING 0.0005 INCHES CONDUCTOR THICKNESS 50.000 OHMS INPUT LINE IMPEDANCE 50.000 OHMS OUTPUT LINE IMPEDANCE SECT ELEMENT NUMB VALUE 1 1.0315 2 1.1474 3 1.0315
Zoe OHMS 52.796 52.946 52.796
0.0060 3 9.9000
GHz BANDWIDTH POLES DIELECTRIC CONSTANT
0.0174 0.0174
INCHES INPUT LINE WIDTH INCHES OUTPUT LINE WIDTH
Zoo Length Width Space OHMS INCHES INCHES INCHES 47.418 0.3396 0.0172 0.0910 47.233 0.3396 0.0172 0.0893 47.418 0.3396 0.0172 0.0917
Zstub StubL StubW OHMS INCHES INCHES 31.436 0.3379 0.0500 31.436 0.3379 0.0500 31.436 0.3379 0.0500
304
CoupledLine Filter Fundamentals
Figure 9.25
(a) Bandstop filter layout. (b) Computed frequency response of filter in part (a).
References [1] [2] [3] [4] [5]
Milligan, T., ‘‘Nomographs and the Filter Designer,’’ Microwaves and RF, Vol. 24, October 1985, pp. 103–107. Saal, R., ‘‘The Design of Filters Using the Catalogue of Normalized LowPass Filters’’ (in German), Telefunken (GMBH), Backang, W. Germany, 1961. Rhodes, J. D., Theory of Electrical Filters, New York: Wiley Interscience, 1976. Skwirzynski, J. K., Design Theory and Data for Electrical Filters, London: Van Nostrand, 1965. Zverev, A. I., Handbook of Filter Synthesis, New York: John Wiley, 1967.
9.7 ParallelCoupled Bandstop Filter [6] [7] [8]
[9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
[27]
[28] [29] [30] [31] [32]
305
Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, Impedance, Matching Networks and Coupling Structures, New York: McGrawHill, 1964. Howe, H., Jr., Stripline Circuit Design, Dedham, MA: Artech House, 1974. Alseyab, S. A., ‘‘A Novel Class of Generalized Chebyshev Low Pass Prototype for Suspended Substrate Stripline Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT30, September 1982, pp. 1341–1347. Mobbs, C. I., and J. D. Rhodes, ‘‘A Generalized Chebyshev Suspended Substrate Stripline Bandpass Filter,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT31, May 1983, pp. 397–402. Bahl, I. J., and P. Bhartia, Microwave SolidState Circuit Design, New York: John Wiley, 1988, Ch. 6. LINMIC+ Computer Program, Jansen Microwave, Ratingen, Germany. em Computer Program, Sonner Software, Liverpool, NY, 2006. EEpal Computer Program, Eagleware Corp., Stone Mountain, GA. IE3D, Zeland Software, Fremont, CA, 2006. SFPMIC+ Computer Program, Jansen Microwaves, Ratigen, Germany. HFSS Computer Program, Agilent, Santa Clara, CA, 2006. Rhea, R., ‘‘PC Tools Simulate and Synthesize RF Circuits,’’ Microwave and RF, Vol. 33, No. 4, pp. 194–199. Ozaki, H., and J. Ishii, ‘‘Synthesis of a Class of Stripline Filters,’’ IRE Trans. Circuit Theory, Vol. CT5, June 1958, pp. 104–109. Davis, W. A., Microwave Semiconductor Circuit Design, New York: Van Nostrand Reinhold Co., 1984, Ch. 3. Richards, P. I., ‘‘ResistorTransmission Line Resonator Filters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT6, April 1958, pp. 223–231. Malherbe, J. A. G., Microwave Transmission Line Filters, Norwood, MA: Artech House, 1990. Cohn, S. B., ‘‘Parallel Coupled Transmission Line Resonant Filters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT6, No. 2, April 1958, pp. 223–232. Moazzam, M. R., S. Uysal, and A. H. Aghvami, ‘‘Improved Performance Parallel Coupled Microstrip Filters,’’ Microwave J., Vol. 34, No. 11, November 1991, pp. 128–135. Mara, J. F., and J. B. Schappacher, ‘‘Broadband Microstrip ParallelCoupled Filters Using MultiLine Sections,’’ Microwave J., Vol. 22, No. 4, April 1979, pp. 97–99. Wong, J. S., ‘‘Microstrip TappedLine Filter Design,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT27, January 1979, pp. 328–339. Tran, M., and C. Nguyen, ‘‘Wideband Bandpass Filters Employing Broadside Coupled Microstrip Lines for MIC and MMIC Applications,’’ Microwave J., Vol. 37, No. 4, April 1994, pp. 210–225. Minnis, B. J., ‘‘Printed Circuit Coupled Line Filters for Bandwidths Up to and Greater Than an Octave,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT29, No. 3, March 1991, pp. 215–222. Ho, C. Y., and J. H. Werdman, ‘‘Improved Design of Parallel Coupled Line Filters with Tapped Input/Output,’’ Microwave J., Vol. 26, No. 18, October 1983, pp. 127–130. Bahl, I. J., ‘‘Capacitively Compensated HighPerformance ParallelCoupled Microstrip Filters,’’ IEEE MTTS Int. Microstrip Symp. Dig., 1989, pp. 679–682. Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House, 1996, Ch. 3. Matthaei, G. L., ‘‘Interdigital BandPass Filters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT10, No. 6, November 1962, pp. 479–491. Wenzel, R. J., ‘‘Exact Theory of Interdigital BandPass Filters and Related Coupled Structures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, No. 5, September 1975, pp. 559–575.
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Matthaei, G. L., ‘‘CombLine BandPass Filters of Narrow and Moderate Bandwidth,’’ Microwave J., Vol. 6, August 1963, pp. 82–91. Hong, J. S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, New York: John Wiley & Sons, 2001. Wenzel, R. J., ‘‘Synthesis of Combline and Capacitively Loaded Interdigital Bandpass Filters of Arbitrary Bandwidth,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT19, No. 8, August 1971, pp. 678–686. Vincze, A., ‘‘Practical Design Approach to Microstrip CombineType Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT22, No. 12, December 1974, pp. 1171–1181. Fukasawa, A., ‘‘Analysis and Composition of a New Microwave Filter Configuration with Inhomogeneous Dielectric Medium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT30, September 1982, pp. 1367–1375. Levy, R., ‘‘Simplified Analysis of Inhomogeneous Dielectric Block Combline Filters,’’ IEEE Int. Microwave Symp. Dig., 1990, pp. 135–138. You, C. C., C. L. Huang, and C. C. Wei, ‘‘SingleBlock Ceramic Microwave Bandpass Filters,’’ Microwave J., Vol. 37, November 1994, pp. 24–35. Hano, K., H. Kohriyama, and K.I. Sawamoto, ‘‘A Direct Coupled /4Coaxial Resonator Bandpass Filter for Land Mobile Communications,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT34, September 1986, pp. 972–976. Yao, H. W., et al., ‘‘FullWave Modeling of Conducting Posts in Rectangular Waveguide and Its Applications to SlotCoupled Combline Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT43, December 1995, pp. 2824–2829. Yao, H. W., C. Wang, and K. A. Zaki, ‘‘QuarterWavelength Combline Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. 44, December 1996, pp. 2673–2679. Cristal, E. G., and S. Frankel, ‘‘HairpinLine and Hybrid HairpinLine/HalfWave ParallelCoupledLine Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT20, No. 22, November 1972, pp. 719–728. Gysel, U. H., ‘‘New Theory and Design for HairpinLine Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT22, No. 5, May 1974, pp. 523–531. Salkhi, A., ‘‘Quick Filter Design and Construction,’’ Appl. Microwave and Wireless, Vol. 6, No. 1, January 1994, pp. 92–100. Pramanick, P., ‘‘Compact 900MHz HairpinLine Filters Using High Dielectric Constant Microstrip Line,’’ Int. J. Microwave MillimeterWave ComputerAided Eng., Vol. 4, No. 3, 1994, pp. 272–281. Rhea, R. W., ‘‘Distributed Hairpin Bandpass, RF Compute!’’ Eagleware Corp., Vol. 3, No. 1, 1991. Matthaei, G. L., et al., ‘‘HairpinComb Filters for HTS and Other NarrowBand Applications,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, No. 9, August 1997, pp. 1226–1231. Schiffman, B. M., and G. L. Matthaei, ‘‘Exact Design of BandStop Microwave Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT12, January 1964, pp. 6–15. WAVECON, Microwave Filter Design Software, Escondado, CA, 2006.
CHAPTER 10
Advanced CoupledLine Filters 10.1 Introduction Traditional parallelcoupled line filters, as discussed in the previous chapter, are easy to design. However, when this class of filters is realized in an inhomogeneous media such as microstrip or coplanar waveguide, it suffers from poor upper stopband performance and typically has spurious passbands centered at harmonics of the fundamental passband center frequency. In the case of parallelcoupled microstrip filters, considerable effort has been directed at suppressing the harmonic spurious passbands, in particular those located near the desired passband. In this chapter we discuss several designs of coupledline filters with enhanced stopband performance. These include designs using unevencoupled stages, periodically nonuniform coupled lines, meandered parallel coupled lines, and defected ground structures. In the previous chapter, we discussed conventional Butterworth and Chebyshev response prototypes. Bandpass filters based on these prototypes may be referred to as directcoupled filters because couplings exist only between adjacent resonators. However, when cross couplings are introduced among nonadjacent resonators, more advanced filtering characteristics such as quasielliptic function and linear phase responses can be obtained. These types of filters are referred to as crosscoupled filters and will be discussed in Section 10.3, together with filters with crosscoupled resonators and filters with sourceload coupling. In addition, filters with asymmetric port excitations can exhibit interesting characteristics and will be described as well. In this chapter, we will also discuss the design of interdigital filters using quarterwavelength stepped impedance resonators (SIR), which can result in a more compact size and a wider upper stopband, compared to conventional interdigital filters, introduced in Chapter 9. Recent advances in wireless communication have created a need for dualband filters, and these will be discussed in Section 10.5. This chapter also provides many design examples using fullwave electromagnetic (EM) simulation. EM simulation tools have become widely available and serve as an invaluable tool for the firstpass design of filters.
10.2 CoupledLine Filters with Enhanced Stopband Performance 10.2.1 Design Using UnevenlyCoupled Stages
For a conventional parallelcoupled line filter introduced in Chapter 9, all the coupledline sections are designed to have an equal electrical length, which is
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90 degrees at the center frequency of the designed passband. This restriction can be relaxed when a more general design approach is used. This means that the electrical lengths of coupled stages are not required to be 90 degrees at the desired center frequency and can be different, resulting in modified parallelcoupled filters with unevenly coupled stages. This helps to suppress the spurious response, especially for microstrip line filters [1–3]. To demonstrate the principle of using unevenly coupled stages, let us consider the singlesection parallelcoupled microstrip filter of Figure 10.1. A halfwavelength resonator of length l R and width W is excited using a coupledline section of length of l C and spacing S. Port 2 is very weakly coupled to the resonator as shown. Figure 10.2 plots responses for different lengths of the coupledline section. The frequency response is obtained using a commercially available EM simulator [4]. The microstrip halfwavelength resonator has a length of 29 mm and a width of 1 mm on the substrate. It can easily be shown that the resonator inherently resonates at a fundamental frequency f 0 = 2 GHz, and at other harmonics around
Figure 10.1
Single coupledline section for parallelcoupled microstrip filter.
Figure 10.2
Transmission response of structure shown in Figure 10.1 for l R = 29.0 mm, W = 1.0 mm, S = 0.2 mm, and W 0 = 1.4 mm on a 1.27mmthick substrate with a dielectric constant of 10.2.
10.2 CoupledLine Filters with Enhanced Stopband Performance
309
2f 0 , 3f 0 , and so on. When l C = 14.6 mm, which corresponds to a 90degree electrical length at f 0 , the resonant frequency response, as indicated by the dotted line, shows the first three resonant peaks and an attenuation pole or finitefrequency transmission zero at about 4.5 GHz. The finite transmission zero results from the coupled line section. The parallelcoupled line structure of Figure 9.9(b) can also be represented by its equivalent circuit depicted in Figure 10.3. As can be seen, the twoport equivalent circuit involves two series opencircuit stubs with an electrical length of . When = 90 degrees, the opencircuit stubs do not affect the signal transmission between ports 1 and 2. For = 180 degrees, however, the opencircuit stubs represent two perfect open circuits along the main signal path, which block the twoport network transmission and thus produce a transmission zero at the frequency where = 180 degrees. Strictly speaking, this equivalent circuit is for homogeneous or pure TEMmode coupled lines such as coupled striplines [5], which have equal even and oddmode phase velocities. This is the reason why a parallelcoupled stripline filter designed using the conventional approach described in Chapter 9 will not produce a spurious response at 2f 0 when each of the coupled sections has an electrical length of 90 degrees at the center frequency f 0 . In the case of a conventional parallelcoupled microstrip filter, an approximate value of electrical length is usually obtained from the average value of the even and oddmode phase velocities. Therefore, if we consider each microstrip coupled section to have an average electrical length of 90 degrees at f 0 , the finite transmission zero associated with this coupling structure will not occur exactly at 2f 0 , due to unequal even and oddmode phase velocities. This is what we observe in Figure 10.2 for l C = 14.6 mm. If we change the length l C , we can shift the finite transmission zero. For example, changing l C from 14.6 mm to 16.6 mm moves the finite transmission zero down in frequency and closer to the spurious response of the resonator at 2f 0 , as shown by the broken line in Figure 10.2. If we further increase the length l C to 17.2 mm, the spurious response of the resonator at 2f 0 can be suppressed as indicated by the unbroken line in Figure 10.2. It is evident that the suppression is due to the spurious resonant frequency coinciding with the frequency of finite transmission zero. Based on
Figure 10.3
Equivalent circuit of the parallelcoupled line structure of Figure 9.9(b).
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this discussion, a modified parallelcoupled microstrip filter design with nonequal coupled sections is presented next. Design Example
Consider a bandpass filter design based on a threepole Chebyshev lowpass prototype with a passband ripple of 0.1 dB. From Table 9.2, we find the lowpass prototype elements as follows: g 0 = g 4 = 1.0 g 1 = g 3 = 1.0315 g 2 = 1.1474 The bandpass filter is designed to have a fractional bandwidth FBW = 0.123 (i.e., 12.3%) at a center frequency f 0 = 1.985 GHz. The filter parameters can be calculated by [6] Q e1 = Q e3 =
g0 g1 = 8.386 FBW
M 12 = M 23 =
FBW = 0.1131 √g 1 g 2
where Q e1 and Q e3 are the external quality factors of the resonators at the input and output, M 12 is the coupling coefficient between resonators 1 and 2, and M 23 is the coupling coefficient between resonators 2 and 3. The filter is to be implemented on a 1.27mmthick substrate with a dielectric constant of 10.2. A practical design approach employing fullwave EM simulation is as follows. The frequency response of the arrangement of Figure 10.1 can be used to extract the external quality factor. Figure 10.4 shows a typical EMsimulated resonant response of the I/O coupled stage. The external quality factor can be extracted using the wellknown formula: Qe =
f0 ⌬ f 3dB
(10.1)
where f 0 is the resonant or center frequency and ⌬ f 3dB denotes the 3 dB bandwidth of the resonant response as shown in Figure 10.4. Note that (10.1) is strictly valid when the structure analyzed is ‘‘lossless.’’ Nevertheless, it is acceptable when the unloaded quality factor of the resonant structure is much larger than the external quality factor. In general for a given W, the external quality factor depends on both S and l C . It can also be shown that the larger the S, the longer the l C needed to suppress the spurious response at 2f 0 . In the design of a practical filter, the spacing S is mainly adjusted for the desired Q e , while the length l C is tuned for suppressing the spurious response at 2f 0 . In this way, the physical dimensions of the input and output coupling sections can be determined.
10.2 CoupledLine Filters with Enhanced Stopband Performance
Figure 10.4
311
Typical frequency response for extracting the external quality factor.
Shown in Figure 10.5(a) is an arrangement to extract the coupling between two resonators using EM simulation. For this particular example, the two resonators are coupled to each other through a common coupled section. The two ports are so arranged in order to weakly excite the coupled resonators. The EMsimulated frequency response of the coupled resonators is plotted in Figure 10.5(b), where two resonant peaks are clearly observed. If fp1 and fp2 denote the two resonant peaks, the coupling coefficient M can be extracted using M=
2
2
2
2
fp2 − fp1 fp2 + fp1
(10.2)
For a given length of coupled section, the desired M can be found by adjusting the spacing. After extracting the desired external quality factor Q e1 = Q e3 and the desired coupling coefficient M 12 = M 23 , the layout and physical dimensions of the designed filter are determined, and are given in Figure 10.6(a). The filter is symmetrical with two identical I/O coupledsections and two identical interresonator coupledsections. However, the lengths of the I/O coupledsections and the interresonator coupledsections are different. The former is 17.2 mm with an electrical length larger than 90 degrees at f 0 , which is designed for suppressing the spurious response at 2f 0 , while the latter is 16 mm with an electrical length of about 90 degrees at f 0 . The two small corner cuts of 0.4 × 0.6 mm on the middle resonator are for frequency tuning. Figure 10.6(b) shows the filter performance, obtained by fullwave EM simulation. The filter exhibits a desired passband at f 0 and the spurious response at 2f 0 has effectively been suppressed below −30 dB. From Figure 10.6(b), it can also be seen that only a single attenuation pole near 4 GHz is present in this case. This attenuation pole or transmission zero is produced by the two identical I/O coupledsections. It is found, however, that if the I/O coupledsections have slightly different lengths, two attenuation poles are
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Figure 10.5
(a) Arrangement for EM simulation to extract the interresonator coupling. The two resonators, which are 29 mm long and 1 mm wide, are coupled to each other through a common coupled section with a length of 16 mm and a spacing of 1 mm on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) Response of coupled resonators.
created, which can suppress the spurious response even more effectively. Figure 10.7(a) illustrates a modified filter where the output coupledsection has a length of 17 mm instead of 17.2 mm. The performance of the modified filter is shown in Figure 10.7(b). As can be seen, the spurious response at 2f 0 is suppressed to about −40 dB. The suppression is evidently enhanced with two closely allocated transmission zeros close to 4 GHz. It is also seen that the effect of a small change in the length of the output coupledsection on the passband is negligible. For a parallelcoupled microstrip filter with unequalcoupled sections, the suppression of the spurious response can be enhanced by using coupled stages with narrower lines or a higherimage impedance. This is because the narrower coupled
10.2 CoupledLine Filters with Enhanced Stopband Performance
Figure 10.6
313
(a) Parallelcoupled microstrip line filter with unevenly coupled stages. All the dimensions shown are in millimeters on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) Fullwave EM simulated performance of the filter.
lines result in a smaller evenmode capacitance so that the ratio of ⑀ re /⑀ ro is reduced, where ⑀ re and ⑀ ro are the even and oddmode effective dielectric constants, respectively. Two filters of this type on the same substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm have been experimentally demonstrated [2]. The two filters, one of which is a thirdorder filter with a fractional bandwidth of 20% and the other of which is a fifthorder filter with a fractional bandwidth of 15%, use a line width about 0.2 mm for the I/O coupled stages. For both filters, the measured attenuation levels at 2f 0 are below −50 dB. Parallelcoupled filters with unequalcoupled sections can also be designed for multispurious suppression since varying the coupling lengths of the coupled sections can finely tune attenuation poles or transmission zeros at 2f 0 , 3f 0 , and so on. Two
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Advanced CoupledLine Filters
Figure 10.7
(a) Parallelcoupled microstrip line filter with unevenly coupled stages where the electrical lengths of the I/O coupled stages are also different. All the dimensions shown are in millimeters on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) Fullwave EM simulated performance of the filter.
experimental filters of this type have been reported in [3], showing the suppression of the spurious response below −30 dB up to 4f 0 . 10.2.2 Design Using Periodically Nonuniform Coupled Lines
A socalled ‘‘wigglyline’’ filter is proposed in [7] to improve the spurious performance of a parallelcoupledline microstrip bandpass filter. Using a continuous perturbation of the width of the coupled lines following a sinusoidal law, the wave impedance is modulated, so that the harmonic passband of the filter is rejected while the desired passband response is maintained virtually unaltered. This strip
10.2 CoupledLine Filters with Enhanced Stopband Performance
315
width perturbation does not require the filter parameters to be recalculated. Thus, the classical design methodology for coupledline microstrip filters can still be used. At the same time, the fabrication of the resulting filter layout is no more difficult than that of a typical coupledline microstrip filter. To test this novel technique, a thirdorder Butterworth bandpass filter has been designed at 2.5 GHz with a 10% fractional bandwidth and different values of the perturbation amplitude. It is shown that for a 47.5% sinusoidal variation of the nominal strip width, a harmonic rejection below −40 dB is achieved in measurement, while the passband at 2.5 GHz is almost unaltered. The details of this filter design are given in [7]. Using the same concept but an alternative design approach, the following microstrip filter design is considered: Center frequency ( f 0 ) Fractional bandwidth (FBW ) Filtering response Substrate
2.5 GHz 0.1 threepole Butterworth ⑀ r = 10.2, thickness h = 1.27 mm
From Table 9.1, the lowpass prototype element values are g 0 = g 4 = 1.0, g 1 = g 3 = 1.0, and g 2 = 2.0. In general, it is difficult to determine exact even and oddmode parameters of nonuniform coupled lines for the filter design. For narrowtomoderatebandwidth filter designs, a more convenient design approach is based on the external quality factor (Q e ) and the coupling coefficient (M). Thus, for the filter to be designed, the desired design parameters are
Q e1 = Q e3 =
g0 g1 = 10 FBW
(10.3)
M 12 = M 23 =
FBW = 0.07 √g 1 g 2
(10.4)
The next step is to work out the structures of the nonuniform line resonators. Figure 10.8 shows the two specific nonuniform line resonators to be used in the filter design. These are similar to the ‘‘wigglyline’’ resonators [7]. Resonator 1 of Figure 10.8(a) is composed of two trapeziumlike structures, which have different sizes as indicated. Resonator 2 of Figure 10.8(b) consists of two identical shapes that are joined asymmetrically, where each shape is obtained by removing a trapezium from a uniform line section. Resonators 1 and 2 can be seen as complementary parts, and they are so designed for implementing couplings discussed later on. For a synchronously tuned filter, as in this case, all the resonators resonate at the same frequency (i.e., at the filter center frequency f 0 ). The EM simulated resonant characteristics of these two nonuniform resonators are plotted in Figure 10.9. It is seen that both the resonators have the same fundamental resonant frequency of 2.5 GHz, but different spurious resonant frequencies. Resonator 1 exhibits an interesting antiresonant behavior around 5.7 GHz, which is 2.28 times the fundamental frequency. Resonator 2 has its first spurious resonant mode at
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Advanced CoupledLine Filters
Figure 10.8
Nonuniform line resonators (not to scale) on a 1.27mmthick substrate with a dielectric constant of 10.2. The dimensions are in millimeters: (a) resonator 1 and (b) resonator 2.
4.4 GHz, which is 1.76 times the fundamental frequency. Thus, there is a separation of 1.3 GHz between the spurious frequencies of the two resonators. We will see later on that this large separation is essential for the suppression of the spurious passband at 2f 0 . Once the nonuniform resonators for the filter design are determined, we can move to the next stage in designing the filter. This involves the determination of two coupling structures. The first coupling structure is the input/output (I/O) coupling structure that provides an external quality factor given by (10.3). The second coupling structure is the interresonator coupling structure that exhibits the desired coupling given by (10.4). Figure 10.10(a) shows the determined I/O coupling structure, where, for clarity, the dimensions of the nonuniform line resonator are not shown, but can be found from Figure 10.8(a). The gap between the feed line and the resonator is 0.55 mm, and changing this dimension results in a different
10.2 CoupledLine Filters with Enhanced Stopband Performance
Figure 10.9
Figure 10.10
317
Resonant characteristics of the nonuniform line resonators in Figure 10.8.
(a) The I/O coupling structure (dimensions are in millimeters) on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) Its transmission response.
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Advanced CoupledLine Filters
value of Q e . To extract Q e , port 2 is weakly coupled, and the transmission response is obtained by fullwave EM simulation, which is plotted in Figure 10.10(b). The same technique as described previously can then be used to find the Q e based on the 3dB bandwidth around 2.5 GHz. We note, from Figure 10.10(b), that the designed I/O coupling structure does not alter the spurious behavior of the nonuniform line resonator. The interresonator coupling structure is shown in Figure 10.11(a), where resonator 1 (on the top) is coupled to resonator 2 through a spacing S. For a given L, the coupling varies with S. For the EM simulation, a twoport arrangement as shown is used, where the interresonator coupling structure is weakly excited by the ports. The coupling coefficient is then extracted from the simulated mode split response in Figure 10.11(b), using (10.2). It is found that for L = 11 mm and S = 0.55 mm, the desired coupling coefficient given by (10.4) is obtained. This completes the design of the filter. The layout of the designed filter is illustrated in Figure 10.12(a). The design is verified by fullwave EM simulation, and the simulated performance is depicted in Figure 10.12(b). It is shown that there is no spurious passband at 2f 0 , which is superior to the conventional parallel coupled line filter. Nevertheless, there are two undesired spikes in the upper stopband, one at 4.36 GHz and the other at 5.76 GHz. A similar response was also observed in the
Figure 10.11
(a) The interresonator coupling structure on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) The mode split response when L = 11 mm and S = 0.55 mm. The dimensions of the nonuniform resonators are the same as those given in Figure 10.8.
10.2 CoupledLine Filters with Enhanced Stopband Performance
Figure 10.12
319
(a) ‘‘Wigglyline’’ microstrip filter on a 1.27mmthick substrate with a dielectric constant of 10.2. All the dimensions shown are in millimeters. (b) Fullwave EM simulated performance of the filter.
filter reported in [7]. A wideband response of the coupled line resonators of Figure 10.11(a) is plotted in Figure 10.13 along with the transmission response of the filter. As can be seen, the coupled resonator section also produces two extra resonant peaks, which coincide with the two spikes in the filter frequency response outside the passband. Recall that, from Figure 10.9, the first spurious resonance of resonator 1 is 5.7 GHz, while the first spurious resonant mode for resonator 2 occurs at 4.4 GHz. Therefore, these two spurious frequencies are the undesired spikes observed in the filter frequency response. The successful suppression of the spurious passband at 2f 0 is due to the separation of the two spurious frequencies. Many other forms of coupledline resonators that have the some fundamental resonant frequency but different spurious mode frequencies can be designed and used to suppress unwanted spurious passbands in a coupled line filter. The method has been extended to reject multiple spurious passbands by employing different perturbation periods in each coupledline section [8]. A sevenpole microstrip ‘‘wigglyline’’ bandpass filter of this type, with a 10% fractional bandwidth centered at f 0 = 2.5 GHz, was fabricated to demonstrate the suppressions
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Figure 10.13
Comparison between the wideband characteristic of the coupled line resonators and the filter transmission response.
of spurious passbands at 2f 0 , 3f 0 , 4f 0 , and 5f 0 , resulting in a very wide upper stopband up to 14 GHz and with a rejection level of 30 dB.
10.2.3 Design Using Meandered ParallelCoupled Lines
For conventional parallel coupled lines in an inhomogeneous medium such as coupled microstrip lines, the even and oddmode phase velocities depend on the line width W and the spacing S as indicated in Figure 10.14(a). It has been found that by meandering the straight parallel coupled lines into a form shown in Figure 10.14(b), the modal phase velocities will depend on the separation of d as well [9]. Thus, for a given W and S, the even and oddmode phase velocities can be manipulated by changing the separation d.
Figure 10.14
(a) Conventional parallel coupled lines. (b) Meandered parallel coupled lines.
10.2 CoupledLine Filters with Enhanced Stopband Performance
321
The meandered parallel coupled line structure of Figure 10.14(b) can be used to suppress the spurious response in a microstrip parallel coupled filter. Figure 10.15 depicts two modified I/O coupled stages of a microstrip line parallel coupled filter. In Figure 10.15(a), the I/O resonator is meandered into the form as shown, which is a halfwavelength resonator with both ends opencircuited. The resonator
Figure 10.15
Modified I/O coupled stages using meandered parallel coupled lines having a line width W = 7 mil, a spacing S = 5 mil on a substrate with a dielectric constant of 10.2 and a thickness of 50 mil. (a) For a separation of d = 107 mil. (b) For a separation of d = 47 mil. (c) Their frequency responses.
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is fed at port 1 via a meandered coupled feed line, as indicated. The meandered feed line and a portion of the meandered resonator forms a meandered parallel coupled line section. Port 2 is weakly coupled to the resonator in order to determine its resonant response. The EMsimulated frequency response of this I/O coupled stage is plotted in Figure 10.15(c). As can be seen, the I/O resonator resonates at a fundamental frequency of 1 GHz. It also resonates at a frequency of 1.83 GHz, which usually causes a spurious passband equivalent to that in a conventional microstrip parallel coupled filter. The harmonic frequency is not exactly twice of the fundamental frequency because of the dispersion of the resonant structure. From the response shown in Figure 10.15(c), we can also observe an attenuation pole or transmission zero at 1.87 GHz. This finite frequency transmission zero exists inherently for a parallelcoupled line structure whose equivalent circuit is given in Figure 10.3. By changing the separation d of the meandered parallel coupled line section, the modal phase velocities are changed, resulting in the reallocation of the finite frequency transmission zero. To demonstrate this, Figure 10.15(b) shows the other I/O coupled stage where d has been changed to 47 mil while the other parameters are kept the same as those of Figure 10.15(a). Its frequency response is also shown in Figure 10.15(c). It is evident that the frequency of finite transmission zero has moved closer to the harmonic frequency, suppressing the harmonic response effectively. Therefore, the effect of tuning the separation of d is similar to that of tuning the coupling length of L C in the I/O coupled stage of Figure 10.1. Based on this mechanism, a parallel coupled filter using meandered parallel coupled lines can be designed to simultaneously suppress the spurious passband and greatly reduce size. An example of this type of filter, following that reported in [9], is illustrated in Figure 10.16. This is a threepole bandpass filter with a center frequency f 0 = 1 GHz, and the filter is designed to suppress the spurious passband around 2 GHz. The substrate is chosen with a dielectric constant of 10.2 and a thickness of 50 mil (1.27 mm). The main reason for choosing this relative high dielectric constant substrate is not only to miniaturize the filter, but also to demonstrate the challenge of the relative large difference between the even and oddmode phase velocities, which makes it difficult to suppress the spurious passband. All the dimensions shown in the filter layout are in mils. The filter is very compact, occupying a circuit area of 625 × 700 mil (about 16 × 18 mm). The filter is symmetrical with two pairs of meandered parallel coupled line sections. For the pair of I/O coupled stages, the separation of meandered parallel coupled lines is 27 mil. For the inner pair of interresonator couplings, the separation of meandered parallel coupled lines is 22 mil. These two separations are critical parameters, and are tuned to achieve a better suppression of the spurious passband at 2f 0 . Figure 10.17 illustrates the EMsimulated filter performance. The filter shows a desired passband at 1 GHz, while the spurious passband at 2 GHz has been suppressed almost below −40 dB. Experimental demonstrations of this type of filter can be found in [9]. Another type of meandered parallel coupled line filter is presented in [10]. In this type of filter, the meandered line design suppresses passband harmonics by using numerousbends and angles to equalize the phase velocities of the even and odd modes. An experimental filter of this type achieved a 40% size reduction and
10.2 CoupledLine Filters with Enhanced Stopband Performance
323
Figure 10.16
Layout of the designed threepole microstrip filter using meandered parallel coupled lines. The filter is on a substrate with a dielectric constant of 10.2 and a thickness of 50 mil. All the dimensions shown are in mils.
Figure 10.17
EMsimulated performance of the filter of Figure 10.16.
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Advanced CoupledLine Filters
70dB rejection at the second harmonic without degrading the desired passband performance [10]. 10.2.4 Design Using Defected Ground Structures
Defected ground structures can be used to equalize the even and oddmode phase velocities in coupled microstrip lines [11, 12]. Figure 10.18 illustrates the cross section of modified coupled microstrip lines with a ground aperture. The width of the aperture is denoted by S a . The characteristics of the modified coupled microstrip lines, that is, even and oddmode effective dielectric constants and characteristic impedances, are affected by the presence of the slot. Figure 10.19 shows the EM simulated results for modified coupled microstrip lines on a 0.635mmthick substrate with a dielectric constant of 10. As can be seen from Figure 10.19(a), the evenmode effective dielectric constant is reduced by an increase in the aperture size, while there is little change in the odd mode. When S a equals zero, which is the case for conventional coupled microstrip lines, the ratio of the evenmode effective dielectric constant to the oddmode effective dielectric constant, ⑀ ree /⑀ reo is 1.177. This ratio is reduced to about 1.0 when S a = 1.6 mm. The reason why the evenmode phase velocity is speeded up faster against the aperture size is that, as S a increases, the capacitance between the conductor strip and ground is decreased more effectively for the even mode. For the same reason, the evenmode characteristic impedance is also increased predominately as shown in Figure 10.19(b). The coupled line structure in Figure 10.18 can be used as a basic building block to design coupledline microstrip filters with enhanced performance. The first spurious passband of a filter can be suppressed efficiently by adjusting the aperture size so as to approximately equalize the even and oddmode phase velocities. Several filters of this type have been demonstrated in [11]. As an additional advantage, the structure of Figure 10.18 provides tight coupling (in comparison with conventional coupled microstrip lines), thus relaxing the requirements on physical dimensions W and S in those cases where tight coupling is necessary. Figure 10.20 shows the cross section of another structure with a floating conductor at the ground aperture. The floating conductor has a width of W f. For W f = 0, the structure degenerates to that of Figure 10.18. The structure with the floating conductor makes filter designs more flexible. In this case, the mode impedances depend mainly on width W, strip separation S, and floating conductor width W f , while the modal phasevelocities depend mainly on W, S, and the
Figure 10.18
Cross section of modified coupled microstrip lines with a defected ground aperture.
10.2 CoupledLine Filters with Enhanced Stopband Performance
325
Figure 10.19
Characteristics of modified coupled microstrip lines as a function of the ground aperture size. The dimensions of the coupled lines are W = 0.4 mm and S = 0.2 mm on a 0.635mmthick substrate with a dielectric constant of 10. (a) Effective dielectric constants. (b) Characteristic impedances.
Figure 10.20
Cross section of modified coupled microstrip lines with a floating conductor at the defected ground aperture.
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Advanced CoupledLine Filters
aperture size S a [12]. Therefore, for each coupled stage in the filter design, one can match the even and oddmode phase velocities and simultaneously obtain the desired modalimpedance values. Following a similar design to that described in [12], Figure 10.21 shows the layout of a threepole parallelcoupled microstrip line filter implemented using defected ground structures with floating conductors. The filter is designed on a dielectric substrate with a dielectric constant of 10 and a thickness of 0.635 mm. The filter structure is symmetrical and the dimensions given in Figure 10.21 are in millimeters. It is noticeable from the bottom view of Figure 10.21(b) that, for each coupled stage, there are three fragmental floating conductors separated by two 50 mwide metal bridges. The metal bridges joining groundplane sides play an essential role in good filter performance because they cancel out undesired groundplane slot modes. The exact position of the metal bridges is not very important, and in this case, the original floating conductor under each coupled stage has been divided into three identical sections.
Figure 10.21
Layout of threepole parallelcoupled microstrip line filter using defected ground structures. The filter is on a substrate with a dielectric constant of 10 and a thickness of 0.635 mm. All the dimensions shown are in millimeters. (a) Top view. (b) Bottom view.
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
327
This is a threepole Chebyshev filter designed to have a fractional bandwidth of 20% at a center frequency f 0 = 3 GHz. Figure 10.22 shows the performance of the filter, obtained by fullwave EM simulation. As can be seen, the spurious passband at 2f 0 has been effectively suppressed by the technique based on defected ground structures. This has also been experimentally verified in [12]. In the simulation, the conductor was assumed to be 17 mthick copper and the loss tangent of the dielectric substrate was taken as 0.002. The cell size used for the simulation was 50 m by 50 m to ensure accuracy, and 600Mb memory was required using Sonnet em [4]. This larger memory requirement was mainly due to the simulation of the defected ground. The time for completing the simulation was 3 hours on a Pentium 4 CPU of 3.2 GHz.
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics 10.3.1 Filters with CrossCoupled Resonators
Conventional Chebyshev or Butterworth filters may also be referred to as directcoupled filters on the basis that there is only direct coupling between adjacent or consecutive resonators. When cross couplings are introduced among nonadjacent resonators, more advanced filtering characteristics such as quasielliptic function and linear phase responses can be obtained, and these types of filters are referred to as crosscoupled filters. Figure 10.23 depicts coupling structures of some typical crosscoupled filters, where S and L denote the source and load, respectively, the black circles represent the resonators, the full lines indicate the direct couplings, and the broken lines denote the cross couplings. Shown in Figure 10.23(a) is the socalled canonical filter coupling structure, where the n directcoupled resonators are folded into two arrays and there is cross coupling between the nonadjacent resonators 1 and n, 2 and n − 1 and so on. This type of filter is sometimes called a folded filter. The coupling structure shown in Figure 10.23(b) is for the socalled cascaded quadruplet (CQ) filter. A CQ filter consists of cascaded sections of four
Figure 10.22
EMsimulated performance of the filter of Figure 10.21.
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Advanced CoupledLine Filters
Figure 10.23
Coupling structures of typical crosscoupled resonator filters. (a) Canonical. (b) Cascaded quadruplet (CQ). (c) Cascaded trisection (CT).
resonators, each with one cross coupling. Similar to the canonical filter, the cross coupling can be arranged in such a way that a pair of attenuation poles or transmission zeros are introduced at the finite frequencies to improve the selectivity, or it can be arranged to result in a linear phase or group delay selfequalization. However, the tuning of a CQ filter is easier because the effect of each cross coupling is independent, which is only responsible for a single pair of zeros. Figure 10.23(c) is a typical coupling structure of a cascaded trisection (CT) filter. Each CT section is comprised of three directly coupled resonators with cross coupling. This cross coupling will only produce a single finite frequency transmission zero, which can be allocated on either side of the passband depending on the implementation of coupling. There are many other coupling structures for crosscoupled filters. For example, a combination of CQ and CT coupling structures leads to a socalled CQT filter [13]. Most applications of crosscoupled filters are as bandpass filters, in particular for narrowband filters. To design crosscoupled filters, a practical approach based on a general coupling matrix and external quality factors is often employed. The general coupling matrix for an ncoupled resonator filter has a form [6]
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
冤
m 11 m 21 [m] = ⯗ m n1
m 12 m 22 ⯗ m n2
329
冥
... ... ⯗ ...
m 1n m 2n ⯗ m nn
(10.5)
which is an n × n reciprocal matrix (i.e., m ij = m ji ) and is allowed to have nonzero diagonal entries m ii for an asynchronously tuned filter. Note that m ij denotes the socalled normalized coupling coefficient, and the required coupling coefficient for a given fractional bandwidth FBW of a bandpass filter centered at f 0 can be obtained from, M ij = m ij ⭈ FBW
(10.6)
The external quality factors, namely, Q e1 that represents the coupling between the source and resonator 1 (the input resonator) and Q en that denotes the coupling between the load and resonator n (the output resonator), are defined as Q ei =
q ei FBW
for i = 1, n
(10.7)
where q ei denotes the scaled external quality factor. For a given filtering characteristic, the coupling matrix and the external quality factors can be obtained using a synthesis procedure. The scattering parameters of the twoport filter network can be computed from [6] S 21 = 2
1 −1 [A]n1 √q e1 ⭈ q en
S 11 = 1 −
(10.8)
2 −1 [A]11 q e1
−1
in which [A]ij denotes the ith row and jth column element of [A]−1 with [A] = [q] + p[U] − j[m] where p=j
1 FBW
冉
f f − 0 f0 f
冊
[U] is the n × n unit or identity matrix and [q] is an n × n matrix with all entries zero, except for q 11 = 1/q e1 and q nn = 1/q en . Crosscoupled resonator filters can be implemented with different forms of microwave resonators. For planar filter realization, microstrip openloop resonators provide great flexibility for implementing a variety of cross coupling structures
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Advanced CoupledLine Filters
[14]. As an example of this realization, a fourpole microstrip crosscoupled filter is designed based on a prescribed general coupling matrix:
冤
0 0.88317 [m] = 0 −0.12355
0.88317 0 0.74907 0
0 0.74907 0 0.88317
−0.12355 0 0.88317 0
冥
with the scaled external quality factor q e1 = q e4 = 0.94908. There is cross coupling between resonators 1 and 4 and the filter may be seen as a canonical or single section CQ filter. The filter is designed for a fractional bandwidth FBW = 0.06 at a center frequency f 0 = 1.17 GHz. Thus, the desired design parameters can be found from (10.6) and (10.7): M 12 = M 34 = 0.05299 M 23 = 0.04494 M 14 = −0.00741 Q e1 = Q e4 = 15.818 Using (10.8), the theoretical frequency responses of the filter are calculated and plotted in Figure 10.24. As can be seen, the pair of finite frequency transmission zeros near the passband improve the filter selectivity significantly. The microstrip filter is realized using the configuration of Figure 10.25 on a substrate with a dielectric constant of 3.38 and a thickness of 0.508 mm, where the four microstrip openloop resonators are numbered to indicate their sequence in the main coupling path. Hence, resonators 1 and 4 are the input and output (I/O) resonators, respectively. Fullwave EM simulations are carried out to extract the desired external
Figure 10.24
Theoretical frequency responses obtained from the prescribed coupling matrix and external quality factors for a fourpole crosscoupled filter having a 6% fractional bandwidth at 1.17 GHz.
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.25
331
Configuration of fourpole microstrip crosscoupled filter using openloop resonators.
quality factors and coupling coefficients using the approach described in Section 10.2.1. Shown in Figure 10.26(a) is an arrangement to extract the external quality factor of the I/O resonator. The microstrip openloop resonator has a line width of 2 mm and a size of 22 mm × 22 mm on the substrate. The resonator is excited at port 1 through a 50ohm tapped line at a location indicated by t. Port 2 is very weakly coupled to the resonator in order to find a 3dB bandwidth of the magnitude response of S 21 for extracting the external quality factor. When t = 0 where there is a virtual grounding of the fundamental mode of the openloop resonator, and the coupling from the source (i.e., port 1) is the weakest, the largest external quality factor is obtained. Increasing t increases the coupling so that the extracted external quality factor becomes smaller. A design curve can be obtained, as shown in Figure 10.26(b), where one can determine the tappedline location for the desired external quality factor. To determine the coupling between resonators 1 and 2 of the filter configuration in Figure 10.25, an arrangement such as that of Figure 10.27(a) is used. The two openloop resonators have the same size and therefore the same resonant frequency. The coupling is mainly controlled by the spacing S between them. An offset d is often required due to the requirement of implementing other couplings for the filter design, as we will see later on. For EM simulation, the coupled resonators are very weakly excited by the two ports as arranged. Two resonant peaks, which result from the mode split due to the coupling between the two resonators, can clearly be observed from the EMsimulated frequency responses. The coupling coefficient can then be extracted using (10.2). Figure 10.27(b) shows the design curve for M 12 for d = 0.4 mm. It is obvious that the coupling decreases as the spacing S increases. It can also be shown that a small offset d has little effect on the coupling. For design of the filter, the physical dimension S can readily be determined from the design curve for the desired M 12 . Figure 10.28(a) depicts an arrangement for the EM simulation to determine the coupling between resonators 2 and 3 of the filter configuration in Figure 10.25. For the given orientation of the two coupled openloop resonators, the coupling between them at the fundamental resonance is dominated by the magnetic field and therefore can be referred to as magnetic coupling [6]. Using the same approach described above for extracting the interresonator coupling, a design curve for M 23 can be produced and is plotted in Figure 10.28(b).
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Advanced CoupledLine Filters
Figure 10.26
(a) Arrangement for extracting external quality factor. All the dimensions are in millimeters on a 0.508mmthick substrate with a dielectric constant of 3.38. (b) Design curve for Q e1 or Q e4 .
To determine the cross coupling between resonators 1 and 4 of the filter configuration in Figure 10.25, we simulate using EM the arrangement of Figure 10.29(a). The orientations of these two coupled openloop resonators are opposite to that of Figure 10.28(a). In this case, the coupling between them is dominated by the electric field and hence this is referred to as electric coupling. This implementation for the cross coupling is necessary because M 14 and M 23 have to be of opposite signs in order to realize a pair of attenuation poles at finite frequencies. Figure 10.29(b) shows the design curve for M 14 for the filter design. From the design curves given earlier, all the physical dimensions associated with the desired design parameters, namely, the external quality factors and coupling coefficients, are readily determined. The layout with dimensions of the designed microstrip openloop resonator filter is illustrated in Figure 10.30(a). Note that
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.27
333
(a) Arrangement for extracting coupling coefficient M 12 . (b) Design curve for M 12 . [The dimensions of the openloop resonators are the same as those given in Figure 10.26(a) on the same substrate.]
the dimensions are rounded off to have a resolution of 0.1 mm. Since the designed filter needs to be synchronously tuned, the open gap of the I/O resonator is slightly tuned to compensate for the frequency shift due to the tapped line I/O arrangement. Figure 10.30(b) shows the EMsimulated performance of the filter, where the two finite transmission zeros as expected can clearly be seen. The slight asymmetric frequency response may be attributed to unwanted coupling as well as frequencydependent coupling.
Miniature CrossCoupled Filter
Refer to the filter topology of Figure 10.30(a). Since there is a virtual grounding in the middle of each openloop resonator at its fundamental resonance, it is feasible
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Advanced CoupledLine Filters
Figure 10.28
(a) Arrangement for determination of coupling coefficient M 23 . (b) Design curve for M 23 . [The dimensions of the openloop resonators are the same as those given in Figure 10.26(a) on the same substrate.]
to introduce a physical via ground in that location without appreciably affecting the primary passband. This, however, leads to a new topology of a miniature crosscoupled filter as shown in Figure 10.31. Each of the four resonators, as numbered, has a short circuit (via ground) at one end and an open circuit at the other end, which is basically half of an openloop resonator. Therefore, the new filter topology requires a circuit size which only amounts to half of that of Figure 10.30(a). The miniature crosscoupled filter can be designed using an approach similar to that used for the design of the openloop resonator filter described earlier. Figure 10.32(a) illustrates the layout of a designed miniature crosscoupled filter on a substrate with a dielectric constant of 3.38 and a thickness of 0.508 mm. The
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.29
335
(a) Arrangement for extracting coupling coefficient M 14 . (b) Design curve for M 14 . [The dimensions of the openloop resonators are the same as those given in Figure 10.26(a) on the same substrate.]
performance of this filter, obtained by EM simulation, is plotted in Figure 10.32(b). It is evident that the miniature filter maintains the primary passband response of the openloop resonator filter in Figure 10.30. Moreover, the miniature crosscoupled filter exhibits a better upper stopband, which is demonstrated in Figure 10.33. It shows that the first spurious only appears at about 3f 0 , resulting in a wide and better upper stopband. This is because the resonators used are only a quarterwave long. 10.3.2 Filters with SourceLoad Coupling
Traditionally, coupled resonator filters have focused on topologies where there is no direct source to load coupling. This is the case for the filter designs described
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Advanced CoupledLine Filters
Figure 10.30
(a) Layout of the designed crosscoupled filter on a substrate with a dielectric constant of 3.38 and a thickness of 0.508 mm. All the dimensions are in millimeters. (b) EMsimulated performance of the filter.
in the previous sections. It is well known that such a topology can produce at most n − 2 finite transmission zeros out of n resonators, where n is the degree of the filter. The addition of a direct signal path between the source and the load allows the generation of n finite frequency transmission zeros instead of n − 2. Filters with sourceload coupling have attracted a lot of attention recently [15–19].
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.31
337
Configuration of a miniature crosscoupled filter.
Figure 10.34 shows the socalled n + 2 or ‘‘extended’’ general coupling matrix for nthdegree coupled resonator filters. The n + 2 coupling matrix has an extra pair of rows at the top and bottom and an extra pair of columns at the left and right surrounding the ‘‘core’’ coupling matrix. The n + 2 coupling matrix is an (n + 2) × (n + 2) reciprocal matrix (i.e., m ij = m ji ). Nonzero diagonal entries m ij are allowed for an asynchronously tuned filter. For a given filter topology, the n + 2 coupling matrix may be obtained using synthesis methods described in [15–18]. Once the coupling matrix [m] is determined, the filter frequency response can be computed in terms of scattering parameters as follows: −1
S 21 = −2j [A]n + 2, 1
(10.9)
−1
S 11 = 1 + 2j [A]11 −1
where [A]ij denotes the ith row and jth column element of [A]−1. The matrix [A] is given by [A] = [m] + ⍀[U] − j[q]
(10.10)
in which [U] is similar to the (n + 2) × (n + 2) identity matrix, except that [U]11 = [U]n + 2, n + 2 = 0, [q] is the (n + 2) × (n + 2) matrix with all entries zeros except for [q]11 = [q]n + 2, n + 2 = 1, and ⍀ is the frequency variable of the lowpass prototype. The lowpass prototype response can be transformed to a bandpass response having a fractional bandwidth FBW at a center frequency f 0 using the wellknown frequency transformation: ⍀=
1 FBW
冉
f f − 0 f0 f
冊
(10.11)
Filter Example I
Consider a fourpole (n = 4) crosscoupled resonator filter that has a coupling structure as shown in Figure 10.35. Each numbered node represents a resonator
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Advanced CoupledLine Filters
Figure 10.32
(a) Layout of the designed miniature crosscoupled filter on a substrate with a dielectric constant of 3.38 and a thickness of 0.508 mm. All the dimensions are in millimeters. (b) EMsimulated performance of the filter.
and the direct coupling between adjacent nodes is indicated by the unbroken line. There is cross coupling between nodes 1 and 4, indicated by the broken line. The coupling between the source and load is also indicated by the broken line between the two nodes. The filtering characteristic of the filter under consideration is prescribed with an n + 2 general coupling matrix given by
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
339
Figure 10.33
Wideband performance of the miniature crosscoupled filter.
Figure 10.34
n + 2 general coupling matrix [m].
Figure 10.35
Coupling structure of a fourpole folded coupledresonator filter with sourceload coupling.
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Advanced CoupledLine Filters
[m] =
冤
0 1.0089 0 0 0 m SL
1.0089 0 0.8514 0 −0.1436 0
0 0.8514 0 0.7380 0 0
0 0 0.7380 0 0.8514 0
0 −0.1436 0 0.8514 0 1.0089
m SL 0 0 0 1.0089 0
冥
(10.12)
The filter possesses a cross coupling m 14 = −0.1436 between resonators 1 and 4, which produces a single pair of finite frequency transmission zeros. It also allows a direct coupling between the source and load, denoted by m SL . The effect of introducing m SL on the filtering characteristic can be seen by varying its value. For example, we consider three cases corresponding to m SL = 0.0, 0.00026, and 0.0028, respectively. The filter is assumed to have a fractional bandwidth FBW of 0.04, centered at f 0 = 2.4 GHz. The computed filter response, obtained by using (10.9)– (10.11), is plotted in Figure 10.36. For m SL = 0.0, the filter only shows two finitefrequency transmission zeros, close to the passband. This pair of transmission zeros results from the cross coupling between resonators 1 and 4, as mentioned before. For m SL = 0.0028, two extra finite frequency transmission zeros are observed. This fourpole filter has a total of four finitefrequency transmission zeros. For the case m SL = 0.00026, the filter characteristic hardly changes from that for m SL = 0.0. This is because the sourceload coupling is too small in this case. Another interesting
Figure 10.36
Frequency responses of a fourpole cross coupled resonator filter allowing a direct sourceload coupling.
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
341
observation in Figure 10.36 is that the return loss (i.e.,  S 11  ) response is almost the same for the three different values of m SL . This indicates that varying the sourceload coupling can effectively adjust the position of the two extra finite transmission zeros with negligible effect on the passband. It is suggested that one can design the symmetric folded coupledresonator filter first and then add some sourceload coupling. This would make the filter implementation easier. The parameters m Si and m Li , which represent the input and output couplings from the source and load to resonator i, can be converted to external quality factors by Q e, Si =
1 m Si ⭈ FBW
Q e, Li =
1 m Li ⭈ FBW
(10.13)
where FBW is the fractional bandwidth of bandpass filter. To determine the direct sourceload coupling m SL , a useful formulation can be derived as follows. Consider an I/O structure that only involves the sourceload coupling. From Figure 10.34, the general coupling matrix for the structure under consideration would have a form [m] =
冋
0
m SL
m SL
0
册
which corresponds to a zeroorder filter case for n = 0. From (10.10), we have [A] = =
冋 冋
0
m SL
m SL
0
−j
m SL
m SL
−j
册 冋 册 冋 册 册 +⍀
0
0
0
0
−j
An analytical solution for [A]−1 can be found to be
[A]
−1
=
冋
j
m SL
m SL
j 2
1 + m SL
Using (10.9), we obtain S 21 = −2j or
m SL 2
1 + m SL
册
1
0
0
1
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Advanced CoupledLine Filters
 S 21  = ±
2m SL 2
1 + m SL
The negative sign is for m SL < 0. Solving the above equation for m SL yields
m SL = ±
1−
√1 −  S 21   S 21 
2
(10.14)
which is a valid solution by taking into account that m SL = 0 for  S 21  = 0 and 0 ≤  S 21  ≤ 1. The choice of a sign is rather relative, which would depend on the sign of other elements in the n + 2 general coupling matrix. In any case, it can easily be determined from the desired frequency response. For example, a microstrip bandpass filter is to be designed to have a 4% fractional bandwidth centered at 2.4 GHz and to have a filtering characteristic prescribed by the coupling matrix given in (10.12) with a sourceload coupling m SL = 0.0028. The filter is to be implemented using openloop microstrip resonators on a 20milthick dielectric substrate with a dielectric constant of 3.38. Figure 10.37(a) is the physical structure designed to implement the required sourceload coupling, where all the dimensions are given in mils. The two input and output (I/O) feed lines are coupled through a 90milwide gap and a short coupling line of 160mil length. While the 90milwide gap is kept constant for the filter design, the sourceload coupling can easily be controlled by the coupling length and spacing of the short coupling line with respect to the two feed lines. Figure 10.37(b) plots the EM simulated results of  S 21  for this I/O coupling structure with the dimensions shown. One can see that, in general, the sourceload coupling is frequencydependent. For a narrowband approximation, we can use the EM simulated result at the center frequency to determine m SL , based on (10.14). Thus, from Figure 10.37(b) it can be found that  S 21  = 0.0056 at the center frequency of 2.4 GHz. Using (10.14) the extracted m SL is equal to 0.0028, which is the desired value. In this way, the dimensions of the structure for the sourceload coupling are determined. The remainder of the filter design can be completed by determining the desired external quality factors and the interresonator couplings following the method described previously. The layout of the design microstrip filter with sourceload coupling is shown in Figure 10.38(a), and its performance, obtained by fullwave EM simulations, is presented in Figure 10.38(b). As can be seen, this fourpole microstrip filter exhibits four finitefrequency transmission zeros, two of which are due to the direct source to load coupling. The frequency response of the filter shows an asymmetrical behavior, which is likely caused by unwanted and/or frequencydependent couplings. A diagnosis method may be used to identify the unwanted effects in the microstrip filter [19].
Filter Example II
Another filter design described here is also based on an n + 2 coupling matrix. The filter uses only three resonators, hence n = 3. It has a coupling topology as
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.37
343
(a) An I/O coupling structure to implement the desired sourceload coupling. All the dimensions are in mils on a 20milthick substrate with a dielectric constant of 3.38. (b) EM simulated transmission response.
shown in Figure 10.39, where the source and the load are coupled to two resonators each. As can be seen, the source is coupled not only to resonator 1, but also to resonator 2 as indicated by the broken line. Similarly, the load is coupled not only to resonator 3, which is usually the output resonator, but also to resonator 2. The coupling structure is basically comprised of two trisections, each of which is able to produce a finitefrequency transmission zero. For the filter, the n + 2 coupling matrix is given by
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Advanced CoupledLine Filters
Figure 10.38
(a) Layout of the designed fourpole crosscoupled filter with sourceload coupling. All the dimensions are in mils on a 20milthick substrate with a dielectric constant of 3.38. (b) EM simulated filter performance.
[m] =
0 1.0103 0.4275 0 0
冤
1.0103 −0.7811 0.8549 0 0
0.4275 0.8549 0.4562 1.0258 0.2334
0 0 1.0258 −0.3857 1.1058
0 0 0.2334 1.1058 0
冥
(10.15)
It is clear from (10.15) that the filter is asynchronously tuned as there are nonzero diagonal elements. The couplings from the source to resonators 1 and 2
10.3 CoupledLine Filters Exhibiting Advanced Filtering Characteristics
Figure 10.39
345
Coupling structure of a threepole coupledresonator filter with the source and the load being coupled to two resonators each.
are defined as m S1 = m 1S = 1.0103 and m S2 = m 2S = 0.4275, respectively. Similarly, the couplings from resonators 2 and 3 to the load can be identified to be m 2L = m L2 = 0.2334 and m 3L = m L3 = 1.1058, respectively. The center frequency and fractional bandwidth of the filter are chosen to be f 0 = 5 GHz and FBW = 0.05. The theoretical response of the filter can be computed using (10.9) and the results are plotted in Figure 10.40. As expected, two finitefrequency transmission zeros occur in the upper stopband, which improves the selectivity on the high side of the passband. To implement the n + 2 coupling matrix of (10.15), a microstrip realization is proposed in [20]. The filter is built on a dielectric substrate with a dielectric constant of 3.58 and a thickness = 20 mils. Figure 10.41(a) shows the layout of the designed filter, where all the dimensions are in mil. The filter is modified from the conventional microstrip parallelcoupled filter by vertically flipping feed lines of the source and the load. Furthermore, two small coupling/shielding lines are added at the ends of the input and output feed lines, allowing the source and the load to be coupled to two resonators each. From Figure 10.41(a), it can be identified that the
Figure 10.40
Theoretical responses of the threepole filter with two finitefrequency transmission zeros.
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Figure 10.41
(a) Threepole microstrip parallelcoupled filter with the source and the load being coupled to two resonators each. The dimensions are in mils on a dielectric substrate with a dielectric constant of 3.58 and a thickness = 20 mils. (b) EM simulated filter performance.
coupling/shielding line at the end of input (source) feed line has a length = 87 mil and a width = 8 mil. It couples to resonator 2 through an 8mil gap to facilitate the required external quality factor Q e, S2 =
1 1 = = 46.78 m S2 × FBW 0.4275 × 0.05
according to (10.13). Similarly, the coupling/shield line, which is 39 mils long and 8 mils wide, at the end of output (load) feed line, is coupled to resonator 2 via a 7mil gap to realize the desired external quality factor Q e, L2 =
1 1 = = 85.69 m L2 × FBW 0.2334 × 0.05
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347
These design parameters can be extracted by EM simulations. The EM simulated performance of the filter is depicted in Figure 10.41(b), showing good agreement with theory. 10.3.3 Filters with Asymmetric Port Excitations
For some filter topologies, it is possible to feed the filters either symmetrically or asymmetrically. Filters designed with an asymmetric port excitation can result in a significantly different frequency response, in particular, in the stopband [21, 22]. For example, Figure 10.42 illustrates a pair of twopole microstrip pseudocombline bandpass filters [6], one with symmetrical port excitation [Figure 10.42(a)] and the other with asymmetrical port excitation [Figure 10.42(b)]. Both the filters use tappedline input and output (I/O) arrangements. The asymmetric port design is obtained by vertically flipping the output resonator associated with the symmetric design. The filter dimensions shown are in millimeters on a dielectric substrate with a thickness of 1.27 mm and a dielectric constant of 10.8. The resonators are halfwavelength long at a fundamental resonant frequency of about 2 GHz. Figure 10.43 plots the EM simulated performance of the two filters. As can be seen, the filter with symmetric ports has a finite frequency transmission zero on the lower side of the passband, whereas the filter with asymmetric ports possesses a finite frequency zero on the upper side of the passband. The allocation of finite
Figure 10.42
Twopole microstrip pseudocombline bandpass filters with tappedline I/O on a 1.27mmthick substrate with a dielectric constant of 10.8. All the dimensions are in millimeters. (a) Symmetric ports. (b) Asymmetric ports.
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Figure 10.43
Frequency responses of the twopole filter structures in Figure 10.42. Circle: symmetric ports. Triangle: asymmetric ports.
frequency transmission zero in opposite directions with respect to the passband leads to different stopband behavior. It can be shown that each finite transmission zero can be tuned by adjusting the tapped point and the spacing between the two resonators, which are currently 10.2 mm and 1.0 mm, respectively, in Figure 10.42. However, in a filter design the tapped point is determined by the required external quality factor and the spacing is decided by the required interresonator coupling. This implies that this type of finite transmission zero cannot be generally independently controlled. As an example, Figure 10.44 shows two fivepole microstrip pseudocombline bandpass filters, with symmetric port excitations and asymmetric port excitations, respectively. The filters are designed on a 1.27mmthick substrate with a dielectric constant of 10.8. The filter structure in Figure 10.44(a) is symmetric with respect to resonator 3 in the middle. If we flip the last two resonators of this symmetric filter vertically, we can obtain the filter with asymmetric port excitations of Figure 10.44(b). Except for this flipping, all the physical dimensions for the two filters are kept the same as shown. With this straightforward modification, the two filters show similar passband performances, but different upper stopband responses, as illustrated in Figure 10.45. Both the filters exhibit a finite transmission zero at 2.2 GHz. This finite transmission zero is inherent due to the topology of parallelcoupled resonators, and is independent of the feed scheme. However, the filter with asymmetric ports shows an extra finite transmission zero near 2.5 GHz, which evidently improves the upper stopband response. The improved asymmetric frequency response is desired for some applications where high selectivity is required. Coupledline I/O arrangements of symmetric and asymmetric port excitations can also show different frequency characteristics. For example, Figure 10.46 depicts two twopole microstrip pseudocombline bandpass filters with coupledline I/O on a 1.27 mm thick substrate with a dielectric constant of 10.8. Again, the filter with asymmetric ports of Figure 10.46(b) is simply obtained by vertically flipping
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349
Figure 10.44
Fivepole microstrip pseudocombline bandpass filters on a 1.27mmthick substrate with a dielectric constant of 10.8. All the dimensions are in millimeters. (a) Symmetric port excitations. (b) Asymmetric port excitations.
Figure 10.45
Frequency responses of fivepole microstrip pseudocombline bandpass filters in Figure 10.44. Thick gray line: symmetric ports. Thin black line: asymmetric ports.
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Figure 10.46
Twopole microstrip pseudocombline bandpass filters with coupledline I/O on a 1.27mmthick substrate with a dielectric constant of 10.8. All the dimensions are in millimeters. (a) Symmetric ports. (b) Asymmetric ports.
the output resonator associated with the symmetric design in Figure 10.46(a). The EMsimulated performance of these two filters are plotted together in Figure 10.47 for comparison. Although each filter exhibits a finite transmission zero, the allocation is completely opposite with respect to the passband. While the finite frequency
Figure 10.47
Frequency responses of the twopole filter structures in Figure 10.46. Circle: symmetric ports. Triangle: asymmetric ports.
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351
transmission zero for the filter with symmetric ports is on the lower side of the passband, the finite frequency transmission zero for the filter with asymmetric port excitations is located on the upper side of the passband. It is also interesting to compare the frequency responses of Figure 10.47 with that of Figure 10.43. It is apparent that coupledline I/O arrangements can lead to a better stopband on the lower side of the passband than the tappedline I/O arrangement. Other filter topologies such as openloop resonator filters can also be fed symmetrically or asymmetrically. Figure 10.48(a) is the I/O arrangement of the fourpole crosscoupled openloop resonator filter discussed in Section 10.3.1, which is a symmetric feed scheme. The corresponding asymmetric feed scheme is depicted in Figure 10.48(b). Figure 10.49 shows the frequency response of these
Figure 10.48
Openloop resonator filter feed structures. (a) Symmetric feed scheme. (b) Asymmetric feed scheme.
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Figure 10.49
Frequency responses of the filter feed structures in Figure 10.48. Circle: symmetric ports. Triangle: asymmetric ports.
two structures. It is clearly seen that the asymmetric feed structure produces two finite frequency transmission zeros whereas the symmetric feed structure produces none. We can replace the feed structure of the fourpole crosscoupled openloop resonator filter of Figure 10.30(a) with the asymmetric one of Figure 10.48(b), resulting in a new fourpole arrangement, as shown in Figure 10.50(a). Its performance is demonstrated in Figure 10.50(b), where two pairs of finite transmission zeros are observed. Compared with the performance shown in Figure 10.30(b) for a similar filter with symmetric port excitations, an extra pair of finite transmission zeros is obtained from the arrangement of asymmetric ports. It is evident that this extra pair of finite transmission zeros improves the stopband performance significantly.
10.4 Interdigital Filters Using Stepped Impedance Resonators The traditional interdigital filter has been described in Chapter 9. Its main advantage is the use of /4 resonators with alternate shortcircuit and opencircuit ends, leading to a compact size when compared to other parallelcoupled filter designs. The use of stepped impedance resonators (SIR) [23–27] further reduces the overall footprint of the conventional design. In addition, a wider upper stopband can be obtained due to the frequency dispersion of stepped impedance resonators. Figure 10.51 shows a grounded /4 stepped impedance resonator, where Z 1 and Z 2 , 1 and 2 are the characteristic impedances and electric lengths of lines 1 and 2, respectively. It assumes that the characteristic impedance of line 1 is higher than that of line 2 (i.e., Z 1 > Z 2 ). Define an impedance ratio R=
Z2 Z1
(10.16)
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353
Figure 10.50
(a) Fourpole crosscoupled openloop resonator filter with asymmetric port excitations on a 1.27mmthick substrate with a dielectric constant of 10.2. (b) EMsimulated performance.
Figure 10.51
Grounded /4 stepped impedance resonator (SIR).
The maximum ratio of the first spurious resonance ( fs1 ) and the fundamental resonance ( f 0 ), when 1 = 2 , can be estimated by [24], fs1 −1 = f 0 tan−1 R √
(10.17)
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Advanced CoupledLine Filters
When R = 1, fs1 = 3f 0 , this yields the case of a traditional grounded /4 uniform impedance resonator (UIR). To have a wider upper stopband, a lower R value (R < 1) should be chosen. Moreover, for a smaller R, the physical length of the SIR can be reduced leading to a size reduction of interdigital filters. 10.4.1 Narrowband Design
For narrowband SIR interdigital filter designs, say for a fractional bandwidth (FBW ) up to 20%, the narrowband design approach discussed before, which is based on external quality factors and coupling coefficients that are directly obtainable from a lowpass prototype, can be used. As an example, a threepole SIR interdigital filter design with FBW = 0.1 at f 0 = 1 GHz is now considered. For a Chebyshev lowpass prototype with 0.1dB ripple, the lowpass element values can be found as follows from Table 9.2: g 0 = g 4 = 1.0 g 1 = g 3 = 1.0315 g 2 = 1.1474 The external quality factors and coupling coefficients are calculated by Q e1 = Q e3 =
g0 g1 = 10.315 FBW
M 12 = M 23 =
FBW = 0.0919 √g 1 g 2
The filter is to be implemented on a 1.27mmthick substrate with a dielectric constant of 6.15. The first spurious response is required to be above 4f 0 . From (10.17), this would require an R smaller than 0.7. However, the formulation of (10.17) is an approximate one that neglects the effect of the physical step discontinuity at the junction of the two lines. More accurate design of the SIR can be obtained by using fullwave EM simulation. Figure 10.52(a) is a structure created for EM modeling, showing a microstrip SIR with its one end grounded through the dielectric substrate. The designed microstrip SIR has dimensions (in millimeters) as shown in Figure 10.52(b), where the lowimpedance (Z 2 ) line has a width of 2.6 mm and the highimpedance (Z 1 ) line has a width of 0.4 mm on the given substrate, which yields an impedance ratio R ≈ 0.42. The EM simulated frequency response of the SIR is plotted in Figure 10.52(c). The SIR has its fundamental resonance at 1 GHz and its first spurious resonance at 4.4 GHz, resulting in fs1 /f 0 = 4.4. Thus, we can expect that the first spurious passband of the filter will occur after 4f 0 . The next step in the filter design is to characterize the input/output (I/O) and interresonator coupling structures. These can be done using EM simulations described previously, and (10.1) and (10.2) can be used to extract the external quality factor and the interresonator coupling, respectively. Figure 10.53(a) shows the I/O coupling stage using a tapped line for the filter design. The tapped line has
10.4 Interdigital Filters Using Stepped Impedance Resonators
Figure 10.52
355
(a) A structure of grounded microstrip SIR for EM modeling. (b) The layout of the designed microstrip SIR with dimensions in millimeters on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. (c) EM simulated resonant frequency responses of the designed SIR.
a width of 1.9 mm on the substrate with a characteristic impedance matching the terminal impedance. The external quality factor Q e depends on the tapping position of t, which is shown in Figure 10.53(b). From this design curve, we can determine a tapping position that yields the desired Q e of 10.315. For the interresonator coupling structure shown in Figure 10.54(a), the two coupled stepped impedance resonators have a fixed offset d = 0.2 mm. The design curve for the interresonator coupling is given in Figure 10.54(b), where one can determine s for the desired coupling coefficient of 0.0919.
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Figure 10.53
(a) The I/O stage for the SIR filter design. (b) Design curve (Q e ).
The final design of the filter is shown in Figure 10.55(a), where all the dimensions are in millimeters. The length of the lowimpedance line for the input and output resonators is 13.8 mm instead of 12.7 mm to compensate for the frequency shifting due to the effect of the tapped I/O lines. The performance of the filter is illustrated in Figure 10.55(b). As can be seen, the desired passband is centered at 1 GHz and the first spurious passband occurs at 4.4 GHz as expected. 10.4.2 Wideband Design
No accurate design procedure exists for SIR filter designs with larger (> 30%) bandwidths. The direct use of the narrowband design approach does not give the correct coupling values required in wideband applications [27].
10.4 Interdigital Filters Using Stepped Impedance Resonators
Figure 10.54
357
(a) Interresonator coupling structure for the SIR filter design. (b) Design curve for the interresonator coupling.
A practical approach has been suggested in [27] for the design of wide bandwidth interdigital filters using SIR. The proposed design approach is based on the concept of parameter mapping between two wideband filters, that is, a conventional interdigital filter with uniform impedance resonators (UIR) and a desired interdigital filter using stepped impedance resonators. The design of a wideband conventional interdigital filter with uniform impedance resonator (UIR) is available as discussed in Chapter 9. The ninepole wideband SIR interdigital filter is designed at a center frequency of 1.5 GHz and a fractional bandwidth of 33.3%, which are the same as those of the conventional interdigital filter design using uniform impedance resonator (UIR) presented in Figure 9.12(a). The SIR filter is also implemented on the same dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. The starting point for the design of the SIR filter is to determine the design parameters from the UIR interdigital filter design. For this purpose, the circuit layout shown in Figure 9.12(a) is broken down into smaller parts. The simulation
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Advanced CoupledLine Filters
Figure 10.55
(a) Layout of the designed narrowband microstrip SIR interdigital filter on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. All dimensions shown are in millimeters. (b) EMsimulated performance of the filter.
10.5 DualBand Filters
359
results on smaller parts can be used to obtain a new set of design parameters that can be applied to the SIR design. For example, the I/O stage of the UIR interdigital filter in Figure 9.12(a) is simulated alone with an arrangement shown in Figure 10.56(a), where the circle with small triangles at one end of the I/O resonator represents the via hole grounding and port 2 is weakly coupled to the I/O stage in order to extract an external quality factor (Q e ) through the simulated twoport transmission (i.e., S 21 ) response. Port 2 should be weakly coupled so that its effect on Q e is negligible. The full line in Figure 10.56(c) indicates the EMsimulated frequency response of the I/O stage of the UIR interdigital filter, where the frequency axis is normalized to the resonant frequency at which a resonant peak is observed. It is interesting to note that the frequency response is not symmetrical with respect to the resonant frequency. This is mainly due to frequencydependent coupling of the excitation port, and is a typical response for the I/O stage of a wideband filter. Nevertheless, from the EM simulated response shown, an external quality factor may simply be extracted using Qe =
1 ⌬ f N_3 dB
where ⌬ f N_3 dB is the normalized 3dB bandwidth. In the case of Figure 10.56(c), Q e = 2.16 is extracted for the I/O stage of the UIR interdigital filter, which will be used as a new design parameter for the SIR filter. In practice, the I/O stage of the SIR filter can be determined with an arrangement as shown in Figure 10.56(b) to find an external quality factor of 2.16. In the next step of the wideband SIR interdigital filter design, each pair of adjacent coupled resonators in the designed UIR interdigital filter of Figure 9.12(a) is modeled in EM software. Figure 10.57(a) demonstrates how to weakly excite a pair of adjacent resonators of the UIR interdigital filter to extract a new coupling coefficient. As an example, the full line shown in Figure 10.57(c) indicates the EM simulated S 21 response for the interresonator coupling between resonators 2 and 3 of the designed UIR interdigital filter. Using this curve, a coupling coefficient of M 23 = 0.198 is found using (10.2). Thus, to determine the spacing between resonators 2 and 3 of the SIR interdigital filter, an arrangement shown in Figure 10.57(b) is employed to find a matching frequency response as shown in Figure 10.57(c) with the broken line. In this way, all the spacings between adjacent resonators of the SIR interdigital filter can be determined. The designed ninepole SIR interdigital filter, based on the concept of parameter mapping, is illustrated in Figure 10.58(a). The measured passband performance is shown in Figure 10.58(b). The wideband response of the SIR interdigital filter is plotted in Figure 10.59 along with that of the UIR interdigital filter. It is evident that the desired passbands of the two filters match closely with each other. The SIR interdigital filter exhibits a better upper stopband with its first spurious response occurring beyond 3f 0 .
10.5 DualBand Filters There is an increasing demand for dualband communication systems [e.g., a wireless local area network (WLAN) device operated at 802.11a (5 GHz) and 802.11b
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Figure 10.56
(a) The I/O stage of the UIR interdigital filter. (b) The I/O stage of the SIR interdigital filter. (c) EMsimulated frequency responses of the I/O stages. (Full line: UIR. Broken line: SIR.)
10.5 DualBand Filters
Figure 10.57
361
(a) Interresonator coupling stage of the UIR interdigital filter. (b) Interresonator coupling stage of the SIR interdigital filter. (c) EMsimulated frequency responses of the interresonator coupling stages. (Full line: UIR. Broken line: SIR.)
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Figure 10.58
(a) Layout of the ninepole microstrip SIR interdigital filter on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. (b) Measured passband performance.
(2.4 GHz)]. Dual band systems can be divided into two categories. The first involves combining two independent bandpass filters with a switch used to choose one signal path at a time; the second category provides dualband operation concurrently. Traditionally, simultaneous operation at different frequency bands can be achieved
10.5 DualBand Filters
Figure 10.59
363
Comparison of wideband responses of the designed ninepole microstrip SIR and UIR interdigital filters.
by combining multiple independent signal paths. This approach needs a large footprint. Recently, a concurrent dualband receiver architecture was proposed to overcome these issues [28]. To obtain a concurrent dualband system, one of the important devices required is a dualband filter. In general, a dual band filter can be achieved by various approaches. One approach is combining two independent bandpass filters with the common input/ output ports [29]. Although the specification of the two passbands can be easily met and designed separately, the area of the filter designed by this method is large. Another approach is cascading a wideband filter with a band stop structure [30]. This approach would seem more suitable when the two passbands are not widely separated, but the size can still be large due to the cascading of the two different filtering structures. A more interesting approach is using stepped impedance resonators to design a single coupledline filter that has two desired passbands [31–34]. The basic idea is to make use of the firstharmonic frequency of the stepped impedance resonator SIR, which is easily adjustable. This will be discussed in greater detail next. Figure 10.60 illustrates a pair of planar stepped impedance resonators, where TT ′ denotes the symmetrical plane. Z 1 and Z 2 are the characteristic impedances of the two transmission lines having widths of W 1 and W 2 , respectively. Define an impedance ratio R = Z 2 /Z 1 . Let f 1 and f 2 be the frequencies of first two resonant modes of the SIR. The first resonant mode at f 1 can be seen as an odd mode because there is a virtual short circuit in the middle of resonator. On the other hand, the second resonant mode at f 2 is an even mode as there is a virtual open circuit in the middle of the resonator. For the SIR in Figure 10.60(a), it can be shown that the frequency ratio f 2 / f 1 > 2 because R < 1. Similarly, it can be shown for the SIR in Figure 10.60(b) that 1 < f 2 / f 1 < 2 since R > 1. In a practical design of a dualband filter, f 1 will be designed to be the midband frequency of the lower passband, while f 2 will be the midband frequency of the higher passband. Thus,
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Figure 10.60
(a) SIR for R < 1 and f 2 /f 1 > 2. (b) SIR for R > 1 and 1 < f 2 /f 1 < 2.
if the midband frequency of the higher frequency filter is greater than twice that of the lower frequency filter, the SIR of Figure 10.60(a) should be used for the filter design. On the contrary, if the midband frequency of the higher frequency filter is smaller than twice that of the lower frequency filter, the SIR of Figure 10.60(b) should be used. The design of SIR usually requires EM simulations for more accurate results. For example, Figure 10.61 shows EMsimulated results of the microstrip SIR of Figure 10.60(a) on a dielectric substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm. The first resonant frequency f 1 and the frequency ratio of f 2 / f 1 are given as a function of W 2 for W 1 = 0.5 mm and l 1 = l 2 = 10 mm in Figure 10.61(a). It is observed from the plot that when W 2 is increased, f 1 decreases whereas the ratio f 2 / f 1 increases and both vary quite linearly with respect to W 2 . In Figure 10.61(b), f 1 and f 2 / f 1 are plotted as a function of l 2 while l 1 + l 2 is kept constant (20 mm) and W 1 = 0.5 mm and W 2 = 1.5 mm. It is interesting to note that while the ratio of f 2 / f 1 increases with an increase in l 2 , the first resonant frequency f 1 remains nearly unaffected. This implies that f 1 is less dependent on the ratio of l 2 /l 1 . To design a dualband filter with the SIR, more physical design parameters are needed than to design a singleband filter. In this case, the ratios W 2 /W 1 and l 2 /l 1 are often used as design parameters, not just for determining f 1 and f 2 , but also for other considerations such as to achieve a wider upper stopband after the second passband as well as to facilitate desired couplings. To demonstrate the characteristics of the SIR of Figure 10.60(b), a microstrip SIR with R > 1 on a substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm is simulated. The EMsimulated results are given in Figure 10.62, where
10.5 DualBand Filters
Figure 10.61
365
EMsimulated results of the first resonant frequency f 1 and the frequency ratio of f 2 /f 1 for a SIR with R < 1 in the form of Figure 10.60(a) on a substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm. (a) As a function of W 2 . (b) As a function of l 2 .
the first resonant frequency f 1 and the frequency ratio of f 2 / f 1 are plotted as a function of W 1 for W 2 = 0.5 mm and l 1 = l 2 = 10 mm. It is clear that when W 1 becomes wider, the characteristic impedance Z 1 of the microstrip line becomes lower. Thus, the impedance ratio R increases, which leads to a decrease of the ratio of f 2 / f 1 as expected. Figure 10.62(a) also shows that f 1 increases as W 1 is increased. Both f 1 and f 2 / f 1 can also be controlled by adjusting l 2 as shown in Figure 10.62(b). To design dualband filters using the SIR, there are two sets of external quality factors and interresonator coupling coefficients that need to be realized for the two desired passbands. In general, to implement the external quality factors for
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Figure 10.62
EMsimulated results of the first resonant frequency f 1 and the frequency ratio of f 2 /f 1 for a SIR with R > 1 in the form of Figure 10.60(b) on a substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm. (a) As a function of W 1 . (b) As a function of l 2 .
the two designed passbands, two or more physical design parameters are needed at the I/O stages. Similarly, the interresonator couplings for the two passbands need to be controlled by at least two physical parameters between adjacent coupled resonators. Some design examples of this type of filter are reported in [32–34]. Figure 10.63(a) illustrates a 5pole microstrip dual band filter design, which uses five stepped impedance resonators on a substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm. The SIR used has an antisymmetrical shape that makes the implementation of interresonator coupling easy. The I/O stages use tapped lines, which also include the dualfrequency transformer. The design details
10.5 DualBand Filters
Figure 10.63
367
(a) Photograph of a fivepole microstrip dualband filter using the SIR on a substrate with a dielectric constant of 2.2 and a thickness of 0.508 mm. (b) Simulated and measured performance. (From: [32]. 2005 IEEE. Reprinted with permission.)
are available in [32]. Figure 10.63(b) plots the responses of the fivepole dualband filter having f 1 = 2.45 GHz, f 2 = 5.8 GHz, and the fractional bandwidths for the two passbands are 12% and 7%, respectively.
References [1] [2]
[3]
[4] [5]
Riddle, A., ‘‘High Performance Parallel Coupled Microstrip Filters,’’ IEEE MTTS Int. Microwave Symp. Dig., 1988, pp. 427–430. Kuo, J.T., S.P. Chen, and M. Jiang, ‘‘ParallelCoupled Microstrip Filters with OverCoupled End Stages for Suppression of Spurious Responses,’’ IEEE Microwave Wireless Comp. Lett., Vol. 13, October 2003, pp. 440–442. Jiang, M., M. H. Wu, and J. T. Kuo, ‘‘ParallelCoupled Microstrip Filters with OverCoupled Stages for Multispurious Suppression,’’ IEEE MTTS Int. Microwave Symp. Dig., 2005, pp. 687–690. Sonnet em, Sonnet Software, Inc., New York. Matthaei, G. L., ‘‘Design of WideBand (and NarrowBand) Bandpass Microwave Filters on the Insertion Loss Basis,’’ IRE Trans. Microwave Theory and Tech., Vol. MTT8, November 1960, pp. 580–593.
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[23]
[24]
Hong, J.S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, New York: John Wiley & Sons, 2001. Lopetegi, T., et al., ‘‘New Microstrip ‘WigglyLine’ Filters with Spurious Passband Suppression,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT49, September 2001, pp. 1593–1598. Lopetegi, T., et al., ‘‘Microstrip ‘WigglyLine’ Bandpass Filters with Multispurious Rejection,’’ IEEE Microwave and Wireless Letters, Vol. 14, November 2004, pp. 531–533. Wang, S. M., et al., ‘‘Miniaturized Spurious Suppression Microstrip Filter Using Meandered Parallel Coupled Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT53, February 2005, pp. 747–753. Vincent, P., J. Culver, and S. Eason, ‘‘Meandered Line Microstrip Filter with Suppression of Harmonic Passband Response,’’ IEEE MTTS Int. Microwave Symp. Dig., 2003, pp. 1905–1908. Vela´zquezAhumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filters with GroundPlane Aperture for Spurious Band Suppression and Enhanced Coupling,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT52, March 2004, pp. 1082–1086. Vela´zquezAhumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filters with Floating GroundPlane Conductor for SpuriousBand Suppression,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT53, May 2005, pp. 1823–1828. Hong, J.S., ‘‘ComputerAided Synthesis of Mixed Cascaded Quadruplet and Trisection (CQT) Filters,’’ 31st European Microwave Conference Proceedings, London, U.K., September 2001, Vol. 3, pp. 5–8. Hong, J.S., and M. J. Lancaster, ‘‘Couplings of Microstrip Square OpenLoop Resonators for CrossCoupled Planar Microwave Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT44, November 1996, pp. 2099–2109. MontejoGarai, J. R., ‘‘Synthesis of NEven Order Symmetric Filters with N Transmission Zeros by Means of SourceLoad Cross Coupling,’’ Electron. Lett., Vol. 36, February 2000, pp. 232–233. Amari, S., ‘‘Direct Synthesis of Folded Symmetric Resonator Filters with SourceLoad Coupling,’’ IEEE Microwave Wireless Comp. Lett., Vol. 11, June 2001, pp. 264–266. Cameron, R. J., ‘‘Advanced Coupling Matrix Synthesis Techniques For Microwave Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. 51, January 2003, pp. 1–10. GariaLamperez, A., et al., ‘‘Synthesis of CrossCoupled Lossy Resonator Filters with Multiple Input/Output Couplings by Gradient Optimization,’’ IEEE APS Int. Symp. Proc., Vol. 2, June 2003, pp. 52–55. Liao, C.K., and C.Y. Chang, ‘‘Design of Microstrip Quadruplet Filters with SourceLoad Coupling,’’ IEEE Trans. Microwave Theory Tech., Vol. 53, July 2005, pp. 2302–2308. Liao, C.K., and C.Y. Chang, ‘‘Modified ParallelCoupled Filter with Two Independently Controllable Upper Stopband Transmission,’’ IEEE Microwave Wireless Comp. Lett., Vol. 15, December 2005, pp. 841–843. Lee, S. Y., and C. M. Tsai, ‘‘New CrossCoupled Filter Design Using Improved Hairpin Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. 48, December 2000, pp.2482–2490. Hong, J.S., and M. J. Lancaster, ‘‘Recent Progress in Planar Microwave Filters,’’ Proc. of 3rd International Conference in Microwave and Millimeter Wave Technology, Beijing, 2002, pp. 1134–1137. Makimoto, M., and S. Yamashita, ‘‘Bandpass Filters Using Parallel Coupled Stripline Stepped Impedance Resonators,’’ IEEE Trans Microwave Theory Tech., Vol. MTT28, December 1980, pp. 1413–1417. Sagaawa, M., M. Makimoto, and S. Yamashita, ‘‘Geometrical Structures and Fundamental Characteristics of Microwave SteppedImpedance Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. 45, July 1997, pp. 1078–1084.
10.5 DualBand Filters [25]
[26] [27] [28] [29]
[30]
[31] [32]
[33] [34]
369
Ting, S. W., K. W. Tam, and R. P. Martins, ‘‘Novel Interdigital Microstrip Bandpass Filter With Improved Spurious Response,’’ IEEE International Symposium on Circuits & Systems (ISCAS) 2004, pp. 984–987. Pang, H.K., et al., ‘‘A Compact Microstrip /4SIR Interdigital Bandpass Filter with Extended Stopband,’’ IEEE MTTS Int. Microwave Symp. Dig., 2004, pp. 1621–1625. Thomson, N., et al., ‘‘Practical Approach for Designing Miniature Interdigital Filters,’’ 35th European Microwave Conference Proceedings, Paris, 2005, pp. 1251–1254. Chang, S.F., et al., ‘‘A DualBand RF Transceiver for Multistandard WLAN Applications,’’ IEEE Trans. Microwave Theory Tech., Vol. 53, March 2005, pp. 1048–1055. Miyake, H., et al., ‘‘A Miniaturized Monolithic Dual Band Filter Using Ceramic Lamination Technique for Dual Mode Portable Telephones,’’ IEEE MTTS Int. Microwave Symp. Dig., 1997, pp. 789–792. Tsai, L. C., and C. W. Hsue, ‘‘DualBand Bandpass Filters Using Equal Length CoupledSerialShunted Lines and ZTransform Technique,’’ IEEE Trans. Microwave Theory Tech., Vol. 52, April 2004, pp. 1111–1117. Chang, S. F., Y. H. Jeng, and J. L. Chen, ‘‘DualBand StepImpedance Bandpass Filter for Multimode Wireless LANs,’’ Electron. Lett., Vol. 40, January 2004, pp. 38–39. Kuo, J. T., T. H. Yeh, and C. C. Yeh, ‘‘Design of Microstrip Bandpass Filters with a DualPassband Response,’’ IEEE Trans. Microwave Theory Tech., Vol. 53, April 2005, pp. 1331–1337. Chuang, M. J., ‘‘Concurrent Dual Band Filter Using Single Set of Microstrip OpenLoop Resonators,’’ Electronics Letters, Vol. 41, September 1, 2005, pp. 1013–1014. Sun, S., and L. Zhu, ‘‘Compact DualBand Microstrip Bandpass Filter Without External Feeds,’’ IEEE Microwave and Wireless Components Letters, Vol. 15, October 2005, pp. 644–646.
CHAPTER 11
Filters Using Advanced Materials and Technologies 11.1 Introduction Advanced materials and technologies have stimulated the development of novel RF/microwave filters for different applications. This chapter aims to cover some recent developments, including superconductor filters, micromachined filters, filters using advanced dielectric materials such as low temperature cofired ceramic (LTCC) and liquid crystal polymer (LCP), filters for emerging ultrawideband (UWB) technology, and filters based on electromagnetic metamaterials.
11.2 Superconductor CoupledLine Filters Superconductors are attractive for use in passive microwave circuits [1]. These materials have very low surface resistance in comparison to normal metals. For example, the hightemperature superconductors, which have a critical temperature above 77K, being used in microwave applications, exhibit a surface resistance at 1 GHz that is three to four orders of magnitude lower than that of copper under equivalent conditions (i.e., at 77K). One important application of high temperature superconductivity is in microwave filters. The extremely low resistance of HTS materials has enabled the realization of miniature thin film filters with low insertion loss and high selectivity. Excellent performance of HTS thin film filters has been demonstrated [2–17]. The driving force behind the development of HTS filters remains wireless applications such as mobile and satellite communications [2–16]. Superconductor filters also find applications in radio astronomy or radio telescope receivers [17]. Filters with crosscoupled resonators discussed in Chapter 10 form the basis for the design of highperformance superconducting filters. Two typical examples are described next. 11.2.1 Cascaded Quadruplet and Triplet Filters
Figure 11.1 is a cascaded quadruplet trisection (CQT) coupling structure for a 10pole crosscoupled resonator filter, where each numbered node represents a resonator; the solid lines indicate direct couplings and the dash lines represent cross
371
372
Filters Using Advanced Materials and Technologies
Figure 11.1
10pole CQT bandpass filter coupling structure.
couplings. Differing from the CQ coupling structure discussed in Section 10.3.1, two trisections, that is, resonators 1 to 3 and 8 to 10 with the cross couplings M 1, 3 and M 8, 10 , respectively, are used to produce and control two transmission zeros close to the passband independently. This makes the filter easier to tune. The only quadruplet section of resonators 4 to 7 with the cross coupling M 4, 7 is used to generate another pair of transmission zeros.
Filter Synthesis
The first step in filter design is to obtain the general coupling matrix and the scaled external quality factors for the CQT filter from the filter synthesis. A computeraided synthesis in [18] or other methods such as [19] can be used. The results of filter synthesis are given by
[m] =
0.00038
0.57723
0.64634
0
0
0
0
0
0
0
0.57723
−0.84209
0.32238
0
0
0
0
0
0
0
0.64634
0.32238
0.09623
0.55502
0
0
0
0
0
0
0
0
0.55502
0.03678
0.52039
0
−0.11031
0
0
0
0
0
0
0.52039
0.01606
0.63437
0
0
0
0 0
0
0
0
0
0.63437
−0.01606
0.52039
0
0
0
0
0
−0.11031
0
0.52039
−0.03678
0.55502
0
0
0
0
0
0
0
0
0.55502
−0.09623
0.32238
−0.64634
0
0
0
0
0
0
0
0.32238
0.84209
0.57724
0
0
0
0
0
0
0
−0.64634
0.57724
−0.0004
q e1 = q e10 = 0.90557
(11.1)
The HTS filter is designed to have a 10MHz bandwidth from 1,950 MHz to 1,960 MHz. Hence, the center frequency f 0 and fractional bandwidth FBW are given by f0 =
√1950 × 1960 = 1954.99361 MHz FBW =
10 = 0.00512 f0
Using (10.6) and (10.7), a set of design parameters including the coupling coefficients M ij and external quality factors (Q e1 and Q e2 ) for the given fractional bandwidth can be obtained.
11.2 Superconductor CoupledLine Filters
373
It should be noted that as a result of using trisections, the CQT filter is not synchronously tuned, which is different from a CQ filter. The synthesized filter is an ideal one and its frequency response can also be computed using the formulation of (10.8). Figure 11.2 displays the theoretical frequency response of the filter. As can be seen from the ideal frequency response, each cascaded trisection in the coupling structure produces a single attenuation pole at a finite frequency either below or above the passband depending upon the polarity of the cross coupling. The crosscoupled quadruplet section produces a pair of attenuation poles at finite frequencies in order to further improve the selectivity. The advantages of the CQT filter are that the quasielliptic function design results in fewer resonators and thus lower passband insertion loss. Additionally, the tuning effort is reduced due to the independent effect of the cross couplings on filter response. Filter Implementation
The second step in the HTS filter design is to implement the coupling scheme of Figure 11.1 with a proper microstrip configuration. For this HTS filter development, an rcut sapphire substrate was used. Since the dielectric properties of sapphire are anisotropic, the dielectric constant is not a single value but a tensor. To make the filter deign simple, an effective dielectric constant of 10.0556 for the rplane sapphire substrate was used [14]. However, in order to make this design approach work, it is important to arrange all microstrip resonators in such a way that they experience the same permittivity tensor on the anisotropic substrate. The implementation of two trisections is illustrated in Figure 11.3, where all the resonators are in principle a halfwavelength long at the center frequency and meandered in a special manner as shown, to facilitate the desired couplings and miniaturize the filter as well. The trisection of Figure 11.3(a) was developed to implement the couplings M 1, 2 , M 2, 3 , and M 1, 3 . For realizing M 8, 9 , M 9, 10 , and M 8, 10 the trisection of Figure 11.3(b) was used. All the direct couplings are realized
Figure 11.2
Theoretical response of the 10pole CQT filter.
374
Filters Using Advanced Materials and Technologies
Figure 11.3
Two microstrip trisections. (a) For producing a transmission zero near the low side of the passband. (b) For producing a transmission zero near the high side of the passband.
with proper proximity only, whereas each cross coupling uses a crossing line with capacitive probes coupled to the two nonadjacent resonators. The difference between the two triplet configurations lies in the locations of capacitive probes as shown, which makes their frequency characteristics distinct. The trisection of Figure 11.3(a) is supposed to produce a transmission zero (attenuation pole) near the low side of the passband, whereas the trisection of Figure 11.3(b) is designed to produce a transmission zero near the high side of the passband. A fullwave EM simulation was carried out to confirm these distinct characteristics as shown in Figure 11.4, where a single transmission zero on either lower or upper side of the passband is observed. In the simulation, each of trisections was weakly excited and the simulation was performed using a commercially available EM simulator [20]. Figure 11.5 shows the overall configuration of the 10pole microstrip CQT bandpass filter.
Sensitivity Analysis
For a narrowband filter it is important to carry out a sensitivity analysis since a narrowband filter tends to be more sensitive to the tolerances in both the design and fabrication. It has been found that the major cause of performance variance for the designed filter is the nonuniformity of the substrate thickness. For a quoted tolerance of ±5 m in the substrate thickness of 430 m, the impact on the filter performance is significant. To demonstrate this, a sensitivity analysis based on the Monte Carlo method was performed using a commercially available microwave design tool [21] and the results for both transmission and reflection responses are plotted in Figure 11.6. The shading in each diagram illustrates the sensitivity of the filter response against a variation of substrate thickness. It is evident from the given results that the distortion in the desired filter performance is severe. Hence, the tuning of this narrowband filter is a must. It can also be shown through a further sensitivity analysis that the tuning of resonator frequencies is much more important than the tuning of couplings. This was used as a guideline for the filter experiments discussed later on.
11.2 Superconductor CoupledLine Filters
375
Figure 11.4
Frequency responses of the microstrip trisections. (a) Exhibiting a transmission zero near the low side of the passband. (b) Exhibiting a transmission zero near the high side of the passband.
Figure 11.5
10pole microstrip CQT bandpass filter configuration.
376
Filters Using Advanced Materials and Technologies
Figure 11.6
Sensitivity analysis against the tolerance in the uniformity of substrate thickness: (a) transmission and (b) reflection.
Experiment
A filter was fabricated on a 0.43mmthick sapphire wafer with doublesided yttrium barium copper oxide (YBCO) superconducting films. The YBCO thin films have a thickness of 300 nm and a characteristic temperature of 87K. Both sides of the wafer were goldplated with 200nmthick gold (Au) for the RF contacts. This wafer is commercially available from THEVA GmbH. The fabricated HTS filter had a size of 47 × 17 mm, which was assembled on a gold plated titanium carrier and placed into a brass test housing as shown in Figure 11.7(a). This assembly was then placed in a cryogenic dewar. As this type of narrowband filter is more sensitive to frequency tuning, sapphire tuners were used to tune the resonant frequencies of all 10 HTS resonators. This is illustrated in Figure 11.7(b). A microwave vector network analyzer was used for all the RF measurements made under cryogenic conditions.
11.2 Superconductor CoupledLine Filters
Figure 11.7
377
Fabricated HTS filter: (a) in the test housing and (b) the lid incorporating sapphire tuners.
Figure 11.8 shows the measured performance of the HTS filter at an operational temperature of 65K. A 10MHz bandwidth was measured with a midband frequency of about 1,973 MHz. The measured midband frequency is higher than the design frequency. This is attributed to the assumed value of dielectric constant for the sapphire substrate and can be corrected in the next iteration of the design. It can be seen that both pairs of transmission zeros are present. The lowest measured insertion loss in the passband is 0.2 dB. This corresponds to a resonator Q of greater than 50,000. The measured return loss (shown at 5 dB per division) is better than −12 dB across the passband. The measured wideband response of the filter is plotted in Figure 11.9, showing the excellent rejection and clean response without harmonics or spurious modes over the entire UMTS transmission band (2,110–2,170 MHz). 11.2.2 HighOrder Selective Filters with GroupDelay Equalization
Highly selective filters are in great demand for applications with stringent selectivity requirements. For instance, the capacity and coverage of a basestation receiver are determined primarily by the receiver selectivity and sensitivity on the uplink. The selectivity can be significantly increased with the use of higherorder filters [10], and there is a trend to develop highorder HTS filters to take advantage of miniature high Q resonators [8, 16].
378
Filters Using Advanced Materials and Technologies
Figure 11.8
Measured results for the 10pole superconducting CQT filter at 65K.
Figure 11.9
Measured wideband response of the 10pole superconducting CQT filter.
In many communication systems, flat group delay of a bandpass filter is also a requirement in addition to its selectivity. The group delay represents the true signal delay between the input and output ports of a communication channel, such as a filter, and is defined in (9.4). The demand for a flat group delay is particularly true for highcapacity communication systems where it is essential for a bandpass filter to have a good linearphase response or flat group delay over the central region of the passband. Due to the requirement of selectivity, some deterioration in the group delay performance at the edges of the passband is allowed. Unfortunately, higherorder highly selective filters tend to result in a poor phase performance even over the band center. To demonstrate this, Figure 11.10 shows a comparison of three different types of highorder filters (i.e., 30pole Chebyshev, 18pole quasielliptic function, and 18pole quasielliptic function with linear phase). All three filters have a passband of 15 MHz from 1,960 MHz to 1,975 MHz with the same ripple level and are supposed to meet a selectivity of 70dB rejection
11.2 Superconductor CoupledLine Filters
Figure 11.10
379
Comparison of performances of three highorder filters: (a) transmission response and (b) group delay response.
bandwidth of about 16 MHz as illustrated in Figure 11.10(a). It is evident that, to meet this requirement, the Chebyshev filter requires 30 resonators compared to 18 required by quasielliptic function filters. The need for the larger number of resonators for the Chebyshev filter not only leads to a larger size, but also results in higher insertion loss (due to finite resonator Q) and greater variation of group delay over the pass band. The group delay response is shown in Figure 11.10(b). Apparently, it is not good practice to realize highly selective filters with the Chebyshev response. While the two quasielliptic function filters are superior in terms of achieving a high selectivity with a lower number of resonators, the one with a linear phase response is more attractive as it exhibits a flatter group delay over the band center, which can be seen from Figure 11.10(b). It is therefore desirable to develop highorder HTS bandpass filters, not only to meet the selectivity requirement, but also to provide capability for selfequalized
380
Filters Using Advanced Materials and Technologies
group delay over the central part of the passband. The design, modeling, and test results of such a highorder HTS microstrip bandpass filter are presented next.
Design and Modeling
The filter model or coupling structure of an eighteenpole filter is shown in Figure 11.11. Each node with a number represents a resonator. Resonators 1 and 18 are coupled to the input and output ports, respectively, denoted by external quality factors, Q e1 and Q e2 . The unbroken lines between adjacent resonators indicate the direct couplings. There are four cross couplings, as indicated by the broken lines, between resonators 2 and 5, 6 and 9, 10 and 13, and 14 and 17. These cross couplings are denoted by coupling coefficients M 2, 5 , M 6, 9 , M 10, 13 , and M 14, 17 . As can be seen, each cross coupling is associated with a quadruplet section, and hence the filter has basically a cascaded quadruplet (CQ) structure. The advantage of this coupling structure lies in that each of the quadruplet sections can be arranged either to produce a pair of transmission zeros (attenuation poles) at finite frequencies in order to achieve higher selectivity for a given number of resonators, or to result in a linear phase performance to achieve a selfequalization of group delay. For our design, only one quadruple section, which consists of resonators 10 to 13, will be used for the group delay equalization, while the other three quadruplet sections are arranged for high selectivity. The cross coupling in each quadruple can be tuned independently, making the tuning easier for such a highorder filter. An 18pole filter was designed to have a 15MHz passband at a center frequency of 1,967.5 MHz. To this end, the circuit model of Figure 11.12 was created with Microwave Office, a commercially available software [21]. In this circuit model, all the LC resonators, which are supposed to resonate at the central frequency f 0 = 1/2 √L 0 C 0 , have an inductance of L 0 = 0.03291 nH and a capacitance of C 0 = 198.83 pF. Each quarterwavelength line has electrical length = ±90 degrees at the central frequency of the passband and functions as an immittance inverter to represent the coupling between the associated pair of resonators. The other
Figure 11.11
Figure 11.12
Coupling structure for an 18pole selective filter with groupdelay equalization.
Circuit model showing the coupling structure of the 18pole CQ filter.
11.2 Superconductor CoupledLine Filters
381
circuit parameters are related, following the formulations given in [22], to the set of desired coupling coefficients M jk and external quality factors Q ei and Q eo . Figure 11.13 shows the simulated frequency responses of the filter, based on the circuit model of Figure 11.12 with desired coupling coefficients and external quality factors. As can be seen, the filter exhibits three pairs of transmission zeros (attenuation poles) at finite frequencies to increase the selectivity, while exhibiting a flat group delay over about the middle 50% of the passband. The design of this type of microstrip filter is based on the procedure described in Chapter 10 or [22]. In order to implement this type of filter in microstrip, two basic quadruplet sections of coupled microstrip resonators, as shown in Figure 11.14, have been investigated. Each quadruplet section is comprised of four meandered microstrip resonators as shown, and the four resonators are arranged in such a way that the direct coupling between adjacent resonators results from the
Figure 11.13
Theoretical responses of the 18pole bandpass filter with selfequalization of group delay: (a) magnitude and (b) group delay.
382
Filters Using Advanced Materials and Technologies
Figure 11.14
Two microstrip quadruplet sections: (a) for the realization of a pair of transmission zeros at finite frequencies, and (b) for the realization of group delay equalization.
proximity of the resonators, while a narrow line coupled to the first and last resonators is used for the cross coupling. By inspecting the quadruplet configurations in Figure 11.14, one can notice that they are almost the same except for the spacing between the two inner coupledresonators in each quadruplet section. This spacing in Figure 11.14(a) is much smaller than that in Figure 11.14(b). As a matter of fact, it is this spacing difference that makes the characteristics of these two quadruplets totally different. It can be shown that the coupling between the two inner coupledresonators in Figure 11.14(a) is dominated by the electric coupling, whereas the magnetic coupling is dominant for the two inner coupledresonators in Figure 11.14(b). Because these two couplings have opposite effects, which tend to cancel out each other, we may denote the electric coupling as a negative coupling, and the magnetic coupling as a positive one. This allows us to use the microstrip configuration of Figure 11.14(a) to implement the desired couplings for those quadruplet sections in Figure 11.11 that produce finite frequency transmission
11.2 Superconductor CoupledLine Filters
383
zeros. On the other hand, the microstrip configuration of Figure 11.14(b) is used for the realization of the required couplings for the quadruplet in Figure 11.11 that results in groupdelay equalization. For example, the desired coupling matrix for the first quadruplet (i.e., coupled resonators 2, 3, 4, and 5) is given by
冤
冥 冤
0
M 23
0
M 25
M 23
0
M 34
0
0
M 34
0
M 45
M 25
0
M 45
0
= 10−2 ⭈
冥
0
0.4089
0
0.1822
0.4089
0
−0.5706
0
0
−0.5706
0
0.3460
0.1822
0
0.3460
0 (11.2)
where the coupling for M 34 is negative. This quadruplet is responsible for a pair of transmission zeros observed in the magnitude response in Figure 11.13(a). A fullwave EM simulation has been performed for a microstrip quadruplet section shown in Figure 11.14(a) to determine physical parameters corresponding to the coupling matrix of (11.2), and the results are plotted (unbroken line) in Figure 11.15(a). Note that the response was obtained by weakly coupling the quadruplet to the input/output ports. The EM simulation was done using a commercially available simulator [20]. A quadruplet resonatorcircuit model based on (11.2) can also be used to compute the theoretical response as shown with the broken line in Figure 11.15(a). As can be seen, there is good agreement between the theory and simulation, and this microstrip quadruplet realization does indeed produce the desired transmission zeros. Similarly, the EM simulation and circuit modeling can be done for the quadruplet which consists of coupled resonators 10, 11, 12, and 13 which has a coupling matrix given by
冤
0
M 10, 11
0
M 10, 13
M 10, 11
0
M 11, 12
0
0
M 11, 12
0
M 12, 13
M 10, 13
0
M 12, 13
0
冥 冤 = 10−2 ⭈
0
0.3419
0
0.1785
0.3419
0
0.2047
0
0
0.2047
0
0.3423
0.1785
0
0.3423
冥
0 (11.3)
In this case, the microstrip configuration of Figure 11.14(b) was used in the simulation. The simulated and theoretical results are plotted in Figure 11.15(b), where no finitefrequency transmission zeros are observable, which is expected as this quadruplet is only for the group delay equalization. Again, the good agreement between theory and simulation validates the microstrip realization of this type of quadruplet section. Figure 11.16 shows the final layout of the designed 18pole superconducting microstrip filter. The substrate is an rcut sapphire and the filter size is 74 mm × 17 mm. It can be recognized in Figure 11.16 that the third quadruplet from the left is for the selfequalization of group delay.
384
Filters Using Advanced Materials and Technologies
Figure 11.15
Frequency responses of quadruplet coupled resonators: (a) for the realization of a pair of transmission zeros at finite frequencies, and (b) for the realization of group delay equalization.
Figure 11.16
Layout of the designed 18pole HTS microstrip filter on sapphire substrate.
Fabrication and Measurement
The filter was fabricated on a 0.43mmthick sapphire wafer with doublesided YBCO films. The YBCO thin films have a thickness of 300 nm and a characteristic temperature of 87K. Both sides of the wafer are goldplated with 200nmthick gold (Au) for the RF contacts. The fabricated HTS filter was assembled into a test housing with two Kconnectors and a lid as shown in Figure 11.17, for measure
11.3 Micromachined Filters
Figure 11.17
385
Photo of the assembled 18pole HTS filter in a testing housing with sapphire tuners on the lid.
ments. One can see clearly that the lid has accommodated a number of sapphire tuners. Most of the tuners were used in the experiment to tune the resonator frequencies as suggested by a sensitivity study, which is similar to that discussed in the previous section for the CQT filter. RF measurements were done using an HP network analyzer and immersing the filter in a cryogenic cooler. Figure 11.18 shows the measured results at 65K and after tuning the filter. The tuning is a must for such a high order and narrowband filter because of the tolerances in both the wafer thickness and fabrication. From Figure 11.18(a) we can see that the measured band center frequency is about 1,970 MHz, which is slightly higher than the designed 1,967.5 MHz. This is because the center frequency is dependent on the orientation of the filter on the sapphire wafer as a result of its anisotropic dielectric property. The measured bandwidth is close to 15 MHz. An insertion loss of 1.4 dB at the band center was measured, including the losses of the connectors. The resonator Q is estimated to be larger than 50,000. The measured return loss shown in Figure 11.18(a) is better than −10 dB across the passband. Furthermore, finer tuning could improve the return loss as well as shift the center frequency to the design frequency. The transmission zeros near the band edges for enhancing selectivity are observed. The measured group delay of the filter is plotted in Figure 11.18(b), showing a flat group delay over the central region of the passband, which is in very good agreement with the theoretical response given in Figure 11.13(b).
11.3 Micromachined Filters With the advent of RF MEMS (microelectromechanical systems) technology [23], there has been a growing interest in micromachined microwave filters [24–31]. For passive components, silicon (Si) has been used as a substrate for choices in micromachined circuits. One of the most popular micromachining techniques consists of etching a Si substrate and suspending the circuit on a thin dielectric membrane. Many planar filter topologies such as popular parallel coupled line filters described in previous chapters may directly be realized on the thin dielectric membrane. The use of additional micromachined substrates can be used to shield the
386
Filters Using Advanced Materials and Technologies
Figure 11.18
The measured responses of the 18pole HTS filter with selfequalization of group delay: (a) magnitude and (b) group delay.
structure to avoid radiation losses. The propagation occurs in air, with almost no dispersion and no dielectric losses. Such a structure is particularly attractive for millimeterwave filters. For example, a silicon micromachined filter with a simple planar integration on another substrate is demonstrated in [24]. The filter is comprised of two endcoupled halfwavelength microstrip resonators supported on an 8 mthick dielectric membrane. The excitation is from the top of the shielding substrate of the membranesupported micromachined filter. Packaging and interconnections are included in the design. Experimental results are presented on a twopole 30 GHz, 4% fractional bandwidth filter with a quality factor of 600 and insertion loss of 1.8 dB. Such a filter can be easily integrated in any circuit using flipchip technology. Using micromachining techniques has also given rise to the development of socalled synthesized substrates that extend the useful range of high and low impedance microstrip values on high dielectric constant materials [25]. In that paper, a silicon
11.3 Micromachined Filters
387
wafer is micromachined to form a ‘‘synthesized substrate’’ that can minimize low impedance and maximize high impedance values, leading to lowpass filter designs with either reduced low impedance or increased high impedance values on the ‘‘synthesized substrate’’ sections, and having high and low impedance values improved by a factor of 1.5. As a result, in the filter response, sharper rejection band edges have been achieved. The use of micromachining technology has also meant that integration with other silicon components on the same wafer is now possible. In particular, the device can be integrated with other RF MEMS components such as switches and varactors for switch filter banks and tunable filters [23, 26].
11.3.1 Miniature Interdigital Filters on Silicon
The footprint of microwave filters is determined not only by the filter topology but also by the dielectric substrate. Using high resistive silicon substrates has enabled the design of small bandpass filters at low microwave frequency bands [30]. Figure 11.19 shows the layout of a fivepole miniature stepped impedance resonator (SIR) interdigital filter designed on a silicon substrate with a dielectric constant of 11.9 and a thickness of 0.525 mm. The high impedance line for the SIR has a width of 0.15 mm while the low impedance line has a width of 0.5 mm on the substrate. The design specification for the given filter design requires a fractional bandwidth (FBW ) of 33%. This can be generally considered as having a large FBW, and hence the technique described in Section 10.4.2 was adopted for the design. Fabrication of the interdigital filter was carried out on high resistive silicon (> 8 k⍀cm) and masks were laid out for 100mmdiameter wafers. The strip conductor and ground plane thicknesses were specified as 3 m of aluminum. Aluminum was chosen as the conductor because this is a standard foundry metal. Standard IC fabrication techniques were used to pattern the conductor layers with deep reactive ion etching used to make via holes to the ground plane. The via holes were metallized with aluminum and were tested for good conductivity. A flash layer of 0.5 m of gold was also applied to the ground plane to ensure a low resistance contact to the test housing. The test pieces were diced and mounted on a brass test fixture using epoxy bonding as shown in Figure 11.20. Two SMA connectors were incorporated for testing. The measured results are shown in Figure 11.21 together with the EM simulated results, and good agreement can be observed. EM simulations were also carried out to investigate conductor, dielectric and radiation losses of the SIR interdigital filter. Figure 11.22 shows the simulated results, where frequencies are normalized with respect to the center frequency. Each curve shows the passband response when only one loss mechanism is considered. To simulate the conductor loss, 3 mthick aluminum (as fabricated) with a conductivity of 3.72 × 107 S/m is assumed. To consider the dielectric loss in the simulation, a dielectric loss tangent of 0.01 for the highresistive silicon substrate was used [30]. From Figure 11.22, one can see that the conductor loss is still dominant in this miniature filter. The dielectric loss from the silicon substrate is also significant, whereas the radiation loss is negligible. In general, the losses in a miniature filter
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Figure 11.19
Layout of fivepole miniature SIR interdigital filter on a silicon substrate with a dielectric constant of 11.9 and a thickness of 0.525 mm. Dimensions are in millimeters.
Figure 11.20
Fabricated fivepole SIR interdigital filter on silicon with two SMA connectors for testing.
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Figure 11.21
Measured and simulated performances of the fivepole miniature SIR interdigital filter on silicon.
Figure 11.22
EMsimulated loss effects of the fivepole miniature SIR interdigital filter on silicon.
not only cause a greater insertion loss, but they also shrink the bandwidth, which needs to be taken into account in the design. The advantage of using silicon is the ability to use IC manufacturing techniques such as silicon under etching to reduce the size of the SIR filter. For example, using bulk micromachining it is possible to reduce the thickness of the silicon substrate under the capacitive element (low impedance line section) of the stepped impedance resonator. The ground plane metal is then profiled to the change in thickness of the substrate as shown in Figure 11.23. The result is an increase in the capacitance of the element, which leads to a larger ratio of high to low impedance and reduces the length of the SIR. For the demonstration, simulations have been carried out to investigate the possible size reduction. For fixed W 1 = 0.15 mm, W 2 = 0.5 mm
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Figure 11.23
Micromachining below the capacitive element of the SIR. Top view: the layout of the SIR. Bottom view: the cross section.
and h 1 = 0.525 mm, it is shown that up to a 40% reduction in overall resonator length L for the same resonant frequency can be obtained when the substrate thickness h 2 under the capacitive element is reduced down to 0.1 mm [30]. 11.3.2 Overlay Coupled CPW Filters
Conventional coplanarwaveguide (CPW), as shown in Figure 11.24(a), suffers from high conductor loss at high and low characteristic impedance extremes due to the narrowing of the center conductor and slot width, respectively [32]. Moreover, very low impedance lines are practically impossible to realize in CPW due to the minimum slot size limit imposed by the fabrication process. Recently,
Figure 11.24
Schematic diagrams of: (a) conventional CPW and (b) overlay CPW.
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micromachining techniques have allowed fabricating a socalled overlay coplanarwaveguide structure of Figure 11.24(b), in which the edges of the center conductor are partially elevated, denoted by E, and overlapped with ground, indicated by O, to achieve a broad ranges of characteristic impedances and to reduce the line losses [31]. The elevated center conductor helps to reduce the conductor loss by redistributing the current over a broad area. It also helps to reduce the dielectric or substrate loss by confining the electric field in the air between the overlapped conductor plates. Compared with conventional CPW lines, the overlay CPW lines show a wider impedance range (25–80 ohms) and lower loss (< 0.95 dB/cm at 50 GHz). Figure 11.25 illustrates the microphotograph of a fabricated overlay CPW line on a quartz substrate, where the elevation of the signal line is about 15 m. The micromachining fabrication processes are detailed in [31]. To demonstrate the practical usefulness of the overlay CPW lines, an Xband steppedimpedance lowpass filter was designed and fabricated using this type of micromachined transmission line [31]. The overlay CPW filter shows distinct advantages over the conventional CPW filter such as lower loss and reduced size, together with improved spurious responses, including improved selectivity and wider stopband characteristics. It is envisaged that various bandpass filters using coupled overlay CPW resonators can be constructed. For example, Figure 11.26 depicts the schematic of a threepole endcoupled overlay CPW resonator filter.
11.4 Filters Using Advanced Dielectric Materials Recent advances in microwave dielectric materials such as lowtemperature cofired ceramic (LTCC) and liquid crystal polymer (LCP) have stimulated a rapid development of multilayer microwave and millimeterwave components including filters for system integration [33–47].
Figure 11.25
SEM photograph of the fabricated overlay CPW line, where CPW pad is for probing measurement. (From: [31]. 2001 IEEE. Reprinted with permission.)
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Figure 11.26
Schematic diagram of endcoupled overlay CPW resonator filter.
Recent developments in LTCC technology are making it more and more attractive for applications in the microwave and millimeterwave frequency bands [33– 35]. The most active areas for high frequency applications include Bluetooth module, frontend modules (FEM) for mobile phones, wireless local access network (WLAN), local multipoint distribution systems (LMDS), and collision avoidance radar. The newer developments in LTCC substrate materials extend the applicable frequency of the technique up to 100 GHz [33]. Owing to low conductor loss, low dielectric loss, and up to 50 laminated layers, LTCC provides a suitable approach for embedded microwave and millimeterwave passive components and accessories including antennas. In addition, LTCC substrate materials possess a wide range of thermal expansion coefficients. It is this feature that makes the LTCC substrates very attractive for integrated packaging solutions. Basic LTCC process consists of tape preparation, punching, via filling, pattern printing, stacking, laminating, debinding, and sintering [36]. LCP is a fairly new and promising thermoplastic material. It can be used as a lowcost dielectric material for highvolume largearea processing methods that provide very reliable highperformance circuits at low cost [37–39]. LCP has a unique combination of properties such as: (1) excellent electrical properties up to millimeterwave frequencies (dielectric constant of 3.16 and a low losstangent of 0.002–0.004 at 60 GHz comparable with ceramics); (2) very good barrier properties [its permeability is (moisture absorption = 0.04%) comparable to that of glass and very close to that of ceramics]; and (3) low coefficient of thermal expansion (CTE) as low as 8 × 10−6/K, adjustable through thermal treatment processes. Material, electrical, and economical considerations make LCP an ideal candidate for all multichipmodule (MCM), systemonpackage (SOP), and advanced packaging technology led by the growing market for digital, RF, and optoRF applications. 11.4.1 LowTemperature Cofired Ceramic Filters LTCC Lumped Element Filter
Figure 11.27(a) shows the physical layout of a multilayer LTCC filter [40]. It consists of a secondorder coupled resonator bandpass filter (with both capacitive and inductive couplings) in parallel with a feedback capacitor. The two resonators
11.4 Filters Using Advanced Dielectric Materials
Figure 11.27
393
(a) Physical LTCC layout of a twopole bandpass filter. (b) Equivalent lumpedelement circuit of the filter. (c) Measured and simulated performance of the prototyped LTCC filter. (From: [40]. 2003 IEEE. Reprinted with permission.)
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are equivalent to parallel LC resonant circuits. Figure 11.27(b) depicts a lumpedelement circuit of the filter, where the element values are CC 1 = CC 2 = 0.79 pF, LL 1 = LL 2 = 1.55 nH, C 1 = C 2 = 2.48 pF, and MM = 9.27 nH. This corresponds to a secondorder Chebyshevtype bandpass filter with 0.2dB ripple, 2.5GHz center frequency, and 0.3GHz equalripple bandwidth. With the multilayer capability of the LTCC technology, the lumpedcircuit elements can be readily realized by using parallel plates for the capacitor and a metallic strip for the inductor. The capacitors are realized on layer 1 with two patches; while the inductors are realized using meandered narrow lines (line width of 8 mil) terminated to the ground plane. All the interconnections are done using conducting vias. The coupling between the two resonators is accomplished by overlaying two inductance strips one above the other on layers 1 and 2 (4.56 mil apart), respectively. Thus, the interresonator coupling is magnetic. The input and output (I/O) are facilitated on layer 2 with two small patches that are coupled to the capacitors of the two resonators on the layer below, so the I/O couplings are capacitive. The feedback capacitor is implemented on layer 3 as indicated, by placing a ‘‘dumbbell’’shaped metal plate directly above the I/O components of the filter. Its purpose is to introduce a pair of finite transmission zeros to the transmission transfer function—one in the lower stopband and another one in the upper stopband. The filter was constructed using six layers of Dupont 951AT material with 3.6mil thickness. The components of the filter were located only at the interfaces between the bottom four layers. A finite ground plane was inserted at the bottom of the substrate for the construction of the grounded resonators. The overall size of the filter is 170 × 80 × 21.6 mil. The measured response of the filter and results from EM simulation are presented in Figure 11.27(c). It can be seen that, due to the zero metallic strip thickness model used in the design, which underestimates the capacitance of the parallel plates, the measured response is slightly shifted toward the lower frequency end. Nevertheless, the correlation of the theoretical and measured results is very good. Notice that the two finite zeros in the transmission response of the filter are located at the prescribed locations. LTCC Cavity Filter
Onpackage integrated cavity filters using LTCC multilayer technology are a very attractive option for threedimensional (3D) RF frontend modules up to the millimeterwave frequency range because of their relatively low loss compared to stripline/microstrip or lumpedelementtype filters [43, 44]. Figure 11.28(a) illustrates the 3D structure of a threepole LTCC cavity filter for 60GHz WLAN narrowband (1 GHz) applications [43]. The design is based on a threepole Chebyshev lowpass prototype filter with 0.1dB inband ripple. The LTCC filter consists of three coupled cavity resonators [i.e., Cavity 1, Cavity 2, and Cavity 3 in Figure 11.28(b)]. The cavity resonator is built utilizing conducting planes as horizontal walls and via fences as sidewalls. The size and spacing of via posts are properly chosen to prevent electromagnetic field leakage and to achieve stopband characteristics at the desired resonant frequency. The cavity height was designed to be 0.5 mm (five substrate layers) to achieve a higher quality factor
11.4 Filters Using Advanced Dielectric Materials
Figure 11.28
395
LTCC threepole cavity BPF employing slot excitation with an open stub: (a) 3D overview and (b) side view of the proposed filter. (From: [43]. 2005 IEEE. Reprinted with permission.)
and, consequently, to obtain a narrower bandwidth. The interresonator couplings from the cavities 1 and 3 and 2 to 3 are accomplished by two internal slots, as indicated on metal layer 7. Microstrip feed lines are utilized to excite the I/O resonators (i.e., cavities 1 and 2) through coupling slots etched in the top metal layer (metal 2), as denoted by the external slots in the figure. In order to maximize the magnetic coupling by maximizing magnetic currents, a virtual short is placed at the center of each slot by terminating the feed lines with quarterwavelength open stubs. The excitation using an open stub contributes to fabrication simplicity with no need to drill viaholes to short the end of the feed lines. It also avoids the loss and inductance effects generated by shorting vias close to the slot, whose
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effects can be significant in the millimeterwave frequency range. The relevant design parameters for the LTCC cavity filter are summarized in Table 11.1. The designed filter was fabricated using LTCC 044 SiO2 – B2O3 glass by the Asahi Glass Company, Kanagawa, Japan. The relative permittivity of the substrate is 5.4 and its loss tangent is 0.0015 at 35 GHz. The dielectric layer thickness per layer is 100 m, and the metal thickness is 9 m. The resistivity of metal (silver trace) is determined to be 2.7 × 10−8 ⍀ ⭈ m. The fabricated LTCC cavity filter exhibits an insertion loss 2.14 dB and a return loss 16.39 dB over the passband. The measurement shows a 3dB fractional bandwidth of approximately 1.53% (0.9 GHz) at a center frequency of 58.7 GHz. The center frequency downshift can be attributed to the fabrication accuracy such as slot positioning affected by the alignment between layers, layer thickness tolerance, and variation in dielectric constant.
11.4.2 Liquid Crystal Polymer Filters Miniature Wideband LCP Filter
A miniaturized wideband filter topology has been developed for LCP implementation based on a circuit for an ideal transmission line filter [45], which is depicted in Figure 11.29. The circuit is symmetrical with respect to the middle where there is a parallel opencircuited stub with an electrical length of 2 and a characteristic impedance of Z 0s . It can be recognized that the symmetrical sections on both side of the opencircuited stub are the same as that of Figure 10.3. This means that they can be implemented with two coupled line sections as shown in Figure 11.30. Thus, Z 0e and Z 0o in Figure 11.29 are the even and oddmode characteristic impedances of the symmetrical coupled lines with a line width of W and spacing s. For this type of filter, the electrical length is required to be 90 degrees at the midband or center frequency f 0 . Therefore, the parallel opencircuited stub has an electrical length of 180 degrees at f 0 . The remaining design parameters for the filter are Z 0e , Z 0o , and Z 0s . Figure 11.31 illustrates the frequency response of a miniature coupled line filter with Z 0e = 133⍀, Z 0o = 51⍀, and Z 0s = 26.5⍀ for a center frequency f 0 = 2 GHz. Since the filter is based on a transmission line filter model, it exhibits a periodical frequency response with multipassbands located at f 0 , 3f 0 , and so forth. We are interested only in the primary passband at f 0 . It can be clearly seen from the S 11
Table 11.1 Design Parameters of the LTCC Cavity Filter Design Parameters
Dimensions (mm)
Effective cavity resonator (L × W × H) External slot position (SPext ) External slot (SL ext × SWext ) Internal slot position (SP int ) Internal slot (SL int × SW int ) Open stub length (OSL) Via spacing Via diameter Via rows
1.95 × 1.31 × 0.5 0.4125 0.46 × 0.538 0.3915 0.261 × 0.4 0.538 0.39 0.13 3
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Figure 11.29
An ideal transmission line filter.
Figure 11.30
A schematic of a miniature coupled line filter with a parallel opencircuited stub in the middle.
response that there are five transmission poles, which implies that this miniature filter topology is equivalent to a fivepole filter, although it only consists of two quarterwavelength coupled line sections and one halfwavelength opencircuited stub when referring to the center frequency. The halfwavelength opencircuited stub also produces two finitefrequency transmission zeros near the passband, which improve the selectivity of the filter. As can be seen, the first transmission zero occurs at 1 GHz. At this frequency, the opencircuited stub has an electrical length of 90 degrees and thus presents a short circuit in the main signal path, blocking the transmission. Similarly, the second transmission zero occurs at 3 GHz because at this frequency the opencircuited stub has an electrical length of 270 degrees that also presents a short circuit in the main signal path. These two transmission zeros also limit the maximum bandwidth of this type of filter. It can be shown that, for a given center frequency f 0 , the two transmission zeros are allocated at 0.5f 0 and 1.5f 0 , respectively. Hence, the fractional bandwidth of this type of filter will be smaller than 100%; nevertheless, it is quite easy to achieve a wide bandwidth of 50% to 60%. A microstrip filter of this type, realized on a dielectric substrate with a dielectric constant of 2.9 and a thickness of 0.33 mm, is demonstrated in Figure 11.32(a).
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Figure 11.31
Frequency response of the miniature coupled line filter with an opencircuited stub for Z 0e = 133⍀, Z 0o = 51⍀, Z 0s = 26.5⍀, and f 0 = 2 GHz.
The coupled line sections have a very small spacing (s = 0.05 mm) and a narrow line width (W = 0.25 mm) for the required coupling. The opencircuited stub is meandered into a loop to make the filter compact. The design center frequency is 2 GHz, and the dimensions shown are in millimeters. The size of the filter is about 0.5 g by 0.25 g , where g is the guided wavelength in the substrate. The filter response is shown in Figure 11.32(b), obtained by EM simulation. A wide passband (> 50%) centered at 2 GHz is achieved with two transmission zeros near to the passband as expected. However, there is a spike near 4 GHz. This unwanted spurious response is due to unequal even and oddmode phase velocities in the microstrip coupled lines. The issue has been addressed in Chapter 10 (see Section 10.2). The techniques discussed there may be adopted to suppress the spike. An LCP filter of this type, with a compact layout, is reported in [45]. The filter was fabricated on LCP substrate characterized by ⑀ r = 2.9, tan ␦ = 0.003, substrate thickness = 330 m and conductor thickness = 18 m. Copperclad LCP dielectric substrates available from Rogers Corporation were used for fabrication. Although LCP substrates are available only with certain thicknesses, it is possible to realize many different substrate configurations due to the two types of LCP substrates available that have different melting temperature. The highmelt LCP (around 315°C) is used as a core layer while the lowmelt LCP (around 290°C) is used as a bonding layer. In this case, a 4mil LCP layer is bonded with an 8mil LCP layer using a 1mil bonding layer to give a total thickness of 330 m. The designed filter is then patterned and measured. The measured passband is from
11.4 Filters Using Advanced Dielectric Materials
Figure 11.32
399
(a) Miniature microstrip coupled line filter with an opencircuited stub on a dielectric substrate with a dielectric constant of 2.9 and a thickness of 0.33 mm. Dimensions are in millimeters. (b) EMsimulated S 21 response.
2.38–4.2 GHz, representing a fractional bandwidth of about 55%. The measured insertion loss within the passband is around 1.2 dB.
60GHz Band LCP Filters and Duplexer
Planar and vialess LCP bandpass filters and duplexers, operating in the 60GHz band or Vband, have been developed [46]. Figure 11.33 shows a photo of the
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Figure 11.33
A photo of the fabricated LCP duplexer with two microstrip square openloop resonator filters. (From: [46]. 2006 IEEE. Reprinted with permission.)
fabricated duplexer. The duplexer consists of two fourpole microstrip square openloop resonator filters with the canonical or single CQ coupling configuration as described in Chapter 10. An optimized Tjunction is used to combine two fourport filters to realize the threeport duplexer. The square openloop resonators allow different coupling mechanisms required to realize desired filtering characteristics with transmission zeros for the duplexer implementation. It was also found that the feed scheme for the filter on the left in Figure 11.33 results in a passband response with steep rolloff on the high side, whereas the feed scheme for the filter on the right gives rise to a higher selectivity on the low side of the passband. These asymmetric characteristics have been developed that are then combined to realize the duplexer. The designed duplexer was fabricated on LCP substrate characterized by ⑀ r = 3.15, tan ␦ = 0.003, substrate thickness 152 m, and conductor thickness 9 m. In this case, two 2mil highmelt (around 315°C) core LCP layers (commercially available as R/flex 3850) are bonded together using two 1mil lowmelt (around 290°C) bond LCP layers (commercially available as R/flex 3600) to give a total thickness of 152 m. Once an LCP substrate of the desired thickness was obtained, the designs were patterned and measured. The filters exhibit a low insertion loss of 2.5 dB and the isolation between the duplexer ports is better than 25 dB.
11.5 Filters for UltraWideband (UWB) Technology Ultrawideband (UWB) technology is being reinvented recently with many promising modern applications [48, 49]. The rapid growth in this field has prompted the development of wideband microwave filters for UWB applications [50–65], described next.
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11.5.1 Optimum Stub Line Filters
A distributed stub transmission line filter, which may be modified and realized in different forms such as the ring or doublesided parallel strip [51, 52], is attractive for designing multipole or highorder selective UWB filters. In particular, the design of an UWB microstrip optimum shortcircuited stub filter whose unit elements or connecting lines are nonredundant [55, 56] is discussed next. Design of Optimum Stub Filter
A general circuit model for a shortcircuited stub filter is shown in Figure 11.34, which is comprised of a cascade of n shunt shortcircuited stubs of electrical length s , separated by connecting transmission lines (unit elements) of electrical length l . Both s and l are determined at some characteristic frequency fc . Z 0 is the terminal impedance. Z 1 , Z 2 , . . . , Z n are the line characteristic impedances of the shortcircuited stubs. The characteristic impedances of the connecting lines are denoted by Z 1, 2 to Z n − 1, n . To demonstrate the advantages of the optimum stub filter, Figure 11.35 shows the frequency response of three shortcircuited stub filters. For the two conventional stub filters, s = l = 90 degrees at fc = 9 GHz, which is also the midband frequency; thus, all the shortcircuited stubs and connecting lines are quarterwavelength long at the midband frequency. It is obvious that the conventional stub filter with 11 stubs (n = 11) has a higher selectivity than the conventional stub filter with 6 stubs (n = 6). Nevertheless, this indicates that a larger number of shortcircuited stubs is usually necessary for this type of UWB filter requiring high selectivity. The design of conventional stub filters is well documented in [66]. For our discussion later on, Table 11.2 lists the designed circuit parameters of the conventional stub filter with 11 shortcircuited stubs. The third filter whose frequency response is also shown in Figure 11.35 is an optimum stub filter, which also has only 6 stubs (n = 6). For this optimum stub filter, the electrical length of the shortcircuited stubs is s = 35 degrees, while the electrical length of the connecting lines or unit elements is l = 2 s = 70 degrees
Figure 11.34
Equivalent circuit of shortcircuited stub filter.
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Figure 11.35
Transmission characteristics of three UWB shortcircuited stub filters with the same passband from 3.5 to 14.5 GHz.
Table 11.2 Circuit Parameters for a Conventional ShortCircuited Stub Filter with 11 Stubs for an UltraWide Passband from 3.5 to 14.5 GHz ( s = l = 90 Degrees at 9 GHz) Stub Line Impedance Z1 Z2 Z3 Z4 Z5 Z6
= = = = = =
Z 11 = 199.07⍀ Z 10 = 101.88⍀ Z 9 = 101.76⍀ Z 8 = 98.49⍀ Z 7 = 97.41⍀ 97.13⍀
Connecting Line Impedance Z 1, 2 Z 2, 3 Z 3, 4 Z 4, 5 Z 5, 6
= = = = =
Z 10, 11 = 38.84⍀ Z 9, 10 = 36.73⍀ Z 8, 9 = 38.95⍀ Z 7, 8 = 39.73⍀ Z 6, 7 = 40.00⍀
at fc = 3.5 GHz. From Figure 11.35, it is notable that the optimum stub filter with six shortcircuited stubs achieves the same selectivity as the conventional stub filter with 11 shortcircuited stubs. This is because the unit elements of the optimum stub filter are not redundant, and they function nearly as effectively as the shortcircuited stubs in improving the selectivity. In general, the circuit parameters of an optimum stub filter are determined using a computeraided synthesis program. As an alternative, tabulated elements can be found in [22] for this type of filter, which is also seen as the optimum pseudohighpass filter. For the optimum stub filter discussed earlier, its circuit parameters are given in Table 11.3. Comparing the circuit parameters in Tables
Table 11.3 Circuit Parameters for an Optimum ShortCircuited Stub Filter with Six Stubs for an Ultra Wide Passband from 3.5 to 14.5 GHz ( s = 35 Degrees and l = 70 Degrees at 3.5 GHz) Stub Line Impedance
Connecting Line Impedance
Z 1 = Z 6 = 103.96⍀ Z 2 = Z 5 = 72.64⍀ Z 3 = Z 4 = 64.48⍀
Z 1, 2 = Z 5, 6 = 48.85⍀ Z 2, 3 = Z 4, 5 = 50.44⍀ Z 3, 4 = 50.82⍀
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11.2 and 11.3, we can see that to achieve the same filtering characteristics shown in Figure 11.35, the first and last stubs of the conventional filter need an extremely high characteristic impedance of 199 ohms. Such a high impedance level will result in a very narrow line, which could be very difficult to realize in microstrip with a lowcost fabrication technology. On the other hand, the optimum stub filter has much more reasonable impedance levels for all the shortcircuited stubs. Moreover, the characteristic impedances of connecting lines are all close to the terminal impedance of 50 ohms. As a matter of fact, with a slight modification, all the connecting lines can be made to have a 50ohm characteristic impedance. Since the unit elements of the optimum stub filter are also accounted for by the order or degree of the filter, another advantage of the optimum stub filter is that it has the fewest number of Tjunctions for the same order (11 in this case) of the conventional stub filter. In addition, the separations between adjacent stubs are wider for l = 70 degrees at 3.5 GHz, resulting in the least interactions between neighboring stubs or junctions. Based on circuit parameters given in Table 11.3, an optimum UWB microstrip filter has been designed on a 1.27mmthick RT/Duroid 6006 substrate with a dielectric constant of 6.15. The layout of the designed microstrip filter is shown in Figure 11.36. All the connecting lines or unit elements have been modified to have a characteristic impedance of 50 ohms, so the filter has a very simple structure of a straight 50ohm line loaded with six shortcircuited stubs. The narrowest stub line has a width of 0.3 mm, which can be easily fabricated using a lowcost printedcircuitboard (PCB) technology. The filter dimensions indicated on the layout have taken into account the effects of Tjunction discontinuities. Also, when determining the electrical length, a nominal characteristic frequency, fc = 4 GHz has been used instead of 3.5 GHz because the substrate used tends to shift down the frequency response. The filter was fabricated using conventional PCB technology and a photo of the fabricated filter is given in Figure 11.37(a). The size of the filter is quite compact, occupying an effective circuit area of only 34 mm × 6 mm. For measurement, the input and output ports are extended to connect to two SMA connectors. The viahole grounds were simply implemented using soldering. The measured results without any tuning or trimming are plotted in Figure 11.37(b). We can see that although the filter has only six stubs along a 50ohm line, its measured frequency response of S 11 does show eleven ripples in the passband, similar to a typical 11pole Chebyshev filter characteristic. The measured bandwidth was 10.65 GHz
Figure 11.36
Optimum UWB microstrip filter on a dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27 mm.
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Figure 11.37
(a) Photo of a fabricated optimum UWB microstrip filter. (b) Measured performance.
at a midband frequency of 8.9 GHz, equivalent to a factional bandwidth of about 120%, which is in good agreement with the design or simulation. The insertion loss at the midband was measured to be 0.75 dB. The insertion loss tended to increase at higher frequencies, which is attributed to the frequencydependent losses of the two SMA connectors, the dielectric material, and the conductor. In addition, the radiation loss is likely higher at higher frequencies.
Optimum Stub Filter with Cross Coupling
Cross coupling may be introduced in an optimum stub filter, which brings about some interesting characteristics. Figure 11.38 depicts a generic circuit topology for an optimum stub filter with cross coupling. In this circuit topology, the conventional optimum stub filter has the same circuit model as that of Figure 11.34 and the cross coupling is introduced between the source and load (i.e., I/O ports), using a coupled line section with a pair of modal impedances denoted by Z 0e and Z 0o and an electrical length eo . There is also a new pair of connecting lines between the coupled lines and the conventional optimum stub filter, and the characteristic impedance and electrical length of the connecting lines are Z c1 and c1 , respectively.
11.5 Filters for UltraWideband (UWB) Technology
Figure 11.38
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Circuit topology of optimum stub filter with cross coupling.
Figure 11.39 illustrates a designed optimum stub filter with cross coupling. Table 11.4 lists all the circuit parameters of this crosscoupled optimum stub filter. The circuit is modeled using Microwave Office, a commercially available computeraided design tool [21]. Figure 11.40(a) shows the primary or desired passband performance of the filter. As can be seen from its return loss or S 11 response, there are nine transmission
Figure 11.39
A crosscoupled optimum stub filter.
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Filters Using Advanced Materials and Technologies Table 11.4 Circuit Parameters for the Optimum Stub Filter with Cross Coupling of Figure 11.39 Coupled Lines
Stub Lines
Connecting Lines
Z 0e = 65.8⍀ Z 0o = 56.7⍀ eo = 50 degrees at 3.8 GHz
Z 1 = Z 3 = 63.4⍀ Z 2 = 48.4⍀ s = 50 degrees at 3.8 GHz
Z c1 = 86.1⍀ Z 12 = Z 23 = 102.6⍀ c1 = l = 100 degrees at 3.8 GHz
poles in the passband, implying the filter response is of order 9. However, the conventional stub filter in this case would only have a degree of 5. In fact, the two connecting lines between the coupledline section and the conventional stub filter add two more poles, and the coupledline section also contributes additional two poles. This makes for a total of nine poles. It is notable that the arrangement of cross coupling, in this case the coupledline section, is able to increase the order of the filter. This is different from crosscoupled narrowband filters discussed in Chapter 10, where the arrangement of cross coupling does not increase the order of the filter. For the filter topology of Figure 11.38, the cross coupling resulting from the coupledline section also produces transmission zeros in the stopband. Almost equalripple responses in both passbands and stopbands are attainable with this arrangement, which is clearly demonstrated by the wideband responses in Figure 11.40(b). In addition, the cross coupling of this type has an effect on group delay equalization, reducing the high peaks of group delay on the passband edges. The group delay response of the designed filter is plotted in Figure 11.40(c).
11.5.2 Multimode CoupledLine Filters
In Section 10.5, we showed how the first harmonic frequency of a stepped impedance resonator (SIR) can be useful in the design of dualband filters. In this section, we will discuss how this concept can be extended to include more harmonic frequencies for the design of UWB filters [57, 58]. Figure 11.41(a) depicts a microstrip SIR that is weakly coupled to two external ports for EM simulation of its resonant frequency responses. The SIR has a configuration similar to that of Figure 10.60(b) with a lowimpedance line in the middle and highimpedance lines on either end. The dimensions shown are in millimeters on a substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. The lowimpedance line has a width of 1.1 mm with a characteristic impedance of 48 ohms at 6 GHz. The high impedance lines are 0.1 mm wide, resulting in a characteristic impedance of 107 ohms at 6 GHz. In general, the characteristic impedances are frequencydependent due to dispersion in the microstrip. Each highimpedance line section is 4 mm long, while the lowimpedance line in the middle is 7.5 mm long. Figure 11.41(b) shows the resonant frequency response of the SIR over a wide frequency range from 2 to 16 GHz, obtained by the EM simulation. As can be seen, there are four resonant modes within this frequency range. The first or fundamental resonant mode resonates at f 1 = 4.5 GHz; the second resonant mode is at f 2 = 7.05 GHz; the third resonant mode at f 3 = 9.65 GHz; and the fourth resonant mode at f 4 = 13.7 GHz. Thus, the ratios f 2 /f 1 , f 3 /f 1 , and f 4 /f 1 are 1.567, 2.144, and 3.044, respectively. These irregular ratios are due to the particular configuration of the SIR and the dispersive nature of microstrip. In
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Figure 11.40
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Performances of the filter of Figure 11.39: (a) magnitude responses of the primary passband, (b) wideband responses, and (c) group delay of the primary passband.
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Figure 11.41
(a) A microstrip SIR on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. (b) Its multimode resonant frequency responses.
practice, the frequency ratios can be manipulated by changing the width and length of the high and lowimpedance lines. For our application, we will use the first three resonant modes to build a UWB filter, which is described next. To design a UWB filter using multimodes of a SIR, we need to excite these modes appropriately. One approach is to use coupled lines at the input and output (I/O) as illustrated in Figure 11.42(a). The I/O feed section is actually comprised of three coupled lines to facilitate the strong coupling required for the UWB design. The width and spacing of the coupled lines are both 0.1 mm. Figure 11.42(b) shows the EMsimulated performance of the UWB filter. It has a 3dB bandwidth from 3.45 GHz to 11 GHz, which is centered at 7.225 GHz. Within the passband,
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Figure 11.42
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(a) A microstrip multimode UWB filter with coupledline I/O feed sections on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) EMsimulated magnitude responses. (c) EMsimulated group delay response.
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there are five transmission poles. It is obvious that three of these transmission poles correspond to the three resonant modes in the SIR. Since the coupled line sections are about a quarterwavelength long at the midband frequency, they contribute the two additional transmission poles in the passband. The group delay response of the designed UWB filter is plotted in Figure 11.42(c). The group delay in the midband is about 0.23 ns. It remains quite flat over most of the central portion of the passband, but increases to about 0.35 ns at the passband edges. From Figure 11.42(b), we note that there is an unwanted spurious response at around 14 GHz. This spurious response is due to the fourth resonant mode that is outside of the desired passband. We also note, from the same figure, that there is a transmission zero at around 15 GHz. This transmission zero is inherent for the coupled line sections used, and it occurs at the frequency where the electrical length of the coupled lines is about halfwavelength long. To suppress the unwanted spurious response, some ideas and techniques described in Section 10.2 may be adopted. For demonstration, a modified UWB filter is shown in Figure 11.43(a). The main technique used here is adding opencircuited stubs at the ends of the outer coupled lines. Each of the opencircuited stubs is 0.25 mm wide and 0.5 mm long. Opencircuited stubs can however deteriorate the inband return loss performance. To compensate for this, tapered coupled lines are introduced [58]. By adding the opencircuited stubs, which corresponds to capacitance loading at the ends of coupled lines, the transmission zero can be effectively shifted down. When the transmission zero is shifted to the frequency that coincides with the resonant frequency of the fourth resonant mode, the spurious response due to this mode is suppressed. This effect can clearly be seen from the EMsimulated response of Figure 11.43(b). As compared with the response of Figure 11.42(b), it is evident that the modified UWB filter has successfully suppressed the spurious response around 14 GHz, improving the selectivity at the high side of the passband. Higher selectivity UWB filters can be constructed using more multimode resonators. Figure 11.44(a) illustrates a coupled line UWB filter, which uses the two identical multimode resonators that are coupled to each other through a twoline coupled section in the middle as shown. The performance of this filter is shown in Figure 11.44(b). Based on a similar concept to that discussed in this section, other alternative implementations of multimode resonator filters are possible for UWB applications. 11.5.3 MicrostripCoplanar Waveguide CoupledLine Filters
A compact UWB filter structure as shown in Figure 11.45 is proposed in [59], which is based on a broadsidecoupled microstripcoplanar waveguide (CPW) structure. As shown in the figure, a single dielectric substrate is used. An openend CPW section is fabricated in the ground conductor of the microstrip line; this then provides a very simple and compact filter configuration. The basic section of the filter has two microstrip lines separated with a gap and broadside coupled to one openend CPW on the ground through the dielectric substrate. The broadsidecoupling and the existence of the dielectric substrate make the coupling between the microstrip line and the CPW very tight. Tight coupling provides a very wide bandpass operation.
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Figure 11.43
411
(a) The modified microstrip multimode UWB filter on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) EMsimulated filter performances.
The operation of this type of UWB filter can be understood by modeling the structure as shown in Figure 11.46. Assume a dielectric substrate with a thickness of 0.508 mm and a dielectric constant of 2.17. The two microstrip line sections on the top have a width of 3.6 mm and a length of L which couple to the bottom CPW resonator. The CPW resonator is 3.6 mm wide and 15.2 mm long with a gap of 0.2 mm to the ground. For EM simulations, the resistance of two ports is matched to the characteristic impedance of the microstrip and the port reference planes are also shifted to eliminate a discontinuity (impedance step) effect, which would only be secondary as far as the filter operation is concerned. Figure 11.47 shows EMsimulated response of the basic UWB filter structure for different values of L over a frequency range from 0.5 GHz to 22 GHz. When
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Figure 11.44
(a) Coupled microstrip line UWB filter using two multimode resonators on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) EMsimulated filter performances.
Figure 11.45
UWB bandpass filter using broadsidecoupled microstripcoplanar waveguide structure. (From: [59]. 2005 IEEE. Reprinted with permission.)
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Figure 11.46
Basic UWB filter structure for EM modeling on a dielectric substrate with a dielectric constant of 2.17 and a thickness of 0.508 mm. (a) Top view (microstrip). (b) Back view (CPW). All dimensions are in millimeters.
Figure 11.47
EMsimulated S 21 responses of the basic UWB filter structure of Figure 11.46.
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L = 0, the CPW on the ground is very weakly excited by the input/output (I/O) ports, and exhibits three resonant modes at frequencies of 7.2, 14.3, and 20.9 GHz, respectively. As L increases, the I/O coupling also increases. A wide passband starts to appear, and the three resonant modes shift towards some lower frequencies. The resonant frequency shift can be attributed to some capacitive loading due to the extended microstrip feed lines on the top of the CPW resonator. From Figure 11.47, it can be observed that there is an attenuation pole or transmission zero at 19.1 GHz for L = 5.2 mm. This results from the microstripCPW coupled line sections. To see this effect, a single microstripCPW coupled line section for the same length of L = 5.2 mm, as depicted in Figure 11.48, is modeled using fullwave simulation. The simulated frequency response is plotted in Figure 11.49. For comparison, the filter response for L = 5.2 mm is also plotted in the same figure. As can be seen, both structures have the same attenuation pole frequency. Based on the results of fullwave EM simulations and discussions presented above, a simple circuit model as shown in Figure 11.50 can be established for the UWB filter structure considered. Each of the microstripCPW coupled line sections is represented by two series opencircuited stubs separated by a unit element (UE); all have an electrical length of . The transmission line in the middle with an electrical length of represents the CPW portion that is not part of coupled line sections. Z c1 , Z c2 , Z c3 , and Z u denote the characteristic impedances of those transmission line elements. Z 0 is the terminal impedance. Apparently, the series opencircuited stubs can result in an opencircuit along the main signal path when is 180 degrees at the first attenuation pole frequency.
Figure 11.48
MicrostripCPW coupled line section on a dielectric substrate with a dielectric constant of 2.17 and a thickness of 0.508 mm. (a) Top view (microstrip). (b) Bottom view (CPW). All dimensions are in millimeters.
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Figure 11.49
Comparison of S 21 frequency responses of the microstripCPW coupled line section and the basic UWB filter for L = 5.2 mm.
Figure 11.50
Equivalent circuit model of the UWB filter structure.
Figure 11.51 plots the theoretical response of the equivalent circuit for two different sets of circuit parameters. The first case in Figure 11.51(a) is when Z 0 = 50⍀, Z u = 69⍀, Z c1 = 10⍀, Z c2 = 25⍀, Z c3 = 56⍀, = 90 degrees, and = 3 degrees at f 0 = 6.5 GHz. In this case, the CPW resonator is about g /2 long at 6.5 GHz, where g is the guide wavelength of the transmission line, and the resultant filtering structure is similar to that shown in Figure 11.45. As can be seen from its theoretical response, there are only three transmission poles in the passband, indicating its equivalence to a threepole filter. For the given bandwidth, the selectivity is mainly set by the two transmission zeros, one at dc and the other at the upper stopband at 2f 0 . As a result, the passband skirt is quite soft with a little selectivity for a single section UWB filter of this type. Nevertheless, a more selective UWB filter can easily be built by cascading more sections of this type of structure. This has been demonstrated in [59] with a selective UWB filter using three sections of that of Figure 11.45. The selective UWB filter was fabricated on a 0.508mmthick Arlon Diclad 880 substrate with a dielectric constant of 2.17 and a loss tangent of 0.00085 at 10 GHz. Each of the CPW resonators is 3.8 mm wide and 14.2 mm long with a gap of 0.2 mm to the ground plane. Each of the microstripCPW coupled lines is 7 mm long, and the microstrip is also 3.8 mm wide. The connecting lines for the three cascaded filtering sections have a characteristic impedance of 50⍀, and each is 1.6 mm wide and 1.8 mm long. This threesection UWB filter has a size of about 50 mm × 5 mm on the substrate. The measured results demonstrate
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Figure 11.51
Theoretical responses of the equivalent circuit model. (a) Case 1: Z 0 = 50⍀, Z u = 69⍀, Z c1 = 10⍀, Z c2 = 25⍀, Z c3 = 56⍀, = 90 degrees, and = 3 degrees at 6.5 GHz. (b) Case 2: Z 0 = 50⍀, Z u = 76.3⍀, Z c1 = 5.75⍀, Z c2 = 80.65 ⍀, Z c3 = 81.7⍀, = 90 degrees, and = 180 degrees at 6.5 GHz.
a passband from 3.0 GHz to 10.63 GHz (−10dB bandwidth) and an upper stopband attenuation larger than 22 dB up to 16 GHz, which can almost meet the FCC’s indoor limit. The measured group delay of the threesection UWB filter is 0.27 ns. For the second case in Figure 11.51(b), we have Z 0 = 50⍀, Z u = 76.3⍀, Z c1 = 5.75⍀, Z c2 = 80.65⍀, Z c3 = 81.7⍀, = 90 degrees, and = 180 degrees at 6.5 GHz. Since = 180 degrees in this case, the CPW resonator is about one guide wavelength or g long at 6.5 GHz. As compared to case 1, the size of the resultant filter for case 2 would be double. However, more selective filtering characteristics can be obtained with a single section. As we can clearly see from the theoretical response of Figure 11.51(b), there are five transmission poles in the passband, which implies that a single filter case 2 section is equivalent to a five
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pole filter with a higher selectivity. In fact, since the CPW resonator is g long at the center frequency, this case actually uses the first three modes of the resonator. This is similar to the multimode resonator filters discussed in Section 11.5.2. The other two transmission poles can be attributed to the two quarterwavelength microstripCPW coupled line sections. Since = 90 degrees at the center frequency, the coupling is strongest for the second resonant mode, which resonates at that frequency. Figure 11.52 shows physical implementation of this type of filter on a substrate with a dielectric constant of 10.8 and a thickness of 0.635 mm, where the top view is the microstrip I/O feed circuit while the bottom view is the CPW resonator. Note that the end CPW line has a much larger gap to the ground (1.2 mm in this
Figure 11.52
MicrostripCPW coupled line UWB filter on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 0.635 mm. (a) Top microstrip view. (b) Bottom CPW view. All dimensions are in millimeters.
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case) than the middle section where the gap is only 0.2 mm. The large gap not only increases the I/O couplings, but also controls the modal resonant frequencies of the CPW resonator [60]. As a matter of fact, we can see this CPW resonator as a stepped impedance resonator (SIR), which we discussed in the last chapter. This is because the CPW with a larger gap to the ground has a higher characteristic impedance whereas the middle section with a smaller gap to ground has a lower characteristic impedance. The length of a microstripCPW coupledline section is denoted by L in the figure. For the filter to be properly tuned, L is chosen so as to have an electrical length of 90 degrees at the center frequency. Figure 11.53(a) shows the EMsimulated performance of this UWB filter for L = 3.7 mm. As expected, five transmission poles appear in the passband with a 3dB bandwidth from 2.8 GHz to 11.05 GHz. The transmission zero occurs at about 12.76 GHz. However, there is spurious response at 13.71 GHz. The unwanted spurious response is due to the dispersion in both the CPW resonator and the microstripCPW coupled
Figure 11.53
EMsimulated performances of the microstripCPW coupled line UWB filter of Figure 11.52 for L = 3.7 mm. (a) Magnitude response. (b) Group delay response.
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lines. The group delay is about 0.22 ns at the midband and increases towards the passband edges. The asymmetrical passband and group delay responses are all due to the location of the transmission zero in the upper stopband, which seems to be too close to the passband. The location of the transmission zero can be controlled by the length of microstripCPW coupled line sections. If L is reduced, the transmission zero will be shifted to a higher frequency. The EMsimulated performances of such a modified filter are shown in Figure 11.54. This modification was simply changing L from 3.7 mm to 3.5 mm, and the main purpose was to move the location of the transmission zero to coincide with the spurious resonant frequency at 13.71 GHz, so as to suppress the spurious response. As can be seen from Figure 11.54, this objective has been achieved and a more symmetrical passband response is obtained, though this simplification affects the passband return loss slightly. Since the equivalent circuit of Figure 11.50 is more general, it can also lead to other implementations. For instance, the CPW resonator can be replaced by a
Figure 11.54
EMsimulated performances of the microstripCPW coupled line UWB filter of Figure 11.52 for L = 3.5 mm. (a) Magnitude response. (b) Group delay response.
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microstrip resonator and the input/output of the filter fabricated on CPW [61]. A filter of this type is illustrated in Figure 11.55. The filter is fabricated on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 0.635 mm. All dimensions shown in Figure 11.55 are in millimeters. From its I/O on the bottom CPW, the filter is composed of two CPWmicrostrip coupled line sections, which excite the microstrip resonator on the top. The microstrip resonator is about a guide wavelength long at the midband frequency, which corresponds to the case of Figure 11.51(b). The performance of the filter is shown in Figure 11.56, which has been optimized to suppress a spurious response caused by the fourth harmonic of the resonator. In a more recent article [62], a compact ultrawideband bandpass filter is proposed based on the composite microstrip–CPW structure illustrated in Figure 11.57. First, the microstripCPW transitions and the CPW shorted stubs are adopted as quasilumpedcircuit elements for realizing a threepole highpass filter prototype. By introducing a crosscoupled capacitance between input and output ports of this highpass filter and suitably designing the transition stretch stubs, a compact threepole ultrawideband bandpass filter is implemented with two transmission zeros located close to the passband edges. Second, to further improve the selectivity with good outofband response, two microstrip shorted stubs are added to form the fivepole ultrawideband bandpass filter. The fivepole UWB bandpass filter was fabricated on a Rogers RO4003C substrate with a dielectric constant of 3.38 and a thickness of 0.508mm. The filter has a very compact size of 8 mm × 11.9 mm. The measured 3dB bandwidth is 108.5% (3.18–10.72 GHz). The implemented filter has a minimum loss of 0.48 dB, and the return loss is greater than 17.2 dB
Figure 11.55
CPWmicrostrip coupledline UWB filter on a dielectric substrate with a dielectric constant of 10.8 and a thickness of 0.635 mm. (a) Top microstrip view. (b) Bottom CPW view. All dimensions are in millimeters.
11.5 Filters for UltraWideband (UWB) Technology
Figure 11.56
421
EMsimulated performances of the CPWmicrostrip coupledline UWB filter of Figure 11.55. (a) Magnitude response. (b) Group delay response.
within the passband. The group delay is below 0.55 ns over the whole passband. Three implemented transmission zeros are found at 2.45, 12.11, and 13.63 GHz. The first and second transmission zeros are generated by the crosscoupled capacitance and stretch stubs, respectively. Note that the third transmission zero is produced by the resonance of the microstrip shorted stubs.
11.5.4 UWB Filters with Notch Band
The UWB radio system can cover a very wide frequency band, which also covers many other existing radio systems. An important issue for UWB systems is to avoid the interference with other existing systems like wireless localarea networks (WLAN), cordless telephones, and IEEE 802.11a WiFi networks. To tackle this problem, very narrow rejection or notched band(s) can be introduced into a UWB bandpass filter [63, 64].
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Figure 11.57
Top/bottomlayer circuit layouts of a fivepole UWB bandpass filter on a dielectric substrate with a dielectric constant of 3.38 and a thickness of 0.508 mm (w 1 = 0.89 mm, w 2 = 2.03 mm, w 3 = 4.318 mm, w 4 = 1.143 mm, w 5 = 0.38 mm, w 6 = 2.29 mm, w 7 = 3.81 mm, w 8 = 0.38mm, w 9 = 6.22 mm, w 10 = 2.03 mm, w 11 = 2.03 mm, d 1 = 0.38 mm, d 2 = 0.635 mm, d 3 = 0.635 mm, and d 4 = 0.28 mm). (From: [62]. 2006 IEEE. Reprinted with permission.)
Figure 11.58 demonstrates the configuration of a microstrip UWB bandpass filter with embedded band notch stubs. The UWB bandpass filter design is based on a circuit model for an optimum bandpass filter of which the connecting lines or unit elements are nonredundant as discussed in Section 11.5.1. In order to introduce a narrow notched band, three different structures in Figure 11.59 were investigated first. These are a conventional opencircuited stub, a spur line, and an embedded opencircuited stub. In principle, to achieve a narrow notch the characteristic impedance of the conventional opencircuited stub will
Figure 11.58
Microstrip UWB bandpass filter with embedded band notch stubs. All the dimensions are in millimeters on a dielectric substrate with a dielectric constant of 3.05 and a thickness of 0.508 mm.
11.5 Filters for UltraWideband (UWB) Technology
Figure 11.59
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Schematic diagrams of (a) conventional opencircuited stub, (b) spurline, and (c) embedded opencircuited stub.
become extremely high which may be difficult to fabricate. Alternatively, a spurline introduced in [65] or embedded opencircuited stub may be utilized. The structures were investigated using EM simulations (Sonnet em). All the three structures are implemented on a dielectric substrate with a dielectric constant of 3.05 and a thickness of 0.508 mm. The main transmission line connecting to the I/O ports is the same for the three cases and has a width W C = 1.3 mm, which corresponds to a 50⍀ line on the substrate. The conventional quarterwavelength ( /4) microstrip opencircuited stub has a width Ws = 0.1 mm. The spurline and embedded opencircuited stub have the same width, W s = 0.1 mm, as well as the same gap G = 0.2 mm. The fullwave EM simulation results are illustrated in Figure 11.60. It is evident that the embedded opencircuited open stub is favorable for implementing extremely narrow notched bands. The narrowband characteristic of the embedded opencircuited stub is attributed primarily to its coupling to the main line, and the notch bandwidth can easily be controlled by adjusting Ws and G. For example, Figure 11.61 illustrates the simulated performance of the embedded stub with varying gap where decreasing the gap reduces the bandwidth. This technique allows us to realize a narrow notch that would not be possible with a conventional opencircuited stub requiring extremely high impedance. For example, to achieve the same bandwidth of a notch produced by a /4 embedded stub with a width of 0.1 mm and a gap of 0.2 mm, the conventional /4 opencircuited stub would need a high impedance of about 700 ohms, which is almost impossible to implement in practice.
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Figure 11.60
The EM simulation for conventional opencircuited stub, spurline, and embedded opencircuited stub with Wc = 1.3 mm, Ws = 0.1 mm, and G = 0.2 mm on a microstrip substrate with a dielectric constant of 3.05 and a thickness of 0.508 mm.
Figure 11.61
The insertion loss of the embedded stub with varying gap for Ws = 0.1 mm and Wc = 1.3 mm.
The achievement of extremely narrow bandwidth is only one advantage of this technique. Additionally, since the majority of the current flows around the edges of the microstrip line, the implementation of this embedded stub technique has less of an effect on the connecting line performance than the spurline over a wide frequency range. Hence, the embedded opencircuited stub structure was found to be the best of the three types of structures that were investigated and was therefore selected to be implemented in the design of a UWB bandpass filter with a band
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notch characteristic. The filter shown in Figure 11.58 has two embedded open stubs in the first and last connecting lines in order to introduce a very narrow notched (rejection) band in the UWB passband. The length of each of the two stubs should be /4 at the desired center frequency of the notched band to ensure that the second resonant harmonic of the embedded stub does not appear in the desired UWB passband. Figure 11.62 illustrates the fullwave EM simulation of the complete layout of the designed UWB bandpass filter with varying equal stub lengths (L). The notched band can be tuned by varying the lengths of the stubs. In the simulation, the width (Ws ) of the stubs and the gap (G) were chosen to be 0.1 mm and 0.2 mm, respectively. It is also possible to produce two different notched bands when desired. In this case, each of the two stubs would have different lengths. It should be noted that the embedded stubs can generate a notched band at the desired frequency with no significant influence on the wide passband performance. Figure 11.63(a) shows the fabricated filter, which has a size of 22.2 mm × 15.1 mm on the substrate used. For the measurement, a microstrip feed line of 5 mm long was added at both the input and the output. Figure 11.63(b) illustrates predicted and the measured results for the filter associated with the FCC indoor mask. The filter exhibited excellent UWB bandpass performance with a FBW of about 110% at a midband frequency of 6.85 GHz. The measured results showed extremely narrow notched bands in the passband with 10dB FBW of 4.6% at a center frequency of 5.83 GHz. The attenuation at the center of the notched band is more than 23 dB. At the midband frequency of each passband, insertion loss of less than 0.5 dB was obtained. The filter also showed a group delay of about 0.5 ns at the midband frequency of each passband. The multimode coupled line UWB bandpass filters described in Section 11.5.2 can be modified to include a notched band as well. Figure 11.64(a) illustrates an
Figure 11.62
The fullwave EM simulation of the complete layout of the designed UWB bandpass filter of Figure 11.58 for Ws = 0.1 mm, G = 0.2 mm, and varying equal stub lengths.
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Figure 11.63
(a) Fabricated microstrip UWB bandpass filter with embedded band notch stubs on a dielectric substrate with a dielectric constant of 3.05 and a thickness of 0.508 mm. (b) Measured and simulated filter performances.
example of this type of filter, one that will exhibit a notch in its otherwise ultrawide passband. The microstrip filter is constructed on a substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. The dimensions shown are in millimeters. The key to generating a notch is to introduce an asymmetric input/output (I/O) coupledline structure as shown in the figure. An opencircuited stub of 1.0mm length and 0.25mm width is added to the end of one I/O coupled line. Owing to this asymmetric modification, the opencircuited stub together with the coupled line section will resonate when their combined electric length is a quarter
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Figure 11.64
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(a) Multimode coupled line UWB filter with notched band on a 1.27mmthick substrate with a dielectric constant of 10.8. (b) EMsimulated magnitude response. (c) EMsimulated group delay response.
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wavelength long. This results in a short circuit at the I/O port, and thus produces a notch at the resonant frequency. Figure 11.64(b) illustrates the EMsimulated magnitude response of the filter, where a notch at 6.6 GHz can clearly be observed. The 10dB rejection bandwidth of the notch is about 5%. The notch frequency and bandwidth can easily be adjusted by changing the length and width of the opencircuited stub. In this case, the notch divides the original UWB into to the two subpassbands, which have the 3dB bandwidth ranging from 3.45–6.3 GHz and 6.8–11 GHz, respectively. The simulated group delay response of the filter is plotted in Figure 11.64(c). The group delay over the most part of each passband is about 0.3 ns. However, it increases rapidly at the band edge adjacent to the notch. This is due to the high selectivity of the notched band. The microstripCPW coupled line UWB bandpass filters discussed in the previous section (11.5.3) of this Chapter can also be modified to facilitate notch band(s). This has been demonstrated in a recent paper [63]. The paper presented UWB bandpass filters with single or multinotched band(s) for the purpose of detect and avoid, to avoid the interference between a UWB radio system and an existing radio system. The filter is modified from the basic filtering structure of Figure 11.45 by integrating stub(s) in the broadsidecoupled conductors to implement the single and multinotched band operation. The resonance of each stub introduces a narrow rejection band in the UWB passband, which then results in single or multinotched band(s). UWB bandpass filters with one or more notched bands are useful and required in practical systems in order to avoid the interference between the UWB radio system and legacy radio systems. The notched band can be easily designed to some specific frequency band(s) by tuning the length of the stub(s). The demonstrated filters used three sections of the modified basic filtering structure and showed both excellent ultrawide bandwidth (from 2.8 GHz to 10.2 GHz) and rejection performance (> 50 dB at central frequencies for a single notched band, or 21 dB and 27 dB for triple bands), as well as outofband performance better than the FCC requirement [48].
11.6 Metamaterial Filters Electromagnetic metamaterials are broadly defined as artificial effectively homogenous electromagnetic structures with unusual properties not readily available in nature [67]. In particular, if such materials exhibit simultaneously negative permittivity and permeability (i.e., ⑀ < 0 and < 0), they are also known as lefthanded (LH) metamaterials. The electromagnetic properties of lefthanded metamaterials were predicted in the late 1960s [68]. However, it was not until 2000 that such media were artificially fabricated and experimentally demonstrated [69]. To this end, a periodic array of metallic posts was combined with a periodic structure consisting of splitrings resonators (SRRs), which were previously proposed in [70]. Figure 11.65 illustrates the configuration of an SRR. The key aspect of these resonators is that they are electrically small and exhibit an effective permeability that is negative in a narrow band above their resonant frequency. Hence, the SRR structure alone may be seen as a metamaterial that exhibits a positive ⑀ and a negative . Furthermore, the
11.6 Metamaterial Filters
Figure 11.65
429
Metal splitring resonator (SRR).
posts produce a negative value of the effective permittivity up to a cutoff frequency (or plasma frequency), and hence an LH metamaterial may be synthesized by designing the composite medium with the SRRs and the posts [69]. It is important to note, however, that the fabricated bulk structure is highly anisotropic, namely, the electric and magnetic field vectors of incident polarization will have significant components in the direction of the posts and rings axes, respectively. Onedimensional lefthanded metamaterials and negative permeability transmission lines based on SRRs have recently been proposed in planar technology [71, 72]. These implementations have been mainly realized in coplanar waveguide (CPW) configurations. Lefthanded transmission lines in microstrip technology have also been reported [73] in which the line is periodically loaded with metallic vias and square shaped splitrings resonators, etched in close proximity to the conductor strip. The SRRs provide the negative effective permeability, , in a narrow band above their resonant frequency, whereas the metallic vias act as shunt connected inductors that make the structure behave as a microwave plasma with negative permittivity, ⑀ , up to the plasma frequency. It is shown that in that region where both and ⑀ are simultaneously negative, lefthanded wave propagation is allowed. Since this occurs in a narrow band above SRRs resonance, and the period of the structure is electrically small, these metamaterial transmission lines can be of interest for filter applications where miniaturization and narrow passbands are of interest. Figure 11.66 is a 3D view of the microstrip LH metamaterial cell proposed in [73]. The cell consists of a section of microstrip line that is shortcircuited with a metallic post realized by viagrounding, and two square SRRs being coupled to the microstrip. The square SRR is a modification from the circular one of Figure 11.65 for enhancing the coupling to the microstrip line. The viaground post plays an essential role in making this structure an LH metamaterial cell. Figure 11.67(a) is a layout of the designed microstrip LH metamaterial cell on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. All the dimensions shown are in millimeters. The outer loop of the SRR has a size of 5 mm × 5 mm and the inner loop has a size of 3.8 mm × 3.8 mm. The line width of the open loops is 0.2 mm and the gap is 0.4 mm. The two square SRRs are coupled to the main signal line, which is a 50⍀ line on the substrate, through 0.2mm spacing. The metal post or via has a diameter of 1.0 mm. Figure 11.67(b) shows the
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Figure 11.66
Structure of a microstrip LH metamaterial cell consisting of SRR and viaground post.
EMsimulated frequency characteristics of the cell, where a narrow passband around 3.1 GHz can be observed. This type of structure can find applications in the design of narrowband bandpass filters, as demonstrated in [73]. For example, Figure 11.68(a) illustrates a microstrip bandpass filter that is comprised of three LH metamaterial cells of Figure 11.67(a). The filter is simply a periodic structure with a distance (side to side) of 3 mm between adjacent cells. The dimensions of the cells are the same as those given in Figure 11.67(a). The magnitude response of the filter is plotted in Figure 11.68(b), obtained by fullwave simulation. The filter exhibits an asymmetric frequency response with a transmission zero on the low side of the passband, which improves the selectivity of that side. The passband is narrow as expected, ranging from 2.992 GHz to 3.077 GHz with a fractional bandwidth of 2.8%. The group delay of the filter is also shown in Figure 11.68(c). Another useful metamaterial structure based on SRRs is depicted in Figure 11.69(a), where two circular SRRs of the form of Figure 11.65 are coupled to a microstrip signal line, although square SRRs may be used. Note that there is no via ground in the microstrip, which is the main difference between this structure and that of Figure 11.67(a). This, however, leads to a completely different frequency characteristic as shown in Figure 11.69(b) with a stopband at the fundamental resonant frequency of the SRRs. The resonant frequency depends on the dimensions of the SRRs. For this particular example, the structure is simulated on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. The microstrip signal line has a width of 1.4 mm on the substrate. The outer radii of the two concentric rings are 2.2 mm and 1.8 mm, respectively; the distance between the rings is 0.2 mm and the ring width is 0.2 mm. It is evident that the basic structure of Figure 11.69(a) can be utilized to design bandstop filters or to suppress unwanted spurious responses in a bandpass filter. The latter application has been demonstrated in [74] and a configuration of this type of filter is illustrated in Figure 11.70, where SRRs have been integrated into a conventional parallel coupled line filter. The dimensions of SRRs can be determined so as to suppress an unwanted spurious, say at 2f 0 , where f 0 is the center frequency of the passband. It is also possible to use different sizes of SRRs to achieve multispurious suppression. It is feasible to develop SRRbased filters in a variety of topologies [75, 76]. For example, compact bandpass filters based on planar structures with three metal
11.6 Metamaterial Filters
Figure 11.67
431
(a) Layout of basic cell of the microstrip LH metamaterial on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) Its frequency characteristics.
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Figure 11.68
(a) Layout of a microstrip LH metamaterial bandpass filter on a dielectric substrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are in millimeters. (b) EMsimulated magnitude response. (c) EMsimulated group delay response.
11.6 Metamaterial Filters
Figure 11.69
433
(a) Layout of basic structure of the microstrip metamaterial with coupled SRRs. (b) Typical frequency response, obtained by fullwave simulation.
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Figure 11.70
Configuration of an SRRbased filter.
levels are reported in [76]. The central layer consists of a coplanar waveguide (CPW) with periodic wire connections between the central strip and ground planes. In the upper and lower metal levels, split ring resonators (SRRs) are etched and aligned with the slots. The wires make the structure behave as a microwave plasma with a negative effective permittivity covering a wide frequency range. SRRs, which are magnetically coupled to the CPW, provide a negative magnetic permeability in a narrow frequency range above their resonant frequency. The result is a bandpass structure that supports wave propagation in a frequency interval where negative permittivity and permeability coexist. The bandwidth of the structure can be controlled by tuning the resonant frequency of the upper and lower SRRs and the distance between SRRs. Other developments in metamaterial filters include the use of complementary splitring resonators (CSRRs) [77, 78]. As a matter of fact, the socalled CSRR is the negative image of an SRR of Figure 11.65. It has been demonstrated that CSRRs etched in the ground plane or in the conductor strip of planar transmission media (microstrip or CPW) provide a negative effective permittivity to the structure, and signal propagation is precluded, which roughly coincides to that of an SRR with identical dimensions etched onto the same substrate. To this end, a new design approach for compact microstrip bandpass filters, based on the use of CSRRs, has been presented in [77]. The basic filter cell consists of a combination of CSRRs, shunt stubs, and series gaps. The addition of shunt stubs to the basic cell provides the required flexibility to synthesize a frequency response with controllable bandwidths. The shunt stubs and series gaps have been modeled by lumped inductors and capacitors, respectively, and CSRRs have been described by parallel resonant tanks (capacitively coupled to the line). A filter design using three cells of this type has been demonstrated. It is a threestage periodic structure, which exhibits a central frequency at f 0 = 1 GHz and a 10% fractional bandwidth. It has been found that the measured frequency response fits the targeted specifications to a good approximation, and the first spurious band does not appear up to 3f 0 . Moreover, it has been demonstrated that the structure behaves as an effective (continuous) medium with lefthanded wave propagation in the allowed band. A wideband filter of this type has also been demonstrated in [78], where the filter, operating at Cband, is realized with simultaneously very small dimensions (area < 1 cm2 ), wide bandwidth (fractional bandwidth > 45%) and high frequency selectivity (transition bands with more than 50 dB/GHz falloff at both band edges).
11.6 Metamaterial Filters
435
Another major development in microwave metamaterial devices is based on the socalled composite right/lefthanded (CRLH) transmission line element [67]. Figure 11.71 is the model for a lossless or ideal CRLH transmission line unit element (cell). It is an equivalent lumpedelement circuit where C R and L R are capacitance and inductance of the righthanded transmission line element and C L and L L are capacitance and inductance of the lefthanded transmission line element. While C R and L R can be extracted from the capacitance/inductance per unit length of a normal homogeneous transmission line (e.g., microstrip or waveguide), C L and L L will have to be implanted or implemented artificially due to the unavailability of real homogeneous LH or CRLH materials. This equivalent circuit itself also implies some filtering characteristics. This is because the C R and L R act as lowpass filtering elements whereas the combination of C L and L L exhibits a highpass characteristic. Define two characteristic frequencies from the equivalent circuit: fse =
1 2 √L R C L
(11.4a)
fsh =
1 2 √L L C R
(11.4b)
where fse and fsh are the series and shunt resonance frequencies, respectively. It can be shown [67] that if f < min ( fse , fsh ), the phase velocity v p and the group velocity v g have opposite signs (i.e., they are antiparallel, v p −  v g ), meaning that the transmission line is LH and therefore the propagation constant is negative. In contrast, in the RH range f > max (fse , fsh ), the phase velocity and group velocity have the same sign (v p  v g ), meaning that the transmission line is righthanded (RH) and the propagation constant is therefore positive. When fse = fsh , the line will be called balanced. Otherwise, it is called unbalanced. Another useful characteristic frequency is defined as f0 =
√ fse fsh
(11.5)
To construct a CRLH transmission line for a microwave device, such as a filter, a large number (N) of cells may need to be included. Figure 11.72 shows balanced CRLH transmission line characteristics for 5 and 10cascaded cells of Figure 11.71. The element values of the cell are C L = 1.0 pf, L L = 2.5 nH, C R = 1.0 pF, and
Figure 11.71
Equivalent circuit model for the ideal CRLH transmission line cell.
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Figure 11.72
Characteristics of balanced CRLH transmission line. (a) 5 cells. (b) 10 cells. Cell element values: C L = 1.0 pF, L L = 2.5 nH, C R = 1.0 pF, and L R = 2.5 nH.
L R = 2.5 nH. Using (11.4) and (11.5), we find fse = fsh = f 0 = 3.18 GHz. Thus, for f < 3.18 GHz, the LH characteristic dominates, whereas the RH characteristic governs in the range for f > 3.18 GHz. In general, the balanced CRLH transmission line itself exhibits a wideband bandpass characteristic. There is always a transmission pole at f 0 , and N − 1 poles on both sides of f 0 . For unbalanced CRLH, Figure 11.73 illustrates frequency characteristics for C L = 0.5 pF, L L = 2.5 nH, C R = 1.0 pF, and L R = 2.0 nH. Figure 11.73(a) is for the 5cell CRLH transmission line. Figure 11.73(b) is for the 10cell case. Again, by using (11.4) and (11.5), we can find fse = 5.03 GHz, fsh = 3.18 GHz, f 0 = 4.0 GHz. Hence, the line is LH for f < 3.18 GHz, but will be RH when f > 5.03 GHz. Between 3.18 GHz and 5.03 GHz, there is an attenuation or reject band and the maximum attenuation occurs at f 0 = 4.0 GHz. This type of frequency selective characteristic may have good use in developing dualband or multiband metamaterial filters.
11.6 Metamaterial Filters
Figure 11.73
437
Characteristics of unbalanced CRLH transmission line. (a) 5 cells. (b) 10 cells. Cell element values: C L = 0.5 pF, L L = 2.5 nH, C R = 1.0 pF, and L R = 2.0 nH.
Physical implementation of CRLH metamaterial could be done in different ways and on different microwave transmission media. Figure 11.74 depicts a typical microstrip CRLH unit cell, which was originally proposed in [79]. Basically, this unit cell consists of an interdigital capacitor and a stub inductor shorted to the ground plane by a metallic via. The contributions C L and L L are provided by the interdigital capacitor and stub inductor, whereas the contributions C R and L R come from their parasitic reactances and have an increasing effect with increasing frequency. The parasitic inductance L R is due to the magnetic flux generated by the currents flowing along the digits of the capacitor and the parasitic capacitance C R is due to the parallelplate voltage gradients existing between the trace and the ground plane. Applications of this type of CRLH metamaterial have been well documented in [67]. A compact multilayered CRLH transmission line is proposed in [80]. This multilayered architecture consists of the periodic repetition of pairs of Ushaped
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Figure 11.74
Unit cell of the microstrip CRLH transmission line.
parallel plates connected to a ground enclosure by meander lines. The parallel plates provide the lefthanded (LH) series capacitance, and the meander lines provide the LH shunt inductance, while the righthanded parasitic series inductance and shunt capacitance are generated by the metallic connections in the direction of propagation and by the voltage gradient from the transmission line to the ground enclosure, respectively. In contrast to planar LH or CRLH transmission lines, such as that based on the cell of Figure 11.74, the multilayered transmission line has its direction of propagation along the vertical direction, perpendicular to the plane of the substrate. This presents the distinct advantage that a large electrical length can be achieved over an extremely short transmission line length and small transverse footprint. The miniaturized multilayered line can find applications in bandpass filters, delay lines, and numerous phaseadvanced components. As an example of application, a 1GHz/2GHz diplexer composed of two multilayered CRLH transmission lines is demonstrated [80]. In summary, electromagnetic metamaterials are new to many of us and their realizations and applications require imagination and new thinking. Nevertheless, innovative engineering approaches to metamaterials will pave the way for new generation of RF/microwave devices including filters. Table 11.5 shows the advantages of each technology. Table 11.5 Comparison of Advanced Filter Technologies HTS
MEMS
LTCC/LCP
Metamaterial
Very low loss High selectivity
High level of integration Millimeterwave applications
Compact and low cost Millimeterwave applications High level of integration (MCM, SOP)
Miniaturization Potential for new functionalities
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Bates, R. N., ‘‘Design of Microstrip SpurLine BandStop Filters,’’ IEE Int. Journal on Microwave, Optics and Acoustics, Vol. 1, November 1977, pp. 209–214. Mattaei, G., L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980. Caloz, C., and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, New York: John Wiley & Sons, 2006. Veselago, V. G., ‘‘The Electrodynamics of Substances with Simultaneously Negative Values of e and f,’’ Sov. Phys. Usp., Vol. 10, 1968, pp. 509–514. Smith, D. R., et al., ‘‘Composite Medium with Simultaneously Negative Permeability And Permittivity,’’ Phys. Rev. Lett., Vol. 84, May 2000, pp. 4184–4187. Pendry, J. B., et al., ‘‘Magnetism from Conductors and Enhanced Nonlinear Phenomena,’’ IEEE Trans. Microwave Theory Tech., Vol. 47, November 1999, pp. 2075–2084. Martin, F., et al., ‘‘Split Ring Resonator Based Left Handed Coplanar Waveguide,’’ Appl. Phys. Lett., Vol. 83, December 2003, pp. 4652–4654. Martin, F., et al., ‘‘Miniaturized CPW Stop Band Filters Based on Multiple Tuned Split Ring Resonators,’’ IEEE Microwave and Wireless Components Letters, Vol. 13, December 2003, pp. 511–513. Gil, I., et al., ‘‘Metamaterials in Microstrip Technology for Filter Applications,’’ IEEE Antenna and Propagation Society International Symposium Digest, 2005, pp. 668–671. Garcı´aGarcı´a, J., et al., ‘‘Spurious Passband Suppression in Microstrip Coupled Line Band Pass Filters by Means of Split Ring Resonators,’’ IEEE Microwave and Wireless Components Letters, Vol. 14, September 2004, pp. 416–418. Garcı´aGarcı´a, J., et al., ‘‘Miniaturized Microstrip and CPW Filters Using Coupled Metamaterial Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. 54, June 2006, pp. 2628–2635. Falcone, F., et al., ‘‘Left Handed Coplanar Waveguide Band Pass Filters Based on BiLayer Split Ring Resonators,’’ IEEE Microwave and Wireless Components Letters, Vol. 14, January 2004, pp. 10–12. Bonache, J., et al., ‘‘Novel Microstrip Bandpass Filters Based on Complementary SplitRing Resonators,’’ IEEE Trans. Microwave Theory Tech., Vol. 54, January 2006, pp. 265–271. Bonache, J., et al., ‘‘Super Compact (< 1 cm2 ) Band Pass Filters with Wide Bandwidth and High Selectivity at CBand,’’ Proceedings of the 36th European Microwave Conference, 2006, pp. 599–602. Caloz, C., and T. Itoh, ‘‘Application of the Transmission Line Theory of Lefthanded (LH) Materials to the Realization of a Microstrip ‘LH Line,’ ’’ IEEE APS Int. Symp., Vol. 2, San Antonio, TX, June 2002, pp. 412–415. Horii, Y., C. Caloz, and T. Itoh, ‘‘SuperCompact Multilayered LeftHanded Transmission Line and Diplexer Application,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. 53, April 2005, pp. 1527–1534.
CHAPTER 12
CoupledLine Circuit Components DC blocks, impedance transformers, interdigital capacitors, and spiral inductors employing coupledline sections are commonly used in microwave circuits. A coupledline section provides the required characteristics for these components over a wide frequency range. This chapter deals with these structures and includes the basic theory, design, and circuit performance to illustrate the design principles.
12.1 DC Blocks A series capacitor is used to isolate the bias voltages applied to various circuits as well as to block dc and lowfrequency voltages while allowing the RF signal to pass through with minimal loss. At microwave frequencies, both a highquality capacitor and a distributed coupled network are used. This section only deals with the coupledline structures used as dc block networks. 12.1.1 Analysis
A 3dB backwardwave coupledline section [1–3] with open circuit terminations as shown in Figure 12.1(a) can be used instead of a series capacitor as a series dc block. The circuit used for analysis is shown in Figure 12.1(b). The scattering matrix of a microstrip backwardwave coupler, when ports are termined in matched loads, is given by
冤冥冤
b1 0 b2 E2 b3 = 0 b4 E4
E2 0 E4 0
0 E4 0 E2
冥冤 冥
E4 0 E2 0
a1 a2 a3 a4
(12.1)
where E2 =
E4 =
jk sin
√1 −
k2
cos + j sin
√1 − k 2
√1 − k 2 cos + j sin 443
444
CoupledLine Circuit Components
Figure 12.1
Quarterwave coupledline section as a dc block: (a) physical layout and (b) circuit schematic.
and
=
1 ( + o ) 2 e
k=
Z 0e − Z 0o Z 0e + Z 0o
where is the electrical line length and subscripts e and o denote even and odd modes, respectively. When ports 2 and 4 are terminated in reflecting loads, then a 2 = b 2 , a 3 = 0, a 4 = b 4 , where is the reflection coefficient at ports 2 and 4. 2 For an ideal coupler when Z 0e Z 0o = Z 0 , the circuit ensures perfect match at the input. As there are reflecting terminations at ports 2 and 4, there is generally not a perfect match at port 1, except when the operating frequency and coupling length are such as to give exactly equal outputs at ports 2 and 4. At the input, the reflected wave amplitude is given by b1 = E2 b2 + E4 b4 Similarly, expressions for b 2 and b 4 are
(12.2)
12.1 DC Blocks
445
b2 = E2 a1
(12.3)
b4 = E4 a1
(12.4)
From (12.2) to (12.4):
in =
1 − k 2 (1 + sin2 )
b1 2 2 = 冠E 2 + E 4 冡 = a1
冋√1 − k 2 cos + j sin 册2
(12.5)
For k = 1/√2 and = 1 Note: = (Z L − Z 0 )/(Z L + Z 0 ), where Z L is the load impedance. VSWR =
1 +  in  1 −  in 
= 1/sin2 When = 90 degrees, VSWR = 1. We note that for a 3dB backwardwave coupler, the direct and coupled waves have equal amplitude when the coupling length is 90 degrees at the coupler’s center frequency. The variation of VSWR with frequency for various coupling values is shown in Figure 12.2. In the calculations, ports 2 and 4 are assumed to be totally reflecting ( = 1), perfect open circuits. From Figure 12.2 we find that for a 2.7dB coupler (k = 0.732), the input VSWR is below 1.2 for 0.65 ≤ f /fo ≤ 1.35, an octave frequency bandwidth. The main disadvantage of this type of dc block is that at lower frequencies, it increases the size of the circuit. At millimeter wave frequencies, where the size becomes small, however, this structure is preferred in comparison with the chip or MIM capacitor because of lower loss. Knowing Z 0e and Z 0o , the physical dimensions can easily be obtained as for any other coupler [4–6]. Simple expressions are also available [7] for Z 0e and Z 0o for a given VSWR and bandwidth, that is: Z 0e =
√S
Z 0o =
√S
冋 √ 冋 √ 1+
1+
−1 +
1+
1+
√1 + ⍀2 ⍀
1+
2
冉 冊册
√1 + ⍀2 ⍀
2
1−
1 S
Z0
冉 冊册 1−
1 S
Z0
(12.6a)
(12.6b)
where S is the voltage standing wave ratio and ⍀ is a normalized bandwidth given by ⍀ = cot
再
[1 − ( f 2 − f 1 )/( f 2 + f 1 )] 2
冎
where f 1 and f 2 are the lower and upper edge of the frequency band.
(12.7)
446
CoupledLine Circuit Components
Figure 12.2
Simulated VSWR versus fractional bandwidth of a coupledline dc block.
The value of coupling coefficient, k, in the dc block design is not very critical as long as it is greater than 0.5. However, for a given coupled length, a tighter coupling coefficient results in lower frequency of operation and larger bandwidth. 12.1.2 Broadband DC Block
A broadband dc block [1] having center frequency of 12 GHz was designed and constructed using a 25milthick alumina substrate (⑀ r ≅ 10). The physical dimensions obtained for Z 0e = 130⍀ and Z 0o = 24⍀ were spacing S = 1 mil, width W = 6.3 mil, and length l = 100 mil. Measured and simulated VSWR and insertion loss responses are shown in Figure 12.3. Over an octave band, the worst case measured VSWR and insertion loss were about 1.4 and 0.2 dB, respectively. More accurate transmission phase agreement between the measured and simulated results was obtained [3] by including openend discontinuity effects in the calculations. 12.1.3 Biasing Circuits
Solidstate circuits require low frequency biasing networks which must be separated from the RF circuit. In other words, when a bias voltage is applied to the device, the RF energy should not leak out through the bias port and also it must isolate the bias voltages applied to various devices. In the case of amplifiers and oscillators,
12.1 DC Blocks
Figure 12.3
447
(a) VSWR versus frequency response of a microstrip interdigital dc block and (b) insertion loss versus frequency response of a microstrip interdigital dc block. (From: [1]. 1972 IEEE. Reprinted with permission.)
the biasing network should not alter stability conditions. In these circuits and many others, the biasing circuitry becomes an integral part of the circuit design. There are many biasing schemes used in practice. Bascially it consists of a dc block and an RF choke as shown in Figure 12.4. A dc block can be either a capacitor or a 3dB backwardwave coupler described earlier. The RF choke at microwave frequencies is generally realized by using a highimpedance /4 line terminated by a RF bypass capacitor or a quarterwave line terminated by another /4 opencircuited line or a radial stub as shown in Figure 12.5. For lowRF leakage through the biasing network, the ratio of the shunt stub impedance and the throughline, impedance must be much greater than unity. In this case, the bandwidth increases when the impedance of the stub increases. For VSWR ≤ 1.05, the bandwidth for Z s = 100⍀ is about 12%. To further increase the bandwidth, two sections of quarterwavelong transmission lines are used. If an open circuit is required across the main line for RF signals, a quarterwave highimpedance line followed by another opencircuited quarterwave lowimpedance line are connected. The configuration is shown in Figure 12.5(a). Assuming that the through line is 50⍀, the normalized admittance with the load is y = 1 + 1/Z in
(12.8)
448
CoupledLine Circuit Components
Figure 12.4
Simplified microwave biasing circuit.
Figure 12.5
Microwave biasing circuits using multisections shunt stubs for large bandwidths. (a) Two /4 sections configuration and (b) a combination of /4 section and radial line configuration.
where Z tan 1 tan 2 − Z 2 Z in = jZ 1 1 Z 1 tan 2 + Z 2 tan 1
(12.9)
Here 1 , Z 1 and 2 , and Z 2 are the electrical line length and characteristic impedance of the first and second line sections, respectively. The VSWR response for various combinations of Z 1 and Z 2 is shown in Figure 12.6. Maximal bandwidth is obtained when Z 1 /Z 2 is large. For example, with Z 1 = 100⍀, Z 2 = 10⍀, Z 0 = 50⍀, and a VSWR = 1.2, the bandwidth is about 40%. A radial line section provides better bandwidth than a /4 opencircuit line section. A broadband biasing network structure was designed on a semiinsulating
12.1 DC Blocks
Figure 12.6
449
Simulated response of several twosection biasing networks as a function of normalized frequency. Impedance values are in ohms.
GaAs substrate with parameters listed in Table 12.1. Other structure parameters are given in Figure 12.7. All dimensions are in micrometers. The structure was analyzed and optimized using a commercial CAD tool. The measurements were made on the wafer using RF probes and TRL calibration techniques. The measured and simulated performance (return loss and insertion loss) of this biasing network are compared in Figures 12.8 and 12.9. Over an octave bandwidth (9–18 GHz), the measured return loss and insertion loss were better than 17 dB and 0.5 dB, respectively [8]. 12.1.4 MillimeterWave DC Block
At millimeterwave frequencies the coupledline dc block has distinct advantages over the conventional dc block, which generally consists of chip or MIM capacitors. Table 12.1 Parameters for the Biasing Network Structure Substrate height, h = 125 m Substrate dielectric constant, ⑀ r = 12.9 Conductor thickness, t = 4.5 m Conductor’s bulk conductivity, = 4.9 × 107 S/m Substrate loss tangent tan ␦ = 0.0005
450
CoupledLine Circuit Components
Figure 12.7
Biasing network using the Figure 12.5(b) configuration. Coupled line length is 2,045 m.
Figure 12.8
Simulated and measured  S 11  versus frequency of the biasing network.
Figure 12.9
Simulated and measured  S 21  versus frequency of the biasing network.
12.1 DC Blocks
451
These structures are broadband and cover full waveguide bands with low insertion loss. Several dc blocks on RT5880 duroid substrates were designed and fabricated with dimensions given in Table 12.2 for several millimeterwave frequency bands [9]. The length of the coupled section is 90 degrees at the center of the band. The measured insertion loss for these structures was between 0.2 to 0.4 dB over the fullwaveguide band.
12.1.5 HighVoltage DC Block
The abovedescribed coupledline dc blocks are capable of handling voltages of less than 200V, depending on the fabrication tolerances and humidity [10]. To increase the protection against voltage breakdown across the gap, an overlay of silicon rubber as shown in Figure 12.10 has been used. Because dielectric loading modifies the electrical characteristics of the coupled line, accurate design or simulation methods such as electromagnetic simulators are required to determine the new parameters. A dc block fabricated on an RT/duroid substrate with spacing S = 50 m and width W = 60 m, achieved a breakdown voltage over 4.5 kV. In general, breakdown occurs at the open ends of the coupled lines [10].
Table 12.2 A Comparison Between Designed and Measured Dimensions of the Coupled Microstrip DC Blocks
Figure 12.10
Band
h (mil)
⑀r
Designed Dimensions (mil) W S
Ka V W
10 5 5
2.2 2.2 2.2
7.0 2.5 2.5
1.0 1.0 1.0
Measured Dimensions (mil) W S 6.1 2.0 2.0
1.7 1.7 1.5
Top and side views of high voltage dc block showing highvoltage insulator dielectric. (From: [10]. 1993 IEEE. Reprinted with permission.)
452
CoupledLine Circuit Components
12.2 CoupledLine Transformers Quarterwavelengthlong tightly coupled lines have been used as filter elements, directional couplers, and dc blocks, and another important application is in broadband impedancematching transformers. There are several configurations one can use for coupledline impedance transformers including simple opencircuit coupledline transformers and transmissionline transformers. These two are discussed briefly in this section.
12.2.1 OpenCircuit CoupledLine Transformers
A simple representation of a coupledline impedance transformer is shown in Figure 12.11. This scheme works very well at millimeterwave frequencies where it eliminates dc blocking capacitors and can handle large impedance transformation without transverse resonances which occurs in conventional /4 lowimpedance microstrip singlesection impedance transformers. The analysis of this structure can easily be performed by using its equivalent circuit shown in Figure 12.12, where the expressions for Z ss and n are given by
Figure 12.11
Coupledline transformer section.
Figure 12.12
Equivalent circuit representation of the coupledline impedance transformer.
12.2 CoupledLine Transformers
453
Z ss = Z L (1 − n 2 )
(12.10)
Z 0e − Z 0o Z 0e + Z 0o
(12.11)
n=k=
Under maximum power transfer from port 1 to port 2: Z 0e = Z 0o + 2 √Z in Z L
(12.12)
If Z L is the load impedance, the input impedance is given by Z in = −jZ ss cot + n 2 Z L
(12.13)
Z in = n 2 Z L
(12.14)
When = 90 degrees:
From (12.11), (12.12), and (12.14): ZL = If Z in = Z S :
Z 0e + Z 0o 2
冉 √ 冊 冉 √ 冊
(12.15)
Z 0e = Z L 1 +
ZS ZL
(12.16a)
Z 0o = Z L 1 −
ZS ZL
(12.16b)
For given Z S and Z L , the even and oddmode impedances of the coupledline transformer can be determined from (12.16a) and (12.16b). Most of commercial CAD tools can synthesize these networks for given Z 0o , Z 0e , and substrate parameters in terms of physical dimensions W, S, and l. If Z S and Z L are the source and load impedances, the transmission coefficient S 21 is given by [7] S 21 =
2(Z 0e csc e − Z 0o csc o )
冉冉√ √ 冊 冋 ZL + ZS 2
+j
ZS ZL
2
Z 0e + Z 0o 2 √Z S Z L
−
(12.17)
(Z 0e cot e + Z 0o cot o )
Z 0e Z 0o (csc e csc o + cot e cot o ) + 2 √Z S Z L √Z S Z L
册冊
where e and o are the electrical lengths of the structure corresponding to the even and oddmode propagation constants. The electrical lenghts are given as
454
CoupledLine Circuit Components
e =
2 2 ᐉ √⑀ ree and o = ᐉ ⑀ 0 0 √ reo
(12.18)
where 0 is the freespace wavelength, and ⑀ ree and ⑀ reo are the effective dielectric constants corresponding to the even and oddmode propagation, respectively. Under firstorder approximation when e = o = /2, and S 21 = 1 (matched condition for lossless network), (12.17) provides Z 0e = Z 0o + 2 √Z S Z L
(12.19)
which is the same as (12.12) when Z S = Z in . The quarterwavelength of the coupled section at the center frequency may be calculated from the following relation [11]: ᐉ=
/2 K + [(Z 0e − Z 0o )/(Z 0e + Z 0o )] ⌬K
(12.20)
where K = (  e +  o )/2 ⌬K = (  e −  o )/2
 e = 2 √⑀ ree / 0  o = 2 √⑀ reo / 0 Table 12.3 summarizes the electrical and physical parameters for typical microstrip coupledline transformers. Parameters from the table were used to compare this type of transformer with a /4 microstrip section to transform 10⍀ impedance to 50⍀ impedance. Three cases (9, 10, and 11 in Table 12.3) were considered for coupled lines with the physical line length used being 2,200 m. For the singlesection case, W = 2,200 m and l = 1,850 m were used. The substrate parameters used are thickness h = 625 m, conductor thickness t = 3 m, ⑀ r = 9.8, tan ␦ = 0.0005, and the conductors were of gold. Figure 12.13 shows the performance comparison. Although a /4 single microstrip transformer gives the best electrical performance in terms of insertion loss and bandwidth, its physical dimension (width) is too large in the transverse direction. A transverse resonance may occur when the width of the conductor is /2. In this example the resonance frequency is about 26 GHz. Symmetrical coupledline section transformers were described above. Asymmetrical coupled sections can also be used as transformers and offer better bandwidths. Coupledline transformers are good candidates for active device impedance matching at millimeterwave frequencies where they also serve the purpose of lowloss dc blocks. Thus, coupledline transformers provide wide bandwidth and eliminate the use of dc blocking capacitors in active circuits.
Parameters for Different Transformation Ratios: ⑀ r = 9.8, t = 3 to 4 m, and h = 635 m
Serial Number
Transformation From: To: ⍀ ⍀
Desired Impedances Z 0o Z 0e ⍀ ⍀
Actual Impedances Z 0e Z 0o ⍀ ⍀
K 0e
K 0o
W/h
S/h
1 2 3 4 5 6 7 8 9 10 11
50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0
162.5 152.5 142.96 132.96 123.67 113.67 110.71 100.71 84.72 74.72 64.72
162.40 153.81 143.06 132.50 123.24 113.75 110.28 100.69 84.66 74.57 65.11
0.404 0.403 0.401 0.399 0.397 0.394 0.394 0.391 0.385 0.381 0.376
0.4300 0.4298 0.4298 0.4296 0.4293 0.4291 0.4287 0.4284 0.4261 0.4253 0.4246
0.120 0.159 0.175 0.240 0.260 0.350 0.335 0.455 0.565 0.790 1.05
0.048 0.018 0.070 0.030 0.135 0.040 0.135 0.055 0.250 0.115 0.01
75.0 75.0 53.0 53.0 35.0 35.0 25.0 25.0 10.0 10.0 10.0
40.0 30.0 40.0 30.0 40.0 30.0 40.0 30.0 40.0 30.0 20.0
40.59 30.24 40.47 30.72 39.83 29.96 40.03 30.07 39.81 30.17 20.14
Velocity Factors
Physical Dimensions
12.2 CoupledLine Transformers
Table 12.3
K 0e = 1/√⑀ ree and K 0o = 1/√⑀ reo
455
456
CoupledLine Circuit Components
Figure 12.13
Simulated performance of several 10⍀ to 50⍀ transformers on 25mil alumina substrates.
12.2.2 Transmission Line Transformers
Transmission line transformers (TLT) using straight or coiled sections of coupled transmission lines are frequently used to realize multioctave impedance transformation at RF and lower microwave frequencies [12–21]. This type of a transformer can be designed employing any multilayer fabrication technology such as a printed circuit board, LTCC, HTCC, and monolithic Si or GaAs ICs. Figure 12.14 shows a 4:1 impedance transformer (Z S = 4Z L ), where Z S and Z L are the source and load impedance, respectively. Here transmission lines A and B are not electromagnetically coupled and their length (L) is typically /4. In this configuration, each line has a characteristic impedance Z 0 . If the lines are coupled, such as the sidecoupled microstrip lines, shown in Figure 12.15 (top view), the bandwidth increases and the transformer length decreases with stronger coupling between the conduc
12.2 CoupledLine Transformers
457
Figure 12.14
Impedance transformer configurations using two uncoupled transmission lines A and B.
Figure 12.15
Impedance transformer configurations using sidecoupled microstrip lines. Ground plane not shown.
tors. In the asymmetric broadsidecoupled microstrip lines shown in Figure 12.16 (side view), the coupling coefficient is much stronger. Therefore, this configuration results in better bandwidth and smaller transformer size. In this case, the length (L) is typically /8. When ports 1 and 2 are switched, then the transformer is designated as 1:4 TLT. The design of an asymmetric broadsidecoupled TLT cannot be performed accurately using conventional circuit simulators, because the substrate and the conductor are of a multilayer type. However, accurate solutions can be obtained by using an electromagnetic simulator. A 50ohm to 12.5ohm (4:1) asymmetric broadsidecoupled TLT was designed [21] using em by Sonnet software. The substrate parameters for impedance transformer are given in Table 12.4. The conductors have a width W and a length L. The characteristics of this transformer were compared with three other transformers: a single section quarterwave microstrip, uncoupled lines TLT, and sidecoupled lines TLT. For these transformers, Figures
Figure 12.16
Asymmetric broadside coupled microstrip lines.
458
CoupledLine Circuit Components Table 12.4 Substrate Parameters for Broadband Impedance Transformers GaAs substrate ⑀ r = 12.9 Substrate thickness h = 75 m Polyimide ⑀ rd = 3.2 Polyimide thickness d = 7 m Gold conductor’s thickness T = 4.5 m
12.17 and 12.18 show the reflection and transmission coefficients versus frequency, respectively, and Table 12.5 summarizes the bandwidth performance. Here, port 1 is terminated in 50 ohms and port 2 is terminated in 12.5 ohms. It may be noted that, among these four transformers, the broadsidecoupled TLT has the largest bandwidth and shortest line length. Table 12.5 compares bandwidths for three cases of return loss: 10, 15, and 20 dB. The bandwidth of a transformer is defined in terms of return loss (RL), that is, the frequency range over which the RL is equal to or greater than a specified value. The fractional bandwidth, FBW, is defined as: FBW =
⌬f f0
⌬ f = f2 − f1 , f0 =
(12.21)
√f1 f2
(12.22)
where f 1 and f 2 are the lower and upper edge of the frequency band. Although this example of a 50ohm to 12.5ohm (both real impedances) transformer demonstrates the unique features, such as the largest bandwidth and shortest line length of an asymmetrical broadsidecoupled microstrip TLT, this transformer can also
Figure 12.17
The reflection coefficient response of four types of 4:1 transformers. Z S = 50 ohms.
12.2 CoupledLine Transformers
Figure 12.18
459
The transmission coefficient response of four types of 4:1 transformers. Z S = 50 ohms.
Table 12.5 Bandwidth Comparison of Several 4:1 Impedance Transformers: 50 Ohms to 12.5 Ohms Performance
Configuration SingleSection Microstrip W = 190 m, L = 2,500 m Uncoupled Microstrip Lines W = 200 m, L = 2,500 m Coupled Microstrip Lines W = 130 m, S = 20 m, L = 2,300 m BroadsideCoupled Microstrip Lines W = 20 m, d = 7 m, L = 1,400 m
Return Loss (dB)
f0 (GHz)
FBW (%)
20 15 10 20 15 10 20 15 10 20 15 10
9.86 9.77 9.47 9.81 9.61 9.07 9.50 9.35 9.25 9.13 8.74 8.01
16.1 31.3 61.3 26.4 47.9 83.8 36.8 67.0 130.9 72.6 130.0 227.5
be used to transform a complex impedance to a real impedance or vice versa. It may also be used to transform one complex impedance to another complex impedance. The size of these transformers is further reduced by folding the lines in a coil or loop shape because the line width is much narrower than the length. By cascading two sections of 4:1 transformers in a series, one can match 50 ohms to 3.1 ohms over larger bandwidths. The selection of suitable structure parameters for a transformer is very important to provide the minimum loss and the required impedance transformation over the desired frequency range. Some of these parameters are described next. The effect of polyimide thickness, d, on the bandwidth of TLT was studied by varying its value and keeping other parameters constant. The microstrip width and length are 30 m and 1,400 m, respectively. Figure 12.19 shows the maximum
460
CoupledLine Circuit Components
Figure 12.19
Maximum fractional bandwidth and source impedance of a 4:1 TLT versus polyimide thickness.
FBW and corresponding source impedance of a 4:1 transformer as a function of polyimide thickness between the two broadsidecoupled conductors. It may be noted that the tighter coupling between the conductors results in a larger bandwidth and lower input impedance. Next we consider the effect of microstrip width on the optimum source impedance to be matched and the corresponding bandwidth. The substrate parameters are given in Table 12.4. The line length in this case is 1,400 m. Table 12.6 gives the fractional bandwidth for 4:1 transformers having different microstrip widths. Here Z S (see Figure 12.15) is the source impedance and W is the microstrip width. As the microstrip width decreases, or the characteristic impedance increases, the bandwidth decreases. In this example, to match 50 ohms to 12.5 ohms, a line width of about 15 m or a characteristic impedance of 74 ohms is required and the resultant fractional bandwidth is about 130%. The fractional bandwidth as a function of source impedance of a 4:1 transformer was calculated for three microstrip widths: 20, 40, and 60 m. This is shown in Figure 12.20. For each microstrip width there is a maximum FBW and it decreases for other impedance value. Table 12.6 Maximum Bandwidth as a Function of Line Width for Several TLTs Where Z S Is the Source Impedance and the Load Impedance Z L = Z S /4 (RL = 15 dB) Line Width W ( m)
Z S (⍀)
Frequency Range (GHz)
Center Frequency, f 0 (GHz)
Fractional Bandwidth, FBW (%)
10 20 40 60 80 100 120
60 40 23 16 12 10.5 8.0
7–21 5–20 3.5–20 3.0–20 2.5–20 2.1–20 2.0–20
12.1 10.0 8.37 7.75 7.07 6.48 6.32
115.5 150.0 197.2 219.5 247.5 276.2 284.6
12.3 Interdigital Capacitor
Figure 12.20
461
Fractional bandwidth of a 4:1 TLT as a function of source impedance.
The advantages of the TLT technique are its compact size, low loss, and wider bandwidth capability. When this transformer is used as part of an active circuit, a dc block capacitor is required between the transformer and ground connection. The design of a high power TLT has been described in [21].
12.3 Interdigital Capacitor The interdigital or interdigitated capacitor is a multifinger periodic structure as shown in Figure 12.21. Interdigital capacitors use the capacitance that occurs across a gap in thinfilm conductors. These gaps are essentially very long and folded to use a small amount of area. As illustrated in Figure 12.21, the gap meanders back and forth in a rectangular area forming two sets of fingers, which are interdigital. By using a long gap in a small area, compact singlelayer smallvaled capacitors can be realized. Typically, values range from 0.05 pF to about 0.5 pF. The capacitance can be increased by increasing the number of fingers, or by putting on an overlay dielectric layer, which also acts as a protective shield. One of the important
Figure 12.21
An interdigital capacitor configuration using seven fingers.
462
CoupledLine Circuit Components
design considerations is to keep the size of the capacitor very small relative to a wavelength so that it can be treated as a lumped element. A larger total widthtolength ratio results in a desired higher shunt capacitance and lower series inductance. This type of capacitor can be fabricated by hybrid technology, used in the fabrication of conventional integrated circuits or monolithic microwave integrated circuit technology and does not require any additional processing steps. 12.3.1 Approximate Analysis
Analysis and characterization of interdigital capacitors have been reported in the literature [22–27]. Earlier analyses [22–24] data when compared with the measured results showed that these analyses were inadequate to describe the capacitors accurately. The analyses were based on lossless microstrip coupled lines [22] and lossy coupled microstrip lines [23]. A more accurate characterization of these capacitors can be performed if the capacitor geometry is divided into basic microstrip sections such as the single microstrip line, coupled microstrip lines, openend discontinuity, asymmetrical gap, 90degree bend, and Tjunction discontinities [26] as shown in Figure 12.22. This model provides better accuracy than the previously reported analyses. This method still provides an approximately solution, however,
Figure 12.22
The interdigitated capacitor and its subcomponents. (From: [26]. 1988 IEEE. Reprinted with permission.)
12.3 Interdigital Capacitor
463
because of several assumptions in the grouping of subsections and does not include interaction effects between basic microstrip sections described above. An approximate expression for an interdigital capacitor is given by [22] C = (⑀ r + 1) ᐉ [(N − 3) A 1 + A 2 ] pF
(12.23)
where A 1 (the interior) and A 2 (the two exterior) are the capacitances of the fingers, N is the number of fingers, and the dimension ᐉ in m is as shown in Figure 12.21. For infinite substrate thickness (or no ground plane), A 1 = 4.409 × 10−6 pF/ m and A 2 = 9.92 × 10−6 pF/ m. For a finite substrate, the effect of h must be included in A 1 and A 2 . In the final design, usually S = W and ᐉ ≤ /4. The total series capacitance can also be written as [6] C = 2⑀ 0 ⑀ re =
K(k) (N − 1) ᐉ F K′(k)
(12.24a)
K(k) 10−11 ⑀ (N − 1) ᐉ × 10−4 F 18 re K′(k)
or
⑀ 10−3 K(k) (N − 1) ᐉ pF C = re 18 K′(k)
(12.24b)
where ᐉ is in m, N is the number of fingers, ⑀ re is the effective dielectric constant of the microstrip line of width W, and
再
1 + √k K(k) 1 = ᐉn 2 K′(k) 1 − √k = 1 + √k′ ᐉn 2 1 − √k′
冋
册
冎
for 0.707 ≤ k ≤ 1
(12.25a)
for 0 ≤ k ≤ 0.707
(12.25b)
and k = tan2
冉 冊
a , a = W /2, and b = (W + S)/2, and k′ = 4b
√1 − k 2 (12.25c)
The series resistance of the interdigital capacitor is given by R=
4 ᐉ R 3 WN s
(12.26)
464
CoupledLine Circuit Components
where R s is the sheet resistivity (⍀/square) of the conductors used in the capacitor. The effect of metalthickness t plays a secondary role in the calculation of capacitance. The Q of this capacitor is given by
Qc =
3WN 1 = CR C4ᐉR s
(12.27)
12.3.2 FullWave Analysis
Quasistatic and fullwave numerical methods have been extensively employed to analyze transmission lines and their discontinuities. The numerical data obtained from these methods have been used to develop analytical and empirical design equations along with equivalent circuit (EC) models to describe the electrical performance of planar transmission lines and their discontinuities, including microstrip, coplanar waveguide, and slot lines. These equations and EC models have been exclusively used in commercial microwave CAD tools. The recent advances in workstations and userfriendly software have made it possible to develop electromagnetic (EM) simulators. These [28–34] have added another dimension to computeraided engineering (CAE) tools. These simulators play an important role in the simulation of singlelayer elements such as transmission lines, patches and their discontinuities, multilayer compenents (namely, inductors, capacitors, packages), and mutual coupling between various circuit elements. Accurate evaluation of the effects of radiation, surface waves, and interaction between components on the performance of densely packed monolithic microwave integrated circuits can only be calculated using EM simulators [28–34]. Table 12.7 lists the physical parameters of a typical interdigital capacitor (Figure 12.21) analyzed using EM simulators, and Figures 12.23 and 12.24 compare the simulated S 11 and S 21 magnitude, and S 11 and S 21 phase performances [35–37], respectively. A detailed treatment of interdigital capacitors is included in [20, Chapter 7].
Table 12.7 Parameters for an Interdigital Structure Substrate height, h = 100 m Substrate dielectric constant, ⑀ r = 12.9 Conductor thickness, t = 0.8 m Conductor’s bulk conductivity, = 4.9 × 107 S/m Substrate loss tangent, tan ␦ = 0.0005 W = 40 m S = 20 m ᐉ = 440 m W ′ = 520 m S ′ = 40 m ᐉ ′ = 60 m Number of fingers = 9 Enclosure: No
12.4 Spiral Inductors
Figure 12.23
465
Interdigitated capacitor  S 11  and  S 21  responses.
12.4 Spiral Inductors Spiral (rectangular on circular) inductors are used as RF chokes, matching elements, impedance transformers, and reactive terminations, and are also found in filters, couplers, dividers and combiners, baluns, and resonant circuits. In the lowmicrowave frequency monolithic approach, lowloss inductors are essential for developing compact and lowcost, lownoise amplifiers and highpower addedefficiency amplifiers. Inductors in MICs are fabricated using standard integrated circuit processing with no additional process steps. The innermost turn of the inductor is connected to other circuitry by using a wire bond connection in the hybrid MICs and through a conductor that passes under airbridges in monolithic MIC technology. The width and thickness of the conductor determines the currentcarrying capacity of the inductor. Typically the thickness is 0.5 to 1.0 m, and the airbridge separates it from the upper conductors by 1.5 to 3.0 m. In dielectric crossover technology, the separation between the crossover conductors may be anywhere from 0.5 to 3 m. Typical inductance values for monolithic microwave integrated circuits working above the Sband fall in the range of 0.5 to 10 nH. The design of spiral inductors for MIC applications is usually based on two approaches: the lumpedelement method and microstrip coupledline method. The lumpedelement approach uses frequency independent formulas for freespace induc
466
CoupledLine Circuit Components
Figure 12.24
Interdigitated capacitor ∠ S 11 and ∠ S 21 responses.
tance with groundplane effects. These formulas are useful only when the total length of the inductor is a small fraction of the operating wavelength and when the interturn capacitance can be ignored. Wheeler [38] presented an approximate formula for the inductance of a circular spiral inductor, with reasonably good accuracy at lower microwave frequencies. This formula has been extensively used for the design of microwave lumped circuits [39–41]. Inductor parameters can also be obtained from twoport Sparameter measurements for the structure. This approach requires fabrication of the structure, however. In the coupledline approach [42, 43], an inductor is analyzed using multiconductor coupled microstrip lines. This technique predicts the spiral inductor’s performance reasonably well for two turns and up to 18 GHz. Inductors with their conductors supported on posts provide lower capacitances from the conductor to ground and between conductors, which results in a higher resonant frequency, thereby extending the frequency range of operation. A 2turn spiral microstrip inductor with opposite sides appropriately connected as shown in Figure 12.25 may be treated as a coupledline section. This figure shows 2turn circular and rectangular spiral inductors and a 1.75turn rectangular spiral inductor with connecting singleline section and the feed lines represented between nodes 1 and 2, and between nodes 4 and 5. In the 2turn case, the parallel coupledline section, which has a total line length equivalent to the spiral length between nodes 2 and 4, is represented between nodes 2, 3, and 4 and constitutes the intrinsic inductor. The length is taken as the average of the outer and inner turn lengths. In the 1.75turn inductor, an additional single line between nodes 3 and 4 is connected, whereas the coupled line is between nodes 2 and 3.
12.4 Spiral Inductors
Figure 12.25
467
Spiral inductors and its coupledline equivalent circuit models (a) circular 2 turns, (b) rectangular 2 turns, and (c) rectangular 1.75 turns.
The electrical equivalent coupledline model shown in Figure 12.25 does not include the crossover capacitance or the rightangle bend discontinuity effects. Figure 12.26 shows a modified equivalent circuit of a 1.75turn rectangular spiral inductor that also includes the crossover capacitor. This figure also shows the physical and electrical parameters of this inductor. Figure 12.27 shows a further subdivision of the inductor that is required to evaluate its performance more accurately. In this case, the inductor is split into three sections representing elements 1, 2, and 3. The performance of these inductors can be calculated by either using commercial CAD tools or by solving cascaded ABCD or Sparameter matrices for these elements. Improved versions of these inductors include chamferred bends or compensated bends [5]. The electrical characteristics of the intrinsic 2turn inductor can be derived from the general fourport network of a coupledline section as shown in Figure
468
CoupledLine Circuit Components
Figure 12.26
Rectangular 1.75turn spiral inductor (a) physical layout and (b) coupledline equivalent circuit model. (From: [42]. 1983 IEEE. Reprinted with permission.)
Figure 12.27
The network model for calculating the inductance of a planar rectangular spiral inductor.
12.4 Spiral Inductors
469
12.28, where the current and voltage relationships of the pair of lines can be described by the admittance matrix equation as follows:
冤冥 冤
I1 Y11 I2 Y21 I3 = Y31 I4 Y41
Y12 Y22 Y32 Y42
Y13 Y23 Y33 Y43
冥冤 冥
Y14 Y24 Y34 Y44
V1 V2 V3 V4
(12.28)
This matrix may be reduced to two ports, by applying the boundary condition that ports 2 and 4 are connected, that is: V2 = V4
(12.29a)
I2 = −I4
(12.29b)
and by rearranging the matrix elements, the twoport matrix may be written as
冋册 冋 I1 I3
=
′ Y11 ′ Y31
册冋 册
′ Y13 ′ Y33
V1 V3
(12.30)
where (Y12 + Y14 ) (Y21 + Y41 ) Y22 + Y24 + Y42 + Y44
(12.31a)
(Y + Y14 ) (Y23 + Y43 ) ′ = Y13 − 12 Y13 Y22 + Y24 + Y42 + Y44
(12.31b)
′ = Y11 ′ Y33
(12.32a)
′ ′ = Y13 Y31
(12.32b)
′ = Y11 − Y11
and
due to symmetry. The admittance parameters for a coupled microstrip line in inhomogenous dielectric medium are given by [44]
Figure 12.28
Fourport representation of a coupledline section of an inductor.
470
CoupledLine Circuit Components
Y11 = Y22 = Y33 = Y44 = −j [Y0e cot e + Y0o cot o ]/2
(12.33a)
Y12 = Y21 = Y34 = Y43 = −j [Y0e cot e − Y0o cot o ]/2
(12.33b)
Y13 = Y31 = Y24 = Y42 = j [Y0e csc e − Y0o csc o ]/2
(12.33c)
Y14 = Y41 = Y23 = Y32 = j [Y0e csc e + Y0o csc o ]/2
(12.33d)
An equivalent ‘‘ ’’ representation of a twoport network is shown in Figure 12.29, where ′ YA = −Y13
(12.34)
′ + Y13 ′ YB = Y11
(12.35)
and
YA = −j
1 2
冦
冋 冉 冋 冉
冊 冉 冊 冉
冊册 冊册
1 − cos e 1 + cos o + Yoo sin e sin o Yoe cot e + Yoo cot o + 1 − cos e 1 + cos o 2 Yoe − Yoo sin e sin o Yoe
冧
(12.36) YB =
2jYoe Yoo (1 − cos e ) (1 + cos e ) [Yoo sin e (1 + cos o ) − Yoe sin o (1 − cos e )]
(12.37)
Because the physical length of the inductor is much less than /4, sin e, o ≅ e, o and cos e, o ≅ 1 − e,2 o /2. Also Yoo > Yoe . Therefore, (12.36) and (12.37) are approximated as follows:
Figure 12.29
Y YA ≅ −j oe 2 e
(12.38)
YB ≅ jYoe e
(12.39)
equivalent circuit representation of the inductor.
12.4 Spiral Inductors
471
which are independent of the odd mode. Thus the ‘‘ ’’ equivalent circuit consists of a shunt capacitance C and a series inductance L, as shown in Figure 12.30. The expressions for L and C may be written as YA =
1 Y = −j oe j L 2 e
(12.40)
2 e Yoe
(12.41)
or L= and YB = j C = jYoe e
(12.42)
Y C = oe e
(12.43)
or
If ᐉ is the average length of the conductor, then
e =
ᐉ ⑀ c √ ree
(12.44)
where c is the velocity of light in free space and ⑀ ree is the effective dielectric constant for the even mode. When Z 0e = 1/Y0e , from (12.41) and (12.43): L=
2ᐉ Z 0e √⑀ ree c
(12.45)
ᐉ √⑀ ree Z 0e c
(12.46)
C=
In a loosely coupled inductor Z oe ≅ Z o and ⑀ ree = ⑀ re for the single conductor microstrip line. The above equations may be used to approximately evaluate induc
Figure 12.30
Equivalent L C circuit representation of the inductor.
472
CoupledLine Circuit Components
tor performance. Figure 12.31 shows measured and modeled S 11 and S 21 responses of a twoturn inductor. A detailed treatment of spiral inductors is included in [20, Chapters 2 and 3].
12.5 Spiral Transformers Classical coil transformers used at low and radio frequencies can be realized by hybrid and monolithic techniques working in the microwave frequency range. A major challenge in printed coil transformers is keeping the parasitic capacitances and series resistances low, to operate these components at higher frequencies with low insertion loss. The transformers can be two, three, or fourport components. The threeport transformers may have 0degree, 90degree, or 180degree phase difference at the output ports. Rectangular spiral transformers fabricated using GaAs MMIC technology have been reported in the literature [45–49]. In active circuits, their impedance transformer ratio and inductance values are used for impedance transformation and resonating out the active device’s capacitance, respectively. Figure 12.32 shows the physical layout of a twoconductor transformer consisting of a series of turns of thin, metallized conductors placed on a dielectric substrate (not shown). The characterization of this structure is not straightforward; however, a multiconductor coupled microstrip line analysis [50, 51] may be used to determine its parameters approximately. A more accurate characterization of this transformer can only be achieved by using fullwave and comprehensive circuit simulators such as EM CAD tools. The twincoil fourport rectangular spiral transformer [46, 48] shown in Figure 12.32 has a ground ring around it. The dimensions of the transformer are, outside ring = 1,020 × 710 m, conductor thickness = 1 m, GaAs (⑀ r = 12.9), substrate thickness = 250 m, dielectric crossover height = 1.3 m, overlay dielectric ⑀ r = 6.8, conductor width = 20 m, and conductor gap = 6 m. Figure 12.33 shows a comparison between the measured and simulated S 11 and S 21 responses
Figure 12.31
Measured and modeled (a) reflection (S 11 ) and (b) transmission (S 21 ) responses for the twoturn inductor. (From: R. Plumb, workshop notes.)
12.5 Spiral Transformers
473
Figure 12.32
Spiral transformer example. (From: [46]. 1989 IEEE. Reprinted with permission.)
Figure 12.33
The  S 11  and  S 21  responses of the twoport configuration of the transformer in Figure 12.32. (From: [46]. 1989 IEEE. Reprinted with permission.)
474
CoupledLine Circuit Components
of this transformer when ports 3 and 4 are shortcircuited as shown in Figure 12.32. The simulated performance was obtained using an electromagnetic simulator. As shown in Figure 12.33, S 21 has a sharp null at 6 GHz that occurs because of the /4 shortcircuited secondary coil at that frequency. In this case, maximum current flows through the grounded port 4 and negligible current flows through port 2. Thus, power flowing into the load at port 2 is negligible and results in a null in the S 21 response. Below 6 GHz, the current in the spiral conductors is nearly constant and the power transfer is provided by magnetic coupling similar to that in classical coil transformers. Above 6 GHz, however, the spiral conductors are electrically long and the current distribution along the conductors has less of a standing wave nature. Their behavior becomes closer to that of coupled transmission lines supporting both magnetic and electric coupling. When ports 3 and 4 are terminated in 50⍀ loads, the measured twoport response of the transformer is shown in Figure 12.34, which shows a gradual increase of S 11 and S 21 . Because in this case there are no standing waves, the power transfer from primary to secondary occurs gradually from magnetic to magnetic and electric coupling. However, 50⍀ terminating loads result in higher loss in the transformer. The above examples show that efficient power transfer in a twoport transformer occurs at high frequencies through both magnetic and electric coupling. At such frequencies, the spiral conductors are longer than /4. This does not permit grounding of the center tap of the secondary spiral conductor to obtain balanced output at ports 2 and 3. On the other hand, at low frequencies, a center tap is possible; as in classical transformers, however, power transfer is inefficient as
Figure 12.34
The  S 11  and  S 21  responses of the fourport configuration, constructed by adding 50⍀ series resistors to ports 3 and 4 in Figure 12.32. (From: [46]. 1989 IEEE. Reprinted with permission.)
12.6 Other CoupledLine Components
475
shown in Figure 12.34. The power transfer can be improved by making these transformers electrically small, minimizing the parasitic capacitances, and increasing the number of tightly coupled turns of the spirals, while maintaining the same spiral length. Efficient power transfer can also be obtained by using more than two coils in a transformer. Figure 12.35 shows a layout of a 3conductor coupledline transformer. Figure 12.36(a) shows a photograph of this structure known as a triformer having 1.5 turns and fabricated on GaAs substrate using conductor width and spacing of 5 m [47] and designed as a twoport matching network. An electrical equivalent circuit is shown in Figure 12.36(b). TRL denotes on wafer probe pads. The triformer structure may be used in the realization of wideband baluns.
12.6 Other CoupledLine Components Other applications of coupledline sections include Schiffman sections [52–54], broadband 180degree bit phase shifters [55], power dividers [56], resonators [57], and delay lines [58]. Coupledline sections are commonly used to improve the bandwidth of phase shifters, and a coupledline resonator is an integral part of bandpass filters. In nway power dividers, they are responsible for reducing component size.
Figure 12.35
Schematic of a 1.5turn rectangular spiral triformer.
476
CoupledLine Circuit Components
Figure 12.36
(a) Photograph of a 1.5turn MMIC triformer; and (b) twoport equivalent circuit of the triformer. (From: [47]. 1989 IEEE. Reprinted with permission.)
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[5]
Lacombe, D., and J. Cohen, ‘‘OctaveBand Microstrip DC Blocks,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT20, August 1972, pp. 555–556. Ho, C. Y., ‘‘Analysis of DC Blocks Using Coupled Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT23, September 1975, pp. 773–774. Free, C. E., and C. S. Aitchison, ‘‘Excess Phase in Microstrip DC Blocks,’’ Electronics Letters, Vol. 20, October 1984, pp. 892–893. Garg, R., and I. J. Bahl, ‘‘Characteristics of Coupled Microstriplines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT27, July 1979, pp. 700–705; also see correction in IEEE Trans. Microwave Theory Tech., Vol. MTT28, March 1980, p. 272. Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House, 1996, Ch. 8.
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[10] [11]
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Bahl, I. J., and P. Bhartia, Microwave SolidState Circuit Design, New York: John Wiley, 2003, Ch. 2. Kajfez, D., and B. S. Vidula, ‘‘Design Equations for Symmetric Microstrip DC Blocks,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT28, September 1980, pp. 974–981. Bahl, I. J., ‘‘Simulation Column,’’ Int. J. Microwave and MillimeterWave ComputerAided Engineering, Vol. 2, July 1992, pp. 204–206. Ho, T. Q., and Y. C. Shih, ‘‘Broadband MillimeterWave EdgeCoupled Microstrip DC Blocks,’’ Microwave Systems News and Communication Technology, Vol. 17, April 1987, pp. 74–78. Koscica, T. E., ‘‘Microstrip QuarterWave HighVoltage DC Block,’’ IEEE Trans. Microwave Theory Tech., Vol. 41, January 1993, pp. 162–164. Easter, B., and B. S. Shivashankaran, ‘‘Some Results on the EdgeCoupled Microstrip Section as an Impedance Transformer,’’ IEE J. Microwaves, Opt. Acoust., Vol. 3, March 1979, pp. 63–66. Rotholz, E., ‘‘TransmissionLine Transformers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT29, April 1981, pp. 327–331. Sevick, J., Transmission Line Transformers, Atlanta, GA: Noble Publishing, 1996. Abrie, P. D., Design of RF and Microwave Amplifiers and Oscillators, Norwood, MA: Artech House, 1999. Davis, W. A., and K. K. Agarwal, Radio Frequency Circuit Design, New York: John Wiley & Sons, 2001. Engels, M., et al., ‘‘Design Methodology, Measurement and Application of MMIC Transmission Line Transformers,’’ IEEE MTTS Int. Microwave Symp. Dig., 1995, pp. 1635–1638. Liu, S.P., ‘‘Planar Transmission Line Transformer Using Coupled Microstrip Lines,’’ IEEE MTTS Int. Microwave Symp. Dig., 1998, pp. 789–792. Horn, J., and G. Boeck, ‘‘Integrated Transmission Line Transformer,’’ IEEE MTTS Int. Microwave Symp. Dig., 2004, pp. 201–204. Ang, K. S., C. H. Lee, and Y. C. Leong, ‘‘Analysis and Design of Coupled Line Impedance Transformer,’’ IEEE MTTS Int. Microwave Symp. Dig., 2004, pp. 1951–1954. Bahl, I., Lumped Elements for RF and Microwave Circuits, Norwood, MA: Artech House, 2003, Chapter 11. Bahl, I. J., ‘‘Broadband and Compact Impedance Transformers for Microwave Circuits,’’ IEEE Microwave Magazine, Vol. 7, August 2006, pp. 56–62. Alley, G. D., ‘‘Interdigital Capacitors and Their Application to LumpedElement Microwave Integrated Circuits,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT18, December 1970, pp. 1028–1033. Hobdell, J. L., ‘‘Optimization of Interdigital Capacitors,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT27, September 1979, pp. 788–791. Esfandiari, R., D. W. Maki, and a M. Siracusa, ‘‘Design of Interdigitated Capacitors and Their Application to GaAs Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT31, January 1983, pp. 57–64. Joshi, J. S., J. R. Cockrill, and J. A. Turner, ‘‘Monolithic Microwave Gallium Arsenide FET Oscillators,’’ IEEE Trans. Electron Devices, Vol. ED28, February 1981, pp. 158–162. Pettenpaul, E., et al., ‘‘CAD Models of Lumped Elements on GaAs up to 18 GHz,’’ IEEE Trans. Microwave Theory Tech., Vol. 36, February 1988, pp. 294–304. Sadhir, V., I. Bahl, and D. Willems, ‘‘CADCompatible Accurate Models for Microwave Passive Lumped Elements for MMIC Applications,’’ Int. J. Microwave and Millimeter Wave ComputerAided Engineering, Vol. 4, April 1994, pp. 148–162. EM, Sonnet Software, Liverpool, NY, 2006. HFSS, ANSOFT, Pittsburgh, PA, 2006.
478
CoupledLine Circuit Components [30]
Microwave Office, AWR, El Segundo, CA, 2006.
[31]
HighFrequency Structure Simulator, Agilent, Santa Rosa, CA, 2006.
[32]
LINMIC + Analysis Program, Jansen Microwave, Ratingen, Germany.
[33]
MSC/EMAS, MacNeal Schwendler, Milwaukee, WI.
[34]
IE3D, Zeland Software, San Francisco, CA, 2006.
[35]
Kattapelli, K., J. Burke, and A. Hill, ‘‘Simulation Column,’’ Int. J. Microwave and MillimeterWave ComputerAided Engineering, Vol. 3, January 1993, pp. 77–79.
[36]
Rautio, J., ‘‘Simulation Column,’’ Int. J. Microwave and MillimeterWave ComputerAided Engineering, Vol. 3, January 1993, pp. 80–81.
[37]
Zhang, J. X., ‘‘Simulation Column,’’ Int. J. Microwave and MillimeterWave ComputerAided Engineering, Vol. 3, July 1993, pp. 299–300.
[38]
Wheeler, H. A., ‘‘Simple Inductance Formulas for Radio Coils,’’ Proc. IRE, Vol. 16, October 1928, pp. 1398–1400.
[39]
Grover, F. W., Inductance Calculations, Princeton, NJ: Van Nostrand, 1946, reprinted by Dover Publications, 1962, pp. 17–47.
[40]
Daly, D. A., et al., ‘‘Lumped Elements in Microwave Integrated Circuits,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT15, December 1967, pp. 713–721.
[41]
Caulton, M., et al., ‘‘Status of Lumped Elements in Microwave Integrated Circuits— Present and Future,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT19, July 1971, pp. 588–599.
[42]
Camp Jr., W. O., S. Tiwari, and D. Parson, ‘‘26 GHz Monolithic Microwave Amplifier,’’ IEEE MTTS Int. Microwave Symp. Dig., 1983, pp. 46–49.
[43]
Cahana, D., ‘‘A New Transmission Line Approach for Designing Spiral Microstrip Inductors for Microwave Integrated Circuits,’’ IEEE MTTS Int. Microwave Symp. Dig., 1983, pp. 245–247.
[44]
Zysman, G. I., and A. K. Johnson, ‘‘Coupled Transmission Line Networks in an Inhomogenous Dielectric Medium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT17, October 1969, pp. 753–759.
[45]
Ferguson, D., et al., ‘‘Transformer Coupled HighDensity Circuit Technique for MMIC,’’ IEEE GaAs IC Symp. Dig., 1984, pp. 34–36.
[46]
Howard, G. E., et al., ‘‘The Power Transfer Mechanism of MMIC Spiral Transformers and Adjacent Spiral Inductors,’’ IEEE MTTS Int. Microwave Symp. Dig., 1989, pp. 1251–1254.
[47]
Boulouard, A., and M. LeRouzic, ‘‘Analysis of Rectangular Spiral Transformers for MMIC Applications,’’ IEEE Trans. Microwave Theory Tech., Vol. 37, August 1989, pp. 1257–1260.
[48]
Chow, Y. L., G. E. Howard, and M. G. Stubbs, ‘‘On the Interaction of the MMIC and its Packaging,’’ IEEE Trans. Microwave Theory Tech., Vol. 40, August 1992, pp. 1716–1719.
[49]
Chen, T. H., ‘‘Broadband Monolithic Passive Baluns and Monolithic DoubleBalanced Mixer,’’ IEEE Trans. Microwave Theory Tech., Vol. 39, December 1991, pp. 1980–1986.
[50]
Marx, K. D., ‘‘Propagation Modes, Equivalent Circuits, and Characteristics Terminations for Multiconductor Transmission Lines with Inhomogeneous Dielectrics,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT21, July 1973, pp. 450–457.
[51]
Djordjevic, A., et al., Matrix Parameters for Multiconductor Transmission Lines, Norwood, MA: Artech House, 1989.
[52]
Schiffman, B. M., ‘‘A New Class of BroadBand Microwave 90Degree Phase Shifters,’’ IRE Trans. Microwave Theory Tech., Vol. MTT6, April 1958, pp. 232–237.
[53]
Quirarte, J. L. R., and J. P. Starski, ‘‘Novel Schiffman Phase Shifters,’’ IEEE Trans. Microwave Theory Tech., Vol. 41, January 1993, pp. 9–14.
12.6 Other CoupledLine Components [54]
[55] [56] [57]
[58]
479
Free, C. E., and C. S. Aitchison, ‘‘Improved Analysis and Design of CoupledLine Phase Shifters,’’ IEEE Trans. Microwave Theory Tech., Vol. 43, September 1995, pp. 2126–2131. Ecom, S. Y., et al., ‘‘Broadband 180‚ Bit Phase Shifter Using a New Switched Network,’’ IEEE MTTS Int. Microwave Symp. Dig., 2003, pp. 39–42. Chiu, J.C., J.M. Lin, and Y.H. Wang, ‘‘A Novel Planar ThreeWay Power Divider,’’ IEEE Microwave Wireless Components Lettr., Vol. 16, August 2006, pp. 449–451. Sharma, A. K., and B. Bhat, ‘‘Spectral Domain Analysis of Interacting Microstrip Resonant Structures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT31, August 1983, pp. 681–685. Fredrick, J. D., Y. Wang, and T. Itoh, ‘‘A New Circuit Topology for Continuous Group Delay Synthesis,’’ IEEE Microwave Wireless Components Lettr., Vol. 12, March 2002, pp. 85–87.
CHAPTER 13
Baluns
13.1 Introduction A balun (balancedtounbalanced) is a transformer used to connect balanced transmission line circuits to unbalanced transmission line circuits. Figures 13.1 and 13.2 show examples of balanced and unbalanced transmission lines, respectively. Two conductors having equal potential with 180degree phase difference constitute a balanced line. In this case, no current flows through a grounded shield (i.e., I 1 = I 2 and I s = I g = 0). When this condition is not satisfied, as shown in Figure 13.2, where I s is finite, the transmission line is termed as unbalanced. In addition to providing a matched transition between a balanced and an unbalanced line, baluns also function as centertap transformers for pushpull applications used in radio frequency applications. Baluns are required for balanced mixers, pushpull amplifiers, balanced frequency multipliers, phase shifters, balanced modulators, dipole feeds, and numerous other applications. This transformation from a balanced medium to an unbalanced one requires special techniques, which are described in this chapter. Because the focus of this book is on coupledline structures, only schemes based on the latter concept are discussed. Both narrowband and broadband circuit designs and their fabrication are described. Over the past halfcentury, several different kinds of balun structures have been developed [1–41]. Examples of baluns are shown in Figures 13.3 to 13.6. Early coaxial baluns were used exclusively for feeding dipole antennas. Later, planar baluns using stripline techniques were developed for balanced mixers and printed antenna feeds. Current interest in transmissionlinetype structures is focused toward making it planar, compact, and more suitable for mixers and pushpull power amplifiers. For highefficiency broadband power amplifier applications, these components are important for enhancing the power added efficiency (PAE) by about 10% or higher. However, the lack of a true RF center tap such as is available in the lowfrequency transformer type baluns, is a problem in transmissionlinetype baluns. Planar baluns shown in Figure 13.5 provide greater flexibility and better performance in mixers and modulators. They are compatible with MIC and MMIC technologies, relatively small in size, can be fabricated on singlesided substrates, have wide bandwidths, and are very useful in surfacemounted packages for largevolume production. The realization of a decade bandwidth mixer in a microstriptype medium, with excellent performance, is now possible with planar balun tech
481
482
Baluns
Figure 13.1
Shielded parallel strip balanced transmission line.
Figure 13.2
Unbalanced transmission lines (a) coaxial and (b) microstrip.
nology [23]. Marchand baluns of the planar and nonplanar variety are widely used at microwave frequencies, especially for mixers. Fourport passive circuits, such as ratrace hybrids and waveguide magic tees, can also be used as baluns. Major limitations of these components are their narrow bandwidth and the lack of a method for centertap grounding. Several kinds of baluns (e.g., ferrite, coaxial) have been successfully used below microwave frequencies. Now, baluns employing new coupledline topologies are becoming more common at microwave frequencies. They provide a lower impedancematching ratio, greater bandwidths, and reduced evenharmonic levels. The principal disadvantages include poor isolation between the two singleended amplifiers in a pushpull configuration and poor VSWR at the input and output. In a pushpull amplifier, the reflected signals from the singleended amplifiers do not cancel as they do in a balanced configuration.
13.2 MicrostriptoBalanced Stripline Balun A smooth transition from a microstrip to a balanced stripline as shown in Figure 13.6(a) works as a broadband balun. When a microstrip is joined to a balanced
13.2 MicrostriptoBalanced Stripline Balun
Figure 13.3
483
Lumpedelement baluns: (a) centertap transformer and (b) 180degree ratrace coupler.
stripline, a step discontinuity between the ground plane of the microstrip line and the bottom conductor of the balanced stripline exists for the same characteristic impedances. A step discontinuity also exists between the top strip conductors of these two lines. But the step discontinuity in the former case is larger than in the latter case. Transmissionline tapers are generally employed to achieve a good match between the two lines.
484
Baluns
Figure 13.4
Coaxial baluns: (a) simple /2 line separation splitter, and (b) Marchand. In (b), Z 1 and Z 4 are the characteristic impedances of the coaxial lines, and Z 2 and Z 3 are the characteristic impedances of the coaxial outer conductors with respect to the outer shield.
An example of a transition from microstrip to balanced stripline when connected back to back, to simplify measurements, is shown in Figure 13.7. For the same impedance, for example, 50⍀, the stripwidth for the microstrip, Wm , is smaller than the stripwidth for the balanced stripline W b . When these two lines are connected, both the stripline conductor and the ground plane are tapered to match the dimensions. Several tapering techniques are available, and we can select a taper shape based on the bandwidth requirement. A Chebyshev tapering contour along the balun’s length yields excellent broadband performance [25]. A simple taper, such as shown in Figure 13.7, can easily achieve an octave bandwidth. For the fabricated example shown in the figure, x = Wm , X = 3Wm , and Y = 6Wm and the transitions were characterized on a 62.5milthick polystyrene substrate (⑀ r = 2.55) and on a 50milthick alumina substrate (⑀ r = 9.7). The complete assembly was 1 inch long and the connectors used were of OSMstriplinetype, which have about a 0.1dB insertion loss per connector at Sband. Figure 13.8 shows typical measured VSWR for the two transitions, as well as for the through line. For the polystyrene transition the maximum VSWR over 2.4 to 4.5 GHz was 1.2, whereas for the alumina case the VSWR was less than 1.2 over the 1.7 to 3.2GHz frequency range. Typical measured insertion loss was less than 0.6 dB and 0.4 dB, respectively, for these two transitions. Another parallelstrip balun [34] is shown in Figure 13.9, in which the input is a microstrip and the output is a broadsidecoupled microstrip [42]. One of the
13.2 MicrostriptoBalanced Stripline Balun
Figure 13.5
485
Planar baluns: (a) simple /2 line separation splitter, (b) multisection halfwave balun, and (c) coupledline (coupledline sections are /4 at center frequency).
advantages of this structure is that it is possible to achieve ground isolation at the balanced output port. That is, the balanced lines have their potential referenced to each other. The simple design equations given by [34] are as follows: Z 0p =
√Z S Z L
(13.1)
Z 0e ≥ 10Z 0o ≥ 5Z 0p
(13.2)
Z 0o = Z 0p /2
(13.3)
where Z S and Z L are the source and load impedances, respectively, and Z 0p is the parallelplate impedance of the two conductors forming broadsidecoupled lines with the housing. Z 0e and Z 0o are the even and oddmode characteristic impedances of the broadsidecoupled lines, respectively.
486
Baluns
Figure 13.6
Nonplanar baluns: (a) simple broadside microstrip, (b) uniplanar, and (c) Marchand.
Consider an example having source impedance Z S = 50⍀ and load impedance Z L = 100⍀. In this case, Z 0p = 70.7⍀ and Z 0o = 35.4⍀. Figure 13.10 shows a balun’s performance when Z 0e = 350⍀ and the structure is lossless. As can be seen, the power split is not the desired 3 dB. In this case, as the Z 0e /Z 0o ratio becomes higher and higher, the power split will approach 3 dB, and when Z 0e is infinite, the power split will be 3 dB.
13.3 Analysis of a CoupledLine Balun The operation of a balun can be easily explained by a pair of coupled microstrip lines of equal width, as shown in Figure 13.11. The length of the line is /4 at the
13.3 Analysis of a CoupledLine Balun
487
Figure 13.7
Unbalancedtobalanced transitions connected back to back.
Figure 13.8
Measured performance of transitions fabricated on polystyrene and alumina substrates.
center frequency. This structure can be analyzed by the even and oddmode excitation technique as shown in Figure 13.12. The admittance matrix for a fourport network consisting of transmission lines having the same propagation constants is given by [11, 43].
冢冣 冢 I1 I2 I3 I4
=
Y11 Y21 Y31 Y41
Y12 Y22 Y32 Y42
Y13 Y23 Y33 Y43
where for a homogenous dielectric medium
冣冢 冣
Y14 Y24 Y34 Y44
V1 V2 V3 V4
(13.4)
488
Baluns
Figure 13.9
Side view of the parallelplate balun. (From: [34]. 1993 Applied Microwave. Reprinted with permission.)
Figure 13.10
Frequency response of the parallelplate balun. (From: [34]. 1993 Applied Microwave. Reprinted with permission.)
Figure 13.11
Coupled microstrip lines.
13.3 Analysis of a CoupledLine Balun
Figure 13.12
489
(a) A coupled line configuration and excitation of (b) even and (c) odd modes on symmetrical coupled lines.
cot 2
(13.5a)
cot 2
(13.5b)
csc 2
(13.5c)
csc 2
(13.5d)
Y11 = Y22 = Y33 = Y44 = −j (Y0o + Y0e ) Y12 = Y21 = Y34 = Y43 = j (Y0o − Y0e )
Y13 = Y31 = Y24 = Y42 = −j (Y0o − Y0e ) Y14 = Y41 = Y23 = Y32 = j (Y0o + Y0e ) At = 90 degrees
Y11 = Y22 = Y33 = Y44 = 0, Y12 = Y21 = Y34 = Y43 = 0
490
Baluns
and Y13 = Y31 = Y24 = Y42 = −j (Y0o + Y0e )/2
(13.6)
Y14 = Y41 = Y23 = Y32 = j (Y0o + Y0e )/2
(13.7)
When port 2 is shortcircuited (i.e., V2 = 0), and the other ports are terminated in matched loads, the voltage transfer from port 1 to port 3 can be obtained by converting the Yparameters into Sparameters, as follows [11]: S 31 =
j2Y0 YA 2 Y0
2
2
+ YA + YB
(13.8)
where Y0 = 1/Z 0 and YA = (Y0o − Y0e )/2, YB = (Y0o + Y0e )/2. Similarly, the voltage transfer from port 1 to port 4 can be obtained as S 41 =
−j2Y0 YB 2 Y0
2
2
+ YA + YB
(13.9)
and S 31 /S 41 = −YA /YB = − (Y0o − Y0e )/(Y0o + Y0e ). The above equation shows that S 31 and S 41 are 180 degrees out of phase at = 90 degrees. A balun works over the required band if  S 31 /S 41  = 1 and  ∠ S 31 − ∠ S 41  = 180 degrees. These conditions can be achieved if Y0e = 0 or Z 0e = ∞, that is, the effect of ground plane on the conductors is negligible. For a perfect match, S 11 = 0, that is: Y0 = YA + YB
2
2
2
(13.10)
Z 0o = Z 0 /√2
(13.11)
or
where Z 0 is the source or load impedance. Figures 13.13 and 13.14 show the balun’s performance where for Z 0o = 35⍀ and Z 0e = 10,000⍀, the amplitude response varied by less than 0.5 dB and the phase difference is within ± 3 degrees over the 2 to 18GHz frequency range. This type of balun has a severe requirement for a very high evenmode impedance, which is not realizable in the microstrip configuration. Thicker and lower dielectric constant value substrates with narrower conductor width result in a higher evenmode impedance. Narrower conductors have also higher insertion loss. These kinds of baluns are therefore not very suitable for highefficiency power amplifiers and lowloss mixer applications.
13.4 Planar Transmission Line Baluns Planar transmission line baluns consist of two sections: the first section divides the signal into two signals having equal magnitude and phase over a broad frequency
13.4 Planar Transmission Line Baluns
491
Figure 13.13
The effects of a ground plane can be analyzed by varying the value of the evenmode impedance to a relatively low level. (From: [11]. 1985 Microwaves and RF. Reprinted with permission.)
Figure 13.14
The degradation in amplitude response for a finite evenmode impedance indicates that a significant amplitude imbalance may result if ground plane effect is not considered. (From: [11]. 1985 Microwaves and RF. Reprinted with permission.)
492
Baluns
range and the second section provides −90degree and +90degree phase shifts for these two signals, so that the balanced output signals have a 180degree phase difference. The power divider section generally uses a Wilkinson inphase power divider, which uses multisections for larger bandwidths. In general, shortcircuited and opencircuited coupled lines are used for the phaseshifter sections. Basic configurations for these baluns are shown in Figure 13.15, where the divider uses only one section. A multisection divider has also been used [28] to obtain 6 to 18GHz bandwidth performance from such baluns. Figure 13.15(a) shows a basic balun where, in order to obtain tight coupling over a multioctave bandwidth, the edgecoupled line sections are commonly replaced with interdigital Lange couplers. Because of the inherent symmetry and broadband characteristics of coupledline sections or Lange couplers, good amplitude and phase balance performance are achievable. As the use of via holes or ground connections through alumina substrate in MICs require additional processing, broadband radial line stubs [44], as shown in Figure 13.15(b), can also be used to simulate RF grounding without additional processing. When the opencircuit coupler is replaced by a ‘‘ ’’ network of transmission lines, as shown in
Figure 13.15
(a–d) Planar balun configurations, where port 1 is unbalanced, and ports 2 and 3 constitute a balanced port.
13.4 Planar Transmission Line Baluns
493
Figure 13.15(c), the 180degree phase shift becomes independent of the electrical length of the two networks [21, 45]. Figure 13.15(d) shows another broadband balun topology [27] suitable for pushpull power amplifiers.
13.4.1 Analysis
An ultra broadband 180degree phase shift can be realized by using the phasereversal property of a tightly coupled (3dB) fourport network. In a coupler when coupled and direct ports are switched from open circuit to short circuit, the transmission phase difference between the input and isolated ports changes from −90 degrees to +90 degrees as shown in Figure 13.16. In the case of the shortcircuited condition, an additional −180degree phase is added, which brings the phase to −270 degrees or +90 degrees. Since tight coupling is required, a Lange coupler is generally used for this application. The Sparameters of a fourport network shown in Figure 13.16 are given by
冢冣冢 b1 b2 b3 b4
S 12 S 22 S 32 S 42
S 13 S 23 S 33 S 43
冣冢 冣 a1 a2 a3 a4
(13.12)
b 1 = S 11 a 1 + S 12 a 2 + S 13 b 3 + S 14 b 4
(13.13a)
b 2 = S 21 a 1 + S 22 a 2 + S 23 b 3 + S 24 b 4
(13.13b)
=
S 11 S 21 S 31 S 41
S 14 S 24 S 34 S 44
or
Figure 13.16
Twoport coupler configuration.
494
Baluns
b 3 = S 31 a 1 + S 32 a 2 + S 33 b 3 + S 34 b 4
(13.13c)
b 4 = S 41 a 1 + S 42 a 2 + S 43 b 3 + S 44 b 4
(13.13d)
In a matched 3dB coupler: S 13 = S 24 = S 31 = S 42 = −j /√2
(13.14)
S 14 = S 23 = S 32 = S 41 = 1/√2
(13.15)
S 11 = S 22 = S 33 = S 44 = 0
(13.16)
S 12 = S 21 = S 34 = S 43 = 0
(13.17)
Therefore, the fourport Smatrix becomes 0 0 1 [S ] = √2 −j 1
冤
0 0 1 −j
−j 1 0 0
1 −j 0 0
冥
(13.18)
When ports 3 and 4 are terminated in an open circuit (i.e., Z T = ∞) and other ports are matched, a 3 = b 3 and a 4 = b 4 and a 1 = b 2 , in this case, (13.13) through (13.17) reduce to b1 =
−j 1 b3 + b4 2 √ √2
(13.19a)
b2 =
1 j b3 − b4 √2 √2
(13.19b)
b3 =
−j j a1 + a2 √2 √2
(13.19c)
b4 =
1 j a1 − a2 √2 √2
(13.19d)
By rearranging these equations, the twoport Sparameters become [S ]open =
冋 册 0 −j
−j 0
(13.20)
Similarly, when ports 3 and 4 are terminated in short circuits, a 3 = −b 3 and a 4 = −b 4 and (13.18) simplifies to
13.4 Planar Transmission Line Baluns
495
[S ]short =
冋 册 0 j
j 0
(13.21)
Equations (13.20) and (13.21) illustrate that open to short switching leads to a 180degree phase shift, signals are combined, and ports 1 and 2 are still matched. This condition holds well over a wide frequency range. When the opencircuited coupler is replaced by a ‘‘ ’’ or equivalent network of transmission lines, the 180degree phase shift becomes independent of the electrical length of the two networks; that is, ‘‘ ’’ and shortcircuited coupler, and this results in a wider bandwidth. Figure 13.17 shows these 180degree phasedifference sections. The two networks are exactly equivalent for all frequencies, except that the transmission phase difference between the two circuits is exactly 180 degrees [45]. This can be seen by developing ABCD matrices for both networks. For the ‘‘ ’’ network:
冋 册 冤 A C
B = D
1 Yoe j tan
0 1
冥
⭈
冤
cos
j (Y 0o − Y 0e ) 2 csc
cos
(Y 0o + Y 0e ) cos (Y 0o − Y 0e ) = (Y 0o + Y 0e )2 cos2 − (Y 0o − Y 0e )2 j 2 (Y 0o − Y 0e ) sin
冤
冥冤
j 2 sin (Y 0o − Y 0e )
⭈
1 Yoe j tan
j 2 sin (Y 0o − Y 0e ) (Y 0o + Y 0e ) cos (Y 0o − Y 0e )
冥
冥
0 1
(13.22)
(13.23)
where Y0o = 1/Z 0o and Y0e = 1/Z 0e . The ABCD matrix of the shorted coupledline section is calculated from the y parameters as
Figure 13.17
(a, b) 180degree phase difference sections.
496
Baluns
冋 册 A C
B D
= coupled
冤
− −
(Y 0o + Y 0e ) cos (Y 0o − Y 0e )
−
(Y 0o + Y 0e )2 cos2 − (Y 0o − Y 0e )2 j 2 (Y 0o − Y 0e ) sin
−
j 2 sin (Y 0o − Y 0e )
冥
(Y 0o + Y 0e ) cos (Y 0o − Y 0e ) (13.24)
From (13.23) and (13.24)
冋 册 冋 A C
B −1 = D 0
册冋 册
0 A ⭈ −1 C
B D
(13.25) coupled
This result is independent of the electrical length, and thus is independent of frequency. For tight coupling, however, = 90 degrees is required. 13.4.2 Examples
Topologies for the planar transmission line baluns shown in Figure 13.15 were analyzed and optimized using a commercial CAD tool and found that the configuration shown in Figure 13.15(c) gives the best results over an octave bandwidth. Physical dimensions and performance results are summarized in Table 13.1. A broadband balun employing Lange couplers as shown schematically in Figure 13.15(a) was fabricated on a 0.635mmthick substrate using conventional photolithography and etching techniques. The coupler has nominal coupling of 2.8 dB and a 0.7dB ripple. The Z 0e and Z 0o values were 126⍀ and 20⍀, respectively
Table 13.1 Design and Performance Summary of a Planar Balun Shown in Figure 13.15(c). Substrate Dimensions: ⑀ r = 9.9, h = 15 mil, t = 0.2 mil, and Conductors of Gold Wilkinson divider dimensions: Line width = 6.2 mil Line length = 96.2 mil Isolation resistor = 88.4⍀ Lange coupler dimensions: Line width = 2.6 mil Line spacing = 1.8 mil Coupler length = 98.0 mil Lowpass section dimensions: Shunt line width = 2.5 mil Shunt line length = 107.0 mil Series line width = 17.2 mil Series line length = 81.7 mil Performance over 8 to 16 GHz: Insertion loss = 3.15 ± 0.05 dB Phase difference = 180 ± 1 degree Amplitude difference = ± 0.01 dB Return loss > 18 dB
13.4 Planar Transmission Line Baluns
497
[29]. The measured performance is plotted in Figure 13.18. Over 5 to 11 GHz, the two output signals have the magnitude of 3.6 ± 0.3 dB and 3.6 ± 0.5 dB. The VSWR and isolation were better than 10 and 14 dB, respectively. The phase difference between the output ports was 170 ± 5 degrees over the 5 to 11GHz band. Another planar balun was constructed on a 10mil alumina substrate using short and opencircuited Lange couplers [28]. The Lange couplers were realized using ionbeam milling techniques to achieve fine 1mil line width and 0.5mil gap dimensions. A photograph of the balun is shown in Figure 13.19. The balun was accurately characterized using thrureflectline (TRL) calibration techniques and the measured amplitude, phase, and returnloss response are shown in Figure 13.20. The maximal loss in each path was about 1.2 dB, and the amplitude and phase imbalance were ± 0.6 dB and ± 7 degrees, respectively, over the 6 to 20GHz
Figure 13.18
Measured performance of a broadband balun of Figure 13.15(a). (From: [29]. 1991 European Microwave Conference. Reprinted with permission.)
498
Baluns
Figure 13.19
Photograph of fabricated balun chip. (From: [28]. 1991 IEEE. Reprinted with permission.)
frequency range. The VSWR and the isolation were better than 10 to 16 dB, respectively.
13.5 Marchand Balun The Marchand balun, which has several versions, is the most commonly used component in broadband doublebalanced mixers. As compared with a shorted coupledline balun, this structure has less stringent requirements for Z 0e ; generally Z 0e ≅ 3 to 5 times of Z 0o is sufficient to obtain good performance for such baluns. Proper selection of balun parameters can achieve a bandwidth of more than 10:1. The balun basically consists of an unbalanced, an opencircuited, two shortcircuited, and balanced transmission line sections. Each section is about a quarterwavelength long at the center frequency of operation. A coaxial version of a compensated Marchand balun is shown in Figure 13.21(a), while its equivalent circuit representation is shown in Figure 13.21(b). The compensation term is used in broadband baluns where the balanced output and reduced phase slope were maintained over a wide bandwidth. As shown in Figure 13.21(a), this structure basically consists of two coaxial lines, each /4 long at the center frequency. The lefthand line has a characteristic impedance of Z 1 . The second line, which has a characteristic impedance of Z 2 , is opencircuited. The outer conductors of these transmission line sections with housing make another two shortcircuited /4 lines that are in series with each other and shunt the balanced lines, having a characteristic impedance of Z B , at locations a and b. As shown in the equivalent circuit [Figure 13.21(b)], the stubs Z S1 and Z S2 are in series and shunt the balanced lines. Their characteristic impedances are made as large as possible. These impedances along with the other transmission line impedances determine the impedance transformation and bandwidth. Figure 13.22 shows a simplified equivalent circuit of the Marchand balun. Because of equal shunting effects on the balanced lines, these stubs
13.5 Marchand Balun
Figure 13.20
499
Measured (a) amplitude and (b) phase balance of planar balun. (From: [28]. 1991 IEEE. Reprinted with permission.)
500
Baluns
Figure 13.21
Marchand compensated balun: (a) coaxial cross section; and (b) equivalent transmissionline model. (From: [22]. 1990 IEEE. Reprinted with permission.)
provide greater bandwidths. The opencircuit stub Z 2 provides a low impedance at the junction of the four different lines and acts like a series resonant circuit. The series resonant circuit with a shunt resonant circuit as shown in Figure 13.23 reduces the phase variation over the designed bandwidth. The ratio of the characteristic impedances of the shortcircuited and opencircuited stubs determines the bandwidth; the higher the ratio, the wider the bandwidth. The transmission line Z B can be designed with a characteristic impedance of the balanced line or can be used as an impedance transformer between the desired impedance and the balancedline impedance.
13.5 Marchand Balun
501
Figure 13.22
Simplified equivalent circuit of a fourthorder Marchand balun.
Figure 13.23
Marchand balun’s series and parallelresonant compensating representation for wider bandwidth.
A Marchand balun is basically comprised of two quarterwave coupled sections and may be realized using printed lines or coaxial lines. The coupled lines require tight coupling with high evenmode impedance for broadband performance. The printed coupled lines may be microstrip lines, multiconductor Lange couplers, coplanar waveguides, multilayer microstrip lines, or spiral coils. A balun fabricated using printedcircuit topology is shown in Figure 13.24(a), and its simplified equivalent circuit is shown in Figure 13.24(b). In such baluns, the performance also depends upon the ground spacing and the housing in which the balun is placed. Like any other distributed balun (coaxial, microstrip, and CPW), the tap center shown in Figure 13.24(b) only works at dc or low frequencies such as IF that are in the RF range. The center tap concept is not valid at microwave frequencies, however, as required for pushpull topology. 13.5.1 Coaxial Marchand Balun
Now we present a simple analysis of a coaxial Marchand balun shown in Figure 13.25(a). It consists of two coaxial lines, a and b, of characteristic impedances Z a
502
Baluns
Figure 13.24
Planar compensated balun fabricated on a lowdielectric substrate: (a) metalization pattern, and (b) lumpedelement equivalentcircuit model. (From: [25]. 1990 John Wiley. Reprinted with permission.)
and Z b , respectively. The signal is fed at the input point, and the balanced signal appears across the O and O ′ nodes. The center conductors of these lines are connected at nodes c and c ′, and the other end of the center conductor of line b is opencircuited. The outer conductor of these two lines are coupled to each other to form the balanced line of characteristic impedance Z ab . The outer conductors of lines a and b are connected at d, and the electrical length at the center frequency is 90 degrees. The equivalent circuit of this balun is shown in Figure 13.25(b). Referring to this, the input impedance at nodes c and c ′, assuming a lossless structure, may be expressed as jZ L Z ab tan ab Z in = −jZ b cot b + Z L + jZ ab tan ab
(13.26)
substituting b = ab = and after simplifying Z L Z ab + j cot 冋Z L 冠Z ab − Z b cot2 冡 − Z b Z ab 册 2
Z in = when
2
2
2
2
Z ab + Z L cot2
(13.27)
13.5 Marchand Balun
Figure 13.25
503
Wideband balun: (a) schematic diagram; and (b) equivalent circuit representation. (From: [4]. 1960 IEEE. Reprinted with permission.)
Z b = Z a and Z ab = Z L Z in = Z L sin2 + j cot (Z L sin2 − Z a )
(13.28)
and the input impedance becomes perfectly matched to Z a at two widely separated frequencies given by the solution of sin2 = Z a /Z L
(13.29)
These frequencies are symmetrically disposed about a center frequency corresponding to = 90 degrees. A larger bandwidth is realized by choosing 2
Z b = Z L /Z ab
(13.30)
and making Z ab as large as possible [2, 4]. Figure 13.26 compares the input VSWR for two cases in which a 50⍀ unbalanced input was transformed to a 70⍀ balanced 2 output. The improvement in the second case, where Z b = Z L /Z ab is quite obvious. The realization of this case in coaxial line form is difficult; however, it can be easily fabricated using coplanar striplines [4]. Figure 13.27 delineates a microstrip
504
Baluns
Figure 13.26
Calculated VSWR versus electrical length for two different balun design criteria. (From: [4]. 1960 IEEE. Reprinted with permission.)
version of this balun. The load impedance Z L is between O and O ′ and is 70⍀ in this case. A printed balun with integrated dipole [12] is shown in Figure 13.28(a), where the unbalanced coaxial line is replaced by a microstrip line section and one end of the balanced line section is connected to the microstrip ground plane conductor, while the other end connects to the printed dipole. The widths of the balanced line conductors must be at least three times that of the microstrip line conductors in order to use microstrip design equations. Narrower widths can also be used if appropriate connections are made. The characteristic impedance Z ab of the balanced line may be calculated by treating it as a pair of coupled microstrip lines on a suspended substrate and excited in the odd mode. The distance between the balance line conductors and the enclosure, which also acts as a ground plane, is kept large as compared with the spacing between the conductors. Under this assumption, the characteristic impedance of the balanced line is approximately twice the oddmode impedance calculated for the coupled line. The effective dielectric constant of the balanced line is the same as that calculated for the coupled lines excited in the oddmode. Lower values of Z ab require narrower spacing and wider conductor widths. Fabrication techniques set the lower limit on spacing that can be realized. Thus a higher limit on Z ab is set by the minimum line widths required to realize Z a and Z b . The effects of the enclosure, discontinuity, and coupling influence the performance of this balun in terms of bandwidth and frequency range. For applica
13.5 Marchand Balun
Figure 13.27
505
Illustration of the construction of a printed circuit balun. (From: [4]. 1960 IEEE. Reprinted with permission.)
tions in the antenna area, the radiating elements are connected to the balanced line by extending it and shaping it into the desired dipole configuration. In the final design, the dipole can be matched over a broadband by treating it as part of the balun design and by tuning out the reactances accordingly [12]. Figure 13.28(b) shows the simlulated and measured input VSWR of a dipole/balun combination. In this example, Z b = Z ab = 80⍀, which is equal to the dipole’s radiation resistance at the resonant frequency which is 13 GHz. b = 105 degrees and ab = 95 degrees and a /4 microstrip having a characteristic impedance of 63⍀ were used to transform 80⍀ to 50⍀ at the input. The structure was printed on a 0.64mmthick fused silica (⑀ r = 3.78) and mounted a quarterwavelength abovetheground plane. Although such baluns have good bandwidth, they are not suitable for ultrabroad band applications. 13.5.2 Synthesis of Marchand Balun
Synthesis of coaxial Marchand baluns is available in the literature [8, 32]. Recent trends focus on planar technology baluns using singlelayer and multilayer microstrip transmission media. A Marchand balun can easily be realized using a pair of coupled lines as shown in Figure 13.29(a). By properly selecting its parameters, we can design a planar balun to meet the desired response. Synthesis techniques of such baluns have been recently described [8, 32] and are summarized in this section.
506
Baluns
Figure 13.28
(a) Printed circuit realization of balun structure with integrated dipole. (b) Measured and calculated input VSWR for the dipole/balun combination. (From: [12]. 1987 Microwave Journal. Reprinted with permission.)
Figure 13.29
(a) A fourport coupled line and (b) its equivalent circuit.
13.5 Marchand Balun
507
An equivalent of a coupledline of equal conductor widths is shown in Figure 13.29(b) where Z 1 and Z 2 are the characteristic impedances of distributed unit elements, and N is the transformation ratio. The characteristic impedance Z 0c and the coupling coefficient k of the coupled line are given by Z 0c =
√Z 0e Z 0o
(13.31a)
k=
Z 0e − Z 0o Z 0e + Z 0o
(13.31b)
and Z 0e = Z 0c Z 0o = Z 0c
√ √
1+k 1−k
(13.32a)
1−k 1+k
(13.32b)
are related to the network equivalent circuit elements as follows: Z1 =
Z 0c
√1 − k
Z 2 = Z 0c
(13.33)
2
√1 − k
2
k2
(13.34)
and N=
1 k
where Z 0e and Z 0o are the even and oddmode impedances, respectively. The balun in Figure 13.30(a) has four reactive elements and is known as a fourthorder Marchand balun. Matching section Z makes it a fourthorder balun with improved broadband performance. Figures 13.30(b) and 13.30(c) show equivalent circuit representations and Figure 13.30(d) shows further simplification when N a = N b = N. In this case, the middle transformers cancel each other. The final equivalent circuit element values Z1′ , Z2′ , Z3′ , Z4′ , and R5′ can be expressed in terms of coupled line parameters Z ac , Z bc , and k as follows: Z1′ =
Z 2a
Z2′ =
Z 2b
N2
N2
= Z ac √1 − k 2
(13.35a)
= Z bc √1 − k 2
(13.35b)
508
Baluns
Figure 13.30
(a–d) Coupledline Marchand balun and its equivalent circuits. (From: [32]. 1992 International Journal of Microwave MillimeterWave ComputerAided Engineering. Reprinted with permission.)
Z3′ =
Z 1a + Z 1b = (Z ac + Z bc ) N2 Z4′ =
R5′ =
Z N
2
R N2
k2
(13.35c)
2 √1 − k
= Zk 2
(13.35d)
= Rk 2
(13.35e)
2
a
a
2
b
b
where R is the load impedance, and Z ac = Z 0e Z 0o and Z bc = Z 0e Z 0o are the characteristic impedances of the coupled lines a and b, respectively. The equations above can be rearranged to solve for the coupledline parameters in terms of the equivalent element values as follows: k=
√
Z3′ Z1′ + Z2′ + Z3′
Z ac =
Z1′ 2 √1 − k
(13.36a)
(13.36b)
13.5 Marchand Balun
509
Z bc =
Z2′ 2 √1 − k
(13.36c)
Choosing element values for Z1′ , Z2′ , Z3′ , Z4′ , and R5′ for the final design of a balun to meet specifications, the coupledline parameters can be calculated from (13.30) and (13.31). Finally, the physical dimensions for the balun are determined using suitable coupledline structures; for example, edgecoupled microstrip [46–48], broadsidecoupled striplines [48, 49], embedded microstrip [50], or any other topology. The design of a balun starts with determining the network parameters given in Figure 13.30(d) to meet design specifications in terms of bandwidth, load impedance, and return loss. This can be achieved by either using nomographs [8] or using synthesis techniques [51], or by using commercial CAD tools. Balun equivalent circuit parameters as a function of return loss for fourthorder Chebyshev filters having 2:1, 3:1, and 5:1 bandwidth are given in Figure 13.31 [32] for a source impedance of 50⍀. Coupledline parameters are calculated using (13.30) and (13.31) and plotted in Figure 13.32. These figures also include the values of Z and the load impedance R. In addition to the above synthesis analysis of planar Marchand baluns, several other analysis and design methods of such baluns have been reported. This includes equivalent circuit model for the twocoupled line sections presented by Tsai and Gupta [33], the method reported by Schwindt and Nguyen [37], that provides analytical expressions for the Sparameters for a planar multilayer Marchand balun based on broadsidecoupled quasiTEM normalmode parameters, and the Engels and Jansen [38] design method based on the mode parameters for multiple coupled strip configurations. These methods provide a reasonably good starting point for the design of complicated multilayer baluns. However, for accurate characterization of these baluns, threedimensional electromagnetic simulators are essential and play a very important role in the development of such baluns. 13.5.3 Examples of Marchand Baluns
Let us consider an example for a balun design with 30dB return loss, R5′ = 95⍀ and 3:1 bandwidth. From Figure 13.31(b), approximate values for filter network are Z1′ = 61.5⍀, Z2′ = 50⍀, Z3′ = 95⍀, and Z4′ = 80⍀. From (13.3) and (13.35), the coupledline parameters are k = 0.678, Z ac = 83.7⍀, Z bc = 68⍀, R = 206⍀, a a and Z = 174⍀. From (13.30) and (13.31), Z 0e = 191.1⍀, Z 0o = 36.7⍀, b b Z 0e = 155.3⍀, and Z 0o = 29.8⍀. Figure 13.32 can also be used directly to obtain these parameters. These values are used to determine the physical parameters for a given substrate thickness and dielectric constant values for coupledline balun [46–48] or broadside coupledline [48, 49]. Tightly coupled sections of this balun are not possible with edgecoupled microstrip lines. The embedded microstrip technique [50] or Lange coupler configuration, however, can be used to realize tightly coupled structures. Realization of such couplers with multilayer techniques on substrates such as alumina, quartz, or GaAs [31] have also been reported. A crosssectional view of a broadsidetype coupled line showing even and oddmode impedance realization is shown in Figure 13.33.
510
Baluns
Figure 13.31
Marchand balun equivalent circuit parameters as a function of load impedance (a) 2:1 bandwidth, (b) 3:1 bandwidth, and (c) 5:1 bandwidth. (From: [32]. 1992 International Journal of Microwave MillimeterWave ComputerAided Engineering. Reprinted with permission.)
13.5 Marchand Balun
Figure 13.32
511
Marchand balun coupledline parameters as a function of load impedance for (a) 2:1 bandwidth, (b) 3:1 bandwidth, and (c) 5:1 bandwidth. (From: [32]. 1992 International Journal of Microwave MillimeterWave ComputerAided Engineering. Reprinted with permission.)
512
Baluns
Figure 13.33
Illustration for the realization of a typical broadsidecoupled line. (From: [31]. 1991 IEEE. Reprinted with permission.)
Marchand Balun Using Lange Coupler
A Marchand balun using two tight couplers having connections as shown in Figure 13.34 has been described by Tsai [52]. The couplers used are slightly undercoupled 3dB Lange couplers, fabricated on 100 mthick GaAs substrate. The linewidth
Figure 13.34
(a) Schematic using crossover couplers and (b) rearranged schematic further simplifying the circuit.
13.5 Marchand Balun
513
of the couplers is 10 m and the spacings between the conductors are 12 m and 10 m for couplers 1 and 2, respectively. The length of the couplers is 1.6 mm. These dimensions are easily realized using MMIC processes. The ground connection was realized using a via hole. A photograph of the chip, which is 1 by 2 mm, is shown in Figure 13.35. Simulated performance of the balun is shown in Figure 13.36. Measured insertion loss of the balun’s output ports is shown in Figure 13.37, and Figure 13.38 shows the amplitude and phase differences between the two outputs over a 0.1 to 20GHz frequency range. In an ideal case, the desired values of amplitude and phase difference are 0 dB and 180 degrees, respectively. This balun demonstrates good performance over the 6 to 20GHz frequency range.
Figure 13.35
Photograph of a balun using Lange couplers. (From: [52]. 1993 IEEE. Reprinted with permission.)
Figure 13.36
Simulated insertion loss and phase difference of the balun. (From: [52]. 1993 IEEE. Reprinted with permission.)
514
Baluns
Figure 13.37
Measured insertion loss of the two output ports. (From: [52]. 1993 IEEE. Reprinted with permission.)
Figure 13.38
Measured amplitude and phase balance of the balun. (From: [52]. 1993 IEEE. Reprinted with permission.)
Marchand Balun Using ReEntrant Couplers
An alternative solution to broadside multilayer coupled lines and Lange couplers is to use reentrant multilayer coupled lines [53–55] to realize the balun circuit. The latter structure has the advantage of providing tight coupling without stringent dimensional requirements in circuit fabrication. Reentrant couplers using
13.5 Marchand Balun
515
multilayer microstrip lines have been reported, and a configuration appears in Figure 13.39. It consists of three conductors in which two conductors are deposited on the top surface of a twolayered dielectric substrate and a third conductor that is floating is sandwiched between the two dielectric layers. The floating conductor beneath the coupled lines helps in realizing higher evenmode impedance and tight coupling. Because there are a total of four conductors, the structure can support three propagation modes instead of the two general even and odd modes in the case of a threeconductor structure. Although multilayer dielectric structures allow much greater freedom in designing the baluns in terms of process and size requirements, they give rise to other structurally related problems. A multilayer microstrip structure is inhomogeneous in character and there are no simple design expressions available for determining the physical dimensions. Commercially available EM simulators can be used to analyze the structure, but they usually require a considerable amount of computation time to optimize the design for best performance. Recently, a generalized network model for coupled multiconductor transmission lines has been reported [36] that can also be used to design reentranttype couplers. A simple analysis of this coupler has been discussed in Section 8.6.3, and the design procedure for the balun can be summarized as follows: 1. Calculate the even (Z 0e ) and odd (Z 0o ) mode impedances for each coupled section as described in the beginning of this section. 2. Because Z 02 = Z 0o , calculate the physical dimensions of the toplayer structure such as ⑀ r2 , d, and W 2 by keeping a large separation between the structure. 3. Calculate Z 01 = (Z 0e − Z 02 )/2 and determine the physical dimensions of the bottomlayer structure such as ⑀ r1 , h, and W 1 . 4. Optimize the balun in terms of the structure’s physical dimensions by finetuning the above calculated dimensions using EM simulators. For example, a Marchand balun covering 3:1 bandwidth with a 41.4dB inband a a b b return loss requires Z 0e = 264⍀, Z 0o = 30⍀, Z 0e = 117⍀, and Z 0o = 13⍀ [32].
Figure 13.39
Microstrip reentranttype coupler.
516
Baluns
The design dimensions are summarized in Table 13.2. Both these examples demonstrate the realization of Marchand baluns using multilayer microstrip structures. Figure 13.40 shows the simulated performance of a balun using reentranttype couplers. Monolithic Marchand Balun
A monolithic Marchand balun has been developed [41] using multilayer dielectric monolithic technology, where polyimide (⑀ r = 3.2) was used as the intermetal dielectric. Figure 13.41 shows the top view and side view of the monolithic Marchand balun. The dimensions for the various line sections are input line A has a 40 m linewidth and is 1,400 m long. The linewidth, length for the opencircuited line B and shortcircuited bottom lines C are 140 m, 1,420 m, and 120 m and 1,427 m, respectively. The unbalanced input impedance is 20⍀ and the balanced output impedance is 50⍀. The balun was designed using electromagnetic simulation. The input return shown in Figure 13.42(a) was measured by terminating the balanced port into a 50⍀ load. The insertion loss of the baluns when connected back to back is shown in Figure 13.42(b). For a single balun, the insertion loss is less than 0.7 dB from 6 to 21 GHz. The measured amplitude balance was within 0.5 dB from 7 to 21 GHz with a corresponding phase difference of 178 to 172 degrees. Table 13.2 Design Parameters for a 3:1 Bandwidth Balun Using ReEntrant Couplers Calculated Using EM Simulator [32] Design #
Z 0e
Z 0o
b
Z 0e
Z 0o
⑀ r2
⑀ r1
h (mil)
d (mil)
W 1a (mil)
W 2a (mil)
Sa (mil)
W 1b (mil)
W 2b (mil)
Sb (mil)
1 2
200 264
28 30
111 117
11 13
9.8 12.9
3.2 3.2
15 25
0.2 0.2
3.0 2.0
0.7 0.7
0.5 0.5
14.0 17.0
2.5 2.0
1.0 1.0
a
Figure 13.40
b
b
Linear circuit simulation of 3:1 bandwidth Marchand balun with ideal element values a a b b Z 0e = 269.3⍀, Z 0o = 29.7⍀, Z 0e = 117.1⍀, Z 0o = 13.2⍀, Z ′4 = 94⍀, and R ′ = 100.4⍀. (From: [32]. 1992 International Journal of Microwave MillimeterWave ComputerAided Engineering. Reprinted with permission.)
13.5 Marchand Balun
517
Figure 13.41
Monolithic Marchand balun (a) top view and (b) side view. (From: [41]. 1997 IEEE. Reprinted with permission.)
Figure 13.42
(a) Measured return loss of four Marchand baluns terminated into a balanced 50⍀ load. (b) Measured loss of four pairs of backtoback Marchand baluns. (From: [41]. 1997 IEEE. Reprinted with permission.)
Compact Marchand Balun
At lower microwave frequencies, the balun size becomes large. Various techniques have been used to reduce the balun size. These include lumped element approach [39, 56–60], spiral transmission line [61, 62], and multilayer LTCC [63] and liquid crystalline polymer LCP [64] technologies. For example, a Marchand balun is realized in different layers of the multilayer substrate and the coupled lines are printed in spiral shapes to reduce drastically the component size. Several other methods, including inductive loading [64], capacitive loading [65], and impedance variation [66], have been used to reduce the balun size. A compact lumpedelement Marchand balun for wireless applications has been described by Jansen et al. [39]. The structure used three metal levels and two polyimide intermetal layers to realize
518
Baluns
broadsidecoupled sections to fabricate a lumpedelement Marchand balun on Si substrate. Figure 13.43 shows the top view of a 1.9GHz balun and occupies only a 2.5mm2 area.
13.6 Other Baluns 13.6.1 Coplanar Waveguide Baluns
Apart from microstrip baluns, coplanar waveguide (CPW) baluns have also been reported [30, 40, 55]. An analysis of such baluns is given in [30], while a physical layout is shown in Figure 13.44. The physical length of the balun section is /4. This balun does not have as large a bandwidth as planar baluns or Marchand baluns. Another CPW balun [55] is shown in Figure 13.45. This balun also has narrowband performance. 13.6.2 Triformer Balun
A triformer balun that is a rectangularspiralshaped inductor using three coupled lines has been developed for microwave monolithic integrated circuit applications [19]. The balun was designed using the multiconductor coupledline transformer chain matrix method. However, this structure can be accurately analyzed using EM simulators. The physical layout, equivalent circuit, and microphotograph of a oneturn triformer are shown in Figure 13.46. The structure was fabricated on GaAs (⑀ r = 12.9, tan ␦ = 0.0003) substrate with a 2 m conductor thickness, a 10 m conductor width, and 5 m spacing between the conductors. The gold conductors have metal resistivity of 0.03 ⍀ m, and the substrate thickness is 100 m. Measured and simulated performance of the triformer is shown in Figure 13.47. The differential phaseshift between the output ports 1 and 6 remains constant at about 182 degrees. A major shortcoming of this approach is the loss
Figure 13.43
Top view of a lumpedelement balun layout. (From: [39]. 1997 IEEE. Reprinted with permission.)
13.6 Other Baluns
Figure 13.44
519
Layout of a planar balun using CPW and coplanar strips. (From: [30]. 1991 Microwave and Optical Technology Letters. Reprinted with permission.)
in the structure because of the thin substrate and narrow conductor width. This can be overcome to some extent by printing thick lines on thick alumina substrates.
13.6.3 PlanarTransformer Balun
Planartransformer baluns consist of two oppositely wrapped twincoil transformers connected in series. In this configuration, one of the two outer nodes in the primary coil and the inner common node in the secondary coil are grounded as shown in Figure 13.48 [31]. Figure 13.48(a) shows the equivalent circuit, and Figure 13.48(b) is a microphotograph of the balun using rectangular spiral transformers. The chip measures about 1.5 mm2, which demonstrates the compactness of this approach. The resonant frequency of the coil transformers due to interturn and other parasitic capacitances with the ground, limits the performance and bandwidth of this type of balun. Usually the balun is operated below the resonant frequency. The lower frequency bound of the bandwidth is set by the inductance, while the upper frequency bound is set by the resonant frequency of the coil. The bandwidth can be increased either by increasing the resonant frequency while maintaining the same inductance or by increasing the inductance while maintaining the resonant frequency. The resonant frequency can be increased by reducing the interturn and parasitic capacitances by employing thick and lowdielectric constant substrates and air bridges in the coil. In addition, the inductance can be increased by optimizing the areatolength ratio.
520
Baluns
Figure 13.45
Top and cross sectional illustrations of a threestrip CPW balun.
Figure 13.49 shows the simulated and measured performance of a planartransformer balun. The amplitude and phase imbalances between the two balanced ports are less than 1.5 dB and 10 degrees, respectively, over the 1.5 to 6.5GHz frequency band. The simulated results shown were obtained using EM analysis. Multiple coupled lines [66–68] have also been used to design baluns, including Marchand baluns. Multiple coupledline based baluns are planar, have coupling tighter than two coupled lines, and are simpler to fabricate than multilayer coupledline baluns. Recently, an improved planar Marchand balun using a patterned ground plane to realize very high evenmode impedance has been reported [69]. We described several kinds of baluns in this chapter. The selection of a particular type depends upon the application, performance, and cost limitations. Among all the baluns described, the Marchand balun achieves the maximum bandwidth and best performance.
13.6 Other Baluns
Figure 13.46
521
Multiconductor coupled line triformer: (a) physical layout, (b) equivalent circuit, and (c) microphotograph. (From: [19]. 1989 IEEE. Reprinted with permission.)
522
Baluns
Figure 13.47
Comparison of computed and measured Sparameters: (a) magnitude, and (b) differential phase shift. (From: [19]. 1989 IEEE. Reprinted with permission.)
13.6 Other Baluns
523
Figure 13.48
(a) Simplified circuit diagram and (b) photograph of a rectangular spiral transformer balun. (From: [31]. 1991 IEEE. Reprinted with permission.)
Figure 13.49
Comparison between simulated and measured performances of a planartransformer balun. (From: [31]. 1991 IEEE. Reprinted with permission.)
524
Baluns
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[65]
[66]
Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House, 1996. Garg, R., and I. J. Bahl, ‘‘Characteristics of Coupled Microstriplines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT27, July 1979, pp. 700–705; also see correction in IEEE Trans. Microwave Theory Tech., Vol. MTT28, March 1980, p. 272. Bahl, I. J., and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed., New York: John Wiley, 2003. Cohn, S. B., ‘‘Characteristic Impedances of BroadsideCoupled Strip Transmission Lines,’’ IRE Trans. Microwave Theory Tech., Vol. MTT8, November 1960, pp. 633–637. Willems, D., and I. Bahl, ‘‘A MMIC Compatible Tightly Coupled Line Structure Using Embedded Microstrip,’’ IEEE Trans. Microwave Theory Tech., Vol. 41, December 1993, pp. 2303–2310. Horton, M., and R. Wenzel, ‘‘General Theory and Design of Optimum QuarterWave TEM Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT13, May 1965, pp. 316–327. Tsai, M. C., ‘‘A New Compact Wideband Balun,’’ IEEE MTTS Int. Microwave and Millimeter Wave Monolithic Circuit Symp. Dig., 1993, pp. 123–125. Cohn, S. B., ‘‘The ReEntrant Cross Section and WideBand 3dB Hybrid Couplers,’’ IEEE Trans. Microwave Theory and Tech., Vol. MTT11, July 1963, pp. 254–258. Lavendol, L., and J. J. Taub, ‘‘ReEntrant Directional Coupler Using Strip Transmission Line,’’ IEEE Trans. Microwave Theory and Tech., Vol. MTT13, September 1965, pp. 700–701. DeBrecht, R. E., ‘‘Coplanar Balun Circuits for GaAs FET HighPower PushPull Amplifiers,’’ IEEE MTTS Int. Microwave Symp. Dig., 1973, pp. 309–312. Chiou, H.K., and H.H. Lin, ‘‘A Miniature MMIC Doubly Balanced Mixer Using Lumped Element Dual Balun for High Dynamic Receiver Applications,’’ IEEE Microwave Guided Wave Lett., Vol. 7, August 1997, pp. 227–229. Chiou, H.K., H.H. Lin, and C.Y. Chang, ‘‘LumpedElement Compensated High/Low Pass Balun Design for MMIC DoubleBalanced Mixer,’’ IEEE Microwave Guided Wave Lett., Vol. 7, August 1997, pp. 248–250. Bakalski, W., et al., ‘‘Lumped and Distributed LatticeType LCBaluns,’’ IEEE MTTS Int. Microwave Symp. Dig., 2002, pp. 209–212. Kuylenstierna, D., and P. Linner, ‘‘Design of BroadBand Lumped Element Baluns with Inherent Impedance Transformation,’’ IEEE Trans. Microwave Theory Tech., Vol. 52, December 2004, pp. 2739–2745. Bahl, I., Lumped Elements for RF and Microwave Circuits, Norwood, MA: Artech House, 2003. Yoon, Y. J., et al., ‘‘Design and Characterization of Multilayer Spiral TransmissionLine Baluns,’’ IEEE Trans. Microwave Theory Tech., Vol. 47, September 1999, pp. 1841–1847. Yoon, Y. J., et al., ‘‘Modeling of Monolithic RF Spiral TransmissionLine Balun,’’ IEE Trans. Microwave Theory Tech., Vol. 49, February 2001, pp. 393–395. Guo, Y. X., Z. Y. Zhang, and L. C. Ong, ‘‘Design of Miniaturized LTCC Baluns,’’ IEEE MTTS, Int. Microwave Symp. Dig., 2006, pp. 1567–1570. Govind, V., et al., ‘‘Analysis and Design of Compact Wideband Baluns on Multilayer Liquid Crystalline Polymer (LCP) Based Substrates,’’ IEEE MTTS, Int. Microwave Symp. Dig., 2005. Ang, K. S., Y. C. Leong, and C. H. Lee, ‘‘Analysis and Design of Miniaturized LumpedDistributed ImpedanceTransforming Baluns,’’ IEEE Trans. Microwave Theory Tech., Vol. 51, March 2003, pp. 1009–1017. Lee, J.W., and K. J. Webb, ‘‘Analysis and Design of LowLoss Planar Microwave Baluns Having Three Symmetric Coupled Lines,’’ IEEE MTTS Int. Microwave Symp. Dig., June 2002, pp. 117–120.
13.6 Other Baluns [67]
[68] [69]
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Cho, C., and K. C. Gupta, ‘‘A New Design Procedure for SingleLayer and TwoLayer ThreeLine Baluns,’’ IEEE Trans. Microwave Theory Tech., Vol. 46, December 1998, pp. 2514–2519. Chen, Y.L., and H.H. Lin, ‘‘Novel Broadband Planar Balun Using Multiple Coupled Lines,’’ IEEE MTTS Int. Microwave Symp. Dig., 2006, pp. 1571–1574. Zhang, Z., et al., ‘‘Improved Planar Marchand Balun Using a Patterned Ground Plane,’’ Int. J. RF and Microwave ComputerAided Engineering, Vol. 15, May 2005, pp. 307–315.
About the Authors
R. K. Mongia received his B.Sc. (Eng.) degree from Delhi College of Engineering, University of Delhi, India, in 1981 and a Ph.D. in electrical engineering from the Indian Institute of Technology (IIT), Delhi, India, in 1989. From 1990 to 1994, he held postdoctoral positions at FAMU/Florida State University, Tallahassee, Florida, and at the University of Ottawa and the Communications Research Center, Ottawa, Canada. He worked at COMDEV Ltd., Ontario, Canada, where he designed dielectric resonators filters and multiplexers for onboard satellite communication equipment. Dr. Mongia presently works for REMEC Defense and Space, Richardson, Texas, where he is involved in the designs of GaAs MMICs, including T/R ICs for phasedarray applications. He has published more than 50 publications in reputed journals and conferences. I. J. Bahl received a B.S., an M.S. in physics, and an M.S. in electronics engineering. In 1975, he received a Ph.D. in electrical engineering from IIT, Kanpur, India. At M/ACOM, as a Distinguished Fellow of Technology, his interests are in the area of device modeling, highefficiency power amplifiers, and MMIC products for commercial and military applications. Dr. Bahl is the author of more than 145 research papers and 12 books, and he holds 16 patents. He is an IEEE Fellow, a member of the Electromagnetic Academy and is also the editor for the International Journal of RF and Microwave ComputerAided Engineering. P. Bhartia graduated with a B. Tech. (Hons.) from IIT, Bombay, and a Ph.D. from the University of Manitoba, Winnipeg. Over a 25year career with the Department of National Defence in Canada, Dr. Bhartia held four directorlevel and two director general level positions. Dr. Bhartia has published extensively with more than 200 publications, 5 patents, and 9 books to his credit. He was appointed to the Order of Canada in 2002 and is a Fellow of the Royal Society of Canada, an IEEE Fellow, and a Fellow of The Engineering Institute of Canada, The Canadian Academy of Engineer, and The Institute of Electronic and Telecommunication Engineers. Dr. Bhartia received the IEEE McNaughton Gold Medal for his contributions to engineering. He is currently the executive vice president of Natel Engineering in Chartsworth, California. J. Hong received a D.Phil. in engineering science from the University of Oxford, United Kingdom, in 1994. Currently, he is a faculty member in the Department of Electrical, Electronic and Computer Engineering, at HeriotWatt University,
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About the Authors
Edinburgh, United Kingdom, leading a team for research into advanced RF/microwave device technologies. He has authored and coauthored more than 130 research papers, as well as a book, Microstrip Filters for RF/Microwave Applications (Wiley, 2001). Dr. Hong is a senior member of the IEEE.
Index 3dB couplers asymmetrical, 153–55 with symmetrical microstrip lines, 151–53 4:1 TLT fractional bandwidth, 461 reflection coefficient response, 458 source impedance, 460 transmission coefficient response, 459 See also Transmission line transformers (TLTs) /4 quadrature phaseshift keying (QPSK) modulation, 12 equivalent, 470, 471 A ABCD matrices filter analysis, 279–80 lumpedelement branchline coupler, 231 of lumpedelement circuit, 226 ABCD parameters, 38–40 defined, 38 illustrated, 44 normalized, 41 properties, 39–40 reflection coefficient and, 40–41 transmission coefficient and, 40–41 unnormalized, 41 Admittance matrix, 27 conversion relations, 43–45 fourport network, 487 lossless network, 27 properties, 27–28 Applications, 11–13 Approximate analysis, 462–64
Asymmetrical coupled lines, 131–32 approximate distributed line parameters, 132–33 backwardwave directional couplers, 136–38 directional couplers using, 133–38 forwardwave directional couplers, 133–36 maximum power coupled between, 162 nonTEM mode, 136 normalmode parameters, 105, 132–33 phase velocities, 133 quasiTEM mode, 136 width, 133 Asymmetrical couplers, 178 defined, 167 multisection, 177 Asymmetrical forward couplers, 153–55 3dB, 153–55 endtoend symmetry, 154 fabrication, 153 microstrip line width, 153 phase difference, 154, 155 strip pattern, 154 theoretical/experimental response, 154 See also Forwardwave directional couplers Asymmetrical nonuniform couplers, 214–17 broadband coupling, 214 equivalent circuit to determine coupling, 215 evenmode equivalent circuit, 214 evenmode reflection coefficient, 215 even/oddmode characteristic impedances, 216
531
532
Asymmetrical nonuniform couplers (continued) illustrated, 214 reflectionless taper, 216–17 scattering matrix, 216 scattering parameters, 214 tandem connection, 216 See also Nonuniform TEM directional couplers Asymmetric port excitation filters, 347–52 EM excitation, 348 fivepole microstrip pseudocombline bandpass, 349 frequency responses, 349, 350 openloop resonator feed structure, 351 symmetrical port excitation versus, 347 twopole microstrip pseudocombline bandpass, 350 Attenuation constant, 54, 55 Attenuation pole, 311 B Backwardwave couplers, 105 with asymmetrical coupled lines, 136–38 conditions, 116–17 defined, 6–7, 167 frequency response, 119 ideal, 118, 119 in inhomogeneous medium, 143 maximum coupling value, 119 maximum value, 120 scattering parameters, 118 TEM, 118 See also Directional couplers Balanced CRLH transmission line, 435, 436 Balance mixers, 481 Baluns, 481–523 analysis, 486–90 broadband, 496 coaxial, 481, 484 coupledline, 486–90
Index
CPW, 518 defined, 481 fabricated chip, 498 lumpedelement, 483 Marchand, 498–518 microstriptobalanced stripline, 482–86 nonplanar, 486 parallelstrip, 484 planar, 481, 485, 490–98 planartransformer, 519–20 printed, 504, 505 spiral transformer, 523 triformer, 518–19 uses, 481, 481–82 VSWR versus electrical length, 504 wideband, 503 Bandpass filters ceramicblock, 295 Chebychev lowpass prototype, 310 fractional bandwidth, 310 frequency response, 278 ideal, 271 response, 271 structure, 278 transformation, 277–78 twopole microstrip pseudocombline, 347 See also Filters Bandstop filters frequency response, 279 parallelcoupled, 300–304 response, 271 structure, 279 transformation, 278–79 See also Filters Biasing circuits, 446–49 with multisection shunt stubs, 448 parameters, 449 simplified, 448 simulated response, 449 Branchline couplers, 219 as 90degree hybrid, 222 broadband, 233 characteristic impedance, 220 conservation of energy, 220
Index
coupling variation, 223 directivity, 225 ideal, isolation, 225 isolation variation, 223 for loose coupling, 223–25 lumpedelement, 230–33 mainline characteristic impedance, 221 modified, 219, 223–25 physical implementation, 222–23 in planar circuit configuration, 220 planes of symmetry, 221 reducedsize, 219, 225–30 scattering parameters, 220 shunt branches, 221 size, 219 VSWR variation, 223 See also Tight couplers Broadband 180degree bit phase shifters, 475, 493 Broadband branchline coupler, 233 Broadband dc block, 446 Broadband forwardwave directional couplers, 149–65 Broadband ratrace coupler, 237–38 Broadsidecoupled lines, 93–99 asymmetric, 10 defined, 1 evenmode field distribution, 94 illustrated, 4 oddmode field distribution, 94 offset striplines, 96–99 striplines, 94–95 suspended microstrip lines, 95–96 Broadside couplers, 255–58 3dB asymmetric, 257, 258 amplitude characteristics, 257 cross section, 256 measured coupling coefficient, 258 multilayer, structure, 256 physical layout, 258 symmetric, 257 Broadside striplines, 2 Butterworth filter, 272 element values, 273 number of sections, 273
533
C Capacitance matrix, 61–62 Capacitances, 61–68 coupled line, 61–68 evenmode, 62 fringing, 66–68, 192 oddmode, 62–65 offset striplines, 141 parallel plate, 66–68 selfcapacitance, 127 total series, 463 wiggly lines, 192 Cascaded quadruplet (CQ) filter, 327–28 coupling structures, 328 defined, 327–28 Cascaded quadruplet trisection (CQT) filters, 371–77 10pole microstrip bandpass configuration, 375 experiment, 376–77 frequency responses, 375 illustrated, 372 implementation, 373–74 measured performance, 378 sensitivity analysis, 374–76 synthesis, 372–73 theoretical response, 373 wideband response, 378 Cascaded trisection (CT) filter coupling structure, 328 defined, 328 Ceramicblock filters, 292–95 crosssectional view, 294 frequency response, 297 schematic, 293 specifications, 295 Characteristic impedance, 32, 54 backwardtraveling mode, 121 branchline couplers, 220 broadsidecoupled suspended microstrip lines, 97 conductorbacked coplanar waveguides, 89 coplanar waveguides, 86 coupled coplanar waveguides, 91
534
Characteristic impedance (continued) even mode, 5, 68 finite strip thickness effect, 70, 71 finitethickness microstrip, 76, 77 inverted microstrip lines, 93 normalized voltages/currents and, 19 odd mode, 5, 68 ratrace couplers, 235 slotcoupled microstrip lines, 100, 101 striplines, 71 suspended microstrip lines, 93 Chebyshev response filter, 272–75 element values, 276 insertion loss, 272 number of sections, 274 passband VSWR maximum, 274–75 Coaxial baluns, 481 illustrated, 484 Marchand, 501–5 synthesis, 505 See also Baluns Coaxial lines, 2 Codirectional couplers. See Forwardwave directional couplers Combline filters, 10, 11, 290–95 ceramic block, 292, 293 coaxial, 291–92 crosssectional view, 294 defined, 290–91 design example, 291–95 design procedures, 291 frequency response, 297 resonator length, 293 See also Filters Compact couplers, 176, 261–64 coupledline, 264 lumpedelement, 262 meander line, 263–64 measured electrical performance, 265 physical dimensions, 264 spiral, 262–63 Compact Marchand balun, 517–19 Complementary SRRs (CSRRs), 434 Composite right/lefthanded (CRLH) transmission line, 435 balanced, 435, 436 metamaterial applications, 437
Index
multilayered, 437–38 structure, 265 unbalanced, 435, 437 unit cell, 438 Conductorbacked coplanar waveguides, 87–88 characteristic impedance, 89 effective dielectric constant, 89 See also Coplanar waveguides (CPW) Conductor loss, 58 quasiTEM mode, 60–61 single microstrip, 79 TEM mode, 60–61 Conservation of power principle, 158 Coplanar waveguides (CPW), 53, 83–88 baluns, 518 characteristic impedance, 86 conductorbacked, 86, 87–88 coupled, 88 effective dielectric constant, 86 on finite thickness dielectric substrate, 85 illustrated, 3 schematic, 390 with upper shielding, 86–87 Coupled coplanar waveguides, 88 characteristic impedance, 91 cross section, 90 effective dielectric constant, 92 Coupledline baluns, 486–90 Coupledline circuit components, 443–76 broadband 180degree bit phase shifters, 475 dc blocks, 443–52 delay lines, 475 interdigital capacitors, 461–65 power dividers, 475 resonators, 475 Schiffman sections, 475 spiral inductors, 465–72 spiral transformers, 472–75 transformers, 452–61 Coupledline filters, 269–304 advanced filtering characteristics, 327–52 with asymmetric port excitations, 347–52
Index
with crosscoupled resonators, 327–35 with defected ground structures, 324–27 with enhanced stopband performance, 307–27 with meandered parallelcoupled lines, 320–24 microstrip CPW, 410–21 multimode, 406–10 with periodically nonuniform coupled lines, 314–20 with sourceload coupling, 335–47 superconductor, 371–85 with unevenlycoupled stages, 307–14 Coupledline Marchand balun, 508 Coupled lines, 1 capacitances, 61–68 distributed equivalent circuit, 126–30 even/oddmode characteristic impedances, 84, 85 illustrated, 2 multiconductor/multilevel configuration, 3 symmetrical, 67–68 uniformly, 105–47 Coupled microstrip lines covered with dielectric overlay, 190 frequencydependent characteristics, 83 illustrated, 82 quasistatic even/oddmode characteristic impedances, 82–83 Coupledmode theory, 149, 156–63 even/oddmode analysis and, 160–61 voltage waves modification, 157 for weakly coupled resonators, 163–65 Coupled spur lines, 192–94 Coupled structures, 1–6 broadside, 1 components based on, 6–11 edge, 1 symmetric, 4 TEM modes, 3 types, 3–4 uniform, 4 weakly, 149
535
Coupling desirable, 1 mechanism, 4–6 parasitic, 1–2 between symmetrical lines, 160, 161–63 weak, 157–58 Coupling coefficients capacitive, 145 determination arrangement, 334 extraction arrangement, 333, 335 inductive, 145 between lines, 158 mutual, 157 nature of, 158 voltage, 6, 171 weakly coupled resonators, 163 Crosscoupled filters, 327–35 applications, 328 defined, 327 external quality factors, 329 implementation, 329–30 layout, 336 miniature, 333–35 openloop resonator, 353 theoretical frequency responses, 330 Crosscoupled optimum stub filter, 404–6 circuit parameters, 406 circuit topology, 405 illustrated, 405 Currents equivalent, 18 normalized, 18–21 unnormalized, 21–22 Cutoff frequency parallelcoupled bandstop filter, 301 striplines, 73 D DC blocks, 443–52 analysis, 443–46 biasing circuits, 446–49 broadband, 446 highvoltage, 451 microstrip interdigital, 447 millimeterwave, 449–51 quarterwave coupledline section as, 444
536
Defected ground structures, 324–27 EMsimulated performance and, 327 floating conductor at, 325 microstrip line characteristics, 325 microstrip line cross section, 324 threepole parallelcoupled microstrip line filters, 326 Delay lines, 475 Desirable coupling, 1 Diagonal conductors, 256 Dielectric loss, 58, 79 Dielectric overlays, 189 Directional couplers, 6–8 with asymmetrical coupled lines, 133–38 backwardwave, 6–7, 105, 116–20 broadside, 255–58 compact, 261–64 coupling, 168 directivity, 168 forwardwave, 105, 114–16 fourport, 167–68 hybrid, with interdigital Lange configuration, 9 ideal, power loss ratio, 180–84 interdigital, 242–52 isolation, 168 lumped capacitor compensation, 9 microwave, 6 multiconductor, 242–52 multilayer, 255–61 multioctave bandwidth, 7 multisection, 177–86 nonuniform broadband TEM, 197–217 parallelcoupled TEM, xxii, 167–94 reentrant mode, 258–61 singlesection, 169–76 spiral, 262–63 tandem, 7, 252–55 tight, 8 uses, 6 wiggly twoline, 9 Directivity, 168 branchline couplers, 225 coupled spur lines, 192–94 dielectric overlays, 189
Index
improvement techniques, 186–94 lumped compensation, 186–89 meandered coupled line sections, 194 modified branchline coupler, 225 shunt inductive feedback, 192, 193 wiggly lines, 189–92 Distributed equivalent circuit, 126–30 Dualband filters, 359–67 approaches, 363 categories, 362 demand, 359 design, 365–66 simultaneous operation, 362–63 with SIR, 365–66 E Edgecoupled striplines, 73–74 characteristic impedances, 74 cross section, 69 See also Striplines Edgecoupled structures, 1 Effective dielectric constants broadsidecoupled suspended microstrip lines, 97 conductorbacked coplanar waveguides, 89 coplanar waveguides (CPW), 86 coupled coplanar waveguides, 92 inverted microstrip lines, 90, 93 slotcoupled microstrip lines, 100, 101 suspended microstrip lines, 90 Electromagnetic metamaterials, 428 Electromagnetic simulators, 91, 222 Electronic warfare (EW), 12 Equal magnitude, 107 Equal ripple function, 208 Even mode broadsidecoupled microstrip lines, 94 capacitances, 62 characteristic impedances, 5, 6, 68 coupledmode theory and, 160–61 excitation, 63, 107, 108–9 impedances, 183 phase velocities, 68 plane of symmetry behavior, 111 reflection coefficients, 117, 215
Index
symmetrical network analysis, 106–11 See also Odd mode External quality factor crosscoupled filters, 329 extraction arrangement, 332 frequency response, 311 unevenlycoupled stages, 311 F Fast wave, 159 Filters, 8–11 with advanced dielectric materials, 391–428 with advanced materials, 371–438 analysis, 279–80 applications, 270–71 bandpass, 271, 277–78 bandstop, 271, 278–79 CAD and, 280 cascaded quadruplet (CQ), 327–28 cascaded trisection (CT), 328 Chebyshev response, 272–75 coaxial line, 10 combline, 10, 11, 290–95 configurations, 11 coupledline, 269–304 coupledline, advanced filtering characteristics, 327–52 coupledline, with enhanced stopband performance, 307–27 crosscoupled, 327–35 crosscoupled optimum stub, 404–6 directcoupled, 327 dualband, 359–67 folded, 327 general network configuration, 269 hairpinline, 11, 295–300 highpass, 271, 277 importance, 8 interdigital, 10, 11, 287–90 LCP, 396–400 lowpass, 271, 272 LTCC cavity, 394–96 LTCC lumped element, 392–94 metamaterial, 428–29, 428–38 micromachined, 385–91 microstrip coupled line, 8–10
537
microstrip CPW coupledline, 410–21 multimode coupledline, 406–10 optimum stub line, 401–6 parallelcoupled line, 11, 283–87 parameters, 269 practical considerations, 280–83 responsetype, 275 sourceload coupling, 335–47 superconductor coupledline, 371–85 theory and design, 271–83 types, 270 UWB, 12, 400–428 UWB with notch band, 421–28 Finitethickness microstrip, 76–77 characteristic impedance, 76, 77 synthesis equation, 77 See also Microstrip lines Folded filters, 327 Forwardwave directional couplers, 105, 114–16, 150–56 3dB, 151–55 asymmetrical, 133–36, 153–55 bandwidth, 149 broadband, 149–65 defined, 149 equations, deriving, 116 perunit wavelength, 156 realization, 149 scattering parameters, 115 symmetrical, 150–53 ultrabroadband, 155–56 See also Directional couplers Fourport directional couplers, 167–68 Fourport network illustrated, 36 reciprocal, 36–38 reduction, 50–51 two ports terminated in arbitrary load, 50 uniform asymmetrical coupled lines, 123 uniform coupled symmetrical lines, 111 Fourthorder Marchand balun, 507 Fractional power maximum, 162 tandem couplers, 254–55 Frequency bandwidth ratio, 169–70
538
Frequency range, 269 Frequency response asymmetric port excitation filters, 349, 350 bandpass filters, 278 bandstop filters, 279 combline filters, 297 CQT filter, 375 for extracting external quality factor, 311 highorder filters, 380 highpass filters, 277 parallelcoupled bandstop filter, 304 parallelplate baluns, 488 singlesection directional coupler, 169–70 sourceload coupling filters, 340 Fringing capacitance, 66–68 defined, 66 oddmode, 192 Fringing fields, 66 Fullwave analysis, 464–65 G Gibb’s phenomenon, 204 Group delay, 269 defined, 270 dependencies, 281 equalization, 377–85 selfequalization, 381 H Hairpinline filters, 295–300 coupling coefficient, 298 defined, 296 design, 297 design example, 297–300 interresonator coupling, 297 layout, 299 measured response, 299 physical layout, 299 See also Filters HighFrequency Structure Simulator (HFSS), 280 Highorder filters coupling structure, 380 design, 380–84
Index
fabrication and measurement, 384–85 frequency response, 380 with groupdelay equalization, 377–85 HTS bandpass, 379–80 layout, 384 modeling, 380–84 transmission response, 379 Highpass filters frequency response, 277 response, 271 schematic, 277 transformation, 276–77 See also Filters High temperature superconductor (HTS) substrates, 12 Highvoltage dc block, 451 Hyperbolic secant, 69 I Impedance matrix, 26–27 conversion relations, 43–45 lossless network, 27 properties, 27–28 Impedance transformers configurations, 457 substrate parameters, 458 Incremental inductance rule, 60, 61 Inductance matrix, 62, 127 Input impedance, 24 filters, 269 normalized, 23 unnormalized, 23–24 Insertion loss, 269 Chebyshev response filter, 272 defined, 179, 269 Marchand baluns, 513, 514 narrow conductor, 490 passband, 281 UWB bandpass filters, 424 Interdigital capacitors, 461–65 approximate analysis, 462–64 configuration, 461 fullwave analysis, 464–65 parameters, 464 responses, 465, 466 series resistance, 463 subcomponents, 462 total series capacitance, 463
Index
Interdigital couplers, 242–52 design, 245–52 design equations, 246 equivalent capacitance network, 244 impedance parameters, 249 length, 247 Nconductor, 246 side view, 244 theory, 243–45 top view, 244 voltage coupling factor, 247 Interdigital filters, 10, 11, 287–90 design equations, 287 design examples, 287–90 measure passband performance, 292 miniature, on silicon, 387–90 narrowband design, 287–89 narrowband microstrip SIR, 358 ninepole, layout, 291 ninepole wideband SIR, 357 popularity, 287 with stepped impedance resonators, 352–59 uniform impedance resonators (UIR), 357, 359 wideband design, 289–90 wideband response, 292 See also Filters Inverted microstrip lines, 88–93 characteristic impedance, 93 cross section, 92 effective dielectric constant, 90, 93 illustrated, 3 See also Microstrip lines Isolation, 168 K K inverters, 283 Kirchhoff’s equations, 279–80 L Lange coupler, 220 for 3dB coupling, 249 design, 245–46 design data, 247–52 dimensional ratios, 250
539
Marchand balun with, 512–14 sixfinger, 251–52 LCP filters, 396–400 60GHz band, 399–400 miniature wideband, 396–99 planar, 399 vialess, 399 Liquid crystal polymer (LCP), 391 duplexer, 400 layers, bonding, 400 uses, 392 See also LCP filters Load impedance, 24 Local multipoint distribution systems (LMDS), 392 Lowpass filters Butterworth, 272 prototype, 272 response, 271 Lowtemperature cofired ceramic (LTCC) substrates, xvii, 12, 264, 391 multilayer capability, 394 technology developments, 392 See also LTCC filters LTCC filters, 392–96 cavity, 394–96 design parameters, 396 layout, 393 lumped element, 392–94 measured/simulated performance, 393 multilayer capability, 394 Lumped capacitors, 186 directivity improvement of, 188 in MMICs, 225 physical length, 187–88 reducedsize branchline couplers, 225 top view, 187 Lumped compensation, 186–89 Lumpedelement branchline coupler, 230–33 ABCD matrix, 231 bandwidth, 232, 233 lumped element values, 232, 233 ‘‘pi’’ network, 230, 231 realization, 230 sections, 233 ‘‘tee’’ network, 230, 231 See also Branchline couplers
540
Lumpedelement compact coupler, 262 Lumpedelement ratrace coupler, 240–41 equivalent lumped circuit, 243 lumped elements, 240–41 See also Ratrace couplers Lumped equivalent circuit, 127 M Magnetic coupling, 331 Marchand baluns, 498–518 amplitude, 514 characteristic impedances ratio, 500 coaxial, 501–5 coaxial cross section, 500 compact, 517–18 coupledline, 508 coupledline parameters, 511 design, 509 elements, 498 equivalent circuit parameters, 510 equivalent transmissionline model, 500 examples, 509–18 fourthorder, 507 insertion loss, 513, 514 with Lange coupler, 512–14 monolithic, 516–17 phase balance, 514 phase difference, 513 realization, 505 with reentrant couplers, 514–16 series/parallelresonant compensating representation, 501 simplified equivalent circuit, 501 simulated performance, 513 synthesis, 505–9 versions, 498 See also Baluns Mathcad, 206, 213 MATLAB, 206 Meandered coupled lines sections, 194 Meandered parallelcoupled lines designed for spurious passband suppression, 322 design with, 320–24 illustrated, 320
Index
layout, 323 passband harmonics suppression, 322 structure, 321 Meander line directional coupler, 263–64 Metalinsulatormetal (MIM) capacitors, 230, 251, 445 Metamaterial filters, 428–38 configuration, 434 frequency response, 433 layout, 432, 433 Metamaterials, 2, 265 CRLH, 435–38 summary, 438 M/FILTER software, 300 Micromachined filters, 385–91 miniature interdigital on silicon, 387–90 overlay coupled CPW, 390–91 synthesized substrate, 387 Micromachining techniques, 264–65 Microstrip CPW coupledline filters, 410–21 CPW resonator, 411 on dielectric substrate, 414, 417, 420 EMsimulated performances, 418, 419, 421 EMsimulated responses, 413 equivalent circuit model, 415 frequency responses, 415 illustrated, 412 operation, 411 physical implementation, 417 structure, 413 theoretical responses, 416 UWB bandpass, 428 Microstrip dispersion, 79–81 Microstrip lines characteristics, 74–83 coupled, 81–83 coupling between, 143 with defected ground aperture, 324 defined, 2 even/oddmode field configurations, 5 illustrated, 3, 75 inverted, 88–93 propagation mode, 74
Index
single, 75–81 slotcoupled, 99–102 suspended, 88–93 types, 3 with unequal impedances, 4 Microstriptobalanced stripline balun, 482–86 Microwave network theory, 17–51 Millimeterwave dc block, 449–51 Miniature crosscoupled filter, 333–35 configuration, 337 defined, 333–34 designing, 334–35 layout, 338 upper stopband, 335 wideband performance, 339 See also Crosscoupled filters Miniature interdigital filters on silicon, 387–90 advantage, 389 EMsimulated loss effects, 389 fabricated fivepole, 388 layout, 388 measured performance, 389 simulated performance, 389 See also Interdigital filters Miniature wideband LCP filter, 396–99 Modified branchline coupler, 219, 223–25 bandwidth, 225 defined, 223–24 directivity, 225 equivalent length, 225 frequency response, 226 highimpedance transmission line section, 224 illustrated, 225 for loose coupling, 223–24 shunt branches, 225 simulated response, 225, 226 See also Branchline couplers Modified ratrace coupler, 237–38 bandwidth, 237 equivalent impedance, 237 illustrated, 237 performance, 237, 238 See also Ratrace couplers
541
Monolithic Marchand balun, 516–17 Monolithic microwave integrated circuits (MMICs), 12 lumped capacitors, 225 reducedsize hybrid implementation, 228 small, lowloss ratrace hybrid, 240 Monte Carlo method, 374 Multiconductor coupled line triformer, 521 Multiconductor couplers, 242–52 design, 245–52 theory, 243–45 Multilayer couplers, 138–47 broadside, 255–58 coupling, 143 design, 140–47 EM simulated response, 147 examples, 145–47 inhomogeneous, 146 reentrant mode, 258–61 Sonnet Lite and, 139–40 tight, 255–61 Multimode coupledline filters, 406–10 coupledline I/O feed sections, 409 EMsimulated group delay response, 409 EMsimulated magnitude responses, 409 modified, 410 resonant frequency responses, 408 UWB, 425–28 Multisection couplers, 177–86 asymmetrical, 177 limitations, 184–86 Nsection, 177, 178 physical layout, 186 power loss ratio, 180 response, 178, 179 specifications, 182 symmetrical, 177 symmetrical nonuniform, 210 synthesis, 177–84 tables of parameters, 182–85 theory, 177–84 Mutual inductance, 164
542
N Networks admittance matrix, 27–28 fourport, 36–38 impedance matrix, 26–28 inputoutput relationship, 17 interconnected, scattering matrix, 45–51 nonreciprocal, 35–36 Nport, 17, 25 power loss ratio, 179 reciprocal, 32, 34–35, 36–38 scattering matrix, 28–32 symmetrical, 106–11 threeport, 34–36 twoport, 33–34, 38–43 Nonplanar baluns, 486 Nonuniform line resonators, 315–16 illustrated, 316 resonant characteristics, 317 structures, 315 Nonuniformly coupled symmetric line, 4 Nonuniform TEM directional couplers, 197–217 asymmetrical, 214–17 defined, 197 design procedure, 210–14 in homogeneous dielectric medium, 211–13 phase constant, 211–12 symmetrical, 197–214 weighting function, 205 See also Directional couplers Normalized currents, 18–21 axial flow, 21 characteristic impedance and, 19 determining, 18 illustrated, 19 total, 20 Normalized input impedance, 23 Normalized voltage, 18–21 characteristic impedance and, 19 determining, 18 illustrated, 19 total, 20
Index
Normal modes interference between, 159 of symmetrical coupled lines, 163 warping, 156 Nport networks, 17, 25 O Odd mode capacitances, 62–65 characteristic impedances, 5, 68 coupled mode theory and, 160–61 fringing capacitance, 192 impedances, 175, 183 phase velocities, 68, 186 propagation velocity, 6 reflection coefficients, 117 symmetrical network analysis, 106–11 See also Even mode Odd mode excitation, 109–11 defined, 108 plane of symmetry behavior, 111 of symmetrical coupled lines, 63 Offset striplines, 7 broadsidecoupled, 96–99 capacitance parameter, 141 illustrated, 141 inductance parameter, 141 See also Striplines Openloop resonator filters, 331 asymmetric feed scheme, 351 crosscoupled, 353 feed structure, replacing, 352 symmetric feed scheme, 351 Optimum stub line filters, 401–6 circuit parameters, 402 with cross coupling, 404–6 design, 401–4 on dielectric substrate, 403 measured performance, 404 photo, 404 transmission characteristics, 402 Organization, this book, xvii–xviii Output impedance, 269 Overlay CPW filters, 390–91 defined, 391 endcoupled, 392
Index
schematic, 390 usefulness, 391 P Parallelcoupled bandstop filter, 300–304 CAD file, 303 cutoff frequency, 301 defined, 300 design example, 301–4 element values, 302 equivalent circuit, 301 frequency response, 304 layout, 300 See also Filters Parallelcoupled line filters, 283–87 design, 283 design example, 285–87 equivalent circuit, 309 illustrated, 284 K inverters, 283 for multispurious suppression, 313 physical dimensions, 284 physical lengths, 285 single coupledline section, 308 threepole, 286 transmission response, 308 unequal coupled sections, 312, 313 See also Filters Parallelcoupled TEM directional couplers, 167–94 coupling, 168 with dielectric overlay compensation, 189 directivity, 168 isolation, 168 multisection, 177–86 parameters, 167–68 singlesection, 169–76 See also Directional couplers Parallelplate baluns, 488 Parallelplate capacitance, 66–68 Parallelplate impedance, 485 Parasitic coupling, 1–2 Periodically nonuniform coupled lines, 314–20 Phase constant, 54, 57–58
543
Phase velocities, 54, 58 asymmetrical coupled lines, 133 backwardtraveling mode, 121 equalizing, 190 even/odd modes, 68, 186 suspended microstrip lines, 93 symmetrical nonuniform couplers, 203 Planar baluns, 481 analysis, 493–96 configurations, 492 examples, 496–98 fabricated on lowdielectric substrate, 502 illustrated, 485 layout, 519 phase balance, 499 sections, 490–92 synthesis, 505–9 topologies, 496 twoport coupler configuration, 493 See also Baluns Planartransformer balun, 519–20, 523 Planar transmission lines, 53–102 Power added efficiency (PAE), 481 Powercoupling coefficient, 137 Power dividers, 475 Power loss ratio, 179 ideal directional coupler, 180–84 multisection coupler, 180 singlesection coupler, 180 Power transfer, 152 Printed baluns, 504, 505 Propagation constant, 24 Q Qfactors, 56–57 conductor, 57 defined, 56–57 dielectric loss, 57 normalized conductor, 72 Quarterwave coupler, 176 QuasiTEM modes asymmetrical coupled transmission lines, 136 characteristics, 58–59 common transmission lines, 54
544
QuasiTEM modes (continued) conductor loss, 60–61 defined, 53 line parameters, 53 R Ratrace couplers, 219, 233–42 as 180degree hybrid, 235 bandwidth, 237 characteristic impedances, 235 conservation of power, 234 design, 235 discontinuities effect, 237 frequency bandwidth requirement, 237 lumpedelement, 240–42 modified, 237–38 in planar circuit configuration, 234 ports, 236 properties, 236–37 reducedsize, 219, 239–40 scattering matrix, 234–35 scattering parameters, 233–34 size, 219 strip conductor layout, 233 threequarterwavelong section, 237 VSWR variation, 236 See also Tight couplers Reciprocal networks, 32 fourport, 36–38 threeport, 34–35 Reducedsize branchline couplers, 219, 225–30 ABCD matrix, 226 bandwidth, 228 calculated phase difference, 228, 229 circuit equivalence, 227 illustrated, 230 implementation, 228 lumped capacitors, 225 mainline length, 228, 230 measured performance, 230 photomicrograph, 230 shunt branches, 228, 230 size, 230 See also Branchline couplers Reducedsize ratrace coupler, 219, 239–40 advantages, 239
Index
defined, 239 illustrated, 240 lowloss MMIC, 240 photomicrograph, 242 port interchange, 241 See also Ratrace couplers Reentrant mode couplers, 258–61 coupling coefficient, 259 cross section, 260 defined, 258 even/oddmode impedances, 258 Marchand balun with, 514–16 microstrip, 515 multilayer microstrip lines, 514–15 performance, 261 schematic view, 259 typical dimensions, 260 Responsetype filters, 275 Return loss, 23, 270 S SATCOM, 270 Scattering matrix, 28–32 asymmetrical nonuniform couplers, 216 conversion relations, 43–45 equivalent circuit determination, 112 interconnected networks, 45–51 normalized, 106 normalized/unnormalized matrices, 32 overall network, 46 ratrace couplers, 234–35 reciprocal networks, 32 reduced networks, 47–48 reduced twoport network, 49 tandem couplers, 253 transformation, 31–32 unitary property, 30–31 unnormalized, 106 Scattering parameters, 29 amplitudes, 37 asymmetrical nonuniform couplers, 214 backwardwave couplers, 118 branchline couplers, 220 forwardwave directional couplers, 115
Index
of reduced networks, 47–48 symmetrical nonuniform couplers, 198 tandem couplers, 253 Schiffman sections, 475 Scope, this book, 13 Selfcapacitance, 127 Selfinductances, 127, 141 Series capacitor, 443 Shunt inductive feedback, 192, 193 Single microstrip, 75–81 conductor loss, 79 dielectric loss, 79 finitethickness, 76–77 microstrip dispersion, 79–81 Singlesection couplers, 169–76 compact, 176 design, 171–75 equivalent circuit, 179 fractional bandwidth, 170 frequency bandwidth ratio, 169–70 frequency response, 169–71 power loss ratio, 180 quarterwave, 176 useful operating bandwidth, 171 variation of coupling, 170 Single stripline, 69–73 Sixfinger microstrip coupler, 251–52 capacitor values, 252 illustrated, 251 line lengths, 252 parameters, 252 See also Lange coupler Slotcoupled microstrip lines, 99–102 characteristic impedance, 100, 101 effective dielectric constant, 100, 101 illustrated, 100 uses, 99 Slow wave, 159 Sonnet Lite, 139–40 defined, 139 geometry, 140 for single transmission line structure, 145 Sourceload coupling filters, 335–47 coupling determination, 341 coupling matrix, 337
545
coupling structure, 339, 345 cross coupling, 338 direct coupling, 338 filtering characteristic, 338 fractional bandwidth, 337 frequency response, 340 I/O coupling structure, 343 layout, 342, 344 n + 2 coupling matrix, 342, 343–44, 345 theoretical responses, 345 threepole microstrip parallelcoupled, 345, 346 topology, 337 Spiral directional couplers, 262–63 size reduction, 262–63 top conductor layout, 263 See also Directional couplers Spiral inductors, 465–72 2turn microstrip, 466 equivalent circuit representation, 470 admittance parameters, 469 coupledline equivalent circuit models, 467 design, 465 electrical characteristics, 467 fourport representation, 469 inductance calculation, 468 in MICs, 465 rectangular, 468 Spiral transformers, 472–75 balun, 523 example, 473 power transfer, 475 rectangular, 472 schematic, 475 twincoil fourport, 472 See also Transformers Splitrings resonators (SRRs), 428 complementary (CSRRs), 434 composite medium with, 429 dimensions, 430 metamaterial structure based on, 430 negative effect permeability, 429 square, 429
546
Stepped impedance resonators (SIR), 307, 352–59 antisymmetrical shape, 366 dualband filters with, 365–66 EMsimulated performance, 358 fivepole microstrip dualband filter, 367 grounded /4, 353 grounded microstrip structure, 355 interdigital filter layout, 362 interdigital filters using, 352–59 interresonator coupling structure, 357 I/O stage, 354 microstrip, 354 narrowband design, 354–56 narrowband layout, 358 ninepole wideband, 357 wideband design, 356–59 wideband responses, 363 Stopband attenuation, 269 Striplines broadsidecoupled, 94–95 characteristic impedance, 71 characteristics, 68–74 cross section, 69 cutoff frequency, 73 defined, 2 edgecoupled, 73–74 lossless, 69 offset, 7, 96–99 single, 69–73 TEM mode of propagation support, 174 Superconductor coupledline filters, 371–85 cascaded quadruplet/triplet filters, 371–77 highorder selective, with group delay equalization, 377–85 Superstar software, 300 Suspended microstrip lines, 88–93 broadsidecoupled, 95–96 characteristic impedance, 93 cross section, 92 effective dielectric constant, 90 illustrated, 3
Index
phase velocity, 93 See also Microstrip lines Symmetrical coupled lines, 67–68 Symmetrical couplers defined, 167 multisection, 177–78 schematic, 178 tables of parameters, 182–85 Symmetrical forward coupler, 150, 151 3dB, 151–53 coupling response, 151 illustrated, 150 strip pattern, 152 theoretical/experimental response, 152 See also Forwardwave directional couplers Symmetrical networks even/oddmode analysis, 106–11 fourport, 106, 108 Symmetrical nonuniform couplers, 197–214 amplitude, 202 coupled signal amplitude, 201 coupling response, 202, 204 design procedure, 210–14 differential reflection coefficient, 200 electrical length, 209 equivalent circuit, 199 evaluation, 206 evenmode characteristic impedance, 199–201 finite length, 201 in homogeneous medium, 211–13 ideal, 202–6 illustrated, 198 multisection, length, 210 nonuniform coupling, 198 phase constant, 211–12 phase velocity, 203 physical length, 208, 209–10 scattering parameters, 198 synthesis, 201–6 weighting function, 205 See also Nonuniform TEM directional couplers Synthesized substrate, 387
Index
T Tandem couplers, 252–55 crossover requirement, 255 defined, 252–53 fractional power, 254–55 physical configuration, 255 reflected voltages, 253 scattering matrix, 253 scattering parameters, 253 schematic, 252 See also Tight couplers TEM modes characteristics, 57–58 common transmission lines, 54 conductor loss, 60–61 line parameters, 53 support, 3 Threeport networks illustrated, 35 nonreciprocal, 35–36 port three terminated in arbitrary load, 48 reciprocal, 34–35 reduction, 48–49 Thrureflectline (TRL) calibration techniques, 497 Tight couplers, 219–65 braided microstrip, 264 branchline, 219, 220–33 combline, 264 compact, 261–64 coplanar waveguide, 264 dielectric waveguide, 264 between edgecoupled lines, 219 embedded microstrip, 264 finline, 264 multiconductor, 242–52 multilayer, 255–61 ratrace, 219, 233–42 size, 219 slotcoupled, 264 tandem, 252–55 uses, 219 vertically installed, 264 wiggly twoline, 264 Tight microwave couplers, 8
547
Transformers coupledline, 452–61 equivalent circuit representation, 452 impedance, 452, 457 opencircuit, 452–56 simulated performance, 456 source/load impedances, 453 spiral, 472–75 symmetrical, 454 transformation ratio parameters, 455 transmission line, 456–61 See also Baluns Transient response, 269, 270 Transmission line transformers (TLTs), 456–61 1:4, 457, 458, 459, 460 broadsidecoupled, 457 defined, 456 fractional bandwidth, 461 line characteristic impedance, 456 maximum bandwidth, 460 polyimide thickness, 459 reflection coefficient, 458 size, 459 See also Transformers Travelingwave tubes (TWTs), 149 Triformer balun, 518–19 Twoport networks ABCD parameters, 38–40 cascade, 39 illustrated, 33 network representation, 45 reduction, 49–50 representative matrices, comparison relationships, 42–43 special representation, 38–43 U Ultrabroadband forwardwave directional couplers, 155–56 Ultrawideband. See UWB Unbalanced CRLH transmission line, 435, 437 Unevenlycoupled stages, 307–14 attenuation poles, 311–12 design example, 310–14
548
Unevenlycoupled stages (continued) external quality factor, 311 principle, 308 Unfolded Lange coupler, 245 Uniform impedance resonators (UIR) interdigital filters, 357, 359 design, 359 interresonator stage, 361 I/O stage, 360 wideband responses, 363 See also Interdigital filters Uniformly coupled asymmetric lines, 120–33 mode, 122–23 capacitive coupling coefficients, 130 characteristic impedances, 122 c mode, 121–22 defined, 4 distributed equivalent circuits, 126–30 fourport network, 123 inductive coupling coefficients, 130 parameters, 121–26 propagation constant, 122 Yparameters, 123–26 Zparameters, 123–26 Uniformly coupled lines, 105–47 asymmetrical lines, 120–33 directional couplers using, 111–20 Unnormalized currents, 21–22 defined, 21 determining, 22 Unnormalized input impedance, 23–24 Unnormalized quantities, 25–26 Unnormalized voltage, 21–22 defined, 21 determining, 22 UWB microstrip CPW coupledline filters, 410–21 multimode coupledline filters, 406–10 optimum stub line filters, 401–6 technology, 400–428 UWB bandpass filters, 421–28 with embedded band notch stubs, 422 EM simulation, 424 fabricated microstrip, 426 fullwave EM simulation, 425
Index
insertion loss, 424 layout, 422 measured/simulated performances, 426 microstrip CPW coupledline, 428 multimode coupled line, 425–28 narrowband characteristic, 423 passband performance, 425 schematic diagrams, 423 V Verylargescale integrated (VLSI) chips, 2 Voltages coupling coefficient, 171 equivalent, 18 normalized, 18–21 reflection coefficient, 22–23 unnormalized, 21–22 Voltage standing wave ratio (VSWR), 2, 23 simulated, 446 variation of, 445 W WAVECON, 301 Weakly coupled resonators coupledmode theory, 163–65 coupling coefficient, 163 Weighting function determination example, 208–9 determination technique, 206–9 for loose couplers, 206 nonuniform couplers, 205 for tight couplers, 206 Wideband baluns, 503 Wiggleline filter coupling structures, 316–18 defined, 314 illustrated, 319 interresonator coupling structure, 318 I/O coupling structure, 317 nonuniform line resonators, 315–16 wideband response, 319 Wiggly lines capacitance parameters, 192 depth, 192 geometrical parameters, 190 length, 191
Index
549
top view, 191 use of, 189–92 Wireless local area network (WLANs), 12
Yttrium barium copper oxide (YBCO) thin films, 376, 384
Y
Zparameters fourport network, 123–26 interdigital twoport network, 125–26
Yparameters, of fourport network, 123–26
Z