RF and Microwave Coupled-Line Circuits, Second Edition

RF and Microwave Coupled-Line CircuitsSecond Edition

For a listing of recent titles in the Artech HouseMicrowave Library, turn to the back of this book.

RF and Microwave Coupled-Line CircuitsSecond Edition

R. K. MongiaI. J. BahlP. BhartiaJ. Hong

Library of Congress Cataloging-in-Publication DataA catalog record for this book is available from the U.S. Library of Congress.

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library.

ISBN-13: 978-1-59693-156-5

Cover design by Yekaterina Ratner

2007 ARTECH HOUSE, INC.685 Canton StreetNorwood, 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 thisbook 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 markshave been appropriately capitalized. Artech House cannot attest to the accuracy of thisinformation. Use of a term in this book should not be regarded as affecting the validity ofany 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 1Introduction 1

1.1 Coupled Structures 11.1.1 Types of Coupled Structures 31.1.2 Coupling Mechanism 4

1.2 Components Based on Coupled Structures 61.2.1 Directional Couplers 61.2.2 Filters 8

1.3 Applications 111.4 Scope of the Book 13

References 13

CHAPTER 2Microwave Network Theory 17

2.1 Actual and Equivalent Voltages and Currents 172.1.1 Normalized Voltages and Currents 182.1.2 Unnormalized Voltages and Currents 212.1.3 Reflection Coefficient, VSWR, and Input Impedance 222.1.4 Quantities Required to Describe the State of a Transmission

Line 242.2 Impedance and Admittance Matrix Representation of a Network 26

2.2.1 Impedance Matrix 262.2.2 Admittance Matrix 272.2.3 Properties of Impedance and Admittance Parameters of a

Passive Network 272.3 Scattering Matrix 28

2.3.1 Unitary Property 302.3.2 Transformation with Change in Position of Terminal Planes 31

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2.3.3 Reciprocal Networks 322.3.4 Relationship Between Normalized and Unnormalized Matrices 32

2.4 Special Properties of Two-, Three-, and Four-Port Passive, LosslessNetworks 322.4.1 Two-Port Networks 332.4.2 Three-Port Reciprocal Networks 342.4.3 Three-Port Nonreciprocal Networks 352.4.4 Four-Port Reciprocal Networks 36

2.5 Special Representation of Two-Port Networks 382.5.1 ABCD Parameters 382.5.2 Reflection and Transmission Coefficients in Terms of ABCD

Parameters 402.5.3 Equivalent T and P Networks of Two-Port Circuits 41

2.6 Conversion Relations 432.7 Scattering Matrix of Interconnected Networks 45

2.7.1 Scattering Parameters of Reduced Networks 472.7.2 Reduction of a Three-Port Network into a Two-Port Network 482.7.3 Reduction of a Two-Port Network into a One-Port Network 492.7.4 Reduction of a Four-Port Network into a Two-Port Network 50References 51

CHAPTER 3Characteristics of Planar Transmission Lines 53

3.1 General Characteristics of TEM and Quasi-TEM Modes 533.1.1 TEM Modes 573.1.2 Quasi-TEM Modes 583.1.3 Skin Depth and Surface Impedance of Imperfect Conductors 593.1.4 Conductor Loss of TEM and Quasi-TEM Modes 60

3.2 Representation of Capacitances of Coupled Lines 613.2.1 Even- and Odd-Mode Capacitances of Symmetrical Coupled

Lines 623.2.2 Parallel-Plate and Fringing Capacitances of Single and Coupled

Planar Transmission Lines 663.3 Characteristics of Single and Coupled Striplines 68

3.3.1 Single Stripline 693.3.2 Edge-Coupled Striplines 73

3.4 Characteristics of Single and Coupled Microstrip Lines 743.4.1 Single Microstrip 753.4.2 Coupled Microstrip Lines 81

3.5 Single and Coupled Coplanar Waveguides 833.5.1 Coplanar Waveguide 843.5.2 Coplanar Waveguide with Upper Shielding 863.5.3 Conductor-Backed Coplanar Waveguide with Upper Shielding 873.5.4 Coupled Coplanar Waveguides 88

3.6 Suspended and Inverted Microstrip Lines 88

Contents ix

3.7 Broadside-Coupled Lines 933.7.1 Broadside-Coupled Striplines 943.7.2 Broadside-Coupled Suspended Microstrip Lines 953.7.3 Broadside-Coupled Offset Striplines 96

3.8 Slot-Coupled Microstrip Lines 99References 102

CHAPTER 4Analysis of Uniformly Coupled Lines 105

4.1 Even- and Odd-Mode Analysis of Symmetrical Networks 1064.1.1 Even-Mode Excitation 1084.1.2 Odd-Mode Excitation 109

4.2 Directional Couplers Using Uniform Coupled Lines 1114.2.1 Forward-Wave (or Codirectional) Directional Couplers 1144.2.2 Backward-Wave Directional Couplers 116

4.3 Uniformly Coupled Asymmetrical Lines 1204.3.1 Parameters of Asymmetrical Coupled Lines 1214.3.2 Distributed Equivalent Circuit of Coupled Lines 1264.3.3 Relation Between Normal Mode (c and p ) and Distributed

Line Parameters 1304.3.4 Approximate Distributed Line or Normal-Mode Parameters of

Asymmetrical Coupled Lines 1324.4 Directional Couplers Using Asymmetrical Coupled Lines 133

4.4.1 Forward-Wave Directional Couplers 1334.4.2 Backward-Wave Directional Couplers 136

4.5 Design of Multilayer Couplers 1384.5.1 Determination of Capacitance and Inductance Parameters

Using Sonnet Lite 1394.5.2 Coupler Design 140References 147

CHAPTER 5Broadband Forward-Wave Directional Couplers 149

5.1 Forward-Wave Directional Couplers 1505.1.1 3-dB Coupler Using Symmetrical Microstrip Lines 1515.1.2 Design and Performance of 3-dB Asymmetrical Couplers 1535.1.3 Ultra-Broadband Forward-Wave Directional Couplers 155

5.2 Coupled-Mode Theory 1565.2.1 Nature of Coupling Coefficient K12 and K21 1585.2.2 Waves on Lines 1 and 2 in the Presence of Coupling 1585.2.3 Coupled-Mode Theory and Even- and Odd-Mode Analysis 1605.2.4 Coupling Between Asymmetrical Lines 161

5.3 Coupled-Mode Theory for Weakly Coupled Resonators 163References 165

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CHAPTER 6Parallel-Coupled TEM Directional Couplers 167

6.1 Coupler Parameters 1676.2 Single-Section Directional Coupler 169

6.2.1 Frequency Response 1696.2.2 Design 1716.2.3 Compact Couplers 1766.2.4 Equivalent Circuit of a Quarter-Wave Coupler 176

6.3 Multisection Directional Couplers 1776.3.1 Theory and Synthesis 1776.3.2 Limitations of Multisection Couplers 184

6.4 Techniques to Improve Directivity of Microstrip Couplers 1866.4.1 Lumped Compensation 1866.4.2 Use of Dielectric Overlays 1896.4.3 Use of Wiggly Lines 1896.4.4 Other Techniques 192References 194

CHAPTER 7Nonuniform Broadband TEM Directional Couplers 197

7.1 Symmetrical Couplers 1977.1.1 Coupling in Terms of Even-Mode Characteristic Impedance 1997.1.2 Synthesis 2017.1.3 Technique for Determining Weighting Functions 2067.1.4 Electrical and Physical Length of a Coupler 2097.1.5 Design Procedure 210

7.2 Asymmetrical Couplers 214References 217

CHAPTER 8Tight Couplers 219

8.1 Introduction 2198.2 Branch-Line Couplers 220

8.2.1 Modified Branch-Line Coupler 2238.2.2 Reduced-Size Branch-Line Coupler 2258.2.3 Lumped-Element Branch-Line Coupler 2308.2.4 Broadband Branch-Line Coupler 233

8.3 Rat-Race Coupler 2338.3.1 Modified Rat-Race Coupler 2378.3.2 Reduced-Size Rat-Race Coupler 2398.3.3 Lumped-Element Rat-Race Coupler 240

8.4 Multiconductor Directional Couplers 2428.4.1 Theory of Interdigital Couplers 2438.4.2 Design of Interdigital Couplers 245

8.5 Tandem Couplers 252

Contents xi

8.6 Multilayer Tight Couplers 2558.6.1 Broadside Couplers 2558.6.2 Re-Entrant Mode Couplers 258

8.7 Compact Couplers 2618.7.1 Lumped-Element Couplers 2628.7.2 Spiral Directional Couplers 2628.7.3 Meander Line Directional Coupler 263

8.8 Other Tight Couplers 264References 265

CHAPTER 9Coupled-Line Filter Fundamentals 269

9.1 Introduction 2699.1.1 Types of Filters 2709.1.2 Applications 270

9.2 Theory and Design of Filters 2719.2.1 Maximally Flat or Butterworth Prototype 2729.2.2 Chebyshev Response 2729.2.3 Other Response-Type Filters 2759.2.4 LC Filter Transformation 2759.2.5 Filter Analysis and CAD Methods 2799.2.6 Some Practical Considerations 280

9.3 Parallel-Coupled Line Filters 2839.3.1 Design Example 285

9.4 Interdigital Filters 2879.4.1 Design Examples 287

9.5 Combline Filters 2909.5.1 Design Example 291

9.6 The Hairpin-Line Filter 2959.6.1 Design Example 297

9.7 Parallel-Coupled Bandstop Filter 3009.7.1 Design Example 301References 304

CHAPTER 10Advanced Coupled-Line Filters 307

10.1 Introduction 30710.2 Coupled-Line Filters with Enhanced Stopband Performance 307

10.2.1 Design Using Unevenly-Coupled Stages 30710.2.2 Design Using Periodically Nonuniform Coupled Lines 31410.2.3 Design Using Meandered Parallel-Coupled Lines 32010.2.4 Design Using Defected Ground Structures 324

10.3 Coupled-Line Filters Exhibiting Advanced Filtering Characteristics 32710.3.1 Filters with Cross-Coupled Resonators 32710.3.2 Filters with Source-Load Coupling 33510.3.3 Filters with Asymmetric Port Excitations 347

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10.4 Interdigital Filters Using Stepped Impedance Resonators 35210.4.1 Narrowband Design 35410.4.2 Wideband Design 356

10.5 Dual-Band Filters 359References 367

CHAPTER 11Filters Using Advanced Materials and Technologies 371

11.1 Introduction 37111.2 Superconductor Coupled-Line Filters 371

11.2.1 Cascaded Quadruplet and Triplet Filters 37111.2.2 High-Order Selective Filters with Group-Delay Equalization 377

11.3 Micromachined Filters 38511.3.1 Miniature Interdigital Filters on Silicon 38711.3.2 Overlay Coupled CPW Filters 390

11.4 Filters Using Advanced Dielectric Materials 39111.4.1 Low-Temperature Cofired Ceramic Filters 39211.4.2 Liquid Crystal Polymer Filters 396

11.5 Filters for Ultra-Wideband (UWB) Technology 40011.5.1 Optimum Stub Line Filters 40111.5.2 Multimode Coupled-Line Filters 40611.5.3 Microstrip-Coplanar Waveguide Coupled-Line Filters 41011.5.4 UWB Filters with Notch Band 421

11.6 Metamaterial Filters 428References 438

CHAPTER 12Coupled-Line Circuit Components 443

12.1 DC Blocks 44312.1.1 Analysis 44312.1.2 Broadband DC Block 44612.1.3 Biasing Circuits 44612.1.4 Millimeter-Wave DC Block 44912.1.5 High-Voltage DC Block 451

12.2 Coupled-Line Transformers 45212.2.1 Open-Circuit Coupled-Line Transformers 45212.2.2 Transmission Line Transformers 456

12.3 Interdigital Capacitor 46112.3.1 Approximate Analysis 46212.3.2 Full-Wave Analysis 464

12.4 Spiral Inductors 46512.5 Spiral Transformers 47212.6 Other Coupled-Line Components 475

References 476

Contents xiii

CHAPTER 13Baluns 481

13.1 Introduction 48113.2 Microstrip-to-Balanced Stripline Balun 48213.3 Analysis of a Coupled-Line Balun 48613.4 Planar Transmission Line Baluns 490

13.4.1 Analysis 49313.4.2 Examples 496

13.5 Marchand Balun 49813.5.1 Coaxial Marchand Balun 50113.5.2 Synthesis of Marchand Balun 50513.5.3 Examples of Marchand Baluns 509

13.6 Other Baluns 51813.6.1 Coplanar Waveguide Baluns 51813.6.2 Triformer Balun 51813.6.3 Planar-Transformer Balun 519References 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 MicrowaveCoupled-Line Circuits.

Sections of coupled transmission structures are critical components in distrib-uted RF and microwave passive circuits. Significance of their role as basic buildingblocks is second only to the sections of single transmission structures. Their applica-tions in design of directional couplers and filters are well known, but equallyimportant is the role played by coupled-line sections in the design of baluns,capacitors, inductors, transformers and dc blocks. Availability of high dielectricconstant materials has extended the usage of coupled-line sections to lower micro-wave and RF frequencies. Traditionally, coupled sections consisting of two singlelines have been used extensively. However, as the circuit designers understandmodeling and characterization of multiple coupled lines, we can look forward tosignificantly larger applications of multiple coupled transmission line structures.Three-line balun structures reported recently are a step in this direction. Also, asthe multilayer RF and microwave circuits become more popular, couplings amongthe transmission lines at different levels of a multilayer structure become a criticaldesign consideration. In some cases this multilayer-multiconductor coupling canbe advantageous as a useful circuit component, while in other cases this couplingcan become an undesirable effect that should be mitigated. Both of these situationsneed the modeling and characterization of multiconductor transmission line struc-tures.

Recognizing the role of coupled lines, it is hard to comprehend why a compre-hensive book on this topic has not been available so far. But then, someone hasto be the leader. Bahl and Bhartia have a history of providing to the microwavedesign community several well-needed ‘‘firsts,’’ and this book is their most recentcontribution. Congratulations Rajesh, Inder, and Prakash on a book which I amconfident will be very well received in the RF/microwave community.

K. C. GuptaUniversity of Colorado at Boulder

November 1998

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Preface to the Second Edition

The first edition of RF and Microwave Coupled-Line Circuits was published in1999. While the fundamentals of coupled line circuits have not changed in the past8 years, further innovations in coupled line filters and other applications haveoccurred with changes in technology and use of new fabrication processes, suchas the use of low temperature cofired ceramic (LTCC) substrates. In this case forexample, it is common to use multilayer structures, with 10–25 layers being quitecommon. Thus, multilayer coupling needs to be better understood and explained forrealizing an optimum three-dimensional design and structure that LTCC permits.

Over the years the first edition of this book has been well received in themarketplace and has been used extensively in industry by microwave engineers.Practitioners have pointed out some errors that crept into print in the first editionand 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 circuitsare 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, commu-nication, and navigation systems, and hence support the need for comprehensivetextbooks in this area.

In the past few years, mainly driven by the desire towards miniaturization,many novel configurations of coupled-line components, such as directional couples,filters, and baluns, have been reported. Most practicing microwave and RF engi-neers are not fully aware of the advancements in this area.

In view of these concerns, the authors, encouraged by the publishers, felt thata revised version of RF and Microwave Coupled-Line Circuits would be useful forthe microwave community. To ensure a fresh look, Dr. Jia-Sheng Hong, an eminentspecialist in the microwave filters area, was invited to become a coauthor for thesecond edition. Dr. Hong has made suggestions on modifications to the book andhas contributed fully in the preparation of some chapters that are closer to his areaof expertise.

With the deletion of some chapters in their entirety and the addition of newchapters and new material in other chapters, overall the book remains the samesize as the first edition. The first few chapters reflect only minor changes, asthese incorporate the fundamentals of microwave transmission lines, networks,and coupled lines, which have not changed. Some additions and changes have beenmade to accommodate the multilayer design of coupled lines for the sake of havinga self-contained, complete text. Most of the major changes occur in the ‘‘Applica-tions’’ 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 softwarepackages used for filter design, together with their limitations and strengths.

Chapters 10 and 11 are new, building on the discussion of filters in Chapter9. These two chapters discuss advanced filter technology, and the design of filtersusing new materials and technologies. Chapter 10 concentrates on coupled linefilters 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 cur-rently important topics such as filters with source-load coupling and asymmetricport excitation are also covered.

Chapter 11 takes a different direction, tackling filters using advanced materialsand technologies. These include superconductor coupled-line filters, micro-machined filters, miniature interdigital filters on silicon, LTCC filters which requiremultilayer coupling, liquid crystal polymer filters, and ultra-wideband filters. Thesetopics encompass the new direction and materials in filter technology that willreplace the older thick film and organic substrate technology over the comingdecade as demands and pressures for smaller, lighter-weight, lower-cost filtersmount.

Chapters 12 and 13 are essentially the former Chapters 10 and 11 with revisionsas appropriate and the inclusion of new material to update the chapters and makethem current.

Thus, Chapter 12 discusses the design of common microwave componentsrequiring coupled line technology. This includes dc blocks, transformers, interdigitalcapacitors, and spiral inductors.

Chapter 13 covers baluns. Baluns in different configurations (e.g., microstripto balanced stripline, planar transmission line, and Marchand type) are discussedin detail. The former Chapters 12 and 13 on high-speed circuit interconnects andmulticonductor transmission lines have been deleted in their entirety. The mainreason for this deletion is that although the material was relevant and useful, mostengineers using the text did not feel that these chapters added much and preferredto see a greater expansion of the coverage on filters, which we have included.

Recognizing the current reality that engineers use software packages for theirdesign and no longer do hand calculations, we have included a short discussion ineach chapter where possible about current software packages that allow one todesign the circuit discussed in that chapter. As an example, in Chapter 9, Section9.2.5 covers the current software packages that are available for designing the typesof filters discussed, together with their strengths, limitations, and shortcomings. Weexpect 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 wehave 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 havebeen replaced by current topics, and the number of figures and their sizes havebeen 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

Preface to the Second Edition xix

used, as well as the lesser-used tables, have been eliminated, and material thatrenders the text more understandable has been added.

The preparation of any text such as this requires significant cooperation andcoordination. The authors express their gratitude to colleagues from several organi-zations for assisting with this work and for providing permission for use of copyrightfigures and tables. We would specially like to thank Dr. Protap Pramanick for hissuggestions and input. The help provided by Dr. James Rautio and other membersof staff at Sonnet Software, Inc., is acknowledged. We appreciate the support ofour work organizations and especially our families for their understanding, support,and encouragement and putting up with us during the writing phase. Last but notleast, we appreciate the input from the reviewers and the support and cooperationthat we received from the Artech House staff, in particular, Mark Walsh, BarbaraLovenvirth, and Rebecca Allendorf.

R. K. MongiaI. J. Bahl

P. BhartiaJ. Hong

May 2007

Preface to the First Edition

There are a number of textbooks on microwave transmission lines. Recent onesinclude extensive information on the modern planar lines such as microstrip, slot-lines, coplanar waveguides, and the like. At the next level of complexity are thevarious functional circuits such as couplers, hybrids, filters, and baluns, which usethe elemental transmission line in different configurations to achieve the desiredfunctionality and meet system performance requirements. Much of this functionalityinvolves coupling between transmission lines, and extensive research has beenconducted in the design and analysis of such structures. Initially, much of theliterature was oriented to coaxial lines and waveguides. With the evolution andthe popularity of planar transmission lines, however, it was felt desirable to puttogether all aspects of coupled circuits using these lines under one cover.

Most current texts, we found, contained perhaps a chapter or two on somespecific components, especially couplers and filters. This text attempts to treat thetopic in its entirety, starting with the fundamental theory of coupled structuresand the application of this to the design and analysis of specific componentssuch as couplers, filters, baluns, and so forth. This treatment emphasizes planartransmission lines, the CAD tools available for the design of these structures, useof full-wave 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 develop-ment specialists who are involved in the area of coupled-line circuit design, analysis,development, and fabrication. The layout of the book facilitates its use as a textfor a graduate course and for short courses on specific component design.

The book is divided into 13 chapters. The first chapter introduces the readerto the topic and covers the nature of coupled structures, the importance of thesestructures in microwave circuits, and some applications. A good introduction tothe principal components using coupled lines (i.e., directional couplers and filters)is also given.

The second chapter establishes the basic circuit parameters and representationof microwave networks. Fundamental network analysis tools such as impedanceand scattering matrix techniques are introduced together with the properties oftwo-, three-, and four-port networks. Relationships between the commonly usedmatrix representation forms such as ABCD, scattering, and impedance are estab-lished to permit the researcher or designer to work in the system of his or herpreference.

The fundamental building blocks for coupled-line circuits (i.e., transmissionlines) 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. Inaddition, the characteristics of coupled lines in these configurations under differentconditions such as broadside coupling, edge coupling, or, in the case of coplanarwaveguides, coupling with shields present are discussed. Whereas Chapter 3 concen-trates on characteristics of physical transmission lines, Chapter 4 presents thegeneral analysis of uniformly coupled asymmetrical lines, including forward andbackward couplers. These fundamentals permit a more in-depth coverage of thecoupling of uniform lines, which is covered in the next chapter. Even- and odd-mode analysis is covered together with an analysis of uniformly coupled asymmetri-cal lines. Forward and backward directional coupler design methods using theaforementioned techniques are also given in the chapter.

The next few chapters are devoted to the design of various types of directionalcouplers. Many directional couplers by their very nature and design have a nar-rowband performance. In a number of applications, broadband performance isessential. The design and performance of forward-directional couplers using asym-metrical coupled lines are the subject of Chapter 5. Coupled-mode theory, alsodiscussed in this chapter, is very useful for the analysis of general weakly coupledsystems.

Parallel-coupled backward TEM directional couplers using a single section ormultisections are discussed in Chapter 6, together with limitations and methodsfor improving the directivity of such couplers. This permits the reader to have agood understanding of how these circuits work and the methods, including lumped-element compensation and dielectric overlay, that can be used to improve directivity.While we dealt with broadband couplers using multisection couplers in Chapter6, one can also obtain this type of characteristic of performance using nonuniformlines. Additional flexibilities, and at the same time complexities, are introducedwith this line, but in many cases it is essential to resort to this process because ofphysical or performance constraints imposed by the overall circuit design. In Chap-ter 7, the design and synthesis procedure for such couplers is outlined, togetherwith some other techniques to obtain broadband performance.

Finally, the last type of coupler that requires special treatment is the tightcoupler. Tight couplers, as described in Chapter 8, can be designed and fabricatedin a number of configurations. Some of the most prevalent forms are the branch-line coupler, rat-race coupler, and lumped-element coupler. These are fully coveredin this chapter, together with a large number of other layouts including the multicon-ductor couplers and tandem couplers. A number of novel designs are also discussed,including the interdigital Lange couplers and compact couplers for wireless applica-tions. The material provided gives the designer a good grasp of the principles andtechniques involved in the design of these coupler types, together with the advan-tages and disadvantages of the specific couplers. This information and understand-ing is critical to the designer in assisting him or her to choose the appropriatecoupler type to meet not only the electrical performance characteristics but alsoto meet any form, fit, and function requirements imposed.

Besides the directional couplers covered in the previous chapters, perhaps themost commonly used form of coupled line circuits is the filter. This is coveredextensively in Chapter 9, starting with a definition of filter parameters and leadingon to filter synthesis, design, and realization. Modern miniature filters are discussed,

Preface to the First Edition xxiii

as they are critical to the wireless communications area, and an assessment of thecapabilities of a number of software packages available for filter design is provided.

The next two chapters delve into a number of other commonly used coupledcircuits. Chapter 10 covers the analysis and design of a number of dc blocks,coupled-line transformers, interdigital capacitors, spiral inductors, and transform-ers, while Chpater 11 treats the design and analysis of baluns. In particular, theMarchand balun is discussed in detail, together with other types of baluns such asthe coplanar waveguide balun, triformer balun, and planar transformer balun.

Whereas the preceding chapters have used coupling as a means of achieving aspecific function and performance, in many cases coupling is not desirable and cancreate problems. This is typically the electromagnetic compatability/electromagneticinterference problem that is encountered by any circuit designer. To try to coverthe topic of coupled circuits in its entirety, we have included high-speed digitalinterconnections in Chapter 12 to bring about an awareness of the cross-talkproblem and provide ways to mitigate this problem. Finally, many of the passivedevices covered could use multiconductor lines for their design. The literature on thistopic is very dispersed. In Chapter 13, we have provided the essential information forthe designer to permit the use of multiconductor lines as the building block forthe type of coupled-line circuit one wished to design.

As with any comprehensive treatment of a topic, one must draw upon theworks of a large number of researchers and authors. We have taken special careto reproduce equations and diagrams and believe that this text is a valuable additionto the microwave circuit designer’s library.

The preparation of this text has depended on a number of very supportiveindividuals and organizations. Naturally, the time spent during evenings and week-ends comes at the expense of time with our families. For their support and under-standing we are eternally grateful. The organizations that we work for have alsosupported 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 feelleft out, we have no hesitation in recognizing the contributions and acknowledgingwith our thanks the assistance of Josie Dunn for typing the manuscript and BobGervais of the Defence Research Establishment, Ottawa, who devoted large blocksof time in preparing the illustrations. Part of the manuscript has been handledefficiently by Tanya Morrision of ITT GaAsTEK, Roanoke. The Artech Houseteam 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 Don-nelly for their patience, support, and cooperation. Finally, we want to thank thereviewers for their thoroughness and excellent suggestions for improving the text.

R. K. MongiaI. J. Bahl

P. Bhartia

C H A P T E R 1

Introduction

In microwave circuits, transmission lines are normally used in two ways: (1) tocarry information or energy from one point to another; and (2) as circuit elementsfor passive circuits like impedance transformers, filters, couplers, delay lines, resona-tors, and baluns. Passive elements in conventional microwave circuits consist mainlyof the distributed type and employ transmission line sections and waveguides indifferent configurations, thereby achieving the desired functionality and meetingperformance specifications. This functionality is largely achieved by the use ofcoupled transmission lines. In this chapter, we briefly describe the various typesof coupled-line structures, the coupling mechanism, and coupled-line componentsand their applications.

1.1 Coupled Structures

When two unshielded transmission lines (as shown in Figure 1.1) are placed inclose proximity to each other, a fraction of the power present on the main line iscoupled to the secondary line. The power coupled is a function of the physicaldimensions of the structure, mode of propagation [TEM (transverse electromag-netic) or non-TEM], the frequency of operation, and the direction of propagationof the primary power. In these stuctures, there is a continuous coupling betweenthe electromagnetic fields of the two lines. These parallel coupling lines are callededge-coupled structures. The structure shown in Figure 1.1(d) is an exception andis called a broadside-coupled structure.

Coupled lines can be of any form, depending on the application and generallyconsist of two transmission lines, but may contain more than two. The lines canbe symmetrical (i.e., both conductors have the same dimensions) or asymmetrical.Both lines are placed in close proximity to each other so that the electromagneticfields can interact. The separation between the lines may be either constant orvariable. The closer the lines are placed together, the stronger the interaction thattakes place. When one port is excited with a known signal, a part of this signalappears at other ports. This interaction effect known as desirable coupling is usedto advantage in realizing several important microwave circuit functions, such asdirectional couplers, filters, and baluns, with the coupled line length usually beingapproximately a quarter-wave long.

In addition, in closely packed hybrid and monolithic integrated circuits, para-sitic coupling can take place between the distributed matching elements or closely

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2 Introduction

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 severalways. It may change the frequency response in terms of frequency range andbandwidth and degrade the gain/insertion loss and its flatness, input and outputvoltage standing wave ratio (VSWR), and many other characteristics includingoutput power, power-added efficiency, and noise figure, depending on the type ofcircuit. This coupling can also result in the instability of an amplifier circuit orcreate a feedback resulting in a peak or a dip in the measured gain response or asubstantial change in a phase-shifter response. In general, this parastitic couplingis undesirable and an impediment to obtaining an optimal solution in a circuitdesign. However, this coupling effect can be taken into account in the design phaseby using empirical equations and by performing electromagnetic (EM) simulations,or it can be reduced to an acceptable level by maintaining a large separation betweenthe matching elements.

Multiconductor microstrip lines (Figure 1.2) are used in very-large-scale inte-grated (VLSI) chips for digital circuit applications and three-dimensional microwaveintegrated circuits. Here, numerous closely spaced interconnection lines in differentplanes are used to integrate the components on a chip. The design of these intercon-nections is very important to satisfy the size, power consumption, clock frequency,and propagation delay requirements. Signal distortion, increase in backgroundnoise, and cross-talk between the lines from coupling are some of the undesirablecharacteristics. Proper design of these interconnections can reduce the distortion

1.1 Coupled Structures 3

Figure 1.2 Cross-sectional view of a multiconductor and multilevel coupled-line configuration.

and cross-talk to acceptable levels and has played a significant role in the evolutionof high-speed VLSI technology.

1.1.1 Types of Coupled Structures

Coupled-line 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 planarforms. In Figure 1.3, cross sections of microstrip coupled lines and microstrip-likelines are shown. In these structures, practical spacing limitations between thelines limits the tight coupling achievable to about 8 dB over l /4 sections. Theseconfigurations are edge-coupled structures. On the other hand, broadside-coupledlines (shown in Figure 1.4) are used extensively to realize tight couplings of theorder of 2 to 3 dB. All three structures support TEM modes in the case of ahomogeneous dielectric medium and quasi-TEM modes in the case of a nonhomoge-neous media.

In Figure 1.5, we show coupled dielectric waveguides. They support non-TEM modes and forward-wave couplers are realized using these structures. Thesestructures are commonly used at millimeter-wave frequencies, and continuous cou-pling occurs from one guide to another when they are placed in close proximityto each other.

The configurations shown in Figure 1.1, 1.3, 1.4, and 1.5 use equal widths forboth conductors and guides and constant spacing between the conductors and

Figure 1.3 Coupled microstrip-like transmission lines: (a) microstrip lines, (b) inverted microstriplines, (c) suspended microstrip lines, and (d) coplanar waveguide.

4 Introduction

Figure 1.4 Coupled broadside transmission lines: (a) broadside-coupled striplines, (b) broadside-coupled inverted microstrip lines, and (c) broadside-coupled suspended microstrip lines.

Figure 1.5 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 withconstant spacing between lines of unequal width. This structure is called a uniformlycoupled asymmetric line. Figure 1.7 shows an example of a symmetric coupledline with variable spacing between the microstrip conductors, called a nonuniformlycoupled symmetric line.

1.1.2 Coupling Mechanism

The symmetric coupled-line structures, as shown in Figure 1.1, support two modes:even and odd. The interaction between these modes induces the coupling betweenthe two transmission lines, and the properties of the symmetric coupled structuresmay be described in terms of a suitable linear combination of these modes. Thefield 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 5

Figure 1.7 Nonuniformly coupled symmetric lines.

shown in Figure 1.8. In even-mode excitation, both microstrip conductors are atsame potential while the odd mode delineates equal but of opposite polarity poten-tials with respect to the ground. The even and odd modes have different characteris-tic impedances, and their values become equal when the separation between theconductors is very large (lines are uncoupled). The even-mode characteristic imped-ance (Z0e ) is the impedance from one line to the ground when both lines are drivenin-phase from equal sources of equal impedances and voltages. The odd-modecharacteristic impedance (Z0o) is defined as the impedance from one line to theground when both lines are driven out of phase from equal sources of equalimpedances and voltages.

Figure 1.8 Even- and odd-mode field configurations in coupled microstrip lines.

6 Introduction

The velocities of propagation of the even and odd modes are equal whenthe lines are embedded in a homogeneous dielectric medium (e.g., stripline). Fortransmission lines such as coupled microstrip lines, however, the dielectric mediumis 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, theeffective dielectric constants (and the phase velocities) are different for the twomodes. This nonsynchronous feature deteriorates the performance of circuits usingthese types of coupled lines. The voltage coupling coefficient of a coupling structureis generally expressed in terms of the even- and odd-mode characteristic impedances,effective dielectric constants, and coupled structure line length. For a quarter-wavecoupled section in a homogeneous dielectric medium, the coupling coefficient k isgiven by

k =Z0e − Z0oZ0e + Z0o

(1.1)

1.2 Components Based on Coupled Structures

There are numerous microwave passive components realized using coupled-linesections. They include directional couplers, filters, baluns, impedance transformernetworks, resonators, inductors, interdigital capacitors, dc blocks, and others, ofwhich directional couplers and filters are the most popular. A brief history of thelatter is presented next.

1.2.1 Directional Couplers

Directional couplers perform numerous functions in microwave circuits and subsys-tems. They are used to sample power for temperature compensation and amplitudecontrol and in power splitting and combining over an ultra-broadband frequencyrange. In balanced amplifiers they help obtain good input and output VSWRs. Inbalanced mixers and microwave instruments (including network analyzers) theyhelp in sampling incident and reflected signals. They have matched characteristicsat all four ports, making them ideal for insertion in a circuit or subsystem.

A historical account of microwave directional couplers including an extensivereference list is given by Cohn and Levy [1]. The first directional coupler using aquarter-wave-long two-wire configuration was reported in 1922, and during the1940s and 1950s, significant progress was made in waveguide couplers usingapertures in the common wall. Directional couplers using planar TEM lines suchas coupled striplines were developed in the mid-1950s. Numerous papers werepublished in the 1950s and 1960s describing the theory, design, fabrication, andmeasured data for TEM-line edge-coupled directional couplers and significantcontributions were made in the development of planar couplers. These couplerscan provide coupling in the 8- to 40-dB range. Early work on these homogeneouscouplers (single and multisections) can be found in [1–5]. These couplers are alsoknown as backward-wave 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 theprimary line when excited with a microwave signal.

In several applications, a tight coupler such as a 3-dB coupler is required andthe cross-section shown in Figure 1.1(b) is difficult to realize as the very tightspacings required are limited by current photo-etch techniques. This problem isalleviated by using the three-dielectric-layer broadside-coupled striplines (includingoffset coupled strips [6]) and tandem-connection directional couplers [7] of Figure1.9 or the vertically installed coupled-line configuration of Figure 1.10.

Multioctave bandwidth in the above-mentioned couplers is realized by usingmultistages of equal-length coupled sections. When these sections are joined, how-ever, abrupt discontinuities in coupling and line widths occur resulting in couplingerror and directivity degradation. Continuously tapered TEM couplers [8, 9] yieldimproved electrical performance including better bandwidth characteristics.

After widespread use of microstrip lines in microwave circuits, attention turnedto microstrip line couplers [10–19]. One of the driving forces for the developmentof microstrip couplers was the higher level of integration of microwave circuitson a single substrate, including both active and passive components. Because a

Figure 1.9 (a) Offset coupled striplines and (b) tandem-connection of directional couplers.

Figure 1.10 Vertically installed coupled-line coupler.

8 Introduction

microstrip is inhomogeneous, the even- and odd-mode propagation velocities fora coupled pair of microstrip lines are not equal, resulting in poor directivity, whichbecomes worse as the coupling is decreased. For example, a 10% difference inphase velocities reduces the directivity of 10-, 15-, and 20-dB couplers to 13, 8,and 2 dB, respectively, from the theoretically infinite value with equal-phase veloci-ties. Thus, the deterioration in directivity is higher for loose coupling and becomesworse with higher phase velocity differences. Therefore, couplers fabricated onlow-dielectric constant substrates such as plastic (er = 2.5), have better directivityperformance than those on alumina or GaAs. The directivity of stripline andhomogeneous broadside directional couplers is much better than that of microstripcouplers.

Figure 1.11 depicts several techniques for equalizing or compensating for thedifference in the modal phase velocities. Of these ‘‘wiggly lines’’ [10], dielectricoverlays [11, 12] and capacitive compensation methods [13, 14, 17] are the mostcommonly used.

Tight microstrip couplers suffer from the same problem as their stripline coun-terparts; that is, requirements of impractically small spacing between the conduc-tors. 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 areconnected 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 re-entrantstructures, asymmetric broadside-coupled microstrip lines, and slot-coupled micro-strip lines.

1.2.2 Filters

Next to directional couplers, filters are the most important passive componentsused in microwave subsystems and instruments. Most microwave systems consistof many active and passive components that are difficult to design and manufacturewith precise frequency characteristics. In contrast, microwave passive filters canbe designed and manufactured with remarkably predictable performance. Conse-quently, microwave systems are usually designed so that all of the troublesomecomponents are relatively wide in frequency response with filters being incorporatedto obtain the precise system frequency response. Because filters are the narrowestbandwidth components in the system, it is usually the filters that limit such systemparameters as gain and group delay flatness over frequency.

The first use of filters was reported in 1937. A historical survey of microwavefilters including an extensive reference list is given by Levy and Cohn [26]. Pioneer-ing work in coupled-line filters using TEM striplines was performed in the 1950s,1960s, and 1970s. The most popular filter configurations are parallel-coupled-line[27, 28], interdigital [29, 30], combline [31, 32] and hairpin-line [33, 34]. Mostfilters are of the bandpass or bandstop type. Coaxial interdigital and comblineconfigurations are shown in Figure 1.14, while stripline coupled-line configurationsare shown in Figure 1.15.

Microstrip coupled line filters are similar to those shown in Figure 1.15. Becausemicrostrip is a nonhomogeneous medium, the different even- and odd-mode phase

1.2 Components Based on Coupled Structures 9

Figure 1.11 (a) Wiggly two-line coupler, (b) parallel coupled microstrip with overlay compensa-tions, and (c) lumped capacitor compensation of microstrip coupler.

Figure 1.12 Ninety-degree hybrid coupler using interdigital Lange configuration.

10 Introduction

Figure 1.13 Other tight coupler configurations: (a) re-entrant structures, (b) asymmetric broadsidecoupled lines, and (c) slot-coupled microstrip lines.

Figure 1.14 Coaxial-line filters: (a) interdigital, and (b) combline.

velocities result in the filter having an asymmetrical passband response, deteriorat-ing the upper stopband performance and moving the second passband (which isat about twice the center frequency) toward the center frequency. To overcomethis problem, phase velocity equalization techniques similar to those employed fordirectional couplers can also be used [35]. Work on coupled line filters can alsobe found in the literature [2, 36–38].

In microstrip filters, temperature variation of er , changes in er from lot to lot,and substrate thickness variations usually mean that the bandwidth has to be widerthan desired to accommodate manufacturing tolerances.

1.3 Applications 11

Figure 1.15 Typical coupled-line filter configurations: (a) parallel coupled, (b) interdigital, (c) comb-line, and (d) hairpin-line.

1.3 Applications

In the past decade, five major areas in the development of coupled-line componentshave been emphasized: (1) development of CAD tools, (2) full-wave analysis andaccurate semiempirical expressions to enhance component designs, (3) the searchfor new structures and configurations, (4) advanced materials and technologies,and (5) the search for new applications. Broader bandwidth, ease of fabricationand integration, compact size, and lower cost have been the driving factors. Forexample, in wireless applications, compact size and lower cost requirements trig-gered investigation of new configurations and the transformation of existing struc-tures into new layouts such as meander line and spiral geometry to realize compactcomponents.

12 Introduction

Other applications of coupled-line sections are in baluns, impedance transform-ers, dc blocks, interdigital capacitors, and spiral inductors. The spiral inductor isthe most popular and is used extensively in hybrid and monolithic microwaveintegrated circuits (MICs). In particular, the compact size and low-cost circuitsused for wireless applications in the L- and S-bands are based on inductors asmatching elements.

Over the past two decades, because of the rapidly growing use of MICs inradar, satellite and mobile communications, electronic warfare (EW) and missiles,couplers and filter technologies have undergone a substantial change in terms ofbandwidth, size, and cost. For example, in wirelsss applications, a 90-degree hybrid/coupler (whose output ports have signals of equal magnitude but with 90-degreephase difference) is needed to determine the phase error of the transmitter usingthe p /4 quadrature phase-shift keying (QPSK) modulation scheme common todigital cellular radio systems. Basic requirements for this coupler are small size,low cost, and tight amplitude balance and quadrature phase between the outputports. This was met by the coupled microstrip line couplers using the meander-line approach [39–41] and spiral configuration [41] on high-dielectric constantsubstrate compatible with MICs, and meander configuration [39] on a GaAs sub-strate compatible with monolithic microwave integrated circuits (MMICs). Thesecouplers have the potential to meet the $2 to $5 price goals when housed in plasticpackages and produced in large quantities.

Satellite, airborne communications, and EW systems require small size, light-weight, low-cost filters. Coupled-microstrip and stripline filters are very suitablefor wideband applications where the demand on selectivity is not severe. Variouskinds of filters, shown in Figure 1.15, can be realized using microstrip-type struc-tures. For wireless applications, however, miniature versions of these filters arerequired because of space and cost constraints. Hairpin-line and combline filtersusing resonators on high-dielectric constant (er = 80 and 90) substrate or embeddedin dielectric cavities have been developed and can be designed using traditionalmethods and/or EM simulators. Each filter shown in Figure 1.15 has several otherversions to make them compact, either by folding or modifying the layout to fitinto a small size. Other applications of filters include dual band communicationsuch as a wireless local area network (WLAN) [42] and ultra-wideband (UBW)communication and imaging [43].

Advanced materials/technologies such as high temperature superconductor(HTS) substrates, micromachining, multilayer monolithic, low-temperature cofiredceramic (LTCC), and liquid crystal polymer (LCP) are commonly used in thedevelopment of advanced coupled-line components [44, 45]. Coupler and filtertechnologies are keeping up with the emerging applications. For example, tightcouplers are designed using multilayer fabrication processes and electromagneticmetamaterials, and UWB filters are being developed with the help of advancedCAD tools and new modified coupled-lines such as defected ground structures(DGSs). Various types of coupled lines, including unevenly coupled [46], periodi-cally nonuniform [47], meandered [48], cross-coupled [49], stepped-impedanceresonators [50], and DGS configurations [51], have been studied to suppress theharmonic spurious passbands in advanced coupled-line filters. Passive components

1.4 Scope of the Book 13

compatible with CMOS technologies are being developed up to millimeter wavefrequencies.

1.4 Scope of the Book

Microwave components based on coupled-line structures have been in use for overhalf a century. This text deals exclusively with these components. The purpose ofthis book is twofold; first, to help the reader understand the theory and workingof coupled-line components, and second, to provide in-depth design informationto supplement commercial CAD tools in the design of microwave integrated circuits.As far as possible, enough information has been included to permit the designof passive components for wireless applications covering radio frequency (RF),microwave, and millimeter wave frequencies.

References

[1] Cohn, S. B., and R. Levy, ‘‘History of Microwave Passive Components With ParticularAttention to Directional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-32,September 1984, pp. 1046–1054.

[2] Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, Impedance-MatchingNetworks and Coupling Structures, New York: McGraw-Hill, 1964 (reprinted by ArtechHouse, Dedham, MA, 1980).

[3] Levy, R., ‘‘Directional Couplers,’’ in Advances in Microwaves, Vol. 1, New York: Aca-demic Press, 1966, pp. 184–191.

[4] Howe, H., Stripline Circuit Design, Dedham, MA: Artech House, 1974.[5] Malherbe, J. A. G., Microwave Transmission Line Couplers, Norwood, MA: Artech

House, 1988.[6] Shelton, J. P., Jr., ‘‘Impedances of Offset Parallel-Coupled Strip Transmission Lines,’’

IEEE Trans. Microwave Theory Tech., Vol. MTT-14, pp. 7–15, January 1966, andcorrection IEEE Trans. Microwave Theory Tech., Vol. MTT-14, May 1966, p. 249.

[7] Shelton, J. P., and J. A. Mosko, ‘‘Synthesis and Design of Wideband Equal-Ripple TEMDirectional Couplers and Fixed Phase Shifters,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-14, October 1966, pp. 462–473.

[8] Tresselt, C. P., ‘‘The Design and Construction of Broadband, High Directivity, 90-DegreeCouplers, Using Nonuniform Line Techniques,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-14, December 1966, pp. 647–657.

[9] Kammler, D. W., ‘‘The Design of Discrete N-Section and Continuously Tapered Symmetri-cal Microwave TEM Directional Couplers,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-17, August 1969, pp. 577–590.

[10] Podell, A., ‘‘A High Directivity Microstrip Coupler Technique,’’ IEEE MTT-S Int. Micro-wave Symp. Dig., 1970, pp. 33–36.

[11] Sheleg, B., and B. E. Spielman, ‘‘Broadband Directional Couplers Using Microstrip withDielectric Overlays,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-22, 1974,pp. 1216–1220.

[12] Paolino, D. D., ‘‘MIC Overlay Coupler Design Using Spectral Domain Techniques,’’ IEEETrans. Microwave Theory Tech., Vol. MTT-26, 1978, pp. 646–649.

[13] Kajfez, D., ‘‘Raise Coupled Directivity with Lumped Components,’’ Microwaves,Vol. 17, No. 3, March 1978, pp. 64–70.

14 Introduction

[14] March, S. L., ‘‘Phase Velocity Compensation in Parallel-Coupled Microstrip,’’ IEEEMTT-S Int. Microwave Symp. Digest, 1982, pp. 410–412.

[15] Davis, W. A., Microwave Semiconductor Circuit Design, New York: Van Nostrand, 1983.[16] Horno, M., and F. Medina, ‘‘Multilayer Planar Structures for High-Directivity Directional

Coupler Design,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-34, December 1986,pp. 1442–1449.

[17] Dydyk, M., ‘‘Accurate Design of Microstrip Directional Couplers with CapacitiveCompensation,’’ IEEE MTT-S Int. Microwave Symposium, digest of papers, 1990,pp. 581–584.

[18] Uysal, S., Nonuniform Line Microstrip Directional Couplers and Filters, Norwood, MA:Artech House, 1993.

[19] Gupta, K. C., et al., Microstrip Lines and Slot Lines, 2nd ed., Norwood, MA: ArtechHouse, 1996.

[20] Lange, J., ‘‘Interdigitated Stripline Quadrature Hybrid,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-17, December 1969, pp. 1150–1151.

[21] Waugh, R., and D. LaCombe, ‘‘Unfolding the Lange Coupler,’’ IEEE Trans. MicrowaveTheory Tech., Vol. MTT-20, November 1972, pp. 777–779.

[22] Ou, W. P., ‘‘Design Equations for an Interdigitated Directional Coupler,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-23, February 1973, pp. 253–255.

[23] Paolino, D., ‘‘Design More Accurate Interdigitated Couplers,’’ Microwaves, Vol. 15,May 1976, pp. 34–38.

[24] Presser, A., ‘‘Interdigited Microstrip Coupler Design,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-26, October 1978, pp. 801–805.

[25] Bhartia P., and I. J. Bahl, Millimeter Wave Engineering and Applications, New York:Wiley, Ch. 7, 1984.

[26] Levy, R., and S. B. Cohn, ‘‘A History of Microwave Filter Research, Design and Develop-ment,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-32, September 1984,pp. 1055–1067.

[27] Cohn, S. B., ‘‘Parallel-Coupled Transmission-Line Resonator Filters,’’ IRE Trans. Micro-wave Theory Tech., Vol. MTT-6, April 1958, pp. 223–231.

[28] Ozaki, H., and J. Ishii, ‘‘Synthesis of a Class of Stripline Filters,’’ IRE Trans. CircuitTheory, Vol. CT-5, June 1958, pp. 104–109.

[29] Matthaei, G. L., ‘‘Interdigital Bandpass Filters,’’ IRE Trans. Microwave Theory Tech.,Vol. MTT-10, November 1962, pp. 479–491.

[30] Wenzel, R. J., ‘‘Exact Theory of Interdigital Bandpass Filters and Related Coupled Struc-tures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, September 1965,pp. 559–575.

[31] Matthaei, G. L., ‘‘Comb-Line Bandpass Filters of Narrow or Moderate Bandwidth,’’Microwave J., Vol. 6, August 1963, pp. 82–91.

[32] Wenzel, R. J., ‘‘Synthesis of Comb-Line and Capacitively Loaded Interdigital BandpassFilters of Arbitrary Bandwidth,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-19,August 1971, pp. 678–686.

[33] Cristal, E. G., and S. Frankel, ‘‘Hairpin Line/Half-Wave Parallel-Coupled-Line Filters,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-20, November 1972, pp. 719–728.

[34] Gysel, U. H., ‘‘New Theory and Design for Hairpin-Line Filters,’’ IEEE Trans. MicrowaveTheory Tech., Vol. MTT-22, May 1974, pp. 523–531.

[35] Bahl, I. J., ‘‘Capacitively Compensated High-Performance Parallel Coupled MicrostripFilters,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1989, pp. 679–682.

[36] Malherbe, J. A. G., Microwave Transmission Line Filters, Dedham, MA: Artech House,1979.

[37] Bahl, I. J., and P. Bhartia, Microwave Solid-State Circuit Design, New York: Wiley, 1988,Ch. 6.

1.4 Scope of the Book 15

[38] Sheinwald, J., ‘‘MMIC Compatible Bandpass Filter Design: A Survey of Applicable Tech-niques,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1989, pp. 679–682.

[39] Arai, S., et al., ‘‘A 900-MHz Degree Hybrid for QPSK Modulator,’’ IEEE MTT-S Int.Microwave Symp. Dig., 1991, pp. 857–860.

[40] Tanaka, H., et al., ‘‘2-GHz One Octave-Band 90 Degree Hybrid Coupler Using CoupledMeandered Line Optimized by 3-D FEM,’’ IEEE MTT-S Int. Microwave Symp. Digest,1994, pp. 906–906.

[41] Tanaka, H., et al., ‘‘Miniaturized 90- Degree Hybrid Coupler Using High DielectricSubstrate for QPSK Modulator,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1996,pp. 793–796.

[42] Tsai, L. C., and C. W. Hsue, ‘‘Dual-Band Bandpass Filters Using Equal Length Coupled-Serial-Shunted Lines and Z-Transform Technique,’’ IEEE Trans. Microwave Theory Tech.,Vol. 52, April 2004, pp. 1111–1117.

[43] Mini-Special Issue on Ultra-Wideband, IEEE Trans. Microwave Theory Tech., Vol. 52,September 2004.

[44] Lancaster, M. J., Passive Microwave Device Applications of High-Temperature Supercon-ductors, Cambridge, U.K.: Cambridge University Press, 1997.

[45] Hong, J.-S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, NewYork: John Wiley & Sons, 2001.

[46] Jiang, M., M. H. Wu, and J. T. Kuo, ‘‘Parallel-Coupled Microstrip Filters with Over-Coupled Stages for Multispurious Suppression,’’ IEEE MTT-S Int. Microwave Symp.Dig., 2005, pp. 687–690.

[47] Lopetegi, T., et al., ‘‘New Microstrip ‘Wiggly-Line’ Filters with Spurious PassbandSuppression,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-49, September 2001,pp. 1593–1598.

[48] Wang, S. M., et al., ‘‘Miniaturized Spurious Suppression Microstrip Filter Using Mean-dered Parallel Coupled Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-53,2005, pp. 747–753.

[49] Hong, J.-S., and M. J. Lancaster, ‘‘Couplings of Microstrip Square Open-Loop Resonatorsfor Cross-Coupled Planar Microwave Filters,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-44, November 1996, pp. 2099–2109.

[50] Pang, H.-K., et al., ‘‘A Compact Microstrip l /4-SIR Interdigital Bandpass Filter withExtended Stopband,’’ IEEE MTT-S Int. Microwave Symp. Dig., 2004, pp. 1621–1625.

[51] Velazquez-Ahumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filterswith Floating Ground-Plane Conductor for Spurious-Band Suppression,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-53, May 2005, pp. 1823–1828.

C H A P T E R 2

Microwave Network Theory

Microwave coupled lines and components can be classified as N-port networkssuch as two-port, three-port, four-port, and so on. If the input-output parametersof an N-port network are known, its behavior under various conditions of excitationand termination can be determined. For example, if two ports of a four-port networkare terminated in open circuit, the input-output parameters of the remaining two-port network can be determined from a knowledge of the parameters of the originalfour-port network. Further, in a microwave system or subsystem, many individualcomponents are cascaded together and the input-output parameters of the cascadednetwork can be determined if those of the individual networks are known.

The input-output relationship of a linear microwave network can be describedin many equivalent ways [1]. In this chapter, we discuss how an N-port networkcan be characterized by its impedance, admittance, or scattering matrix. Althoughany form of matrix can be used to describe a network, one form may be moresuitable then another. In general, the scattering matrix representation has been themost popular way of describing the input-output relationship of a microwavenetwork. Because a network is usually constructed to have specific reflection andtransmission properties, one can directly express the desired response in terms ofa scattering matrix. These quantities can also be easily measured using vectornetwork analyzers. We discuss scattering matrices in more detail in this chaptertogether with the conditions imposed by the losslessness and reciprocity on thevarious representative matrices of a passive network. Some special properties oftwo-, three-, and four-port networks are described. The ABCD representation oftwo-port networks is then discussed. This representation is very useful when anumber of two-port networks are cascaded to form a single two-port network.The relationship among various forms of matrices are also given.

If P ports of an N-port network are connected to P ports of another M-portnetwork, a network with M + N − 2P ports results. Given the scattering matricesof individual networks, the scattering matrix of the overall network can be deter-mined. We give these relationships and apply them to some specific cases to findthe modified scattering parameters of two-, three-, and four-port networks whoseports are not match-terminated.

2.1 Actual and Equivalent Voltages and Currents

For low-frequency networks, one can define (and measure) unique voltages andcurrents 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. Examplesof transmission lines supporting the TEM mode of propagation are a coaxial line,stripline, microstrip line,1 and so forth. Many other commonly used transmissionlines such as hollow waveguides, dielectric guides, and fin lines do not support theTEM mode of propagation. Therefore, one resorts to the concept of equivalentvoltages and currents, and this can be applied to both TEM- and non-TEM-modetransmission lines. Relationships involving equivalent voltages or currents lead tounique physical quantities such as reflection and transmission coefficients, normal-ized input impedance, and the like. Equivalent voltages and currents can be definedon a normalized or unnormalized basis. Because the representative matrix of anetwork may define a relationship between normalized or between unnormalizedquantites, it is essential to understand their meaning.

2.1.1 Normalized Voltages and Currents

Figure 2.1(a) shows a two-port network. The power flows into and out of thenetwork by means of transmission lines connected to the network.2 Each transmis-sion line may carry a wave propagating toward the network defined as the incidentwave or away from the network defined as the reflected wave. If power is incidentin the transmission line connected to port 1, the mode in which the power flowsis a characteristic of the type of transmission line. Associated with a mode areunique electromagnetic fields. The transverse components of electric and magneticfields (transverse to the direction of propagation) have a unique phase associatedwith them, which is the same for both fields. Further, the z-variation of the incidentelectromagnetic wave (assuming that the power flow is in the positive z-direction)can be described by a simple factor e −jb 1z, where b1 is a unique quantity anddenotes the phase constant of the wave in the transmission line of port 1.

To determine the normalized voltage and current waves, we assume that theincident voltage and current waves have the same phase as that of the transverseelectric and magnetic field components of the incident electromagnetic wave.3

Further, the z-variation of voltage and current waves is also given by the samefactor as that for the field components (e −jb 1z ). Mathematically, the normalizedincident voltage and current waves in the transmission line of port 1 can then beexpressed as

V +1 (z) = V +

10e −jb 1z = |V +10 |e jci1e −jb 1z (2.1)

and

I +1 (z) = I +

10e −jb 1z = | I +10 |e jci1e −jb 1z (2.2)

1. Microstrip line supports quasi-TEM mode.2. 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.3. The phase of an electromagnetic wave is unique and can be measured even if the wave is of a non-TEM

type.

2.1 Actual and Equivalent Voltages and Currents 19

Figure 2.1 (a) Normalized and unnormalized voltage and current waves on transmission lines ofa two-port network. (b) A two-port network connected to a source and load.

where c i1 denotes the phase of the incident wave at z = 0, and V +10 and I +

10 arethe complex voltage and current, respectively, at the same terminal plane (z = 0).To reemphasize, the value of c i1 is the same as that of the transverse componentsof the electric and magnetic fields of the incident wave. The symbol ‘‘^’’ has beenadded to denote that the respective quantities are normalized.

When the characteristic impedance of a transmission line is real, the voltageand current waves can be expressed in terms of the incident and reflected power.4

At microwave frequencies, the characteristic impedances of practical transmissionlines are generally real. To compute the values of |V +

10 | and | I +10 | , we force the

condition that the average power flow is given by

|V +10 | | I +

10 | = P +1 (2.3)

where P +1 denotes the incident power, and |V +

10 | and | I +10 | denote the rms quantities.

4. This treatment is, however, not valid if the characteristic impedance is complex [2, 3].


To determine |V +10 | and | I +

10 | , we need to have another relation between them.To define normalized quantities, we choose

|V +10 |

| I +10 |

= 1 (2.4)

and hence from (2.3) and (2.4):

|V +10 | = | I +

10 | = √P +1 (2.5)

Substituting the values of |V +10 | and | I +

10 | from (2.5) in (2.1) and (2.2), weobtain

V +1 (z) = I +

1 (z) = √P +1 e jci1e −jb 1z (2.6)

Equation (2.6) is in a form that aids in understanding the physical meaning ofnormalized voltage and current waves.

When the incident power reaches the network, a part of it (say, P −1 ) is reflected

back. By analogy with incident voltage and current waves, the reflected waves canbe expressed as

V −1 (z) = I −

1 (z) = √P −1 e jcr1e jb 1z (2.7)

where the superscript ‘‘−’’ is used to denote the reflected waves. cr1 is the phaseof the transverse components of the electric and magnetic fields of the reflectedwave at z = 0 and is a unique quantity. Because the reflected wave propagates inthe negative z-direction, its z-dependence is described by the factor e jb 1z.

The total normalized voltage (at any value of z) in the transmission line ofport 1 is thus given by,

V1(z) = V +1 (z) + V −

1 (z)

= |V +10 |e jci1e −jb 1z + |V −

10 |e jcr1e jb 1z (2.8)

= √P +1 e jci1e −jb 1z + √P −

1 e jcr1e jb 1z

On the other hand, the total current at any value of z is given by

I1(z) = I +1 (z) − I −

1 (z)

= V +1 (z) − V −

1 (z) (2.9)

= | I +10 |e jci1e −jb 1z − | I −

10 |e jcr1e jb 1z

= √P +1 e jci1e −jb 1z − √P −

1 e jcr1e jb 1z


Because the current flows in the axial direction, the net current is given by thedifference of the currents flowing in the positive and negative z-directions. We nowshow that the net power flow (into the network) across any z = constant plane inthe transmission line of port 1 is given by the usual low-frequency relation, thatis, 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 V1 I1* can be expressed as

V1 I1* = XV +1 + V −

1 C XV +1 − V −

1 C* (2.10)

After substituting for V +1 and V −

1 from (2.6) and (2.7) and noting that the conjugateof a complex number (A + jB) is equal to (A − jB), we obtain

V1 I1* = |V +1 |2 − |V −

1 |2 + imaginary term

= |V +10 |2 − |V −

10 |2 + imaginary term (2.11)

= P +1 − P −

1 + imaginary term

or

Re XV1 I1* C = P +1 − P −

1 (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 reflectedwave) is chosen to be different from unity (2.4), the resulting quantities are calledunnormalized. For TEM transmission lines, this ratio is generally chosen to beequal to the actual characteristic impedance of the line. In that case, the unnormal-ized voltages and currents reduce to actual quantities on the line. For non-TEMtransmission lines, the characteristic impedance depends on the definition used.Referring to Figure 2.1(a), the unnormalized incident voltage and current wavesin the transmission line of port 1 can be expressed as

V +1 (z) = √Z01P +

1 e jci1e −jb 1z (2.13)

I +1 (z) = √ P +

1Z01

e jci1e −jb 1z (2.14)

where Z01 denotes the characteristic impedance of the transmission line of port 1.In the above equations, the symbol ‘‘^’’ has been dropped to denote unnormalizedquantities.

The unnormalized reflected voltage and current waves in the transmission lineof port 1 can be expressed as


V −1 (z) = √Z01P −

1 e jcr1e jb 1z (2.15)

I −1 (z) = √ P −

1Z01

e jcr1e jb 1z (2.16)

Example 2.1

Consider the circuit shown in Figure 2.1(a). Assume that unit power is incident inthe transmission line of port 1, of which a quarter is reflected back from thenetwork and that the phase of the incident and reflected waves at z = 0 is equalto 0 and p /6 radians, respectively. From (2.6) and (2.7), the normalized voltageand current waves in the input transmission line are

V +1 (z) = I +

1 (z) = √P +1 e jci1e −jb 1z = e −jb 1z (2.17)

V −1 (z) = I −

1 (z) = √P −1 e jcr1e jb 1z = 0.5e jp /6e jb 1z (2.18)

To determine the unnormalized voltage and current waves in the transmissionline, we need to specify a value of the characteristic impedance. Let this value be50 ohms. Using (2.13) and (2.14), the incident unnormalized voltage and currentwaves are then

V +1 (z) = √Z01P +

1 e jci1e −jb 1z = √50e −jb 1z (2.19)

I +1 (z) = √ P +

1Z01

e jci1e −jb 1z =1

√50e −jb 1z (2.20)

Using (2.13) and (2.14), the unnormazlied reflected voltage and current wavesin the transmission line can be expressed as

V −1 (z) = √Z01P −

1 e jcr1e jb 1z = √12.5e jp /6e jb 1z (2.21)

I −1 (z) = √ P −

1Z01

e jcr1e jb 1z =0.5

√50e jp /6e jb 1z (2.22)

2.1.3 Reflection Coefficient, VSWR, and Input Impedance

Referring to Figure 2.1(b), the voltage reflection coefficient G1 in the transmissionline of port 1 is defined as

G1(z) =V −

1 (z)

V +1 (z)

=V −

1 (z)

V +1 (z)

(2.23)


Substituting values of V −1 (z) and V +

1 (z) from (2.6) and (2.7) in (2.23), we have

G1(z) = |V −10 |e jc 1r + jb 1z

|V +10 |e jc 1i − jb 1z (2.24)

= √P −1 /P +

1 e j(c 1r − c 1i )e2jb 1z

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 isreflected back. The ratio of the reflected-to-incident power is commonly referredto as return loss. The return loss (in decibels), which is a positive quantity, is givenby

RL (dB) = −10 log |G1(z) |2 = −20 log |G1(z) | (2.25)

The voltage standing wave ratio (VSWR) in the transmission line of port 1 isgiven by

VSWR =1 + |G1(z) |1 − |G1(z) | (2.26)

More often, it is the practice to use the ratio of total voltage and current, whichcan be termed as input impedance. The ratio of total normalized voltage to currentis defined as the normalized input impedance and is given by

Zin (z) =V1(z)

I1(z)=

V +1 (z) + V −

1 (z)

V +1 (z) − V −

1 (z)=

1 + G1(z)1 − G1(z)

(2.27)

Because G1(z) is a unique quantity, the normalized input impedance is also a uniquequantity.

Similarly, the unnormalized input impedance Zin is given by

Zin (z) = Z011 + G1(z)1 − G1(z)

(2.28)

where Z01 denotes the characteristic impedance of the transmission line connectedto port 1. Using (2.27), the reflection coefficient in terms of normalized inputimpedance is expressed as

G1(z) =Zin (z) − 1

Zin (z) + 1(2.29)

and in terms of the unnormalized input impedance:


G1(z) =

Zin (z)Z01

− 1

Zin (z)Z01

+ 1=

Zin (z) − Z01Zin (z) + Z01

(2.30)

Transformation of Impedance

With Zin (t1) as the input impedance at the terminal plane t1 looking into thenetwork, the iput impedance at the terminal plane t1′ (which is closer to the generatorcompared with terminal plane t1) is

Zin (t1′ ) = Z01Zin (t1) + jZ01 tan b1 lZ01 + jZin (t1) tan b1 l

(2.31)

where l is the distance between terminal planes t1 and t1′ . If Zin (t1) denotes theload impedance ZL , the input impedance Zin at distance l away (toward thegenerator) can be expressed as

Zin = Z01ZL + jZ01 tan b1 lZ01 + jZL tan b1 l

(2.32)

where b1 = 2p /lg denotes the propagation constant of the line, which is assumedto be lossless.

2.1.4 Quantities Required to Describe the State of a Transmission Line

Consider an N-port network as shown in Figure 2.2. The ports are numbered fromm = 1 to m = N. The power is carried into and away from the network by meansof transmission lines connected to each port. The characteristic impedance of thetransmission line of the mth port is denoted by Z0m . Because the voltages andcurrents vary along the length of the transmission line, fictitious terminal planesare located in each transmission line. Voltage or current at port m denotes therespective quantity at the specified terminal plane in the transmission line of portm. We use the notation of the sections above for normalized and unnormalizedquantities and incident and reflected quantities.

Note that V +n and I +

n (similarly, V −n and I −

n ) are not independent quantities.The normalized quantities satisfy the following relationship:

V +n

I +n

=V −

n

I −n

= 1 (2.33)

If the two quantities V +n (or I +

n ) and V −n (or I −

n ) are known, the total voltageand current can be determined using

Vn = V +n + V −

n = I +n + I −

n (2.34)

and


Figure 2.2 An N-port network.

In = I +n − I −

n = V +n − V −

n (2.35)

Therefore, if any two of the four quantities Vn , In , V +n (or I +

n ) and V −n (or I −

n ) areknown, all others can be determined. The same conclusion holds for unnormalizedquantities if the characteristic impedances of all the transmission lines are known.For unnormalized quantities:

V +n

I +n

=V −

n

I −n

= Z0n (2.36)

Vn = V +n + V −

n = Z0n XI +n + I −

n C (2.37)

In = I +n − I −

n =V +

n − V −n

Z0n(2.38)

Relationship Between Normalized and Unnormalized Quantities

The normalized and unnormalized quantities are related by

V ±n =

V ±n

√Z0n(2.39a)

Vn =Vn

√Z0n(2.39b)


I ±n = √Z0n I ±

n (2.39c)

In = √Z0n In (2.39d)

2.2 Impedance and Admittance Matrix Representation of a Network

2.2.1 Impedance Matrix

Consider the N-port network shown in Figure 2.2. In the impedance matrix repre-sentation, the voltage at each port is related to the currents at the different portsas follows:

V1 = Z11I1 + Z12I2 + . . . + Z1NIN

V2 = Z21I1 + Z22I2 + . . . + Z2NIN (2.40)

A A A A A A A A A

VN = ZN1I1 + ZN2I2 + . . . + ZNNIN

In matrix notation, this set of equations can be expressed as

[V] = [Z][I] (2.41)

where

[V] = 3V1

AVN4 (2.42)

[I] = 3I1

AIN4 (2.43)

and

[Z] = 3Z11 Z12 . . . Z1N

Z21 Z22 . . . Z2N

A A A AZN1 ZN2 . . . ZNN

4 (2.44)

The impedance matrix [Z] is unnormalized because it relates unnormalizedvoltages and currents. The impedance matrix relating normalized voltages andcurrents is called normalized, and will be denoted as [Z] with the ‘‘^’’ symbol.The normalized impedance matrix is denoted as

2.2 Impedance and Admittance Matrix Representation of a Network 27

[Z] = 3Z11 Z12 . . . Z1N

Z21 Z22 . . . Z2N

A A A AZN1 ZN2 . . . ZNN

4 (2.45)

2.2.2 Admittance Matrix

In the admittance matrix representation, the current at each port of the networkas 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] = 3Y11 Y12 . . . Y1N

Y21 Y22 . . . Y2N

A A A AYN1 YN2 . . . YNN

4 (2.47)

2.2.3 Properties of Impedance and Admittance Parameters of a PassiveNetwork

For a network not containing any nonreciprocal media (ferrite, plasma, and soforth),

Zmn = Znm (2.48)

and

Ymn = Ynm (2.49)

Similar relationships are also satisfied by the elements of unnormalized impedanceand admittance matrices, that is:

Zmn = Znm (2.50)

and

Ymn = Ynm (2.51)

Note that networks containing dielectrics and conductors are reciprocal and satisfythe above-mentioned properties.

For a lossless network, all the elements of an impedance or admittance matrixare imaginary. This is an expected result because any resistive element would implyloss.


The mnth element of the unnormalized impedance matrix is related to thecorresponding element of the normalized impedance matrix by

Zmn = Zmn√Z0mZ0n (2.52)

where Z0m and Z0n denote the characteristic impedances of transmission lines ofports m and n, respectively. Similarly, the mnth element of the unnormalizedadmittance matrix is related to the corresponding element of the normalized admit-tance matrix by

Ymn = Ymn√Y0mY0n (2.53)

where Y0m = 1/Z0m and Y0n = 1/Z0n denote the characteristic admittance of thetransmission lines of ports m and n, respectively.

2.3 Scattering Matrix

A very popular method of representing microwave networks is by the scatteringmatrix. The scattering matrix is generally represented in a normalized form. In thisrepresentation, the normalized reflected voltage at each port of the network asshown in Figure 2.2 is related to the normalized incident voltages at the ports ofthe network as follows:

V −1 = S11V +

1 + S12V +2 + . . . + S1NV +

N

V −2 = S21V +

1 + S22V +2 + . . . + S2NV +

N (2.54)

A A A A A A A A A

V −N = SN1V +

1 + SN2V +2 + . . . + SNNV +

N

In matrix notation, the above set of equations can be expressed as

[V− ] = [S][V+ ] (2.55)

where

[V− ] = 3V −

1A

V −N4 (2.56a)

[V+ ] = 3V +

1A

V +N4 (2.56b)

2.3 Scattering Matrix 29

and

[S] = 3S11 S12 . . . S1N

S21 S22 . . . S2N

A A A ASN1 SN2 . . . SNN

4 (2.57)

The scattering parameter Smn is therefore given by

Smn =V −

m

V +n

| V +p = 0 where p = 1, . . . , N ; p ≠ n

(2.58)

In terms of the incident power P +n in the nth transmission line, the amplitude

of the normalized incident voltage wave at the nth port is given by

|V +n | = √P +

n (2.59)

Similarly, the amplitude of the normalized reflected voltage wave5 at the mth portis given by

|V −m | = √P −

m (2.60)

where P −n denotes the reflected power at port m.

When the values of |V +n | and |V −

m | from the last two equations are substitutedin (2.58), we obtain

| Smn | = |V −m |

|V +n |

= √P −m

P +n

(2.61)

From the above equation, we see that | Smn |2 denotes the ratio of power coupledfrom port n to port m when V +

p = 0, where p = 1, . . . , N; p ≠ n. The conditionV +

p = 0, where p = 1, . . . , N; p ≠ n is readily esured by exciting only the nth portand terminating all the ports in matched loads. Similarly:

| Snn |2 = |Gn |2 =|V −

n |2

|V +n |2

=P −

n

P +n

(2.62)

where Gn denotes the reflection coefficient at port n. Further, | Snn |2 denotes thefraction of the incident power that is reflected back at port n.

The ports of a typical microwave network are usually match-terminated. There-fore, 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 anywave traveling away from the network is called the ‘‘reflected’’ wave.


power coupled to the other ports of the network can be easily determined if thenormalized 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:

3S*11 S*21 . . . S*N1

S*12 S*22 . . . S*N2

A A A AS*1N S*2N . . . S *NN

4 3S11 S12 . . . S1N

S21 S22 . . . S2N


4 = 31 0 . . . 00 1 . . . 0A A A A0 0 . . . 1

4(2.63)

where the symbol * denotes the complex conjugate. Because the [S] matrix satisfiesthe 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 nonreciprocallossless network is unitary.

In a compact form, (2.63) can be expressed as

[S*]t [S] = U (2.64)

where [S*] denotes the matrix formed by the conjugate of the elements of the [S]matrix and [S*]t denotes the transpose of matrix [S*]. U is a unit matrix of orderN. All the diagonal elements of a unit matrix are 1, while all its nondiagonalelements are 0.

From (2.63), it is seen that if the nth row of the [S*]t matrix is multiplied withthe nth column of the [S] matrix, the following equation results:

S1n S*1n + S2n S*2n + . . . + SNn S*Nn = 1

or

| S1n |2 + | S2n |2 + . . . + | SNn |2 = 1 (2.65)

where n = 1, 2, . . . , N.On the other hand, if the nth row of the [S*]t matrix is multiplied with the

mth column of the [S] matrix with m ≠ n, then the following equation results:

S*1n S1m + S*2n S2m + . . . + S*NnSNm = 0 (2.66)

where m = 1, . . . , N; n = 1, . . . , N; and m ≠ n . The unitary properties of thescattering matrix as given by (2.65) and (2.66) lead to very useful predictions aboutthe properties of a lossless network. For example, the unitary property of a scatteringmatrix leads to the result that it is impossible to match a lossless, three-port

2.3 Scattering Matrix 31

reciprocal network at all its ports simultaneously. We state some special propertiesof two-, three-, and four-port lossless networks derived using the unitary propertyof 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 terminalplanes denoted by tp , p = 1, . . . N as shown in Figure 2.3 is given by (2.57). Ifthe terminal plane in each transmission line is moved to new locations denotedby t ′p where p = 1, . . . N, we define the scattering matrix with the location ofports denoted by t ′p as [S ′], which is related to the scattering matrix [S] as

[S ′] = 3e −jb 1 l1 0 . . . 0

0 e −jb 2 l2 . . . 0A A A A

0 0 . . . e −jbNlN4 3

S11 S12 . . . S1N

S21 S22 . . . S2N


4 (2.67)

× 3e −jb 1 l1 0 . . . 0

0 e −jb 2 l2 . . . 0A A A A

0 0 . . . e −jbNlN4

where bp (p = 1, . . . , N) denotes the phase constant of the wave in the transmissionline of the pth port. From (2.67), it follows that the mnth element of the modified

Figure 2.3 An N-port network. The elements of a representative matrix change when the locationof ports (terminal planes) is changed.


scattering matrix is related to the corresponding element of the original scatteringmatrix by

S ′mn = Smne−jbmlm − jbn ln

(2.68)

2.3.3 Reciprocal Networks

If a network does not contain any nonreciprocal media (e.g., ferrite, plasma), thenthe following relation holds for the elements of its normalized scattering matrix:

Smn = Snm (2.69)

In matrix notation, the condition of reciprocity is stated as

[S] = [S]t (2.70)

2.3.4 Relationship Between Normalized and Unnormalized Matrices

The mnth elements of the unnormalized and normalized scattering matrices arerelated by

Smn = Smn√Z0mZ0n

(2.71)

where Z0m and Z0n denote the characteristic impedances of the transmission linesat ports m and n. The above equation leads to an important conclusion that if all theports of a network have the same characteristic impedance, then the unnormalizedscattering matrix of the network is the same as the normalized matrix. In microwavenetworks, all the ports usually have the same characteristic impedance. Therefore,in these cases it is not necessary to specify whether the scattering matrix is normal-ized or unnormalized. However, if the characteristic impedances of all the portsof a network are not the same, such as in the case of asymmetrical coupled lines,impedance-transforming couplers, baluns, and so forth it should be specifiedwhether the scattering matrix is normalized or unnormalized.

2.4 Special Properties of Two-, Three-, and Four-Port Passive,Lossless Networks

All the elements of a network matrix of a passive lossless network cannot be chosenindependently. For example, if a network does not contain any nonreciprocal media(e.g., ferrite, plasma), then Smn = Snm . Therefore, one cannot design a networkcontaining a passive reciprocal medium having different values of Smn and Snm .As another example, if a network has a plane of symmetry, then Smm = Snn wherem and n are symmetrical ports. The preceding properties described are true ingeneral for any N-port network. There are, however, some special properties of a

2.4 Special Properties of Two-, Three-, and Four-Port Passive, Lossless Networks 33

network depending on the number of ports it has. In the following, we discusssome special properties of passive lossless, two-, three-, and four-port networks.

2.4.1 Two-Port Networks

Figure 2.4 shows a two-port passive and lossless network. The normalized scatteringmatrix of a two-port network can be written as

[S] = FS11 S12

S21 S22G = F| S11 |e ju 11 | S12 |e ju 12

| S21 |e ju 21 | S22 |e ju 22G (2.72)

The unitary property of the normalized scattering matrix as given by (2.65) and(2.66) leads to the following relations:

| S11 |2 + | S21 |2 = 1 (2.73a)

| S12 |2 + | S22 |2 = 1 (2.73b)

S*11 S12 + S*21 S22 = 0 (2.73c)

The solution of this set of equations leads to the following conclusions:

| S11 | = | S22 | (2.74)

| S12 | = | S21 | (2.75)

u11 + u22 = u12 + u21 7 p rad (2.76)

The above relations are true for any two-port network—reciprocal or nonrecipro-cal. Equations (2.74) and (2.75), along with (2.73), show that for a lossless network

Figure 2.4 A passive, lossless, two-port network.


(reciprocal or nonreciprocal) one needs to know the amplitude of one scatteringelement to determine the amplitude of all other scattering elements. Equation (2.75)also shows that even if the two-port network is composed of nonreciprocal media,| S12 | = | S21 | . In physical terms, this relation states that the ratio of the powercoupled from port 1 to port 2 when the power is incident at port 1 is the same asthat coupled from port 2 to port 1 when power is incident at port 2. This is aninteresting result; one might then wonder about how a two-port isolator works.An isolator is supposed to have a very small attenuation between its ports whenthe power is incident at one port, and a large attenuation when the power isincident at the other port. The difficulty is resolved by noting that the unitaryproperty is valid only for lossless networks. On the other hand, an isolator employssome kind of lossy elements to achieve different values of | S12 | and | S21 | .

When the network is reciprocal, S12 = S21 , which leads to u12 = u21 , or from(2.76):

u12 = (u11 + u22 ± p )/2 rad

Using the above equation and (2.74) and (2.75), the scattering matrix of a lossless,reciprocal two-port network can be expressed in the form

[S] = F | S11 |e ju 11 √1 − | S11 |2e j(u 11 + u 22 ± p )/2

√1 − | S11 |2e j(u 11 + u 22 ± p )/2 | S11 |e ju 22G (2.77)

Further, noting that | S11 | ≤ 1, (2.77) reduces to

[S] = F cos ae ju 11 sin ae j(u 11 + u 22 ± p )/2

sin ae j(u 11 + u 22 ± p )/2 cos ae ju 22 G (2.78)

where cos a = | S11 | .For a two-port network, any desired values of u11 and u22 can always be

chosen by changing the location of the terminal planes. If u11 and u22 are chosento be zero by appropriately locating the terminal planes, then the scattering matrixof the two-port network becomes

[S] = FS11 S12

S21 S22G = F cos a ± j sin a

± j sin a cos a G (2.79)

2.4.2 Three-Port Reciprocal Networks

For networks having three ports, the unitary property of the scattering matrix leadsto the result that it is impossible to match a passive, lossless reciprocal networkat all its ports simultaneously. Therefore, a three-port network enclosing reciprocalmedia cannot have a scattering matrix of the form


[S] = 30 S12 S13

S12 0 S23

S13 S23 04 (2.80)

The above condition holds only for a lossless reciprocal network, but by incorporat-ing lossy elements in the network (such as in a Wilkinson’s power divider), a three-port network can be matched at all its ports simultaneously.

For a reciprocal three-port network, the unitary property also leads to anotherimportant 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 scatteringmatrix of this network is then given by

[S] = 30 1 01 0 00 0 1

4 (2.81)

where it is assumed that the phase of nonzero scattering elements has been adjustedto zero.

2.4.3 Three-Port Nonreciprocal Networks

The unitary property leads to the result that a passive, lossless three-port nonrecipro-cal 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 thefollowing form:

Figure 2.5 A passive, lossless, nonreciprocal three-port network. The network acts as a circulatorif it is matched at all its ports.


[S] = 30 0 e ju 13

e ju 21 0 0

0 e ju 32 04 (2.82)

or | S13 | = | S21 | = | S32 | = 1. The lossless circulator has thus an important propertythat if power is incident at port 1 then all the power is transmitted to port 2. Ifthe power is incident at port 2, all the power is transmitted to port 3, and if thepower is incident at port 3, all the power is transmitted to port 1. This is shownschematically in Figure 2.5.

2.4.4 Four-Port Reciprocal Networks

Figure 2.6 shows a four-port network. For a passive, lossless reciprocal four-portnetwork, the unitary property of the scattering matrix leads to the result that it ispossible to match all the four ports of the network simultaneously. If all four portsare matched, the network behaves like a directional coupler. The scattering matrixof a directional coupler is of the form

[S] = 30 0 S13 S14

0 0 S23 S24

S13 S23 0 0S14 S24 0 0

4 (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 four-port network. The network behaves like a directionalcoupler if it is matched at all ports.


| S13 | = | S24 | , | S14 | = | S23 |

Using the results of the above two equations, the elements of the scattering matrixcan be expressed as

S13 = C1e ju 13, S23 = C2e ju 23, S14 = C2e ju 14, S24 = C1e ju 24

where C1 = | S13 | = | S24 | and C2 = | S14 | = | S23 | . Thus, (2.83) can be expressed as

[S] = 30 0 C1e ju 13 C2e ju 14

0 0 C2e ju 23 C1e ju 24

C1e ju 13 C2e ju 23 0 0

C2e ju 14 C1e ju 24 0 04 (2.84)

We can easily show that all the phase factors of the various scattering elementsu13 , u23 , u14 , and u24 cannot be chosen independently. Assume that desired valuesof u13 and u14 have been chosen by varying the positions of ports 3 and 4 respec-tively. The phase factor u23 can be independently chosen by controlling the positionof port 2. The remaining phase factor u24 cannot be changed now because bothports 2 and 4 have already been adjusted. From the unitary property of the scatteringmatrix, one obtains the value of u24 in terms of other phase factors as

u24 = u14 + u23 − u13 ± p rad (2.85)

The amplitudes of scattering elements satisfy

C 21 + C 2

2 = 1 (2.86)

The two forms of matrices to which the scattering matrix of a directional couplercan always be reduced by appropriately locating the position of terminal planesare now derived. Let us first choose u13 = u14 = u23 = 0. From (2.85) we obtainu24 = ±p rad, or the scattering matrix of a directional coupler takes the form

[S] = 30 0 C1 C2

0 0 C2 −C1

C1 C2 0 0C2 −C1 0 0

4 (2.87)

For deriving the second form, we choose u13 = 0 and u23 = u14 = ±p /2 rad.From (2.85), we then obtain u24 = 0, or u24 = ±2p rad. The scattering matrix thusreduces to

[S] = 30 0 C1 ±jC2

0 0 ±jC2 C1

C1 ±jC2 0 0±jC2 C1 0 0

4 (2.88)


We see in later chapters that the scattering matrix of many useful four-port micro-wave networks can be represented either by (2.87) or (2.88). For example, a rat-race hybrid can be represented by (2.87), whereas quadrature hybrids, Langecouplers, and so on can be represented by (2.88).

2.5 Special Representation of Two-Port Networks

A typical microwave subsystem consists of a cascade of two-port networks suchthat the output of one network is connected to the input of the next and so on.The two-port networks can be represented by their impedance, admittance, orscattering parameters. It is often more useful, however, to represent two-portnetworks by ABCD parameters because knowing the ABCD parameters, the matrixof the overall cascaded network can be computed by multiplying the matrices ofthe individual networks.

2.5.1 ABCD Parameters

Figure 2.7 shows a two-port network. In the ABCD matrix representation, thevoltage and current flowing into the network at the input of the network are relatedto 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

FV1

I1G = FA B

C DG FV2

I2G (2.90)

Note that in the ABCD matrix representation, the direction of positive currentflow at the output as shown in Figure 2.7 is taken in an opposite sense than whatis done in the impedance, or admittance matrix representation.

To explain the advantage of representation in terms of ABCD parameters,consider the two-port networks cascaded together in Figure 2.8. The ABCD parame-ters of the individual networks are also shown in the same figure. We are interested

Figure 2.7 A two-port network. In the ABCD matrix representation, the direction of positive currentflow at the output is opposite of that used in impedance and admittance matrixrepresentation.

2.5 Special Representation of Two-Port Networks 39

Figure 2.8 Cascade of two two-port networks and their equivalent representation.

in finding the relationship between the input and output of the overall cascadednetwork. The voltages and currents at the input and output of the first networkare related by the following matrix equation:

FV1

I1G = FA1 B1

C1 D1G FV2

I2G (2.91)

For the second network, V2 represents the input voltage and I2 represents the inputcurrent. So we can write

FV2

I2G = FA2 B2

C2 D2G FV3

I3G (2.92)

Substituting V2 and I2 into (2.91):

FV1

I1G = FA1 B1

C1 D1G FA2 B2

C2 D2G FV3

I3G (2.93)

or the ABCD matrix of the overall network between ports 1 and 3 can be expressedas

FAt Bt

Ct DtG = FA1 B1

C1 D1G FA2 B2

C2 D2G (2.94)

Therefore, the ABCD matrix of the overall network is the product of the ABCDmatrices of the individual networks. The same is true for any number of two-portnetworks connected in cascade.

Properties of ABCD Parameters

Consider the two-port network as shown in Figure 2.7. Let Z01 and Z02 denotethe characteristic impedances of ports 1 and 2, respectively. The normalized andunnormalized ABCD parameters are related as follows:


A = A √Z02Z01

B =B

√Z01Z02(2.95)

C = C√Z02Z01

D = D √Z01Z02

For a passive, lossless, two-port reciprocal network:

AD − BC = 1 (2.96)

This relation is also satisfied by normalized ABCD parameters.For a lossless, two-port symmetrical network:

A = D (2.97)

2.5.2 Reflection and Transmission Coefficients in Terms of ABCDParameters

Consider the reciprocal two-port network shown in Figure 2.9. The characteristicimpedances of transmission lines of the input and output ports are assumed to beZ0 and both ports are assumed to be match-terminated. The reflection coefficientat the input port is given by

Gin =A + B /Z0 − CZ0 − DA + B /Z0 + CZ0 + D

(2.98)

Furthermore, the reflection coefficient at output port is given by

Go =−A + B /Z0 − CZ0 + DA + B /Z0 + CZ0 + D

(2.99)

The return loss (in decibels, which is a positive quantity) is given by (2.25). Thetransmission coefficient between input and output ports is given by

Figure 2.9 Reflection and transmission coefficients of a two-port network.

2.5 Special Representation of Two-Port Networks 41

T =V −

2

V +1

=V −

2

V +1

=2

A + B /Z0 + CZ0 + D(2.100)

The insertion loss between input and output (in decibels, which is a positivequantity) is given by

Insertion loss (dB) = −20 log |T | (2.101)

For a lossless network.

|Gin |2 + T2 = |Go |2 + T2 = 1 (2.102)

In the above example, we have assumed that the characteristic impedances of theinput and output ports are the same. In case they are not the same, the unnormalizedscattering parameters of the network can be obtained from the unnormalized ABCDparameters using the conversion relation given in Table 2.1. The unnormalizedscattering parameters can then be normalized using the equations given in Section2.3.4. The elements S11 and S21 of the normalized scattering matrix directly givethe reflection and transmission coefficients.

The unnormalized and normalized ABCD parameters of some elementary net-works are given in Table 2.2. More complex networks can be obtained by cascadinga number of elementary networks. The ABCD matrix of the overall cascadednetwork can then be determined by multiplying the ABCD matrices of elementarynetworks. It is quite simple to determine the ABCD parameters of elementarynetworks. Consider, for example, the circuit shown in Figure 2.10. Inspecting thecircuit, 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:

A = √Z02Z01

, B =Z

√Z01Z02, C = 0, D = √Z01

Z02

2.5.3 Equivalent T and P Networks of Two-Port Circuits

If the impedance parameters of a two-port reciprocal network are known, thenetwork can be represented as shown in Figure 2.11(a). Similarly, if the admittanceparameters of a two-port reciprocal network are known, the circuit can be repre-sented as shown in Figure 2.11(b). T and P forms are only two of many possible


Table 2.1 Conversion Relationships Between Various Representative Matrices of Two-PortNetworks

FZ11 Z12

Z21 Z22G =

1Y11Y22 − Y12Y21

F Y22 −Y12

−Y21 Y11G

FY11 Y12

Y21 Y22G =

1Z11Z22 − Z12Z21

F Z22 −Z12

−Z21 Z11G

FZ11 Z12

Z21 Z22G =

1C FA AD − BC

1 D GFY11 Y12

Y21 Y22G =

1B FD −(AD − BC)

−1 A GFA B

C DG =1

Z21FZ11 (Z11Z22 − Z12Z21)

1 Z22G

FA B

C DG =1

−Y21F Y22 1

Y11Y22 − Y12Y21 Y11G

FS11 S12

S21 S22G =

1

SZ11Z01

+ 1DSZ22Z02

+ 1D −Z12Z21Z01Z02

× 3SZ11Z01

− 1DSZ22Z02

+ 1D −Z12Z21Z01Z02

2Z12Z02

2Z21Z01

SZ11Z01

+ 1DSZ22Z02

− 1D −Z12Z21Z01Z02

4FS11 S12

S21 S22G =

1

S1 +Y11Y01

DS1 +Y22Y02

D −Y12Y21Y01Y02

× 3S1 −Y11Y01

DS1 +Y22Y02

D +Y12Y21Y01Y02

−2Y12Y01

−2Y21Y02

S1 +Y11Y01

DS1 −Y22Y02

D +Y12Y21Y01Y02

4FY11 Y12

Y21 Y22G =

1(1 + S11)(1 + S22) − (S12S21)

× FY01[(1 − S11)(1 + S22) + S12S21] −2Y01S12

−2Y02S21 Y02[(1 + S11)(1 − S22) + S12S21]G

2.6 Conversion Relations 43

Table 2.1 (continued)

FA B

C DG =1

(2S21) 3[(1 + S11)(1 − S22) + S12S21] Z02[(1 + S11)(1 + S22) − S12S21]

1Z01

[(1 − S11)(1 − S22) − S12S21]Z01Z02

[(1 − S11)(1 + S22) + S12S21]4FS11 S12

S21 S22G =

1(B + CZ01Z02) + (AZ02 + DZ02)

× F(B − CZ01Z02) + (AZ02 − DZ01) 2Z01(AD − BC)

2Z02 (B − CZ01Z02) − (AZ02 − DZ01)GFZ11 Z12

Z21 Z22G =

1(1 − S11)(1 − S22) − S12S21

× FZ01[(1 + S11)(1 − S22) + S12S21] 2Z02S12

2Z01S21 Z02[(1 − S11)(1 + S22) + S12S21]GBy substituting Z01 = Z02 = 1, the above relations can be used for the conversion of normalizedparameters.

ways in which the equivalent circuit of a two-port 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 scatteringmatrices are given. The conversion relations between a scattering matrix and imped-ance and admittance matrices are given by assuming that the respective matricesare normalized. To convert unnormalized matrices, the unnormalized parametersshould 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.

[S] = ([Z] − [U]) ([Z] + [U])−1 (2.106)

Another expression for [S] in terms of [Z] is,

[S] = ([Z] + [U])−1([Z] − [U]) (2.107)


Table 2.2 ABCD Parameters of Elementary Two-Port Networks

Figure 2.10 An impedance Z in series between two transmission lines.

2.7 Scattering Matrix of Interconnected Networks 45

Figure 2.11 (a) T and (b) P network representation of a two-port network.

[Z] = ([U] − [S])−1([U] + [S]) (2.108)

[Y] = ([U] + [S])−1([U] − [S]) (2.109)

In some cases, the conversion formulas cannot be used. For example, if thedeterminant of matrix ([U] − [S]) is zero, the impedance matrix becomes indetermi-nant. Similarly, if the determinant of matrix ([U] + [S]) is zero, the admittancematrix becomes indeterminant.

Frequently, it is required to convert one form of matrix into another. Theconversion relations between two-port matrices are given in Table 2.1 [4]. Theserelations are general and valid for nonreciprocal networks also. By substitutingZ01 = Z02 = 1 in these equations, conversion between normalized parameters canbe obtained.

2.7 Scattering Matrix of Interconnected Networks

A typical microwave system or subsystem results after interconnection of manyintermediate networks. Consider two networks as shown in Figure 2.12. NetworksI and II are assumed to have M + P and P + N ports, respectively. The P portsof each network are directly connected to each other. The overall network hastherefore 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 scatter-ing matrix of the overall network can be determined. Let the ports of network Ibe 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 tonetwork II. All the accessible ports are assumed to be terminated in matched loads.The scattering matrix of network I can be expressed as

SI = F[SAA ] [SAB ][SBA ] [SBB ]G (2.110)


Figure 2.12 Interconnection of two multiport networks.

where [SAA ] and [SBB ] are matrices of order M × M and P × P, respectively.[SAB ] and [SBA ] 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 networkII can be represented as

SII = F[SCC ] [SCD ][SDC ] [SDD ]G (2.111)

where [SCC ] and [SDD ] denote the matrices of order P × P and N × N, respectively.[SCD ] and [SDC ] denote matrices of order P × N and N × P, respectively.

The scattering matrix of the overall network of (M + N) ports can be easilyderived using matrix algebra [5, 6]. It follows that the scattering matrix of theoverall network denoted as SR can be expressed in concise notation as

SR = F[S1] [S2][S3] [S4]G (2.112)

where matrices [S1], [S2], [S3], and [S4] are given by

[S1] = [SAA ] + [SAB ] (U − [SCC ][SBB ])−1[SCC ][SBA ]

[S2] = [SAB ] (U − [SCC ][SBB ])−1[SCD ] (2.113)

[S3] = [SDC ] (U − [SBB ][SCC ])−1[SBA ]

[S4] = [SDD ] + [SDC ] (U − [SBB ][SCC ])−1[SBB ][SCD ]

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


unity, while all its nondiagonal elements are zero. The scattering matrix of theoverall network [SR ] 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 portsare terminated in matched loads. The ports of a network are usually match-terminated when connected in a system. The elements of the scattering matrix thusdirectly give the reflection coefficient at each port and coupling between variousports. When one or more ports of a network are not match-terminated, however,reflections take place from these ports, and the scattering parameters of theremaining network are modified. For example, if the ‘‘direct’’ and ‘‘coupled’’ portsof a backward TEM directional coupler are open-circuited, the resulting two-portnetwork behaves as a bandpass filter. Consider a network of M + P ports as shownin Figure 2.13, whose scattering matrix is assumed to be given by (2.110). Theports 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 ofreflection coefficient GL1 , GL2, . . . GLP , respectively. It is required to find thescattering matrix of the reduced network. The scattering matrix of the reducednetwork will be of the order M × M.

We can assume that the given network and the loads shown in Figure 2.13represent networks I and II of Figure 2.12, respectively, where N = 0. The matrices[SCD ], [SCD ], and [SDD ] are therefore null matrices, and the matrix [SCC ] is asquare diagonal matrix given by

[SCC ] = 3GL1 0 . . . 00 GL2 . . . 0A A A A0 0 . . . GLP

4 (2.114)

The scattering matrix of the reduced network can be found using (2.113). Itis found that the scattering matrix of the reduced M-port network is given by

Figure 2.13 An (M + P) port network whose P ports are terminated in arbitrary loads.


[SR ] = [S1] (2.115)

where

[S1] = [SAA ] + [SAB ] ([U ] − [SCC ][SBB ])−1[SCC ][SBA ] (2.116)

In the above equation [SAA ], [SAB ], [SBB ], and [SCC ] denote the partitioned matricesof the original M + P port network as defined by (2.110). [SCC ] is given by (2.114).

2.7.2 Reduction of a Three-Port Network into a Two-Port Network

The use of the above formulas is first demonstrated by finding the modified scatter-ing matrix of a three-port network, one of whose ports is not match-terminatedas shown in Figure 2.14.

Let the scattering matrix of the three-port network shown in Figure 2.14 begiven by

[S ] = 3S11 S12 S13

S21 S22 S23

S31 S32 S334 (2.117)

Referring to the notation used earlier in this section, we have

[SAA ] = FS11 S12

S21 S22G (2.118)

[SAB ] = FS13

S23G (2.119)

[SBA ] = [S31 S32] (2.120)

[SBB ] = [S33] (2.121)

and

Figure 2.14 A three-port network with its port three terminated in arbitrary load.


[SCC ] = [GL3] (2.122)

where

GL3 =V +

3

V −3

=ZL − Z03ZL + Z03

(2.123)

The matrices [SCD ], [SDC ], and [SDD ] are null matrices. Substituting the valuesof the above matrices in (2.115) and (2.116), we find that the scattering matrixof the reduced two-port network is given by

[SR ] = FS11 S12

S21 S22G + FS13

S23G (1 − S33GL3)−1GL3[S31 S32] (2.124)

which leads to

[SR ] = 3S11 +S13 S31GL3

1 − S33GL3S12 +

S13 S32GL3

1 − S33GL3

S21 +S23 S31GL3

1 − S33GL3S22 +

S23 S32GL3

1 − S33GL3

4 (2.125)

Equation (2.125) thus describes the scattering matrix of the reduced two-portnetwork. Note that the scattering matrix of the reduced two-port does not satisfyunitary conditions if the load connected to port 3 is lossy.

In the following, we give only the final equations describing how the normalscattering parameters of two- and four-port networks are modified when some oftheir ports are not match-terminated.

2.7.3 Reduction of a Two-Port Network into a One-Port Network

Let the scattering matrix of a two-port network as shown in Figure 2.15 be givenby

[S ] = FS11 S12

S21 S22G (2.126)

Figure 2.15 A two-port network with its port two terminated in arbitrary load.


If port 2 of the network is terminated in a load of reflection coefficient GL , thenthe reflection coefficient at port 1 of the network is modified as

S ′11 = S11 +S12GLS21

1 − S22GL(2.127)

where S ′11 denotes the reflection coefficient at port 1 of the reduced one-portnetwork and

GL =ZL − Z02ZL + Z02

(2.128)

denotes the reflection coefficient at port 2.

2.7.4 Reduction of a Four-Port Network into a Two-Port Network

Let the scattering elements of the four-port network as shown in Figure 2.16 beexpressed as

[S ] = 3S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S444 (2.129)

Assume that ports 3 and 4 of the network are terminated in arbitrary loads. Thereflection coefficients are given by

GL3 =V +

3

V −3

=ZL3 − Z03ZL3 + Z03

(2.130)

and

GL4 =V +

4

V −4

=ZL4 − Z04ZL4 + Z04

(2.131)

Figure 2.16 A four-port network with its two ports terminated in arbitrary load.


Using (2.115) and (2.116), the scattering parameters of the resulting two-portnetwork are found as [7]

S ′ij = Sij +GL3 | Si3 S3j −Si3 S34GL4

S4j 1 − S44GL4 | + GL4 | S3j 1 − S33GL3

−Si4 S4j Si4 S43GL3 ||1 − S33GL3 −S34GL4

−S43GL3 1 − S44GL4 |(2.132)

where i = 1, 2 and j = 1, 2 and S ′ij denotes the element of the ith row and jthcolumn of the scattering matrix of the remaining network.

References

[1] Collin, R. E., Foundations for Microwave Engineering, New York: McGraw-Hill, 1966.[2] Kurokawa, K., ‘‘Power-Waves and the Scattering Matrix,’’ IEEE Trans. Microwave Theory

Tech., Vol. MTT-13, March 1965, pp. 194–202.[3] 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.[4] Beatty, R. W., and D. M., Kerns, ‘‘Relationships Between Different Kinds of Network

Parameters, Not Assuming Reciprocity or Equality of the Waveguide or Transmission LineCharacteristic Impedance,’’ Proc. IEEE, Vol. 52, January 1964, p. 84, Corrections:April 1964, p. 420.

[5] Sazanov, D. M., A. N. Gridin, and B. A. Mishustin, Microwave Circuits, Moscow: MirPublishers, 1982.

[6] von Abele, T.-A., ‘‘Uber di streumatrix allgemein zusammengeschalteter mehrpole,’’ (‘‘TheScattering Matrix of a General Interconnection of Multipoles’’), Arch. Elek. Ubertragung,Vol. 14, Pt. 6, 1960, pp. 262–268.

[7] Otoshi, T. Y., ‘‘On the Scattering Parameters of a Reduced Multiport,’’ IEEE Trans.Microwave Theory Tech., MTT-17, September 1969, pp. 722–724.

C H A P T E R 3

Characteristics of Planar TransmissionLines

Transmission lines used at microwave frequencies can be broadly divided into twocategories: those that can support a TEM (or quasi-TEM) mode of propagationand those that cannot. For TEM (or quasi-TEM) modes, the determination ofimportant electrical characteristics (such as characteristic impedance and phasevelocity) of single and coupled lines reduces to finding the capacitances associatedwith the structure. Furthermore, the conductor loss of TEM (or quasi-TEM) modetransmission lines can be determined in terms of variation of the characteristicimpedance with respect to the geometrical parameters. This chapter discusses thegeneral characteristics of single and coupled planar TEM and quasi-TEM modetransmission lines. Further, design equations of some single and coupled popularplanar integrated transmission lines are given. The transmission lines consideredare a stripline, microstrip line, coplanar waveguide, and their variants. Of all theplanar transmission lines, microstrip is still the most popular for realizing micro-wave integrated circuits. We discuss characteristics of the microstrip line thereforein greater detail.

3.1 General Characteristics of TEM and Quasi-TEM Modes

It is an important property of any two-conductor lossless transmission line placedin a homogeneous dielectric medium that it supports a pure TEM mode of propaga-tion. Common examples of these lines are a twin-wire line, coaxial line, and shieldedstripline as shown in Figure 3.1. If a two-conductor transmission line is enclosedin an inhomogeneous dielectric medium, the mode of propagation is pure TEM onlyin the limit of zero frequency. The most common example of such a transmission lineis a microstrip line as shown in Figure 3.2(a). Some other examples of inhomoge-neous transmission lines are a slotline and a coplanar waveguide (CPW) as shownin Figure 3.2(b, c), respectively. If the separation between the conductors of aninhomogeneous transmission line is very small compared to the wavelength, themode of propagation on the line can be considered to be close to TEM. This modeis called a quasi-TEM mode.

The characteristic impedance and complex propagation constant of a TEM ora quasi-TEM mode transmission line can be described in terms of basic parametersof the line (i.e., its per unit length resistance R, inductance L , capacitance C, andconductance 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 quasi-TEM 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, therelations given are valid in the quasistatic limit, which means that the operatingfrequency is assumed to be low enough so that the distance between the conductorsof the transmission line is very small compared to the wavelength (≈ lg /20 orsmaller). For the present discussion, we assume that the conductors of the transmis-sion line have a finite but very high conductivity. It is also assumed that the dielectricloss in the material surrounding the conductors of the transmission line is finitebut small.

The parameters of interest for a transmission line are its characteristic imped-ance Z0 , phase constant b (or phase velocity vp ), and attenuation constant a . In

3.1 General Characteristics of TEM and Quasi-TEM Modes 55

Figure 3.3 Equivalent circuit of a TEM or quasi-TEM transmission line of length dz.

terms of parameters R, G, L, and C expressed per unit length, the characteristicimpedance and the propagation constant g of a transmission line are given by [1]

Z0 = √R + jvLG + jvC

(3.1)

g = √(R + jvL) (G + jvC) (3.2)

At microwave frequencies, low-loss conditions vL @ R and vC @ G are usuallysatisfied for transmission line conductors fabricated out of normal metals andenclosed in a low dielectric loss medium. Equations (3.1) and (3.2) then reduce to

Z0 = √LC

(3.3)

g = jv√LC F1 +R

2jvL+

G2jvC G (3.4)

By substituting (3.3) into (3.4), the complex propagation constant g can also beexpressed as,

g = a + jb =12 S R

Z0+ GZ0D + jv√LC (3.5)

where v = 2p f denotes the angular frequency. From (3.5):

b =vvp

= v√LC rad/unit length (3.6)

where b and vp denote the phase constant and phase velocity, respectively, alongthe direction of propagation. The attenuation constant a is given by

a =12 S R

Z0+ GZ0D Np/unit length (3.7)


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. Theattenuation of the transmission line can therefore also be expressed as

a = 4.343 S RZ0

+ GZ0D dB/unit length (3.8)

Relation Between Characteristic Impedance Z0 , Line Capacitance C, and PhaseVelocity vp

Eliminating L from (3.3) and (3.6) leads to the following very significant result:

Z0 =1

vpC(3.9)

Equation (3.9) shows that the characteristic impedance of a transmission line isrelated to the phase velocity along the transmission line and the capacitance (perunit length) between the conductors of the transmission line. It is also possible toexpress the phase velocity in terms of the ratio of the actual capacitance of thetransmission line to the capacitance of the same transmission line obtained byassuming the dielectric constant of the medium in which it is placed to be unity.Therefore, the problem of determining the characteristic impedance and phasevelocity of the structure reduces essentially to the problem of finding the capacitanceof the structure.

Q-Factor

Equation (3.7) shows that the total line attenuation is due to two factors: the seriesresistance R and shunt conductance G. The total attenuation can therefore beexpressed as

a = a c + ad (3.10)

where

a c =R

2Z0Np/unit length (3.11)

denotes the conductor loss, and

ad =GZ0

2Np/unit length (3.12)

denotes the dielectric loss.The attenuation of a transmission line can also be expressed in terms of the

Q-factor. The Q-factor of a half-wavelength transmission line resonator is givenby


Q =b

2a=

b2(a c + ad )

(3.13)

where the attenuation a is expressed in Np/unit length. We can also define theQ-factors for conductor (Qc ) and dielectric loss (Qd ) separately as

Qc =b

2a c(3.14)

and

Qd =b

2ad(3.15)

Using (3.13) and (3.14), the overall Q-factor can be expressed as

1Q

=1

Qc+

1Qd

For dispersive lines, (3.13) and (3.14) are incorrect and require that the termb in (3.13) and (3.14) be replaced by v /vg , where vg denotes the group velocity[1]. For example, (3.13) then becomes

Q =v

2vga=

v2vg (a c + ad )

(3.16)

Equation (3.16) is more general than (3.13) and is valid for non-TEM modes aswell.

3.1.1 TEM Modes

We now specialize some of the above equations for the case when the transmissionline is placed in a homogeneous dielectric medium. Some examples of these typesof lines are shown in Figure 3.1. For these lines, the velocity of propagation vpalong the transmission line is independent of the type of the transmission line andfrequency of operation and is given by

vp =c

√er(3.17)

where c is the velocity of light in freespace and er denotes the dielectric constant(relative permittivity) of the medium. The phase constant along the transmissionline is therefore given by

b =vvp

=v √er

c= k0√er rad/unit length (3.18)


where k0 = 2p /l0 denotes the free-space propagation constant and l0 denotes thefree-space wavelength.

Substituting the value of phase velocity from (3.18) in (3.9) leads to the follow-ing relation between the characteristic impedance and capacitance of the line:

Z0 = √er

cC= √er

cerC0=

1

c√erC0(3.19)

where C0 denotes the capacitance between the conductors of the transmission lineassuming that the transmission line is placed in a medium of a unity dielectricconstant.

The determination of dielectric loss ad is also straightforward in this case. Itis given by

ad =b2

tan d =k0√er

2tan d Np/unit length (3.20)

or

ad = 4.343b tan d = 4.343k0√er tan d = 27.3√ertan d

l0dB/unit length

(3.21)

where tan d denotes the loss tangent of the dielectric material. In general, the losstangent tan d is also a function of frequency.

On the other hand, conductor loss a c depends on the type of line, the conductiv-ity of the transmission line, the frequency of operation, and geometrical parametersof the line and is discussed in detail in a later section.

3.1.2 Quasi-TEM Modes

Some examples of quasi-TEM mode transmission lines are shown in Figure 3.2.For quasi-TEM modes, the effective dielectric constant ere is defined as follows:

ere =c2

v2p

(3.22)

In qualitative terms, the effective dielectric constant ere takes into account therelative distribution of electric energy in the various regions of the inhomogeneousmedium. The relation between phase constant b , effective dielectric constant ere ,and phase velocity vp is

b =vvp

=v √ere

c= √ere k0 (3.23)


ere is a function of frequency and strictly speaking should be evaluated using(3.22) where the phase velocity vp is computed using some rigorous method basedon Maxwell’s equations. However, in the quasistatic limit, ere can be assumed tobe

ere =CC0

(3.24)

where C denotes the capacitance between conductors of the transmission line inthe inhomogeneous dielectric medium and C0 denotes the capacitance between thesame conductors in a medium of unity dielectric constant. Using (3.9), (3.23), and(3.24), the characteristic impedance of a quasi-TEM mode transmission line canbe expressed as

Z0 = √ere

cC=

1

c√CC0=

1

c√ereC0=

Z0a

√ere(3.25)

where Z0a denotes the characteristic impedance of the same transmission line placedin a medium of unity dielectric constant.

The dielectric loss of a quasi-TEM mode transmission line is given by

ad = 27.3er

√ere

(ere − 1)(er − 1)

tan dl0

dB/unit length (3.26)

The conductance G of a transmission line can be expressed in terms of losstangent tan d as follows:

G =2pZ0

er

√ere

(ere − 1)(er − 1)

tan dl0

(3.27)

Furthermore, it is customary to define the effective filling fraction q of aquasi-TEM mode transmission line as follows:

q =ere − 1er − 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 nearthe outer surface of the conductor. The skin depth of a conductor is defined asthe distance in the conductor (along the direction of the normal to the surface) inwhich the current density drops to 37% of its value at the surface (the currentdecays to a negligible value in a distance of about 4 to 5 skin depths) and is givenby

d s = √ 2vms

(3.29)


where m = m r m0 , m0 = 4p × 10−7 H/m denotes the permeability of free space, ands denotes the conductivity (S/m) of the conductor. Further, m r denotes the relativepermeability of the material. Its value is almost equal to unity except for magneticmaterials. Equation (3.29) shows that the skin depth of a perfect conductor(s = ∞) 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 verysmall at microwave frequencies (e.g., the conductivity of copper is 5.8 × 107 S/mand the skin depth at 10 GHz is 0.66 mm). The tangential electric field at thesurface of a conductor is not zero due to finite conductivity of the conductors ofa transmission line.1 The surface impedance (ohms/square) of a conductor (definedas the ratio of tangential electric and magnetic fields at the surface) is given by

Zs = Rs + jvLs =1 + jsd s

(3.30)

or

Rs = vLs =1

sd 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 Quasi-TEM Modes

An ingenious way to determine the conductor loss of TEM or quasi-TEM modetransmission lines was given by Wheeler [2]. This method of determining theconductor loss is also known as the incremental inductance rule. The rule is validonly if the thickness of conductors of the transmission lines and the radius ofcurvature of conductor surfaces are at least five to six times the skin depth. Theseconditions can usually be satisfied at microwave frequencies except near very sharpedges. According to this rule, the conductor Q-factor of a transmission line is givenby [3]

Qc =Z ′0a

(Z ′0a − Z0a )(3.32)

where Z0a has been defined before and Z ′0a denotes the impedance of the sametransmission line placed in a medium of unity dielectric constant, but assumingthat the thickness of all conductors is reduced by ds /2 from each surface wherethe fields are present. This is shown in Figure 3.4, where solid lines show thesurfaces of the actual microstrip transmission line of a finite strip thickness anddashed lines show the surfaces of the fictitious microstrip transmission line obtainedby removing a depth of ds /2 from each surface of the conductor of the originaltransmission line. Note that the conductor Q-factor (but not the attenuation perunit length) of a TEM or quasi-TEM 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 61

Figure 3.4 Illustration of incremental inductance rule showing the original and perturbed geometryof a microstrip line.

dielectric constant of the medium in which it is placed. Using (3.14), (3.23), and(3.32), the attenuation factor a c can be expressed as

a c = √ere k0(Z ′0a − Z0a )

2Z ′0a=

p √ere f

c(Z ′0a − Z0a )

Z ′0a(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 capaci-tances. It is therefore useful to discuss the representation of capacitances of coupledtransmission lines.

Figure 3.5 shows the cross section of two coupled transmission lines having acommon ground conductor with the capacitances associated with the coupledstructure as shown. If Q1 and Q2 denote the charges and V1 and V2 denote thevoltages of conductors 1 and 2, respectively, the charges Q1 and Q2 can beexpressed in terms of voltages and capacitances as

Q1 = CaV1 + Cm (V1 − V2) = (Ca + Cm )V1 − CmV2 (3.34)

Q2 = Cm (V2 − V1) + CbV2 = −CmV1 + (Cb + Cm )V2 (3.35)

The capacitance matrix of two coupled transmission lines is represented as [4]

Figure 3.5 Representation of capacitances of coupled lines.


[C] = FC11 C12

C21 C22G (3.36)

where C11 and C22 are defined as self-capacitances of lines 1 and 2, respectively,in the presence of each other. The capacitance matrix denotes the relation betweencharges and voltages on the two transmission lines as follows:

Q1 = C11V1 + C12V2 (3.37)

Q2 = C21V1 + C22V2 (3.38)

Hence the capacitance matrix of coupled lines can be expressed as

[C] = FC11 C12

C21 C22G = FCa + Cm −Cm

−Cm Cb + CmG (3.39)

The inductance matrix of a coupled line is given by

[L] = m0e0[C0]−1 (3.40)

where [C0] denotes the capacitance matrix of the transmission lines obtained byassuming that these are placed in a medium of unity dielectric constant.

3.2.1 Even- and Odd-Mode Capacitances of Symmetrical Coupled Lines

When the coupled lines are identical (also called symmetrical coupled lines), theircapacitance matrix can be expressed in terms of even- and odd-mode capacitances.

Even-Mode Excitation

A cross section of uniformly coupled symmetrical lines is shown in Figure 3.6(a).In this case, the capacitance Cm shown in Figure 3.5 has been broken into twocapacitances of values 2Cm each in series. With even-mode excitation, equal andin-phase voltages (V1 = V2 = Ve ) are applied to both lines. Because the geometryunder consideration is symmetrical, it is clear that if equal voltages of the samepolarity are applied to both the lines, the charges on the two lines would also bethe same (i.e., Q1 = Q2 = Qe ). Denoting the ratio Qe /Ve by Ce , (3.34) and (3.35)then reduced to

Ca = Cb =QeVe

= Ce (3.41)

Odd-Mode Excitation

In the odd-mode excitation, equal but out-of-phase voltages (V1 = −V2 = Vo ) areapplied to the two lines as shown in Figure 3.6(b). It follows from the symmetry


Figure 3.6 (a) Even- and (b) odd-mode excitation of symmetrical coupled lines.

of the structure that if equal voltages but of opposite polarity are applied to thesymmetrical lines, equal charges but of opposite polarity will be induced on thetwo lines (i.e., Q1 = −Q2 = Qo ). With the ratio Qo /Vo denoted by Co , (3.34) and(3.35) reduced to

Qo = (Ca + 2Cm )Vo (3.42)

or

Ca + 2Cm =QoVo

= Co (3.43)

Substituting the value of Ca from (3.41) in (3.43), we obtain

Cm =Co − Ce

2(3.44)

Therefore, once the even- and odd-mode capacitance parameters of coupled sym-metrical lines are known, Ca , Cb , and Cm can be determined from (3.41) and(3.44). The capacitance matrix of coupled lines can then be determined using (3.39).


In the even-mode of excitation, the symmetry plane PP ′ as shown in Figure3.6(a) acts as a magnetic wall (open circuit). The determination of the even-modecapacitance reduces to finding the capacitance of either line with the plane ofsymmetry PP ′ replaced by a magnetic wall such as shown in Figure 3.7(a). Thisresults in a great simplification of the problem. Similarly, in the odd-mode ofexcitation, the symmetry plane behaves as an electric wall (short circuit). Thedetermination of the odd-mode capacitance reduces to finding the capacitance ofeither line by replacing the plane of symmetry by an electric wall as shown inFigure 3.7(b).

The relationships between the even- and odd-mode capacitances and imped-ances are given by

Z0e =1

vpeCe=

vbeCe

(3.45)

and

Z0o =1

vpoCo=

vboCo

(3.46)

Figure 3.7 Representation of capacitances of (a) even and (b) odd modes of symmetrical coupledlines.


where Z0e , vpe , and be denotes the characteristic impedance, phase velocity, andphase constant, respectively, of the even mode of the coupled lines; and Z0o , vpo ,and bo denote the same quantities for the odd mode. If the lines are placed in ahomogeneous medium of dielectric constant er , the even- and odd-mode phasevelocities are equal and are given by

vpe = vpo =c

√er(3.47)

However, if the lines are placed in an inhomogeneous dielectric media (suchas coupled microstrip lines), the even- and odd-mode phase velocities are, in general,different and are given by

vpe =c

√eree(3.48)

and

vpo =c

√ereo(3.49)

where eree and ereo are defined as the even- and odd-mode effective dielectricconstants, respectively. These can be determined using

eree =CeC0e

(3.50)

and

ereo =CoC0o

(3.51)

where C0e and C0o denote, respectively, the even- and odd-mode capacitance ofeither line obtained by replacing the relative permittivity of the surrounding dielec-tric material by unity. Ce and Co denote the corresponding capacitances in thepresence of the inhomogeneous dielectric medium. Using (3.48) to (3.51), (3.45)and (3.46) reduce to

Z0e =1

c√CeC0e(3.52)

and

Z0o =1

c√CoC0o(3.53)


3.2.2 Parallel-Plate and Fringing Capacitances of Single and Coupled PlanarTransmission Lines

So far we have discussed the capacitances of single and coupled transmission linesin general. In many cases, it is possible to visualize the various components of thetotal capacitance of the structure. This helps in obtaining a physical understandingof the problem and its analysis. For example, for planar transmission lines, thetotal capacitance can be broken into its various components such as parallel-plateand fringing capacitances. To explain the various components, we consider forsimplicity 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 directlybelow the strip conductor, but extend to the surrounding regions as well. The latterare known as fringing fields. The capacitance that results from the electric fieldsin the region directly below the strip is known as the parallel-plate 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 representedas shown in Figure 3.8(b) and is given by

C = Cp + 2Cf (3.54)

where

Cp =e0erW

h(3.55)

Figure 3.8 (a) Electric field distribution of single microstrip line, and (b) equivalent capacitancerepresentation.


denotes the parallel-plate capacitance and Cf denotes the fringing capacitance fromeither edge of the microstrip line. Once the value of Cf is known, the total capaci-tance of the line can be determined using (3.54) and (3.55). Conversely, if thecharacteristic impedance and effective dielectric constant of a microstrip line areknown, the capacitance C can be found using (3.25), and using (3.54) and (3.55),the fringing capacitance Cf can be determined.

Symmetrical Coupled Lines

A cross-section of symmetrical coupled microstrip lines is shown in Figure 3.9(a)with the electric field distribution in Figure 3.9(b) for one-half of the structure for

Figure 3.9 (a) Cross section of symmetrical coupled lines, (b, c) electric field distribution andcapacitance representation of one-half of the structure for even-mode excitation, and(d, e) electric field distribution and capacitance representation of one-half of the struc-ture for odd-mode excitation.


the case of even-mode excitation. In this case, the normal component of the electricfield at the plane of symmetry PP ′ is zero, because the plane of symmetry behaveslike a magnetic wall. The even-mode capacitance of either of the coupled lines,which can be represented as shown in Figure 3.9(c), is given by,

Ce = Cp + Cf + Cfe (3.56)

where Cp is given by (3.55). If the two lines are not of very narrow width, it canbe assumed that the value of Cf is the same as that of a single microstrip linehaving the same width as that of either of the coupled lines.

The electric field distribution of one-half of the coupled structure is shown inFigure 3.9(d) for the case of odd-mode excitation. The tangential electric field atthe plane of symmetry PP ′ is zero, because in this case, the plane of symmetrybehaves like an electric wall. The odd-mode capacitance of either of the coupledlines is given by

Co = Cp + Cf + Cfo (3.57)

where Cfo denotes the fringing capacitance from the inner edges of the coupledlines. When the spacing between the lines is small (S/2 is small compared to theheight of the substrate h), nearly all the fringing fields that start from the inneredge of one of the lines terminate on the mid plane PP ′. In that case, the capacitanceof either of the coupled lines, which can be represented as shown in Figure 3.9(e),is given by,

Co = Cp + Cf + Cfo = Cp + Cf + Cga + Cgd (3.58)

where the fringing capacitance Cfo is assumed to consist of two capacitances Cgaand Cgd in parallel; that is,

Cfo = Cga + Cgd (3.59)

The characteristic impedances and phase velocities of the even and odd modescan be found by determining the parallel plate and fringing capacitances associatedwith the structure. Conversely, if the characteristic impedances and phase velocitiesof the even and odd modes are known, the corresponding fringing capacitancescan be determined.

The breakup of the total capacitance into various components involves certainapproximations. For example, (3.55) can only be termed as approximate becausethe fields under the strip are not exactly vertical, especially near the edges. Thebreakup of the total capacitance into various components, however, helps in asimple, although approximate design of various coupled-line components such asa Lange coupler.

3.3 Characteristics of Single and Coupled Striplines

A commonly used stripline is shown in Figure 3.10. The strip conductor is sand-wiched between two flat dielectric substrates having the same dielectric constant.

3.3 Characteristics of Single and Coupled Striplines 69

Figure 3.10 Cross section of a stripline.

The outer surfaces of the dielectric substrates are metallized and serve as groundconductors. The signal is applied between the strip conductor and ground. In acommonly used fabrication technique, the strip conductor is etched on one of thedielectric substrates by the process of photolithography. The thickness of both thesubstrates is generally the same, although it is not essential. A lossless stripline cansupport a pure TEM mode of propagation at all frequencies. Two striplines canbe coupled by placing strip conductors side by side as shown in Figure 3.11 in theedge-coupled configuration. In this configuration, both the strip conductors liein the same plane, and although it is very convenient for realizing coupled linecircuits, it has the disadvantage that tight coupling between the lines cannot beachieved. For tight coupling (about 8 dB or tighter), the width of the strip conductorsand the spacing between them becomes quite small, making fabrication difficult.Further, the small dimensions lead to large current densities on the strip conductorleading to higher conductor loss. The edge-coupled configuration is thus suitablefor the design of loose couplers having coupling of 8 to 30 dB. Tighter coupling(e.g., 3 dB) can be obtained using broadside-coupled striplines, as discussed inSection 3.7.

3.3.1 Single Stripline

Based upon the Schwartz-Christoffel transformation, the exact expression for thecharacteristic impedance of a lossless stripline of zero thickness is given by [5, 6]

Z0√er = 30pK(k)K(k ′) (3.60)

where

k = sechSpW2b D; k ′ = √1 − k2

In the above expression, K denotes the complete elliptic integral of the first kindand 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 edge-coupled striplines.


K(k)K(k ′) =

K(k)K ′(k)

=1p

lnS21 + √k

1 − √k D, 0.5 ≤ k2 ≤ 1 (3.61)

=p

lnS21 + √k ′

1 − √k ′D, 0 ≤ k2 ≤ 0.5

Effect of Finite Strip Thickness on Characteristic Impedance

The capacitance between the strip conductor and ground is increased if the thicknessof the strip is finite. This happens because of the additional capacitance resultingfrom the strip conductor edges of finite thickness. The characteristic impedance ofa stripline of finite thickness is given by Wheeler [8] as

Z0√er = 30 lnH1 +12

(16h /pW ′ )F(16h /pW ′ ) + √(16h /pW ′ )2 + 6.27GJ(3.62)

where W ′ denotes the effective width of the stripline. When the thickness of thestrip conductor is zero, the effective width is the same as the physical width. Forfinite strip thickness, the effective width W ′ is given by

W ′ = W + DW (3.63)

where

DWt

=1p

ln2.718

√F 14h /t + 1G

2+ F 1/4p

W /t + 1.1Gm

m =6

3 + t /h

The error in Wheeler’s equation is expected to be less than 1%.Closed-form expressions for the synthesis of a stripline have also been given

by Wheeler [8]. These are as follows:

W = W ′ − DW ′ (3.64)

where

W ′h

=16p

√(e4pr − 1) + 1.568

(e4pr − 1)


r = √er Z0

376.7

DW ′t

=1p

ln2.718

√F 14h /t + 1G

2+ F 1/4p

W ′/t − 0.26Gm

m =6

3 + t /h

The characteristic impedance of a stripline as a function of strip width forvarious values of t is shown in Figure 3.12 together with the characteristic imped-ance of striplines with strip conductor of square and circular cross section (Figure3.13).

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 becomputed in a rather simple manner using Wheeler’s incremental inductance ruleas discussed earlier [2]. For a stripline, the application of the rule leads to thefollowing expression for the conductor loss [9]:

a c =Rs√er

376.7Z0F∂Z0

∂b−

∂Z0∂W

−∂Z0∂t G Np/m (3.65)

Figure 3.12 Characteristic impedance of finite thickness stripline versus W/h for various values oft/h. Curves marked h and s are valid for stripline having strip-conductor of squareor circular cross section, respectively, as shown in Figure 3.13. (From: [8]. 1978IEEE. Reprinted with permission.)


Figure 3.13 Stripline with strip-conductor of square or circular cross section.

where Rs is the surface resistance (ohm/square) given by (3.31) and ∂ denotes thepartial derivative.

The normalized conductor loss (h/Qc ds ) of a stripline is shown in Figure 3.14where Qc and ds denote the conductor Q-factor and skin depth, respectively. Theskin depth ds , is given by (3.29). For given values of W and h, the conductorQ-factor Qc can be determined using Figure 3.14. The conductor loss in Np/mcan then be determined using (3.14). It may be noted that the normalized conductorloss (h/Qc ds ) is independent of the dielectric constant of the substrate.

Figure 3.14 Normalized conductor Q-factor (h/Qcds ) of a stripline versus W/h for various values of t/h(valid for arbitrary value of er ). Curves marked h and s are valid for stripline having strip-conductor of square or circular cross section, respectively, shown in Figure 3.13. (From: [8]. 1978 IEEE. Reprinted with permission.)


For a wide stripline, the cutoff frequency of the first higher order mode is givenby [10]

fc =15

b√er

1(W /b + p /4)

GHz (3.66)

where W and b are in centimeters.

3.3.2 Edge-Coupled Striplines

The even- and odd-mode charateristic impedances of edge-coupled striplines ofzero thickness (Figure 3.11) are given by the following expressions, which are exact[11]:

Z0e√er = 30pK(k ′e )K(ke )

(3.67)

where

ke = tanhSp2

Wb D tanhFp

2(W + S)

b G; k ′e = √1 − k2e

and

Z0o√er = 30pK(k ′o )K(ko )

(3.68)

where

ko = tanhSp2

Wb D cothFp

2(W + S)

b G; k ′o = √1 − k2o

The functions K(ke )/K(k ′e ) and K(ko )/K(k ′o ) can be evaluated using (3.61). Theimpedances (3.67) and (3.68) have been plotted in Figure 3.15. It is seen that whenS/b → ∞, both Z0e and Z0o approach the same value equal to the characteristicimpedance of a single stripline of width W.

Synthesis equations for the design of zero-thickness coupled striplines are alsogiven by Cohn [11]. These are as follows:

Wb

=2p

tanh−1 √keko (3.69)

and

Sb

=2p

tanh−1 S1 − ko1 − ke √ke

ko D (3.70)


Figure 3.15 Characteristic impedances of even- and odd-modes of edge coupled striplines. (From:[11]. 1995 IEEE. Reprinted with permission.)

For given values of Z0e and Z0o , the values of ke and ko can be determinedby numercially solving (3.67) and (3.68), respectively, with the aid of (3.61). Forexample, if (3.67) is plotted as a function of ke with the aid of (3.61), the valueof ke for a given value of Z0e can be found. The values of W /b and S/b canthen be determined by substituting the values of ke and ko in (3.69) and (3.70),respectively.

The propagation constants are equal for the even and odd modes and are givenby (3.18). Similarly, the dielectric losses are also equal for the even and odd modesand are given by (3.21). The conductor losses, however, are different for the twomodes. Expressions for these are given in [12].

The effect of finite thickness of strip conductors on the characteristic impedancesof 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 substrateis chosen to be much smaller than the wavelength in the dielectric. The mode ofpropagation along a microstrip line is not pure TEM. However, because of the

3.4 Characteristics of Single and Coupled Microstrip Lines 75

Figure 3.16 A microstrip line.

simplicity of the structure and ease of fabrication, the microstrip line is the mostpopular transmission line for realizing microwave integrated circuits (MIC). Com-monly used substrates are alumina (Al2O3), teflon-based materials such as RTduroid for hybid MICs, and gallium arsenide (GaAs) for monolithic MICs. Numer-ous methods have been used to determine the characteristics of single and coupledmicrostrip lines. For design purposes, many simple closed-form empirical expres-sions have been reported in the literature [13–22]. A microstrip line is dispersivein nature and its characteristic impedance and effective dielectric constant varywith frequency. The accuracy of various dispersion formulas has also been experi-mentally verified [23, 24].

The characteristics of microstrip lines are generally described by two sets ofequations. One gives the characteristics that are valid in the quasistatic limit, whilethe other set accounts for dispersion. Quasistatic relations are sufficiently accurateas long as the height of the dielectric substrate is very small compared with thewavelength.

3.4.1 Single Microstrip

The quasistatic problem of a single microstrip line was solved analytically byWheeler [14]. Over the years, this model has generally served as the basis forderiving 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]

ere (0) =er + 1

2+

er − 12 S1 +

10u D

−AB

(3.71)

where u = W /h, and

A = 1 +149

lnFu4 + (u/52)2

u4 + 0.432 G +1

18.7lnF1 + S u

18.1D3G

B = 0.564Ser − 0.9er + 3 D0.053


The stated accuracy of (3.71) is better than 0.2% at least for er ≤ 128 and0.01 ≤ u ≤ 100.

The quasistatic characteristic impedance of a microstrip line of zero thicknesscan be expressed as

Z0(0) =60

√erelnF f (u)

u+ √1 + S2

uD2G (3.72)

where

f (u) = 6 + (2p − 6) expF− S30.666u D0.7528G (3.73)

The symbol (0) in ere (0) and Z0(0) denotes that the formula is valid in thequasistatic limit. For er = 1, the stated accuracy of (3.72) is better than 0.01% foru ≤ 1 and 0.03% for u ≤ 1,000. For other values of er , the accuracy of the expressionis determined by the accuracy with which the effective dielectric constant is known.

Finite-Thickness Microstrip

Improved and more general equations for the analysis and synthesis of microstriplines have also been given by Wheeler [3]. The characteristic impedance of amicrostrip line of finite thickness is given by

Z0 =42.4

√er + 1lnH1 + S4h

W ′DFS14 + 8/er11 DS4h

W ′D (3.74)

+ √S14 + 8/er11 D2S4h

W ′D2

+1 + 1/er

2p2GJ

where W ′ denotes the effective width of the microstrip. The effect of finite thicknessof the strip is taken into account by assuming that the effective width of themicrostrip W ′ is greater than its physical width W and is given by

W ′ = W + DW (3.75)

where

DWt

= S1 + 1/er2p D ln

10.872

√S th D

2+ S 1/p

W /t + 1.1D2

The above equations are supposed to be accurate for arbitrary values of er andaspect ratio W /h (within the quasistatic limit). Note that if a general expression


is available to determine the characteristic impedance of a microstrip line as afunction of the dielectric constant of the substrate, it is not necessary to have aseparate formula for the effective dielectric constant. The effective dielectric con-stant can then be determined using (3.25), that is:

ere = SZ0aZ0

D2

where Z0a denotes the characteristic impedance of the microstrip line assumingthat the dielectric constant of the substrate is unity.

The synthesis equations for a microstrip line are as follows:

W = W ′ − DW ′ (3.76)

where

W ′h

= 8√Fe

S Z042.4 √er + 1D

− 1G 7 + 4/er11

+1 + 1/er

0.81

FeS Z0

42.4 √er + 1D− 1G

and

DW ′t

= S1 + 1/er2p D ln

10.872

√S th D

2+ S 1/p

W ′/t − 0.26D2

The characteristic impedance of a microstrip line is shown in Figure 3.17(a) as afunction of effective width W ′ for various values of er [3]. For a microstrip lineof given thickness and width, the effective width can be found using (3.75). Figure3.17(a) can therefore be used to find the characteristics of a microstrip line of finitethickness. The results shown in Figure 3.17(a) can also be used to find the effectivedielectric constant of a microstrip line using (3.25). The characteristics of a micro-strip line printed on some commonly used dielectric substrates are shown in Figure3.17(b).

Example 3.1

Determine the dimension ratio W /h and effective dielectric constant of a microstripline (t = 0) of characteristic impedance of 50V printed on a dielectric substrate ofdielectric constant (er = 2).

From the curve labeled as er = 2 in Figure 3.17(a), we find that W /h shouldbe chosen as equal to 3.3 to obtain a characteristic impedance (Z0) of 50V. Further,


Figure 3.17 (a) Quasistatic characteristic impedance and effective dielectric constant of a microstripline for various values of er . ere = (Z0a /Z0)2. (From: [3]. 1977 IEEE. Reprinted withpermission.) (b) Quasistatic characteristic impedance and effective dielectric constantof a microstrip line for some commonly used dielectric substrates. (From: [12]. 1988John Wiley and Sons. Reprinted with permission.)

for the same value of W /h and er = 1, we find from the graph that Z0a = 66V.Using (3.25), we then find that

ere = S6650D

2= 1.74


Conductor and Dielectric Loss

Once a general expression for the characteristic impedance of a microstrip isavailable in terms of the structural parameters, the conductor loss can be easilycomputed using (3.14) and (3.32). The conductor and other losses of a microstripline on dielectric and ferrite substrates are discussed in [17]. The dielectric loss ofa microstrip line is given by (3.26).

The normalized conductor loss (h/Qc ds ) of a microstrip line (which is indepen-dent of the dielectric constant of the substrate) is shown in Figure 3.18(a) whereQc and ds denote the conductor Q-factor and skin depth, respectively [3]. Forgiven values of W and h, this figure can be used to determine the conductor Q-factorQc and hence the conductor loss in Np/m from (3.14). The conductor and dielectricloss of a microstrip line on some commonly used dielectric substrates is shown inFigure 3.18(b).

Microstrip Dispersion

If the frequency of operation is high, so that the height of the substrate is not verysmall compared with the wavelength in the dielectric, the quasistatic expressionspresented above are not accurate enough. In some other situations also, theseexpressions may be inadequate. For example, if a digital signal with short rise timepropagates along a microstrip line, the signal can be assumed to contain a numberof high-frequency harmonics. An accurate dispersion model for propagation alongmicrostrip lines is therefore required. The frequency dependence of the effectivedielectric constant of a microstrip line is well represented by the following relation[19]:

ere ( f ) = er −er − ere (0)

1 + ( f /f50)m (3.77)

where

f50 =fK, TM0

0.75 + S0.75 −0.332

e1.73r

D Wh

(3.78)

fK, TM0=

c tan−1 Ser√ ere (0) − 1er − ere (0)D

2ph√er − ere (0)(3.79)

m = m0mc

m0 = 1 +1

1 + √W /h+ 0.32 S 1

1 + √W /hD3


Figure 3.18 (a) Normalized conductor Q-factor of a microstrip line (valid for arbitrary value of er ).Curves marked h and s are valid for microstrip line having strip-conductor of squareor circular cross section respectively. (From: [3]. 1977 IEEE, Reprinted with permis-sion.) (b) Conductor and dielectric loss of microstrip line on some commonly useddielectric substrates. (From: [13]. 1996 Artech House. Reprinted with permission.)

mc = 51 +1.4

1 + W /h F0.15 − 0.235 expS−0.45ff50

DG for W /h ≤ 0.7

1 for W /h > 0.7

Here c is the velocity of light in free space. It is claimed that the above modelhas an accuracy of better than 2.7% in the range 0.1 ≤ W /h ≤ 10, 1 ≤ er ≤ 128and for any value of h/l0 .


The dispersion of a microstrip line fabricated on a dielectric substrate of er = 8is shown in Figure 3.19. The characteristic impedance of a microstrip line is alsofrequency dependent and shows a small positive increase with an increase in fre-quency. The following relation describes this dependence quite accurately [18]:

Z0( f ) = Z0(0) √ere (0)ere ( f )

[ere ( f ) − 1][ere (0) − 1]

(3.80)

For many practical purposes, the variation of characteristic impedance withfrequency can be neglected.

3.4.2 Coupled Microstrip Lines

Coupled microstrip lines are shown in Figure 3.20. The equations for the quasistaticcharacteristics of coupled microstrip lines have been given by many authors includ-ing Hammerstad and Jenson [18], Garg and Bahl [13, 21], and others. For theodd-mode case, the Hammerstad and Jenson equations were later modified byKirschning and Jansen [22] to incorporate the effect of dispersion. Although thelatter expressions are lengthy, these are simple to program and believed to beaccurate to within 1% in the range of parameters 0.1 ≤ u ≤ 10, 0.1 ≤ g ≤ 10, and1 ≤ er ≤ 18. The quasistatic even-mode effective dielectric constant for coupledmicrostrip lines for zero conductor thickness is given by

eree (0) = 0.5(er + 1) + 0.5(er − 1) ? (1 + 10/n )−ae (n ) ? be (er ) (3.81)

Figure 3.19 Dispersion characteristics of a microstrip line printed on a dielectric substrate of dielec-tric constant er = 8. (From: [19]. 1988 IEEE. Reprinted with permission.)


Figure 3.20 Edge-coupled microstrip lines.

where

n = u(20 + g2)/(10 + g2) + g ? exp(−g)

ae (n ) = 1 + ln{[n4 + (n /52)2]/(n4 + 0.432)}/49 + ln[1 + (n /18.1)3]/18.7

be (er ) = 0.564[(er − 0.9)/(er + 3)]0.053

and u = W /h, g = S/h. The quasistatic odd-mode effective dielectric constant forzero conductor thickness is similarly given by

ereo (0) = [0.5(er + 1) + ao (u, er ) − ere (0)] ? exp X−cogdoC + ere (0) (3.82)

where

ao (u, er ) = 0.7287[ere (0) − 0.5(er + 1)] ? [1 − exp(−0.179u)]

bo (er ) = 0.747er /(0.15 + er )

co = b0(er ) − [b0(er ) − 0.207] ? exp(−0.414u)

do = 0.593 + 0.694 ? exp(−0.562u)

and ere (0) denotes the effective dielectric constant of a single microstrip of widthW.

Quasistatic Even- and Odd-Mode Characteristic Impedances

The quasistatic even-mode characteristic impedance of coupled microstrip lines isgiven by [22]

Z0e (0) = Z0√ ere (0)eree (0)

1

{1 − [Z0(0)/377][ere (0)]0.5Q4}(3.83)

where

Q4 = (2Q1 /Q2) ? Hexp(−g) ? uQ3 + [2 − exp(−g)] ? u−Q3J−1

3.5 Single and Coupled Coplanar Waveguides 83

with

Q1 = 0.8695 ? u0.194

Q2 = 1 + 0.7519g + 0.189 ? g2.31

Q3 = 0.1975 + [16.6 + (8.4/g)6]−0.387 + ln(g10/[1 + (g/3.4)10]/241

Similarly, the quasistatic odd-mode characteristic impedance of coupled micro-strip lines is expressed by

Z0o (0) = Z0√ ere (0)ereo (0)

1

{1 − [Z0(0)/377][ere (0)]0.5Q10}(3.84)

with

Q5 = 1.794 + 1.14 ? ln[1 + 0.638/(g + 0.517g2.43)]

Q6 = 0.2305 + ln(g10/[1 + (g/5.8)10]/281.3 + ln(1 + 0.598g1.154)/5.1

Q7 = (10 + 190g2)/(1 + 82.3g3)

Q8 = exp[−6.5 − 0.95 ln(g) − (g/0.15)5]

Q9 = ln(Q7) ? (Q8 + 1/16.5)

Q10 = Q−12 ? {Q2Q4 − Q5 ? exp[ln(u) ? Q6 ? u−Q9 ]}

Equations for the frequency dependence of the even- and odd-mode effective dielec-tric constants and characteristic impedances were also given by Kirschning andJansen [22]. These are quite involved, however, and are not repeated here.

The frequency-dependent characteristics of coupled microstrip lines printed onsome typical dielectric substrates are shown in Figures 3.21 and 3.22 [22]. Theresults show that the variation of characteristic impedance with frequency is muchsmaller than the variation of an effective dielectric constant. Further, it is seen thatthe even-mode parameters show a greater variation with frequency than the odd-mode parameters.

3.5 Single and Coupled Coplanar Waveguides

A few coplanar waveguide (CPW) configurations are shown in Figures 3.23 to3.25. In a coplanar waveguide, the signal is applied between the center conductorand 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. Thecoplanar waveguide has some inherent advantages, making it suitable for hybridand monolithic microwave integrated circuits. Because both the center and groundconductors of a CPW lie in the same plane, active devices can be easily placed in


Figure 3.21 Frequency dependent even- and odd-mode characteristic impedances and effectivedielectric constants of coupled microstrip lines printed on a substrate of dielectricconstant er = 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 quasi-TEM. Both numerical andanalytical methods have been used for the analysis in [27–34]. The dispersion inCPW is generally smaller than in a microstrip line [33].

3.5.1 Coplanar Waveguide

Consider the CPW printed on a finite-thickness dielectric substrate (Figure 3.23).The width of the dielectric substrate is assumed to be infinite. The quasistaticparameters of this transmission line are given by [30]:

Z0 =30p

√ere

K(k ′)K(k)

(3.85)

ere = 1 +er − 1

2K(k ′)K(k1)

K(k)K(k ′1)(3.86)

where


Figure 3.22 Frequency dependent even- and odd-mode characteristic impedances and effectivedielectric constants of coupled microstrip lines printed on a substrate of dielectricconstant er = 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.


Figure 3.25 Cross section of a conductor-backed coplanar waveguide with upper shielding.

k = a /b

k ′ = √1 − k2

k1 = sinh(pa /2h)/sinh(pb /2h)

k ′1 = √1 − k21

Further, K is the complete integral of the first kind. The values of functionsK(k)/K(k ′) and K(k1)/K(k ′1) can be found using (3.61).

The effective dielectric constant and characteristic impedance of a CPW areshown in Figure 3.26 for a value of er = 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. Thequasistatic parameters of the transmission line are given by [31]

Figure 3.26 (a) Effective dielectric constant, and (b) characteristic impedance of CPW shown inFigure 3.23 as a function of aspect ration a/b for various values of h/b, er = 13. (From:[13]. 1996 Artech House. Reprinted with permission.)


ere = 1 + q(er − 1) (3.87)

where q is the filling fraction given by

q =

K(k1)K(k ′1)

K(k2)K(k ′2)

+K(k)K(k ′)

(3.88)

In (3.88), parameters k, k1 and k2 are given by

k = a /b

k ′ = √1 − k2

k1 = sinh(pa /2h)/sinh(pb /2h) (3.89)

k2 = tanh(pa /2h1)/tanh(pb /2h1)

k ′i = √1 − k2i

The values of functions K(k)/K(k ′) and K(ki )/K(k ′i ) can be found using (3.61).The characteristic impedance is then obtained from

Z0 =60p

√ere

1K(k2)K(k ′2)

+K(k)K(k ′)

(3.90)

The structure shown in Figure 3.24 reduces to that shown in Figure 3.23 whenh1 → ∞.

3.5.3 Conductor-Backed Coplanar Waveguide with Upper Shielding

The cross section of a conductor-backed CPW with upper shielding is shown inFigure 3.25. This structure has a backside ground plane. The lower ground planeadds mechanical strength to the circuit and increases its power handling capability.The quasistatic effective dielectric constant of this transmission line can also beexpressed in the form

ere = 1 + q(er − 1)

where

q =

K(k3)(k ′3)

K(k3)K(k ′3)

+K(k4)K(k ′4)

(3.91)


In (3.91), parameters k3 and k4 are given by [31]

k3 = tanh(pa /2h)/tanh(pb /2h)

k4 = tanh(pa /2h1)/tanh(pb /2h1) (3.92)

k ′i = √1 − k2i

The expression for the characteristic impedance is

Z0 =60p

√ere

1K(k3)K(k ′3)

+K(k4)K(k ′4)

(3.93)

The characteristic impedance and effective dielectric constant of a conductor-backed CPW with upper shielding are shown in Figure 3.27 as a function of a/bfor various values of h/b and er = 10.

3.5.4 Coupled Coplanar Waveguides

The cross section of coupled CPWs is shown in Figure 3.28. Unfortunately, very fewsimple formulas are available in the literature for the determination of parameters ofcoupled CPWs. For a value of er = 12.9, the even- and odd-mode characteristicimpedances of coupled CPWs are shown in Figure 3.29 [32]. For the same parame-ters of the structure, the even- and odd-mode effective dielectric constants areshown in Figure 3.30.

3.6 Suspended and Inverted Microstrip Lines

The cross section of suspended and inverted microstrip lines is shown in Figures3.31(a) and 3.31(b), respectively. These lines achieve a lower loss (or higer Q) thanpossible with microstrip lines. Further, these lines have a much lower effectivedielectric constant (compared with that of a microstrip line), thus leading to theirperformance being less sensitive to dimensional tolerances at high frequencies.Further, the wide range of impedance values achievable using these lines makesthem particularly suitable in realizing filters. The quasistatic equations for thedesign of suspended and inverted microstrip lines have been given by Pramanickand Bhartia [36] and Tomar and Bhartia [37]. The characteristic impedances ofinverted and supsended microstrip lines (t ! h) are given by the following expression[36]:

Z0(0) =60

√erelnF f (u)

u+ √1 + S2

uD2G (3.94)

where


Figure 3.27 (a) Characteristic impedance, and (b) effective dielectric constant of a conductor-backed CPW with upper shielding shown in inset. (From: [31]. 1987 IEEE. Reprintedwith permission.)


Figure 3.28 Cross section of coupled coplanar lines. (From: [32]. 1996 IEEE. Reprinted withpermission.)

f (u) = 6 + (2p − 6) expF− S30.666u D0.7528G

The parameter is u = W /(a + b) for suspended microstrips and u = W /b forinverted microstrips.

The effective dielectric constant ere of a suspended microstrip is given by

√ere = F1 +ab Sa1 − b1 ln

Wb DS 1

√er− 1DG−1

(3.95)

where

a1 = S0.8621 − 0.1251 lnabD

4

b1 = S0.4986 − 0.1397 lnabD

4

The effective dielectric constant ere of an inverted microstrip is given by

√ere = 1 +ab Sa1 − b1 ln

Wb DX√er − 1C (3.96)

where

a1 = S0.5173 − 0.1515 lnabD

2

b1 = S0.3092 − 0.1047 lnabD

2


Figure 3.29 Characteristic impedance of (a) even and (b) odd modes of coupled coplanar linesshown 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 er ≤ 6. For er ≈ 10, the error is less than ±2%. The characteristicimpedance and phase velocity of a suspended microstrip are shown in Figure 3.32(a)for a value of er = 3.78. The characteristic impedance and effective dielectricconstant of an inverted microstrip line are shown in Figure 3.32(b) for variousvalues of er .

Closed-form formulas are generally not available for coupled suspended andinverted striplines. Electromagnetic simulators, however, can be used to designcircuits using such lines.


Figure 3.30 Effective dielectric constant of even and odd modes of coupled coplanar lines shownin 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 Broadside-Coupled Lines 93

Figure 3.32 (a) Characteristic impedance and phase velocity of suspended microstrip line. er =3.78. (b) Characteristic impedance and effective dielectric constant of inverted micro-strip line. (From: [36]. 1985 IEEE. Reprinted with permission.)

3.7 Broadside-Coupled Lines

In earlier sections, we have considered coupling between lines that lie in the sameplane (edge-coupled lines). Although edge-coupled lines are easier to fabricate, they


are not suitable for tight coupling. Tight coupling can be obtained by placingcoupled lines in a broadside manner. Further, in keeping with the trend towardminiaturization, multilayer microwave circuits that use broadside coupling betweenlines are becoming a reality and are described in Chapter 8.

3.7.1 Broadside-Coupled Striplines

General broadside-coupled microstrip lines are shown in Figure 3.33. For er1 = er2= er , the structure reduces to broadside-coupled striplines. For coupled striplines,the even- and odd-mode effective dielectric constants are the same and are givenby

eree = ereo = er

The characteristic impedance of the odd mode can be found using the followingexpression [38]:

Z0o√er = Z a0∞ − DZ a

0∞ (3.97)

where

Z a0∞ = 60 lnF3S

W+ √S S

WD2 + 1G (3.98)

Figure 3.33 (a) Even-mode and (b) odd-mode field distribution of general broadside coupledmicrostrip lines. (From: [38]. 1988 IEEE. Reprinted with permission.)


and

DZ a0∞ = 5P for

WS

≤ 1/2

PQ forWS

≥ 1/2

P = 270F1 − tanhS0.28 + 1.2√b − SS DG

Q = 1 − tanh−1 30.48√2WS

− 1

S1 +b − S

S D2 4Furthermore, the even-mode characteristic impedance can be found using

Z0e =60p

√er

K(k ′)K(k)

(3.99)

where

k = tanhS293.9S /b

Z0o√erD (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 Broadside-Coupled Suspended Microstrip Lines

The structure shown in Figure 3.33 reduces to broadside-coupled suspended micro-strip lines for er1 = er ≥ 1, er2 = 1. The even- and odd-mode characteristic impedancesof broadside-coupled suspended microstrip lines are given by

Z0e =Z a

0e

√eree(3.101)

Z0o =Z a

0o

√ereo(3.102)

where Z a0e and Z a

0o are the even- and odd-mode characteristic impedances of thecorresponding air-filled homogeneous broadside-coupled striplines (er = 1). Theirvalues can be found using (3.97) and (3.99), respectively. Furthermore, the even-and odd-mode effective dielectric constants are given by [38]


ereo =12

(er + 1) + q(er − 1)

2(3.103)

where

q = q∞qc

q∞ = S1 +5SW D−a(U)b(er )

a(U) = 1 +149

lnFU4 + (U/52)2

U4 + 0.432 G +1

18.7lnF1 + S U

18.1D3G

U = 2W /S

b(er ) = 0.564Ser − 0.9er + 3 D0.053

and

qc = tanhF1.043 + 0.121Sb − SS D − 1.164S S

b − S DG (3.104)

eree = H1 +Sb Fa1 − b1 lnSW

b DG X√er − 1CJ2

where

a1 = [0.8145 − 0.05824 ln(S /b)]8

b1 = [0.7581 − 0.07143 ln(S /b)]8

These equations offer an accuracy of about 1% for er ≤ 16, S/b ≤ 0.4, and W /b≤ 1.2. These conditions are usually met in practice. The even- and odd-modecharacteristic impedances of coupled suspended microstrip lines are shown in Figure3.34(a) for er = 2.32. For the same parameters, the effective dielectric constantsare shown in Figure 3.34(b).

3.7.3 Broadside-Coupled Offset Striplines

Broadside-coupled offset striplines are shown in Figure 3.35. This structure is moregeneral than the broadside-coupled striplines configuration discussed in Section3.7.1 or the edge-coupled stripline configuration shown in Figure 3.11. Shelton[39] has given closed-form expressions for the analysis and synthesis of broadside-coupled offset lines. Here, we present the synthesis equations only as they are more


Figure 3.34 (a) Characteristic impedance and (b) effective dielectric constants of coupled broadsidecoupled suspended microstrip lines. er = 2.32. (From: [38]. 1988 IEEE. Reprintedwith permission.)

Figure 3.35 Broadside coupled off-set striplines.


frequently used. Two sets of equations are given, one for tightly coupled lines andthe other for loosely coupled lines. The conditions for tight and loose coupling aredefined by

Tight coupling case:w′

1 − s′ ≥ 0.35 (3.105)

w′cs′ ≥ 0.7

Loose coupling case:w′

1 − s′ ≥ 0.35 (3.106)

2w′o1 + s′ ≥ 0.85

In these equations, s′ = S/b, w′ = W /b, w′c = Wc /b, and w′o = Wo /b denote thenormalized values. The coupling between TEM lines can be expressed in terms ofeven- and odd-mode characteristic impedances. For a TEM coupler that is matchedat all its ports:

Z0eZ0

=Z0Z0o

(3.107)

Defining r as

√r =Z0eZ0

=Z0Z0o

(3.108)

we obtain the synthesis equation given here:

Tight coupling case: A = expF 60p2

√er Z0S1 − rs′

√r DG

B =A − 2 + √A2 − 4A

2

p =(B − 1)S1 + s′

2 D + √(B − 1)2S1 + s′2 D2 + 4s′B

2

r =s′Bp

Cfo =1p H−

21 − s′ ln s′ +

1s′ lnF pr

(p + s′) (1 + p) (r − s′) (1 − r)GJ

3.8 Slot-Coupled Microstrip Lines 99

Co =120p √r

√er Z0

w′ =s′(1 − s′)

2(Co − Cfo )

w′o =1

2p H(1 + s′) lnpr

+ (1 − s′) lnF(1 + p) (r − s′)(s′ + p) (1 − r)GJ

Loose coupling case: Co =120p √r

√er Z0(3.109)

DC =120p

√er Z0

(r − 1)

√r

K =1

expSpDC2 D − 1

a = √F(s′ − K)(s′ + 1) G

2+ K −

(s′ − K)(s′ + 1)

q =Ka

Cfo =2p F 1

1 + s′ ln1 + a

a(1 − q)−

11 − s′ ln qG

w′c =1p F(s′ ln

qa

+ (1 − s′) lnS1 − q1 + a DG

Cf (a = ∞) = −2p F 1

1 + s′ lnS1 − s′2 D +

11 − s′ lnS1 + s′

2 DGw′ =

1 − s′2

4[Co − Cfo − Cf (a = ∞)]

3.8 Slot-Coupled Microstrip Lines

Slot-coupled microstrip lines are shown in Figure 3.36. This configuration is usefulfor realizing coupling in multilayer MICs. Directional couplers realized using thisconfiguration can achieve both tight and loose coupling values. The quasistaticeven-mode effective dielectric constant and the characteristic impedance of thestructure are given by [40]


Figure 3.36 Slot-coupled microstrip lines.

eree =er

K′(k1)K(k1)

+K(k2)K′(k2)

K′(k1)K(k1)

+K(k2)K′(k2)

(3.110)

Z0e =60p

√eree

1K′(k1)K(k1)

+K(k2)K′(k2)

(3.111)

In (3.112), parameters k1 and k2 are given by

k1 = √ sinh2(pG/4h)

sinh2(pG/4h) + cosh2(pW/4h)(3.112)

k2 = tanh(pW/4ho ) (3.113)

Furthermore, K′(ki ) = K(k ′i ), where k ′i = √1 − k2i .

The quasistatic odd-mode effective dielectric constant and the characteristicimpedance of the structure are given by

ereo =er

K(k3)K′(k3)

+K(k4)K′(k4)

K(k3)K′(k3)

+K(k4)K′(k4)

(3.114)

Z0e =60p

√ereo

1K(k3)K′(k3)

+K(k4)K′(k4)

(3.115)

with parameters k3 and k4 given by

k3 = tanh(pW/4h) (3.116)

k4 = tanh(pW/4ho ) (3.117)


Figure 3.37 Even- and odd-mode (a) characteristic impedances, and (b) effective dielectric con-stants of slot-coupled microstrip lines shown in Figure 3.36. (From: [40]. 1991 IEEE.Reprinted with permission.)

Figure 3.37(a) shows the variation of even- and odd-mode characteristics imped-ances of the structure as a function of strip width for a value of er = 9.9, while Figure3.37(b) gives the variation of even- and odd-mode effective dielectric constants.

In this chapter, we have described some commonly used single and coupledstrip transmission lines. Because it is not in the scope of this book to cover all


transmission lines, readers are referred to the Transmission Line Design Handbookby Wadell [35] and Microstrip Lines and Slotlines by Gupta et al. [13], whichprovide a comprehensive treatment of printed transmission lines.

References

[1] Collin, R. E., Field Theory of Guided Waves, 2nd ed., New York: IEEE Press, 1991.[2] Wheeler, H. A., ‘‘Formulas for the Skin Effect,’’ Proc. IRE, Vol. 30, 1942, pp. 412–424.[3] Wheeler, H. A., ‘‘Transmission Line Properties of Strip on a Dielectric Sheet on a Plane,’’

IEEE Trans. Microwave Theory Tech., Vol. MTT-25, August 1977, pp. 631–647.[4] Kammler, D. W., ‘‘Calculation of Characteristic Admittances and Coupling Coefficients

for Strip Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-16,November 1968, pp. 925–937.

[5] Howe, H., Jr., Stripline Circuit Design, Dedham, MA: Artech House, 1974.[6] Cohn, S. B., ‘‘Characteristic Impedance of Shielded Strip Transmission Lines,’’ IRE Trans.,

Vol. MTT-2, July 1954, pp. 52–55.[7] Hilberg, W., ‘‘From Approximations to Exact Relations for Characteristic Impedances,’’

IEEE Trans. Microwave Theory Tech., Vol. MTT-17, May 1969, pp. 259–265.[8] Wheeler, H. A., ‘‘Transmission Line Properties of a Stripline Between Parallel Planes,’’

IEEE Trans. Microwave Theory Tech., Vol. MTT-26, November 1978, pp. 866–876.[9] Cohn, S. B., ‘‘Problems in Strip Transmission Line,’’ IRE Trans., Vol. MTT-3,

March 1955, pp. 119–126.[10] Vendelin, G. D., ‘‘Limitations on Stripline Q,’’ Microwave J., Vol. 13, May 1970,

pp. 63–69.[11] Cohn, S. B., ‘‘Shielded Coupled Strip Transmission Line,’’ IRE Trans., Vol. MTT-3,

October 1955, pp. 29–38.[12] Bahl, I. J., and Bhartia, Microwave Solid-State Circuit Design, New York: John Wiley &

Sons, 1988.[13] Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House,

1996.[14] Wheeler, H. A., ‘‘Transmission Line Properties of Parallel Strips Separated by a Dielectric

Sheet,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, 1965, pp. 172–185.[15] Schneider, M. V., ‘‘Microstrip Lines for Microwave Integrated Circuits,’’ Bell System

Tech. J., Vol. 48, 1969, pp. 1421–1444.[16] Pucel, R. A., et al., ‘‘Losses in Microstrip,’’ IEEE Trans. Microwave Theory Tech.,

Vol. MTT-16, 1968, pp. 342–350. Corrections: ibid, MTT-16, 1968, p. 1064.[17] Denlinger, E. J., ‘‘Losses in Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech.,

Vol. MTT-28, June 1980, pp. 513–522.[18] Hammerstad, E., and O. Jenson, ‘‘Accurate Models for Microstrip Computer-Aided

Design,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1980, pp. 407–409.[19] Kobayashi, M., ‘‘A Dispersion Formula Satisfying Recent Requirements in Micro-

strip CAD,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-36, August 1988,pp. 1246–1250.

[20] Bianco, B., et al., ‘‘Frequency Dependence of Microstrip Parameters,’’ Alta Frequenza,Vol. 43, 1974, pp. 413–416.

[21] Garg, R., and I. J. Bahl, ‘‘Characteristics of Coupled Microstrip Lines,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-27, July 1979, pp. 700–705.

[22] Kirschning, M., and R. H. Jansen, ‘‘Accurate Wide-Range Design Equations for theFrequency-Dependent Characteristics of Parallel Coupled Microstrip Lines,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-32, January 1984, pp. 83–90. Corrections: IEEETrans. Microwave Theory Tech., March 1985, p. 288.


[23] Veghte, R. L., and C. A. Balanis, ‘‘Dispersion of Transient Signals in Microstrip Transmis-sion Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-34, December 1986,pp. 1427–1436.

[24] York, R. A., and R. C. Compton, ‘‘Experimental Evaluation of Existing CAD Models forMicrostrip Dispersion,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-38,March 1990, pp. 327–328.

[25] Wen, C. P., ‘‘Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonre-ciprocal Gyromagnetic Device Applications,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-17, December 1969, pp. 1087–1090.

[26] Wen, C. P., ‘‘Coplanar-Waveguide Directional Couplers,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-18, June 1970, pp. 318–322.

[27] Ghione, G., and C. Naldi, ‘‘Parameters of Coplanar Waveguides with Lower GroundPlanes,’’ Electronics Letters, Vol. 19, September 1983, pp. 734–735.

[28] Rowe, D. A., and B. Y. Lao, ‘‘Numerical Analysis fo Shielded Coplanar Waveguides,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-31, November 1983, pp. 911–925.

[29] Leong, M. S., et al., ‘‘Effect of a Conducting Enclosure on the Characteristic Impedanceof Coplanar Waveguides,’’ Microwave J., Vol. 29, August 1986, pp. 105–108.

[30] Ghione, G., and C. Naldi, ‘‘Analytical Formulas for Coplanar Lines in Hybrid and Mono-lithic,’’ Electronics Letters, Vol. 20, February 1984, pp. 179–181.

[31] Ghione, G., and C. Naldi, ‘‘Coplanar Waveguides for MMIC Applications: Effect ofUpper Shielding, Conductor Backing, Finite Extent Ground Planes and Line-to-Line Cou-pling,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35, March 1987, pp. 260–267.

[32] Cheng, K. K. M., ‘‘Analysis and Synthesis of Coplanar Coupled Lines on Substrates ofFinite Thickness,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-44, April 1966,pp. 636–639.

[33] Shih, Y. C., and T. Itoh, ‘‘Analysis of Conductor-Backed Coplanar Waveguide,’’ Electron-ics Letters, Vol. 18, June 1982, pp. 538–540.

[34] Kitazawa, T., and R. Mittra, ‘‘Quasi-Static Characteristics of Asymmetrical and CoupledCoplanar-Type Transmission Lines,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-33, September 1985, pp. 771–778.

[35] Wadell, B. C., Transmission Line Design Handbook, Norwood, MA: Artech House, 1991.[36] Pramanick, P., and P. Bhartia, ‘‘CAD Models for Millimeter-Wave Finlines and Suspended-

Substrate Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-33,December 1985, pp. 1429–1435.

[37] Tomar, R. S., and P. Bhartia, ‘‘New Quasi-Static Models for the Computer-Aided Designof Suspended and Inverted Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-35, April 1987, pp. 453–457. Corrections: IEEE Trans. Microwave TheoryTech., November 1987, p. 1076.

[38] Bhartia, P., and P. Pramanick, ‘‘Computer-Aided Design Models for Broadside-CoupledStriplines and Millimeter-Wave Suspended Substrate Microstrip Lines,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-36, November 1988, pp. 1476–1481. Corrections:IEEE Trans. Microwave Theory Tech., October 1989, p. 1658.

[39] Shelton, J. P., ‘‘Impedances of Offset Parallel-Coupled Strip Transmission Lines,’’ IEEETrans. Microwave Theory Tech., Vol. MTT-14, January 1966, pp. 7–15. Corrections:IEEE Trans. Microwave Theory Tech., 1996, p. 249.

[40] Wong, M. F., et al., ‘‘Analysis and Design of Slot-Coupled Directional Couplers BetweenDouble-Sided Substrate Microstrip Lines,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-39, December 1991, pp. 2123–2128.

C H A P T E R 4

Analysis of Uniformly Coupled Lines

Traditionally, two approaches have been used to study coupling between transmis-sion lines (i.e., normal-mode and coupled-mode). The normal-modes of symmetricalcoupled lines are the even and odd modes. The coupling between symmetrical linescan be determined in terms of phase velocities and characteristic impedances ofthe even and odd modes of coupled lines [1–4]. When the coupled lines are asymmet-rical, the even and odd modes are no longer the normal modes of the structureand are now designated as the c and p modes. Knowing the c- and p -modeparameters, the coupling between asymmetrical lines can be determined.

The normal mode theory provides an exact method of analysis for coupledlines. In some cases however, (e.g., when two transmission lines in nonreciprocalmedia are coupled), the task of determining normal-mode parameters is very compli-cated. In this case, another approach known as the coupled-mode theory may proveto be easier and more intuitive. This theory is discussed in the next chapter.

In this chapter, we discuss the normal-mode analysis of uniform symmetricaland asymmetrical coupled lines. We first show how the analysis of a symmetricalfour-port network is reduced to analyzing two two-port networks using the even-and odd-mode analysis. We also examine conditions under which a four-portnetwork composed of symmetrical coupled lines behaves as a forward-wave orbackward-wave directional coupler as well as the unique properties of backward-wave and forward-wave directional couplers. Section 4.3 describes the normal-mode analysis of asymmetrical coupled lines. Also given are the relations betweendistributed line parameters (e.g., L, C, Lm , and Cm ), phase velocities, and character-istic impedances of normal modes along with the Z (impedance) parameters of afour-part network consisting of asymmetrical coupled lines. Once we know theZ-parameters of a linear network, we can determine the response of the networkto any arbitrary excitation and termination.

Determining normal-mode parameters of asymmetrical coupled lines is gener-ally quite complicated. Furthermore, design data are available in the literature foronly a few cases. In Section 4.3, we first describe an approximate method fromwhich the parameters of asymmetrical coupled lines are determined from the data ofsymmetrical coupled lines. We then discuss how these parameters can be determinedusing a popular EM simulator that is available at no cost. We also discuss thedesign of asymmetrical backward-wave and forward-wave 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 Odd-Mode Analysis of Symmetrical Networks

A four-port network (shown in Figure 4.1) is assumed to be symmetrical aboutthe plane PP ′. It is also assumed that the impedances terminating various portsare 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. Thesewaves are referred to as incident or reflected.

The relationship between incident and reflected voltage waves at different portsof the network as shown in Figure 4.1 can be expressed as

3V −

1

V −2

V −3

V −4

4 = [S] 3V +

1

V +2

V +3

V +4

4 (4.1)

where

[S] = 3S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S444 (4.2)

denotes the scattering matrix of the network. Note that because all the ports ofthe network are assumed to be terminated in identical loads (denoted by impedanceZ0), the unnormalized scattering matrix is identical to the normalized scatteringmatrix as discussed in Section 2.3.4.

Not all the elements of the scattering matrix of the network shown in Figure4.1 are independent. For example, because of the assumed symmetry and thereciprocal nature of the structure:

Figure 4.1 A four-port symmetrical network. PP ′ is a plane of symmetry.

4.1 Even- and Odd-Mode Analysis of Symmetrical Networks 107

S21 = S12, S31 = S13, S41 = S14, S32 = S23

S42 = S24, S43 = S34

S33 = S11, S44 = S22, S34 = S12, S23 = S14

The scattering matrix of (4.2) can therefore be expressed as

[S] = 3S11 S21 S31 S41

S21 S22 S41 S42

S31 S41 S11 S21

S41 S42 S21 S224 (4.3)

In a more compact notation, the matrix (4.3) can be expressed as

[S] = F[SA ] [SB ][SB ] [SA ]G (4.4)

where

[SA ] = FS11 S21

S21 S22G (4.5)

and

[SB ] = FS31 S41

S41 S42G (4.6)

(4.1) then becomes

3V −

1

V −2

V −3

V −4

4 = F[SA ] [SB ][SB ] [SA ]G 3

V +1

V +2

V +3

V +4

4 (4.7)

It may be noted that symmetrical structures show some special electrical behav-ior. If the symmetrical ports 1 and 3 are connected to equal magnitude and in-phase sources, the voltages at ports 1 and 3 will be equal in magnitude and inphase. Similarly, the voltages will be equal in magnitude and in-phase at ports 2and 4. This scheme is shown in Figure 4.2(a) and is called an even-mode excitation.Furthermore, if the symmetrical ports 1 and 3 are connected to equal magnitudebut out-of-phase sources, the voltages at ports 1 and 3 will be equal in magnitudebut out of phase. The voltages at ports 2 and 4 will also be equal in magnitude butout of phase. This excitation scheme is shown in Figure 4.2(b) and is called an


Figure 4.2 A four-port symmetrical network excited by (a) even-mode and (b) odd-mode sources.

odd mode excitation. The algebraic sum of these two excitations is equivalent tothe excitation scheme shown in Figure 4.1.

4.1.1 Even-Mode Excitation

Figure 4.2(a) shows a symmetrical structure excited by equal magnitude and in-phase sources at ports 1 and 3. The incident and reflected voltages set up at thedifferent ports are also shown in the same figure. Let V ±

3 = V ±1 = V ±

1e andV ±

4 = V ±2 = V ±

2e , where the suffix e has been used to denote the even mode. Further-more, by the substitution of V ±

1 , V ±2 , V ±

3 , and V ±4 in (4.7), the reflected voltages

can be expressed in terms of incident voltages as follows:

3V −

1e

V −2e

V −1e

V −2e

4 = F[SA ] [SB ][SB ] [SA ]G 3

V +1e

V +2e

V +1e

V +2e

4 (4.8)

From (4.8), we obtain

4.1 Even- and Odd-Mode Analysis of Symmetrical Networks 109

FV −1e

V −2eG = ([SA ] + [SB ]) FV +

1e

V +2eG (4.9)

4.1.2 Odd-Mode Excitation

Figure 4.2(b) shows the symmetrical structure excited by equal magnitude but out-of-phase sources at ports 1 and 3. The incident and reflected voltages set up atvarious ports by the odd-mode sources are also shown in the same figure. LetV ±

1 = −V ±3 = V ±

1o and V ±2 = −V ±

4 = V ±2o , where the suffix o has been used to denote

that the quantities correspond to the odd mode. Furthermore, by substitution ofV ±

1 , V ±2 , V ±

3 , and V ±4 in (4.7), the reflected voltages at various ports can be

expressed in terms of incident voltages as follows:

3V −

1o

V −2o

−V −1o

−V −2o

4 = F[SA ] [SB ][SB ] [SA ]G 3

V +1o

V +2o

−V +1o

−V +2o

4 (4.10)

which gives

FV −1o

V −2oG = ([SA ] − [SB ]) FV +

1o

V +2oG (4.11)

Scattering Matrix in Terms of Even- and Odd-Mode Parameters

Equation (4.9) can be written as

FV −1e

V −2eG = [Se ] FV +

1e

V +2eG (4.12)

where

[Se ] = [SA ] + [SB ] (4.13)

Similarly, (4.11) can be written as

FV −1o

V −2oG = [So ] FV +

1o

V +2oG (4.14)

where

[So ] = [SA ] − [SB ] (4.15)


From (4.13) and (4.15) we obtain

[SA ] =[Se ] + [So ]

2(4.16)

and

[SB ] =[Se ] − [So ]

2(4.17)

Therefore, if the scattering matrices [Se ] and [So ], which are matrices of order2 × 2, are known, then the scattering matrices [SA ] and [SB ] can be determinedusing (4.16) and (4.17), respectively. Furthermore, if the scattering matricies [SA ]and [SB ] are known, the complete scattering matrix of the four-port network canbe determined using (4.4).

Let the elements of the scattering matrices [Se ] and [So ] be given by

[Se ] = FS11e S21e

S21e S22eG (4.18)

and

[So ] = FS11o S21o

S21o S22oG (4.19)

Using (4.16) and (4.17), the 2 × 2 scattering matrices [SA ] and [SB ] are thenfound to be

[SA ] = 3S11e + S11o

2S21e + S21o

2S21e + S21o

2S22e + S22o

24 (4.20)

and

[SB ] = 3S11e − S11o

2S21e − S21o

2S21e − S21o

2S22e − S22o

24 (4.21)

Furthermore, using (4.4), the various elements of the scattering matrix of thefour-port network are given as follows:

4.2 Directional Couplers Using Uniform Coupled Lines 111

S11 =S11e + S11o

2, S12 = S21 , S13 = S31 , S14 = S41

S21 =S21e + S21o

2, S22 =

S22e + S22o2

, S23 = S41 ,

S24 =S22e − S22o

2, S31 =

S11e − S11o2

, S32 = S41 , (4.22)

S33 = S11 , S34 = S21

S41 =S21e − S21o

2, S42 = S24 , S43 = S21 , S44 = S22

In the case of even- and odd-mode excitations, the plane of symmetry behaveslike a magnetic or electric wall, respectively. The reduced circuit for determiningscattering 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 odd-mode analysis, we nowdiscuss the conditions under which a symmetrical four-port network composed ofuniformly 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 odd-modes of thestructure shown in Figure 4.1.

Figure 4.4 A four-port network composed of uniform coupled symmetrical lines.


2, which showed that if all ports of a four-port network are matched, then thenetwork behaves like a directional coupler. Because the network as shown in Figure4.4 is assumed to be symmetrical about the mid-plane PP ′, matching of ports 1and 2 automatically ensures that the ports 3 and 4 are also matched. Therefore,the condition

S11 = S22 = 0

leads to the result that the network is a directional coupler.In terms of even- and odd-mode reflection coefficients, the scattering parameters

S11 and S22 are given by (4.22) as

S11 =S11e + S11o

2(4.23)

S22 =S22e + S22o

2(4.24)

The equivalent circuit for determining the even-mode scattering parametersS11e and S22e is shown in Figure 4.5(a), where Z0e and be denote the characteristicimpedance and propagation constant of the even-mode of symmetrical coupledlines. Similarly, the equivalent circuit for determining the odd-mode scatteringparameters S11o and S22o is shown in Figure 4.5(b), where Z0o and bo denote the

Figure 4.5 Equivalent circuit for determining scattering matrix of the (a) even mode and (b) oddmode of the structure shown in Figure 4.4.


characteristic impedance and propagation constant of the odd mode of symmetricalcoupled lines.

To obtain S11 = S22 = 0, which are the conditions for realizing a directionalcoupler, the following possibilities exist:

Case I

S11e = S11o = S22e = S22o = 0 (4.25)

When the above values are substituted in (4.23) and (4.24), we obtainS11 = S22 = 0. Furthermore, using (4.22), we obtain

S13 = S31 = S42 = S24 = 0 (4.26)

Therefore, in this case [when (4.25) is satisfied], no power is coupled to the back-ward port. For example, if power is incident at port 1, then no power is coupledto port 3 on the coupled line. Similarly, no power is coupled between ports 2 and4. However, power can be coupled between ports 1 and 4 (or between ports 2and 3). These types of couplers are called forward-wave or codirectional couplersand are discussed further in this chapter.

Case II

From (4.23) and (4.24), we see that S11 = 0 and S22 = 0 can also be obtained ifthe following conditions are satisfied:

S11e = −S11o (4.27)

and

S22e = −S22o (4.28)

where S11e , S11o , S22e , and S22o are not equal to zero. Using (4.22), we then findthat

S31 ≠ 0, S42 ≠ 0

In this case, power is thus coupled to the backward port. From the properties ofa directional coupler, we know that if S31 ≠ 0, then either

S41 = 0 or S21 = 0

To ensure that no power is coupled in the forward direction on the coupled line,it is required that S41 = 0. Using (4.22), we find that this condition is satisfiedwhen


S21e = S21o (4.29)

Therefore, when conditions given by (4.27) through (4.29) hold, the structurebehaves as a backward-wave directional coupler.

4.2.1 Forward-Wave (or Codirectional) Directional Couplers

As discussed earlier, the four-port network as shown in Figure 4.4 behaves like aforward-wave directional coupler when (4.25) is satisfied. Referring to the equiva-lent 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 sectionif

Z0e = Z0o = Z0

The above condition can be nearly met in practice by keeping a relatively largespacing between the lines. Substituting

S11e = S22e = 0

in the following equation (this equation follows from the unitary property of thescattering matrix):

|S11e |2 + |S21e |2 = |S22e |2 + |S21e |2 = 1

we obtain

|S21e | = 1

or

S21e = e −jc e (4.30)

where c e denotes the phase difference between ports 1 and 2 for the even-modesignal. Similarly, by substituting

S11o = S22o = 0

in the equation

|S11o |2 + |S21o |2 = |S22o |2 + |S21o |2 = 1

we obtain

S21o = e −jc o (4.31)


where co denotes the phase difference between ports 1 and 2 for the odd-modesignal.

Because the coupled structure is assumed to be uniform, we can further write

c e = b e l (4.32)

and

co = bo l (4.33)

where b e and bo denote the propagation constants of the even and odd-modesignals, respectively, and l is the length of the coupling section.

Furthermore, using (4.22), the scattering parameters of an ideal forward-wavedirectional coupler are given by

S11 = S22 = S33 = S44 = 0 (4.34)

S12 = S21 = S34 = S43 =S21e + S21o

2=

e −jbe l + e −jbo l

2(4.35)

= e

−j(be + bo )l2

cosF(be − bo )l2 G

S14 = S41 = S23 = S32 =S21e − S21o

2=

e −jbe l − e −jbo l

2(4.36)

= −je

−j(be + bo )l2

sinF(be − bo )l2 G

S13 = S31 = S24 = S42 = 0 (4.37)

The fractional power coupled from port 1 to port 4 is thus given by

P4P1

= |S41 |2 = sin2F(be − bo )l2 G (4.38)

while the fractional power coupled from port 1 to port 2 is given by

P2P1

= |S21 |2 = cos2F(be − bo )l2 G (4.39)

Notice that |S21 |2 + |S41 |2 = 1, accounting for all the incident power. It shouldbe apparent that forward-wave directional couplers cannot be obtained using TEMmode lines such as coaxial lines because for the TEM mode, the propagationconstants of the even and odd modes are equal, and therefore as shown by (4.36),


there is no coupling between ports 1 and 4 or between ports 2 and 3. Forward-wave coupling exists only in non-TEM lines such as metallic waveguides, fin lines,and dielectric waveguides and can also exist in quasi-TEM-mode transmission linessuch 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 Forward-Wave or Codirectional Couplers

1. From (4.38), we see that complete power can be transferred between linesif the length l of the directional coupler is chosen as

l =p

|be − bo | =l0

2 | X√eree − √ereo C |(4.40)

This result is significant in the sense that even for arbitrarily small valuesof difference in the propagation constants of even and odd modes, completepower can be transferred between the lines if the length of the coupler ischosen according to (4.40). We show later that it is not possible to completelytransfer power from one line to another in the case of backward-wavedirectional couplers.

2. By comparing (4.35) and (4.36), we find that the phase difference betweenS41 and S21 is 90 degrees. The wave on the ‘‘coupled’’ line is thus 90 degreesout of phase with the ‘‘direct’’ wave.

3. In deriving equations for the forward-wave coupling, we assumed that thecondition given by (4.25) is satisfied, which leads to zero coupling betweenports 1 and 3 or between ports 2 and 4. The directivity and isolation ofthe coupler are thus infinite. In general, however, the above condition cannotbe completely satisfied. Therefore, some finite amount of backward-wavecoupling, however small, always exists between coupled lines. The exactamount of backward-wave coupling (S31 and S42) can be determined using(4.22), if the values of S11e , S11o , S22e , and S22o are known.

4.2.2 Backward-Wave Directional Couplers

As discussed earlier, a symmetrical four-port network as shown in Figure 4.4behaves like a backward-wave directional coupler if the following conditions aresatisfied:

S11o = −S11e

S22o = −S22e

and

S21o = S21e

where S11o , S11e , S22o , and S22e are not equal to zero.


The above conditions are easily satisfied if the coupled lines are of the TEMtype similar to striplines and the even- and odd-mode characteristic impedancesof the lines are properly chosen. The equivalent circuits for the even and odd modesare 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 aregiven, respectively, by

FAe Be

Ce DeG = 3

cos(b l ) jZ0e sin(b l )j sin(b l )

Z0ecos(b l ) 4 (4.41)

and

FAo Bo

Co DoG = 3

cos(b l ) jZ0o sin(b l )j sin(b l )

Z0ocos(b l ) 4 (4.42)

where we assume that the propagation constants are the same for the even andodd modes and are denoted by b . Because the coupled lines are terminated in animpedance of Z0 , the even- and odd-mode reflection coefficients can be shownfrom (2.98) and (2.99) to be

S11e = S22e =jSZ0e

Z0−

Z0Z0e

D sin b l

2 cos b l + jSZ0eZ0

+Z0Z0e

D sin b l(4.43)

and

S11o = S22o =jSZ0o

Z0−

Z0Z0o

D sin b l

2 cos b l + jSZ0oZ0

+Z0Z0o

D sin b l(4.44)

Comparing (4.43) and (4.44), we find that the conditions S11e = −S11o andS22e = −S22o are satisfied for any arbitrary value of length l when

Z0eZ0

=Z0Z0o

or

Z0eZ0o = Z 20 (4.45)


The scattering parameters S21o and S21e can be computed from (2.100) asfollows:

S21e =2

2 cos b l + jSZ0eZ0

+Z0Z0e

D sin b l(4.46)

and

S21o =2

2 cos b l + jSZ0oZ0

+Z0Z0o

D sin b l(4.47)

We see that when (4.45) holds, S21e = S21o as required by (4.29). Therefore,(4.45) gives the necessary condition for TEM backward-wave directional couplers.Once the values of S11o , S11e , S21o , S21e are known, the scattering parameters ofan ideal backward-wave directional coupler can be easily determined using (4.22)as follows:

S11 = S22 = S33 = S44 = 0 (4.48)

S14 = S41 = S23 = S32 = 0 (4.49)

S12 = S21 = S34 = S43 =S21e + S21o

2= S21e = S21o (4.50)

=2

2 cos b l + jSZ0eZ0

+Z0oZ0

D sin b l

S13 = S31 = S24 = S42 =S11e − S11o

2= S11e = −S11o (4.51)

=jSZ0e

Z0−

Z0oZ0

D sin b l

2 cos b l + jSZ0eZ0

+Z0oZ0

D sin b l

From (4.45) and (4.50), we obtain

S21 = √1 − k2

√1 − k2 cos u + j sin u(4.52)

Furthermore, from (4.45) and (4.51),


S31 =jk sin u

√1 − k2 cos u + j sin u(4.53)

where u = b l, and


(4.54)

The maximal amount of coupling between ports 1 and 3 (or between ports 2 and4) occurs when

u = b l =p2

rads (4.55)

or

l =p

2b=

lg

4

where lg denotes the guide wavelength in the medium of the transmission line.The maximum value of coupling is found by substituting u = p /2 in (4.53), whichgives

|S13 | = |S31 | = |S24 | = |S42 | = k (4.56)

Furthermore, when u = p /2,

|S12 | = |S21 | = |S34 | = |S43 | = √1 − k2 (4.57)

Thus, at the frequency where u = b l = p /2, the scattering matrix of a backward-wave coupler can be represented as follows:

[S] = 30 −j√1 − k2 k 0

−j√1 − k2 0 0 k

k 0 0 −j√1 − k2

0 k −j√1 − k2 0

4 (4.58)

This scattering matrix is valid at the frequency where the length of the coupleris a quarter-wave long. We can calculate, however, the frequency response of thebackward-wave coupler at any other frequency using (4.48), (4.49), (4.52), and(4.53). The frequency response of ideal backward-wave couplers of various couplingvalues is given in Chapter 6.


From (4.45) and (4.54), the relationships between even- and odd-mode charac-teristic impedances and the voltage coupling coefficient k are given by

Z0e = Z0√1 + k1 − k

(4.59)

Z0o = Z0√1 − k1 + k

Example 4.1

Compute even- and odd-mode characteristic impedances of a 20-dB quarter-wave,50-ohm backward-wave coupler.

For a 20-dB coupler, k = 10−20/20 = 0.1. Given that the terminal impedance is50 ohms, then from (4.59), Z0e = 55.3 ohms and Z0o = 45.2 ohms.

Coupling k in Terms of Capacitance Parameters

Substituting values of Z0e and Z0o 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 Ca and Cm denote the capacitances of coupled lines as shown in Figure 3.5(Cb = Ca ).

Remarks on Backward-Wave Directional Couplers

1. Equation (4.53) shows that there exists a maximum value of backward-wave coupling that can be achieved between two coupled lines. The maxi-mum value of coupling which is given by (4.56) occurs when the length ofthe coupler is a quarter-wave long (or odd multiples thereof). This is unlikethe symmetrical forward-wave directional couplers case where arbitrarycoupling can be achieved between the lines by properly choosing the lengthof the coupling section.

2. By comparing (4.52) and (4.53), we find that the wave coupled to the‘‘backward’’ port (S31) is 90 degrees out of phase with the wave coupledto the ‘‘direct’’ port (S21). This relationship is independent of the electricallength of the coupling section. These types of couplers are thus capable ofbeing used as quadrature phase-shifters over a wide frequency range.

4.3 Uniformly Coupled Asymmetrical Lines

Symmetrical coupled lines represent a very useful but restricted class of coupledlines. 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 aforward-wave directional coupler using asymmetrical coupled lines is greater thanone formed using symmetrical coupled lines. Also, in some situations, the terminalimpedances 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 characteristicimpedances.

In this section, the analysis and design of asymmetrical coupled quasi-TEMmode lines is described. The normal-mode parameters of asymmetrical coupledlines are first defined. It is shown that six independent parameters are requiredto characterize asymmetrical coupled lines. The relation between normal-modeparameters (i.e., characteristic impedances, phase velocities) and line parameters(i.e., per unit length inductance, capacitance) are derived. Because symmetricallines are a special case of asymmetrical coupled lines, various expressions given inthe following sections can also be used for symmetrical coupled lines. A concisebut excellent analysis of asymmmetrical coupled lines is found in [5], which formsthe 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 ofpropagation (called normal modes). For asymmetrical coupled lines, the two normalmodes of propagation are known as the c and p modes. Strictly speaking, a structurecomposed 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 backward-traveling mode, however, are the same as those of the corresponding forward-wavetraveling mode. The c-mode is an even-like mode, while the p -mode is an odd-like mode.

c Mode

Figure 4.6 shows two uniformly coupled quasi-TEM mode asymmetrical transmis-sion lines. The assumption of quasi-TEM mode is made because it is possible todefine unique voltages and currents in this case as compared with non-TEM modetransmission 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.


lines for the c mode be denoted as shown. The forward-traveling voltage waveson lines 1 and 2 are denoted by V +

1c e−gc z and V +2c e−gc z, respectively; and the

corresponding current waves by I +1c e−gc z and I +

2c e−gc z, respectively. Similarly,V −

1c egc z and V −2c egc z, I −

1c egc z and I −2c egc z denote the corresponding quantities for

the backward-traveling mode.The characteristic impedance of line 1 (in the presence of line 2) can be defined

as

Zc1 =V +

1c

I +1c

=V −

1c

I −1c

(4.61)

Similarly, for the same mode the characteristic impedance of line 2 (in the presenceof line 1) can be defined as

Zc2 =V +

2c

I +2c

=V −

2c

I −2c

(4.62)

Furthermore, let the ratio of voltages on the two lines be defined by a parameterRc as follows:

Rc =V +

2c

V +1c

=V −

2c

V −1c

A c mode is therefore characterized by four parameters: g c , the propagationconstant of the mode; Zc1 and Zc2 , which are, respectively, the characteristicimpedances of lines 1 and 2; and Rc , the ratio of the voltages on the two lines.

p Mode

Similar to the c mode, a p mode is also characterized by four parameters: gp , thepropagation constant of the mode; Zp1 and Zp2 , which are, respectively, thecharacteristic impedances of lines 1 and 2; and Rp , the ratio of the voltages onthe two lines.

Of the eight quantities discussed above (four each for c and p modes), only sixare independent. The currents and voltages of the two modes satisfy the followingrelationships:

V +2c

V +1c

=V −

2c

V −1c

= −I +1p

I +2p

= −I −1p

I −2p

and

V +2p

V +1p

=V −

2p

V −1p

= −I +1c

I +2c

= −I −1c

I −2c


Therefore

Rc =V +

2c

V +1c

=V −

2c

V −1c

= −I +1p

I +2p

= −I −1p

I −2p

(4.63)

and

Rp =V +

2p

V +1p

=V −

2p

V −1p

= −I +1c

I +2c

= −I −1c

I −2c

(4.64)

Using (4.63) and (4.64), the relations between characteristic impedances Zc1 ,Zc2 , Zp1 , and Zp2 and ratio parameters Rc and Rp are found to be

Zc2Zc1

=Zp2Zp1

= −RcRp (4.65)

Therefore, a total number of six quantities (i.e., g c , gp , Zc1 [or Zc2], Zp1 [orZp2], Rc , and Rp ) is required to characterize asymmetrical coupled lines. It is notnecessary to specify both Zc1 and Zc2 , as they are related by (4.65). The sameholds for Zp1 and Zp2 . On the other hand, symmetrical coupled lines are completelycharacterized by four parameters: the even- and odd-mode characteristic imped-ances of any line (as both lines are identical) and the even- and odd-mode phaseconstants. For symmetrical coupled lines, Rc = 1 and Rp = −1.

Z- and Y-Parameters of a Four-Port Network in Terms of Normal-Mode Parameters

Figure 4.7 shows a four-port network composed of asymmetrical coupled lines.When one or more ports of the structure are excited, voltage and current wavesare set up on both the lines. The voltage and current waves can be expressed as alinear sum of forward- and backward-traveling c and p mode waves. For example,the voltage and current waves on the two lines can be represented as

V1(z) = A1e −gc z + A2egc z + A3e −gp z + A4egp z (4.66)

Figure 4.7 A four-port network composed of uniform asymmetrical coupled lines.


V2(z) = A1Rce −gc z + A2Rcegc z + A3Rp e −gp z + A4Rp egp z (4.67)

I1(z) = A1Yc1e −gc z − A2Yc1egc z + A3Yp1e −gp z − A4Yp1egp z (4.68)

I2(z) = A1RcYc2e −gc z − A2RcYc2egc z + A3RpYp2e −gp z − A4RpYp2egp z

(4.69)

where Yci = 1/Zci ; Yp i = 1/Zp i (i = 1, 2) and A1, A2, A3, and A4 are constantsdepending 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, thevoltage at port 2(V2) can be found by substituting z = l in (4.66). The voltagesand currents at various ports are found to be

V1 = A1 + A2 + A3 + A4 (4.70a)

V2 = A1e −gc l + A2egc l + A3e −gp l + A4egp l (4.70b)

V3 = A1Rc + A2Rc + A3Rp + A4Rp (4.70c)

V4 = A1Rce −gc l + A2Rcegc l + A3Rp e −gp l + A4Rp egp l (4.70d)

I1 = A1Yc1 − A2Yc1 + A3Yp1 − A4Yp1 (4.71a)

−I2 = A1Yc1e −gc l − A2Yc1eg c l + A3Yp1e −gp l − A4Yp1egp l (4.71b)

I3 = A1RcYc2 − A2RcYc2 + A3RpYp2 − A4RpYp2 (4.71c)

−I4 = A1RcYc2e −gc l − A2RcYc2egc l + A3RpYp2e −gp l − A4RpYp2egp l

(4.71d)

The set of equations given by (4.71a) to (4.71d) can be solved to obtain thecoefficients Ai in terms of I1 , I2 , I3 , and I4 . Furthermore, substituting these in(4.70a) to (4.70d), the Z-parameters can be determined from the resulting equationsby inspection. The Z-parameters of the four-port network are found in terms ofnormal-mode parameters to be:

Z11 = Z22 =Zc1 coth gc l(1 − Rc /Rp )

+Zp1 coth gp l(1 − Rp /Rc )

Z13 = Z31 = Z24 = Z42 =Zc1Rc coth gc l

(1 − Rc /Rp )+

Zp1Rp coth gp l(1 − Rp /Rc )

= −Zc2 coth gc l

Rp (1 − Rc /Rp )−

Zp2 coth gp lRc (1 − Rp /Rc )

Z14 = Z41 = Z32 = Z23 =RcZc1

(1 − Rc /Rp ) sinh gc l+

Rp Zp1(1 − Rp /Rc ) sinh gp l


Z12 = Z21 =Zc1

(1 − Rc /Rp ) sinh gc l+

Zp1(1 − Rp /Rc ) sinh gp l

Z33 = Z44 = −RcZc2 coth gc lRp (1 − Rc /Rp )

−Rp Zp2 coth gp lRc (1 − Rp /Rc )

=R2

c Zc1 coth gc l(1 − Rc /Rp )

+R2

p Zp1 coth gp l(1 − Rp /Rc )

Z34 = Z43 =R2

c Zc1(1 − Rc /Rp ) sinh gc l

+R2

p Zp1(1 − Rp /Rc ) sinh gp l

(4.72)

Furthermore, the Y-parameters of the four-port network shown in Figure 4.7are given by

Y11 = Y22 =Yc1 coth gc l(1 − Rc /Rp )

+Yp1 coth gp l(1 − Rp /Rc )

Y13 = Y31 = Y24 = Y42 = −Yc1 coth gc l

Rp (1 − Rc /Rp )−

Yp1 coth gp lRc (1 − Rp /Rc )

Y14 = Y41 = Y23 = Y32 =Yc1

(Rp − Rc ) sinh gc l+

Yp1(Rc − Rp ) sinh gp l

Y12 = Y21 = −Yc1

(1 − Rc /Rp ) sinh gc l−

Yp1(1 − Rp /Rc ) sinh gp l

Y33 = Y44 = −RcYc2 coth gc lRp (1 − Rc /Rp )

−RpYp2 coth gp lRc (1 − Rp /Rc )

Y34 = Y43 =RcYc2

Rp (1 − Rc /Rp ) sinh gc l+

RpYp2Rc (1 − Rp /Rc ) sinh gp l

(4.73)

Z -Parameters of Interdigital Two-Port Network

If the two ports (ports 2 and 3) of the four-port network shown in Figure 4.7 areterminated in an open circuit, a two-port network such as the one shown in Figure4.8 results with the Z-parameters given by

FZ11 Z21

Z21 Z22G = −j

Zc1(1 − Rc /Rp ) F cot uc Rc csc uc

Rc csc uc R2c cot uc

G (4.74)

−jZp1

(1 − Rp /Rc ) F cot up Rp csc up

Rp csc up R2p cot up

Gwhere uc = b c l and up = bp l. This subcircuit finds extensive application in planarmicrowave circuits such as bandpass filters.


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 withpermission.)

4.3.2 Distributed Equivalent Circuit of Coupled Lines

The distributed equivalent circuit of two uniformly coupled lossless asymmetricaltransmission lines is shown in Figure 4.9. The voltages and currents on the coupledlines are governed by the following differential equations [6, 7]:

∂V1(z, t)∂z

+ L1∂I1(z, t)

∂t+ Lm

∂I2(z, t)∂t

= 0 (4.75)

∂I1(z, t)∂z

+ C1∂V1(z, t)

∂t− Cm

∂V2(z, t)∂t

= 0 (4.76)

∂V2(z, t)∂z

+ L2∂I2(z, t)

∂t+ Lm

∂I1(z, t)∂t

= 0 (4.77)

∂I2(z, t)∂z

+ C2∂V2(z, t)

∂t− Cm

∂V1(z, t)∂t

= 0 (4.78)


Figure 4.9 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. L1 andC1 denote the (per unit) self-inductance and self-capacitance of line 1 in the presenceof line 2. Similarly, L2 and C2 denote the self-inductance and self-capacitance ofline 2 in the presence of line 1. Furthermore, Lm and Cm denote the mutualinductance and mutual capacitance between the lines, respectively. More specifi-cally, self- and mutual inductance and capacitance parameters are the elements ofinductance and capacitance matrices [L] and [C], where

[L] = FL11 L12

L21 L22G = FL1 Lm

Lm L2G (4.79)

[C] = FC11 C12

C21 C22G = F C1 −Cm

−Cm C2G (4.80)

For a lossless case, the inductance matrix [L] is given by [8]

[L] = e0m0[C]−1 (4.81)


where [C0] denotes the free-space 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)∂z

+ jvL1I1(z) + jvLmI2(z) = 0 (4.82)

∂I1(z)∂z

+ jvC1V1(z) − jvCmV2(z) = 0 (4.83)

∂V2(z)∂z

+ jvL2I2(z) + jvLmI1(z) = 0 (4.84)

∂I2(z)∂z

+ jvC2V2(z) − jvCmV1(z) = 0 (4.85)

By solving the set of coupled equations given by (4.82) to (4.85), the propagationconstants and other parameters of asymmetrical coupled lines defined earlier inthis section can be determined.

The complex propagation constants of the c and p modes are found to be

g 2c =

a1 + a22

+12

[(a1 − a2)2 + 4b1b2]1/2 (4.86)

and

g 2p =

a1 + a22

−12

[(a1 − a2)2 + 4b1b2]1/2 (4.87)

In these equations

a1 = y1z1 + ymzm

a2 = y2z2 + ymzm (4.88)

b1 = z1ym + y2zm

b2 = z2ym + y1zm

where

z1 = jvL1

z2 = jvL2 (4.89)

zm = jvLm

and


y1 = jvC1

y2 = jvC2 (4.90)

ym = −jvCm

By substituting values of a1 , a2 , b1 , and b2 from (4.88) in (4.86) and (4.87),we obtain the phase velocities of the c and p modes:

vc = (4.91)

FL1C1 + L2C2 − 2LmCm + √(L1C1 − L2C2)2 + 4(LmC1 − L2Cm )(LmC2 − L1Cm )

2 G−1/2

and

vp = (4.92)

FL1C1 + L2C2 − 2LmCm − √(L1C1 − L2C2)2 + 4(LmC1 − L2Cm )(LmC2 − L1Cm )

2 G−1/2

where vc, p = v /b c, p and b c, p = −jg c, p . Furthermore, the characteristic impedancesand admittances of the lines for the c and p modes and Rc and Rp parameters aregiven by

Zc1 = S 1gcD z1z2 − z2

mz2 − zmRc

=1

Yc1(4.93)

Zc2 = SRcgcD z1z2 − z2

mz1Rc − zm

=1

Yc2(4.94)

Zp1 = S 1gpD z1z2 − z2

mz2 − zmRp

=1

Yp1(4.95)

Zp2 = SRp

gpD z1z2 − z2

mz1Rp − zm

=1

Yp2(4.96)

Rc =(a2 − a1) + [(a2 − a1)2 + 4b1b2]1/2

2b1(4.97)

and

Rp =(a2 − a1) − [(a2 − a1)2 + 4b1b2]1/2

2b1(4.98)


Inductive and Capacitive Coupling Coefficients

The inductive coupling coefficient between the lines is defined by [6]

kL =Lm

√L1L2(4.99)

whereas the capacitive coupling coefficient between the lines is given by

kC =Cm

√C1C2(4.100)

4.3.3 Relation Between Normal Mode (c and p) and Distributed LineParameters

Symmetrical Lines

The relations between c and p parameters and distributed line parameters (self-and mutual inductances and capacitances) reduce to simple forms for some specialcases. For example, for symmetrical coupled lines, L1 = L2 and C1 = C2 . In thiscase, the c and p modes reduce to even and odd modes, respectively. From (4.86)and (4.87),

gc = jb e = jv √(L1 + Lm ) (C1 − Cm ) (4.101)

and

gp = jbo = jv √(L1 − Lm ) (C1 + Cm ) (4.102)

Furthermore, for symmetrical coupled lines Rc = 1 and Rp = −1 from (4.97)and (4.98). Substituting the values of Rc = 1, Rp = −1 and propagation constantsg c and gp from (4.101) and (4.102) in (4.93) to (4.96), we obtain

Zc1 = Zc2 = Z0e = √(L1 + Lm )(C1 − Cm )

(4.103)

Zp1 = Zp2 = Z0o = √(L1 − Lm )(C1 + Cm )

(4.104)

Using (4.101) to (4.104), the distributed line parameters are found in terms ofcharacteristic impedances and propagation constants of the even and odd modesto be

L1 = L2 =1

2v(b eZ0e + boZ0o ) (4.105)

C1 = C2 =1

2v S boZ0o

+b eZ0e

D (4.106)


Lm =1

2v(b eZ0e − boZ0o ) (4.107)

Cm =1

2v S boZ0o

−b eZ0e

D (4.108)

Note that for lines supporting pure TEM mode of propagation, the even- and odd-mode phase velocities are the same. (4.101) and (4.102) then lead to

LmL1

=CmC1

(4.109)

or from (4.99) and (4.100):

kC = kL (4.110)

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 TEM-mode coupled lines, the propagation constants of both the c andp modes are the same, and are given by

gc = gp = jb = jk0√er (4.111)

The following relations are satisfied by line parameters in this case:

L1C1 = L2C2 (4.112)

Cm

√C1C2=

Lm

√L1L2(4.113)

Furthermore, Rc and Rp are given by

Rc = −Rp = SZ2Z1D1/2

(4.114)

where Z1 = (L1 /C1)1/2 and Z2 = (L2 /C2)1/2.In this case, the Z-parameters of a four-port network as shown in Figure 4.7

are given by


Z11 = Z22 = −j2 SZ1

Z2D1/2

(Zc + Zp ) cot u

Z13 = Z31 = Z42 = Z24 = −j2

(Zc − Zp ) cot u

Z14 = Z41 = Z32 = Z23 = −j2

(Zc − Zp ) csc u (4.115)

Z12 = Z21 = −j2 SZ1

Z2D1/2

(Zc + Zp ) csc u

Z33 = Z44 = −j2 SZ2

Z1D1/2

(Zc + Zp ) cot u

Z34 = Z43 = −j2 SZ2

Z1D1/2

(Zc + Zp ) csc u

where

Zc = (Z1Z2)1/2 F1 + ym /(y1y2)1/2

1 − ym /(y1y2)1/2G1/2

and

Zp = (Z1Z2)1/2 F1 − ym /(y1y2)1/2

1 + ym /(y1y2)1/2G1/2

4.3.4 Approximate Distributed Line or Normal-Mode Parameters ofAsymmetrical Coupled Lines

The performance of a network consisting of coupled asymmetrical lines can bedetermined if the distributed line parameters (i.e., L1 , L2 , C1 , C2 , Lm , and Cm )or the normal-mode parameters (i.e., Zc1 [or Zc2], Zp1 [or Zp2], g c , gp , Rc , andRp ) are known. The computation of distributed line or normal-mode parameters,however, is quite complicated and can only be carried out using field theoreticalmethods [8]. Commercially available programs based on this method are alsoavailable [9, 10]. Ikalainen and Matthaei [11] have given an approximate techniquefrom which the inductance and capacitance parameters of asymmetrical coupledlines can be determined from the characteristic impedances and effective dielectricconstants of the even and odd modes of symmetrical coupled lines. This approachis useful in practice because the even- and odd-mode parameters of symmetricalcoupled lines are generally more readily available.

Consider two coupled lines of width W1 and W2 , each with separation Sbetween them as shown in Figure 4.10(a). It is assumed that the mutual inductanceand capacitance between the lines is the same as that between symmetrical lines

4.4 Directional Couplers Using Asymmetrical Coupled Lines 133

Figure 4.10 (a) Asymmetrical coupled lines of width W1 and W2, (b) symmetrical coupled linesof width (W1 + W2)/2, (c) symmetrical coupled lines of width W1, and (d) symmetricalcoupled lines of width W2.

of width (W1 + W2 /2, each with separation S as shown in Figure 4.10(b). Usingthe even- and odd-mode data of coupled symmetrical lines as shown in Figure4.10(b), the values of Lm and Cm can be computed using (4.107) and (4.108).Furthermore, it is assumed that the self-inductance and capacitance of line 1 inthe presence of line 2 is the same as if line 2 has the same width as line 1. Therefore,by using the even and odd-mode data of coupled symmetrical lines of width W1each and separated by a distance S as shown in Figure 4.10(c), the self-capacitanceand -inductance of line 1 can be computed using (4.105) and (4.106). Similarly,by using even- and odd-mode data of coupled symmetrical lines of width W2 eachand separated by a distance S [Figure 4.10(d)], the self-capacitance and -inductanceof line 2 can be computed. Once the distributed line parameters have been found,the normal-mode 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 Forward-Wave Directional Couplers

It is known that if the phase velocities of the two normal modes of asymmetricalcoupled lines are different, energy is coupled from one line to another in the forwarddirection. Because a microstrip line is essentially a quasi-TEM line, the even- andodd-mode phase velocities of coupled microstrip lines are not equal. Therefore,coupling occurs both in the forward and backward directions. Usually, backward-wave couplers are realized in microstrip configuration by properly choosing thecharacteristic impedances of the even and odd modes. The directivity of microstripbackward-wave couplers is generally poor, however, because of the forward-wavecoupling that takes place because of unequal even- and odd-mode phase velocities.The backward-wave coupling can be reduced to negligibly small values by choosinga relatively large separation between the lines. On the other hand, appreciablepower can be made to couple in the forward direction, if the length of the coupling


section is properly chosen. The bandwidth of an asymmetrical forward-wave cou-pler is larger than that of a symmetrical forward-wave coupler. This makes themuseful in practice [11, 12]. Figure 4.11 shows an asymmetrical microstrip coupler.It is assumed that the backward-wave coupling between the lines is negligible andeach line is terminated in a matched load. With unit power incident at port 1, theforward-traveling voltage waves on the two lines can be expressed as a linearcombination of c- and p -mode voltage waves as follows:

V +1 (z) = A1e −gc z + A2e −gp z (4.116)

V +2 (z) = A1Rce −gc z + A2Rp e −gp z (4.117)

The voltages in these equations denote their actual values. As discussed inChapter 2, the concept of actual voltages and currents is restrictive and is applicableto TEM and quasi-TEM mode transmission lines only. On the other hand, theconcept of normalized voltages is more general and can be applied to non-TEMmode transmission lines as well. Using (4.116), (4.117), and the conversion relationsbetween normalized and actual voltages given by (2.39a) to (2.39d), the normalizedvoltage waves on the two lines can be expressed as

V +1 (z) =

A1

√Zc1e −gc z +

A2

√Zp1e −gp z (4.118)

V +2 (z) =

A1Rc

√Zc2e −gc z +

A2Rp

√Zp2e −gp z (4.119)

Because unit power is assumed to be incident at the input port, the initialconditions are V +

1 = 1 and V +2 = 0 at z = 0. Substituting these conditions in (4.118)

and (4.119), we obtain

A1 = √Zc1

1 −RcRp

(4.120)

Figure 4.11 A forward (codirectional) directional coupler using uniform asymmetrical coupledlines.


A2 = − √Zp1

1 −RcRp

RcRp

(4.121)

Furthermore, substituting the values of A1 and A2 in (4.118) and (4.119), thenormalized voltage wave on line 1 is

V +1 (z) =

1

S1 −RcRp

D e −gc z −1

S1 −RcRp

DRcRp

e −gp z

After some straightforward algebraic manipulations, this equation reduces to

V1(z) = Fcos(bc − bp )z

2− j

1 − p1 + p

sin(bc − bp )z

2 G e −j(bc + bp )z/2 (4.122)

where

p = −RcRp

(4.123)

Similarly, by substituting the values of A1 and A2 from (4.120) and (4.121) in(4.119), the normalized voltage wave on line 2 is given by

V2(z) = −2j√p

1 + psin

(bc − bp )z2

e −j(bc + bp )z/2 (4.124)

Using (4.122) and (4.124), the scattering parameters between different ports of thecoupler 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 S21and S41 are therefore given by

S21 = V +1 (z) |z = l

(4.125)

and

S41 = V +2 (z) |z = l

(4.126)

or

P2P1

= | S21 |2 (4.127)

and


P4P1

= | S41 |2 (4.128)

Note that (4.122) and (4.124) are quite general and are valid for bothquasi-TEM and non-TEM mode asymmetrical coupled transmission lines, or forthat matter any two coupled waves. For non-TEM modes or waves, however, itis not possible to determine p using (4.123). This is because for the non-TEMmodes, Rc and Rp (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 bedetermined as described in [11].

Example 4.2

The design of a 3-dB X |S21 | = |S41 | = √1/2 C directional coupler in microstrip formis now discussed. Let the length of the coupler be chosen as

lg =p

bc − bp(4.129)

From (4.124) and (4.126), we obtain

2√p

1 + p= √1

2

or

−RcRp

= p = 3 ± √8 = 5.828, or 0.1715 (4.130)

The width of the coupled lines and separation between them should be chosensuch that the values of Rc and Rp satisfy (4.130).

4.4.2 Backward-Wave Directional Couplers

It may be of interest in certain applications to design backward-wave couplersusing asymmetrical coupled lines. For example, if the terminating impedances aredifferent for the two lines, it may be advantageous to choose different characteristicimpedances for the two lines. Note, however, that unlike asymmetrical forward-wave couplers, backward-wave asymmetrical couplers do not offer any advantagesover symmetrical couplers in terms of bandwidth [13]. Their main advantage isthat one does not require an additional impedance transformer to match the imped-ance of a low- or high-impedance device to that of the coupler. Cristal [13] hasgiven equations for the design of backward-wave couplers using asymmetricalcoupled lines. Figure 4.12(a) shows such a coupler. Assume that lines 1 and 2 areterminated in conductances Ga and Gb , respectively. Further, assume that the


Figure 4.12 (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 capacitancematrix of the coupled lines can be expressed as (see Section 3.2)

[C] = FC1 C12

C21 C2G = FCa + Cm −Cm

−Cm Cb + CmG (4.131)

If k2 denotes the power coupling coefficient between the lines, then the valuesof Ca , Cb , and Cm should be chosen according to

Cae

=376.7XGa − k√GaGb C

√er √1 − k2

Cbe

=376.7XGb − k√GaGb C

√er √1 − k2(4.132)

Cme

=376.7k√GaGb

√er √1 − k2

where e = e0er , and e0 denotes the permittivity of free space.

Example 4.3

Compute the per-unit length capacitances of a 10-dB asymmetrical coupler whoselines are terminated in loads of 50 ohms and 75 ohms, respectively.


The given quantities are

k = 10(−10/20) = 0.316

Ga = 1/50 = 0.02 ohms

Gb = 1/75 = 0.0133 ohms

From (4.132), we then obtain

Cae

= 5.895

Cbe

= 3.244

Cme

= 2.046

Once the capacitance parameters are known, the required physical dimensionsof the coupler can be determined using data relevant to the transmission line mediain which the coupler is to be realized. Commercially available general programscan be used for this purpose [9, 10].

A general uniform four-port coupler with arbitrary terminations and shuntingcapacitors is shown in Figure 4.13. Formulas useful for the synthesis of a backward-wave directional coupler in the configuration shown in Figure 4.13 are reportedin [14].

4.5 Design of Multilayer Couplers

Recent progress in miniaturization of microwave circuits has resulted from advancesin manufacturing technology such as LTCC and multilayer board fabrication tech-niques. The analysis of multilayer structures is more challenging, as there are morephysical variables (dielectric layers and their thicknesses) compared to single layer

Figure 4.13 A general four-port uniform coupler with shunting capacitors and arbitrary termina-tions. (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 differentlayers of a multilayer circuit, electric symmetry does not exist about the mid plane.Therefore, for analysis and design of multilayer circuits, EM simulation tools aregenerally required.

A systematic design procedure of multilayer asymmetric couplers based on thenormal-mode analysis of c and p modes has been reported [15]. Although themethod is rigorous, it depends on results of EM tools to determine the rathercumbersome normal-mode parameters. Furthermore, the design optimization isbased on an iterative procedure.

For uniformly coupled lines in a multilayer configuration, the inductance andcapacitance parameters can be very useful for the design [16–18].

Investigations on numerical data of coupled transmission lines show some veryinteresting behavior of the capacitive and inductance parameters of coupled lines.This leads to a very simple method for the design of multilayer backward directionalcouplers [19]. For example, using this method, the width of coupled lines can befound using data of single transmission lines. Furthermore, the spacing betweenthe lines can be determined using mutual capacitance and inductance data whichcan be found using Sonnet Lite, as described in the next section or any othersimulation tool (e.g., [10]). The design is considerably simplified, as two of thethree unknowns (the three unknowns are the widths of two lines and the spacingbetween them) are easily determined. Before discussing the design, we discuss atechnique to determine the capacitance and inductance parameters.

4.5.1 Determination of Capacitance and Inductance Parameters UsingSonnet Lite

With the recent progress in the capabilities of EM simulation tools and the increasein speed and memory of personal computers, these tools are now used regularlyfor the analysis of microwave circuits. For planar multilayer circuits, many toolsare now available (generally referred to as 2D or 2.5D EM simulators). One ofthe advantages of 2D or 2.5D EM simulators over 3D EM simulators is that it isquite simple to set up the problem for analysis.

One such EM simulation tool that is available at no cost is Sonnet Lite. It isa limited feature version of the popular Sonnet. Sonnet Lite can be used to directlydetermine 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 bedefined. A sonnet geometry for two uniformly coupled lines is shown in Figure4.14. To determine the inductance and capacitance parameters directly, the inputports of all the lines should be first numbered sequentially. The output ports arethen defined in the same order. For example, for two coupled lines, the ports arelabeled as shown in Figure 4.14. Before the analysis is run, the frequency range ofanalysis needs to be defined. This can be done using Analysis → Setup from themain window. In this case, a very low frequency should be chosen for analysis, aswe are interested in static line parameters. It is generally sufficient to choose ananalysis frequency of about 10 MHz for 10-mm-long lines. After the analysis hasbeen run (which may take less than a few seconds on an average machine), one


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 → N-coupled Line Model Fileto look at the inductance and capacitance parameters. If the metal conductors aredefined to be of finite conductivity and there is dielectric loss defined for thedielectric layers, the program also determines the resistance and conductanceparameters of the coupled lines.

It may be noted that different programs use different formats for the data oninductance and capacitance parameters. For example, for two coupled lines, SonnetLite version 10.511 gives three values each for the inductance and capacitanceparameters. If the inductance parameters from Sonnet Lite are Laa , Lbb , and Lcc ,then the inductance matrix of the coupled lines as defined by (4.79) is given by

[L] = FL11 L12

L12 L22G = FL1 Lm

Lm L2G = FLaa + Lbb Lbb

Lbb Lbb + LccG (4.133)

Similarly, if the capacitance parameters from Sonnet Lite are Caa , Cbb , andCcc , then the capacitance matrix of coupled lines as defined by (4.80) is given by

[C] = FC11 C12

C12 C22G = F C1 −Cm

−Cm C2G = FCaa − Cbb Cbb

Cbb −Cbb + CccG

(4.134)

Sonnet Lite gives the net inductance and capacitance for the length of linesconsidered. To convert the parameters to per unit length, proper scaling can beused.

Once the capacitance and inductance parameters are known, the inductive andcapacitive coupling coefficients kL and kC 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 structureis very useful for the design of high-directivity tight couplers. Table 4.1 shows theparameters of the inductance and capacitance matrices for various values of spacings 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.


Figure 4.15 Offset coupled strip transmission lines.

Table 4.1 Inductance and Capacitance Parameters of Structure Shown in Figure 4.15(er1 = er2 = er3 = 2.2, H1 = H3 = 254 mm, H2 = 50 mm, W1 = W2 = 250 mm)

s L11 L12 L22 C11 C12 C22(mm) (mH/m) (mH/m) (mH/m) (nF/m) (nF/m) (nF/m)

0.07 0.3418 0.2374 0.3418 0.1384 −0.0961 0.13840.11 0.3337 0.2177 0.3337 0.1277 −0.0833 0.12770.15 0.3281 0.1959 0.3281 0.1160 −0.0692 0.11600.19 0.3265 0.1718 0.3265 0.1037 −0.0546 0.10370.23 0.3302 0.1441 0.3302 0.0916 −0.0400 0.09160.25 0.3342 0.1292 0.3342 0.0862 −0.0333 0.08620.29 0.3421 0.1013 0.3421 0.0785 −0.0232 0.07850.33 0.3474 0.0794 0.3474 0.0744 −0.0170 0.07440.39 0.3515 0.0557 0.3515 0.0715 −0.0113 0.07150.49 0.3540 0.0314 0.3540 0.0697 −0.0062 0.06970.59 0.3549 0.0179 0.3549 0.0692 −0.0035 0.06920.79 0.3552 0.0058 0.3552 0.0690 −0.0011 0.06900.99 0.3553 0.0019 0.3553 0.0690 −0.0004 0.0690

of each other (single lines). The line parameters of single lines are shown in Table4.2.

It is interesting to note from Table 4.1 that the self-inductances L11 and L22of 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 Figure4.16(a, b) of the same physical parameters. Furthermore, in Table 4.3, we haveshown values of coupling coefficients determined using (4.99) and (4.100).

Also, shown in Table 4.3 are the variables C11(1 − k2C ) and C22(1 − k2

C ). It isinteresting to note that the values of these variables also remain nearly constantand are nearly the same as the capacitance of the respective single lines shown inTable 4.2.

In a classic paper, Oliver has shown that for the case of a coupler in anhomogeneous medium, the variables L11C11 and L22C22 should vary as (1 − k2)−1,where k = kC = kL . Both Lii and Cii can vary, but the essential variation is due tothe capacitance alone [4]. Furthermore, Oliver has shown that the wave on the


Figure 4.16 Structure of Figure 4.15 with only (a) line 1, and (b) line 2.

Table 4.2 Capacitance, Inductance, and CharacteristicImpedance of Single Lines of Structures Shown in Figure 4.16

Line Line Parameters

Line 1 L10 = 0.3552 mH/m, C10 = 0.0689 nF/m, Z01 = 71.7 ohmsLine 2 L20 = 0.3552 mH/m, C20 = 0.0689 nF/m, Z02 = 71.7 ohms

Table 4.3 Computed Coupling Coefficients and Normalized Self-Capacitances of the Structure Shown in Figure 4.15(er1 = er2 = er3 = 2.2, H1 = H3 = 254 mm, H2 = 50 mm, W1 = W2 =250 mm)

C11(1 − k2C ) C22(1 − k2

C )s (mm) kL kC (nF/m) (nF/m)

0.07 0.6946 0.6946 0.7163 0.71630.11 0.6524 0.6524 0.7336 0.73360.15 0.5971 0.5971 0.7463 0.74630.19 0.5262 0.5261 0.7501 0.75010.23 0.4364 0.4364 0.7415 0.74150.25 0.3866 0.3865 0.7328 0.73280.29 0.2961 0.2960 0.7158 0.71580.33 0.2285 0.2284 0.7050 0.70500.39 0.1585 0.1584 0.6967 0.69670.49 0.0888 0.0887 0.6916 0.69160.59 0.0503 0.0502 0.6900 0.69000.79 0.0163 0.0162 0.6893 0.68930.99 0.0053 0.0053 0.6896 0.6896


coupled line appears only on the ‘‘backward’’ port if the two lines are terminatedin characteristic impedances defined by

ZT1 = √L11C11

(4.135)

ZT2 = √L22C22

(4.136)

It has been demonstrated that backward asymmetric couplers in inhomogeneousmedium can be designed with a very high directivity, if the physical parametersare chosen such that kC ≈ kL and the lines are terminated in impedances given by(4.135) and (4.136).

Next, we consider the case of coupling between asymmetric microstrip lineson different layers of a multilayer substrate as shown in Figure 4.17.

The corresponding results for inductance and capacitance parameters are shownin Tables 4.4 through 4.6. It is seen that the evidence is even more compellingin this case where it is seen that the self-inductances L11 and L22 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(er1 = 9.8, er2 = 3.0, H1 = 254 mm, H2 = 100 mm, W1 = 350 mm, W2 = 150 mm)

s L11 L12 L22 C11 C12 C22(mm) (mH/m) (mH/m) (mH/m) (nF/m) (nF/m) (nF/m)

0.20 0.3587 0.2629 0.5608 0.2682 −0.0599 0.07310.22 0.3581 0.2564 0.5577 0.2651 −0.0576 0.07260.24 0.3573 0.2487 0.5549 0.2616 −0.0549 0.07180.26 0.3566 0.2400 0.5529 0.2577 −0.0517 0.07080.28 0.3558 0.2304 0.5518 0.2537 −0.0483 0.06960.32 0.3551 0.2093 0.5530 0.2453 −0.0408 0.06710.36 0.3554 0.1868 0.5578 0.2376 −0.0334 0.06450.40 0.3562 0.1649 0.5639 0.2316 −0.0269 0.06230.50 0.3582 0.1202 0.5751 0.2234 −0.0159 0.05930.60 0.3591 0.0894 0.5802 0.2205 −0.0098 0.05830.80 0.3594 0.0528 0.5838 0.2190 −0.0040 0.05771.00 0.3596 0.0334 0.5841 0.2187 −0.0018 0.05771.20 0.3593 0.0222 0.5838 0.2187 −0.0009 0.0576


Table 4.5 Capacitance, Inductance and Characteristic Impedanceof Single Lines of Structure of Figure 4.17

Line Line Parameters

Line 1 L10 = 0.3594 mH/m, C10 = 0.2187 nF/m, Z01 = 40.5 ohmsLine 2 L20 = 0.5840 mH/m, C20 = 0.0576 nF/m, Z02 = 100.6 ohms

Table 4.6 Computed Coupling Coefficients and Normalized Self-Capacitances for the Structure of Figure 4.17(er1 = 9.8, er2 = 3.0, H1 = 254 mm, H2 = 100 mm, W1 = 350 mm, W2= 150 mm)

C11(1 − k2C ) C22(1 − k2

C )s (mm) kL kC (nF/m) (nF/m)

0.20 0.5862 0.4278 0.2191 0.05970.22 0.5737 0.4155 0.2194 0.06000.24 0.5585 0.4006 0.2196 0.06020.26 0.5405 0.3830 0.2199 0.06040.28 0.5200 0.3631 0.2202 0.06050.32 0.4723 0.3181 0.2205 0.06030.36 0.4195 0.2699 0.2203 0.05980.40 0.3679 0.2242 0.2200 0.05920.50 0.2648 0.1382 0.2191 0.05820.60 0.1959 0.0863 0.2188 0.05780.80 0.1152 0.0358 0.2187 0.05771.00 0.0730 0.0164 0.2187 0.05761.20 0.0486 0.0083 0.2187 0.0576

stant within about 5% and the same is true of parameters C11(1 − k2C ) and

C22(1 − k2C ). 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, semi-empirically,

L11 ≈ L10 (4.137)

L22 ≈ L20 (4.138)

C11 ≈C10

1 − k2C

(4.139)

C22 ≈C20

1 − k2C

(4.140)

where L10 , L20 , C10 , and C20 denote the parameters of single lines.Substituting results from (4.137) through (4.140) in (4.135) and (4.136), we

obtain:

Z01 ≈ZT1

√1 − k2C

(4.141)


Z02 ≈ZT2

√1 − k2C

(4.142)

where Z01 = √L10 /C10 and Z02 = √L20 /C20 denote the characteristic impedancesof single lines. To reemphasize, L10 , C10 , and Z01 denote the parameters of line1 in the absence of line 2. Similarly, L20 , C20 , and Z02 denote the parameters ofline 2 in the absence of line 1.

In a coupler design, ZT1 and ZT2 should be chosen as equal to the desiredport impedances to have a perfect match. Let the port impedances be Z0 .

Equations (4.137) through (4.140) then show that the width of the lines shallbe chosen to have single line characteristic impedances of

Z01 ≈Z0

√1 − k2C

(4.143)

Z02 ≈Z0

√1 − k2C

(4.144)

For example, for a 6-dB coupler kC = 0.5, which leads to Z01 = Z02 = 57.7 ohms.The widths of the lines can be chosen corresponding to characteristic impedances of57.7 ohms. Once the widths of the lines have been determined, the coupling canbe determined as a function of spacing, using the procedure outlined in Section4.5.1. 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 firstchoose the case of a coupler in the broadside offset coupled stripline configurationas shown in Figure 4.15 using duriod substrate. The parameters were chosenaccording to commercially available thicknesses. The coupler parameters werechosen as er1 = er2 = er3 = 2.2, H1 = H3 = 790 mm, H2 = 127 mm. For a 3-dBcoupler, k = 0.707, which leads to Z01 = Z02 = 70.7 ohms using (4.143) and(4.144). Using Sonnet Lite for a single transmission line structure, we find thatW1 = W2 = 780 mm (Sonnet Lite directly gives the value of characteristic impedancein the response window. A frequency of about 1 GHz or greater should be chosento avoid error messages). Once the widths of the lines have been determined, theinductance and capacitance parameters are found for various values of spacing sbetween lines. The inductive and capacitive coupling coefficients can then be com-puted using (4.99) and (4.100), respectively. The coupling is plotted in Figure 4.18.It is found that for s = 320 mm, k = kC = kL = 0.7, which is close to the desiredvalue.

To determine the accuracy of the method, the whole coupler structure was EMsimulated 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


Figure 4.18 Coupling as a function of spacing between offset coupled strip transmission linesshown in Figure 4.15. er1 = er2 = er3 = 2.2. H1 = H3 = 0.79 mm, H2 = 0.127 mm.W1 = W2 = 0.78 mm.

Figure 4.19 EM simulated response of coupler in configuration of Figure 4.15. er1 = er2 = er3 =2.2. H1 = H3 = 0.79 mm, H2 = 0.127 mm. W1 = W2 = 0.78 mm, length of coupler =10 mm.

the figure that the coupling is very close to 3 dB. Furthermore, the return loss andisolation are both better than about 30 dB.

Example 4.5

Example 4.1 considered was for a homogeneous medium. In practice, multilayercircuits are inhomogeneous. To verify the accuracy of the method, a multilayercoupler of the form shown in Figure 4.17 was considered with same substrateparameters as considered in [15]. The substrate parameters were er1 = 12.8,er2 = 6.8, H1 = 152.4 mm, and H2 = 1.8 mm. Using the procedure as described


Figure 4.20 EM simulated response of coupler in configuration of Figure 4.17. er1 = 12.8,er2 = 6.8. H1 152.4 mm, H2 = 1.8 mm, W1 = 42 mm, W2 = 48 mm, length of coupler= 2 mm.

earlier, we obtain W1 = 42 mm, W2 = 48 mm. Furthermore, using Sonnet Lite,one finds that a spacing s = 0 is required. For this case, kC = 0.67 and kL = 0.72,or k ≈ (kC + kL )/2 = 0.695. Since in this case, kC and kL are slightly different, thedirectivity will not be as good. The EM simulated performance of the coupler oflength 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] Lippmann, B. A., Theory of Directional Couplers, M.I.T. Rad. Lab. Rep., No. 860,December 28, 1945.

[2] Reed, J., and G. J. Wheeler, ‘‘A Method of Analysis of Symmetrical Four-Port Networks,’’IRE Trans., Vol. MTT-4, October 1956, pp. 246–253.

[3] Sazanov, D. M., et al., Microwave Circuits, Moscow: Mir Publishers, 1982.

[4] Oliver, B. M., ‘‘Directional Electromagnetic Couplers,’’ Proc. IRE, November 1954,pp. 1686–1692.

[5] Tripathi, V. K., ‘‘Asymmetric Coupled Transmission Lines in an InhomogeneousMedium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-23, September 1975,pp. 734–739.

[6] Krage, M. K., and G. I. Haddad, ‘‘Characteristics of Coupled Microstrip TransmissionLines-I: Coupled-Mode Formulation of Inhomogeneous Lines,’’ IEEE Trans. MicrowaveTheory Tech., Vol. MTT-18, April 1970, pp. 217–222.

[7] Krage, M. K., and G. I. Haddad, ‘‘Characteristics of Coupled Microstrip TransmissionLines-I: Evaluation of Coupled-Line Parameters,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-18, April 1970, pp. 222–228.

[8] Wei, C., et al., ‘‘Multiconductor Transmission Lines in Multilayered Dielectric Media,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-32, April 1984, pp. 439–450.

[9] Djordjevic, A. R., et al., MULTILIN for Windows: Circuit Analysis Models for Multicon-ductor Transmission Lines, Software and User’s Manual, Norwood, MA: Artech House,1996.


[10] Djordjevic, A. R., et al., LINPAR for Windows: Matrix Parameters for MulticonductorTransmission Lines, Software and User’s Manual, Norwood, MA: Artech House, 1995.

[11] Ikalainen, P. K., and G. L. Matthaei, ‘‘Wideband, Forward-Coupling Microstrip Hybridswith High Directivity,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35,August 1987, pp. 719–725.

[12] Ikalainen, P. K., and G. L. Matthaei, ‘‘Design of Broadband Dielectric Guide 3-dB Cou-plers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35, July 1987, pp. 621–628.

[13] Cristal, E. G., ‘‘Coupled-Transmission-Line Directional Couplers with Coupled Linesof Unequal Characteristic Impedances,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-14, July 1966, pp. 337–346.

[14] Sellberg, F., ‘‘Formulas Useful for the Synthesis and Optimization of General, UniformContradirectional Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-38,August 1990, pp. 1000–1010.

[15] Tsai, C., and K. C. Gupta, ‘‘A Generalized Model for Coupled Lines and Its Applicationsto Two-Layer Planar Circuits,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-40,December 1992, pp. 2190–2199.

[16] Sachse, K., ‘‘The Scattering Parameters and Directional Coupler Analysis of Characteristi-cally Terminated Asymmetric Coupled Transmission Lines in an InhomogeneousMedium,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-38, April 1990, pp. 417–425.

[17] Emery, T., et al., ‘‘Analysis and Design of Ideal Non Symmetrical Coupled MicrostripDirectional Couplers,’’ IEEE MTT-S Int. Microwave Symp. Digest, 1989, pp. 329–332.

[18] Sachse, K., and A. Sawicki, ‘‘Quasi-Ideal Multilayer Two- and Three-Strip DirectionalCouplers for Monolithic and Hybrid MICs,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-47, September 1999, pp. 1873–1882.

[19] Mongia, R. K., ‘‘A Semi-Empirical Method for Design of Multi-Layer Quarter-WaveDirectional Couplers,’’ IEEE Microwave and Wireless Components Letters, submittedfor publication, 2007.

C H A P T E R 5

Broadband Forward-Wave DirectionalCouplers

As discussed in the previous chapter, forward-wave coupling exists between uniformsymmetrical coupled lines if the even- and odd-mode phase velocities of the coupledlines are unequal. Furthermore, the backward-wave coupling between these linescan be reduced to a very small value by keeping a relatively large separation betweenthe lines (such that the even- and odd-mode characteristic impedances of the coupledlines are nearly equal). These types of couplers are known as forward-wave direc-tional couplers and can be realized using non-TEM mode transmission lines suchas metallic waveguides, dielectric guides, and the like. The bandwidth of forward-wave directional couplers realized using symmetrical coupled lines is generallysmall and can be increased by using asymmetrical coupled lines. In this chapter,the design and performance of forward-wave couplers realized using uniform asym-metrical lines is first discussed.

In the previous chapter, normal-mode analysis of symmetrical and asymmetricalcoupled lines was discussed. Although the normal-mode analysis is rigorous, itsapplication may prove to be very tedious in certain cases. Another approach thatcan be used to study coupled structures is known as the coupled-mode theory. Inits simpler form (which is also its most useful form), the theory is valid for weaklycoupled structures [1–3]. The theory has been used extensively in the past innumerous applications for the analysis of both passive and active coupled circuits.In its early development, the theory was used mainly for the analysis of microwavecircuits such as mode conversion in multimoded waveguides, parametric amplifiers,beamwave interaction in TWTs, and so forth. In recent years, the theory has alsobeen used for the design and analysis of fiber optics and optoelectronics circuitsand components. A good review of the coupled-mode theory and its applicationshas been given by Haus and Huang [3]. The theory is quite general and can beused to study coupling phenomenon between any two waves. The two waves mayrepresent two modes of two different transmission lines or of the same trans-mission line. For example, if a transmission line is bent along its length, the coupled-mode theory can be used to study the conversion of power from one mode toanother.

The coupled-mode theory leads to explicit expressions showing how individualwaves are modified in the presence of coupling. The theory also leads to animportant 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. Theequivalence between normal- and coupled-mode theories is also discussed.

149

150 Broadband Forward-Wave Directional Couplers

5.1 Forward-Wave Directional Couplers

The forward-wave coupling between ports 1 and 4 of symmetrical coupled linesshown in Figure 5.1 is given by

|S41 | = |sinSpDneff Lc

fD | (5.1)

where L is the length of the coupler, c is the velocity of light in free space, and fis the operating frequency. Dneff is the difference between the square roots of theeffective dielectric constants of the even and odd modes:

Dneff = √eree − √ereo (5.2)

The direct coupling between ports 1 and 2 can be expressed as

|S21 | = |cosSpDneff Lc

fD | (5.3)

Equation (5.1) shows that the coupling is a function of frequency. Assumingthat Dneff is independent of frequency, the forward-wave coupling becomes asinsusoidal function of frequency. In general, however, Dneff is also a function offrequency. Its variation with frequency depends on the type of transmission lineand its parameters. It is also seen from (5.1) that the maximum coupling that canbe obtained using symmetrical coupled lines by appropriately choosing the lengthL is 0 dB (complete power transfer). Equation (5.1) is plotted in Figure 5.2 whereit is assumed that Dneff is independent of frequency. It is seen from Figure 5.2 thatthe coupling versus frequency curve is flat (zero first derivative) at the frequencywhere the coupling is 0 dB. For any other coupling value, however, the couplingversus frequency curve is not flat at the frequency where the desired coupling isobtained. For example, the coupling versus frequency response is not flat at thefrequency where 3-dB coupling is obtained. It is therefore expected that a 0-dB

Figure 5.1 Forward-wave coupler using symmetrical coupled lines.

5.1 Forward-Wave Directional Couplers 151

Figure 5.2 Coupling response of symmetrical forward coupler as function of frequency ratio f/ f0.f0 denotes the frequency where maximum coupling is achieved.

symmetrical coupler has a wider bandwidth than does a coupler designed for anyother coupling value.

It is known that it is possible to achieve complete power transfer only betweensymmetrical coupled lines. If the lines are symmetrical,1 the maximum couplingthat can be achieved between the lines is less than 0 dB. The amount of maximumcoupling depends on the difference between the phase constants of the asymmetricalcoupled lines. This is described in more detail later in this chapter while discussingthe coupled-mode theory. The asymmetrical couplers will have a flat couplingversus frequency response at the frequency where maximum coupling is obtained.Therefore, if an asymmetrical coupler is designed such that the maximal couplingwhich can be obtained is equal to the desired coupling value, then such a couplerwill have a wider bandwidth than a symmetrical coupler. This principal has beenused to demonstrate wideband 3-dB forward-wave couplers [4]. A more comprehen-sive explanation on the broadband properties of asymmetrical couplers can befound in [5].

5.1.1 3-dB Coupler Using Symmetrical Microstrip Lines

Usually, the microstrip configuration is used to realize quarter-wave backward-wave couplers by choosing suitable values of the even- and odd-mode impedances.

1. By asymmetrical lines, we mean lines that have different phase constants when uncoupled.


It becomes, however, difficult because of fabrication tolerances to achieve verytight backward-wave coupling (e.g., 3 dB) in parallel-coupled microstrip lines.Further, the directivity of backward-wave couplers realized using microstrip linestends to be quite poor. This is because the even- and odd-mode phase velocitiesof coupled microstrip lines are unequal, resulting in forward-wave coupling. Ingeneral, the directivity becomes poorer as the frequency is increased. By choosinga relatively large separation between coupled microstrip lines (such that the even-and odd-mode impedances are nearly equal), the backward-wave coupling can bereduced to a very small value. Further, a desired level of forward-wave couplingcan be obtained by appropriately choosing the length of the coupler, which canbe determined using (5.1).

The strip pattern of a 3-dB coupler using symmetrical coupled microstrip linesis shown in Figure 5.3. The spacing between the lines is tapered toward the endsof the coupler. This is done to avoid any abrupt physical discontinuities in thestructure, which will lead to reflections causing power to couple to port 3, whichis designed to be the isolated port.

The theoretical and experimental results of a 3-dB symmetrical forward-wavecoupler 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 0.762mm. The width of the microstrip line corresponds to an impedance of 50V and isa constant throughout. The length of the straight middle part is 113 mm, whichis equivalent to 5.2 guide wavelengths at 10 GHz. The spacing between the linesin the middle section is twice the substrate thickness. The curved sections have aradius of curvature of 102 mm. The coupling of the straight section was determined

Figure 5.3 Strip pattern of symmetrical 3-dB forward-wave coupler. (From: [5]. 1987 IEEE.Reprinted with permission.)

Figure 5.4 Theoretical and experimental response of a symmetrical 3-dB forward-wave coupler.(From: [5]. 1987 IEEE. Reprinted with permission.)


using (5.1). To determine the coupling between curved sections, they were consid-ered to consist of 20 small, straight segments. The coupling of each small segmentwas determined using (5.1), with the overall coupling determined by summingcoupling from various sections. The coupled microstrip lines were analyzed usingformulas given in [6].

Figure 5.4 shows that the shapes of the measured coupling curves match wellwith the theoretically predicted values, except that the measured center frequencyis somewhat lower. The measured directivity of the coupler is about 40 dB. It isseen that in the frequency band considered, the coupling increases monotonicallywith frequency.

5.1.2 Design and Performance of 3-dB Asymmetrical Couplers

The design equations of a 3-dB directional coupler using asymmetrical coupledmicrostrip lines were given in Section 4.4.1. The length of the coupler is given by

lg =p

bc − bp(5.4)

where bc and bp denote the phase constants of the c and p modes, respectively.Furthermore, the ratio of Rc and Rp should be chosen as

−RcRp

= 3 ± √8 = 5.828, or 0.1715 (5.5)

where Rc and Rp denote the voltage ratios on the two lines for the c and p modes,respectively.

In the design of the coupler, the width of asymmetrical coupled microstriplines was chosen to correspond to (uncoupled) impedances of 50V and 100V. Thedesign is completed by choosing the separation between coupled lines such thatthe ratio of Rc and Rp satisfies (5.5). For given width of the lines and assumedseparation between them, the self and mutual inductance and capacitance parame-ters can be found using the coupled microstrip data [6] and the technique describedin Section 4.3.4. Further, if the self and mutual inductance and capacitance parame-ters of the coupled lines are known, the values of Rc and Rp can be found using(4.97) and (4.98).

The coupler was fabricated on a substrate having the same parameters as thatused for the symmetrical coupler discussed in the last section. The top view of thestrip pattern of the coupled section and the input and output lines are shown inFigure 5.5. It was found that a 1.81-mm separation between the lines is needed toobtain the required value of Rc /Rp . In practice, a 1.65-mm separation was used,which gave more than 3-dB coupling at the center frequency but gave a widerbandwidth for 1-dB amplitude balance. The theoretical and experimental resultsof the coupling and isolation for this coupler are shown in Figure 5.6. The lengthof the coupler was found to be 220 mm using (5.4), where bc and bp were obtainedfrom (4.91) and (4.92), respectively. Because the feed lines also contribute to somecoupling in the fabricated coupler, the length of the uniform coupled section was


Figure 5.5 Strip pattern of asymmetrical forward-wave coupler. (From: [5]. 1987 IEEE. Reprintedwith permission.)

Figure 5.6 Theoretical and experimental response of an asymmetrical 3-dB forward-wave coupler.(From: [5]. 1987 IEEE. Reprinted with permission.)

chosen to be 190 mm and the curved feed lines had a radius of curvature of102 mm. It is seen from Figure 5.6 that the agreement between theory and experi-ment is quite good, considering that the conductor and dielectric losses were notaccounted for in the theory. The coupler has a bandwidth of about 60% for1-dB amplitude balance. The isolation is better than about 40 dB in the completefrequency range. It is easily verified by comparing Figures 5.4 and 5.6 that auniform forward-wave asymmetrical coupler offers more bandwidth than a uniformsymmetrical coupler.

Note, however, that unlike a symmetrical coupler, the phase difference betweenthe output ports of an asymmetrical coupler is not 90 degrees. The theoreticallycomputed and measured phase differences between the coupled and direct port ofthe asymmetrical coupler are shown in Figure 5.7(a). It is interesting to note thatat the frequency where the phase difference between the output ports (ports 2 and4) is 0 degree when port 1 is the driven port, the phase difference between outputports is 180 degrees when the input is at port 3. In general, an asymmetrical couplerthat has end-to-end symmetry satisfies the following phase-difference relationship:

(∠S41 − ∠S21) + (∠S23 − ∠S43) = 180 degrees (5.6)

It is possible to achieve approximately the desired phase difference betweenoutput ports over a reasonably wide bandwidth in an asymmetrical coupler usingan extra length of line as a phase compensating element. For example, if one addsa length of line having a phase shift of 106 degrees at 9.6 GHz to ports 3 and 4,the outputs (S21 and S41) will be in phase quadrature at that frequency. Further-


Figure 5.7 (a) Computed and measured phase difference between output ports of an asymmetrical3-dB forward-wave coupler. (b) Theoretically computed phase difference between thecoupled and through ports of the asymmetrical coupler with reference planes chosento approximate quadrature or magic-T performance. Df1 and Df2 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 afrequency range of 7.0 to 12.2 GHz. By adding another quarter-wave-long linesection at ports 3 and 4, a phase difference of 180 degrees between the outputscan be obtained. This is shown in Figure 5.7(b), where the two cases are labeledas ‘‘quadrature-type’’ and ‘‘magic-T-type,’’ respectively.

5.1.3 Ultra-Broadband Forward-Wave Directional Couplers

The bandwidth of forward-wave couplers realized using asymmetrical coupled linesis greater than those realized using symmetrical coupled lines. It is still not possible,however, to achieve a very broadband coupling (multioctave) using uniform asym-metrical coupled lines. Very broadband coupling can be achieved by continuouslyvarying the phase constants of coupled lines (b1 and b2) and the coupling coefficientbetween them along the length of the structure. This results in a nonuniformstructure. This principle of broadband coupling is known as the normal-modewarping [7–9]. The cross section of such a structure varies continuously along the


length. The essential features of the normal-mode warping can be summarized asfollows [8]:

• Adjust the geometrical parameters at the input of the structure such thatthe input excitation is identical with one of the normal modes of the coupledstructure.

• Gradually warp the normal mode (the mode in which the power is launched)by continuously varying the structure along the longitudinal direction untilthe distribution of the normal mode is identical with the desired output.The normal mode at the cross section of the output now contains power inboth the coupled modes in the desired ratio.

Note that nonuniform couplers tend to be very long (tens to hundreds ofwavelengths). Therefore, they are mainly useful at high millimeter-wave and opticalfrequencies where their physical lengths can be kept reasonably small.

5.2 Coupled-Mode Theory

Consider two lines that are uniformly coupled over a certain length as shown inFigure 5.8. As already discussed, these lines may represent two actual transmissionlines, or in a more general case, any two waves. The lines are assumed to be weaklycoupled. By the term ‘‘weakly’’ coupled, we mean that the impedances of individuallines (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 predomi-nant coupling takes place in the forward direction only. For example, if power isincident at port 1 as shown in Figure 5.8, the power coupled between the linesappears at port 4 only. The forward-wave coupling between the lines, per-unitwavelength of the coupling section, is also assumed to be small.

The forward-traveling waves on the two lines (in the absence of any couplingbetween them) can then be expressed, respectively, as

Figure 5.8 Uniformly coupled asymmetrical coupled lines.

5.2 Coupled-Mode Theory 157

V1 = Ae −jb 1z (5.7)

and

V2 = Be −jb 2z (5.8)

where V1 and V2 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|V1 |2 and |V2 |2, respectively. b1 and b2 denote the respective phase constants oflines 1 and 2. The coupled-mode theory as given by Miller [1] on which the presentdiscussion is based is valid for complex values of b1 and b2 , but for the sake ofsimplicity, we assume that these are real quantities.

By differentiating (5.7) and (5.8) with respect to z, we obtain, respectively,

dV1dz

= −jb1V1 (5.9)

and

dV2dz

= −jb2V2 (5.10)

In the presence of coupling between the lines as shown in Figure 5.8, theexisting voltage waves on both the lines are perturbed. According to coupled-modetheory, (5.9) and (5.10) representing voltage waves on the two lines are modifiedas follows in the presence of coupling:

dV1dz

= −j(b1 + K11)V1 − jK12V2 (5.11)

and

dV2dz

= −jK21V1 − j(b2 + K22)V2 (5.12)

where K11 and K22 are the self-coupling coefficients, and K12 and K21 are themutual-coupling coefficients. Their dimensional unit is per-unit length. When thetwo lines are uncoupled, the propagation constants of the two lines are given byb1 and b2 , respectively. When the two lines are brought closer, the propagationconstant of each line changes because of the presence of the other line. The modifiedpropagation constant of line 1 due to the presence of line 2 is denoted by(b1 + K11). Similarly, (b2 + K22) denotes the modified propagation constant ofline 2 in the presence of line 1. For weak coupling between lines:

|K11 | ! b1

|K22 | ! b2 (5.13)

|K12 | , |K21 | ! b1 , b2


With the above assumptions, (5.11) and (5.12) reduce to

dV1dz

= −jb1V1 − jK12V2 (5.14)

and

dV2dz

= −jK21V1 − jb2V2 (5.15)

Note that while deriving (5.14) and (5.15), the terms containing K12 and K21in (5.11) and (5.12) have been retained, while those containing K11 and K22 havebeen neglected. The reason for this is that although |K12 | and |K21 | are muchsmaller than b1 and b2 , it is not necessary that |K12V2 | ! |b1V1 | or|K21V1 | ! |b2V2 | for all values of z.

5.2.1 Nature of Coupling Coefficients K12 and K21

The total power on the two lines at any cross section is given by

W = |V1 |2 + |V2 |2 (5.16)

The principle of conservation of power requires that if the lines are lossless, thetotal power remains the same at all cross sections. In mathematical terms:

ddz

X |V1 |2 + |V2 |2 C = 0 (5.17)

or

ddz

XV1V1* + V2V2*) = 0 (5.18)

or

V1dV1*

dz+ V1*

dV1dz

+ V2dV2*

dz+ V2*

dV2dz

= 0 (5.19)

Substituting values of first derivatives in (5.19) from (5.14) and (5.15), we obtain

K12 = K21 = 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 wave-carrying unit power on line 1 only; thatis:


V1 = 1, V2 = 0 at z = 0 (5.21)

Solution of coupled equations (5.14) and (5.15) with the initial condition of (5.21)gives

V1 = F12

+(b1 − b2)

2√(b1 − b2)2 + 4K2Ge −jbs z (5.22)

+ F12

−(b1 − b2)

2√(b1 − b2)2 + 4K2Ge −jbf z

and

V2 =K

√(b1 − b2)2 + 4K2e −jbs z −

K

√(b1 − b2)2 + 4K2e −jbf z (5.23)

where

bs =(b1 + b2)

2+ √(b1 − b2)2 + 4K2

2(5.24)

and

bf =(b1 + b2)

2− √(b1 − b2)2 + 4K2

2(5.25)

The above equations show that in the presence of coupling, the waves on thetwo lines can be represented as interference between two waves having differentphase constants from those of the uncoupled waves. One of these waves (whichcan be termed as a slow wave) has a phase constant equal to bs , while the other(which can be termed as a fast wave) has a phase constant equal to bf . The phaseconstants bs and bf depend on the phase constants of individual lines (when the linesare uncoupled) and the coupling coefficient K. These waves with phase constants bsand bf 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 interferencebetween normal modes.

The coupling coefficient K depends on the specifics of the structure, and itsdetermination requires the use of field theory methods [10]. Irrespective of thevalue of the coupling coefficient K, however, (5.22) and (5.23) can be used to drawsome significant conclusions on uniform coupling between two lines (or waves).For example, these equations lead to a significant conclusion that if the phaseconstants of two lines (waves) are different (b1 ≠ b2), it is not possible to completelytransfer power between the two lines (waves).


Coupling Between Symmetrical Lines

Let the two coupled lines be symmetrical (b1 = b2). Using (5.24) and (5.25), weobtain

bs = b0 + K (5.26)

and

bf = b0 − K (5.27)

where

b0 = b1 = b2

Substituting the values of bs and bf from (5.26) and (5.27) in (5.22) and (5.23),we obtain

V1 =(e −jKz + e jKz )

2e −jb 0z (5.28)

= cos(Kz) e −jb 0z

V2 =(e −jKz − e jKz )

2e −b 0z (5.29)

= −j sin(Kz) e −jb 0z

The fraction of power coupled from line 1 to line 2 over a length z of thecoupling section is then given by

r = |V2(z) |2

|V1(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 toline 2 appears at port 4.

5.2.3 Coupled-Mode Theory and Even- and Odd-Mode Analysis

Using (5.26) and (5.27):

K =bs − bf

2(5.31)

and

b0 =bs + bf

2(5.32)


Substituting the values of K and b0 from (5.31) and (5.32) in (5.28) and (5.29),we obtain

V1 = cosF(bs − bf )z2 G e

−j(bs + bf )

2z

(5.33)

and

V2 = −j sinF(bs − bf )z2 G e

−j(bs + bf )

2z

(5.34)

It is interesting to find that (5.33) and (5.34) are identical to (4.35) and (4.36),respectively, if bs and bf are assumed to denote the phase constants of even andodd modes, respectively. Equations (4.35) and (4.36) were derived using the even-and odd-mode approach, while (5.33) and (5.34) were derived using the coupled-mode theory.

5.2.4 Coupling Between Asymmetrical Lines

Let the phase constants of two asymmetrical coupled lines be b1 and b2 , respec-tively. It is assumed that initially a unit amount of power is incident in line 1 (i.e.,V1 = 1 at z = 0). Using (5.22) and (5.23), the voltage wave on line 1 is given by

V1 = e−j

(b1 + b2 )2

zV ′1 (5.35)

where

V ′1 = cosFS√(b1 − b2)2

4K2 + 1DKzG−j

(b1 − b2)2K

1

F√(b1 − b2)2

4K2 + 1G sinFS√(b1 − b2)2

4K2 + 1DKzGThe voltage wave on line 2 is expressed as

V2 = e−j

(b1 + b2 )2

zV ′2 (5.36)

where

V ′2 = −j

√(b1 − b2)2

4K2 + 1

sinFS√(b1 − b2)2

4K2 + 1DKzG


The maximum power transfer from line 1 to line 2 takes place when

S√(b1 − b2)2

4K2 + 1DKz =p2

(5.37)

The maximum fractional power coupled between lines 1 and 2 is given by

rmax = |V2 |2max=

1

(b1 − b2)2

4K2 + 1

(5.38)

It is seen from (5.38) that rmax (which represents the maximum power thatcan be coupled between lines) is less than unity if the values of b1 and b2 aredifferent from each other. The value of rmax is plotted as a function of normalizeddifference in phase velocities of the two lines in Figure 5.9. It is seen that ifthe difference in phase velocities of the two lines is much greater than the couplingcoefficient [(b1 − b2) @ K], only a small amount of power can be coupled betweenthe lines. For example, for a value of (b1 − b2)/K = 10, the maximal coupling thatcan be achieved between the lines is only 14.2 dB.

After having discussed the normal (Chapter 4) and coupled modes, it is usefulto summarize the essential difference between the two. The normal modes of auniform structure are those that can propagate independent of each other alongthe structure. Each normal mode is characterized by a unique phase velocity andfield distribution. For example, the various TEmn and TMmn modes are the normal

Figure 5.9 Maximum power coupled between asymmetrical coupled lines.

5.3 Coupled-Mode Theory for Weakly Coupled Resonators 163

modes of a straight, uniform rectangular metal waveguide. Similarly, the even andodd modes are the two normal modes of symmetrical coupled lines. An importantproperty of normal modes is that there is no conversion of energy from one normalmode to another. For example, if an even mode is launched along a symmetricalcoupled structure, the energy remains in the even mode all along the length of thestructure. On the other hand, energy is continuously exchanged between coupledmodes. The individual waves on two coupled lines are an example of two coupledmodes. There is a continuous exchange of energy between the waves on the twolines. As another example of coupled modes, consider a rectangular waveguidethat is uniformly bent along the direction of propagation. The various TEmn andTMmn modes that are the normal modes of a straight waveguide are no longer thenormal modes of the bent waveguide. The various TEmn and TMmn modes arenow the coupled modes because energy will continuously be exchanged betweenthese modes along the bend of the waveguide. Usually, it is sufficient to considercoupling between two or three modes only to determine the state of a weaklycoupled system.

5.3 Coupled-Mode Theory for Weakly Coupled Resonators

The theory of weakly coupled resonators can be developed in a similar fashion asis for the weakly coupled lines. Consider two isolated resonators having resonantfrequencies as v1 and v2 , respectively. The time-varying amplitudes of twouncoupled resonators are given by [3]

da1dt

= jv1 a1 (5.39)

and

da2dt

= jv2 a2 (5.40)

where a1 and a2 denote the instantaneous normalized amplitudes of resonators 1and 2, respectively. When two resonators are weakly coupled, the governing equa-tions for the resonator amplitudes are modified as

da1dt

= jv1 a1 + jKa2 (5.41)

and

da2dt

= jKa1 + jv2 a2 (5.42)

where K denotes the coupling coefficient between the resonators. The dimensionalunit of K in this case is per-unit time. It may be noted than in case of coupled


lines, the dimensional unit of K is per-unit 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 normalmodes of the coupled system to be different from v1 and v2 . More specifically,the resonant frequencies of normal modes (denoted by va and vb ) are given by

va =(v1 + v2)

2+ √(v1 + v2)2 + 4K2

2(5.43)

and

vb =(v1 + v2)

2+ √(v1 − v2)2 + 4K2

2(5.44)

When two resonators having the same resonant frequency (v0 = v1 = v2) arecoupled, the resonant frequencies of the coupled resonators become

va = veven = v0 + K (5.45)

vb = vodd = v0 − K (5.46)

In microwave and RF circuits, it is common practice to denote coupling betweenresonators using the equivalent circuit approach [11, 12]. For example, couplingbetween two identical resonators can be represented by the circuit shown in Figure5.10 where the resonators are coupled through a mutual inductance M. In theabsence of coupling, the resonant frequency of either resonator is given by

f 20 =

1

4p2LC(5.47)

In the presence of coupling, the effect of the mutual inductance is either additiveor subtractive to the self-inductance. The resonant frequencies of the coupledresonators are then given by

f 2even =

1

4p2(L + M)C=

1

4p2LCS1 +ML D =

1

4p2LC (1 + k)=

f 20

1 + k(5.48)

Figure 5.10 Lumped equivalent circuit of coupled identical resonators.

5.3 Coupled-Mode Theory for Weakly Coupled Resonators 165

f 2odd =

1

4p2(L − M)C=

1

4p2LCS1 −ML D =

1

4p2LC (1 − k)=

f 20

1 − k(5.49)

where k = M /L is the voltage coupling coefficient. For k ! 1, which is usually thecase, (5.48) and (5.49) then lead to the following very useful relations:

f0 ≈feven + fodd

2(5.50)

k ≈fodd − feven

2(5.51)

Equations (5.50) and (5.51) are extensively used in the design of filters asdiscussed later in Chapters 9 to 11.

References

[1] Miller, S. E., ‘‘Coupled Wave Theory and Waveguide Applications,’’ Bell Syst. Tech. J.,Vol. 33, 1954, pp. 661–719.

[2] Louisell, W. H., Coupled-Mode and Parametric Electronics, New York: Wiley, 1960.[3] Haus, H. A., and W. Huang, ‘‘Coupled-Mode Theory,’’ Proc. IEEE, Vol. 79,

October 1991, pp. 1505–1518.[4] Ikalainen, P. K., and G. L. Matthaei, ‘‘Design of Broadband Dielectric Waveguide 3-dB

Couplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35, July 1987, pp. 621–628.[5] Ikalainen, P. K., and G. L. Matthaei, ‘‘Wideband, Forward-Coupling Microstrip Hybrids

with High Directivity,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35,August 1987, pp. 719–725.

[6] Kirsching, M., and R. H. Jansen, ‘‘Accurate Wide-Range Design Equations for the Fre-quency-Dependent Characteristics of Parallel-Coupled Microstrip Lines,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-32, January 1984, pp. 83–90. Corrections,Vol. MTT-33, March 1985, p. 288.

[7] Cook, J. S., ‘‘Tapered Velocity Couplers,’’ Bell Syst. Tech. J., Vol. 34, July 1955,pp. 807–822.

[8] 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 NormalModes,’’ IRE Trans., Vol. 3, December 1955, pp. 2–6.

[9] Louisell, W. H., ‘‘Analysis of Single Tapered-Mode Coupler,’’ Bell Syst. Tech. J., Vol. 34,July 1955, pp. 853–870.

[10] Yariv, A., ‘‘Coupled-Mode Theory for Guided Wave Optics,’’ IEEE J. on QuantumElectronics, Vol. QE-9, September 1973, pp. 919–933.

[11] Cohn, S. B., ‘‘Bandpass Filters Containing High Q Dielectric Resonators,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-16, April 1968, pp. 218–227.

[12] Van Bladel, J., ‘‘Weakly Coupled Dielectric Resonators,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-30, 1982, pp. 1907–1914.

C H A P T E R 6

Parallel-Coupled TEM DirectionalCouplers

In Chapter 4 it was demonstrated that if two identical TEM lines are parallel-coupled as shown in Figure 6.1, then by properly choosing the even- and odd-mode impedances of the coupled lines, a four-port directional coupler can beobtained. The coupler shown in Figure 6.1 is also called a backward-wave direc-tional coupler because the coupling takes place in the backward direction on thecoupled line. For example, if power is incident at port 1, power is coupled to port3. The maximum coupling between ports 1 and 3 (or between ports 2 and 4) takesplace at a frequency where the coupler is a quarter-wave long (or odd multiplesthereof). Because the electrical length of a coupler varies with frequency, thecoupling also varies with frequency. The variation of coupling with frequency canbe reduced by employing multisection TEM couplers. In a multisection coupler, anumber of coupled sections are cascaded. Each coupled section is a quarter-wavelong at the center frequency and has different even- and odd-mode impedancescompared with those of the adjacent sections. Multisection couplers that have end-to-end symmetry are known as symmetrical couplers, while those that do not haveend-to-end symmetry are known as asymmetrical couplers.

This chapter discusses the theory and design of single- and multisection parallel-coupled TEM directional couplers. Simple analytical expressions for the designof single-section couplers exist. Unfortunately, no simple analytical formulas arepossible for the design of optimal multisection couplers. Some design tables areavailable in the literature for equal-ripple symmetrical and asymmetrical couplers[1–3].

The microstrip line is the most popular transmission line for realizing micro-wave integrated circuit (MIC) components. When microstrip is used in backward-wave couplers, however, the directivity is generally poor because the even- andodd-mode phase velocities of coupled microstrip lines are unequal. In this chapter,various techniques that can be used to equalize the even- and odd-mode phasevelocities of coupled microstrip lines are also discussed.

6.1 Coupler Parameters

The schematic of a four-port directional coupler is shown in Figure 6.2. The fourports are labeled as ‘‘input,’’ ‘‘direct’’ (through), ‘‘coupled,’’ and ‘‘isolated.’’ Two

167

168 Parallel-Coupled TEM Directional Couplers

Figure 6.1 A single section TEM coupler.

Figure 6.2 Schematic of a four-port directional coupler.

important factors that characterize a directional coupler are its coupling and direc-tivity, defined here:

Coupling (dB) = 10 logP1P3

(6.1)

Directivity (dB) = 10 logP3P4

(6.2)

where P1 is the power input at port 1 and P3 and P4 are the power outputs atports 3 and 4, respectively. Note that all the ports are assumed to be match-terminated. There is no power at port 4 in the ideal case; in practice, a smallamount of power is always coupled to this port.

If the coupling and directivity are known, the isolation of the coupler can bedetermined. The isolation is defined as

Isolation (dB) = 10 logP1P4

(6.3)

or

Isolation (dB) = Coupling (dB) + Directivity (dB)

All the parameters described above are normally expressed in decibels and aredefined here as positive quantities.

6.2 Single-Section Directional Coupler 169

6.2 Single-Section Directional Coupler

6.2.1 Frequency Response

In Section 4.2.2, it was shown that if two identical parallel TEM lines are coupledover a length l as shown in Figure 6.1, then under the condition

Z0eZ0o = Z20 (6.4)

the scattering parameters of the network are given by

S11 = S22 = S33 = S44 = 0 (6.5)

S14 = S41 = S23 = S32 = 0 (6.6)

S12 = S21 = S34 = S43 = S21e (6.7)

= √1 − k2

√1 − k2 cos u + j sin u

S13 = S31 = S24 = S42 = S11e (6.8)

=jk sin u


where u = b l denotes the electrical length of the coupler and k is given by


(6.9)

Further, S11e and S21e denote the reflection and transmission coefficients, respec-tively, of the coupled lines for the case of even-mode excitation. The maximumamount of coupling between ports 1 and 3 (or between ports 2 and 4) occurs when

u = b l =p2

rads (6.10)

or

l =p

2b=

lg

4

The properties of an ideal parallel-coupled TEM directional coupler weredescribed in Chapter 4.

Frequency Bandwidth Ratio

The frequency bandwidth ratio B of a directional coupler (single- or multisection)is defined as


B =f2f1

(6.11)

where f2 and f1 are the upper and lower frequencies in between which the couplingis within the tolerance amount d compared with its midband value as shown inFigure 6.3. The tolerance amount d can be arbitrarily specified.

Fractional Bandwidth

The fractional bandwidth w of a directional coupler is defined as

w =f2 − f1

f0(6.12)

where

f0 =f1 + f2

2(6.13)

is the center frequency of the coupler.The frequency bandwidth ratio B and the fractional bandwidth w are related

by

w = 2B − 1B + 1

(6.14)

and

B =1 + w /21 − w /2

(6.15)

Figure 6.3 Typical variation of coupling in a single section TEM coupler.


Useful Operating Bandwidth

The variation of coupling to the ‘‘direct’’ and ‘‘coupled’’ ports of an ideal 3 ±0.3-dB coupler is shown in Figure 6.4(a) as a function of frequency ratio f / f0 ,where f0 refers to the frequency where the coupler is a quarter-wave long. Becausewe have assumed a tolerance amount of ±0.3 dB, this coupler is designed to havea coupling (to the ‘‘coupled’’ port) of 2.7 dB at the midband. The performance of6 ± 0.3 and 10 ± 0.5-dB single-section couplers is shown in Figure 6.4(b, c). Thevariation of coupling to the ‘‘coupled’’ port of a 20 ± 0.5-dB coupler is shown inFigure 6.4(d). In this case, the power coupled to the direct port is nearly 0 dB. Theuseful frequency operating range of single-section couplers can be determined byreferring to these plots. For example, it is seen from Figure 6.4(a) that for a tolerancein coupling of ±0.3 dB, a 3-dB coupler can be operated over a frequency bandwidthratio of about 2 ( f2 : f1 ≈ 2 : 1). Single-section couplers are generally useful foroperation over a frequency bandwidth ratio (B) of approximately 2.

6.2.2 Design

From (6.9), we can write Z0e /Z0o in terms of a voltage coupling coefficient k as

Z0eZ0o

=1 + k1 − k

(6.16)

Figure 6.4 (a) Variation of coupling to the direct and coupled ports of a TEM coupler designed for nominalcoupling of 3 dB. (b) Variation of coupling to the direct and coupled ports of a TEM couplerdesigned for 6-dB nominal coupling. (c) Variation of coupling to the direct and coupled portsof a TEM coupler designed for nominal coupling of 10 dB. (d) Variation of coupling to thecoupled port of a TEM coupler designed for a nominal coupling of 20 dB.


Figure 6.4 (continued).

Furthermore, the simultaneous solution of (6.4) and (6.16) gives

Z0e = Z0√1 + k1 − k

(6.17)

and

Z0o = Z0√1 − k1 + k

(6.18)

In a coupler design, for a given voltage coupling coefficient k and characteristicimpedance Z0 , we first determine the even- and odd-mode impedances using (6.17)and (6.18), respectively. The dimensions of the coupler are then calculated usingthe physical data of coupled transmission lines such as those discussed in Chapter3 for various transmission lines. The physical length l of the coupler is chosen as

l =lg

4(6.19)

where lg is the guide wavelength of the TEM wave in the transmission line mediumat the design frequency f0 .



If the coupling from port 1 to port 3 is given as C dB (where C is a positivequantity), then k is related to C as

k = 10−C/20 (6.20)

Example 6.1

A directional coupler of 10 ± 0.5-dB coupling is desired in the configuration asshown in Figure 6.1 at a frequency of 10 GHz. Determine the physical dimensionsof the coupler assuming that ports are terminated in an impedance of 50V andthe coupler is realized in a stripline medium of er = 2.25.

The coupler is designed to have a coupling of 9.5 dB at the midband becausea tolerance of ±0.5 dB in the coupling value has been specified. Using (6.20), thevoltage coupling coefficient k is found to be

k = 10−9.5/20 = 0.335

The even- and odd-mode impedances of the coupled lines are obtained as



Z0e = Z0√1 + k1 − k

= 50√1 + 0.3351 − 0.335

= 70.84V

and

Z0o = Z0√1 − k1 + k

= 50√1 − 0.3351 + 0.335

= 35.29V

Therefore,

√er Z0e = 106.26V

√er Z0o = 52.94V

From Figure 3.15, S/b ≈ 0.03 and W/b ≈ 0.65 if we assume that the thicknessof the strip conductors is negligible. Thus, if the separation between the groundplane 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 ofer = 2.25, the guide wavelength in the medium at a frequency of 10 GHz is givenby


lg =l0

√er=

30

√2.25= 20 mm

The physical length l of the coupler is therefore given by

l =lg

4= 0.005m = 5 mm

The useful bandwidth of the ideal coupler is found to be 62.5% from Figure6.4(c).

Example 6.2

Design a 20 ± 0.5-dB directional coupler in the microstrip configuration at 5 GHz.Determine physical dimensions of the coupler realized on 0.635-mm-thick aluminasubstrate having er = 9.7.

The coupler is designed to have a 19.5-dB midband coupling. From (6.20), thevoltage coupling coefficient k is given by

k = 10−19.5/20 = 0.106

Furthermore, from (6.17) and (6.18), the even- and odd-mode impedances ofthe coupled lines are obtained as

Z0e = 55.6V

and

Z0o = 45.0V

From Figure 3.22, W/h ≈ 0.95 and S/h ≈ 1.3 (approximately extrapolated).For a 0.635-mm-thick substrate, W ≈ 0.6 mm and S ≈ 0.83 mm. The physicallength of the coupler is calculated using

u =ue + uo

2=

2pl0

X√eree + √eree C2

l = 90 deg

or

36060

X√7.2 + √6 C2

l = 90 deg

which gives l ≈ 5.85 mm. The useful bandwidth of the ideal coupler is found tobe 60% from Figure 6.4(d).


6.2.3 Compact Couplers

When size and cost requirements are stringent, compact directional couplers aremandatory. The coupled line approaches for such couplers are reported in theliterature [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 high-dielectric constant (er ≈ 30 − 100) substrates to reduce the size, and second to foldthe coupler length in different shapes such as spiral and meander. MMIC couplersthat generally use GaAs substrates (er = 12.9) employ the latter technique to makecompact couplers. One of the important applications of these couplers is in wirelesscommunications. The design, fabrication, and test results of these couplers arediscussed in Chapter 8.

6.2.4 Equivalent Circuit of a Quarter-Wave Coupler

The coupling to the direct and backward ports of a parallel-coupled TEM coupleris given in terms of even-mode parameters by (6.7) and (6.8), as follows:

S21 = S21e

S31 = S11e

where S11e and S21e denote the reflection and transmission coefficients, respectively,of the coupled lines for the case of even-mode excitation.

The equivalent circuit of an ideal parallel-coupled TEM directional coupler istherefore as shown in Figure 6.5, where Z0e denotes the even-mode characteristicimpedance of the coupled lines [7]. S11e and S21e , which are, respectively, thereflection and transmission coefficients of the two-port circuit shown in Figure 6.5give, respectively, the coupling to the backward (S31) and direct (S21) ports of thefour-port coupler shown in Figure 6.1. We thus see that the analysis of a parallel-coupled directional coupler reduces to analyzing a simple circuit consisting of alength of transmission line of characteristic impedance Z0e terminated by an imped-ance of Z0 at either of its ends. This analogy is extremely useful in the design andsynthesis of parallel-coupled directional couplers, as it is relatively much simplerto analyze single transmission line circuits.

Figure 6.5 Equivalent circuit of an ideal single-section TEM directional coupler.

6.3 Multisection Directional Couplers 177

6.3 Multisection Directional Couplers

6.3.1 Theory and Synthesis

To obtain a near-constant coupling over a wider frequency bandwidth than ispossible using a single-section coupler, a number of coupled sections must becascaded, as shown in Figure 6.6(a). Each section is a quarter-wave long at thecenter frequency. By properly choosing the even- and odd-mode impedances of thevarious sections, the bandwidth of the coupler can be increased. By analogy withthe equivalent circuit of a single-section coupler, the equivalent cascaded transmis-sion line circuit for finding the coupling to the backward and direct ports of themultisection coupler of Figure 6.6(a) is shown in Figure 6.6(b). The reflection andtransmission 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 even-and odd-mode impedances of the ith section of the multisection coupler are relatedby

Z0oi =Z 2

0Z0ei

(6.21)

where Z0 denotes the impedance terminating the ports of the directional coupler.A multisection coupler can be either symmetrical or asymmetrical. In multisec-

tion couplers, the term symmetrical is used to denote a coupler that has end-to-end symmetry. A symmetrical coupler employs an odd number of sections. In a

Figure 6.6 (a) An N-section asymmetrical parallel-coupled multisection directional coupler.(b) Equivalent circuit of directional coupler shown in part (a).


symmetrical coupler, the ith section will be identical to the N + 1 − ith section asshown in Figure 6.7(a). The equivalent circuit for analyzing the symmetrical coupleris shown in Figure 6.7(b). If the coupler does not have end-to-end symmetry [Figure6.6(a)], it is referred to as an asymmetrical coupler. An asymmetrical coupler canemploy an even or odd number of sections. A significant property of symmetricalcouplers is that in their case, the signal coupled to the direct port is 90 degreesout of phase with the signal coupled to the backward port (∠S31 = ∠S21 + 90degrees). This phase relationship is independent of the frequency. Because of thisproperty, 3-dB symmetrical directional couplers find extensive use in diplexers,multiplexers, directional filters, balanced mixers, and in other devices where the90-degree phase difference property is required. Asymmetrical couplers do notexhibit the phase property of symmetrical couplers and are generally used wherecouplers are designed to obtain broadband power division only.

The response of an optimal five-section symmetrical coupler is shown in Figure6.8.

Before discussing the synthesis of multisection TEM couplers, we need to definethe power loss ratio of a directional coupler.

Power Loss Ratio

The power loss ratio is defined as

L =1

|S21 |2(6.22)

Figure 6.7 (a) An N-section symmetrical parallel-coupled multisection directional coupler.(b) Equivalent circuit of directional coupler shown in part (a).


Figure 6.8 Typical response of a symmetrical five-section parallel-coupled TEM directional coupler.

where S21 is the transmission coefficient between the input and direct ports. Thepower 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 powerloss ratio L can be expressed as

L =1

|S21 |2=

1

|S21e |2(6.23)

Note that S21 denotes the scattering parameter between ports 1 and 2 of thefour-port directional coupler, whereas S21e denotes the scattering parameterbetween ports 1 and 2 of the equivalent two-port network.

In decibels:

10 log L = −20 log |S21 | = −20 log |S21e | (6.24)

The quantities on the right-hand side in the above equation denote the insertionloss in decibels between the input and direct ports. The function L is, therefore,also called the insertion loss function. The relationship between the scatteringparameters of an ideal directional coupler (S11 = S41 = 0) is given by

|S21 |2 + |S31 |2 = 1 (6.25)

or using (6.22) and (6.25), we obtain

|S31 |2 = 1 −1L

(6.26)

In terms of ABCD parameters, the power loss ratio L of a network is given by[1, 3]

L = 1 +14 F(A − D)2 − S B

Z0− CZ0D2G (6.27)

The equivalent circuit of a single-section coupler is shown in Figure 6.5. ItsABCD parameters (for a lossless case) are given by,


A = cos u, B = jZ0e sin u, C =j sin u

Z0e, D = cos u

The power loss ratio of a single-section coupler is therefore given by

L = 1 +14 SZ0e

Z0−

Z0Z0e

D2 sin2 u (6.28)

Similarly, the power loss ratio of a symmetrical three section coupler is givenby

L = 1 +14 HF2SZ0e1

Z0−

Z0Z0e1

D + SZ0e2Z0

−Z0

Z0e2DG sin u cos2 u (6.29)

− S Z 20e1

Z0Z0e2−

Z0Z0e2

Z 20e1

D sin3 uJ2

where Z0e1 denotes the even-mode impedance of the first and third sections andZ0e2 denotes the even-mode 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 givenfrequency band, the values of |S21 | and |S31 | be constant. This requires the functionL to be also constant over this frequency band. For example, if it is required tohave a 3-dB flat coupling over 5:1 frequency bandwidth ratio, the form of thecorresponding function L in the frequency band of interest is as shown in Figure6.9. 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 ratioof B = 5.


It is impossible to realize any arbitrary power loss ratio function using physicalnetworks. For example, the power loss ratio function L of the form shown inFigure 6.9 cannot be realized. Siedel and Rosen [8] have stated the necessary andsufficient conditions for the form of power loss ratio function L that can be realizedusing 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 u ) (6.30)

can be realized using an asymmetrical network as shown in Figure 6.6(b), wherePN is a polynomial of degree N whose value is greater than or equal to unity forall real values of u. If the physical network is to be symmetrical as shown in Figure6.7(b), an extra condition is imposed on the power loss function that can berealized. The necessary and sufficient condition that a power loss ratio function Lcan be realized using a symmetrical homogeneous stepped impedance network ofN equal-length sections as shown in Figure 6.7(b) is that it be of the form

L = 1 + [PN (sin u )]2 (6.31)

where PN is an odd polynomial in sin u of degree N.It may be verified that the power loss ratio of a three-section 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 u thatsatisfies 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 optimalin the sense that for a given number of sections, response type (equal-rippleor 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 networksynthesis techniques once the optimal polynomial L has been found.

The above approach has been used by Levy [1] to design optimal asymmetricalcouplers and by Cristal and Young [3], and Toulios and Todd [9] to design symmet-rical couplers. Unfortunately, analytical design expressions tend to be very cumber-some even for couplers with a small number of sections [1, 9]. For aiding designers,Levy has generated design tables for equal-ripple asymmetrical couplers for variousvalues of coupling and bandwidth for up to six sections [2]. For equal-ripple andmaximally flat symmetrical couplers, Cristal and Young have generated similartables for up to nine sections [3]. For equal-ripple response symmetrical couplers,these are reproduced in Table 6.1.


Table 6.1 Tables of Parameters for Symmetrical TEM-ModeCoupled-Transmission-Line Directional Couplers

d Z1 Z2 w B

(a) Normalized even-mode impedances for equal-ripple symmetrical3.01-dB couplers of three sections (Z4 − i = Zi )0.10 1.17135 3.25984 1.00760 3.030630.20 1.20776 3.41242 1.17199 3.830850.40 1.27036 3.66560 1.35225 5.175210.60 1.32964 3.90585 1.46353 6.456160.80 1.38970 4.15648 1.54440 7.779661.00 1.45274 4.43120 1.60798 9.20361

(b) Normalized even-mode impedances for equal-ripple symmetrical6-dB couplers of three sections (Z4 − i = Zi )0.10 1.10298 2.09445 0.91996 2.703560.20 1.12090 2.14693 1.07404 3.319840.40 1.15038 2.22865 1.24518 4.299310.60 1.17680 2.29968 1.35201 5.172910.80 1.20208 2.36724 1.43006 6.018301.00 1.22698 2.43431 1.49150 6.86621

(c) Normalized even-mode impedances for equal-ripple symmetrical8.34-dB couplers of three sections (Z4 − i = Zi )0.10 1.07434 1.71858 0.89286 2.612900.20 1.08644 1.74864 1.04355 3.182110.40 1.10606 1.79461 1.21159 4.073470.60 1.12339 1.83365 1.31688 4.855500.80 1.13973 1.86993 1.39403 5.601031.00 1.15560 1.90510 1.45488 6.33787

(d) Normalized even-mode impedances for equal ripple symmetrical10-dB couplers of three sections (Z4 − i = Zi )0.20 1.06945 1.57423 1.03140 3.129680.40 1.08475 1.60708 1.19816 3.988520.60 1.09817 1.63470 1.30282 4.737380.80 1.11075 1.66014 1.37959 5.447391.00 1.12290 1.68458 1.44020 6.14545

(e) Normalized even-mode impedances for equal-ripple symmetrical20-dB couplers of three sections (Z4 − i = Zi )0.20 1.02070 1.14914 1.00980 3.039580.40 1.02497 1.15617 1.17423 3.843960.60 1.02866 1.16197 1.27772 4.538040.80 1.03208 1.16720 1.35381 5.190111.00 1.03534 1.17213 1.41398 5.82570

Z0e1 = Z1Z0 , Z0e2 = Z2Z0 , Z0e3 = Z3Z0

Example 6.3

Design a symmetrical multisection TEM coupler with the following specifications:

Mean coupling = 6 dBMaximal ripple level = ±0.1 dBFrequency bandwidth ratio B = 8

The schematic of a symmetrical coupler is shown in Figure 6.7(a). To designa symmetrical coupler, we use the design Table 6.1(q) [3]. We find that a 6-dB,nine-section coupler designed for a ripple level of ±0.1 dB exhibits a frequency



d Z1 Z2 Z3 w B

(f) Normalized even-mode impedances for equal-ripple symmetrical 3.01-dB couplersof five sections (Z6 − i = Zi )0.10 1.07851 1.37268 3.97615 1.32559 4.931140.20 1.10921 1.44029 4.21023 1.45184 6.297140.40 1.16266 1.54541 4.57491 1.58152 8.558450.60 1.21370 1.63864 4.90924 1.65791 10.692920.80 1.26555 1.73013 5.25363 1.71196 12.887201.00 1.31988 1.82466 5.62978 1.75370 15.24047

(g) Normalized even-mode impedances for equal-ripple symmetrical 6-dB couplers offive sections (Z6 − i = Zi )0.10 1.04501 1.21972 2.38181 1.25446 4.345220.20 1.06052 1.25302 2.46010 1.37766 5.457380.40 1.08633 1.30203 2.57332 1.50548 7.088660.60 1.10969 1.34262 2.66727 1.58135 8.554620.80 1.13217 1.37978 2.75470 1.63520 9.964821.00 1.15438 1.41542 2.84048 1.67673 11.37370

(h) Normalized even-mode impedances for equal-ripple symmetrical 8.34-dB couplersof five sections (Z6 − i = Zi )0.10 1.03211 1.15690 1.89019 1.23184 4.207270.20 1.04271 1.17918 1.93414 1.35395 5.191500.40 1.06012 1.21142 1.99635 1.48104 6.707670.60 1.07565 1.23760 2.04658 1.55670 8.023230.80 1.09039 1.26114 2.09210 1.61050 9.269590.95 1.10119 1.27785 2.12484 1.64253 10.18985

(i) Normalized even-mode impedances for equal-ripple symmetrical 10-dB couplers offive sections (Z6 − i = Zi )1.00 1.10476 1.28331 2.13562 1.65206 10.496350.20 1.03418 1.14316 1.70922 1.34442 5.101480.40 1.4784 1.16808 1.75305 1.47118 6.564070.60 1.05996 1.18815 1.78805 1.54675 7.825130.80 1.07140 1.20606 1.81943 1.60053 9.013221.00 1.08249 1.22280 1.84912 1.64210 10.17639

(j) Normalized even-mode impedances for equal-ripple symmetrical 20-dB couplers offive sections (Z6 − i = Zi )0.20 1.01016 1.04183 1.17873 1.32734 4.946560.40 1.01406 1.04855 1.18767 1.45350 6.319360.60 1.01747 1.05386 1.19463 1.52888 7.490380.80 1.02066 1.05851 1.20073 1.58261 8.583381.00 1.02371 1.06280 1.20638 1.62420 9.64410

Z0e1 = Z1Z0 , Z0e2 = Z2Z0 , Z0e3 = Z3Z0Z0e4 = Z4Z0 , Z0e5 = Z5Z0

bandwidth ratio B = 7.99. Because this figure of B is very close to the specifiedvalue of B = 8, it is sufficient to employ a nine-section coupler to achieve thedesired specifications. From the same table, we determine even-mode impedancesof the various sections. Once the even-mode impedances are known, we can findthe odd-mode impedances using (6.21). The even- and odd-mode impedances, thevoltage coupling coefficient k, and the coupling in decibels of various sections ofthe coupler are given in Table 6.2. We see that the coupling of the center sectionis 2.17 dB, which is much tighter than the overall coupling for which the couplerhas been designed (6 dB). The coupling of other sections is smaller than the overallcoupling.



d Z1 Z2 Z3 Z4 w B

(k) Normalized even-mode impedances for equal-ripple symmetrical 3.01-dB couplers of seven sections(Z8 − i = Zi )0.10 1.05240 1.18406 1.56753 4.61180 1.49705 6.95310.20 1.07950 1.23581 1.65795 4.90662 1.59539 8.88600.40 1.12798 1.31754 1.79367 5.35611 1.69388 12.06660.60 1.17505 1.39069 1.91172 5.76434 1.75090 15.05780.80 1.22323 1.46258 2.02682 6.18437 1.79087 18.12701.00 1.27399 1.53668 2.14566 6.64407 1.82155 21.4147

(l) Normalized even-mode impedances for equal-ripple symmetrical 6-dB couplers of seven sections(Z8 − i = Zi )0.10 1.02686 1.10756 1.32930 2.62516 1.44052 6.14940.20 1.04246 1.13419 1.37278 2.72038 1.53802 7.65830.40 1.06580 1.17408 1.43416 2.85438 1.63645 10.00260.60 1.08735 1.20755 1.48367 2.96391 1.69378 12.06260.80 1.10831 1.23839 1.52841 3.06523 1.73404 14.03961.00 1.12915 1.26805 1.57104 3.16446 1.76487 16.0119

(m) Normalized even-mode impedances for equal-ripple symmetrical 8.34-dB couplers of seven sections(Z8 − i = Zi )0.10 1.02033 1.07694 1.23301 2.03194 1.42127 5.91170.20 1.02963 1.09518 1.26167 2.08436 1.51889 7.31400.40 1.04538 1.12204 1.30136 2.15641 1.61749 9.45720.60 1.05972 1.14417 1.33267 2.21361 1.67500 11.30770.80 1.07350 1.16423 1.36040 2.26506 1.71543 13.05631.00 1.08702 1.18319 1.38629 2.31408 1.74643 14.7747

(n) Normalized even-mode impedances for equal-ripple symmetrical 10-dB couplers of seven sections(Z8 − i = Zi )0.20 1.02360 1.07622 1.20802 1.81699 1.51198 7.19650.40 1.03597 1.09725 1.23839 1.86715 1.61028 9.26380.60 1.04718 1.11444 1.26213 1.90649 1.66773 11.03830.80 1.05786 1.12991 1.28298 1.94149 1.70815 12.70591.00 1.06834 1.14444 1.30229 1.97446 1.73917 14.3359

(o) Normalized even-mode impedances for equal-ripple symmetrical 20-dB couplers of seven sections(Z8 − i = Zi )0.20 1.00697 1.02256 1.05976 1.20128 1.49853 6.97660.40 1.01052 1.02846 1.06767 1.21112 1.59672 8.91880.60 1.01369 1.03320 1.07372 1.21863 1.65421 10.56780.80 1.01669 1.03740 1.07894 1.22515 1.69472 12.10291.00 1.01958 1.04129 1.08368 1.23116 1.72584 13.5903

Z0ei = Zi Z0

6.3.2 Limitations of Multisection Couplers

One of the major limitations of a multisection coupler is that the coupling of atleast one of the sections is much tighter than the overall coupling as shown in theabove example. This can create fabrication problems in microstrip technologywhere it is difficult to achieve tight coupling. Further, because the even- and odd-mode impedances of each section of a multisection coupler are different from thoseof the adjacent ones, the dimensions of the coupler abruptly change at the startand end of each section. Because of practical considerations, it may be necessaryto join adjacent sections using small lengths of tapered transmission lines as shownin Figure 6.10. If the operating frequency is high, the extra reactances producedby these abrupt discontinuities or extra lengths of joining transmission lines can



d Z1 Z2 Z3 Z4 Z5 w B

(p) Normalized even-mode impedances for equal-ripple symmetrical 3.01-dB couplers of nine sections(Z10 − i = Zi )0.10 1.04112 1.12024 1.29488 1.74863 5.18240 1.6012 9.0300.20 1.06598 1.16366 1.36260 1.85696 5.52654 1.6807 11.5280.40 1.11149 1.23397 1.46548 2.01711 6.04655 1.7594 15.6270.60 1.15624 1.29789 1.55536 2.15551 6.51769 1.8046 19.4750.80 1.20234 1.36117 1.64277 2.29038 7.00316 1.8362 23.4211.00 1.25107 1.42660 1.73250 2.42995 7.53602 1.8604 27.644

(q) Normalized even-mode impedances for equal-ripple symmetrical 6-dB couplers of nine sections(Z10 − i = Zi )0.10 1.02201 1.06888 1.17282 1.42807 2.83542 1.5550 7.9890.20 1.03437 1.09137 1.20736 1.47877 2.94305 1.6345 9.9430.40 1.05615 1.12599 1.25686 1.54902 3.09269 1.7136 12.9690.60 1.07658 1.15561 1.29716 1.60504 3.21427 1.7594 15.6220.80 1.09661 1.18320 1.33370 1.65546 3.32658 1.7913 18.1661.00 1.11663 1.20989 1.36852 1.70342 3.43663 1.8157 20.702

(r) Normalized even-mode impedances for equal-ripple symmetrical 8.34-dB couplers of nine sections(Z10 − i = Zi )0.10 1.01536 1.04904 1.12341 1.30048 2.15200 1.5392 7.6810.20 1.02379 1.06452 1.14687 1.33341 2.21025 1.6190 9.4980.40 1.03846 1.08798 1.17989 1.37809 2.28925 1.6985 12.2650.60 1.05206 1.10771 1.20622 1.41290 2.35152 1.7444 14.6500.80 1.06523 1.12579 1.22966 1.44356 2.40740 1.7766 16.9011.00 1.07823 1.14302 1.25158 1.47211 2.46063 1.8011 19.111

(s) Normalized even-mode impedances for equal-ripple symmetrical 10-dB couplers of nine sections(Z10 − i = Zi )0.20 1.01889 1.05161 1.11743 1.26387 1.90628 1.6133 9.3450.40 1.03041 1.07004 1.14313 1.29777 1.96074 1.6927 12.0160.60 1.04103 1.08543 1.16344 1.32390 2.00313 1.7386 14.3030.80 1.05127 1.09945 1.18139 1.34672 2.04073 1.7708 16.4501.00 1.06133 1.11271 1.19805 1.36779 2.07614 1.7954 18.547

(t) Normalized even-mode impedances for equal-ripple symmetrical 20-dB couplers of nine sections(Z10 − i = Zi )0.20 1.00555 1.01529 1.03447 1.07471 1.21931 1.6024 9.0610.40 1.00886 1.02054 1.04153 1.08328 1.22965 1.6818 11.5710.60 1.01187 1.02485 1.04700 1.08974 1.23748 1.7278 13.6970.80 1.01474 1.02871 1.05175 1.09527 1.24426 1.7601 15.6741.00 1.01753 1.03232 1.05608 1.10028 1.25049 1.7848 17.588

Z0e1 = Zi Z0

Table 6.2 Parameters of a Nine-Section Symmetrical Coupler

Section Z0e /Z0 Z0o /Z0 k C (dB)

1, 9 1.02201 0.97846 0.02177 33.242, 8 1.06888 0.93556 0.06651 23.543, 7 1.17282 0.85265 0.15807 16.024, 6 1.42807 0.70025 0.34197 9.325 2.83542 0.35268 0.77875 2.17


Figure 6.10 Typical physical layout of a symmetrical multisection coupler. Adjacent quarter-wavesections are joined by small length of transmission line sections.

reduce the input match and directivity of the coupler. In this case, a better solutionmay 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 cou-plers [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 shownin Figure 6.11 [10–12]. The lumped capacitor can be added at only one of theends or in the middle of coupled section [13]. With the addition of lumped capaci-tances, the electrical length of the coupler can be made to be equal for the evenand odd modes at the design frequency. The addition of lumped capacitances doesnot affect the even-mode signal, but it affects the odd-mode signal. This can beeasily explained.

In the case of even-mode 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 circuitof one-half of the network for the even mode is shown in Figure 6.11(c). We seethat the lumped capacitance has no effect on the overall capacitance between thestrip and the ground because one of the ends of the capacitor is open circuited.On the other hand, in the case of odd-mode excitation, the midplane PP ′ behaveslike a short circuit. The equivalent circuit of one-half of the network for thecase of odd-mode excitation is shown in Figure 6.11(d). In this case, the lumpedcapacitance is parallel with the capacitance between the strip and ground conduc-tors. 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 additionallumped capacitance. We can show that the electrical length of the coupler can bemade equal for the even- and odd-mode signals by choosing the value of the lumpedcapacitance Cab as [12]

6.4 Techniques to Improve Directivity of Microstrip Couplers 187

Figure 6.11 (a) Top view of lumped capacitor compensated microstrip coupler, (b) top view ofcoupled section showing plane of symmetry pp′ and lumped capacitors, (c) equivalentcircuit for even-mode excitation, and (d) equivalent circuit for odd-mode excitation.

Cab =1

4p f0Z0o tan u0(6.32)

where

u0 =p2 √ereo

ereerad

In the above equation, eree and ereo denote the even- and odd-mode effectivedielectric constants of the coupled microstrip lines, respectively, Z0o denotes theodd-mode impedance of the coupled structure, and f0 is the design center frequency.

The physical length of a capacitor compensated quarter-wave coupler is givenby [10]


lc =

p2

− tan−1 (p f0CabZ0e )

k0√eree(6.33)

where k0 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]. Forweaker coupling, (6.32) and (6.33) may not yield a very accurate design. In thatcase, it is more useful to optimize the value of the capacitance Cab and the lengthlc 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 compensated15.7-dB coupler on an alumina substrate is shown in Figure 6.12 [10].

Example 6.4

Given the parameters of coupled microstrip lines as eree = 6.7713, ereo = 5.5194,Z0e = 88.83V, Z0o = 28.14V, and f0 = 3 GHz, compute the value of compensatingcapacitance Cab and length lc of a quarter-wave coupler.

Figure 6.12 Directivity improvement of a lumped capacitor compensated 15.7-dB microstrip cou-pler on alumina substrate. (From: [10]. 1982 IEEE. Reprinted with permission.)


Using (6.32) and (6.33), we obtain Cab = 0.145 pF and lc = 8.86 mm. On theother 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 shownin Figure 6.13, then by properly choosing the thickness and dielectric constant ofthe layer, near equalization of even- and odd-mode phase velocities can be achievedover a reasonably wide frequency band [14–17]. The dielectric overlay can com-pletely cover the bottom dielectric layer or may only cover the region containingthe strips. The dielectric overlay also affects quite significantly the backward-wavecoupling between the lines. Therefore, the effect of overlay should be consideredwhile computing the dimensions of the strips, the spacing between them, and thelength of the coupler. A successful design of the overlay coupler depends on theavailability of accurate data on the phase velocities and the characteristic imped-ances of the even and odd modes of the overlay structure. Broadband directionalcouplers with high directivity have been demonstrated [14].

Figure 6.14 provides design curves for coupled microstrip lines on aluminasubstrate using alumina for the overlay [15]. The top alumina layer is assumed tocover the bottom alumina layer completely. Here the overlay thickness is the sameas the thickness of the substrate. The strip conductor thickness has been assumedto be zero. Simulated and measured data have shown about 10-dB improvementin the directivity of the 8.34-dB coupler designed using the curves in Figure 6.14.Similar improvements have been demonstrated [17] using low-dielectric constantsubstrate have er = 2.48. For a 10-dB coupler at S-band, the design parameters areer = 2.48, W = 3.2 mm, h = d = 1.42 mm, S = 0.4 mm, and coupled lengthL = 20.5 mm.

6.4.3 Use of Wiggly Lines

Although use of lumped capacitances or dielectric overlay structures lead to animproved directivity of microstrip directional couplers, both techniques complicatefabrication and may undermine the advantages of MICs. Another technique toequalize the phase velocities of even- and odd-mode signals that is compatible withMIC technology is to use wiggly lines instead of straight lines [18–20]. A top viewof wiggly-coupled lines is shown in Figure 6.15(b). It is assumed that by wiggling

Figure 6.13 Parallel-coupled microstrip coupler with dielectric overlay compensation.


Figure 6.14 Design curves for coupled microstrip lines covered with dielectric overlay er = 10.1,d/h = 1.0. (From: [15]. 1978 IEEE. Reprinted with permission.)

the lines, the odd-mode phase velocity is slowed down, whereas the even-modephase velocity is not affected. Further, wiggling affects only the mutual capacitancebetween 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 straight-coupled and wiggly-coupled lines aredefined in Figure 6.15. To consider the effect of wiggling, let us consider thecapacitance of a section of length DL (between reference planes AA and BB) ofstraight- and wiggly-coupled sections. For the straight-coupled section, the odd-mode capacitance between reference planes AA and BB is given by

Co = (Cf + Cp + Cfo )DL (6.34)

In these equations, Cf , Cp , and Cfo denote per-unit length capacitances asdefined in Figure 3.9.

On the other hand, the odd-mode capacitance of wiggly-coupled lines is givenby

Cow = (Cf + Cp )DL + CfoLw (6.35)

The above relation results because in the case of wiggly-coupled lines, the effectivelength seen by the capacitance Cf and Cp between the reference planes AA and BBis DL, which is the same as that for the straight-coupled lines. The effective lengthseen by the odd-mode fringing capacitance Cfo in this case is Lw , however, whichis achieved by wiggling the lines.

To equalize the odd- and even-mode phase velocities, the following relationshould be satisfied:


Figure 6.15 Top view of (a) parallel-coupled straight lines, (b) wiggly-coupled lines, and(c) exploded view of wiggly-coupled lines between planes AA and BB.

Cow =ereeereo

Co (6.36)

where eree and ereo denote the effective dielectric constants for the even and oddmodes, respectively. Using (6.34) to (6.36), we find that the length Lw of wiggly-coupled lines should be chosen as

Lw = DLC ′foCfo

(6.37)

where

C ′fo = (Cp + Cf ) Sereeereo

− 1D +ereeereo

Cfo


To obtain the value of Lw as given by (6.37), the wiggle depth d should bechosen as

d =DL2 √SC ′fo

CfoD2 − 1 (6.38)

The capacitance parameters Cp , Cf , and Cfo are defined in Section 3.2.2. Formicrostrip lines, the capacitances Cp and Cf can be determined as follows:

Cp =e0erW

h

2Cf = √ere

cZ0− Cp

where c is the velocity of light in free space, and Z0 and ere denote the quasistaticcharacteristic impedance and effective dielectric constant, respectively, of a singlemicrostrip line of width W. Furthermore, the odd-mode fringing capacitance Cfocan be determined using

Cfo = Co − Cp − Cf

where

Co = √ereo

cZ0o

In this equation, Co denotes the odd-mode capacitance, and Z0o and ereodenote, respectively, the characteristic impedance and effective dielectric constantof the odd mode.

6.4.4 Other Techniques

Many other techniques to improve the directivity of microstrip couplers have alsobeen reported in [21–23]. Figure 6.16(a) shows a schematic of a microstrip couplerwhere a shunt inductive feedback is used between the direct ports [21]. By properlychoosing the impedance and length of the feedback element, an isolation zero canbe obtained at the desired frequency. The physical layout of a shunt feedbackcompensated coupler is shown in Figure 6.16(b). The design can be carried out bya microwave circuit simulator using equivalent circuits for various discontinuitiesand coupled lines. For more accurate designs especially at high frequencies, EMsimulators should be used. It has been reported that this scheme leads to about a5–30-dB improvement in directivity. The improvement is obtained over a 15%–20% bandwidth.

Figure 6.17(a) shows another technique. In this technique, coupled spur linessections are added close to the various ports [22]. The physical layout of a practical


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 anddiscontinuities, its design should be carried out using microwave circuit simulatorsor 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%.


A disadvantage of this technique, however, is that the return loss bandwidth isreduced considerably. The 20-dB return loss bandwidth is reduced to less thanabout 20%.

It has been reported that the directivity of microstrip couplers can also beimproved by using meandered coupled line sections such as shown in Figure 6.18[23]. The coupling between the parallel sections in the meandered region (controlledby separation D as shown in Figure 6.18) is responsible for increasing the phasevelocity of the even mode, which leads to an improvement in the directivity. Usingthis technique, a measured directivity of greater than 20 dB was reported over anoctave bandwidth, for a 10-dB coupler [23].

Figure 6.18 Coupler with meandered section.

References

[1] Levy, R., ‘‘General Synthesis of Asymmetric Multielement Coupled-Transmission-LineDirectional Couplers,’’ IRE Trans., Vol. MTT-11, July 1963, pp. 226–237.

[2] Levy, R., ‘‘Tables for Asymmetric Multielement Coupled-Transmission-Line DirectionalCouplers,’’ IRE Trans., Vol. MTT-12, May 1964, pp. 275–279.

[3] Cristal, E. G., and L. Young, ‘‘Theory and Tables of Optimum Symmetrical TEM-ModeCoupled-Transmission-Line Directional Couplers,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-13, September 1965, pp. 544–558.

[4] Arai, S., et al., ‘‘A 900-MHZ 90-Degree Hybrid for QPSK,’’ IEEE MTT-S Int. MicrowaveSymp. Dig., 1991, pp. 857–860.

[5] Tanaka, H., et al., ‘‘2-GHz One-Octave-Band 90-Degree Hybrid Coupler Using CoupledMeander Line Optimized by 3-D FEM,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1994,pp. 903–906.

[6] Tanaka, H., et al., ‘‘Miniaturized 90-Degree Hybrid Coupler Using High DielectricSubstrate for QPSK Modulator,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1996,pp. 793–796.

[7] Young, L., ‘‘Stepped Impedance Transformers and Filter Prototypes,’’ IRE Trans.,Vol. PGMTT-10, September 1962, pp. 339–359.

[8] Seidel, H., and J. Rosen, ‘‘Multiplicity in Cascade Transmission Line Synthesis—Part I,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-13, May 1965, pp. 275–283; and PartII, July 1965, pp. 398–407.


[9] Touplios, P. P., and A. C. Todd, ‘‘Synthesis of Symmetrical TEM-Mode DirectionalCouplers,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, September 1965,pp. 536–544.

[10] March, S. L., ‘‘Phase Velocity Compensation in Parallel-Coupled Microstrip Line,’’ IEEEMTT-S Int. Microwave Symp. Dig., 1982, pp. 410–412.

[11] Schaller, G., ‘‘Optimization of Microstrip Directional Couplers with Lumped Capacitors,’’AEU, Vol. 31, July–August 1977, pp. 301–307.

[12] Kajfez, D., ‘‘Raise Coupler Directivity with Lumped Compensation,’’ Microwaves,Vol. 27, March 1978, pp. 64–70.

[13] Dydyk, M., ‘‘Microstrip Directional Couplers with Ideal Performance Via Single-ElementCompensation,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-47, June 1999,pp. 956–964.

[14] Sheleg, B., and B. E. Spielman, ‘‘Broadband Directional Couplers Using Microstrip withDielectric Overlays,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-22,December 1974, pp. 1216–1220.

[15] Paolino, D. D., ‘‘MIC Overlay Coupler Design Using Spectral Domain Techniques,’’ IEEETrans. Microwave Theory Tech., Vol. MTT-26, September 1978, pp. 646–649.

[16] Klein, J. L., and K. Chang, ‘‘Optimum Dielectric Overlay Thickness for Equal Even- andOdd-Mode Phase Velocities in Coupled Microstrip Circuits,’’ Electronics Letters,Vol. 26, 1990, pp. 274–276.

[17] Su, L., T. Itoh, and J. Rivera, ‘‘Design of an Overlay Directional Coupler by a Full-WaveAnalysis,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-31, December 1983,pp. 1017–1022.

[18] Podell, A., ‘‘A High-Directivity Microstrip Coupler Technique,’’ IEEE MTT-S Int. Micro-wave Symp. Dig., 1970, pp. 33–36.

[19] Uysal, S., and H. Aghvami, ‘‘Synthesis, Design and Construction of Ultra-WidebandNonuniform Directional Couplers in Inhomogeneous Media,’’ IEEE Trans. MicrowaveTheory Tech., Vol. MTT-37, June 1989, pp. 969–976.


[21] Chen, J. L., S. F. Chang, and C. T. Wu, ‘‘A High Directivity Directional Coupler withFeedback Compensation,’’ IEEE MTT-S Int. Microwave Symp. Digest, 2002,pp. 101–104.

[22] 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.

[23] Wang, S. M., C. H. Chen, and C. Y. Chang, ‘‘A Study of Meandered Microstrip Couplerwith High Directivity,’’ IEEE MTT-S Int. Microwave Symp. Digest, 2003, pp. 63–66.

C H A P T E R 7

Nonuniform Broadband TEMDirectional Couplers

A major disadvantage of using multisection directional couplers to obtain broad-band coupling is that there is an abrupt change in the transverse dimensions ofthe coupler (e.g., width of the lines and spacing between them) at the start andend of each section. The physical discontinuities that occur from a change in thedimensions lead to poor match and directivity of the coupler. These effects increasein severity with increasing frequency. To avoid the abrupt change, a continuousvariation in the coupling between the lines along the length of the coupler can beimplemented. This requires a continuous variation in the transverse dimensions ofthe coupler, and these types of couplers are called nonuniform couplers.

In this chapter, we discuss the theory and design of symmetrical and asymmetri-cal nonuniform TEM couplers. Symmmetrical couplers have end-to-end symmetryand have a property that the phase difference between the coupled ports is 90degrees. On the other hand, asymmetrical couplers do not have end-to-end symme-try. By proper design, asymmetrical couplers can be designed to have the phaseproperty of a magic-T. The design of symmetrical and asymmetrical couplersreported here is based largely on the work of Tresselt [1] and DuHamel andArmstrong [2].

7.1 Symmetrical Couplers

Figure 7.1 shows a symmetrical nonuniform TEM coupler. The width of the linesand 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 thesignal coupled to the backward port is 90 degrees out of phase with that coupledto the direct port.

Because the coupler is assumed to be symmetrical about the plane PP ′, we cananalyze it in terms of even- and odd-mode parameters. The scattering parametersof a symmetrical four-port network are given by (4.22) as follows:

197

198 Nonuniform Broadband TEM Directional Couplers

Figure 7.1 Symmetrical nonuniform TEM coupler. Even- and odd-mode characteristic impedancesvary continuously along the length of the structure.

S11 =S11e + S11o

2

S21 =S21e + S21o

2(7.1)

S31 =S11e − S11o

2

S41 =S21e − S21o

2

where S11e and S11o denote the reflection coefficients of the even- and odd-modesignals respectively and S21e and S21o denote the transmission coefficients of theeven- and odd-mode signals, respectively.

The nonuniform coupling is achieved basically by varying the even- and odd-mode characteristic impedances of the coupled lines along the length of the struc-ture. If the dimensions of the nonuniform coupler are chosen such that at any crosssection, the values of the even- and odd-mode impedances satisfy the followingcondition:

Z0e (z)Z0o (z) = Z20 (7.2)

then, as in the case of uniform couplers discussed in Section 4.2.2, the scatteringparameters of a nonuniform coupler reduce to

S11 = S41 = 0

S31 = S11e (7.3)

S21 = S21e

7.1 Symmetrical Couplers 199

The equivalent circuit for obtaining the scattering parameters S11e and S21e of anonuniform coupler is shown in Figure 7.2(a) where Zoe (z) denotes the characteris-tic impedance of the nonuniform transmission line as a function of longitudinalcoordinate z.

7.1.1 Coupling in Terms of Even-Mode Characteristic Impedance

We now proceed to find the reflection coefficient S11e at port 1 of the circuit shownin Figure 7.2(a) in terms of the even-mode characteristic impedance of the coupledlines. Because the even-mode characteristic impedance of the coupled lines variescontinuously along the length of the structure, it can be represented as shown inFigure 7.2(b). [The odd-mode characteristic impedance also varies along the lengthof the structure, but because it is always related to the even-mode impedance by(7.2), it is possible to analyze the network in terms of even-mode impedances only.]The incident wave is partially reflected at every step because of a mismatch of theimpedance. The total reflection coefficient at port 1 can be found by using thesmall-signal reflection [3]. Using this theory, the total reflection coefficient at theinput 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.


from each step in proper phase. The differential reflection coefficient at the stepdefined by plane BB ′ is given by

dG =Z0e + dZ0e − Z0eZ0e + dZ0e + Z0e

≈dZ0e2Z0e

=12

d(ln Z0e ) (7.4)

=12

ddz

(ln Z0e ) dz

where Z0e is a function of z, and it is assumed that 2Z0e @ dZ0e . The contributionto the total reflection coefficient at port 1 (z = 0) from the differential reflectioncoefficient dG at plane BB ′ is obtained by multiplying (7.4) by the phase terme −2jbz, or

dS11e = dGe −2jbz =12

e −2jbz ddz

(ln Z0e ) dz (7.5)

where b is the phase constant of the wave along the direction of propagation.Summing reflections from z = 0 to z = d, we obtain

S11e =12 E

d

0

e −2jbz ddz

(ln Z0e ) dz (7.6)

Furthermore, using (7.3), the coupling between ports 1 and 3 of the configurationshown in Figure 7.1 is given by

S31 = S11e =12 E

d

0

e −2jbz ddz

(ln Z0e ) dz (7.7)

This equation is valid if it is assumed that the reflection at any step is quitesmall compared with unity.

The coupler is assumed to be symmetrical about the plane z = d/2. It is usefulto define another coordinate u as

u = z −d2

(7.8)

such that the structure is symmetrical with respect to u = 0. Equation (7.7) thenbecomes


S31 =12

e −jbd Ed /2

−d /2

e −2jbu ddu

(ln Z0e ) du (7.9)

= e −jbd Ed /2

−d /2

e −2jbup(u) du

where

p(u) =12

ddu

(ln Z0e ) (7.10)

The function p(u) is an odd function of u and hence (7.9) reduces to

S31 = −je −jbd Ed /2

−d /2

sin(2bu)p(u) du (7.11)

Inspection of (7.11) shows that the amplitude of the coupled signal is symmetri-cal about b = 0; that is, |S31(b ) | = |S31(−b ) | . Furthermore, the phase of the coupledsignal is given by

∠S31 =p2

− bd rad, for b > 0 (7.12)

= − Sp2

+ bdD rad, for b < 0

Information on the amplitude and phase response for negative values of b arerequired for the synthesis of the coupler.

Using (7.9) or (7.11), we can determine the coupling of a symmetrical nonuni-form coupler, if the distribution of the even-mode characteristic impedance isknown, and (7.9) and (7.11) can be easily evaluated numerically.

7.1.2 Synthesis

For a coupler of finite length d, the even-mode impedance varies only over thelength of the coupler ( |u | ≤ d/2). The function p(u) given by (7.10) is thereforezero for |u | ≥ d/2. The lower and upper limits of integration in (7.9) or (7.11) cantherefore be changed to u = −∞ and u = ∞, respectively, without affecting the valueof the integral. In that case, (7.9) becomes

S31 = e −jbd E∞

−∞

e −2jbup(u) du


or

S31e jbd = E∞

−∞

e −2jbup(u) du (7.13)

It is interesting to note that the quantity on the right-hand side of (7.13) denotesthe Fourier transform of the function p(u). The coupling is therefore given by theFourier transform of the function p(u). In a synthesis problem, however, it isrequired to find the function p(u) for a given coupling response S31 , and this canbe done by using the inverse Fourier transform. Thus, we obtain

p(u) =1

2p E∞

−∞

S31e jbde j2bud(2b ) (7.14)

where d(2b ) denotes that the integration is with respect to 2b . 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 amplituderesponse in the range 2b = 0 to 2b = 1 as shown in Figure 7.3. The synthesis canbe easily extended to a coupler in a different frequency range by scaling the lengthof the coupler as discussed later in the chapter.

Figure 7.3 shows the amplitude of the coupling of an ideal coupler only in therange 2b > 0. As shown by (7.14), however, we need to know the amplitude andphase response of the coupler in the range −∞ < 2b < ∞ to synthesize a coupler.Because the amplitude response of the coupler is symmetrical about b = 0 asdiscussed earlier, we have

Figure 7.3 Coupling response of an ideal directional coupler.


|S31 | = R, for |2b | < 1 (7.15)

= 0, for |2b | > 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 b is related to the frequency f by the relation

b =2p f

v(7.17)

where v is the phase velocity along the direction of propagation.The phase of S31 is given by (7.12). Using (7.12) and (7.15), we obtain

S31 = Re j(p /2 − bd), for 0 < 2b < 1

= Re −j(p /2 + bd), for −1 < 2b < 0 (7.18)

= 0 otherwise

Substituting S31 from (7.18) in (7.14), we obtain

p(u) = −Rp

sin2(u/2)u/2

(7.19)

which is plotted in Figure 7.4 for an arbitrary assumed value of coupling R. Notefrom (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 couplerwill 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.


function p(u) becomes quite small when |u | becomes large. Therefore, if the lengthd of the coupler is chosen as equal to 4np (corresponding to n positive and nnegative lobes of Figure 7.4), the coupling response is not expected to deviate muchfrom the ideal response, if the value of n is sufficiently high. Figure 7.5 shows thevariation in coupling for different lengths of the coupler, for n = 1, . . . , 4; thatis, d = 4p , d = 8p , d = 12p , and d = 16p . From these figures we see that as thelength of the coupler is increased, the response approaches the ideal response. Wealso see that for a coupler of length d = 4np , the number of ripples in the frequencyband of interest is equal to n. It may be emphasized that the stated length corre-sponds to a coupler designed to operate in the range b = 0 to b = 1/2 where b isrelated to frequency by (7.17).

Although increasing the length of the coupler brings the response of the couplercloser to the ideal response, it may be noted that there is an overshoot alwayspresent near b = 0. The value of the overshoot does not decrease in amplitudewith increasing the number of lobes used. This phenomenon is known as Gibb’sphenomenon and occurs when a step function such as shown in Figure 7.3 isapproximated by a finite number of Fourier terms. The problem is resolved byemploying weighting functions to remove the problem of overshoot and to makethe level of ripples equal. In this technique, instead of using function p(u) for thesynthesis as given by (7.19), we use the weighted function

Figure 7.5 Coupling response of a nonuniform coupler of length (a) 4p , (b) 8p , (c) 12p , and(d) 16p .


pw (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 4p of the coupler(e.g., the value of the weighting function is w2 in the intervals −4p < u < −2p and2p < u < 4p ). The weighting function can be determined by a term-by-termcomparison of a known equal ripple-level function with a function containing afew weighted Fourier series terms. A technique for determining weighting functionsis discussed later in Section 7.1.3. Before that, we discuss the final step for thesynthesis of a nonuniform coupler.

Computing Z0e (u) from p(u) or pw (u)

Once the value of p(u) or pw (u) is known, the corresponding value of Z0e (u) needsto be determined for the realization of the physical circuit. Integrating (7.10) withrespect to u, we obtain

12

ln Z0e (u) =12

ln Z0e | u = −d /2+ E

u

−d/2

p(u) du (7.21)

where p(u) is given by (7.19).Furthermore, because

Z0e | u = −d /2 = Z0

(7.21) can also be expressed as

Figure 7.6 Typical weighting function of a nonuniform coupler.


12

lnZ0e (u)

Z0= E

u

−d/2

p(u) du = −Rp E

u

−d/2

sin2 (u/2)u/2

du (7.22)

Unfortunately, the integral in (7.22) cannot be evaluated in closed form. It is,however, quite simple to evaluate the above integrals numerically using availablecomputer software programs such as MATHCAD or MATLAB. If the weightedfunction pw (u) given by (7.20) is used for the synthesis of the coupler, the even-mode impedance is then given by

12

lnZ0e (u)

Z0=

Rp E

u

−d/2

w(u) p(u) du = −Rp E

u

−d/2

w(u)sin2 (u/2)

u/2du (7.23)

Evaluation of (7.23)

Let us assume that in a particular case, the length d of the coupler is chosen as20p (i.e., it extends from u = −10p to u = 10p ) and it is desired to find the even-mode impedance at u = −p . Assuming that the weighted function is of the formas shown in Figure 7.6, the value of Z0e at u = −p can be found using (7.23) asfollows:

12

lnZ0e | u = −p

Z0= −

Rp 3w5 E

−8p

−10p

sin2 (u/2)u/2

du

+ w4 E−6p

−8p

sin2 (u/2)u/2

du + w3 E−4p

−6p

sin2 (u/2)u/2

du (7.24)

+ w2 E−2p

−4p

sin2 (u/2)u/2

du + w1 E−p

−2p

sin2 (u/2)u/2

du4Each 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 withunity. The synthesis technique described earlier is thus strictly valid for realization ofcouplers having coupling of less than about 10 dB. However, (7.23) can also beused to synthesize a ‘‘tight coupler’’ if the weighting function terms wi are chosenproperly. 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 symmetricalmultisection coupler as discussed in the previous chapter is first designed for givencoupler specifications. Let the multisection coupler employ 2n − 1 sections with


impedance levels denoted as shown in Figure 7.7. Such a coupler will have n ripplesin the frequency band of interest. Similarly, a nonuniform coupler that has a lengthequal to 4np1 will have n ripples in the frequency band. Therefore, the responseof a nonuniform coupler can be made identical to the response of a multisectioncoupler of 2n − 1 sections if the length of the nonuniform coupler is chosen equalto 4np . The nonuniform coupler can then be synthesized using (7.23) where theweighting function terms wi are determined by the following method.

Consider the function shown in Figure 7.8. The value of the function for0 ≤ u ≤ p represents the value of desired coupling R. The Fourier series represen-tation of the function in the range 0 < u < p is

g(u ) = R4p Ssin u +

sin 3u3

+sin 5u

5+

sin 7u7

+sin 9u

9+ . . .D (7.25)

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 < u < p .

1. This length is for a coupler designed to operate from 2b = 0 to 2b = 1. For a coupler in a differentfrequency range, the length of the coupler is different.


With a few terms from the above series used to describe the function, theresponse would be similar to that shown in Figure 7.5; that is, there will exist anovershoot near u = 0 and the ripples would be unequal. If only n terms of theseries (7.25) are used, it is more appropriate to construct a new function f (u ) asfollows:

f (u ) = R4p Fw1 sin u + w2

sin 3u3

+ . . . + wnsin(2n − 1)u

(2n − 1) G (7.26)

where w1 , . . . , wn are the weighting function terms. With a suitable choice ofthese terms, the function f (u ) can be made an equal ripple.

A suitable equal ripple function can be easily determined by using the designtables of symmetrical multisection couplers. If (7.11) is used to find the couplingof a symmetrical coupler of 2n − 1 sections shown in Figure 7.7, the coupling(excluding the phase factor) is given by

h(u ) = ∑n

r = 1ln

Z0e (n + 1 − r)Z0e (n − r)

sin(2r − 1)u (7.27)

In (7.27), u denotes the electrical length of each section and is given by

u =bd

2n − 1(7.28)

where b is the phase constant, d denotes the total physical length of the coupler,and

Z0e (n − r) | r = n = Z0 (7.29)

If the various impedances in (7.27) correspond to that of an equal ripplesymmetrical coupler, the function h(u ) is also equal ripple. Comparing term byterm (7.26) and (7.27), we can determine the weighting function terms wi .

Example 7.1

Determine the weighting function terms for the design of a nonuniform couplerhaving a mean coupling C = 3.01 dB, a frequency bandwidth ratio (B) = 8, and aripple tolerance d = ±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 multisec-tion coupler with the above specifications requires seven sections. This gives


N = 2n − 1 = 7

or

n = 4

Also from the same table, the even-mode impedances of the various sections are

Zoe1Z0

= 1.07950

Zoe2Z0

= 1.23581 (7.30)

Zoe3Z0

= 1.65795

Zoe4Z0

= 4.90662

(7.27) then becomes

h(u ) = 1.085 sin u + 0.29385 sin 3u + 0.13522 sin 5u + 0.076497 sin 7u(7.31)

Substituting R = 0.707 and n = 4 in (7.26), we obtain

f (u ) = 0.90034w1 sin u + 0.30011w2 sin 3u (7.32)

+ 0.18006w3 sin 5u + 0.12862w4 sin 7u

Comparing term by term (7.31) and (7.32), the weighting function terms arefound as follows:

w1 = 1.205

w2 = 0.979 (7.33)

w3 = 0.751

w4 = 0.595

7.1.4 Electrical and Physical Length of a Coupler

The synthesis described so far is for a coupler having an equal-ripple response inthe range 2b = 0 to 2b = 1 (the value of b at the center frequency is 1/4). For sucha coupler, the length of the coupler corresponds to those of 2n lobes (n positiveand n negative) and is therefore equal to 4np . The number of ripples for such acoupler is equal to n in the frequency band of interest. The total electrical lengthof the nonuniform coupler at the center frequency is therefore


uc =14

4np = np (7.34)

If bc denotes the phase constant of the wave in the medium of the couplercorresponding to the center frequency of design f0 , then

uc = bcd = np (7.35)

where d is the total physical length of the coupler, or

d =npbc

(7.36)

In (7.36), bc = 2p f0 /v, where v is the velocity of propagation in the mediumof the coupler.

Length of Multisection Coupler

The electrical length of each section of a multisection coupler is p /2 rad at thecenter frequency of design. The electrical length of a multisection coupler of2n − 1 sections is therefore

bcd = (2n − 1)p2

(7.37)

Comparison of (7.34) and (7.37) shows that a nonuniform coupler is a quarter-wave 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 TEMmode coupler can be summarized as follows:

• Step 11. Specify coupling level C(dB), or voltage coupling factor R. Find one from

the other using (7.16).2. Specify ripple level d in dB.3. Specify frequency bandwidth ratio B, and center frequency of design f0

(or frequency range of operation f1 to f2). If the frequency range ofoperation of the coupler is specified, find B and f0 using (6.11) to (6.13),respectively.

• Step 2Using Table 6.1 from Chapter 6, find the minimum number of sectionsrequired for a multisection symmetrical coupler for the given values of C(coupling), d (coupling ripple), and B (bandwidth ratio). Let the number ofsections required be N, then


n =N + 1

2

• Step 3Using the same tables, find Z0e1 , Z0e2 , . . . , Z0en .

• Step 4Using computed values of Z0e1 , Z0e2 , . . . , Z0en , construct the functionh(u ) using (7.27).

• Step 5Construct the function f (u ) using (7.26).

• Step 6Comparing term by term the functions h(u ) and f (u ) computed using steps4 and 5, respectively, find the weighting function terms w1 , . . . , wn .

• Step 7Evaluate Z0e (u) numerically using (7.23) for discrete values of u in the range−2np ≤ u ≤ 2np , and by substituting d/2 = 2np .

• Step 8The values of u for which computations are made in step 7 correspond tothe design of the coupler having a mean value of b = 1/4. To determine thevalues of u for b = bc , divide the values of u computed in step 7 by a factorof 4bc , where bc is the phase constant in the medium of the coupler at thecenter frequency of the design.

Example 7.2

Design a nonuniform coupler in a homogeneous dielectric medium (er = 2.32) withthe following specifications:

Mean coupling (C) = 8.34 dBFrequency of operation = 1–11 GHzRipple level (d ) = ±0.3 dB

• Step 1From these specifications, we have

R = 10−C/20 = 0.3828

f0 =f1 + f2

2=

1 + 112

= 6.0 GHz

B =f2f1

= 11

At 6.0 GHz, the phase constant in free space is given by


b0 =2p f00.3

= 125.67 rad/m

or, the phase constant in the medium of the coupler at the center frequencyof design is given by

bc = √er b0 = 1.524b0 = 191.51 rad/m

• Step 2From Table 6.1(r) in Chapter 6, we find that with a mean coupling level ofC = 8.34 dB, ripple level = ±0.3 dB, a nine-section symmetrical coupler willexhibit a frequency bandwidth ratio of B = 10.96, which is very close tothe specified value. We thus have

n =N + 1

2=

9 + 12

= 5 (7.38)

• Step 3Using the same table, the impedances of the different sections of the nine-section coupler is

Zoe1Z0

= 1.03134

Zoe2Z0

= 1.07697

Zoe3Z0

= 1.16469 (7.39)

Zoe4Z0

= 1.35771

Zoe5Z0

= 2.25315

• Step 4Substituting these values in (7.27), we obtain

h(u ) = 0.5071 sin u + 0.1532 sin 3u + 0.0785 sin 5u (7.40)

+ 0.0433 sin 7u + 0.0309 sin 9u

• Step 5Substituting R = 0.3828 in (7.26):

f (u ) = 0.4873w1 sin u + 0.1624w2 sin 3u + 0.09745w3 sin 5u (7.41)

+ 0.06961w4 sin 7u + 0.05414w5 sin 9u


• Step 6Comparing (7.40) and (7.41) term by term, we obtain the following valuesfor the weighting functions:

w1 = 1.040

w2 = 0.943

w3 = 0.805 (7.42)

w4 = 0.623

w5 = 0.569

• Step 7The values of Z0e (u) were computed for values of u in the range −10p ≤ u≤ 10p using (7.23) by numerical evaluation of the integral using softwarepackage Mathcad. The obtained values of Z0e (u) are shown in Table 7.1.The computed values of u are valid for a coupler having bc = 1/4.

• Step 8The values of u for the specified coupler (bc = 191.51) were determined bydividing the values of u derived in step 7 by a factor 4bc . These values arealso shown in Table 7.1.

Because the coupler is symmetrical about u = 0, it is sufficient to findthe structure parameters for 0 ≤ u ≤ d/2. The overall length of the coupler isd = 2 × 4.1 = 8.2 cm.

Table 7.1 Design Procedure for a Nonuniform TEM Coupler

u (m) u (cm)Normalized Coupler Actual Coupler ln

Z0eZ0

Z0eZ0bc = 1/4 rad/m bc = 191.51 rad/m

10p 4.10 0.00 1.009.5p 3.89 0.00268 1.0029p 3.69 0.01486 1.01498.5p 3.48 0.0277 1.0288.0p 3.28 0.0309 1.0317.5p 3.08 0.0346 1.0357.0p 2.87 0.0517 1.0536.5p 2.67 0.0702 1.0736.0p 2.46 0.0743 1.0775.5p 2.25 0.0806 1.0845.0p 2.05 0.1113 1.1174.5p 1.85 0.1449 1.1564.0p 1.64 0.1532 1.1653.5p 1.44 0.1647 1.17903.0p 1.23 0.2231 1.25002.5p 1.03 0.2911 1.33792.0p 0.82 0.3087 1.36161.5p 0.62 0.3369 1.4001.0p 0.41 0.5087 1.6630.5p 0.21 0.7852 2.19290p 0 0.9263 2.525


Once we determine the even-mode characteristic impedances of the coupler atvarious cross sections of the coupler, we can find the corresponding odd-modecharacteristic impedances using (7.2). Furthermore, the physical dimensions of thelines (i.e., their width and the spacing between them) can be determined. The designand performance of nonuniform couplers using microstrip lines is described in[3, 4].

The design procedure described above requires data of symmetrical multisectionfilters, 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 methodsis described in [5].

7.2 Asymmetrical Couplers

A nonuniform asymmetric coupler which exhibits a broadband coupling and thephase properties of a magic-T is shown in Figure 7.9. The structure does not havean end to end symmetry but has a symmetry with respect to plane PP ′. The couplerconsists of a section of coupled lines of length u extending from z = −d to z = 0and a section of uncoupled lines of length u. The coupling between lines increasesfrom z = −d to z = 0. However, it reduces abruptly to zero at z = 0. The value ofu is 90 degrees at mid-band frequency. The uncoupled section is used to give thestructure of the properties of a magic-T.

The scattering parameters of the coupler can be determined using the even-mode analysis (as discussed earlier in Section 7.1) if the even- and odd-modecharacteristic impedances of the coupled line section at any cross section are chosento satisfy

Z0e (z)Z0o (z) = Z20 (7.43)

The even-mode equivalent circuit of the coupler is shown in Figure 7.10. Interms of even-mode scattering parameters, the scattering parameters of the couplerare given by

Figure 7.9 Asymmetric nonuniform coupler.

7.2 Asymmetrical Couplers 215

Figure 7.10 Equivalent circuit to determine coupling of the asymmetric nonuniform coupler shownin Figure 7.9.

S11 = S22 = S33 = S44 = 0 (7.44)

S41 = S14 = S23 = S32 = 0 (7.45)

S21 = S12 = S34 = S43 = S21e = S12e (7.46)

S31 = S13 = S11e , S24 = S42 = S22e (7.47)

The even-mode reflection coefficient S11e at port 1 can be determined bysumming reflections in proper phase as discussed in Section 7.1. If the even-modeimpedance 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 thecontribution of the taper section to the overall reflection coefficient S11e is zero,the reflection coefficient at port 1 results solely from the abrupt impedance disconti-nuity at z = 0. The reflection coefficient S11e is given by

S11e =Z0 − Z0 /aZ0 + Z0 /a

e −2ju =a − 11 + a

e −2ju (7.48)

where it is assumed that the even-mode characteristic impedance of coupled linesis Z0 /a at z = 0−. Similarly,

S22e =Z0 /a − Z0Z0 + Z0 /a

e −2ju =1 − a1 + a

e −2ju (7.49)

The properties of lossless two-ports were discussed in Chapter 2. Using (2.77),the transmission coefficient between ports 1 and 2 of the network shown in Figure7.10 is found as

S21e = S12e =2√a

1 + ae −2ju (7.50)


Furthermore, using (7.44)–(7.47), the scattering matrix of the coupler can beexpressed as

[S] = 30 b −a 0b 0 0 a

−a 0 0 b

0 a b 04 (7.51)

where

a =1 − a1 + a

e −2ju, b =2√a

1 + ae −2ju (7.52)

For a 3-dB coupler, it is found that a = 0.1717 or Z0e = 5.83Z0 andZ0o = 0.1717Z0 at z = 0−. It is very difficult to obtain coupled lines with suchextreme values of even- and odd-mode characteristic impedances. However, two8.36-dB asymmetric couplers can be connected in tandem as shown in Figure 7.11to achieve a 3-dB coupler. For an 8.36-dB coupler, a = 0.446 or Z0e = 2.24Z0 andZ0o = 0.44Z0 . These values are relatively easier to obtain in practice using planartransmission lines such as broadside-coupled offset striplines discussed in Section3.7.3.

As discussed earlier, the design of a nonuniform asymmetric coupler requiresa 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 notpossible to design a completely reflectionless taper using a finite length. Further-more, 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 whichis quite close to the ideal performance [2].

References

[1] Tresselt, C. P., ‘‘The Design and Construction of Broadband, High-Directivity, 90-DegreeCouplers Using Nonuniform Techniques,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-14, December 1966, pp. 647–656.

[2] DuHamel, R. H., and M. E. Armstrong, ‘‘A Wideband Monopulse Antenna Utilizing theTapered-Line Magic-T,’’ USAF Antenna Research and Development Program 15th Symp.,University of Illinois, 1965, pp. 1–30.

[3] Uysal, S., and H. Aghvami, ‘‘Synthesis, Design, and Construction of Ultra-Wide-BandNonuniform Quadrature Directional Couplers in Inhomogeneous Media,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-37, June 1989, pp. 969–976.

[4] Uysal, S., Nonuniform Line Microstrip Directional Couplers, Norwood, MA: Artech House,1993.

[5] Kammler, D. W., ‘‘The Design of Discrete N-Section and Continuously Tapered SymmetricalMicrowave TEM Directional Couplers,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-17, August 1969, pp. 577–590.

[6] Klopfenstein, R. W., ‘‘A Transmission Line Taper of Improved Design,’’ Proc. IEEE,Vol. 44, January 1954, pp. 31–35.

C H A P T E R 8

Tight Couplers

8.1 Introduction

In planar quasi-TEM transmission line media such as microstrip, it is difficult toobtain tight coupling between lines because it requires a very small spacing betweenthem. Tight couplers, especially 3-dB couplers, are used in many practical circuitssuch as balanced mixers, amplifiers, and so forth. In this chapter, we concentrateon discussing directional couplers specifically suited for obtaining tight couplingvalues. These include branch-line couplers, rat-race couplers, Lange couplers, tan-dem couplers, and several other structures using new concepts and multilayerdielectric configurations. Although conventional branch-line and rat-race couplersdo not use coupled lines, many of their recent modifications employ coupledstructures to enhance their bandwidths or use folded coupled lines or inductors tomake them compact.

Branch-line and rat-race couplers are easily analyzed using the even- and odd-mode approach [1–4]. Only the final design equations for branch-line and rat-racecouplers are given in this chapter. The equations given, however, are valid forarbitrary division of power between the ports of a branch-line or a rat-race coupler.These couplers are inherently narrowband (<20% bandwidth) circuits; some tech-niques used to enhance their bandwidth are described briefly.

Although branch-line and rat-race couplers are more suitable for obtainingtight coupling values (such as 3 dB), they can also be useful for obtaining loosecoupling values in certain applications. For a branch-line coupler designed for loosecoupling, the impedance of the shunt branches becomes very high and cannot beeasily realized using planar transmission lines. A modified branch-line coupler inwhich the high-impedance shunt branches are replaced by coupled lines is described.The modified branch-line coupler can be easily implemented in microstrip configu-ration.

The size of conventional branch-line and rat-race couplers becomes quite largeat low frequencies (below 2 GHz). Further, their size is ‘‘too large’’ for theirimplementation in MMICs even at frequencies as high as 10 GHz. The realizationof branch-line and rat-race couplers using lumped inductive and capacitive elementsis also described. The size of lumped-element couplers is very small compared withthat of conventional couplers. Reduced-size branch-line and rat-race couplers thatuse only lumped capacitors and small sections of transmission lines (smaller thanlg /4) are also described. These couplers are quire suitable for realization in MMICs.

Tight coupling between edge-coupled lines can also be achieved by connectinga 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 designof a general N-conductor interdigital coupler are given. Next, the operation of atandem coupler is explained. We show that connecting two loose couplers in tandemresults in a tight coupler. Finally, compact couplers for wireless applications andtight couplers using multilayer dielectric structures are described.

8.2 Branch-Line Couplers

Figure 8.1 shows a branch-line coupler. It consists of two quarter-wave-long trans-mission line sections of characteristic impedance Z0s each, connected by two shuntbranches. The shunt branches are quarter-wave-long transmission line sections ofcharacteristic impedance Z0p each. By properly choosing the values of Z0s andZ0p , the circuit can be made to operate like a directional coupler. At the centerfrequency the scattering parameters of a branch-line coupler are given by

S21 = −jZ0sZ0

(8.1)

S31 = −Z0sZ0p

(8.2)

S11 = 0 (8.3)

S41 = 0 (8.4)

where Z0 denotes the impedance of various ports of a branch-line coupler. Thescattering parameters of a branch-line coupler also satisfy the following condition,which follows from the principle of conservation of energy:

Figure 8.1 Layout of a branch-line coupler in planar circuit configuration.

8.2 Branch-Line Couplers 221

|S21 |2 + |S31 |2 = 1 (8.5)

where it is assumed that S11 = S41 = 0. Substituting values of scattering parametersfrom (8.1) and (8.2) in (8.5), it is found that Z0s and Z0p should satisfy the followingcondition:

Z20s

Z20

+Z2

0s

Z20p

= 1 (8.6)

A branch-line coupler as shown in Figure 8.1 has two planes of symmetry. Thescattering matrix of a branch-line coupler can therefore be expressed as

[S ] =30 −j

Z0sZ0

−Z0sZ0p

0

−jZ0sZ0

0 0 −Z0sZ0p

−Z0sZ0p

0 0 −jZ0sZ0

0 −Z0sZ0p

−jZ0sZ0

0

4 (8.7)

The characteristic impedance of the main-line and shunt branches of a branch-linecoupler can be computed using (8.1) and (8.2) as

Z0s = Z0 |S21 | = Z0√1 − |S31 |2 (8.8)

and

Z0p =Z0s

|S31 |=

Z0s

√1 − |S21 |2(8.9)

Example 8.1

Compute the characteristic impedances of the main-line and shunt branches of abranch-line coupler given Z0 = 50V and coupling from port 1 to ports 2 and 3 isequal (S21 = S31 = −3.01 dB).

Given that coupling from port 1 to port 2 is 3.01 dB, therefore

|S21 | = 10−3.01/20 = 0.707

Substituting the value of |S21 | in (8.8), we obtain

Z0s = 0.707Z0 = 35.4V

222 Tight Couplers

Further, substituting Z0s = 35.4V and |S21 | = 0.707 in (8.9), the value of Z0p isfound as

Z0p = 50V

The transmission line sections having characteristic impedances of 35 and 50V canbe easily realized using planar transmission lines such as microstrip line. A 3-dBbranch-line coupler is also known as a 90-degree hybrid.

Properties of the Branch-Line Coupler

In a branch-line coupler, the following performance is obtained only at the centerfrequency (i.e., the frequency at which the main-line and shunt branches are eacha quarter-wave long):

S11 = S22 = S33 = S44 = 0 (8.10)

S14 = S41 = S23 = S32 = 0 (8.11)

A branch-line coupler is thus perfectly matched only at its center frequency,at which there is also complete isolation between ‘‘decoupled’’ ports. In the caseof backward parallel-coupled TEM couplers, these properties are satisfied indepen-dent of the frequency (as discussed in Section 6.4). In parallel-coupled TEMcouplers, the relationship of phase quadrature between the signals at direct andcoupled ports is satisfied independent of the operating frequency. In a branch-line coupler, the relationship of phase quadrature between the signals at direct andcoupled ports is satisfied only at the center frequency.

The VSWR, coupling, and isolation of a 3-dB branch-line coupler as a functionof frequency are plotted in Figure 8.2. It is seen that the input VSWR is unity(equivalent to |S11 | = 0) at the center frequency, but increases rapidly away fromthe center frequency and reaches a value of 2 (equivalent to |S11 | = −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 thecenter frequency. Branch-line couplers are useful in applications requiring less thanabout 20% frequency bandwidth only.

Physical Implementation of Branch-Line Couplers

The layout of a branch-line coupler in microstrip form is shown in Figure 8.1. Itis seen that there is a physical discontinuity present at each junction. The effect ofthese discontinuities is to add extra reactances that alter the response of the physicalcoupler compared with the ideal one, and should be considered in evaluating theperformance of the circuit, especially at high frequencies. This can be done usingthe equivalent circuit models of discontinuities, which are available in the literature[5] and also in many commercially available microwave CAD software. A moreaccurate analysis of such couplers is generally performed using electromagnetic(EM) simulators.


Figure 8.2 (a) Variation of VSWR with frequency of a 3-dB branch-line coupler. (b) Variation ofcoupling with frequency of a 3-dB branch-line coupler. (c) Variation of isolation withfrequency of a 3-dB branch-line coupler.

8.2.1 Modified Branch-Line Coupler

Branch-line couplers are generally used for equal power division. For loose coupling,parallel-coupled TEM couplers are preferred. If parallel-coupled directional cou-plers are realized in microstrip configuration, the directivity of the coupler maynot be very high because of unequal even- and odd-mode phase velocities. A branch-line coupler may be a better choice under this situation [6]. When a branch-linecoupler is designed for loose coupling (|S31 | < −10 dB), the characteristic impedanceof the shunt branches becomes quite high. The impedances of the main-line andshunt branches of a 20-dB coupler are found to be nearly 50 and 500V, respectively,using (8.8) and (8.9). However, a high impedance line of 500V cannot be realized

224 Tight Couplers

using microstrip. A similar problem is faced in the design of planar multisectionbranch-line couplers. The impedances of the end sections of a multisection branch-line coupler tend to be quite high.

A high-impedance transmission line section can be realized using coupled linesas shown in Figure 8.3(a). The electrical length of the coupled line is u and eachline is shorted at one of its ends. The equivalent circuit of the coupled lines betweenports 1 and 2 is shown in Figure 8.3(b). When u = 90 degrees (which is equivalentto lines being lg /4 long), the shorting stubs of characteristic admittance Y0e appearas open circuit across the main transmission line. Under this condition, the equiva-lent circuit further reduces to that as shown in Figure 8.3(c) where the characteristicimpedance Zc of the line is given by

Zc =2Z0e Z0oZ0e − Z0o

(8.12)

Note, however, that the length of the equivalent transmission line is 270 degreescompared with 90 degrees for the coupled lines, where an additional 180-degreephase is introduced by the short-circuited ends of the coupled-line section.

Equation (8.12) shows that very high values of Zc can easily be obtained usingcoupled lines (by choosing Z0e ≈ Z0o ). In practice, it is feasible to obtain impedancesabove a certain value only. For example, for coupled lines spaced 1 mil on a 25-milsubstrate, the lowest impedance Zc that can be realized is about 115 to 120V fora dielectric constant of 2. Similarly, on a substrate of dielectric constant 10, thelowest value of impedance Zc that can be realized using coupled lines is about70V.

A coupled section as shown in Figure 8.3(a) can therefore be used to replacethe shunt branches of high-impedance values in a branch-line coupler designed forloose coupling. The modified branch-line coupler is shown in Figure 8.4. Although

Figure 8.3 (a) Shorted coupled-line pair and (b) equivalent circuit. (c) Equivalent circuit of coupled-line pair of part (a) when u = 90 degrees.


Figure 8.4 Modified branch-line coupler.

the equivalent length of the line realized using coupled lines is 180 degrees longerthan that of the coupled lines, the performance of a modified branch-line couplerat the midband frequency is not affected, in comparison to a conventional branch-line coupler. The frequency response of a modified branch-line coupler will besomewhat different from that of a conventional branch-line coupler, however,because the equivalent circuit of the coupled lines shown in Figure 8.3(c) will differfrom that shown in Figure 8.3(b) when the value of u is different from 90 degrees.

The simulated response of a 20-dB conventional branch-line coupler is shownin Figure 8.5(a). The isolation of an ideal conventional branch-line coupler is greaterthan 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 and15 dB over a bandwidth of 12% and 20%, respectively. Figure 8.5(b) shows thesimulated response of a modified branch-line coupler. In the modified design, theshunt branches of 500V impedance have been replaced by coupled sections ofeven- and odd-mode impedances of 100V and 71V, respectively. When these valuesare substituted in (8.12), a characteristic impedance of 490V is obtained, whichis quite close to the desired value. The directivity of the modified branch-linecoupler is greater than 20 dB and 15 dB over a bandwidth of 8% and 14%,respectively. Although the bandwidth of a modified branch-line coupler is narrowerthan that of a conventional coupler, the modified design can be realized in microstripconfirguration. For example, the coupled section can be realized by printing5-mil-wide lines spaced 24 mil apart on a 25-mil alumina substrate. It has beenreported that a simulated response of a modified branch-line coupler matchesclosely with experimental results [6].

8.2.2 Reduced-Size Branch-Line Coupler

In MMICs, lumped capacitors can be easily realized and have become attractivein reducing the size of passive components. Reduced-size branch-line hybrids thatuse only lumped capacitors and small sections of transmission lines (smaller thanlg /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 20-dB branch-line coupler shown in Figure8.1 [6]. (b) Frequency response of modified 20-dB branch-line coupler shown in Figure8.4 [6].

than those for conventional hybrids and is therefore quite suitable for MMICs.Reduced-size branch-line couplers are discussed in the following.

A transmission line section of impedance Zc and electrical length of 90 degreesis shown in Figure 8.6(a). This section serves as a basic building block of a branch-line coupler. Its ABCD matrix is given by (8.13). The quarter-wave section can bereplaced by a section shown in Figure 8.6(b), which comprises a transmission lineof characteristic impedance Z and electrical length u and shunt capacitances Cat either end. By choosing Z > Zc , the electrical length u can be shorter than90 degrees. The ABCD matrix of the circuit shown in Figure 8.6(a) is given by

FA BC DG = 3

0 jZc

jZc

0 4 (8.13)

On the other hand, the ABCD matrix of a lumped-element circuit as shownin Figure 8.6(b) is given by


Figure 8.6 (a) Quarter-wave-long transmission section. (b) Reduced-size circuit equivalent toquarter-wave section.

FA BC DG = F 1 0

jvC 1G 3cos u jZ sin u

jsin u

Zcos u 4 F 1 0

jvC 1G (8.14)

= 3cos u − vCZ sin u jZ sin u

jsin u

Z+ 2jvC cos u − j (vC)2 Z sin u cos u − vCZ sin u4

It can be easily shown that the above matrix becomes identical to (8.13) if thevalues of Z and C are chosen as follows:

Z =Zc

sin u(8.15)

and

vC =cos u

Zc(8.16)

where u denotes the electrical length of the shortened transmission line shown inFigure 8.6(b).

For example, if the value of Z /Zc is chosen to be equal to 2, then using (8.15),we obtain

sin u =12

, or u = 30 degrees

In other words, the length of the shortened transmission line becomes equalto lg /12, which is one-third of the usual quarter-wave section. The relationshipbetween u, Z /Zc , 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 withpermission.)

The characteristic impedances of the main-line and shunt branches of a conven-tional 3-dB branch-line coupler are Z0 /√2 and Z0 , respectively. If the characteristicimpedances of the main-line and shunt branches are replaced by Z, where Z > Z0 ,the lengths of the main-line and shunt branches become less than lg /4. Morespecifically, the lengths of the main-line (u1) and shunt (u2) branches and the valueof capacitance Cb are given by

u1 = sin−1 y (8.17)

u2 = sin−1 y

√2(8.18)

vCbZ0 = √1 − y2 + √2 − y2 (8.19)

where y = Z0 /Z. When y = 1/√2, or Z = √2Z0 , u1 = 45 degrees and u2 = 30degrees. Therefore, if the main-line and shunt branch impedances of a 50V branch-line coupler are chosen equal to 70.7V, the lengths of the main-line and shuntbranches become equal to lg /8 and lg /12, respectively. The value of Cb requiredcan be determined using (8.19). The reduced-size branch-line hybrid is shown inFigure 8.8. The bandwidth of the reduced-size hybrid is a little wider than that ofthe purely lumped hybrid but is narrower than that of the conventional quarter-wavelength hybrid. The calculated phase difference between the signals at the direct(S21) and coupled (S31) ports is shown in Figure 8.9.

Reduced-size hybrids can be easily implemented in MMICs. A photograph ofa reduced-size 25-GHz branch-line hybrid is shown in Figure 8.10(a). The circuit


Figure 8.8 Reduced-size branch-line hybrid.

Figure 8.9 Calculated phase difference between S21 and S31 of the reduced-size hybrid, the conven-tional hybrid, and the purely lumped-element hybrid. (From: [7]. 1990 IEEE. Reprintedwith permission.)

230 Tight Couplers

Figure 8.10 (a) Photomicrograph of the fabricated 25-GHz reduced-size branch-line hybrid. (From:[7]. 1990 IEEE. Reprinted with permission.) (b) Measured performance of the 25-GHzreduced-size branch-line hybrid [7].

is fabricated on a GaAs substrate. All the transmission lines are 70V coplanarwaveguides with a 10-mm center conductor width. The lengths of the main-lineand shunt branches are lg /8 and lg /12, respectively. Metal-insulator-metal (MIM)shunt capacitors are located at the four T-junctions and between the inner conduc-tors and the ground metal. The size of the overall hybrid is 500 mm × 500 mm,representing an 80% savings over a conventional branch line hybrid. Its measuredperformance is shown in Figure 8.10(b).

8.2.3 Lumped-Element Branch-Line Coupler

The size of a branch-line coupler using sections of transmission lines becomes quitelarge at frequencies below about 2 GHz. At these frequencies, a branch-line canbe realized using lumped inductor and capacitor elements. The lumped-element90-degree hybrid can be realized either in a ‘‘pi’’ or a ‘‘tee’’ equivalent network.


In MMICs a ‘‘pi’’ network is preferred to ‘‘tee’’ because it uses fewer inductiveelements which have lower Q and occupy more space.

In lumped-element implementation, basically each transmission line sectionshown in Figure 8.11(a) is replaced by an equivalent ‘‘pi’’ lumped-element networkas shown in Figure 8.11(b). The values of the lumped elements are obtained byequating the ABCD matrix parameters for both these structures. The ABCD matrixof a lossless transmission line section of characteristic impedance Zc and electricallength u is given by

FA BC DG = 3

cos u jZc sin u

jsin uZc

cos u 4 (8.20)

On the other hand, the ABCD matrix of a lumped-element circuit as shownin Figure 8.11(b) is given by

FA BC DG = F 1 0

jvC 1G F1 jvL0 1 G F 1 0

jvC 1G (8.21)

= F 1 − v2LC jvL

2jvC − jv3LC2 1 − v2LCGwhere v denotes the radian frequency. By equating the matrix elements in (8.20)and (8.21) and simplifying, we obtain

L =Zc sin u

v(8.22a)

C =1

vZc √1 − cos u1 + cos u

(8.22b)

Figure 8.11 (a) Quarter-wave section and (b) its lumped equivalent.

232 Tight Couplers

The lumped circuit shown is therefore equivalent to a quarter-wave-long trans-mission line of characteristic impedance Zc , if the elements of the lumped circuitare chosen as follows:

L =Zcv

C =1

Zcv

Note that the circuits shown in Figure 8.11 have identical response at thefrequency at which the transmission line section is quarter-wave long. At otherfrequencies, the response of both the circuits will be different in general. Therefore,the bandwidth of circuits realized using transmission line sections will be differentfrom those realized using lumped elements.

A branch-line coupler as shown in Figure 8.1 uses quarter-wave transmissionline sections of characteristic impedances Z0s and Z0p . By replacing these sectionswith 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 =Z0sv0

, C1 =1

Z0sv0(8.23)

and

Figure 8.12 Lumped-element equivalent circuit model for the 90-degree hybrid shown in Figure8.1.

8.3 Rat-Race Coupler 233

L2 =Z0p

v0, C2 =

1Z0pv0

(8.24)

where v0 denotes the radian frequency corresponding to the center frequency ofthe branch-line coupler. More specifically, the element values of a lumped 3-dBbranch-line coupler are given by

L1 =Z0

√2v0, C1 = √2

Z0v0(8.25)

and

L2 =Z0v0

, C2 =1

Z0v0(8.26)

where Z0 denotes the terminal impedance of the ports of the branch-line coupler.Typical lumped-element values for a 900-MHz coupler designed for 50V

terminal impedance are L1 = 6.3 nH, L2 = 8.8 nH, and Ct = 8.5 pF, whereCt = C1 + C2 . Over 900 ± 45 MHz the calculated value of amplitude unbalanceand 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 moresections of ‘‘pi’’ or ‘‘tee’’ equivalent networks; that is, two sections of 45 degreesor three sections of 30 degrees to realize a 90-degree section, and so on, or byproperly selecting highpass and lowpass networks [8, 9]. In general, two to threesections are sufficient to realize a broadband 90-degree hybrid.

8.2.4 Broadband Branch-Line Coupler

As already discussed, the bandwidth of a branch-line coupler is quite narrow. Toincrease its bandwidth, a multisection branch-line coupler can be used. The synthesisof multisection branch-line couplers has been described by Levy and Lind [10].The synthesis has been described for both maximally flat and Chebyshev type ofresponse of VSWR and directivity of the coupler. Similar techniques can also beapplied to reduced-size and lumped-element couplers [11].

8.3 Rat-Race Coupler

The strip conductor layout of a rat-race coupler in microstrip form is shown inFigure 8.13. In the case of a branch-line coupler as shown in Figure 8.1, the spacingbetween all adjacent ports is lg /4. In the case of a rat-race coupler, however, thespacing between two of the adjacent ports is 3lg /4 and lg /4 between all otheradjacent ports. By properly choosing the values of Z01 and Z02 , the circuit can bemade to operate like a directional coupler. At the center frequency the scatteringparameters of a rat-race coupler are given by

234 Tight Couplers

Figure 8.13 Layout of a rat-race coupler in planar circuit configuration.

S21 = −jZ0Z02

(8.27)

S41 = jZ0Z01

(8.28)

S11 = S31 = 0 (8.29)

Furthermore,

S32 = −jZ0Z01

(8.30)

and

S42 = 0 (8.31)

The impedances Z01 and Z02 should satisfy the condition:

Z20

Z201

+Z2

0

Z202

= 1 (8.32)

The above equation follows from the following condition that needs to be satsifiedbecause of the principle of conservation of power:

|S21 |2 + |S41 |2 = 1 (8.33)

where it is assumed that S11 = S31 = 0. The complete scattering matrix of a rat-race coupler at the midband frequency can be expressed as


[S ] =30 −j

Z0Z02

0 jZ0Z01

−jZ0Z02

0 −jZ0Z01

0

0 −jZ0Z01

0 −jZ0Z02

jZ0Z01

0 −jZ0Z02

0

4 (8.34)

The scattering matrix of a rat-race coupler makes it useful in many applications.For example, consider a rat-race coupler designed for equal power division betweentwo coupled ports. When an input signal is incident at port 2, the power is equallydivided between ports 1 and 3 and no power reaches port 4. The signals arrivingat ports 1 and 3 are in phase. When power is incident at port 1, no power iscoupled to port 3 and the power is equally divided between ports 2 and 4 as inthe previous case. In this case, however, the signals arriving at ports 2 and 4 areout of phase (i.e., 180-degree phase difference). This special property makes therat-race useful in applications, such as in balanced mixers, where the effect of localoscillator AM noise can be canceled by connecting the RF signal to port 3 andlocal oscillator to port 1. A 3-dB rat-race coupler is also known as a 180-degreehybrid.

Design of a Rat-Race Coupler

From the scattering matrix of (8.34), we have

Z02 =Z0

|S21 |=

Z0

√1 − |S41 |2(8.35)

and

Z01 =Z0

|S41 |=

Z0

√1 − |S21 |2

Example 8.2

Compute the characteristic impedances of the various sections of a rat-race couplershown in Figure 8.13, given Z0 = 50V and the coupling coefficient from port 1 toports 2 and 3 equals (S21 = S41 = −3.01 dB).

Given

|S21 | = |S41 | = 10−3.01/20 = 0.707 (8.36)

Using (8.35) and (8.36), we then obtain

236 Tight Couplers

Z01 = Z0 /0.707 = 70.7V

and

Z02 = Z0 /0.707 = 70.7V

Properties of the Rat-Race Coupler

A rat-race coupler has disadvantages similar to that of a branch-line coupler. Forexample, the ports of a rat-race are matched only at the center frequency. Further,there is complete isolation between decoupled ports (such as between port 1 andport 3 in Figure 8.13) at the center frequency only. Furthermore, the phase differenceof 180 degrees between signals arriving at ports 2 and 4 exists only at the centerfrequency. The scattering parameters of a 3-dB rat-race coupler as a function offrequency 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 3-dB rat-race coupler. (b) Variation ofcoupling with frequency of a 3-dB rat-race coupler. (c) Variation of isolation withfrequency of a 3-dB rat-race coupler.


are the direct and coupled ports, respectively. The rat-race coupler finds use inapplications 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 discontinu-ities must be considered in the design, especially at high frequencies.

8.3.1 Modified Rat-Race Coupler

The bandwidth of a rat-race coupler is only about 20% but can be increasedsomewhat by using multisections, or significantly increased (to about an octave)by replacing the 3lg /4 section by an equivalent coupled-line section such as shownin Figure 8.15. The coupled-line section of electrical length 90 degrees is equivalentto a single transmission line of length 270 degrees as described in Section 8.2.1.The three-quarter-wave-long section of a rat-race coupler (of impedance Z01) cantherefore be replaced by a quarter-wave-long coupled-line section. In the modifiedcoupler, and equivalent impedance of Z01 can be obtained using coupled lines ifthe even- and odd-mode impedances of the coupled section are chosen as follows:

Z0e = 2.414Z01 (8.37)

Z0o =Z01

2.414(8.38)

Using the above information, a broadband rat race was simulated. The performanceof 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 thebroadband rat race is that its size is smaller compared to that of a conventionalrat race.

Figure 8.15 Broadband hybrid ring using a shorted parallel-coupled quarter-wave section.

238 Tight Couplers

Figure 8.16 Broadband hybrid response: (a) coupling, (b) isolation, and (c) return loss.


8.3.2 Reduced-Size Rat-Race Coupler

Rat-race hybrids (3-dB rat-race couplers) have unique applications in microwavecircuits because they can divide an input signal into signals that are either in phaseor out of phase. They can also be used to combine two signals (say, A and B) toobtain a ‘‘sum’’ (A + B) signal or a ‘‘difference’’ (A − B) signal. The size of a rat-race hybrid can also be reduced by employing techniques discussed in Section 8.2.2.A conventional 3-dB rat-race coupler is shown in Figure 8.13. It uses a transmissionline section of length 3lg /4 and three sections of length lg /4 each. The characteristicimpedance of each section is √2Z0 . The transmission line section of length 3lg /4can be replaced by a lumped-equivalent circuit as shown in Figure 8.17(a). Further,the lg /4 sections of characteristic impedance √2Z0 can be replaced by transmissionline sections of characteristic impedance 2Z0 and length lg /8 with shunt capaci-tances at the two ends as shown in Figure 8.17(b). The resulting reduced-size hybridis shown in Figure 8.18(a). (In this figure, the port labeling is different from thatshown in Figure 8.13.)

In the reduced-size rat-race hybrid of Figure 8.18(a), note that parallel LCelements (La and Cb ) between port 1 and the ground (and similarly between port2 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) withoutsignificantly affecting the response of the circuit. Alternatively, to compensate forthe effect of removal of these elements, the impedances of the transmission linesections, their length, and the values of other capacitances can be optimized usingCAD tools.

Another advantage of the reduced-size rat-race hybrid is its flexible portarrangement. The usual port arrangement is shown in Figure 8.19(a). In thisarrangement, the right edge of the bottom plate of the metal-insulator-metal (MIM)capacitor is connected to port 1 and the left edge of the top plate is connected toport 2. Port 1 can be connected to the left edge of the bottom plate without affectingthe 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 isshown in Figure 8.19(b). This layout is more convenient for mixer applications.

Figure 8.20(a) shows a photograph of a 25-GHz reduced-size rat-race hybrid[7]. The circuit consists of a 100V coplanar waveguide and MIM capacitors. Thecenter conductor width of the coplanar waveguide is 10 mm. The measured results

Figure 8.17 (a) Lumped-element circuit to replace 3lg /4 section; and (b) reduced-size circuit toreplace lg /4 section of a rat-race hybrid.

240 Tight Couplers

Figure 8.18 (a, b) Reduced-size rat-race hybrids; v0 is the center angular frequency (= 2p f0).

of this hybrid are shown in Figure 8.20(b). Note that the insertion loss of the rat-race hybrid is much smaller than that of the branch-line coupler shown in Figure8.10(b).

A very small, low-loss MMIC rat-race hybrid using elevated coplanar wave-guides has been reported by Kamitsuna [12]. A 15-GHz rat race was developedon a chip size of 0.5 × 0.55 mm.

8.3.3 Lumped-Element Rat-Race Coupler

The design of lumped-element rat-race hybrids [Figure 8.21(a)] is similar to thatof a lumped-element 90-degree hybrid described in the previous section. A lumpedelement equivalent circuit model for the 180-degree hybrid is shown in Figure8.21(b). The 90-degree sections are replaced with a lowpass ‘‘pi’’ network asshown in Figure 8.11 and the 270-degree (or 90-degree) section is replaced withan equivalent highpass ‘‘tee’’ network [13] shown in Figure 8.21(a). Following thesame procedure as described for the 90-degree hybrid, the lumped elements forthe ‘‘pi’’ section and the ‘‘tee’’ can be expressed as

L1 = √2Z0 sin u1

v(8.39a)

C1 =1

√2Z0v √1 − cos u11 + cos u1

(8.39b)


Figure 8.19 Port interchange in the reduced-size rat-race hybrid: (a) usual port layout; and(b) port layout convenient for mixer applications. (From: [7]. 1990 IEEE. Reprintedwith permission.)

L2 = − √2Z0

v sin u2(8.40a)

C2 =1

√2Z0v √1 + cos u21 − cos u2

(8.40b)

When u1 = 90 degrees and u2 = 270 degrees or −90 degrees, element values for a50V system become L1 = L2 = 11.25/ f nH and C1 = C2 = 2.25/ f pF, where f is ingigahertz.

242 Tight Couplers

Figure 8.20 (a) Photomicrograph of the fabricated 25-GHz reduced-size rat-race hybrid. (b) Mea-sured performance of the 25-GHz reduced-size rat-race hybrid. (From: [7]. 1990IEEE. Reprinted with permission.)

8.4 Multiconductor Directional Couplers

The coupling between two TEM lines increases as the mutual capacitance betweenthem is increased. The mutual capacitance between the lines can be increased bydecreasing the spacing between them. However, to obtain tight coupling valuessuch as 3 dB between planar transmission lines such as microstrip lines, the spacingbetween the lines becomes too small to be realized in a convenient manner usingcommonly used photolithographic techniques. Further, very small transverse dimen-sions lead to increased conductor loss. Lange [14] described a scheme using multi-conductors in interdigital configuration in which the mutual capacitance between

8.4 Multiconductor Directional Couplers 243

Figure 8.21 (a) Equivalent lumped circuit of a transmission line section of length 3lg /4.(b) Lumped-element 180-degree hybrid.

the lines can be increased without the need for a small spacing between them. Thissection 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. Cs and Cmu denotethe capacitances of lines (per-unit length) with respect to the ground and the mutualcapacitance between them, respectively. To increase the mutual capacitance, thetwo lines can be divided into two lines of half-width each and arranged in aninterdigital manner as shown in Figure 8.23(a) with alternate conductors connectedtogether at the ends. The spacing between adjacent conductors remains the sameas in the undivided case shown in Figure 8.22(a). The capacitance of each dividedline with respect to the ground is approximately Cs /2 (because of half-width),whereas the mutual capacitance between adjacent lines is somewhat smaller thanbetween the lines in the configuration shown in Figure 8.22(a). The total capacitanceof each line with respect to the ground is therefore approximately Cs as in theoriginal 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) equivalentcapacitance network showing static per-unit-length self-capacitance of the lines andmutual capacitance between them.

Figure 8.23 (a) Top view of an interdigital coupler, (b) side view, and (c) equivalent capacitancenetwork showing static per-unit-length self-capacitance of the lines and mutualcapacitance.

add together to give an overall value of mutual capacitance that is much larger thanbetween the lines in the configuration shown in Figure 8.22(a). This is essentially thebasis of a Lange coupler. The equivalent capacitances of the interdigital configura-tion are shown in Figure 8.23(b, c).


Figure 8.24 shows how a coupler consisting of two parallel-coupled lines can berearranged. For the same spacing between adjacent lines, the coupler configurationsshown in Figure 8.24(b, c) offer larger coupling than the configuration shown inFigure 8.24(a). The configurations shown in Figure 8.24(b, c) are identical electri-cally, but the configuration shown in Figure 8.24(c), which is also known as aLange coupler, offers an extra advantage in that both the direct and coupled portsare 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 N-conductor (N-even) interdigital coupleras 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 parallel-coupled lines, (b) interdigital configuration knownas unfolded Lange coupler, and (c) interdigital configuration known as a Lange coupler.

246 Tight Couplers

Figure 8.25 (a) Top view of an N-conductor interdigital coupler and its (b) side view; and (c) sideview of two parallel-coupled lines having the same width and spacing between themas the N-conductor 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, Z0 (the impedance of various ports) and coupling, thedesign is obtained in terms of even- and odd-mode impedances of a pair of coupledlines as shown in Figure 8.25(c). Once the even- and odd-mode impedances of onepair 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- andodd-mode impedances of one pair of coupled lines because these data are availablein the literature for a large class of transmission lines.

The design equations of an N-conductor (N even) interdigital coupler are givenby [15, 16]

k =(N − 1)(1 − R2)

(N − 1)(1 + R2) + 2R(8.41)

Z =Z0oZ0

= √R[(N − 1) + R] [(N − 1)R + 1](1 + R)

(8.42)

R =Z0oZ0e

(8.43)


where k is the voltage coupling coefficient between the input and coupled ports atthe center frequency of the design and N (N is even) is the total number ofconductors. Z0e and Z0o denote, respectively, the even- and odd-mode impedancesof a pair of coupled lines having the same width and spacing between them as anypair of N-conductor interdigital coupler. It may be worth remarking that the usualrelation Z0 = √Z0e Z0o is valid in the case of an interdigital coupler only whenN = 2. For other values of N, this relation is not satisfied.

The length of the interdigital coupler at the center design frequency is givenby

, =lg

4(8.44)

where lg = 0.5(lge + lgo ) is the wavelength in the medium of the coupler at thecenter frequency of the design. Here lge and lgo are the guide wavelengths for theeven 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 theodd-mode impedance Z0o . Further, the even-mode impedance Z0e can be deter-mined using (8.43). Using now the known values of the even- and odd-modeimpedances, the dimensions of the lines and the spacing between them can bedetermined either by using available nomograms, computer programs [17] or designequations [5].

Equations (8.41) to (8.43) are not exact but are based on the following assump-tions:

• 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 inFigure 8.25(c).

Results

Using (8.41), the value of R has been plotted as a function of coupling in Figure8.26 for values of N = 2 and 4 [15, 16]. Similarly, the normalized odd-modeimpedance has been plotted as a function of coupling in Figure 8.27 for values ofN = 2 and 4. In Table 8.1 numerical values are given for an interdigital couplerfor a few values of coupling and number of conductors N. The input/outputimpedances of the coupler are assumed to be 50V. It may be verified that forN = 2, the values of even- and odd-mode impedances given in Table 8.1 are thesame 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, thenormalized dimensions of the coupler (W /h and S/h) are given as a function of

248 Tight Couplers

Figure 8.26 Impedance ratio R (= Z0o /Z0e ) as a function of coupling. (From: [16]. 1978 IEEE.Reprinted with permission.)

Figure 8.27 Normalized odd-mode impedance (= Z0o /Z0) as a function of coupling. (From: [16]. 1978 IEEE. Reprinted with permission.)


Table 8.1 Impedance Parameters of anN-Conductor Interdigital Coupler

N Coupling (dB) Z0e V Z0o V

3 120.70 20.712 6 86.60 28.87

10 69.37 36.04

3 176.20 52.614 6 142.50 67.96

10 118.30 76.30

3 243.10 82.556 6 204.30 105.10

10 181.11 122.10

Figure 8.28 The dimensional ratios of a Lange coupler printed on a dielectric substrate (er = 10)and Z0 = 50V as a function of coupling. (From: [16]. 1978 IEEE. Reprinted withpermission.)

coupling for Z0 = 50V and printed on a dielectric substrate of er = 10 [16]. Thedata are based on the computation of even- and odd-mode impedance using (8.41)through (8.43). Further, using the computed values of even- and odd-mode imped-ances, the dimension ratios W /h and S/h of the coupler have been found using theprogram for coupled microstrip lines given by Bryant and Weiss [17]. Simpleexpressions given in reference [5] can also be used to obtain the values of W /hand S/h.

A Lange coupler is frequently designed for 3-dB coupling. In Figure 8.29, thedimensions of a 3-dB, 50V Lange coupler are given as a function of the dielectricconstant of the substrate material on which the coupler is printed.

The dimensions given in Figures 8.28 and 8.29 are valid for zero-thicknessconductors. In practice, however, the conductors will always have a finite thickness.Presser [16] has given an empirical formula for the correction factor that was foundby performing a large number of experiments. The effect of thickness can be taken

250 Tight Couplers

Figure 8.29 The dimensional ratios of a 3-dB Lange coupler as a function of dielectric constant ofthe substrate. (From: [16]. 1978 IEEE. Reprinted with permission.)

into account by increasing the separation between adjacent lines and decreasingthe width of the lines as

Sh

=S0h

+DSh

(8.45a)

and

Wh

=W0h

−DSh

(8.45b)

where W0 and S0 denote the width and the separation between lines, respectively,for a coupler employing zero-thickness conductors. The value of DS/h in the aboveequations is given by

DSh

=t /h

p√ereS1 + ln

4pW0 /ht /h D (8.46)

where t denotes the thickness of the metalization and ere can be assumed to be theeffective dielectric constant of a single, uncoupled microstrip line having width W.

Furthermore, Presser [16] found from sensitivity analysis that the most sensitiveparameters in the design of a Lange coupler are the gap dimensions and themetalization thickness, whereas 10% changes in the width and the dielectric con-stant cause practically insignificant deviations in the performance.

Many computer programs are now available, which give the impedance datafor coupled microstrip lines taking the thickness of the conductors into account.With such a program, we can directly compute the dimensions of the interdigitalstructure using the values of even- and odd-mode 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.


In a Lange coupler, the conductor widths and the spacing between the coupler’sconductors can be produced with standard thin film manufacturing processes onthick low-dielectric constant substrates (thickness > 250 mm, er < 10). However,on thin GaAs (er = 12.9) substrates (thickness less than 100 mm), tightly coupledstructures are difficult to realize because the conductor width and the spacingbecome prohibitively small. For example, a 3-dB coupler on a GaAs substraterequires approximately W /h and S/h values on the order of 0.07. Therefore, abroadband 3-dB Lange coupler on a 75-mm-thick substrate requires approximately5-mm conductor width and gap dimensions.

A six-finger 3-dB Lange coupler [20], as shown in Figure 8.30, was designedand tested on a 75-mm 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 distinctfeatures: (1) for a given line impedance it increases the line width, and (2) it reducesthe microstrip line loss. The design parameters for a X-band coupler are given inTable 8.2. Table 8.3 summarizes the line lengths and capacitor values for severalsix-finger Lange couplers working over the 5–37-GHz frequency range. The shuntcapacitors connected between the input and coupled ports, and between the directand 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. Thecapacitors are of the metal insulator metal (MIM) type, using SiN with 300 pF/mm2

Figure 8.30 Six-finger microstrip coupler configuration: (a) top view and (b) side cross-sectionalview.

252 Tight Couplers

Table 8.2 X-BandSix-Finger LangeCoupler Parameters

Number of fingers = 6Line width, W = 9 mmLine spacing, S = 7 mmLength, L = 2,400 mmCapacitor, C = 0.16 pFGaAs substrate, er = 12.9Thickness, h = 75 mmPolyimide, erd = 3.2Thickness, d = 10 mmGold conductors, t = 4.5 mm

Table 8.3 Line Lengths and Capacitor Values for Several Six-FingerLange Couplers

Frequency range (GHz) 5–9 7–13 10–24 16–37Physical length, L (mm) 4,800 3,000 1,800 1,200Capacitor C (pF) 0.22 0.16 0.12 0.07

capacitance density. The measured coupling was 3.3 ± 0.5 dB with a maximumamplitude variation of ±0.6 dB between the coupled and direct ports, over a8.5-GHz bandwidth. The measured return loss at all ports was greater than 13 dBfrom 6 to 15.5 GHz. The measured minimum isolation was 15.5 dB across 6 to16 GHz, and better than 20 dB from 10.5 to 16 GHz. The phase difference was94 ± 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 ofthe form given by (8.47) and (8.48). If the voltage coupling factors for the twocouplers are k1 and k2 , respectively, then the scattering matrix of the first couplerat the center frequency (the center frequency is the one at which the electricallength of the TEM coupler is 90 degrees) is given by,

[S1] = 30 −j cos a1 sin a1 0

−j cos a1 0 0 sin a1

sin a1 0 0 −j cos a1

0 sin a1 −j cos a1 04 (8.47)

where sin a1 = k1 is the voltage coupling coefficient for coupler 1. Similarly, thescattering matrix of the second coupler can be expressed as

[S2] = 30 −j cos a2 sin a2 0



0 sin a2 −j cos a2 04 (8.48)

where sin a2 = k2 denotes the voltage coupling coefficient for the second coupler.With a wave voltage amplitude of unity (V +

1 = 1), incident on port 1 of coupler1, the reflected voltages V −

1 , V −2 , V −

3 , and V −4 are given by

3V −

1

V −2

V −3

V −4

4 = 30 −j cos a1 sin a1 0



0 sin a1 −j cos a1 04 3

10004 (8.49)

which gives

V −1 = 0,

V −2 = −j cos a1 , (8.50)

V −3 = sin a1 , and

V −4 = 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

V +1′ = V −

3 = sin a1 (8.51a)

V +4′ = V −

2 = −j cos a1 (8.51b)

254 Tight Couplers

The output at the second coupler can then be found using

3V −

1′

V −2′

V −3′

V −4′

4 = 30 −j cos a2 sin a2 0



0 sin a2 −j cos a2 04 3

sin a1

00

−j cos a14

(8.52)

which gives

V −1′ = V −

4′ = 0 (8.53a)

V −2′ = −j cos a2 sin a1 − j sin a2 cos a1 = −j sin(a1 + a2) (8.53b)

and

V −3′ = sin a2 sin a1 − cos a2 cos a1 = −cos(a1 + a2) (8.53c)

Now let us choose

a1 = a2 =p8

(8.54)

or

k1 = k2 = sinSp8 D = 0.3827

which is also equivalent to coupling in decibels for the two couplers as C1 andC2 , respectively, where

C1 = C2 = −20 log(0.3827) = 8.34 dB

Substituting the values of a1 and a2 from (8.54) in (8.53), we obtain

V −2′ = −j sinSp

4 D = −j

√2(8.55a)

and

V −3′ = −j cosSp

4 D = −1

√2(8.55b)

The fractional power coupled from port 1 of the first coupler to port 3′ of thesecond coupler is therefore given by

8.6 Multilayer Tight Couplers 255

P −3′

P +1

=|V −

3′ |2

|V +1 |2

=12

(8.56)

which is equivalent to 3-dB coupling. Therefore, by connecting two 8.34-dB cou-plers in tandem, a 3-dB coupler is obtained.

For physical realization of tandem couplers, a scheme such as shown in Figure8.32 is used. Note that crossovers are required in tandem couplers to achieveproper interconnections between the couplers as required by the connection schemesymbolically shown in Figure 8.31.

8.6 Multilayer Tight Couplers

A recent upsurge in MMIC-based system demands and wireless applications has ledto new configurations for tight directional couplers. These are broadside, embeddedmicrostrip, re-entrant, and compact directional couplers. Compact couplers includelumped-element, spiral, and meander line structures, which are described brieflyin Section 8.7.

8.6.1 Broadside Couplers

The asymmetrical broadside-coupled microstrip lines configuration is the simplesttechnique to realize tight coupling. Several different configurations and fabricationtechnologies to design 3-dB couplers have been reported in the literature [24–29].A basic configuration of a MIC/MMIC asymmetric broadside coupler, which con-sists of two conductors separated by a thin layer of polyimide dielectric (er2 = 3.2),is shown in Figure 8.33. This requires multilayer MMIC technology. The 3-dBcoupler design is usually done using an EM simulator and involves an optimalsolution 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 broadside-coupled microstrip line coupler.

substrate thickness. The conductors can be folded or meandered to reduce theoverall chip size.

A coupled-multilayer microstrip line structure as shown in Figure 8.34, whichworks similar to symmetrical broadside-coupled striplines structure, was reportedby Okazaki and Hirota [29]. The coupler was fabricated using multilayer thin-film microstrip line technology and consists of four parallel-strip conductors. Theground plane, which is 1 mm thick, for these conductors is placed on the topsurface of the GaAs. Diagonal conductors as shown in the figure are connected atthe ends, and the structure behaves like symmetrical broadside-coupled striplines,and the couplers can be designed using broadside-coupled stripline formulas givenin Section 3.7. Various dimensions for the coupler reported in [29] are as follows.

Conductors A and C are buried in polyimide layers (er ≅ 3.3) and have widths(W1) of 3 mm. The gap (S1) between them is 5 mm and are placed above theground plane level (h1) at 6.5 mm. The other two conductors B and D have widths(W2) of 5 mm and the gap between them (S2) is also 5 mm. They are placed abovethe ground plane level (h2) at 9 mm. 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.)


connected at both ends. The width of the entire coupled line is only 15 mm, whichallows layout in meander shape (Figure 8.35) to reduce the chip size, which measuresonly 1.3 × 0.4 mm for the X/Ku-band coupler design.

Figure 8.36 shows the measured performance of this coupler; coupling is4.2 ± 0.4 dB, return loss and isolation better than 20 and 15 dB, respectively, overthe 10- to 17.5-GHz frequency range. The measured phase difference between thedirect and coupled ports was about 91 ± 5 degrees over the 10- to 17.5-GHzfrequency range.

Another 3-dB asymmetric broadside coupler (Figure 8.33) was developed usingthe multilevel plating (MLP) process [22] on a 75-mm-thick GaAs substrate. Figure8.37 shows the top-sectional view of the broadband coupler, which operated overthe 6–16-GHz frequency range. The physical length of the coupler is 3,000 mm.The bottom and top conductor line widths are 40 and 60 mm, respectively, andresult in lower dissipated loss than a Lange coupler on GaAs. The dielectricbetween the broadside conductors is 7-mm-thick polyimide. As shown in Figure8.38, the measured coupling to the coupled and direct ports are 3.3 ± 0.5 dB and3.5 ± 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 3-dB asymmetric broadside coupler.

Figure 8.38 Measured coupling coefficient for the 3-dB coupler.

8.6.2 Re-Entrant Mode Couplers

Re-entrant mode couplers have been designed to obtain tight coupling in coaxial[30], stripline [31], and microstrip [32, 33] media. Design procedures of semi-re-entrant [34, 35] and re-entrant [36] couplers have also been discussed. The basictheory of such couplers is very simple and can be described by referring to Figures8.39 and 8.40(a). In Figure 8.39, the top conductor 1 is floating while in Figure8.40(a), the underneath conductor is floating. The latter coupler consists of aparallel-coupled microstrip line (conductors A and B) with another conductor Cfloating underneath. This is again a multilayer configuration, the dielectric constantand the thickness of the dielectric layer between conductors A or B and C aredetermined to achieve the required coupling coefficient.

The even- and odd-mode impedances, like any other symmetrical edge-coupledlines, can be determined by placing magnetic and electric walls at the plane ofsymmetry P–P ′ as shown in Figure 8.40. In the case of even mode shown in Figure


Figure 8.39 (a) A schematic view of a single-section semi-re-entrant coupler. (b) A cross sectionalview of a semi-re-entrant coupled section.

8.40(b), the impedance of floating conductor C (having characteristic impedanceof Z01) is in series with the transmission lines A–C and B–C, each having thecharacteristic impedance of Z02 . By placing a magnetic wall that bisects the cou-pler’s cross section, the even-mode characteristic impedance becomes

Z0e = Z02 + 2Z01 (8.57)

In the case of the odd mode, Figure 8.40(b), the electric wall passing throughthe middle of conductor C sets the conductor at ground plane reference, and theodd-mode characteristic impedance of the coupled structure is

Z0o = Z02

Therefore, the coupling coefficient in this case can be expressed as


=Z01

Z01 + Z02(8.58)

and the termination impedance Z0 = √Z0e Z0o . Equation (8.58) shows that thecoupling coefficient, in this case, does not depend upon the spacing between the

260 Tight Couplers

Figure 8.40 (a) Microstrip re-entrant mode coupler cross section. (From: [33]. 1990 IEEE.Reprinted with permission.) (b) Even- and odd-mode excitations.

conductors A and B, but mostly depends upon the Z02 value, which can easily becontrolled by the parameters er2 and d. Table 8.4 lists typical design parametersfor several 3-dB couplers matched approximately to 50V.

Figure 8.41 shows the measured performance of a re-entrant coupler fabricatedon 0.381-mm-thick alumina substrate. The top gold conductors were about 3 to4 mm thick and coupler worked over the 5- to 19-GHz range.

Table 8.4 Typical Dimensions for Various Re-Entrant Microstrip Couplers

Substrate W1 (mm) W2 (mm) d (mm) h (mm) Z01 (V) Z02 (V)

Alumina (er1 = 9.9; er2 = 3.7) 0.254 0.0685 0.0075 0.381 59.83 18.14

GaAs (er1 = 12.9; er2 = 6.8) 0.075 0.015 0.0023 0.100 49.8 18.000.115 0.015 0.0025 0.150 49.8 18.00

GaAs (er1 = 12.9; er2 = 3.7) 0.115 0.015 0.006 0.150 49.8 18.00

8.7 Compact Couplers 261

Figure 8.41 Coupler performance: (a) coupled and direct power, (b) return loss, and (c) phasedifference between the coupled and direct ports. (From: [33]. 1990 IEEE. Reprintedwith permission.)

8.7 Compact Couplers

In cellular wireless microwave applications, quadrature 3-dB couplers are requiredto determine the phase error of a transmitter using the QPSK modulation scheme.The basic requirements for such couplers include small size, low cost, tight ampli-tude balance, and 90-degree phase difference between the coupled and direct ports.At the L-band, the distributed couplers are big in size and also expensive. Anequivalent lumped-element implementation is compact in size and has the potentialto be low cost. For example, the size of a monolithic coupler on GaAs substratehas to be of the order of about 1 to 2 mm2 to be cost effective for low-frequencywireless applications. Several existing coupler configurations have been transformedinto new layouts to meet size target values. Some of these new configurations such

262 Tight Couplers

as lumped-element couplers [37], the spiral directional coupler [38], and meanderline couplers [39–41] are described briefly in this section.

8.7.1 Lumped-Element Couplers

The coupler shown in Figure 8.42(a) can be modeled as a lumped-element equivalentcircuit as shown in Figure 8.42(b). The values for L, M, Cg and Cc in terms ofZ0e , Z0o , and u are given as follows [37]:

L =(Z0e + Z0o ) sin u

4p foCg =

tan(u /2)Z0e2p fo

(8.59a)

M =(Z0e − Z0o ) sin u

4p foCc = S 1

Z0o−

1Z0e

D tan(u /2)4p fo

(8.59b)

where fo is the center frequency and u = 90 degrees at fo . For a given coupling,using (4.59), the values of Z0e and Z0o are determined and then lumped-elementvalues are calculated using (8.59). The self- and mutual inductors are realized usinga spiral inductor transformer and the capacitors Cg and Cc are of MIM type andtheir partial values are also included in the transformers.

8.7.2 Spiral Directional Couplers

To obtain miniature directional couplers with tight coupling, a coupled structurein a spiral shape (also known as ‘‘spiral coupler’’) is realized. Printing the spiralconductor on high dielectric constant materials further reduces the size of the

Figure 8.42 Two-conductor 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 parallel-coupled microstrip lines placed in a close-proximity spiral configuration. Thisstructure as shown in Figure 8.43 uses two turns and resembles a multiconductorstructure. 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 l0 /8, where l0 is the free-space wavelength at the center frequencyand D ≅ l0 /64 + 4W + 4.5S. Longer lengths result in tighter couplings. Typicalline widths and spacings are approximately 500 mm and 40 mm, respectively, fora 0.635-mm alumina (er = 9.6) substrate. In the spiral configuration, coupling isnot a strong function of spacing between the conductors. The conductors wereabout 5 mm thick. Measured coupled power, direct power, return loss, and isolationat 800 MHz for the two-turn 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 edge-coupled microstriplines [39–41] using high dielectric constant substrates. Figure 8.44 shows a meander-type coupler and a spiral-type 90-degree 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 EMsimulators. The physical dimensions of meander- and spiral-type couplers, designedto work over the S-band, are given in Table 8.5, where W, S, and G are the conductorwidth, separation between the conductors, and gap between the neighboring pair

Figure 8.43 Top conductor layout of a two-turn spiral coupler.

264 Tight Couplers

Figure 8.44 Compact coupled-line 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 (mm) 30 30S (mm) 10 20G (mm) 120 60Line length (mm) 6.5 7.5Substrate thickness (mm) 300 300Chip size (mm) 1.5 × 1.5 1.5 × 1.5

lines. Measured performance of these couplers is summarized in Table 8.6. Thespiral-type 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], slot-coupled [47, 48], combline [49, 50], finline [51, 52],coplanar waveguide [53], wiggly two-line [54], and dielectric waveguide [55–57]directional couplers. Other couplers include three-strip [58], folded line [59],coupled-line hybrid [60], and artificial transmission lines [61].

Monolithic microwave integrated circuit (MMIC), low-temperature cofiredceramic (LTCC) [62], Si-based monolithic integrated circuit [63], micromachining

8.8 Other Tight Couplers 265

Table 8.6 Measured Electrical Performance of TwoCompact Couplers

Configuration Meander Type Spiral Type

Frequency (GHz) 1.8–3.8 1.8–3.8Insertion loss (dB) 4.5–7.5 3.8–5.0Amplitude balance (dB) ±0.7 ±0.5Phase difference (deg) 93 ± 2 90 ± 2Return loss (dB) >15 >15

techniques [64], and metamaterials [65–67] have stimulated a rapid developmentof new coupled line directional couplers. Multilayer and multiconductor technolo-gies have resulted in tight and improved directivity couplers. In order to obtainlower loss in Si-based couplers and improve electrical performance at millimeter-wave frequencies, micromachining techniques are very suitable. A 20-dB micro-machined directional coupler with less than 0.5-dB insertion loss working over the10–60-GHz frequency range has been reported [64].

Recently, several composite right/left-handed (CRLH) transmission line struc-tures composed of highpass (left-handed, LH) and lowpass (right-handed, RH)components cascaded in series have been used to design 3-dB couplers [65–67].Such structures are also known as metamaterials. A major advantage of the CRLH-based 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 suchdetail, readers are referred to the above-mentioned references.

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266 Tight Couplers

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8.8 Other Tight Couplers 267

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C H A P T E R 9

Coupled-Line Filter Fundamentals

9.1 Introduction

Typically, a microwave circuit consists of a number of components, or parts, thefunctions of which depend on the specific application in mind. Engineering thesecomponents 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 ofboth and can usually be designed for the precise frequency response required, atlow cost. Thus, they have been used for a very long time and are popular microwavecomponents, present in virtually every microwave subsystem.

The primary parameters of interest in a filter are the frequency range, band-width, insertion loss, stopband attenuation and frequencies, input and outputimpedance, group delay, and transient response. Consider Figure 9.1, where Pin isthe incident power, PR the power reflected back to the generator, PA the powerabsorbed by the filter, and PL power transmitted to the load:

Pin = PR + PA (9.1)

and if the filter is lossless and there are no reflections, PL = PA and PL = Pin . Theinsertion loss (in decibels) at a particular frequency can be defined as

Figure 9.1 (a) General filter network configuration; and (b) equivalent circuit for power-transfercalculations.

269

270 Coupled-Line Filter Fundamentals

IL = −10 log(PL /Pin ) (9.2)

while the return loss is given by

RL = −10 log(PR /Pin ) = −10 logFVSWR − 1VSWR + 1G

2(9.3)

The group delay (tD ), which is a measure of the time taken by a signal topropagate through the filter, is given by

tD = −1

2pdFT

df(9.4a)

where

fT = arg (S21) (9.4b)

and S21 is the transmission coefficient. For no frequency dispersion, the groupdelay should be constant over the required frequency band.

Finally, the transient and steady-state response of a filter may be different. Thisfeature is an important consideration for certain applications. In general, transienteffects 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 classificationis reflective versus dissipative. In a reflective filter, signal rejection is achieved byreflecting the incident power, while in a dissipative filter, the rejected signal isdissipated internally in the filter. In practice, reflecting filters are used in mostapplications. The most conventional description of a filter is by its frequencycharacteristic such as lowpass, bandpass, bandstop, or highpass. Typical frequencyresponses for these different types are shown in Figure 9.2. In addition, an idealfilter displays zero insertion loss, constant group delay over the desired passband,and infinite rejection elsewhere. However, in practice, filters deviate from thesecharacteristics and the parameters in the introduction above are a good measureof 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 channelizedreceivers. System applications of filters include radars, communications, surveil-lance, ESM receivers, satellite communications (SATCOM), mobile communica-tions, 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 271

Figure 9.2 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 significantreduction in power handling capacity and an increase in the insertion loss. Theformer is not a severe limitation in such systems, however, and the latter can becompensated for by subsequent power amplification.

In this chapter we constrain ourselves to dealing mostly with coupled-linefilters. In addition, a small section on computer-aided design and synthesis softwareis included. Finally, because of the importance of filter miniaturization for someapplications, 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 frequen-cies and total attenuation of all out-of-band frequencies is impractical to realize.In practice, a polynomial transfer function such as Butterworth, Chebyshev, orBessel is used to model the filter response. A combination of inductors and capaci-tors, as shown in Figure 9.3, will obviously result in a lowpass filter, and we candevelop a prototype normalized to 1V and a 1-rad cutoff frequency. From here,it is simply a matter of scaling the g values to obtain the desired frequency responseand insertion loss. In addition, other filter types such as highpass, bandpass, andbandstop merely require a transformation in addition to the scaling to obtain thedesired characteristics.


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 possibleat zero frequency and rise monotonically as fast as possible with increasing fre-quency. With n as the order of the filter (i.e., the number of reactive elementsrequired to obtain the desired response), f1 the defined 3-dB band-edge point, andf the frequency of interest, the insertion loss is given by

IL = 10 log[1 + ( f / f1)2n ] (9.5)

Nomographs, as shown in Figure 9.4, can be used to determine the stopbandattenuation versus number of sections for the desired bandwidth. For example, aneight-section filter gives an attenuation of approximately 48 dB at f / f1 = 2.0 inthe stopband, while it results in an attenuation of 0.35 dB at f / f1 = 0.8 in thepassband.

The Butterworth prototype values can be calculated from the equations belowand are tabulated in Table 9.1 for filters with n = 1 to 10 reactive elements:

g0 = 1

gl = 2 sinF(2l − 1)p2n G, l = 1, 2, . . . , n (9.6)

gn + 1 = 1 for all n

These g values can be scaled for the desired filter input termination resistance Rand cutoff frequency v1 = 2p f1 as

L = gR /v1 (9.7)

C = g /(v1R) (9.8)

9.2.2 Chebyshev Response

In the Chebyshev response filter, the insertion loss remains less than a specifiedripple level Ac , up to a specified frequency v1 , and then rises quickly and monotoni-cally with frequency. For an nth order filter, with Ac the ripple magnitude (indecibels) and v1 , the bandwidth over which the insertion loss has maximum ripple,the insertion loss is given by


Figure 9.4 Nomograph for number of sections of a Butterworth filter for a given insertion loss inthe stopband. (From: [1]. 1985 Microwave and RF. Reprinted with permission.)

Table 9.1 Element Values for a Butterworth Filter with g0 = 1, v1 = 1, and n = 1 to 10

Valueof n g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

1 2.000 1.0002 1.414 1.414 1.0003 1.000 2.000 1.000 1.0004 0.7654 1.848 1.848 0.7654 1.0005 0.6180 1.618 2.000 1.618 0.618 1.0006 0.5176 1.414 1.932 1.932 1.414 0.5176 1.0007 0.4450 1.247 1.802 2.000 1.802 1.247 0.445 1.0008 0.3902 1.111 1.663 1.962 1.962 1.663 1.111 0.3902 1.0009 0.3473 1.000 1.532 1.879 2.000 1.879 1.532 1.000 0.3473 1.000

10 0.3129 0.908 1.414 1.782 1.975 1.975 1.782 1.414 0.908 0.3129 1.000

IL = H 10 log[1 + (10Ac /10 − 1) cos2 (n cos−1 v /v1)] v ≤ v1

10 log[1 + (10Ac /10 − 1) cosh2 (n cosh−1 v /v1)] v ≥ v1(9.9)

As in the case of the Butterworth response, a nomograph as shown in Figure9.5 can be used to determine the filter characteristics. With the cutoff defined asthe ripple value, the lowpass prototype g values are


Figure 9.5 Nomograph for number of sections of a Chebyshev filter for a given ripple and stopbandinsertion loss. (From: [1]. 1985 Microwave and RF. Reprinted with permission.)

g0 = 1

g1 =2a1g

gk =4ak − 1 akbk − 1 gk − 1

k = 2, 3, . . . , n (9.10)

gn + 1 = 1 for n odd

gn + 1 = coth2 (b /4) for n even

The passband VSWR maximum is related to the ripple level Ac by

VSWRmax =1 + A1 − A

where


A = 10[1 − 10−Ac /10]1/2

and

ak = sinF(2k − 1)p2n G k = 1, 2, . . . , n

bk = g2 + sin2 Skpn D k = 1, 2, . . . , n (9.11)

b = lnScothAc

17.37Dg = sinhS b

2nDNotice that for n even, the terminating impedances are not equal. The g values

are tabulated in Table 9.2 for g0 = 1, v1 = 1, and n = 1 to 10 for various ripplevalues. In general, a ripple value in the 0.01- to 0.2-dB range is used.

9.2.3 Other Response-Type Filters

Some other response-type filters are also commonly used, including the ellipticfunction response, the Bessel response, and the generalized Chebyshev response[1–9]. The elliptic function response is a popular type, and some characteristics ofthis are discussed next.

Here the stopband has a series of peaks and a minimal attenuation level Lm .However, no simple equation for the insertion loss is possible. These filters aretreated in detail in [2–5], with specific element values for different n values from3 to 9 given in [7]. These filters can be designed using the coupling matrix approachas discussed in Chapters 10 and 11. This type of filter provides a much steeperstopband skirt for a given n and passband/stopband insertion loss than either theButterworth or Chebyshev response filters.

9.2.4 LC Filter Transformation

As mentioned before, highpass, bandpass, or bandstop filters require a transforma-tion in addition to scaling, and these transformations are discussed below. Forlowpass filters, scaling to the desired frequency band and impedance level is accom-plished by using the equations given next for the series inductors and shunt capaci-tors. In this case

v /v1 = v /vLpb

Lk = gk (Z0 /vLpb ) (9.12)

Ck = gk (1/vLpb Z0) (9.13)


Table 9.2 Element Values for a Chebyshev Lowpass Prototype with g0 = 1, v1 = 1, and n = 1 to 10for Different Ripple Values

Valueof n g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

0.01-dB ripple1 0.0960 1.00002 0.4488 0.4077 1.10073 0.6291 0.9702 0.6291 1.00004 0.7128 1.2003 1.3212 0.6476 1.10075 0.7563 1.3049 1.5773 1.3049 0.7563 1.00006 0.7813 1.3600 1.6896 1.5350 1.4970 0.7098 1.10077 0.7969 1.3924 1.7481 1.6331 1.7481 1.3924 0.7969 1.00008 0.8072 1.4130 1.7824 1.6833 1.8529 1.6193 1.5554 0.7333 1.10079 0.8144 1.4270 1.8043 1.7125 1.9057 1.7125 1.8043 1.4270 0.8144 1.0000

10 0.8196 1.4369 1.8192 1.7311 1.9362 1.7590 1.9055 1.6527 1.5817 0.7446 1.1007

0.1-dB ripple1 0.3052 1.00002 0.8430 0.6220 1.35543 1.0315 1.1474 1.0315 1.00004 1.1088 1.3061 1.7703 0.8180 1.35545 1.1468 1.3712 1.9750 1.3712 1.1468 1.00006 1.1681 1.4039 2.0562 1.5170 1.9029 0.8618 1.35547 1.1811 1.4228 2.0966 1.5733 2.0966 1.4228 1.1811 1.0008 1.1897 1.4346 2.1199 1.6010 2.1699 1.5640 1.9444 0.8778 1.35549 1.1956 1.4425 2.1345 1.6167 2.2053 1.6167 2.1345 1.4425 1.1956 1.000

10 1.1999 1.4481 2.1444 1.6265 2.2253 1.6418 2.2026 1.5821 1.9628 0.8853 1.3554

0.2-dB ripple1 0.4342 1.00002 1.0378 0.6745 1.53863 1.2275 1.1525 1.2275 1.00004 1.3028 1.2844 1.9761 0.8468 1.53865 1.3394 1.3370 2.1660 1.3370 1.3394 1.00006 1.3598 1.3632 2.2394 1.4555 2.0974 0.8838 1.53867 1.3722 1.3781 2.2756 1.5001 2.2756 1.3781 1.3722 1.00008 1.3804 1.3875 2.2963 1.5217 2.3413 1.4925 2.1349 0.8972 1.53869 1.3860 1.3938 2.3093 1.5340 2.3728 1.5340 2.3093 1.3938 1.3860 1.0000

10 1.3901 1.3983 2.3181 1.5417 2.3904 1.5536 2.3720 1.5066 2.1514 0.9034 1.5386

0.5-dB ripple1 0.6986 1.00002 1.4029 0.7071 1.98413 1.5963 1.0967 1.5963 1.00004 1.6703 1.1926 2.3661 0.8419 1.98415 1.7058 1.2296 2.5408 1.2296 1.7058 1.00006 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.98417 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1.00008 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.98419 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0000

10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

where vLpb is the required lowpass cutoff radian frequency and Z0 is the inputand output termination impedance.

Highpass Transformation

Transposing the series inductances into series capacitances and the shunt capaci-tances into shunt inductances transforms the lowpass prototype of Figure 9.3 intoa highpass filter (see Figure 9.6). Thus,


Figure 9.6 Highpass filter schematic and typical frequency response.

vv1

= −vHpb

v(9.14)

where vHpb is the band-edge frequency. The element values obtained are

Ck =1

gkvHpb Z0(9.15a)

Lk =Z0

gkvHpb(9.15b)

Bandpass Transformation

Bandpass filters also require transformation and scaling. Series inductors of thelowpass prototype are transformed into a series combination of an inductor anda capacitor, while the shunt capacitors are transformed into a parallel combinationof an inductor and a capacitor (see Figure 9.7). Hence the bandpass filter has twicethe number of elements. With a lower cutoff frequency fl and upper cutoff frequencyfu defined for the bandpass, the center frequency fo , bandwidth BW, and fractionalbandwidth FBW are defined by

f0 = √ f, fu (9.16)

BW = fu − f, (9.17)

FBW = BW / f0 (9.18)

A typical filter structure that results is shown in Figure 9.7. In this case

vv1

=f0

BW S ff0

−f0f D (9.19)

and the element values obtained are


Figure 9.7 Bandpass filter structure with typical frequency response.

Lk, SR = gkZ0

2pBW; Ck, SR =

2pBW

gk Z0v 20

(Series elements) (9.20)

Lk, SH =2pBWZ0

gkv 20

; Ck, SH =gk

2pBWZ0(Shunt elements) (9.21)

where

v 20 =

1Lk, SRCk, SR

=1

Lk, SHCk, SH(9.22)

Bandstop Transformation

Here, the shunt capacitor of the lowpass prototype is transformed into a seriesinductor and capacitor in shunt to ground and the series inductor is replaced witha parallel inductor-capacitor in series as shown in Figure 9.8. In this case


Figure 9.8 Typical bandstop filter structure and its frequency response.

v1v

=f0

BW S ff0

−f0f D (9.23)

with parallel-tuned circuit element values

v0Ck, SR =1

v0Lk, SR=

v02pBWZ0gk

(9.24a)

and series-tuned circuit element values

v0Lk, SH =1

v0Ck, SH=

v0 Z02pBWgk

(9.24b)

9.2.5 Filter Analysis and CAD Methods

Filter Analysis

While the earlier discussion has covered some analytical aspects of various typesof filters, in general, to account for phase characteristics and finite Q of circuitelements, we resort to the use of either ABCD matrices or Kirchhoff’s equations. The


ABCD matrix method is limited to ladder networks, while Kirchhoff’s equations canbe applied in general to any network. Both of these techniques are well coveredin [10]. Knowing the ABCD matrices for a variety of elements as given in Chapter2, 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 consider-ably simplified by the use of computers. The ABCD matrix method, however, cannothandle re-entrant combinations (i.e., nonladder networks) such as are present inhigh-performance bandpass filters. This limitation is readily overcome through theuse of Kirchhoff’s equations.

Computer-Assisted 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 filtertransfer approximations. Software packages are available to simulate performancebefore prototype construction, thus permitting fine tuning of parameters for optimi-zation, taking into account practical realities and fabrication constraints. Some ofthese packages include LINMIC + [11], which uses a full-wave field solution, andplanar-field simulators such as Sonnet em, IE3D, and SFPMIC [12–15]. Thesecodes can deal with multiple dielectric layers and multiple conductors and areexcellent for design verification. For three-dimensional circuits, other packages,such as the HP High-Frequency Structure Simulator (HFSS) [16], which handlesmultiport structures with unrestricted dielectric and conductor geometry, can beused.

In general, we commence the design process by selecting the appropriate circuitconstruction, modeling the filter response against the physical parameters, and thenusing accurate models for the structure to determine the response and optimizeparameters for the application at hand. Most of these aspects can be performediteratively on a computer using an optimization routine.

One excellent software package for performing the above operation is theGenesys software suite from Eagleware Corporation [17]. The tools are availablein this package for the design of a wide set of LC filter topologies includingconventional, narrowband, flat-delay, symmetric, elliptic, zig-zag, bandpass, low-pass, bandstop, and highpass structures. It also supports a large number of proto-type transfer functions including the most commonly used, such as Butterworth,Chebyshev, and Bessel. The advantage of this package is that there are other filesincluded that permit the design of an overall circuit network including the filter,oscillators, matching networks, and equalizers. The filters can be designed in themedia of interest here (e.g., stripline, microstrip) in most configurations such ashairpin, interdigital and combline.

9.2.6 Some Practical Considerations

In the previous sections we started with the lumped-element filters. From this wecan move into distributed-line filters and from there to coupled-line filters. Thedistributed lines can be realized in any desired form such as waveguide, coaxiallines, or planar transmission lines. Kuroda’s identities [18, 19] allow one to realize


lowpass structures using shunt elements with identical response and with Richardstransformations [19, 20] one can establish the distributed line parameters. Table9.3 provides a listing of transmission line lengths with the equivalent RLC network,which can simplify design considerably. However, as the passband for distributedfilters can reoccur at frequencies of twice or thrice the initial passband frequency,stopband attenuation is severely compromised. In addition, discontinuities such asopen ends, steps in linewidth, and T and cross-junction effects can degrade filterperformance.

A number of practical considerations and limitations often determine the actualfilter 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 overallsystem costs at acceptable levels. Other aspects, such as finite Q , group delay,temperature effects, power-handling capacity, and tunability may also be importantfactors in determining filter configuration and design [10].

For low selectivity, wideband applications, striplines, and microstrip filters areideal. They do suffer from temperature effects, however, and are difficult to tune.Suspended-substrate filters give a higher Q over microstrip, resulting in lower filterloss, sharper band edges, and better temperature stability. Dielectric resonatorfilters have been developed over a wide range of frequencies, and high dielectricconstant 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 passbandinsertion loss is increased or, as in the case of equal-ripple response filters, ripplesare suppressed. The loaded Q of the resonators normally determines the bandwidthof 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 Qvalues 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 specificpower-handling capability depends on the filter topology and will depend on thetransmission media used. In general, the type of media discussed in this text (i.e.,stripline, microstrip) are good for low-power levels but can handle high-pulsepower levels (up to a few kW) and approximately 50-W average power levels.Environmental temperature changes result in changes to the physical dimensionsof the filter structure and therefore changes to the electrical characteristics. Theseeffects are analyzed and discussed in [10].

Finally, the group delay of a filter depends on the selected prototype andnumber of sections with both band center group delay and group delay variationincreasing with an increase in the number of sections. In many applications, aconstant group delay over the desired bandwidth is required. The design of thesefilters is discussed in the literature [21].

There are different types of coupled filters. For example, direct-coupled resona-tor filters tend to have excessive lengths and generally can be reduced significantlyin size by the use of parallel-coupled lines. Since parallel coupling results in muchtighter coupling than with the use of end-coupled structures, greater bandwidthsare possible. The first spurious response occurs at two or three times the center


Table 9.3 Equivalent RLC Networks for Transmission Line Lengths

9.3 Parallel-Coupled Line Filters 283

frequency, depending upon the media, with a larger gap being permitted betweenadjacent strips. A broader bandwidth is also obtained for a given gap tolerance.For compactness, resonator sections are placed side-by-side. Some commonly useddesigns using this concept include the parallel-coupled, interdigital, combline andhairpin line filters, and these configurations are shown in Figure 1.15 (Chapter 1).These parallel-coupled line filters will be discussed in greater detail below. Thetapped input-output structures shown in the figure have a relatively narrow band-width.

9.3 Parallel-Coupled Line Filters

The design of parallel-coupled line filters was formulated by Cohn [22] and hasbeen refined or modified by others for specific design situations or operatingconditions [23–29]. For a typical n section parallel-coupled filter, one starts withcomputing the g1 , g2 , . . . , gn + 1 values of the lowpass prototype for eithermaximally flat or equal-ripple response, using (9.6) or (9.10). As shown in Figure9.9(a), the filter is assembled from n + 1 sections of equal length (l /4) at the centerfrequency, giving a structure of n resonators. The electrical design is specifiedby the even- and odd-mode characteristic impedances Z0e , Z0o on the parallelconductors. Using these impedance values, the transmission line dimensions of eachsection 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 obtainedby substituting u = −90 degrees and Z0 = K in the ABCD matrix of a transmissionline of electrical length u and characteristic impedance Z0 . Therefore, the ABCDmatrix of the filter section is

FA BC DG = 3

cos u jZ0 sin u

j sin uZ0

cos u 4 30 −jK

−jK

0 4 3cos u jZ0 sin u

j sin uZ0

cos u 4(9.25)

The K inverters for the various sections are defined in terms of the g0 , g1 , . . .elements as follows [22]:

Z0K01

= √ pD f2v1g0g1

(9.26a)

Z0Kj, j + 1

=pD f

2v1√gjgj + 1, j = 1 to n − 1 (9.26b)

Z0Kn, n + 1

= √ pD f2v1gngn + 1

(9.26c)

where D f is the fractional bandwidth (FBW), and v1 has been defined previously.After calculating K, the even- and odd-mode impedances of the coupled lines arecalculated from the following relations:


Figure 9.9 (a) Parallel-coupled transmission line resonator filter; (b) parallel-coupled line resonator;(c) K inverter-type equivalent circuit of the coupled line resonator; and (d) resonatorswith end effects.

(Z0e )j + 1Z0

= 1 +Z0

Kj, j + 1+ S Z0

Kj, j + 1D2, j = 0 to n (9.27)

(Z0o )j + 1Z0

= 1 −Z0

Kj, j + 1+ S Z0

Kj, j + 1D2, j = 0 to n (9.28)

The physical dimensions of the filter sections are then calculated [30] for the desiredZ0e and Z0o . An approximate value for the physical length is obtained from theaverage value of the even- and odd-mode velocities, that is:

u =2pl

l =2pl0

√eree + √ereo

2l =

p2

(9.29)

9.3 Parallel-Coupled Line Filters 285

A more accurate value for the length of the resonator is obtained by taking intoaccount the open-end discontinuity capacitance [30], which gives rise to an addi-tional length Dl as shown in Figure 9.9(d). In this case, the line length calculatedfrom (9.29) is shortened at each end by Dl :

Dl = 0.6Co Z01 / X√eree + √ereo C (9.30)

where Dl is in millimeters, the open-end capacitance Co in picofarads [30], andZ01 = √Z0e Z0o .

9.3.1 Design Example

Consider a design of three-pole (n = 3) parallel-coupled microstrip line bandpassfilter that has a fractional bandwidth FBW = 0.15 at a center frequency f0 = 2 GHz.A Chebyshev prototype with a 0.1-dB ripple is chosen for the design. FromTable 9.2, the element values for the lowpass prototype can be found, which areg0 = g4 = 1.0, g1 = g3 = 1.0315, and g2 = 1.1474. Using the design equations (9.26)to (9.28) with terminal impedance Z0 = 50 ohms, the pairs of even- and odd-modeimpedances of coupled sections for the filter design are calculated:

Z0e1 = 85.3181 ohms

Z0o1 = 37.5243 ohms

Z0e2 = 63.1744 ohms

Z0o2 = 41.5163 ohms

Since the filter is symmetrical, only these two pairs of design parameters need tobe considered.

The next step of the filter design is to find the dimensions of coupled microstriplines whose even- and odd-mode impedances match to those above calculated. Forexample, referring to Figure 9.9(a), W1 and S1 are so determined that the resultanteven- and odd-mode impedances match to Z0e1 and Z0o1 given above. Assumethat the microstrip filter is constructed on a dielectric substrate with a dielectricconstant of 10.2 and a thickness of 1.27 mm. The final layout of the filter designedon the substrate is illustrated in Figure 9.10(a). The physical length of each parallel-coupled line section was mainly determined based on (9.29) for an electrical lengthof 90 degrees with some adjustments to take into account the effects of open-endand impedance-step discontinuities. The filter performance, obtained by full-waveelectromagnetic (EM) simulation, is presented in Figure 9.10(b), showing a desiredpassband from 1.85 to 2.15 GHz, which is equivalent to a 15% fractional band-width centered at 2 GHz. Figure 9.10(c) is the wideband response of the filter,showing a spurious passband at approximately 2f0 . The issue of suppressing spuri-ous responses of parallel-coupled line filters will be addressed in Chapter 10.

In parallel-coupled microstrip filters, the physical lengths of the coupled sectionsare the same for both the even and odd modes, and we obtain an asymmetrical


Figure 9.10 (a) Three-pole parallel-coupled microstrip line filter on a dielectric substrate with adielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are in millimeters.(b) EM-simulated passband performance. (c) Wideband response.

9.4 Interdigital Filters 287

passband response with deterioration of the upper stopband from differences inphase velocities in these two modes. To improve the stopband performance, thephase velocities of the two modes should be equalized. Various techniques can beused to do this, including overcoupling the resonators, suspending the substrate,using parallel-coupled stepped impedance resonators or using capacitors at the endof the coupled sections [29]. We can also introduce wiggles in the coupled linesto accomplish this [26]. Some of these techniques are described in Chapter 6.Similarly, we can use other methods such as half-wavelength broadside-coupledmicrostrip lines to enhance bandwidth [23]. These techniques are useful for MICand MMIC applications.

9.4 Interdigital Filters

The interdigital filter is popular because it is compact and uses the available spaceefficiently. It can be designed for both narrow and wide (30–70%) bandwidths. Atypical interdigital bandpass structure is shown in Figure 1.15(b). Accurate designprocedures 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 unloadedQ , thus making it a good choice for narrow-bandwidth filter applications. Forlarge bandwidths, the capacitively-loaded parallel-coupled line filter is ideal. Thisalso reduces the filter size. In addition, the tolerances required in their manufacturecan be relaxed, spurious responses are not present, the rates of cutoff and strengthof the stopbands can be increased by multiple poles of attenuation at dc and ateven multiples of the center frequency of the first passband, and these filters canbe fabricated without dielectrics, thereby eliminating dielectric loss.

The design equations for parallel-coupled line interdigital filters are given in[31, 32] and will not be repeated here. Essentially these equations yield the variousline capacitances per unit length of the line and these can be used to determinethe physical dimensions of the line. Alternately, narrow-to-moderate-bandwidthbandpass filters using resonators can be designed by calculating or measuring thecoupling coefficient between resonators and the external quality factor of the inputand output resonators [10]. These coupling values are then related to a normalizedlowpass prototype value and can be used to realize all possible response shapes.This procedure is the most practical design method when the filter structure iscomplex or its equivalent circuit model is not readily available. The proceduredescribed below is applicable to all types of coupled resonator filters, whetherrealized 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 couplingcoefficients in terms of lowpass prototype element values and design frequencies,as follows:

kn, n + 1 =BW

f0√gngn + 1(9.31)


The next step is to determine physical dimensions of the coupled resonators,depending on the transmission medium used. The final step is to determine theloaded Q of the first and the last resonators required to connect the coupledresonators to input and output terminals.

In a filter design, the singly-loaded Q is calculated from

QL =f0

BWg1 =

f0BW

gn + 1 (9.32)

To illustrate the above-described design method, the following microstripexample is considered:

Center frequency 4 GHzResponse Chebyshev with 0.2-dB rippleBandwidth 0.4 GHz35-dB attenuation points 3.6 and 4.4 GHzSubstrate er = 9.8, h = 1.27 mm

From the nomograph (Figure 9.5), we find that the required number of resona-tors is 5. The prototype values are g0 = 1.0, g1 = g5 = 1.3394, g2 = g4 = 1.337,and g3 = 2.166. From (9.31) the coupling parameters are determined to bek12 = k45 = 0.0747, k23 = k34 = 0.0588, and from (9.32), QL = 13.4. The filtercan be realized using coupled microstrip medium on RT/duroid substrate or aluminasubstrate. Figure 9.11 shows [25] measured coupling coefficients as a function ofS/h for er = 2.22 and W/h = 1.8, which corresponds to a single-strip impedanceZ0I of approximately 70V, and for alumina substrate (er = 9.8) and W/h = 0.7(Z0I = 58V). Similar curves can be obtained for other line widths and dielectricsubstrates. In these filters, the final step in the design is to determine the tap-pointlocation for a given QL . The tap point , /L [Figure 1.15(b)] can be calculated fromthe following equation [25]:

QLZ0 /Z0I

=p

4 sin2 (p, /2L)(9.33)

Thus, the filter dimensions in Figure 1.15(b) on a 1.27-mm-thick alumina substatebecome

h = 1.27 mm

W0 = 1.27 mm (Z0 = 50V)

W = 0.889 mm

S12 = 2.16 mm = S45

S23 = 2.54 mm = S34

, = 2.25 mm

L = 7.5 mm

9.4 Interdigital Filters 289

Figure 9.11 Measured coupling coefficients versus S/h for (a) RT/duroid and (b) alumina. (From:[25]. 1979 IEEE. Reprinted with permission.)

where S12 and S23 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 frequencyf0 = 1.45 GHz and a fractional bandwidth of 0.333. The filter is implemented ona dielectric substrate with a dielectric constant of 6.15 and a thickness of 1.27mm. The design example is based on a nine-pole Chebyshev lowpass prototypewith a return loss of −20 dB (or a ripple of 0.04321 dB). The lowpass elementvalues, which can be obtained from the formulas given in (9.10) and (9.11), are


g0 = g10 = 1, g1 = g9 = 1.0235, g2 = g8 = 1.4619, g3 = g7 = 1.9837, g4 = g6 =1.6778, and g5 = 2.0649.

Using the design equations available in [34], the odd- and even-mode imped-ances are determined, which are given in Table 9.4, where the k values are calculatedfrom

ki, i + 1 =Z0ei, i + 1 − Z0oi, i + 1Z0ei, i + 1 + Z0oi, i + 1

However, these pairs of even- and odd-mode impedances generally cannot bematched for fixed line width resonators as required in the case of symmetricalcoupled line designs. For this design example, all interdigital line resonators havethe same width of 2 mm on the substrate. Therefore, the alternative approach isto determine the coupling coefficients k defined earlier. To this end, the differentset of even- and odd- mode impedance values for the fixed width resonators aredetermined, which are aimed to match the coupling coefficients or k values givenin the Table 9.4. In this way, all the spacings between adjacent coupled resonatorscan be found. The location of tapped-lines at input and output (I/O) resonatorsare also readily determined based on a procedure documented in [34]. Figure9.12(a) shows the layout of the designed filter, where all the physical dimensionsare in millimeters. Each of the resonators is about a quarter-wavelength long atthe center frequency f0 . The I/O resonators are slightly longer to compensate forresonant frequency shift due to the effect of the tapped input and output. Thefilter was fabricated on an RT/Duroid 6006 substrate and the measured passbandperformance is plotted in Figure 9.12(b). The measured center frequency is lowerthan the designed one, which is mainly due to the tolerances in the substrate andmanufacturing process. The wideband frequency response of the filter is demon-strated in Figure 9.12(c), which clearly shows that the first spurious responseappears at about 3f0 because of the use of the quarter-wavelength resonators.

9.5 Combline Filters

A typical combline filter configuration is shown in Figure 1.15(c). Together withthe capacitively loaded interdigital filter, the combline filter is one of the mostcommonly used bandpass structures. This filter is more compact than the interdigi-tal, is generally easier to fabricate, and is an attractive alternative to other filtertypes, especially for narrow bandwidths. Here the resonators consist of TEM modetransmission line elements that are short-circuited at one end and have a lumped

Table 9.4 Calculated Odd- and Even-ModeImpedance and Corresponding k Value

i Z0ei, i + 1 Z0oi, i + 1 ki, i + 1

1 60.399 39.303 0.2122 56.154 41.337 0.1523 55.492 41.703 0.1424 55.310 41.806 0.139

9.5 Combline Filters 291

Figure 9.12 (a) The layout of the designed nine-pole microstrip interdigital filter on a dielectricsubstrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. (b) Measuredpassband performance of the nine-pole microstrip interdigital filter. (c) Widebandresponse of nine-pole microstrip interdigital filter.

capacitance between the other end of each resonator line element and ground.Without these capacitances, the resonator lines would be exactly l /4 at resonanceand the structure would have no passband. Hence the lines must be less than90 degrees long and capacitively loaded to achieve resonance. The capacitancesare made relatively large and the lines 45 degrees or less in length, thereby resultingin a very compact and efficient coupling structure. With the line elements l /8 atthe primary passband, the second passband is far removed and the attenuation isinfinite at a frequency where the line length is l /4. So beyond the passband, theattenuation is very high and cutoff at the upper side of the passband can be verysteep. Further, adequate coupling can be maintained between resonator elementswith sizeable spacing, thereby providing more margin in tolerance requirements.

Design procedures for combline filters are discussed in [33, 35, 36]. The processis parallel to the design process of the interdigital filter discussed above and thereforeis not covered in any detail here. Refer to [33, 35, 36] for the actual design equationsto calculate the distributed line capacitances and from there the dimensions of thestructure.


Coaxial combline filters [Figure 1.14(b)] using ceramic block are commonly usedin applications from 200 to 3,000 MHz. These filters have typically a 2-dB insertion



loss and have bandwidth from 1% to 20%. Ceramic filters are temperature-stable,and their temperature range of operation is normally from −30°C to +85°C. Theyare surface-mountable, and in high volume their cost is $2 to $5.

Figure 9.13 shows a schematic of a three-resonator combline ceramic-blockfilter. The ceramic materials have high-dielectric constant (i.e., er ≅ 40 − 80). Thecoupling between pairs of adjacent resonators is realized by a circular or rectangularair hole. The inhomogeneous interface between the high-dielectic constant ceramicand air hole gives rise to different phase velocities for the even- and odd-modes ofthe coupled lines. This difference provides the required couplings between theresonators to realize a filter. The design of such filters is straightforward butrequires numerical methods, such as EM simulators, to determine coupling betweenthe resonators. Normally, filters are designed empirically and tuned after fabrication

9.5 Combline Filters 293

Figure 9.13 High-er ceramic block combline bandpass filter. All the surfaces are metallized exceptthe top surface. A, B, and C are metallized coaxial resonators. Metallized sidewalls ofthe ceramic block act as outer conductors.

using ceramic grinders and metal scrapers. Analysis, design, and test results forvarious ceramic-block filters have been discussed by many authors [37–42]. In thissection, we describe briefly the design of such filters.

In Figure 9.13, A, B, and C are the metallized center conductors of the coaxialresonators. All resonators are short-circuited at the bottom and open-circuited atthe top and are designed to be l /4 long at the operating center frequency [38].Resonators A, B, and C are coupled to each other for filter action through airholes between them. The first and last resonators are coupled to input and outputports, respectively, by coupling pads P1 and P2 located near them. The capacitivecoupling between the filter and input and output is usually accomplished by a cut-and-try method. Figure 9.14 shows a lumped-element equivalent-circuit for thisfilter; l /4 resonators are represented by parallel resonant circuits (Ci , Li ). Air holesprovide magnetic (inductive) coupling (Lij ), and the filter is connected to the inputand output (usually 50V) through capacitive coupling represented by Cin and Cout .

At the center frequency fo , the length of each resonator is given by

L = 0.25l = 0.25l0

√ere=

0.25c

f0√ere(9.34)

where l0 , c, and ere are the free-space wavelength, velocity of light, and the effectivedielectric constant, ere , is obtained from

ere =eree + ereo

2(9.35)


Figure 9.14 Lumped-element equivalent circuit for the three-resonator filter.

where eree and ereo are, respectively, the even- and odd-mode effective dielectricconstants of the medium in which the coaxial resonators are embedded.

The coupling coefficient is given by [39]

k =2X√eree − √ereo C√eree + √ereo

(9.36)

Figure 9.15 shows a cross-sectional view of an air hole coupled-line structurewith dimensions, and Figure 9.16 shows the calculated values of the even- andodd-mode effective dielectric constants. Finite difference method [39] was used toanalyze the structure, with er = 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 datafor the coupling coefficient has been published by Yao et al. [41, 42].

Figure 9.15 Cross-sectional view of the three-resonator ceramic-block combline filter.

9.6 The Hairpin-Line Filter 295

Figure 9.16 Effective dielectric constant for even and odd modes as a function of air hole radius.er = 80, D = 2.4 mm, H = 6 mm, and S = 0.8 mm.

To illustrate a design example of a bandpass ceramic-block filter, the followingspecifications are chosen:

Center frequency 900 MHzResponse Chebyshev with 0.05-dB rippleBandwidth 30 MHzNumber of resonators 3

This filter can be designed by using normalized coupling coefficients in terms oflowpass prototype element values and design frequencies as given in (9.31) and(9.32). The normalized bandwidth, BW/ f0 = 30/900 = 1/30. From Table 9.2,the lowpass prototype element values are g0 = g4 = 1, g1 = g3 = 0.8794, andg2 = 1.1132. From (9.31), the coupling coefficients k12 = k23 = 0.0339. For thestructure shown in Figure 9.15, and using data in Figure 9.17, the filter designparameters are er = 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 three-resonatorceramic-block filter where the input and output couplings were obtained by experi-ments. In the 50-MHz bandwidth, the measured insertion loss was better than1.5 dB. The ceramic material’s loss tangent (tan d ) was 2.5 × 10−4 and the tempera-ture coefficient was 3 ppm/°C.

9.6 The Hairpin-Line Filter

The interdigital and combline filters described above require ground connections,which may be difficult to achieve when using microstrip lines on ceramic substrates.


Figure 9.17 (a) Coupling coefficient as a function of air hole radius. er = 80, D = 2.4 mm,H = 6 mm, and S = 0.8 mm. (b) Coupling coefficient as a function of separationbetween the metallized resonator and air hole. er = 80, D = 2.4 mm, H = 6 mm, andR = 1.2 mm.

When stripline or microstrip is used, the hairpin-line filter is one of the preferredconfigurations. This is particularly useful when one is interested in MIC or MMICcircuits. The hairpin-line filter can be considered basically to be a folded versionof a half-wave parallel coupled-line filter. It is much more compact, though, andgives approximately the same performance. A typical schematic of this type offilter is shown in Figure 1.15(d). As the frequency increases, the length-to-widthratio is smaller for a given substrate thickness, so that folding the resonator becomes


Figure 9.18 Measured frequency response of the three-resonator combline filter.

impractical. In general, the hairpin-line filter is larger than the combline or interdigi-tal filter. But because no grounding is required, it is amenable to mass productionas a large number of filters can be simultaneously printed on a single substrate,thereby lowering production costs.

The design of hairpin-line filters has been discussed by various researchers[43–47]. Although one could normally use the design expressions for parallel-coupled filters, here, the line between the resonators decreases the length of thecoupling sections, so that the coupled sections are less than a quarter-wavelength.In addition, the bend discontinuities are difficult to handle. Using the parallel-coupled bandpass design and compensating for the end capacitance, however,dissimilar propagation velocities, shortened coupling elements, and bends throughan optimization routine on a computer permits rapid design of such a filter.

We can use the methods given above for the design of hairpin-line filters, sodetails are not provided here. The interarm spacings in the hairpin-line filter designshould 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 designmethod takes the interarm coupling into account, but assumes negligible phaseshift over the line joining the arms of a hairpin. Thus, we must exercise cautionwhen high-dielectric constant substrates are used, as small physical length variationswill result in large phase shifts. Cristal and Frankel’s equations are also not applica-ble for inhomogeneous media and do not account for the effects of right-angledbends and corners. A good comparison of several types of hairpin filters is givenby Matthaei [48].


Hairpin-line filters can also be designed by calculating the inter-resonator couplingas a function of the spacing between resonators for a given set of substrate parame-ters, frequency, and microstrip width. The microstrip width is selected to obtainmaximum Q for the resonators. Figure 9.19 shows measured coupling coefficientsfor hairpin resonators on alumina and high dielectric constant substrates. In thesefilters, the tap point (, /L) is calculated using the following relation [25]:


Figure 9.19 (a) Measured coupling coefficient for hairpin-line resonators, er = 9.8, h = 1.27,W/h = 0.7. (From: [25]. 1979 IEEE. Reprinted with permission.) (b) Computed andmeasured coupling for high dielectric constant resonators. er = 80, h = 2 mm,f0 = 905 MHz; and (c) er = 90, h = 1 mm, f0 = 854 MHz. (From: [46]. 1994International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering.Reprinted with permission.)


QL(Z0 /Z0I )

=p

2 sin2 (p, /2L)(9.37)

The physical layout of a low-frequency hairpin-line filter designed using micro-strip on a high-dielectric constant material is shown in Figure 9.20, while Figure9.21 shows the measured data. The filter was designed having five-pole Chebyshev

Figure 9.20 Hairpin-line filter layout. (From: [46]. 1994 International Journal of Microwave andMillimeter-Wave Computer-Aided Engineering. Reprinted with permission.)

Figure 9.21 Measured response of a 905-MHz hairpin-line filter. (From: [46]. 1994 InternationalJournal of Microwave and Millimeter-Wave Computer-Aided Engineering. Reprintedwith permission.)


response centered at 905 MHz with 46-MHz bandwidth and 20-dB 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 hairpin-line filters, we can use Eagleware’s M/FILTER software[47] for quick results. We start with a filter topology such as edge-coupled, hairpin,interdigital or stepped Z, together with selecting the frequency response, such asButterworth, Chebyshev, or Bessel. The transmission line format is selected amongmicrostrip, stripline, and so on, together with the performance parameters such aslower cutoff frequency, upper cutoff frequency, and passband ripple. Finally, thesubstrate parameters are entered, including dielectric constant and thickness. Enter-ing these, we obtain a layout of the filter, while the frequency response can beevaluated using Eagleware’s Superstar software and the output file of M/FILTER.Should further optimization be required, it can be accomplished by an iterativeprocess of reloading the final values into the filter program to arrive at the finaldimensions of the circuit. The output plot file can be used to create another completefile to drive a numerically controlled milling machine and thereby achieve prototypefabrication.

9.7 Parallel-Coupled Bandstop Filter

Parallel coupled-line bandstop filters are very popular in MIC and planar circuitapplications. They are easy to fabricate and mass-produce using printed circuittechnology. The basic configuration of a parallel coupled-line bandstop filter isshown in Figure 9.22. It consists of a main line attached to a set of L-shaped linesof which the horizontal section is parallel coupled to the main line. Each arm of theL-shaped line has 90 degrees of electrical length. Figure 9.23 shows the equivalenttransmission line circuit of each section of the filter. The element values are obtainedfrom Table 9.5 [49]. Each combination of series section and a shunt section canbe 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 sectionsis one less than the number shunt sections in Figure 9.23, the designer should add

Figure 9.22 Layout of a parallel coupled-line bandstop filter.

9.7 Parallel-Coupled Bandstop Filter 301

Figure 9.23 Equivalent circuit of parallel coupled-line filter with open circuited shunt loading.

an extra series section of impedance ZA right before the first shunt stub, and thenuse the conversion procedure in Figure 9.24 [49].

In Table 9.5 gi (i = 0, 1, . . . N) are the lowpass prototype values given inTables 9.1 and 9.2. Other parameters are defined as

a = cot Sp2

v1v0D (9.38)

where v1 is lower cutoff frequency of the bandstop filter.

L = av1′ (9.39)

where v1′ is the cutoff frequency of the lowpass prototype.


A standard commercial software package like WAVECON [50] can be used toimplement the above procedure. Table 9.6 shows the WAVECON design file ofan L-band stripline bandstop filter. Figure 9.25(a) shows the layout of the filterand Figure 9.25(b) shows the computed frequency response.


Table 9.5 Element Values for Parallel Coupled-Line Bandstop Filters

N = number of stubs

ZA , ZB = terminating impedances

Zj (j = 1 to N) = impedances of open circuit shunt stubs

Zj − 1, j (j = 2 to N) = connecting line impedances

gj = lowpass prototype element values

L = v1′a where v1′ is the cutoff frequency of the lowpass prototype and a is the bandwidthparameter defined in (9.38). The terminating impedance ZA is arbitrary.

Case of N = 1

Z1 =ZA

Lg0g1ZB =

ZAg2g0

Case of N = 2

Z12 = ZA (1 + Lg0g1),Z1 = ZAS1 +

1Lg0g1

D,

ZB = ZAg0g3 .Z2 =

ZAg0Lg2

,

Case of N = 3.

Z1 , Z2 , and Z12 are same as case N = 2.

Z3 =ZAg0

g4S1 +

1Lg3g4

D, Z23 =ZAg0

g4(1 + Lg3g4),

ZB =ZAg0

g4.

Case of N = 4.

Z1 = ZA S2 +1

Lg0g1D, Z12 = ZA S1 + 2Lg0g1

1 + Lg0g1D,

Z2 = ZA S 11 + Lg0g1

+g0

Lg2(1 + Lg0g1)2D, Z23 =ZAg0

SLg2 +g0

1 + Lg0g1D,

Z3 =ZA

Lg0g3, Z34 =

ZAg0g5

(1 + Lg4g5),

Z4 =ZA

g0g5S1 +

1Lg4g5

D, ZB =ZA

g0g5.

Case of N = 5

Z1 , Z2 , Z3 , Z12 , and Z23 are same as case N = 4.

Z4 =ZAg0

S 11 + Lg5g6

+g6

Lg4(1 + Lg4g5)2D, Z34 =ZAg0

SLg4 +g6

1 + Lg5g6D,

Z5 =ZAg6

g0S2 +

1Lg5g6

D, Z45 =ZAg6

g0S1 + 2Lg5g6

1 + Lg5g6D,

ZB =ZAg6

g0.


Figure 9.24 Parallel coupled line and shunt stub line equivalence.

Table 9.6 CAD File for the L-Band Bandstop Filter in Figure 9.25(a)

Stripline Coupled Line Bandstop Filter08/25/2006 09:34:362.7050 GHz CENTER FREQUENCY 0.0060 GHz BANDWIDTH0.1000 dB RIPPLE 3 POLES0.1000 INCHES GROUNDPLANE SPACING 9.9000 DIELECTRIC CONSTANT0.0005 INCHES CONDUCTOR THICKNESS50.000 OHMS INPUT LINE IMPEDANCE 0.0174 INCHES INPUT LINE WIDTH50.000 OHMS OUTPUT LINE IMPEDANCE 0.0174 INCHES OUTPUT LINE WIDTH

SECT ELEMENT Zoe Zoo Length Width Space Z-stub Stub-L Stub-WNUMB VALUE OHMS OHMS INCHES INCHES INCHES OHMS INCHES INCHES

1 1.0315 52.796 47.418 0.3396 0.0172 0.0910 31.436 0.3379 0.05002 1.1474 52.946 47.233 0.3396 0.0172 0.0893 31.436 0.3379 0.05003 1.0315 52.796 47.418 0.3396 0.0172 0.0917 31.436 0.3379 0.0500


Figure 9.25 (a) Bandstop filter layout. (b) Computed frequency response of filter in part (a).

References

[1] Milligan, T., ‘‘Nomographs and the Filter Designer,’’ Microwaves and RF, Vol. 24, October1985, pp. 103–107.

[2] Saal, R., ‘‘The Design of Filters Using the Catalogue of Normalized Low-Pass Filters’’ (inGerman), Telefunken (GMBH), Backang, W. Germany, 1961.

[3] Rhodes, J. D., Theory of Electrical Filters, New York: Wiley Interscience, 1976.[4] Skwirzynski, J. K., Design Theory and Data for Electrical Filters, London: Van Nostrand,

1965.[5] Zverev, A. I., Handbook of Filter Synthesis, New York: John Wiley, 1967.


[6] Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, Impedance, MatchingNetworks and Coupling Structures, New York: McGraw-Hill, 1964.

[7] Howe, H., Jr., Stripline Circuit Design, Dedham, MA: Artech House, 1974.[8] Alseyab, S. A., ‘‘A Novel Class of Generalized Chebyshev Low Pass Prototype for Sus-

pended Substrate Stripline Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-30,September 1982, pp. 1341–1347.

[9] Mobbs, C. I., and J. D. Rhodes, ‘‘A Generalized Chebyshev Suspended Substrate StriplineBandpass Filter,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-31, May 1983,pp. 397–402.

[10] Bahl, I. J., and P. Bhartia, Microwave Solid-State Circuit Design, New York: John Wiley,1988, Ch. 6.

[11] LINMIC+ Computer Program, Jansen Microwave, Ratingen, Germany.[12] em Computer Program, Sonner Software, Liverpool, NY, 2006.[13] EEpal Computer Program, Eagleware Corp., Stone Mountain, GA.[14] IE3D, Zeland Software, Fremont, CA, 2006.[15] SFPMIC+ Computer Program, Jansen Microwaves, Ratigen, Germany.[16] HFSS Computer Program, Agilent, Santa Clara, CA, 2006.[17] Rhea, R., ‘‘PC Tools Simulate and Synthesize RF Circuits,’’ Microwave and RF, Vol. 33,

No. 4, pp. 194–199.[18] Ozaki, H., and J. Ishii, ‘‘Synthesis of a Class of Stripline Filters,’’ IRE Trans. Circuit

Theory, Vol. CT-5, June 1958, pp. 104–109.[19] Davis, W. A., Microwave Semiconductor Circuit Design, New York: Van Nostrand Rein-

hold Co., 1984, Ch. 3.[20] Richards, P. I., ‘‘Resistor-Transmission Line Resonator Filters,’’ IRE Trans. Microwave

Theory Tech., Vol. MTT-6, April 1958, pp. 223–231.[21] Malherbe, J. A. G., Microwave Transmission Line Filters, Norwood, MA: Artech House,

1990.[22] Cohn, S. B., ‘‘Parallel Coupled Transmission Line Resonant Filters,’’ IRE Trans. Micro-

wave Theory Tech., Vol. MTT-6, No. 2, April 1958, pp. 223–232.[23] 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.[24] Mara, J. F., and J. B. Schappacher, ‘‘Broadband Microstrip Parallel-Coupled Filters Using

Multi-Line Sections,’’ Microwave J., Vol. 22, No. 4, April 1979, pp. 97–99.[25] Wong, J. S., ‘‘Microstrip Tapped-Line Filter Design,’’ IEEE Trans. Microwave Theory

Tech., Vol. MTT-27, January 1979, pp. 328–339.[26] 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.

[27] Minnis, B. J., ‘‘Printed Circuit Coupled Line Filters for Bandwidths Up to and GreaterThan an Octave,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-29, No. 3,March 1991, pp. 215–222.

[28] Ho, C. Y., and J. H. Werdman, ‘‘Improved Design of Parallel Coupled Line Filters withTapped Input/Output,’’ Microwave J., Vol. 26, No. 18, October 1983, pp. 127–130.

[29] Bahl, I. J., ‘‘Capacitively Compensated High-Performance Parallel-Coupled MicrostripFilters,’’ IEEE MTT-S Int. Microstrip Symp. Dig., 1989, pp. 679–682.

[30] Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House,1996, Ch. 3.

[31] Matthaei, G. L., ‘‘Interdigital Band-Pass Filters,’’ IRE Trans. Microwave Theory Tech.,Vol. MTT-10, No. 6, November 1962, pp. 479–491.

[32] Wenzel, R. J., ‘‘Exact Theory of Interdigital Band-Pass Filters and Related Coupled Struc-tures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, No. 5, September 1975,pp. 559–575.


[33] Matthaei, G. L., ‘‘Comb-Line Band-Pass Filters of Narrow and Moderate Bandwidth,’’Microwave J., Vol. 6, August 1963, pp. 82–91.

[34] Hong, J. S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, NewYork: John Wiley & Sons, 2001.

[35] Wenzel, R. J., ‘‘Synthesis of Combline and Capacitively Loaded Interdigital BandpassFilters of Arbitrary Bandwidth,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-19,No. 8, August 1971, pp. 678–686.

[36] Vincze, A., ‘‘Practical Design Approach to Microstrip Combine-Type Filters,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-22, No. 12, December 1974, pp. 1171–1181.

[37] Fukasawa, A., ‘‘Analysis and Composition of a New Microwave Filter Configurationwith Inhomogeneous Dielectric Medium,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-30, September 1982, pp. 1367–1375.

[38] Levy, R., ‘‘Simplified Analysis of Inhomogeneous Dielectric Block Combline Filters,’’ IEEEInt. Microwave Symp. Dig., 1990, pp. 135–138.

[39] You, C. C., C. L. Huang, and C. C. Wei, ‘‘Single-Block Ceramic Microwave BandpassFilters,’’ Microwave J., Vol. 37, November 1994, pp. 24–35.

[40] Hano, K., H. Kohriyama, and K.-I. Sawamoto, ‘‘A Direct Coupled l /4-Coaxial ResonatorBandpass Filter for Land Mobile Communications,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-34, September 1986, pp. 972–976.

[41] Yao, H. W., et al., ‘‘Full-Wave Modeling of Conducting Posts in Rectangular Waveguideand Its Applications to Slot-Coupled Combline Filters,’’ IEEE Trans. Microwave TheoryTech., Vol. MTT-43, December 1995, pp. 2824–2829.

[42] Yao, H. W., C. Wang, and K. A. Zaki, ‘‘Quarter-Wavelength Combline Filters,’’ IEEETrans. Microwave Theory Tech., Vol. 44, December 1996, pp. 2673–2679.

[43] Cristal, E. G., and S. Frankel, ‘‘Hairpin-Line and Hybrid Hairpin-Line/Half-Wave Parallel-Coupled-Line Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-20, No. 22,November 1972, pp. 719–728.

[44] Gysel, U. H., ‘‘New Theory and Design for Hairpin-Line Filters,’’ IEEE Trans. MicrowaveTheory Tech., Vol. MTT-22, No. 5, May 1974, pp. 523–531.

[45] Salkhi, A., ‘‘Quick Filter Design and Construction,’’ Appl. Microwave and Wireless,Vol. 6, No. 1, January 1994, pp. 92–100.

[46] Pramanick, P., ‘‘Compact 900-MHz Hairpin-Line Filters Using High Dielectric ConstantMicrostrip Line,’’ Int. J. Microwave Millimeter-Wave Computer-Aided Eng., Vol. 4,No. 3, 1994, pp. 272–281.

[47] Rhea, R. W., ‘‘Distributed Hairpin Bandpass, RF Compute!’’ Eagleware Corp., Vol. 3,No. 1, 1991.

[48] Matthaei, G. L., et al., ‘‘Hairpin-Comb Filters for HTS and Other Narrow-Band Applica-tions,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, No. 9, August 1997,pp. 1226–1231.

[49] Schiffman, B. M., and G. L. Matthaei, ‘‘Exact Design of Band-Stop Microwave Filters,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-12, January 1964, pp. 6–15.

[50] WAVECON, Microwave Filter Design Software, Escondado, CA, 2006.

C H A P T E R 1 0

Advanced Coupled-Line Filters

10.1 Introduction

Traditional parallel-coupled line filters, as discussed in the previous chapter, areeasy to design. However, when this class of filters is realized in an inhomogeneousmedia such as microstrip or coplanar waveguide, it suffers from poor upper stop-band performance and typically has spurious passbands centered at harmonics ofthe fundamental passband center frequency. In the case of parallel-coupled micro-strip filters, considerable effort has been directed at suppressing the harmonicspurious passbands, in particular those located near the desired passband. In thischapter we discuss several designs of coupled-line filters with enhanced stopbandperformance. These include designs using uneven-coupled stages, periodically non-uniform coupled lines, meandered parallel coupled lines, and defected groundstructures.

In the previous chapter, we discussed conventional Butterworth and Chebyshevresponse prototypes. Bandpass filters based on these prototypes may be referredto as direct-coupled filters because couplings exist only between adjacent resonators.However, when cross couplings are introduced among nonadjacent resonators,more advanced filtering characteristics such as quasi-elliptic function and linearphase responses can be obtained. These types of filters are referred to as cross-coupled filters and will be discussed in Section 10.3, together with filters withcross-coupled resonators and filters with source-load coupling. In addition, filterswith asymmetric port excitations can exhibit interesting characteristics and will bedescribed as well.

In this chapter, we will also discuss the design of interdigital filters usingquarter-wavelength stepped impedance resonators (SIR), which can result in a morecompact size and a wider upper stopband, compared to conventional interdigitalfilters, introduced in Chapter 9. Recent advances in wireless communication havecreated a need for dual-band filters, and these will be discussed in Section 10.5.

This chapter also provides many design examples using full-wave electromag-netic (EM) simulation. EM simulation tools have become widely available andserve as an invaluable tool for the first-pass design of filters.

10.2 Coupled-Line Filters with Enhanced Stopband Performance

10.2.1 Design Using Unevenly-Coupled Stages

For a conventional parallel-coupled line filter introduced in Chapter 9, all thecoupled-line sections are designed to have an equal electrical length, which is

307

308 Advanced Coupled-Line Filters

90 degrees at the center frequency of the designed passband. This restriction canbe relaxed when a more general design approach is used. This means that theelectrical lengths of coupled stages are not required to be 90 degrees at the desiredcenter frequency and can be different, resulting in modified parallel-coupled filterswith unevenly coupled stages. This helps to suppress the spurious response, espe-cially for microstrip line filters [1–3].

To demonstrate the principle of using unevenly coupled stages, let us considerthe single-section parallel-coupled microstrip filter of Figure 10.1. A half-wave-length resonator of length lR and width W is excited using a coupled-line sectionof length of lC and spacing S. Port 2 is very weakly coupled to the resonator asshown. Figure 10.2 plots responses for different lengths of the coupled-line section.The frequency response is obtained using a commercially available EM simulator[4]. The microstrip half-wavelength resonator has a length of 29 mm and a widthof 1 mm on the substrate. It can easily be shown that the resonator inherentlyresonates at a fundamental frequency f0 = 2 GHz, and at other harmonics around

Figure 10.1 Single coupled-line section for parallel-coupled microstrip filter.

Figure 10.2 Transmission response of structure shown in Figure 10.1 for lR = 29.0 mm,W = 1.0 mm, S = 0.2 mm, and W0 = 1.4 mm on a 1.27-mm-thick substrate with adielectric constant of 10.2.

10.2 Coupled-Line Filters with Enhanced Stopband Performance 309

2f0 , 3f0 , and so on. When lC = 14.6 mm, which corresponds to a 90-degreeelectrical length at f0 , the resonant frequency response, as indicated by the dottedline, shows the first three resonant peaks and an attenuation pole or finite-frequencytransmission zero at about 4.5 GHz. The finite transmission zero results from thecoupled line section. The parallel-coupled line structure of Figure 9.9(b) can alsobe represented by its equivalent circuit depicted in Figure 10.3. As can be seen,the two-port equivalent circuit involves two series open-circuit stubs with an electri-cal length of u. When u = 90 degrees, the open-circuit stubs do not affect the signaltransmission between ports 1 and 2. For u = 180 degrees, however, the open-circuitstubs represent two perfect open circuits along the main signal path, which blockthe two-port network transmission and thus produce a transmission zero at thefrequency where u = 180 degrees.

Strictly speaking, this equivalent circuit is for homogeneous or pure TEM-modecoupled lines such as coupled striplines [5], which have equal even- and odd-modephase velocities. This is the reason why a parallel-coupled stripline filter designedusing the conventional approach described in Chapter 9 will not produce a spuriousresponse at 2f0 when each of the coupled sections has an electrical length of90 degrees at the center frequency f0 . In the case of a conventional parallel-coupledmicrostrip filter, an approximate value of electrical length is usually obtained fromthe average value of the even- and odd-mode phase velocities. Therefore, if weconsider each microstrip coupled section to have an average electrical length of90 degrees at f0 , the finite transmission zero associated with this coupling structurewill not occur exactly at 2f0 , due to unequal even- and odd-mode phase velocities.This is what we observe in Figure 10.2 for lC = 14.6 mm. If we change the lengthlC , we can shift the finite transmission zero. For example, changing lC from14.6 mm to 16.6 mm moves the finite transmission zero down in frequency andcloser to the spurious response of the resonator at 2f0 , as shown by the brokenline in Figure 10.2. If we further increase the length lC to 17.2 mm, the spuriousresponse of the resonator at 2f0 can be suppressed as indicated by the unbrokenline in Figure 10.2. It is evident that the suppression is due to the spurious resonantfrequency coinciding with the frequency of finite transmission zero. Based on

Figure 10.3 Equivalent circuit of the parallel-coupled line structure of Figure 9.9(b).


this discussion, a modified parallel-coupled microstrip filter design with nonequalcoupled sections is presented next.

Design Example

Consider a bandpass filter design based on a three-pole Chebyshev lowpass proto-type with a passband ripple of 0.1 dB. From Table 9.2, we find the lowpassprototype elements as follows:

g0 = g4 = 1.0

g1 = g3 = 1.0315

g2 = 1.1474

The bandpass filter is designed to have a fractional bandwidth FBW = 0.123(i.e., 12.3%) at a center frequency f0 = 1.985 GHz. The filter parameters can becalculated by [6]

Qe1 = Qe3 =g0g1FBW

= 8.386

M12 = M23 =FBW

√g1g2= 0.1131

where Qe1 and Qe3 are the external quality factors of the resonators at the inputand output, M12 is the coupling coefficient between resonators 1 and 2, and M23is the coupling coefficient between resonators 2 and 3. The filter is to be imple-mented on a 1.27-mm-thick substrate with a dielectric constant of 10.2. A practicaldesign approach employing full-wave EM simulation is as follows.

The frequency response of the arrangement of Figure 10.1 can be used toextract the external quality factor. Figure 10.4 shows a typical EM-simulatedresonant response of the I/O coupled stage. The external quality factor can beextracted using the well-known formula:

Qe =f0

D f3-dB(10.1)

where f0 is the resonant or center frequency and D f3-dB denotes the 3 dB bandwidthof the resonant response as shown in Figure 10.4. Note that (10.1) is strictly validwhen the structure analyzed is ‘‘lossless.’’ Nevertheless, it is acceptable when theunloaded quality factor of the resonant structure is much larger than the externalquality factor.

In general for a given W, the external quality factor depends on both S andlC . It can also be shown that the larger the S, the longer the lC needed to suppressthe spurious response at 2f0 . In the design of a practical filter, the spacing S ismainly adjusted for the desired Qe , while the length lC is tuned for suppressingthe spurious response at 2f0 . In this way, the physical dimensions of the input andoutput coupling sections can be determined.


Figure 10.4 Typical frequency response for extracting the external quality factor.

Shown in Figure 10.5(a) is an arrangement to extract the coupling between tworesonators using EM simulation. For this particular example, the two resonators arecoupled to each other through a common coupled section. The two ports are soarranged in order to weakly excite the coupled resonators. The EM-simulatedfrequency response of the coupled resonators is plotted in Figure 10.5(b), wheretwo resonant peaks are clearly observed. If fp1 and fp2 denote the two resonantpeaks, the coupling coefficient M can be extracted using

M =f 2p2 − f 2

p1

f 2p2 + f 2

p1

(10.2)

For a given length of coupled section, the desired M can be found by adjustingthe spacing.

After extracting the desired external quality factor Qe1 = Qe3 and the desiredcoupling coefficient M12 = M23 , the layout and physical dimensions of the designedfilter are determined, and are given in Figure 10.6(a). The filter is symmetricalwith two identical I/O coupled-sections and two identical inter-resonator coupled-sections. However, the lengths of the I/O coupled-sections and the inter-resonatorcoupled-sections are different. The former is 17.2 mm with an electrical lengthlarger than 90 degrees at f0 , which is designed for suppressing the spurious responseat 2f0 , while the latter is 16 mm with an electrical length of about 90 degrees atf0 . The two small corner cuts of 0.4 × 0.6 mm on the middle resonator are forfrequency tuning. Figure 10.6(b) shows the filter performance, obtained by full-wave EM simulation. The filter exhibits a desired passband at f0 and the spuriousresponse at 2f0 has effectively been suppressed below −30 dB.

From Figure 10.6(b), it can also be seen that only a single attenuation polenear 4 GHz is present in this case. This attenuation pole or transmission zero isproduced by the two identical I/O coupled-sections. It is found, however, that ifthe I/O coupled-sections have slightly different lengths, two attenuation poles are


Figure 10.5 (a) Arrangement for EM simulation to extract the inter-resonator coupling. The tworesonators, which are 29 mm long and 1 mm wide, are coupled to each other througha common coupled section with a length of 16 mm and a spacing of 1 mm on a1.27-mm-thick substrate with a dielectric constant of 10.2. (b) Response of coupledresonators.

created, which can suppress the spurious response even more effectively. Figure10.7(a) illustrates a modified filter where the output coupled-section has a lengthof 17 mm instead of 17.2 mm. The performance of the modified filter is shownin Figure 10.7(b). As can be seen, the spurious response at 2f0 is suppressed toabout −40 dB. The suppression is evidently enhanced with two closely allocatedtransmission zeros close to 4 GHz. It is also seen that the effect of a small changein the length of the output coupled-section on the passband is negligible.

For a parallel-coupled microstrip filter with unequal-coupled sections, the sup-pression of the spurious response can be enhanced by using coupled stages withnarrower lines or a higher-image impedance. This is because the narrower coupled


Figure 10.6 (a) Parallel-coupled microstrip line filter with unevenly coupled stages. All the dimen-sions shown are in millimeters on a 1.27-mm-thick substrate with a dielectric constantof 10.2. (b) Full-wave EM simulated performance of the filter.

lines result in a smaller even-mode capacitance so that the ratio of ere /ero is reduced,where ere and ero are the even- and odd-mode effective dielectric constants, respec-tively. Two filters of this type on the same substrate with a dielectric constant of10.2 and a thickness of 1.27 mm have been experimentally demonstrated [2]. Thetwo filters, one of which is a third-order filter with a fractional bandwidth of 20%and the other of which is a fifth-order filter with a fractional bandwidth of 15%,use a line width about 0.2 mm for the I/O coupled stages. For both filters, themeasured attenuation levels at 2f0 are below −50 dB.

Parallel-coupled filters with unequal-coupled sections can also be designed formultispurious suppression since varying the coupling lengths of the coupled sectionscan finely tune attenuation poles or transmission zeros at 2f0 , 3f0 , and so on. Two


Figure 10.7 (a) Parallel-coupled microstrip line filter with unevenly coupled stages where theelectrical lengths of the I/O coupled stages are also different. All the dimensions shownare in millimeters on a 1.27-mm-thick substrate with a dielectric constant of 10.2.(b) Full-wave EM simulated performance of the filter.

experimental filters of this type have been reported in [3], showing the suppressionof the spurious response below −30 dB up to 4f0 .

10.2.2 Design Using Periodically Nonuniform Coupled Lines

A so-called ‘‘wiggly-line’’ filter is proposed in [7] to improve the spurious perfor-mance of a parallel-coupled-line microstrip bandpass filter. Using a continuousperturbation of the width of the coupled lines following a sinusoidal law, the waveimpedance is modulated, so that the harmonic passband of the filter is rejectedwhile the desired passband response is maintained virtually unaltered. This strip-


width perturbation does not require the filter parameters to be recalculated. Thus,the classical design methodology for coupled-line microstrip filters can still be used.At the same time, the fabrication of the resulting filter layout is no more difficultthan that of a typical coupled-line microstrip filter. To test this novel technique,a third-order Butterworth bandpass filter has been designed at 2.5 GHz with a10% fractional bandwidth and different values of the perturbation amplitude. Itis shown that for a 47.5% sinusoidal variation of the nominal strip width, aharmonic rejection below −40 dB is achieved in measurement, while the passbandat 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 followingmicrostrip filter design is considered:

Center frequency ( f0) 2.5 GHzFractional bandwidth (FBW ) 0.1Filtering response three-pole ButterworthSubstrate er = 10.2, thickness h = 1.27 mm

From Table 9.1, the lowpass prototype element values are g0 = g4 = 1.0,g1 = g3 = 1.0, and g2 = 2.0. In general, it is difficult to determine exact even- andodd-mode parameters of nonuniform coupled lines for the filter design. For narrow-to-moderate-bandwidth filter designs, a more convenient design approach is basedon the external quality factor (Qe ) and the coupling coefficient (M). Thus, for thefilter to be designed, the desired design parameters are

Qe1 = Qe3 =g0g1FBW

= 10 (10.3)

M12 = M23 =FBW

√g1g2= 0.07 (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 thefilter design. These are similar to the ‘‘wiggly-line’’ resonators [7]. Resonator 1 ofFigure 10.8(a) is composed of two trapezium-like structures, which have differentsizes as indicated. Resonator 2 of Figure 10.8(b) consists of two identical shapesthat are joined asymmetrically, where each shape is obtained by removing a trape-zium from a uniform line section. Resonators 1 and 2 can be seen as complementaryparts, and they are so designed for implementing couplings discussed later on.

For a synchronously tuned filter, as in this case, all the resonators resonate atthe same frequency (i.e., at the filter center frequency f0). The EM simulatedresonant characteristics of these two nonuniform resonators are plotted in Figure10.9. It is seen that both the resonators have the same fundamental resonantfrequency of 2.5 GHz, but different spurious resonant frequencies. Resonator 1exhibits an interesting antiresonant behavior around 5.7 GHz, which is 2.28 timesthe fundamental frequency. Resonator 2 has its first spurious resonant mode at


Figure 10.8 Nonuniform line resonators (not to scale) on a 1.27-mm-thick substrate with adielectric 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 separationof 1.3 GHz between the spurious frequencies of the two resonators. We will seelater on that this large separation is essential for the suppression of the spuriouspassband at 2f0 .

Once the nonuniform resonators for the filter design are determined, we canmove to the next stage in designing the filter. This involves the determination oftwo coupling structures. The first coupling structure is the input/output (I/O)coupling structure that provides an external quality factor given by (10.3). Thesecond coupling structure is the inter-resonator coupling structure that exhibits thedesired coupling given by (10.4). Figure 10.10(a) shows the determined I/O couplingstructure, where, for clarity, the dimensions of the nonuniform line resonator arenot shown, but can be found from Figure 10.8(a). The gap between the feed lineand the resonator is 0.55 mm, and changing this dimension results in a different


Figure 10.9 Resonant characteristics of the nonuniform line resonators in Figure 10.8.

Figure 10.10 (a) The I/O coupling structure (dimensions are in millimeters) on a 1.27-mm-thicksubstrate with a dielectric constant of 10.2. (b) Its transmission response.


value of Qe . To extract Qe , port 2 is weakly coupled, and the transmission responseis obtained by full-wave EM simulation, which is plotted in Figure 10.10(b). Thesame technique as described previously can then be used to find the Qe based onthe 3-dB bandwidth around 2.5 GHz. We note, from Figure 10.10(b), that thedesigned I/O coupling structure does not alter the spurious behavior of the non-uniform line resonator.

The inter-resonator coupling structure is shown in Figure 10.11(a), whereresonator 1 (on the top) is coupled to resonator 2 through a spacing S. For a givenL, the coupling varies with S. For the EM simulation, a two-port arrangement asshown is used, where the inter-resonator coupling structure is weakly excited bythe ports. The coupling coefficient is then extracted from the simulated mode splitresponse in Figure 10.11(b), using (10.2). It is found that for L = 11 mm andS = 0.55 mm, the desired coupling coefficient given by (10.4) is obtained. Thiscompletes the design of the filter.

The layout of the designed filter is illustrated in Figure 10.12(a). The designis verified by full-wave EM simulation, and the simulated performance is depictedin Figure 10.12(b). It is shown that there is no spurious passband at 2f0 , which issuperior to the conventional parallel coupled line filter.

Nevertheless, there are two undesired spikes in the upper stopband, one at4.36 GHz and the other at 5.76 GHz. A similar response was also observed in the

Figure 10.11 (a) The inter-resonator coupling structure on a 1.27-mm-thick substrate with adielectric constant of 10.2. (b) The mode split response when L = 11 mm andS = 0.55 mm. The dimensions of the nonuniform resonators are the same as thosegiven in Figure 10.8.


Figure 10.12 (a) ‘‘Wiggly-line’’ microstrip filter on a 1.27-mm-thick substrate with a dielectricconstant of 10.2. All the dimensions shown are in millimeters. (b) Full-wave EMsimulated performance of the filter.

filter reported in [7]. A wideband response of the coupled line resonators of Figure10.11(a) is plotted in Figure 10.13 along with the transmission response of thefilter. As can be seen, the coupled resonator section also produces two extra resonantpeaks, which coincide with the two spikes in the filter frequency response outsidethe passband. Recall that, from Figure 10.9, the first spurious resonance of reson-ator 1 is 5.7 GHz, while the first spurious resonant mode for resonator 2 occursat 4.4 GHz. Therefore, these two spurious frequencies are the undesired spikesobserved in the filter frequency response. The successful suppression of the spuriouspassband at 2f0 is due to the separation of the two spurious frequencies. Manyother forms of coupled-line resonators that have the some fundamental resonantfrequency but different spurious mode frequencies can be designed and used tosuppress unwanted spurious passbands in a coupled line filter.

The method has been extended to reject multiple spurious passbands byemploying different perturbation periods in each coupled-line section [8]. A seven-pole microstrip ‘‘wiggly-line’’ bandpass filter of this type, with a 10% fractionalbandwidth centered at f0 = 2.5 GHz, was fabricated to demonstrate the suppressions


Figure 10.13 Comparison between the wideband characteristic of the coupled line resonators andthe filter transmission response.

of spurious passbands at 2f0 , 3f0 , 4f0 , and 5f0 , resulting in a very wide upperstopband up to 14 GHz and with a rejection level of 30 dB.

10.2.3 Design Using Meandered Parallel-Coupled Lines

For conventional parallel coupled lines in an inhomogeneous medium such ascoupled microstrip lines, the even- and odd-mode phase velocities depend on theline width W and the spacing S as indicated in Figure 10.14(a). It has been foundthat by meandering the straight parallel coupled lines into a form shown in Figure10.14(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 odd-mode phase velocities can bemanipulated by changing the separation d.

Figure 10.14 (a) Conventional parallel coupled lines. (b) Meandered parallel coupled lines.


The meandered parallel coupled line structure of Figure 10.14(b) can be usedto suppress the spurious response in a microstrip parallel coupled filter. Figure10.15 depicts two modified I/O coupled stages of a microstrip line parallel coupledfilter. In Figure 10.15(a), the I/O resonator is meandered into the form as shown,which is a half-wavelength resonator with both ends open-circuited. The resonator

Figure 10.15 Modified I/O coupled stages using meandered parallel coupled lines having a linewidth W = 7 mil, a spacing S = 5 mil on a substrate with a dielectric constant of 10.2and a thickness of 50 mil. (a) For a separation of d = 107 mil. (b) For a separationof d = 47 mil. (c) Their frequency responses.


is fed at port 1 via a meandered coupled feed line, as indicated. The meanderedfeed line and a portion of the meandered resonator forms a meandered parallelcoupled line section. Port 2 is weakly coupled to the resonator in order to determineits resonant response. The EM-simulated frequency response of this I/O coupledstage is plotted in Figure 10.15(c). As can be seen, the I/O resonator resonates ata 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 conventionalmicrostrip parallel coupled filter. The harmonic frequency is not exactly twice ofthe fundamental frequency because of the dispersion of the resonant structure.From the response shown in Figure 10.15(c), we can also observe an attenuationpole or transmission zero at 1.87 GHz. This finite frequency transmission zeroexists inherently for a parallel-coupled line structure whose equivalent circuit isgiven in Figure 10.3. By changing the separation d of the meandered parallel coupledline section, the modal phase velocities are changed, resulting in the reallocation ofthe finite frequency transmission zero. To demonstrate this, Figure 10.15(b) showsthe other I/O coupled stage where d has been changed to 47 mil while the otherparameters are kept the same as those of Figure 10.15(a). Its frequency responseis also shown in Figure 10.15(c). It is evident that the frequency of finite transmissionzero has moved closer to the harmonic frequency, suppressing the harmonicresponse effectively. Therefore, the effect of tuning the separation of d is similarto that of tuning the coupling length of LC in the I/O coupled stage of Figure 10.1.

Based on this mechanism, a parallel coupled filter using meandered parallelcoupled lines can be designed to simultaneously suppress the spurious passbandand greatly reduce size. An example of this type of filter, following that reportedin [9], is illustrated in Figure 10.16. This is a three-pole bandpass filter with acenter frequency f0 = 1 GHz, and the filter is designed to suppress the spuriouspassband around 2 GHz. The substrate is chosen with a dielectric constant of 10.2and a thickness of 50 mil (1.27 mm). The main reason for choosing this relativehigh dielectric constant substrate is not only to miniaturize the filter, but also todemonstrate the challenge of the relative large difference between the even- andodd-mode phase velocities, which makes it difficult to suppress the spurious pass-band. All the dimensions shown in the filter layout are in mils. The filter is verycompact, occupying a circuit area of 625 × 700 mil (about 16 × 18 mm). The filteris symmetrical with two pairs of meandered parallel coupled line sections. For thepair of I/O coupled stages, the separation of meandered parallel coupled lines is27 mil. For the inner pair of inter-resonator couplings, the separation of meanderedparallel coupled lines is 22 mil. These two separations are critical parameters, andare tuned to achieve a better suppression of the spurious passband at 2f0 . Figure10.17 illustrates the EM-simulated filter performance. The filter shows a desiredpassband at 1 GHz, while the spurious passband at 2 GHz has been suppressedalmost below −40 dB. Experimental demonstrations of this type of filter can befound in [9].

Another type of meandered parallel coupled line filter is presented in [10]. Inthis type of filter, the meandered line design suppresses passband harmonics byusing numerous-bends and angles to equalize the phase velocities of the even andodd modes. An experimental filter of this type achieved a 40% size reduction and


Figure 10.16 Layout of the designed three-pole microstrip filter using meandered parallel coupledlines. The filter is on a substrate with a dielectric constant of 10.2 and a thicknessof 50 mil. All the dimensions shown are in mils.

Figure 10.17 EM-simulated performance of the filter of Figure 10.16.


70-dB rejection at the second harmonic without degrading the desired passbandperformance [10].

10.2.4 Design Using Defected Ground Structures

Defected ground structures can be used to equalize the even- and odd-mode phasevelocities in coupled microstrip lines [11, 12]. Figure 10.18 illustrates the crosssection of modified coupled microstrip lines with a ground aperture. The width ofthe aperture is denoted by Sa . The characteristics of the modified coupled microstriplines, that is, even- and odd-mode effective dielectric constants and characteristicimpedances, are affected by the presence of the slot.

Figure 10.19 shows the EM simulated results for modified coupled microstriplines on a 0.635-mm-thick substrate with a dielectric constant of 10. As can beseen from Figure 10.19(a), the even-mode effective dielectric constant is reducedby an increase in the aperture size, while there is little change in the odd mode.When Sa equals zero, which is the case for conventional coupled microstrip lines,the ratio of the even-mode effective dielectric constant to the odd-mode effectivedielectric constant, eree /ereo is 1.177. This ratio is reduced to about 1.0 whenSa = 1.6 mm. The reason why the even-mode phase velocity is speeded up fasteragainst the aperture size is that, as Sa increases, the capacitance between theconductor strip and ground is decreased more effectively for the even mode. Forthe same reason, the even-mode characteristic impedance is also increased predomi-nately as shown in Figure 10.19(b).

The coupled line structure in Figure 10.18 can be used as a basic buildingblock to design coupled-line microstrip filters with enhanced performance. Thefirst spurious passband of a filter can be suppressed efficiently by adjusting theaperture size so as to approximately equalize the even- and odd-mode phase veloci-ties. Several filters of this type have been demonstrated in [11]. As an additionaladvantage, the structure of Figure 10.18 provides tight coupling (in comparisonwith conventional coupled microstrip lines), thus relaxing the requirements onphysical dimensions W and S in those cases where tight coupling is necessary.

Figure 10.20 shows the cross section of another structure with a floatingconductor at the ground aperture. The floating conductor has a width of Wf. ForWf = 0, the structure degenerates to that of Figure 10.18. The structure withthe floating conductor makes filter designs more flexible. In this case, the modeimpedances depend mainly on width W, strip separation S, and floating conductorwidth Wf , while the modal phase-velocities depend mainly on W, S, and the

Figure 10.18 Cross section of modified coupled microstrip lines with a defected ground aperture.


Figure 10.19 Characteristics of modified coupled microstrip lines as a function of the groundaperture size. The dimensions of the coupled lines are W = 0.4 mm and S = 0.2 mmon a 0.635-mm-thick substrate with a dielectric constant of 10. (a) Effective dielectricconstants. (b) Characteristic impedances.

Figure 10.20 Cross section of modified coupled microstrip lines with a floating conductor at thedefected ground aperture.


aperture size Sa [12]. Therefore, for each coupled stage in the filter design, onecan match the even- and odd-mode phase velocities and simultaneously obtain thedesired modal-impedance values.

Following a similar design to that described in [12], Figure 10.21 shows thelayout of a three-pole parallel-coupled microstrip line filter implemented usingdefected ground structures with floating conductors. The filter is designed on adielectric 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 arein millimeters. It is noticeable from the bottom view of Figure 10.21(b) that, foreach coupled stage, there are three fragmental floating conductors separated bytwo 50-mm-wide metal bridges. The metal bridges joining ground-plane sides playan essential role in good filter performance because they cancel out undesiredground-plane slot modes. The exact position of the metal bridges is not veryimportant, and in this case, the original floating conductor under each coupledstage has been divided into three identical sections.

Figure 10.21 Layout of three-pole parallel-coupled microstrip line filter using defected groundstructures. The filter is on a substrate with a dielectric constant of 10 and a thicknessof 0.635 mm. All the dimensions shown are in millimeters. (a) Top view. (b) Bottomview.

10.3 Coupled-Line Filters Exhibiting Advanced Filtering Characteristics 327

This is a three-pole Chebyshev filter designed to have a fractional bandwidthof 20% at a center frequency f0 = 3 GHz. Figure 10.22 shows the performance ofthe filter, obtained by full-wave EM simulation. As can be seen, the spuriouspassband at 2f0 has been effectively suppressed by the technique based on defectedground structures. This has also been experimentally verified in [12]. In the simula-tion, the conductor was assumed to be 17-mm-thick copper and the loss tangentof the dielectric substrate was taken as 0.002. The cell size used for the simulationwas 50 mm by 50 mm to ensure accuracy, and 600-Mb memory was requiredusing Sonnet em [4]. This larger memory requirement was mainly due to thesimulation of the defected ground. The time for completing the simulation was3 hours on a Pentium 4 CPU of 3.2 GHz.

10.3 Coupled-Line Filters Exhibiting Advanced FilteringCharacteristics

10.3.1 Filters with Cross-Coupled Resonators

Conventional Chebyshev or Butterworth filters may also be referred to as direct-coupled filters on the basis that there is only direct coupling between adjacent orconsecutive resonators. When cross couplings are introduced among nonadjacentresonators, more advanced filtering characteristics such as quasi-elliptic functionand linear phase responses can be obtained, and these types of filters are referredto as cross-coupled filters. Figure 10.23 depicts coupling structures of some typicalcross-coupled filters, where S and L denote the source and load, respectively, theblack 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 theso-called canonical filter coupling structure, where the n direct-coupled resonatorsare folded into two arrays and there is cross coupling between the nonadjacentresonators 1 and n, 2 and n − 1 and so on. This type of filter is sometimes calleda folded filter. The coupling structure shown in Figure 10.23(b) is for the so-calledcascaded quadruplet (CQ) filter. A CQ filter consists of cascaded sections of four

Figure 10.22 EM-simulated performance of the filter of Figure 10.21.


Figure 10.23 Coupling structures of typical cross-coupled resonator filters. (a) Canonical.(b) Cascaded quadruplet (CQ). (c) Cascaded trisection (CT).

resonators, each with one cross coupling. Similar to the canonical filter, the crosscoupling can be arranged in such a way that a pair of attenuation poles or transmis-sion zeros are introduced at the finite frequencies to improve the selectivity, or itcan be arranged to result in a linear phase or group delay self-equalization. However,the tuning of a CQ filter is easier because the effect of each cross coupling isindependent, 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 sectionis comprised of three directly coupled resonators with cross coupling. This crosscoupling will only produce a single finite frequency transmission zero, which canbe allocated on either side of the passband depending on the implementation ofcoupling. There are many other coupling structures for cross-coupled filters. Forexample, a combination of CQ and CT coupling structures leads to a so-calledCQT filter [13].

Most applications of cross-coupled filters are as bandpass filters, in particularfor narrowband filters. To design cross-coupled filters, a practical approach basedon a general coupling matrix and external quality factors is often employed. Thegeneral coupling matrix for an n-coupled resonator filter has a form [6]


[m] = 3m11 m12 . . . m1n

m21 m22 . . . m2n

A A A Amn1 mn2 . . . mnn

4 (10.5)

which is an n × n reciprocal matrix (i.e., mij = mji ) and is allowed to have non-zero diagonal entries mii for an asynchronously tuned filter. Note that mij denotesthe so-called normalized coupling coefficient, and the required coupling coefficientfor a given fractional bandwidth FBW of a bandpass filter centered at f0 can beobtained from,

Mij = mij ? FBW (10.6)

The external quality factors, namely, Qe1 that represents the coupling betweenthe source and resonator 1 (the input resonator) and Qen that denotes the couplingbetween the load and resonator n (the output resonator), are defined as

Qei =qei

FBWfor i = 1, n (10.7)

where qei 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 ofthe two-port filter network can be computed from [6]

S21 = 21

√qe1 ? qen[A]−1

n1 (10.8)

S11 = 1 −2

qe1[A]−1

11

in which [A]−1ij denotes the ith row and jth column element of [A]−1 with

[A] = [q] + p[U] − j[m]

where

p = j1

FBW S ff0

−f0f D

[U] is the n × n unit or identity matrix and [q] is an n × n matrix with all entrieszero, except for q11 = 1/qe1 and qnn = 1/qen .

Cross-coupled resonator filters can be implemented with different forms ofmicrowave resonators. For planar filter realization, microstrip open-loop resonatorsprovide great flexibility for implementing a variety of cross coupling structures


[14]. As an example of this realization, a four-pole microstrip cross-coupled filteris designed based on a prescribed general coupling matrix:

[m] = 30 0.88317 0 −0.12355

0.88317 0 0.74907 00 0.74907 0 0.88317

−0.12355 0 0.88317 04

with the scaled external quality factor qe1 = qe4 = 0.94908. There is cross couplingbetween resonators 1 and 4 and the filter may be seen as a canonical or singlesection CQ filter. The filter is designed for a fractional bandwidth FBW = 0.06 ata center frequency f0 = 1.17 GHz. Thus, the desired design parameters can befound from (10.6) and (10.7):

M12 = M34 = 0.05299

M23 = 0.04494

M14 = −0.00741

Qe1 = Qe4 = 15.818

Using (10.8), the theoretical frequency responses of the filter are calculatedand plotted in Figure 10.24. As can be seen, the pair of finite frequency transmissionzeros near the passband improve the filter selectivity significantly. The microstripfilter is realized using the configuration of Figure 10.25 on a substrate with adielectric constant of 3.38 and a thickness of 0.508 mm, where the four microstripopen-loop resonators are numbered to indicate their sequence in the main couplingpath. Hence, resonators 1 and 4 are the input and output (I/O) resonators, respec-tively. Full-wave EM simulations are carried out to extract the desired external

Figure 10.24 Theoretical frequency responses obtained from the prescribed coupling matrix andexternal quality factors for a four-pole cross-coupled filter having a 6% fractionalbandwidth at 1.17 GHz.


Figure 10.25 Configuration of four-pole microstrip cross-coupled filter using open-loop resonators.

quality factors and coupling coefficients using the approach described in Section10.2.1.

Shown in Figure 10.26(a) is an arrangement to extract the external qualityfactor of the I/O resonator. The microstrip open-loop resonator has a line widthof 2 mm and a size of 22 mm × 22 mm on the substrate. The resonator is excitedat port 1 through a 50-ohm tapped line at a location indicated by t. Port 2 is veryweakly coupled to the resonator in order to find a 3-dB bandwidth of the magnituderesponse of S21 for extracting the external quality factor. When t = 0 where thereis a virtual grounding of the fundamental mode of the open-loop resonator, andthe coupling from the source (i.e., port 1) is the weakest, the largest external qualityfactor is obtained. Increasing t increases the coupling so that the extracted externalquality factor becomes smaller. A design curve can be obtained, as shown in Figure10.26(b), where one can determine the tapped-line location for the desired externalquality factor.

To determine the coupling between resonators 1 and 2 of the filter configurationin Figure 10.25, an arrangement such as that of Figure 10.27(a) is used. The twoopen-loop 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 isoften required due to the requirement of implementing other couplings for thefilter design, as we will see later on. For EM simulation, the coupled resonatorsare very weakly excited by the two ports as arranged. Two resonant peaks, whichresult from the mode split due to the coupling between the two resonators, canclearly be observed from the EM-simulated frequency responses. The couplingcoefficient can then be extracted using (10.2). Figure 10.27(b) shows the designcurve for M12 for d = 0.4 mm. It is obvious that the coupling decreases as thespacing S increases. It can also be shown that a small offset d has little effect onthe coupling. For design of the filter, the physical dimension S can readily bedetermined from the design curve for the desired M12 .

Figure 10.28(a) depicts an arrangement for the EM simulation to determinethe coupling between resonators 2 and 3 of the filter configuration in Figure 10.25.For the given orientation of the two coupled open-loop resonators, the couplingbetween them at the fundamental resonance is dominated by the magnetic fieldand therefore can be referred to as magnetic coupling [6]. Using the same approachdescribed above for extracting the inter-resonator coupling, a design curve for M23can be produced and is plotted in Figure 10.28(b).


Figure 10.26 (a) Arrangement for extracting external quality factor. All the dimensions are inmillimeters on a 0.508-mm-thick substrate with a dielectric constant of 3.38.(b) Design curve for Qe1 or Qe4.

To determine the cross coupling between resonators 1 and 4 of the filterconfiguration in Figure 10.25, we simulate using EM the arrangement of Figure10.29(a). The orientations of these two coupled open-loop resonators are oppositeto that of Figure 10.28(a). In this case, the coupling between them is dominatedby the electric field and hence this is referred to as electric coupling. This implemen-tation for the cross coupling is necessary because M14 and M23 have to be ofopposite signs in order to realize a pair of attenuation poles at finite frequencies.Figure 10.29(b) shows the design curve for M14 for the filter design.

From the design curves given earlier, all the physical dimensions associated withthe desired design parameters, namely, the external quality factors and couplingcoefficients, are readily determined. The layout with dimensions of the designedmicrostrip open-loop resonator filter is illustrated in Figure 10.30(a). Note that


Figure 10.27 (a) Arrangement for extracting coupling coefficient M12. (b) Design curve for M12.[The dimensions of the open-loop resonators are the same as those given in Figure10.26(a) on the same substrate.]

the dimensions are rounded off to have a resolution of 0.1 mm. Since the designedfilter needs to be synchronously tuned, the open gap of the I/O resonator is slightlytuned to compensate for the frequency shift due to the tapped line I/O arrangement.Figure 10.30(b) shows the EM-simulated performance of the filter, where the twofinite transmission zeros as expected can clearly be seen. The slight asymmetricfrequency response may be attributed to unwanted coupling as well as frequency-dependent coupling.

Miniature Cross-Coupled Filter

Refer to the filter topology of Figure 10.30(a). Since there is a virtual groundingin the middle of each open-loop resonator at its fundamental resonance, it is feasible


Figure 10.28 (a) Arrangement for determination of coupling coefficient M23. (b) Design curve forM23. [The dimensions of the open-loop resonators are the same as those given inFigure 10.26(a) on the same substrate.]

to introduce a physical via ground in that location without appreciably affectingthe primary passband. This, however, leads to a new topology of a miniature cross-coupled 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 open-loop resonator. Therefore, the new filter topologyrequires a circuit size which only amounts to half of that of Figure 10.30(a).

The miniature cross-coupled filter can be designed using an approach similarto that used for the design of the open-loop resonator filter described earlier. Figure10.32(a) illustrates the layout of a designed miniature cross-coupled filter on asubstrate with a dielectric constant of 3.38 and a thickness of 0.508 mm. The


Figure 10.29 (a) Arrangement for extracting coupling coefficient M14. (b) Design curve for M14.[The dimensions of the open-loop resonators are the same as those given in Figure10.26(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 ofthe open-loop resonator filter in Figure 10.30. Moreover, the miniature cross-coupled filter exhibits a better upper stopband, which is demonstrated in Figure10.33. It shows that the first spurious only appears at about 3f0 , resulting in awide and better upper stopband. This is because the resonators used are only aquarter-wave long.

10.3.2 Filters with Source-Load Coupling

Traditionally, coupled resonator filters have focused on topologies where there isno direct source to load coupling. This is the case for the filter designs described


Figure 10.30 (a) Layout of the designed cross-coupled filter on a substrate with a dielectric constantof 3.38 and a thickness of 0.508 mm. All the dimensions are in millimeters.(b) EM-simulated performance of the filter.

in the previous sections. It is well known that such a topology can produce at mostn − 2 finite transmission zeros out of n resonators, where n is the degree of thefilter. The addition of a direct signal path between the source and the load allowsthe generation of n finite frequency transmission zeros instead of n − 2. Filterswith source-load coupling have attracted a lot of attention recently [15–19].


Figure 10.31 Configuration of a miniature cross-coupled filter.

Figure 10.34 shows the so-called n + 2 or ‘‘extended’’ general coupling matrixfor nth-degree coupled resonator filters. The n + 2 coupling matrix has an extrapair of rows at the top and bottom and an extra pair of columns at the left andright surrounding the ‘‘core’’ coupling matrix. The n + 2 coupling matrix is an(n + 2) × (n + 2) reciprocal matrix (i.e., mij = mji ). Nonzero diagonal entries mijare allowed for an asynchronously tuned filter.

For a given filter topology, the n + 2 coupling matrix may be obtained usingsynthesis methods described in [15–18]. Once the coupling matrix [m] is deter-mined, the filter frequency response can be computed in terms of scattering parame-ters as follows:

S21 = −2j [A]−1n + 2, 1 (10.9)

S11 = 1 + 2j [A]−111

where [A]−1ij denotes the ith row and jth column element of [A]−1. The matrix [A]

is given by

[A] = [m] + V[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 zerosexcept for [q]11 = [q]n + 2, n + 2 = 1, and V is the frequency variable of the lowpassprototype. The lowpass prototype response can be transformed to a bandpassresponse having a fractional bandwidth FBW at a center frequency f0 using thewell-known frequency transformation:

V =1

FBW S ff0

−f0f D (10.11)

Filter Example I

Consider a four-pole (n = 4) cross-coupled resonator filter that has a couplingstructure as shown in Figure 10.35. Each numbered node represents a resonator


Figure 10.32 (a) Layout of the designed miniature cross-coupled filter on a substrate with adielectric constant of 3.38 and a thickness of 0.508 mm. All the dimensions are inmillimeters. (b) EM-simulated 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. Thecoupling between the source and load is also indicated by the broken line betweenthe two nodes.

The filtering characteristic of the filter under consideration is prescribed withan n + 2 general coupling matrix given by


Figure 10.33 Wideband performance of the miniature cross-coupled filter.

Figure 10.34 n + 2 general coupling matrix [m].

Figure 10.35 Coupling structure of a four-pole folded coupled-resonator filter with source-loadcoupling.


[m] = 30 1.0089 0 0 0 mSL

1.0089 0 0.8514 0 −0.1436 00 0.8514 0 0.7380 0 00 0 0.7380 0 0.8514 00 −0.1436 0 0.8514 0 1.0089

mSL 0 0 0 1.0089 0

4(10.12)

The filter possesses a cross coupling m14 = −0.1436 between resonators 1 and4, which produces a single pair of finite frequency transmission zeros. It also allowsa direct coupling between the source and load, denoted by mSL . The effect ofintroducing mSL on the filtering characteristic can be seen by varying its value. Forexample, we consider three cases corresponding to mSL = 0.0, 0.00026, and 0.0028,respectively. The filter is assumed to have a fractional bandwidth FBW of 0.04,centered at f0 = 2.4 GHz. The computed filter response, obtained by using (10.9)–(10.11), is plotted in Figure 10.36. For mSL = 0.0, the filter only shows two finite-frequency transmission zeros, close to the passband. This pair of transmission zerosresults from the cross coupling between resonators 1 and 4, as mentioned before.For mSL = 0.0028, two extra finite frequency transmission zeros are observed. Thisfour-pole filter has a total of four finite-frequency transmission zeros. For the casemSL = 0.00026, the filter characteristic hardly changes from that for mSL = 0.0.This is because the source-load coupling is too small in this case. Another interesting

Figure 10.36 Frequency responses of a four-pole cross coupled resonator filter allowing a directsource-load coupling.


observation in Figure 10.36 is that the return loss (i.e., |S11 | ) response is almostthe same for the three different values of mSL . This indicates that varying thesource-load coupling can effectively adjust the position of the two extra finitetransmission zeros with negligible effect on the passband. It is suggested that onecan design the symmetric folded coupled-resonator filter first and then add somesource-load coupling. This would make the filter implementation easier.

The parameters mSi and mLi , which represent the input and output couplingsfrom the source and load to resonator i, can be converted to external quality factorsby

Qe, Si =1

mSi ? FBW(10.13)

Qe, Li =1

mLi ? FBW

where FBW is the fractional bandwidth of bandpass filter.To determine the direct source-load coupling mSL , a useful formulation can

be derived as follows. Consider an I/O structure that only involves the source-loadcoupling. From Figure 10.34, the general coupling matrix for the structure underconsideration would have a form

[m] = F 0 mSL

mSL 0 Gwhich corresponds to a zero-order filter case for n = 0. From (10.10), we have

[A] = F 0 mSL

mSL 0 G + VF0 0

0 0G − jF1 0

0 1G= F −j mSL

mSL −j GAn analytical solution for [A]−1 can be found to be

[A]−1 =F j mSL

mSL j G1 + m2

SL

Using (10.9), we obtain

S21 = −2jmSL

1 + m2SL

or


|S21 | = ±2mSL

1 + m2SL

The negative sign is for mSL < 0. Solving the above equation for mSL yields

mSL = ±1 − √1 − |S21 |2

|S21 | (10.14)

which is a valid solution by taking into account that mSL = 0 for |S21 | = 0 and0 ≤ |S21 | ≤ 1. The choice of a sign is rather relative, which would depend on thesign of other elements in the n + 2 general coupling matrix. In any case, it caneasily 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 characteristicprescribed by the coupling matrix given in (10.12) with a source-load couplingmSL = 0.0028. The filter is to be implemented using open-loop microstrip resonatorson a 20-mil-thick dielectric substrate with a dielectric constant of 3.38. Figure10.37(a) is the physical structure designed to implement the required source-loadcoupling, where all the dimensions are given in mils. The two input and output(I/O) feed lines are coupled through a 90-mil-wide gap and a short coupling lineof 160-mil length. While the 90-mil-wide gap is kept constant for the filter design,the source-load coupling can easily be controlled by the coupling length and spacingof the short coupling line with respect to the two feed lines. Figure 10.37(b) plotsthe EM simulated results of |S21 | for this I/O coupling structure with the dimensionsshown. One can see that, in general, the source-load coupling is frequency-depend-ent. For a narrowband approximation, we can use the EM simulated result at thecenter frequency to determine mSL , based on (10.14). Thus, from Figure 10.37(b)it can be found that |S21 | = 0.0056 at the center frequency of 2.4 GHz. Using(10.14) the extracted mSL is equal to 0.0028, which is the desired value. In thisway, the dimensions of the structure for the source-load coupling are determined.The remainder of the filter design can be completed by determining the desiredexternal quality factors and the inter-resonator couplings following the methoddescribed previously. The layout of the design microstrip filter with source-loadcoupling is shown in Figure 10.38(a), and its performance, obtained by full-waveEM simulations, is presented in Figure 10.38(b). As can be seen, this four-polemicrostrip filter exhibits four finite-frequency transmission zeros, two of which aredue to the direct source to load coupling. The frequency response of the filter showsan asymmetrical behavior, which is likely caused by unwanted and/or frequency-dependent couplings. A diagnosis method may be used to identify the unwantedeffects 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


Figure 10.37 (a) An I/O coupling structure to implement the desired source-load coupling. All thedimensions are in mils on a 20-mil-thick 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 resonatorseach. As can be seen, the source is coupled not only to resonator 1, but also toresonator 2 as indicated by the broken line. Similarly, the load is coupled not onlyto resonator 3, which is usually the output resonator, but also to resonator 2. Thecoupling structure is basically comprised of two trisections, each of which is ableto produce a finite-frequency transmission zero.

For the filter, the n + 2 coupling matrix is given by


Figure 10.38 (a) Layout of the designed four-pole cross-coupled filter with source-load coupling.All the dimensions are in mils on a 20-mil-thick substrate with a dielectric constantof 3.38. (b) EM simulated filter performance.

[m] = 30 1.0103 0.4275 0 0

1.0103 −0.7811 0.8549 0 00.4275 0.8549 0.4562 1.0258 0.2334

0 0 1.0258 −0.3857 1.10580 0 0.2334 1.1058 0

4 (10.15)

It is clear from (10.15) that the filter is asynchronously tuned as there arenonzero diagonal elements. The couplings from the source to resonators 1 and 2


Figure 10.39 Coupling structure of a three-pole coupled-resonator filter with the source and theload being coupled to two resonators each.

are defined as mS1 = m1S = 1.0103 and mS2 = m2S = 0.4275, respectively. Similarly,the couplings from resonators 2 and 3 to the load can be identified to be m2L =mL2 = 0.2334 and m3L = mL3 = 1.1058, respectively. The center frequency andfractional bandwidth of the filter are chosen to be f0 = 5 GHz and FBW = 0.05.The theoretical response of the filter can be computed using (10.9) and the resultsare plotted in Figure 10.40. As expected, two finite-frequency transmission zerosoccur in the upper stopband, which improves the selectivity on the high side ofthe passband.

To implement the n + 2 coupling matrix of (10.15), a microstrip realization isproposed in [20]. The filter is built on a dielectric substrate with a dielectric constantof 3.58 and a thickness = 20 mils. Figure 10.41(a) shows the layout of the designedfilter, where all the dimensions are in mil. The filter is modified from the conven-tional microstrip parallel-coupled filter by vertically flipping feed lines of the sourceand the load. Furthermore, two small coupling/shielding lines are added at theends of the input and output feed lines, allowing the source and the load to becoupled to two resonators each. From Figure 10.41(a), it can be identified that the

Figure 10.40 Theoretical responses of the three-pole filter with two finite-frequency transmissionzeros.


Figure 10.41 (a) Three-pole microstrip parallel-coupled filter with the source and the load beingcoupled to two resonators each. The dimensions are in mils on a dielectric substratewith a dielectric constant of 3.58 and a thickness = 20 mils. (b) EM simulated filterperformance.

coupling/shielding line at the end of input (source) feed line has a length = 87 miland a width = 8 mil. It couples to resonator 2 through an 8-mil gap to facilitatethe required external quality factor

Qe, S2 =1

mS2 × FBW=

10.4275 × 0.05

= 46.78

according to (10.13). Similarly, the coupling/shield line, which is 39 mils long and8 mils wide, at the end of output (load) feed line, is coupled to resonator 2 via a7-mil gap to realize the desired external quality factor

Qe, L2 =1

mL2 × FBW=

10.2334 × 0.05

= 85.69


These design parameters can be extracted by EM simulations. The EM simulatedperformance of the filter is depicted in Figure 10.41(b), showing good agreementwith theory.

10.3.3 Filters with Asymmetric Port Excitations

For some filter topologies, it is possible to feed the filters either symmetrically orasymmetrically. Filters designed with an asymmetric port excitation can result ina significantly different frequency response, in particular, in the stopband [21, 22].For example, Figure 10.42 illustrates a pair of two-pole microstrip pseudo-comblinebandpass filters [6], one with symmetrical port excitation [Figure 10.42(a)] andthe other with asymmetrical port excitation [Figure 10.42(b)]. Both the filters usetapped-line input and output (I/O) arrangements. The asymmetric port design isobtained by vertically flipping the output resonator associated with the symmetricdesign. The filter dimensions shown are in millimeters on a dielectric substratewith a thickness of 1.27 mm and a dielectric constant of 10.8. The resonators arehalf-wavelength long at a fundamental resonant frequency of about 2 GHz.

Figure 10.43 plots the EM simulated performance of the two filters. As canbe seen, the filter with symmetric ports has a finite frequency transmission zeroon the lower side of the passband, whereas the filter with asymmetric ports possessesa finite frequency zero on the upper side of the passband. The allocation of finite

Figure 10.42 Two-pole microstrip pseudo-combline bandpass filters with tapped-line I/O on a1.27-mm-thick substrate with a dielectric constant of 10.8. All the dimensions arein millimeters. (a) Symmetric ports. (b) Asymmetric ports.


Figure 10.43 Frequency responses of the two-pole filter structures in Figure 10.42. Circle: symmet-ric ports. Triangle: asymmetric ports.

frequency transmission zero in opposite directions with respect to the passbandleads to different stopband behavior. It can be shown that each finite transmissionzero can be tuned by adjusting the tapped point and the spacing between the tworesonators, 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 externalquality factor and the spacing is decided by the required inter-resonator coupling.This implies that this type of finite transmission zero cannot be generally indepen-dently controlled.

As an example, Figure 10.44 shows two five-pole microstrip pseudo-comblinebandpass filters, with symmetric port excitations and asymmetric port excitations,respectively. The filters are designed on a 1.27-mm-thick substrate with a dielectricconstant of 10.8. The filter structure in Figure 10.44(a) is symmetric with respectto resonator 3 in the middle. If we flip the last two resonators of this symmetricfilter vertically, we can obtain the filter with asymmetric port excitations of Figure10.44(b). Except for this flipping, all the physical dimensions for the two filtersare kept the same as shown. With this straightforward modification, the two filtersshow similar passband performances, but different upper stopband responses, asillustrated in Figure 10.45. Both the filters exhibit a finite transmission zero at2.2 GHz. This finite transmission zero is inherent due to the topology of parallel-coupled resonators, and is independent of the feed scheme. However, the filterwith asymmetric ports shows an extra finite transmission zero near 2.5 GHz,which evidently improves the upper stopband response. The improved asymmetricfrequency response is desired for some applications where high selectivity isrequired.

Coupled-line I/O arrangements of symmetric and asymmetric port excitationscan also show different frequency characteristics. For example, Figure 10.46 depictstwo two-pole microstrip pseudo-combline bandpass filters with coupled-line I/Oon a 1.27 mm thick substrate with a dielectric constant of 10.8. Again, the filterwith asymmetric ports of Figure 10.46(b) is simply obtained by vertically flipping


Figure 10.44 Five-pole microstrip pseudocombline bandpass filters on a 1.27-mm-thick substratewith a dielectric constant of 10.8. All the dimensions are in millimeters. (a) Symmetricport excitations. (b) Asymmetric port excitations.

Figure 10.45 Frequency responses of five-pole microstrip pseudocombline bandpass filters inFigure 10.44. Thick gray line: symmetric ports. Thin black line: asymmetric ports.


Figure 10.46 Two-pole microstrip pseudocombline bandpass filters with coupled-line I/O on a1.27-mm-thick substrate with a dielectric constant of 10.8. All the dimensions arein millimeters. (a) Symmetric ports. (b) Asymmetric ports.

the output resonator associated with the symmetric design in Figure 10.46(a). TheEM-simulated performance of these two filters are plotted together in Figure 10.47for comparison. Although each filter exhibits a finite transmission zero, the alloca-tion is completely opposite with respect to the passband. While the finite frequency

Figure 10.47 Frequency responses of the two-pole filter structures in Figure 10.46. Circle: symmet-ric ports. Triangle: asymmetric ports.


transmission zero for the filter with symmetric ports is on the lower side of thepassband, the finite frequency transmission zero for the filter with asymmetric portexcitations is located on the upper side of the passband. It is also interesting tocompare the frequency responses of Figure 10.47 with that of Figure 10.43. It isapparent that coupled-line I/O arrangements can lead to a better stopband on thelower side of the passband than the tapped-line I/O arrangement.

Other filter topologies such as open-loop resonator filters can also be fedsymmetrically or asymmetrically. Figure 10.48(a) is the I/O arrangement of thefour-pole cross-coupled open-loop resonator filter discussed in Section 10.3.1,which is a symmetric feed scheme. The corresponding asymmetric feed scheme isdepicted in Figure 10.48(b). Figure 10.49 shows the frequency response of these

Figure 10.48 Open-loop resonator filter feed structures. (a) Symmetric feed scheme. (b) Asymmet-ric feed scheme.


Figure 10.49 Frequency responses of the filter feed structures in Figure 10.48. Circle: symmetricports. Triangle: asymmetric ports.

two structures. It is clearly seen that the asymmetric feed structure produces twofinite frequency transmission zeros whereas the symmetric feed structure producesnone.

We can replace the feed structure of the four-pole cross-coupled open-loopresonator filter of Figure 10.30(a) with the asymmetric one of Figure 10.48(b),resulting in a new four-pole arrangement, as shown in Figure 10.50(a). Its perfor-mance is demonstrated in Figure 10.50(b), where two pairs of finite transmissionzeros are observed. Compared with the performance shown in Figure 10.30(b) fora similar filter with symmetric port excitations, an extra pair of finite transmissionzeros is obtained from the arrangement of asymmetric ports. It is evident thatthis extra pair of finite transmission zeros improves the stopband performancesignificantly.

10.4 Interdigital Filters Using Stepped Impedance Resonators

The traditional interdigital filter has been described in Chapter 9. Its main advantageis the use of l/4 resonators with alternate short-circuit and open-circuit ends,leading to a compact size when compared to other parallel-coupled filter designs.The use of stepped impedance resonators (SIR) [23–27] further reduces the overallfootprint of the conventional design. In addition, a wider upper stopband can beobtained due to the frequency dispersion of stepped impedance resonators. Figure10.51 shows a grounded l /4 stepped impedance resonator, where Z1 and Z2 , u1and u2 are the characteristic impedances and electric lengths of lines 1 and 2,respectively. It assumes that the characteristic impedance of line 1 is higher thanthat of line 2 (i.e., Z1 > Z2). Define an impedance ratio

R =Z2Z1

(10.16)

10.4 Interdigital Filters Using Stepped Impedance Resonators 353

Figure 10.50 (a) Four-pole cross-coupled open-loop resonator filter with asymmetric portexcitations on a 1.27-mm-thick substrate with a dielectric constant of 10.2.(b) EM-simulated performance.

Figure 10.51 Grounded l/4 stepped impedance resonator (SIR).

The maximum ratio of the first spurious resonance ( fs1) and the fundamentalresonance ( f0), when u1 = u2 , can be estimated by [24],

fs1f0

=p

tan−1√R− 1 (10.17)


When R = 1, fs1 = 3f0 , this yields the case of a traditional grounded l /4 uniformimpedance 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 theSIR 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 isbased on external quality factors and coupling coefficients that are directly obtain-able from a lowpass prototype, can be used. As an example, a three-pole SIRinterdigital filter design with FBW = 0.1 at f0 = 1 GHz is now considered.

For a Chebyshev lowpass prototype with 0.1-dB ripple, the lowpass elementvalues can be found as follows from Table 9.2:

g0 = g4 = 1.0

g1 = g3 = 1.0315

g2 = 1.1474

The external quality factors and coupling coefficients are calculated by

Qe1 = Qe3 =g0g1FBW

= 10.315

M12 = M23 =FBW

√g1g2= 0.0919

The filter is to be implemented on a 1.27-mm-thick substrate with a dielectricconstant of 6.15. The first spurious response is required to be above 4f0 . 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 discontinu-ity at the junction of the two lines. More accurate design of the SIR can be obtainedby using full-wave EM simulation. Figure 10.52(a) is a structure created for EMmodeling, showing a microstrip SIR with its one end grounded through the dielectricsubstrate. The designed microstrip SIR has dimensions (in millimeters) as shownin Figure 10.52(b), where the low-impedance (Z2) line has a width of 2.6 mm andthe high-impedance (Z1) line has a width of 0.4 mm on the given substrate, whichyields an impedance ratio R ≈ 0.42. The EM simulated frequency response of theSIR is plotted in Figure 10.52(c). The SIR has its fundamental resonance at 1 GHzand its first spurious resonance at 4.4 GHz, resulting in fs1 /f0 = 4.4. Thus, we canexpect that the first spurious passband of the filter will occur after 4f0 .

The next step in the filter design is to characterize the input/output (I/O)and inter-resonator coupling structures. These can be done using EM simulationsdescribed previously, and (10.1) and (10.2) can be used to extract the externalquality factor and the inter-resonator coupling, respectively. Figure 10.53(a) showsthe I/O coupling stage using a tapped line for the filter design. The tapped line has


Figure 10.52 (a) A structure of grounded microstrip SIR for EM modeling. (b) The layout of thedesigned microstrip SIR with dimensions in millimeters on a dielectric substrate witha dielectric constant of 6.15 and a thickness of 1.27 mm. (c) EM simulated resonantfrequency responses of the designed SIR.

a width of 1.9 mm on the substrate with a characteristic impedance matching theterminal impedance. The external quality factor Qe depends on the tapping positionof t, which is shown in Figure 10.53(b). From this design curve, we can determinea tapping position that yields the desired Qe of 10.315. For the inter-resonatorcoupling structure shown in Figure 10.54(a), the two coupled stepped impedanceresonators have a fixed offset d = 0.2 mm. The design curve for the inter-resonatorcoupling is given in Figure 10.54(b), where one can determine s for the desiredcoupling coefficient of 0.0919.


Figure 10.53 (a) The I/O stage for the SIR filter design. (b) Design curve (Qe ).

The final design of the filter is shown in Figure 10.55(a), where all the dimen-sions are in millimeters. The length of the low-impedance line for the input andoutput resonators is 13.8 mm instead of 12.7 mm to compensate for the frequencyshifting due to the effect of the tapped I/O lines. The performance of the filter isillustrated in Figure 10.55(b). As can be seen, the desired passband is centered at1 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 thecorrect coupling values required in wideband applications [27].


Figure 10.54 (a) Inter-resonator coupling structure for the SIR filter design. (b) Design curve forthe inter-resonator coupling.

A practical approach has been suggested in [27] for the design of wide band-width interdigital filters using SIR. The proposed design approach is based on theconcept of parameter mapping between two wideband filters, that is, a conventionalinterdigital filter with uniform impedance resonators (UIR) and a desired inter-digital filter using stepped impedance resonators. The design of a wideband conven-tional interdigital filter with uniform impedance resonator (UIR) is available asdiscussed in Chapter 9.

The nine-pole wideband SIR interdigital filter is designed at a center frequencyof 1.5 GHz and a fractional bandwidth of 33.3%, which are the same as those ofthe conventional interdigital filter design using uniform impedance resonator (UIR)presented in Figure 9.12(a). The SIR filter is also implemented on the same dielectricsubstrate 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 designparameters from the UIR interdigital filter design. For this purpose, the circuitlayout shown in Figure 9.12(a) is broken down into smaller parts. The simulation


Figure 10.55 (a) Layout of the designed narrowband microstrip SIR interdigital filter on a dielectricsubstrate with a dielectric constant of 6.15 and a thickness of 1.27 mm. All dimensionsshown are in millimeters. (b) EM-simulated performance of the filter.

10.5 Dual-Band Filters 359

results on smaller parts can be used to obtain a new set of design parameters thatcan be applied to the SIR design. For example, the I/O stage of the UIR interdigitalfilter in Figure 9.12(a) is simulated alone with an arrangement shown in Figure10.56(a), where the circle with small triangles at one end of the I/O resonatorrepresents the via hole grounding and port 2 is weakly coupled to the I/O stage inorder to extract an external quality factor (Qe ) through the simulated two-porttransmission (i.e., S21) response. Port 2 should be weakly coupled so that its effecton Qe is negligible. The full line in Figure 10.56(c) indicates the EM-simulatedfrequency response of the I/O stage of the UIR interdigital filter, where the frequencyaxis 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 respectto the resonant frequency. This is mainly due to frequency-dependent coupling ofthe 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 factormay simply be extracted using

Qe =1

D fN_3 dB

where D fN_3 dB is the normalized 3-dB bandwidth. In the case of Figure 10.56(c),Qe = 2.16 is extracted for the I/O stage of the UIR interdigital filter, which willbe used as a new design parameter for the SIR filter. In practice, the I/O stage ofthe 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 ofadjacent 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 apair of adjacent resonators of the UIR interdigital filter to extract a new couplingcoefficient. As an example, the full line shown in Figure 10.57(c) indicates the EMsimulated S21 response for the inter-resonator coupling between resonators 2 and3 of the designed UIR interdigital filter. Using this curve, a coupling coefficient ofM23 = 0.198 is found using (10.2). Thus, to determine the spacing between resona-tors 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 ofthe SIR interdigital filter can be determined. The designed nine-pole SIR interdigitalfilter, 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 widebandresponse of the SIR interdigital filter is plotted in Figure 10.59 along with that ofthe UIR interdigital filter. It is evident that the desired passbands of the two filtersmatch closely with each other. The SIR interdigital filter exhibits a better upperstopband with its first spurious response occurring beyond 3f0 .

10.5 Dual-Band Filters

There is an increasing demand for dual-band communication systems [e.g., a wire-less local area network (WLAN) device operated at 802.11a (5 GHz) and 802.11b


Figure 10.56 (a) The I/O stage of the UIR interdigital filter. (b) The I/O stage of the SIR interdigitalfilter. (c) EM-simulated frequency responses of the I/O stages. (Full line: UIR. Brokenline: SIR.)


Figure 10.57 (a) Inter-resonator coupling stage of the UIR interdigital filter. (b) Inter-resonatorcoupling stage of the SIR interdigital filter. (c) EM-simulated frequency responses ofthe inter-resonator coupling stages. (Full line: UIR. Broken line: SIR.)


Figure 10.58 (a) Layout of the nine-pole microstrip SIR interdigital filter on a dielectric substratewith a dielectric constant of 6.15 and a thickness of 1.27 mm. (b) Measured passbandperformance.

(2.4 GHz)]. Dual band systems can be divided into two categories. The first involvescombining two independent bandpass filters with a switch used to choose one signalpath at a time; the second category provides dual-band operation concurrently.Traditionally, simultaneous operation at different frequency bands can be achieved


Figure 10.59 Comparison of wideband responses of the designed nine-pole microstrip SIR andUIR interdigital filters.

by combining multiple independent signal paths. This approach needs a largefootprint. Recently, a concurrent dual-band receiver architecture was proposed toovercome these issues [28]. To obtain a concurrent dual-band system, one of theimportant devices required is a dual-band filter.

In general, a dual band filter can be achieved by various approaches. Oneapproach is combining two independent bandpass filters with the common input/output ports [29]. Although the specification of the two passbands can be easilymet 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 widelyseparated, but the size can still be large due to the cascading of the two differentfiltering structures. A more interesting approach is using stepped impedance resona-tors to design a single coupled-line filter that has two desired passbands [31–34].The basic idea is to make use of the first-harmonic frequency of the steppedimpedance resonator SIR, which is easily adjustable. This will be discussed ingreater detail next.

Figure 10.60 illustrates a pair of planar stepped impedance resonators, whereT-T ′ denotes the symmetrical plane. Z1 and Z2 are the characteristic impedancesof the two transmission lines having widths of W1 and W2 , respectively. Definean impedance ratio R = Z2 /Z1 . Let f1 and f2 be the frequencies of first two resonantmodes of the SIR. The first resonant mode at f1 can be seen as an odd modebecause there is a virtual short circuit in the middle of resonator. On the otherhand, the second resonant mode at f2 is an even mode as there is a virtual opencircuit in the middle of the resonator. For the SIR in Figure 10.60(a), it can beshown that the frequency ratio f2 / f1 > 2 because R < 1. Similarly, it can be shownfor the SIR in Figure 10.60(b) that 1 < f2 / f1 < 2 since R > 1. In a practical designof a dual-band filter, f1 will be designed to be the mid-band frequency of the lowerpassband, while f2 will be the mid-band frequency of the higher passband. Thus,


Figure 10.60 (a) SIR for R < 1 and f2 /f1 > 2. (b) SIR for R > 1 and 1 < f2 /f1 < 2.

if the mid-band frequency of the higher frequency filter is greater than twice thatof the lower frequency filter, the SIR of Figure 10.60(a) should be used for thefilter design. On the contrary, if the mid-band frequency of the higher frequencyfilter is smaller than twice that of the lower frequency filter, the SIR of Figure10.60(b) should be used.

The design of SIR usually requires EM simulations for more accurate results.For example, Figure 10.61 shows EM-simulated results of the microstrip SIR ofFigure 10.60(a) on a dielectric substrate with a dielectric constant of 2.2 and athickness of 0.508 mm. The first resonant frequency f1 and the frequency ratio off2 / f1 are given as a function of W2 for W1 = 0.5 mm and l1 = l2 = 10 mm inFigure 10.61(a). It is observed from the plot that when W2 is increased, f1 decreaseswhereas the ratio f2 / f1 increases and both vary quite linearly with respect to W2 .In Figure 10.61(b), f1 and f2 / f1 are plotted as a function of l2 while l1 + l2 is keptconstant (20 mm) and W1 = 0.5 mm and W2 = 1.5 mm. It is interesting to notethat while the ratio of f2 / f1 increases with an increase in l2 , the first resonantfrequency f1 remains nearly unaffected. This implies that f1 is less dependent onthe ratio of l2 /l1 . To design a dual-band filter with the SIR, more physical designparameters are needed than to design a single-band filter. In this case, the ratiosW2 /W1 and l2 /l1 are often used as design parameters, not just for determining f1and f2 , but also for other considerations such as to achieve a wider upper stopbandafter the second passband as well as to facilitate desired couplings.

To demonstrate the characteristics of the SIR of Figure 10.60(b), a microstripSIR with R > 1 on a substrate with a dielectric constant of 2.2 and a thickness of0.508 mm is simulated. The EM-simulated results are given in Figure 10.62, where


Figure 10.61 EM-simulated results of the first resonant frequency f1 and the frequency ratio off2 /f1 for a SIR with R < 1 in the form of Figure 10.60(a) on a substrate with a dielectricconstant of 2.2 and a thickness of 0.508 mm. (a) As a function of W2. (b) As afunction of l2.

the first resonant frequency f1 and the frequency ratio of f2 / f1 are plotted as afunction of W1 for W2 = 0.5 mm and l1 = l2 = 10 mm. It is clear that when W1becomes wider, the characteristic impedance Z1 of the microstrip line becomeslower. Thus, the impedance ratio R increases, which leads to a decrease of theratio of f2 / f1 as expected. Figure 10.62(a) also shows that f1 increases as W1 isincreased. Both f1 and f2 / f1 can also be controlled by adjusting l2 as shown inFigure 10.62(b).

To design dual-band filters using the SIR, there are two sets of external qualityfactors and inter-resonator coupling coefficients that need to be realized for thetwo desired passbands. In general, to implement the external quality factors for


Figure 10.62 EM-simulated results of the first resonant frequency f1 and the frequency ratio off2 /f1 for a SIR with R > 1 in the form of Figure 10.60(b) on a substrate with a dielectricconstant of 2.2 and a thickness of 0.508 mm. (a) As a function of W1. (b) As afunction of l2.

the two designed passbands, two or more physical design parameters are neededat the I/O stages. Similarly, the inter-resonator couplings for the two passbandsneed to be controlled by at least two physical parameters between adjacent coupledresonators. Some design examples of this type of filter are reported in [32–34].Figure 10.63(a) illustrates a 5-pole microstrip dual band filter design, which usesfive stepped impedance resonators on a substrate with a dielectric constant of 2.2and a thickness of 0.508 mm. The SIR used has an antisymmetrical shape thatmakes the implementation of inter-resonator coupling easy. The I/O stages usetapped lines, which also include the dual-frequency transformer. The design details


Figure 10.63 (a) Photograph of a five-pole microstrip dual-band filter using the SIR on a substratewith a dielectric constant of 2.2 and a thickness of 0.508 mm. (b) Simulated andmeasured performance. (From: [32]. 2005 IEEE. Reprinted with permission.)

are available in [32]. Figure 10.63(b) plots the responses of the five-pole dual-bandfilter having f1 = 2.45 GHz, f2 = 5.8 GHz, and the fractional bandwidths for thetwo passbands are 12% and 7%, respectively.

References

[1] Riddle, A., ‘‘High Performance Parallel Coupled Microstrip Filters,’’ IEEE MTT-S Int.Microwave Symp. Dig., 1988, pp. 427–430.

[2] Kuo, J.-T., S.-P. Chen, and M. Jiang, ‘‘Parallel-Coupled Microstrip Filters with Over-Coupled End Stages for Suppression of Spurious Responses,’’ IEEE Microwave WirelessComp. Lett., Vol. 13, October 2003, pp. 440–442.

[3] Jiang, M., M. H. Wu, and J. T. Kuo, ‘‘Parallel-Coupled Microstrip Filters with Over-Coupled Stages for Multispurious Suppression,’’ IEEE MTT-S Int. Microwave Symp.Dig., 2005, pp. 687–690.

[4] Sonnet em, Sonnet Software, Inc., New York.[5] Matthaei, G. L., ‘‘Design of Wide-Band (and Narrow-Band) Bandpass Microwave Filters

on the Insertion Loss Basis,’’ IRE Trans. Microwave Theory and Tech., Vol. MTT-8,November 1960, pp. 580–593.


[6] Hong, J.-S., and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications,New York: John Wiley & Sons, 2001.

[7] Lopetegi, T., et al., ‘‘New Microstrip ‘Wiggly-Line’ Filters with Spurious Passband Suppres-sion,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-49, September 2001,pp. 1593–1598.

[8] Lopetegi, T., et al., ‘‘Microstrip ‘Wiggly-Line’ Bandpass Filters with Multispurious Rejec-tion,’’ IEEE Microwave and Wireless Letters, Vol. 14, November 2004, pp. 531–533.

[9] Wang, S. M., et al., ‘‘Miniaturized Spurious Suppression Microstrip Filter Using Mean-dered Parallel Coupled Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-53,February 2005, pp. 747–753.

[10] Vincent, P., J. Culver, and S. Eason, ‘‘Meandered Line Microstrip Filter with Suppressionof Harmonic Passband Response,’’ IEEE MTT-S Int. Microwave Symp. Dig., 2003,pp. 1905–1908.

[11] Velazquez-Ahumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filterswith Ground-Plane Aperture for Spurious Band Suppression and Enhanced Coupling,’’IEEE Trans. Microwave Theory Tech., Vol. MTT-52, March 2004, pp. 1082–1086.

[12] Velazquez-Ahumada, M., J. Martel, and F. Medina, ‘‘Parallel Coupled Microstrip Filterswith Floating Ground-Plane Conductor for Spurious-Band Suppression,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-53, May 2005, pp. 1823–1828.

[13] Hong, J.-S., ‘‘Computer-Aided Synthesis of Mixed Cascaded Quadruplet and Trisection(CQT) Filters,’’ 31st European Microwave Conference Proceedings, London, U.K.,September 2001, Vol. 3, pp. 5–8.

[14] Hong, J.-S., and M. J. Lancaster, ‘‘Couplings of Microstrip Square Open-Loop Resonatorsfor Cross-Coupled Planar Microwave Filters,’’ IEEE Trans. Microwave Theory Tech.,Vol. MTT-44, November 1996, pp. 2099–2109.

[15] Montejo-Garai, J. R., ‘‘Synthesis of N-Even Order Symmetric Filters with N Trans-mission Zeros by Means of Source-Load Cross Coupling,’’ Electron. Lett., Vol. 36,February 2000, pp. 232–233.

[16] Amari, S., ‘‘Direct Synthesis of Folded Symmetric Resonator Filters with Source-LoadCoupling,’’ IEEE Microwave Wireless Comp. Lett., Vol. 11, June 2001, pp. 264–266.

[17] Cameron, R. J., ‘‘Advanced Coupling Matrix Synthesis Techniques For Microwave Fil-ters,’’ IEEE Trans. Microwave Theory Tech., Vol. 51, January 2003, pp. 1–10.

[18] Garia-Lamperez, A., et al., ‘‘Synthesis of Cross-Coupled Lossy Resonator Filters withMultiple Input/Output Couplings by Gradient Optimization,’’ IEEE AP-S Int. Symp.Proc., Vol. 2, June 2003, pp. 52–55.

[19] Liao, C.-K., and C.-Y. Chang, ‘‘Design of Microstrip Quadruplet Filters with Source-LoadCoupling,’’ IEEE Trans. Microwave Theory Tech., Vol. 53, July 2005, pp. 2302–2308.

[20] Liao, C.-K., and C.-Y. Chang, ‘‘Modified Parallel-Coupled Filter with Two IndependentlyControllable Upper Stopband Transmission,’’ IEEE Microwave Wireless Comp. Lett.,Vol. 15, December 2005, pp. 841–843.

[21] Lee, S. Y., and C. M. Tsai, ‘‘New Cross-Coupled Filter Design Using Improved HairpinResonators,’’ IEEE Trans. Microwave Theory Tech., Vol. 48, December 2000,pp.2482–2490.

[22] 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.

[23] Makimoto, M., and S. Yamashita, ‘‘Bandpass Filters Using Parallel Coupled StriplineStepped Impedance Resonators,’’ IEEE Trans Microwave Theory Tech., Vol. MTT-28,December 1980, pp. 1413–1417.

[24] Sagaawa, M., M. Makimoto, and S. Yamashita, ‘‘Geometrical Structures and FundamentalCharacteristics of Microwave Stepped-Impedance Resonators,’’ IEEE Trans. MicrowaveTheory Tech., Vol. 45, July 1997, pp. 1078–1084.


[25] Ting, S. W., K. W. Tam, and R. P. Martins, ‘‘Novel Interdigital Microstrip BandpassFilter With Improved Spurious Response,’’ IEEE International Symposium on Circuits &Systems (ISCAS) 2004, pp. 984–987.

[26] Pang, H.-K., et al., ‘‘A Compact Microstrip l /4-SIR Interdigital Bandpass Filter withExtended Stopband,’’ IEEE MTT-S Int. Microwave Symp. Dig., 2004, pp. 1621–1625.

[27] Thomson, N., et al., ‘‘Practical Approach for Designing Miniature Interdigital Filters,’’35th European Microwave Conference Proceedings, Paris, 2005, pp. 1251–1254.

[28] Chang, S.F., et al., ‘‘A Dual-Band RF Transceiver for Multistandard WLAN Applications,’’IEEE Trans. Microwave Theory Tech., Vol. 53, March 2005, pp. 1048–1055.

[29] Miyake, H., et al., ‘‘A Miniaturized Monolithic Dual Band Filter Using Ceramic Lamina-tion Technique for Dual Mode Portable Telephones,’’ IEEE MTT-S Int. Microwave Symp.Dig., 1997, pp. 789–792.

[30] Tsai, L. C., and C. W. Hsue, ‘‘Dual-Band Bandpass Filters Using Equal Length Coupled-Serial-Shunted Lines and Z-Transform Technique,’’ IEEE Trans. Microwave Theory Tech.,Vol. 52, April 2004, pp. 1111–1117.

[31] Chang, S. F., Y. H. Jeng, and J. L. Chen, ‘‘Dual-Band Step-Impedance Bandpass Filterfor Multimode Wireless LANs,’’ Electron. Lett., Vol. 40, January 2004, pp. 38–39.

[32] Kuo, J. T., T. H. Yeh, and C. C. Yeh, ‘‘Design of Microstrip Bandpass Filters with aDual-Passband Response,’’ IEEE Trans. Microwave Theory Tech., Vol. 53, April 2005,pp. 1331–1337.

[33] Chuang, M. J., ‘‘Concurrent Dual Band Filter Using Single Set of Microstrip Open-LoopResonators,’’ Electronics Letters, Vol. 41, September 1, 2005, pp. 1013–1014.

[34] Sun, S., and L. Zhu, ‘‘Compact Dual-Band Microstrip Bandpass Filter Without ExternalFeeds,’’ IEEE Microwave and Wireless Components Letters, Vol. 15, October 2005,pp. 644–646.

C H A P T E R 1 1

Filters Using Advanced Materials andTechnologies

11.1 Introduction

Advanced materials and technologies have stimulated the development of novelRF/microwave filters for different applications. This chapter aims to cover somerecent developments, including superconductor filters, micromachined filters, filtersusing advanced dielectric materials such as low temperature cofired ceramic (LTCC)and liquid crystal polymer (LCP), filters for emerging ultra-wideband (UWB) tech-nology, and filters based on electromagnetic metamaterials.

11.2 Superconductor Coupled-Line Filters

Superconductors are attractive for use in passive microwave circuits [1]. Thesematerials have very low surface resistance in comparison to normal metals. Forexample, the high-temperature superconductors, which have a critical temperatureabove 77K, being used in microwave applications, exhibit a surface resistance at1 GHz that is three to four orders of magnitude lower than that of copper underequivalent conditions (i.e., at 77K).

One important application of high temperature superconductivity is in micro-wave filters. The extremely low resistance of HTS materials has enabled the realiza-tion of miniature thin film filters with low insertion loss and high selectivity.Excellent performance of HTS thin film filters has been demonstrated [2–17]. Thedriving force behind the development of HTS filters remains wireless applicationssuch as mobile and satellite communications [2–16]. Superconductor filters alsofind applications in radio astronomy or radio telescope receivers [17].

Filters with cross-coupled resonators discussed in Chapter 10 form the basisfor the design of high-performance superconducting filters. Two typical examplesare described next.

11.2.1 Cascaded Quadruplet and Triplet Filters

Figure 11.1 is a cascaded quadruplet trisection (CQT) coupling structure for a10-pole cross-coupled resonator filter, where each numbered node represents aresonator; the solid lines indicate direct couplings and the dash lines represent cross

371

372 Filters Using Advanced Materials and Technologies

Figure 11.1 10-pole 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 couplingsM1, 3 and M8, 10 , respectively, are used to produce and control two transmissionzeros close to the passband independently. This makes the filter easier to tune. Theonly quadruplet section of resonators 4 to 7 with the cross coupling M4, 7 is usedto generate another pair of transmission zeros.

Filter Synthesis

The first step in filter design is to obtain the general coupling matrix and the scaledexternal quality factors for the CQT filter from the filter synthesis. A computer-aided synthesis in [18] or other methods such as [19] can be used. The results offilter 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 0.63437 −0.01606 0.52039 0 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

qe1 = qe10 = 0.90557 (11.1)

The HTS filter is designed to have a 10-MHz bandwidth from 1,950 MHz to1,960 MHz. Hence, the center frequency f0 and fractional bandwidth FBW aregiven by

f0 = √1950 × 1960 = 1954.99361 MHz

FBW =10f0

= 0.00512

Using (10.6) and (10.7), a set of design parameters including the couplingcoefficients Mij and external quality factors (Qe1 and Qe2) for the given fractionalbandwidth can be obtained.

11.2 Superconductor Coupled-Line Filters 373

It should be noted that as a result of using trisections, the CQT filter is notsynchronously tuned, which is different from a CQ filter. The synthesized filter isan ideal one and its frequency response can also be computed using the formulationof (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 inthe coupling structure produces a single attenuation pole at a finite frequency eitherbelow or above the passband depending upon the polarity of the cross coupling.The cross-coupled quadruplet section produces a pair of attenuation poles at finitefrequencies in order to further improve the selectivity. The advantages of the CQTfilter are that the quasi-elliptic function design results in fewer resonators and thuslower passband insertion loss. Additionally, the tuning effort is reduced due to theindependent effect of the cross couplings on filter response.

Filter Implementation

The second step in the HTS filter design is to implement the coupling scheme ofFigure 11.1 with a proper microstrip configuration. For this HTS filter development,an r-cut sapphire substrate was used. Since the dielectric properties of sapphire areanisotropic, the dielectric constant is not a single value but a tensor. To make thefilter deign simple, an effective dielectric constant of 10.0556 for the r-plane sap-phire substrate was used [14]. However, in order to make this design approachwork, it is important to arrange all microstrip resonators in such a way that theyexperience the same permittivity tensor on the anisotropic substrate.

The implementation of two trisections is illustrated in Figure 11.3, where allthe resonators are in principle a half-wavelength long at the center frequency andmeandered in a special manner as shown, to facilitate the desired couplings andminiaturize the filter as well. The trisection of Figure 11.3(a) was developed toimplement the couplings M1, 2 , M2, 3 , and M1, 3 . For realizing M8, 9 , M9, 10 , andM8, 10 the trisection of Figure 11.3(b) was used. All the direct couplings are realized

Figure 11.2 Theoretical response of the 10-pole CQT filter.


Figure 11.3 Two microstrip trisections. (a) For producing a transmission zero near the low side ofthe 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 linewith capacitive probes coupled to the two nonadjacent resonators. The differencebetween the two triplet configurations lies in the locations of capacitive probes asshown, which makes their frequency characteristics distinct. The trisection of Figure11.3(a) is supposed to produce a transmission zero (attenuation pole) near the lowside of the passband, whereas the trisection of Figure 11.3(b) is designed to producea transmission zero near the high side of the passband.

A full-wave EM simulation was carried out to confirm these distinct characteris-tics as shown in Figure 11.4, where a single transmission zero on either lower orupper side of the passband is observed. In the simulation, each of trisections wasweakly excited and the simulation was performed using a commercially availableEM simulator [20].

Figure 11.5 shows the overall configuration of the 10-pole microstrip CQTbandpass filter.

Sensitivity Analysis

For a narrowband filter it is important to carry out a sensitivity analysis since anarrowband filter tends to be more sensitive to the tolerances in both the designand fabrication. It has been found that the major cause of performance variancefor the designed filter is the nonuniformity of the substrate thickness. For a quotedtolerance of ±5 mm in the substrate thickness of 430 mm, the impact on the filterperformance is significant. To demonstrate this, a sensitivity analysis based on theMonte Carlo method was performed using a commercially available microwavedesign tool [21] and the results for both transmission and reflection responses areplotted in Figure 11.6. The shading in each diagram illustrates the sensitivity ofthe filter response against a variation of substrate thickness. It is evident from thegiven 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 afurther sensitivity analysis that the tuning of resonator frequencies is much moreimportant than the tuning of couplings. This was used as a guideline for the filterexperiments discussed later on.


Figure 11.4 Frequency responses of the microstrip trisections. (a) Exhibiting a transmission zeronear the low side of the passband. (b) Exhibiting a transmission zero near the highside of the passband.

Figure 11.5 10-pole microstrip CQT bandpass filter configuration.


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 0.43-mm-thick sapphire wafer with double-sidedyttrium barium copper oxide (YBCO) superconducting films. The YBCO thin filmshave a thickness of 300 nm and a characteristic temperature of 87K. Both sidesof the wafer were gold-plated with 200-nm-thick gold (Au) for the RF contacts.This wafer is commercially available from THEVA GmbH. The fabricated HTSfilter had a size of 47 × 17 mm, which was assembled on a gold plated titaniumcarrier and placed into a brass test housing as shown in Figure 11.7(a). Thisassembly was then placed in a cryogenic dewar. As this type of narrowband filteris more sensitive to frequency tuning, sapphire tuners were used to tune the resonantfrequencies of all 10 HTS resonators. This is illustrated in Figure 11.7(b). A micro-wave vector network analyzer was used for all the RF measurements made undercryogenic conditions.


Figure 11.7 Fabricated HTS filter: (a) in the test housing and (b) the lid incorporating sapphiretuners.

Figure 11.8 shows the measured performance of the HTS filter at an operationaltemperature of 65K. A 10-MHz bandwidth was measured with a mid-band fre-quency of about 1,973 MHz. The measured mid-band frequency is higher thanthe design frequency. This is attributed to the assumed value of dielectric constantfor 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 measuredinsertion loss in the passband is 0.2 dB. This corresponds to a resonator Q ofgreater than 50,000. The measured return loss (shown at 5 dB per division) isbetter than −12 dB across the passband. The measured wideband response of thefilter is plotted in Figure 11.9, showing the excellent rejection and clean responsewithout harmonics or spurious modes over the entire UMTS transmission band(2,110–2,170 MHz).

11.2.2 High-Order Selective Filters with Group-Delay Equalization

Highly selective filters are in great demand for applications with stringent selectivityrequirements. For instance, the capacity and coverage of a base-station receiverare determined primarily by the receiver selectivity and sensitivity on the uplink.The selectivity can be significantly increased with the use of higher-order filters[10], and there is a trend to develop high-order HTS filters to take advantage ofminiature high Q resonators [8, 16].


Figure 11.8 Measured results for the 10-pole superconducting CQT filter at 65K.

Figure 11.9 Measured wideband response of the 10-pole superconducting CQT filter.

In many communication systems, flat group delay of a bandpass filter is alsoa requirement in addition to its selectivity. The group delay represents the truesignal delay between the input and output ports of a communication channel, suchas a filter, and is defined in (9.4). The demand for a flat group delay is particularlytrue for high-capacity communication systems where it is essential for a bandpassfilter to have a good linear-phase response or flat group delay over the centralregion of the passband. Due to the requirement of selectivity, some deterioration inthe group delay performance at the edges of the passband is allowed. Unfortunately,higher-order highly selective filters tend to result in a poor phase performance evenover the band center. To demonstrate this, Figure 11.10 shows a comparison ofthree different types of high-order filters (i.e., 30-pole Chebyshev, 18-pole quasi-elliptic function, and 18-pole quasi-elliptic function with linear phase).

All three filters have a passband of 15 MHz from 1,960 MHz to 1,975 MHzwith the same ripple level and are supposed to meet a selectivity of 70-dB rejection


Figure 11.10 Comparison of performances of three high-order filters: (a) transmission responseand (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 to18 required by quasi-elliptic function filters. The need for the larger number ofresonators for the Chebyshev filter not only leads to a larger size, but also resultsin higher insertion loss (due to finite resonator Q) and greater variation of groupdelay 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 Cheby-shev response. While the two quasi-elliptic function filters are superior in terms ofachieving a high selectivity with a lower number of resonators, the one with alinear phase response is more attractive as it exhibits a flatter group delay overthe band center, which can be seen from Figure 11.10(b).

It is therefore desirable to develop high-order HTS bandpass filters, not onlyto meet the selectivity requirement, but also to provide capability for self-equalized


group delay over the central part of the passband. The design, modeling, and testresults of such a high-order HTS microstrip bandpass filter are presented next.

Design and Modeling

The filter model or coupling structure of an eighteen-pole filter is shown in Figure11.11. Each node with a number represents a resonator. Resonators 1 and 18 arecoupled to the input and output ports, respectively, denoted by external qualityfactors, Qe1 and Qe2 . The unbroken lines between adjacent resonators indicatethe direct couplings. There are four cross couplings, as indicated by the brokenlines, between resonators 2 and 5, 6 and 9, 10 and 13, and 14 and 17. These crosscouplings are denoted by coupling coefficients M2, 5 , M6, 9 , M10, 13 , and M14, 17 .As can be seen, each cross coupling is associated with a quadruplet section, andhence the filter has basically a cascaded quadruplet (CQ) structure. The advantageof this coupling structure lies in that each of the quadruplet sections can be arrangedeither to produce a pair of transmission zeros (attenuation poles) at finite frequenciesin order to achieve higher selectivity for a given number of resonators, or to resultin a linear phase performance to achieve a self-equalization of group delay. Forour design, only one quadruple section, which consists of resonators 10 to 13, willbe used for the group delay equalization, while the other three quadruplet sectionsare arranged for high selectivity. The cross coupling in each quadruple can betuned independently, making the tuning easier for such a high-order filter.

An 18-pole filter was designed to have a 15-MHz passband at a center frequencyof 1,967.5 MHz. To this end, the circuit model of Figure 11.12 was created withMicrowave Office, a commercially available software [21]. In this circuit model,all the LC resonators, which are supposed to resonate at the central frequencyf0 = 1/2p √L0C0 , have an inductance of L0 = 0.03291 nH and a capacitance ofC0 = 198.83 pF. Each quarter-wavelength line has electrical length u = ±90 degreesat the central frequency of the passband and functions as an immittance inverterto represent the coupling between the associated pair of resonators. The other

Figure 11.11 Coupling structure for an 18-pole selective filter with group-delay equalization.

Figure 11.12 Circuit model showing the coupling structure of the 18-pole CQ filter.


circuit parameters are related, following the formulations given in [22], to the setof desired coupling coefficients Mjk and external quality factors Qei and Qeo .

Figure 11.13 shows the simulated frequency responses of the filter, based onthe circuit model of Figure 11.12 with desired coupling coefficients and externalquality factors. As can be seen, the filter exhibits three pairs of transmission zeros(attenuation poles) at finite frequencies to increase the selectivity, while exhibitinga flat group delay over about the middle 50% of the passband.

The design of this type of microstrip filter is based on the procedure describedin Chapter 10 or [22]. In order to implement this type of filter in microstrip, twobasic quadruplet sections of coupled microstrip resonators, as shown in Figure11.14, have been investigated. Each quadruplet section is comprised of four mean-dered microstrip resonators as shown, and the four resonators are arranged insuch a way that the direct coupling between adjacent resonators results from the

Figure 11.13 Theoretical responses of the 18-pole bandpass filter with self-equalization of groupdelay: (a) magnitude and (b) group delay.


Figure 11.14 Two microstrip quadruplet sections: (a) for the realization of a pair of transmissionzeros 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 lastresonators is used for the cross coupling. By inspecting the quadruplet configura-tions in Figure 11.14, one can notice that they are almost the same except for thespacing between the two inner coupled-resonators in each quadruplet section. Thisspacing in Figure 11.14(a) is much smaller than that in Figure 11.14(b). As a matterof fact, it is this spacing difference that makes the characteristics of these twoquadruplets totally different. It can be shown that the coupling between the twoinner coupled-resonators in Figure 11.14(a) is dominated by the electric coupling,whereas the magnetic coupling is dominant for the two inner coupled-resonatorsin Figure 11.14(b). Because these two couplings have opposite effects, which tendto 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 microstripconfiguration of Figure 11.14(a) to implement the desired couplings for thosequadruplet sections in Figure 11.11 that produce finite frequency transmission


zeros. On the other hand, the microstrip configuration of Figure 11.14(b) is usedfor the realization of the required couplings for the quadruplet in Figure 11.11that results in group-delay equalization.

For example, the desired coupling matrix for the first quadruplet (i.e., coupledresonators 2, 3, 4, and 5) is given by

30 M23 0 M25

M23 0 M34 0

0 M34 0 M45

M25 0 M45 04 = 10−2 ? 3

0 0.4089 0 0.1822

0.4089 0 −0.5706 0

0 −0.5706 0 0.3460

0.1822 0 0.3460 04

(11.2)

where the coupling for M34 is negative. This quadruplet is responsible for a pairof transmission zeros observed in the magnitude response in Figure 11.13(a). Afull-wave EM simulation has been performed for a microstrip quadruplet sectionshown in Figure 11.14(a) to determine physical parameters corresponding to thecoupling matrix of (11.2), and the results are plotted (unbroken line) in Figure11.15(a). Note that the response was obtained by weakly coupling the quadrupletto the input/output ports. The EM simulation was done using a commerciallyavailable simulator [20]. A quadruplet resonator-circuit model based on (11.2) canalso be used to compute the theoretical response as shown with the broken line inFigure 11.15(a). As can be seen, there is good agreement between the theory andsimulation, and this microstrip quadruplet realization does indeed produce thedesired transmission zeros.

Similarly, the EM simulation and circuit modeling can be done for the quadru-plet which consists of coupled resonators 10, 11, 12, and 13 which has a couplingmatrix given by

30 M10, 11 0 M10, 13

M10, 11 0 M11, 12 0

0 M11, 12 0 M12, 13

M10, 13 0 M12, 13 04 = 10−2 ? 3

0 0.3419 0 0.1785

0.3419 0 0.2047 0

0 0.2047 0 0.3423

0.1785 0 0.3423 04

(11.3)

In this case, the microstrip configuration of Figure 11.14(b) was used in thesimulation. The simulated and theoretical results are plotted in Figure 11.15(b),where no finite-frequency transmission zeros are observable, which is expected asthis quadruplet is only for the group delay equalization. Again, the good agreementbetween theory and simulation validates the microstrip realization of this type ofquadruplet section.

Figure 11.16 shows the final layout of the designed 18-pole superconductingmicrostrip filter. The substrate is an r-cut sapphire and the filter size is 74 mm ×17 mm. It can be recognized in Figure 11.16 that the third quadruplet from theleft is for the self-equalization of group delay.


Figure 11.15 Frequency responses of quadruplet coupled resonators: (a) for the realization of apair of transmission zeros at finite frequencies, and (b) for the realization of groupdelay equalization.

Figure 11.16 Layout of the designed 18-pole HTS microstrip filter on sapphire substrate.

Fabrication and Measurement

The filter was fabricated on a 0.43-mm-thick sapphire wafer with double-sidedYBCO films. The YBCO thin films have a thickness of 300 nm and a characteristictemperature of 87K. Both sides of the wafer are gold-plated with 200-nm-thickgold (Au) for the RF contacts. The fabricated HTS filter was assembled into a testhousing with two K-connectors and a lid as shown in Figure 11.17, for measure-

11.3 Micromachined Filters 385

Figure 11.17 Photo of the assembled 18-pole HTS filter in a testing housing with sapphire tunerson the lid.

ments. One can see clearly that the lid has accommodated a number of sapphiretuners. Most of the tuners were used in the experiment to tune the resonatorfrequencies as suggested by a sensitivity study, which is similar to that discussedin the previous section for the CQT filter.

RF measurements were done using an HP network analyzer and immersingthe filter in a cryogenic cooler. Figure 11.18 shows the measured results at 65Kand after tuning the filter. The tuning is a must for such a high order and narrowbandfilter because of the tolerances in both the wafer thickness and fabrication. FromFigure 11.18(a) we can see that the measured band center frequency is about 1,970MHz, which is slightly higher than the designed 1,967.5 MHz. This is because thecenter frequency is dependent on the orientation of the filter on the sapphire waferas a result of its anisotropic dielectric property. The measured bandwidth is closeto 15 MHz. An insertion loss of 1.4 dB at the band center was measured, includingthe 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 acrossthe passband. Furthermore, finer tuning could improve the return loss as well asshift the center frequency to the design frequency. The transmission zeros near theband edges for enhancing selectivity are observed. The measured group delay ofthe filter is plotted in Figure 11.18(b), showing a flat group delay over the centralregion of the passband, which is in very good agreement with the theoreticalresponse 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 inmicromachined circuits. One of the most popular micromachining techniques con-sists of etching a Si substrate and suspending the circuit on a thin dielectric mem-brane. Many planar filter topologies such as popular parallel coupled line filtersdescribed in previous chapters may directly be realized on the thin dielectric mem-brane. The use of additional micromachined substrates can be used to shield the


Figure 11.18 The measured responses of the 18-pole HTS filter with self-equalization of groupdelay: (a) magnitude and (b) group delay.

structure to avoid radiation losses. The propagation occurs in air, with almost nodispersion and no dielectric losses. Such a structure is particularly attractive formillimeter-wave filters. For example, a silicon micromachined filter with a simpleplanar integration on another substrate is demonstrated in [24]. The filter is com-prised of two end-coupled half-wavelength microstrip resonators supported on an8-mm-thick dielectric membrane. The excitation is from the top of the shieldingsubstrate of the membrane-supported micromachined filter. Packaging and inter-connections are included in the design. Experimental results are presented on atwo-pole 30 GHz, 4% fractional bandwidth filter with a quality factor of 600 andinsertion loss of 1.8 dB. Such a filter can be easily integrated in any circuit usingflip-chip technology.

Using micromachining techniques has also given rise to the development of so-called synthesized substrates that extend the useful range of high and low impedancemicrostrip values on high dielectric constant materials [25]. In that paper, a silicon


wafer is micromachined to form a ‘‘synthesized substrate’’ that can minimize lowimpedance and maximize high impedance values, leading to lowpass filter designswith either reduced low impedance or increased high impedance values on the‘‘synthesized substrate’’ sections, and having high and low impedance valuesimproved by a factor of 1.5. As a result, in the filter response, sharper rejectionband edges have been achieved.

The use of micromachining technology has also meant that integration withother silicon components on the same wafer is now possible. In particular, thedevice can be integrated with other RF MEMS components such as switches andvaractors 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 topologybut also by the dielectric substrate. Using high resistive silicon substrates has enabledthe design of small bandpass filters at low microwave frequency bands [30]. Figure11.19 shows the layout of a five-pole miniature stepped impedance resonator (SIR)interdigital filter designed on a silicon substrate with a dielectric constant of 11.9and a thickness of 0.525 mm. The high impedance line for the SIR has a width of0.15 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 hencethe 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 kV-cm) and masks were laid out for 100-mm-diameter wafers. The stripconductor and ground plane thicknesses were specified as 3 mm 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 withdeep reactive ion etching used to make via holes to the ground plane. The via holeswere metallized with aluminum and were tested for good conductivity. A flashlayer of 0.5 mm of gold was also applied to the ground plane to ensure a lowresistance contact to the test housing. The test pieces were diced and mounted ona brass test fixture using epoxy bonding as shown in Figure 11.20. Two SMAconnectors were incorporated for testing.

The measured results are shown in Figure 11.21 together with the EM simulatedresults, and good agreement can be observed.

EM simulations were also carried out to investigate conductor, dielectric andradiation losses of the SIR interdigital filter. Figure 11.22 shows the simulatedresults, where frequencies are normalized with respect to the center frequency. Eachcurve shows the passband response when only one loss mechanism is considered. Tosimulate the conductor loss, 3-mm-thick aluminum (as fabricated) with a conductiv-ity 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 high-resistive silicon substrate was used[30]. From Figure 11.22, one can see that the conductor loss is still dominant inthis 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


Figure 11.19 Layout of five-pole miniature SIR interdigital filter on a silicon substrate with a dielectricconstant of 11.9 and a thickness of 0.525 mm. Dimensions are in millimeters.

Figure 11.20 Fabricated five-pole SIR interdigital filter on silicon with two SMA connectors fortesting.


Figure 11.21 Measured and simulated performances of the five-pole miniature SIR interdigital filteron silicon.

Figure 11.22 EM-simulated loss effects of the five-pole miniature SIR interdigital filter on silicon.

not only cause a greater insertion loss, but they also shrink the bandwidth, whichneeds to be taken into account in the design.

The advantage of using silicon is the ability to use IC manufacturing techniquessuch as silicon under etching to reduce the size of the SIR filter. For example, usingbulk micromachining it is possible to reduce the thickness of the silicon substrateunder the capacitive element (low impedance line section) of the stepped impedanceresonator. The ground plane metal is then profiled to the change in thickness ofthe substrate as shown in Figure 11.23. The result is an increase in the capacitanceof the element, which leads to a larger ratio of high to low impedance and reducesthe length of the SIR. For the demonstration, simulations have been carried outto investigate the possible size reduction. For fixed W1 = 0.15 mm, W2 = 0.5 mm


Figure 11.23 Micromachining below the capacitive element of the SIR. Top view: the layout ofthe SIR. Bottom view: the cross section.

and h1 = 0.525 mm, it is shown that up to a 40% reduction in overall resonatorlength L for the same resonant frequency can be obtained when the substratethickness h2 under the capacitive element is reduced down to 0.1 mm [30].

11.3.2 Overlay Coupled CPW Filters

Conventional coplanar-waveguide (CPW), as shown in Figure 11.24(a), suffersfrom high conductor loss at high and low characteristic impedance extremes dueto the narrowing of the center conductor and slot width, respectively [32]. More-over, very low impedance lines are practically impossible to realize in CPW dueto the minimum slot size limit imposed by the fabrication process. Recently,

Figure 11.24 Schematic diagrams of: (a) conventional CPW and (b) overlay CPW.

11.4 Filters Using Advanced Dielectric Materials 391

micromachining techniques have allowed fabricating a so-called overlay coplanar-waveguide structure of Figure 11.24(b), in which the edges of the center conductorare partially elevated, denoted by E, and overlapped with ground, indicated by O,to achieve a broad ranges of characteristic impedances and to reduce the linelosses [31]. The elevated center conductor helps to reduce the conductor loss byredistributing the current over a broad area. It also helps to reduce the dielectricor substrate loss by confining the electric field in the air between the overlappedconductor plates. Compared with conventional CPW lines, the overlay CPW linesshow a wider impedance range (25–80 ohms) and lower loss (<0.95 dB/cm at50 GHz). Figure 11.25 illustrates the microphotograph of a fabricated overlayCPW line on a quartz substrate, where the elevation of the signal line is about15 mm. The micromachining fabrication processes are detailed in [31].

To demonstrate the practical usefulness of the overlay CPW lines, an X-bandstepped-impedance lowpass filter was designed and fabricated using this type ofmicromachined transmission line [31]. The overlay CPW filter shows distinct advan-tages over the conventional CPW filter such as lower loss and reduced size, togetherwith improved spurious responses, including improved selectivity and wider stop-band characteristics.

It is envisaged that various bandpass filters using coupled overlay CPW resona-tors can be constructed. For example, Figure 11.26 depicts the schematic of a three-pole end-coupled overlay CPW resonator filter.

11.4 Filters Using Advanced Dielectric Materials

Recent advances in microwave dielectric materials such as low-temperature cofiredceramic (LTCC) and liquid crystal polymer (LCP) have stimulated a rapid develop-ment of multilayer microwave and millimeter-wave components including filtersfor system integration [33–47].

Figure 11.25 SEM photograph of the fabricated overlay CPW line, where CPW pad is for probingmeasurement. (From: [31]. 2001 IEEE. Reprinted with permission.)


Figure 11.26 Schematic diagram of end-coupled overlay CPW resonator filter.

Recent developments in LTCC technology are making it more and more attrac-tive for applications in the microwave and millimeter-wave frequency bands [33–35]. The most active areas for high frequency applications include Bluetoothmodule, front-end modules (FEM) for mobile phones, wireless local access network(WLAN), local multipoint distribution systems (LMDS), and collision avoidanceradar. The newer developments in LTCC substrate materials extend the applicablefrequency of the technique up to 100 GHz [33]. Owing to low conductor loss, lowdielectric loss, and up to 50 laminated layers, LTCC provides a suitable approachfor embedded microwave and millimeter-wave passive components and accessoriesincluding antennas. In addition, LTCC substrate materials possess a wide rangeof thermal expansion coefficients. It is this feature that makes the LTCC substratesvery attractive for integrated packaging solutions. Basic LTCC process consistsof 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 alow-cost dielectric material for high-volume large-area processing methods thatprovide very reliable high-performance circuits at low cost [37–39]. LCP has aunique combination of properties such as: (1) excellent electrical properties up tomillimeter-wave frequencies (dielectric constant of 3.16 and a low loss-tangent of0.002–0.004 at 60 GHz comparable with ceramics); (2) very good barrier properties[its permeability is (moisture absorption = 0.04%) comparable to that of glass andvery 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 allmultichip-module (MCM), system-on-package (SOP), and advanced packagingtechnology led by the growing market for digital, RF, and opto-RF applications.

11.4.1 Low-Temperature Cofired Ceramic Filters

LTCC Lumped Element Filter

Figure 11.27(a) shows the physical layout of a multilayer LTCC filter [40]. Itconsists of a second-order coupled resonator bandpass filter (with both capacitiveand inductive couplings) in parallel with a feedback capacitor. The two resonators


Figure 11.27 (a) Physical LTCC layout of a two-pole bandpass filter. (b) Equivalent lumped-elementcircuit of the filter. (c) Measured and simulated performance of the prototyped LTCCfilter. (From: [40]. 2003 IEEE. Reprinted with permission.)


are equivalent to parallel LC resonant circuits. Figure 11.27(b) depicts a lumped-element circuit of the filter, where the element values are CC1 = CC2 = 0.79 pF,LL1 = LL2 = 1.55 nH, C1 = C2 = 2.48 pF, and MM = 9.27 nH. This correspondsto a second-order Chebyshev-type bandpass filter with 0.2-dB ripple, 2.5-GHzcenter frequency, and 0.3-GHz equal-ripple bandwidth.

With the multilayer capability of the LTCC technology, the lumped-circuitelements can be readily realized by using parallel plates for the capacitor and ametallic strip for the inductor. The capacitors are realized on layer 1 with twopatches; while the inductors are realized using meandered narrow lines (line widthof 8 mil) terminated to the ground plane. All the interconnections are done usingconducting vias. The coupling between the two resonators is accomplished byoverlaying two inductance strips one above the other on layers 1 and 2 (4.56 milapart), respectively. Thus, the inter-resonator coupling is magnetic. The input andoutput (I/O) are facilitated on layer 2 with two small patches that are coupled tothe capacitors of the two resonators on the layer below, so the I/O couplings arecapacitive. The feedback capacitor is implemented on layer 3 as indicated, byplacing a ‘‘dumbbell’’-shaped metal plate directly above the I/O components of thefilter. Its purpose is to introduce a pair of finite transmission zeros to the transmis-sion transfer function—one in the lower stopband and another one in the upperstopband.

The filter was constructed using six layers of Dupont 951AT material with3.6-mil thickness. The components of the filter were located only at the interfacesbetween the bottom four layers. A finite ground plane was inserted at the bottomof the substrate for the construction of the grounded resonators. The overall sizeof the filter is 170 × 80 × 21.6 mil. The measured response of the filter and resultsfrom EM simulation are presented in Figure 11.27(c). It can be seen that, due tothe zero metallic strip thickness model used in the design, which underestimatesthe capacitance of the parallel plates, the measured response is slightly shiftedtoward the lower frequency end. Nevertheless, the correlation of the theoreticaland measured results is very good. Notice that the two finite zeros in the transmis-sion response of the filter are located at the prescribed locations.

LTCC Cavity Filter

On-package integrated cavity filters using LTCC multilayer technology are a veryattractive option for three-dimensional (3D) RF front-end modules up to the milli-meter-wave frequency range because of their relatively low loss compared tostripline/microstrip or lumped-element-type filters [43, 44].

Figure 11.28(a) illustrates the 3D structure of a three-pole LTCC cavity filterfor 60-GHz WLAN narrowband (1 GHz) applications [43]. The design is basedon a three-pole Chebyshev lowpass prototype filter with 0.1-dB in-band ripple.The LTCC filter consists of three coupled cavity resonators [i.e., Cavity 1, Cavity2, and Cavity 3 in Figure 11.28(b)]. The cavity resonator is built utilizing conductingplanes as horizontal walls and via fences as sidewalls. The size and spacing of viaposts are properly chosen to prevent electromagnetic field leakage and to achievestopband characteristics at the desired resonant frequency. The cavity height wasdesigned to be 0.5 mm (five substrate layers) to achieve a higher quality factor


Figure 11.28 LTCC three-pole cavity BPF employing slot excitation with an open stub: (a) 3Doverview and (b) side view of the proposed filter. (From: [43]. 2005 IEEE. Reprintedwith permission.)

and, consequently, to obtain a narrower bandwidth. The inter-resonator couplingsfrom the cavities 1 and 3 and 2 to 3 are accomplished by two internal slots, asindicated on metal layer 7. Microstrip feed lines are utilized to excite the I/Oresonators (i.e., cavities 1 and 2) through coupling slots etched in the top metallayer (metal 2), as denoted by the external slots in the figure. In order to maximizethe magnetic coupling by maximizing magnetic currents, a virtual short is placedat the center of each slot by terminating the feed lines with quarter-wavelengthopen stubs. The excitation using an open stub contributes to fabrication simplicitywith no need to drill via-holes to short the end of the feed lines. It also avoids theloss and inductance effects generated by shorting vias close to the slot, whose


effects can be significant in the millimeter-wave frequency range. The relevantdesign parameters for the LTCC cavity filter are summarized in Table 11.1.

The designed filter was fabricated using LTCC 044 SiO2 – B2O3 glass by theAsahi Glass Company, Kanagawa, Japan. The relative permittivity of the substrateis 5.4 and its loss tangent is 0.0015 at 35 GHz. The dielectric layer thickness perlayer is 100 mm, and the metal thickness is 9 mm. The resistivity of metal (silvertrace) is determined to be 2.7 × 10−8 V ? m. The fabricated LTCC cavity filterexhibits an insertion loss 2.14 dB and a return loss 16.39 dB over the passband.The measurement shows a 3-dB fractional bandwidth of approximately 1.53%(0.9 GHz) at a center frequency of 58.7 GHz. The center frequency downshiftcan be attributed to the fabrication accuracy such as slot positioning affectedby the alignment between layers, layer thickness tolerance, and variation indielectric constant.

11.4.2 Liquid Crystal Polymer Filters

Miniature Wideband LCP Filter

A miniaturized wideband filter topology has been developed for LCP implementa-tion based on a circuit for an ideal transmission line filter [45], which is depictedin Figure 11.29. The circuit is symmetrical with respect to the middle where thereis a parallel open-circuited stub with an electrical length of 2u and a characteristicimpedance of Z0s . It can be recognized that the symmetrical sections on both sideof the open-circuited stub are the same as that of Figure 10.3. This means thatthey can be implemented with two coupled line sections as shown in Figure 11.30.Thus, Z0e and Z0o in Figure 11.29 are the even- and odd-mode characteristicimpedances of the symmetrical coupled lines with a line width of W and spacings. For this type of filter, the electrical length u is required to be 90 degrees at themid-band or center frequency f0 . Therefore, the parallel open-circuited stub hasan electrical length of 180 degrees at f0 . The remaining design parameters for thefilter are Z0e , Z0o , and Z0s .

Figure 11.31 illustrates the frequency response of a miniature coupled line filterwith Z0e = 133V, Z0o = 51V, and Z0s = 26.5V for a center frequency f0 = 2 GHz.Since the filter is based on a transmission line filter model, it exhibits a periodicalfrequency response with multipassbands located at f0 , 3f0 , and so forth. We areinterested only in the primary passband at f0 . It can be clearly seen from the S11

Table 11.1 Design Parameters of the LTCC Cavity Filter

Design Parameters Dimensions (mm)

Effective cavity resonator (L × W × H) 1.95 × 1.31 × 0.5External slot position (SPext ) 0.4125External slot (SLext × SWext ) 0.46 × 0.538Internal slot position (SPint ) 0.3915Internal slot (SLint × SWint ) 0.261 × 0.4Open stub length (OSL) 0.538Via spacing 0.39Via diameter 0.13Via rows 3


Figure 11.29 An ideal transmission line filter.

Figure 11.30 A schematic of a miniature coupled line filter with a parallel open-circuited stub inthe middle.

response that there are five transmission poles, which implies that this miniaturefilter topology is equivalent to a five-pole filter, although it only consists of twoquarter-wavelength coupled line sections and one half-wavelength open-circuitedstub when referring to the center frequency. The half-wavelength open-circuitedstub also produces two finite-frequency transmission zeros near the passband,which improve the selectivity of the filter. As can be seen, the first transmissionzero occurs at 1 GHz. At this frequency, the open-circuited stub has an electricallength 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 GHzbecause at this frequency the open-circuited stub has an electrical length of 270degrees that also presents a short circuit in the main signal path. These two transmis-sion zeros also limit the maximum bandwidth of this type of filter. It can be shownthat, for a given center frequency f0 , the two transmission zeros are allocated at0.5f0 and 1.5f0 , respectively. Hence, the fractional bandwidth of this type of filterwill be smaller than 100%; nevertheless, it is quite easy to achieve a wide bandwidthof 50% to 60%.

A microstrip filter of this type, realized on a dielectric substrate with a dielectricconstant of 2.9 and a thickness of 0.33 mm, is demonstrated in Figure 11.32(a).


Figure 11.31 Frequency response of the miniature coupled line filter with an open-circuited stub for Z0e =133V, Z0o = 51V, Z0s = 26.5V, and f0 = 2 GHz.

The coupled line sections have a very small spacing (s = 0.05 mm) and a narrowline width (W = 0.25 mm) for the required coupling. The open-circuited stub ismeandered into a loop to make the filter compact. The design center frequency is2 GHz, and the dimensions shown are in millimeters. The size of the filter is about0.5lg by 0.25lg , where lg is the guided wavelength in the substrate. The filterresponse 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 thepassband as expected. However, there is a spike near 4 GHz. This unwantedspurious response is due to unequal even- and odd-mode phase velocities in themicrostrip coupled lines. The issue has been addressed in Chapter 10 (see Section10.2). 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 filterwas fabricated on LCP substrate characterized by er = 2.9, tan d = 0.003, substratethickness = 330 mm and conductor thickness = 18 mm. Copper-clad LCP dielectricsubstrates available from Rogers Corporation were used for fabrication. AlthoughLCP substrates are available only with certain thicknesses, it is possible to realizemany different substrate configurations due to the two types of LCP substratesavailable that have different melting temperature. The high-melt LCP (around315°C) is used as a core layer while the low-melt LCP (around 290°C) is used asa bonding layer. In this case, a 4-mil LCP layer is bonded with an 8-mil LCPlayer using a 1-mil bonding layer to give a total thickness of 330 mm. Thedesigned filter is then patterned and measured. The measured passband is from


Figure 11.32 (a) Miniature microstrip coupled line filter with an open-circuited stub on a dielectricsubstrate with a dielectric constant of 2.9 and a thickness of 0.33 mm. Dimensionsare in millimeters. (b) EM-simulated S21 response.

2.38–4.2 GHz, representing a fractional bandwidth of about 55%. The measuredinsertion loss within the passband is around 1.2 dB.

60-GHz Band LCP Filters and Duplexer

Planar and via-less LCP bandpass filters and duplexers, operating in the 60-GHzband or V-band, have been developed [46]. Figure 11.33 shows a photo of the


Figure 11.33 A photo of the fabricated LCP duplexer with two microstrip square open-loop resona-tor filters. (From: [46]. 2006 IEEE. Reprinted with permission.)

fabricated duplexer. The duplexer consists of two four-pole microstrip square open-loop resonator filters with the canonical or single CQ coupling configuration asdescribed in Chapter 10. An optimized T-junction is used to combine two four-port filters to realize the three-port duplexer. The square open-loop resonatorsallow different coupling mechanisms required to realize desired filtering characteris-tics with transmission zeros for the duplexer implementation. It was also foundthat the feed scheme for the filter on the left in Figure 11.33 results in a passbandresponse with steep roll-off on the high side, whereas the feed scheme for the filteron the right gives rise to a higher selectivity on the low side of the passband. Theseasymmetric characteristics have been developed that are then combined to realizethe duplexer.

The designed duplexer was fabricated on LCP substrate characterized byer = 3.15, tan d = 0.003, substrate thickness 152 mm, and conductor thickness9 mm. In this case, two 2-mil high-melt (around 315°C) core LCP layers (commer-cially available as R/flex 3850) are bonded together using two 1-mil low-melt(around 290°C) bond LCP layers (commercially available as R/flex 3600) to givea total thickness of 152 mm. Once an LCP substrate of the desired thickness wasobtained, the designs were patterned and measured. The filters exhibit a lowinsertion loss of 2.5 dB and the isolation between the duplexer ports is better than25 dB.

11.5 Filters for Ultra-Wideband (UWB) Technology

Ultra-wideband (UWB) technology is being reinvented recently with many promis-ing modern applications [48, 49]. The rapid growth in this field has promptedthe development of wideband microwave filters for UWB applications [50–65],described next.

11.5 Filters for Ultra-Wideband (UWB) Technology 401

11.5.1 Optimum Stub Line Filters

A distributed stub transmission line filter, which may be modified and realized indifferent forms such as the ring or double-sided parallel strip [51, 52], is attractivefor designing multipole or high-order selective UWB filters. In particular, the designof an UWB microstrip optimum short-circuited stub filter whose unit elements orconnecting lines are nonredundant [55, 56] is discussed next.

Design of Optimum Stub Filter

A general circuit model for a short-circuited stub filter is shown in Figure 11.34,which is comprised of a cascade of n shunt short-circuited stubs of electrical lengthus , separated by connecting transmission lines (unit elements) of electrical lengthul . Both us and ul are determined at some characteristic frequency fc . Z0 is theterminal impedance. Z1 , Z2 , . . . , Zn are the line characteristic impedances of theshort-circuited stubs. The characteristic impedances of the connecting lines aredenoted by Z1, 2 to Zn − 1, n .

To demonstrate the advantages of the optimum stub filter, Figure 11.35 showsthe frequency response of three short-circuited stub filters. For the two conventionalstub filters, us = ul = 90 degrees at fc = 9 GHz, which is also the mid-band frequency;thus, all the short-circuited stubs and connecting lines are quarter-wavelength longat the mid-band frequency. It is obvious that the conventional stub filter with 11stubs (n = 11) has a higher selectivity than the conventional stub filter with 6 stubs(n = 6). Nevertheless, this indicates that a larger number of short-circuited stubsis usually necessary for this type of UWB filter requiring high selectivity. The designof conventional stub filters is well documented in [66]. For our discussion lateron, Table 11.2 lists the designed circuit parameters of the conventional stub filterwith 11 short-circuited stubs.

The third filter whose frequency response is also shown in Figure 11.35 is anoptimum stub filter, which also has only 6 stubs (n = 6). For this optimum stubfilter, the electrical length of the short-circuited stubs is us = 35 degrees, while theelectrical length of the connecting lines or unit elements is ul = 2us = 70 degrees

Figure 11.34 Equivalent circuit of short-circuited stub filter.


Figure 11.35 Transmission characteristics of three UWB short-circuited stub filters with the samepassband from 3.5 to 14.5 GHz.

Table 11.2 Circuit Parameters for a ConventionalShort-Circuited Stub Filter with 11 Stubs foran Ultra-Wide Passband from 3.5 to 14.5 GHz(us = u l = 90 Degrees at 9 GHz)

Stub Line Impedance Connecting Line Impedance

Z1 = Z11 = 199.07V Z1, 2 = Z10, 11 = 38.84VZ2 = Z10 = 101.88V Z2, 3 = Z9, 10 = 36.73VZ3 = Z9 = 101.76V Z3, 4 = Z8, 9 = 38.95VZ4 = Z8 = 98.49V Z4, 5 = Z7, 8 = 39.73VZ5 = Z7 = 97.41V Z5, 6 = Z6, 7 = 40.00VZ6 = 97.13V

at fc = 3.5 GHz. From Figure 11.35, it is notable that the optimum stub filter withsix short-circuited stubs achieves the same selectivity as the conventional stub filterwith 11 short-circuited stubs. This is because the unit elements of the optimumstub filter are not redundant, and they function nearly as effectively as the short-circuited stubs in improving the selectivity.

In general, the circuit parameters of an optimum stub filter are determinedusing a computer-aided synthesis program. As an alternative, tabulated elementscan be found in [22] for this type of filter, which is also seen as the optimumpseudo-highpass filter. For the optimum stub filter discussed earlier, its circuitparameters are given in Table 11.3. Comparing the circuit parameters in Tables

Table 11.3 Circuit Parameters for an OptimumShort-Circuited Stub Filter with Six Stubs foran Ultra Wide Passband from 3.5 to 14.5 GHz(us = 35 Degrees and u l = 70 Degrees at 3.5 GHz)

Stub Line Impedance Connecting Line Impedance

Z1 = Z6 = 103.96V Z1, 2 = Z5, 6 = 48.85VZ2 = Z5 = 72.64V Z2, 3 = Z4, 5 = 50.44VZ3 = Z4 = 64.48V Z3, 4 = 50.82V


11.2 and 11.3, we can see that to achieve the same filtering characteristics shownin Figure 11.35, the first and last stubs of the conventional filter need an extremelyhigh characteristic impedance of 199 ohms. Such a high impedance level will resultin a very narrow line, which could be very difficult to realize in microstrip with alow-cost fabrication technology. On the other hand, the optimum stub filter hasmuch more reasonable impedance levels for all the short-circuited stubs. Moreover,the characteristic impedances of connecting lines are all close to the terminalimpedance of 50 ohms. As a matter of fact, with a slight modification, all theconnecting lines can be made to have a 50-ohm characteristic impedance. Sincethe unit elements of the optimum stub filter are also accounted for by the orderor degree of the filter, another advantage of the optimum stub filter is that it hasthe fewest number of T-junctions for the same order (11 in this case) of theconventional stub filter. In addition, the separations between adjacent stubs arewider for ul = 70 degrees at 3.5 GHz, resulting in the least interactions betweenneighboring stubs or junctions.

Based on circuit parameters given in Table 11.3, an optimum UWB microstripfilter has been designed on a 1.27-mm-thick RT/Duroid 6006 substrate with adielectric constant of 6.15. The layout of the designed microstrip filter is shownin Figure 11.36. All the connecting lines or unit elements have been modified tohave a characteristic impedance of 50 ohms, so the filter has a very simple structureof a straight 50-ohm line loaded with six short-circuited stubs. The narrowest stubline has a width of 0.3 mm, which can be easily fabricated using a low-cost printed-circuit-board (PCB) technology. The filter dimensions indicated on the layout havetaken into account the effects of T-junction discontinuities. Also, when determiningthe electrical length, a nominal characteristic frequency, fc = 4 GHz has been usedinstead of 3.5 GHz because the substrate used tends to shift down the frequencyresponse.

The filter was fabricated using conventional PCB technology and a photo ofthe 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, theinput and output ports are extended to connect to two SMA connectors. Thevia-hole grounds were simply implemented using soldering. The measured resultswithout any tuning or trimming are plotted in Figure 11.37(b). We can see thatalthough the filter has only six stubs along a 50-ohm line, its measured frequencyresponse of S11 does show eleven ripples in the passband, similar to a typical11-pole 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 6.15and a thickness of 1.27 mm.


Figure 11.37 (a) Photo of a fabricated optimum UWB microstrip filter. (b) Measured performance.

at a mid-band frequency of 8.9 GHz, equivalent to a factional bandwidth of about120%, which is in good agreement with the design or simulation. The insertionloss at the mid-band was measured to be 0.75 dB. The insertion loss tended toincrease at higher frequencies, which is attributed to the frequency-dependent lossesof 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 aboutsome interesting characteristics. Figure 11.38 depicts a generic circuit topology foran optimum stub filter with cross coupling. In this circuit topology, the conventionaloptimum stub filter has the same circuit model as that of Figure 11.34 and thecross coupling is introduced between the source and load (i.e., I/O ports), using acoupled line section with a pair of modal impedances denoted by Z0e and Z0o andan electrical length ueo . There is also a new pair of connecting lines between thecoupled lines and the conventional optimum stub filter, and the characteristicimpedance and electrical length of the connecting lines are Zc1 and uc1 , respectively.


Figure 11.38 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 cross-coupled optimum stub filter.The circuit is modeled using Microwave Office, a commercially available computer-aided design tool [21].

Figure 11.40(a) shows the primary or desired passband performance of thefilter. As can be seen from its return loss or S11 response, there are nine transmission

Figure 11.39 A cross-coupled optimum stub filter.


Table 11.4 Circuit Parameters for the Optimum Stub Filter with Cross Coupling of Figure 11.39

Coupled Lines Stub Lines Connecting Lines

Z0e = 65.8V Z1 = Z3 = 63.4V Zc1 = 86.1VZ0o = 56.7V Z2 = 48.4V Z12 = Z23 = 102.6Vueo = 50 degrees at 3.8 GHz us = 50 degrees at 3.8 GHz uc1 = ul = 100 degrees at 3.8 GHz

poles in the passband, implying the filter response is of order 9. However, theconventional stub filter in this case would only have a degree of 5. In fact, the twoconnecting lines between the coupled-line section and the conventional stub filteradd two more poles, and the coupled-line section also contributes additional twopoles. This makes for a total of nine poles. It is notable that the arrangement ofcross coupling, in this case the coupled-line section, is able to increase the orderof the filter. This is different from cross-coupled narrowband filters discussed inChapter 10, where the arrangement of cross coupling does not increase the orderof the filter. For the filter topology of Figure 11.38, the cross coupling resultingfrom the coupled-line section also produces transmission zeros in the stopband.Almost equal-ripple responses in both passbands and stopbands are attainable withthis arrangement, which is clearly demonstrated by the wideband responses inFigure 11.40(b). In addition, the cross coupling of this type has an effect on groupdelay 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 Coupled-Line Filters

In Section 10.5, we showed how the first harmonic frequency of a stepped imped-ance resonator (SIR) can be useful in the design of dual-band filters. In this section,we will discuss how this concept can be extended to include more harmonic frequen-cies for the design of UWB filters [57, 58].

Figure 11.41(a) depicts a microstrip SIR that is weakly coupled to two externalports for EM simulation of its resonant frequency responses. The SIR has a configu-ration similar to that of Figure 10.60(b) with a low-impedance line in the middleand high-impedance lines on either end. The dimensions shown are in millimeterson a substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm. Thelow-impedance line has a width of 1.1 mm with a characteristic impedance of48 ohms at 6 GHz. The high impedance lines are 0.1 mm wide, resulting in acharacteristic impedance of 107 ohms at 6 GHz. In general, the characteristicimpedances are frequency-dependent due to dispersion in the microstrip. Each high-impedance line section is 4 mm long, while the low-impedance line in the middleis 7.5 mm long. Figure 11.41(b) shows the resonant frequency response of the SIRover 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. Thefirst or fundamental resonant mode resonates at f1 = 4.5 GHz; the second resonantmode is at f2 = 7.05 GHz; the third resonant mode at f3 = 9.65 GHz; and thefourth resonant mode at f4 = 13.7 GHz. Thus, the ratios f2 /f1 , f3 /f1 , and f4 /f1are 1.567, 2.144, and 3.044, respectively. These irregular ratios are due to theparticular configuration of the SIR and the dispersive nature of microstrip. In


Figure 11.40 Performances of the filter of Figure 11.39: (a) magnitude responses of the primarypassband, (b) wideband responses, and (c) group delay of the primary passband.


Figure 11.41 (a) A microstrip SIR on a dielectric substrate with a dielectric constant of 10.8 anda thickness of 1.27 mm. (b) Its multimode resonant frequency responses.

practice, the frequency ratios can be manipulated by changing the width and lengthof the high- and low-impedance lines. For our application, we will use the firstthree 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 thesemodes 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 comprisedof 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 EM-simulated performance of the UWB filter. It has a 3-dB bandwidthfrom 3.45 GHz to 11 GHz, which is centered at 7.225 GHz. Within the passband,


Figure 11.42 (a) A microstrip multimode UWB filter with coupled-line I/O feed sections on adielectric substrate with a dielectric constant of 10.8 and a thickness of 1.27 mm.All dimensions are in millimeters. (b) EM-simulated magnitude responses.(c) EM-simulated group delay response.


there are five transmission poles. It is obvious that three of these transmission polescorrespond to the three resonant modes in the SIR. Since the coupled line sectionsare about a quarter-wavelength long at the mid-band frequency, they contributethe two additional transmission poles in the passband. The group delay responseof the designed UWB filter is plotted in Figure 11.42(c). The group delay in themid-band is about 0.23 ns. It remains quite flat over most of the central portionof 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 ataround 14 GHz. This spurious response is due to the fourth resonant mode thatis outside of the desired passband. We also note, from the same figure, that thereis a transmission zero at around 15 GHz. This transmission zero is inherent forthe coupled line sections used, and it occurs at the frequency where the electricallength of the coupled lines is about half-wavelength long. To suppress the unwantedspurious response, some ideas and techniques described in Section 10.2 may beadopted. For demonstration, a modified UWB filter is shown in Figure 11.43(a).The main technique used here is adding open-circuited stubs at the ends of theouter coupled lines. Each of the open-circuited stubs is 0.25 mm wide and 0.5mm long. Open-circuited stubs can however deteriorate the in-band return lossperformance. To compensate for this, tapered coupled lines are introduced [58].By adding the open-circuited stubs, which corresponds to capacitance loading atthe ends of coupled lines, the transmission zero can be effectively shifted down.When the transmission zero is shifted to the frequency that coincides with theresonant frequency of the fourth resonant mode, the spurious response due to thismode is suppressed. This effect can clearly be seen from the EM-simulated responseof Figure 11.43(b). As compared with the response of Figure 11.42(b), it is evidentthat the modified UWB filter has successfully suppressed the spurious responsearound 14 GHz, improving the selectivity at the high side of the passband.

Higher selectivity UWB filters can be constructed using more multimode resona-tors. Figure 11.44(a) illustrates a coupled line UWB filter, which uses the twoidentical multimode resonators that are coupled to each other through a two-linecoupled section in the middle as shown. The performance of this filter is shownin Figure 11.44(b).

Based on a similar concept to that discussed in this section, other alternativeimplementations of multimode resonator filters are possible for UWB applications.

11.5.3 Microstrip-Coplanar Waveguide Coupled-Line Filters

A compact UWB filter structure as shown in Figure 11.45 is proposed in [59],which is based on a broadside-coupled microstrip-coplanar waveguide (CPW) struc-ture. As shown in the figure, a single dielectric substrate is used. An open-end CPWsection is fabricated in the ground conductor of the microstrip line; this thenprovides a very simple and compact filter configuration. The basic section of thefilter has two microstrip lines separated with a gap and broadside coupled to oneopen-end CPW on the ground through the dielectric substrate. The broadside-coupling and the existence of the dielectric substrate make the coupling betweenthe microstrip line and the CPW very tight. Tight coupling provides a very widebandpass operation.


Figure 11.43 (a) The modified microstrip multimode UWB filter on a dielectric substrate with adielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millime-ters. (b) EM-simulated filter performances.

The operation of this type of UWB filter can be understood by modeling thestructure as shown in Figure 11.46. Assume a dielectric substrate with a thicknessof 0.508 mm and a dielectric constant of 2.17. The two microstrip line sectionson the top have a width of 3.6 mm and a length of L which couple to the bottomCPW resonator. The CPW resonator is 3.6 mm wide and 15.2 mm long with agap of 0.2 mm to the ground. For EM simulations, the resistance of two ports ismatched to the characteristic impedance of the microstrip and the port referenceplanes are also shifted to eliminate a discontinuity (impedance step) effect, whichwould only be secondary as far as the filter operation is concerned.

Figure 11.47 shows EM-simulated response of the basic UWB filter structurefor different values of L over a frequency range from 0.5 GHz to 22 GHz. When


Figure 11.44 (a) Coupled microstrip line UWB filter using two multimode resonators on a dielectric substratewith a dielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millimeters.(b) EM-simulated filter performances.

Figure 11.45 UWB bandpass filter using broadside-coupled microstrip-coplanar waveguide struc-ture. (From: [59]. 2005 IEEE. Reprinted with permission.)


Figure 11.46 Basic UWB filter structure for EM modeling on a dielectric substrate with a dielectricconstant of 2.17 and a thickness of 0.508 mm. (a) Top view (microstrip). (b) Backview (CPW). All dimensions are in millimeters.

Figure 11.47 EM-simulated S21 responses of the basic UWB filter structure of Figure 11.46.


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 20.9GHz, respectively. As L increases, the I/O coupling also increases. A wide passbandstarts to appear, and the three resonant modes shift towards some lower frequencies.The resonant frequency shift can be attributed to some capacitive loading due tothe 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 ortransmission zero at 19.1 GHz for L = 5.2 mm. This results from the microstrip-CPW coupled line sections. To see this effect, a single microstrip-CPW coupledline section for the same length of L = 5.2 mm, as depicted in Figure 11.48, ismodeled using full-wave simulation. The simulated frequency response is plottedin Figure 11.49. For comparison, the filter response for L = 5.2 mm is also plottedin the same figure. As can be seen, both structures have the same attenuation polefrequency.

Based on the results of full-wave EM simulations and discussions presentedabove, a simple circuit model as shown in Figure 11.50 can be established for theUWB filter structure considered. Each of the microstrip-CPW coupled line sectionsis represented by two series open-circuited stubs separated by a unit element (UE);all have an electrical length of u. The transmission line in the middle with anelectrical length of f represents the CPW portion that is not part of coupled linesections. Zc1 , Zc2 , Zc3 , and Zu denote the characteristic impedances of thosetransmission line elements. Z0 is the terminal impedance. Apparently, the seriesopen-circuited stubs can result in an open-circuit along the main signal path whenu is 180 degrees at the first attenuation pole frequency.

Figure 11.48 Microstrip-CPW coupled line section on a dielectric substrate with a dielectric constantof 2.17 and a thickness of 0.508 mm. (a) Top view (microstrip). (b) Bottom view(CPW). All dimensions are in millimeters.


Figure 11.49 Comparison of S21 frequency responses of the microstrip-CPW coupled line sectionand 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 twodifferent sets of circuit parameters. The first case in Figure 11.51(a) is whenZ0 = 50V, Zu = 69V, Zc1 = 10V, Zc2 = 25V, Zc3 = 56V, u = 90 degrees, andf = 3 degrees at f0 = 6.5 GHz. In this case, the CPW resonator is about lg /2 longat 6.5 GHz, where lg is the guide wavelength of the transmission line, and theresultant filtering structure is similar to that shown in Figure 11.45. As can be seenfrom its theoretical response, there are only three transmission poles in the passband,indicating its equivalence to a three-pole filter. For the given bandwidth, the selectiv-ity is mainly set by the two transmission zeros, one at dc and the other at the upperstopband at 2f0 . As a result, the passband skirt is quite soft with a little selectivityfor a single section UWB filter of this type. Nevertheless, a more selective UWBfilter can easily be built by cascading more sections of this type of structure. Thishas been demonstrated in [59] with a selective UWB filter using three sections ofthat of Figure 11.45. The selective UWB filter was fabricated on a 0.508-mm-thickArlon Diclad 880 substrate with a dielectric constant of 2.17 and a loss tangentof 0.00085 at 10 GHz. Each of the CPW resonators is 3.8 mm wide and 14.2 mmlong with a gap of 0.2 mm to the ground plane. Each of the microstrip-CPWcoupled lines is 7 mm long, and the microstrip is also 3.8 mm wide. The connectinglines for the three cascaded filtering sections have a characteristic impedance of50V, and each is 1.6 mm wide and 1.8 mm long. This three-section UWB filter hasa size of about 50 mm × 5 mm on the substrate. The measured results demonstrate


Figure 11.51 Theoretical responses of the equivalent circuit model. (a) Case 1: Z0 = 50V,Zu = 69V, Zc1 = 10V, Zc2 = 25V, Zc3 = 56V, u = 90 degrees, and f = 3 degrees at6.5 GHz. (b) Case 2: Z0 = 50V, Zu = 76.3V, Zc1 = 5.75V, Zc2 = 80.65 V,Zc3 = 81.7V, u = 90 degrees, and f = 180 degrees at 6.5 GHz.

a passband from 3.0 GHz to 10.63 GHz (−10-dB bandwidth) and an upper stop-band attenuation larger than 22 dB up to 16 GHz, which can almost meet theFCC’s indoor limit. The measured group delay of the three-section UWB filter is0.27 ns.

For the second case in Figure 11.51(b), we have Z0 = 50V, Zu = 76.3V,Zc1 = 5.75V, Zc2 = 80.65V, Zc3 = 81.7V, u = 90 degrees, and f = 180 degreesat 6.5 GHz. Since f = 180 degrees in this case, the CPW resonator is about oneguide wavelength or lg long at 6.5 GHz. As compared to case 1, the size ofthe resultant filter for case 2 would be double. However, more selective filteringcharacteristics can be obtained with a single section. As we can clearly see fromthe theoretical response of Figure 11.51(b), there are five transmission poles in thepassband, which implies that a single filter case 2 section is equivalent to a five-


pole filter with a higher selectivity. In fact, since the CPW resonator is lg long atthe 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. Theother two transmission poles can be attributed to the two quarter-wavelengthmicrostrip-CPW coupled line sections. Since u = 90 degrees at the center frequency,the coupling is strongest for the second resonant mode, which resonates at thatfrequency.

Figure 11.52 shows physical implementation of this type of filter on a substratewith a dielectric constant of 10.8 and a thickness of 0.635 mm, where the topview 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 Microstrip-CPW coupled line UWB filter on a dielectric substrate with a dielectricconstant of 10.8 and a thickness of 0.635 mm. (a) Top microstrip view. (b) BottomCPW view. All dimensions are in millimeters.


case) than the middle section where the gap is only 0.2 mm. The large gap notonly increases the I/O couplings, but also controls the modal resonant frequenciesof the CPW resonator [60]. As a matter of fact, we can see this CPW resonatoras 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 characteristicimpedance whereas the middle section with a smaller gap to ground has a lowercharacteristic impedance. The length of a microstrip-CPW coupled-line section isdenoted by L in the figure. For the filter to be properly tuned, L is chosen so asto have an electrical length of 90 degrees at the center frequency. Figure 11.53(a)shows the EM-simulated performance of this UWB filter for L = 3.7 mm. Asexpected, five transmission poles appear in the passband with a 3-dB bandwidthfrom 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 responseis due to the dispersion in both the CPW resonator and the microstrip-CPW coupled

Figure 11.53 EM-simulated performances of the microstrip-CPW coupled line UWB filter of Figure11.52 for L = 3.7 mm. (a) Magnitude response. (b) Group delay response.


lines. The group delay is about 0.22 ns at the mid-band and increases towards thepassband edges. The asymmetrical passband and group delay responses are all dueto the location of the transmission zero in the upper stopband, which seems to betoo close to the passband. The location of the transmission zero can be controlledby the length of microstrip-CPW coupled line sections. If L is reduced, the transmis-sion zero will be shifted to a higher frequency. The EM-simulated performancesof such a modified filter are shown in Figure 11.54. This modification was simplychanging L from 3.7 mm to 3.5 mm, and the main purpose was to move thelocation of the transmission zero to coincide with the spurious resonant frequencyat 13.71 GHz, so as to suppress the spurious response. As can be seen from Figure11.54, this objective has been achieved and a more symmetrical passband responseis 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 toother implementations. For instance, the CPW resonator can be replaced by a

Figure 11.54 EM-simulated performances of the microstrip-CPW coupled line UWB filter of Figure11.52 for L = 3.5 mm. (a) Magnitude response. (b) Group delay response.


microstrip resonator and the input/output of the filter fabricated on CPW [61]. Afilter of this type is illustrated in Figure 11.55. The filter is fabricated on a dielectricsubstrate with a dielectric constant of 10.8 and a thickness of 0.635 mm. Alldimensions shown in Figure 11.55 are in millimeters. From its I/O on the bottomCPW, the filter is composed of two CPW-microstrip coupled line sections, whichexcite the microstrip resonator on the top. The microstrip resonator is about aguide wavelength long at the mid-band frequency, which corresponds to the caseof Figure 11.51(b). The performance of the filter is shown in Figure 11.56, whichhas been optimized to suppress a spurious response caused by the fourth harmonicof the resonator.

In a more recent article [62], a compact ultra-wideband bandpass filter isproposed based on the composite microstrip–CPW structure illustrated in Figure11.57. First, the microstrip-CPW transitions and the CPW shorted stubs are adoptedas quasi-lumped-circuit elements for realizing a three-pole highpass filter prototype.By introducing a cross-coupled capacitance between input and output ports of thishighpass filter and suitably designing the transition stretch stubs, a compact three-pole ultra-wideband bandpass filter is implemented with two transmission zeroslocated close to the passband edges. Second, to further improve the selectivity withgood out-of-band response, two microstrip shorted stubs are added to form thefive-pole ultra-wideband bandpass filter. The five-pole UWB bandpass filter wasfabricated on a Rogers RO4003C substrate with a dielectric constant of 3.38 anda thickness of 0.508mm. The filter has a very compact size of 8 mm × 11.9 mm.The measured 3-dB bandwidth is 108.5% (3.18–10.72 GHz). The implementedfilter has a minimum loss of 0.48 dB, and the return loss is greater than 17.2 dB

Figure 11.55 CPW-microstrip coupled-line UWB filter on a dielectric substrate with a dielectricconstant of 10.8 and a thickness of 0.635 mm. (a) Top microstrip view. (b) BottomCPW view. All dimensions are in millimeters.


Figure 11.56 EM-simulated performances of the CPW-microstrip coupled-line UWB filter of Figure11.55. (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 cross-coupled capaci-tance and stretch stubs, respectively. Note that the third transmission zero is pro-duced 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 coversmany other existing radio systems. An important issue for UWB systems is toavoid the interference with other existing systems like wireless local-area networks(WLAN), cordless telephones, and IEEE 802.11a Wi-Fi networks. To tackle thisproblem, very narrow rejection or notched band(s) can be introduced into a UWBbandpass filter [63, 64].


Figure 11.57 Top-/bottom-layer circuit layouts of a five-pole UWB bandpass filter on a dielectricsubstrate with a dielectric constant of 3.38 and a thickness of 0.508 mm (w1 =0.89 mm, w2 = 2.03 mm, w3 = 4.318 mm, w4 = 1.143 mm, w5 = 0.38 mm, w6 =2.29 mm, w7 = 3.81 mm, w8 = 0.38mm, w9 = 6.22 mm, w10 = 2.03 mm, w11 =2.03 mm, d1 = 0.38 mm, d2 = 0.635 mm, d3 = 0.635 mm, and d4 = 0.28 mm).(From: [62]. 2006 IEEE. Reprinted with permission.)

Figure 11.58 demonstrates the configuration of a microstrip UWB bandpassfilter with embedded band notch stubs. The UWB bandpass filter design is basedon a circuit model for an optimum bandpass filter of which the connecting linesor unit elements are nonredundant as discussed in Section 11.5.1.

In order to introduce a narrow notched band, three different structures inFigure 11.59 were investigated first. These are a conventional open-circuited stub,a spur line, and an embedded open-circuited stub. In principle, to achieve a narrownotch the characteristic impedance of the conventional open-circuited stub will

Figure 11.58 Microstrip UWB bandpass filter with embedded band notch stubs. All the dimensionsare in millimeters on a dielectric substrate with a dielectric constant of 3.05 and athickness of 0.508 mm.


Figure 11.59 Schematic diagrams of (a) conventional open-circuited stub, (b) spur-line, and(c) embedded open-circuited stub.

become extremely high which may be difficult to fabricate. Alternatively, a spur-line introduced in [65] or embedded open-circuited stub may be utilized. Thestructures were investigated using EM simulations (Sonnet em). All the three struc-tures are implemented on a dielectric substrate with a dielectric constant of 3.05and a thickness of 0.508 mm. The main transmission line connecting to the I/Oports is the same for the three cases and has a width WC = 1.3 mm, whichcorresponds to a 50-V line on the substrate. The conventional quarter-wavelength(l /4) microstrip open-circuited stub has a width Ws = 0.1 mm. The spur-line andembedded open-circuited stub have the same width, Ws = 0.1 mm, as well as thesame gap G = 0.2 mm. The full-wave EM simulation results are illustrated in Figure11.60. It is evident that the embedded open-circuited open stub is favorable forimplementing extremely narrow notched bands.

The narrowband characteristic of the embedded open-circuited stub is attrib-uted primarily to its coupling to the main line, and the notch bandwidth can easilybe controlled by adjusting Ws and G. For example, Figure 11.61 illustrates thesimulated performance of the embedded stub with varying gap where decreasingthe gap reduces the bandwidth. This technique allows us to realize a narrownotch that would not be possible with a conventional open-circuited stub requiringextremely high impedance. For example, to achieve the same bandwidth of a notchproduced by a l /4 embedded stub with a width of 0.1 mm and a gap of 0.2 mm,the conventional l /4 open-circuited stub would need a high impedance of about700 ohms, which is almost impossible to implement in practice.


Figure 11.60 The EM simulation for conventional open-circuited stub, spur-line, and embeddedopen-circuited stub with Wc = 1.3 mm, Ws = 0.1 mm, and G = 0.2 mm on a microstripsubstrate 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 andWc = 1.3 mm.

The achievement of extremely narrow bandwidth is only one advantage of thistechnique. Additionally, since the majority of the current flows around the edgesof the microstrip line, the implementation of this embedded stub technique hasless of an effect on the connecting line performance than the spur-line over a widefrequency range. Hence, the embedded open-circuited stub structure was found tobe the best of the three types of structures that were investigated and was thereforeselected to be implemented in the design of a UWB bandpass filter with a band


notch characteristic. The filter shown in Figure 11.58 has two embedded openstubs in the first and last connecting lines in order to introduce a very narrownotched (rejection) band in the UWB passband. The length of each of the twostubs should be l /4 at the desired center frequency of the notched band to ensurethat the second resonant harmonic of the embedded stub does not appear in thedesired UWB passband. Figure 11.62 illustrates the full-wave EM simulation ofthe complete layout of the designed UWB bandpass filter with varying equal stublengths (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 be0.1 mm and 0.2 mm, respectively. It is also possible to produce two differentnotched bands when desired. In this case, each of the two stubs would have differentlengths. It should be noted that the embedded stubs can generate a notched bandat the desired frequency with no significant influence on the wide passband perfor-mance.

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 of5 mm long was added at both the input and the output. Figure 11.63(b) illustratespredicted and the measured results for the filter associated with the FCC indoormask. The filter exhibited excellent UWB bandpass performance with a FBW ofabout 110% at a mid-band frequency of 6.85 GHz. The measured results showedextremely narrow notched bands in the passband with 10-dB FBW of 4.6% at acenter frequency of 5.83 GHz. The attenuation at the center of the notched bandis more than 23 dB. At the mid-band frequency of each passband, insertion lossof less than 0.5 dB was obtained. The filter also showed a group delay of about0.5 ns at the mid-band frequency of each passband.

The multimode coupled line UWB bandpass filters described in Section 11.5.2can be modified to include a notched band as well. Figure 11.64(a) illustrates an

Figure 11.62 The full-wave EM simulation of the complete layout of the designed UWB bandpassfilter of Figure 11.58 for Ws = 0.1 mm, G = 0.2 mm, and varying equal stub lengths.


Figure 11.63 (a) Fabricated microstrip UWB bandpass filter with embedded band notch stubs ona 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 ultra-wide passband. The microstrip filter is constructed on a substrate with a dielectricconstant of 10.8 and a thickness of 1.27 mm. The dimensions shown are in millime-ters. The key to generating a notch is to introduce an asymmetric input/output(I/O) coupled-line structure as shown in the figure. An open-circuited stub of1.0-mm length and 0.25-mm width is added to the end of one I/O coupled line.Owing to this asymmetric modification, the open-circuited stub together with thecoupled line section will resonate when their combined electric length is a quarter-


Figure 11.64 (a) Multimode coupled line UWB filter with notched band on a 1.27-mm-thick substrate witha dielectric constant of 10.8. (b) EM-simulated magnitude response. (c) EM-simulated groupdelay response.


wavelength long. This results in a short circuit at the I/O port, and thus producesa notch at the resonant frequency. Figure 11.64(b) illustrates the EM-simulatedmagnitude response of the filter, where a notch at 6.6 GHz can clearly be observed.The 10-dB rejection bandwidth of the notch is about 5%. The notch frequencyand bandwidth can easily be adjusted by changing the length and width of theopen-circuited stub. In this case, the notch divides the original UWB into to thetwo sub-passbands, which have the 3-dB bandwidth ranging from 3.45–6.3 GHzand 6.8–11 GHz, respectively. The simulated group delay response of the filter isplotted in Figure 11.64(c). The group delay over the most part of each passbandis about 0.3 ns. However, it increases rapidly at the band edge adjacent to thenotch. This is due to the high selectivity of the notched band.

The microstrip-CPW coupled line UWB bandpass filters discussed in the previ-ous 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 UWBbandpass filters with single or multinotched band(s) for the purpose of detect andavoid, to avoid the interference between a UWB radio system and an existing radiosystem. The filter is modified from the basic filtering structure of Figure 11.45 byintegrating stub(s) in the broadside-coupled conductors to implement the singleand multi-notched band operation. The resonance of each stub introduces a narrowrejection band in the UWB passband, which then results in single or multinotchedband(s). UWB bandpass filters with one or more notched bands are useful andrequired in practical systems in order to avoid the interference between the UWBradio system and legacy radio systems. The notched band can be easily designedto some specific frequency band(s) by tuning the length of the stub(s). The demon-strated filters used three sections of the modified basic filtering structure and showedboth excellent ultra-wide bandwidth (from 2.8 GHz to 10.2 GHz) and rejectionperformance (>50 dB at central frequencies for a single notched band, or 21 dBand 27 dB for triple bands), as well as out-of-band performance better than theFCC requirement [48].

11.6 Metamaterial Filters

Electromagnetic metamaterials are broadly defined as artificial effectively homoge-nous electromagnetic structures with unusual properties not readily available innature [67]. In particular, if such materials exhibit simultaneously negative per-mittivity and permeability (i.e., e < 0 and m < 0), they are also known as left-handed (LH) metamaterials.

The electromagnetic properties of left-handed metamaterials were predicted inthe late 1960s [68]. However, it was not until 2000 that such media were artificiallyfabricated and experimentally demonstrated [69]. To this end, a periodic array ofmetallic posts was combined with a periodic structure consisting of split-ringsresonators (SRRs), which were previously proposed in [70]. Figure 11.65 illustratesthe configuration of an SRR. The key aspect of these resonators is that they areelectrically small and exhibit an effective permeability that is negative in a narrowband above their resonant frequency. Hence, the SRR structure alone may be seenas a metamaterial that exhibits a positive e and a negative m . Furthermore, the

11.6 Metamaterial Filters 429

Figure 11.65 Metal split-ring 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 bydesigning the composite medium with the SRRs and the posts [69]. It is importantto note, however, that the fabricated bulk structure is highly anisotropic, namely,the electric and magnetic field vectors of incident polarization will have significantcomponents in the direction of the posts and rings axes, respectively.

One-dimensional left-handed metamaterials and negative permeability trans-mission lines based on SRRs have recently been proposed in planar technology[71, 72]. These implementations have been mainly realized in coplanar waveguide(CPW) configurations. Left-handed transmission lines in microstrip technologyhave also been reported [73] in which the line is periodically loaded with metallicvias and square shaped split-rings resonators, etched in close proximity to theconductor strip. The SRRs provide the negative effective permeability, m , in anarrow band above their resonant frequency, whereas the metallic vias act as shuntconnected inductors that make the structure behave as a microwave plasma withnegative permittivity, e , up to the plasma frequency. It is shown that in that regionwhere both m and e are simultaneously negative, left-handed wave propagation isallowed. Since this occurs in a narrow band above SRRs resonance, and the periodof the structure is electrically small, these metamaterial transmission lines canbe of interest for filter applications where miniaturization and narrow passbandsare 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 short-circuited with ametallic post realized by via-grounding, and two square SRRs being coupled tothe microstrip. The square SRR is a modification from the circular one of Figure11.65 for enhancing the coupling to the microstrip line. The via-ground post playsan 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 substratewith a dielectric constant of 10.2 and a thickness of 1.27 mm. All the dimensionsshown are in millimeters. The outer loop of the SRR has a size of 5 mm × 5 mmand the inner loop has a size of 3.8 mm × 3.8 mm. The line width of the openloops is 0.2 mm and the gap is 0.4 mm. The two square SRRs are coupled to themain signal line, which is a 50-V line on the substrate, through 0.2-mm spacing.The metal post or via has a diameter of 1.0 mm. Figure 11.67(b) shows the


Figure 11.66 Structure of a microstrip LH metamaterial cell consisting of SRR and via-ground post.

EM-simulated frequency characteristics of the cell, where a narrow passbandaround 3.1 GHz can be observed. This type of structure can find applications inthe design of narrowband bandpass filters, as demonstrated in [73].

For example, Figure 11.68(a) illustrates a microstrip bandpass filter that iscomprised of three LH metamaterial cells of Figure 11.67(a). The filter is simplya 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). Themagnitude response of the filter is plotted in Figure 11.68(b), obtained by full-wave simulation. The filter exhibits an asymmetric frequency response with atransmission zero on the low side of the passband, which improves the selectivityof that side. The passband is narrow as expected, ranging from 2.992 GHz to3.077 GHz with a fractional bandwidth of 2.8%. The group delay of the filter isalso shown in Figure 11.68(c).

Another useful metamaterial structure based on SRRs is depicted in Figure11.69(a), where two circular SRRs of the form of Figure 11.65 are coupled to amicrostrip signal line, although square SRRs may be used. Note that there is novia ground in the microstrip, which is the main difference between this structureand that of Figure 11.67(a). This, however, leads to a completely different frequencycharacteristic as shown in Figure 11.69(b) with a stopband at the fundamentalresonant frequency of the SRRs. The resonant frequency depends on the dimensionsof the SRRs. For this particular example, the structure is simulated on a dielectricsubstrate with a dielectric constant of 10.2 and a thickness of 1.27 mm. Themicrostrip signal line has a width of 1.4 mm on the substrate. The outer radii ofthe two concentric rings are 2.2 mm and 1.8 mm, respectively; the distance betweenthe rings is 0.2 mm and the ring width is 0.2 mm. It is evident that the basicstructure of Figure 11.69(a) can be utilized to design bandstop filters or to suppressunwanted spurious responses in a bandpass filter. The latter application has beendemonstrated in [74] and a configuration of this type of filter is illustrated in Figure11.70, where SRRs have been integrated into a conventional parallel coupled linefilter. The dimensions of SRRs can be determined so as to suppress an unwantedspurious, say at 2f0 , where f0 is the center frequency of the passband. It is alsopossible to use different sizes of SRRs to achieve multispurious suppression.

It is feasible to develop SRR-based filters in a variety of topologies [75, 76].For example, compact bandpass filters based on planar structures with three metal


Figure 11.67 (a) Layout of basic cell of the microstrip LH metamaterial on a dielectric substratewith a dielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are inmillimeters. (b) Its frequency characteristics.


Figure 11.68 (a) Layout of a microstrip LH metamaterial bandpass filter on a dielectric substratewith a dielectric constant of 10.2 and a thickness of 1.27 mm. All dimensions are inmillimeters. (b) EM-simulated magnitude response. (c) EM-simulated group delayresponse.


Figure 11.69 (a) Layout of basic structure of the microstrip metamaterial with coupled SRRs.(b) Typical frequency response, obtained by full-wave simulation.


Figure 11.70 Configuration of an SRR-based 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 andaligned with the slots. The wires make the structure behave as a microwave plasmawith a negative effective permittivity covering a wide frequency range. SRRs, whichare magnetically coupled to the CPW, provide a negative magnetic permeabilityin a narrow frequency range above their resonant frequency. The result is a bandpassstructure that supports wave propagation in a frequency interval where negativepermittivity and permeability coexist. The bandwidth of the structure can be con-trolled by tuning the resonant frequency of the upper and lower SRRs and thedistance between SRRs.

Other developments in metamaterial filters include the use of complementarysplit-ring resonators (CSRRs) [77, 78]. As a matter of fact, the so-called CSRR isthe negative image of an SRR of Figure 11.65. It has been demonstrated thatCSRRs etched in the ground plane or in the conductor strip of planar transmissionmedia (microstrip or CPW) provide a negative effective permittivity to the structure,and signal propagation is precluded, which roughly coincides to that of an SRRwith identical dimensions etched onto the same substrate. To this end, a new designapproach for compact microstrip bandpass filters, based on the use of CSRRs, hasbeen 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 therequired flexibility to synthesize a frequency response with controllable bandwidths.The shunt stubs and series gaps have been modeled by lumped inductors andcapacitors, respectively, and CSRRs have been described by parallel resonant tanks(capacitively coupled to the line). A filter design using three cells of this type hasbeen demonstrated. It is a three-stage periodic structure, which exhibits a centralfrequency at f0 = 1 GHz and a 10% fractional bandwidth. It has been foundthat the measured frequency response fits the targeted specifications to a goodapproximation, and the first spurious band does not appear up to 3f0 . Moreover,it has been demonstrated that the structure behaves as an effective (continuous)medium with left-handed wave propagation in the allowed band. A wideband filterof this type has also been demonstrated in [78], where the filter, operating atC-band, is realized with simultaneously very small dimensions (area < 1 cm2), widebandwidth (fractional bandwidth > 45%) and high frequency selectivity (transitionbands with more than 50 dB/GHz fall-off at both band edges).


Another major development in microwave metamaterial devices is based onthe so-called composite right/left-handed (CRLH) transmission line element [67].Figure 11.71 is the model for a lossless or ideal CRLH transmission line unitelement (cell). It is an equivalent lumped-element circuit where CR and LR arecapacitance and inductance of the right-handed transmission line element and CLand LL are capacitance and inductance of the left-handed transmission line element.While CR and LR can be extracted from the capacitance/inductance per unit lengthof a normal homogeneous transmission line (e.g., microstrip or waveguide), CL andLL will have to be implanted or implemented artificially due to the unavailability ofreal homogeneous LH or CRLH materials. This equivalent circuit itself also impliessome filtering characteristics. This is because the CR and LR act as lowpass filteringelements whereas the combination of CL and LL exhibits a highpass characteristic.

Define two characteristic frequencies from the equivalent circuit:

fse =1

2p√LR CL(11.4a)

fsh =1

2p√LL CR(11.4b)

where fse and fsh are the series and shunt resonance frequencies, respectively. Itcan be shown [67] that if f < min( fse , fsh ), the phase velocity vp and the groupvelocity vg have opposite signs (i.e., they are antiparallel, vp − || vg ), meaning thatthe transmission line is LH and therefore the propagation constant is negative. Incontrast, in the RH range f > max(fse , fsh ), the phase velocity and group velocityhave the same sign (vp || vg ), meaning that the transmission line is right-handed(RH) and the propagation constant is therefore positive. When fse = fsh , the line willbe called balanced. Otherwise, it is called unbalanced. Another useful characteristicfrequency 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 balancedCRLH transmission line characteristics for 5- and 10-cascaded cells of Figure 11.71.The element values of the cell are CL = 1.0 pf, LL = 2.5 nH, CR = 1.0 pF, and

Figure 11.71 Equivalent circuit model for the ideal CRLH transmission line cell.


Figure 11.72 Characteristics of balanced CRLH transmission line. (a) 5 cells. (b) 10 cells. Cell elementvalues: CL = 1.0 pF, LL = 2.5 nH, CR = 1.0 pF, and LR = 2.5 nH.

LR = 2.5 nH. Using (11.4) and (11.5), we find fse = fsh = f0 = 3.18 GHz. Thus, forf < 3.18 GHz, the LH characteristic dominates, whereas the RH characteristicgoverns in the range for f > 3.18 GHz. In general, the balanced CRLH transmissionline itself exhibits a wideband bandpass characteristic. There is always a transmis-sion pole at f0 , and N − 1 poles on both sides of f0 .

For unbalanced CRLH, Figure 11.73 illustrates frequency characteristics forCL = 0.5 pF, LL = 2.5 nH, CR = 1.0 pF, and LR = 2.0 nH. Figure 11.73(a) is forthe 5-cell CRLH transmission line. Figure 11.73(b) is for the 10-cell case.Again, by using (11.4) and (11.5), we can find fse = 5.03 GHz, fsh = 3.18 GHz,f0 = 4.0 GHz. Hence, the line is LH for f < 3.18 GHz, but will be RH whenf > 5.03 GHz. Between 3.18 GHz and 5.03 GHz, there is an attenuation or rejectband and the maximum attenuation occurs at f0 = 4.0 GHz. This type of frequencyselective characteristic may have good use in developing dual-band or multibandmetamaterial filters.


Figure 11.73 Characteristics of unbalanced CRLH transmission line. (a) 5 cells. (b) 10 cells. Cellelement values: CL = 0.5 pF, LL = 2.5 nH, CR = 1.0 pF, and LR = 2.0 nH.

Physical implementation of CRLH metamaterial could be done in differentways and on different microwave transmission media. Figure 11.74 depicts a typicalmicrostrip CRLH unit cell, which was originally proposed in [79]. Basically, thisunit cell consists of an interdigital capacitor and a stub inductor shorted to theground plane by a metallic via. The contributions CL and LL are provided by theinterdigital capacitor and stub inductor, whereas the contributions CR and LRcome from their parasitic reactances and have an increasing effect with increasingfrequency. The parasitic inductance LR is due to the magnetic flux generated bythe currents flowing along the digits of the capacitor and the parasitic capacitanceCR is due to the parallel-plate voltage gradients existing between the trace and theground plane. Applications of this type of CRLH metamaterial have been welldocumented in [67].

A compact multilayered CRLH transmission line is proposed in [80]. Thismultilayered architecture consists of the periodic repetition of pairs of U-shaped


Figure 11.74 Unit cell of the microstrip CRLH transmission line.

parallel plates connected to a ground enclosure by meander lines. The parallel platesprovide the left-handed (LH) series capacitance, and the meander lines provide theLH shunt inductance, while the right-handed parasitic series inductance and shuntcapacitance are generated by the metallic connections in the direction of propaga-tion 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 thatbased on the cell of Figure 11.74, the multilayered transmission line has its directionof propagation along the vertical direction, perpendicular to the plane of thesubstrate. This presents the distinct advantage that a large electrical length can beachieved over an extremely short transmission line length and small transversefootprint. The miniaturized multilayered line can find applications in bandpassfilters, delay lines, and numerous phase-advanced components. As an exampleof application, a 1-GHz/2-GHz diplexer composed of two multilayered CRLHtransmission lines is demonstrated [80].

In summary, electromagnetic metamaterials are new to many of us and theirrealizations and applications require imagination and new thinking. Nevertheless,innovative engineering approaches to metamaterials will pave the way for newgeneration of RF/microwave devices including filters. Table 11.5 shows the advan-tages of each technology.

Table 11.5 Comparison of Advanced Filter Technologies

HTS MEMS LTCC/LCP Metamaterial

Very low loss High level of integration Compact and low cost MiniaturizationHigh selectivity Millimeter-wave Millimeter-wave Potential for new

applications applications functionalitiesHigh level of integration(MCM, SOP)

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C H A P T E R 1 2

Coupled-Line Circuit Components

DC blocks, impedance transformers, interdigital capacitors, and spiral inductorsemploying coupled-line sections are commonly used in microwave circuits. Acoupled-line section provides the required characteristics for these components overa wide frequency range. This chapter deals with these structures and includes thebasic 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 aswell as to block dc and low-frequency voltages while allowing the RF signal topass through with minimal loss. At microwave frequencies, both a high-qualitycapacitor and a distributed coupled network are used. This section only deals withthe coupled-line structures used as dc block networks.

12.1.1 Analysis

A 3-dB backward-wave coupled-line section [1–3] with open circuit terminationsas shown in Figure 12.1(a) can be used instead of a series capacitor as a series dcblock. The circuit used for analysis is shown in Figure 12.1(b). The scatteringmatrix of a microstrip backward-wave coupler, when ports are termined in matchedloads, is given by

3b1

b2

b3

b44 = 3

0 E2 0 E4

E2 0 E4 00 E4 0 E2

E4 0 E2 04 3

a1

a2

a3

a44 (12.1)

where

E2 =jk sin u


E4 = √1 − k2


443

444 Coupled-Line Circuit Components

Figure 12.1 Quarter-wave coupled-line section as a dc block: (a) physical layout and (b) circuitschematic.

and

u =12

(ue + uo )


where u is the electrical line length and subscripts e and o denote even and oddmodes, respectively. When ports 2 and 4 are terminated in reflecting loads, thena2 = rb2 , a3 = 0, a4 = rb4 , where r is the reflection coefficient at ports 2 and 4.For an ideal coupler when Z0e Z0o = Z2

0 , the circuit ensures perfect match at theinput. As there are reflecting terminations at ports 2 and 4, there is generally nota perfect match at port 1, except when the operating frequency and coupling lengthare such as to give exactly equal outputs at ports 2 and 4. At the input, the reflectedwave amplitude is given by

b1 = E2 rb2 + E4 rb4 (12.2)

Similarly, expressions for b2 and b4 are

12.1 DC Blocks 445

b2 = E2a1 (12.3)

b4 = E4a1 (12.4)

From (12.2) to (12.4):

r in =b1a1

= XE22 + E2

4 Cr =1 − k2(1 + sin2 u )

F√1 − k2 cos u + j sin uG2 r (12.5)

For

k = 1/√2 and r = 1

Note: r = (ZL − Z0)/(ZL + Z0), where ZL is the load impedance.

VSWR =1 + | r in |1 − | r in |

= 1/sin2 u

When u = 90 degrees, VSWR = 1.We note that for a 3-dB backward-wave coupler, the direct and coupled waves

have equal amplitude when the coupling length u is 90 degrees at the coupler’scenter frequency. The variation of VSWR with frequency for various couplingvalues is shown in Figure 12.2. In the calculations, ports 2 and 4 are assumed tobe totally reflecting (r = 1), perfect open circuits. From Figure 12.2 we find thatfor a 2.7-dB 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 dcblock is that at lower frequencies, it increases the size of the circuit. At millimeterwave frequencies, where the size becomes small, however, this structure is preferredin comparison with the chip or MIM capacitor because of lower loss.

Knowing Z0e and Z0o , the physical dimensions can easily be obtained as forany other coupler [4–6]. Simple expressions are also available [7] for Z0e and Z0ofor a given VSWR and bandwidth, that is:

Z0e = √S F1 + √1 +1 + √1 + V2

V2 S1 −1SDG Z0 (12.6a)

Z0o = √S F−1 + √1 +1 + √1 + V2

V2 S1 −1SDG Z0 (12.6b)

where S is the voltage standing wave ratio and V is a normalized bandwidth givenby

V = cotHp2

[1 − ( f2 − f1)/( f2 + f1)]J (12.7)

where f1 and f2 are the lower and upper edge of the frequency band.


Figure 12.2 Simulated VSWR versus fractional bandwidth of a coupled-line dc block.

The value of coupling coefficient, k, in the dc block design is not very criticalas long as it is greater than 0.5. However, for a given coupled length, a tightercoupling 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 andconstructed using a 25-mil-thick alumina substrate (er ≅ 10). The physical dimen-sions obtained for Z0e = 130V and Z0o = 24V were spacing S = 1 mil, widthW = 6.3 mil, and length l = 100 mil. Measured and simulated VSWR and insertionloss responses are shown in Figure 12.3. Over an octave band, the worst casemeasured VSWR and insertion loss were about 1.4 and 0.2 dB, respectively. Moreaccurate transmission phase agreement between the measured and simulated resultswas obtained [3] by including open-end discontinuity effects in the calculations.

12.1.3 Biasing Circuits

Solid-state circuits require low frequency biasing networks which must be separatedfrom 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 isolatethe bias voltages applied to various devices. In the case of amplifiers and oscillators,

12.1 DC Blocks 447

Figure 12.3 (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 manyothers, the biasing circuitry becomes an integral part of the circuit design. Thereare many biasing schemes used in practice. Bascially it consists of a dc block andan RF choke as shown in Figure 12.4. A dc block can be either a capacitor ora 3-dB backward-wave coupler described earlier. The RF choke at microwavefrequencies is generally realized by using a high-impedance l /4 line terminated bya RF bypass capacitor or a quarter-wave line terminated by another l /4 open-circuited line or a radial stub as shown in Figure 12.5. For low-RF leakage throughthe biasing network, the ratio of the shunt stub impedance and the through-line,impedance must be much greater than unity. In this case, the bandwidth increaseswhen the impedance of the stub increases. For VSWR ≤ 1.05, the bandwidth forZs = 100V is about 12%. To further increase the bandwidth, two sections ofquarter-wave-long transmission lines are used. If an open circuit is required acrossthe main line for RF signals, a quarter-wave high-impedance line followed byanother open-circuited quarter-wave low-impedance line are connected. The config-uration is shown in Figure 12.5(a). Assuming that the through line is 50V, thenormalized admittance with the load is

y = 1 + 1/Zin (12.8)


Figure 12.4 Simplified microwave biasing circuit.

Figure 12.5 Microwave biasing circuits using multisections shunt stubs for large bandwidths.(a) Two l/4 sections configuration and (b) a combination of l/4 section and radialline configuration.

where

Zin = jZ1Z1 tan u1 tan u2 − Z2Z1 tan u2 + Z2 tan u1

(12.9)

Here u1 , Z1 and u2 , and Z2 are the electrical line length and characteristic imped-ance of the first and second line sections, respectively. The VSWR response forvarious combinations of Z1 and Z2 is shown in Figure 12.6. Maximal bandwidthis obtained when Z1 /Z2 is large. For example, with Z1 = 100V, Z2 = 10V,Z0 = 50V, and a VSWR = 1.2, the bandwidth is about 40%.

A radial line section provides better bandwidth than a l /4 open-circuit linesection. A broadband biasing network structure was designed on a semi-insulating

12.1 DC Blocks 449

Figure 12.6 Simulated response of several two-section biasing networks as a function of normalizedfrequency. Impedance values are in ohms.

GaAs substrate with parameters listed in Table 12.1. Other structure parametersare given in Figure 12.7. All dimensions are in micrometers. The structure wasanalyzed and optimized using a commercial CAD tool. The measurements weremade on the wafer using RF probes and TRL calibration techniques. The measuredand simulated performance (return loss and insertion loss) of this biasing networkare 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 Millimeter-Wave DC Block

At millimeter-wave frequencies the coupled-line dc block has distinct advantagesover the conventional dc block, which generally consists of chip or MIM capacitors.

Table 12.1 Parameters for the BiasingNetwork Structure

Substrate height, h = 125 mmSubstrate dielectric constant, er = 12.9Conductor thickness, t = 4.5 mmConductor’s bulk conductivity, s = 4.9 × 107 S/mSubstrate loss tangent tan d = 0.0005


Figure 12.7 Biasing network using the Figure 12.5(b) configuration. Coupled line length is2,045 mm.

Figure 12.8 Simulated and measured |S11 | versus frequency of the biasing network.

Figure 12.9 Simulated and measured |S21 | versus frequency of the biasing network.

12.1 DC Blocks 451

These structures are broadband and cover full waveguide bands with low insertionloss. Several dc blocks on RT-5880 duroid substrates were designed and fabricatedwith dimensions given in Table 12.2 for several millimeter-wave frequency bands[9]. The length of the coupled section is 90 degrees at the center of the band. Themeasured insertion loss for these structures was between 0.2 to 0.4 dB over thefull-waveguide band.

12.1.5 High-Voltage DC Block

The above-described coupled-line dc blocks are capable of handling voltages ofless than 200V, depending on the fabrication tolerances and humidity [10]. Toincrease the protection against voltage breakdown across the gap, an overlay ofsilicon rubber as shown in Figure 12.10 has been used. Because dielectric loadingmodifies the electrical characteristics of the coupled line, accurate design or simula-tion methods such as electromagnetic simulators are required to determine thenew parameters. A dc block fabricated on an RT/duroid substrate with spacingS = 50 mm and width W = 60 mm, achieved a breakdown voltage over 4.5 kV. Ingeneral, breakdown occurs at the open ends of the coupled lines [10].

Table 12.2 A Comparison Between Designed and MeasuredDimensions of the Coupled Microstrip DC Blocks

Designed MeasuredDimensions Dimensions

h (mil) (mil)Band (mil) er W S W S

Ka 10 2.2 7.0 1.0 6.1 1.7V 5 2.2 2.5 1.0 2.0 1.7W 5 2.2 2.5 1.0 2.0 1.5

Figure 12.10 Top and side views of high voltage dc block showing high-voltage insulator dielectric.(From: [10]. 1993 IEEE. Reprinted with permission.)


12.2 Coupled-Line Transformers

Quarter-wavelength-long tightly coupled lines have been used as filter elements,directional couplers, and dc blocks, and another important application is in broad-band impedance-matching transformers. There are several configurations one canuse for coupled-line impedance transformers including simple open-circuit coupled-line transformers and transmission-line transformers. These two are discussedbriefly in this section.

12.2.1 Open-Circuit Coupled-Line Transformers

A simple representation of a coupled-line impedance transformer is shown inFigure 12.11. This scheme works very well at millimeter-wave frequencies whereit eliminates dc blocking capacitors and can handle large impedance transformationwithout transverse resonances which occurs in conventional l /4 low-impedancemicrostrip single-section impedance transformers.

The analysis of this structure can easily be performed by using its equivalentcircuit shown in Figure 12.12, where the expressions for Zss and n are given by

Figure 12.11 Coupled-line transformer section.

Figure 12.12 Equivalent circuit representation of the coupled-line impedance transformer.

12.2 Coupled-Line Transformers 453

Zss = ZL (1 − n2) (12.10)

n = k =Z0e − Z0oZ0e + Z0o

(12.11)

Under maximum power transfer from port 1 to port 2:

Z0e = Z0o + 2√Zin ZL (12.12)

If ZL is the load impedance, the input impedance is given by

Zin = −jZss cot u + n2ZL (12.13)

When u = 90 degrees:

Zin = n2ZL (12.14)

From (12.11), (12.12), and (12.14):

ZL =Z0e + Z0o

2(12.15)

If Zin = ZS :

Z0e = ZLS1 + √ZSZLD (12.16a)

Z0o = ZLS1 − √ZSZLD (12.16b)

For given ZS and ZL , the even- and odd-mode impedances of the coupled-linetransformer can be determined from (12.16a) and (12.16b). Most of commercialCAD tools can synthesize these networks for given Z0o , Z0e , and substrate parame-ters in terms of physical dimensions W, S, and l.

If ZS and ZL are the source and load impedances, the transmission coefficientS21 is given by [7]

S21 =2(Z0e csc ue − Z0o csc uo )

SS√ZLZS

+ √ZSZLD (Z0e cot ue + Z0o cot uo )

(12.17)

+ jFZ20e + Z2

0o

2√ZS ZL−

Z0e Z0o

√ZS ZL(csc ue csc uo + cot ue cot uo ) + 2√ZS ZLGD

where ue and uo are the electrical lengths of the structure corresponding to theeven- and odd-mode propagation constants. The electrical lenghts are given as


ue =2pl0

,√eree and uo =2pl0

,√ereo (12.18)

where l0 is the free-space wavelength, and eree and ereo are the effective dielectricconstants corresponding to the even- and odd-mode propagation, respectively.

Under first-order approximation when ue = uo = p /2, and S21 = 1 (matchedcondition for lossless network), (12.17) provides

Z0e = Z0o + 2√ZS ZL (12.19)

which is the same as (12.12) when ZS = Zin .The quarter-wavelength of the coupled section at the center frequency may be

calculated from the following relation [11]:

, =p /2

K + [(Z0e − Z0o )/(Z0e + Z0o )]DK(12.20)

where

K = (be + bo )/2

DK = (be − bo )/2

be = 2p√eree /l0

bo = 2p√ereo /l0

Table 12.3 summarizes the electrical and physical parameters for typical microstripcoupled-line transformers. Parameters from the table were used to compare thistype of transformer with a l /4 microstrip section to transform 10V impedance to50V impedance. Three cases (9, 10, and 11 in Table 12.3) were considered forcoupled lines with the physical line length used being 2,200 mm. For the single-section case, W = 2,200 mm and l = 1,850 mm were used. The substrate parametersused are thickness h = 625 mm, conductor thickness t = 3 mm, er = 9.8, tan d =0.0005, and the conductors were of gold. Figure 12.13 shows the performancecomparison. Although a l /4 single microstrip transformer gives the best electricalperformance in terms of insertion loss and bandwidth, its physical dimension(width) is too large in the transverse direction. A transverse resonance may occurwhen the width of the conductor is l /2. In this example the resonance frequencyis about 26 GHz.

Symmetrical coupled-line section transformers were described above. Asymmet-rical coupled sections can also be used as transformers and offer better bandwidths.Coupled-line transformers are good candidates for active device impedance match-ing at millimeter-wave frequencies where they also serve the purpose of low-lossdc blocks. Thus, coupled-line transformers provide wide bandwidth and eliminatethe use of dc blocking capacitors in active circuits.

12.2C

oupled-LineTransform

ers455

Table 12.3 Parameters for Different Transformation Ratios: er = 9.8, t = 3 to 4 mm, and h = 635 mm

Transformation Desired Impedances Actual Impedances Velocity Factors Physical DimensionsSerial From: To: Z0e Z0o Z0e Z0oNumber V V V V V V K0e K0o W/h S/h

1 50.0 75.0 162.5 40.0 162.40 40.59 0.404 0.4300 0.120 0.0482 50.0 75.0 152.5 30.0 153.81 30.24 0.403 0.4298 0.159 0.0183 50.0 53.0 142.96 40.0 143.06 40.47 0.401 0.4298 0.175 0.0704 50.0 53.0 132.96 30.0 132.50 30.72 0.399 0.4296 0.240 0.0305 50.0 35.0 123.67 40.0 123.24 39.83 0.397 0.4293 0.260 0.1356 50.0 35.0 113.67 30.0 113.75 29.96 0.394 0.4291 0.350 0.0407 50.0 25.0 110.71 40.0 110.28 40.03 0.394 0.4287 0.335 0.1358 50.0 25.0 100.71 30.0 100.69 30.07 0.391 0.4284 0.455 0.0559 50.0 10.0 84.72 40.0 84.66 39.81 0.385 0.4261 0.565 0.250

10 50.0 10.0 74.72 30.0 74.57 30.17 0.381 0.4253 0.790 0.11511 50.0 10.0 64.72 20.0 65.11 20.14 0.376 0.4246 1.05 0.01

K0e = 1/√eree and K0o = 1/√ereo


Figure 12.13 Simulated performance of several 10V to 50V transformers on 25-mil aluminasubstrates.

12.2.2 Transmission Line Transformers

Transmission line transformers (TLT) using straight or coiled sections of coupledtransmission lines are frequently used to realize multi-octave impedance transforma-tion at RF and lower microwave frequencies [12–21]. This type of a transformercan be designed employing any multilayer fabrication technology such as a printedcircuit board, LTCC, HTCC, and monolithic Si or GaAs ICs. Figure 12.14 showsa 4:1 impedance transformer (ZS = 4ZL ), where ZS and ZL are the source andload impedance, respectively. Here transmission lines A and B are not electromag-netically coupled and their length (L) is typically l /4. In this configuration, eachline has a characteristic impedance Z0 . If the lines are coupled, such as the side-coupled microstrip lines, shown in Figure 12.15 (top view), the bandwidth increasesand the transformer length decreases with stronger coupling between the conduc-


Figure 12.14 Impedance transformer configurations using two uncoupled transmission lines Aand B.

Figure 12.15 Impedance transformer configurations using side-coupled microstrip lines. Groundplane not shown.

tors. In the asymmetric broadside-coupled microstrip lines shown in Figure 12.16(side view), the coupling coefficient is much stronger. Therefore, this configurationresults in better bandwidth and smaller transformer size. In this case, the length(L) is typically l /8. When ports 1 and 2 are switched, then the transformer isdesignated as 1:4 TLT.

The design of an asymmetric broadside-coupled TLT cannot be performedaccurately using conventional circuit simulators, because the substrate and theconductor are of a multilayer type. However, accurate solutions can be obtainedby using an electromagnetic simulator. A 50-ohm to 12.5-ohm (4:1) asymmetricbroadside-coupled TLT was designed [21] using em by Sonnet software. The sub-strate parameters for impedance transformer are given in Table 12.4. The conduc-tors have a width W and a length L. The characteristics of this transformer werecompared with three other transformers: a single section quarter-wave microstrip,uncoupled lines TLT, and side-coupled lines TLT. For these transformers, Figures

Figure 12.16 Asymmetric broadside coupled microstrip lines.


Table 12.4 Substrate Parametersfor Broadband ImpedanceTransformers

GaAs substrate er = 12.9Substrate thickness h = 75 mmPolyimide erd = 3.2Polyimide thickness d = 7 mmGold conductor’s thickness T = 4.5 mm

12.17 and 12.18 show the reflection and transmission coefficients versus frequency,respectively, and Table 12.5 summarizes the bandwidth performance. Here, port1 is terminated in 50 ohms and port 2 is terminated in 12.5 ohms. It may be notedthat, among these four transformers, the broadside-coupled TLT has the largestbandwidth and shortest line length. Table 12.5 compares bandwidths for threecases of return loss: 10, 15, and 20 dB. The bandwidth of a transformer is definedin terms of return loss (RL), that is, the frequency range over which the RL isequal to or greater than a specified value. The fractional bandwidth, FBW, isdefined as:

FBW =D ff0

(12.21)

D f = f2 − f1 , f0 = √f1 f2 (12.22)

where f1 and f2 are the lower and upper edge of the frequency band. Althoughthis example of a 50-ohm to 12.5-ohm (both real impedances) transformer demon-strates the unique features, such as the largest bandwidth and shortest line lengthof an asymmetrical broadside-coupled microstrip TLT, this transformer can also

Figure 12.17 The reflection coefficient response of four types of 4:1 transformers. ZS = 50 ohms.


Figure 12.18 The transmission coefficient response of four types of 4:1 transformers. ZS = 50 ohms.

Table 12.5 Bandwidth Comparison of Several 4:1 Impedance Transformers:50 Ohms to 12.5 Ohms

PerformanceReturnLoss f0 FBW

Configuration (dB) (GHz) (%)

Single-Section Microstrip 20 9.86 16.1W = 190 mm, L = 2,500 mm 15 9.77 31.3

10 9.47 61.3Uncoupled Microstrip Lines 20 9.81 26.4W = 200 mm, L = 2,500 mm 15 9.61 47.9

10 9.07 83.8Coupled Microstrip Lines 20 9.50 36.8W = 130 mm, S = 20 mm, L = 2,300 mm 15 9.35 67.0

10 9.25 130.9Broadside-Coupled Microstrip Lines 20 9.13 72.6W = 20 mm, d = 7 mm, L = 1,400 mm 15 8.74 130.0

10 8.01 227.5

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 compleximpedance.

The size of these transformers is further reduced by folding the lines in a coilor loop shape because the line width is much narrower than the length. By cascadingtwo sections of 4:1 transformers in a series, one can match 50 ohms to 3.1 ohmsover larger bandwidths. The selection of suitable structure parameters for a trans-former is very important to provide the minimum loss and the required impedancetransformation over the desired frequency range. Some of these parameters aredescribed next.

The effect of polyimide thickness, d, on the bandwidth of TLT was studied byvarying its value and keeping other parameters constant. The microstrip width andlength are 30 mm and 1,400 mm, respectively. Figure 12.19 shows the maximum


Figure 12.19 Maximum fractional bandwidth and source impedance of a 4:1 TLT versus polyimidethickness.

FBW and corresponding source impedance of a 4:1 transformer as a function ofpolyimide thickness between the two broadside-coupled conductors. It may benoted that the tighter coupling between the conductors results in a larger bandwidthand lower input impedance.

Next we consider the effect of microstrip width on the optimum source imped-ance to be matched and the corresponding bandwidth. The substrate parametersare given in Table 12.4. The line length in this case is 1,400 mm. Table 12.6 givesthe fractional bandwidth for 4:1 transformers having different microstrip widths.Here ZS (see Figure 12.15) is the source impedance and W is the microstrip width.As the microstrip width decreases, or the characteristic impedance increases, thebandwidth decreases. In this example, to match 50 ohms to 12.5 ohms, a linewidth of about 15 mm or a characteristic impedance of 74 ohms is required andthe resultant fractional bandwidth is about 130%.

The fractional bandwidth as a function of source impedance of a 4:1 trans-former was calculated for three microstrip widths: 20, 40, and 60 mm. This isshown in Figure 12.20. For each microstrip width there is a maximum FBW andit decreases for other impedance value.

Table 12.6 Maximum Bandwidth as a Function of Line Width for Several TLTs Where ZS Isthe Source Impedance and the Load Impedance ZL = ZS /4 (RL = 15 dB)

Line Width ZS (V) Frequency Range Center Frequency, Fractional Bandwidth,W (mm) (GHz) f0 (GHz) FBW (%)

10 60 7–21 12.1 115.520 40 5–20 10.0 150.040 23 3.5–20 8.37 197.260 16 3.0–20 7.75 219.580 12 2.5–20 7.07 247.5

100 10.5 2.1–20 6.48 276.2120 8.0 2.0–20 6.32 284.6

12.3 Interdigital Capacitor 461

Figure 12.20 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 widerbandwidth 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 asshown in Figure 12.21. Interdigital capacitors use the capacitance that occurs acrossa gap in thin-film conductors. These gaps are essentially very long and folded touse a small amount of area. As illustrated in Figure 12.21, the gap meanders backand forth in a rectangular area forming two sets of fingers, which are interdigital.By using a long gap in a small area, compact single-layer small-valed capacitorscan be realized. Typically, values range from 0.05 pF to about 0.5 pF. The capaci-tance can be increased by increasing the number of fingers, or by putting on anoverlay dielectric layer, which also acts as a protective shield. One of the important

Figure 12.21 An interdigital capacitor configuration using seven fingers.


design considerations is to keep the size of the capacitor very small relative to awavelength so that it can be treated as a lumped element. A larger total width-to-length 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 fabricationof conventional integrated circuits or monolithic microwave integrated circuit tech-nology and does not require any additional processing steps.

12.3.1 Approximate Analysis

Analysis and characterization of interdigital capacitors have been reported in theliterature [22–27]. Earlier analyses [22–24] data when compared with the measuredresults showed that these analyses were inadequate to describe the capacitorsaccurately. The analyses were based on lossless microstrip coupled lines [22] andlossy coupled microstrip lines [23]. A more accurate characterization of thesecapacitors can be performed if the capacitor geometry is divided into basic micro-strip sections such as the single microstrip line, coupled microstrip lines, open-enddiscontinuity, asymmetrical gap, 90-degree bend, and T-junction discontinities [26]as shown in Figure 12.22. This model provides better accuracy than the previouslyreported 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 includeinteraction effects between basic microstrip sections described above.

An approximate expression for an interdigital capacitor is given by [22]

C = (er + 1) , [(N − 3)A1 + A2] pF (12.23)

where A1 (the interior) and A2 (the two exterior) are the capacitances of the fingers,N is the number of fingers, and the dimension , in mm is as shown in Figure 12.21.For infinite substrate thickness (or no ground plane), A1 = 4.409 × 10−6 pF/mmand A2 = 9.92 × 10−6 pF/mm.

For a finite substrate, the effect of h must be included in A1 and A2 . In thefinal design, usually S = W and , ≤ l /4.

The total series capacitance can also be written as [6]

C = 2e0ereK(k)K′(k)

(N − 1), F (12.24a)

=10−11

18pere

K(k)K′(k)

(N − 1), × 10−4 F

or

C =ere10−3

18pK(k)K′(k)

(N − 1), pF (12.24b)

where , is in mm, N is the number of fingers, ere is the effective dielectric constantof the microstrip line of width W, and

K(k)K′(k)

=1p

,n H21 + √k

1 − √kJ for 0.707 ≤ k ≤ 1 (12.25a)

=p

,n F21 + √k′

1 − √k′Gfor 0 ≤ k ≤ 0.707 (12.25b)

and

k = tan2Sap4b D, a = W /2, and b = (W + S)/2, and k′ = √1 − k2

(12.25c)

The series resistance of the interdigital capacitor is given by

R =43

,WN

Rs (12.26)


where Rs is the sheet resistivity (V/square) of the conductors used in the capacitor.The effect of metal-thickness t plays a secondary role in the calculation of capaci-tance. The Q of this capacitor is given by

Qc =1

vCR=

3WNvC4,Rs

(12.27)

12.3.2 Full-Wave Analysis

Quasistatic and full-wave numerical methods have been extensively employed toanalyze transmission lines and their discontinuities. The numerical data obtainedfrom these methods have been used to develop analytical and empirical designequations along with equivalent circuit (EC) models to describe the electrical perfor-mance of planar transmission lines and their discontinuities, including microstrip,coplanar waveguide, and slot lines. These equations and EC models have beenexclusively used in commercial microwave CAD tools. The recent advances inworkstations and user-friendly software have made it possible to develop electro-magnetic (EM) simulators. These [28–34] have added another dimension tocomputer-aided engineering (CAE) tools. These simulators play an important rolein the simulation of single-layer elements such as transmission lines, patches andtheir discontinuities, multilayer compenents (namely, inductors, capacitors, pack-ages), and mutual coupling between various circuit elements. Accurate evaluationof the effects of radiation, surface waves, and interaction between components onthe performance of densely packed monolithic microwave integrated circuits canonly be calculated using EM simulators [28–34].

Table 12.7 lists the physical parameters of a typical interdigital capacitor (Figure12.21) analyzed using EM simulators, and Figures 12.23 and 12.24 compare thesimulated S11 and S21 magnitude, and S11 and S21 phase performances [35–37],respectively. A detailed treatment of interdigital capacitors is included in [20,Chapter 7].

Table 12.7 Parameters for an InterdigitalStructure

Substrate height, h = 100 mmSubstrate dielectric constant, er = 12.9Conductor thickness, t = 0.8 mmConductor’s bulk conductivity, s = 4.9 × 107 S/mSubstrate loss tangent, tan d = 0.0005W = 40 mmS = 20 mm, = 440 mmW ′ = 520 mmS ′ = 40 mm, ′ = 60 mmNumber of fingers = 9Enclosure: No

12.4 Spiral Inductors 465

Figure 12.23 Interdigitated capacitor |S11 | and |S21 | 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 low-microwave frequency monolithic approach, low-loss inductors are essential fordeveloping compact and low-cost, low-noise amplifiers and high-power added-efficiency amplifiers.

Inductors in MICs are fabricated using standard integrated circuit processingwith no additional process steps. The innermost turn of the inductor is connectedto other circuitry by using a wire bond connection in the hybrid MICs and througha conductor that passes under airbridges in monolithic MIC technology. The widthand thickness of the conductor determines the current-carrying capacity of theinductor. Typically the thickness is 0.5 to 1.0 mm, and the airbridge separates itfrom the upper conductors by 1.5 to 3.0 mm. In dielectric crossover technology,the separation between the crossover conductors may be anywhere from 0.5 to3 mm. Typical inductance values for monolithic microwave integrated circuitsworking above the S-band fall in the range of 0.5 to 10 nH.

The design of spiral inductors for MIC applications is usually based on twoapproaches: the lumped-element method and microstrip coupled-line method. Thelumped-element approach uses frequency independent formulas for freespace induc-


Figure 12.24 Interdigitated capacitor ∠S11 and ∠S21 responses.

tance with ground-plane effects. These formulas are useful only when the totallength of the inductor is a small fraction of the operating wavelength and whenthe interturn capacitance can be ignored. Wheeler [38] presented an approximateformula for the inductance of a circular spiral inductor, with reasonably goodaccuracy at lower microwave frequencies. This formula has been extensively usedfor the design of microwave lumped circuits [39–41]. Inductor parameters canalso be obtained from two-port S-parameter measurements for the structure. Thisapproach requires fabrication of the structure, however. In the coupled-lineapproach [42, 43], an inductor is analyzed using multiconductor coupled microstriplines. This technique predicts the spiral inductor’s performance reasonably well fortwo turns and up to 18 GHz. Inductors with their conductors supported on postsprovide lower capacitances from the conductor to ground and between conductors,which results in a higher resonant frequency, thereby extending the frequency rangeof operation.

A 2-turn spiral microstrip inductor with opposite sides appropriately connectedas shown in Figure 12.25 may be treated as a coupled-line section. This figureshows 2-turn circular and rectangular spiral inductors and a 1.75-turn rectangularspiral inductor with connecting single-line section and the feed lines representedbetween nodes 1 and 2, and between nodes 4 and 5. In the 2-turn case, the parallelcoupled-line section, which has a total line length equivalent to the spiral lengthbetween nodes 2 and 4, is represented between nodes 2, 3, and 4 and constitutesthe intrinsic inductor. The length is taken as the average of the outer and innerturn lengths. In the 1.75-turn inductor, an additional single line between nodes 3and 4 is connected, whereas the coupled line is between nodes 2 and 3.


Figure 12.25 Spiral inductors and its coupled-line equivalent circuit models (a) circular 2 turns,(b) rectangular 2 turns, and (c) rectangular 1.75 turns.

The electrical equivalent coupled-line model shown in Figure 12.25 does notinclude the crossover capacitance or the right-angle bend discontinuity effects.Figure 12.26 shows a modified equivalent circuit of a 1.75-turn rectangular spiralinductor that also includes the crossover capacitor. This figure also shows thephysical and electrical parameters of this inductor. Figure 12.27 shows a furthersubdivision of the inductor that is required to evaluate its performance moreaccurately. In this case, the inductor is split into three sections representing elements1, 2, and 3. The performance of these inductors can be calculated by either usingcommercial CAD tools or by solving cascaded ABCD or S-parameter matrices forthese elements. Improved versions of these inductors include chamferred bends orcompensated bends [5].

The electrical characteristics of the intrinsic 2-turn inductor can be derivedfrom the general four-port network of a coupled-line section as shown in Figure


Figure 12.26 Rectangular 1.75-turn spiral inductor (a) physical layout and (b) coupled-line equiva-lent circuit model. (From: [42]. 1983 IEEE. Reprinted with permission.)

Figure 12.27 The network model for calculating the inductance of a planar rectangular spiralinductor.


12.28, where the current and voltage relationships of the pair of lines can bedescribed by the admittance matrix equation as follows:

3I1

I2

I3

I44 = 3

Y11 Y12 Y13 Y14

Y21 Y22 Y23 Y24

Y31 Y32 Y33 Y34

Y41 Y42 Y43 Y444 3

V1

V2

V3

V44 (12.28)

This matrix may be reduced to two ports, by applying the boundary conditionthat ports 2 and 4 are connected, that is:

V2 = V4 (12.29a)

I2 = −I4 (12.29b)

and by rearranging the matrix elements, the two-port matrix may be written as

FI1

I3G = FY ′11 Y ′13

Y ′31 Y ′33G FV1

V3G (12.30)

where

Y ′11 = Y11 −(Y12 + Y14)(Y21 + Y41)Y22 + Y24 + Y42 + Y44

(12.31a)

Y ′13 = Y13 −(Y12 + Y14)(Y23 + Y43)Y22 + Y24 + Y42 + Y44

(12.31b)

and

Y ′33 = Y ′11 (12.32a)

Y ′31 = Y ′13 (12.32b)

due to symmetry.The admittance parameters for a coupled microstrip line in inhomogenous

dielectric medium are given by [44]

Figure 12.28 Four-port representation of a coupled-line section of an inductor.


Y11 = Y22 = Y33 = Y44 = −j [Y0e cot ue + Y0o cot uo ]/2 (12.33a)

Y12 = Y21 = Y34 = Y43 = −j [Y0e cot ue − Y0o cot uo ]/2 (12.33b)

Y13 = Y31 = Y24 = Y42 = j [Y0e csc ue − Y0o csc uo ]/2 (12.33c)

Y14 = Y41 = Y23 = Y32 = j [Y0e csc ue + Y0o csc uo ]/2 (12.33d)

An equivalent ‘‘p ’’ representation of a two-port network is shown in Figure12.29, where

YA = −Y ′13 (12.34)

YB = Y ′11 + Y ′13 (12.35)

and

YA = −j125Yoe cot ue + Yoo cot uo +

FYoeS1 − cos uesin ue

D + YooS1 + cos uosin uo

DG2FYoeS1 − cos ue

sin ueD − YooS1 + cos uo

sin uoDG6

(12.36)

YB =2jYoeYoo (1 − cos ue ) (1 + cos ue )

[Yoo sin ue (1 + cos uo ) − Yoe sin uo (1 − cos ue )](12.37)

Because the physical length of the inductor is much less than l /4,sinue, o ≅ ue, o and cos ue, o ≅ 1 − u 2

e, o /2. Also Yoo > Yoe. Therefore, (12.36) and(12.37) are approximated as follows:

YA ≅ −jYoe2ue

(12.38)

YB ≅ jYoeue (12.39)

Figure 12.29 p equivalent circuit representation of the inductor.


which are independent of the odd mode. Thus the ‘‘p ’’ equivalent circuit consistsof a shunt capacitance C and a series inductance L, as shown in Figure 12.30. Theexpressions for L and C may be written as

YA =1

jvL= −j

Yoe2ue

(12.40)

or

L =2ue

vYoe(12.41)

and

YB = jvC = jYoeue (12.42)

or

C =Yoeue

v(12.43)

If , is the average length of the conductor, then

ue =v,c √eree (12.44)

where c is the velocity of light in free space and eree is the effective dielectricconstant for the even mode. When Z0e = 1/Y0e , from (12.41) and (12.43):

L =2,Z0e√eree

c(12.45)

C =,√eree

Z0ec(12.46)

In a loosely coupled inductor Zoe ≅ Zo and eree = ere for the single conductormicrostrip line. The above equations may be used to approximately evaluate induc-

Figure 12.30 Equivalent L C circuit representation of the inductor.


tor performance. Figure 12.31 shows measured and modeled S11 and S21 responsesof a two-turn 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 byhybrid and monolithic techniques working in the microwave frequency range. Amajor challenge in printed coil transformers is keeping the parasitic capacitancesand series resistances low, to operate these components at higher frequencies withlow insertion loss. The transformers can be two-, three-, or four-port components.The three-port transformers may have 0-degree, 90-degree, or 180-degree phasedifference at the output ports.

Rectangular spiral transformers fabricated using GaAs MMIC technology havebeen reported in the literature [45–49]. In active circuits, their impedance trans-former ratio and inductance values are used for impedance transformation andresonating out the active device’s capacitance, respectively. Figure 12.32 shows thephysical layout of a two-conductor transformer consisting of a series of turnsof thin, metallized conductors placed on a dielectric substrate (not shown). Thecharacterization of this structure is not straightforward; however, a multiconductorcoupled microstrip line analysis [50, 51] may be used to determine its parametersapproximately. A more accurate characterization of this transformer can only beachieved by using fullwave and comprehensive circuit simulators such as EM CADtools.

The twin-coil four-port rectangular spiral transformer [46, 48] shown in Figure12.32 has a ground ring around it. The dimensions of the transformer are, outsidering = 1,020 × 710 mm, conductor thickness = 1 mm, GaAs (er = 12.9), substratethickness = 250 mm, dielectric crossover height = 1.3 mm, overlay dielectricer = 6.8, conductor width = 20 mm, and conductor gap = 6 mm. Figure 12.33shows a comparison between the measured and simulated S11 and S21 responses

Figure 12.31 Measured and modeled (a) reflection (S11) and (b) transmission (S21) responses forthe two-turn 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 |S11 | and |S21 | responses of the two-port configuration of the transformer inFigure 12.32. (From: [46]. 1989 IEEE. Reprinted with permission.)


of this transformer when ports 3 and 4 are short-circuited as shown in Figure 12.32.The simulated performance was obtained using an electromagnetic simulator. Asshown in Figure 12.33, S21 has a sharp null at 6 GHz that occurs because of thel /4 short-circuited secondary coil at that frequency. In this case, maximum currentflows 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 inthe S21 response. Below 6 GHz, the current in the spiral conductors is nearlyconstant and the power transfer is provided by magnetic coupling similar to thatin classical coil transformers. Above 6 GHz, however, the spiral conductors areelectrically long and the current distribution along the conductors has less of astanding wave nature. Their behavior becomes closer to that of coupled transmis-sion lines supporting both magnetic and electric coupling.

When ports 3 and 4 are terminated in 50V loads, the measured two-portresponse of the transformer is shown in Figure 12.34, which shows a gradualincrease of S11 and S21 . Because in this case there are no standing waves, the powertransfer from primary to secondary occurs gradually from magnetic to magneticand electric coupling. However, 50V terminating loads result in higher loss in thetransformer.

The above examples show that efficient power transfer in a two-port trans-former occurs at high frequencies through both magnetic and electric coupling. Atsuch frequencies, the spiral conductors are longer than l /4. This does not permitgrounding of the center tap of the secondary spiral conductor to obtain balancedoutput at ports 2 and 3. On the other hand, at low frequencies, a center tap ispossible; as in classical transformers, however, power transfer is inefficient as

Figure 12.34 The |S11 | and |S21 | responses of the four-port configuration, constructed by adding50V series resistors to ports 3 and 4 in Figure 12.32. (From: [46]. 1989 IEEE.Reprinted with permission.)

12.6 Other Coupled-Line Components 475

shown in Figure 12.34. The power transfer can be improved by making thesetransformers electrically small, minimizing the parasitic capacitances, and increas-ing the number of tightly coupled turns of the spirals, while maintaining the samespiral length.

Efficient power transfer can also be obtained by using more than two coils ina transformer. Figure 12.35 shows a layout of a 3-conductor coupled-line trans-former. Figure 12.36(a) shows a photograph of this structure known as a triformerhaving 1.5 turns and fabricated on GaAs substrate using conductor width andspacing of 5 mm [47] and designed as a two-port matching network. An electricalequivalent circuit is shown in Figure 12.36(b). TRL denotes on wafer probe pads.The triformer structure may be used in the realization of wide-band baluns.

12.6 Other Coupled-Line Components

Other applications of coupled-line sections include Schiffman sections [52–54],broadband 180-degree bit phase shifters [55], power dividers [56], resonators [57],and delay lines [58]. Coupled-line sections are commonly used to improve thebandwidth of phase shifters, and a coupled-line resonator is an integral part ofbandpass filters. In n-way power dividers, they are responsible for reducing compo-nent size.

Figure 12.35 Schematic of a 1.5-turn rectangular spiral triformer.


Figure 12.36 (a) Photograph of a 1.5-turn MMIC triformer; and (b) two-port equivalent circuit ofthe triformer. (From: [47]. 1989 IEEE. Reprinted with permission.)

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[50] Marx, K. D., ‘‘Propagation Modes, Equivalent Circuits, and Characteristics Terminationsfor Multiconductor Transmission Lines with Inhomogeneous Dielectrics,’’ IEEE Trans.Microwave Theory Tech., Vol. MTT-21, July 1973, pp. 450–457.

[51] Djordjevic, A., et al., Matrix Parameters for Multiconductor Transmission Lines, Nor-wood, MA: Artech House, 1989.

[52] Schiffman, B. M., ‘‘A New Class of Broad-Band Microwave 90-Degree Phase Shifters,’’IRE Trans. Microwave Theory Tech., Vol. MTT-6, 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.


[54] Free, C. E., and C. S. Aitchison, ‘‘Improved Analysis and Design of Coupled-Line PhaseShifters,’’ IEEE Trans. Microwave Theory Tech., Vol. 43, September 1995,pp. 2126–2131.

[55] Ecom, S. Y., et al., ‘‘Broadband 180‚ Bit Phase Shifter Using a New Switched Network,’’IEEE MTT-S Int. Microwave Symp. Dig., 2003, pp. 39–42.

[56] Chiu, J.-C., J.-M. Lin, and Y.-H. Wang, ‘‘A Novel Planar Three-Way Power Divider,’’IEEE Microwave Wireless Components Lettr., Vol. 16, August 2006, pp. 449–451.

[57] Sharma, A. K., and B. Bhat, ‘‘Spectral Domain Analysis of Interacting Microstrip ResonantStructures,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-31, August 1983,pp. 681–685.

[58] Fredrick, J. D., Y. Wang, and T. Itoh, ‘‘A New Circuit Topology for Continuous GroupDelay Synthesis,’’ IEEE Microwave Wireless Components Lettr., Vol. 12, March 2002,pp. 85–87.

C H A P T E R 1 3

Baluns

13.1 Introduction

A balun (balanced-to-unbalanced) is a transformer used to connect balanced trans-mission line circuits to unbalanced transmission line circuits. Figures 13.1 and 13.2show examples of balanced and unbalanced transmission lines, respectively. Twoconductors having equal potential with 180-degree phase difference constitute abalanced line. In this case, no current flows through a grounded shield (i.e.,I1 = I2 and Is = Ig = 0). When this condition is not satisfied, as shown in Figure13.2, where Is is finite, the transmission line is termed as unbalanced. In additionto providing a matched transition between a balanced and an unbalanced line,baluns also function as center-tap transformers for push-pull applications used inradio frequency applications.

Baluns are required for balanced mixers, push-pull amplifiers, balanced fre-quency multipliers, phase shifters, balanced modulators, dipole feeds, and numerousother applications. This transformation from a balanced medium to an unbalancedone requires special techniques, which are described in this chapter. Because thefocus of this book is on coupled-line structures, only schemes based on the latterconcept are discussed. Both narrowband and broadband circuit designs and theirfabrication are described.

Over the past half-century, several different kinds of balun structures havebeen 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 andprinted antenna feeds. Current interest in transmission-line-type structures isfocused toward making it planar, compact, and more suitable for mixers and push-pull power amplifiers. For high-efficiency 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 isavailable in the low-frequency transformer type baluns, is a problem in transmission-line-type baluns.

Planar baluns shown in Figure 13.5 provide greater flexibility and better perfor-mance in mixers and modulators. They are compatible with MIC and MMICtechnologies, relatively small in size, can be fabricated on single-sided substrates,have wide bandwidths, and are very useful in surface-mounted packages for large-volume production. The realization of a decade bandwidth mixer in a microstrip-type 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 usedat microwave frequencies, especially for mixers.

Four-port passive circuits, such as rat-race hybrids and waveguide magic tees,can also be used as baluns. Major limitations of these components are their narrowbandwidth and the lack of a method for center-tap grounding. Several kinds ofbaluns (e.g., ferrite, coaxial) have been successfully used below microwave frequen-cies. Now, baluns employing new coupled-line topologies are becoming more com-mon at microwave frequencies. They provide a lower impedance-matching ratio,greater bandwidths, and reduced even-harmonic levels. The principal disadvantagesinclude poor isolation between the two single-ended amplifiers in a push-pullconfiguration and poor VSWR at the input and output. In a push-pull amplifier,the reflected signals from the single-ended amplifiers do not cancel as they do ina balanced configuration.

13.2 Microstrip-to-Balanced Stripline Balun

A smooth transition from a microstrip to a balanced stripline as shown in Figure13.6(a) works as a broadband balun. When a microstrip is joined to a balanced

13.2 Microstrip-to-Balanced Stripline Balun 483

Figure 13.3 Lumped-element baluns: (a) center-tap transformer and (b) 180-degree rat-racecoupler.

stripline, a step discontinuity between the ground plane of the microstrip line andthe bottom conductor of the balanced stripline exists for the same characteristicimpedances. A step discontinuity also exists between the top strip conductors ofthese two lines. But the step discontinuity in the former case is larger than in thelatter case. Transmission-line tapers are generally employed to achieve a goodmatch between the two lines.

484 Baluns

Figure 13.4 Coaxial baluns: (a) simple l/2 line separation splitter, and (b) Marchand. In (b), Z1and Z4 are the characteristic impedances of the coaxial lines, and Z2 and Z3 are thecharacteristic impedances of the coaxial outer conductors with respect to the outershield.

An example of a transition from microstrip to balanced stripline when con-nected back to back, to simplify measurements, is shown in Figure 13.7. For thesame impedance, for example, 50V, the stripwidth for the microstrip, Wm , issmaller than the stripwidth for the balanced stripline Wb . When these two linesare connected, both the stripline conductor and the ground plane are tapered tomatch the dimensions. Several tapering techniques are available, and we can selecta taper shape based on the bandwidth requirement. A Chebyshev tapering contouralong the balun’s length yields excellent broadband performance [25]. A simpletaper, such as shown in Figure 13.7, can easily achieve an octave bandwidth. Forthe fabricated example shown in the figure, x = Wm , X = 3Wm , and Y = 6Wmand the transitions were characterized on a 62.5-mil-thick polystyrene substrate(er = 2.55) and on a 50-mil-thick alumina substrate (er = 9.7). The completeassembly was 1 inch long and the connectors used were of OSM-stripline-type,which have about a 0.1-dB insertion loss per connector at S-band. Figure 13.8shows typical measured VSWR for the two transitions, as well as for the throughline. For the polystyrene transition the maximum VSWR over 2.4 to 4.5 GHz was1.2, whereas for the alumina case the VSWR was less than 1.2 over the 1.7- to3.2-GHz frequency range. Typical measured insertion loss was less than 0.6 dBand 0.4 dB, respectively, for these two transitions.

Another parallel-strip balun [34] is shown in Figure 13.9, in which the inputis a microstrip and the output is a broadside-coupled microstrip [42]. One of the

13.2 Microstrip-to-Balanced Stripline Balun 485

Figure 13.5 Planar baluns: (a) simple l/2 line separation splitter, (b) multisection half-wave balun,and (c) coupled-line (coupled-line sections are l/4 at center frequency).

advantages of this structure is that it is possible to achieve ground isolation at thebalanced output port. That is, the balanced lines have their potential referencedto each other. The simple design equations given by [34] are as follows:

Z0p = √ZS ZL (13.1)

Z0e ≥ 10Z0o ≥ 5Z0p (13.2)

Z0o = Z0p /2 (13.3)

where ZS and ZL are the source and load impedances, respectively, and Z0p is theparallel-plate impedance of the two conductors forming broadside-coupled lineswith the housing. Z0e and Z0o are the even- and odd-mode characteristic impedancesof the broadside-coupled lines, respectively.

486 Baluns

Figure 13.6 Nonplanar baluns: (a) simple broadside microstrip, (b) uniplanar, and (c) Marchand.

Consider an example having source impedance ZS = 50V and load impedanceZL = 100V. In this case, Z0p = 70.7V and Z0o = 35.4V. Figure 13.10 shows abalun’s performance when Z0e = 350V and the structure is lossless. As can be seen,the power split is not the desired 3 dB. In this case, as the Z0e /Z0o ratio becomeshigher and higher, the power split will approach 3 dB, and when Z0e is infinite,the power split will be 3 dB.

13.3 Analysis of a Coupled-Line Balun

The operation of a balun can be easily explained by a pair of coupled microstriplines of equal width, as shown in Figure 13.11. The length of the line is l /4 at the

13.3 Analysis of a Coupled-Line Balun 487

Figure 13.7 Unbalanced-to-balanced 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 odd-mode excita-tion technique as shown in Figure 13.12. The admittance matrix for a four-portnetwork consisting of transmission lines having the same propagation constants isgiven by [11, 43].

1I1

I2

I3

I42 = 1

Y11 Y12 Y13 Y14

Y21 Y22 Y23 Y24

Y31 Y32 Y33 Y34

Y41 Y42 Y43 Y442 1

V1

V2

V3

V42 (13.4)

where for a homogenous dielectric medium

488 Baluns

Figure 13.9 Side view of the parallel-plate balun. (From: [34]. 1993 Applied Microwave. Reprintedwith permission.)

Figure 13.10 Frequency response of the parallel-plate balun. (From: [34]. 1993 Applied Micro-wave. Reprinted with permission.)

Figure 13.11 Coupled microstrip lines.

13.3 Analysis of a Coupled-Line Balun 489

Figure 13.12 (a) A coupled line configuration and excitation of (b) even and (c) odd modes onsymmetrical coupled lines.

Y11 = Y22 = Y33 = Y44 = −j (Y0o + Y0e )cot u

2(13.5a)

Y12 = Y21 = Y34 = Y43 = j (Y0o − Y0e )cot u

2(13.5b)

Y13 = Y31 = Y24 = Y42 = −j (Y0o − Y0e )csc u

2(13.5c)

Y14 = Y41 = Y23 = Y32 = j (Y0o + Y0e )csc u

2(13.5d)

At u = 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 short-circuited (i.e., V2 = 0), and the other ports are terminated inmatched loads, the voltage transfer from port 1 to port 3 can be obtained byconverting the Y-parameters into S-parameters, as follows [11]:

S31 =j2Y0YA

Y 20 + Y 2

A + Y 2B

(13.8)

where Y0 = 1/Z0 and YA = (Y0o − Y0e )/2, YB = (Y0o + Y0e )/2. Similarly, the voltagetransfer from port 1 to port 4 can be obtained as

S41 =−j2Y0YB

Y 20 + Y 2

A + Y 2B

(13.9)

and S31 /S41 = −YA /YB = − (Y0o − Y0e )/(Y0o + Y0e ). The above equation shows thatS31 and S41 are 180 degrees out of phase at u = 90 degrees. A balun works overthe required band if |S31 /S41 | = 1 and |∠S31 − ∠S41 | = 180 degrees. These condi-tions can be achieved if Y0e = 0 or Z0e = ∞, that is, the effect of ground plane onthe conductors is negligible. For a perfect match, S11 = 0, that is:

Y 20 = Y 2

A + Y 2B (13.10)

or

Z0o = Z0 /√2 (13.11)

where Z0 is the source or load impedance.Figures 13.13 and 13.14 show the balun’s performance where for Z0o = 35V

and Z0e = 10,000V, the amplitude response varied by less than 0.5 dB and thephase difference is within ±3 degrees over the 2- to 18-GHz frequency range. Thistype of balun has a severe requirement for a very high even-mode impedance,which is not realizable in the microstrip configuration. Thicker and lower dielectricconstant value substrates with narrower conductor width result in a higher even-mode impedance. Narrower conductors have also higher insertion loss. These kindsof baluns are therefore not very suitable for high-efficiency power amplifiers andlow-loss mixer applications.

13.4 Planar Transmission Line Baluns

Planar transmission line baluns consist of two sections: the first section divides thesignal 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 even-mode 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 even-mode impedance indicatesthat a significant amplitude imbalance may result if ground plane effect is not consid-ered. (From: [11]. 1985 Microwaves and RF. Reprinted with permission.)

492 Baluns

range and the second section provides −90-degree and +90-degree phase shifts forthese two signals, so that the balanced output signals have a 180-degree phasedifference. The power divider section generally uses a Wilkinson in-phase powerdivider, which uses multisections for larger bandwidths. In general, short-circuitedand open-circuited coupled lines are used for the phase-shifter sections. Basicconfigurations for these baluns are shown in Figure 13.15, where the divider usesonly one section. A multisection divider has also been used [28] to obtain 6- to18-GHz bandwidth performance from such baluns.

Figure 13.15(a) shows a basic balun where, in order to obtain tight couplingover a multioctave bandwidth, the edge-coupled line sections are commonlyreplaced with interdigital Lange couplers. Because of the inherent symmetry andbroadband characteristics of coupled-line sections or Lange couplers, good ampli-tude and phase balance performance are achievable. As the use of via holes orground connections through alumina substrate in MICs require additional pro-cessing, broadband radial line stubs [44], as shown in Figure 13.15(b), can alsobe used to simulate RF grounding without additional processing. When the open-circuit coupler is replaced by a ‘‘p ’’ network of transmission lines, as shown in

Figure 13.15 (a–d) Planar balun configurations, where port 1 is unbalanced, and ports 2 and 3constitute a balanced port.


Figure 13.15(c), the 180-degree phase shift becomes independent of the electricallength of the two networks [21, 45]. Figure 13.15(d) shows another broadbandbalun topology [27] suitable for push-pull power amplifiers.

13.4.1 Analysis

An ultra broadband 180-degree phase shift can be realized by using the phase-reversal property of a tightly coupled (3-dB) four-port network. In a coupler whencoupled and direct ports are switched from open circuit to short circuit, the transmis-sion phase difference between the input and isolated ports changes from −90 degreesto +90 degrees as shown in Figure 13.16. In the case of the short-circuited condition,an additional −180-degree phase is added, which brings the phase to −270 degreesor +90 degrees. Since tight coupling is required, a Lange coupler is generally usedfor this application. The S-parameters of a four-port network shown in Figure13.16 are given by

1b1

b2

b3

b42 = 1

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S442 1

a1

a2

a3

a42 (13.12)

or

b1 = S11a1 + S12a2 + S13b3 + S14b4 (13.13a)

b2 = S21a1 + S22a2 + S23b3 + S24b4 (13.13b)

Figure 13.16 Two-port coupler configuration.

494 Baluns

b3 = S31a1 + S32a2 + S33b3 + S34b4 (13.13c)

b4 = S41a1 + S42a2 + S43b3 + S44b4 (13.13d)

In a matched 3-dB coupler:

S13 = S24 = S31 = S42 = −j /√2 (13.14)

S14 = S23 = S32 = S41 = 1/√2 (13.15)

S11 = S22 = S33 = S44 = 0 (13.16)

S12 = S21 = S34 = S43 = 0 (13.17)

Therefore, the four-port S-matrix becomes

[S ] =1

√2 30 0 −j 10 0 1 −j−j 1 0 01 −j 0 0

4 (13.18)

When ports 3 and 4 are terminated in an open circuit (i.e., ZT = ∞) and otherports are matched, a3 = b3 and a4 = b4 and a1 = b2 , in this case, (13.13) through(13.17) reduce to

b1 =−j

√2b3 +

1

√2b4 (13.19a)

b2 =1

√2b3 −

j

√2b4 (13.19b)

b3 =−j

√2a1 +

j

√2a2 (13.19c)

b4 =1

√2a1 −

j

√2a2 (13.19d)

By rearranging these equations, the two-port S-parameters become

[S ]open = F0 −j−j 0G (13.20)

Similarly, when ports 3 and 4 are terminated in short circuits, a3 = −b3 anda4 = −b4 and (13.18) simplifies to


[S ]short = F0 jj 0G (13.21)

Equations (13.20) and (13.21) illustrate that open to short switching leads to a180-degree phase shift, signals are combined, and ports 1 and 2 are still matched.This condition holds well over a wide frequency range.

When the open-circuited coupler is replaced by a ‘‘p ’’ or equivalent network oftransmission lines, the 180-degree phase shift becomes independent of the electricallength of the two networks; that is, ‘‘p ’’ and short-circuited coupler, and thisresults in a wider bandwidth. Figure 13.17 shows these 180-degree phase-differencesections.

The two networks are exactly equivalent for all frequencies, except that thetransmission phase difference between the two circuits is exactly 180 degrees [45].This can be seen by developing ABCD matrices for both networks. For the ‘‘p ’’network:

FA BC DG

p

= 31 0

Yoej tan u

14 ? 3 cos uj2 sin u

(Y0o − Y0e )j (Y0o − Y0e )

2 csc ucos u 4 ? 3

1 0Yoe

j tan u14 (13.22)

= 3(Y0o + Y0e )(Y0o − Y0e )

cos uj2 sin u

(Y0o − Y0e )

(Y0o + Y0e )2 cos2 u − (Y0o − Y0e )2

j2(Y0o − Y0e ) sin u(Y0o + Y0e )(Y0o − Y0e )

cos u4 (13.23)

where Y0o = 1/Z0o and Y0e = 1/Z0e .The ABCD matrix of the shorted coupled-line section is calculated from the y

parameters as

Figure 13.17 (a, b) 180-degree phase difference sections.

496 Baluns

FA BC DG

coupled

= 3 −(Y0o + Y0e )(Y0o − Y0e )

cos u −j2 sin u

(Y0o − Y0e )

−(Y0o + Y0e )2 cos2 u − (Y0o − Y0e )2

j2(Y0o − Y0e ) sin u−

(Y0o + Y0e )(Y0o − Y0e )

cos u4(13.24)

From (13.23) and (13.24)

FA BC DG

p

= F−1 00 −1G ? FA B

C DGcoupled

(13.25)

This result is independent of the electrical length, and thus is independent offrequency. For tight coupling, however, u = 90 degrees is required.

13.4.2 Examples

Topologies for the planar transmission line baluns shown in Figure 13.15 wereanalyzed and optimized using a commercial CAD tool and found that the configura-tion 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 Figure13.15(a) was fabricated on a 0.635-mm-thick substrate using conventional photoli-thography and etching techniques. The coupler has nominal coupling of 2.8 dBand a 0.7-dB ripple. The Z0e and Z0o values were 126V and 20V, respectively

Table 13.1 Design andPerformance Summary of a PlanarBalun Shown in Figure 13.15(c).Substrate Dimensions: er = 9.9,h = 15 mil, t = 0.2 mil, andConductors of Gold

Wilkinson divider dimensions:Line width = 6.2 milLine length = 96.2 milIsolation resistor = 88.4V

Lange coupler dimensions:Line width = 2.6 milLine spacing = 1.8 milCoupler length = 98.0 mil

Lowpass section dimensions:Shunt line width = 2.5 milShunt line length = 107.0 milSeries line width = 17.2 milSeries line length = 81.7 mil

Performance over 8 to 16 GHz:Insertion loss = 3.15 ± 0.05 dBPhase difference = 180 ± 1 degreeAmplitude difference = ±0.01 dBReturn loss > 18 dB


[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. TheVSWR and isolation were better than 10 and 14 dB, respectively. The phasedifference between the output ports was 170 ± 5 degrees over the 5- to 11-GHzband.

Another planar balun was constructed on a 10-mil alumina substrate usingshort- and open-circuited Lange couplers [28]. The Lange couplers were realizedusing ion-beam milling techniques to achieve fine 1-mil line width and 0.5-mil gapdimensions. A photograph of the balun is shown in Figure 13.19. The balun wasaccurately characterized using thru-reflect-line (TRL) calibration techniques andthe measured amplitude, phase, and return-loss response are shown in Figure 13.20.The maximal loss in each path was about 1.2 dB, and the amplitude and phaseimbalance were ±0.6 dB and ±7 degrees, respectively, over the 6- to 20-GHz

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 withpermission.)

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 usedcomponent in broadband double-balanced mixers. As compared with a shortedcoupled-line balun, this structure has less stringent requirements for Z0e ; generallyZ0e ≅ 3 to 5 times of Z0o 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 open-circuited, two short-circuited, and balanced transmission line sections. Each section is about a quarter-wavelength long at the center frequency of operation. A coaxial version of acompensated Marchand balun is shown in Figure 13.21(a), while its equivalentcircuit representation is shown in Figure 13.21(b). The compensation term is usedin broadband baluns where the balanced output and reduced phase slope weremaintained over a wide bandwidth. As shown in Figure 13.21(a), this structurebasically consists of two coaxial lines, each l /4 long at the center frequency. Theleft-hand line has a characteristic impedance of Z1 . The second line, which has acharacteristic impedance of Z2 , is open-circuited. The outer conductors of thesetransmission line sections with housing make another two short-circuited l /4 linesthat are in series with each other and shunt the balanced lines, having a characteristicimpedance of ZB , at locations a and b. As shown in the equivalent circuit [Figure13.21(b)], the stubs ZS1 and ZS2 are in series and shunt the balanced lines. Theircharacteristic impedances are made as large as possible. These impedances alongwith the other transmission line impedances determine the impedance transforma-tion and bandwidth. Figure 13.22 shows a simplified equivalent circuit of theMarchand balun. Because of equal shunting effects on the balanced lines, these stubs

13.5 Marchand Balun 499

Figure 13.20 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) equivalenttransmission-line model. (From: [22]. 1990 IEEE. Reprinted with permission.)

provide greater bandwidths. The open-circuit stub Z2 provides a low impedance atthe junction of the four different lines and acts like a series resonant circuit. Theseries resonant circuit with a shunt resonant circuit as shown in Figure 13.23 reducesthe phase variation over the designed bandwidth. The ratio of the characteristicimpedances of the short-circuited and open-circuited stubs determines the band-width; the higher the ratio, the wider the bandwidth. The transmission line ZB canbe designed with a characteristic impedance of the balanced line or can be usedas an impedance transformer between the desired impedance and the balanced-line impedance.


Figure 13.22 Simplified equivalent circuit of a fourth-order Marchand balun.

Figure 13.23 Marchand balun’s series and parallel-resonant compensating representation for widerbandwidth.

A Marchand balun is basically comprised of two quarter-wave coupled sectionsand may be realized using printed lines or coaxial lines. The coupled lines requiretight coupling with high even-mode impedance for broadband performance. Theprinted coupled lines may be microstrip lines, multiconductor Lange couplers,coplanar waveguides, multilayer microstrip lines, or spiral coils.

A balun fabricated using printed-circuit 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 whichthe balun is placed. Like any other distributed balun (coaxial, microstrip, andCPW), the tap center shown in Figure 13.24(b) only works at dc or low frequenciessuch as IF that are in the RF range. The center tap concept is not valid at microwavefrequencies, however, as required for push-pull topology.

13.5.1 Coaxial Marchand Balun

Now we present a simple analysis of a coaxial Marchand balun shown in Figure13.25(a). It consists of two coaxial lines, a and b, of characteristic impedances Za

502 Baluns

Figure 13.24 Planar compensated balun fabricated on a low-dielectric substrate: (a) metalizationpattern, and (b) lumped-element equivalent-circuit model. (From: [25]. 1990 JohnWiley. Reprinted with permission.)

and Zb , respectively. The signal is fed at the input point, and the balanced signalappears across the O and O ′ nodes. The center conductors of these lines areconnected at nodes c and c ′, and the other end of the center conductor of line bis open-circuited. The outer conductor of these two lines are coupled to each otherto form the balanced line of characteristic impedance Zab . The outer conductorsof lines a and b are connected at d, and the electrical length u at the center frequencyis 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 losslessstructure, may be expressed as

Zin = −jZb cot ub +jZL Zab tan uab

ZL + jZab tan uab(13.26)

substituting ub = uab = u and after simplifying

Zin =ZL Z2

ab + j cot u FZ2L XZab − Zb cot2 u C − Zb Z2

abGZ2

ab + Z2L cot2 u

(13.27)

when


Figure 13.25 Wideband balun: (a) schematic diagram; and (b) equivalent circuit representation.(From: [4]. 1960 IEEE. Reprinted with permission.)

Zb = Za and Zab = ZL

Zin = ZL sin2 u + j cot u (ZL sin2 u − Za ) (13.28)

and the input impedance becomes perfectly matched to Za at two widely separatedfrequencies given by the solution of

sin2 u = Za /ZL (13.29)

These frequencies are symmetrically disposed about a center frequency correspond-ing to u = 90 degrees. A larger bandwidth is realized by choosing

Zb = Z2L /Zab (13.30)

and making Zab as large as possible [2, 4]. Figure 13.26 compares the input VSWRfor two cases in which a 50V unbalanced input was transformed to a 70V balancedoutput. The improvement in the second case, where Zb = Z2

L /Zab is quite obvious.The realization of this case in coaxial line form is difficult; however, it can beeasily 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 ZL is between O and O ′ and is 70V

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 endof 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 lineconductors must be at least three times that of the microstrip line conductors inorder to use microstrip design equations. Narrower widths can also be used ifappropriate connections are made. The characteristic impedance Zab of the bal-anced line may be calculated by treating it as a pair of coupled microstrip lines ona suspended substrate and excited in the odd mode. The distance between thebalance line conductors and the enclosure, which also acts as a ground plane,is kept large as compared with the spacing between the conductors. Under thisassumption, the characteristic impedance of the balanced line is approximatelytwice the odd-mode impedance calculated for the coupled line. The effective dielec-tric constant of the balanced line is the same as that calculated for the coupledlines excited in the odd-mode.

Lower values of Zab require narrower spacing and wider conductor widths.Fabrication techniques set the lower limit on spacing that can be realized. Thus ahigher limit on Zab is set by the minimum line widths required to realize Zaand Zb . The effects of the enclosure, discontinuity, and coupling influence theperformance of this balun in terms of bandwidth and frequency range. For applica-


Figure 13.27 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 lineby extending it and shaping it into the desired dipole configuration. In the finaldesign, the dipole can be matched over a broadband by treating it as part of thebalun 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, Zb = Zab = 80V, which is equal to the dipole’s radiation resistanceat the resonant frequency which is 13 GHz. ub = 105 degrees and uab = 95 degreesand a l /4 microstrip having a characteristic impedance of 63V were used totransform 80V to 50V at the input. The structure was printed on a 0.64-mm-thickfused silica (er = 3.78) and mounted a quarter-wavelength above-the-ground plane.Although such baluns have good bandwidth, they are not suitable for ultrabroadband applications.

13.5.2 Synthesis of Marchand Balun

Synthesis of coaxial Marchand baluns is available in the literature [8, 32]. Recenttrends focus on planar technology baluns using single-layer and multilayer micro-strip transmission media. A Marchand balun can easily be realized using a pair ofcoupled 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 techniquesof such baluns have been recently described [8, 32] and are summarized in thissection.

506 Baluns

Figure 13.28 (a) Printed circuit realization of balun structure with integrated dipole. (b) Measuredand calculated input VSWR for the dipole/balun combination. (From: [12]. 1987Microwave Journal. Reprinted with permission.)

Figure 13.29 (a) A four-port coupled line and (b) its equivalent circuit.


An equivalent of a coupled-line of equal conductor widths is shown in Figure13.29(b) where Z1 and Z2 are the characteristic impedances of distributed unitelements, and N is the transformation ratio. The characteristic impedance Z0c andthe coupling coefficient k of the coupled line are given by

Z0c = √Z0e Z0o (13.31a)


(13.31b)

and

Z0e = Z0c√1 + k1 − k

(13.32a)

Z0o = Z0c√1 − k1 + k

(13.32b)

are related to the network equivalent circuit elements as follows:

Z1 =Z0c

√1 − k2(13.33)

Z2 = Z0c√1 − k2

k2 (13.34)

and

N =1k

where Z0e and Z0o are the even- and odd-mode impedances, respectively.The balun in Figure 13.30(a) has four reactive elements and is known as a

fourth-order Marchand balun. Matching section Z makes it a fourth-order balunwith improved broadband performance. Figures 13.30(b) and 13.30(c) show equiv-alent circuit representations and Figure 13.30(d) shows further simplification whenNa = Nb = N. In this case, the middle transformers cancel each other.

The final equivalent circuit element values Z ′1 , Z ′2 , Z ′3 , Z ′4 , and R′5 can beexpressed in terms of coupled line parameters Zac , Zbc , and k as follows:

Z ′1 =Z2a

N2 = Zac√1 − k2 (13.35a)

Z ′2 =Z2b

N2 = Zbc√1 − k2 (13.35b)

508 Baluns

Figure 13.30 (a–d) Coupled-line Marchand balun and its equivalent circuits. (From: [32]. 1992 Interna-tional Journal of Microwave Millimeter-Wave Computer-Aided Engineering. Reprinted withpermission.)

Z ′3 =Z1a + Z1b

N2 = (Zac + Zbc )k2

√1 − k2(13.35c)

Z ′4 =Z

N2 = Zk2 (13.35d)

R′5 =R

N2 = Rk2 (13.35e)

where R is the load impedance, and Z2ac = Za

0e Za0o and Z2

bc = Zb0e Zb

0o are thecharacteristic impedances of the coupled lines a and b, respectively.

The equations above can be rearranged to solve for the coupled-line parametersin terms of the equivalent element values as follows:

k = √ Z ′3Z ′1 + Z ′2 + Z ′3

(13.36a)

Zac =Z ′1

√1 − k2(13.36b)


Zbc =Z ′2

√1 − k2(13.36c)

Choosing element values for Z ′1 , Z ′2 , Z ′3 , Z ′4 , and R′5 for the final design of a balunto meet specifications, the coupled-line parameters can be calculated from (13.30)and (13.31). Finally, the physical dimensions for the balun are determined usingsuitable coupled-line structures; for example, edge-coupled microstrip [46–48],broadside-coupled striplines [48, 49], embedded microstrip [50], or any othertopology.

The design of a balun starts with determining the network parameters givenin Figure 13.30(d) to meet design specifications in terms of bandwidth, load imped-ance, and return loss. This can be achieved by either using nomographs [8] orusing synthesis techniques [51], or by using commercial CAD tools. Balun equiva-lent circuit parameters as a function of return loss for fourth-order Chebyshevfilters having 2:1, 3:1, and 5:1 bandwidth are given in Figure 13.31 [32] for asource impedance of 50V. Coupled-line parameters are calculated using (13.30)and (13.31) and plotted in Figure 13.32. These figures also include the values ofZ and the load impedance R.

In addition to the above synthesis analysis of planar Marchand baluns, severalother analysis and design methods of such baluns have been reported. This includesequivalent circuit model for the two-coupled line sections presented by Tsai andGupta [33], the method reported by Schwindt and Nguyen [37], that providesanalytical expressions for the S-parameters for a planar multilayer Marchand balunbased on broadside-coupled quasi-TEM normal-mode parameters, and the Engelsand Jansen [38] design method based on the mode parameters for multiple coupledstrip configurations. These methods provide a reasonably good starting point forthe design of complicated multilayer baluns. However, for accurate characterizationof these baluns, three-dimensional electromagnetic simulators are essential andplay 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 30-dB return loss, R′5 = 95V

and 3:1 bandwidth. From Figure 13.31(b), approximate values for filter networkare Z ′1 = 61.5V, Z ′2 = 50V, Z ′3 = 95V, and Z ′4 = 80V. From (13.3) and (13.35),the coupled-line parameters are k = 0.678, Zac = 83.7V, Zbc = 68V, R = 206V,and Z = 174V. From (13.30) and (13.31), Za

0e = 191.1V, Za0o = 36.7V,

Zb0e = 155.3V, and Zb

0o = 29.8V. Figure 13.32 can also be used directly to obtainthese parameters. These values are used to determine the physical parameters fora given substrate thickness and dielectric constant values for coupled-line balun[46–48] or broadside coupled-line [48, 49].

Tightly coupled sections of this balun are not possible with edge-coupled micro-strip lines. The embedded microstrip technique [50] or Lange coupler configuration,however, can be used to realize tightly coupled structures. Realization of suchcouplers with multilayer techniques on substrates such as alumina, quartz, or GaAs[31] have also been reported. A cross-sectional view of a broadside-type coupledline showing even- and odd-mode 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]. 1992International Journal of Microwave Millimeter-Wave Computer-Aided Engineering.Reprinted with permission.)


Figure 13.32 Marchand balun coupled-line parameters as a function of load impedance for(a) 2:1 bandwidth, (b) 3:1 bandwidth, and (c) 5:1 bandwidth. (From: [32]. 1992International Journal of Microwave Millimeter-Wave Computer-Aided Engineering.Reprinted with permission.)

512 Baluns

Figure 13.33 Illustration for the realization of a typical broadside-coupled line. (From: [31]. 1991IEEE. Reprinted with permission.)

Marchand Balun Using Lange Coupler

A Marchand balun using two tight couplers having connections as shown in Figure13.34 has been described by Tsai [52]. The couplers used are slightly under-coupled3-dB Lange couplers, fabricated on 100-mm-thick GaAs substrate. The linewidth

Figure 13.34 (a) Schematic using crossover couplers and (b) rearranged schematic further simpli-fying the circuit.


of the couplers is 10 mm and the spacings between the conductors are 12 mm and10 mm 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 connectionwas realized using a via hole. A photograph of the chip, which is 1 by 2 mm, isshown in Figure 13.35.

Simulated performance of the balun is shown in Figure 13.36. Measured inser-tion loss of the balun’s output ports is shown in Figure 13.37, and Figure 13.38shows the amplitude and phase differences between the two outputs over a 0.1 to20-GHz frequency range. In an ideal case, the desired values of amplitude andphase difference are 0 dB and 180 degrees, respectively. This balun demonstratesgood performance over the 6- to 20-GHz frequency range.

Figure 13.35 Photograph of a balun using Lange couplers. (From: [52]. 1993 IEEE. Reprintedwith 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. Reprintedwith permission.)

Figure 13.38 Measured amplitude and phase balance of the balun. (From: [52]. 1993 IEEE.Reprinted with permission.)

Marchand Balun Using Re-Entrant Couplers

An alternative solution to broadside multilayer coupled lines and Lange couplersis to use re-entrant multilayer coupled lines [53–55] to realize the balun circuit.The latter structure has the advantage of providing tight coupling without stringentdimensional requirements in circuit fabrication. Re-entrant couplers using


multilayer microstrip lines have been reported, and a configuration appears inFigure 13.39. It consists of three conductors in which two conductors are depositedon the top surface of a two-layered dielectric substrate and a third conductor thatis floating is sandwiched between the two dielectric layers. The floating conductorbeneath the coupled lines helps in realizing higher even-mode impedance and tightcoupling. Because there are a total of four conductors, the structure can supportthree propagation modes instead of the two general even and odd modes in thecase of a three-conductor structure.

Although multilayer dielectric structures allow much greater freedom in design-ing the baluns in terms of process and size requirements, they give rise to otherstructurally related problems. A multilayer microstrip structure is inhomogeneousin character and there are no simple design expressions available for determiningthe physical dimensions. Commercially available EM simulators can be used toanalyze the structure, but they usually require a considerable amount of computa-tion time to optimize the design for best performance. Recently, a generalizednetwork model for coupled multiconductor transmission lines has been reported[36] that can also be used to design re-entrant-type couplers. A simple analysis ofthis coupler has been discussed in Section 8.6.3, and the design procedure for thebalun can be summarized as follows:

1. Calculate the even- (Z0e ) and odd- (Z0o ) mode impedances for each coupledsection as described in the beginning of this section.

2. Because Z02 = Z0o , calculate the physical dimensions of the top-layer struc-ture such as er2, d, and W2 by keeping a large separation between thestructure.

3. Calculate Z01 = (Z0e − Z02)/2 and determine the physical dimensions ofthe bottom-layer structure such as er1 , h, and W1 .

4. Optimize the balun in terms of the structure’s physical dimensions by fine-tuning the above calculated dimensions using EM simulators.

For example, a Marchand balun covering 3:1 bandwidth with a 41.4-dB in-bandreturn loss requires Za

0e = 264V, Za0o = 30V, Zb

0e = 117V, and Zb0o = 13V [32].

Figure 13.39 Microstrip re-entrant-type coupler.

516 Baluns

The design dimensions are summarized in Table 13.2. Both these examples demon-strate the realization of Marchand baluns using multilayer microstrip structures.Figure 13.40 shows the simulated performance of a balun using re-entrant-typecouplers.

Monolithic Marchand Balun

A monolithic Marchand balun has been developed [41] using multilayer dielectricmonolithic technology, where polyimide (er = 3.2) was used as the intermetaldielectric. Figure 13.41 shows the top view and side view of the monolithicMarchand balun. The dimensions for the various line sections are input line A hasa 40-mm linewidth and is 1,400 mm long. The linewidth, length for the open-circuited line B and short-circuited bottom lines C are 140 mm, 1,420 mm, and120 mm and 1,427 mm, respectively. The unbalanced input impedance is 20V andthe balanced output impedance is 50V. The balun was designed using electromag-netic simulation. The input return shown in Figure 13.42(a) was measured byterminating the balanced port into a 50V load.

The insertion loss of the baluns when connected back to back is shown inFigure 13.42(b). For a single balun, the insertion loss is less than 0.7 dB from 6to 21 GHz. The measured amplitude balance was within 0.5 dB from 7 to 21 GHzwith a corresponding phase difference of 178 to 172 degrees.

Table 13.2 Design Parameters for a 3:1 Bandwidth Balun Using Re-Entrant Couplers Calculated UsingEM Simulator [32]

Design h d W1a W2a Sa W1b W2b Sb# Z a

0e Z b0o Z b

0e Z b0o er2 er1 (mil) (mil) (mil) (mil) (mil) (mil) (mil) (mil)

1 200 28 111 11 9.8 3.2 15 0.2 3.0 0.7 0.5 14.0 2.5 1.02 264 30 117 13 12.9 3.2 25 0.2 2.0 0.7 0.5 17.0 2.0 1.0

Figure 13.40 Linear circuit simulation of 3:1 bandwidth Marchand balun with ideal element valuesZ a

0e = 269.3V, Z a0o = 29.7V, Z b

0e = 117.1V, Z b0o = 13.2V, Z ′4 = 94V, and

R ′ = 100.4V. (From: [32]. 1992 International Journal of Microwave Millimeter-Wave Computer-Aided Engineering. Reprinted with permission.)


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 50Vload. (b) Measured loss of four pairs of back-to-back Marchand baluns. (From: [41]. 1997 IEEE. Reprinted with permission.)

Compact Marchand Balun

At lower microwave frequencies, the balun size becomes large. Various techniqueshave 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 liquidcrystalline polymer LCP [64] technologies. For example, a Marchand balun isrealized in different layers of the multilayer substrate and the coupled lines areprinted in spiral shapes to reduce drastically the component size. Several othermethods, including inductive loading [64], capacitive loading [65], and impedancevariation [66], have been used to reduce the balun size. A compact lumped-elementMarchand 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

broadside-coupled sections to fabricate a lumped-element Marchand balun on Sisubstrate. Figure 13.43 shows the top view of a 1.9-GHz balun and occupies onlya 2.5-mm2 area.

13.6 Other Baluns

13.6.1 Coplanar Waveguide Baluns

Apart from microstrip baluns, coplanar waveguide (CPW) baluns have also beenreported [30, 40, 55]. An analysis of such baluns is given in [30], while a physicallayout is shown in Figure 13.44. The physical length of the balun section is l /4.This balun does not have as large a bandwidth as planar baluns or Marchandbaluns. Another CPW balun [55] is shown in Figure 13.45. This balun also hasnarrowband performance.

13.6.2 Triformer Balun

A triformer balun that is a rectangular-spiral-shaped inductor using three coupledlines has been developed for microwave monolithic integrated circuit applications[19]. The balun was designed using the multiconductor coupled-line transformerchain matrix method. However, this structure can be accurately analyzed usingEM simulators. The physical layout, equivalent circuit, and microphotograph ofa one-turn triformer are shown in Figure 13.46. The structure was fabricated onGaAs (er = 12.9, tan d = 0.0003) substrate with a 2-mm conductor thickness, a10-mm conductor width, and 5-mm spacing between the conductors. The goldconductors have metal resistivity of 0.03 V-mm, and the substrate thickness is100 mm. Measured and simulated performance of the triformer is shown in Figure13.47. The differential phase-shift between the output ports 1 and 6 remainsconstant at about 182 degrees. A major shortcoming of this approach is the loss

Figure 13.43 Top view of a lumped-element balun layout. (From: [39]. 1997 IEEE. Reprintedwith permission.)

13.6 Other Baluns 519

Figure 13.44 Layout of a planar balun using CPW and coplanar strips. (From: [30]. 1991 Micro-wave and Optical Technology Letters. Reprinted with permission.)

in the structure because of the thin substrate and narrow conductor width. Thiscan be overcome to some extent by printing thick lines on thick alumina substrates.

13.6.3 Planar-Transformer Balun

Planar-transformer baluns consist of two oppositely wrapped twin-coil transform-ers connected in series. In this configuration, one of the two outer nodes in theprimary coil and the inner common node in the secondary coil are grounded asshown in Figure 13.48 [31]. Figure 13.48(a) shows the equivalent circuit, andFigure 13.48(b) is a microphotograph of the balun using rectangular spiral trans-formers. The chip measures about 1.5 mm2, which demonstrates the compactnessof this approach.

The resonant frequency of the coil transformers due to interturn and otherparasitic capacitances with the ground, limits the performance and bandwidth ofthis 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 theupper frequency bound is set by the resonant frequency of the coil. The bandwidthcan be increased either by increasing the resonant frequency while maintaining thesame inductance or by increasing the inductance while maintaining the resonantfrequency. The resonant frequency can be increased by reducing the interturn andparasitic capacitances by employing thick and low-dielectric constant substratesand air bridges in the coil. In addition, the inductance can be increased by optimizingthe area-to-length ratio.

520 Baluns

Figure 13.45 Top and cross sectional illustrations of a three-strip CPW balun.

Figure 13.49 shows the simulated and measured performance of a planar-transformer balun. The amplitude and phase imbalances between the two balancedports are less than 1.5 dB and 10 degrees, respectively, over the 1.5- to 6.5-GHzfrequency band. The simulated results shown were obtained using EM analysis.

Multiple coupled lines [66–68] have also been used to design baluns, includingMarchand baluns. Multiple coupled-line based baluns are planar, have couplingtighter than two coupled lines, and are simpler to fabricate than multilayer coupled-line baluns. Recently, an improved planar Marchand balun using a patternedground plane to realize very high even-mode impedance has been reported [69].

We described several kinds of baluns in this chapter. The selection of a particulartype depends upon the application, performance, and cost limitations. Among allthe baluns described, the Marchand balun achieves the maximum bandwidth andbest performance.


Figure 13.46 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 S-parameters: (a) magnitude, and (b) differ-ential phase shift. (From: [19]. 1989 IEEE. Reprinted with permission.)


Figure 13.48 (a) Simplified circuit diagram and (b) photograph of a rectangular spiral transformerbalun. (From: [31]. 1991 IEEE. Reprinted with permission.)

Figure 13.49 Comparison between simulated and measured performances of a planar-transformerbalun. (From: [31]. 1991 IEEE. Reprinted with permission.)

524 Baluns

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[45] Boire, D. C., J. E. Degenford, and M. Cohn, ‘‘A 4.5 to 18 GHz Phase Shifter,’’ IEEEMTT-S Int. Microwave Symp. Dig., 1985, pp. 601–604.

526 Baluns

[46] Gupta, K. C., et al., Microstrip Lines and Slotlines, 2nd ed., Norwood, MA: Artech House,1996.

[47] Garg, R., and I. J. Bahl, ‘‘Characteristics of Coupled Microstriplines,’’ IEEE Trans. Micro-wave Theory Tech., Vol. MTT-27, July 1979, pp. 700–705; also see correction in IEEETrans. Microwave Theory Tech., Vol. MTT-28, March 1980, p. 272.

[48] Bahl, I. J., and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed., New York:John Wiley, 2003.

[49] Cohn, S. B., ‘‘Characteristic Impedances of Broadside-Coupled Strip Transmission Lines,’’IRE Trans. Microwave Theory Tech., Vol. MTT-8, November 1960, pp. 633–637.

[50] Willems, D., and I. Bahl, ‘‘A MMIC Compatible Tightly Coupled Line Structure UsingEmbedded Microstrip,’’ IEEE Trans. Microwave Theory Tech., Vol. 41, December 1993,pp. 2303–2310.

[51] Horton, M., and R. Wenzel, ‘‘General Theory and Design of Optimum Quarter-WaveTEM Filters,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-13, May 1965,pp. 316–327.

[52] Tsai, M. C., ‘‘A New Compact Wideband Balun,’’ IEEE MTT-S Int. Microwave andMillimeter Wave Monolithic Circuit Symp. Dig., 1993, pp. 123–125.

[53] Cohn, S. B., ‘‘The Re-Entrant Cross Section and Wide-Band 3-dB Hybrid Couplers,’’IEEE Trans. Microwave Theory and Tech., Vol. MTT-11, July 1963, pp. 254–258.

[54] Lavendol, L., and J. J. Taub, ‘‘Re-Entrant Directional Coupler Using Strip TransmissionLine,’’ IEEE Trans. Microwave Theory and Tech., Vol. MTT-13, September 1965,pp. 700–701.

[55] DeBrecht, R. E., ‘‘Coplanar Balun Circuits for GaAs FET High-Power Push-Pull Amplifi-ers,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1973, pp. 309–312.

[56] Chiou, H.-K., and H.-H. Lin, ‘‘A Miniature MMIC Doubly Balanced Mixer Using LumpedElement Dual Balun for High Dynamic Receiver Applications,’’ IEEE Microwave GuidedWave Lett., Vol. 7, August 1997, pp. 227–229.

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[58] Bakalski, W., et al., ‘‘Lumped and Distributed Lattice-Type LC-Baluns,’’ IEEE MTT-SInt. Microwave Symp. Dig., 2002, pp. 209–212.

[59] Kuylenstierna, D., and P. Linner, ‘‘Design of Broad-Band Lumped Element Baluns withInherent Impedance Transformation,’’ IEEE Trans. Microwave Theory Tech., Vol. 52,December 2004, pp. 2739–2745.

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[61] Yoon, Y. J., et al., ‘‘Design and Characterization of Multilayer Spiral Transmission-LineBaluns,’’ IEEE Trans. Microwave Theory Tech., Vol. 47, September 1999, pp. 1841–1847.

[62] Yoon, Y. J., et al., ‘‘Modeling of Monolithic RF Spiral Transmission-Line Balun,’’ IEETrans. Microwave Theory Tech., Vol. 49, February 2001, pp. 393–395.

[63] Guo, Y. X., Z. Y. Zhang, and L. C. Ong, ‘‘Design of Miniaturized LTCC Baluns,’’ IEEEMTT-S, Int. Microwave Symp. Dig., 2006, pp. 1567–1570.

[64] Govind, V., et al., ‘‘Analysis and Design of Compact Wideband Baluns on MultilayerLiquid Crystalline Polymer (LCP) Based Substrates,’’ IEEE MTT-S, Int. Microwave Symp.Dig., 2005.

[65] Ang, K. S., Y. C. Leong, and C. H. Lee, ‘‘Analysis and Design of Miniaturized Lumped-Distributed Impedance-Transforming Baluns,’’ IEEE Trans. Microwave Theory Tech.,Vol. 51, March 2003, pp. 1009–1017.

[66] Lee, J.-W., and K. J. Webb, ‘‘Analysis and Design of Low-Loss Planar Microwave BalunsHaving Three Symmetric Coupled Lines,’’ IEEE MTT-S Int. Microwave Symp. Dig.,June 2002, pp. 117–120.


[67] Cho, C., and K. C. Gupta, ‘‘A New Design Procedure for Single-Layer and Two-LayerThree-Line Baluns,’’ IEEE Trans. Microwave Theory Tech., Vol. 46, December 1998,pp. 2514–2519.

[68] Chen, Y.-L., and H.-H. Lin, ‘‘Novel Broadband Planar Balun Using Multiple CoupledLines,’’ IEEE MTT-S Int. Microwave Symp. Dig., 2006, pp. 1571–1574.

[69] Zhang, Z., et al., ‘‘Improved Planar Marchand Balun Using a Patterned Ground Plane,’’Int. J. RF and Microwave Computer-Aided 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 theIndian 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 hedesigned dielectric resonators filters and multiplexers for on-board satellite commu-nication equipment. Dr. Mongia presently works for REMEC Defense and Space,Richardson, Texas, where he is involved in the designs of GaAs MMICs, includingT/R ICs for phased-array applications. He has published more than 50 publicationsin 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. AtM/A-COM, as a Distinguished Fellow of Technology, his interests are in the areaof device modeling, high-efficiency power amplifiers, and MMIC products forcommercial and military applications. Dr. Bahl is the author of more than 145research papers and 12 books, and he holds 16 patents. He is an IEEE Fellow, amember of the Electromagnetic Academy and is also the editor for the InternationalJournal of RF and Microwave Computer-Aided Engineering.

P. Bhartia graduated with a B. Tech. (Hons.) from IIT, Bombay, and a Ph.D. fromthe University of Manitoba, Winnipeg. Over a 25-year career with the Departmentof National Defence in Canada, Dr. Bhartia held four director-level and two directorgeneral level positions. Dr. Bhartia has published extensively with more than 200publications, 5 patents, and 9 books to his credit. He was appointed to the Orderof 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 ofEngineer, and The Institute of Electronic and Telecommunication Engineers. Dr.Bhartia received the IEEE McNaughton Gold Medal for his contributions to engi-neering. He is currently the executive vice president of Natel Engineering in Chart-sworth, 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 Departmentof Electrical, Electronic and Computer Engineering, at Heriot-Watt University,

529

530 About the Authors

Edinburgh, United Kingdom, leading a team for research into advanced RF/micro-wave device technologies. He has authored and coauthored more than 130 researchpapers, as well as a book, Microstrip Filters for RF/Microwave Applications (Wiley,2001). Dr. Hong is a senior member of the IEEE.

Index

3-dB couplers Asymmetrical coupled lines, 131–32approximate distributed lineasymmetrical, 153–55

parameters, 132–33with symmetrical microstrip lines,backward-wave directional couplers,151–53

136–384:1 TLTdirectional couplers using, 133–38fractional bandwidth, 461forward-wave directional couplers,reflection coefficient response, 458

133–36source impedance, 460maximum power coupled between,transmission coefficient response, 459

162See also Transmission linenon-TEM mode, 136transformers (TLTs)normal-mode parameters, 105, 132–33p /4 quadrature phase-shift keyingphase velocities, 133(QPSK) modulation, 12quasi-TEM mode, 136p equivalent, 470, 471width, 133

Asymmetrical couplers, 178Adefined, 167

ABCD matrices multisection, 177filter analysis, 279–80 Asymmetrical forward couplers, 153–55lumped-element branch-line coupler, 3-dB, 153–55

231 end-to-end symmetry, 154of lumped-element circuit, 226 fabrication, 153

ABCD parameters, 38–40 microstrip line width, 153defined, 38 phase difference, 154, 155illustrated, 44 strip pattern, 154normalized, 41 theoretical/experimental response, 154properties, 39–40 See also Forward-wave directionalreflection coefficient and, 40–41 couplerstransmission coefficient and, 40–41 Asymmetrical nonuniform couplers,unnormalized, 41 214–17

Admittance matrix, 27 broadband coupling, 214conversion relations, 43–45 equivalent circuit to determinefour-port network, 487 coupling, 215lossless network, 27 even-mode equivalent circuit, 214properties, 27–28 even-mode reflection coefficient, 215

Applications, 11–13 even-/odd-mode characteristicimpedances, 216Approximate analysis, 462–64

531

532 Index

Asymmetrical nonuniform couplers CPW, 518defined, 481(continued)

illustrated, 214 fabricated chip, 498lumped-element, 483reflectionless taper, 216–17

scattering matrix, 216 Marchand, 498–518microstrip-to-balanced stripline,scattering parameters, 214

tandem connection, 216 482–86nonplanar, 486See also Nonuniform TEM directional

couplers parallel-strip, 484planar, 481, 485, 490–98Asymmetric port excitation filters,

347–52 planar-transformer, 519–20printed, 504, 505EM excitation, 348

five-pole microstrip pseudocombline spiral transformer, 523triformer, 518–19bandpass, 349

frequency responses, 349, 350 uses, 481, 481–82VSWR versus electrical length, 504open-loop resonator feed structure,

351 wide-band, 503Bandpass filterssymmetrical port excitation versus,

347 ceramic-block, 295Chebychev lowpass prototype, 310two-pole microstrip pseudocombline

bandpass, 350 fractional bandwidth, 310frequency response, 278Attenuation constant, 54, 55

Attenuation pole, 311 ideal, 271response, 271

B structure, 278transformation, 277–78Backward-wave couplers, 105

with asymmetrical coupled lines, two-pole microstrip pseudocombline,347136–38

conditions, 116–17 See also FiltersBandstop filtersdefined, 6–7, 167

frequency response, 119 frequency response, 279parallel-coupled, 300–304ideal, 118, 119

in inhomogeneous medium, 143 response, 271structure, 279maximum coupling value, 119

maximum value, 120 transformation, 278–79See also Filtersscattering parameters, 118

TEM, 118 Biasing circuits, 446–49with multisection shunt stubs, 448See also Directional couplers

Balanced CRLH transmission line, 435, parameters, 449simplified, 448436

Balance mixers, 481 simulated response, 449Branch-line couplers, 219Baluns, 481–523

analysis, 486–90 as 90-degree hybrid, 222broadband, 233broadband, 496

coaxial, 481, 484 characteristic impedance, 220conservation of energy, 220coupled-line, 486–90

Index 533

coupling variation, 223 Cdirectivity, 225 Capacitance matrix, 61–62ideal, isolation, 225 Capacitances, 61–68isolation variation, 223 coupled line, 61–68for loose coupling, 223–25 even-mode, 62lumped-element, 230–33 fringing, 66–68, 192main-line characteristic impedance, odd-mode, 62–65

221 offset striplines, 141modified, 219, 223–25 parallel plate, 66–68physical implementation, 222–23 self-capacitance, 127in planar circuit configuration, 220

total series, 463planes of symmetry, 221

wiggly lines, 192reduced-size, 219, 225–30

Cascaded quadruplet (CQ) filter, 327–28scattering parameters, 220

coupling structures, 328shunt branches, 221

defined, 327–28size, 219Cascaded quadruplet trisection (CQT)VSWR variation, 223

filters, 371–77See also Tight couplers10-pole microstrip bandpassBroadband 180-degree bit phase shifters,

configuration, 375475, 493experiment, 376–77Broadband branch-line coupler, 233frequency responses, 375Broadband dc block, 446illustrated, 372Broadband forward-wave directionalimplementation, 373–74couplers, 149–65measured performance, 378Broadband rat-race coupler, 237–38sensitivity analysis, 374–76Broadside-coupled lines, 93–99synthesis, 372–73asymmetric, 10theoretical response, 373defined, 1wideband response, 378even-mode field distribution, 94

Cascaded trisection (CT) filterillustrated, 4coupling structure, 328odd-mode field distribution, 94defined, 328offset striplines, 96–99

Ceramic-block filters, 292–95striplines, 94–95cross-sectional view, 294suspended microstrip lines, 95–96frequency response, 297Broadside couplers, 255–58schematic, 2933-dB asymmetric, 257, 258specifications, 295amplitude characteristics, 257

Characteristic impedance, 32, 54cross section, 256backward-traveling mode, 121measured coupling coefficient, 258branch-line couplers, 220multilayer, structure, 256broadside-coupled suspendedphysical layout, 258

microstrip lines, 97symmetric, 257conductor-backed coplanarBroadside striplines, 2

waveguides, 89Butterworth filter, 272coplanar waveguides, 86element values, 273

number of sections, 273 coupled coplanar waveguides, 91

534 Index

Characteristic impedance (continued) multilayered, 437–38structure, 265even mode, 5, 68

finite strip thickness effect, 70, 71 unbalanced, 435, 437unit cell, 438finite-thickness microstrip, 76, 77

inverted microstrip lines, 93 Conductor-backed coplanar waveguides,87–88normalized voltages/currents and, 19

odd mode, 5, 68 characteristic impedance, 89effective dielectric constant, 89rat-race couplers, 235

slot-coupled microstrip lines, 100, 101 See also Coplanar waveguides (CPW)Conductor loss, 58striplines, 71

suspended microstrip lines, 93 quasi-TEM mode, 60–61single microstrip, 79Chebyshev response filter, 272–75

element values, 276 TEM mode, 60–61Conservation of power principle, 158insertion loss, 272

number of sections, 274 Coplanar waveguides (CPW), 53, 83–88baluns, 518passband VSWR maximum, 274–75

Coaxial baluns, 481 characteristic impedance, 86conductor-backed, 86, 87–88illustrated, 484

Marchand, 501–5 coupled, 88effective dielectric constant, 86synthesis, 505

See also Baluns on finite thickness dielectric substrate,85Coaxial lines, 2

Codirectional couplers. See Forward- illustrated, 3schematic, 390wave directional couplers

Combline filters, 10, 11, 290–95 with upper shielding, 86–87Coupled coplanar waveguides, 88ceramic block, 292, 293

coaxial, 291–92 characteristic impedance, 91cross section, 90cross-sectional view, 294

defined, 290–91 effective dielectric constant, 92Coupled-line baluns, 486–90design example, 291–95

design procedures, 291 Coupled-line circuit components, 443–76broadband 180-degree bit phasefrequency response, 297

resonator length, 293 shifters, 475dc blocks, 443–52See also Filters

Compact couplers, 176, 261–64 delay lines, 475interdigital capacitors, 461–65coupled-line, 264

lumped-element, 262 power dividers, 475resonators, 475meander line, 263–64

measured electrical performance, 265 Schiffman sections, 475spiral inductors, 465–72physical dimensions, 264

spiral, 262–63 spiral transformers, 472–75transformers, 452–61Compact Marchand balun, 517–19

Complementary SRRs (CSRRs), 434 Coupled-line filters, 269–304advanced filtering characteristics,Composite right/left-handed (CRLH)

transmission line, 435 327–52with asymmetric port excitations,balanced, 435, 436

metamaterial applications, 437 347–52

Index 535

with cross-coupled resonators, 327–35 Couplingdesirable, 1with defected ground structures,mechanism, 4–6324–27parasitic, 1–2with enhanced stopband performance,between symmetrical lines, 160,307–27

161–63with meandered parallel-coupled lines,weak, 157–58320–24

Coupling coefficientsmicrostrip CPW, 410–21capacitive, 145multimode, 406–10determination arrangement, 334with periodically nonuniform coupledextraction arrangement, 333, 335lines, 314–20inductive, 145with source-load coupling, 335–47between lines, 158superconductor, 371–85mutual, 157with unevenly-coupled stages, 307–14nature of, 158Coupled-line Marchand balun, 508voltage, 6, 171

Coupled lines, 1weakly coupled resonators, 163

capacitances, 61–68 Cross-coupled filters, 327–35distributed equivalent circuit, 126–30 applications, 328even-/odd-mode characteristic defined, 327

impedances, 84, 85 external quality factors, 329illustrated, 2 implementation, 329–30multiconductor/multilevel layout, 336

configuration, 3 miniature, 333–35symmetrical, 67–68 open-loop resonator, 353uniformly, 105–47 theoretical frequency responses, 330

Coupled microstrip lines Cross-coupled optimum stub filter, 404–6covered with dielectric overlay, 190 circuit parameters, 406frequency-dependent characteristics, circuit topology, 405

83 illustrated, 405Currentsillustrated, 82

equivalent, 18quasistatic even-/odd-modenormalized, 18–21characteristic impedances, 82–83unnormalized, 21–22Coupled-mode theory, 149, 156–63

Cutoff frequencyeven-/odd-mode analysis and, 160–61parallel-coupled bandstop filter, 301voltage waves modification, 157striplines, 73for weakly coupled resonators, 163–65

Coupled spur lines, 192–94 DCoupled structures, 1–6 DC blocks, 443–52

broadside, 1 analysis, 443–46components based on, 6–11 biasing circuits, 446–49edge, 1 broadband, 446symmetric, 4 high-voltage, 451TEM modes, 3 microstrip interdigital, 447types, 3–4 millimeter-wave, 449–51uniform, 4 quarter-wave coupled-line section as,

444weakly, 149

536 Index

Defected ground structures, 324–27 improvement techniques, 186–94EM-simulated performance and, 327 lumped compensation, 186–89floating conductor at, 325 meandered coupled line sections, 194microstrip line characteristics, 325 modified branch-line coupler, 225microstrip line cross section, 324 shunt inductive feedback, 192, 193three-pole parallel-coupled microstrip wiggly lines, 189–92

line filters, 326 Distributed equivalent circuit, 126–30Delay lines, 475 Dual-band filters, 359–67Desirable coupling, 1 approaches, 363Diagonal conductors, 256 categories, 362Dielectric loss, 58, 79 demand, 359Dielectric overlays, 189 design, 365–66Directional couplers, 6–8 simultaneous operation, 362–63

with asymmetrical coupled lines, with SIR, 365–66133–38

backward-wave, 6–7, 105, 116–20 Ebroadside, 255–58

Edge-coupled striplines, 73–74compact, 261–64characteristic impedances, 74coupling, 168cross section, 69directivity, 168See also Striplinesforward-wave, 105, 114–16

Edge-coupled structures, 1four-port, 167–68Effective dielectric constantshybrid, with interdigital Lange

broadside-coupled suspendedconfiguration, 9microstrip lines, 97ideal, power loss ratio, 180–84

conductor-backed coplanarinterdigital, 242–52waveguides, 89isolation, 168

coplanar waveguides (CPW), 86lumped capacitor compensation, 9coupled coplanar waveguides, 92microwave, 6inverted microstrip lines, 90, 93multiconductor, 242–52slot-coupled microstrip lines, 100, 101multilayer, 255–61suspended microstrip lines, 90multioctave bandwidth, 7

Electromagnetic metamaterials, 428multisection, 177–86Electromagnetic simulators, 91, 222nonuniform broadband TEM,Electronic warfare (EW), 12197–217Equal magnitude, 107parallel-coupled TEM, xxii, 167–94Equal ripple function, 208re-entrant mode, 258–61Even modesingle-section, 169–76

broadside-coupled microstrip lines, 94spiral, 262–63capacitances, 62tandem, 7, 252–55characteristic impedances, 5, 6, 68tight, 8coupled-mode theory and, 160–61uses, 6excitation, 63, 107, 108–9wiggly two-line, 9impedances, 183Directivity, 168phase velocities, 68branch-line couplers, 225plane of symmetry behavior, 111coupled spur lines, 192–94

dielectric overlays, 189 reflection coefficients, 117, 215

Index 537

symmetrical network analysis, 106–11 microstrip CPW coupled-line, 410–21multimode coupled-line, 406–10See also Odd mode

External quality factor optimum stub line, 401–6parallel-coupled line, 11, 283–87cross-coupled filters, 329

extraction arrangement, 332 parameters, 269practical considerations, 280–83frequency response, 311

unevenly-coupled stages, 311 response-type, 275source-load coupling, 335–47

F superconductor coupled-line, 371–85theory and design, 271–83Fast wave, 159

Filters, 8–11 types, 270UWB, 12, 400–428with advanced dielectric materials,

391–428 UWB with notch band, 421–28Finite-thickness microstrip, 76–77with advanced materials, 371–438

analysis, 279–80 characteristic impedance, 76, 77synthesis equation, 77applications, 270–71

bandpass, 271, 277–78 See also Microstrip linesFolded filters, 327bandstop, 271, 278–79

CAD and, 280 Forward-wave directional couplers, 105,114–16, 150–56cascaded quadruplet (CQ), 327–28

cascaded trisection (CT), 328 3-dB, 151–55asymmetrical, 133–36, 153–55Chebyshev response, 272–75

coaxial line, 10 bandwidth, 149broadband, 149–65combline, 10, 11, 290–95

configurations, 11 defined, 149equations, deriving, 116coupled-line, 269–304

coupled-line, advanced filtering per-unit wavelength, 156realization, 149characteristics, 327–52

coupled-line, with enhanced stopband scattering parameters, 115symmetrical, 150–53performance, 307–27

cross-coupled, 327–35 ultra-broadband, 155–56See also Directional couplerscross-coupled optimum stub, 404–6

direct-coupled, 327 Four-port directional couplers, 167–68Four-port networkdual-band, 359–67

folded, 327 illustrated, 36reciprocal, 36–38general network configuration, 269

hairpin-line, 11, 295–300 reduction, 50–51two ports terminated in arbitrary load,highpass, 271, 277

importance, 8 50uniform asymmetrical coupled lines,interdigital, 10, 11, 287–90

LCP, 396–400 123uniform coupled symmetrical lines, 111lowpass, 271, 272

LTCC cavity, 394–96 Fourth-order Marchand balun, 507Fractional powerLTCC lumped element, 392–94

metamaterial, 428–29, 428–38 maximum, 162tandem couplers, 254–55micromachined, 385–91

microstrip coupled line, 8–10 Frequency bandwidth ratio, 169–70

538 Index

Frequency range, 269 fabrication and measurement, 384–85frequency response, 380Frequency responsewith group-delay equalization, 377–85asymmetric port excitation filters, 349,HTS bandpass, 379–80350layout, 384bandpass filters, 278modeling, 380–84bandstop filters, 279transmission response, 379combline filters, 297

Highpass filtersCQT filter, 375frequency response, 277for extracting external quality factor,response, 271311schematic, 277high-order filters, 380transformation, 276–77highpass filters, 277See also Filtersparallel-coupled bandstop filter, 304

High temperature superconductor (HTS)parallel-plate baluns, 488substrates, 12single-section directional coupler,

High-voltage dc block, 451169–70Hyperbolic secant, 69source-load coupling filters, 340

Fringing capacitance, 66–68 Idefined, 66 Impedance matrix, 26–27odd-mode, 192 conversion relations, 43–45

Fringing fields, 66 lossless network, 27Full-wave analysis, 464–65 properties, 27–28

Impedance transformersGconfigurations, 457

Gibb’s phenomenon, 204 substrate parameters, 458Group delay, 269 Incremental inductance rule, 60, 61

defined, 270 Inductance matrix, 62, 127dependencies, 281 Input impedance, 24equalization, 377–85 filters, 269self-equalization, 381 normalized, 23

unnormalized, 23–24H

Insertion loss, 269Hairpin-line filters, 295–300 Chebyshev response filter, 272

coupling coefficient, 298 defined, 179, 269defined, 296 Marchand baluns, 513, 514design, 297 narrow conductor, 490design example, 297–300 passband, 281inter-resonator coupling, 297 UWB bandpass filters, 424layout, 299 Interdigital capacitors, 461–65measured response, 299 approximate analysis, 462–64physical layout, 299 configuration, 461See also Filters full-wave analysis, 464–65

High-Frequency Structure Simulator parameters, 464(HFSS), 280 responses, 465, 466

High-order filters series resistance, 463coupling structure, 380 subcomponents, 462

total series capacitance, 463design, 380–84

Index 539

Interdigital couplers, 242–52 Marchand balun with, 512–14six-finger, 251–52design, 245–52

LCP filters, 396–400design equations, 24660-GHz band, 399–400equivalent capacitance network, 244miniature wideband, 396–99impedance parameters, 249planar, 399length, 247via-less, 399N-conductor, 246

Liquid crystal polymer (LCP), 391side view, 244duplexer, 400theory, 243–45layers, bonding, 400top view, 244uses, 392voltage coupling factor, 247See also LCP filtersInterdigital filters, 10, 11, 287–90

Load impedance, 24design equations, 287Local multipoint distribution systemsdesign examples, 287–90

(LMDS), 392measure passband performance, 292Lowpass filtersminiature, on silicon, 387–90

Butterworth, 272narrowband design, 287–89prototype, 272narrowband microstrip SIR, 358response, 271nine-pole, layout, 291

Low-temperature cofired ceramic (LTCC)nine-pole wideband SIR, 357substrates, xvii, 12, 264, 391popularity, 287

multilayer capability, 394with stepped impedance resonators,technology developments, 392352–59See also LTCC filtersuniform impedance resonators (UIR),

LTCC filters, 392–96357, 359cavity, 394–96

wideband design, 289–90design parameters, 396

wideband response, 292 layout, 393See also Filters lumped element, 392–94

Inverted microstrip lines, 88–93 measured/simulated performance, 393characteristic impedance, 93 multilayer capability, 394cross section, 92 Lumped capacitors, 186effective dielectric constant, 90, 93 directivity improvement of, 188illustrated, 3 in MMICs, 225See also Microstrip lines physical length, 187–88

Isolation, 168 reduced-size branch-line couplers, 225top view, 187

K Lumped compensation, 186–89Lumped-element branch-line coupler,K inverters, 283

230–33Kirchhoff’s equations, 279–80ABCD matrix, 231bandwidth, 232, 233

Llumped element values, 232, 233

Lange coupler, 220 ‘‘pi’’ network, 230, 231for 3-dB coupling, 249 realization, 230design, 245–46 sections, 233design data, 247–52 ‘‘tee’’ network, 230, 231

See also Branch-line couplersdimensional ratios, 250

540 Index

Lumped-element compact coupler, 262 layout, 323passband harmonics suppression, 322Lumped-element rat-race coupler, 240–41

equivalent lumped circuit, 243 structure, 321Meander line directional coupler, 263–64lumped elements, 240–41

See also Rat-race couplers Metal-insulator-metal (MIM) capacitors,230, 251, 445Lumped equivalent circuit, 127

Metamaterial filters, 428–38M configuration, 434

frequency response, 433Magnetic coupling, 331Marchand baluns, 498–518 layout, 432, 433

Metamaterials, 2, 265amplitude, 514characteristic impedances ratio, 500 CRLH, 435–38

summary, 438coaxial, 501–5coaxial cross section, 500 M/FILTER software, 300

Micromachined filters, 385–91compact, 517–18coupled-line, 508 miniature interdigital on silicon,

387–90coupled-line parameters, 511design, 509 overlay coupled CPW, 390–91

synthesized substrate, 387elements, 498equivalent circuit parameters, 510 Micromachining techniques, 264–65

Microstrip CPW coupled-line filters,equivalent transmission-line model,500 410–21

CPW resonator, 411examples, 509–18fourth-order, 507 on dielectric substrate, 414, 417, 420

EM-simulated performances, 418, 419,insertion loss, 513, 514with Lange coupler, 512–14 421

EM-simulated responses, 413monolithic, 516–17phase balance, 514 equivalent circuit model, 415

frequency responses, 415phase difference, 513realization, 505 illustrated, 412

operation, 411with re-entrant couplers, 514–16series/parallel-resonant compensating physical implementation, 417

structure, 413representation, 501simplified equivalent circuit, 501 theoretical responses, 416

UWB bandpass, 428simulated performance, 513synthesis, 505–9 Microstrip dispersion, 79–81

Microstrip linesversions, 498See also Baluns characteristics, 74–83

coupled, 81–83Mathcad, 206, 213MATLAB, 206 coupling between, 143

with defected ground aperture, 324Meandered coupled lines sections, 194Meandered parallel-coupled lines defined, 2

even-/odd-mode field configurations, 5designed for spurious passbandsuppression, 322 illustrated, 3, 75

inverted, 88–93design with, 320–24illustrated, 320 propagation mode, 74

Index 541

single, 75–81 Monolithic Marchand balun, 516–17slot-coupled, 99–102 Monolithic microwave integrated circuitssuspended, 88–93 (MMICs), 12types, 3 lumped capacitors, 225with unequal impedances, 4 reduced-size hybrid implementation,

Microstrip-to-balanced stripline balun, 228482–86 small, low-loss rat-race hybrid, 240

Microwave network theory, 17–51 Monte Carlo method, 374Millimeter-wave dc block, 449–51 Multiconductor coupled line triformer,Miniature cross-coupled filter, 333–35 521

configuration, 337 Multiconductor couplers, 242–52defined, 333–34 design, 245–52designing, 334–35 theory, 243–45layout, 338 Multilayer couplers, 138–47upper stopband, 335 broadside, 255–58wideband performance, 339 coupling, 143See also Cross-coupled filters design, 140–47

Miniature interdigital filters on silicon, EM simulated response, 147387–90 examples, 145–47

advantage, 389 inhomogeneous, 146EM-simulated loss effects, 389 re-entrant mode, 258–61fabricated five-pole, 388 Sonnet Lite and, 139–40layout, 388 tight, 255–61measured performance, 389 Multimode coupled-line filters, 406–10simulated performance, 389

coupled-line I/O feed sections, 409See also Interdigital filters

EM-simulated group delay response,Miniature wideband LCP filter, 396–99409Modified branch-line coupler, 219,

EM-simulated magnitude responses,223–25409bandwidth, 225

modified, 410defined, 223–24resonant frequency responses, 408directivity, 225UWB, 425–28equivalent length, 225

Multisection couplers, 177–86frequency response, 226asymmetrical, 177high-impedance transmission linelimitations, 184–86section, 224N-section, 177, 178illustrated, 225physical layout, 186for loose coupling, 223–24power loss ratio, 180shunt branches, 225response, 178, 179simulated response, 225, 226specifications, 182See also Branch-line couplerssymmetrical, 177Modified rat-race coupler, 237–38symmetrical nonuniform, 210bandwidth, 237synthesis, 177–84equivalent impedance, 237tables of parameters, 182–85illustrated, 237theory, 177–84performance, 237, 238

See also Rat-race couplers Mutual inductance, 164

542 Index

N Normal modesinterference between, 159Networksof symmetrical coupled lines, 163admittance matrix, 27–28warping, 156four-port, 36–38

N-port networks, 17, 25impedance matrix, 26–28input-output relationship, 17 Ointerconnected, scattering matrix,

Odd mode45–51capacitances, 62–65nonreciprocal, 35–36characteristic impedances, 5, 68

N-port, 17, 25coupled mode theory and, 160–61

power loss ratio, 179fringing capacitance, 192

reciprocal, 32, 34–35, 36–38impedances, 175, 183

scattering matrix, 28–32phase velocities, 68, 186

symmetrical, 106–11propagation velocity, 6

three-port, 34–36 reflection coefficients, 117two-port, 33–34, 38–43 symmetrical network analysis, 106–11

Nonplanar baluns, 486 See also Even modeNonuniform line resonators, 315–16 Odd mode excitation, 109–11

illustrated, 316 defined, 108resonant characteristics, 317 plane of symmetry behavior, 111structures, 315 of symmetrical coupled lines, 63

Nonuniformly coupled symmetric line, 4 Offset striplines, 7Nonuniform TEM directional couplers, broadside-coupled, 96–99

197–217 capacitance parameter, 141asymmetrical, 214–17 illustrated, 141defined, 197 inductance parameter, 141design procedure, 210–14 See also Striplinesin homogeneous dielectric medium, Open-loop resonator filters, 331

211–13 asymmetric feed scheme, 351phase constant, 211–12 cross-coupled, 353symmetrical, 197–214 feed structure, replacing, 352weighting function, 205 symmetric feed scheme, 351See also Directional couplers Optimum stub line filters, 401–6

Normalized currents, 18–21 circuit parameters, 402axial flow, 21 with cross coupling, 404–6characteristic impedance and, 19 design, 401–4determining, 18 on dielectric substrate, 403illustrated, 19 measured performance, 404total, 20 photo, 404

Normalized input impedance, 23 transmission characteristics, 402Normalized voltage, 18–21 Organization, this book, xvii–xviii

characteristic impedance and, 19 Output impedance, 269determining, 18 Overlay CPW filters, 390–91illustrated, 19 defined, 391

end-coupled, 392total, 20

Index 543

schematic, 390 Phase velocities, 54, 58asymmetrical coupled lines, 133usefulness, 391backward-traveling mode, 121

P equalizing, 190even/odd modes, 68, 186Parallel-coupled bandstop filter, 300–304

CAD file, 303 suspended microstrip lines, 93symmetrical nonuniform couplers, 203cutoff frequency, 301

defined, 300 Planar baluns, 481analysis, 493–96design example, 301–4

element values, 302 configurations, 492examples, 496–98equivalent circuit, 301

frequency response, 304 fabricated on low-dielectric substrate,502layout, 300

See also Filters illustrated, 485layout, 519Parallel-coupled line filters, 283–87

design, 283 phase balance, 499sections, 490–92design example, 285–87

equivalent circuit, 309 synthesis, 505–9topologies, 496illustrated, 284

K inverters, 283 two-port coupler configuration, 493See also Balunsfor multispurious suppression, 313

physical dimensions, 284 Planar-transformer balun, 519–20, 523Planar transmission lines, 53–102physical lengths, 285

single coupled-line section, 308 Power added efficiency (PAE), 481Power-coupling coefficient, 137three-pole, 286

transmission response, 308 Power dividers, 475Power loss ratio, 179unequal coupled sections, 312, 313

See also Filters ideal directional coupler, 180–84multisection coupler, 180Parallel-coupled TEM directional

couplers, 167–94 single-section coupler, 180Power transfer, 152coupling, 168

with dielectric overlay compensation, Printed baluns, 504, 505Propagation constant, 24189

directivity, 168Qisolation, 168

multisection, 177–86 Q-factors, 56–57conductor, 57parameters, 167–68

single-section, 169–76 defined, 56–57dielectric loss, 57See also Directional couplers

Parallel-plate baluns, 488 normalized conductor, 72Quarter-wave coupler, 176Parallel-plate capacitance, 66–68

Parallel-plate impedance, 485 Quasi-TEM modesasymmetrical coupled transmissionParasitic coupling, 1–2

Periodically nonuniform coupled lines, lines, 136characteristics, 58–59314–20

Phase constant, 54, 57–58 common transmission lines, 54

544 Index

Quasi-TEM modes (continued) defined, 239conductor loss, 60–61 illustrated, 240defined, 53 low-loss MMIC, 240line parameters, 53 photomicrograph, 242

port interchange, 241RSee also Rat-race couplers

Rat-race couplers, 219, 233–42 Re-entrant mode couplers, 258–61as 180-degree hybrid, 235 coupling coefficient, 259bandwidth, 237 cross section, 260characteristic impedances, 235 defined, 258conservation of power, 234 even-/odd-mode impedances, 258design, 235 Marchand balun with, 514–16discontinuities effect, 237 microstrip, 515frequency bandwidth requirement, 237

multilayer microstrip lines, 514–15lumped-element, 240–42

performance, 261modified, 237–38

schematic view, 259in planar circuit configuration, 234

typical dimensions, 260ports, 236Response-type filters, 275properties, 236–37Return loss, 23, 270reduced-size, 219, 239–40

scattering matrix, 234–35 Sscattering parameters, 233–34

SATCOM, 270size, 219Scattering matrix, 28–32strip conductor layout, 233

asymmetrical nonuniform couplers,three-quarter-wave-long section, 237216VSWR variation, 236

conversion relations, 43–45See also Tight couplersequivalent circuit determination, 112Reciprocal networks, 32interconnected networks, 45–51four-port, 36–38normalized, 106three-port, 34–35normalized/unnormalized matrices, 32Reduced-size branch-line couplers, 219,overall network, 46225–30rat-race couplers, 234–35ABCD matrix, 226reciprocal networks, 32bandwidth, 228reduced networks, 47–48calculated phase difference, 228, 229reduced two-port network, 49circuit equivalence, 227tandem couplers, 253illustrated, 230transformation, 31–32implementation, 228unitary property, 30–31lumped capacitors, 225unnormalized, 106main-line length, 228, 230

Scattering parameters, 29measured performance, 230amplitudes, 37photomicrograph, 230asymmetrical nonuniform couplers,shunt branches, 228, 230

214size, 230backward-wave couplers, 118See also Branch-line couplersbranch-line couplers, 220Reduced-size rat-race coupler, 219,forward-wave directional couplers,239–40

advantages, 239 115

Index 545

of reduced networks, 47–48 coupling structure, 339, 345cross coupling, 338symmetrical nonuniform couplers, 198

tandem couplers, 253 direct coupling, 338filtering characteristic, 338Schiffman sections, 475

Scope, this book, 13 fractional bandwidth, 337frequency response, 340Self-capacitance, 127

Self-inductances, 127, 141 I/O coupling structure, 343layout, 342, 344Series capacitor, 443

Shunt inductive feedback, 192, 193 n + 2 coupling matrix, 342, 343–44,345Single microstrip, 75–81

conductor loss, 79 theoretical responses, 345three-pole microstrip parallel-coupled,dielectric loss, 79

finite-thickness, 76–77 345, 346topology, 337microstrip dispersion, 79–81

Single-section couplers, 169–76 Spiral directional couplers, 262–63size reduction, 262–63compact, 176

design, 171–75 top conductor layout, 263See also Directional couplersequivalent circuit, 179

fractional bandwidth, 170 Spiral inductors, 465–722-turn microstrip, 466frequency bandwidth ratio, 169–70

frequency response, 169–71 p equivalent circuit representation,470power loss ratio, 180

quarter-wave, 176 admittance parameters, 469coupled-line equivalent circuit models,useful operating bandwidth, 171

variation of coupling, 170 467design, 465Single stripline, 69–73

Six-finger microstrip coupler, 251–52 electrical characteristics, 467four-port representation, 469capacitor values, 252

illustrated, 251 inductance calculation, 468in MICs, 465line lengths, 252

parameters, 252 rectangular, 468Spiral transformers, 472–75See also Lange coupler

Slot-coupled microstrip lines, 99–102 balun, 523example, 473characteristic impedance, 100, 101

effective dielectric constant, 100, 101 power transfer, 475rectangular, 472illustrated, 100

uses, 99 schematic, 475twin-coil four-port, 472Slow wave, 159

Sonnet Lite, 139–40 See also TransformersSplit-rings resonators (SRRs), 428defined, 139

geometry, 140 complementary (CSRRs), 434composite medium with, 429for single transmission line structure,

145 dimensions, 430metamaterial structure based on, 430Source-load coupling filters, 335–47

coupling determination, 341 negative effect permeability, 429square, 429coupling matrix, 337

546 Index

Stepped impedance resonators (SIR), 307, phase velocity, 93See also Microstrip lines352–59

antisymmetrical shape, 366 Symmetrical coupled lines, 67–68Symmetrical couplersdual-band filters with, 365–66

EM-simulated performance, 358 defined, 167multisection, 177–78five-pole microstrip dual-band filter,

367 schematic, 178tables of parameters, 182–85grounded l /4, 353

grounded microstrip structure, 355 Symmetrical forward coupler, 150, 1513-dB, 151–53interdigital filter layout, 362

interdigital filters using, 352–59 coupling response, 151illustrated, 150inter-resonator coupling structure, 357

I/O stage, 354 strip pattern, 152theoretical/experimental response, 152microstrip, 354

narrowband design, 354–56 See also Forward-wave directionalcouplersnarrowband layout, 358

nine-pole wideband, 357 Symmetrical networkseven-/odd-mode analysis, 106–11wideband design, 356–59

wideband responses, 363 four-port, 106, 108Symmetrical nonuniform couplers,Stopband attenuation, 269

Striplines 197–214amplitude, 202broadside-coupled, 94–95

characteristic impedance, 71 coupled signal amplitude, 201coupling response, 202, 204characteristics, 68–74

cross section, 69 design procedure, 210–14differential reflection coefficient, 200cutoff frequency, 73

defined, 2 electrical length, 209equivalent circuit, 199edge-coupled, 73–74

lossless, 69 evaluation, 206even-mode characteristic impedance,offset, 7, 96–99

single, 69–73 199–201finite length, 201TEM mode of propagation support,

174 in homogeneous medium, 211–13ideal, 202–6Superconductor coupled-line filters,

371–85 illustrated, 198multisection, length, 210cascaded quadruplet/triplet filters,

371–77 nonuniform coupling, 198phase constant, 211–12high-order selective, with group delay

equalization, 377–85 phase velocity, 203physical length, 208, 209–10Superstar software, 300

Suspended microstrip lines, 88–93 scattering parameters, 198synthesis, 201–6broadside-coupled, 95–96

characteristic impedance, 93 weighting function, 205See also Nonuniform TEM directionalcross section, 92

effective dielectric constant, 90 couplersSynthesized substrate, 387illustrated, 3

Index 547

T Transformerscoupled-line, 452–61Tandem couplers, 252–55equivalent circuit representation, 452crossover requirement, 255impedance, 452, 457defined, 252–53open-circuit, 452–56fractional power, 254–55simulated performance, 456physical configuration, 255source/load impedances, 453reflected voltages, 253spiral, 472–75scattering matrix, 253symmetrical, 454scattering parameters, 253transformation ratio parameters, 455schematic, 252transmission line, 456–61See also Tight couplersSee also BalunsTEM modes

Transient response, 269, 270characteristics, 57–58Transmission line transformers (TLTs),common transmission lines, 54

456–61conductor loss, 60–611:4, 457, 458, 459, 460line parameters, 53broadside-coupled, 457support, 3defined, 456Three-port networksfractional bandwidth, 461illustrated, 35line characteristic impedance, 456nonreciprocal, 35–36maximum bandwidth, 460port three terminated in arbitrarypolyimide thickness, 459load, 48reflection coefficient, 458reciprocal, 34–35size, 459reduction, 48–49See also TransformersThru-reflect-line (TRL) calibration

Traveling-wave tubes (TWTs), 149techniques, 497Triformer balun, 518–19Tight couplers, 219–65Two-port networksbraided microstrip, 264

ABCD parameters, 38–40branch-line, 219, 220–33cascade, 39combline, 264illustrated, 33compact, 261–64network representation, 45coplanar waveguide, 264reduction, 49–50dielectric waveguide, 264representative matrices, comparisonbetween edge-coupled lines, 219

relationships, 42–43embedded microstrip, 264special representation, 38–43finline, 264

multiconductor, 242–52Umultilayer, 255–61Ultra-broadband forward-waverat-race, 219, 233–42

directional couplers, 155–56size, 219Ultra-wideband. See UWBslot-coupled, 264Unbalanced CRLH transmission line,tandem, 252–55

435, 437uses, 219Unevenly-coupled stages, 307–14vertically installed, 264

attenuation poles, 311–12wiggly two-line, 264Tight microwave couplers, 8 design example, 310–14

548 Index

Unevenly-coupled stages (continued) insertion loss, 424layout, 422external quality factor, 311

principle, 308 measured/simulated performances, 426microstrip CPW coupled-line, 428Unfolded Lange coupler, 245

Uniform impedance resonators (UIR) multimode coupled line, 425–28narrowband characteristic, 423interdigital filters, 357, 359

design, 359 passband performance, 425schematic diagrams, 423inter-resonator stage, 361

I/O stage, 360 Vwideband responses, 363

Very-large-scale integrated (VLSI) chips,See also Interdigital filters2Uniformly coupled asymmetric lines,

Voltages120–33coupling coefficient, 171p mode, 122–23equivalent, 18capacitive coupling coefficients, 130normalized, 18–21characteristic impedances, 122reflection coefficient, 22–23c mode, 121–22unnormalized, 21–22defined, 4

Voltage standing wave ratio (VSWR), 2,distributed equivalent circuits, 126–3023four-port network, 123

simulated, 446inductive coupling coefficients, 130variation of, 445parameters, 121–26

propagation constant, 122 WY-parameters, 123–26 WAVECON, 301Z-parameters, 123–26 Weakly coupled resonators

Uniformly coupled lines, 105–47 coupled-mode theory, 163–65asymmetrical lines, 120–33 coupling coefficient, 163directional couplers using, 111–20 Weighting function

Unnormalized currents, 21–22 determination example, 208–9defined, 21 determination technique, 206–9determining, 22 for loose couplers, 206

Unnormalized input impedance, 23–24 nonuniform couplers, 205Unnormalized quantities, 25–26 for tight couplers, 206Unnormalized voltage, 21–22 Wide-band baluns, 503

defined, 21 Wiggle-line filterdetermining, 22 coupling structures, 316–18

UWB defined, 314microstrip CPW coupled-line filters, illustrated, 319

410–21 inter-resonator coupling structure, 318multimode coupled-line filters, 406–10 I/O coupling structure, 317optimum stub line filters, 401–6 nonuniform line resonators, 315–16technology, 400–428 wideband response, 319

UWB bandpass filters, 421–28 Wiggly lineswith embedded band notch stubs, 422 capacitance parameters, 192EM simulation, 424 depth, 192fabricated microstrip, 426 geometrical parameters, 190

length, 191full-wave EM simulation, 425

Index 549

top view, 191 Yttrium barium copper oxide (YBCO)thin films, 376, 384use of, 189–92

Wireless local area network (WLANs), 12Z

Y Z-parametersfour-port network, 123–26Y-parameters, of four-port network,

123–26 interdigital two-port network, 125–26

Documents

RF and Microwave Coupled-Line Circuits, Second Edition