Description

Book Synopsis

An in-depth look at the state-of-the-art in microwave filter design, implementation, and optimization

Thoroughly revised and expanded, this second edition of the popular reference addresses the many important advances that have taken place in the field since the publication of the first edition and includes new chapters on Multiband Filters, Tunable Filters and a chapter devoted to Practical Considerations and Examples.

One of the chief constraints in the evolution of wireless communication systems is the scarcity of the available frequency spectrum, thus making frequency spectrum a primary resource to be judiciously shared and optimally utilized. This fundamental limitation, along with atmospheric conditions and interference have long been drivers of intense research and development in the fields of signal processing and filter networks, the two technologies that govern the information capacity of a given frequency spectrum. Written by distinguished experts wit

Table of Contents

Preface xxiii

1 Radio Frequency (RF) Filter Networks for Wireless Communications—The System Perspective 1

Part I Introduction to a Communication System, Radio Spectrum, and Information 1

1.1 Model of a Communication System 1

1.2 Radio Spectrum and its Utilization 6

1.3 Concept of Information 8

1.4 Communication Channel and Link Budgets 10

Part II Noise in a Communication Channel 15

1.5 Noise in Communication Systems 15

1.6 Modulation–Demodulation Schemes in a Communication System 32

1.7 Digital Transmission 39

Part III Impact of System Design on the Requirements of Filter Networks 50

1.8 Communication Channels in a Satellite System 50

1.9 RF Filters in Cellular Systems 62

1.10 Ultra Wideband (UWB) Wireless Communication 66

1.11 Impact of System Requirements on RF Filter Specifications 68

1.12 Impact of Satellite and Cellular Communications on Filter Technology 72

Summary 72

References 72

Appendix 1A 74

Intermodulation Distortion Summary 74

2 Fundamentals of Circuit Theory Approximation 75

2.1 Linear Systems 75

2.2 Classification of Systems 76

2.3 Evolution of Electrical Circuits: A Historical Perspective 77

2.4 Network Equation of Linear Systems in the Time Domain 78

2.5 Network Equation of Linear Systems in the Frequency-Domain Exponential Driving Function 80

2.6 Steady-State Response of Linear Systems to Sinusoidal Excitations 83

2.7 Circuit Theory Approximation 84

Summary 85

References 86

3 Characterization of Lossless Lowpass Prototype Filter Functions 87

3.1 The Ideal Filter 87

3.2 Characterization of Polynomial Functions for Doubly Terminated Lossless Lowpass Prototype Filter Networks 88

3.3 Characteristic Polynomials for Idealized Lowpass Prototype Networks 93

3.4 Lowpass Prototype Characteristics 95

3.5 Characteristic Polynomials versus Response Shapes 96

3.6 Classical Prototype Filters 98

3.7 Unified Design Chart (UDC) Relationships 108

3.8 Lowpass Prototype Circuit Configurations 109

3.9 Effect of Dissipation 113

3.10 Asymmetric Response Filters 115

Summary 118

References 119

Appendix 3A 121

Unified Design Charts 121

4 Computer-Aided Synthesis of Characteristic Polynomials 129

4.1 Objective Function and Constraints for Symmetric Lowpass Prototype Filter Networks 129

4.2 Analytic Gradients of the Objective Function 131

4.3 Optimization Criteria for Classical Filters 134

4.4 Generation of Novel Classes of Filter Functions 136

4.5 Asymmetric Class of Filters 138

4.6 Linear Phase Filters 142

4.7 Critical Frequencies for Selected Filter Functions 143

Summary 144

References 144

Appendix 4A 145

5 Analysis of Multiport Microwave Networks 147

5.1 Matrix Representation of Two-Port Networks 147

5.2 Cascade of Two Networks 160

5.3 Multiport Networks 167

5.4 Analysis of Multiport Networks 169

Summary 174

References 175

6 Synthesis of a General Class of the Chebyshev Filter Function 177

6.1 Polynomial Forms of the Transfer and Reflection Parameters S21(S) and S11(S) for a Two-port network 177

6.2 Alternating Pole Method for the Determination of the Denominator Polynomial E(S) 186

6.3 General Polynomial Synthesis Methods for Chebyshev Filter Functions 189

6.4 Predistorted Filter Characteristics 200

6.5 Transformation for Symmetric Dual-Passband Filters 208

Summary 211

References 211

Appendix 6A 212

Complex Terminating Impedances in Multiport Networks 212

6A.1 Change of Termination Impedance 213

References 213

7 Synthesis of Network-Circuit Approach 215

7.1 Circuit Synthesis Approach 216

7.2 Lowpass Prototype Circuits for Coupled-Resonator Microwave Bandpass Filters 221

7.3 Ladder Network Synthesis 229

7.4 Synthesis Example of an Asymmetric (4–2) Filter Network 235

Summary 244

References 245

8 Synthesis of Networks: Direct Coupling Matrix Synthesis Methods 247

8.1 The Coupling Matrix 247

8.2 Direct Synthesis of the Coupling Matrix 258

8.3 Coupling Matrix Reduction 261

8.4 Synthesis of the N + 2 Coupling Matrix 268

8.5 Even- and Odd-Mode Coupling Matrix Synthesis Technique: the Folded Lattice Array 282

Summary 292

References 293

9 Reconfiguration of the Folded Coupling Matrix 295

9.1 Symmetric Realizations for Dual-Mode Filters 295

9.2 Asymmetric Realizations for Symmetric Characteristics 300

9.3 "Pfitzenmaier" Configurations 301

9.4 Cascaded Quartets (CQs): Two Quartets in Cascade for Degrees Eight and Above 304

9.5 Parallel-Connected Two-Port Networks 306

9.6 Cul-de-Sac Configuration 311

Summary 321

References 321

10 Synthesis and Application of Extracted Pole and Trisection Elements 323

10.1 Extracted Pole Filter Synthesis 323

10.2 Synthesis of Bandstop Filters Using the Extracted Pole Technique 335

10.2.1 Direct-Coupled Bandstop Filters 338

10.2.1.1 Cul-de-Sac Forms for the Direct-Coupled Bandstop Matrix 341

10.3 Trisections 343

10.4 Box Section and Extended Box Configurations 361

Summary 371

References 371

11 Microwave Resonators 373

11.1 Microwave Resonator Configurations 373

11.2 Calculation of Resonant Frequency 376

11.3 Resonator Unloaded Q Factor 383

11.4 Measurement of Loaded and Unloaded Q Factor 387

Summary 393

References 393

12 Waveguide and Coaxial Lowpass Filters 395

12.1 Commensurate-Line Building Elements 395

12.2 Lowpass Prototype Transfer Polynomials 396

12.3 Synthesis and Realization of the Distributed Stepped Impedance Lowpass Filter 401

12.4 Short-Step Transformers 410

12.5 Synthesis and Realization of Mixed Lumped/Distributed Lowpass Filters 411

Summary 425

References 426

13 Waveguide Realization of Single- and Dual-Mode Resonator Filters 427

13.1 Synthesis Process 428

13.2 Design of the Filter Function 428

13.3 Realization and Analysis of the Microwave Filter Network 434

13.4 Dual-Mode Filters 440

13.5 Coupling Sign Correction 442

13.6 Dual-Mode Realizations for Some Typical Coupling Matrix Configurations 444

13.7 Phase- and Direct-Coupled Extracted Pole Filters 447

13.8 The "Full-Inductive" Dual-Mode Filter 450

Summary 454

References 454

14 Design and Physical Realization of Coupled Resonator Filters 457

14.1 Circuit Models for Chebyshev Bandpass Filters 459

14.2 Calculation of Interresonator Coupling 463

14.3 Calculation of Input/Output Coupling 467

14.4 Design Example of Dielectric Resonator Filters Using the Coupling Matrix Model 468

14.5 Design Example of a Waveguide Iris Filter Using the Impedance Inverter Model 475

14.6 Design Example of a Microstrip Filter Using the J-Admittance Inverter Model 478

Summary 483

References 484

15 Advanced EM-Based Design Techniques for Microwave Filters 485

15.1 EM-Based Synthesis Techniques 485

15.2 EM-Based Optimization Techniques 486

15.3 EM-Based Advanced Design Techniques 496

Summary 513

References 514

16 Dielectric Resonator Filters 517

16.1 Resonant Frequency Calculation in Dielectric Resonators 517

16.2 Rigorous Analyses of Dielectric Resonators 521

16.3 Dielectric Resonator Filter Configurations 524

16.4 Design Considerations for Dielectric Resonator Filters 528

16.5 Other Dielectric Resonator Configurations 531

16.6 Cryogenic Dielectric Resonator Filters 534

16.7 Hybrid Dielectric/Superconductor Filters 536

16.8 Miniature Dielectric Resonators 538

Summary 542

References 543

17 Allpass Phase and Group Delay Equalizer Networks 545

17.1 Characteristics of Allpass Networks 545

17.2 Lumped-Element Allpass Networks 547

17.3 Microwave Allpass Networks 551

17.4 Physical Realization of Allpass Networks 554

17.5 Synthesis of Reflection-Type Allpass Networks 557

17.6 Practical Narrowband Reflection-Type Allpass Networks 558

17.7 Optimization Criteria for Allpass Networks 561

17.8 Dissipation Loss 566

17.9 Equalization Tradeoffs 567

Summary 567

References 568

18 Multiplexer Theory and Design 569

18.1 Background 569

18.2 Multiplexer Configurations 571

18.3 RF Channelizers (Demultiplexers) 575

18.4 RF Combiners 581

18.5 Transmit–Receive Diplexers 601

Summary 606

References 607

19 Computer-Aided Diagnosis and Tuning of Microwave Filters 609

19.1 Sequential Tuning of Coupled Resonator Filters 610

19.2 Computer-Aided Tuning Based on Circuit Model Parameter Extraction 615

19.3 Computer-Aided Tuning Based on Poles and Zeros of the Input Reflection Coefficient 619

19.4 Time-Domain Tuning 622

19.5 Filter Tuning Based on Fuzzy Logic Techniques 627

19.6 Automated Setups for Filter Tuning 637

Summary 639

References 640

20 High-Power Considerations in Microwave Filter Networks 643

20.1 Background 643

20.2 High-Power Requirements in Wireless Systems 643

20.3 High-Power Amplifiers (HPAs) 645

20.4 Gas Discharge 645

20.5 Multipaction Breakdown 651

20.6 High-Power Bandpass Filters 662

20.7 Passive Intermodulation (PIM) Consideration for High-Power Equipment 670

Summary 674

Acknowledgment 675

References 675

21 Multiband Filters 679

21.1 Introduction 679

21.2 Approach I: Multiband Filters Realized by Having Transmission Zeros Inside the Passband of a Bandpass Filter 681

21.3 Approach II: Multiband Filters Employing Multimode Resonators 683

21.4 Approach III: Multiband Filters Using Parallel Connected Filters 700

21.5 Approach IV: Multiband Filter Implemented Using Notch Filters Connected in Cascade with a Wideband Bandpass 701

21.6 Use of Dual-Band Filters in Diplexer and Multiplexer Applications 703

21.7 Synthesis of Multiband Filters 705

Summary 727

References 728

22 Tunable Filters 731

22.1 Introduction 731

22.2 Major Challenges in Realizing High-Q 3D Tunable Filters 733

22.3 Combline Tunable Filters 734

22.4 Tunable Dielectric Resonator Filters 752

22.5 Waveguide Tunable Filters 772

22.6 Filters with Tunable Bandwidth 776

Summary 778

References 779

23 Practical Considerations and Design Examples 785
Chandra M. Kudsia, Vicente E. Boria, and Santiago Cogollos

23.1 System Considerations for Filter Specifications in Communication Systems 785

23.2 Filter Synthesis Techniques and Topologies 796

23.3 Multiplexers 827

23.4 High-Power Considerations 839

23.5 Tolerance and Sensitivity Analysis in Filter Design 851

Summary 858

Acknowledgments 858

Appendix 23A 858

Thermal Expansion 858

References 859

A Physical Constants 861

B Conductivities of Metals 863

C Dielectric Constants and Loss Tangents of Some Materials 865

D Rectangular Waveguide Designation 867

E Impedance and Admittance Inverters 869

E.1 Filter Realization with Series Elements 869

E.2 Normalization of the Element Values 872

E.3 General Lowpass Prototype Case 873

E.4 Bandpass Prototype 874

References 878

Index 879

Microwave Filters for Communication Systems

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    A Hardback by Richard J. Cameron, Chandra M. Kudsia, Raafat R. Mansour

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      View other formats and editions of Microwave Filters for Communication Systems by Richard J. Cameron

      Publisher: John Wiley & Sons Inc
      Publication Date: Publication Date: 25/05/2018
      ISBN13: 9781118274347, 978-1118274347
      ISBN10: 1118274342

      Description

      Book Synopsis

      An in-depth look at the state-of-the-art in microwave filter design, implementation, and optimization

      Thoroughly revised and expanded, this second edition of the popular reference addresses the many important advances that have taken place in the field since the publication of the first edition and includes new chapters on Multiband Filters, Tunable Filters and a chapter devoted to Practical Considerations and Examples.

      One of the chief constraints in the evolution of wireless communication systems is the scarcity of the available frequency spectrum, thus making frequency spectrum a primary resource to be judiciously shared and optimally utilized. This fundamental limitation, along with atmospheric conditions and interference have long been drivers of intense research and development in the fields of signal processing and filter networks, the two technologies that govern the information capacity of a given frequency spectrum. Written by distinguished experts wit

      Table of Contents

      Preface xxiii

      1 Radio Frequency (RF) Filter Networks for Wireless Communications—The System Perspective 1

      Part I Introduction to a Communication System, Radio Spectrum, and Information 1

      1.1 Model of a Communication System 1

      1.2 Radio Spectrum and its Utilization 6

      1.3 Concept of Information 8

      1.4 Communication Channel and Link Budgets 10

      Part II Noise in a Communication Channel 15

      1.5 Noise in Communication Systems 15

      1.6 Modulation–Demodulation Schemes in a Communication System 32

      1.7 Digital Transmission 39

      Part III Impact of System Design on the Requirements of Filter Networks 50

      1.8 Communication Channels in a Satellite System 50

      1.9 RF Filters in Cellular Systems 62

      1.10 Ultra Wideband (UWB) Wireless Communication 66

      1.11 Impact of System Requirements on RF Filter Specifications 68

      1.12 Impact of Satellite and Cellular Communications on Filter Technology 72

      Summary 72

      References 72

      Appendix 1A 74

      Intermodulation Distortion Summary 74

      2 Fundamentals of Circuit Theory Approximation 75

      2.1 Linear Systems 75

      2.2 Classification of Systems 76

      2.3 Evolution of Electrical Circuits: A Historical Perspective 77

      2.4 Network Equation of Linear Systems in the Time Domain 78

      2.5 Network Equation of Linear Systems in the Frequency-Domain Exponential Driving Function 80

      2.6 Steady-State Response of Linear Systems to Sinusoidal Excitations 83

      2.7 Circuit Theory Approximation 84

      Summary 85

      References 86

      3 Characterization of Lossless Lowpass Prototype Filter Functions 87

      3.1 The Ideal Filter 87

      3.2 Characterization of Polynomial Functions for Doubly Terminated Lossless Lowpass Prototype Filter Networks 88

      3.3 Characteristic Polynomials for Idealized Lowpass Prototype Networks 93

      3.4 Lowpass Prototype Characteristics 95

      3.5 Characteristic Polynomials versus Response Shapes 96

      3.6 Classical Prototype Filters 98

      3.7 Unified Design Chart (UDC) Relationships 108

      3.8 Lowpass Prototype Circuit Configurations 109

      3.9 Effect of Dissipation 113

      3.10 Asymmetric Response Filters 115

      Summary 118

      References 119

      Appendix 3A 121

      Unified Design Charts 121

      4 Computer-Aided Synthesis of Characteristic Polynomials 129

      4.1 Objective Function and Constraints for Symmetric Lowpass Prototype Filter Networks 129

      4.2 Analytic Gradients of the Objective Function 131

      4.3 Optimization Criteria for Classical Filters 134

      4.4 Generation of Novel Classes of Filter Functions 136

      4.5 Asymmetric Class of Filters 138

      4.6 Linear Phase Filters 142

      4.7 Critical Frequencies for Selected Filter Functions 143

      Summary 144

      References 144

      Appendix 4A 145

      5 Analysis of Multiport Microwave Networks 147

      5.1 Matrix Representation of Two-Port Networks 147

      5.2 Cascade of Two Networks 160

      5.3 Multiport Networks 167

      5.4 Analysis of Multiport Networks 169

      Summary 174

      References 175

      6 Synthesis of a General Class of the Chebyshev Filter Function 177

      6.1 Polynomial Forms of the Transfer and Reflection Parameters S21(S) and S11(S) for a Two-port network 177

      6.2 Alternating Pole Method for the Determination of the Denominator Polynomial E(S) 186

      6.3 General Polynomial Synthesis Methods for Chebyshev Filter Functions 189

      6.4 Predistorted Filter Characteristics 200

      6.5 Transformation for Symmetric Dual-Passband Filters 208

      Summary 211

      References 211

      Appendix 6A 212

      Complex Terminating Impedances in Multiport Networks 212

      6A.1 Change of Termination Impedance 213

      References 213

      7 Synthesis of Network-Circuit Approach 215

      7.1 Circuit Synthesis Approach 216

      7.2 Lowpass Prototype Circuits for Coupled-Resonator Microwave Bandpass Filters 221

      7.3 Ladder Network Synthesis 229

      7.4 Synthesis Example of an Asymmetric (4–2) Filter Network 235

      Summary 244

      References 245

      8 Synthesis of Networks: Direct Coupling Matrix Synthesis Methods 247

      8.1 The Coupling Matrix 247

      8.2 Direct Synthesis of the Coupling Matrix 258

      8.3 Coupling Matrix Reduction 261

      8.4 Synthesis of the N + 2 Coupling Matrix 268

      8.5 Even- and Odd-Mode Coupling Matrix Synthesis Technique: the Folded Lattice Array 282

      Summary 292

      References 293

      9 Reconfiguration of the Folded Coupling Matrix 295

      9.1 Symmetric Realizations for Dual-Mode Filters 295

      9.2 Asymmetric Realizations for Symmetric Characteristics 300

      9.3 "Pfitzenmaier" Configurations 301

      9.4 Cascaded Quartets (CQs): Two Quartets in Cascade for Degrees Eight and Above 304

      9.5 Parallel-Connected Two-Port Networks 306

      9.6 Cul-de-Sac Configuration 311

      Summary 321

      References 321

      10 Synthesis and Application of Extracted Pole and Trisection Elements 323

      10.1 Extracted Pole Filter Synthesis 323

      10.2 Synthesis of Bandstop Filters Using the Extracted Pole Technique 335

      10.2.1 Direct-Coupled Bandstop Filters 338

      10.2.1.1 Cul-de-Sac Forms for the Direct-Coupled Bandstop Matrix 341

      10.3 Trisections 343

      10.4 Box Section and Extended Box Configurations 361

      Summary 371

      References 371

      11 Microwave Resonators 373

      11.1 Microwave Resonator Configurations 373

      11.2 Calculation of Resonant Frequency 376

      11.3 Resonator Unloaded Q Factor 383

      11.4 Measurement of Loaded and Unloaded Q Factor 387

      Summary 393

      References 393

      12 Waveguide and Coaxial Lowpass Filters 395

      12.1 Commensurate-Line Building Elements 395

      12.2 Lowpass Prototype Transfer Polynomials 396

      12.3 Synthesis and Realization of the Distributed Stepped Impedance Lowpass Filter 401

      12.4 Short-Step Transformers 410

      12.5 Synthesis and Realization of Mixed Lumped/Distributed Lowpass Filters 411

      Summary 425

      References 426

      13 Waveguide Realization of Single- and Dual-Mode Resonator Filters 427

      13.1 Synthesis Process 428

      13.2 Design of the Filter Function 428

      13.3 Realization and Analysis of the Microwave Filter Network 434

      13.4 Dual-Mode Filters 440

      13.5 Coupling Sign Correction 442

      13.6 Dual-Mode Realizations for Some Typical Coupling Matrix Configurations 444

      13.7 Phase- and Direct-Coupled Extracted Pole Filters 447

      13.8 The "Full-Inductive" Dual-Mode Filter 450

      Summary 454

      References 454

      14 Design and Physical Realization of Coupled Resonator Filters 457

      14.1 Circuit Models for Chebyshev Bandpass Filters 459

      14.2 Calculation of Interresonator Coupling 463

      14.3 Calculation of Input/Output Coupling 467

      14.4 Design Example of Dielectric Resonator Filters Using the Coupling Matrix Model 468

      14.5 Design Example of a Waveguide Iris Filter Using the Impedance Inverter Model 475

      14.6 Design Example of a Microstrip Filter Using the J-Admittance Inverter Model 478

      Summary 483

      References 484

      15 Advanced EM-Based Design Techniques for Microwave Filters 485

      15.1 EM-Based Synthesis Techniques 485

      15.2 EM-Based Optimization Techniques 486

      15.3 EM-Based Advanced Design Techniques 496

      Summary 513

      References 514

      16 Dielectric Resonator Filters 517

      16.1 Resonant Frequency Calculation in Dielectric Resonators 517

      16.2 Rigorous Analyses of Dielectric Resonators 521

      16.3 Dielectric Resonator Filter Configurations 524

      16.4 Design Considerations for Dielectric Resonator Filters 528

      16.5 Other Dielectric Resonator Configurations 531

      16.6 Cryogenic Dielectric Resonator Filters 534

      16.7 Hybrid Dielectric/Superconductor Filters 536

      16.8 Miniature Dielectric Resonators 538

      Summary 542

      References 543

      17 Allpass Phase and Group Delay Equalizer Networks 545

      17.1 Characteristics of Allpass Networks 545

      17.2 Lumped-Element Allpass Networks 547

      17.3 Microwave Allpass Networks 551

      17.4 Physical Realization of Allpass Networks 554

      17.5 Synthesis of Reflection-Type Allpass Networks 557

      17.6 Practical Narrowband Reflection-Type Allpass Networks 558

      17.7 Optimization Criteria for Allpass Networks 561

      17.8 Dissipation Loss 566

      17.9 Equalization Tradeoffs 567

      Summary 567

      References 568

      18 Multiplexer Theory and Design 569

      18.1 Background 569

      18.2 Multiplexer Configurations 571

      18.3 RF Channelizers (Demultiplexers) 575

      18.4 RF Combiners 581

      18.5 Transmit–Receive Diplexers 601

      Summary 606

      References 607

      19 Computer-Aided Diagnosis and Tuning of Microwave Filters 609

      19.1 Sequential Tuning of Coupled Resonator Filters 610

      19.2 Computer-Aided Tuning Based on Circuit Model Parameter Extraction 615

      19.3 Computer-Aided Tuning Based on Poles and Zeros of the Input Reflection Coefficient 619

      19.4 Time-Domain Tuning 622

      19.5 Filter Tuning Based on Fuzzy Logic Techniques 627

      19.6 Automated Setups for Filter Tuning 637

      Summary 639

      References 640

      20 High-Power Considerations in Microwave Filter Networks 643

      20.1 Background 643

      20.2 High-Power Requirements in Wireless Systems 643

      20.3 High-Power Amplifiers (HPAs) 645

      20.4 Gas Discharge 645

      20.5 Multipaction Breakdown 651

      20.6 High-Power Bandpass Filters 662

      20.7 Passive Intermodulation (PIM) Consideration for High-Power Equipment 670

      Summary 674

      Acknowledgment 675

      References 675

      21 Multiband Filters 679

      21.1 Introduction 679

      21.2 Approach I: Multiband Filters Realized by Having Transmission Zeros Inside the Passband of a Bandpass Filter 681

      21.3 Approach II: Multiband Filters Employing Multimode Resonators 683

      21.4 Approach III: Multiband Filters Using Parallel Connected Filters 700

      21.5 Approach IV: Multiband Filter Implemented Using Notch Filters Connected in Cascade with a Wideband Bandpass 701

      21.6 Use of Dual-Band Filters in Diplexer and Multiplexer Applications 703

      21.7 Synthesis of Multiband Filters 705

      Summary 727

      References 728

      22 Tunable Filters 731

      22.1 Introduction 731

      22.2 Major Challenges in Realizing High-Q 3D Tunable Filters 733

      22.3 Combline Tunable Filters 734

      22.4 Tunable Dielectric Resonator Filters 752

      22.5 Waveguide Tunable Filters 772

      22.6 Filters with Tunable Bandwidth 776

      Summary 778

      References 779

      23 Practical Considerations and Design Examples 785
      Chandra M. Kudsia, Vicente E. Boria, and Santiago Cogollos

      23.1 System Considerations for Filter Specifications in Communication Systems 785

      23.2 Filter Synthesis Techniques and Topologies 796

      23.3 Multiplexers 827

      23.4 High-Power Considerations 839

      23.5 Tolerance and Sensitivity Analysis in Filter Design 851

      Summary 858

      Acknowledgments 858

      Appendix 23A 858

      Thermal Expansion 858

      References 859

      A Physical Constants 861

      B Conductivities of Metals 863

      C Dielectric Constants and Loss Tangents of Some Materials 865

      D Rectangular Waveguide Designation 867

      E Impedance and Admittance Inverters 869

      E.1 Filter Realization with Series Elements 869

      E.2 Normalization of the Element Values 872

      E.3 General Lowpass Prototype Case 873

      E.4 Bandpass Prototype 874

      References 878

      Index 879

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