# Theory and Design of Broadband Matching Networks

## 1st Edition

### Applied Electricity and Electronics

**Authors:**Wai-Kai Chen

**Editors:**P. Hammond

**eBook ISBN:**9781483159447

**Imprint:**Pergamon

**Published Date:**1st January 1976

**Page Count:**448

## Description

Theory and Design of Broadband Matching Networks centers on the network theory and its applications to the design of broadband matching networks and amplifiers. Organized into five chapters, this book begins with a description of the foundation of network theory. Chapter 2 gives a fairly complete exposition of the scattering matrix associated with an n-port network. Chapter 3 considers the approximation problem along with a discussion of the approximating functions. Chapter 4 explains the Youla's theory of broadband matching by illustrating every phase of the theory with fully worked out examples. The extension of Youla's theory to active load impedance is taken up in Chapter 5. This book will be useful as a reference for practicing engineers who wish to learn how the modern network theory can be applied to the design of many practical circuits.

## Table of Contents

Preface

Chapter 1. Foundations of Network Theory

1. Basic Network Postulates

1.1. Real-Time Function Postulate

1.2. Time-Invariance Postulate

1.3. Linearity Postulate

1.4. Passivity Postulate

1.5. Causality Postulate

1.6. Reciprocity Postulate

2. Matrix Characterizations of n-Port Networks

2.1. The Impedance Matrix

2.2. The Admittance Matrix

2.3. The Hybrid Matrix

2.4. The Indefinite-Admittance Matrix

3. Power Gains

4. Hermitian Forms

5. The Positive-Real Matrix

6. Frequency-Domain Conditions for Passivity

7. Conclusions

Problems

References

Chapter 2. The Scattering Matrix

1. A Brief Review of the Transmission-Line Theory

2. The Scattering Parameters of a One-Port Network

2.1. Basis-Dependent Reflection Coefficients

2.2. Basis-Independent Reflection Coefficient

2.3. The Factorization of the Para-Hermitian Part of z(s)

2.4. Alternative Representation of the Basis-Independent Reflection Coefficient

2.5. The Normalized Reflection Coefficient and Passivity

3. The Scattering Matrix of an n-Port Network

3.1. Basis-Dependent Scattering Matrices

3.2. Basis-Independent Scattering Matrix

3.3. The Scattering Matrices and the Augmented n-Port Networks

3.4. Alternative Representation of the Basis-Independent Scattering Matrix

3.5. Physical Interpretation of the Normalized Scattering Parameters

3.6. The Normalized Scattering Matrix and Passivity

3.7. The Normalized Scattering Parameters of a Lossless Two-Port Network

4. The Bounded-Real Scattering Matrix

5. Interconnection of Multi-Port Networks

6. Conclusions

Problems

References

Chapter 3. Approximation and Ladder Realization

1. The Butterworth Response

1.1. Poles of the Butterworth Function

1.2. Coefficients of the Butterworth Polynomials

1.3. Butterworth Networks

1.4. Butterworth LC Ladder Networks

2. The Chebyshev Response

2.1. Chebyshev Polynomials

2.2. Equiripple Characteristic

2.3. Poles of the Chebyshev Function

2.4. Coefficients of the Polynomial p(y)

2.5. Chebyshev Networks

2.6. Chebyshev LC Ladder Networks

3. Elliptic Functions

3.1. Jacobian Elliptic Functions

3.2. Jacobi's Imaginary Transformations

3.3. Periods of Elliptic Functions

3.4. Poles and Zeros of the Jacobian Elliptic Functions

3.5. Addition Theorems and Complex Arguments

4. The Elliptic Response

4.1. The Characteristic Function Fn(ω))

4.2. Equiripple Characteristic in Passband and Stopband

4.3. Poles and Zeros of Elliptic Response

4.4. Elliptic Networks

5. Frequency Transformations

5.1. Transformation to High-Pass

5.2. Transformation to Band-Pass

5.3. Transformation to Band-Elimination

6. Conclusions

Problems

References

Chapter 4. Theory of Broadband Matching: The Passive Load

1. The Bode-Fano-Youla Broadband Matching Problem

2. Youla's Theory of Broadband Matching: Preliminary Considerations

3. Basic Constraints on ρ(s)

4. Bode's Parallel RC Load

4.1. Butterworth Transducer Power-Gain Characteristic

4.2. Chebyshev Transducer Power-Gain Characteristic

4.3. Elliptic Transducer Power-Gain Characteristic

4.4. Equalizer Back-End Impedance

5. Proof of Necessity of the Basic Constraints on ρ(s)

6. Proof of Sufficiency of the Basic Constraints on ρ(s)

7. Design Procedure for the Equalizers

8. Darlington Type-C Load

8.1. Butterworth Transducer Power-Gain Characteristic

8.2. Chebyshev Transducer Power-Gain Characteristic

8.3. Elliptic Transducer Power-Gain Characteristic

8.4. Equalizer Back-End Impedance

9. Constant Transducer Power Gain

10. Conclusions

Problems

References

Chapter 5. Theory of Broadband Matching: The Active Load

1. Special Class of Active Impedances

2. General Configuration of the Negative-Resistance Amplifiers

3. Nonreciprocal Amplifiers

3.1. Design Considerations for Nα

3.2. Design Considerations for Nß

3.3. Design Considerations for Nc

3.4. Illustrative Examples

3.5. Extension and Stability

4. Transmission-Power Amplifiers

4.1. Tunnel Diode in Shunt with the Load

4.2. Tunnel Diode in Shunt with the Generator

4.3. Stability

4.4. Sensitivity

5. Reciprocal Amplifiers

5.1. General Gain-Bandwidth Limitations

5.2. Cascade Connection

6. Amplifiers Using More than One Active Impedance

6.1. Nonreciprocal Amplifiers

6.2. Reciprocal Amplifiers

7. Conclusions

Problems

References

Appendices

Appendix A. The Butterworth Response

Appendix B. The Chebyshev Response

Appendix C. The Elliptic Response

Symbol Index

Subject Index

## Details

- No. of pages:
- 448

- Language:
- English

- Copyright:
- © Pergamon 1976

- Published:
- 1st January 1976

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483159447

## About the Author

### Wai-Kai Chen

## About the Editor

### P. Hammond

### Affiliations and Expertise

University of Southampton