About the Author xiii

Preface xv

**1 Circuit Theory for Power Transfer Networks 1**

1.1 Introduction 1

1.2 Ideal Circuit Elements 2

1.3 Average Power Dissipation and Effective Voltage and Current 3

1.4 Definitions of Voltage and Current Phasors 5

1.5 Definitions of Active, Passive and Lossless One-ports 6

1.6 Definition of Resistor 6

1.7 Definition of Capacitor 7

1.8 Definition of Inductor 8

1.9 Definition of an Ideal Transformer 11

1.10 Coupled Coils 12

1.11 Definitions: Laplace and Fourier Transformations of a Time Domain Function f(t) 12

1.12 Useful Mathematical Properties of Laplace and Fourier Transforms of a Causal Function 14

1.13 Numerical Evaluation of Hilbert Transform 20

1.14 Convolution 21

1.15 Signal Energy 21

1.16 Definition of Impedance and Admittance 22

1.17 Immittance of One-port Networks 23

1.18 Definition: ‘Positive Real Functions’ 25

**2 Electromagnetic Field Theory for Power Transfer Networks: Fields, Waves and Lumped Circuit Models 35**

2.1 Introduction 35

2.2 Coulomb’s Law and Electric Fields 36

2.3 Definition of Electric Field 37

2.4 Definition of Electric Potential 38

2.5 Units of Force, Energy and Potential 41

2.6 Uniform Electric Field 42

2.7 Units of Electric Field 43

2.8 Definition of Displacement Vector or ‘Electric Flux Density Vector’ D 43

2.9 Boundary Conditions in an Electric Field 46

2.10 Differential Relation between the Potential and the Electric Field 47

2.11 Parallel Plate Capacitor 49

2.12 Capacitance of a Transmission Line 52

2.13 Capacitance of Coaxial Cable 54

2.14 Resistance of a Conductor of Length L: Ohm’s Law 55

2.15 Principle of Charge Conservation and the Continuity Equation 60

2.16 Energy Density in an Electric Field 61

2.17 The Magnetic Field 61

2.18 Generation of Magnetic Fields: Biot–Savart and Ampe`re’s Law 64

2.19 Direction of Magnetic Field: Right Hand Rule 67

2.20 Unit of Magnetic Field: Related Quantities 67

2.21 Unit of Magnetic Flux Density B 68

2.22 Unit of Magnetic Flux 68

2.23 Definition of Inductance L 68

2.24 Permeability m and its Unit 69

2.25 Magnetic Force between Two Parallel Wires 70

2.26 Magnetic Field Generated by a Circular Current-Carrying Wire 71

2.27 Magnetic Field of a Tidily Wired Solenoid of N Turns 73

2.28 The Toroid 73

2.29 Inductance of N-Turn Wire Loops 74

2.30 Inductance of a Coaxial Transmission Line 76

2.31 Parallel Wire Transmission Line 81

2.32 Faraday’s Law 82

2.33 Energy Stored in a Magnetic Field 83

2.34 Magnetic Energy Density in a Given Volume 83

2.35 Transformer 84

2.36 Mutual Inductance 87

2.37 Boundary Conditions and Maxwell Equations 89

2.38 Summary of Maxwell Equations and Electromagnetic Wave Propagation 96

2.39 Power Flow in Electromagnetic Fields: Poynting’s Theorem 101

2.40 General Form of Electromagnetic Wave Equation 101

2.41 Solutions of Maxwell Equations Using Complex Phasors 103

2.42 Determination of the Electromagnetic Field Distribution of a Short Current Element: Hertzian Dipole Problem 105

2.43 Antenna Loss 108

2.44 Magnetic Dipole 108

2.45 Long Straight Wire Antenna: Half-Wave Dipole 109

2.46 Fourier Transform of Maxwell Equations: Phasor Representation 110

**3 Transmission Lines for Circuit Designers: Transmission Lines as Circuit Elements 117**

3.1 Ideal Transmission Lines 117

3.2 Time Domain Solutions of Voltage and Current Wave Equations 122

3.3 Model for a Two-Pair Wire Transmission Line as an Ideal TEM Line 122

3.4 Model for a Coaxial Cable as an Ideal TEM Line 123

3.5 Field Solutions for TEM Lines 123

3.6 Phasor Solutions for Ideal TEM Lines 124

3.7 Steady State Time Domain Solutions for Voltage and Current at Any Point z on the TEM Line 125

3.8 Transmission Lines as Circuit Elements 126

3.9 TEM Lines as Circuit or ‘Distributed’ Elements 127

3.10 Ideal TEM Lines with No Reflection: Perfectly Matched and Mismatched Lines 142

**4 Circuits Constructed with Commensurate Transmission Lines: Properties of Transmission Line Circuits in the Richard Domain 149**

4.1 Ideal TEM Lines as Lossless Two-ports 149

4.2 Scattering Parameters of a TEM Line as a Lossless Two-port 151

4.3 Input Reflection Coefficient under Arbitrary Termination 153

4.4 Choice of the Port Normalizations 154

4.5 Derivation of the Actual Voltage-Based Input and Output Incident and Reflected Waves 154

4.6 Incident and Reflected Waves for Arbitrary Normalization Numbers 157

4.7 Lossless Two-ports Constructed with Commensurate Transmission Lines 165

4.8 Cascade Connection of Two UEs 168

4.9 Major Properties of the Scattering Parameters for Passive Two-ports 170

4.10 Rational Form of the Scattering Matrix for a Resistively Terminated Lossless Two-port Constructed by Transmission Lines 176

4.11 Kuroda Identities 187

4.12 Normalization Change and Richard Extractions 188

4.13 Transmission Zeros in the Richard Domain 196

4.14 Rational Form of the Scattering Parameters and Generation of g(l) via the Losslessness Condition 197

4.15 Generation of Lossless Two-ports with Desired Topology 197

4.16 Stepped Line Butterworth Gain Approximation 211

4.17 Design of Chebyshev Filters Employing Stepped Lines 216

4.18 MATLABCodes to Design Stepped Line Filters Using Chebyshev Polynomials 230

4.19 Summary and Concluding Remarks on the Circuits Designed Using Commensurate Transmission Lines 241

**5 Insertion Loss Approximation for Arbitrary Gain Forms via the Simplified Real Frequency Technique: Filter Design via SRFT 255**

5.1 Arbitrary Gain Approximation 255

5.2 Filter Design via SRFT for Arbitrary Gain and Phase Approximation 256

5.3 Conclusion 267

**6 Formal Description of Lossless Two-ports in Terms of Scattering Parameters: Scattering Parameters in the** **p Domain 277**

6.1 Introduction 277

6.2 Formal Definition of Scattering Parameters 278

6.3 Generation of Scattering Parameters for Linear Two-ports 290

6.4 Transducer Power Gain in Forward and Backward Directions 292

6.5 Properties of the Scattering Parameters of Lossless Two-ports 293

6.6 Blashke Products or All-Pass Functions 300

6.7 Possible Zeros of a Proper Polynomial f(p) 301

6.8 Transmission Zeros 302

6.9 Lossless Ladders 307

6.10 Further Properties of the Scattering Parameters of Lossless Two-ports 308

6.11 Transfer Scattering Parameters 310

6.12 Cascaded (or Tandem) Connections of Two-ports 311

6.13 Comments 313

6.14 Generation of Scattering Parameters from Transfer Scattering Parameters 315

**7 Numerical Generation of Minimum Functions via the Parametric Approach 317**

7.1 Introduction 317

7.2 Generation of Positive Real Functions via the Parametric Approach using MATLAB318

7.3 Major Polynomial Operations in MATLAB321

7.4 Algorithm: Computation of Residues in Bode Form on MATLAB323

7.5 Generation of Minimum Functions from the Given All-Zero, All-Pole Form of the Real Part 335

7.6 Immittance Modeling via the Parametric Approach 349

7.7 Direct Approach for Minimum Immittance Modeling via the Parametric Approach 359

**8 Gewertz Procedure to Generate a Minimum Function from its Even Part: Generation of Minimum Function in Rational Form 373**

8.1 Introduction 373

8.2 Gewertz Procedure 374

8.3 Gewertz Algorithm 377

8.4 MATLABCodes for the Gewertz Algorithm 378

8.5 Comparison of the Bode Method to the Gewertz Procedure 386

8.6 Immittance Modeling via the Gewertz Procedure 392

**9 Description of Power Transfer Networks via Driving Point Input Immittance: Darlington’s Theorem 405**

9.1 Introduction 405

9.2 Power Dissipation PL over a Load Impedance ZL 405

9.3 Power Transfer 406

9.4 Maximum Power Transfer Theorem 407

9.5 Transducer Power Gain for Matching Problems 408

9.6 Formal Definition of a Broadband Matching Problem 408

9.7 Darlington’s Description of Lossless Two-ports 410

9.8 Description of Lossless Two-ports via Z Parameters 423

9.9 Driving Point Input Impedance of a Lossless Two-port 426

9.10 Proper Selection of Cases to Construct Lossless Two-ports from the Driving Point Immittance Function 430

9.11 Synthesis of a Compact Pole 435

9.12 Cauer Realization of Lossless Two-ports 436

**10 Design of Power Transfer Networks: A Glimpse of the Analytic Theory via a Unified Approach 439**

10.1 Introduction 439

10.2 Filter or Insertion Loss Problem from the Viewpoint of Broadband Matching 444

10.3 Construction of Doubly Terminated Lossless Reciprocal Filters 446

10.4 Analytic Solutions to Broadband Matching Problems 447

10.5 Analytic Approach to Double Matching Problems 453

10.6 Unified Analytic Approach to Design Broadband Matching Networks 463

10.7 Design of Lumped Element Filters Employing Chebyshev Functions 464

10.8 Synthesis of Lumped Element Low-Pass Chebyshev Filter Prototype 474

10.9 Algorithm to Construct Monotone Roll-Off Chebyshev Filters 477

10.10 Denormalization of the Element Values for Monotone Roll-off Chebyshev Filters 490

10.11 Transformation from Low-Pass LC Ladder Filters to Bandpass Ladder Filters 492

10.12 Simple Single Matching Problems 494

10.13 Simple Double Matching Problems 499

10.14 A Semi-analytic Approach for Double Matching Problems 500

10.15 Algorithm to Design Idealized Equalizer Data for Double Matching Problems 500

10.16 General Form of Monotone Roll-Off Chebyshev Transfer Functions 511

10.17 LC Ladder Solutions to Matching Problems Using the General Form Chebyshev Transfer Function 517

10.18 Conclusion 526

**11 Modern Approaches to Broadband Matching Problems: Real Frequency Solutions 539**

11.1 Introduction 539

11.2 Real Frequency Line Segment Technique 540

11.3 Real Frequency Direct Computational Technique for Double Matching Problems 571

11.4 Initialization of RFDT Algorithm 599

11.5 Design of a Matching Equalizer for a Short Monopole Antenna 600

11.6 Design of a Single Matching Equalizer for the Ultrasonic T1350 Transducer 611

11.7 Simplified Real Frequency Technique (SRFT): ‘A Scattering Approach’ 616

11.8 Antenna Tuning Using SRFT: Design of a Matching Network for a Helix Antenna 619

11.9 Performance Assessment of Active and Passive Components by Employing SRFT 634

12 Immittance Data Modeling via Linear Interpolation Techniques: A Classical Circuit Theory Approach 691

12.1 Introduction 691

12.2 Interpolation of the Given Real Part Data Set 693

12.3 Verification via SS-ELIP 693

12.4 Verification via PS-EIP 696

12.5 Interpolation of a Given Foster Data Set Xf (!) 698

12.6 Practical and Numerical Aspects 701

12.7 Estimation of the Minimum Degree n of the Denominator Polynomial D(!2) 702

12.8 Comments on the Error in the Interpolation Process and Proper Selection of Sample Points 703

12.9 Examples 704

12.10 Conclusion 716

**13 Lossless Two-ports Formed with Mixed Lumped and Distributed Elements: Design of Matching Networks with Mixed Elements 719**

13.1 Introduction 719

13.2 Construction of Low-Pass Ladders with UEs 725

13.3 Application 727

13.4 Conclusion 731

Index 751