Foundations of Antenna Radiation Theory

Professor Wen Geyi is a Fellow of the IEEE. He has published very widely on theories of microwaves and antennas. He has authored four books and over 100 journal publications and holds more than 40 patents.

About the Author xi

Preface xiii

1 Eigenvalue Theory 1

1.1 Maxwell Equations 3

1.1.1 Wave Equations 3

1.1.2 Properties of Electromagnetic Fields 6

1.1.2.1 Superposition Theorem 7

1.1.2.2 Conservation of Electromagnetic Field Energy 7

1.1.2.3 Equivalence Theorem 12

1.1.2.4 Reciprocity 13

1.2 Methods for Partial Differential Equations 14

1.2.1 Method of Separation of Variables 14

1.2.1.1 Rectangular Coordinate System 15

1.2.1.2 Cylindrical Coordinate System 16

1.2.1.3 Spherical Coordinate System 19

1.2.2 Method of Green’s Function 21

1.2.2.1 Green’s Functions for Helmholtz Equation 22

1.2.2.2 Dyadic Green’s Functions and Integral Representations 24

1.2.3 Variational Method 27

1.3 Eigenvalue Problem for Hermitian Matrix 29

1.3.1 Properties 29

1.3.2 Rayleigh Quotient 30

1.4 Eigenvalue Problems for the Laplace Operator on Scalar Field 32

1.4.1 Rayleigh Quotient 32

1.4.2 Properties of Eigenvalues 36

1.4.3 Completeness of Eigenfunctions 38

1.4.4 Differential Equations with Variable Coefficients 39

1.4.5 Green’s Function and Spectral Representation 41

1.5 Eigenvalue Problems for the Laplace Operator on Vector Field 44

1.5.1 Rayleigh Quotient 45

1.5.2 Completeness of Vector Modal Functions 48

1.5.3 Classification of Vector Modal Functions 54

1.6 Ritz Method for the Solution of Eigenvalue Problem 55

1.7 Helmholtz Theorems 57

1.7.1 Helmholtz Theorem for the Field in Infinite Space 57

1.7.2 Helmholtz Theorem for the Field in Finite Region 59

1.7.3 Helmholtz Theorem for Time-Dependent Field 60

1.8 Curl Operator 61

1.8.1 Eigenfunctions of Curl Operator 62

1.8.2 Plane-Wave Expansions for the Fields and Dyadic Green’s Functions 64

References 66

2 Radiation in Waveguide 69

2.1 Vector Modal Functions for Waveguide 70

2.1.1 Classification of Vector Modal Functions 71

2.1.2 Vector Modal Functions for Typical Waveguides 75

2.1.2.1 Rectangular Waveguide 75

2.1.2.2 Circular Waveguide 76

2.1.2.3 Coaxial Waveguide 77

2.2 Radiated Fields in Waveguide 79

2.2.1 Modal Expansions for the Fields and Dyadic Green’s Functions 79

2.2.2 Dyadic Green’s Functions for Semi-infinite Waveguide 91

2.3 Waveguide Discontinuities 92

2.3.1 Excitation of Waveguide 92

2.3.2 Conducting Obstacles in Waveguide 95

2.3.3 Coupling by Small Aperture 97

2.4 Transient Fields in Waveguide 102

References 107

3 Radiation in Cavity Resonator 109

3.1 Radiated Fields in Cavity Resonator 110

3.1.1 Classification of Vector Modal Functions for Cavity Resonator 111

3.1.2 Modal Expansions for the Fields and Dyadic Green’s Functions 114

3.2 Cavity with Openings 117

3.2.1 Cavity with One Port 118

3.2.2 Cavity with Two Ports 120

3.3 Waveguide Cavity Resonator 125

3.3.1 Field Expansions by Vector Modal Functions of Waveguide 125

3.3.2 Modal Representations of Dyadic Green’s Functions 133

3.4 Vector Modal Functions for Typical Waveguide Cavity Resonators 136

3.4.1 Rectangular Waveguide Cavity 136

3.4.2 Circular Waveguide Cavity 137

3.4.3 Coaxial Waveguide Cavity 139

3.5 Radiation in Waveguide Revisited 140

3.6 Transient Fields in Cavity Resonator 141

References 149

4 Radiation in Free Space (I): Generic Properties 151

4.1 Antenna Parameters 152

4.1.1 Power, Efficiencies, and Input Impedance 152

4.1.2 Field Regions, Radiation Pattern, Radiation Intensity, Directivity, and Gain 155

4.1.3 Vector Effective Length, Equivalent Area, and Antenna Factor 158

4.1.4 Antenna Quality Factor 163

4.2 Theory of Spherical Waveguide 163

4.2.1 Vector Modal Functions for Spherical Waveguide 164

4.2.2 Modal Expansions of Fields and Dyadic Green’s Functions 168

4.2.3 Properties of Spherical Vector Wave Functions 180

4.2.4 Far-Zone Fields 181

4.3 Stored Field Energies and Radiation Quality Factor 182

4.3.1 Stored Field Energies in General Materials 184

4.3.2 Stored Field Energies of Antenna 194

4.3.3 Radiated Field Energy 199

4.3.4 Evaluation of Radiation Quality Factor 202

4.4 Modal Quality Factors 206

4.4.1 Stored Field Energies Outside the Circumscribing Sphere of Antenna 206

4.4.2 Two Inequalities for Spherical Hankel Functions 210

4.4.3 Properties of Modal Quality Factors 212

4.4.3.1 Proof of Properties 2, 4, and 7 213

4.4.3.2 Proof of Properties 1, 3, 6, 8, and 9 215

4.4.3.3 Proof of Property 5 216

4.4.3.4 Proof of Properties 10 and 11 217

4.4.4 Lower Bound for Antenna Quality Factor 218

4.5 Upper Bounds for the Products of Gain and Bandwidth 220

4.5.1 Directive Antenna 221

4.5.2 OmniDirectional Antenna 224

4.5.3 Best Possible Antenna Performance-Guidelines for Small Antenna Design 226

4.6 Expansions of the Radiated Fields in Time Domain 230

References 238

5 Radiation in Free Space (II): Modal Analysis 243

5.1 Basic Antenna Types 245

5.2 Equivalent Current Distributions of Antenna 246

5.3 Antenna as a Waveguide Junction 249

5.4 Integral Equation Formulations 250

5.4.1 Compensation Theorem for Time-Harmonic Fields 251

5.4.2 Integral Equations for Composite Structure 252

5.4.3 Integral Equation for Wire Antenna 254

5.5 Vertical Dipole 257

5.5.1 Fields in the Region r > b 258

5.5.2 Fields in the Region r < b 261

5.6 Horizontal Dipole 261

5.6.1 Fields in the Region r > b 262

5.6.2 Fields in the Region r < b 267

5.7 Loop 267

5.8 Spherical Dipole 269

5.9 Dipole Near Conducting Sphere 271

5.10 Finite Length Wire Antenna 273

5.10.1 Fields in the Region r > l 273

5.10.2 The Fields in the Region r < l 275

5.11 Aperture Antenna 276

5.12 Microstrip Patch Antenna 280

5.13 Resonant Modal Theory for Antenna Design 290

5.13.1 Formulations 291

5.13.2 Applications 293

5.13.2.1 Crossed-Dipole 293

5.13.2.2 Dual-Band Bowtie Antenna 297

References 301

6 Radiation in Free Space (III): Array Analysis and Synthesis 303

6.1 Introduction to Array Analysis 305

6.1.1 Array Factor 305

6.1.2 Linear Array 307

6.1.2.1 Linear Array with Uniform Amplitude 307

6.1.2.2 Linear Array with Nonuniform Amplitude 311

6.1.3 Circular Array 314

6.1.4 Planar Array 316

6.2 Introduction to Array Synthesis with Conventional Methods 318

6.2.1 Array Factor and Space Factor for Line Source 318

6.2.2 Schelkunoff Unit Circle Method 320

6.2.3 Dolph–Chebyshev Method 323

6.2.4 Fourier Transform Method 327

6.2.4.1 Continuous Line Source 327

6.2.4.2 Linear Array 329

6.3 Power Transmission Between Two Antennas 330

6.3.1 The General Power Transmission Formula 331

6.3.2 Power Transmission Between Two Planar Apertures 335

6.3.3 Power Transmission Between Two Antennas with Large Separation 341

6.4 Synthesis of Arrays with MMPTE 343

6.4.1 Power Transmission Between Two Antenna Arrays 344

6.4.1.1 Unconstrained Optimization 346

6.4.1.2 Weighted Optimization 346

6.4.1.3 Constrained Optimization 347

6.4.2 Applications 349

6.5 Synthesis of Arrays with EMMPTE 369

6.5.1 Arrays with Specified Energy Distribution 370

6.5.2 Arrays with Specified Power Distribution 372

6.5.3 Applications 373

References 376

Appendix A Vector Analysis 381

Appendix B Dyadic Analysis 383

Appendix C SI Unit System 385

Appendix D Unified Theory for Fields (UTF) 387

Index 417