Electromagnetics, Microwave Circuit and Antenna Design

for Communications Engineering

Second Edition

Peter Russer

A R T E C H H O U S E BOSTON|LONDON a r t e c h h o u s e . c o m

Contents

Preface xvii

Chapter 1 Introduction 1 References 6

Chapter 2 Basic Electromagnetics 9 2.1 The Electromagnetic Field Concept 9 2.2 Field Intensities 12 2.3 Current and Flux Densities 16 2.4 Constitutive Relations 18 2.5 The Charge Density 23 2.6 The Maxwell Puzzle 24 2.7 The Integral Form of Maxwell's Equations 26 2.8 The Electromagnetic Wave 29

2.8.1 The Wave Equation 35 2.8.2 The Polarization of Electromagnetic Waves 36

2.9 KirchhofF's Laws 38 2.10 Maxwell's Equations in Local Form 41 2.11 Time-Harmonic Electromagnetic Fields 43 2.12 Maxwell's Equations in the Frequency Domain 44 2.13 Curvilinear Coordinates 46 2.14 Boundary Conditions 47 2.15 Problems 56 References 59

vii

Vltt Electromagnetics

Chapter 3 Potentials and Waves 61 3.1 The Electromagnetic Potentials 61 3.2 The Helmholtz Equation 65 3.3 Time-Harmonic Plane Waves 67

3.3.1 Time-Harmonic Plane Waves in Lossless Medium 69 3.3.2 Complex Waves 72

3.4 TM and TE Fields and Waves 74 3.5 Reflection and Transmission of Plane Waves 77

3.5.1 Reflection and Diffraction of a TE Wave at a Plane Boundary 80

3.5.2 Reflection and Diffraction of a TM Wave at a Plane Boundary 83

3.5.3 Total Reflection 86 3.6 Waves in Planar Layered Media 89 3.7 Thin Conducting Sheets 93 3.8 The Vector Wave Equation 94 3.9 Circular Cylindrical Waves 98

3.9.1 Excitation of a Cylindric Wave by a Uniform Current Filament 101

3.10 Spherical Waves 102 3.11 Problems 106 References 107

Chapter 4 Concepts, Methods, and Theorems 109 4.1 Energy and Power 109 4.2 Field Theoretic Formulation of Tellegen's Theorem 116 4.3 Sources of the Electromagnetic Field 118 4.4 The Uniqueness Theorem 120 4.5 The Equivalence Principle 121 4.6 Babinet's Principle 123 4.7 Reciprocity 125

4.7.1 The Lorentz Reciprocity Theorem 125 4.7.2 The Reciprocity Theorem for Impressed Sources 126

4.8 Green's Function 128 4.9 The Integral Equation Method 133 4.10 The Free-Space Green's Dyadic Form 136 4.11 Green's Theorems 136

4.11.1 The Scalar Green's Theorems 136 4.11.2 Green's Theorems in Two Dimensions 138 4.11.3 The Vector Green's Theorems 140

4.12 Integral Formulation of the Equivalence Principle 141

Contents ix

4.13 The Sturm-Liouville Equation 143 4.14 Spectral Representation of Green's Functions 146 4.15 Problems 148 References 148

Chapter 5 Static and Quasistatic Fields 151 5.1 Conditions for Static and Quasistatic Fields 151 5.2 Static and Quasistatic Electric Fields 153

5.2.1 Green's Function for the Static Electric Field 153 5.2.2 Capacitance 155

5.3 Static and Quasistatic Magnetic Fields 161 5.3.1 Green's Function for the Static Magnetic Field 161 5.3.2 Inductance 163

5.4 The Laplace Equation 169 5.4.1 Potential Separation Planes 170 5.4.2 Three-Dimensional Laplace Equation in Cartesian

Coordinates 171 5.5 Conformal Mapping 174

5.5.1 Field ofanEUipticCylindric Line 181 5.5.2 Field ofaCoaxial Line 183 5.5.3 Parallel Wire Line 186

5.6 The Schwarz-Christoffel Transformation 191 5.6.1 The Coplanar Line 193 5.6.2 The Coplanar Stripline 196 5.6.3 The Stripline 197

5.7 Problems 201 References 204

Chapter 6 Waves at the Surface of Conducting Media 207 6.1 Transverse Magnetic Surface Waves 208 6.2 Surface Currents 216 6.3 Surface Current Losses 22" 1 6.4 Induced Surface Currents 224 6.5 Problems 227 References 228

Chapter 7 Transmission-Lines and Waveguides 229 7.1 Introduction 229 7.2 Phase and Group Velocity 232 7.3 The Field Components 233 7.4 Waveguides for Transverse Electromagnetic Waves 235

X Electromagnetics

7.5 Multiconductor Transmission-Lines 249 7.6 Quasi-TEM Modes of Transmission-Lines 254

7.6.1 Quasi-TEM Modes of Two-Conductor Transmission-Lines 254

7.6.2 Quasi-TEM Modes of Multiconductor Transmission-Lines 259

7.7 Planar Transmission-Lines 260 7.7.1 The Microstrip Line 260 7.7.2 Quasistatic Approximation for the Microstrip Line 262 7.7.3 Coplanar Waveguide and Coplanar Stripline 265

7.8 Hollow Waveguides 266 7.8.1 TE Modes 266 7.8.2 TM Modes 270 7.8.3 Modal Expansions in Waveguides 272

7.9 Rectangular Waveguides 276 7.9.1 Transverse Electric Modes 276 7.9.2 Transverse Magnetic Modes 282 7.9.3 Power Flow in the Waveguide 284 7.9.4 Orthogonality of the Waveguide Modes 285 7.9.5 Generalized Currents and Voltages in Waveguides 286 7.9.6 Attenuation Due to Conductor Losses 289 7.9.7 Attenuation Due to Dielectric Losses 291

7.10 Circular Cylindric Waveguides 292 7.10.1 The Circular Waveguide Modes 292 7.10.2 Power Flow and Attenuation in the TE0I Mode 298

7.11 Radial Waveguides 300 7.11.1 Radial Parallel Plate Waveguide 300 7.11.2 Wedged Radial Parallel Plate Waveguide 307

7.12 Spherical Waveguides 309 7.12.1 Conical Waveguide 311 7.12.2 Biconical Waveguide 313

7.13 Dielectric Waveguides and Optical Fibers 316 7.13.1 Homogeneous Planar Dielectric Waveguides 316 7.13.2 Dielectric Slab with Single-Sided Metallization 320 7.13.3 Circular Dielectric Waveguides with Step Index

Profile 322 7.14 Problems 329 References 333

Contents XI

Chapter 8 The Transmission-Line Equations 335 8.1 The Transmission-Line Concept 335 8.2 Generalized Voltages and Currents 337 8.3 Solution of the Transmission-Line Equations 341 8.4 Wave Amplitudes 344 8.5 Reflection Coefficient and Smith Chart 346

8.5.1 Impedance Matching with Lumped Elements 353 8.5.2 Impedance Matching with Stubs 355

8.6 Solution of the Multiconductor Transmission-Line Equations 356

8.7 Multimode Excitation of Uniform HoUow Waveguides 363 8.7.1 The Transverse Field Equations 363 8.7.2 Modal Field Representation 366 8.7.3 Multimode Transmission-Line Equations for

Hollow Waveguides 368 8.7.4 Multimode Transmission-Line Equations of

Lossless Waveguides without Internal Sources 374 8.8 Green's Functions for Transmission-Lines 375

8.8.1 Green's Function for the Transmission-Line with Matched Terminations 378

8.8.2 Green's Function for the Transmission-Line with Arbitrary Linear Passive Terminations 379

8.9 Problems 381 References 384

Chapter 9 Resonant Circuits and Resonators 385 9.1 The Linear Passive One-Port 385 9.2 The Reactance Theorem 387 9.3 Resonant Circuits 389 9.4 The Transmission-Line Resonator 392 9.5 Cavity Resonators 395

9.5.1 The Rectangular Cavity Resonator 395 9.5.2 The Circular Cylindric Cavity Resonator 399

9.6 Couplingof Resonant Circuits and Resonators 402 9.6.1 The Loaded Quality Factor 402 9.6.2 Termination of a Transmission-Line with a

Resonant Circuit 403 9.6.3 Inductive Couplingof Cavity Resonators 405

9.7 Orthogonality of the Resonator Modes 407 9.8 Excitation of Resonators by Internal Sources 409 9.9 Problems 411

XU Electromagnetics

References 412

Chapter 10 Passive Microwave Circuits 413 10.1 Linear Multiports 413 10.2 Source-Free Linear Multiports 414

10.2.1 Impedance and Admittance Representations 414 10.2.2 The Chain Matrix 415 10.2.3 The Scattering Matrix 419 10.2.4 The Transmission Matrix 424

10.3 Tellegen's Theorem 425 10.3.1 Connection Networks 428 10.3.2 Tellegen's Theorem for Discretized Fields 429

10.4 The Power Properties 430 10.5 Reciprocal Multiports 431 10.6 Elementary Two-Ports 433 10.7 Signal Flow Graphs 436 10.8 Lumped Element Equivalent Circuits 439

10.8.1 Foster Representation of Reactance Multiports 439 10.8.2 Cauer Representation of Radiating Structures 445

10.9 Obstacles in Waveguides 450 10.10 The Symmetry Properties of Waveguide Junctions 456

10.10.1 Symmetrie Three-Port Waveguide Junctions 457 10.10.2 Symmetrie Four-Port Waveguide Junctions 460

10.11 Problems 463 References 466

Chapter 11 Periodic Structures and Filters 467 11.1 Periodic Electromagnetic Structures 467

11.1.1 TE Modes in Rectangular Periodic Waveguides 467 11.1.2 Sinusoidal Variation of the Permittivity 472

11.2 Wave Parameter Theory of Two-Ports 474 11.3 Lumped Low-Pass Filter Prototypes 481

11.3.1 The Butterworth Prototype 482 11.3.2 The Chebyshev Prototype 485

11.4 Ladder Filter Networks 488 11.4.1 Butterworth Ladder Networks 489 11.4.2 Chebyshev Ladder Networks 490

11.5 Frequency Transformation 492 11.5.1 Low-Pass to High-Pass Transformation 492 11.5.2 Low-Pass to Band-Pass Transformation 493 11.5.3 Low-Pass to Band-Stop Transformation 495

Contents xm

11.6 Transmission-Line with Periodic Load 497 11.7 Plane Wave Scattering by Periodic Structures 501

11.7.1 Scattering of TE Waves by Periodic Structures 501 11.7.2 Scattering of TM Waves by Periodic Structures 505

11.8 Metamaterials 507 11.9 Problems 515 References 517

Chapter 12 Radiation from Dipoles 519 12.1 The Hertzian Dipole 519 12.2 Aperiodic Spherical Waves 524 12.3 Vertically Oriented Electric Dipole over Lossy Half-Space 528

12.3.1 The Far-Field of the Vertical Dipole over Ground 538 12.3.2 The Surface Wave 539

12.4 Horizontally Oriented Electric Dipole over Lossy Half-Space 540

12.5 Problems 544 References 545

Chapter 13 Antennas 547 13.1 Introduction 547 13.2 Linear Antennas 549 13.3 The Integral Equation for the Linear Antenna 555 13.4 The Impedanceof the Linear Antenna 558 13.5 The Loop Antenna 560 13.6 Receiving Antennas 563

13.6.1 The Hertzian Dipole as Receiving Antenna 563 13.6.2 The Loop Antenna as Receiving Antenna 564 13.6.3 The Linear Dipole Antenna as Receiving Antenna 565

13.7 Gain and Effective Antenna Aperture 569 13.8 Antenna Arrays 575

13.8.1 Linear Antenna Arrays 575 13.8.2 Circular Antenna Arrays 577

13.9 Aperture Antennas 578 13.9.1 Radiating Apertures 578 13.9.2 Hörn Antennas 582 13.9.3 Gain and Effective Area of Aperture Antennas 585 13.9.4 Mirror and Lens Antennas 587 13.9.5 Slot Antennas 589

13.10 Microstrip Antennas 591 13.10.1 Planar Rectangular Patch Antenna 593

XIV Electromagnetics

13.11 Broadband Antennas 595 13.12 Problems 597 References 601

Chapter 14 Numerical Electromagnetics 603 14.1 Introduction 603 14.2 The Methodof Moments 605 14.3 The Transmission-Line Matrix Method 611 14.4 The Mode Matching Method 617 References 623

Appendix A Vectors and Differential Forms 627 A.l Vectors 627 A.2 Differential Forms 631

A.2.1 Products of Exterior Differential Forms 632 A.2.2 The Contraction 633 A.2.3 The Exterior Derivative 634 A.2.4 The Laplace Operator 635

A.3 Stokes' Theorem 636 A.4 Curvilinear Coordinates 640

A.4.1 General Cylindrical Coordinates 646 A.4.2 Circular Cylindric Coordinates 647 A.4.3 Spherical Coordinates 650 A.4.4 Twisted Forms 653 A.4.5 Integration of Differential Forms by Pullback 653

A.5 Double Differential Forms 654 A.6 Relations between Exterior Calculus and Conventional

Vector Notation 656 A.6.1 Differential Operators 656 A.6.2 Maxwell's Equations 656

References 657

Appendix B Special Functions 659 B.l Ordinary Bessel Functions 659 B.2 Modified Bessel Functions 662 B.3 Spherical Bessel Functions 665 B.4 Legendre Polynomials 667 B.5 Spherical Harmonics 670 References 672

Contents xv

Appendix C

Appendix D

Appendix E

Linear Algebra C.l C.2 C.3 C.4

Unitary Vector Space Diagonalization of a Matrix Matrix Functions The Hubert Space C.4.1 Linear Operators in C.4.2 Function Spaces

Hubert Space

C.4.3 Function Spaces with Biorthogonal Basis References

Fourier Series and Fourier Transform D.l D.2 D.3

The Fourier Series The Fourier Integral The Delta Distribution

References

Complex Integration E.l E.2 E.3

Analytic Functions The Residue Theorem The Saddle-Point Method

References

673 673 679 681 683 686 691 693 696

697 697 699 701 704

705 705 707 708 710

List of Symbols 711

About the Author 717

Index 719