Applied Frequency-Domain Electromagnetics

Preface xv

Acknowledgements xvii

1 Background 1

1.1 Field Laws 1

1.2 Properties of Materials 2

1.3 Types of Currents 3

1.4 Capacitors, Inductors 4

1.5 Differential Form 6

1.6 Time-Harmonic Fields 8

1.7 Sufficient Conditions 9

1.8 Magnetic Currents, Duality 9

1.9 Poynting's Theorem 10

1.10 Lorentz Reciprocity Theorem 13

1.11 Friis and Radar Equations 14

1.12 Asymptotic Techniques 16

1.13 Further Reading 17

References 18

Problems 18

2 TEM Waves 21

2.1 Introduction 21

2.2 Plane Waves 22

2.3 Oblique Plane Waves 28

2.4 Plane Wave Reflection and Transmission 29

2.5 Multilayer Slab 36

2.6 Impedance Boundary Condition 38

2.7 Transmission Lines 44

2.8 Transverse Equivalent Network 60

2.9 Absorbers 62

2.10 Phase and Group Velocity 63

2.11 Further Reading 65

References 66

Problems 66

3 Waveguides 71

3.1 Separation of Variables 71

3.2 Rectangular Waveguide 73

3.3 Cylindrical Waves 80

3.4 Circular Waveguide 81

3.5 Waveguide Excitation 84

3.6 2D Waveguides 85

3.7 Transverse Resonance Method 94

3.8 Other Waveguide Types 98

3.9 Waveguide Discontinuities 101

3.10 Mode Matching 107

3.11 Waveguide Cavity 114

3.12 Perturbation Method 121

3.13 Further Reading 127

References 127

Problems 127

4 Potentials, Concepts, and Theorems 135

4.1 Vector Potentials A and F 135

4.2 Hertz Potentials 140

4.3 Vector Potentials and Boundary Conditions 141

4.4 Uniqueness Theorem 148

4.5 Radiation Condition 151

4.6 Image Theory 151

4.7 Physical Optics 153

4.8 Surface Equivalent 154

4.9 Love's Equivalent 158

4.10 Induction Equivalent 161

4.11 Volume Equivalent 162

4.12 Radiation by Planar Sources 164

4.13 2D Sources and Fields 165

4.14 Derivation of Vector Potential Integral 168

4.15 Solution Without Using Potentials 170

4.16 Further Reading 171

References 171

Problems 172

5 Canonical Problems 177

5.1 Cylinder 177

5.2 Wedge 184

5.3 The Relation Between 2D and 3D Solutions 188

5.4 Spherical Waves 192

5.5 Method of Stationary Phase 199

5.6 Further Reading 201

References 202

Problems 202

6 Method of Moments 209

6.1 Introduction 209

6.2 General Concepts 209

6.3 2D Conducting Strip 212

6.4 2D Thin Wire MoM 220

6.5 Periodic 2D Wire Array 224

6.6 3D Thin Wire MoM 228

6.7 EFIE and MFIE 234

6.8 Internal Resonances 236

6.9 PMCHWT Formulation 237

6.10 Basis Functions 238

6.11 Further Reading 240

References 240

Problems 241

7 Finite Element Method 245

7.1 Introduction 245

7.2 Laplace's Equation 246

7.3 Piecewise-planar Potential 246

7.4 Stored Energy 248

7.5 Connection of Elements 248

7.6 Energy Minimization 250

7.7 Natural Boundary Conditions 252

7.8 Capacitance, Inductance 255

7.9 Computer Program 257

7.10 Poisson's Equation 258

7.11 Scalar Wave Equation 262

7.12 Galerkin's Method 266

7.13 Vector Wave Equation 270

7.14 Other Element Types 270

7.15 Radiating Structures 274

7.16 Further Reading 278

References 278

Problems 278

8 Uniform Theory of Diffraction 283

8.1 Fermat's Principle 283

8.2 2D Fields 284

8.3 Scattering and GTD 292

8.4 3D Fields 294

8.5 Curved Surface Reflection 306

8.6 Curved Wedge Face 308

8.7 Non-Metallic Wedge 308

8.8 Slope Diffraction 309

8.9 Double Diffraction 310

8.10 GTD Equivalent Edge Currents 311

8.11 Surface-Ray Diffraction 315

8.12 Further Reading 324

References 325

Problems 326

9 Physical Theory of Diffraction 337

9.1 PO and an Edge 337

9.2 Asymptotic Evaluation 338

9.3 Reflector Antenna 344

9.4 RCS of a Disc 347

9.5 PTD Equivalent Edge Currents 351

9.6 Further Reading 351

References 352

Problems 352

10 Scalar and Dyadic Green's Functions 355

10.1 Impulse Response 355

10.2 Green's Function for A 357

10.3 2D Field Solutions Using Green's Functions 358

10.4 3D Dyadic Green's Functions 362

10.5 Some Dyadic Identities 363

10.6 Solution Using a Dyadic Green's Function 364

10.7 Symmetry Property of G 365

10.8 Interpretation of the Radiation Integrals 367

10.9 Free Space Dyadic Green's Function 367

10.10Dyadic Green's Function Singularity 368

10.11Dielectric Rod 370

10.12Further Reading 372

References 372

Problems 372

11 Green's Functions Construction I 375

11.1 Sturm Liouville Problem 375

11.2 Green's Second Identity 376

11.3 Hermitian Property 376

11.4 Particular Solution 377

11.5 Properties of the Green's Function 377

11.6 UT Method 378

11.7 Discrete and Continuous Spectra 382

11.8 Generalized Separation of Variables 388

11.9 Further Reading 396

References 396

Problems 396

12 Green's Functions Construction II 401

12.1 Sommerfeld Integrals 401

12.2 The Function k(v) = vk²-¿² 402

12.3 The Transformation v= k sin w 405

12.4 Saddle Point Method 406

12.5 SDP Branch Cuts 415

12.6 Grounded Dielectric Slab 417

12.7 Half Space 426

12.8 Circular Cylinder 435

12.9 Strip Grating on a Dielectric Slab 443

12.10Further Reading 455

References 456

Problems 456

Appendix

A Constants and Formulas 461

A.1 Constants 461

A.2 Definitions 461

A.3 Trigonometry 462

A.4 The Impulse Function 462

References 463

B Coordinates and Vector Calculus 465

B.1 Coordinate Transformations 466

B.2 Volume and Surface Elements 466

B.3 Vector Derivatives 468

B.4 Vector Identities 469

B.5 Integral Relations 470

References 472

C Bessel's Differential Equation 473

C.1 Bessel Functions 473

C.2 Roots of H(^1,2)_¿p(x)=0 476

C.3 Integrals 476

C.4 Orthogonality 477

C.5 Recursion Relations 477

C.6 Gamma Function 478

C.7 Wronskians 478

C.8 Spherical Bessel Functions 479

References 480

D Legendre's Differential Equation 481

D.1 Legendre Functions 481

D.2 Associated Legendre Functions 482

D.3 Orthogonality 482

D.4 Recursion Relations 483

D.5 Spherical Form 483

References 483

E Complex Variables 485

E.1 Residue Calculus 485

E.2 Branch Cuts 486

References 487

F Compilers and Programming 489

F.1 Getting Started 489

F.2 Fortran 90 491

F.3 More on the OS 499

F.4 Plotting 501

F.5 Further Reading 502

References 502

G Numerical Methods 503

G.1 Numerical Integration 503

G.2 Root Finding 507

G.3 Matrix Equations 509

G.4 Matrix Eigenvalues 510

G.5 Bessel Functions 511

G.6 Legendre Polynomials 511

References 512

H Software Provided 513

Index 515