Computational Electromagnetics for RF and Microwave Engineering

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface to the second edition......Page 17
Preface to the first edition......Page 19
Acknowledgements......Page 23
Acknowledgements for the second edition......Page 24
To the reader......Page 25
Notation......Page 26
1.1 Introduction......Page 27
1.2 Full-wave CEM techniques......Page 29
1.3 The method of moments (MoM)......Page 34
1.4 The finite difference time domain (FDTD) method......Page 36
1.5 The finite difference time domain (FDTD) method......Page 39
1.6.1 Transmission line matrix (TLM) method......Page 42
1.7 The CEM modelling process......Page 43
1.8 Verification and validation......Page 45
1.8.1 An example: a frequency selective surface......Page 46
1.9 Convergence and extrapolation......Page 49
1.10 Extending the limits of full-wave CEM methods......Page 50
1.11 CEM: the future......Page 51
1.12 A “road map” of this book......Page 54
References......Page 55
2.1 Introduction......Page 58
2.2.1 Partial differential equations......Page 59
2.2.3 Approximating derivatives using finite differences......Page 60
2.3 A very brief history of the FDTD......Page 62
2.4.1 A one-dimensional model problem: a lossless transmission line......Page 63
2.4.2 FDTD solution of the one-dimensional lossless transmission line problem......Page 66
2.4.3 Accuracy, convergence, consistency and stability of the method......Page 72
2.5.1 The Gaussian pulse......Page 78
2.5.3 A polynomial pulse......Page 80
2.5.4 The 1D transmission line revisited from a wideband perspective......Page 83
2.5.5 Estimating the Fourier transform......Page 86
2.5.6 Simulation using Gaussian and Gaussian derivative pulses......Page 88
2.6.1 Dispersion......Page 90
2.6.2 Derivation of the dispersion equation......Page 92
2.6.3 Some closing comments on dispersion in FDTD grids......Page 93
2.7 The Courant stability criterion derived by von Neumann analysis......Page 95
References......Page 97
Problems......Page 98
Assignment......Page 99
3.2 The 2D FDTD algorithm......Page 100
3.2.2 The TEz formulation......Page 101
3.2.3 Including a source: the scattered/total field formulation......Page 105
3.2.4 Meshing the scatterer......Page 107
3.2.5 Absorbing boundary conditions......Page 108
3.2.6 Developing the simulator......Page 110
3.2.7 FDTD analysis of TE scattering from a PEC cylinder......Page 117
3.2.8 Computational aspects......Page 122
3.3.1 An historical perspective......Page 123
3.3.2 A numerical absorber – pre-Berenger......Page 125
3.3.3 Berenger's split field PML formulation......Page 127
3.3.4 The FDTD update equations for a PML......Page 128
3.3.5 PML implementation issues......Page 130
3.3.6 Results for a split field PML......Page 131
3.3.7 Drawbacks of the Berenger PML......Page 132
3.3.8 Uniaxial absorber theory......Page 133
3.3.10 Further reading on PMLs......Page 134
3.4 The 3D FDTD algorithm......Page 135
3.4.1 The Yee cell in 3D......Page 136
3.4.2 An application: determining the resonant frequencies of a PEC cavity......Page 140
3.4.3 Dispersion in two and three dimensions......Page 141
3.5 Commercial implementations......Page 143
3.5.1 An introductory example – a waveguide ''through''......Page 144
3.5.2 A waveguide filter......Page 146
3.5.3 A microstrip patch antenna......Page 147
3.6 Further reading......Page 150
3.7 Conclusions......Page 151
References......Page 152
Assignments......Page 153
4.1 Introduction......Page 156
4.2 An electrostatic example......Page 157
4.2.1 Some simplifying approximations......Page 158
4.2.2 Approximating the charge......Page 159
4.2.3 Collocation......Page 160
4.2.4 Solving the system of linear equations......Page 161
4.2.5 Results and discussion......Page 162
4.3.1 The electrically thin dipole......Page 163
4.4.1 The numerical electromagnetic code (NEC) – method of moments......Page 170
4.4.2 NEC basis functions......Page 171
4.4.4 Junction treatments with piecewise linear basis functions......Page 173
4.5 The method of weighted residuals......Page 176
4.6 Scattering from infinite cylinders......Page 178
4.6.1 General derivation of surface integral equation operators......Page 179
4.6.2 The EFIE for TM scattering......Page 180
4.6.3 MoM solution of EFIE for TM scattering......Page 181
4.6.5 Post-processing: echo width and radar cross-section......Page 183
4.6.6 Discussion, and the Fredholm alternative......Page 185
4.7 Further reading......Page 186
References......Page 188
Assignments......Page 190
5.1 Introduction......Page 192
5.2 An introductor y example: the dipole......Page 194
5.3 A wire antenna array: the Yagi–Uda antenna......Page 198
5.4 A log-periodic antenna......Page 203
5.5 An axial mode helix antenna......Page 211
5.6 A Wu–King loaded dipole......Page 219
References......Page 225
6.1 Electric and magnetic field integral equations......Page 227
6.2 The Rao–Wilton–Glisson (R WG) element......Page 229
6.3.1 The electric field integral equation (EFIE)......Page 232
6.3.2 The RWG basis function revisited......Page 233
6.3.3 The MoM formulation......Page 234
6.3.4 Derivation of the matrix entries......Page 236
6.3.5 Numerical approximation of the matrix entries......Page 237
6.3.6 Coding issues......Page 240
6.3.7 Verification......Page 241
6.3.8 Discussion......Page 243
6.4.1 Scattering from a sphere......Page 244
6.4.2 The analytical solution......Page 248
6.5 Modelling homogeneous material bodies using equivalent currents......Page 250
6.6 Scattering from a dielectric sphere......Page 252
6.7 Computational implications of surface and volume modelling with the MoM......Page 254
6.8.1 Introduction......Page 256
6.8.3 Physical optics and MoM hybridization......Page 257
6.8.4 A FEKO example using the MoM/PO hybrid......Page 260
6.9.1 Background......Page 263
6.9.2 High-performance computing......Page 264
6.9.3 FFT-based methods......Page 274
6.9.4 The fast multipole method......Page 277
6.10 Further reading......Page 284
References......Page 286
Problem......Page 289
7.2 Dyadic Green functions: some introductory notes......Page 290
7.3 A static example of a stratified medium problem: the grounded dielectric slab......Page 292
7.4.1 A brief revision of potential theory......Page 295
7.4.2 The Sommerfeld potentials......Page 296
7.4.3 An example: derivation of G for single-layer microstrip......Page 299
7.4.4 The scalar potential and the mixed potential integral equation......Page 302
7.4.5 Surface waves......Page 303
7.5.1 Approximate evaluation of the Sommerfeld integrals......Page 304
7.5.2 Numerical integration in the spectral domain......Page 305
7.5.3 Locating the pole......Page 313
7.5.4 General source locations......Page 314
7.6 MoM solution using the Sommerfeld potentials......Page 315
7.7 Further reading......Page 323
References......Page 324
Assignments......Page 325
8.1 Printed antenna and microstrip technology: a brief review......Page 326
8.2 A simple patch antenna......Page 327
8.3 Mutual coupling between microstrip antennas......Page 329
8.4 An array with ''scan blindness''......Page 334
8.5 A concluding discussion of stratified media formulations......Page 340
References......Page 341
9.1 Introduction......Page 343
9.2 The variational boundar y value problem: the transmission line problem revisited......Page 344
9.2.1 The model problem......Page 345
9.2.2 The equivalent variational functional......Page 346
9.2.3 The finite element approximation of the functional......Page 347
9.2.4 Evaluating the elemental matrices......Page 349
9.2.5 Assembling the system......Page 351
9.2.6 Rendering the functional stationary and solving the problem......Page 353
9.2.7 Coding the FEM......Page 354
9.2.8 Results and rate of convergence......Page 355
9.3.1 Higher-order elements......Page 357
9.3.2 More general boundary conditions......Page 363
9.4 Further reading......Page 365
References......Page 366
Assignments......Page 367
10.1 Introduction......Page 368
10.2.1 The variational boundary value problem approach......Page 369
10.2.2 Some practical issues: assembling the system......Page 375
10.2.3 An application to microstrip......Page 378
10.2.4 More on variational functionals......Page 381
10.2.6 Discussion......Page 384
10.3 The Galerkin (weighted residual) formulation......Page 385
10.4 Simplex coordinates......Page 390
10.4.1 Simplex coordinates in one, two and three dimensions......Page 391
10.4.2 Some properties of simplex coordinates......Page 392
10.6 The null space of the curl operator and spurious modes......Page 393
10.7.1 An historical perspective......Page 397
10.7.2 Theory of vector elements......Page 398
10.7.3 Vector elements on triangles – the Whitney element......Page 400
10.8.1 The two-dimensional variational functional for an homogeneous waveguide......Page 404
10.8.2 Explicit formula for the elemental matrix entries......Page 405
10.8.3 Coding......Page 408
10.8.4 Results......Page 412
10.8.5 Degenerate modes......Page 415
10.8.6 Higher-order vector elements......Page 417
10.9.1 A vector formulation based on the transverse and axial field components......Page 420
10.9.2 The cut-off eigenanalysis formulation......Page 422
10.9.3 Homogeneously filled guides: TE modes only......Page 423
10.9.5 Results: a half-filled dielectric loaded rectangular waveguide......Page 424
10.10 Further reading......Page 426
10.11 Conclusions......Page 428
References......Page 429
Assignments......Page 432
11.1 The three-dimensional Whitney element......Page 433
11.1.1 Explicit formula for the tetrahedral elemental matrix entries......Page 434
11.1.2 Coding......Page 437
11.2 Higher-order elements......Page 441
11.2.2 Hierarchal vector basis functions......Page 442
11.2.3 Properties of hierarchal basis functions......Page 445
11.2.4 Practical impact of higher-order basis functions in an FEM code......Page 447
11.3 The FEM from the variational boundar y value problem viewpoint......Page 453
11.4.1 Introduction......Page 455
11.4.2 The waveguide formulation......Page 456
11.5.1 Application to a Magic-T......Page 458
11.5.2 Application to a capacitive iris......Page 462
11.6 Open-region finite element method formulations: absorbing boundar y conditions (ABCs)......Page 467
11.6.1 Formulation in terms of the scattered field......Page 468
11.6.2 Formulation in terms of the total field......Page 469
11.7 Further reading......Page 470
References......Page 471
Problems......Page 474
Assignments......Page 476
12.1.1 Introduction......Page 477
12.1.2 Theoretical background......Page 478
12.2.1 Applications of FEM/MoM hybrid formulations......Page 480
12.2.2 Human exposure assessment near GSM base stations......Page 481
12.3 Time domain FEM......Page 483
12.3.1 Basic formulation and implementation......Page 484
12.3.2 Preliminary results......Page 487
12.3.3 The FDTD as a special case of the FETD......Page 490
12.4 Sparse matrix solvers......Page 494
12.4.1 Profile-in skyline storage......Page 495
12.4.2 Compressed row storage......Page 496
12.4.4 Results for sparse storage schemes......Page 497
12.5 A posteriori error estimation and adaptive meshing......Page 499
12.5.1 Explicit, residual-based error estimators......Page 500
12.5.2 An example of the application of an error estimator......Page 502
12.6 Further reading and conclusions......Page 504
References......Page 507
Appendix A: The Whitney element......Page 510
Appendix B: The Newmark-β time-stepping algorithm......Page 512
References......Page 514
Appendix C: On the convergence of the MoM......Page 515
Reference......Page 516
Gradient......Page 517
Appendix E: Web resources......Page 519
MoM codes......Page 522
FEM codes......Page 523
Index......Page 524