Computational Electromagnetics for RF and Microwave Engineering

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
Acknowledgements......Page 17
To the reader......Page 19
Notation......Page 20
1.1 Introduction......Page 23
1.2 Full-wave CEM techniques......Page 26
1.3 The method of moments (MoM)......Page 29
1.4 The finite difference time domain (FDTD) method......Page 31
1.5 The finite element method (FEM)......Page 35
1.6.3 The generalized multipole technique (GMT)......Page 38
1.7 The CEM modelling process......Page 39
1.8 Verification and validation......Page 41
1.8.1 An example: a frequency selective surface......Page 42
1.9 Extending the limits of full-wave CEM methods......Page 44
1.10 CEM: the future......Page 46
1.11 A “road map” of this book......Page 47
References......Page 49
2.1 Introduction......Page 51
2.2.2 Approximating derivatives using finite differences......Page 52
2.3 A very brief history of the FDTD......Page 54
2.4.1 A one-dimensional model problem: a lossless transmission line......Page 55
2.4.2 FDTD solution of the one-dimensional lossless transmission line problem......Page 58
Programming aspects: “in-place” operations......Page 63
Obtaining and evaluating preliminary results......Page 64
2.4.3 Accuracy, convergence, consistency and stability of the method......Page 65
2.5 Obtaining wideband data using the FDTD......Page 70
2.5.1 The Gaussian pulse......Page 72
2.5.2 The Gaussian derivative pulse......Page 73
2.5.3 A polynomial pulse......Page 74
2.5.4 The 1D transmission line revisited from a wideband perspective......Page 76
2.5.5 Estimating the Fourier transform......Page 79
2.6.1 Dispersion......Page 82
2.6.2 Derivation of the dispersion equation......Page 85
2.6.3 Some closing comments on dispersion in FDTD grids......Page 87
2.7 Conclusion......Page 88
References......Page 89
3.1 Introduction......Page 90
3.2.1 Electromagnetic scattering problems......Page 91
3.2.2 The TEz formulation......Page 92
3.2.3 Including a source: the scattered/total field formulation......Page 95
3.2.4 Meshing the scatterer......Page 98
3.2.5 Absorbing boundary conditions......Page 99
3.2.6 Developing the simulator......Page 101
Implementing the update equations......Page 104
Implementing the plane-wave source......Page 105
3.2.7 FDTD analysis of TE scattering from a PEC cylinder......Page 108
3.2.8 Computational aspects......Page 113
3.3.1 An historical perspective......Page 116
3.3.2 A numerical absorber – pre-Berenger......Page 117
3.3.3 Berenger’s split field PML formulation......Page 119
3.3.4 The FDTD update equations for a PML......Page 121
3.3.5 PML implementation issues......Page 123
3.3.6 Results for a split field PML......Page 125
3.3.7 Drawbacks of the Berenger PML......Page 126
3.3.9 Stretched coordinate theory......Page 127
3.4 The 3D FDTD algorithm......Page 128
3.5 Commercial implementations......Page 129
3.5.1 An introductory example – a waveguide “through”......Page 130
3.5.2 A waveguide filter......Page 132
3.5.3 A microstrip patch antenna......Page 134
3.6 Further reading......Page 136
3.7 Conclusions......Page 137
References......Page 138
4.1 Introduction......Page 140
4.2 An electrostatic example......Page 141
4.2.1 Some simplifying approximations......Page 142
4.2.2 Approximating the charge......Page 143
4.2.4 Solving the system of linear equations......Page 145
4.2.5 Results and discussion......Page 147
4.3.1 The electrically thin dipole......Page 148
Approximating the current......Page 151
The incident field......Page 152
Some computed results......Page 153
4.4.1 The numerical electromagnetic code (NEC) – method of moments......Page 154
4.4.2 NEC basis functions......Page 156
4.4.4 Junction treatments with piecewise linear basis functions......Page 158
4.5 The method of weighted residuals......Page 161
4.6 Further reading......Page 164
References......Page 166
5.1 Introductory comments......Page 168
5.2 An introductory example: the dipole......Page 171
5.3 A wire antenna array: the Yagi–Uda antenna......Page 175
5.4 A log-periodic antenna......Page 181
5.5 An axial mode helix antenna......Page 189
5.6 A Wu–King loaded dipole......Page 197
5.7 Conclusions......Page 204
References......Page 205
6.1 Electric and magnetic field integral equations......Page 206
6.2 The Rao–Wilton–Glisson (RWG) element......Page 208
6.3.1 Scattering from a sphere......Page 211
6.3.2 The analytical solution......Page 215
6.4 Modelling homogeneous material bodies using equivalent currents......Page 218
6.5 Scattering from a dielectric sphere......Page 219
6.6 Computational implications of surface and volume modelling with the MoM......Page 221
6.7 Hybrid MoM/asymptotic techniques for large problems......Page 222
6.7.1 Introduction......Page 223
6.7.3 Physical optics and MoM hybridization......Page 224
6.7.4 A FEKO example using the MoM/PO hybrid......Page 227
6.8.1 Background......Page 230
6.8.2 High-performance computing......Page 232
6.8.3 FFT-based methods......Page 238
A two-dimensional FMM prototype......Page 241
The full three-dimensional FMM......Page 244
6.9 Further reading......Page 247
6.10 Concluding comments......Page 249
References......Page 250
7.1 Introduction......Page 253
7.2 Dyadic Green functions: some introductory notes......Page 254
7.3 A static example of a stratified medium problem: the grounded dielectric slab......Page 255
7.4.1 A brief revision of potential theory......Page 259
Preliminaries......Page 260
The spectral domain transform......Page 261
Normal component representation......Page 262
Sommerfeld potentials......Page 263
7.4.3 An example: derivation of Gxx A for single-layer microstrip......Page 264
7.4.4 The scalar potential and the mixed potential integral equation......Page 266
7.4.5 Surface waves......Page 268
7.5.1 Approximate evaluation of the Sommerfeld integrals......Page 269
7.5.2 Numerical integration in the spectral domain......Page 271
7.5.3 Locating the pole......Page 280
7.5.5 Some results for the Sommerfeld potentials......Page 281
7.6 MoM solution using the Sommerfeld potentials......Page 282
7.7 Further reading......Page 290
References......Page 291
8.1 Printed antenna and microstrip technology: a brief review......Page 293
8.3 Mutual coupling between microstrip antennas......Page 295
8.4 An array with “scan blindness”......Page 302
8.5 A concluding discussion of stratified media formulations......Page 308
References......Page 309
9.1 Introduction......Page 311
9.2.1 The weighted residual approach......Page 313
The equivalent variational functional......Page 315
Connecting the elements......Page 317
Connecting the system......Page 320
9.2.4 More on variational functionals......Page 321
Boundary conditions at material interfaces......Page 323
9.2.6 Discussion......Page 324
9.3.1 Simplex coordinates in one, two and three dimensions......Page 325
9.3.2 Some properties of simplex coordinates......Page 326
9.4 The high-frequency variational functional......Page 327
9.5 Spurious modes......Page 328
9.6.1 An historical perspective......Page 331
9.6.2 Theory of vector elements......Page 332
9.6.3 Vector elements on triangles – the Whitney element......Page 335
9.7.1 The two-dimensional variational functional for homogeneous waveguide......Page 339
9.7.2 Explicit formula for the elemental matrix entries......Page 340
Edge and node numbering schemes......Page 343
Meshing......Page 345
Solving the eigenvalue problem......Page 346
Post-processing......Page 347
9.7.4 Results......Page 348
9.8 The three-dimensional Whitney element......Page 350
9.9 Further reading......Page 353
9.10 Conclusions......Page 354
References......Page 355
10 A selection of more advanced topics on the finite element method......Page 358
10.1.1 Complete versus mixed-order elements......Page 359
10.1.2 Hierarchal vector basis functions......Page 360
10.1.3 Properties of hierarchal basis functions......Page 362
10.1.4 Practical impact of higher-order basis functions in an FEM code......Page 364
10.2 The FEM from the variational boundary value problem viewpoint......Page 365
10.3.1 Introduction......Page 367
Formulation overview......Page 368
The waveguide formulation: another perspective......Page 370
Introduction......Page 371
Setting up the problem......Page 372
A rationale for complete basis functions......Page 375
Results......Page 376
10.5.1 Introduction......Page 380
10.5.2 Theoretical background......Page 381
10.6.1 Applications of FEM/MoM hybrid formulations......Page 384
10.6.2 Human exposure assessment near GSM base stations......Page 385
10.7 The time domain FEM......Page 387
Basic finite element formulation......Page 389
10.7.2 Preliminary results......Page 391
10.7.3 The FDTD method as a special case of the FETD......Page 393
10.8 Sparse matrix solvers......Page 394
10.8.1 Profile-in skyline storage......Page 395
10.8.2 Compressed row storage......Page 396
10.8.3 Implementation of matrix solution using these storage schemes......Page 397
10.8.4 Results for sparse storage schemes......Page 398
10.9 A posteriori error estimation and adaptive meshing......Page 400
10.9.1 Explicit, residual-based error estimators......Page 401
10.9.2 An example of the application of an error estimator......Page 402
10.10 Further reading and conclusions......Page 406
References......Page 408
Appendix A The Whitney element......Page 412
Appendix B The Newmark-β time-stepping algorithm......Page 414
References......Page 416
Appendix C On the convergence of the MoM......Page 417
Reference......Page 418
3D FDTD analysis......Page 419
Partial solution......Page 420
2D finite elements......Page 421
Reference......Page 422
Gradient......Page 423
Appendix F Web resources......Page 425
Index......Page 427