Explore the algorithms and numerical methods used to compute electromagnetic fields in multi-layered media
In Theory and Computation of Electromagnetic Fields in Layered Media, two distinguished electrical engineering researchers deliver a detailed and up-to-date overview of the theory and numerical methods used to determine electromagnetic fields in layered media. The book begins with an introduction to Maxwell’s equations, the fundamentals of electromagnetic theory, and concepts and definitions relating to Green’s function. It then moves on to solve canonical problems in vertical and horizontal dipole radiation, describe Method of Moments schemes, discuss integral equations governing electromagnetic fields, and explains the Michalski-Zheng theory of mixed-potential Green’s function representation in multi-layered media.
Chapters on the evaluation of Sommerfeld integrals, procedures for far field evaluation, and the theory and application of hierarchical matrices are also included, along with:
- A thorough introduction to free-space Green’s functions, including the delta-function model for point charge and dipole current
- Comprehensive explorations of the traditional form of layered medium Green’s function in three dimensions
- Practical discussions of electro-quasi-static and magneto-quasi-static fields in layered media, including electrostatic fields in two and three dimensions
- In-depth examinations of the rational function fitting method, including direct spectra fitting with VECTFIT algorithms
Perfect for scholars and students of electromagnetic analysis in layered media, Theory and Computation of Electromagnetic Fields in Layered Media will also earn a place in the libraries of CAD industry engineers and software developers working in the area of computational electromagnetics.
Table of Contents:
Contributors xix
Foreword xxi
Preface xxv
Acknowledgments xxvii
Acronyms xxxi
Introduction xxxv
1 Foundations of Electromagnetic Theory 1
1.1 Maxwell Equations 2
1.1.1 Time-domain Maxwell Equations in differential form 2
1.1.2 Frequency domain Maxwell Equations in differential form 4
1.1.3 Frequency-domain Maxwell Equations in lossy medium 6
1.2 Curl-curl equations for the electric and magnetic fields 7
1.3 Boundary conditions 8
1.4 Poynting Theorem 13
1.4.1 Time-domain Poynting theorem and instanteneous balance of power 13
1.4.2 Frequency-domain Poynting theorem and average balance of energy 15
1.5 Vector and scalar potentials 18
1.5.1 Magnetic vector potential Ae and electric scalar potential ‑e 18
1.5.2 Electric vector potential Am and magnetic scalar potential ‑m 20
1.6 Quasi-electrostatics Scalar potential Capacitance 23
1.7 Quasi-magnetostatics 25
1.7.1 Governing equations for potentials and fields in time domain 25
1.7.2 Governing equations for potentials and fields in frequency (spectral) domain 26
1.7.3 Energy definition of self-inductance, mutual inductance, and resistance 29
1.7.4 Field-based definition of self-inductance and mutual inductance 33
1.8 Theory of DC and AC circuits as a limiting form of Maxwell equations 35
1.9 Conclusions 40
2 Green’s Functions in Free Space 43
2.1 1D Green’s function 43
2.2 3D Green’s function expansion in Cartesian coordinates 48
2.3 3D Green’s function in cylindrical coordinates 50
2.4 Physical Interpretation of Conical Waves Forming Sommerfeld Identity 53
2.5 Integral field representation using Green’s function 57
2.6 Field Decomposition into TE- and TM-waves in Cartesian coordinates 58
2.7 Free-space dyadic Green’s functions of electric and magnetic field 61
2.8 Conclusions 65
3 Equivalence Principle and Integral Equations in Layered Media 67
3.1 Quasi-Electrostatics Reciprocity Relations in Layered Media 68
3.2 Equivalence Principle for the External Electrostatic Field in Layered Media 69
3.3 Integral Equation of Electrostatics for Metal Object in Layered Media 75
3.4 Integral Equation of Electrostatics for Disjoint Metal and Dielectric Objects in Layered Media 75
3.5 Integral Equation of Electrostatics for Metal and Dielectric Objects Sharing a Common Boundary
and Situated in Layered Media 81
3.6 Integral Equation of Electrostatics for Dielectric Objects Sharing a Common Boundary and Situated
in Layered Media 84
3.7 Integral Equations of Quasi-Magnetostatics for Wires in Layered Media 88
3.7.1 Matrix form of the MoM discretized SVS-EFIE 91
3.8 Full-Wave Reciprocity Relations in Layered Media 92
3.9 Integral representations of electromagnetic fields via equivalence principle 99
3.9.1 Equivalence Principle for the external electric field 99
3.9.2 Equivalence Principle for the external magnetic field 107
3.9.3 Equivalence Principle for the internal fields 110
3.10 Electric Field Integral Equation (EFIE) for PEC object in layered medium 113
3.11 Magnetic Field Integral Equation (MFIE) for PEC object 115
3.12 Coupled EFIEs for penetrable object 118
3.13 Coupled MFIEs for penetrable object 120
3.14 Muller, PMCHWT, and CFIE Formulations for Penetrable Object 121
3.15 Volume Integral Equation 124
3.16 Single Source Integral Field Representations and Integral Equations 126
3.17 Conclusions 129
4 Canonical Problems of Vertical and Horizontal Dipoles Radiation in Layered Media 131
4.1 The Electromagnetics of Dipole Currents in Open Planar Multi-Layered Media 131
4.2 Vertical electric dipole above half-space 132
4.3 Vertical Magnetic Dipole in layered media 136
4.3.1 VMD above half-space 136
4.3.1.1 Asymptotic behavior of electric vector potential at k! 1 137
4.4 Vertical magnetic dipole (VMD) in 3-layer medium 138
4.4.1 Ray Tracing Solution 139
4.4.2 The Fields of a Vertical Electric and Magnetic Dipoles: Spectral 1D BVP in General
Layered Media 142
4.5 Horizontal Electric Dipole in Layered Media 144
4.5.1 HED above PEC ground 144
4.5.2 Spatial boundary value problem for HED: Non-uniqueness of vector potential 145
4.5.3 HED spectral domain solution: 2-layer medium 150
4.5.4 HED spectral domain solution: 4-layer medium 155
4.5.5 The Fields of a Horizontal Electric and Magnetic Dipoles: Spectral 1D BVP in General
Layered Media 159
4.6 Integration Paths of Complex Plane k 162
4.6.1 Multi-Valued Sommerfeld Integrands 162
4.6.2 Integration Along Sommerfeld Integration Path and Its Modifications 167
4.6.2.1 Integration over the real axis of k 167
4.6.2.2 Integration on parametric path avoiding singularities and landing on real axis of k 168
4.7 Conclusions 170
5 Computation of Fields Via Integration Along Branch Cuts 171
5.1 Transformation of SIP to Integrals Along Banks of Branch Cuts 171
5.2 Parametrization of the Path Along Branch Cut Banks under 2p -Convention 177
5.3 Parametrization of the Path Along Branch Cut Banks under =2 p Convention 182
5.4 Surface Waves 184
5.4.1 Surface (Zenneck) Waves For VED about Half-Space (Sommerfeld Problem) 184
5.4.2 Analysis of Zenneck’s pole location under 2p branch cut convention 186
5.4.3 Analysis of Zenneck’s pole location under =2 p branch cut convention 190
5.4.4 On Sommerfeld Problem and Existence of Zenneck’s Wave 192
5.4.5 Guided and Leaky Surface Waves 197
6 Computation of Fields Via Integration Along Steepest Descent Path 203
6.1 Definition of Integrand and Spherical Wave SDP S1 206
6.2 Saddle Point on Plane kand SDP in Its Vicinity 208
6.3 Parametrization of Spherical Wave SDP S1 210
6.4 Crossing Point k= k1 sin on the SDP S1 214
6.5 Case 1: SDP S1 Switches Riemann Sheets after Crossing Branch Cut 215
6.5.1 Integral Circumventing Branch Point as Conical Wave SDP S2 218
6.5.2 Parametrization of Path S2 Around Branch Point 219
6.5.3 Analysis of Field Dependence on Radial Coordinate In SDP Integration Approach 222
6.6 Case 2: SDP S1 Remains on Same Riemann Sheet after Crossing Branch Cut 226
6.7 Final Remark on Numerical Integration Along SDP 227
6.8 Reflected Far Field from Saddle Point: Spherical Wave 227
6.9 Reflected Far Field from Branch Point: Lateral (Conical) Wave 229
6.9.1 Physical interpretation of lateral wave 235
7 Computation of Fields Via Angular Spectral Representation 237
7.1 Transformation of SIP to a path on complex plane of angles 237
7.2 Reflected field as integral on complex plane of angles 240
7.3 Modification of integration path on angles plane to the SDP 244
7.4 Accounting For Branch Cut and Surface Wave Poles in Integration Along SDP on Plane 248
7.4.1 Case 1: SDP S1 Switches to Different Riemann Sheet after Crossing Branch Cut 251
7.4.2 Case 2: SDP S1 Remains on Same Riemann Sheet after Crossing Branch Cut 254
7.5 Asymptotic Evaluation of SDP Integrals For k1R 1 255
7.5.1 Reflected Far Field: Spherical Wave 255
7.5.2 Transmitted Field: VED above Half-Space 258
8 Fields in Spherical Layered Media 265
8.1 Scalar Green’s Function in Spherical Coordinates 265
8.2 Electromagnetic Field in terms of Debye potentials 268
8.3 Radial Electric Dipole (RED) in Spherical Layered Media 271
8.4 Tangential Electric Dipole (TED) in Spherical Layered Media 277
8.5 Conclusions 285
9 Mixed Potential Integral Equation 287
9.1 Mixed Potential Integral Equations in Free Space 287
9.2 MPIE Formulation in Layered Medium 292
9.3 Reduction of 3D vector Maxwell’s equations to 1D scalar Telegraphers equations 305
9.4 Telegraphers equations for transmission line voltages and currents and their 1D Green’s functions 318
9.5 Relations of 3D Dyadic Green’s Functions to 1D Transmission Line Green’s Functions 319
9.6 Transmission line formulation of mixed-potential Green’s function components in formulation C 323
9.7 Closed-form expressions for voltages and currents in general layered medium 334
9.7.1 Generalized voltage reflection coefficients 335
9.7.2 Transmission line Green’s function V ejh i , when z is within the source section (the case of p = p0 ) 339
9.7.3 Transmission line Green’s function Iejh i , when z is within the source section (the case of p = p0 ) 346
9.7.4 Transmission line Green’s function V ejh v and Iejh
v , when z is within the source section (the case of p = p0 ) 347
9.7.5 Transmission line Green’s function V ejh and Iejh when z is outside the source section and z > z0 349
9.7.6 Transmission line Green’s function V ejh and Iejh when z is outside the source section and z < z0 352
9.8 Conclusions 355
10 Discretization of the MPIE with Shape Functions Based RWG MoM 357
10.1 MPIE with augmented vector potential dyadic Green’s function 357
10.2 Current expansion over RWG- and half-RWG (ramp) basis functions 358
10.3 Representation of MoM matrix elements in terms of shape function interactions 370
10.4 Delta-gap port model and pertinent discretization 377
10.5 Conclusions 387
11 Computation of Incident Field from Electric Dipole Situated in the Far Zone 389
11.1 Reciprocity theorem application 389
11.2 The method of stationary phase and Green’s function components KA;zz, when dipole is situated in the top layer 391
11.3 The Green’s function components KA;xx, when dipole is situated in the top layer 396
11.4 The Green’s function components KA;zt, when dipole is situated in the top layer 398
11.5 The Green’s function components KA;tz, when dipole is situated in the top layer 401
11.6 The Green’s function components rr0K', when dipole is situated in the top layer 403
11.7 Conclusions 409
12 Surface-Volume-Surface Electric Field Integral Equation 411
12.1 Surface-Volume Equivalence Principle Augmented with Single-Source Representations 411
12.2 SVS-VS-EFIE Formulation: SVS-EFIE Coupled to MPIE and VIE 415
12.3 Method of Moments Discretization of SVS-S-V-EFIE Operators 419
12.3.1 Discretization of the Vector Potential Volume-to-Volume Operator T V;V pp0;a in Layered Media 424
12.3.2 Discretization of the Vector Potential Surface-to-Volume Operator T V;V pp0;a in Layered Media 425
12.3.3 Discretization of the Scalar Potential Volume-to-Volume Operator T V;V pp0;r' in Layered Media 427
12.3.4 Discretization of the Scalar Potential Surface-to-Volume Operator T V;@V pp0;rin Layered Media 429
12.3.5 Assembly of MoM Matrix Elements ZV Lp R1 ;@V Lp0 R3 and ZV Lp R1 ;@V Lp0 Rr 431
12.4 Conclusions 434
13 Electromagnetic Analysis with Method of Moments in Shielded Layered Media 441
13.1 The Electromagnetics of Dipole Fields in Shielded Planar Multi-Layered Media 441
13.1.1 Magnetic Vector Potential of a Horizontal Electric Dipole 442
13.2 Electric and Magnetic Field Dyadic Green’s Functions in Shielded Layered Media 444
13.3 Decomposition of Fields into TEz- and TMz-waves 445
13.3.0.1 Example: HED in Rectangular Waveguide Shorted with Dielectric Grounded
Slab on One Side and Loaded with Free-Space Impedance on the Other 446
13.3.1 The Electric and Magnetic Fields of a Vertical Electric Dipole in Rectangular Waveguide
Filled with Layered Media 450
13.3.1.1 Example: VED in Rectangular Waveguide Shorted with Dielectric Grounded
Slab on One Side and Loaded with Free-Space Impedance on the Other 451
13.4 EFIE and Spectral Domain Method of Moments 454
13.4.1 Electric field integral equation 454
13.4.2 Method of Moment discretization on Manhattan grid 454
13.4.3 Closed-form expression for MoM matrix elements 456
13.4.4 Closed-form expression for Galerkin MoM matrix elements on regular 2D grid 457
13.5 Casting MoM matrix elements into DFT amenable form 459
13.6 Sums re-ordering for robust error-control and minimizing number of FFTs in MoM matrix fill 463
13.6.1 FFT based algorithm for fast error-controlled MoM matrix fill 464
13.7 Construction of Efficient MoM through Combining Basis Functions and Performing Fast Matrix
Vector Products 468
13.8 Space-domain Method of Moments with Manhattan gridded discretization 470
13.8.1 Efficient Calculation of Dipole Fields in Shielded Planar Multi-Layered Media 470
13.8.2 The Discrete Complex Image Method 471
13.8.3 Efficient numerical evaluation of the 4D MoM integrals in space domain 475
13.9 Conclusions 477
14 Method of Weighted Averages (Mosig-Michalski Extrapolation Algorithm) 479
14.1 Introduction 479
14.1.1 Partition of Sommerfeld Integrals on the Complex Plane 480
14.1.2 Finite Integral 481
14.1.3 Infinite Integral Partitioning 482
14.1.4 Sommerfeld Tail Integrals 484
14.1.4.1 Asymptotic Behaviour 484
14.1.4.2 Improper Tail Integrals in Riemann and Abel Sense 485
14.2 Classic First-Order Weighted Average Approximation 488
14.3 Recursive Weighted Average Algorithm 494
14.3.1 Application to Sommerfeld Integral Tails 494
14.3.1.1 General Procedure 494
14.3.1.2 Weights Selection Based on Remainder Estimate 497
14.3.1.3 Estimate for Remainders 497
14.3.1.4 Convergence Classification 502
14.3.1.5 Formulae for the weights 507
14.3.1.6 Residual Attenuation Throughout WA Recursion 510
14.3.1.7 Weights Ratio Throughout WA Recursion 512
14.3.2 Pseudo-Code Implementation of Recursive Weighted Average Algorithm 514
14.3.3 Pseudo-Code Implementation of Partition Extrapolation Algorithm 517
14.4 Conclusions 520
15 Extraction of Quasi-Static Images 521
15.1 Introduction 521
15.2 Prioritized Ray Tracing Algorithm 522
15.3 Static Images for voltages and currents 534
15.4 Static image contributions to Green’s function components in the Michalski-Zheng’s mixed-potential
form: Source and Observer Points are in the same layer 540 15.4.1 Component KA xx 540
15.4.2 Component K' 542
15.4.3 Component KA zz 545
15.4.4 Components KA xz and KA yz 549
15.4.5 Components KA zx and KA zy 555
15.5 Static image contributions to Green’s function components in the Michalski-Zheng’s mixed-potential
form: Source point layer is below observer point layer 559
15.5.1 Component KA
xx 561
15.5.2 Component K' 562
15.5.3 Component KA
zz 563
15.5.4 Components KA
xz and KA
yz 565
15.5.5 Components KA
zx and KA
zy 565
15.6 Static image contributions to Green’s function components in the Michalski-Zheng’s mixed-potential
form: Source point layer is above observer point layer 566
15.6.1 Components KA
xx, K', KA
zz, KA
xz, KA
yz, KA
zx and KA
zy 569
15.7 Conclusions 570
xv
16 Discrete Complex Image Method 573
16.1 Introduction 573
16.2 Complex exponentials fitting 574
16.3 Single-level DCIM 578
16.4 Two-level DCIM 580
16.5 Conclusions 583
17 Extraction of Singular Integrals from MoM Reaction Integrals and Their Analytic Evaluation 585
17.1 Source point and the observation point are in the same layer 586
17.2 Source layer below observation layer 592
17.3 Source layer above observation layer 593
17.4 Conclusions 594
18 Methods Based on Rational Function Approximation of Green’s Function Spectra 595
18.1 Rational Function Fitting Method (RFFM) 596
18.2 Spectral Differential Equations Approximation Method (SDEAM) for Vector Potential Green’s Function 603
18.2.1 VMD in Shielded Layered Media 604
18.2.2 VMD in Open Layered Media 607
18.3 SDEAM for Mixed-Potential Green’s Functions 610
18.3.1 Core 1D Spectral Boundary Value Problems 613
18.3.2 Pole-residual representation of the spectra through numerical solution of the core spectral
1D BVPs 615
18.3.2.1 Finite difference solution of spectral 1D BVP for voltage V h i 616
18.3.2.2 Finite element method solution of spectral 1D BVP for voltage V h
i 618
18.4 Higher-Order SDEAM Solutions and Their Error Bounds 622
18.5 Dependence in number of terms on radial distance 624
18.6 SDEAM for Spherical Layered Media 624
18.6.1 Radial electric dipole (RED) radiation 624
18.6.2 Tangential electric dipole (TED) radiation 628
18.7 Advantages of High-Order SDEAM for Spherical Layered Media 630
18.8 Conclusions 631
A MULTIVALUED COMPLEX FUNCTIONS, BRANCH CUTS, AND RIEMANN SURFACES 633
A.1 Multivalued Complex Functions, Branches, Branch Points, and Branch Cuts 633
A.1.1 Contour mapping from plane k to plane kz 641
A.1.2 Branch cut method for ensuring analyticity of multifunctions 647
A.1.3 Riemann surface representation of multifunctions 648
A.1.4 Practical considerations for evaluation of
q
k2 k2
directly versus as a product
p
k k
p
k + k
: 655
B EVALUATION OF SINGULAR INTEGRALS 659
B.1 Evaluation over Triangles of Integrals Containing ekR=R Green’s Function 659
B.2 Evaluation over Triangles of Integrals Containing Product of ekR=R Green’s Function and a Linear
Function 675
C Reduction of Cos-Cos Series to DFT 683
C.1 Cos-Cos series rearrangement 683
C.2 Casting Cos-Cos series into DFT form 685
D Properties of Vector Potential and its derivatives near a sheet of current 687
D.1 Vector potential near small disk 687
D.2 Tangential derivative of the vector potential 689
D.3 Second tangential derivative of the vector potential 690
D.4 Normal derivative of vector potential 691
D.5 Mixed second order derivative of vector potential over tangential coordinates 691
D.6 Mixed second order derivatives over x and z 692
E Basis definitions of dyadic, tensor, and operations with them 695
F Equivalence Principle for the External Electric Field in Free Space 701
G Physically Consistent Model for the Extraction of Conductance in lossy Dielectrics 707
H Alternative expression of Equivalence Principle for the external magnetic field 711
I Definition of inductance and resistance in frequency-domain 715
J Integral Equations of Electrostatics in Multi-Region Scenarios with Free-Space Green’s Functions719
J.1 Equivalence Principle for the External Electrostatic Field in Layered Media 719
J.2 Integral Equation of Electrostatics for Metal Object in Homogeneous Space 721
J.3 Integral Equation of Electrostatics for Disjoint Metal and Dielectric Objects 722
J.4 Integral Equation of Electrostatics for Metal and Dielectric Objects Sharing a Common Boundary
and Situated in Homogeneous Media 725
J.5 Integral Equation of Electrostatics for Dielectric Objects Sharing a Common Boundary and Situated
in Free Space 726
J.6 Method of Moments Solution of Electrostatic Integral Equations 730
About the Author :
Vladimir Okhmatovski, PhD, is Professor in the Department of Electrical and Computer Engineering at the University of Manitoba in Canada. His research is focused on fast algorithms of electromagnetics, high-performance computing, modeling of interconnects, and inverse problems.
Shucheng Zheng is pursuing a PhD degree in Electrical and Computer Engineering at the University of Manitoba. His current research interests include computational electromagnetics, multilayered media Green’s functions, high-performance computing, the modeling of high-speed interconnects, and transient analysis of power systems.
