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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE/OUP Series on Electromagnetic Wave Theory (formerly IEEE only), Series Editor: Donald G. Dudley.
- Sprache: Englisch
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.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Foreword
Preface
Acknowledgments
Acronyms
Introduction
1 Foundations of Electromagnetic Theory
1.1 Maxwell Equations
1.2 Curl–Curl Equations for the Electric and Magnetic Fields
1.3 Boundary Conditions
1.4 Poynting Theorem
1.5 Vector and Scalar Potentials
1.6 Quasi-Electrostatics. Scalar Potential. Capacitance
1.7 Quasi-Magnetostatics
1.8 Theory of DC and AC Circuits as a Limiting form of Maxwell Equations
1.9 Conclusions
Notes
2 Green’s Functions in Free Space
2.1 1D Green’s Function
2.2 3D Green’s Function Expansion in Cartesian Coordinates
2.3 3D Green’s Function in Cylindrical Coordinates
2.4 Physical Interpretation of Conical Waves Forming Sommerfeld Identity
2.5 Integral Field Representation Using Green’s Function
2.6 Field Decomposition into - and -waves in Cartesian Coordinates
2.7 Free-space Dyadic Green’s Functions of Electric and Magnetic Fields
2.8 Conclusions
Notes
3 Equivalence Principle and Integral Equations in Layered Media
3.1 Quasi-Electrostatics Reciprocity Relations in Layered Media
3.2 Equivalence Principle for the External Electrostatic Field in Layered Media
3.3 Integral Equation of Electrostatics for Metal Object in Layered Media
3.4 Integral Equation of Electrostatics for Disjoint Metal and Dielectric Objects in Layered Media
3.5 Integral Equation of Electrostatics for Metal and Dielectric Objects Sharing a Common Boundary and Situated in Layered Media
3.6 Integral Equation of Electrostatics for Dielectric Objects Sharing a Common Boundary and Situated in Layered Media
3.7 Integral Equations of Quasi-Magnetostatics for Wires in Layered Media
3.8 Full-Wave Reciprocity Relations in Layered Media
3.9 Integral Representations of Electromagnetic Fields via Equivalence Principle
3.10 Electric Field Integral Equation (EFIE) for PEC Object in Layered Medium
3.11 Magnetic Field Integral Equation (MFIE) for PEC Object
3.12 Coupled EFIEs for Penetrable Object
3.13 Coupled MFIEs for Penetrable Object
3.14 Muller, PMCHWT, and CFIE Formulations for Penetrable Object
3.15 Volume Integral Equation
3.16 Single-Source Integral Field Representations and Integral Equations
3.17 Conclusions
Notes
4 Canonical Problems of Vertical and Horizontal Dipoles Radiation in Layered Media
4.1 The Electromagnetics of Dipole Currents in Open Planar Multi-layered Media
4.2 Sommerfeld Problem: Vertical Electric Dipole Above Half-Space
4.3 Vertical Magnetic Dipole in Layered Media
4.4 Vertical Magnetic Dipole (VMD) in 3-Layer Medium
4.5 Horizontal Electric Dipole in Layered Media
4.6 Integration Paths of Complex Plane
4.7 Conclusions
Notes
5 Computation of Fields Via Integration Along Branch Cuts
5.1 Transformation of SIP to Integrals Along Banks of Branch Cuts
5.2 Parametrization of the Path Along Branch Cut Banks Under ‐Convention
5.3 Parametrization of the Path Along Branch Cut Banks Under Convention
5.4 Surface Waves
5.5 Conclusions
Notes
6 Computation of Fields Via Integration Along Steepest Descent Path
6.1 Definition of Integrand and Spherical Wave SDP
6.2 Saddle Point on Plane and SDP in Its Vicinity
6.3 Parametrization of Spherical Wave SDP
6.4 Crossing Point on the SDP
6.5 Case 1: SDP Switches Riemann Sheets After Crossing Branch Cut
6.6 Case 2: SDP Remains on Same Riemann Sheet After Crossing Branch Cut
6.7 Final Remark on Numerical Integration Along SDP
6.8 Reflected Far Field from Saddle Point: Spherical Wave
6.9 Reflected Far Field from Branch Point: Lateral (Conical) Wave
6.10 Conclusions
Notes
7 Computation of Fields Via Angular Spectral Representation
7.1 Transformation of SIP to a Path on Complex Plane of Angles
7.2 Reflected Field as Integral on Complex Plane of Angles
7.3 Modification of Integration Path on Angles Plane to the SDP
7.4 Accounting for Branch Cut and Surface Wave Poles in Integration Along SDP on Plane
7.5 Asymptotic Evaluation of SDP Integrals for
7.6 Conclusions
Notes
8 Fields in Spherical Layered Media
8.1 Scalar Green’s Function in Spherical Coordinates
8.2 Electromagnetic Field in Terms of Debye Potentials
8.3 Radial Electric Dipole (RED) in Spherical Layered Media
8.4 Tangential Electric Dipole (TED) in Spherical Layered Media
8.5 Conclusions
Note
9 Mixed-Potential Integral Equation
9.1 Mixed-Potential Integral Equations in Free Space
9.2 MPIE Formulation in Layered Medium
9.3 Reduction of 3D Vector Maxwell’s Equations to 1D Scalar Telegraphers Equations
9.4 Telegraphers Equations for Transmission Line Voltages and Currents and Their 1D Green’s Functions
9.5 Relations of 3D Dyadic Green’s Functions to 1D Transmission Line Green’s Functions
9.6 Transmission Line Formulation of Mixed-potential Green’s Function Components in Formulation C
9.7 Closed-form Expressions for Voltages and Currents in General Layered Medium
9.8 Conclusions
Notes
10 Discretization of the MPIE with Shape Functions-based RWG MoM
10.1 MPIE with Augmented Vector Potential Dyadic Green’s Function
10.2 Current Expansion Over RWG- and Half-RWG (Ramp) Basis Functions
10.3 Representation of MoM Matrix Elements in Terms of Shape Function Interactions
10.4 Delta-gap Port Model and Pertinent Discretization
10.5 Conclusions
Notes
11 Computation of Incident Field from Electric Dipole Situated in the Far Zone
11.1 Reciprocity Theorem Application
11.2 The Method of Stationary Phase and Green’s Function Components , When Dipole Is Situated in the Top Layer
11.3 Green’s Function Components , When Dipole Is Situated in the Top Layer
11.4 Green’s Function Components , When Dipole Is Situated in the Top Layer
11.5 Green’s Function Components , When Dipole Is Situated in the Top Layer
11.6 Green’s Function Components , When Dipole Is Situated in the Top Layer
11.7 Conclusions
12 Surface-Volume–Surface Electric Field Integral Equation
12.1 Surface–Volume Equivalence Principle Augmented with Single-Source Representations
12.2 SVS-VS-EFIE Formulation: SVS-EFIE Coupled to MPIE and VIE
12.3 Method of Moments Discretization of SVS-S-V-EFIE Operators
12.4 Conclusions
Note
13 Electromagnetic Analysis with Method of Moments in Shielded Layered Media
13.1 The Electromagnetics of Dipole Fields in Shielded Planar Multi-layered Media
13.2 Electric and Magnetic Field Dyadic Green’s Functions in Shielded Layered Media
13.3 Electric Field Integral Equation
13.4 Spectral Domain Method of Moment Discretization on Manhattan Grid
13.5 Space-Domain Method of Moments with Manhattan Gridded Discretization
13.6 Conclusions
Notes
14 Method of Weighted Averages (Mosig–Michalski Extrapolation Algorithm)
14.1 Introduction
14.2 Classic First-Order Weighted Average Approximation
14.3 Recursive Weighted Average Algorithm
14.4 Conclusions
15 Extraction of Quasi‐Static Images
15.1 Introduction
15.2 Prioritized Ray Tracing Algorithm
15.3 Static Images for Voltages and Currents
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
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
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
15.7 Conclusions
Note
16 Discrete Complex Image Method
16.1 Introduction
16.2 Complex Exponentials Fitting
16.3 Single-level DCIM
16.4 Two-level DCIM
16.5 Conclusions
17 Extraction of Singular Integrals from MoM Reaction Integrals and Their Analytic Evaluation
17.1 Source Point and the Observation Point Are in the Same Layer
17.2 Source Layer Below Observation Layer
17.3 Source Layer Above Observation Layer
17.4 Conclusions
18 Methods Based on Rational Function Approximation of Green’s Function Spectra
18.1 Rational Function Fitting Method (RFFM)
18.2 Spectral Differential Equations Approximation Method (SDEAM) for Vector Potential Green’s Function
18.3 SDEAM for Mixed-Potential Green’s Functions
18.4 Higher-Order SDEAM Solutions and Their Error Bounds
18.5 Dependence on Number of Terms on Radial Distance
18.6 SDEAM for Spherical Layered Media
18.7 Advantages of High-Order SDEAM for Spherical Layered Media
18.8 Conclusions
Notes
Appendix A: Multivalued Complex Functions, Branch Cuts, and Riemann Surfaces
A.1 Multivalued Complex Functions, Branches, Branch Points, and Branch Cuts
Notes
Appendix B: Evaluation of Singular Integrals
B.1 Evaluation Over Triangles of Integrals Containing Green’s Function
B.2 Evaluation Over Triangles of Integrals Containing Product of Green’s Function and a Linear Function
Appendix C: Reduction of Cos–Cos Series to DFT
C.1 Cos–Cos Series Rearrangement
C.2 Casting Cos–Cos Series into DFT Form
Appendix D: Properties of Vector Potential and Its Derivatives Near a Sheet of Current
D.1 Vector Potential Near Small Disk
D.2 Tangential Derivative of the Vector Potential
D.3 Second Tangential Derivative of the Vector Potential
D.4 Normal Derivative of Vector Potential
D.5 Mixed Second-order Derivative of Vector Potential Over Tangential Coordinates
D.6 Mixed Second-order Derivatives Over and
Appendix E: Basis Definitions of Dyadic, Tensor, and Operations with Them
Appendix F: Equivalence Principle for the External Electric Field in Free Space
Note
Appendix G: Physically Consistent Model for the Extraction of Conductance in Lossy Dielectrics
Appendix H: Alternative Expression of Equivalence Principle for the External Magnetic Field
Appendix I: Definition of Inductance and Resistance in Frequency Domain
Appendix J: Integral Equations of Electrostatics in Multi-Region Scenarios with Free-Space Green’s Functions
J.1 Equivalence Principle for the External Electrostatic Field in Layered Media
J.2 Integral Equation of Electrostatics for Metal Object in Homogeneous Space
J.3 Integral Equation of Quasi-Electrostatics for Disjoint Metal and Dielectric Objects
J.4 Integral Equation of Quasi-Electrostatics for Metal and Dielectric Objects Sharing a Common Boundary and Situated in Homogeneous Media
J.5 Integral Equation of Quasi-Electrostatics for Dielectric Objects Sharing a Common Boundary and Situated in Free Space
J.6 Method of Moments Solution of Electrostatic Integral Equations
Note
References
Index
IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY
End User License Agreement
List of Tables
10
Table J.1 Capacitance matrix for three-conductor MTL.
List of Illustrations
Preface
Figure 1 Dr. S. Zheng and Prof. V. Okhmatovski at 2022 University of Manitob...
Chapter 1
Figure 1.1 (a) Sub-figure shows cylindrical surface with the top and bottom ...
Figure 1.2 Depiction of electrostatic boundary value problem setups for co...
Figure 1.3 Quasi-magnetostatic boundary value problem setups for realistic
Figure 1.4 Mock-up models of 12-bond wire electronic package demonstrating t...
Figure 1.5 Distributed circuit formed by an open-ended conductor of volume
Figure 1.6 Kirchhoff’s current law as a manifestation of the continuity equa...
Chapter 2
Figure 2.1 Depiction of Dirac’s -function. The height of the pulse tends ...
Figure 2.2 Polar coordinate system of the -space of wavenumbers with positi...
Figure 2.3 Contour plots of magnitude of Hankel function on complex plane
Chapter 3
Figure 3.1 Derivation of the reciprocity relations in layered media for quas...
Figure 3.2 Depiction of the region of layered media and volume used in der...
Figure 3.3 Extraction of capacitance between the nets of IBM Plasma electron...
Figure 3.4 Derivation of integral equation for non-touching conductor and di...
Figure 3.5 Evaluation of the principal value of the singular integral: (a) 3...
Figure 3.6 Derivation of integral equation for the case, when conductor an...
Figure 3.7 To derivation of integral equation for touching dielectrics of di...
Figure 3.8 (a) Interconnect formed by two piece-wise-straight conductor segm...
Figure 3.9 Derivation of the reciprocity relations in layered media for full...
Figure 3.10 Derivation of the external equivalence principle in layered medi...
Figure 3.11 Derivation of the external equivalence principle for the magneti...
Chapter 4
Figure 4.1 Generic layered medium of layers.
Figure 4.2 Vertical dipole radiating in 2-layer non-magnetic medium.
Figure 4.3 Vertical dipole radiating in 3-layer non-magnetic medium.
Figure 4.4 Ray-tracing solution in a 3-layer medium. The tracing is 1D with ...
Figure 4.5 Depiction of electric dipole of arbitrary orientation and dipol...
Figure 4.6 The problem of HED radiation in the presence of 3-layer media wit...
Figure 4.7 HED radiating in the presence of half-space.
Figure 4.8 The problem of HED radiation in the presence of 5-layer media wit...
Figure 4.9 Spectrum versus spectral variable at frequencies GHz (a) an...
Figure 4.10 Spectrum versus spectral variable at frequencies GHz (a) a...
Figure 4.11 Four-sheet Riemann surface of 4-valued reflection coefficient ...
Figure 4.12 Modification of the original SIP in Fourier–Bessel transform (4....
Figure 4.13 Modification of the SIP into the path circumventing the branch p...
Figure 4.14 Modification of the SIP into the path circumventing the branch p...
Chapter 5
Figure 5.1 Two‐sheet Riemann surface of reflection coefficient in (4.21) u...
Figure 5.2 Physical sheet of the Riemann surface (a) and closed‐loop pat...
Figure 5.3 Mapping of the closed path of integration and its distinct segm...
Figure 5.4 Two‐sheet Riemann surface of reflection coefficient under con...
Figure 5.5 Physical sheet of the Riemann surface (a) and closed‐loop pat...
Figure 5.6 Mapping of the closed path of integration and its distinct segm...
Figure 5.7 Magnitude of the integrands in the SIs in (5.18) as a function of...
Figure 5.8 Magnitude of the integrands in the SIs in (5.18) along the real a...
Figure 5.9 Depiction of the branch cuts and and Zenneck’s pole on the ...
Figure 5.10 Snapshot at time s of magnitude of time‐harmonic magnetic vect...
Figure 5.11 Magnitude of magnetic vector potential , thick grey and black ‘...
Figure 5.12 Magnitude of the magnetic vector potential as a function of el...
Figure 5.13 Snapshot at time s of magnitude of time‐harmonic magnetic vect...
Figure 5.14 Graphical solution of coupled equations (5.78) and (5.79). Here
Figure 5.15 Snapshot at time s of magnitude of time‐harmonic magnetic field...
Figure 5.16 Snapshots at time s of time‐harmonic magnetic field components
Chapter 6
Figure 6.1 Formation of closed-loop integration path through augmentation of...
Figure 6.2 Depictions of vertical dipole (electric or magnetic) situated in ...
Figure 6.3 Behavior of real part of in the vicinity of its saddle point ...
Figure 6.4 Behavior of imaginary part of in the vicinity of its saddle poi...
Figure 6.5 Steepest descent path and its change with observation angle . ...
Figure 6.6 SDP with and without SDP circumventing branch point . SDPs
Figure 6.7 Dependence of electric vector potential of reflected field on r...
Figure 6.8 SDP with segments and circumventing branch point (thin bl...
Chapter 7
Figure 7.1 Mapping of the original SIP on complex plane (a) to the integ...
Figure 7.2 Path on angle plane , which the SIP maps to, in the case of si...
Figure 7.3 Regions of convergence (patterned grey (green in color version) a...
Figure 7.4 SDP with (at ) and without (at ) SDP circumventing branch p...
Figure 7.5 SDP with (at ) and without (at ) SDP circumventing branch p...
Figure 7.6 Dependence of electric vector potential of reflected field on r...
Figure 7.7 Depiction of the plane waves propagation path from source point
Figure 7.8 SDP with (at ) and without (at ) SDP circumventing branch p...
Figure 7.9 SDP with (at ) and without (at ) SDP circumventing branch p...
Chapter 8
Figure 8.1 Spherical coordinate system of the -space of wavenumbers with po...
Figure 8.2 Radial electric dipole radiating in the presence of PEC sphere co...
Figure 8.3 Visualization on grid of points in -plane of Debye potential
Figure 8.4 Tangential electric dipole radiating in the presence of PEC spher...
Chapter 9
Figure 9.1 PEC body of an arbitrarily shaped volume with surface current
Figure 9.2 Illustration of vector relations in the surface Gauss theorem (Eq...
Figure 9.3 Gauss divergence theorem applied to the half-sphere surfaces.
Figure 9.4 PEC object of an arbitrary shape situated in a multilayered mediu...
Figure 9.5 Surface of an object crossing two neighboring interfaces of a lay...
Figure 9.6 Contour formed at the intersection of the object surface and laye...
Figure 9.7 The rotated spectral domain (k-space) coordinate system.
Figure 9.8 Transmission line section in derivation of Green’s function.
Figure 9.9 Cascaded connection of three uniform lossless transmission line s...
Figure 9.10 Transmission line excited by 1 shunt current source at in sec...
Figure 9.11 Transmission line excited by current wave at traveling to th...
Figure 9.12 Transmission line excited by current wave at traveling to th...
Figure 9.13 Transmission line excited by 1 V series voltage source at in s...
Figure 9.14 Transmission line section with left terminal at and right te...
Figure 9.15 Voltage at any point in the section when outside the sou...
Figure 9.16 Transmission line section with left terminal at and right te...
Figure 9.17 Voltage at any point in the section when outside the sou...
Chapter 10
Figure 10.1 RWG basis function associated with the th edge of length in...
Figure 10.2 Definitions of the radial vector on the “positive” triangle an...
Figure 10.3 Example of mesh on an open geometry (thin strip) consisting of f...
Figure 10.4 Geometry of a single triangular mesh element with definition of ...
Figure 10.5 The physical dipole antenna excitation model.
Figure 10.6 PEC-filled gap with magnetic frill current wrapped around it....
Figure 10.7 Contour of integration circumventing the cross-section of the ma...
Figure 10.8 Excitation model after the -gap is filled with PEC material and...
Figure 10.9 Depiction of two triangles forming RWG basis function connected ...
Figure 10.10 Triangles of RWG function from Figure 10.9 attached to the port...
Figure 10.11 Depiction of the RWG basis function normal component.
Figure 10.12 The physical electrode excitation model.
Figure 10.13 Triangle attached to the port contour in case of the microstrip...
Chapter 11
Figure 11.1 Plot of the power of exponent (a, c) and the highly oscillatory ...
Figure 11.2 Unit vectors in cylindrical and Cartesian coordinate system.
Chapter 12
Figure 12.1 Scattering problem of a general 3-D composite metal–dielectric o...
Figure 12.2 (a) Derivation of SVS-S-V-EFIE formulation for the scattering of...
Figure 12.3 SLAE obtained from MoM discretization of the SVS-S-V-EFIE formul...
Figure 12.4 Model of 44m Manitoba Hydro tower A-201-A with 4m deep grounding...
Figure 12.5 Magnitude and phase of reflection coefficient for the power to...
Figure 12.6 A LTCC diplexer model is embedded in the seven layer non-magneti...
Figure 12.7 (a) Research icebreaker
CCGS Amundsen
and its depiction on Canad...
Chapter 13
Figure 13.1 Generic layered media situated inside rectangular waveguide with...
Figure 13.2 Example of layered medium inside rectangular waveguide used in s...
Figure 13.3 Single metallization layer interconnect circuit printed on top l...
Figure 13.4 Gridded microstrip geometry non-conformal to uniform grid (a) an...
Figure 13.5 Depiction of the gridded geometry subsectioning agglomerated int...
Figure 13.6 Example of PCB-level interconnect simulation featuring unknown...
Chapter 14
Figure 14.1 Integration path for computation of Sommerfeld integral.
Figure 14.2 Asymptotic periodicity of with respect to variable .
Figure 14.3 Subinterval length is chosen according to the level of attenua...
Figure 14.4 Plot of the imaginary part in (14.14).
Figure 14.5 Scheme representing the procedure of applying first-order weight...
Figure 14.6 Triangular scheme of the procedure for estimation of the integra...
Figure 14.7 Cosine contribution into asymptotic behavior of the Bessel fun...
Figure 14.8 Exponential behavior of convergence classification parameter.
Figure 14.9 Triangular scheme for the first iteration () of the recursive p...
Figure 14.10 Triangular scheme for the second iteration () of the recursive...
Figure 14.11 Triangular scheme for the th iteration of the recursive proced...
Chapter 15
Figure 15.1 The source and observation points reside on the same level ()....
Figure 15.2 Systematic tracing of elementary reflections and transmissions i...
Figure 15.3 Generic scenario with the layer containing the source point situ...
Figure 15.4 Splitting of the source and observation layers for systematic tr...
Figure 15.5 Generic scenario with the layer containing the source point situ...
Figure 15.6 Splitting of the source and observation layers for systematic tr...
Figure 15.7 Three‐layer configuration used in demonstration of the image ext...
Figure 15.8 The magnitude of spectral domain Green’s function with and wit...
Chapter 16
Figure 16.1 Uniform sampling of the interval .
Figure 16.2 DCIM path on and complex planes.
Figure 16.3 DCIM paths and on complex plane.
Figure 16.4 The magnitude of spectral domain Green’s function obtained fro...
Chapter 17
Figure 17.1 Location of image in the term ***(I).
Figure 17.2 Location of image in the term ***(II).
Figure 17.3 Location of image in the term ***(III).
Figure 17.4 Location of image in the term ***(VI).
Chapter 18
Figure 18.1 Magnitude of vector fitting and analytical results of Green’s fu...
Figure 18.2 Magnitude of vector fitting and analytical results of Green’s fu...
Figure 18.3 RFFM-VECTFIT approximation for the spatial Green’s function comp...
Figure 18.4 RFFM-VECTFIT approximation for the spatial Green’s function comp...
Figure 18.5 Distribution of poles on the complex plane computed with VEC...
Figure 18.6 Distribution of poles on the complex plane computed with VEC...
Figure 18.7 Forward wave, backward wave, and their superposition.
Figure 18.8 DCIM and RFFM-VECTFIT approximation for the spatial Green’s func...
Figure 18.9 Structure SDEAM matrix equation in case of layered medium shield...
Figure 18.10 Rational function approximation (18.21) of the inverse square r...
Figure 18.11 Structure of SDEAM matrix equation for the case of excitation b...
Figure 18.12 Magnitude of the vector potential versus distance from the dipo...
Figure 18.13 A layered medium shielded by two perfectly conducting planes (b...
Figure 18.14 Scalar and vector potential Green’s functions versus radial coo...
Figure 18.15 Poles and contours on the complex plane (plot is from (Okhmat...
Figure 18.16 Number of terms with dominant contribution to the series in (49...
Appendix A
Figure A.1 Depiction of two-valued function mapping each value to two va...
Figure A.2 Depiction of function mapping two values and to the same va...
Figure A.3 Partitioning of one complex plane to two complex planes and
Figure A.4 Mapping of complex planes and to plane under the -conventi...
Figure A.5 Mapping of complex planes and to plane under the -conventi...
Figure A.6 Mapping of distinct regions of complex planes and to plane ...
Figure A.7 Mapping of complex planes and to plane under the -conventi...
Figure A.8 Mapping of the closed contour not encircling point or from ...
Figure A.9 Mapping of the closed contour encircling point from complex p...
Figure A.10 Mapping of the closed contour encircling point from complex ...
Figure A.11 Mapping of the closed contour not encircling point or crossi...
Figure A.12 Mapping from complex plane to complex plane of the closed co...
Figure A.13 Mapping from complex plane to complex plane of the closed co...
Figure A.14 Restriction the and values to their branch 1 under -convent...
Figure A.15 Restriction the and values to their branch 1 under -convent...
Figure A.16 Visualization of Riemann surface of multifunction in the form ...
Figure A.17 Visualization of Riemann surface of multifunction in the form ...
Figure A.18 Riemann surface representing multifunction as a 4D object (sub...
Figure A.19 Riemann surface representing multifunction as a 4D object (sub...
Figure A.20 Riemann surface representing multifunction as a 4D object (sub...
Figure A.21 Depiction of vectors and for different value of on the lin...
Appendix B
Figure B.1 Geometry of a flat triangular element and related vector introduc...
Figure B.2 Definitions of the vectors in the surface divergence theorem for ...
Figure B.3 Areas of small disk removed from the triangle element for various...
Figure B.4 Orthogonal projection of vector onto the surface of the triangl...
Figure B.5 Contour integration paths after application of the surface diverg...
Figure B.6 Depiction of the angles of the full small circle arcs when observ...
Figure B.7 Depiction of the angles of partial small circle arcs when observa...
Figure B.8 Depiction of the angles when observation point projection is ou...
Figure B.9 Plane of the triangle and radial function emanating from th ve...
Figure B.10 Derivatives of function .
Figure B.11 Small circle area centered at the observation point located ...
Appendix D
Figure D.1 Small disk of a radius
a
carrying current for determining local...
Appendix F
Figure F.1 Derivation of the external equivalence principle in free space.
Appendix G
Figure G.1 Fitted Debye model for alumina. Source: Data from Pastol (1989)....
Appendix I
Figure I.1 Quasi-magnetostatic boundary value problem setups for realistic c...
Appendix J
Figure J.1 Depiction of the region of homogeneous space with permittivity ...
Figure J.2 Derivation of integral equation for non-touching conductor and di...
Figure J.3 Derivation of integral equation for, the case when conductor in v...
Figure J.4 Derivation of integral equation for touching dielectrics of diffe...
Figure J.5 Cross-section of three-conductor MTL.
Figure J.6 Equipotential contour lines of the electric potential for MTL in ...
Figure J.7 Equipotential contour lines of the electric potential for MTL in ...
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Dedication
About the Authors
Foreword
Preface
Acknowledgments
Acronyms
Introduction
Begin Reading
Appendix A Multivalued Complex Functions, Branch Cuts, and Riemann Surfaces
Appendix B Evaluation of Singular Integrals
Appendix C Reduction of Cos–Cos Series to DFT
Appendix D Properties of Vector Potential and Its Derivatives Near a Sheet of Current
Appendix E Basis Definitions of Dyadic, Tensor, and Operations with Them
Appendix F Equivalence Principle for the External Electric Field in Free Space
Appendix G Physically Consistent Model for the Extraction of Conductance in Lossy Dielectrics
Appendix H Alternative Expression of Equivalence Principle for the External Magnetic Field
Appendix I Definition of Inductance and Resistance in Frequency Domain
Appendix J Integral Equations of Electrostatics in Multi-Region Scenarios with Free-Space Green’s Functions
References
Index
IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY
End User License Agreement
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IEEE Press445 Hoes LanePiscataway, NJ 08854
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Thomas Robertazzi
Diomidis Spinellis
Theory and Computation of Electromagnetic Fields in Layered Media
Vladimir Okhmatovski
University of Manitoba
Winnipeg, Canada
Shucheng Zheng
University of Manitoba
Winnipeg, Canada
Copyright © 2024 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data
Names: Okhmatovski, Vladimir, author. | Zheng, Shucheng, author.
Title: Theory and computation of electromagnetic fields in layered media /
Vladimir Okhmatovski, Shucheng Zheng.
Description: Hoboken, New Jersey : Wiley-IEEE Press, [2024] | Series: IEEE
Press series on electromagnetic wave theory | Includes index.
Identifiers: LCCN 2023040083 (print) | LCCN 2023040084 (ebook) | ISBN
9781119763192 (cloth) | ISBN 9781119763208 (adobe pdf) | ISBN
9781119763215 (epub)
Subjects: LCSH: Electromagnetic fields. | Layer structure (Solids)
Classification: LCC QC665.E4 O55 2024 (print) | LCC QC665.E4 (ebook) |
DDC 621.38101/537–dc23/eng/20231016
LC record available at https://lccn.loc.gov/2023040083
LC ebook record available at https://lccn.loc.gov/2023040084
Cover Image and Design: Wiley
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About the Authors
Vladimir Okhmatovski received the MS degree in Radiophysics and PhD degree in Antennas and Microwave Circuits from the Moscow Power Engineering Institute, Moscow, Russia, in 1996 and 1997, respectively. He was a Post-Doctoral Research Associate with the National Technical University of Athens from 1998 to 1999 and with the University of Illinois at Urbana-Champaign from 1999 to 2003. From 2003 to 2004, he was with the Department of Custom Integrated Circuits, Cadence Design Systems, as a Senior Member of Technical Staff. In 2004, he joined the Department of Electrical and Computer Engineering at the University of Manitoba, where is currently a Full Professor. His research interests are in the fast algorithms of electromagnetics, high-performance computing, modeling of interconnects, and inverse problems. Dr. Okhmatovski was a recipient of the 2017 Intel Corporate Research Council Outstanding Researcher Award. He was also a recipient of Outstanding ACES Journal Paper Award in 2007, Best Paper Award at the 3rd Electronic Packaging Technology Conference in 2001, and 1996 Best Young Scientist Report of the VI International Conference on Mathematical Methods in Electromagnetic Theory. He is a Registered Professional Engineer in the Province of Manitoba, Canada.
Shucheng Zheng, PhD, is Post-Doctoral Fellow in the Department of Electrical and Computer Engineering at the University of Manitoba in Canada. His current research interests include computational electromagnetic, multilayered media Green’s functions, high-performance computing, fast algorithms, the modeling of high-speed interconnects, and transient analysis of power systems.
Foreword
Theory and Computation of Electromagnetic Fields in Layered Media is a worthy and coherent addition to the impressive IEEE Press Series on Electromagnetic Wave Theory. Layered (or stratified) media have been one of the most relevant topics in electromagnetics for more than one century, since the researches of AJW Sommerfeld and his pioneering 1909 Annalen der Physik paper. Then, the now well-known Sommerfeld integrals, mathematically equivalent to Fourier–Bessel (Hankel) transforms, were introduced for the first time as a convenient mathematical expression for the fields and potentials created by a point source embedded in multilayered media. The theoretical work of Sommerfeld was completed (and sometimes hotly debated) in the next decades, culminating in the publication of a long series of books and monographs spanning the second half of the twentieth century (Brekhovskikh, 1960; Wait, 1962; Banos, 1966; Felsen and Marcuvitz, 1973; Kong, 1986; Chew, 1990; Ishimaru, 1991). Most of them are now technical literature classics, and many have been reprinted in this same IEEE Press Series. They firmly established the so-called spectral approach for the analytical computation of stratified media Green’s functions, formulated as combination of Sommerfeld integrals.
While stratified media arise naturally on Earth and a suitable mathematical description is needed when dealing with subjects like ionospheric propagation, geological remote sensing, or ground-penetrating radar, their study remained quite limited and confidential. The thrust for a renewal of interest in the subject was provided by a technological breakthrough: in the 1960s the introduction of microwave planar technologies and printed transmission lines, like the stripline and the microstrip, soon followed in the 1970s by planar components and microstrip antennas, which became ubiquitous elements in RF and microwave technologies. And these circumstances have been enhanced in the twenty-first century by the dramatic growth of integrated optics and photonics.
Suddenly, there was a need to perform a rigorous analysis of these components, which are able to provide accurate predictions for practical quantities like resonant frequencies, scattering parameters, and radiation patterns. Stratified media theory was the sought-after approach to deal with these multilayered structures.
In parallel, there was an intense theoretical activity toward the establishment of integral equation formulations in electromagnetics, both based on fields and on potentials. Pioneering formulations of EFIE and MFIE types (using the electric and magnetic fields, respectively, as unknowns) were established by Poggio and Miller (1973) and ended in the well-known PMCHWT integral equation and its many variants. As for the potentials, the so-called mixed potential integral equation (MPIE) was first used by Harrington (1968), in the frame of wire antennas and scatterers in free space. Its generalization to arbitrary multilayered structures was the subject of a highly cited journal paper by Michalski and Zheng (1990). An important result was that, with minor changes fully explained in this book, the free-space MPIE could be used for multilayered media problems, by just replacing the free-space Green’s functions by the adequate Sommerfeld integrals.
The above-mentioned theoretical developments happened during the dawn of the computer era, and numerical implementations were quickly developed to obtain useful results. Integral equations required to be discretized, and Green’s functions and associated Sommerfeld integrals needed a precise numerical evaluation, either based on some direct quadrature approach, on sophisticated deformations of the integration path in some complex plane, or on a combination of both.
In electromagnetics, the preferred discretization strategy was the so-called method of moments (MoM), a boundary element method transforming the integral equation into a matrix equation, developed by Harrington in 1968. The MoM reached full flexibility thanks to the introduction in 1982 by Rao, Wilton, and Glisson (RWG) of the triangular rooftop basis functions. Since then, most successful computer implementations, including many commercial CEM codes, use some version of these RWG basis functions. As for the numerical integration of Sommerfeld integrals, the first serious attempts also go back to the 1970s in combination with the Lawrence Livermore Lab NEC code (Burke and Poggio, 1977), still widely available. Since then, several successful numerical techniques, like weighted averages, complex images, and rational function fitting methods (RFFM), were used, but their description was until now scattered among some obscure monographs and original research papers in technical journals.
As evident from its contents, this book covers in its more than 700 pages, all the above theoretical and numerical developments, summarizing more than one century of electromagnetic layered media research. The three first chapters review basic electromagnetic theory, tailored to the goal of obtaining the pertinent integral equation formulations. Then the theoretical study of Sommerfeld integrals is thoroughly covered in Chapters 4–7. The computational sides, centered on the MPIE and its solution with MoM techniques, can be found in Chapters 10–12 and 18. An exhaustive coverage of efficient techniques for the numerical evaluation of Green’s functions and Sommerfeld integrals is given in Chapters 14–16. The book included also interesting collateral subjects like spherical (Chapter 8) and shielded (Chapter 13) layered media.
The treatment is both rigorous and exhaustive. All the relevant topics are covered, from an illuminating discussion of the classical (first discussed in 1907!) subject of the Zenneck wave to an exhaustive treatment of the most recent computational developments. There is no lack of original contributions like the introduction of an error-controlled spectral differential equations approximation method (SDEAM) for Green’s functions. Also worth mentioning is the seamless extension to problems including composite metal–dielectric structures with inhomogeneous properties, thanks to a coupled system of integral equations. Didactic material has not been forgotten, like the provided pseudocodes, allowing an immediate computer implementation.
As such, this encyclopedic book is an invaluable and essential source of advanced knowledge, combining tutorial and advanced research aspects. It will be treasured and frequently used, both by students learning the trade and by scientists and engineers concerned with the electromagnetic study of layered media.
Juan R. Mosig, Prof. Emeritus
Electrical Engineering Department
Ecole Polytechnique Fédérale de Lausanne
Lausanne, VD, Switzerland
28 February 2023
Preface
The analysis of electromagnetic fields in layered media is an old but contemporary topic of research in theoretical and computational electromagnetics alike. The interest in these theories and methods has been fueled by a plethora of important practical applications involving objects embedded in layered media. Such applications range from remote sensing in geoscience, to the design of high-speed digital interconnects, analysis of transients in power systems, and beyond. As the research and development on computational tools for such analysis of electromagnetic fields in layered media progressed over the years, it became a necessity to have an organized systematic description of the theoretical foundation and computational techniques going into the construction of such tools based on numerical solutions of pertinent integral equations of electromagnetics. With this book Prof. Okhmatovski and Dr. Zheng (Figure 1) intent to offer the electromagnetics community such a systematic description.
The theories covering the mathematical foundation of the layered media Green’s functions and Sommerfeld integrals, as well as integral equations utilizing them, in many ways overlap with descriptions offered in excellent monographs and original research papers written on the topics by Weng Chew (1999), James Wait et al. (1982), Krzysztof Michalski and Mosig (1997), Juan Mosig and Gardiol (1983), and others. The emphasis in this book compared to the existing literature is two fold. First is to complement the existing literature with textbook-level description of the numerical methods commonly used in practice for evaluation of the layered media Green’s functions and discretization of the integral equations. Second is to make theoretical description understandable and accessible not only to experts but also to graduate students and junior engineers working on development of layered media solvers in the electromagnetic CAD industry.
The selection of numerical techniques for the evaluation of the layered media Green’s functions covered in this book was driven by their popularity and authors’ own areas of expertise. It includes the method of weighted averages, discrete complex image method, and methods based on rational functions representation of the spectra, allowing for analytic evaluation of the Sommerfeld integrals.
Figure 1 Dr. S. Zheng and Prof. V. Okhmatovski at 2022 University of Manitoba Fall Convocation.
Derivations of the integral equations in this book are done using the first principles of the equivalence theorem and reciprocity in layered media. The numerical approaches to their solutions are predominantly based on Michalski–Zheng’s mixed-potential formulation with the exception of the chapter covering electromagnetic analysis in shielded layered media.
Vladimir Okhmatovski
Winnipeg, Canada
26 August 2022
Acknowledgments
Profound heartfelt thanks go to my teachers and mentors who introduced me to the field of computational electromagnetics (CEM) at large and method of integral equations within it. These are my PhD advisor, late Prof. Evgeniy Nikolaevich Vasil’ev, who taught me electromagnetics at the Moscow Power Engineering Institute from 1993 to 1997 and instilled passion for the field in me; Profs. Nicolas Uzunoglu and Konstantina Nikita who supervised me at National Technical University of Athens in 1998–1999; Prof. Andreas C. Cangellaris, who brought me to the Center of Computational Electromagnetics at the University of Illinois at Urbana-Champaign (a.k.a. “Mecca for Electromagnetics”), where he taught and mentored me in the analysis of fields in layered media, theory of Green’s functions, and fast algorithms for CEM from 1999 to 2003; Dr. Feng Ling, who is a long-time friend and was my mentor during our joint work on layered media fast integral equation solver at Neolinear and Cadence Design Systems from 2003 to 2006.
I also would like to express my deep gratitude to my graduate students Shucheng Zheng, Brian Rautio, Xinbo Li, Anton Menshov, Jamiu Mojolagbe, Farhad Sheikh Lori Hossein, Sikander Hossein, Jonatan Aronsson, Ian Jeffrey, Khalid Butt, and Mohammed Al-Qedra. They worked tirelessly during their graduate studies on various aspects of field theory in layered media and significantly contributed to my understanding of the methods described in this book.
Implementation of the methods described in this book would be impossible without the generous support of the industry partners who collaborated with my research group. This includes Dr. Robert Hammond and Dr. Neil Fenzi from Resonant Inc, Dr. Aykut Dengi and Dr. Rodney Phelps from Cadence Design Systems, Dr. James Rautio from Sonnet Software, Dr. Kemal Aygun and Dr. Henning Braunisch from Intel, and Dr. Darshana Muthumuni from Manitoba HVDC Research Center.
My deep appreciation also goes to my collaborators on the research projects related to analysis of fields in layered media, including Prof. Ali Yilmaz from University of Texas at Austin who collaborated with me on development of fast integral equation layered media solvers for analysis of high-speed interconnects, Prof. Behzad Kordi from University of Manitoba, who was my collaborator on electromagnetic analysis of power systems, and Prof. Dustin Isleifson who we collaborated with on remote sensing of Arctic sea ice.
I will be forever grateful to my wife Kristin Okhmatovski Nemetchek and my children Spencer, Kai, Collin, Haydn, Claire, Nash, Ekaterina, and Xenia for their love, support, and sacrifices during the time of this book writing and beyond.
Vladimir Okhmatovski
Acronyms
AC
alternating current
BVP
boundary value problem
CAD
computer-aided design
CFIE
combined field integral equation
DC
direct current
DCIM
discrete complex image method
EFIE
electric field integral equation
FD
finite difference
FEM
finite element method
FFT
fast Fourier transform
GPOF
generalized pencil of function
HED
horizontal electric dipole
HMD
horizontal magnetic dipole
KCL
Kirchhoff’s current law
KVL
Kirchhoff’s voltage law
MFIE
magnetic field integral equation
MoM
method of moments
MPIE
mixed-potential integral equation
ODE
ordinary differential equation
PDE
partial differential equation
PEC
perfect electric conductor
PMCHWT
Poggio–Miller–Chang–Harrington–Wu–Tsai
RFFM
rational function fitting method
RWG
Rao–Wilton–Glisson
SDP
steepest descent path
SDEAM
spectral differential equation approximation method
SI
Sommerfeld integral
SIP
Sommerfeld integration path
SVS-EFIE
surface–volume–surface electric field integral equation
TLGF
transmission line Green’s function
UFFT
unified-FFT
VED
vertical electric dipole
VIE
volume integral equation
VMD
vertical magnetic dipole
WA
weighted averages
Introduction
The purpose of the book is to deliver a synergetic description of the theoretical background in layered media Green’s functions, integral equation formulations with such Green’s functions, and numerical methods used for their evaluation. To reach this goal, the book comprises three parts. First nine chapters are dedicated to the foundations of electromagnetic theory, complex analysis of Sommerfeld integrals, derivation of integral equations for objects in layered media, and their mixed-potential formulations. Subsequent Chapters 10–13 are dedicated to the method of moments (MoM) discretization of the integral equations. The third part, dedicated to the numerical techniques for evaluation of the layered media Green’s functions and their use in the MoM, is covered in Chapters 14–18. Extensive appendices accompany the chapters and provide background as well as complementary information, which is necessary for understanding of the material, but may already be familiar to the readers depending on their background.
Chapter 1 of the book covers foundational equations of electromagnetism, such as Maxwell equations in time and spectral domains, the second-order partial differential equations (PDEs) governing the electric and magnetic fields, such as curl–curl equations and Helmholtz equations, as well as the boundary conditions. It is also meant to facilitate understanding of the static solutions commonly used in capacitance and inductance extractors. Hence, decomposition of the electromagnetic field at low frequencies into electrostatic and magnetostatic components is outlined. A brief discussion of the key circuit equations, such as Kirchhoff’s current and voltage laws as well as constitutive relations on basic lumped circuit components as a limiting case of Maxwell equations at low frequencies, is covered in this chapter also. The concept of vector and scalar potentials is heavily used throughout this book. These potentials and Helmholtz equations governing them are presented in Chapter 1.
Representations of scalar and dyadic Green’s functions for the fields and potentials in free space are introduced in Chapter 2. Spectral formulations of such free space Green’s functions serve as a basis for the construction of their layered media counterparts handled in the subsequent chapters. The chapter starts from the derivation of 1D Green’s function in the space domain as the description of the electromagnetic field produced by an infinite sheet of unidirectional time-harmonic current. The expression is obtained directly as a solution of the second order ordinary differential equation (ODE) with -function in the right-hand side and also by casting the solution of the ODE into the form of an algebraic equation with respect to its spectrum. The spectral domain solution approach is subsequently repeated for obtaining expansion of the 3D scalar Green’s function over the inhomogeneous plane waves and its decomposition over cylindrical and conical waves known as Sommerfeld identity. The scalar 3D Green’s function is subsequently related to the dyadic Green’s function of the vector potential through the identity dyadic. Construction of Green’s functions of the electric and magnetic field as a solution of the dyadic curl–curl equations concludes this chapter.
Chapter 3 uses Green’s theorems, second-order PDEs for the fields, Green’s functions, and the boundary conditions to formulate the equivalence principle for the field representation in multilayered media. The first part of this chapter deals with scalar problems of electrostatic fields in the presence of arbitrary dielectric and composite conductor/dielectric objects embedded in layered media. Derivation of the equivalence principle is performed through the layer-by-layer application of Green’s theorem, bringing it to the form amenable to the formation of the coupled integral equations with respect to the unknown potential on dielectric boundaries and charges on conductor boundaries. Once the derivation patterns for the scalar integral equations with layered media kernels are established in the simpler case of the electrostatic fields, the chapter reviews key reciprocity relations for the full-wave dyadic Green’s functions. It then proceeds to treat the equivalent integral representations for the electromagnetic fields and classical integral equations for the electric, magnetic, and combined fields in layered media. Other traditional superpositions of the surface integral equations forming the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) and Muller formulations are described. Volume equivalence principle and integral equations allowing to handle continuously inhomogeneous objects situated in layered media are derived using the concepts of the dyadic Green’s function and superposition by introducing polarization currents. Overview of the three classes of single-source surface and volume-surface integral equations concludes this chapter.
Chapter 4 treats the vertical and horizontal dipole radiation problems in the presence of layered media. Spectral representations of the vector potential components are invoked to both formulate the 1D BVPs governing their spectra, demonstrate analytic solutions for these components in the spectral domain, and state the Sommerfeld integral representations for them in the space domain. Obtaining the spectra through ray tracing is illustrated to highlight the physical interpretation of the fields. Non-uniqueness in the definition of the vector potential describing the uniquely defined fields of a horizontal dipole in layered media is explained in detail through step-by-step derivations. The concept of quasi-static image extraction from Green’s function as a way to accelerate the convergence of the Sommerfeld integrals is introduced in this chapter. Multi-valued complex functions formalism attributed to the fields spectra in open layered media and methods for dealing with it using concepts of Riemann surfaces and branch cuts finishes the chapter. It provides a bridge to the subsequent chapters on integration techniques.
Chapter 5 illustrates the evaluation of the Sommerfeld integrals through their reduction to the integrals along the branch cuts using the Cauchy theorem. Freedom and constraints in the definitions of the branch cuts and corresponding Riemann surfaces are illustrated using two classical choices. The first choice deals with hyperbolic branch cuts originally introduced by
