Radiating Nonuniform Transmissionline Systems and the Partial Element Equivalent Circuit Method

RADIATING\nNONUNIFORM\nTRANSMISSION-LINE\nSYSTEMS AND THE\nPARTIAL ELEMENT\nEQUIVALENT CIRCUIT\nMETHOD......Page 5
Contents......Page 11
Preface......Page 17
References......Page 19
Acknowledgments......Page 21
List of Symbols......Page 23
Introduction......Page 33
References......Page 40
1.1 Maxwell Equations Derived from Conservation Laws – an Axiomatic Approach......Page 45
1.1.1 Charge Conservation......Page 47
1.1.2 Lorentz Force and Magnetic Flux Conservation......Page 49
1.1.3 Constitutive Relations and the Properties of Spacetime......Page 55
1.1.4 Remarks......Page 56
1.2 The Electromagnetic Field as a Gauge Field – a Gauge Field Approach......Page 57
1.2.1 Differences of Physical Fields that are Described by Reference Systems......Page 58
1.2.2 The Phase of Microscopic Matter Fields......Page 59
1.2.3 The Reference Frame of a Phase......Page 60
1.2.4 The Gauge Fields of a Phase......Page 62
1.2.5 Dynamics of the Gauge Field......Page 65
1.3.2 Minimal Coupling and the Lorentz Force......Page 68
1.3.3 Bianchi Identity and Magnetic Flux Conservation......Page 70
1.3.4 Gauge Approach and Constitutive Relations......Page 71
1.4 Solutions of Maxwell Equations......Page 72
1.4.1.1 Decoupling of Maxwell Equations......Page 73
1.4.1.2 Equations of Motion for the Electromagnetic Potentials......Page 74
1.4.1.3 Maxwell Equations in the Frequency Domain and Helmholtz Equations......Page 75
1.4.1.4 Maxwell Equations in Reciprocal Space......Page 76
1.4.3.1 Helmholtz’s Vector Theorem, Longitudinal and Transverse Fields......Page 77
1.4.3.2 Nondynamical Maxwell Equations as Boundary Conditions in Time......Page 79
1.4.3.3 Longitudinal Part of the Maxwell Equations......Page 80
1.4.3.4 Transverse Part of the Maxwell Equations......Page 81
1.4.4 Electromagnetic Energy and the Singularities of the Electromagnetic Field......Page 83
1.4.5 Coulomb Fields and Radiation Fields......Page 85
1.4.6 The Green’s Function Method......Page 88
1.4.6.1 Basic Ideas......Page 89
1.4.6.2 Self-Adjointness of Differential Operators and Boundary Conditions......Page 90
1.4.6.4 Basic Relations Between Electromagnetic Green’s Functions......Page 93
1.5.1 Surface Integral Equations in Short......Page 94
1.5.2.1 Pocklington’s Equation......Page 95
1.5.2.2 Hallén’s Equation......Page 96
1.5.2.3 Mixed-Potential Integral Equation......Page 97
References......Page 99
2 Nonuniform Transmission-Line Systems......Page 101
2.1.1.1 Local Coordinate System......Page 103
2.1.1.3 Volume and Surface Integrals......Page 105
2.1.2.1 Continuity Equation......Page 106
2.1.2.2 Reconstruction of the Densities......Page 107
2.1.3 Mixed Potential Integral Equation......Page 108
2.1.3.3 Current and Charge Trial Function......Page 109
2.1.3.4 Generalized Telegrapher Equations and TLST......Page 110
2.1.4.1 Parameters......Page 111
2.1.4.3 Solution of the Extended Telegrapher Equations......Page 113
2.1.4.4 Returning to Voltages?......Page 114
2.1.4.5 Discussion of the New Parameters......Page 116
2.1.4.6 Asymmetric Parameter Matrices......Page 117
2.1.5.1 Starting Values for the Iteration......Page 118
2.1.5.2 First Iteration......Page 119
2.1.5.3 Taylor Series Expansion of the Product Integral......Page 120
2.1.5.4 Eigenvalue Decomposition......Page 122
2.2 General Calculation Methods for the Product Integral/Matrizant......Page 123
2.2.1 Picard Iteration......Page 124
2.2.2 Volterra’s Method and the Product Integral......Page 125
2.2.3 Recursion Formulas for Linear Interpolation......Page 127
2.2.4 Approximation by Power Series......Page 129
2.2.5 Interpolation from Diagonalization......Page 132
2.2.6 Numerical Integration......Page 135
2.2.6.2 Integration by trapezoidal rule......Page 137
2.2.6.3 Explicit Runge–Kutta Method......Page 138
2.2.6.4 Hermite Integration......Page 140
2.2.6.5 Improving Accuracy by the Romberg Method......Page 142
2.2.6.6 Controlling Step Size and Error......Page 143
2.2.7 Remarks on Efficiency and the Choice of an Appropriate Method......Page 145
2.3.1 The Straight, Finite Length Wire Above Ground......Page 146
2.3.2 The Semi-Infinite Line......Page 151
2.3.3 Field Coupling to an Infinite Line......Page 153
2.3.4 The Skewed Wire Transmission Line......Page 155
2.3.5 The Periodic Transmission Line......Page 159
2.3.6 Cross Talk in a Nonuniform Multiconductor Transmission Line......Page 162
2.4.1 Classical Telegrapher Equations for Nonuniform Transmission Lines......Page 163
2.4.2.1 The Piecewise-Constant Approximation of the Characteristic Impedance Matrix......Page 166
2.4.2.2 The Continuous Approximation of the Characteristic Impedance Matrix......Page 168
2.4.2.3 Circulant Nonuniform MTLs......Page 172
2.4.2.4 General Approach to Calculate the Matrizant......Page 174
2.4.3 Matrizant Reduction......Page 176
References......Page 179
3.1 The Concept of Electromagnetic Topology......Page 183
3.2 Topological Networks and BLT Equations......Page 187
3.2.1 Wave Quantities......Page 188
3.2.2 BLT 1 Equation......Page 189
3.2.3 BLT 2 Equation......Page 191
3.2.4 Admittance Representation......Page 193
3.3 Transmission Lines and Topological Networks......Page 196
3.3.1 Transformation into a Propagation Matrix......Page 199
3.3.2 Equivalent Scattering Matrices......Page 200
3.4 Shielding......Page 201
3.4.1.1 Transfer Parameters of Cables......Page 202
3.4.1.2 Determination of the Per-unit-length Parameters......Page 204
3.4.1.3 Computation of the Transfer Impedance......Page 207
3.4.2 Shielding of an Anisotropic Spherical Shell......Page 212
3.4.2.2 Spherical Shell......Page 215
3.4.2.3 Application to Thin Conductive Shells......Page 216
References......Page 221
4.1.1 Maxwell Equations and Real Media for Interconnections......Page 223
4.1.2 Mixed Potential Integral Equations (MPIE)......Page 227
4.2.1 The PEEC Equation System in the Frequency Domain......Page 230
4.2.2 Generalized Partial Elements and Circuit Interpretation......Page 235
4.3.1.1 PEEC Models for Conductors......Page 237
4.3.2.1 PEEC Models with Generalized Partial Elements......Page 238
4.3.2.2 PEEC Models with Center-to-Center Retardation......Page 239
4.3.2.3 Quasi-Static PEEC Models......Page 240
4.4 PEEC Models for the Plane Half Space......Page 241
4.5.1 Orthogonal Cells......Page 245
4.5.2 Nonorthogonal Cells......Page 246
4.5.3 Triangular Cells......Page 249
4.6 PEEC Models for the Time Domain and the Stability Issue......Page 253
4.6.1 Standard PEEC Models for the Time Domain......Page 254
4.6.2 General Remarks on Stability of PEEC Model Solutions......Page 255
4.6.3 Stability Improvement of PEEC Models with Center-to-Center Retardation......Page 256
4.6.4.1 Stable Time Domain PEEC Models by Full-Spectrum Convolution Macromodeling (FSCM)......Page 258
4.6.4.2 Stable Time Domain PEEC Models by Macromodeling Using Foster’s Rational Functions and Circuit Synthesis......Page 263
4.7.1 Cross-Sectional Discretization of Wires......Page 264
4.7.2 Skin Effect Modeling by Means of a Global Surface Impedance......Page 265
4.7.3 Skin Effect Modeling by Means of a Local Mean Surface Impedance......Page 266
4.8.1 Motivation......Page 271
4.8.2 The DGFLM–PEEC Method......Page 272
4.8.3 DGFLM–PEEC Model for the Stripline Region......Page 277
4.8.3.1 Green’s Functions for the Stripline Region......Page 278
4.8.3.2 Discussion of the Behavior of the Green’s Functions......Page 279
4.8.3.3 Frequency Domain DGFLM-PEEC Model......Page 280
4.8.3.4 DGFLM–PEEC Models in the Time Domain......Page 287
4.9 PEEC Models and Uniform Transmission Lines......Page 292
4.10.1 General Remarks......Page 296
4.10.2 Power Analysis of Magnetic and Electric Couplings......Page 297
4.10.3 Power Analysis of PEEC Models......Page 299
References......Page 301
A.1 Integration Over a Curve and Covariant Vectors as Line Integrands......Page 305
A.2 Integration Over a Surface and Contravariant Vector Densities as Surface Integrands......Page 307
A.3 Integration Over a Volume and Scalar Densities as Volume Integrands......Page 308
A.4 Poincar Lemma......Page 309
A.6 Lie Derivative......Page 310
References......Page 311
Appendix B: Elements of Functional Analysis......Page 313
B.1.1 Metric Spaces......Page 314
B.1.2 Linear Spaces, Vector Spaces......Page 316
B.1.3 Normed Spaces......Page 317
B.1.4 Inner Product Spaces and Pseudo Inner Product Spaces......Page 318
B.1.5 Hilbert Spaces......Page 320
B.1.6 Finite Expansions and Best Approximation......Page 321
B.2.1 Definition of a Linear Operator, Domain and Range of an Operator......Page 322
B.2.5 The Riesz Representation Theorem......Page 323
B.2.8 Invertible Operators, Resolvent Operator......Page 324
B.3.1 Standard Eigenvalue Problem, Spectrum and Resolvent Set......Page 325
B.3.2 Classification of Spectra by Operator Properties......Page 327
B.4.1 Linear Independence of Eigenfunctions......Page 328
B.4.2 Spectral Theorem for Compact and Self-Adjoint Operators......Page 329
B.4.3 Remarks on the Relation Between Differential and Integral Operators......Page 330
References......Page 331
C.1 Vector Identities......Page 333
Reference......Page 334
Appendix D: Adaption of the Integral Equations to the Conductor Geometry......Page 335
E.2 The Determination of the Product Integral......Page 339
E.3 Inverse Operation......Page 340
References......Page 341
F.2 Integrals Involving Exponential and Power Functions......Page 343
F.3 Integrals Involving Trigonometric and Exponential Functions......Page 345
Reference......Page 346
Index......Page 347