Electromagnetism and Interconnections: Advanced Mathematical Tools for Computer-aided Simulation

Acknowledgements xi

Introduction xiii

Chapter 1. Theoretical Foundations of Electromagnetism 1

1.1. Elements of the theory of distributions applied to electromagnetism 1

1.1.1. Choosing a presentation of the foundations of electromagnetism 1

1.1.2. Linear modeling of physical laws and Green’s kernels 2

1.1.3. Accounting for the “natural symmetries” of physical laws 3

1.1.4. Motivation for using the theory of distributions 4

1.1.5. Quick review of the theory of distributions 5

1.1.6. Application to electromagnetism 9

1.2. Vector analysis review according to the theory of distributions 11

1.2.1. Derivation of discontinuous functions defined on R 11

1.2.2. Derivative of linear mappings 12

1.2.3. Derivation of discontinuous functions on a surface in 􀂃3  12

1.2.4. Derivation of vector distributions in 􀂃3 13

1.2.5. Algebra of the operator 􀂒?n 13

1.3. Maxwell’s equations according to the theory of distributions 14

1.3.1. Symmetries and duality in electromagnetism 14

1.3.2. The symmetry laws of distributions in electromagnetism 14

1.3.3. Application to the first couple of Maxwell’s equations 15

1.3.4. Behavior law of materials by means of the theory of distributions 19

1.3.5. Application to the second couple of Maxwell’s equations 19

1.3.6. Charge density, current density, continuity equations 20

1.3.7. Integral form of Maxwell’s equations 22

1.4. Conclusion 24

Chapter 2. Full Wave Analysis 25

2.1. Discontinuities in electromagnetism 25

2.1.1. Initial and boundary conditions according to the theory of distributions 25

2.1.2. Electromagnetic images, incident and reflected fields 28

2.1.3. Method of moments for the numerical computation of electromagnetic fields 29

2.2. Potentials in electromagnetism 33

2.2.1. Scalar and vector potentials, duality between electrical and magnetic potentials 33

2.2.2. Lossy propagation equations, the Lorentz gauge 35

2.2.3. Green’s kernels for harmonic electromagnetic waves in heterogenous media 39

2.3. Topology of electromagnetic interferences 42

2.3.1. Introduction 42

2.3.2. Topological modeling of electromagnetic interferences 43

2.3.3. Partitioning the electrical network in respect of electromagnetic interferences 45

2.3.4. The tree of electromagnetic interferences and the problem of loops 46

2.4. Conclusion 50

Chapter 3. Electromagnetism in Stratified Media 51

3.1. Electrical and magnetic currents in stratified media 52

3.1.1. Scope of the theory, defining stratified media 52

3.1.2. Integral formulation of the current derivative versus time: general case 53

3.1.3. Integral formula of the current derivative relative to space in the direction of the vector potential 61

3.1.4. Duality between electrical and magnetic currents in lossless media 63

3.2. Straight stratified media 67

3.2.1. Scope 67

3.2.2. Lossy propagation equations and the variational approach 67

3.2.3. Spectral analysis of the longitudinal field 71

3.2.4. From Maxwell’s equations to transmission line equations 76

3.2.5. Generalized transmission line matrix equation 79

3.2.6. Non-existence of the TM and TE modes separately 81

3.2.7. Electrical (or magnetic) currents 84

3.3. Conclusion 84

Chapter 4. Transmission Line Equations 85

4.1. Straight homogenous dielectric media with lossless conductors 86

4.1.1. Hypothesis 86

4.1.2. Electrical current formulae in TM mode of propagation 86

4.1.3. Magnetic current formulae in TE mode of propagation 89

4.1.4. Spectral analysis of electromagnetic fields 89

4.1.5. Modal analysis of electrical current and lineic charge 96

4.1.6. Modal analysis of scalar and vector potentials 101

4.1.7. Transmission line with distributed sources corresponding to a waveguide 103

4.2. TEM mode of wave propagation 104

4.2.1. Defining the TEM mode and the transmission lines 104

4.2.2. Basic existence condition of a TEM propagation mode 105

4.2.3. Variational numerical computation of the lowest wavelength 107

4.2.4. Telegrapher’s equation for current and electrical charge per unit length 109

4.2.5. Lorentz condition and telegrapher’s equation for vector potentials and scalars in TEM mode 111

4.2.6. Lineic distribution of electrical charges and the Poisson equation 112

4.2.7. Transmission line equations for lossy dielectrics and lossless conductors 115

4.2.8. Green’s kernels and the numerical computation of lineic parameters 117

4.3. Quasi-TEM approximation for lossy conductors and dielectrics 122

4.3.1. Foucault’s modal currents of electromagnetic field propagation in lossy media 122

4.3.2. Quasi-TEM approximation of coupled lossy transmission lines 124

4.4. Weakly bent transmission lines in the quasi-TEM approximation 126

4.4.1. Bent lossy heterogenous media with lossless conductors 126

4.4.2. Bent lossy homogenous media with lossless conductors 127

4.4.3. Bent lossless conductors such that en does not depend on q1, and e1 and CH do not depend on qn 128

4.4.4. Lineic capacitance tied to a weak curvature of a transmission line 128

4.5. Conclusion 130

Chapter 5. Direct Time-domain Methods 131

5.1. “Direct” methods in the time domain 132

5.1.1. Defining a “direct” method in the time domain 132

5.1.2. Single lossless transmission lines in homogenous media 132

5.2. Lossless coupled transmission lines in homogenous media 143

5.2.1. Homogenous coupling 143

5.2.2. Heterogenous coupling 150

5.2.3. Bifurcations 151

5.2.4. Complex distributed parameter networks 156

5.2.5. Estimation of the transient state time of signals 159

5.2.6. Numerical computation of the characteristic impedance matrix 161

5.3. Conclusion 162

Chapter 6. Discretization in the Time Domain 163

6.1. Finite difference method in the time domain 163

6.1.1. From full wave analysis to nodal operational matrices 163

6.1.2. Recursive differential transmission line matrix equation of complex networks 167

6.1.3. Estimation of the transient state time 168

6.1.4. Finite difference approximation of differential operators in the time domain 170

6.1.5. Application to lumped quadripole modeling approximation in the time domain 173

6.1.6. Complex distributed and lumped parameter networks approximation 175

6.2. Matrix velocity operator interpolation method 179

6.2.1. Difficulties set by the compounded matrix functions 179

6.2.2. Matrix velocity matrix operator of stratified heterogenous media 181

6.2.3. Matrix velocity operator interpolation method for the matrix drift equation 183

6.3. Conclusion 187

Chapter 7. Frequency Methods 189

7.1. Laplace transform method for lossy transmission lines 190

7.1.1. Transfer matrix in the Laplace domain 190

7.1.2. Transfer impedance matrix, impedance matching, scattering matrix 198

7.2. Coming back in the time domain 202

7.2.1. Inverse Laplace transform for lossy transmission lines 202

7.2.2. Method of the contribution of loops 203

7.2.3. Application to the distortion of a Dirac pulse in lossy media 206

7.2.4. Classical kernel of the convolution methods 207

7.2.5. Diffusion equation and the time-varying “skin depth” 208

7.2.6. Multiple reflections processing 209

7.3. Method of the discrete Fourier transform 210

7.3.1. Fourier transform and the harmonic steady state 210

7.3.2. Discrete Fourier transform and the sampling procedure 211

7.3.3. Application to digital signal processing 213

7.3.4. Bifurcations and complex networks of lossy transmission lines 215

7.4. Conclusion 217

Chapter 8. Time-domain Wavelets 219

8.1. Theoretical introduction 219

8.1.1. Motivation for the time-domain wavelets method 219

8.1.2. General mathematical framework 220

8.1.3. Seed and generator of direct and reverse wavelets family 221

8.2. Application to digital signal propagation 226

8.2.1. Application to lossless guided wave analysis in the time domain 226

8.2.2. Application to the telegrapher’s equation 230

8.2.3. Convergence of wavelet expansions, numerical approximation 233

8.3. Conclusion 241

Chapter 9. Applications of the Wavelet Method 243

9.1. Coupled lossy transmission lines in the TEM approximation 243

9.1.1. Wavelets in homogenously coupled lossy transmission lines 243

9.1.2. Multiple reflections into lossy coupled lines 250

9.1.3. Comparative analysis of frequency and wavelets methods 255

9.2. Extension to 3D wavelets and electromagnetic perturbations 256

9.2.1. Basic second-order partial differential equation of electromagnetic waves 256

9.2.2. Obtaining the wavelet generating equation: Au = 􀀯u. 257

9.2.3. Direct and reverse generators of the wavelet base 258

9.2.4. Spherical seed and wavelets having a zero divergence 260

9.2.5. Modeling electromagnetic perturbations in lossy media 261

9.2.6. Guided propagation in interconnection structures 262

9.3. Conclusion 262

Appendices 263

Appendix A. Physical Data 263

Appendix B. Technological Data 267

Appendix C. Lineic Capacitors 269

Appendix D. Modified Relaxation Method 275

Appendix E. Cylindrical Wavelets 277

Appendix F. Wavelets and Elliptic Operators 281

References 287

Index 291