Table of Contents
Preface vii
1 Introduction to Transmission Lines and Their Application to Electromagnetic Phenomena 1
1.1 Simple Experimental Example 4
1.2 Examples of Impulse Sources 6
1.3 Model Outline 9
1.4 Application of Model to Small Node Resistance 17
1.5 Transmission Line Theory Background 17
1.6 Initial Conditions of Special Interest 22
One Dimensional TLM Analysis. Comparison with Finite Difference Method 24
1.7 TLM Iteration Method 24
1.8 Reverse TLM Iteration 25
1.9 Derivation of Scattering Coefficients For Reverse Iteration 29
1.10 Complete TLM Iteration (Combining Forward and Reverse Iterations) 31
1.11 Finite Difference Method. Comparison with TLM Method 32
Two Dimensional TLM Analysis. Comparison with Finite Difference Method 33
1.12 Boundary Conditions at 2D Node 35
1.13 Static Behavior about 2D Node 37
1.14 Non-static Example: Wave Incident on 2D node 38
1.15 Integral Rotational Properties of Field about the Node 42
1.16 2D TLM Iteration Method for Nine Cell Core Matrix 46
1.17 2D Finite Difference Method. Comparison with TLM Method 50
1.18 Final Comments: Inclusion of Time Varying Signals and Phase Coherence 56
Appendices 57
App. 1A.1 Effect of Additional Paths on Weighing Process 57
App. 1A.2 Novel Applications of TLM Method: Description of Neurological Activity Using the TLM Method 60
2 Notation and Mapping of Physical Properties 65
2.1 1D Cell Notation and Mapping of Conductivity and Field 67
2.2 Neighboring 1D Cells with Unequal Impedance 70
2.3 2D Cell Notation, Mapping of Conductivity and Field 72
2.4 Simultaneous Conductivity Contributions 80
2.5 3D Cell Notation, Mapping of Conductivity and Field 82
Other Node Controlled Properties 88
2.6 Node Control of 2D Scattering Coefficients Due to Finite Node Resistance 89
2.7 Signal Gain 90
2.8 Signal Generation. Use of Node Coupling 91
2.9 Mode Conversion 94
Example of Mapping: Node Resistance in Photoconductive Semiconductor 95
2.10 Semiconductor Switch Geometry (2D) 95
2.11 Node Resistance Profile in Semiconductor 98
3 Scattering Equations 101
3.1 1D Scattering Equation 101
3.2 2D Scattering Equations 104
3.3 Effect of Symmetry on Scattering Coefficients 113
3.4 3D Scattering Equations: Coplanar Scattering 116
General Scattering, Including Scattering Normal to Propagation Plane 124
3.5 Simple 3D Equivalent TLM Circuit 125
3.6 Quasi-Coupling 126
3.7 Neglect of Quasi-Coupling 127
3.8 Simple Quasi-Coupling Circuit. First Order Approximation 129
3.9 Correction to Quasi-Coupling Circuit: Second Order Approximation 133
3.10 Calculation of Load Impedance with Quasi-Coupling 136
3.11 Small Coupling Approximation of Second Order Quasi-Coupling 138
3.12 General 3D Scattering Process Using Cell Notation 140
3.13 Complete Iterative Equations 150
3.14 Contribution of Electric and Magnetic Fields to the Total Energy 153
Plane Wave Behavior 154
3.15 Response of 2D Cell Matrix to Input Plane Wave 154
3.16 Response of 2D Cell Matrix to Input Waves with Arbitrary Amplitudes 163
3.17 Response of 3D Cell Matrix to Input Plane Wave 164
3.18 Final Comments of Uniform Waves versus Plane Waves 167
Appendices 168
App. 3A.1 Consistency of 3D Circuit with the TLM Static Solutions 168
App. 3A.2 3D Scattering Coefficients, Without Quasi-Coupling in Terms of Circuit Parameters 169
App. 3A.3 3D Scattering Coefficients with Both Coplanar and Aplanar Contributions into Unit Cell Lines (yz and zx Planes) 172
App. 3A.4 3D Scattering Equations: with Both Coplanar and Aplanar Contributions into Unit Cell Lines (yz and zx Planes) 174
4 Corrections for Plane Waves and Grid Anisotropy Effects 177
4.1 Partition of TLM Waves into Component Waves 177
4.2 Scattering Corrections for 2D Plane Waves: Plane Wave Correlations Between Cells 179
4.3 Changes to 2D Scattering Coefficients 186
Corrections to Plane Wave Correlations 188
4.4 Correlation of Waves in Adjoining Media with Differing Dielectric Constants 188
4.5 Modification of Wave Correlation Adjacent a Conducting Boundary 190
Decorrelation Processes 192
4.6 Decorrelation Due to Sign Disparity of Plane and Symmetric Waves 192
4.7 Related Scattering Criteria and Sign Conditions for Removal of the Sign Disparity 196
4.8 Minimal Solution Using Differing Decorrelation Factors to Remove Sign Disparities 198
4.9 Decorrelation of Forward and Backward Plane Waves with Same Polarity in Neighboring Series TLM Lines Without Losses 202
4.10 Decorrelation of Forward and Backward Plane Waves with the Same Polarity Occupying the Same TLM Line 205
4.11 Decorrelation Treatment at Boundary Interfaces 208
4.12 Comments on Interaction of Plane Wave Front with a Half-Infinite Conducting Plane 210
4.13 Summary of Correlation/Decorrelation Processes 212
Treatment of Grid Orientation Effects 213
4.14 Dependence of Wave Energy Dispersal on Grid Orientation for Symmetric and Plane Waves 213
4.15 Selection of Grid for Plane Waves 216
4.16 Transformation Properties between Grids 217
4.17 Possible Mini-Plane Wave Fronts Associated with Each Cell. Plane Wave Partitioning 218
Grid(s) Selection. Propagation Vector Independence 220
4.18 Transformation of Fields to Principal Grid 220
4.19 Incorporation of Symmetric Waves 221
4.20 Iteration Method Using Principal Grid Transformations 222
4.21 Treatment of Separate TLM Correlated Wave Sources 224
4.22 Final Comments 226
Appendices 226
App. 4A.1 3D Scattering Corrections of Plane Waves (Plane Wave Correlations) 226
App. 4A.2 Consistency of Plane Wave Correlations with a Simple Quantum Mechanical Model 229
5 Boundary Conditions and Dispersion 233
5.1 Dielectric-Dielectric Interface 234
Node Coupling: Nearest Node and Multi-Coupled Node Approximations 238
5.2 Nearest Nodes for 1D Interface 241
5.3 Nearest Nodes at 2D Interface 242
5.4 Truncated Cells and Oblique Interface 244
5.5 Cell Index Notation at a Dielectric Interface Used in Simulations 245
5.6 Simplified Iteration Neglecting the Nearest Node Approximation 247
5.7 Non-Uniform Dielectric. Use of Cluster Cells 248
Other Boundary Conditions 251
5.8 Dielectric - Open Circuit Interface 251
5.9 Dielectric - Conductor Interface 252
5.10 Input/Output Conditions 254
5.11 Composite Transmission Line 256
5.12 Determination of Initial Static Field by TLM Method 257
Dispersion 260
5.13 TLM Methods for Treating Dispersion 260
5.14 Dispersion Sources 262
5.15 Dispersion Example 262
5.16 Propagation Velocity Dispersion 265
5.17 Node Resistance Dispersion 266
5.18 Anomalous Dispersion 267
Incorporation of Dispersion into TLM Formulation 268
5.19 Dispersion Approximations 268
5.20 Outline of Dispersion Calculation Using the TLM Method 269
5.21 One Dimensional Dispersion Iteration 269
5.22 Initial Conditions with Dispersion Present 280
5.23 Stability of Initial Profiles with Dispersion Present 281
5.24 Replacement of Non-uniform Field in Cell with Effective Uniform Field 284
Appendix 284
App. 5A.1 Specification of Input/Output Node Resistance to Eliminate Multiple Reflections 284
6 Cell Discharge Properties and Integration of Transport Phenomena into the Transmission Line Matrix 287
6.1 Charge Transfer between Cells 288
6.2 Relationship between Field and Cell Charge 291
6.3 Dependence of Conductivity on Carrier Properties 295
Integration of Carrier Transport Using TLM Notation. Changes in Cell Occupancy and Its Effect on the TLM Iteration 296
6.4 General Continuity Equations 296
6.5 Carrier Generation Due to Light Activation 296
6.6 Carrier Generation Due to Avalanching: Identical Hole and Electron Drift Velocities 297
6.7 Avalanching with Differing Hole and Electron Drift Velocities 300
6.8 Two Step Generation Process 303
6.9 Recombination 304
6.10 Limitations of Simple Exponential Recovery Model 306
6.11 Carrier Drift 307
6.12 Cell Charge Iteration. Equivalence of Drift and Inter-Cell Currents Using TLM Notation 310
6.13 Carrier Diffusion 315
6.14 Frequency of Transport Iteration 317
6.15 Total Contribution to Changes in Carrier Cell Occupancy 318
7 Description of TLM Iteration 321
7.1 Specification of Geometry 324
7.2 Use of TLM Matrix 326
7.3 Various Regions which Incorporate Plane Wave Correlation/Decorrelation (PWC Effects) into the Iteration 329
7.4 Simplified Decorrelation Procedure Used for Simulations in Chapter VII 330
7.5 Description of Inputs, Arrays, and Initial Conditions 336
7.6 Plane Wave Correllation (PWC) Inputs 338
7.7 Iteration Outline 339
7.8 Node Resistance R(n,m) Changes. Use of Light Activation 340
Symmetric Scattering Simulations 345
7.9 Symmetric Field Evolution with and without Node Activation 345
PWC Simulations. Comparison of PWC and Symmetric Results 256
7.10 Comparison of Output Waveforms and Static Profiles for Symmetric and PWC Simulations 356
7.11 Comparison of Forward and Backward Waves when Using Wave Correlation 362
7.12 Risetime and Alternating Field Effects in the Guided Region 364
7.13 Field Profile Evolution during Transient Charge-up Phase 366
7.14 Effect of Load Mismatch on Output and Field Profile 370
7.15 Node Recovery and its Effect on Output Pulse and Field Profile 370
7.16 Effects of Risetime on Conductivity 374
7.17 Partial Activation of Nodes and Effect on Profiles and Output 375
7.18 Cell Charge Following Recovery 377
7.19 Role of TLM Waves at Charged Boundary 380
7.20 Incorporation of 3D Scattering Parameters into 2D Iteration. Application to Magnetostatic Solutions 381
7.21 Summary: Comparison of PWC and Symmetric Simulation Results 383
Appendices 385
7A.1 Outline Discussion of Program Statements for Activated Semiconductor Switch 385
7A.2 Program Statements for Optically Activated Semiconductor Switch 397
7A.3 Matching Node Resistors for Input & Output Sections 437
7A.4 Field Decay in Semiconductor Using the TLM Formulation 438
8 Spice Solutions 441
8.1 Photoconductive/Avalanche Switch 441
8.2 Traveling Wave Marx Generator 445
8.3 Traveling Marx Wave in a Layered Dielectric 450
Pulse Transformation and Generation Using Non-Uniform Transmission Lines 451
8.4 Use of Cell Chain to Simulate Pulse Transformer 452
8.5 Pulse Transformer Simulation Results 455
8.6 Pulse Source Using Non-Uniform TLM Lines (Switch at Output) 456
8.7 Radial Pulse Source (Switch at Output) 458
8.8 Pulse Sources with Gain (PFXL Sources) 459
Darlington Pulser 466
8.9 TLM Formulation of Darlington Pulser 469
8.10 SPICE Simulation of Lossy Darlington Pulser 470
Appendices 470
8A.1 Introduction To SPICE Format 470
8A.2 Discussion of Format for Photoconductive Switch 470
8A.3 TLM Analysis of Leading Edge Pulse in a Transformer 477
8A.4 TLM Analysis of Leading Edge Wave in PFXL 480
Biography of Maurice Weiner 487
Index 489
