Circuit Simulation

✍ Scribed by Farid N. Najm

• Publisher: Wiley-IEEE Press
• Year: 2010
• Tongue: English
• Leaves: 344
• Category: Library

✦ Synopsis

A Definitive text on developing circuit simulators

Circuit Simulation gives a clear description of the numerical techniques and algorithms that are part of modern circuit simulators, with a focus on the most commonly used simulation modes: DC analysis and transient analysis. Tested in a graduate course on circuit simulation at the University of Toronto, this unique text provides the reader with sufficient detail and mathematical rigor to write his/her own basic circuit simulator. There is detailed coverage throughout of the mathematical and numerical techniques that are the basis for the various simulation topics, which facilitates a complete understanding of practical simulation techniques. In addition, Circuit Simulation:

• Explores a number of modern techniques from numerical analysis that are not synthesized anywhere else

• Covers network equation formulation in detail, with an emphasis on modified nodal analysis

• Gives a comprehensive treatment of the most relevant aspects of linear and nonlinear system solution techniques

• States all theorems without proof in order to maintain the focus on the end-goal of providing coverage of practical simulation methods

• Provides ample references for further study

• Enables newcomers to circuit simulation to understand the material in a concrete and holistic manner

With problem sets and computer projects at the end of every chapter, Circuit Simulation is ideally suited for a graduate course on this topic. It is also a practical reference for design engineers and computer-aided design practitioners, as well as researchers and developers in both industry and academia.

✦ Table of Contents

CIRCUIT SIMULATION......Page 5
CONTENTS......Page 9
List of Figures......Page 15
List of Tables......Page 21
Preface......Page 23
1 Introduction......Page 27
1.1 Device Equations......Page 28
1.2 Equation Formulation......Page 29
1.3 Solution Techniques......Page 32
1.3.1 Nonlinear Circuits......Page 33
1.4 Circuit Simulation Flow......Page 34
1.4.1 Analysis Modes......Page 35
Problems......Page 36
2.1.1 Passive Elements......Page 39
2.1.2 Active Elements......Page 41
2.1.3 Equivalent Circuit Model......Page 43
2.1.4 Network Classification......Page 44
2.2.1 Network Graphs......Page 45
2.3.1 Current Assignments......Page 49
2.3.3 Orthogonal Spaces......Page 50
2.3.5 Fundamental Circulation......Page 51
2.4 Formulation of Linear Algebraic Equations......Page 53
2.4.1 Sparse Tableau Analysis......Page 54
2.4.2 Nodal Analysis......Page 55
2.4.3 Unique Solvability......Page 56
2.4.4 Modified Nodal Analysis......Page 58
2.5 Formulation of Linear Dynamic Equations......Page 68
2.5.1 Dynamic Element Stamps......Page 69
2.5.2 Unique Solvability......Page 70
Problems......Page 71
3 Solution of Linear Algebraic Circuit Equations......Page 75
3.1.1 Matrix Preliminaries......Page 76
3.1.2 Gaussian Elimination (GE)......Page 80
3.1.3 LU Factorization......Page 86
3.1.4 Block Gaussian Elimination......Page 97
3.1.5 Cholesky Decomposition......Page 99
3.2 Accuracy and Stability of GE......Page 100
3.2.1 Error......Page 101
3.2.2 Floating Point Numbers......Page 104
3.2.3 Norms......Page 106
3.2.4 Stability of GE and LU Factorization......Page 109
3.2.5 Pivoting for Accuracy......Page 112
3.2.6 Conditioning of Ax = b......Page 115
3.2.7 Iterative Refinement......Page 122
3.3 Indirect/Iterative Methods......Page 123
3.3.1 Gauss-Jacobi......Page 124
3.3.2 Gauss-Seidel......Page 125
3.3.3 Convergence......Page 126
3.4.1 Node Tearing......Page 130
3.4.2 Direct Methods......Page 132
3.4.3 Indirect Methods......Page 133
3.5 Sparse Matrix Techniques......Page 135
3.5.1 Sparse Matrix Storage......Page 136
3.5.2 Sparse GE and LU Factorization......Page 138
3.5.3 Reordering and Sparsity......Page 139
3.5.4 Pivoting for Sparsity......Page 141
3.5.5 Markowitz Pivoting......Page 142
3.5.6 Diagonal Pivoting......Page 145
3.5.7 The Symmetric (SPD) Case......Page 146
3.5.8 Extension to the Non-SPD Case......Page 148
Problems......Page 151
4.1 Nonlinear Network Equations......Page 153
4.1.1 Nonlinear Elements......Page 154
4.1.2 Nonlinear MNA Formulation......Page 155
4.2 Solution Techniques......Page 159
4.2.1 Iterative Methods and Convergence......Page 160
4.2.2 Introduction to Newton’s Method......Page 162
4.2.3 The One-Dimensional Case......Page 165
4.2.4 The Multidimensional Case......Page 174
4.2.5 Quasi-Newton Methods......Page 178
4.3.1 Linearization and Companion Models......Page 180
4.3.2 Some Test Cases......Page 182
4.3.3 Generalization......Page 188
4.3.4 Considerations for Multiterminal Elements......Page 192
4.3.5 Multivariable Differentiation......Page 193
4.3.6 Linearization of Multiterminal Elements......Page 197
4.3.7 Elements with Internal Nodes......Page 202
4.4 Quasi-Newton Methods in Simulation......Page 207
4.4.1 Damping Methods......Page 208
4.4.2 Overview of More General Methods......Page 212
4.4.3 Source Stepping......Page 213
4.4.5 Pseudo-Transient......Page 215
4.4.6 Justification for Pseudo-Transient......Page 219
Notes......Page 222
Problems......Page 223
5.1.1 Dynamic Elements......Page 227
5.1.2 Dynamic MNA Equations......Page 229
5.1.3 DAEs and ODEs......Page 230
5.2.1 ODE Systems and Basic Theorems......Page 232
5.2.2 Overview of Solution Methods......Page 235
5.2.3 Three Basic Methods: FE, BE, and TR......Page 237
5.2.4 Quality Metrics......Page 241
5.2.5 Linear Multistep Methods......Page 246
5.3.1 Order......Page 247
5.3.2 Consistency......Page 249
5.3.3 The Backward Differentiation Formulas......Page 250
5.3.4 Local Truncation Error......Page 251
5.3.5 Deriving the LMS Methods......Page 254
5.3.6 Solving Implicit Methods......Page 255
5.3.7 Interpolation Polynomial......Page 257
5.3.8 Estimating the LTE......Page 263
5.4 Stability of LMS Methods......Page 267
5.4.1 Linear Stability Theory......Page 268
5.4.2 The Test Equation......Page 269
5.4.3 Absolute Stability......Page 272
5.4.4 Stiff Systems......Page 278
5.4.5 Stiff Stability......Page 279
5.4.6 Remarks......Page 282
5.5 Trapezoidal Ringing......Page 283
5.5.1 Smoothing......Page 284
5.5.2 Extrapolation......Page 285
5.6 Variable Time-Step Methods......Page 287
5.6.2 Interpolation Methods......Page 288
5.6.3 Variable-Coefficient Methods......Page 290
5.7 Application to Circuit Simulation......Page 291
5.7.1 From DAEs to Algebraic Equations......Page 292
5.7.2 FE Discretization......Page 295
5.7.3 BE Discretization......Page 297
5.7.4 TR Discretization......Page 303
5.7.5 Charge-Based and Flux-Based Models......Page 308
5.7.6 Multiterminal Elements......Page 317
5.7.7 Time-Step Control......Page 322
5.7.8 Enhancements......Page 324
5.7.9 Overall Flow......Page 325
Problems......Page 326
Glossary......Page 331
Bibliography......Page 333
Index......Page 337