Circuit Simulation

This text describes in detail the many numerical techniques and algorithms that are part of modern circuit simulation packages, with an emphasis on the most typically used simulation mode, namely transient analysis. The material is clearly presented, rigorously explained, and sufficient detail is presented to enable the reader to write his/her own circuit simulator. Given how few textbooks are in print on circuit simulation, this text should be a welcome addition to the shelves of many design engineers and computer-aided design practitioners, as well as a graduate-level text.

Farid N. Najm is a professor in the Department of Electrical and Computer Engineering (ECE) at the University of Toronto. He received a BE degree in electrical engineering from the American University of Beirut (AUB) in 1983 and a PhD degree in ECE from the University of Illinois at Urbana-Champaign (UIUC) in 1989. He then worked with Texas Instruments before joining the ECE Department at UIUC as assistant professor, later becoming associate professor. Dr. Najm joined the ECE Department at the University of Toronto in 1999, where he is currently Professor and Chair. His expertise is in the area of computer-aided design for integrated circuits, with an emphasis on circuit-level issues related to power, timing, variability, and reliability. Dr. Najm is a Fellow of the IEEE.

List of Figures	p. xiii
List of Tables	p. xix
Preface	p. xxi
Introduction	p. 1
Device Equations	p. 2
Equation Formulation	p. 3
Solution Techniques	p. 6
Nonlinear Circuits	p. 7
Dynamic Circuits	p. 8
Circuit Simulation Flow	p. 8
Analysis Modes	p. 9
Notes	p. 10
Problems	p. 10
Network Equations	p. 13
Elements and Networks	p. 13
Passive Elements	p. 13
Active Elements	p. 15
Equivalent Circuit Model	p. 17
Network Classification	p. 18
Topological Constraints	p. 19
Network Graphs	p. 19
Cycle Space and Bond Space	p. 23
Current Assignments	p. 23
Voltage Assignments	p. 24
Orthogonal Spaces	p. 24
Topological Constraints	p. 25
Fundamental Circulation	p. 25
Fundamental Potential Difference	p. 27
Formulation of Linear Algebraic Equations	p. 27
Sparse Tableau Analysis	p. 28
Nodal Analysis	p. 29
Unique Solvability	p. 30
Modified Nodal Analysis	p. 32
Formulation of Linear Dynamic Equations	p. 42
Dynamic Element Stamps	p. 43
Unique Solvability	p. 44
Notes	p. 45
Problems	p. 45
Solution of Linear Algebraic Circuit Equations	p. 49
Direct Methods	p. 50
Matrix Preliminaries	p. 50
Gaussian Elimination (GE)	p. 54
LU Factorization	p. 60
Block Gaussian Elimination	p. 71
Cholesky Decomposition	p. 73
Accuracy and Stability of GE	p. 74
Error	p. 75
Floating Point Numbers	p. 78
Norms	p. 80
Stability of GE and LU Factorization	p. 83
Pivoting for Accuracy	p. 86
Conditioning of Ax = b	p. 89
Iterative Refinement	p. 96
Indirect/Iterative Methods	p. 97
Gauss-Jacobi	p. 98
Gauss-Seidel	p. 99
Convergence	p. 100
Partitioning Techniques	p. 104
Node Tearing	p. 104
Direct Methods	p. 106
Indirect Methods	p. 107
Sparse Matrix Techniques	p. 109
Sparse Matrix Storage	p. 110
Sparse GE and LU Factorization	p. 112
Reordering and Sparsity	p. 113
Pivoting for Sparsity	p. 115
Markowitz Pivoting	p. 116
Diagonal Pivoting	p. 119
The Symmetric (SPD) Case	p. 120
Extension to the Non-SPD Case	p. 122
Notes	p. 125
Problems	p. 125
Solution of Nonlinear Algebraic Circuit Equations	p. 127
Nonlinear Network Equations	p. 127
Nonlinear Elements	p. 128
Nonlinear MNA Formulation	p. 129
Preparing for a DC Analysis	p. 133
Solution Techniques	p. 133
Iterative Methods and Convergence	p. 134
Introduction to Newton's Method	p. 136
The One-Dimensional Case	p. 139
The Multidimensional Case	p. 148
Quasi-Newton Methods	p. 152
Application to Circuit Simulation	p. 154
Linearization and Companion Models	p. 154
Some Test Cases	p. 156
Generalization	p. 162
Considerations for Multiterminal Elements	p. 166
Multivariable Differentiation	p. 167
Linearization of Multiterminal Elements	p. 171
Elements with Internal Nodes	p. 176
Quasi-Newton Methods in Simulation	p. 181
Damping Methods	p. 182
Overview of More General Methods	p. 186
Source Stepping	p. 187
Gmin Stepping	p. 189
Pseudo-Transient	p. 189
Justification for Pseudo-Transient	p. 193
Notes	p. 196
Problems	p. 197
Solution of Differential Circuit Equations	p. 201
Differential Network Equations	p. 201
Dynamic Elements	p. 201
Dynamic MNA Equations	p. 203
DAEs and ODEs	p. 204
ODE Solution Techniques	p. 206
ODE Systems and Basic Theorems	p. 206
Overview of Solution Methods	p. 209
Three Basic Methods: FE, BE, and TR	p. 211
Quality Metrics	p. 215
Linear Multistep Methods	p. 220
Accuracy of LMS Methods	p. 221
Order	p. 221
Consistency	p. 223
The Backward Differentiation Formulas	p. 224
Local Truncation Error	p. 225
Deriving the LMS Methods	p. 228
Solving Implicit Methods	p. 229
Interpolation Polynomial	p. 231
Estimating the LTE	p. 237
Stability of LMS Methods	p. 241
Linear Stability Theory	p. 242
The Test Equation	p. 243
Absolute Stability	p. 246
Stiff Systems	p. 252
Stiff Stability	p. 253
Remarks	p. 256
Trapezoidal Ringing	p. 257
Smoothing	p. 258
Extrapolation	p. 259
Variable Time-Step Methods	p. 261
Implementing a Change of Time-Step	p. 262
Interpolation Methods	p. 262
Variable-Coefficient Methods	p. 264
Variable Step Variable Order (VSVO) Methods	p. 265
Application to Circuit Simulation	p. 265
From DAEs to Algebraic Equations	p. 266
FE Discretization	p. 269
BE Discretization	p. 271
TR Discretization	p. 277
Charge-Based and Flux-Based Models	p. 282
Multiterminal Elements	p. 291
Time-Step Control	p. 296
Enhancements	p. 298
Overall Flow	p. 299
Notes	p. 300
Problems	p. 300
Glossary	p. 305
Bibliography	p. 307
Index	p. 311
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