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1 Introduction1

1.1 Device Equations2

1.2 Equation Formulation3

1.3 Solution Techniques6

1.3.1 Nonlinear Circuits7

1.3.2 Dynamic Circuits8

1.4 Circuit Simulation Flow8

1.4.1 Analysis Modes9

Notes10

Problems10

2 Network Equations13

2.1 Elements and Networks13

2.1.1 Passive Elements13

2.1.2 Active Elements15

2.1.3 Equivalent Circuit Model17

2.1.4 Network Classification18

2.2 Topological Constraints19

2.2.1 Network Graphs19

2.3 Cycle Space and Bond Space23

2.3.1 Current Assignments23

2.3.2 Voltage Assignments24

2.3.3 Orthogonal Spaces24

2.3.4 Topological Constraints25

2.3.5 Fundamental Circulation25

2.3.6 Fundamental Potential Difference27

2.4 Formulation of Linear Algebraic Equations27

2.4.1 Sparse Tableau Analysis28

2.4.2 Nodal Analysis29

2.4.3 Unique Solvability30

2.4.4 Modified Nodal Analysis32

2.5 Formulation of Linear Dynamic Equations42

2.5.1 Dynamic Element Stamps43

2.5.2 Unique Solvability44

Notes45

Problems45

3 Solution of Linear Algebraic Circuit Equations49

3.1 Direct Methods50

3.1.1 Matrix Preliminaries50

3.1.2 Gaussian Elimination（GE）54

3.1.3 LU Factorization60

3.1.4 Block Gaussian Elimination71

3.1.5 Cholesky Decomposition73

3.2 Accuracy and Stability of GE74

3.2.1 Error75

3.2.2 Floating Point Numbers78

3.2.3 Norms80

3.2.4 Stability of GE and LU Factorization83

3.2.5 Pivoting for Accuracy86

3.2.6 Conditioning of Ax＝b89

3.2.7 Iterative Refinement96

3.3 Indirect/Iterative Methods97

3.3.1 Gauss-Jacobi98

3.3.2 Gauss-Seidel99

3.3.3 Convergence100

3.4 Partitioning Techniques104

3.4.1 Node Tearing104

3.4.2 Direct Methods106

3.4.3 Indirect Methods107

3.5 Sparse Matrix Techniques109

3.5.1 Sparse Matrix Storage110

3.5.2 Sparse GE and LU Factorization112

3.5.3 Reordering and Sparsity113

3.5.4 Pivoting for Sparsity115

3.5.5 Markowitz Pivoting116

3.5.6 Diagonal Pivoting119

3.5.7 The Symmetric（SPD）Case120

3.5.8 Extension to the Non-SPD Case122

Notes125

Problems125

4 Solution of Nonlinear Algebraic Circuit Equations127

4.1 Nonlinear Network Equations127

4.1.1 Nonlinear Elements128

4.1.2 Nonlinear MNA Formulation129

4.1.3 Preparing for a DC Analysis133

4.2 Solution Techniques133

4.2.1 Iterative Methods and Convergence134

4.2.2 Introduction to Newton’s Method136

4.2.3 The One-Dimensional Case139

4.2.4 The Multidimensional Case148

4.2.5 Quasi-Newton Methods152

4.3 Application to Circuit Simulation154

4.3.1 Linearization and Companion Models154

4.3.2 Some Test Cases156

4.3.3 Generalization162

4.3.4 Considerations for Multiterminal Elements166

4.3.5 Multivariable Differentiation167

4.3.6 Linearization of Multiterminal Elements171

4.3.7 Elements with Internal Nodes176

4.4 Quasi-Newton Methods in Simulation181

4.4.1 Damping Methods182

4.4.2 Overview of More General Methods186

4.4.3 Source Stepping187

4.4.4 Gmin Stepping189

4.4.5 Pseudo-Transient189

4.4.6 Justification for Pseudo-Transient193

Notes196

Problems197

5 Solution of Differential Circuit Equations201

5.1 Differential Network Equations201

5.1.1 Dynamic Elements201

5.1.2 Dynamic MNA Equations203

5.1.3 DAEs and ODEs204

5.2 ODE Solution Techniques206

5.2.1 ODE Systems and Basic Theorems206

5.2.2 Overview of Solution Methods209

5.2.3 Three Basic Methods：FE，BE，and TR211

5.2.4 Quality Metrics215

5.2.5 Linear Multistep Methods220

5.3 Accuracy of LMS Methods221

5.3.1 Order221

5.3.2 Consistency223

5.3.3 The Backward Differentiation Formulas224

5.3.4 Local Truncation Error225

5.3.5 Deriving the LMS Methods228

5.3.6 Solving Implicit Methods229

5.3.7 Interpolation Polynomial231

5.3.8 Estimating the LTE237

5.4 Stability of LMS Methods241

5.4.1 Linear Stability Theory242

5.4.2 The Test Equation243

5.4.3 Absolute Stability246

5.4.4 Stiff Systems252

5.4.5 Stiff Stability253

5.4.6 Remarks256

5.5 Trapezoidal Ringing257

5.5.1 Smoothing258

5.5.2 Extrapolation259

5.6 Variable Time-Step Methods261

5.6.1 Implementing a Change of Time-Step262

5.6.2 Interpolation Methods262

5.6.3 Variable-Coefficient Methods264

5.6.4 Variable Step Variable Order（VSVO）Methods265

5.7 Application to Circuit Simulation265

5.7.1 From DAEs to Algebraic Equations266

5.7.2 FE Discretization269

5.7.3 BE Discretization271

5.7.4 TR Discretization277

5.7.5 Charge-Based and Flux-Based Models282

5.7.6 Multiterminal Elements291

5.7.7 Time-Step Control296

5.7.8 Enhancements298

5.7.9 Overall Flow299

Notes300

Problems300

Glossary305

Bibliography307

Index311