Computer Methods for Circuit Analysis and Design

Second Edition

Jiri Vlach Department of Electrical Engineering University of Waterloo Waterloo, Ontario, Canada

Kishore Singhal AT&T Bell Laboratories Allentown, PA

and

Department of Systems Design University of Waterloo Waterloo, Ontario, Canada

KLUWER ACADEMIC PUBLISHERS BOSTON/DORDRECHT/LONDON

Contents

Preface

Programs

1. Fundamental Concepts

1.1. Basic Network Elements / 1 1.2. Independent Sources / 4 1.3. Capacitors and Inductors in the Laplace Transform

Domain / 5 1.4. Definition of Ports and Terminals / 10 1.5. Transducers (Dependent Sources) / 11 1.6. Elementary Two-Ports / 13 1.7. Thevenin and Norton Transformations / 17 1.8. Network Scaling / 19 1.9. Network Functions, Poles, and Zeros / 20 1.10. Time Domain Response / 24

2. Network Equation and Their Solution

2.1. Nodal Formulation / 32 2.2. Nodal Formulation for Networks with VCTs / 39 2.3. Mesh Formulation / 42 2.4. Linear Equations and Gaussian Elimination / 45 2.5. Triangular Decomposition / 48 2.6. Pivoting / 58 2.7. Sparse Matrix Principles / 59 2.8. Sparse Matrix Implementation / 66

3. Graph Theoretic Formulation of Network Equations

3.1. KVL, KCL, and Oriented Graphs / 95 3.2. Incidence Matrix / 96

Contents

3.3. Cutset and Loopset Matrices / 99 3.4. Orthogonality Relations for the Q and В Matrices /

105 3.5. Independent Currents and Voltages / 106 3.6. Incorporating Sources into Graph Considerations / 107 3.7. Topological Formulation of Nodal Equations / 109 3.8. Topological Formulation of Loop Equations / 112 3.9. State Variable Formulation / 116

General Formulation Methods 123

4.1. Tableau Formulation / 124 4.2. Block Elimination on the Tableau / 133 4.3. Modified Nodal Formulation Using One Graph / 134 4.4. Modified Nodal Formulation by Inspection / 138 4.5. Nodal Analysis of Active Networks / 144 4.6. Separate Current and Voltage Graphs / 155 4.7. Representation of the Graphs on the Computer / 162 4.8. Modified Nodal Formulation Using /- and K-Graphs /

164 4.9. Summary of the Formulation Methods / 173 4.10. Example / 174

Sensitivities 185

5.1. Sensitivity Definitions / 186 5.2. Multiparameter Sensitivity / 192 5.3. Sensitivities to Parasitics and Operational

Amplifiers / 196 5.4. Sensitivities of Active Networks / 202

Computer Generation of Sensitivities 214

6.1. Sensitivity of Linear Algebraic Systems / 2 1 5 6.2. Numerical Solution of the Adjoint System / 220 6.3. Adjoint System Method Applied to Networks / 221 6.4. Sensitivity to OPAMPs and Parasitics / 223 6.5. Applications of the Adjoint System Method / 228 6.6. Higher-Order Derivatives / 247 6.7. Group Delay / 249 6.8. Examples / 258

Contents vii

7. Network Functions in the Frequency Domain 265

7.1. Network Functions, Poles, and Zeros / 266 7.2. Computer Generation of Network Functions / 267 7.3. Unit Circle Polynomial Interpolation / 269 7.4. Condition Numbers for Interpolations / 272 7.5. Algorithm for Symbolic Function Generation / 273 7.6. Roots of Functions and Polynomials / 279 7.7. Root Refinement / 281 7.8. Poles and Zeros from System Equations / 286 7.9. Example / 288

8. Large Change Sensitivity and Related Topics 294

8.1. Large Change Sensitivity / 295 8.2. Differential Sensitivity Using the $ Matrix / 306 8.3. Fault Analysis / 3 1 0 8.4. Symbolic Analysis / 3 1 2

9. Introduction to Numerical Integration of Differential Equations 323

9.1. Simple Integration Methods / 323 9.2. Order of Integration and Truncation Error / 329 9.3. Stability of Integration / 331 9.4. Time Domain Solution of Linear Networks / 336 9.5. Inconsistent Initial Conditions / 343

10. Numerical Laplace Transform Inversion 349

10.1 Development of the Method / 350 10.2. Properties of the Method / 361 10.3. Application / 365 10.4. Stepping Algorithm for Networks and Network

Functions / 371 10.5. Stability Properties / 377 10.6. Inconsistent Initial Conditions / 380

11. Modeling 391

11.1. Diode Models / 392 11.2. Field Effect Transistor Models / 396 11.3. Bipolar Transistor Models / 401 11.4. Macromodels / 409 11.5. Automatic Differentiation / 414

viii Contents

12. DC Solution of Networks

12.1. The Newton-Raphson Algorithm / 427 12.2. Nodal Formulation / 432 12.3. Tableau and Modified Nodal Formulation / 437 12.4. Convergence in Diode-Transistor Networks / 439 12.5. DC Sensitivity / 440 12.6. Piecewise Linearization / 443

13. Numerical Integration of Differential and Algebraic-Differential Equations

13.1. Derivation of LMS Formulae / 456 13.2. Theory of LMS Formulae / 465 13.3. Properties of LMS Formulae / 471 13.4. Systems of Differential Equations / 474 13.5. Backward Differentiation with Variable Step and

Order / 477 13.6. Tableau and Modified Nodal Formulation of Nonlinear

Networks / 482

14. Digital Networks

14.1. Discrete Signals / 492 14.2. г-Transform / 499 14.3. Formulation of Digital Network Equations / 501

15. Switched Capacitor Networks

15.1. Principles / 510 15.2. Time Domain Analysis / 512 15.3. Formulation of Network Equations / 5 1 7 15.4. Frequency Domain Representations / 528 15.5. Analysis of the Digital System / 532 15.6. Frequency Domain Analysis / 535 15.7. Equal Phases / 536 15.8. Single Output and Its Sensitivity / 541 15.9. Solution of the Digital System / 544 15.10. Symbolic Analysis / 547 15.11. Group Delay / 549 15.12. Poles and Zeros / 551 15.13. Application / 551

Contents ix

16. Switched Linear Networks 557

16.1. Modeling of Switches / 558 16.2. Time Domain with Periodic Switching / 560 16.3. Frequency Domain / 562 16.4. Final Conditions / 565 16.5. Sensitivity / 568 16.6. Examples / 572 16.7. Internally Controlled Switches / 575

17. Introduction to Optimization Theory 585

17.1. Basic Definitions / 586 17.2. Classical Minimization / 588 17.3. Basic Iterative Algorithm / 590 17.4. Search Along a Line / 593 17.5. Quadratic Functions in Several Variables / 597 17.6. Descent Methods for Minimization / 603 17.7. Constrained Minimization / 608

18. Time Domain Sensitivities and Steady State 616

18.1. Sensitivity Networks / 617 18.2. Sensitivities of Objective Functions / 622 18.3. Steady State Using Sensitivity Networks / 625 18.4. Steady State on Objective Function / 627 18.5. Steady State by Extrapolation / 630 18.6. Steady State of Switched Networks / 633

19. Design by Minimization 643

19.1. Mean-Square Objective Functions / 644 19.2. Matching of Complex Values / 647 19.3. Minimax Design / 652 19.4. Design by Optimization / 653 19.5. Minimization of Sensitivities / 661 19.6. Monte Carlo Analysis / 667

20. Special Analysis Methods 672

20.1. Iterative Solution Methods / 673 20.2. Simulation Using Relaxation / 677 20.3. Piecewise Constant Approximation / 679

x Contents

Appendix A. Laplace Transforms 686

Appendix В. Partial Fraction Decomposition of Rational

Functions 692

Appendix C. Selected Mathematical Topics 697

Index 705