computer methods for circuit analysis and design
TRANSCRIPT
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