Artificial parameter homotopy methods for the DC operating point problem

Efficient and robust computation of one or more of the operating points of a nonlinear circuit is a necessary first step in a circuit simulator. The application of globally convergent probability-one homotopy methods to various systems of nonlinear equations that arise in circuit simulation is discussed. The coercivity conditions required for such methods are established using concepts from circuit theory. The theoretical claims of global convergence for such methods are substantiated by experiments with a collection of examples that have proved difficult for commercial simulation packages that do not use homotopy methods. Moreover, by careful design of the homotopy equations, the performance of the homotopy methods can be made quite reasonable. An extension to the steady-state problem in the time domain is also discussed. >

[1]  Martin Hasler,et al.  Monotonicity in nonlinear resistive circuits , 1990, IEEE International Symposium on Circuits and Systems.

[2]  Jr. A. Willson.  The no-gain property for networks containing three-terminal elements , 1975 .

[3]  Robert G. Meyer,et al.  Analysis and Design of Analog Integrated Circuits , 1993 .

[4]  L. Chua,et al.  A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits , 1976 .

[5]  Griewank,et al.  On automatic differentiation , 1988 .

[6]  L. Chua,et al.  On the application of degree theory to the analysis of resistive nonlinear networks , 1977 .

[7]  L. Watson.  Solving Finite Difference Approximations to Nonlinear Two-Point Boundary Value Problems by a Homotopy Method , 1980 .

[8]  Fathi M. A. Salam,et al.  Parallel processing for the steady state solutions of large-scale non-linear models of power systems , 1989, IEEE International Symposium on Circuits and Systems,.

[9]  L. Watson,et al.  Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homoto , 1987 .

[10]  J. Munkres,et al.  Calculus on Manifolds , 1965 .

[11]  Kiyotaka Yamamura,et al.  Solving nonlinear resistive networks by a homotopy method using a rectangular subdivision , 1988, 1988., IEEE International Symposium on Circuits and Systems.

[12]  Alberto Sangiovanni-Vincentelli,et al.  Time-Domain Methods , 1990 .

[13]  Bjarne Stroustrup,et al.  C++ Programming Language , 1986, IEEE Softw..

[14]  Ping Yang,et al.  Circuit simulation and modeling , 1987 .

[15]  C. Pan,et al.  A systematic search method for obtaining multiple solutions of simultaneous nonlinear equations , 1975 .

[16]  D. Sattinger,et al.  Calculus on Manifolds , 1986 .

[17]  William J. McCalla.  Nonlinear Equation Solution , 1988 .

[18]  Alan N. Willson,et al.  Nonlinear networks : theory and analysis , 1975 .

[19]  R. Saeks,et al.  Continuation methods in circuit analysis , 1977, Proceedings of the IEEE.

[20]  A. Sard,et al.  The measure of the critical values of differentiable maps , 1942 .

[21]  R.C. Melville,et al.  Finding DC operating points of transistor circuits using homotopy methods , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[22]  Alberto L. Sangiovanni-Vincentelli,et al.  Steady-state methods for simulating analog and microwave circuits , 1990, The Kluwer international series in engineering and computer science.

[23]  G. Wilson.  A monolithic junction FET-n-p-n operational amplifier , 1968 .

[24]  L. Chua,et al.  Tracing solution curves of nonlinear equa-tions with sharp turning points , 1984 .

[25]  L. Trajkovic,et al.  Passivity and no-gain properties establish global convergence of a homotopy method for DC operating points , 1990, IEEE International Symposium on Circuits and Systems.

[26]  Leon O. Chua,et al.  Energy concepts in the state-space theory of nonlinear n-ports: Part I-Passivity , 1981 .

[27]  A. Jimenez,et al.  Algorithms for ASTAP--A network-analysis program , 1973 .

[28]  T. Aprille,et al.  Steady-state analysis of nonlinear circuits with periodic inputs , 1972 .

[29]  Raymond A. DeCarlo,et al.  Continuation methods: Theory and applications , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[30]  H. Taub,et al.  Communication Circuits: Analysis and Design , 1973, IEEE Trans. Commun..

[31]  T. C. Banwell.  Performance limitations of low-voltage regulators using only n-p-n transistors , 1991 .

[32]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[33]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[34]  L. Brouwer.  Über Abbildung von Mannigfaltigkeiten , 1911 .

[35]  Albert E. Ruehli,et al.  The modified nodal approach to network analysis , 1975 .

[36]  L. Watson.  An Algorithm That is Globally Convergent with Probability One for a Class of Nonlinear Two-Point Boundary Value Problems , 1979 .

[37]  L. Watson.  Engineering applications of the Chow-Yorke algorithm , 1981 .

[38]  J. Yorke,et al.  Finding zeroes of maps: homotopy methods that are constructive with probability one , 1978 .

[39]  A. Brokaw,et al.  A simple three-terminal IC bandgap reference , 1974 .

[40]  Yannis Tsividis,et al.  Fully differential operational amplifiers with accurate output balancing , 1988 .

[41]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[42]  W. Mathis,et al.  Finding all DC equilibrium points of nonlinear circuits , 1989, Proceedings of the 32nd Midwest Symposium on Circuits and Systems,.

[43]  D.J. Rose,et al.  CAzM: A circuit analyzer with macromodeling , 1983, IEEE Transactions on Electron Devices.

[44]  Layne T. Watson,et al.  Solving Galerkin approximation to nonlinear two-point boundary value problems bya globally convergent homotopy method , 1987 .

[45]  Kiyotaka Yamamura,et al.  A piecewise-linear homotopy method with the use of the Newton homotopy and a polyhedral subdivision , 1990 .

[46]  Debasis Mitra,et al.  When are transistors passive , 1971 .

[47]  C. W. Yun,et al.  Simple solution curves of non‐linear resistive networks , 1983 .

[48]  R.C. Melville,et al.  Improving DC convergence in a circuit simulator using a homotopy method , 1991, Proceedings of the IEEE 1991 Custom Integrated Circuits Conference.

[49]  Andrew H. Wallace,et al.  Differential topology; first steps , 1968 .

[50]  Joos Vandewalle,et al.  A globally convergent algorithm for solving a broad class of nonlinear resistive circuits , 1990, IEEE International Symposium on Circuits and Systems.

[51]  Martin Hasler,et al.  Nonlinear Circuits , 1986 .

[52]  Joos Vandewalle,et al.  Variable dimension algorithms for solving nonlinear resistive circuits , 1989 .

[53]  T. Ohtsuki,et al.  Existence Theorems and a Solution Algorithm for Piecewise-Linear Resistor Networks , 1977 .

[54]  D. Rose,et al.  Global approximate Newton methods , 1981 .

[55]  Paolo Antognetti,et al.  Semiconductor Device Modeling with Spice , 1988 .

[56]  Kiyotaka Yamamura,et al.  A globally and quadratically convergent algorithm for solving nonlinear resistive networks , 1990, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[57]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.