An Algorithm for Solving Nonlinear Resistor Networks

This article considers an algorithm for solving electrical networks which consist of linear and nonlinear resistors and independent sources and where the current-voltage relation of each of the nonlinear resistors is described by a function (?*(·)> i = Gk(v), where Gk(-) is continuous, monotonically increasing, piecewise linear, and one-to-one from (-- qo , oo) onto (-- and where k is an index which spans all the resistors in the networks. A piecewise linear network of this type can be considered to be an approximation to a more general nonlinear resistor network where the corresponding function Gk(-) is continuous, monotonically increasing, 00 and one-to-one from ) onto (-- c c ) 0 0 ) but not necessarily piecewise linear. 1605 1014 TIIE BELL SYSTEM T E C H N I C A L J O U R N A L , OCTOBER 1905 Networks of the last type were discussed by various authors, 1,2,3>4 ' ' in particular Duffin, who has shown 1 that such networks have a solution which is unique. Various methods were proposed for finding the numerical value of the solution. Birkhoff and Diaz 2 gave an iterative method similar to Seidel's method 5 which is a form of relaxation procedure for solving linear equations. A direct iterative method 5 similar to the "standard" method of solving linear equations by iteration was described by Katzenelson and Seitelman. 6 An exact method (convergence in a finite number of steps) was described by Minty. 7 This latter method approximates a monotonic increasing characteristic by "stairs" and solves the approximating network by a search procedure.