Theorems on the Analysis of Nonlinear Transistor Networks

This paper reports on further results concerning nonlinear equations of the form F(x) + Ax = B, in which F(-) is a "diagonal nonlinear mapping" of real Euclidean n-space En into itself, A is a real n X n matrix, and B is an element of En. Such equations play a central role in the dc analysis of transistor networks, the computation of the transient response of transistor networks, and the numerical solution of certain nonlinear partial-differential equations. Here a nonuniqueness result, which focuses attention on a simple special property of transistor-type nonlinear ities, is proved; this result shows that under certain conditions the equation F(x) + Ax = B has at least twc solutions for some B 6E En. The result proves that some earlier conditions for the existence of a unique solution cannot be improved by taking into account more information concerning the nonlinear ities, and therefore makes more clear that the set of matrices denoted in earlier work by P0 plays a very basic role in the theory of nonlinear transistor networks. In addition, some material concerned with the convergence of algorithms for computing the solution of the equation F(x) + Ax = B is presented, and some theorems are proved which provide more of a theoretical basis for the efficient computation of the transient response of transistor networks. In particular, the following proposition is proved. If the dc equations of a certain general type of transistor network possess at most one solution for all B £ E" for "the original set of as as well as for an arbitrary set of not-larger as", then the nonlinear equations encountered at each time step in the use of certain implicit numerical integration algorithms possess a unique solution for all values of the step size, and hence then for all step-size values it is possible to carry out the algorithms.