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 Eninto itself, A is a real n × 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 nonlinearities, is proved; this result shows that under certain conditions the equation F(x) + Ax = B has at least two solutions for some B ∊ 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 nonlinearities, and therefore makes more clear that the set of matrices denoted in earlier work by P0plays 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 ∊ Enfor “the original set of α's as well as for an arbitrary set of not-larger α's”, 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.