Existence of solutions for the equations of transistor-resistor-voltage source networks

We consider the dc equations of nonlinear networks containing resistors, independent voltage sources (independent current sources are excluded), and certain types of nonlinear devices (such as Ebers-Mollmodeled transistors and diodes) that possess a certain property closely related to passivity. It is proved for the first time that the equations always possess at least one solution. This result complements some of the writers' previous work in which attention was not focused on networks containing only independent sources of the voltage type. In fact it was shown by simple examples given earlier that there exist transistor networks (containing ideal independent current sources) for which the network equations have no solution. Here we also complete a study of conditions under which it is possible to carry out certain implicit numerical integration algorithms for the computation of the transient response of an important class of nonlinear networks containing transistors and diodes. We in fact prove that the assumption of passivity for the transistors and diodes implies that it is always possible to carry out the algorithms (in the sense that for any value of the step size there is always at least one solution of a certain key set of nonlinear equations).