Theorems on the Computation of the Transient Response of Nonlinear Networks Containing Transistors and Diodes

We consider in detail the nonlinear equations encountered at each time step when certain implicit numerical-integration algorithms are used. In terms of only the properties of the Jacobian matrix of the pertinent set of differential equations, we present necessary and sufficient conditions for the existence and uniqueness of the solution of the nonlinear equations for all continuous forcing functions and any given step size. Since engineers often think about dynamic nonlinear transistor network problems in terms of the eigenvalues of the relevant Jacobian matrix, the results described are of immediate conceptual value. In particular, it is possible to carry out the algorithms whenever the conditions presented are satisfied. Several other types of results are also presented. For example, for a special but significant and useful numerical-integration formula, theorems are proved concerning properties of the computed sequence such as the extent to which the sequence is relatively immune to small local errors introduced at each step as a result of the fact that it is ordinarily not possible to compute the solution of a certain equation exactly. All of the results are concerned with network models that are often used in computer simulations. In fact, we heavily exploit some special properties possessed by the nonlinear functions associated with such models.