An Embedding Approach for the Simulation of Electronic Circuits with Widely Seperated Time Scales

Abstract

Widely separated time scales in electronic circuits occur in many cases, especially in radio frequency circuits, making analysis with standard numerical methods very difficult and costly. Low and high frequency parts of the solution are often superimposed enforcing very tiny time steps over a long period in the computation of the numerical solution. In this paper we present a general method of embedding the differential-algebraic equations (DAEs) which describe the electronic circuit in a system of partial differential equations (PDEs), such that a restriction of the solution of the PDEs onto a suitable path yields the desired solution of the DAEs. This allows to treat such different frequency-parts seperately in different dimensions. In many cases the physical background provides knowledge about the frequencies and the solution of the PDEs is periodic in all or some dimensions with often known periodicity lengths. The solution of the PDEs then needs only to be computed on its basic periodicity domain, where it is often a very slowly varying function. Therefore computing the solution of the PDEs can be much more efficient. Here theoretical results are presented as well as new approaches to numerical methods, based on the embedding method.