A General Approach to Circuit Equations

Computer aided circuit analysis is a complex procedure, which may be roughly divided into three steps. The first stage is device modelling, where quantitative properties of the participating electronic devices are determined. In the second step, the circuit’s topology in conjunction with the device equations is being exploited, in order to formulate a linearized nonsingular equation set. Eventually, the equation set is iteratively solved in the third phase, yielding a solution vector of circuit variables. This paper deals with the second and third step, where diverse sparse matrix methods are employed. Its ambition is to provide a general mathematical form, by which most matrix manipulating techniques can be described, thus enabling their classification on a theoretical level. Usually, these methods would be presented descriptively rather than strictly mathematically. The paper introduces a matrix reduction operator, from which a general matrix operator equation is deduced. This operator equation can be used to describe entire analysis approaches. After some necessary definitions, the purpose of this paper is illustrated by studying several classic examples rather than giving lots of mathematical proof. The selected examples involve some well known linear equation set solution methods as well as several typical equation set transformations.