Switched network analysis based on final value theorem of Laplace transform

Abstract

This paper proposes a new method of analysis in the modified nodal analysis of the switched network. In the proposed method, the limit of the switching resistance r for 1/∞ or ∞ is replaced by the limit of the variable “s” of Laplace transform for s 0. Then by applying the final value theorem of the Laplace transform, the solution of the network equation immediately after switching is derived as the limit for t ∞ of the virtual state equation. By this transformation based on the final value theorem of the Laplace transform, the limiting operation of the switching resistance can be transformed into the iterative calculation of the product of the nonsingular matrix, which does not depend on the switching resistance, and a vector. In this new method, the coefficient matrix of the backward Euler method in the numerical integration of the virtual state equation is always nonsingular. This makes it possible to designate the next switching pattern without a particular algorithm, even in the singular problem where the network equation becomes singular for a certain switching pattern. When the state before the switching is the actual solution, as in the SC network, this method can derive the discontinuous solution which jumps from one state to the next state. Since the coefficient matrix of the backward Euler method is always nonsingular, the inverse matrix lemma can be applied to the calculation of the inverse matrix, which can be executed by lowering the order of the matrix from the given matrix to the actual order of the switched pattern.