Time Varying Electrical Networks

Abstract

The operational representation of linear time varying electrical networks is described. A particular case of application is that of ladder networks. A procedure is described for obtainment of an input output function for such structures in such a way that the network components producing the terms are placed in evidence. INTRODUCTION Electrical networks may be employed both as dedicated electrical networks and electrical models of physical systems as models of electrical and of non-electrical systems. Areas of application include electrical power systems, communications, and aerospace. The dynamics of such systems are often of a time varying nature and give rise to time varying differential equations. In particular, as will be the concern here, to linear time varying differential equations. For example, an automatic control system for the loading, transportation and unloading of a body of fluid will have time varying dynamics due to the mass of the fluid varying*during the process. It is well-known that all linear time varying electrical networks can be expressed in a state variable form px = Ax + Bu where A(t) is the state matrix, u(t) is an input vector, B(t) is an input coupling matrix and x(t) is a state vector. p is a differential operator : px = dx/dt and t is time. The state vector may be obtained using a maximally capacitive tree in the graph G of the network. A construction for such a tree and an explicit form for A are given in Bryant, [1]. A direct numerical solution can be obtained computationally from the state equation. However, the structure of the original problem has become dissolved in the state equation, in Transactions on Engineering Sciences vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533