Circuit analysis in the time-domain: Operational approach to form the system-matrix exponential and to obtain the circuit natural response therefrom

A time-domain operational method for the dynamic analysis of force-free Linear Time-Invariant (LTI) lumped-parameter electrical systems is presented. Starting from the state model of the system, a purely time-domain technique for obtaining the matrix exponential (state-transition matrix) has been developed. This approach preserves the time-domain identity of all the network variables (unlike the transformational approaches) and facilitates analytical, closed-form solutions for number-imposed system parameters (unlike numerical techniques). Furthermore, it provides the `algebrization'-advantage in the time-domain itself, in which the system variables are directly observable. This obviates the need for any artificial and cumbersome transformations (both direct and inverse). This approach has been here employed to formulate the solution process of the natural (force-free) response of a generic LTI lumped-parameter electrical network of any given order, starting from the vectorized set of time-domain differential equations that collectively govern the behaviour of the said network. Some crucial aspects regarding system modeling have been highlighted. The proposed method is illustrated by examples which demonstrate the ease with which the natural response of an LTI system could be obtained, for specified initial conditions.