State–space formulation of linear circuits containing periodically operated switches

Abstract

SUMMARYThe state-space formulation of linear circuits containing periodically operated switches requires one to obtain the state and output equations. Further, it requires one to obtain the switching equations where discontinuities in the state variables occur at a switching instant. A new method for the computation of the state and output equations is given. Further, a procedure is described to obtain the switching equations, in the most general form, at a switching instant. Examples are given to illustrate these methods in finding the state, the output and the switching equations.

Determination of initial conditions in active RLC networks

“…The last column vector is calculated as Following exactly the same steps as in Reference 2, (1 1) yields which is the extended state equation and valid for -a < t <a. In fact, with (12), and for t > 0, (13) reduces to the state equation (23) of Reference 2. Comparing (13) with (6) and (12) with (4), and using the result in (lo), the initial value of il, i.e.…”

“…In fact, with (12), and for t > 0, (13) reduces to the state equation (23) of Reference 2. Comparing (13) with (6) and (12) with (4), and using the result in (lo), the initial value of il, i.e. il(O+), can be found in the following form:…”

A computer algorithm to determine the periodic steady-state solutions of nonlinear systems using spectral analysis

“…An important problem in the computer-aided theory of nonlinear systems is the steady-state analysis of nonlinear circuits with periodic responses. There are 2 basic approaches in the literature, i.e., time-domain approaches [1][2][3][4][5] and frequency-domain approaches [6][7][8][9][10][11][12][13]. In principle, the periodic response, if it exists, can always be found by integrating the system differential equations until the transient responses die out, which is known as the 'brute force method' but it is computationally very expensive in slowly settling systems.…”

“…A common property of these algorithms is that they involve the numerical integration of the system equations over one or several periods, which is rather time-consuming for large-scale networks. In the time domain, another approach is Köksal's method [4], which is based upon the use of the previous results about the linear time-invariant networks containing periodically operated switches [5]; this is made possible by treating the nonlinear system as a periodically time-varying one under the steady-state conditions. It does not require numerical integrations, which are necessary at least over 1 period in the shooting methods, and it does not use optimization routine programmes as in some of the frequency domain methods.…”