Analytical non‐Fourier solution of linear time‐invariant circuits with periodic inputs

Abstract

A systematic approach for solving linear time-invariant electric circuits with periodic input signals is presented. The method involves the following steps:(1) The governing differential equation for the desired output is solved for each time segment of the input signal in one period, leaving the constant coefficients of the homogeneous parts of the solution as unknowns. For an nth-order circuit with a source consisting of q segments in one period, there are n  q such unknown coefficients. (2) Using the solution obtained in step 1, the voltage of each capacitor and the current of each inductor are determined.(3) Transition conditions for the voltages and currents found in step 2 at the transition points of successive segments and also periodicity conditions for the beginning and end points of the period are applied. (4) Implementation of step 3 results in a system of n  q equations in terms of the n  q unknown coefficients described in step 1. Solving this system of equations, the circuit response in one period becomes readily available.