A General Theory of Commutated Networks

Abstract

A state-space approach is used to derive an exact closed-form solution for the steady-state and transient response of a general commutated network terminated in a multiport. By expanding the time-varying transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(p, t)</tex> in a Fourier series, the transfer function at input frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{0}(p)</tex> , and the transfer functions at harmonic frequencies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{m}(p)</tex> are then calculated. A necessary and sufficient condition for the recovery of the input signal, without distortion due to the harmonics, is given. From the general analysis, we immediately obtain previously available results on comb filters, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -path filters, sample-data filters, etc., as special cases. The resulting closed-form solutions in terms of element values are most suitable for computer simulation in which the performance of the commutated network is to be evaluated as the element values are varied.