Frequency Analysis of Variable Networks

This paper describes an approach to the analysis of linear variable networks which is essentially an extension of the frequency analysis techniques commonly used in connection with fixed networks. It is shown that a function H(jω; t), termed the system function of a variable network, possesses most of the fundamental properties of the system function of a fixed network. Thus, once H(jω; t) has been determined, the response to any given input can be obtained by treating H(jω; t) as if it were the system function of a fixed network. It is further shown that H(jω; t) satisfies a linear differential equation in t, which has complex coefficients and is for the same order as the differential equation relating the input and the output of the network. Two methods of solution of this equation covering most cases of practical interest are given. In addition to H(jω; t), a network function introduced is the bi-frequency system function Γ(jω; ju) which is shown to relate the Fourier transforms of the input and the output through a superposition integral in the frequency domain. On the basis of the results obtained in this paper it appears that the frequency domain approach using H(jω; t) possesses significant advantages over the conventional approach using the impulsive response of the network.