Some properties of the transfer function of unbalanced RC networks

A method is presented for deducing some RC network theorems, chiefly concerning the transfer function of an unbalanced structure from the properties of pure resistance networks. This is possible mainly since: (a) Zero of the transfer function (voltage ratio) of a 3-terminal R network has a clear topological interpretation, namely in the case of a zero transfer the network disintegrates in two parts (at least), which may have only the reference node in common, and the input and output terminals belong to different parts. (b) The expansion of the RC network transfer function about the values of all its susceptances is a quotient of two polynomials in all these susceptances. The coefficients in this expansion are composed exclusively of cofactors of purely real matrices. These matrices correspond to the original network with some capacitors short-circuited and others open-circuited. (c) Vanishing of some coefficients in the numerator of the expansion of the transfer function of a 3-terminal RC network implies a zero of the transfer function for some of the R networks derived in this way. This fact according to (a) imposes definite conditions upon the topology of the original RC network These limitations show that the only possible form of a degeneration of the transfer function in its non-reduced form is through the disappearance of some left or right consecutive terms of the numerator. For a generalized ladder realization of such degenerated transfer function the discussion offers the limits of the maximum gain.