Chain matrix decompositions, continued fractions and the optimal inhomogeneous ladder networks

Abstract

Studies of two-element-kind ladder networks are well known in the classical literature, among them, the most celebrated ones are due to Cauer. Driving point immittance function synthesis by using continued fractions to obtain the series and shunt arm L-C element values is a standard and routine work. The idea of introducing a class of more general networks, the inhomogeneous ladder networks, was first developed by Lee and Brown, and subsequently the synthesis techniques of such a network were established. In this paper, new results are found such as: (1) the Iff. conditions of the existence of an inhomogeneous ladder network by a given chain matrix of the network satisfying: (a) determinant of the chain matrix is 1; (b) the zeros of A(s) and z −1B(s) or A(s) and y −1C(s) alternate with respect to [z(s)y(s), k] with an appropriate leading set of zeros of A(s); (c) the poles of A(s) and z −1B(s) or A(s) and y −1C(s) are the poles of z(s)y(s) of multiplicity of n and n−1, where n the number of sections of ladder networks; (2) the Iff. condition for the inhomogeneous ladder network to be optimal is that it be antimetrical, whereas for the extended class of inhomogeneous ladder networks it is symmetrical, where an optimal inhomogeneous ladder network is defined as the corresponding network with the minimum sum of immittance levels in the series and shunt arms; (3) algorithms of synthesis procedures were developed as the by-products of the Theorems.