A method for determining the normal modes of foster's reactance networks

Abstract

A method is presented for finding the zeros of the impedance or admittance function of a Foster reactance network. More generally, the method is one of finding the zeros of a rational function having alternate zeros and poles, all of which are simple, if the poles are known; or of finding the poles if the zeros are known. The method consists of writing the function as a sum of partial fractions and replacing all but two of the fractions, depending on the zero to be found, by a constant and solving the resulting quadratic equation. In this way an upper and lower bound of a zero may be found, and, if need be, the method may be applied repeatedly to give a pair of bounds which will be as close to the required zero as necessary. The method has particular value if the rational function is the impedance or admittance function of one of Foster's reactance networks, since these functions are obtained directly from the networks as sums of partial fractions of the form A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> /(Bn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> – w <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Hence, the zeros may be obtained without reducing the function to the polynomial form. The new method may not always be shorter in application than other known methods, but its advantage lies in the facility with which an upper and lower bound of the required zero may be found and in the possibility of dealing with the partial-fraction form of the impedance or admittance function.