Network Representation of Transcendental Impedance Functions

The purpose of the paper is to show that the admittance or impedance of certain continuous structures, such as, for example, a finite length of transmission line of any sort, or resonant cavity, can be represented exactly at all frequencies by a network comprising lumps of constant resistance R, inductance L, conductance G and capacitance C. The network will contain an infinite number of branches, in general, although a finite number may be used if it is desired to represent only certain modes. The procedure is based upon a proposition known to students of f unction theory as "Mittag-Leffler's theorem," which amounts, roughly, to an extension of rational functions to apply to transcendental functions of the type encountered in the theory of continuous structures. Several illustrative examples of the network synthesis are given. GENERAL Students of network theory are familiar with the fact that the impedance at a pair of terminals in a linear network comprising a finite number of resistors, inductors and capacitors, connected in any manner, is a rational function of the frequency having, in general, the fractional form of one polynomial divided by another. They are also familiar with the partial fraction rule whereby the function can bo broken up into a series of elementary fractions, each of which exhibits one of the poles of the original function. This form is sometimes useful in the problem of network synthesis, where the impedance function is given and the object is to find a network having this impedance.