Network Representation of Transcendental 

 Impedance Functions 



BY M. K. ZINN 



(Manuscript received November 5, 1951) 



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 trans- 

 mission line of any sort, or resonant cavity, can be represented exactly at 

 all frequencies hy 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 function 

 theory as "Mittag-Leffler's theorem," which amounts, roughly, to an ex- 

 tension 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 im- 

 pedance 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. Thej'' are also familiar with 

 the partial fraction rule whereby the function can be 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 ob- 

 ject is to find a network having this impedance. 



The purpose of the present paper is to show how a similar procedure 

 can be carried out for certain transcendental impedance functions per- 

 taining to structures having distributed constants, such as, for example, 

 a resonant cavity or a piece of transmission line. The method employ's a 

 well-known proposition of function theory, which is usually referred to 

 as Mittag-Leffler's theorem. This theorem provides a tool for breaking 

 up a transcendental meromorphic function into an infinite series of 

 simple fractions in much the same way as the partial fraction rule is used 

 to break up a rational meromorphic function. The series representation 



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