NETWORKS FOR IMrKDAXCE FUNCTIONS 379 



provides a means of (letormiiiin<2; a netwoik of resistors, inductors and 

 capacitors that will have an inii)e(huicc (M|ual to ihe s|-)(>cifie(l ti-anscen- 

 (lental impedance function. 'I'liis pi-occss will he I'cfci-red to as ol)tainiii.u; 

 a ''network representation" of the function. If the g;i\-en function is the 

 impedance of some continuous (i.e., non-lumped) electric structure, the 

 result ^^'ill be an equivalent network for the structure. For other pur- 

 poses, such as, possibly, analogue methods of computing, the given func- 

 tion may not arise from any electrical structure. In either case, the net- 

 work representations to Ix^ deiixcd are possible only if the function 

 satisfies certain i-esti'ictions, which are stated in the section immediately 

 following. 



The discussion is confined to transcendental impedance functions he- 

 cause of the technological interest in the electromagnetic structures with 

 which they are associated and because they have not received as much 

 attention as rational functions in the literature dealing with network 

 synthesis. The problem with which this paper is concerned can then be 

 stated as follows: gi\-en, a transcendental impedance function satisfying 

 certain conditions: to determine a network comprising elements of 

 constant resistance, inductance and capacitance whose driving-point im- 

 pedance function, at a pair of terminals, will equal the given function at 

 all frequencies, real and complex (except at the poles). 



For illustration of the procediu'e, three examples are given. The first 

 is the impedance of a short-circuited or open-circuited transmission line 

 in which the distributed primary constants, R, L, G and C are assumed 

 to be invariable \\ith frequency. The second and third examples are the 

 impedances of resonant cavities driven in two different modes. In these 

 examples the variation of resistance with frequency, due to "skin-effect," 

 is taken into account. 



IMPEDANCE FUNCTIONS 



The functions inider discussion will he referred to as "impedance 

 functions" with the understanding that the term is meant to include 

 "admittance functions" as well. By reason of the duahty principle that 

 runs through all electric circuit theory, any general proposition devel- 

 oped f(^r one must apply to the other. The functional designation, F(p), 

 will be used to denote either an impedance or an admittance function. 

 When a distinction is necessary, the impedance will he designated by 

 Z(p) and the admittance by Y(p). The independent compk^x variable p 

 is the generalized radian frequency. (For sustained sinusoidal currents 

 and voltages, p = ico = ^rif where / is the real frequency.) 



For the applications contemplated, F(p) is a transcendental mero- 



