386 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



111 the case of the parallel-type iietAVorks (Figs, la and lb), p„ = 

 — an + il^H is a pole of the admittance, F(p), and A„ = a„ + ih^ is the 

 corresponding resicUie. In the case of the series-type network, the same 

 symbols represent a pole and residue of the impedance. Zip). 



The networks specified by Figs. 2a and 2b are duals of the networks 

 of Figs, la and lb, respectively, and are obtained from the latter merely 

 by replacing L„ by C„ , Rn by Gn , and vice versa. 



The formulas are intended to apply to complex poles. They can be 

 applied to real poles by taking 6„ and /?„ equal to zero and doubling the 

 residue, a„ , but this procedui'e is uiniecessary, because the network rep- 



Fig. 3 — Network of the first kind (branches la) 



C, C2 



\[ 



Ro 



— ^^A — 



I 



R, L, 



\^A •— • * 



R2 U 



Fig. 4 — Network of the second kind (branches 2a). 



resent ation of a real pole can be found readily enough by inspection of 

 the impedance terms involved. (See Example 1.) 



The above discussion is intended to sketch a general picture of the 

 procedure. Indi\'idua] cases may involve considerable detail that can be 

 understood more readilv bv reference to the next section. 



APPLICATIONS 



Example la: A transmission line with its far terminals short-circuited 

 affords a simple illustration of the equivalent network theory. Let it be 

 assumed that the parameters, R, L, G and C of the line are constants. In 

 the more advanced examples to follow, the \'ariation of these parame- 

 ters with frequency for a particular kind of line will be taken into con- 

 sideration. 



