NETWORKS FOR IMPEDANCE FUNCTIONS 



389 



Thus, for small dissipation, the resonances would became sharper in di- 

 rect proportion to the frequency (if the parameters R, L, G, C, were 

 invariable with frequency, as assumed). 



The above described branches of the ecjuivalent network account only 

 for the complex poles (n > 1) of the admittance function. Two more 

 branches remain to be calculated. One is for the real pole (n = 0), which 



occurs at po = —R/L, with residue, Aq = - . The required branch for 



Li 



this pole is 



Ao 1 



p — po R + pL 

 The other is the final conductance branch, which is calculated as follows : 



Go = F(0) + f: 4:? = a/^ coth VGR - i 



n=-«, Pn y R R 



(1-13) 



- 2(? E 



nt^oo TT^n^ + GR 







so that, for this example, the conductance branch vanishes. The network 

 is drawn in Fig. 6. 



A series type of network, as shown in Fig. 7, can be determined by 



Fig. 6 — Network of the first kind equivalent to the short-circuited line of 

 Fig. 5. 



(Ro=o) 



Vv\ 



G 



•-^AA/ — —ymu-^ 



2R 2L 



^'iif 



— vw 



G 



i— Vv^ — ^smy-^ 



2R 2L 



Fig. 7 — Network of the second kind ecjuivalent to the short-circuited line of 

 Fig. 5. 



