NETAVORKS FOR IMPEDANCE FUNCTIONS 



387 



The iinpedancc^ of the short-circuited lino (Fig. 5) is 



Z = Zo tanii r 



(i-o; 



where Zo is the characteristic impedance and T is the total i)r()pagati()ii 

 constant of the hnc. We have 



(R + pLY 

 \G + pCj 



.G + pC, 



T = m + pL)(G + pC)f" 



(1-1) 



(1-2) 



R, L, G and G bein"; gix'cn for the total lou/lh of line. 



To obtain a development in terms of network bi'anches of the kind 

 shown in Fig. 1, we consider the admittance function, 



}' = I'o coth r 



(1-3) 



where F = 1/Z and I'o = 1/Zo . Our first task is to find the poles of this 

 function and the residues. Since the complex freciuency variable p occurs 



R,L,G,C 



2=^ 



Fig. 5 — Short-circuited transmission line. 



under square roots in both Zo and T, it might be suspected, offhand, that 

 the singularities of the function are branch points rather than poles. 

 Such is not the case, however. There are no branch points and all the 

 poles are simple. 



The singularities of Y are to lie found among the zeros of tanh F, 

 which occur at 



r = iwn, n = 0, ± 1, ± 2, ± 3, • • • 

 To determine them, we soh^e 



r' = (/? + pL){G + pG) = -ttV 

 and find these roots: 



Pn = —Oin-\- i^n , P-n = Pn = " ^n " t^n 



where 



(1-4) 

 (1-5) 



[n > 0) (1-6) 



