388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



For n = 0, the above would give 



R G 



But if we let T -^ 0, so that tanh T — > F, we find that only the point, 

 —R/L, is a singularity of Y] the other point, —G/C, is a regular point. 

 Therefore Y has only one real singularity. 



To find the nature of the singularities of F, we next calculate 



Limit 



V - Vn 



p-*pn LZiiip) tanh r(p) 

 and find that at each p„ the limit exists and has the value 



= An (1-7) 



1 ^"""l) 



Zo(p„)r'(p„) L ^nL 



where r'(p„) = -— r(p), evaluated at p = Pn ■ The fact that this limit 

 dp 



exists shows that all the singularities are simple poles. The values of ^„ 



are then the residues at these poles. 



When we now apply formulas (6) to determine the elements in the 



general branch of the equivalent network of Fig. la, we obtain, for 



n> I, 



T = - 1 ^ T^ ^ = ? Rn ^ R / qv 



" " 2' LnCn ~ LC Cn~ C L„ " V ^'^^ 



The network then comprises an infinite number of such branches in 

 parallel. Each branch has the same elements 2?„ and L„ , equal, respec- 

 tively, to half the total resistance and inductance of the transmission 

 line, but the elements Gn and C„ decrease from one branch to the next in 

 inverse proportion to the squares of the integers. 



The Q of the n^^ branch, which can be regarded as the Q of the asso- 

 ciated resonance of the short-circuited line, is 



Q_ Wn tOn C*)„ 



n — 



where 



2a„ Gn Rn G R (1-10) 



Cn Ln C L 



- ./I ^n _ ,/^ G' (, 11X 



'" ~ y LnCn CI ~ V LC~ C' ^ ^ 



