396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



The importance of including the effect of the end-plugs in determining 

 Q is shown by the fact that, if they were assumed to have zero impedance, 

 Q at the first resonance would be 12,120 instead of 4250. 

 To determine the residues at the poles, we write 



(1 - p)(l - pe-'-'') _ Fip) 

 ^ - ^° 2(1 - p^e-^y^) G(P) ^^^^ 



and then the residue at a simple pole p„ is 



An = ^!p{ (2-18) 



This limit is found to exist, showing that the poles are, in fact, simple. 

 The value found for the residue, ^„ , is 



An = an -\- ihn, A-n = An = ttn — iK 

 _ HpWQn /, _ 1 \\ 



""^ " "^1^ V 2(i5„ 2Un) (2-19) 



^ ^^oo^, / 1 1 \ 



irn \2dbn 2h8nJ 



When formulas (6) are applied to determine the elements of the tuned 

 branches of the equivalent network of the first kind, the results are, for 

 the n*^ branch, 



_ Koir7l / 1 1 



2won \ 2d8n 2hbn, 



1 2/1,1 2 



= coon 1 + "Tir — 



Gn _ WOn 



Ln " \d8n 2h8n 



In terms of the R, L and C of the piece of coaxial line, the elements of the 

 n^^ branch are as follows: 



^" ~ 2 V 2d5„ ^ 2h8n 



2 \ 2/i/ (2-21) 



" 7r2n2 V 2d8n 2h8n/ 



p _ COOnC / 3 , 3 



" ~ ^^%^\ mn 2h8n 



