578 BELL SYSTEM TECHNICAL JOURNAL 



F.xplicit lormulai- for .1 and B from (\'.\) and (14) are 



A =<^\nh-'^[W{U->rW+V^y+V'\ + {U+W +¥■")], (17) 

 and 



fS = sin-\'2[|V(f/+[^2+F'')2+F2|-(f^+f/='+nj- (18) 



The allow fnniuilae are general and applicable to any laddiT t\pe 

 structure or its equivalent. 



In the case of wave-filters certain approximate formulae are often 

 useful. At frequencies in the attenuating bands away from the 

 critical frequencies and the frequencies of maximum altenualion. 

 and u'herever V" is negligible compared with {U+lP)>0, 



^=sinh-' 2VU+TP, 

 and (19) 



cB = Oor TT. 



;\t the crilical frt'(|uencies and the frec[Uencies of niaxinuini atu-iui- 

 ation, u'here (i'-\-i"-) is negligible compared with W, 



/l=cosh-"(V'l + l^'+l V\), 



and (20) 



rS = cos"'±(vT+F-| V\). 



In the latter the positive sign applies to a critical frequency al which 

 U = 0, and the negativ-e sign to one at which [/= — 1. 



U and V for "Constant k" and M-lype Wave- Filters. Since the 

 wave-filter structures under consideration have "constant k" or 

 derived A/-t\pe terminations, the U and V \-ariables correspoiuling 

 to these wave-filters will always \>e recjuired. Hence, formulae for 

 the variables are given here, limiting them to the four lowest wa\e- 

 filter classes generally used. 



Resistance in an inductance coil of induciance, L\. is taken into 

 account by expressing the total coil impedance as 



(d + i) L,2wf, 



where (/, the "coil dissipation ctjnstanl," is the ratio of coil resist, uui' 

 to coil reactance. The value of d is ordinarily between rf = .004 and 

 d = .04, and it does not vary rapidly with frecjuency. SimilarK', 

 dissipation in a condenser of capacity Ci can be included b>' expressing 

 the total condenser a<linittance as {d'-\-i} Ct2irf, but since d' is usualK' 

 negligible in practice it will here be omitted. 



'{"lie formulae dcrixcd from (0) are based upon those gi\en in this 



I 



