768 BELL SYSTEM TECHNICAL JOURNAL 



The complete expression for the voltage is: 



SI 



n (Jit , 2Vco 



Viiit) = — ^-^+- — p=[cos w/ kei'(2W7rXco) + sin co/ ker'(2.rVxXw)] 

 7rA.V xVttX 



a—v\x^it 



1 1/6 67rXx2 , (ttXx^) 



+ ^^^:^^H- 



W^2 aj2 Xt^ t^ ' f« 



• (12) 



For such values of time where neither of the formulas (8) or (12) 

 give very accurate results it is necessary to perform mechanical 

 integration. 



In so doing it is convenient to introduce a new variable ^ of inte- 

 gration. Let ^ = wT, and the integrals become 



Jo ^c/o 



= - r r e-'i^ cos ^d^ ± j r e-^^ sin ^d^ , 



where 



Now let 



.y = 

 r 



= aco = ttXcoX"^, 1 , -, 



= cot. J ^^-^^ 



A{s, r) = f e-'l^ cos ^d^, (14) 



Jo 



Bis, r) = f e~''^ sin ^d^, (15) 



Jo 



and formula (6) becomes 



where 



, . sin wt V^2 + -B2 



tan^ = -^- (17) 



The values of the functions A and B are given in Figs. 1 and 2 for 

 some values of 5 and r which are frequently met in practice. 



Assuming finally /(/) = e~^' equation (1) gives after simplifications 



Fi2(/) = -4-2 (^~^' - ^~^'''") + ^2 f e-^^^'l^'+^^dr. (18) 

 For small values of / the series expansion (7) may be used. The 



