TRANSIENTS IN GROUNDED CIRCUITS 



767 



The integrals appearing in (6) are apparently not known in closed form. 

 Series expansions holding for small and large values of time may be 

 derived however. 



By successive integration by parts we obtain : 



t/O 



g-(a/r)+/3r^^ = g-(a/0+fl' 



a a a 



24/5 + 36^/6 + 12,32/7 + ^3/8 



a' 



+ 



(7) 



e^' appearing on the right hand side of equation (7) is cancelled by 

 g""^' appearing before the integral, and similarly for the first term in 

 brackets in equation (6). In the complete expression for the voltage 

 odd powers of /3 cancel and we have: 



Vnit) = 



sin o)/ 



CO 



IT 



\X^ 



g-ir\xyt 



2/3 



irXx^ 



24/5 



+ 



6/^ - co2/« 



{wXx^y 



- 12a;¥ 



+ 



For large values of time equation (6) is written as 



(8) 



sm ut . CO 



+ 



CO 



/»« „-(.alT)-\-PT /»oo p-{alT)-pT "I 



(9) 



where the integrals between zero and infinity correspond to the steady 

 state condition while the integrals between / and infinity give the tran- 

 sient distortion. The integral between / and infinity may be evaluated 

 in a manner quite similar to that used above. The result with plus 

 sign for ^ is 



/ 



» g-(a/T)+^T 



dr = f-W0+/3< 





I /6 

 b- 

 12a2 



1 /24 36a: iza^ a- 



6a o2 



(10) 



The integrals between and infinity are evaluated by: 



i 



'00 ^— (a/T)=t/3« 





(11) 



in which the real and imaginary parts of the right hand side may be 

 expressed by the ker' and kei' functions by the relation already given. 



