700 BELL SYSTEM TECHNICAL JOURNAL 



time t = o, be E sin (co^+ff). Substitution in formula (20-b) and 

 rearrangement gives 



I{t)=A{o)E sin (iol+d) 



+ E sin (co/+0) / cos uTA'{T)dT 

 •Jo 



-£cos (i^t + e) /sin wr.l'(r)rfr (21) 



where ,-l'(0 =7-^ (/). 



Now this can lie resolved into two parts 



( /*°° ( 



£ sin (oil + d) -i A{o)+ I cos wT/r(7)rfr . 



-E cos (uit+e) \ J sin or,4'(7-)rfT I 



which is the /!»«/ steady slate, and 



,=0 



— £ sin (a)/ + e) / cos ootA {T)dT 



-f £ cos (co/ + 0) / sin 0)7.4 '(r)(ir 



(22) 



(23) 



which is the transient distortion, which ultimately dies awa\' for suf- 

 ficiently large values of time. 



To correlate the foregoing expressions for the steady slate with 

 the usual formulas we observe that if the symbolic impedance of the 

 network at frequency t<j/27r be denoted by Z{iw), and if we write 



^=a(.) + ./3(.) 



then the steady state current is 



£[«(co). sin (co<-|-9)+/3(co) . cos iyil+e)]. 

 ("onipari^on with (22) gives at once 



a(w) =A{o)-\- f cos ojr /r(r)f/r, (24) 



/3(a)) = - f sin co7.4'(r)rfr. (25) 



