lilJiCIRlC CIKCIIT lllLiOKY 379 



Applications to Electric Circuit Theory 



Let us assume that at time /= —NT, an electromotive force £(/), 

 periodic in fundamental period T, is impressed on a circuit of complex 

 impedance Z(«aj), where co denotes 2iv times the frequency. Required 

 the resultant current /. 



For values of t>—NT the electromotive force (see formula (319)) 

 can be expressed as the Fourier series 



oo 



vhere 



E(t)= yF{incoo)e'"'^-'^ 



F{ina:o) = ^ I E{r)e-'"^'>rd, 

 1 .7o 



The resultant current for t> —NT is therefore 



oo 



Fiinwo) ■ transient oscillations I 



Z{ino:o) ^""'''' "^ i i^tiated at time 



If we are concerned with the current for values of />0, and if .V7" is 

 made sufficiently large, the initial transients will ha\e died away and 

 the complete current for / > 0, will be given by 



I=Y^^\e--< (323) 



This formula implies the periodic character of E{t) for sufficiently 

 large negative values of time. If, howe\er, E{t) is zero for negative 

 values of time, we can employ the Fourier integrals (321) and (322) 

 in precisely the same way and get, as the complete expression for the 

 current for positive or negative values of time : — 



1= r^e-^^d. (324) 



^^ E(r)dT "^j^^dc^. (325) 



The infinite integrals (324) and (325) formulate the current in the 

 network, specified by the impedance function Z{io:), in response to an 

 electromotive force E{t) impressed at time / = 0; they therefore mathe- 

 matically formulate, by aid of the Fourier integral identity, the funda- 

 mental problem dealt with in the preceding chapters and solved by aid 

 of the operational calculus. 



