ELECTRIC CIRCUIT THEORY 7,^ 



The resultant current /(/) may always be written as: 



/(/)=^ l yr s^ (l+p)cos(a;/-5) + (Tsin {u^l-B) 

 Z I Z. \t(ji) I I 



1 , E 



= 2'V(l+p)^ + cr-^7^|COs(co/-5(co)-e) 



where 



6 =tan->r4- 

 1+P 



Evidently the functions p and a, which it is our problem to deter- 

 mine, must be — 1 and respectively for negative values of /, and 

 approach the limits +1 and 0, respectively, as t-^oo. 



In an engineering study of the building-up process we are prin- 

 cipally concerned with the envelope of the oscillations : hence with 



1 / 



-^\/(l + p)^ + a2. 



Our problem is therefore to determine the functions p and a and to 

 examine the effect of the applied frequency co/27r and of the charac- 

 teristics of the circuit, on their rate of building-up and mode of 

 approach to their ultimate steady values. 



The functions p and a can be formulated as the Fourier integrals 



-? 



X 



where 



P=— / lPco(X)-|-Pco(-X)] sin 



--- / [(?o;(X)-(2o.(-X)]cOS/X 



cr = - / [<2a.(X) + (2co(-X)]sin/X^ 



IT Jo A 



+- / [Fco(X)-P.,(-X)]cos/X^, 





