ELECTRIC CIRCL'JT THEORY 367 



circuit is ultimately '" a steady state current of frequency F. This 

 follows from the fact that the definite integral which defines the current 

 Io{t) is resolvable into the ultimate steady state current corresponding 

 to an applied force of fre(iuenc\- I\ and the accompanying transient 

 oscillations which ultimately die away. The fictitious e.m.f. which 

 may he regarded as producing the comj:)()nent current /i(/) is rf{t)IoU); 

 this is ultimately the product of the two frequencies F and /, and 

 therefore resolvable into two terms of frequency F-{-f and F—f re- 

 spectively. Carrying through this analysis, it is easy to show that 

 each component current is ultimately a steady-state but poh-periodic 

 oscillation, as indicated in the following table: 



Component Current Frequency 



/o T, ' 



7, F+f,F-f, 



I. F+2f,F,F-2f, 



h F+3f,F+f,F-f,F-3f, 



I, F+4:f,F+2f,F,F-2f,F-4:f. 



It is of importance to observe that the component currents inxolve, 

 from a mathematical standpoint, multiple integrals of successively 

 higher orders, the wth component /«(/) involving a multiple integral 

 of the n''' order with respect to /o(/). Consequently the successive 

 currents require longer and longer intervals of time to build up to 

 their proximate steady-state values, so that the time required for 

 the resultant steady-state to be reached cannot be inferred from 

 the time constant of the unvaried circuit. 



From the preceding table it will be seen that the ultimate steady- 

 state current is obtained by rearranging the series /o+^^i + ^a and is 



of the form 



+00 



"V An COS (S2 + Maj)/+5„ sin (12 + «w)/ 



n = — oe 



where i} = 27rFand co = 27r/. 



It is interesting to note that this series comes within the definition 

 of a Fourier series onh" when F = or an exact multiple of/. The 

 steady-state solution is of very considerable importance and is con- 

 sidered in more detail in a succeeding chapter. 



From the foregoing we deduce an outstanding distinction between 

 the variable and invariable networks. In the latter the currents are 



'" It hardly seems necessary to remark that the reference time t=o is purely arl)i- 

 trary and that the resistance variation nia\- start at such a time thereafter that /„(/) 

 may be regarded as steady state during the entire time interval in which we are in- 

 terested. Going farther, if we confine our attention to sutiticienlly large values of t, 

 the whole process may be treated as steady state. 



