694 BELL SYSTEM TECUXICAL JOURNAL 



The second term is the conjugate imaginary of the first, so that 



Ij = R 

 = R 





= R 



Zjiiioi) 



F Uu>.-0-4>) 



Zji{iu) 

 F 



i Zf/co) 



COS {wt —6 — (t>). 



We thus arrive at the rule for the steady state solution : 



If the applied e.m.f. is F cos (ut — d), substitute to for d/dt in the 

 differential equations, determine the impedance function 



Z(/co)=Z)(fco)/M(to) (11) 



by the solution of the algebraic equations, and write it in the form 



Z(/o) = I Z(to) I e'*. (12) 



Then the re(|uircd solution is 



7 = , „,^ , , cos {wt-e-<t>}. (13) 



1 Z(lco) I 



This in compact form contains the whole theor\' of the sytiibolic solu- 

 tion of alternating current problems. 



The Complementary Solution 



So far in the solutions which we have discussed the currents are of 

 the same type as the impressed forces: that is to say in ph\-sical 

 language, the currents are "forced" currents and vary with time in 

 precisely the same manner as do the electromotive forces. Such 

 currents are, however, in general only part of the total currents. In 

 addition to the forced currents we have also the characteristic oscilla- 

 tions; or, in mathematical language, the complete solution must 

 include both particular and com])lementary solutions. This may be 



shown as follows: Let // In he solutions of the complementary 



equations. 



(^-^+^"'+i/'^')^-'+ • • +(^4+^""+i/'^')'"'=- 



(14) 



