VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 517 



FM = r e-'^^A'{T)dT, (Ua) 



Jo 



It = e' 





(r)dT, 



(12a) 



to which the more general formulas reduce when ij. — and conse- 

 quently M — \. 



We have now to evaluate Y(iio, t) as given by (11). We shall 

 assume tentatively, at the outset, that ;u = \s{t) has the following 

 properties : 



\s{t) <<C w for all values of /, 

 - 1 < sit) < 1, 



- 1 





sdt < 1. 



With these restrictions the instantaneous frequency lies within the 

 limits CO ± X. 



Let us now replace M{t, t) by the formal series expansion 



M{t, t) = Af{t, 0) + ^, 



Mit, r) 



+ 



2! 



or- 



+•••, (13) 



which converges in the vicinity of all values of / for which 5 has a 

 complete set of derivatives. Then, if we write 



Qn 



(- iycnit) 



(13a) 



and substitute (13) in (11), we get 



F(ico, /) = r e~'-^A'{T)dT + E (- iyCn r -.e~''^^A'(T)dT. (14) 

 Jo 1 Jo ^• 



From (11a) it follows at once that 



/!n /In 



(15) 



X 



-,e-''^^A'{T)dT = -^-f- F(*-co), 



so that 



F(.-a;, /) = F(ico) -f- £ ^ C„(/) ;^ F(t-co). 

 1 n\ aw" 



(16) 



