VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 539 



It remains to establish formulas {20b), (21b) and (22b). From the 

 defining formulas (lOb) and (lib) and the Fourier integral energy 

 theorem, we have 



— I FcMo) = y, I cos^ ( X I sdt j dt, 

 -y I F,-d<^ ^ T i ^^^^ ( ^ r ^(/n dt. 



(24b) 



Adding we get (206). 



Now differentiate (106) and (116) with respect to t and apply the 

 Fourier integral energy theorem ; we get 



-^ / "" ic^Fc-do: = Y^ r^ ^-s^ sin2 ( X T sdt \ dt, 



—^ j (ji^FM^Ji = -^ I X"-J"" COS" ( X j sdt 

 ""Tjo ^Jo \ Jo 



(256) 



and, by addition, we get (216). 

 To prove (226) we note that 



(1 + ijls) cos 





sdt 



r-)-^^-(\f 



= cos ( X I sdt]-\- 



sdt 



-;r[ 



Fc cos (oot -\- dc) -{- - 00 Fs COS (ut + 9s) 



X 



i^c^ + 



+ 2 ^ coF.F, cos (9c - ds) 

 X 



1/2 



cos (co/ + <l>)(/co. (266) 



Consequently, by the Fourier integral energy theorem, 

 r (1 + ns)^ cos2 ( X r sdt\dt 



^ r r F.2 + (^ y'co-7^.2 +2^0,^,7^, cos (^^ - ^s) 



i_ r^ 



(fco (276) 



and 



1 r^ 



Tp \ fjiS- cos- ( X I sdt ] dt 



^T\t)l 



coFcF, COS (0c - ^O^'^- (286) 



