ALTERNATE CURRENT GENERATOR. 



73 



9. As a^ is the vector of length 2>), lying along the axis of y 

 I phase = V I ; ^^^^ ^^ — 



1 1 ^A 



^^= — ^ («o), = ~ '- ('O, 



= - ^^ sin Lf ^ I + lj\= f'' sin L.t - I + K^ 



ai 

 Again- 



that is- 





u,= —L sin (2<of + ^ + /^ + ^,Y 

 Similarly — 



a __rl_ sui (" 8(0/ -"^ + />, + /i. + b\. 



a = — ^!L__ sin (4^,0/ + '^^ + /.. + /3, + 63 + /?,V 



a. = — ^ — «i" ("•"""/ - !; + />». + ^: + ^, +f^. + k), 



and substituting these values in — 



« = a, + a, + a, + &c. 

 $ = Vi + a,+ a^+ &c. 



we obtain the armature and field currents in the usual trigonometrical 

 form of expression. 



It is wortb while drawing attention to the fact that the period of 

 the alternating current induced in the field circuit is half that of the 

 armature current, and that it contains all harmonics, both odd and 

 even, relative to its own fundamental, and so its wave form will in 

 general be unsymmetrical with respect to the time axis. 



10. The total e.m.J'. E generated in the armature circuit being 



equal to — ^ (w ^ cos wf) 



m d 

 -Ei = — — 2 (a,^ - 1 + a,/ + i)„. {See section 4.) 



2 



also as a,^ _ 1 + t,, a, -)- a,/ + 1 =: 



