ALTERNATE CURRENT GENERATOR. 



89 



and proceeding exactly as in section 5 we obtain the infinite series of 

 equations — — ,^^ 



a, + T,a. + a3 =0 /,^ ;■ ^i^ < A 



«. + ''ja,, + a^ = — Jc^ ■.. 



a, + t. a. + a. = — /C- «r" * »v- / - 



&c., &c. \JPv , Cn"^ 



in which a„ = — the vector to the point — , ^ , as befbJ^e-j- ^ '^ -• 

 P P - '"■^<-.-«— -^ 



where Se.y is the applied e.m.f'., and the t and r operators have the same 

 values as in section 5. 



Solving for a, we find that — 



P,a, = - n, (a, + kj - n^^3 - llg^, - &c. 



where P„ IT,, n^, &c., are the infinite determinant operators whose lead- 

 ing terms are f^, t,, t^, &c., respectively as in section 6. 



Eeducing to the continued fraction operators of section 7, we 

 obtain — 



a. = - I; K + *,), - ,_,',; ih). - ,~^ W. - *e., 



and using the equations — 



a, = — f^a, — a^ — k„ 



a, = — T, a, — a,, 



a^ = — ^,a, — u, — A:,, &c., 



the successive harmonics of the armature and field currents can be 

 obtained. 



25. If, in the last example, the e.m.f. inserted in the armature 

 circuit be sinusoidal and equal to E sin (oj#+ h) = e (a. vector), the 

 solution will obviously be identical with that for the simple generator 

 given in sections 5 et seq., when in the latter a„ -f- /< is substituted for 

 a^ where — 



2 '^ 

 K =■ — (, - e, 



and in this case it is important to know the condition which determines 

 whether the machine will run as a motor and develop mechanical power. 

 In section 12 the driving torque T was shown to be given by — 



T = - I I Va,a, + Va,a, + Ya,a, + \a,a^ + &c. | 



and T must be negative for a motor. 



