ALTERNATE CURRENT GENERATOR. 



81 



18. Eeturning to the alternator, if n be the number of armature 



turns, V the number of field turns, and x = a, + a3+ aj + &e., f = "°/ 2 



+ «: + ^4 + &c., the armature and field currents respectively, the 

 magnetizing current-turns M.,- producing flux across the air gap and 

 through the armature in a direction axial to its windings are given by — 



M.r = nx -\- U^ cos 0)t 



and the current-turns M,^ producing flux across the air gap and through 

 the armature in a direction parallel to the planes of the windings and 

 lehind that of JNI, by 90° are given by — • 



M,/ = v^ sin oj^, 

 or in vector notation {see section 4) — 



%+l 



5 (a,,_i — a,, + {) 





which produce the armature fluxes X,, and A,/ given by 



A. = 



t 2 %Gi, (a^/-i — OL,i + i 



where Gr^, = ^(? ^ '' ^^^ 1 a^y odd number. 



[iM'ote that the directions of A,,, and Ay are fixed in the armature.] 



Now, if magnetic leakage be otherwise taken account of, the flux 

 in the stator must be continuous with that in the rotor, so that the 

 flux F looped on the field windings at any instant is given by — 



F = A,r cos Mt -\- Ai^sin od 

 which by means of the relations in section 4 can be reduced to — 



r '^ ~i 



F =r 2 vGp ap -f -^ (G-yj-i a^j-i -f G/,+ia)o+i)io 

 where p is cYen and 2Gr,, = Grp-i + G-p+i. 



19. If Z' be the self inductance in the armature circuit either 

 external to the armature or due to magnetic leakage in it, and if /V be 

 a similar quantity for the field circuit, the equations for the two 

 circuits are — 



dx d 



rx + i' -TT + « — 7 A^ = 



at dt 



^^ d ^ 



where r, p, q, x and f have the same significations as in section 1. 

 p 



