ALTERNATE CURRENT GENERATOR. 83 



Expanding and remembering that — 



3 _2 _2 __ A 



a + y8 = a+;8+ 2ay8 COS a^, 



and that — 



aq-i -j- a,/4-i = — f,i G,, a^i (see section 19), 



we find that the total iron loss is equal o — 



1 . . ( v= -2 _2 _2 I 



— \ 0' sin' 8,/ -j- !? '^ ^' s'li ^'/ cos 8,, > a, 

 where q is any odd number. 



21. An approximate determination of the effect of iron loss on 

 the performance of an alternator can be obtained by taking all the 

 Gr operators for its magnetic circuit as equal to Gr, ; that is, to the 

 one for the fundamental hamnonic of the armature flux. 



Making this simplification in the equations of section 19, we find 

 that a,, a,, &c., a^,, a,, etc., are connected by the two series of equations — 



a-q-y + A/ i^q + O.q + 1 = ^^ 



S'P-i + 1"/) ftp + a^j+i = 

 with ttg = 2rj/p in which— 



2( r - } '^( l^ 8 r- + ^) 



nv(jr / '1'^ ) ^'' ( .9 19^^ \ 



2 S ^P I A. P "I 2 f , , A' 8 p :+^) 



T„ ^ < V (jr -j- A' — -^ f^ > = — ; I'- -)- — t — — — t- 



Jil'Gr 



as Gr = oi 



Putting Z for qrC, A for ^r", and m for gnv, and, remembering that 

 8 is a small angle {see section 17), unity for cos 8, we find that — 



where ,^2 4 r r- Ir . ^1 



^q=~ \ (l + ly + -T",+ 2 - sin 8 [ 



2r 2Z' 



D,, sin /^, = — - sin 8 



•^ q//i<D m 



4 = ±\ (X+ XT- -u J'! +2^smS] 

 m- C poy po) ) 



A/, sm 9p = — '— _ — sin 8. 

 pniM m 



The if and t operators having been calculated from these formulae, 

 the rest of the solution for this case follows in every particular the 

 course for the simple case fully explained in sections 8 and 9. 



