HARMONIC PRODUCTION IN MAGNETIC MATERIALS 793 



If we write 



A = aj2 =^ (ao2lP + aosI~P)/2, 



B ^ 13 + 35/4 = aio + anH + auIP + SasoIP/4, 



C = = - A, ^"^^^ 



D ^ 8/4: = asoH'/4, 



the final form for the loop equations is 



Bi(H cos pt, H) = A -\- Ri" cos pt + C cos 2pt + D cos Zpt, , . 

 Bi{H cos pt, H) = - A + B cos pt - C cos 2/>/ + D cos 3/)^. ^ ^ 



We are now in position to combine the two equations of (21) in a 

 Fourier series valid over the entire cycle as 



B = ^-\- J2ibk cos kpt + ak sin kpt), (22) 



where 



1 r^" 



a/fc = - I Jipt) sin ^/)/ <i(^/), 

 ^ Jo 



1 r^" 



^i- = - /■(i>0 cos kptdipt). 

 T^ Jo 



(22a) 



For our particular case we have, since B = Bi for the first half of the 

 cycle, and B — B^ for the second: 



B^ih, H) sin y^^/ (^(^/) + 1 Bi{h, H) sin kptd{pt), 



ir Jo 



B-2(h, H) COS y^/)/ ^(^/) + I 5i(/i, //) cos kpt d{pt). 



■w Jo 



These integrals may be simplified considerably when we take advantage 

 of the fact that both Bi and Bo are even functions of the time as given 

 by (21). Thus 



TTttk = I IBiQi, H) - B.{h, H)2 sin kpt d{pt), 

 Jo 



rbk = I lBi{h, H) + B^Qi, H)'] cos kpt d(pt). 

 Jo 



Referring to (21) we may then write 



2 r 



ttfc = - I (^ + C cos 2pt) sin kpt d(pt), 



''•^'^ (23) 



2 r 



hi^ = - \ {B cos pt -\- D cos 3pt) cos kpt d{pt). 



TT Jo 



^^ This coefficient is not to be confounded with the general expression for flux 

 density. 



