HARMONIC PRODUCTION IN MAGNETIC MATERIALS 791 



Appendix 1: Simplification of Loop Equations 



From the first of the three properties mentioned we have Ooo = 0, 

 and from the second property 



B,{]i,H) = - B.{- h,II) (3) 



whence, if we write 



Bi{h, H) = aioh + aoiH 



+ aooh" + anhll + 002^^ 



+ azoh' + aojm + anhH^ + ao^H, • • • , (4) 

 then 



B-^ih, H) = aioli — Goii? 



— a^ah? -\- auhH — a^'^.TP 



+ az^¥ - a2ih?H + a^^hR' - aosIP • • • . (5) 



From the third property, the two branches meet at the loop tip which 

 lies on the magnetization curve for which h = H, or 



BriH, H) = B,{H, H). (6) 



From the two equations (4) and (5) we have by virtue of this relation 



a,iH + (aoo + ao2)i?' + (^21 + 003)//' = 



and since this relation holds for every value of H, the coefficients of 

 each power of H must be zero so that 



doi = = ffloO, 



a02 = — ^20, (7) 



dOZ = — fl^2]' 



The final expressions for the branches are then evidently obtained by 

 putting (7) in (4) and (5) which become 



Bi{h, II) = aio/z 



— 002/^^ + anhll + aooIP 



+ asoh' - aoshm + a^hlP + a^^IP, (4a) 



B2{h, II) = aioh 



+ ^02^^ + anhH — aoiIP 



+ azQ^ + dosh^H + anhIP — aosH^. (5a) 



Appendix 2: Rayleigh's Relation 



If we suppose the lower branch of any loop, when referred to the tip 

 of the largest loop considered, to be given by the equation 



Bi = iJiohi + u//r + X/?x3 + cohi* (12) 



