HARMONIC PRODUCTION IN MAGNETIC MATERIALS 769 



consider the loop area to be built up of strips of infinitesimal width 

 based on the /z-axis, the height of any one of the strips is given as 



B = B,Qi, H) - BoQi, H) 

 and we have 



^=^r^^^ (9) 



so that, by Appendix 1, (9) may now be expressed as 



w=^{a,,W + a,^H'-{- '■■). (10) 



OTT 



It is clear, therefore, that at sufficiently low fields the hysteresis loss 

 varies as the cube of the magnetizing force, and diverges when the 

 field is made sufficiently large — it seems to be in general agreement 

 with experimental results on ballistic loops and, as will be pointed out 

 later, is verified by impedance change data under alternating excita- 

 tion. In view of the remanence curve equation, (10) may be rewritten 

 as 



w=^HB{0,H). (11) 



OTT 



Various approximations have been made in the past to hysteresis loop 

 forms in order to obtain convenient expressions for the hysteresis loss. 

 Thus if we consider the loop as an ellipse the loss becomes 



Ihb{o,h), 



while if we consider the loop a parallelogram the loss is 



-HB(0,H). 



IT 



Both these expressions give too large a result since the coefficient 2/3r 

 of the exact equation (11) is 0.212. 



Branch Equations for Materials Obeying Rayleigh's Relation. Ray- 

 leigh's observation enables us to establish relations between the 

 different coefficients involved in our development above when the 

 loops of a family are similar in form. For the sake of completeness a 

 derivation of these relations is given in Appendix 2 ; their validity 

 may then be judged by test in specific cases. In the derivation we 

 assume a power series expansion for one branch of the largest loop 

 of a family referred to the tip, and assume that the smaller loops 

 are of the same form. Then by referring the equations to the origin 

 instead of to the loop tip, we arrive at the hysteresis branch equations. 



