768 BELL SYSTEM TECHNICAL JOURNAL 



1. When both h and H are zero the flux density on either branch 



is zero. 



2. The flux density on one branch with // and h given is equal and 



opposite in sign to the flux density on the other branch corre- 

 sponding to the negative of h and to the same maximum 

 force //. 



3. The two branches corresponding to a definite H meet at the 



normal magnetization curve. 



The application of these properties to the power series enables us 

 to deduce relations between the coefficients as demonstrated in 

 Appendix 1 : 



floi ^= floo ^^ 0, 



a02 = — ^20, (7) 



fl03 = — fl^21. 



We shall find it sufficient to include the third degree terms for our 

 work, so that we need not investigate relations between coefficients 

 of higher degree. The loop equations are simplified by utilizing (7) 

 and we can make a number of interesting deductions. Thus the equa- 

 tion for the normal magnetization curve is given by 



B{H, H) = a,oH + auH' + {an + a3o)H\ (6a) 



In this equation aio will be recognized as the initial permeability 

 usually expressed as no since upon division by H we have for the 

 permeability 



M = aio + auH + (ai2 + a3o)H- • • • . 



According to this equation the change of permeability with magnetizing 

 force is linear at sufficiently small fields. The above equations are, 

 in all rigor, infinite series, but for our purposes it will be found sufficient 

 to consider only coefficients of the third and lower orders, — in some 

 cases the second order will suffice. 



An expression for the remanence curve of the loop family may be 

 obtained by setting h equal to zero in the equation for the upper 

 branch. In that case we find from (4a) 



B{0, II) = ao-2lP + aoJP + •••, (8) 



hence for sufficiently small magnetizing forces the remanence increases 

 as the square of the magnetizing force. 



The hysteresis loss per cycle per unit volume may also be obtained 

 directly from the branch equations (4a) and {5a). The loss in ergs, w, 

 is equal to the area of the hysteresis loop divided by Air. If now we 



