792 BELL SYSTEM TECHNICAL JOURNAL 



we may refer the family of branches to the origin by the transformation 



h = hi — H, , , 



±> — L>1 — JDm, 



if B^nll refer the midpoint of the largest loop to the tip. Then 



B„, = uLoH +2vW~ + 4\H' + 8wH\ (14) 



Putting (13) in (12) 



B + B^ = fxoih + H) + v(h + iiy + x(// + ny + c^{h + iiy, 



whence, subtracting (14), 



B' = fioh + i'(//- + 2hH - iJ2) + x(/z3 + 3PH + 3hH'- - 3H^) 



+ c^{h' + 4¥H + 6F//2 + 4/;j/3 _ 77/4)^ (15) 



which represents the hysteresis branch equation referred to the origin, 

 on the basis of loop similarity. 



The coefficients obtained by the two methods may now be compared. 

 Thus identifying coefficients of (15) with those of the general equation 



(1) 



flio = MO) fill = 2.V, ao2 = — V an = 3X, 



a^o = V, ciii = 3X, aoz = 3X, a^ = 4w, (16) 



flso = X, asi = 4aj, ao4 = — 7w, 022 = 6aj. 



O40 = w, 



Appendix 3: Alternating Magnetization, Sinusoidal 

 Magnetizing Force 



The resulting expression is simplified if we make the following 

 substitutions 



a = ao2H' + ao,IP = B(0,H), 



/3 = aio + anil + anH', (19) 



5 = asoHK 



It may be noted that a is the remanence and that jS is an approxima- 

 tion to the permeability, in fact the permeability is given as the sum 

 of /3 and 5. With (18) and (19) inserted in the branch equations, then, 

 we have 



Bi{II cos pi, II) = a + i3 cos pt - a cos- /?/ + 5 cos^ pt, 

 Boill cos pt, II) = - a + 13 cos pt + a cos"' pt -\- 8 cos^ pt. 



For convenience we shall express these relations in terms of multiple 

 angles, and we have for the equation of the upper loop family 



Bi{II cos pt, I]) = - a/2 + (/3 + 36/4) cos pi - a/2 cos 2pt + 6/4 cos 3pl. 



