FERROMAGNETIC DISTORTION OF A WAVE 331 



The point where the minor loops first vanish and the requirement 

 that they vanish at all may be readily determined. Two adjacent 

 extremes of the magnetizing force approach a common value as the 

 amplitude of the higher frequency component is decreased. When 

 they attain this value, the slope is no longer reversed between them 

 and consequently no minor loop is formed. The instantaneous mag- 

 netizing force is expressed by equation (2), and in this instance p':$> q. 



The minor loops first appear where dh/dt = when pt = — ^ • 



Solved simultaneously, these equations yield 



where the minor loops vanish. They reappear at an equal negative 

 value of sin qt, and other intervals during which they vanish are 

 apparent from symmetry. If {Pp/Qq) > 1, no real solution of 

 equation (3) exists, so the minor loops do not vanish anywhere. The 

 appearance and non-appearance of minor loops is seen to be governed 

 by the ratio of amplitudes in the same way as by the ratio of frequen- 

 cies as long as the restriction on the latter is observed, and the product 

 of these ratios 



pp 



determines the type of hysteresis loop {Id or \e, or an intermediate 

 form) obtained. 



Induction with a Two-Frequency Magnetizing Force 



General Expression for the Induction 



As a function of time, the induction for any type of loop described 

 will consist of intermodulation products of the two fundamental 

 frequencies. Because of the kind of symmetry the characteristic has 

 these products will include only the odd orders, but because it is a 

 loop, quadrature terms must be expected. The induction then will 

 ultimately be in the form 



00 00 



B = Y. Z [c»in sin {mp + nq)t -\- d^n cos (mp + nq)t'] (4) 



m=0 n—x 



with all even order coefficients zero. The odd order coefficients 

 remain to be determined from the hysteresis loop. In doing this only 

 third order products in addition to the fundamentals will be evaluated 

 explicitly, as they are stronger than the higher orders and therefore 



