332 BELL SYSTEM TECHNICAL JOURNAL 



of most importance in measurements of distortion. Any higher order 

 products would be less precisely evaluated, more numerous, and 

 probably of less interest; their computation is clearly evident as an 

 extension of the processes later carried out. 



The Multi-Branched Hysteresis Loop 



Under certain circumstances mentioned earlier, the portions of a 

 complex loop between adjacent reversal points are representable by 



5 = (mo + 2vn)h ± v{ir- - //2). (5) 



Referred to the origin of the B-h plane, the induction on such a portion 

 originating at the jth reversal point from / = is 



Bi = Gy + Mo(// - //;) - (- \yv{h - Hj)\ (6) 



Here 7/y is the magnetizing force and Gj the induction at the jth 

 reversal point. The latter quantity satisfies the difference equation 



Gj = Gy_i + )Uo(/// - Hj~i) + (- ryp{Hi - IIi-,Y, 



arrived at by evaluating the induction on the (j — l)st branch at 

 the jth reversal point. Subject to the initial condition 



Co = (mo + 2j///o)//o, 



this can be solved by the method of successive substitutions; the 

 solution is 



G,- = Mo/Zo + 2vlh'' + MO i: {Hi - Hi-,) 



i=l 



+ vt{- 1) '■(//,- 7/._i)^ (7) 



The foregoing expressions define the induction everywhere on the 

 complex loop. Equations (7) and (6) combined to eliminate Gj give 



Bj - mo// + (- WvHih + (- \yv{n;- - //2). (8) 



The problem remaining is to develop this equation into the equivalent 

 of equation (4). 



The instantaneous magnetizing force is 



//. = P cos />/ + (2 COS qt. (2) 



By a trigonometric transformation this ma>- be put in the form 



