770 



BELL SYSTEM TECHNICAL JOURNAL 



By comparing the coefificients of this equation with those of the general 

 equation (1), we can deduce relations between the coefficients. These 

 are, from Equation 16 of Appendix 2, 



aoi = 0, 



^20 + «02 — 0, flu = la-io, , -. 



O21 + «03 = 0, ai2 = ^21 = 3a;jo, 



040 + 022 + flo4 = 0, 0.31 = an = 4^40 = 6a22- 



The left column is in agreement with Equation (7), while the right 

 column furnishes new relations between the coefficients. Thus the 

 coefficients based on loop similarity are not inconsistent with those 

 deduced under no assumptions, but the former involve additional 

 relationships between coefficients which need not be satisfied in the 

 general case. 



Families of hysteresis loops obtained by the usual ballistic method 

 have been examined for these relationships in the case of several 

 materials with the following results for coefficients up to and including 



the second degree : 



Hysteresis Loop Coefficients 



The values tabulated were obtained by 'smoothing' data obtained 

 from ballistic loops, which leaves a great deal to be desired from the 

 standpoint of precision. The difference between ao2 and — a^o is an 

 index of this lack of precision. The average deviation of aii/2 from 

 the mean of ao2 and a^o (a relation deduced on the basis of similarity) 

 is small for C dust and large for silicon steel while the third order 

 coefficients show much poorer results. The lack of precision in the 

 original data however, is such as to leave unsettled the question of 

 loop similarity for the materials tested at the fields to which the 

 coefficients apply; the experimental error is too great. 



The use of ballistic methods for determining coefficients becomes 

 even less satisfactory for materials with low hysteresis losses and 

 cannot be used for analyses which have any aspirations to precision. 

 It will be shown in the next section, however, that the significant 

 coefficients enter into the impedance of a coil which employs the 

 material under examination as a core, so that the desired coefficients 



'Speed and Elnien, "Magnetic Properties of Compressed Powdered iron," 

 A. LE.E., V. 40. 



