44 BELL SYSTEM TECHNICAL JOURNAL 



The Rayleigh Hysteresis Loop 



The hysteresis loop area for magnetic cycles at low flux densities 

 (i.e. for which Hm. is not more than 10-20 per cent higher than hq) can 

 be calculated from the general shape of such loops, Rayleigh found 

 experimentally ® that the two branches of such loops are parabolas, 

 that the permeability corresponding to the tips of the loop increases 

 in proportion to the peak magnetizing force, thus, 



Mm = Mo + OtHm, (23) 



and that the remanent flux is 



Br=^HJ. (24) 



The loop equation which satisfies the above conditions is 



5 = (mo + CcHrr.)H ± ^ {HJ - H'), (25) 



where the points on the upper branch are obtained by using the + 

 sign and those on the lower branch by using the — sign. Recent 

 ballistic galvanometer measurements of high precision on an iron dust 

 core tend to confirm the reliability of the Rayleigh loop equation for 

 low flux densities.^ 



Integrating HdB around the cycle gives an area AaHm^/S, which 

 can be used in equation (22) to obtain the hysteresis resistance, as 



Defining the permeability variation with flux as 



l^m Mo / •! 7 \ 



— D t \^' ) 



MO-Dm 



the value of a will be 



a = MoMmX, (28) 



and the hysteresis resistance becomes 



Rh = 1 \Hn,i^oL„J. (29) 



<■■ Phil. Mag. |51 23, 225 (1SS7). 



' W. ii. ICIlwood, Physics 6, 215 (\^iS). 



