306 BELL SYSTEM TECHNICAL JOURNAL 



the value He and that for a small variation in // about this point 

 the flux is proportional to // — 11^ the flux threading the elementary 

 circuit is 



The value of dB/dH is obtained from the static loop. It does not 

 depend upon x or / and is for the moment assumed independent of 

 H. The total flux cp in the sample is, therefore, 



{II — lie) I xdx — 8-3 77; -777 I xdx I ixdx (18) 

 Jo a 10 dH Jo J^ 



= cd^^{II-IIe) -<P2. (19) 



The term ip-i may now be expanded by successively substituting in 

 equation (18) the value of ix from equation (16). The result is the 

 series 



.dB,^ „. TT 10-9 (dBY dH . ,_, 



'' = '^dH^^ - ^^^^ - 1 — '"^ jdH)~dr -^ ^- (2^) 



where (pr is the sum of higher terms and where the factors have been 

 simplified by regarding d small compared to c. 



Now the coercive force obtained by the dynamic method is the 

 value Ha of the field at the point of the curve where the total flux 

 (p is zero. The apparent increase in coercive force caused by eddy 

 currents can at this point be obtained from equation (20), giving 



^^ ^ ^ TlO-'.,dBdH 



We may now inquire what, in a qualitative way, is the distortion 

 of the hysteresis loop predicted by equation (21). 



a. The widening of the loop is proportional to the square of the 

 thickness of the lamination. 



b. If the magnetizing force is kept sinusoidal the value of dH/dt is 

 proportional to Hm- Since also dB/dH increases with Hm the value 

 of AHc increases at a greater rate than the first power of H„^. 



c. For narrow loops and sinusoidal magnetizing current coiJ^ is a 

 sufficiently close approximation to dH/dt, wh ile with loo ps which are 

 wide because of hysteresis loss (dH/dt) = o)-\Hm~ — H^^. If now the 

 curve is widened by eddy currents, these values of dH/dt are too 

 large and the correction to be applied is smaller than that given by 

 inserting these values in equation (21). 



