MAGNETIC MEASUREMENTS AT LOW FLUX DENSITIES 47 



The equations for resistance are similarly complicated. The first 

 series gives that part of the eddy current resistance which is due to 

 the constant mo- The coefficient of the second series indicates that 

 this series involves the hysteresis resistance. However, terms in the 

 second series which contain the factor P will be recognized as eddy 

 current components introduced by the fact that the permeability has 

 been increased from the value juo by the factor X5„. 



The complicated form and slow convergence of the above equation 

 (35) for resistance make it difficult for use in interpreting a-c. bridge 

 measurements. Considerable simplification is effected by dividing the 

 observed resistance (eq. 35) by the observed inductance (from eq. 2>Z) 

 for each measuring current and frequency. Performing this operation, 

 and rejecting series terms in X5„ higher than the first power, gives 



Lfm 3p 



1 - 140 (1 + 5X5J + 



+ :^ \Hmflof 



1 - ^ (1 - 5X5J + 



(36) 



The coefficient of the first series is identical with the eddy current 

 expression previously derived (eq. 11), which neglected eddy current 

 shielding and hysteresis. The series itself, which includes these other 

 effects, converges rapidly for < 1, provided that the value of X^^ 

 is not carried too high. 



The coefficient of the second series is identical with the hysteresis 

 expression derived from Rayleigh's equation (29) in which eddy 

 currents were neglected. The second term of this series gives the 

 amount by which eddy current shielding reduces hysteresis resistance 

 at higher frequencies. It appears to converge less rapidly than the 

 series for the eddy current resistance, but this is partly offset by the 

 decrease of its second term with increase of X^^. Thus, the coefficients 

 of ^ become equal to 1/88 in both series if \Bm = 13/110. This 

 value of \Bm is reached when the flux density in the material is large 

 enough to raise the permeability some 10 per cent above no. Evi- 

 dently, this value of X^^ can be exceeded somewhat without making 

 the coefficients excessively large. However, if the measurements are 

 made at too high flux densities, the hysteresis loops diverge more and 

 more from the simple Rayleigh loop, and the present analysis becomes 

 inapplicable. 



In a-c. bridge measurements it is seldom desirable to measure at 

 flux densities which will carry the permeability more than 10 per cent 

 above its initial value. If measurements at higher flux densities are 



