MAGNETIC MEASUREMENTS AT LOW FLUX DENSITIES 59 

 the reciprocal of the value obtained by averaging l/d, namely 



d = ^-^. (60) 



This expression can be converted to the following convenient 

 series 



d = d„ 



U\dm ) 180 I d„ ) 



(61) 



which indicates that the effective diameter is smaller than the arith- 

 metical mean diameter dm, by an amount depending upon Ad, the 

 difference between inside and outside diameters. This series con- 

 verges so rapidly that terms beyond the second may be neglected for 

 all practical purposes. 



A test core of large radial thickness is to be avoided, when accurate 

 measurements are desired, because of the considerable change of 

 flux density from the inside to the outside diameter, with its accom- 

 panying modification of the core permeability. Such variations 

 complicate eddy current and hysteresis behavior, particularly through 

 reaction on the magnetic permeability. If the permeability at every 

 point in the core bears a straight line relationship to the flux density, 

 eq. (27) gives 



fx - Mo(l + XJ5). (62) 



The flux density at diameter y is 



y 



Solving this equation for B, and integrating from di to do gives the 

 total flux in a unit height of core, from which the mean flux density 

 can be calculated. This gives for the mean permeability, approxi- 

 mately 



fim = f^o I i -\- XBm j~r j , (64) 



where Bm is the mean peak flux density in the core, and d is the effective 

 magnetic diameter. Comparison of (64) with (62) shows that the 

 increase of mean permeability in a core of considerable radial thickness 

 is not precisely equal to the ideal increase of permeability for a given 

 flux density. 



The effect of radial thickness of core on losses can be attacked in a 

 similar manner. Inserting the value of H at diameter y in Cauer's 



