MAGNETIC MEASUREMENTS AT LOW FLUX DENSITIES 41 



Simple Analysis of Eddy Current Resistance 



Again, at bridge balance, the resistance of the standard is equated 

 to the resistance of the test coil, which is composed of the copper 

 resistance Re and a resistance which corresponds to the a-c. power P 

 dissipated in the core. Thus 



R = Rc + PIP. (7) 



Power is dissipated in the core through eddy currents and magnetic 

 hysteresis. Although both types of magnetic loss occur simultane- 

 ously, they will first be considered as if occurring alone, after which 

 the details of separating and identifying the two types will be discussed. 

 The resistance due to eddy current power loss depends upon the 

 form of the magnetic core — whether of laminations, wire, or powder — 

 upon the frequency, upon the permeability of the magnetic material, 

 and upon the hysteresis loss, since this modifies the permeability. It 

 is determined with sufficient accuracy for many practical purposes by 

 calculating the eddy current power loss in a volume element consisting 

 of a thin tube so drawn that neither magnetic flux nor eddy currents 

 cross its surfaces, when the flux it encloses varies sinusoidally, and 

 then integrating between proper limits to include the entire cross- 

 section of the lamination. By this method ^ it can be shown that the 

 power consumption per unit volume of sheet core material is 



_2/2f2R 2 



p^^-KtjJ3^ X 10-7 watt, (8) 



op 



where / is the sheet thickness in cm., / is the frequency, and p is the 

 resistivity of the material in e.m.u. 



This relation is derived on the assumption of a very extensive plane 

 sheet with magnetizing force parallel to its surface, but it applies 

 sufficiently well to any sheet material, flat or curved, provided that 

 the magnetizing force is parallel to its surface, and provided that the 

 width of the magnetic sheet is large in comparison to its thickness. 

 These conditions can be fulfilled in a core built up of ring shaped 

 laminations, and wound with an annular winding. 



The total eddy current power loss in a core of volume irAd is then 



P. = f»^BJAd ^ j„_, (,) 



Op 



As already mentioned (eq. 7), such a power loss in the core of a coil 

 *C. P. Steinmetz, "A. C. Phenomena," p. 195 (1908). 



