114 THE BIOLL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



the winding current results in a flux ipc = 3^c/(R, uniformly distril)uted 

 across the core cross section. The leakage paths cause some longitudinal 

 variation in ip, which can be neglected for the present purpose. If the 

 flux is varying, eddy currents flow in circular paths with a current den- 

 sity J, which varies with the radius /■. These give an increment to the 

 total mmf varying from zero at the surface to a maximum at the center, 

 producing a corresponding variation in the density of the total flux. 

 If iFc is large compared with the maximimi magnetomotive force pro- 

 duced by the eddy currents, however, the density is nearly constant, 

 and its rate of change is nearly constant throughout the core cross sec- 

 tion. To the extent that this condition is satisfied, the effect of the eddy 

 currents can be determined as follows. 



dr 



Fig. 2 — Eddy current paths in the core of an electromagnet. 



The current in a cylindrical shell of radius r and differential thickness 

 dr is jt dr. This links a part of the field proportional to the area enclosed, 

 or 4r>/Z)", which varies, by hypothesis, at the same proportional rate 

 as the total field. The resistance of the shell is 2-Krp/{( dr.), where p is 

 the resistivity of the material. The voltage equation for the shell cir- 

 cuit is therefore: 



^^"^^ + 1^="' 



and hence: 



J = 



2r dip 



pirD- dt 



The magnetomotive force of the shell is its current multiplied by 47r. 

 This produces a flux increment dtp in the area enclosed inversely propor- 

 tional to the reluctance of the tubes of induction within this area. This 

 reluctance may be taken as inversely proportional to the area, as would 

 be the case in a closed magnetic circuit of uniform cross section. Then 



