(29) 



278 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1953 



current losses are large compared to hysteresis loss, and then for the 

 situation where hysteresis loss predominates. Referring to Fig. 5, the 

 following relationships are noted. 



t (thickness of shell) is given by {R - t) = R\/r^^P'^ (26) 



A (magnetic cross-section) = ttP^R^ (27) 



I (mean magnetic path) = R{Zy/l - p2 - i j^ 2H) (28) 



W (available winding cross-section) 



= RWi^'P' - P)(2\/l~="P"' - 2 + F) 



X (mean length of turn) = irRi^/l^^P^ + P) (30) 



Vc (coil volume) = 'kR^H (31) 



1. DC Resistance, and Residual or Eddy Current Losses, Predominate 



The dissipation factor, due to the dc resistance, is seen from (4) and 

 (6) to be 



where ku = (p/2Trkkw) is a constant with respect to the core propor- 

 tions, assuming the winding efficiency is not materially affected by 

 changes in shape. 



Since, regardless of shape, we will want to adjust the air gap so that 

 H is optimum, and since we know from the previous discussion that this 

 will occur when the dc resistance is equal to the sum of the core losses, 

 we have, from (10) and (12) 



D,c = De + Dr = (hf + h)fl (33) 



Eliminating n from (32) and (33) : 



Z>.c = A-. y/A^ (34) 



where kn — Vknik^f + A:?) is again a constant with respect to the 

 core dimensions. The dissipation factor is 



D = 2D,e = 2kn |/|^ (35) 



