98 
ME, S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
In the immediate neighbourhood of the sheet the component of the field parallel to 
the sheet is 
k — 2-n-nljb 
and reverses its direction as we pass through the sheet. 
Assuming li to be constant throughout the depth c, the field acting on the first layer 
is (m — 1) k, on the second layer (m — 3) h, and generally on the r th layer (m — 2r — 1) h. 
The mean square value for all the layers is therefore 
h 2 {(m—l) 2 + (m—S) 2 + (m—s) 2 +...}/m — (m 2 —l)h 2 /S = |-7i- 2 (m 2 — l) (nl/b) 2 . 
Upon applying this result to (49) it follows that the added resistance due to the 
action of H /( is 
^7r 2 (m 2 -l) RG (z) (nd/b) 2 . 
Adding these resistances to the skin resistance, the formula for the resistance of a 
many-layered coil is 
It' = R (l + F + -g-7r 2 (2m 2 — l) (nd/b) 2 G}.(92) 
The corresponding formula for a stranded wire coil is obtained by replacing F by 
F 4 - 2s 2 $ 2 Q/d 2 and d by sS. F and G in this case are calculated, using the diameter 
of a single strand. 
Assuming that the correction for curvature for the many-layered coil is of the same 
form as that for the single layer coil, the following formula includes all the previous 
formulae— 
R' = R (1+ F + MG) .(93) 
in which for solid wire coils 
M — u n (2m-—1) (ndfb) 2 .(94) 
and for stranded wire coils 
M = 2 (s Sfd 0 ) 2 + u n (2m 2 —l) (ns S/b 2 ) 2 , .(95) 
(27) Best Conditions for Many-layered Coils .— If different coils are wound with the 
same length and diameter of wire on the same shape of frame, and with the same 
spacing between the wires but with different radii, then the inductance will varv as 
mr- while the resistance will be of the form 
a + 6m 2 , 
in which 
a = R {l+F— u n (nd/b) 2 G}, /3 — 2 u„(nd/b) 2 G. 
At low frequencies F and G are negligibly small, so that increasing the number of 
layers will always improve the time-constant. At high frequencies the best time- 
constant is obtained when 
a = 3 /3m 2 . 
