94 
MR. S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
turns. Using these values in W and adding the skin losses, the following formula is 
obtained for the effective resistance of a stranded wire coil:— 
(84) 
in which 
R 0 (= rjs) is the direct current resistance per unit length of the equivalent solid wire, 
s — No. of strands, 
8 = diameter of each strand, 
a = angle of twist, 
d 0 = over-all diameter of stranded wire, 
D = separation of turns in coil; 
R' = R, sec a 
1 + F (z) + s 2 <fG (z) | sin- a)+ ^ 2 (l — i sin 2 a) 
while in the calculation of F and G, z 2 — irhwS 2 . 
(21) If the twist is so small that sin a = a, (84) becomes 
R' = R„ 
1 + F + s 2 d 2 G 
JL ,Un 
d 2 D 2 
+ W 
1 -t F 4- Lu n G 
sfS 2 
D 2 
(85) 
The correction due to the twist will therefore be less than 1 per cent, if a < 0-14 
radian (8°). In determining the most efficient coil only the main term in (85) need be 
considered. 
(22) The quantities fixed will be taken to be the length of wire, the number of strands, 
and the diameter of each strand. Under these conditions it is clear from (85) that the 
best value of d 0 is d 0 = D, as adjustment of d 0 will have a negligible effect on the 
inductance. Then 
R' = R„ 
1 + F + G (2 + u n ) 
( 86 ) 
The best value of D and shape of coil then follow by a method identical with that for 
the solid wire coil, except that sS replaces d, and 2 + u n replaces u n . 
The method gives as the conditions for the best time-constant 
a/b = 1 '5.(87) 
(D/sS) 2 = 17'76G/(l+F).(88) 
and the value of the time-constant is then 
R'/L = Filly (z) y/fp/lsi, . (89) 
y (z) being calculated from the diameter of a single strand using (81a). 
(23) Limits of Application. —The quantity sS 2 is the diameter of solid copper wire 
having the same section as the copper section of the stranded wire, so that if it were 
possible to pack the circle I) entirely with copper, D 2 /s8 2 could never fall below unity. 
Actually the limit is greater than unity, partly because the wire is circular, but also 
