92 
MR. S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
D = 2d. In estimating, for a given diameter of wire, the wave-length at which 
spacing should be employed, the formula 
K'oX = 172,000 .(83) 
(deduced from (68) with z — 2) is useful. Thus, with No. 28 S.W.G. wire, E/ 0 in ohms 
per 1,000 yards is 140; so that if X is less than 1,230 metres, spaced winding should 
be used. 
Flat Coils.—A similar treatment for flat coils has yielded the following results:— 
Spacing becomes advantageous at practically the same value of z (viz., 2 = 1-6) as 
that for solenoidal coils. At any value of 2 greater than 1 • 6, the best ratio of inner to 
outer radius is r/~R — 0-60, and the best spacing is 1-04 times that for solenoidal coils. 
When both these conditions are satisfied the eddy-current losses in the flat coil are 
5 per cent, greater than those in the solenoidal coil wound in the best way with the same 
wire. When 2 is less than 1 * 6, close winding is the best, and the best ratio of radii 
varies from 0-4 to 0*6. The value r/R = 0-5 will therefore be the best ratio to take 
for a range of working from 2 = 0 to 2 = 1 • 6. 
(E). Single Layer Stranded Wire Coils. 
18. Single Layer Stranded Wire Coils. —If we replace the solid wire in a coil by a 
number of insulated parallel filaments of the same copper section and connected in 
parallel, the direct current resistance of the coil will be unaltered, but for alternating 
currents the distribution of current between the filaments will not be uniform unless 
the filaments are interwoven in such a way that they traverse similar paths. 
As a non-uniform current distribution will cause increased losses, and twisting will 
produce increased length in the filaments, it is a matter for investigation as to what 
gain may be expected by using stranded wire coils. 
It will be assumed that each strand traces out a helix about the axis of stranding, 
the angle of which is a. Actually the radius of the helix will vary along a particular 
strand ; but this need not be considered in getting the average result, as we may pass 
from one strand to its replacing strand, and thus keep at the same distance from the 
axis throughout the length of the stranded Avire. The further assumption that a is 
constant throughout the section Avill assure that there are the same number of strands 
in the same axial length of any layer whatever its distance from the axis. 
19. If r is the direct current resistance per unit length of one strand, the skin resistance 
of each strand in unit axial length of the wire is 
r 0 sec a {1 + F ( 2 )} 
where 2 2 = -/.aA 2 , S being the diameter of one strand. Upon this must be superposed 
the resistances representing the losses due to two fields:— 
(a) A field Hj due to the strands in the same turn of the coil as that in which the 
