58 
MR. S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
for this case is obtained in Section 8 of the present paper. This formula is shown 
(Section 9) to give results in close accordance with Kennelly’s observations. 
In order to reduce the eddy-current losses solid wire is often replaced by stranded 
wire in which a bundle of thin separately insulated wires are interlaced symmetrically 
with each other, the notion being that the sum of the eddy losses in the individual wires 
shall be less than the eddy loss in the corresponding solid wire. Lindemann* verified 
this experimentally at certain frequencies, but also found that if a solid wire coil and 
stranded wire coil were compared at various frequencies, the stranded wire coil increased 
in resistance more slowly at the lower frequencies but less slowly at the high frequencies, 
until above a certain frequency the stranded wire coil had a greater effective resistance 
than the solid wire coil. 
HowEf has treated the problem of straight stranded wire conductors, assuming the 
eddy losses to increase as the square of the frequency, and from his formulae has shown 
that at high frequencies it is difficult to make the resistance of the stranded wire less 
than that of solid wire of equal section. 
In view of the extensive use of stranded wire in the construction of coils for high- 
frequency currents it is important that the limitations of stranded wire in reducing 
effective resistance should be known, so that the present investigation includes the 
consideration of such coils. From the formulae obtained, conclusions are drawn in 
regard to the utility of stranding and in regard to what degree of stranding it is necessary 
to employ, before any improvement over solid wire coils may be expected. 
In formulae hitherto given for the effective resistance of coils, one or other of the 
following limitations occur :— 
(1) The coil is very long. 
(2) The frequency is limited to so low a value that the “ square of frequency” law 
holds. 
(3) The coil is wound with wire of square section. 
The formulae deduced in this paper differ from those already established in that— 
(1) The dimensions of the winding sections of the coils are small compared with the 
coil radii. 
2) There is no limitation imposed upon the frequency. 
(3) The wire is taken to be circular. 
In regard to (1) it is shown that coils of this type have better alternating-current 
time constants than long coils. 
In regard to variation with frequency, the factor governing the upper limit to the 
application of the square law is the magnitude of //R 0 where / is the frequency and 
R 0 is the direct current resistance per unit length of the wire used. If (in C.G.S. units) 
* ‘ Deut. Phys. Gesell.,’ 1909, p. 382 ; 1910, p. 572. ‘ Jahrbueh der Drahtlosen Telegraphie,’ 1911, 
p. 561. 
f ‘ Roy. Soc. Proc.,’ A, vol. 93, p. 468, 1917. 
