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III. Eddy-Current Losses in Cylindrical Conductors , with Special Applications 
to the Alternating Current Resistances of Short Coils. 
By S. Butterworth, M.Sc. 
Communicated by F. E. Smith, F.R.S. 
{From the National Physical Laboratory .) 
Received May 9,—Read June 23, 1921. 
Introduction. 
It is well-known that a considerable proportion of the effective resistance of inductive 
coils when used at radio frequencies is caused by the eddy-currents set up in the wires 
of the coils by the alternating magnetic field in which they are situated, and that in 
extreme cases the alternating current resistance may amount to more than one hundred 
times the direct current resistance. It is therefore important to have reliable formulae 
for the eddy-current resistance of such coils in order to determine the conditions which 
will reduce the eddy-current losses to a minimum. 
The simplest case, that of a long straight cylindrical wire under the action of its own 
current, has been treated by Kelvin,* Rayleigh,! Heaviside,! and others. The 
general effect is known as the “ skin effect,” because the current tends to concentrate 
more and more upon the skin of the conductor as the frequency increases. 
The case of two parallel wires forming a go-and-return circuit has been considered 
theoretically by Nicholson, § and experimentally examined by Kennelly.|| Kennelly 
found that when the wires are close together, the added resistance due to the proximity 
of the wires may be of the same order as that due to the simple skin effect. 
Nicholson’s theoretical treatment includes the possibility that the dimensions of 
the system may be comparable with the wave-length of the disturbance. His formula 
is very complicated and difficult to apply numerically. A formula {formula (47)} 
* ‘ Math, and Phys. Papers,’ vol. 3, 1889. 
t ‘ Phil Mag.,’ vol. 21, 1886. 
J ‘ Electrical Papers,’ vol. 2, p. 64. 
§ ‘ Phil. Mag.,’ vol. 18, p. 417, 1909. 
|| ‘ Trans. A.I.E.E.,’ vol. 35, part 2, p. 1953, 1915. Curtis (‘ Bull. Bureau of Standards,’ 1920) has 
recently published a formula for this case which gives agreement with Kennelly ’s results. 
VOL. CCXXII.-A 596. K [Published September 10, 1921, 
