66 
MR. S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
condition of low frequency is w < Iv 0 /4. When this holds, formula (21a) applies in 
regard to so that by (18) the rate of dissipation of energy is 
W = ^ R 0 (K„) 2 + 
CO 
It, 
] 
5(1 
(K 0 ) 2 + 2 (K„) 2 a 2n /2rr 2 (2n + 2) 
. . (23) 
Now by (17) the terms involving K 0 are due to a held whose components outside 
the cylinder are Q„ = K 0 /r, P 0 = 0. This held can only be due to a current of 
magnitude I = |-K 0 distributed symmetrically round the axis and ho wing parallel to 
the axis. 
Hence the energy dissipation due to such a current is 
W 1 =iRo(1+*j^)T 
(24) 
This is the usual formula for the skin effect at low frequencies. 
If a uniform held H is acting on the cylinder, then H = K,, K 2 = K, = ... =0, so 
that the energy dissipation due to a uniform held H is 
W 2 = KHV/R 0 .(25) 
The remaining terms are due to non-uniformity of the held. 
If the external held is expressed in a Fourier series of the form (17), and if the 
coefficient of the term cos (nQ -j- a n ) in the series for Q 0 has the value L„ at the surface 
of the cylinder, then this portion of the held contributes an amount 
wWL 2 „/2n 2 (2n + 2) R 0 
to the energy dissipation. 
The way in which n occurs in this expression shows how unimportant are the higher 
terms of the Fourier series in producing eddy losses at low frequencies. 
The assumption that the external held is uniform and has its central value will, 
therefore, in most cases give a good approximation to the actual loss when the 
frequency is low. In illustration, suppose the external held to be due to a thin wire 
carrying current I, and stretched parallel to the cylinder at a distance D from the 
axis. The value of Q 0 in the plane common to the axis and the w r ire is 2l/(D — r), 
or, in ascending powers of r, 
so that 
21/ r_ 
D 1 + D + D 2 + 
L 
n 
21 
D D" ’ 
and therefore the energy dissipation is 
