IN CYLINDRICAL CONDUCTORS, ETC. 
59 
//R t) is less than 0-225 the eddy-current losses vary as (//R 0 ) 2 to an accuracy of one 
per cent. At higher frequencies the variation is slower, the ultimate rate of variation 
being as (//R 0 )k A knowledge of these limiting rates of variation enables an immediate 
explanation to be given of Lindemann’s results with stranded wire coils. 
A solid wire in a given alternating field has eddy losses which are a function of 
//R 0 = <p (//R 0 ) say. If the solid wire is replaced by s strands of the same total metallic 
section, the loss per strand in the same field is 0 (//sR„) and the total loss in the 
s strands is s</> (//sR 0 ). 
Thus, as regards the losses due to the general field of the remainder of the coil, we must 
replace <p (//R 0 ) by S 0 (//sR 0 ) in passing from solid to stranded wire. 
At low frequencies <j, (//R 0 ) = C (//R 0 ) 2 where C is a constant independent of the 
stranding, so that the respective losses are C (//R 0 ) 2 and C s (//sR.,) 2 = C (//R 0 ) 2 /s. 
The effect of stranding at low frequencies is thus to reduce these losses in the ratio 1 Js. 
At high frequencies </> (//R 0 ) = C' (//R 0 )® and the losses are C' (f/R 0 f and 
C's (//sRq)® = C's® (f[R 0 f, or the effect of stranding at high frequencies is to increase 
the losses in the ratio s-/ 1. 
Since in inductive coils the general field produces the main losses, Lindemann’s 
results are explained. 
(A). Eddy-Current Losses in a Cylinder in an Alternating Magnetic Field. 
(1) The cylinder is supposed to be non-magnetic and to have electrical conductivity k. 
Its radius is a. The magnetic field is perpendicular to the axis of the cylinder and does 
not vary along the axis ; otherwise its form is general. The field alternates with 
frequency co/27r, and the alternations are so slow that the dielectric current can be 
neglected in comparison with the conductance current. This means that the wave 
length of the disturbance producing the field is large compared with the dimensions 
of the cylinder. On the other hand, the cylinder is supposed to be long enough to 
render its end effects negligible. 
The procedure is to represent the electric and magnetic forces by rotors* E e iu>t , &c. 
The values of these rotors are found at all points in terms of the (given) undisturbed 
field. Then by application of Poynting’s Theorem over unit length of the surface of 
the cylinder, the energy flow into the cylinder is determined. 
This energy flow may be regarded as made up of two portions, one continuous and 
the other alternating. The former portion is the energy dissipated by eddy-currents 
set up in the cylinder. 
(2) Take the axis of the cylinder as the axis of a right-handed system of cylindrical 
* These are the rotating vectors used to represent these quantities on the vector diagram. The term is 
chosen to distinguish them from the space vectors which are also involved in the problem. 
K 2 
