90 
MB, S. BUTTERWOBTH ON EDDY-CUBBENT LOSSES 
If the frequency / is not sufficiently high for the approximation to hold, (81) must be 
replaced by 
R'/L = y (z) .(81a) 
in which y (z) = 3T5 {G- (l + F) 3 }*/ 2 an< ^ Fas the following values :— 
2 = 1-5 2-0 2-5 3-0 3-5 4-0 4-5 5-0 oo 
y(z) = 3-81 3*61 3-53 3-48 3-43 3-41 3-39 3-37 3-15 
It is seen that y (z) does not vary much throughout a large range of frequency, and if 
the formula 
R'/L = 3-4 {fp/ld}* . (81b) 
is used this will represent the best value of alternating current time constant attainable so 
long as 2 is greater than 2. Taking for copper p = 1,600 C.G.S., expressing/ in terms of 
wave-length \ (in metres), and supposing L to be in microhenries, l and d in centimetres, 
R'/L = 2-35 ly/ldk. 
Thus A in Lindemann’s formula cannot be less than 
Kin. - 2-35 L Kid. 
In illustration we have for coils No. 1, 2, 3 of the table of Section 16, 
A m ,„. = 7-1, 8-6, 22 
while 
A rai JA aetual ==0-75, 0-73, 0-86. 
The ratio 2-35 L/A \/ld may be taken as a measure of the efficiency of any coil. 
Condition that Equation (77) may he satisfied .—Since, if the best spacing is used, 
ajb = 1 -6 is always the best shape, we have from (77), with u n — 3*87, 
(D/d) 2 = 11-61G/1+F..(82) 
This gives the following values for D jd :— 
2 
= 1-0 
1-5 
2-0 
2-5 
3-0 
3-5 
4-0 
4*5 
5-0 
D Id 
= 0-425 
0-88 5 
1-36 
l-70 5 
1-89 
1-97 
2-01 
2-04 
2 • 07 5 
2 
= 6-0 
7-0 
8-0 
9-0 
10-0 
inf. 
D /d 
= 2-14 
2-17 
2-20 
2-22 5 
2-24 5 
2-41 
Now D jd must always be greater than unity in practice, so that if 2 is less than 1 •6 1 
close winding is the best. When 2 exceeds 1-61, spacing rapidly becomes advantageous, 
the best spacing at very high frequencies being D = 2 • 4d. As regards departure from 
the best spacing, the table of t/t 0 shows that the time constant will vary by less than 
1 per cent, from the best value if D/D 0 lies between 0 • 85 and 1 • 18, by less than 5 per cent, 
if D/D,, lies between 0*79 and 1 -28. 
