[Elecraft] Optimal Coil Dimensions

[David A. Belsley]

The article cited earlier on this reflector 
<http://www.thebest.net/wuggy/coils.html> dealing with optimal coil 
dimensions presents an interesting problem that, in fact, has a closed 
and simple solution.  Wheeler's formula is used

L = (n^2 r^2)/(9r + 10h),

where L is inductance in uH, n is number of turns, r is radius (in.), h 
is length (in.).  In this form, the relation to the variables of 
interest, such as the total length of the wire and the ratio of r to h 
is not clear.  However, with a change in variables

n = len/(2Pi*r), where len is the total length of the wire
h = a*r, where a is the ratio of the length of the coil to its radius
and
t = n/h = len/(2Pi*a*r^2), the turns per inch,

one gets

len = 2Pi ((L(9+10a))^2/a*t)^(1/3).

If one wishes to find the ratio of r to h (a) that gives shortest wire 
(len) capable of providing a given inductance (L) with a given turns 
per inch (t), one differentiates this expression with respect to 'a', 
sets it equal to zero, and solves.  I expected this to provide a rather 
nasty expression in a, t, and L, and the derivative indeed is.  
However, the numerator of this derivative has a factor

[(20(9+10a)L^2)/at - ((9+10a)^2*L^2)/(ta^2)],

which, when set equal to zero, implies a = .9, regardless of L or t.

In other words, a coil of a given inductance can be created using the 
shortest piece of wire when the coil has a length (h) that is 90% of 
its radius (r).  This is assuming, of course, that the wire gauge is 
such that the given length of wire (len) can indeed be wound within the 
length (h) of the coil.

Consistent with this, the author of the web article cited above did 
some calculations for a special set of assumptions and obtained some 
empirical evidence that a = 1 is somewhat better than a = 2.  Now we 
can see that, regardless of any special assumptions, a = .9 is better 
yet -- indeed best.

best wishes,

dave belsley, w1euy
-------------------------------------
david a. belsley
professor of economics