[Elecraft] Optimal Coil Dimensions

Fri Jan 30 00:41:01 2004

huh?

Note David A. Belsley's stunning brilliance and Chuck's apt reply (gack!)

To:             	[email protected] <[email protected]>
From:           	"David A. Belsley" <[email protected]>
Subject:        	[Elecraft] Optimal Coil Dimensions
Date sent:      	Fri, 30 Jan 2004 00:22:44 -0500

> 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
> 
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