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# RE>Inductance len<.8r bound

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From:  Dale Hall [SMTP:Dale.Hall-at-trw-dot-com]
Sent:  Friday, April 03, 1998 1:03 PM
To:  Tesla List
Subject:  RE>Inductance len<.8r bound

RE>Inductance len<.8r bounds limit  (Lmax -at-H/D=.5)
Erik/List,
Assumption check: (math valid but application boundary validity question)
I believe a derivation of the single-layer air-core coil formula maybe necessary
due to assumptions for the use of  L = (r * r * N * N) / (9 * r + 10 * b):
The '78 ARRL handbook states this formula to be a CLOSE APPROXIMATION
for coils having a length => .8 radius.  Mathematically determined conclusions near/beyond boundary are likely to be flawed.
Additionally some lab confirmation could be performed as a check.

Malcom, what was your purely iterative means, basic formula/assumption ?

Dale

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Date: 3/31/98 11:33 PM
To: Dale Hall
From: Tesla List

---------- Forwarded message ----------
Date: Tue, 31 Mar 1998 18:31:00 +1200
From: Malcolm Watts <MALCOLM-at-directorate.wnp.ac.nz>
To: Tesla List <tesla-at-pupman-dot-com>
Subject: Re: Inductance (fwd)

Hi Erik,

> Date: Mon, 30 Mar 1998 17:35:43 EST
> From: ESchulz531 <ESchulz531-at-aol-dot-com>
> To: tesla-at-pupman-dot-com
> Subject: Inductance
>
> Maximizing the inductance for a cylinder coil.
>
> L : inductance in uH, r : radius in inches, N : number of turns,
> b : length of coil in inches, d : turns per inch, w : length of wire in inches
>
> L = (r * r * N * N) / (9 * r + 10 * b)
>
> Using this and solving for L in terms of r. and using
> (N = b * d and b = w / (2 * Pi * r * d))   (corrected)
>
> we get
>
> L = w / (4 * Pi * Pi * (9 * r + (10 * w) / (2 * Pi * r * d)))
>
> Now I took the derivative in respects to r
>
> - (d * w * w * (9 * d * Pi * r * r - 5 * w)) / (4 * Pi * (9 * d * Pi * r * r +
> 5 * w) ^ 2)
>
> Set this to 0 and solve.
>
> So the most inductance for a cylinder coil is when
> r = sqrt((5 * w) / d) / (3 * sqrt( Pi ))
> or
> w = r * r * Pi * d * 9 / 5
>
> Can somebody check to see if this is right?
> Just having fun with some old fashioned calculus. :)
>
> Erik Schulz
> ESchulz531-at-aol-dot-com

There is a very definite answer for a single layer helix that
involves no variables.  For a given length of wire, Lmax is achieved
in a closewind with an H/D ratio of 0.5 (approx).  I wrote a program
to sus this by purely iterative means. Too lazy to do the calculus.
Check me out on this.

Malcolm

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