# Re: Inductance, h/d ratio

```Subject:   Re: Inductance, h/d ratio
Date:    Mon, 5 May 1997 08:30:33 +0000
From:    "John H. Couture" <couturejh-at-worldnet.att-dot-net>
To:    Tesla List <tesla-at-pupman-dot-com>

>
>On Fri, 2 May 1997 18:53:30 +1200 Malcolm Watts
><MALCOLM-at-directorate.wnp.ac.nz> wrote;
>
>> Hi Alfred, all,
>>                 I wrote a program to calculate the optimum former h/d
>> ratio for a given length of wire. Method: start with a length of wire
>> and h/d = 50, then increase d as l is decreased. The answer the
>> program gave me was just under 0.5. Not great for coiling except with
>> a gigantic coil because of the height. Not the same as maximum Q
>> (without terminal) either. That turned out to be h/d=1, confirmed by
>> measurement.
>
>What were you using as your determinant of optimum? I remember back
>in college that the largest inductance for a given length of wire was
>something close to a h/d ratio of 3. It's not talked about in modern
>say from 1910 through 1950 I'm sure you will see it's close to 3 .
>

>-------------------------------------------------------------------

This problem can be solved by differential calculus.

The derivative  dx/dr = 1- 10A/(18 pi r^2)   where A = 6.283 rh

Using the Wheeler equation for inductance of a cylindrical coil and
also
that the derivative is maximum when equal to zero

r = 10h/9      h = 0.9 radius    h = 0.45 diameter

h/d = 2.22  close to the above results

The complete derivation is shown in Bylund's "Modern Tesla Coil
Theory".

The math says that the maximum inductance for a certain length of wire
is
when the height of the coil is 0.45 times the diameter. For those who
want
to check this result, a 6 inch diameter coil would require a winding 2.7
inches high for maximum inductance. If you have 191 feet of a #24 AWG
enameled wire and close wind 121.5 turns you should get 2.46 mh
inductance
with a 2.7 inch length coil.

With this same size and length of wire wind any other combination of
diameter and length of the winding (close wind) and you should not get
more
than 2.46 mh of inductance. If you do get more inductance, post your
results
so Tesla List can discuss your findings. Use an accurate inductance
meter.

I have never checked this maximum inductance possibility so do not
know
what will be found.

John Couture

```