Re: Mutual Inductance & K Factor

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Paul Nicholson by way of Terry Fritz <twftesla-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

Godfrey Loudner wrote:
> When you set up the Neumann integeral, did you view the conductors
> of the primary and secondary as filaments through the centers of
> their cross sections, or did you take into account the thickness
> of one or both the conductors.

ACMI assumes that the current is concentrated in a central filament.

> Is the mutual inductance of two conductors of circular cross
> section equal to that their central filaments? I feel that this
> is not true in general.

Definately not true in general.  If the B field gradient over the
width of the conductor is a significant fraction of B, then the
central filament approximation starts to break down.  For this 
reason we have to be cautious when applying these methods to 
coils wound with thick wire or tube.

> Have you seen a mathematical demonstration that Wheeler's
> formula approximates the ellipitic integrals?

The exact inductance of a solenoid in which the wire diameter is
negligibly small compared to the turn diameter, is

 L = mu * n^2 * pi * a^2 * K/b          ...Henries

where 

 n = turns
 a = radius                   ...metres
 b = length                   ...meters
 K = shape factor
   =  (1/(3*pi)) * (d * b/a^2 * (F(k)-E(k)) + 4*d*E(k)/b - 8a/b)

 where k = 2a/d; d = sqrt(4*a^2 + b^2);
       F(k), E(k) are the complete elliptic integrals.

This was first worked out by Lorenz in about 1879 I think, and
appeared in Maxwell's Treatise on Electromagnetism at about the
same time.  Nagaoka (1909) calculated a table of 160 values of K
from which by interpolation an accuracy of around 5 or 6 figures
could be obtained.

Wheeler proposed several approximations for K, for example, for
long solenoids, he suggested

 K = 1/(1 + 8/(3*pi)*(a/b))

which gives

 L = mu * n^2 * pi * a^2/( b + 8*a/(3*pi))  ...Henries

and if you convert a and b to inches, you get from this:

 L = a^2 * n^2/(8.465*a + 9.973*b)          ...micro Henries

It is often erroneously stated (on TC websites!) that the Wheeler
formula is an empirical formula:  As you can see, it is not.

BTW, I do hope that no programmers are using this formula in their
TC programs. These cheap and cheerful formulas are just for hand
calculations. As a programmer, you have zero excuse not to use a
more accurate approximation for the shape factor K, eg as per the
Lundin formula.

> What is the formula of Lundin?

Lundin provides a formula for single layer solenoids, which 
involves a polynomial approximation to the geometry factor K.
Good to around 6 figure accuracy, so it is on a par with 
interpolation on Nagaoka's tables.

See http://www.abelian.demon.co.uk/tssp/formulae.html

> To evaluate Neumann integral, I assume that you have employed a
> numerical approximation method for which there exist a proof
> that the computation converges to the desired result for the
> indicated situation.

Nope, I use interpolation over Grover's tables.  This is very much
quicker than computing my own elliptic coefficients for the
coupling between concentric circular filaments, and comes with no
loss of accuracy.  Circa 10^5 coefficients needed per L value.

John Couture wrote:
> I believe a greater number and range of tests are still required
> to determine how accurate any of the programs represent the real
> world of Tesla coils

Absolutely agree. Real windings on TC's depart from the idealised
current sheets and filaments used by the theoretical derivations.
With primary windings, the B field can change by a significant
fraction over the width of a conductor (especially on the inner-
most turns!).  With secondary windings, the current is not uniform
at high frequency.

> I thought you had said in the past that you had to tweak (fiddle
> factors) the program to make it agree with tests that had been
> performed on actual coils

Yes, at one time I tried replacing the central filament with a
bundle of perimeter filaments to represent a tube winding, but the
resulting change in L was less than the accuracy of available
measurements.  Therefore, being unable to assess the effectiveness,
I took that out.  We will continue to run with central filaments 
until the extent of the consequent error becomes clear.

> Do you have any comments on the type of tests that should be
> performed to find the mutual inductance and K Factor?

The largest source of error in estimating M and k comes from the 
primary winding self inductance.  I'd like to see some super-
precision measurements of these so that the effectiveness of the
filament approximation for these coils can be assessed.  Indications
so far suggest that typical TC primaries can be calculated to
better than 5% using the central filament approximation.
 
The coupling coefficient that really matters is that which occurs
at high frequency, ie in the vicinity of secondary Fres.  The
secondary current is non-uniform so we might reasonably expect k to
differ from its low frequency value. I've yet to come up with a good
way to measure these, but it will likely involve measuring the two
beat frequencies.
--
Paul Nicholson
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