Re: Mutual Inductance & K Factor

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

Original poster: "Peter Lawrence by way of Terry Fritz <twftesla-at-qwest-dot-net>" <Peter.Lawrence-at-Sun-dot-com>

Paul,
     is the spiral formula "r^2 * n^2 / (8*r + 11*w)" based on as solid
ground as the helix formula?, is there something better behind this approx?
-Peter Lawrence.



>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.
>

>--
>Paul Nicholson
>--
>
>
>