# Re: Propagation velocity in long helical coils.

```Hi all,

Still checking my results.  Found one error and fixed it.  No major change
in
form. The cyclic velocity at long wavelengths may be a numerical instability
in the numerical integration.   It does now appear to have a mathcad type
closed form. Four page widths long and containing unfamiliar constants,
Sinc type functions and summation of factorials along with regular trig.
It also contains an unexpected  integration factor.

I should point out that this a not a major feat of mathematics.  With
mathcad its just circuit analysis, maybe a feat of circuit analysis and
approximation. One more key point is I used a Wheeler derived coupling
function as I could not face the elliptical integrals this may have enabled
the production of  the closed form. It also only has mag coupling, ie no
internal C.

I have decided that the real test is a comparison with the Medhurst derived
frequency for a secondary coil.  The classical ground plane trick appears to
be valid with a minor variation  The infinity long constant parameter coil
is equivalent to a 1/2 wave coil between to ground planes.  You then just
cut the coil in half  and you have a Tesla coil.  Because the diameter is
significant the EM fringes out of the end of the coil due the  missing other
half.  I estimate this will make the coil  appear to be between zero to one
diameter longer.  I will assume an integrated square effect so I will choose
1/3 diameter longer. (Note this has a dispersion effect) This can be
verified by measurement but as its unlikely to be more than 1/6 or 1/3 in
error this corresponds to only 3.33% or 6.67% for 5:1 H/D coil.  The same
can be done to remove the ground plane for a valid comparison with Medhurst
which I believe is for an isolated coil.  The ground plane effect is said to
be small. This methodology turns the constant parameter case into the
practical case without any analysis of the parameter distribution.

My gaol is near the comparison between the wave and lumped equations.  The
practical applications
for this academic exercise may be a more accurate calculation of  frequency
and voltage with and without a top load.  Although there are much simpler
methods for both. The analysis may be extendible to distributed coupling
and transients using similar methodology but I will leave that for an other
time or
others.

Regards Bob

-----Original Message-----
From: Tesla List <tesla-at-pupman-dot-com>
To: tesla-at-pupman-dot-com <tesla-at-pupman-dot-com>
Date: 05 May 2000 16:09
Subject: Re: Propagation velocity in long helical coils.

>Original Poster: Terry Fritz <twftesla-at-uswest-dot-net>
>
>Hi Bob,
>
>Wonderful!!!  I will bee looking very forward to seeing what you have
>found!  Any model that can give insight into the secondary behaviour at
this
>level of detail will really be a major advance.  Such a model will also
>greatly help issues of dynamic streamer impedance, growth, and such where
>were still know very little and the present analytical tools are weak.  A
>model also allows us to poke and prod it to get a much better idea of the
>true nature of secondary behaviour.
>
>Cheers,
>
> Terry
>
>
>At 10:00 AM 05/05/2000 -0400, you wrote:
>>Hi all,
>>
>>I believe I have solved the propagation equation  numerically.
>>The surprising result is the velocity as a function of wavelength repeats
>>with a similar form to a 1/2 wave resonance. In fact beating between the
>>coupling function and wavelength.
>>This produces several possible modes of propagation at least in very long
>>coils.
>>The effect should readily be detectable in a coil 15 diameters long and
>>in the progression of harmonics in a smaller coil i.e. the frequencies
will
>>be  both higher and lower than a 1/4 wave progression with constant
>>dispersion. I will post the results when I have them tidy.
>>
>>Sorry its only numeric I know this is not much better than simulation but
>>diabolical integrals are hard even for waveophiles.
>>
>>
>>Regards Bob
>>
>>
>>
>
>

```