Re: Propagation velocity in long helical coils.

Hi Bob,
             I'd be very interested to read all the work when it's done. 

> Original Poster: "Robert Jones" <alwynj48-at-earthlink-dot-net> 
> 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.

I've found that a grounded resonator which resonates with some 
frequency F does not resonate at 2F when ungrounded and lying 
horizontal clear of ground. I think from memory it resonated at an 
unexpectedly high frequency well above 2F but would need to check 
again to be sure. It was either high or low but significantly so.

> 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

Many thanks for your work.