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Re: Why Medhurst derived fr is approxiamlty correct and the transmission line equation for the resoance frequency of a 1/4 wave helical resonator or Tesla coil.

Tesla List wrote:
> 
> Original Poster: Terry Fritz <twftesla-at-uswest-dot-net>
> 
> Hi Bob,
> 
> At 02:23 PM 05/17/2000 -0400, you wrote:
> >Hi all,
> >
> >The following is an updated copy of a previous post that was lost.
> 
> Sorry about all that...  Between that virus, some software upgrades, and
> mail bounces from defunct accounts...  The servers choked up.  In
> desperation, the ISP had to dump the mail early last week and start over...
> 
> >
> >I have completed most of my analysis and I have reached my goal of trying
> >to understand what initially appeared to me as violations of circuit laws. I
> >have now even solved the boundary problem for the current and voltage
> >profiles in a Tesla coil.
> 
> Wow!  That would be great!  The number crunching programs (especially mine
> ;-)) could really use such information!
> 
> >
> >I started off with the view that the lumped tuned circuit equation using
> >Medhurst C could not possibly produce the correct frequency. This was due to
> >my belief and apparently that of others that Medhurst C was the true self C
> >of the coil. It is actually the value you must subtract from a much
> >larger(>100x)  tuning capacitor to calculate the measured resonance
> >frequency of a coil with one end grounded and the coil isolated. i.e. a
> >correction factor for true self C and the internal turn to turn capacitance.
> >Where as true self C is the capacitance you would measure with a low
> >frequency (<< fr) LC bridge with the coil isolated.  It is the sum of the
> >distributed capacitance of each turn to ground and is an intrinsic property
> >of the coil.  It is only a function of its geometry and surrounding.  It is
> >independent on the mode of oscillation of the coil. For a close wound coil
> >its value is equal to the capacitance of a hollow cylinder with the same
> >dimensions as the windings.
> >
> >If you differentiate a formula for Med C it confirms that the true self C is
> >approximately constant along the coil with a small increase at the ends.
> >Incidentally it also confirms its for an isolated coil.  Then with simple
> >circuit analysis referring a constant distributed C to one end shows that
> >true self C of the coil is 3 times Med C for very short coils and greater
> >than 10x the 1/4 wave resonant frequency,  and  2 times Med. C for  long
> >coils and at least 10 times the 1/4 wave resonate frequency.  So a quick bit
> >of lumped approximation shows that the lumped equation using Medhurst C
> >could produce approximately the correct answer.  Which has been confirmed by
> >measurements on a variety coils.

	The method you mention for measuring distributed capacitance of a coil
is the one I was taught in school back in the 40's.  You just plot 1/f^2
vs capacitance, and extrapolate to the frequency with external
capacitance equal to zero to get the SRF, and from that and the LF
inductance you can calculate the frequency.  Use of a grid dip meter to
measure self-resonant frequency would be simpler and was known much
earlier, but the classical method was the one they were teaching then. 
This is just one of many methods which can be used.

	As for Medhurst, from the nature of his measurements he was determining
the equivalent capacitance at self-resonance, whatever it was made up
of.  (My wife's an English teacher and would shoot me for that
sentence!)  Since the dimensions and aspect ratio of the coil are
included in the measurement, the question of whether or not the coil is
a loaded transmission line isn't relevant:  just use Medhurst's
capacitance tables or approximate formulae and you'll get an answer
which gives just the thing we want to know.

Ed