Re: Medhurst and E-Tesla thoughts...

[paul-at-xxxxxxxxxxxxxxxxxxx (by way of Terry Fritz <twftesla-at-xxxxxxxxxx>)]

Terry Fritz <twftesla-at-uswest-dot-net> wrote:

> I would guess that Medhurst incorrectly thought the C he was
> measuring was the only factor.  Apparently, the inductance of bare
> coils is affected and lowered by the diminishing currents near the
> top of the coil.  So there is a capacitance involved but also a
> variation in the coil's inductance that is a function of the height
> to diameter ratio.

To be fair to Medhurst [1], his intention was to measure the self
capacitance effective when the RF current in the coil is more or less
uniform, which is very much the case in most HF inductor applications.
Medhurst ensured a uniform current by resonating the inductor with a
large external capacitance, so that the coil current is dominated by
the capacitor current and the variation in current along the coil due
to distributed self capacitance is negligible. He obtained his self
capacitance by measuring Fres with a number of values of the external
capacitance, and then extrapolated the results to find the Cmed.
Therefore the resulting Cmed is the *adjustment* you would need to
make to the C in a lumped LC circuit in order to obtain the correct
Fres. This is only valid as long as C is large enough in the first
place to ensure a reasonably uniform coil current, and as long as this
is true, the L value to use is the DC value of inductance.

However, unlike most HF coil applications, the tesla coil is operated
with very little external lumped capacitance, and as a result the coil
current is far from uniform. Thus, attempting to find Fres based on
Ldc and Cmed cannot be expected to give the right answer. In a tesla
coil, the L effective in the formula for resonance is usually less
than Ldc, since the upper part of the coil sees less current than the
lower part. Likewise, the C effective in the resonance formula will
differ from the Cmed since Cmed is defined, and was obtained, under
conditions of linear voltage rise (due to uniform current) - in other
words, the Cmed is the capacitance which, when charged to Vtop, would
store the same energy as that stored in the distributed coil
capacitance when the coil carries a linear voltage rise.

In a tesla coil, the voltage in the lower part of the coil rises a
little faster than linear, so that, for most of the coil, the
distributed capacitance is charged to a voltage higher than that of a
linear rise. Therefore the equivalent C required to properly represent
the coil's distributed capacitance in an Fres calculation will be a
little higher than Cmed.

So if you use Ldc and Cmed to estimate Fres, the Ldc will be a little
too big, and the Cmed a little too small. To some extent these effects
will cancel and the answer will often come out to within several
percent of the real Fres.

How can we do better than this?  Well, if we knew in advance the
current distribution of a given coil, then we could calculate a 
correction factor for Ldc. If we also new both the voltage
distribution and the physical capacitance distribution, we could also
calculate a suitable equivalent C value. The trouble is, both the
voltage and the current distributions themselves depend on the
distribution of capacitance (and inductance too, but we'll assume
that's uniform). This kind of chicken and egg situation can be dealt
with mathematically without too much trouble, but its takes a computer
quite a long time to crank out the answers.

A reasonable short cut is to *assume* the current and voltage
distributions. Then, you only have to calculate the physical
capacitance distribution from which an effective C can be determined
according to the assumed voltage distribution. Terry's E-Tesla6 [2]
program does just this, and strikes a fair compromise between accuracy
and runtime. This program should be capable of delivering rather
better predictions of Fres than estimates based on Cmed.

No doubt Terry is working on improving accuracy by refining voltage
and current assumptions according to topload, coil size and h/d -
information which, perhaps, tssp [3] will be able to supply one day!

Just one final point. It seems almost traditional in the tesla
community for confusion to arise concerning coil self-capacitance,
often due to that detestably undefined term 'Cself', which by itself
is pretty meaningless. I'll endeavour to type up some notes which will
spell out the full monty on this subject.

[1] R.G. Medhurst, "HF Resistance and Self Capacitance of Single
    Layer Solenoids", Wireless Engineer,
    pp35-43 Feb 1947 and pp80-92 March 1947.
    Available in zipped pdf from Bart Anderson's site,
    http://www.geocities-dot-com/classictesla/Medhurst.zip

[2] http://hot-streamer-dot-com/TeslaCoils/Programs/E-Tesla6.zip

[3] http://www.abelian.demon.co.uk/tssp/

Regards,
--
Paul Nicholson,
Manchester, UK.
--