Re: TESLA COIL REVISED

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

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Hi Jaro, Gary,

Jaro wrote:

 > Let's compare a 1000-turn 3" diam. thin wire secondary with a
 > 50-turn 12" diam. thick wire secondary.
 > ...
 > The 50-turn coil resonates at higher frequency, but it WOULD NOT
 > have a higher resistance than one operating at a lower frequency.

I'm inclined to agree with Jaro on this point.  If we're just
looking at the coil parameters, its easy to see that the theoretical
Q factor of the 50t coil can be much higher than the 1000t coil,
given a few reasonable choices of wire size and coil length.

For eg,

turns        50              1000
diam         12"             3"
length       18"             18"
wire         8 awg           26 awg
Rdc          0.1 ohms        33 ohms
Rac          1.3 ohms        115 ohms
Les          365 uH          9400 uH
Fres         2.2 Mhz         603 kHz

Coil Q       3900            310

The Rac includes a Medhurst phi factor for the proximity loss
estimate.  The HF coil scores well because of the thick wire and
the smaller spacing ratio of the winding.

In practice neither coil would achieve these idealised Q factors
due to other system losses effectively adding to the Rac of the
resonator.  The HF coil Q is likely to go down by a factor of maybe
10 or so, and the LF coil by maybe 2 or 4.  But even so, the HF can
come out the winner for the unloaded system.  Indeed, if we reduce
the turns as far as possible - to a straight tube - and put it inside
a high Q container (a cavity) we obtain a VHF resonator capable of
Q factors well over 1000.  I used to build commmercial VHF cavities,
(large ones!) and casting the memory back more than 20 yrs I seem to
remember *loaded* Q values around a couple of thousand, with
insertion losses of around 1dB.  I'll leave it to the reader to
estimate the unloaded Q from those figures.

As it happens I've been messing with a high Q HF CW coil at the
weekend - I decided to rebuild the PA of my 7Mhz transmitter (a long
story).

The output stage involves a number of L and C components but the
most critical stage is the first L and C stepping the device output
impedance up to higher value, ...

          ------ L ----- ----> to Pi network.
         |              |
      |  | drain        |
      |--               C
      |--  FET          |
      |  |              |
         |              |
      ------------------------- ground

The L and C here are effectively a CW TC, base driven, and loaded at
the 'top' by the next stage - a pi network.  The arrangement steps
up the drain output impedance (a few ohms) up to a higher value for
the next stage.

Now the efficiency is the usual

   100% times (Q_unloaded - Q_loaded)/Q_unloaded

and my problem is to increase Q_unloaded because it's not very good
at the moment.  The test is to operate the thing unloaded, ie I
disconnect the following stage and fire the thing up.  We expect
a large voltage across the C, and the drain should see its lowest
possible load impedance - that being the effective resistance of
the unloaded resonator  (as load is applied to the output, the coil
'base' impedance rises above this value to the normal operating
level, ie the familiar see-saw effect of the 1/4 wave line [*]).

Now if I repeat the above calculations for this circuit, I get
a theoretical Les of 6.5uH and Rac of 0.08 ohms, to give an idealised
Q of about 3600.  The measured value is unsurprisingly very much
lower than this - around 190.  So we have lost a factor of about
20 in going from the theoretical ideal to the real circuit.  The
effective resistance in practice must therefore be around 1.5 ohms
or thereabouts.  I've been trying to get this figure down so that
I can draw more power from the device (by presenting a lower
impedance to the drain).  Some of the extra is the device 'on'
resistance, but much is probably coming from eddy losses in the
poorly shielded circuit.

I mention all this partly to show that the practical Q values are
always much less than the ideal unless considerable effort is put
in, and partly as an example of a very common use for CW TCs with
few turns aimed at high unloaded Q at HF.  You've all seen those
nice shiny silver plated loading coils that HF transmitters
invariably have.

 > Of course the higher frequency will result in shorter sparks, so
 > this coil would be more for people who want to experiment with
 > very high frequencies and perhaps experience beams or walls of
 > light (brush-like discharge), instead of the sparks. And that
 > brush-like discharge would probably be more plasma-like than the
 > usual low-frequency sparks.

Indeed.   This brings us to a more interesting topic, that of
loading the high Q HF coil.

Both the TC and the matching network above behave as a 1/4 wave
line, having a characteristic impedance Z = sqrt(L/C) and the
input and output impedances, Zin and Zout, are related to Z by

    Zin * Zout = Z^2 = L/C

Thus as Zout is reduced (heavier load) Zin increases and vica versa.
For the TC producing a brush discharge, as the output voltage
rises, the size of the discharge increases, so Zout reduces,
which increases Zin, which in turn limits the power which the
driver can put into the TC base.  Thus the system settles to an
equilibrium involving the input power and the load discharge.

The challenge for the designer is to choose an L/C ratio which
allows the driver to transfer its full power to the load discharge.

It remains an open problem to predict the discharge impedance Zout
for a given frequency, voltage, and discharge terminal shape. AFAIK,
very few measurements of Zout are available, so there's plenty of
experimentation to do here.  Without this information - empirical or
theoretic, it is impossible for the CW TC builder to 'close the loop'
at the design stage.

Thus, I would say L/C is at least as important as L/R for CW TC
design, where of course R must be the total effective resistance
(including the discharge) of the resonator, and likewise C must be
the total capacitance, including terminal and discharge load
capacitance.  The L/C ratios of the two example coils above are
roughly

turns        50              1000
L/C          25e6            1260e6
Z            5k ohms         35k ohms

Regardless of Q factor and operating frequency, the low Z of the 50t
coil makes it poorly suited for long streamer formation, and the high
Z of the 1000t coil is probably less suitable for powering a heavy
brush discharge.

It would be easy to prepare a very high L/R coil and find that it
produces feeble discharges due to a gross mismatch of the reflected
load presented to the driver.  Most CW coilers build in some room to
maneuver by installing a ferrite impedance matching transformer
between driver and base with a selection of taps.

I think Jaro is right to point out that HF, low L/C coils are an
under-explored area, as opposed to LF, high L/C coils aimed at
producing long streamers.  While we're very used to dealing with
HV LF streamer-generating coils, it pays to remember that most
'embedded' applications of Tesla coils belong to the other category.

Rather than streamer length, here the figure of merit might be
something like volume of plasma produced per kW input, although
plasma temperature might also be considered equally important, so
maybe volume * temperature would be better indicator.

I also think that Gary has a valid point when he warns us about the
loss resistances rising with frequency.  It is generally harder to
obtain high Q in practice as the frequency goes up, not so much
for the coil itself (as we see above) but from the rest of the
circuit, and from the interaction of the coil E and H fields with
the surroundings.  A high Q HF coil must come equipped with a high
Q shield too - so that the large shield eddy currents will have only
a small impact on effective AC resistance.  It is the overall system
Q that counts, and the coil is only a small part of it.   The VHF
resonator achieves its very high Q largely due to the high Q of the
containing cavity.   A poorly shielded coil such as my PA matching
circuit will fall well short of its theoretical Q factor.

I would encourage Jaro to explore the use of high Q, HF, low L/C
Tesla coils in CW mode loaded by brush discharges. There's lots to
be done!

[*] This 1/4 wave impedance inversion can be regarded, equivalently,
as a conversion from the series LCR impedance seen by the drain to
the parallel LCR impedance seen by the load.
--
Paul Nicholson
--