Re: Resonator base impedance (help needed)

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

Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>" <paul-at-abelian.demon.co.uk>

R.E.Burnett <R.E.Burnett-at-newcastle.ac.uk> asked:

> How can I calculate the input impedance at the base of a Tesla
> Coil at resonance,  before breakout ???...

> ...I would like to know if I can estimate the base impedance at
> the design stage before winding the actual resonator.

Faced with exactly that problem earlier this year I began a project
with the goal of answering that question. I'm afraid that eight months
later the best that can be said is that we now have the full measure
of the difficulty of the problem! 
 
> Gut feeling suggests that the impedance should depend on Ls and Cs.

Yes.

> However it seems that the H/D ratio and wire size may also play a
> significant part in determining Zbase ???

Yes.

> Also,  since a resonator can be modelled as a series RLC circuit in
> which Xl and Xc cancel out at resonance,  then the input impedance
> would seem to be a pure resistance equal to R ???

Er, yes almost. If you're driving the base from a coupling
transformer, the impedance match to the base will likely be slightly
reactive - the Fres of the coupled secondary will be a little
different to that of the freely resonating secondary. (ie the system
will resonate at the frequency at which the coil base Zin is the
complex conjugate of the coupling transformer Zout). To a first
approximation, assume it's a pure resistance, but design your coupling
to accomodate some Fres shift.

BTW, you'll notice that the frequency of minimum |Zin| (ie the freq at
which max measured base current occurs) is slightly different from
that of minumum base resistance (ie max measured Vtop) - a few tens
of Hz difference.

> Does this mean that the usual measures taken to increase Q factor
> (minimise R) will also decrease the base impedance of a given
> resonator ???

Yes.

The resistive part of the input impedance originates from the sum
total of all the resonator losses. Given this, Zin follows almost
immediately. Quantifying all the losses is hard. Even the AC wire
resistance is a bit of an unknown quantity. Medhurst presented tables
derived from empirical results, but these apply to a coil in which a
uniform AC current is flowing, so they are inaccurate for a TC. Add to
this the losses due to E and B field coupling to objects in the
surroundings - very hard to quantify in advance unless you propose
running the coil in a specially constructed environment.

If you are building a CW driver, especially one which is not self-
resonating with the secondary, you will also be interested in knowing
the effective Xl and Xc in the region around Fres, so that you can
determine if your driver will survive detuning by the working load
on the secondary. These input reactances are a lot easier to figure
than the resistive component. Remember that when lossy load is applied
to the top of the secondary, the R component of Zin will go up, not
down.

Richie, the input R depends on everything including the weather!
(yes, ground plane losses are involved too), eg Rin for my big CW
coil:

  My best theoretical estimate: 35 ohms.
  Coil setup over a soggy wet lawn: 80 ohms.
  Coil setup over a dry lawn: 120 ohms.
  Over concrete with a modest wire grid groundplane: 150 ohms.

My advice is to get to work on the design of a variable coupling
transformer that will match your driver to a wide range of the coil's
illusive and variable Zin. Tune and adjust coupling with a small test
signal prior to operation.  I've been working on just such an
adjustable transformer lately, for a 30kW bipolar, but I've not
achieved a workable design yet. 

> I'm very confused.

So was I, now I'm merely stumped.

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