Re: A new Tesla coil and k measurements

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

Original poster: "Paul Nicholson by way of Terry Fritz <teslalist-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

Antonio wrote:

 > I would have to compute the inductance L2 with the same profile
 > to get the correct k. I don't have yet an adequate method for
 > inductance calculations.

Yes.  You can always just compute the self L with the same routine
that does the mutual.  When you come to need the self L of each
element, resort to a formula for the approximately straight line
segments.  Perhaps if you have enough elements, you can just leave
out the self L of each, for only a small error.  The alternative
of using a bundle of filaments is non-trivial.

 > We find then that cos(final_angle)=ctop/(ctop+cself).

Using Ctop = C_disc + Cwhip = 2.7pF + 4.1pF = 6.8pF, and
Cself = 4.4pF, we get final angle = arccos( 6.8/11.2) = 0.91 rad.

Using current cos( S * 0.91) in the acmi model gives

     pri.L|     sec.L| pri-sec.M| sec-pri.M|pri-sec.K|sec-pri.K
  57.22 uH|  24.86 mH| 133.16 uH| 138.15 uH|    0.112|    0.116

These figures differ from tssp, in which Les = 26.5mH and k is
around 0.117.  This may be due to the fact that tssp takes account
of internal C and thus the current profile peaks a little way above
the base.

I think for exploring the difference between low frequency k and
the k at resonance, it might be better to use a system in which the
difference is greater.  I'll have to ponder what sort of geometry
might fit the bill.

 > The different mutual inductances is something strange.

Neumann's integral remains symmetric. The apparent paradox is
resolved by noting that the two M calculations

  show mutual( primary,secondary) as "pri-sec.M" with inductance
  show mutual( secondary,primary) as "sec-pri.M" with inductance

actually refer to different systems now that we have modified the
current in one of the coils.

When computing pri-sec.M, we are looking at the induced Vpri for
the given secondary current profile.  The non-uniform profile of
Isec implies, via Kirchhoff's law, that a certain amount of shunt
conductance is distributed along the coil.

But when we run the opposite calculation, sec-pri.M, we are looking
at the total voltage induced across the secondary with the
assumption that the secondary is completely unloaded, ie it is not
burdened by the distributed conductance of the self C.

In order to restore the symmetry, we would have to take into account
the self induction of the secondary, ie the extra secondary EMF
induced by currents flowing down the shunt paths, in order to obtain
the correct 'effective sec-pri.M'.  I think I can alter acmi to do
this.  But for now, so long as we can neglect distributed C of the
*primary*, it is fair to use the computed pri-sec.M as the overall k
for the coupled resonators, because in the absence of distributed
primary C, the induced Vpri calculated by acmi is correct for the
given secondary current profile.

Ok on the scope plots.  I'll run up some waveforms from tssp and
overlay them, but I think we're a bit out on the two coupled mode
frequencies.

[Section 6 of pn1401]
 > A numerical example of the application of those calculations
 > would be useful.

Yes, I'll work something up for that.  Maybe using geotc, since it
computes a matrix representation of the operator A (for the lossless
case only), and it would just need some functions to generate the
matrices for the operators P, u, and v.  We would have to assume the
resonator was tuned and lossless in order to just use real numbers,
because I don't fancy trying to do complex arithmetic in javascript!

--
Paul Nicholson,
Manchester, UK
--