Re: Measuring self-capacitance directly (Re: flat secondary)

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

Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz <twftesla-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>

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

> Puting in some values, from my large coil model,
> 
>          freq         Zft
>  f1/4   91503 Hz    41851 ohms
>  f3/4  217611 Hz    23684 ohms

> gives the Zin response shown in
> 
>  http://www.abelian.demon.co.uk/tmp/acmq-zin-1.gif
> 
> in which the resonant frequencies match, and the Zft response
> 
>  http://www.abelian.demon.co.uk/tmp/acmq-zft-1.gif
> 
> Notice that the Zft in the acmq LC model intersects the tssp model at
> the two resonant frequencies.  So your LC network does indeed
> successfully reproduces the coil's first two resonant frequencies
> and the relation between Ibase and Vtop at those frequencies.

Ok. Thank you for the verification.

> With the additional degree of freedom provided by C0, the frequency
> of the first pole would have to be given as an input, along with the
> two pairs of (wi,zi).  In this example, the first pole frequency is
> at 125.790kHz.

The value for C0 can be obtained as:

C0=(a*b*c*d*wp^2*(b+d)-(a*b^2+c*d^2))/
   ((a*b*wp^2-1)*(c*d*wp^2-1)*(a*b-c*d)))

wp is the first pole frequency (or any other that you want to match)
in rads/s. a,b,c,d are as in the other formulas.

> In practice, when testing the tssp model, I never bother to attempt
> a match on the Zin poles (aka the even-wave resonances), because their
> frequency is so sensitive to the stray shunt capacitance brought to
> the base by the instrumentation and drive connections.

Ok. C0 has no effect in a regular Tesla coil. But would be 
important in a magnifier.

> The tssp model doesn't find any zeros in Zft, only the broad minima,

Ok. So, no coupling. This is curious, and appears inconsistent
with the physical construction of the coil. Some time ago, models for
secondary coils were discussed here, where it was concluded that
coupling between adjacent coils was needed. But these couplings
invariably create transmission zeros in Vtop/Ibase. 
Ok that they can be at high frequency, beyond the frequency band
where the model works. I will take a new look at this.

> Your LC network, augmented with a C0 cap, seems to give such a good
> account of |Vtop|/|Ibase| that I'd expect to get excellent results,
> even at high k.  I expect that if the network was reduced to a PI
> network C0, L1, C2,  with the values chosed to match w1,z1, and the
> first pole, you would have a very good agreement with the distributed
> model at the two beat frequencies.  I dare say the agreement would
> remain quite good if you dropped the C0 and just ran with L1=Les and
> C2=Ces as a first approximation.  I'll try to run off some plots
> of this situation if I can get to my desk this evening.

You fall again in the second-order model. It works, of course, but
doesn't predict effects of the other resonances.
 
> Without the Ldc-L1-L3 term, the network continues to correctly model
> the unloaded AC Zft of the coil.  With this term, the network also
> accounts for the inductance at very low freqencies.  But what of the
> loaded AC behaviour?  Well the (wi,zi) pairs are computed assuming an
> unloaded resonator, ie they incorporate the effect of toploads, etc,
> but not the effect of any additional loading.  Therefore, if the LC
> network is employed to predict the performance of a TC in which extra
> C or R loading is added, then a certain amount of error will be
> introduced.

This would require correct modeling of the output impedance. The
structure of the model is unique, what hints that the model would
work quite correctly with a load at the top. And indeed it works.
An extra inductor (a bit larger than Ldc-L1-L3) improves the matching of
the two resonances, but creates a false third resonance. This can be 
corrected by increasing the order of the model with an additional 
stage, what moves forward the problem. 
I reached this conclusion comparing the lclc model with a model of 
a transmission line loaded with a capacitor at the output end.
 
> As the loading is increased, the effective series inductance moves
> from its unloaded value Les towards the uniform-current value Ldc.
> The Ldc-L1-L3 term is not the right value to account for the
> change in effective inductance when loaded.  Nor does it account
> for the change (usually a reduction) in Fres which also occurs when
> the coil is loaded.  When faced with a loaded coil, I'm forced to
> recompute the resonator current profiles and establish a revised Les
> that accounts for the loaded condition - I don't know of a better way.

I observe that the model works resonably well (better than the
single LC section model) when loaded. I agree with the small utility
of the Ldc-L1-L3 inductor. It is small, anyway.
 
> This is all very messy and speculative, and for the present, I'd be
> inclined to drop the Ldc-L1-L3 component altogether, so that the
> network is limited to the unloaded AC conditions.  That at least
> gives a solid recipe by which a coiler can take some steady state
> measurements of w1,z1,w2,z2,p1 and put the numbers into a formula to
> obtain the optimum primary values for a given k and coupling mode.
> Since the behaviour of the network is exact for (w1,z1) and (w2,z2)
> and (if account of the first pole p1 is taken) very good for all
> frequencies up to w2, the result seems to make for an excellent
> circuit model.

The fixed lclc structure of the model transforms the coil into a
kind of magnifier , but without the possibility of choosing all
the elements for an optimal design for an arbitrary operation mode. 
And there is the problem of how are the two inductors coupled to the 
primary. A 3 coils transformer? Too complicated. The only connection 
point correctly modeled is the base, and so the model is suitable only 
as model for a third coil of a magnifier. 
Humm... An application for a 8th-order multiple resonance
network, where the final LC section is in the third coil itself.
 
> BTW, anticipating the obvious next question - is k modified from the
> low frequency value by virtue of the non-uniform secondary current?
> In the very few cases that I've looked at, the k derived from modeling
> the distributed resonator has not differed by more than a few percent
> from the value obtained by Neumann integration from the coil geometry.
> What difference there is, has been less than the overall error of the
> modeling so I haven't explored this further, but off-hand I'd say that
> M and Lsec are modified by the secondary current distribution in
> roughly the same way, so that, defining
> 
>    Kdc = Mdc/sqrt(Lp*Ldc)
> and
>    Keff = Meff/sqrt(Lp*Les)
> 
> and then if we assume Meff/Mdc = Les/Ldc, then
> 
>    Keff/Kdc = sqrt(Les/Ldc)
> 
> which would vary from circa 0.95 to 1.05 for average TCs, depending
> on h/d.

Can Les be greater than Ldc? How is the secondary coupled to the
primary, if it is not a single coil anymore?

Antonio Carlos M. de Queiroz