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

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

Antonio, All,

Puting in some values, from my large coil model,

         freq         Zft
 f1/4   91503 Hz    41851 ohms
 f3/4  217611 Hz    23684 ohms

so
 a = z1/w1 = 7.27932e-02
 b = 1/(z1*w1) = 4.15603e-11
 c = z2/w2 = 1.73219e-02
 d = 1/(z2*w2) = 3.08805e-11

and
 L1 = (a*b-c*d)/(b+d) = 3.43784e-02
 C2 = a*c*(b+d)^2/((a+c)*(a*b-c*d)) = 2.94839e-11
 L3 = b*d*(a+c)^2/((a*b-c*d)*(b+d)) = 5.77704e-02
 C4 = (a*b-c*d)/(a+c) = 2.76358e-11

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.

Antonio wrote:
> A problem that I see in the formulation is that the double LC
> circuit presents poles in the input impedance, and this is not 
> considered in the modeling. A grounded capacitor at the base can
> account for one of them, without other effects.

As you say, one of these can be positioned to match an observed Zin
pole by fitting a parallel capacitance across the base of the network,
as in

 o--+--L1---+---L3---+---Ldc-L1-L3---o Vtop
    |       |        |
    C0      C2       C4
    |       |        |
    =       =        =

In this case, a C0 of 46pF gives a response

 http://www.abelian.demon.co.uk/tmp/acmq-zin-2.gif

which gives a pretty good account of the resonator all the way up
to and including the 4/4 wave resonance.  Similarly, the transfer
impedance now matches over a broad range,

 http://www.abelian.demon.co.uk/tmp/acmq-zft-2.gif

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.

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.

> Coupling between the coils would create transmission zeros in
> Vtop/Ibase, that don't exist in practice (don't?)

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

> What would happen in the fast modes (high k), where the frequencies
> in the systems are very diferent?

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.

> If you use a terminal capacitance, and include it in the evaluation
> of the wi, zi, its contribution will be correctly accounted, but
> what about Ldc-L1-L3? Would it naturally disappear?.

I always include the terminal cap when determining zi for that reason.

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.

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.

As a rule of thumb, I'd say that if the secondary is unloaded, or
lightly loaded, use the unloaded Les value, along with Ces. Otherwise,
when the coil is significantly loaded, as in say the 2nd coil of a
three coil system (ie driving a magnifier base), then you may decide
that Ldc,Ces provides a closer description of the 2nd coil than 
would Les,Ces.  Maybe the practical thing to do is to estimate the
fraction of the coils' base current which goes into the load, as 
opposed to the rest of it which charges up the self-C and topload C
of the resonator.  We could define Ibase = Iload + Icap, and use an
effective L of 

  Leff = (Icap * Les + Iload * Ldc)/(Iload + Icap)
    
and then proceed with a=Leff1 instead of a=Les1, etc.  The Ces remain
almost constant with varying load, since the voltage profile of the
coil doesn't alter a great deal between no load and a load resistance
as low as a few times the coil's characteristic impedance.

Further, if the external load was capacitive, say Cload, we could
approximate Icap/(Iload + Icap) with
  Ces/(Cload + Ces),
and similarly for Iload/(Iload + Icap)

Thus, we get an approximation
 Leff = (Ces * Les + Cload * Ldc)/(Cload + Ces)

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.

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.
--
Paul Nicholson
--