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

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

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

Antonio wrote:
> I don't see sense in considering an inductance that is different
> from the DC inductance, if the objective is to model the coil as
> a simple LC parallel circuit.

Well you must if you want the simple LC model to be a good
representation of the resonator at Fres. If you try to go ahead with
just Ldc, you pretty soon notice major discrepancies: 

First, when you compare top volts with base current, you expect
Vtop = 2.pi.f.L.Ibase,  and you only get the right answer if you
plug in Les, not Ldc.  Secondly, with f=1/2.pi.sqrt(L.C) you only
get the right answer when you use Les and Ces, not Ldc and Ces or
Cdc.

Just as the effective shunt capacitance is arrived at by integrating
the stored charge along the physical capacitance, ie 
integral{ C(x).V(x).dx}, similarly, with a non-uniform current, we
have roughly integral{ L(x).I(x).dx} as a term in the expression for
total EMF.  In other words, the contribution of each infinitesimal
inductance and capacitance to the total, is weighted by the current
and voltage, respectively, at that point.

And, as with Ces, you get a different value of Les for each
resonance.

> You can increase the order of the model, adding an extra
> inductance,

Agreed, in which case, with C1=Ces, L1=Les and L2=Ldc-Les you have a 
model which describes the unloaded behaviour at Fres, and also 
matches the measured inductance at DC.

> Add other sections for more resonances.

Yes, agreed.  I suppose you'd need an N-stage LC network to model
the first N resonances, with an extra L at the top end if you wanted
it to also match the DC inductance.  I've never tried to construct
one, but I think I can provide the N pairs of (Les,Ces) that such
a circuit must be equivalent to, independently, at each frequency.

As far as Les goes, I think the concept is physically well justified,
unambiguous, and it is easy to confirm experimentally by means of
simultaneous measurements of base current and top voltage at
resonance. Les = (|Vtop|/|Ibase|) / (2.pi.f)

Further, the formulae which you derived for the coupling modes of
the dual resonator should I think be applied by using these
equivalent reactances, rather than the DC values.

And that's not the whole story.  When you start to look at the
relationship between base input impedance and Q-factor, you need
to use a different set of equivalent reactances, this time weighted
by the square of the distributed voltage and current, so that they
represent, in a simple LC model, the stored energy of the resonator.

See http://www.abelian.demon.co.uk/tssp/pn2511.html for more info on
all this stuff. 
--
Paul Nicholson
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