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>

(after some work...)

> > 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.

I looked at this problem. It is indeed always possible to find this
equivalent LC network from sets of Les,Ces, or sets of frequencies
and transimpedances (|Vtop|/|Ibase|). I list below explicit formulas 
for the case where the two first resonances are considered:

Let the measured resonance frequencies and transimpedances be:
w1,z1, w2,z2
w1,w2 in rads/s, z1,z2 in Ohms.
To simplify the notation, I will use a,b,c,d for the elements:

a=Les1=z1/w1
b=Ces1=1/(z1*w1)
c=Les2=z2/w2
d=Ces2=1/(z2*w2)

The equivalent is:

L1=(a*b-c*d)/(b+d)
C2=a*c*(b+d)^2/((a+c)*(a*b-c*d))
L3=b*d*(a+c)^2/((a*b-c*d)*(b+d))
C4=(a*b-c*d)/(a+c)

Ibase ->
o---L1---+---L3---+---Ldc-L1-L3---o Vtop
         |        |
         C2       C4
         |        |
         =        =

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?.

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

> 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.

This makes sense, at least in the slow modes, where the waveforms
are almost sinusoidal. What would happen in the fast modes (high k),
where the frequencies in the systems are very diferent?

Antonio Carlos M. de Queiroz