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Re: Measuring self-capacitance directly (Re: flat secondary)
Original poster: "Paul Nicholson by way of Terry Fritz <twftesla-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>
Antonio wrote:
> The model reproduces well the input impedance, and quite well
> the output impedance too, even without the Ldc-L1-L3 inductance.
Except for the real part of the input Z, which is zero. Through most
of the frequency range this is negligible error, but becomes
significant near f1, f2, and fp.
> As a lossless model, it doesn't model the Q factor, but with
> a resistor added at the input (series) it would model it too.
Yes, but with these component values, you can choose a resistor to
match a given Q, or to match a given Zin, but not both together.
To match both Q and Zin with a single resistor, we would have to use
different L and C values, derived from the energy storage behaviour of
the resonator.
Let me try to explain why, by using a simplified distributed model of
the base-driven unloaded secondary.
Consider the coil in n sections, eg
Base V1 V2 Vx Vn-1 Vn Top
o---------L1--+--L2--+- ... --Lx--+-- ... --- Ln-1--+--Ln--+---o
| | | | |
C1 C2 ... Cx ... Cn-1 Cn
| | | | |
=== === === === ===
so that the *peak* voltages across each cap C1..Cn are V1..Vn.
Treat the coils L1..Ln as lumped self inductances - the mutual
inductance has no effect on this argument.
At a point in the RF cycle when there is no current flowing, the total
stored energy is
E = 1/2 * sum( Cx * Vx^2) for x = 1..n
and for a real coil this would be an integral rather than a sum, but
the principle is the same.
Then make up a single LC equiv cct,
Base Vn Top
o--------L-----+---o
|
Cee
|
===
and choose Cee to store the same energy given the same peak top
voltage Vn by defining
E = 1/2 * Cee * Vn^2 ...(1)
and equate these to get
Cee = sum( Cx * Vx^2)/Vn^2 for x = 1..n
I've taken to calling this the equivalent energy capacitance to
distinguish it from the equivalent shunt capacitance defined by
Ces = sum( Cx * Vx)/Vn for x = 1..n
We need to employ the correct equivalent for any given application,
for eg, the voltage ratio of the TC is
Vtop/Vpri = sqrt( Cpri/Cee)
whereas the base current for a given topvolts is
Ibase = 2*pi*f*Ces*Vtop ...(2)
Now consider a coil in CW steady state with base input power Pin,
so that the source sees an input resistance Rin at resonat
frequency f.
Then Rin = 2 * Pin/Ibase^2 ...(3)
And by definition, Q = 2 * pi * E / (Pin/f)
where Pin/f is the energy lost per cycle. Putting (1)(2)(3) together
and eliminating Pin (and being careful with a factor sqrt(2) because
Vtop in (1) is a peak and Ibase in (2)(3) is RMS), we get
Q = Cee / (2*pi*f*Ces^2*Rin)
for the relation between Q and Rin.
A similar procedure can be carried out for the inductance, defining
an equivalent series Les, and an energy inductance Lee, by refering
current and energy respectively to the coil base current. We find
Rin = 2 * pi * f * Lee / Q
We can easily go on to derive some informative formulas for the TC,
for example, in CW mode,
Vtop = sqrt( 2 * Q * Pin / (2 * pi * f * Cee))
and
Vtop/Vbase = sqrt( Q / (2 * pi * f * Rin * Cee))
and we have a couple of identities
f = 1/(2 * pi * sqrt( Les * Ces))
Cee * Les = Ces * Lee
Also, the output admittance at resonance is
Ytop = 2 * pi * f * Cee/Q
The values of Cee, Ces, Lee, Les can be computed from the dimensions
of the resonator, for eg my large coil,
http://www.abelian.demon.co.uk/tmp/home.g0006.jpeg
(without the topload)
Lee = 64.4 mH Cee = 36.7 pF
Les = 72.8 mH Ces = 41.6 pF
Ldc = 88.9 mH Cdc = 94.6 pF
Rin = 27.4 ohms; Q = 1352 Fres = 91.5 kHz
Comparing Cee and Ces for this coil gives a rough idea of the error
introduced if you wish to neglect the difference between the two.
This is perfectly reasonable to do in most practical cases, but if you
want to make sense of precision measurements on the coil, then you
must apply the correct equivalents.
For the full monty, see sections 7 to 9 of
http://www.abelian.demon.co.uk/tssp/pn2511.html
Antonio wrote:
> Fp is the frequency where the input impedance has poles, a maximum,
> not a minimum.
Yes, I thought it better to the find the Zin pole by looking for the
Ztop zero.
> Measuring from the other side, with the base open, the same poles
> would appear too,
Yes, but I meant the base to stay grounded, so that the Zin pole
becomes a zero at the top. I just thought it would help to eliminate
the effect of instrument stray C. But then you said
> This measurement is sensitive to parasitic capacitances at the base,
> as you mentioned. The measurement of the top voltage is also
> sensitive to parasitic capacitances.
which I see is correct, so as you say, whichever end you measure the
pole from, it's going to suffer some stray C.
> What do you mean by "9/4"?
Just the 9 quarter wave resonance. That's as high as I dare go with
the distributed model before problems occur which affect accuracy.
[Asymmetric k factors]
> A transformer must be reciprocal. Different coupling coefficients
> in the two directions cause it to be active, generating or
> dissipating energy.
Agreed. I have a self-consistency problem, which I think is due to
the way I'm defining the k. More thought required at my end, but the
apparent problem comes from having an M calculated with a non-uniform
current in one direction (ie not all sec turns fully contribute to
B flux) but calculating M in the other direction using all the sec
turns to collect induced EMF. There is no problem with the full
distributed model, but this self-consistency problem occurs as soon as
you try to define meaningful lumped equivalents for the coupled system.
--
Paul Nicholson
--