From: Paul
Date: Sun, 22 Oct 2000 18:30:25 +0100
Subject: Re: [TSSP] Two questions.
The voltage gain ratio Vtop/Vbase is a tricky one since it depends
on the Q factor which as we know is proving hard to predict.
This is why I'm pushing the use of transimpedance Vtop/Ibase instead,
as this is independent of the Q factor. You may have noticed that
I avoid any mention of Q in the pn1310 notes.
Terrell W. Fritz wrote:
> Normally "i" would think that if a coil has a "Q" of say 200 and
> you apply a 1 volt signal to the base of the resonator, then you
> would get 200 volts at the top.
> Would this still be true for all this distributed parameter case
Thats reasonable, Vtop/Vin = Q for the lumped LC model, and
if the coil current is fairly uniform it would apply also to the
resonator. If the current profile is nearer to a cosine, then we
might expect Vtop/Vin = 2Q/pi, the factor 2/pi being due to
integrating cos(x) over a quarter wave to get the average current.
As a rule of thumb it might be worth trying the average of these
two, on the assumption that the resonator lives between these two
extremes.
Then Vtop/Vin = Q/2 * [1 + 2/pi] = 0.82 Q
There are a couple of catches to this. First, the Q in the above
formula is the bare Q of the resonator, whereas the measured Q
includes the loss in the source impedance of the driver. Therefore
before applying the above formula, you are entitled to scale the
measured Q upwards by a factor (Res + 50)/Res if 50 ohms is your
source impedance, and Res is the equivalent series resistance
for the coil. In the NSVPI results you've provided a measurement
of the drop across the 50 ohms input resistance, and from this we
can estimate the Res as 710 ohms. Thus the measured Qm of the system
can be scaled up to an estimated Qc of the coil,
Qc = Qm * Res/(Res+50) = 67.5 * 760/710 = 72.3
Using the 'rule of thumb' gives a poor answer:
Vtop = Vin * sqrt(2) * 0.82 * Qc = 0.6606 * 1.414 * 0.82 * 72.3
= 55.3 volts peak
where the sqrt(2) converts from rms to peak. We may do better with
the lumped LC assumption,
Vtop = Vin * sqrt(2) * Qc = 0.6606 * 1.414 * 72.3
= 67.5 volts peak
Even this value falls short of the measured, so the top volts
is being raised higher than we might expect, even on the basis
of uniform current. BTW, this ties in with the corresponding
uniform/cosine pair of approximations for Fres, we might expect
Fres to lie between the lumped and cosine formulas, but usually
it's nearer the lumped value, and often Fres is *below* the lumped
(ie uniform current) estimate. Detailed simulation with tsim gives
the right answers for both Fres and transimpedance, but these
approximate formulae seem to be struggling.
> If I bring a high voltage probe is brought down on the
> top of the resonator, the cable to the probe could be modeled as
> a grounded line at X=0 above the coil in the 2D models that are
> stretched into 3D cylindrical coordinates. In other words, the
> situation could be modeled easily. The 2.5 or whatever pF of the
> probe could be added to the end of the resonator and I would then
> think that the top voltage could be measured in a way that could also
> be directly modeled by computer.
If the wire to the probe is thin enough we can neglect it's
effect and simply lump the probe load into the topload. The
bulk loading of the wire and probe can then be determined from
the Fres shift. If the wire came in from the top, the approximation
that its entire effect is through the topload ought to be best of
all.
Its very hard to model a thin wire electrode in the E-field,
so if you wanted to account for it, you might need to use quite a
thick conductor so that it could be accurately modeled. I think
the first option is the best until it proves to be a problem.
> Is that a reasonable thing to
> do and would it help?
I think that both Ibase and Vtop are important measurements,
partly for normalising amplitude profiles, but also, given
simultaneous CW measurements of the two together with an Fres,
you then know a great deal about the resonator. You know its
transimpedance, which I think is a better gain descriptor than
voltage gain (VSWR), and you also then know Les and Cep, which
means you can tell how many volts you'll get for a given net bang
energy. I suspect that precision measurements of corona loading
might also require unequivocal Vtop measurements. For these
benefits I think the presence of a constant extra topload component
is quite acceptable and easily accounted for.
BTW, the trouble with voltage gain VSWR in its role as
Vtop = VSWR * Vin
is that we don't know Vin and its hard to measure. Vin depends
on the effective ground impedance seen by the coil and when
you try to measure it, where do you put the ground terminal
of the voltmeter? Wherever you put it you'll get a reading
dependent on where that 'ground' point is wrt the external
displacement current return path. You can always get a solid
calibrated measurement of Ibase and the transimpedance doesn't
depend on the Q.
Cheers,
--
Paul Nicholson,
Manchester, UK.
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.