From: Paul
Date: Tue, 30 Apr 2002 19:43:30 +0100
Subject: Re: [TSSP] Secondary voltage stress factor
Hi Kurt,
> would the results stay valid, at least in proportion, considering,
> most operating coils are equiped with a toroid?
> How can we deal with this problem?
Good question - that's a tough one alright. The fact is, adding a
toroid completely changes the geometry of the resonator, thus
altering the voltage distribution from which the stress factor
table value was calculated.
I'm hoping that we can use a trick - separating the toroid current
from the coil-only current, dealing with the two separately, then
adding the results. It's only approximate, but better than nothing.
Given a toroided coil with base current Ibase, we can divide Ibase
into a component Ibase * Ctop/Ces which goes on to form the topload
current, ie a 'straight through' uniform current. The remainder of
the base current, ie Ibase * (1 - Ctop/Ces) is the current which
charges up the coil's self capacitance and thus never reaches the
topload. Here I'm using Ctop as the toroid's capacitance to ground
with the coil in-situ, and Ces as the *total* equivalent
shunt capacitance of the resonator (which includes the Ctop).
We can then take Vtop as the sum of the voltages induced by
each current component,
Vtop = w * Ldc * Iu + w * Les * Ic
where Les is that of the bare coil, and w = 2 * pi * Fres.
Then we can apply the table to only the w * Les * Ic component of
the top volts. If T is the relevant stress factor from the table,
the highest volts per unit length due to this component would be
T * w * Les * Ic/length ... volts/metre
and if we assume a uniform gradient for the voltage induced by
the uniform topload current, its gradient is
w * Ldc * Iu/length ... volts/metre
Taking the overall highest gradient to be the sum of these, we
have
Ibase * w/length * (T*Les*(Ces-Ctop) + Ldc*Ctop)/Ces ... V/m
for the combined highest gradient. This is equivalent to an
overall stress factor
T' = w^2 * (T*Les*(Ces-Ctop) + Ldc*Ctop)
because the average volts/metre is Ibase/(w * Ces * length).
Now for another approximation: Let the *coil's* shunt capacitance
be Cxs and approximate it with Ces - Ctop, and noting that
w' = 1/sqrt(Les * Cxs) = the coil's *unloaded* resonant frequency,
w'' = 1/sqrt(Ldc * Ctop) = the notional resonant frequency of the
DC inductance with Ctop
we have the reasonable approximation
T' = w^2 * (T*Les*Cxs + Ldc*Ctop)
= T * (w/w')^2 + (w/w'')^2
Of course, the omegas w, w', and w'', can be replaced by the
frequencies. As Ctop tends to zero, w tends towards w' and
w'' tends towards infinity, so T' tends towards T. For very
large toploads, w becomes approx equal to w'', and w becomes
a lot smaller than w', so T' tends to unity.
> did I overlook something?
No, it was a good question!
--
Paul Nicholson,
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.