From: Paul
Date: Sun, 22 Apr 2001 10:19:42 +0100
Subject: [TSSP] Time domain modeling
Hi All,
Well I'm now pretty certain that we can do a very efficient time
domain model of the secondary with coupled resonant primary, in the
presence of a non-linear load impedance.
Turns out to be not too bad at all. Here's a quick summary:
(I'll assume you've read the section 4 of pn1401)
First we extend the solenoid integral operator to include an extra row
for the primary current. That gives a matrix eigenvector equation for
the combined pri-sec resonant system. (In this formulation there is no
longer a forcing function - everything happens with the normal modes
(eigenfunctions) of the system.) Adding in the primary LCR resonator
like this is very straightforward - we already have an equation which
amounts to a coupled system of several hundred resonators, so adding
one more to account for a lumped primary LCR is a minor step.
We then pre-calculate the complex frequencies of the first N normal
modes of the system by finding the zeroes of the characteristic
equation. Turns out the landscape of the determinant of A-1 in
(A-1)f = 0 is quite well behaved and the roots are easy to home in on.
tsim now has options to compute det(A-1), which it does very quickly,
around 10 seconds to compute F, K, then A, then det(A-1), for each
gamma. Then a further couple of seconds is required to produce the
spatial distribution function corresponding to each root.
It needs to be quick, because the spectrum of complex frequencies must
be computed for every possible load impedance. We choose a set of
discrete impedances to a geometric progression either side of Zft and
map the mode spectrum for each. This is done once, and takes a couple
of hours altogether, but need only be done once for a resonator.
Now, in possession of a mode spectrum map as a function of load
impedance, we simply start off with initial conditions appropriate
for an idle resonator armed with a charged primary cap. We resolve
this initial condition into its components in terms of the mode map
corresponding to zero (or a chosen minimum) load conductance.
We then simply output the secondary top voltage as a function of time,
given by sum( over all modes n; exp( gamma_n * t) * phi_n( h))
which is trivial to calculate.
Assume we have a function Yload( Vtop) which gives the brush discharge
conductance as a function of top voltage.
As the time evolution is generated, eventually Yload( Vtop) will no
longer be close to the initial assumed minimum load conductance, and
will be nearer to another of our load conductances for which we have
precomputed a normal mode spectrum. At this point we freeze the
time evolution, replace the mode map with the new one. Take the
current values of V(x), I(x), Ipri, Vpri, etc to be a new set of
initial conditions. Resolve these into components of the new normal
mode vectors and then restart the time evolution.
Repeat. (Note that mode map changes would be occuring many times
per RF cycle of the lowest mode - eg every few tenths of a radian,
so there is a coding challenge to do this efficiently).
Thus we obtain the time evolution of the entire coupled resonator.
At some point, the top voltage will exceed some chosen threshold
(call it Vdischarge) and we model the consequences of an arc
discharge from the topload.
We assume the discharge dumps all the topload charge in a time much
less than the period of any of the modes we are using.
We stop the time evolution if Vtop exceeds Vdischarge. Vtop is reset
to zero, the initial conditions recomputed in mode amplitudes, and the
time evolution is restarted. This takes proper account of the
excitation of modes due to the discharge transient, and the resulting
behaviour will show us how the coil goes about restoring the topload
charge.
The above is now just a coding exercise, apart from one not-so-minor
point, we need that function Yload( Vtop), along with some guesses of
Vdischarge for a given test coil. Yload can be calculated somewhat
indirectly from measurements of Ibase and Fres, or more directly by
measurements of Vtop and Itop, for various power levels.
Naturally, we can incorporate a primary gap loss in a similar way,
by having an Rgap( Ipri) function, although we then need a 2D array
of normal mode maps.
The remainder of pn1401 covers some of these items,
sect 5. normal modes of the grounded base, computing det(A-1),
det(A-1) landscape,
decomposition of initial conditions into mode amplitudes,
time domain modeling.
sect 6. coupled primary, combined resonance and augmented solenoid
equation.
sect 7. normal modes of the pri-sec coupled resonator. effect of
coupling coefficient.
I'm still typing these up - a slow job as I keep having to digress to
work things out and check things. Probably these next 3 sections will
run to around 20 pages, so I might roll the first two sections, on
impedances, back into pn2511, tacking them on the end.
Comments and questions?
Cheers,
--
Paul Nicholson,
Manchester, UK.
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.