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Re: Optimal Quenching
Subject:
Re: Optimal Quenching
Date:
Wed, 19 Mar 1997 13:07:46 -0500 (EST)
From:
msr7-at-po.cwru.edu (Mark S. Rzeszotarski, Ph.D.)
To:
Tesla List <tesla-at-pupman-dot-com>
Hello All,
Regarding the splitting of frequencies as coupling between the
primary and secondary coil increases, Malcolm Watts said in part:
>.... because side frequency production is a necessary condition of
>the changing Fr amplitude. I'm reluctant to expand too much at this
>point but briefly, consider that when the amplitude is changing
>continuously, the slope d(f(t))/d(t) is also changing and that change
>with respect to time corresponds to a frequency or frequencies that is
>not the resonant frequency. It *is* a matter of degree in my opinion.
>I think it must be happening even on the first quarter cycle but the
>relative amplitudes of the signals will change as the transfer
>proceeds and at that early point, the side frequency signal is almost
>buried in noise beside the main signal.
<snip>
> I came to the conclusion that such an explanation can
>definitely be arrived at and moreover can be be made quite
>understandable without the need for screeds of mathematics (although
>a little will help).
>
I have been looking at this a bit recently both from the
experimental side and the computer simulation side, and agree with
Malcolm
with regards to what is going on. The mathematics provides nice
answers,
but I don't think we are yet asking it the right questions. (The answer
is
still 42, BTW, for you Hitchhiker fans.)
The Corum's TCTUTOR program and accompanying text are
enlightening,
but there are several problems I have found in their analysis. This may
also impact Bert H.'s PSPICE simulations. First, the root finding
algorithm
they use is especially susceptible to roundoff error. Another problem
is
that of specifying the proper initial conditions for the differential
equation solutions. The Corum's assume that the primary capacitor is
fully
charged, and that no current is initially flowing in either the primary
or
secondary. They also specify the initial second derivative at time t=0.
(This was alluded to in Malcolm's comments above.) This scenario may
be
possible to duplicate in single shot lab experiments, but has little to
do
with an operating coil.
Here are the coil parameters system I have built and am dinking
with
both in my basement experimentally and with computer modelling:
Lp= 40.2 microhenries (about 10 turns 1/4 inch copper tubing in a
flat
spiral)
Cp= 23 nanofarads (primary capacitor, mica)
Rp= 0.25 ohms (primary A.C. resistance, estimated from DC value)
Rg= 3.0 ohms (spark gap resistance - assumed to be constant)
Ls= 42 millihenries (6" x 22.25" coil wound with AWG 24)
Cs= 11 picofarads (distributed capacitance of secondary)
Ct= 11 picofarads (toroid capacitance)
Rs= 67.5 Ohms (secondary A.C. resistance estimated from DC
resistance)
Fp= 165.6 kHz Approximate Primary coil resonant frequency in Hz
(simple 1 / 2 x pi x sqrt(L x C) formula)
Fs= 165.6 kHz Approximate Secondary coil/toroid resonant frequency in
Hz
(simple 1 / 2 x pi x sqrt(L x C) formula)
Here are the frequency splitting values in kilohertz predicted from
simulation results for various degrees of coupling k:
k= .01 Flow= 165.5 Fhigh= 165.6
k= .05 Flow= 163.0 Fhigh= 168.3
k= .15 Flow= 154.7 Fhigh= 179.1
k= .20 Flow= 151.4 Fhigh= 184.7
k= .25 Flow= 148.3 Fhigh= 190.9
k= .30 Flow= 145.4 Fhigh= 197.6
k= .35 Flow= 142.6 Fhigh= 205.1
k= .40 Flow= 140.0 Fhigh= 213.5
k= .45 Flow= 137.6 Fhigh= 223.0
k= .50 Flow= 135.3 Fhigh= 233.9
Here are the relative amplitudes for the the harmonics at frequencies
Flow
and Fhigh above:
k= .01 Alow= 1.00 Ahigh= 1.00
k= .05 Alow= .97 Ahigh= 1.00
k= .10 Alow= .91 Ahigh= 1.00
k= .15 Alow= .86 Ahigh= 1.00
k= .20 Alow= .82 Ahigh= 1.00
k= .25 Alow= .78 Ahigh= 1.00
k= .30 Alow= .74 Ahigh= 1.00
k= .35 Alow= .70 Ahigh= 1.00
k= .40 Alow= .66 Ahigh= 1.00
k= .45 Alow= .62 Ahigh= 1.00
k= .50 Alow= .58 Ahigh= 1.00
Note that as coupling increases, the frequency splitting gets
worse,
and the higher frequency harmonic becomes higher in relative amplitude
as
coupling gets tighter. For this coil system, critical coupling occurs
at
k=0.011, using the circuit resistances above. This is a really low
value,
but may be typical of a practical coil. The gap resistance is probably
underestimated, and is proably not constant during a discharge cycle
unless
it can be quenched very rapidly.
What is not shown is the phase relationship between the two
harmonics, their exponential decay. This greatly affects the visual
appearance of the time domain signal we may watch on our 'scopes.
Calculating the phase is essentially useless, since far too many factors
have to be ignored to arrive at a solution. Has anyone looked at this
using
a spectrum analyzer to validate the harmonic amplitudes?
Flames/comments welcomed!
Mark S. Rzeszotarski, Ph.D.