Capacitor charge rate

Greetings! I've been on this mailing list as a lurker for a while, but
haven't yet joined in. Ed Sonderman's note concerning his calculations for
matching the capacitor to the transformer caught my interest because it
intersects some work I've been doing for a while.

I've been working with different high voltage transformers, particularly
neon sign drivers, and reconciling them with various wideband transformer
models. The details of the power transformer have mostly been neglected in
Tesla Coil design, resulting mostly in design by rule-of-thumb in this

I've been hoping to eventually work out a set of procedures that can be
used to determine the various parasitic elements inside the transformer,
and from those measurements, provide some better design equations for
capacitor matching and filter design for kickback prevention. I think
that it should be possible to design in the amount of kickback that
is allowed to get to the transformer, which adds some interesting
possibilities. In any case, I'd be very interested in bouncing my ideas
off the group, and sharing what I've learned so far.

First of all, I think Ed is moving in the right direction to solve this
problem. However, I think it's a first pass approximation, which can take
some refinements. Let me add that a lot of this will be opinion, and I'm
only slowly taking data to verify the models I'm working out. This all
means that I'm an explorer, not an authority, and will probably make some
wrong turns before finally zeroing in on the solution I'm after. The
observations and opinions of the group would be much appreciated. I'm
not sensitive to criticism; if you see a problem with something I say,
rip into it! The sooner I can get to some good answers on this, the better.

Ed's approximation of a 12KV 120mA neon transformer as having a 100Kohm
series resistor seems to work very well for DC loads, and is a good
starting approximation. This "resistor" consists partly of the resistance
of the secondary winding, and mostly due to the effect of the magnetic
shunt inside the transformer. It fairly accurately models the fact that
the secondary voltage of a neon transformer drops like a rock when you try
to draw current from it.

It seems that when selecting a capacitor to match the transformer, you don't
so much want to charge to nearly 100 percent of the peak AC voltage as
you want to maximize the energy stored in the capacitor (.5CV**2). I don't
have my notes here, but the percentage is somewhat below the 97% proposed.
I can share some of my notes when I have them available, but I'd first
appreciate hearing how others have trated the problem of maximum energy in
the capacitor. It's an interesting problem.

This first problem is a very rough first pass, because it assumes that the
capacitor starts discharged. In a real Tesla Coil, the capacitor charges
during the rising quarter of the AC waveform, the spark gap fires somewhere
near the peak of the waveform and discharges it, then the capacitor begins
to charge again in the falling quarter of the waveform. This is a problem,
as any charge in the capacitor when the cycle reverses polarity must be
discharged in the reverse half cycle before the capacitor may be charged
in that half cycle. A significant portion of the secondary current is being
wasted to charge and then discharge the capacitor before it can be
properly charged to fire the spark gap. The problem occurs because the
charging that's done on the downward quarter of the cycle is of the wrong
polarity for the charge of the reverse half-cycle.

Note: I really apologise for the lack of clarity in the previous paragraph.
I can't think of a better way to express it without a a digram of the
charging waveform. If the explanation is too murky, let me know and I'll
try again on it.

Of course, the polarity problem could be eliminated and converted into a
benefit if the capacitor always charged to the same polarity. Charging the
capacitor through a full wave bridge rectifier would allow the residual
charging on one half cycle to be used on the next half cycle, and allow
more energy to be transferred to the capacitor per cycle. Unfortunately,
rectifiers are far more sensitive to kickback than the transformer is,
but if we can predictably control the kickback, we should be able to design
a stack of rectifiers that can handle it. Carefully controlled kickback
was one of the items I'd mentioned earlier as a benefit of the transformer
modeling I'd been working on.

The transformer modelling gets really interesting when you try to include
the effects of a magnetic shunt. The analysis tools are there; they're
just time consuming to use. Most first semester electromagnetic fields texts
show how to analyze a "magnetic circuit", which is analagous to applying
Ohm's Law to a resistor network. Instead of voltage, you use the number of
turns in a winding times the winding current. Instead of current, you
use magnetic flux through the core. Instead of resistance, you use reluctance,
which is a function of the permittivity of the core and its dimensions. The
shunt is another reluctance shorting out the main loop, and is works like
solving current through a resistor loop, but with large shunt resistor,
shorting off some of the current. While we don't have values for the
actual reluctances and things, they lump together as constants in the
final equations, and the lumped constants ought to be able to be determined
from measurements on a real transformer. The equations for behavior of
the transformer show that there are phasing changes that happen when the
size of the transformer load changes. I haven't taken any data at all on
this stuff yet, but it may be useful in better understanding power factor

I should point out that this shunt analysis involves solving some very simple
differential equations (first order linear for a resistive load, second
order linear for a capacitive load) but that if you've taken a differential
equations course at all, they're fairly straightforeward to solve. I've
gone through about a dozen fields texts, and none seem to show this as
either a sample or a homework problem, even though it's quite a basic
problem. The reason may be that the number of pages of paper needed to get
to a solution took about three times as much as most of the homework
problems that I had to do when I took fields, (mumbledy-mumble) years ago.
The point of all this rambling is that I'd greatly appreciate it if someone
else might know of a text that handles the problem, or if someone else might
also try to do what I've just tried to describe. I don't completely trust
the results I've gotten, though I can't find any errors, either. I'd
really love to compare notes with someone in this area.

I guess I should finish up, for now. I appologise for going on in so many
directions, but I'm really fascinated by the not-so-obvious effects that
the power transformer adds to the Tesla circuit, and think ther are some
contributions to still be made there. I'd be really interested in hearing
from others who might be interested in doing some similar analysis and
data gathering, and comparing notes.

Wes B.