Re: theory(?) for long sparks

From: 	Jim Lux[SMTP:jimlux-at-earthlink-dot-net]
Sent: 	Tuesday, December 02, 1997 11:25 AM
To: 	Tesla List
Subject: 	Re: theory(?) for long sparks

I started this line of thought, because what we all want are long sparks
(well, most of us anyway) with minimal cost. So, I started thinking about
and researching what it is that makes for long sparks, and what parameters
you would want to optimize. 

Clearly, the total energy available has to be enough to heat the entire
column, which sets a lower bound on the capacitance and voltage (or L and
I, it's all the same). 

Also, the HV has to be there long enough to keep the leader going, i.e. you
want a low operating frequency.

And, you don't want the voltage to rise too fast (the leader takes some
time to develop, and raising the voltage quickly gets you more leaders that
are shorter). There are some other strange effects that can occur in long
sparks, for instance, the charge flowing back towards the anode down the
leader when there is too much charge at the head of the leader.

Malcom's recent comments about tracking along insulators are well taken,
and well known in the long spark research community. Even for relatively
low voltages (tens of kV), putting an insulating column between two
electrodes can reduce the breakdown by a factor of 3. This may be because
the induced charge on the surface of the insulator helps reduce the
effective gap to where it can be broken down. (This is why insulators have
longer creep paths than flashover paths. That, and the possibility of dust
and contaminants on the insulator, as well.)

Bazelyan describes very long sparks (meters) from low voltages (hundred kV
or so) along the surface of water or thin insulating films. He theorizes
that this is due to the capacitance of the spark channel being increased,
allowing more charge/energy to be stored in the channel during its growth.
Of course, this requires a "stiffer" source to drive it to be able to
supply the current required.

Intermediate conductors provide an energy storage place as well as causing
signficant field non-uniformities, with the latter probably being more
important. The theory for long sparks seems to indicate that the field
non-uniformity at the start is very important, since the leader carries the
nonuniformity "forward".

I don't think anything hangs around between breaks for the sparks in a TC.
The sparks are too small and lose their energy by radiation and conduction
too quickly to remain ionized. It might leave a generally ionized region
with a somewhat lower ionization voltage, or the compounds created by the
spark (e.g. NOx, O3) might ionize easier. The current flow in a stepped
leader is not constant, but is continuous, in that there is always current
flowing from the source into the leader.

I did some quick calculations on a 10 meter long spark: at a spark radius
of .1 cm (typical), the capacitance of the spark itself is around 60 pF.
Gut feel tells me then, that my source has to have a C several times that,
so that the voltage doesn't drop appreciably as the spark channel forms and
fills with charge. The drop along the spark is very low (say 1000 V/meter)
so it is essentially at the same voltage as the anode. Say the source has
to have a C of 300 pF.  This is a sphere some 3 meters in radius (i.e. 19
feet in diameter). And, of course, a sphere that large wouldn't have the
field nonuniformity you need to get things going. Alternately, and this is
where a tesla coil comes in, you could have some means of adding charge to
the anode as the spark channel develops, i.e. the current in the inductance
of the secondary (or the energy in the coiled secondary transmission line,
if you want to analyze it that way).

Now for the energy calculation: I have numbers all over the map for how
much energy it takes to heat a column of air .1 cm in radius to 7000K. Call
it somewhere between 1E-3 and 1E-1 joules per cm length. For a 10 m spark
(1000 cm), we are talking about an energy of 1 to 100 joules, then. (BTW, a
lightning stroke dissipates about 1E5 joules/meter, but has a channel
radius of around a cm, 100 times greater). Using our previous estimate of
100 pF (spark channel, rounded up for ease of computation), how much
voltage does it take to get 1-100 joules?
 E = V(MV)^2*C(pF) => V = SQRT(E/ C)
V = SQRT( 1 / 100) =  0.1 MV for  1 Joule
and  1MV for 100 Joules

It looks like we could get away with a 2 MV terminal at 100 pF for a 10 m
spark, providing the voltage stays there long enough to let the spark
develop, i.e. the resonant frequency of the TC is low enough.

That truly amazing 100+ meter spark in Bazelyan was produced by a
relatively low 5MV but with a rise time in the hundreds of microseconds and
a fall time in the tens of milliseconds.

I will be interested to see what sorts of results Greg gets with his 130
kVA unit, because the resonant frequency is pretty low, so the voltage
pulses should last 10 or more microseconds, which will give appreciable
time to develop a long spark channel.

I'll have a better estimate of spark channel energy in a few days.

All comments desired, good bad or indifferent.