[TCML] Spark models, revisited

```I've tried to make the spark model proposed by Jim Lux

http://www.pupman.com/listarchives/2011/May/msg00081.html

a bit more quantitative by considering only QCW i.e. stationary sparks.

I left out the spark gaps and the inductances in his model
so the spark is described by a series of resistors and capacitors.
Going to the limit of a continuous distribution of
resistors and capacitors, the voltage V along the spark can
be described by a second order differential equation:

d/dx ( 1/R * dV/dx ) = C * dV/dt

where R is the resistance of the arc (ohms/m) and C
the capacitance (F/m).

I've assumed C to be constant and R to be dependent on
the time averaged power dissipation. R depends then on
the position along the arc. A simple guess for R is to make it
inversely proportional to some power of the power dissipation, i.e.

1/R ~ P^n

For n>0 the differential equation has a simple solution.
The voltage amplitude and the power dissipation along the arc
are proportional to

V ~ (L-x)^(1/n)

P ~ (L-x)^(2/n)

where L is the length of the arc and x the position along it,
beginning from the breakout point. The length of the arc
and its total power consumption Ptot are

L ~ Vinp^n * f^(n/2-1/2)

Ptot ~ L^(1+2/n) * f^(1-1/n)

where Vinp is the coils voltage and f its frequency.

The best match to experimental observations seems
to be the case of n=2:

The power consumption is then proportional to L^2, which
is in agreement with the Freau formula and e.g. a measurement
http://www.eece.ksu.edu/~gjohnson/TeslaCoilImpedance.pdf

The power dissipation along the arc drops linearly with
the distance from the breakout point.

The power needed for a given arc length rises with the sqrt(f) making
low frequency coils more efficient regarding spark length.

The arc length grows with the square of Vinp. I believe Steve Ward
has seen a rather sudden growth of spark length with voltage
and my own observations confirm this.

For a given voltage the spark length increases with sqrt(f).
So a lower voltage is sufficient to achieve a certain length
when the frequency is higher.

There are mainly two effects, which generate conductivity
in arcs, thermal ionisation and ionisation by electric fields.
Thermal ionisation has a very strong dependence on
the temperature since the air molecules must have enough energy
to ionise on impact. Conductivity begins at around 6000K and rises
up steeply. See e.g the diagram on plasma conductivity in

http://www.zeuz.ch/doc/itet/sem5/ensy/227-0122-00-autography_systemtechnology.pdf

on page 21. I'm no expert on plasma physics and I don't have
a clear view of where a balance between conductivity, temperature
and power dissipation lies. Maybe some of you can help there.

Ionisation by electric fields requires fields of about 30kV/cm
at room temperature and pressure, which is much more than the
the voltage drop along RF arcs.
But, as Paschens law states, the breakdown voltage is proportional
to the density of the gas and that depends on the temperature.
At 6000K, which is 20 times room temperature, the gas density
is 20 times lower, reducing the field requirement to about 1.5kV/cm,
which is much closer to the observed values. A rough estimate of
this effect is:

V ~ density of gas ~ 1/T (Paschens law and gas law)

P ~ T^4 (Power needed due to loss by thermal radiation)

V^2 / R = P (Power dissipated by the arc)

Putting these together results in:

1/R ~ P^(3/2)

This amounts to a n of 3/2. I have some doubts, though,
whether the T^4 law really holds for gases, which are no black
radiators as the Stefan-Boltzmann law demands. Possibly
the temperature dependence is slower, maybe T^3 or T^2. That
would make n larger. Anyway, at voltage drops in the arc
below 1.5kV/cm electrical ionisation cannot be anymore the
main source of conductivity.
I believe, that there is a mix of both thermal and electrical
ionisation in a typical RF arc.

There are some problems with the n=2 model:

1) I've measured a power consumption of about 20kW at 70kVpk for
a 70cm arc. This is close to Gary Johnsons empirical relation
of L = 0.17 inch * sqrt(Ptot/W). Both of us ran at about 200kHz.
Wards tesla gun achieved (I believe) about 60 inches, which would
result in a power requirement of 120kW, which is not
realistic from what I know about his electronics. From my
discussion above, the higher frequency he works with should make
the power needed even higher.

2) The model predicts a phase shift between the voltage of
the coil and the current drawn by the arc. The phase shift
is by itself of some interest, since it determines the capacitive
load on the coil. The shift is given by

tan(phi) = sqrt(n+1)

For n=2 that would be 60 degrees. I've measured
a value of about 45 degrees. The difference is a lot bigger
than I expect my experimental errors to be.