Re: Breakdown voltages of toroids

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: Bart Anderson <classi6-at-classictesla-dot-com> 

  Hi Antonio,

Tesla list wrote:

>Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br>
>Tesla list wrote:
>
> > For field strength used with Norths equations, Javatc uses:
> >
> > Field Strength = p * ( B / ( C + ln ( p * d)))
> > where
> >     p = pressure in Torr (mm Hg). For air, this value is 760
> >     B = 365 Vcm-1 Torr-1
> >     C = ln( A / ln ( 1 + 1 / gamma))
> >     d = gap width
> > where
> >     gamma = 0.095 (secondary ionization coefficient)
> >     A = 14.6 cm-1 Torr -1
>
>I didn't try this formula. Are you sure that it is in this way? There
>is an apparent dimensional error, because you can't take the ln of
>p*d. And why would the breakdown field depend on the gap distance?

The field strength at breakdown is based on the Townsend affect (the 
generation of successive secondary avalanches to produce breakdown). Part 
of deciding that is defining the distance electrons will travel from 
cathode to anode. The gap is where collisions occur resulting in ionization 
and eventual breakdown of the gap. The gap is a critical element for the 
Townsend breakdown.

This is all better explained in a document by J R Lucas, 2001, Breakdown of 
Gaseous Insulation. Good read!
http://www.classictesla-dot-com/download/breakdown_gi.pdf


> > Trying some values at the end of the table for symmetrical gaps:
> > Spheres with 200 cm:
> > Spacing:               | -----Javatc----- |   (volts)
> > 70        1694 (1560)  | 1226 -at- 17.5kV/cm -at- 391 gradient |
> > 80        1878 (1730)  | 1346 -at- 16.8kV/cm -at- 349 gradient |
> > 90        2051 (1900)  | 1456 -at- 16.2kV/cm -at- 317 gradient |
> > 100       2214 (2050)  | 1559 -at- 15.6kV/cm -at- 291 gradient |
>
>What is the "gradient"?

The gradient was suppose to be the average field strength over the gap 
distance in volts (I should not have "termed" it as gradient, since 
gradient typically refers to the field strength and not the average). But 
for some reason it isn't right. The 70cm distance should have shown 313. I 
would normally fix it, but looking at it, I don't even know why I call out 
the average. It ended up in Javatc from porting equations over, but really 
isn't something needed. I may replace it with the field strength itself. 
(just did it).

> > Your values assume Emax=30kV/cm. Why?
> > I'm assuming Paschens curve ideal?
>
>30 kV/cm is the usual value in normal conditions of temperature and
>pressure. I could add corrections for altitude and temperature.

True, 30kV/cm is the normal field gradient or strength (at breakdown) in 
uniform fields for small gaps on the order of 1cm. But for large gaps of 
several meters can reduce down to 6kV/cm (reference J R Lucas). Paschen's 
curve I think is where the 30kV/cm is pulled as just about every text I've 
read references the curve, but the curve itself is for 1cm spacing at 1 
atmosphere and 25 deg C for conductors of an infinite plane, parallel, and 
perfectly smooth (as quoted from North). If this is case, we know in the 
real world it will always be below 30kV/cm.

The text also notes that under constant atmospheric conditions, it is 
experimentally found that the breakdown voltage of a uniform field gap amy 
be expressed in the form:
V = A*d + B * sqrt(d)
where
d = gap spacing
and for air under normal conditions,
A = 24.4kV/cm
B = 6.29kV/cm^1/2

At Jim Lux's website, the breakdown voltage is expressed as:
Vbreakdown = B * p * d / (C + ln( p * d))
http://home.earthlink-dot-net/~jimlux/hv/paschen.htm 
<http://home.earthlink-dot-net/%7Ejimlux/hv/paschen.htm>

I've made comparisons across a typical range of gap distances and both Jim 
and Lucas follow each other well. Their both based on the Townsend affect 
(as is Paschen's curve). The reason to pull in North is for the electrode 
geometry.

What I do is use Jim's equation for field strength and use it in North's 
equation for Arc Voltage. Works very well I've found. If I were to use 
Jim's arc voltage, it would hold well up until the gap spacing is 1/2 the 
electrode diameter. At that point it takes a fairly steep climb.

Here's a plot showing actual, Jims, Lucas, and then with North's geometry.
http://www.classictesla-dot-com/temp/arccomp.gif

>Something that you or Paul could verify:
>
>When I calculate the charge distribution in a sphere, by decomposing it
>in rings, there is a distinct irregularity in the first and in the last
>rings, the ones that don't have another ring inside. The charge on these
>rings is always somewhat smaller than should be (about 8%), and in the
>potential plot an irregularity can be seen at the poles of the sphere.
>Does this happen also with your formulations?

I'll have to let Paul answer that one.  That's in his area.

Take care,
Bart