Re: Breakdown voltages of toroids

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

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:

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

 > Here's some of your data with Javatc pulled in:
 >
 > Symmetrical gaps:
 > sphere diameter:      5      |Javatc|       10       |Javatc|
 > gap spacing:
 > 0.5              14.3 (17.5) | 16.8 |   15.0 (16.9)  | 17.4 |
 > 1                26.6 (32.2) | 28.9 |   28.6 (31.6)  | 30.8 |
 > 1.5              37.4 (46.1) | 38.8 |   41.3 (45.8)  | 42.7 |
 > 2                47.0 (58.3) | 47.3 |   53.2 (59.3)  | 53.5 |
 >
 > Javatc uses the equations shown in William Norths paper.

Those equations agree well with my simulations, although they don't
say how the potential is applied. With 30 kV/cm as maximum field I get:
        5      10
0.5 14.043  14.511
1.0 26.339  28.087
1.5 37.116  40.790
2.0 46.578  52.678

North says that the equations are "exact", but doesn't say from where
he obtained them. The exact formula that I know is an infinite series.
My program obtains the values by simulation.

 > 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?

 > 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"?

 > 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.

 > Antonio, thanks for doing all you do! You keep us on our toes! BTW, Inca is
 > getting to be quite the program! The field plots are real nice (and fast!).

Thank you. I will work now in coding the exact formulas for potential,
electric field, and breakdown voltage between spheres, and for a toroid
in free space.

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?

Antonio Carlos M. de Queiroz