Re: [TCML] toroids instead of spheres - why?

[Antonio Queiroz <acmdequeiroz@xxxxxxxxx>]

Em 21/08/2013 18:34, Electronic Battle escreveu:
> Hello all
>
> many years ago someone provided a very useful reference which was a 
> document (either a paper or an excerpt from a book). I look to Antonio 
> De Queiroz or Bert Hickman for starters but it may have been one of 
> many similar experts.
>
> If I remember correctly, it explained that for a given breakdown 
> voltage, the dimensions required of a toroid to hold off that voltage 
> meant it presented a smaller capacitive load than a sphere of 
> equivalent hold-off (breakdown) voltage.
>
> I can't find the reference and I have failed to locate it in the 
> archives, can anyone point me in the right direction please?
Let's try some numbers to see what happens:
The breakdown voltage of an isolated sphere is ~3 MV per meter of 
radius, and the
capacitance of a sphere is 111 pF per meter of radius. Consider then a 
sphere with
2 meters of diameter, with breakdown voltage of 3 MV and 111 pf of 
capacitance, and
let's try some toroids with the same breakdown voltage of 3 MV, starting 
with large
diameter and ending with a toroid where the hole at the center almost 
disappears:
Major diameter x minor diameter -> capacitance:

8.20 x 0.5 m -> 280 pF
3.35 x 1.0 m -> 148 pF
2.86 x 1.2 m -> 134 pF
2.69 x 1.3 m -> 129 pF

The conclusion is that a toroid with the same breakdown voltage of a 
sphere has always
GREATER capacitance than the sphere. The conclusion is reasonable, since 
the capacitance
of an isolated conductor is directly proportional to its linear size 
(not to the surface area),
and the breakdown voltage of the sphere is maximum for conductors with 
the same
maximum diameter. Any shape with the same breakdown voltage must have larger
maximum diameter, and almost surely greater capacitance.
An interesting question is if a shape can be found with the same 
breakdown voltage of a
sphere but less capacitance. Probably not.
The calculations are exact, with the Inca program.

Antonio Carlos M. de Queiroz

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