Re: Charge distribution on a Toroid (was spheres vs toroids)

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

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Hi Antonio, All,

 > When the relation is between an element and itself, something
 > must be assumed about the shape of the element.

Yes, that's a fundamental basis to this kind of method - we break
a complicated shape (for which we have no hope of closed form
solution) into a large number of uniform shapes, for each of which
we *do* have a closed expression (or a good approximation).
Then, juggling the matrices stitches it all together to give a
solution for the whole structure.

 > I made an experiment considering a partial toroid, with major
 > and minor radii specified,  ...

 > Ex: That 90 cm x 30 cm toroid:
 > C exact: 40.583973 pF
 > With 20 rings placed at the surface: 41.660502 pF
 > With 20 rings just touching the surface: 39.061591 pF
 > With half of the radius out of the surface: 40.3662370342 pF

Ok, if I understand right you're using a tubular ring, rather than
a flat tape ring.  It seems to get you into the right ballpark.

 > The method appears to work reasonably for toroids, spheres, and
 > half-spheres.

Yes, try it for discs too, because there you have an edge to deal
with.

Tssp's tcap program gives, for the toroid above,

   40.2 pF using 10 rings.
   40.4 pF using 20 rings,
   40.50 pF using 40 rings,
   40.54 pF using 80 rings,
   40.563 pF using 160 rings,
   40.586 pF using 320 rings,

(80 = nominal value for tssp software, for 0.1% error, 0.7 seconds).

The result converges smoothly towards the correct answer as the
number of rings is increased - a good sign!

With more than 300 rings, the result begins to deteriorate, and the
software would need a little fine tuning to do better than this.
Surprising how well the method goes with just 10 rings, isn't it?

 > http://www.coe.ufrj.br/~acmq/programs/inca.zip

I've no means of running this program.  You might want to consider
making the source available.

 > I am using an algorithm that inverts a matrix in place. I am not
 > sure if it would be stable with so many equations.

Yes, it will be stable providing the self potentials are about right.
I've never had any problems with ill-conditioned [P] at the kind of
resolutions we're dealing with here - except when the Pii are not
quite right.

 >> Please someone give me a formula for the potential due to
 >> a tape ring!!!

 > The thin ring approximation is a first step. The integral that
 > would give the inverse of the capacitance of a ring with uniform
 > charge seems solvable, but not trivial.

Yes, perhaps someone has worked this out long ago - buried in an
old textbook somewhere - that often seems to be the case!  It would
sure speed up the calculations.

A challenge facing us is to combine the axial-symmetric model with
a full 3d model of a curved streamer.  I'm considering going back
a step: ditching the rings, and doing the entire thing in 3d with
multipole approximations for the more distant elements of [P] - along
the lines of the program 'fastcap'.

BTW, for your test toroid above, I get the highest surface field
strength in the plane of the toroid (no surprise there!) and the
value is 0.035 volts per metre, per volt on the toroid.  I've
taken to calling this figure the 'specific surface field', for want
of a terminology.

And, to those of you who sat through math lessons thinking - "what
use is all this boring stuff about matrices?"  Well, now you know!
--
Paul Nicholson,
Manchester, UK.
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