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> 

Antonio wrote:

 > I made a document showing the algorithms:
 > http://www.coe.ufrj.br/~acmq/tesla/capcalc.pdf

Thanks  - that summarises the calculations very nicely.  I see you
use AGM for the elliptic integral.  Haven't tried that one.  I
use Carlson's elliptic integral, not sure if it's any quicker.
Both converge quadratically, therefore require very few iterations.
In Carlson's the error reduces by a factor 4096 with each
iteration.  See 'Numerical Recipes in C', section 6.11.

I made an elaboration to deal with the ring self potentials, which
seems to be helpful if, like me, you're evaluating the potential
coefficients at the centroids of the flat rings.  This gives a
little problem when choosing a suitable radius for an equivalent
tube ring, since the centroid may be nearer to one edge of the flat
ring than the other.

When computing the self potential of the flat tape, I decompose the
tape ring into around 10 filament rings and add up their elliptic
coefficients in the usual way.  If a filament ring is close to
the centroid (ie within 1/20th of the tape width), I replace it
with a tube ring self potential based on Antonio's new 1/pi radius
rule.  Thus the effect of ambiguity in the equivalent tube radius
is reduced by some factor.

Antonio wrote:
 > For two concentrical spheres with radii 0.3 and 0.5 m:

 >  rings      tssp        cacal
 >    10      80.341 pF    83.3801277244 pF
 >    20      81.887 pF    83.4395579512 pF
 >    40      82.643 pF    83.4475644027 pF
 >    80      83.045 pF    83.4486029642 pF
 >   200      83.289 pF    83.4487445528 pF

I now get

rings       cacal             tssp
  10      83.3801277244 pF     83.636 pF
  20      83.4395579512 pF     83.429 pF
  40      83.4475644027 pF     83.413 pF
  80      83.4486029642 pF     83.424 pF
200      83.4487445528 pF     83.437 pF

It is now much better at lower resolution.  The convergence is
a bit funny, so something is not quite right in my code.

 >> For two discs, 1 metre diameter, spaced 10cm apart, I get

rings   cacal total      cacal mutual     tssp total   tssp mutual
  10   98.1496072537 pF 78.6827468834 pF   98.13 pF     78.65 pF
  20   99.8002154701 pF 80.1207334151 pF   99.72 pF     80.04 pF
  40  100.6788391600 pF 80.8969570508 pF  100.62 pF     80.84 pF
  80  101.1321593111 pF 81.3000670740 pF  101.092 pF    81.261 pF
200  101.4087787914 pF 81.5468815768 pF  101.389 pF    81.528 pF

As you can see, these figures are also much better.  Overall
about 1 extra digit accuracy with large numbers of rings.

For the 1m radius sphere, (exact = 111.265008)

rings   tssp
  10     111.67 pF
  20     111.34 pF
  40     111.275 pF
  80     111.262 pF
200     111.2618 pF

and with a high resolution

rings    tssp
2000    111.2646 pF

Well, I would say that at last we have a self potential recipe that
works.  My implementation is still not quite right, but it's better
than before, especially at low number of rings.  For this reason
I've incorporated it straight away into Geotc and this has already
propagated through to Bart's JAVATC.   Roughly speaking the error
with default detail setting will be about half or less than it
was before.  Not so much difference at higher detail.

For me, my code is now much tidier! I've been able to dump the last
of the horrible little integrals that approximated the ring self
potential and replace them with Antonio's simple 1/pi rule for the
radius of an equivalent tube ring.  The resulting accuracy is more
than enough for tssp.

So, Thanks Antonio! I think this little discussion delving in to
the fine art of the capacitance problem has been most useful.
--
Paul Nicholson
--