Re: Charge distribution on a Toroid (was spheres vs 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: Paul Nicholson <paul-at-abelian.demon.co.uk>

 >  > 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 saw that algorithm when looking at how to solve elliptic
integrals. I opted for the agm, that is simpler and works well.
The number or iterations depends on the argument k, but never
exceeds 8 cycles in extended precision (with k=last number before 1).
Usually takes 3 to 6 cycles.

 > 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.

I am considering the rings as infinitely thin, and computing the
potentials exactly where the rings are. For the self-potentials,
I am considering a distance = maximum radius/Pi above the ring.

 > 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.

So, you are still using belts, but modeling the belts using rings
instead of rectangles. This saves in the slowest part of the calculation
that is the inversion of the potential matrix.
Really, you have then the problem of where to compute the potentials.
Do you assume identical charges in all the rings of each belt?

 > 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.

Really. A good indication that the algorithm is correct is regular
convergence.

 >  >> 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.

You should get better results using a number of belts identical
to the number of rings that I am using, specially in the case
of spheres, where the charge distribution along a belt is really
almost uniform.

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

I get:

rings   tssp          Inca
   10     111.67 pF    111.2417937469 pF
   20     111.34 pF    111.2619394256 pF
   40     111.275 pF   111.2646089967 pF
   80     111.262 pF   111.2649551919 pF
  200     111.2618 pF  111.2650023882 pF

and with a high resolution

 > rings    tssp
 > 2000    111.2646 pF
With 400 rings: 111.2650052117 pF. Takes several seconds.

A little mistery with spheres: The charge density for the top and bottom
rings is always slightly wrong. With 10 rings in a 2 m sphere, I get:

n      angle      C (pF)          electric field ((V/m)/V)
1    -81.00000    2.5134365815    0.9192994899
2    -63.00000    7.9313316207    0.9995876292
3    -45.00000   12.3521228136    0.9994892507
4    -27.00000   15.5670923546    0.9996504060
5     -9.00000   17.2569135031    0.9996885641
6      9.00000   17.2569135031    0.9996885641
7     27.00000   15.5670923546    0.9996504060
8     45.00000   12.3521228136    0.9994892507
9     63.00000    7.9313316207    0.9995876292
10    81.00000    2.5134365815    0.9192994899

I tried better formulas for the charge density, but continue to get
this 0.92 value. 200 rings lead to 0.9197580409.

This would introduce a systematic error if I try to calculate the
maximum electric field in a two-spheres gap.

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

You are welcome. I am learning a lot of things with these discussions.
I have now good methods to find the capacitance matrix and the
inductance matrix of a set of coils decomposed in rings or groups
of rings. It's easy to integrate everything in a Tesla coil simulator
with adjustable level of distribution. The result would be surely
similar to what you can already do with the tssp software. What is
missing is accurate modeling of losses.

The calculations that I had tested in a separate program are
now integrated in the Inca program:
http://www.coe.ufrj.br/~acmq/programs

Antonio Carlos M. de Queiroz