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,

 > I verified the behavior of the obtained capacitances of the rings,
 > and observed the problem with oscillations near the edges that
 > you mentioned.

Nice work.  It hasn't taken you long to get this up and running!

I hope that Gerry is also following this thread, because I know he
is interested in getting to grips with some modelling too...

 > I noticed then that if I reduce the thickness of the tubes for the
 > Pjj terms to about 1/3 of the distance to the adjacent ring the
 > oscillations disappear, and I obtain very good results with the
 > rings right at the surface.

At some tube radius, presumably, the self potential of the tube
ring becomes equal to the self potential of the tape ring, so I
guess you've found the correct 'equivalent tube' in this instance.

Wonder if the 1/3rd is generally valid?  One might expect it to
depend on curvature of the surface in some way.

 > I get now (C exact: 40.583973 pF):

 > 10 rings: 40.5487 pF
 > 20 rings: 40.5797 pF
 > 40 rings: 40.5835 pF
 > 80 rings: 40.5839 pF

That's much better than before, and better than tssp too.  Let's
hope that it applies to more general shapes...

 > Sphere with 90 cm of diameter:
 > Exact:         50.0692525224 pF
 > With 20 rings: 50.0681169556 pF
 > With 80 rings: 50.0692908972 pF

Still with the same tube_radius/spacing = 1/3 for that one?

 > A very interesting effect happens when I specify a toroid that
 > overlaps itself, as a 90x60 toroid. The program then places some
 > loops inside the figure. The capacitance continues to be
 > calculated correctly, and the capacitances of the internal rings
 > result as almost zero, as should be due to the shielding caused
 > by the external rings.

Yes, you can use this method for looking at some quite general
arrangements of terminals... the 'test page' for Geotc has quite
a few examples, eg of spheres inside toroids, etc.

One problem is that if you succeed in completely isolating one
terminal inside another, the inversion of the potential matrix
will fail - the matrix equation becomes singular because some
objects are decoupled from others.   In practice (with my code at
least) there is always enough 'leakage' to ensure that the [P]
stays non-singular.

 >> Surprising how well the method goes with just 10 rings, isn't it?

 > Enough for practical purposes. Also indicates that a toroid made
 > with a cage of tubes has really practically same capacitance of
 > a solid one.

I have a recent enquiry about just this situation, but I've been
unable to make an informed reply.  The 'C' of a skeleton tube topload
is not the problem, but predicting the effect on breakout could be.
But I don't have the techniques to persue that issue (yet!).

 > My ring formulation adds the difficulty that the electric field
 > can't be evaluated at the surface, at least where the rings are.

You can surely estimate the charge per unit area, which relates
directly to the surface field strength of the modelled smooth
surface, ie not the rings themselves.

Eg you could say that a length L of tube represents an area
L * 1/3 of the modelled surface, and you know the charge Q on that
bit of tube.  So the surface flux per unit area would be
3*Q/L, and therefore the surface field (if you assume that all
the flux radiates outwards from the closed object) is just 3*Q/(L*e0).

I think you've demonstrated the effectiveness of this method, and
shown how quickly it can be made to work (both coding and runtime!).

The method can be extended to include the effects of dielectrics,
but this is a little more complicated.  You have to include extra
surfaces to represent the boundaries between different dielectrics,
and you proceed with what's called the equivalent charge formulation
- a distributed charge is assigned to the boundary 'surfaces' and
these are coupled into the system of matrices.  Unfortunately the
extra equations involve not just Coulomb potentials, but their
differentials with respect to position.  Therefore it would require
quite small element sizes to cope with say a thin layer of
dielectric.  It wouldn't be too bad with big blocks of dielectric
but these are not the ones we see in practice.   For TC modelling
this issue of dielectrics raises its head with secondary coils that
have small radius compared with the tube wall thickness, eg a
2" diam coil on a plastic pipe with wall 1/4" thick, say.  Because
of the extra flux channelling through the dielectric wall, C_internal
calcs start to under estimate the real world, with errors of 20%
or more being typical for the example suggested above.

--
Paul Nicholson,
Manchester, UK.
--