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> 

Gerry wrote:

 > Im planning on decomposing the toroid into tiny surface patches
 > described in (R, phi) coordinates for major dimensions and
 > (r, theta) coordinates for minor dimensions

That seems to give you one too many coordinates - isn't the R co-
ordinate constant for a given toroid - the radius of the tube
center line?

 > charge density approximation will initially be constant over
 > the entire surface and the total toroid charge will be Q.

 > First, I need to redistribute the charge in theta (taking
 > advantage of symmetry) until the summation of all forces on each
 > patch charge is zero tangental in minor curvature.

Ok, and symmetry takes care of the major.

When you say 'redistribute the charge', do you mean some kind of
relaxation algorithm?

 > Second, I need to compute the integral (from infinity to the
 > toroid surface)(E.dl).

I don't think you'll need this step.  The topload potential should
pop out of the first step, in units of Q.  Hmm, where is the E going
to come from in the E.dl?

I would advise to stay as far as possible with just the potentials
during the calculations, rather than bringing in the field strengths.
Only at the end compute E where you need it.

 > I hope this will be sufficient for my purposes in determining
 > the "reach" various toroid sizes have.

I'm not sure it will, but you're embarking on the essential first
step.  I've a horrible feeling that we're going to have to
abandon the exploit of axial symmetry in order to solve for the
charge distribution (and the field) in the presence of an
asymmetric streamer load, eg a single streamer starting out from
the rim.

 > (I don't have a C compiler on my windows98 PC).  I may need to
 > get my hands on a unix machine for this purpose.

It's hard to get any work done with windows - ditch it for a proper
OS!  For C compiling, install gcc.

Antonio wrote:
 > [tube spacing] For the values in the last post I used 31.8372% of
 > the maximum radius (that would make the rings touch), that is the
 > value that results in the right capacitance for the 90x30 toroid
 > with 100 rings. For a sphere with 200 rings, the value is 31.83135%.

Ah, 1/pi then?

 > ...the ideal spacings are a bit different.

That seems to be working rather well, ie very little change in the
spacing ratio between the optimums for the two shapes.  Wonder how
you get on with objects that have edges?  I suspect that the ideal
spacing ratio depends on the curvature of the modelled surface
at that 'latitude', and would therefore vary for each pair of
adjacent rings:  1/pi +/- some curvature-related term?

 > Gauss' law. Really, I don't have to calculate the electric field.

Yup, easy isn't it!

 > Each ring has a charge qi and a length 2*pi*ri.      [ok]
 > A length dl has then a charge dq=qi*dl/(2*pi*ri).    [ok]
 > The spacing of the rings is a*dx, where dx is the angular
 > spacing and a is the minor radius..       [ok]
 > Each segment has then an area a*dx*dl     [ok]
 > The charge density is then p=dq/(a*dx*dl)=qi/(2*pi*ri*a*dx)  [ok]

 > and the electric field is E=p/(2*e0).  [hmm]

If the charge density is p, then the total flux from an infinitesimal
area dA is p*dA/e0.  If this flux were to come out equally on both
faces of the area then the flux out of each face is p*dA/(2*e0) and
the flux density on each face is p/(2*e0).  But here we have a closed
surface, so we are entitled to say that all the flux comes off the
outer surface, thus the surface field strength is actually E=p/e0.

Note that our use of single layer potentials doesn't tell us how the
flux is shared between the two faces, only the total.  For closed
objects we already know the answer, but for partially open objects,
eg the tube of the secondary we would have to calculate some E values
for each side of the surface and apportion the charge pro-rata.
(Or, model two surfaces for each conductor, and put E=0 inside the
  conductor to complete the system of equations).

I wrote (in an earlier post)
 > [for the 90x30cm toroid] the highest surface field ...
 > is 0.035 volts per metre, per volt...

Sorry, I got my units mixed up!  It should be 3.53 V/m/V, I forgot
my program reported in V/cm/V.

Antonio wrote:
 > A sphere with radius=1 meter shall have a surface field of 1 V/m for
 > each Volt of voltage.

Yes, tssp gives 1.01436 V/m/V for the 1m sphere in 135 rings. That's
the highest field that it finds on the surface, so shows, if you like,
the max positive error.  C = 111.185pF (tssp), 111.265pF (4*pi*e0)

Of course, when we're looking at distributed values like C and field
strength, we're going to get less accuracy, by a factor of around
sqrt(Number of rings) because we lose the benefit of the statistical
averaging obtained when computing the total C.  Hence 1.5% error here
is ok.

 > For the 90x30 toroid, I get, with 200 rings:
 > At the largest diameter:   1.7670568794 V/m/V

Something wrong there.  Your're ok for the sphere but not the toroid.

For your Bela simulator runs, I get
  Size       Bela      Tssp C,   Max field,  Breakout
  12 x 3:    302 kV    14.01pF, 9.87 V/m/V,  304kV
  16 x 4:    386 kV    18.68pF, 7.40 V/m/V,  405kV
  20 x 5:    458 kV    23.35pF, 5.92 V/m/V,  507kV
  26 x 6:    532 kV    30.48pF, 4.50 V/m/V,  667kV
  34 x 8.5:  795 kV    39.69pF, 3.48 V/m/V,  862kV
  48 x 12:  1353 kV    56.03pF, 2.47 V/m/V, 1215kV
  12" sphere: 850 kV   33.89pF  3.33 V/m/V,  901kV

The tssp figures are for the toroid in free space, not in a room.
I expect the agreement would be better if we put in your room sizes,
since tssp diverges from Bela when the toroid size increases.
There's something a bit funny about the 48x12 model.

 > [singular potential matrix] ... But the objects are always coupled.
 > The Pij terms never disappear.

True, the Pij terms don't vanish, but things can get a bit tricky
when you try to invert...   Yes, thinking about it, the [P] should
always be invertible for a physically realisable system.
--
Paul Nicholson
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