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Isolated Toroidal Capacitance...

To: teslaatgrendel.objincdotcom

Subject: Isolated Toroidal Capacitance...

From: Tim Chandler <tchandatslipdotnet>

Date: Tue, 26 Mar 1996 23:32:40 0500

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Hi All,
I needed to calculate the capacitance of a toroidal capacitor and could
not find a formula for it anywhere, so I decided to derive one. Not a
good idea, at least using the usual techniques via electromagnetics. The
integrating and such was pretty ugly and very intense. So I decided on
a more novel approach, rather than using Laplace's equations or Gaussian
forms for a toroidal geometry.
If you think about it, a toroidal capacitor is basically a continuous
cylindrical capacitor (with its ends connected). Since it's ends are
connected, the fringe field effects for the toroid are nominal, and can
be neglected. The only problem is that at first I just used the formula
for a cylindrical capacator, plugging the necessary numbers and such. Didn't
work, at all. Then upon closer examination I discovered I had errored in
my primary assumption. The main problem is the field symmetry/geometry in
a cylindrical capacitor is 2 dimensional, so to speak, while the field
symmetry/geometry of a toroidal capacitor is more 3 dimensional (not in
actuality but lets say on paper). I was trying to smash a 3D field into
a 2D field, could never work. I eventually played around with it (using
some rather old techniques I found in a text dated 1929 by Smythe, the
text is really worn out, I don't know what the title is/was...) until
I thought I had finally accounted enough for the difference in field
symmetries/geometries.
I checked them out with Bert's formula and with the values listed in
TESLAC II. They all seemed to jive pretty, with a few exceptions, which
will be noted later.
Here are the revised formulas I came up with:
Actual Formula

C = epsilon[o] * k * 2 * pi * ( 2 * r * R )^0.5
where, C = capacitance of isolated toroid (F)
k = dielectric constant of medium
= 1.000590 at 760mm (for air)
r = radius of the toroid's crosssection (cord) (m)
R = mean radius of the toroid (m)
epsilon[o] = permittivity factor
= 8.85418781762e12 F/m
Easier To Use Formula (more user frieldly)

C = 4.43927641749 * ( 0.5 * (d2 * (d1  d2) ) )^0.5
where, C = capacitance of isolated toroid (pF)
d1 = outer diameter of toroid (inches)
d2 = diameter of crosssection (cord) of toroid (inches)
Comparison Chart

SIZE SURFACE AREA TESLAC CAP. DERIVED EQ. BERT's EQ.
(in) (in^2) (pF) (pF) (pF)

1 x 6 49.3480 5.10 5.4596 6.1670
2 x 8 118.4353 6.80 8.4580 8.8375
3 x 12 266.4793 12.95 12.6870 13.2562
5 x 14 444.1322 16.08 16.3788 15.3302
5 x 20 740.2203 21.58 21.1449 22.0937
7 x 30 1589.0063 32.17 30.9805 32.8954
6 x 36 1776.5288 37.18 *32.7575 37.0018
12 x 36 2842.4661 40.84 41.4354 39.7854
8 x 48 3158.2734 49.60 *43.6767 49.3357
12 x 48 4263.6691 51.80 50.7477 53.0249
12 x 60 5684.8921 63.33 *58.5984 64.2057
20 x 60 7895.6835 68.10 69.0589 66.3090
If you note the * by 3 of the derived equation's calculations, you see that
it's capacitance disagrees with both TESLAC and Bert Pools equation's
calculation. If you look at the surface area, which is the main factor in
determining the capacitance of any capacitor, you see for those calculations
the surface area does not increase enough to give the capacitance given in
TESLAC or by Bert's equation.
(at least I think, if I am sure someone will let me know..)
Well there are the formulas and some figures to support them, draw
your own conclusions...
Thanks,
Tim
oooo
 Timothy A. Chandler  M.S.Physics/B.S.Chemistry 
oooo
 NASALangley Research Center  George Mason University 
 Department of Energy  Department of Physics 
 FRT/Alpha  NASALaRC/DOE JRD/OPM  Department of Chemistry 
 CHOCT FR Designation #82749156/MG09 OPCEFC 
oooo
 Private Email Address: tchandatslipdotnet 
oo