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Isolated Toroidal Capacitance...

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 cross-section (cord) (m)
                   R = mean radius of the toroid (m)

                   epsilon[o] = permittivity factor
                              = 8.85418781762e-12 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 cross-section (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




o------------------------------------oo---------------------------------o
| Timothy A. Chandler                ||   M.S.Physics/B.S.Chemistry     |
o------------------------------------oo---------------------------------o
| NASA-Langley 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||   OPC-EFC                       |
o------------------------------------oo---------------------------------o
|                 Private Email Address:  tchand-at-slip-dot-net               |
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