toroid quesion

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Original poster: "Loudner, Godfrey by way of Terry Fritz <twftesla-at-qwest-dot-net>" <gloudner-at-SINTE.EDU>

Hi All

I did some calculations tonight to compare values of Bert Pool's formula and
the exact formula for the capacitance of an isolated toroid. Let D and d
denote the width and thickness respectively of a toroid. The results of some
randomly chosen values are given below.

D = 1", d = 0.4", C(bert) = 1.06746pF, C(exact) = 1.18129pF

D = 5", d = 2",    C(bert) = 5.33731pF, C(exact) = 5.90647pF

D = 8", d = 3.5", C(bert) = 8.27814pF, C(exact) = 9.60177pF

D = 100", d = 8", C(bert) = 80.6557pF, C(exact) = 90.7548pF

Here are the values for a toroid that sold on ebay tonight for $175.

D = 16", d = 2",  C(bert) = 15.1408pF, C(exact) = 15.615pF

Bert Pool's formula is very good for some values of D and d. The idea of
Bert's formula is to replace a toroid with a sphere of equal surface area.
Then the capacitance of the sphere is tinkered up to agree with a table of
measured values. A simple idea, but the result is quite powerful. 

I've been playing with the exact formula. I'm not saying that C(exact)
depends only on (D - d)/d, but the difficult factor of the formula does. The
difficult part of the formula is an infinite series, and I have been trying
to determine its rate of convergence. This means that I want to know the
optimal number of terms, which have to be added to achieve a given accuracy.
I was hoping to answer this question for all values of D and d such that (D
- d)/d  is greater than 1 with an accuracy of 1 in 1,000,000. I have already
derived an almost such formula for the number of terms, but it is definitely
not optimal. Why add 15 terms when 5 will suffice. The problem seems to be
very difficult. The mathematical difficulty comes from the situation when (D
- d)/d is very close to 1 from above. The closer you get to a toroid with no
hole in the center, the greater the mathematical difficulty. When you get to
toroid with no hole in the center, the formula degenerates to 0 times
infinity. The formula still holds the information about a toroid with no
hole, but one has to find the limit of the formula as (D - d)/d approaches 1
from above to cover the case D = 2d. I have not considered this problem yet.

This brings me to my question. Do coilers have any thoughts about the size
of the hole in the center of a toroid? Do coilers want that hole to be small
or large? 

Eventually I want to provide an approximation formula that is more accurate
than Bert's formula. But without using powerful mathematical techniques,
Bert derived a pretty good formula. My hat is off to Bert Pool.

Godfrey Loudner