Re: topload questions

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Original poster: "Godfrey Loudner by way of Terry Fritz <teslalist-at-qwest-dot-net>" <ggreen-at-gwtc-dot-net>

Hello Antonio

The approximation formula is an observation derived from an exact formula
for the self capacitance of a toroid, which can be found in the "Ideal
Capacitance Chapter" at http://www.eece.ksu.edu/~gjohnson . The main problem
with the exact formula is the slow convergence of the infinite series for
certain values of the parameters. A fantastic number of terms need to be
summed in some applications. In my spare time, I have been investigating the
convergence of the series, with the hope of coming across inequalities which
can be used to derive a useful and very accurate approximation formula over
the entire range of the parameters. The exasperating case is when x = (D -
d)/d is very close to 1 from above. I have derived a number of error
estimates ranging from simple to intractable. Using the simplest of the
error terms, we can entirely neglect the infinite series portion of the
exact formula, yielding the statement below. The bounds in the statement can
be improved by using more accurate estimates. The coefficient is improved
from 35.4 to 35.416 and I make clear what is meant by x "large". The
self capacitance C of a toroid is approximated by the formula with D and d
in inches,

                 35.416 d (x^2 - 1)^(1/2) Q[-1/2, x] / P[-1/2, x] pF

in the following sense:

accurate to 1/10       place value if x = or > 14,
accurate to 1/100     place value if x = or > 59,
accurate to 1/1000   place value if x = or > 271,
accurate to 1/10000 place value if x = or > 1256.

I note again that the above bounds are not optimal.

The definitions of Q[-1/2, x] and P[-1/2, x] can be found at Dr. Johnson's
site above. You might experience problems using software to compute Q and P.
Outputs may vary with the software choosen, depending upon which analytic
continuations are chosen to extend basic definitions of Q and P to the
complex plane. If you use Mathematica, type
Re[LegendreQ[-1/2, x]] and LegendreP[-1/2, x].

Godfrey Loudner







----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
To: <tesla-at-pupman-dot-com>
Sent: Sunday, January 26, 2003 1:35 PM
Subject: Re: topload questions


 > Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz
<teslalist-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>
 >
 > Tesla list wrote:
 >  >
 >  > Original poster: "Godfrey Loudner by way of Terry Fritz
 > <teslalist-at-qwest-dot-net>" <ggreen-at-gwtc-dot-net>
 >  >
 >  > Hello Antonio
 >  >
 >  > I am getting the following:
 >  >
 >  > 16" x 7"------------19.2035 pF
 >  > 16" x 1"------------13.9494 pF
 >  > 16" x 0.1"----------9.87011 pF
 >  > 16" x 0.01---------7.50278  pF
 >  > 16" x 0.001"-------6.03675  pF
 >  >
 >  > Let x = (D - d)/d. If x is large, then a good approximation to C is
given
 >  > below.
 >  >
 >  > C = 35.4 d (x^2 - 1)^(1/2) Q[-1/2, x]/P[-1/2, x], where C is in pF, D
and d
 >  > in inches.
 >  >
 >  > Q and P are the Legendre functions, commonly used in EE.
 >
 > This formula looks better, but still appears to give zero as d -> 0.
 > The term (x^2-1)^(1/2) could be x without difference if x is large.
 > I know the Legendre polynomials Pn(x), that I have seen in some rare
 > cases of filter design, and also the functions Qn(x), that also
 > satisfy Legendre's equation. If the functions that you mention
 > are these, what means the -1/2 in the formulas?
 >
 > Antonio Carlos M. de Queiroz
 >
 >