Re: Flat transmission line capacitance formula?

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:

 > Original poster: Bert Pool <bert.tx-at-prodigy-dot-net>
 >
 > I have formulae for isolated spheres, toroids, flat plate capacitors, etc.,
 > but do not have a formula for calculating the capacitance of a single, long
 > flat plate transmission line.  I use a long strip of aluminum flashing for
 > the transmission line from the driver to the extra coil on my
 > magnifier.  Tesla provides us a formula for a cylindrical transmission
 > line, but nowhere do I find a formula for a flat plate line.  Does anyone
 > out there have such a formula?  I will be conducting some measurements on a
 > real-world line to determine the measured capacitance, but it sure would be
 > a help if I had a starting formula from which to work!

A formula for the capacitance per unit of length for a cylinder above a
ground plane is known:

C/length=4*Pi*e0/(2*Ln((h+sqrt(h^2-a^2))/a))

Where h is the height above ground of the center of the cylinder and a
is
its radius.

For a flat strip above ground, I don't know a closed expression, but
it seems to be a good approximation to consider the strip as a cylinder
with radius equal to 1/4 of the width of the strip, if it is far enough
from the ground.
Running a simulation with the Inca program:
I can describe a flat 10 cm strip making a wide (1 m) loop 20 cm above
a ground plane, having at the other side of the plane a thin toroid, for
comparison, as:

* Transmission line
L top 50 0.95 0.2 1.05 0.2
L ground 100 0 0 1.5 0
C other 50 1 -0.2 0.025 0 360
.v ground 0

The program calculates the capacitances as:
C[top,top]:  123.6100741149 pF <-------
C[top,ground]: -108.9185608288 pF
C[top,other]:   -0.9406758383 pF
C[ground,ground]:  300.4595813461 pF
C[ground,other]: -109.8495626018 pF
C[other,other]:  124.4980846604 pF <-------

The indicated values are the capacitances of the flat strip and of the
toroid, considering the other elements grounded.

The capacitance of a cylinder with length 2Pi m, 0.2 m above ground,
and with radius of 0.025 m would be, by the formula:
C=2*Pi*4*Pi*e0/(2*Ln((0.2+sqrt(0.2^2-0.025^2))/0.025)) = 126 pF

The precision of the approximation seems to be as good as measurements
can be.

Antonio Carlos M. de Queiroz