NEC-LIST: Characteristic impedance of coaxial lines with non-circular inner conductors

The question posed recently by Stanley Lim in a posting to NEC-List
which has already elicited several responses, namely of wanting a
formula for the characteristic impedance of a coaxial line in which
the outer conductor is a cylinder and the inner a symmetrically
placed, infinitely thin strip, has an interesting connection with the
concept of equivalent cylindrical radius for rod antennas of uniform
but non-circular cross-section.

In the instance posed by Lim, several respondents have indicated that,
for sufficiently high characteristic impedances (i.e. where the strip
width is small relative to the diameter of the outer cylinder), the
configuration has the same characteristic impedance as would result
from use of the standard formula for the impedance of a coaxial line
with a cylinder of diameter half the strip width replacing the strip
itself. Half the strip width also happens to be the equivalent
diameter for a linear antenna of the same cross-section. This is not a
coincidence.

Equivalent cylindrical diameter in the case of an antenna is derived
as that of a cylinder which, as part of a structure which extends
infinitely in the axial direction, has the same capacitance per unit
length as the actual antenna conductor. One can imagine determining
this by solution of a two dimensional electrostatic problem in which
the strip is surrounded by a zero potential, circumscribing cylinder
at infinity in the cross-sectional plane. One might then expect to use
the diameter so determined in other situations where the
circumscribing cylinder, though now brought in from infinity to finite
separation, nonetheless remains at all points sufficiently distant
from the central conductor that the electrical lines of force
terminating on it are locally essentially radial and uniformly dense.

This implies that for high impedance coaxial lines, characteristic
impedance for any shaped inner conductor may be approximated using the
normal coaxial line formula with the inner conductor replaced with a
cylinder of the same diameter as the equivalent cylindrical antenna.
Formulas appear in the literature for a number of such shapes
(e.g. rods of elliptical, rectangular, square, etc cross-section) and
from these approximate solutions are available to a variety of coaxial
line problems of the type posed by Lim.

The problem is that one cannot easily put error bounds on the
approximation and therefore know for exactly what range of impedances
the approximate formula can be expected to apply. This might be
guessed at by comparison with other cases as like as possible to that
in question for which either exact solutions or solutions with error
terms are available. Failing that, sketching a few field lines might
help.

Harry E Green,
Adjunct Research Professor,
Institute for Telecommunications Research

Monday 1 May 2000
Received on Tue May 02 2000 - 03:32:46 EDT