NEC-LIST: Wave velocity in TEM lines and Measurement of ground constants

This is intended as a contribution two questions which have been the
subject of some debate on NEC-list in recent times. These are

. velocity of propagation on TEM-mode lines (e.g. Counselman
17/06/98), and

. in situ measurement of ground constants (e.g. Hagn 17/06/98)

Turning to the first and assuming that the dielectric medium is
uniform over the whole cross-section occupied by the field, it can be
shown by separation of the field equations that phase velocity depends
only on the dielectric constant of the medium (which I will assume is
real) and not at all on the conductor geometry (e.g. R E Collin,
"Foundations for Microwave Engineering"). It is equal to the velocity
of light in the medium, meaning that the phase and group velocities
are the same.

It is, however, possible to arrive at this result without recourse to
field theory starting from the distributed circuit theory of
transmission lines in which their properties are described in terms of
the distributed inductance (L) and capacitance (C) per unit length. It
follows from an extension by Caratheodory of the Reimann mapping
theorem that there exists a conformal transformation which will map
any uniformly filled TEM mode transmission line into a coaxial line of
the same characteristic impedance. Since L and C are invariant under
conformal transformation and phase velocity depends on their product,
it follows that the phase velocity in the original line and the
coaxial line are the same. It then remains only to show the
independence of phase velocity and cross-sectional geometry in a
coaxial line to demonstrate that it must be universally true.

It also follows from these considerations that uniformly filled TEM
mode transmission lines of the same characteristic impedance are the
conformal transformations of each other, even though in general
finding the transformation confounds the imagination. I wrote a note
about this to IEEE some years ago; the reference is "Conductor
Geometry Independence of Phase Velocity in TEM-mode Transmission
Lines", IEEE Trans. MTT, Vol. 37, No. 4, p 805, April 1989.

On the ground constants question, I certainly agree on the need for in
situ measurements if trustworthy results are to be obtained. The
reason is simple: water has such a high dielectric constant compared
with just about everything else in the natural environment that the
state of hydration of the sample is critical. It is just about
impossible to get a sample back to the laboratory without some drying
occurring, to say nothing of the effects of disturbing the sample in
the act of obtaining it. In 1982 I and another addressed this problem
by proposing the insertion into the ground of a two-wire transmission
line on which measurements could be made from the surface and the
ground constants deduced. We did this by use of a jig which allowed
boring holes in the ground for insertion of the wires which ensured a
tight fit with otherwise minimal disturbance. The effectiveness of the
method was demonstrated in a number of measurements. Unfortunately we
wrote this up in an Australian Journal, which didn't give it a lot of
exposure. The reference is "In Situ Measurement of Soil Permittivity
and Permeability" JEEEA, Vol. 2, No. 4, pp 202 - 208, December 1982
(JEEEA is the Journal of Electrical & Electronic Engineering,
Australia).

A problem with this method is to know what fringing capacitance to
attribute to the open end of the line. This enters calculation of the
ground constants as a quite significant correction factor. This was
the subject of a later short paper published by me and another in IEEE
which provides a formula for this capacitance. The reference is "End
Effect in Open-Circuited Two-Wire Transmission Lines", IEEE Trans MTT,
Vol. 34, No. 1, pp 180 - 182, January 1986.

It is hoped that the above will prove of some interest and perhaps
even use to persons involved with these problems.

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

23 June 1998
Received on Tue Jun 23 1998 - 09:42:01 EDT