Cecil et al:

[snip]
the stub. I was just wondering what is the nature of your
f(Z0) function.
--
73, Cecil http://www.w5dxp.com
[snip]

I like working with Cecil!

Like a Zen Master's rehtorical approach to facilitating understanding,
Cecil's approach
to this seemingly paradoxical "circuit-to-wave transducer" question is quite
illuminating.

Cecil has neatly sidestepped the fact that, even for the simplest practical
antennas, elementary analytic
formulae for antenna driving point impedances have never been discovered.
Let alone formulae that explicitly
show Zo as an independent variable.

This [obscure?] fact [of delinquint formulae] often comes as a surprise to
most electromagnetic novitiates, I know it did to me.

[Aside: As far as I know, the fact that no one has ever worked out an exact
analytic formula for the driving point
impedance of a simple practical half wave dipole, is not a problem in
practices since other
approximate and/or "sledge hammer" style numerical methods provide
appropriately accurate answers to all
practical Engineering questions about such matters.]

However, as a "seeker of truth", Cecil has noted an easier path as an
approach to the apparently paradoxical
question of the relationship of driving point impedance to the wave or
characteristic impedance of free space
or any other propagating media.

Cecil has zeroed in on an alternative that might give us some insight!

Namely the relatively simple "exact" formula, first revealed by Heaviside
and Kelvin approximately two hundred years ago, the celebrated formula for
the driving point impedance Z = V/I of a lossless transmission line of
characteristic [surge] impedance Zo
terminated in a load impedance ZL. This driving point impedance is given by
the surprising simple relation...

Z(Zo) = Zo[(ZL*cos(theta) + jZo*sin(theta))/(Zo*cos(theta) +
jZL*sin(theta))] (1)

Where theta = 2*pi*(d/lambda) is the relative fractional length of the
transmission line, where d is
the line length and lambda is the wavelength of a sinusoidal signal
supported on the line at the particualr
frequency of interest. Zo of course is the characteristic [surge or wave]
impedance of the line.

It is also well known [Again Kelvin and Heaviside] that Zo can be simply
expressed in terms of the fundamental
transmission line parametric constants [R, L, C, G] by the [equally]
celebrated formula for the characteristic
[surge or wave] impedance of the transmission line as

Zo = sqrt[(R + jwL)/(G + jwC)]; Where, in the lossless case R=G=0.0, Zo -
sqrt(L/C).

[Aside: The terms Impedance, and Reactance were first defined by (Reg
Edward's hero) Oliver Heaviside. I wonder
if an equally simple formula for the driving point impedance in terms of the
Zo of free space for some simple
antenna is lying out there somewhere waiting to be discovered (grin). ]

As can readily be seen, the driving point impedance Z is a function of the
dependent variable Zo and...

although the effect of this relationship is often referred to as an
trasmission line impedance "transformer", the analogy
between the so-called "transmission line transformer" (or should we say
"transducer") described by (1) falls short of
the simple turns ratio relationship where Z = ZL*N^2.

To gain insight here, Cecil has obliquely suggested that, instead of
searching for an antenna formula, that we invert
the celebrated formula (1) and use it to determine unknown characteristic
impedance Zo by assuming ZL known, and
measuring Z.

Inverting formula (1) we obtain the following relationship.

Zo(Z) = ZL[(cos(theta) - jZ*sin(theta))/(Zcos(theta) - jsin(theta))] (2)

[Aside: Apart from the fact that line parameters L,C are also implicit in
the wavelength, Cecil is this right?]

Thus we see that the relationship between Zo and Z is not a simple linear
relationship as for the common transformer,
but instead is, what mathematicians often refer to as, a so-called "bilinear
relationship.

I wonder, is it possible that such a simple relationship exists for some
antennas as well as transmission lines?

An interesting invention... now it will be public domain (smile).

One could clearly construct a sensor to measure unknown Zo's by constructing
a small piece of rigid air dielectric
terminated transmission line and then "immersing" the sensor in substances
of unknown Zo and then determine those unknown
Zo's by measuring the driving point impedance Z. The calibration curve for
this Zo sensor would be the inverse
relationship (2).

Sigh, it's too bad there is not a simple analytical relationship like (1)
for antennas, for perhaps this would
address the OP's question of the relationship between 377 Ohms free space
wave impedance and 73 Ohms driving point
impedance more directly.

On the other hand we now can see that, contrary to Roy's recent assertion up
the thread (grin), that certainly an exact
analytic solution to this problem is likely a challenging Ph.D. thesis
topic.

For... after two hundred or more years [As far as I know...] no one has yet
worked out an exact simple
analytical expression [similar to (1)] for the driving point impedance of
any simple practical antenna.

A Ph.D. thesis indeed!

Thanks Cecil!

Thoughts comments...

--
Pete K1PO
Indialantic By-the-Sea, FL