Slick:

[snip]
And not the complex conjugate of Z0.
:
:
This is ABSOLUTELY WRONG!

The reflection coefficient is zero only when the Zload
is the conjugate of the Zo.

Go look it up in any BASIC RF book!

Slick
[snip]

Easy now boy! You'r almost as bad as me!

It is entirely possible, in fact I know this to be true, that there can be
more than one *definition* of "the reflection coefficient". And so... one
cannot say definitively that one particular defintion is WRONG.

If the definition of the reflection coefficient is given as rho = (Z - R)/(Z
+ R) then that's what it is. This particular definition corresponds to the
situation which results in rho being null when the unknown Z is equal to the
reference impedance R, i.e. an "image match". If the definition is given as
rho = (Z - conj(R))/(Z + conj(R)) then rho will be null when the unknown Z
is equal to the conjugate of the reference impedance conj(R), i.e. a
"conjugate match".

Nothing is WRONG if the definition is first set up to correspond to what the
definer is trying to accomplish. And so one has got to take care when
making statements about RIGHT ways and WRONG ways to define things.
Everyone is entitled to go to hell their own way if they are the onesmaking
the definitions. Just as long as no incorrect conclusions are drawn from
the definitions. That may occur when folks don't accept or agree on a
definition.

OTOH....

Definitions and semantics aside, what we should really be interested in is
what is the physical meaning of any particular definition and what are its'
practical uses.

Clearly if R is a real constant resistance and contains no reactance for all
frequencies [R = r + j0] then the two definitions are equivalent i.e. rho =
(Z - R)/(Z + R) = (Z - conj(R))/(Z + conj(R)) since R = conj(R). This is
the situation for most common amateur radio transmsission line problems and
so in these simple cases it clearly doesn't matter which definition one
takes. But the question of definitions for rho is even broader than that.
We amateurs usually only examine a very small class of problems, and there
are many more and usually much more interesting and challenging problems
that require the use of reflection [scattering] parameters.

Now for broadband problems where the reference impedance R is in general a
complex function of frequency, e.g.
R(w) = r(w) + jx(w), one is faced with the problem of creating a definition
for rho which will be practical and useful and easy to measure...

For an example of a practical consideration, with R a constant resistance it
is easy to manufacture wide ranging reflectometers, like the Bird Model 43
since all it needs inside is a replica of the R, simply a the equivalent of
a garden variety 50 Ohm resistor. But if the reference impedance R needs to
be a complex function of frequency it is not so easy to design an instrument
to measure the reflection coefficient over a broad band. In fact if the
reference impedance R(w) = r(w) + jx(w) corresponds to the driving point
impedance of a real physical system such as 18,000 feet of telephone line
operated over DC to 50MHz, then it can be proved using network synthesis
theory that one cannot exactly physically create the conjugate R(w) = r(w) -
jx(w) for such a line. How then to make a satisfactory reflectometer for
this application? To synthesize the reference impedance for such a "broad
band Bird" one would have to be approximated over some narrow band, etc...
Not an easy problem...

Actually, under the general so-called Scattering Formalism, the reference
impedance can be chosen arbitrarily, and often is, to make the particular
physical problem being addressed easy to solve. Within the general
Scattering Formalism the so-called port "wave variables" a and b [a is the
incident wave and b is the reflected wave] are nothing more than linear
combinations of the port "electrical variables" v and i [v is the port
voltage and i is the port current]. Thus each port on a network has an
electrical vector [v, i]' and a wave vector [a, b]' and these two vectors
are related to each other by a simple linear transformation matrix made up
of the sometimes arbitrarily chosen reference impedance(s).

For example for the "normal" case we are all used to where r is a fixed
constant then... [a, b]' = M [v, i]' where M is the matrix of the
transformation. Specifically...

b = v - ri
a = v + ri

and the 2x2 matrix M relating the "waves" to the "electricals" has the first
row [1, -r] and second row [1, r], i.e. M is equal to:

|1 -r |
|1 +r|

It is easy to show with simple algebra that this definition of the relation
M betweent the waves and the electricals yeilds the common defiintion of the
reflection coefficient rho = b/a = (Z - r)/(Z + r).

The way linear algebraists view this is that the vector of waves a and b is
just the vector of electricals rotated and stretched a bit!

In other words the waves are just another way of looking at the electricals.

Or... the waves and the electricals are just different manifestations of the
same things, their specific numerical values depend only upon your
viewpoint, i.e. what kind of measuring instruements you are using, i.e.
voltmeters and ammeters or reflectometers with a particularly chosen
reference impedance.

All that said, it should be clear that one can arbitrarily chose the matrix
transformation [reference impedance] which relates the waves to the
electricals to give you the kinds of wave variables that makes your
particular physical problem easy to solve. i.e. it dictates the kind of
reflectometer you must use to make the measurements. The Bird Model 43 is
only one such instrument and it is useful only for one particular and common
kind of narrow band set of problems. For broad band problems one needs an
entirely different set of definitions, etc...

And so...

In transmission line problems it us usual to choose the characterisitic
impedance Zo of the transmission media to be the reference impedance for the
system under examination, but that is certainly not necessary, only
convenient. And... if you want a null rho to correspond to a "conjugate
match" you must choose the reference impdance in your reflectometer to be
the conjugate of the reference impedance of the system under examination,
and if you want a null rho to correspond to an "image match" then you must
choose the reference impedance in your reflectometer to be identical to the
reference impedance of the system under examination.

Every one is entitled to go to hell their own way when defining the wave to
electrical variable transformations required to make their measurements and
solve their problems and this will result in a variety of definitions for
the scattering [reflection] parameters. Nothing more nothing less. Others
may not agree with your tools, methodologies and definitions, but just be
careful to follow through and be consistent with your definitions,
measurements, algebra, and arithmetic and you will always get the right
answers.

Thoughts, comments?

--
Peter K1PO
Indialantic By-the-Sea, FL.