Slick:

[snip]
If you believe that there are theoretically no
reflections in a conjugate match, then with

Zl=50+j10 and Zo=50-j10,

the conjugate equation correctly cancels the reactances
giving no reflections, while the non-conjugate still
incorrectly gives a magitude (non zero) for rho.

Slick
[snip]

Oh yes here are voltage reflections at a conjugate match!

Simply put as waves pass across the transition from an impedance
of Zo to an impedance of conj(Zo) they are crossing a boundary
with an impedance discontinuity. Zo on one side and conj(Zo)
on the other side is definitely discontinuous! Unless of course,
Zo = conj(Zo) which occurs only when Zo is real.

There will always reflections at such an impedance
discontinuity where an impedance faces its' conjugate.

If an impedance Zo faces itself Zo there will be no "voltage
reflections", this phenomena was called an "image match" by
Campbell and Zobel way back around 1920 or so, it represents
the basis for the image match design of filters and transmission
systems. Do any reader's here recall image parameter filter
design? I designed more than a few that way myself.

If an impedance Zo faces its' conjugate conj(Zo) then there will be
no "power reflections", but there will in general be voltage reflections,
and this is called a "conjugate match" and represents the basis
for "insertion loss" design of filters and transmission systems.
Darlington and Cauer introduced the insertion loss design
of filters which eventually supplanted the older Campbell,
Zobel image parameter design.

The "classical definition of voltage reflection coefficient from
theoretical physics and the solutions to the wave equations is
the one that does not use the conjugate. This classical
definition corresponds to the calculations made by a reflectometer
configured to measure the reflected voltage at the boundary
between Z and Zo such that if Z=Zo the reflected voltage
(often called the "talker echo") will be zero. This will not
correspond to maximum power transfer in the general case.

i.e. "classical" rho = (Z - Zo)/(Z + Zo) is not zero at a conjugate match.

I also believe that this is "Mother Nature's" reflection coefficient
for it is exactly what she uses as she lets the waves propagate
down her lines of surge impedance Zo following her partial differential
equations at every point along the way. At every infinitesimal
length of line all along it's length the waves are passing from a
infinitesimal region of surge impedance Zo to the next infinitesimal
region of surge impedance Zo and there are no voltage reflections
anywhere along that [uniform] line, although if the line is not lossless
there will be energy lost as the wave progresses.

Slick... On another whole level it simply does not matter which defiinition
of the reflection coefficient one uses to make design calculations though,
as
long as the definition is used consistently throughout any calculations.

One can convert any results based on the non-conjugate version of rho to
results based on the conjugate version of rho and vice versa.

In other words, neither version is "RIGHT" or "WRONG" as long
as the results from using that particular definition are interperted
correctly in terms of the original definition.

In fact one can cook up an [almost] completely ficticious reference
impedance, one which has no relation whatsoever to the Zo of the line.

Just call the ficticious, perhaps complex reference impedance R
[I like to use R since in my mind it stands for "Reference", and it need
not be a pure resistance.] and use this R in a defintition of rho and/or a
Scattering Matrix and then any and all subsequent calculations afterwords
will be correct as long as this ficticious R is used consistently with the
definitions in terms of the electrical port vectors the voltage v, current i
and
wave vector comprising the "waves" a and b.

rho = (Z - R)/(Z + R) = b/a = (v - Ri)/(v + Ri)

This is true simply because the "waves" a and b are mathematically
just simple linear combinations of the voltage v and current i! Look
at how simple the relationship is...

[a, b] and [i, v] are simply related by a simple transformation
matrix as follows:

a = v + Ri
b = v - Ri

or

wave vector = matrix * electrical vector

And the matrix, in this case is:

| 1 R |
| 1 -R|

As long as you choose a reference impedance R such that the transformation
matrix is non-singular then you can go back and forth from a, and b to
v and i any time, any where, etc...

So... who gives a damm about the defintion of rho as long as you are
consistent.

Unless of course, like Roy and others, you insist that the "waves"
correspond to
some preconceived notion of what waves really are... ;-)

Thoughts, comments,
--
Peter K1PO
Indialantic By-the-Sea, FL