Thanks, it's becoming clear why we're getting different results.
When analyzing a transmission line, the forward and reflected voltages
have a meaning that comes from the solutions to the wave equations. One
of the fundamental properties of these waves is that their sum is the
total voltage in the transmission line. (Likewise for current.) We
assume that these are the only voltage waves on the line, so their sum
is necessarily the total voltage. We sum these waves, for example, and
look at the resulting maxima and minima to get the standing wave ratio.
So it was a basic tenet of my analysis that the sum of the forward and
reverse voltage waves at any point on the line (including the end)
equals the voltage at that point. (Again, likewise for current, with
attention paid to the defined direction of positive flow.) This was
explicitly given as equation 5 (for voltage) and 6 (for current) of my
analysis.
Another property of the waves is that the ratio of V to I of either of
the waves equals the Z0 of the cable (equations 1 and 2).
Again making sure I see your model circuit correctly, it's a voltage
source of voltage a or V+, connected to R in series with Z. If that's
correct, then i = a / (Z + R), so your calculated value of b in your
analysis below, (Z  R)i, then equals a * (Z  R) / (Z + R). The sum of
a and b is a * (1 + (Z  R) / (Z + R)) = 2 * Z * a / (Z + R).
While in or at the ends of a transmission line the voltage always equals
the sum of the forward and reflected voltages, the sum of a and b, which
you call forward and reflected voltages, don't sum to the voltage at the
load. The voltage at the load is, from inspection of the circuit, a * Z
/ (Z + R); the sum of a and b is twice that value.
Of course, if we define "forward voltage" and "reflected voltage" to be
something other than the waves we're familiar with on a transmission
line, while maintaining a definition of voltage reflection coefficient
as being the ratio of forward to reverse voltage, we can come up with
any number of formulas for reflection coefficient.
And this seems to be done. I have only two texts which deal with S
parameters in any depth. One, _Microwave Transistor Amplifiers: Analysis
and Design_ By Guillermo Gonzalez, consistently uses forward and reverse
voltage to mean exactly what they do in transmission line analysis.
Consequently, he consistently ends up with the same equation for voltage
reflection coefficient I've been using, and states several places that
the reflection is zero when the line or port is terminated in its
characteristic or source impedance (not conjugate). And this all without
an assumption that Z0 or source Z is purely real. The other book,
however, _Microwave Circuit Design Using Linear and Nonlinear
Techniques_ by Vendelin, Pavio, and Rohde, uses a different definition
of V+, V than either of us does, and different a and b than you do. To
them, a = V+ * sqrt(Re(Zg)) / Zg* where Zg is the source impedance, and
b = V * sqrt(Re(Zg))/Zg. They end up with three different reflection
coefficients, Gammav, Gammai, and one they just give as Gamma. Gammav is
V/V+, Gammai is I/I+, and plain old Gamma, which they say is equal to
b/a, turns out to be equal to Gammai. Incidentally, their equation for
Gammav, the voltage reflection coefficient, is:
Gammav = [Zg(Z  Zg*)]/[Zg*(Z + Zg)]
Which is different from yours, Slick's, and mine.
The whole trick seems to be in defining what forward and reflected or
reverse voltage mean. In transmission line analysis, the meaning is, I
hope, pretty universal and agreed upon. If not, a whole bunch of
equations, statements, assumptions, and definitions have to go out the
window, along with the idea that those are the only voltages on the
line. If you use those widely accepted meanings, you unavoidably end up
with the common equation I've been giving for voltage reflection
coefficient. (At least nobody reputable so far seems to want to define
voltage reflection coefficient as anything but the ratio of reflected to
forward voltage, thankfully.) On the other hand, you and the authors of
at least one book put a different meaning of forward and reflected
voltage (and you two use different meanings), and therefore come up with
correspondingly different formulas for reflection coefficient.
At this point I have to concede that in some S parameter analysis at
least, various different meanings are given to "forward" or "incident",
and "reverse" or "reflected" voltage than are used in transmission line
analysis. And the two books I have disagree with each other, and both
disagree with you, as to what they do mean. With that kind of
nonstandardization, it's useless to argue which is "more right" than
another. It just points out the importance of carefully defining what
you mean by forward and reverse voltages before you begin your analysis
 and being very careful about what conclusions you draw from the
"reflections" or lack of them. When the definitions are different from
those used with transmission lines, the meaning and consequences of
reflections and impedance match are correspondingly different.
Your analysis is self consistent, given your definitions of forward and
reverse voltage. So are the analyses in the two books I have dealing
with S parameters. So all three of you reach different but valid
conclusions. Unless someone comes up with a convincing argument that one
definition of forward and reverse voltages is better or more meaningful
than another in that context, I have to agree that any of these three,
or any of an infinite number of other possibilities, is equally valid
for S parameter analysis.
But not for transmission lines. There, forward and reverse voltages do
have real meaning and a rigorous derivation. So anyone redefining them
in that context is certainly deviating from very well understood usage.
Consequently, I don't see how it would be productive, or add insight to
the problem of voltages, currents, and reflections in a transmission
line, to bring in S parameter terminology which obviously differs and in
an inconsistent way from author to author. I'll be glad to continue
discussing transmission lines, which is what this discussion originally
involved, with the first step being for someone disagreeing with the
formula for voltage reflection coefficient to point out which of the
equations or assumptions preceding it are incorrect. Your analysis
disagrees with equations 5 and 6 and, I believe, 1 and 2, by virtue of
your different definition of forward and reverse voltage. I don't
believe this can be justified within and at the end of a transmission
line  if the forward and reverse voltages on the line don't add up to
the total, then you have to come up with at least one more voltage wave
you assume is traveling along the line (which summed with the others
does equal the total), and justify its existence.
Roy Lewallen, W7EL
Peter O. Brackett wrote:
Roy:
[snip]
b = v  Ri = Zi  Ri = (Z  R)i Volts.
First of all, you're speaking of a circuit with a source impedance R and
load impedance Z, rather than a terminated transmission line. Forward
and reflected wave terminology is widely used in S parameter analysis,
which also uses this model, so I'll be glad to follow along to see if
and how S parameter terminology differs from the transmission line
terminology we've been discussing so far. Please correct me where my
assumptions diverge from yours.
Your "classical definition" of b isn't one familiar to me. v + Ri would
of course be the source voltage (which I'll call Vs). So v  Ri is Vs 
2*Ri. Where does this come from and what does it mean?
[snip]
Yes it should be familiar to you because it is the most common definition
and one you seem to agree with. I presume that you are not used to seeing
the use
of the symbols "a" and "b" for those quantities. The use of "a" and "b" is
widely
used in Scattering Formalism and is less confusing to many than using
subscripts.
In your terminology above the "Vs" symbol is nothing more than the incident
voltage
usually given the symbol "a" in the Scattering Formalism, or the symbol V
with a "+"
sign subscript in many developments. Often you will find authors use a V
with a plus
sign "+" as a subscript to indicate the "a" voltage and a V with a minus
sign "" subscript
to idicate the "b" voltage. Personally I find the use of math symbols ""
and "+" or other
subscripts to variables to be confusing, I much prefer the use of "a" and
"b" for forward
or incident and reflected voltages.
Simply put, if a generator with "open circuit" voltage "a" and "internal
impedance" R
is driving a load Z [Z could be a transmission line driving point impedance,
for instance
Z would be the characteristic or surge impedance Zo of a transmission line
if
the generator was driving a semiinfinite line.] then v is the voltage drop
across Z
and I is the current through Z, and so...
a = v + Ri = Zi  Ri = (Z  R)i
is simply the [usual] forward voltage or incident voltage applied by the
generator to the
to the load Z, which may be a lumped element load or if you prefer to talk
about transmission
lines, Z can be just the driving point impedance of a transmsision line,
whatever you wish.
Then its'just a simple application of Ohms Law tosee that b = v  Ri = Zi 
Ri = (Z  R)i is
the [usual] reflected voltage. b is just the difference between the voltage
across Z which is
calculated as Zi and the voltage that would be across Z if Z was actually
equal to R. i.e. the
reflected voltage b is just the voltage that would exist across Z if there
was an "image match"
between Z and R. [If Z is the Zo of a semiinfinite transmsision line you
could call this a Zo match].
Taking the ratio of "b" to "a" just yeilds the [usual] reflection
coefficient as
b/a = (Z  R)i/(Z + R)i = (Z  R)/(Z + R). A well known result. Simple?
[snip]
From your equation, and given source voltage Vs, i = Vs/(R+Z).
Therefore, your "classical definition" of reflected voltage b is, in
terms of Vs, Vs*((ZR)/(Z+R)).
and the incident voltage a would be the Thevinins equivalent voltage
across
the sum of Z and R, i.e.
a = (Z + R)i
[snip]
Yep you got it all right!
[snip]
Since i = Vs(Z+R), you're saying that a = the source voltage Vs (from
your two equations). So what you're calling the "incident voltage" is
simply the source voltage Vs.
[snip]
Yes, mathematically "a" = "Vs", what else would it be? Nothing mysterious
about
that. The incident voltage is always simply the open circuit voltage of the
source. In
words a is not the source voltage because the source is a Thevinin
equivalent made up
of the ideal voltage generator Vs = a behind the "internal" source impedance
R. A better
way to describe Vs = a in words would be the incident voltage a is the "open
circuit source
voltage".
[snip]
Let's do a consistency check. The voltage at the load should be a + b =
Vs + Vs*((ZR)/(Z+R)) = Vs*2Z/(Z+R). Inspection of the circuit as I
understand it shows that the voltage at the load should be half this
value. So, we already diverge. Which is true:
[snip]
No, the voltage at the load is not (a + b) rather it is [the quite obvious
by Ohms Law] v = Zi.
and the sum of the incident and reflected voltage is simply
a + b = (v + Ri) + (v  Ri) = 2v = 2Zi
Now if there is an "image match" and the "unknown" Z is actually equal to R,
i.e. let Z = R
in all of the above, then...
a = Vs
b = 0
a + b = 2Ri
and i = Vs/2R = a/2R.
[snip]
1. I've goofed up my algebra (a definite possibility)
[snip]
Only a little :)
[snip]
2. I've misinterpreted your circuit, or
[snip]
No you have it correct!
[snip]
3. The voltage at the load is not equal to the sum of the forward and
reflected voltages a and b, as you use the terms "forward voltage" and
"reflected voltage". If v isn't equal to a + b, then what is the
relationship between v, a, and b, and what are the physical meanings of
the forward and reflected voltages?
[snip]
I showed those relationships above. There is nothing new here... these are
the [most] widely accepted definitions of incident and reflected voltages.
[snip]
I'd like to continue with the remainder of the analysis, but can't
proceed until this problem is cleared up.
Roy Lewallen, W7EL
[snip]
OK, let's carry on.

Peter K1PO
Indialantic BytheSea, FL.
