I'm trying to follow this, but have gone astray on the first couple of
steps.

Peter O. Brackett wrote:

The definition of the reflection coefficient is dependent upon what you
define the "reflected voltage" to be.

Consider the classical bridge circuit for measuring reflection coefficients.

If Z is an unknown load presented to a generator of internal impedance R and
that internal impedance R is used as the "reference" impedance to
observe/measure the reflected voltage b and that reflected voltage b is
calculated/measured by observation of the voltage v across the load Z and
the current i through the load Z then the classical definition of a
reflected voltage would be calculated as:

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?

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*((Z-R)/(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

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.

Let's do a consistency check. The voltage at the load should be a + b =
Vs + Vs*((Z-R)/(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:

1. I've goofed up my algebra (a definite possibility)
2. I've misinterpreted your circuit, or
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?

I'd like to continue with the remainder of the analysis, but can't
proceed until this problem is cleared up.

. . .

Roy Lewallen, W7EL