In transmission line analysis, we're not free to rescale the forward and
reverse voltage waves, unless we also scale all the voltages, currents
and powers accordingly. The forward and reverse waves have to add to the
total voltage in the line and at its ends, and the ratio of each
component to the corresponding current component has to equal the Z0 of
the line. It's quite apparent that in S parameter analysis you're quite
free to scale them as you wish, as you have. Vendelin et al didn't just
scale them, but chose a set of V+ and V which aren't even related to a
and b by the same constant.
By defining V+ and V as we wish, we can make the reflection coefficient
V/V+ to be zero when there's a Z0 match, when there's a conjugate
match, or when any other impedance of our choice is used as a
termination. And when we relieve the requirement that the sum of V+ and
V add to the total voltage, we can have any value of V+ we choose, when
V is zero. The various analyses I've seen have made different choices,
and arrived at different V+, V, and voltage reflection coefficient values.
Again, though, when dealing with a transmission line we don't have the
luxury of choosing any definitions of V+ and V we want. Consequently,
in a transmission line, the ratio of V/V+, universally defined at the
voltage reflection coefficient, can be calculated with the familiar
nonconjugate formula. The formula can be derived as I did it, from
basic principles. And from it or other methods, we can conclude that
when a transmission line is terminated in its characteristic impedance,
there is no reflection of the voltage (or current) wave. When it's
terminated in the complex conjugate of its characteristic impedance, or
any other impedance except its characteristic impedance, there is a
reflection.
Roy Lewallen, W7EL
Peter O. Brackett wrote:
Roy:
[snip]
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
[snip]
Thanks for following along with my development. I like to develop stuff
from first principles, instead of quoting often questionable sources.
Roy apart from a few typos and a small factor of 1/2 it seems that we agree
on everything!
Sorry if I messed up a little with typos, and... there is that potentially
confusing but unimportant factor of 1/2 that appears in my definitions of
the incident "waves" a and b.
Your claim as to my definitions being quite different from the "mainstream"
of transmission line theory was a bit hasty, because in fact with a closer
look I believe that you will see that my definitions and yours [Which are
hhe mainstream transmission line definitions and equations] only differ by a
simple numerical factor of 2. I don't use that factor of 1/2 because it
drops out whenever you take a ratio of waves anyway.
Perhaps I should have defined my a to be a = 1/2(v + Ri) and b = 1/2(v  Ri)
instead of a = v + Ri and b = v  Ri as I have done. Including that factor
of 1/2 in my a and b makes them identical to your incident and reflected
voltages.
This factor of 1/2 is a very minor "scaling" difference in the "waves" and
there is absolutely no difference in reflection coefficient, or indeed in
Scattering Matrix definitions since the factor of 1/2 in each of "my" wave
definitions simply cancels out in rho = b/a, when you divide b by a to get
rho or indeed in calculating with wave vectors and Scattering Matrixs of any
order. My definition of the scaling factor [1/2] for the wave variables
works as well as any other as long as consistency is maintained. Like you I
have found that different Scattering Theory books often use a variety of
different scaling factors in defining the waves, for instance some use a
factor of 1/2*sqrt(R) = 1/2*sqrt(Zo) in the definition of the waves, etc...
It simply doesn't matter as long as you are consistent.
In any case, the scaling value in the definition of the "waves" was not my
point.
The point I was trying to make was to address the subtle point initiated by
Slick, i.e. the one about the definition of rho which is the ratio of
reflected to incident waves, where the scaling factors drop out, and whether
the CONJUGATE of the reference impedance should be used in the definition
for rho or not.
My point was that the conventional definition of rho = (Z  R)/(Z + R)
without the CONJUGATE is in fact the "natural" definition for wave motion
whether on transmission lines or in impedance matching to a generator.
And...
That with the conventional definition of rho in the case of a general Zo the
reflected voltage will *NOT* be zero at a conjugate match. At a conjugate
match, the classical rho and b will only be null in the special case of the
reference impedance Zo being a pure real resistance.

Peter K1PO
Indialantic BytheSea, FL.
