There's nothing wrong with the formula or the context. It follows from a
straightforward derivation that begins with the ratio of reflected to
forward waves at the load, and results in satisfying the boundary
conditions and Kerchoff's voltage and current laws at the load. It holds
for any complex values of Zl and Z0. The resulting reflection
coefficient is of course complex, but it's often confused with its
magnitude or with the time domain reflection coefficient. Increasing
this confusion is that there's no standard notation for these terms, so
the complex value in one text might be denoted by the same character as
the magnitude in another text.

There's no problem with the reflection coefficient having any angle in
any of the four quadrants. However, I've frankly had trouble getting
around the notion of the magnitude of the reflection coefficient being
greater than one with a passive load. I know it sometimes happens with
active loads, and there are even Smith chart techniques to deal with it.
You'll find discussions of it in texts on microwave circuit design.

It looks like it's possible to get a reflection coefficient with
magnitude 1 any time Rl*R0 -X*X0. Reg recently posted some values of
Z0 for common coaxial cables that show the angle of Z0 approaching -45
degrees at low frequency. So it wouldn't be hard to envision a cable
with Z0 = 100 - j100 or thereabouts at some very low frequency. If we
were to terminate it with a pure inductor with 100 ohms reactance (Zl =
0 + j100), it looks like the reflection coefficient would be -1 + j2,
which has a magnitude of the square root of three, or about 1.73. What
does this mean? It means that the reflected wave has a greater magnitude
than the incident wave. I'm not sure there's anything wrong with this --
it's sort of like a resonant effect. It would have to be checked to make
sure that the law of conservation of energy isn't violated, and that
Kirchoff's laws are satisfied, but I'd be surprised if there were any
violations. The calculated SWR is negative, but that's pretty
meaningless considering we have a line with a huge amount of attenuation
per wavelength (in order to have such a highly reactive Z0). With that
kind of attenuation there's no danger of having an oscillator with no
power source.

I'd really like to hear from some of the folks who deal more frequently
with reflection coefficient than I do, to see if I'm on the right track,
or if there is some consideration that requires modification of the
equation for very lossy lines. I've got quite a few references that deal
with reflection coefficient. They all give the same formula without
qualifications, but none mentions the possibility of the magnitude
becoming greater than one.

Reg, you've got more experience with very lossy lines (in terms of loss
per wavelength, which is what counts here) than anyone else on this
group. What happens at the load if you terminate a 100 - j100 ohm Z0
line with 0 + j100 ohms?

Roy Lewallen, W7EL

Reg Edwards wrote:
"Dr. Slick" wrote

From Pozar's Microwave Engineering (Pg. 606):

Reflection Coefficient looking into load = (Zl-Zo)/(Zl+Zo)

Where Zl is a purely real load impedance, and Zo is the
purely real characteristic impedance reference.


================================

Either Pozar (who I've never heard of) is not quite correct or Dr Slick has
misquoted him or taken him out of context.

In fact, the equation is also true for complex values of Zl and Zo. The
angle of RC can lie in any of the 4 quadrants. Furthermore, the magnitude of
RC can exceed unity.

I offer no references in support of this statement. It is issued here
entirely on my own responsibility.
----
Reg, G4FGQ