Walter Maxwell wrote:

Consider my two explanations, or definitions of what I consider a virtual short--perhaps it should have a
different name, because of course 'virtual' implies non-existence. The short circuit evident at the input of
the two line examples I presented---do you agree that short circuits appear at the input of the two lines? If
so, what would you call them?

I'd call them "virtual shorts". If they were short circuits, we should
be able to connect a wire across the transmission line at that point
with no change in transmission line operation. But we can't. While
things will look the same on the generator side, they won't be the same
beyond the real short. So they aren't short circuits.

Roy, I'd like for you to take another, but perhaps closer look at the summarizing of the reflection
coefficients below. I originally typed in the wrong value for the magnitude of the resultant coefficients.
With the corrected magnitudes in place, the two paragraphs following the summarization now make more sense,
because the short circuit established at the stub point leads correctly to the wave action that occurs there.

Summarizing reflection coefficient values at stub point with stub in place:
Line coefficients: voltage 0.5 at +120°, current -60° (y = 1 + j1.1547)
Stub coefficients: voltage 0.5 at -120°, current +60° (y = 1 - j1.1547)
Resultant coefficients: voltage 0.5 at 180°, current 0.5 at 0° WRONG
Resultant coefficients: voltage 1.0 at 180°, current 1.0 at 0° CORRECT

Repeating from my original post for emphasis:
These two resultant reflection coefficients resulting from the interference between the load-reflected wave at
the stub point and the reflected wave produced by the stub define a virtual short circuit established at the
stub point.

There's no need to repeat this. I'm well acquainted with transmission
line phenomena, and understand fully what's happening. I have no
disagreement with this analysis. I would draw attention to the fact that
the "virtual short" is, as you say, simply the superposition
(interference) of traveling waves. So there is nothing at that point
except the traveling waves which pass through that point.

The following paragraph shows how the phases of the reflected waves become in phase with the source waves so
that the reflected waves add directly to the source waves, establishing the forward power, which we know
exceed the source power when the reflected power is re-reflected. The same concept applies to antena tuners.

Sorry, I'm not going to divert onto the topic of propagating power,
either instantaneous or average. If that concept is required in order to
show that waves interact with each other, then it simply shows that the
concept is invalid. Let's stick to voltages and currents. If that's not
adequate, then I'll exit at this point, and turn the discussion over to
Cecil. That's his domain, not mine.

Again repeating for emphasis:
Let's now consider what occurs when a wave encounters a short circuit.

Ok.

We know that the voltage wave
encounters a phase change of 180°, while the current wave encounters zero change in phase. Note that the
resultant voltage is at 180°, so the voltage phase changes to 0° on reflection at the short circuit, and is
now in phase with the source voltage wave. In addition, the resultant current is already at 0°, and because
the current phase does not change on reflection at the short circuit, it remains at 0° and in phase with
source current wave. Consequently, the reflected waves add in phase with the source waves,

Ok so far. . .

thus increasing the
forward power in the line section between the stub and the load.

Again, let's leave power out of it, ok?

Keep in mind that the short at the stub point is a one-way short, diode like, as you say, because in the
forward direction the voltage reflection coefficient rho is 0.0 at 0°, while in the reverse direction, rho at
the stub point is 1.0 at 180°, which is why it's a one-way short.

The voltages, currents, waves, and impedances impedances on the line are
just the same as if there were a diode-short at that point. Which is why
it's a useful analytical tool. But all there really is at that point are
some interfering waves, traveling through that point unhindered.

You say that no total re-reflection occurs at the stub point. However, with a perfect match the power rearward
of the stub is zero, and all the source power goes to the load in the forward direction. Is that not total
reflection?

Not from the "virtual short" -- it only looks like it. The re-reflection
is actually occurring from the end of the stub and from the load, not
from the "virtual short". If, for example, you suddenly increased the
source voltage, there would be no reflection as that change propagated
through that "virtual short". (That is, after a delay equal to the
round-trip time to the "virtual short", you'd see no change.) The
apparent reflection from that point wouldn't appear until the change
propagated to the end of the stub and to the load (going right through
the "virtual short" unhindered), reflected from them, and arrived back
at the "virtual short" point. This is one of the ways you can tell that
a "virtual short" isn't a real short. Under steady state conditions, it
looks just like a real one. But it isn't. Waves which seem to be
reflecting from it are really reflecting from the end of the stub and
from the load -- they're passing right through the "virtual short", in
both directions.

Using the numbers of my bench experiment, assuming a source power of 1 watt, and with the
magnitude rho of 0.04, power going rearward of the stub is 0.0016 w, while the power absorbed by the load is
0.9984 w, the sum of which is 1 w. The SWR seen by the source is 1.083:1, and the return loss in this
experiment is 27.96 dB, while the power lost to the load is 0.0070 dB. From a ham's practical viewpoint the
reflected power is totally re-reflected.

Sorry, you're going to have to do this without propagating waves of
average power, or I'm outta here.

In my example using the 49° stub the capacitive reactance it established at its input is Xc = -57.52 ohms.
Thus its inductive susceptance B = 0.0174 mhos, which cancels the capacitive line susceptance B = -0.0174 mhos
appearing at the stub point.

My point is that the 49° stub can be replaced with a lumped capacitance Xc = -57.52 ohms directly on the line
with the same results as with the stub--with the same reflection coefficients.

That's fine, I agree.

In this case one cannot say
that the re-reflection results from the physical open circuit terminating the stub line.

I most certainly can! And do. I don't see how your example furnishes any
proof or even evidence of wave interaction. I can come to the same
conclusion without any assumption of wave interaction, and you have
agreed (in your response to my question about finding an example that
requires interaction for analysis) that this can always be done.

Various posters have termed my approach as a 'short cut'. I disagree. I prefer to consider it as the wave
analysis to the stub-matching procedure, in contrast to the traditional method of simply saying that the stub
reactance cancels the line reactance at the point on the line where the line resistance R = Zo. In my mind the
wave analysis presents a more detailed view of what's actually happening to the pertinent waves while the
impedance match is being established.

I'm sorry, I disagree. It's a less detailed view, and it conceals what's
really going on.

Roy Lewallen, W7EL