Walter Maxwell wrote:

Thank you, Roy, I appreciate your comments, as always. However, I knew that you have always considered that
virtual opens and shorts cannot cause reflections, and I was hoping my discussion would have persuaded you
otherwise.

So I ask you this: What then causes the total re-reflection at the stub point if not a virtual short circuit?
The re-reflection is real, but there is no physical short circuit at the re-reflection point. The resultant of
the reflection coefficients of both the forward and reflected waves of voltage and current possess the exact
reflection coefficients, 0.5 at 180° for voltage and 0.5 at 0° for current, that are present when the short is
a physical short, except that the magnitude would be 1.0 instead of 0.5. The only operational difference is
that a physical short on the line prevents wave propagation in both directions, while the virtual short is
transparent in the forward direction, but opaque in the reverse direction.

I'd think that this diode-like property of virtual shorts would be a
major clue that they're not real, but a mathematical convenience. The
virtual short is a point where the sum of the voltages of all waves,
forward and reflected, add to zero. If this condition causes waves to
reflect when struck from one direction, what possible physical
explanation could there be for it to do absolutely nothing to waves
traveling the other way?

So I repeat the question: If a virtual short circuit cannot cause reflections, then what causes the reflection
at the stub point?

My answer is this: There is no total re-reflection at the stub point. It
only looks that way.

As you've observed, the waves (traveling in one direction, anyway)
behave just as though there was such a re-reflection. But the waves
actually are reflecting partially or totally from the end of the stub
and other more distant points of impedance discontinuity, not from a
"virtual short". The sum of the forward wave and those reflections add
up to zero at the stub point to create the "virtual short", and to
create waves which look just like they're totally reflecting from the
stub point. This has some parallels to a "virtual ground" at an op amp
input. From the outside world, the point looks just like ground. But it
isn't really. The current you put into that junction isn't going to
ground, but back around to the op amp output. Turn off the op amp and
the "virtual ground" disappears. Likewise, waves arriving at the virtual
short look just like they're reflecting from it. But they aren't.
They're going right on by -- from either direction --, not having any
idea that there's a "virtual short" there -- that is, not having any
idea what the values or sum of other waves are at that point. They go
right on by, reflect from more distant discontinuities, and the sum of
those reflections arrives at the virtual short with the same phase and
amplitude the wave would have if it had actually reflected from the
virtual short. Like with the op amp, you can "turn off" the virtual
short by altering those distant reflection points such as the stub end.

Please let me emphasize again that not I or anyone else who has posted
is disputing the validity of your matching methods or the utility of the
"virtual short" concept. The only disagreement is in the contention that
the "virtual short" actually *effects* reflections rather than being
solely a consequence of them.

Incidentally, there has been mention of 'virtual' reflection coefficients. I can't agree with this
terminology. Reflection coefficients are real, and for every reflection coefficient there is an equivalent
real impedance. As such, it is just as valid to use reflection coefficients in transmission-line analyses as
it is to use correspondingly-equal impedances.

I don't use "virtual reflection coefficient" by name or in concept,
although it might have some utility in the same vein as "virtual short".
However, great care would have to be used, as it must with virtual
shorts, to separate analytical conveniences from reality. But I'll leave
that discussion to others, and don't want it to divert us from the
important point at hand.

How now, Roy?

A question: Do you think you can present an example where a "virtual
short" is necessary to explain the impedances, voltages, and currents --
or any other measurable properties -- on a transmission line? Where a
person who assumes that *no* reflection takes place at "virtual shorts"
but only at physical discontinuities would be unable to arrive at the
correct result? If reflections really do occur at "virtual shorts", I
would think that this phenomenon would have a profound effect on
transmission line operation, to the extent that a valid solution
couldn't be obtained if it were totally ignored. I maintain that such an
example can't be found, because in fact reflection takes place only at
physical discontinuities and not at "virtual shorts". Waves in a linear
medium simply don't reflect from or otherwise affect each other. I'm not
saying that you can't apply the analytical concept of "virtual shorts"
to arrive at the same, valid, result. Or that the "virtual short"
approach won't be easier. But I am saying that it's not necessary in
order to fully analyze any transmission line problem, simply because
it's not real. Can you come up with such an example?

Roy