Walter Maxwell wrote in
:

In the thread 'Constructive Interference and Radiowave Propagation',
Owen, on 4-8-07 asserted that my writings in Reflections concerning
the analysis of stub matching procedures using reflection coefficients
are applicable only in cases where the transmission line is either
lossless, or distortionless. I disagree, and in what follows I hope to
persuade those who agree with Owen's position to reconsider.

Hi Walt,

I did not say that, or in my view, imply that, it is your own
interpretation of what I did say. I did make comment limited to Chapter 3
of Reflections II, and I stand by that comment.

Chapter 3 does not discuss stub matching at all, though you may apply
principles that you develop in Chapter 3 to your discussion / analysis in
later chapters, including to stub matching.

Owen

On Fri, 13 Apr 2007 21:39:22 GMT, Owen Duffy wrote:

Walter Maxwell wrote in
:

In the thread 'Constructive Interference and Radiowave Propagation',
Owen, on 4-8-07 asserted that my writings in Reflections concerning
the analysis of stub matching procedures using reflection coefficients
are applicable only in cases where the transmission line is either
lossless, or distortionless. I disagree, and in what follows I hope to
persuade those who agree with Owen's position to reconsider.

Hi Walt,

I did not say that, or in my view, imply that, it is your own
interpretation of what I did say. I did make comment limited to Chapter 3
of Reflections II, and I stand by that comment.

Chapter 3 does not discuss stub matching at all, though you may apply
principles that you develop in Chapter 3 to your discussion / analysis in
later chapters, including to stub matching.

Owen

Hi Owen,

I'm afraid we both got off the the wrong foot along the way. I'm sorry if I misinterpreted what you said in
the post where we got off track.

Quite possibly the misinterpretation arose in your referencing Chapter 3. When I saw that I assumed you had
made a typo, either for 4 or 23, both of which contain the stub discussions. And I thought I had earlier
referenced Chapter 4. I didn't realize you had actually reviewed Chapter 3 instead of 4.

Perhaps also you missed my two responses to your post of 4-7-07 in the earlier thread, in which I accepted
your apology (not needed). Anyway, the issue where I felt you were wrong is my interpretation that you
believed my statements concerning use of reflection coefficients was wrong because they are applicable for use
in analysis only when the transmission lines are either lossless distortionless.

I hope we can resume on a new footing.

Sincerely,

Walt

Walter Maxwell wrote in
:

On Fri, 13 Apr 2007 21:39:22 GMT, Owen Duffy wrote:

Walter Maxwell wrote in
m:

In the thread 'Constructive Interference and Radiowave Propagation',
Owen, on 4-8-07 asserted that my writings in Reflections concerning
the analysis of stub matching procedures using reflection
coefficients are applicable only in cases where the transmission
line is either lossless, or distortionless. I disagree, and in what
follows I hope to persuade those who agree with Owen's position to
reconsider.

Hi Walt,

I did not say that, or in my view, imply that, it is your own
interpretation of what I did say. I did make comment limited to
Chapter 3 of Reflections II, and I stand by that comment.

Chapter 3 does not discuss stub matching at all, though you may apply
principles that you develop in Chapter 3 to your discussion / analysis
in later chapters, including to stub matching.

Owen

Hi Owen,

I'm afraid we both got off the the wrong foot along the way. I'm sorry
if I misinterpreted what you said in the post where we got off track.

Quite possibly the misinterpretation arose in your referencing Chapter
3. When I saw that I assumed you had made a typo, either for 4 or 23,
both of which contain the stub discussions. And I thought I had
earlier referenced Chapter 4. I didn't realize you had actually
reviewed Chapter 3 instead of 4.

Perhaps also you missed my two responses to your post of 4-7-07 in the
earlier thread, in which I accepted your apology (not needed). Anyway,
the issue where I felt you were wrong is my interpretation that you
believed my statements concerning use of reflection coefficients was
wrong because they are applicable for use in analysis only when the
transmission lines are either lossless distortionless.

I hope we can resume on a new footing.

Walt, the last thing I want to do is to upset you.

You have a considerable investment in your publications, and they are a
great service to the amateur community, and a credit to you.

Much of the discussion isn't so much about what happens in the
transmission line, it is about simplified explanations, explanations that
are appealling to learners, and the extension of those simplified
explanations to the more general case. If you look back over the threads,
you and I have both intiated threads where "explanation" was a key word
in the subject line.

My own view is that whilst analysing a simple case that can be seen as
special cases is a good way of introducing the issue that is to be dealt
with (eg showing the inconsistency of the Vf/If=Zo constraint in the
initial wave that travels along a transmission line, and the V/I ratio a
s/c or o/c load), one needs to move on to dealing with the more general
load case, even if in a simplified context (eg lossless line). The
"rules" that are derived have to be clearly qualified with the applicable
limits.

To overemphasis the simple / easy cases and downplay the error of
approximation is at risk of consigning all problem solving to
simplification to a trivial case and applying the solution of that
trivial case to the real problem without appreciation of the leap that
might entail.

Whilst it is no doubt appealing to some to see a virtual s/c or virtual
o/c as an explanation for the single stub tuner example, and it might be
a suitable model for that purpose, it gives the learner a new analysis
tool (without limitations), the virtual s/c or o/c. How perfect does a
virtual s/c need to be to be approximately effective? If I have an
approximately lossless feedline with a VSWR of 100:1, will the virtual
s/c at a current maximum prevent energy propagating in the same way as
the virtual s/c in the stub explanation, or could each virtual circuit
choke of x% of the energy flow? Can you solve a two stub tuner using only
virtual s/c or o/c?

It is a challenge to devise simplified explanations that don't contain
errors that need to be un-learned to develop further. I hate to say to a
learner "throw away what you already know about this, because the
explanation you have learned, understood, and trusted is wrong in part,
and we need to discard it before we move on to a better understanding...
trust me...".

Owen

I have to agree with what Richard and some others have said.

First, that you've done a tremendous job of sharing your extensive
knowledge and experience, and explaining transmission line phenomena in
such a clear and understandable manner. We all owe you a great debt for
this.

But second, that there's something which you do state that I and some
others can't accept. And that is that a "virtual" short (or open)
circuit causes reflections, or that waves reflect from it. I maintain
that for either to happen requires that traveling waves interact with
each other. The "virtual" short or open is only the result of the sum --
superposition -- of traveling waves. Those traveling waves, and hence
their sum, cannot cause a reflection of other waves, or alter those
other waves in any way. Only a physical change in the (assumed linear)
propagating medium can alter the fields in a traveling wave and cause a
reflection. A real short circuit is in this category; a virtual short
circuit is not. It doesn't matter if the waves are coherent or not, or
even what their waveshapes are or whether or not they're periodic -- as
long as the medium is linear, the waves cannot interact.

You have clearly shown, and there is no doubt, that waves behave *just
as though* a virtual short or open circuit were a real one, and this is
certainly a valuable insight and very useful analysis tool, just like
the "virtual ground" at the summing junction of an op amp. But I feel
it's very important to separate analytical tools and concepts from
physical reality. If we don't, we're led deeper and deeper into the
virtual world. Sooner or later, we reach conclusions which are plainly
wrong.

There are many other examples of useful alternative ways of looking at
things, for example differential and common mode currents in place of
the reality of two individual currents, or replacing the actual
exponentially depth-decaying RF current in a conductor with an imaginary
one which is uniform down to the skin depth and zero below. But we have
to always keep in mind that these are merely mathematical tools and that
they don't really correspond to the physical reality.

Unless I've incorrectly read what you've written, you're saying that
you've proved that virtual shorts and opens reflect waves. But in every
example you can present, it can be shown that all waves and reflections
in the system can be explained solely by reflections from real impedance
changes, and without considering or even noticing those points at which
the waves superpose to become virtual short or open circuits. That, I
believe, would disprove the conjecture that virtual shorts or opens
cause reflections. Can you present any example which does require
virtual shorts or opens to explain the wave behavior in either a
transient or steady state condition?

If I've misinterpreted what you've said, I share that misinterpretation
with some of the others who have commented here. And if that's the case,
I respectfully suggest that you review what you've written and see how
it could be reworded to reduce the misunderstanding.

Once again, we all owe you a great deal of thanks for all you've done.
And personally, I owe you thanks for many other things, including
setting such an example of courtesy, civility and professionalism here
in this group (as well in everything else you touch). It's one I strive
for, but continually fall far short of.

Roy Lewallen, W7EL

On Fri, 13 Apr 2007 15:01:19 -0700, Roy Lewallen wrote:

I have to agree with what Richard and some others have said.

First, that you've done a tremendous job of sharing your extensive
knowledge and experience, and explaining transmission line phenomena in
such a clear and understandable manner. We all owe you a great debt for
this.

But second, that there's something which you do state that I and some
others can't accept. And that is that a "virtual" short (or open)
circuit causes reflections, or that waves reflect from it. I maintain
that for either to happen requires that traveling waves interact with
each other. The "virtual" short or open is only the result of the sum --
superposition -- of traveling waves. Those traveling waves, and hence
their sum, cannot cause a reflection of other waves, or alter those
other waves in any way. Only a physical change in the (assumed linear)
propagating medium can alter the fields in a traveling wave and cause a
reflection. A real short circuit is in this category; a virtual short
circuit is not. It doesn't matter if the waves are coherent or not, or
even what their waveshapes are or whether or not they're periodic -- as
long as the medium is linear, the waves cannot interact.

You have clearly shown, and there is no doubt, that waves behave *just
as though* a virtual short or open circuit were a real one, and this is
certainly a valuable insight and very useful analysis tool, just like
it's very important to separate analytical tools and concepts from
physical reality. If we don't, we're led deeper and deeper into the
virtual world. Sooner or later, we reach conclusions which are plainly
the "virtual ground" at the summing junction of an op amp. But I feel
wrong.

There are many other examples of useful alternative ways of looking at
things, for example differential and common mode currents in place of
the reality of two individual currents, or replacing the actual
exponentially depth-decaying RF current in a conductor with an imaginary
one which is uniform down to the skin depth and zero below. But we have
to always keep in mind that these are merely mathematical tools and that
they don't really correspond to the physical reality.

Unless I've incorrectly read what you've written, you're saying that
you've proved that virtual shorts and opens reflect waves. But in every
example you can present, it can be shown that all waves and reflections
in the system can be explained solely by reflections from real impedance
changes, and without considering or even noticing those points at which
the waves superpose to become virtual short or open circuits. That, I
believe, would disprove the conjecture that virtual shorts or opens
cause reflections. Can you present any example which does require
virtual shorts or opens to explain the wave behavior in either a
transient or steady state condition?

If I've misinterpreted what you've said, I share that misinterpretation
with some of the others who have commented here. And if that's the case,
I respectfully suggest that you review what you've written and see how
it could be reworded to reduce the misunderstanding.

Once again, we all owe you a great deal of thanks for all you've done.
And personally, I owe you thanks for many other things, including
setting such an example of courtesy, civility and professionalism here
in this group (as well in everything else you touch). It's one I strive
for, but continually fall far short of.

Roy Lewallen, W7EL

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.

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

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.

How now, Roy?

Walt

Walt

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

Roy Lewallen wrote:

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.


The key word there is "utility" - the virtual short/open concept is
*useful* as a short-cut in our thinking. But concepts are only useful if
they help us to think more clearly about physical reality; and
short-cuts are dangerous if they don't reliably bring us back onto the
main track.

We know that in reality both the forward and the reflected waves take a
side-trip off the main line into the stub, and from the far end of the
stub they are reflected back to rejoin the main line at the junction.
Since an open- or short-circuited stub has a predictable effect at the
junction where it is connected, then we could save a little time by
noting that a stub is present, and simply assuming what its effect will
be.

Within those limitations, I don't have any particular problem about
calling the effect a "virtual short" or "virtual open". As Richard
said, it is only a metaphor. We are using the word "virtual" as a label
to remind ourselves that the effect at the junction is not the same as a
genuine physical short or open circuit on the main line.

Where the concept goes off track is if anyone forgets about the
limitations, and begins to believe that a metaphor has physical
properties of its own. (It doesn't, of course - all of the physical
effects on the main line are caused by the stub, and the stub is the
only place where the root causes can be found.)

If there is any problem in using a short-cut, then simply forget it -
step back and analyse the complete physical system including the stub.


Walt said:
Incidentally, there has been mention of 'virtual' reflection coefficients. I can't agree with this
terminology.
Roy replied:
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".

Agreed. It all comes back to "usefulness" or "utility" again. As I said,
concepts are only useful if they help us to think more clearly about
physical reality - and "virtual reflection coefficient" has exactly the
opposite effect.



--

73 from Ian GM3SEK

Ian White GM3SEK wrote:
Agreed. It all comes back to "usefulness" or "utility" again. As I said,
concepts are only useful if they help us to think more clearly about
physical reality - and "virtual reflection coefficient" has exactly the
opposite effect.

Also, please note that in an S-Parameter analysis, all
reflection coefficients are physical, not virtual.
Since I may have used the term first here, let me
explain what I meant by it. a1, b1, a2, and b2 are
the S-Parameter normalized voltages. Below, a1=10,
b1=0, b2=14.14, and a2=10. s11 is the physical reflection
coefficient encountered by forward wave a1. s11 is
(291.4-50)/(291.4+50) = 0.707. In an S-Parameter,
the reflection coefficient is NOT the ratio of b1/a1.

a1-- b2--
--b1 --a2
100w---50 ohm line---+---1/2WL 291.4 ohm line---50 ohm load
Vfor1=100V-- Vfor2=241.4V--
--Vref1=0V --Vref2=170.7V

Given the actual voltages, someone might say the reflection
coefficient is Vref1/Vfor1 = 0. That is a virtual reflection
coefficient. The physical reflection coefficient at point '+'
remains at 0.707. Vfor1 sees a virtual impedance of 50 ohms
at point '+' during steady-state because of the wave cancellation
that results in a net Vref1=0. But the physical reflection
coefficient doesn't change from power-up through steady-state.
One has to be careful to specify whether the physical rho,
(Z02-Z01)/(Z02+Z01), is being used or whether the virtual
rho, Vref1/Vfor1, is being used. One advantage of an S-
Parameter analysis is that virtual reflection coefficients
are not used and all reflection coefficients are physical.
--
73, Cecil http://www.w5dxp.com

On Apr 13, 11:49 pm, Ian White GM3SEK wrote:
Roy Lewallen wrote:

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.

The key word there is "utility" - the virtual short/open concept is
*useful* as a short-cut in our thinking. But concepts are only useful if
they help us to think more clearly about physical reality; and
short-cuts are dangerous if they don't reliably bring us back onto the
main track.
....

Indeed. I was thinking about this in terms of short-cuts before
reading Ian's post. What if you take a short-cut and it just takes
you off into the woods? I'm not sure my posting about this made it
into the thread in an intelligible way. (I fear Google may have sent
it off on a "short-cut.")

The gist of it was that, although there are examples where considering
points an even number of half-waves from a short as being shorts
themselves work fine, there are plenty of counter examples too. I
fear that people new to the use of stubs will be lulled into a false
sense of security using that concept, when indeed it fails miserably
at times. Especially in this age of computers and readily available
programs to deal with lines, INCLUDING their loss, why would I use a
concept that may take me on a short-cut that turns out to be the long
way around?

What IS useful to me about the concept is NOT the calculation of the
performance of a particular network of stubs, but rather in coming up
with the trial design to test with full calculations. My example was
the use of two stubs to give me a null on one frequency and pass
another frequency; I can get a null by putting a "virtual short" at
that frequency, and that's a line that's a half wave long on that
frequency, shorted at the other end. But on a slightly lower
frequency, it looks capacitive, so I can put another stub that's
inductive in parallel with it to create an open circuit at the
frequency I want to let pass. THEN I pull out the calculations with
line attenuation included, and discover that in some situations it
works fine, and in others, the performance is terrible.

It's a useful visualization tool and design aid; it's a poor analysis
tool at best. At worst, it will lull you into building something that
just won't work, wasting time and resources.

Cheers,
Tom

On Fri, 13 Apr 2007 19:10:18 -0700, Roy Lewallen wrote:

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.
snip

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.
snip
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
Hi R oy,

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?

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.

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.

Again repeating for emphasis:
Let's now consider what occurs when a wave encounters a short circuit. 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, thus increasing the
forward power in the line section between the stub and the load.

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.

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? 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.

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

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.

Walt