Roy Lewallen wrote in message ...

A. The one just posted by Peter, (Zl - Z0conj) / (Zl + Z0conj)


I believe this was a typo on Peter's part, as was my typo in the
original post.


B. Slick's, (Zl - Z0conj) / (Zl + Z0)



This is the correct formula.


C. The one in all my texts and used by practicing engineers, (Zl - Z0) /
(Zl + Z0)



This is correct too, but Zo must be purely real.




What about the seemingly sound logic that the accepted formula doesn't
work for complex Z0 because it implies that a conjugate match results in
a reflection? The formula certainly does imply that. And it's a fact --
a conjugate match guarantees a maximum transfer of power for a given
source impedance. But it doesn't guarantee that there will be no
reflection. We're used to seeing the two conditions coincide, but that's
just because we're used to dealing with a resistive Z0, or at least one
that's close enough to resistive that it's a good approximation. The
fact that the conditions for zero reflection and for maximum power
transfer are different is well known to people accustomed to dealing
with transmission lines with complex Z0.


Wrong. But at least you admit that the "accepted" formula (which
is fine for purely real Zo) implies a reflection.

The absence of reflection is what makes the maximum power
transfer.



But doesn't having a reflection mean that some power is reflected and
doesn't reach the load, reducing the load power from its maximum
possible value? As you might know from my postings, I'm very hesitant to
deal with power "waves". But what's commonly called forward power
doesn't stay constant as the load impedance is changed, nor does the
forward voltage. So it turns out that if you adjust the load for a
conjugate match, there is indeed reflected voltage, and "reflected
power". But the forward voltage and power are greater when the load is
Z0conj than when Zl = Z0 and no reflection takes place -- enough greater
that maximum power transfer occurs for the conjugate match, with a
reflection present.


?? From a theory point of view, when you cancel series reactances
(canceling inductive with capacitive) the series inductor and
capacitor are
resonant, and will thoeretically have zero impedance, allowing the 50
ohms to feed 50 ohms for max power transfer, WITH THEORETICALLY NO
REFLECTIONS.



I'd welcome any corrections to any statements I've made above, any of
the equations, or the calculations. The calculations are particularly
subject to possible error, so should undergo particular scrutiny. I'll
be glad to correct any errors. Anyone who disagrees with the conclusion
is invited and encouraged to present a similar development, showing the
derivation of the alternate formula and giving numerical results from an
example. That's how science, and good engineering, are done. And what it
takes to convince me.

Roy Lewallen, W7EL


It's hard to convince anyone who could never admit that they were
wrong.

Slick

You ARE talking about ME, aren't you Tom, when you say "fine
engineer, with a good education and career"?


Nope. Most engineers don't write like an attitudinous high-school
kid. If you study hard, go to the right schools, get the right job, etc.
you might actually become an engineer when you grow up, Slick.
Arguing with people who know more than you do on the net won't
help you, though.
73,
Tom Donaly, KA6RUH

Slick:

[snip]
And not the complex conjugate of Z0.
:
:
This is ABSOLUTELY WRONG!

The reflection coefficient is zero only when the Zload
is the conjugate of the Zo.

Go look it up in any BASIC RF book!

Slick
[snip]

Easy now boy! You'r almost as bad as me!

It is entirely possible, in fact I know this to be true, that there can be
more than one *definition* of "the reflection coefficient". And so... one
cannot say definitively that one particular defintion is WRONG.

If the definition of the reflection coefficient is given as rho = (Z - R)/(Z
+ R) then that's what it is. This particular definition corresponds to the
situation which results in rho being null when the unknown Z is equal to the
reference impedance R, i.e. an "image match". If the definition is given as
rho = (Z - conj(R))/(Z + conj(R)) then rho will be null when the unknown Z
is equal to the conjugate of the reference impedance conj(R), i.e. a
"conjugate match".

Nothing is WRONG if the definition is first set up to correspond to what the
definer is trying to accomplish. And so one has got to take care when
making statements about RIGHT ways and WRONG ways to define things.
Everyone is entitled to go to hell their own way if they are the onesmaking
the definitions. Just as long as no incorrect conclusions are drawn from
the definitions. That may occur when folks don't accept or agree on a
definition.

OTOH....

Definitions and semantics aside, what we should really be interested in is
what is the physical meaning of any particular definition and what are its'
practical uses.

Clearly if R is a real constant resistance and contains no reactance for all
frequencies [R = r + j0] then the two definitions are equivalent i.e. rho =
(Z - R)/(Z + R) = (Z - conj(R))/(Z + conj(R)) since R = conj(R). This is
the situation for most common amateur radio transmsission line problems and
so in these simple cases it clearly doesn't matter which definition one
takes. But the question of definitions for rho is even broader than that.
We amateurs usually only examine a very small class of problems, and there
are many more and usually much more interesting and challenging problems
that require the use of reflection [scattering] parameters.

Now for broadband problems where the reference impedance R is in general a
complex function of frequency, e.g.
R(w) = r(w) + jx(w), one is faced with the problem of creating a definition
for rho which will be practical and useful and easy to measure...

For an example of a practical consideration, with R a constant resistance it
is easy to manufacture wide ranging reflectometers, like the Bird Model 43
since all it needs inside is a replica of the R, simply a the equivalent of
a garden variety 50 Ohm resistor. But if the reference impedance R needs to
be a complex function of frequency it is not so easy to design an instrument
to measure the reflection coefficient over a broad band. In fact if the
reference impedance R(w) = r(w) + jx(w) corresponds to the driving point
impedance of a real physical system such as 18,000 feet of telephone line
operated over DC to 50MHz, then it can be proved using network synthesis
theory that one cannot exactly physically create the conjugate R(w) = r(w) -
jx(w) for such a line. How then to make a satisfactory reflectometer for
this application? To synthesize the reference impedance for such a "broad
band Bird" one would have to be approximated over some narrow band, etc...
Not an easy problem...

Actually, under the general so-called Scattering Formalism, the reference
impedance can be chosen arbitrarily, and often is, to make the particular
physical problem being addressed easy to solve. Within the general
Scattering Formalism the so-called port "wave variables" a and b [a is the
incident wave and b is the reflected wave] are nothing more than linear
combinations of the port "electrical variables" v and i [v is the port
voltage and i is the port current]. Thus each port on a network has an
electrical vector [v, i]' and a wave vector [a, b]' and these two vectors
are related to each other by a simple linear transformation matrix made up
of the sometimes arbitrarily chosen reference impedance(s).

For example for the "normal" case we are all used to where r is a fixed
constant then... [a, b]' = M [v, i]' where M is the matrix of the
transformation. Specifically...

b = v - ri
a = v + ri

and the 2x2 matrix M relating the "waves" to the "electricals" has the first
row [1, -r] and second row [1, r], i.e. M is equal to:

|1 -r |
|1 +r|

It is easy to show with simple algebra that this definition of the relation
M betweent the waves and the electricals yeilds the common defiintion of the
reflection coefficient rho = b/a = (Z - r)/(Z + r).

The way linear algebraists view this is that the vector of waves a and b is
just the vector of electricals rotated and stretched a bit!

In other words the waves are just another way of looking at the electricals.

Or... the waves and the electricals are just different manifestations of the
same things, their specific numerical values depend only upon your
viewpoint, i.e. what kind of measuring instruements you are using, i.e.
voltmeters and ammeters or reflectometers with a particularly chosen
reference impedance.

All that said, it should be clear that one can arbitrarily chose the matrix
transformation [reference impedance] which relates the waves to the
electricals to give you the kinds of wave variables that makes your
particular physical problem easy to solve. i.e. it dictates the kind of
reflectometer you must use to make the measurements. The Bird Model 43 is
only one such instrument and it is useful only for one particular and common
kind of narrow band set of problems. For broad band problems one needs an
entirely different set of definitions, etc...

And so...

In transmission line problems it us usual to choose the characterisitic
impedance Zo of the transmission media to be the reference impedance for the
system under examination, but that is certainly not necessary, only
convenient. And... if you want a null rho to correspond to a "conjugate
match" you must choose the reference impdance in your reflectometer to be
the conjugate of the reference impedance of the system under examination,
and if you want a null rho to correspond to an "image match" then you must
choose the reference impedance in your reflectometer to be identical to the
reference impedance of the system under examination.

Every one is entitled to go to hell their own way when defining the wave to
electrical variable transformations required to make their measurements and
solve their problems and this will result in a variety of definitions for
the scattering [reflection] parameters. Nothing more nothing less. Others
may not agree with your tools, methodologies and definitions, but just be
careful to follow through and be consistent with your definitions,
measurements, algebra, and arithmetic and you will always get the right
answers.

Thoughts, comments?

--
Peter K1PO
Indialantic By-the-Sea, FL.

Yes indeed. I hope no one has interpreted all this as meaning that I
believe it has any direct relevance to typical amateur antenna
applications. It doesn't. As Bill and quite a few others have stated,
the output Z of the PA isn't important at all for our applications. And
for nearly any calculation you care to do at HF, the assumption that Z0
is purely real is entirely adequate. The precise Z0 might possibly be
important if very precise measurements are being made, but that's not
something done by most amateurs.

But there was information posted that's incorrect, even if it's not
directly relevant to most of us, and that's what prompted my posting in
response.

Roy Lewallen, W7EL

William E. Sabin wrote:

Roger to that. In the special case of conjugate matching generator to
load, via a Z0 line, if we know the generator impedance we can do that.
But for PAs the generator impedance is "who knows what?" so the best we
can do is make the load equal to the complex Z0. Then forward power is
all there is and reflected power is zero. My Bird meter then tells me
that the calculated VSWR is 1.0:1.0. which is what my PA is designed for.

If my coax gets so lossy that I have to worry about stuff like this, I
will buy new (better) coax.

Bill W0IYH

I apologize if it sounded like my analysis was original. I had assumed,
apparently mistakenly, that readers realized it was simply a statement
of very well known principles, and had no intention whatsoever to claim
or imply originality. I did mention in a followup posting that a similar
analysis can be found in many texts.

Please amend my posting from:

"I agree entirely, and it follows from my analysis and my conclusion."

to

"I agree entirely, and it follows from the analysis and conclusion I
posted."

I do take credit for posting it on this newsgroup, something neither Reg
nor anyone else has, to my knowledge, taken the trouble to do.

Roy Lewallen, W7EL

Reg Edwards wrote:
Roy Sez -

That's fine. I agree entirely, and it follows from my analysis and my
conclusion. A similar analysis can be found in many texts. My offering
to provide a large number of references has brought forth no interest
from the most vocal participants, and they've also showed a lack of
willingness to work through the simple math themselves. So I felt that
it might be a good idea to post the derivation before more converts are
made to this religion of proof-by-gut-feel-and-flawed-logic.


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

YOUR analysis !

Oliver Heaviside worked it all out 120 years back.
. . .

I'm eagerly awaiting your analysis showing how and why it's wrong. Or
simply which of the statements and equations I wrote are incorrect, and
what the correct statement or equation should be and why. Or even a
simple numerical example that illustrates the relationship between
reflection and power transfer.

It appears that there are two groups of readers: those who are convinced
by authoritative sounding statements not backed up by any evidence, and
those who require a solid basis for believing a statement. The first
group I can't help at all. But hopefully I've reached at least some
people in the second group.

Roy Lewallen, W7EL

Dr. Slick wrote:
. . .
Wrong. But at least you admit that the "accepted" formula (which
is fine for purely real Zo) implies a reflection.

The absence of reflection is what makes the maximum power
transfer.
. . .

This is interesting. But how did it lead you to the equation you
determined must be correct? That is, what definition of reflection
coefficient did you start with, where did you get it, and how did you
get from there to the reflection coefficient equation you presented?

I assume that, consistent with the admonition in the last paragraph of
your posting, you were "careful to follow through and be consistent with
your definitions, measurements, algebra, and arithmetic". It would be
very instructive for us to be able to follow the process you did in
coming to what you feel is the "right answer".

Roy Lewallen, W7EL

Peter O. Brackett wrote:

Easy now boy! You'r almost as bad as me!

It is entirely possible, in fact I know this to be true, that there can be
more than one *definition* of "the reflection coefficient". And so... one
cannot say definitively that one particular defintion is WRONG.

If the definition of the reflection coefficient is given as rho = (Z - R)/(Z
+ R) then that's what it is. This particular definition corresponds to the
situation which results in rho being null when the unknown Z is equal to the
reference impedance R, i.e. an "image match". If the definition is given as
rho = (Z - conj(R))/(Z + conj(R)) then rho will be null when the unknown Z
is equal to the conjugate of the reference impedance conj(R), i.e. a
"conjugate match".

Nothing is WRONG if the definition is first set up to correspond to what the
definer is trying to accomplish. And so one has got to take care when
making statements about RIGHT ways and WRONG ways to define things.
Everyone is entitled to go to hell their own way if they are the onesmaking
the definitions. Just as long as no incorrect conclusions are drawn from
the definitions. That may occur when folks don't accept or agree on a
definition.

OTOH....

Definitions and semantics aside, what we should really be interested in is
what is the physical meaning of any particular definition and what are its'
practical uses.
. . .

Every one is entitled to go to hell their own way when defining the
wave to
electrical variable transformations required to make their
measurements and
solve their problems and this will result in a variety of definitions for
the scattering [reflection] parameters. Nothing more nothing less.
Others
may not agree with your tools, methodologies and definitions, but just be
careful to follow through and be consistent with your definitions,
measurements, algebra, and arithmetic and you will always get the right
answers.