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

Roy Lewallen wrote in message ...

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.


[s11]**2 + [s21]**2 = 1

For a lossless passive two port network, where the brackets
indicate magnitude only.

If you find a passive network that reflects more voltage than it
receives, let us all know about your free energy device.




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.



Reflection coefficients greater than unity, which go outside the
Smith, only happen with active devices, as you have mentioned above.
Stability circles are a related topic, as their centers are often
based outside the unity RC circle.


Slick

Roy, Chipman on page 138 of "Theory and Problems of Transmission Lines"
makes the statement

" The conclusion is somewhat surprising, though inescapable, that a
transmission line can be terminated with a reflection coefficient whose
magnitude is as great as 2.41 without there being any implication that the
power level of the reflected wave is greater than that of the incident wave.
Such a reflection coefficient can exist only on a line whose attention per
wavelength is high, so that even if the reflected wave is in some sense
large at the point of reflection, it remains so for only a small fraction of
a wavelength along the line away from that point . . . The large reflection
coefficients are obtained only when the reactance of the terminal load
impedance is of opposite sign to the reactance component of the
characteristic impedance."

Chipman makes these remarks after his derivation of the operation of lines
with complex characteristic impedance.


--
73/72, George
Amateur Radio W5YR - the Yellow Rose of Texas
Fairview, TX 30 mi NE of Dallas in Collin county EM13QE
"In the 57th year and it just keeps getting better!"






"Roy Lewallen" wrote in message
...
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

Thanks very much for that additional information about the consequences
of the magnitude of the reflection coefficient exceeding one. I couldn't
find it in any of the electromagnetics or transmission line books on my
shelf, which at last count include about 13 texts. Chipman, alas, isn't
among them. It confirms what I suspected, and provides further evidence
that the posted equation is universally correct.

While I'm mentioning books, I picked up a couple at Powell's Technical
Bookstore yesterday evening that look like real winners. They're
_Engineering Electromagnetics_ by Nathan Ida (2000), and
_Electromagnetic Fields, Energy, and Waves_ by Leonard M. Magid (1972).

The thing that attracted me to Ida was that he explains things in very
clear terms, then follows each section with a number of examples showing
how the principles are applied to real problems. And answers to all the
exercises (separate from the examples) are at the back of the book. This
is a pretty recent book and fairly expensive. I was lucky to have found
a used copy at a reduced price.

Magid has the most rigorous derivation of power and energy flow on
transmission lines I've seen, as well as other extensive transmission
line information. One conclusion that pricked my ears was that on a line
with a pure standing wave (e.g., a lossless line terminated with an open
or short circuit), ". . . power (and therefore, energy) is completely
trapped within each [lambda]/4 section of this lossless line, never able
to cross the zero-power points and thus constrained forever to rattle to
and fro within each quarter-wave section of this line." I had reached
this same conclusion some time ago, but realized I hadn't properly
evaluated the constant term when integrating power to find the energy.
But I didn't want to get into the endless shouting match going on in the
newsgroup, and dropped it before going back and fixing my derivation.
Hopefully some of the participants in power and energy discussions will
read Magid's analysis before resuming. I found this book used at a very
modest price.

Roy Lewallen, W7EL

George, W5YR wrote:
Roy, Chipman on page 138 of "Theory and Problems of Transmission Lines"
makes the statement

" The conclusion is somewhat surprising, though inescapable, that a
transmission line can be terminated with a reflection coefficient whose
magnitude is as great as 2.41 without there being any implication that the
power level of the reflected wave is greater than that of the incident wave.
Such a reflection coefficient can exist only on a line whose attention per
wavelength is high, so that even if the reflected wave is in some sense
large at the point of reflection, it remains so for only a small fraction of
a wavelength along the line away from that point . . . The large reflection
coefficients are obtained only when the reactance of the terminal load
impedance is of opposite sign to the reactance component of the
characteristic impedance."

Chipman makes these remarks after his derivation of the operation of lines
with complex characteristic impedance.


--
73/72, George
Amateur Radio W5YR - the Yellow Rose of Texas
Fairview, TX 30 mi NE of Dallas in Collin county EM13QE
"In the 57th year and it just keeps getting better!"

On Sun, 17 Aug 2003 18:16:42 -0700, Roy Lewallen
wrote:

Interesting stuff snipped

Magid has the most rigorous derivation of power and energy flow on
transmission lines I've seen, as well as other extensive transmission
line information. One conclusion that pricked my ears was that on a line
with a pure standing wave (e.g., a lossless line terminated with an open
or short circuit), ". . . power (and therefore, energy) is completely
trapped within each [lambda]/4 section of this lossless line, never able
to cross the zero-power points and thus constrained forever to rattle to
and fro within each quarter-wave section of this line." I had reached
this same conclusion some time ago, but realized I hadn't properly
evaluated the constant term when integrating power to find the energy.
But I didn't want to get into the endless shouting match going on in the
newsgroup, and dropped it before going back and fixing my derivation.
Hopefully some of the participants in power and energy discussions will
read Magid's analysis before resuming. I found this book used at a very
modest price.


Roy:

Interesting point and I don't recall reading or hearing it elsewhere.

The following is dashed off without fully thinking it through, so no
warranty on its accuracy.

If you think of a sound wave (longitudinal transmission, of course) in
a lossless acoustic transmission line terminated with a short, the
individual air molecules within each 1/4 wave section are likewise
trapped since at the 1/4 wave points there is zero sound pressure.
This may be a useful analogy for the electromagnetic transverse
propagating T-line.

Jack K8ZOA

Magid has the most rigorous derivation of power and energy flow on
transmission lines,
==========================
The following short question is adressed to all contributors to this
newsgroup who feel impelled to bolster their lack of self-confidence by
dragging in the chapter and verse of their favourite worshipped authors and
Gurus, most of whom nobody has ever heard of and highly unlikely ever to get
their hands on.

How do you know that?

"Reg Edwards" wrote in message ...
Magid has the most rigorous derivation of power and energy flow on
transmission lines,
==========================
The following short question is adressed to all contributors to this
newsgroup who feel impelled to bolster their lack of self-confidence by
dragging in the chapter and verse of their favourite worshipped authors and
Gurus, most of whom nobody has ever heard of and highly unlikely ever to get
their hands on.

How do you know that?

Gee, Reg, since you took that out of context, it seems a bit unfair.
Roy wrote it, and after the comma was, "I've seen." I don't know it,
but I'm willing to take Roy at his word on the matter.

Cheers,
Tom

"Tom Bruhns" wrote
"Reg Edwards" wrote
Magid has the most rigorous derivation of power and energy flow on
transmission lines,
==========================
The following short question is adressed to all contributors to this
newsgroup who feel impelled to bolster their lack of self-confidence by
dragging in the chapter and verse of their favourite worshipped authors
and
Gurus, most of whom nobody has ever heard of and highly unlikely ever to
get
their hands on.

How do you know that?
============================

Gee, Reg, since you took that out of context, it seems a bit unfair.
Roy wrote it, and after the comma was, "I've seen." I don't know it,
but I'm willing to take Roy at his word on the matter.
===========================

The question asked was -

"How do you know that Magid has the most rigorous derivation of . . . . . .
".

Of what the derivation was was of no consequence. It was a matter of
judgement of Magid's (or anybody else's) qualifications and authority. What
was the purpose of referring to somebody hardly anybody has ever heard of?
---
Reg.

Reg Edwards wrote:
The following short question is adressed to all contributors to this
newsgroup who feel impelled to bolster their lack of self-confidence by
dragging in the chapter and verse of their favourite worshipped authors and
Gurus, most of whom nobody has ever heard of and highly unlikely ever to get
their hands on.

You never heard of Ramo, the 'R' in TRW? :-)
--
73, Cecil http://www.qsl.net/w5dxp



-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
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Reg Edwards wrote:
Magid has the most rigorous derivation of power and energy flow on
transmission lines,

==========================
The following short question is adressed to all contributors to this
newsgroup who feel impelled to bolster their lack of self-confidence by
dragging in the chapter and verse of their favourite worshipped authors and
Gurus, most of whom nobody has ever heard of and highly unlikely ever to get
their hands on.

How do you know that?

I multiplied v(t) and i(t) in the forward and reverse waves and added
them as a function of position to get the instantaneous power at each
point along the line. Then I integrated to find the energy. As I
mentioned in the part of the posting you excluded from your quote, I
discovered that I hadn't evaluated the constant of integration.
Somewhere along the line, I got sidetracked, and didn't want to get
sucked into the interminable argument going on (which I see I've started
up again -- my sincere apology to all), so didn't go back and clean it
up. I had, however, reached the same conclusion as Magid, so apparently
the constant was zero, or didn't impact the results. Magid follows the
same process, although I haven't yet followed it through completely.

You've now heard of Magid, and you can very likely find a used copy on
the Internet for the price of a couple of bottles of mediocre Pinot Noir
in much less time than it would take to drink it (unless perhaps you're
a speed drinker). You could get your hands on one with even less effort
than I've taken -- I had to walk a few blocks, while you can do it all
from your easy chair, only having to rise and face the Sun when the
postman comes with your book.

Shoot, you can even get it from the same store where I got mine, if they
have another copy just now. http://www.powellsbooks.com.

Roy Lewallen, W7EL
Certified Reg's Old Wife, Nitpicker, Busy-Body, Lacker of
Self-Confidence, Worshipper of Authors and Gurus, and Other Notable
Distinctions and Honours which are Bestowed Almost Daily