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.

When you change Zo, you change the normalized center of the
Smith Chart, and therefore the Reflection Coefficient and SWR, looking
into the same load.



Check it out...


Slick

Among the posters on this topic, I'd be willing to bet a substantial sum
that at least Tom, Ian, Cecil, Bill, and I can recite that formula from
memory. In addition, there are a number of other regular and occasional
newsgroup posters in this category who've been wise enough to not having
posted on this thread, and some who have posted whom I don't know well
enough to put my money on. I'd be willing to further bet that Ian,
Cecil, Bill, and I could have done so at any time for at least the last
20 years. I omit Tom from this second list only because I haven't yet
met him in person and otherwise haven't gotten any hints of his age --
but I'll take a gamble and spot him 10 years at least. Furthermore, we
all know how to use it, and have done so countless times in the process
of designing systems that work.

I'm glad you've discovered this equation. Learn what it means and how to
use it, and you've taken a good first step toward understanding
transmission line phenomena.

Roy Lewallen, W7EL

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.

When you change Zo, you change the normalized center of the
Smith Chart, and therefore the Reflection Coefficient and SWR, looking
into the same load.



Check it out...


Slick

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.

When you change Zo, you change the normalized center of the
Smith Chart, and therefore the Reflection Coefficient and SWR, looking
into the same load.

Correct! What Roy is objecting to is the wording of an assertion of
yours that seemed to imply that changing the reference Z0 on a piece
of Smith Chart paper magically changes the physical Z0 of the transmission
line to a different value. As you have advised me, be careful of what you
say and how you say it.

I referred to "free space" the other day. The image I had in my own
mind was the free space halfway between here and Alpha Centauri. Others
had the image of "free space" as 0.001 WL away from a radiating antenna.
--
73, Cecil http://www.qsl.net/w5dxp



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"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

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!"

W5DXP wrote:

Roy Lewallen wrote:
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."

He's obviously talking about net energy. There is no impedance
discontinuity
in a continuous piece of transmission line so there is nothing to cause
reflections at the zero-power points. Net energy doesn't cross the zero-
power points but equal forward energy and reflected energy must cross the
zero-power points. That is easy to prove by observing ghosting on a TV set
being fed by 1000 feet of ladder-line. If energy is completely trapped
within each 1/4WL section, ghosting would be impossible.

I forgot to add. At the "zero-power points", either voltage or current is
zero. All that means is that all the energy is contained in the opposite
field. If the voltage is zero, the H-field is at a maximum. If the current
is zero, the E-field is at a maximum. The energy still exists, just in one
field or the other. V*I*cos(theta) is NET energy. The power in the forward
wave is Vfwd*Ifwd and the power in the reflected wave is Vref*Iref. Those
values are constant all up and down the line for a lossless feedline.
--
73, Cecil http://www.qsl.net/w5dxp



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W5DXP wrote:

I forgot to add. At the "zero-power points", either voltage or current is
zero. All that means is that all the energy is contained in the opposite
field.

Not quite all. It also means that there is NO power since P = V x I.

There can be lots of energy present but none of it is flowing past
the zero voltage or zero current point; hence no power.

To believe that energy is flowing across a zero voltage or zero current
point requires the rejection of the view that instantaneous power is
equal to instantaneous voltage multiplied by instantaneous current.
Rejection of
P = V x I would have wide impacts on our understanding of electrical
power and energy flows. (My light bulb is not drawing any current but
there could still be energy flowing in it!?)
Think carefully before going there.

....Keith