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-   -   Derivation of the Reflection Coefficient? (https://www.radiobanter.com/antenna/356-derivation-reflection-coefficient.html)

Dr. Slick September 1st 03 11:09 PM

Derivation of the Reflection Coefficient?
 
Hello,

No one has really derived the Reflection Coefficient,
either the "normal" or "conjugate" equation. This would
be key to our understanding of when you can use which equation.


What is not understood is how A/C/F got from:

Voltage R. C.= (Vr/Vi)e**(2*y*z)

where y=sqrt((R+j*omega*L)(G+j*omega*C))
and z= distance from load

To:

Voltage RC=(Z1-Z0)/(Z1+Z0) for purely real Zo
or Voltage RC=(Z1-Z0*)/(Z1+Z0)



Even Kurokawa doesn't show us how he gets the conjugate
equation. Email me to get the paper, his notation is confusing.


I have NO problems with the normalized formula,
AS LONG AS Zo IS PURELY REAL.


Nevertheless, even if you do believe the "normal"
equation is correct even with complex Zo, i'd still like
to see your derivation.

And please give us a derivation with VARIABLES ONLY.
The strong temptation to use specific numbers will only
lead us to incorrect conclusions like:

A**B=A+B, because it's true when A and B are equal to 2.



Slick

Tarmo Tammaru September 2nd 03 01:36 AM

I looked back in one of the earlier chapters, where they originally do
reflection, and
using e**(+/-jyz)= cosyz +/- jsinyz they get

V(z)=(V+ + V-)cosyz -j(V+ - V-)sinyz
and
I(z)=Yo{(V+ - V-)cosyz -j(V+ + V-)sinyz}

a V(z)=V1cosyz + V2sinyz
}
}
b I(z)=-jY0V1sinyz + jY0V2cosyz

It says the equation is divided into two independent solutions for voltage
and current. I do not understand it. The brackets encompass both a and b.

Tam/WB2TT




"Dr. Slick" wrote in message
om...
Hello,

No one has really derived the Reflection Coefficient,
either the "normal" or "conjugate" equation. This would
be key to our understanding of when you can use which equation.


What is not understood is how A/C/F got from:

Voltage R. C.= (Vr/Vi)e**(2*y*z)

where y=sqrt((R+j*omega*L)(G+j*omega*C))
and z= distance from load

To:

Voltage RC=(Z1-Z0)/(Z1+Z0) for purely real Zo
or Voltage RC=(Z1-Z0*)/(Z1+Z0)



Even Kurokawa doesn't show us how he gets the conjugate
equation. Email me to get the paper, his notation is confusing.


I have NO problems with the normalized formula,
AS LONG AS Zo IS PURELY REAL.


Nevertheless, even if you do believe the "normal"
equation is correct even with complex Zo, i'd still like
to see your derivation.

And please give us a derivation with VARIABLES ONLY.
The strong temptation to use specific numbers will only
lead us to incorrect conclusions like:

A**B=A+B, because it's true when A and B are equal to 2.



Slick




David Robbins September 2nd 03 12:30 PM

From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)

then solving from (1) and (3) to get Vn/Vp

multiple (3) by Zload on both sides
Vp*Zload/Zo - Vn*Zload/Zo = Vload
substitute this for Vload in (1) to get:
Vp+Vn = Vp*Zload/Zo - Vn*Zload/Zo
group terms:
Vp-Vp*Zload/Zo = -Vn-Vn*Zload/Zo
factor:
Vp(1-Zload/Zo) = Vn(-1-Zload/Zo)
divide out terms
(1-Zload/Zo)/(-1-Zload/Zo) = Vn/Vp
multiply by Zo/Zo
(Zo-Zload)/(-Zo-Zload) = Vn/Vp
mulitply by -1/-1
(Zload-Zo)/(Zload+Zo) = Vn/Vp

therefo
rho = Vn/Vp = Zload-Zo/Zload+Zo

what could be simpler... apply kirchoff's and ohm's laws and a bit of
algebra.



Roy Lewallen September 2nd 03 07:41 PM

Anyone interested in seeing the same derivation in perhaps slightly
different order can review my posting of 8-23 in the thread " A
subtle detail of reflection coefficient. . .". It includes a numerical
comparison of results using the derived formula with results using a
couple of alternative formulas. Of course, you can find a similar
derivation in nearly any electromagnetics or transmission line text. If
you do look it up, please note that I made an error (later corrected) in
stating that conjugately matching the line results in maximum power
transfer to the load. The condition for maximum power transfer for a
given source impedance is of course that the load impedance be the
complex conjugate of the impedance seen looking from the load back
toward the source.

Roy Lewallen, W7EL


David Robbins wrote:
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)

then solving from (1) and (3) to get Vn/Vp

multiple (3) by Zload on both sides
Vp*Zload/Zo - Vn*Zload/Zo = Vload
substitute this for Vload in (1) to get:
Vp+Vn = Vp*Zload/Zo - Vn*Zload/Zo
group terms:
Vp-Vp*Zload/Zo = -Vn-Vn*Zload/Zo
factor:
Vp(1-Zload/Zo) = Vn(-1-Zload/Zo)
divide out terms
(1-Zload/Zo)/(-1-Zload/Zo) = Vn/Vp
multiply by Zo/Zo
(Zo-Zload)/(-Zo-Zload) = Vn/Vp
mulitply by -1/-1
(Zload-Zo)/(Zload+Zo) = Vn/Vp

therefo
rho = Vn/Vp = Zload-Zo/Zload+Zo

what could be simpler... apply kirchoff's and ohm's laws and a bit of
algebra.




Dr. Slick September 2nd 03 10:18 PM

"David Robbins" wrote in message ...
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)



I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.



then solving from (1) and (3) to get Vn/Vp

multiple (3) by Zload on both sides
Vp*Zload/Zo - Vn*Zload/Zo = Vload
substitute this for Vload in (1) to get:
Vp+Vn = Vp*Zload/Zo - Vn*Zload/Zo
group terms:
Vp-Vp*Zload/Zo = -Vn-Vn*Zload/Zo
factor:
Vp(1-Zload/Zo) = Vn(-1-Zload/Zo)
divide out terms
(1-Zload/Zo)/(-1-Zload/Zo) = Vn/Vp
multiply by Zo/Zo
(Zo-Zload)/(-Zo-Zload) = Vn/Vp
mulitply by -1/-1
(Zload-Zo)/(Zload+Zo) = Vn/Vp

therefo
rho = Vn/Vp = Zload-Zo/Zload+Zo

what could be simpler... apply kirchoff's and ohm's laws and a bit of
algebra.



Nice job David, nobody has done this yet. And done with variables,
as it needs to be done, and not with specific numbers.

I think this is correct for Zo is purely real.

I'd like to see the derivation for the conjugate equation,
which i have seen in Kurokawa, Besser, and the ARRL, among
other sources.


Slick

David Robbins September 2nd 03 11:31 PM


"Dr. Slick" wrote in message
om...
"David Robbins" wrote in message

...
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery

and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to

load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is

positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)



I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.


yes you can. in sinusoidal steady state analysis as discussed in 'Basic
Circuit Theory' by Desoer and Kuh. in chapter 7 and specifically section 5
of that chapter they show the kirchoff current and voltage laws and ohms law
generalized for phasor representations of voltage and current and complex
impedances and admittances. those representations are perfectly valid in
that type of analysis.



David or Jo Anne Ryeburn September 2nd 03 11:57 PM

In article ,
(Dr. Slick) wrote:

"David Robbins" wrote in message

...
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)



I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.


I swore that I wouldn't get into this one, but enough's enough.

Equation (1) is an application of Kirchoff's voltage law.
Equation (2) is an application of Kirchoff's current law.
Equation (3) results from (2) if you apply Ohm's law three times, to the
three terms in Equation (2).

Which of these three principles (Kirchoff's voltage law, Kirchoff's
current law, or Ohm's law) is the one you don't believe? Or do you
disbelieve more than one of the three?

David, ex-W8EZE

--
David or Jo Anne Ryeburn

To send e-mail, remove the letter "z" from this address.

David Robbins September 3rd 03 01:00 AM


"David or Jo Anne Ryeburn" wrote in message
...
In article ,
(Dr. Slick) wrote:

"David Robbins" wrote in message

...
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery

and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to

load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is

positive
so the reflected 'negative' current wave is moving left which gives

the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)



I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.


I swore that I wouldn't get into this one, but enough's enough.

Equation (1) is an application of Kirchoff's voltage law.
Equation (2) is an application of Kirchoff's current law.
Equation (3) results from (2) if you apply Ohm's law three times, to the
three terms in Equation (2).

Which of these three principles (Kirchoff's voltage law, Kirchoff's
current law, or Ohm's law) is the one you don't believe? Or do you
disbelieve more than one of the three?

now, now, take it easy on him... he didn't say he didn't believe kcl or kvl
or ohm's law... he just doesn't understand that they still do apply to
phasor notation used in sinusoidal steady state analysis. an easy
misunderstanding.



Dr. Slick September 3rd 03 06:15 AM

"David Robbins" wrote in message ...


I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.


I swore that I wouldn't get into this one, but enough's enough.

Equation (1) is an application of Kirchoff's voltage law.
Equation (2) is an application of Kirchoff's current law.
Equation (3) results from (2) if you apply Ohm's law three times, to the
three terms in Equation (2).

Which of these three principles (Kirchoff's voltage law, Kirchoff's
current law, or Ohm's law) is the one you don't believe? Or do you
disbelieve more than one of the three?

now, now, take it easy on him... he didn't say he didn't believe kcl or kvl
or ohm's law... he just doesn't understand that they still do apply to
phasor notation used in sinusoidal steady state analysis. an easy
misunderstanding.



Gee, thanks David. I was wrong! This was a little review for me!
Hehe... owww..

But it still doesn't answer my question.

I don't think Kurokawa and Besser and the ARRL just pulled it
out of thin air.

And how do you explain the rho 1 for a passive network?
Shouldn't be possible. And neither should a negative SWR.

I'm not sure what is wrong with your derivation, but there
must be something that they are missing to not have the conjugate
in the numerator. Or there is a particular step that you cannot do
with complex impedances.

Again, the normal equation is only for purely real Zo, or
when Zo*=Zo. If Zo is complex, you have to use the conjugate
equation.

Could you email a scan of some of the pages? Not that it
would absolutely help me too much, but perhaps you are missing
something.


Slick

Cecil Moore September 3rd 03 05:31 PM

William E. Sabin wrote:
4) The determination that rho magnitude in a transmission line can be
greater than 1.0 is correct. In a passively loaded line fed by an
oscillator, where there is no positive feedback from load to oscillator,
there is no problem about a rho magnitude greater than 1.0.


But can |rho|=Sqrt(Pref/Pfwd) ever be greater than 1.0 for a
passive load?
--
73, Cecil http://www.qsl.net/w5dxp



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Reg Edwards September 3rd 03 07:19 PM

Cecil and others, even authors of books, have said -

- - - - |rho|^2 cannot be greater than 1.0 - - - -


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

Would you change your minds if I describe a
reflection-coefficient bridge, which anybody can
construct, which accurately measures values of | rho |
up to its greatest possible value in transmission lines
of 2.414 There's no catch!

For some reason Dr Slick has remained silent to my
acceptance of his challenge to find such an instrument.
Perhaps he's gone away to think about it.
---
Reg, G4FGQ



Roy Lewallen September 3rd 03 07:25 PM

Yet in my example, |rho|^2 *is* greater than one.

Also, in the past, you and others have defined the "forward power" to be
the power calculated from the forward voltage and current waves, namely
Re(fE * fIconj) or |fE| * |fI| * cos(phiE - phiI). This is what you've
consistently been calling the "power of the forward wave" or some such.
Likewise for "reverse power". This is the definition I used for the
substitution for fP and rP in the equation for total average power.
And the result is that the total power *isn't* equal to fP - rP.
What you're doing now is lumping the extra power into fP or rP, now
making those terms mean something else. The additional power term has
two components, one arising from the product of forward current and
reverse voltage, and the other from the product of forward voltage and
reverse current. (I combined the two cosine functions with a trig
identitity into a product of two sine functions, but you should go back
a step or two in the analysis to get a clear idea of their derivation.)
I believe you've chosen to assign each of these, or the sine product, to
either "forward power" or "reverse power", depending on its sign, even
though they're a function of both forward and reverse voltage and
current waves. I can't imagine the justification for doing this, but
then there's quite a lot that people have been doing which I don't
understand. As part of the process, you might consider the consequence
of the sine or cosine function returning a negative value, which either
of course can.

Again, I welcome an alternate solution that accounts for all the
voltages, currents, and powers, including one that does it with rho 1.

Roy Lewallen, W7EL

Cecil Moore wrote:

This seems to me to be somewhat akin to the fact that s11 and
rho can have different values at an impedance discontinuity
where a 'third power' is commonplace. Roy's 'third power' at
the load appears to be analogous to a re-reflection of some
sort as the inductive load tries and fails to dump energy
back into the Z0=68-j39 transmission line. A re-reflection
is another component of forward power.

The ratio of reflected Poynting vector to forward Poynting
vector is |rho|^2. In Roy's example, the total average
Poynting vector points toward the load indicating that
(Pz+ - Pz-) 0. That means |rho|^2 cannot be greater
than 1.0.



Roy Lewallen September 3rd 03 07:39 PM

Well, Cecil, you've redefined Pref and Pfwd. Pref used to be solely a
function of the forward voltage and current waves, and Pref a function
of the reverse voltage and current waves. But now you've chosen to add
an extra term to one or the other of those, or both -- a term which
contains components of both forward and reverse waves. You might recall
from the analysis that I originally had two cosine terms, one arising
from the product of forward voltage and reverse current, and the other
arising from the reverse voltage and forward current. Which of these do
you assign to the "forward power" and which to "reverse power"? If the
choice is based solely on the sign, does the choice automatically change
when the cosine function returns a negative value? (To be truthful, I
haven't checked to see if that is, in fact, possible for possible values
of the argument.) When combined into a product of two sine functions as
I did in the analysis, do you assign this combined function to Pref or
Pfwd? The combined sine functions can, I know, return either positive or
negative values, so what do you do when it returns a negative value? If
I use another trig identitity to convert it to some trig functions
having a different sign, does it then switch from being part of Pref to
part of Pfwd, or vice-versa?

So now when you say Pref and Pfwd, what do you mean?

If you were to stick with the definition you've always used in the past,
i.e., powers calculated from solely forward or reverse voltage and
current waves, the answer is yes. For evidence I offer my derivations.
Roy Lewallen, W7EL

Cecil Moore wrote:
William E. Sabin wrote:

4) The determination that rho magnitude in a transmission line can be
greater than 1.0 is correct. In a passively loaded line fed by an
oscillator, where there is no positive feedback from load to
oscillator, there is no problem about a rho magnitude greater than 1.0.



But can |rho|=Sqrt(Pref/Pfwd) ever be greater than 1.0 for a
passive load?



David Robbins September 3rd 03 09:03 PM


"Dr. Slick" wrote in message
om...
"David Robbins" wrote in message

...


I believe this line (3) is only true if Zo is purely real.

If Zo is complex, i don't think you can apply
this.

I swore that I wouldn't get into this one, but enough's enough.

Equation (1) is an application of Kirchoff's voltage law.
Equation (2) is an application of Kirchoff's current law.
Equation (3) results from (2) if you apply Ohm's law three times, to

the
three terms in Equation (2).

Which of these three principles (Kirchoff's voltage law, Kirchoff's
current law, or Ohm's law) is the one you don't believe? Or do you
disbelieve more than one of the three?

now, now, take it easy on him... he didn't say he didn't believe kcl or

kvl
or ohm's law... he just doesn't understand that they still do apply to
phasor notation used in sinusoidal steady state analysis. an easy
misunderstanding.



Gee, thanks David. I was wrong! This was a little review for me!
Hehe... owww..

But it still doesn't answer my question.

I don't think Kurokawa and Besser and the ARRL just pulled it
out of thin air.

And how do you explain the rho 1 for a passive network?
Shouldn't be possible. And neither should a negative SWR.

I'm not sure what is wrong with your derivation, but there
must be something that they are missing to not have the conjugate
in the numerator. Or there is a particular step that you cannot do
with complex impedances.

Again, the normal equation is only for purely real Zo, or
when Zo*=Zo. If Zo is complex, you have to use the conjugate
equation.

Could you email a scan of some of the pages? Not that it
would absolutely help me too much, but perhaps you are missing
something.


Slick

sorry, no scanner here.

how do you get rho1? please give me the Zo and Zl to try out, i have been
playing for a while with the basic equations and haven't found a case where
either formulat gives rho1.

and of course if |rho|=1 then swr can never be negative.




Cecil Moore September 3rd 03 09:33 PM

Reg Edwards wrote:
Cecil and others, even authors of books, have said -
- - - - |rho|^2 cannot be greater than 1.0 - - - -


Would you change your minds if I describe a
reflection-coefficient bridge, which anybody can
construct, which accurately measures values of | rho |
up to its greatest possible value in transmission lines
of 2.414 There's no catch!


Note that I didn't say |rho| couldn't be greater than one.
I said |rho|^2, the power reflection coefficient, cannot
be greater than 1.0 for a passive load, i.e. you cannot
get more power out of a passive load than you put into it.
It follows that the conservation of energy principle will
not allow the square of rho to be the power reflection
coefficient if rho is greater than 1.0.

For some reason Dr Slick has remained silent to my
acceptance of his challenge to find such an instrument.
Perhaps he's gone away to think about it.


There is an answer here. I suspect you can answer it by
answering the following question about s-parameters.
Consider the following example:

Source--50 ohm feedline--+--1/2WL 150 ohm feedline--50 ohm load

s11 is 0.5 but rho, on the 50 ohm feedline, is zero.

|s11|^2 is defined in the HP AN 95-1 Ap note as the ratio of
the "Power reflected from the network input" to the "Power
incident on the network input" Assuming we have 100 watts of
power incident on the network input, the power reflected from
the network input would have to be 25 watts. But the actual
reflected power on the 50 ohm feedline measures to be zero
watts. Hint: |s12|^2 must also be taken into account.
--
73, Cecil http://www.qsl.net/w5dxp



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Richard Clark September 3rd 03 09:42 PM

On Wed, 3 Sep 2003 18:19:52 +0000 (UTC), "Reg Edwards"
wrote:

Cecil and others, even authors of books, have said -

- - - - |rho|^2 cannot be greater than 1.0 - - - -


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

Would you change your minds if I describe a
reflection-coefficient bridge, which anybody can
construct, which accurately measures values of | rho |
up to its greatest possible value in transmission lines
of 2.414 There's no catch!

For some reason Dr Slick has remained silent to my
acceptance of his challenge to find such an instrument.
Perhaps he's gone away to think about it.
---
Reg, G4FGQ


Ah Reg,

(Too many, like this season's crop of presidential hopefuls, have
usurped the role of clown, sorry to demote you - but you know the
irony in that gesture, you at least gained it honestly. ;-)

No catch? You stand little chance of interest as that would imply an
end to it - what fun when the stream of debate circles endlessly
around simple issues of arithmetic gone bad?

So, in their stead and knowing that anything practical is anathema,
and that anything observed as being cut-and-paste without context is
shunned as a cheap smear, give us the works. [Here's hoping that it
adds to the bottom line of my bountiful discredit.]

73's
Richard Clark, KB7QHC

Cecil Moore September 3rd 03 10:06 PM

Roy Lewallen wrote:

Yet in my example, |rho|^2 *is* greater than one.


If so, |rho|^2 is NOT the power reflection coefficient. The conservation
of energy principle will not allow the power reflection coefficient to
be greater than 1.0.

If you calculate a forward Poynting vector and a reflected Poynting vector
at a passive load, you will find that the forward Poynting vector always has
a larger magnitude than the reflected Poynting vector. Thus,
if Pz-/Pz+ = |rho|^2, as asserted in Ramo & Whinnery,
|rho| cannot be greater than 1.0. I suspect you have stumbled upon a
single-port case where rho and s11 are not equal.

You have apparently calculated an s11 reflection coefficient and called
it "rho" under conditions where s11 doesn't have to equal rho.

Also, in the past, you and others have defined the "forward power" to be
the power calculated from the forward voltage and current waves, namely
Re(fE * fIconj) or |fE| * |fI| * cos(phiE - phiI). This is what you've
consistently been calling the "power of the forward wave" or some such.


ONLY for lossless lines. I never said or implied that it would work for
lossy lines. I have carefully avoided making any assertions about lossy
lines. The only assertion that I will make about lossy lines is that
they obey the conservation of energy principle.

Likewise for "reverse power". This is the definition I used for the
substitution for fP and rP in the equation for total average power.


Well, that's apparently a boo-boo for lossy lines. Apparently, Vfwd*Ifwd*
cos(theta) equals forward power only for lossless lines.

And the result is that the total power *isn't* equal to fP - rP.
What you're doing now is lumping the extra power into fP or rP, now
making those terms mean something else.


Yes, for lossy lines, they apparently do mean something else. It reminds
me of the s-parameter equations for power. Like your calculations, there
are four powers, not just two. They are |s11|^2, |s22|^2, |s21|^2, |s12|^2.
It looks as if you have set fP = |s22|^2 and rP = |s11|^2 and your other
two power components are |s12|^2 and |s21|^2. But in real life, these last
two powers are forced to join either the forward wave or the reflected wave.

The additional power term has
two components, one arising from the product of forward current and
reverse voltage, and the other from the product of forward voltage and
reverse current. (I combined the two cosine functions with a trig
identitity into a product of two sine functions, but you should go back
a step or two in the analysis to get a clear idea of their derivation.)
I believe you've chosen to assign each of these, or the sine product, to
either "forward power" or "reverse power", depending on its sign, even
though they're a function of both forward and reverse voltage and
current waves. I can't imagine the justification for doing this, ...


The justification is two, and only two directions, in a transmission line.
All coherent components are forced to superpose into Total Forward Power
or Total Reflected Power depending on the direction (sign).
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore September 3rd 03 10:28 PM

Roy Lewallen wrote:

Well, Cecil, you've redefined Pref and Pfwd.


Nope, I haven't, Roy. You have somehow arrived at the equations for
a four-port network while dealing with what appears to be a two-port
network. Inadvertently, you seem to have calculated |s11|^2, |s12|^2,
|s21|^2, and |s22|^2 for what appears to be a two-port network. Is a
two-port lossy line network with inductive load really a four-port
network in disguise? Does the delay in the inductor returning energy
to the system constitute an 'a2' term in the s-parameter analysis?

Pref used to be solely a
function of the forward voltage and current waves, and Pref a function
of the reverse voltage and current waves. But now you've chosen to add
an extra term to one or the other of those, or both -- a term which
contains components of both forward and reverse waves.


Roy, that is built right into the s-paramater analysis. For instance,
for a Z0 (image) matched system:

Forward Power = |s11|^2 + |s12|^2 + |s21|^2 + |s22|^2

For a matched system, Forward Power contains four power terms.
In fact, Forward Power can contain from one to four terms depending
on system configuration.

You might recall
from the analysis that I originally had two cosine terms, one arising
from the product of forward voltage and reverse current, and the other
arising from the reverse voltage and forward current. Which of these do
you assign to the "forward power" and which to "reverse power"?


You are talking about |s12|^2 and |s21|^2. The sign and phase of their
power flow vectors will indicate whether they are forward power or
reverse power.

When combined into a product of two sine functions as
I did in the analysis, do you assign this combined function to Pref or
Pfwd?


If the sign is positive, it is flowing toward the load, i.e. it will
superpose with the forward wave. If the sign is negative, it is flowing
toward the source, i.e. it will superpose with the reverse wave. The
conservation of energy principle will not allow the power in the reverse
wave to exceed the power in the forward wave for passive loads, no matter
what the value of rho.

So now when you say Pref and Pfwd, what do you mean?


What I have always meant. Pfwd is the total of all the coherent forward
components. Pref is the total of all the coherent reverse components.

If you were to stick with the definition you've always used in the past,
i.e., powers calculated from solely forward or reverse voltage and
current waves, the answer is yes. For evidence I offer my derivations.


All you have derived is the s-parameter analysis which is known to include
four power parameters. It is known that s11 doesn't always equal rho for
a four-ternimal network. You seem to have proven that to be true for what
appears to be a two-port network.
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore September 3rd 03 11:03 PM

Roy Lewallen wrote:
Again, I welcome an alternate solution that accounts for all the
voltages, currents, and powers, including one that does it with rho 1.


It dawned on me, just now in the shower, what is happening here. When
you introduced the 'x' parameter, the distance from the load, you
introduced a 2-port network analysis, be it an s-, h-, y-, z-, or
whatever-parameter analysis. And of course there are four power
terms in a 2-port analysis. There a

1. The power reflected from the network input back toward the source. |s11|^2

2. The power transmitted through the network port toward the load. |s21|^2

3. The power re-reflected from the network output back toward the load. |s22|^2

4. The power transmitted through the network port toward the source. |s12|^2

These are the four powers you calculated and you consider only |s12|^2 to
be forward power. That is an error. |s22|^2 is also forward power. These
two forward power flow vectors have to be added to obtain the total
forward Poynting vector. I do believe that clears up the confusion.
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore September 3rd 03 11:15 PM

Roy Lewallen wrote:
I didn't, and don't, claim to have derived a "power reflection
coefficient". What I calculated was the ratio of reflected voltage to
forward voltage at the load, and called its magnitude rho. If there's
any step in the analysis that's unclear, I'll be happy to explain it in
more detail.


What you apparently calculated is s11 which is not always equal to rho.

What I have calculated is the ratio of reflected voltage to forward
voltage at the load, no more and no less.


No, you have calculated the ratio of one of the reflected voltages to
one of the forward voltages. I believe you have calculated the ratio
of s21*a1 to s12*a2 when you should be calculating the ratio of
(s11*a1+s12*a2) to (s21*a1+s22*a2). You simply omitted half the terms.

Yet lossy lines are just what we're talking about now, isn't it?


Yes, and I am in the process of trying to understand them.

So what are the "forward power" and "reverse power" for lossy lines? Any
explanation for why they vary (other than with the expected attenuation)
with position along the line?


I don't know, yet.

I'm sure that with enough s parameter and optics references, the facts
of the matter can be satisfactorily obscured.


It is you who is using an s-, h-, y-, z-, or other-parameter analysis
and are inadvertently obscuring the facts. You left out half the voltage
terms that should be included in the forward voltage and reflected
voltage. Add all the reflected voltages together. Add all the forward
voltages together. Divide the total reflected voltage by the total
forward voltage.

Your view of how average powers add and travel do force that
restriction. I'm looking forward to your alternative analysis, which
shows the voltages, currents, and powers at both ends of the line while
simultaneously satisfying your notion of how average powers interact.


I think all that is built into your analysis. When you include all the
necessary terms, I will be surprised if everything doesn't fall out
consistently.
--
73, Cecil http://www.qsl.net/w5dxp



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Roy Lewallen September 3rd 03 11:18 PM

My, it sure didn't take long to get the discussion diverted from the
voltages, currents, and powers in the analysis. I'm sorry to say I
expected that.

Cecil Moore wrote:
Roy Lewallen wrote:

Well, Cecil, you've redefined Pref and Pfwd.



Nope, I haven't, Roy. You have somehow arrived at the equations for
a four-port network while dealing with what appears to be a two-port
network. Inadvertently, you seem to have calculated |s11|^2, |s12|^2,
|s21|^2, and |s22|^2 for what appears to be a two-port network. Is a
two-port lossy line network with inductive load really a four-port
network in disguise? Does the delay in the inductor returning energy
to the system constitute an 'a2' term in the s-parameter analysis?


I'll leave the philosophical question to you of when a transmission line
is an n-port network and when it isn't, and which s parameter I
inadvertently calculated. Was I unclear about what I did calculate? What
part of it don't you understand?

Pref used to be solely a function of the forward voltage and current
waves, and Pref a function of the reverse voltage and current waves.
But now you've chosen to add an extra term to one or the other of
those, or both -- a term which contains components of both forward and
reverse waves.



Roy, that is built right into the s-paramater analysis. For instance,
for a Z0 (image) matched system:

Forward Power = |s11|^2 + |s12|^2 + |s21|^2 + |s22|^2

For a matched system, Forward Power contains four power terms.
In fact, Forward Power can contain from one to four terms depending
on system configuration.


I don't know, and don't really care, where you're trying to go with your
S parameter analysis. But when you're all done, please translate all
that wonderful stuff to voltages, currents, and powers, using a finite
length transmission line, and present your analysis.

Are you having difficulty understanding what I've done simply with
voltages, currents, and powers?

You might recall from the analysis that I originally had two cosine
terms, one arising from the product of forward voltage and reverse
current, and the other arising from the reverse voltage and forward
current. Which of these do you assign to the "forward power" and which
to "reverse power"?



You are talking about |s12|^2 and |s21|^2. The sign and phase of their
power flow vectors will indicate whether they are forward power or
reverse power.

When combined into a product of two sine functions as I did in the
analysis, do you assign this combined function to Pref or Pfwd?



If the sign is positive, it is flowing toward the load, i.e. it will
superpose with the forward wave. If the sign is negative, it is flowing
toward the source, i.e. it will superpose with the reverse wave. The
conservation of energy principle will not allow the power in the reverse
wave to exceed the power in the forward wave for passive loads, no matter
what the value of rho.

So now when you say Pref and Pfwd, what do you mean?



What I have always meant. Pfwd is the total of all the coherent forward
components. Pref is the total of all the coherent reverse components.

So, you mean that the term containing the product of two sine functions
is part of Pfwd when the angles are such that the sine functions return
a positive value, and part of Pref when they return a negative value?

If you were to stick with the definition you've always used in the
past, i.e., powers calculated from solely forward or reverse voltage
and current waves, the answer is yes. For evidence I offer my
derivations.



All you have derived is the s-parameter analysis which is known to include
four power parameters. It is known that s11 doesn't always equal rho for
a four-ternimal network. You seem to have proven that to be true for what
appears to be a two-port network.


No, I did not derive an s parameter analysis. I derived voltages,
currents, and powers. Interpretation of this in terms of s parameters is
strictly your own doing, and it provides wonderful opportunities to
obscure and misinterpret what's really happening. If you're unable to
understand voltages, currents, and powers and want to argue instead
about s parameters (which indeed do represent voltages and powers, but
not necessarily in a one-to-one correspondence to those in the circuit I
analyzed), how many ports the circuit has, and the meaning of the power
reflection coefficient, have at it. But I won't participate. I'll simply
wait until you're done with your philosophising, calculations,
translation back and forth, and post your analysis with V, I, and P as
the variables.

Roy Lewallen, W7EL


Roy Lewallen September 3rd 03 11:32 PM

Clears up what confusion?

Nowhere in my analysis is s11, s12, or s22 mentioned. I don't consider
s12 or s22 to be anything at all, and don't make any claim whatsoevera
about what they are or aren't. Which step or steps of my analysis is/are
incorrect? And in terms of voltages, currents, and powers, why?

Roy Lewallen, W7EL

Cecil Moore wrote:
Roy Lewallen wrote:

Again, I welcome an alternate solution that accounts for all the
voltages, currents, and powers, including one that does it with rho 1.



It dawned on me, just now in the shower, what is happening here. When
you introduced the 'x' parameter, the distance from the load, you
introduced a 2-port network analysis, be it an s-, h-, y-, z-, or
whatever-parameter analysis. And of course there are four power
terms in a 2-port analysis. There a

1. The power reflected from the network input back toward the source.
|s11|^2

2. The power transmitted through the network port toward the load. |s21|^2

3. The power re-reflected from the network output back toward the load.
|s22|^2

4. The power transmitted through the network port toward the source.
|s12|^2

These are the four powers you calculated and you consider only |s12|^2 to
be forward power. That is an error. |s22|^2 is also forward power. These
two forward power flow vectors have to be added to obtain the total
forward Poynting vector. I do believe that clears up the confusion.



Cecil Moore September 3rd 03 11:34 PM

Roy Lewallen wrote:

My, it sure didn't take long to get the discussion diverted from the
voltages, currents, and powers in the analysis. I'm sorry to say I
expected that.


Please calm down, Roy. Disagreeing with you is not a diversion. You made
a simple error. When you introduced the 'x' term, the distance away from
the load, you introduced a 2-port analysis. It is a well known fact that
there are four power terms involved in a 2-port analysis as explained in
another posting.

Was I unclear about what I did calculate?


Yes, you were, but it was inadvertent.

I don't know, and don't really care, where you're trying to go with your
S parameter analysis. But when you're all done, please translate all
that wonderful stuff to voltages, currents, and powers, using a finite
length transmission line, and present your analysis.


I'm just showing you what small error you made when you assumed that only
one of the four power terms was the forward power. There are four power
terms. They divide up and add to obtain the forward power and reflected
power. You neglected to do that.

Are you having difficulty understanding what I've done simply with
voltages, currents, and powers?


Nope, I recognize the tiny error you made and am trying to explain it
to you. You didn't include all the forward voltages in your forward
voltage. There are four voltage terms, two forward and two reflected.
You left out half the terms and got the wrong forward or reflected
voltage or both. The mistake is in assuming that rho = s11. It doesn't
in this case.

So, you mean that the term containing the product of two sine functions
is part of Pfwd when the angles are such that the sine functions return
a positive value, and part of Pref when they return a negative value?


No, after further thought, I think you should NOT have combined those two
terms. Four terms is what exists in the analysis so just leave it at
four terms. All the terms with a plus sign combine and all the terms
with a minus sign combine. Please publish the four term power equation
before you used a trig identity to combine the terms. Two of those terms
are forward power and two of those terms are reflected power.

No, I did not derive an s parameter analysis. I derived voltages,
currents, and powers.


You obviously did a something-parameter analysis (maybe a z-parameter
analysis?). Whatever you did results in four power terms, not two plus
a third. When you introduced 'x' you introduced an analysis that produces
a reflected wave on each side of 'x' and a forward wave on each side of
'x'. That's four waves. You went too far when you combined two of those
waves into one especially since one is a forward wave and one is a
reflected wave.
--
73, Cecil http://www.qsl.net/w5dxp



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Roy Lewallen September 3rd 03 11:43 PM

Cecil Moore wrote:
Roy Lewallen wrote:

I didn't, and don't, claim to have derived a "power reflection
coefficient". What I calculated was the ratio of reflected voltage to
forward voltage at the load, and called its magnitude rho. If there's
any step in the analysis that's unclear, I'll be happy to explain it
in more detail.



What you apparently calculated is s11 which is not always equal to rho.


I calculated the ratio of the reflected to forward voltage at the load,
and called its magnitude rho. If you have some other "rho" you want to
argue about, please call it something else.


What I have calculated is the ratio of reflected voltage to forward
voltage at the load, no more and no less.



No, you have calculated the ratio of one of the reflected voltages to
one of the forward voltages. I believe you have calculated the ratio
of s21*a1 to s12*a2 when you should be calculating the ratio of
(s11*a1+s12*a2) to (s21*a1+s22*a2). You simply omitted half the terms.


Please repeat my analysis, including the voltages or currents which were
omitted, and explain why they should be included. I used standard steady
state analysis, which infers one forward traveling voltage and current
wave, and one reverse traveling voltage and current wave. Although the
physical meaning of multiple traveling forward and reverse waves in
steady state gets a little hazy to me, I don't think there's anything in
principal that prevents you from assuming any number of forward and
reverse voltage an current waves you'd like, calculating reflection
coefficients for each pair, and adding them all up to get the total.
It'll be interesting to see how you choose to do it.

Of course, by choosing the pairs carefully, you can probably assure that
the magnitude of the reflection coefficient for any pair doesn't exceed
one. I'm not sure what that means or proves, but by all means have at it.

. . .

I'm sure that with enough s parameter and optics references, the facts
of the matter can be satisfactorily obscured.



It is you who is using an s-, h-, y-, z-, or other-parameter analysis
and are inadvertently obscuring the facts. You left out half the voltage
terms that should be included in the forward voltage and reflected
voltage. Add all the reflected voltages together. Add all the forward
voltages together. Divide the total reflected voltage by the total
forward voltage.


What the heck are you talking about? Just where in the analysis do you
see any s, h, y, or z parameter? I did calculate an impedance here and
there from voltages and currents -- is that some kind of a no-no in your
eyes?

Again, please show your analysis with the "missing" terms (that is,
voltages and currents) included.

Your view of how average powers add and travel do force that
restriction. I'm looking forward to your alternative analysis, which
shows the voltages, currents, and powers at both ends of the line
while simultaneously satisfying your notion of how average powers
interact.



I think all that is built into your analysis. When you include all the
necessary terms, I will be surprised if everything doesn't fall out
consistently.


Well, good. So show us.

Roy Lewallen, W7EL


Roy Lewallen September 3rd 03 11:49 PM

So do it right and show us how it really should be done.

Roy Lewallen, W7EL

Cecil Moore wrote:

You obviously did a something-parameter analysis (maybe a z-parameter
analysis?). Whatever you did results in four power terms, not two plus
a third. When you introduced 'x' you introduced an analysis that produces
a reflected wave on each side of 'x' and a forward wave on each side of
'x'. That's four waves. You went too far when you combined two of those
waves into one especially since one is a forward wave and one is a
reflected wave.



Roy Lewallen September 3rd 03 11:53 PM

Yuck.

That should, of course, be "principle", not "principal". Sorry, I really
do know better!

Roy Lewallen, W7EL

Roy Lewallen wrote:
. . .
steady state gets a little hazy to me, I don't think there's anything in
principal that prevents you from assuming any number of forward and
reverse voltage an current waves you'd like, . .



Reg Edwards September 3rd 03 11:59 PM

Dear Cec,

Your arithmetic is abominable. ;o) Dr Slick's
vanishing-act was a better tactic.

Your only avenue of escape is to prove the | rho |
meter gives incorrect meter readings.

That's likely to be difficult.

The meter is based on precisely the same simple
principle as your common-or-garden SWR+Fwd Power+Refl
Power meter. In fact, its scale, instead of | rho |,
can be simultaneousy calibrated in terms of SWR from 1
to infinity. And 1 million professional housewives
supported by trusted ARRL handbooks can't be wrong.

By the way, does that Texas vinyard you mentioned have
a website? ;o)

---
Yours, Reg, G4FGQ



Roy Lewallen September 4th 03 12:04 AM

Modify what I believe to be a correct analysis in order to satisfy your
view of reality? You must be kidding again -- sometimes it's hard to tell.

Somehow I expected that an alternative analysis or any specific
correction wouldn't be forthcoming. I'm glad you've got it all sorted
out in your own mind, Cecil.

I'll now bow out, unless a coherent alternative analysis, or specific
corrections to the one I posted, are presented.

Roy Lewallen, W7EL

Cecil Moore wrote:
Roy Lewallen wrote:

Clears up what confusion?

Nowhere in my analysis is s11, s12, or s22 mentioned. I don't consider
s12 or s22 to be anything at all, and don't make any claim whatsoevera
about what they are or aren't.



Those are the reflection and transmission coefficients that represent
the effect the forward waves have on the reflected waves and vice versa.
You said your analysis included that effect so you are performing an
s-parameter-like analysis whether you realize it or not.

Which step or steps of my analysis is/are incorrect? And in terms of
voltages, currents, and powers, why?



Please publish your four term power equation and I will show you exactly
what is wrong. Please don't say rP and fP in that equation but show
the voltage, current, and impedance terms that make up what you think is
rP and fP. Hint: rP is not the total reflected power that you think
it to be. Neither is fP.



Cecil Moore September 4th 03 12:24 AM

Roy Lewallen wrote:
I calculated the ratio of the reflected to forward voltage at the load,
and called its magnitude rho.


No you didn't. The voltage that you think is the reflected voltage
is only one term of two. The voltage that you think is the forward
voltage is only one term of two.

Please repeat my analysis, including the voltages or currents which were
omitted, and explain why they should be included.


I have already done that, Roy. There are four waves. You must combine
the four waves to get the forward wave and the reflected wave. You
didn't do that. You declared one of the four waves to be the forward
wave and one to be the reflected wave and added the other two to get
a "third wave". That is an error.

What the heck are you talking about? Just where in the analysis do you
see any s, h, y, or z parameter? I did calculate an impedance here and
there from voltages and currents -- is that some kind of a no-no in your
eyes?


OK, let me do it in a way that you can understand. When you introduced
'x', you introduced a 2-port analysis whether you realize it or not.
In a 2-port analysis, there are four waves, two forward and two reflected.
The four power waves are proof that you are inadvertently using a 2-port
analysis. There are forward and reflected waves on the left side of 'x'
and there are forward and reflected waves on the right side of 'x'.
Let's look at only the voltages for now where rho is a reflection
coefficient and tau is a transmission coefficient.

V1 = Vfwd1*tau1 similar to s21*a1

V2 = Vref2*rho2 similar to s22*a2

V3 = Vfwd1*rho1 similar to s11*a1

V4 = Vref2*tau2 similar to s12*a2

You are saying that one of these voltages is the forward voltage.
That's just not true.

V1+V2 = forward voltage similar to b2=s21*a1+s22*a2

V3+V4 = reflected voltage similar to b1 = s11*a1+s12*a2

Again, please show your analysis with the "missing" terms (that is,
voltages and currents) included.


Please publish your raw four term power equation, omitting the rP
and fP terms which you are wrong about. If you have already published
that equation, please tell me the date so I can go look it up.
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore September 4th 03 12:28 AM

Roy Lewallen wrote:
So do it right and show us how it really should be done.


Sorry, that diversion won't work. Correct your error first
and effort on my part will be avoided.
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore September 4th 03 12:31 AM

Reg Edwards wrote:
Your only avenue of escape is to prove the | rho |
meter gives incorrect meter readings.


Nope, your only avenue of escape is to prove that a passive
load can reflect more than the incident power. :-)

By the way, does that Texas vinyard you mentioned have
a website? ;o)


I don't know but I will check. Heck, I might even try a
bottle to see if it's worth sending to you.
--
73, Cecil http://www.qsl.net/w5dxp



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Richard Clark September 4th 03 12:32 AM

On Wed, 03 Sep 2003 15:33:56 -0500, Cecil Moore
wrote:

a vast assemblage of text snipped.

Hi Cecil,

So, do you want the bridge description
Or
Not?

This question was even simpler than that of two resistors and the hank
of wire. I can look forward to the amusement of how long a side
thread this may develop into.

73's
Richard Clark, KB7QHC

Cecil Moore September 4th 03 12:33 AM

Roy Lewallen wrote:
I'll now bow out, unless a coherent alternative analysis, or specific
corrections to the one I posted, are presented.


Heh, heh, just discovered how wrong you are, eh? :-) It was an
easy mistake to make, Roy.
--
73, Cecil http://www.qsl.net/w5dxp



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[email protected] September 4th 03 12:39 AM

Cecil Moore wrote:

1. The power reflected from the network input back toward the source. |s11|^2

2. The power transmitted through the network port toward the load. |s21|^2

3. The power re-reflected from the network output back toward the load. |s22|^2

4. The power transmitted through the network port toward the source. |s12|^2

These are the four powers you calculated and you consider only |s12|^2 to
be forward power. That is an error. |s22|^2 is also forward power. These
two forward power flow vectors have to be added to obtain the total
forward Poynting vector. I do believe that clears up the confusion.


Was not the discussion about powers on the source side?
Is not |s22|^2 on the load side?

....Keith

Cecil Moore September 4th 03 12:47 AM

wrote:
And yes, |rho| can be greater than unity for a passive load.


But the power reflection coefficient cannot be greater than 1.0
which is what the argument is all about.
--
73, Cecil
http://www.qsl.net/w5dxp



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Cecil Moore September 4th 03 12:50 AM

Richard Clark wrote:

wrote:

a vast assemblage of text snipped.

Hi Cecil,

So, do you want the bridge description
Or
Not?


Since you snipped my posting, I have no idea what it was all about.
--
73, Cecil http://www.qsl.net/w5dxp



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[email protected] September 4th 03 12:56 AM

Reg Edwards wrote:

Cecil and others, even authors of books, have said -

- - - - |rho|^2 cannot be greater than 1.0 - - - -


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

Would you change your minds if I describe a
reflection-coefficient bridge, which anybody can
construct, which accurately measures values of | rho |
up to its greatest possible value in transmission lines
of 2.414 There's no catch!


I don't think I need to change my mind, but I would like
to see a description of a reflection-coefficient bridge,
if you could be so kind as to post it.

Thanks,

Keith

Cecil Moore September 4th 03 12:57 AM

wrote:

Cecil Moore wrote:

1. The power reflected from the network input back toward the source. |s11|^2

2. The power transmitted through the network port toward the load. |s21|^2

3. The power re-reflected from the network output back toward the load. |s22|^2

4. The power transmitted through the network port toward the source. |s12|^2

These are the four powers you calculated and you consider only |s12|^2 to
be forward power. That is an error. |s22|^2 is also forward power. These
two forward power flow vectors have to be added to obtain the total
forward Poynting vector. I do believe that clears up the confusion.


Was not the discussion about powers on the source side?
Is not |s22|^2 on the load side?


Roy apparently doesn't realize it, but when he introduced 'x' into his
equations, he introduced 2-port analysis terms so, in his math, there are
indeed two powers existing on the load side. That's what got him confused
and he forgot to include those terms in his forward power and reflected
power, i.e. his rP is not all of the reflected power and his fP
is not all of the forward power. It's an easy mistake to make. But to
compound his mistake, he then combined those two terms on the load side,
one reflected and one forward, into one term so it is difficult to determine
which way its power flow vector points.
--
73, Cecil
http://www.qsl.net/w5dxp



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Reg Edwards September 4th 03 01:00 AM

What all you experts have forgotten is that SWR on a
lossless line is the ratio of two voltages, max and
min, SPACED APART BY 1/4-WAVELENGTH. That is if the
line is long enough to contain both a max and a min.

When the line is not lossless, ie., it has appreciable
attenuation in dB per 1/4-wavelength, then the ratio is
'distorted' and has a phase angle. So negative values
of indicated SWR can be expected at some values of |
Vmax | / | Vmin |

SWR is calculated from the square of | rho |. As I've
said before, immediately | rho | is squared, half the
information it contains is junked. Any
discussion/argument about power waves following
rho-squared on a lossy (a real ) line is meaningless
piffle.

Anybody who writes books about power waves, selling
them to make a living, is obtaining money under false
pretences.

On the other hand we should be kind to otherwise
unemployed Ph.D's. They too have wive's and kid's to
clothe, feed and provide a roof over their heads.
That's life!
---
Reg.




Cecil Moore September 4th 03 01:04 AM

Reg Edwards wrote:

Roy wrote -
If you have some other "rho" you want to
argue about, please call it something else.


- - - and while you are about it change the name of
the SWR meter.


Trouble is, (Z2-Z1)/(Z2+Z1) is not always equal to Sqrt(Pref/Pfwd)
What then?
--
73, Cecil http://www.qsl.net/w5dxp



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