Now that the various typo mistakes have been corrected, and putting
aside for the moment the name calling and ad hominem arguments, could
it be that _both_ sides in this discussion are correct? Camp 'A' says
that the reflection coefficient is computed the classical way, without
using Zo conjugate, and offers various mathematical proofs and
discussions of infinitely long lines. Camp 'B' says the reflection
coefficient is computed with Zo* (Zo conjugate) in the numerator, and
offers explanations dealing with the conservation of energy and
maximum transfer of power.

Both sides may be correct since they are talking about _two different_
meanings for the term "reflection coefficient." One has to do with
voltage (or current) traveling waves and the other has to do with
power. Quoting two references:

###

From Chipman, "Theory and Problems of Transmission Lines," 1968:

Section 7.1, Reflection coefficient for voltage waves:
[Discussion and math, then] ...
rho = (ZL - Zo) / (ZL + Zo)

Section 7.6, Complex characteristic impedance
[Various mathematical manipulations, then] ... "the maximum possible
value for |rho| is found to be 1 + sqrt(2) or about 2.41. ... [T]he
principal of conservation of energy is not violated even when the
magnitude of the [voltage wave] reflection coefficient exceeds unity."

[then more math, then] ... "The conclusion is somewhat surprising,
though inescapable, that a transmission line can be terminated with a
[voltage wave] 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."

[then a discussion of a source with internal impedance Zo connected to
a line with characteristic impedance also Zo that is terminated with a
load of impedance ZL, then] "... more power will be delivered to a
terminal load impedance Zo* [conjugate of Zo] that produces a
reflected [voltage] wave on the line than to a terminal load impedance
Zo that produces no reflected [voltage] wave."

So Chipman states quite clearly that zero reflected voltage wave
magnitude does _not_ mean maximum power transfer. On the contrary,
maximum power is transferred only when there is a non-zero voltage
wave reflection (assuming a complex Zo line). Counter arguments along
the lines of "Well that doesn't seem right to me so therefore Chipman
must be wrong" don't carry much weight given Chipman's credentials.

###

From Kurokawa, "Power Waves and the Scattering Matrix," IEEE
Transactions on Microwave Theory and Techniques, March 1965:

Section 2, explanation of and mathematical definition of the concept
of "power waves," explicitly noted by the author to be distinct from
the more commonly discussed voltage and current traveling waves.

Section 3, definition of the reflection coefficient [for power waves]:
s = (ZL - Zo*) / (ZL + Zo)
with a footnote "[Only w]hen Zo is real and positive this is the
voltage wave reflection coefficient." Kurokawa takes pains to make it
clear that his "s" power wave reflection coefficient is not the same
as the (usually rho or Gamma) voltage wave reflection coefficient.

Section 9, comparison with [voltage and current] traveling waves:
"... since the [voltage or current] traveling wave reflection
coefficient is given by (ZL - Zo) / (ZL + Zo) [note no conjugate] and
the maximum power transfer takes place when ZL=Zo*, ... it is only
when there is a certain reflection in terms of [voltage or current]
traveling waves that the maximum power is transferred from the line to
the load."

So Kurokawa agrees with Chipman concerning the condition for maximum
power transfer. Kurokawa also defines two different reflection
coefficients, both in the same paper.

[In some of the above quotes I have altered the subscript letter
assigned to Z, merely for consistency between the two references.]

###

So, it seems to me, everybody can agree as long as it is understood
that there are different meanings for the term "reflection
coefficient." One meaning, and its mathematical definition, applies
to voltage or current waves. The other, with a slightly different
mathematical definition, applies to the power transfer from a line to
a load. They are one and the same only when the reactive portion of
Zo (Xo) is ignored. It may or may not be acceptable to do so,
depending on the attenuation of the line and the frequency. Lossy
lines and lower frequencies yield more negative values for the Xo
component of Zo.

You can use Reg's COAXPAIR or my TLDetails program to do the math and
show concrete examples. Try something like 100 feet of RG-174 at 0.1
MHz, terminated with loads equivalent to Zo and then Zo conjugate, and
compare the rho (or SWR) figures versus the power delivered to the
load for each case. When the termination equals Zo conjugate, note
that the total dB loss is actually _less_ than the matched line loss.
As counter intuitive as this may sound, Chipman offers an explanation
on page 139. (And as others are sure to point out, this makes
absolutely no difference in practical applications and is of academic
interest only.)

Copy of the Kurokawa paper, in pdf format, available on request via
private email. I've obtained copies of Chipman, on two separate
occasions, from Powell's in Portland.

Dan, AC6LA
www.qsl.net/ac6la/

Dan wrote:

So Chipman states quite clearly that zero reflected voltage wave
magnitude does _not_ mean maximum power transfer. On the contrary,
maximum power is transferred only when there is a non-zero voltage wave
reflection (assuming a complex Zo line). Counter arguments along the
lines of "Well that doesn't seem right to me so therefore Chipman must
be wrong" don't carry much weight given Chipman's credentials.

Thanks for that posting, Dan, and for what looks like a very clear
summary.

Just one point, about that very last word: what are Chipman's
"credentials", really? They are not that he is a well-known[*] textbook
author, college professor, PhD, or anything like that. This discussion
is already way overloaded with personal "credentials" of that kind!

Chipman's true credentials are that he has thought about this subject,
noticed the apparent problem, worked it out, and presented clear,
correct conclusions in a way that other people can follow. Those are the
only credentials that really count.


[*] Well-known to some college students in the USA, perhaps? I'd never
heard of him, but Dan's summary suggests this book might be worth
looking for.


--
73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB)
Editor, 'The VHF/UHF DX Book'
http://www.ifwtech.co.uk/g3sek

On Sun, 24 Aug 2003 12:22:34 +0100, "Ian White, G3SEK"
wrote:

[*] Well-known to some college students in the USA, perhaps? I'd never
heard of him, but Dan's summary suggests this book might be worth
looking for.

Hi Ian,

Any university student in course work relating to Transmission lines
would have a copy. It comes from a successful line of tutorials known
as "Schaum's Outlines."

Chipman also discusses the relevancy of the characteristic Z of a
source to SWR, which is tucked away in the unread part. ;-)

73's
Richard Clark, KB7QHC

"Richard Clark" wrote in message
...
Chipman also discusses the relevancy of the characteristic Z of a
source to SWR, which is tucked away in the unread part. ;-)

73's
Richard Clark, KB7QHC

Richard,

There used to be a Dr. Chipman who taught a fields/waves course at the
University of Toledo (OH) in the 60s. Do you know if it is the same guy?

Tam/WB2TT

Dan wrote:
Now that the various typo mistakes have been corrected, and putting
aside for the moment the name calling and ad hominem arguments, could
it be that _both_ sides in this discussion are correct? Camp 'A' says
that the reflection coefficient is computed the classical way, without
using Zo conjugate, and offers various mathematical proofs and
discussions of infinitely long lines. Camp 'B' says the reflection
coefficient is computed with Zo* (Zo conjugate) in the numerator, and
offers explanations dealing with the conservation of energy and
maximum transfer of power.

Both sides may be correct since they are talking about _two different_
meanings for the term "reflection coefficient." One has to do with
voltage (or current) traveling waves and the other has to do with
power.

If a lossy line is terminated with the complex Z0,
there is no reflection from the load, but the
maximum possible power is not delivered. If the
*load* is made equal to the complex conjugate of
Z0 the maximum *forward power* is delivered but
there is a reflected power (VSWR is not 1:1).

It is difficult to say that the maximum *power* is
delivered without knowing the generator impedance,
since it is involved in any so-called "conjugate
match". For a lossy line, the idea of conjugate
match is, at best, very approximate anyway. And
generator impedance is a mystery in most, but not
all, transmitter PA situations. One possible
exception: a large amount of negative feedback
helps to determine, to some extent, output
impedance, for a signal with time-varying
amplitude (e.g. SSB).

It seems to me to be clear that the use of Z0* in
the reflection coefficient equation has not been
corroborated (see Roy's post), but the use of
ZL=ZO* has been. The two ideas are not equivalent.

After looking at some examples, using the exact
complex hyperbolic equations with Mathcad, it is
obvious that a line must be very lossy to make a
significant difference whether ZL or ZL* is used
to terminate the line. Still, it is important to
understand the basic principles involved, so this
exercise is not foolishness at all.

A word about "credentials". We all respect
established and competent authors. But I have
noticed on several occasions that blind faith has
some exceptions. As an experienced author, I am
personally familiar with this problem.

G. Gonzalez (highly respected) "Microwave
Transistor Amplifiers" second edition, has a good
discussion of Power Waves, based on Kurokawa (I
also have his article). There are no transmission
lines, and the term ZS* (ZS=generator impedance)
is used. In particular, a power wave reflection
coefficient is defined:

Gp = (ZL-ZS*)/(ZL+ZS)

which looks quite familiar, with ZS replacing Z0.
Also, a voltage reflection coefficient:

Gv = [Zs/Zs*] x [(ZL-ZS*)/(ZL+ZS)]

and a current reflection coefficient

Gv = Gp.

The author also defines two-port scattering
parameters in terms of power waves, in which ZS*
and ZL* appear.

For the purposes of the present topic, involving
transmission lines, it seems best to stay away
from power waves, without a lot more studying.

Bill W0IYH

William E. Sabin wrote:
Dan wrote:

Now that the various typo mistakes have been corrected, and putting
aside for the moment the name calling and ad hominem arguments, could
it be that _both_ sides in this discussion are correct? Camp 'A' says
that the reflection coefficient is computed the classical way, without
using Zo conjugate, and offers various mathematical proofs and
discussions of infinitely long lines. Camp 'B' says the reflection
coefficient is computed with Zo* (Zo conjugate) in the numerator, and
offers explanations dealing with the conservation of energy and
maximum transfer of power.

Both sides may be correct since they are talking about _two different_
meanings for the term "reflection coefficient." One has to do with
voltage (or current) traveling waves and the other has to do with
power.


If a lossy line is terminated with the complex Z0, there is no
reflection from the load, but the maximum possible power is not
delivered. If the *load* is made equal to the complex conjugate of Z0
the maximum *forward power* is delivered but there is a reflected power
(VSWR is not 1:1).

It is difficult to say that the maximum *power* is delivered without
knowing the generator impedance, since it is involved in any so-called
"conjugate match". For a lossy line, the idea of conjugate match is, at
best, very approximate anyway. And generator impedance is a mystery in
most, but not all, transmitter PA situations. One possible exception: a
large amount of negative feedback helps to determine, to some extent,
output impedance, for a signal with time-varying amplitude (e.g. SSB).

It seems to me to be clear that the use of Z0* in the reflection
coefficient equation has not been corroborated (see Roy's post), but the
use of ZL=ZO* has been. The two ideas are not equivalent.

After looking at some examples, using the exact complex hyperbolic
equations with Mathcad, it is obvious that a line must be very lossy to
make a significant difference whether ZL or ZL*

Correction: Z0 or Z0*

is used to terminate the
line. Still, it is important to understand the basic principles
involved, so this exercise is not foolishness at all.

A word about "credentials". We all respect established and competent
authors. But I have noticed on several occasions that blind faith has
some exceptions. As an experienced author, I am personally familiar with
this problem.

G. Gonzalez (highly respected) "Microwave Transistor Amplifiers" second
edition, has a good discussion of Power Waves, based on Kurokawa (I also
have his article). There are no transmission lines, and the term ZS*
(ZS=generator impedance) is used. In particular, a power wave reflection
coefficient is defined:

Gp = (ZL-ZS*)/(ZL+ZS)

which looks quite familiar, with ZS replacing Z0. Also, a voltage
reflection coefficient:

Gv = [Zs/Zs*] x [(ZL-ZS*)/(ZL+ZS)]

and a current reflection coefficient

Gv = Gp.

Correction: Gi = Gp


The author also defines two-port scattering parameters in terms of power
waves, in which ZS* and ZL* appear.

For the purposes of the present topic, involving transmission lines, it
seems best to stay away from power waves, without a lot more studying.

Bill W0IYH

(Dan) wrote in message . com...

From Kurokawa, "Power Waves and the Scattering Matrix," IEEE
Transactions on Microwave Theory and Techniques, March 1965:

Section 2, explanation of and mathematical definition of the concept
of "power waves," explicitly noted by the author to be distinct from
the more commonly discussed voltage and current traveling waves.

Section 3, definition of the reflection coefficient [for power waves]:
s = (ZL - Zo*) / (ZL + Zo)
with a footnote "[Only w]hen Zo is real and positive this is the
voltage wave reflection coefficient." Kurokawa takes pains to make it
clear that his "s" power wave reflection coefficient is not the same
as the (usually rho or Gamma) voltage wave reflection coefficient.


I've got the same paper. It's a bit confusing, because then he
calls

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2 the "power reflection
coefficent".

Where if you take the square root to the power reflection
coefficient, you should get the voltage r. c.

So perhaps s = (ZL - Zo*) / (ZL + Zo) really IS the voltage R.
C., even in this paper!





Section 9, comparison with [voltage and current] traveling waves:
"... since the [voltage or current] traveling wave reflection
coefficient is given by (ZL - Zo) / (ZL + Zo) [note no conjugate] and
the maximum power transfer takes place when ZL=Zo*, ... it is only
when there is a certain reflection in terms of [voltage or current]
traveling waves that the maximum power is transferred from the line to
the load."



Perhaps he is confused on this point, because he previously
stated:

"When the matching condition (4) is satisfied , the power
reflection
coefficient becomes zero, as expected."

Condition (4) is the Zl=Zo*, which is also called the conjugate
matched condition for maximum power delivered to the load.

It's only when you have a simultaneous conjugate match with a
transistor, that you get the max gain.



So Kurokawa agrees with Chipman concerning the condition for maximum
power transfer. Kurokawa also defines two different reflection
coefficients, both in the same paper.


Again, you failed to mention that he defined:

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2 the "power reflection
coefficent".

Note the squares.


Copy of the Kurokawa paper, in pdf format, available on request
via
private email.


Slick

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

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2 the "power reflection
coefficent".

Note the squares.

yes, please do note the squares.... and remember, just because

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2
does NOT mean that
s = (ZL - Zo*) / (ZL + Zo)

this is the one big trap that all you guys that like to use power in your
calculations fall into. just because you know the power doesn't mean that
you know squat about the voltage and current on the line. you can not work
backwards. that is why it is always better to work with voltage or current
waves and then in the end after you have solved all those waves you can
always calculate power if you really need to know it.

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

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2 the "power reflection
coefficent".

Note the squares.

yes, please do note the squares.... and remember, just because

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2
does NOT mean that
s = (ZL - Zo*) / (ZL + Zo)

this is the one big trap that all you guys that like to use power in your
calculations fall into. just because you know the power doesn't mean that
you know squat about the voltage and current on the line. you can not work
backwards. that is why it is always better to work with voltage or current
waves and then in the end after you have solved all those waves you can
always calculate power if you really need to know it.


yes, but he does say that s = (ZL - Zo*) / (ZL + Zo) , first.

But he foolishly calls it a "power wave R. C."

Then he squares the magnitudes [s]**2 = [(ZL - Zo*) / (ZL +
Zo)]**2

And calls this the "power R. C."


The bottom label is fine, we've all see this before, as the ratio
of the RMS incident and reflected voltages, when squared, should give
you the ratio of the average incident and reflected powers, or the
power R. C.


But to call the voltage reflection coefficient a "power wave R.
C."
is really foolish, IMO.


Slick

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

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2 the "power reflection
coefficent".

Note the squares.

yes, please do note the squares.... and remember, just because

[s]**2 = [(ZL - Zo*) / (ZL + Zo)]**2
does NOT mean that
s = (ZL - Zo*) / (ZL + Zo)

this is the one big trap that all you guys that like to use power in
your
calculations fall into. just because you know the power doesn't mean
that
you know squat about the voltage and current on the line. you can not
work
backwards. that is why it is always better to work with voltage or
current
waves and then in the end after you have solved all those waves you can
always calculate power if you really need to know it.


yes, but he does say that s = (ZL - Zo*) / (ZL + Zo) , first.

But he foolishly calls it a "power wave R. C."

Then he squares the magnitudes [s]**2 = [(ZL - Zo*) / (ZL +
Zo)]**2

And calls this the "power R. C."


The bottom label is fine, we've all see this before, as the ratio
of the RMS incident and reflected voltages, when squared, should give
you the ratio of the average incident and reflected powers, or the
power R. C.


But to call the voltage reflection coefficient a "power wave R.
C."
is really foolish, IMO.


Slick

i don't know what he is refering to as the 'power wave rc' but its not the
voltage or current reflection coefficient, they do not have a conjugate in
the numerator.