If anyone is interested in really getting to the bottom of this endless
jousting, turn to page 136 of "Theory and Problems of Transmission Lines" by
Robert A. Chipman. This is a Schaum's Outline book - mine is dated 1968.
Many professionals acknowledge that this is one of the most succinct and
revealing accounts of t-line theory to be found. Mathematical enough to be
rigorous but readable and highly useful.


Starting in Section 7.6, Chipman derives the full set of equations for lines
with complex characteristic impedance. I will make no effort here to repeat
the development with ASCII non-equation symbols, but the bottom line is that
in the general case, Zo is indeed a complex number which can be highly
frequency-dependent.

Under the condition of certain combinations of physical parameters of the
line, Zo does indeed become actually real - the so-called Heaviside Line
where R/L=G/C where the symbols have the usual meanings - and independent of
frequency. This is the only case wherein a lossy line can have a real Zo.

Finally, he clearly shows how terminating an actual physical line
appropriately can result in a reflection coefficient as large as 2.41.

This revelation DOES NOT imply that the reflected wave would bear more power
than the incident wave. For a line to display this behavior, it must first
of all have a high attenuation per wavelength. Due to this high attenuation,
the power in the reflected wave is high for only a short distance from the
termination.

A couple of surprising consequences of this:

1. in order to terminate a line with complex Zo such that rho is greater
than 1, the reactance of the load must be equal and opposite to the reactive
term of Zo. In other words, the line and the load form a resonant circuit
separated from "the rest of the system" by the very lossy line.

2. calculation of the power at any point on a line with real Zo, lossy or
not, is simply Pf - Pr. But for a complex Zo, this is no longer true and a
much more complex set of equations - given by Chipman - must be used. See
his equations 7.34 and 7.35.

Finally, it should be understood that these effects are found almost
entirely on low-frequency transmission lines. Dealing with complex Zo is
routine with audio/telephone cable circuits and the like.

At HF, the reactive component of Zo for most common lines is so small as to
be safely and conveniently neglected. For example, RG-213 at 14 MHz has a Zo
of 50-j0.315 ohms. The same line at 1000 Hz has a Zo of 50-j35.733 ohms.
(Values taken from the TLDetails program)

When terminated in 50+j0 ohms, the SWR on the line is 2.012.
When terminated in 50-j35.733 ohms, the SWR is 1:1 as would be expected. But
when terminated in 50+j35.733 ohms, the SWR is a whopping 5.985.

RG-213 is nowhere near lossy enough to display the resonant-load effects
Chipman discusses, but these data give some idea of the perhaps unexpected
consequences of using even a common line like RG-213 at a low frequency.

Taken to 100 Hz, we find Zo = 50 - j 113.969 ohms and when terminated in 50
+ j 112.969, rho is determined to be 2.25839. Note that the termination is a
passive circuit in all these examples.

I urge anyone seriously interested in understanding transmission line theory
to include Chipman on their bookshelf. Despite its assumed low station as a
Schaum's Outline book, it provides a source of information and understanding
seldom matched by any text.

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!"







----- Original Message -----
From: "Dr. Slick"
Newsgroups: rec.radio.amateur.antenna
Sent: Wednesday, August 27, 2003 1:18 AM
Subject: Reflection Coefficient Smoke Clears a Bit


Hello,


Actually, my first posting:

Reflection Coefficient =(Zload-Zo)/(Zload+Zo)

was right all along, if Zo is always purely real. No argument there.



However, from Les Besser's Applied RF Techniques:

"For passive circuits, 0=[rho]=1,

And strictly speaking:

Reflection Coefficient =(Zload-Zo*)/(Zload+Zo)
Where * indicates conjugate.

But MOST of the literature assumes that Zo is real, therefore
Zo*=Zo."

This is why most of you know the "normal" equation.


And then i looked at the trusty ARRL handbook, 1993, page 16-2,
and lo and behold, the reflection coefficient equation doesn't have a
term for line reactance, so both this book and Pozar have indeed
assumed that the Zo will be purely real.


Here's a website that describes the general conjugate equation:


http://www.zzmatch.com/lcn.html



Additionally, the Kurokawa paper ("Power Waves and the
Scattering Matrix") describes the voltage reflection coefficient
as the same conjugate formula, but he rather foolishly calls it a
"power wave R. C.", which when the magnitude is squared, becomes the
power R. C.

Email me for the paper.



As Reg points out about the "normal" equation:


"Dear Dr Slick, it's very easy.

Take a real, long telephone line with Zo = 300 - j250 ohms at 1000 Hz.

(then use ZL=10+j250)

Magnitude of Reflection Coefficient of the load, ZL, relative to line
impedance

= ( ZL - Zo ) / ( ZL + Zo ) = 1.865 which exceeds unity,

and has an angle of -59.9 degrees.

The resulting standing waves may also be calculated.

Are you happy now ?"
---
Reg, G4FGQ



Well, I was certainly NOT happy at this revelation, and researched
it until i understood why the normal equation could incorrectly give
a R.C.1 for a passive network (impossible).

If you try the calculations again with the conjugate formula, you
will see that you can never have a [rho] (magnitude of R.C.)
greater than 1 for a passive network. You need to use the conjugate
formula if Zo is complex and not purely real.

How could you get more power reflected than what you put into
a passive network(do you believe in conservation of energy, or do
you think you can make energy out of nothing)? If you guys can tell
us, we could fix our power problems in CA!

Thanks to Reg for NOT trusting my post, and this is a subtle detail
that is good to know.


Slick

On Thu, 28 Aug 2003 06:27:40 GMT, "George, W5YR"
wrote:

... "Theory and Problems of Transmission Lines" by
Robert A. Chipman. This is a Schaum's Outline book - mine is dated 1968.
Many professionals acknowledge that this is one of the most succinct and
revealing accounts of t-line theory to be found. Mathematical enough to be
rigorous but readable and highly useful.


Hi George,

I have notice you recommended this author several times, and yet you
have casually dismissed his rather straightforward coverage relating
to the characteristic Z of a Transmitter:
There is no need to know, since its value, whatever it might be, plays no
role in the design and implementation of the external portion of the system
driven by the transmitter.

How do you reconcile this with his coverage entitled
"9.10. Return loss, reflection loss, and transmission loss."

You may wish to observe the clearly marked figure 9-26 and
specifically the paragraph that follows (or the entire section for
that matter) that quite clearly reveals what is everywhere else
implied: that ALL SWR discussion presumes a Zc matched source. You
may observe that Chapman thus refutes your statement above. Further,
Chapman goes to some length to describe the Smith Chart's appended
line evaluation scales at the bottom to this very matter.

To substantiate this from other sources I have offered a very simple
example that shows this importance that to date has defied "first
principle" analysis (not first principles however, merely the claim of
its being practiced analytically in this regard). I will offer it
again, lest you missed it.

The scenario begins:

"A 50-Ohm line is terminated with a load of 200+j0 ohms.
The normal attenuation of the line is 2.00 decibels.
What is the loss of the line?"

Having stated no more, the implication is that the source is matched
to the line (source Z = 50+j0 Ohms). This is a half step towards the
full blown implementation such that those who are comfortable to this
point (and is in fact common experience) will observe their answer and
this answer a

"A = 1.27 + 2.00 = 3.27dB"

"This is the dissipation or heat loss...."

we then proceed:

"...the generator impedance is 100+0j ohms, and the line is 5.35
wavelengths long."

Beware, this stumper has so challenged the elite that I have found it
dismissed through obvious embarrassment of either lacking the means to
compute it, or the ability to simply set it up and measure it. It
takes two resistors and a hank of transmission line, or what has been
described by one correspondent as:
There is no institutionalized ignorance, just a
lot of skepticism regarding the reliability of the
analysis methods and the measurement methods.
Clearly a low regard for many correspondent's abilities here, and
hardly a prejudice original to me. Imagine the incapacity of so many
to measure relative power loss - a CFA salesman's dream population.

Actually it is quite obvious several recognize that follow-through
would dismantle some cherished fantasies. Chapman clearly knocks the
underpinnings from beneath them without any further effort on my part.
But then, as you offer, they would merely dismiss it by confirming
another prejudice:
its assumed low station as a Schaum's Outline book

I would point out to all, that Chapman's material dovetails with what
would have been then current research and teachings of the National
Bureau of Standards. Prejudice has "refuted" those findings too. :-)

73's
Richard Clark, KB7QHC

Richard Clark wrote:
"...the generator impedance is 100+0j ohms, and the line is 5.35
wavelengths long."

What does the generator impedance have to do with line losses?
--
73, Cecil http://www.qsl.net/w5dxp



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On Thu, 28 Aug 2003 05:58:09 -0500, W5DXP
wrote:

Richard Clark wrote:
"...the generator impedance is 100+0j ohms, and the line is 5.35
wavelengths long."

What does the generator impedance have to do with line losses?

Hi Cecil,

From Chapman (you following this George?) page 28:

"It is reasonable to ask at this point how, for the circuit of
Fig. 3-1(b), page 18, on which the above analysis is based, there
can be voltage and current waves traveling in both directions on
the transmission line when there is only a single signal source.
The answer lies in the phenomenon of reflection, which is very
familiar in the case of light waves, sound waves, and water waves.
Whenever traveling waves of any of these kinds meet an obstacle,
i.e. encounter a discontinuous change from the medium in which
they have been traveling, they are partially or totally
reflected."
...
"The reflected voltage and current waves will travel back along
the line to the point z=0, and in general will be partially
re-reflected there, depending on the boundary conditions
established by the source impedance Zs. The detailed analysis of
the resulting infinite series of multiple reflections is given in
Chapter 8."

The Challenge that I have offered more than several here embody such
topics and evidence the exact relations portrayed by Chapman (and
others already cited, and more not). The Challenge, of course, dashes
many dearly held prejudices of the Transmitter "not" having a
characteristic source Z of 50 Ohms. Chapman also clearly reveals that
this characteristic Z is of importance - only to those interested in
accuracy.

Those hopes having been dashed is much evidenced by the paucity of
comment here; and displayed elsewhere where babble is most abundant in
response to lesser dialog (for the sake of enlightening lurkers no
less). Clearly those correspondents hold to the adage to choose
fights you can win. I would add so do I! The quality of battle is
measured in the stature of the corpses littering the field. :-)

So, Cecil (George, Peter, et alii), do you have an answer? Care to
take a measure at the bench? As Chapman offers, "just like optics."
Shirley a man of your erudition can cope with the physical proof of
your statements. ;-)

The only thing you and others stand to lose is not being able to
replicate decades old work. Two resistors and a hank of line is a
monumental challenge.

73's
Richard Clark, KB7QHC

Richard Clark wrote:
So, Cecil (George, Peter, et alii), do you have an answer?

Years ago, I had a discussion with Jeff, WA6AHL, here on this
newsgroup. I suggested that the impedance looking back into
the source might be Vsource/Isource, i.e. the transformed
dynamic load line. However, I have never taken a strong stand
on source impedance. If reflections are blocked from being
incident upon the source, as they are in most Z0-matched
systems, the source impedance doesn't matter since there
exists nothing to reflect from the source impedance.

My basic approach is to achieve a Z0-match and therefore
forget about source impedance.
--
73, Cecil http://www.qsl.net/w5dxp



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On Thu, 28 Aug 2003 12:18:44 -0500, W5DXP
wrote:

Richard Clark wrote:
So, Cecil (George, Peter, et alii), do you have an answer?

Years ago, I had a discussion with Jeff, WA6AHL, here on this
newsgroup. I suggested that the impedance looking back into
the source might be Vsource/Isource, i.e. the transformed
dynamic load line. However, I have never taken a strong stand
on source impedance. If reflections are blocked from being
incident upon the source, as they are in most Z0-matched
systems, the source impedance doesn't matter since there
exists nothing to reflect from the source impedance.

My basic approach is to achieve a Z0-match and therefore
forget about source impedance.

Hi Cecil,

That's all fine and well. It exhibits a rather standard behavior and
confirms conventional expectations. I take by this response that you
have no interest in the confirmation of interference in both Optical
and RF metaphors being visited at the bench. That is fine too. It is
a rather tough example to replicate - except when stumbled upon, then
we hear cries for exorcism being needed (my cue).

My missives simply offer touchstones of clarity in contrast to the
murky sea of un-fettered statements. We are presented with fantastic
notions that the characteristic source Z of a transmitter is
unknowable, and this statement is usually closely allied to the notion
that this same "unknowable" Z is actually responsible for reflecting
all power arriving at the antenna terminal. Few of those who utter
these witless jokes have any response to the straight line "So what is
this Z that does all that reflecting?" In their chagrin, they fail
even to repeat "it is unknowable...." Absolutely none can venture a
guess that it is either: "much less than 50 Ohms," or it is "much more
than 50 Ohms." This would be two obvious rejoinders and yet neither
is uttered. Such is faith. The universal silence condemns their
specious claims absolutely.

These absurd notions deserve a hearty laugh, because it invalidates
the need for a tuner which is purposely inserted between the source
and load to serve that very purpose (and which you describe as your
typical habit which is a nearly universal application).

But, again, this discussion is generally reserved only for those
interested in accuracy. :-)

73's
Richard Clark, KB7QHC

Richard Clark wrote:
But, again, this discussion is generally reserved only for those
interested in accuracy. :-)

Like I say, my solution is to block any reflections from being
incident upon the source. But I have a question. Since we are
discussing coherent sine waves, it seems to me that any reflection
from the source impedance will become indistinguishable from the
generated wave. In fact, the present convention of generated power
equals forward power minus reflected power is designed to overcome
that very problem.
--
73, Cecil, W5DXP

On Thu, 28 Aug 2003 12:18:44 -0500, W5DXP
wrote:

Richard Clark wrote:
So, Cecil (George, Peter, et alii), do you have an answer?

Years ago, I had a discussion with Jeff, WA6AHL, here on this
newsgroup. I suggested that the impedance looking back into
the source might be Vsource/Isource, i.e. the transformed
dynamic load line. However, I have never taken a strong stand
on source impedance. If reflections are blocked from being
incident upon the source, as they are in most Z0-matched
systems, the source impedance doesn't matter since there
exists nothing to reflect from the source impedance.

My basic approach is to achieve a Z0-match and therefore
forget about source impedance.

Hi Cecil,

That's all fine and well. It exhibits a rather standard behavior and
confirms conventional expectations. I take by this response that you
have no interest in the confirmation of interference in both Optical
and RF metaphors being visited at the bench. That is fine too. It is
a rather tough example to replicate - except when stumbled upon, then
we hear cries for exorcism being needed (my cue).

My missives simply offer touchstones of clarity in contrast to the
murky sea of un-fettered statements. We are presented with fantastic
notions that the characteristic source Z of a transmitter is
unknowable, and this statement is usually closely allied to the notion
that this same "unknowable" Z is actually responsible for reflecting
all power arriving at the antenna terminal. Few of those who utter
these witless jokes have any response to the straight line "So what is
this Z that does all that reflecting?" In their chagrin, they fail
even to repeat "it is unknowable...." Absolutely none can venture a
guess that it is either: "much less than 50 Ohms," or it is "much more
than 50 Ohms." This would be two obvious rejoinders and yet neither
is uttered. Such is faith. The universal silence condemns their
specious claims absolutely.

These absurd notions deserve a hearty laugh, because it invalidates
the need for a tuner which is purposely inserted between the source
and load to serve that very purpose (and which you describe as your
typical habit which is a nearly universal application).

But, again, this discussion is generally reserved only for those
interested in accuracy. :-)

73's
Richard Clark, KB7QHC

On Thu, 28 Aug 2003 12:18:44 -0500, W5DXP
wrote:

My basic approach is to achieve a Z0-match and therefore
forget about source impedance.

Hi Cecil,

This is a cavalier attitude if you can afford it. Otherwise, those
who so desperately hammer out the last 0.1 dB antenna gain are going
to fall to their knees in wrack when they discover that their rig's
characteristic Z of, say, 70 Ohms meeting the discontinuity of their
low pass filter's 50 Ohms turns that effort into heat behind the
antenna jack.

I have long since stopped being surprised by those who spin on like
whirling dervishes over trivial matters in the face of 10 fold losses
in front of them. This, of course, is even more trivial when they
gush on about their premium equipment that behind the knobs
"efficiently" transforms 20 - 25 Amperes of DC current into 100 Watts
RF. Now, that puts perspective to the topic: smoke and reflection
coefficient.

73's
Richard Clark, KB7QHC

Richard Clark wrote:

W5DXP wrote:
My basic approach is to achieve a Z0-match and therefore
forget about source impedance.

This is a cavalier attitude if you can afford it.

It's all part of my "Work Smarter, Not Harder" nature. The elimination
of reflected energy incident upon the source is extremely rewarding
in multiple ways.
--
73, Cecil, W5DXP