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 Dan Posts: n/a It is my opinion that the confusion over whether to use Zo or Zo*
(conjugate of Zo) in the computation for reflection coefficient arises
because there are two different meanings for the reflection
coefficient itself: one applies to voltage or current waves and the
other applies to "power waves." I do not have the Besser text
mentioned above but a 1965 IEEE paper by Kurokawa also uses the Zo*
term to calculate the reflection coefficient. However, Kurokawa makes
it clear that he is referring to "power waves" and not voltage or
current waves.

The Kurokawa paper was given as the justification for what I believe
is an erroneous equation in the 19th edition of the ARRL Antenna Book.
In all previous editions (at least the ones that I have) the formula
for reflection coefficient uses the normal Zo term. In the 19th
edition the formula was changed to use the Zo* (Zo conjugate) term.

I did some research on this and exchanged emails with some smart
folks, including Tom Bruhns and Bill Sabin. Then I wrote a note to
Dean Straw, editor of the Antenna Book, explaining why I thought the
new formula in the 19th edition was wrong. Here's a copy of that
note:

===========================================
Email to Dean Straw, 10/5/01

Dear Dean,

A week or so ago I wrote you concerning the formula for rho in the
19th Antenna Book:

rho = (Za-Zo*)/(Za+Zo) [Eq 6, page 24-7]

where Za is the impedance of the load, Zo is the line characteristic
impedance, and Zo* is the complex conjugate of Zo. You replied that
the justification for using Zo* in the numerator is explained in the
1965 IEEE paper by Kurokawa, and that it didn't really make much
difference whether the "classic" formula (Zo in numerator) or the
"conjugate" formula (Zo* in numerator) was used at SWR levels under
100 or so.

I obtained and studied the Kurokawa paper, did some research on the
Internet, exchanged some emails with some folks who know more about
this stuff than I do, and read through all the other technical
literature I have concerning rho. I'm afraid I disagree with both of
your statements (1. Justified by Kurokawa; 2. Doesn't matter for
normal SWR levels). Here's why:

An -infinitely- long line will have zero reflections (|rho|=0). If a
line of -finite- length is terminated with a load ZL which is exactly
equal to the Zo of the line, the situation will not change, there
should still be zero reflections. So if the formula for rho is
rho = (ZL-Zo)/(ZL+Zo)
then |rho| = 0, since the numerator evaluates to 0+j0.
However, if the formula is
rho = (ZL-Zo*)/(ZL+Zo)
then |rho| evaluates to something other than 0, since the
numerator evaluates to 0-j2Xo.

To use an example with real numbers: RG-174, Ro=50, VF=.66, Freq=3.75
MHz, matched line loss = 1.511 dB/100 ft (at 3.75 MHz), which yields a
calculated Zo (at this Freq) of Zo=50-j2.396. If ZL = Zo = 50-j2.396,
then:
|rho(Zo)| = 0 [classic formula, Zo in the numerator]
and
|rho(Zo*)| = 0.0479 ["conjugate" formula, Zo* in the numerator]

Using SWR=(1+|rho|) / (1-|rho|), these two rho magnitude values
evaluate to SWR 1:1 and SWR 1.10:1, respectively. So if a line is
terminated with a load equal to Zo, which is equivalent to an infinite
line, the "conjugate" formula results in a rho magnitude greater than
0 and an SWR greater than 1. This doesn't seem to make intuitive
sense.

This same anomaly may be extended to loads of other than Zo and to
points other than just the load end of the line. Using the Zo for
RG-174 as stated above (Zo=50-j2.396), and an arbitrary but totally
realistic load of ZL=100+j0 ohms (nominal 2:1 SWR), I used the full
hyperbolic transmission line equation to calculate what the Zin would
be at points along the line working back from the load from 0° to 360°
(one complete wavelength) in 15° steps. I then calculated the
magnitude of both rho(Zo) [classic formula] and rho(Zo*) [conjugate
formula] using the Zin values, and plotted the results. Here's the
plot:

(The scale for rho is on the right. The left scale is normally used
for R, X, and |Z|, but those plot lines have been intentionally hidden
in this case just to reduce the chart clutter.) Note that the plot
line for rho(Zo) [classic formula] progresses downward in a smooth
fashion as the line length increases, as expected. The rho(Zo*)
[conjugate formula] swings around, and even goes above the value at
the load point until a line length of about 75° is reached. Again,
this doesn't seem to make intuitive sense, and I can think of no
physical explanation which would result in the voltage reflection
coefficient magnitude "swinging around" as the line length is
increased.

Of course, the same data may be used to calculate and then plot SWR.
Here's the plot:
Again, this doesn't seem intuitive, and this is for a load SWR much
less than 100.

Now if the intent of the "conjugate" formula was to always force rho
to be = 1 and therefore to avoid the "negative SWR" problem, it
appears that this has the effect of "throwing the baby out with the
bathwater." That is, it may make it possible to calculate a rho value
less than 1 and hence a non-negative SWR value in an "extreme load"
situation like ZL=1+j1000 ohms (even though SWR is pretty meaningless
in that case). However, it also changes the rho and SWR values for
completely reasonable loads, such as the example above. At a line
length of 45°, the impedance at the input end of the line is
41.40-j31.29. Using the "classic" rho formula results in calculated
rho and SWR values of
rho=0.3095 SWR=1.90
while the "conjugate" formula gives
rho=0.3569 SWR=2.11
Note that these results are for a perfectly reasonable load on a
perfectly reasonable line at a perfectly reasonable frequency, but the
results differ by an unreasonable amount.

Another point. In the William Sabin article, "Computer Modeling of
Coax Cable Circuits" (QEX, August 1996, pp 3-10), Sabin includes the
Kurokawa paper as a reference. Even with that reference, Sabin gives
the "classic" formula for rho (called gamma in his paper) as Eq 31.
correct and he stands by the given formula for rho.

Given these various intuitive arguments as to why computing rho with
the Zo conjugate formula doesn't make sense, where did it come from?
Well, so far I have two candidates:

1) A QST Technical Correspondence article by Charlie Michaels (Nov
1997, pg 70). Michaels gave a formula for computing the portion of
the loss on a line that is due to standing waves. That loss formula
involves calculating rho by using the "conjugate" formula. The SWR dB
loss result, when added to the normal matched line loss number, gives
exactly the same figure for total power loss as do other formulas that
use completely different techniques (such as in papers by Sabin and
Witt). However, the Michaels QST article never said that the rho
"conjugate" formula should be used to calculate rho in the general
case, only that it should be used as part of an intermediate step to
calculate a dB number.

2) The 1965 IEEE paper by Kurokawa, "Power Waves and the Scattering
Matrix." Kurokawa does indeed show a formula for -a- reflection
coefficient that uses Z conjugate in the numerator. However, in
Section I of his paper he explains that he is talking about "power
waves" and takes pains to explain that these waves are not the same as
the more familiar voltage and current traveling waves. He then goes
on to give a mathematical description of these power waves. In
Section III he defines the power wave reflection coefficient as

s = (ZL - Zi*) / (ZL + Zi)

where ZL is the load impedance and Zi is the internal impedance of the
source. In a footnote he makes it clear that "s" is equal to the
voltage reflection coefficient only when Zi is real (no jX component).
Finally, in Section IX ("Comparison with Traveling Waves") he
explains that when the line Zo is complex the calculations that apply
to voltage and current waves are not the same calculations used to
determine the power delivered. He ends this section with this
statement: "Further, since the traveling wave reflection coefficient
is given by (ZL-Zo)/(ZL+Zo) [note no conjugate] and the maximum power
transfer takes place when ZL=Zo*, where ZL is the load impedance, it
is only when there is a certain reflection in terms of traveling waves
that the maximum power is transferred from the line to the load."

To put some actual numbers with this statement, consider the RG-174
from above (Zo=50-j2.396), with a load of ZL=Zo*=50+j2.396. Then

|(ZL-Zo)/(ZL+Zo)| = 0.0479
= small voltage traveling wave reflection
coefficient

while

|(ZL-Zo*)/(ZL+Zo)| = 0
= zero power wave reflection, meaning maximum
power transfer.

Now it seems to me that this clears up the confusion. It looks like
it is necessary to consider -two different meanings- for rho. One is
for the voltage (or current) traveling wave reflection coefficient.
The classic formula to compute that still holds (as Kurokawa states),
and that is the rho that should be used when talking about voltage (or
current) standing waves on a line. Specifically, that is the rho that
should be used in the formula for SWR,
SWR = 1 + |rho| / 1 - |rho|
possibly with an explanation that this formula is only applicable when
|rho| 1.

The -other- meaning for rho is used when dealing with "power waves" or
with power and loss calculations as in the Michaels QST formula, and
-that- rho (call it rho prime, or maybe some other letter ala
Kurokawa) may be defined as
rho' = (Z-Zo*)/(Z+Zo)
However, rho' does -not- have to do with voltage (or current)
traveling waves, and may -not- be used to compute SWR. If it is
understood that there are two different "reflection coefficients" then
everything starts to fall in place, including the last part of the
Kurokawa quote above saying that there is a situation when the
"voltage" reflection coefficient is slightly greater than 0 while the
"power wave" reflection coefficient is exactly 0. Note that no matter
what combination of values for Zo and ZL are used, |rho'| will never
be greater than 1 (although it can be equal to 1 for purely reactive
loads), thus satisfying the intuitive understanding that there can
never be more power reflected -from- a (passive) load than is
delivered -to- a (passive) load. But at the opposite extreme, as
shown above, "no reflected power" but does -not- necessarily mean "no
standing waves."

A further example of the importance of making a distinction between
the voltage reflection coefficient and the power reflection
coefficient would be the following: Assume a load of ZL=1+j1000 with
the RG-174 Zo from above, Zo=50-j2.396. Then

|rho| = 1.0047 (voltage wave reflection coefficient)

and

|rho'| = 0.9999 (power wave reflection coefficient)

This shows that the reflected voltage is slightly greater than the
incident voltage, at least at the point of reflection, before the line
loss has caused the calculated (or measured) rho to decay. It further
shows that the reflected power is still less than the incident power,
thus not violating the principal of conservation of energy. Robert
Chipman, "Theory and Problems of Transmission Lines," presents a
mathematical proof of this, including this quote from page 138: "...
a transmission line can be terminated with a [voltage] 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."

In summary, I think a distinction must be made between the "voltage"
reflection coefficient and the "power" reflection coefficient, and
therefore I think the following changes should be made to the 19th
edition of the Antenna Book:

1. Revise the initial equation for rho [Eq 6, page 24-7] back to the
classic "non-Zo*" form, since rho is used in this context as the
voltage reflection coefficient.

2. Equation 11 on page 24-9 is
rho = sqrt(Pr/Pf)
where Pr and Pf are the reflected and forward power levels.
Intuitively this seems to be the "second" definition for rho, namely
the power reflection coefficient, although my math skills are not up
to the task of proving that this formula is the equivalent of the
Kurokawa formula for the power reflection coefficient 's'. Perhaps
this formula should have a footnote indicating that it refers to the
"power" and not "voltage" reflection coefficient, and that the two are
technically equal only when the Xo component of the line Zo is
ignored. (The same point as is made in the Kurokawa footnote referred
to above.) Given the precision to which most amateurs can measure
power, and the fact that under normal circumstances the line loss and
hence the Xo value is much smaller than that of RG-174, of course this
point is moot in a practical sense.

3. If Equation 11 is for the "power" reflection coefficient, then
Equation 12 is a mixing of apples and oranges. Perhaps the second
equal sign could be replaced with an "almost equal" sign.

4. Equation G in Table 2 "Coaxial Cable Equations" on page 24-20
should remain as is, since it obviously is referring to the voltage
reflection coefficient.

Thanks for looking this over, Dean. I would certainly welcome any
comments or feedback you might have.

73,
Dan Maguire AC6LA
===========================================

I don't know if Dean has changed the formula in later printings of the
19th edition or in the upcoming 20th edition. He responded that he
was busy with other matters and would get back to me later. He never
did and I let the matter drop.

I have the Kurokawa paper in pdf format. If anyone would like a copy,
drop me a private email and I'll be glad to send it to you. The two
charts mentioned above were produced with a modified version of the
XLZIZL Excel application. XLZIZL is available free from
www.qsl.net/ac6la.

Dan, AC6LA