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:

http://www.qsl.net/ac6la/adhoc/Rho_C..._Conjugate.gif
(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:

http://www.qsl.net/ac6la/adhoc/SWR_C..._Conjugate.gif
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.

When I asked him recently about this, he stated that the article is

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