(Dan) wrote in message . com...

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.

Whether you find the reflection with Vr/Vi, or (Pr/Pi)**0.5, the

impedances should still be the same.

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.

My 1993 70th ed. of the ARRL handbook assumes the Zo to be always

purely real.

I believe the Zo* version is correct. The purely real Zo version

is correct too, but Zo must be purely real.

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.

I think i does make sense in the sense that if the source

(reference) and the load both have reactance, that there WILL be some

reflections.

If ZL=Zo*=50+j2.396, then the capacitive reactance is indeed

canceled, and you are matching a pure 50 ohms to a pure 50 ohms again,

and the numerator will be zero, as it should be.

This is what impedance matching is all about really, not just

getting the real part of the imedance the same, but cancelling any

reactance.

The conjugate formula is correct.

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.
This would be due to the losses of the line?

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.

I'm not totally sure if you did this right, but if the

transforming transmission line had reactance in it, and you are

measuring everthing from

Zo=50-j2.396, then i would expect the rho to swing up and down the

same way every 1/2 wavelength, as your data shows.

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)

If you had used the conjugate Zo* formula, you get

[rho] = 0.99989 which matches the bottom result.

I stand by the Conjugate formula, even more now.

Slick