A Subtle Detail of Reflection Coefficients (but important to know)
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August 21st 03, 06:44 PM
(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
Whether you find the reflection with Vr/Vi, or (
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
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,
|rho(Zo)| = 0 [classic formula, Zo in the numerator]
|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
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
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
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
(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?
[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
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)
|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.
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