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