On Feb 2, 4:47 am, (J. B. Wood) wrote:
In article .com,

"K7ITM" wrote:
On Jan 30, 11:13 pm, "camelot" wrote:
Hi Tom,
well, after few researches on several books, I found that the formula
I provided by me works only for real Z0. The general formula valid in
case Z0 is complex is the follow:

S11=(Zin-Z0*)/(Zin+Z0)

Hello, and I don't know where you obtained that formula but it's
incorrect. S11 in terms of a reflection coefficient is given by your
formula above but without the complex conjugate of Z0. I have seen
reflection coefficients in technical journals defined with a complex
conjugate Z0 as you have shown but that's not consistent with scattering
or transmission line theory (unless Z0 is real). I know it seems
counterintuitve but a source of complex Z0 would be matched (no
voltage/current reflections) to a transmission line having the same
characteristic impedance. This condition does not in general correspond
to the condition of maximum power transfer from source to line.
Conversely a line of complex Z0 impedance connected to a source of complex
Z0* impedance represents maximum power transfer from source to line but we
still have voltage and current "reflections" RELATIVE to Z0. What you
have to keep in mind is that incident (forward) and reflected
voltages/currents only have meaning when they are referenced to an
impedance, say Z0. And to monitor steady-state incident and reflected
waves you need to separate the steady-state voltage (or current) into
these components using a bridge or directional coupler (sampling devices
that are also designed to use Z0 (e.g. 50 + j0 ohms) as a reference).

In most applications what I've said is moot since line impedance is
usually real or very close to real and we are dealing with sources having
real or very close to real impedances. Under these conditions the matched
condition coincides with maximum power transfer.

If you want an in-depth treatment of what I've attempted to summarize I
recommend the chapter on circuit analysis in the "Electronic Designers'
Handbook", ed. E.J. Giacoletto, published by McGraw-Hill. Sincerely,

John Wood (Code 5550) e-mail:
Naval Research Laboratory
4555 Overlook Avenue, SW
Washington, DC 20375-5337


To amplify just a bit on what John wrote, it's convenient to express
the S-parameters referenced to an impedance that's accepted by other
people you want to exchange data with, so that they don't have to
transform your data to match their reference. 50 ohms resistive is
pretty universally accepted in the non-television RF industry. Bear
in mind that the S parameters work just fine, even though the
transmission lines you are working with may not be exactly 50 ohms
resistive, or even close to 50 ohms. In such a case, of course, S11
will not have the same meaning as a reflection coefficient for that
line.

Part of the "universally accepted" aspect is that the vast majority of
RF test equipment is built to take readings referenced to 50 ohms
resistive. We take great care to make sure that things look as close
to 50+j0 as practical, over a wide frequency range. That said, there
are particular industries where other impedances are used. 75 ohms is
used in the video and television industry more commonly than 50 ohms,
and you can buy vector network analyzers whose native impedance is 75
ohms. We've also made equipment to match the "standard" reference
impedances used by telephone services and the audio industry.

Finally, to head off the nearly inevitable rant that (Z-Z0)/(Z+Z0),
for a complex Z0, allows S11 to have a magnitude greater than unity
for a passive load, yes, we know that's true, and it's really NOT a
problem. Don't try to attach more physical significance to the
reflection coefficient than is actually there.

(John, I do rather wish you'd dropped the line at the beginning of
your post that said I had written the short segment you quoted from my
posting, since I didn't write any of it; it was only lines I'd quoted
from the original by "camelot"...but I trust that will be fairly
obvious to readers.)

Cheers,
Tom