Cecil:

[snip]
In any case, it's good
to know that we cannot use the simplified wave reflection model
on very lossy lines. It appears that a lossy line doesn't yield
a smooth spiral on a Smith Chart.
--
73, Cecil http://www.qsl.net/w5dxp
[snip]

Amen brother and... heh, heh... especially for broad band signals.

Smith Charts are for mono-chomatic signals. Most tough transmission
problems
are broad band and the Smith Chart yeilds no useful insight in those
problems.

A widely applied practical example is the transmission of bi-directional
broad band digital subscriber loop (DSL) signals over telephone twisted
pair.

Telephone twisted pair is very lossy... at the "standard" 18,000 foot
length you can barely tell what is connected on the other end, short
or open. In fact it might just as well be "semi-infinite"! The longest
spans we have built chips for were up to 47,000 feet of #24 AWG
full duplex data transmission at the basic rate with digital echo
cancellation
on both ends using trellis coded pulse amplitude modulation.

I can assure you that 47,000 feetof #24 AWG definitely has a complex
and lossy Zo! I had quite a few big "spools" of such cable in my lab
for the beta tests!

The big problem with such designs is not maximum power transfer,
rather it is hearing the remote end in the presence of the local transmitter
blasting away on the same pair as the receiver [talker echo] and so one
needs to "image match" the transmitter to eliminate as much talker echo
as possible and just take whatever power reaches the receiver at
the other end. Of course you have some control over the spectrum
of the power that reaches the other end by "pre-coding" at the
transmitter, still the optimum strategy at the transmitter is to get
an "image match". i.e. make the generator internal impedance
as close to the complex Zo as you can make it!

And... in those problems you need to differentiate two forward waves
and two reflected waves. Heh, heh... hard to do that using just the
two symbols Vfwd and Vref or V_+ and V_-, you need symbols
for at least two each... Say Vfwd_1 and Vfwd_2 and Vref_1 and
Vref_2, etc... messy to say the least!

For this reason I much prefer the Scattering Formalism symbols
"a" for incident and "b" for reflected, a1 for indicdent on port 1
and a2 for incident on port 2, then b1, b2, etc...

Sometime, when I get some free time from my current
consulting gig, I'll prepare a short example for the group of the
problems inherent in full duplex signalling over complex Zo lines in
situations where the "best" Engineering solution is "image match"
not "conjugate match". ;-)

--
Peter K1PO
Indialantic By-the-Sea, FL.