Roy Lewallen wrote in message ...

A. The one just posted by Peter, (Zl - Z0conj) / (Zl + Z0conj)


I believe this was a typo on Peter's part, as was my typo in the
original post.


B. Slick's, (Zl - Z0conj) / (Zl + Z0)



This is the correct formula.


C. The one in all my texts and used by practicing engineers, (Zl - Z0) /
(Zl + Z0)



This is correct too, but Zo must be purely real.




What about the seemingly sound logic that the accepted formula doesn't
work for complex Z0 because it implies that a conjugate match results in
a reflection? The formula certainly does imply that. And it's a fact --
a conjugate match guarantees a maximum transfer of power for a given
source impedance. But it doesn't guarantee that there will be no
reflection. We're used to seeing the two conditions coincide, but that's
just because we're used to dealing with a resistive Z0, or at least one
that's close enough to resistive that it's a good approximation. The
fact that the conditions for zero reflection and for maximum power
transfer are different is well known to people accustomed to dealing
with transmission lines with complex Z0.


Wrong. But at least you admit that the "accepted" formula (which
is fine for purely real Zo) implies a reflection.

The absence of reflection is what makes the maximum power
transfer.



But doesn't having a reflection mean that some power is reflected and
doesn't reach the load, reducing the load power from its maximum
possible value? As you might know from my postings, I'm very hesitant to
deal with power "waves". But what's commonly called forward power
doesn't stay constant as the load impedance is changed, nor does the
forward voltage. So it turns out that if you adjust the load for a
conjugate match, there is indeed reflected voltage, and "reflected
power". But the forward voltage and power are greater when the load is
Z0conj than when Zl = Z0 and no reflection takes place -- enough greater
that maximum power transfer occurs for the conjugate match, with a
reflection present.


?? From a theory point of view, when you cancel series reactances
(canceling inductive with capacitive) the series inductor and
capacitor are
resonant, and will thoeretically have zero impedance, allowing the 50
ohms to feed 50 ohms for max power transfer, WITH THEORETICALLY NO
REFLECTIONS.



I'd welcome any corrections to any statements I've made above, any of
the equations, or the calculations. The calculations are particularly
subject to possible error, so should undergo particular scrutiny. I'll
be glad to correct any errors. Anyone who disagrees with the conclusion
is invited and encouraged to present a similar development, showing the
derivation of the alternate formula and giving numerical results from an
example. That's how science, and good engineering, are done. And what it
takes to convince me.

Roy Lewallen, W7EL


It's hard to convince anyone who could never admit that they were
wrong.

Slick