William E. Sabin wrote:

I am looking at W.C. Johnson, pages 15 and 16, Equations 1.22 through
1.25, where he shows, very concisely and elegantly, that the
reflection coefficient is zero only when the complex terminating
impedance is identically equal to the the complex value of Z0.

And not the complex conjugate of Z0.

In other words (read carefully), the complex terminating impedance
whose value is Z0 is equivalent to an *infinite extension* of the
coax whose complex value is Z0. This is the "clincher".

That's fine. I agree entirely, and it follows from my analysis and my
conclusion. A similar analysis can be found in many texts. My offering
to provide a large number of references has brought forth no interest
from the most vocal participants, and they've also showed a lack of
willingness to work through the simple math themselves. So I felt that
it might be a good idea to post the derivation before more converts are
made to this religion of proof-by-gut-feel-and-flawed-logic.

Observe also the the reflection coefficient for current is the
negative of the reflection coefficient for voltage.

Likewise.

Bill W0IYH


One other thing. If the reflection coefficient is zero, then all of the
real power that is dumped into the coax is being delivered to the load,
minus the losses in the coax. The system is as good as it can get.

Bill W0IYH

I agree with this only in the sense that the "system is as good as it
can get" means only that the loss in the coax is minimized for a given
delivered power. It doesn't guarantee that the maximum possible power
will be delivered to the load.

After some thought, I see I was in error in stating that terminating the
line in its complex conjugate necessarily results in the maximum power
transfer to the load. If the 50 - j0 had been a source impedance instead
of a transmission line impedance, that would be true. However, what
results in the maximum power from the source in a system like this is
that the *source* be conjugately matched to the impedance seen at the
input of the line. Although terminating the line in its characteristic
impedance minimizes the line loss, it doesn't guarantee maximum load power.

Of course, the conjugate matching theorem says that if the line is
lossless, a load which is the conjugate of the impedance looking back
toward the source from the load will result in a conjugate match at the
source and everywhere else along the line, so that will effect maximum
power transfer. But the impedance looking back toward the source from
the load isn't by any means necessarily equal to the Z0 of the line, so
the conjugate of Z0 isn't necessarily the optimum impedance for power
transfer, as I erroneously stated. And I don't believe that the
conjugate match theorem applies to a lossy line. I certainly don't want
to start up an argument about this topic, though, and will simply state
for certain that the maximum net power will be delivered to the line
when the impedance seen looking into the line is equal to the complex
conjugate of the source impedance.

It should be easy to set up a couple of simple numerical examples to
illustrate this. Unfortunately, I'm pressed for time at the moment and
have to run. I apologize for the error regarding terminating impedance
and maximum power transfer.

Thanks for spurring me to re-think the conditions for maximum power
transfer. I apologize for the error.

Roy Lewallen, W7EL