Yet in my example, |rho|^2 *is* greater than one.

Also, in the past, you and others have defined the "forward power" to be
the power calculated from the forward voltage and current waves, namely
Re(fE * fIconj) or |fE| * |fI| * cos(phiE - phiI). This is what you've
consistently been calling the "power of the forward wave" or some such.
Likewise for "reverse power". This is the definition I used for the
substitution for fP and rP in the equation for total average power.
And the result is that the total power *isn't* equal to fP - rP.
What you're doing now is lumping the extra power into fP or rP, now
making those terms mean something else. The additional power term has
two components, one arising from the product of forward current and
reverse voltage, and the other from the product of forward voltage and
reverse current. (I combined the two cosine functions with a trig
identitity into a product of two sine functions, but you should go back
a step or two in the analysis to get a clear idea of their derivation.)
I believe you've chosen to assign each of these, or the sine product, to
either "forward power" or "reverse power", depending on its sign, even
though they're a function of both forward and reverse voltage and
current waves. I can't imagine the justification for doing this, but
then there's quite a lot that people have been doing which I don't
understand. As part of the process, you might consider the consequence
of the sine or cosine function returning a negative value, which either
of course can.

Again, I welcome an alternate solution that accounts for all the
voltages, currents, and powers, including one that does it with rho 1.

Roy Lewallen, W7EL

Cecil Moore wrote:

This seems to me to be somewhat akin to the fact that s11 and
rho can have different values at an impedance discontinuity
where a 'third power' is commonplace. Roy's 'third power' at
the load appears to be analogous to a re-reflection of some
sort as the inductive load tries and fails to dump energy
back into the Z0=68-j39 transmission line. A re-reflection
is another component of forward power.

The ratio of reflected Poynting vector to forward Poynting
vector is |rho|^2. In Roy's example, the total average
Poynting vector points toward the load indicating that
(Pz+ - Pz-) 0. That means |rho|^2 cannot be greater
than 1.0.