Good comments.

One of my professors made an astute observation that sums up your
conclusion. He said that science isn't set back so much by getting the
wrong results by using the right methods as it is by getting the right
results using the wrong methods. Amen to that.

Roy Lewallen, W7EL

David Robbins wrote:

yes, i still hold that they are different. they are from completely
different realms of electromagnetics. in the transmission line reflection
coefficient you are working with a distributed system that is modeled with
wave equations. there are delays, waves travel and reflections return after
a finite delay. in your example you have coerced the voltage and current
waves to be the same in the transmission line, but that is not the same as
modeling the line itself.

in the generator feeding a load you are looking at a lumped model. these
are normally not solved with wave equations, there are no delays or
reflections. the problem of calculating maximum power transfer from a
generator to a load of a given impedance is a simple problem solved with no
waves and only one pde used to find the max of the power to the load wrt the
generator impedance. it doesn't care what the load is, as long as it is a
linear time invariant impedance. now, you could create that impedance by
putting a load on the end of a transmission line and transforming it back
through the normal transmission line equations, but once you do that you
throw the line out and solve the simple equations in the lumped models.

these discussions always seem to end up in this same quagmire, one group
trying to solve everything with wave equations, sinusoidal steady states,
transmission line transformations, and the other holding on to the lumped
models and trying to make them fit the wrong problem domain. one thing i
did learn in ee classes was to use the right model for the job, choosing the
wrong one will give you wrong answers most of the time.... the real problem
is that sometimes you can get the right answer from the wrong model just
because the numbers work out... and when you have to show your work you lose
credit for using the wrong method but still getting the right numerical
answer. that is what i see here, someone has started with a simplified case
that just happens to work out to the same answer and has tried to generalize
it... -10 points for the wrong method!