Owen Duffy wrote:
. . .

I saw the challenge and note the lack of response.

Let me offer a steady state solution.

In the case of a simple source being an ideal AC voltage generator of Vs
and an ideal series resistance Rs of Ro, and that Zo=Ro, for any
arbitrary load, at the source terminals, Vf=Vs/2, Vl=Vf+Vr=Vs/2+Vr, and
the voltage difference across Rs is Vs/2-Vr (noting that Vr is a complex
quantity and can have a magnitude from 0 to Vs/2 at any phase angle),
therfore dissipation in Rs is given by:

Prs=(Vs/2-Vr)^2/Rs where Vs is the o/c source voltage, Vr is the complex
reflected wave voltage equivalent, Rs is the source resistance.

Clearly, dissipation in Rs is related to Vr, but it is not simply
proportional to the square of Vr as believed by many who lack the basics
of linear circuit theory to come to a correct understanding.

Roy, is that a solution?

Owen

Sorry, I don't think so. The very first equation, Vf=Vs/2, would be true
only if the load = Ro or for the time between system start and when the
first reflection returns. For the specified steady state and arbitrary
load, Vf would be a function of the impedance seen by the source which
is in turn a function of the line length and load impedance.

I'm not saying an equation relating "reflected power" and source
dissipation can't be written, but it does involve load impedance and
line length and Z0. (Equations written with Vf and Vr as variables tend
to conceal the inevitable dependence of these on line length and Z0 and
load impedance, but the dependence is still there.) And the equation
will show that there isn't any one-to-one correspondence between the two
quantities. For a given source voltage and resistance, the source
resistance dissipation depends only on the impedance seen by the source.
This in turn depends on the length and Z0 of the line and the load
impedance, and can be created by an infinite combination of loads and
lines having different "reflected powers". As I illustrated in an
earlier posting, a constant load with various lines having very
different "reflected powers" can present the same impedance to the
source and therefore result in the same source resistor dissipation.
Likewise, different lines having the same "reflected powers" can result
in different impedances seen by the source and therefore different
dissipations. So the two aren't really related, even though you could
write an equation which contains both terms.

Roy Lewallen, W7EL