Chipman, page 138, presents an equation for the power at any point 'z' on a
line that involves Zo and considers the case where Zo = Ro+jXo. He then
derives two equations: one for the real component of the power Pr measured
at any point 'z' on a line and another for the imaginary component of power
Pi at that point. Watts and VARs . . .

Each equation contains three terms. The second equation for Pi and the third
term in both equations vanish when the value of Xo is zero. The condition
that the real part never be negative is shown to be that Xo/R0 is equal to
or less than unity.

He first, however, derives the reflection coefficient for the point 'z'
which is stated as p(z) = (Z(z) - Zo)/(Z(z) + Zo) and Zo is defined to be a
complex number. Some authors have referred to this as "Classical Rho."

Chipman's interpretation of this equation for Pr of three terms is that the
first term represents the real power for the incident wave alone at a point
'z'; the second term relates to the real power in the reflected wave at that
same point; and the third term (which vanishes for real Zo) represents "an
interaction between the reflected and incident waves." This is in effect the
third term that Roy obtained using Vf * Ir and Vr * If.

Thus on a line with real Zo, the net average power in the load is always
given by Pf - Pr. However, with lossy lines having Xo not equal to zero, all
three terms of the equation must be taken into account in determining the
net power at any point along the line. And thus, Pr and Pf alone do not
describe the load power.

He further states that passive terminations exist which can result in
classical rho achieving a value of 2.41 "without there being any implication
that the power level of the reflected wave is greater than that of the
incident wave." The physical example is that of the resonance obtained by
conjugate matching of the Xo-component of the line with the load and the
attendant "resonant rise in voltage.".

The existence of this third term is, I believe, what much of the discussion
has talked around and attempted to avoid confronting by involving all manner
of arcane definitions and interpretations to "prove" that the net power
delivered to a load cannot be other than Pf-Pr and that Pf is always larger
than Pr.

Note that this work is not mine - I am merely reporting the gist of
Chipman's derivations and interpretations due to the scarcity of his book.

This plus Roy's presentation is enough for me . . .

--
73/72, George
Amateur Radio W5YR - the Yellow Rose of Texas
Fairview, TX 30 mi NE of Dallas in Collin county EM13QE
"Starting the 58th year and it just keeps getting better!"






"Roy Lewallen" wrote in message
...
And that's the whole crux of the problem -- the mistaken assumption that
the "reflected power" can never exceed the "forward power". Once you
accept that erroneous idea as a fact, you're stuck with some very
problematic dilemmas that no amount of fancy pseudo-math and alternate
reflection coefficient equations can extract you from. A very simple
derivation, posted here and never rationally disputed, clearly shows
that the total average power consists of "forward power" (computed from
Vf and If), "reflected power" (computed from Vr and Ir), and another
average power term (from Vf * Ir and Vr * If) whenever Z0 is complex.
The only solid and inflexible rule is that these three always have to
add up to the total average power. Not that the "forward power" always
has to equal or exceed the "reflected power". It's in that false
assumption that the problem lies.

Roy Lewallen, W7EL

wrote:

So is kurokawa proposing two completely different rhos?
One for computing voltages and currents and the other for power?

This could work, I supposed, but this discussion started with an
assertion that 'classic' rho was WRONG because it resulted in
more reflected power than incident. My contention is that
'classic' rho is correct and yields the correct voltages regardless
of the results obtained when |rho|^2 is used to predict powers.

If kurokawa wishes to introduce a new rho to solve these problems
in a different manner, that is fine, but he would have reduced
confusion significantly if he had not called it rho.

...Keith