On Jun 28, 5:27*pm, Keith Dysart wrote:
And yet, you have not located any errors in the math or the models.

Superposition of power *IS* an error! You add and subtract powers
willy-nilly as if that mathematical step were valid which it is not.
Two coherent 50w waves do not add up to a 100w wave, even using
average powers, except for the special case of zero interference where
the waves are 90 degrees out of phase with each other.

In order to use power as an energy tracking tool, we must be very
careful to ensure that there is a one-to-one correspondence between
energy and power, i.e. every joule passing a point in one second must
result in one watt of power (no VARs allowed). If that one-to-one
correspondence doesn't exist, no valid conclusion can be drawn from
tracking the power and any valid conclusion must be based on tracking
the energy which is no small task. The key is that there is no such
thing as imaginary energy. All energy is real. Some "power" is not
real.

A one-to-one correspondence does not exist in a standing wave.
Therefore, tracking power as if it were equivalent to energy in
standing waves is invalid. You have made that error for years.

One-to-one correspondence also does not exist over a fraction of a
wave. Therefore, instantaneous power is irrevelent in tracking the
energy. That's your latest error which is the same conceptual error as
before. In general, average power in the traveling waves over at least
one complete cycle (or over many cycles) has a one-to-one
correspondence to the average energy in the traveling waves. But that
one-to-one correspondence is more often than not violated within a
fraction of each cycle.

Here's a quote from "Optics", by Hecht, concerning power density
(irradiance).

"If however, the 'T' is now divided out, a highly practical quantity
results, one that corresponds to the average energy per unit area per
unit time, namely 'I'." - where 'I' is the irradiance (*AVERAGE* power
density).

If I calculate the Z0 of a 1/4WL transformer, I get two roots when I
take the square root of R1*R2. One of the roots is negative. If I ask
you to prove something is in error with the math that yielded a
negative Z0, could you find the math error? If not, does it follow
that you can find the transmission line with the negative
characteristic impedance existing in reality?

That's your argument in a nutshell. There may (or may not) be an error
in your math but it doesn't matter either way. The conclusions that
you reach from your math do not match reality so your math is a moot
point, i.e. there is no one-to-one correspondence between your math
and the real world.

If you have forgotten the importance of the one-to-one correspondence
concept in mathematics, now would be a good time to review that
concept. Without a one-to-one correspondence to reality, math is just
fantasy existing only in your mind.
--
73, Cecil, w5dxp.com