"Antonio Vernucci" wrote in
:

I full agree with your statement:

If the line section is not exactly a half wave, then the real loss
factor might be higher or lower depending on the location of the
current and voltage maxima and minima and the relative contribution
of R and G to loss. So, the formula may have significant error for
short lines that are not exactly a half wave.

But I am not certain about this other statement:

Straight away, that tells you that the VSWR must be almost the same
at both ends for it to not matter which end is the observation point,
so therefore the first assumption is that VSWR is approximately equal
at both ends of the half wave.
If a practical line is very long, it cannot qualify as having a
constant VSWR (unless it is 1, in which case the formula is
unnecessary), so the formula is not suited.

I have a feeling that the ARRL chart makes reference to the SWR at
the antenna, and that it DOES take into account that, for a lossy
line, the line portions closer to the transmitter are subjected to a
lower SWR.

Approximations that depend on VSWR as a load metric can:
1. depend on the integral over a half wave of very low loss line then
apply it as a constant loss loss per unit length; or
2. treat the forward and reflected waves as waves independently subject
to a constant loss per unit length.

I explained the sources of error in extending (1) to the general case in
my earlier post.

Case (2) assumes that the attenuation is the result of (vector) addition
of the power that is lost independently from each travelling wave at any
point, whereas the power lost is due to the effect of currents and
voltage resulting from vector addition of the voltages and currents of
the two waves at each point.

In some scenarios, they may be good approximations, but there are also
scenarios where they are poor approximations. (I gave an example in an
earlier posting where they both fail.)

The reality is that on a practical mismatched line, loss per unit length
is not constant with displacement. See my notes on VSWR at varying
displacement on a practical line, see
http://www.vk1od.net/VSWR/displacement.htm .

Look at Fig 9.

Note that the loss vs displacement line does not have a constant slope,
and anything that ignores that is ignoring an aspect of the problem.

Note that in the example, the red line dips below the blue line (meaning
loss under mismatched conditions is LESS than matched line loss at some
lengths). Any method that prevents that result is ignoring an aspect of
the problem.

The loss under mismatch conditions does depend on load impedance, and if
you throw away some of the detail and reduce it to load VSWR, then you
increase the scope for error.

Is there application for the approximations? Certainly, I use them... but
in the knowledge that they are approximations and an awareness of where
they are not good approximations and may not produce an adequate answer
for the current problem.

Your original posting was about reconciling the chart with some examples,
and I noted that the chart itself is a source of significant error in
some scenarios.

BTW, your calculations seem to fall into case (2), and if so, are subject
to the same errors... though they may reconcile well with a chart based
on that approximation.

Owen