Various authors provide curves or formula for computing the "total
loss" in transmission lines, as opposed to the "matched-line" loss.
Specifically, The ARRL Antenna Book gives an equation in Chapter 24
that seems to give results consistent with other sources (See the
details at the end of this posting). However, there seems to be a
fundamental flaw in the way in which the equation is applied.

In essence, the equation provides a loss factor which is a function of
the matched-loss attenuation and the absolute reflection coefficient.
The matched-loss attenuation is the value normally expressed in dB per
100 ft. and shown in tables or shown in logarithmic plots as a
function of frequency. The reflection coefficient is introduced into
the expression in order to increase the total losses as the SWR on the
line increases.

After calculating a total loss factor it is applied to lines of any
length based on the reflection coefficient at the load. In my opinion,
it makes no sense whatsoever to provide an expression that is to
determine the losses per unit length on a line and have it based on
the reflection coefficient at the end of the line. If there is a
mismatched load, and if the line has losses, then it follows that the
SWR will become lower and lower the further we are from the
termination. That being the case, would it not make more sense to say
that the "additional" losses would be much higher at the load end of
the line, where the SWR is high, than at great distance from the load,
where the SWR is significantly lower? In fact, if the line is long
enough, we know that the SWR approaches 1:1, and in a line with an SWR
of 1:1 there should be no additional losses above the matched-line
losses.

Nonetheless, with that non-sensical approach, the numerical examples
shown at the referenced page and also in a later article on the
subject of Highly Reactive Loads makes it quite clear that the loss
factor is applied uniformly to the entire length of line.

If we take the expression for the total loss and apply it to small
increments of line wherein the SWR is relatively constant, then it not
only makes more sense, but it also predicts noticeably less total loss
in longer lines.

I have embarked on careful measurements of lines severely mismatched
(quarter wave open circuit stubs), and I can find no correlation
between my measurements and the values predicted by the "total loss"
equation. My measurements always show very low losses in comparison to
the model.

I would be interested in corresponding with anyone who has other
models for line losses, or anyone who has made measurements on
quarter-wave stubs.

##########Equation and data taken directly from The ARRL Antenna Book,
17th Ed., page 24-9 ###############
(Eq 10) Total Loss (dB) = 10 log [ {(Alpha * Alpha - (AbsRho *
AbsRho)} / {Alpha * (1- (AbsRho * AbsRho)) } ]

where

Alpha = 10^(ML/10) = matched-line loss ratio

AbsRho = (SWR - 1) / (SWR + 1)

where
ML = the matched-line loss for particular length of line, in
dB

SWR = SWR at load end of line

The text then goes on with a numeric example using a 150 ft. length of
RG-213 coax that is terminated in a 4:1 mismatch (SWR = 4:1, AbsRho =
0.6) at 14.2 MHz. The calculations for total line loss, per the above
equation, results in a total line loss of 2.107 dB.




Bob, W9DMK, Dahlgren, VA
http://www.qsl.net/w9dmk