(J. B. Wood) wrote in
:

....
Hello, and I ran into this issue years ago when trying to measure high
( 10 or 20) VSWR loads (in this case out-of-band shipboard HF antenna
feedpoint impedance) connected via a length of transmission line.
Accurate determination of load resistance is difficult to ascertain
under these conditions. A 1953 AIEE (a progenitor of the IEEE) paper
by W.W. Macalpine, "Computation of Impedance and Efficiency of
Transmission Lines with High Sanding Wave Ratio" describes the
problem.

I was, however, able to obtain accurate results when I included the
small imaginary part (frequency dependent) of the characteristic
impedance that is present in a low-loss line. Outside the high VSWR
load issue the imaginary part can be ignored. Sincerely, and 73s from
N4GGO,

John,

If I understand correctly, Macalpine's work was in finding solutions given
the computation tools readily available.

The approximation of a real transmission line as having Zo that is real can
introduce significant error in some line calcs.

Improving the model to include an estimated X value based on the loss is a
next step that improves calcs. This technique is widely used, and is
revealed by Ro equal to the nominal Zo, whilst Xo is a small negative
quantity for practical low loss cables at HF. This model produces
significantly different results for high line VSWR at lower frequencies.

TLLC (http://www.vk1od.net/calc/tl/tllc.php) goes a step further and
estimates a value for Ro based on the loss. For example, the Zo used for
RG58 at 1MHz is 50.06-j2.31 ohms. This model produces significantly
different results for extreme line VSWR at lower frequencies.

The more detailed models would be computationally unwieldly with log tables
or a slide rule (the tools of Macalpine's day), but TLLC demonstrates that
it is a trivial computational matter today, though not to lose sight of the
fact that mathematical function libraries are often approximations
favouring speed over ultimate accuracy.

Again, we must keep in mind that the uncertainty of the data fed into the
model (including manufacturing tolerances, and through life component
degradation) contributes to uncertainty of the output, and as Roy has
mentioned, the sensitivity of results to input data can be higher than
people might otherwise expect.

Properly implemented, the VNA OSL calibration process and measurement
process should overcome the problem of a sufficiently accurate model of the
actual transmission line used in the 'fixture', though not the phase
stability between calibration and measurement, especially where they are
done under very different environmental conditions.

Owen