"K7ITM" wrote in
oups.com:

....
About Zo being reasonably approximated by Ro, or not:

I made a note to myself some time ago, and I believe it's reasonably
accurate, that neglecting dielectric loss, for a TEM line, given Zo =
Ro+jXo, then to a good approximation

Xo = -0.180*Ro*A*Vf/f

where A = line attenuation in dB/100ft
Vf = line velocity factor
f = frequency in MHz.

So for small diameter 50 ohm polyethylene dielectric line at 1.8MHz,
the worst case for most ham applications, Xo/Ro is about .18. For
that, I used 2.7dB/100ft for RG174 type line. That's getting to be

I make the loss/100' of RG174 to be 1.1dB and from that I get Xo=-3.6
ohms. (Did you get loss/100m from somewhere? This is probably the answer
to Cecil's diligent spot of an apparent error.)

pretty significant, a ten degree phase angle away from purely
resistive. As Owen posted, it's so easy these days to deal with
complex numbers that you may as well just carry them all along. Given
the above formula, it's easy to figure the complex Zo for a line where
you know the nominal attenuation, the velocity factor, and the
frequency, and of course the nominal high frequency Zo value.

Beyond dealing with transmission lines as Zo=Ro:

A common approximation for Xo is -Ro*alpha/beta and Ro=Ro. This
effectively (approximately) attributes all of the effect of the R element
of and RLGC model to Xo. Tom, your method is equivalent to this when
attenuation is converted to nepers/unit-length and frequency is converted
to radians/unit-length.

My line loss calculator at http://www.vk1od.net/tl/tllc.php takes a
different approach. It computes an estimate of the complex characteristic
impedance implied by the loss=k1*f^0.5+k2*f model, nominal Ro, and
velocity factor using an RLGC model. The model assumes:
* R is proportional to square root of frequency;
* L is constant;
* G is proportional to frequency; and
* C is constant.
These are reasonable assumptions for most practical transmission lines
down to about 100kHz. The assumption to become invalid first is the first
assumption (are you still with me) which depends of fully developed skin
effect, hence the low frequency qualification.

The k1 and k2 values are obtained by regression from published
attenuation figures.

I haven't seen this done in other calculators, so it is one of the
reasons why my calculator will give slightly different results to others
such as TLDETAILS for example.

Owen