Likewise, Zo is only sqrt (L/C) at VHF. The more complete equation is
sqrt [ (R+J omega L) /( G + J omega C) ] At VHF, the equation SIMPLIFIES
to sqrt (L/C) At low audio frequencies, and up to VHF, G is insignificant
(leakage) so the complete practical equation is sqrt [(R+ j omega L) / j
omega C] Note that this results in Zo being complex, and a proper
measurement will confirm that this is true. There are MANY references to
complex Zo in the ham literature. Frank Witt published some work about
this, now available in one of the ARRL Anthologies. N6BV's TLW software,
published in the ARRL Handbook, uses complex impedance data for its
transmission line calculations, although it ignores the variability of Vf.
I don't know why you are dragging complex impedance into velocity of
propagation through the cable. It is late summer, and too warm for that.
Everyone knows and agrees, or should know and agree, Zo is complex. That
isn't the issue, it certainly is not news, and you failed to provide any
example at all why Vf varies.
All you said is Zo is complex, which anyone knows, and the Handbook "ignores
Vf variability".
At low audio frequencies, Zo is much, much larger than the VHF value, and
Vf is much, much slower than the VHF value. Both properties begin a rapid
transition to their VHF values and go though at least half of it within
the audio spectrum, approaching the VHF values asymptotically. By 2 MHz,
both are within a few percent of the VHF value.
All of this was WELL KNOWN more than a century ago, and Oliver Heavyside
did a lot of work on applications to equalize lines. While it is often
assumed in modern times that equalization of telephone circuits was done
only for the amplitude response, equalization is equally important for the
TIME response. To get your head around that, consider speech where the
"highs" arrive much sooner than the "lows."
Here's a simple test you can do with any 50 ohm signal source you can read
to an accuracy of at least 0.1 percent and a decent voltmeter across the
source Cut a quarter wave open stub for the lowest frequency you can
observe and measure the first resonance to as many digits as you can, then
repeat for the third, fifth, seventh, and ninth resonances. If you can
hit the precise null and read enough digits, you can plot the variation in
Vf. Or do the same with any vector analyzer, carefully reading the
frequencies of each null.
I already did that. Didn't change.
A hunk of F6 cable about 200 feet long shows about 0.3% variation from 5.199
MHz to 25.977 MHz.
That same hunk of cable shows 0.24% variation from 5.199 MHz to 52.119 MHz
A 100 ft piece of LMR400 type cable shows 0.07 % variation in Vf from 4.023
MHz to 40.261 MHz, and 0.18% from 4.023 to 56.423 MHz.
Where can I find cable like you have, or a meter like you have?
I would expect a large variation at audio, where the skin depth of
conductors is woefully inadequate, or where we are measuring things that are
mismatched, or measuring things too short. Plus, we all know audio things
like silk insulation and oxygen free conductors changing sound. :-)
Rather than telling me what I need to go read, which doesn't seem to apply
unless we start getting into some severe dielectric or inductance change
issues, please give me a reference that shows Vf changes so dramatically
with frequency. I already know Zo is complex.
None of my books, nor the Internet, confirms what you say about a large
variation. Neither does a real network analyzer.
Perhaps I have bad cables of two very different types, and all the dozens of
cables a cable manufacturer sent me over the years were all bad in the same
way.
73 Tom
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