On 9/10/2012 10:44 AM, Tom W8JI wrote:
I firmly do not believe that is true.
Velocity factor in cable is the square root of the inverse of
dielectric constant.
Tom,
Respectfully, I suggest that you go back to your college textbook on the
fundamentals of Transmission Lines. The equations for Zo, velociity of
propagation, and attenuation are COMPLEX -- that is, they contain real
and imaginary components. The "formula" you cite is the result of
simplification to remove those complex elements. It's good at VHF and is
"close" for HF, but becomes increasing erroneous as you go down in
frequency.
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.
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.
73, Jim K9YC
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