Topband: electrical wavelength

[Jim Brown]

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