Newsgroups: rec.radio.amateur.misc

Recently, I posted that to a good approximation for the lines hams use,
at HF and VHF frequencies, the quantity

      sqrt(f)/(loss in dB/unit length)

is essentially constant.
rdewan@casbah.acns.nwu.edu (Rajiv Dewan) disagreed with that, and
said it should be log(f), not sqrt(f), in the numerator.  I'll stand
by the sqrt(f), as shown above; you can check this easily against
tables or charts yourself.  If you work with coax (or balanced line,
for that matter), you may save yourself some time if you just
remember, loss in dB increases as the square root of frequency.

A closer look:
(SKIP THIS IF YOU DON'T WANT TO WADE THROUGH EQUATIONS ;-)  

Though you might intuitively think there needs to
be a log term on the frequency to balance the log inherent in dB,
it isn't so.  A fundamental expression for TEM-mode lines (normal
coax and other "two wire" lines) is the attenuation constant.  It's
usually written as lower-case 'alpha', but let's use 'a':

    a = sqrt(0.5*{sqrt[(R^2+(wL)^2) (G^2+(wC)^2)] + RG - w^2*LC})

where:
    a is attenuation constant, nepers/unit length
	 (= 0.1151* dB/unit length)
    w is radian frequency = 2*pi*f
    R is effective resistance along line at w, per unit length
    G is effective conductance across line at w, per unit length
    L is inductance along line per unit length
    C is capacitance across line per unit length

That mess can be simplified:  the dissipation factor for polyethelene
or Teflon or dry air is so low that up through VHF at least, G is
essentially zero.  Also, at HF and above, R should be small compared
with wL.  (e.g., RG-58 at 1MHz:  wL is about .484 ohms/foot and R
is about .032 ohms/foot.  wL increases linearly with frequency,
but R increases only as the square root of frequency.)  Let's
express R as a function of wL:  R^2 = kwL, where k is a constant.
This just says that because of skin effect, R increases linearly as
the square root of frequency -- given L as a constant for the line and
picking k appropriately.  In fact, k for the case of RG-58 is 2.117e-3
ohms/foot.  OK, we're ready to apply these simplifications to the
equation above:

    a = sqrt(0.5*{sqrt[(kwL+(wL)^2) ((wC)^2)]  - w^2*LC})

But we note that the interior sqrt is operating on something that's
close to a perfect square:  (wL + k/2)^2 = (wL)^2 + kwL + k^2
Note that k<<wL, so k^2 << kwL << (wL)^2, and a very good
approximation for sqrt(kwL+(wL)^2) is just (wL + k/2) :

    a = sqrt(0.5*{ (wL + k/2) wC - w^2*LC})

      = sqrt(0.5*{kwC/2})

      = 0.5 * sqrt(kwC)

Or expressed at dB/unit length:

  dB/u.l. = 4.344 sqrt(kwC)

So there we have it:  the attenuation in dB/unit length (or in
nepers/unit length) is proportional to sqrt of frequency.  The only
assumptions we made:  G is insignificant, R is tiny compared with wL
(so the square root approximation could be used), and R is varying
linearly with the square root of frequency (which is the case where
skin depth is a small fraction of conductor thickness).

Double check:  Does the equation above give a reasonable value for
RG-58??  Plug in k=2.117e-3 ohms/foot, w=2*pi*10MHz, and C=30.8pF/foot,
and you get a=.00101 nepers/foot, or .88dB/100 feet (at 10MHz).
This is a bit lower than the chart I have, but not out of line:  I've
noticed that cable manufacturers often spec attenuations a bit lower
than this particular chart, too.  Also, the resistance figured above
considered only the center conductor.

"Final" note:  you can take the equation above one step further,
noting k=R^2/wL...

    a = 0.5 * sqrt(R^2 wC/wL)

      = 0.5 * R/Zo

or in dB,

    dB/u.l. = 4.344 * R/Zo

where R is the resistance per unit length as above, at operating
freq., and Zo is the line characteristic impedance.  Same caveats
re G being negligible and R << wL.