There's good math for all occasions of antennas and feedlines.
You need to know the feed impedance of the antenna at all frequencies of
interest. And as a first approximation you can figure if the feed line
attaches a quarter wave from an open end, that the feed impedance will
be low. So a half wave dipole on 40 has a low feed impedance, while that
same wire at twice the frequency the feed impedance is high, because
with modification by radiation the antenna acts something like a
transmission line reflecting the impedance and transforming it. On any
line, or antenna (with the radiation modification) a quarter wave
converts and open end to a short (low impedance on the antenna) and
converts a short to an open (shorted quarter wave stub). Then a half
wave of antenna or feedline repeats the load or end impedance. So the
feed impedance of a full wave center fed is high.
Taken to a vertical against ground, the antenna an odd multiple
(1,3,5,7, etc) of a quarter wave has a low (35 to 50 ohms) feed
impedance while the antenna an even multiple of a quarter wave (2,4,6,8,
etc) long or any multiple of a half wave long (two ways of saying the
same thing) has a high feed impedance, like 1000 ohms. In between those
lengths, the impedance has a magnitude between the extremes and
reactance depending on where in between the resistive points.
One transmission line formula that applies is that the feed impedance of
a quarter wave (or odd multiple of that quarter wave) is:
Zin = Z0*Z0 / Zload. Where Z0 is the characteristic impedance of the
line, and Zload is the load on the other end. Works for every load and
every impedance, can be done in complex for a reactive load. There is
another harder to use formula that works for every length, impedance and
phase angle. It says the input impedance is a function of the tangent of
the line length in degrees times the load impedance. Or there's a
messier one using hyperbolic sinh and cosh, equation 12 on page 20.7 of
the 2011 ARRL Handbook for Radio Communications. There is a similar and
maybe more detailed chapter in the ARRL Antenna Handbook, plus numerous
chapters on antennas. There's only one chapter in the ARRL Handbook for
Radio Communications on antennas, but much of what I've banged out here
is covered.
When the load is complex, it is possible to choose a line length to make
it resistive and by choosing the two shortest possibilities to make it
resistive, it can be high or low impedance.
By far the easiest way to calculate is with the Smith Chart rather than
math formulae.
43 feet was chosen as a vertical length that is not resonant on any of
the classic ham bands so you don't hit the extremes of impedance, but
its always reactive. And its tall enough to radiate pretty good on 80
and better than a dummy load on 160. Its easiest to feed on 40m where a
simple series capacitor can compensate for the extra 10 feet of radiator
and make it resistive not far from 72 ohms. So it won't go resonant at a
high impedance and besides when fed through coax that high impedance
transforms to very low. A 1000 ohm load on 50 ohm coax, a quarter wave
away looks like 50 * 50 / 1000 or 2.5 ohms. Which can mean a hot spot in
the coax and a hot tuner if you try to tune. Yet the lowest resistive
component of the vertical that length is likely to be 35 ohms or more
and 35 ohms through a quarter wave of 50 ohm coax is very benign, like
50 * 50 / 35 = 71 ohms.
Just remember that its a quarter wave in space, but in a coax with solid
dielectric you have to figure the velocity factor, and the physical
length of that quarter wave will be a quarter wave in space times the
velocity factor.
73, Jerry, K0CQ
On 2/16/2011 4:09 PM, Richards wrote:
Is there a way to compute this sort of thing for other antenna/feedline
combinations ?
For example, I have a 43 foot monopole with 63 radials, using 45 feet of
LMR-400 coax transmission line. Is there a simple formula or
relationship for this sort of thing ? Is it just a question of feedline
length to antenna wavelength or something that counts. (I am presuming,
without knowing, that the considerations for a 40 dipole might be
different from a 43 ground plane antenna.)
Happy Trails.
======================= Richards / K8JHR =========================
On 2/16/2011 12:35 PM, Dr. Gerald N. Johnson wrote:
No, the SWR does NOT change along the line. The voltage and the current
changes with the SWR is high. The proper term is "where the tuner is
clearly at an extreme VOLTAGE point on the feeder, ..."
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