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Old December 9th 07, 11:32 AM posted to rec.radio.amateur.antenna
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Default Loading Coils; was : Vincent antenna

I think it might be useful to say a little more about standing waves.

Imagine a single lossless transmission line with a sine wave source at
one end and a load at the other. Begin with a load equal to the line's
Z0. Make a graph of the magnitude of the current or voltage as a
function of distance from the source. With the Z0 load, the magnitude
will be the same all along the line so your graph will be a straight
line. This is a "flat" line, with no standing wave. A probe sitting at
one spot would show the instantaneous voltage or current amplitude going
up and down in a sinusoidal manner. A probe a bit farther down the line
would look the same, but delayed; there's a phase difference between the
voltages or currents at the two points. The phase difference is equal to
the line's physical length in degrees divided by the velocity factor.

Now change the load so the line is slightly mismatched. A standing wave
will appear -- the graph of amplitude vs distance won't be flat any
longer, but will have a ripple added to its previous constant value.
(The VSWR is, by definition, the ratio of the highest to the lowest
values of the voltage envelope on a line long enough to have a full
maximum and minimum. The current SWR is the same.) The maxima and minima
of the ripple don't move, hence the name "standing wave". If we look at
the instantaneous voltage or current at a single point, it will go up
and down in step with the source as before. If we also look at the
second point, it'll also go up and down as before, and there will be a
phase angle between the two. But there are two interesting differences
from the flat line: One is that the amplitudes at the two points are now
unequal unless they're an integral number of half electrical wavelengths
apart (or a few other special cases). The other is that the phase shift
isn't the same as before. There's still a phase shift between the two
points, but it's no longer equal to the electrical length of the line
between the points. We'll find that either the voltage has shifted more
and the current less, or vice versa depending on the load and which
points we've chosen. But at every point the current and voltage still
have phase angles which change with position along the line. That is to
say, the voltage or current at one point is delayed compared to the
voltage or current at the other.

As the mismatch gets more extreme (i.e., the SWR increases), the
magnitudes at the two points get more different, and the phase deviates
farther from the electrical length of line between them. (This is why
you can't expect phased array "delay lines" to provide a delay equal to
the lines' electrical lengths when they're not terminated with Z0.)

At the most extreme case of mismatch -- an open, short, or purely
reactive load, resulting in an infinite SWR -- the amplitude of the
standing wave along the line goes from zero to twice the value it had
when the line was flat. And a really interesting thing happens to the
phase of the voltages and currents on the line. Remember how as the
mismatch got worse, the voltage and current phase difference between two
points got farther and farther away from the electrical line length
between them? Well, when the SWR is infinite, it's gotten to the point
where the voltage or current phase remains the same for a distance of a
half electrical wavelength, then abruptly changes 180 degrees, repeating
every half electrical wavelength. Some antennas behave in some (and only
some) ways like transmission lines, and you'll find that modeling
programs report just this behavior of the phase of the current along a
straight wire antenna.

The standing wave and all the characteristics of the voltage and current
(e.g., how their magnitude and phase varies with position along the
line) follow directly from an analysis of forward and reflected
traveling waves on the line. The voltage or current at any point is
simply the sum of the two waves at that point, and they have the
properties I've just described.

I hope this helps in clarifying the meanings of traveling and standing
waves, voltage and current along a transmission line. I'm sure there are
lots of good graphical illustrations available -- but some bad ones too.
Hopefully keeping this explanation in mind when you look at the nice
graphics displays will help you sort the bad ones from the good.

Roy Lewallen, W7EL
 
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