Current across the antenna loading coil - from scratch
Roy Lewallen wrote:
John Popelish wrote:
. . .
It is obvious to me that you are one of them. Every point on a line
carrying a standing wave (except the node points) has AC voltage on
it, and AC current through it. The amplitude and phase of those
voltages and currents can be described as a phasor, with respect to
some reference phase of the same frequency. As you move along the
line, the amplitude changes and when you pass through a node the phase
reverses. So the phasor does not rotate with position change, except
for a step change of 180 degrees at nodes, rather than smooth rotation
with respect to position.
For a traveling wave, every point on the line has an AC voltage on it,
and an AC current passing through it. The amplitude is constant along
the line, but the phasor rotates as you move along the line (the phase
is linearly dependent on position). But at any single point on the
line, a non rotating phasor describes the amplitude and phase with
respect to a reference phase of the same frequency.
There's a potential for ambiguity here, and that ambiguity has been used
a number of times in this thread to cause confusion. So let me try to
clarify things.
All phasors "rotate", in that every one contains an implicit term e^jwt.
That term describes a rotation of the complex phasor quantity at the
rotational frequency w (omega), but no change in amplitude. If a
quantity doesn't include this implicit term, it's not a phasor, by
definition. We can look at any phasor quantity in a system and compare
the phase of its rotation with the phase of a reference, and from this
assign a phase angle to it. In steady state, the phase angle doesn't
change with time -- it's the phase difference between the w - rotating
phasor and the w - rotating reference. Phasors of different rotational
rates (that is, of different frequencies) can't be combined in the same
analysis, unless the implicit term is made explicit, in which case
they're no longer phasors.
The use of "rotation" in John's posting is talking about a change of
phase with physical position. This usage has been confused with the time
rotation of the phasor which comes from the implicit e^jwt term. I'd
prefer to use the term "phase", which doesn't change with time in a
steady state system, directly rather than "rotation" to describe a
change in phase with position.
With that convention, we see that the phase of a pure traveling wave
changes linearly with position. But when we sum forward and reverse
traveling waves together to get a total current (or voltage), the phase
of the total current (or voltage) is no longer a linear function of
position. In the special case of an open or short circuited transmission
line, where the forward and reverse traveling waves are equal in
amplitude, the phase doesn't change with position at all (except for a
periodic reversal in current and voltage direction, which can be
interpreted as a 180 degree phase change). But the phasor representing
total voltage or current (which Cecil refers to as "standing wave
current") at any point, which is the sum of two phasors representing
forward and reverse traveling waves, does indeed rotate at w (omega)
radians/second rate, just like its constituent phasors. The constant
phase with position (of an open or shorted line) simply means that if
you froze time at some instant and looked at the angles of the rotating
phasors representing the total current at each point along the line,
you'd find them all to be at the same angle. They're all rotating.
This isn't revolutionary or controversial -- you can find phasors
discussed in any elementary circuit analysis text.[*] And it's not
difficult to do the summation of forward and reverse traveling waves to
see the result, but if you'd like to see how someone else did it, one of
the clearest discussions I've found is in Chipman's _Transmission
Lines_, a Schaum's Outline book.
[*] You have to be a little careful, though. In most introductions to
phasors, the author introduces the e^jwt term early on, and quickly
drops it from the phasor notation as is customary. So it's easy to
forget it's there. But remembering that it is there is vital to
understanding this topic, and to keep from being misled by misdirection
which takes advantage of confusion and abbreviated notation.
Excellent!
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