A phasor is a replacement of cos(omega * t + phi) with cos(omega * t +
phi) + j * sin(omega * t + phi) = exp(j * (omega * t + phi)) = exp(j *
omega * t) * exp(j * phi). The first of those quantities is understood
but not generally written in phasor analysis, but is nonetheless an
essential part of the definition of a phasor. This shows that a phasor
is a vector which rotates in the complex plane, with a rotational speed
of omega * t radians/sec. The reason the time-dependent rotational term
is left out when speaking of phasors is that phasor analysis is used
only for systems in which only one frequency is present, as you said.
Therefore, all have the identical multiplying term exp(j * omega * t)
and, basically, they all cancel out in phasor equations. Omega is, of
course, 2 * pi * f.

Cecil regularly confuses the change in phase angle of the phasor with
position, with the rotation of the phasor with time.

A proof of the validity of the replacement of the real cos function with
the complex phasor function, as well as a good description of phasors in
general, is given in Pearson and Maler, _Introductory Circuit Analysis_.
A good graphical illustration and description of a phasor as a rotating
vector can be found in Van Valkenburg, _Network Analysis_. Those are the
only two basic circuit analysis texts I have, but I'm sure the topic is
covered well in just about any other one.

Roy Lewallen, W7EL

K7ITM wrote:

Hey, are you guys using a non-standard definition for "phasor"? I'm
really confused by Jim's posting here. To me, a phasor simply
indicates the amplitude and phase of a sinusoidal component, relative
to some reference phase. I'd be comfortable with a "local definition"
that said the amplitude was relative to a reference amplitude, or was
in dB or dBm or dBuV or the like. But I am NOT comfortable with the
idea that a phasor at a particular point in space rotates in time
unless there is some time-varying thing that causes it to rotate,
maybe like a "trombone" section of line that someone is sliding in and
out. I do expect the phasor that represents a sinusoid propagating on
a transmission line to be a function of distance along the line and of
the frequency of the signal, in that it must rotate 360 degrees for
every one wavelength along the line. (More detail on this below.)

For "phasor" to be a useful concept, you'd better be talking about a
system in which there is a single sinusoidal excitation frequency --
or you better be verrrry careful to define what you mean by your
phasor diagrams.

See, for example, the page in Wikipedia on phasors.

Or else please give me enough info or references so I can straighten
out my thinking about them.

If I'm not mistaken, on a lossless line excited by a source at one end
with a reflective load at the far end such that the amplitude of the
forward wave is a1 and the amplitude of the reflected is a2, then the
phasor representing the forward wave, relative to the source end, will
be
forward phasor = a1*exp(-jx/lambda)
and for the reverse, assuming for convenience that the line is just
the right length so that the reverse is in phase with the generator at
the generator end,
reverse phasor = a2*exp(+jx/lambda)
where x is the distance along the line from the generator, lambda is
the wavelenth in the line, and exp() is e to the power(). Then the
phasor of the whole signal, fwd plus refl, at any point x is
net phasor = a1*exp(-jx/lambda)+a2*exp(+jx/lambda)
exp(jy) can be expanded as cos(y)+j*sin(y), so
net phasor = (a1+a2)*cos(x/lambda)+j*(a2-a1)*sin(x/lambda)

This makes is VERY clear that the phasor changes angle along any line
where a2 does not equal a1; in the special case where a2=a1, then the
phase can only be 0 or 180 degrees all along the line. If you pick a
different reference point (e.g. change the load or line lenght or
frequency in a way that moves the generator away from a point where
the return is in phase with the generator at the generator), then that
just adds a constant phase offset. But also notice that if a2 does
not equal a1, the phasor angle along the line goes through all
possible values, zero to 360 degrees. If a2 is almost equal to a1,
that phase shift occurs relatively quickly along the line, centered on
points where cos(x/lambda) goes to zero. I expect the same to be true
on a resonant antenna; the reflected wave is NOT the same amplitude as
the forward, but is similar, so you'll find places where the phase
change is quick but continuous as you move along the wire--this
assumes that the antenna is long enough that you can find such places.

Cheers,
Tom