If we are going to try to use phasors on a
transmission line, then we have to allow for spatial variationas well. This is simple to do, if we just let the phasor be a
function of
, so we have
. How the phasor varies in
is one of the things we now have
to find out.
Let's start with the
Telegrapher's
Equations again.
For
we can now substitute
and for
we plug in
. So we get:
and
We take the derivative with respect to time, which brings down a
and then we cancel the
from both sides of each equation:
and
Viola ! In one simple motion, we have
completely eliminated the time variable,
, from our equations! It is not
really gone, of course, for once we figure out what
is, we have to multiply it by
and then take the real part before we can extract once
again, the actual
that we want. Nonetheless, insofar as the
telegrapher's equations are concerned,
has disappeared from the radar
screen.
To solve these we do just as we did with the
transient problem. We take a derivative with respect to
of
, which
gives us a
on the right hand side, for which we can substitute
, which leaves us with
(- times - is +, but
and so we have a - in front of the
). We then re-write
as
The simplest solution to this equation is
from which we can then get the actual voltage signal
Note that we could factor out an
, from the exponent, which, since it is just a
constant, we could include in
(and call it
, switch the order of
and
, and write
as
which looks a lot like the "general"
solution we were talking about
earlier !
The number
is special. It is usually represented with a Greek
letter
and is called the
propagation coefficient . Thus we have
As previously,a point on the wave of constant phase requires that the
argument inside the parenthesis remains constant. Thus if
is going to equal
(
i.e. what was at point
at
is now at
at time
it must be that
or
Which one again, defines the
phase velocity of the
wave. Other relationships to keep in mind are
The first comes from the fact that the wave varies in
as
. Thus when
, the wavelength,
just increases by
, to get the phasor to go through one full
rotation. Note also, as before, the choice of the minus sign inthein
represents a wave
going in the
direction, while the choice of the + sign will
give a wave going in the
direction. Clearly, by starting out taking the
x-derivative of the equation for
we would end up with
Let's consider the two phasors then, and define the voltage
phasor associated with the positive going voltage wave as
and the negative voltage phasor as
We should keep in mind that both
and
can be, and probably are, complex numbers. (From now
on we will drop the little ~ over the variables because its verytedious to get it to show up with this word processor. You will
just have to keep in mind that any variable we do not explicitlyput inside absolute value markers (
i.e.
) is going to be, in general, a complex number). We
will, of course, have similar expressions for the positive andnegative going current waves.
Let's consider the positive going current and
voltage waves, and plug them into
.
The x-derivative brings down a
, the
's cancel, and we have
But, since
we have
as we had before.
So, what has changed? Not much from the case of
transients on a line. We will now assume we have a
steady state problem. This means we turned
on the generator a long time ago. We assume that it has beenconnected to the line long enough so that all transient behavior
has died away, and that voltages and currents are not changingany more (except oscillating at frequency
, of course).
If the line is semi-infinite (or matched with a
load equal to
)
then it is pretty obvious that
where
is the source impedance, and
is the source voltage phasor.
Receive real-time job alerts and never miss the right job again
Source:
OpenStax, Communications b : filters and transmission lines. OpenStax CNX. Nov 30, 2012 Download for free at http://cnx.org/content/col11169/1.2
Google Play and the Google Play logo are trademarks of Google Inc.
Notification Switch
Would you like to follow the 'Communications b : filters and transmission lines' conversation and receive update notifications?
