Welcome to Electrical Engineering 3300 at
the University of Utah.
Lecture number six, we're going to talk about
lossless transmission lines. The reason these are so
important is because most of our lines can be
considered lossless most of the time. In our
transmission lines, which we call -- we normally model
these as RLGC lines. In a lossless line, that means that
our resistance is zero and our conductance is zero, and
we are left strictly with an inductor and a capacitor;
that's a resonant circuit where the L and the C can self
resonate, and that gives us a transmission line behavior
where the magnitude of the wave will stay exactly the
same down the whole transmission line. The
propagation constant, which is alpha plus J beta, for a
lossless line, alpha will be zero. That gives us J omega
square root of LC. Beta then is omega square root of
LC, and the characteristic impedance, Z knot, is the
square root of L divided by C. The wavelength is always
two pi divided by beta and that gives us two pi divided
by omega square root of LC. The velocity of
propagation is another factor that we need and that is
always omega divided by beta, for a lossless
transmission line, it's one over the square root of LC.
Now, there's another special case of lossless
transmission line and that's the TEM line. Any line that
has two conductors can be considered to be a TEM line
so most of the lines that we are going to work with
follow this case. In that case, LC just happens to be Mu
epsilon -- oops, yeah, just happens to be Mu epsilon
where Mu and epsilon describe the material that's in
between, that's the insulator of the transmission line.
In that case, beta is omega over the square root of Mu
epsilon -- or actually it's not divided by. It's multiplied
by, and the velocity of propagation is one over the
square root of Mu epsilon and for air let’s see what
happens. If we take this as one over Mu knot epsilon
knot square root, it's going to give us C knot which is
about 2.996 times ten to the eighth meters per
second. That's the velocity of the wave in free space
which means that if a TEM transmission line is filled
with air that the velocity of propagation on it going to
be the speed of light.
Remember that, in general, epsilon is epsilon R
epsilon knot but Mu is Mu R, Mu knot. That gives us the
velocity of the propagation is the speed of light
divided by the square root of epsilon R for a lossless
TEM transmission line.
The wavelength is the velocity of propagation
divided by the frequency which ends up being the
wavelength in air divided by the square root of epsilon
R. Now, this is really cool because it means that all of
our major parameters such as these strictly a function
of the material that we're using as the insulation of
the transmission line.
Now, another interesting factor happens on
transmission lines and that's called dispersion. In
dispersion, it means that our velocity of propagation is
a function of frequency. It may mean that high
frequency fields travel faster or that low frequency
fields travel faster. Remember that the velocity of
propagation for a lossless line is C knot divided by the
square root of epsilon R.
Let's imagine what would happen if epsilon R
varied with frequency. This would be very typical in any
material such as water or other fluids because the
polarity of the molecules is very different for
different frequencies. In that case, the velocity of
propagation would vary with frequency and each part
of the wave would travel differently. If we had a
simple sine wave this wouldn't cause any dispersion. It
would mean that this frequency of sine wave and this
frequency of sine wave might propagate with different
velocities but in general, this entire wave would
propagate with one single velocity and this wave would
propagate with another different single velocity, but if
we are using a pulse, let's suppose we've got a
rectangular pulse. This rectangular pulse in time has a
lot of different frequencies in it. In fact, it's kind of
has a sink type of behavior in the frequency domain.
