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Transcript:

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.

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.