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Transcript:

Welcome again to Electrical Engineering 3300

at the University of Utah.

Today, we're going to be talking about the

transmission line or telegrapher’s equations. These

equations are used to derive the voltage and the

current as a function of time on a transmission line.

We're going to derive one equation for voltage and

another equation for current, and we're going to be

using both of them throughout the semester to derive

how the fields propagate on a transmission line.

We're going to begin with a standard RLGC model.

This is the resistor, here is the inductor, right

here is the conductance and here is the capacitance.

This R prime is the resistance per unit meter per unit

length. L prime is in henrys per meter G prime is in

mhos per meter and C prime is in farads per meter.

This little piece of line is a small length. Let's call it

delta Z long. Maybe that's a millimeter maybe that's a

meter. It's a very small piece of a transmission line.

One of the questions that came up in class last

time is, why do I need to use multiple copies of this

RLGC in order to build up an entire transmission line?

Now let's go take a look at that question. Right here is

the transmission line. We're going to put the positive

voltage on the top conductor, we're going to ground

the bottom conductor and what's going to happen is

the wave is going to propagate down this transmission line.

The way the wave propagates is by resonating

between the inductor and the capacitor on this

transmission line so we need a model that represents

inductance and capacitance, that's resonant circuit

right there, and it's going to pass the resonance on to

the next inductor and capacitor and the next inductor

and capacitor and so on. The resistor and the

conductance and the various resistors and conductance

along here represent the loss in the circuit. All they do

is attenuate the wave. So we're going to begin with this

RLGC circuit, which is a resonant circuit that describes

the propagation on the transmission line.

Let's just go draw that again so we have kind

of a clean slate. Here's the resistor, inductor,

conductance, capacitance, there's our little piece of

transmission line and it is a length delta Z.

Now let's see where the voltages and the

currents are within this transmission line piece.

Right here, we're going to call this point Z and this point

right here is going to be Z plus delta Z so right at this

place we have the voltage at location Z and on this side,

we're going to have the voltage at location Z plus delta Z

Now let's draw the currents. We're going to

have one current right here that we're going to call

i(z) and the output current is going to be i(z) plus

delta Z, and then we're going to have a current going

through the conductance and a current going through

the capacitance. So let's call that I of G and I of C.

Now what we're going to do is write two sets

of equations. One of them is going to be the loop

equation and the other is going to be a node equation.

So here's the loop equation that I'm going to write.

Now remember what we do is we start with our voltage

and we add up all of the voltages in a circle and of

course they need to add up to be zero. So we're going

to begin right here. V of Z plus R times the current at

Z minus -- whoops, I'm sorry. That's not a plus. That's

a minus. V of Z minus the resistance times the current

minus L (DI/DT) and that's at location Z minus V of Z plus

delta Z is equal to zero.

Now the other thing that I want to note is

that all of these variables, this voltage, this current,

this derivative and this voltage are all functions of time.

So we could equally well write V of Z plus delta Z and t

but just for simplicity, I'm going to leave this

part out of the equation as I'm writing it and let's

remember some important factors here. R, which is the

resistance in ohms is equal to R prime, which is given in

ohms per meter times delta Z. L, which is given in

henrys is L prime, which is henrys per meter times

delta Z and so forth. So what I'm going to do now is

I'm going to take this entire equation and I'm going to

multiply it by one over delta Z.

at the University of Utah.

Today, we're going to be talking about the

transmission line or telegrapher’s equations. These

equations are used to derive the voltage and the

current as a function of time on a transmission line.

We're going to derive one equation for voltage and

another equation for current, and we're going to be

using both of them throughout the semester to derive

how the fields propagate on a transmission line.

We're going to begin with a standard RLGC model.

This is the resistor, here is the inductor, right

here is the conductance and here is the capacitance.

This R prime is the resistance per unit meter per unit

length. L prime is in henrys per meter G prime is in

mhos per meter and C prime is in farads per meter.

This little piece of line is a small length. Let's call it

delta Z long. Maybe that's a millimeter maybe that's a

meter. It's a very small piece of a transmission line.

One of the questions that came up in class last

time is, why do I need to use multiple copies of this

RLGC in order to build up an entire transmission line?

Now let's go take a look at that question. Right here is

the transmission line. We're going to put the positive

voltage on the top conductor, we're going to ground

the bottom conductor and what's going to happen is

the wave is going to propagate down this transmission line.

The way the wave propagates is by resonating

between the inductor and the capacitor on this

transmission line so we need a model that represents

inductance and capacitance, that's resonant circuit

right there, and it's going to pass the resonance on to

the next inductor and capacitor and the next inductor

and capacitor and so on. The resistor and the

conductance and the various resistors and conductance

along here represent the loss in the circuit. All they do

is attenuate the wave. So we're going to begin with this

RLGC circuit, which is a resonant circuit that describes

the propagation on the transmission line.

Let's just go draw that again so we have kind

of a clean slate. Here's the resistor, inductor,

conductance, capacitance, there's our little piece of

transmission line and it is a length delta Z.

Now let's see where the voltages and the

currents are within this transmission line piece.

Right here, we're going to call this point Z and this point

right here is going to be Z plus delta Z so right at this

place we have the voltage at location Z and on this side,

we're going to have the voltage at location Z plus delta Z

Now let's draw the currents. We're going to

have one current right here that we're going to call

i(z) and the output current is going to be i(z) plus

delta Z, and then we're going to have a current going

through the conductance and a current going through

the capacitance. So let's call that I of G and I of C.

Now what we're going to do is write two sets

of equations. One of them is going to be the loop

equation and the other is going to be a node equation.

So here's the loop equation that I'm going to write.

Now remember what we do is we start with our voltage

and we add up all of the voltages in a circle and of

course they need to add up to be zero. So we're going

to begin right here. V of Z plus R times the current at

Z minus -- whoops, I'm sorry. That's not a plus. That's

a minus. V of Z minus the resistance times the current

minus L (DI/DT) and that's at location Z minus V of Z plus

delta Z is equal to zero.

Now the other thing that I want to note is

that all of these variables, this voltage, this current,

this derivative and this voltage are all functions of time.

So we could equally well write V of Z plus delta Z and t

but just for simplicity, I'm going to leave this

part out of the equation as I'm writing it and let's

remember some important factors here. R, which is the

resistance in ohms is equal to R prime, which is given in

ohms per meter times delta Z. L, which is given in

henrys is L prime, which is henrys per meter times

delta Z and so forth. So what I'm going to do now is

I'm going to take this entire equation and I'm going to

multiply it by one over delta Z.