ECE3300 Lecture 5-1 TL Equations loop

Uploaded by cfurse on 01.09.2009

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.