Now let's look at the time domain concept in this equation. We're
going to be solving the telegrapher's equation in the time domain.
We can see they're in the time domain because these derivatives as
a function of time. These two equations must be solved
simultaneously, which means we can't solve one equation without
solving the other. The reason we can't is because the voltage
generates the current and the current generates the voltage, back
and forth, until the transmission line has transmitted both
voltage and current down the line. We're going to use a specific
layout in order to solve this problem. Let's define the Z to be
the direction of the transmission line. And we're going to break
that transmission line up into little pieces that are each delta Z
long. If we want to know where we are on the transmission line,
we're going to find the value of Z by taking an integer K and
multiplying it by delta Z. K, capital K, will be the largest
index on our transmission line. So the total length of the
transmission line L will be capital K times delta Z. We're also
going to divide the transmission line model up into little pieces that
are delta T long in time. And if I want to know a specific time,
it's going to be N delta T. My maximum time will be capital N
times delta T. Then I'm going to define my voltage right here at
these points. So this will be voltage at K location 0, voltage at
K equal 1 and voltage at K equal 2. Then if I wanted to find the
derivative of the voltage as a function of Z, that would be V1
minus V0 over delta Z or, in a general form, it would be V at K
plus 1 minus V of K divided by delta Z. Now, I want you to notice
that I've taken the subscript and converted it into a parentheses
form just because this is the easier way to be able to do the programming in
the end. Now, let's take a look at the location of this equation.
When I did this derivative, the derivative is defined right here
in between the two values of the voltages. And that's because I'm
using the central difference formula. Let's see where I need to
put the current in order to make sure that the left-hand side and
right-hand side of this equation show up at the same location. So
if I defined my current right here in between my voltages, that
would be at the correct location. So let's call this I0 and this
one I1 and so on. So I'm going to define my current as being
halfway in between my two voltages so that the derivative of the
voltage and the location of the current end up being the same.
Let's check this bottom equation also. If I did the derivative of
the current as a function of Z, that would be taking this value
right here minus this -- let's see, derivative of a function of Z.
This value that would be over here and that would give me the
value of my voltage. So it would be something like V -- sorry, I
of 0 minus I of minus 1 divided by delta Z. This would be I of K
minus I of K minus 1 divided by delta Z. Now, let's see what
location that would be at. That would have given me the value
right here at this indexed voltage. Fortunately, that's where I
need it to be on the right-hand side of my equation. It's very
important to offset our voltages and our currents by half a step
in space, half a delta Z cell, if you will, so that the
derivatives and the locations of the voltages and current end up
at the correct locations for both our top equation and our bottom
equation.
