This module introduces and derives the telegrapher's equations, which describe how electrical signals behave as they move along transmission lines.
Let's look at just one little section of the line, and define
some voltages and currents
.
Applying kirchoff's laws
For the section of line
long, the voltage at its input is just
and the voltage at the output is
. Likewise, we have a current
entering the section, and another current
leaving the section of line. Note that both the voltage and the
current are functions of
time as well as
position.
The voltage drop across the inductor is just:
Likewise, the current flowing down through the capacitor is
Now we do a
KVL around the outside of the section of line and we get
Substituting
for
and taking it over to the RHS we have
Let's multiply by -1, and bring the
over to the left hand side.
We take the limit as
and the LHS becomes a derivative:
Now we can do a
KCL at the node where the inductor and
capacitor come together.
And upon rearrangement:
Now when we let
, the left hand side again becomes a derivative, and on
the right hand side,
, so we have:
and
are so important
we will write them out again together:
These are called the
telegrapher's equations and
they are all we really need to derive how electrical signalsbehave as they move along on transmission lines. Note what they
say. The first one says that at some point
along the line, the incremental
voltage drop that we experience as we move down the line is justthe distributed inductance
times the time derivative of
the current flowing in the line at that point. The secondequation simply tells us that the loss of current as we go down
the line is proportional to the distributed capacitance
times
the time rate of change of the voltage on the line. As youshould be easily aware, what we have here are a pair of
coupled linear differential equations in time and
position for
and
