This module looks at what happens when we excite a transmission line. The module examines this concept in the context of a cable TV co-axial transmission line.
We will now go on and look at what happens when
we excite the line. Let's take a DC voltage source with a sourceinternal impedance
and connect it to our semi-infinite line. The sketch
in
is sort of awkward looking, and will
be hard to analyze, so let's make a more "schematic like"drawing
, keeping in mind that it is a
situation such as
which we trying to
represent.
Exciting a transmission line
Schematic representation
Why have we shown an
and a
but not
or
? The answer is, that if the line is semi-infinite,
then the "other" end is at infinity, and we know thereare no sources at infinity. The current flowing through the
source resistor is just
, so we can do a KVL around the loop
Substituting for
in terms of
using
this
equation :
Which we re-write as
Or, on solving for
:
This
should look both reasonable and
familiar to you. The line and the source resistance are actingas a
voltage
divider . In fact,
is just the usual
voltage divider equation for two resistors in series. Thus, thegenerator can not tell the difference between a semi-infinite
transmission line of characteristic impedance
and a resistor with a resistance of the same value
.
Line is initially a voltage divider!
Have you ever heard of "
twin-lead" or maybe "
co-ax" and wondered why people would want to use wires
with such a high resistance value to bring a TV signal to theirset? Now you know. The
characterization is not a measure of the resistance of
the wire, rather it is a specification of the transmissionline's impedance. Thus, if a TV signal coming from your antenna
has a value of, say,
, and it is being brought down from the roof with
twin-lead, then the current flowing in the wires is
, which is a very small current indeed!
Why then, did people decide on
? An antenna which is just a half-wavelength long
(Which turns out to be both a convenient and efficient choicefor signals in the 100MHz (
) range) acts like a voltage source with a source
resistance of about
. If you remember from ELEC 242, when we have a source
with a source resistance
and a load resistor with load resistance value
, you calculate the power
delivered to the load using the following method.
Power transfer to a load
, the power in the load, is just product of the voltage
across the load times the current through the load. We can usethe voltage divider law to find the voltage across
and the resistor sum law to find the current through
it.
If we take the derivative of
with respect
to
, the load resistor (which we assume we can pick, given
some predetermined
) we have (ignoring the
),
Putting everything on
and then just looking at the numerator:
Which obviously says that for maximum power transfer, you wantyour load resistor
to have the same value as your source resistor
! Thus, people came up with
twin lead so that they could maximize the energy
transfer between the TV antenna and the transmission linebringing the signal to the TV receiver set. It turns out that
for a co-axial transmission line (such as your TV cable)
minimizes the signal loss, which is why that value was
chosen for CATV.
