Unfortunately, since we don't know what value
the phasor
has, these equations do not do us a whole lot of good!
One way to deal with this is to simply divide
this equation into
this equation . That gets
rid of
and the
and so we now come up with a
new variable, which we shall call
line impedance ,
.
represents the ratio of the total voltage to the total
current anywhere on the line. Thus, if we have a line
long, terminated with a load
impedance
, which gives rise to a terminal reflection coefficient
, then if we substitute
and
into
, the
which we calculate will be the "apparent" impedance
which we would see looking into the input terminals to the line!
There are several ways in which we can look at
. One is to try to put it into a more
tractable form, that we might be able to use to find
, given some line impedance
, a load impedance
and a distance,
away
from the load. We can start out by multiplying top and bottom by
, substituting in for
, and then multiplying top and bottom by
.
Next, we use Euler's relation, and substitute
for the exponential. Lots of things will cancel out,
and if we do the math carefully, we end up with
For some people, this equation is more satisfying than
, but for me, both are about equally opaque in
terms if seeing how
is going to behave with various loads, as we move down
the line towards the generator.
does have the nice property that it is easy
to calculate, and hence could be put into MATLAB or aprogrammable calculator. (In fact you could program
just as well for that matter.) You could specify
a certain set of conditions and easily find
, but you would not get much insight into how a
transmission line actually behaves.
