Just a few odds and ends. Consider the
following which is called a "cascaded line"
problem. These are problems where we have two differenttransmission lines, with different characteristic
impedances. Since we will give all of the distances inwavelengths,λ, we will assume that theλwe are
talking about is the appropriate one for the line involved. Ifthe phase velocities on the two lines is the same, then the
physical lengths would correspond as well. The approach isrelatively straight-forward. First let's plot
on the
Smith Chart . Then we
have to rotate
so that we can find
, the normalized impedance at point A, the junction
between the two lines
.
Cascaded line
Smith diagram
Thus, we find
. Now we have to
renormalize the impedance
so we can move to the line with the new impedance
. Since
,
. This is the load for the second length of line, so let's find
, which is easily found to be
, so this can be plotted on the
Smith Chart . Now we have to rotate around
another
so that we can find
. This appear to have a value of about
, so
.
Towards the generator
More smith charts
Even more smith charts
There is one application of the cascaded line problem that is
used quite a bit in practice. Consider the following: We assumethat we have a matched line with impedance
and we connect it to another line whose impedance is
. If we connect the two of them
together directly, we will have a reflection coefficient at thejunction given by
Simplified cascaded line
Now let's imagine that we have inserted a section of line with length
and impedance
. At point A, the junction
between the first line and the matchng section, we can find thenormalized impedance as
Another cascaded line
We take this impedence and rotate around on the Smith Chart
to find
where we have taken advantage of the fact that when we
go half way around the Smith Chart, the impedance we get is justthe inverse of what we had originally (half way around turns
into
).
Thus
If we want to have a match for line with impedence
, then
should equal
and hence:
or
This piece of line is called a
quarter wave matching
section and is a convenient way to connect two lines of
different impedance.
