Often, there are reasons why using a discrete
inductor or capacitor for matching is not such a good idea. At
the high frequencies where matching is important, losses in both
L or C mean that you don't get a good match, and most of the
time (except for some air-dielectric adjustable capacitors) it
is hard to get just the value you want.
There is another approach though. A shorted or open
transmission line, when viewed at its input looks like a pure
reactance or pure susceptance. With a short as a load, the
reflection coefficient has unity magnitude
|Γ|=1.0
Γ
1.0
and so we move around the very outside of the Smith Chart as the length of the line
increases or decreases, and
Z
in
Z
0
Z
in
Z
0
is purely imaginary. When we did the bilinear transformation
from the
Zs
Z
0
Z
s
Z
0
plane to the
rs
r
s
plane, the imaginary axis transformed into the circle of
diameter 2, which ended up being the outside circle which
defined the Smith Chart.
Another way to see this is to go back to
this equation. There we found:
Zs=
Z
0
Z
L
+ⅈ
Z
0
tanβs
Z
0
+ⅈ
Z
L
tanβs
Z
s
Z
0
Z
L
Z
0
β
s
Z
0
Z
L
β
s
(1)
With
Z
L
=0
Z
L
0
this reduces to
Zs=ⅈ
Z
0
tanβs
Z
s
Z
0
β
s
(2)
Which, of course for various values of
ss, can take on any value from
+ⅈ∞
to
-ⅈ∞
. We don't have to go to Radio Shack© and buy a
bunch of different inductor and capacitors. We can just get
some transmission line and short it at various places!
Thus, instead of a discrete component, we can
use a section of shorted (or open) transmission line instead
Figure 2. These matching lines are called
matching stubs. One of the major advantages here
is that with a line which has an adjustable short on the end of
it, we can get any reactance we need, simply by adjusting the
length of the stub. How this all works will become obvious
after we take a look at an example.
Let's do one. In Figure 3 we can
see that,
Z
L
Z
0
=0.2+0.5ⅈ
Z
L
Z
0
0.20.5
, so we mark a point "A" on the Smith Chart. Since we
will want to put the tuning or matching stub in shunt across the
line, the first thing we will do is convert
Z
L
Z
0
Z
L
Z
0
into a normalized admittance
Y
L
Y
0
Y
L
Y
0
by going
180
°
180
°
around the Smith Chart to
point "B", where
Y
L
Y
0
≂0.7-1.7ⅈ
≂
Y
L
Y
0
0.7-1.7
. Now we rotate around on the constant radius,
rs
r
s
circle until we hit the matching circle at point "C". This is
shown in Figure 5. At "C",
Y
S
Y
0
=1.0+2.0ⅈ
Y
S
Y
0
1.02.0
. Using a "real" Smith Chart, I get that the distance
of rotation is about
0.36λ
0.36
λ
. Remember, all the way around is
λ2
λ
2
, so you can very often "eyeball" about how far you
have to go, and doing so is a good check on making a stupid math
error. If the distance doesn't look right on the Smith Chart,
you probably made a mistake!
OK, at this point, the real part of the admittance is unity, so
all we have to do is add a stub to cancel out the imaginary
part. As mentioned above, the stubs often come with adjustable,
or "sliding short" so we can make them whatever length we want
Figure 6.
Our task now, is to decide how much to push or pull on the
sliding handle on the stub, to get the reactance we want. The
hint on what we should do is in
Figure 1. The end
of the stub is a short circuit. What is the admittance of a
short circuit? Answer:
∞
,
ⅈ∞
! Where is this on the Smith Chart? Answer: on the outside, on
the right hand side on the real axis. Now, if we start at a
short, and start to make the line longer than
s=0
s
0
, what happens to
Ys
Y
0
Y
s
Y
0
? It moves around on the outside of the Smith Chart.
What we need to do is move away from the short until we get
Ys
Y
0
=-ⅈ2.0
Y
s
Y
0
2.0
and we will know how long the shorted tuning stub
should be
Figure 7. In going from "A" to "B" we
traverse a distance of about
0.07λ
0.07
λ
and so that is where we should set the position of the sliding
short on the stub
Figure 8.
We sometimes think of the action of the tuning stub as allowing
us to move in along the
ℜYs
Y
0
Y
s
Y
0
to get to the center of the Smith Chart, or to a match
Figure 9. We are not in this case, physically
moving down the line. Rather we are moving along a
contour of constant real part because all
the stub can do is change the imaginary part of the admittance,
it can do nothing to the real part!
