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Now we just talked about how to do stubs by calculating them using
the tangent or co-tangent equation, but we can also use the Smith
chart to calculate them. We know that if we have a short-circuit,
Z of the short-circuit is 0 and that point is right here. That's
Z of a short-circuit. Z of an open-circuit is infinity, and
that's right there. Let's draw a picture of either our
short-circuited case or our open-circuited case, and we can see
that this is the load for a length, for a stub with a length L and
a characteristic impedance Z naught. So if I knew that I wanted
to have some imaginary part right here, all I would need to do is
start at my short-circuit and rotate towards the generator, right
here like this, and, look, I can reach virtually every imaginary
part on this transmission line starting with my short-circuit.
The same thing happens with ZOC. Rotating towards the generator
I'm able to reach virtually every imaginary value. So all I have
to do is change the length of the line and I will be able to
create an imaginary component of any size that I want. So here
again are the steps. Here's my load ZL. And let's say that ZL is
100 plus J 100 ohms on a 50-ohm transmission line. So I'm going
to plot 2 plus J2. Here is the real part of 2. Here's the
imaginary part of 2. There's my Z load. I'm going to draw a line
out from the center through my point like this because I'm going
to use that to tell me how far I want to rotate a distance D.
Remember that distance D is going to match the real part. What
that means is when I get right here to point A, I want the real
part to be 50 ohms. So ZA I want it to be 50, and I don't care
what the imaginary part is yet because I'm going to use a stub to
remove it. If I normalize this by 50 ohms, this would be 1 plus J
and some imaginary value. I know I didn't change this because X
was some random value before. When you divide it by 50, it's just
another random value. So ZA is 1 plus JX. Now, if I want to be
able to get to that value, it means that I'm going to be on the
part where the real circle is 1. This is sometimes called the
matching circle. I'm just going to draw a circle. It's always
the same one. That's where the real part, every place on this
circle is equal to 1. So if I started at my load and I rotated
towards the generator, I should be able to reach a couple of
points that represent the real part being matched. Now, I could
use a protractor and draw a circle right around like this, put the
center of your protractor right there at the center, put the
pencil of your protractor here and start drawing around. But I
don't usually have a protractor with me. So I'm going to do the
same thing I've done before which is use a piece of paper, mark
this distance and then I'm going to line it right up here on the
center of the Smith chart and with my straight piece of paper like
this I'm going to mark right there. That point is ZA. Well, if I
kept going around, there would be another point that would happen
over here on the Smith chart right about there. So I'm going to
do the same thing. Again, take your piece of paper, put it at the
center. I'm kind of going to sketch my paper so you can see it
here. Right there is the point, is another possibility for ZA.
So here's ZA-1. That's the first one I would come to. Here's
ZA-2. Both of those would give me viable matching networks. They
will give me different lengths of line and different distances but
both of them will work. So what I'm going to do is rotate ZL
until it gets to, let's do ZA-1. Draw a straight line out here
from the center through ZA-1 and read off the value. It's about
.32. Then this distance D here is going to be equal to .32 minus
.21. So D equals .32 minus .21 wavelengths.