ECE3300 Lecture 7 2 Input Impedance


Uploaded by cfurse on 07.09.2009

Transcript:
Now let's go back and actually consider what the
input impedance is going to be. Here's our transmission line connected
onto our load, ZL. It has a characteristic impedance of Z knot and it
is L meters long. Right here is the load voltage and this is the load
current. Right here is the input. It's also where we measure the input
impedance which we're going to be calculating, and then we're going to
connect this on to a generator that has a resistance of RG and that
has a voltage of VG. So there's our transmission line connected on to
the generator on the front end, and if we wanted to know what ZN is,
ZN is going to be VN divided by IN. Here's IN right there.
Now let's look at this front end just from the point
of view of a circuit, of our circuits' class like this. Here is VG, here is
RG, and here is VN. When would VN equal VG? The only time that
would happen is if you had an open circuit right here and then there
would be no current in the generator, the input current would be zero
and then the voltage VN would equal VG, but that's not a typical case
because we do in fact have this transmission line connected on here
with its load on the end and as a result, VN is not equal to VG. So ZN
is not simply determined by the generator. It's going to be
determined by the combination of the generator and the transmission line.
So let's go see if we can solve this problem: Here's
VG, right there, here's RG, Z zero and ZL. We're going to write our
standard voltage equation V of Z is equal to V zero plus E to the minus
J beta Z plus V zero minus E to the plus J beta Z. Remember that V
zero minus is the complex reflection coefficient times V zero plus.
Well right here is the location for Z equal to zero and here is the
location for Z equal minus L where L is the length of the transmission
line in meters. So if I wanted to know VL that would be the voltage at
Z equals minus L -- sorry not VL; VN. Right here's VN. So if I want
VN, I look at Z equals minus L. Let's substitute that back in. That
gives us V zero plus E to the minus J beta and then when I put in
minus L that minus changes to a plus, and then let's put in gamma V
zero plus E to the J beta and
when I put in minus L, there we are. So there's VN.
Similarly, let's write an equation for IN. That is going
to be I at Z equals minus L. That's going to give us I zero plus E to
the plus J beta L plus I zero minus E to the plus J beta L and then
this changes to negative. I made a little mistake here on this uh -- I
made a little mistake right there on that sign. That should be a minus
and remember that I zero plus is V zero plus divided by Z zero and I
zero minus is V zero minus divided by Z zero and that gives us that
ZN is going to be equal to V zero plus times E to the plus J beta L
plus gamma E to the minus J beta L. E to the minus J beta L and
that's going to be divided by V zero plus divided by Z zero times E to
the plus J beta L minus gamma E to the minus j beta L.