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Okay, now hopefully you're able to work out those two little problems that I just
gave you. And to wrap up this lecture, I just want
to try to increase your intuition a little bit for what's going on.
Now, these filters, these high pass and low pass filters that we've been talking
about. Are really nothing more than voltage
dividers that happen to be frequency dependent.
So, let's take a look, go back to our, old familiar voltage divider, where we
take a voltage. And we apply it to a string of, it use to
be just resisters, but now we are going to put 2 impedances and now use Z to
represent impedance. But we'll put 2 impedances in series and
then measure the voltage at the mid point.
And so, just like a voltage divider when this use to be a battery and 2 resisters
The voltage here is some fraction of the total voltage there.
Now I can replace Z1 with either a resister, inductor, or capacitor, and the
same thing for Z2. And so if we look at the different cases
of this we can compute the output, magnitude of the output voltage over the
input voltage. It's always just going to be the
magnitude of this impedance divided by the magnitude of the sum of the 2
impedance. So this is just the voltage divider
equation but rewritten with impedances. And since I have these complex numbers to
deal with, we're going to just look at the magnitudes of the output to input
response. Now here's a table that summarizes all of
the cases. Now I solved two of these cases and
hopefully you solved the other two cases. But let's take a minute and look
carefully at this table. And understand all the different cases.
So, in first the column we have Z1 it's whatever I want to put in for Z1.
And the second column is Z2 and then it tells you what kind of filter you've made
and what the cut off frequency is. Now if Z1 is a resistor and Z2 is a
capacitor, then I have the simple RC filter that we solved first and we saw
that that was at low pass filter. And it has a cutoff frequency of 1 over
RC, now that was omega tTo turn that into regular hertz units.
We have to divide by 2 pi, with a relationship between angular frequency
and regular frequency in hertz is angular frequency divided by 2 pi gives me
regular frequency. Now, let's go back and look at this a
little bit more. This is a low pass filter because Z2,
when omega is small. This impedance becomes very large and so
this is like a voltage divider with a small impedance and then a very large
impedance. And you know from the old resistive
voltage divider that you get most of the voltage drop across the larger impedance
and the voltage divider. So now at low frequency Z2, the
capacitor, becomes a very large impedance.
In fact, when omega goes to 0 this goes to infinity.
And all of the voltage appears all of the, the voltage applied to the input
appears at the output. As I go up in frequency, this factor
becomes smaller. And so I have a larger impedance here and
a smaller impedance there. And so this voltage divider then is going
to give you a smaller output voltage. So low frequencies are favored in this,
the output of this RC filter built this way and high frequencies are attenuated
by this filter. Now if I replace Z2 with an inductor,
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And so it's, like if this is the larger impedance, if I move that up there on a
smaller impedance, then this voltage is going to drop.
And so by switching these two, I turn this low pass filter into a high pass
filter. Because at high frequencies, this Z1 is
going to become very small, because omega's large, this is going to be small.
And so, this is a small impedence, that's a large impedence.
Most of the voltage is going to appear here across the large impedance.
So, that's a high pass filter. And then the final case, which hopefully
you worked out, is that when, Z1 is an inductor and this is a resistor.
Then this impedance becomes large as frequency becomes large.
And it's 0 at 0 frequency. And so low frequencies pass through this
filter with very little attenuation. High frequencies ebcause this becomes a
large impedence and this is fixed. This is a resistor, high frequencies are
attenuated, so this is a low-pass filter. So this shows how, just with two simple
components, an R and a C or an R and an L, you can build both low-pass and
high-pass filters. Out of these two simple components.
Robert ClarkProvost and Senior Vice President for Research and Professor
Mark BockoDistinguished Professor and Chair
