Welcome back to Linear Circuits.
This is Dr.
Ferri.
This lesson is on bandpass and notch
filters.
Again, we're at the last topic here and a
very
important application for this material is
the concept of filtering.
In a previous lesson, we introduced
lowpass and
highpass filters In this lesson we will
introduce
the characteristics of notch and bandpass
filters.
To summarize, we looked at RC filters
before in our previous lesson and we found
that we had lowpass filters were found by
taking the output over the capacitor.
Highpass filters we take the output to be
over the resistor.
And the corresponding frequency responses
look like this.
And in both cases we have bandwidth
equal to 1 over rc.
And also in both cases, we have slopes
that are 20 decibels per decade.
This one the slope is going down, and this
one the slope is going up.
And looking at RLC circuits.
And RLC filters.
the Lowpass filter was gotten by taking
the output over
the capacitor, Highpass by taking the
output over the inductor.
In these cases, because they're second
order,
the slopes are 40 decibels per decade.
In this case, the slope is going down.
In this case, the slope is going up.
They're second order because if I wrote a
differential equation
for this, it would be a second order
differential equation.
And in both cases, there's a corner
frequency here.
I'll call it omega 0.
Omega 0 is 1 over the square root of
LC, and this is in radians
per second.
So now
we want to introduce a new type of filter,
and that's called a bandpass filter.
I'm taking my same RLC circuit.
But this time I'm taking the voltage to be
across the resistor and if I look
at this, this is the transfer function of
it, and this is the corresponding
frequency response.
Now in this particular case we see that it
goes up
and back down, and in this range is
Passband just here.
It's just certain frequencies are passed
through without attenuation.
And outside that region, outside the
Passband region, we attenuate the signals.
The width of the passband is R over L.
And that's defined as being the width of
the region between the
points where we're down at minus 3
decibels below the peak value.
Now this passband is centered around the
value of 1 over the square root of LC.
I'll call that same thing I called it
before, omega sub zero,
this is, again, this is a passband filter.
Now let's look at a numerical example of a
passband filter.
Bandpass filter.
Suppose I want, in, in terms of hertz,
I want my center frequency to be at 1,000
hertz.
Well, to change that over to radians per
second, I have to multiply by 2 pi.
So I've got 2 pi times f nought, which
is 6283 radians per second.
And the
other thing I want sort of my
specifications here
in terms of hertz.
I want a 200 hertz passband.
And I have to convert that to radians per
second.
And again I multiply by 2 pi.
And I get 1257 radians per second.
So suppose that.
You know, those are my specifications.
Now I want to do my design.
I want to find specific values of R, L,
and C that give me those characteristics.
Well, let's start with omega naught.
Omega naught is equal to 1 over the square
root of LC.
And, I have to pick.
So I've got two parameters and one known.
I have to pick one.
So let's
just pick L to be some value, I'll call
it, suppose I've got five mill Henry
inductors available.
So I pick it to be that.
And now, then I saw for C, C is equal to
one over omega
naught squared times L, and that is equal
to 5.1
microfarads.
Now we have to solve for
R, and we know that R over L is
equal to that band width.
And we've already
picked L, so that means that R is equal to
6.3 ohms.
So then the frequency characteristics of
this is going to look like this.
At 6283 radians per second.
We've got the peak and then the bandwidth
here is 1257 radians per second.
With these values of R, L, and C.
The other type of filter that I want to
introduce
is the notch filter.
In that particular case we get the notch
filter by taking
the output voltage to be across the
inductor and the capacitor together.
This is a transfer function.
If I plot that transfer function and get
the frequency response, I get something
with this characteristic.
That's why it's called a notch.
Sometimes called the stop band filter
because it
gets rid of signals in a certain frequency
range.
The stop band, or the notch, is centered
about 1 over the square root
of LC.
And we can also look at three
decibels below the dc
value.
And this range here is called the
passband.
And that value is
equal to R over L.
Now
a notch filter is really commonly used to
get rid of line frequency.
Line frequency is what you get in your
power lines.
Like in this country it's 60 hertz.
In other country it's 50 hertz.
So it's very common to use, to, to get
noise in your signal.
That is at 60 hertz or 50 hertz and then
you build a notch filter to get rid of
that signal.
So and then the center frequency would be
60 hertz
in free, in hertz or 2 pi times 60 in
radians
per second.
Let's look at a specific
example of a notch filter and, you
know, speaking of the 60 hertz, we
want to build a 60 hertz notch filter.
So we'll say, f naught will be 60 hertz.
So that means omega naught
is 2 pi f naught, which is 377 radians per
second.
The bandwidth we'll select to be maybe
about 10 hertz.
That means we want to filter out pretty,
pretty much
everything between 55 and 65 hertz.
And in
radians per second.
That's 63 radians per second.
So those are our specifications.
And what we'd like to do now is solve for
R, L, and C.
Omega naught is equal to 1 over the square
root of LC in
radians per second.
And, if we pick, L
to be equal to 100 millihenrys,
then that means that C is equal to 1 over
omega naught squared L is 70.
Microfarads, which is, kind of, large
values here.
And that's,
we need these large values to get down to
that frequency range.
B is equal to R over L,
and that's equal to 63, so we can solve
for R is equal to 6.3 ohms.
And as a result, if I were to plot this,
I get something in terms of hertz.
I get
something that's centered at 60 hertz.
And this is the passband region here.
So this is 65.
And that's 55, down like that.
So pretty much, I'm filtering out
things between 55 and 65 hertz.
So, in summary.
Well, we found that we can get different
filter characteristics from RC and
RLC circuits, just by deciding what we're
going to take the output voltage to be.
A bandpass filter passes frequencies in a
specific frequency band,
and a notch filter rejects frequencies in
a specific frequency band.
In our next lesson, we're going to do a
lab demo.
We're going to go back to the guitar
string experiment
we've seen before and this time we're
going to filter out certain frequencies in
there.
Please make sure that you visit the forum
and that you
ask and answer any questions that you can
on that forum.
Thank you.