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.