Welcome back to Linear Circuits.
This is Dr Ferri.
This lesson in on lowpass and highpass
filters.
We are at the last topic in this module.
In the previous lesson we examined the
frequency responses of RC and RLC
circuits.
In this lesson we want to introduce the
concept of filtering.
Filtering is a really important concept, a
really important
application of circuits when, especially
when you're measuring signals.
We want to show the properties of lowpass
and highpass filters in this lesson as
well.
Let's first define what an analog filter
is.
It's a circuit that has specific shaped
frequency response and the purpose is
either to
attenuate signals in certain frequency
ranges or
perhaps even amplify signals in certain
frequency ranges.
And, the two most common filters are shown
here.
There's a lowpass filter
over here, and in the lowpass filter,
frequency content, so this is plotted
versus frequency.
Low frequency content signals get passed
through without, just get straight passed
through.
But high frequency content is attenuated
or filtered out.
Now high pass filter works exact opposite
way.
Low frequency gets attenuated, but it
passes the high frequency without any
problems.
Lets take a look at a specific low pass
filter example.
Now this, we've looked at this example
before in a previous lesson.
This is an RC filter.
It's a circuit.
And we didn't mention that it was a filter
before.
But if you look at the frequency response
of it, this plotting the magnitude.
So this is the magnitude of H of omega.
And that's versus radians per second.
It has a low frequency characteristics.
And what we saw is that if the signal
looks like this.
It's got two frequency components, a low
frequency, and then a high frequency.
And both of them have equal amplitude.
When we pass it through this lowpass
filter, it attenuates
the high frequency because that's up in
this range right here.
It has a low value, so it attenuates it.
And the lower frequency at 50 hertz, or
50 radians per second, is down here
and it gets passed through without much
attenuation.
And you see the result here.
The low frequency is sent through with
out much attenuation, high frequency is
attenuated.
And that's the characteristics of a low
pass filter.
Now let's look a little bit more
detail about parameters that we might
define here.
First of all this is a DC gain.
DC gain because at omega equals zero
that's a DC
value, that's a DC frequency and that
means if I send an input.
Of a constant value.
If Vin is
equal to some constant C, then Vout is
equal to Kac of DC times C.
It's like a game, a multiplier times that
input.
And then another term that we're going to
identify is a bandwidth, and
that's omega sub b and
that's found by finding, by looking at
this plot and
seeing when it's equal to 0.707 of the DC
value.
We call that the bandwidth and the reason
we say
that is because it's kind of near the the
curve here.
Below the bandwidth the signal is mostly
passed through without much
attenuation so we call that the, the
passband and over here
is the stopband.
That's really the region where we
attenuate the signals.
Now looking at a very specific circuit
example.
This is an RC circuit with these
parameters.
If I look at R times C, that's equal to
0.01.
One over RC.
Is 100 radians per second.
And it turns out that omega sub b is equal
to one over RC.
In this particular case, we look at this
and
we say, all right 0.707, I follow that
across.
That's right here and that's the DC value
is equal
to one, so this is 0.707 of the value of
one.
And drop that down, that is indeed 100
radians per second, is the bandwidth.
So the bandwidth again is calculate 0.707
of the
DC value, in an RC circuit it's one over
RC.
Let's look at a, a specific signal sent
through an RC circuit.
So plotted here is
the magnitude of the transfer function and
has a DC value, okay?
So DC is equal to two here.
Now, if I look at here, the DC value is
equal to ten of
the input signal and that's, you could see
that the average value is ten.
And there's a lot of high frequency in
here that's
quick changes, or fast changes in the
motion in the signal.
And that corresponds to high frequency.
The corresponding output
here looks like this.
Where the out put DC is now equal
to 20, which is the input DC times the DC
gain, gives me 20.
And you see that I've attenuated the high
frequency a great deal.
Because it's a low past filter and the
high frequency
range is somewhere in here that's where
this signal was.
So this is a classic
example of what happens as we pass the
signal through a lowpass filter.
Now, let's look at Bode plots of lowpass
filters.
So this is the same image that we've
seen before and this is plotted on linear
scales.
If I plot this on Bode plot scales then
this is a logrithmic scale right here.
And over here it's in decibels.
We've introduced the idea Bode of of
this sort of plot before when we were
talking about frequency spectrum in a
previous lesson.
So, decibels are defined by taking 20
times the
log base ten of every point along this
curve.
So every point along this curve, we take
20 times the log base ten of it.
So for example at the DC value, we take 20
times the log base ten of the DC gain.
And that's really kind of a low-frequency
ascentote for this signal.
At low frequency it looks like that.
Now the bandwidth is equal to 0.707 of the
DC value.
If I calculate that, I want to see.
What this value is on the log scale and
the Bode Plot scale.
Take 20 times the log of 0.707 times K of
dc.
And with logs, if I take a log of a
product, it turns out it's equal to the
sum of the logs.
Well 20 times the log of 0.707 is minus
three
decibels plus 20 times the log K sub DC.
So, the bandwidth, and that was the point
of which we had the bandwidth here.
So over here that same frequency happens
at three decibels below the DC value.
The other plot, thing that we'll find here
is that at high frequency, we have a roll
off.
And that roll for an RC circuit.
That slope is minus
20 db per decade.
Now
let's look at,
at, cognumerical
example.
This is an RC circuit
and I've plotted the frequency response.
I want to find these values.
I want to find K sub DC and the bandwidth.
K sub DC
is found by looking at this value here.
20 times
the log of K sub DC is equal to zero.
So this value is zero.
And I want to back out what K sub DC is.
So, if I divide both sides by 20, and then
raise it to the tenth,
raise it to the power of 10 to that power.
I have
ten to the zero is equal to one.
So d, DC gain is one.
Now the bandwidth is three decibels below
that
value that looks to be about right there.
That looks about three decibels.
And this is 1000th, so omega sub B is
equal to
1000 radians per second, and that's the
bandwidth.
Now a highpass filter is when
that pass is a high frequency component.
In this case we'll say that this is kind
of the knee of the curve.
It's really that the 0.707 point of the
high frequency.
So this is the high frequency value it's
0.707 of that.
We oftentimes call that the band width.
And this time, this is the pass
band region, and this is the stop band
region.
I get the high pass filter by
interchanging the capacitor and the
resistor positions.
So now I'm taking the voltage across the
resistor.
If I look at the transfer function of
this, it looks like this.
And you can see that as omega goes
to zero, this numerator drops out and
becomes zero.
So that's why the magnitude goes to zero
as omega goes to zero.
And you can also see
that as omega gets very large.
This one becomes negligible, and the
magnitude is going to approach one.
So in this particular circuit, this
magnitude would approach one.
Let's look at a numerical example of
taking
a highpass filter and putting it through
you know using
it to filter out a specific signal.
This is a specific signal that we saw
before.
The DC value was equal to ten, there was a
lot of high frequency on it.
So if we take the magnitude
and plot it here and say this value is
one.
In this particular case we're going to
assume, or we're going
to, in this particular circuit, the
high-frequency content was in this range.
The high-frequency content of this signal
here.
So the corresponding outlet looks like
this.
The DC value is greatly attenuated,
because it's down in this rate region and
then
the high frequency content is hardly
anything happens, hardly any
attenuation at all, because it's the high
frequency content was
out of this frequency rage where we had no
attenuation.
So that's what a highpass filter does.
It's often used to filter out DC values in
measured signals.
So
in summary, we've looked at RC circuits.
And RC circuits were first order.
If I want to get a sharper roll off here,
I would use an RLC filter.
And here's the two examples of RLC
filters, a low
pass and a high pass, and they're just
series circuits.
And the type of response I get depends on
where I'm taking the output voltage.
And then, here, I'm looking at it over the
capacitor.
Here the inductor and they have these very
different frequency responses.
This being low pass and the slope here is
minus 40 decibels per decade
compared to minus 20 for an RC filter.
This is a second order second
order filter, because if I looked at the
differential
equation for this it would be second order
differential equation.
And anything that has second order
differential equation, would have a Bode
plot with minus 40 decibels per decade, as
opposed to minus 20.
And these are the transfer functions for
these two circuits.
this one has plus 40 decibels per decade
slope.
And that means that I've got a short,
sharper
transition between the pass band region,
which is here,
and the stop band over here.
And right up here, I'm going to call these
the corner frequencies.
And in both cases the corner frequency,
sometimes we'll show it as omega zero,
is equal to one over the square root of
LC.
And that's again in radians per second.
So in summary, an analog filter is a
circuit that has a specific shaped
frequency response.
And the two most common.
Types of filters are lowpass and highpass.
The lowpass filter passes low frequency
components in the signals, and
attenuates the high frequency components,
whereas
the highpass filter does the opposit.
It passes through the high frequency
components and attenuates low frequency.
In our next lesson we're going to look at
another
set of common filters, that's the bandpass
and notch filters.