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.