[MUSIC] Welcome back to Linear Circuits. This is Dr. Ferri. This lesson is on lowpass and highpass filters. Let me define what an analogue filter is. It's a circuit that has a specific shaped frequency response. And the purpose of it is to attenuate, or filter out, signals with specific frequency content. For example, right here whenever the frequency response, the magnitude of the frequency response is small, we're going to be filtering out signals at that frequency. So, in that case, this is a, we're filtering out this part. And in this other case, we're filtering out this part. Now, filters are really, really common in terms of filtering out, for example, measurement noise, unwanted measurement noise. Let's look at a little bit more specific at lowpass and highpass filters. A lowpass filter passes low frequency components. So, frequencies in this range are passed through without much attenuation. In this range right here, it filters them out. And a highpass filter filters out, or gets rid of, frequency input that in that range and keep those that are in this range. So, for example, if I have noise that is high frequency, then I might want to use a low pass filter to get rid of that high frequency noise. I want to go back and look at an example that we've studied before when we were looking at frequency response of circuits. So, we've done this example before, but now we want to look at it from the perspective of a filter. So, what we have is an input to an RC circuit and the corresponding output. And we took that input and broke it into its frequency components. There was a 50 Hertz, or 50 radians per second and 800 radians per second. The only difference is, in this particular example I've added a DC component to it. So, if I look at the average here. It has an average of one. And so, I put that right there. So, that's a DC. And I put that into the circuit, as you recall, I take the amplitude at each of these frequencies, multiply it by the amplitude of the transfer function which is plotted. That is, actually, what we're calling the frequency response. And I get the output spectrum, the frequency spectrum of the output signal. And what you can see is this particular one gets multiplied by a value that is much smaller. And so, it is smaller. So, what we see here is that I'm making the input signal, I'm attenuating this high frequency right here. And what you see in the output signal in the time domain is I just don't see as much of this high frequency signal as I did over here. I still have the DC value, so it still has a value of 1, is the offset. And I still have the low-frequency, but I've got rid of this high frequency. This is a low-pass filter. Let me take the same input signal, same input signal with the same frequency spectrum, and instead I'm going to put it through high pass filter. So, remember, I get rid of low frequencies, and I pass through high frequencies. Now, I got this high pass filter by just rearranging the output. So, I took a RC circuit and now I've taken the output over the resistor instead of the capacitor. And this is a corresponding output. And I can get that in the frequency domain, again, by taking these components, and turn by turn, multiply them by the corresponding value in the frequency spectrum, in the frequency response, and I get the output response. And what you see is that I've gotten rid of the DC component because I multiply it by zero. The low frequency is attenuated. The high frequency's passed through without much attenuated. And what you see is something that is predominately high frequency. I've gotten rid of the DC value, you can see it's zero. And I want to compare this to the low frequency filter. So, the high frequency filter got rid of the DC, attenuated the low frequency, and it passes through the high frequency The low-pass filter did just the opposite. It didn't touch the DC value. So, we've still got the DC value. And this is predominantly low frequency because it passed through the low frequency and it attenuated the high frequency. So, that's the difference between a low-pass and a high-pass filter. Let's talk about the properties of these two types of filters. The basic properties of a low pass filter, we talk about the bandwidth, the pass band region, and the pass band gain. So, let's talk first about the pass band region, that's the region where we're passing through the signal. That's the passband region. And it's defined by what we call the bandwidth. Now, that bandwidth is defined as being 0.707 of the DC value. So, this is the DC value. This is, again, the frequency response. Plotted versus omega. The passband gain is defined as that low frequency. This is actually h of zero. So, we could say that k sub DC, or we have defined it in the past as magnitude h of zero. So, that's the passband gain. This is the passband region, and the bandwidth is the frequency at which we're at .707 of that passband gain. That's a linear plot. Let's look at a Bode plot version of this. All I did was take the same characteristics and put it over here. So, in this case, this is the log of that passband gain. Sometimes we'll call it G sub dB is defined this way. And at three decibels below that passband gain, that is our bandwidth. And that corresponds to being at 0.707 of DC. Let's look at why that's true. So, if I took the log of 0.707, of let me call it G because k sub DC is G. And compute that out, that will be the 0.707 goes to minus three db plus 20 times a log of G. And so, that's all we're saying is that we're three decibels below the low pass gain right there. And so, that's the bandwidth. Then, this is the pass band region. So, these are the characteristics in the Bode plot. And these are the characteristics in a linear plot of a low-pass filter. High-pass filter is fairly similar. We no longer define a bandwidth. We define a corner frequency, passband region, and passband gain. Passband gain is the gain at which we really reach our maximum here. The corner frequency is where we're at 0.707 of that gain. And the passband region starts at that corner frequency and goes to the right. That's the linear plot. The Bode plot looks like this. Very similar to the lowpass. We have the corner frequency defined as being three decibels below the passband gained. So, again, a lot of times we'll shorten this to say gain in decibels. So, this is gain in the linear plot. Gain in decibels means it's 20 times the log of G. And then, of course, the passband is to the right. To summarize this lesson, we've introduced two types of filters, the lowpass and the highpass, with these sort of characteristics. The important part being that the lowpass passes through signals that are low frequency and filters out the high frequency components. Whereas the highpass passes through the high frequency and filters out the low frequency components. Now, filters are so, so useful in dealing with circuits, and dealing with censor measurements, actually, because, as I said before, a lot of times sensor measurements come in with measurement noise that we want to filter out. We want to save the particular information that we're interested in. And a lot of times we know what frequency range that is. So, we want to filter out anything that's outside of that range. All right, thank you very much. [MUSIC]
