1 00:00:03,090 --> 00:00:06,810 Welcome back to Linear Circuits. This is Dr Ferri. 2 00:00:06,810 --> 00:00:10,980 This lesson in on lowpass and highpass filters. 3 00:00:10,980 --> 00:00:14,220 We are at the last topic in this module. 4 00:00:16,680 --> 00:00:18,390 In the previous lesson we examined the 5 00:00:18,390 --> 00:00:21,018 frequency responses of RC and RLC circuits. 6 00:00:22,720 --> 00:00:25,790 In this lesson we want to introduce the concept of filtering. 7 00:00:25,790 --> 00:00:29,089 Filtering is a really important concept, a really important 8 00:00:29,089 --> 00:00:33,620 application of circuits when, especially when you're measuring signals. 9 00:00:35,602 --> 00:00:37,640 We want to show the properties of lowpass 10 00:00:37,640 --> 00:00:40,450 and highpass filters in this lesson as well. 11 00:00:42,560 --> 00:00:45,090 Let's first define what an analog filter is. 12 00:00:45,090 --> 00:00:47,500 It's a circuit that has specific shaped 13 00:00:47,500 --> 00:00:50,585 frequency response and the purpose is either to 14 00:00:50,585 --> 00:00:54,730 attenuate signals in certain frequency ranges or 15 00:00:54,730 --> 00:00:58,990 perhaps even amplify signals in certain frequency ranges. 16 00:00:58,990 --> 00:01:03,110 And, the two most common filters are shown here. 17 00:01:03,110 --> 00:01:04,620 There's a lowpass filter 18 00:01:07,170 --> 00:01:10,430 over here, and in the lowpass filter, 19 00:01:10,430 --> 00:01:13,220 frequency content, so this is plotted versus frequency. 20 00:01:13,220 --> 00:01:16,220 Low frequency content signals get passed 21 00:01:16,220 --> 00:01:19,700 through without, just get straight passed through. 22 00:01:19,700 --> 00:01:24,190 But high frequency content is attenuated or filtered out. 23 00:01:24,190 --> 00:01:26,330 Now high pass filter works exact opposite way. 24 00:01:26,330 --> 00:01:28,920 Low frequency gets attenuated, but it 25 00:01:28,920 --> 00:01:31,860 passes the high frequency without any problems. 26 00:01:34,400 --> 00:01:38,580 Lets take a look at a specific low pass filter example. 27 00:01:38,580 --> 00:01:42,440 Now this, we've looked at this example before in a previous lesson. 28 00:01:42,440 --> 00:01:45,840 This is an RC filter. It's a circuit. 29 00:01:45,840 --> 00:01:47,740 And we didn't mention that it was a filter before. 30 00:01:47,740 --> 00:01:52,100 But if you look at the frequency response of it, this plotting the magnitude. 31 00:01:52,100 --> 00:01:59,440 So this is the magnitude of H of omega. And that's versus radians per second. 32 00:01:59,440 --> 00:02:01,920 It has a low frequency characteristics. 33 00:02:01,920 --> 00:02:05,030 And what we saw is that if the signal looks like this. 34 00:02:05,030 --> 00:02:09,650 It's got two frequency components, a low frequency, and then a high frequency. 35 00:02:09,650 --> 00:02:13,200 And both of them have equal amplitude. 36 00:02:13,200 --> 00:02:16,475 When we pass it through this lowpass filter, it attenuates 37 00:02:16,475 --> 00:02:20,030 the high frequency because that's up in this range right here. 38 00:02:20,030 --> 00:02:22,800 It has a low value, so it attenuates it. 39 00:02:22,800 --> 00:02:24,841 And the lower frequency at 50 hertz, or 40 00:02:24,841 --> 00:02:27,700 50 radians per second, is down here 41 00:02:27,700 --> 00:02:31,030 and it gets passed through without much attenuation. 42 00:02:31,030 --> 00:02:32,690 And you see the result here. 43 00:02:32,690 --> 00:02:34,390 The low frequency is sent through with 44 00:02:34,390 --> 00:02:37,650 out much attenuation, high frequency is attenuated. 45 00:02:37,650 --> 00:02:39,850 And that's the characteristics of a low pass filter. 46 00:02:45,920 --> 00:02:47,630 Now let's look a little bit more 47 00:02:47,630 --> 00:02:50,020 detail about parameters that we might define here. 48 00:02:50,020 --> 00:02:55,737 First of all this is a DC gain. 49 00:02:55,737 --> 00:03:00,510 DC gain because at omega equals zero that's a DC 50 00:03:00,510 --> 00:03:05,800 value, that's a DC frequency and that means if I send an input. 51 00:03:07,040 --> 00:03:10,950 Of a constant value. If Vin is 52 00:03:10,950 --> 00:03:20,400 equal to some constant C, then Vout is equal to Kac of DC times C. 53 00:03:20,400 --> 00:03:23,420 It's like a game, a multiplier times that input. 54 00:03:24,710 --> 00:03:29,805 And then another term that we're going to identify is a bandwidth, and 55 00:03:29,805 --> 00:03:36,260 that's omega sub b and 56 00:03:36,260 --> 00:03:40,290 that's found by finding, by looking at this plot and 57 00:03:40,290 --> 00:03:44,720 seeing when it's equal to 0.707 of the DC value. 58 00:03:44,720 --> 00:03:46,740 We call that the bandwidth and the reason we say 59 00:03:46,740 --> 00:03:50,870 that is because it's kind of near the the curve here. 60 00:03:50,870 --> 00:03:56,325 Below the bandwidth the signal is mostly passed through without much 61 00:03:56,325 --> 00:04:01,450 attenuation so we call that the, the passband and over here 62 00:04:01,450 --> 00:04:03,160 is the stopband. 63 00:04:03,160 --> 00:04:06,620 That's really the region where we attenuate the signals. 64 00:04:08,090 --> 00:04:10,520 Now looking at a very specific circuit example. 65 00:04:10,520 --> 00:04:14,970 This is an RC circuit with these parameters. 66 00:04:14,970 --> 00:04:20,175 If I look at R times C, that's equal to 0.01. 67 00:04:20,175 --> 00:04:27,870 One over RC. Is 100 radians per second. 68 00:04:27,870 --> 00:04:32,740 And it turns out that omega sub b is equal to one over RC. 69 00:04:32,740 --> 00:04:34,500 In this particular case, we look at this and 70 00:04:34,500 --> 00:04:38,480 we say, all right 0.707, I follow that across. 71 00:04:40,030 --> 00:04:43,480 That's right here and that's the DC value is equal 72 00:04:43,480 --> 00:04:47,310 to one, so this is 0.707 of the value of one. 73 00:04:47,310 --> 00:04:52,710 And drop that down, that is indeed 100 radians per second, is the bandwidth. 74 00:04:54,060 --> 00:04:57,419 So the bandwidth again is calculate 0.707 of the 75 00:04:57,419 --> 00:05:00,998 DC value, in an RC circuit it's one over RC. 76 00:05:07,570 --> 00:05:13,800 Let's look at a, a specific signal sent through an RC circuit. 77 00:05:13,800 --> 00:05:15,100 So plotted here is 78 00:05:17,340 --> 00:05:21,270 the magnitude of the transfer function and has a DC value, okay? 79 00:05:21,270 --> 00:05:24,330 So DC is equal to two here. 80 00:05:24,330 --> 00:05:27,750 Now, if I look at here, the DC value is equal to ten of 81 00:05:27,750 --> 00:05:32,540 the input signal and that's, you could see that the average value is ten. 82 00:05:32,540 --> 00:05:34,480 And there's a lot of high frequency in here that's 83 00:05:34,480 --> 00:05:38,280 quick changes, or fast changes in the motion in the signal. 84 00:05:38,280 --> 00:05:40,080 And that corresponds to high frequency. 85 00:05:40,080 --> 00:05:42,380 The corresponding output 86 00:05:42,380 --> 00:05:47,660 here looks like this. Where the out put DC is now equal 87 00:05:48,890 --> 00:05:54,290 to 20, which is the input DC times the DC gain, gives me 20. 88 00:05:54,290 --> 00:05:58,580 And you see that I've attenuated the high frequency a great deal. 89 00:05:58,580 --> 00:06:01,490 Because it's a low past filter and the high frequency 90 00:06:01,490 --> 00:06:04,730 range is somewhere in here that's where this signal was. 91 00:06:06,380 --> 00:06:07,720 So this is a classic 92 00:06:07,720 --> 00:06:11,930 example of what happens as we pass the signal through a lowpass filter. 93 00:06:18,090 --> 00:06:20,370 Now, let's look at Bode plots of lowpass filters. 94 00:06:20,370 --> 00:06:24,020 So this is the same image that we've 95 00:06:24,020 --> 00:06:27,360 seen before and this is plotted on linear scales. 96 00:06:27,360 --> 00:06:36,670 If I plot this on Bode plot scales then this is a logrithmic scale right here. 97 00:06:36,670 --> 00:06:43,250 And over here it's in decibels. We've introduced the idea Bode of of 98 00:06:43,250 --> 00:06:45,440 this sort of plot before when we were 99 00:06:45,440 --> 00:06:49,010 talking about frequency spectrum in a previous lesson. 100 00:06:49,010 --> 00:06:53,100 So, decibels are defined by taking 20 times the 101 00:06:53,100 --> 00:06:55,790 log base ten of every point along this curve. 102 00:06:55,790 --> 00:06:59,820 So every point along this curve, we take 20 times the log base ten of it. 103 00:06:59,820 --> 00:07:08,640 So for example at the DC value, we take 20 times the log base ten of the DC gain. 104 00:07:08,640 --> 00:07:12,500 And that's really kind of a low-frequency ascentote for this signal. 105 00:07:12,500 --> 00:07:14,500 At low frequency it looks like that. 106 00:07:14,500 --> 00:07:18,000 Now the bandwidth is equal to 0.707 of the DC value. 107 00:07:18,000 --> 00:07:22,230 If I calculate that, I want to see. 108 00:07:22,230 --> 00:07:26,500 What this value is on the log scale and the Bode Plot scale. 109 00:07:26,500 --> 00:07:29,361 Take 20 times the log of 0.707 times K of dc. 110 00:07:29,361 --> 00:07:33,730 And with logs, if I take a log of a product, it turns out it's equal to the 111 00:07:33,730 --> 00:07:36,810 sum of the logs. 112 00:07:53,290 --> 00:07:58,211 Well 20 times the log of 0.707 is minus three 113 00:07:58,211 --> 00:08:04,116 decibels plus 20 times the log K sub DC. 114 00:08:04,116 --> 00:08:08,060 So, the bandwidth, and that was the point of which we had the bandwidth here. 115 00:08:09,410 --> 00:08:17,280 So over here that same frequency happens at three decibels below the DC value. 116 00:08:19,200 --> 00:08:24,080 The other plot, thing that we'll find here is that at high frequency, we have a roll 117 00:08:24,080 --> 00:08:29,622 off. And that roll for an RC circuit. 118 00:08:29,622 --> 00:08:34,238 That slope is minus 119 00:08:34,238 --> 00:08:39,850 20 db per decade. Now 120 00:08:39,850 --> 00:08:44,830 let's look at, 121 00:08:44,830 --> 00:08:50,640 at, cognumerical 122 00:08:50,640 --> 00:08:55,920 example. This is an RC circuit 123 00:08:55,920 --> 00:08:59,992 and I've plotted the frequency response. I want to find these values. 124 00:08:59,992 --> 00:09:04,968 I want to find K sub DC and the bandwidth. K sub DC 125 00:09:04,968 --> 00:09:10,080 is found by looking at this value here. 20 times 126 00:09:10,080 --> 00:09:15,828 the log of K sub DC is equal to zero. So this value is zero. 127 00:09:15,828 --> 00:09:18,890 And I want to back out what K sub DC is. 128 00:09:18,890 --> 00:09:25,062 So, if I divide both sides by 20, and then raise it to the tenth, 129 00:09:25,062 --> 00:09:29,649 raise it to the power of 10 to that power. I have 130 00:09:29,649 --> 00:09:35,090 ten to the zero is equal to one. So d, DC gain is one. 131 00:09:35,090 --> 00:09:37,630 Now the bandwidth is three decibels below that 132 00:09:37,630 --> 00:09:39,700 value that looks to be about right there. 133 00:09:39,700 --> 00:09:41,530 That looks about three decibels. 134 00:09:41,530 --> 00:09:46,995 And this is 1000th, so omega sub B is equal to 135 00:09:46,995 --> 00:09:52,399 1000 radians per second, and that's the bandwidth. 136 00:09:52,399 --> 00:10:00,289 Now a highpass filter is when 137 00:10:00,289 --> 00:10:04,670 that pass is a high frequency component. 138 00:10:04,670 --> 00:10:07,410 In this case we'll say that this is kind of the knee of the curve. 139 00:10:07,410 --> 00:10:16,230 It's really that the 0.707 point of the high frequency. 140 00:10:16,230 --> 00:10:18,902 So this is the high frequency value it's 0.707 of that. 141 00:10:20,560 --> 00:10:22,850 We oftentimes call that the band width. 142 00:10:22,850 --> 00:10:25,512 And this time, this is the pass 143 00:10:25,512 --> 00:10:31,430 band region, and this is the stop band region. 144 00:10:37,690 --> 00:10:39,125 I get the high pass filter by 145 00:10:39,125 --> 00:10:43,470 interchanging the capacitor and the resistor positions. 146 00:10:43,470 --> 00:10:48,070 So now I'm taking the voltage across the resistor. 147 00:10:48,070 --> 00:10:51,340 If I look at the transfer function of this, it looks like this. 148 00:10:51,340 --> 00:10:53,420 And you can see that as omega goes 149 00:10:53,420 --> 00:10:57,050 to zero, this numerator drops out and becomes zero. 150 00:10:57,050 --> 00:11:01,070 So that's why the magnitude goes to zero as omega goes to zero. 151 00:11:01,070 --> 00:11:02,790 And you can also see 152 00:11:02,790 --> 00:11:04,462 that as omega gets very large. 153 00:11:04,462 --> 00:11:09,450 This one becomes negligible, and the magnitude is going to approach one. 154 00:11:09,450 --> 00:11:13,635 So in this particular circuit, this magnitude would approach one. 155 00:11:13,635 --> 00:11:21,347 Let's look at a numerical example of taking 156 00:11:21,347 --> 00:11:28,400 a highpass filter and putting it through you know using 157 00:11:28,400 --> 00:11:31,190 it to filter out a specific signal. 158 00:11:31,190 --> 00:11:33,920 This is a specific signal that we saw before. 159 00:11:33,920 --> 00:11:38,120 The DC value was equal to ten, there was a lot of high frequency on it. 160 00:11:38,120 --> 00:11:39,950 So if we take the magnitude 161 00:11:42,230 --> 00:11:45,700 and plot it here and say this value is one. 162 00:11:47,110 --> 00:11:49,990 In this particular case we're going to assume, or we're going 163 00:11:49,990 --> 00:11:55,716 to, in this particular circuit, the high-frequency content was in this range. 164 00:11:55,716 --> 00:12:01,200 The high-frequency content of this signal here. 165 00:12:01,200 --> 00:12:04,390 So the corresponding outlet looks like this. 166 00:12:04,390 --> 00:12:08,300 The DC value is greatly attenuated, 167 00:12:08,300 --> 00:12:11,620 because it's down in this rate region and then 168 00:12:11,620 --> 00:12:16,195 the high frequency content is hardly anything happens, hardly any 169 00:12:16,195 --> 00:12:19,770 attenuation at all, because it's the high frequency content was 170 00:12:19,770 --> 00:12:22,598 out of this frequency rage where we had no attenuation. 171 00:12:24,340 --> 00:12:25,740 So that's what a highpass filter does. 172 00:12:25,740 --> 00:12:30,755 It's often used to filter out DC values in measured signals. 173 00:12:30,755 --> 00:12:36,580 So 174 00:12:36,580 --> 00:12:41,820 in summary, we've looked at RC circuits. And RC circuits were first order. 175 00:12:42,840 --> 00:12:48,400 If I want to get a sharper roll off here, I would use an RLC filter. 176 00:12:48,400 --> 00:12:51,370 And here's the two examples of RLC filters, a low 177 00:12:51,370 --> 00:12:54,690 pass and a high pass, and they're just series circuits. 178 00:12:55,980 --> 00:13:00,650 And the type of response I get depends on where I'm taking the output voltage. 179 00:13:00,650 --> 00:13:02,948 And then, here, I'm looking at it over the capacitor. 180 00:13:02,948 --> 00:13:07,970 Here the inductor and they have these very different frequency responses. 181 00:13:07,970 --> 00:13:14,500 This being low pass and the slope here is minus 40 decibels per decade 182 00:13:14,500 --> 00:13:21,430 compared to minus 20 for an RC filter. This is a second order second 183 00:13:21,430 --> 00:13:23,700 order filter, because if I looked at the differential 184 00:13:23,700 --> 00:13:26,410 equation for this it would be second order differential equation. 185 00:13:27,860 --> 00:13:31,600 And anything that has second order differential equation, would have a Bode 186 00:13:31,600 --> 00:13:36,980 plot with minus 40 decibels per decade, as opposed to minus 20. 187 00:13:36,980 --> 00:13:39,410 And these are the transfer functions for these two circuits. 188 00:13:41,070 --> 00:13:44,359 this one has plus 40 decibels per decade slope. 189 00:13:46,320 --> 00:13:48,510 And that means that I've got a short, sharper 190 00:13:48,510 --> 00:13:51,210 transition between the pass band region, which is here, 191 00:13:55,240 --> 00:13:56,670 and the stop band over here. 192 00:14:01,950 --> 00:14:05,366 And right up here, I'm going to call these the corner frequencies. 193 00:14:05,366 --> 00:14:13,120 And in both cases the corner frequency, sometimes we'll show it as omega zero, 194 00:14:15,980 --> 00:14:21,070 is equal to one over the square root of LC. 195 00:14:21,070 --> 00:14:23,394 And that's again in radians per second. 196 00:14:31,440 --> 00:14:34,620 So in summary, an analog filter is a 197 00:14:34,620 --> 00:14:37,850 circuit that has a specific shaped frequency response. 198 00:14:37,850 --> 00:14:39,720 And the two most common. 199 00:14:39,720 --> 00:14:42,140 Types of filters are lowpass and highpass. 200 00:14:42,140 --> 00:14:46,860 The lowpass filter passes low frequency components in the signals, and 201 00:14:46,860 --> 00:14:49,070 attenuates the high frequency components, whereas 202 00:14:49,070 --> 00:14:51,480 the highpass filter does the opposit. 203 00:14:51,480 --> 00:14:54,969 It passes through the high frequency components and attenuates low frequency. 204 00:14:57,440 --> 00:15:00,540 In our next lesson we're going to look at another 205 00:15:00,540 --> 00:15:03,950 set of common filters, that's the bandpass and notch filters.