1 00:00:00,001 --> 00:00:04,734 Welcome back to Linear Circuits. This is Dr. 2 00:00:04,734 --> 00:00:05,410 Ferri. 3 00:00:05,410 --> 00:00:10,430 This lesson is on bandpass and notch filters. 4 00:00:10,430 --> 00:00:12,520 Again, we're at the last topic here and a very 5 00:00:12,520 --> 00:00:17,010 important application for this material is the concept of filtering. 6 00:00:18,510 --> 00:00:21,148 In a previous lesson, we introduced lowpass and 7 00:00:21,148 --> 00:00:25,170 highpass filters In this lesson we will introduce 8 00:00:25,170 --> 00:00:30,596 the characteristics of notch and bandpass filters. 9 00:00:30,596 --> 00:00:38,150 To summarize, we looked at RC filters before in our previous lesson and we found 10 00:00:38,150 --> 00:00:43,665 that we had lowpass filters were found by taking the output over the capacitor. 11 00:00:43,665 --> 00:00:47,860 Highpass filters we take the output to be over the resistor. 12 00:00:47,860 --> 00:00:50,580 And the corresponding frequency responses 13 00:00:50,580 --> 00:00:58,137 look like this. And in both cases we have bandwidth 14 00:00:58,137 --> 00:01:03,480 equal to 1 over rc. 15 00:01:03,480 --> 00:01:10,260 And also in both cases, we have slopes that are 20 decibels per decade. 16 00:01:10,260 --> 00:01:14,280 This one the slope is going down, and this one the slope is going up. 17 00:01:25,060 --> 00:01:29,710 And looking at RLC circuits. And RLC filters. 18 00:01:29,710 --> 00:01:32,860 the Lowpass filter was gotten by taking the output over 19 00:01:32,860 --> 00:01:36,860 the capacitor, Highpass by taking the output over the inductor. 20 00:01:36,860 --> 00:01:38,840 In these cases, because they're second order, 21 00:01:41,510 --> 00:01:46,790 the slopes are 40 decibels per decade. In this case, the slope is going down. 22 00:01:46,790 --> 00:01:48,320 In this case, the slope is going up. 23 00:01:50,420 --> 00:01:54,300 They're second order because if I wrote a differential equation 24 00:01:54,300 --> 00:01:56,450 for this, it would be a second order differential equation. 25 00:01:57,750 --> 00:02:00,280 And in both cases, there's a corner frequency here. 26 00:02:02,080 --> 00:02:08,865 I'll call it omega 0. Omega 0 is 1 over the square root of 27 00:02:08,865 --> 00:02:16,034 LC, and this is in radians 28 00:02:16,034 --> 00:02:23,810 per second. So now 29 00:02:23,810 --> 00:02:28,480 we want to introduce a new type of filter, and that's called a bandpass filter. 30 00:02:28,480 --> 00:02:30,870 I'm taking my same RLC circuit. 31 00:02:34,310 --> 00:02:39,840 But this time I'm taking the voltage to be across the resistor and if I look 32 00:02:39,840 --> 00:02:41,420 at this, this is the transfer function of 33 00:02:41,420 --> 00:02:45,100 it, and this is the corresponding frequency response. 34 00:02:45,100 --> 00:02:47,620 Now in this particular case we see that it goes up 35 00:02:47,620 --> 00:02:51,180 and back down, and in this range is Passband just here. 36 00:02:51,180 --> 00:02:54,780 It's just certain frequencies are passed through without attenuation. 37 00:02:54,780 --> 00:02:59,790 And outside that region, outside the Passband region, we attenuate the signals. 38 00:02:59,790 --> 00:03:02,410 The width of the passband is R over L. 39 00:03:03,450 --> 00:03:08,670 And that's defined as being the width of the region between the 40 00:03:08,670 --> 00:03:12,896 points where we're down at minus 3 decibels below the peak value. 41 00:03:12,896 --> 00:03:19,340 Now this passband is centered around the value of 1 over the square root of LC. 42 00:03:19,340 --> 00:03:25,116 I'll call that same thing I called it before, omega sub zero, 43 00:03:25,116 --> 00:03:34,100 this is, again, this is a passband filter. 44 00:03:34,100 --> 00:03:37,418 Now let's look at a numerical example of a passband filter. 45 00:03:37,418 --> 00:03:44,350 Bandpass filter. Suppose I want, in, in terms of hertz, 46 00:03:44,350 --> 00:03:50,470 I want my center frequency to be at 1,000 hertz. 47 00:03:50,470 --> 00:03:54,845 Well, to change that over to radians per second, I have to multiply by 2 pi. 48 00:03:54,845 --> 00:03:59,960 So I've got 2 pi times f nought, which 49 00:03:59,960 --> 00:04:05,102 is 6283 radians per second. And the 50 00:04:05,102 --> 00:04:10,596 other thing I want sort of my specifications here 51 00:04:10,596 --> 00:04:15,945 in terms of hertz. I want a 200 hertz passband. 52 00:04:15,945 --> 00:04:21,280 And I have to convert that to radians per second. 53 00:04:21,280 --> 00:04:26,530 And again I multiply by 2 pi. 54 00:04:26,530 --> 00:04:31,030 And I get 1257 radians per second. So suppose that. 55 00:04:37,700 --> 00:04:41,200 You know, those are my specifications. Now I want to do my design. 56 00:04:41,200 --> 00:04:46,230 I want to find specific values of R, L, and C that give me those characteristics. 57 00:04:47,710 --> 00:04:51,139 Well, let's start with omega naught. 58 00:04:51,139 --> 00:04:55,915 Omega naught is equal to 1 over the square root of LC. 59 00:04:55,915 --> 00:05:00,690 And, I have to pick. So I've got two parameters and one known. 60 00:05:00,690 --> 00:05:03,030 I have to pick one. So let's 61 00:05:03,030 --> 00:05:06,920 just pick L to be some value, I'll call 62 00:05:06,920 --> 00:05:11,120 it, suppose I've got five mill Henry inductors available. 63 00:05:11,120 --> 00:05:12,260 So I pick it to be that. 64 00:05:12,260 --> 00:05:18,035 And now, then I saw for C, C is equal to one over omega 65 00:05:18,035 --> 00:05:23,130 naught squared times L, and that is equal to 5.1 66 00:05:23,130 --> 00:05:28,114 microfarads. Now we have to solve for 67 00:05:28,114 --> 00:05:31,885 R, and we know that R over L is 68 00:05:31,885 --> 00:05:35,890 equal to that band width. And we've already 69 00:05:35,890 --> 00:05:39,706 picked L, so that means that R is equal to 6.3 ohms. 70 00:05:39,706 --> 00:05:46,884 So then the frequency characteristics of this is going to look like this. 71 00:05:46,884 --> 00:05:56,082 At 6283 radians per second. 72 00:05:58,470 --> 00:06:05,012 We've got the peak and then the bandwidth here is 1257 radians per second. 73 00:06:05,012 --> 00:06:06,882 With these values of R, L, and C. 74 00:06:06,882 --> 00:06:09,740 The other type of filter that I want to introduce 75 00:06:17,450 --> 00:06:19,330 is the notch filter. 76 00:06:19,330 --> 00:06:23,050 In that particular case we get the notch filter by taking 77 00:06:23,050 --> 00:06:26,980 the output voltage to be across the inductor and the capacitor together. 78 00:06:26,980 --> 00:06:28,930 This is a transfer function. 79 00:06:29,960 --> 00:06:32,410 If I plot that transfer function and get 80 00:06:32,410 --> 00:06:35,150 the frequency response, I get something with this characteristic. 81 00:06:35,150 --> 00:06:37,000 That's why it's called a notch. 82 00:06:37,000 --> 00:06:39,870 Sometimes called the stop band filter because it 83 00:06:39,870 --> 00:06:42,700 gets rid of signals in a certain frequency range. 84 00:06:44,220 --> 00:06:48,580 The stop band, or the notch, is centered about 1 over the square root 85 00:06:48,580 --> 00:06:53,003 of LC. And we can also look at three 86 00:06:53,003 --> 00:06:57,249 decibels below the dc 87 00:06:57,249 --> 00:07:01,565 value. And this range here is called the 88 00:07:01,565 --> 00:07:09,410 passband. And that value is 89 00:07:09,410 --> 00:07:12,670 equal to R over L. Now 90 00:07:14,860 --> 00:07:20,070 a notch filter is really commonly used to get rid of line frequency. 91 00:07:20,070 --> 00:07:21,960 Line frequency is what you get in your power lines. 92 00:07:21,960 --> 00:07:26,343 Like in this country it's 60 hertz. In other country it's 50 hertz. 93 00:07:26,343 --> 00:07:30,140 So it's very common to use, to, to get noise in your signal. 94 00:07:30,140 --> 00:07:33,270 That is at 60 hertz or 50 hertz and then 95 00:07:33,270 --> 00:07:36,730 you build a notch filter to get rid of that signal. 96 00:07:36,730 --> 00:07:40,740 So and then the center frequency would be 60 hertz 97 00:07:40,740 --> 00:07:45,747 in free, in hertz or 2 pi times 60 in radians 98 00:07:45,747 --> 00:07:50,846 per second. Let's look at a specific 99 00:07:50,846 --> 00:07:55,760 example of a notch filter and, you 100 00:07:55,760 --> 00:08:00,856 know, speaking of the 60 hertz, we 101 00:08:00,856 --> 00:08:06,530 want to build a 60 hertz notch filter. 102 00:08:06,530 --> 00:08:13,780 So we'll say, f naught will be 60 hertz. So that means omega naught 103 00:08:13,780 --> 00:08:20,360 is 2 pi f naught, which is 377 radians per second. 104 00:08:20,360 --> 00:08:26,235 The bandwidth we'll select to be maybe about 10 hertz. 105 00:08:26,235 --> 00:08:31,565 That means we want to filter out pretty, pretty much 106 00:08:31,565 --> 00:08:36,485 everything between 55 and 65 hertz. And in 107 00:08:36,485 --> 00:08:41,828 radians per second. That's 63 radians per second. 108 00:08:41,828 --> 00:08:45,116 So those are our specifications. 109 00:08:45,116 --> 00:08:50,459 And what we'd like to do now is solve for R, L, and C. 110 00:08:50,459 --> 00:08:56,759 Omega naught is equal to 1 over the square root of LC in 111 00:08:56,759 --> 00:09:03,080 radians per second. And, if we pick, L 112 00:09:03,080 --> 00:09:07,646 to be equal to 100 millihenrys, 113 00:09:07,646 --> 00:09:12,482 then that means that C is equal to 1 over 114 00:09:12,482 --> 00:09:17,200 omega naught squared L is 70. 115 00:09:17,200 --> 00:09:20,980 Microfarads, which is, kind of, large values here. 116 00:09:20,980 --> 00:09:21,690 And that's, 117 00:09:21,690 --> 00:09:24,885 we need these large values to get down to that frequency range. 118 00:09:24,885 --> 00:09:29,450 B is equal to R over L, 119 00:09:30,940 --> 00:09:36,580 and that's equal to 63, so we can solve for R is equal to 6.3 ohms. 120 00:09:39,750 --> 00:09:42,150 And as a result, if I were to plot this, 121 00:09:45,240 --> 00:09:48,860 I get something in terms of hertz. I get 122 00:09:48,860 --> 00:09:52,810 something that's centered at 60 hertz. 123 00:09:52,810 --> 00:09:56,624 And this is the passband region here. So this is 65. 124 00:09:56,624 --> 00:10:04,480 And that's 55, down like that. So pretty much, I'm filtering out 125 00:10:04,480 --> 00:10:12,190 things between 55 and 65 hertz. 126 00:10:12,190 --> 00:10:12,990 So, in summary. 127 00:10:14,110 --> 00:10:17,829 Well, we found that we can get different filter characteristics from RC and 128 00:10:17,829 --> 00:10:23,050 RLC circuits, just by deciding what we're going to take the output voltage to be. 129 00:10:23,050 --> 00:10:27,290 A bandpass filter passes frequencies in a specific frequency band, 130 00:10:27,290 --> 00:10:31,200 and a notch filter rejects frequencies in a specific frequency band. 131 00:10:33,040 --> 00:10:34,940 In our next lesson, we're going to do a lab demo. 132 00:10:34,940 --> 00:10:37,550 We're going to go back to the guitar string experiment 133 00:10:37,550 --> 00:10:39,470 we've seen before and this time we're 134 00:10:39,470 --> 00:10:42,065 going to filter out certain frequencies in there. 135 00:10:43,470 --> 00:10:46,180 Please make sure that you visit the forum and that you 136 00:10:46,180 --> 00:10:49,060 ask and answer any questions that you can on that forum. 137 00:10:49,060 --> 00:10:49,690 Thank you.