1 00:00:00,012 --> 00:00:05,300 This is Doctor Ferry, 2 00:00:05,300 --> 00:00:10,180 and this is an extra work problem, on the frequency response using Bode plots. 3 00:00:12,830 --> 00:00:15,890 Suppose we're given a Bode plot that looks like this right here. 4 00:00:15,890 --> 00:00:20,210 This is the magnitude in decibels, and this is the phase angle. 5 00:00:20,210 --> 00:00:22,650 And we want to understand how. 6 00:00:22,650 --> 00:00:26,030 A input signal, input voltage is processed 7 00:00:26,030 --> 00:00:29,950 through a circuit that has this frequency response. 8 00:00:29,950 --> 00:00:33,835 So suppose this is my input signal here. It's a sine wave, a sinusoid. 9 00:00:35,830 --> 00:00:37,450 And we'll call that V in. And 10 00:00:39,910 --> 00:00:41,960 we want to know what, what is V out. 11 00:00:46,030 --> 00:00:53,100 And in particular, we want to plot V out on the same set of scales as we do V in. 12 00:00:53,100 --> 00:00:56,440 So, the basic relationship that we're going to 13 00:00:56,440 --> 00:00:59,930 need here is that for a linear system. 14 00:01:03,280 --> 00:01:09,110 If I've got an input that's a, a sinusoid, with an input amplitude, A sub i. 15 00:01:13,110 --> 00:01:16,690 The corresponding output is also a sinusoid at the same frequency. 16 00:01:20,000 --> 00:01:22,830 With a change in amplitude and a change in phase. 17 00:01:22,830 --> 00:01:26,130 Now, the amplitude change is A naught is 18 00:01:26,130 --> 00:01:29,310 equal Ai times the magnitude of the transfer 19 00:01:29,310 --> 00:01:33,490 function at that frequency, and the phase is 20 00:01:33,490 --> 00:01:37,140 the angle of the transfer function at that frequency. 21 00:01:37,140 --> 00:01:39,050 Now this is all on a linear scale because 22 00:01:39,050 --> 00:01:42,000 we've got the magnitude, here, on a linear scale. 23 00:01:42,000 --> 00:01:45,300 How to re, re-relate that to a log scale. 24 00:01:45,300 --> 00:01:49,570 Well from the log scale, we have this, all of this information. 25 00:01:49,570 --> 00:01:51,280 The phase is directly. 26 00:01:51,280 --> 00:01:52,620 This is in degrees. 27 00:01:52,620 --> 00:01:56,280 So I can just read off the plot at any given frequency and I read the phase. 28 00:01:56,280 --> 00:02:02,728 But the magnitude, on here, what I have is 20 times a log of 29 00:02:02,728 --> 00:02:08,890 H, in decibels. 30 00:02:08,890 --> 00:02:11,710 And suppose I read that to be some constant. 31 00:02:11,710 --> 00:02:13,020 You know? Some value, C. 32 00:02:15,330 --> 00:02:19,650 You know, at any given particular frequency, I have some value, C. 33 00:02:19,650 --> 00:02:21,640 So I want to find out what H is. 34 00:02:26,370 --> 00:02:32,320 H is equal to, solving for, this magnitude here, I have 10 to the C over 20. 35 00:02:32,320 --> 00:02:37,080 And that tells me what H is, if I've measured C directly off of here. 36 00:02:37,080 --> 00:02:37,810 C in decibels. 37 00:02:39,790 --> 00:02:44,770 So that's the basis that this is, this whole process is, is built upon. 38 00:02:44,770 --> 00:02:48,880 Let's look at, for this particular signal. This is my via input signal. 39 00:02:48,880 --> 00:02:51,600 I've gotta identify the frequency and the amplitude. 40 00:02:51,600 --> 00:02:54,270 Well, the amplitude, A sub i, is equal 41 00:02:54,270 --> 00:02:56,310 to 1, because it's going between one and zero. 42 00:02:56,310 --> 00:03:04,020 There's no DC component to it, because it oscillates about zero volts. 43 00:03:04,020 --> 00:03:07,840 So it's just a single sine wave. And if I look at the period here. 44 00:03:07,840 --> 00:03:13,919 From here to here is 0.01 seconds. So the period of this sine wave is 0.01. 45 00:03:13,919 --> 00:03:16,620 The frequency is 1 over T. 46 00:03:16,620 --> 00:03:21,338 Which is 100 hertz. So this is in hertz. 47 00:03:21,338 --> 00:03:24,600 And my Bode plot's in radians per second. 48 00:03:24,600 --> 00:03:26,532 So I have to convert this to radians per second. 49 00:03:27,920 --> 00:03:32,640 So I'll call it omega sub 1 is equal to 2 pi f. 50 00:03:32,640 --> 00:03:33,570 Call this f sub 1. 51 00:03:33,570 --> 00:03:36,090 This is the particular frequency of my input signal. 52 00:03:37,870 --> 00:03:41,120 That will be 628 radians per second. 53 00:03:43,640 --> 00:03:47,510 Okay. So, I have to figure out what H is 54 00:03:50,010 --> 00:03:53,310 at that particular frequency. So, 628. 55 00:03:53,310 --> 00:03:58,960 Well, this is 100 right here. 200, 300, 400, 500, 600, 700. 56 00:03:58,960 --> 00:04:07,140 So it's between this and this. Going up here to between that and that. 57 00:04:07,140 --> 00:04:15,686 So it's roughly going to be about there. And that is minus three decibels 58 00:04:15,686 --> 00:04:16,652 right there. 59 00:04:18,020 --> 00:04:23,680 And the angle is approximately minus 45 degrees. 60 00:04:26,360 --> 00:04:29,480 So, if I want to find the output, the input 61 00:04:31,200 --> 00:04:35,400 is equal to the cosine amplitude 1 cosine of 628t. 62 00:04:37,500 --> 00:04:43,110 The output is equal to, A out, 63 00:04:43,110 --> 00:04:49,090 cosine of 628 t plus some phase angle. The output 64 00:04:49,090 --> 00:04:55,000 amplitude is equal to the input amplitude which is 1, times H at that. 65 00:04:57,646 --> 00:05:02,850 Frequency times A naught, which is, which is one. 66 00:05:02,850 --> 00:05:04,470 So we have to find what H is. 67 00:05:06,830 --> 00:05:14,200 And going back to this, we measured C to be minus 3 decibels, so H is equal to. 68 00:05:14,200 --> 00:05:21,022 10 to the minus 3 over 20. If I solve that, that is equal to 0.707. 69 00:05:21,022 --> 00:05:27,730 So that's what the amplitude is and the angle we 70 00:05:27,730 --> 00:05:33,020 measure here might be negative 45 degrees. So, our output 71 00:05:33,020 --> 00:05:40,500 is equal to 0.707. Cosine of 628t minus 45 degrees. 72 00:05:42,580 --> 00:05:43,500 Okay. So I need 73 00:05:46,000 --> 00:05:49,350 to plot this on here. Minus 45 degrees. 74 00:05:49,350 --> 00:05:56,630 And an amplitude of 0.707. 75 00:05:56,630 --> 00:06:00,346 Let me mark this. We're going 76 00:06:00,346 --> 00:06:04,061 to be oscillating between plus 0.707 and minus 0.707. 77 00:06:04,061 --> 00:06:13,040 Now another thing to, to notice is that. 78 00:06:13,040 --> 00:06:18,540 When we do, go through a sine wave, we go through 360 degrees for every cycle. 79 00:06:18,540 --> 00:06:23,990 So this is 360 degrees, if this is zero degrees, that's 360. 80 00:06:23,990 --> 00:06:29,810 So this would be 180, 90 and 45 is right there. 81 00:06:29,810 --> 00:06:34,710 So when we have a faze line of 45 degrees. It's about that much. 82 00:06:36,010 --> 00:06:38,780 So, from here, this goes through a zero crossing here. 83 00:06:38,780 --> 00:06:41,050 The output will go through a zero crossing right there. 84 00:06:43,160 --> 00:06:45,400 And this will go through a corresponding 85 00:06:45,400 --> 00:06:49,680 zero crossing with the lag right there, there. 86 00:06:49,680 --> 00:06:50,920 This distance right there. 87 00:06:52,320 --> 00:06:55,780 And correspondingly, the peak and the valleys 88 00:06:55,780 --> 00:06:57,980 will be delayed by about that much. 89 00:06:57,980 --> 00:06:59,550 So this would be, be delay. 90 00:06:59,550 --> 00:07:02,890 This is, this would be 90 degrees, this would be 45. 91 00:07:02,890 --> 00:07:04,430 Should be about right there. 92 00:07:08,710 --> 00:07:11,450 Looks like it's maybe just to the right of these. 93 00:07:11,450 --> 00:07:11,950 Signals. 94 00:07:13,110 --> 00:07:17,390 So all I need to do is then take my 95 00:07:17,390 --> 00:07:21,551 marker, and 96 00:07:21,551 --> 00:07:26,329 fill 97 00:07:26,329 --> 00:07:33,493 this in. 98 00:07:36,590 --> 00:07:37,920 And that's a corresponding V out. 99 00:07:42,520 --> 00:07:48,770 So summarizing what we've done here, is everything is based on this input-output 100 00:07:48,770 --> 00:07:51,480 relationships for linear circuits and linear 101 00:07:51,480 --> 00:07:55,920 systems, where the cosine output is the 102 00:07:55,920 --> 00:07:58,970 sinusoidal output results in sinusoidal output 103 00:07:58,970 --> 00:08:01,110 in steady state at the same frequency 104 00:08:01,110 --> 00:08:06,070 but a change in amplitude and phase and that's given by this relationship. 105 00:08:06,070 --> 00:08:07,760 This being a Bode plot we 106 00:08:07,760 --> 00:08:13,310 have to convert the Bode, the magnitude in decibels back 107 00:08:13,310 --> 00:08:17,990 to a magnitude on a linear scale and we do that with this conversion right here. 108 00:08:17,990 --> 00:08:22,510 Where C is the value in decibels that our plot is. 109 00:08:22,510 --> 00:08:28,090 And looking at our input signal, we had to first find what V in is. 110 00:08:28,090 --> 00:08:30,510 We had to find the, the frequency. 111 00:08:30,510 --> 00:08:32,810 And we had to find the amplitude in order to 112 00:08:32,810 --> 00:08:34,110 get this expression. 113 00:08:34,110 --> 00:08:36,010 Once we have this expression, we have the 114 00:08:36,010 --> 00:08:40,150 input amplitude, and the phase, and the output amplitude. 115 00:08:40,150 --> 00:08:46,310 We have to look at the magnitude at that frequency, covert it to a linear value. 116 00:08:46,310 --> 00:08:53,280 And then we're able to write the output frequency, ou-, output Voltage. 117 00:08:54,600 --> 00:08:58,750 And then just plot it. Remembering of course that going 118 00:08:58,750 --> 00:09:03,560 through one cycle is 360 degrees and whatever this value 119 00:09:03,560 --> 00:09:06,390 is here, we would do that as sort of a proportion. 120 00:09:06,390 --> 00:09:09,370 So, 45 degrees would be here if this is 360. 121 00:09:09,370 --> 00:09:17,459 And then we just plotted it. Alright, thank you.