1 00:00:02,918 --> 00:00:05,284 This is Dr. Ferri and this is an 2 00:00:05,284 --> 00:00:10,290 extra problem worked on frequency response using linear plots. 3 00:00:12,400 --> 00:00:18,030 Let's take a look at this system, this particular circuits got a transfer 4 00:00:18,030 --> 00:00:23,580 function that can be plotted like this to give you this frequency response. 5 00:00:23,580 --> 00:00:25,980 Now, this is on a linear scale. 6 00:00:25,980 --> 00:00:29,160 And, what's plotted is versus frequency and hertz. 7 00:00:30,160 --> 00:00:31,830 And, we're asked to find what is the steady 8 00:00:31,830 --> 00:00:35,540 state response v not to an input like this. 9 00:00:36,570 --> 00:00:37,570 Now, doing a problem like 10 00:00:37,570 --> 00:00:42,810 this, we're basing it all in the basic concept of the input output relationship. 11 00:00:42,810 --> 00:00:43,820 From a transfer function. 12 00:00:47,480 --> 00:00:53,684 So suppose my circuit has this transfer function and 13 00:00:53,684 --> 00:00:59,850 my input is the signusoid like this. Then my output. 14 00:00:59,850 --> 00:01:02,330 Will be a sine wave at that frequency. 15 00:01:06,310 --> 00:01:07,170 With a phase shift. 16 00:01:11,290 --> 00:01:16,250 The output amplitude is related to the input amplitude, and this is in steady 17 00:01:16,250 --> 00:01:19,620 state by this transfer function, in particular 18 00:01:19,620 --> 00:01:23,950 the transfer function magnitude at that frequency. 19 00:01:23,950 --> 00:01:29,690 The angle here, phase angle, is a phase angle of h at that frequency. 20 00:01:31,580 --> 00:01:35,000 Now this particular thing is all done in terms of radians per second. 21 00:01:37,630 --> 00:01:39,110 Frequency or radians per second. 22 00:01:42,350 --> 00:01:50,190 When we deal with hertz, it's, the relationship is omega is equal to 2 pi f. 23 00:01:50,190 --> 00:01:52,600 So f is typically in hertz, and when 24 00:01:52,600 --> 00:01:55,560 we talk about omega, it's in radians per second. 25 00:01:55,560 --> 00:01:57,730 So in this particular case, we just have to realize that 26 00:01:57,730 --> 00:02:01,650 we have to do this conversion between Radians per second, and hertz. 27 00:02:01,650 --> 00:02:04,020 Because this plot is given in hertz. 28 00:02:04,020 --> 00:02:07,400 So we, we're given now, a system with three 29 00:02:07,400 --> 00:02:08,900 different frequency components. 30 00:02:08,900 --> 00:02:15,860 One a DC value, and then this frequency here, and then this frequency here. 31 00:02:15,860 --> 00:02:19,960 So we can apply this whole analysis at each frequency. 32 00:02:19,960 --> 00:02:24,140 And analyze the response individually of these components. 33 00:02:24,140 --> 00:02:26,530 And then at the end, sum them together. 34 00:02:26,530 --> 00:02:29,690 So let's look at them at f 35 00:02:31,710 --> 00:02:37,110 equals zero, which is zero radians per second, this is the DC value. 36 00:02:40,360 --> 00:02:45,088 I look at my plot at that frequency. So that's right here, I 37 00:02:45,088 --> 00:02:49,808 look at the plot at that frequency and look at the 38 00:02:49,808 --> 00:02:54,785 magnitude which is 0.8. So h at zero has a 39 00:02:54,785 --> 00:03:00,069 magnitude of 0.8. And that 40 00:03:00,069 --> 00:03:05,407 means that if I have a DC input of 1, so DC 41 00:03:05,407 --> 00:03:10,588 input of 1, it gets multiplied by that 42 00:03:10,588 --> 00:03:15,570 magnitude, gives me the output. So 43 00:03:15,570 --> 00:03:21,800 that's my DC value. Corresponding to this input DC value. 44 00:03:21,800 --> 00:03:24,260 Now let's look at the next component right here. 45 00:03:24,260 --> 00:03:25,980 That's at f equal to 10,000, so 46 00:03:28,010 --> 00:03:32,730 this is f right there, it gets multiplied by two pi to give me omega. 47 00:03:35,640 --> 00:03:38,920 Let's look at h at that corresponding frequency. 48 00:03:38,920 --> 00:03:46,146 Well this is ten to the fourth right there, so this right here is 10,000. 49 00:03:46,146 --> 00:03:51,350 So H at 10,000 and I'm going to write this in terms of hertz. 50 00:03:55,400 --> 00:03:57,330 The magnitude right there is zero. 51 00:03:59,580 --> 00:04:02,390 Okay. So this 10,000 hertz. 52 00:04:02,390 --> 00:04:04,220 If I were to plot this in radiance per second, 53 00:04:04,220 --> 00:04:07,260 would be 10,000 times 2 pi, which is this omega. 54 00:04:08,690 --> 00:04:12,830 So this particular frequency. We've got a magnitude of zero. 55 00:04:12,830 --> 00:04:17,110 That means my output magni-, my output amplitude is equal to zero. 56 00:04:20,800 --> 00:04:23,450 And since the output amplitudes equal zero, I don't care what 57 00:04:23,450 --> 00:04:26,030 the phase is because this whole thing is going to be zero. 58 00:04:27,660 --> 00:04:31,500 Now let's look at this last term right here, which is a frequency 59 00:04:31,500 --> 00:04:36,229 of 15,000 hertz, or 15,000 times 2 pi in terms of radiance per second. 60 00:04:38,130 --> 00:04:41,762 But we're looking at 15,000 because Hertz is what this plot is. 61 00:04:41,762 --> 00:04:46,477 So at 15,000 is right here, I go up here and 62 00:04:46,477 --> 00:04:51,539 look at this, and that is a value of 0.5. So H 63 00:04:51,539 --> 00:04:56,770 at 15,000 Hertz. 64 00:04:56,770 --> 00:05:01,511 Is equal 0.5. That means my output amplitude 65 00:05:01,511 --> 00:05:06,455 is equal to the input amplitude, which is 1, times this 66 00:05:06,455 --> 00:05:11,651 magnitude of H, which is 0.5, and that will be 0.5. 67 00:05:11,651 --> 00:05:19,103 And my angle. 68 00:05:19,103 --> 00:05:23,420 Is from this angle plot right here, and this is in degrees. 69 00:05:23,420 --> 00:05:27,020 I look at 15,000, and that's right 70 00:05:27,020 --> 00:05:31,636 there. That's minus 71 00:05:31,636 --> 00:05:36,866 130. So, my output phase will be theta 72 00:05:36,866 --> 00:05:43,660 is equal to minus 130 degrees. So let's put them all together, v out 73 00:05:43,660 --> 00:05:49,920 is equal to the response due to this DC, which found to be 0.8. 74 00:05:49,920 --> 00:05:55,500 Plus the response due to this frequency component, which we found to be zero. 75 00:05:56,740 --> 00:05:59,300 Plus response due to this frequency component which 76 00:05:59,300 --> 00:06:02,740 we found to have an amplitude of 0.5. 77 00:06:02,740 --> 00:06:03,800 At that frequency 78 00:06:09,840 --> 00:06:11,600 and then with this angle. 79 00:06:11,600 --> 00:06:16,042 So this is the output there that's the output amplitude and that's a phase. 80 00:06:16,042 --> 00:06:19,387 And this phase was minus 130. 81 00:06:19,387 --> 00:06:21,046 And that's the result. 82 00:06:21,046 --> 00:06:22,989 Now, just to summarize what we did 83 00:06:22,989 --> 00:06:25,937 here, is everything was based on this relationship 84 00:06:25,937 --> 00:06:28,416 between an input sinusoid to a linear system 85 00:06:28,416 --> 00:06:31,710 and how it's related by the transfer function. 86 00:06:31,710 --> 00:06:35,310 It just effects the amplitude and the phase through 87 00:06:35,310 --> 00:06:39,080 this relationship right here. And we use that component by component. 88 00:06:39,080 --> 00:06:41,640 So we had three frequency components here. 89 00:06:41,640 --> 00:06:43,840 And at each frequency component, we looked at the 90 00:06:43,840 --> 00:06:50,220 magnitude and the phase, and we computed them individually. 91 00:06:50,220 --> 00:06:52,920 And then we added up the corresponding responses, 92 00:06:52,920 --> 00:06:55,750 to find the total response due to all three. 93 00:06:55,750 --> 00:06:57,080 Thank you.