1 00:00:00,012 --> 00:00:04,790 In this video, we're going to continue our discussion of general circuits. 2 00:00:04,790 --> 00:00:08,811 Circuits that contain resistors, capacitors, and inductors. 3 00:00:08,811 --> 00:00:12,784 And we're going to interpret the results we've gained already. 4 00:00:12,784 --> 00:00:17,833 We're going to, show, that simple circuits function like what we call 5 00:00:17,833 --> 00:00:21,264 filters. We'll define those in just a second. 6 00:00:21,264 --> 00:00:26,016 we'll extend our ideas of the equivalent circuits to the case where we have 7 00:00:26,016 --> 00:00:31,336 impedances and it's very straightforward and we'll also use impedances to talk 8 00:00:31,336 --> 00:00:35,495 about what's called complex power, makes it very easy to do power 9 00:00:35,495 --> 00:00:41,643 calculations when you use complex power. Okay. So, big question is what is this 10 00:00:41,643 --> 00:00:48,520 circuit doing? We have, we can figure out its output, but let's review how we got 11 00:00:48,520 --> 00:00:55,006 that output first and the solar input, and this will lead us to an understanding 12 00:00:55,006 --> 00:01:01,029 of what the circuit is really doing. So, what we did is that we assume that 13 00:01:01,029 --> 00:01:06,285 the source was a sinusoid of some sort. We assumed a times a cosine 2 pi f naught 14 00:01:06,285 --> 00:01:12,033 t and we wrote that as the real part. This is the critical thing, the real part 15 00:01:12,033 --> 00:01:16,490 of a complex exponential, because once you assume things are 16 00:01:16,490 --> 00:01:22,426 complex exponential, the calculation of the transfer function 17 00:01:22,426 --> 00:01:26,999 is very, very easy. So, we assume it was a conflict 18 00:01:26,999 --> 00:01:31,422 exponential. We derive a relationship between a 19 00:01:31,422 --> 00:01:37,874 complex input to the input and the complex input to the output by using 20 00:01:37,874 --> 00:01:41,442 voltage divider. Very simple calculations, 21 00:01:41,442 --> 00:01:46,167 and this, we're going to concentrate on, this is the transfer function. 22 00:01:46,167 --> 00:01:51,737 But in any case, now that we knew what the complex amplitude of the output was, 23 00:01:51,737 --> 00:01:57,147 it was very easy to find out what the actual waveform in time is, because we 24 00:01:57,147 --> 00:02:00,931 simply put it into the real part for a minute, 25 00:02:00,931 --> 00:02:06,507 because we used real part here. We're going to use real part calculations 26 00:02:06,507 --> 00:02:10,406 down there, and we discovered that the output was 27 00:02:10,406 --> 00:02:15,131 given by this rather complicated looking expression. 28 00:02:15,131 --> 00:02:19,674 But we're going to, figure out what it's really telling us in 29 00:02:19,674 --> 00:02:24,185 just a second. one critical thing we had to do in order 30 00:02:24,185 --> 00:02:30,820 to arrive this result, was we had to convert this expression for the transfer 31 00:02:30,820 --> 00:02:36,403 function into polar form. And it turns out that's the way we think 32 00:02:36,403 --> 00:02:42,339 about transfer functions, not in Cartesian form, but in polar form. 33 00:02:42,339 --> 00:02:46,536 So, here it is. I've written H(f) in terms of its 34 00:02:46,536 --> 00:02:54,323 magnitude, which is the quantity here and in terms of a phase, which was given by 35 00:02:54,323 --> 00:02:58,372 this, and then, those are the pieces that fit 36 00:02:58,372 --> 00:03:03,668 in in here. A little subtle detail here is that here, 37 00:03:03,668 --> 00:03:10,544 we assume the frequency is at zero and if zero occurs down in the expressions for 38 00:03:10,544 --> 00:03:15,286 the final output. And now, I want to think about the 39 00:03:15,286 --> 00:03:21,452 transfer function as a general function of f, as a function of frequency, 40 00:03:21,452 --> 00:03:26,613 that's going to be the key insight into what the circuit is doing. 41 00:03:26,613 --> 00:03:30,515 And so, you'll notice the notation of a change. 42 00:03:30,515 --> 00:03:37,482 So, here's that formula for the transfer function and the form for this circuit 43 00:03:37,482 --> 00:03:42,022 and I'm going to plot it. So, this is what it looks like. 44 00:03:42,022 --> 00:03:48,101 We plot the magnitude and phase, usually, they're plotted one on top of the other. 45 00:03:48,101 --> 00:03:51,642 There is the magnitude, there is the angle. 46 00:03:51,642 --> 00:03:58,005 So you can compare what's happening to the phase and the magnitude at some given 47 00:03:58,005 --> 00:04:00,989 frequency. Okay. So lets look at this. 48 00:04:00,989 --> 00:04:07,516 the general rule of exploring these transfer functions to look at what 49 00:04:07,516 --> 00:04:12,836 happens at very low frequencies, as f is going to zero, and at very high 50 00:04:12,836 --> 00:04:21,111 frequencies, as f is going to infinity. So, at zero frequency this is an easy 51 00:04:21,111 --> 00:04:25,548 calculation, mideterms, it has a value of one. 52 00:04:25,548 --> 00:04:32,812 Okay? As f gets very big, when this term is a whole lot bigger than that, 53 00:04:32,812 --> 00:04:39,492 because the square root, the magnitude is falling down like 1/f, going down like 54 00:04:39,492 --> 00:04:44,089 one over frequency. And in between, it's a nice smooth curve 55 00:04:44,089 --> 00:04:49,893 getting between those two extremes, the constant and behaving like 1/f. 56 00:04:49,893 --> 00:04:55,472 The phase, it is simply an arctangent function with a minus sign. 57 00:04:55,472 --> 00:04:59,699 So this is the arctangent function, very simple. 58 00:04:59,699 --> 00:05:06,382 It's heading toward minus pi over two as frequency gets big, and it's zero at the 59 00:05:06,382 --> 00:05:07,646 origin. Okay. 60 00:05:07,646 --> 00:05:14,332 There's something to notice here. I'm plotting this as a function of both 61 00:05:14,332 --> 00:05:18,472 positive frequency and negative frequency. 62 00:05:18,472 --> 00:05:26,142 As we saw in previous videos turns out sines and cosines actually consist of two 63 00:05:26,142 --> 00:05:32,405 complex exponential terms among the + and one with a -. 64 00:05:32,405 --> 00:05:38,072 So, negative frequency certainly exist, and this, and you should plot your 65 00:05:38,072 --> 00:05:43,605 magnitude and phase as a, realize that they're both positive and negative 66 00:05:43,605 --> 00:05:48,248 frequency components. However, it turns out there's a pretty 67 00:05:48,248 --> 00:05:54,146 easy thing to see in this example and it's going to happen in general. 68 00:05:54,146 --> 00:06:00,591 And that is, for the magnitude, it's a symmetric function about the origin 69 00:06:00,591 --> 00:06:07,188 having what we call even symmetry. And what that means is that a magnitude 70 00:06:07,188 --> 00:06:12,079 of H(f) is equal to the magnitude of H(f), of (-f), 71 00:06:12,079 --> 00:06:19,447 that's called even symmetry. And all magnitudes of transfer functions of 72 00:06:19,447 --> 00:06:24,827 circuits and general systems have this property. 73 00:06:24,827 --> 00:06:31,287 So, you can just tell me what the positive axis looks like and I can plot 74 00:06:31,287 --> 00:06:35,626 for you what happens if negative frequency very easily. 75 00:06:35,626 --> 00:06:39,630 It's a mirror image, so it's a very easy thing to do. 76 00:06:39,630 --> 00:06:45,710 The phase has very similar kind of symmetries and it has what's called odd 77 00:06:45,710 --> 00:06:50,742 symmetry, and that is that the angle of H(f) is 78 00:06:50,742 --> 00:06:58,324 equal to the negative of the angle of H(-f), and that's always true. 79 00:06:58,324 --> 00:07:07,392 So, again, if I tell you what happens at over the positive frequency axis, it has 80 00:07:07,392 --> 00:07:12,648 negative symmetry and that the odd symmetry, negative symmetry in that, this 81 00:07:12,648 --> 00:07:16,585 part is going to be the negative of what it is over here. 82 00:07:16,585 --> 00:07:22,168 So again, I gave, I give you the positive frequency part for both the magnitude and 83 00:07:22,168 --> 00:07:25,971 the phase. You can fill in what happens at negative 84 00:07:25,971 --> 00:07:32,348 frequency easily. I should mention that this is a, the e, 85 00:07:32,348 --> 00:07:40,129 the magnitude being even and the phase being odd, is a way of talking about 86 00:07:40,129 --> 00:07:49,421 what's called conjugate symmetry and that H(f) is equal to H conjugate of -f. 87 00:07:49,421 --> 00:07:55,567 Turns out, another way of thinking that even in, even symmetry for the magnitude, 88 00:07:55,567 --> 00:07:55,567 odd symmetry for the phase is that H(f) equals H conjugate of -f. 89 00:07:55,567 --> 00:07:55,567 And I writed that, again, to make it very clear. 90 00:07:55,567 --> 00:07:55,567 Conjugate symmetry means that H(f) is equal to the conjugate of what is 91 00:07:55,567 --> 00:08:22,053 happening at negative frequency. So all transfer functions have this 92 00:08:22,053 --> 00:08:26,455 property, so it makes it very easy to do your 93 00:08:26,455 --> 00:08:30,784 plotting and everything else. Well, okay, 94 00:08:30,784 --> 00:08:37,404 but what is this curve telling us? Particularly, the magnitude? So, as 95 00:08:37,404 --> 00:08:43,907 frequency increases, the, the output voltage, V out, will be getting smaller 96 00:08:43,907 --> 00:08:49,347 and smaller and smaller, so I'm assuming is that the input voltage, the amplitude 97 00:08:49,347 --> 00:08:52,632 of our sign wave is a constant across all frequency. 98 00:08:52,632 --> 00:08:57,302 And as I change the frequency of that input, the output, just because of the 99 00:08:57,302 --> 00:09:01,102 action of the circuit is going to get smaller and smaller and smaller. 100 00:09:01,102 --> 00:09:05,853 Okay? That's what we call a lowpass filter. 101 00:09:05,853 --> 00:09:12,975 And, I'm going to, give you the reason for the word filter here in just a 102 00:09:12,975 --> 00:09:16,461 second, for the lowpass I think is pretty 103 00:09:16,461 --> 00:09:23,212 straightforward. It's passing through, letting through, things in the low 104 00:09:23,212 --> 00:09:28,842 frequency range, and things at higher frequency ranges tend to be suppressed, 105 00:09:28,842 --> 00:09:33,882 and when we see that, how that happens, through a very simple example. 106 00:09:33,882 --> 00:09:39,612 Suppose our input voltage consisted of a sum of two sinusoids, f1 and f2, and you 107 00:09:39,612 --> 00:09:43,722 can see I've allowed them to have general amplitudes. 108 00:09:43,722 --> 00:09:49,049 What's the output going to be when the input consists of a sum of two 109 00:09:49,049 --> 00:09:55,519 sinusoid's? Well, we're going to use the principle superposition of course, 110 00:09:55,519 --> 00:10:01,301 because the circuits are linear. So, when you have the input is equal to 111 00:10:01,301 --> 00:10:05,062 the first sinusoid, we know what the output is. 112 00:10:05,062 --> 00:10:11,402 And the input is equal to just the second sinusoid, we know what the output is, and 113 00:10:11,402 --> 00:10:17,808 then the output input rather consists of the sum of those two superposition 114 00:10:17,808 --> 00:10:23,850 applies you just add those answers up. That is the beauty of having a linear 115 00:10:23,850 --> 00:10:28,677 circuit, this is the very definition of being 116 00:10:28,677 --> 00:10:32,771 linear. The output for sum is equal to the sum of 117 00:10:32,771 --> 00:10:34,880 the outputs. Okay. 118 00:10:34,880 --> 00:10:44,103 So, what's happening here? Let's assume that f1 is down here somewhere and lets 119 00:10:44,103 --> 00:10:50,969 assume that f2 is out here somewhere. So, let's assume for the sake of 120 00:10:50,969 --> 00:10:57,195 argument, that the amplitudes of these are about the same. 121 00:10:57,195 --> 00:11:05,432 Now, what's going to happen on the output side is that, the transfer function at f2 122 00:11:05,432 --> 00:11:10,243 is much smaller than it is at frequency f1. 123 00:11:10,243 --> 00:11:17,220 So, this term, is going to be attenuated. Filter is the term. 124 00:11:17,220 --> 00:11:23,622 It's going to tend to filter out the high frequency term. 125 00:11:23,622 --> 00:11:31,983 So the terminology is that the low frequency area, the place where you get 126 00:11:31,983 --> 00:11:39,872 frequency region over which things get through is called the passband, 127 00:11:39,872 --> 00:11:45,581 the band of frequencies over signals pass through. 128 00:11:45,581 --> 00:11:52,498 And for frequencies that are much higher, herein, was called the stopband, 129 00:11:52,498 --> 00:11:57,870 the circuit, the filter tends to stop that from coming out. 130 00:11:57,870 --> 00:12:04,759 So this again is a low pass, because it's in the low frequency regions that get 131 00:12:04,759 --> 00:12:11,037 passed through and the high frequencies tend to be attenuated. 132 00:12:11,037 --> 00:12:19,616 So, let's get, so what's, high and what's low? Well, you can see this curve is very 133 00:12:19,616 --> 00:12:27,791 gradual and it's not as if there's, there's was, a clear delineation between 134 00:12:27,791 --> 00:12:34,488 the two. an ideal low pass would have a frequency 135 00:12:34,488 --> 00:12:42,918 characteristic that looks like this. So, let's say it has a gain of one at low 136 00:12:42,918 --> 00:12:49,452 frequencies in and in it, instantly jumps down, and at some frequency, goes to 137 00:12:49,452 --> 00:12:53,291 zero. That would be what we call an ideal low 138 00:12:53,291 --> 00:12:57,105 pass. Well, the RC filter, we have, certainly 139 00:12:57,105 --> 00:13:03,989 doesn't have that characteristic, but we can think about it by defining what's 140 00:13:03,989 --> 00:13:06,445 called a cutoff frequency. So, 141 00:13:06,445 --> 00:13:12,888 the cutoff frequency is generally defined with frequency at which the magnitude at 142 00:13:12,888 --> 00:13:18,126 that frequency is one over the square root of two times the maximum value of 143 00:13:18,126 --> 00:13:22,312 the magnitude, wherever that occurred in the frequency. 144 00:13:22,312 --> 00:13:28,627 So for a lowpass, this maximum is occuring and the orgin has a maximum of 145 00:13:28,627 --> 00:13:31,989 one. And you change frequency till it's, the 146 00:13:31,989 --> 00:13:38,355 output is equal to one over the square root of two of the value at the origin, 147 00:13:38,355 --> 00:13:42,200 which here is one. And the, what, why one over the square 148 00:13:42,200 --> 00:13:48,076 root of two, you might be asking? If you think about that for a second, 149 00:13:48,076 --> 00:13:55,445 what could be so important about the one over square root of two? And the answer 150 00:13:55,445 --> 00:14:01,955 is power, so, at this frequency, because power is proportional to signal squared. 151 00:14:03,013 --> 00:14:09,366 The power of this, of the sinusoid at this cutoff frequency, will be 1/2, of 152 00:14:09,366 --> 00:14:13,788 the value it is at the maximum. So this cutoff frequency, the power at 153 00:14:13,788 --> 00:14:17,287 the, in the output will be 1/2 of what it was at the origin. 154 00:14:17,287 --> 00:14:22,043 If you go to higher frequencies, the power output gets less, and less, and 155 00:14:22,043 --> 00:14:27,900 less, and less at those frequencies. So, for our RC lowpass, we need to look 156 00:14:27,900 --> 00:14:34,531 at the formula to figure out where the cutoff frequency is. And it's pretty easy 157 00:14:34,531 --> 00:14:38,576 in this case, because we know at f = 0 the value is 158 00:14:38,576 --> 00:14:45,390 one. And the frequency at which it's, this square root is going to be two is 159 00:14:45,390 --> 00:14:51,354 when this term equals one. And so solving for that, we get that the 160 00:14:51,354 --> 00:14:54,179 cutoff frequency is equal to one over two pi times R times C. 161 00:14:55,406 --> 00:15:03,115 So, when it comes to designing a circuit, suppose you wanted to design a filter 162 00:15:03,115 --> 00:15:09,936 that would pass yeah, it have a cutoff frequency of 1 kilohertz. 163 00:15:09,936 --> 00:15:16,484 Well, what you do is set this equal to, to a thousand, and what you'd find is 164 00:15:16,484 --> 00:15:24,494 that the only thing that you need is for the product of R and C to be over two pi. 165 00:15:24,494 --> 00:15:36,326 So the important point here is that it's the product of the component names. 166 00:15:36,326 --> 00:15:42,170 We've seen this before. You have, the only thing you need is the 167 00:15:42,170 --> 00:15:47,118 product to have certain property, so you can choose the resistor and capacitor 168 00:15:47,118 --> 00:15:50,423 separately according to whatever criteria you want. 169 00:15:50,423 --> 00:15:55,427 But the, the product of the two is what determines the quality frequency, and so 170 00:15:55,427 --> 00:16:00,218 not the individual component you use. It gives you some latitude in doing the 171 00:16:00,218 --> 00:16:01,467 design. Okay. 172 00:16:01,467 --> 00:16:06,517 So it turns out there are more kind of filters out there of lowpass. 173 00:16:06,517 --> 00:16:11,797 So the lowpass we've talked about. So, and generically, I'm only going to 174 00:16:11,797 --> 00:16:17,757 draw the positive frequency axis, because we know its going to happen through the 175 00:16:17,757 --> 00:16:22,914 magnitude. generally look something like that. 176 00:16:22,914 --> 00:16:29,529 So generic lowpass which has a cutoff frequency somewhere there, 177 00:16:29,529 --> 00:16:35,000 where this value here is one over square root of two times whatever the value is 178 00:16:35,000 --> 00:16:40,302 at the, at the, at zero frequency. You can also have a pass go through. 179 00:16:40,302 --> 00:16:52,047 Highpass, what it looks like is pretty easy to see where the name comes from. It 180 00:16:52,047 --> 00:17:02,077 tends to suppress low frequencies, let through the higher frequencies and its 181 00:17:02,077 --> 00:17:08,857 cutoff frequency will be similar here. And that value is one over the square 182 00:17:08,857 --> 00:17:14,537 root of two times whatever the value is at infinite frequency. 183 00:17:14,537 --> 00:17:25,284 And then finally there's the bandpass filter which, what this means is that it 184 00:17:25,284 --> 00:17:32,396 passes a range of frequencies and suppresses the other ones. 185 00:17:32,396 --> 00:17:40,870 So, if you plotted it, it tends to look something like that and this means it has 186 00:17:40,870 --> 00:17:45,773 two cutoff frequencies, a lower cutoff frequency and an upper 187 00:17:45,773 --> 00:17:49,024 cutoff frequency. We refer to the width, 188 00:17:49,024 --> 00:17:55,248 the distance between the two cutoff frequencies as being the bandwidth of the 189 00:17:55,248 --> 00:18:01,160 filter. And that's an important parameter when 190 00:18:01,160 --> 00:18:05,608 you start talking about band bass filters. 191 00:18:05,608 --> 00:18:12,541 sometimes we refer to the upper and lower cutoff frequencies. 192 00:18:12,541 --> 00:18:19,294 Other times, we might refer to the center of the band, and the bandwith specifying 193 00:18:19,294 --> 00:18:23,216 the bandpass forward whichever way seems most convenient. 194 00:18:23,216 --> 00:18:28,368 We're going to discover, as we go through this course, that all these folders have 195 00:18:28,368 --> 00:18:33,642 very important uses and occur very frequently in real world application. 196 00:18:33,642 --> 00:18:41,285 Okay, I want to explore what really happens when you build a RC lowpass when, 197 00:18:42,479 --> 00:18:50,626 because you can see, I want to use specific elements that I use for, I 198 00:18:50,626 --> 00:18:58,939 really constructed this circuit. The input is going to be a sine wave. 199 00:18:58,939 --> 00:19:08,055 And we know that, from all we've done that the output for a general filter is 200 00:19:08,055 --> 00:19:14,790 going to be the transfer function, magnitude of the transfer function times 201 00:19:14,790 --> 00:19:21,514 the sine plus the sine wave with a given, that has a phase shift given by the volt. 202 00:19:21,514 --> 00:19:27,659 And we have calculated this for the RC lowpass several times, and we know that 203 00:19:27,659 --> 00:19:33,917 what it's going to do, is the magnitude is given by this expression in the phase, 204 00:19:33,917 --> 00:19:35,782 is given by this. Okay. 205 00:19:35,782 --> 00:19:40,970 So, first thing we want to do is calculate exactly where the cutoff 206 00:19:40,970 --> 00:19:45,872 frequency is going to be. To do that, we need to find out where the 207 00:19:45,872 --> 00:19:52,043 magnitude is it's maximum value and it's pretty clear from this expression that 208 00:19:52,043 --> 00:19:55,522 the maximum value occurs at a frequency of zero, 209 00:19:55,522 --> 00:20:00,452 this is after all a lowpass filter. And then, we want to see where the 210 00:20:00,452 --> 00:20:05,837 magnitude is one over the square root of two times that maximum value, which in 211 00:20:05,837 --> 00:20:09,787 this case is one. So, it's pretty clear that it's going to 212 00:20:09,787 --> 00:20:14,742 be equal to one over the square root of two and this expression equals one, 213 00:20:14,742 --> 00:20:19,528 because we had the square root of 1 + 1, will give us what we want. 214 00:20:19,528 --> 00:20:25,629 So, we've calculated this several times and we know that the answer is with the 215 00:20:25,629 --> 00:20:28,889 cutoff frequencies given by one over two pi RC. 216 00:20:28,889 --> 00:20:35,542 And when you stick in the explicit values, for our circuit element values, 217 00:20:35,542 --> 00:20:40,331 we have 10^4 for the resistor, and this is 220 nanofarads. 218 00:20:40,331 --> 00:20:46,987 Like I said, you don't usually have very large value capacitors in circuits, so, 219 00:20:46,987 --> 00:20:52,105 and we get 72.34 Hertz. We'll have to see if that really works, 220 00:20:52,105 --> 00:20:58,389 really happens when you look at the out, output of the circuit that has these 221 00:20:58,389 --> 00:21:04,305 circuit elements. When you plot this, some things that I 222 00:21:04,305 --> 00:21:11,695 want to note before we go to the simulation we predict the cutoff 223 00:21:11,695 --> 00:21:19,612 frequency is going to be 72.34 which is a frequency somewhere like this. 224 00:21:19,612 --> 00:21:26,797 Well, I want to point out something, it's a special thing for this RC lowpass. 225 00:21:26,797 --> 00:21:33,037 Where is the phase, what is the phase at the cut off frequency? 226 00:21:33,037 --> 00:21:37,337 Well, then, the fc in the frequency is equal to one 227 00:21:37,337 --> 00:21:42,306 pi RC. This expression is one, the arctangent of 228 00:21:42,306 --> 00:21:49,113 one is pi over four, 35 degrees, so the phase will be - 45 at that point. 229 00:21:49,113 --> 00:21:53,146 Okay. So here is what we should expect. 230 00:21:53,146 --> 00:21:59,831 When you have frequencies less than fc, the amplitude of the output should be 231 00:21:59,831 --> 00:22:02,967 going up until it gets to very low frequencies, 232 00:22:02,967 --> 00:22:08,350 it should be about the same as the input. The phase shift will become smaller and 233 00:22:08,350 --> 00:22:12,227 smaller and smaller, and then again, as you go toward zero in 234 00:22:12,227 --> 00:22:17,934 frequency, the output in terms of its phase shift will also look like the 235 00:22:17,934 --> 00:22:22,667 input. On the other hand, if you go way beyond 236 00:22:22,667 --> 00:22:31,544 the cutoff frequency, the amplitude of the output will be going down and down 237 00:22:31,544 --> 00:22:38,929 and down and down and the phase shift is headed toward minus 90 degrees, minus pi 238 00:22:38,929 --> 00:22:48,387 over two. So, what is sine of two pi ft minus pi 239 00:22:48,387 --> 00:23:02,312 over two? What is that equal? And what I get is that that is minus cosine. 240 00:23:02,312 --> 00:23:06,805 Okay. So that's what we do, should expect. 241 00:23:06,805 --> 00:23:14,198 As the frequency gets very high, amptitude gets smaller and smaller and 242 00:23:14,198 --> 00:23:21,283 smaller, and the phase is going, the output is going to look more and more 243 00:23:21,283 --> 00:23:27,247 like a minus cosine when all is said and done. 244 00:23:27,247 --> 00:23:37,752 Well, let's see if that really happens. So here is my input, the frequency is, 245 00:23:37,752 --> 00:23:42,377 here is the frequency generator, is setting up a sine. 246 00:23:42,377 --> 00:23:47,491 This output number here is the peak to peak amplitude of the output, 247 00:23:47,491 --> 00:23:53,286 while 80 Hertz is a little bit above the cutoff frequency, so the amplitude should 248 00:23:53,286 --> 00:23:56,524 be 1.414. If we're at the cutoff frequency, 249 00:23:56,524 --> 00:24:02,197 it's a little lower than that, because we're too high in frequency. 250 00:24:02,197 --> 00:24:07,806 I've lowered it to a 74 1/2, getting closer to 1.414, but it's not 251 00:24:07,806 --> 00:24:12,092 quite correct. I've put in 73.2, it's getting closer. 252 00:24:12,092 --> 00:24:19,006 And, you, for this oscilloscope, we could have typed it in all the time. 253 00:24:19,006 --> 00:24:25,970 So we point out that the white line is the input to our circuit, the red line is 254 00:24:25,970 --> 00:24:30,493 the output. So you can see the amplitude is down by 255 00:24:30,493 --> 00:24:35,094 some amount and this is telling exactly what it is. 256 00:24:35,094 --> 00:24:39,747 Right? Well, let's explore what happens as I go to the lower frequency. 257 00:24:39,747 --> 00:24:45,358 So I gotta change the what's called the time base of my oscilloscope. 258 00:24:45,358 --> 00:24:49,250 I'm going to make it go down by a factor of ten in frequency. 259 00:24:49,250 --> 00:24:53,627 Right now, it's 7.3 Hertz. And as you can see, the red line, the 260 00:24:53,627 --> 00:24:59,192 output is almost the same as the white line, the input to the filter, and it's 261 00:24:59,192 --> 00:25:04,262 got very little phase shift and that's exactly what we should predict. 262 00:25:04,262 --> 00:25:09,857 Let's go back to 70, 73 Hertz and we can see how the amplitude's going down like 263 00:25:09,857 --> 00:25:13,272 it should. Now, I'm going to set it up to go up to a 264 00:25:13,272 --> 00:25:18,933 very high frequency and you can see the output is greatly attenuated. And if you 265 00:25:18,933 --> 00:25:24,242 look carefully at that waveform, it looks to me like it looks like a minus cosine 266 00:25:24,242 --> 00:25:27,921 to me, so all the theoretical predictions are correct. 267 00:25:27,921 --> 00:25:32,062 Now, what I'm going to do is go back to our original setting, 268 00:25:32,062 --> 00:25:37,347 original frequency, and I'm going to variate continuously now, so we can see 269 00:25:37,347 --> 00:25:42,947 the go up, in frequency, so the amplitude will be going down as the frequency goes 270 00:25:42,947 --> 00:25:47,567 up just as predicted. The phase shift keeps increasing, getting 271 00:25:47,567 --> 00:25:51,681 more negative. And then, if we go back the other way, we 272 00:25:51,681 --> 00:26:01,442 see the amp, amplitude go up and keep going down, down, down in frequency, 273 00:26:01,442 --> 00:26:10,784 below the cutoff frequency now, well inside, looks more and more like the 274 00:26:10,784 --> 00:26:12,237 input. Okay. 275 00:26:12,237 --> 00:26:17,435 So, let's talk about the big picture. We're going to make sure we really 276 00:26:17,435 --> 00:26:22,947 understand what's been happening here. We started with a circuit. 277 00:26:22,947 --> 00:26:29,485 We assumed complex exponential input, which meant that every voltage and every 278 00:26:29,485 --> 00:26:34,092 current in the circuit was also a complex exponential. 279 00:26:34,092 --> 00:26:40,657 Based on that notion, we defined impedance and converted the circuit to 280 00:26:40,657 --> 00:26:47,975 being an equivalent circuit, so far as complex amplitudes are concerned so that 281 00:26:47,975 --> 00:26:52,959 we had impedances representing the circuit elements. 282 00:26:52,959 --> 00:27:00,502 Because of this use, assuming complex exponential inputs, we can now use all 283 00:27:00,502 --> 00:27:06,837 the shortcuts that we developed for resistor circuits and this makes the 284 00:27:06,837 --> 00:27:12,937 calculations quite easy. and, I think it'd be pretty obvious that 285 00:27:12,937 --> 00:27:18,484 the Mayer-Norton and Thevenin equivalents follow just as well. 286 00:27:18,484 --> 00:27:24,495 So, since they were based simply on using that series imparallel rules and, and 287 00:27:24,495 --> 00:27:29,468 even a bit more generally. The Thevenin equivalent, when you have 288 00:27:29,468 --> 00:27:34,442 impedances, is exactly the same, except this is now an impedance and this is a 289 00:27:34,442 --> 00:27:40,016 complex amplitude. So the frequency of the source is implicit. 290 00:27:40,016 --> 00:27:45,801 All we care about in representing that circuit with a Thevenin equivalent are 291 00:27:45,801 --> 00:27:51,645 the equivalent source and the equivalent impedance, and the Mayer-Norton has 292 00:27:51,645 --> 00:27:53,761 exactly the same properties we decide, drew out the four. 293 00:27:56,133 --> 00:28:03,743 The two impedances are the same and that's the amplitude of the current 294 00:28:03,743 --> 00:28:10,072 source. And that same formula applies for that we 295 00:28:10,072 --> 00:28:15,432 derive, with four impedances, that the equivalent. 296 00:28:15,432 --> 00:28:19,158 amplitude, the amplitude is of the equivalent circuit, 297 00:28:19,158 --> 00:28:25,483 obey this relationship and that's the equivalent impedance, so there's no real 298 00:28:25,483 --> 00:28:29,471 surprise there. And you can now use these equivalents in 299 00:28:29,471 --> 00:28:34,532 RLC circuits just like we did for resistor circuits. 300 00:28:34,532 --> 00:28:41,499 Okay. Well, there's something else that becomes much very easy to talk about, 301 00:28:41,499 --> 00:28:48,519 when we have complex exponential or sinusoidal sources. So, again I'm 302 00:28:48,519 --> 00:28:55,142 going to think about all my circuit elements as being an impedance with some 303 00:28:55,142 --> 00:29:01,070 complex amplitude for the voltage and complex amplitude for the current coming 304 00:29:01,070 --> 00:29:03,361 in. And now what I'm going to do, going to do 305 00:29:03,361 --> 00:29:07,550 is a power calculation, so I'm going to do this in general. 306 00:29:07,550 --> 00:29:13,410 I'm going to assume that the complex amplitudes has a magnitude and a phase 307 00:29:13,410 --> 00:29:17,284 for both the current and the voltage. Okay? 308 00:29:17,284 --> 00:29:26,549 Which means that speaking about them back in, its time-domain quantities the, here 309 00:29:26,549 --> 00:29:32,744 is the magnitude, there, there is the magnitude for the voltage 310 00:29:32,744 --> 00:29:40,212 and there is that phase shift. So I'm thinking about my v, v(t) as being the 311 00:29:40,212 --> 00:29:46,353 real part of this. And for i(t) you take the real part of, of that. 312 00:29:46,353 --> 00:29:50,189 There you go. Well, we know what power is. 313 00:29:50,189 --> 00:29:58,043 Power is always voltage times current. Well, I could plug in these two formulas 314 00:29:58,043 --> 00:30:04,852 into here and try to simplify it. Turns out there is a much easier way, 315 00:30:04,852 --> 00:30:10,802 which I'm going to show you, which actually brings up a very interesting 316 00:30:10,802 --> 00:30:14,628 form. So, I'm going to do that calculation in a 317 00:30:14,628 --> 00:30:19,406 slightly different way. What I want to do is use Euler's formula. 318 00:30:19,406 --> 00:30:24,694 So, in addition to, we can explicitly write out the real part of the rule, 319 00:30:24,694 --> 00:30:30,362 formula, as being just Euler's formula. We've seen how this works before. 320 00:30:30,362 --> 00:30:35,344 Now, is what I want to do is I want to plug that in. I want to plug these 321 00:30:35,344 --> 00:30:40,890 expressions in to that product. Well, it's very easy to multiply this out 322 00:30:40,890 --> 00:30:46,245 and it turns that become, it really simplifies the calculations a lot. 323 00:30:46,245 --> 00:30:51,272 So, the way to think about this is that the voltage is now a sum, 324 00:30:51,272 --> 00:30:57,277 the current is now a sum, so when you multiply the two together, you get all 325 00:30:57,277 --> 00:31:02,377 four possible cross-products. This times this, this times that, that 326 00:31:02,377 --> 00:31:07,097 times that, and that times that. So if we multiply them out, you can 327 00:31:07,097 --> 00:31:12,482 notice that some terms, the complex exponential part cancels, 328 00:31:12,482 --> 00:31:17,376 and for other terms, the complex exponential adds. 329 00:31:17,376 --> 00:31:24,364 And when everything is said and done, this is the expression that you get. 330 00:31:24,364 --> 00:31:30,837 Well, this is quite interesting. Now, if we look at the, these two for 331 00:31:30,837 --> 00:31:35,112 example, this is the complex conjugate of that. 332 00:31:35,112 --> 00:31:41,182 Right? Because the conjugate of a product is the product of the conjugate. 333 00:31:41,182 --> 00:31:46,827 V conjugate, V, the conjugate of V is V conjugate, and the conjugate of I 334 00:31:46,827 --> 00:31:50,367 conjugate is just I, we get it back again. 335 00:31:50,367 --> 00:31:55,835 Same thing happens out here. This is the conjugate of that. 336 00:31:55,835 --> 00:32:01,281 So, those are real parts. And noice, because we multiply by a 337 00:32:01,281 --> 00:32:07,322 quarter, we would have to put a half factor in to get the real part in the 338 00:32:07,322 --> 00:32:12,192 final answer. So, what does this say about the power 339 00:32:12,192 --> 00:32:19,672 when you this is the actual power. There's nothing complex here at all. 340 00:32:19,672 --> 00:32:27,252 Those are actual power values, P(t) is equal to some number and then you get a 341 00:32:27,252 --> 00:32:36,340 term that turns out to have a frequency, which is ft, twice the frequecy of the 342 00:32:36,340 --> 00:32:41,842 source. So if you plot this, this term is a 343 00:32:41,842 --> 00:32:52,614 constant, it's a function of time now and rippling around it, somehow is the time 344 00:32:52,614 --> 00:32:58,649 varying part. Well, we know that energy is the integral 345 00:32:58,649 --> 00:33:07,753 of power, so the time varying part really doesn't contribute over the long term to 346 00:33:07,753 --> 00:33:12,911 energy consumption. Sometimes, it goes to zero, and this 347 00:33:12,911 --> 00:33:19,734 thing, and this term goes to, this pulses goes to zero and sometimes goes positive. 348 00:33:19,734 --> 00:33:26,231 The only thing that contributes long term is that term, and that's called the 349 00:33:26,231 --> 00:33:31,258 average power. this is a general result for all RLC 350 00:33:31,258 --> 00:33:35,550 circuits. And if the source is a exponential, 351 00:33:35,550 --> 00:33:41,821 I'm sorry, is a sinusoid, which we think of as a complex exponential. 352 00:33:41,821 --> 00:33:49,110 So, average power that is consumed is given by the real part, half the real 353 00:33:49,110 --> 00:33:55,309 part of the I conjugate, for these are the complex amplitudes of the circuit 354 00:33:55,309 --> 00:34:00,175 element where, where you met the power consumption flow. 355 00:34:00,175 --> 00:34:03,122 So, this is what is known as complex power. 356 00:34:03,122 --> 00:34:12,087 so in general the only thing that's really important is the P ave term. 357 00:34:12,087 --> 00:34:21,198 I'm sorry, is the P average is the real part of complex power. 358 00:34:21,198 --> 00:34:28,297 And, whenever we have an impedance, it turns out this formula for complex power 359 00:34:28,297 --> 00:34:35,901 can be rewritten in terms of impedance. So, just plugging in the relationship for 360 00:34:35,901 --> 00:34:42,182 impedance in the VI conjugate formula, we can write it either as z times magnitude 361 00:34:42,182 --> 00:34:47,377 of the current squared or one over the impedance times the magnitude of the 362 00:34:47,377 --> 00:34:51,722 voltage squared. I'm going to stick with this one because 363 00:34:51,722 --> 00:34:56,712 it is a little bit simpler. And let's see what it looks like for all 364 00:34:56,712 --> 00:35:02,902 the three common circuit elements. And certainly, we've seen this before. 365 00:35:02,902 --> 00:35:09,745 For the resistor, the power is just 1/2 R times the magnitude of the current 366 00:35:09,745 --> 00:35:16,246 squared and there is no imaginary part, so the real part of that is just what 367 00:35:16,246 --> 00:35:20,268 you've got. Again, stating its resistors that 368 00:35:20,268 --> 00:35:25,218 dissipate power. For the capacitor though, the impedence 369 00:35:25,218 --> 00:35:29,245 is complex. So, it is imaginary value, right here. 370 00:35:29,245 --> 00:35:33,081 So that means when you take the real part, you get zero. 371 00:35:33,081 --> 00:35:39,424 Again, this is saying that the capicators as well as the inductors do not dissipate 372 00:35:39,424 --> 00:35:43,127 power. And, what they contribute to, is that 373 00:35:43,127 --> 00:35:46,872 time-varying part, that part where the, the power 374 00:35:46,872 --> 00:35:52,435 consumption goes up and down. So, sometimes they consume power, 375 00:35:52,435 --> 00:35:56,346 sometimes they give it back in certain times, 376 00:35:56,346 --> 00:36:03,345 but in the long term is they, they don't consume or produce power when you have a 377 00:36:03,345 --> 00:36:04,982 sinusoidal input. Okay. 378 00:36:04,982 --> 00:36:11,482 So, in summary, my thinking of each element as a complex valued resistor, 379 00:36:11,482 --> 00:36:17,982 what we call an impedance. We get a general picture of what circuits do and 380 00:36:17,982 --> 00:36:22,807 how they behave. We can use to solve those circuits, we 381 00:36:22,807 --> 00:36:28,555 can use the series of parallel rules voltage divider, current divider. We can 382 00:36:28,555 --> 00:36:33,796 use the equivalent circuits, we can talk about transfer functions and filters, 383 00:36:33,796 --> 00:36:38,622 these are going to be very important concepts coming up, transfer functions 384 00:36:38,622 --> 00:36:44,304 are going to come all the time filtering thinking about transfer functions is what 385 00:36:44,304 --> 00:36:49,353 kind of categorize them according to what kind of filter they are. We have to plot 386 00:36:49,353 --> 00:36:52,255 the transfer function and see how it works out. 387 00:36:52,255 --> 00:36:57,905 Weou can also talk about power using just the complex amplitudes, we don't have to 388 00:36:57,905 --> 00:37:05,344 really go back solve for what the v and i are, and then multiply now and get it, we 389 00:37:05,344 --> 00:37:10,717 have a very general formula just from using the complex amplitudes. 390 00:37:10,717 --> 00:37:16,596 Now, I want, do want to point out, it's very important, that we, these results 391 00:37:16,596 --> 00:37:23,423 only apply when the source is a sinusoid. Remember, what we did was assume that the 392 00:37:23,423 --> 00:37:29,883 source was the real part of something that looked like a complex exponential, 393 00:37:29,883 --> 00:37:33,611 and so, all these formulas apply only when we 394 00:37:33,611 --> 00:37:38,141 have a sinusoidal input. When it's a sum of sinusoids, 395 00:37:38,141 --> 00:37:42,952 superposition applies and we know how to solve that. 396 00:37:42,952 --> 00:37:47,644 Well, you might be wondering, what happens if it's not a sinusoid? For 397 00:37:47,644 --> 00:37:53,002 example, suppose it's a pulse or a square wave? There's other signals we've been 398 00:37:53,002 --> 00:37:56,657 talking about. Have we gone down a blind alley? Is this 399 00:37:56,657 --> 00:38:01,832 going to go anywhere? as you might think, this is not the end of the story. 400 00:38:01,832 --> 00:38:06,957 We can talk about very general source, source wave forms other than sinusoids. 401 00:38:06,957 --> 00:38:11,732 It turns out these are the building blocks that we're going to use later in 402 00:38:11,732 --> 00:38:12,445 the course.