1 00:00:00,012 --> 00:00:05,997 So in this video, we're going to continue our discussion of circuits that have any 2 00:00:05,997 --> 00:00:11,107 kind of element we want, capacitors, inductors and resistors. 3 00:00:11,107 --> 00:00:16,752 In the previous video, we found the solution of the circuit when the source 4 00:00:16,752 --> 00:00:21,643 was a complex exponential. Well, that's a mathematical We don't 5 00:00:21,643 --> 00:00:26,569 really, we can't really produce forces that look like that in the lab in the 6 00:00:26,569 --> 00:00:29,837 real world. So, is that just a mathematical thing 7 00:00:29,837 --> 00:00:34,836 that's kind of fun to play with, but it turns out it's much more important than 8 00:00:34,836 --> 00:00:37,397 that. We're going to see how to apply that 9 00:00:37,397 --> 00:00:41,672 knowledge we developed last time, and the source is a sinusoid. 10 00:00:41,672 --> 00:00:46,638 Turns out it's an easy extension of what we already have done. 11 00:00:46,638 --> 00:00:52,953 So let's review very quickly the way the so called Impedance method works. 12 00:00:52,953 --> 00:00:57,828 We start with our circuit. We pretend the source is a complex 13 00:00:57,828 --> 00:01:03,248 exponential and that means we can now only worry about complex amplitudes. 14 00:01:03,248 --> 00:01:07,789 We replace every element by, it's appropriate impedance, whatever it may 15 00:01:07,789 --> 00:01:10,537 be, and then we can use, what we learned from 16 00:01:10,537 --> 00:01:14,391 resistor circuits. Voltage divider, current divider, series 17 00:01:14,391 --> 00:01:19,501 parallesl relations, to figure out how the output variables, complex amplituded, 18 00:01:19,501 --> 00:01:22,927 is related to the complex amplitude of the source. 19 00:01:22,927 --> 00:01:26,152 Okay. So that's what we learned last time, 20 00:01:26,152 --> 00:01:32,662 except like now we're going to handle a bit more real world sources than complex 21 00:01:32,662 --> 00:01:36,532 exponentials. So what we did is we applied it to our 22 00:01:36,532 --> 00:01:41,746 simple RC circuit. we replaced it I thought about it as an 23 00:01:41,746 --> 00:01:47,156 impedence kind of circuit and want to make sure we note that this is a 24 00:01:47,156 --> 00:01:51,345 lowercase v and over here's supposed to be v and a t. 25 00:01:51,345 --> 00:01:57,633 This is lowercase v out of t, but over here we're only talking about the complex 26 00:01:57,633 --> 00:02:03,422 amplitude. So I'm thinking about this as being a 27 00:02:03,422 --> 00:02:13,335 complex exponential source. And now everything in this circuit is a 28 00:02:13,335 --> 00:02:19,007 complex amplitude. And what we discovered when we used our 29 00:02:19,007 --> 00:02:25,232 voltage divider for this circuit was that the amplitude of the source amplitude of 30 00:02:25,232 --> 00:02:30,483 the output rather, was equal to this quantity times the amplitude of the 31 00:02:30,483 --> 00:02:36,262 input, and I'm uncancelling the complex exponential so we get the entire answer. 32 00:02:36,262 --> 00:02:42,825 OK, so that's what happens when the source is a complex exponential. 33 00:02:42,825 --> 00:02:50,033 Well what I want now do is talk about the case where the source is a sinusoid. 34 00:02:50,033 --> 00:02:56,512 This sinusoid has a frequency F nought, and it has an amplitude of A. 35 00:02:56,512 --> 00:03:04,942 Okay, well, that's not the same as a complex exponential. So how does this 36 00:03:04,942 --> 00:03:12,784 result apply to the case where that's what the source really is? Well, the 37 00:03:12,784 --> 00:03:22,332 mathematical trick is where it's coming. We know from the Lewis formula that a 38 00:03:22,332 --> 00:03:34,276 cosine is equal to one half times e to the j theta, plus e to the minus j theta. 39 00:03:34,276 --> 00:03:40,410 A complex amp, ampli-, a complex exponential rather at an angle of, theta 40 00:03:40,410 --> 00:03:44,583 and a complex exponential adding angle of minus theta. 41 00:03:44,583 --> 00:03:49,257 That's a cosine. So that applies to the case where we 42 00:03:49,257 --> 00:03:53,602 have, theta here is just cosine 2 pi of naught t. 43 00:03:53,602 --> 00:03:58,318 So I think of this as being a positive frequency, complex exponential, 44 00:03:58,318 --> 00:04:02,169 and this being a negative frequency complex exponential. 45 00:04:02,169 --> 00:04:06,218 This may sound a little strange. Negative frequency things. 46 00:04:06,218 --> 00:04:11,058 But it turns out this is really very important, and the right way to think 47 00:04:11,058 --> 00:04:14,615 about it. So, we can figure out what the output is 48 00:04:14,615 --> 00:04:19,997 for each piece, at least. We know that if the source was that, is 49 00:04:19,997 --> 00:04:29,555 this term, that that's the output, and we know that if it's the negative frequency 50 00:04:29,555 --> 00:04:37,722 term, all we have to do is replace f by minus f, and we get the answer. 51 00:04:37,722 --> 00:04:44,587 So we know about each piece of the original cosine, and what the output is 52 00:04:44,587 --> 00:04:50,302 for each of those, but how about putting them together? How 53 00:04:50,302 --> 00:04:56,910 do we do that? Well, now we get to the really interesting conclusion. 54 00:04:56,910 --> 00:05:02,700 Now, so this is where we are. This, we used the complex exponential 55 00:05:02,700 --> 00:05:06,551 source to hop over to the impedance version. 56 00:05:06,551 --> 00:05:13,517 We used all of the tricks we've learned for resistance circuits, and now when the 57 00:05:13,517 --> 00:05:20,232 source is a I want to point out, that all cir-, the circuit elements we've been 58 00:05:20,232 --> 00:05:27,932 talking about, resistors, capacitors, inductors and the, interconnection laws, 59 00:05:27,932 --> 00:05:33,382 KVL and KCL are linear, which means superposition applies. 60 00:05:33,382 --> 00:05:40,331 Now, what the definition of linear was, that if the input consists of a sum of 61 00:05:40,331 --> 00:05:47,280 terms, the output also, the output consists of the sum of the outputs to 62 00:05:47,280 --> 00:05:55,792 each term considered separately, you just add them up and multiply by a constant if 63 00:05:55,792 --> 00:05:59,928 there are any out there. So, we're done. 64 00:05:59,928 --> 00:06:05,376 Based on what we've already know, we know the output, 65 00:06:05,376 --> 00:06:10,423 voltage. And this, notice this is lowercase v, is 66 00:06:10,423 --> 00:06:17,169 the output to the first term plus the output to the second term. Add it up and 67 00:06:17,169 --> 00:06:22,989 since there's a weighing factor of half, it goes there to, and this is because of 68 00:06:22,989 --> 00:06:28,385 superposition because the circuit elements we've been talking about are 69 00:06:28,385 --> 00:06:31,623 linear. This is why the linear concept is so 70 00:06:31,623 --> 00:06:37,142 important, really makes our ohms easy. Okay, so let's continue this. 71 00:06:37,142 --> 00:06:40,785 I'm going to show you how to simplify that answer. 72 00:06:40,785 --> 00:06:44,739 It looks little daunting as it seems right now, 73 00:06:44,739 --> 00:06:49,588 but there's little, another mathematical thing we can do. 74 00:06:49,588 --> 00:06:53,682 Now, so, this is Oilers formula for the cosine, 75 00:06:53,682 --> 00:07:00,199 and these source was 2 pi f naught t. And like I said I thought about this as a 76 00:07:00,199 --> 00:07:05,406 positive frequency term and a negative frequency term. 77 00:07:05,406 --> 00:07:09,528 There's another way of thinking about this. 78 00:07:09,528 --> 00:07:15,230 Suppose this is a complex number z, this is z conjugate. 79 00:07:15,230 --> 00:07:23,926 I'm adding them up, and dividing by 2. Well, that's the formula for the real 80 00:07:23,926 --> 00:07:32,842 part, and so we're actually writing the source as the real part of a complex. 81 00:07:32,842 --> 00:07:35,070 Pretty much. And on the output. 82 00:07:35,070 --> 00:07:39,660 Looks very similar, right? There's the positive frequency term. 83 00:07:39,660 --> 00:07:45,002 There's the negative frequency term. And that still is, something plus it's 84 00:07:45,002 --> 00:07:48,890 complex conjugate, divided by 2, that's the real part. 85 00:07:48,890 --> 00:07:55,023 So, this, if you will, is the trick. We, can think about, 86 00:07:55,023 --> 00:08:00,841 whne we have a source that is a sinusoid, what you think about is the real part of 87 00:08:00,841 --> 00:08:05,830 the complex exponential. The output is going to be the real part 88 00:08:05,830 --> 00:08:12,051 of the same complex exponential with the amplitude modified by what the circuit 89 00:08:12,051 --> 00:08:15,697 does. That we found using, voltage divider. 90 00:08:15,697 --> 00:08:21,717 Well, how do you find this real part? Well, mathematically, we have a situation 91 00:08:21,717 --> 00:08:26,787 where we have something in cartesan form, and something in polar form. 92 00:08:26,787 --> 00:08:32,142 And you need to, figure out how that real part, the easiest possible way. 93 00:08:32,142 --> 00:08:38,422 And your answer is, is to convert the Cartesian, form to Polar form. 94 00:08:38,422 --> 00:08:44,147 So, that's what I've done. So, the, so this is the ratio of 95 00:08:44,147 --> 00:08:50,332 something in Cartesian coordinates. 1, and the denominator. 96 00:08:50,332 --> 00:08:55,611 J2 pi F nought RC plus 1, and as I showed you, the magnitude of a ratio is the 97 00:08:55,611 --> 00:09:01,438 ratio of the magnitudes. So one obviously has magnitude of one, and the magnitude 98 00:09:01,438 --> 00:09:06,835 at the bottom is the imaginary part squared plus the real part squared, and 99 00:09:06,835 --> 00:09:12,432 you take the square root of result. Okay, that's where that comes from. 100 00:09:12,432 --> 00:09:19,328 The angle of a ratio is the angle of the numerator minus the angle of the 101 00:09:19,328 --> 00:09:26,996 denominator, and the angle of 1 is 0, and so what we're left with is, that's the 102 00:09:26,996 --> 00:09:29,252 angle. Okay, so, 103 00:09:29,252 --> 00:09:34,159 mpw we need to simplify further. All I'm going to do is merge the two 104 00:09:34,159 --> 00:09:41,307 complex exponential terms, and the term, because the exponents add, and now taking 105 00:09:41,307 --> 00:09:45,915 the real part is easy. Because, this is just some real number, 106 00:09:45,915 --> 00:09:52,907 so far as the mathematics is concerned, the only thing complex is, is this, and 107 00:09:52,907 --> 00:09:59,562 that's e to the j theta. And what the real part of the e to the j 108 00:09:59,562 --> 00:10:08,512 theta? That's just a cosine, so our answer is, we're in the source. 109 00:10:08,512 --> 00:10:16,789 It goes a cosine 2 pi f of t Now put is also a sine wave with a same frequency 110 00:10:16,789 --> 00:10:22,349 but with a different phase and a different amplitude. 111 00:10:22,349 --> 00:10:27,344 Very nice, and it turns out that result is quite 112 00:10:27,344 --> 00:10:32,374 general. So let us preview again how we found it. 113 00:10:33,376 --> 00:10:36,822 I really want to make sure we get this down. 114 00:10:36,822 --> 00:10:42,323 So, the source was a sinusoid. I thought about it as the real part. 115 00:10:42,323 --> 00:10:48,645 I know by converting to impedances, that this is the way the amplitude of the 116 00:10:48,645 --> 00:10:55,012 output is related to the amplitude of the source, and over here, that is VN. 117 00:10:55,012 --> 00:11:00,269 Right, because that's the amplitude of that complex exponent. 118 00:11:00,269 --> 00:11:06,453 We now know that the alpha is the real part, where you put in how the circuit 119 00:11:06,453 --> 00:11:13,289 transformed the voltage amplitudes, and just finding real part is just like using 120 00:11:13,289 --> 00:11:18,520 the procedure you did. converting the polar form, gives us our 121 00:11:18,520 --> 00:11:22,530 final answer. Looks a little complicated, but we'll, 122 00:11:22,530 --> 00:11:25,826 understand what this means in just a second. 123 00:11:25,826 --> 00:11:29,014 We'll have a clearer interpretation of it. 124 00:11:29,014 --> 00:11:33,431 Now, let's just handle in a little bit more general case. 125 00:11:33,431 --> 00:11:37,465 Suppose there was a phase, a source out of phase. 126 00:11:37,465 --> 00:11:42,699 Well, the way that works out is now V in is Ae to the j phi. 127 00:11:42,699 --> 00:11:49,755 So it really is a complex number now. Using impedances, we're always going to 128 00:11:49,755 --> 00:11:56,994 find that the complex amplitude of the output is equal to something what that is 129 00:11:56,994 --> 00:12:03,728 the circuit does, times complex amplitude of the source. 130 00:12:03,728 --> 00:12:10,233 This something it does is called a transfer function. 131 00:12:10,233 --> 00:12:17,836 And, if you look back over what we've done, it always depends on frequency. 132 00:12:17,836 --> 00:12:17,836 So that's why I had to put the frequency of the source, f naught. 133 00:12:17,836 --> 00:12:22,441 And I'm just going to define that In general, it's the ratio of the complex 134 00:12:22,441 --> 00:12:28,012 amplitude and the output divided by the complex amplitude of the source. 135 00:12:28,012 --> 00:12:33,755 This captures, the transfer function captures what the circuit does to the 136 00:12:33,755 --> 00:12:39,042 amplitude and what it does to the phase. So, here's what's going on. 137 00:12:39,042 --> 00:12:44,547 We've found using impedances that that's the transfer function in Cartesian form, 138 00:12:44,547 --> 00:12:49,930 and as we've seen it's, much easier to deal with mathematically if we convert 139 00:12:49,930 --> 00:12:53,585 that to Polar form. This is routinely what we do for any 140 00:12:53,585 --> 00:12:56,943 circuit. Using impedances, we're going to wind up 141 00:12:56,943 --> 00:13:01,950 with a result in Cartesian form. We then have to do some Calculations to 142 00:13:01,950 --> 00:13:07,819 figure out what it is in polar form. And then, it gets very easy to figure out 143 00:13:07,819 --> 00:13:12,482 what the output is. So, when the input is that, the output is 144 00:13:12,482 --> 00:13:16,535 always this. For any circuit containing resistors, 145 00:13:16,535 --> 00:13:20,792 capacitors, and inductors. That's always the answer. 146 00:13:20,792 --> 00:13:25,477 This is very very general. So we now can see that assuming this 147 00:13:25,477 --> 00:13:31,097 source was a complex exponential, it actually had a big payoff because the 148 00:13:31,097 --> 00:13:34,942 circuit was linear. Now I'm going to point out some 149 00:13:34,942 --> 00:13:40,672 mathematical details that are very important, and that is, suppose we had a 150 00:13:40,672 --> 00:13:45,764 sine wave, a real sine? Okay, well the easiest way to think about 151 00:13:45,764 --> 00:13:49,349 sin is as the imaginary part of something. 152 00:13:49,349 --> 00:13:56,152 When everything goes through the superposition, still goes through because 153 00:13:56,152 --> 00:14:01,268 of the [INAUDIBLE] formula, and you also use imaginary part, and this is very 154 00:14:01,268 --> 00:14:06,208 important. So, however you define, the, source is 155 00:14:06,208 --> 00:14:13,111 being related to a complex exponential, either by an imaginary part or a real 156 00:14:13,111 --> 00:14:17,568 part. You use the same representation for the 157 00:14:17,568 --> 00:14:21,221 ouput. Sum when it influence that, now put as 158 00:14:21,221 --> 00:14:24,638 this, and and that's a very, again a general result. 159 00:14:24,638 --> 00:14:30,028 You can use either the real part or imaginary part, whichever one is easier, 160 00:14:30,028 --> 00:14:35,686 on the point out that you can actually work this the hard way if you will, you 161 00:14:35,686 --> 00:14:39,707 can assume a cosine is the imaginary part of something. 162 00:14:39,707 --> 00:14:44,114 Notice the presence of this because of the phase shift by power of 2, 163 00:14:44,114 --> 00:14:48,208 represent the sign as the real part of something if you want to. 164 00:14:48,208 --> 00:14:53,601 Should actually go back, over our little example, RC circuit example, once you try 165 00:14:53,601 --> 00:14:58,122 and using the imaginary part and see, see if you get the same answer. 166 00:14:58,122 --> 00:15:04,841 We've got using the real part except we have to change the amplitude of the 167 00:15:04,841 --> 00:15:10,208 source by e to the j pi over 2. You better get the same answer once you 168 00:15:10,208 --> 00:15:14,435 simplify things, and trust me, you will. Okay. 169 00:15:14,435 --> 00:15:20,306 So, here's the big picture. for any circuit we have, pretend the 170 00:15:20,306 --> 00:15:25,949 circuit is a complex exponential [INAUDIBLE] although it really isn't. 171 00:15:25,949 --> 00:15:29,757 Use impedances to find the transfer function. 172 00:15:29,757 --> 00:15:35,324 And now we express the source as the real part or imaginary part of a complex 173 00:15:35,324 --> 00:15:39,282 exponential. This works for sinusoidal sources. 174 00:15:39,282 --> 00:15:46,402 And the output is going to be the real part or the imaginary part, whichever one 175 00:15:46,402 --> 00:15:49,564 we assumed. Don't try to switch them, 176 00:15:49,564 --> 00:15:55,941 of the transfer function times the complex exponential of the source. 177 00:15:55,941 --> 00:16:03,053 That's the procedure using impedances to solve any circuit that has a sinusoidal 178 00:16:03,053 --> 00:16:06,633 source. So, let's go through this for an example, 179 00:16:06,633 --> 00:16:11,657 and we ought to try something bit more interesting than, RC circuit. 180 00:16:11,657 --> 00:16:16,428 And, what I am showing you here, is what's known as a RLC circuit. 181 00:16:16,428 --> 00:16:22,245 So, our circuit now has a capacitor, a resistor, and an inductor. 182 00:16:22,245 --> 00:16:28,069 The source is this, and I'm going to ask you two questions. 183 00:16:28,069 --> 00:16:35,840 I'm going to find the transfer function, and I'm going to find the output for this 184 00:16:35,840 --> 00:16:37,227 input. Okay. 185 00:16:37,227 --> 00:16:41,992 So we now know, we know what to do. So here we go. 186 00:16:41,992 --> 00:16:48,117 So the first thing we do is we take our circuit and replace it by one having 187 00:16:48,117 --> 00:16:52,807 impedances, for again these are conflict amplitudes. 188 00:16:52,807 --> 00:16:57,262 Now I've done a little notational thing for you here. 189 00:16:57,262 --> 00:17:02,939 The impedance of this one ferrite capacity is 1 over j 2 pi f. 190 00:17:02,939 --> 00:17:09,179 Well, to simplify the mathematics and manipulation of these formulas. 191 00:17:09,179 --> 00:17:14,900 As you'll see, going to see in a second, it gets pretty complicated. 192 00:17:14,900 --> 00:17:19,162 I like to think of j2 pi f as a single variable s, 193 00:17:19,162 --> 00:17:25,307 and then write the impedance that way as just being, of the capacitors being 1 194 00:17:25,307 --> 00:17:28,800 over S. What I'm going to do is, use current 195 00:17:28,800 --> 00:17:32,306 divider, voltage divider. those tools. 196 00:17:32,306 --> 00:17:35,560 And then, at the end, replace S as J2 pi F. 197 00:17:35,560 --> 00:17:40,987 So, the impedance of a 1 farad capacitor is 1 over s. 198 00:17:40,987 --> 00:17:50,341 The impedance of R2 of 2 ohm resistors is 2, and we, impedance of our 4 Henry 199 00:17:50,341 --> 00:17:55,800 conductor is 4s. Okay, so we now what to find V out. 200 00:17:55,800 --> 00:18:04,478 Okay, so The complex amplitude V-out , is pretty clear that this is a parallel 201 00:18:04,478 --> 00:18:13,632 combination series with that impedance. So, I use voltage divider, and I find 202 00:18:13,632 --> 00:18:23,504 that V-out over V-in is equal to 2, and parallel with 4s divided 2 and parallel 203 00:18:23,504 --> 00:18:33,288 with 4s plus 1 over s. Okay? So, now we multiply everything all 204 00:18:33,288 --> 00:18:37,918 out. We get 8s over 2 plus 4s divided 8s over 205 00:18:37,918 --> 00:18:45,912 2 plus 4s plus 1 divided s. Bet you didn't know that electrical 206 00:18:45,912 --> 00:18:54,912 engineers have to very fluent with fractions. 207 00:18:54,912 --> 00:18:58,927 This turns out to be one of the things that happens all the time. 208 00:18:58,927 --> 00:19:02,477 You can see now why it did not want to write j2 pi f over it, 209 00:19:02,477 --> 00:19:07,612 because it's complicated enough as it is. This happens all the time obviously. 210 00:19:07,612 --> 00:19:10,902 Alright. So, we can, do some things to simplify 211 00:19:10,902 --> 00:19:14,212 this. The first thing is, notice that this 212 00:19:14,212 --> 00:19:24,162 ratio has some 2's in it, we can cancel. We got 4 we got 1, make that 2, we got 4, 213 00:19:24,162 --> 00:19:28,612 we got 2. Now we can simplify this. 214 00:19:28,612 --> 00:19:36,890 So, I'm going to multiply top and bottom by s. Okay. you get rid of that. 215 00:19:36,890 --> 00:19:41,341 That puts an S square there. Oops, squared. 216 00:19:41,341 --> 00:19:48,096 And that's a squared there, and we multiply top and bottom by 1 plus 217 00:19:48,096 --> 00:19:52,340 2 S, which gets rid of it down and there and 218 00:19:52,340 --> 00:20:01,983 up there and this becomes 1 plus 2s. Okay, so, what we get, I know you can't 219 00:20:01,983 --> 00:20:08,963 read that. It's very difficult to read, so let me 220 00:20:08,963 --> 00:20:15,275 write it for you. It's 2s squared, I'm sorry, 4s squared 221 00:20:15,275 --> 00:20:21,897 divided by 4s squared plus 1 plus 2s. Okay? So, this is the name of the game. 222 00:20:21,897 --> 00:20:28,383 Use the parallel and the series rule. Here we use voltage divider, parallel 223 00:20:28,383 --> 00:20:35,697 rule, series rules and then we simplify it, and really makes your life a hole lot 224 00:20:35,697 --> 00:20:40,162 simpler to use s for j 2pi f. So, this is what you get, 225 00:20:40,162 --> 00:20:47,413 and now I can write it in terms of f, which is what you need to do in order to 226 00:20:47,413 --> 00:20:54,247 solve for our sinusoidal source. We need this h of f, and here it is in 227 00:20:54,247 --> 00:20:58,438 all its glory. So s squared is going to be 4 pi minus 4 228 00:20:58,438 --> 00:21:02,123 pi f squared times the 4 gives us the 16 etc. 229 00:21:03,154 --> 00:21:09,268 So this is what it is. So, at this point, what I would like to 230 00:21:09,268 --> 00:21:17,202 do is find the frequency of the source. So, the, the frequency, this has to be 231 00:21:17,202 --> 00:21:19,430 written as 2 pi. F, t. 232 00:21:19,430 --> 00:21:23,000 So, 2pi f, has to be a half for this example, 233 00:21:23,000 --> 00:21:29,129 which means the frequency, is 1 over 4pi. Kind of a strange frequency, but it 234 00:21:29,129 --> 00:21:33,469 really simplifies the calculation for this example. 235 00:21:33,469 --> 00:21:39,494 I now want to find the value of the transfer function at that frequency. 236 00:21:40,762 --> 00:21:46,797 And, like I said, this is really, makes this assumption of that frequency value, 237 00:21:46,797 --> 00:21:50,532 really makes, the calculations very simple, 238 00:21:50,532 --> 00:21:54,012 because the numerator just turns out to be minus 1. 239 00:21:54,012 --> 00:21:58,152 Put that same term here. This becomes j, and the 1 is 1. 240 00:21:58,152 --> 00:22:02,831 So we get minus 1 over J, and reciprocal of J is minus j. 241 00:22:02,831 --> 00:22:09,366 So we get J for an answer. So our transfer function at the frequency 242 00:22:09,366 --> 00:22:16,735 of the source, that's all we need to know, is what the transfer function is at 243 00:22:16,735 --> 00:22:23,836 the frequency of the source, is J, through the phase shift. 244 00:22:23,836 --> 00:22:31,140 So, now we get to figure out what the output is. 245 00:22:31,140 --> 00:22:41,917 So, I'm going to think about our sinusoid as being the imaginary part Since it's a 246 00:22:41,917 --> 00:22:47,727 sine wave, probably makes it easier. And it's 10, e to the jt over 2. 247 00:22:47,727 --> 00:22:54,582 Well, we know that the output is going to be the imaginary part of 10j, e to the j, 248 00:22:54,582 --> 00:22:58,227 t over 2. Where j is that transfer function 249 00:22:58,227 --> 00:23:05,717 evaluated at the frequency of the source. And, j, again, it's in Cartesian form. 250 00:23:05,717 --> 00:23:10,147 Convert to polar form, and that's our answer. 251 00:23:10,147 --> 00:23:16,217 And, I know from the trig identities, that that's just cosine. 252 00:23:16,217 --> 00:23:22,332 So, we stick in 10, it turns out it's a very special frequency. 253 00:23:22,332 --> 00:23:27,097 because what this really does,at that frequency is convert the sine to a 254 00:23:27,097 --> 00:23:31,737 cosine, and it as the same amplitude. That's really pretty surprising, but 255 00:23:31,737 --> 00:23:36,748 again I have chosen a frequency that makes the calculations easy, and that's 256 00:23:36,748 --> 00:23:41,595 basically why that happened. In general, as we know, it's going to 257 00:23:41,595 --> 00:23:47,288 produce a sine wave at the same frequency with a different phase, and a different 258 00:23:47,288 --> 00:23:50,935 amplitude. We choose a different, frequency, this 259 00:23:50,935 --> 00:23:54,582 amplitude, will, definitely be different than 10. 260 00:23:54,582 --> 00:23:55,393 Okay. Now. 261 00:23:55,393 --> 00:24:01,538 I've taught this course for many years, and I've learned what students do that 262 00:24:01,538 --> 00:24:06,370 isn't correct. So I want to talk little bit about what 263 00:24:06,370 --> 00:24:11,252 not to do. How to use impedances so here's the name 264 00:24:11,252 --> 00:24:16,582 of the game. You start with a circuit. We know the input, 265 00:24:16,582 --> 00:24:21,673 provided it is the imaginary part of something, How do we find the transfer 266 00:24:21,673 --> 00:24:27,516 function, and we evaluate it at the frequency of the source, and we now know 267 00:24:27,516 --> 00:24:33,359 since I used imaginary part here, that that is the imaginary part here and 268 00:24:33,359 --> 00:24:36,914 that's the correct approach. Okay. 269 00:24:36,914 --> 00:24:41,091 Transfer function evaluated at the frequency of the source. 270 00:24:41,091 --> 00:24:46,931 We get that from that formula times the, complex amplitude representing, the, 271 00:24:46,931 --> 00:24:51,185 source, with, along with it's complex exponential frequency, we take the 272 00:24:51,185 --> 00:24:54,438 imaginary part. Well, I have seen students do the 273 00:24:54,438 --> 00:24:58,932 following. They just say well, let's assign this 274 00:24:58,932 --> 00:25:04,257 little source, you told me it's just the transfer function times that. 275 00:25:04,257 --> 00:25:10,932 Well, that's clearly wrong. At least in this example know that the transfer 276 00:25:10,932 --> 00:25:17,286 function of that frequency is j. So why in the world would you put in a real 277 00:25:17,286 --> 00:25:22,437 value input and come out with something that's complex valued? 278 00:25:22,437 --> 00:25:28,959 This is wrong, you have to think of the source as a complex exponential, and then 279 00:25:28,959 --> 00:25:32,429 things go through. So this is not correct. 280 00:25:32,429 --> 00:25:37,902 Another thing that isn't very good is to mix the two things together. 281 00:25:37,902 --> 00:25:42,816 So, you just stick in the source, multiply by the transfer function of the 282 00:25:42,816 --> 00:25:45,927 right frequency, and take the measuring part. 283 00:25:45,927 --> 00:25:50,791 Now that's not what you have to do. There's this intermediate step, remember, 284 00:25:50,791 --> 00:25:54,025 pretending the source is a complex exponential. 285 00:25:54,025 --> 00:25:58,542 That's the right formula. This is wrong. 286 00:25:58,542 --> 00:26:03,142 Okay. So, I hope this helps. 287 00:26:03,142 --> 00:26:12,557 We now know a very general result. The output for any sinusoidal input of a 288 00:26:12,557 --> 00:26:19,089 circuit is going to be the, a sinusoid at the same frequency, and the 289 00:26:19,089 --> 00:26:25,185 way we find it is by converting to complex exponential form and then taking, 290 00:26:25,185 --> 00:26:31,092 multiplying by the transfer function, and taking the real or imaginary part, 291 00:26:31,092 --> 00:26:36,962 depending on how we chose to do. Learn more about, how to use impedances 292 00:26:36,962 --> 00:26:42,546 to figure out what these kind of transfer functions do, and to get a much more 293 00:26:42,546 --> 00:26:45,276 general, broader picture of circuits.