1 00:00:00,920 --> 00:00:03,652 Welcome back, and we'll get back into some linear circuits. 2 00:00:03,652 --> 00:00:07,910 Today we're going to be talking about something called Linearity. 3 00:00:07,910 --> 00:00:12,100 And linearity is probably just really fundamental to this whole course. 4 00:00:12,100 --> 00:00:15,160 It's actually why this course is Linear Circuits. 5 00:00:15,160 --> 00:00:18,615 And so today we'll be describing some, some of these mathematical concepts. 6 00:00:18,615 --> 00:00:21,812 Linearity, super position and something called homogeneity. 7 00:00:21,812 --> 00:00:26,297 So, in the previous lesson, we covered some things about Ohm's Law and leading 8 00:00:26,297 --> 00:00:30,259 to parallel and series combining resistors. 9 00:00:30,259 --> 00:00:34,475 And then there were also some labs identifying the root, the behavior that 10 00:00:34,475 --> 00:00:39,940 we seen mathematically does hold true to physical devices. 11 00:00:39,940 --> 00:00:43,090 Now we're going to get a little bit more into the analytical side of things, and 12 00:00:43,090 --> 00:00:46,040 some analysis techniques to help us to work with some more complicated 13 00:00:46,040 --> 00:00:51,606 electrical systems. In context of the course, we finished 14 00:00:51,606 --> 00:00:55,410 talking about resistors, now we move on to a topic called, Superposition. 15 00:00:55,410 --> 00:00:57,775 And this is going to be broken into two parts. 16 00:00:57,775 --> 00:01:00,960 And this is the first part. where we talk about linearity. 17 00:01:00,960 --> 00:01:03,900 And then the second part is superposition itself. 18 00:01:03,900 --> 00:01:07,500 Then after that we'll look at some systematic ways of getting systems of 19 00:01:07,500 --> 00:01:12,220 equations using the resistor circuits. And the Ohms law and Kirchoff's law that 20 00:01:12,220 --> 00:01:16,667 we've already covered. The objectives for this lesson are to 21 00:01:16,667 --> 00:01:19,769 allow you to identify the ideas of linearity, superposition and homogeneity, 22 00:01:19,769 --> 00:01:22,672 and to be able to describe what they mean. 23 00:01:22,672 --> 00:01:26,454 And then identify whether a system or a function exhibits the property of 24 00:01:26,454 --> 00:01:30,017 linearity. So, first of all, let's talk about why 25 00:01:30,017 --> 00:01:35,246 this course is called Linear Circuits. When we talk about circuits, there's a 26 00:01:35,246 --> 00:01:37,710 lot of different types of things that go into it. 27 00:01:37,710 --> 00:01:41,545 And you might be familiar with things like semiconductors or, or some of these 28 00:01:41,545 --> 00:01:45,242 concepts of transistors. And we don't cover any of those things in 29 00:01:45,242 --> 00:01:49,090 this class, because those types of devices are non-linear. 30 00:01:49,090 --> 00:01:52,090 There's also devices like resistors which are linear. 31 00:01:52,090 --> 00:01:55,775 The reason we covered them separately is that linear systems are a lot simpler to 32 00:01:55,775 --> 00:01:58,930 deal with. But they give us a good practice area to 33 00:01:58,930 --> 00:02:02,230 try using these analysis techniques that can then be applied to the more 34 00:02:02,230 --> 00:02:06,550 complicated systems. In order to also understand the more 35 00:02:06,550 --> 00:02:09,916 complicated systems with the non-linear devices, having a good foundation in 36 00:02:09,916 --> 00:02:14,400 linear devices is important. Since the behavior that we observe in the 37 00:02:14,400 --> 00:02:17,180 non-linear devices is highly dependent on the linear devices. 38 00:02:18,320 --> 00:02:21,606 In the devices that we cover in this class, resistors is the ones that we've 39 00:02:21,606 --> 00:02:24,844 covered so far. Also capacitors and inductors which we 40 00:02:24,844 --> 00:02:29,940 will be covering in subsequent lessons. All of these devices are linear devices. 41 00:02:29,940 --> 00:02:33,246 And so, that gives us some neat mathematical tricks that we can use in 42 00:02:33,246 --> 00:02:38,250 doing our circular analysis, and evaluating our systems. 43 00:02:38,250 --> 00:02:41,474 Any system that is composed of these linear devices and linear elements is 44 00:02:41,474 --> 00:02:45,840 said to be a linear system. These technique that we're going to be 45 00:02:45,840 --> 00:02:48,975 talking about today, apply to all of those type of circuits. 46 00:02:48,975 --> 00:02:56,128 So, lets define what linearity means. Linearity is a concept that requires two 47 00:02:56,128 --> 00:03:02,253 different properties to hold true. And so those two properties are here, 48 00:03:02,253 --> 00:03:07,892 superposition and homogeneity. For superposition, what that means is, if 49 00:03:07,892 --> 00:03:13,844 I have two inputs x1 and x2, and then I combine them by adding them together and 50 00:03:13,844 --> 00:03:22,330 then give those two inputs sum together as an input to some device. 51 00:03:22,330 --> 00:03:26,280 Some circuit, for example, I get this output. 52 00:03:26,280 --> 00:03:31,130 Where it's, whatever my device does, operated on the sum of those two inputs. 53 00:03:32,380 --> 00:03:37,330 Another way I could try and get an output is to take each input individually, and 54 00:03:37,330 --> 00:03:44,740 send it through the system. And then after I get my two results, this 55 00:03:44,740 --> 00:03:48,770 is going to be f of x1. Then here this is f of x2. 56 00:03:48,770 --> 00:03:54,110 I add my two results together to get another result. 57 00:03:54,110 --> 00:04:00,545 If these two values are the same for all inputs, then we say that the property of 58 00:04:00,545 --> 00:04:06,838 superposition holds. For homogeneity, what that means is that 59 00:04:06,838 --> 00:04:11,894 if I have an input x, and I multiply it by some constant, K. 60 00:04:11,894 --> 00:04:17,495 And then give that as an input to my system, my output looks like this. 61 00:04:17,495 --> 00:04:21,091 Another way I could get data is, if I take my input and send it through my 62 00:04:21,091 --> 00:04:24,873 system, and then after I go to results here, this is going to be f of x, I then 63 00:04:24,873 --> 00:04:32,290 multiply it by that same constant. I get another function. 64 00:04:32,290 --> 00:04:35,362 If these two values are equal, then we say that this function has the property 65 00:04:35,362 --> 00:04:41,036 of homogeneity. If both of these properties hold, we say 66 00:04:41,036 --> 00:04:48,595 that this system, this f, is linear. So, superposition to review is, if I add 67 00:04:48,595 --> 00:04:52,625 the two inputs, and then get my output, or if I get my two outputs and add them 68 00:04:52,625 --> 00:04:57,372 together afterwards. I get the same answers, the 69 00:04:57,372 --> 00:05:00,281 superposition. If I multiply by a constant before, or if 70 00:05:00,281 --> 00:05:05,570 I multiply it by a constant after, the same constant for any input. 71 00:05:05,570 --> 00:05:09,300 Then it's said to be, homogeneous. Or it has the property of homogeneity. 72 00:05:10,780 --> 00:05:13,130 Let's look at an example of something that is homogeneous. 73 00:05:14,210 --> 00:05:16,545 And that has the property of superposition. 74 00:05:16,545 --> 00:05:22,820 And that's Ohm's law. So, for this, for Ohm's law, we say that 75 00:05:22,820 --> 00:05:32,520 v, which I will let be a function of the current, i, is equal to i times R. 76 00:05:32,520 --> 00:05:37,008 So, if I want to check the property of superposition, that would be like saying, 77 00:05:37,008 --> 00:05:42,551 I have some i, so let's say that i is equal to 1 amp plus 2 amps. 78 00:05:42,551 --> 00:05:47,249 And I want to see if, if I make this 1 amp plus 2 amps, if that's the same as 79 00:05:47,249 --> 00:05:55,670 doing this one time with a 1 amp source, and again with a 2 amp source. 80 00:05:55,670 --> 00:06:04,776 The way that I can show that is if v of i1 plus i2 is given as an input, then 81 00:06:04,776 --> 00:06:14,196 that gives me i1 plus i2 times R, but I can distribute that out as R times i1 82 00:06:14,196 --> 00:06:29,098 plus R times i2, which is equal to v of i1 plus v of i2. 83 00:06:29,098 --> 00:06:33,386 And, so, it doesn't matter if I add my two currents before, and give those as my 84 00:06:33,386 --> 00:06:37,741 input to my function v, or if I give them each individually to my function v, and 85 00:06:37,741 --> 00:06:43,257 then add up afterwards, I get the same results. 86 00:06:43,257 --> 00:06:47,125 And so it doesn't matter if I add I get the same result. 87 00:06:47,125 --> 00:07:01,127 So, that means the superposition holds. Another thing I could do, is I could do 88 00:07:01,127 --> 00:07:05,549 the exact same thing for homogeneity, but instead of using i, 1 plus i2, we would 89 00:07:05,549 --> 00:07:14,768 say take this current and double it. And so if homogeneity holds then doubling 90 00:07:14,768 --> 00:07:18,130 my current here would mean that my voltage should double. 91 00:07:19,430 --> 00:07:21,870 And so, if you go through the analysis and follow kind of the same example, 92 00:07:21,870 --> 00:07:26,956 you'll see that homogeneity also holds. And so here is kind of an overview of the 93 00:07:26,956 --> 00:07:30,807 work that is used to do that. And so, you can always go through it in 94 00:07:30,807 --> 00:07:33,560 more detail and make sure that you understand all of the steps. 95 00:07:36,410 --> 00:07:38,728 Let's take a look at some things that are linear, and some things which are 96 00:07:38,728 --> 00:07:41,017 non-linear. Somethings that are kind of counter 97 00:07:41,017 --> 00:07:44,045 examples to linearity. So, the first one we looked at is f of x 98 00:07:44,045 --> 00:07:47,210 is equal to 0. And that's almost a trivial case, and so 99 00:07:47,210 --> 00:07:50,344 all of these by the way, I'm not going to derive them. 100 00:07:50,344 --> 00:07:53,644 But it's an excellent exercise to make sure that you understand the material by 101 00:07:53,644 --> 00:07:56,944 checking each of these things to see if you understand why these are linear, and 102 00:07:56,944 --> 00:08:01,939 why these are non-linear. So, f of x is equal to 0 is kind of the, 103 00:08:01,939 --> 00:08:07,971 the trivial example. This multiplying by some constant k, this 104 00:08:07,971 --> 00:08:13,398 is Ohm's Law, essentially. Of course Ohm's Law is actually an 105 00:08:13,398 --> 00:08:18,390 example of this specific case of this constant multiple. 106 00:08:18,390 --> 00:08:21,360 It turns out that if you take the differential, and if you take the 107 00:08:21,360 --> 00:08:25,240 derivative, that also works, is, is also linear. 108 00:08:25,240 --> 00:08:29,682 As is taking an integral. But for the integral, it has to be a 109 00:08:29,682 --> 00:08:34,490 definite integral, has to have these limits of integration. 110 00:08:34,490 --> 00:08:38,585 And I list here the indefinite integral as being a counter example, because if 111 00:08:38,585 --> 00:08:42,415 you do this, there's always this plus C term. 112 00:08:42,415 --> 00:08:44,952 Since this is the class of things, and you have to have that plus C, if you 113 00:08:44,952 --> 00:08:48,950 remember from calculus, you always have to have the plus C. 114 00:08:48,950 --> 00:08:54,200 And so, that kind of makes it non-linear, because x plus c is also non-linear. 115 00:08:54,200 --> 00:08:56,650 Also, things like x squared and sine of x are going to be non-linear. 116 00:08:56,650 --> 00:08:59,920 And so we go through as an example, as an exercise. 117 00:08:59,920 --> 00:09:02,640 And make sure that you understand why all these are linear and non-linear. 118 00:09:02,640 --> 00:09:05,385 And if you get stuck, it's a good, a, good time to ask question on the forms, 119 00:09:05,385 --> 00:09:08,160 to make sure you understand these concepts. 120 00:09:10,000 --> 00:09:12,760 So, summarized we introduced linear operators, superposition and homogeneity, 121 00:09:12,760 --> 00:09:15,320 and identified a little bit of both of those properties hold, then linearity 122 00:09:15,320 --> 00:09:19,264 holds. We identified if the operator is linear 123 00:09:19,264 --> 00:09:22,770 by defining those, or verifying those properties. 124 00:09:22,770 --> 00:09:25,060 And then finally we used linear operators. 125 00:09:25,060 --> 00:09:28,280 You can generate new linear operators by using other linear operators. 126 00:09:29,360 --> 00:09:32,540 I wanted to just remind you that there are quizzes placed at the end of the 127 00:09:32,540 --> 00:09:35,998 lectures. And so, make sure to stay for those 128 00:09:35,998 --> 00:09:38,950 quizzes. And give you a little bit of practice 129 00:09:38,950 --> 00:09:43,040 before you go and, and do the homework. And then finally be the graded quizzes. 130 00:09:44,330 --> 00:09:47,564 In the next lesson, we will be applying all the principles that we covered today 131 00:09:47,564 --> 00:09:51,353 to circuit analysis. And see how superposition can be used as 132 00:09:51,353 --> 00:09:53,580 an analysis technique. Until then.