1 00:00:00,000 --> 00:00:06,201 2 00:00:06,201 --> 00:00:07,310 KENDRA PUGH: Hi. 3 00:00:07,310 --> 00:00:09,800 Today, I'd like to talk to you about circuits. 4 00:00:09,800 --> 00:00:12,290 Last time, we finished up the LTIs, and signals, and 5 00:00:12,290 --> 00:00:16,140 systems, where we learned how to both model existing systems 6 00:00:16,140 --> 00:00:18,950 and predict their long-term behavior. 7 00:00:18,950 --> 00:00:22,620 But we haven't forayed into how to actually create systems 8 00:00:22,620 --> 00:00:23,600 in the physical world. 9 00:00:23,600 --> 00:00:26,510 We've created some amount of systems in software and made 10 00:00:26,510 --> 00:00:27,720 some brains for our robots. 11 00:00:27,720 --> 00:00:29,400 But if we want to make something in the physical 12 00:00:29,400 --> 00:00:32,110 world, then we probably have to come up with ways to model 13 00:00:32,110 --> 00:00:35,770 physical systems or use physical components. 14 00:00:35,770 --> 00:00:37,440 That starts our new model on circuits. 15 00:00:37,440 --> 00:00:40,060 Circuits are going to be our first foray into designing 16 00:00:40,060 --> 00:00:43,630 systems in the physical world, also designing systems using 17 00:00:43,630 --> 00:00:46,080 physical components. 18 00:00:46,080 --> 00:00:48,920 It's worth mentioning now that the information that you learn 19 00:00:48,920 --> 00:00:51,260 about circuits is good for more 20 00:00:51,260 --> 00:00:53,020 things than even circuits. 21 00:00:53,020 --> 00:00:57,770 You can use basic circuit diagrams and properties of 22 00:00:57,770 --> 00:01:01,390 circuits to model all sorts of kinds of systems, especially 23 00:01:01,390 --> 00:01:02,610 ones in the human body-- 24 00:01:02,610 --> 00:01:08,910 circulatory system, neurological system, different 25 00:01:08,910 --> 00:01:12,660 kinds of fluid flow, that kind of thing. 26 00:01:12,660 --> 00:01:16,170 In the next few videos, we'll go over how to represent 27 00:01:16,170 --> 00:01:20,150 circuits, and also cover some of the basic methods by which 28 00:01:20,150 --> 00:01:22,500 people solve circuits. 29 00:01:22,500 --> 00:01:25,150 We'll also introduce an element called an op-amp, and 30 00:01:25,150 --> 00:01:29,020 use that element in order to enable us to do things like 31 00:01:29,020 --> 00:01:30,595 modularity and abstraction from our circuits. 32 00:01:30,595 --> 00:01:33,940 33 00:01:33,940 --> 00:01:35,490 First, let's talk about representation. 34 00:01:35,490 --> 00:01:38,220 35 00:01:38,220 --> 00:01:41,030 In the general sense, when you come across a circuit diagram, 36 00:01:41,030 --> 00:01:42,150 you're going to see-- 37 00:01:42,150 --> 00:01:44,530 at the very broad level-- 38 00:01:44,530 --> 00:01:47,320 a bunch of elements and a bunch of connections between 39 00:01:47,320 --> 00:01:48,670 the elements. 40 00:01:48,670 --> 00:01:54,950 Those things will form loops and nodes. 41 00:01:54,950 --> 00:01:57,500 If you don't actually specify the elements, then your 42 00:01:57,500 --> 00:01:59,085 circuit diagram actually looks a whole lot 43 00:01:59,085 --> 00:02:00,060 like a block diagram. 44 00:02:00,060 --> 00:02:03,590 And in fact, block diagrams and circuit diagrams are very 45 00:02:03,590 --> 00:02:06,550 closely related in part because block diagrams are 46 00:02:06,550 --> 00:02:10,940 used to model feedback systems, which frequently are 47 00:02:10,940 --> 00:02:14,520 implemented using circuits. 48 00:02:14,520 --> 00:02:17,380 49 00:02:17,380 --> 00:02:21,580 In this course, we're going to be focusing on independent 50 00:02:21,580 --> 00:02:24,840 sources and resistors as the two major kinds of elements 51 00:02:24,840 --> 00:02:26,540 that we'll use in our circuits. 52 00:02:26,540 --> 00:02:29,020 We'll also use things like potentiometers, which are 53 00:02:29,020 --> 00:02:31,910 resistors that you can adjust, and op-amps. 54 00:02:31,910 --> 00:02:35,240 And we'll look at op-amps specifically in a later video. 55 00:02:35,240 --> 00:02:37,440 But I have one drawn up here just so you recognize it when 56 00:02:37,440 --> 00:02:38,480 you see it written. 57 00:02:38,480 --> 00:02:42,360 Note that it looks a whole lot like the block diagram symbol 58 00:02:42,360 --> 00:02:42,940 for a gain. 59 00:02:42,940 --> 00:02:46,370 And that's intentional, and we'll cover that later. 60 00:02:46,370 --> 00:02:50,690 But in the meantime, the other sources that we're going to be 61 00:02:50,690 --> 00:02:55,310 using are independent current, and voltage sources. 62 00:02:55,310 --> 00:02:57,710 We're going to use resistors to adjust the amount of 63 00:02:57,710 --> 00:03:00,307 voltage and current that we're actually dealing with and then 64 00:03:00,307 --> 00:03:03,220 sample either the current or the voltage at a particular 65 00:03:03,220 --> 00:03:05,370 point in our circuit to get the desired 66 00:03:05,370 --> 00:03:08,250 values that we're after. 67 00:03:08,250 --> 00:03:10,020 On a circuit diagram, when you're interested in the 68 00:03:10,020 --> 00:03:12,860 voltage drop across a particular element, you'll 69 00:03:12,860 --> 00:03:15,170 indicate it by putting a plus and minus sign. 70 00:03:15,170 --> 00:03:18,810 This also indicates the directionality 71 00:03:18,810 --> 00:03:20,530 of the voltage drop. 72 00:03:20,530 --> 00:03:23,000 Likewise, when you're interested in the current 73 00:03:23,000 --> 00:03:26,480 flowing through a particular element, you'll usually see an 74 00:03:26,480 --> 00:03:30,990 indication of it by labeling the current i, and then maybe 75 00:03:30,990 --> 00:03:35,660 i with some sort of subscript, and an arrow indicating the 76 00:03:35,660 --> 00:03:39,610 direction of current flow through that element so that 77 00:03:39,610 --> 00:03:41,690 you avoid making sign errors with the person that might be 78 00:03:41,690 --> 00:03:42,940 reading or writing your diagram. 79 00:03:42,940 --> 00:03:46,420 80 00:03:46,420 --> 00:03:47,510 A quick note here. 81 00:03:47,510 --> 00:03:50,090 This is the reason that electrical engineers use j to 82 00:03:50,090 --> 00:03:52,630 symbolize values in the complex plane. 83 00:03:52,630 --> 00:03:56,680 It's because i is used in particular 84 00:03:56,680 --> 00:03:57,960 for values of current. 85 00:03:57,960 --> 00:04:05,950 86 00:04:05,950 --> 00:04:08,630 Let's review Kirchhoff's voltage laws and Kirchhoff's 87 00:04:08,630 --> 00:04:09,270 current laws. 88 00:04:09,270 --> 00:04:11,550 You've probably covered this in 8.02, electricity and 89 00:04:11,550 --> 00:04:14,400 magnetism, or possibly in an AP physics class. 90 00:04:14,400 --> 00:04:16,358 But we're going to go over it really fast right now. 91 00:04:16,358 --> 00:04:19,110 92 00:04:19,110 --> 00:04:22,200 Kirchhoff's voltage law is that the voltage drop around a 93 00:04:22,200 --> 00:04:23,940 loop is equal to 0. 94 00:04:23,940 --> 00:04:27,650 Or if you take the voltage drop across a particular loop 95 00:04:27,650 --> 00:04:31,450 in your circuit, the sum of those voltage drop 96 00:04:31,450 --> 00:04:33,430 is going to be 0. 97 00:04:33,430 --> 00:04:35,530 Let's demonstrate on this diagram. 98 00:04:35,530 --> 00:04:36,780 Or, I'll demonstrate on this diagram. 99 00:04:36,780 --> 00:04:43,970 100 00:04:43,970 --> 00:04:45,690 Say the voltage drop across this element 101 00:04:45,690 --> 00:04:46,960 is equal to V, right? 102 00:04:46,960 --> 00:04:48,250 Doesn't matter what it is. 103 00:04:48,250 --> 00:04:49,980 We're going to stick with that. 104 00:04:49,980 --> 00:04:53,440 105 00:04:53,440 --> 00:05:00,400 The voltage drop across these elements, if I were to move 106 00:05:00,400 --> 00:05:07,340 around this loop, is going to sum to 0. 107 00:05:07,340 --> 00:05:10,890 108 00:05:10,890 --> 00:05:16,540 Note that if I'm tracing out my voltage drop across this 109 00:05:16,540 --> 00:05:21,070 loop, I'm actually moving through this voltage source in 110 00:05:21,070 --> 00:05:24,320 the direction opposite of its indicated potential. 111 00:05:24,320 --> 00:05:32,700 So when I move through this voltage source, I'm going to 112 00:05:32,700 --> 00:05:38,630 account for its value as negative V. As I work my way 113 00:05:38,630 --> 00:05:41,440 around the rest of the circuit, the voltage drop 114 00:05:41,440 --> 00:05:46,930 across these elements is going to sum to V. 115 00:05:46,930 --> 00:05:51,640 This is true for all loops in my circuit. 116 00:05:51,640 --> 00:05:59,890 So any loop that includes V, the elements I encounter as a 117 00:05:59,890 --> 00:06:09,130 consequence of moving around that loop are going to have 118 00:06:09,130 --> 00:06:13,330 voltage drop equal and opposite to the value I get by 119 00:06:13,330 --> 00:06:14,580 moving through V in this direction. 120 00:06:14,580 --> 00:06:21,300 121 00:06:21,300 --> 00:06:25,870 This loop counts, too, but it doesn't include V. All this 122 00:06:25,870 --> 00:06:28,790 loop tells me is that the voltage drop across this 123 00:06:28,790 --> 00:06:31,110 element is equivalent to the voltage 124 00:06:31,110 --> 00:06:32,880 drop across this element. 125 00:06:32,880 --> 00:06:36,380 Or, the voltage drop in this direction across that element 126 00:06:36,380 --> 00:06:38,110 is equal to the voltage drop in this 127 00:06:38,110 --> 00:06:41,810 direction across this element. 128 00:06:41,810 --> 00:06:48,120 129 00:06:48,120 --> 00:06:49,380 That's Kirchhoff's voltage law. 130 00:06:49,380 --> 00:06:51,700 Kirchhoff's current law is that the current flow into a 131 00:06:51,700 --> 00:06:53,830 particular node is equal to 0. 132 00:06:53,830 --> 00:06:58,360 Or, if you take all of the current flows in and out of a 133 00:06:58,360 --> 00:07:01,150 particular node and sum them, they should sum to 0. 134 00:07:01,150 --> 00:07:05,930 135 00:07:05,930 --> 00:07:07,790 I've actually got the same set up here. 136 00:07:07,790 --> 00:07:09,700 I'm not going to use a current divider. 137 00:07:09,700 --> 00:07:18,870 138 00:07:18,870 --> 00:07:21,510 I'm interested in the current flowing over this element. 139 00:07:21,510 --> 00:07:23,400 It's actually the same as the current flowing over this 140 00:07:23,400 --> 00:07:26,210 element because resistance doesn't change current, 141 00:07:26,210 --> 00:07:30,740 resistors flowing through a resistor should 142 00:07:30,740 --> 00:07:31,990 not change the current. 143 00:07:31,990 --> 00:07:35,330 144 00:07:35,330 --> 00:07:39,220 So this is still the same i. 145 00:07:39,220 --> 00:07:40,470 Here's my node. 146 00:07:40,470 --> 00:07:42,660 147 00:07:42,660 --> 00:07:45,730 The current flowing in this direction and in this 148 00:07:45,730 --> 00:07:50,520 direction, if I took the linear combination of these 149 00:07:50,520 --> 00:07:55,030 two currents, they would be equal in value to the current 150 00:07:55,030 --> 00:07:56,280 flowing into this node. 151 00:07:56,280 --> 00:08:09,250 152 00:08:09,250 --> 00:08:10,640 When I'm looking at the current flowing through a 153 00:08:10,640 --> 00:08:14,610 particular node, I pick a direction. 154 00:08:14,610 --> 00:08:17,730 It's usually arbitrary. 155 00:08:17,730 --> 00:08:18,450 I pick a direction. 156 00:08:18,450 --> 00:08:21,580 It's arbitrary which direction I pick. 157 00:08:21,580 --> 00:08:24,200 Typically, you pick currents flowing into the 158 00:08:24,200 --> 00:08:25,450 node as being positive. 159 00:08:25,450 --> 00:08:28,130 160 00:08:28,130 --> 00:08:30,960 I sum up all the currents, and I set that equal to 0. 161 00:08:30,960 --> 00:08:32,210 So in this case--. 162 00:08:32,210 --> 00:08:43,059 163 00:08:43,059 --> 00:08:44,309 Or-- 164 00:08:44,309 --> 00:08:51,950 165 00:08:51,950 --> 00:08:53,200 pretty simple. 166 00:08:53,200 --> 00:08:57,670 167 00:08:57,670 --> 00:09:00,390 Let's practice on this particular circuit. 168 00:09:00,390 --> 00:09:03,800 169 00:09:03,800 --> 00:09:06,640 One thing to note is that when you're solving circuits in the 170 00:09:06,640 --> 00:09:10,600 general sense, both when you want TA help and when you're 171 00:09:10,600 --> 00:09:15,620 solving for a mid-term and want partial credit, you want 172 00:09:15,620 --> 00:09:22,980 to label all of your nodes, all of your elements, and all 173 00:09:22,980 --> 00:09:24,940 of the currents that you're interested in solving. 174 00:09:24,940 --> 00:09:28,480 See, I've got my voltage drop across this resistor, this 175 00:09:28,480 --> 00:09:33,760 resistor, and this resistor labeled, as well as these 176 00:09:33,760 --> 00:09:35,520 currents, which I'll also be solving for. 177 00:09:35,520 --> 00:09:40,240 178 00:09:40,240 --> 00:09:42,260 The first thing that I would do when approaching this 179 00:09:42,260 --> 00:09:49,680 problem is attempt to reduce this circuit to something that 180 00:09:49,680 --> 00:09:52,280 is a little bit simpler. 181 00:09:52,280 --> 00:09:56,170 The first thing that I'm going to do is try to figure out how 182 00:09:56,170 --> 00:09:59,670 to change these two resistors in parallel into a single 183 00:09:59,670 --> 00:10:02,830 resistor and still have an equivalent circuit. 184 00:10:02,830 --> 00:10:06,230 That'll allow me to solve for I1. 185 00:10:06,230 --> 00:10:10,700 There will be 0 nodes in my system. 186 00:10:10,700 --> 00:10:12,570 I'll just have one single loop. 187 00:10:12,570 --> 00:10:21,050 And the current through the system will just be V/R. 188 00:10:21,050 --> 00:10:23,550 So if I'm just looking at these two resistors, I have 189 00:10:23,550 --> 00:10:24,800 resistors in parallel. 190 00:10:24,800 --> 00:10:27,260 191 00:10:27,260 --> 00:10:29,230 In the general sense, the way to solve for resistors in 192 00:10:29,230 --> 00:10:39,330 parallel is to take the inverse of the 193 00:10:39,330 --> 00:10:40,580 sum of their inverses. 194 00:10:40,580 --> 00:10:48,750 195 00:10:48,750 --> 00:10:51,140 When you only have two resistors, you can typically 196 00:10:51,140 --> 00:10:58,900 cheat by saying that this is equal to their 197 00:10:58,900 --> 00:11:00,960 product over their sum. 198 00:11:00,960 --> 00:11:06,270 199 00:11:06,270 --> 00:11:08,620 I'm going to redraw my current understanding of the circuit. 200 00:11:08,620 --> 00:11:43,700 201 00:11:43,700 --> 00:11:50,410 The other stuff that I've saved myself is that because 202 00:11:50,410 --> 00:11:52,680 these resistors are in parallel, 203 00:11:52,680 --> 00:11:54,280 they're a current divider. 204 00:11:54,280 --> 00:11:59,780 They take the current in and divide it two ways determined 205 00:11:59,780 --> 00:12:03,650 by the ratio between these two values. 206 00:12:03,650 --> 00:12:05,940 The thing I'm actually interested in expressing is 207 00:12:05,940 --> 00:12:10,740 that V2 and V3 are the same value. 208 00:12:10,740 --> 00:12:13,190 When you have a current divider, the voltage drop 209 00:12:13,190 --> 00:12:17,590 across all elements in the current divider are the same. 210 00:12:17,590 --> 00:12:22,590 So the value of V here is going to be both V2 and V3. 211 00:12:22,590 --> 00:12:37,800 212 00:12:37,800 --> 00:12:41,720 2R plus 6/5 R. I'm going to go with 16/5 R for now. 213 00:12:41,720 --> 00:13:11,510 214 00:13:11,510 --> 00:13:19,330 I've solved for I. 215 00:13:19,330 --> 00:13:22,160 At this point, I have a voltage divider, which means 216 00:13:22,160 --> 00:13:23,870 that the current flowing through this part of the 217 00:13:23,870 --> 00:13:25,830 system is going to be the same. 218 00:13:25,830 --> 00:13:33,150 But the voltage drop across this element versus this 219 00:13:33,150 --> 00:13:35,830 element is going to be proportional to the ratio 220 00:13:35,830 --> 00:13:37,320 between these two values. 221 00:13:37,320 --> 00:14:03,530 222 00:14:03,530 --> 00:14:08,770 V1 is going to be the amount of the total resistance in 223 00:14:08,770 --> 00:14:16,750 this simple circuit that this resistor contributes over the 224 00:14:16,750 --> 00:14:18,450 entire resistance in the system. 225 00:14:18,450 --> 00:14:28,350 Or, 10/5 R over 16/5 R, which is 10/16 R, or 5/8 R. Or, it's 226 00:14:28,350 --> 00:14:52,070 going to be 5/8 V. 227 00:14:52,070 --> 00:14:53,447 Same thing happens with V2. 228 00:14:53,447 --> 00:15:12,840 229 00:15:12,840 --> 00:15:15,530 Note that these two values should sum to V in order to 230 00:15:15,530 --> 00:15:17,000 maintain Kirchoff's voltage law. 231 00:15:17,000 --> 00:15:21,710 232 00:15:21,710 --> 00:15:22,960 We've also found V3. 233 00:15:22,960 --> 00:15:25,900 234 00:15:25,900 --> 00:15:30,390 So the two things that we have to find are I2 and I3. 235 00:15:30,390 --> 00:15:54,640 236 00:15:54,640 --> 00:15:56,310 Here, I've just done Kirchoff's current 237 00:15:56,310 --> 00:15:57,560 law for this node. 238 00:15:57,560 --> 00:16:02,090 239 00:16:02,090 --> 00:16:06,800 Because I'm working with a current divider, I can break 240 00:16:06,800 --> 00:16:11,250 up the total current flowing into that node into the number 241 00:16:11,250 --> 00:16:16,110 of parts equal to the sum of these values and then 242 00:16:16,110 --> 00:16:17,780 distribute them. 243 00:16:17,780 --> 00:16:18,760 And then, that's [? inappropriate ?] 244 00:16:18,760 --> 00:16:22,450 given that less resistance means more current. 245 00:16:22,450 --> 00:16:23,990 What do I mean by that? 246 00:16:23,990 --> 00:16:29,040 Well, I mean that here my current is equal to 5/16 V 247 00:16:29,040 --> 00:16:45,580 over R. I2 is going to be equal to this value over the 248 00:16:45,580 --> 00:16:52,190 sum of these two values times I1. 249 00:16:52,190 --> 00:17:03,340 250 00:17:03,340 --> 00:17:04,590 Likewise--. 251 00:17:04,590 --> 00:17:16,970 252 00:17:16,970 --> 00:17:18,220 And just to simplify--. 253 00:17:18,220 --> 00:17:36,610 254 00:17:36,610 --> 00:17:38,700 That concludes my tutorial on circuits. 255 00:17:38,700 --> 00:17:41,550 Next time, we'll talk about other ways we can solve this 256 00:17:41,550 --> 00:17:44,650 circuit, and then we'll end up talking about op-amps. 257 00:17:44,650 --> 00:17:45,900