1 00:00:02,380 --> 00:00:03,730 Welcome back. I'm Nathan Parrish. 2 00:00:03,730 --> 00:00:07,262 And again, this is linear circuits. Today, we're going to be talking about 3 00:00:07,262 --> 00:00:10,616 power and energy. So first of all, we're going to be trying 4 00:00:10,616 --> 00:00:14,472 to calculate power and energy. Describing the difference between power 5 00:00:14,472 --> 00:00:17,093 and energy. Using the conservation of energy to find 6 00:00:17,093 --> 00:00:20,157 an unknown energy. And using power to calculate an unknown 7 00:00:20,157 --> 00:00:25,060 current or unknown voltage. So from our previous class, we talked 8 00:00:25,060 --> 00:00:28,560 about voltage, this kind of this electric potential based upon the distribution of 9 00:00:28,560 --> 00:00:32,570 charge inside of a device. And then we looked at the battery. 10 00:00:32,570 --> 00:00:35,570 And how it could be charging or discharging, based upon the direction of 11 00:00:35,570 --> 00:00:39,135 the current flow. But in the same context, now we can start 12 00:00:39,135 --> 00:00:42,995 talking about power and energy. And, in the next class, we will look at 13 00:00:42,995 --> 00:00:46,044 some basic circuit diagrams. But actually today, we're going to be 14 00:00:46,044 --> 00:00:51,760 doing our first bit of circuit analysis. Lesson objectives are to calculate power 15 00:00:51,760 --> 00:00:56,150 from an energy function,and to calculate energy from the power function. 16 00:00:56,150 --> 00:00:59,768 To use the conservation of energy to find a power of an unknown device, calculate 17 00:00:59,768 --> 00:01:03,062 power from voltage and current, and finally to find a voltage or a current 18 00:01:03,062 --> 00:01:07,345 for a device with an unknown, for a known power. 19 00:01:07,345 --> 00:01:12,493 So in talking with, about power, power is measured in units of watts and power is 20 00:01:12,493 --> 00:01:18,660 essentially the rate at which energy is being consumed. 21 00:01:18,660 --> 00:01:22,480 Energy being measured in joules. So power is in joules per second. 22 00:01:22,480 --> 00:01:24,975 And one watt is equal to one joule per second. 23 00:01:24,975 --> 00:01:29,915 Now, we can actually use a little bit of calculus to help us do some calculation 24 00:01:29,915 --> 00:01:33,786 about power. Power is going to be the change in energy 25 00:01:33,786 --> 00:01:37,317 with respect to time. Or we can use the chain rule, this sort 26 00:01:37,317 --> 00:01:41,282 of, the change in energy with respect to charge times the change in charge with 27 00:01:41,282 --> 00:01:46,380 respect to time, is going to be equivalent. 28 00:01:46,380 --> 00:01:49,140 And so, if we have energy divided by charge, you might remember from last 29 00:01:49,140 --> 00:01:53,262 class, that's voltage. And if I have charge with respect to 30 00:01:53,262 --> 00:01:58,325 time, that's going to be current. So it turns out that we can calculate 31 00:01:58,325 --> 00:02:02,320 electric power by taking a voltage and multiplying it by a current. 32 00:02:04,190 --> 00:02:07,765 We can also find an energy by integrating the power with respect to time, since 33 00:02:07,765 --> 00:02:12,760 power is the rate at, at which energy is being consumed or generated. 34 00:02:12,760 --> 00:02:15,343 So all we have to do is integrate with respect to time, and then we can find our 35 00:02:15,343 --> 00:02:19,252 energy. In calculations, we're going to be using 36 00:02:19,252 --> 00:02:23,010 w to represent energy. And we're going to use p to represent 37 00:02:23,010 --> 00:02:29,440 power. One of the things that students sometimes 38 00:02:29,440 --> 00:02:32,795 forget that's actually a very useful tool when you're doing analysis, is 39 00:02:32,795 --> 00:02:36,970 remembering that energy cannot instantaneously change. 40 00:02:36,970 --> 00:02:40,751 The power can, it's not a problem. So we can look at this example. 41 00:02:40,751 --> 00:02:44,279 Here we have an energy curve, where the energy is being consumed, the energy 42 00:02:44,279 --> 00:02:48,087 decreases and decreases and decreases until this point, where there's no energy 43 00:02:48,087 --> 00:02:52,600 left. And now, it is regenerated. 44 00:02:52,600 --> 00:02:57,167 And remembering that we can calculate power by taking the derivative of energy. 45 00:02:57,167 --> 00:03:00,400 But we know that this derivative is basically the slope of the slide and so 46 00:03:00,400 --> 00:03:06,721 we are basically just going down one. So oue power is negative 1 watt. 47 00:03:06,721 --> 00:03:11,900 Over here the slope is now positive 1. So we have 1 watt. 48 00:03:11,900 --> 00:03:15,963 At this point in the middle, it's kind of a mathematical problem. 49 00:03:15,963 --> 00:03:19,733 There is no derivative here because if we come from one side or the other, we get 50 00:03:19,733 --> 00:03:23,206 two different answers. And so it's kind of undefined what the 51 00:03:23,206 --> 00:03:26,827 derivative happens to to be here. But this only happens at one isolated 52 00:03:26,827 --> 00:03:30,820 point, and so it really doesn't matter, as far as we're concerned. 53 00:03:30,820 --> 00:03:33,400 So here we just used some circles to identify that we don't have a defined 54 00:03:33,400 --> 00:03:37,490 amount of power. Now if we take an integral, of the power, 55 00:03:37,490 --> 00:03:41,211 we get the energy. Now suppose that we wanted this energy, 56 00:03:41,211 --> 00:03:44,690 to change, instantaneously in time. Or just do jumps. 57 00:03:44,690 --> 00:03:48,520 What would have to happen? Well, over a tiny little slice, if we 58 00:03:48,520 --> 00:03:52,420 remember the definition of an integral, we have to have an inordinately large 59 00:03:52,420 --> 00:03:57,210 amount of power. So large, in fact, that as we make the 60 00:03:57,210 --> 00:04:00,632 area around it smaller and smaller, the value has to overcome that ever 61 00:04:00,632 --> 00:04:05,853 decreasing window of space. Which basically means that we have to 62 00:04:05,853 --> 00:04:09,105 have infinite power. Well infinite power is not something that 63 00:04:09,105 --> 00:04:11,564 we can have. So it turns out that we cannot have the 64 00:04:11,564 --> 00:04:15,181 energy change instantaneously. But power can change instantaneously as 65 00:04:15,181 --> 00:04:18,478 much as it likes. Now suppose that you run a power company 66 00:04:18,478 --> 00:04:23,440 and you want to charge your customers. Well, how are you going to charge them? 67 00:04:23,440 --> 00:04:25,859 Are you going to charge them by the amount of power that they are using or 68 00:04:25,859 --> 00:04:27,916 the amount of energy? And why? 69 00:04:27,916 --> 00:04:30,320 I want you to think about it for a minute. 70 00:04:30,320 --> 00:04:32,185 In fact, I'm going to even stick up a little sign. 71 00:04:32,185 --> 00:04:34,627 A little pause, to let you know that it's okay to pause the video and take a little 72 00:04:34,627 --> 00:04:37,180 bit of time to think about it before you answer. 73 00:04:42,500 --> 00:04:46,198 So what did you come up with? It turns out, if you look at your power 74 00:04:46,198 --> 00:04:49,030 bill, you're probably going to see something about your power company 75 00:04:49,030 --> 00:04:54,108 charging you in units of kilowatt hours. And this is actually a measurement of 76 00:04:54,108 --> 00:04:56,293 energy. It seems to be a little bit easier to 77 00:04:56,293 --> 00:05:01,030 work with than joules, since a kilowatt hour is equal to about 360 joules. 78 00:05:01,030 --> 00:05:03,640 And so it's just an easier number to work with. 79 00:05:03,640 --> 00:05:06,012 But essentially, it's energy. And why energy? 80 00:05:06,012 --> 00:05:08,784 Well that's because they don't really care about what rate you're using the 81 00:05:08,784 --> 00:05:10,988 power. What they want to know is how what, or 82 00:05:10,988 --> 00:05:14,060 how much energy you're actually consuming. 83 00:05:14,060 --> 00:05:16,480 So it's important for them to keep track of that. 84 00:05:16,480 --> 00:05:19,952 But we shouldn't lose complete identity of power because it could become a 85 00:05:19,952 --> 00:05:23,245 problem there as well. Suppose that everybody in the city 86 00:05:23,245 --> 00:05:26,215 simultaneously decided to run their vacuum cleaners and air conditioners all 87 00:05:26,215 --> 00:05:28,993 at once. Well, the power company might have some 88 00:05:28,993 --> 00:05:31,686 problems. They wouldn't be able to generate enough 89 00:05:31,686 --> 00:05:35,730 power that instant to be able to provide for everybody at once. 90 00:05:35,730 --> 00:05:38,396 Even though the energy wasn't too much for it to provide, it's the rate that 91 00:05:38,396 --> 00:05:41,730 becomes a problem. So, in practice, your power company kind 92 00:05:41,730 --> 00:05:45,219 of looks at both. But when you're charged, you're going to 93 00:05:45,219 --> 00:05:49,040 be paying for energy. Another very useful property that we can 94 00:05:49,040 --> 00:05:52,580 use, is something called the conservation of energy. 95 00:05:52,580 --> 00:05:56,399 Just like most closed systems, the total amount of energy that is being provided 96 00:05:56,399 --> 00:06:01,210 or consumed by the system, is a constant value, in time. 97 00:06:01,210 --> 00:06:04,498 No energy is created or destroyed in the system. 98 00:06:04,498 --> 00:06:08,528 But, we actually are going to find it more convenient, generally, to work with 99 00:06:08,528 --> 00:06:12,863 power rather than energy. And so let's take a look at a little bit 100 00:06:12,863 --> 00:06:16,875 of a derivation here. We sum up the total energy in a system. 101 00:06:16,875 --> 00:06:21,228 We're going to get some constant value. Now what we can do then is take the time 102 00:06:21,228 --> 00:06:26,119 derivative of this value and if I take a time derivative of some constant, that's 103 00:06:26,119 --> 00:06:32,242 going to be equal to zero. And it turns out that as long as this sum 104 00:06:32,242 --> 00:06:36,922 is over a finite number of energies, we can interchange the derivative and the 105 00:06:36,922 --> 00:06:43,108 sum to give us this, where we're summing up the derivatives. 106 00:06:43,108 --> 00:06:48,076 Well these derivatives are just powers, individual powers, and we're summing all 107 00:06:48,076 --> 00:06:53,696 of these powers up and we then get zero. So what does this actually mean in 108 00:06:53,696 --> 00:06:56,645 practice? Well, what it really means is that 109 00:06:56,645 --> 00:07:00,803 because energy is conserved, we know that the rate of power which a system is 110 00:07:00,803 --> 00:07:08,000 consuming or generating energy, has to be 0 overall, throughout the system. 111 00:07:08,000 --> 00:07:10,806 So if we sum up all of the powers that are being consumed and all of the power 112 00:07:10,806 --> 00:07:15,930 that's being generated in the system at any point in time, it always has to be 0. 113 00:07:15,930 --> 00:07:17,735 So this is the excellent way to check your work. 114 00:07:17,735 --> 00:07:20,315 It also turns out to be something useful on analysis, if we happen to know 115 00:07:20,315 --> 00:07:25,634 something about the power in the system. Again reference directions are going to 116 00:07:25,634 --> 00:07:28,480 come up. like I said, it's something that students 117 00:07:28,480 --> 00:07:31,270 really have challenges with, especially at first, keeping the reference 118 00:07:31,270 --> 00:07:35,292 directions straight. And because we're taking a product of a 119 00:07:35,292 --> 00:07:38,505 current and a voltage, if we just naively multiply them together, we can get 120 00:07:38,505 --> 00:07:42,405 different answers depending upon how they're related. 121 00:07:42,405 --> 00:07:46,860 And as far as we're concerned, there's only two real configurations to consider. 122 00:07:46,860 --> 00:07:51,404 The first configuration is where our current arrow is pointing from our plus 123 00:07:51,404 --> 00:07:55,600 to our minus. And the other is when our current arrow 124 00:07:55,600 --> 00:08:00,729 is going from our minus to our plus. So, then we'll start off by looking at 125 00:08:00,729 --> 00:08:04,270 the one on the left, where the arrow goes from the plus to the minus. 126 00:08:04,270 --> 00:08:07,450 We're going to calculate power by multiplying the current and the voltage. 127 00:08:09,210 --> 00:08:13,565 What this ends up doing then, is it means that positive power means power's being 128 00:08:13,565 --> 00:08:17,427 consumed, not generated. This might be counterintuitive to what 129 00:08:17,427 --> 00:08:19,875 you might think. Thinking that negative would be that it's 130 00:08:19,875 --> 00:08:23,440 being used and positive would be that it's being generated. 131 00:08:23,440 --> 00:08:26,410 But we're actually going to have positive power being consumed power. 132 00:08:26,410 --> 00:08:29,010 And that's the general trend that we're going to use. 133 00:08:29,010 --> 00:08:32,034 And so consequently, if we flip the references, where the arrow's going the 134 00:08:32,034 --> 00:08:36,970 other way, to keep the values consistent, we have to multiply by a negative 1. 135 00:08:36,970 --> 00:08:39,763 So in this case, power is going to be equal to minus i times v, minus the 136 00:08:39,763 --> 00:08:43,890 current times the voltage. So be very careful when you're 137 00:08:43,890 --> 00:08:47,001 calculating power to look. Is my arrow going from the plus to the 138 00:08:47,001 --> 00:08:49,157 minus? Just multiply them together. 139 00:08:49,157 --> 00:08:52,542 Is it going from the minus to the plus? Remember to switch to the reference 140 00:08:52,542 --> 00:08:55,235 directions so you're multiplying by negative one. 141 00:08:55,235 --> 00:08:58,541 So now we're going to look at a practical, practical problem, and do an 142 00:08:58,541 --> 00:09:02,270 analysis. Now suppose we have a light bulb. 143 00:09:02,270 --> 00:09:06,862 And we have 120 va, volts, of voltage, going to the light bulb to power it. 144 00:09:06,862 --> 00:09:09,647 And this particular light bulb is rated 60 watts. 145 00:09:09,647 --> 00:09:13,510 Now suppose we want to know the current that is flowing through the light bulb. 146 00:09:13,510 --> 00:09:17,110 Well at this point we have all the tools we need to be able to solve this problem. 147 00:09:17,110 --> 00:09:21,540 So, I want you to take a look at the system and try and solve it yourself. 148 00:09:21,540 --> 00:09:24,075 And so I'm going to go ahead and put up a pause button and then we will continue on 149 00:09:24,075 --> 00:09:26,970 after you've had a chance to try to solve it yourself. 150 00:09:32,440 --> 00:09:37,208 Okay, so we have 60 watts of power. And we notice that the arrow for current 151 00:09:37,208 --> 00:09:41,530 is going from the plus to the minus. So in this configuration, power p is 152 00:09:41,530 --> 00:09:44,565 equal to i times v, current times voltage. 153 00:09:44,565 --> 00:09:52,150 So that means the power, 60 watts, is equal to 120 volts times i. 154 00:09:52,150 --> 00:09:56,140 Where i is equal to 60 watts divided by 120 volts. 155 00:09:56,140 --> 00:09:59,150 So the current is equal to one half of an m. 156 00:09:59,150 --> 00:10:01,628 Now, if you were able to solve this problem, it might be a good thing to go 157 00:10:01,628 --> 00:10:04,442 onto the forums and post, because you'll notice that we're not going to actually 158 00:10:04,442 --> 00:10:07,950 show the answer to the solution here on the slide. 159 00:10:10,230 --> 00:10:13,366 To summarize, we've describe the relationship between power and energy and 160 00:10:13,366 --> 00:10:16,658 how to calculate them. We also looked at how voltage and current 161 00:10:16,658 --> 00:10:20,950 interact to give us electric power, and in turn, electric energy. 162 00:10:20,950 --> 00:10:23,850 We presented a derivation for the conservation of power, and how this 163 00:10:23,850 --> 00:10:27,433 property can used in analysis. And we solved our very first simple 164 00:10:27,433 --> 00:10:31,650 analysis problem. Next lesson, we will be taking our first 165 00:10:31,650 --> 00:10:35,450 look at circuit diagrams and seeing how these things come together. 166 00:10:35,450 --> 00:10:37,320 I remind you again, to take a look at the forms. 167 00:10:37,320 --> 00:10:39,110 Answer questions that you know the answers to. 168 00:10:39,110 --> 00:10:41,820 It's great practice, teaching others. As well as ask any questions that you 169 00:10:41,820 --> 00:10:45,680 might have from this lesson. Until then, we will see you next time. 170 00:10:45,680 --> 00:10:46,020 Cheers.