1 00:00:02,790 --> 00:00:04,490 Welcome back to Linear Circuits. 2 00:00:04,490 --> 00:00:08,470 Today we are going to be talking about the root mean square statistic. 3 00:00:08,470 --> 00:00:10,710 So we will introduce it, and show how to calculate it. 4 00:00:12,040 --> 00:00:16,330 This begins Module Five, which covers the concepts of power. 5 00:00:16,330 --> 00:00:19,570 And so today, we'll talk about root mean squared, and then the statistics we 6 00:00:19,570 --> 00:00:22,290 calculate today will be then used our 7 00:00:22,290 --> 00:00:24,750 power factor, and power calculations in subsequent lessons. 8 00:00:26,220 --> 00:00:28,080 The objectives for this lesson are to identify 9 00:00:28,080 --> 00:00:31,150 the equation for calculating root mean square values. 10 00:00:31,150 --> 00:00:32,760 To calculate root mean square values 11 00:00:32,760 --> 00:00:35,960 using this formula for simple periodic functions. 12 00:00:35,960 --> 00:00:39,450 And then to find the peak value given an RMS statistic. 13 00:00:41,430 --> 00:00:43,850 So to motivate us, first we'll look at a sinusoid. 14 00:00:43,850 --> 00:00:47,800 Suppose that we want to know the average voltage of this sinusoid. 15 00:00:47,800 --> 00:00:53,330 Well to calculate the average, we just use this equation right here where the average 16 00:00:53,330 --> 00:00:58,180 value is equal to one over a period, so this is a periodic function. 17 00:00:58,180 --> 00:00:59,130 So the width of the period. 18 00:00:59,130 --> 00:01:03,680 And, then we integrate over one period, of the function. 19 00:01:05,630 --> 00:01:09,030 And, so we can start anywhere we'd like. This probably should be, 20 00:01:11,300 --> 00:01:14,710 T plus the t aught. we could start at anywhere we'd like. 21 00:01:14,710 --> 00:01:16,800 Start here maybe, and so one period is 22 00:01:16,800 --> 00:01:19,750 from this point till we reach that point again. 23 00:01:22,390 --> 00:01:26,130 And so the period here is going to be t. And as we integrate 24 00:01:26,130 --> 00:01:30,040 over this, filling in all of this area, we 25 00:01:30,040 --> 00:01:33,960 recognize that there's as many low regions as high regions. 26 00:01:35,020 --> 00:01:41,760 And so our average happens to be zero. And this turns out to be irr-, 27 00:01:41,760 --> 00:01:48,420 it really doesn't matter, whether this is, five or whether this is a million, 28 00:01:48,420 --> 00:01:53,010 because it a sinusoid, it's centered and the average is always going to be zero. 29 00:01:53,010 --> 00:01:54,630 Consequently just taking the average of 30 00:01:54,630 --> 00:01:58,260 a sinusoid isn't particularly useful information. 31 00:01:58,260 --> 00:02:01,400 So there's some better statistics that we can use when we're dealing with sinusoidal 32 00:02:01,400 --> 00:02:02,840 functions to give us a little bit 33 00:02:02,840 --> 00:02:05,180 more information about the system and the signal. 34 00:02:07,840 --> 00:02:12,050 One of these methods is to use what's known as the quadratic mean, where the 35 00:02:12,050 --> 00:02:19,400 root means square statistic. And here it is defined FRMS. 36 00:02:19,400 --> 00:02:25,910 So, the RMS of the function is equal to the square root, of this one over T 37 00:02:25,910 --> 00:02:32,820 and integrating from tnot to tnot plus T of f squared of tdt. 38 00:02:32,820 --> 00:02:35,290 So, it's called the root mean square because here you have the square root. 39 00:02:37,100 --> 00:02:41,080 And then this one over T integration is the average or the mean. 40 00:02:42,420 --> 00:02:45,090 And then you're going to take the square of the function. 41 00:02:46,650 --> 00:02:47,940 To make it a little bit easier to see how 42 00:02:47,940 --> 00:02:50,180 to calculate this, we'll do a little bit of animation. 43 00:02:50,180 --> 00:02:51,900 But first of all, the thing we wanted to look at here 44 00:02:51,900 --> 00:02:57,500 is if we take a square of this sinusoid, what will happen? 45 00:02:57,500 --> 00:02:57,930 Well, this point 46 00:02:57,930 --> 00:02:59,480 in going to go to VM squared. 47 00:02:59,480 --> 00:03:02,800 When we get to zero we're going to be back at zero. 48 00:03:02,800 --> 00:03:06,790 When we get down here to negative Vm if 49 00:03:06,790 --> 00:03:08,950 you square that, again, your back up to Vm squared. 50 00:03:11,060 --> 00:03:15,010 And, if we take this this function and we square 51 00:03:15,010 --> 00:03:17,740 it, it turns out that there's a useful trig identity. 52 00:03:19,030 --> 00:03:20,860 Which is this. 53 00:03:20,860 --> 00:03:22,140 It's not particularly pretty. 54 00:03:24,230 --> 00:03:27,970 it allows us to basically do this kind of calculation exactly. 55 00:03:27,970 --> 00:03:30,540 But we're going to kind of give an 56 00:03:30,540 --> 00:03:33,500 illustrative example to show what this actually means. 57 00:03:36,320 --> 00:03:40,900 What you end up having is this function, which has a max value at vm squared. 58 00:03:42,460 --> 00:03:43,940 It again is a sinusoidal curve. 59 00:03:46,070 --> 00:03:47,640 So here's cosine. 60 00:03:47,640 --> 00:03:49,950 It's at twice the frequency of the original function. 61 00:03:51,500 --> 00:03:53,020 So you can see that it starts up, it has a 62 00:03:53,020 --> 00:03:56,146 peak here, has a peak here, and goes from high to low. 63 00:03:56,146 --> 00:04:01,930 That's time it comes back to a PQ, or the third peak on the square value. 64 00:04:03,000 --> 00:04:07,590 So it's a useful identity but it's kind of hard to remember, so if you just 65 00:04:07,590 --> 00:04:09,800 kind of remember this illustration, hopefully that will 66 00:04:09,800 --> 00:04:11,750 help you to remember how to calculate it, but. 67 00:04:11,750 --> 00:04:17,408 If I take the average of that, it's going to be one half of VM 68 00:04:17,408 --> 00:04:24,790 squared, because this curve has a max value of VM squared, at a base at zero. 69 00:04:24,790 --> 00:04:28,100 And since it's just a normal cosine function. 70 00:04:28,100 --> 00:04:31,740 The value would be at half of that for the average. 71 00:04:31,740 --> 00:04:33,140 So when we take the square root of that, 72 00:04:35,740 --> 00:04:40,639 we get that the RMS value of the voltage is VM over the square root of two. 73 00:04:41,830 --> 00:04:44,010 It's important to notice a couple things. 74 00:04:44,010 --> 00:04:47,600 First of all, as now, our voltage maximum value, 75 00:04:47,600 --> 00:04:50,840 or our amplitude increases, so does the RMS value. 76 00:04:52,360 --> 00:04:56,690 And, if you double the RM, the the amplitude, double the RMS value. 77 00:04:58,190 --> 00:05:01,350 This is not based 78 00:05:01,350 --> 00:05:02,710 upon anything about the frequency. 79 00:05:02,710 --> 00:05:04,490 You can double the frequency, triple the frequency. 80 00:05:04,490 --> 00:05:05,460 It doesn't matter. 81 00:05:05,460 --> 00:05:09,930 As long as its a sinusoidal function, the shape is all that really matters. 82 00:05:09,930 --> 00:05:10,850 And you'll get the same result. 83 00:05:12,580 --> 00:05:14,430 So it doesn't matter what kind of sinusoid you have. 84 00:05:14,430 --> 00:05:17,860 The RMS value will just be taking the root two of the amplitude. 85 00:05:17,860 --> 00:05:19,330 And that will tell you the RMS value. 86 00:05:20,750 --> 00:05:23,700 Let's look at what happens if we're using a different function. 87 00:05:23,700 --> 00:05:25,190 For example, a triangle wave. 88 00:05:26,300 --> 00:05:29,270 Again, we are going to do the same thing. We're going to start by squaring it. 89 00:05:30,370 --> 00:05:31,830 Then we're going to take the average of that 90 00:05:31,830 --> 00:05:34,100 squared thing and then take the square root of that. 91 00:05:34,100 --> 00:05:36,070 So, first of all we'll square it. 92 00:05:36,070 --> 00:05:41,050 So if I square this triangle function it looks like this series of quadratic 93 00:05:41,050 --> 00:05:46,110 functions is quadratic curves. And if you average, and 94 00:05:46,110 --> 00:05:51,300 to kind of help with illustration. What we're going to be doing is taking 95 00:05:51,300 --> 00:05:53,570 an average of one period, where 96 00:05:55,890 --> 00:05:57,440 we're integrating underneath that curve. 97 00:05:57,440 --> 00:06:03,370 And if you do that you find that the average value is VM squared over three. 98 00:06:03,370 --> 00:06:07,460 When you take the square root of that, you get v m over the square root of 3. 99 00:06:07,460 --> 00:06:09,790 Again, the frequency doesn't matter. 100 00:06:09,790 --> 00:06:14,460 As amplitude increases, so does the amplitude of this this result. 101 00:06:14,460 --> 00:06:15,350 This RMS result. 102 00:06:17,370 --> 00:06:20,070 But the shape matters, before with the sinusoid, it was VM 103 00:06:20,070 --> 00:06:23,560 over root two, with the triangle, it's VM over root three. 104 00:06:23,560 --> 00:06:24,060 Let's 105 00:06:27,270 --> 00:06:31,130 see how to use this as an example, 106 00:06:31,130 --> 00:06:33,250 consider the voltage that goes into your home. 107 00:06:33,250 --> 00:06:35,410 Now, this varies from country to country, but in the 108 00:06:35,410 --> 00:06:40,390 United States, the voltage specified is 120 volts at 60 hertz. 109 00:06:42,010 --> 00:06:47,140 However, this is 120 volts RMS. So that's not the peak amplitude. 110 00:06:47,140 --> 00:06:49,950 So suppose we want to calculate the peak amplitude. 111 00:06:49,950 --> 00:06:51,580 How would we do that? 112 00:06:51,580 --> 00:06:52,040 Well remember, 113 00:06:52,040 --> 00:06:54,160 again, frequency doesn't matter. 114 00:06:54,160 --> 00:06:55,750 It's just an extra little piece of information. 115 00:06:55,750 --> 00:07:00,070 And we know that the voltage coming into homes in sinusoidal. 116 00:07:00,070 --> 00:07:02,550 Which mean that, if I want to calculate VRMS. 117 00:07:03,680 --> 00:07:11,270 Is going to be equal to the amplitude, divided by the square root of two. 118 00:07:11,270 --> 00:07:17,418 We know here, VRMS is 120, so we get 120 times the square 119 00:07:17,418 --> 00:07:20,030 root of two is equal to VM. 120 00:07:24,330 --> 00:07:28,140 So consequently the peak voltage, or the highest voltage that you will see 121 00:07:28,140 --> 00:07:31,010 if you measure the voltage coming out of your home, is 169.7 volts. 122 00:07:31,010 --> 00:07:35,929 With a minimal value of minus 169.7 volts. 123 00:07:39,780 --> 00:07:41,860 So summarize, we define the root mean square calculation. 124 00:07:41,860 --> 00:07:43,350 We calculated the RMS values for 125 00:07:43,350 --> 00:07:46,850 sinusoidal functions, and triangular fun, wave functions. 126 00:07:46,850 --> 00:07:52,330 And then we applied this to find the peak value in residential home power. 127 00:07:52,330 --> 00:07:54,530 You can do the same thing, no matter what country you live in. 128 00:07:54,530 --> 00:07:58,350 Just look in, look up what the voltages are, for the power you use, 129 00:07:58,350 --> 00:08:02,750 and you can use that same calculation to find the peak, the peak values. 130 00:08:02,750 --> 00:08:05,640 Turns out that RMS calculations are very useful 131 00:08:05,640 --> 00:08:06,820 in power calculations. 132 00:08:06,820 --> 00:08:09,910 And we will see the reason for that in subsequent lessons. 133 00:08:11,930 --> 00:08:13,200 In the next lesson, we will be talking 134 00:08:13,200 --> 00:08:16,600 about the concept of power factor, and power triangles. 135 00:08:16,600 --> 00:08:20,390 And this help us to start describing how AC power works. 136 00:08:20,390 --> 00:08:22,800 It has a couple of little nuances that you don't see with 137 00:08:22,800 --> 00:08:27,320 DC power, but that are very useful, and important for us to understand. 138 00:08:27,320 --> 00:08:27,980 Until next time.