1 00:00:02,490 --> 00:00:04,690 Welcome back to Linear Circuits, where, today, we're doing 2 00:00:04,690 --> 00:00:08,360 Part 2 of our Power Factors and Power Triangles lecture. 3 00:00:08,360 --> 00:00:11,640 In the previous lesson, we talked about calculating Complex Power 4 00:00:11,640 --> 00:00:16,170 and identifying what the various aspects of that complex power represented. 5 00:00:17,240 --> 00:00:19,370 Today, we're going to continue on talking about the 6 00:00:19,370 --> 00:00:23,550 power factors and power triangles, most, particularly, about power triangles. 7 00:00:23,550 --> 00:00:27,660 And as we understand those things, we'll then be able to talk more about maximum 8 00:00:27,660 --> 00:00:30,120 power transfer in AC systems. 9 00:00:30,120 --> 00:00:35,090 The objectives of this lesson are to have you use power triangles and to calculate 10 00:00:35,090 --> 00:00:37,260 power angle and power factor, real and 11 00:00:37,260 --> 00:00:39,880 Reactive Power, and Apparent Power for a system. 12 00:00:41,360 --> 00:00:45,900 When calculating the Complex Power we plotted it on the complex plane. 13 00:00:45,900 --> 00:00:51,990 Where we had the real part here, and the imaginary part here. 14 00:00:53,070 --> 00:00:55,110 And we label all of the various values. 15 00:00:55,110 --> 00:00:56,450 What we're going to do is we're going 16 00:00:56,450 --> 00:00:58,890 to take the triangle out of this complex plane, 17 00:01:01,450 --> 00:01:02,770 and draw it like this. 18 00:01:04,630 --> 00:01:06,940 We see that we have all the same basic measurements. 19 00:01:06,940 --> 00:01:10,590 We have the P, the Q, and the absolute value of S, corresponding to 20 00:01:10,590 --> 00:01:12,910 Real Power, Reactive Power, and the Parent 21 00:01:12,910 --> 00:01:16,800 Power, and our angle theta, our power angle. 22 00:01:20,220 --> 00:01:24,010 So to remind us of the equation we use for finding the 23 00:01:24,010 --> 00:01:28,690 Complex Power, I've listed it here and I've put the power triangle. 24 00:01:28,690 --> 00:01:31,590 And, what we're going to do is find out 25 00:01:31,590 --> 00:01:34,800 how to calculate all these things from the equations. 26 00:01:34,800 --> 00:01:38,090 So, here we have the voltage equation, and the current equation would be m cosine 27 00:01:38,090 --> 00:01:42,300 of omega t, plus theta v, and Im cosine of omega t plus theta i. 28 00:01:44,390 --> 00:01:47,280 Now we'll go through, one by one, all of the different 29 00:01:47,280 --> 00:01:50,780 important things that we're going to be defining for these power triangles. 30 00:01:50,780 --> 00:01:52,170 First of all, the power angle. 31 00:01:52,170 --> 00:01:55,290 Power angle theta, is the single right here. 32 00:01:55,290 --> 00:01:55,790 And 33 00:01:57,660 --> 00:02:01,710 that corresponds to theta v minus theta i, which we 34 00:02:01,710 --> 00:02:05,080 can pull immediately from our equations for voltage and current. 35 00:02:07,020 --> 00:02:09,610 We're going to call the cosine of theta the power factor. 36 00:02:09,610 --> 00:02:14,260 And the reason we'll say that is that we can calculate the length of P, given we 37 00:02:14,260 --> 00:02:18,250 know, theta, the Apparent Power, absolute value of S, 38 00:02:18,250 --> 00:02:22,868 as P is equal to modulo S, times cosine 39 00:02:22,868 --> 00:02:24,260 of theta. 40 00:02:24,260 --> 00:02:27,250 And so that's why we called it the power factor because it's how much 41 00:02:27,250 --> 00:02:29,970 we're multiplying by our Apparent Power to 42 00:02:29,970 --> 00:02:32,090 find the average power that's being consumed. 43 00:02:34,350 --> 00:02:38,730 So that means that we can calculate Average Power by taking the RMS voltage. 44 00:02:38,730 --> 00:02:42,390 Multiplying it by the RMS current and multiplying that by cosine theta. 45 00:02:42,390 --> 00:02:43,960 And this just follows immediately. 46 00:02:43,960 --> 00:02:47,540 It consists of our calculation of how we found the Apparent Power. 47 00:02:47,540 --> 00:02:52,020 Then for Reactive Power, Q, we want to find the length of this leg. 48 00:02:52,020 --> 00:02:55,668 And using, again, basic trigonometry, Q is going to be 49 00:02:55,668 --> 00:02:59,040 equal to the Apparent Power times the sine of theta. 50 00:02:59,040 --> 00:02:59,540 Now, 51 00:03:02,770 --> 00:03:06,670 here's where it gets a little bit hairy as far as units are concerned. 52 00:03:06,670 --> 00:03:10,320 Average power is going to be measured in watts, because we're doing work. 53 00:03:10,320 --> 00:03:13,320 And watts is the rate at which energy is being used. 54 00:03:13,320 --> 00:03:17,240 But, when we're talking about Reactive Power, it corresponds to power 55 00:03:17,240 --> 00:03:20,400 that's temporarily being stored up to be used at a later time. 56 00:03:20,400 --> 00:03:22,660 And it's not actually doing any real work. 57 00:03:22,660 --> 00:03:25,220 So to help distinguish it, we're going to be using a different unit. 58 00:03:25,220 --> 00:03:28,760 The AC units are Volt-Amperes Reactive. 59 00:03:28,760 --> 00:03:33,151 Sometimes distinguished as V-A-R-S, or VARS. 60 00:03:33,151 --> 00:03:37,170 It's basically measuring the same thing because we're taking a voltage. 61 00:03:37,170 --> 00:03:38,970 Multiplying it by a current and 62 00:03:38,970 --> 00:03:40,990 then multiplying it by something that's unitless. 63 00:03:43,170 --> 00:03:45,920 which is basically the same thing we have for power. 64 00:03:45,920 --> 00:03:47,230 And so we're just going to call 65 00:03:47,230 --> 00:03:48,700 it something different to help distinguish that 66 00:03:48,700 --> 00:03:53,490 it's not really being used, it's just a temporary storage type of a thing. 67 00:03:53,490 --> 00:03:55,300 And Apparent Power we're again going to use a different unit. 68 00:03:55,300 --> 00:03:58,300 We're going to use Volt-Amperes or VA, and we've 69 00:03:58,300 --> 00:04:02,070 already discussed how to calculate that Apparent Power. 70 00:04:02,070 --> 00:04:08,290 So now we see that we can calculate all these different values, from the, 71 00:04:08,290 --> 00:04:10,940 the voltage and the current equations. 72 00:04:10,940 --> 00:04:12,970 And that they have a relationship that corresponds to 73 00:04:12,970 --> 00:04:15,520 the trigonometric relationship that we see in a triangle. 74 00:04:19,140 --> 00:04:23,950 Now we can look at the relationships between the Apparent Power, the Real 75 00:04:23,950 --> 00:04:25,400 Power, and the Complex Power, or 76 00:04:25,400 --> 00:04:30,150 the, Reactive Power, rather for different devices. 77 00:04:30,150 --> 00:04:33,390 So, first of all, the resistive case, we've already looked at. 78 00:04:33,390 --> 00:04:36,260 The voltage and the currents multiply together 79 00:04:36,260 --> 00:04:39,600 to get something that is non-negative, for power. 80 00:04:39,600 --> 00:04:44,380 Here in the green. And I'm still using the same v is 81 00:04:44,380 --> 00:04:49,440 red, i is blue and green is P. Again, all different units here. 82 00:04:50,970 --> 00:04:57,620 I, here, illustrate the Impedance of this device because it's all real. 83 00:04:57,620 --> 00:05:02,800 It's just a real line out here. Now, this is a triangle, in, the 84 00:05:02,800 --> 00:05:08,050 generative sense, where this phase angle, this angle here in this corner is 0. 85 00:05:08,050 --> 00:05:09,440 And we all 86 00:05:09,440 --> 00:05:14,880 see that the, the Impedance triangles are similar, 87 00:05:16,100 --> 00:05:20,276 in the mathematical sense of similar, to the Complex Power triangles. 88 00:05:20,276 --> 00:05:23,132 All of my power, all of my Parent Power is 89 00:05:23,132 --> 00:05:27,180 Real Power and there is no Reactive Power for a resistor. 90 00:05:28,590 --> 00:05:30,040 Now, as I move this around. 91 00:05:30,040 --> 00:05:34,570 Get more of a proper triangle. Now I have a slightly inductive load, 92 00:05:34,570 --> 00:05:37,970 because I have a resistor, I have an inductor. 93 00:05:37,970 --> 00:05:41,178 Now, you'll notice that, on these plots, my voltages stay the same. 94 00:05:41,178 --> 00:05:44,690 I'm moving currents with respect to the voltages 95 00:05:44,690 --> 00:05:46,860 and then, obviously, the powers are changing as well. 96 00:05:48,810 --> 00:05:50,930 No longer are the voltage and the current in phase. 97 00:05:50,930 --> 00:05:54,200 Now, the voltage, leads the current, a little 98 00:05:54,200 --> 00:05:57,190 bit, or the current lags behind the voltage. 99 00:05:58,260 --> 00:05:59,900 This is my Impedance triangle 100 00:05:59,900 --> 00:06:04,350 for this system, where the real part comes from the resistor. 101 00:06:04,350 --> 00:06:06,360 The imaginary part comes from the inductor. 102 00:06:06,360 --> 00:06:09,458 So the real part and the imaginary part are giving us this. 103 00:06:09,458 --> 00:06:14,770 And this theta, corresponds to this data because these are two similar triangles. 104 00:06:15,970 --> 00:06:22,390 For systems that have, a, an inductive load, my phase 105 00:06:22,390 --> 00:06:24,960 angle is going to be between 0 and pi halves for 106 00:06:24,960 --> 00:06:25,550 my power angle. 107 00:06:25,550 --> 00:06:27,810 And my power is going to be somewhere between the 108 00:06:27,810 --> 00:06:31,280 extremes of 0 and the value of the Apparent Power. 109 00:06:31,280 --> 00:06:33,352 But so is my react, my Reactive Power, 110 00:06:33,352 --> 00:06:36,710 somewhere between 0 and the the Apparent Power. 111 00:06:36,710 --> 00:06:41,740 We're going to say that the system is lagging, because it's inductive. 112 00:06:41,740 --> 00:06:45,310 And we say that it lags, or it's lagging because the current 113 00:06:45,310 --> 00:06:48,820 lags behind the voltage, the voltage hits its maximum point before the current. 114 00:06:50,140 --> 00:06:53,520 We'll go to the extreme case where it is a pure inductor. 115 00:06:53,520 --> 00:06:55,510 Now my triangles are again degenerative because 116 00:06:55,510 --> 00:06:59,110 my angle, my power angles are pi halves. 117 00:06:59,110 --> 00:07:02,530 There's no Real Power, real average power being consumed. 118 00:07:02,530 --> 00:07:05,010 All of the Apparent Power goes to Reactive Power. 119 00:07:08,080 --> 00:07:09,930 I can do the same thing with capacitive loads. 120 00:07:11,220 --> 00:07:12,150 Like this. 121 00:07:12,150 --> 00:07:17,240 Now, we see that the current leads the voltage, so it's going to be leading. 122 00:07:17,240 --> 00:07:21,990 This is my Impedance triangle, which is similar to my power triangle. 123 00:07:21,990 --> 00:07:26,210 My theta, my power angle, is between minus pi halves and 0. 124 00:07:26,210 --> 00:07:29,920 My power is between 0 and the Apparent Power, and 125 00:07:29,920 --> 00:07:33,160 my Reactive Power is now going to be negative because it's 126 00:07:33,160 --> 00:07:34,180 down here in this quadrant. 127 00:07:35,240 --> 00:07:39,110 And it's going to be between the minus value of the Apparent Power and the 0. 128 00:07:39,110 --> 00:07:41,410 We can see all of the relationships here. 129 00:07:41,410 --> 00:07:44,650 And finally, looking at the purely capacitive case, 130 00:07:44,650 --> 00:07:46,690 all of the power is Reactive Power because 131 00:07:46,690 --> 00:07:48,360 the end of the capacitor stores up the 132 00:07:48,360 --> 00:07:50,450 energy temporarily to be used at a later time. 133 00:07:53,160 --> 00:07:54,940 What are the implications of this? 134 00:07:54,940 --> 00:07:58,760 Well, first of all, only Real Power is being transformed as 135 00:07:58,760 --> 00:08:02,610 heat or light or moving a motor or doing real work. 136 00:08:02,610 --> 00:08:04,640 And, typically that's what we're most interested in. 137 00:08:04,640 --> 00:08:07,360 We don't really care about how much power is being temporarily stored up in 138 00:08:07,360 --> 00:08:11,600 the system, we want to know how much is being used to do real work. 139 00:08:11,600 --> 00:08:14,480 But that doesn't mean we can neglect the Reactive Power. 140 00:08:14,480 --> 00:08:16,930 Reactive power causes increased currents. 141 00:08:16,930 --> 00:08:18,320 And so that means that the more Reactive Power 142 00:08:18,320 --> 00:08:21,240 we have in our system, the more currents need 143 00:08:21,240 --> 00:08:23,976 to be used in the lines connecting to your device. 144 00:08:23,976 --> 00:08:24,408 And so you 145 00:08:24,408 --> 00:08:25,002 [INAUDIBLE], 146 00:08:25,002 --> 00:08:28,090 have more loss in the transmission lines, leading to that device. 147 00:08:29,200 --> 00:08:35,280 So if you're a power company, you'll want to give some heat to that Reactive Power. 148 00:08:35,280 --> 00:08:39,590 Now, private customers, for their private homes and residences, typically, 149 00:08:39,590 --> 00:08:42,450 only are charged for the Real Power that they consume. 150 00:08:43,450 --> 00:08:46,610 The power company doesn't really care too much about how much Reactive 151 00:08:46,610 --> 00:08:49,050 Power there is because a resi, residential home doesn't have a lot. 152 00:08:49,050 --> 00:08:50,090 But if you're a big 153 00:08:50,090 --> 00:08:53,230 industrial system, with big motors, there's a lot 154 00:08:53,230 --> 00:08:56,072 of inductive loads in those types of systems. 155 00:08:56,072 --> 00:09:01,350 And so consequently industrial consumers, are often charged at a lower rate 156 00:09:01,350 --> 00:09:05,660 for the Reactive Power because the power companies needs to have more equipment. 157 00:09:05,660 --> 00:09:07,420 They have more losses in their transmission 158 00:09:07,420 --> 00:09:09,980 lines and they need to design their 159 00:09:09,980 --> 00:09:13,360 systems with extra transformers and extra devices, 160 00:09:13,360 --> 00:09:15,092 to be able to handle those increased loads. 161 00:09:15,092 --> 00:09:15,438 So, 162 00:09:15,438 --> 00:09:17,630 [INAUDIBLE] 163 00:09:17,630 --> 00:09:20,920 because of this, big industrial users might want 164 00:09:20,920 --> 00:09:23,830 to give some consideration to their systems and 165 00:09:23,830 --> 00:09:25,990 change their systems a little bit to reduce 166 00:09:25,990 --> 00:09:27,810 the amount of Reactive Power that they have. 167 00:09:27,810 --> 00:09:30,200 To reduce the costs that they pay to the power company. 168 00:09:32,000 --> 00:09:33,890 In summary, we've defined power angle and power 169 00:09:33,890 --> 00:09:37,850 factory, factor, Real and Reactive Power, and Apparent Power. 170 00:09:37,850 --> 00:09:42,560 And showed how they all correlate in a triangle behavior, 171 00:09:42,560 --> 00:09:43,980 using power triangles, we can see 172 00:09:43,980 --> 00:09:45,990 the relationship between all those different values. 173 00:09:47,520 --> 00:09:51,940 In the next lesson, we'll see how we can use reactive elements like capacitors 174 00:09:51,940 --> 00:09:56,320 and inductors to control the amount of Reactive Power that we have in the system. 175 00:09:56,320 --> 00:10:03,180 And we can use, or we can analyze maximum power transfer for AC systems, and look at 176 00:10:03,180 --> 00:10:05,540 how that is different from the maximum power 177 00:10:05,540 --> 00:10:09,270 transfer that was calculated in DC systems, until then.