1 00:00:02,240 --> 00:00:04,330 Welcome back to our class on linear circuits. 2 00:00:04,330 --> 00:00:07,510 Today we're going to be talking about the ideal transformer model. 3 00:00:07,510 --> 00:00:09,948 So we're going to use the ideal transformer model to analyze basic 4 00:00:09,948 --> 00:00:12,846 transformer circuits and then see the relationships between voltages and 5 00:00:12,846 --> 00:00:16,028 currents. So in the previous lesson we looked at 6 00:00:16,028 --> 00:00:18,896 the linear model. It's a fairly complicated model that 7 00:00:18,896 --> 00:00:22,802 makes use of phasor analysis and mutual inductance to do these calculations. 8 00:00:22,802 --> 00:00:26,735 In the ideal model, we're going to make a few assumptions that are never completely 9 00:00:26,735 --> 00:00:30,269 true, but they allow us to get a decent ideal of how a transformer's going to 10 00:00:30,269 --> 00:00:35,870 behave without having to do more complicated analysis. 11 00:00:35,870 --> 00:00:38,795 So first of all, we need to identify the asusmptions that are used for the ideal 12 00:00:38,795 --> 00:00:41,675 transformer model, and then use the ideal transformer model to do some basic 13 00:00:41,675 --> 00:00:45,622 circuit analysis. Then we will describe the importance of 14 00:00:45,622 --> 00:00:49,377 transformers in power transmission. First of all we need to define the 15 00:00:49,377 --> 00:00:52,514 coefficient of coupling. So the coefficient of coupling is going 16 00:00:52,514 --> 00:00:56,925 to be represented by a lower case k. And the coefficient of coupling is the 17 00:00:56,925 --> 00:01:01,345 value that gives this relationship between the mutual inductance, M, and the 18 00:01:01,345 --> 00:01:05,901 self inductance is L1 and L2 in the transformer. 19 00:01:05,901 --> 00:01:12,659 So, it's always possible to find this k, but some k's are valid in a physical 20 00:01:12,659 --> 00:01:19,179 sense and some k's are not. K's basically have to be similar between 21 00:01:19,179 --> 00:01:22,494 0 and 1, 0 meaning that there's no mutual inductance between the two coils, and 22 00:01:22,494 --> 00:01:25,950 they're completely independent of each other. 23 00:01:25,950 --> 00:01:29,060 And one, meaning that they're very tightly coupled. 24 00:01:29,060 --> 00:01:33,090 There's a limit to, a physical limitation to how tightly this coupling can be. 25 00:01:33,090 --> 00:01:40,140 So to do an example problem, consider that L1 is 4mH, L2 is 9mH and M is 2mH. 26 00:01:40,140 --> 00:01:45,264 Or, we can simply use this equation to see that k is equal to M divided by the 27 00:01:45,264 --> 00:01:54,360 square root of L1 times L2. So that equals 2mH divided by 4mH times 28 00:01:54,360 --> 00:02:03,162 9mH, so it's 36mH squared. Take the square root of that to get 6mH 29 00:02:03,162 --> 00:02:06,554 under 2mH. So that means that k is going to be equal 30 00:02:06,554 --> 00:02:09,543 to one third. So it is quite simple to calculate simple 31 00:02:09,543 --> 00:02:13,215 to calculate the coefficient of coupling. it's going to be very important when we 32 00:02:13,215 --> 00:02:15,921 start talking about ideal transformers, though, because we are going to make a 33 00:02:15,921 --> 00:02:19,852 certain assumption about what this coefficient of coupling happens to be. 34 00:02:19,852 --> 00:02:23,097 In the ideal transformer case the coupling coefficient k is equal to 1, 35 00:02:23,097 --> 00:02:27,078 which means that these two coils are very tightly coupled. 36 00:02:27,078 --> 00:02:32,447 L2 and L1 are assumed to go to infinity, which means that so is our mutual 37 00:02:32,447 --> 00:02:37,008 inductance. Now, this is a limit, it's not that it's 38 00:02:37,008 --> 00:02:40,976 equal to the value infinity, that their going to approach very, very large 39 00:02:40,976 --> 00:02:44,758 numbers. And the reason that we make this 40 00:02:44,758 --> 00:02:48,044 assumption, is that, it allows the analysis that leads to the transformer 41 00:02:48,044 --> 00:02:51,999 equations. We're going to skip over the actual 42 00:02:51,999 --> 00:02:53,570 analysis. It's available. 43 00:02:53,570 --> 00:02:57,900 You can find it if you're interested. But we just need to, to make use of it. 44 00:02:57,900 --> 00:03:03,490 But we clearly know that having this infinite in-, inductance is not possible. 45 00:03:03,490 --> 00:03:08,840 Finally, we assume that losses from coil [UNKNOWN] resistances are negligible. 46 00:03:08,840 --> 00:03:13,064 Sometimes when we're doing analysis of transformers, we're going to stick a 47 00:03:13,064 --> 00:03:17,288 resistor here and a resistor here, to correspond to the resistance of the two 48 00:03:17,288 --> 00:03:24,012 wires that are placed here and here. it gives a better representation, but the 49 00:03:24,012 --> 00:03:27,745 ideal transformer case we're going to assume that both of those go to zero. 50 00:03:27,745 --> 00:03:33,284 Now, the implications of this ideal transformer model are the following. 51 00:03:33,284 --> 00:03:37,920 First of all v1 over N1 is equal to v2 over N2 where v1 and v2 are the voltages 52 00:03:37,920 --> 00:03:43,485 across the primary and secondary coils respectively. 53 00:03:43,485 --> 00:03:46,845 And then N1 and N2 are the number of rotations of the coil, or how many reps 54 00:03:46,845 --> 00:03:52,360 of the coil in the primary and the secondary coils respectively. 55 00:03:52,360 --> 00:03:58,030 We also have the relationship that N1 times i1 is equal to N2 times i2. 56 00:03:58,030 --> 00:04:02,556 And these equations come from the uh,Faraday's law of induction so what we 57 00:04:02,556 --> 00:04:09,146 have is we're making a circle like this. Then we have a changing B field through 58 00:04:09,146 --> 00:04:12,080 that surface. But as I wrap coils around again, and 59 00:04:12,080 --> 00:04:15,635 again, and again, that corresponds to additional surfaces. 60 00:04:15,635 --> 00:04:21,710 And so every time we have another wrap it's just another one of these surfaces. 61 00:04:21,710 --> 00:04:25,301 And so we take the number of these wraps and multiple it by the voltage that's 62 00:04:25,301 --> 00:04:30,110 induced, by the changing magnetic flux, in each of these sections. 63 00:04:31,320 --> 00:04:36,850 Which is why we get this, N [INAUDIBLE] behavior. 64 00:04:36,850 --> 00:04:39,660 But this equation is very simple to work with. 65 00:04:39,660 --> 00:04:43,540 They are quite easy to, to use for analysis. 66 00:04:43,540 --> 00:04:45,430 And so that's the most important thing that we're going to need to take out of 67 00:04:45,430 --> 00:04:48,090 this. So let's do an example. 68 00:04:48,090 --> 00:04:50,460 Here we're going to start by writing down our two equations. 69 00:04:50,460 --> 00:04:59,078 That v1 over N1 is equal to v2 over N2, and that N1 times i1 is equal to N2 times 70 00:04:59,078 --> 00:05:04,164 i2. We have a phasor voltage here, and I'm 71 00:05:04,164 --> 00:05:07,702 going to do all of my analysis for this problem in terms of RMS voltages and 72 00:05:07,702 --> 00:05:11,936 currents, because it's entirely possible to do that without any kind of confusion, 73 00:05:11,936 --> 00:05:19,781 but I want to point that out. So, we have 120 kilo Volts rms here, 74 00:05:19,781 --> 00:05:25,240 leading to 120 Volts rms here. Now we don't know the number of coils 75 00:05:25,240 --> 00:05:27,782 because we don't have a number for either of these, but what we do know is the 76 00:05:27,782 --> 00:05:32,436 relationship between them. So frequently, with ideal transformers 77 00:05:32,436 --> 00:05:37,024 will show something like this, a ratio of the number of coils from one to the 78 00:05:37,024 --> 00:05:40,768 other. Because the actual number themselves 79 00:05:40,768 --> 00:05:44,610 don't really matter as the relationship between them. 80 00:05:44,610 --> 00:05:48,081 And in this case, we see that there's 100 coils to one over here. 81 00:05:48,081 --> 00:05:55,697 Because, if I take 120 and multiply it by 100, we get 12 kilo-volts, over here. 82 00:05:55,697 --> 00:06:00,945 And if I want to find the current, it's going through over here, we had 120 volt 83 00:06:00,945 --> 00:06:06,578 rms, 240 ohms. So that means that i2 is going to be 84 00:06:06,578 --> 00:06:12,519 equal to half an amp with phase angle of zero. 85 00:06:12,519 --> 00:06:18,824 On this side i1, we're going to have the same relationship using this i2 and i1, 86 00:06:18,824 --> 00:06:26,455 that means that i1 is going to be equal to 5mA, yeah, 5mA rms. 87 00:06:26,455 --> 00:06:31,767 And so what we can then do is we can find the power that's being consumed in each 88 00:06:31,767 --> 00:06:37,477 of these devices. So first of all, we'll look at the 89 00:06:37,477 --> 00:06:47,300 resistor, i2 is one half, amp RMS and the voltage it crosses is equal to 120. 90 00:06:47,300 --> 00:06:51,708 And since this is a completely real device, p is going to be equal to the 91 00:06:51,708 --> 00:06:56,496 apparent power which is one half times 120 the rms current in the rms voltage 92 00:06:56,496 --> 00:07:03,015 together, so 60 watts. So this is basically 60 watts. 93 00:07:03,015 --> 00:07:07,110 If I calculate the power that's being consumed here we find that p is equal to 94 00:07:07,110 --> 00:07:10,945 minus 60 watts based on the reference structure and the current and the 95 00:07:10,945 --> 00:07:16,115 voltage. So this transformer is generating 60 96 00:07:16,115 --> 00:07:20,145 watts of power and that's because this side if we take this voltage and this 97 00:07:20,145 --> 00:07:26,710 current and multiply them together. We find that that p is also equal to 60 98 00:07:26,710 --> 00:07:32,280 watts. So we'll call this p of the secondary. 99 00:07:32,280 --> 00:07:34,510 And we'll call this p of the primary. And this is p of the resistor. 100 00:07:34,510 --> 00:07:41,629 And that's because this source is going to be generating 60 watts as well. 101 00:07:43,730 --> 00:07:46,980 So we see that that's why we need to have this relationship between voltage and 102 00:07:46,980 --> 00:07:51,530 current because the power is being transferred from one side to the other. 103 00:07:51,530 --> 00:07:54,411 And that's one of the nice things about ideal transformer is it's converting all 104 00:07:54,411 --> 00:07:57,120 of the power that's going in over here, pushing it to the other side to be used 105 00:07:57,120 --> 00:08:01,310 to do some type of work. So essentially what we're doing is we're 106 00:08:01,310 --> 00:08:05,110 allowing ourselves to change the voltage to operate at a different voltage. 107 00:08:05,110 --> 00:08:09,078 So consequently transformers are often used in computers where you have 120 108 00:08:09,078 --> 00:08:14,680 volts coming out of the wall but you need 5 volts DC to run your components. 109 00:08:14,680 --> 00:08:16,505 You need a transformer to change the voltage. 110 00:08:16,505 --> 00:08:20,384 It's also used in power applications. In this case, this is what we're 111 00:08:20,384 --> 00:08:23,606 illustrating here. This is like the power that being sent by 112 00:08:23,606 --> 00:08:27,194 the power company, this transformer is a big power transformer that's owned by the 113 00:08:27,194 --> 00:08:30,418 power company that changes the voltage they're transmitting to a smaller 114 00:08:30,418 --> 00:08:34,374 voltage. The reason for this is that if you have 115 00:08:34,374 --> 00:08:39,020 high currents flowing through wires, it leads to, basically, heaters. 116 00:08:39,020 --> 00:08:42,530 And you lose a lot of power as heat lost in the wires. 117 00:08:42,530 --> 00:08:46,066 Power companies don't want to lose all of that heat in their wires, so they operate 118 00:08:46,066 --> 00:08:49,498 their transmission lines at very high voltages so that the currents can remain 119 00:08:49,498 --> 00:08:52,880 small. And they use transformers to go from 120 00:08:52,880 --> 00:08:55,856 those very high voltages to smaller voltages that are used in residential 121 00:08:55,856 --> 00:08:59,720 homes. So what are the implications? 122 00:08:59,720 --> 00:09:02,423 Well transformers allow change from one voltage to another voltage. 123 00:09:02,423 --> 00:09:07,031 With a high-voltage and low-current power and transform it into using long-distance 124 00:09:07,031 --> 00:09:11,511 power distribution through transformers into something that is a low-voltage and 125 00:09:11,511 --> 00:09:17,110 high-current so you can still get the power to all of the homes. 126 00:09:17,110 --> 00:09:19,710 And it turns out that this is a very important phenomenon. 127 00:09:19,710 --> 00:09:23,858 Before this event it was required for power stations to be placed very close to 128 00:09:23,858 --> 00:09:29,318 where the power was being consumed. By being able to use transformers the 129 00:09:29,318 --> 00:09:32,856 power can now be sent over very long distances, so things like the Niagara 130 00:09:32,856 --> 00:09:36,684 Falls power plant can now send power all across the United States and Canada 131 00:09:36,684 --> 00:09:43,140 without having to have a whole bunch of power stations all over the place. 132 00:09:44,640 --> 00:09:48,357 So in summary, we showed the ideal transformer model, and used this model to 133 00:09:48,357 --> 00:09:52,310 solve an example system, and identified that transformers are useful for power 134 00:09:52,310 --> 00:09:56,204 transmission because they make possible for sending power using very high 135 00:09:56,204 --> 00:10:00,098 voltages and low currents to be able to avoid getting high heat through high 136 00:10:00,098 --> 00:10:07,215 currents, through transmission links. In the next lesson we'll be looking at a 137 00:10:07,215 --> 00:10:10,955 sensor that makes use of this concept of mutual inductance, linear-variable 138 00:10:10,955 --> 00:10:15,282 differential transformer. And see places that they can find useful 139 00:10:15,282 --> 00:10:17,320 applications. Until then.