Welcome back to the course on magnetics for power electronic converters. Now that we know how to analyse inductors, let's turn our attention to transformers. Again we're going to use our knowledge of magnetic circuits. In this and the next few lessons, our goal will be to develop electric circuit models for transformers. Models that we can use in analyzing bulk orders, would like to be able to use these models also in spy simulations. In this lesson, we'll begin by looking at the ideal transformer, and it's models. Let's begin by reviewing some basic concepts that you're probably already familiar with. Here is a simple transformer with two windings. The first winding has N1 turns and has current i1 going into that winding. And has a voltage v1 across that terminal. It has a second winding with n2 turns. And a current going into that winding, and a voltage v2 across its terminals. The behavior of a transformer depends on how the two windings are wound. Especially the relative way the windings are wound is important. A winding can be wound in one of two ways. Let's consider the simple case, Here's our core, And I'll draw another core, and then we'll look at two ways to wind this. So in the first case, I can wind the winding, so that it first goes in front of the core. And then behind the core, and then in front of the core, and then behind. So it starts off by going in front of the core. I could have also formed this, so that it goes first behind the core, then in front of the core, and then behind the core and so forth. These two windings are different, because the direction in which it will flux is different. In the first case, if I have current going into the top terminal of the winding, then using the right hand rule, which means I curl my fingers in the direction of the current, then my thumb points in the direction of the flux, the flux will flow in the clockwise direction. On the other hand, in the winding at the bottom, again if I have current going into the top terminal, then again by using the right hand rule, this time the flux flows in the anti-clock direction. The way the windings are wound does not matter in the case of an inductor, because in either case I'll get the same inductance as long as I have the same number of currents. However, in the case of a transformer where I'm going to have a second winding, the direction of the winding does matter, because at the very least it's going to determine the polarity of the voltage that appears on the second winding if I have a current going into the first winding. As we will soon see. So it is important to us to be able to know how the winding is wound on a transformer. One way to do that is to each time draw a 3D diagram showing how the winding is wound as I've shown here as well as in this picture right here. However, when you have to do this repeatedly, this becomes tedious. So engineers have come up with a clever way to indicate how the windings are wound. In fact, what you really care about is the relative direction of how the windings are wound, and that can be done by using what's known as a Dot Convention. The dot convention works as follows. So if I want to show this transformer, rather then draw each time a three dimensional picture of it, showing exactly how the winding are wound, I can take this transformer and simply show it as two windings, which are magnetically coupled. This double line indicates that they are magnetically coupled. And then I simply put dots on the windings, on the sides of the terminals that indicates how they're wound, or how their relatively wound. What is this dot convention exactly mean? Well, the way I've drawn these dots in this two dimensional drawing, not just how the winding's are gone in the three dimensional wheel. Not in the precise where the winding's revolve, but in the way the windings are wound relative to one another, what that means is, that if I have current going into the dart of one winding, then that winding is going to throw flux in a certain direction. And the Dot Convention tells me that if I also have current going into the dot of the second winding, the second winding is going to throw flux in the same direction as the first winding when it had current going into it's dot. This means I could have put the dots on the opposite terminals as long as I did that for both windings. Let's go back and take a look at our 3D diagram to understand exactly how this works. So let's consider now the 3D diagram of this 2D schematic, and here you can see that I've again put dots on the top terminals for these two windings. Now if what we just said is true, the following should happen, when I have current going into the first winding, let's check how the flux is thrown. Again using the right hand rule, I can apply that. I put my fingers in the direction of the current, my thumb points in the direction of the flux, which in this case is going to be in the clock wise direction. For the second winding, again, if I've got it going into the dot, then again, putting my fingers of the right hand in the direction of the current, my thumb points in the direction, the flux is strong. And clearly this flux is also in the clockwise direction, and so the way I have indicated the dots on this structure is correct. Again, I could have put the dots at the bottom for each of the windings and that would have worked out just as well. So let's review what the dot convention is saying. It's simply saying that if you have currents going into the dot of the windings of a transformer, those windings will flux in the same direction. It's not saying that you will always have currents going into the dots off the windings. It's just the way to figure out how the windings are bound. In reality, how the currents flow in or out of the dots depends on how the transformer is connected to the rest of the circuit. There are a few consequences of the dot convention. The first thing it says is that if you have a voltage applied to the first winding with a positive polarity of the voltage such that the positive sign aligns with the dot of the first winding. Then the voltage that appears on the second winding Also has polarity such the positive side of that voltage aligns with the dot of the second winding. So you can also interpret the dot convention as that if you apply a positive voltage on the first winding with the positive side on the dot, then the voltage that appears on the second winding will also have it's positive side on this side of the dot. This comes straight out of Maxwell's equations. Another consequence of the dot convention is that if we have a load on the secondary winding. Let's say resistor. And we have a current going into the first winding. Then the second winding will have current coming out of the dot. So the real direction of current will actually be out of the dot, which means that i2 as shown will actually turn out to be negative. Remember that our convention of having currents go into the dots was simply only to figure out which way the windings are wound. And had nothing to do with the actual directions of current flow. The reason why the current will come out of the dot in this case is because when current comes out of the dot in the second winding, it throws flux in a direction that opposes the flux from the first winding, and that also comes from Maxwell's equations. In fact, it's commonly known as law, which says that the second winding acts in such a way so as to oppose the change in flux. Now that we know how the dot dimension works, let's figure out the voltage conversion ratio of a simple ideal transformer. Recall that the voltage is related to the flux linkage, and it's essentially the time derivative of the flux linking a particular winding. So to find the voltage V 1 across the first winding, we simply need to find what is the flux linking. This will cross to winding, lets say we have as a result of this current i1 bringing to this winding, of flux for ic, that is generated and will be generated, in the direction shown, using the right hand rule, in that case the flux that's linking the first winding Will simply be given by phi c multiplied by N1. So lambda 1 = N1 phi c. Recall that this flux is linking back into each of the turns of the winding hence we have this factor N1 multiplying phi c to find the flux linkage lambda one hence v1 is simply given by N1 times d phi c dt. To find the voltage on the second terminal, again, that's simply given by the rate of change of lambda 2 with respect to time, with lambda 2 is the flux linking the second winding. And since we've assume that phi c is not leaking out of the core, the same flux is going to link the second winding as well except now we have to multiply 5c with the number of turns in the second winding. So lamda 2 is going to equal n2(5c), and therefore v2 will be n2 Times d phi c over dt. Once we have this, it's easy to see how we can find v1 over v2. We'll just divide v2 by v1 and we get N2 over N1, since they both share d Phi c dt. So here you have the relationship between the voltages of the two windings, v2 over v1 is equal to N2 over N1, which is something that you've probably seen before. So the two voltages are simply related by the terms ratio of their windings. Next, let's find out the relationship between the curves of the two windings. For dot we will use the magnetic circuit model for our transformer. Recall how we did the magnetic circuit model. Each of our windings produces an MMF or a magneto motor force who's value is simply given by the current times the number of turns and is presented here as N1, I1 as a voltage source of the first winding and to I do again as a voltage source for the second winding. Note the polarity of these voltage sources. These have to match up with our dot convention. Note that the first winding with the current going as shown throws flux in the clockwise direction, and the second winding with also the current going to it's dot to those flux as shown. In fact, both windings through flux in the same direction. And so in our magnetic circuit model, you also have to have those sources in such a way that they throw flux in the same direction as in the original circuit. Note that it's not important to have known that the flux was going in the clockwise direction or the counterclockwise direction. All you know is that there are two windings through a flux in the same direction, and that's all that is needed in order to draw the magnetic circuit model. So as long as we had these voltage sources with the polarities in such a way that the true flux in the same direction, that would be fine. Now that we have this magnetic circuit model, we can figure out what is the actual flux, just simply using Ohm's Law. And that will be given my Phi c is equal to the total MMF are essentially equivalent to the potential that is across this reluctance RC, divided by the reluctance RC. Now if we assume that our transformer is ideal we can say that its permeability is infinite. That is one of the prerequisites of a transformer being ideal. We can now calculate the reluctance of the score, which recall is given by simply the length of the core divided by the permeability of the core and the flexibility of the core. So if the permeability of the core goes to infinity, the reluctance of the core goes to zero, and the flux will then all stay within the core because it is facing no impedance to its flow. While, if it went outside into the air it would have some finite impedance. With the reluctance of the core equal to 0, then with any finite amount of flux the MMF drop across the core must be 0. Which means that N 1 i 1 + N 2 i 2 must add up to 0. We can also see this from an expression for phi c, because if we cross multiply, we will simply get that N1i1 plus N2i2 equals phi c times Rc, but with Rc going to zero with Phi c being finite. And Phi c has to be finite, because we only have finite amount of energy in any of our sources, then N1i1 plus N2i2 must also be zero which says then i2 over i1 must equal minus N1 over N2. And that result is also expected, as you may recall that the currents in an ideal transformer are related by the inverse turns ratio. And the negative sign is also expected as all it says is that if I have current going into the dot of the first winding, then by Lens' Law I have current coming out of the dot in the second winding. Now that we have the basic relationships for an ideal transformer relating to voltages across its windings, and the currents through its windings, we are in a position to determine a circuit model for this transformer. Schematically, we show the transformer with the two windings coupled together magnetically and the dots showing how the two windings are wound. In the case of an ideal transformer, we will sometimes emphasize that by adding a dashed box around the schematic. And we would write the number of turns of the first winding and the second winding on top of those respective windings. Also by the associated variables convention, we often show the current for each winding going into the positive terminal associated with that winding. That does not mean that is the direction of the current actual flow, as that will be determined by how the transformer is connected to the rest of the circuit. For analytical analysis of a power converter using a transformer, we can simply use the equations that we've derived here for an ideal transformer to determine how the power converter works. However, if you want to model it in spice or another circuit simulator, we can use a dependent voltage source and a dependent current source as follows. The secondary side, we can model using a dependent voltage source of value V2, where V2 is equal to N2 over N1 times V1, where V1 is the voltage across the terminals of the primary side. And then we can model the primary winding with a dependent current source of value i1. Where the value of i1 is given by N2 over N1 times i2, where i2 is the current flowing out of the secondary side. Note here that I don't have negative signs because I've shown the current in the secondary coming out of the terminal rather than going into the positive terminal. It is also useful to see what the expressions for the ideal transformer are telling us in terms of the power flowing into and out of the transformer. Note that if we compute the output power of the transformer, v2 times i2 all that simply equal to N2 over N1 times v1 times minus N1 over N2 times i1. The N1 and N2s cancel and so we get V2, i2 simply equal to minus we won, I won. It simply says that the instantaneous power going into the transformer is equal to the instantaneous power coming out of the transformer. The negative sign here again, is simply there because of the way we've defined the current i1 and i2. Note that as expected, this ideal transformer has no losses, because all the power that's going in is coming out. And it also stores no energy, because instantaneously the power that's going in is coming out. Let's also quickly revise how impedances can be transformed across a transformer, as that's a useful skill to know as we go forward. If we have an Impedance z connected on the secondary side of a transformer. And we can move this impedance to the primary side. Again, simply using the expressions for the voltage conversion and the current conversion that we have already derived. When we move this impedance Z from the secondary side to the primary side of the transformer, its value becomes the original impedance Z times the square of the turns ratio. Note how the turns ratio multiplication goes when we're moving from the secondary side to the primary side it's getting multiplied by the turns ratio of the primary side divided by the turns ratio of the secondary side, the whole thing squared. The multiplication would be reversed if we were moving an impedance from the primary side to the secondary side. In that case you would multiply it by N2 over N1 squared. That's all for this lesson. In our next lesson, we will start to introduce non-idealities into the transformer model. {BLANK_AUDIO]