Welcome back to the course on Magnetics for Power Electronic Converters. In our last lesson, we developed an equivalent circuit model for transformers that included the magnetizing inductance, as well as the two leakage inductances. We developed this model using two equations that relate how the terminal voltages of the transformer are related to the terminal currents. In this lesson, we will see how the same equations can be used to develop alternate models for the transformer. Circuit models are useful if you are interested in simulating a power converter with a transformer inside it. However, if you are only interested in a mathematical model for a transformer, there is an alternate way to use the same equations that we derived earlier to get an alternate model for our transformer. Instead of combining these two current turns into one that we did previously, we can combine the turns that have the current i1 in them into a single turn, and similarly combine the terms with i2 into a separate term. And we can do that for both of these equations, again, combining the terms with i1 and combining the terms with i2 into separate terms. If we do that, we get these equations for v1 and v2. Here, v1 is now written explicitly in the form with a term multiplying the time derivative of i1 and another term multiplying the time derivative of i2. Similarly, v2 is written with a term that multiplies the time derivative of i1 and another term that multiplies the time derivative of i2. We can rewrite these two equations in the form of a matrix equation that represents how v1 and v2 depend on the time derivatives of i1 and i2. This representation is known as the inductance matrix representation. For a two-binding transformer, we have a two-by-two matrix with four elements in the inductance matrix. The first term is known as L11. The other diagonal term is L22. And the two cross-diagonal terms are L12 and L21. You will notice that L12 and L21 are both equal, so for simplicity, we can simply call both of them L12 and L12. The reason the two off diagonal terms are equal is because our equivalent circuit model for the transformer is reciprocal since we've only used linear passive components in our equivalent circuit model. Now let's compare the mathematical model we have developed for the transformer to the equivalent circuit model we developed earlier. Note that the mathematical model tells us how the terminal voltages relate to the terminal currents. Also note that the problem circuit model for the transformer that we developed is a physically based So good model. Since in developing it, each of the items that we used had a physical manifestation in the original device. The turns ratio expressed the actual number of turns that we had on the primary winding, as well as the number of turns we had on the secondary winding. The magnetizing inductance captured the effect of the energy that was being stored in the core and the two linkage inductances captured. The energy stored in the leakage flux, Ll1, was capturing the energy that was in the flux generated by the first winding that was not coupling to the second winding. And Ll2 represents the energy in the flux that's generated by the second winding that is not coupling to the first winding. So each of these entities that is in this equivalent circuit model has a physical meaning. On the other hand, the mathematical model that we have developed simply tells us what is the relationship between the terminal voltages of the transformer and it's terminal current. The physical model also tells us that, but let's see if there's a difference between the mathematical model and the physical model. Let's ask the question, how many independent parameters are there in our physical model? I'll let you think for a second. What do you think? Well, let's start counting. We have the magnetizing inductance which is one. We have one leakage inductance, two. A second leakage inductance, three, and then we have at least a turns ratio, which makes it four. And if we were to count the actual number of turns on each winding, then we could possibly say we have five parameters. But at the very least, we have four independent parameters that define this physically based circuit model. Note that the voltage conversion ratio from the primary side or the secondary side only depends on the ratio of these turns, not the actual value of the winding turns. So we can say there are four independent parameters in the physically based circuit model for our transformer. Now let's take a look at our mathematical model. We have L11 as a parameter, L22 as another independent parameter, and then we have L12. Note L21 is the same as L12. So we only have three independent parameters in our mathematical model. Note that both of these models, the mathematical model and the physically based circuit model, from the perspective of its terminals, have identical behavior, since we derive both of them from the same mathematical equations. Therefore, if I gave you a transformer inside a black box with only its terminals exposed, the only things that you could measure would be those that are represented by its mathematical equations. So by doing different experiments on those terminals, Such as measuring inductors, while keeping the other winding's either open or short. You could compute at most 3 independent parameters. What does that mean for our physically based circuit model? Here we have at least 4 independent parameters. How do we figure out the fourth parameter? It turns out that there is no way for us to find all four parameters of the physically based circuit model if all that is available to us are the terminals of this transformer, and all we can do is make measurements on those terminals. If you really wanted to find out all the parameters, you would have to open the box up, pull the transformer out, unwind it to figure out how many actual turns it has, and then you could make some more measurements to figure out all the parameters. But most of the time that's unnecessary, because what we're really interested in is taking our transformer and putting it inside a simulation of our power converter, and figuring out how the power converter works. From the perspective of the rest of the circuit, all that matters is the terminal relationships between the voltages and the current at the terminals of this transformer. So why even bother? For use of this transformer in a simulation, of a larger circuit or analysis of a larger circuit, we could simply assume one of these parameters as any arbitrary value within certain physical constraints, and figure out the other three remaining parameters. And that would, from a thermal perspective, be just as good a model as this physically based circuit model. Let's look at an example of that. In this example of an alternate transformer model, I have arbitrarily picked the turns ratio of the transformer to be 1:1. Since there are four parameters here, and there are only three that I can determine from external measurements, I'm free to choose any one of these four as an arbitrary value. So I pick the turns ratio to be one is to one. I could've just as easily have picked a value for one of the other parameters in the circuit model. For example, I could have pick l c to be equal to zero and started with that as my starting point. But let's work on this example with respect the turns ratio of this transformer to be one to one. And then let's assume we can do the terminal measurements and figure out. The inductances that go in the inductance matrix representation of this transformer. We know what is the value of L11, L12 and L22. What we'd like to know is what are the values of LA, LB and LC in this model, once I've assumed that the turns ratio is 1 is to 1. To do that, we can simply write a inductors matrix in a presentation of this model, and then just map that to the inductors matrix that we actually have calculated. So let's begin by writing what it is, simply equal to the sum of the voltage across LA, plus the voltage across LB. Well, the voltage across LA is nothing more than LA times. DT and the voltage across LB will be LB times the derivative of the current through it. And the current through LB is simply going to be the sum of two currents. I 1 which is coming from here. And another current which comes from the other side. This current, because its transformer has a 1 to 1 turns ratios is simply equal to i2. And so the current through LB is simply i1 plus i2. So this becomes a time to derivative of i 1 plus i 2. We can combine terms, rewrite this as LA + LB times di1/dt. + LB times di2, dt. Similarly, we can write an expression for V2. V2, again, is the sum of the voltages of across LC, plus the voltage across the secondary of the IU transformer. LC times di2 over DT. And the voltage across the secondary the transformer is simply the same as voltage across ld, is to one transformer. So that's the same as the voltage that we already figured out and that's just LB times d i1 plus i2 over dt. Again I can write this in a way that I combine the terms for I 1 and I two, separately. And so I get LB DI one DT plus LC plus LB times DI two. DT. Once we have these expressions for V1 and V2, we can rewrite these in matrix form, as shown here. By simply combining these two equations, we get V1 and V2 in the form of an inductance matrix expression. By comparing the two inductance matrices, we can figure out how LA, LB and LC relate to the known values L11, L12 and L22. And we can simply then say that we have LB then equal to just by comparing these off-diagonal terms, LB is equal to L one two, then we have LA plus LB Equal to L11, which simply gives us that La is nothing more than L11 minus L12. And then finally by comparing This diagonal term with L22 we have that LC + LB = L22. And hence, LC is simply equal to L22- L12. Now, we have all the three parameters needed for our circuit model. This is no longer a physical model, however, it captures the terminal behavior of the transformer just as well as the physical model, or any other model. This model is actually a fairly useful model because we can simply ignore the ideal transformer since it has a turns ratio of one is to one and can work this model into simply a model that looks like this. Here we have La, here we have Lc, and here we have LB. And this is what would also be known as a T model for the transformer. And this is a fairly simple model to use as in this model, we can model the transformer with simply three inductances. Note again, that we developed this model by arbitrarily chosing the turns ratio to be one is to one. We could have also chosen the turns ratio to be something else, in which case we would have gotten a different model. We could have also chosen one of the leakage inductances, for example, to be zero or picked a different value, or some value, for a leakage inductance or the magnetizing inductance. Hence, the total number of models that you could potentially develop for a transformer is infinite, and all of these models will be equally good as far as modeling the transformer in terms of its terminal characteristics is concerned. So, we can choose any of the models that we please. Finally, let me go over some terminology that is commonly used in describing transformers and their models. The on diagonal terms of the inductance matrix L 1-1. An L22 are also referred to as self-inductances of the transformer. L11 would the the self-inductance of the primary winding of the transformer, and it would be the inductance that you would see looking into the primary of the transformer with the secondary winding open circuited, i.e. It's the inductors that you see here with I 2 set equal to zero. And what you would then see is simply the sum of the leakage inductance on the primary side, plus the magnetizing inductance of the transformer. So this expression tells you how L one one relates to the parameters in the physical model of the transformer. Similarly, L two two is the self inductance of the secondary winding and you would find that while looking in through the secondary winding with the primary side open circuited, that is how to do the inductance looking in here with the current i1 = 0 and its value. You can easily see it is going to be the sum of the leakage inductance on the secondary side plus the magnetizing inductance reflected to the secondary side, which will be L mu 1 multiplied by N 2 over N 1, squared. And L l 1 would not show up, as there is no current in the primary winding when we are looking inside the second rewinding and asking what does L2 do. The cross diagonal terms of the inductance matrix, which are equal, are known as the mutual inductance. Mathematically, the mutual inductance is equal to the magnetizing inductance reflected on the primary side multiplied where the turns ratio, N2 over N1. You could ask what does mutual inductance physically mean. There's a mathematical way to you think about mutual inductance where you can see it provides the component through the voltage, from the current that is flowing in the other winding of the transformer, but you can also think of the mutual inductance as a form of the magnetizing inductance that is neither on the primary side, nor on the secondary side, but sort of in the middle. If I were to, for example, create a circuit model for the transformer that used two ideal transformers, I could by placing an inductance in the middle of these dual ideal transformers. I could call the inductance that is in the middle here. As the mutual inductance L12 if I have a turns ratio for each of these transformers, which is equal to the square root of the actual turns ratio Note that if I moved this L12 to the primary side it would just become Lmu1. And if I moved this to the secondary side, it would just become L mu 2, which would be L mu 1 times N2 over N1, squared. This is just one way to understand mutual inductance that works for a two-winding transformer. Another commonly used term to describe a transformer is the effective turns ratio. The effective turns ratio is equal to l22 over l11 square root. So it's essentially the square root of the ratio of the self inductance's of the transformer. Note that if the transformer has no leakage ie if Ll1 is 0 and Ll2 os 0, then the effective terms ratio ne simply becomes equal, the actual trans-ratio N2 over N1. Note, that in circuit similarities like spice and Ld spice, which have a couple inductor modeled that you can use to model a transformer, it is the inductance values L 2 2 and L 1 1, which set the transformer turns ratio. So knowing this relationship would be helpful to you there. Another common term used in describing transformers is the coupling coefficient. The coupling coefficient is the ratio of the mutual inductance to the square root of the product of the self inductances. Again, in the case where you have no leakage, this expression simplifies. So again, if we have Ll1 equal to 0 and Ll2 equal to 0, the coupling coefficient is going to become equal to either plus 1 or minus 1. The minus 1 comes because you have square root and the square root can take either a positive or a negative value. Physically, what that means is, it depends on how you've defined the relative polarities of the voltages, V1 and V2, or how you've marked the dots in your transformer. That's all for this lesson. We'll talk about multi winding transformers, in our next lesson.