Welcome back to the course on magnetics for power electronic converters. In this lesson we will learn about core losses. Core losses arise because of two phenomena in the core. One is hysteresis and the other are Eddy currents. We will look at losses associated with each of these phenomena in detail. But, in summary hysteresis loss is because of the energy required to rotate the magnetic domains in the core material. And the eddy current loss is a resistive loss, or an ie squared r loss because of the induced Eddy currents that flow in the core. Let's consider Hysteresis loss first. The magnetic course we typically use are made of ferromagnetic material. Ferromagnetic materials are made up of tiny magnetic domains. You can think of a magnetic domain as a tiny magnet that can rotate when a magnetic field is applied to it. Sort of like a compass rotates when in the Earth's magnetic field. It is the presence of these magnetic domains that enables magnetic cores to have high permeability. Recall that permeability is essentially the ratio between magnetic flux density and the magnetic field. Just like a magnet, the magnetic domains produce magnetic flux because of their internal electron spins. Under normal conditions, without an externally applied magnetic field, the magnetic domains are randomly oriented and therefore produce no net flux. However, when a magnetic field is applied to them, the magnetic domains start to rotate to align themselves with the applied magnetic field. Hence resulting in much larger magnetic flux than would have been predicted by simply mu naught times h. This amplification of the b field is the reason why the apparent permeability of cores is much higher than the permeability of free space. However, it takes energy to rotate these magnetic domains. And if the applied magnetic fields is varying and possibly changing direction then the magnetic domains also have to rotate along with it. For example if you are applying a sinusoidal current to a winding wrapped around the core, the h field in the core is also varying sinusoidally. As a result the magnetic domains will also have to rotate their alignment at the frequency of the applied current. However, to rotate the magnetic domains have to overcome frictional forces and that requires energy. Hence there are losses because of this effect. This effect is what causes the BH loop to be not a single-valued function. And in fact creates a hysteresis in the bh gov which is why this effect is also known as hysteresis. And the losses associated with it are known as hysteresis losses. Let's try and develop a model for this hysteresis loss in terms of the physical parameters of our core. To do this, let's do the following thought experiment. Let's assume that all of the losses associated with our magnetic device are due to hysteresis loss. In that case, if you would to measure the energy going into the winding of our magnetic structure over one complete cycle then if there was any net energy, that energy must be associated with the hysteresis loss. Note that if there was no loss in our magnetic device, then the energy that goes into our magnetic device and part of the cycle would be equal to the energy that flows out in the rest of the cycle. So if there is any loss that loss must be measurable by simply measuring the average energy going into our magnetic device over a complete cycle. Lets complete this thought experiment by computing what this energy would be. Lets consider the core shown here with a winding, with N turns. The core itself has a cross section area of Ac and it has a mean length in the core through which the flux flows of lc and a permeability of mu c. The flux that flows in the core is v and this flux flows and the result of the current i flowing in the winding. And we call the voltage across the winding terminals, v. Now the power that goes into the winding is simply v times i. So if we integrate v times i over one complete cycle we get the energy that goes into our magnetic device over one cycle. We can convert this expression into one that involves the physical properties of the core by writing v and i in terms of the core properties. Recall that the voltage of the terminals is simply equal to d lambda dt, where lambda is the flux linkage. And the flux linkage in this case is simply n times phi. So we can write this as n d phi dd. Phi in turn is just the flux density b times the cross sectional area Ac. So this can be further written as N times Ac times dB dt. Similarly, we can find an expression for i. Recall from Amperes law that the integral of H around this loop which will be H times lc must equal the total current cutting the surface of this loop which is just N times i. So Hl is nothing but Ni and therefore i is can be written as H times lc over N. Now that we have expressions for v and i, we can substitute these into an expression for the energy flowing into the winding over one cycle to get an expression that is in terms of the physical properties of the core. So we get the W is the integral gain over one cycle of simply NAc dB dt times Hlc over N integrated over dt. We can cancel out the N. We can also change this to an integral in dB by canceling out the dt. And so this expression can be rewritten as simply W Equal to Ac times lc since they're both constants, I can pull that out of the integral and so, the integral is again over 1 cycle. And now, it's just an integral of H with respect to B. So, here's a final expression for the energy going into the winding over one cycle, which is also the loss due to hysteresis. Let's see what it is saying. Note, that Ac x Lc is nothing but the volume of the core. So, we can simply define that as a script Vc see which is the volume of the core So what is this integral, HdB? To integrate H with respect to B over one cycle, we simply have to integrate H as a function of B as we transcend this BH curve. Let's say we have some starting value which corresponded to some minimum value of B, let's say Bmin. This has some corresponding value of H and so, to find the integral of H with respect to B, we integrate which is essentially the area under this curve as we move along this curve. And note, we're moving along the curve to the right, since that's the curve to follow when H or B are increasing. So, we move along this curve up, until we reach the maximum value of P, let's say that's some value B max. Then, the area that we covered while we were going up would be essentially the area of this shaded region shown right here. To complete the cycle, we have to integrate all the way back to Bmin. Since this time, the value of B and H is decreasing, we have to follow the curve to the left and so, we go down on this spot until we reach Bmin. And this time, the area is the area between the dB axis and this line, so that's this shaded part. So what is, then, the value of this integral? HdB over one cycle. Well, using that example, we can rewrite this as an integral, which goes as follows. So, we integrate from Bmin to Bmax, H, and the H we follow is the one to the right, so we'll just call that HrdB. Now, we also have integrate on the way back and that's starting from Bmax coming back to Bmin. So, that's an integral from Bmax to Bmin of HDB, but this is now the H to the left, so we'll call it HlDB. Note, that the second integral is integral from Bmax to Bmin, so to put it back in the same form as the first integral. I can simply swap the sign here and rewrite this integral as simply minus the integral Bmin to Bmax of HldB. Therefore, our original integral HdB, or one cycle Is nothing but the difference between the area of the first integral which is the area under the H curve to the right minus the area of the H curve to the left. On this figure, that simply the area between the two curves. So, this area here is simply our integral HdB over one cycle. If the range of flux density B over which we moved along this curve was narrower, then the area would be smaller and our loop would be smaller. If on the other hand, the values of B over which we moved extended to a much larger region of this PH loop, then the Hysteresis Loss would certainly be bigger. Therefore, to summarize, a Hysteresis Loss is given by the area enclosed by the BH loop over the values of B and H that we move over, multiplied by the volume of the core. So the bigger the core, the bigger the loss. Now, this is the energy lost in one cycle. To determine what is our average power lost, you must multiply it by the number of cycles we transcend in the second, which is essentially an operating frequency. So to determine the average power loss, you must multiply the result By the operating frequency f. And so this expression then gives us the expression for the average power loss due to Hysteresis. Note, based on this formulation, the Hysteresis Loss is proportional to our operating frequency. That is assuming, that this integral HdB has no frequency dependents. In reality, that is not going to be true because the shape of the BH curve itself depends on the frequency at which you're operating. Note that this full BH loop that's shown here, is only a snapshot of the BH loop that you may actually go over and it depends on the range of B and H values over which you move, as well as the actual operating frequency. If we assume tha the shape of the BH loop does not depend on the operating frequency, then we can come up with an approximate model for the average power loss due to Hysteresis that has a linear dependence on frequency. We can also roughly model the area of the BH loop that we surround by a parameter delta b which is the flux swing. The flux swing, Delta B is simply the average to peak value of the flux swing we have. So, if our swing in B is from some Bmin to some Bmax So that we are essentially covering the loop as shown. Then, we can define an average value. B average, which is in the middle of B min and B max, and then delta B is simply the distance from the average to the maximum value, and that's how we define delta B on the flux ring. I this model k h and alpha are fitting parameters. In theory, you could try and find their values by fitting this expression to measure data, but in practice, measuring hysteresis loss by itself is extremely difficult unless you have no other losses in your core. Let's turn our attention then to the other type of losses that occur in cores The other loss in the core is due to Eddy currents. Eddy currents are currents that flow in a conductor whenever there is h ending flux in that material. It turns out that many of our magnetic materials are also conductors. At least have some level of conductivity. Recall that iron is a good magnetic material but is also conductive. Therefore, when we have changing flux in the magnetic core we will also have induced Eddy currents. By Maxwell's equations, eddy currents flow so as to oppose a change in flux. Let's see how eddy currents will flow in this example structure. Here we have some time varying current applied to a winding around a core. This current produces a flux that goes through the core. And this flux is also changing with time. The Eddy Currents will flow in a plain perpendicular to the direction of the flux. Now here, the flux generated by the winding current, both flow from right to left as shown. The Eddy Currents will flow so as to cancel this changing flux. Through that, they will flow in the plane perpendicular to this flux, and by using the right hand rule, we can determine that the Eddy currents would flow in the direction shown. This current is going to produce a field that is in the opposite direction from the original flux and will tend to cancel the original flux in the center of the core. You can also see that in this picture right here, we have the original flux is flowing from left to right. And the eddy currents are flowing in a direction, such that, they are producing a field which is going in the opposite direction and trying to cancel the original field. Mathematically, this comes straight from Maxwell's equation or from Faraday's law in particular. You can see when you have a changing magnetic field, it's going to result in an electric field and that electric field in turn, will generate a current depending on the conductivity of the material in which the electric field is present. Eddy currents cause two problems. One, they clearly cause I squared R losses because of the resistivity of the material in which they flow. In addition, they redistribute how the flux is flowing in the core. Note that in the center of the core, they're cancelling the original flux. However, near the surface of the core, they're actually reinforcing the original flux, because if you draw the full flux for this loop formed by the Eddy current, the flux lines will actually look something as shown. So near the surface, the flux lines created by the eddy current are actually adding to the original flux lines, while in the middle of the core, they're cancelling them. As a result, the flux distribution is nonuniform inside the core. And the flux density increases around the surface of the core and is essentially zero in the middle. So essentially, the cross-sectional area available for the flux to flow on the core is substantially reduced, because now, the same amount of flux is essentially flowing just near the surface of the core. So if you design your magnetic device assuming a certain cross sectional area of available for flux flow, that is probably no longer going to be true. And so your magnetic device may actually start to saturate. We can also come up with an expression for the Eddy current loss from first principles. The Eddy current loss is essentially an i squared R type of loss. So the instantaneous Eddy current loss is equal to the instantaneous Eddy current squared times the resistance of the material through which they're flowing. To determine the average power loss, you will simply average this expression over one time period. To determine the functional dependence of Eddy current losses on the physical properties of the core, let's see what the eddy currents are proportional to. In a material whose impedance is purely resistive, the strength of the Eddy currents will be proportional to the voltage that's driving it. From Faraday's law, this driving voltage itself is proportional to the rate of change of the flux. That is in the core, which is d phi d t. Which can in turn be written simply as, A c, which is the cross of the core, times db dt would be is the flux density in the core. Let's assume that our driving current is sinusoidal in which case our flux density will also be changing sinusoidally. Let's say B is equal to some B-naught times some sin of 2 pi ft, where f is the frequency of the sinusoid. Therefore, dB/dt is simply equal to to 2 pi f B nought, Cosine of 2 pi ft. Therefore the voltage, VE which is driving the Eddy current, is proportionate to the frequency as well as the amplitude of the flux density B naught. Since the current ie is proportional to ve, we also have that the current ie is proportional to the frequency f and proportional to the magnitude of the flux density. And since power loss is proportional to the square of the current, then the power loss is also proportional to the square of the frequency and the square of the magnitude of the flux density. Here, delta B is essentially the same as B nought. So our approximate model for the average power loss due to Eddy current is given here. And notice that unlike hysteresis loss, Eddy Current Loss goes by the square of the frequency while the hysteresis loss was proportionate to frequency. Also the dependents on delta B can be different from the case of hysteresis loss. Note that this model is for core materials that have purely resistant impedance. Some core materials have partially capacitance looking impedance in which case the functional dependence can be quite different. Although we have separately modeled, hysteresis loss and Eddy Current Loss using approximate models. In practice, it is very difficult to measure hysteresis loss and Eddy Current Loss separately. So it is near impossible to determine the fitting parameters for our approximate models. In practice, if you measure the core loss by measuring the energy that's going into the core over one cycle. What you would really be measuring is the sum of the hysteresis loss and the Eddy Current Loss. In practice, we use a single expression to model the combined effect of hysteresis loss and Eddy Current Loss and just simply call it the Total Core Loss. This Total Core Loss is modeled by the well known Steinmetz equation that is shown here. In the Steinmetz equation, the Total Core Loss is written as a function of the operating frequency F and the flux swing Delta B. And the broadened core loss is also proportional to the volume of the core. This equation has three fitting parameters. K nought, alpha and beta. F is raised to the power of alpha and delta b is raised to the power of beta. These fitting parameters can be determined from core loss data that is typically provided by core manufacturers. At a given frequency, this version of the Steinmetz equation can be simplified to the one shown below. Here, the dependence on f has been absorbed into the parameter gear one, and so there are now only two fitting parameters, K one and delta which is the power to which delta B is raised. The type of core loss data that core manufacturers provide is also shown here. Here you see a plot of Power loss density. With power loss density is the power loss per unit volume of the core essentially equal to, The actual power loss in the core, Pc, divided by the volume of the core, Vc, which would be equal to. If we were to use the second version of the Steinmetz equation, K1 times delta B raise to the power of Beta. Note that in this chart the power loss density is giving in watts for centimeter cube. So if you're working in SI units or in watts per meter cubed. You want to make sure you do the units conversion before you utilize the data from this chart. The power loss density in this chart is given as a function of the flux swing delta-b for a range of values of delta-b. And also for a range of different operating frequencies from 20 kilohertz down here, all the way up to 1 megahertz. So, if you want to know what your power loss density is at a delta b of, say, 0.03 tesla, which would be here, then you would simply go up to the frequency of interest. Let's say that was 500 kilohertz so that's this first line here and then you can read off what your power loss density would be for that delta B and 500 kilohertz operating frequency. Now, you would have to multiply the power loss density by the actual volume of your core to figure out what your actual power loss in the core is. Note that you can also use this chart to find the values of the two filling parameters key one and beta for example, if you wanted to know the values of k 1 and beta for an operating frequency of 200 kilohertz. You would go to the 200 kilohertz line and pick two points on that and let's say these are the two points, then for each point you can read off the value of delta B and the corresponding value for power loss. That gives you one equation, which has two unknowns and then from the other point, you get another equation, which also has two unknowns. And by solving these two equations simultaneously you can find the values for K1 and beta that would be suitable for 200 kilohertz operation. You can repeat the process and get different values of K1 and beta for 500 kilohertz or 1 megahertz or whatever frequency you are interested in Now that we know the origin of Hysteresis loss and Eddy Current Loss let's see if there are ways to minimize them. For a given core material and a given core volume there's not much you can do to reduce hysteresis loss. The only things you could do is reduce your operating frequency and reduce your flux swing, if that's a possibility. For Eddy current losses, we can do one more thing, and that is to use laminations. So instead of using a solid core, we'll slice up the core Into thin laminations as I'm showing here. Essentially the core is built through thin slices of core material that are stacked together with insulation between them. When the core is sliced up like this with insulation between each of our slices, It blocks the natural flow of the eddy current and the eddy currents are forced to flow in much smaller loops. We can see that pictorially here where the top picture here shows the original core without any laminations. And in this case, You have the flux going down the center of the core and the Eddy currents are flowing in a way so as to oppose that flux. Since the Eddy currents can flow in large loops in this case, corresponding to a large amount of flux enclosed by them. Then by Faraday's law, we can see that we have essentially the integral of flux density over a large cross sectional area. Which results in a large electric field being generated which, subsequently means, a large Eddy current being produced. On the other hand, if we slice up the core, as shown in the pictorial below, then the Eddy currents flow in much tighter loops, corresponding to this area becoming small. And so, the corresponding electric fuse becomes small and so do the generated Eddy currents. You can also see this from a more visual perspective by seeing that as you slice the core into thinner and thinner slices. The Eddy currents are essentially being forced to flow back and forth on top of one another. Hence, there is not much effective eddy current flowing at all. This method of reducing Eddy currents using laminations is commonly used when the core material is an iron alloy which has high conductivity and hence high Eddy currents. You will commonly find 50 or 60 hertz transformers that are built with laminations. Note that the thinner the laminations, the smaller the Eddy Current Loss will be. Before we close our discussion on core losses, let's quickly review some commonly used core materials. At low frequencies such as at 50 or 60 hertz, commonly used core materials are laminated iron alloys. The most popular amongst these is silicon steel. Iron alloys are good because they have high saturation flux density. As high as two tesla but the problem is that they have relatively high core losses especially because their high conductivity and high eddy currents. At higher frequencies, powered iron or powdered molypermalloy are used as the name suggest, powdered iron cores are made from small particles of iron embedded in epoxy. They have low saturated flux density, relative to laminated iron but they have also low core losses. At even higher frequencies, especially frequencies from 20 kHz up to 20 MHz, the material of choice are ferrites. Ferrites have even lower saturation flux density, but they have the lowest core loss. Ferrites are essentially a ceramic material containing iron oxide and other metals. There are very hard resistivities, and therefore, resent very low Eddy Current Loss until you get to very high frequencies. Commonly used ferrites are generally classified into manganese-zinc ferrites or nickel-zinc ferries. The manganese-zinc ferrites have relatively high permeability, but the resistivity relative to nickel-zinc ferrites. Is lower and therefore, they're useful up to frequencies in megahertz or a few megahertz. Above a few megahertz of operating frequency, Nickel-Zinc ferrites are the preferred choice. Once you get to frequencies much above 20 megahertz, the losses in the available ferrites also become too high, and you're best off using air core magnetics. That is to say that you have no magnetic core at all and typically, you wind your winding either on some on non-magnetic plastic material, or they're just made in the PCB itself. That's all for this lesson. In our next lesson, we will start discussing winding losses.