Welcome back to the course on Magnetics for Power Electronic Converters. In this lesson, we will start discussing winding losses. In general, calculating winding losses can be quite complicated. However, when we're dealing with dc or very low frequency currents, calculating winding loss is actually quite simple. In that case, you can calculate winding loss simply by the square of the rms current times the resistance of the wire. Where the resistance of the wire is really just the dc resistance which is the resistivity of the wire times the length of the wire divided by the cross section area of the wire. Typically the wire is made out of copper, so the resistivity to use is that of copper. The thing to remember is that the resistivity of copper depends on temperature. Here's the value of the resistivity of copper at 25 degree C or room temperature. The resistivity of copper increases with temperature at a 100 degree C, here's the value of the resistivity of copper. It is generally safer to assumed that your magnetic device will be running hot. So it's safe to use the resistivity of copper in a 100 degrees C. So determining winding loss at low frequencies or at dc is clearly quite simple. However at high frequencies we have two additional effects that come into play, which make calculation of winding loss far more complicated. The first is what's known as skin effect And the second is known as proximity effect. Both of these effects will alter the resistance of the wire and essentially, what you have to compute then is what's known as the ac resistance of the wire. Once you have the ac resistance of the wire, you can go back and use the same formula for power calculation which is the square of the automus current times this ac resistance. We'll begin by looking at skin effect, and how that impacts the calculation of the ac resistance. When we calculated the dc resistance of the wire, we assumed that the current was flowing uniformly across the entire cross sectional area of the wire. So we could use the entire cross sectional area of the wire in the formula for calculating resistance. Let's see what happens when we crank up the frequency of the current going through this as well. Let's start first with a dc current through this wire. And let's say, we have a current that's flowing in this direction as shown and it's flowing uniformly through the entire cross sectional of this wire. We'll only show the current on one plane to keep the drawing simple, whereas you can imagine, the same current is flowing through this entire cross sectional area. If we were to sketch the current density through this wire along this plane, then that would come out to be flat as shown on this chart here. So this would be the current density at dc already low frequencies. Now the current that's flowing this wire is also going to generate a magnetic field that surrounds it. So there will be a magnetic field that surrounds this wire and flows in the direction shown using again the right hand rule, and this field at dc will also penetrate the wire. Now let's see what happens as we increase the frequency of this current. As we increase the frequency of this current, the frequency of the H field that's outside as well as that's inside is also going to increase. And as you recall, a changing magnetic field inside a conductor is going to induce Eddy currents. And the Eddy currents are going to flow so as to try and repel this field or cancel this field inside a conductor. The Eddy currents will flow in such a way, so as to generate a field that is in the opposite direction from the field that already here. And so that field that is generated by the Eddy current will flow as shown and will basically be in the direction that's opposite from the original field. In order to generate that filed that's cancelling the original field, the Eddy currents will have to flow in circles as shown. And the direction of the Eddy currents will be in such a way that they're generating this new H field, so the Eddy currents will flow in the direction shown. As you can see, in the center of the conductor, the Eddy currents are flowing in a direction opposite from the original current direction. While near the surface of the conductor, the dEdy currents are flowing in the same direction as the direction of the original current. So near the center of the conductor, the net current will diminish, while the current will increase near the surface of the conductor. In essence, the current has redistributed towards the surface of the conductor due to these Eddy currents. So if we were to again sketch the current distribution along this plane in the conductor. We will see that the current is close to if not exactly 0 at the center and it has much higher value around the surface. So it will have some kind of a shape as shown and it is flowing near the surfaces where it's essentially 0 in the center. This would be the shape of the current at high frequencies. We can determine the exact shape of this current distribution by solving Maxwell's equations inside a conductor. Doing that for a cylindrical geometry is fairly complex, however if we ignore the curvature of the wire, then solving Maxwell's equations inside the conductor give a solution for current density that looks like this. Basically the current density falls off exponentially from the surface. So both of these falls from the surface are going to be exponential. The spatial constant that defines the rate at which this fall occurs is known as delta or skin depth. To a first order, we can approximate that all of the current in the vial. Flaws in this skin depth delta from the surface of the wire. Now if you've drawn the current density carefully for these three cases, then the area under the third for each of these cases must be the same since in all cases we carry the same current through the wire. When we model the current flowing through the wire as if it flowing in only a skin depth from the surface, essentially what we're saying is that the current is flowing in a tube like conductor with a thickness equal to delta. And so the current is only flowing In this shaded region. So in our skin depth model, you can model the conductor like a hollow tube with current only being able to flow in the thickness of the tube. Note that, just like the current has been constrained just to the surface of the conductor. The magnetic fields inside the center of the conductor have also been cancelled and the magnetic fields now are also only present near the surface of the conductor. When we solve Maxwell's equation inside the conductor, we also get an expression for the skin depth delta in terms of the physical parameters of the conductor. Let's look at that next. Solving Maxwell's equations inside the conductor, give the following equation for the skin depth delta. The skin depth depends on the permeability of the material, which for copper is essentially the same as free space. It depends on the conductivity of the material and it depends on the frequency. Since conductivity is just the inverse of resistivity, you can also express it as shown here. Substituting the known values of conductivity and permeability for copper at a 100 degrees C, we can rewrite the expression for the skin depth as approximately equal to 7.5 centimeters divided by the square root of the frequency. Note that skin depth gets small fairly quickly at frequency equal to 60 hertz delta is approximately equal to 1 centimeter and at 20 kilohertz. Delta is only about half a millimeter and by the time you get the 1 megahertz, the value of delta is close to 75 microns. Here you can also see a plot of skin depth delta as a function of frequency. Note that delta falls as the square root of f, so when it is plotted on a log log plot, you get a straight line. This chart also shows you the wire gauge, which is a diameter corresponding to the value of skin depth at that frequency. Note that the skin depth at 500 kilohertz Corresponds to a wire of roughly 40 gauge. That is an extremely thin wire, and would not be able to carry much current. However, if you wanted the entire cross sectional area of the wire to be used for carrying current then at 500 kilohertz, you need a wire that's roughly a 40 gauge wire. So far we've looked at current redistribution in a wire due to skin effect when the wire is free standing. Let's see what happens to the current redistribution when the wire is part of a winding and inside a core. Here, you see a core with a winding with some current i going through this winding. To see what kind of fields exist in the proximity of the wire, let's model this winding as shown here. Here, we consider a winding with four turns of round wire, and the current is going into the screen to indicate that we show arrow tails indicating that the current direction is into the screen, because we're seeing the back of the arrow. To determine the fields in the vicinity of these wires, we can apply Ampere's Law. To do that we can select a loop that's in the proximity of the wire and encloses the wires. Then the strength of the H field will be given again by Ampere's Law by the total current that passes through the surface suspended by this loop. I would note that the magnetic field edge would not be the same in the part where the field is passing through the air in the four window area versus the field that's passing through the core itself. That's because the flux density bead has to be continuous around this loop. The value of H in the core window area is going to equal to flux density B divided mu not, while the field H in the core itself is going to equal to B divided by mu of the core, mu c. If we assume that the permeability of the core is much, much larger than the permeability of free space or air, then the value of H inside the core will be much, much smaller than the value of H in the core window area. If we assume mu c to be really really large, then we can approximate that H, inside the core, is approximately equal to 0. If we make that approximation then H only exists on the right side of these wires, and there is no H on the left side of the wires. By the symmetry of this winding, you can also convince yourself that the H field in the window is all pointing vertically in this case, vertically downwards. Because any field that tries to come in horizontally is cancelled by the field generated by the other winding which is pointed in the opposite directions. While there will be some courager of the field around these windings as they're coveing, we can approximate the field to be essentially going straight down. To further simplify the problem of finding the current distribution inside these windings, we can model these separate wires as a single rectangular wire that carries the same net current. We'll discuss a little later how you can model round wires using a single rectangular wire in a systematic way. The main takeaway here is that the fields impinging on these wires are only impinging from the right side. Here we consider the same winding in the core. And as we saw, the fields in this case exist on the right side of these windings. And so they are impinging on these wires from the right side, but there is no field inside the core. And so, there is no field impinging on the wires from the left side. The situation is similar on the other side of the winding, where you now have no field on the right side, but only field on the left side. Again, only in the window. For analysis purposes, we'll only concentrate on modeling the right side of the core but everything we do also applies to the left side of the core as well. To simplify the problem, let's model these separate wires as a single rectangular conductor. Here, we show an exploded view of that. Again, we are looking at the right hand side window of the core, and so the core structure itself is to the left of the wire, as well as to its top. For clarity, the core to the top and core to the bottom is not shown. As noted, the fields that are impinging on this wire are flowing along its side pointing downwards. When the current flowing through this wire, which is also producing these fields, is AC, then the fields, because they're time varying, when they try to enter the wire itself, they will be repelled by eddy currents flowing in the wire. The eddy currents will again flow in such a way so as to produce a magnetic field that is in the opposite direction from the field shown here. And hence, the field from the eddy currents will be in the upwards direction. The effect of the eddy currents will be such that it will tend to cancel the current in the interior of the wire and towards its left. And the current will redistribute itself towards the surface where the impinging field exists. The resultant current density distribution through this wire as plotted across its width, which we call h, is as shown below. Again, it'll have an exponential drop of current density as we move away from the surface of the wire, eventually getting to 0. Notice that in this case, there is no current that is flowing on the left hand side of the wire, since there is no h field on the left side of the wire. All of the current is now concentrated on one surface of the wire. Again, we can model this as if all the current is flowing within a skin depth of the surface of the wire. Note that if drawn carefully, the total area under the blue curve should match the total area under the red curve as both represent the total current through the wire, which has to be the same. This model is accurate when the width of the wire, h, is much, much greater than the skin depth, delta. In that case, since the current is flowing only in a skin depth delta from the surface, rather than the entire width h. The effective cross-sectional area of our wire has reduced by a factor delta over h. And since resistance is inversely proportional to cross-sectional area, the resistance in this case has increased by a factor h over delta, which is what you can use as a simple model to calculate the effective resistance which we call the ac resistance. So our ac resistance can be approximated by just the dc resistance multiplied by h over delta. Recall that the dc resistance is simply the length of the wire, which is the total length of your winding, divided by the cross-sectional area of your wire multiplied by the resistivity of the material, presumably copper. Once you had the ac resistance of the winding, you can simply calculate the loss in the winding by multiplying the ac resistance with the square of the autonomous current. This formula for ac resistance is an approximate one, and only holds when h, the width of the conductor, is much greater than the skin depth, delta. For this Cartesian geometry, you can solve Maxwell's equations inside this conductor. Calculate the current density distribution, use that to calculate a loss density distribution, integrate that to compute the total losses, and hence, calculate an effective value for Rac. That will take a few pages of mathematics, but in the end, the result will be as shown here. This expression here gives you an equation for the ac resistance for an arbitrary value of h relative to delta. Note that it has sine hyperbolic functions, cosh hyperbolic functions, sines and cosines in it. Note that in the limit that h is much greater than delta, this expression simplifies to the same expression that we are assumed for h much greater than delta. Also note that in the limit that h is much smaller than delta, this expression simplifies to simply that the ac resistance is equal to the dc resistance. You should expect this answer since in this case when h is much less than delta, effectively, you'll have uniform current distribution throughout the wire, and so your ac should be expected to be equal to the dc resistance. So depending on the regime you're in, whether h is much greater than delta, somewhere in the vicinity of delta, or much less than delta, you can use the appropriate value for Rac substituted in this expression. Knowing the rms value for the current that's flowing in your winding, you can compute winding losses. Note however, that if you want to utilize the full cross-sectional area of your winding, then you'd like h to be roughly equal to delta or less than delta. However, if h is smaller than delta, then the dc resistance starts to go up because we have less cross-sectional area in the first place. So with skin effect, the optimum value of h is roughly equal to delta. I said earlier that we could model multiple round wires by a single flat wire. Let's see how we can do that in a systematic fashion. Let's say, we have N round wires, each of diameter d. To begin, we will convert each of these round wires into a corresponding square wire. In making this conversion, we will ensure that the cross-sectional area of the square wire is the same as the cross-sectional area of the round wire that it corresponds to. This means that we will set the width of the square wire, h, as follows. We will make h square the cross-sectional area of the squared wire is equal to pi times d over 2 squared, where d is the diameter of the round wire. Solving this simply gives us that h is equal to the square root of pi over 4 times d. In our next step, we will simply take all of the square wires and squish them together to form a single rectangular wire. The single rectangular wire also has width h. But because h is less than the diameter of the round wire d The total height of this rectangular wire is less than the total height of the original round wires. To solve that problem, we'll take our rectangular wire and stretch it to match its length with the original winding. However, to insure that the cross sectional area available to current flow in this rectangular wire is the same as the original cross sectional area, we'll punch holes through this wire. Effectively, we made this flat wire from a porous material and they can define what is known as the porosity factor eta. In such a way that eta times the cross sectional area of this flat wire is equal to the total cross sectional area of our original windings. When you solve this, we get an expression for the porosity factor eta. The porosity factor is effectively telling us the amount by which the density of the material has decreased. We can then use the porosity factor to calculate an effective conductivity of the material in this larger conductor. The new conductivity, sigma prime, will simply equal the porosity factor time the original conductivity of the material used for the winding. Now we can use this effective conductivity of this winding to calculate the effective skin depth of this winding. The effective skin depth is simply given by a skin depth formula except we replace the conductivity with the effective conductivity. If you express this in terms of the original skin depth, it turns out that the new or the effective skin depth is simply the original skin depth divided by the square root of the porosity factor. So if we actually have a winding made up of round wires and remodel it using a flat wire. Then we can go back and adjust the value of skin depth delta by the porosity factor eta, as calculated here. We find the effective skin depth that should be applicable. Note that eta will always be a number less than one. So the effective skin depth will be larger than the skin depth that you originally thought. We've seen that due to skin effect, the optimal conductor thickness that we would like to use is equal to the skin depth of the conducting material at the the frequency at which we are operating. But we also saw that at medium to high frequencies that are very typical for bioelectronic converters, the skin depth Can be very small. So how do we deal with the case when we have to carry large currents at high frequencies. In this case we have a number of options. We could use what's known as Litz wire or we could use Foil winding. Litz wire is basically standard wire where you choose each of the strands to have a thickness roughly equal to the skin depth at the frequency at which you are planning to operate. Each individual strand is insulated from the other strands because if it was not this bundle of strands would essentially look like a single wire and the current would simply flow on its external strands. However, that is not enough to prevent the current from flowing only in the outer strands, since the strands are electrically connected on the outside of the magnetic device. To ensure that all of the strands carry the same amount of current, the strands have to be twisted in a special way. In a Litz wire, each strand takes on different positions along the length of the wire itself. For part of its length, a strand may be on the outer side of the Litz. Then moving to an inner position then to the inner most position and then moving out again. This v thing pattern is followed for all strands, so that each strand sees the same magnetic field along the length of the winding as any of the others. This ensures that the impedance of any of the strands is the same as that of the others and the current has no preference to flow through one strand or another and hence it flows through all of them equally. By having multiple strands you are able to carry larger currents, even at higher frequencies, while having the current flow uniformly across the entire conductor. Unfortunately, Litz wire can be very expensive, as manufacturing it is a substantial effort. A relatively inexpensive alternative is Foil winding. A Foil winding is essentially a rectangular wire that can be made very thin. Here you can see an example of a Foil winding and you can wind a magnetic structure as shown here. Again, we're using the tail of the arrow to indicate that the current is going into the screen and an arrowhead to indicate that the current is coming out. So far we've only seen how current redistributes in the winding when we have a single layer of the winding. In our next lesson, we'll see what happens when we have multiple layers of windings wrapped on itself.