Section 10.4 is about AC losses in winding conductors through the mechanisms called the skin effect and the proximity effect. These AC losses in the windings can be one of the largest sources of loss in a switching converter. And often, the transistor switching loss and the magnetics proximity loss are the largest sources of loss and inefficiency. Let's talk first about the skin effect. This comes from Eddy currents and lenses law in Maxwell's equations. So let's suppose we have, say some round copper wire or cylindrical wire, and we put an AC current, I have T through it in this direction. Maxwell's equations tell us that there will be magnetic flux around this wire, and by the right-hand rule it goes up, and over, and around like that. So there's a magnetic flux, F of t induced by the AC current i of t, and it's drawn here inside the wire. So if we have current flowing inside, the wire there will be magnetic flux inside the wire. Now, just as in the core loss or Eddy current loss inside magnetic cores, we have similar kind of Eddy current loss inside the winding conductors. And what happens is that this magnetic flux induces currents in the conductors, and the currents align in a way that tend to oppose changing flux. And ultimately, they end up opposing the changing current, so we see them going around in small loops like this. It's true cross-section is drawn over here, and you can see that the Eddy currents tend to cancel out the current in the middle of the wire, while they tend to reinforce or increase the current on the surfaces. We can solve Maxwell's equations, I'm not going to take the time to do it here. But we can solve Maxwell's equations to find how the current actually distributes itself inside the wire under these AC circumstances. And what we find is that the current density in the wire is a decaying exponential function of the depth or distance into the wire from the surface. So say we have a plot here, we got a plot of distance x into the wire versus current density J. And this function right here is some function that goes like e to the minus x over delta where the characteristic length delta is called the skin depth or the penetration depth, okay? So we have a decaying exponential function for current density versus depth into the wire. In other words, what happens is that the current crowds at the surface of the wire, and if we have a wire that's much thicker than one skin depth, then there's very little current flowing in the middle of the wire. What we're interested here is in the effect of this on the loss in the wire or what we call the copper loss. So if we had the DC case where we just have a constant DC current, and we have copper wire, then we can write that the copper loss is the current squared times the DC resistance. And the DC resistance follows the classic formula that it's the resistivity of the wire of row multiplied by the length of the wire divided by the cross-section area of wire. In the case of AC currents, the skin effect which causes the current to crowd at the edges effectively increases this resistance. And so we talked about an AC resistance of the wire that tells us the copper loss within an AC current, say a sinusoidal AC current flowing in the wire with sinusoidal currents. We can write this as the Ims current squared times this effective AC resistance. We can talk about the AC resistance following a similar kind of function. The resistivity times the length divided by some effective cross-sectional area of the wire. But the cross-sectional area of the wire effectively is smaller because the current is not flowing down the middle of the wire. It can be shown really by integrating the current density function squared times the resistivity. We can show that the AC resistance or this effective cross-sectional area of the wire is given by really an annulus, it's like we have a hollow copper pipe instead of wire. It says if the current is just flowing on the surface of the wire, or we have a copper pipe with a thickness that's equal to the skin depth and it's hollow in the middle. So this effective cross-sectional area, one can show is the area of this annulus or the copper part of this hollow pipe, having a thickness of one skin depth. So then we can talk about the AC resistance as being equal to the DC resistance times the ratio of these effective areas. The skin depth, delta can be shown, again by solution of Maxwell's equations to be given by this expression. It's the square root of the resistivity of the wire row divided by pi and the permeability, mu of the wire. The permeability of copper is basically mu naught, the permeability of free space, and then also times the frequency in hertz. So for copper wire at room temperature, if we evaluate this expression, it turns out that the skin depth is 7.5 divided by the square root of the frequency in hertz and this is in centimeters. Here's a plot of that versus frequency. So we often build switching converters with switching frequencies of maybe 100 kilohertz, or megahertz, or some something like that. Here's what this formula predicts for the skin depth, and we're getting skin depths that are what? Less than a millimeter at 100 kilohertz at room temperature, the skin depth is about 0.2 millimeters right here, and so it's pretty small. In fact, if we compare it to wire sizes, a wire that has one skin depth of diameter at 100 kilohertz is about number 33 gauge wire. This is American wire gauge, 33 gauge wire is pretty small, usually wouldn't put more than maybe a tenth of an ampere or so through that. If we have larger currents in an amp, or tens, or hundreds of amps, we have much larger wires that are maybe up here on this scale. So these wires have diameters much, much greater than one skin depth. So the skin effect is very important. In fact, it's even important at utility frequencies 60 or 50 Hz. Of course, the skin depth is larger, but skin depth is just a weak function of frequency, goes like square root. Plus often at these utility-scale applications, we have very large currents. And so we have large conductors that are even larger than this. They are big conductors and the skin effect even at 60 Hz can be important. So the skin effect increases, the copper loss, we have this effective AC resistance that can be much larger than the DC resistance. Now, if that were the only problem, we could probably deal with it, but actually there's an even worse thing that happens that is called the proximity effect. The proximity effect we can think of as a coral area of the skin effect. And it's what happens when you have adjacent conductors that have AC current and they interact with each other. For simplicity I'm going to draw the conductor as a square instead of round now. And we're not going to worry about the effect of round wires, we'll come back to that later. But we can think of, say, a square or rectangular cross section wire that is going in and out of the page here and it has some thickness h that is much greater than the skin depth. So let's consider what happens when h is much greater than the skin depth, delta. Okay, and this conductor, conductor 1 here, has current i flowing down it. And so we know from the skin effect then that the current density inside this wire is this decaying exponential function of depth into the wire. And so we have some function like this for the current density. And that current, I've drawn it here is coming out of the page. By the right hand rule, we'll have flux going around the wire like this. And now, let's consider what happens if we bring a second conductor up next to the first conductor. And let's say our second conductor has no current at all flowing in it. So net current of zero. Well, what happens when you put the second conductor next to the first one is that this magnetic field induces eddy currents in the second conductor. And in fact, if we have the same flux fee on the surface of the first conductor also near the surface of the second conductor, we'll get the same kind of current distribution. So in fact, there will be current on the second conductor that flows in a way to try to cancel out the magnetic field by Lenz's law or oppose changes to the magnetic field. And so the current on the surface of the second conductor will flow in the opposite direction. So it will go into the board in the opposite direction of the current and the first conductor to try to cancel the magnetic field out. So again, an equal and opposite current on the surface of the second conductor. Now, the second conductor has no net current. Yet if we have current i flowing out of the board, we must have some opposite current somewhere in the second conductor, so that the net current in the second conductor is zero. And what happens is that we get this return current flowing on the other surface, the opposite surface, of the second conductor. So we have current circulating into the board on the left side and out of the board on the right side of the second conductor. There is no net current in the conductor, they add up to zero, but we have current circulating around the second conductor. How much power loss is here? Well, the first conductor has a power loss, call it Pcu1 for the power loss in the first conductor, and that's equal to the rms current squared times the AC resistance that we talked about previously. The second conductor has two of those. It's got this equal and opposite current on one surface and an equal magnitude return current on the other surface, and so Pcu2, the second conductor, really has two of these. So there's a net copper loss now of three times. So the increased Rac comes from the skin effect, and now I have this factor of three that is coming from the proximity effect. So our copper losses are going up fast. Okay, let's think about a transformer, so let's suppose we're trying to build a transformer that has a primary and secondary winding, and let's say it's three turns to three turns, and we have current I, say, flowing in the primary, that's AC. And so we'll have a secondary current going in the opposite direction that is also I. Okay, here, I've drawn a structure with a magnetic core and then with some turns inside of it. Okay, let's talk for just a minute about how we do that. Here is a, what's called a ferrite pq core. It has a magnetic core, a ferrite core, and then inside of it, there is a bobbin. That's a plastic bobbin with some pins that let you put it into a printed circuit board. And we wind wire on the bobbin and then we get the windings on the core. So here's what the core halves looked like. There's a round center post and then some outside posts and the magnetic flux will go down the round center post and then out the outside posts. Okay, the bobbin fits inside the core. And what you do is you take your copper magnet wire and you'll wind it on the bobbin. So you wind your turns like this and then you put it inside the core. So what I'm trying to draw here, really, is just one part of something topologically equivalent to that. So our primary winding is, say, three turns. Our first turn goes around the core, so I've got it going around this way. Comes to the second turn which is put on top of the first turn, goes around this outside leg, goes to the third turn, and so on. So really we have turns going around this center leg of the core. The secondary layers are then wound on top of the primary layers, so they go around as well. So what we're drawing here is a cross section really of the winding. Okay? So in a transformer then, our primary turns or layers will have, say, current i, our secondary turns will have opposite currents so -i, as drawn here, okay? What happens with the skin and proximity effects? So let's start with the first turn or first layer. We have a current i flowing in it, it will flow on the outside of that layer, with this decaying exponential function, okay? So basically it's flowing in this right hand side of layer 1 with a characteristic depth of a skin depth. This will induce flux on its surface. That flux will induce an equal and opposite current on the adjacent surface of layer 2. So I'm calling that -i. But layer 2 has a net current +i. So if we have -i on the left surface of layer 2, we must have +2i on the right surface so that we get a total current in this layer of i. So there's 2i on the far side or right hand surface of layer 2. That makes twice the flux, 2 phi, on that surface and that 2 phi induces -2i on the adjacent surface of layer 3, okay? But then layer 3 has net current i, and so it must have 3i on its opposite surface to get a net of 1i. So then there's three times the flux on the surface of layer 3, and that will induce -3i on the left hand side of layer 3. Layer 3 has a net current of -i, so there's 2i on the right hand side, we'll get -2i in layer 2 of the secondary and -i on the left side of layer 1 of the secondary. So we have these currents building on the surfaces, the adjacent surfaces of the windings from the proximity effect. Okay, let's estimate the loss. So in the first layer, we have a current i. And we'll have an effective AC resistance, Rac, from the skin affect. And so we get a total loss of IRMS squared Rac. In layer 2, we've got that same loss on the left surface of layer 2, And on the right surface we have twice the current. So we'll have 2IRMS squared Rac on the right surface, and the total copper loss on layer 2 then is 1 + 2 squared, so 5, IRMS squared Rac. So five times the loss on layer 2 as we had in layer 1. The proximity effect makes the currents build up and it makes the copper loss go up very quickly. Let's look at layer 3. The left hand surface of layer 3 has 2i in current, so we'll get 2IRMS squared Rac. And the opposite surface has 3i, so we'll have 3 times IRMS squared Rac there. And so that's 2 squared + 3 squared. Let's see 4 + 9 is 13. So 13 times the loss in layer 3 is what we had in layer 1. So it's like 1, 5 and 13. Every time we add more layers, we can think of this as maybe flat copper wire or copper ribbon that's wound around the core. The loss is really going up geometrically with the number of turns. In fact, we can generalize this to say if we have layer m, lowercase m, these expressions are the RMS current squared times m- 1 squared + m squared. And this is the AC resistance. We can add up the losses in all the layers, so 1 + 5 + 13 would be 19. So with our three turn or three layer primary winding, the total copper loss then is 19 IRMS squared Rac. So we're getting big numbers here from the proximity loss. You can sum this series, if we have little m is the layer number, if we make capital M the total number of layers, so capital M was 3 in this case, we can sum the series and this is what we get. So in the case where M is 3, this term here turns out to be 19. Let's compare this with what you get at DC. At DC, first of all we have a resistance Rdc not Rac. And second, the number of layers or turns is capital M. And so the total copper loss at DC is the current squared times the number of turns times a DC resistance of one turn. So the ratio of this AC resistance to the DC resistance, we can divide this expression by this one and we call that the F sub R, it's the proximity loss factor by which loss is increased. And it turns out to be this function. It goes like the number of turns or layers, the number of layers squared here, and so it can go up very quickly. So any currents in winding conductors can significantly increase the copper loss in the magnetics in our switching converters, and it's worth learning how to model and control these losses, to understand how the skin effect or proximity effect work, to design good magnetics for our high frequency converters.
