Welcome back to the course on Magnetics for Power Electronic Converters. In this lesson we will learn how to reduce the impact of proximity effect in transformers using a technique known as interweaving. Recall from our earlier lesson where we looked at the power loss in the windings of a two winding transformer. This was the case where we had assumed that the width of the conductor, h, was much greater than delta. In that case, we had been able to come up with the expressions for power loss in each of the layers. And then by summing all of these power losses in the different layers, you're able to find the power loss in the primary winging and be equal to 19 times P1. Where P1 was the ac loss in the first layer which was equal to h over delta times the dc loss. Due to the symmetry of the structure, the power loss in the secondary winding was also equal to the power loss in the primary winding and equal to 19P1. So our total power loss in the winding in this example came out to be 38P1. We had also noted that if we did not have proximity effect, then the power loss in the winding would simply have been 6 times P1. Hence, if there is a way to reduce to proximity effect, there is opportunity for considerable reduction in loss. Note that this is an extreme example, since under the assumption that H is much greater than delta, which gives us the worst case loss in the windings. Also note that even after we've gotten rid of proximity effect, our loss would still be higher than the DC loss. Since they're still going to be skin effect. And that loss could be as high as h over delta times the power loss in the dc case. To see how we could potentially reduce proximity effect and the loss associated with it. Let's see what was causing proximity effect in the first case. Here again is the same example that we just looked at showing the current distributions in each of the layers. The excessive loss was caused by the extra eddy currents that were flowing in the layers that were in the middle of the winding. The reason we had large eddy currents in the layers in the middle of the winding was because the field impinging on these layers was building up from the sides of the core window towards the center. We can see the strength of the impinging fields on each of the layers from the MMF diagram for this core. In our original transformer the MMF between the layers built up from a value of i between layer 1 and layer 2 to a value of 2i between layer 2 and layer 3, to a value of 3i between layers 3 of the primary and layer 3 of the secondary until they start to go back down again as we go towards the end of the window area. Since the field between the layers is proportional to the MMF, and in fact given by H multiplied by the height of the window is equal to MMF. We can reduce the strength of the H field that is impinging on the layers by reducing the value of the MMF. In the area between the layers. To see how we could reduce the value of MMF between adjacent layers, consider an alternate way of winding the same transformer. Instead of putting all the primary layers next to one another, one after another followed by the secondary layers. What if instead, we have one layer of primary, followed by one layer of secondary, then followed by another layer of primary. Followed by a layer of secondary. And then a primary. And then finally the secondary. This way of winding a transformer is known as interleaving. Because we have interleaved the two different windings, the primary and the secondary, by winding one, then the other, then the first, and then repeating that process. Let's see if this winding pattern can be beneficial. To do that, let's determine the MMF between the windings in this case. Again in our first primary, we have a current I, that will be flowing close to its right surface. Because on its left surface is the core, where we assume that there is no H field. And so to expel the H field that is impinging on layer one from the right, the current is situated on the right surface of layer one. We can determine the MMF between the first primary and the first secondary layer by simply doing a loop around as shown. Since this loop encloses. One current of value I, the MMF at this location is also simply equal to 1 I. So the MMF between the first primary and the first secondary has not changed from the MMF we had in our original finding. But now lets consider what happens to the current distribution in the first secondary. Since we have an H field of strength, proportional to this current I, impinging on the left side of the secondary, the secondary would like to have a current flowing on its left surface. That is directed into the surface of the screen. This current will also have strength i. Note, however, that the secondary is already getting a current, i directed into the screen so it does not need to generate another eddy-current in order to block this time wearing field on the left side of it from entering its center. As a result, there is no current on the right side of the secondary. Hence, there is no field on the right side of the secondary. We can also confirm this by determining the MMF between the right side of the first secondary and the left side of the second primary. To do that, we simply try another loop which passes through the region of interest as shown. Since this loop encloses a current directed into the screen, and a current directed out of the screen, it has no net current enclosed within it. Since there's no net current within this loop there is no MMF along this region. Hence, the MMF between the right side secondary one and the left side of primary two is zero. Compare this to the case of original winding where the MMF had build up to 2i in the region between the second and the third layer. Here the second and third layers have no MMF between them and hence no H fields. We can repeat this thought process to figure out the current distribution in the second primary. Since there's no field impinging the second primary from its left side for that as we will soon see there will be field impinging it from its right side. The current in the second primary will flow, flows through its right surface. And again, since it's a primary this current will be directed coming out of the screen. We can again confirm that there will be field and MMF to the right side of the second primary by again doing another loop through this region. And since that loop would then enclose a net current i there will be a field in this region directed upwards. Since the net current enclosed in this loop was equal to i, the MMF value between the second primary and the second secondary is again equal to i. Again, compared to what was going on in our original finding this is a much smaller MMF. In our original finding the MMF by this time had risen to a value of 3i, while now we still have only an MMF of i. We can repeat this process to show that the MMF between the next set of secondary and primary is again back to zero. And that the MMF between the third primary and the third secondary rises back up and only up to i. So the highest the MMF ever gets in this interleaved version of the winding is i. Compared to our original winding where the MMF started off at both ends being equal to i but then rose to 2i and then finally rose up to as high as 3i. The high MMF in this case resulted in large impending fields on the layers, resulting in large in the currents which in turn resulted in the large losses. By winding the same transformer differently, basically by interleaving the primary and the secondary windings, we're able to ensure that the MMF value never exceeds i. Note that the MMF value of i is what you would have gotten if we had only had skin effect in play. Since that's associated with a field strength equal to the field strength that would have been produced by the current flowing in a single layer by itself. In effect by interleaving the primary and secondary, we have completely eliminated the proximity effect and all that we got left is the skin effect. Each layer now only sees a field proportional to its own current eye impinging on one of our surfaces. And each layer has a loss equal to simply P1. Resulting in a total loss for the whole binding equal to 6P1. Note that this is the loss we expected if there was no proximity effect. Compared to this the loss we had in our original winding was equal to 38P1. So we've considerably reduced the total loss in our winding by using interleaving. Note that by reducing the MMF in the window of our core, we've also reduced the leakage flux associated with our transformer. Therefore, designing your transformer using interleaving techniques is a good way to minimize the leakage inductions in your transformer. However, manufacturing a transformer that is interleaved requires more effort as you have to first wind, one winding then add a second winding, then again the first winding and repeat that process. Also, if you need large insulation between the primary and the secondary you may have to add some extra tape in between these layers to ensure that your insulation requirement is being met. However, from a proximity effect perspective and from the perspective of minimizing the total winding losses, interleaving is a big win. In the transformer example we just looked at, interleaving was relatively easy to do since we had an equal number of primary and secondary layers. So we could simply put one primary followed by a secondary. Then a primary, a secondary, a primary and a secondary. These form of interleaving is known as a fully interleaved structure. And in this case the worst case MMF will only rise to as high as the current in one of the layers. For a fully interleaved winding, we can determine the loss in each layer for the case where the width of the layer H is arbitrary relative to the skin depth delta by using the charts we had presented earlier. In using those charts, we should use the curve that was associated with M being equal to one. We can use the M equal to one curve for finding the losses in these layers since each of these layers essential just has the skin effect and no proximity effect. And the current distribution in each layer is simply the same as that would have been in the first layer of a multi layer transformer. If the number of primary and secondary layers is not equal but is related to one another through an integer multiple, then we can still quite easily interleave these windings. For example, if we have twice the number of secondaries compared to the primaries, we would simply put one primary followed by two secondaries. The other primary and two secondaries, a primary then two secondaries. The MMF diagram in that case would look fairly similar except the height to which it grows corresponds to the current in one of the primaries or two of the secondaries. We could call that case still a fully interleaved case. If the number of primary layers and secondary layers is different and not related by an integer multiple, then we can still do interleaving, but our interleaving would then not be considered fully interleaved. In that case what we would have would be a partially interleaved situation Here is an example of a partly interleave transformer. In this case we have three layers of primary, each carrying a current I and we have four layers of secondary, each carrying three fourths of the current I. The currents have to be related in this fashion since the transformer has an effective threes to four Dunn's ratio. One way to interleave the transformer with three primaries and four secondaries is as shown here. To determine if this is the best way to interleave in this case, we would want to know the total power lost in the winding. Your objective, again, would be to minimize the MMF that develops between the different layers. You can find the MMF between layers by, again, making a loop that goes between those two layers and figuring out what is the current enclosed in that loop. For an example, to determine the MMF between the first and the second layer, we would make a loop that surrounds the first layer. And we would find that the value of the MMF is minus three-fourths i. Note that we're interested in minimizing the magnitude of the MMF and the polarity is not important. As the polarity simply tells us the direction in which the magnetic field is pointing. We can follow the same procedure to figure out the MMF values between all of the different layers and we would get a chart similar to the one shown here. Note that in plotting this MMF, we have not paid attention to the details of how the MMF varies within the layer itself. That's because the formulas and the charts that we looked at earlier for finding the loss in a winding only depend on the MMF between the layers and not within the layers. If you want to use the charts that we have seen earlier that showed the loss in a layer for a given layer number and for an arbitrary value of h relative to the skin depth. Then what we really need to calculate is a specific layer number for each layer. Recall that the way we have defined layer number was using the formula presented here. Again, in this formula F(h) is the MMF on one side of the layer and F(0) is the MMF on the other side of that layer. Since we've calculate the MMF across each layer, we can plug those values in this formula and get a value for M for each of these layers. Note that in some cases, if we do that arbitrarily, we may get values of M that come out to be negative. But as discussed earlier it does not matter which side of the winding we call F of h and which side we call F of 0. And we can do either way and it would work out to give us the same loss. Since our original charts only catered for positive values of m, we can swap F(h) and F(0) if m is coming out to be negative. that way we will get a positive value for M in terms of losses will be equal to the negative value we got. If you apply this method to this example here, the values of M that we get for each of these layers is written here. Notice that some of the M values turn out to be non integers. That's okay as M is defined by this formula does not have to be an individual value. Even in our original power loss chart, we had sketched power loss for some of the non-integer values of M. Once you've used the power loss chart, with the small M value given here to determine the loss in each of the layers, you can simply add up all of these losses to find the total loss in winding. In general, you can try different interleaving strategies until you found a pattern that minimizes the total winding loss. Based on what we have learned, here are some tips on how to minimize proximity loss in your windings. First of all, minimize the total number of layers in your winding structure. If at all possible use a single layer. Sometimes you can achieve this by using a different code geometry. For example, if you're using a core with a window that's not very tall, you will be forced to essentially have multiple layers. This is not a great choice. A better choice would be to pick a core which has a window that's taller. In this case, you could potentially fit enough copper on a single layer to get all the current in the turns that you need. So real possible you can try this instead. In cases where you have multiple windings, such as in a transformer, you can use interleaving. Note that interleaving will not work for an inductor, since you only have a single winding. Not only do you need multiple-windings to achieve effective interleaving, but you also need the currents in those mutliple-windings to be flowing in such a direction that they can produce flux that cancels each other. In some transformers, that may not happen. For example, in the flyback transformer, even though you have two windings, only one of the two windings is getting current at a time. Therefore interleaving the two windings of a flyback transformer will not do anything to reduce the proximity effect. Another thing you can do to minimized the loss due to proximity effect is to use in optimal thickness of the copper use in each layer. Be so what the optimal value of the conductor thickness was in an earlier lesson. If you only had one layer, the optimum value of h was slightly greater than the skin depth, delta. However, with multiple layers h is either close to delta, or below delta, and sometimes substantially below delta when we have a large number of layers. Finally as a general rule it is always good to try and minimize the amount of conducting material or copper in the vicinity of high fields, or high MMF as that will minimize the Edi currents or the loss of the winding. With that we have come to the end of our discussion on winging losses. And loses in general, in magnetic devices. In our next lesson, we'll pick up a new topic.
