Welcome back to the course on Magnetics for Power Electronic Converters. In this lesson we will learn how to design filter inductors. As in the design of any magnetic device, the design of a filter inductor means that we need to select a particular core. Specifically we need to select the size of the core. We also need to figure out how many turns we need in the winding, and you also need to pick the size of the wire to use in the winding. And in the case of the inductor since it has an air gap, we also need to specify the length of the air gap. Typically in the design of a filter inductor we will either be given or required to calculate the following specifications. First of all, we would need to know what is the required inductance for our application. If this is not given to us then we can figure out the value of L we need from the current ripple that is allowed in our application. Note that the smaller the ripple, the larger the value of L that you will need. In order to design the inductor we must also know the maximum current that the inductor is required to carry. You can determine this by adding the average to P in the inductor to the average current through the inductor. In the case of the Buck Converter, where the filter inductor is at its output, you can figure out the average value of the inductor current i. Simply from i equal to the output power of this converter divided by its output voltage. If the ripple in the filter inductor is very small, you can also approximate the maximum current through the inductor as simply being equal to the average value of the current. But for a more precise calculation, you probably want to add the average to peak ripple current to this average value to get the maximum value of the current through the inductor. The final specification we need to design the inductor is related to the power losses in the inductor. In a filter inductor, since the ripple current through the inductor is typically very small. We can assume that the losses in the inductor are heavily dominated by the DC losses in the winding. Therefore we can assume that the power loss in the inductor is given by the square of the rms current through the winding times the DC resistance of the winding. Basically, we're ignoring the core losses and we're also ignoring skin and proximity effect. Typically, we'll be given a power loss budget for our inductor. To be safe, we can leave some of that budget for potential core and easy losses in the winding. And we can use the rest of the power loss budget and calculate the value of R. For example we could leave 20% of our power loss budget for losses in the cord, and AC losses in the winding. We can then use the rest to calculate a value for the winding resistance R using this formula. Hence, our inductor design problem reduces to finding these four different things. The core size, the number of turns of wire, the wire size, the air gap length, given these three parameters. The inductance, the maximum current through the inductor and the inductor's winding resistance. In order to determine these design quantities, we need to have some design equations. There is no single one best design method. What I will do in this lesson, is teach you a method that was first proposed by Colonel McLyman. This is a non-iterative, step-by-step method that will give you a first-order design of your inductor. It is not guaranteed to give you the most optimal design. However, it will give you insights into what parameters affect the design in which way. To come up with a highly optimized design, it is generally advisable to use a iterative approach in which case you could use the method taught here to come up with the first order design. And then iterate from there. You can also write a computer script to go through all possible designs and figure out which design meets the criteria that you're most interested in. Whether it's minimizing losses or minimizing size or coming up with a tradeoff between those two. So let's begin by developing this step by step inductor design procedure first proposed by Colonel McLyman. The design equations that go into inductor design come from our design constraints. The first design constraint that we want to meet it to avoid saturation. To avoid saturation, the maximum flux density to our core must not exceed the saturation flux density we set. However, it is always good engineering practice to leave some margin. So we will ensure that the flux density in the core never exceeds a value Bmax which will be set lower than Bsat. Typically, we may want to set Bmax as equal to 80% of Bsat. The value of Bsat or saturation flux density depends on the core material. The core material that we will use will mostly depend on the switching frequency of our power converter. If the switching frequency is greater than 20 kilohertz, we will most likely use a ferrite material. We can look up the value of Bsat from the manufacturer's data sheet for the material that we choose. Once we know the value of Bsat and hence the value of Bmax to use, we can proceed with coming up with our first design equation. For the design example in this lesson, I will use the simple core geometry shown here. This core has a uniform cross sectional area equal to AC. It has an air gap of length LG, it has a mean part length within the core of LC, the permeability of the core is mu C. The core has n turns of winding, a current i through the winding and a voltage v across it. We can model this magnetic structure using the magnetic circuit model shown here. We have an MMF source of value n times i. A reluctance Rc that models the reluctance of the code. And a reluctance R G that models the reluctance of the air gap. From what you know already the reluctance of the core can simply be written as the length in the core divided by the permeability of the core and the cross sectional area of the core. Similarly the reluctant in the air gap can be written as the length of the air gap divided by the probability of free space and the core. Note that this is an approximation since the actual area in the air gap maybe slightly larger than the core cross sectional area due to the fringing fields. However for a first order design, this is a good enough approximation. Using this magnetic circuit model we can write an equation that involves the flux find through the core. This equation is ni = the flux times the total reluctance of the part. This equation is simply the equivalent of Ohm's Law for magnetic circuits. Now to simplify this further we can assume that the permeability of the core mu c is much, much greater than the probability of free space mu not. If we make the assumption which will be true for hypermobility materials, then Rc will be much smaller than Rg. In that case we can ignore. RC in this equation. And our equation simplifies to simply Ni approximately equal to the flux fee, times Rg. The flux fee can further be written simply as the flux density through the core B Times the core cross sectional area Ac, yielding that In is approximately equal to B times Ac times Rg. Now we don't want the flux density in the core to ever exceed B max. And the maximum value of flux density in the core when we have the largest amount of current flowing through the winding. The largest value of current through the winding, is equal to Imax. So when I equals the Imax we would write B to equal Bmax, and that equation becomes simply Ni max equal B max AcRg. We can substitute the value of Rg into this expression and that's simply gives us B max Ac lg / mu not Ac, the Ac's cancel away and so what we get then is that NImax = Bmax lg/mu not. Hence, we have our first design equation. Which we will call Constraint 1. In this equation, we know the value of Imax Since that was one of our specifications. We also know the value of B max since that's something we picked based on being less than B sat. We also know the value of U knot since that's a universal constant. The two unknowns in this equation are n, the number of turns in the winding, and lg, which is the air gap in the core. We have one equation but two unknowns. So if we want a unique design, we need another equation with just these two unknowns in it. Our second constraint comes from the inductance value that we need. Our initial specifications To require us to have an inductance of value l. Now we can also determine the value of l in terms of the code parameters using the magnetic circuit model. Recall that the inductance for any magnetic structure is given by The square of the number of turns divided by the reluctance in the part. In this design the reluctance in the part is Rc plus Rg. Again assuming that the permeability of the core, mu c Is much greater than the permeability of free space, mu not. We get that the reluctance of the core rc is much smaller than the reluctance of the air gap, rg. Hence, we can again ignore the reluctance of the core in this expression. And our inductance expression simplifies to simply N squared over Rg. Substituting the value of Rg, we can get an expression for inductance in terms of The physical parameters of the core. You should already be quite familiar with this expression as we've seen it multiple times before. This is then our second design equation and we'll call this constraint 2. In this equation we know the value of L since that's specified. We also know the value of nu naught since that's a constant. However we don't know the values of A C N and lg. N and lg were already unknowns from our first constraint equation, but now we have a third unknown, which is Ac, or the cross-sectional area of the core. Hence, now although we have two equations Since we have three unknowns, we're still In order to have a unique design, you must keep going until we have the same number of equations as we have unknowns. Our next design constraint comes from the fact That we need to fit the winding through the window area of the core. The window area of the core is this space in the center of the core through which the winding must pass. We defined this area, As WA which stands for the area of the window. It's presumed that each wire in the winding has a conductor in the cross-section area equal to Awire, and if we have N turns in our winding then the total area that is consumed by all the conductors is going to be NAwire. However this is just the area of the conductors themselves. It does not account for any area that is consumed by the insulation around each conductor. Also, if our conductor's around, there will be gaps between the conductors even though we may packed them very tightly. Also, if you recall, the way a magnetic device is constructed, Is by putting the winding on a bobbin, and then the bobbin goes and sits in the core. So there will be some space that will be consumed by the bobbin itself. Finally, if you're building a high voltage inductor, ie an inductor that's going to see a large voltage across its terminals. And you may need to add insulating tape between the different layers of the winding. This tape is also going to use up window area. Hence, all these factors result in reducing the effective window area that is available to get a current. To take this reduction in available window area into account, we introduce a factor, Ku, which is known as the window fill factor. The window fill factor will have values between 0 and 1. For a typical inductor made from round wire, the window fill factor, Ku Maybe about 0.5. So, as a starting point, you may want to simply use 0.5 as the value for Ku, and see at the end if that came out to be roughly the right number for your design. In case your actual Fill Factor comes out to be higher or lower. Then you can do another iteration on your design. The value for the window fill factor Ku will be slightly higher if you're using foil-winding instead of round wire. As in the case of foil winding, you will not lose space between the conductors as you do in the case of the round wire. If you need to add insulation between layers, your fill factor will be even lower. For example, in an offline transformer where it has to withstand hundreds of volts, the fill factor will only be 0.25. For even higher voltage transformers, The fill factor can be even smaller. Now, from a design perspective, what we must ensure is that our total conductor area, which is given by n times a wire is less than the available area in the window. The available area in the window is just K-u times W-A. So K-u times W-A must be greater than or equal to N times A-wire. And our third design equation, which is in fact an inequality, is this one. And we'll call this Constraint 3. In this design inequality we know the value of Ku since we'd simply pick a reasonable value such as 0.5. However, we don't know the value of the window area WA. We also don't know the number of turns, N, and we don't know the cross-sectional area of the wire, Awire. N was an unknown that we already had from before, but now, we've added two more unknowns, WA Which is the window area. And A wire, which is the cross-sectional area of the wire. So now while we have three design equations, we have five unknowns. So we must keep going in order to find a unique design. Our fourth design constraint comes from our last budget specification. If we assume that our core losses and our winding Ac losses are small as will typically be the case for if we'll turn inductor. Then we can assign most of our lost budget To the DC losses. Our typical number may be 80% of our loss budget to the DC losses. We can compute the value of the winding resistance from the portion of the loss budget that we've assigned to the DC losses. Then we can use this winding resistance value R To come up with our fourth design equation. From the physical geometry of our magnetic structure we can calculate the value for the winding resistance. The winding resistance R will equal the conductivity of the conducting material, typically that would be copper. Multiplied by the total length of the wire that is used in our winding and divided by the cross sectional area of the wire. We can also express the total length of wire used in our winding in terms of the total number of terms in our winding times a factor known as the mean length per turn. The mean length were done, or MLT is simply the average length of a single turn of wire in our binding. Assuming that we've designed our magnetic structure in such a way, That we fully utilize the window area of the core. Then the mean length of turn MLT will be equal to the length of a turn that is sitting sort of in the middle of the winding. Expressing the length of wire in terms of the number of turns. And the mean length per turn, we can rewrite the resistance of the winding in the form shown here, where the resistance now equals the resistivity of the conducting material times the total number of turns in the winding, times the mean length per turn of the winding Divided by the cross sectional area of the wire. We'll call this our Constraint 4. In this expression we know the value of R since that we calculated from our last budget We also know the conductivity of the conductor raw. However, the other three terms N, MLT and Awire are unknowns. Of these, N was already an unknown as was Awire. However, we have now introduced another unknown which is MLT. So now we have a total of six unknowns and four design equations. We don't seem to be catching up with our number of unknowns. And we seem to be running out of constraints that we can impose. So before we proceed further, let's take stock of our situation. Let's summarize where we are right now. We have four design constraints. Our first design constraint Came from the requirement to make sure that our core does not saturate. Our second design constraint came from the requirement you have a specific inductance value. Our third design constraint came from the requirement that we must fit our winding in the window area of the core. And our final design constraint came from the requirement to stay within our power loss budget. While we have four design constraints, we have a total of six unknowns. Everything else that's used in these equations or inequalities is, either a given specification or is a known quantity or constant. So, given that we have more unknowns that we have equations or inequalities, how do we proceed? This is where we leverage the inside of Colonel McCleman and utilize his methodology to go forward. Let's look at the first three unknowns. AC, which is the core cross sectional area. WA, which is the window area, and MLT, which is the mean length per turn. What's common about all three of these unknowns? Well, they all relate to the core. Ac is the area of the core, WA is the area of the window. And MLT Is the mean length of return through the window. Since all three quantities relate to the geometry of the core, once a core is selected. All three of these quantities are automatically specified. Therefore, we could potentially club all three of these as a single unknown, since they all come from the core or the core geometry. If we could club these three unknowns into a single unknown, then we have a total of four unknowns. And we have four design equations, so we could potentially find a unique solution for our inductor design. However, it is still not clear how we can combine these three unknowns. Into a single meaningful unknown that allows us to do the design. We'll see how to do that next. In order to come up with the methodology, you'll find a unique in doctor design. Let's do the following. Consider again our four design constraints. In these design constraints, we have six unknowns. However, since we have four equations, we could eliminate three of the unknowns and come up with a single equation. The unknowns that we will eliminate are the length of the air gap. The total number of turns on the winding, and the cross sectional area of the wire. When we eliminate these three unknowns, then the unknowns that are left behind will be the cross sectional area of the core. The window area of the core, and the mean length of turn for our core. I will not show you the algebra for bring this elimination, which is fairly straight forward. But once you do the elimination, what you're left with is the expression shown here. Note that this is an inequality, since one of our design constraints was also an inequality. In this expression, everything that's on the right-hand side is, either a given specification or a quantity that we already know. The three terms that are on the left side of this expression are the cross-sectional area of the core. The window area of the core and the mean length of turn. Since all of them relate purely to the core, we will define this term as kg, are the core geometrical constant. By defining kg in this manner, our design inequality reduces to simply having kg greater than or equal to. This expression on the right-hand side. Hence, in order to find a suitable core for our magnetic design. All we need to ensure is, that the Kg value for that core as defined by this expression. Is greater or equal to this expression on the left-hand side. We could choose to pick the smallest available core that meets this design criteria and let that core set the rest of our design. Hence, we can now have a procedure that yields a unique design. So, let's construct a step-by-step design procedure that allows us to have a unique design for our inductor. In the first step, we will select our core and we will do that by selecting the smallest core. Who's kg value satisfies this expression on the right side of this inequality. All the terms on the right hand side of this expression were either specifications or knowns. To determine whether a core meets this criteria, we can determine the kg value for all that code. Using the expression for Kg which is Kg = Ac squared times WA divided by MLT. We could also pre-compute the Kg values for all the cores we're interested in and put them in a sorted list. Then, we can simply pick up the core whose KG value is the smallest but greater than the value on the right. The values for AC WA and MLT are typically given in the data sheet for the core. Once you've selected a code using this procedure. You can note down the values for Ac, WA and MLT for that code for to use in the next steps. In the next step, we can determine the length of the air gap, using the expression shown here. This expression comes from combining some of our earlier constraint expressions. Next, we can determine the number of terms on the winding. This expression also comes from combining some of our earlier constraint expressions. Once the number of turns is known, we can proceed to find our lost unknown, Which is the cross sectional area of the wire. Using this expression given here. Note, this expression is an inequality. Since the y cross sectional area is selected using an inequality, we should do a check. To make sure that you've not picked a very thin wire, so that the resistance is within specifications and we do not have A power loss that exceeds a power loss budget. To reduce your losses in a filter inductor, you probably want to use the thickest wire that you can fit in the available window area. You may also want to do other checks once you're done with your design, such as the factor Ku that you assumed is close to what you finally get in your final design. Also using your final design you can compute the core losses and the AC losses in your winding, and if those losses are either much smaller or much larger than the losses that you assumed while doing the power loss budget allocation to DC losses, you may want to do another iteration on your design. Finally, if you're planning to have your core gapped by having one of its cores ground down, then rather than giving the company that is going to create the gap the gap length, it would be better to instead give them the AL value that you need. The AL value is the inductance per unit turn of your core and it is measured in nanohenries, hence you can calculate it using this expression here. The reason to give the AL value instead of the LG value to the company that's going to do the grinding is that they can typically measure the inductance of the core as they're grinding it down. The LG value you computed is only approximate because it did not take into account fringing effects and also ignored the permeability of the core material itself. On the other hand, if you get a core with the correct AL value you will be guaranteed to get the correct inductance when you put the right number of turns on it. Hence to summarize, we now have a step by step procedure which we can follow to get a unique inductor design. Again, I must emphasize that the design that this will yield may not be the optimum design, however, this design does yield insights. For example, you can see that you could reduce the size of your core if you allowed for a larger value for the maximum flux density. Similarly, your inductor size would decrease if you allowed for a larger winding resistance, which basically means you're allowing larger losses. On the other hand, if you need a larger inductance value, then the size of the inductor will increase. Similarly the size of the inductor will increase if you have to carry a logic current through that inductor. With these insights and this design procedure you can get your first order design and if you're not satisfied with it you can iterate the design further until you get to an optimized design. With that we come to the end of this lesson, as well as to the end of this course. I hope you enjoyed this course and I look forward to having you as a student in some of my future courses. Goodbye. [BLANK AUDIO]