In the last chapter, we discussed how to design an inductor given the constraints of allowable DC copper loss and maximum flux density to avoid saturation. Now I want to extend that discussion to consider optimization of transformers where the core loss is important as well. In these applications, we need to minimize the sum of the total copper plus core loss. The operating flux density is no longer simply constrained by saturation, rather we choose a smaller flux density that manages the core loss to optimize the total loss. So in this chapter now, the flux density is no longer given, but rather as a variable to be found such that the total loss is minimized. We'll consider a multiple winding transformer and we'll use the results from the last chapter of how to optimize the window area and its allocation among the various windings. We're going to write the constraints for transformer design in this lecture. And then we'll do some examples and look at how increasing the frequency can reduce the core size. So that 100 kiloHertz 100 watt transformer might be only this large, whereas a 60 Hertz 100 watt transformer is much larger and requires a large amount of of iron Again, this lecture covers the basic constraints of transformer design. We'll use these constraints to develop a first pass design procedure similar to the kg method from the last chapter. We'll work some examples. We'll also look at AC inductor design where the ripple is large and we'll summarize. What are the equations that are the basic constraints of a transformer design? Since we're considering core loss, first we need to write the equation for core loss. We previously talked about this. We will use the empirical Steinmetz type equation, in which the core loss depends on a constant multiplied by delta B, the Ac component of flux density, and it's raised to some power beta, that is an empirical number. Multiplied by the volume of the core, or its cross sectional area times its path length. Typical values of beta for ferrite might be 2.6 or 2.7. Generally increasing delta B increases the core loss very quickly and greater than a square law. Will use this equation as the first design constraint. Second, we need to work delta B. Let's assume that for winding 1 we have some arbitrary voltage waveform V1 of T applied and maybe it looks like this. Since this is the winding of a magnetic element the volt seconds have to balance and we're going to define lambda 1 as the volt second supplied to the winding during the positive half of the waveform. So this area here are the net volt seconds which make the flux and magnetizing current increase. Of course, during the second half of the cycle, we have equal and opposite or negative volts seconds that bring the flux and the magnetizing current back down to where they started. We can then define lambda one as this area so it's the integral of the voltage between the zero crossings. We know from Faraday's law, V1 is equal to N1 AC times times dB dT. Therefore, the change in B over this half cycle is 1 over n1Ac times the integral of the voltage. And this is from 1,0 crossing to the next, so from t1 to t2. It's traditional to call delta B, the peak rather than the peak to peak, so we need to divide by 2. This here is our lambda 1. And so we get the delta B is lambda 1 over 2 in 1Ac. This is our second constraint and you can see that the volts seconds applied to the primary winding control the flux in the core and in particular, they control delta B. To obtain a given delta B then this tells us how to choose n1 when we can solve for n1 and get this. So that if we want to reduce delta B we can increase n1 and therefore, we can control the delta B at which we design our transformer to run. The third constraint is the copper loss. In the previous chapter, we found how to allocate the window area among the various windings to minimize the total copper loss. And this was the result that we got. Here this quantity I tot is the sum of the RMS currents of the windings all reflected through to the primary winding. When we optimally allocated the window areas among the different windings, then we got this total copper loss. And so we'll assume that we do that and that we use the alphas from that previous chapter result. The copper loss depends on n1, actually on n1 squared. We can go back to the previous slide where we solve for n1 and we can plug that into this equation to eliminate n1. When we do that, here's what we get for the total copper loss. So the copper loss varies like 1 over delta B squared. As you increase delta B, the copper loss goes down very quickly and in fact it goes down like the square. What we want to do is minimize the total power loss. So the sum of the core plus the copper loss. Here's our equation for core loss from Steinmetz equation which increases as delta B to this beta power, or beta is a number typically greater than 2. And here's the copper loss from the last slide that goes down like 1 over delta B squared. And so the sum is the sum of these two and it looks like this. If there's a minimum, in fact, the total core plus copper loss occurs at some optimum value of delta B, where the sum of these two quantities is minimized. If you look at this plot, p tot has zero slope. Where the negative slope of the copper loss cancels the positive slope of the core loss. They have equal and opposite slopes and there's where the minimum occurs. It's not necessarily where their values are the same here, but it's where their slopes are equal and opposite. We can solve for that optimum B. So again, we take these derivatives and where the slopes have equal and opposite values, we find the minimum. Here are the equations to do that. Here's our core loss equation. Here's our copper loss equation. Take the derivatives of those to get these equations. We can equate them or plug them into the previous formula. When we solve them, delta B turns out to be this expression. Given that solution for the optimum delta B, now we can find the total loss. If we plug this delta B into our Pfe and Pcu equations, and add them together to get the total loss, this is the result. The equations are getting a little bit big, but we can still manage this equation. We can rearrange this in a way similar to the kg design method where we put all the quantities that depend on the core geometry on the left hand side, we put all the specifications on the right hand side. And then we can derive a first pass design procedure for selecting the core that's similar to the kg design method. I'm going to call the left hand side of that equation Kgfe. So it's a function of the core geometry, which is the window area, the cross sectional area, the mean length per turn, and now the magnetic path length also. And it also depends on beta which is a function of the core material properties. So, all of these things we can put on the left hand side of the equation, the right hand side are the quantities that are specifications and that we take as given. The core tables and appendix D also lists Kgfe as well. They listed for a given value of beta I believe 2.7. Thus we find the optimum delta B that minimizes the total core plus copper loss. And delta B now is a variable in our design rather than a given specification. We can calculate the Kgfe required for a given application using the formulas on this slide to select the core that is large enough to do the job. Once we found that core, then we can go back from the earlier equations and compute the terms and wire gauge and so on.
