In this brief lecture, I'm going to explain how to derive the averaged AC equations of a switching converter. And we're going to do this by simple extension of the ideas of volt second balance and charge balance that you already know how to do. We're going to extend those to write the equations of the average voltages and currents under transient conditions. I'm going to explain just how to do it, but not derive where it comes from. this is what you need to know to do the homework, but in the next lecture, which really is perhaps optional we're going to discuss where it comes from, and give a little more idea of the mathematical foundations. So I'm going to explain with reference to this buck boost converter. So, the usual buck boost converter with an input voltage vg of t, an inductor current i of t, and an output voltage v of t. So as us, we start in the usual way, with the switch in position one, here is the circuit, and we can write the equations for the inductor voltage and capacitor currents. so the inductor voltage, which is L di dt in the first position is equal to the input voltage vg. Likewise the capacitor current, which is C dv dt, is equal to minus the load current, during this first interval. at this point what we previously did, was to replace these quantities on the right hand side of the equation with their DC components, and make the small ripple approximation to ignore their ripples. So what we want to do today is still make the small ripple approximation, but don't assume that the quantities are purely DC, allow them to vary. And so what we will do is replace the quantities on the right hand side of the equation with their averaged values, where the angle brackets here denote averaging. And I mentioned that on the previous lecture, so say the average value of v of t, where we average over a period Ts is equal to 1 over the period times the integral over one period of v of t dt. And the integrating period we can there are different choices we can make for the integrating period. The one that I described in the last lecture was that we integrate over a period that is centered at time t, and we integrate a half a period before and after, like this. And what the point of this is, is that if removes the switching ripple by averaging over exactly one period. And what is left is the underlying low frequency component including the DC component, but also this, this average can vary during transients. So vg of t here is replaced by its average, and v of t is replaced by its average. In fact, what this amounts to is instead of writing the DC component, we write the average value. Okay, same thing for the switch in position two, here is our usual circuit with the usual equations for the inductor voltage and capacitor current for this interval. we have v of t, and we also have I of t on the right hand sides of these equations, so we will replace I with its average and v with its average. So we have equations for what the inductor voltage is during each interval. And from those equations as usual we can write the inductor voltage wave form, which is one value during the first interval and a different value during the second. the solid dark line here is the actual wave form including the ripples. And the dashed lines here have ignored the ripple, making the small ripple approximation. And, for example, during the second interval, the dark line, or solid line, is v of t, while the dash line is the averaged v. So, what we do now is we calculate the average inductor voltage. Previously, the average inductor voltage equaled zero by volt second balance, but now we're going to let the average inductor voltage equal L di dt. Where here we're taking the derivative of the average inductor current with the ripple removed or ignored. So this average inductor voltage is simply d times the value during the first interval, plus d prime times the value during the second interval, here and here. And we equate this then to L times the derivative of the average i, and this is our averaged equation that describes the inductor during transient as well as DC conditions. And what this shows, is how the low frequency component of the inductor current with the ripple removed will vary over time. Okay, we can do the same thing for the capacitor. Here is the capacitor current waveform. The solid lines include the ripple, and the dash lines employ the small ripple approximation. So, the average capacitor current, again, can be written as d times the value during the first interval, here, plus d prime times the value during the second interval. We collect terms and equate this to C times the derivative of the average v, and we get this equation that describes how the low frequency components of the capacitor voltage will change with time. Finally, we know that to complete the model of the converter we may have to write an equation for the average input current. And in the DC case, we found the DC component of ig, the current coming out of vg, and so for the AC model we will need to do the same thing. So, we express ig during each interval and write the ig of t waveform. And we apply the small ripple approximation to it. So when a switch is in position one, ig is equal to the inductor current, which we can write by ignoring the ripple in the inductor current, and say ig is the average i during the first interval. And it's zero during the second interval, when the transistor's off, so the average ig then, here, which is the average, or low frequency component of the switched waveform ig. that average is equal d times the average i, plus d prime times 0, so here is our equation that we get for the average inductor current. So to summarize, we can generalize the ideas of inductor volt second balance and capacitor charge balance to write the equations of the averaged or low frequency components of the wave forms without requiring that they're DC, but rather they can vary with time. we employ the small ripple approximation. The equations look very similar to those you've already been writing, but now we allow the waveforms to have AC variations. Okay. In the next lecture, which is optional, we're going to discuss some of the mathematical details and foundations behind averaging. And then in the following lectures, which aren't optional, we're going to linearize these equations that we derived in this lecture, and then construct small sig, signal AC equivalent circuit models.