So, this is what we explained is the block diagram of the control system around a converter that is operating as a low harmonic rectifier. So, remember the input voltage to the converter comes from a full wave rectified AC line voltage. So, V g of t, is going to be a full wave rectified sine wave. And the first goal of the control system of the control loop that we see right here is to shape the input current to follow the input voltage so that ideally we get ij equal to Vg over Re, Re is the emulated resistance. That is the purpose of the inner current control loop that we're going to design first. Then we discussed how that inner current control loop receives a reference value that's proportional to the input voltage. But that constant of proportionality is then dependent on the control input that really determines the value of the emulated resistance Re. And it also in some cases modified by the inverse proportionality to the peak of the input line voltage to achieve what is called a feed forward compensation of the input voltage make the power flow independent of the input voltage amplitude. And so we have the power taken from the AC line the average power taken from the AC line directly proportional to the control input. And that control input is going to be the value that comes out of the outer voltage control loop that we will design next after we are done with the discussion about designing the inner current control loop first. So let's do the design of the inner current control loop. So, design of the inner current control loop will follow first the most usual approach that based on application of the average current mode control technique. And to do that, of course, we do need to know Gid. Gid is what we are looking for the control to current transfer function response, keeping in mind that we are talking about the rectifier application as opposed to just the dc-dc application. So a starting point is our average model for a boost converter. That average model here is shown in the form of an average switch network. Keep a note of that. This is a boost average switch network tells us on the input side, the average value of the voltage that we see on the fuse the input port of that boost average switch network is really just equal to d prime times the output voltage. So this here is equal to d primes times the average value of the output voltage. The average value of the output voltage right here is the one that hopefully we have essentially equal to a DC value. We will follow a little bit later on the energy storage capacitor C is the one that is going to have to absorb AC line harmonics, in particular the second harmonic of the AC line we'll see more details about it coming up. But that large value of the output filter capacitor means that when we discuss the current control loop, we can in fact assume that this voltage V on average at the output is approximately equal to just the DC value. And dynamic variations with respect to the duty cycle changes in order to control the input current are going to be relatively small. So this is the assumption that we're going to make in the derivation of the model that applies to the control of the input current. So our model on the input side looks basically like this. The current that you're interested in is equal to input current, which in turn is equal to the average current through the inductor. If you look at this now, and the goal is to find out what is Gid, right for the in that control loop, we approach that by basically doing the linearization. We say d is going to be equal to d plus d hat, and so on. We are going to have a circuit model that looks like this. So if we multiply these, we're going to have 1 minus d times v plus 1 minus d, times v hat, minus d hat times v and minus d hat times v hat. We're going to as usual just neglect this part here entirely. This is the product of small single quantities. The second thing that we are going to do, is to neglect this part again because of this assumption that v hat is approximately 0. That the output voltage is really held essentially constant, not just DC wise by dynamically wise at frequencies that are of interest with respect to controlling the input current. Then becomes simply equal to 1 minus d times v minus d hat times capital V. And if you want for the expression now for Gid, you see that that's going to simply be equal to V over SL. And so this here is the result that we were looking for it gives us the Gid for controlling the input current. And you see that that really follows almost the same discussion we had earlier in the context of controlling the average current in a dc-dc boost converter. The difference that we made right here is this. So, the difference is that we are not concerned about the dynamics of what happens with respect to the output filter cap because presumably the output filter cap is so large that at frequencies of interest for the control of the input current, that voltage across the capacitor can be considered constant and we just take that out from the consideration. The control of the input current in a large signal sense looks like this model right here. In a small signal sense, it looks like simply a response that's a constant over s V over L over S or V equal to the DC value of the output voltage. Looks really simple. One thing to keep in mind for a moment is that we are not in a DC DC situation. So Vg of t, on average, is going to be doing this. It is going to be changing in time significantly as we operate our current control loop. In response, durecycle is going to be changing in time substantially d prime or d if you wish, is going to be changing substantially in the form of this type of free form as we go through the line cycle. Our system is never in steady state DC operating point, the small signal linearization around a DC operating point is a term that we have to be careful about right here. Fortunately in the boost case, we have that the model here is in fact a large signal linear and so the linearization that we are performing right here turns out to be fully justified. But the next two page is really just going through the same thing but you have it as a summary. In the end, we have a large signal model, large signal, non linearized model that looks like this. And with respect to d, that model is linear, which really gives us an opportunity to say that what we have in a small signal sense, holds for large signal variations as well and we can work with confidence on designing the control loop that looks like this. In other words at every single point in time that you have along that AC line cycle going up and down on the input, every single point the model looks the same, the same model applies. That's why we say this is linear in a large signal sense. With that important note on a side, we realize these two points down here. This model looks exactly the same, as the model we used earlier, when we designed the average current mode control for a boost dc-dc converter. So, the high frequency asymptote of Gid in average current mode control boost dc-dc converter has exactly the same response. So, as a result, the simple PI compensated design strategy that we applied earlier to the dc-dc case of a boost converter applies directly.
