From the state equations that are describing the operation of the converter in time domain, you can certainly do very quickly steady-state analysis by applying the principles of, what do you do? This is like a mantra and polenta. That is about second bonds, charge bonds for steady-state analysis and that's what we do. We basically, average out these state equations so that we weigh the voltage applied across an inductor during the first time interval of dts. The second time interval of d prime ts, d prime being 1 minus d. We obtain a dynamic averaged equation for the inductor current in the following form, ld per dt of the average value of the current is equal to d times the voltage applied during the first subinterval plus d prime or 1 minus d times the voltage applied during the second interval. So this is the state-space average dynamic equation for the inductor. That's perhaps is maybe one of the most important steps we are doing modeling power electronics is this averaging step. You can certainly do exactly the same thing with respect to the capacitor current. You have the capacitor current is C d per dt of the average value of the capacitor voltage. That's simply equal to the average value of the inductor current minus the average value of the load current. Notice that since the capacitor equation does not depend at all on the position of the switches, we don't see any ds in the dynamic equation on average for the capacitor current. Finally, the input current. That's one of the output equations in the average domain, is simply equal to the product of the duty cycle and the average value of the inductor current. The output voltage average value is equal to the average value of the capacitor voltage plus the R_esr times the difference between average values of the inductor current and the load current. Now, in a dynamic equation, this is not necessarily equal to zero. Dynamically, we can have an imbalance between the average value of the inductor current and the load current, that imbalance is going to charge or discharge the capacitor dynamically. Okay, so now we have these averaged equations into state-space form for the converter. Why did we write these equations so much better than these equations right here? I mean, we write them all, but what is the reason that this averaging step was applied in the first place? What is the main reason of doing that? What is the fundamental reason of going through this step of averaging state-space equations into the form? All right, so the answer to this was that the first set of equations are going to be giving us only the dc operating point, for the second set of equations are giving us the dynamics of the converter. I would not agree with that statement. Why would these equations not be able to give us dynamics of the converter? They contain complete behavior in time domain or the converter. In fact, as we go through the end of this review material at the beginning, we'll do a spice simulation of the time domain equations and all kinds of dynamics there. It's not just steady-state. So wrong. The switching model equations are the equations that describes the converter as is. Switching behavior including dynamics and including the possibility to round that dynamic into steady-state if you wish to do so. But it is incorrect to say that these equations cannot possibly describe dynamics of the converter. They can and they do. All right, so let's answer that. Is much more computationally efficient to handle averaged equations as opposed to the switching model equations? I would say that's actually correct. This is entirely true. You'll actually see later on when we do simulations of a converter circuit model using switching circuit model, it takes a certain amount of time to do that. If you do that with an average model, it takes like towards the magnitude last time. All right, so there's less computational effort. Although true is not fundamentally why we do this. If the computing power were the only problem, we would not have done this step here because we have really plenty of computing power available. We're going to remove the switching ripple. By doing this step of averaging, I think we're getting closer. So that's again true. Still not the fundamental answer I'm looking for. What is the fundamental benefit of having this set of equations compared to this set of equations? This is definitely non-linear, but it's not just non-linear, it's also time-varying. This is a non-linear time-varying system. The only thing that you have available right there is to run transient simulations. You can actually, if you have skill and you apply the volt second balance and charge balance, you can solve steady-state, that's great. But by having to work with the non-linear time-varying system is very not just cumbersome, it's fundamentally difficult. We don't have good tools to do that. By performing averaging, we are getting what type of system? Time-invariant. So the time-invariance is really the reason. Going from time-varying to time-invariant system is why we do averaging in the most fundamental sense. Alongside with everything else that you said are related to that, within most fundamental sense we do the averaging step to remove the time-varying nature of the converter and get to the point where we have a time-invariant is that opens up the possibilities of applying a whole slew of different techniques for analysis purposes. The beauty of averaging is that it does retain this important low-frequency dynamics that we're interested in when we design control systems around converters. Yes, we are going to actually lose some information here. We are going to lose information about the ripples. We are going to lose information about switching transient. But we are going to retain the dynamic information about the low-frequency behavior of the converter in a non-linear timely and form, and that's the important step of performing the averaging in the first place. Okay, the mechanics of applying the averaging tool pulse-width modulated converters. As you see from this example here is extremely simple. You just take the two sets of state equations. You say d times 1 plus d prime times the other, and you get it. It's really really simple. But again, the fundamentally there is a lot more to it than meets the eye. Again, most important aspect being that we now have a large-signal average model that is time-invariant. All right, and that time-invariant large signal average model now, we going to solve these capabilities of studying dynamics that we are really most concerned with is the low-frequency dynamics of the converter and allows us to do yet another step after that which is also important fundamentally, which is going to be what? What can we do with a large signal nonlinear but time-invariant model that we couldn't do with switching model? We could do Laplace transform, but we need to first do one step to allow us to do that. Yes, this thing can now be linearized. So that's really where things are going. Once you get to this point here, you have large signal dynamics in a time-invariant. You can then linearize that system and obtain linear time-invariant model of the system. For this particular buck converter, I'm not going to go to that derivation, I'm going to ask you to review that. But conceptually, the most important point is that the linearization at an operating point give us an LTI system model in a circuit diagram form that we really prefer as shown right here. That gives us now complete small-signal AC dynamics of the converter in the form that is best suited for analysis and modeling techniques because for linear time-invariant systems, we can apply Laplace transform and many tools from frequency domain analysis of systems in general, circuits in particular, and we can then go into launch into finding transfer functions, analyzing salient features of those transfer functions, discussing how do we actually reduce output impedance. Very fundamental information becomes suddenly available to us because we have linear time-invariant model to work with. So that's the small signal model right here. If you look at the structure of the small signal model, just a little bit of a reminder. We have here what looks like really an ideal transformer in the model. We have a voltage source. Here it's proportional to d hat, d hat being small-signal perturbation in the duty cycle. In response, we are going to have small-signal perturbation in everything around, including small-signal perturbation in the inductor current and output voltage and so on. So we have now complete circuit model with everything presented in the small-signal average sense. That's easily suitable for analysis and design in frequency domain, which is really what we like to do. We rely on this wealth of knowledge in frequency domain techniques. So we can find out what is in Laplace domain that responds from d hat perturbation to the output voltage. That Gvd transfer function is the key to being able to design a controller around this power stage. Then you have line to output transfer function. How does the output voltage perturbation depend on perturbation in the input voltage? What's the value in knowing line to output transfer function? Yeah, because our input voltage is not necessarily well regulated and it has dynamics and it can vary up and down. We want to see how much those variations which are really disturbances to our system are affecting the output voltage that typically we would like to regulate tightly. Then finally, there is the last of the three classical transfer functions that we would be interested in. Would be the output impedance, which is the response of the output voltage with respect to the small signal variations in the load current, and the output impedance is negative, we had to over, I load. Why do we care about that? Because again, the dynamic changes in the load are a disturbance to our system affects our output voltages. In most cases, we are interested in regulating that output voltage, and that's what the output impedance is going to give us.