In this course, so far, we have discussed converter circuits and their DC equivalent circuits. Beginning this week, we will spend the rest of the course discussing converter control systems. What we're going to do is to extend the DC equivalent circuit models that you've learned to construct so far to include the converter dynamics, and we'll be able to solve these AC models of switching converters then to find the important converter transfer functions. Lastly, we're going to use those transfer functions and models to design the control systems of some switching converters. This week we're discussing chapter 7 which is titled AC Equivalent Circuit Modeling. We will extend our previous equivalent circuits to incorporate now the inductors and capacitors of the converter and we're going to develop small signal AC equivalent circuit models that can be solved with standard circuit analysis to find things like transfer functions of the switching converters. Here is a sketch of a simple feedback system that controls the output voltage of a buck converter. What's illustrated on top is the switching converter power stage as we've discussed so far and the bottom half of this diagram is a feedback or a control circuit that adjusts the duty cycle automatically to regulate the output voltage. The object in such a control circuit is to regulate a voltage such as the output voltage v of t of the converter, so that it is equal to a constant and well controlled value capital V. Now, you can't just set the duty cycle to one value and leave this circuit alone running at that value and expect the output voltage to always be the desired capital V. The reason for this is that there are disturbances and signals such as in the input voltage Vg. It may be, for example, that we derive Vg by rectifying the AC power line voltage or perhaps Vg has an input battery voltage or some other voltage that is not well controlled. There are variations in Vg of T and these variations propagate through the switching converter and cause variations in v. There can also be variations in the load. Here we simply have a load resistor R but in fact the load can be something whose current changes and we know that the output voltage does depend on the load current. Certainly in continuous mode, if we include losses in the circuit, the output voltage does become a function of the load resistance or the load current. In discontinuous mode, the output voltage has an even stronger and larger dependence on the load current. We have variations in load current and in input voltage and these lead to variations in the output voltage. In addition, we have uncertainties. All the components and element values have tolerances. We may have plus or minus 10 percent tolerances which could cause some variation in the output voltage just from that. When we account for all the tolerances of all of the elements and for the value of Vg and so on, this also causes variations in the steady state value of the output voltage. What we do is we build a control system that senses the output voltage through this connect feedback connection, this voltage is put into a summing node that generates what's called an error signal. We subtract the measured output voltage from a desired reference value called V ref and the objective here is to make the output voltage equal to V ref. The difference is called the error signal and this signal coming out of the summing node is a measure of how much the output voltage differs from the desired value. That error signal is put through what's called a compensator circuit that typically is realized with an op amp or a similar circuitry to produce a control signal Vc that controls the duty cycle of the transistor. The pulse-width modulator, we're going to talk about it later in this week, is a circuit that takes an analog input voltage Vc in this case and produces a switched wave form whose duty cycle is proportional to the control voltage. This control waveform then goes through the gate driver and controls the MOSFET, turning on and off. This circuit, if properly designed, can automatically adjust the duty cycle to attempt to drive the error signal to zero and make the output voltage equal to the reference. We're going to talk a lot more about design of this feedback loop in chapter 9. Some applications of control. We nearly always have a control system in any power converter. In DC-DC converter, typical application is to regulate the DC output voltage as I've just described. The control system will adjust the duty cycle automatically again to make the output voltage accurately follow a reference signal. In a DC to AC inverter, we may want to regulate an AC output voltage. This is similar to the DC-DC case except that the desired output voltage now is an AC, typically a sinusoid, and so we want that AC voltage to accurately follow an AC reference signal but otherwise it's the same. In a grid tied solar inverter, our inverter may actually produce an output current instead of a voltage but it's the same idea, we adjust the duty cycle to make the current that we put into the grid follow a reference signal. Another case is an AC to DC rectifier and we often now have requirements on our rectifiers that draw power out of the AC power line regarding the quality of their AC current waveform. In many locations we're not allowed to build peak detection diode capacitor circuits that draws spikes of current out of the power line at the top of the sine wave. These pollute the power system with harmonics and disrupt the operation of neighboring devices. There often are regulations that limit the kind of waveform we can draw out of the power line and generally require it to be a sinusoid that is in phase with the voltage. Then we build a control system to regulate the current that our rectifier draws out of the power line and again we will control the duty cycle to make that current follow a reference. So in order to design these control systems, the first thing we need to know is we need to model the dynamics of the converter. So in order to model those dynamics we need good AC equivalent circuit models, and then we need to be able to solve and understand what the models are saying to guide us to do a good design of the control system. So some applications then up-converter modeling. We want to do worst-case analysis, is one example where we need to prove that our circuit will work under worst-case conditions. So what are the worst-case values of the applied signals and of all the components with their tolerances. So we may need to do this for aerospace. Often it's a requirement or in a commercial high-volume production, we need to do a similar design to get high reliability and high yield from our production line. So for high-quality design, we already know steady-state models and one thing we need to do with our steady-state model is to ensure that under steady-state conditions with all of these variations and tolerances in steady state, that the circuit will perform acceptably well with a high enough efficiency, low enough losses that the temperature rise is not too high, adequate steady-state voltage regulation and so on. Then for our AC control system design, we need AC models that we can analyze and prove that we have stability and a good transient response, a well behaved transient response under these worst case conditions. So the way a good design practice goes then is we will simulate our equivalent circuit model during preliminary design to really verify that the design works under these worst case conditions. We'll construct laboratory prototypes and make those prototypes work under nominal conditions or with whatever component values are present in the laboratory prototype. We'll then compare the performance of this prototype to what our AC model predicts, and we will have to refine our model until it predicts the observed behavior adequately well. Once we have such a model then we use it to predict behavior under worst-case conditions and improve the design untill we can show that the circuit will always work and meet our goals for reliability and production yield in performance. So the objective here then is to develop the tools that we need for this modeling analysis and design of the converter control system. Chapter 7 is about the modeling, chapter 8 is about analysis of our models, and chapter 9 continues analysis for the feedback system and then discusses how to design the feedback system using the models from this chapter. So our models of the power converter need to predict how AC variations in things like the input voltage or the load or the duty cycle affect the output voltage under transient and not DC conditions. Another way to say that is what are the small signal transfer functions of the converter from these input quantities to the output. So what we're going to do in this chapter then is to extend the steady-state converter models that you've already learned to include the dynamic elements such as the inductor and capacitor. In the next chapter, next week we will talk about how to construct the transfer functions of these converters using the models from this week, and in the final week of the course we'll discuss design of the converter control systems or chapter 9. In fact we're going to take four weeks to cover these chapters 7 through 9. Okay, modeling. So modeling is the representation of the physical behavior of some system by mathematical means. One thing you have to understand is that models are not exact, they're an approximation of what's going on at some level that is adequate to describe what we need to know, but is no more complex than that so that we don't get bogged down in unimportant details. So we certainly for example don't attempt to model what every electron and every wire is doing. We really want the simplest model we can that describes the important things of interest and ignores everything else. So we make approximations formally to ignore what's not important. It's best to start with a simple model. The simplest model is best to begin with. Later we can refine the model and add in things that we find are important, if we have the time and if those things are found to be necessary to model. But certainly we don't attempt to model everything. Here's an example, and when we talk about modeling the dynamics of a switching converter, here's exactly what the waveforms actually look like. So on the top plot here is the MOSFET gate drive signal of a switching converter. This is actually a buck-boost converter. This Vc of t is the control voltage at the input to the pulse with modulator. So these top two waveforms, we go back to my diagram, is it's the Vc of t continuous signal here that looks like this. Then this control signal that switches up and down and looks like this. So there is some AC variation in our control signal, and you can see that that causes some variation in the duty cycle. Now the AC variation is slower than the switching frequency, and we're generally talking about variations that happen much slower than the switching of the converter. So this slow variation in Vc causes a slow variation in duty cycle. The second waveform here is the observed inductor current. It has some switching ripple, but there's an average that if we were working in steady-state we would have a constant average that is the DC component of the inductor current. In this diagram though we have some AC variation and that's causing the underlying average to vary. So I've sketched here both the dark waveform is the actual inductor current and this lighter more smooth line is what we call the average of the inductor current. It's a quantity that varies and basically it's the waveform without its ripple. The bottom waveform is the similar waveform for the output voltage. So this is the output capacitor voltage of our converter. It has some switching ripple and underneath that is some underlying variation in the low-frequency component. So we can think about this as saying our control signal has some DC or constant value capital Vc. Here might be some DC value plus some sinusoidal variation that makes Vc vary sinusoidally about this DC value. In response to that, the pulse width modulator makes the duty cycle vary also. So there's some steady state value of duty cycle plus some slow AC variation in the duty cycle. Here's what a typical waveforms such as the actual output voltage waveform of the converter looks like when we have this sinusoidal variation in the duty cycle. So what's shown here first of all is we have a spectrum. So these are the amplitudes of frequency components of the waveforms versus frequency. This frequency right here is the switching frequency. So we have switching harmonics in the waveform and so we see a frequency component at the switching frequency and also add its harmonics. So this is twice the switching frequency, three times the switching frequency and so on. In addition to that, there is a low-frequency variation at this low frequency modulation frequency. So when we vary the control input sinusoidally at frequency omega sub m, we observe some output variation at frequency Omega sub m. Now the converter power stage is nonlinear in general and so it generates harmonics. So we have not only omega sub m, but we also see the harmonics of omega sub m from this non-linear process. Then finally modulation causes sidebands. So we see sidebands of the switching frequency. So there's one right here at frequency omega s plus omega m. There'll be sidebands including the harmonics as well and they extend on both sides of each of the switching harmonics. So we see the spectrum all the way up here. So it's pretty complicated waveform that comes out of our converter. Now it's time to simplify our model. We're not going to try to predict all of this, and frankly, we don't care about it at all. Generally, when we're designing a control system, what we care about is this. If you vary the duty cycle at 100 hertz, what is the resulting 100 hertz variation in the output voltage? This is what we want to predict, and as we've done so far in this course, we generally don't want to predict the switching harmonics. We want to remove the switching ripple and just model the underlying low-frequency variations in the system. We're going to throw these out, and we will do that by a process called averaging that is a generalization of the way that we have calculated the DC components so far in the course. This approach does continue to apply the small ripple approximation, but we need to discuss in more detail exactly how to do that in the AC model. Then we also have these harmonics, that are low frequency harmonics, that are multiples of the modulation frequency. We are going to ignore these or throw them out also, we'll not model them. The reason for that is that, otherwise are differential equations that we get for the converter dynamics are non-linear, and we can't simply take the Laplace transform and find transfer functions. What we're going to do is similar to what is done in an electronics class in modeling other nonlinear elements such as transistors and diodes that we construct a small signal model, in which we ignore these low-order harmonics and linearize the circuit. Then this is in the end, the only thing that we want to model. The business of averaging to remove the switching ripple. So far in the course, we found the DC component of the wave form by integrating the wave form, really over one switching period and dividing by the period. Although that definition assumes that the circuit is in steady state and that the waveforms are periodic. Here, what we're going to do is generalize that, and talk about a moving average. If we go back here and look at say, the output voltage, we will integrate over one switching period. But then make this period of integration, this one switching period integral move forward in time along the waveform, and what we'll get by doing that is the underlying variation with the switching ripple are smoothed out or removed. So formally, this is how we define the average of an AC signal. The average value of some signal x of t is found with this moving average where at time t, we integrate over one period that's centered at time t. So it extends from t minus half the switching period to t plus half the switching period, and we define the average of the waveform at time t is found by integrating the waveform over this interval, and then dividing by the period. What we're going to find when we do that, and we're going to show that in the next lecture, is that we can write the equations of the converter like this. So for every inductor, we're going to have an equation L d, idt is the average voltage on the inductor where the waveforms in this equation are the averaged values of the inductor, current, and voltage similar for each capacitor. Now, this turns out to be a simple generalization of what you already know how to do in the DC case. In fact, if you look at what happens to these equations in steady state when the transients have settled down, then the derivatives of the inductor, currents, and capacitor voltages go to zero, and that's what happens when we reach steady state. In that case, the left-hand sides of these equations go to zero, and what we're left with is the right-hand side equal to zero, and the right-hand side contains an average inductor voltage or an average capacitor current, and these are in fact the volt-second balance and charge balance equations that we already know how to find. In fact, the bottom line is all you have to do is take the equations that you already know from volt-second balance and charge balance, and instead of equating these voltages and currents to zero, we equate them to Ld, i dt or C d, v dt. So in fact, averaging is a small step from what you already know. Now, I'm going to give an optional lecture where we're going to talk about this approximation in more detail and give it some mathematical foundation. But you can apply it without even understanding that foundation. The more you know the better, and I hope many of you will choose to watch that lecture. Here's the equations we get. It turns out that the right-hand side is nonlinear. It involves the products of time varying quantities such as duty cycle times output voltage, for example, and these are nonlinear processes is multiplication of time varying quantities. So we have to linearize this circuit if we want linear differential equations that we can solve as in basic circuit analysis. So the next step is to construct a small signal converter model that is linearized about a quiescent operating point. This is something we already do or already know how to do in undergraduate electronics courses. For example, if we think about the equation of a diode. The diode, physical diode has this exponential relationship between current and voltage. If we operate the diode at some quiescent value of current, meaning a DC current, I which might be right there at two amps, then we can find the resulting DC voltage from the IV characteristic. There would be this value, and then on top of that quiescent value, if we add some small AC variation, that here is called i hat of t. So we vary the current sinusoidally about this quiescent value like this, then in result, in a vat variation, we will see some AC variation in the voltage. That's illustrated here, and we call that variation v hat. In the small signal model, what we're trying to do is simply find what v hat is in terms of i hat. In your electronics classes, I hope that you learned how to do that, we linearize the non-linear i-v characteristic of the diode using its slope at the quiescent operating point, and we define a small-signal model using a resistor whose resistance depends on the slope. We can then from this model, represent the relationship between the small signal AC current and the resulting small signal AC voltage. Power converter has the same thing going on really. Here's an example of the buck-boost converter, where we know that the output voltage is the input voltage multiplied by D over 1 minus D, and there is a minus sign there also. So the output voltage is a non-linear function of D. This curve here of output voltage versus duty cycle is not a straight line. You might then try to do a similar thing. We could say, operate the converter with a quiescent duty cycle or steady state value of one-half, which from solving this expression tells us that we'll have a resulting quiescent output voltage of minus V_g, and then on top of that, we could say, what if we also added some small variation, D hat variation in the duty cycle? From this slope, we can calculate the resulting small AC variation V hat in the output voltage that would result. To linearize this about the quiescent operating point, we would find the slope of this curve, and from that slope, we would know the gain, the small signal gain from a duty cycle variation to an output voltage variation. It turns out that the buck-boost converter is actually more complicated than that. Linearizing this DC characteristic is not enough. We actually have to get into the details of the model and how the L's and C's interact with each other to to correctly do this. We're going to learn how to do that this week. What I'll say to that is here is the answer for the buck-boost converter, and we're going to derive this answer. So this may look similar to the DC model of the buck-boost converter that you've already seen. It has the same transformers, one to D transformer and a D prime to one transformer. But it also includes the values of L and C, and it also includes some extra generators. These are d hat sources that model how variations in the duty cycle excite variations in the voltages and currents of the converter power stage. We can add losses into this if we like as well. I haven't added them in this particular example. So in the next several lectures, I'm going to develop this model and teach you how to derive an equivalent circuit model such as this.
