Having developed AC models of switching converters, we're now in a position to use the results of these models, to draw their transfer functions and bode plots and then, to design their control systems. So in chapter eight, which we'll cover this week and part of next week, we're going to discuss, the transfer functions of, that are predicted by the small signal models of our power converters. So we began with a review of Bode plots. Now in my experience, usually our electrical engineering undergraduates have seen Bode plots in their beginning circuits classes, however, the depth at which it's covered and the abilities to construct Bode plots, are highly variable. And we really need to be good at these, in order to use them and actually design the control systems. So what I'm going to do, is start at the beginning, but cover Bode plots fairly quickly. Those of you who feel that you are are good at these, can skip the first several lectures. However, often people comment that they learn something from watching them. So I would suggest, that if you have time you, you take a look. we'll also talk about second order systems. We're going to concentrate as well, on constructing phase asymptotes, not just the magnitude asymptotes, and we'll talk about some approximations, that can be made, that help us develop analytical expressions for fairly complicated transfer functions and let us control the algebra. After that, we're going to talk about converter transfer functions and so, we'll apply the Bode plots to construct the transfer functions, of some switching converters and discuss their important features. Next, we're going to talk about the graphical construction technique to construct Bode plots. And, this is an approximate method that actually lets us construct transfer functions, nearly by inspection. So, we can take a circuit that could be fairly complex and develop, without doing a lot of algebra the Bode plot of its frequency response. This gives a lot of physical insight, into what's going on and it actually allows us to function as engineers on a higher level, to know what we're doing and get nice quantitative results, without getting bogged down in the algebra. So, I think this is a fun section to discuss and we'll do that third. Finally I'm going to talk about a couple of practical issues, regarding measurements of transfer functions in the lab. As an example, here is the buck-boost converter model, that we developed in chapter seven and so, we identified a number of quantities of interest, some transfer functions, such as GBG, which is the transfer function from the VG hat variations to V hat, or output variations. We also identified GBD of S, which is the control to alpha transfer function, from D hat to V hat. We may be interested in the output impedance of our converter, to see how load current variations affect the output voltage. And we may also be interested, even in the input impedance, in a larger system application. So these are examples of transfer functions, that we want to construct. Okay, this is a fairly complex circuit. It's actually not that bad, but if we start refining it with resistor, loss elements and other things, it can grow very quickly. Now and, in teaching undergraduate circuits, we faculty often have a rule, that if you give a problem with more than three elements, most of the class gets it wrong. Certainly it gets difficult to handle circuits that have a lot of complexity and many elements and yet interesting circuits that do important things, do have many elements, hundreds or thousands or more. And so, how do you handle that? We need to be able to workout the transfer functions of interest, gain an understanding of what's, what's going on in them and yet, gain that understanding without getting bogged down in algebra. And if you have very many algebra elements, then the algebra can get long and fill up many pages, and it's unlikely, that the average engineer will get through that algebra, without making algebra mistakes. So we have to manage and handle the complexity of real systems. And how do we deal with that? In the electrical engineering field, there is a movement among certain faculty, to adopt what is known as design oriented analysis. We power electronics faculty at the University of Colorado, are among those faculty. What we are trying to do, is teach our students how to handle this complexity In the systems, and be able to function as engineers. There's an old story that really, about a situation that has been going on for decades, in which the typical electrical engineering graduate will get a new job at a company, and be given a task involving, say the current design of some system and they'll be given the, the you know, set of schematics, with many, many components and fairly complex system. Classically, the new graduate will start writing loop and node equations to try to analyze this system, if it's an analog system and they'll make a giant mess. They'll get equations that are pages long and get nowhere. And then you start hearing comments like, everything I learned in college was useless and to the student at that point in their career, they're right. Even worse, they then walk down the hall and ask the technician, say the engineering technician, what's happening and the technician can explain it. Even though the technician didn't take college, perhaps and can't write all the equations, the technician could explain in paragraph and with physical explanations, how the circuit works, which is even more frustrating to the new graduate. The seasoned experienced engineer though, can write those equations, know what they're doing and function to design an, a new system, that works well. So, how does the, the experienced engineer do it, and what have they learned, that the new graduate doesn't know? Well, what we're trying to do with design oriented analysis, is teach some of those things. This week, we're going to talk about a few of those things. there are many components of design oriented analysis, but this is at least, a good start. So, we want to teach how to approach a real and complicated system, here in the context of transfer functions, and solving the AC models of power converters and then, using those models to design control systems. So what invariably happens, is we have complicated derivations, long equations, and multitude of algebra mistakes. We want to avoid those things or conquer them. And what we would like to do in our design process, is obtain physical insight that leads the engineer to synthesize a good design, which might mean, choosing element values, or it might mean, gaining insight to change the circuit in some way, to make it better. As far as choosing element values go, we want to obtain simple equations, that can be inverted. So, with these simple equations, we can solve for the element values that are needed, to get the circuit to behave the way we want. If we have a long, complicated equation that you can't invert to solve for the element values, then it's really pretty useless. So design-oriented analysis is a structured approach to analysis, that attempts to avoid these problems. Another major component of design oriented analysis, is learning how to make engineering approximations. Unwittingly, in academia, we train students to prefer exact solutions and the students have some kind of feeling that, an approximation is, is a failure. The opposite is in fact, the case. You have to realize that the exact solution is in fact, the exact equation that describes the approximate model, that the professor gave you in class, that has neglected all kinds of things. The model in the first place, is approximate. So the best equation to begin with, is a simple one, that gives insight and makes all kinds of approximations. You have to realize that, a simple approximation is, is really painless. If you can figure out how to make a simple approximation, it saves you from doing all that work, until you can get one quickly. And if that approximation gains some insight, then it was, it was worthwhile. And you have nothing to lose, because you can still do the algebra later, if you really want to and in the meantime, you've gained insight, with a small amount of work. In fact, the way the process goes, is we refine our approximation to make our model or our equations, somewhat more complex, to account for whatever is significant, but we will never go all the way and exactly solve everything. That’s an impossible task and the result is worthless anyway because it’s a big mess. So we will keep refining our equations and our approximations, until at some point, we decide that's enough and it's not worth expending any more effort to, to further improve the accuracy of, of the result. Specifically, here are some of the things we're going to talk about, in this chapter. We're going to write transfer functions in normalized form and what this does, is it directly exposes the salient or important features of our frequency response. This is also a way to manufacture multiple constraints, out of one equation. So for example if, we may have several specifications for our system, but we may have hundreds of components. So how do we get hundreds of equations, to solve for those components, out of a small number of specifications? The salient features in fact, are sub-equations that come out of, say a larger transfer function, that give us equations for each of the important features of the circuit and how those features depend on individual element values. We'll talk about how to find simple analytical expressions for the asymptotes, and corner frequencies, and similar kinds of salient features then that, let us choose the element values. I'm going to introduce the ideas of frequency inversion, which we'll also call inverted poles and zeros, that let us mold our expressions into ones, that are the most meaningful, and are written in terms of the important features, of what we want. We're going to talk about analytical approximation, when we have higher order polynomials and then, we're going to generalize that into graphical construction of Bode plots, that actually lets us construct Bode plots, nearly by inspection, directly on the plot, without doing Algebra. And if you can do that, then you know what you're doing, you have a lot more physical insight, into what's going on and it saves a lot of time. This graphical construction is actually equivalent to this analytical approximation of the roots. Here's an example of the, what we want to get, at the end of this chapter. This is the control to output transfer function of that buck boost converter example and this is the bode plot of its magnitude and phase, versus frequency. So we're going to talk about what, what the Bode plot is. Hopefully some of you know what it is, but I'm not assuming everybody does. this also has analytical expressions for the salient features, so there is a DC gain that is written here, in terms of the duty cycle and the output voltage, there is a Q factor and a resonance, with some complex poles. There's a right half plane zero, or non-minimum phase zero and we have analytical expressions for the corner frequencies, and the asymptotes, that are the kinds of things we need to do, or know, to design the control system. We also have asymptotes, for the phase response, so we have zero degrees of phase shift in this control to output transfer function, at low frequency. There are break frequencies that depend on things like, the resonant frequency, and there are phase asymptotes with a, near a high frequency phase asymptote, of minus 270 degrees, for the buck boost converter. So again, we need to know this, in order to determine the stability of our feedback loop, that we will use to control this, converter example.
