1 Switched Mode Power Conversion Prof. L. Umanand Department of Electronics Systems Engineering Indian Institute of Science, Bangalore Lecture - 19 Modeling DC-DC convertors Good day to all of you. Today, we shall study about modeling switched mode power convertors. Though it is a vast topic and we will not be able to cover in just one hour, we will be taking more, more classes, we will start towards being able to model switched mode power convertors. Today, we will be discussing more on the modeling basics, so that is why the title of today s lecture is called modeling basics. (Refer Slide Time: 01:03) So, this is what we will be dealing in the major portion of today s lecture. Till now you have been discussing DC-DC convertors, the steady state analysis, non- idealities and to some extend the dynamic models of the various isolated and non-isolated convertors. We shall now look at, how we try to control the switched mode power convertor? You have looked at the switched mode convertor as an open loop system, we need to control it, develop controllers such that we get outputs with a certain performance specification. So, that is the intent and towards this intent, we will develop the, evolve the subject. Before we go, before we start even trying to control the switched mode power convertor, we need to model them, the dynamic model which you have been familiarized by previous lectures.
2 We will dwell today in this class on trying to look at the generic way of developing these models and then apply it to the specific DC-DC convertors, so towards this goal I would like to bring your attention to one important aspect. In control system terminology, what is a plant? The plant, this plant is what you have been discussing in the previous lectures on DCDC convertors, but let us bring it into a format, which is compatible for control systems design, it has an input energy port as shown here. Very important, I am going to hash it. This is an input power port, energy port where in power is fed into this plant. You have an output from the plant. We will call it as the output power port-these two are the major energy or power level ports in the DC in any DC-DC convertor. The input is normally obtained from the input power supply DC power supply, which has been termed as V g in the previous classes and the output, which is termed as V 0 having a voltage V 0, so Vg Ig gives the input power V0 I0, the output power apart. Apart from this there are two other ports not power ports, but signal ports which you would already be familiar with-one is called the control input, control input it is normally given a symbol u and the sensed or controlled output. So, this controlled output has a symbol y, so internally this controlled output is generally linked to the actual power output by means of a sensing circuitry. So, the sensing circuitry does the function of capturing the actual power signal converting it into compatible low, low level signal voltage and giving it as signal with which we have named it as y. So, for example, in the case of a DC-DC convertor the output voltage if you want to feed it back, you will be passing it through a device like an optocoupler or probably a hall sensor and the output of this is signal conditioned and brought out as the controlled output signal, which is what will be fed back. So, with these, with these 4 ports, a controller is generally designed, which is basically interfacing with the control input and the control output. So, the control process is generally linking these two aspects of the plant ports; so this is our control process which we will see what goes to make that up. Now, if we look at the case of specific convertor like the buck convertor.
3 (Refer Slide Time: 08:16) Now, we know that the buck convertor is of this form we indicate the switches in this manner, so the single pole double throw switches are replaced by two single pole single throw switches. You have an inductor, then a capacitor. So, this LC form the filtering mechanism and followed by a load Rl. Now here at the input side, you have an unregulated or uncontrolled source, V g; you have switch S1 and switch S2 and these are mutually exclusive in the switching. when S1 is on, S2 is off. When S2 is on S1 is off and you say that this is on for a period of time DT, where T is the switching T S is the switching period and S2 is on for a period 1 - D TS, where D is the on time of S1 by T S. The total switching period so to make it a bit more clear, if we say that time proceeds along this axis. We have this much time allocated for the switching period T S. Then we can say that this portion of the time, where D is the fraction less than 1, less than or equal to 1. This fraction D T S and this fraction (1 D) T S are the two mutually exclusive intervals of time, during which time S1 is on during D T S. S2 is on during (1 D) TS. Now this particular circuit if we look at this, how do we map this circuit into the plant configuration that we have been looking at? So, if we look at the plant, let us say that we have one input energy port. We have one output energy or power port, power output. So, what is the input and the output? So, if you look at this portion, this is the power input to the buck convertor, so that basically constitutes the input power. So, let us rewrite that as coming from here and the voltage here is V g, the
4 generated voltage and the output you see is on this side and we shall bring that out in this fashion, where the output here is V0 and the load is connected external to this plant, because it is decided by not the plant. But, it is decided by the user. Now we have the input and output power ports. We need to define two more ports which is the control input, u. (Refer Slide Time: 14:01) And another port, which comes out of the plant, which is the controlled output y. Now what is the control input for this plant. We have seen the steady state analysis V0 =DVG. Now, D what is D? D is basically the fraction of the time the switch S1 and switch S2 is on and off within a period T S. So, D actually can be termed as a control input, if you reduce the value of D, then the output reduces. If you increase the value of D, the output also increases, therefore the output in this case which is V 0.
5 (Refer Slide Time: 15:53) So, if we, if we measure the output voltage, signal condition it by passing it through appropriate optocoupler circuit and then pass it on to the output port. So, V0 is the output signal and this output signal can be controlled by input. What is that input D? In the steady state we say it is upper case D, so this D which basically implies that you are in some manner trying to adjust the times for which these two switches are on and off. So, in affect it is actually one control input because these two switches are directly coupled. They have only single variable D to control it and their mutual exclusive. So, in affect we have actually one control variable D and that is what is brought out, as the plant control input. So, if you see this buck convertor is mapped into control plant block diagram, control plant block diagram.
6 (Refer Slide Time: 17:26) By saying that the control input is u, which in this case is d, this lower case d is used to represent both variations plus steady state. There is a controlled output y, which is to be controlled which in this case is V0. We have the power inputs, the power input being obtain from a supply Vg, unregulated Vg and Ig being the input currents and an output power output, which is actually supplying a load I0 at a voltage V0. So, this is our DC-DC convertor, which needs to be modeled. Now, this DC-DC convertor a practical circuit needs to mathematically represented, in a way which is amenable to develop the controllers. Now, the process of control is that we need to have a comparator and we need to have a controller. So, the basic minimum we need to have these two important components, this is the controller and a comparator. So what is done with these two? So, the controlled output which is sensed and then given as controlled output signal low power, low level signalinformation signal is fed back to this comparator. Now, to this comparator, we need to also give a reference, so it could be an output reference signal or a value, which is given as a positive or in phase. To that reference we are subtracting the fed back signal, to get the error. So, what you get here is the error between the fed back signal and the reference and the objective of the controller is to give an appropriate value of control input. Whatever be the control input d d, in this case such that the error goes to 0, so that is the job of this controller. So, this is what we want to ultimately achieve, so in order to design this controller, we need
7 to have knowledge of the DC-DC converter in a mathematical form. That representation is called the dynamic model. (Refer Slide Time: 22:26) So, this dynamic model can be represented in two basic ways; the states space representation and the other transfer function representation. However, it may be pointed out at this point that when we develop the dynamic model of a physical system from the physics of its operation, we actually arrive at differential equations, which are directly represented in this form called the state space representation and from the state space representation, you can extract the transfer function representation. So you could say that the state space representation is the one which you will first obtain. Then convert it into transfer function representation for further analysis or you could continue to do your controller design in the state space representation itself. Anyway first let us see how we will obtain the state space representation of the system. The system in our case is the DC-DC converter.
8 (Refer Slide Time: 24:41) So, in control balance the whole system including the controller and plant controller plus plant, we generally call it as the system. The open loop uncontrolled part, that we have here in blue, is called the plant, so keep that in mind. Let us proceed for then trying to see how we can obtain the state space representation of a given plant. (Refer Slide Time: 25:27) So, before we go into obtaining the state space representation of a DC-DC converter, because the DC-DC converter is a switched power converter, it has switches which are non-linear in nature, in the sense that they have two states. So, superposition principle may not be directly
9 valid. We shall look at a generic method for first obtaining the state space representation of a general sis plant. Then we apply it to the specific and special case of DC-DC converter. (Refer Slide Time: 26:25) So, let us look at the case of a general plant wherein we have a controlled input u and control input u and a controlled output y. I will not be writing this power input and power output explicitly every time. It is understood that these two ports are there, but to make the discussion faster, these two will be understood to be present in the plant. I will be using only u and y from the control point of view. These two are control signals these two are only control signals, they do not contain any power, whereas these two are power signals So, the job of the control signals is to control the flow of power in the power signals. So, even if I do not show these two ports, bear in mind that these two ports exist. They are part of the plant, it is only that I may want to draw your attention to this part well, while we are talking on the controller design or the controller portions. Now, let us say that we have a plant now like this a black box, we do not know what is inside? The nature of the transfer curve, let us say input is on the x axis- independent variable and the dependent variable y,output, is on the y axis can be linear, as shown like this. It could be in this fashion or it could also be partly linear and non linear with the saturation.
10 (Refer Slide Time: 28:59) It is a non linear characteristic or it could also have memory hysteresis and various other characteristics. So, let us take a general non linear case and see, how we go about trying to model the system? So, here I would like to mention that all the control signal analysis is done for a linear system. (Refer Slide Time: 29:55) Analysis and design is done considering that the plant is a linear system. However, most of the systems are non linear in nature. Therefore, we need to perform a linearization, if we need
11 to have an effective model of the plant. So, we will discuss these issues. (Refer Slide Time: 30:40) Therefore, consider a general plant which is not linear and let us say for this input-output transfer curve, there can be various operating points like that. Let us say a specific operation point. Let me shade it a solid, the specific operating point shown like that and if it continues to stay in that specific operating point, then we say it is in the steady state. We say the steady state values are upper case values like this. Now, this transfer curve as you see, gives you an idea of the operating point. This is the operating point. However, this transfer curve does not give an idea of the evolution time evolution of the variables u and y. You should know that both u and y are functions of time, both u and y are functions of time. So, let us say, for example, that we would also like to see the time evolution of the two variables- the input and the output variable. Let me put another graph oriented differently. This is u t along that axis and along another orthogonal axis let us say is time and you project this value of u along. Now for this operating point to have reached this stable operating point, it would probably have gone through these various intermediate operating points over short may be then come back and then settled at this particular operating, operating point. So, to see the time evolution, it probably may look like this. It may have started from 0 over short the nominal operating point, under shoot and then probably settle down to this, which will give this corresponding operating point as you are seeing there. Likewise, one could also
12 have a time evolution of the output variable -the y variable versus time on another graph with two orthogonal coordinates. Let us say if you were, if if you are interested in seeing the time evolution, it may look like this starting from 0 over shoots, under shoots and then settles down at the steady state value as it is shown by this operating point and this over shoots, and under shoots all corresponds to these various points as shown here. So, this time evolution is the one which is of interest to us and that is what gives you an idea of the dynamics. So, dynamics of the input and the dynamics of the output and this is what we need to study and model, such that these dynamics are taken care by the control input, which we are showing here. So, we need to control the way the time signals evolve with time and ultimately settle in the steady state point- upper case U and upper case Y. The design of course most of the time is going stay at the steady state point. Therefore, the design is done for the steady state operating point, which is what you have been discussing and studying till now, in the previous classes. (Refer Slide Time: 36:24) So, looking at this specific graph, so let us say that we have u and y input and output with the input-output transfer cover like that. Let us we are interested and a specific operating point upper case U and Y like this. Now, this is a non-linear curve superposition principle not exactly valid for this particular curve. So, you have to treat it as a non-linear system. The problem here is that designing and analyzing control of non- linear system is much more tedious and non intuitive. So, most of the time, the controllers are design for linear system,
13 but then if the controllers are designed for the linear systems and their plants are non-linear, how do they become compatible? So, we make a modification something like this, we call linearization. We take a small portion, such has this for deviations of u and deviations, expected deviations of y. We have a small portion segment of the transfer curve and this segment of the transfer curve can be considered as linear. So, if we say that the system is operating at this operating point than deviations in the neighborhood of operating point can be considered as linear and the controller can address or take of this deviations in the neighborhood of the operating point. Then a linear mathematical model and controller addressing the taking care of the small deviations in the neighborhood of the operating point can be designed and built. So, that is basically the approach that we will be using in building the controllers. Now, how do we go about saying that we have a linear model. Now, if if you look at the generic model you say the y is a function of the input variable u. Now this is the general model for the complete transfer curve. Therefore, we call it as a large signal model because this model, even though non-linear, handles the complete transfer curve u versus y of a particular system. Now, for this specific case, for the specific case of the input U, upper case U at the operating point you get a response output, which is upper case Y and that is called the model for that specific operating point, which is in the equilibrium state. This is called the steady state model, this called the steady state model. Notice that in the steady state model both the upper case input and the upper case response output, they are in equilibrium. They do not vary with time. They are constant with respect to the time or the differential quantity would be 0. Now, let us consider the case the generic case y =f(u). We could use Taylor series and expand it about the operating point, so we know that the output y has to be y plus or - something plus or - some portion, which is given by the Taylor series as differential with respect to u the function u - U, the Δ u d (u U). 2! d u2 All these evaluated at all the derivatives evaluated at u equal to the operating point value plus so on, it is going. Now, if you if you take a very small deviation where is Δ u, which is u - the operating point. If you take a very small deviation about the operating point and consider them to be linear, then we could say that because Δ u is small, u - U is very small, all second
14 order and higher order terms can be neglected Δ u2 will be still smaller. We can take only this portion and as we say that it is linear the slope, we can say is equal to a constant k. (Refer Slide Time: 44:04) So, this becomes, y becomes k ( u U) or y - Y, steady state value is. Now, this is the small signal deviation in the output. This is the small signal deviation in the input, so the small signal deviation the output. We will give it as a symbol input, we will give it as a symbol u^. So, we say ^y ^y and the small signal deviation in is equal to some constant times u^ and this is a linear equation. This is the linearized small signal model of y = f (u), which is the large signal model. So, you see that we have seen three ways of representing a particular system. Firstly, you have the transfer input-output transfer curve and this whole transfer curve can be represented as y = f (u). That is called the large signal model because it handles the whole large signal of u and y. It can address. And then the equilibrium value, the operating point value is given by this Y = f ( U). This is called the steady state model, it does not have any derivatives and thirdly you have the small signal model which is basically in the neighborhood of the operating point. So, in the neighborhood of operating point we take a very small deviation about the operating point. That small deviation is so small that we can consider that segment of the transfer curve as linear and thereby have a model which is a linear model, called the small signal model of y = f (u). Now, it is this small signal model, which is used for designing the controller. The
15 controller handles these deviations the small signal deviations, the steady state operating point requirement is met by separately by a feed forward value. (Refer Slide Time: 47:35) So, for example, in the case of a DC-DC converter, you could say the input u comprises of a steady state value of duty cycle plus a small signal deviation in the duty cycle. So, the steady state duty cycle is a constant which is given directly to the open loop plant and the controller output, which handles the deviations about this steady state value is coming from the controller. So, the controller output provides this small signal corrections to the plant. So, this is the approach that we will be taking even for the DC-DC converter. Now, from this what we gain is that, whatever may be the plant linear or non-linear the first step is to convert the plant into a linearized small signal model. So it is this linear small signal model that we further use for developing the controller design. So, if you look at our controller, it would look something like this. Now, let us say we have a comparator, we have a feedback portion, you have a controller as shown here and the output of the controller is actually given as the controlled input to the plant in this case it is the duty cycle. So, let us say that at the plant I have an adder here plus and plus and to this you give the steady state D. This goes in as d and let the controller output give the deviation. So, this becomes d plus d^, which is actually going into the plant and controls the output y. which is now a small signal value v^0. So, out of the sensed signal from the actual power output, which is at v0 this is having the steady state component V0.
16 So, let me write it in this following the v0 is containing the steady state portion plus small signal deviations above the steady state. So, this is the one which is fed back here and this is one which is fed back here. What should be the deviations? We want ultimately that the deviations should right die down, should be 0 in the steady state, we do not want any deviation about the operating point. It should state the operating point, which means the reference for the deviations should be 0. So, if this error e is equal to 0, then this d would represent the d^ corresponding to the deviations and this D represents the steady state value. Together they are given to the converter and the power of course, coming from the input V g. (Refer Slide Time: 52:17) So, this is how the control block diagram will look like and this is how we will try to attack the problem. So, we try to develop the small signal model for the converter, linearized and then try to see, how we go about designing the controller? So, that being the main objective. So in order to develop the small signal model of the controller, we adopt the following strategy. Any system, at least power electronic systems, are composed of three basic or elementary components, which is R, this is dissipative.
17 (Refer Slide Time: 53:11) Second one we call it as the L, kinetic storage, kinetic energy storage or current based energy storage. As you know that the energy stored in 1 2 L I by virtue of the current flowing 2 through the inductor L and third another energy storage component called C capacitance, which is potential energy storage component. This is actually a voltage based energy storage component. As you know that the energy in the capacitance is 1 C V 2 The energy is stored 2 by virtue of a potential charge within the capacitor. Now, these are the three basic components, which are used in power electronic equipments and to that we have one more component, which is the switch. So, switch is actually a non linear version of a resistor, a resistor has an I-V characteristic, which is a linear, whereas in the case of the switch it as an I -V characteristics, which is nonlinear. So, when the switch is off, we have it along the x axis and when the switches on, we have the y axis as the operating points, in this fashion. Therefore, we need to address this component also which is the switch component also into our dynamic models.
18 (Refer Slide Time: 56:03) How we go about doing this is what we will look at and study in the classes to come. (Refer Slide Time: 56:21) So to summarize now, what I would suggest now is that you get some practice on what are what are the inputs? What are the output, control output? What is the power output the power output? What is the power input of various DC-DC convertors. What are the input variables? What are the output variables? What are the power in variables and power out variables of the three non -isolated converters? The isolated converters like the forward, fly back and the derived forward convertors like the push pull, half bridge, full bridge convertors.
19 So, for these converters get some practice in trying to phrase your block diagram in this fashion. So, that it becomes compatible for controller design and understanding the controllers. In the next class, we shall proceed continue from where we left of today in trying to obtain these state space equations, the dynamic model for the DC-DC converters. Thank you.
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