Modelling And Control Of A Buck Converter - Diva-portal

Modelling and control of a Buck converter Bachelor thesis by Shun Yang Karlskrona, Blekinge, Sweden Date: 2011-06-22 This thesis is presented as part of Degree of Bachelor of Science in Electrical Engineering Blekinge Institute of Technology School of Engineering Department of Electrical Engineering Supervisor: Anders Hultgren Examiner: Jörgen Nordberg i

Modelling and control of a Buck converter Bachelor thesis at Blekinge Institute of Technology By Shun Yang 870924-6030 Examiner: Jörgen Nordberg. Supervisor: Anders Hultgren Karlskrona, Blekinge, Sweden Date: 2011-06-22 ii

Acknowledgement First and foremost I offer my sincerest gratitude to my supervisor, Anders Hultgren, who has supported me throughout my thesis with his patience and knowledge whilst allowing me the room to work in my own way. I attribute the level of my Bachelor’s degree to his encouragement and effort and without him this thesis, too, would not have been completed or written. One simply could not wish for a better or friendlier supervisor. In my daily work I have been blessed with a friendly and cheerful group mate --Gaojun Chen. He has provided good arguments about controller theory and helps me regain some sort of fitness: healthy body, healthy mind. He is a good companion and has had the good grace to pester me much less than average with computer questions. He provided an experienced ear for my doubts about writing a thesis. The Department of Electrical Engineering has provided the support and equipment I have needed to produce and complete my thesis. My deepest gratitude also extend to all lecturers of at Blekinge Institute of Technology (BTH), for their guidance, ideas, and support in completing this final year’s thesis project. This project had opened my eyes on solving real problems and I was able to relate them with what I’ve been studied in BTH during the past 3 years. Finally, I thank my parents for supporting me throughout all my studies at University and providing financial fund to help me to complete my bachelor education. iii

Abstract DC/DC buck converters are cascaded in order to generate proper load voltages. Rectified line voltage is normally converted to 48V, which then, by a bus voltage regulating converter also called the line conditioner converter, is converted to the bus voltage, e.g. 12V. A polynomial controller converter transforms the 12V into to a suitable load voltage, a fraction of or some few voltages. All cascaded converters are individually controlled in order to keep the output voltage stable constant. In this presentation focusing on the polynomial controller converter implemented as Ericsson’s buck converter BMR450. In this paper modeling, discretization and control of a simple Buck converter is presented. For the given DC-DC-Converter-Ericsson BMR 450 series, analyzing the disturbance properties of a second order buck converter controllers by a polynomial controller. The project is performed in Matlab and Simulink. The controller properties are evaluated for measurement noise, EMC noise and for parameter changes. Keyword: polynomial controller, second order, buck converter iv

Table of Contents Acknowledgement . iii Abstract . iv List of Tables . vi List of Figures . vii List of Symbols . viii Chapter 1. Introduction . 1 Chapter 2. Modeling of Buck Converter. 3 Chapter 3. Polynomial Controller Design. 11 Chapter 4. Simulation on MATLAB . 15 Chapter 5. Results and Conclusion . 19 References . 30 Appendix . 31 Appendix A Simulink Models . 31 Appendix B MATLAB Code . 33 Appendix C Interesting Phenomenon found in the simulation . 38 Appendix D Transfer function . 42 v

List of Tables Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Parameters of Buck Converter . 2 Parameters of system . 10 Polynomial coefficients of system. 10 Pole Placement of Polynomial Controller . 14 Coefficients of Polynomial controller . 14 Parameters of system . 27 Appendix Table 1 Appendix Table 2 Appendix Table 3 Appendix Table 4 Capacitor’s Resistor equals Inductor’s . 38 Capacitor’s Resistor is bigger than Inductor’s . 39 Capacitor’s Resistor is smaller than Inductor’s . 40 Capacitor’s Resistor equals Inductor’s . 40 vi

List of Figures Fig. 1 Overall system circuit . 3 Fig. 2 PWM Wave . 4 Fig. 3 PWM controlled input voltage dVg . 4 Fig. 4 Circuit with Voltage Source . 5 Fig. 5 Circuit without Voltage Source . 7 Fig. 6 Controller with Buck Converter System [2] . 11 Fig. 7 The shadow region inside the unit cycle is stable for system . 13 Fig. 8 Buck Converter System with polynomial Controller [2] . 15 Fig. 9 System block . 15 Fig. 10 Polynomial controller block. 16 Fig. 11 PWM with polynomial controller . 16 Fig. 12 Relational operator. 17 Fig. 13 Switch . 17 Fig. 14 Saturation condition . 18 Fig. 15 Discrete model without saturation, system output and controller output. 19 Fig. 16 Continue model, system output and controller output . 20 Fig. 17 Simulation result, without disturbance, without noise . 21 Fig. 18 Noise Magnitude . 22 Fig. 19 Output Voltage with noise . 22 Fig. 20 L 0.9μH, C 150μF , Simulation result, with disturbance, without noise . 23 Fig. 21 10%, L 0.99μH, C 165μF, Simulation result, with disturbance, without noise . 24 Fig. 22 -10%, L 0.81μH, C 135μF, Simulation result, with disturbance, without noise . 25 Fig. 23 Simulation result, with disturbance, with noise . 26 Fig. 24 RC 50mΩ, C 500μF, Simulation result, with disturbance, without noise . 26 Fig. 25 Output Voltage before and after disturbance when RL 2mΩ, RC 50mΩ, C 500uF . 27 Fig. 26 Output Voltage filtered noise when RL 2mΩ, RC 50mΩ, C 500uF . 28 Fig. 27 Output Voltage with noise when RL 2mΩ, RC 50mΩ, C 500uF . 29 Appendix Fig. 1 Continuous Time Model without Saturation . 31 Appendix Fig. 2 Discrete Time Model without Saturation . 31 Appendix Fig. 3 Discrete Time Model with Saturation . 32 Appendix Fig. 4 PWM Model with Controller . 32 Appendix Fig. 5 Output Voltage before and after disturbance when RL 2mΩ and RC 2mΩ . 38 Appendix Fig. 6 Output Voltage before and after disturbance when RL RC . 39 Appendix Fig. 7 Output Voltage before and after disturbance when RL RC . 40 Appendix Fig. 8 Output Voltage before and after disturbance when RL 10mΩ and RC 10mΩ . 41 vii

List of Symbols % V L C R d DC fs iL yref PWM Percentage Volt Inductor Capacitor Resistor Duty Cycle Direct Current Switching Frequency Switch/Inductor Current Reference Voltage Pulse Width Modulation viii

Chapter 1. Introduction This chapter describes the thesis background, objectives, scopes, and summary. In the chapter, it briefs the description of the buck converter and the voltage-mode controller as well as the objectives and the scopes. At the end, outline of this thesis is given in this chapter. 1.1 Thesis Background Direct current to direct current (DC-DC) converters are power electronics circuits that converts direct current (DC) voltage input from one level to another. DC-DC converters are also called as switching converters, switching power supplies or switches. DC-DC converters are important in portable device such as cellular phones and laptops. Why DC-DC converter is needed? Just image that when wanting to use a device with low voltage level, if connect the device such as laptop or charger directly to the rectified supplied from the socket at home, the device might not functioning properly or it might be broken due to overcurrent or overvoltage. The voltage level needs to be converted to suitable voltage level for the equipment to function properly. In this project, the configuration of DC-DC converter chosen for study was buck configuration. Buck converter converts the DC supply voltage to a lower DC output voltage level. The control method chosen to maintain the output voltage from the buck converter was polynomial controller. Polynomial controller technique compares the actual output voltage with the reference voltage. The difference between both voltages will drive the control element to adjust the output voltage to the fixed reference voltage level. This is called as voltage regulation. 1.2 Thesis Objectives The main objective of this project is to design a buck converter controller based on the theory for discrete polynomial controllers. A basic introduction can be found in [1].The converter control signal is implemented as a Pulse Width Modulated signal making the system into a switched system. It is then interesting to find the nonlinear system properties by simulations. The control design should be evaluated by means of accuracy in output voltage level given some different kind of disturbances. The disturbances are load current changes, measurement noise and parameter variations. 1.3 Thesis Scopes The scopes of this thesis are: i. Study the operation of buck converter. ii. Design the mathematical system model. iii. Design the polynomial controller. iv. Simulation of buck converter and controller circuit using Simulink and MATLAB in order to test the properties of the system. v. Get the conclusion from simulation result. 1

The parameters used in the project are as below, see Table 1. Table 1 Parameters of Buck Converter Topology DC-DC Buck Converter Inductance L 0.9μH Capacitance C 150μF / 500μF Resistance of Inductor RL 2mΩ / 10 mΩ Resistance of Capacitor RC 2mΩ / 10mΩ / 50mΩ DC Input Voltage Vg 12V Reference Voltage (Expected DC Output Voltage) yRef 3.3V Wanted Inductor Current iI 0-20A Sample Frequency fs 330kHz Sample interval h 3μs 1.4 Outline of Thesis This thesis consists of 5 chapters. In the first chapter, it discusses thesis background, objectives, and scopes. In Chapter 2, building of mathematical system model and theories on buck converter are displayed. The polynomial controller design is showed in Chapter 3 while the simulation implementation on Matlab and Simulink are offered in Chapter 4. In Chapter 5, it discusses the simulation result and conclusions obtained upon successfully completing thesis project. Finally, the last part in the thesis provides the references and appendices used in the thesis project. 2

Chapter 2. Modeling of Buck Converter A Buck converter consists of a transistor and diode that applies the supply voltage on an inductor capacitor, LC, circuit. The output voltage is the voltage across the capacitor. The input voltage u on the LC circuit is controlled by pulse width modulation, PWM. i.e. part of a cycle time, the voltage applied on the LC circuit is Vg and the rest of the cycle time the input voltage is zero. The range of duty cycle, d [0, 1], is the relative time of the cycle time that the supply voltage is connected to the circuit, i.e. the input is Vg. The duty cycle d is the control signal to the Buck converter. A circuit diagram can be seen in Fig. 1. Switch RL L node 1 Vg RC node 2 iI C Fig. 1 Overall system circuit The system need to be modeled to get the relationship between the input signal and output. The Buck converter system is a switched system. It consists of two switching states: firstly, when the switch is connected to node 1, charging status; secondly, when the switch is connected to node 2, discharging status. The switched system can be modeled as an averaged system. The averaged system describes the switched system up to about one tenth of the PWM frequency. In this chapter deriving the two system models of these two switching state, and then take the average of the two systems according to the duty cycle. PWM is the method of choice to control modern power electronics circuits. The basic idea is to control the duty cycle of a switch such that a load sees a controllable average voltage. To achieve this, the switching frequency (repetition frequency for the PWM signal) is chosen high enough that the LC filter cannot follow the individual switching events. In this case, the sampling interval is 3μs. So the sample frequency is 330 kHz, see Fig. 2. 3