# Fundamentals of Power Electronics. Robert W. Erickson University of Colorado, Boulder

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1 Robert W. Erickson University of Colorado, Boulder 1

2 1.1. Introduction to power processing 1.2. Some applications of power electronics 1.3. Elements of power electronics Summary of the course 2

3 1.1 Introduction to Power Processing Power input Switching converter Power output Control input Dc-dc conversion: Ac-dc rectification: Change and control voltage magnitude Possibly control dc voltage, ac current Dc-ac inversion: Produce sinusoid of controllable magnitude and frequency Ac-ac cycloconversion: Change and control voltage magnitude and frequency 3

4 Control is invariably required Power input Switching converter Power output feedforward Control input Controller feedback reference 4

5 High efficiency is essential η = P out P in η 1 P loss = P in P out = P out 1 η High efficiency leads to low power loss within converter 0.4 Small size and reliable operation is then feasible Efficiency is a good measure of 0.2 converter performance P loss / P out 5

6 A high-efficiency converter P in Converter P out A goal of current converter technology is to construct converters of small size and weight, which process substantial power at high efficiency 6

7 Devices available to the circuit designer linearmode switched-mode Resistors Capacitors Magnetics Semiconductor devices DT s T s 7

8 Devices available to the circuit designer linearmode switched-mode Resistors Capacitors Magnetics Semiconductor devices DT s T s Signal processing: avoid magnetics 8

9 Devices available to the circuit designer linearmode switched-mode Resistors Capacitors Magnetics Semiconductor devices DT s T s Power processing: avoid lossy elements 9

10 Power loss in an ideal switch Switch closed: v = 0 Switch open: i = 0 In either event: p = v i = 0 Ideal switch consumes zero power v i 10

11 A simple dc-dc converter example V g 100V Dc-dc R converter 5Ω I 10A V 50V Input source: 100V Output load: 50V, 10A, 500W How can this converter be realized? 11

12 Dissipative realization Resistive voltage divider V g 100V P in = 1000W 50V P loss = 500W R 5Ω I 10A V 50V P out = 500W 12

13 Dissipative realization Series pass regulator: transistor operates in active region 50V I 10A V g 100V P in 1000W linear amplifier R and base driver 5Ω P loss 500W V ref V 50V P out = 500W 13

14 Use of a SPDT switch 1 I 10A V g 100V 2 v s R v 50V v s V g V s = DV g 0 DT s (1D) T s t switch position:

15 The switch changes the dc voltage level v s V g V s = DV g D = switch duty cycle 0 D 1 DT s (1 D) T s t switch position: T s = switching period f s = switching frequency = 1 / T s DC component of v s = average value: V s = 1 T s 0 T s v s dt = DV g 15

16 Addition of low pass filter Addition of (ideally lossless) L-C low-pass filter, for removal of switching harmonics: V g 100V P in 500W i 1 L 2 v R s C P loss small v P out = 500W Choose filter cutoff frequency f 0 much smaller than switching frequency f s This circuit is known as the buck converter 16

17 Addition of control system for regulation of output voltage Power input Switching converter Load i v g v H(s) sensor gain δ transistor gate driver δ pulse-width modulator v c G c (s) compensator error signal v e Hv dt s T s t reference input v ref 17

18 The boost converter 2 V g L 1 C R V 5V g 4V g V 3V g 2V g V g D

19 A single-phase inverter 1 v s 2 V g 2 v load 1 v s H-bridge t Modulate switch duty cycles to obtain sinusoidal low-frequency component 19

20 1.2 Several applications of power electronics Power levels encountered in high-efficiency converters less than 1 W in battery-operated portable equipment tens, hundreds, or thousands of watts in power supplies for computers or office equipment kw to MW in variable-speed motor drives 1000 MW in rectifiers and inverters for utility dc transmission lines 20

21 A computer power supply system regulated dc outputs v ac i ac Rectifier Dc-dc converter ac line input Vrms dc link loads 21

22 A spacecraft power system Dissipative shunt regulator Solar array v bus Battery charge/discharge controllers Dc-dc converter Dc-dc converter Batteries Payload Payload 22

23 A variable-speed ac motor drive system 3øac line 50/60Hz Rectifier v link Inverter variable-frequency variable-voltage ac Dc link Ac machine 23

24 1.3 Elements of power electronics Power electronics incorporates concepts from the fields of analog circuits electronic devices control systems power systems magnetics electric machines numerical simulation 24

25 Part I. Converters in equilibrium Inductor waveforms Averaged equivalent circuit v L V g V R L D Ron D' V D D' R D D' : 1 DT s D'T s V t V g I V R switch position: i L I i L (0) V g V L i L (DT s ) V L 0 DT s T s i L t Predicted efficiency 100% 90% 80% 70% % 0.05 η 50% R L /R = 0.1 Discontinuous conduction mode Transformer isolation 40% 30% 20% 10% 0% D 25

26 Switch realization: semiconductor devices The IGBT collector Switching loss transistor waveforms i A Q r gate V g v A i L emitter 0 0 t n Gate p n Emitter n p n diode waveforms i L 0 area Q r i B v B 0 V g t n - minority carrier injection t r p p A = v A i A Collector area ~Q r V g area ~i L V g t r t 0 t 1 t 2 t 26

27 Part I. Converters in equilibrium 2. Principles of steady state converter analysis 3. Steady-state equivalent circuit modeling, losses, and efficiency 4. Switch realization 5. The discontinuous conduction mode 6. Converter circuits 27

28 Part II. Converter dynamics and control Closed-loop converter system Averaging the waveforms Power input Switching converter Load gate drive v g v R feedback connection t transistor gate driver δ δ pulse-width modulator v c compensator G c (s) v c voltage reference v ref v actual waveform v including ripple averaged waveform <v> Ts with ripple neglected t dt s T s t t Controller Small-signal averaged equivalent circuit v g Id V L g V d 1 : D D' : 1 Id C v R 28

29 Part II. Converter dynamics and control 7. Ac modeling 8. Converter transfer functions 9. Controller design 10. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 11. Current-programmed control 29

30 Part III. Magnetics transformer design i 1 n 1 : n 2 i M L M R 1 R 2 i 2 i k the proximity effect layer 3 layer 2 3i 2i 2i i 2Φ Φ layer 1 i d : n k R k current density J transformer size vs. switching frequency Pot core size B max (T) 25kHz 50kHz 100kHz 200kHz 250kHz 400kHz 500kHz 1000kHz Switching frequency 0 30

31 Part III. Magnetics 12. Basic magnetics theory 13. Filter inductor design 14. Transformer design 31

32 Part IV. Modern rectifiers, and power system harmonics Pollution of power system by rectifier current harmonics A low-harmonic rectifier system boost converter i g i ac L D 1 i v ac v g Q 1 C v R Harmonic amplitude, percent of fundamental 100% 80% 60% 40% 20% 0% 100% 91% 73% 52% THD = 136% Distortion factor = 59% 32% 19% 15% 15% 13% 9% Harmonic number Model of the ideal rectifier v control multiplier v ac i ac ac input v g i g PWM R s X v a v err G v ref c (s) = k x v g v control compensator controller R e (v control ) Ideal rectifier (LFR) p = v ac 2 / R e i v dc output v control 32

33 Part IV. Modern rectifiers, and power system harmonics 15. Power and harmonics in nonsinusoidal systems 16. Line-commutated rectifiers 17. The ideal rectifier 18. Low harmonic rectifier modeling and control 33

34 Part V. Resonant converters The series resonant converter Q 1 D 1 Q 3 D 3 L C 1 : n V g R V Q 2 D 2 Q 4 D 4 Zero voltage switching 1 Q = 0.2 v ds1 V g Dc characteristics M = V / V g Q = Q = 20 Q = conducting devices: Q 1 X D 2 Q 4 turn off Q 1, Q 4 D 3 commutation interval t F = f s / f 0 34

35 Part V. Resonant converters 19. Resonant conversion 20. Quasi-resonant converters 35

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