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power electronics

fundementale of power electronics

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power electronics

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  1. Fundamentals of Power Electronics Second edition Robert W. Erickson Dragan Maksimovic University of Colorado, Boulder 1 Fundamentals of Power Electronics Chapter 1: Introduction

  2. Chapter1: Introduction 1.1. 1.2. 1.3. Introduction to power processing Some applications of power electronics Elements of power electronics Summary of the course 2 Fundamentals of Power Electronics Chapter 1: Introduction

  3. 1.1 Introduction to Power Processing Switching converter Power input Power output Control input Dc-dc conversion: Ac-dc rectification: Dc-ac inversion: Change and control voltage magnitude Possibly control dc voltage, ac current Produce sinusoid of controllable magnitude and frequency Ac-ac cycloconversion: Change and control voltage magnitude and frequency 3 Fundamentals of Power Electronics Chapter 1: Introduction

  4. Control is invariably required Switching converter Power input Power output Control input feedforward feedback Controller reference 4 Fundamentals of Power Electronics Chapter 1: Introduction

  5. High efficiency is essential 1 η =Pout η Pin 0.8 Ploss= Pin– Pout= Pout1 η– 1 0.6 High efficiency leads to low power loss within converter Small size and reliable operation is then feasible Efficiency is a good measure of converter performance 0.4 0.2 0 0.5 1 1.5 Ploss / Pout 5 Fundamentals of Power Electronics Chapter 1: Introduction

  6. A high-efficiency converter Pin Pout Converter A goal of current converter technology is to construct converters of small size and weight, which process substantial power at high efficiency 6 Fundamentals of Power Electronics Chapter 1: Introduction

  7. Devices available to the circuit designer + – DTs Ts Linear- mode Switched-mode Resistors Capacitors Magnetics Semiconductor devices 7 Fundamentals of Power Electronics Chapter 1: Introduction

  8. Devices available to the circuit designer + – DTs Ts Linear- mode Switched-mode Resistors Capacitors Magnetics Semiconductor devices Signal processing: avoid magnetics 8 Fundamentals of Power Electronics Chapter 1: Introduction

  9. Devices available to the circuit designer + – DTs Ts Linear- mode Switched-mode Resistors Capacitors Magnetics Semiconductor devices Power processing: avoid lossy elements 9 Fundamentals of Power Electronics Chapter 1: Introduction

  10. Power loss in an ideal switch Switch closed: v(t) = 0 + i(t) Switch open: i(t) = 0 v(t) In either event: p(t) = v(t) i(t) = 0 Ideal switch consumes zero power – 10 Fundamentals of Power Electronics Chapter 1: Introduction

  11. A simple dc-dc converter example I 10A + Dc-dc converter Vg + – R 5Ω V 50V 100V – Input source: 100V Output load: 50V, 10A, 500W How can this converter be realized? 11 Fundamentals of Power Electronics Chapter 1: Introduction

  12. Dissipative realization Resistive voltage divider I 10A + + 50V – Vg + – Ploss = 500W R 5Ω V 50V 100V – Pin = 1000W Pout = 500W 12 Fundamentals of Power Electronics Chapter 1: Introduction

  13. Dissipative realization Series pass regulator: transistor operates in active region I 10A + 50V – + Vref linear amplifier and base driver Vg +– + – R 5Ω V 50V 100V Ploss≈ 500W – Pin≈ 1000W Pout = 500W 13 Fundamentals of Power Electronics Chapter 1: Introduction

  14. Use of a SPDT switch I 10 A 1 + + Vg 2 + – vs(t) v(t) 50 V R 100 V – – vs(t) Vg Vs= DVg 0 (1 – D) Ts t DTs switch position: 1 2 1 14 Fundamentals of Power Electronics Chapter 1: Introduction

  15. The switch changes the dc voltage level vs(t) Vg D = switch duty cycle 0 ≤D≤ 1 Vs= DVg 0 Ts = switching period (1 – D) Ts t DTs switch position: fs = switching frequency = 1 / Ts 1 2 1 DC component of vs(t) = average value: Ts Vs=1 vs(t) dt = DVg Ts 0 15 Fundamentals of Power Electronics Chapter 1: Introduction

  16. Addition of low pass filter Addition of (ideally lossless) L-C low-pass filter, for removal of switching harmonics: i(t) 1 + + L 2 Vg + – vs(t) v(t) C R 100 V – – Pin≈ 500 W Pout = 500 W Ploss small • Choose filter cutoff frequency f0 much smaller than switching frequency fs This circuit is known as the “buck converter” • 16 Fundamentals of Power Electronics Chapter 1: Introduction

  17. Addition of control system for regulation of output voltage Power input Switching converter Load + i + – vg v Sensor gain H(s) – Transistor gate driver Error signal vc ve Hv Pulse-width modulator δ Gc(s) –+ δ(t) Compensator Reference input vref dTs Ts t 17 Fundamentals of Power Electronics Chapter 1: Introduction

  18. The boost converter 2 + L 1 + – C R V Vg – 5Vg 4Vg 3Vg V 2Vg Vg 0 0 0.2 0.4 0.6 0.8 1 D 18 Fundamentals of Power Electronics Chapter 1: Introduction

  19. A single-phase inverter vs(t) 1 2 + – + – Vg + v(t) – 2 1 load “H-bridge” Modulate switch duty cycles to obtain sinusoidal low-frequency component vs(t) t 19 Fundamentals of Power Electronics Chapter 1: Introduction

  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 Fundamentals of Power Electronics Chapter 1: Introduction

  21. A laptop computer power supply system Inverter Display backlighting iac(t) Charger Microprocessor Buck converter PWM Rectifier vac(t) Power management ac line input 85–265 Vrms Boost converter Disk drive Lithium battery 21 Fundamentals of Power Electronics Chapter 1: Introduction

  22. Power system of an earth-orbiting spacecraft Dissipative shunt regulator + Solar array vbus – Battery Dc-dc converter Dc-dc converter charge/discharge controllers Batteries Payload Payload 22 Fundamentals of Power Electronics Chapter 1: Introduction

  23. An electric vehicle power and drive system ac machine ac machine Inverter Inverter control bus battery µP + system controller 3øac line Battery charger DC-DC converter vb 50/60 Hz Vehicle electronics – Low-voltage dc bus Inverter Inverter Variable-frequency Variable-voltage ac ac machine ac machine 23 Fundamentals of Power Electronics Chapter 1: Introduction

  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 Fundamentals of Power Electronics Chapter 1: Introduction

  25. Part I. Converters in equilibrium Inductor waveforms Averaged equivalent circuit D' VD RL D Ron D' RD vL(t) D' : 1 Vg – V DTs + – + D'Ts + – t Vg V R I – V switch position: – 1 2 1 iL(t) Predicted efficiency iL(DTs) ∆iL I iL(0) 100% Vg– V L – V L 0.002 90% 0.01 80% t 0 Ts DTs 0.02 70% 0.05 60% 50% RL/R = 0.1 η 40% Discontinuous conduction mode Transformer isolation 30% 20% 10% 0% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D 25 Fundamentals of Power Electronics Chapter 1: Introduction

  26. Switch realization: semiconductor devices iA(t) The IGBT Switching loss transistor waveforms collector Qr Vg gate iL vA(t) 0 0 emitter t iB(t) Emitter diode iL waveforms vB(t) Gate 0 0 t area –Qr –Vg n n n n p p minority carrier injection n- tr p pA(t) =vA iA area ~QrVg Collector area ~iLVgtr t t0 t1t2 26 Fundamentals of Power Electronics Chapter 1: Introduction

  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 Fundamentals of Power Electronics Chapter 1: Introduction

  28. Part II. Converter dynamics and control Closed-loop converter system Averaging the waveforms Power input Switching converter Load gate drive + + – vg(t) v(t) R feedback connection t – transistor gate driver actual waveform v(t) including ripple compensator vc v pulse-width modulator δ(t) Gc(s) –+ averaged waveform <v(t)>Ts with ripple neglected voltage reference δ(t) vc(t) vref t dTs Ts t t Controller Vg– V d(t) L 1 : D D' : 1 + – Small-signal averaged equivalent circuit + + – vg(t) v(t) I d(t) I d(t) C R – 28 Fundamentals of Power Electronics Chapter 1: Introduction

  29. Part II. Converter dynamics and control 7. Ac modeling 8. Converter transfer functions 9. 10. Controller design Input filter design 11. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 12. Current-programmed control 29 Fundamentals of Power Electronics Chapter 1: Introduction

  30. Part III. Magnetics n1 : n2 transformer design the proximity effect iM(t) i2(t) i1(t) 3i layer 3 LM –2i 2Φ R1 R2 2i layer 2 –i ik(t) Φ i layer 1 d : nk Rk current density J 4226 transformer size vs. switching frequency 3622 0.1 0.08 Pot core size 2616 2616 2213 2213 0.06 Bmax (T) 1811 1811 0.04 0.02 0 25kHz 50kHz 100kHz 200kHz 250kHz 400kHz 500kHz 1000kHz Switching frequency 30 Fundamentals of Power Electronics Chapter 1: Introduction

  31. Part III. Magnetics 13. Basic magnetics theory 14. Inductor design 15. Transformer design 31 Fundamentals of Power Electronics Chapter 1: Introduction

  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(t) ig(t) + + iac(t) L D1 vac(t) vg(t) Q1 C v(t) R – – vcontrol(t) vg(t) ig(t) PWM Rs multiplier X va(t) verr(t) +– Gc(s) vref(t) = kx vg(t) vcontrol(t) compensator controller 100% 100% 91% percent of fundamental THD = 136% Distortion factor = 59% Harmonic amplitude, Ideal rectifier (LFR) 80% iac(t) i(t) 73% + + 60% 2 / Re p(t) = vac 52% Model of the ideal rectifier 40% 32% vac(t) v(t) Re(vcontrol) 19%15% 15%13%9% 20% – – 0% 1 3 5 7 9 11 13 15 17 19 ac dc Harmonic number input output vcontrol 32 Fundamentals of Power Electronics Chapter 1: Introduction

  33. Part IV. Modern rectifiers, and power system harmonics 16. Power and harmonics in nonsinusoidal systems 17. Line-commutated rectifiers 18. Pulse-width modulated rectifiers 33 Fundamentals of Power Electronics Chapter 1: Introduction

  34. Part V. Resonant converters The series resonant converter L C Q1 Q3 1 : n D1 D3 + + – Vg R V – Zero voltage switching Q2 Q4 D2 D4 vds1(t) 1 Vg Q = 0.2 0.9 Q = 0.2 0.8 0.35 0.7 0.5 t Q1 Q4 D2 D3 X 0.35 0.6 conducting devices: M = V / Vg 0.75 0.5 1 0.5 turn off Q1, Q4 0.4 commutation interval 0.75 1 1.5 0.3 1.5 2 2 0.2 Dc characteristics 3.5 3.5 5 5 0.1 10 10 Q = 20 Q = 20 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 F = fs / f0 34 Fundamentals of Power Electronics Chapter 1: Introduction

  35. Part V. Resonant converters 19. 20. Resonant conversion Soft switching 35 Fundamentals of Power Electronics Chapter 1: Introduction

  36. Appendices A. RMS values of commonly-observed converter waveforms B. Simulation of converters C. Middlebrook’s extra element theorem D. Magnetics design tables L iLOAD 1 2 3 + 50 µH 1 2 Vg C CCM-DCM1 + – R1 R v 500 µF 11 kΩ 28 V 20 dB 5 4 || Gvg|| – Open loop, d(t) = constant R2 0 dB 4 3 85 kΩ C2 Xswitch L = 50 µΗ fs = 100 kΗz R3 –20 dB R = 3 Ω C3 2.7 nF 1.1 nF 120 kΩ –40 dB +12 V 8 5 7 6 – R = 25 Ω –60 dB Closed loop vx –vy + LM324 vz VM = 4 V vref R4 47 kΩ –80 dB Epwm 500 Hz 5 Hz 50 Hz 5 kHz 50 kHz + – value = {LIMIT(0.25 vx, 0.1, 0.9)} f 5 V .nodeset v(3)=15 v(5)=5 v(6)=4.144 v(8)=0.536 36 Fundamentals of Power Electronics Chapter 1: Introduction

  37. Chapter 2 Principles of Steady-State Converter Analysis 2.1. Introduction 2.2. Inductor volt-second balance, capacitor charge balance, and the small ripple approximation 2.3. Boost converter example 2.4. Cuk converter example 2.5. Estimating the ripple in converters containing two- pole low-pass filters 2.6. Summary of key points 1 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  38. 2.1 Introduction Buck converter 1 SPDT switch changes dc component + + 2 + – vs(t) v(t) Vg R – – vs(t) Vg Switch output voltage waveform D'Ts DTs 0 Duty cycle D: 0 ≤ D ≤ 1 0 DTs Ts t Switch position: complement D′: D′ = 1 - D 1 2 1 2 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  39. Dc component of switch output voltage vs(t) Vg 〈vs〉 = DVg area = DTsVg 0 0 DTs Ts t Fourier analysis: Dc component = average value Ts vs =1 vs(t) dt Ts 0 vs =1 Ts(DTsVg) = DVg 3 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  40. Insertion of low-pass filter to remove switching harmonics and pass only dc component L 1 + + 2 + – Vg vs(t) v(t) C R – – V Vg v ≈ vs = DVg 0 0 1 D 4 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  41. Three basic dc-dc converters (a) 1 M(D) = D L 1 0.8 + iL(t) 0.6 Buck M(D) 2 + – 0.4 Vg C R v 0.2 – 0 0 0.2 0.4 0.6 0.8 1 D (b) 5 L 1 2 M(D) = 1 – D 4 + iL(t) Boost 3 M(D) 1 + – Vg C R v 2 1 – 0 0 0.2 0.4 0.6 0.8 1 D D (c) 0 0.2 0.4 0.6 0.8 1 0 Buck-boost + 1 2 –1 iL(t) –2 M(D) + – Vg C R v L –3 M(D) =– D –4 – 1 – D –5 5 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  42. Objectives of this chapter G Develop techniques for easily determining output voltage of an arbitrary converter circuit G Derive the principles of inductor volt-second balance and capacitor charge (amp-second) balance G Introduce the key small ripple approximation G Develop simple methods for selecting filter element values G Illustrate via examples 6 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  43. 2.2. Inductor volt-second balance, capacitor charge balance, and the small ripple approximation Actual output voltage waveform, buck converter iL(t) L 1 Buck converter containing practical low-pass filter + + vL(t) – iC(t) 2 + – v(t) Vg C R – Actual output voltage waveform Actual waveform v(t) = V + vripple(t) v(t) V v(t) = V + vripple(t) dc component V 0 t 7 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  44. The small ripple approximation Actual waveform v(t) = V + vripple(t) v(t) V v(t) = V + vripple(t) dc component V 0 t In a well-designed converter, the output voltage ripple is small. Hence, the waveforms can be easily determined by ignoring the ripple: vripple< V v(t) ≈ V 8 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  45. Buck converter analysis: inductor current waveform iL(t) L 1 + + vL(t) – iC(t) original converter 2 + – v(t) Vg C R – switch in position 1 switch in position 2 L iL(t) L + + + vL(t) – + vL(t) – iC(t) iC(t) + – iL(t) + – v(t) Vg C R v(t) Vg C R – – 9 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  46. Inductor voltage and current Subinterval 1: switch in position 1 iL(t) L Inductor voltage + + vL(t) – iC(t) vL= Vg– v(t) + – v(t) Vg C R Small ripple approximation: – vL≈ Vg– V Knowing the inductor voltage, we can now find the inductor current via vL(t) = LdiL(t) dt Solve for the slope: diL(t) dt ≈Vg– V =vL(t) ⇒ The inductor current changes with an essentially constant slope L L 10 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  47. Inductor voltage and current Subinterval 2: switch in position 2 L Inductor voltage + + vL(t) – iC(t) vL(t) = – v(t) + – iL(t) v(t) Vg C R Small ripple approximation: – vL(t) ≈ – V Knowing the inductor voltage, we can again find the inductor current via vL(t) = LdiL(t) dt Solve for the slope: diL(t) dt ⇒ The inductor current changes with an essentially constant slope ≈ –V L 11 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  48. Inductor voltage and current waveforms vL(t) Vg– V DTs D'Ts t – V Switch position: 1 2 1 vL(t) = LdiL(t) dt iL(t) iL(DTs) ∆iL I iL(0) Vg– V L – V L 0 Ts t DTs 12 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  49. Determination of inductor current ripple magnitude iL(t) iL(DTs) ∆iL I iL(0) Vg– V L – V L 0 Ts t DTs (change in iL) = (slope)(length of subinterval) Vg– V L 2∆iL= DTs L =Vg– V ∆iL=Vg– V DTs DTs ⇒ 2∆iL 2L 13 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

  50. Inductor current waveform during turn-on transient iL(t) Vg– v(t) L iL(nTs) iL((n + 1)Ts) – v(t) L iL(Ts) iL(0) = 0 t 2Ts 0 DTsTs (n + 1)Ts nTs When the converter operates in equilibrium: iL((n + 1)Ts) = iL(nTs) 14 Fundamentals of Power Electronics Chapter 2: Principles of steady-state converter analysis

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