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

Transcription

1 Fundamenals of Power Elecronics Second ediion Rober W. Erickson Dragan Maksimovic Universiy of Colorado, Boulder

2 Chaper 1: Inroducion 1.1. Inroducion o power processing 1.2. Some applicaions of power elecronics 1.3. Elemens of power elecronics Summary of he course

3 1.1 Inroducion o Power Processing Power inpu Swiching converer Power oupu Dc-dc conversion: Ac-dc recificaion: Dc-ac inversion: Conrol inpu Change and conrol volage magniude Possibly conrol dc volage, ac curren Produce sinusoid of conrollable magniude and frequency Ac-ac cycloconversion: Change and conrol volage magniude and frequency

4 Conrol is invariably required Power inpu Swiching converer Power oupu feedforward Conrol inpu Conroller feedback reference

5 High efficiency is essenial 1 = P ou P in 0.8 P loss = P in P ou = P ou 1 1 High efficiency leads o low power loss wihin converer Small size and reliable operaion is hen feasible Efficiency is a good measure of converer performance P loss / P ou

6 A high-efficiency converer P in Converer P ou A goal of curren converer echnology is o consruc converers of small size and weigh, which process subsanial power a high efficiency

7 Devices available o he circui designer inearmode Swiched-mode Resisors Capaciors Magneics Semiconducor devices DT s T s

8 Devices available o he circui designer inearmode Swiched-mode Resisors Capaciors Magneics Semiconducor devices DT s T s Signal processing: avoid magneics

9 Devices available o he circui designer inearmode Swiched-mode Resisors Capaciors Magneics Semiconducor devices DT s T s Power processing: avoid lossy elemens

10 Power loss in an ideal swich Swich closed: v = 0 Swich open: i = 0 i In eiher even: p = v i = 0 v Ideal swich consumes zero power

11 A simple dc-dc converer example 100V Dc-dc R converer 5 I 10A V 50V Inpu source: 100V Oupu load: 50V, 10A, 500W How can his converer be realized?

12 Dissipaive realizaion Resisive volage divider 50V I 10A 100V P loss = 500W R 5 V 50V P in = 1000W P ou = 500W

13 Dissipaive realizaion Series pass regulaor: ransisor operaes in acive region 50V I 10A 100V linear amplifier R and base driver 5 P loss 500W V ref V 50V P in 1000W P ou = 500W

14 Use of a SPDT swich 1 I 10 A 100 V 2 v R s v 50 V v s DT s (1 D) T s swich posiion: V s = D

15 The swich changes he dc volage level v s V s = D D = swich duy cycle 0 D 1 DT s (1 D) T s swich posiion: T s = swiching period f s = swiching frequency = 1 / T s DC componen of v s = average value: V s = 1 T s 0 T s v s d = D

16 Addiion of low pass filer Addiion of (ideally lossless) -C low-pass filer, for removal of swiching harmonics: 100 V P in 500 W 1 v R s C Choose filer cuoff frequency f 0 much smaller han swiching frequency f s This circui is known as he buck converer 2 P loss small i v P ou = 500 W

17 Addiion of conrol sysem for regulaion of oupu volage Power inpu Swiching converer oad i v g v H(s) Sensor gain Transisor gae driver Pulse-widh modulaor v c G c (s) Compensaor Error signal v e Hv dt s T s Reference inpu v ref

18 The boos converer 2 1 C R V 5 4 V D

19 A single-phase inverer 1 v s 2 2 v load 1 v s H-bridge Modulae swich duy cycles o obain sinusoidal low-frequency componen

20 1.2 Several applicaions of power elecronics Power levels encounered in high-efficiency converers less han 1 W in baery-operaed porable equipmen ens, hundreds, or housands of was in power supplies for compuers or office equipmen kw o MW in variable-speed moor drives 1000 MW in recifiers and inverers for uiliy dc ransmission lines

21 A lapop compuer power supply sysem Inverer Display backlighing v ac i ac Charger PWM Recifier Buck converer Microprocessor Power managemen ac line inpu Vrms ihium baery Boos converer Disk drive

22 Power sysem of an earh-orbiing spacecraf Dissipaive shun regulaor Solar array v bus Baery charge/discharge conrollers Dc-dc converer Dc-dc converer Baeries Payload Payload

23 An elecric vehicle power and drive sysem ac machine ac machine Inverer Inverer conrol bus 3øac line 50/60 Hz Baery charger baery v b DC-DC converer µp sysem conroller ow-volage dc bus Vehicle elecronics Inverer Inverer Variable-frequency Variable-volage ac ac machine ac machine

24 1.3 Elemens of power elecronics Power elecronics incorporaes conceps from he fields of analog circuis elecronic devices conrol sysems power sysems magneics elecric machines numerical simulaion

25 Par I. Converers in equilibrium Inducor waveforms Averaged equivalen circui v V R D Ron D' V D D' R D D' : 1 DT s D'T s V I V R swich posiion: i I i (0) V i (DT s ) V 0 DT s T s i Prediced efficiency 100% 90% 80% 70% % % R /R = 0.1 Disconinuous conducion mode Transformer isolaion 40% 30% 20% 10% 0% D

26 Swich realizaion: semiconducor devices The IGBT collecor Swiching loss ransisor waveforms i A Q r gae v A i emier 0 0 Gae Emier diode waveforms i 0 i B v B 0 n p n n p n area Q r n - minoriy carrier injecion r p Collecor p A = v A i A area ~Q r area ~i r

27 Par I. Converers in equilibrium 2. Principles of seady sae converer analysis 3. Seady-sae equivalen circui modeling, losses, and efficiency 4. Swich realizaion 5. The disconinuous conducion mode 6. Converer circuis

28 Par II. Converer dynamics and conrol Closed-loop converer sysem Averaging he waveforms Power inpu Swiching converer oad gae drive v g v R feedback connecion ransisor gae driver pulse-widh modulaor v c compensaor v c G c (s) volage reference v ref v acual waveform v including ripple averaged waveform <v> Ts wih ripple negleced dt s T s Conroller Small-signal averaged equivalen circui v g Id V g V d 1 : D D' : 1 Id C v R

29 Par II. Converer dynamics and conrol 7. Ac modeling 8. Converer ransfer funcions 9. Conroller design 10. Inpu filer design 11. Ac and dc equivalen circui modeling of he disconinuous conducion mode 12. Curren-programmed conrol

30 Par III. Magneics ransformer design i 1 n 1 : n 2 i M M R 1 R 2 i 2 i k he proximiy effec layer 3 layer 2 3i 2i 2i i 2 layer 1 i d : n k R k curren densiy J ransformer size vs. swiching frequency Po core size B max (T) 25kHz 50kHz 100kHz 200kHz 250kHz 400kHz 500kHz 1000kHz Swiching frequency 0

31 Par III. Magneics 13. Basic magneics heory 14. Inducor design 15. Transformer design

32 Par IV. Modern recifiers, and power sysem harmonics Polluion of power sysem by recifier curren harmonics A low-harmonic recifier sysem boos converer i g i ac D 1 i v ac v g Q 1 C v R v conrol muliplier X v g R s i g v a PWM v ref = k x v g v conrol v err G c (s) compensaor Harmonic ampliude, percen of fundamenal 100% 80% 60% 40% 20% 0% 100% 91% 73% 52% THD = 136% Disorion facor = 59% 32% 19% 15% 15% 13% 9% Harmonic number Model of he ideal recifier v ac i ac ac inpu R e (v conrol ) conroller Ideal recifier (FR) p = v ac 2 / R e i v dc oupu v conrol

33 Par IV. Modern recifiers, and power sysem harmonics 16. Power and harmonics in nonsinusoidal sysems 17. ine-commuaed recifiers 18. Pulse-widh modulaed recifiers

34 Par V. Resonan converers The series resonan converer Q 1 D 1 Q 3 D 3 C 1 : n R V Q 2 D 2 Q 4 D 4 Zero volage swiching 1 Q = 0.2 v ds1 Dc characerisics M = V / Q = Q = Q = 20 conducing devices: Q 1 X D 2 Q 4 urn off Q 1, Q 4 D 3 commuaion inerval F = f s / f 0

35 Par V. Resonan converers 19. Resonan conversion 20. Sof swiching

36 Appendices A. RMS values of commonly-observed converer waveforms B. Simulaion of converers C. Middlebrook s exra elemen heorem D. Magneics design ables µh i OAD G vg 20 db Open loop, d = consan 0 db 20 db R = 3 40 db 60 db Closed loop R = db 5 Hz 50 Hz 500 Hz 5 khz 50 khz f 28 V V M = 4 V CCM-DCM1 3 X swich = 50 µ f s = 100 k z v x v z E pwm value = {IMIT(0.25 v x, 0.1, 0.9)} 7 v y C 500 µf R 3 C k M V.nodese v(3)=15 v(5)=5 v(6)=4.144 v(8)= R 2 85 k 2.7 nf v ref 5 V 4 R 1 11 k 5 R 4 47 k R C nf v 2 1

37 Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer example 2.5. Esimaing he ripple in converers conaining wopole low-pass filers 2.6. Summary of key poins

38 2.1 Inroducion Buck converer SPDT swich changes dc componen 1 2 v R s v Swich oupu volage waveform v s DT s D'T s Duy cycle D: 0 D 1 complemen D : D = 1 - D 0 DT s T s Swich posiion:

39 Dc componen of swich oupu volage v s area = DT s v s = D 0 DT s T s 0 Fourier analysis: Dc componen = average value v s = T 1 T s v s d s 0 v s = 1 T s (DT s )=D

40 Inserion of low-pass filer o remove swiching harmonics and pass only dc componen 1 2 v s C R v V v v s = D D

41 Three basic dc-dc converers Buck (a) 1 2 i C R v M(D) M(D) =D D Boos (b) i 1 2 C R v M(D) M(D) = 1 1D D Buck-boos (c) 1 2 i C R v M(D) D M(D) = D 1D

42 Objecives of his chaper Develop echniques for easily deermining oupu volage of an arbirary converer circui Derive he principles of inducor vol-second balance and capacior charge (amp-second) balance Inroduce he key small ripple approximaion Develop simple mehods for selecing filer elemen values Illusrae via examples

43 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion Acual oupu volage waveform, buck converer Buck converer conaining pracical low-pass filer 1 2 i v C i C R v Acual oupu volage waveform v Acual waveform v = V v ripple v=v v ripple V dc componen V 0

44 The small ripple approximaion v=v v ripple v V Acual waveform v = V v ripple dc componen V 0 In a well-designed converer, he oupu volage ripple is small. Hence, he waveforms can be easily deermined by ignoring he ripple: v ripple < V v V

45 Buck converer analysis: inducor curren waveform original converer 1 2 i v C i C R v swich in posiion 1 swich in posiion 2 i v i C v i C C R v i C R v

46 Inducor volage and curren Subinerval 1: swich in posiion 1 Inducor volage v = v i v i C Small ripple approximaion: C R v v V Knowing he inducor volage, we can now find he inducor curren via v = di d Solve for he slope: di d = v V The inducor curren changes wih an essenially consan slope

47 Inducor volage and curren Subinerval 2: swich in posiion 2 Inducor volage v =v Small ripple approximaion: v i C i C R v v V Knowing he inducor volage, we can again find he inducor curren via v = di d Solve for he slope: di d V The inducor curren changes wih an essenially consan slope

48 Inducor volage and curren waveforms v V DT s D'T s V Swich posiion: i I i (0) V i (DT s ) V i v = di d 0 DT s T s

49 Deerminaion of inducor curren ripple magniude i i (DT s ) I i (0) V V i 0 DT s T s (change in i )=(slope)(lengh of subinerval) 2 i = V DT s i = V 2 DT s = V 2 i DT s

50 Inducor curren waveform during urn-on ransien i i (T s ) i (0) = 0 0 DT s T s v v i (nt s ) 2T s nt s (n 1)T s i ((n 1)T s ) When he converer operaes in equilibrium: i ((n 1)T s )=i (nt s )

51 The principle of inducor vol-second balance: Derivaion Inducor defining relaion: v = di d Inegrae over one complee swiching period: i (T s )i (0) = 1 0 T s v d In periodic seady sae, he ne change in inducor curren is zero: T s 0= v d 0 Hence, he oal area (or vol-seconds) under he inducor volage waveform is zero whenever he converer operaes in seady sae. An equivalen form: 0= 1 T s v T d = v s 0 The average inducor volage is zero in seady sae.

52 Inducor vol-second balance: Buck converer example Inducor volage waveform, previously derived: v V DT s Toal area V Inegral of volage waveform is area of recangles: T s = v d =( V)(DT s )(V)(D'T s ) 0 Average volage is v = T = D( V)D'( V) s Equae o zero and solve for V: 0=D (D D')V = D V V = D

53 The principle of capacior charge balance: Derivaion Capacior defining relaion: i C =C dv C d Inegrae over one complee swiching period: v C (T s )v C (0) = 1 C 0 T s i C d In periodic seady sae, he ne change in capacior volage is zero: 0= T 1 T s i C d s 0 = i C Hence, he oal area (or charge) under he capacior curren waveform is zero whenever he converer operaes in seady sae. The average capacior curren is hen zero.

54 2.3 Boos converer example 2 Boos converer wih ideal swich i v 1 i C C R v D 1 Realizaion using power MOSFET and diode i v DT s T s Q 1 i C C R v

55 Boos converer analysis 2 original converer i v 1 i C C R v swich in posiion 1 swich in posiion 2 i v i C i v i C C R v C R v

56 Subinerval 1: swich in posiion 1 Inducor volage and capacior curren v = i C =v / R i v i C Small ripple approximaion: C R v v = i C =V / R

57 Subinerval 2: swich in posiion 2 Inducor volage and capacior curren v = v i C = i v / R i v i C Small ripple approximaion: C R v v = V i C = I V / R

58 Inducor volage and capacior curren waveforms v DT s D'T s V i C I V/R DT s D'T s V/R

59 Inducor vol-second balance Ne vol-seconds applied o inducor over one swiching period: v DT s D'T s 0 T s v d =( ) DT s ( V) D'T s V Equae o zero and collec erms: (D D') VD'=0 Solve for V: V = D' The volage conversion raio is herefore M(D)= V = 1 D' = 1 1D

60 Conversion raio M(D) of he boos converer 5 4 M(D)= 1 D' = 1 1D M(D) D

61 Deerminaion of inducor curren dc componen i C I V/R Capacior charge balance: DT s D'T s T s 0 i C d =( V R ) DT s (I V R ) D'T s V/R Collec erms and equae o zero: V R (D D') ID'=0 I /R 8 Solve for I: I = V D' R Eliminae V o express in erms of : I = D' 2 R D

62 Deerminaion of inducor curren ripple Inducor curren slope during subinerval 1: di d = v = Inducor curren slope during subinerval 2: i I V 0 DT s T s i di d = v = V Change in inducor curren during subinerval 1 is (slope) (lengh of subinerval): 2 i = DT s Solve for peak ripple: i = 2 DT s Choose such ha desired ripple magniude is obained

63 Deerminaion of capacior volage ripple Capacior volage slope during subinerval 1: dv C d Capacior volage slope during subinerval 2: dv C d = i C C = i C C = V RC = I C V RC v V V RC I C V RC 0 DT s T s v Change in capacior volage during subinerval 1 is (slope) (lengh of subinerval): 2 v = V RC DT s Solve for peak ripple: v = V 2RC DT s Choose C such ha desired volage ripple magniude is obained In pracice, capacior equivalen series resisance (esr) leads o increased volage ripple

64 2.4 Cuk converer example Cuk converer, wih ideal swich 1 C 1 2 i 1 i v C 2 v 2 R Cuk converer: pracical realizaion using MOSFET and diode 1 C 1 2 i 1 i v 1 2 Q 1 D 1 C 2 v 2 R

65 Cuk converer circui wih swich in posiions 1 and 2 Swich in posiion 1: MOSFET conducs Capacior C 1 releases energy o oupu 1 i 2 i 1 v 1 i v C1 2 2 v 1 C 1 C 2 i C2 v 2 R Swich in posiion 2: diode conducs Capacior C 1 is charged from inpu 1 i 1 i 2 v 1 i C1 2 C 1 v 1 C 2 v 2 i C2 v 2 R

66 Waveforms during subinerval 1 MOSFET conducion inerval Inducor volages and capacior currens: v 1 = 1 i 2 i 1 v 1 i v C1 2 2 v 1 C 1 C 2 i C2 v 2 R v 2 =v 1 v 2 i C1 = i 2 i C2 = i 2 v 2 R Small ripple approximaion for subinerval 1: v 1 = v 2 =V 1 V 2 i C1 = I 2 i C2 = I 2 V 2 R

67 Waveforms during subinerval 2 Diode conducion inerval Inducor volages and capacior currens: v 1 = v 1 1 i 1 i 2 v 1 i C1 2 v 2 C 1 v 1 C 2 i C2 v 2 R v 2 =v 2 i C1 = i 1 i C2 = i 2 v 2 R Small ripple approximaion for subinerval 2: v 1 = V 1 v 2 =V 2 i C1 = I 1 i C2 = I 2 V 2 R

68 Equae average values o zero The principles of inducor vol-second and capacior charge balance sae ha he average values of he periodic inducor volage and capacior curren waveforms are zero, when he converer operaes in seady sae. Hence, o deermine he seady-sae condiions in he converer, le us skech he inducor volage and capacior curren waveforms, and equae heir average values o zero. Waveforms: Inducor volage v 1 v 1 DT s D'T s Vol-second balance on 1 : v 1 = D D'( V 1 )=0 V 1

69 Equae average values o zero Inducor 2 volage v 2 V 2 DT s D'T s V 1 V 2 Average he waveforms: Capacior C 1 curren i C1 I 1 v 2 = D(V 1 V 2 )D'( V 2 )=0 i C1 = DI 2 D'I 1 =0 DT s I 2 D'T s

70 Equae average values o zero Capacior curren i C2 waveform i C2 DT s I 2 V 2 / R (= 0) D'T s i C2 = I 2 V 2 R =0 Noe: during boh subinervals, he capacior curren i C2 is equal o he difference beween he inducor curren i 2 and he load curren V 2 /R. When ripple is negleced, i C2 is consan and equal o zero.

71 Cuk converer conversion raio M = V/ D M(D) -2-3 M(D)= V 2 = D 1D -4-5

72 Inducor curren waveforms Inerval 1 slopes, using small ripple approximaion: di 1 d di 2 d = v 1 1 = 1 = v 2 2 = V 1 V 2 2 i 1 I 1 i 1 1 V 1 1 DT s T s Inerval 2 slopes: DT s T s di 1 d di 2 d = v 1 1 = V 1 1 = v 2 2 = V 2 2 I 2 i 2 V 1 V 2 2 V 2 2 i2

73 Capacior C 1 waveform Subinerval 1: dv 1 d Subinerval 2: dv 1 d = i C1 C 1 = I 2 C 1 = i C1 C 1 = I 1 C 1 v 1 v 1 V 1 I 2 C 1 DT s I 1 C 1 T s

74 Ripple magniudes Analysis resuls i 1 = DT s 2 1 i 2 = V 1 V v 1 = I 2DT s 2C 1 DT s Use dc converer soluion o simplify: i 1 = DT s 2 1 i 2 = DT s 2 2 v 1 = D 2 T s 2D'RC 1 Q: How large is he oupu volage ripple?

75 2.5 Esimaing ripple in converers conaining wo-pole low-pass filers Buck converer example: Deermine oupu volage ripple 1 2 i i C i R C v C R Inducor curren waveform. Wha is he capacior curren? i I i (0) V i (DT s ) V 0 DT s T s i

76 Capacior curren and volage, buck example i C Mus no neglec inducor curren ripple! Toal charge q T s /2 i DT s D'T s If he capacior volage ripple is small, hen essenially all of he ac componen of inducor curren flows hrough he capacior. v C V v v

77 Esimaing capacior volage ripple v i C v C Toal charge q DT s T s /2 D'T s i Curren i C is posiive for half of he swiching period. This posiive curren causes he capacior volage v C o increase beween is minimum and maximum exrema. During his ime, he oal charge q is deposied on he capacior plaes, where V v v q = C (2 v) (change in charge)= C (change in volage)

78 Esimaing capacior volage ripple v i C Toal charge q The oal charge q is he area of he riangle, as shown: T s /2 i q = 1 2 i T s 2 DT s D'T s Eliminae q and solve for v: v C v = i T s 8 C V v v Noe: in pracice, capacior equivalen series resisance (esr) furher increases v.

79 Inducor curren ripple in wo-pole filers Example: problem Q 1 i T i 1 i 2 2 C C 1 v C1 2 D 1 R v v Toal flux linkage v T s /2 DT s D'T s can use similar argumens, wih λ = (2 i) i I i i λ = inducor flux linkages = inducor vol-seconds

80 2.6 Summary of Key Poins 1. The dc componen of a converer waveform is given by is average value, or he inegral over one swiching period, divided by he swiching period. Soluion of a dc-dc converer o find is dc, or seadysae, volages and currens herefore involves averaging he waveforms. 2. The linear ripple approximaion grealy simplifies he analysis. In a welldesigned converer, he swiching ripples in he inducor currens and capacior volages are small compared o he respecive dc componens, and can be negleced. 3. The principle of inducor vol-second balance allows deerminaion of he dc volage componens in any swiching converer. In seady-sae, he average volage applied o an inducor mus be zero.

81 Summary of Chaper 2 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a swiching converer. In seadysae, he average curren applied o a capacior mus be zero. 5. By knowledge of he slopes of he inducor curren and capacior volage waveforms, he ac swiching ripple magniudes may be compued. Inducance and capaciance values can hen be chosen o obain desired ripple magniudes. 6. In converers conaining muliple-pole filers, coninuous (nonpulsaing) volages and currens are applied o one or more of he inducors or capaciors. Compuaion of he ac swiching ripple in hese elemens can be done using capacior charge and/or inducor flux-linkage argumens, wihou use of he small-ripple approximaion. 7. Converers capable of increasing (boos), decreasing (buck), and invering he volage polariy (buck-boos and Cuk) have been described. Converer circuis are explored more fully in a laer chaper.