Lecture Set 6 Brushless DC Machines

Transcription

1 Lectue Set 6 Bushless DC Machines S.D. Sudhoff Sping 2018

2 Reading Chapte 8, Electomechanical Motion Devices, 2 nd Edition 2

3 A Bushless DC Machine 3

4 Sample Applications Low Powe: Disk dive motos Medium Powe: Robot manipulatos Sevo systems Hybid/electic vehicles High Powe: Ship and submaine populsion Wind tubines 4

5 Disk Dive Moto 5

6 4 Hp BDC Machine 6

7 Chaacteistics The Good The Bad 7

8 Pemanent Magnet Synchonous Machines Radial Vesus Axial Suface Mounted Vesus Buied Magnet Sinusoidal Vesus Non-Sinusoidal 8

9 Radial Suface Mounted PMSM bs-axis Path of Integal fo Section q-axis φ m θ m φ sm as-axis cs-axis d-axis 9

10 3-Phase PMSM 10

11 3-Phase PMSM Notation ( fabcs) = [ fas fbs fcs ] Voltage equations T dλ v i as as = s as + dt d v i bs bs = s bs + dt dλ v i cs cs = s cs + dt v = i + pλ abcs s abcs abcs 11

12 12 3-Phase PMSM Flux Linkage Equations whee m abcs s λ λ + = i L abcs + = = π θ π θ θ 3 2 sin( ) 3 2 sin( sin m csm bsm asm m λ λ λ λ λ = ms ls ms ms ms ms ls ms ms ms ms ls s L L L L L L L L L L L L L

13 PM Tems Intuitive Appoach 13

14 PM Tems Intuitive Appoach 14

15 PM Tems Intuitive Appoach 15

16 Inductances We will assume the following n n n as bs cs = N s sin( Pφsm / 2) = N sin( Pφsm / 2 2π/3) s = N sin( Pφsm / 2 + 2π/3) s It follows that w w as w bs cs 2N = s cos( Pφsm / 2) P 2N = s cos( Pφsm / 2 2π / 3) P 2Ns = cos( Pφ sm / 2 + 2π / 3) P 16

17 Inductances Recall λ α, m i β = L m, αβ = µ L Fom which we obtain 0 2π 0 w α ( φ) w β g( φ) ( φ) dφ L asbs 2 s 4πµ LN L asas = 0 = 2 P g 2πµ 0LN = 2 P g 2 s L ms = 1 2 L ms Doesn t include leakage 17

18 PM Tems Analytical Appoach 18

19 PM Tems Def. of Elec. Quantities θ ω φ φ s = P = P = P = P θ ω φ φ m m sm m / 2 / 2 / 2 / 2 19

20 PM Tems Suppose the B field due to the PM may be expessed B and suppose w due to PM as = 2N P Bpm 0 φ π = Bpm π φ 2π s cos( Pφ / 2) = sm 2N P s cos( φ ) s 20

21 PM Tems It can be shown that λ as due to PM = λ m sinθ whee λ = m 8LBpmNs P 21

22 PM Tems 22

23 PM Tems 23

24 PM Tems 24

25 PM Tems 25

26 PM Tems Comment: The sinusoidal tuns distibution gives ise to a sinusoidal flux linkage vesus electical oto position chaacteistic 26

27 Expession fo Toque It can be shown that T e P 2 [ i cos( θ ) + i cos( θ 2π / 3) + i cos( θ 2π / 3) ] = m as bs cs λ + 27

28 Expession fo Toque 28

29 Expession fo Toque 29

30 Expession fo Toque 30

31 Expession fo Toque 31

32 Expession fo Toque 32

33 Machine Equations in Roto Refeence Fame Conside the tansfomation Why? qd 0s f = K s f abcs Whee T ( fqd s) = [ fqs fds f0 K s = 0 s cosθ 2 sinθ cos( θ π ) 3 2 sin( θ π ) ] 2 cos( θ + π ) 3 2 sin( θ + π )

34 Machine Equations in Roto Refeence Fame 34

35 Machine Equations in Roto Refeence Fame Voltage v v qs ds = si = i Flux Linkage λ qs Toque s = L qs ds ss i ds +ω λ + qs ω λ + qs λ = L i + λ T e ds ss ds m 3 P = λ 2 2 i m qs pλ qs pλ ds 35

36 36 Aside: Some Shothand 3) / 2 sin( 3) / 2 sin( ) sin( 3) / 2 cos( 3) / 2 cos( ) cos( π θ π θ θ π θ π θ θ + = = = + = = = + + s s s c c c

37 37 Aside: Some Tig IDs ) sin( 2 3 3) / 2 3) cos( / 2 sin( 3) / 2 / 3)cos( 2 sin( ) )cos( sin( ) cos( 2 3 3) / 2 3) sin( / 2 sin( / 3) 2 3) sin( / 2 sin( ) )sin( sin( ) cos( 2 3 3) / 2 3) cos( / 2 cos( 3) / 2 3) cos( / 2 cos( ) )cos( cos( 0 3) / 2 sin( 3) / 2 sin( ) sin( 0 3) / 2 cos( / 3) 2 cos( ) cos( y x y x y x y x y x y x y x y x y x y x y x y x x x x x x x = = = = = π π π π π π π π π π π π π π π π

38 Tansfomation of Voltage Equations 38

39 Tansfomation of Voltage Equations 39

40 Tansfomation of Voltage Equations 40

41 Tansfomation of Voltage Equations 41

42 Tansfomation of Voltage Equations 42

43 This yields v Whee Tansfomation of Voltage In expanded fom Equations qd 0s = si qd 0s + ωλ dqs + pλ qd 0s ( λ v v v qs ds dqs = = ) T s s ds = [ λ λ i i qs ds + ωλ qs 0s = si0 s + pλ 0s ωλ ds qs + + pλ pλ 0] qs ds 43

44 Tansfomation of Flux-Linkage Equations 44

45 Tansfomation of Flux-Linkage Equations 45

46 Tansfomation of Flux-Linkage Equations 46

47 Tansfomation of Flux-Linkage Equations 47

48 Tansfomation of Flux-Linkage Equations 48

49 Tansfomation of Flux-Linkage Equations 49

50 Tansfomation of Flux-Linkage Equations This yields 3 L + L 0 0 ls ms 2 iqs 0 3 qd 0 s = 0 Lls + Lms 0 ids + λ m 1 2 i0s L ls λ O in expanded fom λ qs = L Whee ss i qs 3 Lss = Lls + 2 L ms λds = ss ds + λ m λ 0 s = Llsi 0 s L i 50

51 Tansfomation of Toque Equation Stat with T e P 2 [ i cos( θ ) + i cos( θ 2π / 3) + i cos( θ 2π / 3) ] = m as bs cs λ + 51

52 Tansfomation of Toque Equation Finally, we aive at 3 P Te = λ 2 2 m i qs 52

53 Zeo Sequence 53

54 Zeo Sequence 54

55 Relationship of RMS Value and Phase to QD Components 55

56 Relationship of RMS Value and Phase to QD Components 56

57 Relationship of RMS Value and Phase to QD Components 57

58 Relationship of RMS Value and Phase to QD Components 58

59 Voltage Souce Opeation In this mode, idealized voltage applied is vas = 2v s cosθesv 2 v bs = 2v s cos( θesv π ) 3 2 v cs = 2v s cos( θ esv + π ) 3 Whee θ = θ + φ esv v 59

60 Applied Voltage in QD Vaiables We can show that vqs = 2vs vds = 2vs cos sin φ φ v v 60

61 Applied Voltage in ABC Vaiables 61

62 Applied Voltage in QD Vaiables 62

63 Analysis of Steady State Opeation Pediction of Q- and D-Axis Cuents 63

64 Analysis of Steady State Opeation 64

65 Analysis of Steady State Opeation 65

66 Analysis of Steady State Opeation 66

67 Example 1 Conside a machine with the following paametes s = 3.1 Ω P = 4 L ss = 12.1 mh λ m = Vs N =3 Futhe suppose V s = 100 φ v = 0 ω m = 1800 RPM Find the toque and efficiency 67

68 Example 1 68

69 Example 2 Conside the machine with paametes of example 1. Plot the toque speed and ms cuent speed cuves 69

70 Example T e ( 0, ω i ) T e T e π, 4 ω i π, 2 ω i ( ( ), ω i ) T e φ vmt ω i ω i

71 Example i s ( 0, ω i ) i s i s π, 4 ω i π, 2 ω i ( ( ), ω i ) i s φ vmt ω i ω i

72 Optimization of Phase Advance 72

73 Optimization of Phase Advance 73

74 Cuent Souce Opeation Intepetation 1 (ABC Vaiable) 74

75 Cuent Souce Opeation Intepetation 2 (Toque Tansduce) 75

76 Desied D-Axis Cuent 76

77 Example 3 Conside a machine with the following paametes s = 3.1 Ω P = 4 L ss = 12.1 mh λ m = Vs N =3 Plot the voltage equied and efficiency fo the following conditions Toque command: 2 Nm, d-axis cuent 0 A Toque command: 6 Nm, d-axis cuent 0 A Toque command: 6 Nm, d-axis cuent -6 A 77

78 Example 3 78

79 Example ( ) (, ( ), 0.0 ) (, ( ), 6 ) v s ω i, i q ( 2), 0.0 v s ω i i q 6 v s ω i i q ω i 79

80 Example ( ) (,, 0) ( ) η ω i, 2, 0 η ω i η ω i, 6, ω i 80

81 Effect of D-Axis Cuent on Voltage 81

82 D-Axis Injection 82

83 D-Axis Injection 83

84 D-Axis Injection 84

85 D-Axis Injection 85

86 D-Axis Injection 86

87 Example 4 At 2000 pm, the zeo-to-peak line-to-line voltage has a 100 V amplitude and a fequency of 100 Hz. Compute λ m and P. At standstill and at 60 Hz, the impedance looking into the a- to b-phase is 0.2+2j. Find s and L ss. 87

88 Example 4 Pat 1 88

89 Example 4 Pat 1 89

90 Example 4 Pat 1 90

91 Example 4 Pat 1 91

92 Example 4 Pat 2 92

93 Example 4 Pat 2 93

94 Example 4 Pat 2 94