Lecture Set 8 Induction Machines

Transcription

1 Lecture Set 8 Induction Machine S.D. Sudhoff Spring 2018

2 Reading Chapter 6, Electromechanical Motion Device, Section ,

3 Sample Application Low Power: Shaded pole machine (mall fan) Permanent Split Capacitor Machine (Reidential HVAC) Medium Power: Indutrial HVAC Pump Vehicle Propulion Drive Application High Power: Ship, Train Propulion Wind Power Generation 3

4 Characteritic The Good The Bad The Ugly 4

5 Type of IM Machine Wound Rotor Squirrel Cage Solid Rotor 5

6 1.5 MW Induction Generator 6

7 RPM 7

8 RPM 8

9 RPM (Stator) Spring 2010 ECE321 9

10 RPM (Rotor) Spring 2010 ECE321 10

11 Two-Phae IM 11

12 Stator MMF 12

13 Stator MMF 13

14 Stator MMF 14

15 Stator MMF 15

16 Stator MMF 16

17 Stator MMF 17

18 Rotor MMF 18

19 Rotor MMF 19

20 Rotor MMF 20

21 Rotor MMF 21

22 Rotor MMF 22

23 Rotor MMF 23

24 Comparion of MMF A viewed from tator A viewed from rotor 24

25 IM Torque Production 25

26 Our Analyi Machine Variable Model Referred Machine Variable Model QD Model Steady-State Model 26

27 Machine Variable Model Voltage equation v ab = r i ab + pλ v = r i + pλ abr r abr abr ab Flux Linkage Equation λab L Lr iab = T abr ( r ) λ L Lr iabr Inductance Matrice L L L r Ll + Lm 0 = 0 Ll L + m Llr + Lmr 0 = 0 Llr + L mr r coθr inθr = Lr inθr coθ r Torque equation P Te = Lr [( iaiar + ibibr )inθr 2 + (i i i i )co θ ] a br b ar r 27

28 Referred Machine Variable Model Referral of variable N i i r abr = abr N N v abr = v abr Nr N λ abr = λ abr Nr Voltage equation v ab = r i ab v abr = rr i abr + pλ abr 2 N r r = rr N r + pλ ab Flux linkage equation λab L L r iab = λ T abr ( Lr ) L r i abr L lr + Lm 0 L r = 0 L lr + L m L r = N coθr Lr = Lm N r inθr Torque equation T e P = L 2 m [( i a ar + (i i i + i a br b br i i b ar )inθ i r )coθ ] inθr coθ r r 28

29 QD Model 29

30 Phaor Model P 2 Torque: T = 2 L Re[ ji I ] e m ~ ~ * a ar 30

31 31 Machine Variable IM Model Voltage Equation Voltage Equation dt d i r v a a a λ + = dt d i r v b b b λ + = dt d i r v ar ar r ar λ + = dt d i r v br br r br λ + = abr abr r abr ab ab ab p p λ λ + = + = i r v i r v ] [ ) ( b a T ab f f = f ] [ ) ( br ar T abr f f = f

32 Machine Variable Machine Model Flux Linkage Equation In Scalar Form λ a = L i + L i + L i + aa a ab b aar ar L abr i br λ b = L i + L i + L i + ba a bb b bar ar L bbr i br λ ar = L i + L i + L i + ara a arb b arar ar L arbr i br λ br = L i + L i + L i + bra a brb b brar ar L brbr i br 32

33 Machine Variable Machine Model Flux Linkage Equation In Matrix-Vector Form Or λ λ λ λ ab abr ab abr = L i + L r ab T ab r = ( L ) i + L = L ( L r ) T i abr L L r r i r abr i i ab abr 33

34 Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance 34

35 Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 35

36 Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 36

37 Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 37

38 Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 38

39 Summary Machine Variable Model Flux Linkage Equation L 0 = 0 L L L = I Lrr 0 Lr = = Lrr L I 0 rr L = r L r coθr inθr inθr coθ r 39

40 Where Machine Variable Model Flux Linkage Equation L = L + rr l L = L + L L L m mr r = lr 2 N = R N = R N r R m 2 r m N m L L m mr 40

41 Let tart with Machine Variable Model Torque Equation λ λ ab abr = L ( L r ) T L L r r i i ab abr 41

42 Machine Variable Model Torque Equation 42

43 Machine Variable Model Torque Equation 43

44 Referred Machine Variable Model Why?? 44

45 Referred Machine Variable Model Flux Linkage Equation Starting Point λ λ ab abr = L ( L r ) T L L r r i i ab abr L 0 = 0 L L L = I Lrr 0 Lr = = Lrr L I 0 rr L = r L r coθr inθr inθr coθ r 45

46 Referred Machine Variable Model Flux Linkage Equation 46

47 Referred Machine Variable Model Flux Linkage Equation 47

48 48 Referred Machine Variable Model Flux Linkage Equation Reult Where = abr ab r T r r abr ab i i L L L L ) ( λ λ = = rr rr r r r L L N N L L m lr mr r lr rr L L L N N L L + = + = 2 = = r r r r m r r r L N N θ θ θ θ co in in co L L

49 Referred Machine Variable Model Voltage Equation Start with v v ab abr = = r r r i i ab abr + + pλ pλ ab abr 49

50 Referred Machine Variable Model Voltage Equation Finally, we get Where v v ab abr = = r r i r i ab abr + + pλ pλ ab abr r r = N N r 2 r r 50

51 Referred Machine Variable Model It can be hown that Torque Equation T e = P 2 L m [( i a i ar + i b i br )inθ + ( i i i i r a br b ar )coθ ] r 51

52 52 Next Step: Stationary Reference Frame Stator tranformation Rotor tranformation = b a d q f f f f ab qd f f = K qd ab f K f 1 ) ( = = br ar r r r r dr qr f f f f θ θ θ θ co in in co abr r qdr f K f = qdr r abr f K f = 1 ) (

53 Next Step: Stationary Reference Frame 53

54 Geometrical Interpretation 54

55 Tranformation of Voltage Equation 55

56 Tranformation of Voltage Equation 56

57 Tranformation of Voltage Equation 57

58 Tranformation of Voltage Equation 58

59 Tranformation of Voltage Equation Thi yield qdr v qd r = r i qdr qd r + pλ dqr qd v = r i ωλ + pλ λ = [ λ λ ] T dqr dr qr qdr 59

60 Tranformation of Flux Linkage Equation Recall λ ab = L i + L i ab r abr λ abr = ' r ab + L rri abr L T i 60

61 Tranformation of Flux Linkage Equation 61

62 Tranformation of Flux Linkage Equation 62

63 Tranformation of Flux Linkage Equation 63

64 Tranformation of Flux Linkage Equation Thi yield λ qd = L qd i + L i m qdr λ qdr = L m i qd + L ' rr i qdr L L ' rr = = L L l ' lr + + L L m m 64

65 Tranformation of the Torque Equation Start with T P in co T θr θr = L i 2 co θr in θ i r e r ab abr 65

66 Tranformation of the Torque Equation 66

67 Tranformation of the Torque Equation 67

68 Tranformation of the Torque Equation Finally, we get P T e = Lm ( iqidr idi 2 qr Which can be hown to be equal to P ' T e = ( λqr i dr λ dri 2 P Te = ( λdi q λqi 2 qr d ) ) ) 68

69 QD Equivalent Circuit 69

70 QD Equivalent Circuit 70

71 QD Equivalent Circuit 71

72 Comment on QD Machine When i it valid? What i it ued for? 72

73 Balanced Steady-State Operation Steady-tate form We can how F = 2F co[ ω t + θ (0)] a e ef F = 2F in[ ω t + θ (0)] b e ef ~ ~ jθef Fq = Fa = Fe ~ Fd = Fb = ( jfe ~ ~ F = jf (0) ~ jθef (0) q d ) 73

74 Balanced Steady-State Operation 74

75 Balanced Steady-State Operation 75

76 Balanced Steady-State Operation Rotor relationhip F = 2F co[( ω ω ) t + θ (0)] ar r e r erf F = 2F in[( ω ω ) t + θ (0)] br r e r erf We can how ~ ~ F = Far ~ ~ F = F ~ ~ F = jf qr dr qr br dr 76

77 Balanced Steady-State Operation 77

78 Balanced Steady-State Operation 78

79 Balanced Steady-State Operation 79

80 Balanced Steady-State Phaor Equivalent Circuit (2-Phae) P 2 Torque: T = 2 L Re[ ji I ] e m ~ ~ * a ar 80

81 Balanced Steady-State Phaor Equivalent Circuit (2-Phae) Voltage Equation ~ ~ ~ ~ V = ( r + jω L ) I + jω L ( I + I a ~ V ar rr = + e e l jω L ' lr a ~ I ar + e e m jω L m a ~ ( I a ar ~ + I ar ) ) Slip Torque ω e ω = r ω P ~ ~ T e = 2 L m Re[ ji a I 2 e * ar ] 81

82 Derivation of Balanced Steady-State Phaor Equivalent Circuit 82

83 Derivation of Balanced Steady-State Phaor Equivalent Circuit 83

84 Derivation of Balanced Steady-State Phaor Equivalent Circuit 84

85 Derivation of Balanced Steady-State Phaor Equivalent Circuit 85

86 Derivation of Balanced Steady-State Phaor Equivalent Circuit 86

87 Chief Problem with Model Magnetizing Saturation Leakage Saturation Ditributed Sytem Effect Thermal Effect 87

88 Balanced Steady-State Phaor Equivalent Circuit (3-Phae) Where T e = 3 P L 2 M ~ Re[ ji * a ~ I ar ] L M = 3 2 L m 88

89 Delta Connected Machine 89

90 Wye Connected Machine 90

91 Typical Operating Situation Utility grid Fixed voltage, fixed frequency voltage ource Inverter (power electronic control) Voltage ource baed inverter (variable voltage, variable frequency) Volt-per-hertz control Current ource baed inverter (variable current, variable frequency) Maximum torque per amp control Field-oriented control Direct torque control 91

92 Derivation of Rotor Current 92

93 Torque for a Given Stator Current We can how T e = P N 2 2 ( r ) r 2 ω LM I + ( ω L rr 2 ) r 2 r 93

94 Torque for a Given Stator Current 94

95 Torque for a Given Stator Current 95

96 Torque for a Given Stator Current 96

97 Operation from Current Source Machine Parameter 460 V, l-l, rm; rpm Stator reitance: 72.5 mω Rotor reitance: 41.3 mω Stator and referred rotor leakage: 1.32 mh Magnetizing Inductance: 30.1 mh Operating Condition Armature current: 50 A, rm Frequency: 60 Hz 97

98 Current Source Torque - Speed ( ) T e ω ri ω ri

99 99 Maximum Torque Per Amp Control (Operation from Current Source) Control Summary To how thi, tart with rr M rr r e r L NP L r T I + = * ) ) ( ( 2 ω ω rr r L r = ω ) ( ) ( 2 rr r r M e L r r I L P N T + = ω ω

100 Maximum Torque Per Amp Control (Operation from Current Source) 100

101 Maximum Torque Per Amp Control (Operation from Current Source) 101

102 Maximum Torque Per Amp Control (Operation from Current Source) 102

103 MTPA Control Example Conider the 50 Hp example Suppoe we want 200 Nm at a peed of 1000 rpm Compute the magnitude of the a-phae current, A Compute the required voltage Compute the efficiency 103

104 MTPA Control Example 104

105 MTPA Control Example 105

106 MTPA Control Example 106

107 Operation From Voltage Source: Prediction of Stator Current 107

108 Operation from Fixed Voltage Source: Example 1 Conider our 50 Hp machine Fed from 460 V l-l rm ource Load torque of 200 Nm Objective: Compute peed and current 108

109 Steady-State Operating Point ( ) 300 T e V, ω e, ω ri T L ω ri, ω ri

110 Steady-State Operating Point 110

111 Operation from Fixed Voltage Source: Example 2 Conider our 50 Hp machine Fed from 460 V l-l rm, 60 Hz ource Let look at machine propertie veru peed 111

112 Current v Speed ( ) (,, ω ri, r ) r0 (,, ω ri, r ) rb I a V, ω e, ω ri, r r I a V ω e I a V ω e ω ri

113 Torque v Speed ( ) (,, ω ri, r ) r0 ( ) T e V, ω e, ω ri, r r 300 T e V ω e T e V, ω e, ω ri, r rb ω ri

114 Efficiency v Speed ( ) (,, ω ri, r ) r0 ( ) η V, ω e, ω ri, r r η V ω e 0.6 η V, ω e, ω ri, r rb ω ri

115 Efficiency v Output Power ( ) (,, ω ri, r ) r0 ( ) η V, ω e, ω ri, r r 0.6 η V ω e η V, ω e, ω ri, r rb ( ) P out( V, ω e, ω ri, r ) r0 ( ) 0 P out V, ω e, ω ri, r r,, P out V, ω e, ω ri, r rb

116 Parameter Identification DC Tet 116

117 Parameter Identification Blocked Rotor Tet 117

118 Parameter Identification Blocked Rotor Tet 118

119 Parameter Identification No Load Tet 119

120 Volt Per Hertz Control The idea: 120

121 Volt Per Hertz Control The idea: 121

122 Volt Per Hertz Control ( ) (, ω ri ) (, ω ri ) (, ω ri ) T e ω eb, ω ri T e ω eb 0.75 T e ω eb 0.5 T e ω eb ω ri