Lecture Set 8 Induction Machine S.D. Sudhoff Spring 2018
Reading Chapter 6, Electromechanical Motion Device, Section 6.1-6.9, 6.12 2
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
Characteritic The Good The Bad The Ugly 4
Type of IM Machine Wound Rotor Squirrel Cage Solid Rotor 5
1.5 MW Induction Generator 6
150 Hp @ 1800 RPM 7
5 Hp @ 1800 RPM 8
1 Hp @ 1800 RPM (Stator) Spring 2010 ECE321 9
1 Hp @ 1800 RPM (Rotor) Spring 2010 ECE321 10
Two-Phae IM 11
Stator MMF 12
Stator MMF 13
Stator MMF 14
Stator MMF 15
Stator MMF 16
Stator MMF 17
Rotor MMF 18
Rotor MMF 19
Rotor MMF 20
Rotor MMF 21
Rotor MMF 22
Rotor MMF 23
Comparion of MMF A viewed from tator A viewed from rotor 24
IM Torque Production 25
Our Analyi Machine Variable Model Referred Machine Variable Model QD Model Steady-State Model 26
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
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
QD Model 29
Phaor Model P 2 Torque: T = 2 L Re[ ji I ] e m ~ ~ * a ar 30
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
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
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
Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance 34
Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 35
Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 36
Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 37
Machine Variable Model Flux Linkage Equation Derivation of Magnetizing Inductance (Cont) 38
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
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
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
Machine Variable Model Torque Equation 42
Machine Variable Model Torque Equation 43
Referred Machine Variable Model Why?? 44
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
Referred Machine Variable Model Flux Linkage Equation 46
Referred Machine Variable Model Flux Linkage Equation 47
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 0 0 2 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
Referred Machine Variable Model Voltage Equation Start with v v ab abr = = r r r i i ab abr + + pλ pλ ab abr 49
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
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 Next Step: Stationary Reference Frame Stator tranformation Rotor tranformation = b a d q f f f f 1 0 0 1 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 ) (
Next Step: Stationary Reference Frame 53
Geometrical Interpretation 54
Tranformation of Voltage Equation 55
Tranformation of Voltage Equation 56
Tranformation of Voltage Equation 57
Tranformation of Voltage Equation 58
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
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
Tranformation of Flux Linkage Equation 61
Tranformation of Flux Linkage Equation 62
Tranformation of Flux Linkage Equation 63
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
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
Tranformation of the Torque Equation 66
Tranformation of the Torque Equation 67
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
QD Equivalent Circuit 69
QD Equivalent Circuit 70
QD Equivalent Circuit 71
Comment on QD Machine When i it valid? What i it ued for? 72
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
Balanced Steady-State Operation 74
Balanced Steady-State Operation 75
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
Balanced Steady-State Operation 77
Balanced Steady-State Operation 78
Balanced Steady-State Operation 79
Balanced Steady-State Phaor Equivalent Circuit (2-Phae) P 2 Torque: T = 2 L Re[ ji I ] e m ~ ~ * a ar 80
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
Derivation of Balanced Steady-State Phaor Equivalent Circuit 82
Derivation of Balanced Steady-State Phaor Equivalent Circuit 83
Derivation of Balanced Steady-State Phaor Equivalent Circuit 84
Derivation of Balanced Steady-State Phaor Equivalent Circuit 85
Derivation of Balanced Steady-State Phaor Equivalent Circuit 86
Chief Problem with Model Magnetizing Saturation Leakage Saturation Ditributed Sytem Effect Thermal Effect 87
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
Delta Connected Machine 89
Wye Connected Machine 90
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
Derivation of Rotor Current 92
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
Torque for a Given Stator Current 94
Torque for a Given Stator Current 95
Torque for a Given Stator Current 96
Operation from Current Source Machine Parameter 460 V, l-l, rm; 50 Hp @ 1800 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
Current Source Torque - Speed 250 216.266 200 ( ) T e ω ri 150 100 50 1.508 0 0 100 200 300 400 0 ω ri 376.992 98
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 2 2 * ) ) ( ( 2 ω ω rr r L r = ω 2 2 2 2 ) ( ) ( 2 rr r r M e L r r I L P N T + = ω ω
Maximum Torque Per Amp Control (Operation from Current Source) 100
Maximum Torque Per Amp Control (Operation from Current Source) 101
Maximum Torque Per Amp Control (Operation from Current Source) 102
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
MTPA Control Example 104
MTPA Control Example 105
MTPA Control Example 106
Operation From Voltage Source: Prediction of Stator Current 107
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
Steady-State Operating Point 500 493.596 400 ( ) 300 T e V, ω e, ω ri T L 200 100 0.499 0 0 100 200 300 400 0 ω ri, ω ri 376.984 109
Steady-State Operating Point 110
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
Current v Speed 300 271.655 250 ( ) (,, 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 200 150 100 50 22.45 0 0 100 200 300 400 0 ω ri 376.615 112
Torque v Speed 500 493.595 400 ( ) (,, 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 200 100 4.981 0 0 100 200 300 400 0 ω ri 376.615 113
Efficiency v Speed 0.985 0.8 ( ) (,, 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 0.4 0.2 0 0 0 100 200 300 400 0 ω ri 400 114
Efficiency v Output Power 0.985 0.8 ( ) (,, 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 0.4 0.2 0 0 0 1.10 4 2.10 4 3.10 4 4.10 4 5.10 4 6.10 4 7.10 4 8.10 4 9.10 4 1.10 5 ( ) 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 9.216 10 4 115
Parameter Identification DC Tet 116
Parameter Identification Blocked Rotor Tet 117
Parameter Identification Blocked Rotor Tet 118
Parameter Identification No Load Tet 119
Volt Per Hertz Control The idea: 120
Volt Per Hertz Control The idea: 121
Volt Per Hertz Control 500 493.583 ( ) (, 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 0.25 400 300 200 100 0 0 0 100 200 300 400 0 ω ri 376.615 122