EE595S: Class Lecture Notes Chapter 2* / QD Models for Permanent Magnet Synchronous Machines

Transcription

1 EE595S: Class ectue Notes Chapte * / QD Models fo Peanent Magnet Synchonous Machines S.D. Sudhoff Fall 5 *Analysis and Design of Peanent Magnet Synchonous Machines S.D. Sudhoff, S.P. Pekaek B. Fahii

2 .1 SMPM: Configuation bs-axis Peanent Magnet q-axis Shaft φ θ Roto Coe φ s as-axis Space Stato Stato Slot cs-axis d-axis Fall 5 EE595S Electic Dive Systes

3 .1 Position Relationships φ θ + φ s P θ P ω θ ω φ P φ φ s P φ s φ θ + φ s Fall 5 EE595S Electic Dive Systes 3

4 .1 Tuns Distibution n as ( φ ) s n p sin ( φ ) s (.1-7) ( φ / 3) n bs ( φ ) n sin π s p s ( φ / 3) n cs ( φ ) n sin + π s p s (.1-8) (.1-9) Fall 5 EE595S Electic Dive Systes 4

5 .1 Winding Function w w w as bs cs ( φ ) s n P p cos ( φ ) np ( φs ) cos s π P np ( φ s ) cos s + π P s ( φ / 3) ( φ / 3) (.1-1) (.1-11) (.1-1) Fall 5 EE595S Electic Dive Systes 5

6 .1 Coents on Winding Function Popeties Fall 5 EE595S Electic Dive Systes 6

7 .1 Coents on Winding Function Deivation Fall 5 EE595S Electic Dive Systes 7

8 . ABC Model Voltage Equation v i + abcs s abcs pλ abcs (.-1) Flux inkage Equation λ + abcs λabcs, l λabcs, (.-) Fall 5 EE595S Electic Dive Systes 8

9 . ABC Model eakage Flux λ abcs, l lp l l l lp l l l lp i abcs (.-3) Magnetizing Flux π λas, was ( φs ) B( φs ) ds dφs (.-4) Fall 5 EE595S Electic Dive Systes 9

10 . ABC Model Magnetic Mateial α p π Stato Ion and Slots Peanent Magnet (Noth Pole) Roto Coe Peanent Magnet (South Pole) π Space φ Fall 5 EE595S Electic Dive Systes 1

11 . ABC Model In Magnetic Mateial B H + µ µ B (.-5) B p α p π B µ s µ p µ π φ Fall 5 EE595S Electic Dive Systes 11

12 . ABC Model In the agnetic ateial: In the gap: B µ H g B H + µ µ B (.-5) (.-6) Thus H 1 1 H g B µ µ o (.-7) Fall 5 EE595S Electic Dive Systes 1

13 . ABC Model Recall that MMF dop is defined as Thus F stato oto g Hd F H g + H d (.-8) (.-9) Fall 5 EE595S Electic Dive Systes 13

14 . ABC Model Afte Manipulation, whee B cbf F + B, eff c bf µ µ g µ + d (.-1) (.-11) B, eff d g µ + d B (.-1) Fall 5 EE595S Electic Dive Systes 14

15 . ABC Model Deivation: Fall 5 EE595S Electic Dive Systes 15

16 . ABC Model B p, eff α p π B, eff cbfs cbf c bf π φ Fall 5 EE595S Electic Dive Systes 16

17 . ABC Model Note: we have c bf µ µ p g µ + d p (.-13) c bfs µ µ s g µ + d s (.-14) B p, eff d g µ + d p B p (.-15) Fall 5 EE595S Electic Dive Systes 17

18 . ABC Model Recapping λ as π, w φ ) B( φ ) d dφ (.-4) as ( s s s s B cbf F + B, eff (.-1) Also, we have F + wasias + wbsibs wcsics (.-16) Fall 5 EE595S Electic Dive Systes 18

19 . ABC Model Substituting (.-16) into (.-1) and esult into (.-4) λ d π ( c B ) ( w i + w i + w i ) was bf as as bs bs cs cs + dφ as, s, eff s λ d π ( c ( w i + w i + w i ) B ) was bf as as bs bs cs cs + dφ as, s, eff (.-17) (.-18) λ as, d s P π / P ( c ( w i + w i + w i ) B ) was bf as as bs bs cs cs +, eff dφ (.-19) Fall 5 EE595S Electic Dive Systes 19

20 . ABC Model Evaluating (.-19) yields λ + λ as, asasias + asbsibs + ascsics λas, p asy π ds waswycbf dφ d π w B dφ as, p s as, eff (.-) (.-1) (.-) Fall 5 EE595S Electic Dive Systes

21 . ABC Model Inductance te asy π ds waswycbf dφ (.-1) asy 1 (1+ α p π ds cbfs waswydφ + ( cbf cbfs ) w 1 (1 α p ) π as ) π w y dφ (.-3) Fall 5 EE595S Electic Dive Systes 1

22 . ABC Model Self Inductance using (.-3) asas 4d n P s p [ c π + ( c c )( πα sin( πα ) cos(θ ))] bfs bf bfs p p (.-4) asas A + B cos( θ ) (.-5) A B 4πn pds bfs α P 4d P ( c + ( c c )) p bf s ( c )sin( bfs cbf πα p ) bfs (.-6) (.-7) Fall 5 EE595S Electic Dive Systes

23 . ABC Model Mutual inductance using(.-3) asbs d snp P c bfs π + ( c bf c bfs ) πα p sin( πα p ) cos θ π + 3 (.-8) asbs 1 + cos + π A B θ 3 (.-9) ascs 1 + cos π A B θ 3 (.-3) Fall 5 EE595S Electic Dive Systes 3

24 . ABC Model PM te λ d π w B dφ as, p s as, eff λ λ sin( θ ) as, p (.-) (.-31) λ as, p 8n p d s P B p (.-3) Fall 5 EE595S Electic Dive Systes 4

25 Fall 5 EE595S Electic Dive Systes 5. Final ABC Model Finally, we aive at ( ) ) / sin( 3) / sin( ) sin( 3) / cos( ) cos( 3) / cos( ) cos( 3) / cos( 3) / cos( 3) / cos( 3 / cos ) cos( , π θ π θ θ λ π θ θ π θ θ π θ π θ π θ π θ θ abcs B abcs A abcs i i λ (.-33)

26 . ABC Model Coents on ABC Model Fall 5 EE595S Electic Dive Systes 6

27 .3 Standad QD Model As usual qds f K s f abcs (.3-1) f abcs [ f f f ] T as bs cs (.3-) f qd s [ ] f f T fqs ds s (.3-3) K s 3 cos( θ ) sin( θ ) 1 cos( θ sin( θ π / 3) π / 3) 1 cos( θ cos( θ + π / 3) + π / 3)` 1 (.3-4) Fall 5 EE595S Electic Dive Systes 7

28 .3 Standad QD Model Tansfoation of voltage equation qds si qds + ωsλqds pλqds v + (.3-5) S 1 1 (.3-6) Fall 5 EE595S Electic Dive Systes 8

29 .3 Standad QD Model Tansfoation of flux-linkage equation λ + (.-) abcs λabcs, l λabcs, qd s λqd s, l λqd s, λ + (.3-7) Fall 5 EE595S Electic Dive Systes 9

30 .3 Standad QD Model Tansfoation of leakage flux linkage In qd vaiables λ lp l l abcs, l l lp l iabcs (.-3) l l lp λ qds, l ls ls iqds (.3-8) ls lp l lp + l (.3-9) (.3-1) Fall 5 EE595S Electic Dive Systes 3

31 Fall 5 EE595S Electic Dive Systes 31.3 Standad QD Model Tansfoation of agnetizing flux linkage In qd vaiables ( ) ) / sin( 3) / sin( ) sin( 3) / cos( ) cos( 3) / cos( ) cos( 3) / cos( 3) / cos( 3) / cos( 3 / cos ) cos( , π θ π θ θ λ π θ θ π θ θ π θ π θ π θ π θ θ abcs B abcs A abcs i i λ (.-33) , s qd B s qd A s qd λ i i λ (.3-11)

32 Fall 5 EE595S Electic Dive Systes 3.3 Standad QD Model In qd vaiables, it is convenient to wite + 1 s qd d q s qd λ i λ ) ( 3 B A ls q + + ( ) B A ls d + 3 (.3-1) (.3-13) (.3-14)

33 .3 Standad QD Model Toque T e 3 P ( ) λ i λ i ds qs qs ds (.3-15) See Chapte 5 of Techniques fo Analysis and Design of Electoechanical Systes, S.D. Sudhoff, S. P. Pekaek, onogaph in pepaation fo publication. Fall 5 EE595S Electic Dive Systes 33

34 .4 Steady State Opeation Steady state voltage equations Fall 5 EE595S Electic Dive Systes 34

35 .4 Steady State Opeation Steady state toque equation Fall 5 EE595S Electic Dive Systes 35

36 .4 Steady State Opeation Steady state odel suay: v qs s i qs d ds + ω i + ω λ (.4-1) v ds s i ds ω q i qs (.4-) ( ) i + ( i i 3 P T e λ qs d q ) qs ds (.4-3) Fall 5 EE595S Electic Dive Systes 36

37 .4 Steady State Opeation Opeation fo a VSI v v v as v s v cos( θ + φ ) bs v s v cos( θ π / 3 + φ ) cs v s v cos( θ + π / 3 + φ ) (.4-4) (.4-5) (.4-6) In qd vaiables v v qs ds vs v cos( φ ) vs v sin( φ ) (.4-7) (.4-8) Fall 5 EE595S Electic Dive Systes 37

38 .4 Steady State Opeation Solving fo the qd cuents Fall 5 EE595S Electic Dive Systes 38

39 Fall 5 EE595S Electic Dive Systes 39.4 Steady State Opeation q d s d qs s qs v i ) ( ω ω λ ω + Thus q d s ds s ds q ds v v i ) ( ω λ ω ω + + (.4-9) (.4-1)

40 .4 Steady State Opeation Othe handy elationships ( ) ( ) i i ( ) ( ) v s vqs + v 1 ds i s qs + in P out ωt P e ( ) v i v i 3 P + qs qs ds ds η P P P P 1 ds out in in out P P P out out in > <, P, P, P in in out > < Fall 5 EE595S Electic Dive Systes 4

41 .4 Steady State Opeation Conside a achine s.6 W q d 1.4 H λ 86 Vs P4 Rated values Cuent 3.3 A s Voltage 3 V l-l s Powe: 746 W at p Speed: 55 p (ax) Fall 5 EE595S Electic Dive Systes 41

42 .4 Steady State Opeation Opeation at ated voltage (no phase shift) pe unit 4 i s T e P in P out η ω, RPM Fall 5 EE595S Electic Dive Systes 4

43 .4 Steady State Opeation Opeation fo a CSI i i as i s i cos( θ + φ ) bs i s i i In qd vaiables i i cos( θ π / 3 + φ ) cs i s i ds qs cos( θ + π / 3 + φ ) is i sin( φ ) is i cos( φ ) (.4-11) (.4-1) (.4-13) (.4-14) (.4-15) Fall 5 EE595S Electic Dive Systes 43

44 .4 Steady State Opeation Opeation at ated cuent (no phase shift) pe unit v s T e P in P out η ω, RPM Fall 5 EE595S Electic Dive Systes 44

45 .5 Paaete Identification Fo no load conditions Thus v qds ωλ 1 (.5-1) λ v abcs pk fnd 3ω (.5-6) P ound( ω / ω ) (.5-7) Fall 5 EE595S Electic Dive Systes 45

46 .5 Paaete Identification Deivation Fall 5 EE595S Electic Dive Systes 46

47 .5 Paaete Identification Measueent of d-axis paaetes Ipedance Mete v ~ ab Z ~ ias + v ab - i as b a c / θ π Fall 5 EE595S Electic Dive Systes 47

48 .5 Paaete Identification Petinent elationships Z d 3 v ~ ~ i ab as 3 Z (.5-17) s Re( Zd ) 1 d I( Z d ω ) (.5-) (.5-1) Fall 5 EE595S Electic Dive Systes 48

49 .5 Paaete Identification Coent on positioning the oto Fall 5 EE595S Electic Dive Systes 49

50 .5 Paaete Identification Deivation of d-axis esults Fall 5 EE595S Electic Dive Systes 5

51 .5 Paaete Identification Measueent of q-axis paaetes Ipedance Mete v ~ bc Z ~ ibs i bs b v + bc a θ π / - c Fall 5 EE595S Electic Dive Systes 51

52 .5 Paaete Identification Petinent esults Z q 1 v ~ ~ i ab as 1 Z (.5-9) 1 q Re( Z q ) ω (.5-3) Fall 5 EE595S Electic Dive Systes 5

53 .6 Siulation Deivation of state odel Fall 5 EE595S Electic Dive Systes 53

54 .6 Siulation Fall 5 EE595S Electic Dive Systes 54

55 .6 Siulation Model suay: qd s vabc, x ut v K ω pi pi qs ds P ω ( v v qs ds d s s i i qs q ds ) ω λ ( ) i + ( i i 3 P T e λ qs d q ) pω pθ T e P ω T J l qs ds Fall 5 EE595S Electic Dive Systes 55