Electric Machines. Leila Parsa Rensselaer Polytechnic Institute

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1 Elctc Machn la Paa Rnla Polytchnc Inttut 1

2 Hybd Vhcl

3 Elctc Shp Applcaton

4 4

5 Elctc Shp Pow Sytm 5

6 6

7 Populon Moto 19MW 7

8 Elctc Mnng Tuck On of th haul tuck fttd wth a pantogaph fo th tolly-at ytm, whch u lctc pow fom th man gd to mo th tuck up th tp pt amp. 8

9 Elctc Shol and Tuck Elctc hol load th bokn matal nto tuck fo tanpot fom th pt. Intal pt qupmnt ncludd hol wth 11 cubc yad buckt and 85 ton poducton tuck. Fo th 199, th mn contng t flt to 17 and 19 ton tuck. 9

10 Unmannd Aal Vhcl 1

11 Mannd Aal Vhcl Sonx Acaft E-Flght Intat buhl dc moto ffcncy: 96% moto wght: 5 lb. flght tm: 1 h. moto pow: 1 hp. oltag: 7 Vdc 11

12 Rfnc Fam Thoy Intoducd by R.H. Pak n 199 to modl ynchonou machn 1

13 Th-Pha Tanfomaton to th Abtay Rfnc Fam f qd K f abc f f f q qd d f f abc f f f a b c f = oltag, cunt, o flux lnkag q = q-ax (quadatu ax) d = d-ax (dct ax) = zo qunc a = a-pha b = b-pha c = c-pha 1

14 14 Th Rfnc Fam Tanfomaton n n n co co co K = fnc fam pd (ad/c) = fnc fam poton (ad)

15 15 1 n co 1 n co 1 n co 1 K qd abc f K f 1 Th In Tanfomaton

16 Exampl: Th-Pha St of Voltag V co a V co b V co c t f f V - lctc fquncy (Hz) - lctc adan fquncy (ad/c) - lctcal poton (ad) - m Voltag (V) - pha hft (ad) 16

17 Tanfom to th Abtay Rfnc Fam K qd abc co co co Vco q d n n n Vco Vco 17

18 q-ax oltag co co co co co co q V ung th dntty, coxco ycox coy cox coy cox y q V Voltag n Abtay Rfnc Fam co d-ax oltag n co n co n co d V ung th dntty, n xcoyn x coy n x coy n x y d V n zo qunc oltag 1 co co co V 18

19 8 V, -pha, f = 6 Hz (8 V ln-to-ln m) 8 V V 1 V 1. Statonay fnc fam V co q V n d Numcal Exampl. Synchonou fnc fam q V V co d V n 19

20 Ax Sktch wth = q d V V co n

21 Ax Sktch wth = V co 17V q d V n 1

22 = and = o V co 147 V q V n 85V d

23 Commonly Ud Rfnc Fam Abtay,, fq fd f K Statonay fq, fd, f K Synchonou fq, fd, f K Roto fq, fd, f K compact notaton Not: In all fnc fam K qd abc f 1 f a f b f c K qd abc

24 Ral Pow n th q-d Rfnc Fam P n a a b b c c T a b c b abc abc a c T T T 1 n qd qd qd qd P K K K K K 1 T 1 K qq dd Pn 4

25 Rfnc Fam Tanfomaton Dlopd by R.H. Pak n 199 fo analy of ynchonou machn. Allow tatmnt of balancd th-pha ac ytm a two-pha dc ytm. Th lad to applcaton of clacal contol thoy Alo mplf contol quaton of om ytm 5

26 Tanfomng Ccut Elmnt: R- Exampl oltag quaton a R a p a b R b p b c R c p c not: p d dt flux lnkag quaton a a b b c c 6

27 Tanfom Flux nkag Equaton 1. Comp quaton abc abc. Tanfom quaton K abc K abc K abc qd qd. Expand quaton q q d d 7

28 Tanfom Voltag Equaton 1. Comp quaton abc R abc p abc. Tanfom quaton K KR K p abc abc abc 1 qd Rqd K p{ K qd } 1 1 Rqd K K pqd K p{ K } qd K K 1 I 8

29 9 co n co n co n } { 1 K p } { 1 K p K dq qd qd qd p R d dq q. Expand quaton d q q q p R q d d d p R p R

30 Equalnt Ccut q-ax d-ax zo qunc q R q p q d d R d p d q R p q q d d

31 1 abc abc abc p R dq qd qd qd p R q d dq Coupld Inducto oltag quaton

32 c b a m l m m m m l m m m m l c b a flux lnkag quaton Coupld Inducto abc abc abc abc K K qd qd K K 1

33 K K 1 l m l m l xpandd fom q d l m l m q d l

34 Tanfomaton of Ccut Elmnt Balancd th-pha ac ccut tanfomd to two-pha dc ccut (nglctng th zo qunc and aumng analy n th ynchonou fnc fam) Flux lnkag quaton fo nduct ccut w ud fo gnalty to oth ccut; ncludng lctc machn Couplng tm btwn th q- and d-ax ult fom th tanfomaton. Th wll lat b wd a back-mf tm whn obng lctc machny n th ynchonou fnc fam. 4

35 Balancd Stady-Stat Voltag V co t a b V cot c Vcot V a V V b c j V V V j j j j ~ V V a 5

36 Synchonou Rfnc Fam q-d Voltag Synchonou: = V V co q V V n d not: V V V q d 1 V d V q tan 6

37 Stady-Stat Calculaton (= ) tady-tat quaton V q RI q I d V d RI d I q 7

38 8 d q d q V V R R R I I 1 ol fo cunt d q d q I I R R V V tady-tat quaton n matx fom

39 Stady-Stat q-d Calculaton In th ynchonou fnc fam, th q-d ccut a uppld fom dc and th copondng dc oluton tady-tat (nducto tatd a hot-ccut, capacto tatd a opn-ccut) Can b ud to analyz tady-tat opaton of lctc machn Equaton can b lnazd about th dc opatng pont fo applcaton of contol thoy 9

40 PMSM Roto and Stato 4

41 Roto and Stato Dgn 41

42 Som PMSM Applcaton Elctc bk Elctc aplan Rmot opatd hcl All-tan hcl 4

43 Pmannt Magnt Moto D Pmannt magnt AC moto (PMAC) : qua-ctangula fd BDC Moto Snuodally fd PMSM BDC moto: Two pha a conductng at ach ntant of tm A low oluton poton no (Hall Sno) nough fo podng commutaton ntant Hgh toqu dnty compad to PMSM at low and mdum pd Commutaton toqu pulaton Th pfomanc dtoat at hgh pd PMSM : Poton nfomaton ndd at ach ntant of tm Mo adancd contol tchnqu wth fat tannt pon applcabl to th knd of d o th whol pd ang.

44 BDC Moto Back-EMF and pha cunt BDC Cunt Wafom

45 BDC Moto c b a c b a c b a c b a dt d M M M R R R ) ( 1 c c b b a a T B dt d J T T Equaton gonng th pha BDC moto: wh a, b, and c a tapzodal back- EMF. Th lctomagntc toqu xpd a: And, th ntacton of T wth th load toqu dtmn how th moto pd buld up:

46 Hgh Spd Opaton of BDC Moto Th BDC opaton abo atd pd bng pfomd by adanc angl tchnqu At a gn toqu o a gn pd, t dffcult to fnd th xact adancangl to b appld. At hgh pd th tato wndng nductanc' can cau th pha cunt to dat gnfcantly fom th dal ctangula wafom whch wll duc th toqu poducton at hgh pd

47 PMSM Tanfomaton fom tatonay fam of fnc to otatng fam of fnc Equaton gonng th pha PMSM moto: d q d q q d d dt ( ) q d d Toqu quaton: T P n m q q m d d q d d q dt q B V t t Elctomchancal moton quaton P/T T T J d dt B PMSM t

48 Hgh Spd Opaton of PMSM Th PMSM opaton abo atd pd bng pfomd by njctng ngat d (fld waknng) and condng cunt and oltag lmtaton: ( d ) ( q ) I atd ( m d d ) ( q q ) V ( atd ) Bad on th abo two lmt th amount of d ax cunt to b njctd known at any gn pd Pow d I q E q ψt Ba Spd Toqu Spd ψm d ψ

49 Pmannt Magnt Synchonou Machn (PMSM) wndng conncton moto contucton aum: 1. Round oto. Snuodal flux lnkag mq md m 49

50 Bac Magntc Equaton N N l A o B H N N dfn nductanc N flux lnkag and cunt a popotonal to B and H H N l B A N A 5

51 Bac Magntc Dfnton = MMF (A t) = luctanc (1/H) l = flux path lngth (m) A = co ctonal aa (m ) μ o = 4π*1-7 H/m pmablty of f pac μ = lat pmablty μ = 4 on, μ = tl Φ = magntc flux (Wb=V c) λ = Flux lnkag (V c) = nductanc (H=Ω c) B = flux dnty (V c/m ) H = magntc fld ntnty (A t/m) 51

52 Inductanc Tm cond a cunt n th a-pha wndng lf nductanc aa aa l m l m - tato lakag nductanc (H) - tato magntzng nductanc (H) mutual nductanc ab (dtmn th componnt of a du to b ) manng nductanc tm o 1 ab co 1 m m ab ac ba bc aa ca bb cb cc 1 m l m 5

53 Inductanc Rlatd to Machn Dmnon m N g l 5

54 54 m am ' n n m bm ' n m cm ' Magnt Flux magnt flux lnkng th a-pha magnt flux lnkng th b- and c-pha

55 machn col aabl PMSM Equaton a a p a b b p b c c p c flux lnkag 1 1 a a l m m m n 1 1 b m l m m b m' n 1 1 m m l m n c c 55

56 Rducd Flux nkag Equaton 1 1 a l m a m b m c m 1 ' ' n n l m a m a b c m fo a wy conncton, a + b + c = n ' a a m wh = l + (/) m co p ' a a a m p a a a a - a-pha back-mf 56

57 machn aabl Col Voltag Equaton a a p a b b p b c c p c oto fnc fam, q-d aabl p q q d q p d d q d p 57

58 1 1 a a l m m m n 1 1 b m l m m b m n 1 1 m m l m n c c comp quaton Flux nkag Expon abc abc abcm 58

59 Tanfom Flux nkag K K K abc abc abcm 1 K K qd qd qd m nductanc K K 1 l m l m l dfn l m magnt flux lnkag qd m K abcm m 59

60 Flux nkag Equaton n th Roto Rfnc Fam q q d d ' m l compad to a-b-c aabl: no coupld tm and no oto poton dpndnc q q d d m' l 6

61 Equalnt Ccut Modl ubttut flux lnkag quaton nto oltag quaton q q p q ' d m d d p d q q-ax ccut d-ax ccut 61

62 Gnal Machn wth P Pol P m P m m - lctcal oto poton (ad) - mchancal oto poton (ad) - lctcal oto pd (ad/c) m - mchancal oto pd (ad/c) 6

63 6 q m d q q m q d out ' ' P m out T P T P out P P T q m ' P T Toqu Equaton fom Pow

64 Toqu Contol n q-d- aabl T P m' q toqu contol 64

65 Mchancal Equaton T J d m dt T B m m T Jp m alo m p m T - lctcal toqu (N m) B m - fcton contant (Kg m /c) J - total nta (Kg m ) m - mchancal pd (ad/c) T - load toqu (N m) m - mchancal oto poton (ad) 65

66 PMSM Modl Th PMSM can b modld wth a faly taghtfowad tanc, nductanc, and back-mf tm n ach pha. If th zo qunc cunt can b nglctd, th q-d modl pnt th machn a a two-pha ccut wth dc quantt n th tady-tat. Th two ccut a coupld by back-mf tm. Th toqu quaton n th q-d modl mplfd compad to th machn-aabl (a-b-c) modl. Th popty wll b ud fo dlopng a toqu contol. Th q-d modl alo lad to mpl contol n oth ytm uch a nducton machn, act ctf, and act flt. 66

67 PMSM Stady-Stat Equaton ound oto machn q d contant q d q d u notaton V q V d I q I d d wth p dt V q I q I ' d m V T d I d P m'i q I q B m m T 67

68 68 olng fo oltag ' I I V V m d q d q d m q d q V ' V I I olng fo cunt ' d m q q V V I d m q d V ' V I Stady-Stat Soluton

69 PMSM wth D (Stady-Stat) nt oltag a V b V c V V V q d V V co co co co n ultng cunt a I co b I co c I co I I q d I I co n V V V 1 q d tan 1 V Vq d I I I 1 q d 1 I d I q tan 69

70 Idal D Calculaton wth = moto paamt P.9 opatng condton 11.4 mh ' m.156 V V 9 V ad m 6 RPM P m m 68 ad 68 ad q- and d-ax oltag and cunt V q V co V d V n V q V d 17V V I q I d 1 V q V d ' m I q 1.4A I d.51a 1 I I q I d I.7A I.79A atan I d I q 68dg P T ' m I q T.Nm 7

71 PMSM Stady-Sat Exampl = 71

72 Idal D Calculaton wth = put all cunt n th q-ax dg I q I co I q.79a I d I n I d A V q I q ' m V d I d V q 19V V d 7.1V 1 V V q V d V 79.4V V d atan V 14dg q P T ' m I q T.89Nm V 11V 7

73 PMSM Stady-Sat Exampl wth = 7

75 Rfnc Fam Thoy A th-pha nduct load connctd to a th-pha oltag ouc. Th nductanc o 1mH and th ouc ha paamt of V 1V, 45, and f 6 Hz (wh t f t ). Calculat th tady-tat q- and d-ax oltag and cunt n th ynchonou fnc fam. Sktch th quantt and th otatng oltag and cunt cto on th gaph blow. Sktch th q- and d-ax oltag n th ynchonou fnc fam fo two cycl of.

76 Pmannt-Magnt Ac D A PMAC machn wth th paamt P 4 1 5mH.95V opatng wth nglgbl fcton and wthout load. Gn that th appld oltag and pha angl V 7V o Dtmn th tady-tat no-load mchancal oto pd. At no-load pd, comput th lctcal nput pow, mchancal output pow, and pow lo. Comput th q- and d-ax oltag and cunt. Sktch th quantt and th otatng oltag and cunt cto on th gaph blow. ' m I th machn opatng wth a ladng o laggng pow facto?

Applications of Lagrange Equations

Applications of Lagrange Equations Applcaton of agang Euaton Ca Stuy : Elctc Ccut ng th agang uaton of oton, vlop th athatcal ol fo th ccut hown n Fgu.Sulat th ult by SIMI. Th ccuty paat a: 0.0 H, 0.00 H, 0.00 H, C 0.0 F, C 0. F, 0 Ω, Ω