Complex Dynamic Models of Star and Delta Connected Multi-phase Asynchronous Motors

Transcription

1 Complx Dynamic Modls of Sar and Dla Conncd Muli-pas Asyncronous Moors Robro Zanasi Informaion Enginring Dparmn Univrsiy of Modna Rggio Emilia Via Vignols 95 4 Modna Ialy robrozanasi@unimori Giovanni Azzon Informaion Enginring Dparmn Univrsiy of Modna Rggio Emilia Via Vignols 95 4 Modna Ialy giovanniazzon@unimori Absrac Tis papr addrsss complx modling of a muli-pas asyncronous moor wi sar-conncd and dlaconncd saor pass T modl is obaind using a complx rcangular ransformaion a rducs numbr of complx inrnal variabls and is grapically rprsnd using Powr- Orind Graps cniqu T dynamic quaions av bn dfind considring saor pass sar-conncd and dlaconncd proving a in bo cass inrnal complx modl dos no cang Finally som simulaion rsuls av bn prsnd o sow ffcivnss of modld sysm and conribuion of sar-dla ransformaion in rms of providd orqu I INTRODUCTION T advanags and propris of asyncronous moors in muli-pas vrsion ar wll known and dscribd in liraur s for insanc 4 ogr wi ons rfrrd o sar-dla conncion pass s 5 T main focus of is papr is o dfin a complx rducd dynamic modl of a muli-pas asyncronous moor in bo sar-conncd and dla-conncd saor pass cas and o analyz possibl sysm modl diffrncs bwn wo conncion cass T dynamic quaions of sysm av bn obaind using a complx sa spac ransformaion and grapically rprsnd using Powr-Orind Graps modling cniqu A nw ransformaion a imposs sar or dla conncion as bn includd ino complx modl T papr is organizd as follows: Scion II prsns a brif dscripion of basic propris of POG cniqu in complx cas In Scion III complx rducd dynamic quaions of considrd sysm ar dfind and dscribd in sar-conncd and dla-conncd saor pass cass Las Scion IV sows som simulaion rsuls puing in vidnc diffrncs bwn wo pass conncions in rms of providd orqu II POWER-ORIENTED GRAPHS TECHNIQUE T Powr-Orind Graps cniqu s and is suiabl for modling pysical sysms T POG is basd on sam nrgic idas of Bond Graps cniqu s 3 bu i uss a diffrn and spcific grapical rprsnaion T POG ar normal block diagrams combind wi a paricular modular srucur ssnially basd on Fig a) laboraion block x y G(s) x y b) conncion block x y K x K POG basic blocks: a) laboraion block; b) conncion block us of wo blocks sown in Fig a and Fig b: laboraion block (b) sors and/or dissipas nrgy (i springs masss damprs capaciis inducancs rsisancs c); conncion block (cb) rdisribus powr wiin sysm wiou soring or dissipaing nrgy (i any yp of gar rducion ransformrs c) T cb ransforms powr variabls imposing consrain x y = x y T b and cb ar suiabl for rprsning bo scalar and vcorial sysms In vcorial cas G(s) and K ar marics: G(s) is always a squar marix of posiiv ral ransfr funcions; marix K can also b rcangular im varying and funcion of or sa variabls T circl prsn in b is a summaion lmn and black spo rprsns a minus sign a muliplis nring variabl T main faur of Powr-Orind Graps is o kp a dirc corrspondnc bwn dasd scions of graps and ral powr scions of modld sysms: ral par of scalar produc x y of wo powr vcors x and y involvd in ac dasd lin of a powr-orind grap s Fig as pysical maning of powr flowing roug a paricular scion Anor imporan aspc of POG cniqu is dirc corrspondnc bwn POG rprsnaions and corrsponding sa spac dscripions For xampl POG scm sown in Fig can b rprsnd by sa spac quaions givn in () wr nrgy marix L is symmric and posiiv dfini: L = L > Wn an ignvalu of marix L nds o zro (or o infiniy) sysm () dgnras owards a lowr y

2 u B L - y B s x Fig A { Lẋ = Ax + Bu y = B x (x = Tz) { Lż = Az+Bu y = B z POG block scm of a gnric dynamic sysm () () I s I s I s3 I sms I r I r I r3 I rmr V I V M s I V 3 M s3 I 3 V ms M s3ms I ms M sr M r V rr M r3 M r3mr R r L r dimnsion dynamic sysm In is cas dynamic modl () of rducd sysm can b dircly obaind from () by using a simpl congrun ransformaion x = Tz (marix T can also b complx and/or rcangular) wr L = T LT A = T AT and B = T B If marix T is im-varying an addiional rm T LṪz appars in ransformd sysm Wn marix T is rcangular sysm is ransformd and rducd a sam im T POG scms mainain irs pysical maning vn wn dynamic sysm is dscribd using complx variabls : numbr of saor pass; m r : numbr of roor pass; p : numbr of roor and saor polar xpansions; γ s : saor angular pas displacmn (γ s = π ); γ r : roor angular pas displacmn (γ r = π m r ); θ m : roor angular posiion; m : roor angular vlociy; θ s : saor volag angular posiion; s : saor volag frquncy; θ : lcric angl (θ = p θ m); : saor pass rsisanc; : saor pass slf inducanc; M s : maximum muual inducanc of saor pass; R r : roor pass rsisanc; L r : roor pass slf inducanc; M r : maximum muual inducanc of roor pass; M sr : maximum valu of muual inducanc bwn saor and roor pass; J m : roor inria momnum; b m : roor linar fricion cofficin; τ m : lcromoiv orqu acing on roor; : xrnal load orqu acing on roor τ TABLE I ELECTRICAL AND MECHANICAL PARAMETERS OF A MULTI-PHASE ASYNCHRONOUS MOTOR A Noaions In is papr following noaions ar usd o dno full diagonal column and row marics rspcivly: R R R m i j R R R m R ij = :n :m i Ri = R :n R R n R n R n nm V s Saor V r Roor Fig 3 Elcric scm of a muli-pas asyncronous moor wi sarconncd saor and roor pass I I V I sms V ms I ms I s V sms V s I sms V Saor V s I s V s4 V s3 I s3 V 4 I 4 V 3 I 3 Fig 4 Elcric scm of saor of muli-pas asyncronous moor wi dla-conncd pass i R i :n = R R R n T j R j = R R R m :m T symbol δ(n) m k dno following funcion: { if n k k ± m k ± m δ(n) m k = in or cass wr nkm Z T symbol I m dnos an idniy marix of ordr m III ELECTRICAL MOTORS MODELLING L us considr muli-pas sar-conncd asyncronous moor sown in Fig 3 wi saor pass m r roor pass and caracrizd by lcrical and mcanical paramrs sown in Tab I T dynamical bavior of is conncion cas of moor will b compard wi bavior of sam moor wn saor pass ar dlaconncd as sown in Fig 4 L I l and V l dno currn and volag saor-lin vcors: I l = I I I ms T V l = V V V ms T

3 and l I s V s I r and V r dno currn and volag saor-pas and roor-pas vcors: I s = I s I s I sms T V s = V s V s V sms T I r = I r I r I rmr T V r = V r V r V rmr T Wn saor pass ar sar-conncd s Fig 3 volag and currn saor-pas vcors V s and I s ar rlad o saor-lin vcors V l and I l as follows: I s = I l V s = V l V s (3) wr V s = T V s On or and wn saor pass ar dla-conncd as sown in Fig 4 vcors V s I s V l and I l ar rlad as follows: I s = T I l wr marix T R ms ms is srucurd as: T = V s = T T V l (4) Rfrring o following sa-spac vcor and inpu vcor rspcivly: I s I r I V l = V r V = m τ m τ and using Lagrangian approac as widly dscribd in 6 and 7 dynamic quaions S of muli-pas asyncronous moor xprssd wi rspc o fixd fram Σ ar following: ( d L ) I R = + F K I V d J m m + K T b m m τ (5) wr marics R and L ar dfind as follows: R R = s L L R = s M T sr r M sr L r wr = I ms R r = R r I mr and slf and muual inducanc marics L r and M sr ar supposd o assum following srucur: = I ms +M s L r =L r I mr +M r M sr =M sr i j ms a s n cos(n(i j)γ s ) n=: : : i j mr a r n cos(n(i j)γ r ) n=: :m r :m r i j msr a sr n cos(n(θ+iγ r jγ s )) n=: :m r : wr M sr = M sr (θ) r = min{ m r } = M s and L r =L r M r T cofficins a s n a r n and a sr n of Fourir sris saisfy following consrains: s n=: a s n r n=: a r n sr n=: a sr n Noic a in is way odd ordr armonic injcion as bn considring a mans a moor is supposd o av concnrad-windings saor and roor pass T orqu vcor K T and marix F in (5) ar dfind as: K T = I T L θ m F = L (θ m ) L T N C (ms+mr+) (ms+mr+) dno following complx marix: TN (m T N = s θ s ) TN (m r θ p ) T (m = s θ s ) Nms T (m r θ p ) N mr }{{}}{{} N T wi θ p = θ s θ and marics T (mθ) C m (m+)/ and N m C (m+)/ (m+)/ dfind as follows: T (mθ) = m k j k(θ γm) :m I m N m = ::m :m wr γ m = π m Applying psudo-ransformaion I = T N I o sysm (5) on obains following ransformd and rducd sysm S in roaing fram Σ : L Ī R + = F + Ω K J m m K b m wr R = T R T = diag(i ms+ marix L = T L T is dfind as follows: L = k k ::ms Ī m R ri mr+ l k M srk ::ms ::mr l L rk k l M srk ::mr ::ms ::mr L r V + τ (7) ) wil

4 + ms M sa s M sr a T sr Ī s +j s k ms ( + ms M sr a sr L r + mr M M sa s ) j s M sr k ms a T sr ra r Ī r = j p M sr k mr a sr R r +j p k mr (L r + mr M ra r ) J m m j p M sr Ī rk mr a sr j p M sr Ī sk ms a T sr b m Ī s Ī r m Vs + (6) τ Fig 5 Complx dynamic quaions of a muli-pas asyncronous moor in ransformd fram Σ Morovr sum of marics F = T F T and Ω = Ṫ L T in (7) assums following srucur: k l k j s kk j s km srk ::ms ::ms ::mr F + Ω = k l l j p km srk j p kl rk ::mr ::ms ::mr wr k = + ms M s a s k L r k = L r + mr M r a r k and M srk = M sr a sr k δ(k) l T ransformd vcors V = T N V and Ī = T N I av following srucur: Vs V = = Vr Vs Ī Ī = s = Ī r V sms Ī s I sms Ī r I rmr (8) Noic a Vr = bcaus roor pass ar sorcircuid Vcors Vs Ī s and Ī r ar dfind as: k k Vs = Vsk = V dsk + j V qsk :: :: :: k k Ī s = Ī sk = I dsk + j I qsk ::m r :: k k Ī r = Ī rk = I drk + j I qrk ::m r wil las componns V sms I sms and I rmr of vcors V s I s and I r in (8) av following srucurs: V sms = I sms = V s = = I s = (V l V s ) = I rmr = m r m r I r = T saor and roor componns K s and K r of ransformd orqu vcor K = K T N = K s K r assum following srucurs: K s = j p K r = j p Ī k l r M srk ::mr ::ms Ī l k s M srk ::ms ::mr I rmr I sms Finally mcanical orqu τ m can b xprssd as follows: τ m = R ( K Ī ) = R K K s r sr = pm sr k=: ka sr Ī s I sms Ī r I rmr k (I drk I qsk I dsk I qrk ) (9) L us now considr las saor quaion of sysm (7) a is rprsnd by following diffrnial quaion: Isms = I sms + V sms () Wn saor pass ar considrd sar-conncd following rlaion olds: I sms = I s = () = Imposing condiion () quaion () rducs o: V sms = (V l V s ) = = and xprssion of cnr-sar-volag is givn: V s = = V l Wras if saor pass ar considrd dla-conncd following rlaion olds: V sms = V s = () = Subsiuing () in () following diffrnial quaion is obaind: Isms + R s I sms = (3) Rmark Equaion (3) is a sabl dynamic a nds o zro and dos no produc any soliciaion o providd orqu: on can n rmov las saor quaion from (7) Considring a roor pass ar sar-conncd

5 V I T Sar-Dla Conncion V T T R( ) T V N R( ) I Complx/Ral convrsion T N Σ Σ 3 s L- Ī R Elcrical par (complx variabls) Ē K R( ) 4 Enrgy Convrsion 5 Complx/Ral convrsion 6 m J m Ω F s K R( ) τm Fig 6 POG grapical rprsnaion of a muli-pas asyncronous moor in ransformd roaing fram Σ τ r b m Mcanical par (ral variabls) 7 τ and sor-circuid rlaions V ri = V rr V r = and mr = I r= old: las roor quaion from (7) can n b rmovd as wll T rsuling rducd dynamic sysm as now ( + m r )/ complx inrnal variabls insad of ( + m r + )/ Tis dmonsras ow sar or dla conncion of saor pass dos no cang inrnal modl of macin T rsuling rducd sysm S in roaing fram Σ as following srucur: L Ī R + = F + Ω K Ī V J m m + K b m m τ (4) wos xpandd form is sown in Fig 5 wr M sr = (M sr ms m r )/ p = s k m = k k and wr ::m a s a r and a sr ar ral consan marics dfind as follows: a s = k a s k i ::ms a r = k a r k i ::mr k δ(k) l i ::mr ::ms a sr = k l a sr Noic a is sysm dscribs dynamics of sysm in bo sar-conncion cas and dla-conncion on A POG grapical rprsnaion of sysm (4) is sown in Fig 6 Scion - rprsns sar-dla ransformaion marix T a can assum following wo xprssions rfrrd o sar or dla conncion cas rspcivly: Ims T T = T I = mr I mr and wr I T = I T s I T r and V T = Vs T Vr T Scion - 3 rprsns ral-complx sa spac ransformaion Σ Σ wr funcion R( ) dnos complx o ral convrsion of inpu Scion 3-4 rprsns Elcrical par of sysm a in is cas is dscribd only by complx marics and complx variabls ( ligly sadd scion of Fig 6) T Mcanical par of moor is dscribd by scion 6-7 wic is caracrizd only by ral valus and ral variabls Scion 4-6 rprsns nrgy and powr convrsion (wiou accumulaion nor dissipaion) bwn lcrical and mcanical domains Elcrical paramrs = 5 m r = 5 p = = 7 mh M s = 4 mh = 3Ω L r = 5 mh M r = 3 mh R r = Ω M sr = mh V max = V s = 8π rad/s Mcanical paramrs J m = 75 Kg m b m = 45 N/rad τ = Nm TABLE II ELECTRICAL AN MECHANICAL PARAMETERS OF THE SIMULATED MOTOR MODEL IV SIMULATION RESULTS T modl of muli-pas asyncronous moor rprsnd in Fig 6 wos quaions ar dscribd in Fig 5 as bn implmnd in Malab/Simulink T simulaion rsuls prsnd in is scion av bn obaind using lcrical and mcanical paramrs lisd in Tab II T inpu balancd saor volag vcor a as bn considrd is: V l = V l :5 = 3 k=: V mk cos(k (θ s ( )γ s )) :5 (5) wr k indicas injcd armonic ordr L us considr a s = a r = a sr = diag 7 3 as slf and muual inducancs cofficin marics T mod adopd o dfin V mk volags of (5) as bn o coos fundamnal volag ampliud V m and n scaling V m3 using a prcnag cofficin K 3 = V m3 /V m = V ds3 /V ds = 5% a indicas prcnag of 3 rd armonic componn ampliud wi rspc o fundamnal In Fig 7(a) im bavior of saor volag vcor V s saor currn vcor I s and roor currn vcor I r in rang s is sown considring saor pass sar-conncd wil in Fig 7(b) i is sown sam vcors voluion bu in dla-conncion cas: noic a dla-conncion is caracrizd by a igr currn absorpion compard wi

6 Volags V s Mcanical orqu τ m V s V Currns I s τm Nm 4 3 Sar conncion Dla conncion I s A I r A V s V I s A I r A Currns I r Tim s (a) Sar-conncion Volags V s Currns I s Currns I r Tim s (b) Dla-conncion Fig 7 Sar-conncd and dla-conncd saor pass: saor volag vcor V s saor currn vcor I s and roor currn vcor I r in original rfrnc fram Σ sar-conncion Tis aspc xplains ow dynamic mcanical orqu incrass wn saor pass ar dlaconncd (Fig 8(a) and Fig 8(b)) vn if τ m as sam xprssion (9) in bo saor pass conncion cass In Fig 8(a) mcanical orqu τ m is sown as funcion of im in rang s and in Fig 8(b) as funcion of angular vlociy in rang m 5 rad/s wi sar-conncd saor pass (blu plos) and dla-conncd saor pass (rd plos) V CONCLUSIONS In papr a gnral complx dynamic modl of a mulipas asyncronous moor wi sar-conncd and dlaconncd saor pass as bn obaind and grapically τm Nm Tim s (a) Mcanical orqu VS Tim Mcanical orqu τ m 5 5 m rad/s (b) Mcanical orqu VS Angular vlociy Sar conncion Dla conncion Fig 8 Mcanical orqu τ m wi sar-conncd and dla-conncd saor pass as funcion of im (a) and as funcion of angular vlociy (b) rprsnd using POG cniqu A complx ransformaion as bn usd in ordr o rduc numbr of complx inrnal variabls and a nw ransformaion marix a imposs sar or dla conncion of saor pass as bn dfind and includd in modl I as bn dmonsrad a sar/dla ransformaion dos no cang inrnal complx quaions modl of moor T modl as n bn implmnd in Malab/Simulink and simulaion rsuls av sown and compard sar and dla conncion cass REFERENCES R Zanasi Powr Orind Modlling of Dynamical Sysm for Simulaion IMACS Symp on Modlling and Conrol Lill Franc May 99 R Zanasi T Powr-Orind Graps Tcniqu: sysm modling and basic propris 6 IEEE VPPC Lill Franc Sp 3 D C Karnopp DL Margolis R C Rosmbrg Sysm dynamics - Modling and Simulaion of Mcaronic Sysms Wily Inrscinc ISBN rd d 4 E Lvi R Bojoi F Profumo HA Toliya S Williamson Mulipas inducion moor drivs - A cnology saus rviw IET Elcr Powr Appl vol no 4 pp July 7 5 P Pillaya RG Harlya EJ Odndala A Comparison Bwn Sar and Dla Conncd Inducion Moors Wn Supplid by Currn Sourc Invrrs Elcric Powr Sysms Rs vol 8 no pp 4-5 April R Zanasi F Grossi G Azzon T POG cniqu for Modling Muli-pas Asyncronous Moors 5 IEEE Inrnaional Confrnc on Mcaronics Málaga Spain 4-7 April 9 7 R Zanasi G Azzon Complx Dynamic Modl of a Muli-pas Asyncronous Moor ICEM - Inrnaional Confrnc on Elrical Macins Rom Ialy 6-8 Sp 8 G Azzon Modling and Conrol of Muli-pas Asyncronous Elcrical Moors PD Tsis Fb