A CONTROL STRATEGY FOR NOISE REDUCTION IN SWITCHED RELUCTANCE MOTORS

Transcription

1 A CONROL SRAEGY FOR NOISE REDUCION IN SWICHED RELUCANCE MOORS Erdal Bzvlc UBIAK-BİLEN Informaton chnologs and Elctroncs Rsarch Insttut Anara ury Kmal Lblbcoğlu and H. Bülnt Ertan Mddl East chncal Unvrsty Dpartmnt of Elctrcal and Elctroncs Engnrng Anara ury Abstract hs study dscrbs an approach to lmnat th torqu pulsatons and hnc acoustc nos of SR motors. o control th forc pulsatons phas voltag s slctd as th control varabl contrary to what s oftn ncountrd n ltratur. h authors blv that cost ffctv mplmntaton of ths choc s now tchncally possbl. Slcton of approprat phas voltags s achvd by Sldng Mod Control n ordr to produc th pr-dfnd torqu rfrnc for ach phas. In ths approach pay currnt wavforms ar avodd. In ths mannr th ratng of th powr supply s rducd. Smooth currnt wavforms also contrbut to th rducton of th nos lvl of th drv systm. Effctvnss of th approach s llustratd wth xprmntal rsults. INRODUCION Swtchd rluctanc SR motor drvs ar basd on ruggd SR motors whch provd a wd rang of spd control and proms to b cost ffctv. Howvr thy ar nown to hav hghr acoustc nos lvls than som of th othr drv typs. Shaft torqu varaton s on of th sourcs of acoustc nos n SR motors. Howvr radal forcs xctng th mchancal structur ar also vry mportant. Rducton of th acoustc nos lvl s possbl by payng attnton to th mchancal dsgn of th motor and mprovng ts magntc crcut dsgn so that th motor has a smoothr shaft torqu output. Howvr onc th motor s dsgnd to satsfy th rqurmnts mntond abov th dsgn of th lctronc controllr rmns th acoustc nos prformanc of th motor drv. h torqu producd by as SR motor s dpndnt on both th poston of th shaft and th currnt n th phass. Furthrmor th torqu producd s not a lnar functon of currnt. So th problm s hghly nonlnar. h motor torqu s rmnd by th turn on angl ON of th phass th conducton angl COND and by th phas currnt or voltag. Dspt ths t may b argud that th problm s that of placng th rght amount of currnt to th xctd phass so that th dsrd smooth torqu s obsrvd at th shaft. Howvr attnton must b pad to ma sur that th powr supply ratng s not rqurd to b xcssv th currnt wav shap s smooth to avod suddn radal forc pulsatons. At low spds th control s asr as thr s tm to adjust th currnt lvl. At hghr spds th nducd mf n th phas wndngs furthr complcat th problm. It s obvous that xctng two of th phass smultanously to obtan a smooth shaft torqu s lly to b mor advantagous than xctng on phas at a tm. In ths mannr powr ratng of th powr supply may b rducd a smoothr phas currnt wavform can b obtand tc. An mportant ssu n mplmntng a two phas on xctaton s how to shar th torqu btwn two phass to avod xcssv and pay phas currnts. h control varabls ar th advanc angl conducton prod and th phas currnt or voltag magntud as mntond abov. Evn wth a vry fast procssor thr s not

2 nough tm for an on-ln optmzaton to mnmz phas currnts and obtan th dsrd torqu. Such an approach would lmt th uppr spd that th control s achvabl. For ths rason oftn a prdfnd torqusharng functon s mployd n th prvous studs []. Flux [] [3] currnt [4] [5] or voltag [6] [7] may b usd as th ntrmdat control varabl to follow th dsrd torqu rfrnc. Voltag controllrs nd varabl voltag suppls. Flux controllrs rqur a flux obsrvr [3] or ψ-- loo-up tabls [] for not suffrng th ntgraton drft and nos problms. Whn th ltratur s nvstgatd t s obsrvd that currnt controllrs ar gnrally usd wth hgh-gan hgh-bandwh amplfrs [4]. Voltag controllrs ar not much favord so far as thy would ntroduc adonal cost. Howvr authors blv that prsnt tchnology prmts a cost ffctv soluton at last for som problms. For ths rason ths papr concntrats on dsgnng a torqu controllr whch also lmnats torqu rppl whl mang sur that th convrtr ratng s pt at a mnmum and phas currnt varatons ar mad as smooth as possbl. At ths stag to dntfy th y factors of th problm advanc angl s tan as constant th conducton angl s pt at unty and only th phas voltag s tan as th control varabl. CONROLLER DESIGN In th approach hr th dsrd shaft torqu s producd by two phass. h contrbuton of ach phas to th total torqu s dsgnd such that th sum of th phas torqus must b qual to th dsrd torqu rf at any rotor poston. In ordr to p th total torqu constant just l th othr torqu sharng mthods n ltratur an arbtrary contour functon g s dfnd such that rf = m = g m = numbr of phass In ths cas ndvdual phas rfrncs g s chosn sgmodal as sn n Fg.. hs phas torqu rfrncs may b calld orqu Sharng Functons. hs functon must b ralstc so th actual phas torqu must b abl to follow th contour dfnd by th torqu sharng functons wth a fnt DC ln voltag.. torqu sharng functons otal shaft torqu rf. phas torqu Nm Phas D Rfrnc g 4thta Phas A Rfrnc g thta Phas B Rfrnc g thta phas angl dgr Fg. Indvdual phas torqu rfrncs and th total shaft torqu to gv th rf. For a gvn rf onc th sharng of th torqu btwn phass s dcdd for ach of th actv phass th torqu rror s calculatd as follows: σ = rf hn a control stratgy s usd to mnmz th ndvdual phas torqu rror. In th xprsson abov s th poston at whch th calculaton s mad s th shaft spd and s th th phas currnt.

3 Controllr mplmntaton In th applcaton hr t s assumd that torqu-poston-currnt and flux lnag-poston-currnt data s avalabl. Although xprmntally obtand data s usd hr ths data can also b computatonally producd [9 ]. It s also dcdd that a Sldng Mod Control stratgy s vry sutabl for th applcaton hr. SRM Modl Stat quatons of an m phas SRM ar gvn as follows [8]: d = dw m = { } J = l d ψ u r.. = =... m l whr l s th load torqu J s th total rotatonal nrta of th motor and th load s th ndvdual phas torqu r s th phas rsstanc u s th appld phas voltag and ψ s th flux-lnag of th th phas. l s th ncrmntal nductanc and gvn by ψ l = Sldng Mod Control SMC SMC forcs th systm stats to sld on a dsrd surfac n th stat spac. hs surfac s calld as th sldng surfac. If th stat varabls of th systm sld on th dsrd surfac t mans that th stat varabls hav th dsrd valus and th systm s undr control. o p th systm n th sldng rgm a propr functon of th stat varabls must b dfnd to p th systm varabls on th sldng surfac. hs functon s namd as swtchng functon and shown by th symbol of σ x r. hs s a nd of rror functon and th controllr forcs ths functon towards zro durng opraton σ x r. Hr x r s th stat vctor of th rlvant systm. If th numrcal valu of σ x r s qual to zro th systm s on th sldng surfac othrws not. h numrcal valu of σ x r s calculatd by usng th actual valus of th stat varabls. Whn σ x r s gratr than zro ths mans that a control sgnal must b appld to dcras th valu of σ x r. h oppost occurs whn σ x r s lss than zro. hs s a nd of closd-loop stat fdbac systm to control th valu of th swtchng functon σ x r [7]. If th valu of σ x r s dffrnt from zro th varaton n th SMC brngs t bac to zro. h xstnc conon of th sldng rgm s d r. σ x < r r dσ x σ x. < σ. σ < 3

4 hs nqualty s transformd to an qualty f th rght hand sd s always ngatv. r r dσ x r r σ x. = f σ x f σ x > σ σ = f σ f σ > 4 f f σ rf σ = = 5 σ rf Equaton 5 can b calld as Sldng Mod Functon SMF whch wll b usd to dsgn dffrnt typs of SMC. Equvalnt nput voltag vctor u q can b calculatd by substtutng th stat quatons nto th quaton 5. By slctng dffrnt f σ valus dffrnt typs of SMC can b achvd. hs subjct wll b consdrd n th nxt sctons. SMC Dsgn Our am s to p th shaft torqu at a constant rfrnc valu. o st th phas torqu at th rfrnc valu w nd to dvlop a control sgnal to mnmz th torqu rror. orqu rror s th dffrnc btwn th total shaft torqu and th torqu rfrnc. hs rror valu can b consdrd as a swtchng functon. hs swtchng functon s dsgnd to p th systm stat varabls on th sldng surfac and dfnd as follows Substtutng 6 nto 3 w obtan σ = 6 rf d rf < hs nqualty s transformd to qualty by sttng th rght hand sd to a ngatv valu as shown n 3. 7 d = f rf rf f > rf d f rf = rf rf d d f rf = rf 8.a rf d rf d d d rf rf d = = = rf.. 8.b d d

5 + = d d d 8.c If w substtut stat quatons nto 8.c and thn substtut 8.b and 8.c nto 8.a w obtan;. ψ rf f rf r u l rf = + 9 Whr th partal drvatvs of th torqu and th flux-lnag wth rspct to poston and currnt ar calculatd numrcally by usng th masurd statc torqu and flux-lnag curvs Fg.. Control voltag u can b solvd as a functon of th stat varabls from ths quaton as follows r l rf rf rf f u = ψ hs s th control voltag for SMC to p th systm n th sldng rgm. In othr words f ths voltag s appld to th phas th torqu rror wll b forcd to b zro. Equaton ncluds th partal drvaton of rf wth rspct to tm whch mans that th quaton s vald for also dynamc conons. At stady stat rf dos not chang wth tm so th drvatv of rf wth rspct to tm wll b zro. = rf d In ths study stady stat conons ar consdrd so ths trm dsappars from and th control voltag s calculatd as follows: r l rf rf f u + + = ψ Not that th frst trm of ths quaton s th Sldng Mod Functon SMF whch s dfnd n 5. In ths study two dffrnt functons ar trd for ths purpos n ordr to s th ffct of th slcton of ths functon. Som of th rsults ar gvn blow. SIMULAION RESULS In ths study th drv systm whch s composd of SRM and controllr s smulatd by usng Matlab Smuln. h smulatons ar don at stady stat conons whch man that th motor producs constant torqu at constant spd. For th SRM modl usd n smulatons masurd statc torqu-currnt-poston and flux lnag-currnt-poston curvs ar usd Fg.. Statc torqu curvs ar masurd up to 3 A wth currnt stps of.5 A. For ach currnt lvl phas torqu s masurd at vry dgrs of lctrcal phas angl. Flux lnag curvs ar masurd from OU to 8 IN poston wth th poston stps of

6 3 dgrs. For th ntrmdat phas currnt and phas postons both torqu and flux-lnag data s obtand by cubc spln ntrpolaton Flux lnag Wb phas torqu Nm A.5 A A.5 A A.5 A 3 A phas currnt A phas poston lctrcal dgr Fg. Masurd flux-lnag charactrstcs of th tst motor wth 3 dgrs of masurng stps and masurd statc torqu curvs wth currnt stps of.5 A. An ntgraton stp of μs s usd for th smulatons. h control s also appld at vry μs. DC ln voltag whch s shown by th symbol V DC s assumd to vary contnuously btwn ±3 V. o b abl to mplmnt ths control phas currnts and th rotor angl nformaton s rqurd. In smulatons phas currnts ar calculatd from th motor stat quatons and th rotor angl s obtand from th ntgraton of th assumd constant spd for th studd cas. Frcton and wndag s not tan nto account. wo of th cass studd va smulaton for mplmntng SMC ar dscussd blow. Cas d σ / = α. sgn σ α > h drvatv of th torqu rror n ths cas s assgnd a sgnum functon of as shown n Fg. 3. Substtutng ths drvatv conon nto th phas voltag can b calculatd from u = α sgn σ ψ. l + + r whr σ x r = rf sgn σ = rf rf If α s chosn to b a vry bg postv numbr th othr trms of xcpt for th on whch ncluds α can b nglctd. Hnc th control voltag bcoms; u = α sgn σ l

7 5 dsgma/ sgma Fg. 3 Sgnum functon of sgma h DC ln voltag s fnt whch has a maxmum valu of V DC. If w choos α suffcntly bg for xampl 4 th phas voltag valu u wll saturat for th valus gratr than V DC or smallr than -V DC. Equaton can b usd to calculat th phas voltag at both postv and ngatv torqu rgon of th motor. If s nvstgatd t s asly sn that th sgn of th phas voltag s dpndnt of th sgns of th torqu drvatv and th ncrmntal nductanc. If th motor s usd at postv torqu rgon thn th drvatv of th torqu wth rspct to currnt and th ncrmntal nductanc drvatv of th flux-lnag wth rspct to currnt wll always b postv. In ths spcal cas whch s nvstgatd n ths study may b smplfd mor. h smplfd quaton s gvn blow whch gvs us u + V = V u = α sgnσ DC DC σ σ < > In ths study snc advanc angl control s not ncludd th SRM s assumd to oprat always at postv torqu rgon. So th smplfd quaton 3 s usd nstad of to calculat th phas voltag. hs mans that f th torqu rror s lss than zro th DC ln voltag must b appld to th phas n ordr to ncras th actual torqu. If th torqu rror s gratr than zro ngatv DC ln voltag must b appld to dcras th torqu producton so th torqu rror gos to zro. Fg. 44 dsplays th shaft torqu and th phas torqus of th motor obtand from smulaton wth th choc of sgnum SMC functon. h motor s opratng at 785 rpm wth a Nm torqu rfrnc. It can b obsrvd from Fg. 4 that th rfrnc lvl of Nm could b mantand wthn a narrow band. h wh of th band of cours would wdn n a practcal applcaton du to th computatonal dlays n th mplmntaton. Whn ths fgur s closly nvstgatd a chattrng problm s notcd. In othr words th appld voltag fluctuats btwn ± V DC. A smoothr opraton s dsrabl wth a gradual chang btwn voltag lvls Nm.6 Nm tm sc tm sc Fg. 4 a Shaft torqu tracng a rfrnc valu of Nm n= 785 rpm b Indvdual phas torqus to gv Nm total shaft torqu n= 785 rpm

8 3.5 Ampr tm sc Fg. 5 Phas currnts for sgnum functon at 785 rpm Nm Cas d σ / = α. sgmod σ α > In ths cas th drvatv of th torqu rror s assgnd to a sgmod functon and ths s as shown n Fg. 6. h SMC functon whch ld to bst rsults s found to b dσ/ = -α sg σ. h smulatons ar mad for a rfrnc torqu of Nm at 785 rpm wth μsc ntgraton stp sz and μsc control cycl. Of cours updatng th controllr output at μsc s not possbl n ral lf but ths smulaton s mad to obsrv th dal prformanc of th controllr. In Fg. 7 th rqurd phas voltag and currnt ar gvn for a smooth torqu opraton for th conons mntond abov. It can b obsrvd that th top of th currnt wavforms s vrtually flat. hrfor th powr supply ratng s not rqurd to b unncssarly hgh. Furthrmor th dynamc rspons of th powr supply dos not nd to b fast. Not that ths smooth currnt shaps also assur that th radal forcs on th rotor dos not hav pulsd shaps. It s qut asy to conclud that any acoustc nos du to such forcs s also lmnatd for all practcal purposs. h shaft torqu obtand from smulatons s gvn n Fg dsgma/ sgma torqu rror Fg. 6 Sgmod functon sgma Volt tm sc Ampr tm sc Fg. 7 Phas voltaga and currnt b wavform for th smulaton at Nm 785 rpm usng sgmod functon Smulaton stp sz: μsc

9 Nm tm sc Fg. 8 Smulaton rsults for a total shaft torqu of Nm at 785 rpm. Smulaton stp tm μsc Mor ralstc smulaton rsults Onc th choc of th control structur s dcdd a mor ralstc smulaton of th controllr and th motor s mad. In ths cas th dc ln voltag V DC s fxd at 3 V th ntgraton stp s tan as μs as bfor. h calculatd rfrnc sgnal s assumd to b producd by a Hz PWM modulator actng on th DC ln voltag. h phas currnt valus ar agan solvd from rlatd quatons. h poston nformaton s calculatd from th constant opratng spd. h smulator acqurs currnt and poston data to calculat th nxt rfrnc voltag. It s assumd that th tm th mcroprocssor would ta to do th rqurd calculatons shall b n th ordr of μs so th control s appld at vry μs. Howvr ths rfrnc lvl s fd to th controllr aftr μs. Untl ths prod s compltd th smulaton s contnud wth th prvous rfrnc voltag lvl. Control paramtr α s chosn as 3 aftr som trals. h choc of ths paramtr dpnds on th systm dynamcs and th controlld paramtrs so durng smulatons som xprmntaton wth th valu of α s ncssary to rmn a sutabl valu for th control systm. h rsults of ths typ of smulaton ar gvn n Fg. 9 for Nm torqu rfrnc at 785 rpm. It can b obsrvd from ths fgur that th shaft torqu fluctuaton s largr about 4% n ths cas as compard to th rsult n Fg. 8. hs s an xpctd fndng snc thr s now a ralstc dlay on th control acton. h phas currnt wavforms for th conons mntond abov ar gvn n Fg. 9. h currnt magntud and shap s qut smlar to th rsults found n arlr smulatons. Ampr tm sc Nm tm sc Fg. 9 a Phas currnts wth Hz PWM for Nm at 785 rpm b otal shaft torqu wth Hz PWM for Nm at 785 rpm

10 Hardwar Implmntaton Wth ths undrstandng of th problm th dvlopd control schm s mplmntd on a dspace DS4 controllr board. h phas voltag control s achvd by th sam procssor controllng 4 PWM outputs at Hz wth th rfrnc voltags provdd by th control stratgy. h swtchng rqurd by th controllr s usd to drv th transstors of an SR motor drvr. Hnc th phas voltags could b st as computd by th controllr. h DC ln voltag of th SRM drv s obtand by rctfyng a sngl-phas 5 Hz AC supply and th DC bus voltag s st to 3V. h drvr s usd to drv a 5 V four-phas 8/6 SRM whch producs a low spd runnng torqu of approxmatly.3 Nm at th ratd currnt of 3 A. h controllr algorthm s dvlopd on Matlab Smuln and thn mplmntd on a dspace DS4 controllr board. For th controllr mplmntaton masurd statc torqu-currnt-poston and flux lnagcurrnt-poston curvs ar usd as dscrbd n th smulatons. h motor s coupld to a magntc powdr bra wth a rotatng torqu transducr n btwn th two shafts. h motor runs wth a Nm torqu rfrnc at 785 rpm as n th smulatons. Exprmntal Rsults In Fg. xprmntal rsults for th phas currnts ar gvn at th xprmnt conons. In ths cas torqu rfrnc s chosn as Nm at a rotor spd of 785 rpm. Hz PWM sgnals ar usd just as th smulatons. Rsults obtand from th xprmnts dsplay a good torqu rfrnc tracng capablty and rducton prcvd n nos lvl. If th smulatd and th xprmntd currnt wavforms n Fg. 9 and Fg. ar compard t can b obsrvd that th currnt magntud and shap ar qut smlar to prdctons. h shaft torqu s obsrvd to fluctuat btwn.88 and.93 Nm. h controllr dos not nclud th frcton and wndag losss so th masurd torqu s smallr than th xpctd valu. Furthrmor th controllr rls on th accuracy of th data and th modl usd n th controllr. Fg. Currnt and voltag wavforms at 785 rpm Nm rac : Phas voltag controlld by PWM Volts/dv rac : Phas currnt A/dv Conclusons In ths study t s shown that Sldng Mod Control can b usd to obtan smooth lctromchancal torqu from SR motors provdd that SMC functon s wll chosn. Furthrmor t s obsrvd that th currnt

11 wavforms ar not fast changng and do not hav pay wavforms contrbutng to th rducton of nos. Both th smulatons and xprmnts showd that a 4 phas 5 V SRM shaft torqu could b controlld to follow th rfrnc lvl st. It s also shown that th torqu rppl s vry small. In th applcaton hr PWM modulatd phas voltags ar usd for control purpos. Dspt ths.. th voltags appld ar formd of pulss rducton n audo nos lvl s prcvd. h smulatons and th mplmntaton hr hav crtan smplfcatons mposd to dntfy whthr th ntal da was vald. In th lght of ths promsng fndngs thr s nd for furthr nvstgaton. For xampl th spd rang n whch th control can b achvd nds to b rmnd. h control stratgy can b furthr dvlopd to nclud th advanc angl control to xtnd th spd rang. Of cours th masurmnt of acoustc nos lvls and vbraton wth and wthout th controllr would also b vry usful. h computaton tm on th DSP board s not nown at ths stag as th controllr softwar s downloadd from Matlab Smuln and th cod s by no mans optmzd. Rfrncs [] M. Ilc-Spong. J. E. Mllr S. R. MacMnn and J. S. horp Instantanous torqu control of lctrc motor drvs IEEE rans. Powr Elctron. vol. pp Jan [] P. G. Barrass B.C. Mcrow Flux and torqu control of swtchd rluctanc machns IEE Proc. Elctr. Powr Appl. Vol. 45 No.6 Nov. 998 pp [3] Ahmd M. Osama Hossam A. Abdl Fattah Ahmd M. Sar Varabl structur flux lnag controllr for torqu rppl mnmzaton n swtchd rluctanc motors Amrcan Control Confrnc. Procdngs of th Vol. 4 pp [4] D. G. aylor An xprmntal study on compost control of swtchd rluctanc motors IEEE Control Syst. Mag. 99 pp [5] Husan and M. Ehsan orqu rppl mnmzaton n swtchd rluctanc motor drvs by PWM currnt control IEEE rans. Powr Elctron vol. pp Jan [6] M. Ilc-Spong R. Marno S.M. Prsada D.G. aylor Fdbac lnarzng control of swtchd rluctanc motors IEEE rans. Auto. Control vol.ac-3 no pp [7] Guspp S. Buja Robrto Mns Mara I. Valla Varabl structur control of an SRM drv IEEE ransactons on Ind. Elctroncs Vol. 4 No. Fbruary 993 pp [8] J. M. Stphnson J. Corda Computaton of torqu and currnt n doubly salnt rluctanc motors from nonlnar magntsaton data Proc. IEE Vol.6 No.5 May 979 pp [9] Ertan H.B. Daşdmr B. An Approach to Analyss and Dsgn of Swtchd Rluctanc Motors ISEF - Sptmbr Cracow-Poland pp [] Ertan H.B. "Analytcal Prdcton of orqu and Inductanc Charactrstcs of Idntcally Slottd Doubly- Salnt Rluctanc Motors" IEE Proc. Pt B 33 No [] Krshnan R. Swtchd Rluctanc Motor Drvs: Modlng Smulaton Analyss Dsgn and Applcatons CRC Prss.