INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 72102, DECEMBER 27-29, 2002 25 Modeling nd Simultion of Permnent Mgnet Brushless Motor Dries using Simulink Mukesh Kumr, Bhim Singh nd B.P.Singh Astrt: Permnent Mgnet Brushless Motors (PMBL) re ille in two types nmely Permnent Mgnet Synhronous Motor (PMSM) with sinusoidl flux distriution, nd Permnent Mgnet Brushless DC motor (PMBLDC) with trpezoidl flux distriution. This pper presents simulink-sed modeling of the omplete drie system onsisting of the urrent ontrolled oltge soure inerter (CC-VSI), PI speed ontroller, motor nd the lod. The simultion results re gien oth for trnsient nd stedy stte operting onditions of PMSM nd PMBLDC drie systems. I. Introdution The sene of the field exittion, rushes nd the ommuttor in Permnent Mgnet Brushless Motors (PMBL) offers mny dntges oer other motors, suh s high effiieny, smll size nd quik response time [1-4]. The ontroller of the PMBL motors onsists of three-phse inerter, position, speed nd urrent sensors nd ontrol (whih proesses the rious signls to produe the swithing signls for the deies of the inerter). This pper presents the modeling of the omplete drie systems with PMSM nd PMBLDC motors. The modeling equtions re represented y simulink loks nd the simultion results re gien oth for trnsient nd stedy stte operting onditions. II. PM Brushless Motor Drie System Figs.1 nd 2 show the drie systems with PMSM nd PMBLDC motors respetiely. It onsists of threephse Current Controlled Voltge Soure Inerter (CC- VSI), the motor nd the ontroller. The inerter, whih is onneted to the d supply, feeds ontrolled power to the motor. The mgnitude nd frequeny of the inerter output oltge depends on the six swithing signls, whih re generted y the ontroller. The stte of these swithing signls t ny instnt is determined y the rotor position, speed error nd winding urrents. The ontroller synhronizes the winding urrents with the rotor position. It lso filittes the rile speed opertion of the drie, nd mintins the motor speed t the referene lue een during lod nd supply flututions. From these figures it is oious tht the drie systems for PMSM nd PMBLDC motors re lmost identil expet for some differene in the ontroller. Hene, the modeling of these two drie systems will he mny things in ommon, suh s inerter nd prts of the ontroller. Therefore, the modeling of omplete drie system omprising of PMSM motor is onsidered first nd for PMBLDC motor drie the modeling of only those prts is gien whih re different from PMSM drie system. Mukesh Kumr, Bhim Singh nd B.P.Singh Eletril Engineering Deprtment,Indin Institute of Tehnology, New Delhi,Indi 110016 III. Modeling of PMBL Motors Drie System () PMSM Drie System Fig.1 shows the lok digrm of the PMSM drie system. Eh omponent of the drie system is represented y set of equtions. These equtions re then represented y suitle simulink loks to form the omplete model of the drie system s shown in Fig.. The rious prts of the drie system re: Controller: The referene speed (ω r ) nd rotor speed (ω r ) re the inputs to the PI ontroller, whih together genertes the referene lue of the torque-produing omponent of sttor urrent (i q ). The output of PI ontroller t ny instnt (k) is i q (k) =k p {ω e (k) ω e(k-1) } + k i {ω e (k) } (1) Where i q is the referene q xis urrent. k p nd k i re the proportionl nd integrl gins of the PI ontroller The speed error is: ω e (k) = ω r (k) ω r (k) dq to xis urrent trnsformtion: The referene lues of the d nd q xis urrents i d ( whih is set to zero in this study s the drie is not operted in the field wekening region ) nd i q re onerted to three phse referene urrents ( i, i, i ) whih re used y the inerter to determine its output. This trnsformtion uses the inerse Prk trnsformtion defined elow: os( ) sin( ) i 2 2 i q i = os sin (2) i d i 2π 2π os + sin + Where is the rotor postion. The Inerter: This lok models the IGBT sed three-phse Voltge Soure Inerter (VSI). Three phse sttor urrents re synhronized with the rotor position y proiding proper gting signls to the deies of the inerter. The referene lue of phse urrent (determined y the preious lok) nd winding urrent of tht phse re used to find the oltge of tht phse s shown elow: if ( i + H) i then S = 1 else S = 0 () i + H i if ( ) S = 0 (4) if ( ) then S = 1 else i + H i then S = 1 else S = 0 (5) d = 2 ( S ) (6)
254 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 d = ( 2 S ) (7) d = ( 2 S ) (8) Where, H is the hysteresis nd. S, S, S re swithing funtions (whih re either 1 or zero).,, nd d re oltges of phse winding,, nd d link, respetiely. to dq xis oltge trnsformtion: Three phse oltges (,, ) of the inerter re onerted to dq xis oltges ( d, q ) y using Prk trnsformtion shown elow: 2 2 os( ) os os + q 2 = () 9 2 2 d sin ( ) sin sin + The PMSM: The sttor of the PMSM omprises of lned three phse winding similr to the wound rotor synhronous motor. The mthemtil model of PMSM is deried from tht of wound rotor mhine under the ssumption tht the rmture emf is indued y the permnent mgnets in ple of d exittion. Assuming tht the indued emf is sinusoidl nd eddy urrent nd hysteresis losses re negligile, the sttor oltge equtions in the rotor referene frme re [5]: q = Ri q + pλ q + ω r λ d (10) d = Ri d + pλ d - ω r λ q (11) Where λ q = L q i q (12) λ d = L d i d +λ f (1) Where, q nd d re the d,q xis sttor oltges. i q nd i d re the d,q xis sttor urrents. L q nd L d re the d,q xis indutnes. λ f is sttor flux linkges due to permnent mgnets. R is sttor-winding resistne per phse. ω r is rotor speed in rd/se (eletril). p is the d/dt opertor. The eletromgneti torque produed is [5]: T e = (/2) P{λ f i q + (L d L q ) i d i q )} (14) Where, P is the numer of pole pirs. The eletromgneti torque is lned y the lod torque, elerting torque nd dmping torque of the system nd n e represented s: T e = T L +Bω r +Jpω r (15) Where, T L is the lod torque,b is the dmping oeffiient nd J is the moment of inerti. For the purpose of dynmi simultion the equtions (10-15) n e rerrnged to gie first order nonliner differentil equtions in terms of riles s: p i d = ( d R i d + ω s L q i q ) / L d (16) p i q = ( q Ri q - ω s L d i d -ω s λ f )/L q (17) p ω r = (T e T L B ω r )/J (18) p = ω r (19) The simulink-sed model of the PMSM motor drie system is deeloped on the sis of these equtions nd is shown in Fig. () The Modeling of the PMBLDC Motor Drie System It n e seen from the lok digrm of PMBLDC motor drie system shown in Fig.2 tht in this se the PI ontroller nd the inerter loks re the sme s in the PMSM drie system nd hene these loks re not disussed gin nd only the modeling of PMBLDC motor is disussed. The sttor of PMBLDC motor is similr to PMSM nd the rotor hs permnent mgnets. The only differene etween them is due to the shpe of the ir gp flux. The ir gp flux distriution is sinusoidl in PMSM nd trpezoidl in PMBLDC motor. Hene the d q rile pproh nnot e diretly pplied to model the PMBLDC motor. Therefore, the PMBLDC motor is modeled using the -phse riles. The olt-mpere eqution for eh phse of PMBLDC motor is expressed s follows [6]: n = Ri + pλ +e n (20) n = Ri + pλ +e n (21) n = Ri + pλ +e n (22) Where n, n, n re the phse oltges. e n, e n, e n re the phse k emfs. The flux linkges of eh phse (λ, λ, λ ) n e expressed s: λ = L s i M (i +i ); λ = L s i M (i +i ); λ = L s i M (i +i ) (2) Where L s nd M re self nd mutul indutnes, respetiely. The PMBLDC motor hs n isolted neutrl whih mens tht the sum of three urrents (i, i nd i ) is zero. The flux linkges n now e written s: λ = (L s +M)i ; λ = (L s +M)i ; λ = (L s +M)i (24) The oltge of eh phse n e expressed in terms of the potentil of tht phse w.r.t the mid point of d link ( o, o,, o ) nd the potentil of neutrl w. r. t the mid point of d link ( no ). Hene we get the expressions for the phse oltges s: n = o - n ; n = o - n ; n = o - n (25) The oltges ( o, o, o ) ssume one of these three lues ( d +, d, 0) 2 2 depending on whether the upper swith, lower swith or no swith is turned on. The potentil of neutrl point n e expressed s: no = o + o + o +(e n +e n +e n )/ (26) The k emfs depend on the rotor position, nd n e expressed s: e = K ω r f (); e = K ω r f (); e = K ω r f () (27) Where K is the k emf onstnt nd f (), f (), f () re funtions of rotor position () nd f () is defined s: f ( ) = 1 0º< 120º (28) 6 f = π 120º< 180 (29) ( ) ( ) 1 π ( ) = 1 6 = 2π + π f 180º< 00º (0) ( ) ( ) 1 f 00º< 60º (1) The funtions f () nd f () will e similr to f () with phse differene of 120 nd 240 respetiely. As shown in Fig.4 these k emf s re represented y using S-funtion
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 72102, DECEMBER 27-29, 2002 255 lok (user defined funtion), euse the ille lirry loks of simulink ould not model these k emf funtions. The eletromgneti torque produed is [6]: T e = (e n i + e n i + e n i )/ω r (2) Whih is simplified to: T e =K {f ()i +f ()i + f ()i } () Whih is lned y lod torque (T L ), frition torque (Bω r ) nd elerting torque (Jpω r ) s: T e = T L +Bω r +Jpω r (4) For the purpose of dynmi simultion equtions (20-4) n e rerrnged to gie first order nonliner differentil equtions in terms of riles s: pi = ( n - -e n i R) / (L s + M) (5) pi = ( n - -e n i R) / (L s + M) (6) pi = ( n - -e n i R) / (L s + M) (7) p ω r = ( T e T L B ω r ) / J (8) p = ω r (9) The simulink-sed model of the PMBLDC motor drie system is deeloped on the sis of these equtions nd is shown in Fig 4. IV. Results nd Disussion Fig. 5 shows the simultion results of PMSM motor (prmeters gien in Appendix) drie system. The motor speed nd urrent weforms re gien for rious operting onditions suh s strting, reersl, nd lod perturtion. During trnsient onditions (strting nd reersl) the motor drws high urrent to ttin the referene speed. The rotor speed remins onstnt with the sudden pplition nd remol of lod, showing the losed loop opertion of the drie system. The simulted response of the PMBLDC motor (prmeters gien in Appendix) drie system during strting, lod perturtion nd speed hnge is shown in Fig.6. Similr to the response of PMSM motor, the PMBLDC motor lso drws high urrent during strting nd reersl. The four-qudrnt opertion of the drie systems is demonstrted through these simulted weforms. V. Conlusion Modeling nd simultion of the PMBL motors he een rried out in MATLAB through Simulink Toolox. Eh omponent of the drie systems hs een modeled nd represented y seprte susystem. These susystems he een onstruted using simulink lirry loks. The simulted response of oth PMSM nd PMBLDC motor drie systems for trnsient nd stedy stte onditions he onfirmed the effetieness of simulink sed models for these PMBL motors. VI. Referenes [1] T. Kennnjo nd S. Ngmori, Permnent Mgnets Brushless DC Motors, Clrendon Press, Oxford, 1985. [2] T.J.E. Miller, Brushless Mgnets nd Relutne Motor Drie, Clrendon Press, Oxford, 1989. [] J.F. Gieers, Permnent Mgnets Motor Tehnology Design nd Applitions, Mrel Dekker, In., New York, 1997. [4] Bhim Singh, Reent Ades in Permnent Mgnets Brushless DC Motors, Sdhn, Vol. 22, Prt 6, Deemer 1997, pp 87-85. [5] P.Pilly nd R. Krishnn, Modeling, Simultion, nd Anlysis of Permnent Mgnets Motor Dries, Prt I: The Permnent Mgnet Synhronous Motor Drie, IEEE Trns. on Industry Applitions, Vol. 25, No. 2, Mrh/April 1989, pp. 265-27. [6] P.Pilly nd R. Krishnn, Modeling, Simultion, nd Anlysis of Permnent Mgnets Motor Dries, Prt II: The Brushless DC Motor Drie, IEEE Trns. on Industry Applitions, Vol. 25, No. 2, Mrh/April 1989, pp. 274-279. VII. Aknowledgements Authors express their sinere thnks to the Deprtment of Siene & Tehnology (DST) for proiding the finnil ssistne to rryout this work t IIT Delhi. VIII. Appendix Permnent Mgnet Synhronous Motor (PMSM) Voltge: 40 Volts Power: 1.9 kw Current: 8.64 A Phse: : 27 rpm. Pole: 4 Permnent Mgnet Brushless DC Motor (PMBLDC) Voltge: 400 Volts Power: 1.5 kw Current: 4 A Phse: : 1500 rpm Pole: 4 DC supply -Phse Voltge Soure Inerter PM SM Position Sensor Current ontroller i i Controller dq to xis trnsformtion Ref. speed + - ontroller ω ω r i q i d =0 Position to Conersion Fig.1 Blok digrm of Closed Loop Control Sheme of PMSM Motor Drie System
256 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 DC supply -Phse Voltge Soure Inerter PM Brushless Motor Position Sensor Controller Current ontroller i i Ref. Current Genertion Ref. speed + - ontroller ω ω r Position to Conersion Fig.2 Blok digrm of Closed Loop Control Sheme of PMBLDC Motor Drie System 1 Referene PI 0 iqref idref ioref the ir i iref i Wr the d.mt Dt file dq to VSI Inerter 2 Lod TL PMSM Te Fig. Simulink Bsed Model of PMSM Motor Drie System
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 72102, DECEMBER 27-29, 2002 257 m d.m t Dt file i ire f V SI Inerter V e i 2 e1 i T L the1 T e wr the1 T het Thet kemf kem f m PM BLDC M otor Lod T orque & -K - 1 R eferene Sp eed PI Fig.4 Simulink Bsed Model of PMBLDC Motor Drie System Fig.5 Simulted Response of PMSM Drie System for Strting, Reersl nd Lod Perturtion.
258 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 Fig.6 Simulted Response of PMBLDC Drie System for Strting, Reersl nd Lod Perturtion.