Robust Sensorless Control of BLDC Motor using Second Derivative Function of the Sum of Terminal Voltages

Transcription

1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 1, No., Jun 13, UDK: : DOI: 1.98/SJEE131144B Robust Snsorlss Control of BLDC Motor using Scond Drivativ Function of th Sum of Trminal Voltags Abdlali Boughaba 1, Mabrouk Chaaban, Said Bnaggoun 3 Abstract: This papr proposs a nw snsorlss control mthod for th spd and position control of a BLDC Motor. This snsorlss driv tchniqu calculats th commutations instants (duration of commutation) by driving th sum of th trminal voltags of th motor (SigV i ). Thus, it is possibl to stimat th rotor position (and back EMF of th motor) by only using masurmnts of th stator lin currnts and voltags. Th implantation of ths dtctors is asy and chap. This mthod is quit robust across variations in stator rsistanc du to changs in tmpratur or frquncy. With this mthod th motor can b startd without nding th initial position of th rotor. This proposd mthod is validatd through xtnsiv simulations at diffrnt spds, and a vry satisfactory prformanc has bn achivd. Kywords: Snsorlss, Driv, Control, Brushlss DC Motor, Trminal Voltags, Trapzoidal Back EMF. 1 Introduction Prmannt magnt synchronous motors, and particularly thos known as bing without a collctor (brushlss motors), hav an incrasingly important us. This dvlopmnt is du to thir high fficincy, high powr dnsity and larg torqu to inrtia ratio [1]. Brushlss DC (BLDC) motor is inhrntly lctronically controlld, and rquirs rotor position information for th propr commutation of th currnt []. Th tchniqus for control dvlopd until now ar numrous. Svral among thm ar basd on th dtction of lctromotiv forc wavforms (back EMF) [3]. Th implantation of ddicatd snsors is an opration which is xtrmly dlicat and rlativly xpnsiv [4]. Howvr th problms rlatd to th cost and rliability of rotor position snsors hav motivatd rsarchrs to dvlop th position-snsorlss BLDC motor driv. 1 LRPI Rsarch Laboratory, Institut of Halth and Industrial Safty, Batna Univrsity, El Hadj Lakhdar Campus, 5 Batna, Algria. abdlali_boughaba@yahoo.fr LEB Rsarch Laboratory, Dpartmnt of Elctrical Enginring, Batna Univrsity, Chahid Md El Hadi Boukhlouf, Batna 5, Algria. mabrouk.chaban@univ-batna.dz 3 LSTE Rsarch Laboratory, Dpartmnt of Elctrical Enginring, Batna Univrsity, Chahid Md El Hadi Boukhlouf, Batna 5, Algria. s_bnaggoun@yahoo.fr 75

2 A. Boughaba, M. Chaaban, S. Bnaggoun Various tchniqus of control for th BLDC motor hav bn dvlopd ovr th last fw yars. Among thm ar a control stratgy basd on th common DC signal [], control by a nw function of flux stimation [1], and a snsorlss commutation intgratd circuit (IC) for a BLDC motor [5]. This papr prsnts th basis of a nw snsorlss position stimation mthod by th application of a nw concpt. This tchniqu is basd on th drivativ function of th sum of th stator phas voltags. Th rsults validation has bn carrid out on a simulation modl for th snsorlss control of a BLDC motor. Stabilization is takn into account by intgrating a PI spd rgulator. Basic Equations of a BLDC Motor.1 Synchronous motor (PMSM) modl For controlling and analysing a BLDCM, th structur of a prmannt magnt synchronous motor (PMSM) is usually usd. That mans that th distribution of th magntic fild is takn as sinusoidal [6]. Th circuit quations ar givn by:. X AX BU, (1) whr X Id Iq, and d Id Vd Id Iq, dt L d Iq Vq Iq K Id, dt L L d pk f p I q T L dt J J J d, dt T 1 with Y CX, whr Y [ I ] T d Iq, C 1, and V d, V q, I d, I q voltags and currnts on a (d,q) fram, T L torqu load, lctrical angular vlocity, K factor torqu, L inductanc, R rsistanc, lctric tim-constant. 76

3 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function....1 Modl of BLDC Motor (Trapzoidal Back EMF Form) A BLDC motor with trapzoidal back EMF is convntionally modlld in th stationary fram using thr phass, a, b, c, bcaus EMFs ar difficult to transform into th d-q rfrnc fram. W considr a motor with a prmannt magnt mountd on th surfac (without salincy ffcts). Fig. 1 givs th quivalnt circuit of th BLDCM. Fig. 1 BLDC motor quivalnt circuit trapzoidal Back EMF's Ea Eb Ec 5 Eabc (Volts) tim (s) Fig. Trapzoidal back EMF. Th gnral voltag quations of th BLDC motor ar givn as follows: d i v R i L, () dt whr R is th stator rsistanc pr phas: 77

4 A. Boughaba, M. Chaaban, S. Bnaggoun R R R, R L is th matrix of inductanc ( L s, M : slf and mutual inductanc): LS M M L M LS M M M L S and T a b c is th vctor of th trapzoidal back EMF (Fig. ). Equation () bcoms: 1 LT Ia Van R Ia an d 1 I b V bn R I b bn dt L, (3) T I c Vcn R Ic cn 1 LT whr LT Ls M. Th mchanical quation of motion is: dr J Tm TL f r, (4) dt with: 1 Tm ( ania bnib cnic ), (5) r whr: r mchanical spd [rad/s], T l torqu load [N m], J motor shaft and load inrtias [kg m ] f frictional damping cofficint [N m s/rad m], T m lctromagntic torqu [Nm]. In a BLDC motor without a nutral connction w hav: Ia Ib Ic, (6) (Evry tim w hav on phas th currnt is st to zro, and th two othrs ar opposd.) 78

5 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... 3 Proposd Mthod for Estimation of Rotor Position Th proposd mthod is basd on th dcomposition of th phas currnts and back EMF by th Fourir transformation. Th dcomposition givs: 4Im 1 ia( ) cosn sinn, n 6 n Im 1 ib( ) sinn sinn cosn cosn sinn n n 6 6, (7) Im 1 ic( ) sin n sin n cos n cos n sin n n n 6 6. Th quations of th Fourir transformations of a trapzoidal back EMF ar givn as follows: 4E 1 a( ) sinn sinn, n 6 n1,,3... 1E 1 b( ) sinn sinn sinn cos cos n n n1,,3,... n 6 6, (8) 1E 1 c( ) sinn sinn sinn cos cos n n n1,,3,... n 6 6, whr: E f, p, (9) p, lctrical rotor position, lctrical spd, fild flux linkag, f p pol pair numbr. In ordr to stimat th rotor position it is ncssary to us systm (), which givs: d Vk RS ik L ik k, (1) kabc,, kabc,, dt kabc,, kabc,, whr: ik. kabc,, 79

6 A. Boughaba, M. Chaaban, S. Bnaggoun Thus, (11) combind with th abov condition givs: V From (3) w obtain:. (11) k k kabc,, kabc,, for n 1,5,7,11,81E sin 3 for n 3 (1),9 sin 9 for n 9.34Esin15 for n15 k ka, b, c E Fig. 3 illustrats th dcomposition of th sum of back EMFs, which is clarly nonzro. 1 Dcomp into harmonics Amplitud tim (s) Fig. 3 Sum of back EMF dcomposition. It is clar from Fig. 3 that th sum of th back EMFs contains information about th rotor position. From th simulation w can s in Fig. 4 that th rror btwn th sum of th trminal voltags (sumv i ) and sum of th back EMFs (sume i ) is narly zro. Thus w can us th sum of trminal voltags, which is xactly qual to th sum of back EMFs, to stimat th rotor position. Th proposd mthod is basd on th scond drivation of th function of th sum of trminal voltags: d sigv F i. dt 8

7 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... SigVi - SigEi (Volts) 1.5 x 1-13 Error sumvabc and sumeabc tim(s) x 1-14 Zoom of Error sumvabc and sumeabc SigVi - SigEi (Volts) tim(s) Fig. 4 Error SumV i SumE i. Figs. 5a and 5b show th structur of th block diagram, giving th principl of th snsorlss commutation of BDCM control. Th block (tim calculation switching) gnrats th control signal of th invrtr (conditions givn by Tabl 1 ar takn into account). Signals from function F rquir procssing and adaptation. Th maximum valu of th rfrnc currnt is obtaind from th PI block. Th block (currnt control) is a hystrsis currnt controllr. V dc I a,b,c * r Spd control (PI Rgulator) r I * max currnt control Instant Commutation calcul Invrtr BLDC MOTOR I * abc Gnrator d F ( i a, b, c V ) dt Fig. 5a Block diagram of BDCM control with snsorlss driv. 81

8 A. Boughaba, M. Chaaban, S. Bnaggoun +V dc K 1 K 3 K K 4 K 5 K 6 Ra La a R b L b ~ b ~ R c L c c ~ V a V b V c Procssing and Adaptation F d ( Vi a, b, c dt ) (b) Fig. 5b Dscription of invrtr block with BDCM. 4 BLDC Motor Starting Mod Th procdur for starting a snsorlss driv for th BDCM is so difficult, bcaus th rotor has a prmannt magnt and its position is unknown. In th rfrncs w can found svral possibl procdurs for starting a BDCM [7]: 1 auxiliary snsor; opn loop control; 3 spcific gat pattrn; 4 arbitrary starting; 5 salint-pol motor; Th arbitrary start-up is th mthod chosn for this simulation, but in practic this mthod cannot b achivd bcaus, in som cass, it may b accompanid by tmporary rvrsd rotation. 5 BLDC Motor Paramtrs (Bosch SE-B.4.6) Paramtrs of th BDCM usd in th simulations ar all in ral units [6]: R = 1.43 Ω, L w = H, f =.158 Wb, J = kgm, f = 1 3 Ns/rad, p = pol pairs, K t =.41 Nm/A, Ratd currnt = 9.7 A, Ratd voltag = 38 V, Pak currnt = 58 A. 6 Snsitivity Study and Simulation Rsults As dscribd in th block diagram (Fig. 5), th spd rgulation provids th modul of th currnt phas. Th nw proposd snsorlss driv has bn succssfully simulatd on a BLDC Motor with th paramtrs as givn abov (Sction 5). 8

9 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... In this sction, th ffctivnss of th proposd mthod of snsorlss control for a BLDC motor is vrifid by computr simulation. Svral tsts wr conductd to chck th prformanc of th proposd mthod. In all th figurs th tim axis is scald in sconds. Th procdur for nrgizing th invrtr gats ar as follows in Tabl 1. Tabl 1 Commutation Squncs. FG I k Thn I k (k = a,b,c) If > If > > I * If < I * If < < I * If > < I * If < > I * I * 1 Va 1 Constant=1 Sign Vb 3 Vc du/dt first drivation du/dt scond drivation K Adaptation Gain u Abs Switch Product 1 output F Constant= Fig. 6 Block schm of procssing circuit. Fig. 6 illustrats th block diagram schm for th procssing signals ( d(sig Vi )dt ) which contains a rippl nar zro, whr V a, V b and V c ar th masurd trminal voltags and F givs th command signals from th scond drivativ function of SigV i. K is an attnuation cofficint (K < 1) of th scond drivativ function of SigV i. 6.1 Variation in spd and load torqu application Fig. 7 prsnts th spd rspons and lctrical rotor position for th variabl spd rfrnc. Th raction is quit, fast and of high accuracy. Morovr, in Fig. 8 th tracking prformancs wr improvd whn a load torqu is suddnly applid and rmovd. Howvr, th torqu of th BLDC motor contains a rippl. 83

10 A. Boughaba, M. Chaaban, S. Bnaggoun Ths proprtis mak this nw mthod suitabl for applications in which accuracy and spd ar rquird. 6. Tst undr constant spd Fig. 9 shows th spd rspons, rotor position, sum of thr trminal voltags (SigV i ), back EMF phas currnts, and th crossing zro signal (corrsponding to signals ncssary to nrgis th gats). Th squnc from th back EMF zro crossing commutation is clarly dmonstratd. As w can s, th commutation will occur approximatly 3 lctric dgrs aftr th zro crossing of th back EMF. 6.3 Variation in stator rsistanc Th ffct of changs in th stator rsistanc on th nw mthod is xamind. Fig. 1 dscribs th prformanc of th nw mthod for a wid variation of stator rsistancs, and illustrats th rsults of th simulation with a rfrnc spd qual to 1 rad/s, load torqu qual to Nm and two valus for th stator rsistanc (R = R n and R = R n + 15% R n ). This variation in rsistanc will not significantly affct th prformanc of this nw mthod. From this figur w dtrmin a dlay tim of 4 s. Fig. 9 dmonstrats th accuracy of th commutation timing. This mthod of calculation of commutation instants (zro crossing) has a vry good rsolution, particularly at low spd whn th back EMF amplitud is vry low Spd, Rfrnc Spd Wr Wrf Wr, Wrrf [Rad/s] tim(s) Fig. 7a Rspons obtaind with variabl rfrnc spd: Spd, Rfrnc Spd. 84

11 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... X:.3614 Y: 6.83 Elctrical Rotor Position 6 X:.484 Y: tta [Rad] tim(s) Fig. 7b Rspons obtaind with variabl rfrnc spd: Elctrical rotor position élctromagntiqu Torqu - Rfrnc Torqu T T_Rf 5 T-Tr(N.m) tim (s) Fig. 8 Rspons undr load torqu charg. 85

12 A. Boughaba, M. Chaaban, S. Bnaggoun Spd, Spd Rf Wr, Wrrf (Rad/s) tim (s) Fig. 9a Rspons of driv undr constant spd (with zoom): W r, W rf. Elctrical Rotor Position 6 X:.3614 Y: 6.83 X:.6777 Y: tta(rad) tim(s) Fig. 9b Rspons of driv undr constant spd (with zoom): tta. 86

13 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... Iabc(A) ; tta (Rad/s) ; FG Phas currnts (iabc),elctrical Rotor Position(tta) and Firing Gats(FG) ia ib ic tta 1*FG tim (S) Fig. 9c Rspons of driv undr constant spd (with zoom): i abc, tta, FG. EMF(a), Phas currnt(ia), Trminal Voltags Sum (SigVi),Firing-Gat(FG)and Tta 1 a 3*ia 55*FG SumVoltags 3*Tta Ea,SigVi(V),ia(A), Tta( rad)fg tim (s) Fig. 9d Rspons of driv undr constant spd (with zoom): a, i a, tta, SigV i, FG. 87

14 A. Boughaba, M. Chaaban, S. Bnaggoun Phas currnts iabc ia ib ic Iabc (A) tim (s) Fig. 9 Rspons of driv undr constant spd (with zoom): i abc. 1 1 Spd, Spd Rf Wr, Wrrf (Rad/s) Wr Wrf Wr, Wrrf (Rad/s) (ZOOM) R R+15%R tim(s) x 1-3 Fig. 1a Rspons undr stator rsistanc chang (R = R n, R = R n + 1.5R n ): W r, W rf. 88

15 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... Ea,SigVi(V),ia(A), Tta( rad)fg Ea,SigVi(V), Tta(Rad),FG EMF(a), Phas currnt(ia), Trminal Voltags Sum (SigVi),Firing-Gat(FG)and Tta Ea 4 R+15%R R Ia tim (s) a 3*ia 55*FG SumVoltags 3*Tta Fig. 1b Rspons undr stator rsistanc chang (R = R n, R = R n + 1.5R n ): a, ia, tta, SigV i, FG (with zoom). Iabc(A) ; tta (Rad/s) ; FG 3 1 Phas currnts (iabc),elctrical Rotor Position(tta) and Firing Gats(FG) ia ib ic tta 1*FG Iabc(A) ;tta (Rad/s) ;FG R R+15%R tim (S) Fig. 1c Rspons undr stator rsistanc chang (R = R n, R = R n + 1.5R n ): iabc, tta, FG. 89

16 A. Boughaba, M. Chaaban, S. Bnaggoun 8 Conclusion A novl robust snsorlss driv for a BLDC motor by th calculation of commutation instants using th scond drivativ function of th sum of trminal voltags has bn prsntd. This is a diffrnt way to us th information containd in th trminal voltags. Th tst rsults, by simulation, vrify th analysis and dmonstrat th advantags of this nw mthod. Ths rsults show clarly that th tru zro crossing back EMF is xactly containd in th sum of trminal voltags. It was shown that th nw mthod posssss good robustnss against paramtr variation, in this cas stator rsistanc. SigV i has a fw discrt valus, and w not that SigV i rprsnts th sum of th trminal voltags avragd ovr ach PWM cycl, and that th masurd voltags in ral application should b filtrd. Nvrthlss, as with most snsorlss mthods, th motor start up procdur still has to b intgratd. 9 Rfrncs [1] T. Kim, M. Ehsani: Snsorlss Control of th BLDC Motors From Nar-zro to High Spds, IEEE Transactions on Powr Elctronics, Vol. 19, No. 6, Nov. 4, pp [] C.L.P. Swamy, B. Singh, B.P. Singh: Dynamic Prformanc of a Prmannt Magnt Brushlss DC Motor Powrd by PV Array For Watr Pumping, Solar Enrgy Matrials and Solar Clls, Vol. 36, No., Fb. 1995, pp [3] N. Matsui: Snsorlss PM Brushlss DC Motor Drivs, IEEE Transactions on Industrial Elctronics, Vol. 43, No., April 1996, pp [4] B. Trzic, M. Jadric: Dsign and Implmntation of th Extndd Kalman Filtr for th Spd Rotor Position Estimation of Brushlss DC Motor, IEEE Transactions on Industrial Elctronics, Vol. 48, No. 6, Dc, 1, pp [5] G.H. Jang, J.H. Park, J.H. Chang: Position Dtction and Start-up Algorithm of a Rotor in a Snsorlss DLDC Motor using Inductanc Variation, IEE Procdings -Elctric Powr Applications, Vol. 149, No., March, pp [6] M.T. Wishart, R.G. Harly, G. Diana: Th Application of Fild Orintd Control to th Brushlss DC Machin, 4th Europan Confrnc on Powr Elctronics and Applications, Firnz, Italy, 3 6 Spt. 1991, Vol. 3, pp [7] J.X. Shn, Z.Q. Zhu, D. How: Snsorlss Flux-wakning Control of Prmannt Magnt Brushlss Machins using Third Harmonic Back EMF, IEEE Transactions on Industry Applications, Vol. 4, No. 6, Nov/Dc. 4, pp [8] L.W. Dixon, I.A. Lal: Currnt Control Stratgy for Brushlss DC Motors Basd on a Common DC Signal, IEEE Transactions on Powr Elctronics, Vol. 17, No., March, pp [9] K.Y. Chng, Y.Y. Tzou: Dsign of a Snsorlss Commutation IC for BLDC Motors, IEEE Transactions on Powr Elctronics, Vol. 18, No. 6, Nov. 3, pp [1] J. Figuroa, C. Brocart, J. Cros, P. Viaroug: Simplifid Simulation Mthods for Polyphas Brushlss DC Motors, Mathmatics and Computrs in Simulation, Vol. 63, No. 3-5, Nov. 3, pp

17 Robust Snsorlss Control of BLDC Motor Using Scond Drivativ Function... [11] P.M. Plczwski, U.H. Kunz: Th Optimal Control of a Constraind Driv Systm with Brushlss DC Motor, IEEE Transactions on Industrial Elctronics, Vol. 37, No. 5, Oct, 199, pp [1] M. Tursini, R. Ptrlla, F. Parasiliti: Initial Rotor Position Estimation Mthod for PM Motors, IEEE Transactions on Industry Applications, Vol. 39, No. 6, Nov/Dc. 3, pp [13] Z. Zhng, Y. Li, M. Fadl: Snsorlss Control of PMSM basd on Extndd Kalman Filtr, 1th Europan Confrnc on Powr Elctronics and Applications, Aalborg, Dnmark, 5 Spt. 7. [14] B.K. L, M. Ehsani: Advancd BLDC Motor Driv for Low Cost and High Prformanc Propulsion Systm in Elctric and Hybrid Vhicls, 4th IEEE Intrnational Elctric Machins and Drivs Confrnc, Cambridg, MA, USA, 17 Jun 1, pp [15] I. Colak, M. Sahin: Snsorlss Control of a Brushlss DC Motor using a Slf-tuning PID, Intrnational Symposium on Powr Elctronics, Elctrical Drivs, Automation and Motion, Sorrnto, Italy, Jun 1, pp [16] T.L. Chm, P.L. Pan, Y.L. Chm, D.M. Tsay: Snsorlss Spd Control of BLDC Motor using Six Stp Squar Wav and Rotor Position Dtction, 5th IEEE Confrnc on Industrial Elctronics and Applications, Taichung, Taiwan, Jun 1, pp [17] D. Makila: Snsorlss Control of High-spd PM BLDC Motor, Intrnational Symposium on Industrial Elctronics, Gdansk, Poland, 7 3 Jun 11, pp