Control of a Linear Permanent Magnet Synchronous Motor using Multiple Reference Frame Theory

Transcription

1 Cotrol of a Liear Permaet Maget Sychroou Motor uig Multiple Referece Frame Theory Jawad Faiz ad B. RezaeiAlam Departmet of Electrical ad Computer Egieerig Faculty of Egieerig Uiverity of Tehra, Tehra, Ira. faizj@ofe.ece.ut.ac.ir Key word: Feedig Curret Miimizatio, Liear Motor, Multiple Referece Frame. Abtract: Thi paper ue a multiple referece frame theory (MRF) i order to etimate ad cotrol the curret harmoic of a liear ychroou motor with permaet maget tator tructure (PMLSM). Thi i doe i uch a way that the rm upply curret i miimized, therefore the loe will be lower. Alo relatiohip betwee the tractio force ad curret of the motor i tudied i dq axe. The uggeted techique i robut agait timevariable parameter uch a iductace variatio. Fially imulatio reult of the uggeted tructure are preeted. INTRODUCTION Liear motor are coveiet for rapid traport with o acoutic oie ad high performace. The major advatage of uig the liear motor i traferrig the applied force o the movig object without a mechaical iterface. Vector cotrol techique i atifactorily applied to rotatig motor ad there are alo report cocerig the vector cotrol applicatio i liear motor []. Geerally i uch applicatio it i aumed that the electrical quatitie vary iuoidally, while the edeffect ad iductace variatio are oiuoidal. Therefore dq model of liear motor i ychroou flux caot be ued. Meawhile it i difficult to compeate the logitudial ed effect alog the vehicle, while the flux produced by the primary curret repeat periodically ad it caot compeate the flux reductio at the begiig of the liear motor. Thee are the reao that it i difficult to ivetigate the harmoic caued by the logitudial ed effect ad oiuoidal iductace. Alo there are higher loe i liear motor becaue of logitudial ad cro ed effect, large leakage flux ad aturatio of the teeth. I [2] multiple referece frame (MRF) theory ha bee uggeted for etimatio ad cotrol of the curret harmoic i the bruhle direct curret motor (BLDCM) ad good reult have bee obtaied. Thi ca reduce the ripple of the developed torque ad improve the efficiecy of the motor. Thi method doe ot require Fourier erie coefficiet of the curret wave. I [3] the MRF theory ha bee recommeded for modelig a PMLSM with oiuoidal lietolie iductace. Harmoic of the curret i obtaied uig the erie expaio ad every harmoic trafer to the relevat ychroou referece frame ad eparate cotrol i applied. The preet paper ue the model give i [3] ad a cotrol tructure i uggeted for a PMLSM with oiuoidal backemf waveform. A miimum upply curret i achieved by ijectig proper curret harmoic which reduce the loe. The recommeded techique doe ot require Fourier erie coefficiet of the curret waveform ad it i robut agait timevariable parameter uch a iductace chage uder aturatio. Fially the imulatio reult are preeted for the propoed tructure.

2 2 MODELING OF A PMLSM Fig. how the propoed PMLSM with pecificatio give i Table. The ytem voltage equatio of the motor circuit are a follow: Where V abc =R i abc.d/dt(λ Λ m ) R=diag[r r r] i abc =[i a i b, i c ] T, V abc =[V a V b V c ] T Λ m =λ m k q [i(θ) i((θφ)) i((θφ))] T =,3,5 Λ=LI, L=[L L 2 L 3 L 2 L 22 L 23 L 3 L 32 L 33 ] L =L N L B co(2(θφ)) L C co(4(θφ)) L 22 =L N L B co(2θ) L C co(4θ) L 33 =L N L B co(2(θφ)) L C co(4(θφ)) L 2 =L 2 =L v L B co(2(θφ)) L C co(4(θφ)) L 3 =L 3 =L v L B co(2θ) L C co(4θ) L 23 =L 32 =L v L B co(2(θφ)) L C co(4(θφ)) Θ=2πx/3, φ=2π/3, L N =L l L A, L v =L A /2 TABLE. SPECIFICATIONS OF THE PROPOSED MOTOR Quatity Number of phae Number of pole, P Widig reitace, R Iductace, L A Leakage iductace, L l Poitive equece iductace, L B Negative equece iductace, L C Ma, m Frictio coefficiet, B v t Fourier coefficiet, K q 3 d Fourier coefficiet, K q3 5 th Fourier coefficiet, K q5 λ m Value Ω 2.2 mh.2 mh.9 mh.5 mh.2 kg V/rad widig maget Fig.. Propoed liear PM LSM with a hort primary the electric force equatio i a liear motor ca be obtaied a follow [4]: F=.5P{.5(i abc ) T / x[ll l I] i abc ( i abc ) T / x[λ m ]} 2 The mechaical equatio are: F= mab v V x F L = mdv x /dtb v V x F L 3 Now the Park traformatio ca be applied ad depedecy of the iductace o the poitio i elimiated. By defiig matrix K a follow: K = 2/3 [ co((θπ/3)) co((θπ)) co((θπ/3)) Si((θπ/3)) i((θπ)) i((θπ/3)) / 2 / 2 / 2 ] 4

3 th harmoic of the curret ca be traferred to the correpodig ychroou flux axe ad obtai i d ad i q. Aumig i d =, force equatio for ay harmoic will be a follow: Ad the total average force i: F =(π/τ)(λ m K q i q ) 5 F= F 6 3 MINIMIZATION OF SUPPLY CURRENT Now a curret vector i obtaied where it rm value i miimal over differet mode of operatio. Miimizatio of the rm curret lead to the reductio of the iverter ad primary widig loe. It i eceary to defie two curret ad fluxlikage coefficiet a follow: I q =[ i q i q2 i q ] K=[ K q K q2 K q ] 7 Baed o Eq. 56, the followig equatio ca be obtaied: F=( π/τ)λ m KI q T 8 The i rm mut be miimized ad the optimal curret vector ca be obtaied uig the orm miimum olutio [5,6] a follow: I Mc =K T F/( π/τ)kk T 9 4 CONTROL STRUCTURE The tructure of the drive ha bee how i Fig. 2. Compoet of curret i Mc i calculated uig Eq. 9 ad coiderig the mea value of the force. I thi ectio, Fudametal ad harmoic compoet of the curret are etimated uig MRF theory ad they are cotrolled i uch a way that they lead to the deirable value [2]. The multiple referece frame etimator (MRFE) block ha bee repreeted i Fig. 3. I thi block, the threephae curret are ampled ad compoet of i dq are etimated. Firt, i each brach, i dq compoet of the relevat harmoic are determied by the Park traformatio ad the i dq compoet coverge the actual value by the help of a itegral loop with high gai. The MRFR block i Fig. 2 regulate compoet of i dq curret i uch that i dq follow i dq *. Thi block ha bee how i Fig. 4. The itegral of the curret error i ay referece frame are traferred to the threeaxe frame. Their ummatio give the deirable primary widig curret i abc *. Thi i applied to the voltage ource iverter (VSI) with curret upply. Stability of the abovemetioed techique ha bee proved i [2].

4 Speed Referece V* Speed Cotroller F* Curret Harmoic Geeratio i*dq MRFR ia* ib* ic* Hyteri Curret Cotroller Tabc Iverter idq V Bv frictio compeator / X (poitio) MRFE ia ib V (peed) PMLSM Fig. 2. Block diagram of the drive i abc K Ge i dq K K o Ge i dq K Fig. 3. Multiple referece frame etimator (MRFE) i * dq i dq Gc o K i * abc i * dq i dq Gc K Fig. 4. Multiple referece frame regulator (MRFR)

5 5 SIMULATION RESULTS The uggeted algorithm ha bee applied to the propoed PMLSM. Takig ito accout the fudametal, 3 rd ad 5 th harmoic imulatio reult have bee obtaied. It i oted that the techique ca be applied to ay umber of harmoic. The flux waveform of the maget with Fourier coefficiet K q =, K q3 =.2 ad K q5 =.6 ha bee how i Fig. 5. The tep repoe of peed ad the actual phae curret i a, covergig the deirable i a *, have bee preeted i Fig. 6. The deirable value if i d i equal to zero, ad a how i Fig. 7, the actual value of i d approache zero. i q ha bee coverge it fial value. Fig. 8 idicate the ifluece of 3% reductio of iductace L A due to aturatio. Whe the Fourier erie i ued for the curret harmoic thi iductace chage icreae the torque ripple. However there i a light torque ripple whe MRF theory i ued. Alo thi iductace reductio ha o effect upo the upply curret miimizatio, becaue the iductace have ot bee ued i the cotrol ytem..6.4 Volt.Sec/Rad Rad Fig. 5. Waveform of maget flux Speed (Cm/Sec) (A) (a) Sec Fig. 6. Step repoe of the ytem peed (b) Sec

6 6 CONCLUSION A cotrol algorithm for a PMLSM ha bee recommeded which ca compeate the oliear characteritic of the motor by ijectig the coveiet harmoic of curret baed o MRF theory. Therefore, ome obtaclee i applicatio of vector cotrol i the liear motor are removed ad the loe are coiderably reduced by reductio of the rm curret. The imulatio reult cofirm the proper performace of the uggeted techique. 7 Referece. Gu B. G. ad Nam K., A vector cotrol for a PMLSM coiderig a ouiform flux ditributio, IEEE APEC, 2, pp Chapma P. L. ad Sudhoff S. D., A multiple referece frame ychroou etimator, IEEE Tra. O Eergy Coverio, Vol.5, No. 2, pp. 9722, Liu C. T. ad Hu S. C., Aalyi of liear electromagetic motio device by multiplereferece frame theory, IEEE Tra. o Magetic, Vol. 34, No. 4, pp , Kraue P. C., Aalyi of electric machiery, McGrawHill, Chapma P.L., Sudhaff S. D. ad Whitcomb C. A., Optimal curret cotrol trategy for urfacemouted permaetmaget ychroou machie drive, IEEE Tra. O Eergy Coverio, Vol. 4, pp. 435, Broga W. L., Moder Cotrol Theory, Pretice Hall, 99. (A) iq 5.2 id.2.. iq id3.2 iq5 id5... (A) Sec Fig. 7. Harmoic curret repoe

7 with fourier erie ad LA with MRF ad LA Fe ( N) with fourier erie ad.7*la. 5 with MRF ad.7*la Fe (N) Sec Fig. 8. Ifluece of 3% reductio of iductace L A