Efficiency Maximization of the Air Core Double-Sided Permanent Magnet Linear Synchronous Motor

Transcription

1 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 Efficiency Maximization of the Air Core Double-Sided Permanent Magnet Linear Synchronou Motor A Darabi 1 R Ghanaee 1 A Shariati 1 M Siahi 1 1 Electrical Engineering Department Azad Univerity Garmar Branch Garmar Iran darabi_ahmad@hotmailcom Abtract Air core permanent magnet linear ynchronou motor (ACPMLSM recommended for application in which the accurate control of peed and poition i required Omiion of the core in the primary of the motor reduce the detent force and increae the controllability of the motor In thi paper the flux denity of different part of an ACPMLSM i calculated uing both the Maxwell equation and finite element method ( A precie flux denity model i preented and the deign of the air core double ided permanent magnet linear ynchronou motor i optimized for the efficiency maximization uing the genetic algorithm method Index Term Air core permanent magnet linear ynchronou motor flux denity efficiency genetic algorithm finite element method I INTRODUCTION The permanent magnet linear ynchronou motor (PMLSM have taken more attention than linear induction motor becaue of high force denity and efficiency low loe atifactory dynamic performance and eay control [1] [6] However the detent force generated by the lotted tructure and the end effect caued by the limited length of the moving part are the main diadvantage of lotted linear motor Some technique uch a kewing the lot and optimizing the permanent magnet (PM and the width of the primary winding have been already uggeted to reduce the detent force [5] [7] Poition and peed control of a PMLSM can be improved by mitigation or alleviation of the detent force Since uing the air-core intead of the iron core in a permanent magnet linear ynchronou motor coniderably reduce the detent force; the air core machine i uually recommended for high precie application Some invetigation are carried out on the optimal deign of the PMLSM It ha been hown that ome increae in the thrut can be achieved by modifying the hape or dimenion of the permanent magnet and width of the winding [7] [8] The ripple reduction ha been the other concern of earlier tudie [7] Author in [9] have been optimized the motor deign in order to reduce the electrical time contant In [1] increaing the motor thrut and reducing the magnet Manucript received June 15 1; accepted February conumption have been the goal for the optimization In [11] the output power maximization or lo minimization ha been done for a tubular permanent magnet linear ynchronou motor In general the efficiency of a mall PMLSM employed in a control ytem i not conidered a the main objective and o the mot effort a mentioned earlier have been conducted to improve the tranient repone of the machine However the efficiency i one of the mot important criteria when employing a motor for the energy converion purpoe Due to the increaingly development of the linear motor in the indutry the efficiency optimization need more attention In thi paper efficiency optimization of a four pole ACPMLSM i invetigated The paper i organized a the following: in ection the baic tructure of the ACPMLSM i preented and the procedure to evaluate the air gap flux denity ditribution are dicued The flux denity ditribution are obtained by the ue of both Maxwell equation a an analytical method and the finite element method a a numerical method Finally a comparion between the reult i given in ection Section 3 involve the deign formulation and modelling tak of the ACPMLSM defining the exiting relationhip among ome performance characteritic and the deign parameter The optimization algorithm and the reult are preented in ection 4 Finally the concluion and dicuion are preented in ection 5 II FLUX DENSITY DISTRIBUTION OF ACPMLSM A Motor tructure The tructure of an ACPMLSM i hown in Fig 1 in which the moving part of motor i a hort primary and conit of a three phae air core winding g h c Fig 1 The tructure of ACPMLSM τ h m τ p g h b 11

2 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 The econdary part involve the N and S permanent magnet located on the urface of the back iron The primary and the econdary part are eparated by an air gap g The parameter and the dimenion of the motor are preented in Table I Suppoing that the primary winding i not excited an ACPMLSM a hown in Fig contain two layer of iron extended along x axi two layer of PM and a layer of air gap The following aumption are made for the magnetic field calculation [3] [4]: 1 All the region (hown in Fig are extended along ± x ; The end effect along z axi are neglected; 3 The permeability of the PM i conidered equal to vacuum permeability µ ; 4 The permeability of the econdary yoke i infinite g Fig Analyi model of an ACPMLSM τ p µ = h m µ = TABLE I PARAMETERS OF THE STUDIED MOTOR Parameter Symbol Value Number of turn N 1 58 Height of the econdary yoke h b 1 mm Width of the motor L 1 mm Pole pitch τ 465 mm Reidual flux denity Br 1 Tela Width of the PM τ p 4 mm Height of the PM hm 5 mm Height of the winding hc 1 mm Air gap length g 5 mm Width of the winding w c 155 mm Relative permeability of PM µ r 15 B Flux denity ditribution Conidering Coulomb criteria A = Maxwell equation to calculate the magnetic potential of the PM can be expreed by Laplace and Poion partial differential equation a below [4]: A1 A1 + = x y A A + = µ J x y where A i the magnetic vector potential Flux denity B i obtained by uing the vector potential a B (1 = A ( The ubcript determine the region and J i the equivalent current denity defining the whole permanent magnet effect that can be expreed a the Fourier erie [4] with coefficient J = αn in( nkx (3 n= 13 4Br 1 αn = in nkτ p τµ (4 where k =π τ i the patial frequency Conidering Maxwell equation ( H = J the boundary condition that mut be atified at the border urface of two material i a below [4] nˆ ( H1 H = (5 where nˆ i a unit vector normal to the boundary urface directed from region to region 1 and H i the magnetic field intenity Equation (5 can be expreed a below [4]: H1x = y = g H1x = Hx and B1 y = B y y = hm Hx = y = Given the boundary condition (6 and the excitation J the X-Y component of the flux denity can be expreed a below [4]: ( nkx ( nkhm ( nkg α inh n B1 x = µ inh( nk ( g y n= 13 ( nk inh in ( nkx ( nkhm ( nkg α inh n B1 y = µ coh( nk( g y n= 13 ( nk inh co In the ret of thi ection the flux denity of the primary winding i calculated uing the method in [1] In thi method at the firt tep the magnetic potential of the primary winding A i calculated baed on the magneto motive force (MMF of the traveling magnetic field At the next tep the ditribution of the flux denity B i calculated uing the magnetic potential According to the method in [1] the amplitude of the ν th harmonic of MMF produced by the primary winding i equal to (6 (7 (8 6 1 ν = 1 a wν π pkc ν (9 f N I k where I a i the RMS value of the phae current p i the th number of pole pair k wν i the winding factor for the ν harmonic and K c i the Carter' coefficient According to Laplace equation and the boundary condition the component of the flux denity generated by the primary winding in the middle of the air gap can be expreed a follow: 1

3 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 A1 = y = hm + g A1 = fv y = g A1 B1x = µ = x νπ inh ( h + g y = µ ν (1 m π f τ π ν co x (11 ν = 1ντ νπ τ inh ( hm + g A1 B1y = µ = y τ νπ coh ( hm g y π + τ π = µ f ν in ν x ν = 1ντ νπ τ (1 inh ( hm + g τ C Analytical and reult of flux denity In thi ection the reult of the flux denity obtained by the analytic method ie Maxwell equation i compared to thoe evaluated by the X and Y component of the air gap flux denity ditribution are preented a a function of diplacement in Fig 3 and Fig 4 when the motor i open circuited Y component of the flux denity in the middle of the magnetic air gap i illutrated in Fig 5 method and are nearly imilar In the ACPMLSM the effective length of the magnetic air gap i large Therefore a hown in Fig 4 and 5 the ditribution of the Y-axi component of the air gap flux denity in the middle of the mechanical and magnetic air gap are nearly rectangular and inuoidal repectively The flux line of the motor are hown in Fig 6 Auming balanced three-phae current where I a = co( ωt current for the time ω t = will be I a = I b = I c = 1 Magnetic flux denity(t Analythical Method Diplacement(mm Fig 5 Y component of the flux denity in the middle of the magnetic air gap 54 5 Fig 6 Flux line at the center of the motor 8 6 Analytical Prediction Fig 3 X component of the flux denity in the middle of the air gap Magnetic flux denity(t Analytical Method Diplacement(mm Fig 4 Y component of the flux denity in the middle of the air gap Nonlinearity and aturation of the iron core are conidered by analyi uing the correponding B-H curve of the ferromagnetic material In addition tructural complexity uch a lotting can be imply imulated by A depicted in Fig 3 and 4 the reult obtained by the analytic B1y produced by pole (T Diplacement (mm Fig 7 The firt harmonic of vertical component of the air gap flux denity produced by the PM uing analytical method and B1x produced by pole (T Analytical Prediction Diplacement(mm Fig 8 The firt harmonic of horizontal component of the air gap flux denity produced by the PM uing analytical method and Fig 7 to Fig 9 illutrate a few ample of the firt 13

4 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 harmonic of the vertical and horizontal air gap flux denitie generated individually by the PM and primary current which are evaluated uing Maxwell equation and A een in the figure the air gap flux denity i mainly produced by the PM Alo a comparion between the firt harmonic of the flux denitie calculated by the analytical model and evaluated by reveal the validity of the analytical modeling approach of the flux denitie B1x B1y for wt= (T B1y by Analytical Prediction B1y by B1x by Analytical Prediction B1x by Diplacement (mm Fig 9 The vertical and the horizontal component of the air gap flux denity generated by the primary current at ωt= uing analytical method and III BASIC EQUATIONS OF SYNCHRONOUS MOTOR The phaor diagram of a ynchronou motor i hown in Fig 1 The following equation are deduced from the vector diagram of Fig 1 [1]: 1 in δ = d 1 + q q (13 V i R i X 1 co δ = q 1 + d d + f (14 V i R i X E where δ i the load angle which i the angle between the terminal phae voltage V 1 and the no-load voltage E f Parameter X d and X q are the d and q axi reactance repectively and R 1 i the reitance per phae of the primary winding q V1 ( X q coδ R1 in δ E f X q id = X d X q + R1 (15 V1 ( R1 coδ + X d in δ E f R1 iq = (16 X d X q + R1 The rm value of the primary current can be evaluated veru V 1 E f X d X q δ and R 1 uing A Thrut calculation = d + q (17 I i i Enduring d-q model of the machine in a ynchronouly rotating reference frame i ued for the optimal deign purpoe In thi model the iron aturation i neglected and the flux denity ditribution along the air gap aumed to be inuoidal Therefore the motor thrut can be calculated by [1] in which PM and ( ( 3 π Fav = λ PM + Ld Lq id iq (18 τ λ PM i the linkage flux per phae caued by the L d and Lq are the d axi and q axi inductance repectively For a PM motor made by the rare earth material the relative permeability i omewhat equal to 1 and thu the material i far to be aturated o that L L A mall difference between L d and L q yield to a mall component of the reluctance force which i neglected here Therefore the mean value of the thrut i imply written a B Loe calculation 3 π Fav = λ PM iq (19 τ For an ACPMLSM the reitance of the primary winding i coniderable compared with the leakage reactance The copper lo of an ACPMLSM i obtained by d q R 1 i d V 1 inδ Pcu 3 R1 I = ( I R 1 V1 i q R 1 Iron lo conit of the hyterei lo and eddy-current lo It can be etimated uing [13] ji d X d ji d Xad ji q X 1 ji q X aq δ E f φ ψ i d X d V 1 co δ I i q β 3 Pt = Ph + Pe = kh Bm f + ke Bm f [ W m ] (1 where B m i the maximum flux denity k h ke and β are contant Contant parameter β ha a value from 18 to depending on the laminated material Variation of the flux denity in the yoke can be approximated by Fig 11 B B ymax Fig 1 The vector diagram of the ynchronou motor According to (13 and (14 the d and q current are a below [1]: i d d τ p τ τ p v v Fig 11 Flux denity variation in the yoke [14] t 14

5 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 At the firt tage of deigning an initial etimated value i choen for the maximum flux denity of the yoke The eddycurrent lo denity of the yoke i expreed a [14] where τ v 3 P = 8 k B [ W m ] ey e ymax τ p τ v i the linear peed and ( B y max i the maximum value of the yoke flux denity a hown in Fig 11 In ( the longitudinal component of the eddy-current lo ha been ignored Thi component i imply taken into account by a contant k c with the value choen within 11 to 1 [14] The eddy-current lo of the motor i then written a [14] τ v P k k B V W e = 8 e c ( ymax y ( τ p τ (3 C Efficiency calculation Given the motor loe and the output power the efficiency i calculated a follow Pout η = Pout + Pcu + Padd + Pt (4 in which P out = Fav v i the output power and P add i the additional loe coniting the mechanical and tray loe IV OPTIMIZATION The optimization problem with n number of variable m number of contraint and the objective function f ( x i defined a [15] Maximize f ( x x K (5 where the parameter K i defined a n { i ( } K = x R : g x i = 1 P (6 In the previou equation g i ( x determine the limit of the deign variable The deign parameter are optimized conidering the objective function defined uch a thrut per volume or ma maximization volume of the PM minimization and power per volume or ma maximization In thi paper the deign parameter are optimized for the efficiency maximization uing the genetic algorithm For the purpoe of maximizing the efficiency of the motor four deign variable are optimized Thee four variable are height of the PM air-gap height of the primary winding and height of the econdary yoke The width of the motor current denity and pole pitch are aumed contant pecified properly A Contraint Thermal tre and demagnetizing boundary of the PM are conidered a contraint In addition to prevent aturation of the yoke the minimum value of it dimenion are conidered The thermal tre which depend on the total loe of the motor i given a [16] T = loe T hs TH max (7 where h S TH and Tmax are the heat exchange coefficient of the urface heat exchange or cooling urface and maximum rie of the operating temperature repectively Maximum flux denity produced by the primary current B ha to be maller than the value that demagnetize the PM B i given by [16] 3 4µ kw1n1 B = I (8 gπ To avoid demagnetizing the PM primary current mut be limited o that [16] hm B in α Br B D ge (9 where B D i the critical or minimum allowable magnetic flux denity of the PM The value of B D i about - tela for the Nd-Fe-B magnet A minimum value for the height of the econdary yoke can be obtained by uing the following equation B Simulation reult of optimization h B 1g ymin = τ (3 Bymax A mentioned earlier efficiency maximization i the objective of the optimization problem of thi paper The rated power P out and linear peed ν of the motor are aumed 1 W and 465 m/ at 5 Hz upply frequency repectively The additional lo i aumed 4% of the output power The deign variable and their variation range are lited in Table II The temperature limit of the motor i aumed 1 C and the heat exchange coefficient of the urface i aumed h=46 (W/m C The maximum flux denity of the yoke i B 1 4 tela y max = Optimized deign variable obtained by the genetic algorithm are given in Table III The reult how that the efficiency of the optimized motor ha been increaed by 43% in comparion with the initial deign Alo depite the copper lo ha reduced by 597% the iron lo ha increaed by 14% However the influence of the copper lo on the efficiency i more than that of the iron lo o the efficiency of the optimized machine i higher Harmonic content of the flux denity produced by the PM are evaluated uing Thee Harmonic for the initial deign and optimized machine are hown in Fig 1 The amplitude of the firt harmonic of the flux denity before and after the optimization i 6 and 78 Tela repectively howing an increae of the flux denity for the optimized machine However a expected an increae of the 15

6 ELEKTRONIKA IR ELEKTROTECHNIKA ISSN VOL 19 NO 7 13 flux denity yield to a higher iron lo TABLE II DESIGN VARIABLES AND THEIR VARIATION RANGES Parameter (unit Min Max Air gap length (mm 5 1 Height of the PM (mm 4 7 Height of the econdary yoke (mm 5 3 Height of the winding (mm 5 15 TABLE III PARAMETERS OF THE OPTIMIZED MACHINE Parameter (unit Initial deign GA output Air gap length (mm 5 51 Height of the PM (mm 5 7 Height of the econdary yoke (mm 1 6 Height of the winding (mm 1 1 Cupper lo (W Iron lo (W Efficiency Fig 1 Harmonic content of y component flux denity produced by the PM before and after optimization V CONCLUSIONS A modeling and optimally deign procedure of the aircore double-ided permanent magnet linear ynchronou motor are preented in thi paper Flux denity MMF and other parameter of motor are calculated uing the Maxwell equation baed analytical model The reult obtained from the analytical model are compared with thoe obtained from the Thi comparion confirm the accuracy of the preented model Therefore the analytical model propoed in thi paper can be applied for analyi deign and optimization of the ACPMLSM with ome confidence At the lat ection of the paper a maximum efficiency of ACPMLSM i optimally deigned uing the propoed model coupled with genetic algorithm baed optimization approach The imulation reult of the optimally deigned motor indicate that the iron lo increae by more than 14% due to the increaing of the firt harmonic of the flux denity from 6 to 78 tela Although the iron lo increae the copper lo reduce up to 6% Since the copper lo ha more effect on the efficiency the overall efficiency of the optimally deigned motor i increaed magnetizing current IEEE Tran on Magnetic vol 37 pp Sept 1 [4] N Chayopitak D G Taylor Performance aement of air core linear permanent magnet ynchronou motor IEEE Tran on Magnetic vol 44 pp Oct 8 [Online] Available: [5] K C Lim J K Woo G H Kang J P Hong G T Kim Detent force minimization technique in permanent linear ynchronou motor IEEE Tran on Magnetic vol 38 pp Mar [Online] Available: [6] M Inoue H Sato An approach to a uitable tator length for minimization the detent force of permanent magnet linear ynchronou motor IEEE Tran on Magnetic vol 36 no 4 pp July [Online] Available: / [7] A Haanpour Ifahani S Vaez Zadeh Deign optimization of a linear permanent magnet ynchronou motor for extra low force pulation Journal of Energy Converion and Management vol 48 pp [Online] Available: jenconman66 [8] G H Kang J P Hong G T Kim A novel deign of an air core type permanent magnet bruhle motor by pace harmonic field analyi IEEE Tran on Magnetic vol 37 pp Sept 1 [Online] Available: [9] D H Im J P Hong I S Jung S B Yoon The optimum deign of permanent magnet linear ynchronou motor in Proc of IEEE CEFC 96 pp [1] S Vaez Zadeh A Haanpour Ifahani Multi objective deign optimization of air core linear permanent magnet ynchronou motor for improved thrut and low magnet conumption IEEE Tran on Magnetic vol 4 pp Mar 6 [Online] Available: [11] J Wang M Wet D Howe H Zelaya W M Arhad Deign and experimental verification of a linear permanent magnet generator for a free piton energy convertor IEEE Tran on Energy converion vol pp June 7 [1] Z Q Zhu D Howe Intantaneou magnetic field ditribution in bruhle permanent magnet dc motor part III: effect of tator lotting IEEE Tran on Magnetic vol 9 pp Jan 1993 [Online] Available: [13] C Mi G R Slemon R Bonert Modeling of iron loe of permanent magnet ynchronou motor IEEE Tran Ind App vol 39 pp May/Jun 3 [Online] Available: [14] B Sheikh Ghalavand S Vaez Zadeh A Haanpour Ifahani An improved magnetic equivalent circuit model for iron core linear permanent magnet ynchronou motor IEEE Tran on Magnetic vol 46 Jan 1 [15] N Bianchi S Bolognani Analytical optimization of inet permanent magnet machine baed on a genetic algorithm Electrical Machine and Power Electronic pp Sept 7 [16] T Sebatian G Slemon Tranient modeling and performance of variable peed permanent magnet motor IEEE Tran Ind App vol 5 pp Jan/Feb 1989 [Online] Available: REFERENCES [1] J G Giera Z J Pich Linear Synchronou Motor Boca Raton FL: CRC [] I Boldea S A Naar Linear Electromagnetic Device New York Taylor & Franci 1 [3] G H Kang J P Hong G T Kim Deign and analyi of lotle type permanent magnet linear bruhle motor by uing equivalent 16