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[doi 10.1109_icelmach.2012.6349866] Prieto, D.; Daguse, B.; Dessante, P.; Vidal, P.; Vannier, J.-C. -- [IEEE 2012 XXth International Conference on Electrical Machines (ICEM) - Marseille, France (2012.09.2-20.pdf

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Φ ΦΦ Φ Abstract -- This paper presents the design of Synchronous Reluctance Motors (SynRM) with four flux-barriers. The study is focused on the use of ferrite magnets into flux- barriers and its impact on average torque, torque ripple and power factor. The analysis uses Finite Element Method (FEM) for different pole pair numbers in order to choose an efficient structure. Index Terms—Electric motors, Permanent magnet motors, Power factor, Torque ripple
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    ΦΦΦΦ  Abstract -- This paper presents the design of Synchronous Reluctance Motors (SynRM) with four flux-barriers. The study is focused on the use of ferrite magnets into flux-barriers and its impact on average torque, torque ripple and power factor. The analysis uses Finite Element Method (FEM) for different pole pair numbers in order to choose an efficient structure.  Index Terms —Electric motors, Permanent magnet motors, Power factor, Torque ripple, Synchronous reluctance machines. I. N OMENCLATURE   d q ii ,  : stator electromagnetic current magnitudes d q V V  ,  : stator electromagnetic voltage magnitudes d q  ΦΦ ,  : stator flux linkage d q  L L ,  : quadrature and direct axis inductances  p  : pole pair number α   : current angle II. I NTRODUCTION  ermanent magnet synchronous motors are known for their electromagnetic performances and their compact size. Rare earths like NdFeB are the kind of material usually used for permanent magnets, but now their cost is incremented and therefore the machine becomes more expensive. A usual solution is the replacement of the motor by a synchronous reluctance motor which is used in industrial applications and has a good torque-density [1]. The torque is produced by the saliency of rotor. Several investigators have compared performances of synchronous reluctance motors and induction motors [1]-[4]. Common drawbacks of such motors are the low power factor and the high torque ripple [5]. In [6] a formula is deduced for calculating the power factor and its behavior versus the saliency ratio (  L q  /   L d  ). To improve performances of synchronous reluctance motor, we must design a rotor with a large inductance difference (  L q -L d  ) and a large saliency ratio (  L q  /   L d  ) [7]. Fig. 1(a) shows a transversally laminated rotor, it is also called multiple-flux barrier rotor. The interest of that rotor Dany Prieto is with the Ecole Supérieure d’Electricité, F-91192 Gif sur Yvette CEDEX, France (e-mail: dany.prieto@supelec.fr). Benjamin Dagusé is with the Ecole Supérieure d’Electricité, F-91192 Gif sur Yvette CEDEX, France (e-mail: benjamin.daguse@supelec.fr). Philippe Dessante is with the Ecole Supérieure d’Electricité, F-91192 Gif sur Yvette CEDEX, France (e-mail: philippe.dessante@supelec.fr). Pierre Vidal is with the Ecole Supérieure d’Electricité, F-91192 Gif sur Yvette CEDEX, France (e-mail: pierre.vidal@supelec.fr). Jean-Claude Vannier is with the Ecole Supérieure d’Electricité, F-91192 Gif sur Yvette CEDEX, France (e-mail: jean-claude.vannier@supelec.fr). is its high saliency ratio. The barriers limit the d-axis flux without obstructing the q-axis flux. We will call this motor “SynRM”. If permanent magnets are inserted into the flux-barriers of SynRM rotor, Fig. 1(b), the torque-density and power factor of SynRM can be increased [8]. The magnet flux is lower than that produced by stator excitation. This structure is called Permanent Magnet Assisted Synchronous Reluctance Motor (PMA-SynRM). (a) SynRM (b) PMA-SynRM Fig. 1. Sketch of (a) a synchronous reluctance motor and (b) permanent magnet assisted synchronous reluctance motor This paper proposes the use of ferrite permanent magnets. This material is another alternative and the paper aims to assess the impact of these magnets on the synchronous reluctance motors. It studies a methodology to compare the performances (average torque, torque ripple and power factor) of the SynRM and the PMA-SynRM for different pole pair number with a same stator imposed for all structures. The assumption adopted here, is to use the same geometry (flux-barriers thickness and opening angles of flux-barriers) and same volume of magnets for all PMA-SynRM structures treated in order to keep same price at the rotor. III. M ATHEMATICAL M ODEL  The compared structures are a Synchronous Reluctance Motor (SynRM) and a Permanent Magnet Assisted Synchronous Reluctance Motor (PMA-SynRM), both with the same number of flux-barriers per pole, the pole pair number being a parameter.  A. Axis d-q The d-q reference frame is shown in Fig. 2. The d-axis is aligned with the permanent magnet flux of the PMA-SynRM, and the same convention is used for the SRM. Effect of Magnets on Average Torque and Power   Factor   of    Synchronous   Reluctance   Motors D. Prieto, B. Dagusé, P. Dessante, P. Vidal, J.-C. Vannier   P 978-1-4673-0142-8/12/$26.00 ©2012 IEEE 213   Fig. 2. Sketch of d-q reference frame  B. Mathematical model of SynRM The Park’s equations for a synchronous reluctance machine are expressed by [6]: qsdt d d d isr d v  Φ−Φ+=  ω   (1) d sdt qd qisr qv  Φ+Φ+=  ω   (2) In steady-state the currents, i d   and i q , are constant, so the derivative of flux, Φ  d   and Φ  q , is zero. The equations (1) and (2) lead to equation (3). q I q jX d  I d  jX  I s RV   ++=  (3) The equivalent electrical phasor diagram is depicted in Fig. 3. This diagram shows the low power factor of synchronous reluctance motors. Fig. 3. Electrical phasor diagram of SynRM. α  is the current angle, δ  is the torque angle, ϕ   is the phase angle Equations (4) and (5) give the SynRM torque. It depends on the large difference between L d  and L q . )(23 d iqqid  pT   Φ−Φ=  (4) ( )  ( )  −= d iqiq Lqid id  L pT  23 (5) C. Mathematical model of PMA-SynRM By inserting permanent magnets into the flux-barriers of SynRM rotor, the magnet flux linkage is in the d-axis flux path. So the flux linkage expressions become: ad id  Ld   Φ+=Φ  (6) qiq Lq =Φ  (7) Equation (8) gives the torque of a PMA-SynRM. It increases with the magnet flux linkage. ( )  ( ) { } d iqiq Lqiad id  L pT   −Φ+= 23 (8) The equivalent electrical phasor diagram is represented on Fig. 4, where  E   is the electromotive force produced by the permanent magnets. It shows the improvement of power factor with the insertion of permanent magnets. Fig. 4. Electrical phasor diagram of PMA-SynRM, α  is the current angle, δ  is the torque angle, ϕ  is the phase angle IV. R EQUIREMENTS  Motor’s characteristics are reported in Table I. The stator is the same as that of an industrial motor. The magnets are ferrite type and the amount of magnets is the same for all designed motors. The pole pair number is the parameter to be defined in order to have an efficient structure. TABLE   I R EQUIREMENTS OF S YNCHRONOUS R ELUCTANCE M OTOR  Quantity Value Power [kW] 630 Speed [rpm] 1 500 External diameter [mm] 600 Inner stator diameter [mm] 425 Shaft diameter [mm] 180 Air gap [mm] 1.5 Number of slots 72 V. A NALYSIS  The analysis uses Finite Element Method (FEM).  A. PMA-SynRM Structures In a first approach, if we increase the number of flux-barriers, the flux in the d-axis decreases, consequently it is possible to get a higher value for the reluctance torque. But if we think of increasing the pole pair number, the size of a pole becomes smaller, so it generates a geometric constraint. Fig. 5 shows PMA-SynRM structures. The study compares four structures with: 2, 3, 4 and 6 pole pair numbers. For this paper we choose four flux-barriers per pole for all motors, the structure with  p =6 is almost geometric limit. All motor structures have the same stator because it is imposed. All flux-barriers have the same thickness. The opening angles of flux-barriers are uniform d-axis q-axis I I q  I d   jX d I d   jX q I q  R s I V δ   α   ϕ   d-axis q-axis I I q  I d   jX d I d   jX q I q  R s I V δ   α   ϕ   E 214    and the magnets have the same thickness as the barriers, Fig. 6. (a) (b) (c) (d) Fig. 5. Structure motor (a) 2 pole pairs (b) 3 pole pairs (c) 4 pole pairs (d) 6 pole pairs Fig. 6. Flux-barriers Geometry    B. Calculation Method For each structure we set a current angle for the calculation of average torque, torque ripple and power factor. Fig. 7 shows the torque behaviors versus rotor position for an electrical period for a PMA-SynRM four pole pairs and a current angle α =50°. The periodicity for the torque calculation is equal to the sixth part of the electrical period, so the average torque and torque ripple will be calculated on T  e  /6  . It is shown in Fig. 8 and in the equations (9) and (10). Fig. 7. Torque versus rotor position – PMA-SynRM, p=4 and α =50°   Fig. 8. Torque versus rotor position on Te/6– PMA-SynRM, p=4 and α =50° Average torque: )( T meanaveT   =  (9) Torque ripple: aveT T T rippleT  )min()max(  −=  (10) Fig. 9 shows the flowchart on the power factor computation process. First we compute the flux vector Φ  abc  for each phase, then Φ  dq  is obtained with Park’s transformation. The voltage vector and the torque angle δ  are computed from Φ  dq . Fig. 9. Power factor computation flowchart VI. R ESULTS  All simulations are carried out with the same current equivalent to maximal thermal current in one slot. For each current angle, the average torque, the torque ripple and the power factor are computed. The gap between SynRM and PMA-SynRM is expressed with the expression (11). %100)()()( )(  ×−= SynRM  X SynRM  X SynRM PMA X  X Gap  (11)  A. Average torque comparison Fig. 10 and Fig. 11 show respectively the SynRM and the PMA-SynRM average torque ( T  ave ) versus current angle ( α ) for each value of pole pairs. Table II gives the values of maximum average torque and their optimal current angle for each pole pair number. It also shows the impact of magnets on average torque. The increase percentage is between 34 and 45%. The best results are for the PMA-SynRM with 3 and 4 pole pair number. For the PMA-SynRM with  p =4, the maximum average torque is 4257N.m 󰁆󰁅󰁍   󰁃󰁯󰁭󰁰󰁵󰁴󰁡󰁴󰁩󰁯󰁮 󰁛 Φ 󰁡󰁢󰁣 󰁝   󰁐󰁡󰁲󰁫 󰁴󰁲󰁡󰁮󰁳󰁦󰁯󰁲󰁭󰁡󰁴󰁩󰁯󰁮   󰁛 Φ 󰁤󰁱 󰁝󰀽󰁛󰁐󰁝󰁛 Φ 󰁡󰁢󰁣 󰁝   󰁃󰁯󰁭󰁰󰁵󰁴󰁡󰁴󰁩󰁯󰁮   󰁛󰁖 󰁤󰁱 󰁝 󰁡󰁮󰁤 δ   󰁃󰁯󰁳 ϕ 󰀽󰁣󰁯󰁳 󰀨δ 󰀭 α󰀩   215    and its optimal current angle is 50°. PMA-SynRM with  p =3 also has a good maximum average torque and its optimum current angle, α = 58°, is slightly higher than that of the four pole pairs PMA-SynRM. Therefore structures PMA-SynRM  p =3 and  p =4 have a good operation range. PMA-SynRM with  p =2 has a high optimal current angle, α =60°, however it presents a low average torque. Fig. 10. Average torque versus current angle, SynRM. Fig. 11. Average torque versus current angle, PMA-SynRM TABLE   II I MPACT OF THE N UMBER OF P OLE P AIRS  Structure Quantity 2 3 4 6 SynRM T ave_max (N.m) 2160 3162 3146 2206 α opt  (°) 70 60 54 50 PMA-SynRM T ave_max (N.m) 3083 4224 4257 3208 α opt  (°) 66 58 50 46 Gap (%) 43 34 35 45 The torque for the structures with  p =2 is penalized because of saturation in stator especially the yoke, therefore the stator lamination consumes amperes-turns. Increasing the number of pole pairs produces a decrease in the airgap flux density and therefore the stator yoke is less saturated. It is possible to increase the number of pole pairs to reduce the yoke magnetic saturation and the magnetomotive force (MMF) is also diminished in consequence, to restore the expected value of the torque. But as shown in Fig. 10 and 11, the torque of the structures with  p =6 is lower than those of the p=4 because flux-barriers are beginning to take more space (Fig. 5d) and hamper the flux passage, so the flux decreases and torque also.  B. Torque ripple comparison Fig. 12 and Fig. 13 show respectively for the SynRM and the PMA-SynRM, the torque ripple ( T  ripple ) versus current angle ( α ) for each value of pole pairs. Table III gives the values of minimum torque ripple and their optimal current angle. The percentage of reduction is between 2 and 40%. It is evident that PMA-SynRM with  p =4 has a low torque ripple for current angle between 10° and 65°. PMA-SynRM with  p =2 has around 15% of torque ripple for a current angle between 20° and 65°, these percentages are quite satisfactory. On the other side, PMA-SynRM structure with three pole pairs has the values of torque ripple around 25%. PMA-SynRM with  p =6 has a low torque ripple, but this value is for a small range around 36°. Fig. 12. Torque ripple versus current angle, SynRM Fig. 13. Torque ripple versus current angle, PMA-SynRM TABLE   III I MPACT OF THE N UMBER OF P OLE P AIRS  Structure Quantity 2 3 4 6 SynRM T ripple_min (%) 13.69 26.39 7.37 21.14 α  (°) 48 30 44 20 PMA-SynRM T ripple_min (%) 13.47 22.10 6.67 12.68 α  (°) 54 44 42 36 Gap (%) -2 -16 -9 -40 C. Power factor Fig. 14 and Fig. 15 show respectively for the SynRM and the PMA-SynRM, the power factor ( cos ϕ  ) versus current angle ( α ) for each value of pole pair. Table IV gives the values of maximum power factor and their optimal current angle. All PMA-SynRM structures have improved power factor with the addition of permanent magnets. The percentage of increase is between 40 and 56%. PMA-SynRM with  p =2, 3, and 4 have the best results; the power 216
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