Minimization of Cogging Force in Fractional-Slot Permanent Magnet Linear Motors with Double-Layer Concentrated Windings

Transcription

1 energies Article Minimization Cogging Force in Fractional-Slot Permanent Magnet Linear Motors with Double-Layer Concentrated Windings Qian Wang, Bo Zhao *, Jibin Zou and Yong Li Department Electrical Engineering, Harbin Institute Technology, Harbin 151, China; (Q.W.); (J.Z.); (Y.L.) * Correspondence: zhaobo_ren@163.com; Tel.: Academic Editor: David Wood Received: 27 July 216; Accepted: 3 November 216; Published: 5 November 216 Abstract: Permanent magnet linear motors (PMLMs) with double-layer concentrated windings generally show significant s due to introduction auxiliary teeth for eliminating end-effect induced phase unbalance, even when fractional-slot technology is applied. This paper presents a novel approach to reduce by adjusting armature core dimensions in fractional-slot PMLMs with double-layer concentrated windings, toger with magnet skewing. It is shown that proposed technique is capable reducing motor in an effective way, with peak value minimized to less than.4% rated thrust in case study. Such a technique can also be applicable to or linear motors with appropriate changes. Keywords: ; concentrated winding; linear motor; permanent magnet motor 1. Introduction Fractional-slot permanent magnet linear motors (PMLMs) are a key enabling technology in modern applications for ir favorable features high -density, improved efficiency, and excellent dynamic performance [1 5]. However, interactions permanent magnets (PM) with finite-length cores and slotted armatures result in s, which would deteriorate performances. As a result, minimization is essential importance, and development new linear motor topologies and technologies (with reduced s) continues [6,7]. In fractional-slot PMLMs, according to production mechanism, re are two types components that may exist, i.e., tooth-ripple component due to slotting effect, and end-effect component associated with finite length armature core. According to [8 1], end-effect can be minimized effectively by installing chamferings, adopting a suitable stator length, or introducing auxiliary poles on armature core. It can be furr reduced by employing skewed or stepped end faces [11,12]. As for tooth-ripple, generally speaking, by employing fractional-slot technology, it can be reduced to quite a low level [13]. According to [14], however, for fractional-slot PMLM with double-layer concentrated windings, a significant phase unbalance exists due to essentially open magnetic circuits, and auxiliary teeth are usually imported at both armature core ends to provide flux return path, resulting in presence an additional slot, as illustrated in 1. As has been proven in [14], in this configuration, an undesirable large still exists, and refore well-known fractional-slot design method loses its expected strong capability tooth-ripple reduction. Energies 216, 9, 918; doi:1.339/en

2 Energies 216, 9, Energies 216, 9, Schematic illustration a permanent magnet linear motor (PMLM) with double-layer 1. Schematic illustration permanent magnet linear motor (PMLM) with double-layer concentrated concentrated windings. windings. To address this problem, extensive research work has been undertaken. Although several To address this problem, extensive research work has been undertaken. Although several controller-based techniques have also been described [15,16], it is generally preferable to minimize controller-based techniques have also been described [15,16], it is generally preferable to minimize by improving motor design [17]. The simplest method regarding design is by improving motor design [17]. The simplest method regarding design parametric optimization, i.e., re-selecting values leading dimensions and/or parameters. For is parametric optimization, i.e., re-selecting values leading dimensions and/or parameters. example, values tooth width and magnet dimensions have been optimized in [18] to reduce For example, values tooth width and magnet dimensions have been optimized in [18] to reduce in an 8-pole/9-slot linear motor. The key problem with this method is that, in general, in an 8-pole/9-slot linear motor. The key problem with this method is that, in general, it shows slightly reduced over initial values, since it still has fundamental it shows slightly reduced over initial values, since it still has fundamental problem problem tooth-ripple arising from additional slot being generally high. A tooth-ripple arising from additional slot being generally high. A more efficient more efficient solution is proposed in [19], where linear machine adopts a 9-pole/1-slot structure. solution is proposed in [19], where linear machine adopts a 9-pole/1-slot structure. However, However, three phases are asymmetric in this topology, which leads to a relatively larger thrust three phases are asymmetric in this topology, which leads to a relatively larger thrust ripple, which is ripple, which is less desirable in practice. less desirable in practice. This paper presents a novel motor-based technique for reducing in a fractionalslot PMLM equipped with double-layer concentrated windings. Based on investigation This paper presents a novel motor-based technique for reducing in a fractional-slot PMLM equipped with double-layer concentrated windings. Based on investigation tooth-ripple in motor, a technique which combines optimization armature tooth-ripple in motor, a technique which combines optimization core dimensions with skewed magnets is presented to minimize. The effectiveness armature core dimensions with skewed magnets is presented to minimize. approach is confirmed by detailed finite element (FE) computations, as well as an analytical The effectiveness approach is confirmed by detailed finite element (FE) computations, as analysis. well as an analytical analysis Tooth-Ripple Tooth-Ripple Cogging Cogging Force Force in in Double-Layer Double-Layer Concentrated-Winding Concentrated-Winding Design Design In In a PMLM PMLM with with double-layer double-layer concentrated concentrated windings, windings, due to due incorporation to incorporation an additional an additional slot, slot, amplitude is mainly amplitude determined is mainly by determined component by arising component from additional arising from slot [14]. Therefore, additional slot [14]. Therefore, is considerably higher is than considerably that in higher single-layer-winding than that in single-layer-winding PMLM. PMLM. 2 shows shows FE FE model model a PMLM PMLM with with double-layer double-layer windings, windings, where where Dirichlet Dirichlet boundary boundary condition condition is is imposed imposed at at outer outer surfaces. surfaces. The The primary primary parameters parameters machine machine are listed are in listed Table in 1. Table Finite element model PMLM.

3 Energies 216, 9, Energies 216, 9, Table 1. Specification and Leading Design Parameters a permanent magnet linear motor (PMLM). Parameters Data Parameters Data Table 1. Specification and Leading Design Parameters a permanent magnet linear motor (PMLM). Rated (N) 25 Pole No. 14 Parameters Data Parameters Data Pole pitch (mm) 1 Slot No. 12 Rated (N) 25 Pole No. 14 Permanent Pole pitch magnets (mm) (PM) 4 1PM remanence Slot (T) No Permanent magnets thickness (PM) (mm) thickness (mm) 4 PM remanence (T) 1.5 Air-gap length (mm) 1. PM relative permeability 1.5 Air-gap length (mm) 1. PM relative permeability 1.5 Slot opening width (mm) 4 Halbach ratio.65 Slot opening width (mm) 4 Halbach ratio shows tooth-ripple waveform and its its harmonic spectrum. Cogging Force (N) Slotted Slotless Tooth-ripple Magnitude (N) Displacement (mm) (a) Harmonic order (b) Cogging Cogging in in a PMLM a PMLM with with double-layer double-layer windings. windings. (a) Waveform; (a) Waveform; (b) Harmonic (b) spectrum Harmonic spectrum tooth-ripple tooth-ripple.. It can be seen that introduction auxiliary slot in fractional-slot design gives rise to It can be seen that introduction auxiliary slot in fractional-slot design gives rise an additional tooth-ripple with a fairly high amplitude, accounting for 17.9% to an additional tooth-ripple with a fairly high amplitude, accounting for 17.9% rated rated thrust thrust,, which which makes makes fractional-slot fractional-slot design design lose lose its its strong strong capability capability reduction. reduction. Therefore, Therefore, extra extra effort effort is is necessary necessary for for PMLMs PMLMs with with double-layer double-layer concentrated concentrated windings, windings, to to minimize s. s. Furr Furr observationon on harmonic spectrum in in 3b 3b reveals reveals that that for for tooth-ripple tooth-ripple, fundamental and and 2nd 2nd order order harmonics harmonics are dominant. are dominant. Therefore, Therefore, in order to minimize order to minimize tooth-ripple tooth-ripple, practice, it isin generally practice adequate it is generally to take effective adequate countermeasures to take effective to countermeasures only eliminate to 1st only andeliminate 2nd order harmonic 1st and components. 2nd order In harmonic following components. sections, two In techniques, following sections, namely two armature techniques, length optimization namely armature and magnet length skewing, optimization will beand employed magnet to skewing, reduce will two be employed harmonicto components, reduce two respectively. harmonic components, respectively Elimination Elimination 1st-Order 1st-Order Harmonic Cogging Force Since Since tooth-ripple tooth-ripple has has same same fundamental fundamental period period as as end-effect end-effect, viz. one pole pitch, it is important to properly design armature dimensions, so that, viz. one pole pitch, it is important to properly design armature dimensions, so that endeffect component is in an equal and opposite fashion to tooth-ripple component, thus eliminating end-effect component is in an equal and opposite fashion to tooth-ripple component, thus eliminating 1st-order component resultant. In a more comprehensive way, 1st-order component resultant. In a more comprehensive way, motional motional length steel core can be optimized to ensure that 1st-order magnitude length steel core can be optimized to ensure that 1st-order magnitude end-effect end-effect be equal to that tooth-ripple, and at same time, be equal to that tooth-ripple, and at same time, tooth widths tooth widths at steel core ends are appropriately designed so that ir initial phases have an fset at 18 steel electrical core ends degrees. are appropriately designed so that ir initial phases have an fset 18 electrical degrees Influence Armature Core Length on End-Effect Cogging Force

4 Energies 216, 9, Influence Armature Core Length on End-Effect Cogging Force It has been reported in [8] that magnitude end-effect is a function Energies armature Energies 216, 216, 9, core 918 9, 918 length. In view this, armature length should be properly designed to maintain amplitudes 1st-order component end-effect and tooth-ripple It has It ashas equal. been been reported in in [8] [8] that that magnitude end-effect is a is function a function armature Using core core length. FE length. model In In view inview this, this, 4, end-effect armature length length should should waveforms be be properly were designed obtained to and to maintain are shown amplitudes in 5, where 1st-order L refers component to armature incremental end-effect length. and and tooth-ripple More specifically, as as equal. equal. 6 shows variation 1st-order end-effect amplitude along Using Using with L. FE Particularly, FE model model in in when L 4, 4, is equal end-effect to 5.2 mm, 1st-order waveforms end-effect were were obtained amplitude and and isare are shown approximately shown in in 5, equal where 5, where to that ΔL ΔL refers refers tooth-ripple armature incremental. length. length Finite 4. Finite Element Element (FE) (FE) (FE) model model model slot-less slot-less slot-less PMLM PMLM PMLM with with with an an auxiliary an auxiliary auxiliary steel steel steel core. core. core. 8 8 End-effect (N) End-effect (N) ΔL ΔL (mm) (mm) Displacement (mm) (mm) (a) (a) 5. Cont.

5 Energies 216, 9, ΔL=2mm Energies 216, 9, Energies 216, 9, 918 ΔL=4mm ΔL=6mm 4 ΔL=8mm 8 ΔL=1mm ΔL=2mm 2 ΔL=4mm 6 ΔL=6mm 4 ΔL=8mm -2 ΔL=1mm Displacement (mm) End-effect (N) End-effect (N) End-effect 1 2vs. mover 3 4displacement 5 6 7and ΔL. 8 (a) 9 3-D 1view; (b) 2-D view. Displacement (mm) More specifically, 6 shows variation (b) 1st-order end-effect amplitude along with ΔL. Particularly, when ΔL is equal to 5.2 mm, 1st-order end-effect amplitude End-effect End-effect vs. vs. mover mover displacement displacement and and L. ΔL. (a) 3-D (a) 3-D view; view; (b) 2-D (b) view. 2-D view. is approximately equal to that tooth-ripple. More specifically, 8 6 shows variation 1st-order end-effect amplitude along with ΔL. Particularly, when ΔL is equal to 5.2 mm, 1st-order end-effect amplitude is approximately equal to 7 that tooth-ripple. Fundamental amplitude (N) Fundamental amplitude (N) (b) ΔL (mm) First-order end-effect amplitude as as aa function function L. ΔL Phase Phase Adjustment Adjustment by byoffsetting Offsetting Auxiliary CoreΔL (mm) In In order to to eliminate 6. First-order 1st-order 1st-order harmonic end-effect harmonic, amplitude apart, as from a function apart magnitude from ΔL. requirement, magnitude requirement, initial phase initial phase 1st-order harmonic 1st-order end-effect harmonic end-effect should be designed should withbe andesigned fset with an Phase fset electrical Adjustment 18 degrees electrical by tooffsetting that degrees tooth-ripple Auxiliary that Core tooth-ripple, which can, be achieved which can bybe fsetting achieved by fsetting auxiliary core, auxiliary as shown core, in as shown 7. in 7. In order to eliminate 1st-order harmonic, apart from magnitude Assuming that initial phases 1st-order end-effect and tooth-ripple components are θ requirement, initial phase 1st-order harmonic end-effect should be designed 1 and θ 2 (in electrical degrees), respectively, fset auxiliary core can be calculated as with an fset 18 electrical degrees to that tooth-ripple, which can be achieved by fsetting auxiliary core, as shown in 7. o f f set = θ 2 (θ 1 ± 18 ) 36 τ p (1) In this case, θ 1 is 9.2, and θ 2 is when L is designed as 5.2 mm. Accordingly, fset auxiliary core should be 2.62 mm.

6 Energies 216, 9, Energies 216, 9, Initial phase adjustment end-effect by fsetting auxiliary core. ssuming that initial phases 1st-order end-effect and tooth-ripple components are θ1 2 (in electrical degrees), respectively, fset auxiliary core can be calculated as = ( ± 18 ) 36 (1) this case, θ1 is 9.2, and θ2 is when ΔL is designed as 5.2 mm. Accordingly, 7. Initial phase adjustment end-effect by fsetting auxiliary core. auxiliary core should be Initial mm. phase adjustment end-effect by fsetting auxiliary core. Assuming that initial phases 1st-order end-effect and tooth-ripple components are θ1 lidations 3.3. Validations and θ2 (in electrical degrees), respectively, fset auxiliary core can be calculated as igure 8 gives waveform 8 gives resultant waveform resultant = PMLM ( ± after 18 ) performing PMLM both after performing both mature core optimization armature and coreauxiliary optimization 36 (1) Energies core 216, fset. and9, 918 auxiliary core fset. In this case, θ1 is 9.2, and θ2 3 is when ΔL is designed as 5.2 mm. Accordingly, 3 fset auxiliary core should be 2.62 mm. Resultant Cogging Force (N) Validations 1 8 gives waveform resultant 2 PMLM after performing both armature core optimization and auxiliary core fset Resultant Cogging Force (N) (a) Displacement (mm) -1-2 Magnitude (N) Harmonic order (b) 8. Resultant 8. after Resultant performing armature after core performing optimization armature and core auxiliary optimization and auxiliar core fset. (a) Waveform; (b) core Harmonic fset. (a) spectrum. Waveform; (b) Harmonic spectrum. It can be seen that by adopting techniques both optimizing armature lengt It can be seen that by adopting techniques fsetting auxiliary 3 core, 4 5 1st-order 6both7 optimizing 8component 9 1 armature length and resultant has been re fsetting auxiliary core, 1st-order component Displacement (mm) resultant has been reduced drastically, thus verifying validity above proposed method. drastically, thus verifying validity above(a) proposed method. 4. Elimination 2nd-Order Harmonic Cogging Force 4. Elimination 2nd-Order Harmonic Cogging Force After utilizing techniques in Section 3, 1st-order component ha After utilizing techniques in Section 3, 1st-order component has been greatly reduced. The dominant component is now 2nd-order harmonic, and it can be elim greatly reduced. The dominant component is now 2nd-order harmonic, and it can be eliminated by by skewing magnets by τp/2, i.e., wavelength 2nd-order harmonic, since sk magnets reduces magnetic reluctance variation seen by armature, and thus minimiz. 9 illustrates schematic a linear machine equipped with skewed ma

7 It can be seen that by adopting techniques both optimizing armature length and fsetting auxiliary core, 1st-order component resultant has been reduced drastically, thus verifying validity above proposed method. 4. Elimination 2nd-Order Harmonic Cogging Force Energies 216, 9, After utilizing techniques in Section 3, 1st-order component has been greatly reduced. The dominant component is now 2nd-order harmonic, and it can be eliminated by skewing skewing magnets magnets by τ p by /2, τp/2, i.e., i.e., wavelength wavelength 2nd-order 2nd-order harmonic, harmonic, since skewing since skewing magnets magnets reduces reduces magnetic reluctance magnetic reluctance variation seen variation by seen armature, by and armature, thus minimizes and thus minimizes. 9. illustrates 9 schematic illustrates aschematic linear machine a linear equipped machine wiquipped skewed magnets. with skewed magnets Schematic use skewed magnets for reduction in a linear machine Analytical AnalyticalAnalysis Analysis The Theanalytical analyticalanalysis analysis in in PMLM, PMLM, which which employs employs skewed skewedmagnets, is is derived derived below. Firstly, Firstly, if if magnets magnets are are straight, straight, resultant resultant F Fr, which comprises r, which comprises both both tooth-ripple Fslot tooth-ripple F and end-effect Fend, can be expressed by slot and end-effect F end, can be expressed by x τ p /4 ( ( ) 2 ) 2πn F r (x) = F n sin + x + θ n n=1,2,,, where x is mover displacement, and F where is mover displacement, and n and θ Fn and n are amplitude and initial phase n θn are amplitude and initial phase th -order th -order component, respectively. component, respectively. By skewing magnets by τ By skewing magnets by τp/2, p /2, resultant can be calculated as resultant can be calculated as F rsk (x) = 2 x+τp /4 2 ( nπ ) ( ) 2πn F r (x) dx = τ p nπ sin F n sin x + θ n (3) 2 n=1,2, As can be seen from Equation (3), technique magnet skewing has following impacts, namely: (1). When n = 2, 4, 6,..., sin(nπ/2) =, which means that even-order components s can be eliminated by skewing magnets by τ p /2. (2). The magnitudes or harmonic -s can be brought down with a factor (2/nπ) Validations In order to account for skewing effect on reduction, 3-dimensional (3D) FE analysis was applied to calculate PMLM with skewed magnets. 1 displays result, toger with analytically predicted waveform for comparison, which was derived from Equation (3). It can be observed that by skewing magnets by τ p /2, has been greatly reduced; amplitude decreased from 27.6 N initially to nearly.9 N, accounting for less than.4% rated. There is a small error between FE calculations and analytical predictions, which may be mainly attributed to computation error and transverse edge effect that are not taken into account in analytical derivation. Neverless, results have confirmed effectiveness this approach. τ p τ p (2) (2)

8 4.2. Validations In order to account for skewing effect on reduction, 3-dimensional (3D) FE analysis was applied to calculate PMLM with skewed magnets. 1 displays Energies 216, 9, result, 918 toger with analytically predicted waveform for comparison, which 8was 1 derived from Equation (3). Resultant Cogging Force (N) Analytical 3D FE Displacement (mm) 1. Cogging waveform PMLM with skewed magnets. 1. Cogging waveform PMLM with skewed magnets. It can be observed that by skewing magnets by τp/2, has been greatly 5. Practical Considerations reduced; amplitude decreased from 27.6 N initially to nearly.9 N, accounting for less than.4% 5.1. Skewing rated. Magnets There by τis p A a small Higherror Pricebetween FE calculations and analytical predictions, which may be mainly attributed to computation error and transverse edge effect that are not taken into account At this stage, in one analytical might wonder derivation. that Neverless, why not directly results skew have PMs confirmed by one pole pitch, effectiveness instead this half approach. a pole pitch, to eliminate both tooth-ripple and end-effect s, since both have a fundamental period τ p. Truly, skewing PMs in this manner would achieve expected 5. elimination Practical Considerations, but cost an excessive loss back electromotive, and hence power, must be paid. From this point view, skewing PMs by half a pole pitch proves 5.1. to be Skewing a bettermagnets solutionby which τp A compromises High Price and power density. Energies 216, 9, Incorporation At this stage, one Cooling might Ducts wonder into Armature that why Cores not directly skew PMs by one pole pitch, instead half It a should pole pitch, be pointed to eliminate out that both due to tooth-ripple introduction and end-effect auxiliary teeth, s, since both density have a fundamental linear It should PM be motor pointed period would out τp. Truly, be that slightly due to skewing decreased. introduction PMs In in order this to manner minimize auxiliary teeth, would this achieve impact, ducts density expected can be elimination incorporated linear PM into motor would armature, be slightly cores but to cost decreased. reduce an excessive weight, In order loss as toillustrated minimize back in this electromotive impact, 11. Furrmore, ducts, can and be hence incorporated ducts power, can into also must be armature utilized be paid. for cores From cooling to this reduce point and are weight, conducive view, skewing as illustrated to improving PMs in by half heat 11. a pole transfer Furrmore, pitch from proves to ducts cores be a can and better also windings. solution be utilized which However, for compromises cooling and dimensions are conducive and to positions improving and power heat ducts density. transfer should from be carefully cores and optimized windings. to have However, minimal effect dimensions magnetic and positions field distribution, ducts should which is bebeyond carefully optimized scope this to 5.2. have paper. Incorporation minimal effect Cooling on Ducts magnetic into field Armature distribution, Cores which is beyond scope this paper. 11. Incorporation cooling ducts into armature cores. 11. Incorporation cooling ducts into armature cores. 6. Conclusions 6. Conclusions This paper has studied minimization techniques in fractional-slot PMLM with double-layer This paper has studied minimization techniques in a fractional-slot PMLM with double-layer concentrated windings. The following key observations can be drawn: concentrated windings. The following key observations can be drawn: (1). The PMLM with double-layer concentrated windings exhibits significant tooth-ripple even with fractional-slot technology. (2). By combining optimized design armature core dimensions with magnet skewing, 1stand 2nd-order s have been eliminated, and in this way, resultant can be greatly minimized. (3). The approach described in paper is capable reducing in an effective way, which is conducive to expanded use PMLMs in a range applications. Such techniques

9 Energies 216, 9, (1). The PMLM with double-layer concentrated windings exhibits significant tooth-ripple even with fractional-slot technology. (2). By combining optimized design armature core dimensions with magnet skewing, 1stand 2nd-order s have been eliminated, and in this way, resultant can be greatly minimized. (3). The approach described in paper is capable reducing in an effective way, which is conducive to expanded use PMLMs in a range applications. Such techniques can be readily extended to or linear PM machines. Acknowledgments: This work was supported in part by Natural Science Foundation China under Grant , by National Key Basic Research Program China under Grant 213CB356, and by China Postdoctoral Science Foundation under Grant 214M Author Contributions: The paper was a collaborative effort between authors. The authors contributed collectively to oretical analysis, modeling, simulation, and manuscript preparation. Conflicts Interest: The authors declare no conflicts interest. References 1. Kou, B.; Luo, J.; Yang, X.; Zhang, L. Modelling and analysis a novel transverse-flux flux-reversal linear motor for long-stroke application. IEEE Trans. Ind. Electron. 216, 1, [CrossRef] 2. Teo, T.J.; Zhu, H.; Pang, C.K. Modeling a two degrees--freedom moving magnet linear motor for magnetically levitated positioners. IEEE Trans. Magn. 214, 12, [CrossRef] 3. Ji, J.; Yan, S.; Zhao, W.; Liu, G.; Zhu, X. Minimization in a novel linear permanent-magnet motor for artificial hearts. IEEE Trans. Magn. 213, 7, [CrossRef] 4. Gandhi, A.; Parsa, L. Thrust optimization a flux-switching linear synchronous machine with yokeless translator. IEEE Trans. Magn. 213, 4, [CrossRef] 5. Tavana, N.R.; Shoulaie, A. Pole-shape optimization permanent-magnet linear synchronous motor for reduction thrust ripple. Energy Convers. Manag. 211, 52, [CrossRef] 6. Wang, Q.; Zou, J.M.; Zou, J.B.; Zhao, M. Analysis and computer-aided simulation characteristic a linear electromagnetic launcher with tubular transverse flux machine. IEEE Trans. Plasma Sci. 211, 1, [CrossRef] 7. Kim, Y.J.; Watada, M.; Dohmeki, H. Reduction at outlet edge a stationary discontinuous primary linear synchronous motor. IEEE Trans. Magn. 27, 1, [CrossRef] 8. Zhu, Z.Q.; Xia, Z.P.; Howe, D.; Mellor, P.H. Reduction in slotless linear permanent magnet motors. IEE Proc. 1997, 4, [CrossRef] 9. Inoue, M.; Sato, K. An approach to a suitable stator length for minimizing detent permanent magnet linear synchronous motors. IEEE Trans. Magn. 2, 4, [CrossRef] 1. Zhu, W.; Lee, S.; Chung, K.; Cho, Y. Investigation auxiliary poles design criteria on reduction end effect detent for PMLSM. IEEE Trans. Magn. 29, 6, [CrossRef] 11. Wang, J.; Inoue, M.; Amara, Y.; Howe, D. Cogging--reduction techniques for linear permanent-magnet machines. IEE Proc. 25, 3, [CrossRef] 12. Jung, I.; Hur, J.; Hyun, D. Performance analysis skewed PM linear synchronous motor according to various design parameters. IEEE Trans. Magn. 21, 5, Hor, P.J.; Zhu, Z.Q.; Howe, D.; Jones, J. Minimization in a linear permanent magnet motor. IEEE Trans. Magn. 1998, 5, [CrossRef] 14. Wang, Q.; Wang, J. Assessment --reduction techniques applied to fractional-slot linear permanent magnet motors equipped with non-overlapping windings. IET Electr. Power Appl. 216, 8, [CrossRef] 15. Zhao, S.; Tan, K.K. Adaptive feed forward compensation ripples in linear motors. Control Eng. Pract. 25, 13, [CrossRef] 16. Bianchi, N.; Bolognani, S.; Cappello, A. Back EMF improvement and ripple reduction in PM linear motor drives. In Proceedings 24 IEEE 35th Annual Power Electronics Specialists Conference, Aachen, Germany, 2 25 June 24; pp

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