Designing a Permanent Magnet Linear Generator Multi Air-gap Structure using the Finite Element Method.

Transcription

1 Designing a Peranent Magnet Linear Generator Multi Air-gap Structure using the Finite Eleent Method. Pierre Kenfack, Daniel Matt, and Philippe Enrici Institute of Electronics and Systes, University Montpellier, CC rue Saint Priest, Montpellier cedex 5, France (e-ail: first-nae.nae@ies.univ-ontp2.fr). Abstract This paper deals with the design of a Peranent Magnet Linear Generator Multi Air-gap Structure (MALG) using the Finite Eleent Method (FEM). The cogging force, which occurs due to the interaction between stator teeth and the Peranent Magnet (PM), is reduced by the MALG diensions. The generated voltage is analyzed for both noload on the ulti-air gap flux density. This paper predicts the thrust force and flux linkage. 1. Introduction Linear electric achines are electroagnetic, electrostatic, piezoelectric, and agneto-striction force devices capable of producing progressive or oscillatory translational (linear) otion directly [1, 2, 3]. A Peranent Magnet Linear Generator Multi Air-gap Structure is an electroagnetic device that develops short travel linear otion. The MALG is intended for an electroacoustic converter. An electroacoustic achine converts acoustic energy into electrical energy, and vice versa, converts electric energy into acoustic energy Fig 1. An acoustic wave is created as "input" using a linear otor. This acoustic wave is aplified by the heat recovered as "output" using a linear generator. The advantages of the Peranent Magnet Linear Generator Multi Air-gap Structure over a traditional structure are its copactness, very high dynaic range due to the lightness of the oving part, and higher efficiency. Generator Multi Air-gap Structure generally has to deal with a high cogging force. The double-sided linear electric generator has a cogging force that is uch lower copared to the single and double-sided iron core generator. The cogging force is produced by the interaction between the peranent agnet and the slotted iron structure. The cogging force produces a pulsating force ripple resulting in vibrations and acoustic noise that is detriental to the MALG Structure. This paper presents investigations on ethods to reduce cogging force. In this paper, a MALG Structure is designed using a 2D finite eleent ethod. The ain design paraeters are the electroagnetic force (ef) across the stator bars and the cogging force. The slot opening width length is chosen to give a iniu cogging force. The peranent agnet thickness was varied over a wide range and the voltage and cogging force are calculated each tie by FEM. Fig 2 shows the structure chosen to ipleent the linear electric generator using 2D FEM ANSYS Maxwell software. Fig.2 Magnetic structure, three-phase linear generator. 2. Calculating ain paraeters Fig. 1.Electroacoustic converter. In the ulti-air gap concept, the ai is to increase, within a given volue, the nuber of air-gap surfaces generating electroagnetic conversion [4, 5]. Despite these advantages, the Peranent Magnet Linear The first step in designing and analyzing a Peranent Magnet Linear Generator Multi Air-gap Structure is to deterine the airgap field produced by the agnets. The fields ay be deterined precisely by finite eleent analysis. Generally, haronic displaceent is used to describe the oscillatory otion of an electric achine or oving part Electroagnetic equations Whereas fields ay be deterined precisely by finite eleent analysis, a agnetic circuit approach yields approxiate results that are acceptable for practical purposes. Thus, neglecting

2 saturation and leakage, we have: = = H (1) * g μ0 g Neglecting the reluctance of core, Apere s law yields: 2gH + H h = 0 (2) g Assuing rare-earth agnets, their deagnetization characteristic can be written: = + μ H (3) r rc Cobining (1), (2) through (3) yields. ( π ) 2 (8) x= 2 f * x The speed of synchronis ( V ) or displaceent is independent of the nuber of the peranent agnets pole pairs. V = 2τ f (9) L s : the stroke; f : the echanical frequency andτ : the pole pitch. The electroagnetic force is calculated as follows: () F t et () it () = xt () (10) μ rc 2g = r 1+ μ0 h 1 (4) The highest interaction electroagnetic force per given current occurs when e(t) and i(t) are in phase with each other. 3. Carried out siulations : PM operation flux density, T H : PM operation field intensity, A/ g : Air-gap, r : Residual flux density, T μ rc : PM recoil pereability, H/ g : Air gap flux density, T H g : Air gap agnet field intensity, A/ h : Magnet thickness, The electroagnetic force (ef) is proportional to linear speed. The ain characteristics of the MALG being studied were obtained as in [6, 7, 8]. For our purposes here, ANSYS-Maxwell software was used with the Finite Eleent Method. Maxwell Transient is able to evaluate interactions between transient electroagnetic fields and otion [9, 10, 11]. The MALG Structure has equal slot and pole pitches. The cross section of the linear electric generator is shown in Fig 3 and specification design data is given in Table 1. e = kx (5) e : ef, V; k : function with PM flux density and generator width; x : speed Mechanical equations The obtained echanical values prohibit the use of traditional echanical systes because acceleration is too high. Displaceent of the oving part is written in the following for: ( π ) x x sin 2 ft = ; Speed is the derivative of displaceent Ls x = (6) 2 L s (7) ( π ) x= 2π f cos 2 ft 2 Acceleration is the derivative of speed. Fig. 3 MALG slot diensions. The ain variable diensions are optiized to iniize the cogging force value. The studied variables are the air-gap thickness, PM thickness and slot opening width. The slot pitch of the structure is: τ s = b11 + b21 (11) τs: slot pitch, b11: width of rectangular slot, b21: width of tooth.

3 The slot factor bs and bw are used to draw the cogging force curve with the position, peranent agnet thickness, and airgap thickness. bs = This paraeter gives inforation on the evolution of the cogging force ([0.20, 0.80]). bw = This paraeter gives inforation on the optiization of winding and the decrease of iron ass ([0.40, 0.60]). Slot width is varied to optiize the bars diensions Air-gap TALE 1: SLOT DIMENSIONS Eleent Sybol Value () Slot opening height h Slot wedge height h Slot body height h Slot opening width b Slot wedge width b Slot body botto width b Slot pitch Pole pitch τ 5.0 PM length PM thickness h 5.0 Air gap g 0.8 Generator length Generator width - 50 Stroke Ls 20 Generator power 1 kw In the design paraeters of the MALG, air-gap thickness is the paraeter that has the ost influence on perforance. The air-gap is very iportant in the electroagnetic paraeterization of a linear generator, because it affects perforance. When the airgap thickness increases, the air-gap agnetic flux density decreases siilarly. This helps decrease the cogging force greatly. We analyzed the cogging force aplitude characteristic with different air-gap thicknesses. The cogging force aplitude characteristic with different air-gap thicknesses and positions is shown in Fig 4. The axiu flux density in the teeth is T PM thickness We chose NdFe to ake our prototype, with agnetic properties provided in Table 2. The PM volue is varied with different PM thicknesses and using siulation calculation. Decreasing PM thickness causes a reduction in the equivalent current density of the PM that results in decreasing the cogging force. Reducing PM thickness has ore effects, the largest of which is enhancing the linear electric generator output voltage. The oving plate generates varying flux and induces voltage in the windings. The cogging force aplitude characteristic with different peranent agnet thicknesses and positions is shown in Fig 5. Fig. 4 Cogging force versus position and air-gap thickness. TALE 2: MAGNETIC PROPERTIES OF MAGNETS Characteristic Units Min Noinal Max r, Residual T Induction Hc, Coercivity ka/ HcJ, Intrinsic ka/ 2, Coercivity Hax, Maxiu Energy Product kj/ Slot opening width We chose to use M300 35A to ake our prototype, with typical provided in Table 3. The agnetic properties of a ferroagnetic aterial are described graphically in ters of the aterial s -H curves and core losses. When the slot opening width varies, the flux linking the bars changes as the peranent agnet passes through the slot, affecting the cogging force and output voltage. The sei-open slots in the fixed part can decrease the cogging force, because the cogging force is the interaction between the peranent agnet edges and fixed part slot edges.

4 constraints can be coputed using a nuerical field coputation approach (FEM). Fig. 5 Cogging force versus position and PM thickness. TALE 3: TYPICAL DATA FOR M300-35A (T) H(A/) at 50 Hz P(W/kg) at 50 Hz Fig. 6 Cogging force versus position and bs. MALG optiization can be forulated as a constrained optiization proble with ore than one objective (iniization of cost, iniization of the aount of PM aterial, axiization of efficiency and output power). The independent variables used in MALG design diensions are the PM thickness (h), air-gap (g), relative slot opening width (bs), and relative slot width (bw). The objective function is not easily expressed in ters of the independent variables. We therefore did use the objective function for odeling directly, but rather for odelling the perforance characteristics of the MALG, used in the constraints, in ters of the independent variables. Table 4 shows the final optiized MALG details, with resistive losses using FEM, 70 W (aluinu aterials). The cogging force aplitude is zero in average value per unit area air-gap Fig 7. Despite a very arked decrease of the cogging force in the case of sei-open slots, the leakage flux is higher, which can reduce the efficiency of the linear electric generator. The cogging force increases when the relative opening width slot (bs) increases Fig Optiization The 2D finite eleent ethod is sufficient to analyze the Peranent Magnet Linear Generator Multi Air-gap Structure. Optiization ethods try to find the axiu or iniu of a function where restrictions or constraints on the independent variables ay exist. In optiizing the MALG, the objective function and Fig. 7 Cogging force three-phase MALG versus position. The parasitic contact force between stators and the oving part depends on the air-gap. At position g = 0.3, the plate is oved laterally. Plate oveent increases the area of the lower stator, and decreases the area of the upper stator by an equal aount. The cogging force through the two stators is still equal.

5 Cogging force is strictly linked to the utual position of all peranent agnets and all winding. TALE 4: MALG OPTIMIZATION Eleent Sybol Value () Slot opening height h Slot wedge height h Slot body height h Slot opening width b Slot wedge width b Slot body botto width b Slot pitch Pole pitch τ 5.0 PM length PM thickness h 3.0 Air gap g 0.3 Relative slot opening bs 0.20 Relative slot width bw 0.50 Generator length 230 Generator width 50 Stroke Ls 20 PM ass kg Generator power 1 kw The voltage aplitude characteristic with the position is shown in Fig 8. Moving part speed is 0.5 /sec. Fig. 8 No-load voltage versus position. In Peranent Magnet (PM) electric linear generator the noload voltage has a shape of a square (trapezoidal) wavefor. Given the current of each phase Fig.9, we studied the thrust force in the electric linear peranent agnet generator. Fig. 9 Three-phase current versus position Fig. 10 shows the thrust force versus displaceent characteristic, which gives the relationship between the tangential force and displaceent fro the equilibriu position. Fig. 10 Thrust force three-phase MALG versus displaceent. 4. Conclusions The Peranent Magnet Linear Generator Multi Air-gap Structure is designed and siulated using 2D FEM ANSYS Maxwell software. A prototype is being built at this tie. We investigated the effects of paraeters including air-gap width, peranent agnet thickness, and slot opening width upon the cogging force. After analysis, the best paraeters to decrease the cogging force were chosen. 5. References [1] I., oldea: Linear Electric Machines, Drives, and MAGLEVs Handbook, CRC Press, Taylor & Francis Group, 2013, 635 p. [2] I., oldea, and Syed A. Nasar: Linear Electric Actuator and Generator, Cabridge University Press, 1997, 237 p. [3] J, Prudell, M, Stoddard, and all: A Peranent-Magnet Tubular Linear Generator for Ocean Wave Energy Conversion, IEEE Transactions on Industry Applications, Vol.48, NO.6, Noveber- Deceber 2010 [4] P.-E, Cavarec, and H., Ahed and, Multon: Force density iproveents fro increasing the nuber of airgap surfaces in synchronous linear actuators, Electric Power Applications, IEE Proceedings, 2003, 150, (1), pp [5] D, Matt, and R, Goyet, and N, Haddad, and J, Lucidare, and C, Rioux: Study of original structures of ulti-airgap reluctance achines. Proc. International Conference on Electrical Machines, Pise, Italy, 1988 [6] Young-shin Kwon, and Won-jong Ki: Detent-Force Miniization of Double-Sided Interior Peranent-Magnet Flat Linear rushless Motor, IEEE Transactions on Magnetics, , 2015 [7] Eui-Jong Park, and Yong-Jae Ki: A Study on Reduction of End Cogging Force by Arature Offset Distance of Doublesided PMLSM, 9 th International Conference on Power Electronics ECCE Asia, June 1-5, 2015 [8] M.W, Zouaghi, I, Abdennadher, and all: Prediction of Air Gap Flux Density Distribution of a T-LSM with Quasi-Halbach Magnetized PMs: Application of the cogging Force Miniization, 10 th International Conference on Ecological vehicle and Renewable Energies, 2015 [9] G, Cipriani, V, Di Dio, and all: A Ferrite Tubular Linear Peranent Magnet Generator (FTLPMG) analysis and design, /14, 2014 [10] Y, Huang, S, Zhou, and all: Design and Optiization for Unilateral Flat Peranent Linear Motor, /12, 2012 [11] J, Faiz, M, Ebrahii-Salary, and all: Cogging Force Alleviation in Linear Peranent Magnet Generator, IEEE AFRICON 2009, Septeber 2009.