PERMANENT MAGNET LINEAR GENERATOR DESIGN USING FINITE ELEMENT METHOD Hamzuh Arof Ahmad M Eid Khalid M. Nor ahmazah@um.edu.my ahmadeid@,um.edu.my khalid@,~im.edu.my Electrical Engineering Department, Faculty of Engineering, University of Malaya 50603 Kuala Lumpur, Malaysia Abstract - This paper presents a general proposal to design and calculate the performance of a tubular permanent magnet linear generator using the Finite Element Method. The cogging force, which occurs due to the interaction between stator teeth and the permanent magnets, is reduced by optimizing the linear generator dimensions. The generated voltage is analyzed for both no load and load cases to take the armature reaction efsects on the air gap flux density. A repetitive routine is followed to calculate the output voltage from the change offlux and the speed of the generator. The output voltage is calculated for dferent resistive load, and hence the generator load characteristic is obtained. The designed linear generator is capable to generate an output power of 5.3kW with output voltage of 222V and the generator eficiency of 96.8%. Index Terms-Tubular linear generator, finite element method, cogging force, electromotive force, permanent magnet, harmonic. I. INTRODUCTION Linear electric generators are electromagnetic devices which develop short travel progressive linear motion. The main applications of the linear generator is the free piston applications, pen recorders, machine tool sliding tables, compressors, factory automation and material control systems [l], [2]. The advantages of a linear generator (LG) over a conventional one are its compactness and higher efficiency. Among the various configurations of linear generators, the tubular permanent magnet (PM) is superior to the flat design in terms of volume compactness and symmetry. The tubular structure has less leakage flux and is better in utilizing the material which leads to a higher electromotive force (emf). This is one pf the main design factors. In spite of these advantages, the PM-LG’s in generals have a high cogging force to deal with, The cogging force is produced by the interaction between the PM’s and the slotted iron structure and the finite length of the stator core. These factors have different periods of one-slot pitch and one-pole pitch, respectively [4] [ 5 ] . The cogging force produces a pulsating force ripple resulting in vibrations and acoustic noise which is detrimental to the linear generators. Fortunately, the cogging force problem can be alleviated with proper optimization of the dimensions 0-7803-8575-6/04/$20.00 02004 IEEE of stator and translator. Consequently, investigations of methods to reduce the cogging force are presented in this paper. In this paper, a tubular PM-LG is designed using a 2D finite element method (FEM). The main design parameters are the emf across the stator coils and the cogging force. To reduce the cogging force, the stator and the translator elements are optimized separately to get the desired values of the output voltage. The stator length is chosen to give a minimum cogging force. The PM length, the PM depth, the coil volume and teeth depth are varied over a wide range and in each time the output voltage and the cogging force are calculate by FEM. The proposed tubular PM-LG is of slotted structure and axially magnetized PM’s as shown in Figure 1. The stator of the PM-LG consists of three non-open slots which contain the coils and the core laminations which have a fill factor of 95%. The translator part consists of four PM’s of NdFeB material, which are magnetized in the axial direction (in the motion direction), a mild steel spacers between the PM’s acting as a pole shoes and a non-magnetic material (stainless steel) of the shaft which is connected to the external device in the generator operation to provide the oscillatory motion. It can be seen from Figurel, that the PM-LG has equal slot and pole pitches (7s equals 7). The main dimensions of the PM-LG are listed in table I. The stator slots are semi-closed with constant slot depth sd and opening slot width of so. The slot opening height is 2 mm and a slope height of 3 mm. This construction of the stator slots enhances the PM-LG performance. It results in a reduction in the cogging force as it will be demonstrated in the following sections. Due to symmetry of the generator around z-axis, the 2D FEM is sufficient to analyze it. 893 Figure 1 The axi-symmetric sketch of the tubular PM-LG the external surface of the PM-LG (nD02pr). The winding temperature rise 8, is related to the copper loss pcu in the coil by means of the overall heat transfer coefficient h, [9] as, Table I The actuator Specifications Slot pitch Slot width I Turndcoil Back-iron htcik: PM force Translator Air gap “le itch PM length PM width Mechanical air gap 1 1 1 ll(mm) c: 883 k N m 55 (mm) hm Variable variable 6 1 ‘lm where Do is the outer diameter of the stator, 2p is the number of poles, pcuis the copper resistivity, V,, is the volume of copper and J is the current density. Solving 5 and 6, one can get the appropriate current density in the coil conductors. 11. FIELDANDFORCECALCULATIONS The governing equation of the PM-LG is given by using the magnetic vector potential, A as [6]: V x [v(Vx A)]= J , +J , where J, is the exciting current density of the coil which is zero for generator, J, is the equivalent magnetization current density of the PM and can be written as: J, = V x (vp0M) where v is the magnetic reluctivity, po is the free space permeability, and M is the magnetization vector intensity of the PM. Using the FEM, the cogging force densities are calculated using Maxwell Stress Method as [71: f , =:-(. 1 2PO -49 1 f , =-BB,., Po (3) where B, is the normal magnetic flux density to the integral surface, B, is the tangential magnetic flux density to the integral surface. The generated electromotive force (em0 at stator coils terminals is calculated from Faraday’s law of magnetic induction as: d# dz emf =- N - dz dt (4) where N is the number of turns per coil, 4 is the flux passing in each turn in time t, z is the position of the translator. 111. CURRENT DENSITY CALCULATION The current of the output circuit of the PM-LG depends on the cross-section of the coil wire. For most electrical machine, the insulation class F is assumed for the coil windings with a temperature rise of 0, equals 115 “C [8]. The heat is transferred to the environment through IV. COGGING FORCE REDUCTION The main variable dimensions are optimized in order to minimize the cogging force value. The studied variables are the PM dimensions, the coil dimensions and the slot opening length. These variables are studied separately in the following sections as: A. PM Dimensions Effects The PM length is varied over a wide range with different PM width h, and the cogging force is calculated each time and plotted in Figure 3. The corresponding root mean square of the no-load back emf induced in the stator coils are plotted in Figure 4. Examining Figure 3, one can observe that the cogging force curve has a minimum for each value of the PM width h,. This minimum value always occurs when the PM length (7,) equals 22 mm. The cogging force in general increases with the increase of the PM width. The generated output voltage increases with the increase of the PM dimensions. For each magnet width, increasing PM length causes the emf to saturate to a constant value as shown in Figure 4. It can be seen that the optimal PM length (rm) equals 22 mm. B. Slot Opening Eflects The effect of slot opening so on the cogging force (FJ and the generated output emf (V,) is shown in Figure 5. The voltage, V, is directly proportional to the slot opening so, while there is fluctuations for the F, values. When so varies, the flux linking the coil changes as the PM passes through the slot, affecting the output voltage Vo. It is observed that decreasing the length of the slot opening leads to the increase of the iron path for the magnetic flux. This in turn decreases the F,. It is found that for this application the suitable the slot opening is around 12 mm C. Slot Width Effects To optimize the coil dimensions, the slot width s, is varied and its effects on the cogging force F, as well as on the output voltage V, is shown in Figure 6. Changing the slot width, s, inevitably affects the number of the coil turns. The cogging force F, has a minimum at a slot 894 width of 34 mm as shown in Figure 6. The corresponding emf and cogging force are 258 V and 97.5 N, respectively. V. WEIGHTANDCURRENT CALCULATIONS 5 10 15 20 25 30 35 After optimizing the overall dimensions of the PM-LG, its weight is calculated. The stator weighs 20.85kg, while the translator weighs 5kg. It is clear that the moving part is not heavy. The moving part in any machine should be as light as possible to enhance the machine performance. The weight of the PM’s should be small, as the PM material is the most expensive one in the design. Using 5, 6, and taking the copper conductivity of 58 MS/m, h, equals 24.3 W/(OC.m2) and 8, equals 115 “C [9], the current density of the copper conductor can be calculated. The calculated current density J is 3.5 A/mm2. The coil cross-section area equals 1020 mm’, and taking the coil fill factor equals 0.7, the net copper cross-section area equals 714 mm2. If the conductor cross-section is considered to be 1.6x4.25 mmz (for design purpose), the number of turns per coil is about 105 turns. From the calculated current density of the conductor and the conductor cross-section, the appropriate full load current ofthe generator is 23.8 A. 40 magnet length (mm) Figure 3 The cogging force versus PM length 350 300 250 1 200 150 100 50 0 5 10 15 20 25 30 40 35 VI. PM-LGPERFORMANCE magnet length (rnrn) The PM-LG is simulated under load conditions to calculate the generator performance and taking the armature reaction effect into consideration. After completing the fist loop, the internal back emf is calculated and from the equivalent circuit, the load current can be calculated. The output power and copper and core losses are then determined so the efficiency of the alternator can be calculated. Next the flux is calculated again from PM’s and currents of the winding. This newly flux is compared with the original flux. If the flux change exceeds a predetermined threshold, the entire process with the voltage calculation is repeated [IO]. This routine flow chart is shown in Fig. 7. The calculation routine is repeated with different resistive loads. The terminal voltage and output power of the generator are plotted in Fig. 8. As it can be seen that, when the load resistance value increases, the output power is decreased due to the decrease of the load current. At the same time, the output voltage is going to the open circuit value with decreasing the load. For different load resistance, the load characteristics of the generator are shown in Fig. 9. For the calculated full load current of 23.8A, the output power of the generator is 5.3kW, with a terminal voltage of 222V. From the no load and full load voltage values the generator regulation is 11.7%. The corresponding efficiency of the generator is 96.8%. The considered losses are copper and core loss only of 147W and 26W, respectively. It is worth to mention that the core losses are calculated for power and higher frequency values. Figure 4 The output voltage versus PM length 140 +Fc (N) 110 80 4 6 8 10 12 14 16 18 20 22 24 slot opening (mm) Figure 5 Slot opening effects, T~ 22 26 30 = 22 34 mm,hm = 20 mm,bi = 11 mm 38 42 46 50 coil length (mm) Figure 6 Effects of slot width, so = 12 m m 895 VII. CONCLUSIONS A tubular permanent magnet linear generator is designed using a two dimensional FEM Maxwell software by Ansoft Corporationo. The design based on minimizing the cogging force and maximizing the output induced voltage of the generator. The cogging force is reduced by optimizing the generator dimensions and using the optimal length for both the stator core and permanent magnet. The current density of the permanent magnet linear generator is calculated based on the thermal capacity of the copper wire; and hence the output power of the generator is calculated. The generator is analyzed under various resistive loads to take the armature reaction effect. A repetitive routine is used for each load value to calculate the output voltage and hence, the load characteristic is obtained. The efficiency of the designed tubular linear generator is calculated to be 96.8% at full load current of 23.8A with output voltage of 222V. 6 1 ,(*)=E@ t=t+dt ACKNOWLEDGEMENTS (+) The authors gratefully thank the Ministry of Science, Technology and Environment, Malaysia for supporting this project under IRPA Grant No. 33-02-03-3013. d VIII. REFERENCES Figure 7 Output voltage algorithm routine 240 12 225 10 - -210 * - Boldea and S . A. Nasar, “Linear electric actuators and generators”, IEEE Trans. Energy Conversion, Vol. 14, No. 3, pp. 712-716, 1999. M. Inoue and K. Sato, “An approach to a suitable stator length for minimizing the detent force of permanent magnet linear synchronous motors”, IEEE Trans. Magn., Vol. 36, No. 4, pp. 1890-1893,2000. J. Wang, G. W. Jewel1 and D. 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Theodore Wildi, “Electrical machines, drives, and power systems” 5th edition, Pearson Education New Jersey, 2002. N. Bianchi, S . Bolognani, D. Dalla, and F. Tonel, ‘‘Tubular linear permanent magnet motors: an overall comparison”, IEEE Trans. Mam.. DD. 466-475.2003. , Vol. 39. No. 24.... [lo] W. R. Cawthorne, “Optimization of a brushless permanent magnet linear alternator for use with a linear internal combustion engine”, Ph. D. West Virginia University, USA, 1999. 8 - k : 6 - 195 :180 4 165 2 150 % 0 3 7 11 15 19 23 27 31 35 load re61stance (ohm) Figure 8 Simulated output power and terminal voltage 12 10 8 z 6 ; 3 4 2 0 5 10 15 20 25 30 35 40 45 50 current (A) Figure 9 Load characteristics of the generator 896