Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Generator

Transcription

1 Journal of Magnetics 0(4), (015) ISSN (Print) ISSN (Online) htt://dx.doi.org/10.483/jmag Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Generator Hassan Moradi Cheshmeh Beigi 1 and Sohrab Akbari * 1 Electrical Engineering Deartment, Faculty of Engineering, Razi University, Kermanshah 67149, Iran Electrical Power Distribution Comany Kermanshah, Iran (Received 4 June 015, Received in final form 11 Setember 015, Acceted 17 Setember 015) In this study, we describe the effects of changing the magnet shae of ermanent magnets (PMs) in a rotor Halbach-array PM generator for recirocating free iston generator alications. More secifically, the rectangular-shaed magnets were relaced by the traezoidal-shaed magnets. The initial design, an analytical magnetic field solution of rectangular shaed magnets, is resented and air-ga magnetic flux density and thrust force were estimated. The results were comared to the finite element analysis (FEA) showing excellent agreement. Using FEA, the effect of the shae of the magnets on the flux density and thrust force waveforms is analyzed. Moreover, the roortion of the Halbach array in the machine was otimized by the means of a arametric search. The results obtained from the analytical calculations and FEA were validated by comaring to those of Radial-array PM generator. Keywords : Dual-Halbach Array, free iston generator, finite element analysis, arametric search. 1. Introduction In this study, we describe the merits of changing the PM shae in Halbach arrays alied to tubular generators. These generators are increasingly used owing to their high efficiency, high ower/force density, and down rile force characteristics [1]. This class of machines is found to be suitable for recirocating free iston generator alications such as in an imroved outut ower and a voltage for free iston generator, warranting the utmost average thrust force and least value of cogging force []. Although a few core-tye Halbach-array PMs [3, 4] have been reorted, the ossibility of a coreless configuration utilizing a dual Halbacharray PM has not been investigated in detail. PM tubular generators have been roved to rovide the best all-around erformance for recirocating free iston generator alications in comarison to induction-tye generator and switched reluctance generator, because of their high force and high efficiency [5-7]. A radial array, which is a secial case of PM generators, is less used for The Korean Magnetics Society. All rights reserved. *Corresonding author: Tel: Fax: , Sohrab.akbari7@yahoo.com recirocating free iston generator alications, because of its high rile force roduction. The magnitude of the detent force resented in the radial array is almost 13% of the develoed force [8]. Core-tye Halbach array PM generators also give rise to rile force including the cogging force roduced by the machine. Cogging is one of the disadvantages faced in the slotted generator design, as it causes a rile in the force generated by the generator [9]. The advantages of slotless Permanent Magnet Tubular Generator (PMTG) have numerous attributes making them an ideal choice for a lot of alications [10]: Zero-cogging torque for smooth oeration and minimal vibration. Excellent ower-to-weight ratio enabling the design of comact, light-weight hand tools. Dual Array PMTG A dual array PMTG utilizes a slotless stator winding unlike the conventional slotted-tye PMTG. The concentrated tye of winding is used owing to the advantage of no overlaing of hase windings. Being air cored, this machine ossesses zero cogging thrust force. The introduction of horizontal magnetized magnet eliminates the flux through the rotor iron core, and hence the core loss. 015 Journal of Magnetics

2 406 Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Hassan Moradi Cheshmeh Beigi and Sohrab Akbari ) The back iron is infinitely ermeable. 3) The ermeability of PM material is equal to the ermeability of free sace. Therefore, the governing field equations, in terms of magnetic vector otential, lead to Lalace and Poisson equations as follows [11]: A1 0 in region 1, A. µ 0JM in region (1) () The vector otential A has only one comonent, A(θ), which is indeendent of θ in the cylindrical system. Therefore, Equations (1) and () can be written as Fig. 1. PMTG with dual Halbach array. In addition to that, the air-ga flux density also increases, resulting in high thrust force. With these advantages, this machine is an ideal candidate for low-seed high-force recirocating free iston generator alications requiring outut ower of more than the conventional slotted-tye PMTG. Figure 1 shows the PMTG with a dual Halbach array. 3. Formulation of Magnetic Field and Thrust Force Magnetic field modeling is a recondition of thrust force formulation, and analytical field model is a owerful tool for designing and analyzing linear machines. From the Maxwell equation and magnetic roerty of different materials, the governing equations of magnetic field, i.e., Lalace's and Poisson's equations are significant for the solution of magnetic field. The general schematic of the machine shown in Fig. is divided into two regions, in which region is the Halbach magnetized magnets and the other region is air/winding region. To establish an analytical solution for the field distribution roduced in a multiole Halbach machine, the following assumtions were made: 1) The length of the machine is extended to infinity. Fig.. (Color online) General schematic diagram of a coreless dual Halbach-array PMTG ( 1 ) ( 1 ) 0 in region 1,(3) raθ + ra θ z r z r r r ( ra θ) ( ra θ ) + µ 0JM in region.(4) z r z r r r The remanent magnetization vector can be written in the form of its comonents M M r e r + M z e z, and the flux density B obtained from A is reresented by B z 1/r. / r( ra θ ) and B r A θ / z. In the Fourier series form, the magnetization vector M is decomosed into harmonics as shown by Equation (5). Mr Mrnsin mnz, Mz Mznsin mnz, where τ is the PM ole itch and B r is the remanence. Therefore, Equation () can be written as ( ) ( ), (7) ra1 θ + ra1 θ 0 in region 1 z r z r r r ( ra θ) ( ra θ) + Pn cos mnz in region, (8) z r z r r r where (5) (6)

3 Journal of Magnetics, Vol. 0, No. 4, December B Pn τ mnτ mτ sin sin rem n mr. (9) As a result, the general solutions to the Lalace and Poisson equations are: θ1 ( 1n 1( n ) + 1n 1( n )) ( n ) (( n ( n ) n ( n )) ( n ) ( )) A a BI m r b BK m r cos m z Region 1 A a BI m r + b BK m r cos m z + S r, z Region θ 1 1,(10).(11) Equation (11) is the general solution of Equation (8), which is a nonhomogeneous Bessel s differential equation. Therefore, Equation (11) contains Struve functions [1] that exist in the unknown term S(r, z). Solving Equation (8) for S(r, z) yields: (, ) S r z ( ) cos( ) 1 π StruveL1 m r P m z n n n mn, (1) where Struve L 1 is the modified Struve function of order one. The flux distributions in all the two regions are derived from Equations (10) and (11), yielding to the following equations. ( ) ( ) Br1 mn a1 nbi1( mnr) + b1 nbk1( mnr) sin mnz Region 1,(13) z1 n 1n 0 n 1n 0 n n r n( n 1( n ) + n 1( n )) ( n ) + z n( n 0( n ) n 0( n )) ( n ) + ( ) ( ) B m a BI ( m r) b BK ( m r) cos m z Region 1,(14) B m a BI m r b BK m r sin m z S( r, z) Region B m a BI m r b BK m r cos m z S( r, z) Region,(15),(16) where BI 0 and BI 1 are the modified Bessel function of the first kind of order 1; BK 0 and BK 1 are the modified Bessel function of the second kind of order 1; and a n and b n are constants. Equations (13) to (16) reresent the field density of the whole machine. Br1 and Bz1 reresent the radial and axial magnetic field in the winding region, whereas B r and B z are in the magnetic regions. The uer scrit, 1;, reresents the external and internal PMs, resectively. The boundary conditions to be satisfied by the solution to Equations (7) and (8) are: B µ M, B µ M, B B, z r R 0 z z 0 z r 1 r R r s r Rr b r Rb z r R z r r z z b r R r Ra r R r Rb r R H H, B B, H H, b a a The thrust force outut of electromagnetic linear machine is governed by the Lorentz law. The current in the windings interacts with magnetic field to generate an axial thrust force reresented by Equation (17). ( ) F J B dv, (17) w V where J is the current density. Figure 3 shows the structure of a single hase machine. Each winding occuies an axial width of τ w. The two coils of one hase are searated by a distance of τ w. The thrust force exerted on one coil is as follows. τ w FW 4π J Krnsin mn cos( ), (18) 1 mnz where R i is the outer radius of the armature. K rn is given by: Ri rn 1n 1( n ) + 1n 1( n ) r Ra K r a I mr b K mr d. (19) Therefore, the thrust force exerted on a single winding is reresented by the following equation. τ w Fw Fw ( z) Fw ( z τw ) 8πJ Kn sin mn z, (0) 1 where K n is given by τ w τ w Kn sin mn sin mn Krn. (1) Substituting J J rms cos ωt into equation (0) gives τ w Fw 8 Jrms Kn sin mn z cosωt, () 1 where J rms is the root mean square of the current density; and τ w is the winding itch. R i is defined as R i R a + g, where R a is the outer radius of the internal PMs; R b is the inner radius of the external PMs; g is the size of the air ga, and τ w is the axial width of the hase winding er ole. Given v T --, where v is the velocity of the τ mover, following equation is obtained. v t π π ω t ( z x), (3) τ τ where x and z are the starting osition and the current osition of the hase winding, resectively. Equation (3) shows that the starting osition and current osition affect the instantaneous current and thus thrust force outut. Substituting into Equation () yields the thrust force roduced by a single-hase winding. τ w π F 8 sin cos ( ).(4) w Jrms Kn mn z z x 1 τ

4 408 Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Hassan Moradi Cheshmeh Beigi and Sohrab Akbari Fig. 3. Single-hase winding structure. The machine thrust force is not only related to the current motion osition z, but also the starting osition x. 4. Analytical Results The closed form solution derived in the revious section is used for comuting the radial comonent of the air-ga flux density and thrust force for the rectangular shaed dual Halbach-array PMTG. Figure 4 shows the air-ga flux density variation of dual quasi-halbach array and radial array, which was calculated by Equation [13]. The amlitude of the flux density of dual Halbach-array PMTG follows the higher amlitude obtained for the radial-array PMTG. Thrust force is imortant for the analysis of machine erformance. Based on the derived mathematical model, thrust force for a PM linear machine was studied. The result of the thrust force for dual quasi-halbach array and radial array toologies is shown in Fig. 5. As shown, the Fig. 5. (Color online) Thrust force of radial array and dual quasi-halbach array toologies. amlitude of the flux density of dual Halbach-array PMTG is higher than the amlitude obtained for the radialarray PMTG. 5. FE Modeling and Analysis of the Machine 5.1. FE Modeling and Analysis A -D nonlinear time-steing FE method was used to verify the accuracy of the results obtained from the analytical results. The imagery of the machine was carried out using a COMSOL software ackage. Figure 6 shows the flux lines in the PMTG obtained by the FE methods. Clearly, the horizontal magnetized magnet acts as a ath for the flux between the adjacent oles and reduces the flux in the shaft. This effect will be clearer with an increased roortion between the horizontal and vertical magnetized magnets. Fig. 4. (Color online) A comarison of the flux density of the dual Halbach array and radial array toologies. Fig. 6. (Color online) Flux distributions in dual quasi-halbach array.

5 Journal of Magnetics, Vol. 0, No. 4, December Fig. 9. (Color online) Quasi-Halbach magnetization attern. (a) Rectangular shaed PMs. (b) Traezoidal shaed PMs. Fig. 7. (Color online) Comarison between the analytically redicted and FE methods calculated distributions of the flux density. Fig. 10. (Color online) Effect of τ mz on the waveform of the Radial comonent of the thrust force in the air ga. Fig. 8. (Color online) Comarison of analytically and FE methods-redicted thrust force waveforms. A comarison between the analytically redicted and FE methods calculated distributions of the flux density comonent, as a function of z osition in dual quasi- Halbach array, is shown in Fig. 7. A slight difference was observed between the two methods. Figure 8 comares the analytical and FE redicted thrust force in dual quasi-halbach array. As shown, the analytically redicted thrust force waveforms agree well with those deduced from the FE methods analysis. 5.. FE Otimization of Dual Quasi-Halbach Array PMTG In this study, the first FE methods are utilized for designing otimization rectangular shaed PMs, and then FE methods are carried out to design otimized traezoidal shaed PMs. Figure 9(a) shows the Quasi-Halbach magnetization attern with rectangular shaed PMs, and Fig. 9(b) shows the Quasi-Halbach magnetization attern with Traezoidal shaed PMs RECTANGULAR SHAPED PMs In this section, this model describes the merits of Fig. 11. (Color online) Effect of τ mz on the waveform of the radial comonent of the flux density in the air ga.

6 410 Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Hassan Moradi Cheshmeh Beigi and Sohrab Akbari Table 1. Variation in the erformance arameters with the change in horizontal magnetized magnet roortion. τ mr τ mz Average Force (K.N) Peak Force (K.N) Force Rile Average Flux density (T) Peak Flux density (T) Flux density Rile changing the rectangular shaed PM in Halbach arrays. The variation in the develoed thrust force attern, with the change in roortion of the vertical magnetized magnet (τ mr ) and the horizontal magnetized magnet, (τ mz ) is shown in Fig. 10. Here, the rile and eak thrust forces increase and average force decrease with increasing roortion of the horizontal magnetized magnet to the vertical magnetized magnet. The flux density distribution in the air ga was calculated for the variation in the vertical magnetized magnet and the horizontal magnetized magnet. Figure 11 shows the magnetic flux density in the air ga of the rectangular shaed PMs. The variation in the eak value, average value, and rile for the thrust force and flux density of PMTG is listed in Table 1. You are kindly advised to use this temlate for editing your submission, as you have nothing to change in terms of aer and text format. Simly alying the styles defined here will be sufficient TRAPEZOIDAL-SHAPED PMs In this section, this model is extended to be able to investigate the effect of the angle α m of the traezoidal magnet on the thrust force and flux density. In order to increase the average thrust force and decrease the rile, the vertical magnetized magnet is increased to the horizontal magnetized magnet (τ mr 0.6 τ m ). The flux density distribution in the air ga and thrust force were calculated for several values of α m. Figure 1 shows a comarison among the simulation results of the thrust force of the three roosed designs. The amlitudes of the thrust force of the PMTG follow the higher amlitude obtained for the roosed design α m 75; however, the average value of the thrust force in α m 105 is higher than other cases. Figure 13 shows a comarison of the simulation results Fig. 1. (Color online) Effect of α m the waveform of the Radial comonent of the thrust force in the air ga. Fig. 13. (Color online) Effect of α m the waveform of the Radial comonent of the flux density in the air ga. of the ermanent flux density for the three roosed designs. The variation in the eak value, average value, and rile for the thrust force and flux density of the Table. Variation In The erformance arameters with the change in α m roortion. α m (degree) Average Force (K.N) Peak force (K.N) Force Rile Average Flux density (T) Peak Flux density (T) Flux density Rile

7 Journal of Magnetics, Vol. 0, No. 4, December the traezoidal toology for α m 10 has more uniformity and lower flux leakage than the rectangular one, thus decreasing the rile force (1.57%) and increasing the average flux density in the air ga. Consequently, the electrical characteristic of the generator imroved. Figure 15 demonstrates the distribution of the flux for the traezoidal and rectangular toologies. 6. Outut Power Fig. 14. (Color online) 3D FE methods investigation of the dual quasi-halbach-array PMTG. To obtain outut ower for the PMTG, the FE methods were simulated by moving the mesh techniques [14]. The motion of the translator was modeled by moving translator mesh without any modification to the mesh toology. The model recirocating motion mover was simulated by the force balance equation, as shown in Equation (5) and reeated here as Equation (5) over one comlete stroke [13]. n n n r x x AP B mx, (5) r+ 1 xm x m Fig. 16. (Color online) Comarison of outut ower radial array and dual quasi-halbach array toology. Fig. 15. (Color online) Flux distributions in generator (a) Traezoidal toology (b) Rectangular toologiy. PMTG is listed in Table. As shown, the average thrust force increases and rile force decreases with increasing roortion of the vertical magnetized magnet to the horizontal magnetized magnet. The FE model of the designed dual quasi-halbach-array PMTG in τ mr 0.6τ m and α m 105 is shown in Fig. 14. According to the results obtained from the FE methods, Table 3. Generator arameters Number of active rotor oles P 4 Poles Pole itch τ Mm Diameter D Mm Number of windings N s 96 Turns Magnet Height h m Mm Magnet width l w Mm Magnet Ga W mg Mm Air Ga g Mm Magnet Inner Diameter ID m.6035 Mm Magnet outer Diameter OD m Mm

8 41 Dual-Halbach Array Permanent Magnet Tubular Generator for Free-Piston Hassan Moradi Cheshmeh Beigi and Sohrab Akbari where A B is the bore area; P 1 is the inut intake ressure, r is the comression ratio; n is the ratio of secific heats; X is the translator osition; X m is the maximum half stroke; and m is the mass of the translator. The load testing was conducted using a urely resistive variable load. Figure 16 shows the exerimental and simulated generator outut ower used in the exerimental of the generator and the geometric arameters [13]. The geometric arameters of the generator are listed in Table 3. The engine generator system was caable of sulying a eak ower of 315 W for radial array and 340 W for dual quasi-halbach array. 7. Conclusions In this study, a modest attemt was erformed to design a novel dual quasi-halbach-array PMTG in recirocating free iston generator system. An analysis based on the Maxwell equations was derived to redict the air ga magnetic flux density distribution and thrust force. The otimization of the machine was erformed by the FE methods. The traezoidal dual quasi-halbach array magnet configuration hels in achieving high thrust force, increasing outut ower, and uniform flux density along the air ga of the machine, thereby decreasing the thrust force rile. The designed machine is simulated, and the results obtained from the analytical calculations were validated with those of the FE methods. References [1] S.-M. Jang and J.-Y. Choi, Journal of Electrical Engineering & Technology, 1 (007). [] J. Wang, M. West, D. Howe, Hector Zelaya-De La Parra, and W. M. Arshad, IEEE Trans. Energy Convers., 99 (007). [3] J. Wang and D. Howe, IEEE Trans. Magn. 41, 479 (005). [4] W. H. Arof, and H. W. Ping, IET Electr. Power Al. 4, 69 (010). [5] J. Faiz1, B. Rezaeealam1, and S. Yamada, IEEE Trans. Magn. 4, 17 (006). [6] J. Pan, Y. Zou, and G. Cao, IET Renew. Power Gener, 7, 98 (013). [7] W. M. Arshad, Ph.D. dissertation, Royal Inst. Technol., Stockholm, Sweden, (003). [8] S.-M. Jang, S.-H. Lee, and I.-K. Yoon, IEEE Trans. Magn. 38, 361 (00). [9] J. Wang, D. Howe, and G. W. Jewell, IEEE Trans. Magn. 39, 3517 (003). [10] L. Yan and L. Zhang, Progress In Electromagnetics Research 136, 83 (013). [11] J. Wang and D. Howe, IEEE Trans. Magn. 41, 470 (005). [1] M. Abramowitz and I. A. Stegun, National Bureau of Standards, Washington, DC, 10th Printing, 496 (197). [13] W. R. Cawthorne, P. Famouri, J. Chen, N. N. Clark, T. I. McDaniel, R. J. Atkinson, S. Nandkumar, C. M. Atkinson, and S. Petreanu, IEEE Trans. Veh. Technol. 48, 1797 (1999). [14] S. L. Ho, N. Chen, and W. N. Fu, IEEE Trans. Magn. 47, 947 (011).