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1 Powered by TCPDF ( Ths s an electronc reprnt of the orgnal artcle. Ths reprnt may dffer from the orgnal n pagnaton and typographc detal. Author(s): Ttle: Martn, Floran & Belahcen, Anouar & Lehkonen, Antt & Raslo, Paavo Homogenzaton Technque for Axally Lamnated Rotors of Synchronous Reluctance Machnes Year: 2015 Verson: Post prnt Please cte the orgnal verson: Martn, Floran & Belahcen, Anouar & Lehkonen, Antt & Raslo, Paavo Homogenzaton Technque for Axally Lamnated Rotors of Synchronous Reluctance Machnes. Volume 51, Issue DOI: /TMAG Rghts: 2015 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other work. All materal suppled va Aaltodoc s protected by copyrght and other ntellectual property rghts, and duplcaton or sale of all or part of any of the repostory collectons s not permtted, except that materal may be duplcated by you for your research use or educatonal purposes n electronc or prnt form. You must obtan permsson for any other use. Electronc or prnt copes may not be offered, whether for sale or otherwse to anyone who s not an authorsed user.

2 IEEE TRANSACTIONS ON MAGNETICS, VOL.?, NO.?, Homogenzaton Technque for Axally Lamnated Rotors of Synchronous Reluctance Machnes Floran Martn, Anouar Belahcen, Antt Lehkonen, and Paavo Raslo Aalto Unversty, Department of Electrcal Engneerng and Automaton, Otakaar 5, Espoo, Fnland In ths paper, we propose a homogenzaton technque to model the axally lamnated rotor of synchronous reluctance machnes. Thus, the computatonal effort can be sgnfcantly reduced by replacng the lamnated parts of the rotor by some equvalent ansotropc meda. The proposed method s valdated n terms of flux densty and electromagnetc torque. Some small dscrepances can be notced due to the ar-gap fluctuatons caused by the steel sheets and the nterlamnar nsulaton sheets of the rotor. Wth the test machne, the homogenzaton method reduces by the number of elements to one fourth and the computaton tme to one thrd. Index Terms Electromagnetc torque, fnte element analyss, homogenzaton technque, magnetc felds, magnetc flux, reluctance motors, synchronous machnes I. INTRODUCTION AXIALLY lamnated synchronous reluctance machnes have receved growng nterest n recent years [1], [2]. Due to ther nherent smplcty and low cost, these machnes are vable canddates for varous general applcatons wth adjustable speed. Wth axally lamnated rotors, sychronous reluctance machnes ncrease the electromagnetc torque by rasng ther salency rato [1], [2]. Moreover, ther qute smooth rotor decreases sgnfcantly ar frcton losses whch are of great sgnfcance for hgh speed applcatons [3], [4]. The constructon of the axally lamnated rotor conssts of assemblng a stack of steel lamnatons wth nterlamnar nsulaton sheets n order to enhance the salency rato wth dfferng reluctances on drect and quadrature axes. Nonmagnetc bolts and pole holders are mounted to fx the lamnatons. The stack of steel lamnatons can be composed of non-orented steel or gran-orented steel. Wth gran orented lamnatons, the effcency s slghtly hgher but the core losses n stator and rotor also ncrease [5]. The calculaton of magnetc quanttes such as the electromagnetc torque can be performed wth analytcal models [1], [6] or numercal methods [1], [5], [7]. The fnte element method s generally appled when a hgh degree of accuracy s needed, but t requres a detaled characterzaton of the geometry [6], [7]. Every steel lamnaton and nterlamnar nsulaton sheet are dscretzed nto fnte elements. When the lamnaton thckness s small compared to the rotor radus, the computatonal effort s huge because the rotor geometry dscretzaton contans a hgh number of nodes and elements. In ths paper, we propose to reduce the computatonal effort and tme by modelng the axally lamnated rotor parts wth a homogenzaton technque so that a coarser mesh can be used to dscretze the axal lamnatons. Usually, Manuscrpt receved March 20, 2015; revsed????, Correspondng author: F. Martn (emal: floran.martn@aalto.f). homogenzaton methods are appled on the stator wndngs [8], or on test ron stacks [9], [10], [11], [12]. Thus, the homogenzaton technque nvolves both magnetc and electrc propertes of the dfferent materals. In case the thckness of the axal lamnatons s lower than twce the smallest skn depth, the eddy current flowng n these lamnatons can be neglected [13] and so the axally lamnated parts can be homogenzed only n terms of magnetc propertes. In ths paper, the homogenzaton technque of the axally lamnated parts s frst detaled, then the fnte element method s brefly descrbed. Fnally, the homogenzaton method s valdated n terms of ar gap flux densty components and electromagnetc torque by comparng computatons for models of fully lamnated rotor parts and homogenzed rotor parts. II. METHOD OF ANALYSIS A. Homogenzaton technque The stack of steel sheets wth nterlamnar nsulaton sheets s homogenzed and the consttutve relaton between magnetc quanttes s derved. Ths stack of lamnatons can be homogenzed wth an ansotropc medum (Fg. 1). Its equvalent reluctvty can be calculated by consderng a seral and a parallel assocaton of both the steel sheets and the nsulaton sheets, respectvely for the lamnaton drecton u and the orthogonal drecton v of the lamnaton [12]. The magnetc feld components, h u and h u, n the homogeneous medum can be computed by: h u (b u ) = ν 0 ν (b u ) (1 α) ν (b u ) + α ν 0 b u h v (b v ) = α ν (b v ) b v + (1 α) ν 0 b v (1) where b u and b v are the components of the magnetc flux densty n the lamnaton coordnate system, ν s the reluctvty of the steel sheet wth the reluctvty of the nterlamnar nsulaton sheets consdered as the reluctvty of vacuum ν 0, and α s the rato of steel lamnatons n the stack.

3 2 IEEE TRANSACTIONS ON MAGNETICS, VOL.?, NO.?, 2015 #» v #» u Interlamnar nsulaton sheet (ν ν 0 ) Steel lamnaton (ν = ν ) Homogenzed ansotropc medum ν u (ν 0, ν, α) ν v (ν 0, ν, α) ν v Fg. 1. Prncple of homogenzaton of a stack of steel sheets wth nterlamnar nsulaton sheets. Upper fgure shows the fully lamnated model, lower one presents the homogenzed model. ν s the reluctvty of the materal wth the ndex as ron, u and v ndces relate to the respectve drectons. B. Fnte element method The cross secton S of the axally lamnated synchronous reluctance machne s dscretzed nto second order tranglular elements and the magnetostatc problem s formulated n terms of the magnetc vector potental A. Wthn the consdered 2D geometry, the axal component of the magnetc vector potental A s governed by the partal dfferental equaton gven by: ν u α d ν (A e z ) = J e z (2) where ν s the reluctvty dependng on both the consdered regon and the components of the magnetc flux densty, J s the current densty source n the conductors, and e z the unt vector n the z drecton. Ths equaton s solved wth the Galerkn s method by mnmzng the energy functonal F gven by [14]: F = S [ ] B H T db J T A ds (3) 0 where B and H are the magnetc flux densty and the magnetc feld respectvely. Snce ron based regons are non-lnear, ths energy functonal s also non-lnear and ts mnmzaton requres an teratve process. The estmated magnetc vector potental A k s updated after each k teraton wth the Newton-Raphson method. Its expresson s gven by: A k+1 = A k P k 1 R k (4) d where the resdual vector R and the Jacoban matrx P are composed of the terms gven by: R = (N e z ) T H J N ds S [ ] H P j = (N e z ) T (N j e z ) ds S B wth N the shape functon of the fnte element method, and j are node numbers. Except for homogenzed regon, every materal s consdered sotropc so the ncremental reluctvty tensor H/ B s a dagonal matrx. Wthn the homogenous ansotropc meda, the magnetc feld components h u and h v are expressed n the lamnaton coordnate system (O, u, v). In order to mplement the homogenzed model, we should express these magnetc feld components n the global coordnate system (O, x, y). Snce the homogeneous meda are located n the rotor, the relaton between the lamnaton drecton u and the x axs depends on the angular poston of the rotor. Moreover, n case the lamnaton sheets are bended, the lamnaton drecton can be consdered as the tangent of the curved lamnaton and ths relaton also depends on the coordnate of the ntegraton ponts (Fg. 2). Thus, we can calculate h x and h y by rotatng the magnetc feld components h u and h v : [ ] [ ] [ ] [ ] hx hu cos θ sn θ hu = R (θ) = (6) sn θ cos θ h y h v where θ s the angle between the x axs and the lamnaton drecton u. #» v #» y Insulaton sheet Steel lamnaton θ #» u #» x Fg. 2. Relaton between the lamnaton coordnate system (O, u, v) and the global coordnate system (O, x, y). The left fgure presents a rotor at an arbtrary angular poston wth straght lamnaton. The rght fgure shows a rotor wth curved lamnaton. The ncremental reluctvty tensor s calculated wth a smlar manner by: h x h x h u b x b y 0 = R (θ) b u R T (θ) (7) h y b x h y b y 0 h v b v The Newton-Raphson method converges f the ncremental reluctvty tensor s postve defnte. #» v #» y h v θ (5) #» u #» x

4 HOMOGENIZATION TECHNIQUE FOR AXIALLY LAMINATED ROTOR OF SYNCHRONOUS RELUCTANCE MACHINES C. Torque calculaton Wth constant flux Ψ, e. constant magnetc vector potental, the electromagnetc torque Γemg can be determned by dfferentatng the magnetc energy Wmag wth respect to the rotor dsplacement φ by : Γemg = Wmag (8) Ψ=constant Wth a smlar method descrbed n [15], [16], the electromagnetc torque s computed n the ar-gap wth the method of vrtual dsplacement wth constant magnetc vector potental : Nag Z X G 1 G Γemg = ν0 L B T G 1 B G B 2 ds (9) 2 e=1 Se where L s the actve length of the machne, Nag s the number of elements n the ar-gap layer, G s the Jacoban matrx of the transformaton from (x, y) coordnate to (ξ, η) coordnate of the reference trangle, G s ts determnant, and Se s the surface of the reference trangular element. G s defned for soparametrc transformaton by : N N y x Np X ξ ξ (10) G = N N =1 y x η η 3 TABLE I PARAMETERS OF THE TEST MACHINE Number of pole pars Number of phases Outer radus of the stator core [mm] Core length [mm] Inner radus of the stator [mm] Outer radus of the rotor [mm] Number of stator slots Thckness of rotor lamnaton [mm] Rato of steel n the lamnaton Appled current per phase [A] A. Magnetc flux densty In Fgs. 3-5, we can notce that the flux lnes follow the same drecton n both models. The ampltude of the flux densty n the homogenzed rotor s equal to the average flux densty n the steel sheets and the nterlamnar nsulaton sheets. Moreover, the ampltude of the magnetc flux densty presents smlar values n the stator. Besdes the fact that the flux lne and the average flux densty are the same, n Fg. 4, the ansotropy effect can be apprecated n the behavor of the magnetc flux. In Fg. 5, the stator wndngs produces flux n the transverse drecton of the rotor stack that heavly reduces the total flux n the machne. Results of homogenzed and non-homogenzed models are very smlar. where Np s the number of nodes n the tranglular element (Np = 6 for second order trangles). The dfferentatons G / are calculated by : N x N y Np ξ φ G X ξ φ (11) = N x =1 N y η φ η φ The dfferentaton of the determnant of the Jacoban wth respect to the vrtual dsplacement can easly be computed wth the terms of G and G /. (a) Lamnated rotor In case all the nodes of one element are vrtually movng from the same angle, the terms of G / and G / become nl so only the elements whch contan at least one movng node and not all of them can be consdered to compute the electromagnetc torque. Ths means that only dstorted elements from the ar-gap are needed n ths computaton. In our models, these elements are confned wthn a predefned layer located n the mddle of the ar-gap. III. R ESULTS AND DISCUSSION In order to valdate the proposed method, an axally lamnated synchronous reluctance machne s analyzed by modelng both a fully lamnated rotor and a homogenzed ansotropc rotor wth the same mesh for dfferent load angles and the same current densty. The machne s meshed wth second order trangles and nodes, among whch nodes are n the rotor. The man parameters of the test machne are reported n Table I. (b) Homogeneous rotor Fg. 3. Flux densty ampltude and flux lnes n the reluctance machne wth rotor at 0 deg. The flux lnes follow the same drecton n both models. The ampltude of the flux densty n the homogenzed case s equal to the average flux densty n the full model. In Fgs. 6-8, we can notce that the components of the magnetc flux densty n the mddle of the ar-gap are smlar wth the homogeneous rotor and the axally lamnated rotor. However, the ar gap flux densty computed wth the fully modeled lamnatons presents some small addtonal fluctuatons. They are caused by the magnetc ar-gap varatons due to the alternaton of the steel sheets and the nsulaton sheets. Snce the homogenzaton technque does not consder ths phenomenon, the ar-gap flux densty does not nclude these small fluctuatons n ths case. In Fg. 9, we can perceve that, for 45 deg. of load angle, the crcumferental components of magnetc flux densty presents some hgher dfference (29.6 %) than the radal component (19.2 %).

5 4 IEEE TRANSACTIONS ON MAGNETICS, VOL.?, NO.?, 2015 (a) Radal component of the magnetc flux densty (a) Lamnated rotor (b) Homogeneous rotor Fg. 4. Flux densty ampltude and flux lnes n the reluctance machne wth rotor at 45 deg. Besdes the fact that the flux lnes and the average flux are the same, the ansotropy effect can be observed n the behavor of the flux. (b) Crcumferental component of the magnetc flux densty Fg. 7. Flux densty components n the ar-gap wth rotor at 45 deg. Increasng the load ange causes a decrease of ar-gap flux densty, whch s correctly modeled by both the homogenzed and the full model. (a) Lamnated rotor (b) Homogeneous rotor Fg. 5. Flux densty ampltude and flux lnes n the reluctance machne wth rotor at 90 deg. The fact that the stator produced flux s now n the transverse drecton of the rotor stack reduces heavly the total flux n the machne. (a) Radal component of the magnetc flux densty (a) Radal component of the magnetc flux densty (b) Crcumferental component of the magnetc flux densty Fg. 8. Flux densty components n the ar-gap wth rotor at 90 deg. The flux practcally does not flow as t has to go along the lowest reluctvty drecton correspondng to the transverse lamnaton. Once more, the homogenzed model takes correctly ths effect nto account. (b) Crcumferental component of the magnetc flux densty Fg. 6. Flux densty components n the ar-gap wth rotor at 0 deg. Except for the hgh spatal harmonc caused by the rotor lamnatons, the results of the homogenzed and full models are very smlar. In Fg. 7, ncreasng the load angle causes a decrease of the ar-gap flux densty, whch s correctly modeled by both the homogenzed and the full model. In Fg. 8, the flux practcally does not flow as t has to go along the lowest reluctvty drecton correspondng to the transverse lamnaton. Once more the homogenzed model takes correctly ths effect nto account.

6 HOMOGENIZATION TECHNIQUE FOR AXIALLY LAMINATED ROTOR OF SYNCHRONOUS RELUCTANCE MACHINES 5 (a) Radal component (b) Crcumferental component Fg. 9. Zoom of the flux densty components n the ar-gap wth rotor at 45 deg. Some bgger dscrepances are reached for the crcumferental components (29.6 %). The radal component presents at maxmum a relatve error of 19.2 %. B. Electromagnetc torque The electromagnetc torque was computed n the ar-gap of the axally lamnated synchronous reluctance machne for dfferent rotor postons, from -90 to 90 by step of 5. Wth the method of vrtual dsplacement descrbed n II-C, t s mportant that the vrtually movng elements reman the same for any rotor poston, so the torque s computed n an annulus layer of the ar-gap [16]. In Fg. 10, we can notce that the electromagnetc torque presents the well-known torque snusodal dependance on the load angle. Moreover, the rotor wth fully modeled lamnatons and the homogeneous rotor wth the same mesh and a coarse mesh present smlar torque computatons regardless the fact that the coarse mesh presents one fourth of elements. Furthermore, the dfference n the torque computaton between homogenzed and non-homogenzed models s only 8.02 % at ts maxmum. These dscrepances are manly caused by the magnetc ar-gap fluctuatons due to alternaton of the steel sheets and the nterlamnar nsulaton sheets that were neglected n the homogenzed medum. In Table II, the accuracy and computaton tme of the torque computaton are compared for the rotor wth fully modeled lamnatons and for the homogeneous rotor wth the same mesh and a coarse mesh. Wth the same mesh, the computaton tme wth the homogeneous rotor s slghtly larger than wth the fully modeled lamnatons. Ths can be explaned because more elements are non-lnear n the homogenzed rotor compared wth the fully modeled lamnatons composed of nterlamnar nsulaton sheets whch are lnear. However, snce the homogeneous rotor can be dscretzed wth a coarse mesh, the number of elements can be reduced to one fourth and the computaton tme can be reduced to one thrd. Moreover, the accuracy of the torque computaton remans smlar wth a coarse mesh and wth the fully lamnated mesh. Thus, most of the dscrepances arse from the neglected magnetc ar-gap fluctuatons due to the alternaton of the steel sheets and the nterlamnar nsulaton sheets. TABLE II COMPARISON OF MODELS ACCURACY AND THEIR COMPUTATION TIMES Rotor model Fully modeled lamnatons Homogeneous wth fully lamnated mesh Homogeneous wth coarse mesh Number of Number of Computaton Relatve elements nodes tme [s.] error [%] These magnetc ar-gap fluctuatons are manly dependng on the steel rato α of the lamnaton stack. The fgure 11 shows the mpact of ths rato on the electromagnetc torque for a load angle of 45 deg. Frst, we can notce that the maxmum electromagnetc torque s reached for 60 % of steel n the lamnaton stack. Wth ths rato, the relatve error between homogenzed and non-homogenzed computaton s 7.93 % only. When the steel rato s ether 0 or 1, the electromagnetc torque should be nl snce the smooth rotor would be composed of exclusvely nterlamnar nsulatng materal or steel respectvely. In those cases, the nductance n the drect axs would be equal to the nductance n the quadratc axs, resultng n a nl reluctance torque. Fnally, we can perceve that the torque dscrepances (27.4 %) between homogenzed and non-homogenzed computaton are maxmum wth a steel rato of 20 %. Even f ths relatve error s qute small, t would be reasonable to keep the steel rato near 50 % snce the desgn of an axally lamnated synchronous rotor ams to ncrease the rato between the quadratc and the drect nductances. Fg. 10. Electromagnetc torque for dfferent rotor postons. The homogenzed models compute the torque wth smlar accuracy regardless of the fact that one of them has half the numbers of elements. Furthermore, the dfference between homogenzed and non-homogenzed computaton s only 8% at ts maxmum. IV. CONCLUSION In ths paper, we proposed a homogenzaton technque for an axally lamnated rotor of a synchronous reluctance machne. The axally lamnated rotor s represented by an equvalent ansotropc medum n order to ease the computatonal effort. The proposed method was valdated by

7 6 IEEE TRANSACTIONS ON MAGNETICS, VOL.?, NO.?, 2015 comparng the flux densty and the electromagnetc torque computed wth lamnated and homogeneous rotors. Even f some small dscrepances were notced due to the alternaton of the steel sheets and the nterlamnar nsulaton sheets, the components of the flux densty and the electromagnetc torque can be accurately predcted wth the homogenzaton method. Moreover, the computaton tme can be reduced to one thrd wth the test machne when the homogeneous parts of the rotor are dscretzed wth a coarse mesh. In future work, the homogenzaton technque wll be mproved by consderng the magnetc ar-gap fluctuatons due to the alternaton of the steel sheets and the nterlamnar nsulaton sheets. These fluctuatons could be taken nto account by modulatng the components of the magnetc flux densty n the ar-gap by a relatve ar-gap permeance. Moreover, the homogeneous model wll be ncluded n an optmzaton process for desgn purpose. Fg. 11. Electromagnetc torque for dfferent steel rato α n the stack at the same rotor poston of 45. The maxmum torque s reached wth α = 0.6. For ths rato, the relatve error between homogenzed and non-homogenzed computaton s 7.93 %. However, the relatve error ncreases for lower steel rato untl 27.4% that s reached for α = 0.2. ACKNOWLEDGMENT The research leadng to these results has receved fundng from the European Research Councl under the European Unon s Seventh Framework Programme (FP7/ ) / ERC grant agreement n The Academy of Fnland s acknowledged for fnancal support. REFERENCES [1] N. Banch and B. Chalmers, Axally lamnated reluctance motor: analytcal and fnte-element methods for magnetc analyss, IEEE Trans. Magn., vol. 38, no. 1, pp , Jan [2] Y. H. Km, J. H. Lee, and J. K. Lee, Optmum desgn of axally lamnated ansotropc rotor synchronous reluctance motor for torque densty and rpple mprovement, n Computaton n Electromagnetcs (CEM 2014), 9th IET Internatonal Conference on, March 2014, pp [3] A. Arkko, T. Joknen, and E. Lantto, Inducton and permanent-magnet synchronous machnes for hgh-speed applcatons, n Proceedngs of the 8 th Internatonnal Conference on Electrcal Machnes and Systems. ICEMS 2005, 2005, pp [4] M. Lamghar-Jamal, Modélsaton magnéto-thermque et optmsaton de machnes rapdes, Ph.D. dssertaton, Unversté de Nantes, [5] F. Isaac, A. Arkadan, and A. El-Antably, Characterzaton of axally lamnated ansotropc-rotor synchronous reluctance motors, IEEE Trans. Energy Convers., vol. 14, no. 3, pp , Sep [6] E. Obe, Calculaton of nductances and torque of an axally lamnated synchronous reluctance motor, Electrc Power Applcatons, IET, vol. 4, no. 9, pp , Nov [7] F. Isaac, A. Arkadan, and A. El-Antably, Magnetc feld and core loss evaluaton of ala-motor synchronous reluctance machnes takng nto account materal ansotropy, IEEE Trans. Magn., vol. 34, no. 5, pp , Sep [8] G. Meuner, V. Charmolle, C. Guern, P. Labe, and Y. Marechal, Homogenzaton for perodcal electromagnetc structure: Whch formulaton? IEEE Trans. Magn., vol. 46, no. 8, pp , Aug [9] P. Dular, J. Gyselnck, C. Geuzane, N. Sadowsk, and J. Bastos, A 3-d magnetc vector potental formulaton takng eddy currents n lamnaton stacks nto account, IEEE Trans. Magn., vol. 39, no. 3, pp , May [10] P. Dular, J. Gyselnck, and L. Krhenbhl, A tme-doman fnte element homogenzaton technque for lamnaton stacks usng skn effect subbass functons, COMPEL - The nternatonal journal for computaton and mathematcs n electrcal and electronc engneerng, vol. 25, no. 1, pp. 6 16, [11] L. Cheng, S. Sudo, Y. Gao, H. Dozono, and K. Muramatsu, Homogenzaton technque of lamnated core takng account of eddy currents under rotatonal flux wthout edge effect, IEEE Trans. Magn., vol. 49, no. 5, pp , May [12] H. Kamor, A. Kamear, and K. Fujwara, Fem computaton of magnetc feld and ron loss n lamnated ron core usng homogenzaton method, IEEE Trans. Magn., vol. 43, no. 4, pp , Aprl [13] F. Martn, M. E. Zam, A. Tounz, and N. Bernard, Improved analytcal determnaton of eddy current losses n surface mounted permanent magnets of synchronous machne, IEEE Trans. Magn., vol. 50, no. 6, pp. 1 9, June [14] G. Meuner, The Fnte Element Method for Electromagnetc Modelng. Wley-ISTE, [15] J. L. Coulomb, A methodology for the determnaton of global electromechancal quanttes from a fnte element analyss and ts applcaton to the evaluaton of magnetc forces, torques and stffness, IEEE Trans. Magn., vol. 19, no. 6, pp , Nov [16] B. Slwal, P. Raslo, L. Perkko, M. Oksman, A. Hannukanen, T. Erola, and A. Arkko, Computaton of torque of an electrcal machne wth dfferent types of fnte element mesh n the ar gap, IEEE Trans. Magn., vol. 50, no. 12, pp. 1 9, Dec 2014.