Analytical Calculation of Rotor Magnet Eddy-Current Losses for High Speed IPMSM

Transcription

1 Analytcal Calculaton of Rotor Magnet Eddy-Current Losses for Hgh Speed IPMSM Adel Bettayeb, Xaver Jannot, Jean-Claude Vanner To cte ths verson: Adel Bettayeb, Xaver Jannot, Jean-Claude Vanner. Analytcal Calculaton of Rotor Magnet Eddy- Current Losses for Hgh Speed IPMSM. ICEM 010 XIX Internatonal Conference on Electrcal Machnes, Sep 010, Rome, Italy. pp.cd-rom Proceedngs, 010. <hal > HAL Id: hal Submtted on 4 Nov 010 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 XIX Internatonal Conference on Electrcal Machnes - ICEM 010, Rome Analytcal Calculaton of Rotor Magnet Eddy- Current Losses for Hgh Speed IPMSM Adel Bettayeb, Xaver Jannot and Jean-Claude Vanner Abstract -- For hgh-speed machnes applcatons, eddycurrent losses n the nteror permanent magnet of synchronous machne (IPMSM) form a porton of the total losses whch can be sgnfcant. Indeed, the magnets are exposed to the harmonc felds whch rotate wth respect to the rotor. The nduced losses n the magnets provoke temperature arsng that must be lmted to avod the rsk of demagnetzaton. The study carres out a predcton of eddy current losses n PM where the skn effect s consdered. A complete analytcal model s presented and compared to 3D Fnte Element (FE) harmonc computatons. The results gven by the proposed model are n agreement wth the FEA results for local electromagnetc quanttes and loss calculatons. Ths approach can be useful for losses estmaton n magnets when desgnng machnes by analytcal method. Index Terms-- Hgh speed, Permanent magnet machnes, Eddy-currents, power losses, skn effect, fnte-element method, analytcal modelng I I. INTRODUCTION N many applcatons such as compressors, machne tools, vacuum pumps, turbne generators, nerta wheels, hgh speed electrcal machnes are of great nterest [1]. Most of the tme, these machnes are used qute contnuously. So they need to have a good effcency to avod over electrcal consumpton. For ths reason, the permanent magnet synchronous machnes are recommended for hgh rotatonal speeds. The desgn of these machnes s very senstve due to the hgh rotatonal speed aspects such as thermal behavor, especally when consderng magnets demagnetzaton due to an overheatng. To avod such mpedments, the losses n the magnets must be evaluated and kept below a suffcently low level durng the desgn of the machne. Ths s emphaszed by the dffculty to evacuate the heat generated n the rotor. Hgh speed machnes nvolve hgh electrcal frequency. When fed by a Voltage Source Inverter (VSI) under Pulse Wdth Modulaton (PWM) technque, the maxmal swtchng frequency can be lmted because of the temperature rse n the semconductors. Ths can lead to sgnfcant current harmoncs n the machne and consequently to hgher ron losses [] and to hgher PM losses at hgh frequences [3]. Thus, due to the hgh rotaton speed, attenton wll be pad to the frequency effect on losses. The total losses n permanents magnets of synchronous machne are manly due to nduced currents created by the macroscopc varaton of magnetc flux densty B(t) n A. Bettayeb s wth the Department of Electrcal Machnes & Power Systems of Supélec, 9119 Gf-sur-Yvette, France (adel.bettayeb@supelec.fr) X. Jannot s wth the Department of Electrcal Machnes & Power Systems of Supélec, 9119 Gf-sur-Yvette, France (xaver.jannot@supelec.fr) J-C. Vanner s wth the Department of Electrcal Machnes & Power Systems of Supélec, 9119 Gf-sur-Yvette, France (jean claude.vanner@supelec.fr) conductve materal. In lterature, these losses are often calculated assumng that the behavor law for the magnet s lnear; the permanent magnet can be modeled by a sngle sold conductor, wth conductvty σ and permeablty µ almost equal to ar permeablty [3-6]. Ths hypothess was verfed n the case of magnet submtted to an alternatve flux for a range of frequency from 100 Hz to 100 khz [7]. An analytcal equaton of the eddy-current losses n magnets of PMSM, due to tme harmoncs of stator currents, has been developed [4] to consder the effect of PM crcumferental segmentatons and has been proved by comparng calculated and measured mpedances of PM machne for dfferent harmoncs at 1~10kHz. However, ths equaton was based on assumpton that the pole-arc dmenson of magnet s small enough to consder a unform flux densty over t. In [5][8-9], both analytcal and fnte element (FE) methods for predctng the eddy-current losses n magnets of PMSM, due to space harmoncs of magnetomotve force (MMF), have been presented n order to quantfy the effcency of segmentatons of PM. The reported analytcal technques to evaluate the eddy-current losses, for the cases when skn depth at the frequences of nterest s greater than both pole-arc and radal dmensons of magnets, were valdated by D tme-steppng fnte element (FE) analyss [9-10], and by 3D magneto-statc FE analyss [5]. However, the condton that skn depth s greater than both pole-arc wdth and radal heght, s not always satsfed. Recently, usng 3D tme-steppng fnte element nonlnear analyss, the authors of reference [6] have studed the effect of axal segmentaton on loss of Interor PM Motor. To explan these results, authors ntroduced theoretcal solutons of the eddy-current losses wth the nfluence of skn effect n thn conductor, when a unform magnetc feld s appled. The study we propose s based on analytcal modellng for eddy currents n magnets at low and hgh frequency. Ths model s used to calculate the losses whatever the skn effect. Consequently ths knd of model can be used on desgn procedures, nvolvng the need for an accurate calculaton of rotor losses to defne the machne effcency for example. II. MACHINE UNDER STUDY Ths study wll focus on a partcular knd of IPMSM whose rotor uses the flux focus prncple n whch magnets are located wth a crcumferental magnetzaton (Fg. 1). It s composed of a shaft whch s often made of magnetc ron, wth a nonferromagnetc hub n order to prevent magnetc flux leakage through the shaft. The magnets are nserted between the pole peces. Table I shows the specfcatons of the 3-phase hgh speed IPMSM. The am of ths paper s then to propose an estmaton of PM losses whch n hgh speed IPMSM. Because of the rotor structure, the losses due to the stator slots are assumed to be /10/$ IEEE

3 neglgble compared to those generated by tme-harmonc currents whch are manly consdered n the paper. magnet due to stator current, H 0, t s necessary to consder the three phases suppled. Therefore the total fundamental MMF can be expressed by: Perodc condton Z drecton Parallel flux Perodc condton θ drecton Parallel flux Perodc condton θ drecton Fg.. Flux lnes when stator current reacton s n lne wth q-axs. Fg. 1. Geometry of fnte element model one pole-par of the PM synchronous machne under study TABLE I MACHINE DIMENSIONS AND PARAMETERS Quantty Value Pole number 8 Tooth number 7 Stack length [mm] 00 Ar gap [mm] 1.4 Magnet wdth [mm] 7 Magnet heght [mm] 47.5 Bore radus [mm] 130 Slot openng [mm] 5.03 Tooth wdth [mm] 6.4 SmCo17 Magnet resstvty [Ωm] 90e-8 Current densty [A/m] 1 Fg. 3. Flux lnes when stator current reacton s n lne wth d-axs. III. ANALYTICAL MODELS FOR FIELD AND LOSS CALCULATIONS A. Magnet s magnetc flux densty computaton: Ampere s law + Gauss law The magnetc flux densty n the magnets s computed as the sum of the feld generated by d-axs stator current and the feld generated by q-axs stator current [10]. The magnetc materal s supposed to be unsaturated and has lnear behavor. Ths assumpton s qute always satsfed n hgh speed machnes to avod excessve ron losses [11]. A FE analyss of the magnetc flux densty dstrbuton nsde the machne, n both cases of q-axs stator current reacton and d-axs stator current reacton, s performed. The magnetc flux densty lnes are depcted n Fg. and Fg. 3. Notceably the flux lnes do not cross the magnets n q-axs confguraton. Thus only the d-axs stator currents and ther tme-varatons lead to the magnet s eddy-current losses. Consequently the analytcal model for magnet magnetc flux densty computaton wll be focused on d-axs stator current confguraton. In ths confguraton, a set of balanced three phase currents s njected n the stator wndng n such a way that the resultng stator current phasor s n lne wth the d-axs. The schema for computaton prncple s gven n Fg. 4. The magnets are supposed to be unmagnetzed, havng a relatve magnetc permeablty of 1.1, and no electrcal conductvty. In order to compute the magnetc feld strength n the Fg. 4. Scheme for computaton prncple when stator current reacton phasor s n lne wth d-axs MMF d 3 I k N cosp (1) w spp Where k w s the global wndng factor, p s the pole par number, and N spp the number of turns per pole par and per phase. The system of equatons consdered s composed of the Ampere s law () and the Gauss law (3). Ths latter neglects the reacton of the magnet nduced eddycurrents n feld calculaton. Ths system of equatons s completed by the consttutve relatons of the materals (4). H. h H. k. e MMF () 0 ag c d Where k c s the Carter s coeffcent. ag m m Bm. t. l (3) p ag l. Bag. Rd 0

4 1 h Wth:. R Bm. H 0 (4) Bag 0. H ag A combnaton of (), (3), and (4) allows wrtng the equaton drvng the magnetc feld n the magnets: 1. R. kw. NsppId H0 4. kc. e. t.. p h.. R (5) B. Eddy-current loss estmatons In the case of parallelepped shape magnets submtted to an alternatve magnetc feld, eddy-current occurs as descrbed n Fg. 5. The goal of the followng secton s to compute these eddy-currents and the assocated losses at low and hgh frequency. Fg. 5. Scheme of a magnet wth eddy current paths B1. Formula wthout skn effect The eddy-current losses can be computed analytcally usng varous expressons dependng on the workng condtons. A low frequency analytcal formulaton for tme average eddy-current per unt volume computaton s proposed n [1]: Pe f B m t l (6) VOL 8 t l When the skn effect s to be consdered at hgher frequences, equaton (6) s not adequate, because the flux densty n the magnet s no longer homogeneous [13]. B. Formula wth skn effect The skn effect occurs n the magnet when the overall dmensons are larger than the skn depth: t/ > δ and l/ > δ the skn depth s defned by: h (H,B) t/ o dz l/t*x It s assumed that the dstrbuton of local electromagnetc quanttes s ndependent of the magnet thckness. The analytcal resoluton to get magnetc feld n the magnet can be smplfed to the resoluton of D Maxwell s equatons, n the plane (Oxz) presented n Fg. 5. Takng nto account the nduced currents, the theoretcal expresson of the magnetc feld s gven by [6][14]. x Je z dx l/ x (7) H x, y, t H0 snt cost snt m n m, n odd odd (8) m, n 16H0 mx ny sn sn mn m, n a b m n Wth m, n t l Then eddy-current dstrbuton can be determned by dfferentatng (8) and the total per volume eddy current losses are derved by ntegraton. Pe VOL 1 m, n 4H0 16H0 m n mn mn m, n odd odd Formula (9) can be used to estmate power loss n a sngle magnet excted by homogeneous feld, and then t wll be appled n the case of a magnet nserted n the machne. IV. FINITE ELEMENT METHOD FOR FIELD AND LOSS CALCULATIONS A. Harmonc method wth locked rotor 3D harmonc FE analyss s used to valdate the analytcal model proposed n equaton (9). Only one of the 4 pole-pars of the studed PM synchronous machne has been modeled, wth perodc boundary condtons n and Z drecton (Fg. 1). In ths method, the prncple of superposton of harmonc fluxes can be appled, each of the relevant harmoncs (Table II), generates correspondng eddy-currents n magnets. The rotor s consdered at standstll and equvalent travellng current sheet rotates around t [15]. The total eddy-current losses n the magnet are gven by the sum of the losses of each relevant harmonc. TABLE II STATOR AND ROTOR TIME HARMONIC MMF S AND ASSOCIATED SKIN DEPTHS FOR THE MACHINE UNDER STUDY Harmonc order stator rotor stator rotor The magnetc flux densty feld n the 3D harmonc FE analyss s derved from the magnetc vector potental A and solvng the followng equaton [16]: J e Skn depth , , , ,6 (9) 1 rot ( rot A) J (10)

5 s the eddy- Where J s source current densty, current and µ s the permeablty. J e The current densty n the magnet s gven by Faraday- Maxwell s equaton solved wth A and V (V s electrc scalar potental) and the Ohm s law: A J e ( gradv ) (11) t Wth σ s the conductvty of the magnet. B. Eddy-current losses computaton The losses are evaluated n every element volume of permanent magnet as followng: n 1 Pe ( J J *). V (1) 1 Wth n s number of element, V element volume of each element n the magnet, J et J * are current densty and conjugate current densty n the magnet. (a) Eddy-current wthout sgnfcant skn effect at f =1Hz V. COMPARISONS BETWEEN ANALYTICAL AND FE RESULTS A. Effect of the mesh on loss estmaton The skn effect s accurately consdered n the FE model f the element sze of mesh s lower than half the penetraton depth δ. When skn effect can be neglected, Fg. 6.a, the refned mesh sze has not an nfluence on the results. Contrary, as shown n Fg. 6.b, the mesh sze should be consdered n the presence of skn effect due to a steep varaton of the electrcal values along the geometrcal dmensons. As shown n Fg. 7, for parallelepped mesh of dmensons e x * e z, wth e x and e z representng respectvely the mesh szes along Ox and Oz, the skn effect can be consdered precsely when both the ratos (δ/e x ) and (δ/e z ) exceed. Each curve for each rato, P EF FE1 for δ/e x and P EF FE for δ/e z, tend to an asymptote when the ratos ncrease above. The local flux densty s homogeneous along the thckness of the magnets. Ths has been verfed n FE model. B. Study of a sngle magnet excted by a homogeneous feld In ths part, a sngle magnet s consdered. It has the same dmensons and propertes of the one n the machne. It s excted by a harmonc external homogenous magnetc feld at varous frequences. The local quanttes, B and J, are computed wth analytcal formulas derved from the one (8) developed for magnetc feld calculaton. A comparson between the analytcal calculaton of these local quanttes and ther FE computaton s carred out to check the valdty of the analytcal method. The results plotted n Fg. 8 and Fg. 9 show the good agreement between the obtaned values. One can hope good agreement concernng the loss evaluaton when local quanttes match well. (b) Eddy-current wth skn effect at frequency f = 8 khz Fg. 6. Scheme of a magnet wth the nduced eddy current paths at 1Hz and 8 KHz. PM dmensons : t*h*l=47.5*7*50[mm3]. δ/e Fg. 7. Effect of mesh sze on eddy-current losses n the magnet at harmonc frequency f = 8 khz. The rato of skn depth (δ=5.1mm) versus element sze (e) s varyng along Ox and Oz. Fg. 8. Comparson between analytcal and FE computaton of magnetc flux densty along Ox axs n the magnet, at 8 khz

6 Fg. 9. Comparson between analytcal and FE computaton of nduced eddy-currents along Ox axs n the magnet, at 8 khz Fg. 10-a. Comparson between analytcal and FE computaton of magnetc flux densty along the Ox axs of the magnet, at 8 khz The results gven by the dfferent methods presented for the evaluaton of eddy-current losses are presented n Table III. The comparson between the results at dfferent frequences clearly shows the very good behavor of the formulaton (9) that ncludes skn effect. C. Study of the magnet nsde the machne The prevous statement wth analytcal loss estmaton s now consdered wth magnets as a part of the whole machne. The magnetc materals are taken lnear, whch s not an exaggerated hypothess n the case of hgh speed machne that are often unsaturated to avod excessve heatng provoked by ron losses. A 1A/m² current densty s njected n the stator slots of a synchronous machne. In the followng the formulaton (6) wll not be consdered any longer. Therefore ths part of the study wll only be based on formulaton (9). To avod meshng mpedments, a set of smulatons s frst performed where the length of the machne s chosen wth respect to the skn depth. In ths case, the length of the machne wll be equal to three tmes the skn depth. Ths permts to mantan at least 3 elements along the skn depth lmtng the computaton tme below 1h30. The local electromagnetc quanttes and the eddy-current losses n the magnets are computed wth the analytcal method and wth the FE method for varous frequences. A comparson between local electromagnetc quanttes s carred out and presented n Fgs. 10 and 11. Fg. 10-b. Comparson between analytcal and FE computaton of magnetc flux densty along the Oz axs of the magnet, at 8 khz Fg. 11-a. Comparson between analytcal and FE computaton of nduced eddy-current denstes along the Ox axs of the magnet, at 8 khz TABLE III LOSSES IN A SINGLE MAGNET Formula (6) Formula (9) Fnte Element e e e e e e Fg.11-b. Comparson between analytcal and FE computaton of nduced eddy-current denstes along the Oz axs of the magnet, at 8 khz

7 It can be observed a good agreement between analytcal and 3D-FE-computed magnetc flux denstes and nduced currents along two cross lne n the magnets. The skn effect s thus well taken nto account. The losses are summarzed n Table IV. The presented results show that the analytcal estmaton of eddy-current losses n the magnets are about 5% less than those estmated wth FE. The reference machne, defned n Table I, has a larger stack length. Therefore the system dmenson n FE analyss s greater and leads to more mportant computaton tme, about 48h. The losses n the magnets are computed for varous frequences lsted n Table V. The comparson between analytcal and FE results shows some devaton n case of varable stack length wth frequency or n the case of the reference machne. Several reasons can explan a part of ths devaton. Frst the contrbuton of q-axs current n eddy-current losses contrbuton s neglected n the analytcal approach. However ths contrbuton can explan why the absolute devaton ncreases wth the frequency, the losses due to q- axs current also ncrease wth frequency. Second the magnetc feld computed wth (5) can be mprecse due to smplfyng hypothess needed to set t up. Fnally the hypothess of feld homogenety across the magnet thckness s not fully satsfed n the machne, there are some sde effects. TABLE IV LOSSES IN THE MAGNETS OF THE IPMSM: VARIABLE LENGTH WITH SKIN EFFECT Formula (9) Fnte Element VI. CONCLUSION Devaton [%] e e % e e % TABLE V LOSSES IN THE MAGNETS OF THE IPMSM Formula (9) Fnte Element Devaton [%] e e e e In ths paper, two ways of computng eddy-current losses n the IPMSM have been proposed. A complete analytcal approach s descrbed and compared to a numercal approach nvolvng 3D-FE harmonc computatons wth the locked rotor test. The results show an acceptable devaton between both approaches n the case where skn effect appears. Ths phenomenon must be well addressed by both the analytcal and 3D-FE method. It partcularly concerns the mesh of 3D-FEM and the harmonc number n the analytcal model. An advantage of the analytcal method s precsely to account for that skn effect and to estmate the eddy-current losses wth a suffcent precson for a desgn stage. Moreover ts very low computaton tme, less than one second, makes ths approach very sutable for an teratve optmal desgn procedure. However, the feld computaton presents some devatons because of the smplfed geometry and the neglect of the magnet eddy-current. Furthermore, the varaton of the magnetc flux densty n the q-axs s not consdered for loss calculaton. On the other hand the 3D-FE harmonc approach based on locked rotor method s more precse and can be nvolved n the detaled analyss of exstng structures. In addton to, there s no need of large amount of CPU tme or n storage space. VII. REFERENCES [1] N. Banch, S. Bolognan, and F. Luse, Potentals and Lmts of Hgh-Speed PM Motors, IEEE Trans. Industry Applcatons, vol. 40, no. 6, pp , Nov./Dec [] Yamazak, K., Seto, Y., Iron loss analyss of nteror permanentmagnet synchronous motors-varaton of man loss factors due to drvng condton, IEEE Transactons on Industry Applcatons, v 4, n 4, p , July-Aug. 006 [3] D. Ishak, Z. Q. Zhu, D. Howe, Eddy-Current Loss n the Rotor Magnets of Permanent-Magnet Brushless Machnes Havng a fractonal Number of Slots Per Pole, IEEE Transactons on Magnetcs, vol. 41, n 9, September 005, p [4] H. Polnder and M.J. Hoejmakers, Eddy-current losses n the Permanent magnets of a PM machne, EMD97, Sep. 1997, pp [5] J. D. Ede, K. Attalah, G.W. Jewell, J. B. Wang, D. Howe, Effect of Axal Segmentaton of Permanent Magnets on Rotor Loss n Modular Permanent Magnet Brushless Machnes, IEEE Transactons on Industry Applcatons, vol. 43, n 5, September 007, p [6] Y. Katsum, A. Atsush, Loss Investgaton of Interor Permanent- Magnet Motors Consderng Carrer Harmoncs and Magnet Eddy Currents IEEE Transactons on Industry Applcaton, vol. 45, n 4, Aprl 009, p [7] A. Benabou, S. Georges, S. Clenet, Permanent magnet modelng for dynamc applcatons, Journal of magnetsm and magnetc materal, Vol.30, N 6, p , 008. [8] K. Altallah et al, Rotor loss n Permanent-magnet brushless AC machnes, IEEE Trans. on Industry Applcatons, vo. 36, no. 6, Nov. 000, pp [9] H. Toda et al, Rotor eddy-current loss n permanent magnet brushless machnes, IEEE Trans. on Magnetcs, vo. 40, no. 4, July 004, pp [10] X. Jannot, J-C. Vanner, J. Sant-Mchel, M. Gabs, C. Marchand, D. Sadarnac,»An analytcal model for pm synchronous machne wth crcumferental magnetzaton desgn,n Proc. Electromoton 009, Llle, 009 [11] A. Bnder, T. Schneder, Hgh-speed nverter-fed AC drves, Source: 007 Internatonal Aegean Conference on Electrcal Machnes and Power Electroncs (ACEMP '07), p , 008 [1] W. Huang, A. Bettayeb, R. Kaczmarek and J. C. Vanner, Optmsaton of magnet segmentaton for reducton of eddy-current losses n Permanent Magnet Synchronous Machne, IEEE Trans. on Energy converson, vo. 5, no., pp. June 010, pp [13] S. M. Abu Sharkh, M. R Harrs, N. Taghzadeh Irenj, Calculaton of rotor Eddy current loss n Hgh-speed PM alternators, electrcal Machnes and Drves, pp , Sep.1997 [14] R. L. Stoll, The Analyss of Eddy Current, Oxford Unversty press, chap4, pp 35-44, [15] J. C. Vanner, R. Kaczmarek, Z. Wang, A. Randra, Rotor loss n PMSM by calculaton, smulaton and measurements, mutual verfcaton of methods, Electrcal Power Qualty and Utlzaton, Journal Vol XII, N, 006, P [16] Y. Kawase, T. Ota, and H. Furkunaga, 3-D Eddy current Analyss n permanent Magnet of nteror Permanent Magnet Motors, IEEE Transactons on Magnetcs, vol. 36, n 4, July 000, p Adel Bettayeb receved the Ingéneur d Etat Degree n electrcal power engneerng from the U.S.T.H.B Unversty, Algers, Algera, n 004, and the Master Degree from the Unversty of Paul Sabater, Toulouse, F.R.., n 007. In December 007 He s currently a PhD student n Ecole Supéreure d Electrcté SUPELEC (France). Hs research nterests are on the rotor ron losses n synchronous machnes Xaver Jannot s workng towards the Ph.D. degree n Electrcal Engneerng wth the Department of Electrcal Power Systems n Supelec. Hs man research nterests nclude multdscplnary desgn of permanent magnet synchronous machne wth ther power electronc supply. Jean-Claude Vanner s Professor and head of Department of Electrcal Power Systems n the Ecole Supéreure d Electrcté (Supélec) n France. Hs research nterest s wth energy converson systems (motors, actuators, generators) and concerns the modelng, the desgn and the optmzaton of these equpments for specfc applcatons.