Modeling and Analysis of Halbach Magnetized Permanent-Magnets Machine by Using Lumped Parameter Magnetic Circuit Method

Transcription

1 Progress In Electromagnetcs Research M, Vol. 41, , 215 Modelng and Analyss of Halbach Magnetzed Permanent-Magnets Machne by Usng Lumped Parameter Magnetc Crcut Method Guoha Lu, Mngmng Shao, Wenxang Zhao *, Jnghua J, Qan Chen, and Qan Feng Abstract Permanent-magnets (PMs) wth tangental and parallel magnetzaton drectons are combned n the Halbach PM (HPM) machne, whch can offer hgh performances. However, the exstng lumped parameter magnetc crcut (LPMC) model can only calculate one PM magnetzaton drecton, namely ether tangental drecton or parallel drecton. The key of ths paper s to propose a method to dvde and establsh equvalent magnetc motve force (MMF) for HPM machne wth both tangental and parallel magnetzatons. Then, a LPMC model, usng equvalent MMF, s developed to predct the electromagnetc performances for a four-phase HPM machne. In order to verfy the effectveness of the proposed LPMC model, a 6-pole/8-slot 15 kw HPM prototype s bult. The comparatve results of the proposed LPMC model, two-dmensonal (2D) FEM and the experments verfy the effectveness of the proposed LPMC model. 1. INTRODUCTION Halbach permanent-magnet (HPM) can nherently offer snusodal ar gap feld, neglgble coggng torque, snusodal ar-gap feld dstrbuton, back-electromotve force (back-emf) and needless of rotor back-ron, due to self-sheldng magnetzaton [1]. Hence, these HPM machnes have been extensvely appled n many electrc drve felds. The lumped parameter magnetc crcut (LPMC) models have been used to predct the electromagnetc performances n many PM machnes, except HPM machne, at the desgn stage because of ts many merts [2 22]. A LPMC model of a swtched reluctance machne was presented n [2 7]. Ths LPMC model constructon s smple because there s no PM n swtched reluctance machne. In [8 15], the LPMC models for doubly salent PM, axal-flux PM, spoke-type PM and flux-swtchng PM machnes were bult. The magnetzaton drectons of ther PMs are tangental. Approxmatvely, these PMs can be consdered as rectangle sources. Also, the surface-mounted PM (SPM) and the nteror PM machnes were analyzed by usng LPMC method n [16 22]. The magnetzaton drecton of ther PMs s radal or parallel. In summary, the above-mentoned LPMC models have only one PM magnetzaton (tangental or parallel). Snce ther PMs are separated, the magnetc motve force (MMF) of PMs can be consdered as a constant. However, for the HPM machne, both tangental and parallel magnetzatons are adopted, and the MMF of PMs changes wth the rotor poston. Hence, the exstng LPMC models are not sutable for HPM machne. The novelty of ths paper s to develop a LPMC model for HPM machne. Frstly, the model of PMs s dvded and nvestgated to obtan the equvalent MMF of the HPM machne. Then, the branches of the LMPC are dvded and establshed for the HPM machne based on a generalzed approach [18]. In fact, ths generalzed approach was proposed to buld the LPMC model for desgn of SPM machne, and the PM constructon of HPM machne s smlar to that of the SPM machne. The PM and ar-gap are equvalent to the branches. Based on the developed branches, a nonlnear adaptve LPMC model Receved 22 January 215, Accepted 6 March 215, Scheduled 23 March 215 * Correspondng author: Wenxang Zhao (zwx@ujs.edu.cn). The authors are wth the School of Electrcal and Informaton Engneerng, Jangsu Unversty, Zhenjang 21213, Chna.

2 178 Lu et al. Fgure 1. HPM machne. Cross-sectons. Prototype machne. h 2 B s1 w sy 2 o 4 o B s w st h R sy R s R o R sh R pm Fgure 2. Detal parameters. Stator. Rotor. for HPM machne s bult to predct the electromagnetc performances, such as flux lnkage, back-emf, and torque. In Secton 2, the parameters of HPM machne wll be presented. Then, the methods to dvde PMs, establsh the PMs model, obtan equvalent snusodal MMF of every branch and nonlnear adaptve LPMC model of HPM machne wll be presented n Secton 3. The HPM machne wll be analyzed by usng fnte element method (FEM) and the proposed LPMC model, and the results wll be compared and verfed by the experment n Secton 4. Fnally, some conclusons wll be drawn n Secton TEST MACHINE Fgure 1 shows the cross-secton of the HPM machne. It can be seen that there are four armature teeth (AT) and four fault-tolerant teeth (FTT) n the stator. The sngle-layer concentrated wndng s ntroduced nto ths machne to reduce copper loss. The sngle-layer concentrated wndng has shorter end length than dstrbuted feld wndng, hence savng copper wndng. So, ts copper loss decreases n the same current stuaton. Stanless steel sleeve s adopted to strengthen mechancal robustness of HPM. The 6-pole/8-slot 15 kw HPM machne s bult as shown n Fgure 1. Fgure 2 shows the detaled parameters of the rotor and stator. The desgn dmensons and materals are lsted n Table LUMPED PARAMETER MAGNETIC CIRCUIT MODEL In fact, the LPMC has been a common technque for the analyss and desgn of electrc machnes by lnkng the materal characterstcs to the machne behavor. The magnetc feld characterstcs can be techncally obtaned usng electrc crcut prncples, e.g., Krchhoff s voltage law and Krchhoff s current law. The basc prncple of LPMC model can be expressed as Φ=FG (1)

3 Progress In Electromagnetcs Research M, Vol. 41, Table 1. Desgn specfcatons. Items Halbach machne Rated power 15 kw Number of PM poles (p) 6 Number of stator slot (N s) 8 Based speed n b (rpm) 14 Stack length, l s (mm) 73 Outsde stator radus, R o (mm) 43 Insde stator yoke radus, R sy (mm) 36.5 Insde stator radus, R s (mm) Outsde PM radus, R pm (mm) 22 Insde shaft radus, R sh (mm) 11 Slot openng heght h (mm) 3. Slot wedge heght h 1 (mm) 1. Slot body heght h 2 (mm) 8.87 Slot openng wdth Bs (mm) 3.3 Slot wedge wdth Bs 1 (mm) 13.6 Wdth of teeth w st (mm) 8.6 Ar gap length h g (mm).75 Stator yoke wdth, w sy (mm) 6.5 Col turns per slot, N 45 PM type SmCo24 B r, (T) 1.1 H c,(a/m) 756 where Φ, F and G are flux, MMF, and permeance of an element n the magnetc crcut, respectvely. The permeance of a rectangular element can be gven as S G = μ r μ (2) L where μ r and μ are the relatve permeablty and the permeablty of ar, respectvely, and L and S are the effectve length and cross-sectonal area of the element, respectvely. The permeance of the PM, the ar-gap, the stator and the yoke of stator can be gven as { Spm = w pm l pm S G m = μ r μ pm (3) h pm ( ) S g = l s R s hg 2 α (4) S G (...7)g = μ g h g h sb = h h st = h 2 ) h sy = θ s ( Ro+R sy 2 S sb = l s R s α S st = l s w st S sy = l s w sy S G (...7)sb = μ r μ sb h sb S G (...7)st = μ r μ st h st S G (...7)sy = μ r μ sy h sy (5)

4 18 Lu et al. where h pm, w pm and l pm are the magnet thckness, wdth and heght, respectvely. The μ pm s the relatve permeablty of the PM. h g, h sb, h st and h sy are effectve heght of the ar-gap, stator boot, stator teeth and stator yoke, respectvely. S g, S sb, S st and S sy are effectve cross-sectonal area of the ar-gap, stator boot, stator teeth and stator yoke, respectvely. G (...7)sb and G (...7)st are the permeance of teeth-boot and teeth, respectvely. α and θ s are the arc of boot at teeth and slot ptch, ther values are 4 and 45, respectvely. l s s stack length. Snce the varaton of each phase magnetc crcut s dentcal, only one phase of the machne needs to be modeled. Here, phase-a s chosen as example, and ts MMF can be obtaned as F a = N a (6) where N s the col turns per stator slot, and a s the njected current of phase A PMs Model The feld dstrbuton of HPM machne by FEM s shown n Fgure 3. The arrows n each block ndcate the magnetzaton drectons of PMs. It can be observed that the rght half of PM, the left half of PM2 and PM1 form one magnetc flux loop. Then, the PM can be dvded nto left and rght parts. Snce the magnetzaton drecton of PM s parallel, each part of PM can be also dvded nto two segments, marked as segment A and segment B (shown n Fgure 4). As shown n Fgure 4, the drecton and value of the MMF are nvarable n segment A because of the unchanged wdth of the PM. The drecton of the MMF s nvarable n segment B, but the value of the MMF s decreasng wth the poston because of the changed wdth of the PM. The value of MMF of segment B s zero at E and F ponts, whle the value of the MMF of PM1 ncreases outward along the tangental drecton, and the drecton of PM1 s constant, as shown n Fgure 4. Magnetzaton (M) of PMs was resolved nto radal and tangental components usng the analytcal model n [23 25]. Especally, a general analytcal model of magnetzaton (M) was proposed to predct PM11 PM1 PM PM1 PM2 E F PM9 PM3 B A A B PM8 PM4 PM7 PM6 PM5 Fgure 3. Feld dstrbuton. Fgure 4. Drecton and value of PMs. PM. PM1. F m B Fm Br θ2 B P F m A F m Ar θ1 A A B M f m P 6+θ f m r N F m Bθ θ F m Aθ Fgure 5. MMF resultant vector of PMs. PM. PM1.

5 Progress In Electromagnetcs Research M, Vol. 41, electromagnetc performance of PM brushless machnes havng segmented halbach array n [26]. Hence, n the LPMC model, the MMF of PMs should be resolved nto the radal and tangental components. The MMFs of PM and PM1 are shown n Fgure 5. Due to magnetc lne of force s dsjont, pont P s the crtcal separaton pont for adjacent magnetc crcut. There s only tangental component for M and N ponts, as shown n Fgure 5. The drecton of the radal component between M and N ponts s outward, whch can be termed as one postve pole, and else are negatve poles. The MMF, the radal and tangental components n segment A, segment B and PM1 can be obtaned as F m A = F m Ar + F m Aθ F m Ar = F m A cos θ 1 F m Aθ = F m A sn θ 1 F m B = F m Br + F m Bθ F m Br = F m B cos θ 2 F m Bθ = F m B sn θ 2 f m = f m r + f m θ f m r = f m cos(θ + 6) f m θ = f m sn(θ + 6) F m r = F m Ar + F m Br + fmr F m θ = F m Aθ + F m Bθ + f m θ where F m A, F m B and f m are the MMF of segment A, segment B and PM1, respectvely. F m Ar and F m Aθ are the radal and tangental components of F m A, respectvely. F m Br and F m Bθ are the radal and tangental components of F m B, respectvely. f m r and f m θ are the radal and tangental components of f m, respectvely. θ, θ 1 and θ 2 are dfferent mechancal angle at dfferent poston, respectvely. F m r and F m θ are the MMF of the radal and tangental components of HPM, respectvely LPMC Model Fgure 6 shows the open-crcut magnetc feld dstrbuton of test machne at and 9 postons. The correspondng smplfed actual magnetc crcuts are shown n Fgure 7. It can be known that each stator tooth s connected wth segments A, B and one of assstant PM (.e., PM1). Thus, the MMF of all PMs lnked wth one tooth can be equvalent to one effectve MMF. G y, G fm, G t and G Fm are the permeance of the yoke, PM1, the tooth and half of PM, respectvely. In ths secton, the branches of the LPMC are dvded from stator teeth, and the PM and the ar-gap are equvalent to every stator branch. Then, the value of the MMF vares at dfferent postons. PM and PM1 are separated nto 4 and 3 parts. As a result, one mechancal angle ndcates one poston. In ths way, when the machne operates one electrcal degree, the waveform of the equvalent MMF of PM (F m curve) can be ganed by (7), and shown n Fgure 8. In order to verfy ts snusodal shape, an deal snusodal fttng F m curve s mported and compared wth the predcted F m curve. It (7) Fgure 6. Open-crcut magnetc feld dstrbuton. Poston =.Poston=9.

6 182 Lu et al. G y G fm G t G Fm G y G fm G t G Fm Fgure 7. Smplfed actual magnetc crcut. Poston =.Poston= Fm Fttng Fm Fm (A) Poston (elec.deg.) Fgure 8. Equvalent MMF of PM n one electrcal degree perod. θ s nth tooth θ s nth tooth (n+1)θ θ s1 θ s nθ (n+1)θ θ s1 nθ θ s (n-1)θ Fgure 9. Equvalent MMF n a slot ptch. One pole. Two adjacent opposte poles. can be concluded that there s only small dscrepancy between the actual F m and fttng F m curve, and that ths dscrepancy s reasonable and expected because of partton and smplfcaton of PMs. So, the equvalent MMF of PM can be gven as F m (θ) =F m cos(pθ) (8) where F m (θ) s the value of the MMF under dfferent electrcal degree θ. p s number of pole-pars. F m s the maxmum value of MMF. Although the equvalent MMF of the stator teeth s changed by the relatve poston when the HPM machne s under operaton, the relatve postons of the PM and stator can be classfed as two types. Fgure 9 shows the relatve poston of the PM and the stator. Frst, a slot ptch s wthn one postve or negatve pole, the flux produced by the PM flow throw the stator wndngs entrely, as shown n Fgure 9. The equvalent MMF n a slot ptch can be obtaned as MMF = θs1 θ s F m (θ) θ s1 θ s dθ (9)

7 Progress In Electromagnetcs Research M, Vol. 41, where θ s and θ s1 are angles of both ends of nth tooth, respectvely. F m (θ) s the value of the MMF under dfferent electrcal degree θ. Second, porton of two adjacent opposte pole are wthn a slot ptch, the contrbuton of flux from the magnet s mtgated due to the gap between the two adjacent opposte pole, as shown n Fgure 9. The equvalent MMF n a slot ptch can be obtaned as MMF = θs1 nθ F m (θ) θ s1 nθ dθ nθ θ s F m (θ) nθ θ s dθ (1) where nθ s the center angle of nth tooth. The equvalent LPMC model of HPM machne s shown n Fgure 1. It can be concluded that t s consttuted by 8 branches accordng to stator teeth. In ths model, the PM (...7) are the models of equvalent PM under a slot ptch. G g(...7) are the models of the ar gap. AT (...4) and FT (...4) are the models of AT and FTT, respectvely. Also, t can be known that there are four layers n the LPMC model, and they stand for the layers of PM, the ar, the stator and the yoke of stator along the radal drecton from nsde to outsde, respectvely. The adjacent layer connects wth each other by permeance Fnal Soluton It can be known from Fgure 1 that the node number of the equvalent LPMC model s 33. Meanwhle, the total node number keeps nvarable wth the poston. For a magnetc crcut wth N nodes, f some one node s chosen to be as reference node, the other (N 1) nodes can be treated as ndependent. The MMF of reference node s zero, MMF dfference between ndependent node and reference node are called node MMF, and thus, 32 ndependent nodes correspond to 32 nodes MMF. Meanwhle, 32 2 permeances exst n the LPMC model correspondng to 32 ndependent nodes, formng a permeance matrx G(, j), (, j =, 1, 2,...,31). When = j, permeance G(, j) s called self-permeance, otherwse, permeance G(, j) s called mutual-permeance. Accordng to the foregong developed LPMC model, the G sy7 27 G sy AT 28 G sy6 17 FT G g7 PM 7 G g FT G sy1 25 G sy5 AT FT 2 G g6 PM 6 6 PM 5 G g AT 2 PM PM 1 PM 4 PM G g4 22 PM 2 G g3 G g1 2 FT 1 G g AT G sy2 G sy4 31 G sy3 Fgure 1. Proposed lumped parameter magnetc crcut model.

8 184 Lu et al. Intal permeablty Rotaton angle θ Permanent magnet Current data Wndng confguraton Permeance of stator & ar-gap Calculaton of Equvalent F m & G m Calculaton of F a Calculaton of matrx G & Φ New Permeablty μ Calculaton of matrx F n Materal operatng ponts Materal B-H profle Update of μ No Error check Yes Step check Yes No Results calculaton No deal results Fgure 11. Iteratve process of LPMC model. Accomplshment node potental equaton can be obtaned by usng (11). G(, ) G(, 1)... G(, 31) F n () Φ n () G(1, ) G(1, 1)... G(1, 31) F. n (1) = Φ n (1). (11) G(31, ) G(31, 1)... G(31, 31) F n (31) Φ n (31) vz. as GF n = Φ (12) where G ((...31), (...31)) s the permeance of node number, F n (...31) the MMF of the node number, and Φ n (...31) the flux of the node number. G, F n and Φ are the matrx of permeance, MMF and the flux, respectvely. The teratve process of the LPMC model s shown n Fgure 11. Frstly, the permeances of the stator and ar-gap are calculated by usng (4) and (5) based on the ntal permeablty. Secondly, the equvalent MMF (F m ) and permeance of PM (G m ) of every branch are obtaned by rotaton angle θ and PMs. Meanwhle, the MMF of each phase can be obtaned by the current and wndng confguraton by usng (6) when the HPM machne s under load. After all permeances n the LPMC model are prepared, the node potental equaton can be bult as (11). Snce nonlnear materal WGT-2 s adopted n the stator core, a consecutve B-H table of the nonlnear materal s lsted to consder saturaton effect. The Gauss-Sedel teratve method s adopted to mprove the teratve speed. A new flux Φ (k) and new flux densty B (k) of the stator (.e., AT )can be derved by usng (13) and (14) after the ntal permeablty of the stator s appled to solve (11). A new magnetc ntensty H (k) can be ganed by consderng where B (k) s located. Snce the measured ponts are lmted, when the value of B (k) s equal to the pont of the table, H (k) s replaced by the value of the pont. Else, when B (k) s located between two samplng ponts (N and N + 1) of the nonlnear B-H table, the curve between two ponts s treated as a straght lne, then, the H (k) can be wrtten as (15). The correspondng relatve permeablty μ (k) can be obtaned as (16). ( ) =Φ (k) t = G st F (k) 18 F (k) 27 F a (k) (13) Φ (k)

9 Progress In Electromagnetcs Research M, Vol. 41, B (k) H (k) μ (k) max ( = Φ(k) (14) S st =H n +(H n+1 H n ) B(k) = B(k) H (k) B (k) B (k 1) B (k) B n (15) B n+1 B n (16) ) ε (17) where ε s the set error precson. Fnally, the μ (k) s fed back to the LPMC model to replace the ntal permeablty of the stator, thus one crculaton of the teraton s acheved. If the values of kth and (k 1)th teraton flux denstes (B (k) and B (k 1) ) are satsfed by (17), the teratve process s over. Otherwse, t wll start next crculaton before the teratons k s over the set calculaton step, whch s appled to prevent the process droppng nto the endless loop. 4. SIMULATION AND EXPERIMENT VERIFICATIONS To verfy the accuracy of the proposed LPMC model for HPM machne, the tme-steppng FEM and the experment are employed to analyze the electromagnetc performances, such as flux densty of the AT, the phase back-emf, and the output torque. The flux densty of the armature tooth can be calculated by usng proposed LPMC model, as shown n Fgure 12. The FEM-based one s shown n Fgure 12. It can be concluded that there s a good agreement between the FEM and the LPMC results. Based on the soluton of (13) and (14), and the back-emf can be gven as e a = dψ a = N dφ dt dt = NS db t (18) dt where S t s the effectve cross-sectonal area of the AT and Ψ a the flux of the phase A. Fgure 13 compares the LPMC-based and FEM-based back-emfs of the HPM machne at the rated speed of 14 rpm wth the measured one. It can be concluded that there s a good agreement n the shape and ampltude. Furthermore, the harmonc analyses of the back-emf by usng the LPMC model, the FEM and the experment are shown n Fgure 14, n whch ther THD are.52%, 4.6% and 1.6%, respectvely. It can be known that the major harmonc order s 3th n the FEM result and 9th n the measured result. Although there s major dscrepancy n thrd harmonc of the THD among the LPMC model, the FEM and the measured, t s acceptable by consderng the saved-tme because there.5 PhaseA PhaseB PhaseC PhaseD.5 PhaseA PhaseB PhaseC PhaseD Flux densty (T) Flux densty (T) Poston (elec. deg.) Poston (elec. deg.) Fgure 12. Flux densty of AT.LPMCmodel.FEM.

10 186 Lu et al. 2 PhaseA PhaseB PhaseC PhaseD 2 PhaseA PhaseB PhaseC PhaseD 1 1 EMF (V) -1 EMF (V) Poston (elec. deg.) Poston (elec. deg.) 2 PhaseA PhaseB PhaseC PhaseD 1 EMF (V) Poston (elec. deg.) Fgure 13. Back-EMF. LPMC model. FEM. (c) Measured. (c) 5% 12 4% 9 A /A 1 n 3% 2% 1% LPMC FEM Measured Torque (Nm) 6 3 LPMC FEM % Harmonc order Fgure 14. Harmonc analyss of back-emf Poston (elec. deg.) Fgure 15. Torque. are some predgestons n the LPMC model, and some errors exst n manufacture and nstallaton of HPM machne and the effects of vbraton and nose n the experment. The phase flux lnkage and back-emf of the machnes can easly be obtaned from the LPMC model. The electromagnetc torque can be predcted n terms of back-emf and the current of each phase. e =a...d T e = (19) w Fgure 15 shows the predcted output torque when the snusodal phase current wth 4 RMS s njected. Although the rpples of these results are dfferent, the average torque of these results s almost same. The average value of the torque by FEM s 1.35 Nm, and the LPMC one s 1.37 Nm. It can be concluded that the proposed LPMC model s effectve.

11 Progress In Electromagnetcs Research M, Vol. 41, CONCLUSIONS In ths paper, a new method s proposed to dvde and establsh equvalent MMF for HPM machne, and the LPMC model of HPM machne s bult based on equvalent MMF. The electromagnetc performances, e.g., the flux lnkage, back-emf, and torque performance, are obtaned by the proposed LPMC model. From the comparson of the results among the LPMC model, the FEM and the experment, the effectveness of proposed LPMC model s verfed. ACKNOWLEDGMENT Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Projects , and ), by the Specalzed Research Fund for the Doctoral Program of Hgher Educaton of Chna (Project ), by the Natural Scence Foundaton of Jangsu Provnce (Project BK21311), and by the Prorty Academc Program Development of Jangsu Hgher Educaton Insttutons. REFERENCES 1. Zhu, Z. Q. and D. Howe, Halbach permanent magnet machnes and applcatons: A revew, IEE Proc. Electr. Power Appl., Vol. 148, No. 7, , Jul Popa, D.-C., V.-I. Glga, and L. Szabó, Theoretcal and expermental study of a modular tubular transverse flux reluctance machne, Progress In Electromagnetcs Research, Vol. 139, 41 55, Dng, W., Z. Yn, L. Lu, J. Lou, Y. Hu, and Y. lu, Magnetc crcut model and fnte-element analyss of a modular swtched reluctance machne wth E-core stators and mult-layer common rotors, IET Electr. Power Appl., Vol. 8, No. 8, , Aravnd Vathlngam, C., N. Msron, I. Ars, M. H. Marhaban, and M. Nre, Electromagnetc desgn and FEM analyss of a novel dual-ar-gap reluctance machne, Progress In Electromagnetcs Research, Vol. 14, , Ln, D., P. Zhou, S. Stanton, and Z. J. Cendes, An analytcal crcut model of swtched reluctance motors, IEEE Trans. on Magn., Vol. 45, No. 12, , Dec Kokernak, J. M. and D. A. Torrey, Magnetc crcut model for the mutually coupled swtchedreluctance machne, IEEE Trans. on Magn., Vol. 36, No. 2, 5 57, Mar Xu, Z., S. Xe, and P. Mao, Analytcal desgn of flux-swtchng hybrd exctaton machne by a nonlnear magnetc crcut method, IEEE Trans. on Magn., Vol. 49, No. 6, 32 38, Jun Zhu, Z. Q., Y. Pang, D. Howe, S. Iwasak, R. Deodhar, and A. Prde, Analyss of electromagnetc performance of flux-swtchng permanent-magnet machnes by nonlnear adaptve lumped parameter magnetc crcut model, IEEE Trans. on Magn., Vol. 41, No. 11, , Nov Zhou, S., H. Yu, M. Hu, C. Jang, and L. Huang, Nonlnear equvalent magnetc crcut analyss for lnear flux-swtchng permanent magnet machnes, IEEE Trans. on Magn., Vol. 48, No. 2, , Feb Cheng, M., K. T. Chau, C. C. Chan, E. Zhou, and X. Huang, Nonlnear varyng-network magnetc crcut analyss for doubly salent permanent-magnet motors, IEEE Trans. on Magn., Vol. 36, No. 1, Part 2, , Kano, Y., T. Kosaka, and N. Matsu, A smple nonlnear magnetc analyss for axal-flux permanent-magnet machnes, IEEE Trans. on Ind. Electron., Vol. 57, No. 6, , Jun Tarmer, İ., Desgnng an effcent permanent magnet generator for outdoor utltes, Int. J. Eng. Scence Inno. Tech., Vol. 3, No. 3, , Chen, Q., G. Lu, W. Zhao, and M. Shao, Nonlnear adaptve lumped parameter magnetc crcut analyss for spoke-type fault-tolerant permanent-magnet motors, IEEE Trans. on Magn., Vol. 49, No. 9, , Sep. 213.

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