Slotless PM machines with skewed winding shapes

Transcription

1 Sotess PM machines with skewed winding shapes Jumayev S.; Boynov K.; Pauides J.J.H.; Lomonova E.; Pyrhonen J. Pubished in: IEEE DOI: 1.119/TMAG Pubished: 1/11/16 Document Version Accepted manuscript incuding changes made at the peer-review stage Pease check the document version of this pubication: A submitted manuscript is the author's version of the artice upon submission and before peer-review. There can be important differences between the submitted version and the officia pubished version of record. Peope interested in the research are advised to contact the author for the fina version of the pubication or visit the DOI to the pubisher's website. The fina author version and the gaey proof are versions of the pubication after peer review. The fina pubished version features the fina ayout of the paper incuding the voume issue and page numbers. Link to pubication Citation for pubished version APA): Jumayev S. Boynov K. Pauides J. J. H. Lomonova E. & Pyrhonen J. 16). Sotess PM machines with skewed winding shapes: 3D eectromagnetic modeing. IEEE 511) [8181]. DOI: 1.119/TMAG Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-making activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta? Take down poicy If you beieve that this document breaches copyright pease contact us providing detais and we wi remove access to the work immediatey and investigate your caim. Downoad date: 1. Oct. 18

2 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 1 Sotess PM machines with skewed winding shapes: 3D eectromagnetic semi-anaytica mode S. Jumayev1 K.O. Boynov1 J.J.H. Pauides1 E.A. Lomonova1 and J. Pyrho nen 1 Eectrica Engineering Department Eindhoven University of Technoogy Eindhoven 561 AZ The Netherands Engineering Department Lappeenranta University of Technoogy Lappeenranta 5385 Finand Eectrica The 3D modeing technique presented in this paper predicts with high accuracy eectromagnetic fieds and corresponding dynamic effects in conducting regions for rotating machines with sotess windings e.g. sef-supporting windings. The presented modeing approach can be appied to a wide variety of sotess winding configurations incuding skewing and/or different winding shapes. It is capabe to account for induced eddy-currents in the conductive rotor parts e.g. permanent magnet eddy-current osses abeit not iron and winding AC osses. The specific focus of this paper is to provide the reader with the compete impementation and assumptions detais of such a 3D semi-anaytica approach which aows mode vaidations with reativey short cacuation times. This mode can be used to improve future design optimiations for machines with 3D sotess windings. It has been appied in this paper to cacuate fixed parameter Fauhaber Rhombic and Diamond sotess PM machines to iustrate accuracy and appicabiity. Index Terms 3D harmonic modeing Fourier BLDC PM machine sotess winding Fauhaber rhombic sef-supporting winding rotor eddy-current osses high-speed. I. I NTRODUCTION LOTLESS rotating permanent magnet PM) machines are being empoyed in ow to medium power industria appications such as medica aerospace factory automation etc. Their 3D design evauation woud benefit from a fast eectromagnetic modeing too that does not suffer from ong computationa times when accuracy is paramount. Indeed especiay for design optimiation procedures computationa time is even more important due to the required evauation of a arge fied of design soutions. Thus anaytica or semianaytica modeing techniques are superior over numerica inasmuch as they usuay require shorter computationa times. Semi-)Anaytica modeing of sotess AC machines is not a trivia task considering the fact that sotess windings have a arge variety of configurations with compex geometries. Athough in some configurations e.g. toroida concentrated distributed overapping winding machines the modeing can be reduced to a -dimensiona D) probem due to the soe presence of an axia current component [1] [] [3] [4]. However for some sotess winding machines this D simpification is not appicabe due the presence of skewing and/or different winding shapes e.g. Fauhaber aso caed heica and skewed) rhombic and diamond winding which are shown in Fig. 1. Here the conductors can be skewed for haf an eectrica period which impies both and axia and circumferentia current component. This necessitates a 3dimensiona 3D) eectromagnetic mode to be impemented. These 3D magnetic fied modeing techniques have been reported in severa papers. For exampe the authors of [5] have presented the magnetic fied modeing of a Fauhaber winding based on the numerica soution of the Biot-Savart aw. In [6] [7] anaytica modes of magnetic fied in machines with Fauhaber heica) winding have been presented. The S Corresponding author: S. Jumayev e-mai: s.jumayev@tue.n). a) b) c) d) Figure 1: 3D representation of a 3-phase a) Fauhaber b) rhombic and c) diamond windings for rotating machines d) photo of a potted 3-phase Fauhaber winding. resuts obtained by these modes are accurate however do not consider induced rotor eddy-currents osses. These effects have been presented in [8] however the mode contains an assumption which introduces an error in the rotor eddy-current c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

3 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE cacuations if rotor overhang is present. A possibiity to overcome this assumption is to add air regions at the rotor sides on front and rear) and impement a 3D mode-matching technique [9]. However the mode in this case is extremey compex to be derived and soved which dramaticay increases the cacuation time. In this paper a generaied and reativey fast 3D approach to mode the armature reaction fied of sotess winding PM machines is presented. The eectromagnetic fied cacuation is based on the so-caed Harmonic modeing which assumes the direct soution of the quasi-static Maxwe s equations. The winding space distribution is assessed through D Fourier series which convenienty describes the distribution of compex winding geometries. Moreover D Fourier series aows to improve the accuracy of the modeing for exampe compared to [8] by extending the axia boundaries of the winding distribution which is difficut with 1D Fourier series. The D Fourier series contains D and 3D components simutaneousy that forces to impement D or 3D Harmonic modeing for the same mode which is thoroughy expained in this paper. An additiona feature of the presented approach is abiity to account for the eddy-current osses and the eddy-current reaction fied. A = µr µ σ A t µ rµ J Brem ) where µ is the magnetic permeabiity of vacuum [H/m] µ r the reative permeabiity of the materia σ the eectric conductivity [S] J the current density [A/m ] and B rem the remanent fux density of a materia [T]. It shoud be stated that the rotor eddy-current component caused by the moving conducting rotor in the magnetic fied is not present in this equation but accounted for by considering the probem in the rotor reference frame which is introduced in Section III. Equation ) is the so-caed governing equation which reates the magnetic vector potentia to the two magnetic fied sources in PM machines i.e. the excitation magnetic fied originated from the PM as defined by B rem and the armature reaction fied source as described by J. This paper concentrates on cacuating the armature reaction hence the PM fied can be omitted in the cacuations as it does not generate any rotor eddy-currents in sotess PM machines. It is however reativey simpe to assess the PM fied using [1] [] [3] [4] if needed. The current density in ) can be represented as a inear current density on a boundary and therefore is reduced to the Hemhot equation II. HARMONIC MODELING IN CYLINDRICAL COORDINATES In order to derive a 3D Harmonic Mode HM) the foowing assumptions have been considered: The HM mode is based on the quasi-static Maxwe s equations; Materias are isotropic i.e. both reative permeabiity and conductivity remain the same in different directions); Soft-magnetic materia parts iron) are infinitey permeabe; Winding current is modeed by inear current density which assumes infinitey thin winding paced on a boundary; The mode is periodic in both axia and circumferentia direction; Skin and proximity effects in conductors are not accounted for; Baanced 3-phase currents in the windings are considered; Back iron and rotor ength are equa; The interna PM magnetiation is ero. In HM the magnetic fied is derived using the magnetic vector potentia. Keeping in mind that the magnetic fied is the soenoida vector fied B = ) the magnetic vector potentia is introduced as B = A 1) where B is the magnetic fux density [T] and A is the magnetic vector potentia [Wb/m]. Combining the quasi-static Maxwe s equations and constitutive equations in eectromagnetism under the Couomb gauge condition A = ) resuts in A = µr µ σ A t. 3) This represents the governing equation in the conducting region that faciitates modeing of induced eddy-currents. For the air region where the conductivity σ = the governing equation takes the form of Lapace equation A =. 4) In genera the HM assumes the division into regions with different eectromagnetic properties i.e. with different conductivity and permeabiity where the fied behavior in each region is governed by 3) or 4). Soutions of these governing equations are obtained by separation of variabes. The soutions contain unknown coefficients which are determined by soving a system of equations based on the boundary information between considered regions as discussed further. A. D Harmonic modeing In radia fux rotating machines eectromagnetic torque is created by axia current components thus if the mode is imited to axia current components ony axia magnetic vector potentias have to be present. Therefore a D HM can be used where the magnetic fux density in a cyindrica poar) coordinate system is expressed as B = 1 A r θ e r A r e θ. 5) where e r and e θ are the unit vectors in the radia and circumferentia aimutha) directions. The compete derivation of the D HM in cyindrica poar) coordinates r θ) is thoroughy reported in [1] [] and [4] c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

4 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 3 1) Soution of the Lapace and Hemhot equations The compex form soution of 3) in cyindrica poar) coordinate system with ony axia component of the vector potentia is given as A ν) = { Cν)I ν βr) + Dν)K ν βr))e jνθ+ωrt) if β Cν)r ν + Dν)r ν )e jνθ+ωrt) if β = β = jµ r µ σω r where I and K are the modified Besse functions of the first and second kind C and D the unknown coefficients ν the space harmonic order j the imaginary unit and ω r the reative anguar veocity between the media and magnetic fied. The soution of the Lapace equation 4) in compex form is written as 6) A ν) = Cν)r ν + Dν)r ν )e jνθ+ωrt). 7) ) Boundary conditions The boundary conditions between two regions are used to cacuate the unknown coefficients of the soutions of Lapace 7) and Hemhot 6) equations expressed as n B 1 B ) = 8) n H 1 H ) = K 9) where subscript 1 and indicate variabe of the first and second interfacing regions K the inear current density [A/m] n the surface norma and H the magnetic fied strength [A/m]. The atter can be determined by the foowing reation where W 1 and W are the magnetic scaar potentia functions [Wb] and [Wbm] respectivey and e the unit vector in axia direction. Substituting 1) into 3) separate expressions for W 1 and W are W 1 = β W 1 W = β W 13) β = jµ r µ σω r. These equations are the governing equations in the conducting region of the 3D HM. Using 11) 1) and 13) the magnetic vector potentia is expressed as A = 1 r W 1 θ + W r ) e r + 1 W r θ W 1 r ) e θ+ + W β W ) e. 14) The expression for the vector potentia is substituted into 1) and the magnetic fux density inside the conducting region is given as B = W 1 r β W r θ ) e r + 1 W 1 r θ + W β r ) e θ+ + W 1 β W 1 ) e. 15) The governing equation for the air region is obtained by simpy setting β to ero. Then from 15) it can be observed that when β is ero the scaar potentia W does not contribute to the fux density. Therefore using 13) the governing equation for the air region is H = B. 3D Harmonic modeing B µ r µ. 1) If the equivaent inear current density is not imited to the axia current components as e.g. in Fauhaber rhombic and etc. windings aso circumferentia current components are present. Thus two components of the magnetic vector potentia have to be introduced which necessitates a 3D HM impementation. Unfortunatey in cyindrica coordinates the soutions of 3) and 4) cannot be determined by the magnetic vector potentia [1]. However the soution of these equations can be obtained by introducing a second order vector potentia. Keeping in mind A = this second order magnetic vector potentia W is introduced as [11] A = W. 11) W consists of two orthogona scaar potentias W 1 and W and in cyindrica coordinates is given by [1] W = W 1 e + W e 1) W 1 =. 16) The expression for the fux density in the air region is given by B = W 1 r e r + 1 W 1 r θ e θ + W 1 e. 17) 1) Soution of the Lapace and Hemhot equations The soution of 13) is aso determined by separation of variabes which resuts in [9] W 1 ν ω ) = C 1 ν ω )I ν ξr) + D 1 ν ω )K ν ξr)) F 1 ν ω ) cosω ) + G 1 ν ω ) sinω ))e jνθ+ωrt) W ν ω ) = C ν ω )I ν ξr) + D ν ω )K ν ξr)) F ν ω ) cosω ) + G ν ω ) sinω ))e jνθ+ωrt) 18) ξ = ω + β where C D F G are unknown constants cacuated using boundary conditions and ω is the spatia frequency in the -direction. For the Lapace equation 16) the soution takes the form [9] c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

5 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 4 W 1 ν ω ) = Cν ω )I ν ω r) + Dν ω )K ν ω r)) F ν ω ) cosω ) + Gν ω ) sinω ))e jνθ+ωrt). 19) ) Boundary conditions The boundary conditions for 3D HM incude 8) 9) and the boundary information of the eectric fied strength expressed as K K K K K K n E 1 E ) = ) where E is eectric fied strength [V/m] which can be cacuated using E = A t. 1) It needs noting that ony one component of eectrica fied axia or tangentia) requires a boundary condition ). For exampe constraining the axia eectrica fied strength omits the use of the circumferentia one. C. Rotor eddy-current osses As the rotor PM fied is negected the power transfered to torque production is ero. However the armature reaction causes rotor eddy-current osses which can be cacuated by integrating the Poynting vector over the rotor surface. For radia fux rotating machines where the radia component of the vector potentia is ero the rotor eddy-current osses are cacuated as P = 1 ReE Hθ E θ H )ds rotor ) S rotor where H is the conjugated magnetic fied strength [A/m] and S rotor the rotor surface [m ]. s Figure : A phase coi contour of a Fauhaber winding. The bue arrows indicate the inear current density and back arrows the axia and circumferentia components correspondingy. A. Axia component of inear current density The axia component of the inear current density in a singe phase can be written as K θ t) = Nit) sr f θ ) 4) where N is the phase turn number i the instantaneous vaue of phase current [A] r the radius at which the inear current density is defined [m] s the phase spread in circumference [rad] t time [s] and f θ ) the distribution function dependent on the geometry of the winding. This distribution function can be generay approximated through D Fourier series as III. DISTRIBUTION OF LINEAR CURRENT DENSITY Modeing winding regions with imposed currents in cyindrica coordinates does not seem possibe by means of first and second order vector potentias [13]. Thus the winding is modeed as an air region and inear current density imposed on the iron bore surface. If this inear current density is soey distributed aong the axia direction a D HM can be used e.g. for machines with toroida or concentrated sotess windings. However if an axia and circumferentia components are present see Fig. ) the inear current density can be written as f approx θ ) = [ css ν m) sinω θ θ) sinω )+ ν= m= + c sc ν m) sinω θ θ) cosω )+ + c cs ν m) cosω θ θ) sinω )+ + c cc ν m) cosω θ θ) cosω ) ] 5) ω θ = ν T θ ω = m T K = K e + K θ e θ 3) where K and K θ are the axia and circumferentia inear current density components respectivey [A/m]. where ω θ and ω are the spacia frequencies in the θ- and -direction and T θ and T the periodicities in the θ- and - direction. The coefficients c ss c sc c cs and c cc are derived by the foowing integras c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

6 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 5 c ss ν m) = 4T θ T c sc ν m) = 4T θ T c cs ν m) = 4T θ T c cc ν m) = 4T θ T T T θ T T θ T T θ T T θ f θ ) sin ω θ θ) sin ω ) dθd f θ ) sin ω θ θ) cos ω ) dθd f θ ) cos ω θ θ) sin ω ) dθd f θ ) cos ω θ θ) cos ω ) dθd 6) where 4 for ν > m > for ν = m > =. 7) for ν > m = 1 for ν = m = The integra imits in 6) can be set by inear equations to describe winding skewing [14]. The practica impementation of the D Fourier series is given in the Appendices. For three phase rotating machines the phase cois are mutuay shifted by /3 hence the phase currents are given as it) = x = Îk) coskωt + x)) 8) ) where Î is the current ampitude of each time harmonic order [A] k the time harmonic order and ω the anguar frequency [rad/s]. Substituting 5) and 8) into 4) resuts in a Fourier series representation of the axia inear current density K θ t) = m= [ ˆKss m k L)e j3lk)θ+kωt) sinω )+ + ˆK sc m k L)e j3lk)θ+kωt) cosω )+ + ˆK cs m k L)e j3lk)θ+kωt) sinω )+ + ˆK cc m k L)e j3lk)θ+kωt) cosω ) ]. 3) This compex inear current density can be transformed from the compex domain back to 9) by taking the rea or imaginary part of each component. B. Circumferentia component of inear current density The circumferentia component of the inear current density can be derived using the same procedure aternativey from the continuity equation as K = 1 r K θ θ. 31) However there is no need for the circumferentia component of the inear current density for the fied cacuation. This statement resuts from the continuity equation with absence of the free charge conservation and Ampere s aw. If on the boundary with inear current density the transition conditions for the norma and a singe tangentia axia or circumferentia) magnetic fied components have been satisfied the second tangentia component is aso satisfied [1]. C. Linear current density in rotor reference frame K θ t) = m= [ ˆKss m k L) sin3l k)θ + kωt) sinω )+ + ˆK sc m k L) sin3l k)θ + kωt) cosω )+ + ˆK cs m k L) cos3l k)θ + kωt) sinω )+ + ˆK cc m k L) cos3l k)θ + kωt) cosω ) ] 9) where L is an integer number and ˆK the ampitude of Fourier series components. Since the soutions of the governing equations 6) 7) 18) and 19) are in the compex form the inear current density expression 9) for the simpicity reasons is transformed to the compex form as The axia component of the inear current density 3) is given in the stator reference frame. This means that the harmonics are traveing with respect to the stator. However for the rotor eddy-current cacuation the expression shoud be transferred to the rotor reference frame since ony the harmonics rotating with respect to the rotor induce eddycurrents. Therefore the anguar position of the rotor is inked to the anguar position of the stator through the anguar veocity of the rotor as θ = θ r + ωt 3) where θ r is the anguar position in the rotor reference frame [rad]. Substituting 3) equation into 3) resuts in the expression of the axia component of the inear current density in the rotor reference frame c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

7 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 6 K θ r t) = m= [ ˆKss m k L)e j3lk)θr+3lωt) sinω )+ + ˆK sc m k L)e j3lk)θr+3lωt) cosω )+ + ˆK cs m k L)e j3lk)θr+3lωt) sinω )+ + ˆK cc m k L)e j3lk)θr+3lωt) cosω ) ]. 33) input a) output IV. THREE-DIMENSIONAL MODEL SYNTHESIS OF SLOTLESS WINDING MACHINES This section describes the reaiation procedure of the 3D HM. For vaidation the mode is impemented for fixed parameter Fauhaber rhombic and diamond shaped windings as shown in Fig. 3. The axia component of the inear current density of these windings is given in section Appendix. Ony the 3D HM synthesis for a PM machine with rhombic winding is derived in detai in this paper i.e. on the sotess PM machine as shown in Fig. 4a. Here the rotor is competey made of a diametricay magnetied singe PM and as discussed before the PM excitation fied is not considered thus the PM is modeed as a conducting region. The back iron is infinitey permeabe and the winding is represented by an air region with inear current density on its boundary between air and back-iron). The resutant mode consists of two regions: the PM and air gap as shown in Fig. 4b. Finay an important mode detai is the axia periodicity. In [8] where the 3D HM of the Fauhaber winding machines is presented the mode has an axia periodicity of a singe winding ength. However the rotor usuay overhangs the winding which infuences the induced eddy-currents distribution thus the assumption in [8] eads to an error in the rotor eddycurrent estimation and magnetic fied. To overcome this the axia periodicity of the mode is set to T = as shown in Fig. 4b. However this axia periodicity automaticay assumes that the back-iron ength equas the axia periodicity. The authors reaie that this is not a competey correct impementation since ideay the axia periodicity shoud be equa to the rotor ength. Nevertheess this assumption simpifies the derivation of the inear current density expression and the obtained resuts are more accurate than the resuts shown in [8]. A. HM mode of PM machine with rhombic winding The impementation of the 3D HM starts with the derivation of the inear current density which is for the rhombic winding given as input input b) c) output output Figure 3: One phase coi of a) Fauhaber heica) b) rhombic and c) diamond sotess winding types. K θ r t) = K 1 + K 34) K 1 = ˆKsc1 m k L) cosω 1 ) e j3lk)θr+3lωt) K = ˆKsc m k L) cosω ) e j3lk)θr+3lωt) 3L k m ) ω 1 = 3L k + m ) ω = /4 / / c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

8 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 7 r si r m r r Winding PM Air Iron a) Air region PM region r r r m r w r si W 1PM m k L) = C 1PM m k L)I 3Lk ξ PM r) sinω )e j3lk)θr+3lωt) W PM m k L) = C PM m k L)I 3Lk ξ PM r) cosω )e j3lk)θr+3lωt) 36) β PM = j3lωσ PM µ PM ξ PM = ω + β PM. r b) Figure 4: a) Cross-sections of a sotess PM machine with a shaftess rotor b) Representation of the regions in the anaytica mode iron is assumed to be infinitey permeabe). where ˆK sc1 and ˆK sc are inear current density component ampitudes [A/m]. These ampitude expressions are given in Appendix A as we as the derivation of inear current density for the rhombic winding. Equation 34) contains two components thus the superposition principe is used to obtain the fied soution. 1) 3D harmonic mode Eectromagnetic fied behavior in the air region is described by the Lapace equation 16) and in the conducting region by the Hemhot equation 13). The soution of these equations needs to compy with the inear current density distribution. Thus the magnetic scaar potentias 18) in the PM region shoud aso consist of two components as the expression of the inear current density where the first component can be written as W 1PM1 m k L) = C 1PM1 m k L)I 3Lk ξ PM1 r) sinω 1 )e j3lk)θr+3lωt) W PM1 m k L) = C PM1 m k L)I 3Lk ξ PM1 r) cosω 1 )e j3lk)θr+3lωt) 35) β PM = j3lωσ PM µ PM ξ PM1 = ω 1 + β PM and the second component as Since the PM region inner radius is ero the component that consists the modified Besse function of the second kind is aso ero. The soution of the Lapace equation aso has two components where the first component is W a1 m k L) = C a1 m k L)I 3Lk ω 1 r)+ + D a1 m k L)K 3Lk ω 1 r)) sinω 1 )e j3lk)θr+3lωt) 37) and the second component W a m k L) = C a m k L)I 3Lk ω r)+ + D a m k L)K 3Lk ω r)) sinω )e j3lk)θr+3lωt). 38) ) D harmonic mode A D HM is empoyed when the inear current density becomes -independent e.g. the current distribution is inherenty independent of as one of the components of the diamond winding inear current density see Appendix C) or when ω =. Simiary to the 3D mode and taking 6) and 34) the soution in the PM region is A k L) = C PM k L)I 3Lk β PM r) e j3lk)θr+3lωt) if L C PM k L)r abs3lk) e j3lk)θr+3lωt) if L = 39) β PM = jω r µ PM σ PM = j3lωµ PM σ PM. This soution contains both the first and second component of the inear current density distribution. For the air region the soution is c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

9 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 8 A k L) = C a k L)r 3Lk) + D a k L)r 3Lk ) e j3lk)θr+3lωt). 4) 3) Determination of unknown coefficients To determine the unknown coefficients in 35)-4) a system of equations based on the boundary condition information is composed for the 3D HM as B rpm B ra = for r = r m H θpm H θa = for r = r m 41) H PM H a = for r = r m H θa = K for r = r si This system of equations consists of four equation which equas to the number of unknowns in the PM and air regions in the 3D HM. The system of equations for the D HM is B rpm B ra = for r = r m H θpm H θa = for r = r m 4) H θa = K for r = r si Consequenty soving these system of equations for both components of the axia inear current density the unknown coefficients cacuated. B. Cacuation of rotor eddy-current osses Equation ) is used to cacuate the rotor osses due to the induced eddy-currents however anayticay this equation is difficut to impement. This difficuty is reated to the fact that magnetic and eectric fied strengths in ) can have severa components due to the inear current density distribution which makes the anaytica soution of the oss cacuation rather compicated. Therefore a numerica approach is impemented that assumes the division of the rotor surface into sma areas/eements S e ) as shown in Fig. 5. The rotor eddy-current osses are cacuated for each eement separatey and eventuay summaried together. The vaues of the eectrica and magnetic fied strengths are determined in the center of these eements. Mathematicay for the case of rhombic winding this can be formuated as E P = E vw θ V W v=1 w=1 S e ReEvw k)h vw θ k) k)h vw k)) 43) S e = r mt V W E vw k) = H vw k) = E vw 1 m k L) + E vw m k L)) H vw 1 m k L) + H vw m k L)) where v and w indicate the eement position in and θ directions 1 and are indexes corresponding to the quantities resuting from the first and second component of the rhombic winding inear current density and V and W are number of eements in and θ directions thus V W resuts in the number of eements the rotor is divided into. V. MODEL VALIDATION The 3D semi-anaytica mode vaidation is done by comparison of radia fux densities and rotor eddy-current osses obtained by the semi-anaytica and 3D FEM modes. The materia and geometrica properties of the benchmark machines are summaried in Tabe I. The software used to buid the 3D FEM modes is Fux 1 from Cedrat. The iron core permeabiity in the 3D FEM mode is set to infinity and the iron axia) ength is equa to the winding ength. The windings in FEM modes are modeed by means of non-meshed cois which represent nonconducting conductor voumes with imposed current density. Tabe I: Materia and geometrica properties of the benchmark PM machines Parameter [symbo] Vaue Speed [n] 1 5 rpm Reative magnetic permeabiity of PM [µ rpm ] 1.5 Eectric conductivity of PM [σ PM ] S/m Number of turns per coi [N] 16 Winding spread [s] /3 rad Machine active ength [] mm Magnet radius [r m].75 mm Stator bore inner radius [r si ] 5 mm Stator bore outer radius [r so] 8 mm Air-gap ength [δ].5 mm L m 1 1 V W 1 H Figure 5: A sampe of the area/eement S e ). H E A. Magnetic fux density The magnetic fux density radia components of the benchmark PM machines obtained by the semi-anaytica mode are iustrated in Fig. 6. These are extracted for the first time harmonic component with a phase current peak vaue of 1A. Figure 7 shows the reative difference of the radia magnetic fux densities obtained by the anaytica mode and the 3D FEM for the benchmark machines. The sma c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

10 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE B. Rotor eddy-current osses [mm] θ [deg] SANA FEM a) [mm] θ [deg] 3 b) 3 4 This paper presents a compete 3D mode derivation for accurate modeing of sotess PM machines with different winding shapes incuding skewed. This semi-anaytica approach does aow to cacuate the magnetic fied in a regions and eectric fied in regions with an assigned conductivity. More specificay this paper discussed extraction of the induced rotor eddy-current osses using semi-anaytica mode. This approach is based on a magnetic fied derivation from the Maxwe equations by means of magnetic vector potentia and magnetic second order vector potentia in the cyindrica coordinates. The source of the armature reaction is impemented using a inear current density which is expressed as Fourier series. To vaidate the mode three benchmark sotess PM machines with fixed parameter Fauhaber rhombic and diamond windings have been modeed. The obtained resuts Br [mt] VI. C ONCLUSION Br [mt] 1 1 Br [mt] SANA FEM Tabe II: Rotor eddy-current osses versus the time harmonic order semi-anaytica - SANA 3D FEM - FEM) PP P k P [W] PP P Fauhaber Rhombic Diamond Br [mt] To summarie the resuts of the rotor eddy-current oss cacuation Tabe II provides the osses per harmonic for the rotating speed of 1 rpm with both the semi-anaytica and FEM anaysis respectivey. Within this simuation the phase current ampitude for a time harmonic components for the convenience is assumed to be 1A. This however represents a theoretica vaue since it is important to account for the rotor eddy-current osses in PM machines especiay in high-speed ones to avoid therma overoading of the rotor. This summary iustrates that the difference between the resuts of the semi-anaytica and FEM anaysis does not exceed 5% which vaidates the 3D semi-anaytica mode. It needs noting that this mode does aow the eddy-current oss cacuation for a integer time harmonics however without tripet harmonics due to the considered three phase system. SANA FEM Br [mt] Br [mt] difference indicates a good agreement between these modes where the maximum reative difference is ess than 1% and the weighted mean vaue over the active part of the rotor is about 3% respectivey. Further it can be seen that the difference repicates the winding geometries. Additionay to give a visua impression of the magnetic fux density difference between the semi-anaytica and 3D FEM modes where D pots of radia fux densities of the benchmark machines are iustrated in Fig [mm] θ [deg] 3 c) Figure 6: Radia component of the armature reaction magnetic fux density of the benchmark PM machines with a) Fauhaber b) rhombic and c) diamond sotess winding types at r = rm c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

11 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE = 5 mm SANA FEM [mm] [mm] θ[deg] θ[deg] 15 1 a) b) Br[mT] Br[mT] Br [mt] Br [mt] θ [deg] a) = 5 mm θ [deg] 4 3 b) = 5 mm SANA FEM SANA FEM [mm] θ[deg] c) Figure 7: Reative difference between radia component of the armature reaction fux densities obtained semi-anayticay and by 3D FEM for the benchmark PM machines with a) Fauhaber b) rhombic and c) diamond sotess winding types at r = r m. Br[mT] Br [mt] θ [deg] c) Figure 8: Reative difference between radia component of armature reaction fux densities obtained by a semi-anayticay SANA) and a 3D FEM FEM) for the benchmark PM machines with a) Fauhaber b) rhombic and c) diamond sotess winding types at r = r m and = 5 mm c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

12 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 11 have been compared with those obtained by 3D FEM and this comparison showed a good agreement. The discrepancy between magnetic fux density and rotor eddy-current osses had a maximum of 1% and 5% respectivey between the 3D semi-anaytica and 3D FEM modes for the benchmark machines. The cacuation time of semi-anaytica approach for a winding configurations is ess than a minute where for the 3D FEM it takes a few days to reach the steady-state point assuming equa hardware conditions. For sotess PM machines with comparabe winding shapes and physics as the benchmark topoogies 3D harmonic modeing has proven to be an exceent too for modeing and design optimiation in terms of computationa time as we as accuracy. c ss = 4 / θ ν m sin θ sin dθd + θ 1 / θ 4 θ 3 θ 6 / θ 5 θ 8 / θ 7 ν m sin θ sin dθd+ ν m sin θ sin dθd ν m sin θ sin dθd 44) APPENDIX A. Linear current density of rhombic winding To cacuate the axia component of the inear current density distribution the rhombic winding is divided into four current carrying fiaments as shown in Fig. 9. Foowing the coefficients in the distribution function 5) are derived using 6) as c sc = 4 / θ ν m sin θ cos dθd + θ 1 / θ 4 θ 3 θ 6 / θ 5 θ 8 / θ 7 ν m sin θ cos dθd+ ν m sin θ cos dθd ν m sin θ cos dθd 45) / -/ s Figure 9: A phase coi of a rhombic winding represented as the four current carrying strips. The bue arrows represent the current direction. c cs = 4 / θ ν m cos θ sin dθd + θ 1 / θ 4 θ 3 θ 6 / θ 5 θ 8 / θ 7 ν m cos θ sin dθd+ ν m cos θ sin dθd ν m cos θ sin dθd 46) c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

13 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 1 where c cc = 4 / θ ν m cos θ cos dθd + θ 1 / θ 4 θ 3 θ 6 / θ 5 θ 8 / θ 7 ν m cos θ cos dθd+ ν m cos θ cos dθd ν m cos θ cos dθd θ 1 = 4 3 θ = 3 / ) θ 3 = θ 4 = / ) θ 5 = θ 6 = / + ) θ 7 = 3 / + ) θ 8 = 47) ) These coefficients contain four integras which correspond to the number of current carrying fiaments and the sign preceeding the integras shows the directions of current with respect to the axia direction. The imits of the inner integras are the equations of the fiament border ines where these dependencies can be derived as fθ ) = ν=1 c ss + c sc + c ss c sc sin νθ + x)) cos sin νθ + x)) cos n m ) n + m ) c ss = 16 cosn) sinm/4) sinn/3) mn 8 sinm/) sinn/3) sinn) c sc = mn x = ). + 51) Substituting this distribution function into 4) and having in mind the foowing trigonometric identity: sinα) cosβ) = sinα + β) + sinα β) 5) the axia component of the inear current density for the rhombic winding is K θ t) = ˆKsc1 m k L) sin 3L k)θ + kωt)) cosω 1 )+ θ = ) θ 1 θ 1 + θ. 49) These coefficients shoud be cacuated for the foowing cases 1 st case for ν > m > nd case for ν = m > 3 rd case for ν > m = 4 th case for ν = m = 5) which eventuay resuts in nine coefficients. In the case of the rhombic winding ony coefficients c ss and c sc for ν > m > ) do not resut in a ero. Foowing some mathematica simpifications the distribution function of three phases for the rhombic winding is + ˆK ) sc m k L) sin 3L k)θ + kωt)) cosω ) ) ω 1 = ω = ˆK sc1 = 3L k m 3L k + m ) 3NÎ 16 cos3l k)) sinm/4) sin3l k)/3) sr si m3l k) ˆK sc = 53) 3NÎ sr si 8 sinm/) sin3l k)/3) sin3l k)) m3l k) c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.

14 This artice has been accepted for pubication in a future issue of this journa but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TMAG IEEE 13 B. Linear current density of Fauhaber winding K θ t) = sin ω ˆK sc sin 3L k)θ + kωt)) 3L k) ) + sin ω + ˆK s sin 3L k)θ + kωt)) sin 3L k + m) ω 1 = 3L k m) ω = 3L k) ω 3 = ˆK sc = 3NÎ 4 sinm/) sin3l k)/3) sr si m3l k) ˆK s = 3NÎ sin3l k)/3). sr si 3L k) )) 3L k) + ω 3 + C. Linear current density of diamond winding K θ t) = 3L k) 54) ˆKss sin 3L k)θ + kωt)) cosω 1 ) + cosω )) + + ˆK m )) sc sin 3L k)θ + kωt)) cos + + ˆK s sin 3L k)θ + kωt)) 55) ω 1 = ω = 3L k m 3L k + m ) ) ˆK ss = 3NÎ sr si cosm/4)) cos3l k)) m3l k) sinm/8) sin3l k)/3) ˆK sc = 3NÎ sr si 8 sinm/4) sin3l k)/3) sin3l k)/) m3l k) ) REFERENCES [1] B. L. J. Gysen K. J. Meessen J. J. H. Pauides and E. A. Lomonova Genera formuation of the eectromagnetic fied distribution in machines and devices using fourier anaysis Magnetics IEEE Transactions on vo. 46 pp Jan 1. [] A. Borisavjevic Limits Modeing and Design of High-Speed Permanent Magnet Machines. Springer-Verag Berin Heideberg 13. [3] P.-D. Pfister and Y. Perriard Sotess permanent-magnet machines: Genera anaytica magnetic fied cacuation Magnetics IEEE Transactions on vo. 47 pp June 11. [4] S. Hom Modeing and optimiation of a permanent magnet machine in a fywhee. PhD thesis Deft University of Technoogy 3. [5] P. Ragot M. Markovic and Y. Perriard Anaytica determination of the phase inductances for a brushess dc motor with fauhaber windings Industry Appications IEEE Transactions on vo. 45 pp [6] E. Matagne B. Dehe and K. Ben-Naoum Exact specia d spectra expression of the magnetic fied in air gap with skewed source in Eectrica Machines ICEM) 1 XXth Internationa Conference on pp Sept 1. [7] A. Anderson J. Bumby and B. Hassa Anaysis of heica armature windings with particuar reference to superconducting a.c. generators Generation Transmission and Distribution IEE Proceedings C vo. 17 no. 3 pp [8] S. Jumayev J. J. H. Pauides K. O. Boynov J. Pyrhonen and E. A. Lomonova Three-dimensiona anaytica mode of heica winding pm machines incuding rotor eddy-currents Magnetics IEEE Transactions on vo. PP no. 99 pp [9] K. Weiget Über die dreidimensionae anaytische Berechnung von Wirbeströmen in gekoppeten Kreisyinderschaen. PhD thesis Technischen Universität München [1] B. Hague The Principes of Eectromagnetism Appied to Eectrica Machines. Dover Pubications inc [11] W. R. Smythe Static and dynamic eectricity. New York : McGraw-Hi 3rd ed ed Incudes bibiographies. [1] P. Hammond Use of potentias in cacuation of eectromagnetic fieds Physica Science Measurement and Instrumentation Management and Education - Reviews IEE Proceedings A vo. 19 pp March 198. [13] T. Theodouidis C. S. Antonopouos and E. E. Krieis Anaytica soution for the eddy current probem inside a conducting cyinder using the second order magnetic vector potentia COMPEL - The Internationa Journa for Computation and Mathematics in Eectrica and Eectronic Engineering vo. Vo. 14 No. 4 pp [14] K. J. W. Puk J. W. Jansen and E. A. Lomonova Modeing of noncuboida magnetic sources in 3-d fourier modeing Magnetics IEEE Transactions on vo. 51 pp. 1 4 Nov 15. ˆK s = 3NÎ cos3l k)/6) + cos3l k)/)) sr si 3L k) sin3l k)/6) c) 16 IEEE. Persona use is permitted but repubication/redistribution requires IEEE permission. See for more information.