IEEE TRANSACTIONS ON MAGNETICS, DATE 1 Generalized Slip Transformaions and Air Gap Harmonics in Field Models of Elecrical Machines Ville Räisänen, Saku Suuriniemi, Lauri Keunen Absrac In numerical field analysis of seady-saes of elecrical machines, frequency-domain mehods are ofen much faser han ime-domain mehods. Frequency-domain echniques ofen involve simplificaions ha impac heir accuracy on harmonic effecs due o sloing, sauraion and ime harmonics in winding currens. The naure of hese inaccuracies and heir relaionship o air gap field harmonics is no ha well covered in he lieraure. Correc predicion of air gap field harmonics can be based on careful use of specral Dirichle-o-Neumann (D-o-N) mappings and generalized slip ransformaions (GST). We show how nonzero harmonics in he air gap fields can be reliably prediced and explain he naure of inaccuracies in cerain common seady-sae soluion echniques. Index Terms Boundary value problems, Fourier series, harmonics, roaing machines. I. INTRODUCTION THE seady-sae operaion of roaing elecrical machines can be prediced in a number of ways. Approximae soluion by Finie Elemen Mehods (FEM) in ime domain is a sandard approach, bu full soluion of hese compuaionally inensive problems is no always feasible. Then frequencydomain or saic mehods are ofen used. Many moor analyses depend on a mehod ha accouns for he relaive movemen beween he roor and he saor. In he case of inducion machines, ofen only a single imefrequency is used in he roor equaions. This roor frequency is he single saor frequency muliplied by he roor slip he simples example of a slip ransformaion. The res of problem is solved as usual a he synchronous frequency 1]. While he single slip ransform approach may provide a useful firs orque esimae, i is inadequae for mos oher asks: Inverers higher harmonics and he roaion of a sloed roor induce addiional ime-frequencies ino he saor fields. When he roor slip is no an ineger, hese ime-frequencies are no always ineger-muliples of he synchronous frequency 5]. The fundamenal slip ransformaion is incorrec for mos oher space and ime harmonics 2], 3]: boh he ime and space harmonic indices influence he appropriae slip ransformaion, as found in previous work 4] on seady-sae problems. The generalized slip ransformaions (GST) arise from his need. GSTs describe how he ime-frequencies of he air gap field harmonics ransform beween he roor and saor coordinae sysems in relaive moion. They are he key o accuraely accoun for he air gap field harmonics in wo-dimensional V. Räisänen was wih he Deparmen of Elecrical Engineering, Tampere Universiy of Technology, Tampere 3311, Finland. S. Suuriniemi and L. Keunen are wih he Deparmen of Elecrical Engineering, Tampere Universiy of Technology, Tampere 3311, Finland. Corresponding auhor: Saku Suuriniemi (e-mail:saku.suuriniemi@u.fi) frequency-domain BVPs for elecrical machines. The applicaion of he ransformaion needs aenion o deail, no well covered in he lieraure on frequency-domain soluion mehods. The roor and he saor can be seen o respond o he air gap field harmonics. Cerain air gap sream funcion harmonic excies a unique combinaion of harmonics in he normal derivaive of he sream funcion. This is reducible o air gap harmonic coefficiens, and compleely described in erms of specral Dirichle-o-Neumann (D-o-N) maps 7], 8], 9]. The marix represenaions of linear specral D-o-N maps exhibi sparsiy paerns ha sem from he roaional symmery of sloed geomeries. The sparsiy paerns, ogeher wih he GSTs, explain how roor and saor subproblems creae harmonics ino he air gap fields 8]. The non-zero air gap field harmonics can be hen accuraely prediced. The specral D- o-n mappings and GSTs also provide a solid foundaion for accurae and efficien frequency-domain mehods. Prior frequency-domain echniques ha he auhors know of suffer from over-simplified slip ransformaions. The permeance wave heory, a popular air gap harmonics analysis mehod, describes he effec of slos by a permeance relaion beween he magneomoive force over he air gap and he air gap flux densiy 1]. While some permeance relaions can be derived direcly from simple boundary value problems, hey rely on very resricive assumpions abou slo shapes, curren densiies, and maerial relaions 6], 1], 11]. In conras, he sparsiy paerns for specral D-o-N mappings follow formally from roaional symmery of maerial parameers in sloed geomeries and apply in any such BVP, regardless of he shape or maerial parameers of he slos. The Muli-Layer Transfer Marix (MLTM) mehods build on analyical soluions of roor subproblems in erms of individual space harmonics 12]. However, since he higher space harmonics in MLTM are due o he space harmonics in he curren shee expansion of he saor winding currens, he saor models in MLTM fail o accoun for he influence of he saor sloing 4], 13]. The roor and saor subproblems can be coupled also in FE models by a space harmonic analyical model of he air gap 14]. Such an air gap model can be inerpreed as a specral D-o-N mapping from he viewpoin of he roor and he saor subproblems 15]. In his paper, he specral D-o-N mappings of he roor and saor subproblems are expressed in co-moving coordinae sysems, and hey do no couple differen ime-frequencies. Consequenly, he effec of moion is isolaed o he slip ransformaions.
IEEE TRANSACTIONS ON MAGNETICS, DATE 2 Ω s Ω r Figure 1. Half of he problem geomery. The air gap widh is exaggeraed. Λ s Λ s Λ s θ Γ dr Λ r Γ sr Λ r Γ ds Λ r θ Figure 2. An example of he ransformaion of harmonics beween he saor and he roor coordinae sysems. II. AIR GAP FIELD HARMONICS Le us denoe by Ω he cross secion of a machine. This is furher divided ino he saor and he roor regions Ω s and Ω r, whose common inerface Γ sr = Ω s Ω r is a he cener of he air gap (see Fig. 1). The ouer boundary of he saor is denoed by Γ ds and he inner boundary of he roor region by Γ dr. We assume ha he magneic flux densiy has no componen in he axial direcion and ha he curren densiy is purely axial. An A z -formulaion wih a sream funcion A corresponding o he magneic vecor poenial is used 9]. As in publicaion 9], magneic sream funcions A s and A r are used in he regions Ω s and Ω r in heir respecive comoving coordinae sysems. The change of coordinaes from he saor o he roor sysem is u : R 2 R 2 : rs θ s ] rr θ r ] := r s θ s (1 s) ωs p where s is he roor slip, p is he number of pole pairs and ω s is he synchronous frequency. If A is he magneic sream funcion in he whole problem domain expressed in he saor coordinae sysem, he sream funcion expressed in he roor coordinae sysem is A r := A Ωr u. A. Dirichle and Neumann daa The magneic sream funcion and is radial derivaive on Γ sr are expanded in boh coordinae sysems as a Fourier series expansion of wo variables: ime and respecive mechanical angle θ k. Le us denoe he se of non-zero air gap harmonics in he subproblem k {s,r} by Λ k and index he harmonics in Λ k wih naural numbers. In order o faciliae he use of marix algebra and numerical compuaions, he series expansions for boh subproblems k mus be runcaed ] (1) o respecive combinaions over finie ses of harmonics. Tha is, we wrie Λ k := { ( k n, k n) : n = 1,..., Λ k } Z 2, (2) where n k and n k are he ime and space harmonic indices of he n:h non-zero harmonic of he subproblem k. The selecion of he ses Λ k will be discussed in he Subsecion IIIc. The se of space and ime harmonics in he se Λ k are denoed by Λ k θ and Λk, respecively. The se of space harmonics Λ k θ is independen of he coordinae sysem whereas he relaion beween Λ s and Λ r is dependen on he roor slip (see Fig. 2). We assume ha he magneic sream funcions in boh coordinae sysems are ime-periodic a some fundamenal imefrequency ω f. An appropriae ω f can be always deermined, and is choice is discussed in secion IIID. The ime-harmonic index of he synchronous frequency is denoed by h = ω s /ω f and he harmonics ±(h, p) are called he working harmonics. Thus, we expand A k (θ k,) = D k e j( ω f θ k ) (3) (,) Λ k A k r (θ k,) = Nη k ηe j(η ω f ηθ k ). (4) (η,η) Λ k The expansions (3) (4) are called he Dirichle and Neumann daa in coordinae sysem k, respecively. The coefficiens D k and Nk η η are he Dirichle and Neumann coefficiens, respecively. Since A k and is radial derivaive mus be real, conjugae symmery D( k, ) = Dk (,) mus hold for all (,) Λ k, where denoes he complex conjugae. Le us define inner producs, θ and, on θ,2π] and,2π/ω f ], respecively, so ha f,g θ := 1 2π f,g := ω f 2π 2π 2π/ωf f(θ)g(θ) dθ (5) f()g() d (6) hold. Thereafer, he Dirichle and Neumann coefficiens can be obained from he Dirichle and Neumann daa (3) (4) Ak D k = k,e jθ ω f (7) θ,ej Nη k A k η = r,e jηθ k,e jη ω f. (8) For convenience, he harmonic indices for he harmonics in he Dirichle and Neumann daa are denoed wih differen symbols so ha ( k n, k n) = (η k n, η k n) (9) hold for all n = 1,2,..., Λ k. Then, he coefficiens are assembled ino he vecors ] D k = D k, D k... D k T 1 kk 1 2 kk 2 k (1) Λ k Λ ] N k = N k, N k... N k T η 1 kηk 1 η 2 kηk 2 η k Λ ηk Λ. (11) θ
IEEE TRANSACTIONS ON MAGNETICS, DATE 3 Roor Slip s 2 1.8 1.6 1.4 1.2 (, 37) 1.8.6.4.2 (, 35) (, 5) (, ) (1,1) (1,5) (, 7) (1,7) Opposie (1, 35) (1, 37) Locked Synchronous 5 3 1 9 7 5 3 1 3 5 7 9 11 13 15 Generalized Slip s 1 Figure 3. Slip ransformaions for he ime-frequencies in he Dirichle daa for some harmonics (,) when ω f = ω s and p = 1. Change in he ime harmonic number corresponds o a horizonal shif in he generalized slip s p. B. Slip Transformaions and Inerface Condiions When he roor is no locked, he ime-frequencies of he space harmonics in he Dirichle and Neumann daa are no equal in differen coordinae sysems. The relaionships beween he ime-frequencies of he fields of he roor and he saor subproblems are known as slip ransformaions, which we shall derive here. Equivalen ransformaions have been widely discussed in he lieraure 2], 4], 5]. Slip ransformaions arise from he inerface condiions on Γ sr for he Dirichle and Neumann daa: ( A s θ s + 1 s ) p ω s, = A r (θ s,) (12) A s r ( θ s + 1 s ) p ω s, = A r r (θ s,). (13) Dirichle daa in he roor coordinae sysem can be obained by he subsiuion of (3) ino (12) ( A r = A s θ r + 1 s ) p ω s (14) = D s expj(s p ω f θ r ) (,) Λ s where we have defined he generalized slip s p := ω s ω f 1 s p. (15) In Fig. 3, he ransformaion (15) is demonsraed for a se of harmonics. From he inerface condiions (12) (13), i follows ha he number of non-zero air gap harmonics is independen of he coordinae sysem. The harmonics of differen subproblems are relaed by D s = D r s p (16) N s = N r s p. (17) The ses Λ r and Λ s used for he expansions (3) (4) should be seleced so ha(,) Λ s holds if and only if(s p,) Λr holds. The Dirichle and Neumann coefficiens for boh subproblems k are assembled ino vecors D k,n k C Λk so ha roor and saor coefficiens relaed by (16) (17) share common rows. Thereafer, (16) (17) are equivalen o C. Winding Currens D s = D r (18) N s = N r. (19) The curren densiy in each subproblem can consis of winding currens and eddy-currens in massive conducors. Tha is, he curren densiy is expanded as J k = Q k q=1 Λ k γ= Q k /2 I k qj q k ej ω f σ A k (2) where J q k is he source curren densiy corresponding o uni curren in he winding of he slo q in he subproblem k. For simpliciy, we assume ha slos conain only a single layer. Wihou loss of generaliy, we assume ha he slos are ordered in ani-clockwise direcion according o mechanical angle. Thereafer, we expand he slo winding currens as a discree Fourier series Q k /2 ( I k q = O k γ exp j 2πγq ) (21) Q k where O γ is called he harmonic (,γ) in he slo winding currens of subproblem k. The equaion (21) can wrien in he marix form I k = ζ k] O k. (22) D. Specral Dirichle-o-Neumann Mappings From he viewpoin of he inerface Γ sr a he cener of he air gap, he elecromagneics of each subproblem is compleely characerized by a mapping from he Dirichle daa o he Neumann daa. Such mappings are known as Dirichle-o- Neumann (D-o-N) mappings. We call represenaions of D-o- N mappings in erms of Dirichle and Neumann coefficiens specral D-o-N mappings 8], 9]. They compacly express e.g. differen roor varians in a design siuaion. If he fundamenal frequency ω f and he ses Λ r,λ s have been chosen appropriaely, he specral D-o-N mapping for he subproblem k in seady sae can be expressed as a funcion N k = S(D k,i k ). (23) If sauraion can be negleced, here exis marices B k] C Λk Λ k and F k] C Λk Q k such ha N k = B k] D k + F k] I k (24) holds (see Appendix A). The marix B k characerizes he behavior of he subproblem in he absence of winding currens. The marix F k characerizes he behavior of he subproblem when no flux crosses Γ sr. Combinaion of (24) wih he inerface condiions (18) (19), yields B s B r] D s = F r ζ r] O r F s ζ s] O s. (25)
IEEE TRANSACTIONS ON MAGNETICS, DATE 4 III. PREDICTION OF AIR GAP HARMONICS This secion explains how he sparsiy paerns for he marices B k] and F k ζ k] ogeher wih he GSTs (15) can applied o (25) in order o predic he air gap field harmonics, which can be non-zero. A. Sloed Geomeries and Sparsiy Paerns When sauraion can be negleced, he maerial parameers of sloed geomeries of he roor and he saor subproblems are roaionally symmeric wih respec o roaions by a single slo. Le e k be he sandard basis vecor corresponding o he harmonic(,) in he subproblemk. The roaional symmery leads o sparsiy paern in he marix B k] such ha ] B k e k = ] B k (η,η)(,) ek η η (26) (η,η) Z 2 = n Z B k ] (,+nq k )(,) ek (+nq k ) (,) (s p,) Dirichle (s p,) (,) Dirichle D-o-N D-o-N ( +n r, +nq r ) η (s p, +nq r) Neumann (s p, +nq s) ( n s, +nq s ) Neumann Figure 4. The influence of roor moion o he specral Dirichle-o-Neumann map for he roor (above) and he saor (below) subproblems from he viewpoin of he oher subproblem. η η η holds for every (,) 8]. The erms in he series (26) ouside Λ k are runcaed. Tha is, he marix B k] maps harmonic (,) in he Dirichle daa o he harmonics {(η,η) = (, +nq k ) : n Z} in he Neumann daa. In a similar way, each harmonic (,γ) in he slo winding currens is mapped by he marix F k ζ k] o he harmonics {(,γ +nq k ) : n Z} in he Neumann daa. Tha is, F k ζ k] e k γ = F k ζ k] (,γ+nq k )(,γ) ek (γ+nq k ) (27) holds. n Z B. Coordinae Sysems and Movemen The sparsiy paerns (26) (27) imply ha specral D- o-n maps consruced for he roor and saor subproblems in heir own coordinae sysems do no couple differen ime-frequencies in he Dirichle and Neumann daa. Insead, differen ime-frequencies become coupled due o he relaive movemen of he roor wih respec o he saor. In order o undersand his coupling, le us invesigae he combined effec of he slip ransformaions and he specral D- o-n mapping for he roor from he viewpoin of he saor. Consider a single harmonic (,) in he saor Dirichle daa. In he roor coordinae sysem, he corresponding harmonic is (s p,), where he change in he ime-frequency depends on he space harmonic index according o (15). In Fig. 4, his change in ime-frequency is depiced as a verical shif from he saor line o he roor line. From (26), i follows ha he specral D-o-N mapping for he roor maps he harmonic (s p,) in he roor Dirichle daa o he harmonics {(s p, +n rq r ) : n r Z} in he roor Neumann daa. The corresponding harmonics in he saor sysem are obained wih (15) s p (s p )( (+n rq r)) = s p +( +n rq r ) ω s 1 s ω f p = +n r r (28) r s Q r +( r,q r Q s) +( r,q r) +( r,q r +Q s) +( r,q r +2Q s) (,Q s) (,γ) +(,Q s) +(,2Q s) ( r,q r) ( r,q r Q s) ( r,q r Q s) Q s +( s, Q r Q s) +( s, Q s) +( s,q r Q s) +( s,2q r Q s) (,Q r) (,γ) +(,Q r) +(,2Q r) +( s,q s Q r) Q s +( s,q s) Q r +( s,q r +Q s) +( s,2q r +Q s) Figure 5. Equivalence classes,γ] s and,γ] r of he harmonic (,γ) in he slo winding currens in he saor (above) and roor (below) coordinae sysems, respecively. where we have defined he laice cell heigh k := Q k ω s ω f 1 s p. (29) Thus, he combined effec of he specral D-o-N map for he roor and he relaive roaion can couple differen imefrequencies in he saor subproblem. In a symmeric way, from he viewpoin of he roor, differen ime-frequencies can be coupled due o he combined effec of relaive roaion and he specral D-o-N mapping for he saor (see Fig. 4). C. Dirichle and Neumann Laices Expansion of he row (η,η) of (25) in erms of he saor Dirichle daa wih(28) and applicaion of he sparsiy paerns
IEEE TRANSACTIONS ON MAGNETICS, DATE 5 (26) (27), yields ] B s (η,η)(η,η+nq Ds s) (η,η+nq s) (3) n Z Λ k n Z B r ] (s p η η,η)(sp η η,η+nqr)ds (η +n r,η+nq r) = n Z F s ζ s] (η,η)(η,η+nq s) Os (η,η+nq s) n Z F r ζ r] (s p η η,η)(sp η η,η+nqr)or (s p η η,η+nqr). The sysem of equaions (3) is linear wih respec o he source curren vecors O s and O r. Thus, he Dirichle coefficien vecor D s solved from (3) wih O s and O r is he sum of he Dirichle coefficien vecors solved from (3) wih individual rows of he source curren vecors. The se of non-zero Dirichle coefficiens is herefore a subse of he union of non-zero Dirichle coefficiens obained from (3) wih individual winding curren harmonics. If O s = e s γ and Or = hold, i follows from (27) ha on he righ-hand side of (3), only he rows (η,η) = (,γ +nq s ) : n s Z can be non-zero. Each row (η,η) on he lef-hand side of (3) is dependen on he saor Dirichle coefficiens (η,η + n s Q s ) : n s Z and (η + n r r,η + n r Q r ) : n r Z. Thus, he harmonic (,γ) in he saor slo winding currens and he harmonic (s p γ,γ) in he roor winding currens is coupled only o he harmonics,γ] s := {( +n r r, γ +n s Q s +n r Q r ) : n s,n r Z} (31) in he saor Dirichle daa, where he laice cell heigh r = ω Q s 1 s r ω f p (see Fig. 5). In a similar way, harmonic (,γ) in he roor slo winding currens and he harmonic (s p γ,γ) in he saor winding currens are coupled o he harmonics,γ] r := {( n s s, γ +n r Q r +n s Q s ) : n s,n r Z} (32) in he roor Dirichle daa, where he laice cell heigh s = ω Q s 1 s s ω f p (see Fig. 5). The same harmonics can be non-zero in he Neumann daa. Harmonics in he Dirichle and Neumann daa can be pariioned ino equivalence classes (31) (32). Le us denoe he se of non-zero winding curren harmonics in he subproblem k wih Λ k I. Harmonics in each equivalence class,γ] k form a laice (see Fig. 5) and he ses of non-zero harmonics Λ s and Λ r saisfy Λ s,γ] s s p ( γ),γ] s (33) Λ r respecively. (,γ) Λ s I (,γ) Λ r I,γ] r (,γ) Λ r I (,γ) Λ s I s p γ,γ] r (34) Figure 6. Pars of equivalence classes for wo coil curren ime harmonics (black and whie nodes). Time harmonics in orque arise for nodes wih equal space harmonic indices (a few illusraed by dashed lines). Heigh of he line corresponds o he ime harmonic index γ of he orque (righ). ω f should be chosen o make boh h = ω s /ω f and h(1 s)/p inegers in(15) heoreically possible for raional values of slip only. However, any slip can be approximaed o arbirary precision by a raional number. When he roor slip is an ineger and he number of slos in boh subproblems is divisible by he number of pole pairs, k Z holds for boh subproblems k and he synchronous frequency is suiable as he fundamenal frequency. E. Immediae Applicaions The air gap harmonics uniquely correspond o roor and saor fields. Therefore, loss esimaes can be compued from he Dirichle coefficiens and he soluions of he field problems ha are needed o consruc he D-o-N map 4]. The laice (33) (34) of non-zero air gap harmonics relaes simply o he orque ime harmonics. Le r δ be he radius of he circle Γ sr and L he lengh of he acive area of he roor. Subsiuion of (3) (4) o he orque formula obained wih he Maxwell Sress Tensor in 9] is T = Lr 2π δ D k µ N k η η e j( +η )ω f (+η)θ k dθ k (, )(µ,µ ) (35) The inegral is nonzero only for η =, and applicaion of he complex conjugae propery N η k η = Nk η η yields T = 2π Lr δ µ ( ) η Λ k D k N k η e j( +η )ω f. (36) The ime harmonic coefficien of index γ in he orque is T,e jγ ω f = 2π Lr δ D k µ N ( k γ ). (37) (,) Λ k Thus, orque harmonics arise from Dirichle and Neumann coefficiens wih equal space harmonic indices. The orque harmonic index γ = η is he difference of he ime harmonic indices of he coefficiens (see Fig. 6). If, for example, a orque frequency componen causes an objecionable noise, one can idenify he air gap harmonics ha cause i. D. Selecion of he Fundamenal Frequency In order o Fourier series expansions (3) (4) of he Dirichle and Neumann daa o exis, he fundamenal frequency ω f has o be seleced so ha he ime-frequencies of all non-zero harmonics are ineger-muliples of ω f. IV. EXAMPLES This secion applies he framework discussed in secions II- III o wo realisic examples. Fourier series expansion (21) of he winding currens is used o predic a se of laices. The unions (33) (34) of hese laices are compared o a Fourier
IEEE TRANSACTIONS ON MAGNETICS, DATE 6 log 1 D s, s= 86 84 43 Hz 42 Hz 69 67 345 Hz 335 Hz 52 5 26 Hz 25 Hz 35 33 175 Hz 165 Hz 18 16 9 Hz 8 Hz 1 5 Hz 5 Hz 44 8 72 36 36 72 18 144 log 1 D s, s=.1 775 3875 Hz 755 3775 Hz 622 311 Hz 62 31 Hz 469 2345 Hz 449 2245 Hz 316 158 Hz 296 148 Hz 163 815 Hz 143 715 Hz 1 5 Hz 5 Hz 44 8 72 36 36 72 18 144 log 1 D s, s = 1 log 1 D r, s = 9 45 Hz 72 36 Hz 54 27 Hz 36 18 Hz 18 9 Hz Hz 362 68 34 34 68 12 136 log 1 D r, s=.1 811 89 455 Hz 445 Hz 649 647 3245 Hz 3235 Hz 487 485 2435 Hz 2425 Hz 325 323 1625 Hz 1615 Hz 163 815 Hz 161 85 Hz 1 5 Hz 5 Hz 362 68 34 34 68 12 136 log 1 D r, s = 1 3 5 3 5 1 5 Hz 5 Hz 72 36 36 72 1 5 Hz 5 Hz 72 36 36 72 Figure 7. The Dirichle coefficiens prediced and compued from a ime-domain FE soluion. Solid lines are drawn for he he equivalence classes ±h,p] s and ±s p hp,p]r of he working harmonic ±(h,p) in he saor and roor coordinae sysems, respecively. The remaining equivalence classes in Λs I and Λr I are drawn wih dashed lines. Circles are drawn for he Dirichle coefficiens compued from he FE soluion. Circles darkness correspond o ampliudes of he harmonics wihin he displayed range. series coefficiens exraced from ime-domain FE soluions wih (7) (8). In all es cases, 4-pole inegral-slo saor windings are driven wih balanced 3-phase currens. Thus, he slo winding currens conain harmonics 8] Λ s I = {±(h,2),±(h,14),±(h,)}. (38) By subsiuion of (38) ino (33) (34), we obain he predicions Λ s ±h,2] s ±h,14] s ±h,] s (39) Λ r ±s p h2,2] r ±s p h14,14] r ±s p h,] r. (4) A. Squirrel-Cage Inducion Moor Consider a 4-pole squirrel-cage inducion moor wih Q s = 36 saor slos and Q r = 34 roor slos depiced in Fig. 1. The es case is a simplified version of he moor presened in 17]. A no-load, he slip s = and we selec ω f := ω s. Thereafer, h = 1, r = 17, s = 18 and h,] s = {(h+17n r, +34n r +36n s ) : n r,n s Z} (41) h,] r = {(h8n s, +34n r +36n s ) : n r,n s Z} (42) hold. A s =.1, we selec ω f := ω s /1. Thereafer, h = 1, r = 153, s = 162 and h,] s = {(h+153n r, +34n r +36n s ) : n r,n s Z} (43) h,] r = {(h62n s, +34n r +36n s ) : n r,n s Z} (44) hold. When he roor is locked, he slip s = 1 and we selec ω f := ω s. Thereafer, h = 1, r = s = and h,] r = h,] s = {(h, +2n) : n Z} (45) hold. In Fig. 7, he predicions (38) (45) are compared o resuls obained wih (7) (8) from a ime-domain FE simulaion using he code discussed in 9] wih he assumpion of linear maerials. B. Synchronous Relucance Moor Consider a 4-pole synchronous relucance moor wih Q s = 36 saor slos and Q r = 4 roor poles depiced in Fig. 8. Since he roor roaes a he synchronous speed, ω f = ω s, r = Q r /p = 2 and s = Q s /p = 18 hold. Thus,,] s = {( +2n r, +4n r +36n s : n r,n s Z)} (46),] r = {( 8n s, +4n r +36n s : n r,n s Z)} (47) hold. In Fig. 9, he predicions (46) (47) are compared o resuls obained wih (7) (8) from a ime-domain FE simulaion using he code discussed in 9]. V. INACCURACIES IN PRIOR METHODS This secion uses he approach of Secions II-III o gain insigh ino inaccuracies in wo formulaions used wih prior
IEEE TRANSACTIONS ON MAGNETICS, DATE 7 d-axis Figure 8. Geomery of he synchronous relucance machine wih he roor and he saor drawn separaely. log 1 D s 29 145 Hz 27 135 Hz 25 125 Hz 23 115 Hz 21 15 Hz 19 95 Hz 17 85 Hz 15 75 Hz 13 65 Hz 11 55 Hz 9 45 Hz 7 35 Hz 5 25 Hz 3 15 Hz 1 5 Hz 1 2 log 1 D r 36 18 Hz 3 15 Hz 24 12 Hz 18 9 Hz 12 6 Hz 6 3 Hz Hz 2 4 Figure 9. Dirichle coefficiens prediced wih (46) (47) and compued from a ime-domain FE soluion for he synchronous relucance machine wih (7) (8). Lines and circles are used as in Fig. 7. seady-sae compuaional mehods. Boh cases follow he approach used in Secion IIIB. The formulaion for he roor subproblem is inerpreed as a specral D-o-N map and he coupling echnique used o couple he roor and saor subproblems is inerpreed as wo disinc slip ransformaion rules for he Dirichle and Neumann daa, respecively. By comparison o he correc consrucion of he specral D-o-N maps and slip ransformaions developed in Secions II-III, we can undersand error sources in he discussed mehods. A. Single-Slip Approach The mos widely applied approach for inducion machines is presened in 1]. The problem is formulaed as a imeharmonic eddy-curren problem a he synchronous frequency. To accoun for he relaive movemen of he roor wih respec o he saor, he ime-frequency used for he roor equaions is muliplied wih he slip. Muliplicaion of he ime-frequency used in he roor subproblem wih he slip corresponds o applicaion of he same slip ransformaion s p hp = s o all space harmonics ±(h,) in he Dirichle daa of he saor subproblem. In Fig. 1, his is depiced as a verical shif from he saor line o he roor line. Thus, if he slip is small, slo harmonics wih large space harmonic indices are given he usually smaller ime-frequency 3 5 3 5 s p hp ω f = sω s of he working space harmonic = p. This can lead o severe inaccuracies in eddy-curren loss esimaes 4]. The roor subproblem can be inerpreed as a specral D-o- N map, which maps harmonics (sh,) in he roor Dirichle daa o harmonics (sh, + nq r ) in he roor Neumann daa (see Fig. 1). Coninuiy of he Neumann daa on Γ sr implies ha hese harmonics are ransformed o harmonics (h, + nq r ) in he saor Neumann daa wih he slip ransformaion appropriae only for (sh, p). (h,) (s p hp,) D-o-N Dirichle η (h, +nq r) (s p hp, +nqr) η Neumann Figure 1. Inerpreaion of he common approach of using a single slip ransformaion. B. One-Way Generalized Slip In 2] 4], separae ime-harmonic BVPs are formulaed for differen space harmonics in he roor subproblem wih he appropriae slip ransformaion of ime-frequency used in each roor BVP. Only he synchronous frequency is considered in he saor subproblem. The saor Dirichle daa is expanded as a Fourier series wih respec o he mechanical angle and differen space harmonics are coupled o appropriae roor models. Each roor BVP can be inerpreed as par of a specral D-o-N map for he roor, which maps harmonic(s p h,) in he roor Dirichle daa o he harmonics {(η,η) = (s p h, +n rq r ) : n r Z} in he roor Neumann daa (see Fig. 11). In order o saisfy he coninuiy of he Neumann daa on Γ sr, Neumann daa for he saor subproblem is assumed o be equal o he sum of he Neumann daa from individual roor models. However, his corresponds o he assumpion ha he slip ransformaion for n r = is appropriae for all n r Z (compare Fig. 11 and Fig. 4). Dirichle (h,) (s p h,) D-o-N Neumann η (h, +nq r ) Figure 11. Inerpreaion of he approach discussed in 2] 4]. (s p h, +nq r) η Thus, all space harmonics are coupled back o he saor model a he synchronous ime-frequency. This is incorrec for he new space harmonics wihn r, creaed o he Neumann daa due o sloing of he roor. In ime-domain simulaions of inducion machines, effecs of hese ime harmonics creaed by roor sloing are ofen observed as addiional ime harmonics in he saor winding currens and as oscillaions in he orque. VI. CONCLUSION Analysis of air gap field harmonics in seady-sae analysis of wo-dimensional elecromagneic BVPs for elecrical machines was presened. From he viewpoin of he air gap, he
IEEE TRANSACTIONS ON MAGNETICS, DATE 8 roor and he saor responses can be compleely represened by specral Dirichle-o-Neumann maps in co-moving coordinae frames. The effec of moion is hen isolaed o he GSTs ha describe how ime-frequencies ransform beween he roor and saor subproblems. The main limiaion of he presened framework is he assumpion of linear maerial relaions. Nonnegligible sauraion in a subproblem requires a nonlinear specral D-o-N mapping, and he convenien lineariy of equaions (24) (25) is los. Addiional sauraion harmonics hen also appear. The roaional symmery of maerial parameers in sloed geomeries leads o saic sparsiy paers in he marix represenaions of he specral D-o-N maps. The analysis exhausively explains which air gap and orque ripple harmonics arise and enables efficien and generally applicable frequencydomain compuaion mehods. APPENDIX A CONSTRUCTION OF SPECTRAL D-TO-N MAPPINGS Consider he boundary value problem: Find A k defined in Ω k ha saisfies µ A k = σ A k Q k I k qj q ω f k ej Λ k q=1 A k = D k e j( ω f θ k ) (,) Λ k (48) (49) A k Γdk = (5) wih σ he elecric conduciviy and µ he magneic permeabiliy. The soluion o (48) (5) is expanded as he sum A k (r,) = + Λ k q=1 Q k I k qa q ω f ik (r)ej (,) Λ k D k à dk (r)e j ω f where A q ik for each q = 1,2,..,Q k is he soluion o (51) µ A q ik = J k q (52) A q ik Γ sr = (53) A q ik Γ dk = (54) and à dk for each (,) Λ k is he soluion o µ à dk = j ω f σã dk (55) à dk = e jθ k (56) =. (57) Γdk à dk The radial derivaive of A k on Γ sr can be expanded as A k = Q k I k A q ik r q r e j ω f + Λ k q=1 (,) Λ k D k à dk r e j ω f. (58) Le δ denoe he Kronecker dela. Applicaion of he inner producs(5) (6) o(58) and definiion of he marices B k] C Λk Λ k and F k] C Λk Q k such ha (24) follows: B k ] (η,η)(,) F k ] (η,η)(,q) à dk,e jηθ k r A q ik := δ η r,e jηθ k := δ η REFERENCES θ θ (59) (6) 1] E. Vassen, G. Meunier, A. Foggia, Simulaion of Inducion Machines Using Complex Magneodynamic Finie Elemen Mehod Coupled Wih he Circui Equaions, IEEE Trans. Magn., vol. 27, no. 5, pp. 4246-4248, 1991. 2] H. De Gersem, K. Hameyer, Air-Gap Flux Spliing for he Time- Harmonic Finie-Elemen Simulaion of Single-Phase Inducion Machines, IEEE Trans. Magn., vol. 38, no. 2, pp. 1221-1224, 22. 3] Y. Ouazir, N. Takorabe, R. Ibiouen, O. Touhami, S. Mezani, Consideraion of Space Harmonics in Complex Finie Elemen Analysis of Inducion Moors Wih an Air-Gap Inerface Coupling, IEEE Trans. Magn., vol. 42, no. 4, pp. 1279-1282, 26. 4] V. Räisänen, S. Suuriniemi, S. Kurz, L. Keunen, Rapid Compuaion of Harmonic Eddy-Curren Losses in High-Speed Solid-Roor Inducion Machines, IEEE Trans. Energy Convers., vol. 28, no. 3, pp. 782-79, Sep. 213. 5] K. Oberrel, Die Oberfeldheorie des Käfigmoors uner Berücksichigung der durch die Ankerrückwirkung verursachen Saorobersröme und der parallelen Wicklungszweige, Archiv für Elekroechnik, vol. 49, no. 6, pp. 343-364, 1965. 6] K. Oberrel, Losses, orques and magneic noise in inducion moors wih saic converer supply, aking muliple armaure reacion and slo openings ino accoun IET Elecric Power Applicaions, vol. 1, issue 4, pp. 517-531, 27. 7] D. Givoli, Recen advances in he DN FE Mehod, Arch. Compu. Meh. Eng. vol. 6, no. 2, pp. 71-116, 1999. 8] V. Räisänen, Air Gap Fields in Elecrical Machines: Harmonics and Modeling of Movemen, PhD hesis, Tampere Universiy of Technology, 215. 9] V. Räisänen, S. Suuriniemi, S. Kurz, L. Keunen, Subdomain Reducion by Dirichle-o-Neumann Mappings in Time-Domain Elecrical Machine Modeling, IEEE Trans. Energy Convers., vol. 3, no. 1, pp. 34-348, 215. 1] B. Heller, V. Hamaa, Harmonic Field Effecs in Inducion Machines, 1977: Elsevier. 11] W. Gibbs, Conformal Transformaions in Elecrical Engineering, Chapman & Hall, London, 1958. 12] E. Freeman, Travelling waves in inducion machines: inpu impedance and equivalen circuis, Proceedings of he IEEE, vol. 115, no.12, 1968. 13] J. Gieras, Analysis of mulilayer roor inducion moor wih higher space harmonics aken ino accoun, IEE Proceedings B: Elecric Power Applicaions, vol. 138, no. 2, pp. 59-67, 1991. 14] A. Razek, J.L. Coulomb, M. Feliachi, J. Sabonnadiere, Concepion of an air-gap elemen for he dynamic analysis of he elecromagneic field in elecric machines, IEEE Trans. Magn., vol. 18, no. 2, pp. 655-659, 1982. 15] H. De Gersem, T. Weiland, A compuaionally efficien air-gap elemen for 2D FE machine models, IEEE Trans. Magn., vol. 41, no. 5, pp. 1844-1847, 25. 16] R. Merens, U. Pahner, K. Hameyer, R. Belmans, Force Calculaion Based on a Local Soluion of Laplace s Equaion. 17] A. Arkkio, Analysis of inducion moors based on he numerical soluion of he magneic field and circui equaions,, Ph.D. disseraion, Helsinki Univ. Technol., Helsinki, Finland, 1987.