Finite-Element Electrical Machine Simulation

Transcription

1 Lectue Seies D.-Ing. Hebet De Gesem Finite-Element Electical Machine Simulation in the famewok of the DFG Reseach Goup 575 High Fequency Paasitic Effects in Invete-fed Electical Dives D.-Ing. Hebet De Gesem summe semeste 2006 Technische Univesität Damstadt, Fachbeeich Elektotechnik und Infomationstechnik Schloßgatenst. 8, Damstadt, Gemany - URL:

2 Geneal Infomation D.-Ing. Hebet De Gesem 2 Contact: Hebet De Gesem (pefeed): [email protected] oom 133 in this building (S2/17) Schedule almost evey Thusday: 15:00-16:40 (also at Thusday: 17:00-18:00 Semina Computation Engineeing) exact schedule + contents of the lectues website: Examination: on demand

3 Geneal Infomation D.-Ing. Hebet De Gesem Schedule next Thusday 27.4: no lectue!! next lectue: Thusday 4.5 see website: 3

4 Electomagnetic field theoy - vecto algeba + gad/div/cul - Maxwell laws + potentials - analytical solution techniques fo PDEs Foeknowledge D.-Ing. Hebet De Gesem Electical machine theoy - DC, induction and synchonous machines - otating field theoy, equivalent cicuits, DQ-axes - feomagnetic mateials Numeical simulation - linea algeba, systems of equations - analyse, appoximation theoy 4

5 Stuctue D.-Ing. Hebet De Gesem 5 Simulation techniques oveview FE/FD/FIT discetisation static simulation non-linea mateials time-hamonic and tansient simulation modelling of motion pemanent magnet mateial field-cicuit coupling hysteesis models coil models optimisation lectue seies Examples DC machine tansfome induction machine linea machine synchonous machine single-phase moto magnetic beaing eluctance machine magnetic bake

6 Methodology D.-Ing. Hebet De Gesem 6 at evey simulation step machine-theoetical consideations (e.g.) elevant unelevant phenomena linea nonlinea behaviou field-theoetical consideations (e.g.) fomulations (magnetoquasistatic, full Maxwell equations,...) spatial effects ( cicuit and/o field simulation) skin depth ( gid esolution) altenating and/o otating fields ( scala o vectoial hysteesis model) numeical consideations (e.g.) compute configuation, algebaic solution methods discetisation eo (space/time) loss of accuacy fo deived quantities (toque,...)

7 Related Couses (1) D.-Ing. Hebet De Gesem electical machines SS Elektische Maschinen, Antiebe und Bahnen (Binde) SS Elektische Maschinen und Antiebe I und II (Binde) WS Electical Machines and Dives I (Binde) SS/WS Design of Electical Machines and Actuatos with Numeical Field Simulation (Binde, Funieu) 7

8 Related Couses (2) D.-Ing. Hebet De Gesem electomagnetic field theoy & field simulation WS Technische Elektodynamik (Weiland) SS: Vefahen und Anwendungen de Feldsimulation (Weiland, Ackemann) WS Electomagnetic Field Simulation (De Gesem, Gjonaj) WS Finite Elements in Electomagnetism (Munteanu) 8

9 Liteatue (1) D.-Ing. Hebet De Gesem 9 intenational jounal IEEE Tansactions on Magnetics IEEE Tansactions on Enegy Convesion Achiv fü Elektotechnik intenational confeences ICEM : Int. Conf. on Electical Machines (2006: Cete) Compumag : Int. Conf. on the Computation of EM Fields (2007: Aachen) CEFC : IEEE Conf. on EM Field Computation (2006: Miami) EMF : Int. Wokshop on Electic and Magnetic Fields (2006: Fance) SPEEDAM : Symposion on Powe Electonics and Electical Dives (2006: Capi) IEMDC : IEEE Int. Electic Machines and Dives Conf.

10 Liteatue (2) D.-Ing. Hebet De Gesem books J.P.A. Bastos, N. Sadowski, Electomagnetic Modeling by Finite Element Methods, K. Hameye, R. Belmans, Numeical Modelling and Design of Electical Machines and Devices, M. Kaltenbache, Numeical Simulation of Mechatonic Sensos and Actuatos, E. Kallenbach et al., Elektomagnete,

11 Electomagnetic field theoy - vecto algeba + gad/div/cul - Maxwell laws + potentials - analytical solution techniques fo PDEs Foeknowledge D.-Ing. Hebet De Gesem Electical machine theoy - DC, induction and synchonous machines - otating field theoy, equivalent cicuits, DQ-axes - feomagnetic mateials Numeical simulation - linea algeba, systems of equations - analyse, appoximation theoy 11

12 Softwae D.-Ing. Hebet De Gesem 12 semi-analytical - SPEED field simulation (commecial tools) - Ansys TUD-EW - Maxwell (Ansoft) - MagNet (Infolytica) - Flux2d/Flux3d (Cedat) - Opea (VectoFields) - EMStudio (CST) TUD-TEMF field simulation (tools at univesity) - FEMAG (ETH Züich) TUD-EW - MEGA (Univ. Bath) TUD-EW - Olympos (K.U. Leuven) TUD-TEMF - Dido (TUD-TEMF) TUD-TEMF

13 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 13

14 D.-Ing. Hebet De Gesem 14 Magnetic Equivalent Cicuit (1) Γ magnetic flux [Wb=Vs] φ = B da Ω electic cuent [A] Ω magnetic voltage [A] m = V H ds Γ electic voltage [V]

15 Magnetic Equivalent Cicuit (2) S l µ Vm N t V m V m D.-Ing. Hebet De Gesem 15 φ eluctance = magnetic esistance φ coil = magnetic voltage Vm = Rmφ Vm = t R m = l µs N I I φ induced cuents = magnetic inductance dφ V = L dt L m m = m 1 R e

16 D.-Ing. Hebet De Gesem 16 Shaded-Pole Moto (1) stato bidge squiel cage shot-cicuited ing coil

17 D.-Ing. Hebet De Gesem 17 Shaded-Pole Moto (2)

18 D.-Ing. Hebet De Gesem 18 ( L + L ) st t d φ dt Shaded-Pole Moto (3) L st R t L t ai gap ai gap R st φ V m ( ) + Rst + Rt φ + R ag φ = V m

19 D.-Ing. Hebet De Gesem 19 Magnetic Equivalent Cicuit (3) but - feomagnetic satuation? - eddy-cuent effects? - motional pats?

20 D.-Ing. Hebet De Gesem 20 Magnetic Equivalent Cicuit (4) feomagnetic satuation B µ 1 ν B 1 H 1 H R m ν = l S B φ = BS

21 D.-Ing. Hebet De Gesem 21 Ampèe Ohm magnetic S φ = π R 2 l µ J E B H z J θ z θ θ θ z Magnetic Equivalent Cicuit (5) H = z = ρ J θ = µ H V m z H φ R z V = l m Faaday-Lenz 1 H ρ z = jωµ Hz 2 2 H + H jωµσ H = 0 z z z modified Bessel equation I I 0 0 ( ξ) ( ξ R) ( ξ ) ( ξ R) Vm 2π Rµ I1 R = l = ξ m 2 I j ξ = jωµσ = δ ( ξ ) ( ξ R) l 1+ jri0 R µπ R 2 δ I skin depth

22 D.-Ing. Hebet De Gesem 22 Magnetic Equivalent Cicuit (4) 15 x 105 eluctance + eddy cuents fequency (Hz) fequency (Hz) angle(rm) abs(rm) (A/Wb)

23 D.-Ing. Hebet De Gesem 23 slotting m,tooth = m ( ) θ δ µ 0 l z m,slot Ai-Gap Reluctance (1) = δ + µ 0 h slot l z θ

24 D.-Ing. Hebet De Gesem 24 Ai-Gap Reluctance (2) m( ) ct θ a λ cos λθ = + 1 = + 2 ct m,1 m,2 aλ = m,2 λ 0 λα sin t 2 λ m ( ) θ α t m,2 m,1 θ

25 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 25

26 D.-Ing. Hebet De Gesem 26 Rotating-Field Theoy (1) ( ) j z θ z θ θ cuent shield ai-gap (magnetic) field ( θ ) j = z j e z ( θ ) λ λ j λθ ( ) = j t j j e ω λθ λ λ

27 D.-Ing. Hebet De Gesem 27 Rotating-Field Theoy (2) ( ) H θ ( ) j z θ δ = µ µ = µ 0 z θ = µ Ampèe H ( θ + dθδ ) H( θδ ) = jz( θ) Rdθ dh R = jz ( θ ) d θ δ ai-gap (magnetic) field b θ b e ω λθ ( ) ( ) = j t λ λ

28 D.-Ing. Hebet De Gesem 28 Rotating Fields (1) { } e im jω t It () = Re Ie = I cos ωt I sin ωt

29 D.-Ing. Hebet De Gesem 29 θ Rotating Fields (2) u ( θ, t ) u( θ, t) = Re { u( θ) e j ω t } u j (, ) Re ( t t a e ω λθ ) θ = λ λ Λ ω angula fequency syn λ wave numbe ω ω = wave velocity λ

30 Angula Slip Fequency ω m D.-Ing. Hebet De Gesem 30 θ same amplitudes same wave numbes diffeent fequencies θ j (, ) Re ( t t a e ω λθ ) u θ = λ λ Λ θ=θ +ω m t j( ( ω λωm ) t λθ (, t) Re a e ) u θ = λ λ Λ ω s,λ angula slip fequency

31 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 31

32 D.-Ing. Hebet De Gesem 32 induction machine stato stato end windings oto shaft (omitted) oto ing Equivalent Cicuits (1) stato slot cooling ducts oto slot

33 D.-Ing. Hebet De Gesem 33 equivalent cicuit R 1 U_ 1 I_ 1 Equivalent Cicuits (2) X σ1 X ' σ2 ' R2 _ I RFe R Fe I_ 0 I_ µ X h1 ' I_ 2 ' (1-s) s R2

34 D.-Ing. Hebet De Gesem 34 no-load test P 0 3 U 0,line 3 Equivalent Cicuits (3) R 1 X σ1 I 0 E R Fe X h

35 D.-Ing. Hebet De Gesem 35 Equivalent Cicuits (4) shot-cicuit test R 1 X σ 1 X σ2 R' 2 ' R k X k P I U k k k,line 3 3

36 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 36

37 D.-Ing. Hebet De Gesem 37 Reluctance Machine

38 D.-Ing. Hebet De Gesem 38 Diect- & Quadatue Axis seen fom one of the phases θ diect axis quadatue axis 2 2 µ S N t t l R m L= N = L high L low

39 Model dψ () t ut () = Rit () + voltage in a coil dt D.-Ing. Hebet De Gesem 39 di() t dl( θ ) ut () = Rit () + L( θ ) + it () dt dt di() t dl( θ ) ut () = Rit () + L( θ) + ωm() t it () dt dθ inductance L(θ) dependent on oto angle dw d toque T co ( θ, i) = θ i= ct mechanical velocity electomagnetic field simulation

40 Appoach (1) D.-Ing. Hebet De Gesem (fist ty) magnetic field simulation magnetic vecto potential fomulation tansvesal symmety 2D model lamination no eddy cuents static simulation impotant feomagnetic satuation expected nonlinea simulation 40

41 D.-Ing. Hebet De Gesem 41 applied cuents nonlinea mateial 2D FE Model electic bounday conditions (Diichlet) electical bounday conditions (floating potential)

42 D.-Ing. Hebet De Gesem 42 Simulation (1) spatial esolution fo the pemeability not sufficient 1.27 T

43 Appoach (2) D.-Ing. Hebet De Gesem (second ty) magnetic field simulation magnetic vecto potential fomulation tansvesal symmety 2D model lamination no eddy cuents static simulation impotant feomagnetic satuation expected nonlinea simulation local satuation adaptive mesh efinement till e.g. the elative change of the magnetic enegy < 1% 43

44 D.-Ing. Hebet De Gesem 44 2D FE Model 4.25 T

45 3D End Effects (1) coil D.-Ing. Hebet De Gesem 45 yoke length l z end pats finging effect leakage inductance assumptions laktiv = γ lz leakage inductance L σ independent of the satuation and the oto angle magnetically active length l > l aktiv L σ z

46 Appoach (3) D.-Ing. Hebet De Gesem (thid ty) magnetic field simulation magnetic vecto potential fomulation tansvesal symmety 2D model lamination no eddy cuents static simulation impotant feomagnetic satuation expected nonlinea simulation local satuation adaptive mesh efinement till e.g. the elative change of the magnetic enegy < 1% end effects, compute L σ and γ compae 3D and 2D models linea simulation (smalle gids) 46

47 D.-Ing. Hebet De Gesem 47 3D End Effects (2) linea simulation

48 3D End Effects (3) adapted scaling D.-Ing. Hebet De Gesem 48 leakage flux

49 3D End Effects (4) linea models 1 Wmagn,3D = L3D i Wmagn,2D = L2D i 2 2 D.-Ing. Hebet De Gesem L3D,d = γ L2D,d + L σ L3D,q = γ L2D,q + L σ di() t dl2d( θ ) di() t ut () = Rit () + γl2d( θ) + γ ωm() t it () + Lσ dt dθ dt 49

50 Appoach (4) D.-Ing. Hebet De Gesem 50 (fouth ty) magnetic field simulation magnetic vecto potential fomulation tansvesal symmety 2D model lamination no eddy cuents static simulation impotant feomagnetic satuation expected nonlinea simulation local satuation adaptive mesh efinement till e.g. the elative change of the magnetic enegy < 1% end effects, compute L σ and γ compae 3D and 2D models linea simulation (smalle gids) automate the whole pocedue in ode to cay out paamete vaiation and optimisation steps

51 D.-Ing. Hebet De Gesem 51 Enegy A 3 A 5 A 7 A 9 A 11 A 13 A 15 A oto angle (degees) magnetic enegy (J)

52 D.-Ing. Hebet De Gesem 52 Coenegy A 3 A 5 A 7 A 9 A 11 A 13 A 15 A oto angle (degees) magnetic coenegy (J)

53 D.-Ing. Hebet De Gesem 53 Toque lowe accuacy 1 A 3 A 5 A 7 A 9 A 11 A 13 A 15 A oto angle (degees) toque (Nm)

54 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 54

55 D.-Ing. Hebet De Gesem 55 wave equation W elec W magn τp loss EM Field Simulation full Maxwell equations W elec W magn τp loss magnetoquasistatics W magn τp loss electoquasistatics W elec W elec τp loss Wmagn

56 Magnetoquasistatics (1) D.-Ing. Hebet De Gesem 56 neglect displacement cuents with espect to conducting cuents Ampèe-Maxwell magnetic vecto potential D H = J + t consevation of magnetic flux B = 0 electic scala potential (voltage) Faaday-Lenz B E = = t A t φ A B = 0 + A A E = φ t W magn τp loss W elec

57 Magnetoquasistatics (2) D.-Ing. Hebet De Gesem 57 Ampèe H = ( ν B) J = κ E ( ) A ν A + κ = κ ϕ t 123 J s 1 B = µ H = H ν J = κ E paabolic patial diffeential equation elliptic PDEs (e.g. electostatics, magnetostatics) hypebolic PDEs (e.g. wave equation) pemeability eluctivity conductivity souce cuent density

58 Oveview D.-Ing. Hebet De Gesem semi-analytical techniques (oveview) magnetic equivalent cicuit otating-field theoy equivalent cicuits + standad tests analytical model suppoted by field simulation e.g. eluctance machine magnetoquasistatic fomulation discetisation in space 58

59 Spatial Discetisation (1) D.-Ing. Hebet De Gesem 59 Weighted esidual appoach Ω ( ) A ν A + κ = J s t ( ) A ν A + κ wi dω = Js wi dω t w ( x, y, z) scala poduct : i ( uv, ) Ω in Ω w vectoial weighting functions vectoial test functions = u vdω Ω i ( x, y, z)

60 Spatial Discetisation (2) D.-Ing. Hebet De Gesem 60 weak fomulation Ω Ω ( ) A ν A + κ wi dω = Js wi dω t Ω v w = v w + v w ( ) ( ) w i ( x, y, z) ( ) A ν A w i + ν A w i + κ w i dω = s dω J w i t Gauss A ν A w dγ+ ν A w + κ w dω = J w dω i i i s i Ω Ω t Ω H only fist deivative equied weak fomulation Ω

61 Spatial Discetisation (3) Diichlet BC at Γ di A n = A n di B n = B n Γ di J B n = 0 ν 0 D.-Ing. Hebet De Gesem 61 homogeneous Neumann BC at H A n ( ) t = ν = 0 Γ neum = 0 Γ neum di Γ neum H ν A w dγ+ ν A w dγ natual bounday condition i Γ i ν 1 H t = 0 ν 2 w i Ω ( x, y, z) = 0 wi : wi n = 0 at Γdi essential bounday condition

62 Spatial Discetisation (4) D.-Ing. Hebet De Gesem discetization A = u j v j u j j v Ritz-Galekin method Petov-Galekin method j ( x, y, z) v ( x, y, z) n = 0 at Γ j di shape/fom functions, tial functions unknowns, degees of feedom v v j j ( x, y, z ) = w ( x, y, z) ( x, y, z) w ( x, y, z) j j 62

63 D.-Ing. Hebet De Gesem 63 Spatial Discetisation (5) nodal shape functions ( x, y, z) = ϕ 1 u 1 N ( x, y) + u 2 N 2 ( x, y) + u 3 N 3 ( x, y) edge shape functions n n N m m k p N n v( x) = Nm N n N n N m m

64 Spatial Discetisation (6) D.-Ing. Hebet De Gesem 64 discetization Ω A ν A vi + κ vi dω = s dω J vi t Ω A = u jv j u w = k ij = m ij = fi du j + kij u j mij = f dt j j j ν vj vi κvj vi Js vi dt Ω Ω Ω [ ] i i ( x, y, z) du dω+ dω = dω j K and M symmetic, semi-positive-definite

65 Spatial Discetisation (7) u du dω+ dω = ndω j j j ν vj vi κvj vi Js vi dt Ω Ω Ω D.-Ing. Hebet De Gesem 65 vj = c q jq m zq p du dω+ dω = dω j uj cipcjq νzq zp κvj vi Js vi dt j p q Ω Ω Ω ) a j FE M ν, pq, ) FE FE ) % ) da CMν Ca + Mκ = j dt FE M κ,, i j js, i s )

66 Lectue Seies D.-Ing. Hebet De Gesem Finite-Element Electical Machine Simulation NEXT LECTURE : THURSDAY May 4th D.-Ing. Hebet De Gesem summe semeste 2006 Technische Univesität Damstadt, Fachbeeich Elektotechnik und Infomationstechnik Schloßgatenst. 8, Damstadt, Gemany - URL: