Finite-Element Electrical Machine Simulation

Transcription

1 Lecture Series Finite-Element Electrical Machine Simulation in the framework of the DFG Research Group 575 High Frequency Parasitic Effects in Inverter-fed Electrical Drives summer semester 2006 Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik Schloßgartenstr. 8, Darmstadt, Germany - URL:

2 2 Contents Considering Time Dependence Solving Nonlinear Problems

3 full Maxwell equations Problem Classes wave equation (non)linear materials frequency dependence (un)structured grids time- and frequencydomain explicit time integration magnetoquasistatics electroquasistatics nonlinear materials magnitude dependence unstructured grids time- and frequencydomain implicit time integration 3

4 Overview higher harmonics in electro/magnetoquasistatic models higher harmonics due to excitations, switches nonlinear/hysteretic materials moving parts simulated by time integration frequency domain (multi-harmonic) (time-harmonic) combinations (e.g. envelope method) 4

5 Overview higher harmonics in electro/magnetoquasistatic models higher harmonics due to excitations, switches nonlinear/hysteretic materials moving parts simulated by time integration frequency domain (multi-harmonic) (time-harmonic) combinations (e.g. envelope method) 5

6 Phasors () { j t = Re y e ω } y t ( ) = cos( ω ) sin ( ω ) y t y t y t re im ( ) = ( ω ϕ ) y t y t peak cos ( ) = 2cos( ω ϕ ) y t y t eff () Re{ 2 j ω = t } y t y e eff real time instant y ( t ) y re y im t 6 imaginary time instant

7 Energy and Power 7 ( ) 2cos( ω ϕ ) = eff u u t u t ( ) 2cos( ω ϕ ) = eff i i t i t ( ) = cos( ϕ ϕ ) + cos( 2ω ϕ ϕ ) p t u i u i t eff eff () { j t = Re } eff 2 u t u e ω () { j t = Re } eff 2 i t i e ω av * eff eff P = u i i u eff eff i u

8 Phasor Fields (1) 8 r A x y z t A x y z t A x y z t A x y z t (,,, ) = (,,, ), (,,, ), (,,, ) r A x y z t ( ) x y z ( ) { j t,,, = Re A( x, y, z) e ω } r r r A t A t A t ( ) = cos( ω ) sin ( ω ) re r r A t A t r im ( ) ( ω ϕ ) peak cos r A t A cos t, () = ( peak, x ( ω ϕx) A peak, y A peak, z ( ωt ϕ ) cos, cos y ( ωt ϕ )) z

9 9 Phasor Fields (2) r r = ( ) { j t,,, Re (,, ) } A x y z t A x y z e ω r r r d A dt ( A ) J s ν +σ = r r r ν A + j ωσ A = J ( ) s

10 10 Az ( R, θ, t) = α cosθ B y Static Field real imaginary A ( x, y, t) = z y (,, ) B x y t α x R A = z α = x R

11 11 Alternating Field (1) B y real imaginary A ( R, θ ) = α cosθ z A ( R, θ, t) = α cosθ cos( ωt) z A ( x, y) = z α (, ) = By x y y x R α R R α (,, ) = cos( ω ) B x y t t

12 12 Alternating Field (2) B y real imaginary magnitude phase

13 13 Rotating Field (1) B y real imaginary Az ( R, θ) = α e j θ Az ( R, θ, t) = α cos( ωt θ) x y A x y j j (, ) = ( cos sin ) = z α θ θ α α R R r α r α r B = j ex + ey R R B x

14 14 Rotating Field (2) B y real imaginary magnitude phase B x

15 15 z Coil (1) θ r

16 16 Coil (2) massive conductor stranded conductor real imaginary real imaginary I U 34.8 A 60 = = 1mV o I U = 1mA = 29.2 nv 62 o

17 17 Coil (3) massive conductor stranded conductor real imaginary real imaginary

18 Levitation Device (1) 18 H. Karl, J. Fetzer, S. Kurz, G. Lehner, W.M. Rucker, Description of TEAM workshop problem 28: an electrodynamic levitation device,

19 19 Levitation Device (2) real imaginary

20 Overview higher harmonics in electro/magnetoquasistatic models higher harmonics due to excitations, switches nonlinear/hysteretic materials moving parts simulated by time integration frequency domain (multi-harmonic) (time-harmonic) combinations (e.g. envelope method) 20

21 θ-methods 21 e.g. magnetoquasistatics da ( ν A ) +σ = J s dt K Ku du + M = dt f ( 1 ) ( n+ 1) ( n) u θ u + θ u ( n+ 1) ( n) du u u dt t n+ f θ f + 1 θ f ( ) ( 1) ( n) ( n) u ( n) t M u = f + f + Mu θ t θ θ t θ t ( n+ 1) ( n+ 1) θ ( n) ( n) θ ( n) Ku t u ( n + 1) ( n + 1)

22 Runge-Kutta Methods 22 Ku s U i U i U u i du + M = dt number of stages stage values stage derivatives ( n) f s = u + t a ( n+ 1) ( n) j = 1 ij s = u + t U j = 1 embedded solution j b U j j Butcher table c s % ( n+ 1) ( n) % u = u + t b j U j j = 1 Nicolet, Delincé, Benderskaya, Clemens, Wilke, research at TEMF A b b T %T e.g. Singly Diagonally Runge-Kutta SDIRK 3(2) order 3 order 2

23 Adaptive Time Step error vector n+ + = % n e u u ( 1) ( 1) accelerating factor (e.g. 1.2) 23 yes reject time step e err >µε tol no time step predictor user specified tolerance ε t + = ρ tol n 1 p e err 1/ safety factor (e.g. 0.9) p order

24 24 E A Step-Up Converter diode an aus switch an aus Zeit (s) voltage (V) Benderskaya, Clemens, De Gersem time (s) smaller time steps

25 Bushing transformer bushing 25 Feigh, Benderskaya, Clemens, De Gersem Air part tank wall oil part simplified test configuration Pictures: SIEMENS

26 Example High Voltage Bushings: Transient Switching- and Polarity Reversal Processes Φ1( t) Condenser bushings 0 Φ() t Zeit Φ 0 Φ() t Zeit 1min. >90min. 1min. >90min. r εκ, ( E) Insulator oil Pressboard Epoxy insulator material Φ2( t) ε 26

27 27 Φ 0 Φ () t 1 min. 90 min. t Polarity Reversal

28 Time Step Selection Transient simulation by Singly Diagonally Implicit Runge-Kutta time integration (SDIRK 3(2)) 28 voltage / Volt stationary start configuration Φ( t ): = Φ time / minutes duration of the polarity reversal process = 1 min. default time step accuracy ε tol =

29 Overview higher harmonics in electro/magnetoquasistatic models higher harmonics due to excitations, switches nonlinear/hysteretic materials moving parts simulated by time integration frequency domain (multi-harmonic) (time-harmonic) combinations (e.g. envelope method) 29

30 30 H Nonlinearities Ht () µ () t = Bt () µ B µ 1 H t t t 1 B 1 B B 1 µ ( ) ( ) ( ) ω µ ω = ω H B H 1 H

31 Harmonic Balance 31 A ( ν A) +σ = t ( ν A) +σ Τ A= J s ( K + L Τ) x = f discretisation J s Fourier transform real compact equivalent notation (Yamada, 2. complex equivalent Bessho, Lu) ν 0 +ν2cg+ jωσ + 3 jωσ ν 2 re ν ν ν0 2 ν ν + ωσ ν j re re ν 2 ν 0ν + ν 2 2 im im ν 2 ν 2 ωσ im 3ωσ ν 2 0 ν constant and 1 harmonic for A σ time-harmonic simulation ( ) formulation (Yamada, Bessho, Lu) dedicated Krylov-subspace solver (De Gersem, Hameyer) FEM-BEM coupling, motion (Gyselinck, Vandevelde, Dular) ν A + jωσ A = J s ν2 a1 b1 im = + 3j νωσ a 3 b 2 3ωσ Re 3 a im im 0 jωσ ν 2 ν + ωσ ν Re a 2 2 re re ν ν ν ν Im a jωσ re ν ν 0 Im a 2 a 3 b 3 a Re b 3 1 = b 1 1 a 1 Re b b1 = 1 a Im b 3 b 1 3 Im b

32 32 Saturated Inductor ferromagnetic core winding air gap 1st harmonic B (T) 3rd harmonic 5th harmonic saturation at the corners 7th harmonic higher harmonics (transient = 130 min)

33 Overview higher harmonics in electro/magnetoquasistatic models higher harmonics due to excitations, switches nonlinear/hysteretic materials moving parts simulated by time integration frequency domain (multi-harmonic) (time-harmonic) combinations (e.g. envelope method) 33

34 34 Envelope Method (1) () { jω t = } y t y t e Re ( ) 2 eff () y t () yt t

35 Envelope Method (2) 35 r r A x y z t A x y z t e ( ) { jωt,,, = Re } eff (,,, ) 2 r r A r A Re eff jω t = jω Aeff + 2 e t t r r da ν +σ = dt ( A) Js r r r d A r ν + ωσ +σ = dt ( A) j A Js r

36 Summary higher harmonics in electro/magnetoquasistatic models higher harmonics due to transient excitations, switching elements nonlinear and hysteretic materials moving parts simulated by low- and high-order time integration frequency domain (harmonic balance) methods (multi-/time-harmonic) combinations 36

37 37 Solving Nonlinear Problems Contents

38 Contents 38 Magnetostatics Successive substitution Newton method Generalisation to unconstrained minimisation quadratic model descent directions line search methods trust region methods trust region subproblem trust region step control Examples switched reluctance machine magnetic brake superconductive magnet

39 39 Literature (1)

40 40 Literature (2) Hans Vande Sande

41 ( ν ( A) A) Js = r Successive Substitution (1) r r ( ) 0, G i u = i 41 r A 0 = 0 for n = 0,1,... until convergence compute B r n = A r n ν r evaluate ( B n ) r r r solve ( ν ( ( A ) ) n r An+ 1r = J ( ) * ) s solve ν An An + = r * An+ 1 = r αan ( 1 r α) An end r J 1 s

42 42 B B 1 B 3 B sol B 4 B 2 wp 0 B 0 wp 4 wp 2 Successive Substitution (2) ν 0 ν ν 1 3 ν 2 wp 1 wp 3 solve ν ( B) B = Hgiven iterate ν ( Bn) Bn+ 1 = Hgiven H given H

43 43 Newton (1) i ( ) 0, G u = i Taylor series ( ( n) ) G ( ( ) ) ( + 1) + i n G + ( 2 ) i u u n δu j O δu = 0, u j j + + = i G i ( n + 1) δ u = u j j ( n ) u= u [ G ] ( n ) i u= u Newton method =

44 Newton (2) solve ( B) B Hgiven ν = 44 solve G( B) = ν ( B) B Hgiven = 0 dg iterate n+ 1 db B= B B= B n δ B = G dν = ( Bn) Bn +ν ( Bn) db = dν +ν d B ( ) ( ) 2 B 2 ( ) 2 n Bn Bn dh B B db ( ) ( ) = n =ν n n chord reluctivity differential reluctivity

45 45 B B 1 B 2 B sol wp 0 B 0 ν,2 ν 0 = ν,0 Newton (3) ν ν 1,1 wp 1 wp 2 solve ν ( B) B = Hgiven iterate dh ( B ) δ B + H ν ( B ) B db = n n 1 given n n H given H

46 Contents 46 Magnetostatics Successive substitution Newton method Generalisation to unconstrained minimisation quadratic model descent directions line search methods trust region methods trust region subproblem trust region step control Examples switched reluctance machine magnetic brake superconductive magnet

47 r r r g A A J Variational Formulation (1) ( ) = ( ν ) s = 0 minimise functional r r r r F( A) = w ( ) m A dω A Js dω Ω stored magnetic energy Ω magnetic enthalpy work done by current sources 47

48 48 Variational Formulation (2) r r r r r r F( A) w ( ) discretisation: A = = m A dω A Js dω uw j i ( ) Ω r r r df ( A) dw ( ) m A r r G A J w Ω = = dω s i dω dui du Ω i Ω r B r r (( ) ) 1 r r wm A = ν H dab A r 2 0 dw dw (( ) r r ) m A r r db r da dν rr r r = νh A = Hw i + 2= ν AA wwi B du i dui du ddub ( ) i i i r r ( ) r r r Gi A = ν A wi dω Js wi dω Ω Ω 2 j j

49 Unconstrained Minimisation find that minimises r r r r = m Ω dω A r F( A) w ( A) d A Js Ω Ω given A r 0 for n = 0,1,... until convergence r r compute step s = α d check model r put A + 1 = r A + r s end n n n n n descent direction e.g. succ. subst. r r r d = A n * A n+ 1 relaxation factor ] 0,1] n 49

50 50 Quadratic Model (1) ( ) ( ) F u qm n + s F n s qm T 1 T 2 ( ) ( ) ( ) ( ) Fn s = F un + s F un + s F un s 2 = + + contours of F(u) u n quadratic model qm ( ) n F s

51 Quadratic Model (2) qm T 1 T 2 Fn ( s) = F( un) + s F( un) + s F( un) s 2 51 descent direction dn 1. steepest descent direction d SD n ( n ) ( u ) F u = F 2. Newton direction N 2 1 n = n n 3. quasi-newton direction n ( F( )) F( ) d u u ( ) QN 1 n = n F n d B u d T n ( u ) F < n 0 contours of F(u) quadratic model F qm ( s) n

52 Line Search Method (1) given u k 2 ( u ) F( u ) evaluate F, compute descent direction ( ) { } solve min F uk + αd α ( 0,1 ) set α α ; s α d k k k k k k k d k 52

53 Line Search Method (2) ( ) { } solve min F uk + αdk α ( 0,1) 53 Wolfe conditions : 0< c < c < ( + αd ) F u k k OK ( u + αd ) ( u ) + αd T ( u ) F F c F k k k 1 k k T ( α ) ( ) T k k k 2 k k d F u + d c d F u α OK

54 Newton direction k u k trust region Trust Region Method (1) given u, < 2 ( uk) F( uk) F ( s ) evaluate F, solve Quadratic model TR step Contours of F(u) k s min k k k { } qm k 54

55 Contents 55 Magnetostatics Successive substitution Newton method Generalisation to unconstrained minimisation quadratic model descent directions line search methods trust region methods trust region subproblem trust region step control Examples switched reluctance machine magnetic brake superconductive magnet

56 Reluctance Machine (1) 8/6 Switched reluctance motor Windings Stator iron 56 B (T) 5157 nodes elements M H (A/m) Rotor iron max ( ) B = 2.6 T

57 Reluctance Machine (2) 57 Numerical results k, s k Flops (x 10 8 ) F ( A ) k TR NR α = 1 NR Flops (x 10 8 ) ρ k < 0.25 reject ρ > 0.75 and s = k k k ρ > 0.75 and s < k k k 0.25 < ρ k < 0.75 sk

58 Contents 58 Magnetostatics Successive substitution Newton method Generalisation to unconstrained minimisation quadratic model descent directions line search methods trust region methods trust region subproblem trust region step control Examples switched reluctance machine magnetic brake superconductive magnet

59 Magnetic Brake (1) no mechanical contact noiseless no maintenance ω m model no force at standstill force decreases at high velocities heating of the rotor 59

60 60 Magnetic Brake (2) 100 rad/s 10 rad/s better flux linkage 250 Nonlinear model more leakage flux linear ω m Torque (Nm) Linear model Angular velocity (rad/s) 1 rad/s non-linear ω m ω m ω m ω m ω m

61 Contents 61 Magnetostatics Successive substitution Newton method Generalisation to unconstrained minimisation quadratic model descent directions line search methods trust region methods trust region subproblem trust region step control Examples switched reluctance machine magnetic brake superconductive magnet

62 62 Frankfurt Darmstadt Heavy-Ion Accelerator Gesellschaft für Schwerionenforschung

63 63 Frankfurt Darmstadt FAIR Project Facility for Antiproton and Ion Research Gesellschaft für Schwerionenforschung

64 64 SIS100 Synchrotron 4 T/s B / T Pictures: GSI dipole magnets time / s ramped excitation

65 Superconductivity conventional magnet superconductive magnet 65 B J c 1 m T + higher current density + smaller and lighter + no DC power loss + smaller operation cost - AC power losses - cooling (liquid He) - complex design - higher investment cost

66 copper wire 66 Pictures: GSI Nuclotron Magnet superconductive filament

67 67 Time-Varying Magnetic Fields flux perpendicular to the lamination eddy currents + power losses flux through superconductive cable

68 68 Influence of the Lamination 1. Lamination z y 2 δ µ xy ωµ xy κxy = µ 0 higher skin depth µ Fe Permeability 2. Ferromagnetic saturation µ δ 4 T/s B / T Zeit / s

69 69 Homogenisation ( ) + t A ν A κ = J t t s κ t κ = κ 0 y µ 0 µ Fe ν xy t ν = ν xy ν z

70 70 standard Computed Configurations + SMP blocks + horizontal cuts + deep hor.cuts alternative coil original coil

71 m δ eddy 2 = = ωµ κ z xy m Skin depth has to be resolved! Skin depth has to be resolved! graded mesh in z-direction Meshing

72 Power Losses (1) original coil alternative coil time (s) time (s) power loss (W) magnetic flux density (T)

73 73 Power Losses (2) standard yoke SMP block horizontal cuts deep horizontal cuts time (s) time (s) power loss (W) magnetic flux density (T)

74 Loss Distribution original yoke SMP block original yoke length along magnet axis (m) loss distribution (J/m)

75 Power Losses (3) magnet axis (m) power loss (W/m)

76 Lecture Series Finite-Element Electrical Machine Simulation NEXT LECTURE : THURSDAY, June 8th 2006 summer semester 2006 Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik Schloßgartenstr. 8, Darmstadt, Germany - URL: