Finite Element Models for Eddy Current Effects in Windings:

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Computational Electromagnetics Laboratory Finite Element Models for Eddy Current Effects in Windings: Application to Superconductive Rutherford Cable 1 2, Thomas Weiland 1 1 Technische Universität

1 Computational Electromagnetics Laboratory Finite Element Models for Eddy Current Effects in Windings: Application to Superconductive Rutherford Cable 1 2, Thomas Weiland 1 1 Technische Universität Darmstadt (Germany), Computational Electromagnetics Laboratory 2 Katholieke Universiteit Leuven (Belgium) Dep. ESAT, Div. ELECTA MACSI-NET NET-13 & 2: Electromagnetics in Telecommunication Workshop Coupled problems / model reduction 2-33 May 2003, Zürich,, Switzerland

2 1. Introduction: field-circuit coupling 2. Voltage/current-driven branches 3. Circuit description + consistency check + partial loop/cutset transformations + topological changes 4. Field-circuit coupling + examples 5. Algebraic system properties + selection of iterative solution techniques 6. Superconductive Rutherford cable + adjacency eddy currents + cross-over eddy currents 7. Conclusions Overview

3 field-circuit coupling FE/FIT model geometrical details ferromagnetic saturation (non-linear!!) (motional) eddy currents circuit external sources/loads, (e.g. power electronic equipment) parts outside the FE model (e.g. end windings/rings) representing (linear) parts for which an equivalent circuit suffices (e.g. homopolar shaft flux) z y end-rings R m Introduction x end-windings

4 B = A magnetic vector potential pulsation ( ν A) σv ( A) + jωσa + σ = σ V reluctivity A z ν x x velocity A ν z y y conductivity 2D model + σv [ k m + l ][ A ] = [ f ] ij + ij ij z, j i x A z x discretisation Quasistatic formulation voltage A = + σv large & sparse system of equations A t ( 0,0, A z ) y A z y + jωσa z finite element A + σ t A z = Az, j N j ( x, y) j k ij Ω z = σ N i z ( x, y) N N N i j Ni j = ν + d Ω x x y y V

5 z I sol V sol J sol V sol port sol voltage-driven conductor no circuit equation if voltage drop is known no fill-in in the FE matrix part if the voltage drop is an independent degree of freedom Solid (massive) conductor magnetic equation I A ( A) + σ = Vsol ν total current sol I source t σ z Ω I sol = JdΩ σ A = dω Vsol σ dω z t Ωsol Ωsol Gsol equivalent scheme Gsol Ieddy sol I eddy

6 N str z I str V str J 1w current-driven conductor no circuit equation if current is known no fill-in in the FE matrix part if the current is an independent degree of freedom Stranded (filamentary) conductor str V str port str magnetic equation N st r I ( A) = str ν total voltage V V str str = N str str z str Ω ( V ) str V dω N str z A = R str I str + dω str t V Ω res str equivalent scheme Rstr Vind ind

7 K B T x f = B C y g unknowns keep the FE matrix part unchanged sparsity preconditioners (multigrid) possible benefits thanks to structured grids (FIT) preserve symmetry Krylov subspace solvers for symmetric systems (CG, MINRES, QMR) storage preserve positive definiteness solvers (CG) preconditioners (IC) (compacted) modified nodal analysis nodal voltages (+a few currents) Coupling requirements (compacted) loop analysis loop currents (+a few voltages) no (yes) no (yes) yes yes yes yes hybrid analysis twig voltages link currents yes (no) yes (no) no [yes]

8 Circuit description (1) priorities for tree branches 1. voltage sources 2. solid conductors (field-circuit coupling) 3. capacitors (largest capacitance first) 4. resistors (largest conductance first) 5. inductors (smallest inductance first) 6. stranded conductors (field-circuit coupling) 7. current sources 1. Trace a tree through the circuit Priority list highest priority, preferably twig n1 1 Ω n3 10 V -2 A n0# 3 Ω twig link starting from the circuit node n0#, the twigs are selected in the order 1. voltage source 10 V 2. resistor 1 Ω 3. resistor 3 Ω voltage sources solid conductors (coupled) capacitors (largest capacitance first) resistors (largest conductance first) inductors (smallest inductance first) stranded conductors (coupled) current sources smallest priority, preferably link 4 Ω n4 Technische Universität Darmstadt, Computational Electromagnetics Laboratory

9 Circuit description (2) 2. Determine fundamental cutsets and fundamental loops a n1 n0# b c e n3 d n4 A fundamental cutset is formed by 1 twig and the unique set of links completing the set of branches which would upon removal result in two disconnected circuit parts. Property: priority(twig) priority(branch), branch fundamental cutset fundamental cutset fundamental loop The orientation of the fundamental cutset/loop is determined by the orientation of the corresponding twig/link A fundamental loop consists of 1 link and the unique path through the tree closing the loop. Property: priority(link) priority(branch), branch fundamental loop

10 Circuit description (3) 3. Construct the fundamental cutset and fundamental loop matrices n1 a n0# b n3 e d fundamental cutset matrix D = fundamental loop matrix n c B = remark: B ln, tw = D T tw, ln

11 Circuit description (4) 4. Partition the fundamental incidence matrices b n1 n3 e a d D = n0# n4 c twv twigs at which the voltage is known (voltage sources) two twigs at which an unknown voltage is assigned ( free twigs ) twu eliminated twigs ( eliminated twigs ) (not in this example) lnu eliminated links ( eliminated links ) (not in this example) lno links at which an unknown current is assigned ( free links ) lni links at which the current is known (current sources) D twv,lno D twv,lni D two,lni D two,lno

12 Circuit description (5) 5. Write impedance/admittance matrices and voltage/current vectors a n1 n0# b c e n3 d n4 1. admittance matrix for the free twigs Y two = 1/1 2. impedance matrix for the free links Z lno = v twv = [ 4] [ 10] 1/ 3 3. voltage vector for the voltage sources 4. current vector for the current sources i lni = [ 2]

13 6. Write system of equations a n1 n0# Y B two lno,two b c D e two,lno Z lno n3 d n4 v i two lno remark: symmetric because B D = B lno, two Circuit description (6) intuitive approach: 1. write the Kirchhoff current law for each fundamental cutset associated with a free twig 2. write the Kirchhoff voltage law for each fundamental loop associated with a free link vb vc id two,lni lno,twv v = D i lni twv T two, lno 0 = 2 10

14 Circuit description (7) 7. Solve system of equations & propagate the circuit solution 0.5 V 0.5 A 10 V b 9.5 V n1 n3 solution 10 V -2.5 A 2 A 2 V 0.5 A e a d 2.5 V n0# n4 0 V c -2.5 A 7.5 V -7.5 V twig currents: link voltages i i two twv = D = D two,lno twv,lno i i lno lno D D two,lni twv,lni i i lni lni v v lno lni = B = B v v i lno,two lni,two b c d v v 0.5 = two two B B lno,twv lni,twv v v twv twv

15 n4 n1 n3 n0# 3 V 5 Ω -1 A 7 Ω 2 V n5 selfloop n2 dangling node 1. Distinct circuit parts 2. Dangling nodes 4 Ω 3. Self-loops Particularities a branch to a dangling node always a twig associated fundamental cutset only contains the twig a self-loop is always a link the associated fundamental loop only contains the link

16 1. Fundamental loop consisting of voltage sources v1 v0 v2 Consistency check (1) Problem: a voltage source is necessarily selected as link Treatment: check the Kirchhoff voltage law in the associated fundamental loop ` e.g. v0-v1+v2 = 0?? if valid omit the voltage source link if not valid the circuit has no solution

17 2. Fundamental cutset consisting of current sources i0 i2 i1 Consistency check (2) Problem: a current source is necessarily selected as twig Treatment: check the Kirchhoff current law in the associated fundamental cutset e.g. i0 + i1 - i2 = 0?? if valid replace the current source twig by a short-circuit connection if not valid the circuit has no solution

18 1. Stranded conductor being selected as twig istr i2 i1 Partial transformation (1) Problem: a stranded conductor (current-driven branch) is necessarily selected as twig Property: priority(twig) priority(branch), branch associated fundamental cutset Treatment: apply the Kirchhoff current law in the associated fundamental cutset to express the stranded-conductor current in terms of link currents e.g. istr = i1 + i2 ~ small and independent Schur complements

19 2. Solid conductor being selected as link v1 vsol v2 Partial transformation (2) Problem: a solid conductor (voltage-driven branch) is necessarily selected as link Property: priority(link) priority(branch), branch associated fundamental loop Treatment: apply the Kirchhoff voltage law in the associated fundamental loop to express the solid-conductor voltage in terms of twig voltage e.g. vsol = v1 + v2 ~ small and independent Schur complements

20 Topological changes (1) 1. Switching elements closes (switch, diode, thyristor, ) Ut () Ut () R R = 1/ G 2 2 = 1/ G 2 2 R R = 1/ G 1 1 = 1/ G 1 1 Problem: the priority of a branch increases during (transient) simulation G1 0 v1 0 0 G = v Treatment: consider associated fundamental loop and possibly change link/twigmode with the branch with the lowest priority G1 1 v1 0 = 1 R 2 i 2 U

21 Topological changes (2) 1. Switching element opens (switch, diode, thyristor, ) Ut () Ut () R R = 1/ G 2 2 = 1/ G 2 2 R R = 1/ G 1 1 = 1/ G 1 1 Problem: the priority of a branch decreases during (transient) simulation G1 1 v1 0 = 1 R 2 i 2 U 2 2 Treatment: consider associated fundamental cutset and possibly change link/twigmode with the branch with the highest priority G1 0 v1 0 0 G = v 0

22 i t KCL applied to the fundamental cutsets i t + Dil FE equations: = 0 cutset equations: loop equations: KVL applied to the fundamental loops v l + Bvt = 0 Field-circuit coupling (3) magnetodynamic PDE magnetic vector potential reluctivity conductivity kij + jωlij qit pil Azj 0 qsj χgm χd vt = χi pkj χb χrm il χv symmetrisation factor & B = D ( ν A) + jωσa = σ V T = Gmvt jωqmj Azj vl = Rmil + jωpmj Azj twig branch relations symmetric! χ app app link branch relations

23 Coupled system matrix symmetric/indefinite FE/FIT part sparse (no fill-in) changes due to nonlinearities Circuit part dense changes due to nonlinearities & switching Coupling blocks dense does not change spectrum Algebraic solution (1) #positive-eigenvalues = #FE/FIT-dofs + #cutset-equations #negative-eigenvalues = #loop-equations loop equation cutset equation finite element discretisation

24 K B T x f = B C y g Schur complement 1 ( T ) 1 T 1 K B C B x= f B C g Schur complement 2 ( T ) 1 1 C BK B y = g BK f Algebraic solution (2) solver preconditioner MINRES SSOR QMR SSOR MINRES/QMR BJac (AMG,LU) CG(LU)? SSOR/AMG? DD LU(AMGCG)? LU? DD & Domenico Lahaye: Algebraic multigrid for field-circuit coupled systems

25 Algebraic multigrid for field-circuit coupled systems (Dr. Domenico Lahaye, CRS4, Cagliari, Italy) fine grid system: K B T x f = B C y g Algebraic solution (3) prolongation: restriction: PH h T Rh H = PH h coarse grid system: T T T P H h K x H B H P H h H P f H h = I B y H C I H I g smoother: block-jacobi with 1. Gauss-Seidel for the FE/FIT unknowns 2. no smoothing or an exact solution for the circuit unknowns

26 Algebraic solution (4) Performance of an indefinite and definite preconditioner error SSOR MINRES SSORQMR iteration steps Technische Universität Darmstadt, Computational Electromagnetics Laboratory

27 Algebraic solution (5) Preconditioning of the Schur-complement system 10 0 error AMGCG SSORCG CG iteration steps Technische Universität Darmstadt, Computational Electromagnetics Laboratory

28 Algebraic solution (6) Iteration counts and computation times of the iterative system solution for 1 time step of a transient simulation. iteration steps computation time (s) MINRES QMR without preconditioning GMRES SSOR MINRES non-symmetric SSORQMR symmetric solver SSORGMRES JAC(SSOR,LU)GMRES block JAC(AMG,LU)QMR preconditioning CG* SSORCG* Schur complement AMGCG* indefinite definite preconditioning & Domenico Lahaye: Algebraic multigrid for field-circuit coupled systems

29 Examples (1) Time-harmonic simulation of a three-phase induction machine FEM Twig Link three-phase voltage excitation end-winding resistances & inductances end-rind resistances & inductances

30 1. magnetic field + magnetic circuit 2. magnetic field + analytical model + electric circuit 3. electrokinetic field + magnetic circuit + electric circuit U ~ R C L external electric circuit line of symmetry φ app Θ app electrode Λ 0 Λ 1 Λ 2 magnetic short-circuit connection dielectricum Γ w,1 Γ g,1 Γ w,2 φ 2 Γ g,2 φ 0 φ 1 Γ g,0 Examples (2) Λ φ 0 0 φ app Γ w,1 Λ φ 1 1 Γ w,0 Θ app Γ w,2 Λ 2 φ 2

31 More general conductor systems? d y δ IBC 1 foil solid h x multi-conductor δ δ stranded foil δ h 1 y δ = 1 π f σµ d x skin depth / δ Technische Universität Darmstadt, Computational Electromagnetics Laboratory magnetic field

32 foil winding transformer y insulation foil x J V Foil windings cooling current channels distribution x y voltage distribution y x Technische Universität Darmstadt, Computational Electromagnetics Laboratory

Electromagnetics Laboratory foil conductor model De Gersem, Hameyer, CEFC2000 Dular, Geuzaine,

33 wrong skin effect large mesh (a) (b) (c) (d) h y y Foil conductor model V M p ( x ) ( x ) M q n f ( x ) = V ( ) = q 1 q M q x z h x 50 foils Technische Universität Darmstadt, Computational Electromagnetics Laboratory foil conductor model De Gersem, Hameyer, CEFC2000 Dular, Geuzaine, Compumag2001 foil shape functions foil mesh magnetic finite element mesh x foil winding geometry

foil winding V A/m 2 Foil winding transformer

10 y 10-3 0 5 10 10 time (s) J y x Technische

Electromagnetics Laboratory error 10 10 0-1 error

34 foil winding V A/m 2 Foil winding transformer voltage distribution V y time (s) J y x Technische Universität Darmstadt, Computational Electromagnetics Laboratory error error wire winding x degrees of freedom current distribution

B fast ramped magnets 1 T/s yoke: eddy currents & hysteresis

Rutherford cable: cable eddy currents/magnetisation t SIS100

accelerator facilities in the rest of this presentation are

35 B fast ramped magnets 1 T/s yoke: eddy currents & hysteresis superconductive filaments: persistent & coupling currents Rutherford cable: cable eddy currents/magnetisation t SIS100 Superconductive magnets (1) all photographs and figures of accelerator facilities in the rest of this presentation are copyright of the Gesellschaft für Schwerionenforschung, Darmstadt SIS300

36 Superconductive magnets (2) Technische Universität Darmstadt, Computational Electromagnetics Laboratory

37 Rutherford Cable (1) strand Rutherford cable twisted strands copper wire superconductive filament adjacency loop cross-over loop Technische Universität Darmstadt, Computational Electromagnetics Laboratory

38 perpendicular flux B r θ p (, ) φ pc parallel flux B ( r, θ ) φ 2b 1 2c r 1 keystoning: Rutherford Cable (2) 2b 2 r 2 adjacency resistance cross-over resistance J N I r r r r turns app sz, 2 ( r2 r1) 2 1 = + 2b 2b 1 2 Technische Universität Darmstadt, Computational Electromagnetics Laboratory

39 B ( r, θ ) parallel magnetic field Magnetic flux density perpendicular magnetic field p (, ) B r θ Technische Universität Darmstadt, Computational Electromagnetics Laboratory

40 B p Ω q σ pa due to perpendicular magnetic field E zq, Adjacency eddy current (1) ψ 3. netto current through Ω q = 0 Izq, = Jpa, z(, r θ) dω = 0 Ω q 1. additional discretisation for unknown electric field E zq, : E ( xy, ) = E M ( xy, ) z z, q q q 2. adjacency eddy current density : Az Jpa, z(, r θ ) =σpa Ez, q σpa t additional constraint!

41 additional load term for magnetic FE model u g = M Z e t pa pa pa pa Adjacency eddy current (2) additional constraint u Z + G e = t T pa pa pa 0 degrees of freedom for E zq, M pa 0 u K Zpa u f T Zpa 0 t e + = pa 0 Gpa e pa 0 Mpa, = σpa N ( x, y) N ( x, y)dω ij i j Ω Zpa, iq = σpa Ni ( x, y) Mq ( x, y)dω Ω Gpa, pq = σpa M p ( x, y) Mq ( x, y)dω Ω

42 Ω q B due to parallel magnetic field σ pa E zq, Adjacency eddy current (3) ψ shape functions related (but not necessarily equal) (, ) M xy to the zones of current redistribution q θ q (, ) M xy r Technische Universität Darmstadt, Computational Electromagnetics Laboratory

43 adjacency eddy currents due to perpendicular magnetic field Adjacency eddy current (, ) B r θ p J pa,z Technische Universität Darmstadt, Computational Electromagnetics Laboratory

44 Az ( r2, θ ) φ p φ pc τ pc r θ z Az ( r1, θ ) Cross-over eddy current (1) θ ψ 1. coupled flux ( ) ( ) φ p θ = z Az( r2, θ) Az( r1, θ) z 2. magnetisation φp θ φpc ( θ ) = τpc t ( ) time constant Technische Universität Darmstadt, Computational Electromagnetics Laboratory

45 J pc,z (, ) B r θ p cross-over eddy currents due to perpendicular magnetic field Cross-over eddy current cross-over magnetisation Technische Universität Darmstadt, Computational Electromagnetics Laboratory

46 Conclusions hybrid method for circuit analysis convenient field-circuit modelling solution of the coupled algebraic systems of equations specialised conductor models complicated eddy current phenomena applications to electrical machine foil winding transformer superconductive magnets, Rutherford cable

Algebraic Multigrid as Solvers and as Preconditioner

Ò Algebraic Multigrid as Solvers and as Preconditioner Domenico Lahaye domenico.lahaye@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ domenico/ Department of Computer Science Katholieke Universiteit Leuven