Accounting for Remanence and Hysteresis in Finite-Element Magnet Models

Home , Remanence

from 1st of July 2014

Herbert De Gersem

+ contributions from B. Masschaele, A. Mierau, S. Koch, J. Yuan T. Roggen, U. Römer, S. Schöps, J. Trommler, T. Weiland

CERN, Geneva, 20nd March 2014 Example: GSI-SIS-100 magnet

SIS100 dipole (prototype) yoke

coil

brackets photo: J. Guse, GSI (www.gsi.de)

end plate length: 3 m

Prof. Dr. Ir. Herbert De Gersem 2 Example: GSI-SIS-100 magnet excitation profile

110 100 extraction 90 80 injection 70 60 50 12%→ 0.24 T 40 30 20 10 acceleration fraction of maximum current / % 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time / s

Prof. Dr. Ir. Herbert De Gersem 3 Magnetoquasistatic formulation   ∂A  differential equation: ∇×(νσ ∇×AJ) + = s ∂t

Prof. Dr. Ir. Herbert De Gersem 4 Discretisation in space   ∂A  differential equation: ∇×(νσ ∇×AJ) + = s ∂t    spatial discretisation AA≈=FE ∑ ajwj j    da  semi-discrete system: Kaνσ+= M j dt s

shape functions:  edge finite elements wj (curl-conforming)

Prof. Dr. Ir. Herbert De Gersem 5 Discretisation in time B /T differential equation ∆t

time / s spatial discretisation    da  semi-discrete system: Kaνσ+= M j dt s

temporal discretisation

 discrete system: (Kνσ+=α Ma) k+1 RHS

Prof. Dr. Ir. Herbert De Gersem 6 Results

magnetic field eddy currents in the end plane

simulation by CST EMStudio®

Prof. Dr. Ir. Herbert De Gersem 7 Overview • magnet simulation (standard 3D FE solver) • challenges o geometrical details o materials o transient effects o high accuracy • magnet simulation (dedicated 3D FE solver) • hysteresis modelling • conclusions

Prof. Dr. Ir. Herbert De Gersem 8 Overview • magnet simulation (standard 3D FE solver) • challenges o geometrical details o materials o transient effects o high accuracy • magnet simulation (dedicated 3D FE solver) • hysteresis modelling • conclusions

Prof. Dr. Ir. Herbert De Gersem 9 Challenge 1: Detailed geometry yoke

o length (meter) vs. lamination thickness (mm)

o shimming, holes beam tube

o < 1mm thick end-winding parts

o determine the eddy currents in the end plates

Prof. Dr. Ir. Herbert De Gersem 10 Challenge 2: Materials yoke iron:

o anisotropic (rolling & transverse direction)

ν  rol νν= T  ()BRtrans R

ν trans

ν rol reluctivity in the rolling direction

ν trans reluctivity in the transversal direction R local rotation matrix

Prof. Dr. Ir. Herbert De Gersem 11 Challenge 2: Materials yoke iron:

o anisotropic (rolling & transverse direction) o nonlinear (saturation)

3 Newton method  2 ν ()B

B /T 1 Bc =1.9 T

0 1 2 5 0 3 4 5 x 10 H / A/m

Prof. Dr. Ir. Herbert De Gersem 12 Challenge 2: Materials yoke iron:

o anisotropic (rolling & transverse direction) o nonlinear (saturation) o hysteretic (remanent field)

Jiles-Atherton model Preisach model estimation of losses by Steinmetz-Bertotti

Prof. Dr. Ir. Herbert De Gersem 13 Challenge 2: Materials yoke iron: stacking factor o anisotropic (rolling & transverse direction) γst ≈≤0.95 ~ 1 o nonlinear (saturation) o hysteretic (remanent field) coating o composite (lamination) iron

(simple) homogenisation 1 γγst1− st along lamination direction = + νν ν xy Fe 0 perpendicular to laminates νz = γνst Fe +−(1 γ st) ν 0

Prof. Dr. Ir. Herbert De Gersem 14 Challenge 2: Materials yoke iron:

o anisotropic (rolling & transverse direction) o nonlinear (saturation) o hysteretic (remanent field) o composite (lamination) o variability

stochastics, sensitivity (see recent research of Ulrich Römer and Sebastian Schöps, TU Darmstadt)

Prof. Dr. Ir. Herbert De Gersem 15 Challenge 2: Materials

2 7 - yoke iron: 6 o anisotropic (rolling & transverse direction) 5 o nonlinear (saturation) 4 o hysteretic (remanent field) 3 2 o composite (lamination) Current density kAmm o variability 1 superconductor: 2 2 4 4 o critical current 6 6 8 8 o temperature 10 10 12 o magnetic field 14 16

Prof. Dr. Ir. Herbert De Gersem 16 Challenge 3: Transient phenomena lamination

o hysteresis + remanence

Jiles-Atherton model Preisach model estimation of the remanence (based on data from material vendor)

Prof. Dr. Ir. Herbert De Gersem 17 Challenge 3: Transient phenomena lamination   ∂  o hysteresis + remanence A ∇×(νσ ∇×AJ) + = s o eddy currents ∂t

eddy current term + (simple) homogenisation σxy = γσst Fe

σ z = 0

or + multi-scale model (hand-shaking)

Prof. Dr. Ir. Herbert De Gersem 18 Challenge 3: Transient phenomena

0.3mm lamination δ o hysteresis + remanence o eddy currents beam tube shell elements

o eddy currents additional matrix contributions K δ and Mδ assembling into system matrix by Q T +α KMνσ+σ QK( δδ+α MQ) KMfull full + JensTrommler

Prof. Dr. Ir. Herbert De Gersem 19 Challenge 3: Transient phenomena lamination

o hysteresis + remanence o eddy currents beam tube

o eddy currents superconductor persistent currents o fig. courtesy C. Völlinger

Bean model  magnetisation (Christine Völlinger) implemented in ROXIE

fig. courtesy C. Völlinger

Prof. Dr. Ir. Herbert De Gersem 20 Challenge 3: Transient phenomena dBx dt M lamination x

o hysteresis + remanence J z ()θ o eddy currents beam tube R

o eddy currents M z superconductor Jrθ () persistent currents o dBz o coupling currents dt o cable eddy currents   ∂∂AA ∇×(ν ∇×AJ) + σ +∇× ντ0 cb ∇× = s ∂∂tt additional magnetisation

Prof. Dr. Ir. Herbert De Gersem 21 Challenge 4: High accuracy requirements losses

o dimensioning of the cooling system o hot spots o quench aperture field

o multipoles during injection, ramping and extraction o + influence of eddy currents

huge models parallelisation, multi-core computers

Prof. Dr. Ir. Herbert De Gersem 22 Overview • magnet simulation (standard 3D FE solver) • challenges o geometrical details o materials o transient effects o high accuracy • magnet simulation (dedicated 3D FE solver) • hysteresis modelling • conclusions

Prof. Dr. Ir. Herbert De Gersem 23 Dedicated Simulation Tool

FEMSTER, TRILINOS, LLNL Sandia Labs

1. ® CST Studio Suite 2. own software • CAD modelling • FE assembly • meshing (higher order FEs) 4. • transient solver • visualisation • nonlinear materials 3. • system solver Matlab • postprocessing parallelisation • visualisation + Stephan Koch, Jens Trommler

Prof. Dr. Ir. Herbert De Gersem 24 Results: Eddy-Current Losses 40 γ pk = 93% 35 eddy-current 13 J γ pk = 96%

losses over one 30 γ pk = 98% cycle 25 10 J for different γ stacking factors pk 20

15 B /T

power power losses W / 10 ∆t 8 J 5 time / s T 0 loss energy: = W∫ Ptd 0 0 . 25 0. 50 0 . 75 1. 00 1. 25 1 . 50 0 time / s + Stephan Koch, Jens Trommler

Prof. Dr. Ir. Herbert De Gersem 25 Results: Loss Energy 17  tv,(0) ▪ discretization: FE, wj 16  tv,(1) - increase number of FE, wj elements 15 - increase order of approximation 14   tv 13 A≈= AFE ∑ awjj j 12 11 reference solution: degrees of freedom: loss energy / J 1.000.000 dofs + 2 per face 10 9 8  4 5 6 7 8 tv,(0) n = 6 10 10 10 10 10 wj dof number of dofs  tv,(1) = + S. Koch, J. Trommler wj ndof 20 Prof. Dr. Ir. Herbert De Gersem 26 Convergence: Loss Energy 10 0 ▪ relative error with respect to reference solution 10 -1 1% error 0.4 billion dofs 0.3 million dofs degrees of freedom: -2 relativeerror 10 x 1000

 tv,(0) + 2 per face FE, wj  tv,(1) FE, wj  -3 wtv,(0) 10 j 10 4 10 5 10 6 10 7 10 8  tv,(1) number of dofs wj + S. Koch, J. Trommler

Prof. Dr. Ir. Herbert De Gersem 27 Comparison: Shape Functions

0  tv,(0) 10 solution of 110 x 4 = 440 wj linear systems of equations

10- 1 4.4 h @ 132 CPUs  tv,(1) wj

-2 relativeerror 10

 tv,(0) FE, wj 7.3 h @ 72 CPUs  tv,(1) FE, wj 10- 3 + 2 per face 10 4 10 5 10 6 10 7 10 8 number of dofs + S. Koch, J. Trommler

Prof. Dr. Ir. Herbert De Gersem 28 Overview • magnet simulation (standard 3D FE solver) • challenges o geometrical details o materials o transient effects o high accuracy • magnet simulation (dedicated 3D FE solver) • hysteresis modelling • conclusions

Prof. Dr. Ir. Herbert De Gersem 29 Hysteresis turning point • Ferromagnetic materials reversible area o remanence ft() o hysteresis losses • Superconductive materials major loop • Friction

ut()

minor loop

Prof. Dr. Ir. Herbert De Gersem 30 Hysteresis and Field Modelling

• transient field simulation + hysteresis model • transient field simulation + simple hysteresis model e.g. hysteresis losses: complex permeability ˆ Bˆ Bt( )= B cos(ω t) µ = e jϕ ˆ Ht( )= Hˆ cos(ωϕ t − ) H • post-processing step + simple hysteresis model e.g. hysteresis losses: Steinmetz formula  2 f B pk= σ  hyst hyst hyst 50 Hz 1T  • post-processing step + simple remanence model e.g. remanence: look-up table

Prof. Dr. Ir. Herbert De Gersem 31 Hysteresis Models

• Preisach model • Jiles-Atherton model • Stone-Wolfarth model • ...

Preisach model

I.D. Mayergoyz, „Mathematical Models of Hysteresis“, Springer-Verlag, New York, 1991, pp. 1-44.

Prof. Dr. Ir. Herbert De Gersem 32 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 33 Elementary Hysteresis Operator f() t= γˆαβ ut ()

β α ut()

−1 ut() ft() γˆαβ

Prof. Dr. Ir. Herbert De Gersem 34 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 35 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 36 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 37 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 38 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 39 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 40 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 41 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 42 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 43 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 44 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 45 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 46 Elementary Hysteresis Operator ut()

f() t= γˆαβ ut () α +1 β

β α ut() t ft() −1 t

Prof. Dr. Ir. Herbert De Gersem 47 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 48 Combining Hysteresis Operators f() t= mγˆ ut() ∑ p αβpp p

α α α β3 β1 3 β2 1 2 ut()

Prof. Dr. Ir. Herbert De Gersem 49 Combining Hysteresis Operators f() t= mγˆ ut() ∑ p αβpp p

α α α β3 β1 3 β2 1 2 ut()

Prof. Dr. Ir. Herbert De Gersem 50 Combining Hysteresis Operators f() t= mγˆ ut() ∑ p αβpp p

α α α β3 β1 3 β2 1 2 ut()

Prof. Dr. Ir. Herbert De Gersem 51 Combining Hysteresis Operators f() t= mγˆ ut() ∑ p αβpp p

α α α β3 β1 3 β2 1 2 ut()

Prof. Dr. Ir. Herbert De Gersem 52 Combining Hysteresis Operators f() t= mγˆ ut() ∑ p αβpp p

α α α β3 β1 3 β2 1 2 ut()

Prof. Dr. Ir. Herbert De Gersem 53 Combining Hysteresis Operators

f() t= mγˆ ut() ∑ p αβpp p

m 1 γˆ αβ11 m ut() γˆ 2 ft() αβ22 m γˆ 3 αβ33

Prof. Dr. Ir. Herbert De Gersem 54 Combining Hysteresis Operators

f() t= mγˆ ut() ∑ p αβpp p

continuous analogue

=Γ=ˆ µαβγ α β f() t ut ()∫∫ ( ,) ˆαβ ut () d d αβ≥ Preisach model

Prof. Dr. Ir. Herbert De Gersem 55 γˆ ut() αβ11 Preisach Model +1 α (αβ00, ) β1 α1 ut() (αβ11, ) −1

(αβ22, )

γˆ ut() β αβ22 +1

µαβ ( , ) : finite function with β α2 ut() support within T 2 −1 limiting triangle closed major loops

Prof. Dr. Ir. Herbert De Gersem 56 Preisach Model α (αβ00, )

St− ()

Lt() β St() + = µαβγ α β f() t ∫∫ ( ,) ˆαβ ut () d d T ±1 ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β St+ () St− ()

Prof. Dr. Ir. Herbert De Gersem 57 Magnetic Preisach Model α (αβ00, )

St− ()

Lt() Bt()=µ Ht () + M () t β 0

St+ () Bt()=µ0 Ht () + f( Ht ())

=µ + µαβγ α β Bt()0 Ht ()∫∫ ( ,) ˆαβ (Ht ()) d d T ±1

Prof. Dr. Ir. Herbert De Gersem 58 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 59 Complete Saturation negative saturation positive saturation all hysteresis all hysteresis operators „down“ operators „up“ fneg = −∫∫ µαβ( , ) dd α β fpos=−= f neg ∫∫ µαβ( , ) dd α β T α T α (αβ00, ) (αβ00, )

St− ()

β St+ () β

Prof. Dr. Ir. Herbert De Gersem 60 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 61 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 St− () t

β2

β1

β0

Prof. Dr. Ir. Herbert De Gersem 62 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 St− () t

β2 St+ ()

β1

β0 increasing input

Prof. Dr. Ir. Herbert De Gersem 63 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 α1 St− () t

β2 St+ ()

β1

β0

Prof. Dr. Ir. Herbert De Gersem 64 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 St− ()

β2 St+ ()

β1

β0 decreasing input

Prof. Dr. Ir. Herbert De Gersem 65 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− ()

β β1 β2 St+ ()

β1

β0

Prof. Dr. Ir. Herbert De Gersem 66 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− () t

β2 St+ ()

β1

β0 increasing input

Prof. Dr. Ir. Herbert De Gersem 67 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− () α t 2

β2 St+ ()

β1

β0

Prof. Dr. Ir. Herbert De Gersem 68 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− () t

β2 St+ ()

β1

β0 decreasing input Prof. Dr. Ir. Herbert De Gersem 69 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− () (αβ22, ) t

β2 β β2 St+ ()

β1

β0

Prof. Dr. Ir. Herbert De Gersem 70 Memory Formation ut() ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β α1 St+ ( ) St− ( ) α 2 α (αβ, ) α3 00 (αβ11, ) St− () (αβ22, ) t

β2 St+ ()

β1

β0 increasing input

Prof. Dr. Ir. Herbert De Gersem 71 Memory Formation state of the hysteretic material determined by staircase interface L(t) increasing input decreasing input α α (αβ00, ) (αβ00, )

(αβ11, ) (αβ11, ) St− () St− () (αβ22, ) Lt() Lt() β β

St+ () St+ ()

Prof. Dr. Ir. Herbert De Gersem 72 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 73 (Non)local Memory local memory nonlocal memory slope only determined slope determined by by position position and history

ft() ft()

ut() ut()

Prof. Dr. Ir. Herbert De Gersem 74 Nonlocal Memory suppose ft()==−=− f′ () t ∫∫µµµµ ∫∫ ∫∫ ∫∫ St++() St−−() S′ (t) S′ ()t

α α (αβ00, ) (αβ00, )

(αβ11′′, ) St− () St−′ () (αβ11, )

Lt() Lt′() β β

= ′ St+ () ut() u () t St+′ ()

Prof. Dr. Ir. Herbert De Gersem 75 Nonlocal Memory

but ∆SS++′ ≠∆ fttftt()()+∆ ≠′ +∆

α α (αβ00, ) (αβ00, ) αβ′′ St− ()+∆ t ( 11, ) St−′ ()+∆ t (αβ11, ) Lt()+∆ t Lt′()+∆ t uttutt()()+∆ =′ +∆ β β ′ St+ ()+∆ t ∆S+ St+′ ()+∆ t ∆S+

Prof. Dr. Ir. Herbert De Gersem 76 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 77 Wiping-Out Property ut()

α1 α3 α 2 α (αβ00, )

(αβ11, ) St− () (αβ22, ) t Lt() β

β2 St+ ()

β1

β0 increasing input

Prof. Dr. Ir. Herbert De Gersem 78 Wiping-Out Property ut() each local input maximum wipes out the α vertices of L(t) whose α-coordinates are 1 below this maximum α3 α 2 α (αβ00, ) (αβ, ) 11 St() α3 − Lt() t (αβ22, )

β2 St+ ()

β1

β0

Prof. Dr. Ir. Herbert De Gersem 79 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 80 Congruency Property ut() ut′() α1 α1′ α2′ uu++= ′ t t uu−−= ′ ′ α β1 α (αβ00, ) (αβ00, )

(αβ11, ) (αβ11′′, ) St− () St− () (αβ22′′, )

u+ u+′ β β St() St() + ∆Tt() + ∆Tt′()

u− u−′

Prof. Dr. Ir. Herbert De Gersem 81 Congruency Property

∆=ft() 2∫∫ µαβ( , ) d α d β ft() ∆Tt() ft′() ∆=ft′() 2∫∫ µαβ( , ) dd α β ∆Tt′()

u− u+ ut() ∆=∆T′() t Tt ()

∆=∆f′() t ft ()

Prof. Dr. Ir. Herbert De Gersem 82 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 83 First-Order Transition Curves ft()

fαβ′′ fα′

β ′ α′ ut()

Prof. Dr. Ir. Herbert De Gersem 84 Identification ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β St+ ( ) St− ( )

ffα′− αβ ′′ α F(,αβ′′ )= 2 St− () α′ F(,α′′ β )= ∫∫ µαβ( ,) dd α β T (,αβ′′ ) T (,)αβ′′

β ∂2F(,αβ′′ ) µα( ′′, β) = − St+ () β ′ ∂∂αβ′′ 2 1 ∂ fαβ′′ µα( ′′, β) = 2 ∂∂αβ′′

Prof. Dr. Ir. Herbert De Gersem 85 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 86 Representation Theorem The wiping-out property and the congruency property constitute the necessary and sufficient conditions for a hysteresis phenomenon to be represented by the Preisach model based on a set of piecewise monotonic inputs ft() congruency

ut() every minor loop is congruent to a first-order transition curve

Prof. Dr. Ir. Herbert De Gersem 87 Representation Properties 1. symmetry for physical reasons : µ(−− β′,, α ′) = µα( ′′ β)

2. demagnetized state : ft()= 0 α

ut() St− ()

t β

St+ ()

Prof. Dr. Ir. Herbert De Gersem 88 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 89 Numerical Implementation ft()= ∫∫ µαβ( , ) dd α β− ∫∫ µαβ( , ) dd α β St+ ( ) St− ( ) ft()=−+∫∫ µαβ( ,) dd α β 2∫∫ µαβ( , ) dd α β decreasing T St+ () α (αβ00, ) F (αβ00, ) nt( ) (αβ11, ) Lt() St− () µαβ( , ) dd α β (αβ, ) ∑ ∫∫ 22 = k 1Qtk ( ) Qt1() β ∫∫ =∫∫ − ∫∫ Qtk ( ) T (,αβkk−1 ) T (,αβkk−1 ) St+ ()

Prof. Dr. Ir. Herbert De Gersem 90 Numerical Implementation ∫∫ µ(,)αβddα β= ∫∫ µαβ( ,)dd α β− ∫∫ µ(,αβ) dd α β Qk ()t T (αβkk,)−1 T (,αβkk−1 )

F (αβkk, −1) F (αβkk, ) decreasing α (αβ00, )

(αβ11, ) Lt() St− () (αβ, ) 22 ft()= − F (αβ00 , ) nt()− 1 Qt1() β +−2∑ [FF (,αβkk−1 ) (, αβ kk )] k=1 St+ () +−2(,[Fαβnn−1 ) F (,()) α n ut ]

Prof. Dr. Ir. Herbert De Gersem 91 Numerical Implementation n−1 ft()=−+ f+ 2 f − f + f − f ∑ ( αβkk αβ kk−−11) α nut() αβ nn k=1

set of first-order transition curves decreasing only for particular αβ and α (αβ00, )

(αβ11, ) St− () represent f αβ by bilinear (αβ, ) 22 interpolation

Qt1() Lt() fαβ =+++ cc01α c 2 β c 3 αβ β

St+ () store αβ kk and as representation of the material state

Prof. Dr. Ir. Herbert De Gersem 92 Roadmap • Preisach model o elementary hysteresis operator o combining hysteresis operators o Preisach model • Properties o complete saturation o memory formation o nonlocal memory o wiping-out property o congruency property • Representation o first-order transition curves o representation theorem • Numerical Implementation • Example

Prof. Dr. Ir. Herbert De Gersem 93 Material Characteristic

working point

B ν µ   B B=µ+ HBr   remanence Br H=ν+ BHc

H H Hc coercitivity

Prof. Dr. Ir. Herbert De Gersem 94 Example Test Problem: hysteretic material: measured hysteretic first order transient curves

non-hysteretic nonlinear material

+Jing Yuan Prof. Dr. Ir. Herbert De Gersem 95 Vector Hysteresis

transformator flux density at position C3 flux density at position C1 C1

C3 C3

y y C3 C1 x x simplified vector-Preisach model = scalar hysteresis model + direction factor Prof. Dr. Ir. Herbert De Gersem 96 Jiles-Atherton model and measurements

XXXXXX measurementsMeasured

Scalar Hysteresis Models Jiles comparison: Preisach model with Jiles-Atherton model flux density (T) flux density

field strength (A/m)

Preisach model and measurements

XXXXXX Measuredmeasurements

PreisachPreisach

flux density (T) flux density flux density (T) flux density field strength (A/m)

field strength (A/m) • magnetoquasistatische simulations with BDF1 • 70 time steps

Prof. Dr. Ir. Herbert De Gersem 97 Vector-Hysteresis Model 3D example of a vector-hysteresis model t=4s with hysteresis

Y Z Z Y

hysteretic material +Z X

without hysteresis current (A) current

Prof. Dr. Ir. Herbert Detime ZeitGersem (s) / s 98 TEAM Benchmark Problem 32 cross-section of the TEAM Benchmark Problem 32

y C2 C1 175 winding 30 30 winding 30 30

120 175 hysteresis curves x

C3 y Z x

(dimensions in mm, z=2.5mm)

Prof. Dr. Ir. Herbert De Gersem 99 current excitation: TEAM Benchmark Problem 32 comparison simulation to measurements (at position C3) current (A) current

time (s)

flux density (T) simulated (scalar)

simulated (vector)

measured

time (s) C3

• magnetoquasistatische simulation with BDF1 time integrator • inverse Preisach model • Conjugate-Gradient method with SSOR preconditioning • linearisation by the Newton method

Prof. Dr. Ir. Herbert De Gersem 100 C1

C2 TEAM Benchmark Problem 32 comparison simulation to measurements C1 (at position C1) simulated (scalar)

simulated (vector)

measured

current excitation: current (A) current

• magnetoquasistatische simulation with BDF1 time integrator • inverse Preisach model • Conjugate-Gradient method with SSOR preconditioning time (s) • linearisation by the Newton method

Prof. Dr. Ir. Herbert De Gersem 101 SIS100: Remanence at Injection

injection field injection field (without remanence) (with remanence)

Prof. Dr. Ir. Herbert De Gersem 102 SIS100: Remanence at Injection injection field remanent field (with remanence)

injection field (without remanence) region with heavy saturation acts as „source“ of magnetisation

Prof. Dr. Ir. Herbert De Gersem 103 SIS100: Remanence at Injection

-4 injection field x 10 (with remanence) 3.5 without remanence 3 with remanence

2.5

1.5 injection field (T/T) multipoles 1

(without remanence) sextupole decapole 0.5

0 1 3 5 7 9 harmonic order

Prof. Dr. Ir. Herbert De Gersem 104 Conclusions • nonlinear 3D transient magnetic simulation feasible with of-the-shell software • challenges remain and are problem specific o geometrical details o materials o transient effects o high accuracy • hysteresis modelling (Preisach) necessity of accurate material models and measurement data

Prof. Dr. Ir. Herbert De Gersem 105