Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik Schloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de Dr.-Ing. Herbert De Gersem Institut für Theorie Elektromagnetischer Felder 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” http://www.ew.e-technik.tu-darmstadt.de/FOR575 Dr.-Ing. Herbert De Gersem summer semester 2006 Institut für Theorie Elektromagnetischer Felder
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Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de
Dr.-
Ing.
Her
bert
De
Ger
sem
In
stitu
t für
The
orie
Ele
ktro
mag
netis
cher
Fel
der
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”http://www.ew.e-technik.tu-darmstadt.de/FOR575
Dr.-Ing. Herbert De Gersemsummer semester 2006
Institut für Theorie Elektromagnetischer Felder
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rGeneral Information
• Contact: Herbert De Gersem– email (preferred): [email protected]– 06151-164801– room 133 in this building (S2/17)
• Schedule– almost every Thursday: 15:00-16:40– (also at Thursday: 17:00-18:00
Seminar Computation Engineering)– exact schedule + contents of the lectures
→ website: http://www.ew.e-technik.tu-darmstadt.de/FOR575
• Examination: on demand
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rGeneral Information
• Schedule– next Thursday 27.4: no lecture !!– next lecture: Thursday 4.5– see website: http://www.ew.e-technik.tu-darmstadt.de/FOR575
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rForeknowledge
• Electromagnetic field theory- vector algebra + grad/div/curl- Maxwell laws + potentials- analytical solution techniques for PDEs
• Electrical machine theory- DC, induction and synchronous machines- rotating field theory, equivalent circuits, DQ-axes- ferromagnetic materials
• Numerical simulation- linear algebra, systems of equations- analyse, approximation theory
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rStructure
Simulation techniques• overview• FE/FD/FIT discretisation• static simulation• non-linear materials• time-harmonic and
transient simulation• modelling of motion• permanent magnet material• field-circuit coupling• hysteresis models• coil models• optimisation
Examples• DC machine • transformer• induction machine• linear machine• synchronous
machine• single-phase motor• magnetic bearing• reluctance machine• magnetic brake
lecture series
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rMethodology
at every simulation step• machine-theoretical considerations (e.g.)
– relevant ↔ unrelevant phenomena– linear ↔ nonlinear behaviour
• field-theoretical considerations (e.g.)– formulations (magnetoquasistatic, full Maxwell equations, ...)– spatial effects (→ circuit and/or field simulation)– skin depth (→ grid resolution)– alternating and/or rotating fields
(→ scalar or vectorial hysteresis model)• numerical considerations (e.g.)
– computer configuration, algebraic solution methods– discretisation error (space/time)– loss of accuracy for derived quantities (torque, ...)
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rRelated Courses (1)
electrical machines– SS Elektrische Maschinen, Antriebe und Bahnen
(Binder)– SS Elektrische Maschinen und Antriebe I und II
(Binder)– WS Electrical Machines and Drives I (Binder)– SS/WSDesign of Electrical Machines and Actuators with
Numerical Field Simulation (Binder, Funieru)
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rRelated Courses (2)
electromagnetic field theory & field simulation– WS Technische Elektrodynamik (Weiland)– SS: Verfahren und Anwendungen der Feldsimulation
(Weiland, Ackermann)– WS Electromagnetic Field Simulation
(De Gersem, Gjonaj)– WS Finite Elements in Electromagnetism (Munteanu)
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rLiterature (1)
international journal– IEEE Transactions on Magnetics– IEEE Transactions on Energy Conversion– Archiv für Elektrotechnik
international conferences• ICEM : Int. Conf. on Electrical Machines (2006: Crete)
• Compumag : Int. Conf. on the Computation of EM Fields (2007: Aachen)
• CEFC : IEEE Conf. on EM Field Computation (2006: Miami)
• EMF : Int. Workshop on Electric and Magnetic Fields (2006: France)
• SPEEDAM : Symposion on Power Electronics and Electrical Drives (2006: Capri)
• IEMDC : IEEE Int. Electric Machines and Drives Conf.
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rLiterature (2)
books– J.P.A. Bastos, N. Sadowski, „Electromagnetic Modeling by Finite
Element Methods“, 2003.– K. Hameyer, R. Belmans, „Numerical Modelling and Design of
Electrical Machines and Devices“, 1999.– M. Kaltenbacher, „Numerical Simulation of Mechatronic Sensors
and Actuators“, 2004.– E. Kallenbach et al., „Elektromagnete“, 2003.– ...
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rForeknowledge
• Electromagnetic field theory- vector algebra + grad/div/curl- Maxwell laws + potentials- analytical solution techniques for PDEs
• Electrical machine theory- DC, induction and synchronous machines- rotating field theory, equivalent circuits, DQ-axes- ferromagnetic materials
• Numerical simulation- linear algebra, systems of equations- analyse, approximation theory
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rSoftware
• semi-analytical- SPEED
• field simulation (commercial tools)- Ansys → TUD-EW- Maxwell (Ansoft)- MagNet (Infolytica)- Flux2d/Flux3d (Cedrat)- Opera (VectorFields)- EMStudio (CST) → TUD-TEMF
• field simulation (tools at university)- FEMAG (ETH Zürich) → TUD-EW- MEGA (Univ. Bath) → TUD-EW- Olympos (K.U. Leuven) → TUD-TEMF- Dido (TUD-TEMF) → TUD-TEMF
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
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rMagnetic Equivalent Circuit (1)
Ω
= ⋅∫∫rr
B dAφ
mΓ
= ⋅∫r rV H ds
Γ
Ω
magnetic flux [Wb=Vs]
electric current [A]
magnetic voltage [A]
electric voltage [V]
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rMagnetic Equivalent Circuit (2)
µ
S
l
φ
mV mV mVtN
I
φ φ
reluctance= magnetic resistance
coil= magnetic voltage
induced currents= magnetic inductance
m m=dV Ldtφ
m m=V R φ m t=V N I
me
1=L
Rm =lRSµ
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rShaded-Pole Motor (1)
coil
short-circuited ring
squirrel cage
stator bridge
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rShaded-Pole Motor (2)
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rShaded-Pole Motor (3)
stL rtL
( )st rtφ
+dL Ldt
mV
m=V
stRrtR
( )st rt+ + φR R
φ
air gap
air gap
ag+ φR
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rMagnetic Equivalent Circuit (3)
• but- ferromagnetic saturation ?- eddy-current effects ?- motional parts ?
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rMagnetic Equivalent Circuit (4)
B
H
1B
1H
µ1ν
BmR
Sν
=l
B Sφ =
ferromagnetic saturation
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rMagnetic Equivalent Circuit (5)
Ampère∂
= −∂
zHJrθ
=E Jθ θρ=z zB Hµ
Faraday-Lenz1 ∂∂ ⎛ ⎞− = −⎜ ⎟∂ ∂⎝ ⎠
zz
Hr j Hr r r
ρ ωµ
2 2 0′′ ′+ − =z z zr H rH j r Hωµσ
Ohm
magnetic
l µ
2=S Rπ
φ
mV
zH
Jθ
rz θ
modified Bessel equation
( )( )
0m
0=
lz
I rVHI R
ξξ
1+= =
jjξ ωµσδ
skin depth( )( )
1m
0
2=
l
I RV RI R
ξπ µφξ ξ
( )( )
0m 2
1
12+
=l I Rj RR
I RR
ξδ ξµπ
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rMagnetic Equivalent Circuit (4)
0 50 100 150 200 250 300 350 400 450 5000
5
10
15x 105
frequency (Hz)
abs(
Rm
) (A
/Wb)
0 50 100 150 200 250 300 350 400 450 5000
10
20
30
40
50
frequency (Hz)
angl
e(R
m)
reluctance + eddy currents
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rAir-Gap Reluctance (1)
θ
( )mr θ
m,tooth0
=l z
r δµ
slotm,slot
0
+=
l z
hr δµ
slotting
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rAir-Gap Reluctance (2)
θ
( )mr θ
ct m,1 m,212
= +r r r
t
m,2
sin2=a rλ
λα
λ
( )m ct0
cos≠
= + ∑r r aλλ
θ λθ
tα
m,1r
m,2r
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
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rRotating-Field Theory (1)
( )zj θ
θ
current shield air-gap (magnetic) field
rz
θ
( ) −= ∑ jzj j e λθ
λλ
θ
( ) ( )−= ∑ j tzj j e ω λθ
λλ
θ
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rRotating-Field Theory (2)
= ∞µ
0=µ µ
( )zj θ
( )rH θ
= ∞µ
r
zθδ
Ampère
( ) ( ) ( )+ − =r r zH d H j Rdθ θ δ θ δ θ θ
( )=rz
dH R jd
θθ δ air-gap (magnetic) field
( ) ( )−= ∑ j trb b e ω λθ
λλ
θ
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rRotating Fields (1)
re im( ) Re cos sinω= = ω − ωj tI t I e I t I t
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rRotating Fields (2)
( )( , ) Re j tt a e ω −λθλ
λ∈Λ
⎧ ⎫⎪ ⎪θ = ⎨ ⎬⎪ ⎪⎩ ⎭∑u
( , )tθu ( , ) Re ( ) j tt e ωθ = θu u
θ
ω angular frequency
wave numberλ
synω
ω =λ
wave velocity
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rAngular Slip Frequency
mω
′θ
mt′θ = θ +ω
( )( )m( , ) Re j tt a e ′ω−λω −λθλ
λ∈Λ
⎧ ⎫⎪ ⎪θ = ⎨ ⎬⎪ ⎪⎩ ⎭∑u
s,λω
angular slip frequencysame amplitudessame wave numbersdifferent frequencies
( )( , ) Re j tt a e ω −λθλ
λ∈Λ
⎧ ⎫⎪ ⎪θ = ⎨ ⎬⎪ ⎪⎩ ⎭∑u
θ
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
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rEquivalent Circuits (1)
induction machine
stator
stator endwindings
rotor
rotor ring
shaft (omitted)
cooling ducts
stator slot
rotor slot
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rEquivalent Circuits (2)
equivalent circuit
XR1
U_ 1
I_1
R2'1σX
h1X
I_ 0
2σ'
I_ 2'
RFe
I RFe_ (1-s)
sR2'______I_µ
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rEquivalent Circuits (3)
no-load test
R1 X
X hE
I 0
U0,line3
P03
1σ
RFe
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rEquivalent Circuits (4)
short-circuit test
R1 X 1σ R'2Xσ2'
Rk Xk
I kPk3
Uk,line3
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
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rReluctance Machine
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rDirect- & Quadrature Axis
direct axis quadrature axis
seen from one of the phases
22 tt
m
NSL NR
µ= =
lL high L low
θ
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rModel
( )( ) ( ) d tu t R i tdtψ
= + voltage in a coil
( ) ( )( ) ( ) ( ) ( )di t dLu t R i t L i tdt dt
θθ= + +
m( ) ( )( ) ( ) ( ) ( ) ( )di t dLu t R i t L t i t
dt dθθ ωθ
= + +
mechanical velocity
inductance L(θ) dependenton rotor angle
electromagneticfield simulation
torque ( ) co
ct,
i
dWT id
θθ =
=
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rApproach (1)
(first try)• magnetic field simulation
→ magnetic vector potential formulation• transversal symmetry
→ 2D model• lamination→ no eddy currents → static simulation
• important ferromagnetic saturation expected→ nonlinear simulation
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r2D FE Model
electric boundary conditions (Dirichlet)
electrical boundary conditions (floatingpotential)
nonlinear material
applied currents
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rSimulation (1)
1.27 T
spatial resolution for the permeabilitynot sufficient
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rApproach (2)
(second try)• magnetic field simulation
→ magnetic vector potential formulation• transversal symmetry
→ 2D model• lamination
→ no eddy currents → static simulation• important ferromagnetic saturation expected
→ nonlinear simulation
• local saturation→ adaptive mesh refinement till e.g. the relative change
of the magnetic energy < 1%
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r2D FE Model
4.25 T
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r3D End Effects (1)
coil
magneticallyactive length
end parts→ ‚fringing‘ effect→ leakage inductance
yoke length zl
aktiv z>l l
assumptions→→ leakage inductance independent of thesaturation and the rotor angle
aktiv zγ=l l
Lσ
Lσ
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rApproach (3)
(third try)• magnetic field simulation
→ magnetic vector potential formulation• transversal symmetry
→ 2D model• lamination
→ no eddy currents → static simulation• important ferromagnetic saturation expected
→ nonlinear simulation• local saturation
→ adaptive mesh refinement till e.g. the relative change of the magnetic energy < 1%
• end effects, compute and→ compare 3D and 2D models→ linear simulation (smaller grids)
γLσ
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r3D End Effects (2)
linear simulation
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r3D End Effects (3)
leakage flux
adapted scaling
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r3D End Effects (4)
linear models
2magn,3D 3D
12
W L i= 2magn,2D 2D
12
W L i=
3D,d 2D,dL L Lσγ= +
2D2D m
( )( ) ( )( ) ( ) ( ) ( ) ( )dLdi t di tu t R i t L t i t Ldt d dtσ
θγ θ γ ωθ
= + + +
3D,q 2D,qL L Lσγ= +
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rApproach (4)
(fourth try)• magnetic field simulation
→ magnetic vector potential formulation• transversal symmetry
→ 2D model• lamination
→ no eddy currents → static simulation• important ferromagnetic saturation expected
→ nonlinear simulation• local saturation
→ adaptive mesh refinement till e.g. the relative change of the magnetic energy < 1%
• end effects, compute and→ compare 3D and 2D models→ linear simulation (smaller grids)
• automate the whole procedure in order to carry out parameter variation and optimisation steps
γLσ
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rEnergy
-30 -25 -20 -15 -10 -5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
rotor angle (degrees)
mag
netic
ener
gy(J
)1 A3 A5 A7 A9 A11 A13 A15 A
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rCoenergy
-30 -25 -20 -15 -10 -5 00
0.5
1
1.5
2
2.5
rotor angle (degrees)
mag
netic
coe
nerg
y(J
)1 A3 A5 A7 A9 A11 A13 A15 A
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rTorque
-30 -25 -20 -15 -10 -5 0-2
0
2
4
6
8
10
12
rotor angle (degrees)
torq
ue(N
m)
1 A3 A5 A7 A9 A11 A13 A15 A
lower accuracy
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
55
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rEM Field Simulation
full Maxwell equations
wave equationWelec Wmagn
τPloss
Welec
Wmagn
τPloss
electroquasistatics
Welec
Wm
agn
τPloss
magnetoquasistatics
Welec
Wmagn
τPloss
56
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rMagnetoquasistatics (1)
• neglect displacement currents with respect to conducting currents– Ampère-Maxwell
• magnetic vector potential– conservation of magnetic flux
• electric scalar potential (voltage)– Faraday-Lenz
DH Jt
∂∇× = +
∂
rr r
0= +∇×rr
B A0B∇⋅ =r
B AEt t
∂ ∂∇× = − = −∇×
∂ ∂
rrr
Ar
φ
AEt
φ∂= − −∇
∂
rr
Welec
Wmagn
τPloss
57
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Ampère
parabolic partial differential equation↔ elliptic PDEs (e.g. electrostatics,
magnetostatics)↔ hyperbolic PDEs (e.g. wave equation)
H J∇× =r r
( )B Eν κ∇× =r r
( )s
∂∇× ∇× + = − ∇
∂ r
rr
123J
AAt
ν κ κ ϕ
1B H Hµν
= =r r r
J Eκ=r r
conductivity
permeability
reluctivity
Magnetoquasistatics (2)
source current density
58
Dr.-
Ing.
Her
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De
Ger
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itut f
ür T
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rom
agne
tisch
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rOverview
• semi-analytical techniques (overview)• magnetic equivalent circuit• rotating-field theory• equivalent circuits + standard tests
• analytical model supported by field simulatione.g. reluctance machine
• magnetoquasistatic formulation• discretisation in space
59
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Ing.
Her
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• Weighted residual approach
• scalar product :
Spatial Discretisation (1)
Ωin
vectorial „weighting functions“vectorial „test functions“
( , , )iw x y zr
( ) sd dΩ Ω
⎛ ⎞∂∇ × ∇× + ⋅ Ω = ⋅ Ω⎜ ⎟∂⎝ ⎠
∫ ∫r r
rr r
i iJt
wA wAν κ ( , , )iw x y z∀r
( ) s∂
∇ × ∇× + =∂
rr r
JAt
Aν κ
( ),u v u v dΩ
= ⋅ Ω∫r r r r
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• → weak formulation
( ) sd dΩ Ω
⎛ ⎞∂∇ × ∇× + ⋅ Ω = ⋅ Ω⎜ ⎟∂⎝ ⎠
∫ ∫r r
rr r
i iJt
wA wAν κ ( , , )iw x y z∀r
( ) sd dΩ Ω
⎛ ⎞∂∇ ⋅ ∇ × × + ∇× ⋅∇ × + ⋅ Ω = ⋅ Ω⎜ ⎟∂⎝ ⎠
∫ ∫r
r rr r rrri i i iw AA A w w wJ
tν ν κ
( ) ( )v w v w v w∇× ⋅ = ∇ ⋅ × + ⋅∇ ×r r r r r r
Gauss
only first derivative required„weak“ formulation
sd d d∂Ω Ω Ω
⎛ ⎞∂∇× × ⋅ Γ + ∇× ⋅∇ × + ⋅ Ω = ⋅ Ω⎜ ⎟∂⎝ ⎠
∫ ∫ ∫r r r r
rr r r r
i i i iw w w wA JtAAν ν κ
Hr
Spatial Discretisation (2)
61
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rSpatial Discretisation (3)
0ν
1ν
2ν
neumΓ
dirΓ
Ω
Jr
t 0H =r
n 0B =
dirA n A n× = ×r rr r
( )t 0H A nν= ∇× × =rr r
dirΓ
neumΓ
Dirichlet BC at
homogeneous Neumann BC at
nB n B⋅ =r r
0=
( , , )iw x y z∀r
0= dir0: ati iw w n∀ × = Γrr r
neum dir
d di iA wAwν νΓ Γ
∇× × ⋅ Γ + ∇× × ⋅ Γ∫ ∫r rrrr r
Hr
„natural“boundary condition
„essential“boundary condition
62
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rSpatial Discretisation (4)
• discretization
• Ritz-Galerkin method
• Petrov-Galerkin method
= ∑r r
j jj
uA v
„shape/form functions“, „trial functions“
dir( , , 0 at)jv x y z n× = Γrr
( , , ) ( , , )j jw x y zv x y z =rr
( , , ) ( , , )j jw x y zv x y z ≠rr
( , , )jv x y zr
unknowns, degrees of freedomju
63
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rSpatial Discretisation (5)
• nodal shape functions
• edge shape functions
( ), , =x y zϕ 11 ( , )Nu x y 2 2 ( , )+ Nu x y 3 3( , )+ Nu x y
( ) m n n mv x N N N N= ∇ − ∇r r
n
p
m km
n
nN∇
mN−∇
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• discretization( , , )iw x y z∀
r
jik= if=
[ ]⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎣ ⎦j i
jji ij
duk u m f
dt K and M symmetric,semi-positive-definite
sd dΩ Ω
⎛ ⎞∂∇ × ⋅∇ × + ⋅ Ω = ⋅ Ω⎜ ⎟∂⎝ ⎠
∫ ∫r r
rr rr
i i iv v vAtA Jν κ
jj
jA vu= ∑r r
sd d dΩ Ω Ω
⎛ ⎞⎜ ⎟∇ × ⋅∇ × Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠
∑ ∫ ∫ ∫r r rr r ri ij j i
jj
j
dv v vv v
uu J
dtν κ
jim=
Spatial Discretisation (6)
65
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rSpatial Discretisation (7)
sd d dΩ Ω Ω
⎛ ⎞⎜ ⎟∇ × ⋅∇ × Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠
∑ ∫ ∫ ∫r r rr r ri ij j i
jj
j
dv v vv v
uu J
dtν κ
∇× = ∑r rj qqj
qcv z
sd d dΩ Ω Ω
⎛ ⎞⎜ ⎟⋅ Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠
∑ ∑∑ ∫ ∫ ∫r r rr rr
p i iq jj
j ip jqj p q
z vdu
z v vu c c Jdt
ν κ
m
n
p
FE, ,p qνM FE
, ,i jκM s,))
ij)
ja
FE FE+ =) )))% s
ddtν κaCM Ca M j
Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de
Dr.-
Ing.
Her
bert
De
Ger
sem
In
stitu
t für
The
orie
Ele
ktro
mag
netis
cher
Fel
der
Lecture Series
Finite-Element Electrical Machine Simulation
http://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : THURSDAY May 4th
Dr.-Ing. Herbert De Gersemsummer semester 2006
Institut für Theorie Elektromagnetischer Felder
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