finite-element electrical machine simulation · −im{u 2} 4 t t = re im alternating fields...
TRANSCRIPT
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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 Effectsin 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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rRotor Geometry/Motion
rotation: rt st mtθ θ ω= −
static/dynamic eccentricity( )rt mstst rt
j tj jr e r e deθ ωθ γ+= +
rt st m skewt zθ θ ω γ= − −skewing
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
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rAir-Gap Fields (1)
( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
=θ λθ−ωtjtot eata~
223Re),(
{ }tjju eIei ω°−= 02ReuI
( ){ }°−λθ=θ ω°− 0cos~2Re),( 0 tjju eeata
{ }tjjv eIei ω°−= 1202RevI
( ){ }°−λθ=θ ω°− 120cos~2Re),( 120 tjjv eeata
{ }tjjw eIei ω°−= 2402RewI
( ){ }°−λθ=θ ω°− 240cos~2Re),( 240 tjjw eeata
alternating fields, detuned in time & space rotating field
alternating currents, detuned in time
windings detuned in space
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rAir-Gap Fields (2)
0=t
{ }2Re u
{ }2Im u−
4Tt =
re
im
rotating fieldalternating fields
t
)(tu
t
)(tu
t
)(ture
re
re
-im
-im-im
t
)(ture
-im
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rAir-Gap Fields (3)
air gap field
7-3 5-1 1 3-7 -5
field spectrum
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rAir-Gap Fields (4)
every stationary air-gap field can be expressed by a series of rotating fields
( ),( , )
j ta t a e+∞ +∞
ω −λθω λ
ω=−∞ λ=−∞θ = ∑ ∑
syn,λω
ω =λ
synchronous speed:
*, ,a a−ω −λ ω λ=
-2 -10 1
2-2
-10
12
0
0.2
0.4
angular frequencpole pair number
mag
nitu
de
wave spectrum ispoint symmetric:
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rAir-Gap Fields (5)
symmetrical variant
( ){ },0
( , ) Re j ta t a e+∞ +∞
ω −λθω λ
ω= λ=−∞θ = ∑ ∑
-2-1
01
2
-2-1
01
2
0
0.5
1
angular frequencypole pair number
mag
nitu
de
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rAir-Gap Fields (6)
Example 1:rotating field with angular frequency and pole pair number :ω λ
( )rot ˆ( , ) cosa t a tθ = ω −λθ−ϕ( ) ( )
rotˆ ˆ
( , )2 2
j jj t j tae aea t e e
− ϕ ϕω −λθ −ω +λθθ = +
{ }rot ˆ( , ) Re j ta t ae e− ϕ ω −λθθ =
-2 -10 1
2-2
-10
12
0
0.2
0.4
angular frequencpole pair number
mag
nitu
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rAir-Gap Fields (7)
Example 2:alternating field with angular frequency and pole pair number :ω λ
( ) ( )rot ˆ( , ) cos cosa t a tθ = ω λθ( ) ( ) ( ) ( )
rotˆ ˆ ˆ ˆ
( , )4 4 4 4
j t j t j t j ta a a aa t e e e eω −λθ −ω +λθ ω +λθ −ω −λθθ = + + +
-2-1
01
2
-2-1
01
2
00.05
0.10.15
0.2
angular frequencypole pair number
mag
nitu
de
( ) ( )rot
ˆ ˆ( , ) Re
2 2j t j ta aa t e eω −λθ ω +λθ⎧ ⎫θ = +⎨ ⎬
⎩ ⎭
= +forward
rotating fieldbackward
rotating fieldalternatingair gap field
decompositionω
λω λ
ω−
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rAir-Gap Fields (8)
Example 3:elliptical air-gap field with angular frequency and pole pair number :ω λ
( ) ( ) ( ) ( )rot ˆˆ( , ) cos cos sin sina t a t b tθ = ω λθ + ω λθ( ) ( ) ( ) ( )
rotˆ ˆ ˆ ˆˆ ˆ ˆ ˆ
( , )4 4 4 4
j t j t j t j ta b a b a b a ba t e e e eω −λθ −ω +λθ ω +λθ −ω −λθ+ + − −θ = + + +
-2 -10 1
2-2
-10
12
0
0.2
0.4
angular frequencpole pair number
mag
nitu
de
( ) ( )rot
ˆ ˆˆ ˆ( , ) Re
2 2j t j ta b a ba t e eω −λθ ω +λθ
⎧ ⎫+ −⎪ ⎪θ = +⎨ ⎬⎪ ⎪⎩ ⎭
&φ −φ
+φ
elliptical
= +
forward rotating field
backwardrotating field
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
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r(A)synchronous Operation
S
synchronous operation
mωω = λ
mω
N
S
λω
S
asynchronous operation
mωω ≠ λ
N
S
λω
mω
rotating field in stator/rotorstatic field in rotor/stator
rotating field in both stator and rotor
synchronous machines& DC machines& reluctance motors
induction (asynchronous) machines& single-phase machines& magnetohydrodynamic pumps
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r(Non-)Uniform Configuration
uniform configuration non-uniform configuration
mωmω
v v
v
v geometryexcitationsboundary conditions
with respect to the direction of motionuniform
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rEuler Approach
x
y
xv
φ
• single (standstill or laboratory) coordinate system
• current (modified Ohms law)
• partial differential equation
( ) sv A AA Jt+ ×∇×∂
∇× ∇× + =∂
ν σσ
J E v B= + ×σ σ
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rLagrange Approach
x
y x~y~ xv
φ
• separate coordinate system attached to every solid body• standard Maxwell equations
• partial differential equation:
• relation:
( ) sAA Jt∂
∇× ∇× + =∂
ν σ
euler lagrangeE BE v+= ×
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
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Ing.
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rSolid-Rotor Machines
10 m/sxv =
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rSolid-Rotor Machines
0 rad/s 1 rad/s
adap
tive
refin
emen
t
10 rad/s 100 rad/s
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rHomogenization
rφ
θφI
rφ
θφI
homogenization
mωslotσ
toothσ toothν
slotν
slotµ
toothµ
)(rθν
)(rrµ)(rzσ
mω
∫τ
θθστ
=σz
zz rr
0
d),(1)(
∫τ
θθµτ
=µz
zr rr
0
d),(1)(
∫τ
θ θθντ=ν
z
zrr
0
d),(1)(
simulation of motional eddy currents in one single step
• anisotropy• non-linearities• external circuit coupling• inaccurate in practice
torque ofinductionmachine
293.0 Nm225.7 Nm
approach
transientEuler+hom
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rCoordinate Transformation
stator
rotor rtθ stθcoordinate transformationst rt mtθ = θ +ω
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r(Brushless) DC Motors
mechanical/electronic commutation
PM
armaturewinding
xy
1. current load seen from stator :( ) ( )( )st st armˆ, cosi t i p≈ −θ θ ϕ
stθrtθ
armϕ
current load seen from rotor :
( ) ( )( )st m rt armˆ, cosi t i p t p≈ + −θ ω θ ϕ
( ) ( )st r st,B t B≈θ θ
• attach single coordinate system to the stator
2. PM field seen from stator :
• neglects effects at higher spatial and temporal harmonics, e.g., slotting
• static simulation
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rSynchronous Machines
current load observed from the stator :
( ) ( )st stˆ, cosi t i t p≈ −θ ω θ
2. PM field observedfrom the rotor :
mωxy
stθ
rtθ
1. current load observed from the rotor :
( ) ( )rt r rt,B t B≈θ θ
• static simulation
( ) ( )st rtˆ, cosi t i p≈θ θ
• attach coordinate systemto the rotor
• neglects effects at higher spatial and temporal harmonics, e.g., slotting
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rSlip Transformation (1)
mω
stator
rotor
( ) Vj ∇σ−=ωσ+×∇ν×∇ uu
( ) ,sj Vλν ω σ σ∇× ∇× + = − ∇u u
λω
( ) ( ){ }stst ˆ, Re j tu t a e −= ω λθθ air gap field= rotating wavecoordinate transformationrtθ stθ
st rt mtθ = θ +ω
,s λω
( ) ( )( ){ }rtrt ˆ, Re mj tu t a e − −= ω λω λθθ,s mλω ω λω= −
slip pulsation
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rTorque Computation (1)
mω
stator
rotor
λω
( ) ( )ˆ, cosz z aE t A tθ = ω ω −λθ+ ϕ
θ′ θ
,s λω
( ) ( )ˆ, cos hH t H tθ θθ = ω −λθ+ ϕ
( ) ( ) ( ), , ,r zS t E t H t− θθ = θ θ
( )stator ˆ ˆ sinz z a hP R A Hθ= π ω ϕ −ϕ
( )rotor , ˆ ˆ sinz s z a hP R A Hλ θ= π ω ϕ −ϕ
( )mech ˆ ˆ sinz m z a hP P R A Hθ∆ = = π ω ϕ −ϕmechP
rotorP
statorP
( )mech ˆ ˆ sinz z a hT R A Hθ= π ϕ −ϕ
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rTorque Computation (2)
machine operation modes
S
mω
N
S
S
mω
N
S
S
mω
N
S
S
mωN
S
mech 0T > mech 0T
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rSlip Transformation (2)
( ) Vj ∇σ−=ωσ+×∇ν×∇ uu
( ) Vsj ∇σ−=σω+×∇ν×∇ λ uu
θ′ θmω
λω
stator
rotor
mechanical speedof the rotor
speed of the rotatingair gap field
slip pulsation ≈ speed difference betweenrotating air gap field and rotating rotor
λω
mω
λω
simulation of three-phaseinduction machines incorporating motional eddy current effects in a single computation step
ωλω−ω
=λmsslip
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r3ph Induction Machines
( ) ( ){ }, Re j tu t a e ω −λθθ ≈induced currents
induced torques
only approximately:
butλ1~
21~λ
no component with 1−=λ
time-harmonic FE analysis with external circuit coupling and slip transformation yields acceptable results for 3-ph induction machines
☺
torque ofinductionmachine293.0 Nm292.6 Nm225.7 Nm
finite elementformulationtransienttime-harmonicEuler
treatment ofmotional eddycurrentsmoving bandslip transformation+ homogenization
negligible
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
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rModelling Solid-Body Motion
Modelling rotation in electrical-machine models
mω
1. sliding-surface techniqueslocked step approachlinear/quadratic interpolationmortar projectiontrigonometric interpolation
rtΩ
stΩ
rt stΓ = Γ
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rModelling Solid-Body Motion
Modelling rotation in electrical-machine models
1. sliding-surface techniqueslocked step approachlinear/quadratic interpolationmortar projectiontrigonometric interpolation
2. air-gap modelssingle layer of finite elements(moving-band technique)boundary elementsdiscontinuous Galerkin techniqueair-gap element (spectral elements)
rtΩ
stΩ
agΩ
stΓ
rtΓ
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rSliding-Surface Technique
non-matching grids at the interface (e.g. by linear interpolation or mortar elements
∆θ
( )1− ε ∆θε∆θ
α
eccentricityconsistency errortorque ripple
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rMoving-Band Technique
stator moving band
rotor
stator moving band
rotor
stator moving band
rotor
stator moving band
rotor
stator moving band
rotor
stator moving band
rotor
stator moving band
rotor
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
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rInterface Conditions
stΩ
stΓ
str
stustu
st rt 0+ =g g
3. continuity of the tangential componentof the magnetic field strength
fictitious surface currents vanish
2. continuity of the normal componentof the magnetic flux density
magnetic vector potential continuous
st rt=u u
st st st st
rt rt rt rt
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
K u g fK u g f
1. decoupled FE/FIT systems
select components at the interface:
st st st=u Q u
st st stT=g Q g
prolongate interface components:
stQ
stTQ
nB
tH
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rLocked-Step Approach
rt stshiftq=u k u
shift
0 1 0 00 0 1 00 0 0 11 0 0 0
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
k
∆θ
qα = ∆θ
α
rotation over an integral number of mesh steps
cyclic permutation
but: mesh equidistant at the interfacerestriction on the time step
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rLinear Interpolation
rt stshiftq
ε=u k k u
shift
0 1 0 00 0 1 00 0 0 11 0 0 0
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
k
1 0 00 1 00 0 1
0 0 1
ε
− ε ε⎡ ⎤⎢ ⎥− ε ε⎢ ⎥=⎢ ⎥− ε ε⎢ ⎥ε − ε⎣ ⎦
k
∆θ
( )1− ε ∆θε∆θ
qα = ∆θ+ ε∆θ
α
reduces to the locked-step approach when 0ε =
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rCoupled Formulation
shiftq
α ε=H k krotation operators forward rotation operator
shiftqTH T
αα ε −α= =H k k H backward rotation operator
saddle-point formulationst stst st st st
rt rt rt rt rt rt
st st rt st
00 00 0 00 0 0 0 0
T H
Tddt
α
α
⎡ ⎤−⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥+ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
K Q HM u u fM u K Q u f
g H Q Q g
projected system (eliminate )H Hd
dt⎛ ⎞+ =⎜ ⎟⎝ ⎠
P M K Pu P f
rt st rt rt
0T T
α
⎡ ⎤= ⎢ ⎥
−⎢ ⎥⎣ ⎦
IP
Q H Q I Q Qwith projector
rtu
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rOverview
alternating, rotating and elliptical air-gap fields
classificationsynchronous ↔ asynchronous motionuniform ↔ non-uniform geometriesEuler ↔ Lagrange formulations
implicitly considering motionEuler formulationstatic/time-harmonic simulationslip transformation technique
explicitly considering motion (Lagrange)sliding-surface ↔ moving-band techniqueslocked-step approach, polynomial interpolation, mortar-element method, trigonometric interpolation
-
40
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Ing.
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heor
ie E
lekt
rom
agne
tisch
er F
elde
rMoving-Band Technique (1)
d
mrω
d
mrω
small rotation small displacement d
change of the mesh topology
mω
classical moving-band discretization is not stablewith respect to rotation and eccentricity
-
41
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rMoving-Band Technique (2)
consistent change of the mesh according to eccentricity
possibly bad meshes
difficulties when rotation has to be considered
d+d−
-
42
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rAir-Gap Element
Air-gap element (Razek et al. 1982)
ag, 0 0rt rt rt0
( , ) log jzr r rA r er r r
λ −λ− λθ
λ λλ≠
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟θ = + + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∑a b a b
0 0 0ag,
rt rt0( , ) jr rH r e
r r r r
λ −λ− λθ
θ λ λλ≠
⎛ ⎞⎛ ⎞ ⎛ ⎞ν ν λ ⎜ ⎟θ = + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∑b a b
rtΩagΩ
rtΓ
rtr
α
rtu
stΩ
stΓ
str
stustu
rtu
harmonic coefficients
-
43
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rStandard Air-Gap Element
ddt+M K
nz=224172
-
44
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rStandard Air-Gap Element
ddt+M K
agddt+ +M K K
introduce air-gap element
nz=809040
nz=224172
system solution time
-
45
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rInterface Conditions
stΩ
stΓ
str
stustu
st ag, st st( , )2 0H r rθ+ θ π =g
st ag, st( , )zA r= θu
st st st st st st
rt rt rt rt rt rt
0 00 0
ddt
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ − =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
M u K u g fM u K u g f
1. decoupled FE systems
select components at the interface:
st st st=u Q u
st st stT=g Q g
prolongate interface components:
stQ
stTQ
2. continuity of
magnetic vector potential continuousnB
3. continuity of
fictitious surface currents vanishtH
rt ag, rt( , )zA r= θu
rt ag, rt rt( , )2 0H r rθ+ θ π =g
-
46
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rWeak Interface Conditions
stΩ
stΓ
str
stustu
( ) ( )st, ag,( , ) ( , ) d 0z z pA r A r wΓ
θ − θ θ Γ =∫
( ) ( )st, ag,( , ) ( , ) d 0H r H r vθ θ ςΓ
θ − θ θ Γ =∫
3. weak form of the interface conditions
2. additional discretization for st, ( )H θ θ( )st, ( ) q q
qH x hθ θ = θ∑
( ) ( )st
st,st, , st, , st, st, st,
ji j i j j i i
j
dH N d
dt θΓ
⎛ ⎞+ + θ θ Γ =⎜ ⎟
⎝ ⎠∑ ∫
uM K u f
1. magnetodynamic weak formulation
st,ig
( )pw∀ θ
( )vς∀ θ
-
47
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCollocation and FFT
4. choice of test/trial functions( ) ( )q qh θ = δ θ
( ) ( )p pw θ = δ θ
( ) jv e ςθς θ =
= point-wise matchingat the FE nodes(collocation)
( )dj pe− λθΓ
δ θ Γ∫hybrid integrals of the form
can be carried out using FFT
but : degenerated convergenceof the consistency error !?
-
48
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stu
rtu
rtu
-
49
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stQ
stu
rtu
rtQ
rtu
-
50
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stQ
stu
F
stc
F
rtc
rtu
rtQ
rtu
-
51
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stQ
stu
F
F
rtc
stc1−
λT ab
11
1 1
−λ −λ−λ
⎡ ⎤ξ ξ= ⎢ ⎥⎢ ⎥⎣ ⎦
T
rtu
rtQ strt
rr
ξ =
rtu
-
52
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stQ
stu
F
F
rtc
stc
rtd
std1−
λT
rtu
rtu
rtQ
ab
λG
0 1 1
λ −λ
λ⎡ ⎤ξ ξ= ν λ ⎢ ⎥−⎢ ⎥⎣ ⎦
G
-
53
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
stu
stQ
stu
F
stg
1−F
F
rtc
stc
rtd
std1−
λT
rtu
rtu
rtQ
ab
1−F
rtg
λG
-
54
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure
rtu
stu
stg
rtg
stu
rtd
std
1−F
1−F
stg
rtg
rtg
stg
rtTQ
stTQstQ
stu
F
F
rtc
stc1−
λT
rtu
rtu
rtQ
ab
λG
-
55
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
stu
stu
rtu
rtu
rtQ
stQ
F
F
rtc
stc
ab
rtd
std
1−F
1−F
stg
rtg
rtg
stg
rtTQ
stTQ
1−λT λG
Procedure
rtu
stu
stg
rtg
agK
-
56
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCoupled System
1st st st st1
1rt rtrt rt
T
T
−−
−
⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦
g Q Q uF FG T
g uFQ QF
Dirichlet-to-Neumann map:
1 1ag
T − −=K Q F GT FQAir-gap contribution to the stiffness matrix:
agddt
+ + =M u Ku K u fSystem to be solved
reluctance of the air gap
-
57
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rRotation
rtu rtu
rtc
1−FF
αRrtc
, ,je λαα λ λ =R
-
58
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rExploiting Symmetries
rtu
-
59
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rExploiting Symmetries
rtu
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IW
I
rt,extu
-
60
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rExploiting Symmetries
rtu
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IW
I
rt,extu rt,extg
1 1− −F GT F
-
61
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rExploiting Symmetries
rtgrtu
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IW
I12
TW
rt,extu rt,extg
1 1− −F GT F
-
62
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rStator/Rotor Skew (1)
R
L R
LU
skewγmulti-slice technique
-
63
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rStator/Rotor Skew (2)
rtu rtu
F
rtc rtc
1−F
skewS
skew
skew, ,skew
sin2
2
λ λ
λγ⎛ ⎞⎜ ⎟⎝ ⎠=λγ
S
-
64
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rEccentricity (1)
electrical machines magnetic bearing
Fnon-centeredrotor position
unbalancedmagnetic pullvibrationsnoise
-
65
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rEccentricity (2)
str
rtρ
dγ
j j jre e deθ ϕ γρ= +
rt rt rt
j j jr de e eθ ϕ γρρ ρ ρ
= +
je γε ε=
insert this transformation into the air-gap elementneglect all terms of order and higher2ε
modified operators andεT εG
-
66
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r+Rotation, Skew, Eccentricity
stu stg
stTQstQ
rtd
std
1−F
1−F
stĝstû
F
F
rtu
rtû
rtQ
rtc
stc
ab
αR
rtĝ
1−εT εG
−αR
I
skewS
I
skewS
rtTQ
rtg
-
67
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCoupled System
ag
1 1skew skew
T H Tddt
− −α ε ε α+ + =
K
M u Ku Q F R S G T S R FQ u f
3. stable discretization of eccentricity and rotationno remeshing requiredapplication for any combination of static/dynamic/whirling motion
2. accounting for skew even with only 1 slicesmaller “skew discretization error”
4. stator and rotor models may havedifferent geometrical periodicitydifferent number slices (multi-slice models)
1. only re-assemble FE systems for non-linearitieschange of rotor position only requires change of αRchange of eccentricity only requires change of andεT εG
5. accurate force and torque computation
-
68
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCoupled System
System to be solved
1 1skew skew
T H Tddt
− −α ε ε α
⎛ ⎞+ + =⎜ ⎟⎝ ⎠M K u Q F R S G T S R FQ u f
-
69
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCoupled System
System to be solved
represented by an algebraic matrix(compressed row storage)
represented by an operation
returning the Neumann data (vector ) for given Dirichlet data (vector )u
g
ag⎯⎯⎯→Ku g
1 1skew skew
T H Tddt
− −α ε ε α
⎛ ⎞+ + =⎜ ⎟⎝ ⎠M K u Q F R S G T S R FQ u f
function g=airgap_element(u,alpha,gamma_skew,epsilon,gamma)
α skewγ ε γ
-
70
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
stu
stu
rtu
rtu
rtQ
stQ
F
F
rtc
stc
ab
rtd
std
1−F
1−F
stg
rtg
1−λT λG
Inverse Procedure
rtg
stg
rtTQ
stTQ
-1agK
-1agK
-
71
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rPreconditioning
1 rt agΩ =Ω ∪Ω
2Ω1Ω
2 rtΓ = Γ1 stΓ = Γ2 st agΩ =Ω ∪Ω
2Γ
1Γ
1. two overlapping FE models
2. additive/multiplicative Schwarz
3. restrictions on andinvolve FFT, rotation, skewand eccentricity operations
1Γ 2Γ
-
72
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rCapacitor Motor
main windingauxiliary windingrotor bar capacitor ~~
2D-FEM2D-FEMFEM2D-
2D-FEM
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10-8-6-4-202468
10
time (s)
cur
rent
(A)
-
73
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
TEAM workshop model 30
1+1− 1− 1+1− 1+
j±
j∓ϕ∓e
ϕ±eϕ±2e
ϕ2∓e
+=
2. split the single-phase excitation in two poly-phase excitations (but winding harmonics?!)
4. air gap flux splitting approach
3. apply the Eulerian formulation(with homogenization if the rotor is slotted!)
FeAl
Air
Fe
CuCu
mω
stator yoke
rotor
coil 1. transient FE simulation with FEM-BEM coupling, moving band, ... (expensive!)
☺
1ph Induction Machines (1)
-
74
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r1ph Induction Machines (2)
+=
+=
alternating forward rotating backward rotatingre
al ti
me
inst
ant
imag
inar
y ti
me
inst
ant
-
75
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r1ph Induction Machines (3)
Eulerian formulation:only exact for solid-rotor devices
non-motional formulation with excitation splitting:introduction of non-physical winding harmonics
analyticalsolution (x)
0 50 100 150 200 250 300 350 400-0.4
-0.2
0
0.2
0.4
0.6
0.8
forward rotating field
backward rotating field
torque (Nm)
speed (rad/s)
pull-out speed
alternating field
-
76
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rPhasor Fields (2)
stator
rotor
rotor bars
windings
mω
contour in the air gap
Γ
-
77
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rStatic Field
Γ
consider separate stator/rotor models
first harmonic air gap field higher harmonic air gap field components
mωΓ
mω mωΓ Γ
, 1sω − ,3sω
rotor model 1,1sω
rotor model 2 rotor model 3
-
78
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rAlternating Field (1)
first harmonicair gap field
1R
&φ
backward rotatingair gap fields
R−
air gap field
F
remaining set of forward rotating air gap fields
R+
mω mω mω, 1sω − ,3sω
rotor model 1
1φ−φ +φ
1F−1F−1F−
rotor model 2 rotor model 3,1sω
-
79
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
~
stat
or m
odel
forwardrotor model
sRadd−2
sRadd
main winding
shadingrings
stator bridgebackwardrotor model
&φ
−φ
s
s−2+φ
FE
cros
s-se
ctio
n
FE
cros
s-se
ctio
n
endwinding
voltagesource
end ring
end
ring
-
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 MachineSimulation
http://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : THURSDAY, June 22th 2006
Dr.-Ing. Herbert De Gersemsummer semester 2006
Institut für Theorie Elektromagnetischer Felder
-
81
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rAlternating Field (2)
-
82
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
fundamentalair gap field
7th harmonicair gap field
5th harmonicair gap field
fR
7R5−Rair gap field
F&φ
mω mω7ω
mω5ω−
σ σ
static rotor model dynamic rotor models
fφ 7φ 5−φ
1F−1F−1F−
eddy
cu
rren
tsin
the
PM
s ?
-
83
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rRotating Field (1)
fundamentalair gap field
11th harmonicair gap field
13th harmonicair gap field
1R 11R− 13Rair gap fieldF
&φ
mω
, 11sω −
mω
,13sω
dynamicrotor models
1φ 13φ11φ− 1F−1F−1F−
mωmωstaticrotor model
-
84
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
&φ −φ
+φ
mω
mω
t
( )tAz
4T
T{ }2Im u−
{ }2Re u
{ }2Re u
s
s−2
+φ−φ
mω
mω{ }2Im u−
-
85
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
r
~
stator modelbackwardrotor model
forwardrotor
model
FE cross-section
sRadd−2 s
Radd
&φ−φ +φ
ss−2
main winding auxiliary winding
capacitor
-
86
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stustu
-
87
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
stQstu stû
-
88
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
stQstu stû
F
stc
-
89
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
stQstu stû
F
stc
rtc
αRskewS
skew
skew, ,skew
sin2
2
λ λ
λγ⎛ ⎞⎜ ⎟⎝ ⎠=λγ
S, ,je λαα λ λ =R
-
90
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
stQstu stû
F
stc
rtc
αRskewS
1−F
rtû
-
91
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
stQstu stû
F
stc
rtc
αRskewS
rtu
1−F
rtTQ
rtûrtu
-
92
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rProcedure 1
stu
rturtu
stu
rtTQ
stQstû
F
αHstc
rtc
αRskewS
1−F
rtû
-
93
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rHybrid Formulation
1rt skew stT −
α α=H Q F S R FQrotation operator
saddle-point formulationstst st st st
rt rt rt rt rt
st st
00 00 0 00 0 0 0 0
Hddt α
α
⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ − =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
K IM u u fM u K H u f
g I H g
projected system( )H Hβ + =P M K Pu P f
rt rt
0T
α
⎡ ⎤= ⎢ ⎥
−⎢ ⎥⎣ ⎦
IP
H I Q Qwith projector
Lecture SeriesFinite-Element Electrical Machine Simulationin the framework of the DFG Research Group 575„High Frequency POverviewOverviewOverviewOverviewModelling Solid-Body MotionOverviewOverviewAir-Gap ElementStandard Air-Gap ElementStandard Air-Gap Element+Rotation, Skew, EccentricityLecture SeriesFinite-Element Electrical Machine Simulationhttp://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : T