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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V08:Modelling and Simulation of
Induction Machines
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Her
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rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
4
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rLiterature
[1] S. Williamson, "Induction motor modelling using finite elements", ICEM 1994, Paris, 5-8 Sept, 1994, Vol. 1, pp. 1-8.
[2] E. Vassent, G. Meunier, J.C. Sabonnadiere, "Simulation of induction machine operation using complex magnetodynamic finite elements", IEEE Trans. Magn., Vol. 25, No. 4, 1989, pp. 3064-3066.
[3] A. Arkkio, "Finite element analysis of cage induction motors fed by static frequency convertors", IEEE Trans. Magn., Vol. 2, No. 2, 1990, pp. 551-554.
[4] D. Dolinar, R. De Weerdt, R. Belmans, E.M. Freeman, "Calculation of two-axis induction motor model parameters using finite elements", IEEE Trans. Energy Conversion, Vol. 12, No. 2, June 1997, pp. 133-140.
8
Dr.-
Ing.
Her
bert
De
Ger
sem
Inst
itut f
ür T
heor
ie E
lekt
rom
agne
tisch
er F
elde
rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
9
Dr.-
Ing.
Her
bert
De
Ger
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rEquivalent Scheme
statorresistance
rotorresistance
XR1
U_ 1I_1
R2'1σX
h1X
I_ 0
2σ'
I_ 2'
RFe
I RFe_ (1-s)
sR2'______I_µ
rotor leakage inductance
slip
stator leakage inductance
main inductance
additionallosses
transformation
C.P. Steinmetz, "The alternating current induction motor", Trans. Am. Inst. Elect. Eng., 1897, pp. 185-217.
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rCoupled Inductance Model
U
V
W
1
2
3
r
r
r
iiiiii
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
i
st
st
st
rt
rt
rt
RR
RR
RR
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
R
U
V
W000
uuu
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
u
( )ddt
= +u Ri Li
( )
( )
( )
( )
st stst st, m m m
st stst st, m m m
st stst st, m m m
m m
2 2cos cos cos2 2 3 3
2 2cos cos cos2 2 3 3
2 2cos cos cos2 2 3 3
2cos cos3
p p p
p p p
p p p
p p
σ
σ
σ
⎛ ⎞ ⎛ ⎞+ − − θ θ+ π θ− π⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− + − θ− π θ θ+ π⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞− − + θ+ π θ− π θ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠=
θ θ − πL
l ll l l l l
l ll l l l l
l ll l l l l
l l
( )
( )
rt rtm rt rt,
rt rtm m m rt rt,
rt rtm m m rt rt,
2cos3 2 2
2 2cos cos cos3 3 2 22 2cos cos cos3 3 2 2
p
p p p
p p p
σ
σ
σ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥θ + π + − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥θ + π θ θ− π − + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥
⎛ ⎞ ⎛ ⎞⎢ ⎥θ − π θ+ π θ − − +⎜ ⎟ ⎜ ⎟⎢⎢ ⎝ ⎠ ⎝ ⎠⎣ ⎦
l ll l l
l ll l l l l
l ll l l l l
⎥⎥
st st,σ+l lst2
−l
rt rt,σ+l l rt2−l
( )m cos pθl m2cos3
p⎛ ⎞θ + π⎜ ⎟⎝ ⎠
l
=L
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rd-q-Axis Model
coupled inductance model
Park transformation
a U
b V
0 W
1 112 2
2 3 303 2 2
2 2 22 2 2
i ii ii i
⎡ ⎤− −⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦
d d d d d
q q q q q
00
u R i L M idu R i M L idt
⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠
R.H. Park, "Two reaction theory of electrical machines, generalised method of analysis, part 1", AIEE Trans., Vol.48, July1929, pp. 716-727.
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rRequired FE Software
linea
r
exte
rnal
ci
rcui
t
mot
ion
static
2D time-harmonic
transient
static
3D time-harmonic
transient
X
X X
X
X
X
X
X
X
X X Xno
nlin
ear
X
X
X
X
X
X
13
Dr.-
Ing.
Her
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De
Ger
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Inst
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ür T
heor
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lekt
rom
agne
tisch
er F
elde
rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
14
Dr.-
Ing.
Her
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De
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rNo-Load Operation (1)
XR1
U_ 1I_1
R2'1σX
h1X
I_ 0
2σ'
I_ 2'
RFe
I RFe_ (1-s)
sR2'______I_µ
0s = 21 sR
s−′ = ∞
1 FeR R
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rNo-Load Operation (2)
R1 X
X hE
I 0
U0,line3
P03
1σ
RFe
compute stator resistance analytically1R
1Xσneglect with respect to h1X
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rNo-Load Operation (3)
simulation features• static simulation should be
sufficient• nonlinear simulation
expected phenomena• no induced currents in the
rotor bars• ferromagnetic saturation
simulation approach• 2D magnetostatic simulation:• nonlinear BH-characteristic
(+ adaptive mesh refinement for achieving a sufficient resolution)
• instantaneous current distribution in the stator windings
( )A J∇× ν∇× =r r
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rNo-Load Model (1)
electric boundary conditions
periodic boundary conditions
region labels
2 of 4 polesto be modelled
48 stator slots58 rotor slots
GeometryBoundary conditions
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rNo-Load Model (2)
Materials
0µ = µ0µ = µ( )Bµ = µ
air :Cu :Fe :
0 2000 4000 6000 8000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8B-H characteristic
Magnetic field H (A/m)
Mag
netic
indu
ctio
n B
(T)
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rNo-Load Model (3)
U+ eff 2i U=
V+ eff1 22
i U= −
W+ eff1 22
i U= −
U- eff 2i U= −
V- eff1 22
i U=
W- eff1 22
i U=
Excitations
V+ V+ V+
V+
V+
V+V+
V+
U- U-U-
U-
W-
W-
W-
W-
U+tr
V+tr
W+tr
+ winding functions
U-tr
V-tr
W-tr
6
1q q
qJ t i
== ∑
r rexcitation current
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rNo-Load Results (1)
real time instant
imaginary time instant
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rNo-Load Results (2)
R1 X
X hE
I 0
U0,line3
P03
1σ
RFe
compute flux linked to e.g. phase U:
U+ U-
U U+ U-d dA t A tΩ Ω
ψ = ⋅ Ω − ⋅ Ω∫ ∫r rr r
Uh1 U
UX j L j
iψ
= ω = ω
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rNo-Load Results (3)
R1 X
X hE
I 0
U0,line3
P03
1σ
RFe
compute hysteresis losses by the Steinmetz formula2
hyst hyst hyst 50 Hz 1 T
Bfp k⎛ ⎞⎜ ⎟= σ⎜ ⎟⎝ ⎠
r
integrate for the stator iron (not for the rotor)
effFe
hyst
3UR
P=
hyst
hyst hyst zP p dΩ
= Ω∫ l resistance
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rNo-Load Results (4)
( )
( )
( )
( )
st stst st, m m m
st stst st, m m m
st stst st, m m m
m m
2 2cos cos cos2 2 3 3
2 2cos cos cos2 2 3 3
2 2cos cos cos2 2 3 3
2cos cos3
p p p
p p p
p p p
p p
σ
σ
σ
⎛ ⎞ ⎛ ⎞+ − − θ θ+ π θ− π⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞− + − θ− π θ θ+ π⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞− − + θ+ π θ− π θ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠=
θ θ − πL
l ll l l l l
l ll l l l l
l ll l l l l
l l
( )
( )
rt rtm rt rt,
rt rtm m m rt rt,
rt rtm m m rt rt,
2cos3 2 2
2 2cos cos cos3 3 2 22 2cos cos cos3 3 2 2
p
p p p
p p p
σ
σ
σ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥θ + π + − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥θ + π θ θ− π − + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥
⎛ ⎞ ⎛ ⎞⎢ ⎥θ − π θ+ π θ − − +⎜ ⎟ ⎜ ⎟⎢⎢ ⎝ ⎠ ⎝ ⎠⎣ ⎦
l ll l l
l ll l l l l
l ll l l l l
⎥⎥
st st,σ+l lst2
−l
rt rt,σ+l l rt2−l
( )m cos pθl m2cos3
p⎛ ⎞θ + π⎜ ⎟⎝ ⎠
l
=L
• define "three-phase system" at the rotor sideby linear combination of rotor bar winding functions
• excite 1 phase of the system (either rotor or stator)
• compute fluxes linked to all phases
• mutual inductance
• permutations for other phases• multiply by cos(pθ) to introduce motion
dy
y x yA tΩ
ψ = ⋅ Ω∫r r
yyx
xM
iψ
=
24
Dr.-
Ing.
Her
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De
Ger
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Inst
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ür T
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lekt
rom
agne
tisch
er F
elde
rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
25
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Ing.
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rShort-Circuit Operation (2)
21 0sR
s−′ =1s =
XR1
U_ 1I_1
R2'1σX
h1X
I_ 0
2σ'
I_ 2'
RFe
I RFe_ (1-s)
sR2'______I_µ
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rShort-Circuit Operation (2)
R1 X 1σ R'2Xσ2'
Rk Xk
I kPk3
Uk,line3
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Inst
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rom
agne
tisch
er F
elde
rShort-Circuit Operation (3)
expected phenomena• induced currents in the rotor bars• ferromagnetic saturation
(especially for closed rotor slots)• currents in e.g. rotor ring
simulation features• time-harmonic simulation• nonlinear simulation
(effective saturation characteristic)
• external circuit coupling
simulation approach• 2D time-harmonic simulation:• effective BH-characteristic
(+ adaptive mesh refinement for achieving a sufficient resolution in the air gap and in wedges)
• current or voltage excitation of the stator through external circuit
• possible source of discrepancy with measurements:measurements : under lower voltage (nominal current)simulation : possibly under nominal voltage
( )A j A J∇× ν∇× + ωσ =r r r
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rShort-Circuit Model (1)
Effective material charactistic
0µ = µ0µ = µ
( )eff Bµ = µair :Cu :Fe :
0 2000 4000 6000 8000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8B-H characteristic
Magnetic field H (A/m)
Mag
netic
indu
ctio
n B
(T)
( ) ( ) { }( )2*eff eff eff eff eff0
1 1 1 Re 22 2
Tj tB B B B B e dt
Tωµ = µ∫
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rShort-Circuit Model (2)
field-circuit coupling
z
x
end-windingsend-rings
y
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rShort-Circuit Model (3)
when an even number of poles are modelled
U+
2D FE
...
U-
V+
V-
W-
W+
1
2
29
Rbar
Rring
Rbar
R3D X3DuU
uV
uW
part of the stator windingsin the magnetic model
part of the rotor barsin the magnetic model
part of the rotor barsoutside the magnetic model
end windings
rotor ring
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rShort-Circuit Model (4)
external circuit parameters
3DR linear, analytical computation
3DX linear, analyticalor 3D FE computation
barR frequency and temperturedependent, analytical or 2D linear time-harmonic FE computation
ringR frequency and temperturedependent, analytical or 3D linear time-harmonic FE computation
picture: PhD Ronny Mertens
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rShort-Circuit Model (5)
multi-slice technique
slice 1
slice 2
slice 3
i4
i4
i4
picture: PhD Ronny Mertens
34
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rShort-Circuit Results (1)
real time instant
imaginary time instant
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rShort-Circuit Results (2)
R1 X 1σ R'2Xσ2'
effU avI
120 120U V W
av 3
j jI I e I eI
° − °+ +=simulation result:
( )1 2 1 2 aveffU R R jX jX Iσ σ′= + + +
11
2 2
XRR X
σ
σ=
′
36
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De
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Inst
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rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
37
Dr.-
Ing.
Her
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De
Ger
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Inst
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tisch
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elde
rLoad Operation (1)
expected phenomena• induced currents in the rotor bars• ferromagnetic saturation
(especially for closed rotor slots)• currents in e.g. rotor ring
simulation features• time-harmonic simulation• nonlinear simulation
(effective saturation characteristic)
• external circuit coupling• slip frequency at the rotor• torque computation
simulation approach• 2D time-harmonic simulation:• effective BH-characteristic• current or voltage excitation of the stator through external circuit• impedance of the rotor circuit scaled by s !!
( )A j s A J∇× ν∇× + ω σ =r r r
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rLoad Results (1)
0 500 1000 15000
0.5
1
1.5
2x 104
speed (rpm)
torq
ue (N
m)
39
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Her
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De
Ger
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Inst
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ür T
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agne
tisch
er F
elde
rOverview
literature overview
induction machine modelsequivalent schemecoupled inductance modeld-q-model
computation of stationary operation (equivalent scheme)no-load operationshort-circuit operationload operation
computation of dynamic operation
40
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Ing.
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rTransient Simulation (1)
mechanical equation of motion
explicit time-stepping scheme
e.g. moving-band techniquefor implementing rotor displacement
LM TTtC
tJ −=θ+θ
dd
dd
2
2
( ) 11 1 −− ω∆α−+ω∆α+θ=θ nnnn tt
41
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rTransient Results (1)
4
stationary charactistic3
torq
ue/ n
omin
al to
rque 2
1
0
-1
-2
-3
-4
0 0.2 0.4 0.6 0.8 1 1.2
speed / nominal speed
picture: PhD Ronny Mertens
42
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rTransient Results (2)
0
0.2
0.4
0.6
0.8
1
1.2
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picture: PhD Ronny Mertens
43
Dr.-
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Her
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picture: PhD Ronny Mertens
Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de
Dr.-
Ing.
Her
bert
De
Ger
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In
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The
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mag
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Lecture Series
Finite-Element Electrical Machine Simulation
http://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : THURSDAY, July 13th 2006
V09: Modelling of hysteresis
Dr.-Ing. Herbert De Gersemsummer semester 2006
Institut für Theorie Elektromagnetischer Felder
Lecture SeriesFinite-Element Electrical Machine Simulationin the framework of the DFG Research Group 575„High Frequency PV08:Modelling and Simulation of Induction MachinesOverviewLiteratureOverviewOverviewOverviewOverviewOverviewLecture SeriesFinite-Element Electrical Machine Simulationhttp://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : T