Finite-Element Electrical Machine Simulation · 2021. 1. 31. · mrtrt, rt rt mmm rtrt, rt rt mmm...

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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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

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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.

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rOverview

literature overview




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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



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

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rOverview

literature overview




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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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R1 X

X hE

I 0

U0,line3

P03

1σ

RFe

compute stator resistance analytically1R

1Xσneglect with respect to h1X

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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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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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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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( )

( )

( )

( )

st stst st, m m m

st stst st, m m m

st stst st, m m m

m m




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

=

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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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R1 X 1σ R'2Xσ2'

Rk Xk

I kPk3

Uk,line3

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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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field-circuit coupling

z

x

end-windingsend-rings

y

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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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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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multi-slice technique

slice 1

slice 2

slice 3

i4

i4

i4


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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=

′

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rOverview

literature overview




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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)

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rOverview

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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

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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


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0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3Tijd [s]

Snel

heid

/ no

min

ale

waa

rde

MetingSimulatie

time (s)

measurementsimulation

spee

d / n

omin

al sp

eed


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-4-3-2-101234

0 0.5 1 1.5 2 2.5 3Tijd [s]

Lijn

stro

om /

nom

inal

e w

aard

e

SimulatieMeting

measurementsimulation

time (s)

line

curr

ent/

nom

inal

line

cur

rent


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, 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

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