Finite-Element Electrical Machine Simulation

(1)

Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik

Schloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de

Dr.-Ing. Herbert De Gersem Institut für Theorie El

ektromagnetischer 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

(2)

2

General 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:

(3)

3

General Information

• Schedule

– next Thursday 27.4: no lecture !!

– next lecture: Thursday 4.5

(4)

Foreknowledge

• 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

(5)

5

Structure

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

(6)

Methodology

at every simulation step

• machine-theoretical

considerations (e.g.)

– relevant

↔

unrelevant phenomena

– linear

↔

nonlinear behaviour

• field-theoretical

– 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

– computer configuration, algebraic solution methods

– discretisation error (space/time)

(7)

7

Related 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/WS Design of Electrical Machines and Actuators with

Numerical Field Simulation (Binder, Funieru)

(8)

Related 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)

(9)

9

Literature (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)

(10)

Literature (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.

– ...

(11)

11

Foreknowledge

• 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

(12)

12

Software

• 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

(13)

13

Overview

• semi-analytical techniques (overview)

• magnetic equivalent circuit

• rotating-field theory

• equivalent circuits + standard tests

• analytical model supported by field simulation

e.g. reluctance machine

• magnetoquasistatic formulation

• discretisation in space

(14)

Magnetic Equivalent Circuit (1)

Ω

=

∫∫

B dA

r

⋅

r

φ

m

Γ

=

∫

r

⋅

r

V

H ds

Γ

Ω

magnetic flux [Wb=Vs]

electric current [A]

magnetic voltage [A]

(15)

15

Magnetic Equivalent Circuit (2)

µ

S

l

φ

m

V

N

t

V

m

V

m

I

φ

reluctance

= magnetic resistance

coil

= magnetic voltage

induced currents

= magnetic inductance

m

=

m

d

V

L

dt

φ

m

=

m

V

R

φ

V

m

=

N I

t

m

e

1

=

L

R

m

=

l

R

S

µ

(16)

Dr.-Ing. Herbert De Gersem Institut für Theorie El ektromagnetischer Felder

Shaded-Pole Motor (1)

coil

short-circuited ring

squirrel cage

stator bridge

(17)

17

(18)

Shaded-Pole Motor (3)

st

L

rt

(

L

st

+

L

rt

)

d

φ

dt

m

V

m

=

V

st

R

rt

R

(

st

rt

)

+

R

+

R

φ

a

ir

g

a

p

a

ir

g

a

p

ag

+

R

φ

(19)

19

• but

- ferromagnetic saturation ?

- eddy-current effects ?

(20)

B

H

1

B

1

H

µ

1

ν

B

m

R

S

ν

=

l

B S

φ =

ferromagnetic saturation

(21)

21

Magnetic Equivalent Circuit (5)

Ampère

= −

∂

z

H

J

r

θ

=

E

θ

ρ

J

θ

=

z

B

µ

H

Faraday-Lenz

1

∂ ⎛

−

∂

⎞

= −

⎜

⎟

∂

⎝

∂

z

⎠

z

H

r

j

H

r r

ρ

r

ωµ

2

′′

+

′

−

2

=

₀

z

r H

rH

j

ωµσ

r H

Ohm

magnetic

l

µ

2

=

S

π

R

φ

m

V

z

H

J

θ

r

z

θ

modified Bessel equation

( )

0

m

0

=

l

z

I

r

V

H

I

R

ξ

1

+

=

j

=

j

ξ

ωµσ

δ

skin depth

( )

1

m

0

2

=

l

I

R

V

R

I

R

ξ

π µ

φ

ξ

( )

0

m

2

1

2

+

=

l

j R

I

R

I

R

ξ

δ

ξ

µπ

(22)

0 50 100 150 200 250 300 350 400 450 500 0 5 10 15x 10 5 frequency (Hz) abs (R m ) ( A /W b) 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 frequency (Hz) angl e(R m )

(23)

23

Air-Gap Reluctance (1)

θ

( )

m

r

θ

m,tooth

0

=

l

z

r

δ

µ

slot

m,slot

0

+

=

l

z

h

r

δ

µ

slotting

(24)

24

Air-Gap Reluctance (2)

θ

( )

m

r

θ

ct

=

m,1

+

1

2

m,2

r

t

m,2

sin

2

=

a

λ

r

λα

λ

( )

m

ct

0

cos

≠

=

+

∑

r

a

λ

θ

λθ

t

α

m,1

r

m,2

r

(25)

25

Overview

• rotating-field theory

(26)

Rotating-Field Theory (1)

( )

z

j

θ

current shield

air-gap (magnetic) field

r

z

θ

( )

=

∑

−

j

z

j

j e

λ

λθ

λ

θ

( )

=

∑

j

(

t

−

)

z

j

j e

λ

ω λθ

λ

θ

(27)

27

Rotating-Field Theory (2)

= ∞

µ

0

=

µ µ

( )

z

j

θ

( )

r

H

θ

= ∞

µ

r

z

θ

δ

Ampère

(

+

)

−

( )

=

( )

r

z

H

θ

d

θ δ

H

θ δ

j

θ

Rd

θ

( )

=

r

z

dH

R

j

d

θ

δ

θ

air-gap (magnetic) field

( )

=

∑

j

(

t

−

)

r

b

b e

λ

ω λθ

λ

(28)

Rotating Fields (1)

{

}

re

im

( ) Re

=

j t

ω

=

cos

ω −

sin

ω

I t

I e

I

t

I

t

(29)

29

Rotating Fields (2)

(

)

( , ) Re

t

a e

λ

j

ω −λθ

t

λ∈Λ

⎧

⎫

⎪

θ =

⎨

⎬

⎪

⎩

∑

⎭

u

( , )

θ

t

u

( , ) Re ( )

θ =

t

{

u

θ

e

j t

ω

}

θ

ω

angular frequency

wave number

λ

syn

ω

=

λ

wave velocity

(30)

30

Angular Slip Frequency

m

ω

′

θ

m

t

′

θ = θ + ω

(

)

(

m

)

( , ) Re

t

a e

λ

j

ω−λω

t

−λθ

′

λ∈Λ

⎧

⎫

⎪

θ =

⎨

⎬

⎪

⎩

∑

⎭

u

s,

λ

ω

angular slip frequency

same amplitudes

same wave numbers

different frequencies

(

)

( , ) Re

t

a e

λ

j

ω −λθ

t

λ∈Λ

⎧

⎫

⎪

θ =

⎨

⎬

⎪

⎩

∑

⎭

u

θ

(31)

31

Overview

• rotating-field theory

• analytical model supported by field simulation

(32)

Equivalent Circuits (1)

induction machine

stator

stator end

windings

rotor

rotor ring

shaft (omitted)

cooling ducts

stator slot

rotor slot

(33)

33

Equivalent Circuits (2)

equivalent circuit

X

R

1

U

_

1

I_

1

R

'

2

1

σ

X

h1

X

I_

₀

2

σ

'

I_

2

'

R

Fe

I

RFe

_

(1-s)

s

R

'

2

______

I_

µ

(34)

no-load test

R

1

X

h

E

I

₀

U

0,line

3

P

₀

3

1

σ

R

Fe

(35)

35

short-circuit test

R

1

X

σ

1

X

σ2

'

R'

2

R

k

X

k

I

k

P

k

3

U

k,line

3

(36)

Overview

(37)

37

(38)

38

Direct- & Quadrature Axis

direct axis

quadrature axis

seen from one of

the phases

2

t

m

N

S

L

N

R

µ

=

l

L high

L low

θ

(39)

39

Model

( )

d

t

u t

R i t

dt

ψ

=

+

voltage in a coil

( )

di t

dL

( )

u t

R i t

L

i t

dt

θ

=

+

m

( )

di t

dL

( ) ( )

u t

R i t

L

t i t

dt

d

θ

ω

θ

=

+

mechanical velocity

inductance L(

θ

) dependent

on rotor angle

electromagnetic

field simulation

torque

( )

co

ct

,

i

dW

T

i

d

θ

=

(40)

Approach (1)

(first try)

• magnetic field simulation

→

magnetic vector potential formulation

• transversal symmetry

→

2D model

• lamination

→

no eddy currents

→

static simulation

• important ferromagnetic saturation expected

(41)

41

2D FE Model

electric boundary

conditions (Dirichlet)

electrical boundary

conditions (floating

potential)

nonlinear

material

applied currents

(42)

42

Simulation (1)

1.27 T

spatial resolution

for the permeability

not sufficient

(43)

43

Approach (2)

(second try)

•

magnetic field simulation

→

•

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%

(44)

2D FE Model

(45)

45

3D End Effects (1)

coil

magnetically

active length

end parts

→

‚fringing‘ effect

→

leakage inductance

yoke length

l

z

aktiv

>

z

l

assumptions

→

leakage inductance independent of the

saturation and the rotor angle

aktiv

=

γ

z

l

L

σ

(46)

Approach (3)

(third try)

•

→

•

→

2D model

•

lamination

→

no eddy currents

→

static 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)

γ

L

σ

(47)

47

3D End Effects (2)

linear

simulation

(48)

3D End Effects (3)

leakage flux

(49)

49

3D End Effects (4)

linear models

2

magn,3D

1

2

3D

W

=

L

i

magn,2D

1

2D

2

W

=

L

i

3D,d

2D,d

L

=

γ

L

+

L

σ

2D

( )

m

( )

di t

dL

( ) ( )

di t

u t

R i t

L

t i t

L

dt

d

σ

dt

θ

γ

θ

γ

ω

θ

=

+

3D,q

2D,q

L

=

γ

L

+

L

σ

(50)

Approach (4)

(fourth try)

•

→

•

→

2D model

•

lamination

→

no eddy currents

→

static 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

γ

L

σ

(51)

51

Energy

-30

0

-25

-20

-15

-10

-5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

rotor angle (degrees)

magnetic

energy

(J)

1 A

3 A

5 A

7 A

9 A

11 A

13 A

15 A

(52)

52

Coenergy

-30

0

-25

-20

-15

-10

-5

0

0.5

1

1.5

2

2.5

magnetic coenergy

(J)

1 A

3 A

5 A

7 A

9 A

11 A

13 A

15 A

(53)

53

Torque

-30

-25

-20

-15

-10

-5

0

-2

0

2

4

6

8

10

12

torq

ue

(Nm)

1 A

3 A

5 A

7 A

9 A

11 A

13 A

15 A

lower accuracy

(54)

Overview

e.g. reluctance machine

(55)

55

EM Field Simulation

full Maxwell equations

wave equation

W

_elec

W

_magn

τP

_loss

W

_elec

W

_magn

τP

_loss

electroquasistatics

W

_elec

W

magn

τP

_loss

magnetoquasistatics

W

_elec

W

_magn

τP

_loss

(56)

Magnetoquasistatics (1)

• neglect displacement currents with respect to

conducting currents

– Ampère-Maxwell

• magnetic vector potential

– conservation of magnetic flux

• electric scalar potential (voltage)

– Faraday-Lenz

D

H

J

t

∂

∇ ×

= +

∂

r

0

= + ∇×

r

B

A

0

B

∇ ⋅ =

r

B

A

E

t

∂

∇× = −

= −∇×

∂

r

A

r

φ

A

E

t

φ

∂

= −

− ∇

∂

r

W

_elec

W

_magn

τP

_loss

(57)

57

Ampère

parabolic

partial differential equation

↔

elliptic PDEs (e.g. electrostatics,

magnetostatics)

↔

hyperbolic PDEs (e.g. wave equation)

H

J

∇ ×

r

=

r

( )

ν

B

κ

E

∇ ×

r

=

r

(

)

s

∂

∇× ∇×

+

= − ∇

∂

r

123

J

A

t

ν

κ

κ ϕ

1

B

µ

H

ν

=

r

J

r

=

κ

E

r

conductivity

permeability

reluctivity

Magnetoquasistatics (2)

(58)

Overview

• magnetoquasistatic formulation

(59)

59

• Weighted residual approach

• scalar product :

Spatial Discretisation (1)

Ω

in

vectorial „weighting functions“

vectorial „test functions“

( , , )

i

w x y z

r

(

)

d

s

d

Ω

⎛

∂

⎞

∇ × ∇ ×

+

⋅

Ω =

⋅

Ω

⎜

∂

⎟

⎝

⎠

∫

r

i

∫

J

r

i

t

w

A

w

ν

κ

∀

w x y z

r

i

( , , )

(

)

∂

s

∇ × ∇ ×

+

=

∂

r

J

A

t

A

ν

κ

( )

u v

,

u v d

Ω

=

∫

⋅

Ω

r r

(60)

•

→

weak formulation

(

)

d

s

d

Ω

⎛

∂

⎞

∇ × ∇ ×

+

⋅

Ω =

⋅

Ω

⎜

∂

⎟

⎝

⎠

∫

r

i

∫

J

r

i

t

w

A

w

ν

κ

∀

w x y z

r

i

( , , )

(

)

d

s

d

Ω

⎛

∂

⎞

∇ ⋅ ∇ × ×

+ ∇ × ⋅∇ ×

+

⋅

Ω =

⋅

Ω

⎜

∂

⎟

⎝

⎠

∫

r

w

r

i

r

i

r

i

∫

r

i

A

w

J

w

t

ν

κ

(

∇ ×

v

r

)

⋅ = ∇ ⋅ ×

w

r

(

v

r r

w

)

+ ⋅∇ ×

v

r

w

r

Gauss

only first derivative required

„weak“ formulation

s

d

∂Ω

Ω

⎛

∂

⎞

∇ × ×

⋅ Γ +

⎜

∇ × ⋅∇ ×

+

⋅

⎟

Ω =

⋅

Ω

∂

⎝

⎠

∫

A

r

w

r

i

r

∫

r

w

r

i

r

w

r

i

∫

J

r

w

r

i

t

A

ν

κ

H

r

(61)

61

0

ν

1

ν

2

ν

neum

Γ

dir

Γ

Ω

J

r

t

0

H

r

=

n

0

B

=

dir

A n

r

× =

r

A

r

×

n

r

(

)

t

0

H

r

=

ν

∇ ×

A

r

× =

n

r

dir

Γ

neum

Γ

Dirichlet BC at

homogeneous Neumann BC at

n

B n

r r

⋅ =

B

0

=

( , , )

i

w x y z

∀

r

0

=

∀

w

r

i

:

w

r

i

× =

n

r

0

at

Γ

dir

neum

dir

d

i

A

w

A

w

ν

Γ

∇ × ×

⋅ Γ +

∇ × ×

⋅ Γ

∫

r

∫

r

H

r

„natural“

boundary condition

„essential“

boundary condition

(62)

• discretization

• Ritz-Galerkin method

• Petrov-Galerkin method

=

∑

r

j

u

A

v

„shape/form functions“, „trial functions“

dir

( , ,

)

0

at

j

v

r

x y z

× =

n

r

Γ

( , , )

j

w

j

x y z

v

r

x y z

=

r

( , , )

j

w

j

x y z

v

r

x y z

≠

r

( , , )

j

v

r

x y z

unknowns, degrees of freedom

j

(63)

63

Spatial Discretisation (5)

• nodal shape functions

• edge shape functions

(

x y z

, ,

)

=

ϕ

u

1

N

1

( , )

x y

+

u

2

N

2

( , )

x y

+

u

3

N

3

( , )

x y

( )

m

n

m

v x

r r

=

N

∇

N

−

N

∇

N

n

p

m

k

m

n

N

∇

m

N

−∇

(64)

64

• discretization

( , , )

i

w x y z

∀

r

j

i

k

=

f

i

[ ]

⎡

⎤

⎡

⎤ ⎡ ⎤ ⎡

+

⎤

⎢

⎥

=

⎣

⎦ ⎣ ⎦ ⎣

⎦

⎣

⎦

j

i

j

i

j

i

du

k

u

m

f

dt

K and M symmetric,

semi-positive-definite

s

d

Ω

⎛

∂

⎞

∇ × ⋅∇ × +

⋅

Ω =

⋅

Ω

⎜

∂

⎟

⎝

⎠

∫

A

r

v

r

i

r

v

r

i

∫

r

v

r

i

t

A

J

ν

κ

j

A

r

=

∑

u

v

r

s

d

Ω

⎛

⎞

⎜

∇ ×

⋅∇ ×

Ω +

⋅

Ω =

⎟

⋅

Ω

⎜

⎟

⎝

⎠

∑

j

∫

r

j

r

i

∫

r

i

∫

r

i

j

d

v

u

v

u

J

dt

ν

κ

j

i

m

=

(65)

65

s

d

Ω

⎛

⎞

⎜

∇ ×

⋅∇ ×

Ω +

⋅

Ω =

⎟

⋅

Ω

⎜

⎟

⎝

⎠

∑

j

∫

r

j

r

i

∫

r

i

∫

r

i

j

d

v

u

v

u

J

dt

ν

κ

∇ ×

r

j

=

∑

j

q

r

q

c

v

z

s

d

Ω

⎛

⎞

⎜

⋅

Ω +

⋅

Ω =

⎟

⋅

Ω

⎜

⎟

⎝

⎠

∑ ∑ ∑

∫

r

q

r

p

∫

r

j

r

i

∫

r

i

j

ip jq

j

p

q

z

du

v

u

c c

J

dt

ν

κ

m

n

p

FE

, ,

p q

ν

M

κ

FE

, ,

i j

))

j

s,

i

)

j

a

FE

)

+

FE

) ))

=

%

d

s

dt

ν

κ

a

CM

Ca M

j

(66)

Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik

Schloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de

Lecture Series

Finite-Element Electrical Machine

Simulation

NEXT LECTURE : THURSDAY May 4th

Dr.-Ing. Herbert De Gersem

summer semester 2006