Dynamic simulation of electromagnets · 2009. 1. 29. · Division Chassis & Safety Electronic Brake Systems Simulation 1 / Dr. Harald Biller / printed 21 October 2008 © Continental

Division Chassis & SafetyElectronic Brake Systems Simulation

1 / Dr. Harald Biller / printed 21 October 2008 © Continental AG

Dynamic Simulation of Electromagnets

European COMSOL Conference

5 November 2008

Presented at the COMSOL Conference 2008 Hannover



Contents

1. Electronic brake systems by Continental Automotive Systems

ABS

ESC

2. COMSOL for electromagnetic actuators

Magnetic force

Armature movement by mesh deformation

Fast calculation of system dynamics

3. Conclusion



Contents


ABS

ESC


Magnetic force



3. Conclusion



Anti-lock brake system (ABS)

control slip between wheel and road

“stick friction > slip friction”

maximize force between wheel and road

maintain lateral guiding force

vehicle remains stable

vehicle can be steered

shorter stopping distance in most situations

targets:



ABS components



ABS hydraulic layout

brake pedal,booster,master cylinder

wheel brake

outlet valve

inlet valve



ABS active control

inlet valve closed outlet valve open

constant pressure phase pressure decrease phase



with ABS

Antilock Brake System ABS – µ-Split Braking



without ABS

Antilock Brake System ABS – µ-Split Braking



Electronic Stability Control ESC – Active Yaw Control



Electronic Stability Control ESC

ABS: Anti-lock Brake System+ EBD: Electronic Brake force Distribution+ TCS: Traction Control System+ AYC: Active Yaw Control

= ESC: Electronic Stability Control



Contents


ABS

ESC


Magnetic force



3. Conclusion



Electronic Stability Control ESC – Exploded View

MK60 ESC



Electromagnetic inlet valve



Magnetization of steel

densityflux magnetic1

field magnetic0

BHr

rr

µµ=

COMSOL > Physics > Equation System > Subdomain Settings

normB_qa = sqrt( eps + abs(Br_qa)^2 + abs(Bz_qa)^2 )

COMSOL > Physics > Subdomain Settings

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

magnetic field [kA/m]

flu

x d

ensity [

T]



Materials, mesh, solution

saturation !

force : 23.1 N



Contents


ABS

ESC


Magnetic force



3. Conclusion



Geometry and deformation domain

simplified geometry armature with deformation domain armature with rigid hull

win

din

g

arm

atu

re

core

yoke



Idea of ALE (“arbitrary Lagrangian-Eulerian”) mesh deformation

physical equation for u(r, z)

→ partial diff´l equation (PDE) for U(R,Z):

R

Z

r

R

Z

z

reference coordinates space coordinates

),(

),(

ZRzz

ZRrr

=

=

(r, z)

U u

),(),(def

zruZRU =

R

z

z

u

R

r

r

u

R

U

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

Z

z

z

u

Z

r

r

u

Z

U

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

chain rule

R

z

Z

r

Z

z

R

r

∂

∂

∂

∂−

∂

∂

∂

∂=∆with

∂

∂

∂

∂−

∂

∂

∂

∂

∆=

∂

∂

R

z

Z

U

Z

z

R

U

r

u 1

∂

∂

∂

∂+

∂

∂

∂

∂−

∆=

∂

∂

R

r

Z

U

Z

r

R

U

z

u 1

solve system of equations

insert into PDE for u



Determine the functions r(R, Z) and z(R, Z)

1. prescribed

a) physical deformation (e.g. structural mechanics)

b) explicit formula

2. from boundary conditions

a) Laplace smoothing:

b) Winslow smoothing:

02

2

2

2

=∂

∂+

∂

∂

Z

z

R

r

02

2

2

2

=∂

∂+

∂

∂

z

Z

r

R

recommended here

(purely axial motion of rigid parts)



Explicit deformation z(Z)

-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z

?



Continuous / differentiable deformations z(Z)

-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z



-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z





-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z




-0.25 0 0.25 0.5 0.75 1 1.25-0.25

0

0.25

0.5

0.75

1

1.25

Z

z

recommended here:

linear deformation



Geometry and deformation domain

simplified geometry armature with deformation domain armature with rigid hull

win

din

g

arm

atu

re

core

yoke



Voltage step response – mesh deformation and magnetic field



Voltage step response – eddy currents



Voltage step response – armature motion

0 5 10 150

0.5

1

1.5

2

solenoid current [A]

with eddy currents

without eddy currents

0 5 10 150

10

20

30

40

50

magnetic force [N]

0 5 10 150

0.5

1

1.5

eddy current power [W]

time [ms]

armature

core

yoke

total



Voltage step response – armature motion

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

gap [mm]

with eddy currents

without eddy currents

0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

velocity [m / s]

time [ms]



Comparison to experiment

Inlet valve - force versus gap for 250 ampere turns

0

2

4

6

8

10

12

14

16

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

gap [mm]

forc

e [

N]

COMSOL

test bench



Contents


ABS

ESC


Magnetic force



3. Conclusion



Static dependence of field on solenoid current (up to eddy currents)

coil

extrot

S

eInBevHE z

ϕσσ

rrrrr

=×−+−

Ar

rot=

0 (neglect eddy currents from field change)

t

AE

∂

∂−=

rr

JHrr

=rot

coil

ext

0

rotrot1

rotS

eInAevA

t

Az

r

ϕσ

µµσ

rrrr

r

=×−

+

∂

∂

),,( xvIAA ext

rr=

coil

ext)(

S

eInBevE z

ϕσ

rrrr

+×+=

ext)( JBvErrrr

+×+= σ BHr

rr=µµ0



Dynamics of solenoid current: integral parameter magnetic flux

)(1

indextext UUR

I +=

+= ∫∫

coil

21

coilext

S

dzdrErS

nU

R

rπ

Ψ

−= ∫∫

4444 34444 21

r

coil

21

coilext

S

dzdrArS

n

dt

dU

Rπ

,1

Ψ−=

dt

dU

RI extext )(A

rΨ=Ψ ( )),,( xvIA ext

rΨ=



System dynamics

,1

Ψ−=

dt

dU

RI extext ),,( xvIextΨ=Ψ

),,( xvII Ψ= extext

),,(extext xvIRUdt

dΨ⋅−=

Ψ

+ armature equation of motion

+ simple effective eddy current model

0 1 2 3 4 50

10

20

30

40

50

60

Iext

[A]

ΨΨ ΨΨ [

mV

s]

4 m/s

2 m/s

0 m/s

-2 m/s

-4 m/s

• COMSOL solves

for

• look-up table / interpolation

prescrext )( Ψ=Ψ I

extI



Effective eddy current model

( ))(eddy ti+)(tψ+solenoid resistance R

effective eddy current resistance Reddy

effective eddy current inductance Leddy

external voltage Uext(t)

( ))()()(' statext tIRtUt Ψ−=Ψ

)(stat ΨI

eddyi

)()(')(' eddyeddyeddyeddy tiRttiL −Ψ=

≈ ms 1

eddy

eddy

R

L

( )ms 02.01 ≈τ)()(')(' 21 ttt ψτψτ −Ψ=

determine Leddy , Reddy , τ1 , τ2 from least squares fit to Comsol result



Generation of small systems of ordinary differential equations

),( yufy =&

y

f(u, y) Integrator

1/su

yy

u

look-up table

parameter fit

reduced linear model

field problem

(e.g. COMSOL)

dynamic system

(e.g. Simulink)

• fast solution

• parameter variation

• integration into comprehensive model



Conclusion – simulation in industrial development

1. efficiency

fewer prototypes

better experiments

2. product performance

more variants

automatic optimization

functional insight

• “observe” new quantities

• isolate parameter influence

3. robust products

create limit configurations

study tolerances statistically

Documents

Dynamic simulation of electromagnets · 2009. 1. 29. · Division Chassis & Safety Electronic Brake Systems Simulation 1 / Dr. Harald Biller / printed 21 October 2008 © Continental