Dynamic simulation of electromagnets · 2009. 1. 29. · Division Chassis & Safety Electronic...

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

2. COMSOL for electromagnetic actuators

Magnetic force

Armature movement by mesh deformation

Fast calculation of system dynamics

3. Conclusion

Contents

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

Magnetic force

3. Conclusion

Electronic Stability Control ESC – Exploded View

MK60 ESC

Electromagnetic inlet valve

Magnetization of steel

densityflux magnetic1

field magnetic0

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

magnetic field [kA/m]

ensity [

Materials, mesh, solution

saturation !

force : 23.1 N

Contents

Magnetic force

3. Conclusion

Geometry and deformation domain

simplified geometry armature with deformation domain armature with rigid hull

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

physical equation for u(r, z)

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

reference coordinates space coordinates

(r, z)

),(),(def

zruZRU =

chain rule

∂−

∂=∆with

∂−

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:

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

Continuous / differentiable deformations z(Z)

-0.25 0 0.25 0.5 0.75 1 1.25-0.25

recommended here:

linear deformation

Geometry and deformation domain

simplified geometry armature with deformation domain armature with rigid hull

Voltage step response – mesh deformation and magnetic field

Voltage step response – eddy currents

Voltage step response – armature motion

0 5 10 150

solenoid current [A]

with eddy currents

without eddy currents

0 5 10 150

magnetic force [N]

0 5 10 150

eddy current power [W]

time [ms]

armature

Voltage step response – armature motion

0 0.5 1 1.5 2 2.5 30

gap [mm]

with eddy currents

without eddy currents

0 0.5 1 1.5 2 2.5 3-1

velocity [m / s]

time [ms]

Comparison to experiment

Inlet valve - force versus gap for 250 ampere turns

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

gap [mm]

COMSOL

test bench

Contents

Magnetic force

3. Conclusion

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

extrot

eInBevHE z

ϕσσ

=×−+−

0 (neglect eddy currents from field change)

∂−=

rotrot1

eInAevA

µµσ

=×−

),,( xvIAA ext

eInBevE z

ext)( JBvErrrr

+×+= σ BHr

rr=µµ0

Dynamics of solenoid current: integral parameter magnetic flux

indextext UUR

+= ∫∫

coilext

dzdrErS

−= ∫∫

4444 34444 21

coilext

dzdrArS

Ψ−=

RI extext )(A

rΨ=Ψ ( )),,( xvIA ext

System dynamics

Ψ−=

RI extext ),,( xvIextΨ=Ψ

),,( xvII Ψ= extext

),,(extext xvIRUdt

dΨ⋅−=

+ armature equation of motion

+ simple effective eddy current model

0 1 2 3 4 50

ΨΨ ΨΨ [

-2 m/s

-4 m/s

• COMSOL solves

• look-up table / interpolation

prescrext )( Ψ=Ψ I

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

)()(')(' eddyeddyeddyeddy tiRttiL −Ψ=

≈ ms 1

( )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 =&

f(u, y) Integrator

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

Dynamic simulation of electromagnets · 2009. 1. 29. · Division Chassis & Safety Electronic...

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