Simulation Based Design ofO2 MEMS Sensor

Z. Andjelic, P. Krippner, A. VogelABB Corporate Research Center, Ladenburg, Germany

e-mail: zoran.andjelic@de.abb.com

ABSTRACT

The paper presents the methodology for theSimulation Based Design (SBD) of MEMSsensors influenced by mechanical andelectromagnetic phenomena. The complexityof the problems mostly asks for both theefficient 3D numerical technologies and thesophisticated tools to capture the physicalphenomena appearing in the operationalconditions. In this paper we present theapplication of the Multipole Based IntegralTechnology (MBIT) for the simulation ofMEMS-based paramagnetic oxygen sensors.A special procedure for the calculation of theforces caused by the molecular pressure isalso presented.

Keywords: BEM, Magnetostatic, MBIT, IntegralMethods, Sensors

PROBLEM DESCRIPTION

The traditional design of magneto-mechanical MEMS sensors like, forexample, paramagnetic O2 sensors, assumesthe magnetic circuit (excited either bypermanent magnets or by current-curryingcoils) and movable indicating organ. Theworking principal of a paramagnetic O2

sensor is based on the fact that in thepresence of the magnetic field, theparamagnetic O2 molecules tend toconcentrate in the area of higher magneticflux density. An eccentrically positionedglass-made probe immersed in such anenvironment exhibits then the differentspatial molecular pressures. This pressuredifference results further in a force, i.e.torque that causes the probe rotation. Tokeep the probe in the initial position anopposite force is generated by current-

carrying filaments attached to the probesurface. Thus, by measuring the currentrequired to keep the probe in the initialposition, one can measure the content of O2

in the surrounding media.

MBIT

MBIT [2], [3], [4] is applied to theBoundary Element Method with integralformulations of type

( ) ( )( )∫∑=yee

dyyyxkx σϕ ,)( (1)

where k (x,y) for the present type of

calculation is of a form 3r

nrrr ⋅

. Here rr

is the

distance between collocation point x andintegration point y and n

r is a normal vector

in the collocation point. MBIT is used toapproximate the kernel k(x,y) for largerelative distances r, so called ‘far field’approximation.. For the ‘near-field’ regionswith small r the standard BEM is applied.Using series Taylor series expansion orSpherical Harmonics the above kernel canbe, in the case of ‘far-field’, replaced by:

( ) ( ) ( )

( ) ( )∑∫

∑ •=ϕ

e y

e

ccentercluster

nen

dyycyY

cccxXx

σ

κ

ν

µνµ

;

.,;

(2)

Here cn denotes the cluster centers ofnodes/vertices and c e the cluster centers ofelements.

( ) ( )( ) ( ) ( )∑=

≈

νµνµµνκ

,

;;,

,;,,

enne

neMBIT

cyYcxXcc

ccyxkyxk(3)

The terms κµν, Xµ and Y ν in SphericalHarmonics have a form:

( ) ( )

−−

−

==

+++++

+

+

ne

ne

ne

nene

cc

ccY

ccC

cccc

221111122

114

1

:,:,

µµνµνµµµ

νµ

νµµν

π

κκ

( )

−−

−= −

n

nnn cx

cxYcxC:c,xX 2

1

121

µµ

µµµµ

( ) ( )! pl!

1

+−=

pl:C p

l

( ) ( ) φθ ippl

pl ecosP:xY =

Here ( )en c,cκ is a correlation matrixbetween element and node clustertrees. Finally we can evaluate thecomplete ‘far field’ integral bymultiplying and summing up all pre-calculated values of the correlationfunction κ, of X and Y. We only have toevaluate the X(x,cn) ( )en c,cκ Y(y,ce)σ(y)for every cluster center pair (cn,ce) andcharge density σ(y).

FORCES ON A GLASS-MADE PROBE

Above described technique is applied to finda magnetic field distribution in the sensorstructures given in Fig. 1. The force actingon the glass-made probe is caused by theinhomogeneity of the molecular pressureacting on the probe surfaces. The spatialenergy density for such problem can becalculated as:

∫=B

m HdBW0

(3)

The energy in the small volume dV is than

∫=B

m HdBdVdW0

(4)

The energy change due to movement forsmall increment dr is than

drdH

dVdrdWm 2

05.0 χµ ∆= (5)

Here 0µ is the magnetic permeability of

the air and χ∆ is a difference in themagnetic susceptibility. This resultsfurther in a force acting on the probesurface S:

SdHFrr

∫ ∆−= 205.0 χµ (6)

Here H is a module of a magnetic field

Hr

calculated as:

)()(4

1

)(4

1)(

03

3

xHdVr

rN

dSr

rxxH

NxN

xN

x

S x

x

V

rr

rr

+

+=

∫

∫

ρπ

σπ ς

ς

(7)

In Eq. (7) )(iσ is the magnetic surfacecharge density, )(xρ is the magneticvolume charge density, n

r is the normalvector in the calculation node x, ς−= xr

and 0nH

ris the normal component of the

known excitation field caused by thepermanent magnets. More about theprocedure for the calculation of staticmagnetic field in the non-linear materialsusing BEM can be found in [1].

SOME RESULTS

The calculation is performed on thesimplified model of an O2 sensor. It consistsof four permanent magnets serially orientedto produce maximal flux density in the gapsbetween the steel poles, Fig. 1. Fig. 2 showsthe calculated flux density distributionacting on the O2 molecules. The force on aglass-made probe is calculated using Eq. (6),Fig. 3.

N

N

S

S

Glass-made indicator

Yoke

Fig. 1: Model of the MEMS O2 sensor with fourpermanent magnets connected in ‘series’ toproduce maximal flux density around the glass-made probe

B

NNNN

NNNN

SSSS

SSSS

O2

molecules

B

FO2 molecules

Fig. 2: Calculated flux density distribution on themagnetic structures. The paramagnetic O2

molecules tend to concentrate in the regions withmaximal flux density.

Fr

Fig. 3: The force distribution on the glass-made probe is caused by the inhomogeneityof the molecular pressure acting on theprobe surfaces

BEM VS. MBIT: SOMECOMPARISONS

The above analysis is done using our 3D-simulation software POLOPT. Both thestandard BEM and MBIT are applied forelectromagnetic field analysis. The diagramsin Fig. 4 show the difference in the memoryrequirements between these two. The MBITcalculation is performed for two differentvalues of parameter η (0.5; 0.9) thatrepresents the ration between “near field”and “far field”.Also, the calculations are repeated fordifferent expansions orders m=3,4,5 and 6. Fig. 5 shows the comparison in CPUbetween standard BEM and MBIT.Application of MBIT has enabled more than10 times faster simulation using 10 timesless memory simultaneously.For the present test it can be seen that theincreasing of the expansion order m leads tothe increasing of the memory, Fig. 4, but doesnot lead to the increasing of the CPU, Fig. 5.

0.0E+001.0E+082.0E+083.0E+084.0E+085.0E+086.0E+087.0E+088.0E+089.0E+081.0E+09

3 4 5 6 m

mem

ory

/byte

MBIT eta=0.5MBIT eta=0.9PoloptPolopt/10

Fig. 4: Memory requirements for standard BEM(POLOPT) and MBIT for different “distance”parameter η and different expansion orders m.

Finally, we can say that application of MBITrequires more than 10 time less memory incomparison with standard BEM. The reasonis that when using MBIT it is necessary tostore only “near-field” matrix coefficients.This results in memory reduction from O(n2)(standard BEM) to O(nlog4n) for MBIT.

0.0E+002.0E+024.0E+026.0E+02

8.0E+021.0E+031.2E+031.4E+031.6E+031.8E+032.0E+03

3 4 5 6 m

tim

e/s

ec

.

MBIT eta=0MBIT eta=0Polopt

Fig. 5: CPU time measurement: BEM versusMBIT

CONCLUSION

The paper presents a procedure for asimulation of the paramagnetic O2 MEMSsensors. Applied MBIT technique enablesefficient treatment of fully 3D geometry.Application of MBIT has enabled more than10 times faster simulation using 10 timesless memory simultaneously. A special procedure for analysis of theforces caused by a molecular pressure on aglass-made probe immersed into magneticfield is presented.

REFERENCES:

[1] B. Krstajic, Z. Andjelic, S. Milojkovic, S. Babic,S. Salon: “Nonlinear 3DMagnetostatic FieldCalculation by the Integral Equation Method withSurface and Volume Magnetic Charges”, IEEETransaction on Magnetics, vol.28, N0.2, March1992

[2] Greengard, L.:”The Rapid Evaluation ofPotential Fields in Particle Methods”,MIT Press,Cambridge Ma., 1988

[3] Lage, C.:”Softwareentwicklung zurRandelementmethode:Analyse und Entwurfeffizienter Techniken”; DissertationChristian-Albrechts-Universität Kiel, 1995

[4] Fischer, U.: “MBIT-POLOPT”, ABB internalReport , C.01.T.54, July 2000.