Numerical determination of the electromechanical field for a micro servosystem

Sensors and Aczuarors, AZI-A23

215

(1990) 215-218

Numerical Determinationof the ElectromechanicalField for a Micro Servosystem HIROYUKI FUJITA and TOSHIAKl IKOMA Imtlrute of Industnal Scwnce, Unrversrtyof Tokyo, Mmatoku, Tokyo 106 (Japan)

Abstract

The present paper deals wth the calculation of a combmed electroelastlc field The Raylagh-ktz method IS applied to obtam an approximate analytlcal solution The total potential energy of both electnc and elastic energy 1s mmmuzed wth respect to the parameters of an approximate compatible function assumed for the displacement The function was chosen to satisfy boundary condltlons The bendmg of a slhcon square dlaphragm by the electrostatic force was determmed The result predicts a larger displacement than the simple model m which the enhancement of the electrostatic field was neglected Introduction

The calculation of the electrostatic field 1s essential m the design of electrostatic nucroactuators Analytical methods such as conformal mapping [ 1,2] and numerical methods such as the iimte element method [3] and the charge-superposltlon method [l] have been used In some cases, an estimation of Just the capacitance between the electrodes 1s sticlent to determme the dnvmg force of a vanable capacitance motor [2,4] by using the followmg relation F=

_av=v2 ac ax

2

element changes the electrostatic field Equation (1) shows that the displacement tends to enhance the field The enhanced field attracts the element more, that results m a larger displacement In a very simple system of a parallel capacitor and a spnng, the problem 1s easily solved It has been shown that the gap length can be adjusted down to two-thirds of the tmtlal gap length and that once the movable electrode moves one-third of the way down mto the gap, its posltlon becomes unstable and it w111spontaneously displace the rest of the way as a result of the rapid bmld-up of electrostatic force NaJafi et al [6] used this fact to measure the mechanical properties of thin slhcon beams There were few results gven for the more general case except for a cantilever [5j The present paper deals v&h the calculation of the combmed electroelastlc field The Rayhghfitz method 1s apphed to obtam an approximate analyUca1 solution The total potential energy of both electnc and elastic energy 1s mmlrmzed Hrlth respect to the parameters of an approximate compatible function assumed for the displacement The function 1s called the tnal function and 1s chosen to satisfy boundary condltlons The bending of a &con square diaphragm by the electrostatic force was determmed The result predicts a larger displacement than the simple model m which the enhancement of the electrostatic field was neglected

ax

where F IS the force, U 1s the energy of a system, C 1s the capacitance, x 1s the distance along the dnvmg dlrechon, and V is the applied voltage If the force depends on the position of the movmg electrode, it IS necessary to calculate the electrostatic fields at several posltlons [l] The dependence 1s called the force (or torque) npple The problem IS still pure determmation of the field In some apphcatlons, however, an elastically movable element such as a deflection nurror [5] must be pomt~oned by the electrostatic force The position 1s determmed by the eqmhbnum between the dnvmg force and the elastic restoration force The difficulty arises when the dtsplacement of the 0924-4241/90/$3 50

2. Electroelastic Field of the Micro !Servosystem 2 I Formulation The mm0 servosystem [7j consists of a smgle-

crystal silicon &aphragm and a plane dnvmg electrode (Fig 1) The diaphragm was made by amsotroplc etchmg using KOH It was roughly 10 x 10 mm square with a thickness of 10 to 100 pm The gap length between the diaphragm and the electrode was 30pm In order to deduce an analytical solution, the followmg assumptions were made (1) all the edges of the diaphragm were fully bmlt-in, 0 Elsevler Sequola/Pnnted m The Netherlands

216

(n) the elastic energy of a thin plate wth small deflection was employed, (m) the electnc field was quasi-umform m the z-dlrectlon at each point on the electrode, (iv) amsotropy of the silicon was neglected Let us introduce the coordinate shown in Fig 2 The dimensions of the structures are also shown m the Figure From the assumption (1) the boundary condltlons are Bven as follows w(*a,y)

=w(x, +a) =o

=wx,

w+Q,y)

ax

(2)

*:a) =.

Rg 2 Coordmates and dtmenslons of the configuratton

9

Therefore, it 1s appropnate to use the following function as a tnal function

w(x,Y) = f

k=l

w,

(a - 1)x cos* (2k- 1lY

co52

D = 12(1 -v’)

The total potential energy, J[w], IS the sum of the electrical energy, U,, and the elastic energy, U,,, From the assumption (11) U, 1s Bven by the followmg formula [8]

(5)

“..., I

Sllicon

Eh’

(4)

2x

2x

Here, from the assumption (IV) the flexural ne;ldlty, D, IS gven as follows

where E 1s Young’s modulus and v 1s Poisson’s ratio The electrical energy, U,, 1s equal to the Maxwell’s strain energy m the gap space and the potential energy of the charge on the high voltage electrode, the voltage of wluch 1s kept at a 6xed voltage, V Let us denote the electnc field strength by Ef and the pernuthvlty of the vacuum by e,,, then U, IS gven by

From’the assumption (in) the electnc field in the gap 1s quasi-umform (see AppenQx) and grven by

I

[51 Y)) wx, Y)= V/M- WC%

(8)

Substltutmg eqn (8) mto eqn (7), we integrate the first term with respect to z Noting that the mtegrand IS mdependent of z, we have u,=

&dY

+P

d - 4x, y) -n

-a

As mentioned before, the potential energy J[w] IS the sum of U,,, and ZJ, In the Rayltigh-Rttz w, are chosen method, such parameters w1, w2 that rnmmuze J Namely, aJ G’O (z=lcVn) (10) I

22 An Analytrcal Solution Let us consider the simplest tnal function as follows Fig 1 Cross-sectlonal wew of the micro servosystem [7]

WC?Y)

=

w

1

?.x

c33s2 TEcos2 k

h

(11)

217

Rg 3 Companson between the present result (the sohd hne) and the result obtained by a ample calculation m wluch the field enhancement IS Ignored (the dashed lme) 2u = 148mm,h=84~m,d=30pm,E=19x10”,v=O2

Substltutmg eqn (10) into eqns (5) and (9), we have U,,,= x4 Dw12/(2a)2

IS not desirable m electrostatic actuators, the field Lsdesigned to be as umform as possible Therefore the assumption (in), the quasi-umfonmty of that field, holds well (as shown m the Appendix) Also the assumption (n) holds well except for extremely thm diaphragms The boundary condltlon must be treated carefully The expenmental results [7j showed that the boundary IS not fully buned especially when the thickness of the diaphragm IS large compared to the mtltial wafer thckness Because the boundary condmon lies m between the simple support and the fully built-m edge, an actual case can be represented by the linear superposltlon of these extreme cases [8] Let us introduce a coefficient, b, which depends on the ratio between thickness of the diaphragm and the wafer The tnal function can be Bven as follows

(12)

w,cos2(2k-

U,= -:

V+(k)

(13)

Here, C, = &,,4a*/dIS the mltlal value of the capacitance without deformation, and K(k) IS the elhp tic Integral of the first kmd wth the modulus k = (wl/d)“* Noting that the denvatlve of the elhptlc function with respect to its modulus [9] IS equal to {E(k)/( 1 - k*) -K(k)}/k, the partial derivative of J with respect to wI 1s gven by

n4Dwl +

x

lW

2a

k-1

cos*(2k - ‘)”+(1 -

x cos

2a

(2k - 1)nx 2a

b) f

WI&

k-l

cos (2k - lhV 2a

(15)

The amsotropy of the single-crystal sdicon IS very difficult to treat by such an analytical method [lo] The present solution IS smtable for estlmatmg the performance very qmckly The more exact numerical simulation, say by the finite-element method, should he adopted m a detailed design

2a2

Setting the nght-hand side equal to zero and solvmg in terms of w,, we obtain the maxlmum deflection of the diaphragm at the voltage V 2 3 Numerical Results Figure 3 shows the numerical results The sohd hne represents the solution of eqn (14) by the linear mterpolatlon method The dashed line IS Bven by the simple solution m which the field enhancement 1s neglected [7j At low voltages and Hrlth small deflection, they are very close At lugher voltages, values obtamed by the present method exceed those by the simple formula The simple formula gave smaller displacement than the expenmental value at large displacements [7j The present method explams the discrepancy 3. Discussion

Let us examme the assumption employed m Section 2 1 Because an electnc field concentration

4. Codusions In order to analyze the combined field of both electrical and mechamcal fields m nucro electrostatic actuators, the Raylagh-R& method was apphed As an example, deflection of a thm dlaphragm by a quasi-umform electnc field was examined Although the present analysis IS a rather simple one, It Bves a better estimation of the displacement Ddlicultles m the analysis have been discussed and possible solutions suggested Acknowledgements This work was partly supported by the Special Coordination Funds for Promoting Science and Technology of the Japanese Government The authors wsh to thank Professors Y TOI, T Kouno and K Hldaka of the Umverslty of Tokyo for their helpful dlscusslons

218

References 1 H FUJI~~and A Omodaka, Electrostatx actuators for nucromechatromcs, Proc IEEE Mtcro Robots mrd Teleoperators Workshop, Hyanms, MA, US A , Nov 1987 2 R Mahadevan, Capacitance calculations for a smglestator, smgle-rotor electrostatic motor, Proc IEEE Mmo Robots and Teleoperators Workshop, Hyannts, MA, U S A , Nov 1987 3 S C Jacobsen, R H PI-ICC, J E Wood, T H Ryttmg and

M Rafaelof, A design ovemew of an eccentnc-mouon electrostatic nucmactuator ( the Wobble Motor), Sensors and Acfuators, 20 (1989) l-15 4 W S N Tnmmer and R Jebens, Harmomc eloctrostahc motors, Sensors and Actualors, 20 (1989) 17-24 5 K Petersen, Dynarmc nucromech&cs’on s~bcon techmques and devxes, IEEE Tram Electron Dewces, ED-25 (1478) 1241-1250 6 K Nalafi and K Suzulu, A novel techmaue and structure for thi measurement of intrmslc stress aid Young’s modulus of thm films. Proc IEEE Macro EIectro Mechanrcal Systems Workshop, Salt L.uke Ctty, UT, US A pp 96-91

, Feb 1989,

7 M Harada and H FUJI& Micro servocontrol of ticon &aphragms dnven by electrostatx force. Proc IEEE IEt?ON’88, Smgapore,wOtt 1988, pp 489-&J 8 S Xmoshenko. Theorv of Plates attd SItells. McGraw-H111.

New York, 1959 _ 9 A G Greenhdl, Apphcatton of Elhpttc Functtons, MacnulIan, London, 1892, p 175 10 B Puers, E Peeters and W Sansen, CAD tools IOmechanical sensor design, Sensors and Actuators, I7 (1989) 423429

Let the electnc field strength at the center of the diaphragm lx E,,, and the average strength be E., = V/(d - IV) A field mtenstfy factor, FZ, IS defined as ErnIEa, If the ratio FZ 1s nearly equal to umty, the assumption of quasi-umform field holds In the spheroidal coordmates x=A

smhrsm0,

z = A cash r cos 0

The surface 8 = 6J1represents the daphragm The field can be solved analytically and FZ IS gven as follows [Al] FZ = I/(sm fll tan Br In cot 0,)

Glvmg (x=O,z=d-W) have 8, as follows cos 8, =

(A2)

and (x=a,z=d),

[~*/(~(2d - WI> + ii- 1’2

we (A3)

In the present case, where u 1s much larger than w and d, FZ can be wntten FZ N l/sin* 0, N 1 + w(2d - w)/a*

Therefore, if the aspect ratlo,

d/a,

(A4) IS

3%, we have

FZ = 1 001, which means the devlatlon 1s only

0 1% Al H Pnntz, Hocbspannungsfel&r, R Oldenbourg-Verlag, Mumch. 1969

Appendix

In order to estimate the deviation from the locally umform field, the shape of the &con diaphragm 1sreplaced by a hyperboloid (Fig A 1)

(Al)

Rg Al Hyperboloid representmg the daphragm