static modeling of a bi-axial micro-mirror with …

Proceedings of the ASME International Mechanical Engineering Congress and ExpositionIMECE 2014

November 14-20, 2014, Montreal, CANADA

IMECE2014-38834

STATIC MODELING OF A BI-AXIAL MICRO-MIRROR WITH SIDEWALLELECTRODES

Shahrzad Towfighian ∗Department of Mechanical Engineering

State University of New York at BinghamtonBinghamton, New York 13902

Email:[email protected]

Mehmet OzdoganDepartment of Mechanical Engineering

State University of New York at BinghamtonBinghamton, New York 13902

Email: [email protected]

ABSTRACTThe static modeling of a bi-axial torsional micro-mirror with

sidewall electrodes is investigated. The mirror and sidewall elec-trodes experience different voltages. As a result of the potentialdifference, the mirror rotates about x and y axes with the torquesgenerated from electrostatic forces by the sidewall and bottomelectrodes. The rotations about two axes are made possible us-ing a gimbal frame and serpentine springs. An analytical modelis presented that is a simplified model of a previous study basedon independent excitation of the two rotation angles. The sim-plified model enables prediction of the rotation angles with goodaccuracy and faster computation time.

1 INTRODUCTIONMicro-electromechanical systems (MEMS) are becoming

mainstream using recent microfabrication methods (e.g Siliconbulk micro-machining, surface micro-machining [1]). Biaxialmicro-mirrors are among MEMS devices adopted in numerousareas such as laser imaging, image digitizing, projection dis-plays [2] and medical applications, e.g. endoscopy and tomog-raphy. Features to be mostly considered for micromirrors in-clude simple fabrication processes, low driving voltage, largetilt angle and linearized angular scan [3]. Actuation methodsused for micromirrors include electrostatic [4–10], piezoelec-tric [11–17](PZT),electrothermal [18–20] and electromagnetic[21–24] actuators. Electrostatic actuation is the most popular ac-tuation methods due to easy fabrication, low power consumption

∗Address all correspondence to this author.

and high driving speed [25]. However, they require large oper-ation voltages to have large tilting angles. Also for the MEMSdevices that use parallel plate capacitors, pull in instability is abig concern [11, 26]. Electrothermal actuation mechanism pro-vides large deflection angle for low voltage values, but their basicdisadvantages are low driving speed and large energy consump-tion in addition to the change of optical behavior due to tempera-ture [27,28]. Piezoelectric actuators have small device sizes, bet-ter response of driving speed compared to other actuators and lowdriving voltages [14–17]. However, they have the disadvantageof difficult fabrication. Electromagnetics actuators can achievelarge mechanical tilting angle [23, 29] but at the price of largersize. Among these actuation types electrostatic actuator is pre-ferred due to ease of fabrication in large volumes. Common elec-trostatic actuator types are comb-drive and parallel plate. Parallelplate actuators are preferred due to their smaller size comparedto comb-drive actuators.

In this study an analytical model is investigated for a bi-axialmicro-mirror. The model was introduced by Bai et al. [30], whodeveloped an electrostatic micro-mirror with sidewall electrodes.They have presented simulations and experimental results. Con-tribution of this study is to simplify the analytical model pre-sented [30] based on the independent excitation of the two ro-tation angles. we also changed the boundaries of the integral toaccount for the changes in the forces that become significant asthe rotation angles increase.

The sidewall electrodes are used to decrease voltages and toincrease rotation angles. The reason for adding sidewall elec-trodes is to increase the area of electrostatic force according to

1 Copyright c© 2014 by ASME

the electrostatic force equation F =ε0.A.V 2

d2 . Increasing forceson the mirror leads to larger torque around its axis of rotationand consequently larger rotation angle for the same amount ofvoltage. The micro-mirror structure consists of five main com-ponents; mirror plate, gimbal frame, sidewall electrode, bottomelectrode and serpentine spring (torsion bar). The mirror is madeof silicon and is connected to the ground and the voltage is ap-plied to the sidewall and the bottom electrodes. Thus, the mirroris actuated by electrostatic forces generated from different poten-tials between the sidewall, the bottom electrodes and the mirrorplate.

2 OPERATION PRINCIPLESA schematic of the micro-mirror is shown in Figure 1.

The micro-mirror is suspended by the double-gimbal structure,which has two pairs of serpentine springs (Figure 2) for 2degree-of-freedom (DOF) scans: α and β scan which are aboutx-axis and y-axis, respectively. The mirror actuators consist offour equivalent electrode quadrants (also sidewall electrodes)numbered as in Figure 1. The parameters of the micro-mirror arelisted in Table 1. Different voltage potential between the mirrorplate and the sidewall and bottom electrodes creates electrostatictorques that rotate the mirror. To operate the mirror, the twoangles are excited independently. Voltages are applied to thesidewall and bottom electrodes of each quadrant according tothe differential drive method suggested by Hao et al. [31] Forthe rotation about Y-axis (β )

V1 =V2 =Vbias +Vdi fV3 =V4 =Vbias−Vdi f

(1)

where Vbias is the bias voltage and Vdi f is the differential voltage,and V1...4 is the voltage on each quadrant. For the rotation aboutX-axis (α)

V1 =V3 =Vbias−Vdi fV2 =V4 =Vbias +Vdi f

(2)

In other words, quadrants Q1 and Q2 will have equal and largervoltage compared to other quadrants to make the rotation angleβ . Similarly, quadrants Q2 and Q4 will have equal and largervoltage compared to other quadrants to make the rotation angleα .

FIGURE 1: Mirror Elements and Quadrants

FIGURE 2: Serpentine Spring

3 SYSTEM MODEL3.1 Electrostatic Forces and Torques

The global coordinate is XYZ and fixed at the center ofthe initial position of mirror center. The body fixed coordinatexyz is fixed at the center of the mirror plate and is rotating withthe micro-mirror. Rotations about x and y axes are done inde-pendently. Using the defined coordinate systems, equations forforces and torques on the mirror from sidewall electrodes, gim-bal frame and bottom electrodes are obtained, respectively in thissection for angle α . The derivation for angle β is presented inthe Appendix.


3.1.1 Sidewall electrodes Figure 3 shows the projec-tion of the mirror plate on the YZ plane. The center of the mirrorplate passes through the center of the global coordinate. The fig-ure shows the mirror rotation about the X axis when quadrants 2and 4 have equal and higher voltage than quadrants 1 and 3. InFigure 3, since the torques about the Y axis balance each otherin this case, there will not be any rotation about the Y axis, soβ = 0.

The model for electrostatic forces and torque here follow theprocedure explained in Bai et. al [30], which consider the elec-trostatic forces from the sidewall 22 and 12 on the bottom surfaceof the mirror. However, we simplified the model based on the op-eration principle in the previous section (the two angles are oper-ated independently). We also considered the electrostatic forcesfrom the sidewall electrode on the top surface of the mirror asthey become significant in large rotation angles. To incorporatethis effect at large angles, the integral boundaries are defined thatdepend on the rotation angles obtained in the previous voltagestep of the simulations.

For an element of dy on the top layer of the mirror plate,the electrostatic flux from sidewall 22 on the top surface of themirror is written as;

Ee22t =V2_AB

(3)

where V2 is the voltage applied on quadrant 2 and

_AB= AC.(π/2−α) =

ADsin(π/2−α)

(π/2−α)

= (Le

2−Y )

π/2−α

sin(π/2−α)

(4)

Substituting Y = ycosα for a body fixed point A, the fluxequation 3 is written as;

Ee22t =V2 sin(π/2−α)

(Le2 − y.cosα)(π/2−α)

(5)

The electrostatic pressure is

P =ε0E2

2(6)

Therefore the electrostatic force on on element with dimensionsof dx and dy is

dF =ε0E2dxdy

2(7)

TABLE 1: Parameters for micro-mirror structure [30] in Figures 1 and2

Parameter Symbol Value

Mirror Plate

Mirror width lm 1000µ m

Mirror length lm 1000µ m

Gimbal Frame

Outer width Lgw 1400µ m

Inner width Lgwi 1200µ m

Outer length Lgl 1700µ m

Inner length Lgi 1200µ m

Thickness tg 35 µ m

Width in Y direction gwy 250 µ m

Serpentine Spring

Width w 1.5-3 µ m

Thickness tb 12 µ m

l0 10 µ m

lp 110 µ m

l f 120 µ m

li 120 µ m

w f 160 µ m

Electrode Dimensions

Bottom edge width Le 1060 µ m

Gap g 290 µ m

Sidewall thickness ts 40µ m

Sidewall height h 250 µ m

Material Properties

Silicon density ρ 2329 kg/m3

Young’s Modulus E 79 Gpa

Shear Modulus G 73 Gpa

Air permittivity ε0 8.854e−12 Fm

The force in the Z direction from sidewall e22 is then found


FIGURE 3: The mirror rotated about X axis by α , and the sidewalls 12and 22 projected on the YZ plane.

from integration

S1 =ε0.sin2(π/2−α)

2

Fze22t =S1V 2

2(π/2−α)2 .

∫ x2x1

∫ y2y1

{ 1Le2 − y.cosα

}2.dxdy

(8)

where y1 =Le

2cos(α0)− Le

2 tan(α0), y2 =lm2 , x1 =

w f2 and x2 =

lm2 ,

and α0 is the rotation angle in the previous voltage step. In otherwords, the boundaries of the integral changes as the rotation an-gle changes. At small angles, that does not make a difference inthe results, but as the rotation angles grow, this effect becomesmore significant.

From the force equation, one can find the torque about the xaxis from sidewall e22 on top surface of the mirror plate.

T xe22t =−S1V 2

2(π/2−α)2 .

∫ x2

x1

∫ y2

y1

{ 1Le2 − y.cosα

}2.ydxdy (9)

For an element of dy at the bottom surface of the mirror plate(3), the electrostatic flux from sidewall e22 is written as;

Ee22b =V2_

AB′(10)

where

_

AB′= AC.(π/2+α) =AD

sin(π/2−α)(π/2+α)

= (Le

2−Y )

π/2+α

sin(π/2−α)

(11)

Substituting Y = ycosα , the flux equation (10) is written as;

Ee22b =V2 sin(π/2−α)

(Le2 − y.cosα)(π/2+α)

(12)

The force in the Z direction from sidewall e22 on the bottom partof the mirror is

Fze22b =−S1V 2

2(π/2+α)2 .

∫ x4x3

∫ y4y3

{ 1Le2 − y.cosα

}2.dxdy (13)

The boundaries of the integral can be found in Appendix B. Thetorque about the x axis from sidewall e22 at the bottom of themirror plate is then

T xe22b =S1V 2

2(π/2+α)2 .

∫ x4

x3

∫ y4

y3

{ 1Le2 − y.cosα

}2.ydxdy (14)

Electrostatic forces from sidewall 12 also changes the rota-tion angle α . As it can be seen from Figure 3, the electrostaticflux on the bottom surface of the mirror plate from side wall 12is

Ee12 =V1_

A′′B′′=

V3

A′′C′′.(π/2−α)(15)

where V1 is the voltage of quadrant 1. The force and torque on themirror plate from sidewall 12 can be derived similarly to equation(9) with different boundary for the integral.

Fze12 =−S1V 2

1(π/2−α)2 .

∫ x6x5

∫ y6y5

{ 1Le2 − y.cosα

}2.dxdy

T xe12 =−S1V 2

1(π/2−α)2 .

∫ x6x5

∫ y6y5

{ 1Le2 − y.cosα

}2.ydxdy

(16)

Now we consider the effect of electrostatic forces from thesidewall electrodes on the gimbal frame which contributes to therotation of the mirror. The gimbal frame does not rotate about Y


axis based on Figure 4, and it can only rotate about X axis. Thatmeans the electrostatic forces generated between sidewalls andgimbal frame changes α angle only. Figure 4 shows the projec-tion of the mirror plate and the gimbal frame on a plane parallelto YZ plane. Due to larger width of the gimbal frame in the ydirection, only the electrostatic torques and forces caused by thesidewalls 12,22,32 and 42 are considered in the simulations.

The resulting electrostatic force in the Z direction and torquearound the X axis caused by sidewall 12 forces on the top surfaceof the gimbal can be written:

S2 =ε0.sin2(π/2−α)

2

Fzeg22t =S2V 2

2(π/2+α)2

∫ x8x7

∫ y8y7

{ 1ycosα− Le

2 − ts

}2.dxdy

T xeg22t =−S2V 2

2(π/2+α)2

∫ x8x7

∫ y8y7

{ 1ycosα− Le

2 − ts

}2.ydxdy

(17)It should be noted that the above equations are only valid whenthe angle of α is large enough so that integral boundaries y8 =Le/(2.cosα0)+Le/2. tanα0 > y7 = Lgl/2− gwy and that hap-pens when α0 > 0.06 radians. In other words, the forces on thetop layer of gimbal frame only applies when the rotation angle α

reaches certain value of 0.06 radians.The sidewall 22 also exerts electrostatic forces on the bot-

tom surface of the gimbal frame. The corresponding electrostaticforce and torque are

Fzeg22b =−S2V 2

2(π/2−α)2

∫ x10x9

∫ y10y9

{ 1ycosα− Le

2 − ts

}2.dxdy

T xeg22b =S2V 2

2(π/2−α)2

∫ x10x9

∫ y10y9

{ 1ycosα− Le

2 − ts

}2.ydxdy

(18)Similarly the electrostatic force and torque caused by side-

wall 12 on the gimbal frame are

Fzeg12 =−S2V 2

1(π/2+α)2

∫ x12x11

∫ y12y11

{ 1ycosα− Le

2 − ts

}2.dxdy

T xeg12 =−S2V 2

1(π/2+α)2

∫ x12x11

∫ y12y11

{ 1ycosα− Le

2 − ts

}2.ydxdy

(19)

3.1.2 Bottom Electrodes The electrostatic forces andtorques caused by bottom electrodes should also be added to thesimulations. For the rotation angle α , as it is shown in Figure 5,the flux can be obtained from

Eb =V_AB

(20)

where

_AB= AC.(α) =

ADsin(α)

(α)

= (g+Y tanα)α

sin(α)

(21)

Substituting Y = ycosα , and using the similar procedure as be-fore, the electrostatic force and torque caused by voltages onquadrants 1 or 3 is derived

S3 =ε0.V 2

12

sin2(α)

α2

Fzeb1a =−S3.∫ x14

x13

∫ y14y13

(1

g+ y.sin(α)

)2

.dxdy

T xeb1 =−S3.∫ x14

x13

∫ y14y13

(1

g+ y.sin(α)

)2

.ydxdy

(22)

similarly, the electrostatic force and torque caused by bottomelectrodes of quadrants 2 or 4 are

S4 =ε0.V 2

22

sin2(α)

α2

Fzeb2a =−S3.∫ x14

x13

∫ y14y13

(1

g+ y.sin(α)

)2

.dxdy

T xeb2 = S3.∫ x14

x13

∫ y14y13

(1

g+ y.sin(α)

)2

.ydxdy

(23)

For rotation angle β , the equations are derived in a similarfashion, except that the forces on the gimabl frame do not con-tribute to the rotation angle β , and therefore are not considered(Appendix A).

3.2 Mechanical Spring ForceWhen the micro-mirror is actuated by sidewall electrodes,

the electrostatic forces and torques will be balanced by mechan-ical restoring forces and torques.

2.Kα α = 2(T xe22t +T xe22b +T xe12 +(T xeg22t +T xeg22b)+T xeg12 +∑

2m=1 T xebm)

2.Kβ β = 2(Tye13t +T xe13b +Tye33 +Tyeb1 +Tyeb3)

KzZ = 2(Fze12 +Fze22b +Fze22t +Fzge12 +Fzge22b +Fzge22t+Fzeb1a +Fzeb2a +Fze33 +Fze13b +Fze13t +Fzeb1b +Fzeb3b)

(24)In the above equations, the stiffness coefficients are doubled toaccount for the two springs resisting the motion about the two


FIGURE 4: The Projection of the mirror plate, and gimbal and Sidewall electrodes on the YZ Plane

FIGURE 5: Bottom electrode effect for α scanning angle

axes. The serpentine spring has the stiffness [30] of

Kα = Kβ =

l f +64

lp +2li4

G.Jr+

2l0EI

−1

(25)

where for either of rotation angles, polar moment of inertiaJr, and moment of inertia of the spring rectangular cross section

I are

Jr =t3b .w3

.

(1− 192tb

w.π5 . tanh(π.w2tb

)

)I =

w.t3b

12

(26)

For the simulations, displacement in the z directions are not illus-trated and only the rotation angles are simulated using the equa-tions 24 as described in the next section.

4 SIMULATION RESULTSFor obtaining the rotation angles, the summation of all the

torques in corresponding directions of x and y are found consid-ering the voltages of each quadrant. Equations 24 were solvedat different differential voltages as indicated in equations 1 and2. The displacement in the z direction has not been presented asthe experimental data was not available for comparison. In thesimulations, the fringe field effect is also considered negligible.Dimensions and material properties for the simulations, are listedin Table 1.

Figure 6 shows the comparison of the rotation angle α ver-sus differential voltage based on the simplified analytical modeland the experimental and simulation results [30]. Vbias is appliedas 55V and Vdi f f ranges from 0-150V. It is noted the simplified


model reveals close results for α scanning angle in Figure 6 com-pared with the reported results. However, unlike experiments,our model does not respond nonlinearly to the applied voltageat large angles. Nonlineartity in the response can be due to thegap decrease as the voltage increases. In our simulation, we as-sumed the gap is constant. More accurate model should accountfor changing of the gap as the voltage increases and using thecalculated gap in the simulated rotation angles. This approachshould make the curve nonlinear and similar to a regular pull-incurve.

Figure 7 shows the rotation angle β versus differential volt-age when bias voltage vbia = 55V . It can be seen that the simpli-fied model underestimate the scanning angles. For both rotationangles, we used the same mechanical stiffness for the springs andthat can be the cause for deviation seen in the simulation of therotation angle β . It is more likely that the mirror experiencesmore resistance in rotation about the X axis compared to the ro-tation about the Y axis. Overall, the model predicts the rotationangle α with more accuracy compared to the rotation angle β ,which is underestimated.

FIGURE 6: Alpha Scanning Angle when the Vbias = 55V

5 CONCLUSIONSAn analytical model for a bi-axial micro-mirror with side-

wall electrodes is presented, which is a simplified model of aprevious study [30] based on independent excitation of the tworotation angles. The mirror rotates about X and Y axes as a re-sult electrostatic torques from different voltages on the mirrorplate, side-wall electrodes, gimbal frame and the bottom elec-trodes. Actuation of the mirror is done using differential volt-age excitation for the four quadrants and therefore, the rotationsabout axes are done independently which enabled simplifying the

FIGURE 7: Beta Scanning Angle when the Vbias = 55V

equations. The integral boundaries are also chosen such that theyaccount for forces applied on the mirror as the rotation angle in-creases. The simulation results of rotation angle about the X axisare in close proximity of the experiments, while simulation forrotation about Y axis underestimate the angles. Our future workaccounts for the stiffness differences for rotations about the twoaxes. The simplified model with less computation time enablesadding optimization algorithm, which was the motivation of thisstudy to reduce the operating voltage and to increase the rotationangles.

ACKNOWLEDGMENTThe authors would like to thank Turkish Military Academy

(Ankara) for a PhD scholarship to support Mehmet Ozdogan forhis research study.

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Appendix A: Equations for rotation angle β

Sidewall 13 creates electrostatic forces on the top and bot-tom parts of the mirror plate. However, similar to rotation angleα , the top forces only appear when the rotation angle β¿0.06 ra-dians. For an element of dx on the top layer of the mirror plate(8), the electrostatic flux from sidewall 13 is written as;

Ee13t =V_AB

(27)

where

_AB= AC.(π/2−β ) =

ADsin(π/2−β )

(π/2−β )

= (Le

2−X)

π/2−β

sin(π/2−β )

(28)

FIGURE 8: The mirror rotated about Y axis by β , and the sidewalls 13and 33 projected on the XZ plane.

Substituting X = xcosβ for a body fixed point A, the fluxequation 27 is written as;

Ee13t =V.sin(π/2−β )

(Le2 − x.cosβ )(π/2−β )

(29)

The electrostatic pressure is

P =ε0E2

2(30)

Therefore the electrostatic force on on element with dimensionsdx and dy is

dF =ε0E2dxdy

2(31)

The electrostatic force in the Z direction and the torqueabout the Y axis from sidewall e13 on top surface of the mirrorplate are then obtained.

S5 =ε0.sin2(π/2−β )

2

Fze13t =S5V 2

1(π/2−β )2 .

∫ y16y15

∫ x16x15

{ 1Le2 − x.cosβ

}2.dxdy

Tye13t =−S5V 2

1(π/2−β )2 .

∫ y16y15

∫ x16x15

{ 1Le2 − x.cosβ

}2.xdxdy

(32)

For an element of dx at the bottom surface of the mirror plate(8), the electrostatic flux from sidewall 13 is written as;

Ee13b =V1_

AB′(33)

where

_

AB′= AC.(π/2+β ) =AD

sin(π/2−β )(π/2+β )

= (Le

2−X)

π/2+β

sin(π/2−β )

(34)

Substituting X = xcosβ for a body fixed point, the fluxequation (33) is written as;

Ee13b =V1 sin(π/2−β )

(Le2 − x.cosβ )(π/2+β )

(35)


The force and torque equations are then derived

Fze13b =−S5V 2

1(π/2+β )2 .

∫ y18y17

∫ x18x17

{ 1Le2 − x.cosβ

}2.dxdy

Tye13b =S5V 2

1(π/2+β )2 .

∫ y18y17

∫ x18x17

{ 1Le2 − x.cosβ

}2.xdxdy

(36)

Electrostatic forces from sidewall 33 also changes the rota-tion angle β . The force and torque on the mirror plate from side-wall 33 can be derived similarly to equation (32) with differentboundary for the integral.

Fze33 =−S5V 2

3(π/2−β )2 .

∫ y20y19

∫ x20x19

{ 1Le2 − x.cosβ

}2.dxdy

Tye33 =−S5V 2

3(π/2−β )2 .

∫ y20y19

∫ x20x19

{ 1Le2 − x.cosβ

}2.xdxdy

(37)

The forces and torques generated by bottom electrode ofquadrants 1 or 2 to change β angle (9) are

S6 =ε0.V 2

12

sin2(β )

β 2

Tyeb1 = S6.∫ y22

y21

∫ x22x21

(1

g− x.sin(β )

)2

.xdxdy

Fzeb1b =−S6.∫ y22

y21

∫ x22x21

(1

g− xsin(β )

)2

.dxdy

(38)

FIGURE 9: Bottom electrode effect for β scanning angle

Appendix B: Integral Boundaries

y1 =Le

2cos(α0)− Le

2tan(α0) y2 =

lm2

x1 =w f

2x2 =

lm2

y3 =Le

2cos(α0)−g+

Le2

tan(α0) y4 =lm2

x3 =w f

2x4 =

lm2

y5 =w f

2y6 =

lm2

x5 =w f

2x6 =

lm2

y7 =Lgl2−gwy y8 =

Le2cos(α0)

+Le2

tan(α0)

x7 =w f

2x8 =

Le2

+ ts

y9 =Lgl2−gwy y10 =

Le2cos(α0)

+g− Le2

tan(α0)

x9 =w f

2x10 =

Le2

+ ts

y11 =Lgl2−gwy y12 =

Le2cos(α0)

+Le2

tan(α0)+g

x11 =w f

2x12 =

Le2

+ ts

y13 =w f

2y14 =

lm2

x13 =w f

2x14 =

lm2

x15 =Le

2cos(α0)− Le

2tan(α0) x16 =

lm2

y15 =w f

2y16 =

lm2

x17 =−Le

2+g x18 =

lm2

y17 =w f

2y18 =

lm2

x19 =Le2−g x20 =

lm2

y19 =w f

2y20 =

lm2

x21 =w f

2x22 =

lm2

y21 =w f

2y22 =

lm2


static modeling of a bi-axial micro-mirror with …

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