Large displacement parallel plate electrostatic actuator with saturation type characteristic

Sensors and Actuators A 130–131 (2006) 497–512

Large displacement parallel plate electrostatic actuator with saturation type characteristic Slava Krylov a,∗ , Yacov Bernstein b,1 a

Department of Solid Mechanics Materials and Systems, Tel Aviv University, Ramat Aviv 69978 Tel Aviv, Israel b Teraop Ltd., Israel Received 14 June 2005; received in revised form 4 November 2005; accepted 25 November 2005 Available online 19 January 2006

Abstract We report on a long travel electrostatic actuator operated by parallel plate electrodes. The actuator architecture is based on a folded suspension composed of initially curved beams with an actuating force applied along the beams. The straightening of the beams, which are used simultaneously as motion amplifiers and suspension elements, is transformed into the long stroke lateral motion of the actuator and much smaller axial displacement of the beams’ ends. Small distances between electrodes are therefore possible, improving actuator effectiveness. Steep increase in the stiffness of the straightened beams improves actuator stability and leads to the saturation type voltage–displacement dependence. In the vicinity of the inflection point the voltage–displacement dependence is practically linear and therefore of importance for applications. The exact extensible elastica and approximate reduced order models of the actuator are used for the stability analysis. For some range of parameters multistability of the actuator is possible. Actuators of varying configurations were fabricated using SOI wafers and deep reactive ion etching (DRIE). The comparison between the experimental and model results reveals good agreement between the two. © 2005 Elsevier B.V. All rights reserved. Keywords: Electrostatic actuator; Curved beam; Parallel plate electrode; Pull-in; Multistability

1. Introduction Long motion electrostatic actuators are important components of various microsystems. As an example, we can mention variable optical attenuators [1,2], which, due to the high sensitivity of the optical setup, should utilize long stroke actuators that exhibit preferably linear voltage–displacement dependence and low sensitivity to mechanical shock and vibrations. The lateral comb drive actuator [3] is one of the most widely used components in optical devices where long stroke areaefficient actuators are required. Displacements of comb drive actuators are limited by side pull-in instability [4]. In order to enlarge the stable displacement, prebent [5] or tilted [6] suspensions were suggested. It should be noted that although the force, which is generated by a comb drive transducer, is independent ∗

Corresponding author. Tel.: +972 36405930; fax: +972 36407617. E-mail address: [email protected] (S. Krylov). 1 Present address: Center of Nanotechnology, Ben-Gurion University of the Negev, Beer-Sheva, Israel. 0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.11.063

of displacement, it is non-linear (quadratic) with respect to voltage, resulting in non-linear voltage–displacement dependence. In order to achieve linear voltage–displacement dependence, linearly engaging [5] or shaped [7] comb teeth were suggested. Motion amplifiers and compliant mechanisms (see [8–10], and references therein) allow for very large motions but require higher actuating forces since amplification of the displacement is usually accompanied by a reduction in force. In addition, actuators incorporating motion amplifiers or mechanical leverage are usually assembled of additional structural elements (e.g. links and hinges [8,9,11]), which increase actuator foot print, demonstrate complex mechanical behavior and operate at relatively low natural frequencies. Parallel plate electrostatic actuators are widely used due to their simplicity and their ability to generate high forces. However, parallel plate actuators have limited range of stable displacements due to pull-in instability. In addition, they exhibit a highly non-linear dependence between the actuating force and displacement. Various approaches were exploited in order to enlarge the stable range of travel of parallel plate actuators. These include the use of curved electrodes [12], leveraged

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Fig. 1. Operation principle of the actuator: (a) configuration before the application of the actuation force; (b) after the application of the force.

bending and strain stiffening [13] as well as feedback control algorithms [14]. It should be noted that the stretching force arising in a double clamped beam actuated by an electrostatic force results in an increase of the stable stroke of the actuator as compared to the simplest spring capacitor model, usually at the expense of higher actuation voltages [15–17]. The actuator design presented in this work is perhaps the simplest structure for providing motion amplification combined with an area-efficiency typical to the traditional comb drive actuator. It is based on a folded suspension composed of initially curved beams with an actuating force, produced by the parallel plate electrodes, applied along the beams, Fig. 1. The straightening of the beams, which are used simultaneously as motion amplifiers and suspension elements thereby reducing the actuator dimensions and mass, is transformed into the long stroke lateral motion of the actuator. In contrast to traditional comb drive actuators used in conjunction with motion amplifiers (e.g. [8,9]), the actuator does not contain any additional rigid or deformable elements like links or hinges. The actuator benefits from the steep increase in the mechanical stiffness of the straightened beam which improves actuator stability and leads to the voltage–displacement dependence of the saturation type. The goal of the present work is to demonstrate, both theoretically and experimentally, the feasibility of actuation based on the straightening of an initially curved beam. We investigate the main features of this type of actuation and the influence of various design parameters. Additionally, we also investigate the actuator’s stability properties. In the next section, a generic non-linear stiffness element, namely an initially curved beam with forces applied to its end, is introduced followed by the development of the exact and the reduced order models of the beam. Next, these models are used for the construction of the electromechanical model of the actuator as well as for the investigation of its stability. Note that the models of the curved beam and of compliant suspension developed in this work are of general interest and can be used by designers of large deflection actuators and other micro devices incorporating geometrically non-linear flexible suspension elements (e.g. see [18]). The description and analysis of experimental results is followed by the final section which summarizes the main conclusions, discusses possible applications and offers recommendations for future work.

Fig. 2. Configuration of the curved beam: (a) initial configuration; (b) after deformation.

2. Model of the curved beam In order to evaluate the range of the actuator’s design parameters, as well as to provide better insight into its behavior, a model of the actuator was built using various approaches. The suspension element is modelled as a curved beam loaded by a force Px applied along the x-axis, as well as lateral force, Py (Fig. 2). The elevation of the beam in its initial configuration is H while h denotes the elevation in the deformed configuration. The distance between the beam ends is L and l in the initial and deformed configurations, respectively. 2.1. Extensible elastica model The deformation of the beam is described using extensible elastica theory. Note that accounting for extensibility can play an important role in the adequate modelling of the actuator stiffness and stability especially at high deformations when the beam is almost straightened. It is assumed that Euler–Bernoulli’s hypothesis is valid and that the deformed central axis always remains on the xy-plane. Each point of the central axis of the beam in the undeformed configuration is identified by its material coordinates X and Y (see Fig. 3). The following geometric relations are valid (we follow the general lines of description in [19]): λ=

ds dS

dX = cos(Θ), dS dx = cos(θ), ds dx = λ cos(θ), dS

(1) dY = sin(Θ) dS dy = sin(θ) ds dy = λ sin(θ) dS

(2) (3) (4)

Here, S and s are the arc length of the beam in the undeformed and deformed configurations, respectively, λ is stretching of the central axis, Θ the angle made by the tangent to the undeformed beam axis with a X-direction, and θ the angle made by the tangent to the deformed beam axis with a x-direction. The strains  and

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499

form: Px Py cos(θ) + sin(θ) (11) EA EA Differentiating the second equilibrium Eq. (10) with respect to S, using constitutive Eq. (8) and geometric relations Eq. (4) combined with Eq. (11) we obtain the non-linear second order ordinary differential equation: λ=1+

EI

(Px2 − Py2 ) d2 θ − (P sin(θ) − P cos(θ)) − sin(2θ) x y dS 2 2EA

+

θ  − π2 (Pˆ x sin(θ) − Pˆ y cos(θ)) −

µ are defined as follows: (5)

dθ dΘ µ= − (6) dS dS The resultant axial tension force N and bending moment M are expressed in terms of the strains using constitutive relations: N = EA

(7)

M = EIµ

(8)

Here, E is Young’s modulus of the beam material, A and I are the cross-sectional area and moment of inertia, respectively. Taking into account the fact that the bending moment is equal to zero at the inflection point O located at one-half of the length x0 = l/2, y0 = h/2, the equilibrium equations are as follows (see Fig. 4): N = Px cos(θ) + Py sin(θ) M = −Px (y0 − y) + Py (x0 − x)

(9) (10)

Using the equilibrium Eq. (9), constitutive Eq. (7) and strain definition Eq. (5) we express the stretching of the beam in the

π4 2 ˆ 2 ˆ 2 rˆ (Px − Py ) sin(2θ) 2

+ π4 rˆ 2 Pˆ x Pˆ y cos(2θ) = Θ

(13)

Here, Px Pˆ x = , PE rˆ =

r , L

Py Pˆ y = , PE
I r= A

PE =

π2 EI , L2 (14)

In accordance with Eq. (14), forces Px and Py are normalized by the Euler-buckling force PE while the arclength in the initial configuration S and gyration radius r are normalized by the distance L between the beam ends in its initial configuration. Hereafter, () = d/dSˆ denotes the derivatives with respect to the non-dimensional arc length Sˆ = S/L. Solving Eq. (13), we can obtain the dependence between the forces Px and Py applied at the end of the beam and the lateral deflection of the end point, H − h. Note that in the case of the initially straight beam or beam of constant initial curvature, Θ = 0, Eq. (12) is autonomous (does not explicitly contain S) and can be solved in terms of elliptic integrals (see [20,21]). In our case, due to the presence of non-homogeneous term, the solution of Eq. (12) cannot be obtained in terms of elliptic integrals. In order to solve Eq. (13), the initial shape of the beam should first be specified. Consider the beam of the initial shape such that ˆ L ˆ ˆ S − S) Θ = Θ0 4S(

Fig. 4. Equilibrium of the beam element.

(12)

For convenience we re-write Eq. (12) in non-dimensional form:

Fig. 3. Geometry of the deformation.

=λ−1

Px Py d2 Θ cos(2θ) = EI 2 EA dS

(15)

ˆ 2 is the slope at the inflection point. On the other where Θ0 L S hand, it is more convenient to specify the initial shape of the beam in the form Y = Y(X). In order to obtain the dependence Y = Y(X) corresponding to Eq. (15), the values of Θ0 and the total non-dimensional length of the beam in its initial configuration ˆ S = LS /L should be obtained. Using geometric relations Eq. L ˆ S = LS /L are obtained from the system of the (2), Θ0 and L following two non-linear equations:  Lˆ S cos(Θ) dSˆ = 1 (16) 0

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ˆ = N/PE and the axial force N is related to the angle θ Here, N ˆ 0 is the and applied forces Pˆ x and Pˆ y by equilibrium Eq. (9); N value of the non-dimensional axial force at the inflection point. Integrating Eq. (20) with respect to the initial arclength Sˆ and satisfying second boundary condition θ(0) = 0 we obtain:  θ0 dθ ˆS = L (21)  0 θ (θ)

Fig. 5. Elongation of the beam given by the approximation Eq. (18). Dots correspond to exact solution.



where θ  is given by Eq. (20). Using Eq. (21) we can find the relation between the applied forces Pˆ x and Pˆ y and the slope at the inflection point θ 0 for given initial shape defined by Θ0 , Eq. (15). When θ 0 is found, a displacement of the end point of the beam as well as lateral deflection, can be found using following expressions (see Eq. (4)):  Lˆ S /2 ˆ x = ˆl − 1 = 2 ∆ λ cos(θ) dSˆ − 1 (22) 0

ˆS L

ˆ sin(Θ) dSˆ = H

(17)

0

ˆ and Yˆ (S) ˆ are evaluated using inteˆ = H/L. Then, X( ˆ S) where H ˆ S replaced grals similar to Eqs. (16) and (17) with upper limit L ˆ Note that the initial shape given by the by variable upper limit S. ˆ − cos(πX)]/2 ˆ expression Yˆ = H[1 can serve as a good approximation of the shape corresponding to Eq. (15) even for high ˆ of the beam. elevations H Expanding integrands in Eqs. (16) and (17) into Taylor series up to quadratic order in terms of Θ0 , we obtain the approximation for the initial length of the curved beam:  
ˆ 1 12 3H ˆS ≈ ˆ2 , Θ0 ≈ 1+ 1+ H (18) L ˆ3 2 5 2L S

ˆ MAX ˆ S − 1 is actually the maximal =L Note that the value ∆ x displacement of the beam end which can be achieved if the beam axis, considered to be inextensible, is completely straightened. ˆ MAX The value ∆ corresponding to approximation Eq. (18) is x shown in Fig. 5. Dots correspond to the expression calculated ˆ S . Note also that for H ˆ  1, from Eq. (18) using exact value of L MAX 2 2 ˆ ˆ ˆ ˆ ˆ we have ∆x ≈ 3H /(5LS ) ≈ 3H /5  H. The advantage of the shape defined by Eq. (15) is that the initial curvature varies linearly along the beam and Θ = −8Θ0 = const. As a result, Eq. (13) is of autonomous form and can be integrated once:

ˆy =H ˆ − hˆ = H ˆ −2 ∆



ˆ S /2 L

λ sin(θ) dSˆ

(23)

0

or, taking into account Eq. (20):  θ0 λ cos(θ) ˆx =2 dθ − 1 ∆ θ  (θ) 0  θ0 λ sin(θ) ˆ ˆ dθ ∆y = H − 2 θ  (θ) 0

(24) (25)

Here (see Eq. (11)): λ = 1 + π2 rˆ 2 (Pˆ x cos(θ) + Pˆ y sin(θ))

(26)

and the symmetry of the displacements relative to the inflection point is exploited. Note that integrals appearing in Eqs. (21), (24) and (25) cannot be reduced to elliptic integrals. Special quadrature formulas were used in order to handle weak singularities appearing in these integrals [22]. ˜x = Non-dimensional axial displacement of the beam’s end ∆ ˜ ∆x /∆MAX and non-dimensional lateral deflection ∆ = ∆ y /H, y x both normalized by the values corresponding to the completely straightened inextensible beam, are shown in Fig. 6 for differing ˆ and gyration radius rˆ and for the case of zero lateral values of H force Pˆ y = 0. Non-dimensional data correspond to the beam length L = 1000 ␮m and cross-section 3 × 30 ␮m (ˆr = 0.87 × 10−3 ) and 5 × 30 ␮m (ˆr = 1.14 × 10−3 ). For comparison, we also present the result for a beam with an inextensible axis (the

1 2 π4 2 ˆ 2 ˆ 2 π4 2 ˆ ˆ (θ ) = −π2 (Pˆ x cos(θ) + Pˆ y sin(θ)) − rˆ (Px − Py ) cos(2θ) + rˆ Px Py sin(2θ) − 8Θ0 θ + C(19) 2 4 2 The integration constant C is found from the condition that at the ˆ S /2) = 0. This condition inflection point the moment is zero µ(L is equivalent to θ  = 0 at θ = θ 0 since, in accordance with Eq. (15), Θ = 0 at the inflection point. Substituting the value of the integration constant into Eq. (19), obtain:    π2 2 ˆ  2 ˆ ˆ ˆ θ (θ) = 2π (N0 − N) 1 + rˆ (N0 + N) + 16Θ0 (θ0 −θ) 2 (20)

case of inextensible axis can be obtained formally from Eqs. (21), (24) and (25) by setting rˆ = 0). One observes that while the influence of the extensibility of the beam axis on the lateral ˜ y is minor, it has strong influence on deflection of the beam ∆ ˜ x , which in the case of the axial displacement of the beam, ∆ the beam with an extensible axis can be larger than unity. The influence of the extensibility is especially pronounced in the case ˆ  1 as shown in Fig. 6(a). One observes of shallow beams H also that the non-linearity of the beam under tension in the xˆ

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˜ x = ∆x /(LS − L) (a) and lateral deflection ∆ ˜ y = ∆/H (b) of the beam for the case of zero Fig. 6. Dependence of the axial end force Pˆ x on the axial displacement ∆ ˆ and gyration radius rˆ . Continuous curve corresponds to the inextensible model rˆ = 0. lateral force Pˆ y = 0 and for differing initial elevations H

direction is of stiffening type, Fig. 6(a). In contrast, the nonlinearity of the beam under compression (Pˆ x < 0 in Fig. 6(a)) is of the softening type. The significant difference in the beam stiffness under tension and compression suggests that the beam can be viewed, in a sense, as a one directional (e.g. contact type) constraint. The exact solution presented in this section will be used for the validation of the approximate reduced order model of the curved beam. 2.2. Reduced order model Although the exact solution for the curved beam under end forces can be built using extensible elastica theory, the dependence between the applied forces and deflection is given by the exact solution in an implicit form which complicates the analysis. In order to simplify calculations and provide better insight into non-linear behavior of the beam, a reduced order model of the beam is built using the Rayleigh–Ritz method. Potential energy of the beam is given by the expression (see [19,20]):  LS  LS 1 1 U = EI µ2 dS + EA 2 dS − Px ∆x + Py ∆y 2 2 0 0 (27) which for convenience is re-written in non-dimensional form using notations given by Eq. (14):  Lˆ S /2  Lˆ S /2 1 1   2 ˆ ˆ (θ − Θ ) dS + 2 2 U= 2 2 dSˆ π 0 π rˆ 0 ˆ x + Pˆ y ∆ ˆy − Pˆ x ∆

(28)

We approximate the deformed shape of the beam by the same function which defines the initial shape of the beam Eq. (15), namely ˆ L ˆ ˆ ≈ θ0 4S( ˆ S − S) θ(S)

One can see that this function satisfies the boundary conditions of Eq. (13), namely θ(0) = 0, θ  (LS /2) = 0. Then, in accordance with the Rayleigh–Ritz method, for given forces Pˆ x and Pˆ y , the value of the angle at the inflection point θ 0 can be found from the condition: ∂Uˆ =0 (31) ∂θ0 Note that after substituting Eqs. (15) and (30) into Eq. (29) the integrals appearing in the expression for the potential energy can be calculated exactly (commercially available computer algebra package was used for this purpose), expressed in terms of a special function (Fresnel Sine and Cosine integrals) without assuming that θ 0 is small. Expanding the resulting expression into Taylor series under condition θ 0  1 and solving in terms of θ 0 yields θ0 ≈

Θ0 + Pˆ y (π2 /8)(1 + π2 rˆ 2 Pˆ x ) ˆ 2 (π2 /10)[Pˆ x + π2 rˆ 2 (Pˆ x2 − Pˆ y2 )] 1+L

(32)

S

The displacements of the end point of the beam can be expressed in terms of Fresnel integrals using Eqs. (22), (23), where θ is given by Eq. (30) and θ 0 is given by Eq. (32). The Taylor series expansion of the resulting expressions up to quadratic order under condition Θ0  1 gives:

ˆ x and ∆ ˆy Here, Uˆ = U/(PE L) and end point displacements ∆ ˆx ≈L ˆ 3 π2 rˆ 2 Pˆ y θ0 ˆS −1+L ˆ S π2 rˆ 2 Pˆ x + 2 L ∆ are given by Eqs. (22) and (23), respectively. Using Eqs. (5), (6) 3 S and (11) and Eq. (21), the potential energy of the beam, Eq. 4 ˆ5 2 2ˆ 2 (28), can be written in the form (see [20] for the case of initially − L S (1 + 2π rˆ Px )θ0 15 straight beam):   Lˆ S /2     Lˆ S /2  Lˆ S /2 1   2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ U= 2 (θ − Θ ) dS − Px 2 cos(θ) dS − 1 + Py H − 2 sin(θ) dS π 0 0 0  Lˆ S /2 2 + π2 rˆ 2 (Pˆ x cos(θ) + Pˆ y sin(θ)) dSˆ 0

(30)

(33)

(29)

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˜ x = ∆x /(LS − L) (a) and lateral deflection ∆ ˜ y = ∆y /H (b) of the beam described by an Fig. 7. Dependence of the axial end force Pˆ x on the axial displacement ∆ extensible elastica (continuous curves) and reduced order (dashed curves) models for differing lateral forces Pˆ y = αPˆ x . Initial elevations and gyration radius are ˆ = 0.1 and rˆ = 0.87 × 10−3 , respectively. H

ˆy ≈H ˆ − 2L ˆ 3 (1 + π2 rˆ 2 Pˆ x )θ0 − 8 L ˆ 5 π2 rˆ 2 Pˆ y θ02 ∆ 3 S 15 S

(34)

The expressions for the initially straight beam can be obtained ˆ S = 1, H ˆ = 0. by setting Θ0 = 0, L The comparison of the axial and lateral deflections given by the exact solution Eqs. (21)–(26) and reduced order model Eqs. (32)–(34) is presented in Fig. 7 for the case of lateral force increasing proportionally to the axial force in such a manner that Pˆ y /Pˆ x = α = const. One observes that the application of the lateral force has a very strong influence on the axial forcedeflection dependence and can be used for the tuning of the characteristic of the actuator suspension. One observes also that ˆ the reduced order even for relatively large initial elevations H model describes the deflection of the beam with high accuracy. ˜ y = 0.8 the relative error in ˆ = 0.5 and ∆ For example, for H lateral displacement is 1.5%. Note that Eq. (34) combined with Eq. (32) can be used also for the calculation of the relation between the axial Pˆ x and latˆ y. eral Pˆ y forces for a given value of the lateral deflection ∆ In the simplest case of an inextensible beam and in the initial ˆ y = 0, we have: configuration ∆ ˆ 4H Pˆ Pˆ y = ˆS x 5L

(35)

In the case that only axial force is applied such that Pˆ y = 0 we obtain: Θ0 θ0 ≈ (36) 2 2 ˆ 1 + LS (π /10)Pˆ x (1 + π2 rˆ 2 Pˆ x ) ˆx ≈L ˆS −1+L ˆ S π2 rˆ 2 Pˆ x − 4 L ˆ 5 (1 + 2π2 rˆ 2 Pˆ x )θ02 ∆ 15 S

(37)

ˆy ≈H ˆ 3 (1 + π2 r 2 Pˆ x )θ0 ˆ − 2L ∆ 3 S

(38)

One observes that in the case Pˆ y = 0 the beam cannot be comˆy =H ˆ in Eq. (38)) pletely straightened (which corresponds to ∆ by finite axial force Pˆ x since the second term in Eq. (38) is always positive. However, the axial displacement of the end of

ˆ MAX due to the extenthe extensible beam can be larger than ∆ x sibility of the beam’s axis which manifests itself in the positive third term in Eq. (37) (see also Fig. 6(a)). One observes also from Eq. (37) that the influence of extensibility becomes more pronounced for less slender (larger rˆ ) and more shallow (smaller Θ0 ) beams. It should be noted that typical suspension elements used in large displacement micro devices are usually realized as very slender beams. Taking into account that for slender beams rˆ  1 and using Eq. (18), we can write Eqs. (37) and (38) in the form:   1 ˆ x ∼ (L ˆ S − 1) 1 − ˆ S π2 rˆ 2 Pˆ x ∆ +L 2 2 2 ˆ ˆ (1 + L (π /10)Px ) S

(ˆr Pˆ x  1) 2

 ˆy ∼H ˆ ∆

1 1− 2 ˆ 1 + LS (π2 /10)Pˆ x

(39)  (ˆr 2 Pˆ x  1)

(40)

One observes that the first term in Eq. (39)) corresponds to the model of a curved beam with inextensible axis while the second term is actually elongation of a straight rod under axial tension by force Pˆ x . For small axial forces Pˆ x  1 the second term is negligible and deformation of the beam is predominantly due to the bending. Under large applied force Pˆ x  1 (but rˆ 2 Pˆ x  1) when the beam is almost completely straightened (i.e. ∆x ≥ ∆MAX but ∆y < H) the contribution of the first term in Eq. (39) x ˆ MAX . At this stage to the elongation is constant and equal to ∆ x the displacement of the end of the beam is predominantly due to the extensibility of its axis (second term in Eq. (39)). Note that this result can also be obtained by setting θ 0 = 0 in Eq. (37) or θ = 0 directly in an exact solution Eq. (22). We can conclude that an initially curved beam with an extensible axis can be viewed approximately as a serial connection of two elements—a curved beam with an inextensible axis and a straight extensible beam. Note also, that, as follows from Eqs. (39) and (40) the influence of the extensibility of the beam’s axis on the axial deflection ˆ x is more pronounced than its influence on the lateral deflection ∆ ˆ y (see also Fig. 6(a)). ∆

S. Krylov, Y. Bernstein / Sensors and Actuators A 130–131 (2006) 497–512

One should mention that one of the advantages of the representation Eq. (39) is that it can be reduced to a cubic equation in terms of Pˆ x

(41)

and solved analytically in order to obtain an explicit dependence ˆ x and applied force Pˆ x . In the case of between the elongation ∆ the beam with inextensible axis the second term in Eq. (39) vanishes and the elongation-force dependence takes the form:   1 10 −1 (ˆr = 0) (42) Pˆ x = 2 2  ˆ ˜x π L 1−∆ S ˜ x = ∆x /∆MAX where ∆ . x In closing this section, a remark should be made about the level of stresses arising in the beam during the straightening. The stresses were calculated as a combination of maximal bending b and tension σ t stress in the following way: σxx xx b t σxx = σxx + σxx ,

b σxx =

Ma , 2I

t σxx =

N A

in initially more shallow beams than in beams with higher initial elevation for the same displacement of the beam’s end. 3. Electro mechanical model

ˆ MAX ˆ x ) + 20r 2 )Pˆ x2 ˆ 5S π6 rˆ 2 Pˆ x3 + π4 L ˆ 3S (L ˆ S (∆ L −∆ x ˆ MAX ˆ x ) + 5ˆr 2 )Pˆ x − 100∆ ˆx =0 ˆ S (L ˆ S (∆ + 20π2 L −∆ x

503

(43)

First, we consider a single initially curved beam under electrostatic force Px applied at the beam’s end and generated by a parallel plate electrode. The goal of the analysis is to receive voltage-deflection dependence and investigate the stability properties of the beam for various initial configurations. Note that some of the phenomena presented below are not described in literature and can be useful for implementation in various electrostatically actuated micro devices incorporating deformable mechanical elements with stiffening type non-linearities. 3.1. Curved beam under electrostatic force applied to the beam’s end The electrostatic force Px applied to the end of the beam is calculated using the parallel capacitor formula: Pe =

1 ε0 bLe V 2 2 (d0 − ∆x )2

(46)

where a is the width of the beam. Using constitutive relation for the beam Eq. (8) combined with the expressions for the strain (6) and the approximation for the slope Eq. (30) as well as equilibrium Eq. (9) we obtain the bending and tensile stresses (written here for convenience in terms of non-dimensional parameters) at the root Sˆ = 0 of the beam in terms of the applied force Pˆ x :

Here, b is the height of the beam (height of the device layer), Le the length of the electrode, ε0 = 8.854 × 10−12 F/m is the permittivity of the free space, V voltage and d0 the distance between the stationary and movable electrodes (electrostatic gap). For convenience we introduce the voltage parameter:

b ˆ S (Θ0 − θ0 )E σxx = 2ˆaL

(44)

β=

t σxx = π2 rˆ 2 Pˆ x E

(45)

Here, aˆ = a/L and θ0 (Pˆ x ) is given by Eq. (36). The estimation of maximal stresses at the root of the 1000 ␮m × 30 ␮m × 3 ␮m beam are shown in Fig. 8 for different initial elevations. One observes that when the beam is straightened the stresses remain relatively low (below 200 MPa). Higher stresses are developed

Fig. 8. Maximal axial stress σ xx at the root of the beam S = 0 for different initial configurations of the beam. Gyration radius rˆ = 0.87 × 10−3 corresponds to the 1000 ␮m × 30 ␮m × 3 ␮m beam.

ε0 bLe LS V 2 2EAd03

(47)

such that the non-dimensional force Pˆ x = Px /PE can be expressed in the form: Pˆ x =

β dˆ 0 2 2 ˆ LS π rˆ (1 − ∆ ˆ x /dˆ 0 )2

(48)

Substitution of Eq. (36) into Eq. (37) and using the expression of the axial force given by Eq. (48) leads to the fifth order polynomial equation in terms of the voltage parameter β. Solution of this equation gives the voltage parameter for given values ˆ x. of the displacement ∆ Results of the calculations are presented in Figs. 9–11 where the dependencies between the axial and lateral displacements of the beam’s end and voltage parameter β (bifurcation diagrams) are shown. The axial displacement ∆x of the beam’s end is normalized by the electrostatic gap d0 . The lateral deflection ∆y is normalized by the initial elevation H. Results are presented for differing initial elevations of the beam and differing electrostatic gaps normalized by ∆MAX in such a way that η = d0 /∆MAX . x x All results correspond to the 1000 ␮m × 30 ␮m × 3 ␮m beam. For the sake of definiteness, the length of the electrode is taken to be equal to the length of the beam Le = L. The influence of the initial elevation and electrostatic gap on the character of beam deformation is illustrated in Figs. 9–10. Note that for shallow initial configurations the displacement ˆ MAX ∆ is relatively smaller than for more curved beams. For x

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Fig. 9. Dependence of the voltage parameter β on the normalized axial displacement ∆x /d0 (a) and lateral deflection ∆y /H (b) of the beam under electrostatic force ˆ = 0.01 and rˆ = 0.87 × 10−3 , respectively. Dashed curve corresponds to for differing electrostatic gap η = d0 /(Ls − L). Initial elevations and gyration radius are H ˆ = 0. Dashed–dotted curve corresponds to the inextensible beam rˆ = 0. the straight beam H

Fig. 10. Dependence of the voltage parameter β on the normalized axial displacement ∆x /d0 (a) and lateral deflection ∆y /H (b) of the beam under electrostatic force ˆ = 0.05 and rˆ = 0.87 × 10−3 , respectively. Dashed curve corresponds to for differing electrostatic gap η = d0 /(Ls − L). Initial elevations and gyration radius are H ˆ the straight beam H = 0. Dashed–dotted curve corresponds to the inextensible beam rˆ = 0.

ˆ MAX ˆ = 0.01 we have ∆ example, for H = 6 × 10−5 which gives x ∆MAX = 0.06 ␮m for 1000 ␮m long beam. Since the minimum x space which can be achieved by micro-machining technology is limited, the situation η  1 is more typical for shallow initial

configurations. In this case, the applied force is practically constant during the straightening of the beam and the displacement in the interval ∆MAX < ∆x < d0 , (∆MAX  d0 ) mainly due to x x extensibility (Fig. 9). For higher initial elevations the separation

˜ y = ∆y /H (b) of the beam under electrostatic Fig. 11. Dependence of the voltage parameter β on the normalized axial displacement ∆x /d0 (a) and lateral deflection ∆ ˆ = 0.05 and rˆ = 0.87 × 10−3 , respectively. Dash–dot curve corresponds to force for the electrostatic gap η = d0 /(Ls − L). Initial elevations and gyration radius are H the beam with an inextensible axis.

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into “inextensible” and “extensible” stages is more pronounced (Fig. 10). The fact that in the case η  1 the pull-in instability takes place on the stage when the beam is straightened (see also Eq. (39)) suggests that the pull-in value of the voltage parameter can be calculated using the model of straight beam under electrostatic force applied to its end. The equilibrium equation for such a beam is given by the expression: ˆx = ∆

dˆ 0 β ˆ MAX /dˆ 0 − ∆ ˆ x /dˆ 0 )2 (1 − ∆

(49)

x

and therefore (see [15]): ∆PI x =

1 ), (d0 − ∆MAX x 3

βPI =

4 27

1−

∆MAX x d0

(50)

Voltage parameter-displacement dependencies obtained using Eq. (49) are shown in Figs. 9(a) and 10(a) (dashed lines). Good agreement with results provided by Eq. (37) is observed. The accuracy of the approximation Eq. (49) is improved with an increase of η. It should be noted that the results provided by the approximation Eq. (39) and by Eq. (36) are in excellent agreement and indistinguishable in the graphs. However, the estimation of the location of the pull-in point based on the use of Eq. (49) is much simpler since it does not require the solution of the cubic Eq. (41). With the displacement ∆x in the axial direction calculated, the lateral deflection ∆y can be obtained using Eq. (38) (or Eq. (40)) and Eq. (47). It follows from Figs. 9(b) and 10(b) that the lateral deflection of the beam is stable almost within the entire displacement interval 0 ≤ ∆y < H. The reason is that the lateral motion covers almost the entire interval at the “straightening” stage when the electrostatic force is small and the system is far from the pull-in point. The stable interval of the lateral motion is larger (relative to H) for larger values of the relative electrostatic gap η and higher initial elevations. It follows from Figs. 9(a) and 10(a) that when the relative electrostatic gap η is decreased, the location of the pull-in point is shifted toward the electrode. However, when the electrostatic

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/d0 , i.e. η is of the order of gap becomes comparable with ∆MAX x unity, the non-linearity of the electrostatic force manifests itself not only during the “extensible” stage of the deformation but also during the “straightening” stage. The situation is illustrated in Fig. 11. For larger values of the electrostatic gap (η = 1.3) the pull-in instability is due to the extensibility of the axis. For smaller values of electrostatic gap (η = 1) the pull-in instability takes place at much smaller values of displacements and voltage parameter. In this case, the instability is also observed for the beam with inextensible axis. The pull-in values of the voltage parameter and the displacement can be calculated using the inextensible model with high accuracy (Fig. 11(a)). It follows from Fig. 11(b) that in the case that the instability takes place during the “straightening” stage, the maximally achievable stable lateral deflection is smaller than the initial elevation H. It is interesting to note that due to the interaction between mechanical and electrostatic non-linearities the range of electrostatic gap exists such that both “bending” pull-in and “extensibility” pullin are present. In the case that the pull-in value of the voltage parameter corresponding to the “extensibility” pull-in is higher, multistability of the beam is observed, i.e. two different stable equilibrium states are possible for the same value of the voltage parameter [23]. As follows from Fig. 11(a) even in the case of small electrostatic gap the location of the second pull-in point calculated using the straight beam model is of reasonable accuracy. In conclusion, a remark should be made about the level of voltages necessary for the straightening of the beam and achievement the large stable lateral deflections. The voltage-deflection dependencies for typical dimensions of the 1000 ␮m × 30 ␮m × 3 ␮m beam are shown in Fig. 12. The length of the electrode is taken to be equal to the distance between the ends of the beam in the undeformed configuration Le = L (see Fig. 2). Since in the actuator considered in the present work is fabricated using SOI wafers and DRIE, the electrostatic gaps are taken to be within the typical range of a few micrometers. The influence of the electrostatic gap on the deflection of a beam with relatively high elevation of 50 ␮m is illustrated in Fig. 12(a). One observes that the beam can be straightened and

Fig. 12. Dependence of the actuation voltage on the lateral deflections ∆y of the beam under electrostatic force applied to its end (a) for the electrostatic gap d0 and H = 50 ␮m (b) for different initial elevations H and electrostatic gap d0 = 1.5 ␮m. Gyration radius rˆ = 0.87 × 10−3 corresponds to 1000 ␮m × 30 ␮m × 3 ␮m beam. The length of the parallel plate electrode is Le = L = 1000 ␮m.

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Pˆ x(1) − Pˆ x(2) = 0 Pˆ y(1) + Pˆ y(2) = 0 ˆ (1) ˆ (3) ˆ (2) ˆ (4) −∆ x + ∆x = ∆x − ∆x ˆ (1) ˆ (3) ˆ (2) ˆ (4) ∆ y + ∆y = ∆y + ∆y Here, positive directions of the non-dimensional relative disˆ (i) ˆ (i) placements ∆ x and ∆y between the ends of the ith beam as well as positive directions of the end forces Pˆ x(i) and Pˆ y(i) are chosen to be consistent with notation of Fig. 2 and Eqs. (22) and (23). Non-dimensional electrostatic forces Fˆ R and Fˆ L acting on the right and left trusses, respectively, are given by the expressions: Fig. 13. Model of the suspension as an assembly of rigid and flexible elements.

lateral deflections of tens of microns can be achieved by voltages of tens of volts. In the case of more shallow beams and typical electrostatic gaps of order of a few micrometers, higher voltages are required in order to reach the deflection comparable with the initial elevation of the beam (Fig. 12(b)). It is possible to argue that in this case the deflection is limited more by the voltage limitations (or electrical break down) rather than by pull-in instability. On the other hand, in the case of more shallow beams, the interval of stable deflection—as related to the initial elevation—is higher. 3.2. Stability analysis of the actuator The electromechanical model of a curved beam under electrostatic force applied to its end considered in the previous section permits, in principle, the calculation of the displacement of the actuator of the configuration presented in Fig. 1. To this end, the total length of the parallel plate electrode should be subdivided by the number of beams to get the length of the electrode per each beam. Then the displacement of the shutter is calculated as twice deflection of the beam. However, this approach cannot describe the post-buckling behavior of the actuator. In addition, it cannot predict the influence of imperfections, which are always present due to fabrication process, on the actuator stability. The actuator is modelled as an assembly of a rigid shutter and truss and flexible suspension elements (beams), Fig. 13. It is assumed that rotational motion of the truss and of the shutter around z-axis is precluded. Equilibrium and compatibility equations of the assembly have the form: Pˆ x(1) + Pˆ x(3) = Fˆ R −Pˆ y(1) + Pˆ y(3) = 0 Pˆ x(2) + Pˆ x(4) = Fˆ L Pˆ y(2) + Pˆ y(4) = 0

(51)

Fˆ R =

β dˆ 0 ˆ S π2 rˆ 2 (1 − δ/d − uˆ /dˆ )2 L 0 R 0

(52)

Fˆ L =

β dˆ 0 ˆ S π2 rˆ 2 (1 + δ/d − uˆ /dˆ )2 L 0 L 0

(53)

ˆ (3) ˆ (4) ˆL = ∆ where uˆ R = ∆ x and u x are the displacements of the right and left trusses, respectively, in the directions toward the electrodes. The imperfection δ is introduced as deviation of the electrostatic gap from its nominal value. Using the expressions ˆ (i) ˆ (i) for end displacements ∆ x and ∆y in terms of forces Eqs. (33) and (34) in view of Eq. (32) the system (51) of eight non-linear equations with respect to end forces is solved numerically and end forces Pˆ x(i) , Pˆ y(i) , i = 1, . . ., 4 are found. Next, the displacements of the shutter u, ˆ υˆ in x and y directions, respectively (see Fig. 13) are calculated. This procedure permits the evaluation of stable equilibrium states of the actuator. In order to describe unstable equilibrium states, displacement control loading is used in calculations instead of force control. The system of Eqs. (51) is modified in such a way that the first ˆ (3) of Eqs. (51) is replaced by the equation uˆ R = ∆ x , where the displacement of the right truss is viewed now as a parameter. The first of Eqs. (51) in its turn combined with Eq. (52) is used in order to express the voltage parameter β in terms of end forces and of right truss displacement. Results of the calculations are shown in Figs. 14–16. One observes that initial imperfections of the gap between electrodes have a strong influence on the stability of the actuator and its post-buckling behavior. Even for very small imperfections the instability takes place at significantly smaller values of the voltage parameter than predicted by a single beam model. Calculations show that for relatively shallow beams or large distance between electrodes, the instability is associated with the extensibility of the beams and second “extensibility related” pull-in. For deeper beams, or, alternatively, smaller distance between electrodes, the stable stroke of the actuator is limited by the pull-in associated with the bending compliance of the suspension elements. Similarly to the case of a single beam, the stable displacement limited by the “bending related” pull-in is much smaller than in the case of “extensibility related” pull-in (Figs. 14 and 15). In both cases, with decreased imperfections

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Fig. 14. Equilibrium paths for different relative electrostatic gaps η = d0 /(Ls − L) and different imperfections δd0 . Axial u/d0 (a) and lateral v/H (b) displacements ˆ = 0.05, gyration radius rˆ = 0.87 × 10−3 corresponds to the 1000 (m × 30 (m × 3 (m beam. of the shutter are shown. Initial elevation is H

Fig. 15. Equilibrium paths on the (u/d0 ) − (v/H) plane for different relative electrostatic gaps η = d0 /(Ls − L) and different imperfections δd0 . Initial ˆ = 0.05, gyration radius rˆ = 0.87 × 10−3 corresponds to the elevation is H 1000 ␮m × 30 ␮m × 3 ␮m beam.

the voltage parameter corresponding to instability approaches the value predicted by single beam model (Fig. 14). The pull-in values of the actuation voltages and displacements of the single beam can be used for the estimation of the pull-in voltages and stable travel of the actuator from above. Results of calculations suggest that in order to reach the stable travel of the actuator approaching a value twice as large as initial elevation, the distance between the electrodes should be chosen larger than the “critical” value when the “bending related” instability appears. Analysis of post-buckling behavior of the actuator reveals that at the post-buckling stage the distribution of forces between suspension beams as well as displacements of the right and left connecting trusses are essentially not symmetric. The asymmetry is more pronounced for larger initial imperfections, whereas in the case of ideally perfect structure the displacements of the connecting trusses can be described by a single beam model (see Fig. 16). Note that as a result of this asymmetry end forces acting in the directions perpendicular to the beams axis (Py(i) in our notations) should be taken into consideration. It is also interesting to note that for relatively small electrostatic gaps which are however larger than a “critical” value

ˆ = 0.05, gyration radius Fig. 16. Axial uR /d0 , uL /d0 (a) and lateral vR /H, vL /H (b) displacements of the connecting truss of the actuator. Initial elevation is H rˆ = 0.87 × 10−3 corresponds to the 1000 ␮m × 30 ␮m × 3 ␮m beam. Relative electrostatic gap is η = 1.3.

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corresponding to the appearance of “bending related” pull-in, a situation is possible where a displacement of the shutter in x direction is observed that is comparable to the distance between electrodes while the motion of the actuator remains stable (Fig. 14). This behavior is due to a steep increase in the stiffness of most stressed suspension beam—the beam anchored to the substrate (beams #3 and #4 in our notations, Fig. 13 and Eq. (52)). When the right truss approaches the electrode due to “bending” instability, the beam is straightened, the steep increase of its stiffness prevents further displacement of the right frame toward the electrode. Since the suspension beam on the other side (#4) is less straightened and its axial stiffness is lower, the increase in voltage causes a displacement of the left side truss resulting in the displacement of the shutter in the forward, y, direction and lateral displacement of the shutter in the −x direction toward left electrode, i.e. the self centralization of the shutter (Figs. 14 and 15). This mechanism explains the reason why the actuator considered in this work does not exhibit side pull-in instability observed in comb drive actuators [4]. 4. Experiment and discussion Actuators of various configurations were fabricated of highly doped single crystal Si using SOI wafers and deep reactive ion etching (DRIE) based process. Typical suspension beam dimensions were: length 700–1000 ␮m, width 3–5 ␮m, height 30 ␮m, initial elevation is in the range of 10–50 ␮m. Overall dimensions of the actuators were typically in the range of 1.5 mm × 1.5 mm to 2 mm × 2 mm. Wet release using HF acid followed by drying in vacuum was used. In order to minimize the danger of stiction between the device and handle layers the SOI wafers with relatively thick (2 ␮m) SiO2 insulator layer were used. Actuators were fabricated in two different FABs and tested in two different laboratories. Despite some differences in fabrication processes used by two FABs resulting in slight variation of device dimensions (mainly suspension beam width), the main effects described in this work were demonstrated for both cases. Two types of electrodes were implemented. In the first configuration a single electrode of the length equal to or larger than the length of the connecting truss (typically 1500–2000 ␮m) is located near the truss (Fig. 17(a)). An advantage of this configuration is the simplicity and low mass of the actuator. In the second configuration, the parallel plate electrode is realized as an array of off-set comb-like fingers (Figs. 17(b), 18 and 19). The advantage of this configuration is the much larger total electrode area that can be achieved within the actuator footprint. In addition, this configuration was found to provide better uniformity of the DRIE process. The configuration of the parallel plate electrodes implemented in the actuator and realized as an array of offset fingers, eliminates the “comb effect”. As a result the actuating force acts on the truss connecting suspension beams in the direction perpendicular to the direction of the shutter motion and along the suspension beams. The typical off-set ratio was between 1:2 and 1:3 (e.g. see [24]). In order to improve the circulation of plasma during the etch within the narrow (typically 3–4 ␮m wide and 30 ␮m deep) trench between the truss and electrode, a tooth-like configuration of the electrode was

Fig. 17. Configurations of parallel plate electrodes (a) single electrode located near the connecting truss (b) array of off-set combs parallel plate electrodes.

used (Fig. 19). Note that the decrease in the electrode area is not accompanied by proportional decrease in force due to fringe effects of the electrostatic field. The actuators were operated statically using dc voltage. The response was registered by CCD camera and image processing was used for quantitative characterization of the motion. The stages in the loading of the actuators of different configurations are illustrated in Figs. 20 and 21. The straightening of the suspension beams resulting in large motion of the shutter is observed. The analysis of the experimental results reveals that while the saturation type character of the voltage–displacement curve is observed, some quantitative discrepancy exists between the model and experimental results. The reason for this discrepancy is the high level of uncertainty in the stiffness parameters of the actuators, mainly the width of the suspension beams and the distance between parallel plate electrodes. Direct measurements of the device geometry by destructive methods (dicing followed by the polishing of the cross-sections) reveal the reduction in the

Fig. 18. SEM image of the DRIE fabricated actuator. Initially curved suspension beams and off-set combs parallel plate electrodes are observable.

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developed by combs can be considered to be independent of displacement (see [4]), the ratio between the displacement of the comb drive actuator in the v direction and square of the actuation voltage of ith actuator is a constant value: Si =

Fig. 19. Off-set parallel plate electrodes configuration. Due to asymmetric position and presence of two sets of fingers, the combs operate in parallel plate mode.

width of beams and combs originating in the overetch during the DRIE process. Since direct measurement of the actuator geometry by destructive methods is not always available, a simple parameter estimation procedure was implemented using traditional comb drive actuators fabricated on the same wafer. The force

Fig. 20. Configuration of the actuator operated by the off-set comb electrodes before (a) and after (b) application of the actuating voltage (80 V). Straightening of the 700 ␮m long and 3 ␮m wide suspension beams with initial elevations of H = 50 ␮m and displacement of the shutter are observed.

vi ε0 Ni L3i = 2 Vi 2E(d0i + δ)(ai − δa)3

(54)

where δ and δa are deviations of the electrostatic gap and of the width of the suspension beam (which are usually different) from their nominal values and N is number of combs. It is assumed that the averaged overetch values δ and δa are identical for different actuators. Parameter Si can be obtained using experimental voltage-deflection curves. Note that it was observed in experiments that this ratio is indeed close to constant for long stroke comb drive actuators. The ratio between the values of parameter S of actuators with differing geometries: γij =

Si Ni L3i (d0j + δ)(aj − δa)3 = Sj Nj L3j (d0i + δ)(ai − δa)3

(55)

leads to the system of algebraic equations in terms of overetch parameters δ and δa. For example, in accordance with Eq. (55), the width of the suspension beams is extracted using voltage–displacement curves of two actuators with identical parameters with exception of the width of the beams. Since in our case the actuation voltage is affected by various parameters (e.g. asymmetry in the gap between electrodes, deviation of the DRIE fabricated beam cross-section from the rectangular shape, high roughness of the surfaces) this method does not demonstrate high accuracy and is used for rough parameter estimation. However, a comparison of extracted and measured values reveals satisfactory agreement between the two. It was found that the actual width of the suspension beams is usually lower than nominal values. The voltage-deflection characteristic corresponding to the actuator with 700 ␮m × 15 ␮m × 3 ␮m suspension beams and 3 ␮m electrostatic gap is presented in Fig. 22. The thickness of the beam used in calculations is fitted to meet experimental data (2.15 ␮m instead of nominal value of 3 ␮m). One observes that the characteristic is of the saturation type. The important feature of the characteristic presented in Fig. 22(b) is that in the vicinity of the inflection point of the curve the voltage–displacement dependence is close to linear. This almost linear characteristic is observed during the interval of motion of 10–15 ␮m. The slope of this linear part of the characteristic is defined by the initial elevations, i.e. initial geometry of suspension beams. This property is beneficial in many applications, especially in the insertion type variable optical attenuators (VOA). In accordance with conclusions of Section 3, no side pull-in instability was observed in the actuators with relatively small initial elevations (10–20 ␮m for 700–1000 ␮m long beams) up to high actuation voltages. The displacement of the actuator are limited by the applied voltages, namely abilities of the voltage source or electrical break down. However, actuators with higher elevations were found to demonstrate “bending-type” pull-in instability. The nominal values of relative electrostatic gap of these actuators η = d0 /(Ls – L) although usually larger that the

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Fig. 21. Configuration of the actuator operated by two long parallel plate electrodes before (a) and after (b) application of the actuating voltage (150 V). Straightening of the 1000 ␮m long and 3 ␮m wide suspension beams with initial elevations of H = 50 ␮m and displacement of the shutter are observed.

Fig. 22. Voltage–displacement curve for the actuator with off-set comb electrodes. Suspension beams are 700 ␮m long and 3 ␮m wide, initial elevation is H = 15 ␮m. The effective length of parallel plate electrodes is 3300 ␮m per beam, electrostatic gap is 3 ␮m. The width of the beam is fitted to meet experimental data (2.15 ␮m is used in calculations instead of nominal value of 3 ␮m). (a) Saturation-type characteristic (b) linear part of the characteristic in the vicinity of the inflection point.

“critical” values where this kind of instabilities is possible are comparable to the “critical” values. In addition, one can argue that secondary compliances of the off-set combs and truss can contribute to the reduction of the actual distance between elec-

trodes and cause instability. Results for the actuator suspended on 700 ␮m long beams with initial elevation 50 ␮m and width 3 ␮m and 4 ␮m are shown in Figs. 23 and 24, respectively. The non-smooth motion predicted by the analysis of the global

Fig. 23. Voltage–displacement curve for the actuator with off-set comb electrodes. Suspension beams are 700 ␮m long and 3 ␮m wide, initial elevation is H = 50 ␮m. The effective length of parallel plate electrodes is 1395 ␮m per beam, electrostatic gap is 4 ␮m.

Fig. 24. Voltage–displacement curve for the actuator. Suspension beams are 1000 ␮m long and 4 ␮m wide, initial elevation is H = 50 ␮m. The length of the parallel plate electrode located along the connecting truss is 1850 ␮m, electrostatic gap is 4 ␮m. The width of the beam is fitted to meet experimental data.

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stability of the actuator, Fig. 15 was observed in a number of cases. 5. Conclusions The present work reports on a large displacement electrostatic actuator operated by parallel plate electrodes. It is based on a folded suspension composed of initially curved suspension beams while an actuating force produced by the parallel plate electrodes is applied along the beams. The models of the actuator are built based on the exact extensible elastica and approximate reduced order models of the curved beam. The behavior of the actuator is studied theoretically and experimentally. In the case of a curved beam actuated by an electrostatic force applied to its end, for a typical lateral displacement of tens of microns, the axial elongation of the beam is in the submicron range allowing small distances between the electrodes and making the use of parallel plate electrodes very effective. The fact that the axial displacement of the beam’s end is comparable with the initial distance between electrodes results in a significant increase of the non-linear, distance dependent electrostatic force during the deformation which compensates for the reduction in force common to motion amplifying structures. It is found theoretically, and confirmed experimentally, that for typical values of parameters the voltage–displacements dependence is of a saturation type. Due to the interaction between mechanical and electrostatic non-linearities, the voltage–displacement dependence is practically linear in the vicinity of the inflection point of this characteristic. This feature is useful in various applications. Moreover, using the initial curvature of the beam and distance between the electrodes as parameters, the characteristic of the actuator can be tuned over a very large range. The increase of the mechanical stiffness of the straightened beam improves the actuator stability. The actuator does not exhibit pull-in instability until the suspension beams are almost completely straightened. For typical configurations of the actuators, the pull-in instability is possible due to the extensibility of the beam’s axis and only at very high actuation voltages. Note that, theoretically, an infinite force should have to be applied in order to completely straighten the beam. The analysis of the behavior of an initially curved beam and of its stability properties is performed primarily in connection with the long motion actuator reported in the present work. However, the curved yet shallow beam straightened by an electrostatic end force can be viewed as a generic non-linear stiffness element which can be used in various micro devices. It is possible to say that the actuator presented here is only one example of implementation of this type of element. The obtained results are general and relevant to a broad variety of applications. The fact that the stable travel of the beam can be extended to almost the entire distance between parallel plate electrodes can be used, for example, in capacitive RF devices. The possibility to tune the voltage-dependence characteristic and the ability to provide linear dependence in a particular range of displacements

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are useful in optical and RF devices as well as high precision micro instruments. Due to the interaction between mechanical and electrostatic non-linearities, for a certain range of parameters the actuator can demonstrate multisatbility which is useful, for example, in switching and latching devices. In the future we plan to extend the study to dynamics and enhance the effectiveness of the actuator through improved electrode design and implementation of curved beams of different configurations. Acknowledgments The experimental part of the work was partially performed at Teraop Ltd. The authors would like to thank the staff of the Tel Aviv University Micro-Fabrication Facility. We thank Colton Foundation, NY, for the support of this work. References [1] H. Toshiyoshi, K. Isamoto, A. Morosawa, M. Tei, H. Fujita, A 5volt operated MEMS variable optical attenuator, in: Proceedings of 12th International Conference on Solid-State Sensors, Actuators and Microsystems, “Transducers’03”, Boston, MA, 8–12 June 2003, pp. 1768–1771. [2] J.H. Leea, Y.Y. Kima, S.S. Yuna, H. Kwona, Y.S. Hongb, J.H. Leeb, S.C. Jungb, Design and characteristics of a micromachined variable optical attenuator with a silicon optical wedge, Opt. Commun. 221 (2003) 323–330. [3] W. Tang, T. Nguyen, R. Howe, Laterally driven polysilicon resonant microstructures, Sens. Actuators A 20 (1989) 25–32. [4] R. Legtenberg, A.W. Groenveld, M. Elwenspoek, Comb drive actuators for large displacements, J. Microelectromech. Syst. 6 (1996) 320– 329. [5] J.D. Grade, H. Jerman, T.W. Kenny, Design of large deflection electrostatic actuators, J. Microelectromech. Syst. 12 (2003) 335–343. [6] G. Zhou, P. Dowd, Tilted folded-beam suspension for extending the stable travel range of comb drive actuators, J. Micromech. Microeng. 13 (2003) 178–183. [7] W. Ye, S. Mukherjee, N.C. MacDonald, Optimal shape design of an electrostatic comb drive in microelectromechanical systems, J. Microelectromech. Syst. 7 (1998) 16–26. [8] S. Kota, J. Joo, Z. Li, S.M. Beamgers, J. Sniegowski, Design of compliant mechanisms: applications to MEMS, Analog Integrat. Circuits Signal Process. 29 (2001) 7–15. [9] Y.-C. Tsai, S.H. Lei, H. Sudin, Design and analysis of planar compliant microgripper based on kinematic approach, J. Micromech. Microeng. 15 (2005) 143–156. [10] N. Lobontiu, E. Garcia, Analytical model of displacement amplification and stiffness optimization for a class of flexure-based compliant mechanisms, Comput. Struct. 81 (2003) 2797–2810. [11] N.R. Tas, T. Sonnenberg, R. Molenaar, M. Elwenspoek, Design, fabrication and testing of laterally driven electrostatic motors employing walking motion and mechanical leverage, J. Micromech. Microeng. 13 (2003) N6–N15. [12] R. Legtenberg, J. Gilbert, S.D. Senturia, Electrostatic curved electrode actuators, J. Microelectromech. Syst. 6 (1997) 257–265. [13] E.S. Hung, S.D. Senturia, Extending the travel range of analog-tuned electrostatic actuators, J. Microelectromech. Syst. 8 (1999) 497–505. [14] J.I. Seeger, B.E. Boser, Dynamics and control of parallel-plate actuators beyond the electrostatic instability, in: Proceedings of 10th International Conference on Solid-State Sensors and Actuators, “Transducers’99”, Sendai, Japan, 7–10 June 1999, pp. 474–477. [15] S.D. Senturia, Microsystems Design, Kluwer Academic Publishers, Boston/Dordrecht/London, 2001.

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Biographies Slava Krylov received the MSc degree in 1989 in structural mechanics and PhD degree in 1993 in applied mechanics, both from the State Marine Technical University of St. Petersburg, Russia. Since he moved to Israel in 1993, he was postdoctoral fellow at the Department of Solid Mechanics, Materials and Systems, Tel Aviv University, worked as a R&D engineer for Israel Aircraft Industries and as a principal scientist and co-founder of a start up company developing optical MEMS. He joined Department of Solid Mechanics, Materials and Systems, Tel Aviv University in 2002. His research interests include design and modeling of MEMS, especially optical MEMS, large displacement actuators and dynamics of micro- and nano-electromechanical devices. Yacov Bernstein received the MSc degree in 1978 in radio-electronic engineering from Ulyanovsk Polytechnic Institute, Russia, and PhD degree in 1982 in physics of semiconductor devices and opto-electronics from Ioffe Physical-Technical Institute, St. Peterburg, Russia. Since he moved to Israel in 1993, he worked as research affiliate at The Hebrew University of Jerusalem and Technion-Israel Institute of Technology as well as in several start up companies developing optical micro devices. He is currently process engineer at the Center of Nanotechnology, Ben-Gurion University of the Negev, where he is responsible for fabrication and characterization of micro and nano devices.