A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage

7/30/2019 A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage

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Nonlinear Dyn (2011) 63: 719734DOI 10.1007/s11071-010-9833-0

O R I G I N A L P A P E R

A double microbeam MEMS ohmic switch

for RF-applications with low actuation voltage

Hatem Samaali Fehmi Najar Slim Choura

Ali H. Nayfeh Mohamed Masmoudi

Received: 21 December 2009 / Accepted: 1 September 2010 / Published online: 21 September 2010 Springer Science+Business Media B.V. 2010

Abstract In this paper, we propose the design of anohmic contact RF microswitch with low voltage ac-tuation, where the upper and lower microplates aredisplaceable. We develop a mathematical model forthe RF microswitch made up of two electrostaticallyactuated microplates; each microplate is attached tothe end of a microcantilever. We assume that the mi-crobeams are flexible and that the microplates arerigid. The electrostatic force applied between the twomicroplates is a nonlinear function of the displace-

ments and applied voltage. We formulate and solvethe static and eigenvalue problems associated with theproposed microsystem. We also examine the dynamicbehavior of the microswitch by calculating the limit-cycle solutions. We discretize the equations of motionby considering the first few dominant modes in the mi-crosystem dynamics. We show that only two modesare sufficient to accurately simulate the response of the

H. Samaali () S. Choura M. MasmoudiMicro Electro Thermal Systems Research Unit, NationalEngineering School of Sfax, BP 3038, Sfax, Tunisiae-mail: [email protected]

F. NajarApplied Mechanics and Systems Research Laboratory,Tunisia Polytechnic School, BP 743, La Marsa 2078,Tunisia

A.H. NayfehDepartment of Engineering Science and Mechanics,MC 0219, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061, USA

microsystem under DC and harmonic AC voltages. Wedemonstrate that the resulting static pull-in voltage andswitching time are reduced by 30 and 45%, respec-tively, as compared to those of a single microbeam-microplate RF-microswitch. Finally, we investigatethe global stability of the microswitch using differentexcitation conditions.

Keywords MEMS Microswitch Pull-in Pull-out Reduced-order model

1 Introduction

In recent years, the application of micro-electro-mechanical systems (MEMS) to radio frequency (RF)components has been developed, especially for RF-MEMS switches. Traditional microelectronic switch-es, such as silicon FETs (field effect transistors)and PIN (positive-intrinsic-negative) diode switches

present inadequate switching characteristics when thesignal frequency is greater than 1 GHz [1]. Theseswitches present high insertion loss and poor isolationduring the ON and OFF switching, whereas switchesfabricated by MEMS technology can overcome theselimitations [2, 3]. For instance, RF-MEMS switchesprovide improved insertion loss, isolation, and linear-ity, but they are limited because of their high actu-ation voltage (up to 30 V) and slow switching time(near 300 microseconds). Recent studies focused on
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720 H. Samaali et al.

improving the switching time, minimizing the actua-tion voltage, and integrating RF-MEMS switches withintegrated circuits (IC) [4].

In the literature, different types of RF-microswitch-es with a variety of actuation mechanisms (electro-static, magnetostatic, piezoelectric, or thermal), con-

tact modes (capacitive or ohmic contact), and cir-cuit implementation (shunt or series) are identified.Electrostatic actuation is the most used type in RF-microswitches. However, this type requires relativelyhigh DC voltage (up to 30 V), and thus requires anadditional CMOS integrated up-converter to raise thetypical 5 Volt control voltage to the required level.

All electrostatically actuated RF-microswitches arebased on an out-of-plane suspension bridge or can-tilever type [5]. The cantilever type, which is char-acterized by less rigidity as compared to the suspen-

sion bridge, yields a reduced actuation voltage but in-creases the switching time. The ohmic contact modeis suitable for this type of switches since it leads tovery low ON state insertion loss and very high OFFstate isolation. However, they are highly susceptibleto corrosion, stiction, and microscopic bonding of thecontact electrode metal surfaces [5].

Integration of RF-microswitches with IC has beenone of the important trends in the last decades [68].This integration requires that the RF-microswitch be(a) very small size, (b) of low actuation voltage, and

(c) of low power consumption.RF MEMS switches generate lower intermodula-

tion (IM) as compared to their equivalents semicon-ductors [9]. In particular for ohmic switches, the gen-erated IM is significantly low because of the verysmall capacitance at the OFF-state and the linearity ofthe contact at the ON-state [3, 10].

Static and dynamic pull-in instabilities of MEMSdevices have been key issues in the literature. Sta-tic pull-in, identified by Nathanson et al. [11], occurswhen the DC voltage exceeds a threshold value. Stud-

ies on static pull-in reveal that the maximum static sta-ble deflection varies from 33% to 41% of the originalelectrode gap distance [12, 13].

The dynamic pull-in phenomenon have been also ofmajor interest in the literature. Dynamic pull-in takesplace when the system is excited using DC and/or ACvoltages. In fact, The transient behavior of MEMS de-vices is important for RF switch and for optical ap-plications [3]. Gupta et al. [14] and Krylov and Mai-mon [15] showed that pull-in occurs at voltages below

the static pull-in value due to transient effects for mi-crobeams actuated by a step voltage. Both studies in-dicate that the dynamic pull-in voltage can be as lowas 91% of the static pull-in voltage. In the presenceof squeeze-film damping, the dynamic pull-in voltageis shown to approach the static pull-in voltage [15].

A recent study by Krylov [16] shows that, using Lya-punov exponents, the system may become unstable be-fore reaching the static pull-in voltage due to dynamiceffects when the system is excited using a DC volt-age. Under the same excitation Nielson and Barbas-tathis [17] concluded that dynamic pull-in occurs athalf of the electrostatic gap and 92% of the static pull-in voltage.

Nayfeh and coworkers [1820] generated frequencyand force-response curves for electrostatic microactu-ators whose main component is a clamped-clamped

microbeam. They showed that dynamic pull-in occursunder voltages lower than the static pull-in voltage, aslow as 25%, when the frequency of the AC compo-nent is in the neighborhood of a resonant frequency.On the other hand, Najar et al. [21] and Lenci andRega [22] studied the basin of attraction of boundedmotions and showed that the erosion of the basin ofattraction is the principle reason for the occurrenceof dynamic pull-in by homoclinic bifurcation. Theyshowed that smoothness of the boundary of the basinof attraction of bounded motions can be lost and re-placed with fractal tongues as the excitation amplitudeis increased. Similar results were reported by Nayfehet al. [23] for a microcantilever with a rigid plate at-tached to its free end.

The decrease of actuation voltage of electrostaticRF-MEMS switches can be accomplished by (i) us-ing different materials and hinges to reduce the mi-crobeam rigidity, (ii) increasing the area of the elec-trostatic field, and/or (iii) decreasing the gap. Thesevariations degrade the principal parameters of the RF

MEMS switches, such as isolation. Abbaspour-Saniand Afrang [24] proposed to decrease the equivalentrigidity of the microswitch by using a structure com-posed of two displaceable microplates This structurepreserves the microswitch parameters while increas-ing its lifetime. Similarly, Chaffey and Austin [25] de-creased the equivalent rigidity of the microsystem andconcluded that the use of two cantilever microbeams,when compared to a single microbeam structure, re-duces significantly the pull-in voltage.


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A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage 721

The present paper examines the static and dynamicbehaviors of an electrostatically actuated ohmic con-tact RF microswitch. The proposed design consists ofa pair of cantilevered microbeams. At the free endof each microbeam, a rigid microplate is clamped.An electrostatic force is applied between the two mi-

croplates (electrodes) causing the deflections of bothmicroplatemicrobeam subsystems. These deflectionscontinue to grow up to a point where the electrosta-tic force exceeds the elastic force of the microbeams.This leads to the collapse of the upper microplate ontothe lower one when the pull-in voltage is reached. Themeans by which this instability occurs and the selec-tion of the microsystem parameters that affect this in-stability are of paramount importance in the designof MEMS electrostatic devices. In practice, pull-in in-stability of microswitches is suitable for changing the

state of an electric circuit from open to close or viceversa. In this study, we develop an accurate mathemat-ical model that accounts for significant nonlinearitiesof the microswitch. We first investigate the static andtransient responses of the RF microswitch, and thenwe study its openclosed cycle and compare it withother designs. Here, stiction and Casimir forces will

be neglected due to the dominance of the electrosta-tic force. To gain more insight into its dynamical be-havior, we also simulate the frequency response of theproposed RF microswitch. Finally, we investigate theglobal stability of the microsystem by estimating thesize of the resulting basin of attraction of bounded mo-

tions.

2 Problem formulation

We propose the design of RF microswitches withohmic contact, as shown in Fig. 1. The proposed de-sign consists of two cantilevered microbeams; eachone is attached to a rigid microplate (electrode) at itsfree end, and clamped to the substrate at the otherend. The transmitted signal is applied to a transmis-sion line located between the two electrodes. On top of

this transmission line an insulator layer is deposited inorder to provide separation between the actuation andthe transmitted voltages. The thickness of the trans-mission line with its insulator layer is 0.8 m. An elec-trical voltage, composed of DC and AC components, isapplied between the two electrodes. Sufficiently largevoltages cause the pull-in instability and, thus, the ON

Fig. 1 Ohmic contact RF-microswitch


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state of the microswitch, whereas zero or low voltagesrelease the microbeams to establish the OFF state. Themicrobeams are modeled as EulerBernoulli beams ofdensity , modulus of elasticity E, width b, thick-ness h, cross-section area A = bh, and second mo-ment of area I = bh312 . The microplates are modeledas rigid bodies of masses M1 and M2 and moments ofinertia J1 = 13 M1L2C and J2 = 13 M2L2C about the y1-axis. Due to their small thicknesses, only the dominantterms of the moments of inertia of the microplates areconsidered in this study.

We derive the equations of motion using Hamil-tons principle, which states thatt2

t1

(T VD VE ) dt= 0. (1)

The total kinetic energy of the microswitch is given

by

T = 12

A

L10

w1x1, t2 dx1+ 1

2A

L20

w2x2, t2 dx2+ 1

2M1

w1L1, t+ Lc w1L1, t2+ 1

2M2

w2L2, t+ Lcw2L2, t

2

+ 12

J1 w1L1, t2 + 12 J2

w2L2, t2 (2)where w1(x1, t) and w2(x2, t) are, respectively, the de-flections of microbeams 1 and 2 at time t about the z1and z2 axes and at locations x1 and x2, respectively.The dot denotes the derivative with respect to time tand the prime designates the derivatives with respectto the spatial variables x1 and x2 for microbeams 1and 2, respectively. The total elastic energy of the mi-crosystem is given by

VD =1

2EIL1

0

w1

x1, t2

dx1

+ 12

EIL2

0

w2

x2, t2

dx2. (3)

Finally, the potential energy due to the electrostaticfield between the microplates is

VE = bp

2(VDC + VAC)2

2Lc

0

ds

dn w1(L1, t)s w2(L2, t)(2Lc s)

= bp(VDC + VAC)2

2(w2(L2, t ) w1(L1, t))

ln dn 2 Lc w1

(

L1,

t)

dn 2Lcw2(L2, t)

(4)

where dn = d w1(L1, t) w2(L2, t), is the permit-tivity of air, VDC and VAC are, respectively, the DC andAC voltage differences between the microplates and sis the local coordinate attached to the microplate in thex directions.

Substituting (2)(4) into (1), incorporating viscousdamping, and using the following nondimensionalvariables:

w1 = w1d

, w2 = w2d

, x1 = x1L1

,

x2 =x2

L2, t= t

T, T =

AL41

EI,

= L2L1

we obtain the nondimensional equations of motion

wiv

1 (x1, t) + c1w1(x1, t) + w1(x1, t) = 0, (5)wiv2 (x2, t) + c2w2(x2, t) + 4w2(x2, t) = 0 (6)

where c1 = c1 L41

EITand c2 = c2

4L41

EITare the nondimen-

sional damping coefficients of the microbeams. Now,the prime and dot denote the derivatives with respectto x and t, respectively.

Equations (5) and (6) are subject to the followingnondimensional boundary conditions:

w1(0, t) = 0, (7)w1(0, t) = 0, (8)w2(0, t) = 0, (9)w2(0, t) = 0, (10)w1 (1, t) = M1

w1(1, t) + Lc1w1(1, t)

2Qw,(11)

w2 (1, t) = 4M2

w2(1, t) + Lc2w2(1, t ) 22Qw,

(12)


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w1 (1, t) = M1Lc1

w1(1, t) +4

3Lc1w

1(1, t )

1Qw1 , (13)

w2 (1, t) = 4M2Lc2

w2(1, t) +4

3Lc2w

2(1, t)

12Qw2 , (14)where

Qw =(VDC + VAC)2

(dn 2Lc1w1(1,t))(dn 2Lc2w2(1,t))

Qw1= (VDC + VAC)

2

(w2(1,t)

w1(1,t))2

ln

dn 2Lc1w1(1, t)dn

2Lc2w2(1, t)

+ 2Lc1(w2(1,t)

w1(1,t))

dn 2Lc1w1(1, t)

,

Qw2 =(VDC + VAC)2

(w2(1,t)

w1(1,t))2

,

ln

dn 2Lc1w1(1, t)dn 2Lc2w2(1, t)

2Lc1(

w2(1,t)

w1(1,t))dn 2Lc2w2(1, t) ,

dn = 1 w1(1, t) w2(1, t), Lc1 =Lc

L1,

Lc2 =Lc

L2, 1 =

bpL41

2EId3, 12 = 14,

M1 =2bpLc

bL1, M2 =

2bpLc

bL2,

2

=bpLcL

31

EId3, 22

=2

3.

3 Static analysis

The static problem can be formulated by setting thetime derivatives and the AC forcing terms in (5)(14)equal to zero. The solutions of the static equations aregiven by

ws1 = Ax31 + Bx 21 + Cx1 + D, (15)

ws2 = Gx32 + F x22 + Ex2 + H (16)

where ws1(x1) and ws2(x2) are the static deflectionsfor microbeams 1 and 2, respectively.

Using boundary conditions (7)(10) yields C =D = E = H = 0. Use of the remaining boundary

conditions leads to the following nonlinear algebraicequations:

6A = 2(VDC)2

1

Dn 2Lc1D1

1

Dn 2Lc2D2

,

(17)

6G = 22(VDC)2

1

Dn 2Lc1D1

1

Dn 2Lc2D2

,

(18)

6A

+2B

=

1(VDC)2

(

D2

D1)2ln

Dn 2Lc1D1

Dn 2Lc2D2 2Lc1(

D2

D1)Dn 2Lc1D1

, (19)

6G + 2F = 12(VDC)2

(D2

D1)2

ln

Dn 2Lc1D1Dn 2Lc2D2

+ 2Lc1(D2

D1)Dn 2Lc2D2

, (20)

where Dn=

1

A

B

G

F, D1

=3A

+2B and

D2 = 3G + 2F.The microbeam-microplate subsystems deflect un-

der an applied electric field. We examine these de-flections by developing closed-form solutions for thestatic deflections, in (15) and (16), whose coefficientsare determined using the NewtonRaphson methodin Mathematica from (17)(20). The geometric andphysical parameters of the microswitch are given inTable 1. Figure 2 shows variation of the static de-flections of microbeams 1 and 2 at x1 = x2 = 1 withthe applied DC voltage. For a given DC voltage each

microbeam has two equilibrium solutions; one is sta-

Table 1 Geometric and physical parameters of the microswitch

L1 L2 b h d

250 m 250 m 5 m 1.5 m 4 m

Lc bp E

25 m 20 m 2300 kg/m3 160 GPa 8.851 1012 F/m


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ble (lower branch) and the other is unstable (upperbranch). As the voltage is increased, these solutionsmeet at the pull-in point which is characterized by avoltage VDC = Vp = 5.89 V and maximum deflectionequal to 0.1536. Since the microbeam microplate sub-systems are alike, the deflections ws1(x1) and ws2(x2)

are identical. We also simulate, in Fig. 2, the static re-sponse of the system using the commercial softwareANSYS and show very good agreement between bothsolutions. The pull-in voltage obtained by ANSYS is5.9 V at a maximum deflection equal to 0.144. Weshow in Fig. 3 the deflected configurations of the mi-

Fig. 2 Microbeams deflection under an applied DC Voltage

crobeams using ANSYS for VDC = 5.5 V, the simula-tion validate the rigid plate assumption adopted in ourmodel even for a relatively high DC voltage.

Figure 4 shows the influence of the length of thesecond microbeam L2 on the pull-in voltage Vp. If

= 0, which corresponds to L2 = 0 while the lowermicroplate remains under the upper one, the systembecomes a conventional microswitch with a singlecantilevered microbeam. This corresponds to the mi-croswitch studied by Nayfeh et al. [23] with the pull-involtage Vp = 8.3 V, which is in agreement with Fig. 4for = 0. As shown in Fig. 4, the pull-in voltage Vp isreduced as increases, and attains its lowest value at = 1, which is adopted for the rest of the simulations.

4 Natural frequencies and mode shapes

The deflections of the microbeams, subjected to anelectrostatic force, can be decomposed into the sumof static components due to the DC voltage and dy-namic components due to the AC voltage, denoted byu1(x1, t ) and u2(x2, t); that is,

w1(x1, t ) = ws1(x1) + u1(x1,t), (21)w2(x2, t ) = ws2(x2) + u2(x2,t). (22)

Expanding the nonlinear electrostatic force terms us-ing Taylor series about ui = 0 (i = 1, 2) yields theproblem describing the dynamics of the system aboutits static equilibrium. We drop the nonlinear forcing

Fig. 3 Static solution usingAnsys for VDC = 5.5 V


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Fig. 4 Pull-in voltage variation with

and damping terms in (5)(14), use (21)(22), and letui (xi , t) = i (xi )ej t (i = 1, 2), where i (xi ) is themode shape and is the associated nondimensional

natural frequency. The linear eigenvalue problem isgiven by

iv1 (x1) 21(x1) = 0, (23)iv2 (x2) 22(x2) = 0 (24)

and the boundary conditions

1(0) = 0, (25)2(0) = 0, (26)

1(0)

=0, (27)

2(0) = 0, (28)

1 (1) = M1Lc121(1) +4

3M1L

2c1

21(1)

1V2DC

11(1) + 11(1) + 22(1)+ 22(1)

, (29)

2 (1) = 4M2Lc222(1) +4

34M2L

2c2

22(1)

12V2DC11(1) + 1

1(1) + 22(1)

+ 22(1), (30)1 (1) = M121(1) M1Lc121(1)

2V2DC

11(1) + 11(1) + 12(1)+ 22(1)

, (31)

2 (1) = 4M222(1) 4M2Lc222(1) 22V2DC

11(1) + 11(1) + 12(1)

+ 22(1)

, (32)

where

1 =1

1

1

12+ 1

11+ 2Lc1

21

,

1 =1

12 ln( 1

2)

2

1

+ 4Lc111

+ 4L2c1

2

1, 2 = 1,

2 =1

1

2 ln( 12

)

21

2Lc111

2Lc212

,

1 =1

1

1

12 1

11 2Lc1

22

,

1 =1

1

2 ln( 12

)

21

2Lc112

2Lc111

, 2 = 1,

2 =1

12 ln( 1

2)

2

1

+ 2Lc112

+ 2Lc212

4Lc1Lc22

2,

1 =

1

21 2+ 1

122

, 1 =

2Lc121 2

,

2 =2Lc222 1

, 1 =ws2(1)

ws1(1),

1 = 1 2Lc1ws1(1) ws1(1) ws2(1),2 = 1 2Lc2ws2(1) ws1(1) ws2(1).

The general solutions of the eigenvalue problem

can be expressed by1(x1) = 1 cos(x1) + 2 sin(x1)

1 cosh(x1) 2 sinh(x1), (33)2(x2) = 1 cos(x2) + 2 sin(x2)

1 cosh(x2) 2 sinh(x2), (34)

where = and the boundary conditions (25)(28) are taken into account. The rest of the coeffi-cients are determined by solving the system of non-

linear algebraic equations that result from substitutionof (33)(34) into the boundary conditions (29)(32).A NewtonRaphson technique is used in Mathemat-ica to calculate three of these coefficients in terms ofthe fourth coefficient, and thus to determine the firstfew natural frequencies and mode shapes. In Fig. 5,we show the first four mode shapes and correspondingnatural frequencies for VDC = 3 V. The mode shapesshow the in phase and out of phase motion of the twomicrobeams for each natural frequency of the system.


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Fig. 5 First four modeshapes and correspondingnatural frequencies forVDC = 3 V

Figure 6 shows variation of the first natural fre-quency 1 with the applied DC voltage. We observea significant drop in 1 as the DC voltage approaches

the static pull-in voltage. In Table 2, we show the ef-fect of varying the DC voltage on the first ten naturalfrequencies. In this case, the frequencies 2 to 10 arerelatively insensitive to the DC voltage.

5 Reduced-order dynamic model

A reduced-order model (ROM) is derived to simulatethe dynamic response of the microsystem. This ROM

is obtained by applying the Galerkin method by using

the first two global mode shapes associated with the

first two natural frequencies. In addition, we let

=1,

Lc1 = Lc2 = Lc , L1 = L2 = L and M1 = M2 = M.The Lagrangian is used here to derive the discretized

equations of motion in nondimensional form; that is,

=1

0w21(x1, t ) d x1 +

10

w22(x2, t ) d x2

1

0

w1 (x1, t)

2dx1

10

w2 (x2, t )

2dx2


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Table 2 Variation of thefirst ten natural frequencieswith VDC

VDC (V) 0 2 3 4 5.89

1 1.51787 1.49077 1.45075 1.37900 0.46613

2 1.51787 1.51807 1.51830 1.51869 1.52198

3 13.18123 13.1814 13.1784 13.1795 13.1739

4 13.18123 13.2266 13.1817 13.1822 13.1864

5 39.53288 39.5330 39.5311 39.5314 39.52816 39.53288 39.5629 39.5330 39.5333 39.5350

7 80.56617 80.5642 80.5653 80.5647 80.5651

8 80.56617 80.584 80.5677 80.5684 80.5694

9 139.011 138.938 138.947 138.988 138.945

10 139.011 139.069 139.031 139.037 139.07

Fig. 6 Variation of the first natural frequency 1 with VDC

+ R(VDC + VAC)2

(w2(1, t) w1(1,t))ln

dn 2Lcw1(1, t)dn 2Lcw2(1, t)

+ Mw2(1, t) + Lcw2(1, t)2+ ML2c3

w1(1, t )2

+ ML2c

3

w2(1, t)

2+ Mw1(1, t) + Lcw1(1, t )2(35)

where R = bpL4

EId3.

Next, we approximate the microsystem deflectionsby

wi (xi , t) wsi (xi ) +2

j=1qi (t)ij(xi ) i = 1, 2

where the ij(xi ) are the mode shapes and the qi (t)are the associated modal amplitudes. Substituting

the above approximation into (35), using the EulerLagrange equation and adding viscous modal dampingterms, we obtain the following reduced-order modeldescribed by two second order ODEs in time and rep-resented in matrix form as:

[M][Q] + [C][Q] + [KL][Q]

+ R(VDC + VAC sin t )2

BF[KN][Q]

+ [F1] +R(VDC + VAC sin t )2

AF[F2] = [0] (36)

where

[M] =

m11 m12m12 m22

, [C] =

c1 00 c1

,

[KL] =

k11 k12k12 k22

, [KN] =

0 K12

K12 0

,

[F1] =

f11f12

,

f11 =1

0ws1(x1)

11(x1) dx1

+

1

0ws2(x2)

21(x2) dx2,

f12 =1

0ws1(x1)

12(x1) dx1

+1

0ws2(x2)

22(x2) dx2,

[F2] =

f21f22

=

21(1) 11(1)22(1) 12(1)

,


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[Q] =

q1(t)

q2(t)

and

m11 =1

0211(x1) dx1 +

10

221(x2) dx2

+M211(1) +

221(1)

+ 2Lc

11(1)11(1) + 21(1)21(1)

+ 4

3L2c

112

(1) + 212(1)

,

m12 =1

011(x1)12(x1) dx1

+1

021(x2)22(x2) dx2

+ M11(1)12(1) + 21(1)22(1)+ Lc

12(1)

11(1) + 11(1)12(1)

+ 22(1)21(1) + 21(1)22(1)

+ 43

L2c

11(1)12(1) + 21(1)22(1)

,

m22 =1

0212(x1) dx1 +

10

222(x2) dx2

+ M212(1) + 222(1)+ 2Lc

12(1)

12(1) + 22(1)22(1)

+ 4

3L2c

122

(1) + 222(1)

,

k11 =1

0211 (x1) dx1 +

10

212

(x2) dx2,

k12 =1

011(x1)

12(x1) dx1

+

1

021(x2)

22(x2) dx2,

k22 =1

0212 (x1) dx1 +

10

222 (x2) dx2,

K12 = Lc

22(1)11(1) 11(1)12(1)

21(1)12(1) + 12(1)

11(1) 21(1)

22(1)21(1) + 11(1)22(1)+ 21(1)22(1)+ 2Lc

11(1)

22(1) 12(1)21(1)

,

AF =2(ws1(1) ws2(1) +

2j=1 qj(t)(

1j(1) 2j(1)))2

Ln[1

+2j

=1 qj(t)(1j(1)

+2j(1)

+2Lc1j(1))

2+

2j=1 qj(t)(1j(1)+2j(1)+2Lc2j(1)) ]

,

BF =

1 +2

j=1qj(t)

1j(1) + 2j(1)

+ 2Lc1j(1)2 + 2

j=1qj(t)

1j(1)

+ 2j(1) + 2Lc2j(1)

ws1(1) ws2(1)

+2

j=1qj(t)

1j(1) 2j(1)

.

6 Static and dynamic responses

6.1 Static response

We calculate the static response of the microsystem bysolving (15)(20) with different applied DC voltages.If the microswitch is at its ON state, the DC voltagemust be lowered to recover its OFF state. The voltage,at which the microswitch electrodes loose contact, isknown as the pull-out voltage. We estimate the pull-out voltage by conducting a transient analysis. We as-sume that the ON state (both electrodes are in contact)is the initial state of the transient response. The elec-trodes are in contact, with the transmission line and theinsulator layer, when the microbeam deflections attain0.4, where a distance of 0.2 is kept as a separation gap.Thus, the pull-out voltage can be estimated by solvingthe dynamic equations (36) by means of long-time in-

tegration using the RungeKutta technique in Mathe-matica.

A comparison of the pull-in and pull-out voltagesassociated with the proposed model and that of Nayfehet al. [23] ( = 0) is shown in Fig. 7. We remark thatthe computed pull-in voltages associated with the sin-gle beam (Nayfehs model) and the proposed double-beam microstructure are 8.297 V and 5.89 V, respec-tively. Consequently, a 30% reduction of the actuationvoltage is obtained.


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Fig. 7 The responses of the single microbeam (gray curve) anddouble microbeam (black curve) designs subjected to a DC volt-age

6.2 Dynamic response

We use the Finite Difference Method (FDM) [15] toexamine the limit cycles of the microsystem model bytwo modes subject to DC and AC voltages. The ACexcitation is harmonic with period T = 2/ , where is the forcing frequency. We discretize the orbitinto m + 1 points and enforce the periodicity condi-tion qi0 = qi (t0) = qi (tm) = qim . This condition im-plies that the first and last points of the orbit (point 0

and m) are identical. At each of these points, we haveqip = qvip,qvip = f (qip, qvip, VDC, VAC(tp))

(37)

where p = 1, 2, . . . , m, i = 1, 2, tp = pt, t = Tm ,qip = qi (tp),and qvip = qvi (tp) = qi (tp). The state func-tion f is given by

f

qip, qvip, VDC, VAC(tp)

= [M]1[C]

qv1p

qv2p+ RVDC + VAC sin(2tp)2

[KN]

BF

q1pq2p

+ [F2]

AF

+ [KL]

q1pq2p

+ [F1]

.

The FDM can now be applied to system (37) toyield a set of nonlinear algebraic equations. In thiscase, a two-step explicit central-difference scheme is

Table 3 Loading cases

Study cases VDC 1 2 VAC Q

Case 1 3 V 1.450 1.518 1 V 100

Case 2 3 V 1.450 1.518 0.05 V 100

used to approximate the time derivatives. Therefore,for an m+1 FDM discretized orbit, the microstructuredynamics can be approximated by a set of 4m non-linear algebraic equations in 4m unknown displace-ments and velocities. These equations can be solvedfor the unknowns using the NewtonRaphson method.The stability of the orbits can then be analyzed bymeans of long-time integration (LTI). Next, we exam-ine the frequency-response curves of the microbeam-microplate system, whose parameters are given in Ta-ble 1 and simulate its time response subject to twodifferent loading cases described in Table 3. We sim-ulate the frequency response of the first microbeamtip wmax = w1(1, t) in the neighborhood of the nondi-mensional fundamental frequency 1 while fixing thenumber of FDM time steps per period to m = 100. Forsimplification we use the same modal damping coef-ficient for both mode shapes by defining c = 1/Q,where Q is the quality factor. This assumption is ver-ified by the fact that the two first natural frequenciesare very close to each others.

In Fig. 8, we use the loading case 1 to validate the

convergence of the proposed two modes approxima-tion by plotting the frequency-response curve usingtwo, three and four mode shapes in the approxima-tion of the beams deflections. Figures 9 and 10 displaythe frequency-response curves of the system subject toloading Cases 1 and 2, respectively. For loading Case 1(Fig. 9), we find two resonance regions associated withthe first symmetric mode and the first antisymmetricmode. The symmetric and antisymmetric modes corre-spond to the in-phase and out-of-phase motions of themicrobeams, respectively. For loading Case 2, the res-

onance corresponding to the antisymmetric mode al-most disappears because the AC voltage being appliedis small. In both figures, the solid and dashed curvesdenote stable and unstable limit cycles, respectively,and the gray dashed line represents the limit of stabil-ity given by the unstable equilibrium solution. Furtherdetails on the stability of these branches can be foundin [18] and [20].

We also investigate the force-response curve fromwhich the minimum applied AC voltage is assigned to


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Fig. 8 Convergence of the proposed solution using a two modesapproximation

Fig. 9 Frequency response curve for loading case 1

Fig. 10 Frequency response curve for loading case 2

the proposed microswitch design. The use of a combi-nation of DC and AC voltages can lead to the phenom-

Fig. 11 Force response-curve of the microswitch for = 1

Fig. 12 Force response curve of the microswitch for = 1.38

enon of dynamic pull-in [18, 20], which correspondsto setting up the microswitch to its ON state. Fig-ures 11 and 12 show the force-response curves for dif-ferent values of the AC voltage VAC by solving (37)for = 1 and = 1.38. The resulting bifurcation curvesshow that the minimum AC voltage to be applied to themicroplates for dynamic pull-in is, respectively, 2.9

and 1.013 V for = 1 and = 1.38. We concludethat dynamic pull-in is better obtained for excitationfrequencies close to the fundamental frequency.

7 Switching time estimation using static

and dynamic pull-in

In the proposed design, as the electrostatically actu-ated microbeams deflect, each microplate travels only


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Fig. 13 Influence of DC voltage on the switching time for sim-ple and double cantilever beam designs

50% of the gap distance to reach the ON state. Weexpect that in the proposed design the switching time

is shorter when compared to the single beam design.To confirm this, we solve the dynamic equations (36)and estimate the transient time needed for the mi-crobeams to travel from the equilibrium position tothe ON state under an applied DC or a DCAC volt-ages. The switching time is an important parameterin RF-MEMS since it represents the main limitationfor reaching high frequencies [26]. In this section, weexamine the effect of reducing the switching time byincreasing the applied voltage and its correspondingpower requirement for a given design using first DC

voltage only and second using a combination of DCand AC voltages.

7.1 Switching time using DC voltage: static pull-in

Here, we use the pull-in voltages calculated in Sect. 6.1to measure the switching time for simple and doublecantilever beam designs.

Figure 13 shows the influence of DC voltage on theswitching time for both single and double cantileverbeam designs. It is clear that, when using the DC volt-

age alone to actuate the microswitch, the double beamdesign offers a significant improvement of the switch-ing time, especially when high actuation frequenciesare required. In Fig. 14, we simulate the correspondingrequirements on power characterizing the electrostaticenergy of the system given by (4). We note that thedouble cantilever beam design requires a very low ac-tuation power when compared to the single cantileverbeam design. The difference can reach one order ofmagnitude.

Fig. 14 Influence of DC voltage on the power consumptionsfor simple and double cantilever beam designs

7.2 Switching time using DC and AC voltages:dynamic pull-in

When we use a combination of DC and AC voltagesto actuate the microswitch from the OFF state to theON state, a voltage lower than the DC voltage can beapplied to force the collapse of the microplates. Thisphenomenon is known as dynamic pull-in. This latteris a homoclinic bifurcation that occurs as the AC volt-age is increased [21] when stable and unstable mani-folds of the saddle approach each other, touch and thenintersect transversely infinitely many times as a resultof a homoclinic entanglement [27].

Figure 15 displays the influence of the AC voltageon the switching time associated with the simple anddouble-beam microswitch designs with different actu-ation frequencies for VDC = 3 V and Q = 100. Wevary the AC voltage amplitude and calculate the cor-responding switching time. Figure 15 depicts that theswitching time is low at frequencies in the neighbor-hood of the first natural frequency for both designs.In order to reduce the electrostatic voltage, we selectactuation frequencies that belong to the pull-in band,within which no stable solutions exist (see Fig. 9), orwe apply an AC voltage higher than the smallest volt-

age corresponding to a stable solution in the corre-sponding force-response curve. We conclude that dy-namic pull-in is not an appropriate alternative to DCvoltage actuation since it results in large switchingtimes for both simple and double beam designs. How-ever, compared to the simple beam design the pro-posed design considerably reduces the actuation ACvoltage by 33% to 50%.

In Fig. 16, we simulate the electrostatic power asso-ciated with Fig. 15 for different excitation frequencies.


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Fig. 15 Influence of AC voltage on the switching time for sim-ple and double cantilever beam designs at different actuationfrequencies

For both designs, we show that actuation frequenciescloser to the first natural frequency lower the electro-static power. The double beam design depicts one or-der of magnitude reduction in the electrostatic power.However, the resulting electrostatic power using dy-namic pull-in is larger as compared the case of staticpull-in for the same switching time. This confirms theresults obtained in Fig. 15.

In case of initial conditions corresponding to anequilibrium position, the dynamic pull-in actuation for

a double beam design yields lower switching time andelectrostatic power. In the next section, we investigatethe influence of other initial conditions on the globalstability of the microsystem and the correspondingswitching time for the unstable case.

8 Stability of the microswitch under small and

large perturbations

To study the global stability of the limit-cycle solu-tions obtained by solving system (37) for a singlemode, we investigate the motion of the microswitchfor a set of initial conditions in the proximity of eitherthe stable or unstable fixed points. In the beginning,we study the behavior of the system for the undampedand unforced case by examining the separatrices asso-ciated with the microswitch dynamics. Then we incor-porate damping and forcing and examine their influ-ence on the region of stable motion in the phase space.

Fig. 16 Influence of AC voltage on the power consumptions forsimple and double cantilever beam designs at different actuationfrequencies

Fig. 17 Separatrix for VDC = 3 V

8.1 Unforced and undamped case

In the absence of damping and forcing, we investigatethe region in the phase space that leads to bounded mo-tion. In Fig. 17, we show the separatrix for VDC = 3 Vby integrating system (36) in forward and backwardtime using LTI, for the undamped and unforced case(c = 0 and VAC = 0), starting from the unstable staticsolutions corresponding to displacements ws1(1) andws2(1). The dashed curves represent the separatricescorresponding to both microbeams. These separatri-ces are perfectly symmetric about half distance of theelectrostatic gap d.

8.2 Forced and damped case

The integration of AC-forcing and damping in study-ing the microsystem global stability requires a differ-


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Fig. 18 Basin of attraction of safe motion VDC = 3 V, = 1.38and VAC = 0.05 V

ent approach. In fact, we use LTI to determine the sta-bility of the system by assuming a set of initial con-ditions belonging to the phase space of the system. Todetermine these initial conditions, associated with thefirst microbeam, we divide the phase plane using a gridcomposed of 500 500 lines. The grid points are cho-sen as initial conditions to solve system (36).

In Figs. 18 and 19, we show the basin of attractionof bounded solutions of the microswitch for = 1.38and VAC = 0.05 V (Fig. 18) and VAC = 0.9 V (Fig. 19)using a one-mode approximation. The use of only sin-gle mode for approximating the dynamic solution isrelated to the reduction of the computational time andjustified by the choice of the excitation frequencies,which are far from the second resonance frequency of

the microsystem. In both figures, the red regions cor-respond to initial conditions that lead to bounded mo-tions. These regions correspond to the case in whichthe pull-in dynamics fails to set the microswitch toits ON state. Outside the basin of attraction, wheredynamic pull-in occurs, the color levels indicate themagnitude of the switching time. More contrasted col-ors correspond to smaller switching times. In caseVAC = 0.9 V, we note that the choice of the initial con-ditions is crucial for determining the switching time.This may be considered as a drawback of the proposed

design because its performance depends on both mi-crobeams being able to recover their initial stable po-sitions in order to initiate the next switching cycle.

9 Conclusions

We proposed the design of a RF microswitch with low-voltage actuation. The microsystem is composed oftwo displaceable microcantilever electrodes at which

Fig. 19 Basin of attraction of safe motion VDC = 3 V, = 1.38and VAC = 0.9 V

two rigid microplates generate the nonlinear electro-static force to actuate the microswitch. A mathemati-cal model was developed to analyze the static, eigen-

value, and dynamic problems where the limit-cyclesolutions of the system are calculated under DC andharmonic AC voltages. Using a static and transientanalysis, we demonstrated that the resulting static pull-in voltage and switching time are reduced by 30 and45%, respectively, as compared to the design madeof a single microbeam-microplate system. We alsoshowed that microswitches can pull-in at voltages be-low the static pull-in voltage due to the transient ef-fects when the system is excited using a combinationof AC and DC voltages. The frequency- and force-

response curves indicate the use of a set of AC voltagefrequency at which no stable positions exists.

This fact is suitable for switching applications.Then we studied and compared the switching times forDC and AC actuation. We showed that using a combi-nation of DC and AC voltages minimize the switchingelectrical power, however, in that case the switchingtime is much higher than a standard DC forcing case.This fact allows an optimization of the choice of theactuation voltage with respect to the required switch-

ing performance. Finally, the global stability of the mi-crosystem was studied in order to examine the varia-tion of the switching time as the initial conditions arevaried. For small actuation voltages, we found a largesensitivity to initial conditions. This fact influences theperformance of the switching time.

Acknowledgements The authors would like to thank the re-search team of Dr. Mohammad Younis of Binghamton Univer-sity for their assistance in the ANSYS simulations.


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A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage

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