Design, Simulation and Bifurcation Analayis of a Novel ... · DESIGN, SIMULATION AND BIFURCATION...

DESIGN, SIMULATION AND BIFURCATION ANALYSIS OF A NOVEL MICROMACHINEDTUNABLE CAPACITOR WITH EXTENDED TUNABILITY

Hamed Mobki1, Kaveh Rashvand2, Saeid Afrang3, Morteza H. Sadeghi1 and Ghader Rezazadeh21Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran

2Mechanical Engineering Department, Urmia University, Urmia, Iran3Electrical Engineering Department, Urmia University, Urmia, Iran

E-mail: [email protected]; [email protected]; [email protected]

Received March 2013, Accepted September 2013No. 13-CSME-115, E.I.C. Accession 3573

ABSTRACTIn this paper, a novel RF MEMS variable capacitor has been presented. The applied techniques for increasingthe tunability of the capacitor are the increasing of the maximum capacitance and decreasing of the minimumcapacitance. The proposed structure is a simple cantilever Euler–Bernoulli micro-beam suspended betweentwo conductive plates, in which the lower plate is considered as stationary reference electrode. In thisstructure, two pedestals are located in both tips of the cantilever beam. In the capacitive micro-structures,increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system toan unstable condition (pull-in phenomenon). By deflecting the beam toward the upper (lower) plate theminimum (maximum) capacitance decreases (increases) and tunability increases consequently. The locatedpedestals increase and decrease the maximum and minimum capacitance respectively. The results show thatthe proposed structure increases the tunability of cantilever beam significantly. Furthermore, bifurcationbehavior of movable electrode has been investigated.

Keywords: tunable capacitor; tenability; instability; saddle node bifurcation; pull-in voltage.

ANALYSE DU DESIGN, DE LA SIMULATION ET DE LA BIFURCATION D’UN NOUVEAUMICRO-CONDENSATEUR ACCORDABLE À RÉGLAGE ÉTENDUE

RÉSUMÉDans cet article, un nouveau condensateur variable RF MEMS est présenté. Les techniques appliquées pouraugmenter le réglage du condensateur sont l’augmentation de la capacité maximale et la diminution de lacapacité minimale. La structure proposée est une simple micro-poutre cantilever Euler–Bernoulli, suspendueentre deux plateaux conductifs, dans lequel le plateau inférieur est considéré comme une électrode station-naire de référence. Dans cette structure, deux piédestaux sont situés aux deux bouts d’une poutre cantilever.Dans les micro-structures condensables, l’augmentation du voltage appliqué retire à la structure une rigi-dité équivalente et mène le système à un état instable (phénomène pull-in). En déviant la poutre vers leplateau supérieur (inférieur), la capacité minimale (maximale) décroît (augmente) et le réglage augmenteconséquemment. Les piédestaux augmentent et décroissent respectivement les capacités maximales et mini-males. Le résultat montre que la structure proposée augmente de façon significative le réglage de la poutrecantilever. En outre, le comportement de bifurcation d’électrodes mobiles a été étudié.

Mots-clés : condensateur réglable ; stabilité ; instabilité ; bifurcation selle nœud ; voltage pull-in.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014 15

1. INTRODUCTION

Nowadays, micro-electro mechanical systems (MEMS) have shown tremendous popularity in the engineer-ing industry because of their several advantages such as order of magnitude, smaller size, better performancethan other solutions, possibilities for batch fabrication and cost effective integration with electronic systems,virtually zero DC power consumption and potentially large reduction in power consumption [1].

MEMS-based tunable capacitors are the key component in RF integrated circuits such as tunable filter,resonators, wireless communication applications, voltage control oscillators and phase shifters [2–7].

Tunable capacitor and most of MEMS devices are generally actuated by electrostatic force due to theirsimplicity, as they require few mechanical components and small voltage levels for actuation [8]. Most ofthe MEMS which operate with this actuation, include a conductive flexural beam/plate that is suspended ona fixed conductive plate. To design and evaluate the operating condition of these structures, it is necessaryto analyze the mechanical behaviors of this flexural beam under different conditions. Therefore, electrostat-ically actuated MEM/NEM devices such as micro-phones [9] micro-switches [1], sensors [10], resonators[11], oscillators [12], and tunable capacitors [13] are widely designed, fabricated, used and analyzed.

Tunability and tuning ratio are most important parameters in the design of MEMS capacitors [14]. Thetuning range of electrostatical capacitor is limited due to the nonlinear nature of electrostatic force and pull-in phenomenon. Many structures are proposed to overcome or delay of this phenomenon and increasingthe tuning ratio. Shavezipour et al. [15] presented a novel poly-silicon-based capacitor with segmentedmoving electrode. Tunability of this structure is over 50%. In other works, they presented parallel platecapacitors to increase the tunability or linearity [7, 16]. One of the most common techniques for increasingthe tunability is to adopt different gaps for the actuation and sense [14, 17, 18]. In this case, Rijks et al.[3, 19] developed designs with tuning ranges of 700% to over 1700%. Similar designs provide tunability ashigh as 500%, where the ratio of the actuation gap and sense gap are different [14, 20]. The main drawbackfor these designs is the high sensitivity of their C-V curves to the voltage changes, especially at the pull-in voltage. At this point, the device behaves like a capacitive switch and loses its fine tunability. Largersense gaps lead to less sensitive responses and lower tunabilities [14, 21]. In most of the tunable capacitors,designers improve tunability of the capacitor by increasing the maximum capacitance and rarely designersuse decreasing of minimum capacitance method for improvement of tunability [22].

In this paper, a novel MEMS-based variable capacitor is presented. The applied techniques for improve-ment of the tunability are based on increasing of the maximum capacitance and decreasing the minimumcapacitance of the capacitor. For these aims two pedestals are located at the tip of the suspended cantileverbeam. It must be noted that the reference electrode is the lower plate. By deflecting the beam toward theupper plate, the minimum capacitance decreases and by deflecting toward the lower plate, maximum capac-itance is increased and tunability increases consequently. The located pedestals cause more increasing anddecreasing of the maximum and minimum capacitance, respectively. These conditions are accomplishedby changing the boundary condition of movable electrode from fixed-free to fixed-simple support and con-sequently, increasing the resultant stiffness. The governing nonlinear equation for static deflection of themicro-beam, based on Euler–Bernoulli beam theory has been obtained and presented. The results show thatthe proposed structure increases the capacitance tuning range, significantly.

2. MODEL DESCRIPTION

Figure 1 shows a model of an electro-mechanically tunable capacitor which consists of a suspended can-tilever over stationary conductor plate with length L, and initial gap of G1. When the voltage V1 is appliedacross the capacitor, the cantilever beam is attracted toward the stationary plate due to the resultant elec-trostatic force. As the beam is balanced between electrostatic attractive force and mechanical (elastic)restoring force, both electrostatic and elastic restoring forces are increased, when the bias voltage increases.

16 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

Fig. 1. Schematic view of a micro-beam based tunable capacitor.

Fig. 2. Schematic view of a proposed micro-structure-based tunable capacitor.

When the voltage and distance of the beam from fixed plate reaches the critical value, pull-in instabilityoccurs. In this condition, the capacitance of the capacitor reaches to maximum amount. Further increasingin the voltage will cause the structure to have sudden displacement jump, causing structural collapse andfailure.

Figure 2 shows a conceptual model for increasing the tunability of the capacitor as shown in Fig. 1. Asshown in Fig. 2 a cantilever beam is suspended between two stationary conductive plates. The dielectricmaterial with the thickness of t1 and t2 is deposited over bottom and upper plates, respectively. One end ofthe micro-beam is suspended over two pedestals with distances of G1− d1 and G2− d2 from bottom andupper plates, respectively.

Applying the bias voltage V1 in the condition of V2 = 0 causes the moving of suspended beam toward thebottom plate. Similarly, if the bias voltage V2 in condition of V1 = 0 is applied, the beam moves toward theupper plate.

The nonlinear nature of the electrostatic force causes pull-in instability in the micro-beam at a specificdistance from the upper or bottom plate. In this condition, the micro-beam loses its equivalent stiffness. Inthe proposed structure, by increasing the applied voltages, the pedestals located under or upper the micro-beam tip hold the free end of the micro-beam and change the fixed-free boundary condition of the beamto fixed-simple support condition. This change increases the stiffness of the micro-beam and allows it tocontinue moving toward plates and increasing maximum capacitance or decreasing minimum capacitanceof capacitor as a consequence.


3. MATHEMATICAL MODELING

In the initial condition and without applied voltage, the cantilever beam is in a parallel position with respectto Reference Stationary Electrode (RSE) (in this paper RSE is the bottom plate). When the voltage is applied,the beam moves. Therefore, to calculate the capacitance between the beam and RSE, it is not possible toconsider them as two parallel plates. Hence, it is necessary to find the beam shape after displacement and tocalculate the capacitance. The capacitance of the structure is calculated using the following equation:

C(V1,V2) =

L∫0

bε0εr

d(V1,V2,x)dx, (1)

where d(V1,V2,x)=G1−w(x) is the voltage dependence gap with respect to RSE and C(V1,V2) is the voltagedependence capacitance between them, b is width of the micro-beam, ε0 = 8.854×10−12C2 N−1m−2 is thepermittivity of vacuum and εr is the dielectric constant of the material within G1.

Considering the Euler–Bernoulli beam theory, the governing equation for static deflection of the movableelectrode with distributed parameters, can be presented as [23]:

EId4wdx4 = qelec(V1,w)+qelec(V2,w′), (2)

where w(x) and w′(x) are transversal deflection of the micro-beam, with respect to bottom and upper plates,respectively. qelec(V1,w) and qelec(V2,w′) are electrostatic forces, imposed from bottom and upper plates.For a wide micro-beam with thickness h, and b ≥ 5h, the effective modulus E can be approximated by theplate modulus E/(1−ν2); otherwise, E is the Young modulus E.

The applied electrostatic forces per unit length from lower and upper plates can be computed using astandard capacitance model [24] as the following equations, respectively:

qelec(V1,w) =b(εr)1ε0V 2

12(G1−w)2 , (3)

qelec(V2,w′) =b(εr)2ε0V 2

22(G2−w′)2 . (4)

Considering that the deflection of the micro-beam with respect to the lower plate equals the negative valueregarding the upper plate, hence:

w =−w′. (5)

The governing equation for static deflection of the mentioned micro-beam is

EId4wdx4 =

bε0

2

((εr)1V 2

1(G1−w)2 −

(εr)2V 22

(G2 +w)2

)≡ Felec(V1,V2,w), (6)

where Felec(V1,V2,w) indicates resultant electrostatic forces per unit length of the micro-beam. In this equa-tion, the first and second right-hand terms indicate the imposed forces from lower and upper plates to micro-beam, respectively.

The region between beam and each plate are divided to two sections. The first one is between beam anddielectric, which named by capacitor C1 and the second one, is dielectric thickness, which it is named bycapacitor C2. These capacitors are in series and the total structure capacitance between beam and bottomplates is

1Ceq

=1

C1+

1C2

,

(G1

(εr)1ε0bL

)eq=

t1ε0εrbL

+(G1− t1)

ε0bL,

(G1

(εr)1ε0

)eq=

1ε0

(G1− t1 +

t1εr

). (7)


If we consider that G1− t1 +(t1/εr) = G′1, then for the case of variable capacitor Eq. (7) changes to(G1

(εr)1ε0

)eq=

1ε0(G′1−w). (8)

Therefore, the electrostatic force between the beam and lower plate is given by

qelec(V1,w) =ε0bV 2

12(G′1−w)2 . (9)

By pursuing this procedure for beam and upper plate, the resultant force qelec(V2,w) is

qelec(V2,w) =−ε0bV 2

22(G′2 +w)2 , (10)

where G′2 is equal to

G′2 = G2− t2 +t2εr. (11)

Hence, Eq. (6) is changed as follows:

EId4wdx4 =

ε0b2

(V 2

1(G′1−w)2 −

V 22

(G′2 +w)2

)= Felec(V1,V2,w). (12)

4. NUMERICAL METHOD

Due to the nonlinear nature of electrostatic forces, an analytical solution is impractical to obtain and there-fore, a numerical solution must be implemented. A solution of the electrostatic deflections problems bynumerical techniques currently produces a cost effective design and an analysis for a variety of applications.The choice of the numerical technique, as well as the choice of difference operators, plays a major rolein the accuracy of the solutions. One method to solve the nonlinear equations is to change the governingequations into linear ones. Because of the sensitivity of the value of the electrostatic force with respect tothe micro-beam deflection, especially when the applied voltage to the electrostatic areas is increased, thelinearization with respect to the initial position may cause considerable errors. Therefore, to minimize theseerrors, a step-by-step increase of applied voltage is used and the governing equation is linearized at eachstep [25, 26]. The obtained equation is a linear ordinary differential equation that represents the variation ofdeflection along the micro-beam. The mentioned equation can be solved using Galerkin’s method [27].

5. RESULTS AND DISCUSSIONS

5.1. Validation of the Numerical MethodThe validation of the results may be investigated by comparing them with those given by Osterberg [28].The considered case study is a cantilever capacitive beam, without residual stress, and its properties aregiven in Table 1. The result of this comparison is presented in Table 2. As shown in this table, the resultsshow good agreements.

5.2. Modeling Results and TunabilityIn this section, the tunability results of the proposed structure are presented. Figure 3 shows capacitanceversus voltage for the cantilever capacitive micro-structure, which its schematic view is depicted in Fig. 1.Table 3 shows the geometrical and material properties of this beam and proposed capacitor. As shown inFig. 3, the tunability ((Cmax−Cmin)/Cmin) [14] for the cantilever beam is about 30%.


Table 1. Geometrical and material properties of micro-switch.Properties ValueLength 150 µmWidth 50 µmHeight 3 µmYoung’s modulus 169 GPaInitial gap 1 µm

Table 2. Calculated pull-in voltages for the cantilever micro-beam.Current paper CoSolve simulation [28] Closed form 2D model [28]

Vpull−in [V] 16.92 16.9 16.8

Table 3. Geometrical and material properties of the proposed capacitor.Properties ValueLength 400 µmWidth 100 µmHeight 2 µmYoung’s modulus 169 GPaG1 3 µm

Fig. 3. Capacitance versus applied voltage V1 for cantilever beam suspended over stationary conductor plate.

Figure 4 shows a C-V diagram for the proposed structure without upper stationary plate. As shown inFigs. 3 and 4, the minimum capacitance for both figures is the same. However, maximum capacitanceof Fig. 4 is more than that obtained for cantilever beam, so locating the bottom pedestal in the proposedstructure increases the maximum capacitance and tunability, consequently. Tunability for Fig. 4 is 56%.

Based on the equation (Cmax−Cmin)/Cmin, tunability of the capacitor arises with increasing the maximumcapacitance. As shown in Fig. 4 tunability of capacitor can reach high values by increasing the maximumcapacitance. Furthermore, tunability can be increased with decreasing of the minimum capacitance. In theproposed capacitor, we try to achieve the most tunability with both mentioned procedures. For the proposed


Fig. 4. Capacitance versus applied voltage for proposed capacitor without upper stratum and without dielectric layer(d1 = 1.28 µm).

Fig. 5. Capacitance versus applied voltage for proposed capacitor without dielectric layers (d1 = d2 = 1.28 µm).

capacitor, the maximum capacitance occurs when the deflection of the movable electrode is in the maximumamount with respect to the lower plate and minimum one occurs when the movable electrode is in closerlocation with respect to the upper plate. Figure 5 shows a C-V diagram of the proposed structure. Com-paring of Figs. 4 and 5 it is clear that the maximum capacitances of both diagrams are equal but minimumcapacitance of Fig. 5 is smaller. In the case of G1 = G2 = 3 µm and without dielectric layers, the tunabilityof the proposed structure is 105%.

As is known, in the electrostatically capacitors the tunability is independent from geometrical and materialparameters. For example, by decreasing the initial gap in two parallel plates of the capacitor, the maximumand minimum capacitances increase but tunability remains fixed. However, in the proposed structure, bydecreasing G1 and increasing G2, the maximum capacitance increases and the minimum capacitance de-


Fig. 6. Capacitance versus applied voltage for proposed capacitor without dielectric layers (d1 = 1.28 µm, d2 =2.56 µm).

Fig. 7. Capacitance versus applied voltage for proposed capacitor with upper dielectric layer (d1 = 1.28 µm, d2 =2.6 µm, t2 = 2 µm).

creases, respectively. This condition is phenomenal in the micro-capacitor world. In the rest of this paperthe condition that reduces Cmin is discussed. It is obvious that decreasing in G1 arises Cmax and tunability.Figure 6 presents a C-V diagram for the proposed capacitor with G2 = 6 µm. From Figs. 5 and 6 it is notedthat Cmin decreases from 9.14×10−13F to 7.44×10−13F and tunability increases from 105 to 147%.

As shown in Fig. 6, by increasing G2 the amount of applied voltage V2 is raised. In order to decrease theamount of V2, the upper dielectric with a thickness of t2 is applied.

Figure 7 shows a C-V diagram of the capacitor with G2 = 6 µm, εr = 100 and t2 = 2 µm. As shown inthis figure, utilizing the upper dielectric layer decreases the maximum applied voltage V2 from 67.8 to 36.7.

According to Fig. 7, utilizing the dielectric layer causes a trivial increase of Cmin. This phenomenonoccurs due to instability of the movable electrode in closer (further) distance from the lower (upper) plate.


Fig. 8. Capacitance versus applied voltage for proposed capacitor with upper dielectric layer (d1 = 1.28 µm, d2 =2.58 µm, t2 = 1 µm).

This condition is proved below in the discussion of the results.Locating dielectric layers over two fixed plates reduces the maximum amount of applied voltages V1 and

V2. Figure 8 shows a C-V curve of the capacitor with G1 = G2 = 3 µm and t1 = t2 = 1 µm. From Figs. 5and 8 it is noted that the maximum amount of applied voltages decreases from 24.5 to 16.1.

5.3. Stability Analysis of Movable ElectrodeIn this section, the stability analysis of the micro-beam without lower pedestal is investigated. For simplicity,the structure which is shown in Fig. 2 may be considered as lumped model. Hence, Eq. (12) is changed to

keqy =ε0bLV 2

12(G1− t + t/εr− y)2 , (13)

where keq is the equivalent elasticity stiffness of the micro-beam. The equivalent stiffnesses for fixed-fixedand cantilever micro-beams are equal to 384EI/L3 and 8EI/L3, respectively, and y is the deflection of thebeam in the lumped model [29]. Considering t− (t/εr) = β > 0 and G1−y = d, the above equation may berewritten as

keq(G1−d) =ε0bLV 2

12(d−β )2 , (14)

where d is the beam distance with respect to the lower plate. The applied voltage according to d is

V1 =√

α(G1−d)(d−β )2, (15)

where α = 2keq/ε0bL.The instability distance of the micro-beam with respect to the lower plate is obtained by setting

d(V1)/d(d) = 0 in Eq. (16). Solving this relation, the instability distance of the micro-beam from thelower plate is extracted, which is

d =2G1

3+

β

3, (16)


and the corresponding applied voltage is obtained by substituting Eq. (16) in Eq. (15) as follows:

V1 =

√8keq

27ε0bL(G1−β )3. (17)

For capacitance without dielectric layer, because of εr = 1, Eq. (13) changes into

keqy =ε0bLV 2

12(G1− y)2 . (18)

By perusing the above procedures for the case without dielectric layer, the instability distance and corre-sponding applied voltage is obtained as

d =2G1

3, (19a)

V1 =

√8keq

27ε0bL(G1)3, (19b)

which are the same as the results reported by Shavezipur [14].Comparing Eq. (19b) with Eq. (17), it is obvious that by locating the dielectric layer, the threshold volt-

age is decreased and comparing Eq. (19a) with Eq. (16), it is clear that by utilizing this layer the minimumdistance of beam with respect to the lower plate is increased and maximum capacitance is decreased, con-sequently. Generalizing this theorem for the upper plate, it is found that with locating dielectric layer andimposing V2, the maximum distance of beam respect to lower plate is decreased and minimum capacitanceis increased, consequently. This theorem may be proved from bifurcation view point. To this end, the mo-tion equation of the micro-beam based lumped model must be obtained. Adding the term m(d2y/dτ2) to theleft-hand side of Eq. (13), the dynamic equation is found as

md2ydτ2 + keqy =

ε0bLV 21

2(G1− t + t/εr− y)2 ≡ε0bLV 2

12(G′1− y)2 , (20)

where m represents mass of the beam and τ is time. By setting w = y, Eq. (20) may be transformed into thefollowing form:

dydτ

= w,

mdwdτ

=ε0bLV 2

12(G′1− y)2 − keqy.

(21)

At the equilibrium points, the micro-beam is at rest, hence considering Eq. (21), the equilibrium pointsare obtained by

w = 0,

ε0bLV 21

2(G′1− y)2 − keqy = 0.(22)

According to Eq. (22), in order to obtain fixed points, the following equation must be solved:

f (V1,y) = ε0bLV 21 −2keqy(G′1− y)2 = 0. (23)

The equilibrium points for the capacitor, with α = 636.25×1018, t1 = 1 µm, G1 = 3 µm and (εr)1 = 100has been obtained using the above mentioned procedure. Position of the fixed points in the state-control


Fig. 9. Bifurcation diagram for the capacitor with dielectric layer.

Fig. 10. Bifurcation diagram for the capacitor without dielectric layer.

space versus applied voltage of V1 as a control parameter (bifurcation diagram); for the case of with andwithout dielectric layer are illustrated in Figs. 9 and 10, respectively.

In order to check the stability in the vicinity of each equilibrium point, the following Jacobian matrix isused [30]:

J =

[0 1

εobLV 21

(G′1−y)3 − keq 0

], (24)


Fig. 11. Phase diagram with given V1 = 2V with dielectric layer.

where eigenvalues of the Jacobian satisfy

λ2− εobLV 2

1(G′1− y)3 + keq = 0. (25)

For λ 2 < 0, it has two pure imaginary roots, which means that the equilibrium point (y1,V1) is a center point.Applying the same method to the other equilibrium point (y2,V1), its eigenvalues satisfy λ 2 > 0, then it hastwo real eigenvalues, one is positive, and the other one is negative. This means that the equilibrium pointis an unstable saddle point [30]. Using this method, the stability in the vicinity of each equilibrium pointcan be identified. In Figs. 9 and 10, dashed and continuous curves represent unstable and stable branches,respectively.

As shown in Fig. 9 for a given V < Vpull−in there exist three fixed points in which the third one cannotexist physically due to position of dielectric layer. Based on Eq. (24), the first and third fixed points arestable centers and the second one is an unstable saddle node. As shown in Fig. 10, two fixed points exist,where the first one is a stable center and the other is an unstable saddle node. As shown in these figures,by increasing the controlling parameter V1, the distance between two physically fixed points decreases andfor a certain voltage, which is called a “pull-in voltage” in the MEMS literature, they meet in a saddle nodebifurcation.

Figures 11 and 12 present motion trajectories of the micro-beam for V1 = 2V with different initial valuesfor the case with and without dielectric layer.

As shown in these figures, there are a basin of periodic set and a region of repulsion of unstable saddlenode. Of course, it must be noted that the substrate position and dielectric layer act as a singular point andvelocity of the system near these singular points tends to infinity, for cases with and without dielectric layer.The basin of periodic of the stable center is bounded by a closed orbit (homoclinic orbit). Depending on thelocation of the initial condition, the system can be stable or unstable. For these figures, continuous, dashedand bold curves represent periodic, unstable and Homoclinic trajectories for the phase portraits, respectively.

6. CONCLUSIONS

A new structure to increase the tunability of RF variable MEM capacitor was presented. The governingequation of static deflection for capacitive beam was obtained and presented. Due to the nonlinearity of


Fig. 12. Phase diagram with given V1 = 2V without dielectric layer.

governing equation, it was linearized and solved using SSLM and Galerkin weighted residual method. Tun-ability results of novel structure was obtained and presented. It was shown that by decreasing the minimumcapacitance and increasing the maximum capacitance, more tunability could be achieved. Furthermore, itwas shown that locating of upper and lower pedestals could be effective for further increasing and decreas-ing the maximum and minimum capacitance. It was also shown that, in spite of many tunable capacitors,tunability of the proposed structure is dependent on lower and upper gap size. The effect of increasing theupper gap on tunability was investigated and it was shown that by increasing this parameter, the tunabilityand applied voltage V2 were increased. Furthermore, it was noted that with a decrease of the lower gap,the tunability was increased too. To decrease the applied voltages, two dielectric layers were depositedover stationary plates. Results showed that employing dielectric layers decreased the applied voltages ofthe capacitor. Instability distance of movable electrode from lower plate in the presence of dielectric layerand without it was investigated from a bifurcation viewpoint. Bifurcation results proved that saddle nodebifurcation occurs by applying threshold voltage.

ACKNOWLEDGEMENT

The authors are grateful for the helpful comments of Ms. Aseman Sabet, PhD student at Université deMontréal for the French version of the abstract and useful editorial changes in the article.

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