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

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1 DESIGN, SIMULATION AND BIFURCATION ANALYSIS OF A NOVEL MICROMACHINED TUNABLE CAPACITOR WITH EXTENDED TUNABILITY Hamed Mobki 1, Kaveh Rashvand 2, Saeid Afrang 3, Morteza H. Sadeghi 1 and Ghader Rezazadeh 2 1 Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran 2 Mechanical Engineering Department, Urmia University, Urmia, Iran 3 Electrical Engineering Department, Urmia University, Urmia, Iran hamedmobki@live.com; kaveh.rashvand@gmail.com; g.rezazadeh@urmia.ac.ir Received March 2013, Accepted September 2013 No. 13-CSME-115, E.I.C. Accession 3573 ABSTRACT In this paper, a novel RF MEMS variable capacitor has been presented. The applied techniques for increasing the tunability of the capacitor are the increasing of the maximum capacitance and decreasing of the minimum capacitance. The proposed structure is a simple cantilever Euler Bernoulli micro-beam suspended between two conductive plates, in which the lower plate is considered as stationary reference electrode. In this structure, 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 to an unstable condition (pull-in phenomenon). By deflecting the beam toward the upper (lower) plate the minimum (maximum) capacitance decreases (increases) and tunability increases consequently. The located pedestals increase and decrease the maximum and minimum capacitance respectively. The results show that the proposed structure increases the tunability of cantilever beam significantly. Furthermore, bifurcation behavior 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 NOUVEAU MICRO-CONDENSATEUR ACCORDABLE À RÉGLAGE ÉTENDUE RÉSUMÉ Dans cet article, un nouveau condensateur variable RF MEMS est présenté. Les techniques appliquées pour augmenter le réglage du condensateur sont l augmentation de la capacité maximale et la diminution de la capacité minimale. La structure proposée est une simple micro-poutre cantilever Euler Bernoulli, suspendue entre deux plateaux conductifs, dans lequel le plateau inférieur est considéré comme une électrode stationnaire 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 rigidité équivalente et mène le système à un état instable (phénomène pull-in). En déviant la poutre vers le plateau supérieur (inférieur), la capacité minimale (maximale) décroît (augmente) et le réglage augmente conséquemment. Les piédestaux augmentent et décroissent respectivement les capacités maximales et minimales. Le résultat montre que la structure proposée augmente de façon significative le réglage de la poutre cantilever. 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,

2 1. INTRODUCTION Nowadays, micro-electro mechanical systems (MEMS) have shown tremendous popularity in the engineering industry because of their several advantages such as order of magnitude, smaller size, better performance than 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 their simplicity, as they require few mechanical components and small voltage levels for actuation [8]. Most of the MEMS which operate with this actuation, include a conductive flexural beam/plate that is suspended on a fixed conductive plate. To design and evaluate the operating condition of these structures, it is necessary to analyze the mechanical behaviors of this flexural beam under different conditions. Therefore, electrostatically 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]. The tuning range of electrostatical capacitor is limited due to the nonlinear nature of electrostatic force and pullin phenomenon. Many structures are proposed to overcome or delay of this phenomenon and increasing the tuning ratio. Shavezipour et al. [15] presented a novel poly-silicon-based capacitor with segmented moving electrode. Tunability of this structure is over 50%. In other works, they presented parallel plate capacitors to increase the tunability or linearity [7, 16]. One of the most common techniques for increasing the 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 as high as 500%, where the ratio of the actuation gap and sense gap are different [14, 20]. The main drawback for these designs is the high sensitivity of their C-V curves to the voltage changes, especially at the pullin voltage. At this point, the device behaves like a capacitive switch and loses its fine tunability. Larger sense 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 designers use 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 improvement of the tunability are based on increasing of the maximum capacitance and decreasing the minimum capacitance of the capacitor. For these aims two pedestals are located at the tip of the suspended cantilever beam. It must be noted that the reference electrode is the lower plate. By deflecting the beam toward the upper plate, the minimum capacitance decreases and by deflecting toward the lower plate, maximum capacitance is increased and tunability increases consequently. The located pedestals cause more increasing and decreasing of the maximum and minimum capacitance, respectively. These conditions are accomplished by changing the boundary condition of movable electrode from fixed-free to fixed-simple support and consequently, increasing the resultant stiffness. The governing nonlinear equation for static deflection of the micro-beam, based on Euler Bernoulli beam theory has been obtained and presented. The results show that the 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 cantilever over stationary conductor plate with length L, and initial gap of G 1. When the voltage V 1 is applied across the capacitor, the cantilever beam is attracted toward the stationary plate due to the resultant electrostatic 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

3 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 instability occurs. In this condition, the capacitance of the capacitor reaches to maximum amount. Further increasing in the voltage will cause the structure to have sudden displacement jump, causing structural collapse and failure. Figure 2 shows a conceptual model for increasing the tunability of the capacitor as shown in Fig. 1. As shown in Fig. 2 a cantilever beam is suspended between two stationary conductive plates. The dielectric material with the thickness of t 1 and t 2 is deposited over bottom and upper plates, respectively. One end of the micro-beam is suspended over two pedestals with distances of G 1 d 1 and G 2 d 2 from bottom and upper plates, respectively. Applying the bias voltage V 1 in the condition of V 2 = 0 causes the moving of suspended beam toward the bottom plate. Similarly, if the bias voltage V 2 in condition of V 1 = 0 is applied, the beam moves toward the upper plate. The nonlinear nature of the electrostatic force causes pull-in instability in the micro-beam at a specific distance from the upper or bottom plate. In this condition, the micro-beam loses its equivalent stiffness. In the proposed structure, by increasing the applied voltages, the pedestals located under or upper the microbeam tip hold the free end of the micro-beam and change the fixed-free boundary condition of the beam to fixed-simple support condition. This change increases the stiffness of the micro-beam and allows it to continue moving toward plates and increasing maximum capacitance or decreasing minimum capacitance of capacitor as a consequence. Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

4 3. MATHEMATICAL MODELING In the initial condition and without applied voltage, the cantilever beam is in a parallel position with respect to 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 to consider them as two parallel plates. Hence, it is necessary to find the beam shape after displacement and to calculate the capacitance. The capacitance of the structure is calculated using the following equation: L bε 0 ε r C(V 1,V 2 ) = dx, (1) d(v 1,V 2,x) 0 where d(v 1,V 2,x) = G 1 w(x) is the voltage dependence gap with respect to RSE and C(V 1,V 2 ) is the voltage dependence capacitance between them, b is width of the micro-beam, ε 0 = C 2 N 1 m 2 is the permittivity of vacuum and ε r is the dielectric constant of the material within G 1. Considering the Euler Bernoulli beam theory, the governing equation for static deflection of the movable electrode with distributed parameters, can be presented as [23]: ẼI d4 w dx 4 = q elec(v 1,w) + q elec (V 2,w ), (2) where w(x) and w (x) are transversal deflection of the micro-beam, with respect to bottom and upper plates, respectively. q elec (V 1,w) and q elec (V 2,w ) are electrostatic forces, imposed from bottom and upper plates. For a wide micro-beam with thickness h, and b 5h, the effective modulus Ẽ can be approximated by the plate modulus E/(1 ν 2 ); otherwise, Ẽ is the Young modulus E. The applied electrostatic forces per unit length from lower and upper plates can be computed using a standard capacitance model [24] as the following equations, respectively: q elec (V 1,w) = b(ε r) 1 ε 0 V 2 1 2(G 1 w) 2, (3) q elec (V 2,w ) = b(ε r) 2 ε 0 V 2 2 2(G 2 w ) 2. (4) Considering that the deflection of the micro-beam with respect to the lower plate equals the negative value regarding the upper plate, hence: w = w. (5) The governing equation for static deflection of the mentioned micro-beam is ẼI d4 w dx 4 = bε ( 0 (εr ) 1 V1 2 2 (G 1 w) 2 (ε r) 2 V2 2 ) (G 2 + w) 2 F elec (V 1,V 2,w), (6) where F elec (V 1,V 2,w) indicates resultant electrostatic forces per unit length of the micro-beam. In this equation, the first and second right-hand terms indicate the imposed forces from lower and upper plates to microbeam, respectively. The region between beam and each plate are divided to two sections. The first one is between beam and dielectric, which named by capacitor C 1 and the second one, is dielectric thickness, which it is named by capacitor C 2. These capacitors are in series and the total structure capacitance between beam and bottom plates is 1 C eq = 1 C C 2, ( G 1 (ε r ) 1 ε 0 bl ) eq = t 1 ε 0 ε r bl + (G 1 t 1 ) ε 0 bl, ( G1 (ε r ) 1 ε 0 ) eq = 1 ( G 1 t 1 + t ) 1. (7) ε 0 ε r 18 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

5 If we consider that G 1 t 1 + (t 1 /ε r ) = G 1, then for the case of variable capacitor Eq. (7) changes to ( ) G1 = 1 (G 1 w). (8) (ε r ) 1 ε 0 ε 0 Therefore, the electrostatic force between the beam and lower plate is given by eq q elec (V 1,w) = ε 0bV1 2 2(G. (9) 1 w)2 By pursuing this procedure for beam and upper plate, the resultant force q elec (V 2,w) is q elec (V 2,w) = ε 0bV2 2 2(G, (10) 2 + w)2 where G 2 is equal to G 2 = G 2 t 2 + t 2 ε r. (11) Hence, Eq. (6) is changed as follows: ẼI d4 w dx 4 = ε ( 0b 2 4. NUMERICAL METHOD V 2 1 (G 1 w)2 V 2 2 (G 2 + w)2 ) = F elec (V 1,V 2,w). (12) Due to the nonlinear nature of electrostatic forces, an analytical solution is impractical to obtain and therefore, a numerical solution must be implemented. A solution of the electrostatic deflections problems by numerical 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 role in the accuracy of the solutions. One method to solve the nonlinear equations is to change the governing equations into linear ones. Because of the sensitivity of the value of the electrostatic force with respect to the micro-beam deflection, especially when the applied voltage to the electrostatic areas is increased, the linearization with respect to the initial position may cause considerable errors. Therefore, to minimize these errors, a step-by-step increase of applied voltage is used and the governing equation is linearized at each step [25, 26]. The obtained equation is a linear ordinary differential equation that represents the variation of deflection 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 Method The 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 are given in Table 1. The result of this comparison is presented in Table 2. As shown in this table, the results show good agreements Modeling Results and Tunability In this section, the tunability results of the proposed structure are presented. Figure 3 shows capacitance versus 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 in Fig. 3, the tunability ((C max C min )/C min ) [14] for the cantilever beam is about 30%. Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

6 Table 1. Geometrical and material properties of micro-switch. Properties Value Length 150 µm Width 50 µm Height 3 µm Young s modulus 169 GPa Initial gap 1 µm Table 2. Calculated pull-in voltages for the cantilever micro-beam. Current paper CoSolve simulation [28] Closed form 2D model [28] V pull in [V] Table 3. Geometrical and material properties of the proposed capacitor. Properties Value Length 400 µm Width 100 µm Height 2 µm Young s modulus 169 GPa G 1 3 µm Fig. 3. Capacitance versus applied voltage V 1 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 in Figs. 3 and 4, the minimum capacitance for both figures is the same. However, maximum capacitance of Fig. 4 is more than that obtained for cantilever beam, so locating the bottom pedestal in the proposed structure increases the maximum capacitance and tunability, consequently. Tunability for Fig. 4 is 56%. Based on the equation (C max C min )/C min, tunability of the capacitor arises with increasing the maximum capacitance. As shown in Fig. 4 tunability of capacitor can reach high values by increasing the maximum capacitance. Furthermore, tunability can be increased with decreasing of the minimum capacitance. In the proposed capacitor, we try to achieve the most tunability with both mentioned procedures. For the proposed 20 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

7 Fig. 4. Capacitance versus applied voltage for proposed capacitor without upper stratum and without dielectric layer (d 1 = 1.28 µm). Fig. 5. Capacitance versus applied voltage for proposed capacitor without dielectric layers (d 1 = d 2 = 1.28 µm). capacitor, the maximum capacitance occurs when the deflection of the movable electrode is in the maximum amount with respect to the lower plate and minimum one occurs when the movable electrode is in closer location with respect to the upper plate. Figure 5 shows a C-V diagram of the proposed structure. Comparing of Figs. 4 and 5 it is clear that the maximum capacitances of both diagrams are equal but minimum capacitance of Fig. 5 is smaller. In the case of G 1 = G 2 = 3 µm and without dielectric layers, the tunability of the proposed structure is 105%. As is known, in the electrostatically capacitors the tunability is independent from geometrical and material parameters. For example, by decreasing the initial gap in two parallel plates of the capacitor, the maximum and minimum capacitances increase but tunability remains fixed. However, in the proposed structure, by decreasing G 1 and increasing G 2, the maximum capacitance increases and the minimum capacitance de- Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

8 Fig. 6. Capacitance versus applied voltage for proposed capacitor without dielectric layers (d 1 = 1.28 µm, d 2 = 2.56 µm). Fig. 7. Capacitance versus applied voltage for proposed capacitor with upper dielectric layer (d 1 = 1.28 µm, d 2 = 2.6 µm, t 2 = 2 µm). creases, respectively. This condition is phenomenal in the micro-capacitor world. In the rest of this paper the condition that reduces C min is discussed. It is obvious that decreasing in G 1 arises C max and tunability. Figure 6 presents a C-V diagram for the proposed capacitor with G 2 = 6 µm. From Figs. 5 and 6 it is noted that C min decreases from F to F and tunability increases from 105 to 147%. As shown in Fig. 6, by increasing G 2 the amount of applied voltage V 2 is raised. In order to decrease the amount of V 2, the upper dielectric with a thickness of t 2 is applied. Figure 7 shows a C-V diagram of the capacitor with G 2 = 6 µm, ε r = 100 and t 2 = 2 µm. As shown in this figure, utilizing the upper dielectric layer decreases the maximum applied voltage V 2 from 67.8 to According to Fig. 7, utilizing the dielectric layer causes a trivial increase of C min. This phenomenon occurs due to instability of the movable electrode in closer (further) distance from the lower (upper) plate. 22 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

9 Fig. 8. Capacitance versus applied voltage for proposed capacitor with upper dielectric layer (d 1 = 1.28 µm, d 2 = 2.58 µm, t 2 = 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 V 1 and V 2. Figure 8 shows a C-V curve of the capacitor with G 1 = G 2 = 3 µm and t 1 = t 2 = 1 µm. From Figs. 5 and 8 it is noted that the maximum amount of applied voltages decreases from 24.5 to Stability Analysis of Movable Electrode In 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 k eq y = ε 0 blv 2 1 2(G 1 t +t/ε r y) 2, (13) where k eq is the equivalent elasticity stiffness of the micro-beam. The equivalent stiffnesses for fixed-fixed and cantilever micro-beams are equal to 384ẼI/L 3 and 8ẼI/L 3, respectively, and y is the deflection of the beam in the lumped model [29]. Considering t (t/ε r ) = β > 0 and G 1 y = d, the above equation may be rewritten as k eq (G 1 d) = ε 0bLV 2 1 2(d β) 2, (14) where d is the beam distance with respect to the lower plate. The applied voltage according to d is V 1 = α(g 1 d)(d β) 2, (15) where α = 2k eq /ε 0 bl. The instability distance of the micro-beam with respect to the lower plate is obtained by setting d(v 1 )/d(d) = 0 in Eq. (16). Solving this relation, the instability distance of the micro-beam from the lower plate is extracted, which is d = 2G β 3, (16) Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

10 and the corresponding applied voltage is obtained by substituting Eq. (16) in Eq. (15) as follows: 8k eq V 1 = 27ε 0 bl (G 1 β) 3. (17) For capacitance without dielectric layer, because of ε r = 1, Eq. (13) changes into k eq y = ε 0bLV 2 1 2(G 1 y) 2. (18) By perusing the above procedures for the case without dielectric layer, the instability distance and corresponding applied voltage is obtained as d = 2G 1 3, (19a) V 1 = 8k eq 27ε 0 bl (G 1) 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 voltage is decreased and comparing Eq. (19a) with Eq. (16), it is clear that by utilizing this layer the minimum distance of beam with respect to the lower plate is increased and maximum capacitance is decreased, consequently. Generalizing this theorem for the upper plate, it is found that with locating dielectric layer and imposing V 2, the maximum distance of beam respect to lower plate is decreased and minimum capacitance is increased, consequently. This theorem may be proved from bifurcation view point. To this end, the motion equation of the micro-beam based lumped model must be obtained. Adding the term m(d 2 y/dτ 2 ) to the left-hand side of Eq. (13), the dynamic equation is found as m d2 y dτ 2 + k eqy = ε 0 blv 1 2 2(G 1 t +t/ε r y) 2 ε 0bLV 1 2 2(G, (20) 1 y)2 where m represents mass of the beam and τ is time. By setting w = ẏ, Eq. (20) may be transformed into the following form: dy dτ = w, m dw dτ = ε 0bLV 1 2 (21) 2(G 1 k eqy. y)2 At the equilibrium points, the micro-beam is at rest, hence considering Eq. (21), the equilibrium points are obtained by w = 0, ε 0 blv 1 2 2(G 1 k (22) eqy = 0. y)2 According to Eq. (22), in order to obtain fixed points, the following equation must be solved: f (V 1,y) = ε 0 blv 2 1 2k eq y(g 1 y) 2 = 0. (23) The equilibrium points for the capacitor, with α = , t 1 = 1 µm, G 1 = 3 µm and (ε r ) 1 = 100 has been obtained using the above mentioned procedure. Position of the fixed points in the state-control 24 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

11 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 V 1 as a control parameter (bifurcation diagram); for the case of with and without 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 is used [30]: [ ] 0 1 J = ε o blv 1 2, (24) k (G eq 0 1 y)3 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

12 Fig. 11. Phase diagram with given V 1 = 2V with dielectric layer. where eigenvalues of the Jacobian satisfy λ 2 ε oblv 2 1 (G 1 y)3 + k eq = 0. (25) For λ 2 < 0, it has two pure imaginary roots, which means that the equilibrium point (y 1,V 1 ) is a center point. Applying the same method to the other equilibrium point (y 2,V 1 ), its eigenvalues satisfy λ 2 > 0, then it has two real eigenvalues, one is positive, and the other one is negative. This means that the equilibrium point is an unstable saddle point [30]. Using this method, the stability in the vicinity of each equilibrium point can 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 < V pull in there exist three fixed points in which the third one cannot exist physically due to position of dielectric layer. Based on Eq. (24), the first and third fixed points are stable 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 V 1, the distance between two physically fixed points decreases and for a certain voltage, which is called a pull-in voltage in the MEMS literature, they meet in a saddle node bifurcation. Figures 11 and 12 present motion trajectories of the micro-beam for V 1 = 2V with different initial values for 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 saddle node. Of course, it must be noted that the substrate position and dielectric layer act as a singular point and velocity 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 the location of the initial condition, the system can be stable or unstable. For these figures, continuous, dashed and 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 governing equation of static deflection for capacitive beam was obtained and presented. Due to the nonlinearity of 26 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

13 Fig. 12. Phase diagram with given V 1 = 2V without dielectric layer. governing equation, it was linearized and solved using SSLM and Galerkin weighted residual method. Tunability results of novel structure was obtained and presented. It was shown that by decreasing the minimum capacitance and increasing the maximum capacitance, more tunability could be achieved. Furthermore, it was shown that locating of upper and lower pedestals could be effective for further increasing and decreasing 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 the upper gap on tunability was investigated and it was shown that by increasing this parameter, the tunability and applied voltage V 2 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 deposited over stationary plates. Results showed that employing dielectric layers decreased the applied voltages of the capacitor. Instability distance of movable electrode from lower plate in the presence of dielectric layer and without it was investigated from a bifurcation viewpoint. Bifurcation results proved that saddle node bifurcation occurs by applying threshold voltage. ACKNOWLEDGEMENT The authors are grateful for the helpful comments of Ms. Aseman Sabet, PhD student at Université de Montréal for the French version of the abstract and useful editorial changes in the article. REFERENCES 1. Mobki, H., Rezazadeh, G., Sadeghi, M., Vakili-Tahami, F. and Seyyed-Fakhrabadi, M-M., A comprehensive study of stability in an electro-statically actuated micro-beam, International Journal of Non-Linear Mechanics, Vol. 48, pp , Fang, D.M., Jing, X.M., Wang, P.H. and Zhao, X.L., Fabrication and dynamic analysis of the electrostatically actuated MEMS variable capacitor, Microsystem Technologies, Vol. 14, No. 3, pp , Rijks, T.G., van Beek, J., Steeneken, P.G., Ulenaers, M., De Coster, J., Puers, R. and. Res, P., RF MEMS tunable capacitors with large tuning ratio, 17th IEEE international conference, Eindhoven, The Netherlands, Abbaspour-Tamijani, A., Dussort, L. and Rebeiz, G., Miniature and tunable filters using MEMS capacitors, IEEE Transactions on Microwave Theory and Techniques, Vol. 51, No. 7, pp , Rebeiz, G., Guan-Leng, T. and Hayden, J., RF MEMS phase shifters: Design and applications, IEEE Microwave Magazine, Vol. 3, No. 2, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1,

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