CHAPTER 5 FIXED GUIDED BEAM ANALYSIS

77 CHAPTER 5 FIXED GUIDED BEAM ANALYSIS 5.1 INTRODUCTION Fixed guided clamped and cantilever beams have been designed and analyzed using ANSYS and their performance were calculated. Maximum deflection and pull-in voltages were also calculated and compared. Marc Dequesnes et al (2002) estimated the pull-in voltage characteristics based on a lumped model was derived to compute the pull-in voltage of cantilever and fixed fixed switches. Jeong-Soo Lee (2005) has characterized a sub-micron high aspect ratio metallic electrothermal actuators using a combination of electron beam lithography and electroplating techniques. Electro-thermo-mechanical FEA of the actuator using ANSYS was carried out and the FEA results were measured. Patrick Chu et al (1995) have demonstrated the dynamics of a polysilicon parallel-plate electrostatic actuator using a laser interferometer. Preliminary data had been used to verify the accuracy of a nonlinear model for the actuator. Fabricated structures having 100 µm gaps can be closed electrostatically with voltages. The fixed guided clamped and cantilever microbeams are shown in the Figure 5.1(a) and (b) respectively. They are two metal structures separated by an air gap. A bias voltage is applied between the metal structures, which

78 results in a separation of charges between them. This produces an electrostatic force that can be used to decrease the gap between the plates (Stephen D.Senturia 1997). Movable electrode Fixed electrode Figure 5.1(a) Structure of fixed guided clamped beam V Figure 5.1 (b) Structure of fixed guided cantilever beam The two plates have an overlapping area of A and spacing of d. The dielectric constant or relative electrical permittivity of the media between the two plates is denoted r. The permittivity of the media is = r o, where o is the permittivity of the vacuum. With the voltage applied between the two fixed beams (plates), an electrostatic attraction force is developed. For each beam, with W as the distributed force, Young s modulus E and moment of inertia I, the maximum displacement is given by

79 3 Wl d (5.1) 192EI And for cantilever beams is given by 3 Wl d (5.2) 3EI developed. When a voltage is applied, an electrostatic force F electric is Movable beam Fmechanical A F electrical V Anchored beam Figure 5.2 Schematic of coupled electromechanical model of fixed guided beam This force will tend to decrease the gap and gives displacement and mechanical restoring force. Under static equilibrium, the mechanical restoring force has an equal magnitude but opposite direction as the electrostatic force (Yang Xu 2009) as shown in Figure 5.2. At a particular bias voltage, mechanical restoring forces k m and the electrostatic force balance each other. The magnitude of electric force constant equals the mechanical force constant. The effective force constant of the spring is zero. The bias voltage invokes such a condition is called the pullin voltage V p. If the bias voltage is increased beyond V p, the equilibrium

80 position disappears. The electrostatic force continues to rise while the mechanical force increases linearly only. The two beams are pulled against each other and they make contact. This condition is called pull-in or snaps in. At the pull-in conditions shown in the Figure 5.3(a) and (b) the magnitudes of electrical force and mechanical force and can be equated as Figure 5.3(a) Deformed condition of fixed guided clamped beam Figure 5.3(b) Deformed condition of fixed guided cantilever beam The pull-in voltage for fixed guided microbeam of length L, width w and air gap g is given by Marc Dequesnes et al (2002). V p 3 8 Kx (5.3) 27 wl 0 5.2 COMPUTATIONAL TECHNIQUE-REDUCED ORDER MODELING There is a lack for comprehensive models and efficient design tools to simulate the behavior of electrically actuated microbeams in MEMS

81 devices. The use of FEM method in this field remains limited for two reasons. First, the use of FEM codes to simulate MEMS devices is prohibitively cumbersome, expensive and time consuming. Consequently, it is very expensive to close the loop on an FEM model of a device to allow for the design of feedback control laws or to use the model in system-level simulations. As a result, FEM models are mostly used to analyze the performance of finished devices rather than to design them. Second, FEM models use numerous variables to represent the device state. This approach makes the process of mapping the design space complex. Also, the relationship between each of these variables and the overall device performance is not clear to designers. It would be easier and more intuitive for the designer to explore the design space if the model had only a few variables with a clear relationship between them and the overall device performance. The mechanical deformations can be computed using either linear or non linear theory (Gang Li et al 2001). Reduced-order models, also called macro models, lend themselves very well to these purposes. Models constructed using this approach seeks to capture the most significant characteristics of a device behavior in a few variables governed by a few ordinary-differential equations of motion. The variables can be selected to represent physically meaningful quantities. The resulting system is typically easy to simulate as a standalone model or integrate into system-level simulations. A novel technique to generate reduced- order models in MEMS devices geared for efficient, accurate and fast simulation is used. Design parameters are included in the model by lumping them into non dimensional parameters, thereby allowing for an easier

82 understanding of their effects and the interaction between the mechanical and electrical forces. The model treats the devices as distributed-parameter systems and accounts for moderately large deflections, dynamic loads and coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and nonlinear electric forces generated by capacitors. In reduced-order model, the basis functions (modes) can be extracted from the simulated data and then reformulate the problem in terms of these modes to significantly reduce the complexity of the original problem. The system s dynamic behavior is usually dominated by only a few modes, the total number of degrees of freedom can be reduced significantly in a ROM simulation. By approximating the unknown solution in terms of the few modes that are known to be dominant, the ROM simulation is able to drastically reduce the computation cost without significantly compromising on the accuracy of the solution. The development of reduced-order models involves two important steps. The first is the generation of global basis functions and the second is the solution of the governing equations using the global basis functions. The basis functions can be obtained from the solution of a certain eigen-problem or by Fourier decompositions. Here in the simulation, the basis functions are generated using Eigen values (Mohammad I. Younis et al 2003). 5.3 EIGEN VALUE ANALYSIS The most common type of dynamic analysis for structures is the natural frequency or eigen value analysis. The natural frequencies of vibration

83 and the corresponding mode shapes for a structure are of interest and for fixed-fixed beams is given by Chang Liu et al (2006). f 1 13.86 EIG (5.4) 3 2 Fl For cantilever beams natural frequencies is given by f 1 1.732 EIG (5.5) 3 2 Fl given by With the inclusion of weight of the beams, the natural frequency is f k EIG Wl n (5.6) n 4 2 where K 1 =22.4 and K 2 =61.7 for fixed-fixed beams K 1 =3.52 and K 2 =22.0 for cantilever beams. K 1 =15.4 and K 2 =50.0 for fixed guided beams. The macromodel is used to calculate the natural frequencies of the resonant microbeams. For a given voltage, the static solution corresponding to the lower branch is substituted into the Jacobian matrix and the corresponding eigen values are found. Then by taking the square root of the magnitudes of the individual eigen values, the natural frequencies of the device are obtained. Using ROM and the ANSYS, the natural frequency can be obtained to estimate the performance of both the actuators to find their capacity to use them in optical MEMS switches.

84 5.4 RESULTS AND DISCUSSION The electrostatic actuators with fixed guided clamped-clamped beam and the cantilever beam structures have been simulated using ANSYS and the maximum deflection, pull-in voltages and natural frequencies have been estimated using ROM analysis. The inplane deflection was carried out for both fixed-fixed and cantilever beams. Flexibility is achieved by modeling the beam with hinges of length 10 m and 2 m thickness at both sides of the fixed microbeam and at one side for cantilever beam of length 100 m respectively. The non linear analysis is carried out for both the beams to study the performance of the microbeams. Different materials that are suitable in the wavelength region of optical MEMS are considered to analyze the performance of the microbeams. Silicon, gold and aluminium are the materials used and the pull-in voltages are compared. The results shows that in fixed guided microbeams, the pull-in voltages have reduced and is shown in Table 5.1 for fixed guided clamped microbeam and Table 5.2 fixed guided cantilever beams as compared with normal beams. The simulated outputs of the flexible clamped-clamped beam and cantilever electrostatic actuators are shown in Figure 5.4(a) and (b) respectively. The capacitance between the beams is extracted which forms base to estimate the peak displacement and pull-in voltages. The capacitance between the electrodes under electrostatic actuation for flexible fixed-fixed and cantilever beams with aluminium and gold as materials are shown in the Figure 5.5-5.8.

85 Figure 5.4(a) Fixed guided clamped beam simulated output Figure 5.4(b) Fixed guided cantilever beam simulated output Figure 5.5 Fixed guided clamped beam capacitance with aluminium as the material

86 Figure 5.6 Fixed guided clamped beam capacitance with gold as the material From the Figure 5.4 and Figure 5.5, it is clear that the capacitance between the beams builds up as time increases. This is due the forces acting on the beams. The eigen values and eigen vector which are necessary to validate the beams are obtained from the capacitance matrix. Figure 5.7 Fixed guided cantilever beam with aluminium as the material

87 From the graph for capacitance between the fixed guided beams with aluminum as the material, we can say that the capacitance value is high when compared with silicon and gold as the material. This causes the pull-in voltage to decrease when compared with the other materials. Figure 5.8 Fixed guided cantilever beam with Gold as the material. The Figure 5.8 gives the capacitance between the fixed guided cantilever beams with gold as the material. Here it is observed that the capacitance value is high and rises quickly when compared with the other beams causing the pull- in voltage to decrease. The maximum displacement and natural frequencies are obtained by running the simulations using ANSYS. The comparison of these parameters is given the Table 5.1 for various dimension and material of the fixed guided beams.

88 Table 5.1 Maximum displacement and natural frequencies in clamped and fixed guided clamped beam for the beam width of 2 µm and Air gap of 4 µm Dimensions Silicon Gold Aluminium L B Y max f n Y max f n Y max f n m m µm MHz m MHz m MHz Clamped beam 100 15 2.939 10.292 2.939 0.38 2.939 1.247 100 1 2.939 10.284 2.939 0.41 2.939 1.2461 95 15 2.9389 0.10.294 2.9449 0.4 2.9449 1.3624 95 1 2.945 0.11.332 2.945 0.54092 2.945 1.3613 90 15 2.9508 12.566 2.9508 0.59485 2.9508 1.4911 90 1 2.9508 12.566 2.9508 0.59439 2.9508 1.4959 85 15 2.9564 14.011 2.9564 0.65794 2.9564 1.6558 85 1 2.9564 13.999 2.9564 0.6574 2.9564 1.6545 80 15 2.9617 15.730 2.9617 0.73303 2.9617 1.8448 80 1 2.9618 15.716 2.9618 0.73420 2.9618 1.8432 75 15 2.9668 17.797 2.9668 0.82343 2.9668 2.0723 75 1 2.9668 17.781 2.9668 0.82269 2.9668 2.0705 Fixed guided clamped beams 100 15 2.9609 1.3275 2.9609 0.31528 2.9609 0.79346 100 1 2.9609 1.337 2.9609 0.31528 2.9609 0.79346 95 15 2.9712 1.489 2.9712 0.4266 2.9712 0.801248 95 1 2.9712 1.492 2.9712 0.4266 2.9712 0.801367 90 15 2.9832 1.635 2.9802 0.5437 2.9751 0.86754 90 1 2.9832 1.699 2.9802 0.5437 2.9751 0.86881 85 15 2.9887 1.751 2.9901 0.612 2.9822 0.881589 85 1 2.9887 1.774 2.9968 0.7676 2.9871 0.893561 80 15 2.9919 1.821 2.9968 0.7618 2.9871 0.89471 80 1 2.9919 1.841 2.9987 0.8152 2.9914 0.91021 75 15 2.9938 1.952 2.9987 0.8152 2.9914 0.9113 75 1 2.9938 1.941 2.9968 0.7676 2.9871 0.893561

89 Table 5.2 Maximum displacement and natural frequencies of cantilever and fixed guided cantilever beam for the width of 2 µm and air gap of 4 µm Dimensions Silicon Gold Aluminium L B Y max f n Y max f n Y max f n m m µm MHz m MHz m MHz Cantilever beams 100 15 3.3517 4.8965 3.3517 0.41059 3.3517 0.1687 100 1 3.3517 1.7325 3.3517 0.42156 3.3517 0.1692 95 15 3.3517 5.4233 3.3517 0.48593 3.3517 0.1793 95 1 3.3517 1.9193 3.3517 0.48612 3.3517 0.1794 90 15 3.3517 6.0395 3.3517 0.5079 3.3517 0.19986 90 1 3.3517 2.1381 3.3517 0.5112 3.3517 0.19986 85 15 3.3516 6.7670 3.3516 0.569 3.3516 0.22404 85 1 3.3516 2.3965 3.3516 0.572 3.3516 0.22413 80 15 3.3516 7.6340 3.3516 0.60192 3.3516 0.2529 80 1 3.3516 2.7046 3.3516 0.60382 3.3516 0.2536 75 15 3.3515 8.6784 3.3515 0.7307 3.3515 0.28775 75 1 3.3515 0.30762 3.3515 0.7312 3.3515 0.2889 Fixed guided cantilever beams 100 15 3.3615 0.19706 3.3615 0.31412 3.3615 0.11779 100 1 3.3615 0.19824 3.3615 0.31891 3.3615 0.11779 95 15 3.3615 0.19886 3.3615 0.34786 3.3615 0.11785 95 1 3.3615 0.19912 3.3615 0.39176 3.3615 0.11785 90 15 3.3613 0.19956 3.3623 0.40231 3.3614 0.11789 90 1 3.3613 0.19971 3.3623 0.4213 3.3614 0.11789 85 15 3.3613 0.19985 3.3623 0.4776 3.3614 0.11791 85 1 3.3613 0.19989 3.3623 0.4901 3.3614 0.11791 80 15 3.3612 0.19991 3.3646 0.50122 3.3613 0.11801 80 1 3.3612 0.1997 3.3646 0.55712 3.3613 0.11801 75 15 3.3612 0.20100 3.3646 0.61213 3.3613 0.11822 75 1 3.3612 0.20111 3.3646 0.66718 3.3613 0.11822

90 2500 pull in votage in V 2000 1500 1000 fixed-fixed silicon fixed-guided silicon fixed-fixed gold fixed-guided gold fixed-fixed aluminium fixed guided aluminium 500 0 100 95 90 85 80 75 length of the beams in m Figure 5.9 Variation of pull-in voltages in fixed guided clamped beam for different materials Comparison of pull-in voltages for fixed fixed and fixed guided microbeams with the variation in dimensions and materials is given in the Figure 5.9.It is seen that the fixed- fixed silica beam has highest pull in voltage when compared with lowest of fixed guided aluminium. pull in voltages 300 250 200 150 100 50 fixed cantilever si fixed guided si fixed cantilever gold fixe guided gold fixed cantilever aluminium fixed guidedcantilever 0 100 95 90 85 80 75 length in m Figure 5.10 Variation of pull-in voltages in fixed guided cantilever beam for different materials

91 It is seen from the Figure 5.11, that a decrease in the pull- in voltage is evident for fixed guided cantilever beams when compared with clamped beams of all kinds. 5.5 CONCLUSION It is seen from the simulation results that the pull-in voltages is less in aluminum compared with gold and silicon. Further the pull-in voltage of fixed guided micro beams has greatly reduced by 1.65 times for clamped beams and 1.45 times for cantilever beams compared with normal micro beams. It is also found that for the same length, pull-in voltage decreases with decrease in beam width and also pull-in voltage increases with decrease in beam length.