61 CHAPTER 4 DESIGN AND ANALYSIS OF CANTILEVER BEAM ELECTROSTATIC ACTUATORS 4.1 INTRODUCTION The analysis of cantilever beams of small dimensions taking into the effect of fringing fields is studied and the parameters are estimated. Extracting the mechanical properties of MEMS devices has always been a challenge in terms of test time and test cost due to difficulties associated with accurate characterization of thin films. A technique for diagnosing the mechanical parameters of a cantilever-beam accelerometer using purely electrical test stimulus was given by Vishwanath Natarajan et al (2006). Governing equations of an electrostatically actuated clamped clamped and a cantilever narrow microbeam; and an expression for the distributed electrostatic force that simulates well the fringing field effects was derived by Romesh C. Batra (2006). The dependence of the critical tilting angle and pullin voltage on the sizes of structure was investigated by Jian-Gang Guo (2004). Patrick B. Chu et al (2005) showed that, in the ANSYS modeling design, short spring elements have insignificant contribution to the torsional resonant frequency but have more effect on the higher modes. A method to overcome pull-in effect of electrostatic actuators was developed for an electrostatic micromirror. The method was applied to both parallel- plate and torsional actuators and performance was studied by Jinghong Chen et al (2005).
62 Nitin Agarwal et al (2009) reported the variations in various design parameters such as material properties, geometrical features, and/or operating conditions on the performance of electrostatic MEMS devices. For rapid computational prototyping of such devices, it is required to accurately model the interaction of various physical fields such as mechanical and electrical. Typical MEMS structures consist of arrays of thin beams with cross sections in the order of microns ( m) and lengths in the order of ten to hundreds of microns. Sometimes, MEMS structural elements are plates. An example is a small rectangular silicon plate with sides in the order of mm and thickness of the order of microns that deforms when subjected to electric fields. Owing to its small size, significant forces and/or deformations can be obtained with the application of low voltages (Arik 2005). Computational analysis of deformation of MEMS parallel plate actuators due to electrostatic stresses has also been the main objective. MEMS cantilevers are the major components for micro sensors and actuators. Deformable structures are frequently encountered in MEMS. Micro Structures undergo deformations because of electrostatic forces caused by applied potentials. The cantilever beam is shown in Figure 4.1 is fixed in the x-y plane at z = 0. The beam in Figure 4.1 is fixed in the x-y plane at z = 0 and z = l. the beam is subjected to a uniform force (F) that is caused by acceleration. The beams have length (l), width (w), and height (h). The force on the beam is determined by F = ma (4.1) where m is the mass of the beam and a is the acceleration.
63 x y Figure 4.1 Structure of Cantilever beam with Dimensions Figure 4.2 shows the initial and deformed configuration of the parallel plate actuator. The upper plate is fixed at the left end and free at the other end. The lower plate is fixed and hence cannot deform. The upper plate is held at a voltage V with respect to the fixed plate. Positive charges are induced on the upper plate and negative charges on the fixed plate. This results in an attractive force between the two plates. Hence the upper plate bends downwards due to electrostatic stresses. V Figure 4.2 Schematic of deformed Cantilever beam As seen in Figure. 4.2, when a voltage is applied, static charges are induced on the surfaces of the plates. Since the upper plate is held at a positive voltage, the charges induced on the upper plate will be positive so as to keep the voltage at any point on the upper plate equal to applied voltage V (conductor surface is an equipotential surface).the fixed plate is held at zero potential. Now, the positive charges on the upper plate will result in positive potentials at the surface of the fixed plate. Hence, negative charges will be
64 induced on the surface of the fixed plate so as to nullify the positive potentials due to positive charges on the upper plate and maintain any point on the fixed plate at zero potential. On the whole, the effect of applying an external voltage to the system is to induce positive charges on the upper plate and negative charges on the fixed plate. The distribution of these charges depends on applied voltage and geometry of the system. The geometry of the system is the dimensions of the plate, distance between plates etc., and deflection of the upper plate from current position to a new position. The opposite polarity charges on the plates gives rise to attractive forces between the plates. The lower plane being fixed cannot deform. The upper plate is fixed at left end and free at the other end. It deflects according to mechanical properties of the material used for plates. 4.2 ELECTROSTATIC ANALYSIS When the upper plate deflects the geometry of the plate changes and the charge distribution changes consequently. Thus, the forces also change and therefore the plate cannot remain at the current (deflected) position. The new force will have a corresponding deflection relative to initial configuration and the plate will make a transition to this position. This position is not stable as well. For the same reasons as before the plate again deflects to a new position and this process continues and can have two possible end states. 1. The successive changes in deflection become smaller and smaller and further deflection become negligible. This state as is referred to as equilibrium position. 2. The deflection becomes large enough so that the upper plate touches the fixed plate. This state is referred as pull-in.
65 The pull-in voltages for cantilever beam is given by V pi 0.529 3 Ew d 4 l o 3 (4.2) 4.3 NATURAL FREQUENCY ANALYSIS If the MEMS variable capacitor operating frequency is the same as the mechanical resonance frequency, then the variable capacitor is capable of introducing unwanted distortion at that frequency. The first mechanical resonance frequency of a cantilever beam for bending oscillations is given in Equation (4.3). An increase in elastic modulus or the moment of inertia will tend to increase the mechanical resonance frequency. Increasing the beam width or the beam height will increase the moment of inertia. Increasing the beam mass or length will decrease the frequency. f R 3.52 2 E I 4 l (4.3) In the above equations µ is the mass per unit length of the beam. For typical MEMS variable capacitor designs, the mechanical resonant frequencies are normally at khz range. Since the operating frequencies are at least 10000 times the mechanical bandwidth, these devices are unlikely to produce a significant amount of harmonic content. As the voltage is increased, the force increase, which decreases the air gap, further increases the force. In the case of the cantilever beam, Figure 4.1, the force causes a maximum deflection at the tip of the cantilever. For the cantilever beam of Figure 4.1 the maximum deflection is
66 y max Fl 8El 3 (4.4) where E is the effective modulus. This deflection occurs at Z l. For narrow beams w 5h Young s modulus. For wide beams 2 the plate modulus E 1 v, the effective modulus E is equal to w 5h, where v is Poisson s ratio., the effective modulus becomes 4.4 RESULTS AND DISCUSSION Finite Element Analysis with ROM has been completed successfully to understand the physical nature of the micro cantilever beam. Since most of the MEMS devices do not have the same characteristics as they were designed and predicted and always needed a trail and error method using simulations with the support of software tools to get the actual characteristics. Microbeams of different lengths, widths and height are simulated thoroughly to understand the change in the characteristics of the beams as with device geometry. The simulated output is shown in Figure 4.3. Figure 4.3 Simulated Output of cantilever beam
67 As described in the previous chapters, the goal of the reduced-order modeling is to generate a fast and accurate description of the coupled-physics systems to characterize their static or dynamic responses. Reduced Order Modeling (ROM) substantially reduces time since the dynamic behavior of most structures can be accurately represented by a few eigen modes. With advantages of reduced order modeling microcantilevers beams are simulated using ANSYS finite element software. First the capacitance between the beams is extracted and results for maximum deflection and pull-in voltages are obtained through non linear modal analysis. The results obtained for the capacitance with the dimension of beam length of 80 m, width of 15 m, height of 4 m is shown in the Figure 4.4. Then maximum deflection for a load step increase in the voltage and the pull-in voltages are also calculated. A detailed comparison is made for different dimensions of length and width of the beams. Since only simulations help the designers to predict accurately the characteristics of the beams parameters of the beam. The comparisons for pull-in voltage for various widths of the beams are shown in the Figure 4.5. Figure 4.4 Ansys output of cantilever beam for capacitance
68 pull in voltages in volts 900 800 700 600 500 400 300 200 100 0 100 95 90 85 80 75 Length in m 15 m width of Beam 10 m width of Beam 5 m width of Beam 2 m width of Beam 1 m width of Beam Figure 4.5 Variation of pull-in voltage in cantilever beam for different beam widths It is seen form the Figure 4.5, there is no drastic deviation in the pull-in voltages for the width b=15,10,5 m of the micro beams. But the pull-in voltage drastically increases for b=2 m and b=1 m. A non linear modal analysis that is used to determine the vibration characteristics, such as natural frequencies and mode shapes of the cantilever beams. It was also a starting point for the harmonic analyses. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions (Arik et al 2005). The best way of determining the vibrational extracting modes is determining the specified frequency at which the devices will display range amplitudes of vibration. A solution method for efficiently solving coupled-field problems involving flexible structures is ROM. This ROM method is based on a modal representation of the structural response. The deformed structural domain is described by a factored sum of the mode shapes (eigenvectors). The resulting ROM is essentially an analytical expression for the response of a system to
69 any arbitrary excitation. This methodology has been implemented for coupled electrostatic-structural analysis and is applicable to micro-electromechanical systems (MEMS). The solver tool enables essential speed up for two reasons: A few eigen modes accurately represents the dynamic behavior of most structures. This is particularly true for MEMS. Modal representations of electrostatic-structural domains are very efficient because just one equation per mode and one equation per conductor are necessary to describe the coupled domain system entirely. This modal method can be applied to nonlinear systems. Geometrical nonlinearities, such as stress stiffening, can be taken into account if the modal stiffness is computed from the second derivatives of the strain energy with respect to modal coordinates. Capacitance stroke functions provide nonlinear coupling between Eigen modes and the electrical quantities if stroke is understood to be modal amplitude. The process involves the following distinct steps and is shown as a flow chart Figure 4.6. The model preparation step creates the necessary finite element model for the generation pass. The generation pass executes a modal analysis of the structure. It also executes an optional static analysis to determine the deformation state of the structure under operating conditions. Using this information, the generation pass then selects the modes and performs computations to create a reduced order model. The use pass uses the reduced order model in an analysis. The reduced order model is stored in a ROM database and a polynomial coefficients file, and utilized by a ROM element.
70 The expansion pass extracts the full DOF set solution and computes stresses on the original structure created in the model preparation phase. A VHDL-AMS mathematical model of the ROM structure may be exported for use in electrical design automation (EDA) system simulators. The ROM method is applicable to 2-D and 3-D models. The generation pass requires multiple finite element solutions of the structural and electrostatic domains, where the structure is displaced over its operating range. Model Preparation Generation pass ROM Use pass Include to system environment Expansion pass Use pass Figure 4.6 Flowchart of Reduced Order Modeling scheme To support both morphing and remesh operations for the multiple finite element solutions, PLANE121, SOLID122, or SOLID123 elements must model the electrostatic domain. INFIN110 or INFIN111 elements can model the open boundary of electrostatic fields. 2-D or 3-D structural or shell elements can model the structural domain. Care was taken while preparing the model of the electrostatic domain to ensure that morphing or remeshing will succeed over the deflection range of the structure. The ROM characterization requires that the device operate primarily in one dominant direction (X, Y, or
71 Z in the global Cartesian system). This includes not only the transversal shift of most rigid bodies (inertial sensors), but also cantilever and plate bending (RF filters, pressure gauges, ultrasonic transducers) and swivel motions (micromirrors). Material properties must be elastic and temperature independent. Stress stiffening and prestress effects are available. First two modes are extracted, Figures 4.7 to 4.9 show typical results of the analysis for the first and second modes for different dimensions. Eigen values Figure 4.7 Variation of the first two modes of cantilever beam with beam length 100 m The modal outputs for the first two modes are shown in the Figure 4.7. It is observed from the Figure 4.7, a large variation is observed for the second mode and almost the same output is maintained for first mode even for variation in the width of the beams.
72 3.00E-06 2.50E-06 Eigen values m o des 2.00E-06 1.50E-06 1.00E-06 MODE 1 MODE 2 5.00E-07 0.00E+00 15 10 5 2 width of the beams( m) Figure 4.8 Variation of the first two modes of cantilever beam with beam length 80 m From the Figure 4.8, curves are similar to l =100 m. So we can say that there is no change in the mode shape for change in the length of the beams. 3.50E-06 Eigen values m odes 3.00E-06 2.50E-06 2.00E-06 1.50E-06 1.00E-06 MODE 1 MODE 2 5.00E-07 0.00E+00 100 95 90 85 80 Length of the beams( m) Figure 4.9 Variation of the first two modes of cantilever beam with beam width 15 m
73 It was found that for breadth of 15 m, 10 m and 5 m there is no change in the mode shape. A change was observed for breadth of 2 m. This shape change may be due to small dimension and need to be analysed. 3.00E-15 2.50E-15 EIGEN m VALUES odes 2.00E-15 1.50E-15 1.00E-15 5.00E-16 MODE 1 MODE 2 0.00E+00 100 95 90 85 Length of the beams( m) Figure 4.10 Variation of the first two modes of cantilever beam with beam width 1 m When compared with fixed beams, cantilever beam s first mode changes are not prominent as shown in the Figure 4.9.but for mode 2; it is similar to that of clamped beam. It is found that both the modes of peaks at l = 95 µm, b = 15 µm and h = 2 µm. It was also observed that for mode 2, the changes at the maximum value are prominent. 2.00E+07 1.80E+07 natural frequencies in MHz 1.60E+07 1.40E+07 1.20E+07 1.00E+07 8.00E+06 6.00E+06 4.00E+06 2.00E+06 0.00E+00 100 95 90 85 80 75 length in micrometers w idth 15 mm m w idth 10 mm m w idth 55mm m w idth 22mm m w idth 11mm m Figure 4.11 Variation of natural frequencies with the length and width of the cantilever beam
74 The variation in the natural frequency obtained is given in the Figure 4.11. It is seen from the Figure 4.11 a large variation is seen for width of beams 2 m and 1 m. Table 4.1 Effect on Natural frequencies and Eigen values in cantilever beam for beam height 2 µm Beam Width ( m) Beam Length m) Natural Frequencies (MHz) Eigen values Mode 1 Mode 2 15 100 0.4.8965 9.86E-08 3.07E-06 10 100 0.4.8.96 6.66E-08 2.32E-06 5 100 4.8957 3.37E-08 1.22E-06 2 100 1.7325 8.81E-19 1.04E-15 1 100 1.7325 8.74E-19 6.75E-16 15 95 5.4232 3.04E-06 2.21E-08 10 95 5.4226 2.03E-06 1.39E-08 5 95 5.4222 1.20E-06 7.58E-09 2 95 1.9193 9.33E-19 1.14E-15 1 95 1.9193 1.92E-18 2.46E-15 15 90 6.0395 2.82E-06 3.16E-08 10 90 6.0389 1.91E-06 2.00E-08 5 90 6.0384 9.64E-07 1.03E-08 2 90 2.1381 7.97E-19 3.00E-16 1 90 2.1381 6.04E-19 7.85E-16 15 85 6.7670 2.78E-06 4.49E-08 10 85 6.7662 1.81E-06 3.19E-08 5 85 6.7657 1.04E-06 1.52E-08 2 85 2.3965 2.55E-18 3.52E-16 1 85 2.3965 1.21E-18 6.97E-16 15 80 7.6340 1.21E-18 6.93E-08 10 80 7.6330 2.58E-06 4.63E-08 5 80 7.6324 2.07E-06 2.30E-08 2 80 2.7046 9.80E-07 6.91E-16 1 80 2.7046 2.93E-18 3.07E-06
75 The results show that the fundamental frequency increases with decreasing beam length. It is worth noting that the fundamental frequency increases dramatically below the 100 m beam length mark. As a result, careful design is needed for small structures of small length and comparison of various parameters of cantilever beam is shown in Table 4.2. Table 4.2 Comparison of various parameters in cantilever beam for Length m) beam height 2 µm Breadth m) Lower and upper bound displacement for 1 and 2 Modes ( m) Pull-in Voltage (V) Mode 1 Mode 2 Minimum Maximum 100 15 ±3.3517 ±0.084594 150 151 100 10 ±3.3517 ±0.084593 135 136 100 5 ±3.3517 ±0.084593 128 129 100 2 ±3.0198 ±0.24178 404 405 100 1 ±2.1313 ±0.68601 504 505 95 15 ±3.3517 ±0.084620 166 167 95 10 ±3.3517 ±0.084620 159 160 95 5 ±3.3517 ±0.084620 142 143 95 2 ±3.0199 ±0.24175 448 449 95 1 ±2.1319 ±0.68571 529 530 90 15 ±3.3517 ±0.084651 185 186 90 10 ±3.3517 ±0.084650 177 178 90 5 ±3.3517 ±0.084650 156 157 90 2 ±3.0199 ±0.24171 490 491 90 1 ±2.1325 ±0.68543 586 587 85 15 ±3.3516 ±0.084687 207 208 85 10 ±3.3516 ±0.084687 198 199 85 5 ±3.3516 ±0.084687 178 179 85 2 ±3.0200 ±0.24168 550 551 85 1 ±2.1330 ±0.68516 650 651 80 15 ±3.3516 ±0.084730 234 235
76 Table 4.2 (Continued) Length m) Breadth m) Lower and upper bound displacement for 1 and 2 Modes ( m) Pull-in Voltage (V) Mode 1 Mode 2 Minimum Maximum 80 10 ±3.3516 ±0.084730 224 225 80 5 ±3.3516 ±0.084730 201 202 80 2 ±3.0201 ±0.24164 641 642 80 1 ±2.1336 ±0.68490 752 753 75 15 ±3.3515 ±0.084782 266 267 75 10 ±3.3515 ±0.084782 255 256 75 5 ±3.3515 ±0.084782 229 230 75 2 ±3.0201 ±0.24161 713 714 75 1 ±2.1341 ±0.68464 832 833 4.5 CONCLUSION The performance of cantilever beam was estimated for various dimensions of the beams. The effect of fringing field was taken in to account and analysis has been carried out using reduced order modeling. It was found that pull-in voltage is reduced compared to clamped beams. The mode shapes were also obtained to analyze and estimate the performance of the beams. The maximum deflection for the cantilever beam structure has been determined as 3.3517 m.