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MEMS pressure sensor, Tunable Capacitor design and simulation using COMSOL 4.0a

A project report

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- 1. DEPARTMENT OF INSTRUMENTATIONCOCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY 1
- 2. SUBMITTED BY BABUL KUMAR GOUTAM KUMAR DEPARTMENT OF INSTRUMENTATION COCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY KOCHI – 22 2008-2012 CERTIFICATE This is to certify that the project report entitledsubmitted by BABUL KUMAR and GOUTAM KUMAR to the Department ofInstrumentation, CUSAT as mini project is a bona fide record of the work carried outby them under my supervision and guidance. 2
- 3. 3
- 4. Abstract The tunable capacitor or variable capacitor is one of the most important component in filters, Phase shifters, VCO etc. present the simulation and analysis of a MEMS variable capacitor. The variable capacitor iscomposed of the components:( 1) the capacitor built by two squared plates, which one plate is mechanically fixed to the substrateand the other is a moving plate.(2) A mechanical suspension using a spring of known spring constant.Depending on the voltage applied to these plates the spring would pull down the variable plate fromits sides to the substrate for equilibrium. This work includes the simulation and analysis usingCOMSOL Multiphysics, and provides tables for the resulting values of the variable capacitors. Theresults accomplished show that the variable capacitor has potential for automatic compensation ofcapacitances and for integration into frequency oscillators and filters. This example presents a model of a micro-scale square inductor, used for LC bandpass filters in microelectromechanical systems (MEMS).The purpose of the model is to calculate the self-inductance of the microinductor. Given the magneticfield, we can compute the self-inductance, L. The model uses the Terminal boundary condition, whichsets the current to 1 A and automatically computes the self-inductance. Finally examine the operation of a capacitive MEMS pressure sensor, treating 3D model using COMSOL Multiphysics.The operating principle of a capacitive pressure sensor is to measure the change in capacitancebeten two electrodes when a change in pressure displaces one of the electrodes, located on athin diaphragm. The diaphragm separates a reference compartment kept at vacuum pressureand a pressurized compartment.At the bottom of the pressurized compartment is a fixed base (with one electrode), while thediaphragm (with a counter-electrode) is located at its top. As the pressure changes, thediaphragm that separates the two compartments is displaced, and the change in separationbeten the two electrodes results in a corresponding change in the capacitance. 4
- 5. CONTENTS1. Introduction to RF MEMS2. About COMSOL Multiphysics3. Tunable MEMS Capacitor 3.1. Introduction 3.2. Model Definition 3.3. Modeling Instructions 3.4. Computation and Results4. Integrated Square-Shaped Spiral Inductor 4.1. Introduction 4.2. Model Definition 4.3. Modeling 4.4. Computation and Results5. Capacitive Pressure Sensor 5.1. Introduction 5.2. Model Definition 5.3. Modeling 5.4. Computation and Results6. GLOSSARY7. References 5
- 6. 1. Introduction to RF MEMSThe Radio Frequency microelectromechanical system (RF MEMS) acronym refers to electroniccomponents of which moving sub-millimeter-sized parts provide RF functionality. RF MEMStechnology can be used to implement high quality switches, varactors (variable reactors),inductors, resonators, filters and phase shifters. Among the broad range of applications theMEMS technology gives a unique possibility to implement micromechanical resonators andfilters with high performance regarding selectivity and Q-factors. When combining thesemechanical structures with microelectronics, central parts in wireless systems, RF systems(Radio Frequency systems) can be implemented. Examples can be various types of oscillators,VCOs (Voltage Controlled Oscillators), mixers and sharp filters. The MEMS structures canthereby replace traditional costly and large off-chip discrete components by making possibleintegrated solutions that can be batch processed. Vibrating MEMS resonators and filters thathave been implemented so far are based on mechanical vibrations in lateral or verticaldirections on Silicon wafers. Different types of beams, comb structures and disks can be used.Research is focused on areas of RF MEMS and the possibility of integrating electronics (CMOScircuits) and MEMS on the same chip. Thereby system designers could benefit from the highperformance and flexibility given by introducing micromechanical components into thesystems. RF MEMS can replace the bulky, off-chip components that are necessary in todays RFtranceivers to obtain sufficient performance. Thus, oscillators, mixers and RF filters are veryimportant units that can take advantage from being implemented by their mechanicalcounterparts.Even if the MEMS parts would not have as good performance separately as units made in aspecialized MEMS process, the overall system performance could be very good due to the lowparasitics, impedances and short wires between the mechanical parts and the neighbouringelectronics.Works are being done on combining MEMS with circuitry in ultra low power, robust andradiation tolerant CMOS for future space applications. The MEMS parts will typically implementsensors, actuators, or central parts in the radio transmission system in future pico-satellites.Telecommunications and Radio Frequency are fast growing technological areas. Recentdevelopment of computers, PDAs, wireless phones, GPS systems, even toys are rapidly evolvingand supporting complex computational and signal processing. Most of the advances incommunications are thanks to the advances in the development of MicroElectro-MechanicalSystems (MEMS.) Generally, MEMS brings together different kinds of technology whiledeveloping devices. Some examples of this are combinations of electrical-mechanical,mechanical-optical, optical-electrical, mechanical-fluidics, electrical-fluidics, electro-thermal,and so forth. One set of specific devices that are supporting the advances of communicationsthrough MEMS technology is composed of capacitors, resonators and filters. These devices canbe built to get high Qs of above 1 GHz ranges 6
- 7. About COMSOL MultiphysicsCOMSOL Multiphysics is a powerful interactive environment for modeling and solving all kindsof scientific and engineering problems. With this software can easilyextend conventional modelsfor one type of physics into multiphysics models that solve coupled physics phenomena—anddo so simultaneously. Accessing this power does not require an in-depth knowledge ofmathematics or numerical analysis.Using the built-in physics interfaces and the advanced support for material properties, it ispossible to build models by defining the relevant physical quantities—such as materialproperties, loads, constraints, sources, and fluxes—rather than by defining the underlyingequations. can always apply these variables, expressions, or numbers directly to solid and fluiddomains, boundaries, edges, and points independently of the computational mesh. COMSOLMultiphysics then internally compiles a set of equations representing the entire model.Using these physics interfaces, can perform various types of studies including:• Stationary and time-dependent (transient) studies• Linear and nonlinear studies• Eigenfrequency, modal, and frequency response studiesWhen solving the models, COMSOL Multiphysics uses the proven Finite Element Method (FEM).The software runs the finite element analysis together with adaptive meshing (if selected) anderror control using a variety of numerical solvers. The studies can make use of multiprocessorsystems and cluster computing, and can run batch jobs and parametric steps.Partial differential equations (PDEs) form the basis for the laws of science and provide thefoundation for modeling a wide range of scientific and engineering phenomena.Many real-world applications involve simultaneous couplings in a system of PDEs—multiphysics. For instance, the electric resistance of a conductor often varies with temperature,and a model of a conductor carrying current should include resistive-heating effects. Manypredefined multiphysics interfaces provide easy-to-use entry points for common multiphysicsapplications.Different modules of COMSOL Multiphysics-•AC/DC Module •Heat Transfer Module•Acoustics Module •MEMS Module•Batteries and Fuel Cells Module •Plasma Module•CFD Module •RF Module•Chemical Reaction Engineering Module •Structural Mechanics Module•Earth Science Module can build models of all types in the COMSOL Multiphysics user interface. For additionalflexibility, COMSOL also provides LiveLink for MATLAB, a seamless interface to MATLAB. Thisgives us the freedom to combine multiphysics modeling, simulation, and analysis with othermodeling techniques. For instance, it is possible to create a model in COMSOL and then export itto Simulink as part of a control-system design. 7
- 8. Tunable MEMS CapacitorIntroductionIn an electrostatically tunable parallel plate capacitor, we can modify the distance beten the twoplates when the applied voltage changes. For tuning of the distance beten the plates thecapacitor includes a spring that attaches to one of the plates. If we know the characteristics ofthe spring and the voltage beten the plates, we can compute the distance beten the plates. Thismodel includes an electrostatic simulation for a given distance. A postprocessing step thencomputes the capacitance.The capacitor in this model is a typical component in various microelectromechanical systems(MEMS) for electromagnetic fields in the radio frequency range 300 MHz to 300 GHz.Figure : The tunable MEMS capacitor consists of two metal plates. The distance beten the plates istuned via a spring connected to one of the plates.Model DefinitionTo solve the problem, use the 3D Electrostatics physics interface in the AC/DC Module. Thecapacitance is available directly as a variable for postprocessing.DOMAIN EQUATIONSThe electric scalar potential, V, satisfies Poisson’s equation,where ε0 is the permittivity of free space, εr is the relative permittivity, and ρ is the space chargedensity. The electric field and the displacement are obtained from the gradient of V: 8
- 9. BOUNDARY CONDITIONSPotential boundary conditions are applied to the capacitor plates and bars. A port conditionmaintains the potential 1 V at the upper plate and the connecting bars, whereas the lor plate iskept at ground potential. For the surface of the surrounding box, apply conditionscorresponding to zero surface charge at the boundary,Modeling InstructionsMODEL WIZARD1. Go to the Model Wizard window.2. Click Next.3. In the Add Physics tree, select AC/DC>Electrostatics (es).4. Click Next.5. In the Studies tree, select Preset Studies>Stationary.6. Click Finish.GEOMETRY 11 . In the Model Builder window, click Model 1>Geometry 1.2. Go to the Settings window for Geometry.3. Locate the Geometry Settings section. Find the Units subsection. From the Length unit list, select µm. The snapshot of the model in progress, after building the above mentioned 10 blocks. 9
- 10. Cylinder 11. Right-click Geometry 1 and choose Cylinder.2. Go to the Settings window for Cylinder.3. Locate the Size and Shape section. In the Radius edit field, type 5.5.4. In the Height edit field, type 38.5. Locate the Position section. In the x edit field, type 11.6. In the y edit field, type 250.7. In the z edit field, type 8.8. In the Model Builder window, right-click Cylinder 1 and choose Build Selected.Union 11. Right-click Geometry 1 and choose Boolean Operations>Union.2. Click the Select Box button on the Graphics toolbar.3. Select the objects blk1, blk2, blk3, blk4, blk5, blk6, blk7, blk8, blk9, blk10, and cyl1.We can do this by first copying the text “blk1, blk2, blk3, blk4, blk5, blk6, blk7, blk8, blk9,blk10, and cyl1” and then clicking the Paste Selection button next to the Selection box orclicking in the box and pressing Ctrl+V.4. Go to the Settings window for Union.5. Locate the Union section. Clear the Keep interior boundaries check box.6. In the Model Builder window, right-click Union 1 and choose Build All.Block 111. Right-click Geometry 1 and choose Block.2. Go to the Settings window for Block.3. Locate the Size and Shape section. In the Width edit field, type 176.4. In the Depth edit field, type 262.5. In the Height edit field, type 8.6. Locate the Position section. In the x edit field, type 62.7. In the y edit field, type 19.8. In the z edit field, type 8.9. In the Model Builder window, right-click Block 11 and choose Build Selected.Block 121. Right-click Geometry 1 and choose Block.2. Go to the Settings window for Block.3. Locate the Size and Shape section. In the Width edit field, type 181.4. In the Depth edit field, type 22.5. In the Height edit field, type 8.6. Locate the Position section. In the x edit field, type 139.7. In the y edit field, type 139.8. In the Model Builder window, right-click Block 12 and choose Build Selected.Union 21. Right-click Geometry 1 and choose Boolean Operations>Union.2. Select the objects blk11 and blk12 only.3. Go to the Settings window for Union.4. Locate the Union section. Clear the Keep interior boundaries check box.5. In the Model Builder window, right-click Union 2 and choose Build Selected. 10
- 11. The model as seen after the union of both the plates separately.Block 131. Right-click Geometry 1 and choose Block.2. Go to the Settings window for Block.3. Locate the Size and Shape section. In the Width edit field, type 360.4. In the Depth edit field, type 340.5. In the Height edit field, type 94.6. Locate the Position section. In the x edit field, type -40.7. In the y edit field, type -20.8. In the z edit field, type -20.9. In the Model Builder window, right-click Block 13 and choose Build Selected.DEFINITIONSSelection 11. In the Model Builder window, right-click Model 1>Definitions and choose Selection.2. Right-click Definitions>Selection 1 and choose Rename.3. Go to the Rename Selection dialog box and type Electrode in the New name edit field. Click OK.4. Go to the Settings window for Selection.5. Locate the Geometric Scope section. From the Selection output list, select Adjacent boundaries.6. Select Domain 2 only.Selection 21. In the Model Builder window, right-click Definitions and choose Selection.2. Right-click Definitions>Selection 2 and choose Rename.3. Go to the Rename Selection dialog box and type Ground Plane in the New name edit field. Click OK.4. Go to the Settings window for Selection.5. Locate the Geometric Scope section. From the Selection output list, select Adjacent boundaries.6. Select Domain 3 only. 11
- 12. Selection 31. In the Model Builder window, right-click Definitions and choose Selection.2. Right-click Definitions>Selection 3 and choose Rename.3. Go to the Rename Selection dialog box and type Dielectric in the New name edit field. Click OK.4. Go to the Settings window for Selection.5. Locate the Geometric Scope section. From the Selection output list, select Selected domains.6. Select Domain 1 only.MATERIALSMaterial 11. In the Model Builder window, right-click Model 1>Materials and choose Material.2. Right-click Materials>Material 1 and choose Rename.3. Go to the Rename Material dialog box and type Dielectric in the New name edit field. Click OK.4. Go to the Settings window for Material.5. Locate the Geometric Scope section. From the Selection list, select Dielectric.6. Click to expand the section.7. Locate the Material Contents section. In the Material contents table, enter the followingsettings:ELECTROSTATICS1. In the Model Builder window, click Model 1>Electrostatics.2. Go to the Settings window for Electrostatics.3. Locate the Domains section. From the Selection list, select Dielectric.Terminal 11. Right-click Model 1>Electrostatics and choose Terminal.2. Go to the Settings window for Terminal.3. Locate the Boundaries section. From the Selection list, select Electrode.4. Locate the Terminal section. From the Terminal type list, select Voltage.Ground 11. In the Model Builder window, right-click Electrostatics and choose Ground.2. Go to the Settings window for Ground.3. Locate the Boundaries section. From the Selection list, select Ground Plane.MESH 1The default mesh gives a sufficiently accurate solution for the purposes of this example. To examinemesh convergence we can optionally go back later and re-solve the model with a finer mesh. Asshown below- the mesh is obtained using Free Tetrahedral Meshing. 12
- 13. STUDY 1In the Model Builder window, right-click Study 1 and choose Compute.RESULTSData Sets1. In the Model Builder window, expand the node Results>Data Sets, then right-click Solution 1 and choose Add Selection.2. Go to the Settings window for Selection.3. Locate the Geometric Scope section. From the Geometric entity level list, select Boundary.4. From the Selection list, select All boundaries.5. Select Boundaries 3 and 5–78 only. This is easiest done by removing boundaries 1, 2, and 4 from the list once we have selected all. 13
- 14. 3D Plot Group 1Figure: The electric potential is shown as a surface plot while the cones indicate the strength andorientation of the electric field.Derived ValuesHaving solved the model, can now extract the capacitance.1. In the Model Builder window, right-click Results>Derived Values and choose Global Evaluation.2. Go to the Settings window for Global Evaluation.3. In the upper-right corner of the Expression section, click Replace Expression.4. From the menu, choose Electrostatics>Capacitance (es.C11).5. In the Model Builder window, right-click Global Evaluation 1 and choose Evaluate. The capacitance evaluates to about 0.09 pF. 14
- 15. Integrated Square-Shaped Spiral InductorIntroductionThis example presents a model of a micro-scale square inductor, used for LC bandpass filters inmicroelectromechanical systems (MEMS).The purpose of the model is to calculate the self-inductance of the microinductor. Given themagnetic field, we can compute the self-inductance, L, from the relationwhere Wm is the magnetic energy and I is the current. The model uses the Terminal boundarycondition, which sets the current to 1 A and automatically computes the self-inductance. Theself-inductance L becomes available as the L11 component of the inductance matrix.Model DefinitionThe model geometry consists of the spiral-shaped inductor and the air surrounding it. Figurebelow shows the inductor and air domains used in the model. The outer dimensions of themodel geometry are around 0.3 mm. Figure : Inductor geometry and the surrounding air.The model equations are the following:In the equations above, σ denotes the electric conductivity, A the magnetic vector potential, Vthe electric scalar potential, Je the externally generated current density vector, μ0 thepermittivity in vacuum, and μr the relative permeability.The electric conductivity in the coil is set to 106 S/m and 1 S/m in air. The conductivity of air isarbitrarily set to a small value in order to avoid singularities in the solution, but the errorbecomes small as long as the value of the conductivity is small.The constitutive relation is specified with the expression 15
- 16. where , H denotes the magnetic field.The boundary conditions are of three different types corresponding to the three differentboundary groups; see Figure (a), (b), and (c) below. Figure : Boundaries with the same type of boundary conditions.The boundary condition for the boundary highlighted in Figure (a) is a magnetic insulatingboundary with a port boundary condition. For the boundaries in Figure (b), both magnetic andelectric insulation prevail. The last set of boundary conditions, Fig(c), are magneticallyinsulating but set to a constant potential of 0 V (ground).Modeling:Materials:Material 1 is selected and renamed to "Conductor" with domain 2 assigned to it.Material 2 is selected and renamed to "Air " with domain 1 assigned to it. 16
- 17. Setting the conductivity to zero in the air would lead to a numerically singular problem. We canavoid this problem by using a small non-zero value. As 1 S/m is much less than the electricconductivity in the inductor, the fields will only be marginally affected.Magnetic and Electric FieldsTerminal 1 Select Boundary 5 as terminal 1 and specified Io=1Ground 1 Boundaries 75 and 76 are taken as GROUNDS. This concludes the boundarysettings. Here the boundaries that have not assigned are electrically and magnetically insulatedby default.MESH 1 choose Free Tetrahedral Meshing and modify the size to Coarse. Afterwardsgo on to select Build All, so as to get the model ready for further computation. Now, move onto STUDY1 and select compute .The magnetic insulation condition on the exterior boundaries causes the field lines to bend andfollow the contours of the box. This inevitably introduces a systematic error to the inductancecomputation. It would be possible to reduce this error by increasing the size of the box, orintroducing an infinite element domain. Nevertheless, since the field is comparatively small nearthe surface of the box, the result is reasonably accurate already. Try visualizing the localmagnitude of the field by having it decide the color of the streamlines.RESULTS3D Plot Group 1The default plot shows the electric potential distribution on the surface of the model domain.There are plenty of better ways of visualizing the solution. The following instructions detail howto combine an electric potential distribution plot on the surface of the inductor with astreamline plot of the magnetic flux density in the air surrounding it. 17
- 18. Fig. Shows the electric potential in the inductor and the magnetic flux lines. The color of the flowlines represents the magnitude of the magnetic flux. As expected this flux is largest in the middleof the inductor.Derived ValuesThe inductance evaluates to 0.755 nH.Capacitive Pressure SensorIntroductionThis example examines the operation of a capacitive MEMS pressure sensor, treatingboth 2D and 3D model versions.The operating principle of a capacitive pressure sensor is to measure the change incapacitance beten two electrodes when a change in pressure displaces one of theelectrodes, located on a thin diaphragm. The diaphragm separates a referencecompartment kept at vacuum pressure and a pressurized compartment; see Figure.Figure : One quarter of the pressure sensor (the two vertical cross-sectional planes are symmetryplanes). The vacuum compartment looks like a small ashtray with a thin diaphragm at its bottom.Beten this diaphragm and the fixed base is a very thin pressurized compartment.At the bottom of the pressurized compartment is a fixed base (with one electrode), while thediaphragm (with a counterelectrode) is located at its top. As the pressure changes, thediaphragm that separates the two compartments is displaced, and the change in separationbeten the two electrodes results in a corresponding change in the capacitance. 18
- 19. Although the deformation of the sensor is primarily caused by the applied pressure, anyinitial stresses in the material also affect the deformation. Therefore, the manufacturingprocess and the selected materials directly influence sensor operation. For example, in somestructures the membrane and cavities are engraved onto silicon and sealed with layers ofglass.Because the materials are bonded together at a high temperature, cooling them down to thesensor’s normal operating temperature produces undesirable stresses in the material thataffect device performance.The sensor here measures static pressures of a magnitude from zero to atmospheric pressure.The model first computes the initial stresses from the manufacturing process; then it accountsfor the structure’s mechanical deformation resulting from an applied pressure. It finallycalculates the sensor’s capacitance for the deformed shape: the 3D model simply integratesinfinitesimal capacitance contributions over the electrode boundary.Model DefinitionMODEL GEOMETRYThe pressure sensor consists of a silicon structure that includes a micrometer-thick diaphragmsituated beten two glass layers. Figure shows the geometry and the dimensions are given inTable 1. In addition, two 1 mm2 rectangular plates at the pressurized compartment’s top(applied potential) and bottom (grounded) form the electrodes. 19
- 20. Figure 3: Materials used in the structural analysis of the capacitive pressure sensor in 3D.STRESS AND DEFORMATIONDuring manufacturing, the sensor is bonded together in a vacuum and at a high temperaturebefore it is cooled down. Therefore, during this process no external forces act on the sensor’sboundaries, but internal stresses appear because the two materials have different coefficientsof thermal expansion. This process also produces a vacuum in the upper cavity that serves asthe reference pressure.During regular operation, the sensor is fixed to a solid surface, and ambient pressure pusheson all outer boundaries. The temperature also changes, which produces extra stresses due tothermal expansion.For a linear elastic material, the stress-strain relationship—taking into account initial stress,σ0, initial strain, ε0, and thermal strain, εth—iswhere D is the elasticity tensor, and the 6-dimensional vectors σ and ε give the normal andshear values of the stresses and strains.Initially only thermal expansion is active. It is given by 20
- 21. where αvec are the coefficients of thermal expansion, T is the ambient temperature, and Tref isthe reference temperature. The manufacturing stage produces the initial stress for normaloperation, where further thermal expansion takes place. This model assumes that the sensor isclose to its initial geometry after manufacturing, so that the initial strain equals zero.Furthermore, we solve the first application mode using the small deformation assumption butallow large deformations for the second one.CAPACITANCETo compute the sensor’s capacitanceThe 3D model makes use of calculus by integrating over the surface of the capacitoraccording towhere h denotes the local distance across the capacitor and ε is the permittivity of air. Thiscalculation rests on the assumption that the lor glass block does not deform much, so that thelocal separation only depends on the initial distance and the diaphragm deformation. In themodel this expression is further multiplied by 4 to get the full capacitance of the model.Modeling Using the Graphical User Interface—3DModeling is done using MEMS Module>Structural Mechanics>Solid, Stress-Strain .OPTIONS AND SETTINGSSelect constants and define there values according to following table: 21
- 22. Selected Materials/Coefficients library and then defined the values for " Glass Hoya" asfollos:GEOMETRY MODELINGThen made the Reacting/Square according to following dimensions:PHYSICS SETTINGSSubdomain Settings Select Subdomains 2 and 6, Select Glass HOYA from the Library material list. Select Subdomains 3, 7, and 9–11. Select Silicon from the Basic Material Properties list. Select Subdomains 2 and 6, and choose Silicon from the Library material list.On the Initial Stress and Strain page, select the Include initial stress. 22
- 23. Boundary Settings Select boundaries 2, 8, 12, 47, 51, 53, 55, 57, 59, and 61, then choose x-z symmetry plane. Select boundaries 1, 7, 11, 15, 21, 25, 28, 31, 34, and 37, then choose y-z symmetry plane. Select boundaries 3, 17, and 27, then choose Fixed. Select boundaries 2, 8, 12, 47, 51, 53, 55, 57, 59, and 61, then choose x-z symmetry plane.INTEGRATION COUPLING VARIABLESThe capacitance can be estimated by assuming that the gap operates as a plate capacitor. Thenwe can integrate the infinitesimal capacitance C(x, y) = ε0/gap(x, y) over the surface of themembrane electrodes:Integration Coupling Variables>Boundary Variables.Select Boundary 9 and assign it the expression :4*epsilon0/(5[um]+w2)MESH GENERATION 23
- 24. The mesh contains about 1440 brick elements. Figure: Element mesh for 3D pressure sensor model.COMPUTING THE SOLUTIONThe following steps generate a solver sequence that uses a static solver to first solve thebonding phase and the parametric solver to then solve the sensor operation fordifferent pressures.P_ambient - range(0,2e4,1e5)POSTPROCESSING AND VISUALIZATIONStress DistributionFigure . shows the results from the 3D model when the sensor is in operation: it is exposed toa pressure of one atmosphere at 15 °C. The largest stress in the diaphragm appears near theposition where the diaphragm connects the surrounding material. 24
- 25. Figure: Sensor deformation and von Mises Stress (MPa) when exposed to ambient pressure.The capacitance for the full sensor modeled in the 3D model is shown in Figure Thiscapacitance corresponds to condition A in Table 1. can also compare the computed capacitance values to those for a plate capacitor with a platesize of 1 mm times unity and a gap of 5 μm when not deformed. The capacitance for thisplate capacitor is C = ε0A/d = 1.771·10−9 F; the corresponding value from the COMSOLMultiphysics model is 1.7760763·10−9 F computed with settings for Condition C and zeropressure. 25
- 26. Figure : Capacitance of the sensor as the function of the operating pressure from the 3D model. 26
- 27. GLOSSARY of some typical terms and methods usedMeshingThe finite element method approximates the solution within each element, using someelementary shape function that can be constant, linear, or of higher order. Depending on theelement order in the model, a finer or coarser mesh is required to resolve the solution. Ingeneral, there are three problem-dependent factors that determine the necessary meshresolution:• The first is the variation in the solution due to geometrical factors. The mesh generator automatically generates a finer mesh where there is a lot of fine geometrical details. Sometimes have to remove such details if they do not influence the solution, because they produce a lot of unnecessary mesh elements.• The second is the skin effect or the field variation due to losses. It is easy to estimate the skin depth from the conductivity, permeability, and frequency. We need at least two linear elements per skin depth to capture the variation of the fields. If we do not study the skin depth, we can replace regions with a small skin depth with a boundary condition, thereby saving elements.• The third and last factor is the wavelength. To resolve a wave properly, it is necessary to use about 10 linear (or 5 2nd-order) elements per wavelength. Keep in mind that the wavelength might be shorter in a dielectric medium.Creating MeshesA mesh is a discretization of the geometry model into small units of simple shapes, referred toas mesh elements.Mesh ElementsELEMENTS FOR 3D GEOMETRIESThe mesh generator discretizes the domains into tetrahedral, hexahedral, prism, or pyramidmesh elements whose faces, edges, and corners are called mesh faces, mesh edges, and meshvertices, respectively.The boundaries in the geometry are discretized into triangular or quadrilateral boundaryelements.The geometry edges are discretized into edge elements. Similar to 2D, the geometry vertices arerepresented by vertex elements.Meshing TechniquesThe following meshing techniques for creating a mesh are available in COMSOL Multiphysics:• Free meshing• Mapped meshing• Boundary layer meshing• Spt meshingThe mesh generator used for free meshing, also referred to as the free mesher, is the onlymesher that can be used on all types of geometry objects.In a 3D geometry, choose beten:• Free meshing generating an unstructured mesh with tetrahedral elements• Spt meshing generating a structured mesh (at least in the direction of the sep) with prism or hexahedral elements. 27
- 28. ABOUT FREE MESHINGThe free mesher is available in all dimensions, and we can use it for all types of geometriesregardless of their topology or shape. If we have not defined or generated a mesh, the freemesher automatically creates an unstructured mesh and adds a corresponding node to theModel Builder window when we compute a study.When we use the free mesher:•The number of mesh elements is determined by the shape of the geometry and various mesh parameters.•We control mesh parameters for the free mesher by Size and Distribution feature nodes in the meshing sequences.We can also control the size of the mesh generated by a specific Free Triangular, FreeQuadrilateral, or Free Tetrahedral node by adding a Size or Distribution subnode.ABOUT 3D SWEPT MESHESThe spt mesher operates on a 3D domain by meshing a source face and then seping the resultingface mesh along the domain to an opposite target face. A spt mesh is structured in the sepdirection and can be either structured or unstructured orthogonally to the sep direction.For straight and circular sep paths, we can use several connected faces as source faces. Each faceabout a domain that is to be operated on by the spt mesher is classified as either a source face, atarget face, or a boundary face. The boundary faces are the faces linking the source and targetface .The spt mesher can handle domains with multiple boundary faces in the sep directionFinite ElementsOnce we have a mesh, we can introduce approximations to the dependent variables. For thisdiscussion, concentrate on the case of a single variable, u. The idea is to approximate u with afunction that we can describe with a finite number of parameters, the so-called degrees offreedom (DOF). Inserting this approximation into the ak form of the equation generates a systemof equations for the degrees of freedom. 28
- 29. ACKNOWLEDGEMENTWe would like to express our heartfelt gratitude to our project guide Dr. K. N. Madhusoodanan for his unflinching support and guidance towards thecompletion of this mini project. His timely guidance and monitoring motivated us andkept us in pursuit of the goal.Furthermore, our sincere thanks go to our Head of Department Dr. StephanRodrigues, who kindly allowed us to make use of all the facilities in the laboratory.. BABUL KUMAR GOUTAM KUMAR 29
- 30. References:1. http://www.comsol.co.in/2. COMSOL 4.0a/documentation2. IEEE paper on mems tunable capacitor.3. http://www.darpa.mil/mto/programs/mems/index.html4. www.mtl.mit.edu/researchgroups/mems-salon/xueen_rebeiz_03.pdf5. www.memsinvestorjournal.com/rf_mems/6. http://www.youtube.com/watch?v=2xiX6_wbb-U 30

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