Pressure Dependent Quality Factor of Micron Scale Energy Harvesters A.T. Brimmo, M.I. Hassan, A. N. Chatterjee Version 1 1
Overview Problem statement Approach Experimentation Computational Model Validation Conclusion 2
Problem Statement Fluidic damping in piezoelectric based MEMs energy harvesters causes the device to loss harvested energy Fluidic damping is a function of cavity pressure Reducing cavity pressure comes at a cost Designers of micro cantilever energy harvesters (EH) need a tool which can demonstrate the effect of pressure levels on the degree of harvested energy (Q-factor) in an expedient manner. The explicit FSI approach (Navier Stokes Structural mechanics coupling) of modeling MEMs damping is very time intensive. 3
Approach To minimize result s turn-over time, Reynolds Equation (Lubrication theory) is coupled with the Structural Mechanics equations to model the effect of pressure on fluidic damping Two experimental setups (simple cantilever and complex structured magnetometer) are used to validate the model s accuracy A single mass cantilever EH model is used to compare results and computational time using COMSOL and ANSYS implicit and explicit FSI simulations 4
Simple Cantilever Resonator Experiments Figure 1: Experimental setup for simple cantilever EH (Pandey et al.) 5
Simple Cantilever Resonator Model Reynolds Equation: t ρh + t. (hρv av ) = 0 v av = 1 2 v w,t + v b,t ( h2 12μ eff ) t. P f μ eff = μ 1+9.638Kn 1.159 - Low pressure Regime (Kn (ʎ 0 /h) >0.1) Figure 2: Model setup for simple cantilever EH (Pandey et al.) 6
Mesh Sensitivity Figure 3: Mesh sensitivity analysis for (a) Modal Fq (b) Q factor 7
Q Factor Time Step Sensitivity Analysis 200 180 160 140 120 100 80 60 40 20 0 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 Time Step (sec) Figure 4: Mesh sensitivity analysis for (a) Modal Fq (b) Q factor 8
Model Time Stepping Figure 3: Effect of time step and time step formulation on dynamic response of un-damped resonator 9
Simple Cantilever Model Validation: Q factor as a function of mode frequency Magnitude (m/s/pa) Pandey et al Thin Film-Structural Dynamics model 0.0008 0.0007 0.0006 D=5.25um 0.0005 0.0004 0.0003 0.0002 0.0001 0 0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05 1.00E+06 Frequency (Hz) Figure 5: Simple cantilever resonator validation: Modal Q factors Model's Resonant Fq (Hz) Measured Resonant Fq(Hz) Frequency s Percentage error Model's Q Factor Streamline Measured Q Factor Q Factor Error First Mode 50000 43000 16% 1.21 1.20 1% Second Mode 290000 245000 18% 7.92 7.58 5% Third Mode 790000 690000 14% 19.55 18.52 6% 10
Simple Cantilever Model Validation: Q factor as a function of cantilever length Q facotr (Ambient Pressure) 8 7 D=5.25um Length Microns Experiment Q Model %error 350 1.204819277 1.195057-1% 300 1.5625 1.513789-3% 250 2.487562189 2.419897-3% 200 4 3.892728-3% 150 7.042253521 7.26456 3% 6 5 4 3 2 1 Experiment Model 0 0 100 200 300 400 Cantilever length (micro meters) Figure 6: Simple cantilever resonator validation (cantilever length) Streamline 11
Magnetometer Resonator Model fixed Springs fixed fixed Direction of Motion fixed Parallel Plate Fingers Figure 7: Magnetometer resonator model boundary conditions 12
MAGNETOMETER RESONATOR MODAL MODEL 1 3 2 Figure 8: Magnetometer resonator modal analysis 13
Magnetometer Model Validation: Q factor as a function of Cavity pressure Q-Factor 10000 1000 100 Experimental Measurmetnts Model 10 1 0.01 0.1 1 10 Pressure (Torr) Figure 9: Magnetometer resonator Q factor Validation 14
Single mass resonator model SiO AlN Silicon Figure 10: Single Mass Cantilever Model 15
Q factor Single Mass Cantilever Models Comparisons 10000000 1000000 100000 COMSOL ANSYS 10000 1000 100 10 1 0.00001 0.001 0.1 10 1000 Pressure (Torrs) Figure 11: Single Mass Cantilever Q factor Model 16
CONCLUSION Pressure dependent damping characteristics of MEMs resonator have been modeled using the Structural mechanics - thin Film models coupling. The models were validated for modal, geometry and pressure dependent Q factor estimates These solutions are 15 times faster than explicit FSI simulations with only 7% deviation in Q factor calculations Care must be taken to ensure time stepping formulations are adequate and time step selections are strict to avoid numerical damping COMSOL and ANSYS estimates are identical but the COMSOL modeling approach presents many advantages for this type of model 17
Thank you! 18