Transcript mirror
PULL-IN IN OF A TILTED MIRROR Jan Erik Ramstad and Osvanny Ramos • Problem: How to find pull-in • Geometry shown in the figures • Objective: Run simulations with Coventor and try to find pull in. Compare simulated results with analytical approximations CoventorWare Analyzer Mirror Design Fmec g Felect • Before simulations, we wanted to find formulas to compare simulations with. The parallell plate capacitor analogy • The parallell plate capacitor formulas are analog to how the mirror actuation works. • Mechanical force must be equal to electrical force to have equilibrium Fnet Felect Fmec 1 C ( g )V 2 2 • Storing of energy in capacitor W (g) • Energy formula used to derive electrical force Felect W ( g ) g Q CoventorWare Analyzer Mirror Design Fmec g The parallell plate capacitor analogy (continued) • Using parallell plate capacitor formula with F gives • Fmech comes from the spring and gives net force • By derivating net force we can find an expression to find stable and unstable equilibrium. • The calculated k formula will give us the pull in voltage and pull in gap size if inserted in Fnet formula 3 8kg0 V pi 27A g pi 2 g0 3 Felec Fnet k > Felect AV 2 2g 2 AV 2 k(g0 g) 2 2g AV 2 2g 3 CoventorWare Analyzer Mirror Design Derivation of formulas for the mirror design mec elect • By using parallell plate capacitor analogy formulas we can find formulas for mirror design • The forces are analogous with torque where distance x is now replaced with Θ Tilted angle z gx x • Formulas for torque calculations shown below net elect mec 1 W ( ) C ( )V 2 2 elect W ( ) Q mec CoventorWare Analyzer elect Mirror Design z Derivation of formulas for the mirror design (continued) gx x • Hornbecks analysis computes torque directly treating tilted plate as parallell plate. • Eletric torque formula is analogous to electric force: 2 2 Felect net V A 2g 2 V xdx elect 2 A 2g 0 x tan( ) 2 V 2 2 g x tan( ) A 0 xdx k ( 0 ) ...and analyzing the stability of the equilibrium Difficult analytically! Mirror Design Alternative analytical solution: • Using Hornbecks electrical torque formula will be difficult to calculate. By running simulation, capacitance and tilt values can be achieved • Using the values from simulation can be used to make a graph. This graph is a result of normalized capacitance and angle Normalized capacitance CoventorWare Analyzer 3,0 2,5 2,0 1,5 1,0 0,00 0,04 0,08 Angle 0,12 • General formula from graph can be of the following third polynomial formula; C ( ) C 0 (1 a1 a3 3 ) • Using the same formulas as earlier, but now with the new formula for capacitance is used to find electric torque: 1 W ( ) C ( )V 2 2 elect W ( ) 1 C0V 2 (a1 3a3 2 ) Q 2 mec k • From mechanical torque formula, we can find the spring constant (stiffness of ”hinge”) 1 k C0V 2 (a1 3a3 2 ) 2 CoventorWare Analyzer Mirror Design Alternative analytical solution (continued): • The spring constant formula has our variable Θ. By rearranging this formula, Θ is a second degree polynomial, which must be solved for positive roots: k 1 C0V 2 (a1 3a3 2 ) 2 2 a1 k k 3a C V 2 3a 3a3C0V 2 3 3 0 • The root expression must be positive for a stable solution. This will give us a formula for pull in voltage 2 a1 k 2 3a3 3a3C0V • Now that we had a formula to calculate pull in voltage, we attempted to run Coventor simulations k V pi 2 3 a a C 1 3 0 2 1/ 4 CoventorWare Analyzer Graph of normalized capacitance vs angle Mirror Design Original geometry: 0.4 1.5 20V Normalized capacitance 3,0 2,5 47V Data: w04old_Capacitance Model: polin_3 Equation: y=1+a1*x+a3*x^3 Weighting: y No weighting Chi^2/DoF = 0.02063 R^2 = 0.96587 2,0 a1 a3 1 ±0 759.22603 ±48.23932 1,5 40V 20V 1,0 -0,02 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Angle 40V Graph: Red line is analytical approximation Dotted points are measured results from Coventor • Only one electrode has applied voltage • No exaggeration is used 47V 0,14 • Mesh is 0,4 micrometer, equal to hinge thickness Mesh was not changed when changing geometry parameters. Results: k 1.97 10 10 Nm V pi 62V CoventorWare Analyzer Graph of normalized capacitance vs angle 2,8 2,6 0.2 1.5 10V Normalized capacitance Varying k by reducing hinge thickness 2,4 2,2 2,0 1,8 1,6 20V Data: w02new_Capacitance Model: polin_3 Equation: y=1+a1*x+a3*x^3 Weighting: No weighting y = 0.00988 Chi^2/DoF = 0.98199 R^2 a1 a3 1.2164 ±1.21089 ±86.56382 639.2894 1,4 15V 10V 1,2 1,0 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 Angle 15V 20V Graph: Red line is analytical approximation Dotted points are measured results from Coventor Reducing hinge thickness resulted in: • Decreased k k 1.94 1011 Nm • Decreased pull in voltage V pi 19V CoventorWare Analyzer Graph of normalized capacitance vs angle 2,8 0.2 2.5 20V Normalized capacitance Varying the distance from the electrodes 2,4 2,0 1,6 35V Data: height_Capacitance Model: polin_3 Equation: y=1+a1*x+a3*x^3 Weighting: y No weighting Chi^2/DoF = 0.00646 R^2 = 0.98704 a1 a3 0.35615 180.28928 ±0.69503 ±21.05237 30V 20V 1,2 0,8 0,00 0,05 0,10 0,15 0,20 Angle 30V Graph: Red line is analytical approximation Dotted points are measured results from Coventor Increasing gap size resulted in: • Small deacrease in k k 1.59 1011 Nm • Increased pull in voltage V pi 37V 35V Pull in not found CONCLUSIONS - We didn’t find pull-in regime in our simulations. 2 - Instead of the parallel capacitor where g pi 3 g 0 , in the tilted capacitor the pull-in depends on the characteristics of the system. - The fitting of the curve was not easy. Our measured results were very sensitive to how the curve looked. The curve might have something different than a third degree polynomial dependency on the angle. - Nonlinearities of the forces not taken into account for the analytic calculations. - Problems with the solution when this happens -> - Suggestion to find pull in : Increase hinge thickness Decrease mesh size 50 V