1. 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

2. CoventorWare Analyzer Mirror Design 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 Storing of energy in capacitor Energy formula used to derive electrical force

3. CoventorWare Analyzer Mirror Design 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 The parallell plate capacitor analogy (continued) 2 2 2 g AV F elec    2 3 2 g AV k  

4. CoventorWare Analyzer Mirror Design Derivation of formulas for the mirror design 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 Formulas for torque calculations shown below

5. ...and analyzing the stability of the equilibrium CoventorWare Analyzer Mirror Design Derivation of formulas for the mirror design (continued) Hornbecks analysis computes torque directly treating tilted plate as parallell plate. Eletric torque formula is analogous to electric force: Difficult analytically!

6. CoventorWare Analyzer 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 Using the same formulas as earlier, but now with the new formula for capacitance is used to find electric torque: General formula from graph can be of the following third polynomial formula; From mechanical torque formula, we can find the spring constant (stiffness of ”hinge”)

7. 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: The root expression must be positive for a stable solution. This will give us a formula for pull in voltage Now that we had a formula to calculate pull in voltage, we attempted to run Coventor simulations

8. CoventorWare Analyzer Mirror Design 0.4 1.5 20V 40V 47V 20V 40V 47V Original geometry: Graph of normalized capacitance vs angle Only one electrode has applied voltage No exaggeration is used Mesh is 0,4 micrometer, equal to hinge thickness Mesh was not changed when changing geometry parameters. Results: Graph: Red line is analytical approximation Dotted points are measured results from Coventor

9. CoventorWare Analyzer Varying k by reducing hinge thickness 0.2 1.5 10V 15V 20V 10V 15V Graph of normalized capacitance vs angle Graph: Red line is analytical approximation Dotted points are measured results from Coventor Reducing hinge thickness resulted in: Decreased k Decreased pull in voltage

10. 2.5 CoventorWare Analyzer Varying the distance from the electrodes 20V 30V 35V 20V 30V 35V 0.2 Graph of normalized capacitance vs angle Graph: Red line is analytical approximation Dotted points are measured results from Coventor Increasing gap size resulted in: Small deacrease in k Increased pull in voltage  Pull in not found

11. CONCLUSIONS - We didn’t find pull-in regime in our simulations. - Instead of the parallel capacitor where, 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