PULL-IN IN OF A TILTED MIRROR
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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.

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PULL-IN IN OF A TILTED MIRROR

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Pull in in of a tilted 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


Pull in in of a tilted mirror

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


Pull in in of a tilted mirror

-

e

e

2

2

AV

AV

=

>

F

k

elec

2

3

2

2

g

g

CoventorWare Analyzer

Mirror Design

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


Pull in in of a tilted mirror

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


Pull in in of a tilted mirror

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:

...and analyzing the stability of the equilibrium

Difficult analytically!


Pull in in of a tilted mirror

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”)


Pull in in of a tilted mirror

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


Pull in in of a tilted mirror

0.4

CoventorWare Analyzer

Graph of normalized capacitance vs angle

Mirror Design

47V

Original geometry:

1.5

40V

20V

20V

Graph: Red line is analytical approximationDotted points are measured results from Coventor

40V

  • Only one electrode has applied voltage

  • No exaggeration is used

  • Mesh is 0,4 micrometer, equal to hinge thicknessMesh was not changed when changing geometry parameters.

  • Results:

47V


Pull in in of a tilted mirror

CoventorWare Analyzer

Graph of normalized capacitance vs angle

20V

Varying k by reducing hinge thickness

0.2

1.5

15V

10V

10V

15V

Graph: Red line is analytical approximationDotted points are measured results from Coventor

  • Reducing hinge thickness resulted in:

  • Decreased k

  • Decreased pull in voltage

20V


Pull in in of a tilted mirror

0.2

CoventorWare Analyzer

Graph of normalized capacitance vs angle

Varying the distance from the electrodes

35V

2.5

20V

30V

20V

30V

Graph: Red line is analytical approximationDotted points are measured results from Coventor

  • Increasing gap size resulted in:

  • Small deacrease in k

  • Increased pull in voltage

35V

 Pull in not found


Pull in in of a tilted mirror

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


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