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# PULL-IN IN OF A TILTED MIRROR PowerPoint PPT Presentation

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.

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

• 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

-

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

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

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!

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”)

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

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

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

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

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