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PULL-IN IN OF A TILTED MIRRORPowerPoint Presentation

PULL-IN IN OF A TILTED MIRROR

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Presentation Transcript

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

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

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

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!

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”)

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

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

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

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

- 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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