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

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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
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

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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 T ? Tilted angle
• Formulas for torque calculations shown below

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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!
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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)

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CoventorWare Analyzer
Mirror Design
Alternative analytical solution (continued)

• The spring constant formula has our variable T.
  By rearranging this formula, T 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

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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
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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
10
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
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CONCLUSIONS
- We didnt 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 -gt
• Suggestion to find pull in
• Increase hinge thickness
• Decrease mesh size

50 V