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MECH 466Microelectromechanical Systems
University of VictoriaDept. of Mechanical Engineering
Lecture 6:Electrostatic Sensors and Actuators
© N. Dechev, University of Victoria
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Equilibrium Positions of Electrostatic Devices
Pull-in Voltage
Case-Studies of Electrostatic Devices
Overview
© N. Dechev, University of Victoria
3© N. Dechev, University of Victoria
Equilibrium Position of Electrostatic Devices
Consider the following parallel plate capacitor system:
(a) Without electrical bias
anchored plate
moveable plate
km
anchored plate
km
(b) With electrical bias
* Note, that C is also a function of d
We can describe the mechanical force as:
We can describe the electrical force as:
4© N. Dechev, University of Victoria
Equilibrium Position of Electrostatic Devices
From the diagram, let:
Therefore, the expression for the electrostatic force becomes:
To find an expression for the equilibrium position, we must equate:
Therefore, we have the expression:
5© N. Dechev, University of Victoria
Equilibrium Position of Electrostatic Devices
The previous equation can be expanded, which yields the quadratic equation for x:
*Note, this equation yields two solutions
6© N. Dechev, University of Victoria
Equilibrium Position of Electrostatic Devices
A plot of the mechanical force and electrical force shows:Fmechanical =-kmx
Felectrical =εAV2/(2d2)
- The linear mechanical function intersects with the electrical function at two points.- Note: Only the solution closest to the starting position (right side) is realizable.
As we increase voltage, the previous graph will change as follows:
7© N. Dechev, University of Victoria
Concept of ‘Pull-in’ Voltage:
Fig 4.5 Balance between mechanical forceand electrical force [Chang Liu]
Eventually, we reach a ‘Critical Point’ voltage where there is only one solution, where the mechanical force is equal to the electrical force.
This is called the ‘Pull-in’ voltage.
8© N. Dechev, University of Victoria
Concept of ‘Pull-in’ Voltage:
Fig 4.6 Pull-in Voltage [Chang Liu]
Consider the consequences of applying a voltage higher than the ‘Pull-in’ voltage.
To conclude: Electrostatic devices should be designed or operated such that the applied voltage remains below the ‘Pull-in’ voltage to avoid ‘snap-in’.
‘Snap-in’ may damage the mechanism or cause burn-out due to contact under high applied voltage.
9© N. Dechev, University of Victoria
Concept of ‘Pull-in’ Voltage:
10
Review Examples 4.2 & 4.3 in Textbook
Homework:
© N. Dechev, University of Victoria
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Pull-in Voltage
Calculate ‘Pull-in’ voltage, Vp:
We know that at equilibrium conditions:
Fe = Fm where: and
(eq. 4.12)
© N. Dechev, University of Victoria
Moving PlateWhere: V - Voltage applied between the two plates Km - Mechanical Spring Constantxo
Anchored Plate
xFmechanical
FelectricalKm
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Upon review of the Force vs. Displacement graphs, we know that at Vpull-in the Fe curve touches the Fm curve at one point.
At this point, the tangent (slope) of Fe is equal to the slope of Fm.
The slope of Fe at this point can be defined as:
And the slope of Fm is defined as:
Therefore, this can be written as:
© N. Dechev, University of Victoria
Pull-in Voltage
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Substituting in the expression (4.12) for have:
Therefore:
Which gives an expression for displacement, x, as:
© N. Dechev, University of Victoria
Pull-in Voltage
Note: This means that the plate is displaced by 1/3rd the total separation distance d, when V = Vpull-in
This displacement value for x (at Vpull-in) can be substituted back into equation 4.12, which allows us to determine Vpull-in to be:
14© N. Dechev, University of Victoria
Pull-in Voltage
Note: That C is the Capacitance at the “pull-in” displacement point. Hence, it must be recalculated at that point.
Note: Therefore, it is simpler to replace C, with 1.5Co , i.e. the original capacitance times 1.5, to represent capacitance value at xo/3.
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Example of Equilibrium Position:
Given two round plates of diameter d, directly overtop one-another, separated by a 2 um gap.
Q: What is the displacement of the upper plate if a voltage of 5 V is applied between the plates?
© N. Dechev, University of Victoria
Parallel Plate Capacitors
where: d = 100 umL = 100 umw = 5 umt = 2 um
Material: PolysiliconE: 160 GPa
16© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution:
Step 1: Calculate the mechanical spring constant Km.
- This is a “fixed-guided” beam
- From Appendix B, the maximum deflection for a fixed-guided beam is given by:
- Establish the deflection formula for the beam, by determining the appropriate model in terms of end boundary conditions
17© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution-Continued:
- Note that I for a beam with rectangular cross-section is:
-therefore:
18© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution-Continued:
- Since
- we have for each beam.
- Since the beams are identical and loaded symmetrically, the total system stiffness is:
19© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution-Continued:
Step 2: Find the displacement
- Using equation (4.12) find an expression for displacement and solve:
* However, C varies with displacement d, therefore, for parallel plates, this expression can be re-written as:
where:
20© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution-Continued:
- the previous equation can be re-arranged with respect to x as:
-substituting values:
- this equation yields three possible solutions for x:
<-- physically realizable
<-- past pull-in deflection
<-- impossible
21© N. Dechev, University of Victoria
Parallel Plate Capacitors
Solution-Continued:
Step 3: Check value for Pull-in Voltage, to ensure it is not exceeded.
Therefore:
Answer: Upper plate will be displaced by 0.014 um when 5 V is applied between the plates.
Where:
22
Review Case Studies 4.1, 4.3 & 4.6 in Textbook
Homework:
© N. Dechev, University of Victoria
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Example: “Case Study 4.2” from Textbook
Torsional Parallel Plate Capacitive Accelerometer
© N. Dechev, University of Victoria
Torsional Parallel Plate Capacitors
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Example: “Case Study 4.2” from Textbook
© N. Dechev, University of Victoria
Torsional Parallel Plate Capacitors
See Class Notes
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