lec 6 and 7 electrostatic

7/22/2019 LEC 6 and 7 Electrostatic

http://slidepdf.com/reader/full/lec-6-and-7-electrostatic 1/33

Electrostatic Actuation andDetection

Department of M echanical Engineer ing

Sherry Towfighian

ME521 Dynamics of MEMS



Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling2

Electrostatic Transduction

It is the most common actuation and sensing method in MEMS

because of its simplicity and high efficiency.

It relies on simple capacitors of parallel plate electrodes. These can

be easily fabricated using surface micromachining.

It does not require any special material, deposition of any patches,

or any external field sources. It only requires a voltage source.

Two classical MEMS devices :

-The Analog Devices accelerometers

for airbag deployments,which use capacitive detection.

-The digital mirror display DMD

for projection displays

by TI, which relies on electrostatic actuation.




Features

It is characterized by very low power consumption (parallel-plate

capacitor is an open-circuit component).

Electrostatic actuation offers high energy density, high mechanical

flexibility, and well controlled force. Also, it is considered fast actuation method.

In electrostatic sensing, a physical quantity, such as acceleration,

changes the capacitance values of the parallel-plate capacitor,

which can be related to the physical quantity being measured.




Disadvantages

•Its main disadvantages is the nonlinearity, especially for out-of-

plane capacitors. This limits the controlled travel range of actuators

and can result in unexpected collapse and functional failure of

sensors.

•Also, the large driving voltage for actuation applications, such as

in RF switches, is another limitation.




Parallel-Plate Capacitor

The most common forms of electrostatic sensing and actuation are

based on either simple parallel-plate capacitors or comb-drive

configuration of multiple interdigitated or non-interdigitated fingers.

Parallel-plate capacitor derivations:

Assume infinite plates (negligible fringe effect).

Because of the power source, the capacitor

is charged according to

where Q is the electrical charge, C is capacitanceand is a variable representing the degree of

freedom of the moveable electrode of the capacitor.

d

V

Stationary

Moveable

( )Q C V



Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling 6

Parallel-Plate Derivations

The potential energy E c stored in the capacitor can be expressed as

The initial energy of the battery that is hooked-up to the capacitor

is assumed to be E 0. When connected to the capacitor, the battery provides the electrical

charge Q and an electrical potential – QV , which is negative since it

is lost from the battery. Hence the new potential energy of the

battery E b is reduced to

The total potential energy E is equal to that from the battery and the

capacitor, that is

21( )

2c E C V

20 0b E E QV E CV

2 2 2

0 0

1 1

2 2b c E E E E CV CV E CV


http://slidepdf.com/reader/full/lec-6-and-7-electrostatic 7/33Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling7

Parallel-Plate Force

The attractive force F between the two electrodes of the capacitor

can be obtained according to

For electrodes of simple geometries, an analytical expression can

be found for C .

For electrodes of complicated geometry, finite-element or

boundary-element analysis can be used to characterize the

dependence of C on the displacement . Then, curve fitting is

used to extract a simple analytical expression for that can be

used in Eq. (*) .

2( ) 1 ( ) (*)

2

E C F V

( )C

Si l ll l l l



Simple Parallel-Plate Rectangular

Capacitors

C ( x) above is obtained by solving

the Laplace equation of the electrostatic

potential between the two rectangular plates.

A is the overlap area between the two electrodes ( A=Lb), is the

relative permittivity of the gap space medium with respect to the free

space (equal unity in air), and is the dielectric constant in free

space, .

Substituting for C ( x)into Eq. (*) yields

Minus: Attractive force, opposite to x direction.

Fixed Moveable

b L

-

+

x

t x

( ) r o r o

A LbC x

x x

2

2 (**)

2

r o AV F

x

o

r

12 2 28.85 10 ( / )o C Nm



Pull in

Note from Eq. (**) that the parallel-plate electrostatic force is inherently

nonlinear in terms of the electrodes separation. Typically in MEMS, the movable electrode is a flexible structure, such as

a beam or a plate. Hence, when deflected with a distance x due to the

electrostatic force, the resorting force of the structure tends to oppose the

electrostatic force (acts as a spring). If the electrostatic force is not too

big, equilibrium between the restoring and electrostatic force is achieved.

However, if the voltage load is increased too much, then the electrostatic

force overcomes the resorting force of structure leading to its collapse,

where it hits the stationary electrode. This phenomenon is known in

electrostatic MEMS as the pull-in instability.

VDCVDC VDCVDCVDCVDCVDC VDCVDC

(a) V DC =0 (b) V DC < V pull-in (c)V DC > V pull-in



Electrostatic Sensing

In sensing application, pull-in is undesirable. It leads to short

circuit, stiction, and un-functionality of the sensor.

Diaphragm

Pressure

VLower electrode

Diaphragm Upper electrodeLower electrode

(a) Taken apart. (b) Assembled.

(c) Schematic.

Cantilever beamsAttachment point

(a) Taken apart. (b) Assembled.

Proof mass

(upper electrode)

VLowe electrode

motion

(c) Schematic.

Accelerometer. Pressure sensor.

(Sensata Technologies)



Electrostatic Actuation

Many actuators rely on pull-in to achieve reliable actuation in short

time. RF switches and micro-relays use this actuation method. RF

switches are commonly used for wireless, communications, and radar

applications. Pull-in provides fast actuation method with large stroke. Another example is the actuation of the Deformable Mirror Device

DMD or the micromirror of Texas Instruments. The micromirroir is

actuated beyond pull-in in angular motion to deflect and reflect

incident lights .

V

V

Switch “On”

Switch “Off”

Gate Drain

Micromirror

VVActuation padLanding pad



Pull-in Model

The above represents a model for a parallel-plate capacitor, where

the moveable electrode of mass m is a flexible structure, such as a

beam, of stiffness k , which forms the upper electrode.

k

x

m

d V DC

m

kx

2

22( )

r o AV F d x

2

2

2( )

AV F

d x

2

2 (1)

2( )

AV kx

d x

Equilibriumr o



Pull-in Model

Equation (1) is a cubic equation in x with three possible

solutions:

(a) One predicts x>d , and hence it is nonphysical and discarded.

(b)One represents unstable solution, meaning that practically the

structure cannot hold into it.

(c)One solution ( x<d ) that is stable, which represents the “real”

deflection of the upper electrode in response to the DC load. It

is smaller than (b).

One easy approach to explore the solutions of Eq. (1) is to

plot the right-hand and left-hand sides of the equation and

then look for their intersection.

2

2 (1)2( )

AV

kx d x



Pull-in Expression

Note that at V DC = V pull , x=d/3 .

0 0.5 10

0.2

0.4

0.6

0.8

1

F o r c e / ( k

d )

x/d 0 0.5 1

0

0.2

0.4

0.6

0.8

1

F o r c e

/ ( k d )

x/d

0 0.5 10

0.2

0.4

0.6

0.8

1

F o r c e / ( k d )

x/d

d/3

0 0.5 10

0.2

0.4

0.6

0.8

1

F o r c e

/ ( k d )

x/d

(a) V DC << V pull . (b) V DC < V pull .

(c) V DC = V pull . (d) V DC > V pull .

38

27 pull

kd V

A

Subst. in Eq. (1), solve for V


http://slidepdf.com/reader/full/lec-6-and-7-electrostatic 15/33Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling 15

Pull-in Plot

Another way to view the solution of Eq. (1) is to solve for V in terms

of x as One can then plot this equation for several values of x.

Of the two curves, the smaller one represents stable solutions and

the larger one represents unstable solutions. Both meet at the pull-in.

22 ( ) / ( ) DC V kx d x A

0.2 0.4 0.6 0.8 1.0

x

d

0.2

0.4

0.6

0.8

1.0

V DC

V pull

2

2 (1)2( )

AV

kx d x

T i l A t t d


http://slidepdf.com/reader/full/lec-6-and-7-electrostatic 16/33Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling 16

Torsional Actuators and

Micromirrors

Electrostatically actuated torsional micromirrors form the backbone of

the technology of video and image projections and flat-screen TV’s of

Texas Instrument (www.dlp.com).

Each mirror, also called Deformable Mirror Device DMD, representsone image pixel on the chip. The chip contains approximately two

million of these. The mirror reflects the incident light thousands of tim

per second in a digital on-off fashion.

The mirror is brought to the on position by actuating it beyond pull-in.

http://www.dlp.com/

http://www.dlp.com/


http://slidepdf.com/reader/full/lec-6-and-7-electrostatic 17/33Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling

17

Modeling Torsional Actuators

Electrostatic force on a small segment.

V DC

a3

x

a2

d

t

a1

dx

a1

b

w

t

a3

a2

Electrode

l

2

22( )

bV dxdF

d x

2 2

1 1

2

22( )

a a

DC e

a a

bV x M xdF dxd x

2

2

2

2 1 1

ln2

DC e

bV d ad d M

d a d a d a

Electrostatic torque



18


A more convenient form of the electrostatic torque can be obtained

by normalizing with respect to its maximum possible value

and defining

max 3/d a

max

;

1 2

3 3

;a a

a a

2

2 2

max

1 1 1ln

2 1 1 1 DC

e

bV M

In the special case that the electrode spans the whole length of the

actuator a1= =0 , a2 = a3 and =1. Then

2

2 2

max

ln 12 1

e

bV M



19


When an electrostatic torque is applied on the upper plate of the

actuator, an opposing restoring torque M mech from the torsional

springs is induced to counter this torque:

where K t is the effective torsional spring coefficient of the two

torsional springs. Thus, the actuator reaches an equilibrium state at

the tilted angle . The equation of equilibrium can be written as

Each side of Eq. (2) can be plotted for a given voltage.

Alternatively, an expression for V can be solved:

maxmech t t M K K

2

max 2 2

max

1 1 1ln (2)

2 1 1 1

DC t

bV K

3 3

max2

1 1 1ln1 1 1

t DC

K V

b



20

Torsional Pull-in

For the special case of a full-electrode actuator, pull-in occurs at

More generally, it occurs at

The pull-in voltage can be calculated according to

For the special case of a full-electrode actuator:

0.4404

1 22.117

0.4404; /

1 0.322

pull a a

3

0.32442 0.5 1.931

3

2

0.8275(1 ) 1 0.6735t

pull

K d V

ba

3

3

2

0.90967 t pull

K d V

ba



21

Example

Consider the torsional actuator of the parameters in the table below.

Evaluate the normalized angle for V DC

=12 V

Plot V DC versus .

Calculate V pull and analytically and compare to the plot.

K t 310 (N. m) a1 ( m ) a2 ( m ) a3 ( m ) d ( m ) b ( m )

0.0143087 430 580 600 3.42 1300

pull

V DC

a3

x

a2

d

t

a1

dx

a1

b

w

t

a3

a2

Electrode

l



22

Example Solution

First, we plot both sides of Eq. (2) to show the intersection of the

curves and the two possible solutions. Using the command

“FindRoot” in the software Mathematica with an initial guess from

the figure gives

Using the analytical expressions gives

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Torques K t max

0.2 0.4 0.6 0.8 1.0

5

10

15

20V DC V

0.075064

0.376 pull

19.5 V pull V

2

max 2 2

max

1 1 1ln (2)

2 1 1 1 DC

t

bV K




Comb-Drive Devices

They are commonly used in MEMS for sensing and actuation. They have

been implemented in accelerometers, microgrippers, resonators, filters, andgeneric force actuators.

Comb-drive devices rely on two comb structures, a stationary and a

moveable one that is connected to flexible structures (springs or tethers).

The flexible structures are made to be compliant in the direction of the

desirable displacement and stiff in the orthogonal directions to limit motion

in these directions.

Each comb consists of interdigitated fingers. The stationary and moveable

fingers form parallel-plate

capacitors among each other.




Comb-Drive Types

Comb-drive devices can have many configurations, such as rotary, out-of-

plane, and in-plane devices. The most common are the in-plane devices.

In-plane devices can be classified into two categories: transverse and

longitudinal. In either case, each moveable finger is sandwiched between

two fixed fingers, where portion of its surface overlap with those of the fixed

fingers. This causes capacitive and electrostatic interaction among thefingers resulting in an actuation and/or detection mechanism.

y x

z

d

l g

t

b




Transverse Comb-Drive

The moveable comb moves in transverse direction to the longitudinalaxis of the fingers (along the x-axis in the figure). As seen in the

figure, the moveable finger represents an electrode for two parallel-

plate capacitors with two stationary fingers, thereby forming one

capacitor to its right and another one to its left.

y

x

z

Fixed electrode

Fixed electrode

Moveable electrode

Anchors

Tether(spring)

Tether




Transverse Comb-Drive Force

If the finger moves a positive distance x (to the left), then the

capacitance of the left capacitor C left

and right capacitor C right

are

expressed as

y x

z

d

l g

t

b

( );

( );

left

right

lt C

g x

lt C

g x



Mohammad Younis MEMS ModelingMohammad Younis MEMS Modeling 27

Transverse Comb-Drive Force

Note that in the case of no displacement, x=0, and C left = C right .

Otherwise,

This differential capacitance can be used to sense the motion of the

moveable comb. To increase the value of the induced capacitance,

the number of comb fingers is increased.

If the capacitors are biased by a voltage V , then two electrostatic

forces will be generated on the moveable finger:

If x=0, the electrostatic forces acting on the

moveable finger cancel each other. Otherwise a

resultant force F right -F left will act on the

moveable finger.

This represents the backbone of the ADXL-50 accelerometer of Analog Devices.

2 2

2 ( )left right

lt xC C

g x

2

2

2

2

( );

2

( );

2

left

right

lt V F

g x

lt V F

g x




The ADXL-50 Accelerometer of Analog Devices




Longitudinal Comb-Drive

In the case of a longitudinal comb-drive, the moveable comb moves

parallel to the longitudinal axis of the fingers (along the y-axis).

V

Moveable Electrode x

y b

z

t

d Fixed

Fixed electrode

Fixed electrode

Moveable electrode

y

x z

Tether Tether




Longitudinal Comb-Drive Force

One can note from that the electrostatic force along the length of

the finger is independent from its length and depends on its width.

This force tends to increase the overlap among the fingers while

being constant in magnitude. The fact that this force remains

constant during the actuation and that is independent of the

displacement makes the longitudinal comb-drive actuators capable

of providing linear well-controlled actuation.

( ) r o A ybC yd d

V

Moveable Electrode

x y b

z

t

d Fixed2

21 ( )

2 2 y

C bV F V

d




Longitudinal Array

For the case of a longitudinal moveable finger sandwiched between

two stationary fingers, there are two capacitive forces generated on

the moveable finger from the two nearby stationary electrodes with

the electrode area of each capacitor is ty and the gap width g . For n

number of moveable fingers, the total electrostatic force acting on

the moveable comb drive is

y x

z

d

l g

t

b

2 2

2

2

Tot

y

tV n tV F n

g g




Example

Example: A comb-drive actuator has 10 moveable fingers with t=2 µm and g=2 µm. The stiffness of the tethers holding the moveable

comb was measured and found to be k=0.01 N/m. Calculate and plot

the deflection of the comb when actuated by a voltage load ranging

from 0-10 V .

Solution:

It is clear from that the deflection is linearly proportional to the

voltage squared and to the electrostatic force. This is unlike the case

of parallel-plate capacitive actuation, where the deflection and force

are related nonlinearly

2

n tV ky g

2 1 Tot

y

n tV y F

kg k



33

Example

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Voltage (V)

Displacement(

m)

lec 6 and 7 electrostatic

Documents