V F V The Parallel Plate Capacitor as a Simplified Representation of Electrostatic Actuators

(1)

Brussels, September 10-19, 2001

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté Institut des Microtechniques de Franche Comté

5 Silicon-Based Microactuators

5.1 IC-Compatible Microactuators using

Electrostatic Field Interactions

NATO Advanced Study Institute RESPONSIVE SYSTEMS … / MEMS CourseBrussels, September 10-19, 2001

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté Institut des Microtechniques de Franche Comté

The Parallel Plate Capacitor as a Simplified Representation of Electrostatic Actuators

V

Fw

d

h w

x d

V

h

w

Fd

F d V

F

w

Normal Electrostatic Field Interactions Tangential Electrostatic Field Interactions F

d

>> F

w

NATO Advanced Study Institute RESPONSIVE SYSTEMS … / MEMS Course Brussels, September 10-19, 2001

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté

Normal Electrostatic Force Acting Within Parallel Plate Capacitors

d

V

h w

Fd

k

V

k

F d

Capacitance:

d C = ε _r ε ₀ wh

Electrostatic Energy:

d CV whV

U ^r

2 2

1 ₂ ε ε ₀ ²

−

=

−

=

Electrostatic Force that Pulls the Plates Together:

0 2 2

0 2

2 1 2

1 SE

d whV d

F _d U ₌ ε ^r ε ₌ ε _r ε

∂

− ∂

=

Design Assumption S = 50 x 50 µm

D = 2 µm

V = 100 Volts (p

el

= 10

^-2

Mpa)

Restricted Displacement

(Pull-in Effect & Driving Voltage)

F d = 25 µN

(2)

Tangential Electrostatic Force Acting Within Parallel Plate Capacitors

V

Fw

d

w h

x

Electrostatic Energy:

d CV whV

U ^r

2 2

1 ₂ ε ε ₀ ²

−

=

−

=

Electrostatic Force that Pulls the Plates Together:

0 2 0 2

2 1 2

1 hdE

d hV w

F _w U ε ^r ε ε _r ε

=

∂ =

− ∂

=

Design Assumption S = 50 x 50 µm

D = 2 µm F w = 1 µN (F w << F d )

V = 100 Volts

Unrestricted Displacement (Except Comb Drive Actuators)

Electrostatic Force Remains Constant V x (Fringing Effects)

F

w

Driving Torque Reference on the Micrometer scale

L = 1 meter !

F = 1 µN Γ = 1µNm

How Shall we Solve Industrial Needs using Micrometer Size Electrostatic Actuators?

- Desperate Situation?

- Even worse taking into account structural height limitation !..

Lever

V

Fw

d

w h

x

h!

Side-Drive Electrostatic Actuators

(a) (b)

From [FAN 88] / Berkeley Sensor & Actuator Center

Side-Drive Electrostatic Actuators

Capacitance:

( ) ( )

d n S

C ϑ = ε _r ε ₀ ϑ

Electrostatic Energy:

( ) ( ) ( )

d V S V n

C

U ^r

2 2

1 ₂ ε ε ₀ ϑ ²

ϑ

ϑ = − = −

( ) d

V hR

U n ^r

2 0 ϑ 2

ε ϑ = − ε

Theoretical Electrostatic Torque

0 2 2

0 2 0 2

2 1 2

2 n hRdE

d hRdV n

d hRV n

U

r r

r ε ε ε ε ε

ε

ϑ ⁼ ⁼ ⁼

∂

− ∂

= Γ

Where:

d E = V

From LAAS - CNRS

(3)

Side-Drive Electrostatic Actuators: Driving Limitations

φ : 120 µm Ω

exp

= 50 rpm << Theoretical Speed h: 1 µm; d: 4 µm Γ

elec

= 2.5 10

^{- 6}

µNm >> External Torque V: 200 Volts P

_mec

= 10

^–12

Watt !

- Restricted Structural Thickness (h)

- Gap Limitation (d)

- Driving Voltage Limitation (V)

- Restricted Electrostatic Energy (E=V/d)

- Material’s Permittivity ( ε

r

) - Parasitic Friction

Incompatibility in Size with Industrial Needs

0 6 12 18

0 50 100 150

External Diameter Φ (µm)

Driving T orque Γ (pN.m)

R.S. Müller Micromotor : Γ = 9pN.m

Γ = 1µN.m ⇒ Φ > 30,000 microns ! Γ = 1µNm φ > 30,000 microns !

0 500 1000 1500 2000 2500 3000 3500 4000

0 500 1000 1500 2000

4 10

^{– 3}

µNm

Structural Thickness Restricted to a few microns !

External Diameter Limit according to the Structural Height Limitation of Surface Micromachined Actuators

!

Driving Force Amplification on the Micrometer Scale:

Classical Approaches

Integrated Speed Reduction using “Harmonic Drive” Microactuators Comb-Drive Electrostatic Actuators

High Aspect Ratio Polysilicon Microactuators Integrated Polysilicon Gear Reductions

Top-Drive Electrostatic Microactuators

Top-Drive Electrostatic Actuators

(a) (b)

d R1

R0

Largeur radiale d’entrefer : w = R₁–R₀

From [MEH 90] / Massachusetts Institute of Technology

(4)

Top-Drive Electrostatic Actuators

Capacitance:

( ) ( ) ( ) 



 



 +

−

=

= 2

0 0 1

0 1 0

R R R

d R n d n S

C _r ϑ ε ^r ε ϑ

ε ε ϑ

Theoretical Electrostatic Driving Torque:

( ) ( 2 0 ² )

1 0 2 2

4 2

1 R R

d V n

V C = ^r −

∂

= ∂

Γ ε ε

ϑ ϑ

( ₁ ₀ )

0 2

2 R R

d RV

n _r −

= Γ ε ε

where:

( R ₁ R ₀ ) / 2 R = +

R

1

R

0

(a) (b)

d R1

R0

Largeur radiale d’entrefer : w = R1–R0

φ : 300 µm Ω

exp

= 150 rpm Ω

_theo

= 120, 000 rpm!

d: 2 µm Γ

elec

= 450 10

^{- 6}

µNm >> External Torque!

V: 200 Volts P

_mec

= 5 10

^–10

Watt

0 2

2 1 n ε _r ε hRdE

= Γ

Torque Amplification:

h R G R ¹ − ⁰

= ( ₁ ₀ )

0 2

2 R R

d RV

n _r −

= Γ ε ε

Parasitic Friction!

Comparing Side-Drive &Top-Drive Actuators

Harmonic-Drive Electrostatic Actuators

(a)

(b)

(c)

(d) Stato

RotorJeu / tGapinter-

t

Cham tournan t Axe

Sens de t ti Point de

t t

Sens de t ti

Cham tournan t Gapinter-

t Point de Rotor t t Stato

Isolan t

- Integrated Speed Reduction:

r r

r N R

S

R δ

ω

ω − =

=

- Dual Torque Amplification:

δ r G = N 1 =

- Practical Limitation:

10 2

≈ G

Γ

elec

= 250 10

^{- 6}

µNm

⁽

*

⁾

From [MEH 90] / Massachusetts Institute of Technology

Harmonic-Drive Electrostatic Actuators

From web site: mems.colorado.edu

(5)

NATO Advanced Study Institute RESPONSIVE SYSTEMS … / MEMS CourseBrussels, September 10-19, 2001

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté Institut des Microtechniques de Franche Comté

Harmonic-Drive Electrostatic Actuators From [HIR 95] / IIS-The University of Tokyo

Comb-Drive Electrostatic Actuators

Masse

electrode 1

electrode 2 y

x L w SHUFFLE ANCHOR

g/2 δ

- Driving Force per Unit Area : 100 µN/mm

²

V F V The Parallel Plate Capacitor as a Simplified Representation of Electrostatic Actuators

5 Silicon-Based Microactuators

5.1 IC-Compatible Microactuators using

Electrostatic Field Interactions

The Parallel Plate Capacitor as a Simplified Representation of Electrostatic Actuators

V

d

h w

x d

h

w

F d V

F

Normal Electrostatic Field Interactions Tangential Electrostatic Field Interactions F

>> F

Normal Electrostatic Force Acting Within Parallel Plate Capacitors

d

h w

V

k

F d

Capacitance:

d C = ε r ε 0 wh

Electrostatic Energy:

d CV whV

U r

2 2

1 2 ε ε 0 2

−

=

−

=

Electrostatic Force that Pulls the Plates Together:

0 2 2

0 2

2 1 2

1 SE

d whV d

F d U = ε r ε = ε r ε

∂

− ∂

=

Design Assumption S = 50 x 50 µm

D = 2 µm

V = 100 Volts (p

= 10

Mpa)

Restricted Displacement

(Pull-in Effect & Driving Voltage)

F d = 25 µN

Tangential Electrostatic Force Acting Within Parallel Plate Capacitors

V

d

w h

x

Electrostatic Energy:

d CV whV

U r

2 2

1 2 ε ε 0 2

−

=

−

=

Electrostatic Force that Pulls the Plates Together:

0 2 0 2

2 1 2

1 hdE

d hV w

F w U ε r ε ε r ε

=

∂ =

− ∂

=

Design Assumption S = 50 x 50 µm

D = 2 µm F w = 1 µN (F w << F d )

V = 100 Volts

Unrestricted Displacement (Except Comb Drive Actuators)

Electrostatic Force Remains Constant V x (Fringing Effects)

F

d C = ε _r ε ₀ wh

U ^r

1 ₂ ε ε ₀ ²

F _d U ₌ ε ^r ε ₌ ε _r ε

U ^r

1 ₂ ε ε ₀ ²

F _w U ε ^r ε ε _r ε

C ϑ = ε _r ε ₀ ϑ

U ^r

1 ₂ ε ε ₀ ϑ ²

U n ^r

ϑ ⁼ ⁼ ⁼