EE C245 – ME C218 Introduction to MEMS Design Fall 2007ee245/fa08/lectures/Lec... · 2008. 11. 21. · ªSince it derives from V P, we call it the electrical stiffness given by:

EE C245 – ME C218Introduction to MEMS Design

Fall 2007Fall 2007

Prof Clark T C NguyenProf. Clark T.-C. Nguyen

Dept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley

Berkeley, CA 94720y

L t 23 El t i l Stiff

EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 1

Lecture 23: Electrical Stiffness

Lecture Outline

• Reading: Senturia, Chpt. 5, Chpt. 6g p p• Lecture Topics:

Energy Conserving TransducersCharge ControlCharge ControlVoltage Control

Parallel-Plate Capacitive TransducerspLinearizing Capacitive ActuatorsElectrical Stiffness

Electrostatic Comb DriveElectrostatic Comb-Drive1st Order Analysis2nd Order Analysis


Linearizing the Voltage-to-Force T f F tiTransfer Function


Linearizing the Voltage-to-Force T.F.

• Apply a DC bias (or polarization) voltage VP together with the intended input (or drive) voltage vi(t), where VP >> vi(t)

(t) V (t)EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 4

v(t) = VP + vi(t)

Differential Capacitive Transducer

• The net force on the d d suspended center

electrode is


Remaining Nonlinearity(El t i l Stiff )(Electrical Stiffness)


Parallel-Plate Capacitive Nonlinearity

• Example: clamped-clamped laterally driven beam with balanced electrodes kElectrode

Conductive Structure

•Nomenclature: kmElectrode

Va or vAva=|va|cosωt

VAd1

xt

V V +

m

Fd1

Va or vA = VA + va

Total Value AC or Signal Component

V

v1Total Value

DC Component(upper case variable;

g p(lower case variable; lower case subscript)


V1 VP

(upper case variable; upper case subscript)


• Example: clamped-clamped laterally driven beam with balanced electrodes kElectrode


• Expression for ∂C/∂x: kmElectrode

d1

xm

Fd1

V

v1


V1 VP


• Thus, the expression for force from the left side becomes: kElectrode


kmElectrode

d1

xm

Fd1

V

v1


V1 VP


• Retaining only terms at the drive frequency:CVCVF oo i|||| 121 txd

Vtvd

VF oo

Poo

Pdo

ωωω sin||cos|| 21

1211

1

111 +=

P i l Drive force arising from the input excitation voltage at

the frequency of this voltage

Proportional to displacement

90o phase-shifted from the frequency of this voltage 90 phase-shifted from drive, so in phase with

displacement• Th t t th m th t thi • These two together mean that this force acts against the spring restoring force!g

A negative spring constantSince it derives from VP, we call it the electrical stiffness given by: 212 AVCVk o ε

==


the electrical stiffness, given by: 31

121

1 dV

dVk PPe ==

Electrical Stiffness, ke

• The electrical stiffness kebehaves like any other stiffness kElectrode


• It affects resonance frequency: kmElectrode

−′ kkk em

21⎞⎛

==mm

emoω d1

x1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

kk

mk

m

em m

Fd1

21

3

211 ⎟⎟

⎞⎜⎜⎛−=′

AkVP

ooεωω

V

v131⎟⎠

⎜⎝ dkm

oo

Frequency is now a


V1 VPFrequency is now a

function of dc-bias VP1

Voltage-Controllable Center Frequency

Gap


Microresonator Thermal Stability

−1.7ppm/oC

Poly-Si μresonator -17ppm/oC

h l b l f l h l

pp


• Thermal stability of poly-Si micromechanical resonator is 10X worse than the worst case of AT-cut quartz crystal

Geometric-Stress Compensation• Use a temperature dependent mechanical stiffness to null frequency shifts due to Young’s modulus thermal dep.

[Hsu et al IEDM’00] [W.-T. Hsu, et al., IEDM’00][Hsu et al, IEDM 00]

• Problems:stress relaxationcompromised design flexibility


flexibility

[Hsu et al IEDM 2000]

Voltage-Controllable Center Frequency

Gap


Excellent Temperature StabilityR t Top Metal

ElectrodeResonator

[Hsu[Hsu et alet al MEMS’02]MEMS’02]

−1.7ppm/oCElect.-StiffnessCompensation

Top Electrode-to-Resonator Gap

[Hsu [Hsu et alet al MEMS’02]MEMS’02]

pp Compensation−0.24ppm/oC

AT-cut Quartz

p

Elect. Stiffness: ke ~ 1/d3 Q

Crystal at Various

Cut Angles

Frequency:fo ~ (km - ke)0.5

Uncompensated resonator

Angles

Counteracts reduction in

frequency due to On par with


μresonator100

[Ref: Hafner]

frequency due to Young’s modulus temp. dependence

quartz!

Measured Δf/f vs. T for ke-Compensated μResonators

150016V

VP-VC

mp μ

500

100016V

14V12V10V

0

500

7V8V9V10V

-1000

-500 6V4V2V0V

Design/Performance:f =10MHz Q=4 000

-1500

1000

300 320 340 360 380

0VCC

fo =10MHz, Q=4,000VP=8V, he=4μm

do=1000Å, h=2μmW =8μm L =40μm

−0.24ppm/oC

300 320 340 360 380

• Slit h l t l th t t d b l t l

Temperature [K]Wr=8μm, Lr=40μm

[Hsu et al MEMS’02]


• Slits help to release the stress generated by lateral thermal expansion linear TCf curves –0.24ppm/oC!!!

Can One Cancel ke w/ Two Electrodes?

•What if we don’t like the dependence of frequency on VP?C l k i diff i l k

xF• Can we cancel ke via a differential

input electrode configuration?• If we do a similar analysis for F

kmFd1

Fd2

If we do a similar analysis for Fd2at Electrode 2: d1 d2

Subtracts from the de 2

e 1

tvCVF o ωcos||2=

Fd1 term, as expected m

Elec

trod

ectr

ode

C

tvd

VF oPdo

ωω cos||

22

22

22 −= E

Ele

V

v1

V

v2txdCV o

oP ωsin||2

2

222+


V1 VPV2Adds to the quadrature term → ke’s add,

no matter the electrode configuration!

Problems With Parallel-Plate C Drive

•Nonlinear voltage-to-force transfer function

Resonance frequency becomes kx

FResonance frequency becomes dependent on parameters (e.g., bias voltage VP)Output current will also take on

kmFd1

Fd2

Output current will also take on nonlinear characteristics as amplitude grows (i.e., as x

h d )

d1 d2

de 2

e 1

approaches do)Noise can alias due to nonlinearity

m

Elec

trod

ectr

ode

y• Range of motion is small

For larger motion, need larger gap but larger gap weakens

E

Ele

gap … but larger gap weakens the electrostatic forceLarge motion is often needed (e g by gyroscopes V

v1

V

v2


(e.g., by gyroscopes, vibromotors, optical MEMS)

V1 VPV2

Electrostatic Comb DriveElectrostatic Comb Drive


Electrostatic Comb Drive

• Use of comb-capacitive tranducers brings many benefitsLinearizes voltage-generated input forces(Ideally) eliminates dependence of frequency on dc bias(Ideally) eliminates dependence of frequency on dc-biasAllows a large range of motion y Stator Rotor

xz


Comb-Driven Folded Beam Actuator

Comb-Drive Force Equation (1st Pass)

T Vi

x

Top View

y

Lfxz

Side View


Lateral Comb-Drive Electrical Stiffness

T Vi

x

Top View

y

Lfxz

Side View

• Again: ( ) hNChxNxC εε 22=

∂→=g

•No (∂C/∂x) x-dependence → no electrical stiffness: ke = 0!

( )dxd

xC∂

→


p e

• Frequency immune to changes in VP or gap spacing!

Typical Drive & Sense Configuration

y

xz


Comb-Drive Force Equation (2nd Pass)

• In our 1st pass, we neglectedFringing fieldsP ll l l i b d Parallel-plate capacitance between stator and rotorCapacitance to the substrate

• All of these capacitors must be included when evaluating the All of these capacitors must be included when evaluating the energy expression!

Stator Rotor

Ground


Ground Plane

Comb-Drive Force With Ground Plane Correction

• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures

d h d l difi h i i and the ground plane → modifies the capacitive energy

[Gary Fedder Ph D


[Gary Fedder, Ph.D., UC Berkeley, 1994]

Capacitance Expressions

• Case: Vr = VP = 0V• Csp depends on whether or not p pfingers are engaged

Capacitance per l h

Region 2 Region 3

unit length



Comb-Drive Force With Ground Plane Correction

• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures

d h d l difi h i i and the ground plane → modifies the capacitive energy



Simulate to Get Capacitors → Force

• Below: 2D finite element simulation

20-40% reduction of Fe,x


Documents

EE C245 – ME C218 Introduction to MEMS Design Fall 2007ee245/fa08/lectures/Lec... · 2008. 11. 21. · ªSince it derives from V P, we call it the electrical stiffness given by: