ee c245 - me c218 introduction to mems design fall 2003 · 4 7 ee c245 – me c218 fall 2003...

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EE C245 – ME C218 Fall 2003 Lecture 26

EE C245 - ME C218Introduction to MEMS Design

Fall 2003

Roger Howe and Thara SrinivasanLecture 26

Micromechanical Resonators I

2EE C245 – ME C218 Fall 2003 Lecture 26

Today’s Lecture

• Circuit models for micromechanical resonators

• Microresonator oscillators:

sustaining amplifiers, amplitude limiters,and noise

• Resonant inertial sensors:

accelerometers and gyroscopes

2


Reading/Reference List• C. T.-C. Nguyen, Ph.D. Thesis, Dept. of EECS, UC Berkeley, 1994.• T. A. Roessig, R. T. Howe, A. P. Pisano, and J. H. Smith, “ Surface-

micromachined resonant accelerometer,” (Transducers ’97), Chicago, Ill., June 16-19, 1997, pp. 859-862.

• A. A. Seshia, R. T. Howe, and S. Montague, “An integrated microelectromechanical resonant-output gyroscope,” IEEE MEMS 2002,Las Vegas, Nevada, January 2002.

• C. T.-C. Nguyen, “Transceiver front-end architectures using vibrating micromechanical signal processors,” Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, Sept. 12-14, 2001, pp. 23-32.

• J. Wang, Z. Ren, and C. T.-C. Nguyen, “Self-aligned 1.14 GHz vibrating radial-mode disk resonator,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 947-950.

• B. Bircumshaw, et al, “The radial bulk annular resonator: towards a 50Ω RF MEMS filter,” Transducers ’03, Boston, Mass., June 8-12, 2003.

• M. U. Demirci, M. A. Abdelmoneum , and C. T.-C. Nguyen, “Mechanically corner-coupled square microresonator array for reduced series motional resistance,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 955-958.

• V. Kaajakari, et al, “Square-extensional mode single-crystal silicon micromechanical RF-resonator,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 891-894.

next

lect

ure


Comb-Drive Lateral Resonator

Typical bias:

VI = VO = 0 V

DC voltage across drive and sense electrodes to res-onator = VP

Anchor connectsground plane andresonator

C. T.-C. Nguyen, Ph.D. Thesis, EECS Dept., UC Berkeley, 1994

3


The Lateral Resonator as a “Two-Port”

C. T.-C. Nguyen, Ph.D. Thesis, EECS Dept., UC Berkeley, 1994


Input CurrentInput current i1(t) is the derivative of the charge q1 = C1vD

dtdC

vdt

dvCti D

D 111 )( +=

The capacitance C1 has a DC component and a time-varying component due to the motion of the structure

)()( 111 tCCtC mo += )()( 11 tx

xC

tCm ∂∂

= (linearized case)

Substitute to find the input current:

tx

xC

vtx

xC

Vdtdv

Cdtdv

Cti Pmo ∂∂

∂∂

+∂∂

∂∂

−++= 11

11

1111 )()(

)()()( 111 tvVVtvVtv PPID +−=−+=

)(1 ti x

4


Input Motional Admittance Y1x(jω)

Phasor form of the motional current i1x:

∂∂

−==)()(

)()(

)(1

11

1

11 ω

ωω

ωω

ωjVjX

jxC

VjVjI

jY Px

x

The displacement-to-voltage ratio can be re-expressed in terms of the drive force Fd(jω)

The input motional admittance (inverse of impedance) is the ratio of the phasor motional current to the ac drive voltage:

)()( 111 Xj

xC

VjI Px ωω∂

∂−=

∂

∂−=

)()(

)()(

)(1

111 ω

ωωω

ωωjVjF

jFjX

jxC

VjY d

dPx

∂

∂−=

)()(

)()(

)(1

111 ω

ωωω

ωωjVjF

jFjX

jxC

VjY d

dPx


Input Admittance (Cont.)The electrostatic force component at the drive frequency ω is:

xC

tvVxC

tvtf PDd ∂∂−=

∂∂= 1

1112

, )()(21

)(ωω

The mechanical response of the resonator is (Lecture 9):

→ xC

VjVjF

Pd

∂∂

−= 11

1 )()(

ωω

( ) ( )ood Qjk

jFjX

ωωωωωω

//1)()(

2

1

+−=

−

The input admittance is:

( ) ( )

∂∂−

+−

∂∂−=

−

xC

VQj

kj

xC

VjVjI

Poo

Px 1

12

11

11

1

//1)()(

ωωωωω

ωω

( ) ( )oo

P

x

Qjx

CVkj

jVjI

ωωωω

ω

ωω

//1)()(

2

21

121

1

1

+−

∂∂

=

−

5


Series L-C-R Admittance

The current through an L-C-R branch is:

C

L

R

→I+

-

V

( ) ( )RCjCj

jVjI

o ωωωω

ωω

+−= 2/1)(

)(

LCo =−2ω

Match terms in motional admittance à find equivalent elements


Equivalent Circuit for Input Port

kCx

2

1η

=

A series L-C-R circuit results in the identical expression àfind equivalent values Lx1, Cx1, and Rx1

21 ηmLx =

21 ηQkmRx = =

∂∂

=x

CVP

11η electromechanical

coupling coefficient

Cx1

Lx1

Rx1

Co1

→Ix1

+

-

V1

At resonance, the impedances of the inductance and the capacitance cancel outà

1

11

xx R

VI =

6


Output Port ModelConsider first the current due to driving the input (set v2 = 0 V)

tx

xC

Vt

CVti PP ∂

∂∂

∂−=∂

∂−= 22

222 )(

In phasor form,

( ) ( ))(

//1)()( 12

2121

1

222 ω

ωωωω

ωωωω jV

Qjx

Cx

CVVkj

jXx

CVjjI

oo

PP

P +−

∂∂

∂∂

=∂

∂=

−

I2 and Ix1 are related by the forward current gain φ21:

xC

V

xC

V

jIjI

P

P

x

∂∂∂

∂

==1

1

22

1

221 )(

)(ωω

φ → model by a current-controlledcurrent source


Two-Port Equivalent Circuit (v2 = 0)

Cx1

Lx1

Rx1

Co1

→Ix1

+

-

V1φ21Ix1

+

-

V2= 0 V

I2←

7


Complete Two-Port Model

Cx1

Lx1

Rx1

Co1

→

Ix1+

-

V1φ21Ix1

+

-

V2φ12Ix2

Cx2

Lx2

Rx2

Ix2

→

Symmetry implies that modeling can be done from port 2, with port 1 shorted à superimpose the two models

Co2


Equivalent Circuit forSymmetrical Resonator (φ21 = φ12 = 1)

C. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994

8


455 kHz Comb-Drive Resonator Values

C. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994

Lx

Cx

← assumes vacuum

← huge!

← not small

← mind-boggling!


Double-Ended Tuning Fork Resonators

Current through structure à more resistance (decreases Q)more feedthrough to substrate

i ≈ 0

T. Roessig, Ph.D.,ME ,UC Berkeley, 1997

9


Ideal Tuning Fork Two-Port Response

Phase change of 180o

at resonance “pins” thefrequency, with driftsin the feedback amplifierhaving little effect

Response assumes nofeedthroughcapacitancebetween input and outputports



Tuning Fork Response withCapacitive Feedthrough Cf

+

vd

Leq Ceq Req

Co Cint

structure node - -

+

is

drive Co

Rint

Cint

Rint

sense

Cf

Feedthroughcapacitanceresults in a null in the amplitude response andan added sense currentwhich increases with fre-quency … and which canobscure the resonance en-tirely!

Next lecture: Cf and itscontrol


10


Microresonator Oscillator

C. T.-C. Nguyen and R. T. Howe, IEEE J. Solid-State Circuits, 34, 440-454 (1999).


Current-to-Voltage(or Transresistance) Amplifier

-

+vout ≈ -Rf iin

iin

Rf

i- ≈ 0

The feedback resistor can be implementedusing a MOSFET biased in the triode region

11


Microresonator Oscillator Schematic

Transresistance amplifier: M3 implements a variable resistance RfM1-M2 implement a simple inverting amplifierM6-M7 implement a second amplifying stage



Integrated 16.5 kHzMicroresonator Oscillator


CMOS with tungsten metallization

Poly-Si lateral resonator

Erratic (chaotic) behavior observed for high DC biases in this and other MEMS oscillators was later explained by Kim Turner (Ph.D. Cornell, 1999, now UCSB)

12


Pierce Oscillator Schematic

crystal = double-ended tuning fork

Advantage over trans-Rconfiguration:

capacitive impedances determine loop gain àlower noise, higher gain

A. A. Seshia, et al, MSM-02,San Juan, Puerto Rico


Tuning-Fork OscillatorNear-Carrier Spectrum (Pierce Topology)

outp

ut p

ower

(dB

c/H

z)

frequency (x 105 Hz)

thermal electronic noise

Measured rms noise

A. A. Seshia, et al,IEEE MEMS-02.

13


Differential Resonant Accelerometer

Inertial force is coupled from a proof mass through a leverage system to two DETF oscillators in a “push-pull” manner

tension compression



Leverage Mechanism


DETF oscillators are extremely stiff to forces along their length,which makes mechanical amplification feasible

In the ideal case of a perfect pivot, Archimedes à

Fout / Fin = rin / rout

14


Resonant Accelerometer Performance

Fractional RAV measures instability of an oscillator as a function of integration time. RAVmin = 6 x 10-8 at τ = 2 sec for 70 kHz DETF oscillators à ∆fmin ≈ 0.004 Hz.Sensitivity = 45 Hz/g à amin ≈ 90 µg



x

yz

outer frame

lever arm

frame suspension

sense direction

fixedfree

drive flexure

tuning forkoscillator

tuning forkoscillator

proof massoscillator

directionof motion

A. A. Seshia, Ph.D. ThesisEECS Dept., UC BerkeleyMay 2002

FcΩz

Resonant-Output Rate Gyroscope

15


proof mass

outer framelever arm

tuning forkforce sensor

tuning forkforce sensor

referenceresonator

proof massflexure

self-test electrodes

error correction


Resonant-Output Gyro: Mechanical Element


4.5 mmProof MassDrive Electronics

Mechanical Structure

Tuning Fork DriveElectronics

x

yz


Sandia IMEMS “MEMS-first” process

Resonant-Output Gyroscope Die Shot

16


Sideband Modulation by Coriolis Force

-15

-10

-5

0

5

10

15

20

25

30

35

40

1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640

Frequency offset from carrier (Hz)

Out

put s

ideb

and

pow

er (d

Bµ

V) Offset

Rotation rate signal

sideband output in presence of an applied 12 deg/sec rotation rate at 6 Hz.

DETF oscillator output

-15

-10

-5

0

5

10

15

20

25

30

35

1520 1540 1560 1580 1600 1620 1640

sideband output in the absence of rotation

Out

put s

ideb

and

pow

er (d

Bµ

V)

Frequency offset from carrier (Hz)

Nominal peak

Coriolis offset Coriolis offset

Frequency (x105 Hz)

Osc

illat

or o

utpu

t pow

er (d

Bm)


ee c245 - me c218 introduction to mems design fall 2003 · 4 7 ee c245 – me c218 fall 2003...

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