EE C245 - ME C218 Introduction to MEMS Design Fall 2003 · 4 7 EE C245 – ME C218 Fall 2003...
Transcript of 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
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3EE C245 – ME C218 Fall 2003 Lecture 26
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
4EE C245 – ME C218 Fall 2003 Lecture 26
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
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5EE C245 – ME C218 Fall 2003 Lecture 26
The Lateral Resonator as a “Two-Port”
C. T.-C. Nguyen, Ph.D. Thesis, EECS Dept., UC Berkeley, 1994
6EE C245 – ME C218 Fall 2003 Lecture 26
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
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7EE C245 – ME C218 Fall 2003 Lecture 26
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
8EE C245 – ME C218 Fall 2003 Lecture 26
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
+−
∂∂
=
−
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9EE C245 – ME C218 Fall 2003 Lecture 26
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
10EE C245 – ME C218 Fall 2003 Lecture 26
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 =
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11EE C245 – ME C218 Fall 2003 Lecture 26
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
12EE C245 – ME C218 Fall 2003 Lecture 26
Two-Port Equivalent Circuit (v2 = 0)
Cx1
Lx1
Rx1
Co1
→Ix1
+
-
V1φ21Ix1
+
-
V2= 0 V
I2←
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13EE C245 – ME C218 Fall 2003 Lecture 26
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
14EE C245 – ME C218 Fall 2003 Lecture 26
Equivalent Circuit forSymmetrical Resonator (φ21 = φ12 = 1)
C. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994
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15EE C245 – ME C218 Fall 2003 Lecture 26
455 kHz Comb-Drive Resonator Values
C. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994
Lx
Cx
← assumes vacuum
← huge!
← not small
← mind-boggling!
16EE C245 – ME C218 Fall 2003 Lecture 26
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
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17EE C245 – ME C218 Fall 2003 Lecture 26
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
T. Roessig, Ph.D.,ME ,UC Berkeley, 1997
18EE C245 – ME C218 Fall 2003 Lecture 26
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
T. Roessig, Ph.D.,ME ,UC Berkeley, 1997
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19EE C245 – ME C218 Fall 2003 Lecture 26
Microresonator Oscillator
C. T.-C. Nguyen and R. T. Howe, IEEE J. Solid-State Circuits, 34, 440-454 (1999).
20EE C245 – ME C218 Fall 2003 Lecture 26
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
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Microresonator Oscillator Schematic
Transresistance amplifier: M3 implements a variable resistance RfM1-M2 implement a simple inverting amplifierM6-M7 implement a second amplifying stage
C. T.-C. Nguyen and R. T. Howe, IEEE J. Solid-State Circuits, 34, 440-454 (1999).
22EE C245 – ME C218 Fall 2003 Lecture 26
Integrated 16.5 kHzMicroresonator Oscillator
C. T.-C. Nguyen and R. T. Howe, IEEE J. Solid-State Circuits, 34, 440-454 (1999).
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)
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23EE C245 – ME C218 Fall 2003 Lecture 26
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
24EE C245 – ME C218 Fall 2003 Lecture 26
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.
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25EE C245 – ME C218 Fall 2003 Lecture 26
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
T. Roessig, Ph.D.,ME ,UC Berkeley, 1997
26EE C245 – ME C218 Fall 2003 Lecture 26
Leverage Mechanism
T. Roessig, Ph.D.,ME ,UC Berkeley, 1997
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
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27EE C245 – ME C218 Fall 2003 Lecture 26
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
T. Roessig, Ph.D.,ME ,UC Berkeley, 1997
28EE C245 – ME C218 Fall 2003 Lecture 26
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
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29EE C245 – ME C218 Fall 2003 Lecture 26
proof mass
outer framelever arm
tuning forkforce sensor
tuning forkforce sensor
referenceresonator
proof massflexure
self-test electrodes
error correction
A. A. Seshia, et al,IEEE MEMS-02.
Resonant-Output Gyro: Mechanical Element
30EE C245 – ME C218 Fall 2003 Lecture 26
4.5 mmProof MassDrive Electronics
Mechanical Structure
Tuning Fork DriveElectronics
x
yz
A. A. Seshia, et al,IEEE MEMS-02.
Sandia IMEMS “MEMS-first” process
Resonant-Output Gyroscope Die Shot
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31EE C245 – ME C218 Fall 2003 Lecture 26
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)
A. A. Seshia, et al,IEEE MEMS-02.