EE C245 - ME C218
Introduction to MEMS Design
Fall 2003
Roger Howe and Thara Srinivasan
Lecture 26
Micromechanical Resonators I
EE 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
EE C245 – ME C218 Fall 2003 Lecture 26
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1
Reading/Reference List
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next lecture
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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, “ Surfacemicromachined 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 812, 2003, pp. 891-894.
EE C245 – ME C218 Fall 2003 Lecture 26
3
Comb-Drive Lateral Resonator
Anchor connects
ground plane and
resonator
Typical bias:
VI = VO = 0 V
DC voltage across
drive and sense
electrodes to resonator = VP
EE C245 – ME C218 Fall 2003 Lecture 26
C. T.-C. Nguyen, Ph.D. Thesis,
EECS Dept., UC Berkeley, 1994
4
2
The Lateral Resonator as a “Two-Port”
C. T.-C. Nguyen, Ph.D. Thesis,
EECS Dept., UC Berkeley, 1994
EE C245 – ME C218 Fall 2003 Lecture 26
5
Input Current
Input current i1(t) is the derivative of the charge q1 = C1vD
i1 (t ) = C1
dvD
dC
+ vD 1
dt
dt
vD (t ) = VI + v1 (t ) − VP = −VP1 + v1 (t )
The capacitance C1 has a DC component and a time-varying
component due to the motion of the structure
C1( t ) = Co1 + Cm1 (t )
Cm1 (t ) =
∂C1
x(t )
∂x
(linearized case)
Substitute to find the input current:
dv
dv
∂C ∂x
∂C ∂x
i1 (t ) = Co1 1 + Cm 1 + ( −VP1 ) 1
+ v1 1
dt
dt
∂x ∂t
∂x ∂t
i1x (t )
EE C245 – ME C218 Fall 2003 Lecture 26
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3
Input Motional Admittance Y1x(jω)
Phasor form of the motional current i1x :
∂C1
( jωX )
∂x
The input motional admittance (inverse of impedance) is the
ratio of the phasor motional current to the ac drive voltage:
I1 x ( jω ) = −VP 1
Y1x ( jω ) =
 X ( jω ) 
I x1 ( jω )
∂C
= −VP 1 1 j ω

V1( j ω)
∂x
 V1( j ω) 
The displacement-to-voltage ratio can be re-expressed in
terms of the drive force Fd(jω)
Y1x ( jω ) = −VP1
∂ C1  X ( jω )  Fd ( jω ) 


jω
∂x
 Fd ( jω )  V1 ( j ω) 
EE C245 – ME C218 Fall 2003 Lecture 26
7
Input Admittance (Cont.)
The electrostatic force component at the drive frequency ω is:
f d ,ω (t ) =
1 2
∂ C1
∂C
v D (t )
= −VP 1v1 (t ) 1
ω ∂x
2
∂x
→
Fd ( jω)
∂C
= −VP1 1
V1 ( jω )
∂x
The mechanical response of the resonator is (Lecture 9):
X ( jω )
k −1
=
Fd ( jω ) 1 − (ω / ωo )2 + j (ω / Qωo )
The input admittance is:

I1 x ( j ω ) 
∂C
k −1
∂C1 

=  − VP1 1 jω  

 − VP1
2
V1 ( j ω ) 
∂x
∂x 
 1 − (ω / ωo ) + j (ω / Qωo )
 ∂C 
jωk −1V 2 P1  1 
I1 x ( j ω )
 ∂x 
=
V1 ( j ω ) 1 − (ω / ωo )2 + j (ω / Qωo )
2
EE C245 – ME C218 Fall 2003 Lecture 26
8
4
Series L-C-R Admittance
The current through an L-C-R branch is:
+
I
L
I ( jω )
jω C
=
V ( jω ) 1 − (ω / ω o )2 + j(ωRC )
C
ω o = LC
→
V
−2
R
-
Match terms in motional admittance à find equivalent elements
EE C245 – ME C218 Fall 2003 Lecture 26
9
Equivalent Circuit for Input Port
A series L-C-R circuit results in the identical expression à
find equivalent values Lx1, Cx1, and Rx1
L x1 =
m
η2
+
V1
-
C x1 =
η2
k
→
Ix1
Co1
R x1 =
km
Qη 2
Lx1
η = V P1
∂C1
= electromechanical
∂x
coupling coefficient
At resonance, the impedances
of the inductance and the
capacitance cancel out à
Cx1
Rx1
EE C245 – ME C218 Fall 2003 Lecture 26
I x1 =
V1
Rx1
10
5
Output Port Model
Consider first the current due to driving the input (set v2 = 0 V)
∂C2
∂C ∂x
= −VP 2 2
∂t
∂x ∂t
i2 (t ) = −VP 2
 ∂C  ∂C 
j ωk −1VP1VP2  1  2 
∂C
 ∂x  ∂x  V ( j ω )
I 2 ( jω ) = j ωVP 2 2 X ( jω ) =
1
2
∂x
1 − (ω / ω o ) + j (ω / Qω o )
In phasor form,
I2 and Ix1 are related by the forward current gain φ 21:
∂C 2
I ( jω )
∂x
φ 21 = 2
=
∂
I x1 ( jω ) V C1
P1
∂x
VP 2
→ model by a current-controlled
current source
EE C245 – ME C218 Fall 2003 Lecture 26
11
Two-Port Equivalent Circuit (v2 = 0)
+
V1
-
→
Ix1
Co1
Lx1
Cx1
Rx1
EE C245 – ME C218 Fall 2003 Lecture 26
←I
2
+
φ 21Ix1
V2= 0 V
-
12
6
Complete Two-Port Model
-
Lx1
Co1
Cx1
Lx2
φ 12Ix2
Rx1
φ 21Ix1
+
Ix2
→
V1
Ix1
→
+
Cx2
Co2 V2
Rx2
-
Symmetry implies that modeling can be done
from port 2, with port 1 shorted à superimpose
the two models
EE C245 – ME C218 Fall 2003 Lecture 26
13
Equivalent Circuit for
Symmetrical Resonator (φ 21 = φ 12 = 1)
C. T.-C. Nguyen, Ph.D.,
UC Berkeley, 1994
EE C245 – ME C218 Fall 2003 Lecture 26
14
7
455 kHz Comb-Drive Resonator Values
← assumes vacuum
← not small
← huge!
← mind-boggling!
Lx
Cx
C. T.-C. Nguyen, Ph.D.,
UC Berkeley, 1994
EE C245 – ME C218 Fall 2003 Lecture 26
15
Double-Ended Tuning Fork Resonators
i ≈0
Current through structure à more resistance (decreases Q)
more feedthrough to substrate
EE C245 – ME C218 Fall 2003 Lecture 26
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
16
8
Ideal Tuning Fork Two-Port Response
Phase change of 180o
at resonance “pins” the
frequency, with drifts
in the feedback amplifier
having little effect
Response assumes no
feedthrough capacitance
between input and output
ports
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
EE C245 – ME C218 Fall 2003 Lecture 26
17
Tuning Fork Response with
Capacitive Feedthrough Cf
Feedthrough capacitance
results in a null in the
amplitude response and
an added sense current
which increases with frequency … and which can
obscure the resonance entirely!
Cf
+
Req
R int
Leq
Ceq
R int
is
Next lecture: Cf and its
control
+
drive vd
Cint
Co
Co
Cint
sense
-
-
structure node
EE C245 – ME C218 Fall 2003 Lecture 26
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
18
9
Microresonator Oscillator
C. T.-C. Nguyen and R. T. Howe,
IEEE J. Solid-State Circuits, 34,
440-454 (1999).
EE C245 – ME C218 Fall 2003 Lecture 26
19
Current-to-Voltage
(or Transresistance) Amplifier
Rf
i- ≈ 0
iin
-
vout ≈ -Rf iin
+
The feedback resistor can be implemented
using a MOSFET biased in the triode region
EE C245 – ME C218 Fall 2003 Lecture 26
20
10
Microresonator Oscillator Schematic
C. T.-C. Nguyen and R. T. Howe,
IEEE J. Solid-State Circuits, 34,
440-454 (1999).
Transresistance amplifier:
M3 implements a variable resistance Rf
M1-M2 implement a simple inverting amplifier
M6-M7 implement a second amplifying stage
EE C245 – ME C218 Fall 2003 Lecture 26
21
Integrated 16.5 kHz
Microresonator Oscillator
CMOS with tungsten metallization
Poly-Si lateral resonator
C. T.-C. Nguyen and R. T. Howe,
IEEE J. Solid-State Circuits, 34,
440-454 (1999).
EE C245 – ME C218 Fall 2003 Lecture 26
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)
22
11
Pierce Oscillator Schematic
crystal = doubleended tuning fork
Advantage over trans -R
configuration:
capacitive impedances
determine loop gain à
lower noise, higher gain
A. A. Seshia, et al, MSM-02,
San Juan, Puerto Rico
EE C245 – ME C218 Fall 2003 Lecture 26
23
output power (dBc/Hz)
Tuning-Fork Oscillator
Near-Carrier Spectrum (Pierce Topology)
Measured rms noise
thermal electronic noise
EE C245 – ME C218 Fall 2003 Lecture 26
A. A. Seshia, et al,
IEEE MEMS-02.
frequency (x 105 Hz)
24
12
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
EE C245 – ME C218 Fall 2003 Lecture 26
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
25
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
EE C245 – ME C218 Fall 2003 Lecture 26
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
26
13
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
EE C245 – ME C218 Fall 2003 Lecture 26
T. Roessig, Ph.D.,ME ,
UC Berkeley, 1997
27
Resonant-Output Rate Gyroscope
frame suspension
outer frame
lever arm
direction
of motion
proof mass
oscillator
tuning fork
oscillator
Fc
x
z
Ωz
y
fixed
free
tuning fork
oscillator
sense direction
drive flexure
EE C245 – ME C218 Fall 2003 Lecture 26
A. A. Seshia, Ph.D. Thesis
EECS Dept., UC Berkeley
May 2002
28
14
Resonant-Output Gyro: Mechanical Element
reference
resonator
proof mass
flexure
tuning fork
force sensor
tuning fork
force sensor
proof mass
error correction
outer frame
lever arm
self-test electrodes
EE C245 – ME C218 Fall 2003 Lecture 26
A. A. Seshia, et al,
IEEE MEMS-02.
29
Resonant-Output Gyroscope Die Shot
Tuning
Fork Drive
Electronics
Proof Mass
Drive
Electronics
4.5 mm
Mechanical
Structure
A. A. Seshia, et al,
IEEE MEMS-02.
x
z
EE C245 – ME C218 Fall 2003 Lecture 26
y
Sandia IMEMS
“MEMS-first” process
30
15
Sideband Modulation by Coriolis Force
A. A. Seshia, et al,
IEEE MEMS-02.
Oscillator output power (dBm )
Nominal peak
Coriolis offset
Coriolis offset
5
Frequency (x10 Hz)
EE C245 – ME C218 Fall 2003 Lecture 26
Offset
Rotation
rate signal
40
35
30
25
20
15
10
5
0
-5
-10
-15
1540
1550
1560
1570
1580
1590
1600
1610
1620
1630
1640
Frequency offset from carrier (Hz)
Output sideband power (dBµV)
DETF oscillator output
Output sideband power (dBµV)
sideband output in presence of an applied
12 deg/sec rotation rate at 6 Hz.
sideband output in the absence of rotation
35
30
25
20
15
10
5
0
-5
-10
-15
1520
1540
1560
1580
1600
1620
Frequency offset from carrier (Hz)
1640
31
16