Download - Design and Analysis of MEMS Accelerometers
Sponsored by the IEEE Sensors Council, www.ieee-sensors.org
SENSORS 2013Tutorials: November 3, 2013 Conference: November 4-6, 2013
SENSORS 2013
Source: Yole Développement, “Inertial Sensors in Mobile Products”, 2012
SENSORS 2013
• • •
• •
• • • •
• •
SENSORS 2013
M. Judy, Proc. Solid-State Sensors, Actuators, and Microsystems Workshop, Hilton Head Island, SC, Jun. 2004
Source: Yole Développement, “MEMS Packaging sample report”, 2012
SENSORS 2013
The Evolution of Compact Three Axis Accelerometers, St.J. Dixon-Warren, Chipworks Inc. Source: www.bosch-sensortec.com
Source: System Plus Consulting – Reverse costing sample reports
SENSORS 2013
LGM-118 Peacekeeper ICBM IMU
SENSORS 2013
main = −k x
x
ain= −
m
k
SENSORS 2013
dC
dx=
ε ⋅w ⋅ t
g0 − x( )2dC
dx=ε ⋅2n ⋅ tg0
l
g0
g0
t: thickness, n: # of fingers
SENSORS 2013
J. Chae, et. al., Journal of Microelectromechanical Systems, Vol. 14, No 2, Apr. 2005
Electrode fixed to substrate
Electrode fixed to mass
SENSORS 2013
SENSORS 2013
Parameter Units Expression Description
Resonance Frequency Hz Free-vibration frequency of accelerometer
Scale Factor F/(m/s2) Linear term of the acceleration to capacitance change sensitivity
Quadratic non-linearity F/(m/s2)2 Quadratic term of the acceleration to capacitance change sensitivity
Brownian Noise (m/s2)/√Hz Mechanical noise equivalent acceleration
Pull-in Voltage V Voltage required to snap-down moving device to parallel
Bandwidth Hz Approximate 3 dB Bandwidth in over-damped second-order system
ω0 =k
m
MNEA =4k B ⋅T ⋅ω0
Q ⋅m
ΔCx ≈ Sx ⋅ax + Sx2 ⋅ax2 + Sy ⋅ay + Sz ⋅az +C0SxS ⋅ SxS 2 + Sy ⋅ + SzS +C0CC
Scale Factor Non-linearity Cross-Axis Sensitivity Offset
SF2 ≈1
ω04
ε ⋅Aelec
g03
SF ≈1
ω02
ε ⋅Aelec
g02
Vpi =8ω0
2 ⋅m ⋅ g03
27ε ⋅Aelec
ω3dB ≈Q ⋅ω0
SENSORS 2013
m = ρ ⋅V = ρ ⋅ h ⋅w ⋅ l( )
k =Cn ⋅E ⋅ Ilt3 I =
1
12b3 ⋅h
l
w
h
lt
lt
ρ E I
Cn ≈ 48 Cn ≈ 384 Cn ≈ 36
lt1
lt2
for: lt1 = 2 lt2
b
h
SENSORS 2013
X( jω)Ain ( jω)
≈1
ω02=m
k
md 2x
dt2+ b
dx
dt+ k x =main
X(s)
Ain (s)=
1
s2 +ω0
Qs+ω0
2
ω << ω0
Q =km
bω0 =
k
m
SENSORS 2013
1
Q=
1
QSFD
+1
QTED
+1
Qanchor
+1
Qmaterial
1
Q≈
1
QSFD
Plate
Air flow Air flow
ΔP
x
h: Gap size
P: Pressure
µ: Coefficient of viscosity
∂2P∂x2
+∂2P∂y2
=12μeff
h3dh
dt
SENSORS 2013
b ≈μeff ⋅ l ⋅ t
3
h3
Plate
λ =μP
2kBT
m
Ks ≈λh
μeff ≈μ0
1+ 9.638Ks1.159
Q∝h3
SENSORS 2013
SENSORS 2013
b ≈nelec ⋅μeff ⋅ l ⋅ t
3
h3
Q =km
b
⋅ l ⋅ t3
h3nelec nelec
µefflth
Dam
ping
coe
ffici
ent (
Ns/
m)
Electrode area (um2) Gap size (nm)
Gap , Area Larger damping
SENSORS 2013
Vpull−in ≈8
27
k ⋅ g03
ε ⋅nsens ⋅Aelec
SF =ΔCain
≈1
ω02
ε ⋅Aelec
g02
BW3dB ≈Q ⋅ω0
Felec ≈1
2
ε ⋅Aelecg02
V 2
b = μeff ⋅ l ⋅t
g03
⎛
⎝⎜
⎞
⎠⎟
3
⋅ nsens + ndamp( )+ ndadd mpm )nsens(
SENSORS 2013
Out
put V
olta
ge (m
V)
Applied acceleration (g) Applied acceleration (g) Applied acceleration (g)
Out
put V
olta
ge (m
V)
SENSORS 2013
MUX
X
DEMUX
Y
Z
ReferenceCapacitor
Z-AXIS GYRO
Y-AXIS GYRO X-AXIS GYRO
3-AXIS ACCEL
SENSORS 2013
f0 =1
2πk
meff
SENSORS 2013
f0 =1
2πk +Δkmeff
f0ff =1
2
k +Δk
SENSORS 2013
Felec =1
2
∂C∂xV 2 =
1
2
ε A(g0 ∓ x)
2V 2
Felec ≈1
2
ε Ag0V 2 ⋅
1
g0±2
g02x +
3
g03x2 ±...
⎛
⎝⎜
⎞
⎠⎟
FelecTOT = Felec1 − Felec2
FelecTOT ≈ε Ag0
⋅ VAC ⋅VP +2 ⋅VP
2
g02x
⎛
⎝⎜
⎞
⎠⎟+
2 ⋅VPVV2
g02x⎞
⎠
SENSORS 2013
md 2x
dt2+ b
dx
dt+ k x =main +FelecTOT
Mechanical Transfer Function
Input Acceleration
Electrostatic Force
md 2x
dt2+ b
dx
dt+ k −
2ε Ag3
VP2
⎛
⎝⎜
⎞
⎠⎟x =main +
ε Ag2Vac VP
ωTOT =k − 2
ε Ag3VP2
m
∂ωTOT
∂ain≈ −
3
2
ε Akω0 g
4VDC2
g0 ain
SENSORS 2013
TCF CTE α TCE
f =1
2πk
meff
∝w
ρ ⋅ l2
E = E0 1+TCE ⋅ ΔT( )
w = w0 1+α ⋅ ΔT( ) l = l0 1+α ⋅ ΔT( ) ρ = ρ0 1−3α ⋅ ΔT( )
TCFα =1
f0
df
dT≈α2
f =1
2πk
meff
∝ E
TCFTCE =1
f0
df
dT≈TCE
2TCF ≈ -30 ppm/ºC SF ≈ 500-1500 ppm/g
SENSORS 2013
f = f0 1+NL2
EI
SF ≈ 200 Hz/g @ 83 KHz
SF ≈ 7.7 Hz/g @ 2.6 KHz
Differential FM Accelerometer
SENSORS 2013
proof-mass
Vac
ΔC+ ΔC-
x
ain RF
RF
TIA
ΔC-
ΔC+
Vac
VoutΔC
≈ 2 ⋅RF ⋅ jωac ⋅Vacω3dB ≈Q ⋅ω0 Q <1
ωac >>ω3dB
F ⋅ jωj ac ⋅VaVV c
Mag
nitu
de [d
B]
Frequency [rad/s]
ωac
ω3dB
SENSORS 2013
Amplifying Phase
OTA OTA
Sampling Phase
proof-mass
Vcm
CP2 CN2
x
ain
CP1 CN1
Vout =1
2
VDDCF
CTOT
(CTOT =CP1 −CN1 +C P2−CN 2 )
CP2
CP1
CN1
CN2
SENSORS 2013
SENSORS 20
SENSORS 2013
dC
dx=
ε ⋅A
g0 − x( )2 Feedback electrodes
y =FelecFB
≈FextFB
=m ⋅ainFB
Ferr ∝ x ≈ 0
SENSORS 2013
atest
aref
Programing Interface
SENSORS 2013
SENSORS 2013
Amp
x
go
proof-massΔC
Felec
VtestAmp
x
go
proof-mass
ain
ΔC
ΔCain
≈1
ω02
ε ⋅Aelec
g02
ΔCVtest2≈
1
2M ⋅ω02
ε ⋅Aelec( )2
g04
Cha
nge
in
Cap
acita
nce
[F]
Input Stimulus
SENSORS 2013
SC-AMP
proof-mass
CN1CP1
Integrated Self-Test
Ccal
SENSORS 2013
Vout =1
2
VDDCF
(CP1 −CN1 +CP2 −CN 2 )
=2VDDCF
ΔC +VDD2CF
(CS.P1 −CS.N1 +CS.P2 −CS.N 2 )
SENSORS 2013
Ceq ≈b0Cb0 + b1Cb1 + b2Cb2 + b3Cb3
(Cb0 +Cb1 +Cb2 +Cb3)+Ct1
Ct2
Ct2 +Ct3
⎛
⎝⎜
⎞
⎠⎟Ct4
SC-AMPproof-mass
CN1CP1
Ceq
Ceq
SENSORS 2013
Vout =2VDDCF
ΔC +VDD2CF
Cmismatch +0.5VDD −Vcal
CF
Coffset
SENSORS 2013
SOUT ≈ FIN1
FB+COFF
k
α ⋅FB+COFF
kmodα ⋅FB
SENSORS 2013
SENSORS 2013
SENSORS 2013
SENSORS 2013
Sponsored by the IEEE Sensors Council, www.ieee-sensors.org
SENSORS 2013Tutorials: November 3, 2013 Conference: November 4-6, 2013