Sponsored by the IEEE Sensors Council, www.ieee-sensors.org

SENSORS 2013Tutorials: November 3, 2013 Conference: November 4-6, 2013

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Source: Yole Développement, “Inertial Sensors in Mobile Products”, 2012

 

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M. Judy, Proc. Solid-State Sensors, Actuators, and Microsystems Workshop, Hilton Head Island, SC, Jun. 2004

Source: Yole Développement, “MEMS Packaging sample report”, 2012

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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

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LGM-118 Peacekeeper ICBM IMU

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main = −k x

x

ain= −

m

k

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dC

dx=

ε ⋅w ⋅ t

g0 − x( )2dC

dx=ε ⋅2n ⋅ tg0

l

g0

g0

t: thickness, n: # of fingers

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J. Chae, et. al., Journal of Microelectromechanical Systems, Vol. 14, No 2, Apr. 2005

Electrode fixed to substrate

Electrode fixed to mass

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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

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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

 

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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

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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

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b ≈μeff ⋅ l ⋅ t

3

h3

Plate

λ =μP

2kBT

m

Ks ≈λh

 

  

 

μeff ≈μ0

1+ 9.638Ks1.159

Q∝h3

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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

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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(

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Out

put V

olta

ge (m

V)

Applied acceleration (g) Applied acceleration (g) Applied acceleration (g)

Out

put V

olta

ge (m

V)

 

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MUX

X

DEMUX

Y

Z

ReferenceCapacitor

Z-AXIS GYRO

Y-AXIS GYRO X-AXIS GYRO

3-AXIS ACCEL

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f0 =1

2πk

meff

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f0 =1

2πk +Δkmeff

f0ff =1

2

k +Δk

  

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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⎞

⎠

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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

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  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

 

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f = f0 1+NL2

EI

SF ≈ 200 Hz/g @ 83 KHz

SF ≈ 7.7 Hz/g @ 2.6 KHz

Differential FM Accelerometer

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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

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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

     

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 dC

dx=

ε ⋅A

g0 − x( )2 Feedback electrodes

   

 

y =FelecFB

≈FextFB

=m ⋅ainFB

Ferr ∝ x ≈ 0

 

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atest

aref

Programing Interface

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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

  

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SC-AMP

proof-mass

CN1CP1

Integrated Self-Test

Ccal

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Vout =1

2

VDDCF

(CP1 −CN1 +CP2 −CN 2 )

=2VDDCF

ΔC +VDD2CF

(CS.P1 −CS.N1 +CS.P2 −CS.N 2 )

  

   

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Ceq ≈b0Cb0 + b1Cb1 + b2Cb2 + b3Cb3

(Cb0 +Cb1 +Cb2 +Cb3)+Ct1

Ct2

Ct2 +Ct3

⎛

⎝⎜

⎞

⎠⎟Ct4

SC-AMPproof-mass

CN1CP1

Ceq

Ceq

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Vout =2VDDCF

ΔC +VDD2CF

Cmismatch +0.5VDD −Vcal

CF

Coffset

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SOUT ≈ FIN1

FB+COFF

k

α ⋅FB+COFF

kmodα ⋅FB

  

 

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Sponsored by the IEEE Sensors Council, www.ieee-sensors.org

SENSORS 2013Tutorials: November 3, 2013 Conference: November 4-6, 2013