ee c245 – me c218 introduction to mems design fall...
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 1
EE C245 – ME C218Introduction to MEMS Design
Fall 2007
Prof. Clark T.-C. Nguyen
Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley
Berkeley, CA 94720
Lecture 25: Sensing Circuits
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 2
Lecture Outline
• Reading: Senturia Chpts. 6, 14 (op amp parts), 19• Lecture Topics:
Sensing CircuitsIdeal Op AmpsVelocity Sensing CircuitsPosition Sensing CircuitsNon-Ideal Op Amps
MEMS/Transistor Integration
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 3
Complete Electrical-Port Equiv. Circuit
V1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
I1C1 C2
k
x
m
b
ηe2:1cxlx rx
Co1
1:ηe1
Co2
V2
I2
+
-V1
I1
+
-V2
I2
x
mlx =
kcx
1=
brx =
1
111 d
CVxCV o
PPe =∂∂
=η
2
222 d
CVx
CV oPPe =
∂∂
=η
Static electrode-to-mass overlap
capacitance
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 4
Condensed Equiv. Circuit (Symmetrical)
ηe:1cxlx rx
Co1
1:ηe
Co2
+
-V1
I1
+
-V2
I2
x
CxLx Rx
Co1 Co2
+
-V1
I1
+
-V2
I2
2e
xmLη
=
kC e
x
2η=
2e
xbRη
=
If ηe1 = ηe2, then …
where
Holds for the symmetrical case, where port 1 and
port 2 are identical
Holds for the symmetrical case, where port 1 and
port 2 are identical
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 5
Phasings of Signals
• Below: plots of resonance electrical and mechanical signals vs. time, showing the phasings between them
V1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
I1C1 C2
k
x
m
b
V2
I2
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 6
Port 1 to 2 TransG Across the Circuit
ηe2:1cxlx rx
Co1
1:ηe1
Co2
+
-V1
I1
+
-V2
I2
x
•What is the transconductance from port 1 to port 2 with port 2 shorted to ground?
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 7
Port 1 to 2 vi-to-io Transfer Function
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 8
Sensing Circuits
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 9
Sense Electrodes
Tuning Electrodes
Sense Electrodes
Tuning Electrodes
Drive Electrode
zΩr
Drive
Sense
[Zaman, Ayazi, et al, MEMS’06]
Drive Voltage Signal
(-) Sense Output Current
(+) Sense Output Current
Drive Oscillation Sustaining Amplifier
Differential TransRSense
Amplifier
MEMS-Based Tuning Fork Gyroscope
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 10
Detecting Velocity Versus Position
• Transfer Function:
• Detect position when the output is varying slowly, i.e., at low frequencies
Position(i.e., displacement)
22
2
)(1
)()(
oo
o
d sQsksFsX
ωωω
++=
ω
)()(
sFsX
dLowpassBiquad
ωo
Velocity
• Transfer Function:
• Detect velocity when the output is at resonance or when a bandpass response is required
22
2
)(1
)()(
oo
o
d sQss
ksFs
ωωωυ
++=
ω
)()(sFs
d
υ
ωo
BandpassBiquad
6
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 11
Detecting Velocity Versus Position
• Transfer Function:
Position(i.e., displacement)
22
2
)(1
)()(
oo
o
d sQsksFsX
ωωω
++=
Velocity
• Transfer Function:
22
2
)(1
)()(
oo
o
d sQss
ksFs
ωωωυ
++=
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 12
Output Current Measures Velocity
ηe2:1cxlx rx
Co1
1:ηe1
Co2
+
-V1
I1
+
-V2
I2
x
• Relationship between output current and velocity:
• To turn current into voltage (for a voltage output), send the current into a resistor RD
• To get position, must integrate → send the current into a capacitor CD
xi eo &2η= Output current is proportional to velocity, and thus, directly measures velocity
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 13
Velocity-to-Voltage Conversion
• To convert velocity to a voltage, use a resistive load
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 14
Position-to-Voltage Conversion
• To sense position (i.e., displacement), use a capacitive load
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 15
Ideal Operational Amplifiers
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 16
Ideal Op Amp
• Equivalent Circuit of an Ideal Op Amp:
• Properties of Ideal Op Amps:
01 =i1v
-
+02 =i
( )−+ − vvA
0R
( )−+ −= vvAv0
( )12 vvA −=
2v
inR
+-
+-+-+v
−v
Voltage-Controlled Voltage Source (VCVS)
Single-ended outputDifferential input
∞=inR00 =R∞=A
1.2.3.
0== −+ ii
finite assuming , 0 == −+ vvv
4.
5. Why?
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 17
Ideal Op Amp (cont)
∞=inR00 =R∞=A
1.2.3.
0== −+ ii
finite assuming , 0 == −+ vvv
−+−+ =→=−∴ vvvv 0
ground) (virtualcircuit short virtual 0 ⇒∞v
4.
5.
( ) finite 0 ==−∞ −+ vvv
Why? Because for
( )finite 0 =v
• Properties of Ideal Op Amps:
• Big assumption!• How can we assume this? We can assume this only when there is an appropriate negative feedback path!
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 18
Negative Feedback
0
0
SSSaSS
i βε
ε
−==
[ ] finite! 10 ==≈⇒∞→ββa
aSSa
i
finite! 10 ==∴ iSS
β
Where S could be a current, voltage, displacement, etc.,…
Negative feedback acts to oppose or subtract from input.
( )
( )β
β
β
aa
SSaSaS
SSaS
ii
i
+=→=+
−=
1 1 0
0
00
Overall transfer function.
(When there is negative FB around the amplifier.)
↑Δ βS
↑Δ iS↑Δ εS
εS+ a
β
↓Δ εS
↑Δ 0S0SiS
12
5
4
3
+-
10
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 19
Negative Feedback
• Comments:1. Negative FB can insure S0 = finite even with a = ∞.2. Overall gain dependent (or overall T.F.) dependent only
on external components. (e.g., β).3. Overall (Closed-loop) gain (So/Si) is independent of
amplifier gain a.→ Very important, since amplifiers using transistors
can be designed to have large gain, but it’s hard to get an exact gain. i.e., if you’re shooting for a = 50,000, you might get 47,000 or 60,000 instead.
→ Comment 3 makes this less consequential.
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 20
Positive Feedback
• Contrast with Positive Feedback:
• But for a bounded, controllable function, need negative FB around the op amp
↑Δ βS
↑Δ iS ↑↑Δ SS+ a
β
↑↑Δ 0S0SiS
+
+Output blows up!
(for aβ > 1)
Will be the case for a = ∞.
If β is a bandpassbiquad transfer function → get oscillation at the
resonance frequency ωωο
β
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 21
Inverting Amplifier
0viv-
++- 1i 0=−i
2i1R
2R
Virtual ground
OV
1. Verify that there is negative FB.2. node attached to (-) terminal is virtual
ground.3.
→=→=∴ −+ finite 0 vvv
21 0 iii =∴=−
22220
211
1
0
0
RiRiv
iRv
Rvi ii
−=−=
==−
=
1
20
1
22
10
RR
vvv
RRR
Rvv
ii
i −=∴−=⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇒
NOTE: Gain dependent only on R1 & R2 (external components), not on the op amp gain.
Benefit: Any shunt C at this node will be grounded out.
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 22
• Put the feedback around the (+) terminal and see what happens
• This is not (-) FBCannot analyze using the ideal op amp methodCircuit will “rail out”
Flip the Op Amp Around
2R
0viv +
-
1R
−+ ≠≠∴ vvv finite, 0
( ) −+ →−= Lv
( ) ++ →+= Lv
conditions initialon depending or 0−+= LLv
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 23
Transresistance Amplifier
1. Verify that there is neg. FB → yes, since same FB as inverting amplifier
2. Thus, vo = finite → v+ = v- → (-) terminal is virtual ground3. i- = 0 → i1 = i2
0vii-
+
1i
0=−i
2i
2R
Virtual ground
OV
2220 RiRiv i−=−= ⇒ 20 R
iv
i−=
An inverting amplifier is just a transresistanceamplifier with an R1 to
convert voltage to current!
• Take R1 away
Again, shunt C at this node will be grounded out.
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 24
Velocity Sensing Circuits
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 25
Problems With Purely Resistive Sensing
v1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
io
i1 C1 C2
k
mCp
x b
vo
RD
Includes Co, line C, bond pad C, and next stage C
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 26
Problems With Purely Resistive Sensing
v1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
io
i1 C1 C2
k
mCp
x b
vo
RD
• In general, the sensor output must be connected to the inputs of further signal conditioning circuits → input Riof these circuits can load RD
Ri
Dx
iD
i
o
ss
RRR
svv
ω+⋅Θ⋅=1
1)()(( ) piDD CRR
1=ω
Now:
,
These change w/ hook-up → not good.
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 27
The TransR Amplifier Advantage
0v-
+
2i
2R
• The virtual ground provided by the ideal op amp eliminates the parasitic capacitance Cp and Ri
v1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
i2
i1 C1 C2
k
m Cp
x b
• The zero output resistance of the (ideal) op amp can drive virtually anything
Ω= 0oR
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 28
Position Sensing Circuits
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 29
Integrator
1. Verify that there is neg. FB → need a large resistor R2 to DC connect the output and (-) terminal
2. With R2, vo = finite → v+ = v- → (-) terminal = virtual gnd3. i- = 0 → i1 = i2
0vii-
+
1i
0=−i
2i
Virtual ground
OV
22
20 sC
isCiv i−=−= ⇒
2
0 1 sCi
v
i−=
• Replace R2 with C2
Again, shunt elements at this node will be grounded out.
C2
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 30
Problems With Pure-C Position Sensing
v1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1
io
i1 C1 C2
k
mCp
x b
vo
CD
16
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 31
The Op Amp Integrator Advantage
• The virtual ground provided by the ideal op amp eliminates the parasitic capacitance Cp
v1
VP
Fd1
d1 d2
Elec
trod
e 2
Elec
trod
e 1 io
i1 C1 C2
k
mCp
x b
0v-
+
C2
R2
(for biasing)2
21
sCR >>
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 32
Non-Inverting Amplifier
2R1R
+
- 0v
iv1i
0=−i
2i
iv
21 0 iii =∴=−
21
1 iRvi i ==
⇒
Again, gain depends only on R1 & R2(external or ratioedcomponents), not on the op amp gain.
1. Verify that there is negative FB → same feedback circuit as for inverting amplifier, so neg. FB
−+ =→=∴ vvv finite 02.3. Analysis:
( )211
22110 RRRvRiRiv i +=+= 1
20 1 RR
vv
i+=
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 33
Unity Gain Buffer
+
- 0v
iv
0=−iiv
1
20 1 RR
vv
i+=
2R1R
+
- 0v
iv1i
0=−i
2i
iv
1 0 =iv
v
R1=∞R2=0
Non-Inverting Amplifier
Unity Gain Buffer
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 34
Differential Position Sensing
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EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 35
Differential Position Sensing
Suspension Beam in Tension
Proof Mass
Sense Finger
C2
Fixed Electrodes
Tethers with fixed ends
C1
C2
C1Vo
VP
-VP
• Example: ADXL-50
EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 36
Buffer-Bootstrapped Position Sensing
+VP
-VP
+
-0v
Unity Gain Buffer
Cp
Includes capacitance from interconnects, bond pads, and Cgs of the op amp
Cgd
0v
0v
Cgd = gate-to-drain capacitance of the input MOS transistor
• Bootstrap the ground lines around the interconnect and bond pads
No voltage across CpIt’s effectively not there!
1×
InterconnectGround Plane