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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 1
EE C247B – ME C218Introduction to MEMS Design
Spring 2017
Prof. Clark T.-C. Nguyen
Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley
Berkeley, CA 94720
Lecture Module 11: Equivalent Circuits I
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 2
Lecture Outline
• Reading: Senturia, Chpt. 5
• Lecture Topics:Lumped MassLumped StiffnessLumped DampingLumped Mechanical Equivalent CircuitsElectromechanical Analogies
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 3
Lumped Parameter Mechanical Equivalent Circuit
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 4
Equivalent Dynamic Mass
•Once the mode shape is known, the lumped parameter equivalent circuit can then be specified
• Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity
Maximum Kinetic Energy
Maximum Velocity @ location x
Equivalent Mass =
Location x
Density
Maximum Velocity Function
z
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 5
Equivalent Dynamic Mass
• For the folded-beam structure, we’ve already determined the maximum kinetic energy
• And in our resonance frequency analysis, we’ve already determined expressions for velocity
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 6
Equivalent Dynamic Stiffness & Damping
• Stiffness then follows directly from knowledge of mass and resonance frequency
• And damping also follows readily from knowledge of Q or other loss measurands
•With mass, stiffness, and damping lumped parameter equivalent circuit
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 7
Get Potential Energy & Frequency
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
h = thickness
Folding truss w/
mass Mt\2
Meq(shuttle) = 2.16x10-11 kg
Keq(shuttle) = 4.8 N/m
Ceq(shuttle) = 1.02x10-10 kg/s
Meq(truss) = 8.64x10-11 kg
Keq(truss) = 19.2 N/m
Ceq(truss) = 4.08x10-10 kg/s
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 8
Electromechanical Analogies
meq
keq
ceq
lx rxcx
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 9
Electromechanical Analogies (cont)
•Mechanical-to-electrical correspondence in the current analogy:
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 10
Bandpass Biquad Transfer Function
meq
keq
ceq
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 11
-60
-50
-40
-30
-20
-10
0
8.7 8.9 9.1 9.3Frequency [MHz]
Tra
ns
mis
sio
n [
dB
]
Pin=-20dBm
In Out
VP
Sharper roll-off
Sharper roll-off
Loss PoleLoss Pole
Performance:fo=9MHz, BW=20kHz, PBW=0.2%
I.L.=2.79dB, Stop. Rej.=51dB20dB S.F.=1.95, 40dB S.F.=6.45
Performance:fo=9MHz, BW=20kHz, PBW=0.2%
I.L.=2.79dB, Stop. Rej.=51dB20dB S.F.=1.95, 40dB S.F.=6.45
Design:Lr=40m
Wr=6.5m hr=2m
Lc=3.5mLb=1.6m VP=10.47VP=-5dBm
RQi=RQo=12k
[S.-S. Li, Nguyen, FCS’05]
3CC 3/4 Bridged Mechanical Filter
[Li, et al., UFFCS’04]
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 12
Micromechanical Filter Circuit
1/krmr cr 1/krmr cr 1/krmr cr-1/ks -1/ks
1/ks
-1/ks -1/ks
1/ks
1/kb 1/kb
-1/kb
Co Co
1:e e:1
1:c 1:cc:1 c:1
1:b b:1
/4
/4
/4Input
Outputvi
RQ
RQ
vo
VP
Bridging Beam
Coupling Beam
Resonator
vovi
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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 13
Micromechanical Filter Circuit
1/krmr cr 1/krmr cr 1/krmr cr-1/ks -1/ks
1/ks
-1/ks -1/ks
1/ks
1/kb 1/kb
-1/kb
Co Co
1:e e:1
1:c 1:cc:1 c:1
1:b b:1
/4
/4
/4Input
Outputvi
RQ
RQ
vo
VP
Bridging Beam
Coupling Beam
Resonator
vovi
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 14
Micromechanical Filter Circuit
1/krmr cr 1/krmr cr 1/krmr cr-1/ks -1/ks
1/ks
-1/ks -1/ks
1/ks
1/kb 1/kb
-1/kb
Co Co
1:e e:1
1:c 1:cc:1 c:1
1:b b:1
/4
/4
/4Input
Outputvi
RQ
RQ
vo
VP
Bridging Beam
Coupling Beam
Resonator
vovi
8
EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 15
vovi
Micromechanical Filter Circuit
1/krmr cr 1/krmr cr 1/krmr cr-1/ks -1/ks
1/ks
-1/ks -1/ks
1/ks
1/kb 1/kb
-1/kb
Co Co
1:e e:1
1:c 1:cc:1 c:1
1:b b:1
/4
/4
/4Input
Outputvi
RQ
RQ
vo
VP
Bridging Beam
Coupling Beam
Resonator
All circuit element values determined
by CAD layout
All circuit element values determined
by CAD layout
Amenable to automated circuit
generation
Amenable to automated circuit
generation
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 16
-60
-50
-40
-30
-20
-10
0
8.7 8.9 9.1 9.3Frequency [MHz]
Tra
ns
mis
sio
n [
dB
]
Pin=-20dBm
In Out
VP
Sharper roll-off
Sharper roll-off
Loss PoleLoss Pole
Performance:fo=9MHz, BW=20kHz, PBW=0.2%
I.L.=2.79dB, Stop. Rej.=51dB20dB S.F.=1.95, 40dB S.F.=6.45
Performance:fo=9MHz, BW=20kHz, PBW=0.2%
I.L.=2.79dB, Stop. Rej.=51dB20dB S.F.=1.95, 40dB S.F.=6.45
Design:Lr=40m
Wr=6.5m hr=2m
Lc=3.5mLb=1.6m VP=10.47VP=-5dBm
RQi=RQo=12k
[S.-S. Li, Nguyen, FCS’05]
3CC 3/4 Bridged Mechanical Filter
[Li, et al., UFFCS’04]
9
EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 17
Beam Resonator Equivalent Circuits(Pretty Much the Same Stuff)
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 18
Equivalent Dynamic Mass
•Once the mode shape is known, the lumped parameter equivalent circuit can then be specified
• Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity
h
W
Maximum Kinetic Energy
Maximum Velocity @ location x
Equivalent Mass =
Location x
Density
Maximum Velocity Function
z
10
EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 19
Equivalent Dynamic Mass
•We know the mode shape, so we can write expressions for displacement and velocity at resonance
h
W
Location xz
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 20
Equivalent Dynamic Stiffness & Damping
• Stiffness then follows directly from knowledge of mass and resonance frequency
• And damping also follows readily
h
W
Location xz
11
EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I
CTN 3/16/17
Copyright © 2017 Regents of the University of California
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 21
Equivalent Lumped Mechanical Circuit
h
W
Location x
Meq(x)
Keq(x)
Ceq(x)
2
2
)]([
)]([)(
xu
xdxuAxM
l
oeq
)()( 2 xMxK eqoeq
Q
xMxC
eqoeq
)()(
z
EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 22
Equivalent Lumped Mechanical Circuit
h
W
Meq(0) = 1.03x10-13 kg
Keq(0) = 19,927 N/m
Ceq(0) = 5.66x10-9 kg/s
Meq(l/2) = 2.78x10-13 kg
Keq(l/2) = 53,938 N/m
Ceq(l/2) = 1.53x10-8 kg/s
Meq(node) = ∞
Keq(node) = ∞
Ceq(node) = ∞
Example: Polysilicon w/ l=14.9m, W=6m, h=2m 70 MHz
z