EE245: Introduction to MEMSModule 10: Resonance Frequency
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Copyright @2009 Regents of the University of California 1
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 1
EE C245 – ME C218Introduction to MEMS Design
Fall 2010Prof. Clark T.-C. Nguyen
Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley
Berkeley, CA 94720
Lecture Module 10: Resonance Frequency
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 2
Lecture Outline
• Reading: Senturia, Chpt. 10: §10.5, Chpt. 19• Lecture Topics:
Estimating Resonance FrequencyLumped Mass-Spring ApproximationADXL-50 Resonance FrequencyDistributed Mass & StiffnessFolded-Beam Resonator
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 3EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 3
Estimating Resonance Frequency
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 4EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 4
Lr hWr
VPvi
Clamped-Clamped Beam μResonator
ωωο
ivoi
Q ~10,000
vi
Resonator Beam
Electrode
io
]cos[ tVv oii ω= ]cos[ tFf oii ω=
Voltage-to-Force Capacitive Transducer
Sinusoidal Forcing Function
Sinusoidal Excitation
• ω ≠ ωo: small amplitude• ω = ωo: maximum amplitude → beam reaches its maximum potential and kinetic energies
EE245: Introduction to MEMSModule 10: Resonance Frequency
CTN 10/25/10
Copyright @2009 Regents of the University of California 2
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 5EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 5
Estimating Resonance Frequency
• Assume simple harmonic motion:
• Potential Energy:
• Kinetic Energy:
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 6EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 6
Estimating Resonance Frequency (cont)
• Energy must be conserved:Potential Energy + Kinetic Energy = Total EnergyMust be true at every point on the mechanical structure
• Solving, we obtain forresonance frequency:
Maximum Potential Energy
Maximum Kinetic Energy
StiffnessDisplacement Amplitude
MassRadian
Frequency
Occurs at peak displacement
Occurs when the beam moves through zero displacement
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 7EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 7
Example: ADXL-50• The proof mass of the ADXL-50 is many times larger than the effective mass of its suspension beams
Can ignore the mass of the suspension beams (which greatly simplifies the analysis)
• Suspension Beam: L = 260 μm, h = 2.3 μm, W = 2 μm
Suspension Beam in Tension
Proof Mass
Sense Finger
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 8EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 8
Lumped Spring-Mass Approximation
•Mass is dominated by the proof mass60% of mass from sense fingersMass = M = 162 ng (nano-grams)
• Suspension: four tensioned beamsInclude both bending and stretching terms [A.P. Pisano, BSAC Inertial Sensor Short Courses, 1995-1998]
EE245: Introduction to MEMSModule 10: Resonance Frequency
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Copyright @2009 Regents of the University of California 3
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 9EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 9
ADXL-50 Suspension Model
• Bending contribution:
• Stretching contribution:
• Total spring constant: add bending to stretching
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 10EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 10
ADXL-50 Resonance Frequency
• Using a lumped mass-spring approximation:
•On the ADXL-50 Data Sheet: fo = 24 kHzWhy the 10% difference?Well, it’s approximate … plus …Above analysis does not include the frequency-pulling effect of the DC bias voltage across the plate sense fingers and stationary sense fingers … something we’ll cover later on …
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 11EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 11
Distributed Mechanical Structures
• Vibrating structure displacement function:
• Procedure for determining resonance frequency:Use the static displacement of the structure as a trial function and find the strain energy Wmax at the point of maximum displacement (e.g., when t=0, π/ω, …)Determine the maximum kinetic energy when the beam is at zero displacement (e.g., when it experiences its maximum velocity)Equate energies and solve for frequency
ŷ(x)Maximum displacement function (i.e., mode shape function)
Seen when velocity y(x,t) = 0
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 12EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 12
Maximum Kinetic Energy• Displacement:
• Velocity:
• At times t = π/(2ω), 3π/(2ω), …
The displacement of the structure is y(x,t) = 0The velocity is maximum and all of the energy in the structure is kinetic (since W=0):
]sin[)(ˆ),(),( txyt
txytxv ωω−=∂
∂=
]cos[)(ˆ),( txytxy ω=
0),( =txy
Velocity topographical mapping
)(ˆ))2()12(,( xynxv ωωπ −=+
EE245: Introduction to MEMSModule 10: Resonance Frequency
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Copyright @2009 Regents of the University of California 4
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 13EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 13
Maximum Kinetic Energy (cont)
• At times t = π/(2ω), 3π/(2ω), …
•Maximum kinetic energy:
0),( =txy
)(ˆ))2()12(,( xynxv ωωπ −=+Velocity:
2)],([21 txvdmdK ⋅⋅=
)( dxWhdm ⋅= ρ
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 14EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 14
The Raleigh-Ritz Method
• Equate the maximum potential and maximum kinetic energies:
• Rearranging yields for resonance frequency:
ω = resonance frequencyWmax = maximum potential
energyρ = density of the structural
materialW = beam widthh = beam thicknessŷ(x) = resonance mode shape
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 15
Example: Folded-Beam Resonator
• Derive an expression for the resonance frequency of the folded-beam structure at left.
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
h = thickness
Folding truss w/
mass Mt\2
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 16
Get Kinetic Energies
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
h = thickness
Folding truss w/
mass Mt\2
EE245: Introduction to MEMSModule 10: Resonance Frequency
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Copyright @2009 Regents of the University of California 5
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 17
Folded-Beam Suspension
Comb-Driven Folded Beam Actuator
Folding Truss
x
y
z
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 18
Get Kinetic Energies (cont)
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
h = thickness
Folding truss w/
mass Mt\2
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 19
Get Kinetic Energies (cont)
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
Folding truss w/
mass Mt\2
h = thicknessEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 20
Get Potential Energy & Frequency
Folded-beam suspension
Anchor
Shuttle w/ mass Ms
h = thickness
Folding truss w/
mass Mt\2
EE245: Introduction to MEMSModule 10: Resonance Frequency
CTN 10/25/10
Copyright @2009 Regents of the University of California 6
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 21
Brute Force Methods for Resonance Frequency Determination
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 22
Basic Concept: Scaling Guitar StringsGuitar String
Guitar
Vibrating “A”String (110 Hz)Vibrating “A”
String (110 Hz)
High Q
110 Hz Freq.
Vib.
Am
plitu
de
Low Q
r
ro m
kfπ21
=
Freq. Equation:
Freq.
Stiffness
Mass
fo=8.5MHzQvac =8,000
Qair ~50
μMechanical Resonator
Performance:Lr=40.8μm
mr ~ 10-13 kgWr=8μm, hr=2μmd=1000Å, VP=5VPress.=70mTorr
[Bannon 1996]
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Anchor Losses
Q = 15,000 at 92MHz
Fixed-Fixed Beam Resonator
GapAnchor AnchorElectrode
Problem: direct anchoring to the
substrate anchor radiation into the
substrate lower Q
Problem: direct anchoring to the
substrate anchor radiation into the
substrate lower Q
Solution: support at motionless nodal points
isolate resonator from anchors less
energy loss higher Q
Solution: support at motionless nodal points
isolate resonator from anchors less
energy loss higher Q
Lr
Free-Free Beam
Supporting Beams
λ/4
Anchor
Anchor
Elastic WaveRadiation
Q = 300 at 70MHz
Free-Free Beam Resonator
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 24
92 MHz Free-Free Beam μResonator• Free-free beam μmechanical resonator with non-intrusive supports reduce anchor dissipation higher Q
EE245: Introduction to MEMSModule 10: Resonance Frequency
CTN 10/25/10
Copyright @2009 Regents of the University of California 7
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 25
Higher Order Modes for Higher Freq.2nd Mode Free-Free Beam 3rd Mode Free Free Beam
Anchor
Support Beam
Electrodes
Anchor-72
-69
-66
-63
-60
-57
101.31 101.34 101.37 101.40
Frequency [MHz]
Tran
smis
sion
[dB
]
-180
-135
-90
-45
0
45
90
135
180
Phas
e [d
egre
e]
Q Q = 11,500= 11,500
Distinct Mode Shapes
h = 2.1 μm
Lr = 20.3 μm
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 26
Flexural-Mode Beam Wave Equation
• Derive the wave equation for transverse vibration:
u
L
x
y
F
dxxFF∂∂
+
uTransverse Displacement = ma
h
W = widthz
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 27
Example: Free-Free Beam
• Determine the resonance frequency of the beam• Specify the lumped parameter mechanical equivalent circuit• Transform to a lumped parameter electrical equivalent circuit
• Start with the flexural-mode beam equation:
h
W
4
4
2
2
xu
AEI
tu
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂
ρ
z
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 28
Free-Free Beam Frequency
• Substitute u = u1ejωt into the wave equation:
• This is a 4th order differential equation with solution:
• Boundary Conditions:
(1)
(2)
EE245: Introduction to MEMSModule 10: Resonance Frequency
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Copyright @2009 Regents of the University of California 8
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 29
Free-Free Beam Frequency (cont)
• Applying B.C.’s, get A=C and B=D, and
• Setting the determinant = 0 yields
•Which has roots at
• Substituting (2) into (1) finally yields:
Free-Free Beam Frequency Equation
(3)
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 30
Higher Order Free-Free Beam Modes
Fundamental Mode (n=1)
1st Harmonic (n=2)
2nd Harmonic (n=3)
More than 10x increase
EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 31
Mode Shape Expression• The mode shape expression can be obtained by using the fact that A=C and B=D into (2), yielding
• Get the amplitude ratio by expanding (3) [the matrix] and solving, which yields
• Then just substitute the roots for each mode to get the expression for mode shape
Fundamental Mode (n=1)[Substitute ]