EE C245 ME C218 Introduction to MEMS Design Fall 2012

Transcription

1 EE C245 ME C218 Introduction to MEMS Design Fall 2012 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA Lecture EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 1 Lecture Outline Reading: Senturia, Chpt. 10: 10.5, Chpt. 19 Lecture Topics: Estimating Resonance Frequency Lumped Mass-Spring Approximation ADXL-50 Resonance Frequency Distributed Mass & Stiffness Folded-Beam Resonator EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 2 Regents of the University of California 1

2 Estimating Resonance Frequency EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ W r v i Clamped-Clamped Beam μresonator Resonator Beam Electrode Sinusoidal Excitation vi L r Voltageto-Force Capacitive Transducer EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ V P h i o Sinusoidal Forcing Function Q ~10,000 i v o i vi = Vi cos[ ωot] fi = Fi cos[ ωot] ω ω o : small amplitude ω = ω o : maximum amplitude beam reaches its maximum potential and kinetic energies ω ο ω Regents of the University of California 2

3 Estimating Resonance Frequency Assume simple harmonic motion: Potential Energy: Kinetic Energy: EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Estimating Resonance Frequency (cont) Energy must be conserved: Potential Energy + Kinetic Energy = Total Energy Must be true at every point on the mechanical structure Occurs at peak displacement Occurs when the beam moves through zero displacement Maximum Potential Energy Stiffness Displacement Amplitude Maximum Kinetic Energy Mass Radian Frequency Solving, we obtain for resonance frequency: EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Regents of the University of California 3

4 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 Proof Mass Sense Finger Suspension Beam in Tension EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Lumped Spring-Mass Approximation Mass is dominated by the proof mass 60% of mass from sense fingers Mass = M = 162 ng (nano-grams) Suspension: four tensioned beams Include both bending and stretching terms [A.P. Pisano, BSAC Inertial Sensor Short Courses, ] EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Regents of the University of California 4

5 Bending contribution: ADXL-50 Suspension Model Stretching contribution: Total spring constant: add bending to stretching EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ ADXL-50 Resonance Frequency Using a lumped mass-spring approximation: On the ADXL-50 Data Sheet: f o = 24 khz Why 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 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Regents of the University of California 5

6 Distributed Mechanical Structures Vibrating structure displacement function: Maximum displacement function (i.e., mode shape function) Seen when velocity y(x,t) = 0 ŷ(x) Procedure for determining resonance frequency: Use the static displacement of the structure as a trial function and find the strain energy W max 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 EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Displacement: Velocity: Maximum Kinetic Energy y( x, t) = yˆ( x)cos[ ωt] y( x, t) v( x, t) = = ωyˆ( x)sin[ ωt] t At times t = π/(2ω), 3π/(2ω), y( x, t) = 0 Velocity topographical mapping The displacement of the structure is y(x,t) = 0 The velocity is maximum and all of the energy in the structure is kinetic (since W=0): v( x,(2n + 1) π (2ω )) = ωyˆ( x) EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Regents of the University of California 6

7 Maximum Kinetic Energy (cont) At times t = π/(2ω), 3π/(2ω), y( x, t) = 0 Velocity: v( x,(2n + 1) π (2ω)) = ωyˆ( x) 1 2 dk = dm [ v( x, t)] 2 dm = ρ( Wh dx) Maximum kinetic energy: EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ The Raleigh-Ritz Method Equate the maximum potential and maximum kinetic energies: Rearranging yields for resonance frequency: ω = resonance frequency W max = maximum potential energy ρ = density of the structural material W = beam width h = beam thickness ŷ(x) = resonance mode shape EE C245: Introduction to MEMS Design LecM 910 C. Nguyen C. Nguyen 11/4/08 9/28/ Regents of the University of California 7

8 Example: Folded-Beam Resonator Folded-beam suspension Derive an expression for the resonance frequency of the folded-beam structure at left. Shuttle w/ mass M s h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 15 Get Kinetic Energies Folded-beam suspension Shuttle w/ mass M s h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 16 Regents of the University of California 8

9 Folded-Beam Suspension Folding Truss y z x Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 17 Get Kinetic Energies (cont) Folded-beam suspension Shuttle w/ mass M s h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 18 Regents of the University of California 9

10 Get Kinetic Energies (cont) Folded-beam suspension Shuttle w/ mass M s h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 19 Get Potential Energy & Frequency Folded-beam suspension Shuttle w/ mass M s h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 20 Regents of the University of California 10

11 Brute Force Methods for Resonance Frequency Determination EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 21 Basic Concept: Scaling Guitar Strings Guitar String μmechanical Resonator Vib. Amplitude Low Q High Q Guitar Freq. 110 Hz Freq. Vibrating Vibrating A A String String (110 (110 Hz) Hz) Stiffness Freq. Equation: 1 kr fo = 2π m r Mass [Bannon 1996] f o =8.5MHz Q vac =8,000 Q air ~50 Performance: L r =40.8μm m r ~ kg W r =8μm, h r =2μm d=1000å, V P =5V Press.=70mTorr EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 22 Regents of the University of California 11

12 Fixed-Fixed Beam Resonator Losses Gap Electrode Q = 300 at 70MHz Free-Free Beam Resonator Supporting Beams Elastic Wave Radiation Problem: direct anchoring to to the the substrate anchor radiation into into the the substrate lower Q Solution: support at at motionless nodal points isolate resonator from anchors less less energy loss loss higher Q λ/4 L r Free-Free Beam Q = 15,000 at 92MHz EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/ MHz Free-Free Beam μresonator Free-free beam μmechanical resonator with non-intrusive supports reduce anchor dissipation higher Q EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 24 Regents of the University of California 12

13 Higher Order Modes for Higher Freq. 2 nd Mode Free-Free Beam 3 rd Mode Free Free Beam Distinct Mode Shapes Electrodes L r = 20.3 μm h = 2.1 μm Support Beam Frequency [MHz] EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 25 Transmission [db] Q = 11, Phase [degree] Flexural-Mode Beam Wave Equation yz u Transverse Displacement W = width u = ma L h x F F F + x dx Derive the wave equation for transverse vibration: EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 26 Regents of the University of California 13

14 Example: Free-Free Beam z W h 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: 2 u 2 t = EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/ EI u 4 ρa x Free-Free Beam Frequency Substitute u = u 1 e jωt into the wave equation: (1) This is a 4 th order differential equation with solution: Boundary Conditions: (2) EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 28 Regents of the University of California 14

15 Free-Free Beam Frequency (cont) Applying B.C. s, get A=C and B=D, and (3) Setting the determinant = 0 yields Which has roots at Substituting (2) into (1) finally yields: Free-Free Beam Frequency Equation EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 29 Higher Order Free-Free Beam Modes More than 10x increase Fundamental Mode (n=1) 1 st Harmonic (n=2) 2 nd Harmonic (n=3) EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 30 Regents of the University of California 15

16 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 ] EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 31 Regents of the University of California 16