Comb Resonator Design (1)

Transcription

1 Lecture 5: Comb Resonator Design (1) Sh School of felectrical ti lengineering i and dcomputer Science, Si Seoul National University Nano/Micro Systems & Controls Laboratory dicho@snu.ac.kr URL:

2 Principle : Why is Comb Drive Type? Interlacing comb fingers create large capacitor area; electrostatically actuated suspended microstructures Features : Fairly linear relationship between capacitance and displacement Higher surface area/ capacitance than parallel plate capacitor Electrostatic actuation: low power (no DC current) Electrostatic Comb Drive Type usage and possession is in violation of copyright laws 2

3 What is Comb Drive Resonator? Comb drives combine mechanical and electrostatic issues : Mechanical issues Elasticity Stress and strain Resonance (natural frequency) Damping Electrical issues Capacitance Electrostatic forces Electrostatic work and energy Tang, Nguyen and Howe JMEMS usage and possession is in violation of copyright laws 3

4 Normal Stress and Strain Stress: force applied to surface σ = F / A measured in N/m 2 or Pa, compressive or tensile Strain: ratio of deformation to length ε =Δl / l measured in %, ppm, or microstrain Young s Modulus: E = σ / ε Hooke s Law: K = F/ Δ= l EA/ l usage and possession is in violation of copyright laws 4

5 Shear Stress and Strain Shear Stress: force applied parallel to surface τ = F / A measured in N/m 2 or Pa, Share Strain: ratio of deformation to length γ =Δl/ l Shear Modulus: G = τ / γ usage and possession is in violation of copyright laws 5

6 Poisson s Ratio Tensile stress in x direction results in compressive stress in y and z direction (object becomes longer and thinner) Poisson s Ratio: < Materials > εy transverse strain ν = = ε longitudinal strain x Metals : ν 0.3 Rubbers : ν Cork : ν 0 Homogeneous -Isotropic -Anisotropic Heterogeneous usage and possession is in violation of copyright laws 6

7 Mechanical Property for Poly Crystalline Silicon Young s modulus for different deposition processes Doping Conditions PSG diffusion doping POCl 3 diffusion doping Deposition temp min min min min min min min min min min min ± ± ± ± ± ± ± ± ± ± ±2 Young s modulus (GPa) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±2 [Ref] S. Lee et al., IOP Journal of Micromechanics and Microengineering, vol. 8, no. 4, pp , 337, Dec usage and possession is in violation of copyright laws 7

8 Mechanical Property for Single Crystal Silicon Silicon Crystallography Silicon (100) Silicon (110) Plane ( ) { } Direction < > [ ] Silicon (111) usage and possession is in violation of copyright laws 8

9 Mechanical Property for Single Crystal Silicon (cont d) Young s modulus Silicon (100) Silicon (110) Silicon (111) usage and possession is in violation of copyright laws 9

10 Mechanical Property for Single Crystal Silicon (cont d) Shear modulus Silicon (100) Silicon (110) Silicon (111) usage and possession is in violation of copyright laws 10

11 Mechanical Property for Single Crystal Silicon (cont d) Poisson s ratio Silicon (100) Silicon (110) Silicon (111) [Ref] J. Kim et al., Proceedings of Transducers 2001, pp , Munich, Germany, June usage and possession is in violation of copyright laws 11

12 Mechanical Property for Single Crystal Silicon (cont d) Torsional stiffness (1) Schematic of torsional spring z T y θ x The dimension splits of torsional springs - Thickness(t) : 10 μm, 20 μm - Length(l) : 40 μm - Width(w) : 2 μm, 4 μm usage and possession is in violation of copyright laws 12

13 Mechanical Property for Single Crystal Silicon (cont d) Torsional stiffness (2) Silicon (110) The torsional stiffness on silicon (110) varies from % to % usage and possession is in violation of copyright laws 13

14 Mechanical Property for Single Crystal Silicon (cont d) Torsional stiffness (3) Silicon (100) The torsional stiffness on silicon (100) varies from -1.6 % to 9.6 %. usage and possession is in violation of copyright laws 14

15 Mechanical Property for Single Crystal Silicon (cont d) Torsional stiffness (4) Silicon (111) The torsional stiffness on silicon (111) varies from 1.7 % to 2.3 %. usage and possession is in violation of copyright laws 15

16 Mechanical Property for Single Crystal Silicon (cont d) Torsional stiffness (5) Summary for results The average for the normalized maximum torsional stiffness variation ratio (%) [Ref] D. Kwak et al., Proceedings of IMECE 2003, Washington D.C, USA, November usage and possession is in violation of copyright laws 16

17 Electrostatic Forces Parallel Plate Capacitor: Capacitance: C = Q/ V = ε ε A/ d 0 0 ε ; dielectric constant of free space 12 ( F / m) ε ; dielectric permittivity r r Stored energy: Q ε A W = CV = C = 2 2 C d ε = εε 0 r Electrostatic force between plates: F x W 1ε A 1CV x 2 d 2 d 2 = = V = 2 2 usage and possession is in violation of copyright laws 17

18 Electrostatic Actuation Positioning of capacitor plate: 1 = εε 2 2 Fel 0 r AV / x < 0 2 F = K( x d ) < 0 S where d 0 0 :distance at rest (no applied voltage) Stable equilibrium when Fel = FS usage and possession is in violation of copyright laws 18

19 Pull-In Point The higher V, the closer the plate is pulled in. F el when d 0. What is the closest stable distance x min? F el and F S must be tangential: εε A x V K 0 r 3 2 / =, so 2 3 V = Kx / εε 0 r A Substitute into F el =F S to get x = 2/3d min 0 can control x only from 2/3d 0 to d 0 usage and possession is in violation of copyright laws 19

20 Electrostatic Comb Drive Capacitance is approximately: C = ε 0 ε A/ d r = 0 2 nεεlh/ d r (L>>d) (ignore fringing field effect) Change in capacitance when moving by x : Δ C = εε 0 r Δ A / d = 2 nεεδxh/ d 0 r Electrostatic force : F = 1/2 V 2 dc / dx = nεεh/ dv 2 Electric field distribution in el 0 r com-finger gaps Note: F el is independent of x over wide range (fringing g field), and quadratic in V. usage and possession is in violation of copyright laws 20

21 Comb Drive Design [Folded d suspension type] [U spring suspension type] [Serpentine suspension type] [Fishhook suspension type] usage and possession is in violation of copyright laws 21

22 Comb Drive Design (cont d) [Bent beam serpentine suspension type] [Spiral spring suspension type] usage and possession is in violation of copyright laws 22

23 Comb Drive Failure Modes Comb drives require low stiffness in x direction but high stiffnessiny,zdirectionaswellasrotations. in rotations. Note: comb fingers are in unstable equilibrium with respect to the y direction. usage and possession is in violation of copyright laws 23

24 Summary of Translation Motion Acceleration, velocity, distance a = v = x Force, momentum F p = mv = Ft Kinetic energy 1 2 E = mv 2 Dynamic (spring, damper, mass) F = Kx + bx + mx Oscillation (assume b=0) f = 1 K ( Hz) ω= 2π m K m usage and possession is in violation of copyright laws 24

25 Summary of Rotation Motion Angular acceleration, angular velocity, angle α = ω = φ Torque, angular momentum T = r F L = r p = Iω Kinetic energy 1 2 E = Iω 2 Dynamic (moment of inertia) T = Kφ + βφ + I φ Oscillation (assume b=0) = 1 K f 2π I usage and possession is in violation of copyright laws 25

26 Lumped-Parameter Model The dynamic of motion is based on the lumped-parameter model Input: external fore F, output: displacement x mx () t + bx () t + Kx () t = F 0 sin( wt ) where m:mass, b :damping, K :stiffness Schematic microdevice Lumped-parameter model usage and possession is in violation of copyright laws 26

27 Lumped-Parameter Model (cont d) Transfer function: Hs () x = = F s 2 1 m b + s + m K m Schematic microdevice Lumped-parameter model usage and possession is in violation of copyright laws 27

28 Resonators Analogy between mechanical and electrical system: Mass m Inductivity L Spring K Capacitance 1/C Damping b Resistance R (depending where R is placed in circuit) Solution to 2 nd order differential equation: 2 2 ω0 ω0 Hs ( ) = or 2 ω 0 2 s + 2 ξω + ω + + ω 0s s s 0 0 Q where ω = 2 πf : natural frequency 0 o 2 2 K ω 0 = : mechanical system, m 1 ω0 = : electrical system LC Q: quality factor ( Q = 1 / 2 ξ, ξ : damping ratio) usage and possession is in violation of copyright laws 28

29 Mechanical Resonators Frequency and phase shift under damping: xt ( ) Ae cos( t ϕ) t /2τ = ω1 + m where τ = :damping time b 2 1 b 2 ω1 = ω0 1 = ω0 1 = ω0 1 ξ = ω 2 2 d 4ω τ 4 Km ϕ: phase shift 0 t / Energy dissipation: Et () = Ee τ 0 usage and possession is in violation of copyright laws 29

30 Quality Factor Definition: Quality factor (Q factor) Ratio of stored energy and lost energy: E τ Q = 2π = 2π = ω0τ ΔE T m Mechanical system: Q = ω0 = b Km b Similar for electric systems: (a) Q L 1 L = ω0 = R R C C (b) Q = ω0rc = R L usage and possession is in violation of copyright laws 30

31 Quality Factor (cont d) How fast does energy dissipate? Q m τ = = (mechanical) ω b 0 What is the maximum amplitude for a given frequency? : At resonance, amplitude is Q times the DC response usage and possession is in violation of copyright laws 31

32 Summary: Mechanical/Electrical Resonator Mechanical resonator : Torsional resonator : Electrical resonator : mx () t + bx () t + Kx() t = 0 m : mass, b: damping, K: stiffness natural frequency ω = K / m (for small b) 0 I θ() t + b θ() t + kθ() t = 0 I: moment of inertia, b: damping, k: stiffness natural frequency ω = k/ I 0 1 Lqt () + Rqt () + qt () = 0 C L: inductivity, R: resistance, C: capacitance natural frequency ω = 0 1/ LC usage and possession is in violation of copyright laws 32

33 Measurement V P V d I : output current V dc / dt 0 : DC bias : di drive signal at ω Feedback via transimpedance amplifier d P V c carrier signal at ωc Output signal: frequency spectrum includes ω, ω, but also ω ± ω basis for frequency transfer d c c d usage and possession is in violation of copyright laws 33

34 Application of Comb Drive Accelerometer (NML SNU) Gyroscope (NML SNU) usage and possession is in violation of copyright laws 34

35 Application of Comb Drive (cont d) Micromirror (UC Davis & Stanford) x/y stage (MiSA SNU) usage and possession is in violation of copyright laws 35

36 Reference Lee, S. W., Cho, C. H., Kim, J. P., Park, S. J., Yi, S. W., Kim, J. J., and Cho, D. I., "The Effects of Post-deposition Processes on Polysilicon Young's Modulus, IOP Journal of Micromechanics and Microengineering, g vol. 8, no. 4, pp , Dec Kim, J. P., Cho, D. I., and Muller, R. S., "Why is (111) Silicon a Better Mechanical Material for MEMS," Proceedings of Transducers 2001: 11th International Conference on Solid State Sensors and Actuators, pp , Munich, Germany, June 10-14, Kwak, D. H., Kim, J. P., Park, S. J., Ko, H. H., and Cho, D. I., "Why is (111) Silicon a better Mechanical Material for MEMS: Torsion Case," 2003 ASME International Mechanical Engineering Congress (IMECE 2003), Washington D.C., USA, Nov , Lee, S. W., Park, S. J., Yi, S. W., Lee, S. C., Cho, D. I., Ha, B. J., Oh, Y. S., and Song, C. M., "Electrostatic Actuation of Surface/Bulk Micromachined Single-crystal Silicon Microresonators," Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 2, pp , Kyeongju, Korea, Oct , Ko, H. H., Kim, J. P., Park, S. J., Kwak, D. H., Song, T. Y., Setiadi, D., Carr, W., Buss, J., and Cho, D. I., "A High-performance X/Y-axis Microaccelerometer Fabricated on SOI Wafer without Footing Using the Sacrificial Bulk Micromachining (SBM) Process," 2003 International Conference on Control, Automation and Systems (ICCAS 2003), pp , Gyeongju, Korea, Oct , Kim,C.H.,Jeong,H.M.,Jeon,J.U.,Kim,Y.K., Silicon Jeon, J. K., micro XY-stage with a large area shuttle and no-etching holes for SPM-based data storage, Journal of Microelectromechanical Systems, vol. 12, issue 4, pp , Aug Chang Liu, Foundations of MEMS, Pearson, Nicolae O. Lobontiu, Mechanical ca design of microresonators, s McGraw-Hill, usage and possession is in violation of copyright laws 36