Lec23p.ElectricalSti..
Transcription
Lec23p.ElectricalSti..
EE C245 – ME C218 Introduction to MEMS Design Fall 2007 Prof Clark T Prof. T.-C. C Nguyen Dept of Electrical Engineering & Computer Sciences Dept. University of California at Berkeley Berkeley, y CA 94720 L t Lecture 23 Electrical 23: El t i l Stiffness Stiff EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 1 Lecture Outline • Reading: g Senturia, Chpt. p 5, Chpt. p 6 • Lecture Topics: ª Energy Conserving Transducers (Charge Control (Voltage Control p Transducers ª Parallel-Plate Capacitive (Linearizing Capacitive Actuators (Electrical Stiffness ª Electrostatic Comb-Drive Comb Drive (1st Order Analysis (2nd Order Analysis EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 2 Linearizing the Voltage-to-Force T Transfer f F Function ti EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 3 Linearizing the Voltage-to-Force T.F. • Apply a DC bias (or polarization) voltage VP together with the intended input (or drive) voltage vi(t), where VP >> vi(t) v(t) (t) = VP + vi(t) EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 4 Differential Capacitive Transducer • The net force on the suspended d d center electrode is EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 5 Remaining Nonlinearity (El t i l Stiff (Electrical Stiffness)) EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 6 Parallel-Plate Capacitive Nonlinearity • Example: clamped-clamped laterally driven beam with balanced electrodes • Nomenclature: Va or vA Conductive Structure Electrode km va=|va|cosωt d1 VA m x t Va or vA = VA + va AC or Signal g Component p Total Value (lower case variable; v1 DC Component lower case subscript) (upper case variable; V1 upper case subscript) EE C245: Introduction to MEMS Design Lecture 23 Fd1 VP C. Nguyen 11/20/08 7 Parallel-Plate Capacitive Nonlinearity • Example: clamped-clamped laterally driven beam with balanced electrodes • Expression for ∂C/∂x: Conductive Structure Electrode km d1 m Fd1 v1 V1 EE C245: Introduction to MEMS Design Lecture 23 x VP C. Nguyen 11/20/08 8 Parallel-Plate Capacitive Nonlinearity • Thus, the expression for force from the left side becomes: Conductive Structure Electrode km d1 m Fd1 v1 V1 EE C245: Introduction to MEMS Design Lecture 23 x VP C. Nguyen 11/20/08 9 Parallel-Plate Capacitive Nonlinearity • Retaining only terms at the drive frequency: Fd 1 ω o Co1 2 Co1 = VP1 | v1 | cos ωot + VP1 2 | x | sin i ωo t d1 d1 Drive force arising from the input excitation voltage at the frequency of this voltage • These Th two t together t th mean m that th t this thi Proportional P i l to displacement force acts against the spring g force! restoring ª A negative spring constant ª Since it derives from VP, we call it the electrical stiffness, stiffness given by: EE C245: Introduction to MEMS Design Lecture 23 90o phase-shifted from drive, so in phase with displacement ke = VP21 C. Nguyen Co1 2 εA = VP1 3 2 d1 d1 11/20/08 10 Electrical Stiffness, ke • The electrical stiffness ke behaves like any other stiffness • It affects resonance frequency: Conductive Structure Electrode k m − ke k ωo = = m m ′ km d1 12 k m ⎛ ke ⎞ ⎜⎜1 − ⎟⎟ = m ⎝ km ⎠ m 12 ⎛ ε A⎞ ′ ⎟ ωo = ωo ⎜⎜1 − 3⎟ k d m 1 ⎠ ⎝ 2 VP1 Fd1 v1 Frequency is now a function of dc-bias VP1 EE C245: Introduction to MEMS Design x Lecture 23 V1 VP C. Nguyen 11/20/08 11 Voltage-Controllable Center Frequency Gap EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 12 Microresonator Thermal Stability −1.7ppm/oC Poly-Si μresonator pp oC 17ppm/ • Thermal h l stability bl of f poly-Si l micromechanical h l resonator is 10X worse than the worst case of AT-cut quartz crystal EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 13 Geometric-Stress Compensation • Use a temperature dependent mechanical stiffness to null frequency shifts due to Young’s modulus thermal dep. [Hsu et al al, IEDM IEDM’00] 00] [W.-T. Hsu, et al., IEDM’00] • Problems: ª stress relaxation ª compromised design flexibility EE C245: Introduction to MEMS Design Lecture 23 [Hsu et al IEDM 2000] C. Nguyen 11/20/08 14 Voltage-Controllable Center Frequency Gap EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 15 Excellent Temperature Stability R Resonator t Top Metal Electrode [Hsu et al MEMS’02] Top Electrode-toResonator Gap p× −1.7ppm/ pp oC Elect. Stiffness: ke ~ 1/d3 Ø Elect.-Stiffness Compensation −0.24ppm/oC AT-cut Quartz Q Crystal at Various Cut Angles Frequency: fo ~ (km - ke)0.5 × Counteracts reduction in frequency due to Young’s modulus temp. dependence EE C245: Introduction to MEMS Design Uncompensated μresonator resonator Lecture 23 On par with quartz! C. Nguyen 100 [Ref: Hafner] 11/20/08 16 Measured Δf/f vs. T for kemp μResonators μ Compensated VP-VC 1500 16V 1000 14V 12V 10V 9V 8V 7V 6V 4V 2V 0V CC 500 0 -500 Design/Performance: fo =10MHz, =10MHz Q=4,000 Q=4 000 VP=8V, he=4μm do=1000Å, h=2μm Wr=8μm, =8μm Lr=40μm [Hsu et al MEMS’02] -1000 1000 −0.24ppm/oC -1500 300 320 340 360 Temperature [K] 380 • Slits Slit help h l tto release l th the stress t generated t d b by lateral l t l thermal expansion Ö linear TCf curves Ö –0.24ppm/oC!!! EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 17 Can One Cancel ke w/ Two Electrodes? • What if we don’t like the x Fd1 d1 o Ele ectrode e 1 Subtracts from the Fd1 term, as expected Fd 2 ω Co 2 = −VP 2 | v2 | cos ωot d2 2 + VP 2 Co 2 | x | sin ωot 2 d2 d2 m v1 v2 V add, 1 V2 Adds to the quadrature term → ke’s no matter the electrode configuration! EE C245: Introduction to MEMS Design km Fd2 Electrod E de 2 dependence of frequency on VP? • Can C we cancell ke via i a diff differential i l input electrode configuration? • If we do a similar analysis for Fd2 at Electrode 2: Lecture 23 VP C. Nguyen 11/20/08 18 Problems With Parallel-Plate C Drive • Nonlinear voltage-to-force EE C245: Introduction to MEMS Design Lecture 23 x d1 d2 m Electrod E de 2 Fd1 km Fd2 Ele ectrode e 1 transfer function ª Resonance frequency becomes dependent on parameters (e.g., bias voltage VP) ª Output current will also take on nonlinear characteristics as amplitude grows (i.e., as x approaches h do) ª Noise can alias due to nonlinearity y • Range of motion is small ª For larger motion, need larger gap … but larger gap weakens the electrostatic force ª Large motion is often needed (e g by gyroscopes, (e.g., gyroscopes vibromotors, optical MEMS) v1 v2 V1 V2 VP C. Nguyen 11/20/08 19 Electrostatic Comb Drive EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 20 Electrostatic Comb Drive • Use of comb-capacitive tranducers brings many benefits ª Linearizes voltage-generated input forces ª (Ideally) eliminates dependence of frequency on dc-bias dc bias y ª Allows a large range of motion Stator Rotor z x Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 21 Comb-Drive Force Equation (1st Pass) x T View Top Vi y z x Lf Side View EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 22 Lateral Comb-Drive Electrical Stiffness x T View Top Vi y z x Lf Side View 2 Nεhx ∂C 2 Nεh • Again: g C (x ) = → = d ∂x d • No (∂C/∂x) x-dependence p → no electrical stiffness: ke = 0! • Frequency immune to changes in VP or gap spacing! EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 23 Typical Drive & Sense Configuration y z x EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 24 Comb-Drive Force Equation (2nd Pass) • In our 1st pass, we neglected ª Fringing fields ª Parallel-plate P ll l l capacitance i between b stator and d rotor ª Capacitance to the substrate • All of these capacitors must be included when evaluating the energy expression! Stator Rotor Ground Plane EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 25 Comb-Drive Force With Ground Plane Correction • Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and d the h ground d plane l → modifies difi the h capacitive i i energy [Gary Fedder, Fedder Ph.D., Ph D UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 26 Capacitance Expressions • Case: Vr = VP = 0V • Cspp depends p on whether or not fingers are engaged Capacitance per unit length l h Region 2 Region 3 [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 27 Comb-Drive Force With Ground Plane Correction • Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and d the h ground d plane l → modifies difi the h capacitive i i energy [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 28 Simulate to Get Capacitors → Force • Below: 2D finite element simulation 20-40% reduction of Fe,x EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 29