EE C245 – ME C218Introduction to MEMS DesignFall 2007Fall 2007Prof Clark TC NguyenProf. Clark T.-C. NguyenDept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720yLt 23 El t i l StiffEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 1Lecture 23: Electrical StiffnessLecture Outline• Reading: Senturia, Chpt. 5, Chpt. 6gpp• Lecture Topics:ª Energy Conserving Transducers(Charge Control(Charge Control( Voltage Controlª Parallel-Plate Capacitive Transducersp( Linearizing Capacitive Actuators( Electrical StiffnessªElectrostatic CombDriveªElectrostatic Comb-Drive( 1stOrder Analysis( 2ndOrder AnalysisEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 2Linearizing the Voltage-to-Force Tf FtiTransfer FunctionEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 3Linearizing the Voltage-to-Force T.F.• Apply a DC bias (or polarization) voltage VPtogether with the intended input (or drive) voltage vi(t), where VP>> vi(t)(t) V(t)EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 4v(t) = VP+ vi(t)Differential Capacitive Transducer• The net force on the dd suspended center electrode isEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 5Remaining Nonlinearity(El t i l Stiff )(Electrical Stiffness)EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 6Parallel-Plate Capacitive Nonlinearity• Example: clamped-clamped laterally driven beam with balanced electrodeskElectrodeConductive Structure• Nomenclature:kmElectrodeVaor vAva=|va|cosωtVAd1xtVV+mFd1Vaor vA = VA+ vaTotal ValueAC or Signal ComponentVv1Total ValueDC Component(upper case variable; gp(lower case variable; lower case subscript)EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 7V1VP(upper case variable; upper case subscript)Parallel-Plate Capacitive Nonlinearity• Example: clamped-clamped laterally driven beam with balanced electrodeskElectrodeConductive Structure• Expression for ∂C/∂x:kmElectroded1xmFd1Vv1EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 8V1VPParallel-Plate Capacitive Nonlinearity• Thus, the expression for force from the left side becomes:kElectrodeConductive StructurekmElectroded1xmFd1Vv1EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 9V1VPParallel-Plate Capacitive Nonlinearity• Retaining only terms at the drive frequency:CVCVFooi||||121txdVtvdVFooPooPdoωωωsin||cos||2112111111+=Pil Drive force arising from the input excitation voltage at the frequency of this voltageProportional to displacement90ophase-shifted from the frequency of this voltage90phase-shifted from drive, so in phase with displacement•Th t t th m th t thi •These two together mean that this force acts against the spring restoring force!gª A negative spring constantª Since it derives from VP, we call it the electrical stiffness given by:212AVCVkoε==EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 10the electrical stiffness, given by:311211dVdVkPPe==Electrical Stiffness, ke• The electrical stiffness kebehaves like any other stiffnesskElectrodeConductive Structure• It affects resonance frequency:kmElectrode−′kkkem21⎞⎛==mmemoωd1x1⎟⎟⎠⎞⎜⎜⎝⎛−=kkmkmemmFd1213211⎟⎟⎞⎜⎜⎛−=′AkVPooεωωVv131⎟⎠⎜⎝dkmooFrequency is now a EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 11V1VPFrequency is now a function of dc-bias VP1Voltage-Controllable Center FrequencyGapEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 12Microresonator Thermal Stability−1.7ppm/oCPoly-Si μresonator -17ppm/oChl bl f l hl ppEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 13• Thermal stability of poly-Si micromechanical resonator is 10X worse than the worst case of AT-cut quartz crystalGeometric-Stress Compensation• Use a temperature dependent mechanical stiffness to null frequency shifts due to Young’s modulus thermal dep.[Hsuet alIEDM’00][W.-T. Hsu, et al., IEDM’00][Hsu et al, IEDM 00]• Problems:ª stress relaxationª compromised design flexibilityEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 14flexibility[Hsu et al IEDM 2000]Voltage-Controllable Center FrequencyGapEE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 15Excellent Temperature StabilityRtTop Metal ElectrodeResonator[Hsu[Hsuet alet alMEMS’02]MEMS’02]−1.7ppm/oCElect.-StiffnessCompensationTop Electrode-to-Resonator Gap ×[Hsu [Hsu et alet alMEMS’02]MEMS’02]ppCompensation−0.24ppm/oCAT-cut Quartz pElect. Stiffness: ke~ 1/d3ØQCrystal at Various Cut AnglesFrequency:fo~ (km- ke)0.5 ×Uncompensated resonatorAnglesCounteracts reduction in frequency due toOn par with EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 16μresonator100[Ref: Hafner]frequency due to Young’s modulus temp. dependencequartz!Measured Δf/fvs. Tfor ke-Compensated μResonators150016VVP-VCmpμ500100016V14V12V10V05007V8V9V10V-1000-5006V4V2V0VDesign/Performance:f=10MHzQ=4 000-150010003003203403603800VCCfo=10MHz, Q=4,000VP=8V, he=4μmdo=1000Å, h=2μmW=8μmL=40μm−0.24ppm/oC300320340360380•Slit h l t l th t t d b l t l Temperature [K]Wr=8μm, Lr=40μm[Hsu et al MEMS’02]EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 17•Slits help to release the stress generated by lateral thermal expansion Ö linear TCfcurves Ö –0.24ppm/oC!!!Can One Cancel kew/ Two Electrodes?• What if we don’t like the dependence of frequency on VP?C l ki diff il kxF•Can we cancel kevia a differential input electrode configuration?•If we do a similar analysis for FkmFd1Fd2If we do a similar analysis for Fd2at Electrode 2:d1d2Subtracts from the de 2e 1tvCVFoωcos||2=Fd1term, as expectedmElectrodectrodeCtvdVFoPdoωωcos||222222−=EEleVv1Vv2txdCVooPωsin||22222+EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 18V1VPV2Adds to the quadrature term → ke’s add, no matter the electrode configuration!Problems With Parallel-Plate C Drive• Nonlinear voltage-to-force transfer functionªResonance frequency becomes kxFªResonance frequency becomes dependent on parameters (e.g., bias voltage VP)ªOutput current will also take on kmFd1Fd2ªOutput current will also take on
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