EE C245 – ME C218 Fall 2003 Lecture 12EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 12Capacitive Position Sensing2EE C245 – ME C218 Fall 2003 Lecture 12Today’s Lecture• The capacitor as a position sensor• Capacitive sensing configurations:divider with unbalanced, balanced drive• Practical issues: other capacitors are always present!ultimate position resolution• The capacitor as a velocity sensor• Reading:Senturia, S. D., Microsystem Design, Kluwer AcademicPublishers, 2001, Chapter 19, pp. 500-502, Chapter 16, 427-429.3EE C245 – ME C218 Fall 2003 Lecture 12The Simple Capacitor DividerC(x)Crefvout)cos(ˆ)( tVtvω=1XCin= 0Why modulate v(t)?()⎟⎟⎠⎞⎜⎜⎝⎛+=xZZZVVrefrefoutˆ⎟⎟⎠⎞⎜⎜⎝⎛+=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+=oxrefrefrefoutCCVxCjCjCjVV/11ˆ)(111ˆωωωIdeal buffer: Cin= 04EE C245 – ME C218 Fall 2003 Lecture 12A Capacitive Divider from the Early DaysK. E. Petersen (IBM Almaden), IEEE IEDM, 1980.Question: how is the potential VBset?Metal gate of MOSFET is directlyconnected to the top plate of thesense capacitor VB.. other capacitorCpis parasiticAnswer: with an external probe tip!…the two-transistor amplifier wouldremain biased for high gain for afew minutes after VBwas set Cp5EE C245 – ME C218 Fall 2003 Lecture 12Matched Air-Gap Reference CapacitorsxgAxCoo+=ε)(oorefgACε=compliant suspension (vertical azsensitivity)stiff suspension: insensitiveWeijie Yun, P. R. Gray, and R. T. Howe, Hilton Head Workshop, 1992, pp. 21-25.6EE C245 – ME C218 Fall 2003 Lecture 12Simple Capacitor Divider (Cont.)⎟⎟⎠⎞⎜⎜⎝⎛+=⎟⎟⎠⎞⎜⎜⎝⎛++=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛++=xggVgxgVxgAgAVVooooooooout2ˆ/)(11ˆ)/(/11ˆεε⎟⎟⎠⎞⎜⎜⎝⎛−≅⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=oooooutgxVVgxggVV22ˆ2ˆ2/112ˆoffsetsignalmatched air-gap reference capacitorOffset signal is undesirable for buffer amplifier andfor downstream signal processing7EE C245 – ME C218 Fall 2003 Lecture 12Capacitor Divider With Differential ExcitationC(x)Crefvout)cos(ˆ)( tVtvω=+)cos(ˆ)( tVtvω−=−1XCin= 0Why modulate v+and v-?Ideal buffer: Cin= 0Impedance divider withsuperposition:)cos(ˆ)( tVtvω−=−() ()⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎠⎞⎜⎜⎝⎛+=xZZxZVxZZZVVrefrefrefout)(ˆˆ8EE C245 – ME C218 Fall 2003 Lecture 12Improved Capacitive Divider (Cont.)()()()()⎟⎟⎠⎞⎜⎜⎝⎛+++−=⎟⎟⎠⎞⎜⎜⎝⎛+−=⎟⎟⎠⎞⎜⎜⎝⎛+−=−−−−oooorefrefrefrefoutgxgxggVxCCxCCVxZZxZZVVˆˆ)(ˆ1111⎟⎟⎠⎞⎜⎜⎝⎛−≅⎟⎟⎠⎞⎜⎜⎝⎛+−=oooutgxVxgxVV2ˆ2ˆno offset!)cos(...2)(2)(2ˆ)cos()()(2ˆ)(2tgtxgtxVttxgtxVtvooooutωω⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−−=⎟⎟⎠⎞⎜⎜⎝⎛+−=distortion9EE C245 – ME C218 Fall 2003 Lecture 12Parasitic Electrostatic Forcefor Differential ExcitationC(x)Crefvout)cos(ˆ)( tVtvω=+)cos(ˆ)( tVtvω−=−1XCin= 0vout≈ 0 V for smalldisplacements() ()⎟⎟⎠⎞⎜⎜⎝⎛−≅⎟⎠⎞⎜⎝⎛−=+2222cosˆ2121)(oooutgAtVdxdCvvtfεωForce has both DC and 2ω components:pull-in and resonant excitation can happen!small displacements10EE C245 – ME C218 Fall 2003 Lecture 12The Capacitive Half-BridgevoutC+(x))cos(ˆ)( tVtvω=+)cos(ˆ)( tVtvω−=−1XCin= 0C-(x)C+(x) = εoA/(go+ x)C-(x) = εoA/(go- x)() ()⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎠⎞⎜⎜⎝⎛+=+−−−++xZZxZVxZZxZVVout)(ˆ)(ˆImpedance divider withsuperposition:11EE C245 – ME C218 Fall 2003 Lecture 12Capacitance Half Bridge (Cont.)⎟⎟⎠⎞⎜⎜⎝⎛==⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛+=oooooooutgxVgxVgxgVgxgVVˆ22ˆ2ˆ2ˆSimplify expression:no offset;2X signalincreaseElectrostatic force:() ()⎟⎠⎞⎜⎝⎛−−⎟⎠⎞⎜⎝⎛−=−−++dxdCvvdxdCvvtfoutout222121)(() ()⎟⎟⎠⎞⎜⎜⎝⎛+−−⎟⎟⎠⎞⎜⎜⎝⎛−−=−+oooutoooutgCvvgCvvtf222121)(12EE C245 – ME C218 Fall 2003 Lecture 12Electrostatic Force (Cont.)()()⎥⎦⎤⎢⎣⎡−−−⎟⎟⎠⎞⎜⎜⎝⎛=22cosˆcosˆ21)( tVvvtVgCtfoutoutooωω()()[]tVtVvvvtVvtVgCtfoutoutoutoutooωωωω222222cosˆcosˆ2cosˆ2cosˆ21)( +−−+−⎟⎟⎠⎞⎜⎜⎝⎛=()tVvgCtfoutooωcosˆ2)(⎟⎟⎠⎞⎜⎜⎝⎛=tVgxvooutωcosˆ⎟⎟⎠⎞⎜⎜⎝⎛=Output voltage is proportional to the displacement (for x << go)xtVgCtVgxgCtfooooo⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=ωω22222cosˆ2cosˆ2)(DC and 2ω terms13EE C245 – ME C218 Fall 2003 Lecture 12Electrostatic Spring Constant keC+(x))cos(ˆ)( tVtvω=+)cos(ˆ)( tVtvω−=−C-(x)xxkxtVgCtfeoo−=⎟⎟⎠⎞⎜⎜⎝⎛=ω222cosˆ2)(note direction: springapplies force oppositeto displacementtVgCkooeω222cosˆ2−=both DC and 2ω components: usesquare wave excitation to yield constant ke14EE C245 – ME C218 Fall 2003 Lecture 12Parasitic CapacitancesGary Fedder, Ph.D. thesis,EECS, UC Berkeley, 1994Surface micromachined z-axis parallel-plate capacitor15EE C245 – ME C218 Fall 2003 Lecture 12Equivalent CircuitGary Fedder, Ph.D. thesis, EECS, UC Berkeley, 1994Cpp(x): nominal || plate sense capacitorCf1(x): fringe capacitance (varies with plate displacement)Cf2: fringe capacitance between upper plate (connected toanchor plane) and lower plate … slight dependence on xCpu: parasitic capacitance from upper plate to substrateCpl: parasitic capacitance from lower plate to substrate16EE C245 – ME C218 Fall 2003 Lecture 12Velocity SensingFundamental current-voltage relationship for a time-varying capacitor:[]dtdCtvdtdvtCtvtCdtddtdqissssss)()()()( +===Consider special case: v = VP= constantdtdCVisP=… used in high-quality capacitance microphones17EE C245 – ME C218 Fall 2003 Lecture 12Velocity Sensing (Cont.)Sense capacitor’s time variation:vdxdCdtdxdxdCdtdCsss==Parallel-plate sense capacitor with gap go:osoxsgCdxdC−==0Harmonic motion:txtxωcosˆ)(=()tgxCVtxgCVdtdCViosoPosoPsPsωωωωsinˆsinˆ⎟⎟⎠⎞⎜⎜⎝⎛=−−==18EE C245 – ME C218 Fall 2003 Lecture 12Some NumbersSurface micromachined capacitors:ADXL-50 sense capacitorCs≈ 100 fFgo= 1 μmV+=-V-= 2.5 VxSvxout=mVgVSoxμ/5.2==+Vvoutμ100min=mxmVVSvxxoutμμμ6minmin1040/5.2100−===…noise in buffer amppmx 40min=is this real?19EE C245 – ME C218 Fall 2003 Lecture 12World Record CapacitivePosition-Sense Resolution**surface micromachining classaudio frequency bandAnalog Devices ADRS-150 vibratory rate gyroscopeJohn Geen, Steve