1EE C245 – ME C218 Fall 2003 Lecture 26EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 26Micromechanical Resonators I2EE C245 – ME C218 Fall 2003 Lecture 26Today’s Lecture• Circuit models for micromechanical resonators• Microresonator oscillators:sustaining amplifiers, amplitude limiters,and noise• Resonant inertial sensors:accelerometers and gyroscopes23EE C245 – ME C218 Fall 2003 Lecture 26Reading/Reference List• C. T.-C. Nguyen, Ph.D. Thesis, Dept. of EECS, UC Berkeley, 1994.• T. A. Roessig, R. T. Howe, A. P. Pisano, and J. H. Smith, “ Surface-micromachined resonant accelerometer,” (Transducers ’97), Chicago, Ill., June 16-19, 1997, pp. 859-862.• A. A. Seshia, R. T. Howe, and S. Montague, “An integrated microelectromechanical resonant-output gyroscope,” IEEE MEMS 2002,Las Vegas, Nevada, January 2002.• C. T.-C. Nguyen, “Transceiver front-end architectures using vibrating micromechanical signal processors,” Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, Sept. 12-14, 2001, pp. 23-32.• J. Wang, Z. Ren, and C. T.-C. Nguyen, “Self-aligned 1.14 GHz vibrating radial-mode disk resonator,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 947-950.• B. Bircumshaw, et al, “The radial bulk annular resonator: towards a 50Ω RF MEMS filter,” Transducers ’03, Boston, Mass., June 8-12, 2003. • M. U. Demirci, M. A. Abdelmoneum, and C. T.-C. Nguyen, “Mechanically corner-coupled square microresonator array for reduced series motional resistance,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 955-958.• V. Kaajakari, et al, “Square-extensional mode single-crystal silicon micromechanical RF-resonator,” Transducers ’03, Boston, Mass., June 8-12, 2003, pp. 891-894.next lecture4EE C245 – ME C218 Fall 2003 Lecture 26Comb-Drive Lateral ResonatorTypical bias:VI= VO= 0 VDC voltage across drive and sense electrodes to res-onator = VPAnchor connectsground plane andresonatorC. T.-C. Nguyen, Ph.D. Thesis, EECS Dept., UC Berkeley, 199435EE C245 – ME C218 Fall 2003 Lecture 26The Lateral Resonator as a “Two-Port”C. T.-C. Nguyen, Ph.D. Thesis, EECS Dept., UC Berkeley, 19946EE C245 – ME C218 Fall 2003 Lecture 26Input CurrentInput current i1(t) is the derivative of the charge q1= C1vDdtdCvdtdvCtiDD 111)( +=The capacitance C1has a DC component and a time-varying component due to the motion of the structure)()(111tCCtCmo+=)()(11txxCtCm∂∂=(linearized case)Substitute to find the input current:txxCvtxxCVdtdvCdtdvCtiPmo∂∂∂∂+∂∂∂∂−++=11111111)()()()()(111tvVVtvVtvPPID+−=−+=)(1tix47EE C245 – ME C218 Fall 2003 Lecture 26Input Motional Admittance Y1x(jω)Phasor form of the motional current i1x:∂∂−==)()()()()(111111ωωωωωωjVjXjxCVjVjIjYPxxThe displacement-to-voltage ratio can be re-expressed in terms of the drive force Fd(jω)The input motional admittance (inverse of impedance) is the ratio of the phasor motional current to the ac drive voltage:)()(111XjxCVjIPxωω∂∂−=∂∂−=)()()()()(1111ωωωωωωjVjFjFjXjxCVjYddPx∂∂−=)()()()()(1111ωωωωωωjVjFjFjXjxCVjYddPx8EE C245 – ME C218 Fall 2003 Lecture 26Input Admittance (Cont.)The electrostatic force component at the drive frequency ω is:xCtvVxCtvtfPDd∂∂−=∂∂=11112,)()(21)(ωωThe mechanical response of the resonator is (Lecture 9):→xCVjVjFPd∂∂−=111)()(ωω( ) ( )oodQjkjFjXωωωωωω//1)()(21+−=−The input admittance is:( ) ( )∂∂−+−∂∂−=−xCVQjkjxCVjVjIPooPx 11211111//1)()(ωωωωωωω( ) ( )ooPxQjxCVkjjVjIωωωωωωω//1)()(22112111+−∂∂=−59EE C245 – ME C218 Fall 2003 Lecture 26Series L-C-R AdmittanceThe current through an L-C-R branch is:CLR→I+-V( ) ( )RCjCjjVjIoωωωωωω+−=2/1)()(LCo=−2ωMatch terms in motional admittance à find equivalent elements10EE C245 – ME C218 Fall 2003 Lecture 26Equivalent Circuit for Input PortkCx21η=A series L-C-R circuit results in the identical expression àfind equivalent values Lx1, Cx1, and Rx121ηmLx=21ηQkmRx==∂∂=xCVP11ηelectromechanical coupling coefficientCx1Lx1Rx1Co1→Ix1+-V1At resonance, the impedances of the inductance and the capacitance cancel out à111xxRVI =611EE C245 – ME C218 Fall 2003 Lecture 26Output Port ModelConsider first the current due to driving the input (set v2= 0 V)txxCVtCVtiPP∂∂∂∂−=∂∂−=22222)(In phasor form,( ) ( ))(//1)()(1221211222ωωωωωωωωω jVQjxCxCVVkjjXxCVjjIooPPP+−∂∂∂∂=∂∂=−I2and Ix1are related by the forward current gain φ21:xCVxCVjIjIPPx∂∂∂∂==11221221)()(ωωφ→ model by a current-controlledcurrent source 12EE C245 – ME C218 Fall 2003 Lecture 26Two-Port Equivalent Circuit (v2= 0)Cx1Lx1Rx1Co1→Ix1+-V1φ21Ix1+-V2= 0 VI2←713EE C245 – ME C218 Fall 2003 Lecture 26Complete Two-Port Model Cx1Lx1Rx1Co1→Ix1+-V1φ21Ix1+-V2φ12Ix2Cx2Lx2Rx2Ix2→Symmetry implies that modeling can be done from port 2, with port 1 shorted à superimpose the two modelsCo214EE C245 – ME C218 Fall 2003 Lecture 26Equivalent Circuit forSymmetrical Resonator (φ21= φ12 = 1) C. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994815EE C245 – ME C218 Fall 2003 Lecture 26455 kHz Comb-Drive Resonator ValuesC. T.-C. Nguyen, Ph.D.,UC Berkeley, 1994LxCx← assumes vacuum← huge!← not small← mind-boggling!16EE C245 – ME C218 Fall 2003 Lecture 26Double-Ended Tuning Fork ResonatorsCurrent through structure à more resistance (decreases Q)more feedthrough to substratei ≈ 0T. Roessig, Ph.D.,ME,UC Berkeley, 1997917EE C245 – ME C218 Fall 2003 Lecture 26Ideal Tuning Fork Two-Port ResponsePhase change of 180oat resonance “pins” thefrequency, with driftsin the feedback amplifierhaving little effectResponse assumes nofeedthroughcapacitancebetween input and outputportsT. Roessig, Ph.D.,ME,UC Berkeley, 199718EE C245 – ME C218 Fall 2003 Lecture 26Tuning Fork Response withCapacitive Feedthrough Cf + vd Leq Ceq Req Co Cint structure node - - + is drive Co Rint Cint Rint sense Cf Feedthroughcapacitanceresults in a null in the amplitude response andan added sense currentwhich increases with fre-quency… and which canobscure the resonance en-tirely!Next lecture: Cfand itscontrolT. Roessig,
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