EE C245 ME C218 Introduction to MEMS Design Fall 2003 Roger Howe and Thara Srinivasan Lecture 26 Micromechanical Resonators I EE C245 ME C218 Fall 2003 Lecture 26 Today s Lecture Circuit models for micromechanical resonators Microresonator oscillators sustaining amplifiers amplitude limiters and noise Resonant inertial sensors accelerometers and gyroscopes EE C245 ME C218 Fall 2003 Lecture 26 2 1 Reading Reference List next lecture 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 Surfacemicromachined 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 50 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 812 2003 pp 891 894 EE C245 ME C218 Fall 2003 Lecture 26 3 Comb Drive Lateral Resonator Anchor connects ground plane and resonator Typical bias VI VO 0 V DC voltage across drive and sense electrodes to resonator VP EE C245 ME C218 Fall 2003 Lecture 26 C T C Nguyen Ph D Thesis EECS Dept UC Berkeley 1994 4 2 The Lateral Resonator as a Two Port EE C245 ME C218 Fall 2003 Lecture 26 C T C Nguyen Ph D Thesis EECS Dept UC Berkeley 1994 5 Input Current Input current i1 t is the derivative of the charge q1 C1vD i1 t C1 dvD dC vD 1 dt dt vD t VI v1 t VP VP1 v1 t The capacitance C1 has a DC component and a time varying component due to the motion of the structure C1 t Co1 Cm1 t Cm1 t C1 x t x linearized case Substitute to find the input current EE C245 ME C218 Fall 2003 Lecture 26 6 3 Input Motional Admittance Y1x j Phasor form of the motional current i1x C1 j X x The input motional admittance inverse of impedance is the ratio of the phasor motional current to the ac drive voltage I1 x j VP 1 Y1x j I x 1 j V1 j The displacement to voltage ratio can be re expressed in terms of the drive force Fd j Y1x j VP1 C1 X j Fd j j x Fd j V1 j EE C245 ME C218 Fall 2003 Lecture 26 7 Input Admittance Cont The electrostatic force component at the drive frequency is f d t 1 2 C1 C v D t VP 1v1 t 1 x 2 x Fd j V1 j The mechanical response of the resonator is Lecture 9 X j Fd j The input admittance is I1 x j V1 j EE C245 ME C218 Fall 2003 Lecture 26 8 4 Series L C R Admittance The current through an L C R branch is I L I j j C V j 1 o 2 j RC C o LC V 2 R Match terms in motional admittance find equivalent elements EE C245 ME C218 Fall 2003 Lecture 26 9 Equivalent Circuit for Input Port A series L C R circuit results in the identical expression find equivalent values Lx1 Cx1 and Rx1 L x1 m 2 V1 C x1 2 k Ix1 Co1 R x1 km Q 2 Lx1 V P1 C1 electromechanical x coupling coefficient At resonance the impedances of the inductance and the capacitance cancel out Cx1 Rx1 EE C245 ME C218 Fall 2003 Lecture 26 I x1 V1 Rx1 10 5 Output Port Model Consider first the current due to driving the input set v2 0 V C2 C x VP 2 2 t x t i2 t VP 2 C C j k 1VP1VP2 1 2 C x x V j I 2 j j VP 2 2 X j 1 2 x 1 o j Q o In phasor form I2 and Ix1 are related by the forward current gain 21 C 2 I j x 21 2 I x1 j V C1 P1 x VP 2 model by a current controlled current source EE C245 ME C218 Fall 2003 Lecture 26 11 Two Port Equivalent Circuit v2 0 V1 Ix1 Co1 Lx1 Cx1 Rx1 EE C245 ME C218 Fall 2003 Lecture 26 I 2 21Ix1 V2 0 V 12 6 Complete Two Port Model Lx1 Co1 Cx1 Lx2 12Ix2 Rx1 21Ix1 Ix2 V1 Ix1 Cx2 Co2 V2 Rx2 Symmetry implies that modeling can be done from port 2 with port 1 shorted superimpose the two models EE C245 ME C218 Fall 2003 Lecture 26 13 Equivalent Circuit for Symmetrical Resonator 21 12 1 C T C Nguyen Ph D UC Berkeley 1994 EE C245 ME C218 Fall 2003 Lecture 26 14 7 455 kHz Comb Drive Resonator Values assumes vacuum not small huge mind boggling Lx Cx C T C Nguyen Ph D UC Berkeley 1994 EE C245 ME C218 Fall 2003 Lecture 26 15 Double Ended Tuning Fork Resonators i 0 Current through structure more resistance decreases Q more feedthrough to substrate EE C245 ME C218 Fall 2003 Lecture 26 T Roessig Ph D ME UC Berkeley 1997 16 8 Ideal Tuning Fork Two Port Response Phase change of 180o at resonance pins the frequency with drifts in the feedback amplifier having little effect Response assumes no feedthrough capacitance between input and output ports T Roessig Ph D ME UC Berkeley 1997 EE C245 ME C218 Fall 2003 Lecture 26 17 Tuning Fork Response with Capacitive Feedthrough Cf Feedthrough capacitance results in a null in the amplitude response and an added sense current which increases with frequency and which can obscure the resonance entirely Cf Req R int Leq Ceq R int is Next lecture Cf and its control drive vd Cint Co Co Cint sense structure node EE C245 ME C218 Fall 2003 Lecture 26 T Roessig Ph D ME UC Berkeley 1997 18 9 Microresonator Oscillator C T C Nguyen and R T Howe IEEE J Solid State Circuits 34 440 454 1999 EE C245 ME C218 Fall 2003 Lecture 26 19 Current to Voltage or Transresistance Amplifier Rf i 0 iin vout The feedback resistor can be implemented using a MOSFET biased in the triode region EE C245 ME C218 Fall 2003 Lecture 26 20 10 Microresonator Oscillator Schematic Transresistance amplifier C T C Nguyen and R T Howe IEEE J Solid State Circuits 34 440 454 1999 EE C245 ME C218 Fall …