1EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 1EE C245 – ME C218Introduction to MEMS DesignFall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture 25: Sensing CircuitsEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 2Lecture Outline• Reading: Senturia Chpts. 6, 14 (op amp parts), 19• Lecture Topics:ª Sensing Circuits( Ideal Op Amps( Velocity Sensing Circuits( Position Sensing Circuits( Non-Ideal Op Ampsª MEMS/Transistor Integration2EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 3Complete Electrical-Port Equiv. CircuitV1VPFd1d1d2Electrode 2Electrode 1I1C1C2kxmbηe2:1cxlxrxCo11:ηe1Co2V2I2+-V1I1+-V2I2xmlx=kcx1=brx=1111dCVxCVoPPe=∂∂=η2222dCVxCVoPPe=∂∂=ηStatic electrode-to-mass overlap capacitanceEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 4Condensed Equiv. Circuit (Symmetrical)ηe:1cxlxrxCo11:ηeCo2+-V1I1+-V2I2xCxLxRxCo1Co2+-V1I1+-V2I22exmLη=kCex2η=2exbRη=If ηe1= ηe2, then …whereHolds for the symmetrical case, where port 1 and port 2 are identicalHolds for the symmetrical case, where port 1 and port 2 are identical3EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 5Phasings of Signals• Below: plots of resonance electrical and mechanical signals vs. time, showing the phasings between themV1VPFd1d1d2Electrode 2Electrode 1I1C1C2kxmbV2I2EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 6Port 1 to 2 TransG Across the Circuitηe2:1cxlxrxCo11:ηe1Co2+-V1I1+-V2I2x• What is the transconductance from port 1 to port 2 with port 2 shorted to ground?4EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 7Port 1 to 2 vi-to-ioTransfer FunctionEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 8Sensing Circuits5EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 9Sense ElectrodesTuning ElectrodesSense ElectrodesTuning ElectrodesDrive ElectrodezΩrDriveSense[Zaman, Ayazi, et al, MEMS’06]Drive Voltage Signal(-) Sense Output Current(+) Sense Output CurrentDrive Oscillation Sustaining AmplifierDifferential TransRSense AmplifierMEMS-Based Tuning Fork GyroscopeEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 10Detecting Velocity Versus Position• Transfer Function:• Detect position when the output is varying slowly, i.e., at low frequenciesPosition(i.e., displacement)222)(1)()(ooodsQsksFsXωωω++=ω)()(sFsXdLowpassBiquadωoVelocity• Transfer Function:• Detect velocity when the output is at resonance or when a bandpass response is required222)(1)()(ooodsQssksFsωωωυ++=ω)()(sFsdυωoBandpassBiquad6EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 11Detecting Velocity Versus Position• Transfer Function:Position(i.e., displacement)222)(1)()(ooodsQsksFsXωωω++=Velocity• Transfer Function:222)(1)()(ooodsQssksFsωωωυ++=EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 12Output Current Measures Velocityηe2:1cxlxrxCo11:ηe1Co2+-V1I1+-V2I2x• Relationship between output current and velocity:• To turn current into voltage (for a voltage output), send the current into a resistor RD• To get position, must integrate → send the current into a capacitor CDxieo&2η=Output current is proportional to velocity, and thus, directly measures velocity7EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 13Velocity-to-Voltage Conversion• To convert velocity to a voltage, use a resistive loadEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 14Position-to-Voltage Conversion• To sense position (i.e., displacement), use a capacitive load8EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 15Ideal Operational AmplifiersEE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 16Ideal Op Amp• Equivalent Circuit of an Ideal Op Amp:• Properties of Ideal Op Amps:01=i1v-+02=i()−+−vvA0R()−+−=vvAv0()12vvA−=2vinR+-+-+-+v−vVoltage-Controlled Voltage Source (VCVS)Single-ended outputDifferential input ∞=inR00=R∞=A1.2.3.0==−+iifinite assuming , 0==−+vvv4.5.Why?9EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 17Ideal Op Amp (cont) ∞=inR00=R∞=A1.2.3.0==−+iifinite assuming , 0==−+vvv−+−+=→=−∴vvvv 0ground) (virtualcircuit short virtual 0⇒∞v4.5.()finite 0==−∞−+vvvWhy? Because for()finite 0=v• Properties of Ideal Op Amps:• Big assumption!• How can we assume this? We can assume this only when there is an appropriate negative feedback path!EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 18Negative Feedback00SSSaSSiβεε−==[]finite! 10==≈⇒∞→ββaaSSaifinite! 10==∴iSSβWhere S could be a current, voltage, displacement, etc.,…Negative feedback acts to oppose or subtract from input.()()βββaaSSaSaSSSaSiii+=→=+−=1 10000Overall transfer function.(When there is negative FB around the amplifier.)↑ΔβS↑ΔiS↑ΔεSεS+aβ↓ΔεS↑Δ0S0SiS12543+-10EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 19Negative Feedback• Comments:1. Negative FB can insure S0= finite even with a = ∞.2. Overall gain dependent (or overall T.F.) dependent only on external components. (e.g., β).3. Overall (Closed-loop) gain (So/Si) is independent of amplifier gain a.→ Very important, since amplifiers using transistors can be designed to have large gain, but it’s hard to get an exact gain. i.e., if you’re shooting for a = 50,000, you might get 47,000 or 60,000 instead.→ Comment 3 makes this less consequential.EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 20Positive Feedback• Contrast with Positive Feedback:• But for a bounded, controllable function, need negative FB around the op amp↑ΔβS↑ΔiS↑↑ΔSS+aβ↑↑Δ0S0SiS++Output blows up!(for aβ > 1)Will be the case for a = ∞.If β is a bandpassbiquad transfer function → get oscillation at the resonance frequencyωωοβ11EE C245: Introduction to MEMS Design Lecture 25 C. Nguyen 11/19/07 21Inverting Amplifier0viv-++-1i0=−i2i1R2RVirtual groundOV1. Verify that there is negative FB.2. node attached to (-) terminal is virtual ground.3.→=→=∴−+ finite 0vvv21 0 iii =∴=−22220211100RiRiviRvRviii−=−===−=12012210 RRvvvRRRRvviii−=∴−=⎟⎟⎠⎞⎜⎜⎝⎛−=⇒NOTE: Gain dependent only on R1& R2(external
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