1EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 1EE C245 – ME C218Introduction to MEMS DesignFall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture 27: NoiseEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 2Announcements• Project emailsª Main theme: do micro-to-macro comparison using a table• Project Outbrief sign-up sheets will be on my door by this afternoon• HW#6:ª Now due this Thursdayª Let’s go through it since there’s been no discussion section for the past two weeks2EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 3AnnouncementsyxzEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 4HW#6 (cont)• SPICEª Why use it? Why make equivalent electrical circuits for mechanical devices?ª Answer: Noise analysis is needed to determine the minimum detectable signal for a MEMS-based sensorª Simulink will not do this nearly as conveniently3EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 5EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 6Lecture Outline• Reading: Senturia Chpt. 16, 19• Lecture Topics:ª Non-Ideal Op Amps( Input Offset Voltage, VOSª Determining Sensor Resolution( Noise( Noise Sources( Equivalent Input-Referred Noise Sources( Example: Gyro MDS Calculation4EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 7Back to Op Amp Non-IdealitiesEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 8Actual Op Amps Are Not Ideal• Actual op amps, of course, are not ideal; rather, they …ª Generate noiseª Have finite gain, Aoª Have finite bandwidth, ωbª Have finite input resistance, Riª Have finite input capacitance, Ciª Have finite output resistance, Roª Have an offset voltage VOSbetween their (+) and (-) terminalsª Have input bias currentsª Have an offset IOSbetween the bias currents into the (+) and (-) terminalsª Have finite slew rateª Have finite output swing (governed by the supply voltage used, -L to +L)• And what’s worse: All of the above can be temperature (or otherwise environmentally) dependent!5EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 9Input Offset Voltage V0SInput Offset Voltage, V0S:()−+−=vvAv0+-)out! railsit :or (usually, 000−+=≠ LLvv00=vIdeal case: Reality:Why? Internal mismatches within the op amp → cause a dc offset.Model this with an equivalent input offset voltage V0S.0v+-+-SV0Typically, V0S= 1mV – 5mVEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 10Effect of V0Son Op Amp CircuitsExample: Non-Inverting AmplifiermVVmVVRRS505 ,9 e.g.,0012=→==⎟⎟⎠⎞⎜⎜⎝⎛+=12001RRVVS2R1R+-+-SV0(not so bad …)6EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 11Effect of V0Son Op Amp Circuits (cont.)fRTo fix this, place a resistor in shunt with the C → then: ⎟⎟⎠⎞⎜⎜⎝⎛+=RRVvfS100Example: Integrator0vtWill continue to increase until op amp saturatesdtiCVvtS∫+=010010000011=+⎟⎠⎞⎜⎝⎛+=+=∫tCStSSvRCtVdtRVCVR+-+-SV0SV0-+CvC1iEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 12Noise7EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 13Determining Sensor ResolutionEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 14Minimum Detectable Signal (MDS)• Minimum Detectable Signal (MDS): Input signal level when the signal-to-noise ratio (SNR) is equal to unity• The sensor scale factor is governed by the sensor type• The effect of noise is best determined via analysis of the equivalent circuit for the systemSensor Scale FactorSensed SignalCircuit GainSensor NoiseCircuit Output NoiseSensor Signal Conditioning CircuitOutputIncludes desired output plus noise8EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 15NoiseEE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 16Noise• Noise: Random fluctuation of a given parameter I(t)• In addition, a noise waveform has a zero average valueAvg. value (e.g. could be DC current)IDI(t)t• We can’t handle noise at instantaneous times• But we can handle some of the averaged effects of random fluctuations by giving noise a power spectral density representation• Thus, represent noise by its mean-square value:()∫−=−=∞→TDTDdtIITIIi02221limDItIti −= )()(LetThen9EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 17Noise Spectral Density• We can plot the spectral density of this mean-square value:fiΔ2[units2/Hz]One-sided spectral density→ used in circuits→ measured by spectrum analyzersTwo-sided spectral density (1/2 the one-sided)Often used in systems courses2i= integrated mean-square noise spectral density over all frequencies (area under the curve)EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 18Circuit Noise Calculations• Deterministic:• Random:InputsOutputs)(ωjH)(ωjvi)(ωiS )(ωoS)(ωjvoLinearTime-Invariant SystemDeterministicRandom)(tvot)(ωjvoωoωπ2ωο⇔)(tSot)(ωjSoωωο⇔Mean square spectral density)()()(ωωωjvjHjvio=[])()()()()()(2*ωωωωωωiioSjHSjHjHS ==)()()(ωωωioSjHS =Root mean square amplitudesHow is it we can do this?10EE C245: Introduction to MEMS Design Lecture 27 C. Nguyen 11/27/07 19Handling Noise Deterministically• Can do this for noise in a tiny bandwidth (e.g., 1 Hz))(tvottAoωcos)(ωjSnωωοBioSSωoωB1~τWhy? Neither the amplitude nor the phase of a signal can change appreciably within a time period 1/B.[This is actually the principle by which oscillators work →oscillators are just noise going through a tiny bandwidth filter])(121fSfvn=ΔωωοBfSvn⋅= )(11Can approximate this by a sinusoidal voltage generator (especially for small B, say 1
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