1EE C245 – ME C218 Fall 2003 Lecture 23EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 23Capacitive Position Sensing:Electronic and Mechanical Noise2EE C245 – ME C218 Fall 2003 Lecture 23Today’s Lecture• Basic CMOS buffer amplifiers for position sensing• Electronic noise sources: transistors and resistors• Brownian noise• Signal-to-noise ratio• Reading:Gray, P.R., and Meyer, R. G., Analysis and Design of Analog Integrated Circuits, 3rdEd., 1993.23EE C245 – ME C218 Fall 2003 Lecture 23The Capacitive Half-Bridge, Revisited( ) ( )+−+=+−−−++xZZxZVxZZxZVVout)(ˆ)(ˆC+(x) = εoA/(go+ x)C-(x) = εoA/(go- x)==oinoutgxVVVˆvoutC+(x))cos(ˆ)( tVtv ω=+)cos(ˆ)( tVtv ω−=−1XCin≈ 0C-(x)4EE C245 – ME C218 Fall 2003 Lecture 23Precision Unity Gain via Feedback+-v+v-vout=Ad(v+- v-), Adis large+-v+=vinv-=voutVDDvout= Ad(vin– vout)+-vin35EE C245 – ME C218 Fall 2003 Lecture 23Basic CMOS Differential AmplifierSimplified analysis:v+= - v-= (v+- v-)/2v+v-ISSISS / 2VDDvoutABM1M2v-gm2vgs2aro2ro,cs→ ∞b+-vout+-vgs26EE C245 – ME C218 Fall 2003 Lecture 23Differential Gain Adv-gm2vgs2ro2+-vout+-vgs2vout= (-gm2vgs2)ro2 vout= -gm2 ro2v-Ad= 100Typical valuesgm2 ro2 = 20047EE C245 – ME C218 Fall 2003 Lecture 23Basic Unity-Gain BuffervinInput resistance Rin= ∞ ΩInput capacitance = Cin= ?ISSISS / 2VDDvoutM1M2+-+-Zin8EE C245 – ME C218 Fall 2003 Lecture 23Input Capacitance CinVingm1Vgs1ro1ro,cs→ ∞+-+-Cgs1Cgd1Vgs1Cinro2ro,cs→ ∞gm2Vgs2 =gm2Vds2gate-drain short on transistor M2R´Resistor R´=Therefore, Vgs1≈Input capacitance59EE C245 – ME C218 Fall 2003 Lecture 23Improved Unity-Gain BufferVDDvout+-+ISSISS / 2M1M2-ZinM3M4vinWeijie Yun, Ph.D. EECS, 1992Result:10EE C245 – ME C218 Fall 2003 Lecture 23Setting the DC BiasC+(x))cos(ˆ)( tVtv ω=+)cos(ˆ)( tVtv ω−=−C-(x)ISSISS / 2M1M2-M3M4VDDvoutANode A has no path to ground;so it’s called a “floating node”Solutions:611EE C245 – ME C218 Fall 2003 Lecture 23Electronic Noise Sources1. Thermal noise in resistorsgenerated by random motion of electrons or holeswhite spectral density (up to 10 THz)fTRkvBn∆= 42spectral density≅∆fvn/2Example: R = 1kΩ, T = 300 KIn a 1 MHz bandwidth,12EE C245 – ME C218 Fall 2003 Lecture 23Thermal Noise (Cont.)Thermal noise current: find Norton equivalentRv2Ri2Direction of arrow is arbitaryRTkfiBn4/2=∆Example: R = 1 kΩ, T = 300 K, BW = 1 MHz== BWRTkiBn42713EE C245 – ME C218 Fall 2003 Lecture 23Resistor Noise in MEMSInterconnect resistance Rintto capacitive position sensors> routing in polySi0 can lead to a high resistances,due to relatively high sheet resistance of this layer> inertial MEMS often have compliant suspensions à large number of squares in polySi1and significant contribution to RintvoutC+(x))cos(ˆ)( tVtv ω=+)cos(ˆ)( tVtv ω−=−1XC-(x)Rint+Rint-vR+214EE C245 – ME C218 Fall 2003 Lecture 23Flicker (1/f) NoiseNoise mechanism requires a DC current, in contrast to thermal noiseOrigin of 1/f noise in MOSFETs: surface statesffKIian∆=2Near DC, the noise current diverges!815EE C245 – ME C218 Fall 2003 Lecture 23MOSFET Noise SourcesWhen biased in saturation, (VDS> VDS,sat), the noise can be represented by an input noise voltage and an input noise currentOrigin of 1/f noise in MOSFETs: surface statesffWLCKfgTkvoxfmBin∆+∆=13242inverse dependence on gate area àlarger transistors have lower 1/f noisechannel resistance insaturation]1324[2222ffKIfgTkgCiDmBmgsin∆+∆≅ω(neglecting DC gate current and it’s shot noise)16EE C245 – ME C218 Fall 2003 Lecture 23MOSFET Noise Sources (Cont.)Equivalent MOSFET small-signal model with input-referred noise sources gmvgs+-CgsCgdroidiin2vin2Crossover frequency between thermal and flicker noise can range from 1 kHz to 1 MHz917EE C245 – ME C218 Fall 2003 Lecture 23Buffer Equivalent Input Noise Substitute noise voltage and current for each MOSFET in buffer and find the total equivalent noise at the inputK+∆+= fgggTkvmmmBin162113441flicker noise termsvoutC+(x))cos(ˆ)( tVtv ω=+)cos(ˆ)( tVtv ω−=−1XC-(x)Rint+Rint-vin2vR+2vR-218EE C245 – ME C218 Fall 2003 Lecture 23Mechanical (Brownian) NoiseImpinging molecules give rise to a Brownian noise force:fTbkfBn∆= 42b = (Mω1)/Q = damping coefficientNoise force applied to M-k-b system results in random Brownian motion with a frequency-dependent power spectrum:()( )[ ]212212/1/4+−∆=QfffffkTbkxBnImplications: f << f1 →1019EE C245 – ME C218 Fall 2003 Lecture 23Combining Noise Sources22212nnnvvv +=If noise sources are uncorrelated, then their powers can be added. For two voltage noise sources:22212nnrmsnvvvv +==For a sensor, it is convenient to refer all noise sources to the input, by scaling them appropriately. For a capacitive divider position sensor, the position noise due to electronicnoise at the input of the buffer is:=22,22ˆoenngxVv→20EE C245 – ME C218 Fall 2003 Lecture 23SNR and DR, DefinedSignal-to-noise ratio = SNR====rmsnsnsnsnsvvvvvvPPSNR,222log20log20log10log10Note 1: Pnoiseis calculated over a limited bandwidthDynamic range = DRNote 2: vn,rmsis taken as the minimum detectablesignal, in the absence of special coding orsignal processing===rmsnsnsnoisesvvvvPPDR,max,22max,max,log20log10log101121EE C245 – ME C218 Fall 2003 Lecture 23Signal and Noise WaveformsSinusoid with amplitudenormalized to 1Gaussian noise with rmslevel normalized to 1; notethat peak-peak noise level is occasionally as high as 6!22EE C245 – ME C218 Fall 2003 Lecture 23SNR = 1 and 10 for Gaussian