# Berkeley ELENG C247B - Capacitive Position Sensing: Electronic and Mechanical Noise (11 pages)

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## Capacitive Position Sensing: Electronic and Mechanical Noise

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- Pages:
- 11
- School:
- University of California, Berkeley
- Course:
- Eleng C247b - Introduction to MEMS Design

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EE C245 ME C218 Introduction to MEMS Design Fall 2003 Roger Howe and Thara Srinivasan Lecture 23 Capacitive Position Sensing Electronic and Mechanical Noise EE C245 ME C218 Fall 2003 Lecture 23 Today 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 3rd Ed 1993 EE C245 ME C218 Fall 2003 Lecture 23 2 1 The Capacitive Half Bridge Revisited v t V cos t C x C x oA go x C x oA go x Cin 0 1X vout C x v t V cos t Z x Z x Vout V V Z Z x Z Z x x Vout Vin V go EE C245 ME C218 Fall 2003 Lecture 23 3 Precision Unity Gain via Feedback VDD v vout Ad v v vv vin vin Ad is large v vout vout Ad vin vout EE C245 ME C218 Fall 2003 Lecture 23 4 2 Basic CMOS Differential Amplifier VDD Simplified analysis B v v v v 2 ISS 2 vout M1 v M2 b v ro cs vout A ro2 ISS g m2vgs2 vgs2 v a EE C245 ME C218 Fall 2003 Lecture 23 5 Differential Gain Ad vout gm2vgs2 ro2 vout ro2 g m2vgs2 vgs2 v vout gm2 ro2 v Typical values gm2 ro2 200 Ad 100 EE C245 ME C218 Fall 2003 Lecture 23 6 3 Basic Unity Gain Buffer VDD ISS 2 Zin M1 M2 vout Input resistance Rin Input capacitance Cin vin ISS EE C245 ME C218 Fall 2003 Lecture 23 7 Input Capacitance Cin Cin Vin Vgs1 ro cs Cgd1 Cgs1 ro1 ro2 ro cs R g m1Vgs1 gate drain short on transistor M2 g m2Vgs2 gm2Vds2 Resistor R Therefore Vgs1 Input capacitance EE C245 ME C218 Fall 2003 Lecture 23 8 4 Improved Unity Gain Buffer VDD Weijie Yun Ph D EECS 1992 ISS 2 M3 M4 Zin vin M1 M2 vout ISS Result EE C245 ME C218 Fall 2003 Lecture 23 VDD 9 Setting the DC Bias ISS 2 v t V cos t M3 Node A has no path to ground so it s called a floating node M4 C x vout A M1 M2 Solutions C x ISS v t V cos t EE C245 ME C218 Fall 2003 Lecture 23 10 5 Electronic Noise Sources 1 Thermal noise in resistors generated by random motion of electrons or holes white spectral density up to 10 THz vn 2 4k BTR f spectral density Example R 1k T 300 K vn 2 In a 1 MHz bandwidth f EE C245 ME C218 Fall 2003 Lecture 23 11 Thermal Noise Cont Thermal noise current find Norton equivalent Direction of arrow is arbitary R v2 i2 R in 2 f 4 k BT R Example R 1 k T 300 K BW 1 MHz in 2 4k B T BW R EE C245 ME C218 Fall 2003 Lecture 23 12 6 Resistor Noise in MEMS Interconnect resistance Rint to 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 large number of squares in polySi1 and significant contribution to Rint v t V cos t vR 2 Rint C x 1X Rint vout C x v t V cos t EE C245 ME C218 Fall 2003 Lecture 23 13 Flicker 1 f Noise Noise mechanism requires a DC current in contrast to thermal noise Origin of 1 f noise in MOSFETs surface states in 2 KI a f f Near DC the noise current diverges EE C245 ME C218 Fall 2003 Lecture 23 14 7 MOSFET Noise Sources When biased in saturation VDS VDS sat the noise can be represented by an input noise voltage and an input noise current Origin of 1 f noise in MOSFETs surface states vin 2 2 1 Kf f 4k BT 3 gm WLCox inverse dependence on gate area larger transistors have lower 1 f noise channel resistance in saturation iin 2 f f 2C gs 2 4 k T 2 1 f KI D f neglecting DC gate gm2 B 3 gm f current and it s shot noise EE C245 ME C218 Fall 2003 Lecture 23 15 MOSFET Noise Sources Cont Equivalent MOSFET small signal model with inputreferred noise sources vin2 Cgd id iin2 Cgs g mvgs ro Crossover frequency between thermal and flicker noise can range from 1 kHz to 1 MHz EE C245 ME C218 Fall 2003 Lecture 23 16 8 Buffer Equivalent Input Noise Substitute noise voltage and current for each MOSFET in buffer and find the total equivalent noise at the input 4 1 1 vin2 4 k B T 3 g m 1 g m6 g m1 f K flicker noise terms v t V cos t C x vR 2 Rint vin2 vout 1X vR 2 Rint C x v t V cos t EE C245 ME C218 Fall 2003 Lecture 23 17 Mechanical Brownian Noise Impinging molecules give rise to a Brownian noise force fn 4k BTb f 2 b M 1 Q damping coefficient Noise force applied to M k b system results in random Brownian motion with a frequency dependent power spectrum xn 2 4k BTb k f 1 f f 2 2 1 f Qf 1 2 Implications f f1 EE C245 ME C218 Fall 2003 Lecture 23 18 9 Combining Noise Sources If noise sources are uncorrelated then their powers can be added For two voltage noise sources vn vn1 vn 2 2 vn 2 vrms 2 2 v n1 v n 2 2 2 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 electronic noise at the input of the buffer is vn 2 x n e 2 2 V g o2 EE C245 ME C218 Fall 2003 Lecture 23 19 SNR and DR Defined Signal to noise ratio SNR v2 P SNR 10 log s 10 log s2 vn Pn 20 log vs vn 2 vs 20 log v n rms Note 1 Pnoise is calculated over a limited bandwidth Note 2 vn rms is taken as the minimum detectable signal in the absence of special coding or signal processing Dynamic range DR v2 P 20 log vs max DR 10 log s max 10 log s max 2 v vn Pnoise n rms EE C245 ME C218 Fall 2003 Lecture 23 20 10 Signal and Noise Waveforms Sinusoid with amplitude normalized to 1 Gaussian noise with rms level normalized to 1 note that peak peak noise level is occasionally as high as 6 EE C245 ME C218 Fall 2003 Lecture 23 21 SNR 1 and 10 for Gaussian Noise EE C245 ME C218 Fall 2003 Lecture 23 22 11

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