EE C45 ME C18 Introducton to MEMS Desgn Fall 007 Prof. Clark T.C. Nguyen Dept. of Electrcal Engneerng & Computer Scences Unversty of Calforna at Berkeley Berkeley, CA 9470 Lecture 8: Mnmum Detectable Sgnal (MDS) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 1 Lecture Outlne Readng: Sentura Chpt. 16, 19 Lecture Topcs: Determnng Sensor Resoluton Nose Nose Sources Equvalent InputReferred Nose Sources Example: Gyro MDS Calculaton Fnal Exam Informaton Wrap Up EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 1
Inputs Random Determnstc: Random: Crcut Nose Calculatons Determnstc Outputs v ( jω) v o ( jω) H ( jω) S (ω) S o (ω) Lnear TmeInvarant System v o (t) S o (t) v ( jω) H ( jω) v ( jω) o = EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 3 π ω o t t v o ( jω) ω ο S o ( jω) ω ο Mean square spectral densty * [ H ( jω) H ( jω) ] S ( ω) H ( jω) S ( ω) S ( ω) = = o S ( ω) = H ( jω) S ( ω) o Root mean square ampltudes How s t we can do ths? ω ω Handlng Nose Determnstcally Can do ths for nose n a tny bandwdth (e.g., 1 Hz) 1 f v n = S1( Δf ω ο S n ( jω) ω ο B ω ) ω v n 1 = S f ) B S S [Ths s actually the prncple by whch oscllators work oscllators are just nose gong through a tny bandwdth flter] o 1 ( v o (t) A cosωot EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 4 ω o B ω Can approxmate ths by a snusodal voltage generator (especally for small B, say 1 Hz) τ ~ 1 B Why? Nether the ampltude nor the phase of a sgnal can change apprecably wthn a tme perod 1/B. t
Systematc Nose Calculaton Procedure H ( j ω ) H5( jω) General Crcut Wth Several Nose Sources v n n1 v n3 n5 n4 v n6 v on H1( jω) Assume nose sources are uncorrelated 1. For n1, replace w/ a determnstc source of value n1 n1 = Δf (1Hz) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 5 Systematc Nose Calculaton Procedure. Calculate von 1( ω) = n1( ω) H ( jω) (treatng t lke a determnstc sgnal) 3. Determne von1 = n1 H ( jω) 4. Repeat for each nose source: n1, v n, v n3 5. Add nose power (mean square values) v ontot = v on1 v on v on3 v on4 L v ontot = v on1 von von3 von4 L Total rms value EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 6 3
Nose Sources EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 7 Thermal Nose Thermal Nose n Electroncs: (Johnson nose, Nyqust nose) Produced as a result of the thermally excted random moton of free e s n a conductng medum Path of e s randomly orented due to collsons Thermal Nose n Mechancs: (Brownan moton nose) Thermal nose s assocated wth all dsspatve processes that couple to the thermal doman Any dampng generates thermal nose, ncludng gas dampng, nternal losses, etc. Propertes: Thermal nose s whte (.e., constant w/ frequency) Proportonal to temperature Not assocated wth current Present n any real physcal resstor EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 8 4
Crcut Representaton of Thermal Nose Thermal Nose can be shown to be represented by a seres voltage generator v R or a shunt current generator R R actual R R noseless or R v R noseless Note: These are onesded meansquare spectral denstes! To make them sded, must dvde by. kt R = 4 Δf R 4kT = 1.66x10 v R = 4 ktr Δf 0 V C where and where these are spectral denstes. EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 9 Nose n Capactors and Inductors? Resstors generate thermal nose Capactors and nductors are noseless why? L C v v Can oscllate forever t Now, add a resstor: L C R v v Decays to zero t But ths volates the laws of thermodynamcs, whch requre that thngs be n constant moton at fnte temperature Need to add a forcng functon, lke a nose voltage to keep the moton gong and ths nose source s assocated wth R EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 10 v R 5
Why s v R = 4kTRΔf Why 4kTR? (a heurstc argument) The Equpartton Theorem of Statstcal Thermodynamcs says that there s a mean energy (1/)kT assocated w/ each degree of freedom n a gven system An electronc crcut possesses two degrees of freedom: Current,, and voltage, v Thus, we can wrte: 1 L 1 = kbt, 1 Cv = Smlar expressons can be wrtten for mechancal systems For example: for dsplacement, x Sprng constant 1 k x = Energy EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 11 1 k B T 1 k B T Why 4kTR? (cont) Why s v R = 4kTRΔf? (a heurstc argument) Consder an RC crcut: R C v R R C v C EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 1 6
Why 4kTR? (cont) v R R C v C EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 13 Shot Nose Assocated wth drect current flow n dodes and bpolar juncton transstors Arses from the random nature by whch e s and h s surmount the potental barrer at a pn juncton The DC current n a forwardbased dode s composed of h s from the p regon and e s from the nregon that have suffcent energy to overcome the potental barrer at the juncton nose process should be proportonal to DC current Attrbutes: Related to DC current over a barrer Independent of temperature Whte (.e., const. w/ frequency) Nose power ~ I D & bandwdth h p h EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 14 I D V D h e e n n = qi D Δf Charge on an e (=1.6x10 19 C) pnjuncton DC Current 7
Flcker (1/f) Nose In general, assocated w/ random trappng & release of carrers from slow states Tme constant assocated wth ths process gves rse to a nose sgnal w/ energy concentrated at low frequences Often, get a meansquare nose spectral densty that looks lke ths: n Δf 1 ~ f ω b 1/f Nose ω Shot Nose 1/f Nose Corner Frequency n Δf = qi D I K f a D b I D = DC current K = const. for a partcular devce a = 0.5 b ~ 1 EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 15 Example: Typcal Nose Numbers Hookup the crcut below and make some measurements R v R 1kΩ C vn vo 1pF Low Nose Amplfer 100x Measure w/ AC voltmeter Measure w/ spectrum analyzer EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 16 8
Example: Typcal Nose Numbers Hookup the crcut below and make some measurements R v R 1kΩ C vn vo 1pF Low Nose Amplfer 100x Measure w/ AC voltmeter Measure w/ spectrum analyzer EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 17 Back to Determnng Sensor Resoluton EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 18 9
MEMSBased Tunng Fork Gyroscope Electrodes Ω r z Tunng Electrodes Drve Voltage Sgnal Tunng Electrodes Drve Electrode () Output Current () Output Current Electrodes Drve Drve Oscllaton Sustanng Amplfer Dfferental TransR Amplfer [Zaman, Ayaz, et al, MEMS 06] EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 19 Drve Axs Equvalent Crcut 180 o 1:η e l x c x r x η e :1 o x d C o1 180 o C o Drve Voltage Sgnal Drve Oscllaton Sustanng Amplfer Generates drve dsplacement velocty x d to whch the Corols force s proportonal To Amplfer (for synchronzaton) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 0 10
Gyro Crcut Electrodes Ω r z Tunng Electrodes () Output Current Tunng Electrodes Electrodes Drve Electrode Drve () Output Current [Zaman, Ayaz, et al, MEMS 06] Dfferental TransR Amplfer EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 1 Drveto Transfer Functon x& d x& d = ω x d d Ampltude Drve/ Response Spectra: Drve Response Response Drve Mode Ω Drven Velocty f o (@ T 1 ) ω x& s = ω x s s x& s Mode Velocty EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 11
r F c r = ma c Gyro Readout Equvalent Crcut (for a sngle tne) Nose Sources r r = m ( x & Ω) d f F c l x c x f rx r x η e :1 o v a x 0 a C p v Gyro Element Output Crcut Sgnal Condtonng Crcut (Transresstance Amplfer) Easest to analyze f all nose sources are summed at a common node EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 3 Mnmum Detectable Sgnal (MDS) Mnmum Detectable Sgnal (MDS): Input sgnal level when the sgnaltonose rato (SNR) s equal to unty d Sgnal Sensor Scale Factor Sensor Nose Sensor Crcut Gan Crcut Output Nose Sgnal Condtonng Crcut Output Includes desred output plus nose The sensor scale factor s governed by the sensor type The effect of nose s best determned va analyss of the equvalent crcut for the system EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 4 1
Move Nose Sources to a Common Pont Move nose sources so that all sum at the nput to the amplfer crcut (.e., at the output of the sense element) Then, can compare the output of the sensed sgnal drectly to the nose at ths node to get the MDS d Sgnal Sensor Scale Factor Crcut Gan Output Sensor Nose Sensor Crcut Input Referred Nose Sgnal Condtonng Crcut Includes desred output plus nose EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 5 r F c r = ma c Gyro Readout Equvalent Crcut (for a sngle tne) r r = m ( x & Ω) d Nose Sources Noseless l x c x f rx r x η e :1 o F x v 0 c eq C p Gyro Element Output Crcut Sgnal Condtonng Crcut (Transresstance Amplfer) Here, and eq are equvalent nputreferred voltage and current nose sources EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 6 13
Move Nose Sources to a Common Pont Move nose sources so that all sum at the nput to the amplfer crcut (.e., at the output of the sense element) Then, can compare the output of the sensed sgnal drectly to the nose at ths node to get the MDS d Sgnal Sensor Scale Factor Crcut Gan Output Sensor Nose Sensor How can we get ths? Crcut Input Referred Nose Sgnal Condtonng Crcut Includes desred output plus nose EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 7 Equvalent InputReferred Voltage and Current Nose Sources EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 8 14
Equvalent Input v, Nose Generators Take a nosy port network and represent t by a noseless network wth nput v and nose generators that generate the same total output nose Nosy Network eq Noseless Remarks: 1. Works for lnear tmenvarant networks. and eq are generally correlated (snce they are derved from the same sources) 3. In many practcal crcuts, one of and eq domnates, whch removes the need to address correlaton 4. If correlaton s mportant easer to return to orgnal network wth nternal nose sources EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 9 Calculaton of a) To get for a twoport: and eq Case I Case II Nosy Network v 0I eq Noseless v 0II v 0I 1) Short nput, fnd (or ) 0I ) For eq. network, short nput, fnd v 0II (or 0II ) v = v 0I 0II ( v ) f eq f ( ) = 0I 0II 3) Set solve for (or ) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 30 15
eq Calculaton of and (cont) b) To get for a port: eq Nosy Network v 0I eq Noseless v 0II v 0I 0I 1) Open nput, fnd (or ) ) Open nput for eq. crcut, fnd v0ii (or 0II ) 3) Set solve for (or ) v = v ( ) 0I 0II eq eq 0I = 0II ( eq ) Once the equvalent nputreferred nose generators are found, nose calculatons become straghtforward as long as the nose generators can be treated as uncorrelated EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 31 Cases Where Correlaton Is Not Important There are two common cases where correlaton can be gnored: 1. Source resstance R s s small compared to nput resstance R.e., voltage source nput. Source resstance R s s large compared to nput resstance R.e., current source nput 1) R S = small (deally = 0 for an deal voltage source): R S v S eq Noseless eq Current shorted out! For R S = small, eq can be neglected only s mportant! (Thus, we need not deal wth correlaton) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 3 16
Cases Where Correlaton Is Not Important ) R S = large (Ideally = for an deal current source) S R S eq Noseless Voltage R n effectvely opened out! v Rn R = veq n = 0! For R S = large, can be neglected! only eq s mportant! ( and agan, we need not deal wth correlaton) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 33 Example: TransR Amplfer Nose f Inputreferred current nose: Case I v a a () v () R v oi av Case II eq () v () R av v oii EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 34 17
Example: TransR Amplfer Nose f Inputreferred current nose: Case I v a a () v () R v oi av Case II eq () v () R av v oii EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 35 Example: TransR Amplfer Nose (cont) f Inputreferred voltage nose: Case I v a a () v () R v oi av Case II eq () v () R av v oii EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 36 18
Example: TransR Amplfer Nose (cont) To summarze, for a transresstance amplfer, the equvalent nputreferred current and voltage nose generators are gven by: eq Noseless v 0 eq va = a f = v a EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 37 Back to Gyro Nose & MDS EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 38 19
r F c r = ma l x c c x Example: Gyro MDS Calculaton r r = m ( x & Ω) f rx r x d η e :1 o F x& 0 c s eq C p Noseless v The gyro sense presents a large effectve source mpedance Currents are the mportant varable; voltages are opened out Must compare o wth the total current nose eqtot gong nto the amplfer crcut EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 39 r F c Example: Gyro MDS Calculaton (cont) r = ma l x c r r = m ( x & Ω) c x f rx r x d η e :1 o F x& 0 c s eq C p Noseless v Frst, fnd the rotaton to o transfer functon: EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 40 0
Example: Gyro MDS Calculaton (cont) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 41 r F c Example: Gyro MDS Calculaton (cont) r = ma l x c r r = m ( x & Ω) c x f rx r x d η e :1 o F x& 0 c s eq C p Noseless v Now, fnd the eqtot enterng the amplfer nput: EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 4 1
Example: Gyro MDS Calculaton (cont) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 43 LF356 Op Amp Data Sheet EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 44
Example ARW Calculaton Example Desgn: Sensor Element: m = (100μm)(100μm)(0μm)(300kg/m 3 ) = 4.6x10 10 kg ω s = π(15khz) ω d = π(10khz) k s = ω s m = 4.09 N/m x d = 0 μm Q s = 50,000 V P = 5V h = 0 μm d = 1 μm Sensng Crcutry: = 100kΩ a = 0.01 pa/ Hz v a = 1 nv/ Hz Tunng Electrodes Electrodes Electrodes z Tunng Electrodes Drve Electrode EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 45 Ω r Drve Example ARW Calculaton (cont) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 46 3
Example ARW Calculaton (cont) EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 47 What f ω d = ω s? EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 48 4
Wrap Up Go through fnal exam handout Sgn up for project brefs Schedules posted on my door Choces: Wednesday, Dec. 1; Sunday, Dec. 16, Wednesday, Dec. 19 Upcomng courses n MEMS: BoMEMS (Mcrofludcs): already a few of these from other departments, but ths wll be an EECS verson RF MEMS: wll show up as an EE 90 course Undergraduate MEMS course: a work n progress EE C45: Introducton to MEMS Desgn Lecture 8 C. Nguyen 11/9/07 49 5