1、xLxIEFwBB362222BBddBdzzb: Iwith:xwdBLBFz(187,1)(187,2)Rectangular beam:123BBdb: I (187,3)I := Area moment of inertia of the beam B := Strain at the surface of the beamF := Force acting at the end of the beam dB := Thickness of beam w := Deflection of beam bB := Width of beam LB := Length of beam EB

2、:= Youngs modulus of beam187* W. Beitz, K.-H. Grote, Dubbel, Taschenbuch fr den Maschinenbau”*FdbELLxw:wBBBBB3304(187,4)03304wLdbEwFBBBBB(187,5)(188,1)(188,2)Rectangular:123BBdbI (188,3)I := Area moment of inertia of the beam EB := Youngs modulus of the beam FB := Elastic force of the beam at its en

3、d w := Deflection of the beam dB, bB, LB, RB := Thickness, width, length, and radius of the beam, respectively188More aria momentums of inertia are found in books like: W. Beitz, K.-H. Grote, Dubbel, Taschenbuch fr den Maschinenbau” orR.D. Blevins, Formulas for Natural Frequency and Mode Shape“, Kri

4、eger, Malaba, FL (1987)FIELLw:wBBB3300303wLIEwFBBBbBdB(187,3)Circular:RB44BRI Trapezoid shaped:dBebB,1bB,2212221213436,B,B,B,B,B,BBbbbbbbdI(188,4)212123,B,B,B,BBbbbbdewith:(188,5)xLdbEFBBBBB26xwdBLBFz(189,1)Rectangular beam:(187,3) :(189,2)189* W. Beitz, K.-H. Grote, Dubbel, Taschenbuch fr den Masch

5、inenbau”*xLIEFdBBBB2I := Area moment of inertia of the beam EB := Youngs modulus of the beam F := Force acting at the end of the beam B := Strain at the surface of the beam w := Deflection of the beam dB, bB, LB : Thickness, width, and length of the beam, respectivelyxwdBLBFzxLdbEFBBBBB26(189,2):x m

6、 x mw mxLxdbEFwBBBB3223(187,1), (187,3):LB=800 mbB=40 mdB=20 mEB=140 GPaF=1 mN190EB := Youngs modulus of the beam B := Strain at the surface of the beam F := Force acting at the end of the beamdB, bB, LB, w := Thickness, width, length, and deflection of the beam, respectivelyThe strain at the surfac

7、e of a beam clamped at one end and loaded at the other end in transversal direction is largest at the fixed end.Strain and deflection are not a functions of an initial stress of the beam. Rectangular beam:xLdbEFBBBBB26(189,2):Strain and deflection are proportional to the force (linear characteristic

8、 curve).xLxdbEFwBBBB3223(187,1), (187,3):191xwdBLBFzEB := Youngs modulus of the beam B := Strain at the surface of the beam F := Force acting at the end of the beamdB, bB, LB, w := Thickness, width, length, and deflection of the beam, respectivelyRectangular beam :Because of the transverse strain th

9、e beam gets narrower on the side with tensile stress and wider on the opposite side. (With the exception of the region next to the clamping)Cross-section of the beam:Without loadWith loadbBbB (1 B B)dB192xwdBLBFzxLdbEFBBBBB26(189,2):EB := Youngs modulus of the beam B := Poissons ratio of the beam B

10、:= Strain at the surface of the beam F := Force acting at the end of the beamdB, bB, LB, w: Thickness, width, length, and deflection of the beam, respectivelyOnly isotropic materials have been considered so far. *J.J. Wortman, R.A. Evans, Youngs Modulus, Shear Modulus, and Poissons Ratio in Silicon

11、and Germanium“, J. Appl. Phys. 36 (1965) 153 - 156Youngs modulus GPa of silicon and germanium as a function of the orientation in the (100)-plane*However, membranes from mono-crystalline silicon are anisotropic.115*J.J. Wortman, R.A. Evans, Youngs Modulus, Shear Modulus, and Poissons Ratio in Silico

12、n and Germanium“, J. Appl. Phys. 36 (1965) 153 - 156(100)-plane*(110)-plane1501005005010015015010050050100150116*J.J. Wortman, R.A. Evans, Youngs Modulus, Shear Modulus, and Poissons Ratio in Silicon and Germanium“, J. Appl. Phys. 36 (1965) 153 - 156(100)-plane*(110)-plane00,10,20,30,30,20,100,30,20

13、,10,10,20,3117(100)Freely stretched membraneStrain gauges from p-siliconThe edges of v-grooves in (100)-wafers are orientated in -direction.In -direction the piezo effect of p-silicon is largest. r ,elRr ,elRt ,elRt ,elR140 計算最大的電阻變化率及其電壓變化率，假定梁及電阻的分布如上頁所示Capacity C of a capacitor:CCr0eldACCel := El

14、ectrical capacity 0 := Absolute permittivity r := Relative permittivity AC := Inner area of capacitor plates dC := Distance of capacitor platesCapacity CPressure differenceExample pressure sensor:The capacitive measurement of the deflection of a membrane results in no linear signal.The characteristi

15、c curve of a membrane is much more complex than the one of a capacitor.147(147,1)Characteristic curve of a pressure sensor calculated by Finite ElementsCapacity CPressure difference* L. Rosengren, J. Sderkvist, L. Smith, ”Micromachined sensor structures with linear capacitive response”, Sensors and

16、Actuators A 31 (1992) 200 - 205Membrane touches the substrate*148* L. Rosengren, J. Sderkvist, L. Smith, ”Micromachined sensor structures with linear capacitive response”, Sensors and Actuators A 31 (1992) 200 - 205*Top end of the comb structure is conductive.149Characteristic curve of a pressure se

17、nsor calculated by Finite ElementsBending moments are dominating.w0 dM(82,1)dMw082r(82,2)Circular plate bulged up by a pressure difference: 22201MRrwrww(r) := Deflection of membrane w0 := Deflection of the center of the membrane dM := Thickness of membrane 2 RM := Diameter of membrane2 RMw0Mechanica

18、l stress is dominating.w(r) := Deflection of membrane w0 := Deflection of the center of the membrane dM := Thickness of membrane 2 RM := Diameter of membranew0 dM(84,1)84(84,2)Circular membrane bulged up by a pressure difference: 2201MRrwrw114RM, M, dM, EM, w0, 0 := Radius, Poissons ratio, thickness

19、, Youngs modulus, central deflection, and initial stress of the membrane, respectively aM := Length of a square membrane p := Pressure drop over the membraneThin, circular,exactlyThin, square,exactlyThick, circular, without 0, exactlyThick, square,without 0, exactlyIn general, circular,rough approxi

20、mation22200222201105641344MMMMMMMMMERwERdRwdp2MMM2M2002MM0233. 0793. 0026. 1ERw32Rdw4p2MMM2M2002MM021. 070. 01Eaw33. 2adw6 .13ppEdRwwERdpMMMMMMMM2340024311631316pEdawwEadpMMMMMMMM2340024316611660022202,122eqeegW V gAFVggFkzAFggzggVkkgWAqCVVVgVoltage increaseGap decreaseForce increase2320223()22netne

21、tnetAVkgAVFk gggFAVdFdgk dggg Range of stability: examine net (attractive) force on plateIf we increase the gap by dg,the increment 0 or the plate collapsesnetdF2320222023000300202231122827PIPIPInetPIPIPIPInetPIPIPIPIPIPIPIPIAVkgAVFk gggAVAVFggggggggggggkgVASolve for point at which plate goes unstab

22、le:Substitution for k leads to: coscosEquation of Motion: using phasor concepts: eqeqeqeqeqeqF tFtx txtm xC xk xF tkFj m xxC xj coscosImpedance looking in:1 1/ xxxxxxv tvti tItvj lrij CCvj l iir ijParameter Relationships by anology: in the Current Analogy1 eqxeqxeqxFvmlCrxikCMechanical-to-electrical

23、 correspondence in the current analogy:Converting to full phasor form:F= jeqeqeqeqeqeqkFj m xxC xjkj x mj xCj xj1200j=1eqkxFjQ112200020001111 ,eqeqeqeqeqeqeqeqeqeqeqeqeqeqmCxjjjFkkkkQkmkkQQmCCC Reading: Senturia, Chpt. 6, Chpt. 14Lecture Topics:Input ModelingForce-to-Velocity Equiv. CircuitInput Equ

24、ivalent Circuit.Current ModelingOutput Current Into Ground Input CurrentComplete Electrical-Port Equiv. CircuitImpedance & Transfer Functions coscosEquation of Motion: using phasor concepts: eqeqeqeqeqeqF tFtx txtm xC xk xF tkFj m xxC xj coscosImpedance looking in:1 1/ xxxxxxv tvti tItvj lrij CC

25、vj l iir ijParameter Relationships by anology: in the Current Analogy1 eqxeqxeqxFvmlCrxikCThe relationship between input voltage v1 and force Fd1:When displacement x is the mechanical output variable:When velocity v is the mechanical output variable:111dpCFVvx 20221001/dX sFsk sQ s 202211001/ddv ssX

26、 ssFsFsk sQ sCombine the previous lumped LCR mechanical equivalent circuit with a circuit modeling the capacitive transducer circuit model for oltage-to-velocityA TransducerConverts energy from one domain (e. g., electrical) to another (e. g., mechanical)has at least two portsis not generally linear

27、, but is virtually linear when operated with small signals (i.e., small displacements)For physical consistency, use a transformer equivalent circuit to model the energy conversion from the electrical domain to mechanical domain2121 010 -eeffE2=Fd1, e1=v1, just need 1:From the matrix: e2=e1111dpCFVvx

28、 11pCVxWhen the mass moves with time-dependent displacement x(t), the electrode-to-mass capacitors C1(x,t) and C2(x,t) vary with timeThis generates an output current: 2222222222222,In phasor form: I IpppppdqVCqCViCVdtttdVtdCx titCx tVtdtdtdCCxVtViVVdtxtCjVj xxCjj Vxx 22o290 phase log t t when x= -1p

29、pCCIjj VxVvxxI 2112221222221, pfffffIfvIvCVx Again, model with a transformer: 111111111111111111111111 111111 1Get I: ppppppjdV tdCx ti tCx tV tdtdtdVCxV tvViCvvdtxtCCIjCj VVj xVj xxxCCj CVj Vxj VxxxCvVIjj CVj Vxx 111 Feedthrough Motional Current CurrentC1DC: xresdpFxVVkk d111QFonance: x=jkpCQVvjkx

30、21101 101201 101o11Thus: resonance : 90 phase-shifted In phase /V from V This is a cpCQIjjCVjVVxjkQjCVVke01112221011apacity This is an effective in shunt /k input resistance seen looking into Electrode 1Motional Resistance:The equivalent ckf. better get txxmVkbRRIQQhis right!Static electrode-to-mass

31、 overlap capacitance1111oeppCCVVxd2222oeppCCVVxd212112222221221122112e 0211From our transformer model: 0 -1/12112111111iixixeexixxxeeeeeeeffffffveeeFzzfffixlzj lrjj C 22111xeexrjC What is the impedance seen looking into port 1 with port 2 shorted to ground?22222222111ixxixxixeeeexvlrzj lrjij CjC Wha

32、t is the impedance seen looking into port 2 with port 1 shorted to ground?Note: there are not the same as Lx1,Cx1,1/2Rx1!222011112112012121212121212121111121 1eieieeiexxxxxeeeexxxeexixxxxxxeexveivzjlrxjCellijjlrCCvjCjlrrjCR What is the transconductance from port 1 to port 2 with port 2 shorted to gr

33、ound? 0220000/0:00:1:0sQsssQssjjs 121120121212121212121211xxeeeexxxeexixxxxxxeelLijj LrCCvj Cj lrrj CR 02211111xxixxxxxxxxxxxsCisCsvs L CsC RRsLRsssCL CL Separate freq. response f magnitude: 0200002200/111,/xxxxxxixxsQLRiQsL CRLQvRRssQHolds for the symmetrical case, where port 1 and port 2 are ident

34、ical222wherexeexxemLCkbRBelow: plots of resonance electrical and mechanical signals vs. time, showing the phasings between them 01011000212121201ddeiieeeieexQxssFkQxFvssvkiQixssvkQsRm 0012120121 LixLixLixLviRssvRRvRvRssvRRTo convert velocity to a voltage, use a resistive load2x22Since this structure

35、 has completely symmetrical I/O port: C exxeebmRLkTo convert velocity to a voltage, use a resitive load00what resonance: (to simplify the analysis) resonanceThen, generate to off resonance: , where DixDxDixDxDvRvRRvRRs QQQvRRRR 0220022200111/,1/DDxxDxDixxxxDxxxDxxxxxDxDDDxDxDixDxDxxxRsvsR CLRsRRvsR

36、Cs L CsR CRsLRsssCLL CRRssQLvRRRss QRRRRvRRRRssQssLL C To convert velocity to a voltage, use a resistive loadSince this structure has completely symmetrical I/O ports:000 xxxDxxDxLLRRQQRRRLQBrute force approach: 000111ixxxvsCsvRsLsCsC 0022022000200/11/111/1/11/ 1/1/1/ Q,1/xxDxxxxxixDxxxxxDxDxDxDxxxD

37、xxxDxxDxxxxxxxxsCvsCCCssCR CL CvCCsR Cs L CssCCCCCCCCCL CCCL CCCRCCLRssQQCCLL CRLQ DTo sense position (i.e., displacement), use a capacitive loadBrute force approach: 2002200/1/xDixDvCCsvCCssQTo shense position (i.e., displacement), use a capacitive load 000Note: Can we similar shut-cut to the R cas

38、eGet DC responce Cs dominateThen:1DC Gain,ivssQQvs2x22Since this structure has completely symmetrical I/O port: C exxeebmRLkTo convert velocity to a voltage, use a resitive load00what resonance: (to simplify the analysis) resonanceThen, generate to off resonance: , where DixDxDixDxDvRvRRvRRs QQQvRRRR01pxpRRC00N o w , W e g et: ap p ro x im ately1,1DixDpvRssQsvRR In general, the sensor output must be connected to the inputs of further signal conditioning circuits input Ri of these circuits can load RDThese change w/ hook-up not goodProblem: need a sensing circuit that is immu