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:=Youngs modulus o
2、f 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 end w:=Deflection of the beam dB,b
3、B,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“,Krieger,Malaba,FL(1987)FIELLw:wBBB3300303wLIEwFBBBbBdB(1
4、87,3)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 Maschinenbau”*xLIEFdBBBB2I:=Area moment of inertia of the beam EB:=Y
5、oungs 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 x mw mxLxdbEFwBBBB3223(187,1),(187,3):LB=800 mbB=40 mdB=20 mEB=140 GPaF=1 mN190
6、EB:=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 surface of a beam clamped at one end and loaded at the other end in transversal direction is largest a
7、t 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 curve).xLxdbEFwBBBB3223(187,1),(187,3):191xwdBLBFzEB:=Youngs modulus of the beam B:=Strain at the
8、 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 the beam gets narrower on the side with tensile stress and wider on the opposite side.(With the exception of the regi
9、on 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:=Strain at the surface of the beam F:=Force acting at the end of the beamdB,bB,LB,w:Thickness,width,length,and deflection
10、 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 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*How
11、ever,membranes from mono-crystalline silicon are anisotropic.115*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)-plane1501005005010015015010050050100150116*J.J.Wortman,R.A.Evans,Youngs Modulus,Shear Modulus,
12、and Poissons Ratio in Silicon and Germanium“,J.Appl.Phys.36(1965)153-156(100)-plane*(110)-plane00,10,20,30,30,20,100,30,20,10,10,20,3117(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-silic
13、on is largest.r,elRr,elRt,elRt,elR140 计算最大的电阻变化率及其电压变化率，假定梁及电阻的分布如上页所示Capacity C of a capacitor:CCr0eldACCel:=Electrical capacity 0:=Absolute permittivity r:=Relative permittivity AC:=Inner area of capacitor plates dC:=Distance of capacitor platesCapacity CPressure differenceExample pressure sensor:
14、The capacitive measurement of the deflection of a membrane results in no linear signal.The characteristic 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.
15、Sderkvist,L.Smith,”Micromachined sensor structures with linear capacitive response”,Sensors and 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
16、end of the comb structure is conductive.149Characteristic curve of a pressure sensor 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 membr
17、ane dM:=Thickness of membrane 2 RM:=Diameter of membrane2 RMw0Mechanical 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:2201MRrw
18、rw114RM,M,dM,EM,w0,0:=Radius,Poissons ratio,thickness,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,exa
19、ctlyIn general,circular,rough approximation22200222201105641344MMMMMMMMMERwERdRwdp2MMM2M2002MM0233.0793.0026.1ERw32Rdw4p2MMM2M2002MM021.070.01Eaw33.2adw6.13ppEdRwwERdpMMMMMMMM2340024311631316pEdawwEadpMMMMMMMM2340024316611660022202,122eqeegW V gAFVggFkzAFggzggVkkgWAqCVVVgVoltage increaseGap decrease
20、Force increase2320223()22netnetnetAVkgAVFk 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
21、 at which plate goes unstable: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 eqxeqxeqxFvmlCrxikCMe
22、chanical-to-electrical 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.C
23、ircuitInput Equivalent 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 lr
24、ij CCvj 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
25、 ssX 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,bu
26、t 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 11pCVxWhe
27、n 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=-1ppCCIjj VxVvxxI 2
28、112221222221,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 21101 101201 101o11Thu
29、s: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 overlap capacitance1111oepp
30、CCVVxd2222oeppCCVVxd212112222221221122112e 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 What is the impedance seen looki
31、ng 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 ground?0220000/0:00:1:0sQsssQssj
32、js 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 identical222wherexeexxemLCkbRBelow:plot
33、s 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 has completely symmetrical I/O port:C
34、 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 Cs L CsR CRsLRsssCLL CRRssQLvRRRss QRRRRvRRRR
35、ssQssLL 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/xxDxxxxxixDxxxxxDxDxDxDxxxDxxxDxxDxxxxxxxxsCvsCCCssCR CL CvCCsR Cs L CssCCCC
36、CCCCCL 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 caseGet DC responce Cs dominateThen:1DC Gain,ivssQQvs2x22Sin
37、ce 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
38、 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 immune to parasitics or loadingSolu:use op amps.The virtual ground provided by the ideal op a
39、mp eliminates the parasitic capacitance Cp and RiThe Zero output resistance of the(ideal)op amp can drive virtually anythingLDp Integration yields displacementTo maximize gain,minimize f Problem:parastic capacitane C/DC Gain:1/Remedy:suppress C Via use of op ampsDPipbxDPipbxDPipbCCCCCCCCCCCTo sense
40、position(i.e.,displacement),use a capacitive load 000/1,1/xDixDvCCssQQvCCs 0002222111ixsxsvvviRsssCRsCvR Cs The virtual ground provided by the ideal op amp eliminates the parasitic capacitance Cp1211120121212121222Issue:Parasitic capacitance:As before,Cp reduces gainsoln:use op amp!ppppppDDppV CV CV
41、 CCCCvvvVVCCCCCCCCVVCCC Example:ADXL-50Includes capacitance from interconnects,bond pads,and Cgs of the op ampBootstrap the ground lines around the interconnect and bond padsNo voltage across CpIts effectively not there!00000000000i000000p111Get Z:1111 112Ex:100,C220101Not negligible compared/iiiiiipipipipiieffpieffvAvAvvAvvvAA vvAvAivvsCvsCvsCiAACvZCCiAsApApFCfFfof ADXL-SO C100fF012121212000121 A seenningly perfect differential sensor/amplifier output!but only when op amp is idealppppFFpFiiivsCvsCv s CCCCvivsCCvCCvC Can drive next stages Ri w/o interference to transfer function!
