EE C245 – ME C218Introduction to MEMS DesignFall 2007Fall 2007Prof Clark TC NguyenProf. Clark T.-C. NguyenDept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720yLt 24 It/OttMdliEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 1Lecture 24: Input/OutputModelingLecture Outline• Reading: Senturia, Chpt. 6, Chpt. 14gpp• Lecture Topics:ª Electrostatic Comb-Drive(1stOrder Analysis(1stOrder Analysis( 2ndOrder Analysisª Input/Output Modelingppgª I/O Equivalent Circuit Models( Electromechanical Coupling(Mechanical Coupling(Mechanical Couplingª Detection Circuits( Position Sensing( Velocity Sensing( Operational AmplifiersEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 2Comb-Drive Force Equation (2ndPass)• In our 1stpass, we neglectedª Fringing fieldsªPllll i b d ªParallel-plate capacitance between stator and rotorª Capacitance to the substrate•All of these capacitors must be included when evaluating the All of these capacitors must be included when evaluating the energy expression!StatorRotorGround EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 3Ground PlaneComb-Drive Force With Ground Plane Correction• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures d h d l difi h i i and the ground plane → modifies the capacitive energy[Gary Fedder Ph D EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 4[Gary Fedder, Ph.D., UC Berkeley, 1994]Capacitance Expressions• Case: Vr= VP= 0V• Cspdepends on whether or not ppfingers are engagedCapacitance per lhRegion 2Region 3unit lengthEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 5[Gary Fedder, Ph.D., UC Berkeley, 1994]Comb-Drive Force With Ground Plane Correction• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures d h d l difi h i i and the ground plane → modifies the capacitive energyEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 6[Gary Fedder, Ph.D., UC Berkeley, 1994]Simulate to Get Capacitors → Force• Below: 2D finite element simulation20-40% reduction of Fe,xEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 7Vertical Force (Levitation)()222111rsrpspVVdCVdCVdCWF−++=′∂=•F V 0V ( h )(),222rsrszeVVdzVdzVdzzF++∂()2,1rsespVCCdNF⎥⎤⎢⎡′+′EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 8•For Vr= 0V (as shown):()2,,2srsespzeVdzNxF⎥⎦⎢⎣=Simulated Levitation Force• Below: simulated vertical force Fzvs. z at different VP’s [f/ Bill Tang Ph.D., UCB, 1990] ªS h Fi hl i l f ªSee that Fzis roughly proportional to –z for z less than zo→ it’s like an electrical stiffness that adds to the mechanical ()()zz−2that adds to the mechanical stiffness()()zzkzzzVFoeooPzz−=≈2γElectrical StiffnessEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 9Vertical Resonance Frequencyezzkkk +=ωω2Vzkze⎟⎟⎠⎞⎜⎜⎝⎛=γwhereVertical resonance frequencyzzokωzo⎠⎝ωz/ωzoVertical resonance frequencyLateral =resonance frequency at VP= 0Vresonance frequency=• Signs of electrical stiffnesses in MEMS:Comb (x-axis) → ke= 0Comb (z-axis) → ke> 0Parallel Plate →k< 0Parallel Plate →ke< 0EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 10Suppressing Levitation• Pattern ground plane polysilicon into differentially excited electrodes to minimize field lines terminating on top of combEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 11fgpf• Penalty: x-axis force is reducedForce of Comb-Drive vs. Parallel-Plate• Comb drive (x-direction)ª V1= V2= VS= 1V• Differential Parallel-Plate (y-direction)(y)ª V1= 0V, V2= 1VParallel-plate generates a much larger force; but at the cost of EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 12linearityInput ModelingInput ModelingEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 13Electromechanical AnalogieskeqlxrxcxmeqcceqEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 14Bandpass Biquad Transfer FunctionkeqmeqcceqEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 15Force-to-Velocity Relationshipbkxb• The relationship between input voltage v1and force Fd1:kFd1111vxCVFPd∂∂−≈d1 1• When displacement x is the mechanical output variable:mectrode 222)(1)()(osQsksFsXωωω++=Elei1C1• When velocity υ is the mechanical output variable:1)()(oodsQsksFωω++v1i11mechanical output variable:21)()(osssXsωυ==EE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 16VP2211)()()(ooddsQsksFsFωω++==Force-to-Velocity Equiv. Ckt.b• Combine the previous lumped LCR mechanical equivalent circuit with i it d li th iti kxba circuit modeling the capacitive transducer → circuit model for voltage-to-velocitykFd1gyd1 1VelocityCurrentVoltage+I1U = -xmectrode lx=m rx=bCurrentLinearTwo-Port Element++V1Fd1Elei1C1cx=1/kElement--El lMh lv1i11ForceEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 17ElectricalMechanicalVPEquiv. Circuit for a Linear Transducer• A transducer …ª converts energy from one domain (e.g., electrical) to th ( h i l)another (e.g., mechanical)ª has at least two portsªis not generally linear, but is virtually linear when ªgy, yoperated with small signals (i.e., small displacements)VelocityCurrentLi++IU = -xCurrentVoltageLinearTwo-Port Element--VFForceVoltage--ElectricalMechanicalForceEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 18ElectricalMechanicalEquiv. Circuit for a Linear TransducerlLinear++IU = -xVelocityCurrentVoltageLinearTwo-Port Element--VFForce•F hil it t f ilt Electrical Mechanical•For physical consistency, use a transformer equivalent circuit to model the energy conversion from the electrical domain to mechanical domain1:ηf1Flowf2⎤⎡⎥⎤⎢⎡⎤⎡0eeη+e1Efforte2+⎥⎦⎤⎢⎣⎡⎥⎥⎦⎢⎢⎣−=⎥⎦⎤⎢⎣⎡112210fefeηηEE C245: Introduction to MEMS Design Lecture 24 C. Nguyen 11/24/08 19--⎦⎣ηDescribing MatrixElectromechanical Equivalent Circuitb• e2=Fd1, e1=v1, just need η1:• From the matrix:
View Full Document
Unlocking...