1EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 1EE C245 – ME C218Introduction to MEMS DesignFall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture 21: Capacitive TransducersEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 2Lecture Outline• Reading: Senturia Chpts. 5, 6• Lecture Topics:ª Energy Conserving Transducers( Charge Control( Voltage Controlª Parallel-Plate Capacitive Transducers( Linearizing Capacitive Actuators( Electrical Stiffnessª Electrostatic Comb-Drive( 1stOrder Analysis( 2ndOrder Analysis2EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 3Energy Conserving TransducersEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 4Basic Physics of Electrostatic Actuation• Goal: Determine gap spacing g as a function of input variables• First, need to determine the energy of the system• Two ways to change the energy:ª Change the charge qª Change the separation gΔW(q,g) = VΔq+ FeΔgdW = Vdq + Fedg• Note: We assume that the plates are supported elastically, so they don’t collapse3EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 5Charge-Control Case• Here, the stored energy is the work done in increasing the gap after charging capacitor at zero gap• Find force and voltage:ª Need stored energyª Can find by recognizing that the energy in the final state is just the energy stored in capacitor charged to qEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 6Charge-Control Case• Having found stored energy, we can now find the force acting on the plates and the voltage across them:+-V4EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 7Voltage-Control Case• Practical situation: We control Vª Charge control on the typical sub-pF MEMS actuation capacitor is difficultª Need to find Feas a partial derivative of the stored energy W = W(V,g) with respect to g with V held constant? But can’t do this with present W(q,g) formulaª Solution: Apply Legendre transformation and define the co-energy W′(V,g)EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 8Co-Energy Formulation• For our present problem (i.e., movable capacitive plates), the co-energy formulation becomes5EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 9Electrostatic Force (Voltage Control)• Find co-energy in terms of voltage• Variation of co-energy with respect to gap yields electrostatic force:• Variation of co-energy with respect to voltage yields charge:EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 10Spring Suspended Capacitive Plates6EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 11Charge Control of a Spring-Suspended CIEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 12Voltage Control of a Spring-Suspended C7EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 13Stability Analysis• Net attractive force on the plate:• An increment in gap dg leads to an increment in net attractive force dFnetEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 14Pull-In Voltage VPI• VPI= voltage at which the plates collapse• The plate goes unstable when• Substituting (1) into (2):and(1) (2)When a gap is driven by a voltage to (2/3) its original spacing, collapse will occur!When a gap is driven by a voltage to (2/3) its original spacing, collapse will occur!8EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 15Voltage-Controlled Plate Stability Graph• Below: Plot of normalized electrostatic and spring forces vs. normalized displacement 1-(g/go)Normalized DisplacementForcesSpring ForceElectrical ForcesIncreasing VStable Equilibrium PointsEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 16Advantages of Electrostatic Actuators• Easy to manufacture in micromachining processes, since conductors and air gaps are all that’s needed → low cost!• Energy conserving → only parasitic energy loss through I2R losses in conductors and interconnects• Variety of geometries available that allow tailoring of the relationships between voltage, force, and displacement• Electrostatic forces can become very large when dimensions shrink → electrostatics scales well!• Same capacitive structures can be used for both drive and sense of velocity or displacement• Simplicity of transducer greatly reduces mechanical energy losses, allowing the highest Q’s for resonant structures9EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 17Problems With Electrostatic Actuators• Nonlinear voltage-to-force transfer function• Relatively weak compared with other transducers (e.g., piezoelectric), but things get better as dimensions scaleEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 18Linearizing the Voltage-to-Force T.F.• Apply a DC bias (or polarization) voltage VPtogether with the intended input (or drive) voltage vi(t), where VP>> vi(t)v(t) = VP+ vi(t)10EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 19Differential Capacitive Transducer• The net force on the suspended center electrode isEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 20Remaining Nonlinearity11EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 21Parallel-Plate Capacitive Nonlinearity• Example: laterally driven spring suspended plate (eventually with balanced electrodes)• Nomenclature:V1v1VPkmElectrodeConductive StructurexFd1tVaor vAva=|va|cosωtVAVaor vA= VA+ vaTotal ValueDC Component(upper case variable; upper case subscript)AC or Signal Component(lower case variable; lower case subscript)d1mEE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 22Parallel-Plate Capacitive Nonlinearity• Example: clamped-clamped laterally driven beam with balanced electrodes• Expression for ∂C/∂x:V1v1VPkmElectrodeConductive StructurexFd1d1m12EE C245: Introduction to MEMS Design Lecture 21 C. Nguyen 11/8/07 23Parallel-Plate Capacitive Nonlinearity• Thus, the expression for force from the left side becomes:V1v1VPkmElectrodeConductive
View Full Document
Unlocking...