EE C245 – ME C218 Fall 2003 Lecture 10EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 10Electrostatic Actuators I2EE C245 – ME C218 Fall 2003 Lecture 10Today’s Lecture• Energy in electromechanical systems define carefully• Parallel-plate electrostatic actuators: snap-down• Why is electrostatic actuation important in MEMS?• Linearization of the square-law electrostatic force:differential electrostatic actuation• Reading:Senturia, S. D., Microsystem Design, Kluwer AcademicPublishers, 2001, Chapter 6, pp. 125-137.3EE C245 – ME C218 Fall 2003 Lecture 10Basic Physics of Electrostatic ActuationH. H. Woodsen and J. R. Melcher,Electromehcanical Dynamics Part I,Chapter 3, Wiley, 1968.+-V+++++++-------+q -qFeFegNote: we assume that the platesare supported elastically, so theydon’t collapse. ΔW(q,g) = V Δq + FeΔgTwo ways to change the energy:1. Change the charge q2. Change the separation x4EE C245 – ME C218 Fall 2003 Lecture 10Charge-Control Case• Stored energy: work done in increasing gap aftercharging capacitor at zero gapgdFWge′+=∫00• Electrostatic force as a functionchargeρ(x)xg0g¸q´W(q,g)gqzero gap zero stored energyE(x)xg0Gauss: E = q/(εA), 0 < x < gFe= (q/2)[q/(εA)] = constant (?)?5EE C245 – ME C218 Fall 2003 Lecture 10Charge-Control Case (Cont.)• Stored energy:AgqgdAqWgεε22202=′=∫constant• Force (attractive, internal):AqggqWFqeε2),(2=∂∂=• Voltage:CqAqgqgqWVg==∂∂=ε),(Independent of the gap!6EE C245 – ME C218 Fall 2003 Lecture 10Voltage-Control Case• Practical situation: we control V (since charge-control on the typical sub-pF MEMS actuation capacitor is a major challenge for electronics designers)• Can we find Feas a partial derivative of the stored energy W = W(V,g) with respect to g with V held constant? No!• Answer: apply Legendre transformation and define the co-energy W´(V,g)WqVgVW−=′),(dWVdqqdVgVWd−+=′),(dgFqdVgVWde−=′),(7EE C245 – ME C218 Fall 2003 Lecture 10Voltage-Control Case (Cont.)• Evaluate the co-energy:• Voltage in terms of charge at fixed gap:g¸V´W´(V,g)gVzero voltage zero force, zero integralVdVgqWV′′+=′∫0),(0gAqgEdVg⎟⎠⎞⎜⎝⎛=′−=′∫ε0VAgCVgAVgq′=′=′),(),(εcapacitance8EE C245 – ME C218 Fall 2003 Lecture 10Electrostatic Force (Voltage Control)2222121),(VgCVgAggVWFVe=⎟⎟⎠⎞⎜⎜⎝⎛−−=∂′∂−=ε• Variation of co-energy with respect to gap yields e.s. force:• Find co-energy in terms of voltage20021),( VgAVdVgAVdVgqWVV⎟⎟⎠⎞⎜⎜⎝⎛=′′⎟⎟⎠⎞⎜⎜⎝⎛=′′=′∫∫εεstrong function of gap!• Variation of co-energy with respect to voltage yields charge:CVVgAVgVWqg=⎟⎟⎠⎞⎜⎜⎝⎛=∂′∂=ε),(as expected9EE C245 – ME C218 Fall 2003 Lecture 10Spring-Suspended Capacitor:Charge-Control Case+-V+++++++-------+q-qFegkzfixed fixedAqggqWFqeε2),(2=∂∂=kzFe=AkqgkFgzggoeooε2/2−=−=−=)2(/2AkqgAqCqVoεε−==Can increase q anddrive g to zeroV decreases as gapdecreases10EE C245 – ME C218 Fall 2003 Lecture 10Spring-Suspended Capacitor:Voltage-Control Case2221),(VgAggVWFqeε=∂′∂=kzFe=+-V+++++++-------+q-qFegkzfixed fixed222/ VkgAgkFgzggoeooε−=−=−=Voltage increases Gap decreases Force increases …VgACVVWqgε==∂′∂=11EE C245 – ME C218 Fall 2003 Lecture 10Stability AnalysisRange of stability: examine net (attractive) force on plate()ggkgAVFonet−−=222εIf we increment the gap by dg, the increment dFnet> 0 orthe plate collapsesdgkgAVdggFdFnetnet⎟⎟⎠⎞⎜⎜⎝⎛+−=∂∂=32ε32gAVkε>12EE C245 – ME C218 Fall 2003 Lecture 10Pull-In Voltage VPI• Solve for point at which plate goes unstable:32PIPIgAVkε=)(2022PIoPIPInetggkgAVF −−==ε• Substitution for k leads to:)(203222gggAVgAVFoPIPIPIPInet−−==εε21=−PIPIoggg21=−PIPIogggAkgVoPIε2783=oPIgg32=13EE C245 – ME C218 Fall 2003 Lecture 10Graphical Solution for Plate Stability• Plot normalized electrostatic and spring forces vs. normalized displacement 1-(g/go)0.20.2 0.4 0.6 0.8 10.40.60.8100forcesdisplacementV1V2VPI14EE C245 – ME C218 Fall 2003 Lecture 10So Why are Electrostatic Actuators Important in MEMS, Anyway? Easy to make in micromachining processes, since conductors and air gaps are all that’s needed Energy-conserving  only parasitic energy loss through i2R losses in conductors and interconnects Pull-in phenomenon can be exploited to make a hysteretic actuator  simplifies control Multiple plate structures (combs, 3D) can be used to tailor the force(displacement,voltage) function Scaling of the electrostatic force is favorable due to Paschen’s curve Same structure can be used for position sensing15EE C245 – ME C218 Fall 2003 Lecture 10Paschen’s Curvevacuum limit: field emission at E ≈ 1 V/nm = 1000 V/μm16EE C245 – ME C218 Fall 2003 Lecture 10Linearizing the Voltage Square-Law• Polarize the capacitor by applying a DC offset voltage VPtogether with a (small) signal voltage vsig(t) << VP+-v(t) = VP+vsig(t)+++++++-------+q(t) -q(t)Fe(t)g22))((2121tvVgCvgCFsigPe+==)()(2))((222tvtvVVtvVsigsigPPsigP++=+DC offsetneglect(small))(2)(2tvVgCgCVtfFFsigPPsigDCe+=+≈17EE C245 – ME C218 Fall 2003 Lecture 10The Differential Electrostatic Actuator+-vL(t) = VP-v(t)-----++++++ql(t)-q(t)Fel(t)g++++++++qr(t)Fer(t)-------gfixed fixed-+vR(t) = VP+v(t)Net force on suspendedcenter electrode is thedifferenceFnet=Fer(t)-Fel(t)()()])()(2)()(2[2)(2222tvtvVVtvtvVVgCtFPPPP+−−++=)(2)( tvVgCtFP=18EE C245 – ME C218 Fall 2003 Lecture 10Second-Order Effects• Assumption was that the gaps were both g … correct only if the fabrication technology is perfect and the moving electrode is stationary• Allow for gap difference: residual DC and v2arepresent• Is there a way to ensure linearization?19EE C245 – ME C218 Fall 2003 Lecture 10Pulse Linearization • Applied voltage is limited to discrete levels (e.g., V1= 0 V and V2= 3 V) applied at a clock rate much higher than the mechanical resonant frequency• The electrostatic force is either zero or a value that is precisely set by the capacitance and the voltage• The density of pulses determines the force, with the system bandwidth filtering out the high frequency components• EE: sigma-delta feedbackME: bang-bang