EE C245 ME C218 Introduction to MEMS Design Fall 2007

Transcription

1 EE C45 ME C8 Introduction to MEMS Design Fall 007 Prof. Clark T.C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 9470 Lecture 3: Input Modeling & Gyros EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 Announcements MakeUp Lecture: Friday, /6, from :00 p.m. to :30 p.m. Room 45 McCone Still trying to get a room for a Monday, /9, makeup lecture Trying for 0 a.m., but will take what I can get Updates to HW#6 posted Basically, you can ignore the masses of the springs, since the proof masses are so large Also, just identify the electrodes in part (c) Times for project outbriefs Wednesday, Dec. (morning), 9:30 noon Sunday, Dec. 6? Wednesday, Dec. 9? Thursday, Dec. 0? EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07

2 Lecture Outline Reading: Senturia Chpts. 5, 6, Handouts: Electrostatic Comb of Lateral Polysilicon Resonators, Electrostatic Comb Levitation And Control Method, Micromachined Inertial Sensors Lecture Topics: Electrostatic Comb st Order Analysis nd Order Analysis Levitation Electrical Stiffness Input Circuit Modeling Gyroscopes EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 3 Comb Force Equation ( nd Pass) In our st pass, we neglected Fringing fields Parallelplate capacitance between stator and rotor Capacitance to the substrate All of these capacitors must be included when evaluating the energy expression! Stator Rotor Ground Plane EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 4

3 Comb Force With Ground Plane Correction Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and the ground plane modifies the capacitive energy [Gary Fedder, Ph.D., UC Berkeley, 994] EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 5 Case: V r = V P = 0V C sp depends on whether or not fingers are engaged Capacitance Expressions Capacitance per unit length Region Region 3 [Gary Fedder, Ph.D., UC Berkeley, 994] EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 6 3

4 Comb Force With Ground Plane Correction Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and the ground plane modifies the capacitive energy [Gary Fedder, Ph.D., UC Berkeley, 994] EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 7 Simulate to Get Capacitors Force Below: D finite element simulation 040% reduction of F e,x EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 8 4

5 Vertical Force (Levitation) F e, W = = dc d sp For V r = 0V (as shown): V s + F e, dcrp V d = Nx dcrs r + d d C + C ( V V ) ( sp, e rs ) EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 9 d s V s r Simulated Levitation Force Below: simulated vertical force F vs. at different V P s [f/ Bill Tang Ph.D., UCB, 990] See that F is roughly proportional to for less than o it s like an electrical stiffness that adds to the mechanical stiffness F γ V P ( ) o o = k e ( ) o Electrical Stiffness EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 0 5

6 ω /ω o Vertical resonance frequency Vertical Resonance Frequency Vertical resonance frequency at V P = 0V k + k ω e = k e = V o k γ where o ω = Lateral resonance frequency Signs of electrical stiffnesses in MEMS: Comb (xaxis) k e = 0 Comb (axis) k e > 0 Parallel Plate k e < 0 EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 Suppressing Levitation Pattern ground plane polysilicon into differentially excited electrodes to minimie field lines terminating on top of comb Penalty: xaxis force is reduced EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 6

7 Force of Comb vs. ParallelPlate Comb drive (xdirection) V = V = V S = V Differential ParallelPlate (ydirection) V = 0V, V = V Parallelplate generates a much larger force; but at the cost of linearity EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 3 Input Modeling EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 4 7

8 Electromechanical Analogies k eq l x c x r x m eq c eq EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 5 Bandpass Biquad Transfer Function k eq m eq c eq EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 6 8

9 F d d Electrode i C ForcetoVelocity Relationship x m b k The relationship between input voltage v and force F d : F C x d VP v When displacement x is the mechanical output variable: X ( s) ωo = Fd ( s) k s + ( ωo Q) s + ωo When velocity υ is the mechanical output variable: v V P ( s) sx ( s) ωo s = = d ( s) Fd ( s) k s + ( ωo Q) υ F o s + ω EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 7 F d d Electrode i C v V P ForcetoVelocity Equiv. Ckt. x m b k + V Combine the previous lumped LCR mechanical equivalent circuit with a circuit modeling the capacitive transducer circuit model for voltagetovelocity Voltage Velocity Current I U = x l x =m r x =b Electrical Linear TwoPort Element + F d Mechanical c x =/k Force EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 8 9

10 Equiv. Circuit for a Linear Transducer A transducer converts energy from one domain (e.g., electrical) to another (e.g., mechanical) has at least two ports is not generally linear, but is virtually linear when operated with small signals (i.e., small displacements) Voltage + + V Velocity Current I U = x Linear TwoPort Element F Force Electrical Mechanical EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 9 Equiv. Circuit for a Linear Transducer Current Voltage + + V I U = x Linear TwoPort Element F Velocity Force Electrical Mechanical For physical consistency, use a transformer equivalent circuit to model the energy conversion from the electrical domain to mechanical domain Flow Effort + e f :η f EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/ e e f η = 0 0 e f η Describing Matrix 0

11 F d Electromechanical Equivalent Circuit d x b k e =F d, e =v, just need η : From the matrix: e =ηe F C x d VP v C x η = V P Electrode i C v V P m Voltage Velocity Current + V I U = x :η l x =m r x =b + F d Force Electrical Mechanical c x =/k EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 Output Modeling EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07

12 Types of Capacitive Outputs Low Frequency Example: accelerometer Acceleration inputs are generally at low frequency, e.g., near DC High Frequency Example: gyroscope Rotationderived sense forces are generally sinusoidal, e.g., resonance Ω r Tuning Tuning Electrode Differential Capacitive Pickoff ADXL50 Capacitive Pickoff Capacitive EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 3 Gyroscopes EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 4

13 Classic Spinning Gyroscope A gyroscope measures rotation rate, which then gives orientation very important, of course, for navigation Principle of operation based on conservation of momentum Example: classic spinning gyroscope Rotor will will preserve its its angular momentum (i.e., will will maintain its its axis axis of of spin) despite rotation of of its its gimbled chassis EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 5 Vibratory Gyroscopes Generate momentum by vibrating structures Again, conservation of momentum leads to mechanisms for measuring rotation rate and orientation Example: vibrating mass in a rotating frame y Mass at rest y Get an x component x x of motion n into vibration along the yaxis C(t) C(t ) > C(t ) Rotate 30 o C(t ) y x ydisplaced mass Capacitance between mass and frame = constant C(t ) EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 6 3

14 Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c () Response Torque Basic Vibratory Gyroscope Operation Ω r n f o x v r y Amplitude / Response Spectra: Response Response Acceleration Force Displacement n Velocity f o (@ T ) r r r a c = v Ω r r r r Fc mac a x = = = k k ω Beam Stiffness Rotation Rate EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 7 c r Beam Mass ω Frequency Basic Vibratory Gyroscope Operation Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c () Response Ω r n f o v r Torque x y EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 8 4

15 Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c () Response Torque Vibratory Gyroscope Performance Ω r n f o x v r y r r Fc x = k Beam Mass r ma = k r r r a c = v Ω EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 9 c r a = ω Beam Stiffness c r Frequency n Velocity To maximie the output signal x, need: Large senseaxis mass Small senseaxis stiffness (Above together mean low resonance frequency) Large drive amplitude for large driven velocity (so use combdrive) If can match drive freq. to sense freq., then can amplify output by Q times MEMSBased Gyroscopes Vibrating Ring Gyroscope Tuning Fork Gyroscope [Ayai,, GA Tech.] Tuning Fork Gyroscope [Draper Labs.] 3. mm Nuclear Magnetic Resonance Gyro [NIST] [Najafi, Michigan] Laser Polarier Rb/Xe Cell Photodiode mm mm EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 30 θ & t 5

16 MEMSBased Tuning Fork Gyroscope Electrode Tuning Tuning Electrode Mode Quadrature Cancellation Inplane drive and sense modes pick up axis rotations Modematching for maximum output sensitivity From [Zaman, Ayai, et al, MEMS 06] Mode EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 3 MEMSBased Tuning Fork Gyroscope Ω r Tuning Voltage Signal () Output Current (+) Output Current Tuning Electrode EE C45: Introduction to MEMS Design Lecture 3 C. Nguyen /5/07 3 [Zaman, Ayai, et al, MEMS 06] Oscillation Sustaining Amplifier Differential TransR Amplifier 6