EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 20: Equivalent Circuits it I & Gyroscopes EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 1
Lecture Outline Reading: Senturia, Chpt. 5, Chpt. 6, Chpt. 21 Lecture Topics: Lumped Mechanical Equivalent Circuits Project: Gyroscopes Deep Reactive-Ion Etching (revisited) Lossless Transducers EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 2
Free-Free Beam Frequency (cont) Applying B.C. s, get A=B and B=D, and (3) Setting the determinant = 0 yields Which has roots at Substituting (2) into (1) finally yields: Free-Free Beam Frequency Equation EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 3
Higher Order Free-Free Beam Modes More than 10x increase Fundamental Mode (n=1) 1 st Harmonic (n=2) 2 nd Harmonic (n=3) EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 4
Mode Shape Expression The mode shape expression can be obtained by using the fact that A=B and C=D into (2), yielding Get the amplitude ratio by expanding (3) [the matrix] and solving, which yields Then just substitute the roots for each mode to get the expression for mode shape Fundamental Mode (n=1) [Substitute ] EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 5
Lumped Parameter Mechanical Equivalent Circuit it EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 6
Equivalent Dynamic Mass Once the mode shape is known, the lumped parameter equivalent circuit can then be specified Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity z Location x W h Maximum Kinetic Energy Density Equivalent Mass = Maximum Velocity @ location x Maximum Velocity Function EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 7
Equivalent Dynamic Mass z Location x W h We know the mode shape, so we can write expressions for displacement and velocity at resonance EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 8
Equivalent Dynamic Stiffness & Damping z Location x W h Stiffness then follows directly from knowledge of mass and resonance frequency And damping also follows readily EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 9
Equivalent Lumped Mechanical Circuit z Location x W h K eq (x) C eq (x) K M eq (x) 2 eq ( x ) = ω o M eq M eq ( x) ( x ) = ρ ωo Meq( x) C eq ( x ) = Q 2 A l [ u ( x )] dx o 2 [ u( x)] EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 10
Equivalent Lumped Mechanical Circuit z Example: Polysilicon w/ l=14.9μm, W=6μm, h=2μm 70 MHz W h K eq (0) = 19,927927 N/m M eq (0) = 1.03x10-13 kg K eq (node) = M eq (node) = C eq (node) = K eq (l/2) = 53,938 N/m M eq (l/2) = 2.78x10-13 kg C eq (0) = 5.66x10-9 kg/s C eq (l/2) = 1.53x10-8 kg/s EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 11
3CC 3λ/4 Bridged μmechanical Filter Performance: f o =9MHz, BW=20kHz, PBW=0.2% IL=2 I.L.=2.79dB, Stop. Rej.=51dB 20dB S.F.=1.95, 40dB S.F.=6.45 V P In Out 0 Transmis ssion [db B] -10-20 -30-40 -50-60 P in =-20dBm [S.-S. Li, Nguyen, FCS 05] Sharper roll-off Loss Pole 8.7 8.9 9.1 9.3 [Li, et al., UFFCS 04] Frequency [MHz] Design: L r =40μm W r =6.5μm h r =2μm L c =3.5μm 35 L b =1.6μm V P =10.47V P=-5dBm R Qi =R Qo =12kΩ EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 12
Electromechanical Analogies k eq l x c x r x m eq c eq EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 13
Electromechanical Analogies (cont) Mechanical-to-electrical correspondence in the current analogy: EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 14
Bandpass Biquad Transfer Function k eq m eq c eq EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 15
Gyroscopes EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 16
Classic Spinning Gyroscope A gyroscope measures rotation rate, which then gives orientation very important, of course, for navigation Principle i of operation based on conservation of momentum Example: classic spinning gyroscope Rotor will preserve its angular momentum (i.e., will maintain its axis of spin) despite rotation of its gimbled chassis EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 17
Vibratory Gyroscopes Generate momentum by vibrating structures Again, conservation of momentum leads to mechanisms for measuring rotation ti rate and orientation ti Example: vibrating mass in a rotating frame y Driven into vibration along the y-axis Mass at rest y Get an x component x x of motion C(t) Rotate 30 o C(t 2 ) > C(t 1 ) C(t 1 ) C(t 2 ) y-displaced mass Capacitance between mass and frame = constant EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 18
Basic Vibratory Gyroscope Operation Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c Coriolis (Sense) Response z Ω r Driven Vibration @ f o v r Coriolis Torque x z y EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 19
Basic Vibratory Gyroscope Operation Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c Coriolis (Sense) Response Coriolis Torque z Ω r Driven Vibration @ f o x v r Am mplitude Coriolis Acceleration Coriolis Force Drive/Sense Response Spectra: Drive Response r x Driven Velocity r r r a c = 2v Ω r F k r ma f o (@ T 1 ) c c c z 2 k ω r y Coriolis Displacement = = Beam Stiffness = Sense Response Rotation Rate r a Beam Mass ω Sense Frequency EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 20