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Unformatted text preview: UNIVERSITY OF CALIFORNIA IRVINE STABLE ELECTROSTATIC ACTUATION OF MEMS DOUBLE-ENDED TUNING FORK OSCILLATORS THESIS submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in Mechanical and Aerospace Engineering by Andreu Fargas-Marques Thesis Committee: Professor Andrei M. Shkel, Chair Professor Kenneth D. Mease Professor Athanassios Sideris 2001 c 2001 Andreu Fargas-Marques The thesis of Andreu Fargas-Marques is approved: Committee Chair University of California, Irvine 2001 ii Contents List of Figures List of Tables Acknowledgments Abstract of the Thesis 1 Introduction 1.1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi xv xvi xvii 1 2 5 7 7 10 11 12 16 17 20 2 Double-Ended Tuning Fork Design 2.1 2.2 Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2.3 Damping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatic Driving and Sensing . . . . . . . . . . . . . . . . . . . . 3 DETF Simulation and Stability Analysis 3.1 3.2 Dynamic Behavior of a DETF with Lateral Actuation . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 3.3 Pull-in in Parallel Plate Actuators . . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 3.3.3 Static Pull-in Voltage . . . . . . . . . . . . . . . . . . . . . . . Dynamic Pull-in Voltage . . . . . . . . . . . . . . . . . . . . . AC Dynamic Pull-in Voltage . . . . . . . . . . . . . . . . . . . 25 27 31 38 49 51 51 52 55 56 59 62 65 65 70 70 71 75 80 83 83 85 3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 DETF Prototypes 4.1 MUMPs Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.2 Process Description . . . . . . . . . . . . . . . . . . . . . . . . ........................ DETF Prototypes Design 4.2.1 4.2.2 Structural Design . . . . . . . . . . . . . . . . . . . . . . . . . Actuation and Sensing . . . . . . . . . . . . . . . . . . . . . . 4.3 4.4 Final Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Elements Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5 DETF Testing 5.1 5.2 5.3 5.4 Experimental Evaluation Set-up . . . . . . . . . . . . . . . . . . . . . Analysis of Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . AC Dynamic Pull-in Analysis . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and Future Work 6.1 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 6.2.1 6.2.2 6.2.3 AC Dynamic Pull-in Characterizations . . . . . . . . . . . . . Oscillator Electronics . . . . . . . . . . . . . . . . . . . . . . . Frequency Shift Detection . . . . . . . . . . . . . . . . . . . . 6.2.3.1 6.2.3.2 Tunable DETF Oscillators . . . . . . . . . . . . . . . Angular Accelerometers . . . . . . . . . . . . . . . . 85 86 87 88 89 93 97 97 Bibliography A Matlab listings A.1 snap detf.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 eq snap detf.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B Maple listings 102 B.1 tk lat bal 1b.mws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 B.2 AC dynamic pull-in.mws . . . . . . . . . . . . . . . . . . . . . . . . . 104 C Ansys listings 107 C.1 lb1b beam4.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 C.2 lb1b beam4 sens.inp . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 v List of Figures 1.1 1.2 Schematic of a beam oscillating with electrostatic actuation and sensing Schematic of a DETF oscillator with electrostatic actuation and sensing 1.3 .................................. 3 2 Scheme of the dynamics of an electrostatic actuator. The force that appear between the plates depends on the voltage (V ) and the position of the plate (y ). ............................. ... 4 8 2.1 2.2 Vibrating beam oscillating under the influence of an axial force Force balance in a differential of a laterally oscillating beam. Adapted from [Rao, 1990] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 11 13 14 17 2.3 2.4 2.5 3.1 3.2 (a) Couette flow damping; (b) Squeeze film damping . . . . . . . . . (a) Lateral electrostatic actuator; (b) Parallel electrostatic actuator . (a) Lateral comb fingers actuation; (b) Parallel plate actuation. . . . Beam with lateral electrostatic actuation and sensing . . . . . . . . . Frequency response of a DETF. The gain at the resonant pick is proportional to the quality factor. ................... 19 vi 3.3 Different configurations for driving and sensing with electrostatic parallel plates. (a) Beam driven with lateral combs and motion sensed with parallel plates; (b) The driving and sensing is done with parallel plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 (a) Amplitude of oscillation of a beam driven with lateral combs; (b) Output current of a beam sensed with parallel plates . . . . . . . . . 21 3.5 Illustration of DETF response when the system is driven into oscillation using DC+AC signals combination. The DETF is driven using an AC signal at its mechanical natural frequency (fn =148 kHz, Q=100). Due to mismatch between driving frequency and fundamental frequency of the electro-mechanical system, increase of the DC bias does not always translate into increase of the amplitude. (a) Driven with 10 V DC and 15 V AC; (b) Driven with 15 V DC and 15 V AC; (c) Driven with 25 V DC and 15 V AC; (d) Driven with 50 V DC and 15 V AC. .... 22 3.6 (a) DETF Oscillating at its resonant frequency under the action of the control loop; (b) 5 V square wave signal that drives the DETF to resonance. ................................ 24 3.7 Amplitude and frequency of oscillation of the DETF (Q=30). (a) Relationship between the amplitude and the DC bias for a fixed AC voltage; (b) Stiffness softening translates in shift of the frequency of oscillation with the increase of the DC bias. . . . . . . . . . . . . . . 25 26 3.8 Scheme of a parallel plate actuator . . . . . . . . . . . . . . . . . . . vii 3.9 Evolution of the potential energy along the gap between the electrodes. (a) Low voltages present stable and unstable static equilibrium; (b) The equilibrium points are points where the force of the capacitor is compensated by the restoring spring force. .............. 28 3.10 Evolution of the potential energy along the gap between the electrodes at Pull-in Voltage. (a) The energy only presents an inflexion point; (b) The force of the capacitor is only compensated by the restoring spring force at one third of the gap, what corresponds to the inflexion point. 3.11 Evolution of the energy profile in a DETF with a gap of 2 µm when changing the voltage between the plates. (a) Low voltages present stable and unstable static equilibrium; (b) At high voltages both equilibrium points get closer until they disappear at pull-in voltage (61.9 V in the example). . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Evolution of the system when the voltage is applied as a step function. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode overshoots the stable equilibrium before it settles down; (b) Evolution of the energy for the same trajectory. The initial energy is dissipated due to the damping forces until it reaches the stable equilibrium. ............. 31 30 29 viii 3.13 Evolution of the system when the voltage applied leads the system to snap. (a) Displacement of the moving electrode when the voltage is applied. It can be seen how the position of the electrode reaches the unstable equilibrium. At that point, the evolution becomes unstable; (b) Evolution of the energy for the same trajectory. In the plot, the energy losses are small, so the energy is almost constant, reaching over the unstable equilibrium and leading the system to pull-in. Energy levels above this line all lead over the unstable equilibrium. ..... 33 3.14 Pull-in Voltage as a function of the Quality Factor. The Dynamic Pull-in Voltage is 56.9 V for our example. . . . . . . . . . . . . . . . 3.15 (a) Evolution of the position of a DETF for a 56 V step input (Q=500); (b) Evolution of the same DETF showing snapping at 57 V. (Vpin = 61.9 V, Q=500) ............................. 36 35 3.16 Evolution of the energy for the examples in Figure 3.15. (a) Smooth energy evolution using a step of 56 V; (b) Unstable energy evolution using a step of 57 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 The evolution of the energy including the dynamic component is compared to the static energy depending on the position in the gap for the examples in Figure 3.15. (a) The smooth evolution using a step of 56 V is bounded by the static energy; (b) The unstable evolution condition is achieved using a step of 57 V because the energy has reached over the unstable equilibrium. . . . . . . . . . . . . . . . . . 37 37 ix 3.18 Potential energy bounds when the system is oscillated with a DC bias voltage and an AC voltage. ....................... 39 3.19 Calculation of the maximum amplitude of oscillation for the simulated DETF depending on the combination of DC and AC voltages. (a) Plot of the amplitude for each voltage over which the DETF is at risk of snapping (Calculated for VD + VAC ); (b) Plot of the amplitude for each voltage over which the DETF is pulled-in (Calculated for VD − VAC ). 3.20 Stable oscillation of a DETF. The potential energy curves define a steady-state oscillation or limit cycle. ................. 41 42 40 3.21 Oscillation of a DETF using 20 V DC and 5 V AC. . . . . . . . . . . 3.22 Evolution of the energy using 20 V DC and 5 V AC. (a) The energy of the system increases until it reaches the steady state oscillation; (b) Evolution of the energy in the gap. In stable oscillation, the energy is confined between the static energy limits, without overshooting the unstable equilibrium. .......................... 42 3.23 Unstable oscillation. Beginning from the initial position, the amplitude of oscillation of the system increases until it reaches over the unstable equilibrium point for VDC + VAC . Then, the system snaps. . . . . . . 3.24 Unstable evolution of a DETF using 30 V DC and 5 V AC. (a) The amplitude of oscillation at the resonant frequency leads the DETF to the unstable region; (b) Close-up of the point where the unstable region is reached. The cross indicates the position where the unstable equilibrium point has been overshot. . . . . . . . . . . . . . . . . . . 44 43 x 3.25 Energy evolution when using 30 V DC and 5 V AC. (a) The energy increases without settling until pull-in is reached; (b) The increase of the energy leads the position to overshoot the unstable equilibrium limit what leads to snapping. . . . . . . . . . . . . . . . . . . . . . . 3.26 AC Dynamic Pull-in algorithm. The energy distribution is stable because amplitudes close to the unstable region decrease. ...... 46 44 3.27 AC Dynamic Pull-in algorithm. The energy distribution is unstable because amplitudes close to the unstable region increase, leading the system to snapping. ........................... 46 3.28 AC Dynamic Pull-in Voltage Curves. (a) Minimum DC bias and AC voltage that when applied produces snapping for a fixed Quality Factor; (b) Evolution of the minimum bias that produces snapping for a fixed AC signal in function of the Quality Factor. . . . . . . . . . . . . . . 4.1 4.2 4.3 MUMPs Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section of a DETF designed in MUMPs ................ 49 52 54 The inclusion of the actuation mechanism increases the mass of the structure. (a) Lateral comb fingers actuation; (b) Parallel plate 57 actuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 (a) Double-ended tuning fork prototype with electrostatic lateral comb driving and sensing; (b) Close-up of the axial force actuator placed at one end of the DETF to test axial force sensitivity and frequency tunability. 4.5 ................................ ........ 64 65 Model using BEAM4 element for the prototype DL1B xi 4.6 First mode of oscillation for the prototype DL1B. It can be observed the out-of-plane motion. . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7 Second mode of oscillation for the prototype DL1B. Desired mode of in-plane oscillation. ........................... 68 4.8 Third mode of oscillation for the prototype DL1B. Torsional mode of the actuation comb. ........................... 69 4.9 Fourth mode of oscillation for the prototype DL1B. Torsional out-ofplane oscillation of the actuation comb. ................ 69 5.1 Scanning Electron Microscope (SEM) picture of the 1cm × 1cm chip fabricated in the Cronos MUMPs run 42. In the image, the sector where the double-ended tuning forks are located. ........... 71 5.2 (a) Microscope probe-station available in the Microsystems Lab at UCI; (b) Close-up of the precision-positioners used to supply the power to the chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 72 5.3 5.4 Experimental set-up used to test the prototypes. ........... (a) SEM picture of one of the double-ended tuning forks; (b) Picture of a double-ended tuning fork taken with the camera embedded in the microscope probe-station. . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 (a) SEM of a beam testing structure with parallel plate actuation after snapping; (b) A laterally-actuated double-ended tuning fork burned after snapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 xii 5.6 Comparison between the electrostatic stiffness softening predicted by the model when increasing the DC bias and the experimental results. (a) Spring softening for 3 V AC; (b) Spring softening for 5 V AC; (c) Spring softening for 7 V AC; (d) Spring softening for 10 V AC. ... 76 5.7 Comparison between the amplitude of oscillation predicted by the model when increasing the DC bias and the experimental results. (a) Amplitude for 3 V AC; (b) Amplitude for 5 V AC; (c) Amplitude for 7 V AC; (d) Amplitude for 10 V AC. . . . . . . . . . . . . . . . . . . 78 5.8 Evolution of the amplitude of oscillation when increasing the DC bias; (a) Beam at rest; (b) Beam oscillating with 15 V DC and 7 V AC; (c) Beam oscillating with 18 V DC and 7 V AC; (d) Beam after snapping with 20 V DC and 7 V AC. . . . . . . . . . . . . . . . . . . . . . . . 79 5.9 Comparison of the values for which the experimental DETF was pulledin. The trend of the experimental results agree with that of the 80 simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Comparison between the experimental amplitude of oscillation and the maximum stable amplitude for the different combination of voltages. (a) Evolution for 3 V AC. The amplitude does not reach the unstable region; (b) Evolution for 5 V AC. Snapping occurs when the amplitude reaches the unstable region at 29 V DC; (c) Evolution for 7 V AC. Snapping occurs for 19 V DC; (d) Evolution for 10 V AC. Snapping occurs for 15 V DC; 6.1 ........................... ........................ 82 86 Vacuum testing chamber. xiii 6.2 6.3 6.4 6.5 DETF Oscillator electronics . . . . . . . . . . . . . . . . . . . . . . . Vibrating beam frequency shift under the action of an axial force . . DETF with force tuning in one of the ends. . . . . . . . . . . . . . . (a) Schematic of a linear accelerometer using a DETF; (b) Schematic of an angular accelerometer using the same principle. . . . . . . . . . 87 88 89 90 6.6 (a) SEM of the Asymmetric Vibratory Angular Accelerometer; (b) Close-up of the proof mass of the device. ............... 91 6.7 (a) SEM of the Symmetric Vibratory Angular Accelerometer; (b) Close-up of a DETF attached to the proof mass. ........... 92 xiv List of Tables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.1 MUMPs layers and attributes [Koester et al., 2001] . . . . . . . . . . Polysilicon properties .......................... .............. 55 56 58 60 61 63 63 64 67 Example of a DETF structural parameters Estimated DETF lateral combs parameters in air . . . . . . . . . . . Estimated DETF lateral combs parameters in vacuum at 1 Torr . . . Design parameters for testing prototypes . . . . . . . . . . . . . . . . Testing prototypes structural parameters ............... ....... Testing prototypes actuation and sensing voltages in air Modes of oscillation for the Actuation Prototypes. .......... Frequencies and voltages of actuation of the fabricated prototypes in air 73 xv ACKNOWLEDGMENTS I want to thank the Balsells family, the Generalitat de Catalunya and the University of California for their support through the Balsells Fellowship. Without their help this thesis wouldn’t have been written. My gratitude also to Professor Roger H. Rangel, coordinator of the Balsells Fellowship program, for his invaluable help before and after the fellowship was awarded. I would also like to thank Professor Shkel for his support and guidance, and the rest of students in the Microsystems Laboratory for their companionship and help. Thanks to my Balsells fellows for their support and friendship. Their presence here in UCI made more pleasant this year of work. Finally, thanks to my girlfriend and parents for their patience while I have been far away. xvi ABSTRACT OF THE THESIS STABLE ELECTROSTATIC ACTUATION OF MEMS DOUBLE-ENDED TUNING FORK OSCILLATORS by Andreu Fargas-Marques Master of Science in Mechanical and Aerospace Engineering University of California, Irvine, 2001 Professor Andrei M. Shkel, Chair In this thesis, the study of stable electrostatic actuation is presented for doubleended tuning fork oscillators. This thesis proposes to use the analysis of the energy of the electro-mechanical system to deduce the conditions of stable electrostatic actuation. Two cases are considered: (i) Static case, when balance of the energy stored in the elastic deformation and the energy stored in the capacitive drive are contemplated; (ii) Dynamic case, when in addition to the energy balance of the static case kinetic energy is accounted for. Step function as well as harmonic excitation are considered in the second case. Understanding of the energy evolution of the system leads to the implementation of the AC Dynamic Pull-in Algorithm. This algorithm allows to determine the stability of the electrostatic actuation of a double-ended tuning fork at resonance. The conditions of stability are validated experimentally using designed surface-micromachined double-ended tuning forks. The issues involved in the fabrication are discussed. Applications of the devices for on-chip frequency references and angular acceleration sensing are introduced. xvii Chapter 1 Introduction The use of oscillators as a frequency source is common in many applications ranging from communications systems to wrist-watches. To build an oscillator, a frequency source must be provided. A common source for these frequency-dependent devices is a mechanical resonator. Micro Electro Mechanical Systems (MEMS) technologies such as quartz micromachining, silicon bulk micromachining, or polysilicon surface micromachining, all have demonstrated their ability to fabricate mechanical resonators. The stability of the frequency output is one of the main challenges when designing mechanical oscillators. A resonator like the double-ended tuning fork consists of a beam that is driven into resonant oscillation (Figure 1.1). The characteristics of the output signal are defined by the mechanical parameters of the beam and the dynamics of its deformation. St...
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