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Thermally-driven angular rate sensors in standard CMOS

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Thermally-driven angular rate sensors in standard CMOS

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Content THERMALLY-DRIVEN ANGULAR RATE SENSORS IN STANDARD CMOS by Michael Dean Pottenger A Thesis Presented to the FACULTY OF THE SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Mechanical Engineering) December 1995 © 1995 Michael Dean Pottenger T his thesis, 'Written by Michael Dean Pottenger under the guidance o f F acu lty C om m ittee and a p proved by a ll its m em bers, has been presented to and accepted by the School of E ngineering in pa rtia l fu lfillm e n t o f the re quirem ents fo r the degree o f Master of Science Mechanical Engineering Date Au8.u s .t ...1 . ^ . 5.................... Faculty (Committee Chairman DEDICATION To Ingrid — fo r helping me keep my eyes on the horizon without losing sight o f my feet “There are moments when a m an’s imagination, so easily subdued to what it lives in, suddenly rises above its daily level, and surveys the long windings of destiny.” — Edith Wharton The Age o f Innocence ACKNOWLEDGEMENTS This thesis is the result of work which began as my design project for EE250b, “MEMS System Design", taught by Professor Kris Pister at UCLA. My thanks to Kris for suggesting this project, and for his patience through countless questions as I worked my way through my first solo design task in MEMS. And I think we had a cup of coffee or two along the way as well. With the mechanical system somewhat in place, I undertook more of a system-level analysis in EE250c, “Microsensors and Microinstruments’’, taught by Professor Bill Kaiser. Most of the material on frequency dependence and error sources of the sensor system described here was done at that time. My thanks to Bill for his help in understanding resonant systems. It has been my good fortune to be able to work with people as talented, friendly and cooperative as Professors Pister and Kaiser, and the enthusiasm they show for their work has certainly been instrumental in keeping myself focused to the task at hand. The design described here was fabricated through the MOSIS brokerage service, submitted under a UCLA contract. All laboratory and experimental work described in this thesis was performed at UCLA, and would not have been possible without the cooperation and sponsorship of Kris Pister and his research group. I have had the good fortune to work at USC’s Information Sciences Institute, and would like to acknowledge the benefits that my research duties there have afforded me. It is through my activities at ISI that I was introduced to the work in MEMS underway at UCLA. For specific contributions, I would like to thank Eric Hoffman for teaching me how to use the Xenon difluoride etching system. All SEM photographs were graciously taken by Gisela Lin. Thanks also to Professor Frederick Browand at USC for his help in understanding viscous damping and the calculation of Q factors. Finally, I would like to thank my thesis committee at USC— Professors Phil Muntz, Henryk Flashner and especially Geoff Shiflett, who offered to chair this endeavor— for their efforts in bringing this work to completion. V TABLE OF CONTENTS I ABSTRACT................................................................................................. 1 II INTRODUCTION A. Angular Rale Sensors........................................................................3 B. Principle of O p eration......................................................................8 C. Resonant Systems.............................................................................14 1. Forced Harmonic Oscillator with Viscous Damping ..15 2. Quality Factor....................................................................... 18 3. Mechanical Response Time................................................ 20 D. CMOS Fabrication P ro cess..........................................................23 III DESIGN A. Thermal A ctuators..........................................................................25 B. Spring-Mass Resonators.................................................................30 C. Frequency Dependence and Resonant B ehavior......................33 1. Resonance o f Drive, Sense and Transducer A xes 34 2. Multiple Resonances........................................................... 38 3. Q as a Design P aram eter.................................................. 43 4. Mechanical Response Time................................................53 TABLE OF CONTENTS (continued) DESIGN (continued) C. Frequency Dependence and Resonant Behavior (continued) 5. Force T ransm issibility.......................................................54 6 . R esp o n sivity......................................................................... 56 7. Transducer Interface Circuit N o is e ................................62 8. S en sitivity ..............................................................................63 9. Phase o f Feedback Control L o o p .................................... 65 D. Noise and O ther Error S o u rc e s ................................................... 67 1. Electrical Noise in Piezoresistors.................................... 70 2. Electrical Noise in the Interface C ircuitry.................... 73 3. Thermal Motion o f P roof M a sses.................................... 75 4. Temperature D ependence.................................................. 77 5. Errors due to M anufacturing Lim itations...................... 79 6. Use o f Feedback Control ......................................... 83 E. Design of Complex S y s te m s .......................................................... 88 1. Design Hierarchy o f Complex System s...........................91 2. Opportunities fo r Invention................................................97 vii TABLE OF CONTENTS (continued) IV ANALYSIS AND RESULTS A. Thermal A ctuators...........................................................................102 B. Spring-M ass Resonators..................................................................108 C. Resonant B ehavior...........................................................................110 1. Quality Factor...................................................................... 110 2. Mechanical Response Time............................................... 114 3. R esp o n sivity.........................................................................115 4. S en sitivity............................................................................. 117 V C O N C L U S IO N ....................................................................................... 120 VI R E F E R E N C E S ........................................................................................121 L IST or FIG U R E S Figure I M echanical System D esig n .......................................................................... 5 Figure 2 Mechanical System M odel.............................................................................6 Figure .1 SEM Photograph of Mechanical System D esign..................................... 7 Figure 4 Transducer Interface C irc u it........................................................................8 Figure 5 Derivation of Coriolis Acceleration............................................................ 9 Figure 6 Coriolis Effect on Tuning Fork V ibration...............................................13 Figure 7 Cross-Sectional View of CMOS Technology Layer Structure . . . .24 Figure 8 Thermal A ctuator..........................................................................................25 Figure 9 Thermal and Restoring Forces...................................................................32 Figure 10 Cross-Section View of Pit Feature............................................................41 Figure 11 Variation of Geometry Defined by Pit F e a tu re .....................................41 Figure 12 Natural Frequency Mismatch as Function of Beam W id th ................42 ix LIST OF FIGURES (continued) Figure 13 Problem of Stokes’ Oscillating Plate...................................................... 46 Figure 14 Force Trunsm issibility............................................................................... 56 Figure 15 Responsivity................................................................................................. 61 Figure 16 Transducer Interface Circuit N o ise ........................................................ 62 Figure 17 S en sitiv ity ................................................................................................... 65 Figure 18 Phase of Feedback Control Loop.............................................................67 Figure 19 Torsional Imbalance due to Nonplanar O scillations...........................82 Figure 20 Overview of Open-Loop Sensor Operation...........................................84 Figure 2 1 Overview of Closed-Loop Sensor Operation........................................ 86 Figure 22 Example of Design H ierarchy................................................................. 93 Figure 23 Design Hierarchy for Thermal Actuator Subsystem ...........................94 Figure 24 Design Hierarchy for Spring-Mass Subsystem ....................................95 X LIST OF FIGURES (continued) Figure 25 Design Hierarchy of Transducer S ubsystem .........................................97 Figure 26 Thermal Actuator Buckling Force and Residual Stress...................... 103 Figure 27 Residual Stress in Thermal A ctu ato rs.................................................. 105 Figure 28 Thermal Actuator Buckled Due to Residual S tress............................106 Figure 29 Coupled Actuator-Spring Mass Test S tru c tu re s................................107 Figure 30 Proof Mass D isplacem ent........................................................................107 Figure 31 Thermal Actuator Time C o n sta n t..........................................................108 Figure 32 Natural Frequency of Spring-Mass S ystem ..........................................110 Figure 33 Quality Factor versus Natural F req u en cy ............................................I l l Figure 34 Quality Factor versus Spring L ength..................................................... 113 Figure 35 Mechanical Response Time versus Natural Frequency...................... 114 Figure 36 Mechanical Response Time versus Spring L e n g th ............................115 xi LIST OF FIGURES (continued) Figure 37 Responsivity versus Natural Frequency................................................ 116 Figure 38 Responsivity versus Spring L en g th ....................................................... 117 Figure 39 Sensitivity versus Natural F req u e n c y ...................................................118 Figure 40 Sensitivity versus Spring L ength............................................................119 xii LIST OF TABLES Table 1 Summary of CMOS L ay ers........................................................................24 Table 2 Beams on Pit L a y e r.....................................................................................27 Table 3 Beams on First Oxide...................................................................................28 Table 4 Beams on Field Oxide and First O x id e ...................................................28 Table 5 Beams on First and Second O x id es......................................................... 28 Table 6 Summary of Transfer F u n c tio n s.............................................................. 66 Table 7 Material Properties for Silicon................................................................. 101 Table 8 Other Material Properties.......................................................................... 101 xiii LIST OF SYMBOLS Symbol Meaning Page A Cross-sectional area of beam (A = a x b) 26 A P Area of proof mass plate 31, 48 a Beam or layer thickness — ac Coriolis acceleration 12 b Beam width — t> d Damping coefficient 51 c Thermal capacitance 29 CP Specific heat at constant pressure — E Young’s modulus — Es Energy stored in spring per cycle of oscillation 18 Ed Energy required to overcome damping per cycle of oscillation 18, 48 Fc Coriolis force 12 Fs Spring restoring force 55 Flh Thermal force 26 Fv Viscous damping force 48,51 f Circular frequency 29 G Gage factor — g Thermal conductance 29 I Moment o f inertia 53 k a Spring constant for shorter support spring (length L„) 59 xiv LIST OF SYMBOLS (Continued) Symbol Meaning Page k.n Spring constant for longer support spring (length Lm) 31 La Shorter support beam or spring length — Liu Longer support beam or spring length — Lpr Length of piezoresistor — m Proof mass 31 n Number of cycles of free oscillation 21 Pcrit Critical Euler buckling load 102 Q Quality factor 19, 50 R Responsivity 56, 60 Rc Bending moment arm for Coriolis force 59 R< ), Rs Resistance of reference and sensor piezoresistors 27 r Polar coordinate 9 rA Proof mass deflection due to thermal driving force 10 S Sensitivity 63, 73 T Temperature, period o f oscillation — , 18 AT Change in temperature due to resistive heating of actuator 27 Tc Temperature coefficient of resistance 27 t Time — u Velocity of damping fluid 47 V Voltage — V Volume of proof mass 31 vn Electrical noise voltage 72 XV LIST OF SYMBOLS (Continued) Symbol Meaning Page w Thermal actuator length — Aw Deflection of thermal actuator 26 wp Width o f proof mass plate (Ap = wp2) — *n Proof mass displacement during /j,h cycle of free oscillation 20 Z Distance to neutral axis of beam — a Thermal conductivity 29 P Coefficient of thermal expansion — X Magnitude of displacement for harmonic oscillator solution 17 An Logarithmic decrement o f free oscillation of n cycles 20 e Strain 57 n. y. K $ Scaling factors — 4 > Arbitrary phase angle — K Natural logarithm of Xi/xn 20 p Viscosity of damping fluid — V Dynamic viscosity o f damping fluid (v = \i/px) — e Phase angle between driving force and displacement of harmonic oscillator, polar coordinate 17,9 p Radius o f curvature 57 Px Mass density of material x — <*r Residual stress 105 Shear stress exerted by damping fluid on oscillating plate 47 xvi LIST OF SYMBOLS (Continued) Symbol Meaning Page Mechanical response time of free oscillation 22, 53 *th Thermal time constant 29 Q Rate of rotation 5 (0d Angular drive frequency of oscillation 30 (% fti()X Natural angular frequency, natural angular frequency in x-direction 30 c Modified damping coefficient = b/2m0}(,) 17 I. ABSTRACT The complete design cycle for micromachined angular rate sensors is presented. Original analysis is combined with previously published results culled from a diverse spectrum of sources, and form a complete treatment o f the pertinent design issues. Main functional components (actuator, oscillating mass, transducer and interface circuit) are introduced with design analysis and methodologies, and the dependence of component and sensor performance on operating frequency is discussed in detail. System-level issues such as electrical noise, mechanical and thermal error sources, and the use of feedback control are also addressed. A general introduction to resonant systems, and angular rate sensors in particular, is also presented. A discussion on the design of complex systems is given, and the angular rate sensor system is used to illustrate the design principles conveyed. The concept of design hierarchies is used to decompose the complex functional system of the angular rate sensor in a manner similar to that used in VLSI. System-level and component designs are presented for the mechanical portion of a micromachined angular rale sensor fabricated in a standard commercial CMOS process. The sensor consists of a thermal actuator coupled to a spring-mass system. Actuator performance is characterized, and generated force and displacements are correlated to theory as functions of actuator dimensions. Generated forces are quite high (typically 100-200(iN), while actuator deflections are small (on the order of tens of nanometers) due to a very high structural stiffness of the actuator for axial deformations. Residual stress in the actuator structure is also examined. A thorough analysis of sensor dependence on drive frequency is undertaken, and preliminary design analysis is presented for the spring-mass system. Resonant frequency of the spring-mass oscillator is calculated for various configurations, and a novel analysis relating Q factor to system design parameters and device operating conditions is presented. Finally, a brief discussion of future work is given. II. INTRODUCTION 3 A. Angular Rale Sensors Angular rate sensors constitute a class of instruments that are well understood on the macro scale [ 1], and that are well-suited to implementation on the micro scale. Among the benefits of implementing such devices using silicon micromachining techniques are the high Q performance of polysilicon as a structural material for micro-resonators [2], the ability to implement resonators without contact friction [3], and a drastic reduction in overall device size. Regardless of the method of manufacture, high-precision angular rate sensors remain a difficult design task. Currently available conventionally-machined sensors may cost in excess of $ 1000 per unit for navigation-grade devices. These devices are typically made using traditional precision manufacturing processes and are very intensive in skilled labor, and often are produced on a one-at-a-time basis. Clearly the ability to micromachine angular rate sensors in a standard process would result in a dramatic decrease in per-unit cost. Angular rate sensors pose such a design challenge due their extreme sensitivity to errors in manufacture or wear in use, which directly impair the sensor’s ability to measure its objective: the rate o f rotation of the sensor and the body to which it is attached; however, due to their widespread use as navigation aids (in which the objective is angular position, not necessarily the rate at which it is changing), the errors incurred in an angular rate sensor’s operation are integrated along with the true sensor output to yield an angular position. Thus, small errors in angular rate detection can lead to substantial errors in the calculation of angular position. For this reason, it is especially important to the designer of such a device to minimize errors. Typical angular rate sensor designs consist of an actuation device coupled to a resonating spring-mass system. The rate at which the resonator is driven is determined by the proof mass, sizing of the springs, and the amount of damping that is desired in the system. The presence of contact friction as well as other significant drawbacks results in the tuning fork variety to be the most natural form of angular rate sensor for implementation as a micromachined device [3). This has been the approach taken here. A schematic overview of the mechanical system is given in Figure 1. The mechanical system can be modeled analytically as a driven harmonic oscillator as shown in Figure 2. Figure 3 shows an SEM photograph of a typical test structure, including a thermal actuator, support springs and a proof mass, with the component parts labeled. A complete, functional angular rate sensor system will consist of the actuator and proof masses, piezoresistive transducers to convert the mechanical measure o f angular rate to an electrical output, an interface circuit and associated processing electronics. As the first step towards realizing such a system, the scope of the initial work is limited to demonstration o f a thermal actuator and the ability to establish and maintain resonant response of the proof masses. Plans for future work include incorporating the test structures detailed here into a complete sensor system as described above. 5 Analysis and design considerations will be presented for alt aspects involved in the design of an angular rate sensor, although only a portion of what is presented in this thesis has been implemented in silicon at the present time. m m m m Thermal Actuator Output y Torque/ ~ 7 Piezoresist j vey Strain Gauges Figure 1: Mechanical System Design (a) As current is pulsed through the thermal actuator beam, a poly silicon resistor is successively heated and cooled, thus causing a rhythmic expansion and contraction of the actuator. Due to the coupling of the spring-mass elements at each end of the actuator, a resonant oscillation is established in the plane of the paper. The shaded areas indicate where the substrate will be removed to render the structure free. 6 m m x + bdx + kx - F{ )cos ( coy + < j> ) Figure 2. Mechanical System Model The resonant frequency of this oscillation for the lumped parameter model is a function of the size of the proof mass (m), stiffness of the supporting springs (*„,), air damping and actuator dimensions. To illustrate the operation of the system shown in Figure 1, consider the proof masses oscillating in the plane of the paper as shown. The rotation vector Q is orthogonal to the velocity vector of the proof masses; as shown, the sensor would detect angular rates about the Q-axis. For this reason, the direction of proof mass deflection is referred to as the drive axis, and the £2-axis is known as the sense axis. For a nonzero rotation rate Q the deflections of the proof masses will deviate from being strictly in the plane of the paper in Figure I, and will deflect an amount normal to the plane of the paper in Figure 1. This normal deflection is directly proportional to rotation rate Q and proof mass deflection rA, and is the result of the Coriolis force. (The Coriolis effect will be addressed in detail in the following section.) These deflections normal to the substrate will induce a torque in the piezoresistive strain gauges mounted in the base o f the sensor. As piezoresistive materials undergo a change in resistance when subjected to strain, by monitoring the fluctuation in 7 resistance of the strain gauge elements, a measurement of rotation rate can be obtained. A natural implementation for the piezoresistors is in the form of a Wheatstone bridge, whereby the bridge is normally balanced for no change in the piezoresistors (i.e., no strain or rotation). The transducer interface circuit is shown in Figure 4. Shorter Support Beam Etch Holes Bonding Pads Thermal 'Actuator Longer Support Beam Figure 3: SEM Photograph of Mechanical System Design SEM photograph showing the implementation of test structures in the manner of Figure I . These test structures only contain one proof mass per actuator; a full sensor test structure will have a single actuator driving two proof masses as in Figure 1 . 8 To Processing Circuitry Figure 4: Transducer Interface Circuit Circuit schematic for the transducer interface, a Wheatstone bridge circuit balanced upon the two piczoresistive strain gauge elements (labeled Rs). Two reference resistors complete the bridge circuit. All four resistors are geometrically identical and made from polysilicon. The change in resistance is measured as a voltage difference, which is then input to an operational amplifier to increase the sensor signal so it can be processed by additional downstream circuitry (not shown). Gyroscopes and angular rate sensors utilize an effect peculiar to rotating frames of reference in order to measure, respectively, angular position and velocity. Named after the French mathematician who first presented a complete analysis of the phenomenon, the Coriolis effect results from the apparent differences in a body’s motion as witnessed by observers in rotating and nonrotating reference frames (4J. For example, if two people are standing on a rotating merry-go-round and wish to toss a ball back and forth between them, the ball will appear to travel in different paths to the two people on the merry-go-round and someone standing stationary on the ground who is not rotating with them. The Coriolis effect is purely an artifact of one’s particular reference frame. Thus, the people moving with the merry-go-round associate the ball’s B. Principle o f Operation 9 flight with their rotating coordinate system, whereas the ball and stationary observer relate the ball’s path to their fixed reference frames of the Earth. The people rotating see the ball as tracing a curved trajectory, and yet the ball and the stationary observer report a straight line path. In reality, the ball does travel in a straight line, and the merry-go-round is merely rotating out from under it. However, to the people on the merry-go-round it appears as if a force has acted on the ball, changing its direction. In order to quantify this inertial force which most certainly exists to observers in the rotating frame, consider the diagram given in Figure 5. Successive time derivative of position will yield velocity and acceleration terms. Once the expression for the angular acceleration is known, the force can be computed from Newton’s second law of motion [5|. Figure 5: Derivation of Coriolis Acceleration The position of a point C which is moving along an arbitrary path can be expressed through three generalized reference frames as follows: Frame I (axes Xl-Yl, with unit vectors i 1 , j 1 and k 1 ) can be any inertial frame (i.e., translating at constant velocity, but not rotating or accelerating); frame 2 can be any non-inertial, non-rotating frame; frame 3 is rotating relative to frame 2. PAB is the position of point B (origin of frames 2 and 3) relative to A (origin of frame I); PBC is the position of point C relative to point B. / / / / / * ~ X1 A 10 We begin with the position vector used to describe the location of point C: P AC = P AB + P BC = P AB + P rel ^ where PAli is defined relative to unit vectors ?f , y , and Jt( ; is defined with respect to unit vectors i^.j^ and The position vector PAii is the location of the point in reference frame I, and any coordinate system may be chosen to represent this point. For example, the vectors PAii and P ^t in cartesian coordinates are ? ah = * 'i (2) ?rel = x h + y J l + z h (3) The first derivative of position with respect to time represents the velocity of C: -C4 ~ -AC ~ -AB + ?rel and substituting the expressions for PAB and PKi yields yC a = * ' 1 + yJ i + ik \ + + yh + z * 3 + xii + yJi+ **3 (5 ) The important distinction between frames 1 and 3 is seen in the difference of their lime derivatives; the unit vectors /] , y'j and are constant, and so their derivative with respect to time is zero; however, the unit vectors i-,, 7 3 and £ 3 do have nonzero derivatives, as seen in (5). The following simplifications can be made at this point: = xi, + y j, + zk, {6) Frri = + >'h + + x iy + v/3 + z 'k) = vrej + 12, x Pfe{ (7) And so the velocity of point C is given by vrA = ''d i + V , + £2-. x P . (8 ) ~C 4 *ti 3 3 ~r#’/ ' 7 The time derivative o f velocity will produce acceleration: ~C4 = ^C4 = -H3 + -ret + ^ 3 X ^ rel + ^ 3 X ^ r e t ^ Again utilizing the fact that the time derivatives of the unit vectors i], j j and k i are zero and noting the following relationships: = a 3 ( a is the angular acceleration of O and P rel = yrft + £23 x P t as before results in the acceleration of point C relative to an inertial frame: -C 4 ~ -S 3 + a 3 X ?ret + ^ 3 X ^ 3 X ^ ret^ + ^ ^ 3 X - r el + -ret ^ ^ Recall from the discussion on the Coriolis effect on page 8 that the Coriolis force (and therefore the Coriolis acceleration also) are only evident to an observer in a rotating frame of reference; an observer in an inertial frame (i.e., frame 1) or an accelerating frame that is not rotating (frame 2) will not report the presence of the Coriolis force. Thus, the Coriolis force is due to an acceleration only present in frame 3. By inspection of the expression for the general acceleration of point C given in equation (10), the Coriolis acceleration then is simply « , = 2ft , x vref (II) where vre{ = for proof mass displacement of rA driven at a frequency o)(/. Thus the Coriolis force is F = ma = 2m il, x v = 2m r, a) 12 (12) " r -r S - rtt A 3 v * Note that the Coriolis force is directly proportional to the linear velocity of C and yet acts orthogonally to it. (A convenient way to visualize the Coriolis effect is to remember that v, f t and ac form a coordinate system of their own, and so must be mutually orthogonal to one another.) In order to illustrate the Coriolis force in a system similar to that which is of interest here, Figure 6 shows a tuning fork being driven and rotated in much the same manner as the system in Figure 1 is expected to operate. Strictly speaking, tuning fork angular rate sensors do not directly measure angular rate, but rather the Coriolis force exerted on the oscillating proof masses. As stated in (12), Coriolis force is directly proportional to angular rate ft, and what remains is the problem of accurately calculating m, rA and The difficulties in the manufacture and determination of theoretical and actual values of m, rA and (a(i with high fidelity is of utmost importance in both the design and characterization of angular rate sensors. This topic will be more deeply addressed in the section Errors due to Manufacturing Limitations on page 79 as part of the chapter on device design. A Input Rotation Rate 12 Coriolis Acceleration ac Proof Mass Deflection r^ Driving Force F(t) Figure 6: Coriolis Effect on Tbning Fork Vibration The Coriolis acceleration acts to deflect the proof masses in a direction normal to their driven oscillations.The Coriolis force will always lag behind the driving excitation in phase, with a maximum difference of 90° at resonance. Illustration after figure from [10], 14 C. Resonant System s In the broadest terms, oscillators are merely objects, mechanisms or quantities which vary periodically in time. The study of oscillatory behavior is fundamental to a wide number of diverse scientific fields, and a brief list of well-understood systems which exhibit oscillatory behavior bears this out: a weight bouncing on a spring; oscillations of charge in an electrical circuit; vibrations of a tuning fork, resulting in acoustic waves; vibrations of electrons, giving rise to electromagnetic waves; and so on. While periodicity is certainly an appealing feature which has beneficial implications in many applications, much of the interest in oscillators is due to the phenomenon known as resonance. Resonance is simply the preference of an entity to oscillate at a particular frequency or frequencies. When driven at a so-called resonant frequency, an oscillatory system exhibits a tremendous increase in its response to the driving force. Thus, the displacement of a mass attached to a spring, when driven at a resonant frequency, is much larger than the displacement for the same system “off-resonance”. It is this highly selective response to input frequency, as well as any output signal’s additional dependence on frequency and phase, which further enhance resonant system s’ capabilities as sensors [6]. Alternatively, less work is required to sustain an oscillation if it is driven on resonance than otherwise. It is from this perspective that tuning fork angular rate sensors are most often scrutinized. The parameter most often used to 15 characterize such a resonant system is Q, known as the quality factor, figure of merit, or simply Q factor. Q may be defined as the ratio of energy stored per oscillation (£ ,) to work done in overcoming damping per oscillation (£ / + kmx = x ^ o cos ( ° V + (14) 16 The solution of (14) will in general depend on the amount o f damping present in the oscillating system. The case of interest in the design of resonant sensors is for subcritical damping, as this allows for a sustained oscillation. (Critically- and supercritical ly-damped systems do not oscillate, as the motion decays in a single period or less than a single period, respectively.) Note that (14) is a second order nonhomogeneous ordinary differential equation; as such, the solution may he written as the sum of the solution to the homogeneous equation and a particular solution of the nonhomogeneous equation: In the study of systems modeled as harmonic oscillators, the solutions given in (15) represent two distinct physical conditions: The solution to the equation of motion in the absence of a driving force is known as the transient solution, whereas the particular solution represents what is referred to as the steady-state motion o f the system. The transient solution describes the system ’s transition between a state of no motion and steady-state motion; the system behaves symmetrically whether the system is initially at rest, or comes to rest after the driving force is removed. The transient solution of the driven harmonic oscillator is [7] X(t) = Xh +Xp (15) (16) 17 The term £ is introduced for convenience and is related to the damping coefficient hd Once the oscillation has completed the transition from no motion to harmonic motion the transient behavior expressed by (16) will no longer be evident in the system; the motion is then said to be in steady-state form, and will remain in steady- state until the driving force is removed. At that time the transient solution will again appear in the response of the system. The steady-state solution of (14) for subcritical damping is given by (7] * „ (0 = x ^ o c o s (<v + < t > - e ) (17) where x^o > s (he modulus of x, tod the driving frequency, < > is an arbitrary phase angle, and 0 is the phase between the driving force and the displacement. X = m - x + iy (18) 0 = tan' 1 tan -1 = -ta n -t 2 2 C O ,, - to v 0 d' (19) 18 Note that the displacement of the proof mass will always lag behind the driving force, as the argument of 0 is always negative. For tod = too, we have (20) and thus the maximum phase shift between the driving force and displacement occurs at resonance. 2. Quality Factor Each term in (14) represents a force, and so to calculate the energy stored by the spring or the work done in overcoming damping we need only integrate those terms in (14) over one complete cycle of the oscillation. The energy stored by the spring per cycle is given by Es: where the parameter T denotes the time required for one complete cycle of oscillation: (% 7 = 2k . This integral may be evaluated by substituting the solution for xp (t) given in (21 ) (17): E s = (P^o) 2J^sin0J0/cosco0/c// = 0) 2 (22) 19 In a similar manner, the work done in overcoming damping per cycle of oscillation is found to be e “ = i M s ) 2* = ,23) Q may be written in terms of the basic coefficients for the equation of motion as follows: Q = l i t \ EdJ - r ~ < 24> hd where m is the proof mass, bj is the damping coefficient in (14), and is the natural 2 km frequency of the system. Noting that ion = — , it is evident immediately that Q is a u m function of the coefficients of (14); furthermore, km and m are known functions of device geometry (being the spring constant and proof mass, respectively), and are thus design parameters themselves. Thus we need only determine hd in terms of design parameters in order to know how Q will vary as a design parameter. The derivation of Q as a design variable is presented in the section on Q as a Design Parameter on page 43 as part of the chapter on device design. A value of Q is often quoted to indicate the efficiency o f an oscillating system; due to the inherent dependence of Q on operating frequency, it is also sometimes used to quantify a resonant system ’s resolution [8 ]. J. Mechanical Response Time From the definition of Q given earlier, it is apparent that there is a relationship between the length of time a system exhibits transient behavior and the amount of damping present in the system. Intuitively, a high Q system will continue to oscillate for a longer period of time after the driving force has been removed than a low Q system, as each successive cycle will consume a smaller amount of the total stored energy. Thus we expect the response time o f the system to be proportional to Q, The response time for the system to reach steady stale amplitude can be derived in the following manner. The decay in amplitude of a freely oscillating system which is under the influence of viscous damping is typically measured by a quantity known as the logarithmic decrement: A = I n - (25) x 2 where x t and x 2 represent the amplitude of motion of any two successive maxima, and x(t) is given by (16). For systems with small damping, the logarithmic decrement for displacement amplitudes which are separated by an integral number of cycles n is (9) For a system which the amplitude is decaying from steady-state to rest, the ratio of the first amplitude (steady state motion) to the last (system comes to rest) will be infinite. (Strictly speaking, for the purposes of establishing the mechanical response time of the system, the minimum amplitude need not be zero, but can be made arbitrarily small to avoid an infinite argument for the natural logarithm function). Formally, we may define the following quantity to represent the decaying amplitude: tc = iim In x . ->» (27) where rA is the amplitude of motion in the steady-state condition. Solving for n, the number of cycles need to bring the system to rest (or as close to rest for which a displacement measurement may be made) gives the following; n = (28) A„ n An alternative definition for A is For the case of small damping, the denominator in (29) may be taken as unity, and the following expression results for the number of cycles needed to bring the system to rest: n n (30) The mechanical response time may be written as Here we see that the mechanical response time is directly proportional to Q. The effect of high Q on system response time serves to increase both the start up and wind-down times, and is the only true penalty for operation of a high Q system at resonance. Note that the expression for mechanical response time which is commonly stated in the literature (e.g., [ 10]) is an approximation of (31): (32) where the factor K has been om itted. 23 D. CM O S Fabrication Process In addition to lowering the cost o f manufacture, utilizing a commercially-available CMOS fabrication process [ 11J also offers integration of the mechanical sensor with commercial quality on-chip signal conditioning circuitry. Furthermore, once an angular rate sensor has been demonstrated in a standard CMOS process, the design may be scaled as CMOS technology improves and progresses to smaller feature sizes. Although micromachined angular rate sensors have previously been reported (3] [I0 | [12] [13], it is my goal to demonstrate in CMOS the feasibility of such devices despite the lack of control over specific processing issues. The previously demonstrated angular rate sensors [3] [10] [12] [13] all represent custom “hand-made” devices in which full control over fabrication parameters is maintained by the designer. In these cases it is often difficult to distinguish between device design and fabrication process development, as there are trade-offs frequently made between the two domains. The emphasis here is on standard CMOS processes in which the designer does not have control over the specific issues of the fabrication process. A cross-sectional view of the physical layers available in the CMOS process for which the design described here will be fabricated is given in Figure 7. 24 Insulators Overglass 1.0pm Second Oxide 0.65 pm First Oxide 0.85pm Poly Oxide 0.04pm Field Oxide 0.60pm Substrate Figure 7: Cross-Sectional View of CMOS Technology Layer Structure Layer names and thicknesses for a standard CMOS pmcess. Layers are deposited in the order shown, starting from the substrate and ending with the Overglass layer The total thickness of all layers combined is approximately 5.65pm. Table 1: Summary of CMOS Layers Layer Abbreviations Function Thickness (pm) Field Oxide FOx Insulator 0.6<) First Polysilicon Poly 1, PI Conductor 0.40 Polysilicon Oxide PolyOx. POx Insulator 0.04 Second Polysilicon Poly 2. P2 Conductor 0.40 First Oxide Oxl Insulator 0.85 First Metal Metal 1. M l Conductor 0.60 Second Oxide 0x2 Insulator 0.65 Second Metal Metal2, M2 Conductor 1.15 Overglass OG Insulator Planarization 1.00 Conductors Second Metal 1.15pm First Metal 0.60pm Second Poly 0.40pm First Poly 0.40pm 25 III. DESIGN A. Thermal Actuators The spring-mass system is driven to resonance by a thermal actuator (Figure 8). Similar actuators have previously been reported [14] [15], in which a polysilicon resistor is used to create a temperature gradient in the actuator structure. Deflection results due to differential thermal expansion between hot and cold elements of the actuator. However, in the present case the actuator structure is not a cantilever beam free to expand and deflect both parallel and normal to the substrate (e.g., as in [14] [15]). Figure 8: Thermal Actuator SEM photograph of typical thermal actuator. The structural materials are polysilicon and silicon dioxide. The actuator is bordered on either side by cuts through the oxide layers, where the exposed silicon substrate can be seen to have been removed. The oxide layers normally provide insulation between the four conductive layers (two polysilicon and two metal) present in the CMOS process. Rather, the present design is a clamped-clamped beam, constrained from motion normal to the substrate for thermal forces less than the critical Euler buckling load, and therefore forced to expand axially (parallel to the substrate). The effective spring constant of the structure is much greater in the axial direction, and therefore a much greater force is generated in exchange for smaller deflections. Thermal isolation is achieved by stacking oxide cuts next to the actuator, and the exposed silicon is etched during a maskless post-processing release step [16] [17]. The performance of the thermal actuator was characterized for two parameters (maximum generated force and displacement) as functions of beam geometry. Thermal force is linearly dependant on actuator cross-sectional area, and deflection relies linearly on beam length. Both characteristics are directly proportional to changes in temperature due to resistive healing: Flh = A E $ (AT) (33) Aw = wP(A7*) (34) where A is the actuator cross-sectional area, E is Young’s modulus, w is the beam length, p is the thermal coefficient o f expansion and AT is the change in temperature of 27 the beam due to resistive heating. The use of resistors as temperature sensors is widely known, and AT is easily calculated for a known initial resistance [14] [18]: R(T) , *o AT = — ^ ----- (35) < * where R() is a reference value of resistance and Tc is the temperature coefficient of resistance for polysilicon. Test structures were designed in two ways to investigate both residual stress and thermal buckling force. First, actuator length was varied from 50pm to 250pm in 50pm increments while keeping the structural layers constant at 5.6pm thick; these are the structures shown in Figure 26. Secondly, actuator length was kept fixed at 50pm , and the structural layers were varied. Examples of this type of structure are shown in Figure 8 and the foreground of Figure 27. Table 2 through Table 5 list the actuator test structures of this second type which were fabricated. Tbble 2: Beams on Pit Layer* Layers Thickness Metal2 1.10pm Metal 1 + Metal 2 1.65 pm Polyt + Poly 2 + Metal 1 1,35pm Poly 1 + Poly 2 + Metal 2 1.90pm Poly 1 + Poly 2 + Metal 1 + Metal 2 2.45pm a. The pit layer is described in Figure 10 on page 41. Table 3: Beams on First Oxide Layers Thickness Ox 1 + Metal2 1.95 Jim Ox 1 + Metal 1 + MetaI2 2.50 Jim Oxl + Poly 1 + Poly2 + Metal 1 2.20pm Ox 1 + Poly 1 + Poty2 + Melal2 2.75pm Ox 1 + Poly 1 + Poly2 + Metal 1 + Metal2 3.3pm Table 4; Beams on Field Oxide and First Oxide Layers Thickness FOx + Ox 1 + Metal2 2.55 pm FOx + Ox 1 + Meta] 1 + Metal2 3.10pm FOx + Ox 1 + Poly 1 + Poly2 + Metal 1 2.80pm FOx + Ox 1 + Poly 1 + Poly2 + Metal2 3.35pm FOx + Ox 1 + Poly 1 + Poly 2 + Metal 1 + Metal 2 3.90pm Table 5: Beams on First and Second Oxides Layers Thickness Ox 1 + 0 x 2 + Melal2 2.60pm Ox 1 + 0x 2 + Metal 1 + Metal 2 3.15pm Ox 1 + 0 x 2 + Poly 1 + Poly2 + Metal 1 2.85pm Oxl + 0 x2 + Poly] + Poly2 + Metal2 3.40pm Ox 1 + 0 x2 + Poly 1 + Poly2 + Metal 1 + Meta)2 3.95 pm Clearly the spring-mass system can only be driven as fast as the actuator can be cycled through the heating and cooling process. The thermal time constant of the actuator is the ratio of thermal capacity to thermal conductance: c _ c j f i s i abw _ cPPs,w 2 (36) a where if, is the specific heat at constant pressure, py, is the material density, and a is the thermal conductivity for a resistor having thickness a, width b, and length w. All polysilicon material properties use values quoted for single crystal silicon. (Refer to Table 7 and Table 8 on page 101 at the end of this chapter for a list of material property values used in design calculations). Values of T,h ranged from microseconds to milliseconds for the actuator configurations considered. These time constants translate into frequencies in the range of kHz to MHz, according to the following relationships: 30 The desired range of operation for the angular rate sensor i s / = 1-10 kHz, and so all actuator configurations fabricated are acceptable as drivers for the oscillator. The net result of the previous discussion is that actuators with short beam lengths generate less displacement, while having shorter cycle times, than do actuators with longer beam lengths. Note that angular rate detection is linearly proportional to the displacement of the proof masses. Thus, actuators with shorter thermal time constants (i.e., lower thermal displacements) will produce smaller output signals for a given angular rate than actuators with longer thermal time constants. The design of the spring-mass resonator elem ents was guided by the desired drive frequency of the oscillations, (0j. This parameter is related to the natural frequency of the spring-mass element by the expression where ro0 is the natural frequency o f the spring-mass element, hd is the damping coefficient, and m is the proof mass. Recall B. Spring-M ass Resonators (40) to, (41) 'o m where km is the effective spring constant of the structure for in-plane motion and is found to be k E ah3 (42) m for a beam of thickness u, width h and length Lm. For small amounts of damping, the common approximation = to0 is made. The proof mass m is the product of the average densities of the materials composing m and its volume; the volume of the proof mass is taken to be the product o f plate thickness and area: (43) ignores contributions of the support springs to the proof mass. This approximation is consistent with the lumped-parameter model pictured in Figure 2. For comparison, the mass of a support spring 5 pm wide, 5 pm thick and 500pm long is 8 % that of a plate 5 pm thick and 250pm on a side; for a plate 500pm on a side the same support spring is only 2 % of the mass of the plate. Factors affecting the deflection of the proof mass are dimensions of the actuator and both support springs (labeled km and ka in Figure 1). Deflection of the proof mass is the net result o f the balance of forces and displacements that occurs among the actuator and both support springs. The structural stiffness of the actuator is very high in the axial direction, and the design trade-off between thermal force and proof mass (43) 32 displacement can be visualized as in Figure 9, where the spring forces of the actuator and smaller support springs are shown. Figure 9 is an illustration of the trade-off between force in the thermal actuator and actuator displacement: The balance point where the thermal force of the actuator and restoring force of the support spring are equal will determine the net force and displacement for the structure. The points of interest in Figure 9 are intersections between the thermal force (plotted with a negative slope, decreasing from left to right) and the restoring force of the support beams. Thermal force for AT = 25 °C 400 = I0 |im 3(X) 250 £ 200 150 100 Actuator Deflection (nm) Figure 9; Thermal and Restoring Forces illustration of design trade-off involving thermal force, actuator deflection and sizing of the support springs. Shorter support spring lengths are shown to be 10pm and 25pm. The force balance between the actuator and two candidate support spring lengths is shown by the two paints of intersection between the thermal force and restoring spring forces. These points of intersection determine the net force and displacement of the actuator and spring-mass system, i.e., if the actuator shown were coupled to a shorter support spring of length 10pm, the net force and actuator displacement would be approximately 200pN and I4nm, respectively. Equation (42) represents the stiffness of the support springs in the direction of the displacement of the thermal actuator, i.e., the plane of the paper of Figure 1. To obtain the stiffness of the support springs relative to out-of-plane (i.e., normal to the paper in Figure 1), we can merely interchange a and b in (42). The dimension a is the structural thickness in either case, and thus we can see that k increases linearly with a for in plane deflections, hut increases as a * for out-of-plane movement. For this reason, the support springs were made of the 4 field oxide layers, the two available layers of polysilicon, and two layers of metallization in an attempt to maximize the thickness of the beam. This maximum beam thickness is determined by the fabrication process, and for the process in question is approximately 5pm . Finally, in order to obtain the low natural frequency described earlier, the longer support springs were sized between 500pm and 1 mm. C. Frequency Dependence and Resonant Behavior Frequency analysis o f an oscillatory system is typically most concerned with the system response to a narrow range of drive frequencies close to the expected resonant frequencies for the system. But there are important facets of resonant system behavior for drive frequencies either well above or below the expected resonances as well. This section will explore the response of a tuning fork angular rate sensor as a function of frequency for various mechanical and electrical quantities o f interest. These include deflection o f the proof masses due to both drive and Coriolis forces, sensor 34 responsivity and sensitivity, electrical noise, and some insight on feedback control as a function of system drive frequency. The issue of noise and the use of feedback control will be addressed in more depth in the following section, Noise and Other Error Sources on page 67, which will also discuss the role of resonance on noise and error sources. /. Resonance o f Drive, Sense and Transducer Axes A key design decision which must be made early in the process of tuning fork angular rate sensor design is to establish which (if any) system component will be designed to be operated in a resonant condition. For the design presented here, there are three oscillating elements, and in practice each will have a distinct natural frequency. (The discussion that follows will only address the lowest resonant frequency of the tuning fork system— referred to as the natural frequency— although the conclusions and method of analysis also applies to higher modes of the system. These higher modes are usually referred to as the harmonics of the system.) The three oscillating elements in the current design are the drive and sense axes of the tuning fork, and the flexure containing the piezoresistive sensor elements themselves. Operation at any o f these resonant frequencies will result in the appropriate quantity (proof mass deflection due to drive excitation, deflection due to Coriolis force, or change in resistance o f the piezoresistors due to increased strain) being amplified by a factor of Q as stated in the introductory discussion Resonant Systems on page 14. Each 35 of these three resonant conditions has important implications. In general the principle of operation at resonance is attractive due to the mechanical amplification of sensor output. Because the amplification is mechanical in nature, operation in a resonant mode does not similarly increase the electrical noise present in the system, as would occur if the electrical measure of rotation rate were input to an operational amplifier. For a surface micromachined device, the size of the proof mass is limited by the number and thicknesses of the process layers, given that the lateral dimensions cannot be made arbitrarily large. This results in a value for the proof mass value in the range of 10' 10 to I O'9 kilograms. Thus, the magnitude of the Coriolis force will also be partly limited by the fabrication process, and for this reason it is expected that the design currently considered must be operated in at least one resonance condition. Operation at the resonant frequency of the drive axis results in the actual deflection of the proof mass due to the drive excitation being enhanced by a factor of Q. (This is the case illustrated by equation (13)), and involves the frequency of the driven deflection of the proof masses matching the natural frequency of the spring-mass system. Recall from the discussion of the harmonic oscillator model that the deflection of the mass will always lag behind the driving force by the phase angle 0 as given by equation (34). If the resonant frequency of the drive axis is Q )0x, the drive frequency of the excitation force should be: = (0»X + Q (44) The magnitude of the Coriolis force is also increased by a factor o f Q, as it depends linearly on the deflection of the proof mass. Thus, operating the sensor at the resonant frequency of the drive axis directly enhances the mechanical quantity being measured (Coriolis force) by a factor of Q before it is converted to an electrical signal by the sensor transducer. In this way, the smallest detectable signal which the sensor can reliably report is decreased by a factor of Q from its nominal value for off-resonance operation; the minimum signal is known as the device sensitivity. The current design is expected to be operated at the resonant frequency of the drive axis in order to utilize this phenomena of “noise-free gain” due to mechanical resonance of the sensor structure. Operation at the resonant frequency of the sense axis does not increase the deflection of the proof mass due to the driving force, yet at the same time amplifies the deflection of the proof mass due to the Coriolis force. Thus, the Coriolis force remains unchanged, and only the system ’s response to the Coriolis force is increased by Q. In this mode, the sensor reports a rotation rate consistent with the Coriolis force required to deflect the proof masses, and the output must be scaled by a factor of Q in order to attain the true rotation rate £1 This is fundamentally different from the previous case, in which resonance caused the physical quantity being measured (the Coriolis force) to be scaled by a factor of Q; in the case of operating at the natural frequency of the sense axis, only the system ’s response to the Coriolis force is enhanced. The required driving frequency for resonant operation along the sense axis is determined by the effective spring constant of the longer support springs in the direction normal to the substrate. (Refer to page 33 for a discussion of the relationship between the effective spring stiffnesses for motion parallel and normal to the substrate). Operation of the sensor at the resonant frequency o f the sense axis, however, can introduce unwanted difficulties in the interface electronics; by the very nature of resonant operation, these are typified by large, nonlinear responses which can overwhelm the system electronics in either size or complexity (i.e., nonlinearily) [19]. Operation at resonance of the drive axis is preferred for these reasons. Operation at the resonant frequency of the transducer flexure also generates sensor output for an “apparent” Coriolis force which is larger than the actual Coriolis force by a factor of Q. Resonant response o f the transducer flexure again merely amplifies the true signal through the increased bending of the piezoresistive beams which convert the mechanical measure of rotation rate (the torque induced by the Coriolis force) to an electrical quantity (the change in resistance o f the piezoresistors, or the equivalent change in voltage across the W heatstone bridge). Thus, the change in resistance reported corresponds to a Coriolis force that is a factor of Q greater than the actual Coriolis force, and so the reported rotation rate O is also a factor of Q too high. The resonant frequency for this mode is determined by the stiffness and mass of the entire mechanical structure in an m anner identical to that in the section Spring Mass Resonators on page 30. Operation in this mode of resonance is not expected. 38 2. Multiple Resonances Recall that the required frequency of the driving excitation for resonant operation along both the drive and sense axes depends only on the effective spring constants (kx and ky) and the value of the proof mass m (see equation (42)). Since m is a common factor, for both modes to have the same resonant frequency requires only that k K = J k v . The definition of k given by equation (42) then implies that if the longer support springs have square cross-sections then both modes will have the same resonant frequency. One immediate implication of this operating condition (a “double resonance") is that the sensor output will be multiplied not only by a single factor of Q, but by the Q factors appropriate for the respective modes: G„„„, - «?,> < gv> (45) It is for this reason that the longer support springs were designed to be square in cross- section, and this limits the width of the beams to be no more than the total thickness of all process layers combined; as stated previously (refer to Table 1), the total layer thicknesses of the CMOS process used for fabrication is approximately 5.65pm. It is also for this reason that it is a generally accepted design goal to match the natural frequencies of the drive and sense axes for a tuning fork angular rate sensor [ 10] [13]. Also note that the Q factors for the drive and sense axes will in general be distinct, as suggested by (45). This seemingly curious result will be explained in the section Q as 39 a Design Parameter on page 43, and is mainly due to the differences in viscous damping between the drive and sense axes. It should be noted that there are several practical considerations which limit the designer’s ability to fabricate an oscillator with matched resonant frequencies, whether in micromachining or another manufacturing method. As will be discussed further in the following sections, for an oscillator with low damping (i.e., medium to high Q), there exists an increasingly small window of operating frequencies for which the system will exhibit resonant behavior; furthermore, the gain o f Q which has thus far been discussed as the main objective of resonant operation only applies to operation at the resonant frequency, and will decrease rapidly as the drive frequency deviates from this value. W hile parts can theoretically be machined or manufactured without error, in practice the fabricated part will deviate from the dimensions specified by the designer by some small tolerance. This error tolerance is a function of the manufacturing process on one hand, and also depends on how critical a given dimension is to the final function of the part. This relationship between tolerance and function explains the cost of high-precision angular rate sensors manufactured using high-precision machining techniques given in the opening remarks of this thesis; in conventional machining, as the number of critical dimensions increases, so does the time and cost required to manufacture the given part. Thus, for the tuning fork angular rate sensor design to operate at double resonance requires the natural frequencies for the longer support springs to be identical for in plane and out-of-plane motion; this very idea was the driving force behind the design of these springs, and yet manufacturing tolerances will prohibit the cross-sectional area of the beams from being perfectly square. The thicknesses of the beams is determined by the manner in which the structural layers are deposited in the CMOS process, and are in close agreement with the data given in Table 1 on page 24. The width of the beams, however, is just as critical of a dimension from the perspective of functionality as the beam thickness, but is defined during fabrication by a process with much poorer control than that which defines the beam thickness. The proof masses, support springs and thermal actuators are rendered free from the substrate through the use of the “pit” or “open” feature [16]. A pit is essentially the stacking of the four contact layers available in the CMOS process; a contact layer is used where the designer wishes to remove the insulating oxide layer between two conductive layers (and thus make “contact” between them) in order to pass a current through them. By placing all four contact cuts on top of each other, without geometry on any other layer, the end result is the exposure of the native silicon substrate at the end of the fabrication process (see Figure 8). Figure 10 shows a comparison between an ideal pit which would result from perfect manufacturing processes with pit geometry that is more commonly encouniered in practice. Figure 11 shows the effect this poor dimensional control can have on structure definition. Layout View L ayout View * * Pit Layer = Active Active Contact Metal 1 - Metal2 Contact Overglass * Overglass = 1.00 pm Second Oxide = 0.65 pm First Oxide = 0.85pm Field Oxide = 0.60pm Substrate (a) Ideal Pit (b) Typical Pit Figure 10: Cross-Section View of Pit Feature As the CMOS process progresses, the overetch of oxide, poly and metal layers results in the substrate becoming recessed. This further degrades the next layer’s patterning fidelity due to incomplete exposure o f the photoresist in the pit area. The lack of precise control over features defined with the pit feature is due in general to this reliance of feature size on a process under poor control such as etching. Figure 11: Variation of Geometry Defined by Pit Feature SEM photograph showing the poor definition of geometry around the perimeter of a plate, and in the exposed area in the plate center. The extra material is due to incomplete etching of the oxide and/or metal layers in the CMOS process. While variations in dimensions are not always this severe, the photo illustrates the potential lack of control over dimensions defined by the pit feature in the CMOS fabrication process. Photograph and structure taken from [20|. 42 The effect this lack of dimensional control has on matching of resonant frequencies can be quantified by recalling the discussion on Spring-Mass Resonators on page 30; The natural frequency of the drive axis depends on the cube of the beam width, whereas the natural frequency of the sense axis is a linear function o f beam width. Thus, small changes in beam width will produce a more dramatic change in natural frequency for the drive axis than for the sense axis. A plot of the difference in natural frequencies for the drive and sense axes as a function of variation in beam width for a longer spring length o f 500pm , thickness of 5pm and a proof mass 300pm on a side is given in Figure 12. 1 £ I (X X ) i £ 500 4 5 4.2 4.4 4.6 4.8 5.2 5.4 5.6 5.8 6 Beam Width b (pm) Figure 12: Natural Frequency Mismatch aa Function of Beam Width Beam thickness is held constant at 5pm and the difference in natural frequency for the drive and sense axes are calculated as the width varies from 4pm to 6pm. For example, if the longer spring length were fabricated at 5.2pm wide rather than 5.0pm wide, the drive and sense axes would differ in natural frequencies by approximately 500 rad/sec. The spring length is 500pm and the proof mass is 300pm on a side (I x 10 kg). If the expected deviance from the design dimension of 5pm is known or can be estimated, then the mismatch in natural frequencies for the two axes can be calculated. This establishes a window within which the two natural frequencies are expected to fall, and this frequency range can then be used to estimate the effective QtoSa( for operation at the double resonance condition. j. Q as a Design Parameter Recall from the introductory discussion of Resonant Systems on page 14, Q is easily expressed as a function of the coefficients of the equation of motion: This is a general expression for Q, and is not dependent on the particular design, operating conditions or type of damping which is present other than the assumption that the damping force is proportional to the velocity of the proof mass. Clearly, Q will depend on these quantities, and it is the purpose of this section to derive an expression for the quality factor as a function of design variables and operating conditions. It is important to note at the outset that even for a system designed to operate at a double resonance, the Q factors for each resonance will in general be distinct even if the geometry of the springs is identical in the respective axes of resonance. Hence, even if the natural frequencies of the respective directions were perfectly matched, the response of the oscillations at resonance will be distinct unless the damping mechanisms are also identical. For the tuning fork angular rale sensor design being considered here, although the drive and sense axes were designed to be as closely matched as possible, expressions for the Q factors are expected to be very different due to the difference in damping conditions between the two axes of resonance: For the drive axis, damping of the proof mass may be modeled as a standard problem from fluid mechanics (Stokes’ oscillating plate [21]), where the main dissipative mechanism is shear stress exerted by the damping fluid on the moving plate; for the sense axis, however, energy dissipation is predominantly due to squeeze film damping between the plate and the excavated substrate beneath it (e.g., as in [22]). The reduction of displacement due to damping is expected to be much more dramatic in the latter case. The current design is expected to be operated at the natural frequency of the drive axis. While the natural frequencies of the drive and sense axes were designed to be the same, the fabrication limitations described earlier—coupled with the differences in damping mechanisms for the two cases— will result at best in a small amplification along the sense axis due to operation at “near resonance” . The discussion to follow will present a derivation of Q for resonance along the drive axis (i.e., Qx). The assumption that the damping force is proportional to proof mass displacement is the basis for much o f the work previously published on micromachined resonators [2] [13] [23] [24] [25], However, as nearly all of these works acknowledge, the effect of damping is a strong function of the ambient pressure in which the resonator is 45 operating. As such, some researchers have reported three distinct realms for Q factor analysis as a function of operating pressure [26] [27]: intrinsic, for pressures 10'2Pa (equivalent to IO~7atm) and below; molecular, for pressures higher than the intrinsic level but less than atmospheric (105Pa or latm ); and viscous, for ambient pressure being atmospheric or higher. The dominant mechanism in the intrinsic regime is the internal damping within the support spring’s polycrystalline lattice, and is thus present in the system regardless of ambient pressure. However, polysilicon is inherently a high-(7 material, and any amount of intrinsic damping will be negligible compared to other sources of energy dissipation [2] [3] [6] [8]. The results here are derived from a viscous damping model, and if applied to ambient pressures much lower than atmospheric the flow past the plate will transition from viscous to rarefied flow. Damping in that case will be due to the collision of noninteracting fluid molecules with the structure, and must be analyzed using techniques from the kinetic theory of gases. One further remark must be made before we begin the derivation of Qx regarding apparent disagreements in previously published research on the topic of viscous damping of micromachined resonators. As stated in [27], Q in general may be regarded as a function of ambient temperature; however, for the molecular region previously defined, they report Q appears to "plateau” in this region, and is thus apparently independent of ambient pressure. At the same time, the expression for Q for this region is shown to depend on temperature and density of the damping fluid, 46 and so clearly there is a relationship between the quality factor Q and ambient pressure. Furthermore, (26] states that for the viscous region Q is also independent of ambient pressure; the expression given for Q, however, clearly shows a dependence on both density and viscosity of the damping fluid, and these both fluctuate with varying pressure and temperature. Finally, there is some disagreement between published accounts on whether the effects of mass loading have an appreciable impact on calculating shift in resonant frequency and Q (23] [28], u (y) « exp (-y ) u ( 0 = «0 sin ( o y ) Z Z \ Substrate 7 7 7 7 7 7 7 Figure 13: Problem of Stokes’ Oscillating Plate Viscous damping of the proof mass for oscillations parallel to the substrate are modeled after the problem of Stokes' oscillating plate. The flow velocity u(y,r) oscillates in time with the plate, but decays exponentially as the distance from the plate increases. An identical flow pattern exists beneath the plate, and both sides of the plate must be used to calculate the area over which the viscous shear stress acts. The energy dissipation mechanism illustrated in Figure 13 is viscous damping due to the shear stress exerted by the ambient fluid on the plate as it oscillates. We have already calculated the energy stored by the springs per cycle in equation (21); we must now calculate the energy used to overcome damping due to the shear stress. This will be done by writing down an expression for the shear stress, multiplying the shear 47 stress by the plate area (both sides) to obtain the viscous force present and then integrating this force over one complete cycle of oscillation. The shear stress exerted on the plate by the fluid is governed by N ewton’s law of viscosity, which states: where ft is the viscosity of the damping fluid, and u is the fluid velocity. As we are concerned with obtaining an expression for Qx, the system is assumed to be operating at resonance. Thus, the oscillations o f the proof masses have reached the steady-state condition. The steady-state solution to Stokes’ problem is given by [21]: where u0 is the initial velocity determined by the boundary condition at r=0; the initial velocity is equal to the product of the frequency at which the plate oscillates and the displacement for one cycle of oscillation (as given by equation (17)): / \ / (48) (49) 48 As both terms in (48) are dependent on y, the expression for shear stress is obtained by applying the chain rule for differentiation; the exponential function remains the same except for a scalar multiplier, but differentiation of the sine term will produce both sine and cosine terms in the expression for T = — - = (sin to ,,/+ costonf) (50) /2v V“ o The viscous damping force is the shear stress multiplied by the area over which it is acting: 's . = (51) The work done in overcoming damping, then, is the viscous force integrated over one cycle of oscillation: ( . T J ( sin2G )0r + costo()/sin (aQ i)d t (52) 0 ' 49 The integral of (sino>0/cosa)0f) over one complete period is zero; the integral over one complete period of ( sin2U )()r) is: f sin2a)(1 fJf = j () 1 sin2to()r 2 2k U ) , 4(0, ( I T t (0,, (53) Inserting the results of (53) in (52) yields the expression for the energy required to overcome damping per cycle of oscillation for the proof mass driven at the resonant frequency of the drive axis: E , = to fiv W “ o- 2n\iA „w0 (pF0) (54) Recall from the discussion on Quality Factor on page 18 that Q is the ratio of the energy stored per oscillation to the work required to overcome damping: Qx = 2 jc V d j - 2 k 2k (Pf o) 2k\iA co0 (pF0) a K f — U p/ , m (55) 50 The expression given for Qx in (55) can be expressed in a variety of ways by simple substitution; for example, the proof mass is related to the area of the proof mass plate by equation (43), which results in the following alternative expression for Qx: f P N V Pf '2vJ (56) In this way, a designer may obtain an expression for Qx as a function of design parameters. One of the more fundamentally important expressions for Q as a function of design variables is to reduce an expression such as the final result given in equation (55) to “leaf-level” design variables and material constants. This may be accomplished by further substitution into (56) for the parameter Q r = K Pf ) ( j f c X s f - w f f E 1 f i. p b ravg 3N \ ( (4w p i t) 2j \ ) / (57) It is also worth noting that comparison between the second-to-last expression in (55) and either of equations (24) or (46) reveals the damping coefficient bj as a function of design variables and operating conditions, as alluded to in the introductory remarks on quality factor: / \ h (58) J & W hile this formally concludes the derivation of Q as a design parameter, the preceding derivation reveals an interesting (and subtle) discrepancy in the viscous damping model assumed for the oscillating plate and that which actually occurs. Recall the expression given for the shear stress in equation (50) is multiplied by the area over which it is acting in order to obtain an expression for the viscous force. To show this explicitly: (59) This can in turn be rewritten as — E r ^ s,or « / + 4 j (60) 52 where the identity ( sinoi,/ + cosco,,/) = */2 sin ^ io0/ + (61) has been used to combine the sine and cosine terms. The phase of the viscous force is ^ (0()r + ^ in the damping model used to calculate £>; however, recall in the equation of motion for the harmonic oscillator that the damping term is proportional to the product of the damping coefficient b and proof mass velocity: bjX = /^sintUjjf (62) Thus there is a phase shift between the forces in the harmonic oscillator model and the viscous damping model used to calculate Q [291 , However, as Q is always calculated over an entire cycle of oscillation and not instantaneously this discrepancy can be disregarded. 53 4, Mechanical Response Time Recall from the introduction to resonant systems on page 20 that the mechanical response time is directly proportional to Q for drive axis proof mass displacement: If we wish to interpret Tm in terms of design variables, we may substitute in (63) the expression for Q obtained in the preceding discussion. As before, this expression can also be obtained in terms of the fundamental design parameters and material properties by making the proper substitutions (i.e., Qx from (57)). Further substitution yields mechanical response time as a function of longer spring length and size of the proof mass plate are readily found to be: ( 2 k V m \ I®o _ ' u k J 2 ' » UJUp^A^vJ * (64) 5 \ 4 (65) Thus, the length of the time interval for transitioning to or from steady-state motion increases for corresponding increases in beam length or size of the proof mass; 54 alternatively, the response time decreases for increases in the density or kinematic viscosity of the damping fluid, a somewhat intuitive result. While Xm is directly proportional to Q, it is rare that a resonant system has high enough efficiency that the mechanical response time is too long to be acceptable for a given application. However, it is worth noting that the response time for an oscillating system is a maximum at resonance in a manner precisely analogous to any other system response (displacement, force, and so on); furthermore, the response time is also increased by a factor of Q at resonance. 5. Force Transmissibility The frequency analysis of an oscillating system which is expected to be driven at resonance naturally focuses on system response for frequencies at or very close to the calculated resonant frequencies of the system. As was alluded to in the section Multiple Resonances on page 38, however, whether an oscillator exhibits resonant behavior or not can depend on very small changes in operating frequency. For this reason, a complete investigation of operating frequency for an oscillating system must include the full frequency domain. This is accomplished by plotting the transfer function for the system in question for the expected amount of damping. The transfer function is sometimes referred to as the system response factor, and is a dimensionless measure of the effect the oscillatory nature of a phenomenon (force, displacement, velocity, and so on) has on the system. Thus, the transfer function 55 essentially normalizes the frequency dependence of these phenomena to their corresponding result if applied statically rather than being allowed to vary with time. One of the fundamental quantities of interest here is the ratio of transmitted force to applied force as a function of frequency. As we have already seen, there is a dramatic response to an applied force whose frequency closely matches the natural frequency of the system; however, we have not yet defined the system response to stimuli well below or above the resonant domain of frequencies. The ratio of force transmitted by the structure to the applied force is known as the transmissibility of the system. The transmitted force is the resultant o f the restoring force of the spring and the viscous force which is the source of energy dissipation; as the restoring force is proportional to displacement, and energy dissipation is proportional to velocity, these two forces are 90° out of phase. Substituting for x(t) from the solution to the harmonic oscillator equation of motion (14) results in the following expression for transmissibility [30): (66) T (67) 56 0) \i Figure 14: Force lYansmissibility Frequency dependence of sensor responsivily can be interpreted for three ranges of drive frequency: (0^ « Q)q , (0 . ~ (0 ^, and (0^ » (0q . The modified damping coefficient £ is the width of the curve at tne halbpower points shown. The quality factor Q is the height of the resonance peak as shown. 6. Responsivity Responsivily of a sensor is defined as the ratio of change in output signal per given change in input signal. In the case of an angular rate sensor, the input signal is the rotation rate 12 which is to be measured; the output signal measured by the W heatstone bridge interface circuit is the voltage required in order to maintain the balance of the bridge (refer to Figure 4 for the definition of the bridge circuit and symbols). Hence, the responsivity of the tuning fork angular rate sensor will be given in terms of Volts per radian per second, or V/(rad/sec): In order to calculate responsivity we must obtain an expression for the output voltage as a function of input rotation rate. The change in voltage across the bridge is due to the change in resistance of the piezoresistors as they are strained; the strain in the sensor resistors is caused by the deflection of the proof masses (magnitude y(.) due to the Coriolis force; the Coriolis force is in turn a response to the forced oscillations of the proof masses (magnitude rA) in a frame of reference uniformly rotating at angular frequency £1 In this way, then, we can relate the behavior of the output voltage to the input rotation rate. First we must relate output voltage to the change in resistance of the piezoresistors: AV AR - (69) K 2 R „f But the change in resistance for a piezoelectric material is given by the relation ARs = {GE)Rref, for a material with initial resistance Rr e f and gage factor G, experiencing a strain of e. From strength of materials theory, strain in a beam under pure bending can be expressed as the ratio of the distance to the neutral axis of the 7 beam to its radius of curvature: e = - , which yields the intermediate result P 58 Another result from strength of materials relates radius of curvature of a beam under pure bending to the moment, beam geometry and material properties: o = V i {71) p El where / is the moment of inertia for the beam, which is purely a function of geometry; ( a ) for a beam of rectangular cross section is given by / = — —— , where apr is the thickness and bpr is the width of the beam containing the piezoresistors. The moment is the magnitude of the Coriolis force multiplied by the distance between its line-of- action and the bending axis of the piezoresistors. (For the moment of inertia, the dimension which is cubed is that in the direction of the force; the dimension normal to the force appears linearly in the expression for I.) Substituting for M, 1 and (71) into the expression given in (70): AV (>Gz(FcRc) (72) W e E{a ) yb pr The key step in this analysis is relating the spring constant for the piezoresistor elements to that of the proof masses for the driven oscillations (magnitude rA). Using the definition given in (42), we define the respective spring constants as follows: The 59 spring constant pertinent to proof mass displacement is k as given in (42); the spring constant for bending of the piezoresistors is kpr and is given by the following k m = Eab 4 a y and . = £ < y > V P T 4 (L ) 3 v nr' (73) Next we introduce the scaling factors X, y and r): = Xu, h = y b , and L/)r = which leads to the following relationship between km and k/ir\ pr 3 \ r\ J (if (74) For the present design, the ratio of proof mass beam thickness to width is unity (see Multiple Resonances on page 38). Comparison of (72) and the second expression in (73) gives AV _ 3 V, = 2 kPr ^ py ) (75) Relating the length of the piezoresistive beam Lpr to the moment arm R(. through another scaling factor (L = £/?(, ) produces the following result: AV _ 3 K = 2 f G zF c \ (76) 60 Utilizing the relationship between km and kpr given by (74) tT i 1 II <1 ( 3 \ T 1 K 2 (77) Finally, substituting for the Coriolis force (12), and solving (42) for km we have an expression for the output voltage as a function of input rotation rate: A V (ft) = f - y L U t V VeG zrAu dL r % * - 2 - (78) The responsivity of the sensor is now easily obtained by simply taking the derivative of (78) with respect to Q: R = dV 3r| d a i -V1 U 4 Y' 2 _ 2 (79) At resonance, the responsivity is a maximum: dV r , 3 ) 331 (Q V eGzrA) d a w „ U \ y) I w0Rc2 ' (80) 61 In order to examine responsivity at resonance as a function of design variables, the expression for Qx obtained on page 50 may be substituted into (80) as follows: = f i - M ' P ^ ravg ( V ,G ” a ) U \ y) { P / J 1 * 2 ) j 2 V(D(j) (81) Or in terms of basic geometry and material constants: 3t ) R a w V fC fn { Eb (82) Figure 15 illustrates responsivity as a function of operating frequency for a system with subcritical damping. & 1 > Angular Drive Frequency (rad/sec) Figure 15: Responsivity Responsivity as a function of drive frequency of the forced oscillations shows the same general trends as that for force (ransmissibility. 62 7. Transducer Interface Circuit Noise Flicker noise (commonly referred to as Mf noise) is a phenomenon present in a wide variety of physical (and also nonphysical, such as reasoning and psychology) systems. For CMOS technology, the comer frequency below which flicker noise will become prominent is approximately 1 kHz (34]. For operation above the corner frequency, electrical noise is "flatter” with respect to drive frequency. This is the trend illustrated in Figure 16, in which the low frequency domain is dominated by 1 If noise, and the broadband region is seen to level off with respect to frequency. A practical rule of thumb for estimation o f interface circuit noise for a typical CMOS operational amplifier is in the range of 10 to 100 nano Volts per root Hertz [34]. This is value which has been used in design calculations, with 100 nanoVolts/rtHz being considered conservative or worst-case. N I t B > Si o z Angular Drive Frequency (rad/sec) Figure 16: Transducer Interface Circuit Noise At low frequency (below the corner frequency), the noise in the interface circuit is dominated by the flicker or I// noise. For f>fc, the noise level flattens out. This is referred to as broad band noise. Estimates for noise are 10- lOOnV/rtHz. 63 S. Sensitivity Device sensitivity is defined as the minimum signal which can be resolved from background noise present in the system from all sources. In terms of symbols, a convenient representation of sensitivity is the ratio of electrical noise to responsivity Based on the preceding two sections, then, we should be able to construct a plot of device sensitivity by taking a composite of the frequency dependent plots of responsivity and noise given in Figure 15 and Figure 16, respectively. It is a matter of convenience that across the spectrum of operating frequencies, one of the two quantities on which sensitivity depends is a constant with respect to drive frequency. For low frequencies, responsivity is flat and noise is predominantly l//in nature; for high frequencies, responsivity drops off sharply but electrical noise is in the broadband region and is relatively flat with respect to frequency. Thus, the ratio of noise to responsivity is (83) -3-3 2 D 2 U ^ y A 0> oRr ) VeG z r A®d) (84) 64 At resonance, the sensitivity is a minimum, i.e., the minimum detectable signal is at a minimum: 51 = A . = w " R L t > IQT)*VGzrA (85) Substituting for Q with equation (56) gives = x _ § j l 3r|3 ) *L \ P avg J c V a , vnf i ™ o { V, G ^ A (86) This expression for sensitivity can subsequently be given in terms of basic design variables by substituting for the natural frequency term. This result is SI = I CO,. 3t|3 ) v R K V .G z r J ,/) Actuator Exert force on proof mass to hold in place; magnitude of force determined by position sensor X Mixer Add signals — 67 m -1 8 0 (Oj 'n Figure 18: Phase of Feedback Control Loop Phase between position sensor measurement and actuator deployment for force rebalance closed-loop feedback system. Note that phase shifts sharply from perfectly in-phase to perfectly out-of-phase for shift in frequency on the order of Xm 1 rad/sec above resonance. If the device is operated in this region, the feedback measurement and implementation will be exactly out of phase, and rather than returning proof mass to the preset null position, the actuator will continue to move the proof mass further away from nullity, progressively increasing the error incurred in the output signal as a result. D. Noise and Other E rror Sources There are two facets of real-world design which inevitably introduce error into the design of any system: Inaccuracies in the analytical models used to simulate product or system performance, and statistical variation in the expected results due to random fluctuation of certain quantities (dimensions, material properties, etc.) [31]. There is a subtle though important difference in these two “corrections” to the results predicted by preliminary design analysis: Flaws or simplifications in analytical models will always generate erroneous performance data, regardless of how well controlled or 68 characterized the fabrication process may be. It is the responsibility of the designer to justify any assumptions made in the process of modeling a system under design, in very much the same manner that it is the responsibility of the manufacturing and quality control personnel to ensure the expected values for error tolerances furnished to the designer lor use in his or her calculations are as accurate as is possible. Errors in design which are the result of bad design choices arc correctable and (more importantly) avoidable; errors which are due to random or statistical variances in certain parameters are typically not able to be fully corrected, and so must be taken into account in the design phase itself. In this way, noise and other expected errors act as design parameters of a sort in that they are also modeled along with other parts of the system, and the results of such simulation are then used as the basis for subsequent design decisions. As an example, in the section on Multiple Resonances on page 38, it was stated that regardless of design intent no two spring elements will have perfectly matched resonant frequencies. To interpret this information within the framework outlined above, we must evaluate both the modeling of the spring elements as well as considering how well the fabrication process of the springs can be controlled. The analytical model used to calculate resonant frequency is characterized by equation (41), and results from the lumped-parameter model of the actuator-proof mass system being modeled as a driven harmonic oscillator with viscous damping. By the definition of a lumped-parameter model, the springs are assumed to be massless, and so will not contribute to the mass of the system. Similarly, the support spring is an 69 idealized element which is solely responsible for the storing and translation of energy from potential to kinetic as the mass oscillates. In reality, the springs will have a small amount of mass (estimates are given on page 31), and the longer support spring is not the only elastic element of the system. Thus, the resonant frequency calculated according to (41) is in reality an approximation, given the limitations of the analytical model just discussed. There are similar examples for other design quantities previously discussed (i.e., in the calculation of Q, the oscillating plate is analyzed using Stokes’ method, in which one of the assumptions is an infinite plate). These idealizations will introduce an error into the expected performance of the angular rate sensor; the burden of such analysis is for the designer to convince himself or herself that the error are within acceptable limits of actual performance. The role that manufacturing imperfections will have on matching the resonant frequencies of the proof mass support springs has been discussed in some detail previously (see Figure 12). However, manufacturing limitations are only one example of statistical or stochastic "noise” which will affect device performance. Other important examples are fatigue and wear of the device in service and environmental effects [31 ]. W hile fatigue and/or hysteresis is not anticipated to be a problem for the polysilicon structure [32], the effect of temperature variation on device performance could be considerable. For example, Young’s modulus is known to be a slight function of temperature; thus, the resonant frequency of the springs will change for sufficient changes in ambient temperature. O f course, viscous damping effects (already shown to 70 he a function of pressure, fluid density and temperature) will show strong variation with changes in properties of the damping fluid. To close this introduction to noise and error sources, it should be stressed that more robust designs are by definition less susceptible to the influence of noise. For the design and fabrication process being considered, such insensitivity to stochastic interference is best obtained by, as much as is feasible, minimizing the dependence of device performance on dimensions which are poorly controlled during fabrication. As has been shown, this rule of thumb is not always practical. The remainder of this section will address the significance of various sources of error which stem from this second category. Assumptions, simplifications and limitations of any analytical models used in the design of the angular rate sensor are addressed either in the derivations given earlier in this chapter or in the appropriate section of the next. The use of feedback control will also be considered, particularly with respect to the minimization or elimination of the error sources discussed here. I. Electrical Noise in Piezoresistors The phenomena of noise is common to all measurement systems. Regardless of what quantity is to be measured, there is inevitably an error introduced during the measurement process; this unfortunate truism has its roots in the uncertainty principle of quantum mechanics, which asserts that position and velocity cannot be simultaneously known with arbitrary precision. 71 However, noise is not relegated to quantum systems alone. Rather, it is most commonly seen in any system where energy is being converted from one state to another, and in the process of changing state a fluctuation or variance occurs in the nature of the system. Thus, as in the case to be discussed at the present, for current flowing through a resistor there is an inevitable loss in the electrical energy in the form of heat dissipated by the resistor: There is a fluctuation in current flowing through the resistor (or in the voltage across its terminals), as the released heat will effect changes in the local material structure to due the change in temperature. Thus, we expect the amount of fluctuation to be dependent on the internal energy (i.e., temperature) of the current-carrying material. In fact, the expression for Johnson noise in a resistor is: r '^‘ 1 where v , is given in - = , kh is Boltzmann’s constant (1.3810 J/°K), T is the JJT z absolute temperature and R is the resistance in Ohms, For a 1 kf2 resistor at room temperature, the Johnson noise would be This is the expected noise level for each resistor in the W heatstone bridge, resulting in ( 88 ) ft V a total noise contribution o f approximately 16----- The noise in the piezoresisiors also can be related to the minimum resolvable signal of the sensor system. The operating point at which the output voltage of the bridge due to change in resistance o f the sensor resistors (i.e., a result of the strain in the resistor beams) equals the Johnson noise in the resistors defines the minimum resolvable signal. It is important to note that this quantity (the sensitivity discussed previously on page 63) is defined prior to the sensor signal being input to the system processing electronics. This is because the processing electronics will amplify the voltage it is given— including the sensor signal and any noise present in the transducer elements— and so the signal to noise ratio will remain unchanged. The minimum resolvable signal is given by equating the sensor output voltage and Johnson noise terms: (90) (91) G V e in which the bandwidth A/- has been explicitly included. From the derivation of responsivity starting on page 56, we may substitute for E 0 as follows: 73 Further substitution for Coriolis force, moment of inertia and the scaling relationships between the piezoresistive beam and the proof mass support spring given on page 59, and also assuming operation on resonance, we have 3 3 T| (QzrAil) U w Finally we may equate (91) and (93) and solve for £2; this will translate the minimum resolvable strain into the minimum resolvable rotation rate which the sensor can faithfully detect: This represents a second, independent method for deriving the sensitivity of the sensor, and the results are in good agreement with those found on page 64. 2. Electrical Noise in the Interface Circuitry The output voltage of the Wheatstone bridge will be input to an operational amplifier, and from there will be passed through various stages o f processing. While such processing serves a necessary purpose (the raw signal will most likely be weak and will contain other information which must be filtered out), unfortunately each additional operation will increase the noise present in the signal. Fortunately, it has (94) 74 been shown that all downstream processing circuitry adds a negligible amount of noise from a system perspective— with the exception of the very first stage of electronics [33]. Thus, in order to minimize electrical noise in the system, only the initial stage of circuitry must be optimized for low noise. As discussed in the previous section, all discrete electronic devices will dissipate a finite amount of power during operation. This power is converted from electrical energy to thermal energy and typically dispersed as heat. As a result, there will be a component of the total noise in the first stage of circuitry due to Johnson noise, and this contribution can be quantified using (88). Additionally, however, there will be noise contributions which are dependent on operating frequency. Flicker noise (commonly referred to as 1 if noise) is a phenomenon present in a wide variety of physical (and also nonphysical, such as reasoning and psychology) systems. For CMOS technology, the com er frequency below which flicker noise will become prominent is approximately 1 kHz [34], For operation above the com er frequency, electrical noise is “flatter” with respect to drive frequency. This is the trend illustrated in Figure 16, in which the low frequency domain is dominated by 1// noise, and the broadband region is seen to level off with respect to frequency. A practical rale o f thumb for estimation of interface circuit noise for a typical CMOS operational amplifier is in the range of 10 to 100 nano Volts per root Hertz [34]. This is value which has been used in design calculations, with 100 nanoVolts/rtHz being considered conservative or worst-case. 75 3. Thermal Motion o f Proof Masses In the introductory remarks on noise, quantum mechanics was mentioned to play a defining role in the existence of noise in all physical systems. This is true also for the spontaneous vibration of all matter due to thermal excitation. From thermodynamics, we know that this quantity of thermal motion {known as the internal energy) must be nonzero for any nonzero absolute temperature. The amount of thermally-induced vibrational motion can be calculated by equating the thermal energy with the energy stored in the spring due to these spontaneous oscillations [26] [35]: This expression can then in turn be solved for the amount of thermal displacement each proof mass will undergo due to their internal thermal energy, and this value subsequently substituted for the driven displacement rA to determine the thermal noise equivalent rotation rate which is reported by the transducer for the thermal motion: The true cause of the thermal displacement is transparent to the angular rate sensor; it cannot discriminate between motion of the proof mass resulting from the applied driving force and motion due to any other impetus. Thus, the thermally-induced (95) (96) vibrations in the tuning fork will give rise to a pseudo Coriolis force, and to the sensor system the thermal motion will appear to add to the net rotation rate being measured. It is this single fact of tuning fork angular rate sensors which poses the largest design challenge, as virtually any input to the system is readily converted to behavior which is physically identical to the operating principle of the device. Regardless of the source of such erroneous stimuli, the tuning fork angular rate sensor will interpret them and incorporate their effect in the measured output of rotation rate. (This topic will be addressed in more detail in the section Errors due to Manufacturing Limitations on page 79.) (97) S n (98) To normalize the signal due to thermal motion, the ratio of — ^ is: S U 77 At room temperature (300°K), for a typical value of km (with the longer support spring of length 500pm), we have <” >>. , .8x,0-" <100. or approximately 2 A. From Figure 34 Quality Factor versus Spring Length, we see that for a beam 500pm long, Q will range from approximately 200 to 300, for proof mass displacement rA of a few microns. Thus, the equivalent rotation rate reported due to thermal motion is given by (99): ^11 8 2 * 1 0 10 7 - * - 2x10 = 4 x 1 0 7 (101) Q rA (250) (2x10 6) This is a sensitivity of one part per million (ppm). 4. Temperature Dependence In addition to thermal motion, there are additional fluctuations due to temperature that must be accounted for in angular rate sensor design. Thermal motion is present regardless of how nearly the temperature is maintained constant; for constant temperature, thermal motion will be constant as shown in (96). However, there are additional difficulties encountered in regulating sensor performance if the ambient operating temperature of the device is allowed to fluctuate. Analogously named for the 78 temperature behavior which causes them, these fluctuations are known as sources of drift. Gradients in operating temperature are responsible for drift in device performance mainly due to changes in material properties and performance of the processing electronics. Nearly all material properties used in micromachining are functions of temperature; naturally, certain materials or properties will show greater variation over temperature than others. O f principle interest is Young’s modulus, as it is present in nearly all of the pertinent design equations for the sensor, from thermal force in the actuator to spring constants of the beams. Quantifying the effects of temperature on variation on Young’s modulus serves to make a difficult problem even harder; there is wide variation in the accepted values for E at room temperature (ranging from 90 to 140GPa) (2] [36] [37], and literature on how these values deviate with temperature is scarce. Estimates may still be made for errors introduced in this way. For calculation of natural frequency, for example, there is a square root dependence on E ; thus, a 1% change in E due to temperature variation will result in a 10% change in natural frequency. This is rather significant, and potentially will have much more influence on device performance than thermal motion. Change in operating temperature will most certainly cause a change in the transport properties of the damping fluid, and such variations are well understood for any damping fluid of interest to micromechanics. The expressions for quality factor derived earlier showed a dependence on factors such as mass density and viscosity of 79 the damping fluid; and so the design parameters dependent on these properties of the damping fluid will show a variation in performance due to temperature change. Temperature dependence of this sort is typically decided early in the design cycle, as ambient conditions may influence design decisions early in the process (i.e., whether to encapsulate the device and operate and low pressure). Drift in the processing electronics due to temperature variations, however, may be removed rather easily through the use of feedback control (see Use o f Feedback Control on page 83). A closed-Ioop control system will inevitably experience phase delays between the position sensor and feedback actuator (refer to Figure 21), and prior to filtering the signal to correct this phase lag the feedback signal will be fed into an operational amplifier. By introducing a bias voltage at the input to the opamp (such as one corresponding to the drift in voltage due to temperature change), the change in signal due to change in temperature is effectively nulled. The signal that is subsequently filtered or output will not show the effects of a change in operating temperature. 5. Errors due to Manufacturing Limitations The issue of dimensional variations was introduced in relation to the designer’s ability to match resonant frequencies for the orthogonal vibrations in the drive and sense axes (refer to M ultiple Resonances on page 38). At that time, the difficulty in reliably matching the resonances given the nature of the fabrication process was 80 shown to be considerable. However, at that time the only disadvantage this bore the designer was a narrower window with which to achieve any resonant response; furthermore, the higher the offset in resonant points, the more degraded the resonant peak was found to be. In this way, the difficulty of matching resonance points for the drive and sense axes is not seen to degrade actual performance of the sensor, but rather to detract from the theoretical operating potential which could be realized given proper fabrication; the inability to match resonant frequencies was not shown to further introduce erroneous signals or other errors into the measurement system (i.e., as thermal vibrations do). The full impact that such manufacturing variability has on tuning fork angular rate sensor design is much more considerable than the difficulty in matching resonant frequencies. In addition, physical imperfections in the tuning fork itself violate the inherent balance in the design, thus giving rise to an entire class of sensor errors; due to the manner in which they are evident in the system, these errors are commonly referred to as torsional unbalances [381- In the simplest case, imagine that the two proof mass plates are fabricated such that one plate is slightly larger than the other due to poor control over the geometry defined by use of the pit feature. Thus, rather than two masses each of size m/2, the proof masses are now sized m/2 and (m/2)+e. As the two masses are driven by the actuator, the Coriolis force acting on each is proportional to the size o f the proof mass (recall from equation (12) that |F .| = 2m rA(O dSl). Thus, each proof mass will experience a 81 different magnitude Coriolis force, with the difference being the mass discrepancy between the two. The effect of the mismatched Coriolis force on the proof masses is to cause a net torque about the piezoresistive sensors and centered on each muss, and as mentioned earlier this erroneous torque is in every way physically indistinguishable from the “true" torque from which the rotation rate is calculated. A mismatch in the proof masses, then, is coupled directly into the output as an error signal. As the tuning fork is expected to be driven at its natural frequency, the displacement of each proof mass will be increased by a factor of Q, thus, the error signal due to the mismatch in proof mass sizes is also subject to enhancement by a factor o f Q. While it is possible to remove some error sources through the use of feedback control, note that most imbalance errors are impervious to correction even when operated in closed-loop form using a technique such as force rebalance. (Feedback control is addressed in detail in the following section.) There is no means to separate the output signal as the physical behavior of the device which generates the output is identical, whether from the driven excitation o f the proof mass or otherwise. As a second example of an error signal which is due to manufacturing imperfections (but not due to difficulties related to the pit layer), consider the presence of residual stress in the longer support springs. The length of these beams was determined partly by the desired natural frequency for the spring-mass system, and in part out of considerations related to magnitude o f the proof mass displacement; in either case, longer support springs are preferred over shorter ones (lower to0 and higher rA). 82 However, due to residual stress in the thin films used in a CMOS process, there is a limit to the length of a clamped-clamped beam which can be fabricated and will not be not buckled upon release. Figure 3 and Figure 27 show spring-mass test structures in which the longer support springs have buckled due to residual stress, and this will result in the oscillation of the proof masses to deviate from their ideal in-planc motion. Figure 19 illustrates the origins of this type of error signal. (a) Ideal Proof Mass Displacement m/2 | (a) Nonplanar Proof Mass Displacement Figure 19: Torsional Imbalance due to Nonplanar Oscillations (a) Edge-on view of proof mass displacement in ideal and nonideal conditions. Given perfect fabrication ability, uniform and defect-free materials, and ideal spring elements connecting the actuator and proof masses, the oscillation of the proof masses would be perfectly symmetrical, (b) Errors in fabrication, materials and non-ideal spring elements can result in the oscillation becoming nonplanar. Thus, there is a net torque with moment arm AR, and this torque appears to the transducer elements as a rotation rate; the transducer elements respond to any torque applied to them, and cannot distinguish between torque resulting from the Coriolis force and any other source. Illustration after figure from |38|. 83 The list o f performance characteristics which are affected by dimensional variation for the spring-mass system is all-inclusive; from simple traits {proof mass value), mid level parameters (spring constant) up to higher-level quantities (natural frequency and proof mass displacement), the entire operating principle of the tuning fork sensor relies on an extreme level of manufacturing precision for functionality. Since this single source of errors has been classified as the most dangerous and difficult to avoid, it is only logical that endeavoring to create a design sufficiently robust to perform satisfactorily with respect to these errors is the top design challenge facing the state- of-the-art today in angular rate sensors. 6. Use o f Feedback Control The angular rate sensor design described thus far could be symbolically represented by a diagram such as the one given in Figure 20. The top-level functions of (he actuator subsystems are shown in a functional block diagram, connected by an arrow representing the signal traveling through the various components of the system, and subsequently is measured as the output; the output measurement represents angular rate, The sensor system is seen to consist of four main functional components: the thermal actuator, spring-mass system, piezoresistor transducer circuit, and processing electronics. For open-loop operation o f the sensor, the actuator responds to an applied input voltage, causing the deflection of the proof masses. This leads to a deflection in 84 response to the Coriolis force, and it is this secondary deflection which is responsible for the strain in the piezoresistor transducer elements. The change in resistance of two of the resistors forming the W heatstone bridge will correspondingly change the voltage required to keep the bridge balanced, and monitoring the value of this voltage provides the sensor with the measure of angular rate. In both cases, Vi/( is the voltage applied to the actuator to produce displacement, and Vout is the voltage required to maintain balance of the Wheatstone bridge. is the sensor’s actual measure of angular rate. OpAmp Actuator Strain Gauge Proof Mass Q ; in (a) Open-Loop System Block Diagram ^ o u t (Actuator) ♦ (Proof Mass) (Strain Gauge) ► AR — ► A (OpAmp) I ^ o u t (b) Open-Loop System Functional Diagram Figure 20: Overview of Open-Loop Sensor Operation (a) The angular rate sensor for open-Ioop operation contains four components as shown: thermal actuator, spring-mass oscillating elements, piezoresistive transducers and processing electronics for signal conditioning, (b) Each of the elements shown in (a) can be represented parametrically by the element's function or output: actuator throw (k), proof mass deflection due to Coriolis force (8V ). change in resistance of piezoresistors (AR) due to strain, and gain of operational amplifier (A). 85 Open-loop operation is the easier system to implement, and it is recommended that an open-loop system be undertaken prior to incorporating feedback control; such a progression will provide necessary understanding of the error sources inherent in the tuning fork configuration, and a fundamentally thorough grasp of both ideal and practical tuning fork sensor behavior is required for competent design of such a system. There are several key disadvantages, however, to open-loop operation of the tuning fork angular rate sensor system. To begin with, there are some imbalance errors which may be removed by keeping the proof mass stationary as done in the force rebalance method of closed-loop operation. For example, while it is true that the imbalance torque due to mismatch in proof masses cannot be removed with force rebalance, non planar oscillations due to residual stress in the support springs can be suppressed by not allowing the proof mass to oscillate. In addition, the open-loop configuration potentially suffers from unrestrained behavior at resonance. The substantial increase in response to the driving force which occurs at resonance can potentially overwhelm the tuning fork sensor system, with both mechanical and electrical responses out o f range from the behavior the sensor is designed to experience. The ability to restrain the proof mass removes this over-range and nonlinear response problem while still providing the mechanical noise-free gain o f Q due to resonance. 86 in OpAmp Actuator Filter Position Sensor Proof Mass (a) Closed-Loop System Diagram Q.I in* © — 6 3 , AVp s (Position Sensor) A (OpAmp) (Actuator) ^ u t (b) Closed-Loop System Functional Diagram P (Filter) Figure 21: Overview of Closed-Loop Sensor Operation (a) The angular rale sensor for closed-loop forcc-rcbalance operation contains five components as shown: spring-mass oscillating elements, position sensor, operational amplifier, filter, and thermal actuator Note the transducer element used in open-loop systems has been replaced by a position sensor, (b) Each of the elements shown in (a) can be represented parametrically by the element’s function or output: proof mass deflection due to Coriolis force (8y). change in voltage of position sensor (AV^) required to keep proof mass stationary, gain of operational amplifier (A), output signal of filter (3) and actuator throw (A .). The parameter X is somewhat misleading, as the actuator does not move the proof mass, but rather supplies a force F^ which would cause the proof mass to deflect by X if not for the force-rebalance control loop. Via is the voltage applied to the actuator to return the proof mass to its null position, and Voul is the sensor's actual measure of angular rate. Closed-loop operation of the sensor, however, offers several key advantages over open-loop operation, but at the expense of considerably more complex integration requirements with control and processing electronics. The system-level block and 87 functional diagrams are given in Figure 21. M ajor components are again connected by an arrow representing the input signal, travels through the various components of the system, and subsequently is measured as the output; the output voltage measurement again represents angular rate. In the case of closed-loop operation, the sensor system contains five main functional components: spring-mass system, position sensor, operational amplifier, processing electronics, and an actuator. Closed-loop operation of the sensor is based on a feedback control technique known as the force rebalance method. In this mode of operation, the deflection of the proof mass due to the Coriolis force is suppressed, and thus the signal which travels through the system is not the system ’s response to the behavior of the proof mass as in open-loop systems, but in reality is the force required to keep the proof mass from moving. Thus, a closed-loop angular rate sensor implemented with this feedback control approach will directly measure the Coriolis force. The force required to keep the proof mass stationary is measured by a position sensor, and this signal is input to an operation amplifier and output to indicate rotation rate. However, by definition the measurement o f the position sensor must be fed back into the actuator-proof mass system in order to “close the loop", and so part o f the signal is then filtered and input to the actuator. The filter corrects phase lag which develops due to non-ideal components; for a system with instantaneous response between the proof mass and actuator, there would be no need to filter the signal. In addition, prior to filtering the second input to the operational amplifier can be set to a reference or bias voltage, and then the difference between the 88 sensor and bias signals is input to the filter and also output as In this way, the sensor can be set with a bias built in to account for a known error signal which cannot be compensated or removed by other means; drift in the electronics due to temperature variations can be handled in this manner. The actuator which appears in the open-loop system is responsible for supplying a driving force to the proof masses. The actuator which appears in the diagrams for the closed-loop system, however, has an altogether different function: To apply the force (i.e., voltage) supplied by the position sensor in an attempt keep the proof masses from moving in response to the Coriolis force. Thus, another actuator system in needed to supply the proof masses with the driving force from which the Coriolis force will result; this may be done with a single actuator, but two or more actuators are permissible. The open-loop system has room for only one actuator. E. Design of Complex Systems The design methodologies presented in the previous section (particularly Figure 20 and Figure 21) are fundamentally different than that given by the expressions derived for quality factor Q (equation (56)) or responsivity (equation (79)); the former are examples of system design, whereas the latter represent detail or component design. Various definitions can be found for both levels of design, with detail design somewhat erroneously being placed subordinate^ under system design, and in traditional design cycles this arrangement may be functionally and historically accurate. System design is concerned with an overview of how distinct functional blocks in a system may be connected; the components themselves are only recognized at the top-level of their function. For example, the system design for the angular rate sensor is typified by Figure 20, where parts of the system are labeled according to their function. Detail design, on the other hand, cannot be rendered in schematic block- diagram form as easily, but rather is best represented by design equations and engineering drawings from which the part will be manufactured; this design function is focused on individual parts of the system, and lacks the “big picture” impact for which system design is valued. Intuitively, these two branches of design each independently fulfill a separate function, and that a complete design should merely be the seamless conjunction of the two; the design cycle would initially consist mostly of system-level decisions, gradually shifting as the project progressed until the “overview" was finalized and all that remained were details to be worked out. An interesting question is raised if one considers that in many actual design projects what drives the system-level decision-making process are performance, reliability and cost goals which are realty facets of the detail-design phase. For example, the angular rate sensor system design presented here was determined partly by true system-level considerations (i.e., manufacturability and ease of implementation in a standard process), yet the stated design goal is inherently tied to component-level considerations (i.e., acceptable drift rate will limit the frequency domain of operation, 90 yet drive frequency is already partly limited by realistic proof mass values for surface micromachined structures). One of the main ideas of this thesis has been the consideration of Q as a design variable for resonant systems. In the perspective o f system versus component design, tradiL illy Q has been treated more as a system feature than a tunable parameter which may be manipulated through changes in geometry. However, as was derived earlier in this chapter in the section Q as a Design Parameter on page 43, Q most certainly is under the control of the designer. Thus, Q serves as an excellent illustration of the connection between system and component design: Q is often cited as an indication of device performance, yet in resonant sensor applications quoting a certain value of Q reveals more information about design decisions than merely stating a system specification. This dual meaning of Q is evident in quantities such as sensor responsivity and sensitivity, which are often stated both as design goals as well as indicators of measured device performance; for resonant sensors, both of these quantities depend on Q. As Q in turn is a known function of geometry and operating conditions, it is possible to affect changes to a system-level quantity such as sensitivity through the manipulation of fundamental component design parameters. As a final note, device characteristics such as dynamic range (defined as the ratio of maximum to minimum detectable signal), which is dependent on device sensitivity, can also be viewed as design parameters and indicators of measurable sensor performance. When viewed as a design quantity, consider that dynamic range is at the top of the tree of design variables, i.e., there are no design quantities which are functions of dynamic range. However, a quantity such as sensitivity (which has been shown to depend on responsivity, Q and electrical noise in the system, all of which are “upper level" quantities in the sense that they can in turn be expressed as functions of other design variables) is one level down in the design tree from dynamic range, as the latter is a function of sensitivity. In this way, a hierarchy of design variables may be established to allow the designer to better track design options and their subsequent effects on performance. This idea will be explored more formally in the next section. /. Design Hierarchy o f Complex Systems The concept of a design hierarchy is well-defined in certain fields, such as VLSI for integrated circuit design, where complex system functions are progressively decomposed into smaller portions. The subdivision is complete when all essential facets o f functionality in the top-level device are present in the sum of these smaller pieces. If the subdivided elements are still too complex to be manageable, the process is repeated until the level of detail is more tractable [39], In this way, the functional dependence of a complex system may be related to the functionality o f the component parts which together form the larger system. The concept of hierarchy as an organizational tool is common to many diverse fields, from engineering to business management. However, the important aspect of hierarchical representations to be stressed here is that the function of a higher-level element is dependent on the details 92 o f the lower-level components to which it is connected in the hierarchical tree; conversely, the performance of a higher-level element can only be changed by implementing changes to the lower-level elements on which it depends. Figure 22 illustrates the use of design hierarchy in VLSI. The system-level description is shown at the top of (a}, and the detail equivalent is shown beneath, in progressively lower levels of the hierarchy. Thus, the ring oscillator is one level above the inverter from which it is made, and two levels above the transistors and capacitors which form the inverter. This is shown in part (b) of the figure. Again, the essential information conveyed by the dependency shown in part (b) reduces to the realization that only through changes in the transistors and capacitors can changes in the performance of the oscillator be effected. This very same idea is applicable (and arguably more illuminating) to the angular rate sensor system. 93 Oscillator Oscillator O ut Gnd Gnd (a) Functional Hierarchy of Oscillator Circuit (b) Design Hierarchy of Oscillator Figure 22: Exam ple of Design H ierarchy (a) The ring oscillator circuit is built from instances of identical logic blocks (in this case a CMOS inverter). The function of the top-level element (the oscillator) is dependent on the function of the building blocks (the inverter). The inverter in turn is built from instances of n-type and p-type transistors, and the function of the inverter is dependent on the function of these "leaf-level" elements, (b) An abstract design tree showing symbolically the same dependency as in (a). Design hierarchy could also be expressed using variable names in place of symbols, and can be applied to complex nonelectrical systems such as the angular rate sensor. In that case, each element in the hierarchy would be identified by the quantity which defines it’s function: Resonant frequency (oscillator), gain (amplifier), and geometry factors such as length and width (transistors). The angular rale sensor system presented in this thesis is dramatically more complex than the simple electrical oscillator shown in Figure 22; a cursory glance through the list of design parameters used in the course o f this work (which is found following the table o f contents, starting on page xiii) will bear this assertion out. For this reason, the ability to clearly delineate which subset of design variables must be 94 responsible for effecting a desired change in performance is not only more useful, but also more necessary. The first angular rate sensor system to be implemented will be open-loop for the reasons discussed in the previous section of this chapter Use o f Feedback Control on page 83; the following figures contain sample design hierarchies for the open-loop system described in Figure 20. Aw AT Actuator Force Cross-Sectional Area (A) Actuator thickness (a) Actuator width (A) Change in Temperature (AT) Reference Resistance (Rr) Variable Resistance (R) Actuator width ()>) Actuator length (w) Thermal Actuator Subsystem Actuator Deflection Actuator Time Constant Change in Temperature (AT) Actuator length (w) Thermal capacitance ( r ) Actuator thickness (a) Actuator width (f> ) Actuator length (w) Thermal conductance (g) Actuator thickness (a) Actuator width (/>) Actuator length (w) Figure 23: Design Hierarchy for Thermal Actuator Subsystem Design hierarchy for thermal actuator subsystem, showing three top-level performance functions: thermal force (Fllt), actuator displacement (Aw) and thermal time constant (x/A ). The dashed lines indicate the connected quantities are dependent on identical lower-level parameters, i.e., both c and g (thermal capacitance and conductance, respectively) are functions of a, b and w (actuator thickness, width and length, respectively). 9 5 Note that in the design hierarchy diagrams, only geometry dependence is shown; material constants and ambient conditions have been omitted for brevity. A verbal description is given below the symbolic hierarchy of the thermal actuator subsystem to help convey the meaning, but yet also shows the power of the condensed symbolic depiction. Clearly, a similar verbal description for the more complicated spring-mass and transducer subsystems would be much more cumbersome. m m m m m m "m m m Notes: F,h, Aw and t,a are detailed on thermal actuator hierarchy diagram. Key to subscripts on beam dimensions: a = shorter support spring m = longer support spring pr = piezoresistor beam Q appears on for resonant operation; otherwise is should be omitted. Figure 24: Design Hierarchy for Spring-Mass Subsystem Design hierarchy for spring-mass subsystem, showing three top-level performance functions: magnitude (rA) and phase (8) of proof mass displacement, and mechanical response time (tm) of system to reach steady-state operation. The dashed lines connect equal quantities. Note that thermal force, actuator displacement and thermal time constant design hierarchies are shown in Figure 23. 9 6 A diagram for the interface electronics subsystem is not shown; of the four main components which form the open-loop sensor system, the interface circuit consists entirely of discrete electrical components, and so is an independent design problem from the three other components which together form the sensor. The only design parameter of the interface circuit is the input voltage it receives from the transducer output. Changes in geometry intended to effect a change in this input voltage to the operational amplifier must be made in one of the three "sensor” subsystems, as the geometry of the opamps will only alter the performance of the amplifier itself (i.e., the opamp is “downstream” from the sensor signal). 97 w. m 'pr m m rn Note: rA is detailed on thermal actuator hierarchy diagram. Key to subscripts on beam dimensions: a = shorter support spring m = longer support spring pr » piezoresistor beam Q appears on for resonant operation; otherwise is should be omitted. Figure 25: Design Hierarchy of Transducer Subsystem Design hierarchy for transducer subsystem, showing one top-level performance function: change in resistance of piezoresistors in Wheatstone bridge. The dashed lines connect equal quantities. Note that proof mass displacement hierarchy is shown in Figure 24. 2, Opportunities fo r Invention As a final note on the topic of angular rate sensor design, it should be explicitly stated that all o f the difficulties presented here in obtaining a functional system. 98 whether imposed by a fundamental physical limit or merely inadequate knowledge of the design space, are to be regarded as opportunities for further invention. The remark was made in the introduction to this thesis that angular rate sensors pose a considerable design challenge, although no attempt was made to clarify or embellish the claim at that time. Rather than attempting to be overtly cavalier, this chapter on angular rate sensor design, if nothing else, should have presented a fairly substantial testimony to this claim. There are many specific design issues, in both the mechanical and electrical domains, which immediately present themselves as design challenges. First o f all, the very method of manufacture imposes perhaps the most fundamental of all design constraints: Scale. The effect that manufacture on the microscale can potentially have on a successful angular rate sensor design can be seen merely by inspection of the Coriolis force (12): Fc = 2m rAi In this expression, there are three design variables (m, rA, torf) and a single measurand (12); as all three design variables will scale with reduced device dimensions, so will the sensor output. Infinitesimal mechanical measures o f angular rate— from proof mass displacement rA to strain e in the piezoresistors— transform to infinitesimal electrical measures of angular rate. This is a trademark of surface micromachined resonant sensors; due to the limitations on size and scale imposed by fabrication considerations, sensor design and operation is inevitably pushing the envelope of existing performance limits. As mentioned earlier, angular rate sensors do not literally detect angular rate as much as Coriolis force; by 99 inspection of the approximate values for m, rA, and G )^ one immediately sees that the force detection being considered here is on the order of 10' 1 1 N/{rad/sec) for (m, rA, (Oj) = {lO'^kg, I pm, 104 rad/sec). The first consequence of designing a small-signal sensor is concern for the signal- to-noise ratio (SNR) for the system. Given that the forces which must be measured and reported with fidelity are on the order just mentioned ( lO 1 1 N/(rad/scc)), this clearly indicates that a functional angular rate sensor system must be inherently as noise-free as possible. While reducing unwanted fluctuation in sensor output is an accepted goal for any sensor design, in this case having unnecessary noise will not only degrade sensor performance, but in fact may render the sensor impotent if the noise levels rise above the angular rate threshold which is the minimum signal resolvable from noise from all sources (see Noise and Other Error Sources on page 67). This need to minimize or remove noisy elements (both electrical and mechanical) of the sensor system is the fundamental design challenge which currently faces the state-of-the-art today in angular rate sensor design; the same claim was made on page 83 regarding error signals introduced by imperfections in the manufacture of the sensor, which in fact are merely the main embodiments of mechanical noise. Much of the current research and development that is currently being done on gyroscopes and angular rate sensors are geared towards producing commercial products. Traditionally, the market for such sensors has been limited to high-end navigation and control applications, such as for spacecraft and the military. However, 100 ihere is a developing market for more consumer-oriented products which would require such a sensor; clearly the consumer market will not bear the costs (nor require the precision which drives them) of the traditional rate sensors. The potential cost benefits of micromachining sensors was mentioned in the opening remarks of this thesis, and it is these potential gains which fuels the considerable development currently directed towards angular rate sensors. These benefits depend on adequate solutions not only to the design challenges pertaining to the operation and functionality of the sensor, but also on other issues which the consumer-oriented applications will engender: manufacturability; testability; ease of validation and certification; self-test and redundancy features (or self-failure alert capability); packaging, both for operation and ease of interfacing with an existing infrastructure of the larger system of which the sensor is a part (e.g., car, pointing device or virtual reality headset). Many of these topics have not yet been solved, in some cases because they are not applicable to traditional sensors or the angular rate sensor industry has not yet faced the burden of finding more economical solutions. 101 Tfcble 7: Material Properties for Silicon Property Symbol Value Units Density P 2330 kg/m1 Specific heat at constant pressure cp 728 J/kg°C Temperature coefficient of resistance T, 1.1 x 10 1 l/°C Thermal coefficient of expansion P 2.3 x 10 6 (m/m)/°C Thermal conductivity a 157 W /m°C Young's modulus E 2 x 10" N/m2 Table 8: Other Material Properties Property Value Units Density of Air (1 atm, 25 °C ) 1.204 kg/m1 Density of Aluminum (Al) 2700 kg/m1 Density of Silicon Dioxide {SiOj) 2500 kg/m1 Kinematic viscosity of Air ( latm, 25 °C ) 1.5 x 10 1 m2/sec Thermal coefficient of expansion (S i0 2) 0.5 x 10'6 (m/m)/°C Thermal conductivity (S i0 2) 14 W /m°C Viscosity of Air (latm , 2 5 °C ) 1.8 x 10 5 N-sec/m2 Young's modulus (Al) 7 x I0 1 1 N/m2 Young’s modulus (S i0 2) 7 x I0n N/m2 102 IV. ANALYSIS AND RESULTS A. Thermal Actuators Test structures were fabricated to study the effects of both beam length and width on actuator force. Clamped-clamped beam structures made of various combinations of oxides, metal and a polysiticon resistor were tested (refer to Table 2-Table 5 for a complete listing of actuator test structures). Actuator deflections are expected to be small (nanometers), and so thermal force can be characterized by evaluating the critical Euler buckling load for each actuator configuration. The maximum buckling force expected is 396 mN, for the beam with the greatest width and shortest length (24pm and 50pm respectively). Full results are given in Figure 26. It should be noted that forces given in Figure 26 represent the minimum force generated by the thermal actuator, assuming there is no residual stress in the actuator after fabrication and release. This is due to the fact that the Euler buckling formula ~ 2 0 0 2 ) w accounts only for the load in the beam itself and does not consider any load taken up by the clamped supports. Thus, for actuators which do not possess residual stress, the generated thermal force for the various beam geometries could be substantially higher than that indicated in Figure 26. 103 3(M ) 7(XX) 250 b = 16pm i — £ 41M X ) 50 K M ) b = 8pm 1000 0 0 5 0.15 0 2 0.25 0.3 l/L2 (l/p m 2 * 10 ^ 0 4 5 0.35 0.4 Figure 26: Thermal Actuator Buckling Force and Residual Stress Plot of thermal force as a function of beam length. Here force is the calculated Euler buckling load for various beam lengths and widths. Beam lengths ranged from 50pm to 250pm in 50pm steps. Beam widths were 8pm , 16pm and 24pm. Beam length is plotted as (l/L 2), resulting in a straight-line plot. All test structures consisted of two layers of polysilicon and four layers of oxide, resulting in a beam thickness of approximately 5pm. Residual stress for thermal actuators which were found buckled after release. Stress was calculated using the buckling data given in Figure 26, for the same conditions and dimensions. Using the plot above, residual stress levels can be estimated for actuators which were not buckled due to residual stress alone. The buckling formula given in (102) contains certain limitations which will result in a buckling force calculation which will not agree with observable results from testing. The amount o f residual stress in the beams is of paramount importance, and will be discussed in more detail shortly. However, there are two other sources of error which should be addressed: end conditions and the structural composition of the beams. First of all, the factor of 4 in (102) represents the theoretical increase in load that a c lamped- clamped beam can withstand before buckling; in practice this increase is much less. Suggested values typically do not exceed 1 for the end-condition factor [40]. Secondly, the formula in (102) assumes a homogeneous beam composition; in reality, nearly all of the test structures will consist of several layers of oxide, polysilicon and possibly metal (see Table 2— Table 5 beginning on page 27 for exact composition of each beam test structure). For more accurate modeling, the moment of inertia / in (102) should be evaluated using composite beam theory [41], and the Young’s modulus E of each material used where appropriate. For the materials of interest here, there is fairly close agreement among the values of E. The assumption of zero residual stress is an idealization, as the materials in question will inevitably possess some residual stress. (Notably silicon dioxide, which produces thin-film structures which are typically highly compressed). This is verified by a visual inspection of the test structures, as some actuators were found to be partly buckled after release (see Figure 27 and Figure 28). This is due to residual stress in the thin films which compose the actuator structure. 105 Figure 27: Residual Stress in Thermal Actuators SEM photo showing variation in residual stress among actuator designs. Note some of the shorter actuator structures in the foreground that appear buckled, as well as the larger actuator in the top right comer (a close-up view of this actuator is shown in Figure 28). Actuators in the foreground are 50pm long. The spring-mass test structure in the middle of the photo has a beam length of 500pm from actuator to proof mass. Using the data given in Figure 26, the residual stress in non-buckled beams may be estimated by interpolation. The beams may then be subsequently forced to buckle under an applied thermal load, and thus a window of applicable force may be constructed, as residual stress will reduce the applied thermal force required for buckling. In this way an estimate for the effective critical thermal load may be obtained. For beams free o f residual stress, the critical thermal load will equal the critical Euler load given by (102); however, for beams with residual stress that are not yet buckled, the difference between the critical Euler and residual stress loadings results in a reduced "critical” thermal load which the actuator structure can carry before buckling. Buckling of the actuator test structures is the goal of testing, as it 106 offers a readily observable and measurable criteria for characterization. In practice, however, buckling of the actuator structure is in general undesirable. Figure 28: Thermal Actuator Buckled Due to Residual Stress SEM photo showing actuator structure partly buckled due to residual stress. The structure is 250pm long and 24 pm wide, and is composed of one layer of polysilicon and the four insulating oxide layers. Beam thickness is approximately 4 pm. Since the maximum calculated actuator deflection was only 58nm, actuator deflection is best studied by measuring the DC deflection of the proof mass. The clamped-clamped beam test structures can also be used to characterize proof deflection when appropriately coupled to additional test structures. Thermal actuators were connected to two support beams as suggested by Figure 1. The design of the support springs has previously been discussed. Figure 29 shows a typical test structure for measuring proof mass deflection, resonant frequency and Q factor. Deflection results are given in Figure 30. 107 Figure 29: Coupled Actuator-Spring Mass Test Structures SEM photograph showing thermal actuator attached to spring-mass elements. Debris can be seen in the etching holes of the proof mass, due to incomplete etching of (he pit feature during the fabrication process. w = 100pm E I 2.5 q = a 3 5 w = 50pm • 0 5 300 350 400 450 500 550 600 650 700 750 800 Longer Spring Length (pm) Figure 30: Proof Mass Displacement Deflection of the proof mass for actuator lengths of 50pm and 5100pm. AT = 100°C. Thickness for the actuator and both support beams was held constant at 5 pm, and the length of the shorter support spring was 100pm. 108 B. Spring-Mass Resonators Preliminary characterization of the spring-mass system resulted in the data shown in Figure 32 and Figure 34. Structures created for testing purposes consisted o f an actuator coupled to a spring-mass test fixture (see Figure 29). Resonant frequencies were calculated according to equations (41) and (42), assuming a square proof mass of constant thickness (5pm ). As mentioned in the section on design considerations (notably equations (37) - (39)), the resonant frequency of the spring-mass system should be less than the thermal frequency o f the actuator system. 7(X) 600 500 200 too 20 0 1 0 30 50 40 Beam Length2 (pin^lO'-') Figure 31: Thermal Actuator Time Constant Plot of thermal time constant as a function of actuator beam length. Beam lengths again ranged from 50pm to 250pm in 50pm steps. Length is plotted as (Lz), resulting in a straight-line plot. Beam thickness was 5pm. The connection between T,h and W q given by (39) is the fundamental design constraint for the tuning fork angular rate sensor design presented, linking the actuator and sensor components. The analytical model used to calculate resonant frequency is characterized by equation (41), and results from the lumped-parameter model of the actuator-proof mass system being modeled as a driven harmonic oscillator with viscous damping. Such a model inevitably is based at least in part on certain key assumptions or simplifications regarding the expected operation of the device being modeled. The simplifications introduced by the lumped parameter model have been briefly discussed previously in the chapter on device design under the section Noise and Other Error Sources on page 67, and the pertinent points will be reiterated here. By the definition of a lumped-parameter model, the springs are assumed to be massless, and so will not contribute to the mass of the system. Similarly, the support spring is an idealized element which is solely responsible for the storing and translation of energy from potential to kinetic as the mass oscillates. In reality, the springs will have a small amount of mass (estimates are given on page 31), and the longer support spring is not the only elastic element of the system. Thus, the resonant frequency calculated according to (41) is in reality an approximation, given the limitations of the analytical model just discussed. It should be noted here that in the lumped parameter model there is a clean division between elements which store energy and those which dissipate energy (symbolized respectively by the spring and dashpot in Figure 2), and natural frequency is a parameter which is associated with elements responsible for storing energy (the support springs of the proof mass). Thus, the assumptions of this model just discussed 110 will not affect the physical elements associated with energy dissipation; conversely, the limitations of the viscous damping model used to analyze energy dissipation (and notably the calculation of Q) will not affect the modeling o f elements for energy storage. 2 1 K X X ) & 1 a 3 Z 5<X X ) 11 < X > 500 «X> 700 lin g e r Spring Length Lm (pm) I (X X ) K00 900 Figure 32: Natural Frequency of Spring-Mass System Resonant frequencies for proof mass and support spring systems. Notice decreases dramatically for small increases in Lm and is more weakly dependent on increases in m. Proof mass dimensions are for one side of a square mass of width wp and maximum thickness (a= 6pm .) Actual values of m range from 9xlO '10kg (WpslSOgm) to 4 x l()'9kg (wp=500pm). C . R esonant B ehavior /. Quality Factor One widely-cited benefit of polysilicon resonators is the high intrinsic Q factor of the structural material (e.g., [2] [3] [6 ] [8]). Depending on the viewpoint, the Q factor Ill is an indication of either how much gain or relative damping a system possesses. Insofar as damping is concerned, most work relating damping to Q factor has involved the study of various flow-types between narrow gaps separating a resonant-plate structure from the substrate. In these prior studies, the fluid thickness for damping calculations is typically taken to be the thickness of the sacrificial layer whose removal renders the structure free [23] [24], In the present case, however, rather than a narrow 2pm gap separating the resonant proof mass plates from the substrate, the substrate itself will be removed resulting in large, scalloped-botlomed cavities in excess of 30pm . The impact that such a difference in operating environment will have on damping is best determined by direct observation. 250 200 150 100 5000 10000 15000 Natural frequency (rad/sec ) Figure 33: Quality Factor versus Natural Frequency Plot of the relationship between Q and (U q given in equation (56). The expression in (56) is proportional to the square root of natural frequency, but is linearly dependent on changes in the proof mass. Since i% is also dependent on m (tty varies inversely with the square root of m), the curve of Q versus is closer to being a linear relationship than a square root. For each set of symbols corresponding to a distinct size of the proof mass plate, the spring lengths from left to right are 400pm, 500pm, 750pm, and 1000pm. 112 The limitations in the use of Stokes’ oscillating plate solution for planar structures resonating parallel to the substrate has been addressed elsewhere [23]. The assumption of an infinite plate results in the simplification of the Navier-Stokes equation for the flow of the damping fluid past the plate, and the terms such a simplified analysis ignores were indeed shown to be negligible in [23] for geometries where the width of the plate is much greater than its thickness. This is the case for the present design as well. One physical phenomenon which is not accounted for in the viscous damping model used here is the effect of the damping fluid on the effective mass of the resonating structure. This loading effect serves to increase the effective mass of the oscillating structure, and thus lower its natural frequency as well as the Q of the system. There is a divergence of opinion in the current research literature as to whether this effect is significant or not, although there is unanimous agreement that it is present and will cause a shift in (o0 and Q. This disagreement has been discussed previously in the course of deriving Q as a design parameter on page 45. A brief discussion of possible error sources involved in calculating Q for a resonating structure using Stokes’ model is presented in [23]. The data presented for Q in Figure 33 is numerically identical to that shown in Figure 34. The variation in the m anner in which the data is presented illustrates visually the concept of design hierarchy introduced near the end of the previous chapter. Q{(o0) appears as a single curve, despite the fact that three quantities were 113 varied to produce the plot: (%, Lm and wp. Spring constant km and proof mass m depend on Lm and wp, respectively; afy in turn is a function of km and m. Thus, any changes in Lm and wp appear rather generically in a plot o f Q (1 %) as merely changes in w0. Figure 34, however, shows Q (Lm), and is in terms of leaf-level design variables and material properties. Thus, as both Lm and wp appear explicitly in the expression for Q (Lm), the data becomes more recognizably interpretable based on given variations in Lm and wp. (Note that plotting Q(wp) and varying Lm and wp would have produced an analogous plot to that shown, except each of the four curves would represent a different beam length rather than plate width). hi o & LU o 2(X) 150 500 700 800 400 600 1000 1100 Longer Spring Length 1^, (pm) Figure 34: Quality Factor versus Spring Length Quality factor as a function of length for the spring supporting the proof mass, based on equation (57). Width of the proof mass plate was also varied, which produced the rather uniform family of curves shown. (57) gives Q(Lm)» 1 !Lm, with changes in wp serving to shift the plot of Q(Lm) as illustrated by the figure. 114 2. Mechanical Response Time The expressions obtained in Mechanical Response Time on page 53 are plotted in a similar manner to Figure 33 and Figure 34 for quality factor Q. Substitution of Q(ongcr Spring Length Lm (pm) Figure 36: Mechanical Response Time versus Spring Length Mechanical response time as a function of length for the spring supporting the proof mass, based on equation (65). Behavior is similar to that shown in Figure 34 for Q, except increases in either Lm or wp result in an increase in t m. Response time was calculated for the resonant condition only. 3. Responsivity The expression for responsivity obtained in (81) is dependent on three main parameters: the scaling factor T|, which related the length of the piezoresistor beam (Lpr) to the length of the longer support spring (Lm), proof mass displacement rA and natural frequency (%. Thus, R(ti, rA, to0) is plotted in Figure 37; the figure is roughly a plot of R 116 30 25 20 15 1 0 10000 15000 N atural frequency (rad/sec) Figure 37: Responsivity versus Natural Frequency Plot of the relationship between R and Up given in equation (81). The expression in (81) is inversely proportional to natural frequency, but is linearly dependent on changes in the displacement of the proof mass. Proof mass displacement is in turn proportional to the length of the longer support spring (as suggested by the figure), and so the length of the support spring was also varied as shown. Responsivity was calculated for the resonant condition only. R(T|, rA, tOfl) can be further reduced to the basic design parameters by substituting for natural frequency. Thus, responsivity is seen to fundamentally depend on the length of the longer support spring (Lm) and size of the proof mass (w^). R(wp, Lm) is shown in Figure 38. 117 . 16 p * j X c > c 24 > M X) 330 400 450 500 550 600 650 700 750 K00 Longer Spring Length Lm (pm) Figure 38: Responsivity versus Spring Length Plot of the relationship between R. Wp, and L m given in equation (82). The expression in (82) is directly proportional to both size of the proof mass(square root) and length of the longer support spring (3/4 power). Responsivity was calculated for the resonant condition only. 4. Sensitivity Expressions for sensitivity as a function o f higher-level (Q ), middle-level ((%) and leaf-level variables (wp and Lm) are given respectively in equations (85), (86) and (87). The relationships in equations (86) and (87) are correspondingly plotted in Figure 39 and Figure 40. 118 20 IK 16 14 1 2 1 0 6 4 2 10000 15000 20000 Natural frequency (rad/sec) Figure 39: Sensitivity versus Natural Frequency Plot of the relationship between S and given in equation (86). The expression in (86) is proportional to the square root of natural frequency, but is linearly dependent on changes in the proof mass. Since c% is also dependent on m (tty varies inversely with the square root of m), the curve of S versus (tyj is closer to being a linear relationship than a square root. The length of the support spring was also varied. For each set of symbols corresponding to a distinct size of the proof mass plate, the spring lengths from left to right are 4(X)(im, 500pm, 750pm, and 1000pm. Sensitivity was calculated for the resonant condition only. 119 6 5 q > £ 4 5 3< X ) 350 400 450 500 550 600 650 7(X > 750 K00 [xmger Spring Length Lm (pm) Figure 40: Sensitivity versus Spring Length Sensitivity as a function of spring length and size of proof mass. Sensitivity is inversely proportional to the square root of wp, and the three-fourth root of l.m. giving the figure above its general i/x characteristic. Sensitivity was calculated for the resonant condition only. 120 V. CO NCLUSIO N The complete design process for angular rate sensors has been presented and discussed in some detail. Major topics of focus were found to be frequency dependence, time constants for thermal cycling and transient motion, modeling of viscous damping and the calculation of quality factor as a function of design variables (i.e., geometry, operating conditions, and material properties), and consideration of the prominent noise and error sources inherent to the system. A system-level overview of a novel thermally-driven angular rate sensor fabricated in a standard commercial CMOS process has been presented. Also, performance curves for the thermal actuator have been derived as functions of beam geometry, and preliminary design analysis of the spring-mass system has been shown. Validation of these expected results is the first step towards realizing a functional angular rate sensor fabricated in a standard CMOS process. Future work on this device will include the integration and optimization of the systems introduced here. The main goal of this paper has been to validate the concept through a careful characterization of the driving actuator and a demonstration that the spring-mass system can be effectively driven by such a mechanism. Subsequent work will build on these mechanical foundations and investigate such a system’s ability to deliver a readable and reliable signal [19]. 121 VI. REFERENCES [ 1 ] G.C. Newton, Jr., “Comparison of Vibratory and Rotating-Wheel Gyroscopic Rate Indicators,” Proc. AIEE, July 1960, pp. 143-150. [2] W.C. Tang, T.-C. Nguyen, and R.T. Howe, “Laterally Driven Polysilicon Resonant Microstructures," Sensors and Actuators A, vol. 20, 1989, pp. 25-32. [31 J. Bernstein, S. Cho, A.T. King, A. Kourepenis, P. Maciel, and M. Weinburg, "A Micromachined Comb-Drive Tuning Fork Rate Gyroscope," Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘93), 1993, pp. 143-148. [41 J.E. McDonald, “The Coriolis Effect", Scientific American, May 1952, pp. 72-78. 1 5] R. Resnick and D. Halliday, Physics, Part /, John Wiley and Sons, New York, 1977, p. A l. [6] R.M. Langdon, “Resonant Sensors - A Review”, J. o f Phys. E.: Sci. Instrum., vol. 18, 1985, pp. 103-115. [7] R.P. Feynman, et al, The Feynman Lectures on Physics, Vol. I, Addison-Wesley Publishing Company, Reading, MA, pp. 23-3— 23-4. Any college physics or vibrations textbook will contain a solution to the harmonic oscillator, but the approach taken in the Feynman Lectures is the most straightforward I have encountered. [8] P. Hauptmann, “Resonant Sensors and Applications", Sensors and Actuators A, vol. 25-27, 1991, pp. 371-377. [9] R.A. Anderson, Fundamentals o f Vibrations, The Macmillan Company, New York, 1967, p. 43. [10] M.W. Putty and K. Najafi, “A Micromachined Vibrating Ring Gyroscope,” Tech. Digest o f IEEE Solid-State Sensor and Actuator Workshop (Hilton Head ‘94), 1994, pp. 213-220. [11] Orbit Semiconductor 2.0jim process as offered through the MOSIS (MOS Implementation Service) brokerage. Details available through World Wide Web at http://info.broker.isi.edu/mosis; or send electronic mail to mosis@ mosis.edu. [12] P. Greiff, B. Boxenhom, T. King, and L. Niles, “Silicon Monolithic Gyroscope,” Proc. Int. Conf. on Solid-State Sensors and Actuators (Transducers ‘91 J, 1991, pp. 966-969. 122 [13] K. Tanaka, Y. Mochida, S. Sugimoto, K. Moriya, T. Hasegawa, K. Atsuchi, and K. Ohwada, “A Micromachined Vibrating Gyroscope,” Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘95), pp. 278-281. [14] M. Parameswaran, Lj. Ristic, K. Chau, A.M. Robinson, and W. Allegretto, “CMOS Electrothermal Microactuators,” Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘90), pp. 128-131. [15] M. Ataka, A. Omodaka, N. Takeshima, and H. Fujita, “Fabrication and Operation of Polyimide Bimorph Actuators for Ciliary Motion System,” J. o f Microelectromech. Systems, vol. 2, no. 4, December 1993, pp. 146-150. [16] J. Marshall, et al, "Realizing Suspended Structures on Chips Fabricated by CMOS Foundry Processes Through the MOSIS Service”, NISTIR 5402, U.S. National Institute of Standards and Technology, Gaithersburg, MD 20899, June 1994. [17] E. Hoffman, B. Warneke, E. Kruglick, J. Weigold, and K.S.J. Pister, “3D Structures with Piezoresistive Sensors in Standard CMOS,” Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘95), pp. 288-293. [18] M. Parameswaran, A.M. Robinson, Lj. Ristic, K. Chau, and W. Allegretto, “A CMOS Thermally Isolated Gas Flow Sensor," Sensors and Materials, vol. 2, no. 1, 1990, pp. 17-26. [19] K.S.J. Pister, “Designing Accelerometers in Standard CMOS”, EE250b course notes, Winter 1995, University of California, Los Angeles. [20] P. Will, W. Liu, M. Pottenger, “Microrobot Assembly Systems Semiannual Report”, July 1995, prepared under ARPA contract DABT 63-92-C-0052. [21 ] R.L. Panton, Incompressible Flow, John Wiley and Sons, New York, 1984, pp. 266-270. [22] R.T. Howe, R.S. Muller, “Resonant-Microbridge Vapor Sensor”, IEEE Trans, on Elec. Dev., vol. ED-33, no. 4, April 1986, pp. 499-506. [23] Y.-H. Cho, A.P. Pisano, R.T. Howe, “Viscous Damping Model for Laterally Oscillating M icrostructures,"/ o f Microelectromech. Systems, vol. 3, no. 2, June 1994, pp. 81-87. [24] X. Zhang, W.C. Tang, “Viscous Air Damping in Laterally Driven Microstructures,” Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘94), 1994, pp. 199-204. 123 [25 J H. Hosaka, K. Itao, S. Kuroda, “Evaluation o f Energy Dissipation Mechanisms in Vibrational M icroactuators”, Proc. IEEE Workshop on Microelectromech. Systems (MEMS ‘95), pp. 193-198. [26| W.E. Newell, “Miniaturization of Tuning Forks”, Science, vol. 161, September 1968, pp. 1320-1326. [27] F.R. Blom, S. Bouwstra, M. Elwenspoek, J.H.J. Fluitman, “Dependence of the Quality Factor of M icromachined Silicon Beam Resonators on Pressure and Geometry", J. o f Vac. Sci. Tech. B , vol. 10, no. 1, January/February 1992, pp. 19- 26. [28] M. Christen, “Air and Gas Damping of Quartz Tuning Forks”, Sensors and Actuators A, vol. 4, 1983, pp. 555-564. [29] F Browand, personal communication, August 1995. [30] C.M. Harris, C.E. Crede, Shock and Vibration Handbook, vol. 1, M cGraw-Hill, New York, pp. 2-7— 2-15. [31 ] D.G. Ullman, The M echanical Design Process, McGraw-Hill, New York, 1992, pp. 229-233. [32] K.E. Petersen, "Silicon as a Mechanical Material", Proc. o f the IEEE, vol. 70, May 1982, pp. 420-457. [33] F. Bordoni, A. D'Am ico, “Noise in Sensors", Sensors and Actuators A, vol. 21- 23, 1990, pp. 17-24. [34] Notes, “Short Course on M icroElectroMechanical Systems”, University of California, Los Angeles, February 21-24, 1995. [35] T.B. Gabrielson, “Mechanical-Thermal Noise in M icromachined Acoustic and Vibration Sensors”, IEEE Trans, on Elec. Dev., vol. ED-40, no. 5, May 1993, pp. 903-909. [36] R.A. Brennan, M.G. Lim, A.P. Pisano, A.T. Chou, “Large Displacement Linear Actuator”, Tech. Digest o f IEEE Solid-State Sensor and Actuator Workshop (Hilton Head ‘90), pp. 135-139. [37] Y.C. Tai, R.S. Muller, “M easuring Young’s M odulus on M icrofabricated Structures Using a Surface Profiler", Proc. IEEE Workshop on Microelectromech. Systems (MEMS *90), pp. 147-152. 124 [38] G.C. Newton, Jr., “Theory and Practice in Vibratory Rate Gyros,” Control Engineering, June 1963, pp. 95-99. [39] N. Weste, K. Eshraghian, Principles o f CMOS VLSI Design, Addison-Wesley Publishing Company, Reading, MA, p. 239. [40] J.E. Shigley, C.R. Mischke, Mechanical Engineering Design, McGraw-Hill Book Company, New York, p. 123. [41 ] D. Burgreen, P.J. Manitt, “Thermal Buckling of a Bimetallic Beam”, Proc. Amer. S< )c. Civil Eng., vol. 95, no. EM2, April 1969, pp. 421-432. INFORMATION TO USERS This manuscript has been reproduced from the microfilm m aster. UMI film* the text directly from the original or copy submitted. 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Asset Metadata

Creator Pottenger, Michael Dean (author)

Core Title Thermally-driven angular rate sensors in standard CMOS

School School of Engineering

Degree Master of Science

Degree Program Mechanical Engineering

Degree Conferral Date 1995-12

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,engineering, mechanical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Shiflett, Geoff R. (committee chair), Flashner, Henryk (committee member), Muntz, Eric Phillip (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-3737

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