Inertial MEMS: Principles and Practice

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Inertial MEMS Principles and Practice A practical and systematic overview of the design, fabrication, and testing of MEMS-based inertial sensors, this comprehensive and rigorous guide shows you how to analyze and transform application requirements into practical designs, and helps you to avoid potential pitfalls and to cut design time. With this book you’ll soon be up to speed on the relevant basics, including MEMS technologies, packaging, kinematics and mechanics, and transducers. You’ll also get a thorough evaluation of different approaches and architectures for design and an overview of key aspects of testing and calibration. Unique insights into the practical difficulties of making sensors for real-world applications make this up-to-date description of the state of the art in inertial MEMS an ideal resource for professional engineers in industry, managers, and application engineers, as well as for students looking for a complete introduction to the area. Volker Kempe has more than 40 years of experience in research and development in both academia and industry. He led the microelectronics engineering department at Austria Mikro Systems for over 10 years. In 2003 he co-founded, and became Vice President of, SensorDynamics AG, and his current interests focus on the functionality, technology, and application of inertial MEMS.

Inertial MEMS Principles and Practice Volker Kempe Sensor Dynamics AG, Austria

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ a o Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521766586 ° C

Cambridge university Press 2011

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Kempe, Volker. Inertial MEMS : principles and practice / Volker Kempe. p. cm. Includes bibliographical references and index. ISBN 978-0-521-76658-6 (hardback) 1. Microelectromechanical systems. 2. Inertial navigation systems. TK7875.K46 2011 629.04′ 5 – dc22 2010037668 ISBN

3. BioMEMS.

978-0-521-76658-6 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

I. Title.

Contents

Preface Acknowledgments Notation

page xiii xv xvi

1

Introduction 1.1 A short foray through the pre-MEMS history 1.2 Applications and market 1.3 The ingredients of inertial MEMS References

1 1 6 9 11

2

Transducers 2.1 Anisotropic material properties, tensors, and rotations 2.1.1 Stress, strain, and piezoresistivity Hooke’s law Normal stresses Shear stresses Stress and strain tensors The stress–strain relation for anisotropic materials The piezoresistance of silicon 2.1.2 Rotation of coordinate systems Coordinate frames The rotation tensor Transformation of tensors of second order 2.2 Piezoresistive transducers 2.2.1 Piezoresistors 2.2.2 Piezoresistors on silicon Thin piezoresistors Temperature compensation in piezoresistors 2.2.3 Piezoresistors on polysilicon 2.3 Piezoelectric transducers 2.3.1 The piezoelectric effect 2.3.2 Piezoelectric equations Piezoelectric sensors in MEMS

13 13 14 14 14 15 17 21 24 25 26 27 30 33 33 34 35 36 38 39 39 42 43

vi

Contents

2.4

Capacitive transducers 2.4.1 Electrostatic forces 2.4.2 Parallel-plate capacitors Capacitance sensing The pull-in effect 2.4.3 Tilting-plate capacitors Nonlinear distortions Instabilities of a singular tilting plate Instabilities of the tilting capacitance pair 2.4.4 Comb capacitors Unidirectional linear combs Bidirectional actuation Radial combs Frame-based capacitors 2.4.5 Levitation Comb levitation Levitation in drive combs Reduction of levitation forces References 3

Non-inertial forces 3.1 Springs 3.1.1 Beams 3.1.2 The stiffness matrix 3.1.3 The bending equation for beams Internal forces and moments Differential relations of a bent beams 3.1.4 Cantilever beams Cantilevers under different loads Skew beam bending and asymmetric suspensions Residual stress in bending beams 3.1.5 Torsion springs Cylindrical torsion bars Torsion bars with arbitrary cross-section Rectangular bars Cylindrical bars 3.1.6 Stress concentration 3.1.7 Suspensions Parallel and serial spring connections Beam chains Plate suspension 3.2 Damping forces 3.2.1 Fluid-flow models

47 49 51 51 53 55 56 58 58 62 62 65 66 67 69 69 71 73 75 79 79 80 81 82 83 83 87 87 90 93 96 96 97 100 101 101 103 103 104 105 108 108

Contents

Continuous viscous flow Viscosity of gases Continuous-flow equations Slide damping Couette flow for slowly moving plates Stokes flow for rapidly oscillating plates Squeeze damping Reynolds’ equation Low-frequency squeeze damping High-frequency squeeze damping The impact of perforation Drag forces Free molecular flow Structural damping

111 112 114 117 118 120 124 125 127 131 137 143 145 147 148

MEMS technologies 4.1 Microfabrication of inertial MEMS 4.1.1 Basic microelectronic fabrication steps Deposition Patterning Doping 4.1.2 Etching Isotropic wet etching Anisotropic wet etching Electrochemical etch stop 4.1.3 Dry etching Reactive-ion etching Deep reactive-ion etching 4.2 Wafer bonding 4.2.1 Zero-level packaging and wafer bonding 4.2.2 Wafer-bonding processes Fusion bonding Anodic bonding Glass-frit bonding Metallic-alloy seal bonding Polymer bonding Thermocompression bonding 4.3 Integrated processes 4.3.1 Bulk micromachining 4.3.2 Surface micromachining A thick polysilicon process Cavity sealing using SMM

152 153 154 155 159 161 162 162 164 167 168 168 170 172 172 174 175 176 177 179 180 180 180 182 184 185 188

3.2.2

3.2.3

3.2.4 3.2.5 3.2.6 References 4

vii

viii

Contents

4.3.3

SOI-MEMS processes MEMS prototyping processes 4.3.4 CMOS-MEMS Pre-CMOS MEMS Intra-CMOS MEMS Post-CMOS MEMS References

189 192 192 193 195 196 199

5

First-level packaging 5.1 FLP packages 5.2 FLP technologies 5.2.1 Dicing and die separation 5.2.2 Die attachment Packaging materials Die-attachment-induced stress 5.2.3 Electrical interconnection 5.2.4 Encapsulation Overmolded plastic packages Pre-molded plastic packages References

205 206 209 210 211 212 213 217 220 221 223 225

6

Electrical interfaces 6.1 Sensing electronics – building blocks 6.1.1 The MOS transistor Drain current The small-signal model 6.1.2 Operational and transconductance amplifiers A simple transconductance amplifier Models of operational and transconductance amplifiers The real Op Amp Instrumentation amplifiers 6.2 Sensor interfaces 6.2.1 Resistive interfaces 6.2.2 Piezoelectric interfaces 6.2.3 Capacitive interfaces Principles of capacitive sensing Current sensing Voltage sensing Charge sensing Comparison and improvements Switched-capacitor sensing 6.3 Data converters 6.3.1 Sampling and hold

227 228 229 229 231 234 234 236 240 246 247 247 249 253 254 255 258 259 260 260 266 266

Contents

6.3.2 6.3.3

7

Single-sample conversion in the amplitude domain Time-domain conversion Pulse-width and pulse-density modulation Σ∆ Converters

ix

References

268 269 269 271 280

Accelerometers 7.1 General measurement objectives 7.2 The spring–mass system 7.2.1 The transfer functions The trade-off between sensitivity and bandwidth 7.2.2 Accelerometer imperfections A simplified accelerometer model with imperfections Cross-coupling 7.2.3 Accelerometer feedback control The linearized feedback model The signal-to-noise ratio Closed-loop dynamics 7.2.4 Feedback control with nonlinear actuators Bidirectional capacitive actuators Single-sided actuators Linearization and embedded Σ∆ converters 7.3 Resonant accelerometers 7.3.1 Resonant beams Resonance vibration – exact solution Resonance frequencies by the energy method 7.3.2 Resonant accelerometer systems 7.4 Beam accelerometers 7.4.1 Beam dynamics The principle of virtual work Eigenmode expansion Damping and electrostatic forces Static deflection 7.4.2 Model implementation The impact of nonlinear damping Feedback control 7.5 Various other accelerometer principles 7.5.1 Tunneling accelerometers 7.5.2 Convective and bubble accelerometers 7.6 From 1D to 6D accelerometers 7.6.1 1D accelerometers Piezoresistive accelerometers Capacitive accelerometers

283 283 284 286 291 292 296 297 298 300 305 307 309 310 312 313 319 320 320 323 324 326 328 329 330 332 333 333 335 336 337 337 339 341 342 343 346

x

Contents

Piezoelectric accelerometers 2D and 3D accelerometers Parallel implementation 2D and 3D accelerometers with multi-DOF sensing elements 7.6.3 6D accelerometers References

348 350 350 353 355 358

Gyroscopes 8.1 Some basic principles 8.2 Kinematics of gyroscopes 8.2.1 Platform rotation and angular velocity 8.2.2 Body rotation in a non-inertial system 8.2.3 The angular-momentum theorem 8.2.4 The momentum equation 8.2.5 The small-angle approximation 8.3 The performance of gyroscopes 8.4 Rate-integrating gyroscopes 8.4.1 Two-DOF gyroscopes 8.4.2 The principle of angular gyroscopes 8.4.3 An imperfection model 8.4.4 Imperfection in angular gyroscopes 8.4.5 Gyroscope control 8.5 Rate gyroscopes 8.5.1 System architecture The drive resonator Sensing 8.5.2 Resonance sensing 8.5.3 Non-resonant sensing 8.5.4 Noise 8.5.5 The zero-rate output Mechanical bias sources Q-bias The impact of transducer imperfections R-bias Other bias sources 8.5.6 Bias stability 8.5.7 Acceleration suppression and tuning forks Anti-phase-driven identical gyroscopes Tuning-fork gyroscopes 8.5.8 Drive-motion control and spring nonlinearities The phase-locked loop The amplitude loop Spring nonlinearities and the resonator transfer function

364 364 367 369 371 373 375 376 378 380 380 381 385 387 388 389 391 393 393 395 398 400 402 403 405 408 409 410 411 413 414 415 418 418 419 421

7.6.2

8

Contents

9

xi

8.6

Gyroscope architectures 8.6.1 Mode-decoupling architectures 8.6.2 z-Gyroscopes Mode decoupling by frame-based architectures Doubly decoupled z-gyroscopes 8.6.3 In-plane-sensitive gyroscopes In-plane-sensitive linear gyroscopes Linear–rotatory gyroscopes 8.6.4 Torsional gyroscopes 1D torsional gyroscopes Decoupled torsional gyroscopes 8.7 Non-planar MEMS gyroscopes 8.7.1 Beam gyroscopes 8.7.2 Quartz tuning forks 8.7.3 Ring gyroscopes 8.7.4 Bulk acoustic-wave gyroscopes 8.8 2D and 3D gyroscopes and ways towards a 6D IMU 8.8.1 Single-mass multiple-DOF inertial sensors Gyroscope-free, single-mass IMUs Single-mass, gyroscope-based IMUs 5D inertial sensors 8.8.2 2D gyroscopes 8.8.3 3D gyroscopes A fully decoupled 3D gyroscope and extension towards an IMU References

424 424 424 424 427 429 429 430 431 432 433 438 438 440 441 443 444 445 445 445 447 448 449 450 452

Test and calibration References

460 464

Concluding remarks References

466 467

Index

468

Th e co l o r p la t es a re t o be fo und bet w een pa ges 240 a nd 241.

Preface

Inertial microelectromechanical sensors – commonly abbreviated to inertial MEMS – have a history of more than two decades of intense research, development, and commercialization. Sometimes unperceived, they left the shadow of military and space-related utilities and entered daily life hidden in products surrounding us. Cars with airbag-release sensors and electronic stability control have become a matter of course. Activity monitoring of pacemaker patients and stabilization of platforms such as transport robots and cameras are now improving our quality of life. The creation of easy-to-use human–machine interfaces has helped many people to conquer complicated equipment around us, not only computer games. The penetration of inertial MEMS – often merged with other sensor systems – into new application areas is a trend that is still gaining momentum. The intention of this book is to reflect the interdisciplinary complexity of inertial MEMS. It will try to give a systematic survey of the design, fabrication, and performance evaluation of MEMS-based inertial sensors, with emphasis on the practical problems arising from the impact of technological imperfections and of often harsh environmental conditions. A product going to the market has to be guaranteed to have a certain level of reliability against failure throughout its lifetime. The basic concepts and the theoretical background of inertial measurements will be presented. However, the book has evolved not from academic activity but rather from conceptual and development work within industry. It is intended to address the symbiosis of practice and theory. Consequently, the analysis and transformation of application requirements into design concepts plays a significant role. Considerable space is devoted to the analysis and modeling of parasitic effects, of shock and vibration robustness, of the stability of the main performance parameters and so on, since this is necessary for practical work. The book contains nine chapters. Six of them – including the introduction – cover various aspects of MEMS, with a special focus on inertial MEMS. The first chapter describes the most important transducers and their properties. The second one is dedicated to non-inertial forces such as spring forces and damping forces that play a crucial role for designing inertial MEMS. The next two chapters cover the main MEMS technologies, including packaging, while the electronic interfaces are presented in a further chapter. These six chapters may be

xiv

Preface

interesting not only for people working with inertial MEMS but also for everybody who is looking for a general introduction into mechanical MEMS. The following two main chapters are devoted to the two representatives of inertial MEMS – accelerometers and gyroscopes. Here the focus is on the basic principles, the methods and models to describe the dynamic behavior, and a comprehensive presentation of different approaches and architectures, including their pros and cons. A short overview on test and calibration is added as a separate chapter. The book is written on an engineering level. Where possible, effects and processes are described analytically by mathematical models in order to impart a feeling for the order of magnitude of different effects. The book should be useful not only for specialists developing, manufacturing, and using inertial sensors but also for people working in the application field, for product managers, and for sales people looking for background knowledge in their area. The book can serve as a starting point for further academic investigations, for instance in the area of shock-impact analysis of an entire packaged gyro, including the effect of signal processing. In the experience of the author, many engineers, physicists, and mathematicians are thankful for an exact but comprehensible presentation of the complex and difficult world of MEMS-based inertial sensors, where the effects and models behind the practical problems are reflected without improper simplifications or phenomenological descriptions. The book is a modest attempt to meet some of these challenges. Having worked with many specialists in the production, testing, and design of inertial sensors, the author is convinced that the book can meet actual needs, and hopes to elicit the broad interest of practitioners and scientists in this area. For interested people, including students, the book may also serve as an introduction to the world of mechanical MEMS.

Acknowledgments

I would like to express my gratitude to all my colleagues at SensorDynamics AG (Austria) and the Institut f¨ ur Silizium Technologie (ISIT) of the Fraunhofer Society (Germany) for creating an atmosphere that has helped to solve the manifold problems of MEMS industrialization. I would like to thank the ‘Inertial Micro Sensor Systems’ team, with whom I have had the great privilege of working even during the childhood of the newly founded company SensorDynamics AG. This time was most fruitful, flooding us all with new problems and insights into how to solve them. My thanks go to my colleagues from SensorDynamics for providing me with such necessary illustrative material for the book as SEM photographs and measurement plots. Gottfried Frais, Manfred Heller, Christian Rossadini, J¨ org Sch¨ onbacher, Ute Stotter, and Johann Wagner prepared a lot of material from which I could select the most appropriate items. Gerd Radl and his team accompanied me during all my mistakes with new hardware and software. Drago Strle from the University of Ljubljana deserves my special thanks for the close cooperation throughout all Σ∆-related issues. Peter Merz from the ISIT gave me invaluable feedback on all technology-related questions. Professor Karl Wohlhart from the University of Graz helped me to gain a deeper understanding of the kinematics of gyroscopes. Hanno Hammer took on the burden of proofreading the first chapters and supplying me with valuable feedback. Last but not least, warmest thanks go to my family for supporting me despite all the personal loads that each of us has had to carry. Julia and Oded helped me to resolve quickly the countless difficulties in adapting appropriate documentcreating tools to my needs. Vera, Ian, and Marius gave valuable advice on English phrasing. My dear wife took on additional duties despite her own excessive workload in working with very ill patients. Thank you.

Notation

1. A convention employed in this book is the slightly lax usage of “s” as differential operator, s = d/dt, as argument of the Laplace transformation, and as argument of the Fourier transformation, s = jω. The case-dependent unambiguous or multivalent meaning is usually clear from the context. Accordingly, a filter function is described by f = f (s), which means that in a transfer function this expression has to be interpreted as a Laplace or Fourier transformation and within a differential equation as a rational fraction of two polynomial differential operators. Correspondingly, a variable like x has to be treated as a representant in the time domain if s = d/dt is supposed, or as a Laplace/Fourier-transformed function if s is meant as the argument of such a transformation. 2. Unless stated otherwise, coordinate systems pertain to the platform carrying the inertial sensor. In this case the x- and y-axes lie in the plane of the platform, while z is the out-of-plane axis. Out-of-plane and z-axis orientation are used synonymously.

1

Introduction

An inertial sensor is an observer who is caught within a completely shielded case and who is trying to determine the position changes of the case with respect to an outer inertial reference system. Inertial sensors exploit inertial forces acting on an object to determine its dynamic behavior. The basic dynamic parameters are acceleration along some axis and the angular rate. External forces acting on a body cause an acceleration and/or a change of its orientation (angular position). The rate of change of the angular position is the angular velocity (angular rate). A speedometer is not an inertial sensor because it is able to measure a constant velocity of a body that is not exposed to inertial forces. An inertial sensor is unable to do so; however, if the initial conditions of the body are known, their evolution can be calculated by integrating the dynamic equation on the basis of the measured acceleration and rate signals. In the overwhelming majority of practical applications, such as vibrational measurements, active suspension systems, crash-detection systems, alert systems, medical activity monitoring, safety systems in cars, and computer-game interfaces, the short-term dynamic changes of the object are of interest. But there are also many applications where inertial sensors are used for determination of the positions and orientations of a body, as in robotics, general machine control, and navigation. Owing to the necessity of integrating the corresponding dynamic equations, the accuracy requirements in these applications are usually higher because the measurement errors and instabilities of the sensors are accumulated over the integration time. Often inertial sensors are used in conjunction with other measurement systems, as in the case of robotics, where they are used together with position and force/torque sensors, or in the case of the integration of Inertial Navigation Systems (INS) with Global Positioning Systems (GPS) in cars. The accuracy of INS measurements can be improved significantly by correcting them with the GPS data using Kalman filtering procedures. The INS can then aid navigation even when the GPS is degraded or interrupted because of jamming or interference.

1.1

A short foray through the pre-MEMS history The history of inertial sensors is relatively short. Despite the fundamental role played by inertial sensors in controlling the movement of a body, very little is

2

Introduction

known about early applications. This is even more remarkable given that most of the ingredients for building acceleration and angular-rate sensors, such as fine mechanics and precise spring technologies, were available from the late Middle Ages on and were used in the construction of, for instance, beautiful precision clocks.

Accelerometers One of the most likely reasons for the late appearance of acceleration sensors (or “accelerometers” for short) was the lack of indicator technologies, or, in modern phrasing, the lack of interfaces. This is certainly the reason why some former applications of acceleration switches that needed only very simple mechanical interfaces can be found. An acceleration switch initiated an action at a certain level of acceleration, as in the activation of a detonator in some bombs during the First World War. The first commercial accelerometer for broader application is credited to B. McCollum and O. S. Peters and was developed around 1920 [McCullom and Peters 1924]. It was based on a tension–compression resistance of a Wheatstone half-bridge where the resistances were formed by carbon rings. The next technological step was the use of strain-gauge transducers starting from around 1938, followed by the introduction of piezoelectric and piezoresistive transducers at the end of the 1940s. These transducers could capture the forces caused by the displacement of an elastically mounted mass within the sensor structure. Miniaturization and the high robustness of this type of sensor paved the way for broad applications in industry, terrestrial transport, aerospace, military uses, seismology, science, and so on. The piezoelectric and piezoresistive transducer principles were also among the first to be employed at the beginning of the entry into the world of inertial microelectromechanical systems (MEMS) – the world of the combination of micrometer- and nanometer-scale mechanical elements, sensors, actuators, and electronic circuits on one carrier or even on one chip. This entry was prepared in the late 1970s, for instance with the demonstration of a batchfabricated silicon accelerometer with piezoresistive transducers. The silicon bulk micromachined proof mass was bonded between two glass wafers [Roylance and Angell 1979]. The commercialization of similar devices began around 10 years later and was very soon based on a variety of available transducer principles such as the sensing of capacitance changes between fixed and movable plates, the frequency measurement of resonant devices, the stabilization of a tunneling current by a closed-loop system, the sensing of thermal changes between a heater and a movable heat sink, and the sensing of changes of the thermal distribution within an air bubble. This broad invasion of new and old ideas in the world of microelectronic technologies has opened the way to inexpensive mass applications of inertial sensors in industry, cars, medicine, consumer goods, and so on. Everybody knows the pioneering role of MEMS-based 50g accelerometers used in airbag ignition devices, which became the first high-volume product in the area of inertial MEMS. It was especially encouraging that within these successful highvolume products an example of the full monolithic integration of sensor and

1.1 The pre-MEMS history

3

signal processing on one chip could be found. Analog Devices, supported by strong governmental funding, developed a special BiCMOS-MEMS process combining a known microelectronic process with a polysilicon deposition, etching, and release technology. Various inertial sensors were developed on the basis of this process, of which the first was the 50g accelerometer [Analog Devices 1993].

Gyroscopes The stabilizing effect of rapidly rotating disks has been known for a long time and was used in ancient times for yo-yo-like toys and for ceremonies. Real angularrate sensors emerged quite late but have had a remarkably long history compared with accelerometers. This obviously is due to the much lower early requirements on the speed of the interfaces and, of course, to the moderate values of the signals to be measured. For instance, the Earth’s rotation is characterized by a rate signal of 360◦ /24 hours or 0.1◦ /s. The first technical realization of an angular-rate sensor took place around 1817 with the mechanical gyroscope designed by Johann Gottfried Friedrich von Bohnenberger in T¨ ubingen in Germany. It was not a true sensor but a demonstrator of rotational effects. The system was based on the spinning top and – not surprisingly – was used to demonstrate the mechanism of the Earth’s axis’ very slow precession accelerated in duration from a full cycle of 25 800 years to a small amount of seconds or minutes. Similar demonstrators were built around 1830 by the American Walter Johnson (Johnson’s Rotascope). The principle underlying the emergence of Coriolis forces within a rotating non-inertial coordinate system was demonstrated in 1851 by the French scientist Leon Foucault by building a 67-m-long pendulum with a mass of 28 kg within the Paris Pantheon (Fig. 1.1). This was the first real rotation sensor which measured the rotation of the Earth. Incidentally, a similar experiment was first performed in 1661 by the Italian physicist Vincenco Viviani and, after Foucault, implemented in countries all over the world. Under the influence of the Earth’s rotation the oscillation plane of such a pendulum changes by 360◦ sin ϕ in 24 hours. The angle ϕ is the geographic latitude of the experimental location. At the equator, the Foucault Pendulum does not show any reaction; at the poles, the rotation would be the full 360◦ /24 hours. At all other places the tip of the pendulum will draw nice rosettes on the floor. The Coriolis force was introduced by Gaspard-Gustave Coriolis, a French scientist, who described it in 1835. The Coriolis force appears in the equation of motion of an object in a rotating frame of reference and depends on the linear or angular velocity of the moving object. It will be considered in more detail in Chapter 8. Using this principle, in 1852 Foucault built a spinning-top gyroscope (“Meridiankreisel”), which can be considered the basis of modern spinning-top gyros. The term “gyroscope” was introduced at this time (“gyros” – rotation, “skopein” – vision). However, the rise of rotation sensors was preceded by the use

4

Introduction

Figure 1.1 Foucault’s Pendulum, in the Pantheon, Paris.

of the stabilizing ability of spinning wheels in torpedoes and cannon ammunition. Only in 1904 was the technical principle of the fast-spinning-top gyroscope patented by the German art historian Hermann Ansch¨ utz-K¨ ampfe [Schell 2005], who developed the idea of using the gyroscope within a compass (in 1908). The spinning wheel in a gyroscope is mounted on gimbals so that the wheel’s axis is free to orient itself. The key element for a compass was the introduction of a mechanism that results in an applied torque whenever the compass’ axis is not pointing North. The precession returns the compass’ axis towards the true North if it is disturbed towards another orientation. Other inventors such as the American Elmer Ambrose Sperry (1910) followed Ansch¨ utz-K¨ ampfe, and the acceptance of the gyro-compasses by the navy led to a quick penetration first on large ships and later on smaller ones and on airplanes. Inertial navigation and platform stabilization in aircraft and naval vessels remained the domain of spinning-top gyroscopes for a very long time. Driven by the requirements for cost reduction and miniaturization, around 1960 new types of gyros like the vibrating-string gyro [Quick 1964], the tuning-fork gyro [Hunt and Hobbs 1964] and the vibrating-shell resonator emerged and opened the way to a drastic size and weight reduction, which finally ended with the transfer of these principles into the world of MEMS. The vibrating-string principle is based on the action of the Coriolis force on a simple oscillator such as a mass on a string or a vibrating beam; the tuning-fork principle is based on balanced oscillators; and the vibrating shell uses the two vibration modes of a ring or a cylinder as in the classic wineglass effect. These principles will be presented in Chapter 8.

1.1 The pre-MEMS history

5

The new miniaturized vibrating gyros captured marked shares of the market step by step. However, for high-precision applications a rival for the classical gyros emerged around the end of the 1970s: the optical gyroscope. These gyros based on optical ring resonators and later on fiber optics have dominated, for instance, the aircraft navigator market since 1980. The transition from miniaturized gyros to the MEMS gyros was smooth. In miniaturized gyros it has become more and more difficult to realize bearings for endlessly rotating objects. The same problem applies for the early MEMS technologies and – with respect to specific friction, wear, and reliability – is still present today despite all the progress in this area. Only in 2008 did the first rotating MEMS gyro, developed by the Japanese company Tokimec, emerge on the market. Previous MEMS gyros used not the spinning-top principle, but rather the vibration or oscillation of masses within small linear or angular intervals as prepared by the miniaturized mechanical constructions. These so-called Coriolis vibratory gyroscopes (CVGs) – irrespective of whether or not they are based on MEMS – use at least two vibration modes of the structure, in which the Coriolis force excited by the interaction of the external rotation with the socalled primary mode causes an energy transfer to the secondary vibration mode. The vibrating elements are joined to hinges or anchors via springs. Such springs can be outstandingly formed in silicon because silicon possesses not only excellent mechanical and thermal properties in comparison with classical metals but also outstanding machinability. However, the full implementation of a complete gyroscope structure by using only microelectronic or emerging MEMS technologies was not the first step towards MEMS-based gyroscopes. Instead the designers first tried to use MEMS technologies to create key components for miniaturized gyros, such as the quartz tuning forks with piezoelectric actuators and piezoresitive transducers developed by Systron Donner [Soderkvist 1990] and the silicon-based rings including the spring suspension for the vibrating-shell systems developed by British Aerospace System and Equipment [Hopkin 1997]. Such components were then mounted on appropriate carriers. Concurrently, typical MEMS-technology-based devices formed and bonded on wafer level were proposed around 1986 by the Charles Draper Laboratory [Greiff et al. 1991] and demonstrated first in 1991 with a bulk-micromachined tuning-fork gyroscope and a little bit later, in 1993, with a silicon-on-glass tuning-fork gyroscope [Weinberg et al. 1994]. In 1998, researchers at the University of Michigan demonstrated a polysilicon-ring gyroscope produced with a trench-refill technology [Ayazi and Najafi 1998]. Predecessors of different batch-fabricated gyros that used more exotic technology exist, but these designs could not prevail against products based on technologies that were becoming mainstream MEMS technologies. One of the most interesting of the non-mainstream products was the vibrating-ring gyroscope of the University of Michigan produced using metal electroforming of nickel into a thick polyimide mold on a silicon substrate, which was demonstrated in 1994 [Putty and Najafi 1994].

6

Introduction

There were hundreds of different demonstrators from the University of Berkeley, from Samsung and Murata, from the Hahn-Schickard Gesellschaft– IMIT (Germany) and many others, but only a few were commercially successful. An excellent overview of the emerging MEMS-based inertial sensors can be found in Yazdi et al. [1998], which can be complemented by reading Shkel [2001].

1.2

Applications and market The applications of the classical accelerometers are vibrometry, shock detection, tilt measurement, dynamometry, seismology and other areas related to test and evaluation of devices exposed to inertial forces. Some of these applications coincide with the main application areas of classical gyroscopes – inertial navigation and platform stabilization. MEMS technologies dramatically changed this relatively peaceful picture. Nearly every month a new application is created and checked for commercial attractiveness and realizability.

Some general trends The trends of the inertial-MEMS market’s development are not significantly different from those of the entire MEMS market if one excludes the two leading and very old and stable products – ink-jet heads for printers and write/read heads for magnetic and optical memory disks. These two market segments alone occupied around an estimated 25% of the about 10 Bn $ MEMS market in 2010. MEMS addressed very fragmented markets that have had predominantly low-volume and only a few large to truly high-volume applications. This market is transforming more and more into a high-volume market with steadily expanding size, and, crucially, with a growing number of different applications. Compound annual growth rates (CAGRs) of 5% to 20% are typical for these different applications. At present the number of companies manufacturing MEMS products is around 270 in addition to 150 fab-less companies. Around 90 R&D industrial facilities, which are able to develop prototypes and to perform small-volume production, round off the picture of today’s MEMS community. It should be borne in mind that in the 1990s three to five years were needed to develop new MEMS designs and five to eight years from the prototypes to volume production. In the case of safety-critical applications the time lapse was even longer. Now the overall time from design start to volume production has decreased by a factor of two to three and is reducing further. Consequently, the interest in research and development is enormous and is still growing. It will continue to grow as long as MEMS products penetrate all areas of human activity. Today the distance which has thus far been covered on the way to all-encompassing applications of MEMS and inertial sensors is almost negligible in comparison with the distance still to go.

1.2 Applications and market

7

The inertial-MEMS market Within the MEMS market modern inertial sensors – accelerometers and gyroshave gained a considerable share, exceeding 20% of the expected 12 Bn $ MEMS market in 2011 (∼6 Bn $ in 2009). In 2005 nearly 80% of all applications were related to automotive safety functions such as automatic break systems (ABSs), airbag sensors, rollover sensors, electronic stabilization systems (ESP), and other anti-skid systems as well as to navigation. Starting with 50g accelerometers in airbag safety systems, the next step – the introduction of electronic stabilization control (ESC) by Bosch and Systron Donner in 1994 – was significantly accelerated by the disastrous elk test of the newly invented Mercedes-Benz A-Class in 1997. ESC had to rescue the reputation of the brand name of one of the leading makers of high-quality cars. In ESC, yaw-rate gyroscopes and low-g sensors are usually the decisive information sources for controlling the finely allotted brake forces on the different wheels to avoid accidents. Rollover protection, highly sophisticated front and side airbags combined with safety-belt control, and suspension control, especially for trucks, and many other applications have not only expanded the market but also forced the development of new low-g accelerometers, of gyroscopes sensitive in different axes and with different accuracy levels, and – importantly – the co-integration of two or more sensors in one package or even on one chip. Remarkably, the market shares of gyroscopes and accelerometers nearly equalized around 2005. New accelerometer applications mainly in the consumer market have shifted the relation to a stable 40% share of gyroscopes within the inertialMEMS market. After 2012–13 the picture may change because the killer applications in the consumer market, which with an expected 1 Bn $ contribution will then be at least comparable in size to the other segments, have not yet been defined and are hard to predict. Today, large companies are fighting for their share in the market of automotive inertial MEMS sensors. Among these, the world’s largest MEMS-sensor manufacturer is Bosch in Germany, which has put considerable effort into the production of MEMS gyroscopes for automotive applications. However, British Aerospace Equipment Silicon Sensor Systems (BAE SSS), BEI Systron Donner, DelphiDelco, Murata, Matsushita, and Samsung have also long been very successful and delivered many millions of gyros to component manufacturers in the car industry. Within the accelerometer market the Norwegian company Sensonor, with its 50g accelerometers, and the Swedish VTI Technolgies, which was long the leader in the low-g accelerometer market, have made surprisingly dominant contributions to the emerging killer applications within the automotive industry. However, the number of high-volume players in the inertial sensor market is quite limited, encompassing Bosch, Analog Devices, Freescale, and almost a dozen others. Besides the automotive industry, the remainder of the market is dictated by consumer applications, which in recent years have shown dramatic growth, with CAGR 25%–30%. Accepted and growing applications are related to the use of

8

Introduction

inertial sensors in hand-held cameras for picture stabilization, in personal computers for hard-disk protection against mechanical shocks, in pedometers for motion sensing, and in more exotic products such as the two-wheel Human Transporter of Segway. Probably one of the most interesting applications is the motion sensing integrated into mobile phones, game controllers, toys and other human–machine interfaces. The Nintendo Wii’s motion-sensing remote control has attracted the broad interest of the public to inertial MEMS. The consumer and information technology (IT) sector has increased from about 10% in 2005 to about a 45% share in 2009/10. Within the consumer market the companies Analog Devices, Kionix, ST Microelectronics, and MEMSIC are the dominating players within the accelerometer business, while Panasonic and Murata have long led the gyro market. However, every year new companies are entering the market for inertial sensors, and the established manufacturers of inertial MEMS as well as newcomers are focusing their attention more and more on the consumer market. New systems such as very small and cheap one-axis sensors as well as more highly integrated multidimensional accelerometers and gyroscopes are now on the market. ST Microelectronics introduced in 2009 a three-axis high-performance gyroscope, whereas VTI announced a three-dimensional (3D) accelerometer combined with a 1D gyroscope [MEMSentry 2009]. Sensordynamics introduced in 2008 a combined one-axis accelerometer and one-axis gyroscope [Micronews 2009] and announced a three-axis accelerometer plus one-axis gyro combination. The race into the world of multi-axis inertial sensors, including completely integrated (six-axis) inertial measurement units (IMUs), is fully under way. An IMU measures the accelerations and rotation rates on all three axes and, in principle, represents the (functionally) ultimate inertial sensor. Applications are numerous, ranging from medical 3D gesture and motion recognition via human– machine interfaces (HMIs) for game controllers and mobile phones to personal navigation systems. The high-volume application of inertial MEMS in the automotive and consumer markets was for a long time in some contrast with aeronautic and defense applications. Here the high added value guaranteed a large benefit for the customer, and, consequently, the low quantities have been to some extent compensated for by good prices. Remarkably, within the last few years many inertialMEMS products developed for the aeronautic and defense sector have reached commercialization. This sector has doubled within the last five years, approaching now around 50% of the size of the automotive segment. As with aeronautics and defense markets, the industrial and medical segments of inertial MEMS’ applications are also at the stage of entry to high-volume markets. Applications such as activity monitoring in pacemakers have considerable dissemination throughout the world. It can be expected that the inertial control of robots and machine parts with more than one degree of freedom (DOF) will achieve a quite broad distribution.

1.3 The ingredients of inertial MEMS

1.3

9

The ingredients of inertial MEMS Inertial sensors convert the inertial forces caused by the input acceleration or rate signal into some physical changes such as deflection of masses or deviations of stresses, which then are captured by a corresponding transducer and transformed into an electrical signal. The electrical signal is subjected to some estimation procedures such as linear or nonlinear filtering in order to derive an estimate of the input signal. The final output represents the calibrated value of the measured acceleration or rate. Of course, not only electrical output signals are feasible; however, only in such exceptional cases as for instance in highly explosive environments are other forms of the output, such as optical signals, used. Within this book only sensors with electrical output signals are treated. Accelerations and angular velocities are vectorial signals possessing absolute values and orientations. If only one component of the vector should be measured the sensor is denoted 1D or one-axis. If two or all three components of the acceleration or the rate signal should be captured, the sensor is called a 2D or 3D accelerometer, or a rate sensor. Today a MEMS-based accelerometer or gyroscope is understood as a complete product that is packaged, calibrated, and tested, and has to be delivered to the customer, who wants to integrate this component with minimal effort into a higher-level measurement or control system. The level of accuracy required depends on the application. The environmental conditions for the integration of inertial MEMS at the customer site may be also quite different and usually are divided into classes with respect to the applicable temperature range and the exposure to humidity and aggressive chemicals as well as to vibration and shocks. The length of the lifetime, the reliability, and the safety against failures during operation may vary significantly, and to a large extent determine the product’s price. Consequently, orders of magnitude may separate the complexity and the price of an inertial MEMS for different applications even if the underlying sensor principles are identical. In order to make the following explanations and terminology systematic, it is meaningful to sketch a general structure of a sensor system with emphasis on inertial sensors. The system consists of not only the sensor itself but also additional components such as transducers and electronics. In Fig. 1.2 a very crude representation of the whole system is shown. The intrinsic sensor transforms the input signal – acceleration or rate – into a physical objective, which can be gathered by the transducer and transformed into an electrical signal. The sensor and transducer are subject to interactions with the package. In inertial MEMS the main interactions are stress and heat transfer. Environmental factors may be transferred via the package to the sensor and transducer, changing their behavior. The electronic part consists of an input stage that amplifies the transformed signal into a conveniently manageable form. The electronics also may generate

10

Introduction

Electronics Input signal

Sensor

Transducer

Output signal Input stage

Actuator

Figure 1.2 The general architecture of a sensor system.

excitation and control signals that are necessary for bias setting and for operational conditioning. Actuating stimuli may be used, for instance, for feedback control, as well as for test signals. In practice the borders between these generic blocks are quite fuzzy. The transducer and the electronic input stage often form an indivisible object where the input stage provides the necessary biasing and excitation for the transducer and, vice versa, parts of the input stage may act as components of the transducer. The transducer is often directly integrated into the intrinsic sensor. In general, the application of a certain transducer principle usually entails not only the choice of a certain transducer element but also the adaption of the sensor and the electronics. Nevertheless, the generic structure shown allows us to systematize the understanding of the main interactions between the components.

r Feedback control is beneficial with respect to linearity and optimization of transfer characteristics. However, it requires actuators and, thus, additional effort. Therefore, not all sensors have feedback components and, where possible, sensors operate in an open-loop mode. r The system performance is to a large extent determined by noise and disturbances. Typical noise sources exist within inertial MEMS. So, the intrinsic inertial sensor exhibits mechanical noise caused mainly by friction with the usually gaseous environment. For not-too-ambitious performance targets the dominant noise stems from the second noise source – the transducer and electronic stages, and here mainly from the electronic input stage. In accord with the system structure illustrated, the various basic sensor principles, the transducer mechanisms, the corresponding governing models, and, last but not least, the typical error sources and performance parameters of the overall system will be covered by this book. The seemingly central role of the sensor principle within inertial MEMS quickly dissolves when one looks at the many stages of creating the final product.

r A working environment for the sensor is usually established by the so-called zero-level packaging.

r The sensor signals must be acquired and processed.

References

11

r In many cases, as for instance in all vibratory gyroscopes, the sensor has to be driven in a primary movement that must be controlled to high accuracy.

r For high-performance applications, on-line-monitoring functions are integrated.

r The output signals for the external communication – including, if applicable, the results of in-build tests – and monitoring must be formed.

r The sensor and the signal-processing unit have to be packaged and bonded to connections with the external pins or solder balls of the (first-level) package.

r The whole product must be calibrated, which requires special equipment in order to create the necessary accelerations or rate signals with high accuracy. Selected representative functions are tested, usually within the entire temperature range of application. r Last but not least, more or less comprehensive qualification procedures must be passed before production may be initiated. Similarly to most other MEMS applications, the key for success within the area of inertial MEMS is the comprehensive development and integration of all the mentioned components necessary for these multidisciplinary products: MEMS technologies, first- and second-level packaging, application analysis, system design considering the specific effects of technology tolerances and forces in a micro- to nano-scale environment, test methods for high-volume production, and, finally, reliability and lifetime guarantees on an unprecedented scale, especially for automotive and medical applications. Huge investments were necessary to build up high-volume fabs and to run a high-yield production. Partial solutions have reached some maturity. For instance, selected surface and bulk micromachining technologies of leading manufacturers and of R&D facilities have become something like standard processes and are free to be used by the scientific community for prototype development. However, most of the subareas are still in a phase of rapid growth and improvement. Correspondingly, product cycles are changing rather quickly despite the stringent stability requirements of the automotive industry. The third generation of acceleration sensors and gyroscopes is now on the market, the next generation ante portas.

References Analog Devices (1993). ADXL50 – monolithic accelerometer with signal conditioning. Data sheet, Analog Devices, Norwood, MA. Ayazi, F. and Najafi, K. (1998). Design and fabrication of a high performance polysilicon vibrating ring gyroscope, in Proceedings of the IEEE Micro Electro Mechanical Systems Workshop (MEMS ’98), Heidelberg, pp. 621–626. Greiff, P., Boxenhorn, B., King, T., and Niles, L. (1991). Silicon monolithic micromechanical gyroscope, in Proceedings of the IEEE 1991 International

12

Introduction

Conference on Solid State Sensors and Actuators, San Francisco, pp. 966– 968. Hopkin, I. (1997). Performance and design of a silicon micromachined gyro, in Proceedings of the Symposium on Gyro Technology, Stuttgart, pp. 1.0–1.10. Hunt, G. W. and Hobbs, A. E. W. (1964). Development of an accurate tuningfork gyroscope, in Symposium on Gyros, Proceedings of the Institute of Mechanical Engineers (London), 1964–65, 179(3 E). McCullom, B. and Peters, O. (1924). A new electric telemeter. Technology Papers of the National Bureau of Standards, 17(247). MEMSentry (2009). ST’s new 3-axis analog gyroscope. MEMSentry, December 2009, 47:5. Micronews (2009). Inside the first combination inertial sensor. Micronews, May 2009, 80:7. Putty, M. and Najafi, K. (1994). A micromachined vibrating ring gyroscope, in Technical Digest Solid-State Sensor Actuator Workshop, Hilton Head Island, SC, pp. 213–220. Quick, W. H. (1964). Theory of the vibrating string as an angular motion sensor. Transactions of the ASME Journal of Applied Mechanics, 523–534. Roylance, L. M. and Angell, J. A. (1979). A batch-fabricated silicon accelerometer. IEEE Trans. Electron Devices, 26:1911–1917. Schell, B. (2005). 100 Years of Anschuetz gyro compasses – 100 years of innovations in nautical technology, in Symposium Gyro Technology 2005, Stuttgart, pp. 1–20. Shkel, A. M. (2001). Micromachined gyroscopes: challenges, design solutions, and opportunities. Proceedings of the SPIE, 4334:74–85. Soderkvist, J. (1990). Design of solid-state gyroscopic sensor made of quartz. Sensors and Actuators A, 21/23:293–296. Weinberg, M., Bernstein, J., Cho, S. et al. (1994). A micromachined comb-drive tuning fork gyroscope for commercial applications. Proceedings of the Sensor Expo, Cleveland, OH, pp. 187–193. Yazdi, N., Ayazy, F., and Najafi, K. (1998). Micromachined inertial sensors. Proceedings of the IEEE, 86(8):1640–1659.

2

Transducers

The basic building blocks of inertial MEMS are sensing elements to acquire the reaction of the measurement system, actuators to excite the mechanical system, and other components of the mechanical system such as proof masses, beams, springs, and suspensions. The properties and dimensions of all these components are decisive for their application within a given sensor system. Silicon, with its outstanding mechanical properties (e.g. Gad-el-Hak [2002] and Franssila [2004]), plays a key role for building these blocks. It is as strong as steel. It is ideally elastic, not exhibiting plastic deformations up to the yield point as do most metals. The elastic modulus E may be as large as 190 GPa, depending on crystal orientation, and the yield strength is about 7 GPa. With appropriate doping (boron, phosphorus) concentration the resistivity can be changed by eight orders of magnitude between 10−4 and 104 Ω cm; thus structures such as conductive plates or comb fingers can be manufactured. The density of silicon is 2300 kg/m3 , the thermal conductivity is 1.57 W/(cm K), and the thermal expansion coefficient CT = 2.33 × 10−6 /K [Kovacs 1998]. Polycrystalline silicon, or “polysilicon” for short, which is made up of small single-crystal domains of silicon (grains), has similar properties and is the most popular building material for surface-micromachined devices (SMMs). Monocrystalline silicon is anisotropic and exhibits pronounced orientationdependent properties such as piezoresistivity and piezo-Hall effects. The mechanical and electrical properties of polysilicon are slightly inferior to those of monocrystalline silicon but are very similar to those of an isotropic material; thus, the material is much easier to handle than monocrystalline silicon. While the main focus of this chapter is on transducers, some material properties that are relevant to both transducers and general mechanical components of inertial MEMS will be summed up first in order to better understand the relevant transducer mechanisms.

2.1

Anisotropic material properties, tensors, and rotations Anisotropic material properties play a prominent role in inertial MEMS. A suitable mathematical description of these properties is provided by the concepts of stress and strain. Considering the fundamental role of these terms in the

14

Transducers

(a)

(b)

Figure 2.1 Axial and shear forces acting on a body. (a) Beam elongation under axial force. (b) Cube deformation under shear load.

engineering literature, a short introduction to the matter is given. Readers familiar with these concepts can skip this section.

2.1.1

Stress, strain, and piezoresistivity Hooke’s law The concept of stress and strain was developed in order to describe the interrelation between external forces, acting on a solid body, and the internal forces between the volume elements of the body, which cause the deformations. Another objective was to reflect the effects a strained condition can induce on material properties such as resistivity. Stress is a force distribution and strain is the distribution of deformations.

Normal stresses The stress–strain relation is a generalization of Hooke’s law, which, when applied to a metal bar orientated along the x-axis and with length lx0 and cross-section Ax , can be expressed as (see Fig. 2.1(a)): Fx = EAx

∆lx lx0

or

σx = Eεx ,

(2.1)

where σx is the axial stress, i.e. the applied axial force per unit area that is normal to the surface, σx =

Fx , Ax

(2.2)

and εx is the (normal) strain, i.e. the dimensionless relative elongation εx =

lx1 − lx0 ∆lx = . lx0 lx0

(2.3)

E is the elastic modulus or Young modulus of the given material. It expresses the material’s resistance to elastic deformations. A large elastic modulus characterizes a stiff material and a small E characterizes a highly elastic object. If

2.1 Anisotropic material properties

15

the applied forces cause an elongation of the object, the resulting stress is called tensile; in the opposite case the stress is called compressive. For ordinary materials, the elongation of the bar is accompanied by a reduction of its lateral dimensions (contraction of the bar), i.e. by a transverse strain εy y = εz z =

∆ly ∆lz = . ly lz

(2.4)

The ratio of the lateral to the axial strain εt /εa is Poisson’s ratio ν=−

εz z εy y =− . εx εx

(2.5)

Within the elastic range of the medium, ν is constant, and for most materials its value is between 0.2 and 0.35. An upper limit of 0.5 exists and pertains to ideally incompressible media. For an isotropic elastic object, which is subject to applied stresses along all three orthogonal axes, the strain in any of the three directions is the sum of the elongation in that direction minus the contraction caused by the stresses in the two remaining orthogonal axes, for instance 1 [σx − ν(σy + σz )]; E 1 εy = [σy − ν(σx + σz )]; E 1 εz = [σz − ν(σx + σy )]; E

εx =

E [(1 − ν)εx + ν(εy + εz )], (1 + ν)(1 − 2ν) E σy = [(1 − ν)εy + ν(εx + εz )], (1 + ν)(1 − 2ν) E σz = [(1 − ν)εz + ν(εx + εy )]. (1 + ν)(1 − 2ν) (2.6)

σx =

On the right-hand side the equation is resolved for the case in which the strains are known. For “non-standard” materials such as foams and synthetic compounds Poisson’s ratio can take negative values; however, such materials are not considered in this book.

Shear stresses Besides the normal stresses, which are perpendicular to the body’s surfaces, shear stresses exist, particularly in twisted bodies. A shear force acts parallel to a given surface as shown in Fig. 2.1(b). For instance, Fz x denotes a force in the x-direction acting parallel to a surface whose normal vector is oriented along the z-direction. The shear stress is the shear force per unit area. Shear forces create shear strain, i.e. deformations along the direction of the shear force. In Fig. 2.1(b) the shear strain εz x is the ratio between the displacement ∆lx and the height of the cube lz . Shear stress and shear strain are related to each other by the shear modulus G: Fz x = GAz

∆lx lz

or

σz x = Gεz x .

(2.7)

16

Transducers

(a)

(b)

Figure 2.2 Torsion of a hollow cylinder. (a) A twisted cylinder. (b) Cross-sections of

untwisted and twisted hollow cylinders.

The basic experiment to determine the relation between shear stress and shear strain is shown in Fig. 2.2(a). A homogeneous hollow cylinder with thin walls is exposed to a torsional moment at both sides, and the resulting shear angle γ is measured. This angle determines the torsion or twist angle ϕ of the front face of the cylinder according to the obvious relation L , (2.8) R where L is the length of the cylinder, t is the thickness of the wall, and R is the radius of the cylinder. Assuming that the stress across the cross-section of the cylinder, σt , which is orientated in the tangential direction, is constant, the moment created is Rσt A, where A is the area of the cross-section. This moment is balanced by the applied Mt , ϕ=γ

Mt = 2πR2 tσt .

(2.9)

By measuring Mt and the shear angle γ, the ratio σt /γ can be determined. Within the limits of elastic deformation without warping it has been found that σt = Gγ.

(2.10)

The shear angle γ is the deformation angle of the elementary rectangles of the cylinder as shown in Fig. 2.2(b). For the marked rectangle (in the middle of the bottom part of the cylinder) the stress lies in the plane x = constant and is directed along the y-axis: σt = σxy .1 The shear angle γ is exactly double the shear strain γ = 2εxy . Using this definition and applying the results to an arbitrary 1

The natural coordinate system is here a cylindrical system in which σ t = σ x , ϕ .

2.1 Anisotropic material properties

17

volume element of an isotropic body, the shear stress–strain relation can be written in the more common notation εxy =

σxy ; 2G

εxz =

σxz ; 2G

εy z =

σy z , 2G

(2.11)

which together with Eq. (2.6) represents the generalized form of Hook’s law. It is valid for small deformations, where the superposition principle remains in force, i.e. the resulting deformation caused by different forces can be substituted by the sum of the deformations caused by these forces. A deeper insight into the fundamentals of the theory of elasticity reveals that for isotropic materials the original tension test of Hooke without any additional torsion test is sufficient to derive not only the deformations caused by normal stresses, but also the relation between shear forces and corresponding deformations. This leads to a well-defined interrelation between Young and shear moduli presented below (Eq. (2.36)). However, the torsion test is illustrative with respect to torsional components of inertial sensors and allows us to introduce the basic ideas for the analysis of torsional springs.

Stress and strain tensors Usually, objects subject to external forces are not built as simply as the ones considered so far, i.e. bars or torsional cylinders. They may have a complicated geometry and may be exposed to a combination of normal and shear stresses. To analyze the reaction of the body, the external forces have to be related to the internal forces and deformations at different inner points. Consequently, the concepts of stress and strain must be brought down to infinitesimal length scales, where balance of force and momentum is applied on each face of such an entity. On the basis of such a representation, partial differential equations for the relation among stress, strain, and global and local forces within the body can be derived; as usual, they must be complemented by appropriate boundary conditions. These equations can be numerically simulated or – in some exceptional cases – solved analytically.

The stress tensor To derive an infinitesimal description of the stress or strain distribution, infinitesimal volume elements of the medium, pertaining to a Cartesian coordinate frame, are considered; for such coordinates, the volume elements are cubes, as can be seen in Fig. 2.3 [Popov 1968, 1999], The faces of the cube have areas dA. The medium is supposed to be in a state of mechanical equilibrium; this implies that forces and moments either originating from interaction with neighboring volume elements or being imparted by external sources must be balanced at each face of each volume element. At the boundary of the body, external load forces and reaction forces of the supports have to be added.

18

Transducers

Figure 2.3 The stress tensor.

For convenience the x-, y-, and z-axes are often defined by the vectors e¯1 , e¯2 , and e¯3 , and the corresponding coordinates are x ⇒ x1 ,

y ⇒ x2 ,

z ⇒ x3 .

(2.12)

A force or a stress component acting on the face with normal vector e¯i and oriented along direction e¯j is termed ∆Fij . The stress is the limit σij = ∆Fij /dAi |dA i →0 . The cube under consideration is in mechanical equilibrium with adjacent cubes so that the total force exerted on a face of the cube is compensated for by the corresponding force of the adjacent cube. The stress at a face σ¯i = ∆F¯i /dA is decomposed into the orthogonal components σii , σij , and σik . The normal stress σii is caused by the force perpendicular to the face and tends to compress or elongate the cube. Shear stresses act parallel to the face and tend to shift the corner points of the face, thus distorting the cube into a rhomboid. Mechanical equilibrium implies that the total force and the total torque acting on the cube must be zero. This has two implications. 1. The normal stresses at opposite faces have the same values but opposite directions σii′ = −σii . 2. Zero total torque about any axis requires that the stress components must satisfy a symmetry condition, σij = σj i .

(2.13)

Stress components at opposite faces have opposite orientations by definition. A complete description of the state of stress at point P (x, y, z) is then given by nine quantities σij (x, y, z), and is subject to the symmetry condition (2.13), which reduces the number of independent quantities to six. This collection of quantities σij (x, y, z) at P (x, y, z) defines the coordinates of the so-called stress tensor at P (x, y, z). The stress tensor represented within the given coordinate system can be represented by the coordinate matrix σ with nine components

2.1 Anisotropic material properties

19

Table 2.1. Index transfer by Voigt’s notation

Matrix indices Vector index

11 1

22 2

33 3

23, 32 4

∂v dy ∂y

13, 31 5

12, 21 6

∂u dy ∂y ∂u dx ∂x

β x, y

α

∂u ∂v ; ∂x ∂y

1 − 2 2

(

x, y

∂v dx ∂x

) = 12 (α + β )

Figure 2.4 Deformation of an elementary cube.

σij ; i, j = 1, 2, 3: σ = {σij }.

(2.14)

In this book, tensor quantities are usually represented in a given coordinate system. As a consequence, matrix algebra can be used to express tensor relations. Since the symmetry property (2.13) reduces the number of independent stress components from nine to six, it is convenient to express the stress status not by a 3 × 3 stress matrix but by a stress vector σ ¯ using Voigt’s notation, which is based on the equivalence of indices (shown in Table 2.1): σ ¯ T = (σ1 = σ11 , σ2 = σ22 , σ3 = σ33 , σ4 = σ23 , σ5 = σ13 , σ6 = σ12 ).

(2.15)

The superscript T denotes the transpose of a vector or a matrix.

The strain tensor The strain tensor can be introduced in the same way as the stress tensor. Again, a non-deformed cube is considered at point P (x, y, z) (see Fig. 2.4). The deformation of the cube is described by the displacements each of its corner points

20

Transducers

is undergoing. Thus, a¡ point with coordinates (x, y,¢z) in the undeformed state will take the location u(x, y, z), v(x, y, z), w(x, y, z) after deformation. A point (x + ∆x, y + ∆y, z + ∆z) nearby will move to        ∂u/∂x ∂u/∂y ∂u/∂z ∆x u u(x + ∆x, y + ∆y, z + ∆z)  v(x + ∆x, y + ∆y, z + ∆z)  =  v  + ∂v/∂x ∂v/∂y ∂v/∂z   ∆y  ∂w/∂x ∂w/∂y ∂w/∂z ∆z w w(x + ∆x, y + ∆y, z + ∆z)     u ∆u =  v  +  ∆v  . (2.16) w ∆w

Deformations are supposed to be small so that the principle of superposition holds. From Eq. (2.16) it follows that the elongation of the face at z = 0 in the xdirection is (∂u/∂x)dx, and accordingly that in the y-direction is (∂v/∂y)dy. The diagonal strain components are now defined as the relative elongations in the respective directions: εxx = ε11 =

∂u ; ∂x

εy y = ε22 =

∂v ; ∂y

εz z = ε33 =

∂w . ∂z

(2.17)

The shear strain characterizes the deformation of the quadratic faces into parallelograms. It is defined as the average of the two angles β ∼ = ∂u/∂y and α∼ = ∂v/∂x of the resulting parallelogram (see Fig. 2.4), i.e. as the average of the angular distortions of the z–x-plane and the z–y-plane: µ ¶ 1 1 ∂u ∂v εxy = ε12 = (α + β) = + . (2.18) 2 2 ∂y ∂x Accordingly, for the other faces µ ¶ 1 ∂u ∂w + , 2 ∂z ∂x µ ¶ 1 ∂v ∂w = + . 2 ∂z ∂y

εxz = ε13 =

(2.19)

εy z = ε23

(2.20)

Again, the symmetry relation εij = εj i

for

i, j = 1, 2, 3

(2.21)

holds because for non-rotating solid bodies (i.e. for bodies not exposed to additional centrifugal, centripetal or Coriolis forces) ∂u ∂v ∂u ∂w ∂w ∂v − = − = − = 0. ∂y ∂x ∂z ∂x ∂y ∂z

(2.22)

In other words, without rotation, εxy and εy x are the average deformation angles for the same face and thus identical. Consequently, in Eq. (2.16) the mixed partial derivatives can be substituted by the average shear stresses, e.g. µ ¶ ∂u 1 ∂u ∂v = + = εxy . ∂y 2 ∂y ∂x

2.1 Anisotropic material properties

Thus, the deformation equation (2.16) can be rewritten         ∆x ∆x ε1 ε6 ε5 ∆u εxx εxy εxz  ∆v  =  εxy εy y εy z   ∆y  =  ε6 ε2 ε4   ∆y  . ∆z ε5 ε4 ε3 ∆z εxz εy z εz z ∆w

21

(2.23)

The 3 × 3 strain matrix ε = {εij } is the representation of the strain tensor and – owing to the symmetry relation – can be reduced to the six-dimensional strain vector using again Voigt’s index correspondence (Table 2.1) ε¯T = (ε1 , ε2 , ε3 , ε4 , ε5 , ε6 ).

(2.24)

In engineering disciplines like structural mechanics the shear strain components are defined as twice the values used in physics and correspond to the values used in the torsion test of the section “Shear stresses”: γxy = 2εxy = 2ε6 ;

γxz = 2εxz = 2ε5 ;

γy z = 2εy z = 2ε4 .

(2.25)

The stress–strain relation for anisotropic materials Assuming small deformations, the general relation between the strain and stress vectors (2.15) and (2.24) can be described by the linear equation σ ¯ = E · ε¯ + α ¯ ∆T,

(2.26)

where E = {Eij } is the elasticity matrix with 6 × 6 coefficients. The first contribution to the stress derives from the strain, but in the case of temperature differences ∆T a temperature-induced stress term α ¯ ∆T must be added. The vector α ¯ represents the thermal expansion coefficients α ¯ T = (αxx , αy y , αz z , αy z , αxz , αxy ).

(2.27)

The incremental change of the elastic energy δU of a unit cube is given by the sum of the deformation work 2 δU = Fxx δεxx + Fy y δεy y + Fz z δεz z + Fy z δεy z + Fxz δεxz + Fxy δεxy . (2.28) The forces are given by the stress components Fij = σij dA. Owing to the identity of the mixed partial derivatives 2 ∂ 2 U/∂εij ∂εk l = −∂Fij /∂εk l = −∂Fk l /∂εij

(2.29)

the matrix of elastic moduli is symmetric. Moreover, it has to be noted that the initial relation between the two 3 × 3 strain and stress tensors of second order should be represented by a tensor of fourth order with 9 × 9 = 81 coefficients. The symmetry relation of stress and strain has reduced this complexity to a tensor with 36 non-zero coefficients or to the equivalent compliance matrix with 6 × 6 coefficients. The last symmetry consideration has decreased this number to 21. In the light of desired analytical expressions such a large number of entities is still difficult to handle. Fortunately, for most of the relevant materials used in MEMS technologies, additional symmetry relations hold, allowing us to reduce the number of non-zero coefficients further.

22

Transducers

Figure 2.5 Silicon orientation on wafer.

The elasticity matrix of silicon As is well known, silicon has a diamond structure based on a cubic crystal lattice. The wafers used in microelectronics are usually cut along the (100) plane for CMOS and bulk micromachined devices.2 As shown in Fig. 2.5 the surface of the wafer is in the (001) plane, and the orientation flat is directed in the [110] direction. This means that a device on the wafer that is orientated perpendicularly to the flat has an angle of 45◦ with the cube orientation axis [100]. If the different rotational and mirror symmetries of a cubic lattice are considered, the number of independent coefficients of the elasticity matrix reduces from 21 to 3. The elasticity matrix for a material under stress, orientated along the [100] axis, is   E11 E12 E12 0 0 0  E E E  12 11 12 0 0 0    0 0 0  E E E (2.30) E =  12 12 11   0 0 0 E44 0 0     0 0 0 0 E44 0  0 0 0 0 0 E44

with only 12 non-zero moduli of elasticity. This structure reflects a general property of a parameter matrix of anisotropic materials with cubic symmetry and holds, for instance, also for the parameters describing the piezoresistivity effect. For stress orientations differing from the [100] axis, the coefficients of the rotated matrix have to be calculated. This will be done in Section 2.1.2. The inverse relation between stress and strain can be described by the compliance matrix, which is the inverse of the matrix of elastic moduli: ε¯ = S · (¯ σ−α ¯ ∆T ). 2

(2.31)

(100) Planes are perpendicular to [100] directions. The set of six equivalent directions is denoted by h100i, and the set of equivalent planes by {100}.

2.1 Anisotropic material properties

S = {sij } has the same structure as the E matrix,   s11 s12 s12 0 0 0 s s s   12 11 12 0 0 0    0 0 0  s s s S =  12 12 11  = E−1 ,  0 0 0 s44 0 0     0 0 0 0 s44 0  0 0 0 0 0 s44

23

(2.32)

with the following coefficients: s11 =

1 ; E

s12 = −

ν ; E

s44 =

1 . 2G

(2.33)

E and ν are the Hookean Young modulus and Poisson ratio. G is the shear modulus or modulus of rigidity, defined by the ratio σij G= , i 6= j. (2.34) 2εij For silicon, the following data, referred to the [100] crystal orientation, are usually quoted [Gad-el-Hak 2002, Brantley 1973]: E=

1 = 131 GPa; s11

ν=−

E11 = 166 GPa;

E12 = 64 GPa;

s12 = 0.28; s11

G=

1 = 80 GPa; 2s44

E44 = 80 GPa.

(2.35)

Despite its significantly lower yield and fracture strength, a mechanically isotropic polysilicon layer has elastic properties similar those of to bulk silicon. The Young modulus is usually specified as Ep oly = 160 MPa, and the Poisson ratio νp oly = 0.23; however, the spread within materials with different grain structures may be quite significant, and also the spread within different dies need not be negligible. To get a feeling for the orders of magnitude, a very simple example is considered. A silicon cube of dimensions 1 cm × 1 cm × 1 cm is loaded with 1000 kg on the top face. The cube is then compressed by 0.07% or 7 µm, and the lateral extension is 0.02% or 2µm.

Elasticity of isotropic materials For isotropic materials the structure of Eqs. (2.26) and (2.31) does not change. However, G is no longer an independent parameter, but given by the relation [Macke 1962, Chou and Pagano 1967] G=

E . 2(1 + ν)

(2.36)

This equation follows from the anisotropy definition αanisotropy =

2E44 , E11 − E12

(2.37)

24

Transducers

which for isotropic materials (αanisotropy = 1) fixes the interrelation among E11 , E12 , and E44 . It can also be derived from the identity of the principal stress and strain axes in isotropic bodies.3

Elastic strain energy If stress and strain within a material are known, it is often convenient to represent the status by the elastic strain energy Uelastic within a given volume V . Equilibrium states can be found by minimizing the energy, forces can be derived as gradients of the energy, and so on. The elastic energy Uelastic is the accumulated work for the generation of all displacements, i.e. the sum of all displacements multiplied by the forces needed for this displacement: Z Z Z 1X Uelastic = εij σij dx dy dz 2 i,j V Z 1 = dV (σx εx + σy εy + σz εz + 2σxy εxy + 2σxz εxz + 2σy z εy z ). (2.38) 2 V For isotropic materials the application of Hooke’s law yields Z n1 1 Uelastic = dV [σ 2 + σy2 + σz2 − 2ν(σx σy + σx σz + σy σz )] 2 V E x o 1 2 2 + (σxy + σxz + σy2 z ) . G

(2.39)

The piezoresistance of silicon Piezoresistors are excellent stress transducers and have found broad application within inertial MEMS. They change their resistance depending on the applied stress. ¯ and The material laws of Maxwell’s equation state that the electrical field E current density J¯ in an isotropic medium relate to each other as ¯ = ρJ¯ E

and

¯ J¯ = σ E

where

σ=

1 . ρ

(2.40)

Here ρ is the resistivity and σ the conductivity of the material. In silicon or other crystals resistivity and conductivity are anisotropic; ρ and σ become tensors of second order, e.g.      Jx ρxx ρxy ρxz Ex ¯ ¯      (2.41) E = ρJ ⇒ Jy  . Ey = ρxy ρy y ρy z Jz ρxz ρy z ρz z Ez

Here the symmetry relations ρij = ρj i , i 6= j, were used [Smith 1958, Nye 1985]. Using again Voigt’s notation ρ1 = ρxx , ρ2 = ρy y , ρ3 = ρz z , ρ4 = ρy z , ρ5 = ρxz , 3

The principal stress (strain) axes are defined by coordinate systems at the point P (x, y, z), for which the shear-stress (shear-strain) components vanish.

2.1 Anisotropic material properties

25

Table 2.2. Piezoresistivity coefficients for high-resistivity silicon in units of (100 GPa)−1

p-Silicon n-Silicon

ρ0 (Ω cm) 7.8 11.7

π11 6.6 −102.2

π12 −1.12 53.4

π44 138.1 −13.6

and ρ6 = ρxy , the resistivity can be expressed as the sum of the resistivity component in the absence of stress plus the changes induced by the stress4        ρ1 ρ0 π11 π12 π12 0 0 0 σ1 ρ  ρ  π π π     2  0  12 11 12 0 0 0   σ2          ρ3   ρ0   π12 π12 π22 0 0 0   σ3  (2.42)   =   + ρ0   .  ρ4   0   0 0 0 π44 0 0   σ4          ρ5   0   0 0 0 0 π44 0   σ5  ρ6 0 0 0 0 0 0 π44 σ6

As before, the symmetries of a cubic crystal with its axis aligned along the h100i axis were taken into consideration. The πij are the coefficients of the piezoresistivity tensor and are usually cited from Smith’s early work [Smith 1954], as shown in Table 2.2. For p-Si the approximation π11 = π12 = 0 is often used, and for n-Si π11 = −2π12 ; π44 = 0. At high temperatures doped piezoresistor structures on bulk silicon are corrupted by leakage currents to the substrate, which limits their applicability. Isolated polysilicon layers eliminate this drawback and are a useful alternative to crystalline silicon. The polysilicon layer always has a random grain distribution in the x–y-plane. There may be preferred z-orientations of the grains. In any case, the piezoresistivity coefficients for a planar piezoresistor with arbitrary orientation have to be calculated by taking the average of the piezo-coefficients over all grains and grain orientations. This task can be simplified considerably by considering only thin resistors with negligible height, as is the case in piezoresistive MEMS transducers (see Section 2.2.1).

2.1.2

Rotation of coordinate systems There are two occasions when the mathematical tools for describing the rotation of coordinate systems are needed.

r The orientation-dependency of stress, strain, and anisotropic material properties. Here, the two coordinate systems have fixed, time-independent orientations. 4

The components of the stress tensor σ i should not be confused with the notation for the conductivity of the isotropic medium σ.

26

Transducers

Figure 2.6 A vector in two different coordinate systems.

r The rotation as source of virtual inertial forces, e.g. for the description of the dynamics of gyroscopes. Here, one coordinate system rotates with respect to another with a certain, non-zero, speed. In this subsection the static transformation rules for vectors and tensors of second order in rotated coordinate systems are considered. In view of dynamic effects this subsection is formulated in a way that allows a natural extension towards rotating coordinate systems as considered in Chapter 8.

Coordinate frames In Fig. 2.6 the vector r¯ is represented in two different coordinate systems ¯x , E ¯y , E ¯z and e¯x , e¯y , e¯z , which have the same origin but are rotated with respect E to each other. More precisely, the two coordinate frames before – ΣE – and after – Σe – rotation are defined by their basis vectors ¯ T = [E ¯1 , E ¯2 , E ¯3 ] ΣE : E

and

Σe :

e¯T = [¯ e1 , e¯2 , e¯3 ],

(2.43)

¯T E ¯j = δij , and satisfy the right-handedness conwhich are orthonormal, e.g. E i T ¯2 × E ¯3 = +1. A¯ · B ¯ = A¯T B ¯ is the scalar or dot product of two ¯ E dition: e.g. E 1 T ¯ ¯ ¯ is their vector prodvectors A = [A1 , A2 , A3 ] and B = [B1 , B2 , B3 ]T . A¯ × B uct. The superscript T denotes a vector or matrix transposition, and δij is the Kronecker symbol. Underlined letters characterize a column arrangement of corresponding items, in Eq. (2.43), of the vectors e¯i . The product of a matrix D with an underlined vector, Du, is interpreted as a usual matrix multiplication irrespective of whether the elements of u are numbers or vectors. The orthonormality of the base yields ¯·E ¯T = E ¯T E ¯ = e¯ · e¯T = e¯T e¯ = I, E where I is the unit matrix.

(2.44)

2.1 Anisotropic material properties

27

¯ T = A¯ ⊗ B ¯ is interpreted as the dyadic product of the vectors A¯ and Here, A¯B ¯ ¯ T = [B1 , B2 , B3 ] is defined by B. The dyadic product of A¯T = [A1 , A2 , A3 ] and B   A1 B1 A1 B2 A1 B3 ¯ =  A2 B1 A2 B2 A2 B3  . (2.45) A¯ ⊗ B A3 B1 A3 B2 A3 B3

The dot sign between the vectors in Eq. (2.44) means that the components of the dyadic product are connected by scalar multiplication. The vector r¯ is seen as the same entity in both coordinate systems and remains unchanged. Thus X X ¯i = re,i e¯i . (2.46) rE ,i E r¯ = i

i

For convenience the column vectors rE and re are introduced: rE = [rE ,1 , rE ,2 , rE ,3 ]T ;

re = [re,1 , re,2 , re,3 ]T .

(2.47)

They represent the vectors of coordinates of r¯ in ΣE and in Σe , respectively, and should not be confused with the actual vector r¯ itself, which can now be represented in a compact form: ¯ T r¯E = e¯T r¯e , r¯ = E

¯ r¯ and r¯e = e¯r¯. r¯E = E

(2.48)

The tensor concept should be briefly evoked. A tensor K of second order is an arrangement of vectors according to the following scheme: o nXX o nXX ¯ T KE E ¯= ¯j = e¯T Ke e¯ = ¯i E ¯j . (2.49) ¯i E K=E KijE E Ke E ij

i

j

i

j

It allows us to express vector relations independently of their representation in different coordinate systems. The corresponding coordinate matrices KE and Ke ¯ and e¯ are defined by the equality (2.49). On applying a tensor in two different E ¯ : r¯ = E ¯ T rE , one gets to a vector r¯, say within the coordinate system E T

T

T

¯ KE E ¯·E ¯ rE = E ¯ KE r E . K · r¯ = E

(2.50)

This means that the application of a tensor to a vector within a given coordinate system transforms the coordinate vector within this system by matrix multiplication with the coordinate matrix of the tensor represented within the same coordinate system.

The rotation tensor The tensor of rotation represented by its coefficient matrix in Σe , namely S = e, rotates the unit vectors of ΣE into the unit vectors of Σe . Strictly speaking, e¯T S¯ the rotation matrix S should be labeled by the superscript E, which will be omitted.

28

Transducers

The rotation is determined by ¯ e¯ = SE

or

e¯i =

X

¯j . Sˆij E

(2.51)

j

T

¯ the rotation By means of right-hand multiplication of the first equation with E matrix S can be explicitly written ¯T . S = e¯ E

(2.52)

Equations (2.48) and (2.52) deliver immediately rE = ST re

and correspondingly

re = SrE .

(2.53)

The coefficients of the matrix S = {Sˆij } follow from the second equation in ¯k , (2.51) after transposition and right-hand multiplication with E ¯ Sˆi,j = e¯T i Ej .

(2.54)

¯ ¯j . The scalar product e¯T ¯i onto E i Ej is the direction cosine, i.e. the projection of e T T −1 T T ¯ ¯ The inverse rotation is, by definition, S = E e¯ = (¯ e E ) = S . Thus, S−1 = ST .

(2.55)

Additionally, it is easy to show that det S = +1. Analogous equations are valid for all tensors of rotation. Since ST S = I, where I is the unit matrix, it follows that X X X 2 Sˆi,k = 1. (2.56) Sˆk2 ,i = ⇒ Sˆk ,i Sˆk ,j = δij k

k

k

This equation may be useful; it reduces the nine directional cosines to the three independent values.

Separation of partial rotations – Euler angles In order to derive the coefficients of the matrix S, the projections according to (2.54) have to be calculated. The most convenient way to do this goes back to Euler. Euler partitioned the whole rotation into three partial rotations, each partial rotation taking place about a well-defined axis by a certain angle. The fact that any rotation can be partitioned into three subsequent rotations shows that the matrix S has only three independent parameters. There are different ways to define the sequence of partial rotations to get a total rotation. Euler introduced the so-called standard Euler angles φ, θ, and ψ for the following (standard) sequence of rotations.

r First, counterclockwise rotation about the z-axis (E¯3 -axis) of the initial coordinate system by the angle φ, transforming the coordinates of a vector endpoint {1} {1} {1} (r1 , r2 , r3 ) into the new coordinates (r1 , r2 , r3 = r3 ). r Second, counterclockwise rotation about the new E¯ {1} -axis by the angle θ. This 2 {1} {1} {1} {2} {2} {1} {2} rotation transforms the coordinates (r1 , r2 , r3 ) into (r1 , r2 = r2 , r3 ).

2.1 Anisotropic material properties

29

Figure 2.7 z-Rotation of the x–y-coordinate system. x′ = x cos φ + y sin φ, y ′ = −x sin φ + y cos φ, z ′ = z, and φ = θ3 .

r Third, counterclockwise rotation about the new E¯ {2} -axis by the angle ψ. This {2}

{2}

3 {2}

rotation transforms the coordinates (r1 , r2 , r3 ) into the final coordinates {2} {3} {3} {3} (r1 = r1′ , r2 = r2′ , r3 = r3 = r3′ ). The described sequence converts the coordinate frames step by step according to {3} {2} {1} ¯ {2} ) is called ¯3 , E ¯ {1} , E ΣE → ΣE → ΣE → ΣE = Σe . The triad of vectors (E 3 2 the Euler basis.5

Bryan angles Throughout this book the Bryan or Cardan angles (e.g. Wittenburg [2008]) are used. They are represented by slightly modified, non-standard Euler angles ψ1 , ψ2 , ψ3 . Their corresponding Euler basis is ¯ {2} ), ¯1 , E ¯ {1} , E (E 3 2

(2.57)

i.e. the first counterclockwise rotation is about the x-axis by ψ1 = ψx , the second about the y-axis of the rotated frame by ψ2 = ψy , and the third about the z-axis of the secondly rotated coordinate system by ψ3 = ψz . In a rotation sequence – starting with rotation about the x-axis – only the third rotation differs from a standard Euler sequence. All coordinate transformations follow the same rules. For example, the third rotation about the z-axis by the angle ψ3 , shown in Fig. 2.7, transforms the coordinates of a given vector endpoint according to    {1}   x cθ 3 sθ 3 0 x  y {1}  =  −sθ 3 cθ 3 0   y  = S3 (ψ3 ). (2.58) {1} 0 0 1 z z Throughout this text the abbreviations sα for sin α and cα for cos α shall be used. Correspondingly, in cases of unambiguity sin ψi = si and cos ψi = ci . The

5

There are different standard Euler bases. So, the sequence of rotations may start from any of the axes. For the second rotation there are always two possibilities. The only requirement is the identity of the first and third rotational matrices.

30

Transducers

following transformations can be derived by a simple cyclic permutation of the axes of the coordinate system, i.e. for the second rotation by the permutation z ⇒ y; y ⇒ x; x ⇒ z. The following sequence of transformations results: r{1} = S1 (ψ1 )r, r{2} = S2 (ψ2 )r{1} ,

(2.59)

r{3} = S3 (ψ3 )r{2} = r′ , where the partial transformations are defined by     c3 s3 0 c2 0 −s2 S3 (ψ3 ) =  −s3 c3 0  ; S2 (ψ2 ) =  0 1 0  ; s2 0 c2 0 0 1

 1 0 0 S1 (ψ1 ) =  0 c1 s1  . 0 −s1 c1 (2.60) 

The overall coordinate transformation can now be written as re = r{3} = S3 (ψ3 )S2 (ψ2 )S1 (ψ1 )rE = SrE ,

(2.61)

with  c3 c2 c3 s2 s1 + s3 c1 −c3 s2 c1 + s3 s1 S(ψ1 , ψ2 , ψ3 ) = {Sˆij } =  −s3 c2 −s3 s2 s1 + c3 c1 s3 s2 c1 + c3 s1  s2 −c2 s1 c2 c1   l1 m1 n1 (2.62) =  l2 m2 n2  . l3 m3 n3 

In contrast to Euler angles, the Bryan angles give one the possibility to linearize the rotation matrix R about the zero angles, while the Euler transformation consists of members of order ψi ψj and higher with missing linear terms. Instead of doubly subscripted coefficients Sˆi,j , it is convenient to introduce the factors li , mi , and ni , for the sake of visual clarity (e.g. Gad-el-Hak [2002]). The reader should be reminded that, according to Eq. (2.56), the sum of the squares along a column or a row is equal to one.

Transformation of tensors of second order Up to now the transformation of vectors between rotated frames has been considered. However, the consideration of elasticity, piezoelectricity, moments of inertia, and so on requires the description of material properties governed by second-order tensors (or matrices) within rotated coordinate systems. Piezoresistivity may serve as a representative example for the transformation of material properties from one coordinate system into another. According to (2.40), ¯ = ρJ, ¯ E

(2.63)

2.1 Anisotropic material properties

31

¯ and current density J¯ within an the relation between the electrical field E anisotropic material, is valid in a given coordinate frame Σ. In the case of orientation-dependent properties such as piezoresistance this coordinate system is inseparably linked to the material orientation, here to the (100) plane and [100] axis of the cubic silicon crystal. If the material which possesses orientationdependent properties is rotated with respect to the applied field or current (or vice versa), the impact of the material properties changes. In order to describe ¯ and J¯ have to be expressed in the rotated coordinate system. these changes, E ¯ ′ = SE ¯ and J¯′ = SJ¯ in the rotated frame Σ′ , Eq. (2.63) can With the fields E be rewritten using property (2.55) as ¯ ′ = SρST J¯′ = ρ′ J¯′ . E

(2.64)

Thus, the coordinate matrix of the resistivity tensor ρ′ in the rotated coordinate system is ρ′ = SρST .

(2.65)

Since the new matrix {ρ′ij } is also symmetric and, thus, has only six known independent components, Voigt’s notation can be applied, transforming ρ′ into the vector representation ρ¯′T = [ρ′1 = ρ′xx , ρ′2 = ρ′y y , ρ′3 = ρ′z z , ρ′4 = ρ′y z , ρ′5 = ρ′xz , ρ′6 = ρ′xy ]. On performing this slightly cumbersome transformation for the matrix equation (2.65), one obtains ρ¯′ = Γ¯ ρ,

(2.66)

where Γ is defined by   2m1 n1 2l1 n1 2l1 m1 n21 m21 l12  l2  2m2 n2 2l2 n2 2l2 m2 n22 m22  2   2  2m3 n3 2l3 n3 2l3 m3 n23 m23  l3  Γ=  . (2.67)  l2 l3 m2 m3 n2 n3 m2 n3 + m3 n2 n2 l3 + n3 l2 m2 l3 + m3 l2     l1 l3 m1 m3 n1 n3 m3 n1 + m1 n3 n3 l1 + n1 l3 m3 l1 + m1 l3  l1 l2 m1 m2 n1 n2 m1 n2 + m2 n1 n1 l2 + n2 l1 m1 l2 + m2 l1 The matrix of piezo-coefficients in Eq. (2.42),  π11 π12 π12 0 0 π π π  12 11 12 0 0  0 0 π π π Π =  12 12 22  0 0 0 π44 0   0 0 0 0 π44 0 0 0 0 0

 0 0    0  , 0   0  π44

(2.68)

represents a tensor of fourth order relating the two tensors of second order, namely the stress tensor σ and the resistivity tensor ρ; more precisely it represents the tensor of resistivity changes ∆ρ under stress: ∆ρ =

ρ − ρ0 I = Πσ. ρ0

(2.69)

32

Transducers

The vector representation of the tensors of second order σ and ρ by σ ¯T = (σ1 , σ2 , σ3 , σ4 , σ5 , σ6 ) and ρ¯ allows us to describe the transformation of the tensor coefficients for a rotation of the coordinate system by a simple matrix relation: ρ¯′ = Γ¯ ρ;

¯ ′ ρ = Γ∆ρ; ¯ ∆

σ ¯ ′ = Γ¯ σ.

(2.70)

On substituting these expressions into Eq. (2.69) it follows that ∆′ ρ = ΓΠΓ−1 σ ′ .

(2.71)

Consequently, the piezoresistive tensor in the rotated coordinate system is given by the matrix Π′ = ΓΠΓ−1 .

(2.72)

Unfortunately, during the transfer of second-order tensors to vectors the nice unitarity property of the rotation tensor S−1 = ST has not been handed over to the corresponding matrix Γ. Γ−1 must be calculated straightforwardly:   2l2 l3 2l1 l3 2l1 l2 l32 l22 l12   m2 m2 m2 2m2 m3 2m1 m3 2m1 m2   3 2 1   2 2 2 2n2 n3 2n1 n3 2n1 m3 n3 n2   n1 −1 Γ = .  m1 n1 m2 n2 m3 n3 m2 n3 + m3 n2 m3 n1 + m1 n3 m1 n2 + m2 n1     l1 n1 l2 n2 l3 n3 n2 l3 + n31 l2 l1 n3 + l3 n1 n1 l2 + n2 l1  l1 m1 l2 m2 l3 m3 m2 l3 + m3 l2 l1 m3 + l3 m1 m1 l2 + m 2 l1 (2.73) In summary, the rotated matrix of piezo-coefficients Π′ no longer has only 12 non-zero components, but, rather, all 36 are now non-vanishing. The calculation of Π′ is simplified by the block structure of Γ as well as of Π. For the most important cases, where the coordinate system is rotated only about the axis perpendicular to the (100) plane of silicon by the angle ψ3 , the angles are ψ2 = 0 and ψ1 = 0, so that ′ {πij }=  π11 − 21 π0 sin2 (2ψ3 ) π11 + 21 π0 sin2 (2ψ3 ) π12  π + 1 π sin2 (2ψ ) π − 1 π sin2 (2ψ ) π 3 11 3 12  11 2 0 2 0  π12 π12 π11    0 0 0   0 0 0 1 − 41 π0 sin(4ψ3 ) π sin(4ψ ) 0 0 3 2

0 0 0 π44 0 0

 0 − 21 π0 sin(4ψ3 )  1 0  2 π0 sin(4ψ3 )  0 0  ,  0 0   π44 0 0 π44 + π0 sin2 (2ψ3 ) (2.74)

where π0 = π11 − π12 − π44 .

(2.75)

2.2 Piezoresistive transducers

2.2

33

Piezoresistive transducers Transducers are the ultimate sensing and actuating elements and as such responsible for the transformation of movement of measuring components into an electrical signal or vice versa. There is a great variety of transducer principles for inertial MEMS such as piezoresistive, capacitive, thermal [Dauderstadt et al. 1995], optical [Hall et al. 2008], electromagnetic [Abbaspour-Sani et al. 1994], piezoelectric [Weinberg 1999, Zou et al. 2008, Yu and Lan 2001, DeVoe and Pisano 2001], tunneling-current sensing [Kubena et al. 1996, Yeh and Najafi 1997], floatinggate field-effect-transistor (FET) sensing [Kniffin et al. 1998], and strain FET sensing [Haronian 1999]. The most often used transducers in inertial MEMS are capacitors shaped as combs or plates. The piezoresistive transducer also plays a significant role, in particular in MEMS accelerometers. Piezoelectric devices are used more infrequently in MEMS but are again gaining popularity. All three principles sometimes are coupled with resonance excitations of the inertial structure, which results in high sensitivity and performance. Thus far the techniques for implementation of the remaining principles are not yet mature enough to find broad application. Within this chapter the focus will be on piezoresistive, piezoelectric, and capacitive transducers, including the actuating mechanisms. Some introductory remarks on other principles will be relegated to Chapter 7.

2.2.1

Piezoresistors A very popular, small-sized, and sensitive transformer of stress into electrical signals is the piezoresistor. Piezoresistivity is the (linear) change of the electrical resistivity under stress; it is not restricted to anisotropic materials such as those considered in Section 2.1.1. Piezoresistivity has a long history, starting with the discovery by Lord Kelvin that some metals increase their resistance under tension. However, there were no practical applications of the phenomenon before the first half of the twentieth century, starting with the development of strain gauges – metal foils that are mounted on elastic carriers, which are then glued onto the surface of the device undergoing deformation. Strain gauges were also the first sensors used for commercially available accelerometers [McCullom and Peters 1924]. Their sensitivity was characterized by the so-called gauge factor GF =

∆R/R ∆R = , ∆L/L εR

(2.76)

where ∆R/R is the relative change of resistance and ε = ∆L/L is the relative change of the length, which is, by definition, the applied strain. This classical definition is based on materials whose properties do not change when they are subjected to stress. This means that changes in resistivity are caused by the elastic deformation of the material.

34

Transducers

In general, considering a block of material (e.g. Fig. 2.1(a)), the resistance is given by l R=ρ , A

(2.77)

where A is the cross-sectional area. On using Eqs. (2.3), (2.4), and (2.5), the change of resistance dR can then be expressed as dR dρ dl dr dρ = + −2 = + εa (1 + 2ν). R ρ l r ρ

(2.78)

Thus the gauge factor is GF =

dρ + (1 + 2ν). εa ρ

(2.79)

The first term describes the change of the material’s specific resistivity under stress, while the second term characterizes the strain-dependent change in resistance owing to the geometric deformation of the material. Materials with predominantly strain-dependent resistivity change are termed strain gauges. Their gauge factors lie between 1.4 and 1.7 (ν = 0.2–0.35). In piezoresistive materials such as the semiconductors silicon and germanium, heterogeneous solids, thin-metal films, metal–insulator–metal (MIM) structures, and Schottky-barrier junctions as well as in superconductors, the term dρ/(εa ρ) sometimes exceeds the strain-gauge effect by orders of magnitude. For example, silicon may reach absolute values of gauge factor up to 200, while polysilicon can have values on the order of 30 [Beeby et al. 2004]. These are among the most widely used materials for piezoresistors in inertial MEMS.

2.2.2

Piezoresistors on silicon Piezoresistors are very popular as transducers in pressure sensors and bulk silicon accelerometers. They can be implemented by selective bulk doping (diffusion, ionimplantation), which creates junction-isolated resistors. In n-doped silicon, stress increases the effective mobility of majority charges (electrons) and consequently reduces the resistivity. The gauge factor becomes negative. For p-doping the opposite holds. Leakage currents through the junction diodes, which increase exponentially with increasing temperature, are an inherent drawback of these devices. Moreover, the sheet resistance of the junction-embedded resistor depends on the local bias across the isolation diode. The temperature coefficient is also very high (around 0.25%/◦ C), exceeding by far the stress-induced resistance changes. Optimized stress-dependent-arrangement techniques for the resistors in order to reduce the temperature impact, plus additional electronic compensation techniques, are required in order to overcome these deficiencies. Sometimes materials with crystalline silicon on an insulator substrate are used in order to reduce the isolation problems, especially at high temperature, namely over 100 ◦ C. However, a more cost-efficient alternative is the use of polysilicon

2.2 Piezoresistive transducers

35

Figure 2.8 The coordinate system and dimensions of a piezoresistor.

to form piezoresistors. It can be deposited onto the SiO2 /Si substrate and has a much lower temperature coefficient than that of doped silicon, typically around 0.05%/◦ C [Middelhoek and Audet 1989]. The design of piezoresistive transducers depends on the construction of the stress-creating mechanism. At a given location, this mechanism should translate the local stress into a measurement quantity (acceleration, pressure), by means of a well-defined functional relationship between the two. In any case, the piezoresistor should be as small as possible in order to help to eliminate the impact of stress gradients.

Thin piezoresistors A very simple model of a piezoresistor is shown in Fig. 2.8. The resistor is usually a thin, rectangular layer of silicon or polysilicon. The resistance change under stress can be derived by measuring the voltage and current changes. In practice, the applied voltage or the feed-in current is kept constant (low-ohmic voltage supply or high-ohmic current source) and the corresponding change of current, or voltage, respectively, is measured. The resistance layer has its own coordinate system (x′ , y ′ , z ′ ) that may be rotated with respect to the reference system of the material. Since H ≪ B and H ≪ L, the vertical field components Ez′ and Jz′ can be set to zero. The fielddensity relation corresponding to Eq. (2.41) simplifies to µ

Ex′ ′ Ey′ ′

¶

=

µ

ρ′1 ρ′6 ρ′6 ρ′2

¶µ

Jx′ ′ Jy′ ′

¶

.

(2.80)

where ρ′1 = ρ′x ′ x ′ , ρ′2 = ρ′y ′ y ′ , and ρ′6 = ρ′x ′ y ′ . If there are no contacts on the sidewalls of the resistor, no current can flow in the y ′ -direction. Thus, Jy′ ′ = 0 and Ex′ ′ = ρ′1 Jx′ ′ ,

(2.81)

Ey′ ′

(2.82)

=

ρ′6 Jy′ ′ .

36

Transducers

With the notation Vl = LEx′ ′ for the voltage difference along the resistor and i′ = i′x ′ = BHJx′ for the current through the resistor, the first equation in (2.81) is transformed into6 i′ = RVl ,

(2.83)

where the resistance R is given by R=

L ′ ρ . BH 1

Since R0 = R = [L/(BH)]ρ0 the resistance change becomes ∆R ρ′ − ρ0 = 1 = ∆′ ρ. R0 ρ0

(2.84)

The stress-induced resistance change should be calculated according to Eq. (2.71). However, since the resistor is thin, the significant stress components σi′ can develop in the surface plane only, so that σ3′ = σ4′ = σ5′ = 0. In this situation the resistance change is described by ∆R ′ ′ ′ = π11 σ1′ + π12 σ2′ + π16 σ6′ . R0

(2.85)

Very often, the in-plane shear stress σ6′ can also be neglected, and only the longitudinal and transverse stresses determine the variation of R.

Temperature compensation in piezoresistors For permanent operating loads, such as vibrations in accelerometers, the maximal stress should be more than two orders of magnitude less than the fracture strength, or the yield point, in order to avoid fatigue effects. That is, stresses larger than some 107 Pa must be avoided. Thus, since the maximal value of the πs is about 10−9 /Pa, the maximal resistance change will be on the order of some few percent. In contrast, the resistance change caused by a temperature variation of, say, 100 ◦ C is about 25% and will mask the much smaller stress-induced changes.

Temperature compensation in p-doped silicon An elegant temperature compensation may be applied on the basis of the large differences among the piezo-coefficients of p-doped silicon (see Table 2.2). In this case the approximation π11 ≈ π12 ≈ 0 holds. If two piezoresistors are arranged in the (100) plane of silicon, one, R1 , parallel to the [110] direction along the resistor axis, and the second one, R2 , perpendicular to it, then their resistance 6

For doped layers the doping profile is not constant. Thus, the specific conductivity changes depending on z ′ . Consequently, ρ0 = ρ0 (z ′ ), and the coefficients π i′ j depend on z ′ . However, the dependency of π i′ j (z ′ ) on z ′ is weak. Neglecting it, the integration of the current density over z ′ leads to slightly modified total currents for the stressed and unstressed situations, which cancels out for the relative resistance change ∆R/R.

2.2 Piezoresistive transducers

37

i

Vout

R − ∆R

V

R + ∆R

R + ∆R

R − ∆R

Figure 2.9 A full Wheatstone Bridge implemented within a silicon plate.

changes are approximately equal but have opposite signs. Indeed, the first resistor is rotated with respect to the [100] axis by ψ3 = 45◦ , the other one by −45◦ . The corresponding piezo-coefficients are, according to Eq. (2.74), 1 π44 ′ = π11 − π0 ≈ π11 ; 2 2

1 π44 ′ = π11 + π0 ≈ − π12 ; 2 2

′ = 0. π16

(2.86)

For R2 the places of normal stresses in Eq. (2.85) are interchanged, hence the resistance changes of the two resistors become ∆R1 π44 = (σ1 − σ2 ), R 2 ∆R2 π44 ∆R1 =− (σ1 − σ2 ) = − . R 2 R

(2.87) (2.88)

A well-known method for the measurement of small resistance changes with opposite signs is the usage of a full Wheatstone bridge as shown in Fig. 2.9. Four identical resistors are arranged in a rectangular layout: two along the [110] orientation and two perpendicular to it. As illustrated in Fig. 2.9, the stressinduced resistivity changes unbalance the bridge by ∆R, resulting in non-zero output signals that are temperature-independent: Vout =

∆R π44 V = (σ1 − σ2 )V R 2

(2.89)

in the case of a voltage source and Vout =

∆R π44 V = (σ1 − σ2 ) iR R 2

(2.90)

in the case of a current source. In practice, the bridge will have to be electronically complemented by small alignment resistors inside the bridge in order to correct the misalignment between the four not completely matching resistors, and by external resistors in order to compensate for the remaining temperature coefficient.

38

Transducers

Temperature compensation in n-doped silicon In n-doped silicon such efficient temperature compensation is not possible, since, according to Table 2.2, the piezo-coefficients π11 , π12 , and π44 are comparable in terms of their absolute values. In this case, two out of the four piezoresistors of the Wheatstone bridge are located on an unstressed piece of the substrate and a Wheatstone half-bridge with approximately half the sensitivity is used.

Hall-like piezoresistors An alternative way to reduce the impact of the large temperature coefficient of R is measurement of the lateral voltage along the resistor and analysis of the ratio of the lateral and longitudinal voltages. Two additional contacts in the middle of the sidewalls of the resistors (in Fig. 2.8, in the y ′ -direction) have to be implemented. The four-contact design forms a sensing element that resembles a Hall sensor and is therefore often called “Hall-like.” The ratio of the longitudinal ′ and lateral fields depends only on the piezo-coefficients πij , not on the strongly temperature-sensitive specific resistance ρ0 . It can be shown that such a “Halllike” element in (100) silicon has maximum sensitivity if it is oriented along the [100] direction (see Fig. 2.5).

2.2.3

Piezoresistors on polysilicon If polysilicon piezoresistors are used, the stress in the polysilicon layer is different from the stress in the underlaying Si substrate. The intermediate insulator layer may also distort the stress transfer. However, the thicknesses of the polysilicon and isolator layers are much smaller than that of the substrate. Thus, the strain in the polysilicon layer is practically the same as the substrate strain. It follows that the stress is transferred according to the stress–strain relations of the two materials: εp olysi = εsi

or

σp olysi = σsi

Ep olysi . Esi

(2.91)

The theoretical calculation of the piezo-coefficients of polysilicon is more complicated than for an undisturbed crystal lattice. Usually the polysilicon is modelled by randomly or partly orientated grains with “isolating” boundary regions in between, and with depletion layers at the grain boundaries. Owing to the quite different technologies used in their fabrication, the grain size, the dominant orientation, and the boundary regions may differ significantly for polysilicon layers of different manufacturers. It is of utmost importance that even for a completely random orientation of grains a significant piezoresistive effect remains [Bao 2000]. It has the same order of magnitude as the piezo-coefficients of doped silicon. Bao [2000] estimates the piezoresistive effect of polysilicon, on average, as 50% lower than that in silicon. The actual values of the πij for a given polysilicon technology should be derived from experimental data.

2.3 Piezoelectric transducers

39

Figure 2.10 The direct piezoelectric effect.

2.3

Piezoelectric transducers Since piezoelectric transducers require special materials or technologies they are not used so frequently as piezoresistors or capacitive transducers. However, they were present at the advent of early industrial MEMS gyroscopes based on a quartz tuning fork (e.g. from BEI/Systron-Donner, Madni et al. [2003]) and occupy a considerable share within inertial MEMS for consumer applications. The technological improvements in integrating piezoelectric devices in MEMS structures led also to a growing interest in the area of 3D accelerometers in particular (i.e. DeVoe and Pisano [2001], Yu et al. [2003], Bohua and Rui [2005], and Huang et al. [2003]), and recently also in the area of 3D angular-rate sensors (i.e. Kagawa et al. [2006] and Wakatsuki et al. [2004]).

2.3.1

The piezoelectric effect Piezoelectric devices produce charges under applied stress (the direct piezoelectric effect) and deform under applied electrical fields (the converse or inverse piezoelectric effect). The direct piezoelectric effect is illustrated in Fig. 2.10. The stressed piezoelectric material generates charges Q at the surfaces, which are collected by the two electrodes, consequently creating a voltage V = Q/C. C is the capacitance between the electrodes. Conversely, if a voltage is applied to the electrodes, the piezoelectric material undergoes a deformation according to the stress/strain produced. Thus, a piezoelectric sensor measures the charges or voltages generated while the actuator transforms the applied voltage into a deformation of the piezoelectric material. The direct piezoelectric effect was discovered in 1880 by the brothers Jacques and Pierre Curie, and the converse effect was predicted in 1881 by Gabriel Lippmann on the basis of thermomechanical considerations, and immediately verified by the Curies.

Natural piezoelectrics The separation of charges under stress and the deformation of materials under electrical fields have the same root cause: the non-centro-symmetric charge

40

Transducers

Figure 2.11 SiO2 in a quartz crystal.

distribution within a crystal lattice. In Fig. 2.11 the mechanism is demonstrated for a natural piezoelectric – the quartz crystal. The silicon oxide in a quartz lattice is arranged in such a way that the four valency electrons of the silicon atom each bind one valency electron of one of the four surrounding oxygen atoms, while the four second valency electrons of those oxygen atoms (not shown) are bound by the neighboring silicon atoms. The geometric structure of this interconnection determines how the quartz will crystallize – in a stable α-quartz configuration or in other possible forms, e.g. at very high temperatures in the β-quartz configuration. The unit cell of the quartz crystal is electrically neutral. If a force is applied, the cell is deformed, and the negative anions are shifted in an asymmetric way with respect to the central Si cation, because the crystal does not exhibit centro-symmetry (corners 1, 3, 5 and 7 are empty). In this case the charge redistribution leads to a polarization: an electrical dipole is created. This example shows that a necessary precondition of piezoelectricity is the noncentro-symmetry of the crystal. Silicon itself is centro-symmetric and thus not piezoelectric. Overall, of the 32 crystal classes, 21 are centro-symmetric and 20 are piezolectric. Ten classes possess a non-vanishing electrical dipole moment associated with their unit cell even in the absence of external stress (pyroelectric polar materials). Despite the numerous applications of quartz as frequency normals, accelerometer sensors etc., piezoelectric materials based on single crystals are not particularly attractive for MEMS applications. Their technological integration into MEMS devices is very difficult. If used, they act as standalone materials like in tuning-fork gyroscopes.

Ferroelectric materials Fortunately, there is another class of piezoelectrics that are MEMS-friendly and may exhibit much higher efficiency. The ferroelectric materials were discovered and developed during the 1940s and quickly found their way into such applications as microphones and loudspeakers in music, sound and ultrasound sensors

2.3 Piezoelectric transducers

41

Figure 2.12 Domain orientation before and after poling.

and generators in material science, medicine, and military uses, high-precision actuators in optics and microscopy, including scanning probe microscopy, highvoltage generators and piezoelectric transformers in consumer goods. The most often used ferroelectric materials are PZT (lead zirconate titanate – Pb[Zrx Ti1−x ]O3 ; 0 < x < 1) and barium titanate (BaTiO3 ). A wide variety of other metal oxides is known. Since they are polar materials, the unit cells of these materials exhibit electrical dipole moments below the Curie temperature TC , which – like in ferromagnetic materials – interact locally and create the so-called Weiss domains: areas with dipoles aligned more or less randomly, as indicated in the left-hand picture of Fig. 2.12 (see, for instance, Draganovic [1998]). This effect is called spontaneous polarization, which vanishes for temperatures higher than the Curie point. For instance, above TC PZT materials change their tetragonal structure, with a displacement of the central cation of the unit cell into the geometry of a cubic cell with a central cation and thus no net dipole moment. Typical Curie temperatures of PZT are around 200 ◦ C. All ferroelectric materials must be poled in order to obtain the piezoelectric behavior. A high voltage is applied at temperatures below the Curie point, which orientates the Weiss domains and in this way polarizes the material. After release, the domains are polarized as indicated in the right-hand picture of Fig. 2.12. The polarization is a typical hysteresis process – see Fig. 2.13. The corresponding strain changes are shown in Fig. 2.13 by dashed lines. It follows that poling always creates the same compression or elongation, independently of the polarity of the poling voltage. After poling, the remanent polarization determines the piezoelectric parameters of the material. The operating point is one of the two remanent polarization points on the hysteresis curve. Accordingly, if the material is compressed, the voltage generated has the same polarity as the poling voltage. If, on the other hand, the applied voltage has the same polarity as the poling voltage, the probe elongates. Depending on the movability of the domain walls, the remanent polarization may deteriorate over the lifetime of the device. The domain-wall motion can be manipulated by doping (e.g. Bassiri-Gharb et al. [2007]). Hard materials with

42

Transducers

Polarization and strain vs. electrical field 1

Remanant polarization

P-Polarization; S-Strain

0.8

0.6

Strain

0.4

0.2

0

-E

E

0

-0.2

0

-0.4

-0.6

-0.8

-1

-1.5

-1

-0.5

0

0.5

1

1.5

E - Electrical field

Figure 2.13 Polarization and strain hysteresis of a typical ferroelectric material.

low wall movability and hysteresis curve close to a rectangular shape have good longevity but lower efficiency. Ferroelectric materials are usually manufactured as ceramics by sintering fine powders. Within the MEMS environment the piezo-ceramic materials can be deposited onto silicon carriers by sputtering, sol–gel deposition (manufacturing of non-metallic, inorganic or hybrid polymer material starting from a colloidal dispersion – the sol), screen-printing processes, and other means. Some polymers also feature high piezoelectric constants. The best-known representative is polyvinylidene fluoride (PVDF). The piezoelectric effect of polymers is caused by the mutual interaction of the long-chain intertwined molecules which attract or repel each other under the impact of an electrical field.

2.3.2

Piezoelectric equations Once polarized, the piezoelectric material properties can be described in linear approximation by an extension of Maxwell’s and Hook’s equations: ¯ = ǫσ¯ E ¯+dσ D ¯ E¯

¯ ¯ + dT E ε¯ = S σ

direct piezoeffect,

(2.92)

inverse piezoeffect.

¯ is the vector of the electrical displacement, the matrix ǫ contains the coeffiD ¯ is the electrical field vector, ε¯ and σ ¯ are the cients of electrical permittivity,7 E strain and stress vectors, respectively, in the usual convention of Voigt’s nota¯ indicate that the corresponding matrices are tion. The superscripts σ ¯ and E ¯ respectively. d is determined for a zero or constant stress σ ¯ or electric field E, 7

For better distinction from the strain components, permittivity is denoted here by the symbol ǫ instead of the more common notation ε.

2.3 Piezoelectric transducers

43

the matrix of direct piezoelectric coefficients or piezoelectric charge constants and describes the polarization caused by unit stress, or the strain generated by unit applied electrical field. As a consequence of using Voigt’s notation for the stress and strain tensors, d is not quadratic but a 6 × 3 matrix. For correct interpretation of the piezoelectricity equations the convention regarding the axes is very important. The dominant axis is given by the direction of poling and denoted by e¯z = e¯3 . The data sheets provided by manufacturers usually contain the most important piezo-coefficients and permittivities. For ferroelectrics based on materials of the ditetragonal-pyramidal class (4mm group) (with four vertical mirror planes and one four-fold polar rotation axis) or of the dihexagonal-pyramidal class (6mm) the equations in (2.92) take the forms   σ1        σ2   0 0 0 0 d15 0   E1 ǫ11 0 0 D1  σ3   D2  =  0 ǫ22 0   E2  +  0 0 0 d24 0 0     (2.93) σ  d31 d32 d33 0 0 0  4  E3 0 0 ǫ33 D3  σ5  σ6 and    E¯ s11 ε1  ε   sE¯  2   21    E¯  ε3   s31  =  ε4   0     ε5   0 ε6 0

¯

sE 12 ¯ sE 22 ¯ sE 32 0 0 0

¯

sE 13 ¯ sE 23 ¯ sE 33 0 0 0

0 0 0 ¯ sE 44 0 0

   0 0 0 σ1     0 0   σ2   0    0 0   σ3   0   +    σ4   0 0 0    ¯   σ5   d15 sE 0 55 ¯ ¯ E E σ6 0 0 2(s11 − s12 )

0 0 0 d24 0 0

 d31 d32  E   1 d33     E2 . 0   E3 0  0 (2.94)

PZT and barium titanate belong to the described crystal classes. In view of the important role these materials play for MEMS applications, in particular for stress/strain sensing, their properties should be considered in more detail.

Piezoelectric sensors in MEMS If the piezoelectric material is used as a sensor, the electrical displacement field created by the direct piezoelectric effect has to be measured [Li Yang 2009, Wang et al. 2004, Iula et al. 1999, Pereyma 2008].

Different load conditions Assuming that the material is shaped as as H shown in Fig. 2.14, the free charges on ¯ ¯ S D dS = −D3 ∆A, according to Gauss’ surface of a small box enclosing the area

a bar with electrodes in the z-plane an area ∆A of the plates are Q∆A = law. Here S is the chosen Gaussian ∆A, and the vector dS¯ is orientated

44

Transducers

Figure 2.14 Different load conditions for a piezoelectric bar.

normal to the surface. Thus, D3 is the (negative) charge density at the capacitor plate. If the electrical displacement is constant, the charge of the capacitor is Q = −LBD3 .

(2.95)

Some of the different load conditions are illustrated in Fig. 2.14. If, for instance, a force F33 = LBσ3 in the z-direction is present at the bar in the absence of an electrical field (a longitudinal load in the poling direction), and if all other stress components are zero, the total charges are, in light of Eq. (2.93) (d33 mode) D3 = d33 σ3

⇒ Q = −d33 F33 .

(2.96)

If a transverse load is present, e.g. F11 = BHσ11 , and all other stresses are zero, the charges to be measured are (d31 mode) D3 = d31 σ1

⇒Q=−

L d31 F11 . H

(2.97)

In the case of shear stresses along the sidewalls, the piezoelectric bar will not react (d34 = d35 = d36 = 0). In order to measure shear stresses, the capacitor plates must be placed perpendicularly to the poling direction on the sidewalls under load. If, for instance, the shear stress is applied along the right and left crosssections (F13 = BHσ5 ), and the electrodes are placed at these cross-sections, the free charges are equal to D1 = d15 σ5

⇒ Q = −d15 F13 .

(2.98)

The actual values of dij for a given material may have quite a large spread and depend on the doping materials, the manufacturing technology etc. The coefficients d33 and d31 for PZT and barium titanate can be roughly estimated, with absolute values on the order of some hundred 10−12 C/N = 10−12 m/V up to 1000 × 10−12 C/N. The coefficients of the d matrix are strongly temperature-dependent and have the tendency to decrease with increasing temperature.

2.3 Piezoelectric transducers

45

Instead of dij coefficients, other sets of parameters are often used, such as, for instance, the e = eij matrix, which is defined as ¯ ¯ ∂Di ¯¯ ∂σi ¯¯ eij = =− (2.99) ∂εj ¯E¯ = constant ∂Ej ¯ε= ¯ constant

In view of Eq. (2.32), the eij terms can easily be calculated: X dik Ek l . eij =

(2.100)

k

For applications in MEMS devices the d31 mode is the most important one, because the piezoelectric material can be deposited most easily as a planar layer, which normally is poled in the z-direction. Poling in the plane of the wafer is difficult, because vertical electrodes are needed.

Piezoelectric sensor layers The use of the direct piezoelectric effect for sensing purposes is usually based on uniform cantilever-beam structures or membranes, which fully or locally are covered by a piezoelectric layer. The carrier layer, which forms the beam or membrane, transfers the stress from the movable mass to the deposited piezoelectric material. Usually the strain transfer is assumed to occur without losses within a possibly used adhesion layer. If this is not the case a correction taking into account the so-called shear-lag effect has to be considered [Sirohi and Chopra 2000]. Both layers – carrier and piezoelectric – are embedded between electrodes, or the carrier layer forms one of the electrodes. Such an arrangement in the form of a beam corresponds to Fig. 2.14 with laminated conducting, piezoelectric, and carrier layers instead of the homogeneous piezoelectric material. One or two ends of the console beam may be fixed to the substrate. The described structure may occupy only part of the beam at the place where the stress should be measured. The beam may be stressed in the x- and y-directions (σ2 = σ4 = σ6 = 0) and experiences strain ε1 = εxx , ε3 = εz z , and ε5 = εxz . The object of interest is the electrical displacement DT = [D1 , D3 ] as a function of the strain produced. On expressing the stress by the strain in Eq. (2.93) and using Eq. (2.94), the displacement current becomes   µ ′ ¶µ ¶ ¶ µ ¶ ε1 µ ǫ11 0 E1 0 0 e15   D1 (2.101) = ε3 + E3 0 ǫ′33 e13 e33 0 D3 ε5 with ∆ = s11 s33 − s13 s31 , e15 =

d15 ; s55

e13 =

d31 s33 − d33 s31 ; ∆

e33 =

−d31 s13 + d33 s11 ∆

(2.102)

and ǫ′11 = ǫ11 −

d215 ; s55

ǫ′33 = ǫ33 −

d231 s33 + d233 s11 − d31 d33 (s31 + s13 ) . (2.103) ∆

46

Transducers

(a)

(b)

Figure 2.15 Equivalent models of a piezoelectric sensor. (a) Charge model. (b) Voltage

model.

Piezosensor

Beam

Figure 2.16 A piezoelectric sensor for beam strain measurement.

This incorporates the impact of the electrical field, in particular between the plate electrodes. If a constant strain is applied, the free charges generated will inevitably be lost over time via unavoidable leakage currents. Hence piezoelectric sensors have a bandpass characteristic. However, present-day IC and packaging technologies allow one to reduce the leakage currents to values that guarantee hold times for the charges of minutes to hours. For most practical applications this is fully sufficient.

A model of a piezoelectric sensor In Fig. 2.15(a) an electrical model of the piezoelectric sensor is shown, where QP are the charges generated by the piezoelectric effect, CP is the shunt capacitance between the collecting electrodes complemented by possible parasitic capacitances, and RP is the unavoidable shunt resistance, which is responsible for the dissipation of static charge. To avoid confusion with a current source, the charge generator is represented by a triple circle. Figure 2.15(a) is the so-called charge model, which is equivalent to the voltage model shown in Fig. 2.15(b). A typical example for measuring strain within a bent beam using a piezoelectric sensor is demonstrated in Fig. 2.16. Assuming that the sensor is small, thin, and tightly attached to the beam so that the strain in the x-direction, ε1 , is transferred to the sensor and can be assumed constant along all sensor dimensions, the charge generated is, according to Eq. (2.97), QP = −d31 σ1 BL = −d31 EP BLε1 .

(2.104)

2.4 Capacitive transducers

47

EP is here Young’s modulus of the piezoelectric material, which for simplicity is assumed to be mechanically isotropic (e.g. PZT). The sensor with length L, width B, and thickness H forms the capacitance ǫ33 LB , H

(2.105)

QP d31 EP H = ε1 . CP ǫ33

(2.106)

CP = the voltage across which is VP =

Piezoelectric actuation The converse piezoelectric effect is often characterized by the matrix of piezoelectric voltage constants g = {gij }, which is defined as the electrical field generated by unit mechanical stress, or as the strain generated by unit electrical displacement field: ¯ ¯ X ∂εi ¯¯ ∂Ei ¯¯ gij = = − (2.107) dk i ǫ−1 = ik . ∂Dj ¯ ∂σj ¯ ¯ σ=0 ¯

D =0

k

ǫ−1 ik

are the coefficients of the inverse matrix ǫ−1 . The coefficients The use of the inverse piezoelectric effect in a MEMS environment requires relatively high voltages. The reason is the low value of the voltage constant, which for applied voltages in the range up to some volts causes deformations in the nanometer range only. This is often not sufficient for efficient actuation in the micrometer range. However, piezoelectric MEMS actuators are well suited for nano and sub-nano technologies like scanning probe microscopy. High-voltage processes allowing one to realize driving voltages on the order of dozens of volts are normally a precondition for the implementation of piezoelectric actuators in inertial MEMS.

2.4

Capacitive transducers Capacitive transducers are easily customizable interlinks between inertial sensors and electronic signal processing. Unlike piezoresistors or piezoelectric sensors, they do not require additional technology steps. They enjoy wide popularity due to their insensitivity to temperature changes and low drift, which allows one to design high-performance products with very small changes in offset and sensitivity with temperature. The size of the capacitances can be adapted to the necessary transfer characteristics in a broad range. In-plane and out-of-plane capacitance sensing and actuation is possible in all MEMS technologies. A great variety of electronic input stages, such as charge amplifiers, transconductance stages, switch-capacitor devices and embedded sigma–delta converters, is at the disposal of the system designer for various sensitivity levels and various requirements for power consumption and size [Kraft et al. 1998, Petkov and Boser 2005,

48

Transducers

(a)

(b)

Flexures

Figure 2.17 Plate capacitance. (a) A generic plate capacitance. (b) A capacitance with

a movable plat.

Wu et al. 2004, Yazdi et al. 2004, Kulah et al. 2006, Lei et al. 2005]. Resolutions down to the sub-atto-farad region (< 10−18 F) are typical for modern inertial MEMS. Capacitive transducers comprise capacitive sensors and electrostatic actuators. The basis of all capacitive transducers is the plate capacitor as shown in Fig. 2.17(a), which usually consists of two in-plane electrodes or is arranged as a connected and interdigitated vertical comb structure. The capacitance of a plate capacitor is given by the well-known expression CP = εε0

A LB = εε0 . D D

(2.108)

ε0 = 8.854 × 10−12 F/m is the electrical permittivity in vacuum and ε is the relative permittivity of the medium between the plates. Since the capacitances in inertial sensors are usually located in cavities filled with inert gas, the relative permittivity is ε = 1. To simplify matters, this assumption is applied in what follows. In inertial MEMS four basic arrangements are used:

r plate capacitor with parallel, in-plane electrodes; one electrode movable in the z-direction (see Fig. 2.17(b));

r plate capacitor with one tilting electrode (see Fig. 2.19); r linear comb capacitor with one comb movable in plane (see Section 2.4.4 and Fig. 2.21);

r radial comb capacitor with one comb pivotable about the z-axis (see Fig. 2.22). Plate capacitors are characterized by small distances between the electrodes compared with the lateral plate dimensions. This makes them very sensitive with respect to tiny changes of distance. To measure or to generate such distance changes, one of the electrodes is typically fixed to the substrate and the other one is part of the moving structure of the inertial sensor. The simple parallelplate capacitor is used, if the sensed or actuated motion is out-of-plane, i.e. if one of the electrodes is moving in the z-direction as indicated in Fig. 2.17(b).

2.4 Capacitive transducers

49

The parallel-plate capacitor is invoked here as the introductory example for sensing and actuating mechanisms. It must be pointed out that any capacitor with a fixed electrode and a movable electrode acts as sensor and actuator at the same time. In particular, on applying a voltage to the electrodes of a sensing capacitance, a charge is created that is proportional to the capacitance value. This charge can be sensed by electronic means. Simultaneously the applied voltage creates a force, which may displace the movable electrode and consequently falsify the initial capacitance value. Capacitive sensing requires careful measures in order to avoid or to compensate for such effects.

2.4.1

Electrostatic forces In order to derive the forces acting on the electrodes of any capacitor, the combination “capacitance plus voltage generator” has to be considered as a whole. The potential energy of both is U , 1 U = UC + UV = − CV 2 , 2 where the potential energy UC stored in a capacitor with charge Q is

(2.109)

1 1 QV = CV 2 , (2.110) 2 2 and the potential energy UV of the voltage generator – with a voltage that is independent of the status of the capacitor – is UC =

UV = −QV = −CV 2 .

(2.111)

The negative sign indicates that the voltage generator is discharged on charging a capacitor. The force field of the charges is conservative. Thus, the force acting on the electrodes is the gradient of the potential energy of the system, where the gradients are taken with respect to the coordinates of admissible movement q1 , q2 , q3 : 1 F¯ = V 2 ∇C(q1 , q2 , q3 ). (2.112) 2 If, for instance, the upper plate electrode in Fig. 2.17(b) is movable in the z-direction, the capacitance according to (2.108) is CP = ε0

A , D+z

(2.113)

where z is the deviation from the nominal distance D. The corresponding electrostatic force becomes 1 ∂C(D + z) 1 A FPz = V 2 =− V2 (2.114) 2 ∂z 2 (D + z)2 and acts in the negative z-direction, creating an attractive force between the two electrodes. If the dimensions of the capacitance are scaled down in all three

50

Transducers

dimensions, Eq. (2.114) implies that the force does not change as long as the fringe fields do not corrupt the validity of the model (2.108). More generally speaking, the force vector is · ¸ 1 ∂C(¯ q) ∂C(¯ q) ∂C(¯ q) F¯ = V 2 e¯q 1 + e¯q 2 + (2.115) e¯q 3 , 2 ∂q1 ∂q2 ∂q3 where the unit vectors are denoted by e¯q i . For the sake of clarity the Cartesian coordinates are used, · ¸ 1 ∂C(x, y, z) ∂C(x, y, z) ∂C(x, y, z) F¯ = V 2 e¯x + e¯y + e¯z 2 ∂x ∂y ∂z = Fx e¯x + Fy e¯y + Fz e¯z ,

(2.116)

so that after linearization around the point x, y, z (for instance around the rest position x = 0, y = 0, z = 0) the force changes ∆F¯ can be described by    ∆x kxx kxy kxz r. (2.117) ∆F¯ = −  kxy ky y ky z   ∆y  = −K ∆¯ ∆z kxz ky z kz z

K = {kx i x j } is the symmetric spring-rate matrix kx 1 x 2 = −

V 2 ∂2 C = kx 2 x 1 . 2 ∂x1 ∂x2

(2.118)

As can be seen from Eq. (2.114), the spring rate for the plate capacitor kz z = − 21 V 2 A/(D + z)3 is negative. This effect is typical for movable-plate capacitances and is called “spring softening,” because the electrostatic attractive forces act against the mechanical spring. Sometimes the forces generated by induced or intentionally applied charges are of interest. If, for instance, the device is switched off, all voltage generators have zero energy, but the plates of different capacitances may still be charged or subjected to different charge-redistribution processes. Charges may have been induced also by external charge-separation processes. In all these cases the force is given by 1 Q2 F = −∇UC = −∇ QV = −∇ . 2 2C

(2.119)

As an example, a charged plate capacitor movable in the z-direction is subjected to the force FC z = −∇

(D + z)Q2 Q2 =− , 2εA 2εA

(2.120)

which is again an attractive force. It may cause undesirably large movements up to the point of mechanical contact between the electrodes, and subsequent sticking. Care must be taken in order to avoid undesired charging effects (e.g. ESD protection).

2.4 Capacitive transducers

51

Figure 2.18 The principle of differential sensing.

2.4.2

Parallel-plate capacitors The plate capacitor with parallel plates and one electrode movable in the zdirection is used for sensing small deviations of the movable electrode from some zero position z0 = 0. The capacitance change ∆CP for small deflections z < D can be expressed as a Taylor series, ∆CP = ε0

AX zk (−1)k k = SP1 z + SP2 z 2 + SP3 z 3 · · · . D D

(2.121)

k=1

A is the value of the overlapping area of the two parallel electrodes. As can be seen from Eq. (2.121), the distance measurement via ∆CP is not linear in z, but contains nonlinear distortions. The coefficients SPk are the sensitivities with respect to the kth power of z. The relative magnitude of the kth distortion is ¯ ¯ µ ¶k −1 ¯ SPk z k ¯ ¯ = |z| γk = ¯¯ . (2.122) SP1 z ¯ D

For example, if it is desired that the quadratic distortion for a singular plate measurement be less than 1% over the entire operating range |z| < zop m ax , the operating range should not exceed 1% of the distance D.

Capacitance sensing In order to detect a deflection z, an excitation voltage V must be applied to the capacitor in order to generate charge Q. Capacitance measurements are usually performed as relative measurements or differential measurements, where the capacitance is measured with respect to a reference. An idealized block diagram of a charge-amplifier measurement is presented in Fig. 2.18. It is assumed that the operational amplifier has infinite gain, bandwidth, and input impedance, so that both the input voltage and the input current are negligible.8 The noise 8

In practice the potential of the operational amplifier is not at ground, but shifted by some reference voltage V A G N D , so that the voltage V A G N D has to be added to all voltages V i and to U o u t . Additionally, the feedback capacitance in CMOS implementations must be

52

Transducers

of the amplifier is represented by the input noise voltage VNin . All parasitic capacitances of the movable plate to ground, including electrically connected parts of the sensor, wiring connections, and so on, are summarized under CPar . Thus, the current generated by the two voltages applied to the capacitances C1 and C2 , namely voltages V1 and V2 , flowing through the feedback capacitance of the charge amplifier CFB , is i = Q˙ 1 + Q˙ 2 =

d (C1 V1 + C2 V2 ). dt

Under the idealizations described above the output voltage Uout is Z 1 1 1 Uout = i(t)dt = (Q1 + Q2 ) = (C1 V1 + C2 V2 ). CFB CFB CFB

(2.123)

(2.124)

Let C1 = CP0 + ∆CP be the capacitance to be measured, and let the voltages be set in anti-phase, V2 = −V1 = −V ; then Uout =

V V (C1 − C2 ) = (∆CP + CP0 − C2 ). CFB CFB

(2.125)

The reference measurement allows one to compensate for the large contribution of the rest capacitance CP0 (CP0 − C2 → 0) and to get a much higher dynamic range and resolution. However, optimal conditions are achieved when the reference capacitance C2 is part of the mechanical sensor and arranged in such a way that it changes in anti-phase with C1 : C2 = CP0 − ∆CP ;

⇒ Uout = 2V

∆CP Az = −2ε0 V . CFB CFB D2

(2.126)

Compliance with the condition C1,2 = CP0 ± ∆CP

(2.127)

is part of nearly all designs of inertial sensors.

Noise The resolution of a sensor is strongly determined by the electronic noise component, which is represented here by the noise voltage VNin . The noise voltage creates fluctuating charges in all capacitances, including the parasitic capacitance CPar , QNin = VNin (C1 + C2 + CPar + CFB ),

(2.128)

which is transferred to the output noise voltage UNout , µ ¶ C1 + C2 + CPar + CFB C1 + C2 + CPar UNout = VNin = VNin 1 + . (2.129) CFB CFB shunted by some resistance in order to limit the impact of uncontrollable leakage currents etc. These deficiencies do not change the general relation presented in what follows, but must be considered in the real design.

2.4 Capacitive transducers

53

This equation illustrates the utmost importance of small parasitic capacitances. Furthermore, it shows that the signal-to-noise ratio depends only weakly on the value of the feedback capacitance, which should be chosen to be as small as possible but large enough to avoid the impact of leakage currents on the integrating capability of the operational amplifier. Typical values in sub-micrometer-CMOS technologies are in the range of 1 pF.

Excitation voltages The excitation voltages V1 and V2 can be DC voltages or periodic functions of time. The different approaches to choosing the sensing voltages and the kind of charge transfer and processing of the charges (not necessarily by a charge amplifier) have spawned a great variety of electronic solutions. Some of the principles will be discussed later (Chapter 6). Here it should be noted that periodic high-frequency excitation voltages, for instance, in the form of harmonic voltages V = VDC ± V0 sin(ωt) offer a lot of benefits.

r The charge flow ∆CP V is transferred by modulation into the high-frequency domain that is not affected by the impact of low-frequency noise of the input amplifier (1/f noise). r According to Eq. (2.114), the generated force has a DC component and highfrequency components with frequencies ω and 2ω that can be adjusted in order to have little impact on the mechanical behavior of the single plate. The impact of the DC component can be taken into account by considering the corresponding deflections or bending effects, or can be mitigated by using stiff constructions in the case of differential arrangements (see, for example, Section 2.4.3).

The pull-in effect As can be seen from Eqs. (2.124) and (2.126), the sensitivity of the capacitive transducer increases with the applied voltages. Apart from the nonlinear distortions of ∆CP (z) there are, however, two more limits: first, the limited voltage level in CMOS circuits; and second, the intrinsic pull-in effect of capacitors with a movable plate. The last effect should be kept in mind for any capacitive interface.

Stability limits The pull-in effect is the result of instability in the system “plate plus spring plus voltage source feeding the capacitance plates.” A movable plate according to Fig. 2.17(b) is subjected to two forces: the first is the elastic force Fz = −kz, created by the suspending springs and acting on the plate as part of the inertial sensor; the second one is the electrostatic force according to (2.114). In an equilibrium state they balance each other: FΣ = Fz + FPz = 0;

1 V2 ⇒ FΣ = − ε0 A − kz = 0. 2 (D + z)2

(2.130)

54

Transducers

The equilibrium state is stable when the total potential energy of the system “spring plus capacitance plus voltage generator” has a minimum. Hence, the second derivative should be positive or, according to Eq. (2.112), the first derivative of the force should be negative: ∂FΣ < 0; ∂z

⇒ ε0 A

V2 − k < 0. (D + z)3

(2.131)

The last expression can easily be interpreted by rewriting it using the definition of the spring rates according to Eq. (2.117): −kz z < k.

(2.132)

The negative electrostatic spring rate must be smaller than the (positive) mechanical spring rate in order to keep the equilibrium state z according to (2.130) stable. Since −kz z < 0, there exist displacements z for which this condition will be violated. With Eq. (2.130) it follows that the displacement in the negative z-direction (towards the fixed plate) must be smaller than D/3 in order to keep the equilibrium point z stable: a stable state exists for D . (2.133) 3 For larger deviations the movable plate becomes unstable, moving beyond the point of unstable static equilibrium towards the fixed plate, where it is stopped. This undesired contact may cause sticking effects, which have to be avoided. In any case, the mechanical operating range of the sensor should be much less than the stability limit −z < −zm ax =

D . (2.134) 3 If for a given elastic spring rate k the voltage V is increased, the electrostatic force grows until it can no longer be balanced by the spring force for any z. The corresponding pull-in voltage Vpull-in can be obtained from Eq. (2.130). Since µ ¶ ε0 AV 2 z z2 =− 1+ 2 , (2.135) 2kD3 D D |zop

m ax |

≪

the maximum of the right side is reached at z/D = − 31 , and, correspondingly, a stable dynamics bound to the domain |z| ≤ |zm ax | is possible only as long as s 8kD3 V ≤ Vpull-in = . (2.136) 27ε0 A According to Eq. (2.135), V 2 (z) is a cubic function of z. Such nonlinear dependencies, embedded in dynamic systems, typically cause a hysteresis behavior, which here manifests itself in an increase of |z| with V 2 until the instability point is reached, and in a subsequent jump of z into the position −D, i.e. into a contact position of both electrodes. Decreasing the voltage does not release the moving electrode (even in the absence of contact forces, causing the usual

2.4 Capacitive transducers

55

sticking). The capacitor must be completely switched off in order to release the moving plate.

Pull-in during switch-on If the capacitor is driven with a constant voltage V0 < Vpull-in , the corresponding deflection z can be calculated by solving Eq. (2.130). However, keeping the amplitudes of stationary plate motions within the indicated stability region does not yet guarantee stable dynamics since during the switch-on process an overshoot appears that may drive the movable plate into the unstable condition. Neglecting the damping effects and assuming a step-like switch-on process, the overshoot distance can be found from the fact that, at the overshoot distance, the velocity vP is zero and, consequently, the accumulated kinetic energy EP is zero. The kinetic energy is the work performed by the total force, thus Z zm in 1 EP = mP vP2 = (Fz + FP z )dz = 0. (2.137) 2 0

The integration of the forces results in

AV0 2 . (2.138) kD However, even if the condition −zm in ≤ D/3 is also fulfilled, the stability condition may still be violated due to additional forces caused by external shocks and vibration. A large safety margin is needed, at least for failure-critical applications. A hard switch-on effect with large overshoot should be avoided by implementing a soft electronic ramp-up procedure. zm in (D + zm in ) = −ε0

Increased stability region Theoretically, the stability region can be increased significantly by adding a capacitor connected in series to the plate capacitance and by compensating the resulting voltage loss by enlarged drive voltages [Puers and Lapadatu 1996, Bao 2000]. The two serially connected capacitors can be envisaged as an equivalent capacitor with the same plate area, but an increased distance between moving and fixed plate. A serial capacitor CS < 21 CP0 corresponds to a distance that is increased by a factor of more than three, in which case the tendency for instability disappears for all voltages. The moving plate may now contact the fixed plate of the initial capacitor, remaining in a stable state. However, due to the limitations on the drive voltages for cost-efficient CMOS circuits, this method is usually regarded as unfavorable.

2.4.3

Tilting-plate capacitors Paired capacitances with tilting plates are often used to sense or actuate rotary motions about an in-plane axis. A typical arrangement is shown in Fig. 2.19. Two movable plates are stiffly connected with a torsional spring suspension, which permits a tilting movement of both plates about the y-axis. The tilting

56

Transducers

(a)

(b)

θ

Figure 2.19 A tilting-plate capacitance. (a) Top view. (b) Side view.

angle of the right plate in Fig. 2.19(b) is θ, while the left plate is rotated by −θ. The capacitance of the right plate CT1 is Z y U (x) Z L2 dy dx CT1 = ε0 , (2.139) D + z(x, y) y L (x) L1 where yL (x) is the lower border line in the x–y-plane of the plate capacitance, and, correspondingly, yU (x) is the upper border. L1,2 are the minimal and maximal dimensions along the x-axis, respectively. The z-deflection at a given point x, y does not depend on y: z(x, y) = x tan θ.

(2.140)

The plates are usually symmetric with respect to the x-axis, yL (x) = −yU (x), so that Z L2 dx yU (x) CT1 = 2ε0 . (2.141) D + x tan θ L1 Since the second capacitance plate is stiffly connected to the first one, and since θ2 = −θ1 , the second capacitance changes in the opposite direction: Z L2 dx yU (x) CT2 = 2ε0 . (2.142) D − x tan θ L1

Nonlinear distortions In MEMS implementations the gap between the movable plate and the fixed electrode D is much smaller than the lateral dimension L2 . Consequently, the maximum tilting angle θm ax is very small, µ ¶ D θm ax = arctan ≪ 1, (2.143) L2 and the tangent can be approximated by the angle θ. Typical values are L2 > 0.5 mm and D < 2 µm. Thus θm ax < 0.2◦ ; however, θm ax values on the order of 1◦ are not unusual. Owing to the pull-in effect, the operating angles must be chosen to be even significantly smaller than θm ax . Typical values for capacitive sensors are in the

2.4 Capacitive transducers

57

range of milli-degrees and less. Thus, the actual capacitance changes can be calculated assuming |xθ| ≪ D,

(2.144)

which allows one to represent the capacitance as a Taylor series with only a few terms: X (−1)k θk Z L 2 CT1 (θ) = 2ε0 yU (x)xk dx (2.145) Dk + 1 L1 k=0

or

∆CT1 = ST1 θ + ST2 θ2 + ST3 θ3 + · · ·

(2.146)

with STk = 2ε0

(−1)k Dk + 1

Z

L2

yU (x)xk dx.

(2.147)

L1

The sensitivity integrals STk can be easily derived for rectangular or trapezoidal plates, for which yU (x) = B/2 or y√ U (x) = χx. For disk-shaped plates with radius R and upper boundary yU (x) = R2 − x2 analytical expressions can also be derived using the substitution x/R = sin ϕ and integrating over ϕ. In the case of harmonic high-frequency, differential sensing, the excitation voltages are V1 = VDC + V0 sin(ωt) and V2 = VDC − V0 sin(ωt), and the highfrequency component of the total charge flow ∆QΣ = V1 ∆CT1 + V2 ∆CT2 = 2V0 sin(ωt)(ST1 θ + ST3 θ3 ) + 2VDC ST2 θ2 · · · (2.148) exhibits a suppressed quadratic nonlinearity, i.e. the total capacitance change is linearized. The “DC”-component 2VDC ST2 θ2 varies slowly with θ and is proportional to the second-order sensitivity ST2 . It can be suppressed by the signalprocessing electronics. For rectangular plates with areas (L2 − L1 )B the sensitivities are STk = ε0 B

(−1)k 1 − β k +1 . k +1 k + 1 θm ax

(2.149)

β is the length ratio, β = L1 /L2 . The corresponding relative distortions µ ¶k −1 STk θk 2 1 − βk + 1 θ γTk = = . (2.150) ST1 θ k+1 1−β θm ax reach their theoretical maximum for β = 1: µ ¶k −1 k θ γTk m ax = . 2 θm ax

(2.151)

In the case of differential sensing, according to (2.148), the first distortion of interest is γT3 . If, for instance, a cubic distortion of 1% is allowed, the operating range should be limited to θ < 0.08θm ax .

Transducers

-0.32

-0.34

θ / θmax Pull-in

58

-0.36

-0.38

Rectangular Plate Trapezoidal Plate

-0.4

-0.42

-0.44 0

0.2

0.4

0.6

0.8

1

L1 / L2

Figure 2.20 Stability limits of rectangular and trapezoidal plates.

Instabilities of a singular tilting plate As in the case of parallel-plate capacitances, the tilting capacitor may be subjected to the pull-in effect. For a singular tilting plate the spring torque Mθ of the torsional suspension is Mθ = −kθ θ, and the electrostatic moment of a single plate is Z L2 1 ∂CP1 2 yU (x)x dx MC θ = V = −ε0 V 2 (1 + θ2 ) . 2 ∂θ (D + xθ)2 L1

(2.152)

(2.153)

The total torque MΣ = MC θ + Mθ = 0 is zero, and stability is guaranteed by the condition ∂MΣ /∂θ < 0. Performing the same calculations as in Section 2.4.2, the stability condition can be represented as Z 1 L 2 yU (x)x(D + 3xθ) dx < 0, θ 6= 0. (2.154) θ L1 (D + xθ)3 Even for a rectangular plate with yU (x) = B/2, the integral must be evaluated numerically, and the stability limit must be determined iteratively. In Fig. 2.20 the stability limits of rectangular and trapezoidal plates are presented for various plate dimensions. L1 /L2 is the ratio of the shortest to the largest distance of the plate edges according to Fig. 2.19(b). The shape of the trapezoidal plates is here determined by boundaries orientated radially with respect to the pivotal point, whereby the radial edges enclose an angle of 90◦ (i.e. yU (x) = x; yL = −x).

Instabilities of the tilting capacitance pair For differential sensing as introduced in Section 2.4.2, paired or balanced plates with central suspension as in Fig. 2.19(a) are preferably used. They are excited by voltages V1 = VDC + VAC and V2 = VDC − VAC . VDC is supposed to be constant and VAC may be taken as a time-varying periodic or harmonic

2.4 Capacitive transducers

59

excitation, applied in anti-phase. The plate pair is then subject to the total moment MBPθ = 21 [V12 ∂CP1 /∂θ + V22 ∂CP2 /∂θ]: MBPθ = 2ε0

Z

L2

yU (x)x

L1

2 2 ) − 2VDC VAC (D2 + x2 θ2 ) + VAC Dxθ(VDC Dx dx. (D2 − x2 θ2 )2 (2.155)

If θ 6= 0, a moment is created even in the case of equal voltages (VAC = 0); it can be represented by a negative spring force. The spring rate is easily derived from kBPθ = −∂MBPθ /∂θ, yielding kBPθ = Z −2ε0

L2

yU (x)x2

L1

2 2 ) − 4xθ(3D2 + x2 θ2 )VDC VAC + VAC D(D2 + 3x2 θ2 )(VDC dx. 2 (D − x2 θ2 )3

For small angles θ the spring rate is approximated by kBPθ = −2ε0

2 2 VDC + VAC D3

Z

L2

yU (x)x2 dx.

(2.156)

L1

Spring softening The last expression reflects again the spring-softening effect. Since kBPθ < 0, the total spring rate kΣθ = kθ + kBPθ of the suspended plate pair is smaller than kθ , and the resulting suspension is “softer” than in the absence of electrostatic forces. This effect is often used for electrostatic trimming of resonance frequencies. If the mass of the plate pair including stiffly associated bodies is denoted by mBP , the corresponding modal frequency of the system is 2 ωBP =

kΣθ . mBP

(2.157)

Increasing the voltages reduces the resonance frequency. This effect has to be taken into account during the design flow. It allows also a fine tuning of the resonance frequency. In the case of high-frequency, harmonic sensing the AC voltage is not constant but a high-frequency sinusoidal function VAC = V0 sin(ωt). The expression 1 1 2 2 2 VDC + VAC = VDC + V02 − V02 cos(2ωt) 2 2

(2.158)

contains a high-frequency component. Usually the mechanical system is designed 2 + 21 V02 not to respond to such high-frequency excitations. Only the term VDC remains effective. The spring softening as discussed above pertains to small excitation voltages and small tilting angles for which system stability is guaranteed. However, care has to be taken to remain within the stability regime.

60

Transducers

Stability limits For the plate pair there are two stability limits: one on the right side and one on the left. When these limits are exceeded, the right or left plate will be pulled in and will contact the substrate. Since different voltages may be applied to the right and the left plate, pull-in will occur on the side with the higher voltage. In order to get a rough estimate of the stability limits, a simplification is made using Eq. (2.145), and truncating the series after the fourth power of θi , ∆CTi = ST1 θi + ST2 θi2 + ST3 θi3 + ST4 θi4 ,

i = 1, 2.

(2.159)

The moments of both plates, 21 V1,2 ∂C1,2 /∂θ, have to be added, keeping in mind that, for the second plate, the tilting angle is θ2 = −θ1 = −θ. The total moment is balanced by the torque −kθ θ, 1 2 V [(1 − α2 )ST1 + 2(1 + α2 )ST2 θ + 3(1 − α2 )ST3 θ2 + 4(1 + α2 )ST4 θ3 ] − kθ θ = 0. 2 1 (2.160) Here α2 = V22 /V12 is the squared ratio of the driving voltages. The stability regime is determined – as above – by the negative slope of the total torque: ∂MΣ = V12 [(1 + α2 )ST2 + 3(1 − α2 )ST3 θ + 6(1 + α2 )ST4 θ2 ] − kθ < 0. (2.161) ∂θ If equal absolute voltages are applied, then α2 = 1, and the solutions of Eq. (2.160) can be found analytically: s kθ − 2V 2 ST2 θ1 = 0, θ2,3 = ± . (2.162) 4V 2 ST4 According to Eq. (2.161), the corresponding slopes of the moments obey the equations ∂MΣ ¯¯ θ1 = 0 : (2.163) ¯ = 2V 2 ST2 − kθ ; ∂θ θ 1 ¯ ∂MΣ ¯ θ2,3 : = −2(2V 2 ST2 − kθ ). (2.164) ¯ ∂θ θ 2 , 3

Consequently, as long as 2V 2 ST2 < kθ , the equilibrium point θ1 = 0 is stable and the points θ2,3 are unstable. If the voltage is increased to exceed the stability limit V 2 = kθ /(2ST2 ), the equilibrium point θ1 = 0 becomes unstable, and the two additional states θ2,3 disappear (become imaginary values). In this case, tiny deviations from the zero position result in a collapse. Under these circumstances the tilting-plate pair can no longer be operated. It is important to check this worst-case condition for any design. A plausibility check can be based on the fact that ST2 is proportional to the second moment of inertia Iθ of the plate pair, Z L2 x2 yU (x)dx. (2.165) Iθ = 2ρH0 L1

2.4 Capacitive transducers

61

ρ is the density of the plate material – for silicon, ρ = 2300 kg/m3 . H0 is the thickness of the plate. As is well known, the tilting pair has a resonance frequency of 9 kθ ωT2 = . (2.166) Iθ The stability condition for θ1 according to (2.163) can now be expressed as ρ V 2 < ωT2 H0 D3 . (2.167) ε0 For a typical gap between the electrodes of D = 1.75 µm, a plate thickness of H0 = 10 µm, and a resonance frequency fT = 5 kHz, the voltage must be less than 3.7 V. This example shows how critical this condition may be, especially for small gaps and low resonance frequencies. The case of equal voltages as considered above is, of course, not very representative for the tilting plates. In order to drive or sense the pair of capacitances, different time-varying voltages are applied. For a simplified analysis they are considered as acting so slowly in comparison with the transfer processes which start once an unstable point has been reached that they can be approximated by constant values. If different absolute voltages are applied, the central stationary state θ1 will be shifted from the zero position. In a crude approximation, the equilibrium point θ1 can be determined by neglecting the higher-order terms in Eq. (2.160), θ1 ≈

(V12 − V22 )ST1 . 2[kθ − (V12 + V22 )ST2 ]

The moment’s slope is calculated to the same level of accuracy: ∂MΣ ¯¯ −[kθ − V12 (1 + α2 )ST2 ]2 + 1.5(1 − α2 )2 V14 ST1 ST3 . ¯ ≈ ∂θ θ 1 kθ − V12 (1 + α2 )ST2

(2.168)

(2.169)

If the system has been stable for V12 = V22 (i.e. according to (2.163) kθ − (V12 + V22 )ST2 > 0), and V1 is kept constant while V2 is decreased, then the denominator remains positive since α2 ≤ 1. In this case the stability criterion ∂MΣ /∂θ|θ 1 < 0 can be simplified to √ · ¸ √ ST1 ST3 V12 ST2 1 + α2 + 1.5(1 − α2 ) < kθ . (2.170) ST2 √ The actual change of the pull-in voltage depends on the ratio ST1 ST3 /ST2 , which for various plate geometries is on the order of 1. Consequently, the pull-in voltage remains on the same order as for the symmetric voltage condition. If, on the other hand, the voltage V2 is increased, stability loss happens if the denominator becomes negative. To preserve stability the condition 9

Usually the tilting plates are connected to some additional co-moving masses, so that the actual moment of inertia is larger. Consequently, the sensitivity S T 2 is overestimated, and the stability limit for the voltage in reality is higher than that estimated here.

62

Transducers

kθ − (V12 + V22 )ST2 > 0 must be satisfied, which again leads to the same order of average pull-in voltages. In summary, the stability check for symmetric voltages is a qualified starting point for an estimate of the pull-in point. The unstable states θ2,3 as well as their existence limits have to be determined numerically for any given capacitance configuration and spring torque kθ . The limits beyond which stability is no longer possible can be derived analogously to Eq. (2.154) from 1 θ

Z

L2

L1

·

D + 3xθ D − 3xθ yU (x)x − α2 (D + xθ)3 (D − xθ)3

¸

< 0,

θ 6= 0.

(2.171)

It is sufficient to investigate the stability limit θpull-in only for 0 < α2 < 1 (corresponding to V22 < V12 ), because for α2 > 1 the absolute value of the corresponding pull-in angle is the same as for 1/α2 , but with opposite sign. In other words, for V22 < V12 the right side of the tilting plate pair is going to collapse, whereas for V22 > V12 the left plate loses stability for angles outside the stability region.

2.4.4

Comb capacitors To sense or actuate in-plane movements of inertial proof masses, capacitors with plates orientated normal to the plane of the proof mass are needed. These socalled vertical plates have the drawback that the extent of the electrodes in the z-direction (the direction normal to the sensor plane) is strongly limited by constraints of the MEMS technologies to some 10 to 100 µm if surface micromachining (SMM) is used. Thus, for the design of sufficiently large capacitors in SMM technologies, multiple structures are required.

Unidirectional linear combs The interdigitated comb structure as shown in Fig. 2.21 is the most common building block of such multiple arrangements. It consists of two half-combs carrying finger electrodes, which – together with the counter-electrodes of the second half-comb – form the capacitance. The operating motion is in the y-direction and can be employed for actuation or for sensing (operating motions in the x-direction will be discussed in Section 2.4.4). However, in contrast to the tilting-plate capacitor, where a high stiffness against orthogonal movements can be assumed, the suspending springs for the comb usually yield in more than one direction. The limited stiffness, especially in the z-direction, has to be taken into consideration. As long as the linear dimensions of the interleaving fingers are much larger than the distance between them, the expression for the plate capacitance can be used. In a first approximation the capacitances of the end-faces of the fingers to the opposite walls are neglected. If the movable half-comb in Fig. 2.21 is displaced in the x-direction from the zero position by the value x, in the y-direction by the value y and in the z-direction by the value z, the total capacitance of the

2.4 Capacitive transducers

63

Figure 2.21 Comb structure.

comb is CC = ε0 N (H0 − f (z))(Y0 + y)

µ

1 1 + D0 + x D0 − x

¶

.

(2.172)

N is the number of fingers in the movable half-comb. The number of fingers in the fixed half-comb is usually N + 1. The expression (2.172) has to be interpreted with care: the overlapping area of the movable fingers with the fixed electrodes is simply (H0 − |z|)(Y0 + y). However, here the parallel-plate capacitor model is too crude, because a displacement in the z-direction will change the fringe capacitances provided by the upper and lower finger surfaces. A better-suited model should reflect the fact that, around the zero position z = 0, the capacitance change should be approximated by a function like f (z) = κz 2 , which, for increasing |z|, quickly approaches the linear region f (z) = |z|. The exact shape of the function f (z) around z = 0 is not relevant for the following discussion of the stability limits.

Forces in unidirectional combs According to Eq. (2.115), the force vector acting on the movable comb is given by · (H0 − f (z))(Y0 + y)2x 2 N D0 ¯ F C = ε0 V e¯x D02 − x2 D02 − x2 ¸ ∂f (z) + (H0 − f (z))¯ ey − (Y0 + y) e¯z ∂z = F¯Cx e¯x + F¯Cy e¯y + F¯Cz e¯z . (2.173) The attractive forces between the individual fingers in the x-direction correspond to face-to-face forces as in the plate capacitances considered above. They balance each other, so that the combs – if movable in the x-direction – take the average

64

Transducers

position x = 0 between the fingers of the non-movable half-comb. However, again, the stability condition must be considered. The movable half-comb is subject to the sum of the spring force Fx = −kx x and the electrostatic force FCx in the x direction, which balance each other at the equilibrium point x x FΣx = 2V 2 ε0 N D0 HY − kx x = 0. (2.174) 2 (D0 − x2 )2 H = H0 − f (z) and Y = Y0 + y are the actual overlapping dimensions. Stability holds for ∂FΣx /∂x < 0, or, in more detail, for 2V 2 ε0 N D0 HY

D02 + 3x2 − kx < 0. (D02 − x2 )3

(2.175)

As in the case of tilting plates in the previous paragraph, the first equilibrium point is x = 0, which is stable for 2V 2 ε0 N

HY < kx . D03

(2.176)

As the comb is moving preferably in the y-direction, the spring is stiff in the x-direction and the spring constant kx is very large, so this condition usually holds for standard designs.

Driving forces At the stable equilibrium point x = 0, the force in the y-direction, acting on the movable half-comb, is, according to Eq. (2.173), equal to FCy =

1 2 ∂C(Y0 + y) H 2 V = ε0 N V . 2 ∂y D0

(2.177)

It is orientated in the positive y-direction and, thus, is also an attractive force towards the fixed half-comb. ε0 N (H/D0 )V is the voltage-to-force gain of the driving comb. As for the horizontal plate capacitors, the independency of the generated force of dimensional scaling remains intact. Remarkably, the force is independent of y as long as the comb capacitance can be approximated by a collection of individual plate capacitances. Thus, the comb is an excellent actuator with quadratic voltage-to-force transformation.

Out-of-plane forces According to Eq. (2.173), the force in the z-direction FCz = −ε0 V 2

N Y D0 ∂f (z) D02 − x2 ∂z

(2.178)

follows the slope ∂f (z)/∂z. It is balanced by the spring force of the comb, Fz = −kz z, or −ε0 V 2

N Y D0 ∂f (z) − kz z = 0, D02 − x2 ∂z

(2.179)

2.4 Capacitive transducers

65

Figure 2.22 A schematic view of a radial comb structure.

and does not cause any instability. Indeed, the stability condition ∂FΣz /∂z < 0 or −ε0 V 2

N Y D0 ∂ 2 f (z) < kz D02 − x2 ∂z 2

(2.180)

is always satisfied because, in accordance with the properties of the function f (z) introduced above, the relation ∂ 2 f (z)/∂z 2 > 0 holds for all z.

Bidirectional actuation Applied voltages on plates or unidirectional combs create unidirectional attractive forces. In order to drive an object in both directions, a bidirectional force is needed. To create it, at least two plate capacitors or two combs are required. For instance, two combs can be arranged as a common, movable beam with rows of fingers on both sides, complemented by two corresponding interdigitated finger rows fixed on the substrate. The design principle is shown in Fig. 2.22, which at the same time illustrates the radial comb structure. For practical applications the mechanical cross-coupling and stability conditions are of interest.

The spring matrix of a bidirectional comb If the two identical linear combs with capacitances µ ¶ 1 1 CC1(2) = ε0 N H(Y0 + (−)y) + D0 + x D0 − x

(2.181)

are driven by two different voltages V1 = VDC + VAC and V2 = VDC − VAC , the resulting force of the balanced comb can be represented as ( N D0 4xH 2 2 ¯ FBC = ε0 2 [Y0 (VDC + VAC ) + 2yVDC VAC ]¯ ex + 4HVDC VAC e¯y 2 D0 − x D02 − x2 ) ∂f (z) 2 2 −2 (2.182) [Y0 (VDC + VAC ) + 2yVDC VAC ]¯ ez . ∂z

66

Transducers

The spring-rate matrix ki,j = ∂FBCi /∂xj becomes  2H(D02 + 3x2 )[Y V 2 ] 4HxVDC VAC − − 2 − x2 )2  (D D02 − x2 0    4HxVDC VAC N D0   − 0 KBC = 2ε0 2 D02 − x2 D0 − x2      2f ′ x[Y V 2 ] − 2 2f ′ VDC VAC D0 − x2

 2f ′ x[Y V 2 ] − 2 D0 − x2      ′ 2f VDC VAC  ,     ′′ 2 f [Y V ]

(2.183)

2 2 )+ + VAC where the following abbreviations were introduced: [Y V 2 ] = Y0 (VDC ′ ′′ 2 2 2yVDC VAC , f = ∂f (z)/∂z, and f = ∂ f (z)/∂z . Balanced combs are usually suspended by elastic beam-shaped springs. The beams are yielding in the y-direction, but also inevitably exhibit some elasticity in the z-direction. As mentioned above, the stiffness in the x-direction is much larger, so that, for an estimate of cross-coupling terms, x can be set to zero and the rates kxy and kxz vanish. The spring rate kBCxx leads to a light softening of the comb suspension in the x-direction as described in Section 2.4.1 and can usually be neglected. Remarkable is the cross-coupling term ky z = 2f ′ VDC VAC . As discussed above, the slope f ′ should be set to zero for small z-deflections, i.e. under normal operating conditions of the comb. Thus, this term is usually negligible. However, for larger z, caused, for instance, by shock or vibrations, the cross-coupling term may become relevant and induce an additional parasitic modulation in the z-direction. The spring rate kBCz z = f ′′ [Y V 2 ] is proportional to f ′′ , which is practically zero outside a very small interval around the zero position. However, within this interval, f ′′ and the corresponding spring rate become very large, keeping the movable comb close to the position z = 0.

The driving force of a bidirectional comb The desired driving force in the y-direction depends linearly on the excitation voltage VAC : FBCy = 4ε0

N H0 VDC VAC . D0

(2.184)

The voltage-to-force gain is now independent of the applied driving function VAC . If the AC voltage is a sinusoid, the resulting force creates vibrational oscillations of the mass linked to the driving combs. This is the basic principle for the driving of vibrational gyroscopes.

Radial combs Radial combs as shown in Figs. 2.22 and 2.23 are used in order to create torques. The finger lengths are proportional to the distance to the pivotal point so that

67

2.4 Capacitive transducers

Figure 2.23 A radial comb structure manufactured in a surface micromachining

process.

the overlap area of an individual finger with its neighbors is proportional to the common overlap angle δk = δ defined in Fig. 2.22. The capacitance of one comb, rotated from the zero position by an angle θ < 21 δ, is µ ¶ X H0 θ CCR = 2ε0 rk = CCR0 1 + (δ + θ) , (2.185) D0 δ k

where CCR0 is the comb capacitance in the zero position. Since the fingers are positioned equidistantly as shown in Fig. 2.22, rk = r0 + k ∆r, the sum over rk can be easily calculated, and one · H0 H0 CCR0 = 2ε0 δN Ravg = 2ε0 δN r0 + D0 D0

(2.186) obtains ¸ 1 (N − 1)∆r , 2

(2.187)

where Ravg is the average distance between fingers and pivot. On the basis of expression (2.185), the torque Mθ generated about the z-axis is 1 ∂CCR 2 1 Mθ = V = CCR0 V 2 . (2.188) 2 ∂θ 2δ As in the case of linear combs, the combination of two radial comb structures according to Fig. 2.22 with voltages applied in anti-phase creates a driving moment 2 MΣθ = CCR0 VDC VAC , (2.189) δ which for VAC = V sin(ωt) is a clean sinusoidal excitation.

Frame-based capacitors As mentioned above, in-plane sensing and actuation require interdigitated structures, where the opposite faces of the movable fingers see electrodes with different voltages. Thus, between two movable fingers there must be placed two fixed fingers that carry two different excitation voltages. Each group of fixed fingers with

Transducers

(a)

Top view

(b) Fixed Plates

Moving Frame

Moving Box Shield

x

Fixed Plates

Wiring

Contacts

...

68

Side view Figure 2.24 A sensing box formed by vertical walls of a frame. (a) A bird’s-eye view of a sensing box. (b) A separated sensing box.

one of the applied excitation voltages constitutes a half-comb. In summary, three half-combs – two fixed and one movable – are necessary in order to create a comb sensitive with respect to in-plane movements. Usually such an arrangement is mirrored: one movable comb with fingers on both sides and two fixed combs on each side of the movable comb. The electrical interconnection of such multiple structures is not a simple task. A more advantageous solution is presented in Fig. 2.24. Here, the electrodes of the moving capacitances are formed by the vertical walls of the moving frame. The counter-electrodes are arranged as pairs and can be supplied with different voltages. They are fixed mechanically on the substrate, but are electrically wired to different voltage sources V1 and V2 . Additional shields with the same potential as the moving box may be implemented. They shield the working capacitances from possible horizontal box deflections which may take place in gyroscopes. The opposite horizontal walls of an individual box see electrodes with different voltage loads. If the deflection of the frame from the zero position upwards in Fig. 2.24(a) is x, then the capacitances CF1,2 between frame and upper fixed electrodes and between frame and lower fixed electrodes, respectively, are CF1,2 = ε0

H0 L . D0 ∓ x

L is the length of the fixed, vertical electrode plates. The total force of both capacitances is · ¸ V12 V22 FFx = ε0 H0 L − . (D0 − x)2 (D0 + x)2

(2.190)

(2.191)

The stability considerations are identical to the explanation in Section 2.4.4 and result in a formula equivalent to (2.176): H0 L < kx . (2.192) D3 The unidirectional combs, considered in the section “Unidirectional linear combs,” are preferentially sensitive in the direction of the fingers (denoted by y) and stiff in the in-plane direction normal to the faces of the fingers (x). For 2V 2 ε0 N

2.4 Capacitive transducers

(a)

69

(b)

te

Figure 2.25 (a) A cross-section of comb fingers and electrical field lines.

(b) A simplified model of fringe impact.

frame-based combs the sensitivity direction is orientated perpendicular to the faces of the moving walls. Consequently, the spring rate kx is much lower by design, which increases the susceptibility to a pull-in effect. Thus, a careful check of tolerable voltages is mandatory during the design. The frame-based capacitance shown in Fig. 2.24 can be divided in two halfparts by cutting it along the x-axis. The resulting construction of one part is the mentioned three-subcomb design.

2.4.5

Levitation In microelectronics, fringe capacitances, i.e. capacitances associated with the edges or outer surfaces of comb- or frame-based capacitances, increase the nominal value of a capacitance but do not change its basic properties. This is not always the case for MEMS capacitances, because the movability of one of the electrodes may cause unexpected movements due to forces originating in the fringe capacitances. This is true in particular for comb capacitances, which were described in the preceding section.

Comb levitation Comb levitation in MEMS is the result of an asymmetric field configuration around the comb fingers. Let’s consider a comb with N fingers per movable halfcomb and with an overlapping area Hy Y = H0 (Y0 + y) as shown in Fig. 2.21. A cut across three fingers is shown in Fig. 2.25(a). For driving combs the applied voltages V1 and V2 are, of course, equal to each other: V = V1 = V2 .10 The movable comb is kept at a potential VAGND , commonly called analog ground. This potential is higher than the potential of the substrate, because 10

The case of different voltages is relevant for sensing combs or frames, where the movable finger is encased between the fingers of two different half-combs.

70

Transducers

modern integrated circuits are powered by one voltage only and therefore are not able to efficiently deliver positive and negative voltages around some external zero potential that could be connected to the substrate. Thus, the neutral potential of the integrated circuit has to be shifted by some voltage that is called the “analog ground”; all positive and negative signals are referenced to this potential. As a consequence, the moving structure is biased by VAGND with respect to the substrate. Since VAGND is on the order of volts, the attractive forces may become so high that the moving structure may collapse and contact the substrate. A common countermeasure is therefore the insertion of a conducting shield layer between the moving structure and the substrate, which is supplied by VAGND , so that the potential difference between moving structure and underlying surface becomes zero. This layer is electrically insulated with respect to the substrate. This grounded “shield” prevents the field lines between the fingers in the upper and in the lower space closing symmetrically. In a very crude approximation, the impact of the fringe field can be represented by some additional capacitances CF1 and CF2 between the top of the movable fingers and the fixed neighbors as shown in Fig. 2.25(b), and correspondingly between the bottom and the neighbors (not shown due to limited drawing space). These fringe capacitors represent the impact of field lines that are directed out of the upper or lower plane of the movable structure. In the zero position, z = 0, the bottom fringe capacitors are small, because the ground plane in close proximity to the bottom of the moving structure blocks the creation of field lines between the bottoms of the moving and fixed structures. Thus, the larger top capacitors create attractive forces in the positive z-direction – the levitation forces. With increasing displacement in the positive direction the bottom fringe capacitances increase, because more and more field lines starting at the bottom of the moving structure end at the fixed comb, until a point z0 > 0 where the upwards and downwards forces balance each other is reached. The exact analysis of the levitation forces requires the calculation of the charge densities at the finger surfaces and the integration of charges according to Coulomb’s law in order to derive the force distribution. Only a little work on analytical expressions for the resulting forces is available (e.g. Johnson and Warne [1995] and Yeh et al. [2000a]), because the inevitable idealization of the model entails extremely large inaccuracies. Most of the work is based on FEM and BEM simulations that allow a remarkably good insight into the consequences of the levitation effect. The starting point was the fundamental work of W. C. Tang and others [Tang et al. 1989, 1992, Tang 1990], who found a nearly linear dependency of the levitation force on the z-displacement. On the basis of the recent work of Chyuan [Chyuan 2008, Chyuana et al. 2004], the dependency of the force density per micrometer length on the overlapping area was verified for a test example with comb fingers of width 4 µm and height 2 µm (i.e. h = layer thickness) and with 2-µm gaps between combs and ground layer. The result is presented in Fig. 2.26. It shows that the levitation force FLz can be modeled by

71

2.4 Capacitive transducers

250 200

Force density

150 100 50 0 –50 –100 –150 0

2

4

6

8

10

12

14

16

18

20

z

Figure 2.26 Levitation force density fL z in ε0 /µm. Adapted from Chyuana et al.

[2004].

the equation FLz = fLz N Y,

fLz = χz ε0 V 2

z0 − z . z0

(2.193)

fLz is the force density per unit length of the overlapping area. N Y denotes the total length of the overlapping area and z0 the zero-levitation point. ε0 χz expresses the vertical drive capacity per unit length in micrometers. For the example presented, the value of χz is about 0.15 (≈ 0.19 according to Tang et al. [1992]). The zero-levitation point is at z0 ∼ 1.2 µm. The proportionality factor χz decreases with decreasing finger width and increases with decreasing fingerto-finger gaps. For a given comb configuration the exact values should be derived by an appropriate FEM simulation. The electrostatic force FLz corresponds to a positive spring constant kLz = −

∂FLz χz ε 0 = NY V 2 ∂z z0

(2.194)

and does not cause stability problems. It shifts the equilibrium point of a comb, which is governed by a mechanical spring rate Fz = −kz z, according to the force balance FLz + Fz = 0 from z = 0 to a state between z = 0 and z = z0 , z=

N Y χz ε0 V 2 z0 . N Y χz ε0 V 2 + kz z0

(2.195)

The z-shift reduces the capacitance between the fingers. As long as the voltage is constant, stable, and not too large, the effect is not really troublesome, particularly for combs with z0 < (0.1–0.2)H0 and for voltages in the range of some volts only. However, the situation is more critical for variable voltages.

Levitation in drive combs A drive comb, consisting of a movable central comb and two fixed half-combs, is presented in Fig. 2.27. With γz = N Y χz ε0 , the upper fingers in the figure

72

Transducers

Figure 2.27 A comb drive with turnable half-combs.

generate the levitation force FLz 1 = γz V12

z0 − z1 z0

(2.196)

and, correspondingly, the lower fingers provide FLz 2 = γz V22

z0 − z2 . z0

(2.197)

As long as the movable beam is yielding in the z-direction, but is very stiff with respect to rotations about the x-axis (i.e. z1 = z2 ), the forces can be added and create a levitation force proportional to V12 + V22 . For sinusoidal excitation the voltages are given, as usual, by V1,2 = VDC ± VAC with VAC = V0 sin(ωt); in this case the total force contains no component at frequency ω, but one at the double frequency 2ω. This component, together with the sinusoidal force in the y-direction, may create deflections in the z-direction at frequency ω, which, for instance, disturb the correct operation of vibratory gyroscopes. It can be suppressed by choosing a non-sinusoidal alternating voltage with the shape of a meander: V = V0 h(t), where ( µ ¶ X 1, for 0 ≤ t < T /2, T k (−1) 10 t − k h(t) = (2.198) , 10 (t) = 2 0, otherwise, k so that V 2 and the corresponding levitation force are constant. The limited stiffness with respect to torsional movements is another root cause for parasitic movements of the comb. It causes a time-varying reaction on the total torque M = (FLz 1 − FLz 2 )r. The torque is proportional to VDC V0 sin(ωt): µ · ¶ ¸ ¡ 2 ¢ z1 + z2 2 z1 − z2 2 M = rγz 4VDC V0 1 − sin(ωt) − VDC + V0 sin (ωt) . 2z0 z0 (2.199)

73

2.4 Capacitive transducers

(a)

V

(b)

-V

-V

V

Figure 2.28 (a) Differential drive with a biased shield. (b) Differential drive strips.

r denotes the average distance from the top or the bottom fingers to the xaxis, and z1 and z2 are the average deflections of the upper and lower fingers, respectively, in Fig. 2.27 during torsional movements. The tilt angle about the x-axis, θx , is θx =

z1 − z2 . 2r

(2.200)

The last term in Eq. (2.199) lies outside the operating frequency range of an oscillating drive. However, the first term causes tilting movements of the drive comb, which change the capacitances of the upper and lower half-combs. Since in resonant structures the torsional resonance is usually far away from the drive frequency, the driving excitation is transferred by the mechanical transfer function into oscillatory tilt movements that are approximately in phase with the excitation voltages. Being transferred mechanically to sensing combs, such inphase movements may cause distortions of the sensing capacitors and create contributions to the so-called quadrature error in gyroscopes.

Reduction of levitation forces The reduction of levitation forces is desirable, particularly for high-voltage drives. There are various techniques to reduce the field asymmetries or to compensate for levitation by mechanisms attracting the moving structure towards the shield. Tang et al. [1992] proposed differential biasing. This idea is based on the fact that the driving forces between two capacitor fingers are proportional to the square of the voltages. On applying on one of the fixed fingers the voltage V and on the adjacent fixed finger −V , the field configuration of the finger triple changes as shown in Fig. 2.28(a). The upper field lines of a first fixed finger no longer terminate on the top face of the movable finger, but do so on the adjacent fixed finger. Therefore, the upper fringe capacitance is split into two

74

Transducers

Fixed half-comb

Movable half-comb Figure 2.29 A comb drive with a partially biased shield.

parts: a main contribution between the fixed fingers and a significantly reduced part between fixed and movable fingers. The fringe fields, which are still present at the lower finger faces, can be reduced further by dividing the former ground shield into stripes that are alternately grounded and biased with V and −V , respectively. The fact that all lower surfaces on the fingers now are facing shield stripes loaded with the same potential completely eliminates direct finger-to-shield field lines in the intermediate space. There remains a weak residual force directed towards the substrate that is generated by field lines that connect lower surfaces of adjacent fingers; see Fig. 2.28(b). The last solution is technologically difficult, if not impossible, because it requires very narrow stripes and stripe-wiring. Another solution is based on the balance between levitation and attractive forces [Geen and Carow 2000], which is schematically presented in Fig. 2.29. The former ground shield is subdivided into areas that are grounded and some smaller areas that are biased with the driving voltage V and consequently attract the moving fingers. If the comb is stiff enough, the areas can be chosen in such a way as to balance the forces. Remarkably, the force balance is valid for any applied voltage. This method, obviously, was used in some industrial gyroscopes. A dynamic suppression of the drive modulation due to levitation forces is also possible. If the impact of the levitation on the transversal forces FCi , particularly due to the loss of overlap in the z-direction, is taken into account, the drive force for a half-comb can be approximated by FCy i = N ε0 Vi2

H0 − χz , D0

(2.201)

where χ should be determined experimentally. The drive-comb motion is balanced by the spring forces acting in the y-direction (ky y) and in the z-direction (kz z). Thus, voltages V1 and V2 can be determined for which a predetermined constant z-level (0 < z < zm ax ) and a desired motion y(t) can be established without distortion of the intended drive motion! (See, for instance, Timpe et al. [2008].)

References

75

It should be mentioned that fringe fields are not always undesirable. They may be used, for instance, in order to build asymmetric combs. In this case, one half-comb consists of fixed fingers with height H0 , whereas the fingers of the opposite row have only a fraction of this height. If the center of gravity of the second half-comb is additionally shifted a little bit upwards and the halfcomb is suspended in a way that allows one to perform small rotations about the lateral axis, the fringe fields create a large torque, which can be used to create considerable deflection angles. Such approaches were discussed for optical switches and similar applications [Yeh et al. 2000b]. For inertial sensors this technique is applied for some types of gyroscopes with out-of-plane excitation (see Chapter 8 and Adams et al. [2003]).

References Abbaspour-Sani, E., Huang, R. S., and Kwok, C. Y. (1994). A novel electromagnetic accelerometer. IEEE Electron Device Letters, 15(8):272–273. Adams, S. G., Groves, J., Cardarelli, D., Caroll, R., and Dauwalter, C. R. (2003). Electrically decoupled silcon gyroscope. US Patent 6,626,039 B1. Bao, M. H. (2000). Handbook of Sensors and Actuators, Micro Mechanical Transducers, volume 8. Amsterdam: Elsevier. Bassiri-Gharb, N., Fujii, I., Hong, E. et al. (2007). Domain wall contributions to the properties of piezoelectric thin films. Journal of Electroceramics, 19(1):47– 65. Beeby, S., Ensell, G., Kraft, M., and White, N. (2004). MEMS Mechanical Sensors. Boston: Artech House, Inc. Bohua, S. and Rui, Z. (2005). MEMS accelerometer with two thin film piezoelectric read-out, in International Conference on MEMS, NANO, and Smart Systems, p. 318. Brantley, W. A. (1973). Calculated elastic constants for stress problems associated with semiconductor devices. Journal of Applied Physics, 44:534–535. Chou, P. C. and Pagano, N. J. (1967). Elasticity: Tensor, Dyadic, and Engineering Approaches. New York: Dover Publications, Inc. Chyuan, S. W. (2008). Computational simulation for MEMS combdrive levitation using FEM. Journal of Electrostatics, 66:361–365. Chyuana, S. W., Liaoa, Y. S., and Chen, J. T. (2004). Computational study of variations in gap size for the electrostatic levitating force of MEMS device using dual BEM. Microelectronics Journal, 35:739–748. Dauderstadt, U. A., de Vries, P. H. S., Hiratsuka, R., and Sarro, P. M. (1995). Silicon accelerometer based on thermopiles. Sensors and Actuators A, 46/ 47(1–3):201–204. DeVoe, D. and Pisano, A. (2001). Surface micromachined piezoelectric accelerometers (PiXLs). Journal of MicroElectroMechanical Systems, 10(2):180–186.

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Draganovic, D. (1998). Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Reports on Progress in Physics, 61(9):1267– 1324. Franssila, S. (2004). Introduction to Microfabrication. New York: Wiley. Gad-el-Hak, M. (2002). The MEMS Handbook. Boca Raton, FL: CRC Press. Geen, J. and Carow, D. (2000). Micromachined gyros. US Patent 6,122,961. (Assignee: Analog Devices, Inc.) Hall, N. A., Okandan, M., Littrell, R. et al. (2008). Micromachined accelerometers with optical interferometric read-out and intgrated electrostatic actuation. Journal of MicroElectroMechanical Systems, 17(1):37–44. Haronian, D. (1999). Direct integration (DI) of solid state stress sensors with single crystal micro-electro-mechanical systems for integrated displacement sensing, in Technical Digest IEEE Workshop on Micro Electro Mechanical Systems (MEMS ’99). New York: IEEE, pp. 88–93. Huang, P. S., Ren, T. L., Lou, Q. W. et al. (2003). Design of a triaxial piezoelectric accelerometer. Integrated Ferroelectrics, 56(1):1115–1122. Iula, A., Lamberti, N., and Pappalardo, M. (1999). Analysis and experimental evaluation of a new planar piezoelectric accelerometer. IEEE/ASME Transactions on Mechatronics, 2:207–212. Johnson, W. and Warne, L. (1995). Electrophysics of micromechanical comb actuators. Journal of MicroElectroMechanical Systems, 4(1):49–59. Kagawa, Y., Wakatsuki, N., Tsuchiya, T., and Terada, Y. (2006). A tubular piezoelectric vibrator gyroscope. IEEE Sensors Journal, 6(2):325– 330. Kniffin, M. L., Wiegele, T. G., Masquelier, M. P., Fu, H., and Whitfield, J. D. (1998). Modeling and characterization of an integrated FET accelerometer, in Technical Digest IEE International Conference on Modeling and Simulation of Microsystems, Sensors and Actuators. Cambridge, MA: Computational Publications, pp. 546–551. Kovacs, G. T. A. (1998). Micromachine Transducers Sourcebook. New York: McGraw-Hill. Kraft, M., Lewis, C., Hesketh, T., and Szymkoviak, S. (1998). A novel micromachined accelerometer capacitive interface. Sensors and Actuators A, 68:466– 473. Kubena, R. L., Atkinsin, G. M., Robinson, W. P., and Stratton, F. P. (1996). A new miniaturized surface micromachined tunneling accelerometer. IEEE Electron Device Letters, 17(6):306–308. Kulah, H., Chae, J., Yazdi, N., and Najafi, K. (2006). Noise analysis and characterization of Σ∆ capacitive silicon micro-accelerometers. IEEE Journal of Solid-State Circuits, 41(2):352–361. Lei, S., Zorman, C. A., and Garverick, S. L. (2005). An oversampled capacitanceto-voltage converter IC with application to time-domain characterization of MEMS resonators. IEEE Sensors Journal, 5(6):1353–1361.

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Li Yang, S. Z. (2009). Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature. Composite Structures, 87:257– 264. Macke, W. (1962). Mechanik der Teilchen – Systeme und Kontinua. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. Madni, A., Costlow, L., and Knowles, S. (2003). Common design techniques for BEI GyroChip quartz rate sensors for both automotive and aerospace/defense markets. IEEE Sensors Journal, 3(5):569–578. McCullom, B. and Peters, O. S. (1924). A new electric telemeter. Technology Papers of the National Bureau of Standards, 17(247). Middelhoek, S. and Audet, S.A. (1989). Silicon Sensors. New York: Academic Press. Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Oxford University Press. Pereyma, M. (2008). Mathematical model of piezoelectric MEMS accelerometer. MEMSTECH 2008, Polyana, Ukraine, 87:257–264. Petkov, V. P. and Boser, B. E. (2005). A fourth-order Σ∆ interface for micromachined inertial sensors. IEEE Journal of Solid-State Circuits, 40(8):1602–1609. Popov, E. P. (1968). Introduction to Mechanics of Solids. Englewood Cliffs, NJ: Prentice-Hall, Inc. (1999). Engineering Mechanics of Solids. Upper Saddle River, NJ: PrenticeHall Inc. Puers, R. and Lapadatu, D. (1996). Electrostatic forces and their effects on capacitive mechanical sensors. Sensors and Actuators A, 56(3):203–210. Sirohi, J. and Chopra, I. (2000). Fundamental understanding of piezoelectric strain sensors. Journal of Intelligent Material Systems and Structures, 11:246– 257. Smith, C. S. (1954). Piezoresistive effect in germanium and silicon. Physical Review, 94:42–49. (1958). Macroscopic symmetry and properties of crystals. Solid State Physics, Advances in Research and Applications, 6:175–249. Tang, W. (1990). Electrostatic comb drive for resonant sensor and actuator applications. Ph.D. thesis, Dept. EECS, University of California, Berkeley. Tang, W., Lim, M., and Howe, R. (1992). Electrostatic comb drive levitation and control method. Journal of MicroElectroMechanical Systems, 1(4):170–178. Tang, W., Nguyen, T. C., and Howe, R. (1989). Laterally driven polysilicon resonant microstructures. Sensors and Actuators, 20:25–32. Timpe, S., Hook, D., Dugger, M., and Komvopoulos, K. (2008). Levitation compensation method for dynamic electrostatic comb-drive actuators. Sensors and Actuators A, 143:383–389. Wakatsuki, N., Kagawa, Y., and Haba, M. (2004). Tri-axial sensors and actuators made of a single piezoelectric cylindrical shell. IEEE Sensors Journal, 4(1):102–107.

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Wang, Q. M., Yang, Z., Li, F., and Smolinski, P. (2004). Analysis of thin film piezoelectric microaccelerometer using analytical and finite element modeling. Sensors and Actuators A, 113:1–11. Weinberg, M. S. (1999). Working equations for piezoelectric actuators and sensors. Journal of MicroElectroMechanical Systems, 8(4):529–533. Wittenburg, J. (2008). Dynamics of Multibody Systems. 2nd edn. Berlin: Springer. Wu, J., Fedder, G. K., and Carley, √L. R. (2004). A low-noise low-offset capacitive sensing amplifier for a 50-µg/ Hz monolithic CMOS MEMS accelerometer. IEEE Journal of Solid-State Circuits, 39(5):722–730. Yazdi, N., Kulah, H., and Najafi, K. (2004). Precision readout circuits for capacitive microaccelerometers, in IEEE Sensors 2004 (Proceedings). Yeh, C. and Najafi, K. (1997). Micromachined tunneling accelerometer with a low-voltage CMOS interface circuit. Technical Digest IEEE International Conference on Solid-State Sensors and Actuators (Transducers ’97), pp. 1213– 1216. Yeh, J. L. A., Hui, C. Y., and Tien, N. C. (2000a). Electrostatic model for an asymmetric combdrive. Journal of MicroElectroMechanical Systems, 9(1):49– 59. (2000b). Electrostatic model for an asymmetric combdrive. Journal of MicroElectroMechanical Systems, 9(1):126–135. Yu, H., Zou, L., Deng, K., Wolf, R., Tadigadapa, S., and Trolier-McKinstry, S. (2003). Lead zirconate titanate MEMS accelerometer using interdigitated electrodes. Sensors and Actuators A, 107(1):26–35. Yu, J. and Lan, C. (2001). System modeling of microaccelerometer using piezoelectric thin films. Sensors and Actuators A, 88(2):178–186. Zou, Q., Tan, W., Kim, E. S., and Loeb, G. E. (2008). Single- and triaxis piezoelectric-bimorph accelerometers. Journal of MicroElectroMechanical Systems, 17(1):45–57.

3

Non-inertial forces

Force compensation is the basic principle of inertial MEMS. For instance, accelerations acting on a movable mass create deflections that are limited by counter forces. Without restoring forces the inertial mass would bump into the walls of the sparingly calculated measurement cavity. Hence, the exploitation of restoring forces is the key principle of all inertial sensors. For stationary accelerations or rate signals they allow the system to reach a spatially stationary equilibrium state of moving masses within the measuring room. Restoring forces typically are generated by elastic deformations.1 If the movable mass is suspended by cantilever beams, tethers, hinges or other elastic members with negligible masses, and if the deformation of the inertial object itself can be neglected, then the system can be described by a lumped-element approximation consisting of stiff members and elastic springs or hinges, which are described in terms of their spring rates. In most cases this approximation holds for the operating range of applied inertial forces. For overloads like shocks and large vibrations the deformation of all elements of the mechanical system – including the inertial masses – should be taken into account. Furthermore, for some systems, such as an elastic beam under acceleration, the mass of the beam itself creates inertial forces, so a lumped-element approximation is impossible. Some such systems will be considered later (see, for example, Section 7.4). Besides the intended restoring forces, inevitable damping forces caused by interaction of movable parts with a gaseous environment or by mechanical friction within the moving parts complement the following triad of forces acting within inertial sensors: inertial forces, proportional to the acceleration of the body; damping forces, typically proportional to the velocity of movable parts; and restoring forces, proportional to the mass deflection.

3.1

Springs There exists a great variety of elastic members used within micromechanical sensors. A comprehensive description can be found, for instance, in Lobontiu 1

In levitating disks, electrostatic compensating forces are used (e.g. Houlihan et al. [2001] and Kraft and Evans [2000]).

80

Non-inertial forces

and Garcia [2004]. The general theory of compliant mechanisms is presented in Howell [2001] (see also Timoschenko and Goodier [1970]). The most important compliant members in inertial MEMS are springs. Plates and membranes are also very popular for many MEMS, especially for pressure sensors, but they are rarely used for inertial MEMS. An ideal spring should have high compliance for one degree of freedom and infinite stiffness in the remaining five degrees of freedom (DOF). For instance, a beam should be compliant to forces along one of the axes perpendicular to the beam axis and should not respond to other forces and moments. It is desirable that a linear relation between force and displacement or between moment and torsion angle holds: Fz = kz z ∆z

Mθ x = kθ x θx ,

or

(3.1)

where kz z is the bending spring rate for deflections in the z-direction and kθ x is the torsional spring constant for a rotation about the x-axis. The ratios ∆z/Fz = 1/kz z and θx /Mθ x = 1/kθ x are called compliances.

3.1.1

Beams Since spiral-shaped or z-folded springs are difficult to manufacture, the basic spring types in inertial MEMS are made up of beams, normally with uniform cross-section. The planar nature of most of the MEMS technologies limits the z-dimensions of beams. They can be long and wide, but not high. The length may be limited by the action of electrostatic forces that attract the beam towards the substrate. The minimal width is limited by technological restrictions. A wide beam can be made quite insensitive against in-plane forces and torques about the z-axis. In general, the limited aspect ratios and the necessarily small widths of in-plane bending beams make them susceptible mainly to moments about the beam axis and to forces in the out-of-plane direction. Therefore, multiple beam structures based on singular beams or beam-chains are necessary in order to limit the compliance of a suspension in undesired directions. However, it must be noted that cross-coupling effects in beams are generally unavoidable. For instance, the relation between forces and moments acting on a beam on the one side, and linear deflections and rotations on the other, is illustrated in Fig. 3.1. Assuming that the beam bends and twists in the x−zplane, only Fz and My are different from zero. Accordingly, Fz = kz z ∆z + kz θ y ∆θy

and

My = kz θ y ∆z + kθ y ∆θy ,

(3.2)

or µ

Fz My

¶

=

µ

kz kz θ y kz θ y kθ y

¶µ

∆z ∆θy

¶ .

(3.3)

3.1 Springs

(a)

81

(b)

Figure 3.1 Beams under different loads. (a) A beam under a force load. (b) A beam

under a moment load.

One sees that the same deformation can be generated by a force (Fig. 3.1(a)) or by a moment (Fig. 3.1(b)). kz reflects the desired effect of linear deflection under force, whereas kθ y represents the intended rotation under an applied moment. The rate kz θ y characterizes the cross-coupling effect: an applied force generates not only a deflection but also a rotation; and vice versa, an applied torque creates besides the rotation also a displacement. In any case, a singular beam is a far cry from a one-DOF idealization. At best, it can approximate it for kz z ∆z ≫ kz θ y ∆θy or for kθ y ∆θy ≫ kz θ y ∆z. The acceptability of the inherent cross-coupling has to be checked for any spring design.

3.1.2

The stiffness matrix For more complicated suspensions consisting of many springs, movements in one or more directions may be intended. Consequently, the full stiffness matrix is required. This consists of the intentionally non-zero spring rates as well as the cross-coupling terms: µ ¶ µ ¶ ∆¯ r F¯ (3.4) ¯ = KF ,M ∆θ¯ , M       ∆x kxx kxy kxy kxθ x kxθ y kxθ z Fx    F  k  y   xy ky y ky z ky θ x ky θ y ky θ z   ∆y         Fz   kxz ky z kz z kz θ x kz θ y kz θ z   ∆z  (3.5) . = =   Mx   kxθ x ky θ x kz θ x kθ x kθ x θ y kθ x θ z   ∆θx         My   kxθ y ky θ y kz θ y kθ x θ y kθ y kθ y θ z   ∆θy  ∆θz kxθ z ky θ z kz θ z kθ x θ z kθ y θ z kθ z Mz

In accordance with the theorem of Maxwell and Betti, the matrix is symmetric. It should be noted that the coefficients kij have different dimensions, depending on whether they are related to bending or to torsion, or to both. For practical applications two cases are of special interest: movement of a body within the x–y-plane that include rotations about the z-axis; and out-ofplane movements with rotations about both in-plane axes and deflection in the z-direction. The corresponding spring matrixes are given by   kxx kxy kxθ z (3.6) Kin-plane =  kxy ky y ky θ z , kxθ z ky θ z kθ z

82

Non-inertial forces

(a)

(b)

(c)

Figure 3.2 (a) A fixed–guided beam. (b) The deformation principle. (c) A symbolic

representation.

Kout-of-plane

 kz z kz θ x kz θ y =  kz θ x kθ x kθ x θ z . kz θ y kθ x θ z kθ y 

(3.7)

The off-diagonal elements of the stiffness matrix play a crucial role, especially in gyroscopes, where they are responsible for some very important parasitic effects. In the case of parallel or serial beam connections their emergence is related to imperfections in forming a constant, rectangular cross-section. Manufacturing flaws generate axis misalignment leading to skewed beam bending, or sidewall tilt with a similar effect. Often analytical models are missing and careful FEM simulations must be carried out for in-depth investigations of the most important secondary effects.

3.1.3

The bending equation for beams An elastic beam can be exploited under different load conditions. It can be clamped2 at one or both ends and can be subjected to distributed and concentrated point loads. Hence, different boundary conditions and load applications must be used. In inertial MEMS the typical applications are

r a cantilever with a free end r a cantilever with a guided end (fixed-guided beam) r a bridge (beam, clamped at both ends). The cantilever beam is clamped only at one end, whereas the fixed–guided beam, as shown in Fig. 3.2, is clamped at one end and guided at another. The guidance changes the boundary condition by forcing the displacement slope to be zero not only at the clamped end but also at the guided end. 2

Clamping fixes the position and the slope of the beam at the clamped end, whereas a support limits only the position.

83

3.1 Springs

(a)

(b)

Figure 3.3 Differential relations along a beam. (a) Definitions of cutting forces and loads. (b) Differential relations.

Internal forces and moments In order to derive a differential equation governing beam bending, the internal forces and moments have to be considered first. Figure 3.3(a) illustrates their definition. A virtual cut creates a left positive and a right negative cutting edge. For the positive cutting edge the coordinates correspond to (positive) coordinate axes. At the negative cutting edge the coordinates are inverted. Positive forces and moments – the normal force, transverse (shear) force, and bending moment – are orientated in line with the corresponding coordinate axes. Forces and moments at the cutting edges, together with load forces and reaction forces in the supports, keep the two parts in static equilibrium. As usual in technical mechanics the z-axis is orientated “down,” i.e. in the direction of a reference fiber, which is drawn on the bottom of the framework under consideration.3 The bending moment is defined positive if the reference fiber is strained. Correspondingly, the bending torque on a positive cutting edge acts counterclockwise (right-handed about the y-axis) and that on a negative cutting edge acts clockwise.

Differential relations of a bent beams The differential relations along the beam can be easily understood considering an infinitesimally small slice as shown in Fig. 3.3(b). The force and moment balances deliver

FS ∆x +

q(x)∆x + ∆FS = 0

⇒

1 ∆FS ∆x − ∆Mby = 0 2

⇒

∂FS = −q(x), ∂x ∂Mby FS = , ∂x

(3.8) (3.9)

where q(x) is the distributed load. The index y in the notation Mby is added to emphasize the moment orientation, which acts about the y-axis. 3

For lateral bending in the y-direction the indices “z” and “y” must be interchanged.

84

Non-inertial forces

(a)

(c)

(b)

Figure 3.4 (a) A cross-section of a beam. (b) An unbent beam. (c) A bent beam.

Bending moment and internal stress A bending moment Mby about the y-axis can be created only by normal forces within the beam’s cross-section, not by shear forces. In Fig. 3.4 a bent beam is presented. The bending moment is the sum of all partial moments dMby = σxx z dS = σz dS generated by the stress distribution over the cross-section S shown in Fig. 3.4(a): Z zσ(z)dS. (3.10) Mby = S

The bending moment is equivalent to a pair of forces within the upper and lower parts of the cross-section. Hence, a positive bending moment produces compressive stress in the upper part and tensile stress in the lower part. Between these regions there exists a neutral line where the normal stress remains zero. In an unloaded case, the neutral line is located at z = 0 if the origin of the z-axis is located at the cross-section’s center of gravity, i.e. if Z z dS = 0. (3.11) S

The stress distribution σ(z) is found by assuming the validity of Bernoulli’s hypothesis. Bernoulli argued that all of the points which initially are located in a planar cross-section perpendicular to the longitudinal beam axis x will, after pure bending (no axial torsion), remain in a plane normal to the deformed beam axis as indicated in Figs. 3.4(b) and (c). In other words, cross-sectional planes behave like rigid planes that rotate when a bending moment about the y-axis is applied. Therefore, the elongation of longitudinal fibers in the beam is linearly proportional to the distance from the neutral line. Since strain and stress are linked by the linear form of Hooke’s law, the stress also changes linearly with z: σ(z) = constant × z.

(3.12)

85

3.1 Springs

The constant can be found by considering Eq. (3.10): Z Mby constant = z 2 dS. , Iy y = Iy = Iy y S

(3.13)

Iy y = Iy is the geometric moment of inertia of the beam’s cross-section (i.e. R 2 z dS, I one of the area moments defined in the y–z-plane with I = z = y S R R 2 yz dS). y dS, and I = − yz S S The stress distribution can now be written as σ(z) =

Mby z. Iy

(3.14)

It has to be noted that this equation is valid under the following conditions:

r the origins of the y- and z-axes are located at the center of gravity of the beam’s cross-section; and

r the cross area moment is zero: Iy z = − R yz dS = 0. S

Iy z is the second-order biaxial moment (centrifugal moment). The last condition defines the main axes of the cross-section and simply states that bending stress does not create a moment about the z-axis. If this condition is not fulfilled, the bending moment is skewed and can be projected onto the main axes: σ(z) =

Mby Mbz z+ y. Iy y Iz z

(3.15)

In the linear approximation, the bending in the z- and y-directions can be calculated as a superposition of two orthogonal bending processes. Here the case of bending about the y-axis is considered first.

The Bernoulli equation To analyze the beam displacement, pure bending is assumed (with a constant bending moment). According to Bernoulli’s hypothesis, an arbitrary longitudinal fiber with distance z from the neutral axis and initial length ∆x = |R|∆ϕ elongates under bending by the increment ∆l (see Figs. 3.4(b) and (c)): ∆l = (|R| + z)∆ϕ − ∆x = z ∆ϕ.

(3.16)

while the neutral fiber retains the initial length ∆x = |R|∆ϕ. R is the radius of curvature of the neutral axis under bending. The strain of fiber elongation is4 ε=

z ∆ϕ z =− . ∆x R

(3.17)

On the other hand, with Eq. (3.14) and Hooke’s law the following is obvious: σ=

4

Mby z z = Eε = −E , Iy R

Note that with the axis convention specified above dx = −R dϕ.

(3.18)

86

Non-inertial forces

which yields the relationship between bending moment and deformation Mby = −EIy

1 . R

(3.19)

The curvature of the neutral line is proportional to the bending moment. The proportionality factor is the bending stiffness, which is the product of the modulus of elasticity and the moment of inertia: EIy . For practical applications it is more convenient to use displacements of beams w(x) instead of curvatures. The mathematical relationship between the displacement curve w(x) and the radius of curvature R is given by the formula 1 w′′ = 3 . R [1 + (w′ )2 ] 2

(3.20)

Substitution into Eq. (3.19) delivers a nonlinear differential equation for the beam displacement, w′′ (x) +

3 Mby [1 + (w′ )2 ] 2 = 0. EIy

(3.21)

In inertial MEMS spring displacements are small, and so is the slope w′ (x). This justifies the simplification EIy (x)w′′ (x) = −Mby (x),

(3.22)

which is the well-known Bernoulli equation (often called the Euler–Bernoulli equation). This differential equation is normally used in the context of MEMS. It is common to allow for variable moments of inertia in this equation, although this is not fully consistent with Bernoulli’s hypothesis. However, the longer the beam is relative to the lateral beam extensions, the smaller the warping of the cross-sectional planes becomes. Thus, Bernoulli’s hypothesis is the more accurate the longer the beam is in comparison with the typical deflection. The deflection error exceeds 10% if the deflection becomes on the order of 30% of the length [Gere and Timoshenko 1984]. ′′ = FS′ = −q(x)), the second-order differential equaIn light of Eq. (3.8) (Mby tion can be transformed into [EIy (x)w′′ (x)]′ = FS , and by a further differentiation into [EIy (x)w′′ (x)]′′ = q(x).

(3.23)

It is a matter of convenience which of Eqs. (3.22) and (3.23) is used for solving a given problem. Equation (3.23) requires the determination of four boundary conditions, whereas Eq. (3.22) needs a-priori determination of the bending moment for all x. From Eqs. (3.14) and (3.22) the relation between stress and displacement follows: σ(z) = −Ezw′′ (x).

(3.24)

87

3.1 Springs

If the acting force has to be estimated by stress measurement, e.g. by using piezoresistors at the beam’s surface, this relation allows one to link the measured variable with the acting load.

3.1.4

Cantilever beams Cantilevers under different loads A cantilever beam of length L can be subjected to a distributed load q(x) and a point load. As illustrated in Fig. 3.3(a), point loads usually are applied at the free end at x = L as a force Fz or as a bending moment about the y-axis.

Boundary conditions Assuming that the beam’s cross-section does not depend on x (Iy = constant), differentiation of Eq. (3.22) yields ∂Mby = −FS (x), ∂x ∂FS EIy w′′′′ (x) = − = q(x). ∂x EIy w′′′ (x) = −

(3.25) (3.26)

These equations allow one to derive boundary conditions for a point-force or point-torque load, for instance, at the free end. In general, the differential equation (3.23) has to be complemented by boundary conditions at the clamped end, w(0) = 0,

w′ (0) = 0,

(3.27)

and at the free end, w′′ (L) = 0, ′

w (L) = 0, ′′

EIy w (L) = −M, ′′

w (L) = 0,

EIy w′′′ (L) = −Fz

for force loading of the cantilever,

′′′

for force loading at the guided end,

′′′

for moment loading of the cantilever,

′′′

for the bridge (double-clamped beam). (3.28)

EIy w (L) = −Fz w (L) = 0 w (L) = 0

In the case of a distributed load the boundary conditions are homogeneous, w(0) = w′ (0) = w′′ (L) = w′′′ (L) = 0. Bent bridges must obey symmetric boundary conditions, w(0) = w′ (0) = w(L) = w′ (L) = 0. The bridge load is usually a distributed force or a point load at position x0 , which can be represented by a delta function q(x) = Fz δ(x − x0 ).

Load–deflection relations R If f1 (x) is the integral function of q(x) (f1 (x) = q(x)dx), f2 (x) the integral function of f1 (x), and f3 (x) the integral function of f2 (x), integration of

88

Non-inertial forces

Eq. (3.26) generates a set of equations EIy w′′′ (x) = f1 (x) + C1 , EIy w′′ (x) = f2 (x) + xC1 + C2 , x2 C1 + xC2 + C3 , (3.29) 2 x3 x2 EIy w(x) = f4 (x) + C1 + C2 + xC3 + C4 , 6 2 where the integration constants have to be determined using boundary conditions (3.27) and (3.28). The corresponding solutions are summarized in Table 3.1. For the bridge the integral functions fi (x) have a discontinuity in order to fulfill Eq. (3.26): µ ¶ L (x − L/2)i−1 fi (x) = Fz h x − , i ≥ 1, 2 (i − 1)! EIy w′ (x) = f3 (x) +

where h(x) is the unit-step (Heaviside) function ( 0 for x < 0, h(x) = 1 for x > 0.

(3.30)

The spring rate for the guided beam is four times higher than for a free-end cantilever and, as is intuitively clear, is highest for a doubly clamped bridge. Where desired, the cross-coupling stiffness according to Eq. (3.2) can be easily derived by calculating the ratio between the slope at the beam’s tip and the applied load.

Rectangular beams The derived expressions have to be specified for the most common cross-sections. For a cantilever beam with height h and z-dependent width b(z), the moment of inertia is Z h 2 Iy = z 2 b(z)dz. (3.31) − h2

A rectangular beam of width b possesses the moments of inertia Iy = bh3 /12 and Iz = b3 h/12 and the corresponding stiffness values along the three axes 1 bh3 1 b3 h bh E 3, ky = E 3 , kx = E . (3.32) 4 L 4 L L Therefore, the ratio between the spring rates for z- and y-deflections of a freeend or a guided beam is kz /ky = h2 /b2 . To make the beam much stiffer in the y-direction than in the z-direction one requires b ≫ h. The axial stiffnes, kx = 4 (L2 /h2 ) kz , is orders of magnitude larger then the lateral stiffness (L ≫ h, L ≫ b). kz =

Bridge with point load Fz at x = L/2

Moment M at the free end

Force Fz at the guided end

Force Fz at the free end

Constant distributed load q(x) = q0

Load

Table 3.1. A beam under different loads

x2 (3L − 2x) 12EIy

x2 (3L − x) 6EIy

M

x2 2EIy " ¶3 # ¶µ µ L L Fz 2 3 x− 3Lx − 4x + 8h x − 2 2 48EIy

Fz

Fz

Deflection w(x) x2 (6L2 − 4xL + x2 ) q0 24EIy

w

L3 3EIy

µ ¶ L = Fz 1 2

L2 2EIy

L3 12EI M

Fz

Fz

Tip deflection L q0 EIy 8E

90

Non-inertial forces

(a)

(b)

(c)

Figure 3.5 Different cross-sections of MEMS beams: (a) rectangular, (b) trapezoidal,

and (c) with tilted sidewalls.

Non-rectangular beams The specific features of isotropic etching of silicon bulk material entail that crosssections of beams that originally were designed to be rectangular emerge as trapezoids or with tilted sidewalls, as illustrated in√Fig. 3.5. The parameter d of the bulk-micromachined trapeze is given by d = ( 2/2)h, and θ is the sidewall inclination. To keep the notation of the axes in accordance with the previously used indices, the cross-section of the real manufactured beam with a small inclination θ of the vertical sidewalls is rotated by 90◦ , resulting in a horizontally inclined sidewall. The area moments of the trapezoidal beam can be easily calculated taking into consideration that the center of gravity is shifted by z0 = −dh/6b, µ ¶ µ ¶ bh3 d2 hb3 d2 Iy ,trap ez = 1− 2 , Iz ,trap ez = 1+ 2 , Iy z ,trap ez = 0, 12 3b 12 b (3.33) where b = (b1 + b2 )/2 and d = (b1 − b2 )/2. For the with tilted sidewall beam the main area moments are Iy ,sidewall tilt =

bh 2 bh3 (h + tan2 θ b2 ) ∼ , = 12 12

Iz ,sidewall tilt =

b3 h 12

(3.34)

and Iy z b2 b2 ∼ = tan θ 2 θ . = 2 Iy h + tan θ b2 h2

(3.35)

The mixed moment of inertia of a beam with height b = 10 µm, thickness 2 µm and sidewall tilt of 0.1◦ amounts to 4.4% of the moment Iy . This causes a coupling of 4.4% of the lateral beam’s deflection into an out-of-plane movement. Summarizing, symmetric cross-sections may cause a shift of the central beam axis, whereas asymmetric cross-sections cause a mixed moment of inertia that is the root cause for off-diagonal elements in the stiffness matrix.

Skew beam bending and asymmetric suspensions Skew beam bending and the more general case of skewed deflections of complex suspensions is one of the most dangerous imperfections in designing highperformance gyroscopes. To derive some quantitative relations the cross-coupling

3.1 Springs

91

Figure 3.6 The cross-section and main axes of a skewed beam.

of the intended motion into a parasitic motion can be analyzed on the singular beam level, or – in the best case – on the level of simple spring chains according to Fig. 3.12. For complex suspensions few general relations are available. Skew beam bending is an effect where deflections along two orthogonal axes are caused by only one force or torque component along one of these axes. Let’s consider first the case of two orthogonal forces. The equations in Table 3.1, derived for deflections in the z-direction, can be applied to the orthogonal direction y by substituting Iy by Iz . Since superposition holds, arbitrary beam bending can be calculated, projecting the applied force onto the main axes y and z and adding the corresponding responses. No cross-effects arise.

Principal axes and principal moments of inertia However, for asymmetric cross-sections the biaxial moment Iy z is not zero, and the main axes do not coincide with the coordinate axes. In this case, the main axes y ′ and z ′ of an asymmetric cross-section can be determined by rotating the coordinate system by the angle ϕ as illustrated in Fig. 3.6. The coordinates of a point in the rotated coordinate system are given by y ′ = y cos ϕ + z sin ϕ,

z ′ = −y sin ϕ + z cos ϕ.

(3.36)

Using this fact, moments of inertia in the rotated coordinate system Iy ′ , Iz ′ , and Iy ′ z ′ , Z Z Z y ′ z ′ dS, (3.37) y ′2 dS, Iy ′ z ′ = − z ′2 dS, Iz ′ = Iy ′ = dS

dS

dS

can be expressed in terms of the corresponding moments in the initial system. Simple calculations show that 1 (Iy + Iz ) + 2 1 = (Iy + Iz ) − 2

Iy ′ = Iy ′ y ′ = Iz ′ = Iy ′ y ′

1 (Iy − Iz )cos(2ϕ) + Iy z sin(2ϕ), 2 1 (Iy − Iz )cos(2ϕ) − Iy z sin(2ϕ), 2

1 Iy ′ z ′ = − (Iy − Iz )sin(2ϕ) + Iy z cos(2ϕ). 2

(3.38) (3.39) (3.40)

92

Non-inertial forces

To define the main axes, ϕ has to be chosen to bring the biaxial moment to zero, Iy ′ z ′ = 0, tan(2ϕ) =

2Iy z . Iy − Iz

(3.41)

The inertial moments with respect to the main axes are correspondingly " # q 1 2 2 Iy ′ (z ′ ) = Iy + Iz ± (Iy − Iz ) + 4Iy z . (3.42) 2 The sign “−” corresponds to the moment Iz ′ . The non-principal moments can be expressed in terms of the principal moments and the tilt angle ϕ of the principal axes: Iy =

Iy ′ + Iz ′ Iy ′ − Iz ′ Iy ′ + Iz ′ Iy ′ − Iz ′ + cos(2ϕ), Iz = − cos(2ϕ), 2 2 2 2 Iy ′ − Iz ′ Iy z = sin(2ϕ). (3.43) 2

The same relation holds for the spring rates.

The stiffness matrix for rotated principal axes Indeed, for forces and displacements of an arbitrary suspension that act along the principal axes, the relation ¶ µ µ ¶ µ ¶ kz ′ 0 ∆z Fz = = Fy 0 ky ′ ∆y holds. No cross-coupling terms exist. Rotating first the acting forces by the angle ϕ on the principal axes and re-projecting the reactions onto the actual coordinates yields the real stiffness matrix, µ ¶µ ¶µ ¶ c −s kz ′ 0 c s K= , s c 0 ky ′ −s c where c = cos ϕ and s = sin ϕ. On performing the matrix multiplications one gets   kz ′ + ky ′ kz ′ − ky ′ kz ′ − ky ′ − cos(2ϕ) − sin(2ϕ)   2 2 2 K=  kz ′ − ky ′ kz ′ + ky ′ kz ′ − ky ′ − sin(2ϕ) + cos(2ϕ) 2 2 2 ¶ µ 3E Iz −Iy z = 3 . (3.44) −Iy z Iy L The first relation of Eq. (3.44) between principal stiffness and real stiffness is valid for any suspension because it was derived without using any information on the nature of the suspension. It can be used for investigating the impact of cross-coupling stiffness elements (anisoelasticity) on parasitic effects in gyroscopes.

3.1 Springs

93

Figure 3.7 The moment Mσ0 created by residual stress.

If applied to a single beam, the second equation of (3.44) shows that the cross-coupling terms are proportional to the cross moment of inertia Iy ,z , i.e. to the asymmetries in the beam’s cross-section with respect to the force–deflection axes. For well-designed, complex suspensions consisting of multiple springs or springchains, the dominant reason for the emergence of off-diagonal stiffness elements is differences in the constituting springs, caused by manufacturing imperfections, including varying cross-sections over the length of the individual beams, sidewall tilts etc.

Residual stress in bending beams If elastic members are formed out of stressed materials like polysilicon or thinfilm silicon, the “frozen” residual stress may significantly change the compliance. The origins of residual stress include thermal mismatch during production, lattice mismatch between layers or between the material and its sacrificial layer, substitutional or interstitial impurities disturbing the homogeneous structure, non-uniform plastic deformation and so on. The stress may vary from MPa up to some hundreds of MPa and – if balanced by the reaction forces of supports – may lead to buckling or cracking. Here, structures below the buckling limit will be considered. Besides frozen stress, stress gradients within a structural layer may exist. Stress gradients lead, for instance, to static bending of free-standing beams. Apart from some exceptions such as processes based on active layers with embedded metal stripes, modern MEMS processes used for manufacturing of elastic members exhibit small stress gradients below 0.05–0.5 MPa/µm. Hence, the impact of stress gradients is not considered here.

The beam equation with residual stress A prestressed cantilever beam is shown in Fig. 3.7. The residual stress in the axial direction is represented by the force arrow Fσ0 : Fσ0 = σ0 S,

(3.45)

94

Non-inertial forces

where σ0 is the uniform residual stress and S the area of the cross-section. σ0 is positive in the case of tensile stress and negative in the case of residual compression. The force Fσ0 generates an additional bending moment Mσ0 = Fσ0x w(x), whereas Fσ0x = Fσ0 cos(arctan w′ (x)) ∼ = Fσ0 is the projection of the residual stress force onto the x-axis. Since the bending slope is very small, the approximation Mσ0 = Fσ0x w(x) holds. It should be noted that, in the case of large residual stress, the full nonlinear relation must be used. The bending moment Mσ0 is acting against the moment Mby and, consequently, has to be taken with negative sign. At a given position the local moment Mby is MbΣ = Mby − σ0 Sw(x).

(3.46)

Since for a local bending moment the Euler equation (3.22) remains in force, the resulting expression has the form EIy (x)w′′ (x) = −Mby + σ0 Sw(x)

(3.47)

or w′′ (x) − κ2 w(x) = −

Mby , EIy

κ2 =

|σ0 |S . EIy

(3.48)

For a homogeneous beam the fourth-order differential equation yields w′′′′ (x) − κ2 w′′ (x) =

q . EIy

(3.49)

(See also, for example, Landau and Lifschitz [1986], Clark [1997], Kuehnel [1995], and Osterberg and Senturia [1997].)

The impact of residual stress First, the case of residual tensile stress is considered: σ0 > 0. Thereby the parameter κ is a real number and the solution of the fourth-order homogeneous equation can be written as w′′ (x) = C1 sinh(κx) + C2 cosh(κx). Further integration leads to the demanded function w(x) =

C1 C2 sinh(κx) + 2 cosh(κx) + C3 x + C4 . κ2 κ

(3.50)

In the case of residual compressive stress σ0 < 0, and κ has to be substituted by jκ. The coefficients of the total solution should be determined in accordance with boundary conditions (3.27) and (3.28) (the first two equations). For example, the solution for a fixed guided beam is given by  1 − cosh(κL)   (1 − cosh(κx)) − sinh(κx) + κx, σ0 ≥ 0 Fz  sinh(κL) . (3.51) w(x) = 3 1 − cos(κL) κ EIy    (1 − cos(κx)) + sin(κx) − κx, σ0 ≤ 0 sin(κL)

3.1 Springs

The maximal deflection at x = L is  Fz 1 − cosh(κL)    κ3 EI × 2 sinh(κL) + κL, y w(L) =  F 1 − cos(κL) z   3 ×2 − κL, κ EIy sin(κL)

95

σ0 > 0, (3.52) σ0 < 0.

and the spring rate results with kz = Fz /w(L). On substituting Iy by Iz one gets the spring constant ky . Serial expansion reveals the order of magnitude particularly for small amounts of stress. The first three members of the maximal deflection at x = L are " # µ ¶ 2 2 2 F σ SL σ SL z 0 0 3 w(L) ∼ 1 − 0.1 + 0.0101 − ··· . (3.53) =L 12EIy EIy EIy If the residual stress is zero, the equation is in accordance with the third equation in Table 3.1. Equation (3.53) is valid for positive and negative stress values. Thus, for tensile stress the spring stiffness increases, whereas for compressive stress the spring becomes weaker. For compressive stress the limits of validity of Eq. (3.52) are given by κL ≤

π 2

or

|σ0 |S ≤

π2 EIy . 4L2

(3.54)

Limits of the linear approximation For larger compressive stresses the denominator of the second equation in (3.52) decreases and the deflection starts to grow, eventually approaching instability limits; this indicates that the limits of validity for a linear approximation have been reached and that the onset of buckling is incipient (see Section 3.1.6). The limit at which buckling takes place is four times larger than the indicated buckling onset value and is called the Euler buckling limit. It corresponds to the value κL = π at which reaction forces can no longer balance the stress associated with a cosine-shaped deflection. The tendency for instability is the result of reaction forces at the second end of the guided beam and is also present for bridges. Thus, not only residual stress may cause such behavior. The same effect takes place if a compressive axial force is applied. Such a force is equivalent to the stress-induced force Fσ0 = σ0 S and causes the same bending moments. In contrast, tensile stress does not cause any instability or buckling problems. An example should demonstrate the impact of residual stress. Assuming a rectangular beam’s cross-section with length-to-height ratio L/h ∼ 50, the expression for κL becomes r σ0 L2 κL = 12 . (3.55) E h2 In standard MEMS processes residual stress on the order of 50 MPa could be expected. With the elasticity of polysilicon E ∼ 160 GPa and σ0 ∼ 50 MPa,

96

Non-inertial forces

Figure 3.8 A slice of a torsion bar.

the value of κL is approximately 3.0. Compared with an unstressed beam, the corresponding displacement reduction is 47%. This example shows that residual stress may increase the spring rate remarkably. The improvements of polysilicon deposition and annealing processes in the 1990s, however, have further reduced the residual stress, pushing the impact on the spring rate down to the percent level. The impact of residual stress can be mitigated by structural measures. For instance, they can aim at avoiding reaction forces at the second end of a spring construction. So stress relief is facilitated by using folded beams instead of fixed guided ones.

3.1.5

Torsion springs Cylindrical torsion bars The simplest torsional flexure would be a full cylinder similar to the hollow cylinder used for the shear test described in Chapter 2, in the section on “Shear stresses,” Fig. 2.2(a). It can be assumed that, under an applied external moment, the cross-sectional planes of the bar remain planar and are only rotated with respect to each other, creating the constant shear angle γ. Thus, no bowing exists. As illustrated in Fig. 3.8, the torsion angle θ along the bar slice with radius r changes by dθ according to the relation γ dx = r dθ

⇒

γ=r

dθ . dx

(3.56)

The shear stress is subject to Hooke’s law, σt = Gγ, and, since it is constant over the cross-section, it generates the moment Z Z dθ rGr dS = GIP θ′ . rσt dS ⇒ Mt = Mt = (3.57) dx S S Here IP is the polar geometric moment of inertia, Z r2 dS = Iy y + Iz z , IP = S

(3.58)

3.1 Springs

97

and GIP denotes the torsional stiffness of the cylindrical bar. The polar moment of a full cylinder is π IPcyl = R4 2 and that of an annular cylinder is π IPannul = (R24 − R14 ), (3.59) 2 where R2,1 are the outer and inner radii, respectively. One obtains the torsion angle of the bar with length L by integration over x, Mt =

GIP θ, L

(3.60)

which leads to a torsional spring rate ∂Mt GIP = . (3.61) ∂θ L The cylindrical bar presented here is a nice example for introducing torsional flexures but is of little interest for implementation in MEMS technologies. Technologically, cylindrical cross-sections are possible only within the plane of a wafer and, thus, have lengths that are too short in the out-of-wafer-plane direction. Therefore, most torsional springs that are designed for twisting about in-plane axes have a rectangular or near-rectangular cross-section. In addition, the presented results on the cylindrical bar are based on the assumption that there is no warping of cross-sectional planes. In the next section this assumption will be weakened and substituted by presupposing constant warping along the beam according to Saint-Venant’s theory. Only full cylinders remain without bulging, and IPcyl delivers correct results. For hollow cylinders an improved relation will be given in the section on “Cylindrical bars.” kθ =

Torsion bars with arbitrary cross-section The cross-sectional planes in torsion bars without central symmetry undergo bowing. Ideally, the bow is identical for any cross-section. However, if one or two ends of the flexure are clamped, the bulge is disturbed or eliminated at the clamped ends. In this case the full three-dimensional deformation has to be found, by solving the exact equations of elasticity. For long flexures, disablement of the end bulge can be neglected, assuming uniform torsion according to the theory of Saint-Venant. Thus, the introduction of an external moment does not disturb the bulge shape. Consequently, the twisting angle γ is constant, which leads to a torsion angle that increases in proportion to x, dθ(x) x = θ′ x. (3.62) dx The analysis is based on consideration of warped cross-sections that are independent of x, and basically follows Wittenburg and Pestel [2001]. θ(x) =

98

Non-inertial forces

(a)

(b)

Figure 3.9 A segmental cut through a torsion bar and a corresponding slice.

After torsion, the point P(x, y, z) is shifted to P(x + u, y + ν, z + w). This is shown in Fig. 3.9(a). Taking into account the similarity of the triangle 0– A–P(x, y, z) to the triangle P(x, y, z)–B–P(x + u, y + ν, z + w) constituted by the deflections w(x, y, z), −ν(x, y, z), and rθ(x), displacements w and ν can be related to (small) torsion angles θ by the equations −ν z = , rθ(x) r

w y = , rθ(x) r

(3.63)

which, by comparison with Eq. (3.62), yields ν = −θ′ xz,

w = θ′ xy.

(3.64)

Now, shear strains εxy and εxz in the y–z-plane can be determined and related to shear stresses σxy and σxz according to Hooke’s law: µ ¶ µ ¶ 1 ∂ν ∂u 1 ∂u σxy ′ εxy = + = −θ z + = , 2 ∂x ∂y 2 ∂y 2G (3.65) µ ¶ µ ¶ 1 ∂w ∂u 1 ∂u σxz ′ εxz = + = θy+ = . 2 ∂x ∂z 2 ∂z 2G The dislocation u(y, z) represents the cross-sectional bulge. It changes the torsional stiffness in comparison with that of a non-warped bar. For further calculation the axial deformation u has to be either determined or excluded. To do so, a natural equilibrium condition for shear stress along an arbitrary smooth curve that cuts the cross-section is used. In Fig. 3.9(b) the shear stress σxn denotes stress orientated perpendicular to the cutting curve L(l) which cuts free the lower part of the dx-bar slice. Since the cut-free part is in equilibrium, the stress σxn along the x-axis is equal to the shear stress in the x-plane as indicated in Fig. 3.9 (no resulting moment). The total force in the x-direction must be zero: µZ C ¶ Z C σxn (l)dl = 0. (3.66) σxn (l)dl dx = 0, ⇒ A

A

3.1 Springs

99

The integral from point A to point P, shown in Fig. 3.9(b), represents the socalled stress function. More precisely, the function Φ, defined as Z P(y ,z ) 1 Φ=− σxn (l)dl, (3.67) 2Gθ′ A is the stress function which fulfills the following Poisson equation: ∂2 Φ ∂2 Φ + =1 ∂y 2 ∂z 2

(3.68)

Φ(boundary) = 0.

(3.69)

with boundary condition

Indeed, it first has to be noted that Φ depends not on the form of the cutting curve L(l), in particular on the starting point A, but only on the coordinates of the final point P(y, z). This follows from the fact that a substitution of curve L(l)A→P by L(l)B→P does not change the balancing integral over L(l)P→C : Z P(y ,z ) Z C Z P(y ,z ) σxn (l)dl. (3.70) σxn (l)dl = σxn (l)dl = − P(y ,z )

A

B

Secondly, it is obvious by definition that Φ(boundary) = 0, where the boundary is defined by arbitrary endpoints A and C. Thirdly, differentiating Eq. (3.67) with respect to the curve parameter l gives ∂Φ 1 =− σxn (l) dl 2Gθ′

⇒

∂Φ 1 = σxz , dy 2Gθ′

−

∂Φ 1 = σxy . (3.71) dz 2Gθ′

Here the direction of the normal of a curve extending along the positive ydirection is downwards so that σxn = −σxz . For a curve extending along the positive z-direction, the normal is pointed in the positive y-direction so that σxn = σxy . If now Eqs. (3.71) and (3.65) (the last right-hand equations) are combined, one gets ∂u ∂Φ = −2θ′ , ∂y ∂z ∂u ∂Φ θ′ y + = 2θ′ . ∂z ∂y

−θ′ z +

(3.72) (3.73)

Differentiation of the first equation with respect to z and of the second with respect to y, and subtraction of one of the results from the other immediately leads to the Poisson equation (3.68). Now the interrelation between shear stresses and the applied moment Mt must be determined. If Eq. (3.71) is taken into consideration, the balance of moments can be presented as follows: ¶ Z Z µ ∂Φ ∂Φ y Mt = (yσxz − zσxy )dS = 2Gθ′ +z dS. (3.74) ∂y ∂z S S

100

Non-inertial forces

It is easy to show by partial integration with boundary conditions Φ(boundary) = 0 that Z Z Z ∂Φ ∂Φ y z dS = (3.75) dS = − Φ dS. ∂y ∂z S S S Substitution into Eq. (3.74) leads to the final equation for torsion of flexures with arbitrary but constant cross-sections in Saint-Venant’s approximation Z Mt θ′ = (3.76) , It = −4 Φ dS. GIt S It is called the areal moment and plays the same role as the polar moment for annular bars. For a bar of length L the torsion spring rate is then kθ =

∂Mt GIt = . ∂θ L

(3.77)

Rectangular bars The Poisson equation (3.68) with homogeneous boundary conditions can be analytically solved for a rectangular bar with height h and width b (b ≤ h), using a known Green function G(y, z, η, ζ) Z b Z h dζ G(y, z, η, ζ) dη Φ=− 0

0

with G(y, z, η, ζ) =

∞ 2 X sin(pn y) sin(pn η) Hn (z, ζ), b n = 1 pn sinh(pn h)

where Hn (z, ζ) =

( sinh(pn ζ)sinh[pn (h − z)]

sinh(pn z)sinh[pn (h − ζ)]

for for

0 ≤ ζ < z < h,

0 ≤ z < ζ ≤ h,

with pn = πn/b. The integration is straightforward and must be completed by a second integration in order to obtain the areal moment It . Assuming b ≤ h the final result is " # ∞ hb3 192 b X 1 − cosh[(2n + 1)πb/h] It = 1+ 5 3 π h 0 (2n + 1)5 sinh[(2n + 1)πb/h] " # µ ¶5 hb3 b b = 1 − 0.63 + 0.052 + ··· . (3.78) 3 h h For b ≥ h the rectangle edges should be renamed. If the applied moment is skewed with respect to the bar axis, it has to be projected onto the three axes of the coordinate system. The moments which are orthogonal to the bar axis create a slope of the beam tip, which can be calculated according to Section 3.1.4.

3.1 Springs

101

Cylindrical bars In the case of hollow cylindrical bars with inner and outer radii R1 and R2 the Poisson equation (3.68) can be presented more conveniently in cylindrical coordinates, accounting for the fact that axial symmetry entails the condition ∂Φ/∂θ = 0: µ ¶ 1 ∂ ∂Φ r = 1. (3.79) r ∂r ∂r The solution pertaining to zero-boundary conditions takes the form · µ ¶ ¸ R22 r2 R12 ln(r/R2 ) Φ(r) = −1+ 1− 2 . 4 R22 R2 ln(R1 /R2 ) RR The areal moment Itcyl = R 12 2πrΦ dr of an annular bar is therefore · ¸ π (R22 − R12 )2 4 4 Itcyl = R2 − R1 + 2 ln(R1 /R2 )

(3.80)

(3.81)

and will be used later for the calculation of squeeze damping forces in Section 3.2.3. If the thickness of the cylinder wall is small, (R2 − R1 )/R2 = t/R2 ≪ 1, the areal moment becomes Itcyl = 2πtR23 , and substitution into (3.77) shows that the result fits with the initial defining Eqs. (2.9) and (2.10) in Chapter 2. A full cylinder has the areal moment Itcyl full = (π/2)R24 .

3.1.6

Stress concentration The maximal stress calculated for bent beams or torsional bars does not reflect the worst-case stress conditions which usually emerge at the corners of clamping supports. However, such calculations are useful in order to estimate the operational stress to be measured by piezoresistors or piezoelectric sensors. Furthermore, they give a first orientation for estimating worst-case values. The maximal stress in a rectangular beam follows from Eq. (3.14) on setting zm ax = h/2: σb,m ax =

Mby Mby zm ax = 6 2 . Iy bh

(3.82)

The maximal shear stress for a rectangular torsion bar has to be determined using Eq. (3.71): ¯ ¯ ¯ ¯¸ · ¯ ¯ ¯ ¯ ′ ∂Φ ¯ ′ ∂Φ ¯ ¯ ¯ σt,m ax = max max ¯2Gθ , max ¯2Gθ . (3.83) dy ¯ dz ¯

It can be shown that

· ¸ 0.65(t/h)3 hMt σt,m ax = 1 − 1 + (t/h)3 It

(3.84)

102

Non-inertial forces

(a)

(b)

Figure 3.10 (a) Stress relief by fillets. (b) Stress relief by forking.

At clamped ends of beams the real maximum stress values can reach multiples of values calculated for homogeneous members. Over the lifetime of a sensor, either fractures or, in cases of overload, complete immediate breaks may occur. In practical MEMS processes, the sharp corners at the clamped ends become slightly smoothed out by fillets even without additional design measures, since these fillets arise naturally during the etch process. A typical natural shape is shown in Fig. 3.10(a). The curvature of fillets can be purposely decreased in order to further reduce stress concentration at the corners. For critically stressed beams, shapes formed in accord with archetypes of natural evolution, such as forks or multiple forks, should be used (see Fig. 3.10(b)). The exact stress loads and spring constants for such complicated elastic members must be determined by FEM simulations.

Buckling Inertial MEMS usually are based on mature technologies with low residual stress and well-matched temperature behavior of different substructures. However, situations with high loads in the lateral direction of beams may occur. The result of axial loads may be not only reduced spring stiffness but also buckling. This is an indication of high stress concentration and should be avoided at all cost.5 The theory of buckling is related to the nonlinear analysis of deformation and presented for instance in Timoschenko and Gere [1961] and Fang and Wickert [1994]. To check the onset of buckling for a fixed guided beam the following equation, which has already been mentioned in Section 3.1.4, can be used: π 2 EI (3.85) 4 L2 where for I the smaller of the two area moments should be taken. As indicated in Section 3.1.4 the limit at which buckling takes place is four times larger than the value at which the onset of buckling occurs. Fx,onset =

5

Owing to fatigue risk, applications of buckling beams, for instance, in the form of mechanical switches are not considered in this book.

3.1 Springs

(a)

103

(b) (c)

Figure 3.11 (a) Serial, (b) anti-parallel, and (c) parallel connection of springs.

For a doubly clamped beam (bridge) it can be calculated by solving Eq. (3.49) for a point load using an eigenfunction expansion. The Euler buckling limit for a bridge is four times that for a fixed guided beam: Fx,buckling ≤ 4

3.1.7

π2 EI. L2

(3.86)

Suspensions In inertial MEMS the suspension usually should constrain the motion to rectilinear directions and, thus, can be created by straight beams. The suspension may be formed by connecting various springs with each other and/or with the rigid subject of suspension. Such spring combinations allow one to increase compliance in desired directions and stiffness for all other motions.

Parallel and serial spring connections Basic spring combinations are serial and parallel connections as illustrated in Fig. 3.11. According to Fig. 3.11(a), in a serial spring connection the forces acting on the two springs are equal, F1 = F2 = F . Consequently, the partial displacements of both springs add up to ∆yΣ = ∆y1 + ∆y2 =

F F + k1 k2

(3.87)

and the resulting spring rate for serial spring connection is given by 1 1 1 = + . kΣ k1 k2

(3.88)

For parallel and anti-parallel connections the two deflections are equal, and the spring forces add up to F = F1 + F2 = k1 ∆yΣ + k2 ∆yΣ ,

(3.89)

kΣ = k1 + k2

(3.90)

resulting in

for parallel spring connection.

104

Non-inertial forces

Figure 3.12 Different types of rectilinear beam chains: from folded beam to serpentine.

Beam chains Rectilinear connections of beams are used in order to implement comparable compliances in two directions or to mitigate the impact of compressive stress and axial loads. The most common types of beam connections are shown in Fig. 3.12. The exact calculation of spring constants requires the consideration of force/torque propagation through the chain. A more convenient approach is based on Castigliano’s theorems [Lobontiu and Garcia 2004, Wittenburg and Pestel 2001], which state that ∂UF = ∆i ∂Fi

(first theorem),

∂U∆ = Fi ∂∆i

(second theorem),

(3.91)

where UF is the elastic energy expressed as a function of the generalized forces, Fi , and U∆ is the same energy expressed as a function of the generalized deflections, ∆i . In particular, the second theorem allows one to calculate the forces and moments Fi at the connecting points of a beam chain. Since the calculations should be performed by a formula-manipulation program and lead to long expressions, the results are cited only for a folded beam and a crab-leg structure.

Folded beam For the estimation of the main stiffness elements the folded beam according to Fig. 3.12 is considered simply as a serial connection of three cantilever beams. If the x-axis is along the anchored beam axis and the z-axis is the in-plane axis, the spring rate kz ,FB is given by 1 kz ,FB

=

L31 L2 L33 + + , 3EIy 1 ES2 3EIy ,3

(3.92)

whereby the axial stiffness of the linking member with length L2 has been added. In many practical designs the intermediate beam is made very short and stiff, and can be neglected. If both beams have the same length, L1 = L3 = L, and identical cross-sections, the total spring rate kz ,FB = 23 EIz /L3 is eight times smaller than that of a guided beam with the same length. In order to get the same spring rate, two springs each, with length 2L, have to be used. Nevertheless, folded beams are preferable if residual stress or temperature-dependent displacements of load

3.1 Springs

105

points (axial stress) give rise to concern. A folded beam mitigates such stress considerably because the beam ends are free to move. Folded beams have also a smaller nonlinearity than fixed guided or doubly clamped beams. The reason is the smaller stress concentration. Therefore, tolerable deflections are significantly smaller in fixed guided beams than in folded beams. If the folded beam features different lengths L1 and L3 , cross-coupling between the two in-plane axes (x and z; y is the out-of-plane axis) emerges as a result of the transfer of torques at the connecting points of the sub-beams. For L1 ≃ L3 the off-diagonal stiffness becomes (derived as a special case of U-shaped springs from Iyer [2003]) kxz ,FB = 3E

Iy ,2 (L1 − L3 ) L2 L31

⇒

kxz ,FB ∼ Iy ,2 L1 − L3 . =2 kz ,FB Iy ,1 L2

(3.93)

Crab-leg spring Crab-leg springs can also reduce the maximal stress and buckling tendency by virtue of the added “thigh” fraction. Iyer [2003] gives the following expression for the cross-coupling stiffness: kxz ,CL = 9E

Iy ,1 Iy ,2 . L1 L2 (L1 Iy ,2 + L2 Iy ,1 )

(3.94)

The basic stiffness values are, according to Fedder [1994], kz ,CL = 3E

Iy ,1 4L1 Iy ,2 + L2 Iy ,1 , L31 L1 Iy ,2 + L2 Iy ,1

kx,CL = 3E

Iy ,2 L1 Iy ,2 + 4L2 Iy ,1 . (3.95) L32 L1 Iy ,2 + L2 Iy ,1

As may be expected, the in-plane cross-coupling is incomparably higher then for folded springs, for which it can be brought to zero.

Plate suspension Plate or disk suspensions have to facilitate linear or rotatory motions in one (accelerometers with one DOF) or two directions (gyroscopes or gyroscope members).

Linear suspensions Plate suspensions supporting linear motions are usually designed as combinations of four beams or four beam-chains as shown in Fig. 3.13. The beams are connected individually to the plate and anchored to the substrate at four points. The anchors may be located outside or inside the plate (frame). The springs may extend along the plate to save footprint area rather than diverging as in the figure. Since the springs are connected in parallel, the resulting main stiffness values of the suspension are four times larger then the individual spring rates, provided that identical springs are used, kx(y )(z ),susp ension = 4kx(y )(z ),spring .

(3.96)

106

Non-inertial forces

Figure 3.13 Standard suspensions for supporting orthogonal linear motions.

Small differences in the main stiffness values and asymmetries in the individual springs lead to different deflections at the plate-suspension points. The suspension deviates from a parallel-spring connection, which makes an exact analysis very difficult. The main characteristics of the different suspension types are as follows.

r The simplest four-beam suspension according to Fig. 3.13(a) consists of four fixed guided beams. It is susceptible to residual stress or external axial loads that may lead to buckling. Since the lateral endpoints have a fixed distance, it features also an increased nonlinearity. With growing deflections there emerges a nonlinearly increasing axial load that requires increasing additional deflection forces. Thus, this kind of suspension is not suited for large deflections as are necessary, for instance, for driving a gyroscope. r An excellent countermeasure is the use of folded beams as illustrated in Fig. 3.13(b). Owing to the increased overall length and the possibility of shifting freely one of the beam ends, it provides good linearity, albeit at the price of increased sensitivity to accelerations in the out-of-plane direction. For large deflections the pendulous effect increases the compliance in the axial direction. Using doubly folded beams avoids this effect. The U-shaped suspension is basically a folded beam suspension with an elongated middle fraction of the folded beam. It has similar properties to the foldedbeam suspension.

r The crab-leg suspension used for a one-DOF system (with short “thigh”) may tolerate residual stress, but the main advantage is a better symmetry of the overall construction, because cross-couplings in one direction are to some extent absorbed by the compliance in the orthogonal direction. A crab-leg

3.1 Springs

107

Figure 3.14 Typical suspensions for rotating plates or disks.

suspension may be designed to support comparable sensitivity in the x- and y-directions (two-DOF movement). In this case an improved symmetrization may be achieved by connecting the break points by relatively stiff bars as shown in Fig. 3.13(e), forming an H-type suspension. r The five-beam serpentine suspension (Fig. 3.13(d)) exists in two versions: the intrinsically serpentine suspension and the hairpin suspension. In a hairpin suspension the serpentine length LS is considerably smaller than the outer length of the spring chain members and often located at one of their ends. Owing to small asymmetries any motion in one direction in crab-leg and serpentine suspensions causes deflections of the orthogonal beams that constitute an additional potential root cause of increased cross-coupling. This is why folded beams or U-shaped suspensions are preferred in mode-couplingsensitive applications, such as gyroscopes.

Torsional suspensions For rotating plates the basic design consists of one- or two-anchor constructions as illustrated by Fig. 3.14. Inner suspensions as in Figs. 3.14(a) and (c) are one-anchor suspensions. Outer suspensions require two anchors. In reality anchor areas for inner suspensions are often divided into sub-anchors that are symmetrically arranged around the central anchor point. Torsional springs in inertial MEMS are typically shorter then linear beams. Therefore, according to Eq. (3.86) residual stress is more critical. For stress release the torsion springs should include compensating members, for instance, formed as tridents without the middle jag or as inserted loops as illustrated in Figs. 3.14(a) and (b). Torsional suspensions supporting rotations about the out-of-plane axis as in Fig. 3.14(c) are usually made from two or more bending beams. If the dimension of the anchor can be neglected, i.e. the beam lengths L are equal to the radius of rotation of the load points, L = ri , the torsional spring rate can be calculated as parallel connection of N beams with loads Mθ Z /ri at the free end: kθ = 3EIy /(N L). The real beams attached to an anchor with non-negligible dimensions are shorter: L < ri . A rotation of the outer load points causes not only a bending force but also axial stresses, and changes the boundary conditions at the loaded end.

108

Non-inertial forces

A suspension as in Fig. 3.14(a) is compliant with respect to torsion about the beam axis and about the out-of-plane axis. Correspondingly, a suspension with four beams linked in two orthogonal pairs can be used to design a given compliance about all three rotation axes. The design of an optimized suspension for a given application is a challenging task, especially for gyroscopes. It usually requires a careful refinement and validation of the analytical top design by extensive FEM simulations.

3.2

Damping forces

3.2.1

Fluid-flow models Damping forces acting on oscillating masses determine the quality factor of spring–mass systems according to Q = 2π

Utot . Udiss

(3.97)

Utot is the total system energy and Udiss is the average energy loss during one period at the resonance frequency. The energy dissipation is the combination of intrinsic energy losses Uint within the structure and the loss caused by interaction of surfaces with surrounding fluid Us : Q = 2π

Utot 1 = . Uint + Us 1/Qint + 1/Qs

(3.98)

It is well known that, as the device shrinks, surface phenomena become more and more important in comparison with volumetric effects, because the volumeto-surface ratio scales down with linear dimension. In MEMS devices the impact of surface effects such as damping increases accordingly. Moreover, since internal energy losses are usually related to friction in the most stressed device parts and can be reduced by proper design geometry, the internal losses often become small or negligible, particularly at not-too-low pressures. They can be determined by Q-factor measurements in vacuum. In general, the determination of damping forces in MEMS requires analysis of the interaction between a moving body and the surrounding fluid – usually air or an inert gas at various pressure levels. To some extent, the fluid flow around an object follows the object’s movement due to normal (pressure) and tangential (shear) forces acting on the fluid. Forces at the boundary are translated into volumes of fluid creating a velocity field. Pressure and density may change locally. Inner friction of the gas creates temperature changes, which will be transferred to the surrounding walls. A complete description of all effects requires that the velocity field of flow, together with its pressure and density as well as its temperature, must be analyzed. Even if tiny thermal effects are neglected, however, such an analysis may be quite complicated due to the numerous different interacting

3.2 Damping forces

109

Figure 3.15 Damping forces acting on a moving comb.

geometries and different surface-to-fluid interactions as well as fluid-flow models. For instance, bodies in continuum flow can steadily move through fluid far away from other objects and experience drag forces. Or, the body can interact with fluid that is bound by another, fixed surface. If the moving and fixed surfaces are parallel to the direction of motion, sliding forces occur. In a different scenario, two parallel surfaces move towards each other and squeeze out the gas between them. As a result, squeeze damping takes place. In Fig. 3.15 some of the different situations are illustrated. If the comb is moving in the x-direction, the indicated slide-film forces occur at all surfaces parallel to the x-axis and orthogonal to the z-axis. Drag forces occur at the front and back surfaces, which open and close the encasing flow, and mix with slide forces along the surfaces. If the comb is moving in the y-direction, squeeze-film forces arise between opposite fingers of moving and fixed combs. It is clear that substitution of the entire damping process by superposition of different damping mechanisms neglects complicated corner effects and leads to approximate solutions only. When this is not sufficient, numerical simulations of the Navier–Stokes equations should refine qualitative results. The classes of interaction models specified above must be adapted accordingly for non-continuum fluids.

Fluid flows and Knudsen number A careful differentiation of the nature of the fluid flow is necessary. Different regimes can be identified according to the nature of the gas–particle interaction. This nature is determined by the ratio between the mean free path λ of gas particles and a characteristic device length LK (including distances of interest to other surfaces), which is known as the Knudsen number Kn: Kn =

λ . LK

(3.99)

110

Non-inertial forces

It is important to recognize that there may be different Knudsen numbers for one and the same device depending on what fluid flow is modeled. If, for instance, the flow in a narrow gap between a moving plate and a substrate is of interest, the characteristic length is the gap height. If the flow on top of the plate is analyzed, the characteristic length is the distance between the plate and the top of the cavity (the upper gap). If λ ≪ LK ⇒ Kn ≪ 1, the flow can be considered as a continuum where the molecular nature of the fluid can be neglected. This regime is well described by the Navier–Stokes equation mentioned earlier. Vice versa, if λ ≫ LK ⇒ Kn ≫ 1, then collisions of gas particles with each other and with device walls determine the nature of the fluid. Here the general Boltzmann transport equation must be analyzed. The mean free path of molecules in air under standard conditions (atmospheric pressure P0 ) is about λ0 = 65 nm. For minimal characteristic dimensions of 2 µm this results in a Knudsen number of 0.033, where the continuum approximation holds. However, the mean free path is inversely proportional to pressure, λ = λ0

P0 . P

(3.100)

Since gyroscopes and resonators are often driven under low-pressure conditions, down to 0.1 mbar and below, Knudsen numbers of some hundreds may be encountered, indicating the regime of free molecular flow. Transition zones exist between the regimes of continuum and free molecular flow. They are characterized by an intermediate gas status where features of continuum behavior are present and must be completed by free molecular idiosyncrasies. The choice of the correct flow model is not trivial. To select a correct approach, the flow models are grouped into four regions [Beskok 2002, Bird 1996], according to the corresponding Knudsen numbers:

r r r r

Kn ≤ 0.01, 0.01 ≤ Kn ≤ 0.1, 0.1 ≤ Kn ≤ 10, 1 ≤ Kn,

continuum flow with no-slip boundary condition;6 continuum flow with slip boundary condition; transitional flow; free molecular flow.

Flow analysis in any of these regions requires further constraints. In inertial MEMS two classes of movement can be considered in order to simplify the models:

r slow motion like in accelerometers; and r fast harmonic oscillation like in vibratory gyroscopes or resonators. The general case of arbitrary motions with a broad spectrum is relevant, for instance, in cases of slowly or harmonically driven objects under the impact 6

No relative movement between the surface and adjacent fluid at the boundary.

3.2 Damping forces

(a)

111

(b)

Figure 3.16 (a) Viscosity and Couette flow. (b) Stokes flow.

of shock and vibration. The models usually are based on convolution of an impulse transfer function and an input excitation, assuming linearity of the damping equation. Nonlinear extensions, which are necessary for shock-induced large amplitudes of motion, are complicated. So far, they have attracted little attention in the literature and, thus, still represent a challenge for further investigations.

Continuous viscous flow A continuum flow is regarded as a collection of fluid volume elements that interact by shear and normal forces. To introduce basic properties of fluid a twodimensional flow model wherein a fluid is located between two infinite parallel plates (see Fig. 3.16(a)) is used. The upper plate is moving with velocity vxP ; the lower one is at rest. Friction between wall and fluid generates a (steady) shear force σz x that forces the fluid to flow. The fluid can be imagined as layers, which are moving with different velocities; the topmost layer has the highest velocity, equal to the plate velocity (vx (0) = vxP – no slip), while the lowermost remains at rest (vx (d) = 0). This corresponds to a changing strain with a strain rate −vx (0)/(2d). If the distance d between plates is small, the steady velocity distribution of layers is linear, ³ z´ vx (z) = vx (0) 1 − . (3.101) d

The ratio between the velocity change perpendicular to the flow, vx (0)/d, and the shear force is a material property called viscosity and is denoted by µ, σz x = −µ

vx (0) . d

(3.102)

Since the shear force of fluid must oppose plate motion, the sign is negative in order to have a positive viscosity constant. On decreasing the distance d one obtains the general expression for the shear stress of an infinitesimal thin layer, σz x = µ

∂vx (z) . ∂z

(3.103)

112

Non-inertial forces

For a large class of fluids – called Newtonian fluids – the viscosity µ (dimension: Pa · s)7 is constant. Non-Newtonian fluids like polymers, blood, starch and so on exhibit a nonlinear dependency between shear stress and strain rate ∂vx (z)/∂z.

Viscosity of gases The viscosity of gases can be estimated using the common formula µ=

1 ̺hviλ, 3

(3.104)

where ̺ is the density, hvi the average velocity of molecules, and λ the mean free path. It is obvious that the viscosity is proportional to density. The proportionality to velocity and free path can be understood as a result of interaction between molecules of adjacent layers, which are accelerated as they cross the border between layers in the direction of the faster layer and hence retard it; this interaction increases with hvi and λ. Molecules crossing the border in the direction of the slower layer are decelerated, transferring their impulse to the layer and speeding it up. p From the kinetic theory of gases the three relations ̺ = mn, hvi = 8kT /(πm) and λ = 1/(nσm ) follow, where n is the particle density, m the the mass of the molecule, k the Boltzmann constant, T the absolute temperature, and σm = π(2rm )2 the scattering cross-section of the molecule (rm is the radius of molecule). Substitution into Eq. (3.104) yields s 1 8kmT µ= . (3.105) 2 3 πσm Therefore, viscosity depends on temperature, but not on pressure. The inclusion of the temperature dependency of the scattering cross-section can be managed successfully by use of a semi-empirical relation – the Sutherland equation [White 1974]: µ = µ0

1 + TS /T0 T . 1 + TS /T T0

(3.106)

T0 = 273.16 K is room temperature and µ0 the viscosity at T0 . TS is the Sutherland temperature and T the actual temperature. For the most relevant fill gases, N2 , air, argon, and neon, the corresponding values (from Bao [2000], Gudeman et al. [1998], and Thornton and Baker [1962]) are shown in Table 3.2.

Slip at borders Viscosity was introduced on assuming that the velocity of the wall and of gas flow at the border are equal (no slip). This is true for small Knudsen numbers. 7

µ is the dynamic viscosity, in contrast to the kinematic viscosity µ ∗ = µ/̺, which is normalized by the density ̺.

3.2 Damping forces

113

Table 3.2. Viscosity at T0 and TS for N2 and air

−6

µ0 (10 TS K

Pa · s)

N2 16.6 104

Air 17.2 124

Argon 22.2

Neon 30.8

For larger mean free paths a slip between the wall and the adjacent fluid layer appears. In other words, the flow velocity vx (d) of a moving fluid layer is not zero at the border with a stationary wall at z = d. The flow velocity of the boundary layer is proportional to the velocity gradient in the z-direction at the wall and to the mean free path. The velocity gradient is responsible for the intensity of velocity transfer in adjacent layers, particularly between the boundary layer and the surface. For ∂vx (z)/∂z = 0 there is no transfer of wall velocity to fluid or vice versa. Slip means that particles approaching the wall have different velocities than particles leaving it. This behavior can be modeled by the following Maxwellian boundary conditions [Chapman and Cowling 1970, Arkilic et al. 2001, Sattler and Wachutka 2004]: ¯ 2 − σ ∂vx (z) ¯¯ vx (d) = −λ σ ∂z ¯z =d

or at a moving plate

vx (0) = vxP

¯ 2 − σ ∂vx (z) ¯¯ +λ , σ ∂z ¯z = 0

(3.107)

where 0 ≤ σ ≤ 1 is the fraction of diffusely reflected molecules and is called the tangential momentum accommodation coefficient. It represents the average tangential momentum exchange between impinging gas molecules and the solid surface. Experiments have shown that, for smooth silicon, σ is about 0.7 [Arkilic et al. 2001]. For engineering surfaces σ is close to 1, i.e. the particle loses all of its normal momentum. σ = 0 indicates an ideal reflection of particles without momentum transfer.

Effective viscosity Slip of a boundary layer can be accounted for by substituting the viscosity, µ, by a reduced effective viscosity, µeff , keeping the non-slip condition: µeff =

µ . 1 + f (Kn)

(3.108)

f (Kn) must be determined using the introduced slip condition. However, there is an urgent need to further extend the powerful models of continuum flow towards higher Knudsen numbers, even exceeding the value Kn = 0.1 of the slip area and entering into the region of transitional flow of

114

Non-inertial forces

rarefied gases. This is due to the complicated nature of transitional flow models that makes analysis very cumbersome. Different approaches to defining the function f (Kn) for high Knudsen numbers were summarized by the broadly accepted work of Veijola et al. [1995], who presented an empirical approximation of the effective viscosity, which is applicable also in the lower region of transitional flow, µeff =

µ . 1 + 9.658Kn1.159

(3.109)

Boltzmann’s transport equation provides a basis for similar approximations. Simulation results of Boltzmann’s equation can be fitted with the Navier–Stokes model. Such numerical calculations include the slip region and refine the function f (Kn), derived for the non-slip region, for higher Knudsen numbers. Strictly speaking, Eq. (3.109) was deduced in this way for squeeze-film damping and hence should be applied to other flows only with caution. Taking into account the described and other possible extensions, the continuum-flow model, which will be briefly recapitulated in the next section, may serve as a basis for damping analysis up to Knudsen numbers significantly exceeding unity.

Continuous-flow equations The two-dimensional fluid flow between two plates with constant velocity gradient which was described above and used to introduce viscosity is called Couette flow. Assuming that edge effects of the – finitely sized – plates can be neglected, the force acting on the plate and caused by a steady or slowly oscillating flow is A FCouette = −vx (0)µ , d

(3.110)

where A = bL denotes the area of the moving plate. This is one of the rare examples where the damping force can be directly estimated on the basis of strong assumptions on the flow’s nature. In more general cases, especially in the case of non-steady flows, the interrelation of velocity, pressure, and density must be found in order to calculate the stress distribution on the object’s surface.

Generalization of the Couette model Therefore, a generalization of the simple layer model is necessary. First, a strain (or deformation) rate for arbitrary fluid configurations is introduced. In analogy to the shear-strain definition (see Chapter 2) it is given by µ ¶ 1 ∂vi (¯ x) ∂vj (¯ x) ε˙ij = + (3.111) 2 ∂xj ∂xi so that with Eq. (3.103) the shear stress becomes σij = 2µε˙ij .

(3.112)

3.2 Damping forces

115

Comparison with elastic solid objects – where shear stress is generated by shear strain – reveals that in fluids shear strain is replaced by a shear-strain rate, which is responsible for the appearance of shear stress. Normal stress components also depend on the velocity field of fluid. Clearly, a velocity component, which is, for instance, normal to the surface x = constant, will increase the stress by 2µ ∂vx /∂x. A further stress component is caused by velocity sources or sinks at a given location of fluid. This contribution is related to the continuity equation, which states that the mass M of a fluid with local density ̺(¯ x, t), encased in an initial volume V , is constant even if the volume is changing.8 Z Z I dM d ∂̺ = ̺(¯ x, t)dV = ̺¯ v · dS¯ = 0. (3.113) dV + dt dt V (t) V (t) ∂t S (t) S is the surface of the volume V (t) which moves with the velocity of the flow v¯(¯ x, t). dS¯ is a vector that is normal to the surface and possesses area dS. Applying Gauss’ theorem and taking into account that the resulting volume integral must vanish for any volume, one obtains the continuity equation ∂̺ ∂̺ + div(̺¯ v) = + v¯ · grad ̺ + ̺ div v¯ = 0. (3.114) ∂t ∂t A fluid is incompressible if div v¯ = 0. In general, gases are compressible so that sinks and sources of velocity field emerge, which contribute to normal stresses. In the theory of fluid mechanics it has been shown that this contribution is η div v¯, where η is the so-called bulk viscosity of the fluid, which for gases is related to the shear viscosity according to 2 1 η=− µ ⇒ µ + η = µ. (3.115) 3 3 To summarize, the components of the stress tensor can be expressed as ∂vi σi = σii = −p + 2µ + η div v¯, ∂xi µ ¶ ∂vi ∂vj σij = µ + = 2µε˙ij , (3.116) ∂xj ∂xi 1 p = − (σ11 + σ22 + σ33 ). 3 p denotes the pressure of fluid, defined as the average of the negative normal stresses, directed into a volume element.

The Navier–Stokes equation The density, pressure, and velocity field are connected by the Navier–Stokes equation, which expresses Newton’s law applied to an elementary volume element. Indeed, the principle of linear momentum states that dP /dt = F . The 8

The well-known R relation for R the time derivative H of a volume integral with time-dependent ¯ volume (d/dt) V A¯ dV = V ((d/dt)A)dV + S A¯ · (V¯ · dS) is used.

116

Non-inertial forces

momentum P¯ = m¯ v is related to a fluid enclosed in a small volume V (t); and the forces F are inner forces F¯V like gravity (dF¯V g = g¯̺ dV ) or Coriolis forces, plus stress forces σ dS¯ acting on the surface S of the volume element. σ is the stress tensor, the components of which are given by Eq. (3.116). The volume element is occupied by an elementary portion of fluid and may change with time (Eulerian description of fluid): Z Z Z d ¯ ¯ σ dS. (3.117) d FV + ̺¯ v dV = dt V (t) S V Substitution of the stress tensor according to Eq. (3.116) leads, after some transformations, to the famous Navier–Stokes equation (which was derived in 1827 by Claude Louis Marie Henry Navier, in 1843 by Jean Claude de Saint-Venant, and in 1845 by George Gabriel Stokes): ̺

∂¯ v 1 + (¯ v · ∇)¯ v = −∇p + µ △¯ v + µ ∇(∇ · v¯) + f¯. ∂t 3

(3.118)

f¯ = dF¯V /dV is the force density, describing the impact of gravity or Coriolis forces. For the sake of compactness the nabla operator is used (∇a = grad a; ∇ · ¯ and △ = ∇ · ∇-Laplace operator). A¯ = div A; Equation (3.118) is also called the momentum equation, because it is derived from the principle of linear momentum. The left-hand side represents inertial forces of fluid including the convection term (¯ v · ∇)¯ v . On the right-hand side µ △¯ v + 31 µ ∇(∇ · v¯) are viscous forces. Together with the continuity equation (3.114), the nonlinear vectorial Navier– Stokes equation establishes four relations for five variables ̺, p and vx , vy , vz . A missing fifth relation must be added, which is an equation of state, linking density and pressure at a given temperature. Usually, the pressure–density relation of ideal gases is used: p=̺

R T. M

(3.119)

Here R = 8.314 N m/(K mol) is the universal gas constant and M is the molecular mass in kg. If there are no integrated heaters, fluids in cavities of inertial MEMS can be considered isothermal, because local temperature changes due to frictional, viscous forces or gas compression are very small and are immediately balanced by the heat flow to the environment. In other words, at any given location of the fluid p = c2 ̺,

(3.120)

where c is the isothermal speed of sound. The set of equations (3.114), (3.118), and (3.119) must be complemented by corresponding boundary conditions, which may, for instance, represent a pressure distribution at a given plane, motions of boundary fluid layers at moving walls, etc.

3.2 Damping forces

117

To date no analytical solutions of the general Navier–Stokes equation are known. Moreover, there is no evidence for their existence. However, there exists a huge number of investigations and corresponding literature for special cases and assumptions. One of the first simplifications for inertial MEMS is omission of turbulence. Turbulence may take place for large velocities along large dimensions and for high densities and small viscosity. The Reynolds number Re connects these factors by Re =

̺L|v| , µ

(3.121)

where L is a characteristic dimension and |v| is a characteristic velocity. For Re < 1000 the flow is laminar, following smooth streamlines. In inertial MEMS not only does this condition hold, but usually Re ≪ 1, which indicates that the inertial forces are small to negligible in comparison with viscous forces. For an incompressible flow this leads to the Stokes equation µ △¯ v = ∇p − f¯.

(3.122)

which describes the so-called Stokes or creeping flow. The Stokes equation is particularly suited for analyzing highly viscous flows with low velocities and small lengths. In the next sections some fluid models will be analyzed that are typical for inertial MEMS: slide damping of a plate with small and large distances to surrounding surfaces, drag forces, and squeeze damping between two plates with a small gap between them. The plate motions are assumed to be slow or, if fast, oscillatory. The focus is on qualitative understanding of damping forces rather than on quantitative accuracy, because the effort required for high-accuracy analytical models is huge and usually not justified by the achievement of a deeper insight. Where high accuracy is needed, numerical methods should be used, for instance FEM tools like the 3D Navier–Stokes solver ANSYS-CFX, FLOTRAN etc.

3.2.2

Slide damping Slide and squeeze damping are the two basic damping mechanisms in inertial MEMS. Slide damping arises when an elastically suspended plate is moving parallel to the substrate. It is assumed that the plate with height h maintains a constant gap D to the substrate and performs a linear (see Fig. 3.16) or rotatory motion along the x- or about the z-axis. The motions are governed by the following differential equations: m¨ x + cx x˙ + kx x = Fx (t)

or

Iz θ¨ + cθ θ˙ + kθ = Mθ (t).

(3.123)

118

Non-inertial forces

The resonance frequencies are given by ω02 = kx /m or ω02 = kθ /Iθ , and the quality factors for not-too-large damping are (the index “s” stands for slide damping) Qsx = ω0

m cx

or

Qsθ = ω0

Iθ cθ

(3.124)

(see Chapter 7).

Couette flow for slowly moving plates The objective of this section is to derive an estimate of the dependency of the quality factor on ambient pressure and temperature in the cavity: Q(p, T ). A steady or slowly oscillating Couette flow as in Fig. 3.16(a) is assumed. For fluid between plate and substrate there is no gradient of pressure. External forces are negligible. Furthermore, the fluid is not compressed and can be considered as incompressible (div v¯ = 0). In a first step, motion of the plate in the x-direction is considered, so that the velocity components of the fluid in the y- and z-directions can be neglected (vy ,vz ≪ vx ). Under this assumption the Navier–Stokes equation (3.118) collapses to ∂vx ∂vx µ ∂ 2 vx + vx = . ∂t ∂x ̺ ∂z 2

(3.125)

The equation is still nonlinear and not easy to solve. For the intended rough estimate the dependency of vx (x) on x will be neglected, which causes a larger error the shorter the plate is. The steadiness of flow and the assumed slow changes in time allow one to neglect the term ∂vx /∂t. This results in the following equation: ∂ 2 vx = 0, ∂z 2 which has to be solved with slip boundary conditions (3.107) ¯ 2 − σ ∂vx (z) ¯¯ vx (D) = −λ , σ ∂z ¯z =D ¯ 2 − σ ∂vx (z) ¯¯ vx (0) = vxP + λ . σ ∂z ¯z = 0

(3.126)

(3.127)

vxP is the velocity of the plate. With zero mean free path λ = 0 the non-slip condition is included. The slip condition is used to extend the validity of the flow model to lower pressures (higher Knudsen numbers). Despite the fact that such a model is limited by Kn ≤ 0.1, it will be extended towards more rarefied gases, assuming that the quality factor will be dominated by internal friction of plate suspensions at not-too-low pressures. Consequently, for low pressure the free molecular regime is partially overridden by the mechanical damping. Equation (3.126) with (3.127) can easily be solved: vx (z) = vxP

D + λ′ − z . D + 2λ′

(3.128)

3.2 Damping forces

119

Here the abbreviation λ′ = λ

2−σ σ

(3.129)

is used. Since the shear stress in the x-direction is σz x = µ ∂vx /∂z, the shear force of the boundary layer becomes Fz x = Aσz x |z = 0 = −A

µ vxP , 1 + 2λ′ /D D

(3.130)

where µ/(1 + 2λ′ /D) is the effective viscosity. The proportionality factor between the damping shear force and the velocity is the damping coefficient, or, more precisely, the part of cx which is caused by fluid damping. With (3.124) one finds after substituting the mass m by its volume–density product m = Ah̺P (A is the plate area, h its height, and ̺P its density) µ ¶ hD̺P λ′ Qsx = ω0 1+2 . (3.131) µ D The same formula holds for a rotatory oscillation of a plate with angular velocity ˙ Indeed, the governing equation for fluid velocity v¯ = vr e¯r + vθ e¯θ + vz e¯z vϕP = θ. becomes, with vr , vz ≪ vθ and vθ = vθ (r, z), ∂vθ vθ ∂vθ ∂ 2 vθ + = . ∂t r ∂r ∂z 2

(3.132)

Neglecting again radial dependency and assuming slow motion, the resulting equation is identical with (3.126). The moment caused by a surface element with radius r and corresponding velocity rvθ |z = 0 is then dM = r2 [µ/(1 +R2λ′ /D)](vxP /D)dA and can be expressed by the inertial moment Iz = h̺P A r2 dA, so that the quality factor Qsθ = Qsx . In Fig. 3.17 the dependency of the total quality factor according to Eqs. (3.97) and (3.98) is calculated for the following parameter values: quality factor in vacuum Qi = 200 000; tangential momentum accommodation coefficient σ = 0.7; D = 1.6 µm; h = 11 µm; ω0 = 2π × 2 kHz; and ̺P = 2300 kg/m3 . For air viscosity µ the Sutherland relation (3.106) is used and, finally, the free path λ is calculated according to (3.100). Furthermore, it is assumed that the damping on the top surface of the plate is much smaller than the slide-film damping in the gap. This is true for a large distance between the top of the plate and the upper cavity wall. Regardless of all these limitations, Fig. 3.17 reveals typical properties and orders of magnitude of slide damping in inertial MEMS: at atmospheric pressure quality factors of some tens can be expected; at 1 mbar the quality factor can exceed 1000, and for very low pressures vacuum quality factors on the order of 105 and higher are common. Clearly, the quality factor decreases with increasing temperature.

Non-inertial forces

10

10

Quality factor

120

10

10

6

5

T=20°C

4

3

T= - 40°C T=120°C

10

10

2

1

10

-6

10

-4

10

-2

10

0

P in bar

Figure 3.17 The quality factor for a slowly moving plate at various pressure: Couette

flows.

It should be noted that Eq. (3.131) assumes slow oscillations and quickly becomes invalid for higher frequencies. Therefore, the impact of the inertial forces requires separate analysis.

Stokes flow for rapidly oscillating plates Plates in inertial MEMS are often driven at high frequencies on the order of tens of kHz (vibratory gyroscopes) up to MHz (resonators). In this case, the inertial fluid terms can no longer be neglected, and Eq. (3.125) or (3.132) must be taken into account. Owing to the difficulties of solving such nonlinear equations, the nonlinear term is neglected, assuming a small dependency of flow on the lateral dimension. Thus, the starting point is ∂vx µ ∂ 2 vx = . ∂t ̺ ∂z 2

(3.133)

It can be solved by separation of variables. However, since the focus here is on damping of oscillating plates, vxP (t) = v0P cos(ωt),

(3.134)

the solution is a harmonic function and can be represented as vx (t, z) = 2ℜ[exp(jωt)Zx (z)] with ℜ the real part. Substitution of vx (t, z) into Eq. (3.133) yields an ordinary differential equation for Zx : d 2 Zx ̺ − jω Zx = 0 2 dz µ

(3.135)

with solutions Zx = A exp

³p z ´ ³ p z´ 2j + B exp − 2j . δ δ

(3.136)

121

3.2 Damping forces

The factor 2 is for the sake of representational simplicity, keeping in mind that √ 2j = 1 + j. The parameter δ has the dimension of length, r 2µ δ= , (3.137) ω̺ and characterizes how fast the fluid velocity is changing with growing z. With growing oscillation frequency the characteristic length δ decreases because the gas layers are not able to follow the fast motion of the plate. For estimating the effective Knudsen number, the characteristic length δ has to some extent replaced the geometric distance D because it represents the effective thickness of the fluid film near the boundary of the moving plate. Therefore, the ratio between the mean free path and the effective distance represents the dominant fluid regime: r λ ω̺ ′ K n = = λ 0 P0 . (3.138) δ 2µP 2 Since the gas density is ̺=

P , Rg T

(3.139)

where Rg denotes the individual gas constant (Rg = R/M ; see (3.119)), one finds, for instance, for dry air (Rg = 287.05 N m/(kg K)) at cavity pressure 1 mbar √ and room temperature an effective K ′ n = 3.05 × 10−3 f . For frequencies f = 20 kHz this means K ′ n = 0.136. In order to include such low-pressure conditions, which often arise in vibratory gyroscopes, the slip condition must be applied. Hence, the constants A and B are determined so as to satisfy Eq. (3.127). With the following notation, κ=

λ′ , δ

α = 1 + κ + jκ,

β = 1 − κ − jκ,

z′ =

z , δ

D , δ (3.140)

D′ =

the solution can be written as ½ ¾ −β exp(1 + j)(z ′ − D′ ) + α exp(−(1 + j)(z ′ − D′ )) vx (t, z) = v0P ℜ exp(jωt) . α2 exp((1 + j)D′ ) − β 2 exp(−(1 + j)D′ ) (3.141) After some simple but boring transformations, the force exerted by the boundary layer on the moving plate with surface area A is Fs = 2Aµ ∂vx (t, z)/∂z|z = 0 and can be expressed in the following form: Fs = Fsd cos(ωt) + Fse sin(ωt)

(3.142)

122

Non-inertial forces

with v0P [(1 + 6κ2 ) sinh(2D′ ) + 4κ(1 + κ2 )cosh(2D′ ) + (1 − 6κ2 ) sin(2D′ ) δC + 4κ(1 − κ2 )cos(2D′ )], (3.143) v0P 2 ′ ′ 2 ′ = −µA [−(1 + 2κ )sinh(2D ) − 2κ cosh(2D ) + (1 − 2κ ) sin(2D ) δC + 2κ cos(2D′ )], (3.144)

Fsd = −µA Fed

where C = (1 + 8κ2 + 4κ4 )cosh(2D′ ) + 4κ(1 + 2κ2 )sinh(2D′ ) − (1 − 8κ2 + 4κ4 )cos(2D′ ) + 4κ(1 − 2κ2 )sin(2D′ ).

The component Fsd cos(ωt) is in phase with the velocity of the plate oscillation and is therefore responsible for damping. The second term, Fse sin(ωt), is in phase with the plate motion x(t) and thus represents an elastic force. The damping constant csx is the proportionality factor between Fsd and vP (t): csx =

Fsd mω = . v0P Qsx

(3.145)

Pure viscous flow On inserting the damping constant into Eq. (3.124), one obtains for an oscillating plate in pure viscous flow (κ = 0) Qsx =

h̺P δ cosh(2D/δ) − cos(2D/δ) ω . µ sinh(2D/δ) + sin(2D/δ)

(3.146)

For large characteristic lengths δ ≫ d or small frequencies, the quality factor and damping coefficient can be approximated by Qsx =

h̺P Dω, µ

csx =

µA , D

(3.147)

which agrees with the result for viscous Couette flow according to Eq. (3.131) (λ = 0). From a physical point of view, large δ means that the flow decays slowly and the boundary conditions determine the velocity profile. For the other extremal case of very large frequencies, the characteristic length δ is small, δ ≪ D, and replaces the gap size D, so that the quality factor and damping coefficient become Qsx =

h̺P δω, µ

csx =

µA . δ

(3.148)

In this case, the velocity profile, which is derived from Eq. (3.141), will be expressed by ³ z´ vx (t, z) = v0P exp − cos(ωt). (3.149) δ

3.2 Damping forces

123

Hence, to a good approximation the fluid flow follows the plate oscillation but decays exponentially with distance from the plate. This is the typical behavior of Stokes flow, as indicated in Fig. 3.16(b).

General Stokes flow The results of the previous paragraph are generalized to low values of pressure, where κ > 0. Again, two extremal cases are considered first. For large characteristic lengths δ ≫ D the damping force can be approximated by Fsd = −v0P

A µ A µ = −v0P , ′ D 1 + 2λ /D D 1 + 3.71λ/D

(3.150)

which shows that both the damping force and the effective viscosity are identical to those calculated for Couette flow (see Eq. (3.130)). The expression on the right is related to a tangential accommodation coefficient of σ = 0.7. If the plate is oscillating with high frequencies the characteristic length δ is small, and the movement of a fluid is concentrated near the plate. The damping forces for δ ≪ D can be approximated by A µ δ 1 + 2κ2 (1 + 2κ2 )/[(1 + 2κ)(1 + 2κ + 2κ2 )] A µ ∼ . = −v0P δ 1 + 0.952(λ/δ)1.4

Fsd = −v0P

(3.151)

The correcting term for the effective viscosity depends not on the gap size, but on the effective length δ. For a tangential accommodation coefficient of a silicon surface, σ = 0.7, the formula can be fitted by a simplified expression, which is shown on the lower line of Eq. (3.151). Notably, the effective viscosity here, too, differs significantly from the values which are usually cited for squeeze-film damping (see Eq. (3.109)). For arbitrary characteristic lengths Eqs. (3.142) and (3.143) must be used. For instance, for a plate oscillating at room temperature at 10 kHz, the dependency of the quality factor on pressure is shown in Fig. 3.18 for various gap sizes D and arbitrary δ. It is derived from Eq. (3.143) by using expressions for the mean free path, Eq. (3.100), for λ′ , Eq. (3.129), and for the gas density, Eq. (3.139). At high pressures, the gap size D has a significant impact because the characteristic length at 1 bar is still 21 µm and a condition close to D ≪ δ holds. In the mbar region where the air density is low and the characteristic length is even larger (at 41 mbar it is δ = 670 µm), the result is determined by the slip boundary conditions, and the damping forces for different D equalize. Figure 3.18 represents the case of dry air around a moving plate of height 11 µm and with the mass density of silicon. It can easily be scaled to other plate thicknesses and densities. If, at high pressure, the plate is driven parallel to the substrate with a gap size of, say, 2 µm, and with a distance from its top to the upper cavity wall

124

Non-inertial forces

10

5

Quality factor

Dry air at room temperature f=10kHz 10

4

D= 25 µm

10

D= 5 µm

3

D= 2 µm D= 1 µm 2

10 -4 10

10

-3

-2

10 pressure in bar

10

-1

10

0

Figure 3.18 The quality factor for an oscillating plate at various pressures: Stokes flow.

of 25 µm, the damping force at the top surface according to Fig. 3.18 is at least one order of magnitude smaller than that at the gap-faced surface. Thus, for high pressures, top-surface forces can be neglected. With decreasing pressure the forces equalize, because the fluid flow is governed more and more by the slip conditions which overcompensate the impact of increasing effective length δ. This example demonstrates that the extremal condition with small δ ≪ D is usually not relevant for the behavior of fluid under real conditions. Only for very high frequencies (and pressures, which is unrealistic for MEMS devices) does the condition δ ≪ D apply. Plates in MEMS are often perforated. Perforation holes are needed in order to remove the underlying sacrificial layer via gas-phase etching and, in this way, to release the plate. Perforation holes are arranged in regular grids and may occupy about 25% of the area. An exact analysis of how the fluid within the holes changes the surface forces is complicated. However, for small diameters and high pressure, the co-moving gas columns within the plate do not significantly disturb the flow and can be neglected. The holes should be accounted for by a reduced plate density ̺P in Eq. (3.142). With decreasing pressure the picture becomes very complicated. Some of the analytical approaches can be found in Chen and Kuo [2003] and Martin et al. [2008], although most of the work on perforated plates is related to squeeze-film damping, where the impact of perforation is very much larger.

3.2.3

Squeeze damping Squeeze-film damping occurs when a parallel plate is moving face to face with respect to a substrate or another wall. Motion towards the wall increases the

125

3.2 Damping forces

Figure 3.19 Squeeze-film damping.

pressure and squeezes the gas out of the gap; motion away from the wall decreases the pressure and draws the gas into the gap. Figure 3.19 demonstrates the situation. The drag forces of gas, flowing in or out of the gap, create damping forces at the plate surface. Furthermore, compression or decompression of the gas film also generates elastic forces. The mechanism described operates also with rotating plates oscillating, for instance, about an in-plane axis x or y. The results for parallel plates can be extended to include this case.

Reynolds’ equation A common assumption for deriving the governing equation is a small Reynolds number Re = ̺D|v|/µ ≪ 1, which allows one to neglect inertial forces and to start with the Stokes equation (3.122). Even for plates oscillating with small amplitudes about 0.1 µm and frequencies up to 100 kHz, this assumption is true as long as the gap size is smaller than 2 µm. For larger gaps and amplitudes the condition may be violated even at atmospheric pressure and it must be checked whether it applies. Assuming small Reynolds numbers, the flow can be treated as steady flow. Then, for lateral dimensions much larger than the gap size – b ≫ D, L ≫ D – a pressure difference in the lateral direction leads to a so-called Poiseuille flow with quadratic velocity profile along the vertical z-axis as indicated in Fig. 3.20: vx (z) =

1 ∂p [z(z − D) − λ′ D] , 2µ ∂x

vy (z) =

1 ∂p [z(z − D) − λ′ D] . 2µ ∂y (3.152)

The profile of pressure-driven, steady horizontal Poiseuille flow follows immediately from the Stokes equation adapted for small gaps, µ

∂ 2 vx ∂p − = 0, ∂z 2 ∂x

(3.153)

126

Non-inertial forces

Figure 3.20 Pressure-driven Poiseuille flow.

with slip boundary conditions according to Eq. (3.127) at the plate surfaces z = 0, D. The flow rates qx and qy in the x- and y-directions through a rectangle with height D and widths dx and dy, respectively, are qx =

Z

0

D

vx dz = −

D3 (1 + 6λ′ /D) ∂p , 12µ ∂x

qy = −

D3 (1 + 6λ′ /D) ∂p , (3.154) 12µ ∂y

and the corresponding mass flows are ̺qx dy and ̺qy dx. The balance of mass flow within the considered rectangular column with height D and footprint dx dy requires ∂(̺qx ) ∂(̺qy ) ∂(̺D) + + = 0, ∂x ∂y ∂t

(3.155)

where the first two terms are the differences between entering and leaving flow in the x- and y-directions, respectively, and the last term is the mass change within the volume dx dy D. On substituting the mass flows and assuming isothermal conditions (̺ = (̺0 /p0 )p) one gets the famous Reynolds equation for squeezefilm damping: 12µ ∂(Dp) = D3 ∂t

µ

1+6

λ′ D

¶·

∂ ∂x

µ

p

∂p ∂x

¶

+

∂ ∂y

µ

p

∂p ∂y

¶¸

.

(3.156)

The term (1 + 6λ′ /D) = 1 + f (Kn) accounts for the change of viscosity with increasing Knudsen number. For tangential accommodation coefficient σ = 0.7 the equivalent mean free path λ′ is given by λ′ = [(2 − 0.7)/0.7]λ. Thus, 1 + 6λ′ /D = 1 + 11.1λ/D, which is not so far away from Veijola’s approximation 3.109 (made for σ = 1). Reynolds’ equation usually is derived from the Navier–Stokes equation, assuming small Reynolds numbers; however, the way presented here, which follows ideas of Bao [2000, 2005], seems to be more illustrative.

127

3.2 Damping forces

Linearization of Reynolds’ equation The Reynolds equation is substantially nonlinear. Fortunately, in nearly all MEMS applications the plate motion is limited to small amplitudes D(t) = D0 + ∆D(t),

∆D(t) ≪ D0 .

(3.157)

Consequently, induced pressure variations are small as well: p(t) = pa + ∆p(t),

∆p(t) ≪ pa .

(3.158)

pa stands for the ambient pressure within the working cavity but outside the gap. It makes sense to normalize the dependent variables: ∆D d˜ = , D0

p˜ =

∆p . pa

(3.159)

If, furthermore, the effective viscosity µ′ =

µ 1 + 6λ′ /D(t)

(3.160)

is used, the linearized Reynolds equation becomes ∂ p˜ D02 pa ∂ d˜ = △˜ p − . ∂t 12µ′ ∂t

(3.161)

It has to be solved with the boundary condition p˜ = 0

(3.162)

at the boundary of the plate. Equation (3.161) corresponds to the well-known diffusion equation which describes, for instance, heat flow transfer.

Low-frequency squeeze damping Changes of pressure with time ∂ p˜/∂t become negligible for slow plate motions. ˜˙ This condition is fulfilled ˜ The term “slow” means that |∂ p˜/∂t| ≪ |∂ d/∂t| = |d|. if p˜ ≪ d˜ or ∆p/pa ≪ ∆D/D0 . Since, in this case, in Reynolds’ equation (3.161)

the term ∂ p˜/∂t can be neglected, the following estimate can be performed: D02 D02 ∆p ∂∆D △˜ p∼ = , ′ 12µ 12µ′ l2 pa pa D0 ∂t

(3.163)

since for instance ∂ 2 p/∂x2 ∼ ∆p/l2 . l is the characteristic plate length for which the pressure changes monotonically. For a rectangular plate this is half of the smallest lateral dimension; for a disk it is the radius. The estimate becomes especially illustrative when a harmonic plate excitation is applied, ∆D(t) = ∆0 cos(ωt). Then ∆p 12µ′ l2 ω ∆0 sin(ωt) 12µ′ l2 ω ∆D = = . 2 pa pa D0 D0 pa D02 D0

(3.164)

128

Non-inertial forces

The coefficient σs =

12µ′ l2 ω pa D02

(3.165)

is called the “squeeze number.” It indicates how large pressure variations are in comparison with gap changes, i.e. how strongly the gas film is squeezed. Small squeeze numbers are typical for low frequencies. That is, with decreasing frequencies, the error in skipping the term ∂ p˜/∂t becomes negligible. The linearized Reynolds equation for slow plate excitations △˜ p=

12µ′ ∂ d˜ pa D02 ∂t

(3.166)

is a Poisson equation as used in Section 3.1.5 in order to determine torsional spring constants. The force on the plate (the index “q” stands for squeeze damping) Z Z pa p˜(x, y, t)dS (3.167) ∆p(x, y, t)dS = Fq = S

S

R can be expressed using the areal moment It = −4 S Φ dS from Saint-Venant’s theory of torsional bending. Indeed, the solution of Eq. (3.166) is the scaled solution of the special Poisson equation (3.68) with the same boundary conditions: p=

12µ′ ∂ d˜ Φ. D02 ∂t

By virtue of this analogy the force exerted on the plate is Z 12µ′ ∂ d˜ 3µ′ ∂ d˜ Fq = Φ dS = − 2 It . 2 D0 ∂t D0 ∂t

(3.168)

The damping coefficients are cq =

mω Fq =− . Qq ∆D˙

With Eq. (3.78) a rectangular plate possesses a damping coefficient " # ∞ µ′ 192 b X 1 − cosh[(2n + 1)πb/L] 3 cqrectangle = 3 Lb 1 + 5 D0 π L 0 (2n + 1)5 sinh[(2n + 1)πb/L] " # µ ¶5 µ′ b b = 3 Lb3 1 − 0.63 + 0.052 + · · · , b ≤ L. D0 L L

(3.169)

(3.170)

With Eq. (3.58) a cylindrical plate with radius R features the damping coefficient cqcircle =

3πµ′ 4 R 2D03

(3.171)

3.2 Damping forces

(a)

129

(b)

Figure 3.21 (a) An asymmetric torsion plate. (b) A symmetric torsion plate.

and an annular plate with inner and outer radii R1 and R2 has, according to Eq. (3.81), the coefficient cqannul

" ¡ 2 ¢2 # R2 − R1 3πµ′ 4 4 =− R2 − R1 + . 2D03 ln(R1 /R2 )

(3.172)

Areal moments for a great variety of other plate geometries can be found in handbooks on technical mechanics. Remarkably, in Eq. (3.168) the areal moment It is fluid-independent. Hence, the complete pressure dependency of the quality factor is included in the effective viscosity and therefore exhibits the same qualitative behavior as for Couette flow; the latter was analyzed in the section on “Couette flow for slowly moving plates.” A comparison of damping coefficients for vertical and lateral motions of rectangular plates shows that, according to Eqs. (3.124) and (3.131), the ratio between squeeze damping coefficients and Couette damping is given by µ′squeeze b2 cqrectangle = ′ csx µCouette D02

µ

1 − 0.63

b b5 + 0.052 5 + · · · L L

¶

.

(3.173)

Since typically b ≫ D0 , it is intuitively clear that squeeze damping at low frequencies is orders of magnitude larger than slide damping. It has to be noted that the absolute values of squeeze-damping coefficients are very high. Assuming a quadratic plate with side length 200 µm and a gap of 2 µm, the damping coefficient at atmospheric pressure is cp = 0.0014, which corresponds, for a silicon plate with height 11 µm resonating at 10 kHz, to a Q-factor of 0.5. Such large damping constitutes serious barriers for creating resonating structures. Lower pressure and plate perforation are needed in order to overcome this obstacle.

Torsion plates Torsion plates are often used in inertial MEMS acting as asymmetric accelerometers or as sensing plates in vibratory gyroscopes. Assuming that the origin of the coordinate system resides at the left-hand corner (see Fig. 3.21(a)), the motion of a torsion plate for small angles θ is

130

Non-inertial forces

described by µ ¶ b D(t) = D0 + θ(x − b2 ) = D0 + θ x − + θ ∆b, 2

∆b =

b − b2 . 2

(3.174)

b2 is the distance of the pivotal point to the left corner. The motion is described as a superposition of symmetric rotation plus vertical plate motion θ ∆b. For symmetric motion, b2 = b/2 and ∆b = 0. Even though the plates are no longer parallel, for small torsion angles the linearized Reynolds equation (3.161) still holds. Since for asymmetrically driven accelerometer plates the motion is slow, the term ∂ p˜/∂t can be neglected, and one gets µ ¶ θ˙ b θ˙ D 3 pa ∆˜ p= x− + ∆b, γ = 0 ′ . (3.175) γ 2 γ 12µ Within the plate area 0 ≤ x ≤ b, 0 ≤ y ≤ L, the solution can be presented in light of zero boundary conditions as µ ¶ µ ¶ X X x y p˜ = ck ,l sin kπ sin lπ . (3.176) b L k ,l= 1

If the right-hand side of Eq. (3.175) is periodically continued one gets results in the form ·µ ¶ ¸ · µ ¶ µ ¶¸ θ˙ b θ˙ 4 X 4 ∆b x b x x− + ∆b = sin (2k − 1) π − sin 2kπ γ 2 γ π2 2k − 1 b 2k b k=1 µ ¶ X 1 y × sin (2l − 1) π . (3.177) 2l − 1 L l= 1

Substitution of (3.176) into (3.175) and comparing coefficients at equal sine functions yields ck ,2l = 0, c2k −1,2l−1 = −

c2k ,2l−1 =

˙ 3 4 θb 1 , 4 π γ 2k(2l − 1)∆2k ,2l−1

16 θ˙ ∆b3 1 , π 4 γ (2k − 1)(2l − 1)∆2k −1,2l−1

∆k ,l = (k 2 + l2 β 2 ).

(3.178) (3.179)

β = b/L is the aspect ratio. R The torque exerted on the plate, Mqt = S (x − b2 )pa p˜ dS, can now be calculated. The ratio −Mqt /θ˙ = cqtorsion is the damping coefficient cqtorsion =

12 · 16 ′ Lb4 µ 3 π6 D0 " # X X b − ∆b ∆b × + . (2k)2 (2l − 1)2∆2k ,2l−1 (2k − 1)2 (2l − 1)2∆2k −1,2l−1 k ,l= 1

(3.180)

131

3.2 Damping forces

For L ≫ b (β ≈ 0) the series can be easily summed up, cqtorsion |L ≫b = µ′

Lb4 (b + 14 ∆b), 60D03

(3.181)

and for ∆b = 0 the expression coincides with the results of Veijola et al. [2005]. For the other extremal case, b ≫ L, the damping coefficient changes to µ ¶ L3 b2 15 cqtorsion |b≫L = µ′ b − ∆b . (3.182) 120D03 16 Since the series in (3.179) converges rapidly, for practical calculations with arbitrary dimensions the first three terms are sufficient.

High-frequency squeeze damping The analysis of squeeze damping for fast plate motions requires the solution of the full (linearized) Reynolds equation D02 pa ∂ p˜ △˜ p− = f (t), ′ 12µ ∂t

f (t) =

∂ d˜ . ∂t

(3.183)

If the pressure response to an applied delta-pulse f (t) = f0 δ(t) is known and denoted by Fδ (t), the response to arbitrary f (t) is, by virtue of the principle of R∞ superposition, F (t) = −∞ Fδ (t − τ )f (τ )dτ . Fδ (t) is the transient response and represents the response to a step excitation.9 The initial condition can be derived by integrating Eq. (3.183) over the small time interval [0− , ∆t]: ¶ Z ∆t µ 2 Z ∆t D0 pa ∂ p˜ △˜ p − f0 δ(t)dt = f0 . (3.184) dt = 12µ′ ∂t 0− 0− For ∆t → 0 the left-hand side converges towards −˜ p(0), so that the initial condition is given by p˜(0) = −f0 , which has to be supplemented by the boundary condition p˜(boundary) = 0. Using the principle of separation of variables, the pressure is presented as p˜ = pˆ(x, y)exp(−αt),

(3.185)

where, according to the typical behavior of first-order systems (in the time domain), the time-dependent factor for t > 0 is assumed to take the form exp(−αt). The solution is zero for t < 0. Thus, D02 pa △ˆ p + αˆ p = 0. 12µ′

(3.186)

Equation (3.186) is the Helmholtz equation that usually emerges after separation of variables in a wave equation. Methods of solving it for a long, rectangular plate and then for arbitrary rectangular plates are briefly recalled in the next section. 9

δ(t) = (d/dt)h(t); h is the Heaviside function h(t) = 1 for t > 0 and h(t) = 0 for t < 0.

132

Non-inertial forces

Long, rectangular plate A plate with b ≪ L is considered; see Fig. 3.19. To a good approximation, squeeze damping is determined by the flow parallel to the plate, i.e. in the x-direction. Thus, assuming ∂ 2 pˆ/∂y 2 ≪ ∂ 2 pˆ/∂x2 , Eq. (3.186) reduces to D02 pa ∂ 2 pˆ + αˆ p = 0, 12µ′ ∂x2

(3.187)

which must be solved with boundary conditions pˆ(0) = pˆ(b) = 0 and initial conditions pˆ = −f0 . Apparently, Griffin et al. [1966] were the first to analyze the squeeze-film damping problem of long plates using the approach described. p ′ 2 With pγ = D0 pa /(12µ ), the solution of (3.187) takes the form p A sin( α/γ x) + B cos( α/γ x) and satisfies the boundary conditions with αk /γ b = kπ. The coefficient αk obeying the last equation corresponds to a set of eigenfunctions sin(kπx/b), so that, for t > 0, µr µ ¶ ¶ X X αk x Ak sin p˜ = Ak sin kπ x exp(−αk t) = exp(−αk t), (3.188) γ b k=1

k=1

where

αk = γ

µ

kπ b

¶2

=

D02 pa 12µ′

µ

kπ b

¶2

.

(3.189)

In order to determine the coefficients Ak , the initial condition p˜(t) = −f0 = −d˜0 can be used: µ ¶ X x Ak sin kπ p˜(0) = = −d˜0 (x). (3.190) b k =1

This condition must hold for arbitrary x within the interval [0 ≤ x ≤ b]. Since the initial plate displacement d˜0 (x) is constant within this interval, it can be periodically continued outside, so that 4 X sin(2k − 1)πx/b d˜0 (x) = d˜0 . π 2k − 1 k=1

Comparison with the left-hand side of Eq. (3.190) shows that A2k + 1 = −

4 d˜0 , π 2k − 1

A2k = 0.

The final solution is therefore, with h(t) the Heaviside function, ³ 4 X d˜0 x´ p˜ = − sin (2k − 1) π h(t)exp(−α2k −1 t). π 2k − 1 b

(3.191)

k=1

The force acting on the plate can easily be derived by integration over the whole plate: 8 ∆0 X exp(−α2k −1 t) FqStep (t) = −pa bL 2 h(t) . (3.192) π D0 (2k − 1)2 k=1

3.2 Damping forces

133

The viscous properties of the fluid are hidden in the transient terms exp(−α2k −1 t). The step response FqStep may serve as the basis for a better illustration and for calculating arbitrary R ∞ responses. A convenient way is to work with Laplace transforms F˜qStep = 0 exp(−st)FqStep (t)dt or Fourier transforms (s = jω). 96 F˜qStep (s) = −∆0 µ′ 4 L π

µ

b D0

¶3 X k=1

1 1 . (2k − 1)4 1 + s/α2k −1

(3.193)

The response F˜δ of the plate motion to a delta-impulse (not for f (t) but for d(t)) is sF˜qStep (s). Hence, the response to arbitrary plate motions with Laplace ˜ transform ∆D(s) becomes µ ¶3 X b 1 s ′ 96 ˜ ˜ Fqh (s) = −µ 4 L ∆D(s). (3.194) 4 π D0 (2k − 1) 1 + s/α2k −1 k=1

The index “h” is added in order to denote high-frequency plate changes. A harmonically oscillating plate ∆D(t) = ∆0 cos(ωt) is particularly interesting. The force can be written in the following way: µ ¶3 b ′ 96 Fqh (t) = µ 4 L π D0 µ ¶ X 1 1 ω ×ω ∆0 sin(ωt) − ∆0 cos(ωt) . 2 (2k − 1)4 1 + ω 2 /α2k α2k −1 −1 k=1

(3.195)

It consists of two components: firstly, the damping force proportional to velocity; and secondly, the elastic force proportional to position: ˙ Fqh (t) = cqhd ω∆0 sin(ωt) − kqhe ∆0 cos(ωt) = −cqhd ∆D(t) − cqhe ∆D. (3.196) Since the series converges very fast – the second term is about 34 = 81 times smaller than the first – the damping coefficient cd and elastic spring constant ke are represented in the following forms: µ ¶3 µ ¶3 96 b 1 b 1 ′ ∼ cqhd = µ′ 4 L µ L , (3.197) = π D0 1 + ω 2 /ωc2 D0 1 + ω 2 /ωc2 ¯ ¯ ¯ ω2 4 Lb ¯¯ kqhe = cqd , kqhe ¯¯ = 2 pa , (3.198) ωc π D0 ¯ω =ω c ω =ω c

with

ωc = α1 = π 2

D02 pa pa D02 = 0.8225 . b2 12µ′ µ′ b2

(3.199)

fc = ωc /(2π) is the cut-off frequency at which the damping coefficient cqhd has decreased by a factor of two. At the cut-off frequency, the elastic force becomes

134

Non-inertial forces

equal to the damping force. At significantly larger frequencies the elastic force dominates and the fluid acts as a spring. This corresponds to the following physical scenario: at low frequencies, gas is squeezed out or sucked in, and lateral viscous forces acting on the surface dissipate energy synchronously with the plate velocity; at high frequencies, the fluid film can no longer follow, and gas is compressed or decompressed, generating elastic counterforces. For a typical long plate with width 200 µm and an air gap of 2 µm, the cutoff frequency at atmospheric pressure is 77 kHz. Since the viscosity µ′ increases with decreasing pressure, it decreases with pressure at a faster rate than does the pressure itself. Usually, the elastic force has negligible impact. Indeed, the shift of the resonance frequency of the plate (ω0 + ∆ω)2 = (k + kqhe )/m relative to an undisturbed resonance ω02 = k/m is given by ∆ω 96 ′ b3 1 6 · 96 ′2 b4 1 1 ≈ µL 3 = µ . 2 4 2 ω0 2π D0 mωc (1 + ω0 /ωc ) π6 D04 hD0 ̺P pa 1 + ω02 /ωc2 (3.200) As earlier, the mass is represented by the plate height and its density ̺P . For an 11-µm-thick silicon plate with a gap of 2 µm and width 200 µm the frequency shift at atmospheric pressure is then on the order of 10−4 . For lower pressures, the viscosity drops, compensating partly for the term 1/pa , and the resulting shift remains on the same order. A comparison of damping coefficients for fast vertical movement of plates with corresponding lateral movement reveals that, according to Eqs. (3.124) and (3.151), the ratio between squeeze-damping coefficients and Couette damping is given by µ′squeeze b2 δ cqhd 1 = ′ . csx µCouette D02 D0 1 + ω 2 /ωc2

(3.201)

As can be seen, for high frequencies, especially above the cut-off frequency, the dominance of slide-damping effects over squeeze-damping effects, expressed, for instance, by Eq. (3.173), diminishes. Considering that, for low pressure, the cutoff frequency is small, for practical low-pressure applications of fast oscillating plates it must be expected that squeeze- and slide-damping coefficients become comparable in magnitude.

Rectangular plate In order to drop the limitation of a long plate, the full linearized Reynolds equation (3.161) must be solved, expediently again by separation of variables. This is a simple, but a little bit daunting, procedure. It was performed by Blech [1983]. Rewriting his results in terms of damping and spring coefficients leads to

135

3.2 Damping forces

the following expressions: cqhd,rectangle =

12 · 64 ′ b3 X X k 2 + β 2 l2 µ L , 3 π6 D0 k 2 l2 [(k 2 + β 2 l2 )2 + σ 2 /π 4 ]

(3.202)

k ,l= o dd

kqhe,rectangle =

144 · 64 ′2 2 b5 X X 1 µ ω L 5 , 8 2 2 2 2 π pa D0 k l [(k + β l2 )2 + σ 2 /π 4 ] k ,l= odd

(3.203)

with β=

b ≥ 1, L

σ=ω

12µ′ b2 . pa D02

(3.204)

σ is the squeeze number. The dependencies of both coefficients on frequency are hidden within the squeeze number. Owing to the slower convergence of the series, the cut-off frequency at which the elastic and damping forces equalize (ωcqhd = kqhe ) cannot be determined as easily as for very long plates. However, a very crude approximation can be obtained, accounting only for the first terms of the series expansions: ¡ ¢ ¢ pa D 2 ¡ σc = π 2 1 + β 2 ⇒ ωc = 0.8225 ′ 20 1 + β 2 . µb

(3.205)

For L → ∞ this expression tends to (3.199). The exact values are about 10% higher than those given by the approximate solution [Bao and Yang 2007, Bao 2005].

Circular plate For circular plates the Helmholtz equation (3.186) takes the form µ ¶ D02 pa ∂ ∂ pˆ r + αˆ p = 0, (3.206) 12µ′ ∂r ∂r p which after the coordinate transformation r′ = α/γr becomes the p equation for the Bessel function of zeroth order, generating eigenfunctions J0 ( αk′ /γr). The coefficients αk′ are determined by the boundary conditions for pˆ at r = R: αk′ =

pa D02 2 η . 12µ′ R2 k

(3.207)

ηk are the zeros of the Bessel function of zeroth order (η1 = 2.4048, η2 = 5.5201, η3 = 8.6537, . . .). Using the same methodology as for long plates [Griffin et al. 1966], the transient response in Laplace representation is then 4

R F˜qcirc (s) = −∆0 48πµ′ 3 D0

X

η4 k=1 k

s . (1 + s/αk′ )

(3.208)

Similarly to the case of a long plate, the series converges quite fast and can be approximated by its first term. This leads to the following damping coefficient

136

Non-inertial forces

and cut-off frequency: cqhd,circ = 4.51

µ′ R4 , D03

ωc = 0.482

pa D02 . µ′ R2

(3.209)

Oscillating torsion plate Fast torsional motions in inertial MEMS are usually symmetric oscillations according to Fig. 3.21(b). The tilting angles are very small, in contrast to noninertial MEMS like mirrors. The motion can be described as µ ¶ b D(t) = D0 + θ0 x − cos(ωt), (3.210) 2 or as the real part of the excitation D(t) = D0 + θ0 (x − b/2)exp(jωt). In the linear approximation the pressure is also a harmonic function: p˜ = pˆ exp(jωt). Therefore the linearized Reynolds equation (3.161) is transformed into10 µ ¶ ωD0 ω b D 3 pa ∆ˆ p−j pˆ = jθ0 x− , γ= 0 ′ . (3.211) γ γ 2 12µ The coefficient γ and the more commonly used squeeze number σ are related to each other by σ=

ωD0 b2 12µ′ ωb2 = . γ pa D02

(3.212)

Equation (3.211) can be solved analogously to the case of slow torsion. The resulting complex pressure must be integrated in order to get the torque, which has two components: one proportional to the cosine and correspondingly to the plate motion; and one proportional to the sine, i.e. to the angular velocity of the plate. Pan et al. [1998] carried out this calculation, which leads to the following expression: 192 µ′ ωLb5 X X 1 3 6 2 π D0 (2k) (2l − 1)2 ∆′2k ,2l−1 k ,l= 1 " # ´ σ θ˙ ³ 2 2 2 × θ 2 + , (2k) + β (2l − 1) π ω

Mqt = −

(3.213)

where ¢2 σ 2 ¡ ∆′k ,l = k 2 + β 2 l2 + 4 . π

For ω → 0 this corresponds to Eq. (3.179). 10

As emphasized, the nonlinear damping effects are not really important for inertial MEMS. Readers interested in deeper analysis of possible nonlinear effects should refer to Hao et al. [2002] and Bao et al. [2006]

137

3.2 Damping forces

(a)

(b)

(c)

Figure 3.22 (a) Flow in a perforated plate. (b) Separation in cells. (c) A single cell.

The first term of the quickly converging series may serve for the determination of the cut-off frequency σc = π 2 (4 + β 2 ) or ωc = π 2

¢ pa D02 ¡ 4 + β2 , ′ 2 12µ b

(3.214)

which is significantly larger than that of a vertically oscillating rectangular plate actuator (see Eq. (3.205)). Since the two sides of the plate are moving in opposite directions, this result is not surprising.

The impact of perforation Large lateral structures in inertial MEMS must be released from the underlying sacrificial layer in order for them to become movable. However, etch technologies for surface micromachining are not selective enough to work with pure underetching commencing from the sides. Therefore, etching holes are implemented to attack the sacrificial layer at many points. Sometimes etching holes are implemented purposely to reduce squeeze damping. In any case, the perforation holes are arranged in periodic lattice structures as indicated in Fig. 3.22(b). Etching holes change the behavior of plates. The basic mechanism is illustrated in Fig. 3.22(a). If the plate is moving down, fluid underneath is pushed into the holes and creates a pipe flow. The holes act like a “chimney,” and the pipe flow develops shear stress at the wall. If the plate is very thin, another effect dominates: the pressure at the hole’s entry is equal to the ambient pressure, which leads to the absence of a pressure difference ∆p at the hole’s entry, i.e. the boundary conditions change periodically over the plate. A general analysis of squeeze damping of perforated plates is difficult, and a comprehensive literature exists for different geometries, motion speeds, pressures and, most importantly, approaches to solve the problem (e.g. Bao and Yang [2007], Homentcovschi and Miles [2005], Kwok et al. [2005], Mohite et al. [2008], and Pandey et al. [2007]). Only a few of the results are well prepared for use by engineers; among them, Bao [2005] should be mentioned. For the analysis the plate is approximated by a superposition of cells as shown in Figs. 3.22(b) and (c). Owing to difficulties with rectangular cells, mostly cylindrical cells, arranged in a square or hexagonal lattice, are used.

138

Non-inertial forces

Furthermore, the hole is assumed to be cylindrical in order to simplify the flow description within the cell and hole to a one-dimensional problem. The radius of cylindrical holes is chosen in such a way that it substitutes for rectangular holes with the same area. Between cells, continuous pressure is assumed, with ∂p(r, t)/∂r|r =r c = 0. In this way, the agglutination of cells is guaranteed. The approaches used to model squeeze damping of perforated plates are partially based on additional intuitive approaches, for instance, on the assumption that different contributions from pipe flow in the hole and from flow within the cell to the hole can be added with equal weight etc.

Slow plate movement at atmospheric pressure A powerful approach to describe the impact of perforation on pressure distribution was developed by Bao et al. [2003]. They assumed that the locally different flow can be substituted by an average flow over a great number of cells. The flow through the holes can be considered as flow uniformly penetrating the plate. In this case the continuity equation (3.155) can be extended to ∂(̺qx ) ∂(̺qy ) ∂(̺D(t)) + + ̺Qz + = 0, ∂x ∂y ∂t

(3.215)

where, as before, qx = −

D3 (1 + 6λ′ /D) ∂p , 12µ ∂x

qy = −

D3 (1 + 6λ′ /D) ∂p . 12µ ∂y

(3.216)

Qz is the penetration rate. The penetration rate was calculated by Bao et al., assuming a fully developed Poiseuille flow through holes, and adding to the corresponding damping force the contribution of flow from the border of the cell to the hole. The last contribution can easily be calculated using the model of a cell belonging to a thin plate. In this calculation, no slip conditions were considered, which limits the applicability to low Knudsen numbers or not-too-low pressures. The resulting modified Reynolds equation is ∂ 2 p˜ ∂ 2 p 3βc r02 12µ′ ∂∆D + 2 − p˜ = , 3 2 ′ ∂x ∂y 2D0 h η(βc ) pa D03 ∂t

(3.217)

where βc =

µ

r0 rc

¶2

,

η(βc ) = 1 +

´ p 3r04 ³ 2 4β − β − 4 ln β − 3 . c c c 16hD03

h denotes the plate thickness, as before. The cell size rc should be chosen so as to give best coverage with balanced negative and positive overlap. A hexagonal lattice has a cell size rc = 0.525ds , where ds is the distance between the holes. The introduction of an effective plate thickness h′ = h +

3πr0 8

(3.218)

3.2 Damping forces

139

1

h= D

0

relative damping

0.8

h=2 D

0

b=100 D

0.6

0

h=20 D

0

h=2 D

0

b=50 D

0.4

0

b=20 D0 0.2

0 0

5

10 15 fraction of hole area in %

20

Figure 3.23 The relative damping coefficient of a perforated plate.

allows one to account for the entrance length of the flow when r0 becomes comparable to h and the Poiseuille flow is not fully developed at the pipe ends. In order to get a feeling for the impact of perforation holes, a long, rectangular plate will be considered. This example was also analyzed by Bao et al. [2003]. The governing equation is ∂ 2 p˜ p˜ 12µ′ ∂∆D − = , ∂x2 l2 Pa D03 ∂t

(3.219)

where l2 =

2D03 h′ η(βc ) . 3βr02

On solving as usual with zero boundary conditions, the pressure distribution becomes µ ¶2 · ¸ ˙ 48µb2 ∆D(t) l cosh[2x/(2l)] ∆p(x) = − 1− . (3.220) D03 2b cosh[b/(2l)] ˙ By integrating over the plate area and dividing by the excitation velocity ∆D, one gets the damping coefficient µ ¶2 · µ ¶¸ 3µLb3 2l 2l b cqp erf = 1 − tanh . (3.221) 3 D0 b b 2l The first factor is three times the damping coefficient for non-perforated plates according to Eq. (3.170) (L ≫ b). For large l the ratio approaches unity; for small values, it can be approximated by cqp erf /cqrectangle = 3(2l/b)2 . In Fig. 3.23 the dependency of the ratio of the two coefficients with and without perforation, cqp erf /cqrectangle , on area coverage by holes, βc , is presented for a √ typical hole-to-gap ratio r0 = ( π/2)D0 and for the following plate dimensions:

140

Non-inertial forces

b = 20D0 , b = 50D0 , and b = 100D0 ; plate thickness h = D0 , h = 2D0 , and h = √ 20D0 . The ratio r0 = ( π/2)D0 corresponds to a quadratic hole with length 2D0 and the same area as a cylindrical hole. With increasing plate height the pipe flow resistance increases, equalizing more and more the impact of the boundary conditions within the hole with those of the cell. In other words, damping decreases. Increasing the plate width b also decreases relative damping, because the mitigating border effects become more and more negligible. On the basis of the fact that, for low characteristic lengths l ≪ b, the pressure underneath the plate equalizes within a large, central interval along the x-axis and drops with exp[−(b ± 2x)/(2l)] at borders (see (3.220)), Bao et al. [2003] proposed that one should generally substitute the dimensions of arbitrary plates by reduced values b − 2l and L − 2l, leading to the following equations: cqp erf ,rect ≈ cqp erf ,circle

8µh′ η (b − 2l)(L − 2l) βc r02

8µh′ η ≈ π (R − l)2 βc r02

for rectangular plates, (3.222) for circular plates.

The impact of perforation on rapidly oscillating plates at low pressure The fast decrease of damping with increasing hole area may be surprising. However, the example presented is valid for low frequencies (steady flow within the holes) and continuous flow (no slip, no rarefaction) only. The situation becomes more critical for low pressure and is mitigated for rapidly oscillating plates. The analysis usually includes annoying algebraic transformations and leads to formulas that are not well suited for engineering applications [Mohite et al. 2008, Homentcovschi and Miles 2005]. However, some simplified expressions can be derived, by generalizing the modified Reynolds equation for low pressures and fast plate motions as in Pandey et al. [2007]. To account for the slip conditions within the holes and along the plate, the function η(β) in Eq. (3.217) must be adapted according to Veijola [2006] and Hwang et al. [1996] by incorporation of the factors µ ′ ¶1.17468 λ′ λ Qgap = 1 + 0.108 42 + 9.3593 for the flow in the gap, D0 D0 λ′ Qh = 1 + 4 for the flow in the hole, (3.223) r0 to become η(βc ) = 1 +

³ ´ p 3r04 Qh 2 4β − β − 4 ln β − 3 . c c c 16D03 h (1 + 3πr0 /h) Qgap

(3.224)

Qgap can be considered as the common correction factor for efficient viscosity of the lateral flow in squeeze films: µ′ = µ/Qgap . The modified Reynolds equation with corrected characteristic length l′ and time-dependent pressure term ∂ p˜/∂t now accounts for higher Knudsen numbers

3.2 Damping forces

141

2D03 h′ η(βc ) . 3βc r02

(3.225)

and fast motions: ∂ 2 p˜ ∂ 2 p˜ p˜ 12µ ∂ p˜ 12µ ∂∆D + 2 − ′2 − 3 = , ∂x2 ∂y l D0 ∂t pa D03 ∂t

l′2 =

Pandey et al. [2007] solved it for a rectangular plate. After transformation into the well-known diffusion equation by substitution of p˜ = U exp(−κ2 t) by κ2 , chosen so as to eliminate the term constant · p˜, they applied Green’s function and obtained, after back-transformation, p˜ = − jω ∆D0 exp(jωt)

16 π4 m−n−2

×

X X

k ,l=o dd

2 2

k l

(−1) 2 #. µ ¶2 b 12µb2 2 2 2 k +β l + ′ + jω 2 lπ D0 pa Qgap π 2

"

(3.226)

Integration and separation of real and imaginary parts leads to expressions for the damping coefficient and spring constant11 c′qhp erf =

′ kqhp erf =

12 · 64 b3 µL 3 π6 D0 Qgap X X k 2 + β 2 l2 + [b/(l′ π)]2 ·³ ¸, × ´2 k ,l= odd k 2 l2 k 2 + β 2 l2 + [b/(l′ π)]2 + σ 2 /π 4

144 · 64 2 2 b5 µ ω L 5 π 8 pa D0 X X 1 ·³ ¸, × ´2 k ,l= odd k 2 l2 k 2 + β 2 l2 + [b/(l′ π)]2 + σ 2 /π 4

(3.227)

with β=

b , L

σ=ω

12µ b2 . pa Qgap D02

The terms with b/l′ are responsible for the impact of plate perforation, modified for low pressures, while terms with σ reflect inertial forces, caused by fast oscillations. An estimate for the cut-off frequency is obtained by taking the first term of the rapidly converging series µ ¶2 ¢ ¡ b σc = ′ (3.228) + π2 1 + β 2 . l 11

Pandey et al. [2007] subtracted a contribution, which they extracted by integration of average pressure over all holes. In the author’s opinion, this overestimated the impact of holes and hence is not considered here.

142

Non-inertial forces

To illustrate the impact of perforation for low pressures/high frequencies, it is useful to compare damping coefficients without and with perforation, according to Eqs. (3.227) and (3.202). Arguably, it can be assumed that the effective viscosity in (3.202) and µ/Qgap in (3.227) are identical. For simplicity, a long plate (L ≫ b) is considered, for which summation over l can be performed. A first-term approximation of the ratio of the two damping coefficients is ih h ¢2 i ¡ 1 + [b/(l′ π)]2 1 + σ/π 2 c′qhp erf = h . (3.229) i2 cqhd rectangle 1 + [b/(l′ π)]2 + (σ/π 2 )2

Clearly, for large characteristic lengths the ratio converges to unity. For very small l′ the relative damping decreases according to the full Eq. (3.227) with µ ¶ c′qhp erf π 4 l′ 2 = . (3.230) cqhd rectangle 8 b

In contrast, for very high frequencies σ ≫ (b/l′ )2 and arbitrary pressures the ratio of the damping of perforated and unperforated plates converges to unity: the impact of perforation for rapidly oscillating plates becomes negligible. The high-frequency condition is given by µ ¶2 b pa Qgap βc r02 σ≫ ′ >1 ⇒ ω≫ , (3.231) l 8µ D0 h′ η and is rarely fulfilled for inertial MEMS. For instance, in the example given in the following, the high-frequency condition means f ≫ 850 kHz at a pressure of 10 mbar and f ≫ 85 kHz at 1 mbar. However, the impact of decreasing pressure for low frequencies is the most dramatic and important effect. The following example aims at demonstrating this. In the case of air pressure equal to 10 mbar the damping of the plate with √ D0 = 2 µm, r0 = (2/ π)D0 = 2.26 µm, b = 200 µm, and a perforation ratio βc = 0.05 drops to 3.8% of the value for an unperforated plate of thickness 4 µm (l′ = 12.3 µm, Qgap = 38.2) and to 3.7% for h = 2 µm (l′ = 11.1 µm).12 That is, the impact of perforation at low frequencies decreases further with decreasing pressure in comparison with the continuous-flow conditions. Thus, at low frequencies, perforation is an efficient measure for achieving high quality factors for plates with squeeze-film damping in small gaps. For the example presented both factors, namely slip conditions and the inertia of the gas, determine the resulting damping coefficient for frequencies in the multi-kHz region. In general, increasing the frequencies mitigates the dramatic drop of damping at low frequencies. If more accurate estimates of damping values for complicated geometries are needed it is meaningful to use 3D or 2D Navier–Stokes solvers. For squeeze-film 12

Surface-micromachined plates typically feature perforation rates four times higher than that, which further aggravates the demonstrated effect.

3.2 Damping forces

143

problems, simulations with 2D solvers that use the equivalence of the linearized Reynolds equation to the heat-transfer equation are suggested. The pressure can be simulated when the boundary conditions within the hole area are known. Taking into consideration slip effects at the cell as well as within the hole, Kwok et al. [2005] derived the following boundary condition: Ã  µ ¶2 !  3   hD r0 0     1−    rc4  rc 1 (3.232) p˜|within hole = × ×σ µ ¶4 ′ /r   1 + 4λ 0 r0     6     r c

with squeeze number

σ=

12µb2 , pa D02

which can be applied to cover the area of all holes. Summarizing, the methods presented are far from being perfect. Plate-edge effects are not included. This can be justified for not-too-thin plates. Nonlinearities and associated amplitude effects are not considered, which is permissible for small excitations in inertial MEMS. The slip conditions are of first order and neglect higher-order dependencies. Thermoelastic effects are excluded by assuming isothermal behavior etc. However, for the purpose of engineering system design such approximations, revealing the basic dependencies on geometry and pressure, are often preferable to the refined approaches which should be applied during design verification of accelerometers or gyroscopes.

3.2.4

Drag forces Drag forces act on a body moving through a fluid. To differentiate drag forces from slide or squeeze forces, they are interpreted as forces arising on an object moving in a fluid that is not disturbed by boundaries near the body. In MEMS any moving body is surrounded by some walls or other objects, so that pure drag-force conditions are rarely encountered. However, sometimes partial flows around a body can be approximated by drag-force conditions, for instance, the flow against the front side of a plate, circulating around it, or the flow on top of a plate that is far away from the opposite wall. Drag forces were first analyzed by G. Stokes, who considered a sphere with radius R and velocity v¯ in a steady flow. At infinity the flow should not be disturbed. By solving Stokes’ equation (3.122) with this boundary condition, he obtained the formula for the flow-resistance force which later on was slightly corrected by Oseen: µ ¶ 3 ̺R|¯ v| F¯DSp = −6πµR¯ v 1 + Re , Re = . (3.233) 8 µ

144

Non-inertial forces

With Ossen’s correction the equation is nonlinear in |v|. For small Reynolds numbers Re it is negligible. Interestingly, drag forces depend only weakly on body shape, which makes them attractive for rough-and-ready estimates. In Landau and Lifschitz [2000] can be found the expressions for a circular dish moving in various directions: F¯DDnorm al = −16µR¯ v 32 F¯DDinplane = − µR¯ v 3

for dish moving in its normal direction, for dish moving in its in-plane direction.

(3.234)

Despite the different shapes, the drag forces differ by less than a factor of two. For fast-moving objects the drag forces become dependent on flow density ̺, for instance according to Landau and Lifschitz [2000], for a fast-moving sphere, µ ¶ µ ¶ R 2π̺R3 9 δ d¯ v ¯ FDSp = −6πµR 1 + v¯ + 1+ . (3.235) δ 3 2 R dt p δ = 2µ/(̺ω) is the characteristic length according to Eq. (3.137), which is used for the analysis of slide damping under the impact of slip forces. Equation (3.235) can be applied also to rapidly moving dishes, correcting low- and high-frequency terms with corresponding weight coefficients, partially derived from the ratio of drag forces of the sphere and dish for low frequencies. In contrast to squeeze damping, there are few publications dealing with drag forces on plates or beams in MEMS. This obviously is due to the better suitability of other models such as slide and squeeze damping models. The difficulties of analyzing the drag force associated with 3D models may be another reason. Finally, the drag forces are linked to body geometries rather than to characteristic distances, which usually leads to much smaller counter-forces than, for instance, for squeeze-film damping. For lack of better alternatives, a common approach used to estimate drag forces is to substitute a moving plate or beam by an equivalent dish or superposition of dishes with the same total outer dimension or area with respect to the dominant motion. For instance, a beam moving normal to the surface is substituted by a series of dishes with 2R = b or πR = b, where b is the width of the beam. The dishes contact each other in the direction of the beam axis. The drag force is supposed to be the sum of forces acting on individual dishes. As a second example, if a quadratic plate is undergoing in-plane motion, it is substituted √ again by a dish with radius R = b/ π, where b is the dimension of the plate’s front side. To check the consistency of the drag-force approach with the slide-force analysis, a plate with area A ⇐ πR2 is considered. To apply the drag-force condition, the distance to the closest surfaces should be larger than the characteristic length δ which characterizes the attenuation of flow disturbances. Thus, Eq. (3.148), csx = µA/δ, must be considered. The total slide damping coefficient is twice the cited value, because the calculation in Section 3.2.2 was carried out for forces on gap-facing surfaces only. Since δ ≪ D is assumed, R/δ ≫ 1 holds even more

3.2 Damping forces

(a)

145

(b)

Figure 3.24 (a) Head-on collision of molecules with the plate. (b) Collisions of gap-entering molecules.

stringently, and according to Eqs. (3.234) and (3.235) the damping coefficient created by a drag force can be estimated to be µ ¶ µ ¶ 32 R 32 1 1 32ζ cD,plate = µR 1 + ζ = µA +ζ ≈ , (3.236) 3 δ 3π R δ 3πδ where ζ is a correction factor mapping the shape differences according to the drag forces for slow movement. R is chosen so as to lead to the same area as that of the embedded rectangle. It is intuitively clear that ζ < 1, because a rapidly dragged dish exhibits lower resistance than does a sphere. The estimated ratio of the drag-force and slide-force damping calculations is 1.7ζ. Considering the approximate nature of the applied assumptions, this is an acceptable result.

3.2.5

Free molecular flow At low pressures, fluid flow more and more becomes a motion of separate gas molecules. With growing Knudsen number Kn > 0.1 there are still elements of viscous flow in the transition regime, decreasing with increasing Kn and being substituted by molecules that collide with each other and with the surrounding walls, and finally being turned into free molecular flow (Kn > 10), where molecules collide only with surrounding walls. Since well-developed analytical tools for the transition regime are lacking, our focus is directed at free molecular flow. Methods often are extended towards lower Knudsen numbers in order to include features of the transition regime, as was analogously done in the case of continuous-flow models extended to higher Knudsen numbers. In order to analyze forces in the free molecular regime, the basic idea is simply to determine the reaction forces between the wall and isolated molecules and to sum them up according to their statistical distribution. First, a plate is considered, moving in the z-direction as shown in Fig. 3.24(a). The plate experiences head-on collisions with molecules of mass mg . The velocity component responsible for the momentum transfer of molecules to the plate is vz . Any collision on the front side of the plate causes a momentum transfer of ∆P+ = 2mg (vz + z), ˙ and correspondingly on the back side: ∆P− = 2mg (vz − z). ˙ If the number of molecules in a unit cube is N , and the fraction possessing velocities between vz and vz + dvz is dNv z , the number of molecular

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Non-inertial forces

collisions on a unit area in unit time is (vz + z)dN ˙ v z . Thus, the pressure exerted by molecules with velocities within the interval vz + dvz can be written as ˙ 2 dNv z . According to the kinetic theory of gases the density ∆P+dv z = 2mg (vz + z) of velocity components dNv z /N at temperature T follows a Maxwellian distribution: r µ ¶ mg mg vz2 f (vz ) = exp − . (3.237) 2πkT 2kT Therefore the pressure due to front-side collisions is µ ¶ Z ∞ 2mg N mg vz2 2 P+ = √ (vz + z) ˙ exp − . 2kT 2πkT −z˙

(3.238)

Analogously, for the back side, integration extends over (vz + z) ˙ 2 from z˙ to ∞. The difference between the two represents the acting pressure and can be easily calculated, r r µ ¶ Z ∞ 8mg N mg vz2 2 M ∼ P =√ vz z˙ exp − =4 pa z, ˙ (3.239) 2kT π RT 2πkT −z˙ where the relation z˙ ≪ vz , which is obvious for not-too-low temperatures, as well as the relations among the pressure pa of gas, its density, and N are taken into account. M is the molar mass in kg and R the universal gas constant (see also Eq. (3.119)). With cfree = AP /z, ˙ the damping coefficient is therefore r r r 2 M 1 cfree = 4pa A ; or cfree = 0.188pa A for dry air (3.240) π RT T with T in degrees Kelvin. The discussion presented here goes back to the early work of Christianson [1966]. The results cannot be directly applied to squeeze-film damping because the existence of the opposite surface is nowhere taken into account. Bao and Yang [2007] extended Christianson’s model towards squeeze-film damping. In addition to the head-on collisions of molecules with the plate as described above, Bao and Yang [2007] included energy loss of the plate caused by molecules entering the gap with some lateral velocity component and colliding multiple times with the plate, as indicated in Fig. 3.24(b). The extra energy gained by a molecule traveling through the gap can be calculated to be " # 1 2bvz20 b2 vz20 ∆E = mg z˙ + (3.241) z˙ 2 , 2 2 (D0 − z)vxy 0 (D0 − z)2 vxy 0 where vz 0 and vxy 0 are the velocity components of the molecule entering in the zand in-plane directions, respectively. The number of collisions during the dwell time within the gap, ∆T = b/vxy 0 , is ∆N = ∆t vz 0 /[2(D0 − z)], so the energy loss during one oscillation cycle of the plate can be derived. With some further approximations, Bao and Yang [2007] obtained the following expression for the

3.2 Damping forces

147

Q-factor: Qfree

3 ω D0 ∼ = (2π) 2 h̺p pa L

r

RT . M

With Q = mω/cfree an equivalent damping coefficient can be derived: r 2 2 bL pa M 3 csfree = (2π) . D0 RT

(3.242)

(3.243)

In comparison with Christianson’s damping coefficient the squeeze damping coefficient is L/(16πD0 ) times greater. Cutcherson and Ye [2004] overcame some of the limitations in the model of Bao and Yang [2007]. They assumed that the in-plane velocity of molecules may change after each collision within the gap, resulting in a modified travel time. Figure 3.24(b) illustrates possible changes of the path between collisions in cases where the vertical velocity increases after each collision. Furthermore, they abandoned the assumption of large ratios between the oscillation period and the traveling time. Finally, they also assumed larger oscillation amplitudes than in the model of Bao and Yang [2007]. No analytical expressions could be derived; however, numerical simulations showed that the assumption of a constant velocity change after each collision might be responsible for underestimating the damping coefficient by a factor of about two. On the other hand, the model of Bao and Yang [2007] works well also for lower Knudsen numbers, extended into the transition regime.

3.2.6

Structural damping Structural damping is mainly caused by heat transfer and associated heat losses within vibrating structures. According to the laws of thermodynamics a variation of strain in a solid body is accompanied by a variation of temperature. The temperature variations cause an irreversible heat flow and, thus, dissipation of vibration energy. The model of thermoelastic damping is usually based on a mechanical vibrating model of, say, a rectangular beam of width Rb, complemented by the therb/2 mal forces acting across the beam: F∆T = Eα −b/2 ∆T dy. α is the thermal coefficient of expansion which is part of the stress–strain relation according to Eq. (2.26) of Chapter 2. By calculating the volume changes caused by the vibration of the beam and inserting them into the heat-transfer equation with corresponding adiabatic boundary conditions the thermal distribution and, thus, the thermal force can be found. By re-inserting the result into the coupled vibrationmode equations one can derive the complex mode frequency, the imaginary part of which represent the damping term. The first resonance-mode analysis of a vibrating beam was obviously that performed by Zener [1937, 1938], in which just the first mode was considered. It was complemented later by Lifshitz and Roukes [2000] (see also Nayfeh and

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Non-inertial forces

Younis [2004] and Tang et al. [2008]). To get an impression of the impact factors, the result of Zener that is typical for resonators with resonance frequency ω is cited here: Qint =

ρCB 1 + ω 2 τ 2 , Eα2 T0 ωτ

τ=

b2 , π2 χ

(3.244)

where CB is the heat capacitance, ρ the density, and χ the thermal diffusivity (for silicon χ = 0.8 × 10−4 m2 /s). The main factors limiting thermal virulence – a large heat capacitance CB and a small thermal coefficient α – are responsible for large Q-factors. Well-designed gyroscopes exhibit small structural damping, resulting in quality factors Qint above 10 000 and up to 200 000. The first value was, for instance, calculated and measured for a ring gyroscope by Hao and Ayazi [2005b]. Structural damping with large Q-factors, Qint , is relevant under low-pressure conditions only where the viscous damping at the body’s surface, 1/QS , can be neglected. Besides the structural damping within the vibrating structure itself, anchor losses may play a significant role. They are determined by the vibration energy dissipated by transmission through the support. Any resonator creates vibrating shear forces and moments on its clamped ends. These will excite elastic waves propagating into the support, which absorbs some of the vibration energy of the resonator. If the resonator is suspended so that the exciting forces at the clamped ends compensate each other, low damping losses result. Otherwise support losses can exceed the intrinsic losses of the vibrating structure. Support losses require permanent attention in designing vibrating gyroscopes for vacuum conditions. Most of the support-loss models based on the theory of thermoelastic damping assume symmetric and regular supports, and do not incorporate additional crosscoupling terms between the vibrating modes (e.g. Hao et al. [2003] and Hao and Ayazi [2005a]). However, the latter are especially important for the emergence of disturbing offset effects, which will be considered in Chapter 8. An efficient way to minimize isotropic and anisotropic support losses is by reduction of the stress concentration within all supports and flexures.

References Arkilic, E., Breuer, K., and Schmidt, M. (2001). Mass flow and tangential momentum accomodation in silicon micromachined channels. Journal of Fluid Mechanics, 437:29–34. Bao, M. (2005). Analysis and Design Principles of MEMS Devices. Amsterdam: Elsevier. Bao, M., Sun, Y., Zhou, J., and Hang, Y. (2006). Squeeze film air damping of torsion mirror at finite tilting angle. Journal of Micromechanics and Microengineering, 16(11):2330–2335.

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Bao, M. and Yang, H. (2007). Squeeze film air damping in MEMS. Sensors and Actuators A, 136:3–27. Bao, M., Yang, H., Sun, Y., and French, P. J. (2003). Modified Reynolds equation and analytical analysis of squeeze-film air damping of perforated structures. Journal of Micromechanics and Microengineering, 13:795–800. Bao, M. H. (2000). Handbook of Sensors and Actuators – Micromechanical Transducers, volume 8. Amsterdam: Elsevier. Beskok, A. (2002). Molecular-based microfluidic simulation models, in The MEMS Handbook, ed. M. Gad-el-Hak. Boca Raton, FL: CRC Press, pp. 8.1– 8.28. Bird, G. (1996). Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Oxford University Press. Blech, J. (1983). On isothermal squeeze films. Journal of Lubrication Theory, 105:615–620. Chapman, S. and Cowling, T. (1970). The Mathematical Theory of Non-uniform Gases. Cambridge: Cambridge University Press. Chen, C. S. and Kuo, W. J. (2003). Squeeze and viscous dampings in micro electrostatic comb drives. Sensors and Actuators A, 107:193–203. Christianson, R. (1966). The theory of oscillating vane–vacuum gauges. Vacuum, 16:175–178. Clark, W. (1997). Micromachined vibratory rate gyroscopes. Ph.D. thesis, University of California at Berkeley. Cutcherson, S. and Ye, W. (2004). On the squeeze-film damping of microresonators in the free-molecule regime. Journal of Micromechanics and Microengineering, 14:1726–1733. Fang, W. and Wickert, J. (1994). Post-buckling of micromachined beams. Journal of Micromechanics and Microengineering, 4(3):116–122. Fedder, G. K. (1994). Simulation of micromechanical systems. Ph.D. thesis, EECS Department, University of California at Berkeley. Gere, J. M. and Timoshenko, S. P. (1984). Mechanics of Materials. Belmont: Wadsworth, 2nd edn. Griffin, W., Richardson, H., and Yamanami, S. (1966). A study of fluid squeezefilm damping. Transactions of the AMSE, Journal of Basic Engineering, 88:451–456. Gudeman, C., Staker, B., and Daneman, M. (1998). Squeeze film damping of double supported ribbons in noble gas atmospheres, in Technical Digest, Solid-State Sensors and Actuators Workshop, Hilton Head, SC, pp. 288– 291. Hao, Z. and Ayazi, F. (2005a). Support loss in micromechanical disk resonators, in 18th IEEE International Conference on Micro Electro Mechanical Systems (MEMS 2005), Miami, pp. 137–141. (2005b). Thermoelastic damping in flexural-mode ring gyroscopes, in 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando, FL, IMECE2005–79965, pp. 1–9.

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Hao, Z., Clark, R., Hammer, J., and Whitley, M. (2002). Modeling air-damping effect in bulk micromachined 2D tilt mirror. Sensors and Actuators A, 102:42– 48. Hao, Z., Erbil, A., and Ayazi, F. (2003). An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations. Sensors and Actuators A, 109:156–164. Homentcovschi, D. and Miles, R. (2005). Viscous damping of perforated planar micromechanical structures. Sensors and Actuators A, 119:544–552. Houlihan, R., Kukharenka, A., Gindila, M., and Kraft, M. (2001). Analysis and design of a capacitive accelerometer based on a electrostatically levitated micro-disk, in Proceedings of the SPIE Conference on Reliability, Testing and Characterization of MEMS/MOEMS, San Francisco, CA, pp. 277–286. Howell, L. L. (2001). Compliant Mechanisms. New York: Wiley. Hwang, C., Fung, R., Yang, R., Weng, C., and Li, W. (1996). A new modified Reynolds equation for ultra-thin film gas lubrication. IEEE Transactions on Magnetics, 32(2):344–347. Iyer, S. V. (2003). Modeling and simulation of non-idealities in a Z-axis MEMSCMOS gyroscope. Ph.D. thesis, Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburg. Kraft, M. and Evans, A. (2000). System level simulation of an electrostatically levitated disk, in Proceedings of the 3rd Conference on Modeling and Simulation of Microsystems, San Diego, CA, pp. 130–133. Kuehnel, W. (1995). Modelling of the mechanical behaviour of a differential capacitor acceleration sensor. Sensors and Actuators A, 48:101–108. Kwok, P., Weinberg, M., and Breuer, K. (2005). Fluid effects in vibrating micromachined structures. Journal of Microelectromechanical Systems, 14(4):770– 781. Landau, L. and Lifschitz, E. (1986). Theory of Elasticity. London: ButterworthHeinemann, 3rd edn. (2000). Fluid Mechanics. London: Butterworth-Heinemann, 2nd edn. Lifshitz, R. and Roukes, M. L. (2000). Thermoelastic damping in micro- and nanomechanical systems. Physical Review B, 61:5600–5609. Lobontiu, N. and Garcia, E. (2004). Mechanics of Microelectromechanical Systems. Amsterdam: Springer. Martin, M., Houston, B., Baldwin, J., and Zalalutdinov, M. (2008). Damping models for microcantilevers, bridges, and torsional resonators in the freemolecular-flow regime. Journal of Microelectromechanical Systems, 17(2):503– 511. Mohite, S., Sonti, V., and Pratap, R. (2008). A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures. Journal of Microelectromechanical Systems, 17(3):709–723. Nayfeh, A. H. and Younis, M. I. (2004). Modeling and simulations of thermoelastic damping in microplates. Journal of Micromechanics and Microengineering, 14:1711–1717.

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Osterberg, P. M. and Senturia, S. D. (1997). M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures. Journal of MicroElectroMechanicalSystems, 6:107–118. Pan, F., Kubby, J., Peeters, E., Trany, A. T., and Mukherjee, S. (1998). Squeeze film damping effect on the dynamic response of a MEMS torsion mirror. Journal of Micromechanics and Microengineering, 8(3):474–479. Pandey, A. K., Pratap, R., and Chaub, F. S. (2007). Analytical solution of the modified Reynolds equation for squeeze film damping in perforated MEMS structures. Sensors and Actuators A, 135:839–848. Sattler, R. and Wachutka, G. (2004). Compact models for squeeze-film damping in the slip flow region, in Proceedings of Nanotech 2005, Boston, MA, pp. 243–246. Tang, H., Yi, Y. B., and Matin, M. A. (2008). Predictive modeling of thermoelastic energy dissipation in tunable MEMS mirrors. Journal of Micro/Nanolithography, MEMS, and MOEMS, 7(2):023004–1–8. Thornton, E. and Baker, W. A. D. (1962). Viscosity and thermal conductivity of binary gas mixtures: argon–neon, argon–helium, and neon–helium. Proceedings of the Physical Society, 80:1171–1175. Timoschenko, S. and Gere, S. (1961). Theory of Elastic Stability. New York: McGraw-Hill Book Company. Timoschenko, S. and Goodier, J. (1970). Theory of Elasticity. New York: McGraw-Hill Book Company, 3rd edn. Veijola, T. (2006). A new modified Reynolds equation for ultra-thin film gas lubrication. Microfluid Nanofluid, 2(3):249–260. Veijola, T., Kuisma, H., Lahdenpera, J., and Ryhanen, T. (1995). Equivalentcircuit model of the squeezed gas film in a silicon accelerometer. Sensors and Actuators A, 48:239–248. Veijola, T., Pursula, A., and Raback, P. (2005). Extending the validity of squeezed-film damper models with elongations of surface dimensions. Journal of Micromechanics and Microengineering, 15(9):1624–1636. White, F. (1974). Viscous Fluid Flow. New York: McGraw-Hill Book Company. Wittenburg, J. and Pestel, E. (2001). Festigkeitslehre – Ein Lehr- und Arbeitsbuch. Berlin: Springer, 3rd edn. Zener, C. (1937). Internal friction in solids I. Theory of internal friction in reeds. Physical Review, 52:230–235. (1938). Internal friction in solids II. General theory of thermoelastic internal friction. Physical Review, 53:90–99.

4

MEMS Technologies

MEMS technologies are used for building complete sensor systems including intrinsic sensor elements, transducers, signal-processing blocks, packaging, and testing. Neglecting the interface between the test equipment and the device under test, testing is not ostensibly related to miniaturization. The creation of sensor elements and transducers, however, is based on generic microfabrication technologies, which are exploited to manufacture both integrated circuits and microsystems. Microsystems, in the broad sense of MEMS, bio-MEMS, microfluidics, optoelectronics (MOEMS), nano-devices and others, have borrowed their basic fabrication steps from IC technologies and extended their use towards micromolding, laser machining, wire electrodischarge machining, diamond milling and other techniques. The ultimate basis for judging the suitability of microfabrication technologies is economics. Economics is strongly determined by the unit volume and reusability of existing equipment and technology. Even automotive sensors reach, in the best case, multimillion unit sales per year. This number sounds impressive; however, considering a typical yield of some thousand units per wafer, the volume requirement reduces to some thousand wafers per year – a very moderate number for modern wafer fabs. Dedicated fabrication lines would be far too expensive. Common use of basic technologies for industrial ICs and MEMS is the basic paradigm. If they are economically feasible, specialized processing steps like machining of quartz are organized in small and relatively cheap fabrication lines complementing existing technologies. Generally speaking, a high degree of compatibility of fabrication steps with mainstream microelectronic equipment and technology recipes is decisive for a successful MEMS product and its long-term advancement.1 Hence, not only the technical pressure on co-integration of sensors and electronics, but mainly economic facts, dictate the indispensable conjunction of MEMS technologies and microelectronics. Furthermore, given the relatively low number of wafers to be fabricated, even for high-volume MEMS, standardization plays a key role. Flexible process flows 1

The present exceptions, which involve dedicated fabrication lines, are print heads and magnetic or optic read–write heads. Bio-MEMS may follow.

4.1 Microfabrication of inertial MEMS

153

suitable for the design and fabrication of a variety of different sensors are preferable. They correspond not only to recent market trends of foundries in offering MEMS-fabrication support for fab-less design houses but also to the traditional quest of big players to unify their own process portfolios.

4.1

Microfabrication of inertial MEMS The success of microelectronics is based on the batch fabrication of a great number of identical components on wafers. MEMS fabrication basically follows this paradigm. Despite the use of some non-planar, three-dimensional devices such as vertical bipolar or power transistors, and of stacked devices, the world of ICs and MEMS is primary two-dimensional,2 following the multiple recurrence of film- or layerfocused fabrication steps. These involve

r film deposition (physical and chemical vapor deposition, electrodeposition, spin casting, sol–gel deposition, etc.)

r pattern transfer (optical proximity and projection step-and-repeat lithography, direct electron-beam and laser writing techniques)

r structural change (oxidation, doping, ion implantation, drive-in diffusion) r etching and cleaning (isotropic and anisotropic wet etching, vapor- and plasma-assisted dry etching, deep reactive-ion etching, planarization, lift-off techniques, chemical–mechanical polishing, wafer cleaning, ashing). Using classical IC technologies and given their planar nature, a variety of microsensors such as magnetic Hall sensors, piezoresistive and some piezoelectric stress sensors, among others, can be fabricated without appreciable changes of the existing processes. However, most of the MEMS technologies are aimed at breaking the barriers into the truly three-dimensional world. They are accompanied by similar efforts within the IC world to create three-dimensional structures such as deep trenches for DRAM isolation. In this regard, the genuinely new feature in MEMS is the creation of moving objects and mass-flow environments. The creation of moving objects within planar structures is inevitably linked to removal of material around the target object. If the target object is embedded in a silicon substrate, it can be released by bulk micromachining, by which bulk material is etched away, leaving the released structures attached to the surrounding substrate or to layers covering them. If the target object is a section of a deposited layer or a layer-stack and has to be released from the underlying layer(s), surface micromachining is used to etch away a sacrificial layer under the 2

There have always been attempts to break through the barrier into real three-dimensional structures such as levitating balls for accelerometers [Toda et al. 2002, Takeda 2000]; however, the potential for commercial success of such approaches is more than questionable.

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intended structure. Both techniques are usually complemented at least by the integration of transducer elements such as piezoresistors, piezoelectric sensors or capacitive interfaces. These two techniques can be combined and executed separately or within an IC-fabrication flow. In the last case one is talking about monolithic integration, placing the moving structure and corresponding transducers together with signal-processing electronics on one chip. This book is not about technology. Therefore, the aforementioned basic technology steps will be touched on only insofar as they are relevant for the shape and properties of the building blocks of inertial MEMS: proof masses, suspensions, transducers, etc. The intention is to impart a basic understanding for non-experts, presenting some technological basics as well as typical surface and bulk micromachining processes. Selected integrated process flows, are representative for inertial MEMS fabrication and seem promising for further development, will be discussed. Of course, nobody knows whether or when these flows may become obsolete, being substituted by unexpected innovations. This is the risk of any view into the future. For readers who are interested in acquiring basic knowledge on IC fabrication, excellent descriptions can be recommended, such as, for instance, the classical work of the Fairchild Corporation [1979] or later books [Campbell 1996, Colclaser 1980, Sze 1988, Wolf and Tauber 2000]. Details of microfabrication technologies are, for example, presented in books by Franssila [2004], Kovacs [1998], Madou [2002], and Fukuda and Menz [1998], and review papers by Bustillo et al. [1998], Kovacs et al. [1998], and Schmidt [1998].

4.1.1

Basic microelectronic fabrication steps CMOS or BiCMOS integrated circuits are manufactured by alternating steps of film deposition, patterning, doping, and etching away unnecessary material. p–n-junctions, bipolar transistors, source or drain areas of MOS transistors, and conducting-well resistors on the surface of bulk silicon are created by local doping. They are isolated from the environment, especially from superimposed structures, by isolating layers, usually formed of SiO2 . Electrical connections are made by deposited metallic strip-lines, mostly aluminum or copper, which later became necessary for deep sub-micrometer technologies. Despite its inferior electrical properties, tungsten is sometimes used in order to improve thermal stability against subsequent high-temperature processing steps. Silicon nitride is a favored material for protecting structures during subsequent processing steps. Silicon nitride passivation is, among others, the final step of wafer processing, being aimed at protecting all structures from humidity and possible chemical reactions, as well as from mechanical damage. Any creation of a structure, be it a doped region or a selected layer pattern, includes a preceding mask-deposition and mask-patterning step. Lithography is the method of choice for patterning deposited photoresist layers, followed by

4.1 Microfabrication

155

etching away the light-exposed (or unexposed) areas. If a SiO2 or silicon nitride hard mask is needed, it is created by etching through a preceding photoresist mask. Sub-micrometer IC technologies on the one hand add the creation of deep isolation trenches and deep vias to the horizontal-layer paradigm, and, on the other hand, extend it by stacking more and more layers, mainly for interconnect purposes. The standard diameters of silicon wafers are four, six, eight, and twelve inches. Most of the MEMS devices have been fabricated on four- and six-inch wafers; however, the transfer to eight inches is proceeding rapidly and is nearly finished, at least for high-volume markets. By contrast, the overwhelming fraction of electronic circuitry is fabricated in eight- and twelve-inch fabs, with emphasis on digital circuitry. Mixed-signal ICs would be the ideal environment for MEMS, and, since they are preferably manufactured in eight-inch fabs, the past gap between MEMS and IC equipment has been closed. Start wafers for IC production are low-doped in the case of digital electronics and high-doped for mixed-signal applications. High doping may reduce etching rates for bulk-etched MEMS structures (p-doping slows down KOH etching). This is only one example of many factors requiring compromise, if co-integration of MEMS devices and IC circuitry is desired. Microelectronic technologies have developed rapidly while maintaining the validity of Moore’s law. This has been possible because many of the basic process principles have formed a stable technological base extended and complemented by a continuous flow of innovations.

Deposition The most common thin-film deposition method is chemical vapor deposition (CVD), which is based on the reaction of vaporous species at a hot surface. It is used to deposit silicon oxide, silicon nitride, and polysilicon or amorphous silicon. Low-pressure CVD (LPCVD), plasma-enhanced CVD (PECVD), and atmospheric-pressure CVD (APCVD) are the most widely used types.

Silicon The deposition of silicon and here especially of polysilicon as the basic structural material for MEMS devices needs special attention. Amorphous silicon is deposited in a LPCVD furnace at deposition temperatures of 570 ◦ C using silane, SiH4 , as the precursor. Polysilicon requires deposition temperatures up to 650 ◦ C. The grain size of polysilicon increases with the deposition temperature. The large stress within the deposited layers makes annealing necessary in order to adjust the final grain size and the residual stress. Typical annealing temperatures are about 1100 ◦ C in an atmosphere of N2 . Annealing reduces the residual stress by one to two orders of magnitude, and the stress gradient becomes very small.

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Crystalline silicon can be grown in LPCVD or APCVD systems in the form of a so-called epitaxial deposition. Epitaxy is the process of growing a single-crystalline layer on top of a single-crystalline substrate. The growing layer follows the underlying crystal lattice, provided that the lattices of the two layers match closely and surface impurities or insufficient smoothness do not prevent the transfer of the lattice information. Since the resulting structures have the same or related lattice structures and temperature coefficients of expansion (CTE), the stress level in the grown layer is low. Operational temperatures lie between 900 and 1250 ◦ C for APCVD epitaxy and between 700 and 900 ◦ C for LPCVD. Remarkably, epitaxial silicon can be grown selectively, suppressing growth in certain areas by covering them with non-matching materials, for instance with SiO2 or silicon nitride. Normally amorphous silicon or polysilicon would be deposited in these areas; however, this deposition can be prevented by the addition of HCl to the reaction gases, which prevents nucleation at the SiO2 or silicon nitride surfaces. This feature makes selective epitaxy attractive for forming MEMS structures.

Silicon dioxide Natural silicon oxide or “native oxide” is formed as a very thin layer (∼1 nm) when silicon is exposed to air under ambient conditions. Clearly, the thickness is insufficient for electrical isolation between layers. If its presence is unwanted, the native oxide can easily be removed by a slight etching step. To avoid oxidation, silicon wafers can be transported in an inert-gas atmosphere. Silicon dioxide is the most used material for depositing isolating layers, which are usually of the order of micrometer thickness. It is either grown thermally on a silicon surface at high temperatures, between 750 and 1200 ◦ C, in a wet or dry oxygen environment (thermal oxide), or deposited in a CVD apparatus (CVD oxide) at temperatures between 300 and 900 ◦ C. High-temperature oxides have better quality. Low-temperature oxides can be deposited in APCVD systems and are often used in the form of phosphorus-doped oxides (PSG, phosphosilicate glass) and boron phosphosilicate glass (BPSG), which can be reflowed at T > 800 ◦ C, in this way smoothing the coverage over vertical layer steps. Owing to its good etchability, PSG is used as a sacrificial material in surface micromachining. A popular source of silicon dioxide is TEOS (tetraethylorthosilicate, or, equivalently, tetraethoxysilane, Si(OC2 H4 )4 ). In TEOS the silicon atom is already oxidized. The conversion to silicon dioxide is basically a rearrangement rather than an oxidation. The process proceeds in an inert atmosphere. However, addition of oxygen increases the deposition rate, mainly by removing intermediate ethyl groups from the surface. TEOS is generally performed in tube reactors at pressures of a few torr and temperatures above 600 ◦ C. To carry out deposition at lower temperatures, it is necessary to add more aggressive oxidants, such as ozone.

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Silicon nitride Silicon nitride is a reaction product of silane or dichlorosilane (SiH2 Cl2 ) with ammonia in a CVD system. It is used as mask during local oxidation and for device passivation. SiNx exhibits outstanding resistivity against wet chemical etchants, but it is produced in LPCVD systems at 700 to 850 ◦ C. Since aluminum–silicon contacts cannot be exposed to temperatures higher than 450 ◦ C, at which the dissolution of silicon into aluminum starts, LPCVD silicon nitride is incompatible with wafers carrying aluminum interconnects. Silicon nitride layers exhibit large stress, on the order of 0.5 to 1 GPa, which prohibits their use as structural MEMS elements. Low-stress LPCVD nitride films that are usable in a MEMS environment have been developed. They feature a larger silicon content in order to produce siliconenriched nitride. Alternatively, N2 O can be added to the reaction gases, producing the oxynitride. Also low-temperature (250–350 ◦ C) PECVD-deposited silicon nitride has been developed, which is compatible with aluminum interconnects. However, stress control is difficult and pinhole problems may prevent its use as an etching mask.

Metals Physical vapor deposition (PVD) includes evaporation and sputter techniques. It is widely used for metal-layer deposition. Metals such as aluminum, titanium, and titanium nitride (TiN) are usually sputtered. Titanium acts as an adhesion layer and TiN prevents diffusion. As pointed out above, aluminum interconnects should not be exposed to temperatures higher than 450 ◦ C in order to avoid degradation, especially in contact areas to doped silicon. Other sputtered metals are gold, platinum, silver, and various alloys. Metal layers can be deposited also by CVD methods, which, inter alia, lead to improved step coverage. Sputtered metal layers exhibit tensile stress as a result of thermal mismatch with surrounding dielectric or silicon material. In general, LPCVD and CVD are suitable for not-too-thick films in the range from ten nanometers to some micrometers. This is partially due to the low deposition rates of from 1 to 100 nm/min. Most of the thicker polycrystalline or amorphous films accumulate unacceptably high stress levels. Liquid-phase deposition methods such as electroless deposition and electroplating (galvanic deposition) are well known from the printed-circuit-board (PCB) industry and frequently used in MEMS technologies. Liquid deposition of metals, such as gold, nickel, and copper, is also an essential component of sub-micrometer technologies.

Resists and polymers One of the most widely used methods for resist spinning is spin-coating. This is growing in popularity for other materials also, such as resistant polymers, e.g. for spin-on dielectrics (SODs), and for silicon-containing polymers (spin-on

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Figure 4.1 Access directions of depositing material.

Figure 4.2 Step and bottom coverage. Adapted from Franssila [2004].

glasses – SOGs), which, after curing, are similar to silicon dioxide glassy layers. The film thicknesses may reach 100 µm or more.

Deposition on patterned substrates Most thin films are deposited not onto planar surfaces, but onto previously deposited patterned films. The film thickness here depends on the surface orientation: on vertical steps the deposited film is thinner than on the horizontal area. This is especially true for PECVD, where an orientated particle flow is present. The angular region for the arriving particles is largest at convex corners and smallest at concave corners (Fig. 4.1). Correspondingly, the largest deposition is at the convex corners, as depicted in Fig. 4.2. The ratio of the deposition thicknesses, B/H, is called the step coverage, and the ratio HB /H is the bottom coverage. High-temperature CVD processes for polysilicon and nitride produce nearly ideal step and bottom coverage, whereas PECVD usually leads to poor step coverage. Clearly, step coverage becomes more critical the higher the aspect ratio. Insufficient step coverage of deposited conducting metal films increases the current density at the narrowed regions and may lead to problems of reliability. The different coverage of sidewalls and horizontal areas, as well as the different quality of the deposited materials at these locations, may have a large impact on the corresponding subsequent etching rates (see Section 4.1.2) and needs meticulous control in order to prevent residuals at the sidewalls after etching (stringers).

Molding In principle, molds can be made from nearly any material, be it resist mold for electroplating of various metals, or metal for polymers such as nickel for

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polydimethylsiloxane (PDMS), etc. Traditionally, the material is cast into a master mold. However, many alternative methods are available and used in modern MEMS fabrication: electroplating of metals, CVD of polysilicon, injection molding of thermoplastics, sol–gel processes of PZT, and others. A photoresist is most commonly used for mold deposition, especially for electroplating into a resist structure. Mold techniques in early MEMS have opened up some new features, especially impressive aspect ratios of moving structures. Thick resists, such as polymethyl methacrylate (PMMA), are used in LIGA (LIGA is a German acronym for Lithographie, Galvanoformung, Abformung) – a process developed in the early 1980s by the nuclear research center in Karlsruhe in order to structure layers up to some millimeters thick [Becker et al. 1982, Bley et al. 1991]. Exposure of the resist was performed not by optical lithography, but by using hard-X-ray synchrotron radiation. Later an ultraviolet-based alternative was developed (see the section on “Patterning”). The resist mold can be removed simply; however, there exist many alternative methods to release the molded structure, for instance by coating the mold with material, eliminating reactions between mold and deposited fill material, or by anti-stiction surface coating of the mold.

Patterning The tremendous progress in photolithography is one of the pillars on which the success of microelectronics rests. Patterning means the transfer of a designed pattern onto the substrate or onto a layered material. The geometric pattern is first created in the form of a mask, which is a quartz plate with a patterned opaque layer. At this point the polygonal representation of the pattern generated during computer-aided design (CAD) is transformed into the geometric mask pattern by using electron-beam lithography. The mask is repeatedly used for all wafers of a given lot, reducing the fabrication costs considerably. The spincoated resist on a wafer is exposed to ultraviolet (UV) light through the mask as illustrated in Fig. 4.3. In the case of positive photoresist, the exposed areas are removed during the development process, opening the areas for the etch attack to the film-layer to be patterned. In the case of negative resist the unexposed areas are removed. After etching the desired pattern is transferred to the target layer as shown in the lower part of Fig. 4.3. The remaining photoresist can be removed, usually by ashing (resist stripping). Instead of layer structuring, the patterned photoresist may also serve as a mask for ion implantation. Since device implementation constitutes a repeated flow of patterning processes, masks and wafers have to be aligned to previous patterns. Sophisticated and expensive mask aligners perform this task. The mask can either be close to or in direct contact with the wafer (contact or proximity printing), or a mask image is projected onto the photoresist (projection printing). Imaging and alignment accuracies, especially in the case of deep sub-micrometer technologies, are

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Figure 4.3 Photolithographic thin-film structuring.

Figure 4.4 Edge smoothing over vertical steps. Adapted from Beeby et al. [2004].

astonishingly exceeding by far the normally required accuracies for MEMS devices, becoming, however, relevant for nano-devices. Another problem is always present during repeated layer structuring. Since the etched layers represent a step pattern, the next thin-film deposition process is performed with some conformity, which leads to a step transfer into the next layer. The step coverage, i.e. the thickness of the layer at the edge, is often less then 50% of the flat-layer thickness. This usually undesired vertical step behavior can be improved and smoothed, for instance by using PSG or BPSG and adding a reflow step as shown in Fig. 4.4. The remaining unevenness is sometimes unacceptable, especially if wafer bonding is intended. In this case, after deposition of a sufficiently thick final layer, planarization using chemical– mechanical polishing (CMP) is applied. Resists are usually between fractions of a micrometer and some hundred micrometers thick. Thick resists are readily used in MEMS fabrication, acting usually as molds for electroplated metal structures. The resist structuring

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within the LIGA technique was initially based on expensive hard-X-ray lithography, which allows high-precision structures to be formed with unprecedented aspect ratios. A cheap alternative – the so-called “poor man’s” LIGA – was found in the form of a special epoxy-resin-based optical resist, called SU-8. It can be spun on in thick layers of up to 0.5 mm and, most importantly, patterned with commonly used UV-contact lithography. This makes LIGA-like techniques commercially attractive. Microcontact printing also enjoys growing popularity for MEMS design; however, it is rather seldom used for inertial MEMS. For rapid prototyping of wafers such as, for instance, during R&D cycles, electron-beam lithography may be used, in which an electron beam is swept across the wafer and switched on and off according to the required resist mask pattern. The process is slower than optical lithography; however, its use avoids the need for preparation of a costly mask.

Doping Doping changes the concentration of donor or acceptor impurities in a substrate like silicon. The additional electrons or holes in the crystal lattice increase the electrical conductivity. p–n-junctions are formed at the borders separating parts of the material with different dopant concentrations. Thus, doping is the underlying key process for fabricating active electronic devices. Donor doping (phosphorus, arsenic) yields n-type silicon. Boron doping leads to p-silicon. Dopants are driven into a substrate by ion implantation or by diffusion. The diffusion gradient stems from a gaseous, liquid, or solid source. For instance, during epitaxial growth the appropriate source gases (phosphine, arsine, or diborane) can be added in the epitaxy reactor, leading to a nearly homogeneous dopant concentration. In general, diffusion is carried out in a furnace at temperatures between 800 and 1200 ◦ C. In the case of ion implantation, accelerated ionized dopant atoms are fired into the substrate. The location and amount of the dopants can be precisely controlled. The penetration depth of the atoms follows a nearly Gaussian distribution. This distribution is usually flattened by a subsequent thermal “drive-in” diffusion process, in which the dopants diffuse until the desired profile has been achieved. The annealing step is also necessary in order to heal the damage to the crystal lattice caused by the bombardment, and to move the dopant atoms into substitutional sites of the lattice. Ion implantation is used for fabrication of inertial MEMS if moderate penetration depths are required, as in the case of piezoresistors. For deeper doping profiles, such as for implementing etch-stops, diffusion in furnaces should be chosen. For deep profiles exceeding a dozen micrometers, the epitaxial growth of a doped layer becomes mandatory.

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4.1.2

Etching As noted above, etching is one of the most widely used fabrication steps for forming movable structures on Si wafers. It includes the old-fashioned wet etching as well as modern dry-etching technologies. All etching processes are based on chemical reactions between a solid like silicon and a liquid or gaseous etchant that transforms the solid into soluble or volatile products. Etching is a surface process, requiring transport of the etchant to the surface and removal of the etch products. Normally it is desirable that etching of MEMS structures exhibits a high degree of compatibility with standard metal–oxide CMOS processes. The compatibility is twofold: first, the materials, etchants, and equipment used should be compatible with a CMOS manufacturing environment; and, secondly, the etch process should not affect CMOS structures such as aluminum or dielectric layers in cases where an on-chip co-integration of MEMS and CMOS substructures (e.g. piezoresistors, capacitances) or with complete signal-conditioning ICs is intended. However, etch-sensitive CMOS structures can often be protected by temporarily deposited layers. Despite aggressive etch chemistry, an appropriate process flow can guarantee the desired CMOS compatibility.

Isotropic wet etching The substrate with exposed patterns is immersed into a tank with liquid chemicals. Etch chemistry is a science in itself. The basic etch reaction is a chargetransfer process, for instance, hole injection into the Si valence band by an oxidant in the case of the Si-etch process. The silicon–silicon bonds are broken and OH groups inserted. The OH− groups delivered by the etchant, together with the oxidized Si4+ , create new silicon oxide and H2 according to the basic etch reaction Si4+ + 4OH− → SiO2 + H2 ,

(4.1)

where the oxidation products are subsequently dissolved in HF. Similar reactions apply for other solids. The typical parameters of an etch process are the etch rates within a given material, etch selectivity with respect to other materials such as masking and non-masking layers, the dominant etch direction, and so on. Isotropic etching has no preferred etch direction – the etching propagates independently of the orientation of the material. The result of isotropic wet etching of a dielectric layer is shown in Fig. 4.5(a). Layers of silicon oxide (SiO2 ) or silicon nitride (Si3 N4 ) are usually etched using hydrofluoric acid, HF, buffered with ammonium fluoride (NH4 F). At 25 ◦ C SiO2 etch rates of 120 nm/min are common. Etching propagates not only towards the substrate, but also sidewards under the etch mask, resulting in a tapered sidewall of the dielectric layer with some curvature. The undercut of the mask layer illustrates the loss of pattern-transfer accuracy.

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(b)

Figure 4.5 (a) Isotropic layer etching. (b) Isotropic substrate etching.

-

Figure 4.6 Surface micromachining with sacrificial-layer etching.

Unfortunately, the sidewall shape is sensitive to changes of etchants, the flow of etch products at the surface, temperature inhomogeneity, etc., which are not easy to control. The undercut can be compensated for by mask biasing; however, the spread of the undercut and its shape must be carefully considered if mechanical structures are involved. The picture described remains valid also for isotropic dry-etching processes. Figure 4.5(b) illustrates isotropic wet etching deep in the substrate. As mentioned above, such a process is referred to as bulk micromachining. A typical acidic etchant, like buffered HF (5 : 1 NH4 F : concentrated HF) or the most commonly used “HNA” (a mixture of HF, nitric acid, HNO3 , and acetic acid, CH3 COOH), is very aggressive and overcomes all activation barriers related to the different plane orientations in Si. Etching rates > 10 µm/min are possible. Owing to the fast etching rates, the choice of the mask layer can be critical. Thick SiO2 is usable for shallow etching; silicon nitride (Si3 N4 ) or gold masks are preferred for deep bulk micromachining. The typical curved and not readily controllable sidewalls of the etched cavity are acceptable for release processes, if, for instance, an overlying material block is to be released from a substrate. If the undercut is larger than half the lateral block dimension, an accurate release takes place. Isotropic wet and dry etching is a basic release process in surface micromachining. Figure 4.6 illustrates the principal flow of a surface-micromachining process.

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An isolation layer is first deposited, followed by the sacrificial or spacer layer. Pattern transfer – in order to define anchor regions – is followed by deposition of the structural layer – often polysilicon. The structural layer is then also patterned and the sacrificial layer removed by isotropic etching, in order to release the structure outside the anchor region. Of course, instead of one structural layer a stack of thin films can be used. This simple picture indicates the typical problems of sacrificial-layer removal: the underetching must be pronounced in order to remove all sacrificial material between the structural layer and the substrate. If the lateral dimensions are large relative to the sacrificial-layer thickness, long etching times result and the structural layer may be affected due to the limited etch selectivity. One way out is to introduce etch holes as discussed in conjunction with squeeze damping in Chapter 3. Another effect is related to the configuration of the anchor region. The adhesion of a structural layer to the ground layer (isolation layer, substrate, or conducting layer) must be large enough to avoid enhanced etching and subsequent structural changes, including stress accumulation. Similar effects have to be carefully considered in the design of anchor regions in inertial MEMS. However, the main difficulty is caused by capillary forces within the small gap between the structural and ground layers. During the long rinse and dry steps following the etch process, ionized water is present within the gap. With progressive drying the surface tension of the rinse water attracts the microstructure towards the ground layer until stiction occurs. To release the structure, large forces are needed, which usually destroy the microstructure [Maboudian and Howe 1997, Mastrangelo and Hsu 1993a, 1993b]. Stiction was one of the basic problems which, for a long time, prevented the introduction of surface micromachining for elastic beams or membranes. It has been overcome by gas-phase dry etching and by the introduction of anti-sticking agents such as hydrophobic coatings, by sublimation-drying processes, by the use of temporary stiffening structures, by using permanent contact-reducing bumps (dimples), and by other means [Abe et al. 1995, Mastrangelo and Saloka 1993, Mulhern et al. 1993]. Contact-reducing dimples as well as anti-sticking agents and atomic-layer deposition [Hoivik et al. 2002] are also common for preventing in-operation sticking.

Anisotropic wet etching Anisotropic etching behavior is caused by different surface-orientation-dependent activation barriers of silicon. Silicon has a diamond structure, as illustrated in Fig. 4.7(a). It can be viewed as two interleaved, face-centered cubic lattices that are shifted with respect to each other along the diagonal by one quarter of the diagonal length. Hence, the outer planes of the cube are occupied by four corner atoms and by one atom in the center of the planes. Four inner atoms, marked gray and black, are bonded with each other and with atoms positioned in the planes. The distances of the inner atoms to the planes are indicated for one of

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(b)

Figure 4.7 (a) The diamond structure of silicon. (b) (100) and (111) Si wafer planes.

the inner atoms, which is colored black. It is positioned in the center of the cube drawn with dashed lines, thus being a/4 away from the next cube plane. a is the lattice constant a = 0.543 nm. Grown silicon crystals are preferably cut along a {100} plane in order to get {100} wafers for IC and MEMS fabrication. The angle√ between an exposed {100} surface plane and the inner {111} plane is arctan(1/ 2) = 35.26◦ ; the complementary angle is 54.74◦ . One can see that the {100} and {110} planes are occupied by atoms, two bonds of which are directed into the cube and two bonds outwards with unfulfilled valence on the surface. In contrast, atoms on the {111} planes feature three bonds directed into the cube and only one – free for attack – towards the outside [Seidel et al. 1990]. The resulting etch selectivity for the different planes may be as much as 200 : 1. Anisotropic etching of a single Si crystal is performed preferably by strong bases such as KOH (potassium hydroxide), EDP (ethylenediamine pyrocatechol) and TMAH (tetramethylammonium hydroxide). The toxic EDP has etching rates for Si comparable with those of KOH, but an excellent selectivity against SiO2 of the order of 1 : 5000, which is more then ten times higher than for KOH. TMAH has excellent etch selectivity against SiO2 (larger then 1500 : 1) and other dielectrics, and is common in a CMOS clean room, whereas KOH and other agents have been banished from it. Figure 4.8(a) shows the basic mechanism of anisotropic etching. A rectangular mask is perfectly aligned with one of the equivalent in-plane h110i directions. In this case, four {111} planes intersect the [100] plane along one of the h110i directions, i.e. along the mask edges. Etching perpendicular to the exposed {100} planes proceeds rapidly, leaving nearly untouched {111} planes. The etch process, starting at any of the mask edges, opens a {111} plane, which stops further propagation in the direction normal to the {111} plane. Revealing {111} planes is tantamount to forming etch slopes at 54.74◦ to the (100) plane. It is known that, at the convex corners of {111} planes, the etch behavior changes rapidly. Convex corners appear to open two bonds to the outer space

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(a)

(b)

[110]

Convex corners

[100]

[111]

Figure 4.8 (a) Anisotropic etching with an aligned mask. (b) The undercut at convex

corners.

per surface atom instead of one as the {111} surfaces do [Senturia 2001]. In Fig. 4.8(b) the two convex corners, which are marked in the central picture, develop as soon as etching starts. Corner etching accelerates, leading to a growing undercut. It stops only if the whole cavity is limited by the four {111} planes shown in the right-hand picture. Once convex corners can emerge, the undercut is not dependent on the form of the mask, as long as the {111} planes do not meet. This may be used in order to undercut arbitrarily shaped beams or membranes. To prevent fast etching initiated by convex corners, compensation techniques can be used. Additional masking structures with advanced convex corners are arranged around the convex corner like singular or ring-positioned outposts. They are designed to be scarified (underetched) until the protected corner is met, when the desired etch depth is reached. Thus, any required etch depth needs its own compensation structures [Puers and Sansen 1990, Mukhiya et al. 2006]. Surface roughness after anisotropic etching is significantly enlarged, preventing, for instance, wafer-to-wafer bonding. Often a slight isotropic etch step is added in order to polish the surface. Anisotropic wet etching is often used for back-side etching of substrates covered on the front side by isolating and structural layers, which should form membranes, beams, isolated hot plates, etc. The cavity shape corresponds to Fig. 4.8(a); however, since the opening is on the substrate’s back side, it occupies a significantly larger area than is required for the front-side release. It is possible to form vertical sidewalls, using anisotropic wet etching. The etch mask must be aligned along two {100} planes that enclose a 45◦ angle with the wafer flat. In this case etching propagates perpendicular to all three {100} planes at the same rate. The result is a heavily underetched rectangular cavity, where the vertical sidewalls are shifted laterally by the same amount as is the bottom {100} plane. The method is consequently used rather seldom.

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V

SiO2 n-well p-substrate

V

possible source/drain implants

Figure 4.9 The principle of an electrochemical etch stop.

However, for h110i wafers, the mask can be aligned so that the slow-etching {111} planes form the sidewalls.3 In this case perfect vertical etching without an undercut is possible. It is important to note that the high orientation dependence of etch rates depends on etchant concentration and temperature. The relative rates of different planes can be balanced by a proper choice of etchant and etch conditions, opening the way to numerous combinations of mask alignment and etch borders.

Electrochemical etch stop There are many methods to control the etch depth, which depends on the etch time, etchant composition and aging, etchant temperature, stirring of the bath, type of substrate (doping and impurities), surface preparation, and so on. Nonuniformity of wafers may be a significant source of taper and variation in etch depth. These different factors notwithstanding, depth and shape can be controlled accurately, using structural features to stop an etch. A popular method is the boron etch stop. It is based on the fact that the etch rate of heavily boron-doped silicon decreases by a factor of 100 and more when the doping level increases from about 1019 cm−3 to 1020 cm−3 . The application is limited by the thickness of the heavily doped layer, which is usually less than 15 µm. It has to be noted that heavily boron-doped silicon is incompatible with standard CMOS processes. Another widely used etch-stop technique is the electrochemical etch stop. When a positive potential is applied to a wafer in an alkaline etchant, it starts to oxidize if the applied potential is larger than the passivation potential. The oxide film stops dissolution of silicon (electrochemical passivation). In an analogous manner, an n-type layer of a p–n-structure can be protected. The substrate is p-doped with an n-Si layer or a diffused n-well on top. A positive voltage 3

In h110i wafers four of the eight {111} planes are perpendicular to the (110) surface.

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above the passivation potential is applied to the n-layer. In practice the classical four-electrode setup of Kloeck et al. [1989], with additional counter and reference electrodes within the etch sink, is used [Mueller 1999]. Etching of p-silicon stops when the p–n-junction is destroyed and the n-type silicon starts to become passivated. The principle is illustrated in Fig. 4.9. The negative potential can be applied directly to the etchant or indirectly via the p-doped substrate. The process is excellent for CMOS postprocessing, where the structural layers and electronics are implemented during the CMOS process, as indicated by the source/drain implants shown in Fig. 4.9, and release is performed as an additional process outside the wafer-fabrication line. During release, the front side of the wafer is protected by a passivation layer or by a special chuck. A process variant with wafer-level wiring to supply the required potentials was developed in cooperation between the ETH Z¨ urich and Austria Microsystems AG and offered for commercial use [Mueller et al. 2000].

4.1.3

Dry etching Dry etching is usually understood as anisotropic plasma etching capable of producing vertical sidewalls. This is not fully correct, because dry etching covers all methods by which physical or chemical reactions at a surface take place in the gas or vapor phase. It includes two basic types, vapor etching and plasma etching. The basis in each group comprises chemically reactive vapors, supplemented or substituted in the case of plasma etching by ionization and acceleration. The corresponding surface actions are chemical reactions through reactive species, physical impact by ion bombardment, or both. Pure vapor etching in micromachining is used mainly for etching silicon. Xenon difluoride, XeF2 , exhibits excellent selectivity for silicon, insofar as it does not attack SiO2 , Si3 N4 , metals or other materials. It is best suited for release etching of surface-micromachined structures over polysilicon sacrificial layers. Dry, reduced-pressure, gas-phase anhydrous HF/alcohol etching is the standard process for removing sacrificial SiO2 . It eliminates stiction and is compatible with a wide range of metals, especially with unprotected Al contact pads.

Reactive-ion etching Plasma-assisted dry etching is performed either as a glow-discharge or as an ion-beam process. In the first case a substrate is positioned within a vacuum chamber, where the plasma is generated by DC or RF excitation (Fig. 4.10). The plasma in reactive-ion-etching (RIE) systems is usually initiated by strong electromagnetic RF fields applied to the wafer platter (isolated bottom plate) within the vacuum chamber. The electrical field, oscillating with a typical frequency of 13.56 MHz, ionizes the gas molecules. The accelerated electrons, which are absorbed in the wafer platter, charge them down to negative voltages of some

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gas feed electrode

V~

plasma

substrate platter

Figure 4.10 A reactive-ion etch chamber.

hundred volts. The upper plate is grounded and thus does not change its potential. Positive ions, however, are accelerated towards the wafer platter, eventually colliding with it and causing the desired reactions. Newer systems use an improved scheme called ICP (inductively coupled plasma), which is able to generate high electron densities of around 1015 cm−3 at low pressures. The high density is responsible for a high etch efficiency. An RF current through a coil around a quartz chamber creates time-dependent magnetic fields, which in turn induce vertical electrical currents within the gas. Sufficiently large currents form a stable plasma with temperatures between 5000 and 10 000 K. In contrast to the diode systems described above, in a triode system an ion beam is generated within a two-chamber setup, which is commonly called remote plasma. The first chamber is responsible for plasma generation. The substrate is placed on a third electrode in a second chamber, and the ion flux is directed towards the substrate by appropriate focusing coils. At low pressure, a noble gas such as argon is ionized, and the ions are accelerated towards the substrate. If the ions have the right energy, they sputter the substrate. The process is called ion-beam milling. It can also be arranged by using a well-focused ion beam, which strikes the substrate only within a small region (focused-ion-beam milling) and can be guided to perform a desired pattern transfer. Plasma etching (RIE) performed in a vacuum chamber according to Fig. 4.10 is a complex mixture of chemical etching, ion-enhanced etching, physical etching, and deposition of non-volatile materials on sidewalls (sidewall passivation) and on horizontal surfaces, from which they are removed by ion bombardment. It is the reaction of reactive gases, excited by RF fields, with the substrate surface. The plasma contains ions, excited neutral, species, and electrons. The source gas constitutes the majority of species in the reactor, followed by the volatile reaction products, with a much lesser amount of excited neutral species, and a few, but important, ions. Chemical etching reactions take place when the reaction bonds are stronger than the Si—Si bonds. The choice of etch gases is made taking this consideration into account. The ion bombardment transfers energy

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after etch 1

after passivation

after etch 2

Figure 4.11 Deep reactive-ion etching with the Bosch process.

to the horizontal surfaces, causing ion-activated chemical reactions, ion-induced desorption, and damage. Vertical anisotropy is developed more strongly the lower the vacuum pressure in the chamber. At lower pressure the physical etch rate increases since the ions have a longer mean free path. The chemical etch rate drops because the ion density within the plasma decreases. Altogether, the process shifts from predominantly chemical to predominantly physical etching. The anisotropy increases with decreasing pressure; however, the etch selectivity against the resist and the stop oxide decreases. The gain in physical etch rate is smaller than the loss of chemical rate – therefore the total etch rate decreases.

Deep reactive-ion etching Deep reactive-ion etching (DRIE) is an enhancement of the older RIE processes. If the horizontal areas and sidewalls of the etched cavity are covered by films, which may result from reactions of etchant gases with erosion products of mask resist, bombardment removes the film on horizontal areas, leaving the sidewall films untouched. With an appropriate combination of film deposition on sidewalls and its chemical removal, the sidewall protection allows processes with astonishing aspect ratios and nearly perfect vertical sidewalls to be designed. In the early 1990s the German company Bosch patented a process according to the principle described by Laermer and Schilp [1994] [Gunn et al. 2009], which has become one of the most important tools in surface and bulk micromachining. The deposited film is a polymer, which results from the chemical reaction between SFx and CFx . The process gases SF6 and C4 F8 are used for the socalled “Bosch process.” In the first step, plasma etching is performed by sulfur hexafluoride, SF6 , which opens the cavity (the left-hand picture in Fig. 4.11). The next step is passivation with C4 F8 whereby the polymer film covers the sidewalls as well as the horizontal bottom. The film on the bottom is removed rapidly by physical etching at the beginning of the next etch step, while the sidewall film is removed slowly, preventing etching of the sidewall. At the end of the etch cycle the film is completely removed and the process starts again. Typical cycle times are less than 10 s and the process is repeated many times. The Bosch process features excellent properties. Fast etching rates of about 5 to 10 µm/min are common. The deviation from 90◦ is typically 0.1–0.2◦ . Deep etching through a wafer, with sidewall accuracies on the order of a few

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sidewall angle 0,1° notching ~200x400nm

Figure 4.12 Polysilicon over a sacrificial layer, etched with DRIE. Courtesy of ISIT.

micrometers, is possible. The repetitive nature of the process leads, however, to slight scalloping of between tens of nanometers and fractions of a micrometer (0.2–0.5 µm), which is usually not critical. Figure 4.12 shows an SEM photo of a DRIE-etched, 11-µm-thick polysilicon layer, grown over a sacrificial SiO2 layer, which acts as an etch-stop layer with a material selectivity between Si and SiO2 of 100 : 1. The sidewalls are inclined by less than 0.1◦ . The notching or footing effect at the bottom is caused by charging of the silicon oxide or polysilicon oxide interface, which, in the case of overetching, leads to ion reflection and attack of sidewall passivation. The dimensions in the right-hand figure serve as references for an estimate of the undulation amplitude, which here is on the order of 10 nm. Another effect that has to be controlled is “bowing,” i.e. a stronger lateral sidewall etch near the mask openings. This effect is clearly caused by changing charges along the sidewall that disturb the isotropic ion flux. It occurs only in the case of overetching of a (poly)silicon/dielectric interface at the bottom of the trench. For some applications, such as springs in gyroscopes, the asymmetry of the cross-section causes cross-coupling between different vibration modes. The crosssection asymmetry and inclination are the most critical issues for such coupling effects. The radial dependency of the ion density and, correspondingly, of the offaxis ion trajectories within the plasma, as exaggeratedly illustrated in Fig. 4.10, creates such sidewall tilts that are larger the greater the distance of the spring from the wafer center. Sensor dies in the wafer center are usually affected much less than outside dies. The resulting spring-rate changes were considered in Chapter 3. The impact on gyroscopes will be considered in more detail in Chapter 8. Countermeasures are presented in Merz et al. [2007].

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4.2

Wafer bonding Etching is removal of material. Wafer bonding is complementary, involving addition of material. The latter has numerous applications: manufacturing silicon-oninsulator (SOI) wafers, forming MEMS structures, and/or providing zero-level packaging. It is a key technology for inertial MEMS. SOI wafer production is a bonding process, in which the entire oxidized wafer area is bonded with an Si counter-wafer. SOI wafers are the starting material for micromachining movable structures with high aspect ratio – a technique that is rapidly gaining acceptance. In contrast, wafer bonding for building MEMS structures or performing zero-level packaging requires patterned substrates in which only selected areas around the intended cavities, called bond frames, are bonded together. Thus the wafer–wafer composite constitutes a regular mosaic of dies bonded with their counterparts and containing cavities enclosed between the dies, as well as between the substrates in the surrounding areas. Alignment accuracies between two wafers differ depending on positioning requirements for MEMS substructures or for die-to-die and the respective wafer-to-wafer alignment.

4.2.1

Zero-level packaging and wafer bonding Zero-level packaging is one of the driving forces for wafer-level bonding technologies. Indeed, most of the MEMS structures require a special working environment to protect fragile structures during final dicing and/or during operation (e.g. pressure-sensor membranes) or for creating hermeticity and vacuum conditions. Protection against particle contamination and mechanical overload, against corrosion and stiction, against material change, e.g. under the impact of water molecules, and against changes of the gaseous environment is required. Zero-level packaging is the packaging step that protects the fragile sensor structure but does not ensure mechanical and electrical assembly of the entire system on the component level. It consists of creating a cavity and electrical connections to the outside world. Zero-level packaging is typical for MEMS, and antedates the first-level packaging, which affords a sensor system ready for placement on and connection to a PCB. Individual sealing of every die by a cap is possible and was sometimes used during the early MEMS phase; however, it is normally far too expensive for volume production. However, the technique was reborn with mature surface micromachining processes and will be discussed in Section 4.3.2 under the heading “Cavity sealing using SMM.” Another form of protection is made possible by placing MEMS sensors, together with their ASICs, directly into hermetic or vacuum packages that are equipped with external leads. Zero-level and first-level packaging merge in this case. To ensure hermeticity, the ceramic packages or carriers are enclosed in a

4.2 Wafer bonding

173

Sawing cap wafer Sealing ring

Bond pads

isolation

beam

patterned conducting layer

carrier wafer

Figure 4.13 The principle of zero-level packaging.

metallic can as, for instance, in the case of some commercial gyroscopes. However, zero-level packaging should already be present during die separation (dicing) in order to prevent the impact of particles or humidity that makes it necessary to implement at least temporary protection. For zero-level packaging of inertial MEMS the following aspects have to be considered:

r mechanical protection during dicing and operation r hermeticity over the lifetime of the component r creation of a chemically benign gaseous environment with the required working pressure

r “stress-free” packaging to avoid a disturbing impact on the sensor structure r (partial) compensation of the add-on costs by simplified first-level packaging r robustness with respect to the first-level packaging (for instance within an overmolded plastic package)

r electrical connection to the outside world with low feedthrough resistance and good isolation of components from each other

r the provision of an electromagnetically quiet cavity (shielding). A chemically benign environment means, inter alia, that in the case of hermetic encapsulation, the fill gases must be chosen properly. For instance, sealing small silicon cavities in an atmosphere at normal pressure (1 atm) will, at high temperatures, lead to a pressure change inside the cavity due to the thermal oxidation at the silicon surface. After the 20% of the air constituted by oxygen has been consumed, a pressure of 0.8 atm will be established. Using noble-gas environments avoids such effects. A typical zero-level packaging approach is demonstrated in Fig. 4.13. The sensor, here represented by a simple acceleration-sensitive beam, is encased by a cavity formed by the slightly doped device wafer (bottom) and the top cap wafer. Both wafers, thus, exhibit some conductivity, which benefits electrical shielding, provided that a conducting bond interface is established. A special bond-frame area has been introduced, where the two wafers are bonded together.

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The sensor has a capacitive interface represented by some electrodes patterned within the conducting layer. Electrical connection to the outer bond pads can be created in various ways. For instance, the electrical interconnection to the outside world may be simply formed by a highly doped, conductive well (p–njunction feedthrough) within the substrate, which does not disturb the surface flatness of the substrate. Other feedthroughs are based on buried electrodes, on sealed feedthrough channels, and on thermomigration of aluminum [Ko 1995]. Lead-through of the inner connection points through vertically etched vias within the substrate wafer is also possible. It is area-consuming and expensive and should, if possible, be substituted by low-ohmic horizontal feedthroughs, which lead the interconnects from the inner space to outside as shown in Fig. 4.13. If necessary, isolating layers can be used to prevent contact with conducting substrates. Various materials and layer combinations, such as aluminum between insulating films, are used, some with more than one conducting layer. The resulting surface profile may be uneven. In any case, direct bonding of the device wafer and the cap wafer has to tolerate such possible unevenness or the unevenness has to be eliminated before bonding. Wafer bonding for zero-level packaging is performed in special bonding equipment that is able to perform wafer-to-wafer alignment with the accuracy needed for the exact alignment of opposing dies. The wafers to be bonded can be exposed to well-defined pressures and temperatures by heating and compressing chucks, which carry the wafers. Karl Suess (Germany, now SUSS Micro Tec) and Electro Vision Group (EVG, Austria) offer high-quality equipment for nearly all waferbonding processes. After bonding the wafer must be diced in order to separate the individual chips. Dicing is usually performed in two steps. In the first, the outer bond pads are opened, by sawing through only the upper cap wafer along the two marked lines shown in Fig. 4.13. By accessing the opened bond, wafer probing can be carried out. The final dicing along the middle lines then separates the individual dies. Edge phasing can also be performed using various types of saw blade. This may be required for better second-level packaging, if the die has to be covered by some coating material.

4.2.2

Wafer-bonding processes Summarizing the requirements of zero-level packaging, appropriate wafer bonding processes are decisive for MEMS-device packaging. There are five classes of wafer bonding:

r silicon direct bonding (silicon-fusion bonding, SFB) for Si–Si, SiO2 –Si, Si– glass, and glass–glass bonding

r anodic bonding (field-assisted bonding) for Si–glass, glass–Si–glass, and glass– metal bonding

4.2 Wafer bonding

r r r r

175

low-temperature glass-frit bonding metallic-alloy seal bonding (eutectic, liquid transient phase) polymer bonding (adhesive polymer bonding) thermocompression bonding (TCB) (solder, soft metal thin films).

SFB is typically a high-temperature process with temperatures above 800 ◦ C whereas the other technologies apart from eutectic bonding are low-temperature processes with temperatures T < 450 ◦ C. With respect to the total thermal budget low-temperature processes are compatible with standard CMOS processes, in particular with sensitive aluminum–silicon contacts, whereas high-temperature process steps cannot be performed on wafers with typical CMOS structures without potential deterioration.

Fusion bonding Direct silicon-to-silicon bonding is a high-temperature process proceeding in two steps. The first is a joining process at room temperature with weak adhesive forces. The joining forces develop between cautiously pressed-together, hydrated (hydrophilic or hydrophobic) surfaces [Bengtsson 1992], which are created during the preparation process [Vansant et al. 1995]. The second step is a hightemperature fusion process. For Si–Si bonding, the preparation includes the standard RCA cleaning process4 or boiling in nitric acid or ammonium hydroxide, which leaves the wafer surfaces in a hydrophilic condition with Si—OH groups. Two wafers brought into close contact are bonded together by hydrogen bonds. The reaction products are hydrogen in the case of hydrophobic bonding (wafer preparation in HF solution) and water in the case of hydrophilic bonding. The second step is the fusion process at temperatures of about 1000 ◦ C, whereby a chemical reaction between silicon and oxygen forms strong siloxane Si—O—Si bonds. It is based on the reaction Si—O—H + H—O—Si → Si—O—Si + H2 O,

(4.2)

which starts above 300 ◦ C. Significantly above 300 ◦ C, water dissociates and the additional, non-bonded oxygen bonds together with Si, forming Si—O bonds. Liberated hydrogen diffuses onto intermediate lattice positions. Depression takes place at non-bonded places, pushing the wafers together, which strengthens the fusion. Maximal applicable shear loads of 20 MPa are reached at temperatures of about 1000 ◦ C. The bond strength may exceed the fracture strength of a silicon crystal. The identical temperature coefficients of the two wafers ensure that practically no stress is introduced into the interface and, importantly, stress does 4

The standard RCA clean was developed by Werner Kern from RCA in 1965 and includes organic clean, oxide strip, and ionic clean stages.

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not appear during subsequent high-temperature processes such as oxidation or diffusion. Fusion bonding is most frequently used to bond silicon with oxidized silicon wafers in order to manufacture SOI wafers. The resulting wafer stack is thinned from one side, leaving a silicon-on-insulator layer of thickness from some tens of micrometers to more than 100 µm. The oxide layer may be up to a few micrometers thick. The key element for fusion bonding of silicon and silicon-related materials such as silicon oxide, silicon nitride, and polysilicon is surface roughness. A roughness of less than 1 nm is mandatory. Brief chemical–mechanical polishing (CMP) before bonding can reduce the roughness to < 1 nm and improve bond quality significantly, even for poorly bondable materials [Gui et al. 1997]. Fusion bonding is attractive for wafer-level packaging. The superior bond strength and the absence of intermediate adhesive layers, which would possibly be sensitive to aging and corrosion, make fusion bonding well suited for hermetic vacuum packaging. However, the need for vigorous cleaning and the high temperature are often in conflict with the preceding MEMS-fabrication processes (implementation of metal wires, setting of doping profiles) or with the robustness of the unprotected MEMS structure. Many efforts to reduce the bonding temperature to below 400 ◦ C have been made. The approaches are focused on increasing the surface bond energy, e.g. by the creation of “designed” monolayers or by plasma treatment after the RCA clean. Wafer-level packaging using fusion bonding with a thin intermediate glass layer is possible. The glass layer is melted and then cooled, forming a hermetic bond. Bonding of two oxidized wafers can also be performed.

Anodic bonding Field-assisted bonding is a joining method that is widely used to bond chips or entire wafers hermetically with a glass carrier. Native oxides or thin deposited oxides on silicon do not retard bonding. Glass materials such as Pyrex (Corning #7740) and Tempax (Schott), both with thermal expansion coefficients (CTEs) of 3.3 × 10−6 /◦ C, were developed to match the CTE of silicon at the bonding temperature, as well as within the operating range around room temperature (2.5 × 10−6 /◦ C at 25 ◦ C and 4.0 × 10−6 /◦ C at 450 ◦ C ). Both materials are heated up by a grounded hotplate to temperatures between 300 and 450 ◦ C in a normal or noble-gas atmosphere or in vacuum. At this temperature the sodium oxide Na2 O separates into mobile Na+ and negative oxide ions (O2− ). On applying a negative potential between some hundreds of volts and 1000 V to the glass wafer, the Na+ ions start to move towards the outside glass surface, where they are neutralized. The negative O2− ions travel towards the positive silicon wafer, creating a depletion layer. The emerging electrical field causes strong attractive forces that pull the two wafers together. They are able to squeeze and to eliminate small residual gaps between the two wafers.

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177

The initiation of bonds can be supported by applied pressure. However, in vacuum the pressure-assisted initiation may be dispensed with. The fusion process relies on oxygen ions leaving the glass under the impact of the strong electrical field within the depletion layer. At the silicon–glass boundary they form Si—O bonds. Thus, the fusion process is a chemical reaction, and the generation of electrostatic attraction is a necessary preliminary stage. Reliable bonds require polished and particle-free surfaces. The roughness should be less than 1 µm. The bonds become less stressed the more closely the thermal expansion coefficients match over the entire temperature range – from the bonding temperature down to operating temperatures. However, some residual stress is unavoidable, though it is lower the smaller the fusion temperature. Thus, there has to be a compromise between residual stress and bond strength. Another drawback is the typically large bond-frame width, on the order of 200 µm, which consumes valuable die area and causes additional costs. In extreme cases, the bond width can be brought down to approximately 20 µm with much additional effort. Thin metallic feedthroughs (<0.2–0.3 µm) can be accepted without significant deterioration of hermeticity or bond strength. This is a big advantage of the method, and allows the integration of simple electrical interconnects. Also thin oxide layers on silicon can be bonded to glass. Bonding of multilayer structures such as glass–silicon–glass can be performed within a single anodic bonding process, provided that electrical contacts to all three layers and appropriately uniform heating are established. Similar arrangements were used for manufacturing the first batch-fabricated MEMS accelerometer [Roylance and Angell 1979] as well as successors with large proof masses (e.g. Matsumoto and Eshai [1993] and Takao et al. [2001]). Another possibility is silicon-to-silicon bonding with an intermediate thin glass layer evaporated or sputtered onto one of the wafers. However, it is not easy to achieve uniformity of the deposited glass layer and low-stress interfaces. Silicon-to-quartz and glass-to-metal anodic bonding can be performed without any intermediate layer.

Glass-frit bonding This bonding process is similar to anodic bonding of glass and silicon, except that the pressure is mechanical rather than electrostatic. The technique is used on the wafer level as well as for singular chips to be assembled on a substrate. Low-melting-point glasses are used for glass-frit bonding. They are available as paste, which allows screen printing to be applied to deposit and to structure the paste on one of the wafers to be bonded. Alternatively, sputtering of a glass layer is possible, but this approach is rather seldom used in recent MEMSencapsulation techniques. The paste consists of glass powder, organic binders, and solvents. The wetting temperature of a melting glass compound is typically less than 450 ◦ C. Aluminum

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glass frit bond interlayer bonded wafers

Figure 4.14 A glass-frit bond interface. From Knechtel [2004], with permission.

interconnects do not suffer at such temperatures. The CTE of the glass can be adapted to the desired value of the bonding surface by adding mineral filler particles. After the paste has been printed, thermal conditioning is needed before the bonding process. During this preconditioning the solvent is burned out and the glass is pre-melted. In the final step the two aligned wafers are pressed together and the glass is heated up to the wetting temperature, at which its viscosity decreases until it wets both surfaces. The subsequent atomic reactions are similar to the silicon–glass anodic bonding process, forming strongly fused regions on both silicon surfaces. The pressure has an equalizing function only and is aimed at eliminating wafer warping and supporting the initial wetting. The cooling-down profile after fusion has to prevent thermal shock and cracking. Care has to be taken to prevent uncontrolled glass flow into the cavity. If necessary, flow stoppers must be implemented. In Fig. 4.14 a cross-sectional view of the bond interface is shown, illustrating “impressive” vertical bond dimensions. The height of the printed structure is about 20 to 30 µm, but the bond frame has a width typically equal to or larger than 200 µm. For widths > 250 µm Glien et al. [2004] investigated the fracture strength of the bond frame, revealing a linear dependency on the bond-frame width. Thus, reduction of the bond-frame width corrupts reliability. Such large bond frames cause serious additional area consumption, which may be critical for fitting in the final package and for costs. The height after bonding is reduced with a relatively large spread, not allowing the realization of accurate gaps. However, it is large enough to tolerate lead-through with height up to a few micrometers. Normally the bond strength is higher than the fracture strength of glass. But the bond reliability is sensitive to large, rapid thermal shocks [Knechtel et al. 2006]. Vacuum encapsulation of structures in glass-frit-bonded cavities is reliable down to moderate pressures of the order of a few mbar. Many millions of gyroscopes were fabricated by the German company Bosch in glass-frit-bonded

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179

cavities. However, the lower pressures are strongly compromised by the (organic) outgassing products of glass and at least require the use of getter material. Glass-frit bonding can be used for nearly all common surface materials in MEMS, including polysilicon, insulator and passivation layers, metals, and special layers such as polyimide.

Metallic-alloy seal bonding Eutectic bonding is the most important representative of seal bonding using metallic alloys. During a eutectic bonding process an alloy is created at a temperature that is significantly lower than the melting points of each of the bond partners.5 Silicon and gold constitute a eutectic alloy product with 97.1% Au and 2.85% Si, which forms at a eutectic temperature of 363 ◦ C. If two silicon surfaces have to be bonded together, the interface material, for instance preformed gold, is placed or deposited onto one of the wafers. The wafers are then brought into intimate contact by applying external pressure, and heated up to above the eutectic temperature. Now the Au atoms start rapidly diffusing into the atomic structure of both Si wafers, forming the eutectic alloy. The initially liquid gold film vanishes. Subsequently the structure is cooled down. Instead of pure gold, gold–tin alloy (Au 80%/Sn 20%) can be used. To improve the control of bonding more sophisticated metallic multilayers are used instead of a single gold layer, especially if silicon is to be bonded with SiO2 . The stack consists of an adhesion layer (usually Ti), a diffusion barrier (e.g. Pt or Ti) to prevent undesired diffusion of gold, and a plating base for the basic gold layer. With optimized temperature profiles, such structures increase adherence and prevent anodic dissolution of adhesive layer and other effects endangering the longevity of the hermetic bond in a harsh environment. The choice of an optimized temperature profile for heating up and cooling down has a large impact on the bond quality (e.g. Lani et al. [2005]). Despite some residual local stress within the eutectic alloy and the possibility of subsequent creep caused by the CTE mismatch, eutectic bonding is one of the most attractive solutions for wafer-level vacuum packaging of inertial MEMS, especially for vibrating gyroscopes: it is a low-temperature process; the bond frame around the cavity can be as small as 50 µm; the requirements concerning substrate roughness and planarity are less strong than for direct bonding techniques and allow bonding over non-planar surfaces; it does not need high voltages like anodic bonding, which can destroy electrostatically embedded structures; outgassing and hermeticity performances are superior in comparison with bonding processes using glass or organic films; and the dwell time for one bond cycle is comparable to or less than that for fusion and anodic bonding.

5

Wolffenbuettel [1997] states that it is most likely that the initiation phase is not purely eutectic but includes silicidation.

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The tin–lead alloy (Sn 62%/Pb 38%) is another well-known eutectic alloy, having a eutectic temperature of 138 ◦ C. It is cheaper than Au–Si alloy and can bond nearly all metallic surfaces. If non-metallic surfaces have to be bonded together, a thin film of metal has to be sputtered first before the tin–lead alloy is exposed.

Polymer bonding Adhesive bonding using polymers such as epoxy resins, polyamides, negative photoresists, and so on is a simple, benign, and low-cost process. It tolerates some particle contamination and surface unevenness as is typical for structured wafers. Only low temperatures, < 170 ◦ C, must be applied for curing. The dispensing process, which must provide an exact amount of adhesive on the bonding area, may be problematic. Usually spin coating is used. After deposition the adhesive is exposed to an initial bake to burn out solvents. In the next step the wafers are joined under a well-defined pressure, and finally the polymer is cured over typically 0.5–1 h. The curing temperature must be above the glass temperature, at which the polymer becomes subject to plastic deformation. Polymers age and, thus, feature a limited stability over their lifetime. The maximal storage temperature is below a few hundred degrees Centigrade. They undergo outgassing and are open for diffusion. Hence, hermeticity or vacuum conditions cannot be guaranteed. All this limits their application for wafer-level bonding. However, they are the materials of choice for chip-to-carrier bonding, for which simplicity and cost are dominant factors.

Thermocompression bonding The term “thermocompression bonding” (TCB) reflects not a common physical principle, but rather the conditions under which bonding is performed. Often eutectic bonding is subsumed under TCB. Sometimes also polymeric bonding is considered as a kind of TCB. Here TCB is treated much more narrowly, being taken to include only bonding based on solder or soft metal films. Solder bonds and TCB of soft metal films are similar to the adhesive bonding described above. Deposition of solder or soft metal is possible using preforms or layer deposition and patterning. The mating surfaces have to be pre-coated with material that the solder can easily wet. The bonds are stable and do not outgassing exhibit, which makes them suitable for hermetic and vacuum packaging.

4.3

Integrated processes The design and fabrication of MEMS products is a process that usually starts from system design based on a target specification. A first geometric and electrical blueprint is created by system designers in interaction with packaging and test experts and with process specialists. The result is a basic concept including

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181

r a high-level system description (physical and functional system partitioning) r the geometry, material, and fabrication environment of the basic components r fabrication, subassembly, and test stages. The fabrication, assembly, and test stages are usually subdivided into two sections: before and after zero-level packaging. In this context, an integrated MEMS process is understood as the section encompassing fabrication and subassembly of all components including zero-level packaging. The integration of micromachining process steps in order to fabricate a zerolevel packaged MEMS device is a complicated process. This is due to the nonstandard nature of combining different fabrication steps, such as bulk or surface micromachining, with wafer-level packaging or, even more so, with co-integration of ICs. Further, MEMS devices exploit a great variety of new materials and technologies not known before from the CMOS portfolio. This is not only related to additional materials such as nickel and platinum or sensitive films like PZT, but sometimes concerns the basic material itself, as for instance in the case of using ceramic, glass, quartz, or metallic carriers. The complicated 3D nature of MEMS devices formed by high-aspect-ratio structures and wafer bonding with different, sometimes double-side-processed materials leads to a multitude of special process steps and process sequences, which often represent only specific interests. Technically there are no limits to the creativeness of process experts in realizing MEMS structures. Such a situation is termed technology-driven and is limited only by restrictions exposed by the efficiency and reliability requirements of existing or new killer applications. The latter lead to a more focused selection of possible materials and process flows.

MEMS processes and CMOS-co-integration MEMS processes can be divided roughly into

r bulk micromachining r surface micromachining r SOI processes. Bulk micromaching and surface micromachining are often used in combination, as for instance in a SOI process. They can be performed as independent processes or integrated with ASIC manufacturing on the same die. In the first case the product is a hybrid component consisting of a MEMS die and an IC die, which then have to be packaged together during first-level packaging. The advantage of a hybrid approach is the freedom to choose a well-suited MEMS technology independently of compatibility with a CMOS process. Furthermore, the yield is usually higher, because the MEMS process can be optimized, and inoperative MEMS or ASIC dies will not be assembled together as is the case with MEMS–ASIC co-integration. The different MEMS–ASIC-co-integration approaches are classified according to the location of MEMS fabrication within the CMOS process [Baltes 1997,

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Implemented piezoresistor

lead

silicon

200 m silicon rim

glass cover

glass cover

conducting epoxy

silicon – seismic mass

Figure 4.15 The principle of the bulk-micromachined piezoresistive accelerometer according to Roylance and Agnell. Adapted from Roylance and Angell [1979].

Baltes and Brand 2002, Baltes et al. 2005, Ghosh and Bayoumi 2005] as

r pre-CMOS r intra-CMOS r post-CMOS. Co-integration means not only permanent stress control during the whole process flow but mainly overcoming possible incompatibilities between MEMS and IC fabrication. Such incompatibilities may be caused by the type of materials used, by the thermal budget, or by non-conformity of the layers. For instance, some electrically active materials, such as gold and copper, are rapid diffusers and are thus not allowed in IC fabrication lines. The same is true for some adhesives or polymers desirable for MEMS packaging but capable of introducing contamination. The thermal budget is determined predominantly by the CMOS front-end process, which includes all of the high-temperature steps and is performed before interconnection and packaging (the back-end process). If a subsequent MEMS process changes the thermal budget, diffusion profiles may change. If, in contrast, high-temperature steps are performed after MEMS structuring, the structure of polycrystalline materials such as polysilicon may change and the additional stress may become critical. However, the possibly strictest limitation is the use of low-temperature metal interconnects before high-temperature steps, which is absolutely prohibited.

4.3.1

Bulk micromachining The first batch-fabricated accelerometer in MEMS technology was the bulkmicromachined 50-g accelerometer of Roylance and Angell [1979]. It is a glass– silicon–glass sandwich device of length 3 mm with a seismic mass as shown in Fig. 4.15. Acceleration sensing is performed by piezoresistors implemented within the beam, which carries the seismic mass. Although the device is the veteran of inertial MEMS, the fabrication steps contain many elements that are still

4.3 Integrated processes

after back-side etch

183

2h

front-side window for final etch

before final etch

Figure 4.16 The etch sequence to define beam and free-space regions. Adapted from Roylance and Angell [1979].

present for more advanced products. The fabrication flow comprises the following steps.

r Etching of widely spaced alignment holes through the wafer. r Thermal oxide growth on both sides, patterning and diffusion of ppiezoresistors and p+ -contacts on the front side.

r Protection of the front side by thick SiO2 , photolithography of the back side, and anisotropic KOH etch of areas where the silicon must be removed or thinned down to double the beam thickness h (see the upper picture in Fig. 4.16). r A final etch through opened front-side windows opposing the desired etchtrough regions (see the lower picture in Fig. 4.16). Since the intended freespace areas are etched from both sides, they open when the desired beam thickness h is reached. r Final oxide stripping. The one-sided polished Pyrex-glass wafers are prepared by etching the required cavities. After stripping off the masking layers, the aluminum pads are deposited onto the top wafer, and the wafers are bonded together by anodic bonding at 400 ◦ C and 600 V. Dicing, electrical connection, and protective coating of the hermetically sealed dies are performed after bonding. Similar bulk-micromachining flows have been used to fabricate accelerometers with capacitive interfaces. The electrodes were implemented on the glass wafers opposing the movable structure. No well-etching within the glass wafers is required, because the small gaps between the electrodes and the movable mass can be defined by additional distance layers on the silicon wafer. Anodic bonding tolerates the unevenness caused by the aluminum interconnects through the bond frame. More advanced bulk-micromachining processes often include more complicated suspension structures, etch-stop formation, and piezoelectric transducers. Most of the bulk-micromachined inertial MEMS are accelerometers, because they benefit most from the bulky seismic masses. However, there are also successful

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gyro implementations using bulk micromachining. For instance, BAE systems presented in 2005 a ring gyroscope (see Chapter 8) manufactured in a Pyrex– silicon–Pyrex sandwich [Eley et al. 2005], where structuring of the silicon wafer is performed after the first anodic bond. The first glass wafer consists of through holes for contacting and cavities for the moving structure; the second glass wafer has cavities for the moving structure and for getter deposition. Another interesting approach uses skewed beams for gyroscope suspensions, the cross-sections of which follow the anisotropic etch angles 35.27◦ of the {111} planes. In this way a combined vertical and horizontal excitation of vibrating masses is easily realized [Andersson et al. 1999]. Bulk-micromachined wafers are fragile after deep etch. Dicing (sawing), packaging, and wafer-level testing are difficult. Shaping of structures is often limited by the anisotropic wet-etching processes. Generally speaking, the benefits of bulk micromachining are transferred continuously to SOI technologies, which are much more CMOS-friendly and allow similar spring–mass systems and vertical sidewall capacitances to be realized.

4.3.2

Surface micromachining Surface micromachining is a layering technique in which the 3D structures are created by film deposition, layer patterning by DRIE etching, and release from underlying layers by wet or vapor etching. The fabrication steps are very similar to those for standard CMOS processing. Film thicknesses are between fractions of a micrometer and several micrometers if CMOS process steps are used. Extensions of up to some tens of micrometers are popular. The structural layer is usually polysilicon. Despite the fact that the material properties of polysilicon, such as its yield strength and piezoresitivity, are inferior to those of crystalline silicon, they are superior to those of metal films. Importantly, fine-grained and texture-free polysilicon is mechanically isotropic, which simplifies the design. Annealing of polysilicon at temperatures around 580 ◦ C is necessary in order to transform the LPCVD-deposited amorphous silicon into grainy polysilicon and release the accumulated stress. This temperature regime fits well with the high-temperature CMOS front-end processes. Not surprisingly, a great variety of CMOS integrated surface micromachining processes are known and some of them will be discussed in Section 4.3.4. Surface micromachining with extended layer thicknesses of about 10 µm is of great interest, especially for inertial MEMS. In the mid 1990s Lange et al. [Lange et al. 1995, Marek et al. 2003] developed a deposition process for thick polysilicon layers using an epi batch reactor with growth rates around 0.5 µm/min. On the basis of this approach, some commercially successful, integrated surface micromachining processes were developed and are being exploited, for instance, by the German company Bosch and by ISIT/FhG.6 6

ISIT/FhG is the abbreviation for the Institut f¨ u r Siliziumtechnology of the Fraunhofer Gesellschaft (FhG).

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185

Figure 4.17 A released epi-poly structure within a vacuum cavity. Courtesy of ISIT.

(This figure also appears in the color-plate section.)

A thick polysilicon process This process is used, for instance, for the fabrication of vacuum-encapsulated vibratory gyroscopes and accelerometers [Offenberg et al. 1996]. An example of a final device manufactured for SensorDynamics AG using the ISIT process PSM-X2 is shown in Fig. 4.17 [Merz et al. 2005]. The heart is the released moving structure in the center, formed by DRIE etch of a thick “epi-poly” layer. This layer is grown at around 1100 ◦ C in an epitaxial reactor, using dichlorosilane as the process gas and phosphine as the doping source. After annealing, the stress behavior is excellent, with light compressive stress below 5 MPa and stress gradients less than 0.1 MPa/µm. The poly film is deposited on top of one or two layers: directly onto the sacrificial oxide or onto a buried poly layer, which is deposited over the sacrificial oxide. The doped buried poly layer forms the interconnects, so that fixed and moving structures can be easily connected to any potential. The entire moving structure is hermetically sealed by a cap wafer. The bottom and cap wafers are connected by eutectic bonding, which allows pressure levels between several parts of a microbar and atmospheric pressure to be realized. The bond frame of the cap wafer is covered by oxide, which serves as a diffusion barrier. The metal layers consist of a plating base for the gold frame. A sophisticated getter-deposition [Moraja et al. 2003] and bonding procedure, together with an individual leakage control, guarantees stable vacuum conditions over a lifetime of more than 17 years. The ultra-fine leak test [Reinert et al. 2005a, 2005b] exploits internal high-quality resonators, which can be part of the product or can be implemented as additional test structures. Quality-factor changes can be measured during wafer sorting and are a highly sensitive indicator for vacuum-level changes. After bombing the wafer in pressurized neon for some hours, the cavity pressure changes according to the individual leak rate, because noble gases are not absorbed by the getter. The corresponding Q-factor changes are used to calculate the individual leakage rate in air. Fine leaks up to a critical leakage rate of 0.5 × 10−14 mbar l/s can be detected. The PSM-X2 process includes three basic components:

r sensor-wafer processing, with a surface-micromachined MEMS structure r cap-wafer processing, including patterning with a high-efficiency getter film r eutectic bonding of the sensor and cap wafers.

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LPCVD buried poly-Si thick isolation oxide Si substrate deposition protection layer deposition sacrificial oxide 1 Litho 1 + RIE buried poly deposition sacrificial oxide 2 Litho 2 + RIE protection layer

Litho 4 + RIE isolation oxide Litho 3 + RIE sacrificial oxide

Litho 5 + dimple wet etch

thick epi-poly deposition

epi-poly polishing

Litho 6 + AlCu wet etch deposition AlCu

Litho 7 + DRIE epi-poly

HF vapor-phase etch

Figure 4.18 PSM-X2 process flow 1. Courtesy of ISIT. (This figure also appears in the color-plate section.)

The detailed process flow for both the bottom wafer and the cap wafer is illustrated in Figs. 4.18 and 4.19. For better correspondence of the layer and deposition steps within a subfigure, a later step is noted on top of the previous step (e.g. Litho 4 on top of Litho 3).

4.3 Integrated processes

187

Litho 1 + back-side RIE wet oxidation

Litho 2 + front-side RIE wafer turnover

Oxide removal KOH cavity etch

Litho 3 + gold electroplating plating base deposition wet oxidation

getter deposition

Figure 4.19 PSM-X2 process flow 2. Courtesy of ISIT. (This figure also appears in the color-plate section.)

The fabrication of the bottom sensor wafer requires seven masks. One mask is dedicated to structuring the so-called protection layer, which is introduced intentionally to protect open buried poly areas outside the cavity. Since the open areas are in trenches, as shown on the left-hand side in Fig. 4.17, they may be affected by conducting particles generated during the dicing process. Another mask is used to implement dimples on the underside of the moving structure to avoid stiction. The cap wafer is a double-sided processed substrate. Alignment marks on the back side are implemented first, then a KOH etch of the front-side cavity follows. The resulting surface is wet oxidized and patterned with a titanium adhesion layer and a subsequent gold frame. The patterning lithography of gold is performed over the deep-cavity topography using a thick photoresist with corresponding viscosity. The oxide on the cap wafer would isolate the two wafers from each other; however, the plating base is deposited over the entire cap wafer and creates, via the metallic bond-line, a Faraday cage together with the sensor wafer. This surrounding shield protects the structure against external electromagnetic fields. The gas-absorbing getter material is sputtered by the SAES Getters Group S.p.A., which offers a patterned thin-film getter deposition by the proprietary PaGeWaferTM technology. A shadow mask ensures a 30-µm positional accuracy for the 3-µm-thick getter film. Getters are transition-metal-based materials that, after activation, are highly reactive with nitrogen, oxygen, carbon, and other substances. The getter material acts as a small vacuum pump. The SAES’s getter

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is a zirconium-based alloy that, after activation at T > 200 ◦ C, absorbs active gases including H2 O, CO, CO2 , O2 , H2 , and N2 . The desired pressure level below 1 mbar is adjusted using a backfill process: the bond chamber is filled with an inert gas, which also enters the cavity. After bonding, the outgassing products of the cavity are absorbed as soon as the getter is activated, leaving the noble gas inside at the desired initial pressure. The sorption capacity of the deposited getter layer is designed to hold the vacuum level over 17 years, provided that the leakage rates and expected amounts of outgassing are within the set limits. This PSM-X process is just one of the many similar thick-poly processes that are used in surface micromachining of inertial MEMS. It comprises most of the typical process steps and integration tasks. The sensor-wafer process remains unchanged if, instead of eutectic bonding, glass-frit or other bonding techniques are used, which is especially interesting for not-too-high vacuum levels. Thick epi-poly processes like the PSM-X process are cost-efficient and robust. The entire costly front process up to the epi-poly deposition (first five masks) can run in nearly any CMOS fab. This is a great advantage, and keeps the position of thick epi-poly processes within the range of dominant technologies for inertial MEMS. The co-integration of such processes with CMOS fabrication has to face two problems. First of all, the CMOS devices should be not affected by the hightemperature poly deposition. This problem can be solved by performing the CMOS fabrication after epi-poly deposition and polishing, and adding DRIE and release etching at the end of the process. Secondly, CMOS processing requires planar wafer surfaces. One solution would be to bury the epi-poly and underlying layers in deep trench areas, so as to achieve planarity of the polished epi-poly and wafer surfaces. However, the thick-poly SMM processes are not close enough to CMOS processing to call for the additional co-integration effort, and not far enough away to be a specialized process with unchallenged advantages for realization of inertial MEMS. SOI processes are strong competitors here, a prototype of which will be presented in the next but one section.

Cavity sealing using SMM Surface micromachining offers the unique possibility of building sealed cavities over micromechanical structures. An example of the so-called reactive sealing based on a nitride diaphragm is shown in Fig. 4.20. In the first step the MEMS structure is surrounded by an oxide stack (Figs. 4.20(a) and (b)) that is covered by the LPCVD diaphragm nitride (Fig. 4.20(b)). The oxide is removed by a selective isotropic etch through the small openings left within the nitride (Fig. 4.20(c)), and the open cavity is sealed in a final step by an 850-◦ C LPCVD nitride. The nitride coverage is excellent and fast so that only a small amount of nitride can enter the cavity.

4.3 Integrated processes

(a)

(b) sacrificial oxide

189

diaphragm nitride

MEMS structure oxide

silicon substrate

(c)

silicon substrate

(d)

silicon substrate

sealing nitride

silicon substrate

Figure 4.20 Integrated cavity sealing using SMM techniques: (a) the MEMS structure,

(b) the embedding of an oxide stack and nitride coverage, (c) oxide removal, and (d) final nitride sealing.

Alternatively, a polysilicon cap over a polysilicon MEMS structure can be used. After removal of the sacrificial PSG oxide, the cap is sealed to the substrate by thermal oxidation at 1000 ◦ C. Such sealed caps are small and do not require additional space. The disadvantages of reactive cavity sealing are the relatively high temperatures and long etch times. One way to overcome this problem is to deposit polysilicon layers with permeable windows onto the openings of the diaphragm nitride. The pore size is between 5 and 20 nm and, after etch, allows subsequent safe sealing by the LPCVD nitride. Since the windows can be large, the HF etch of the sacrificial PSG runs very fast [Lebouitz et al. 1995]. There are other SMM-based cavity-sealing methods. Epitaxial cavity sealing, for instance, consists of the alternating deposition of epitaxial silicon layers with p+ - and p+ + -doping and, hence, with different etch rates. The p+ + -layers with the MEMS structure remain after etch, while the removed p+ -layers leave the cavity. If the sealing is performed in vacuum, stable low pressure levels within the cavity can be achieved. The integrated sealing processes described here are, however, more complicated and less versatile than wafer-bonding techniques.

4.3.3

SOI-MEMS processes SOI wafers have three layers: a support wafer of standard thickness, say 500 µm, an insulating silicon oxide layer, and a surface layer of silicon. The oxide layer is some 100 nm thick, while the surface silicon can cover thickness ranges from fractions of a micrometer to a hundred micrometers according to the application. The basic idea of SOI technology for inertial MEMS is to structure the surface

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silicon layer and release by etching the oxide layer, which acts like a sacrificial layer in SMM. The considerable price difference between SOI wafers and standard wafers has been for some time a barrier to introducing SOI into IC and MEMS fabrication. However, the general use of SOI wafers in the microelectronics industry, starting in the year 2000, has lowered the barrier considerably. SOI-MEMS is a fast growing area, especially for inertial MEMS. SOI-MEMS combine the advantages of bulk and surface micromachining: superior material properties of crystalline silicon, with low tolerances and basically no residual stress, respectable vertical dimensions and correspondingly large proof masses, freedom in shaping the structures within the x–y-plane, and, last but not least, a high degree of CMOS compatibility. Fewer process steps are often required than in SMM. The technology is a leading candidate for highperformance inertial MEMS. The following example illustrates the typical process steps for inertial SOIMEMS [Geiger et al. 2002]. The process is running at the HSG-IMIT (Institut f¨ ur Mikro- und Informationstechnik der Hahn-Schickard-Gesellschaft e.V., Germany) and is used, among other things, for prototyping industrial gyroscopes. The process, called SCRESOI, from single-crystalline reactive etching using SOI, has been extended from a 15-µm-thick silicon layer, described here, to a 50-µm structure [Gaisser et al. 2003, Link et al. 2005]. The principal flow is shown in Fig. 4.21. Here the process steps are arranged top down according to their sequence. The process starts with structuring aluminum bond pads. Next, the mask for deep reactive-ion etching is prepared by patterning a PECVD oxide. The DRIE is performed, and a CVD oxide is deposited onto the sidewalls to protect the trenches during the subsequent isotropic etch, which takes place after opening the CVD and the buried oxide in the trenches by DRIE. The mild isotropic plasma etch removes the “sacrificial” buried oxide underneath the moving structure, avoiding the formation of electron traps at the silicon surfaces and surface stress. Only two masks are needed for the sensorwafer patterning. The sensor element is then protected by glass-frit bonding the sensor wafer with a prefabricated cap wafer. In a later process, isolating trenches within the structure layer have been introduced, which are refilled and, after closing them on top by an isolating layer, can be bridged by aluminum. In this way electrically isolated areas can be created and supplied by a needed potential, avoiding the need for area-consuming silicon interconnect structures. The process is applicable if all moving structures have mechanical and electrical connections to the area outside the etched cavity where they are anchored. If a structure that is anchored within the cavity is needed, then the process must be further extended in order to introduce an additional electrical connection to the anchor point. This is, for instance, necessary for ring gyros or torsional gyros with anchor points within the cavity.

4.3 Integrated processes

bond pad

buried oxide

191

active layer

- deposition and structuring of Al metallization for bond pads PECVD oxide

photoresist

- deposition and structuring of PECVD oxide mask for trench etching - DRIE etching of trenches to the buried oxide PECVD oxide

CVD oxide

- deposition of CVD oxide sidewall passivation - RIE opening of buried oxide - freeing of movable structure using isotropic etching - removal of oxide

thermal oxide

glass frit

Cap wafer

- hermetic encapsulation using glass frit of a prestructured cap wafer

Figure 4.21 The process flow for a typical SOI-MEMS process. Adapted from Geiger et al. [2002].

thermally grown oxide after cavity DRIE device wafer handle wafer

Figure 4.22 A customized SOI wafer for thick silicon structures.

Since it is difficult to under-etch a 50-µm-thick structure through small gaps, the 50-µm version is based on customized SOI wafers. They are fusion-bonded after trench areas have been etched in the sensor wafer, which form the necessary cavities for the moving structure. No further release etch is required. The principle is shown in Fig. 4.22. Of course, alignment marks have to be implemented before DRIE, and the non-etched sensor surface is protected from particle bombardment using oxide and photoresist layers.

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The SOI wafers, be they customized or standard versions, are structured in one DRIE etch step, which determines the accuracy and spread of the resulting geometry. After structuring, the wafers are thermally resistant and can serve for further CMOS implementation steps, provided that the sensitive structures are temporarily and appropriately protected. Processes similar to the 15-µm process described above are now available at many industrial CMOS fabs and provide a sound platform for the industrialization of SOI-based inertial MEMS.

MEMS prototyping processes Surface micromachining processes are widely supported by specialized MEMS fabs. For instance, Multi-User MEMS Processes, or MUMPs, is a commercial program that was initially developed by CRONOS Integrated Microsystems at the North Carolina Research Center (NCRC) and later continued by MEMSCAP, France. It offers proof-of-concept MEMS fabrication to industry, universities, and governments worldwide. MEMSCAP now offers three standard processes as part of the MUMPs program: PolyMUMPsTM , a three-layer polysilicon surface micromachining process using sacrificial oxide layers to suspend structures made from 2-µm-thick poly 2 and 1.5-µm-thick poly 3 layers; MetalMUMPsTM , an electroplated-nickel process with a 20-µm-thick gold-plated nickel layer; and SOIMUMPsTM , a 10-µm silicon-on-insulator process with back-side etching for the release of MEMS structures. Sandia’s Ultra-planar Multi-level MEMS Technology V (SUMMiT V) with five polysilicon layers, one of them for ground-plane and electrical interconnects, and four mechanical layers from 1.5 to 2.25 µm thick for MEMS structures, is also a widely used process for MEMS prototyping. R ) The commercial services of MEMSCAP and CMP (Circuit Multi-Projects° also support post-CMOS bulk micromachining of standard CMOS/BiCMOS wafers fabricated in selected foundries.

4.3.4

CMOS-MEMS There has been very little work on integrating moving structures directly into a CMOS process using existing polysilicon or dielectric layers. The company Siemens together with the University of Kaiserslautern developed a capacitive accelerometer that uses the doped gate polysilicon (350 nm thick) as the structural layer. The field oxide (600 nm) serves as a sacrificial layer [Hierold et al. 1996]. Doped n-well substrate regions form the fixed counter-electrodes. However, the strong geometric limitations of a standard CMOS process have made such approach an exception rather than the rule. The introduction of additional MEMS processes into a CMOS flow must take careful account of the thermal budget (e.g. O. Brand’s contribution in Baltes et al. [2005]). If long-lasting, additional high-temperature MEMS-process steps with temperatures ≥ 800 ◦ C are necessary, they must be introduced prior to the

4.3 Integrated processes

193

implementation of active CMOS devices. Medium-temperature processes, such as LPCVD deposition of polysilicon at about 600 ◦ C, should be inserted after doping of active devices but prior to the interconnect metallization. Short, hightemperature annealing steps to relieve stress in polysilicon layers can often be tolerated after the CMOS front-end process, but must be evaluated in order to estimate the possible impact on p–n-junction profiles. On integrating bulk micromachining into a CMOS process, the starting wafer material has to be adjusted to the needs of wet etching. Heavily p-doped wafers feature unacceptably low etch rates. Epi-wafers7 with reduced p-substrate doping less than 5 × 1018 cm−3 , or low-p-doped non-epi wafers, should be chosen [Mueller 1999]. It has to be noted that, for a long time, many CMOS fabs did not tolerate the additional passing in and out of wafers through airlock doors, or modifications of the design rules, which are often necessary in order to form a MEMS structure. However, the growing market value of MEMS has changed this situation. Some fabs are now ready to accept such changes and offer well-defined possibilities, at least for process modifications. The variety of CMOS MEMS processes is exemplified, for instance, by 60 different approaches found in the year 2005 in the literature, and summarized by Baltes et al. [2005]. Only some selected approaches, representing typical features of pre-, post-, and intra-CMOS-MEMS, and relevant for inertial MEMS, can be given here.

Pre-CMOS MEMS Most of the pre-CMOS processes integrate thick polysilicon layers, which require stress reduction by high-temperature annealing. The anneal step is performed before the CMOS front-end process. In order to provide a standard CMOS process, the MEMS structure is usually buried and sealed. One of the first demonstrations of this approach was Sandia’s M3 EMS process [Smith et al. 1995, Allen et al. 1998], which is presented schematically in Fig. 4.23. It follows the concept of concealing the MEMS structure during CMOS processing. In the case of single-level mechanical polysilicon, a 6-µm-deep trench is first anisotropically wet etched into the substrate wafer. Multilayer polysilicon structures need deeper trenches. The trench is first isolated by nitride. The poly 0 layer is used for electrical interconnects, and the poly-plugs serve as contacts to the outside CMOS-world. After structuring the trench is refilled by a series of oxide depositions that are optimized to avoid void formation. Subsequent planarization by CMP, stress relief by annealing, and sealing the entire MEMS structure by a silicon nitride cap forms a wafer ready for CMOS processing. After standard manufacturing of the CMOS part, including formation of metal 7

Wafers with a doped epitaxial layer on top of a heavily p-doped substrate.

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CMOS area

CMOS-poly 1

CMOS-poly 2

MEMS area

PECVD TEOS

PECVD nitride

metal

pad

sac oxide

n-well

p-well

nitride seal

poly stud

mechanical poly

field oxide/TEOS/BPSG

epitaxial silicon layer

nitride

poly 0

n-type silicon substrate

Figure 4.23 Sandia’s M3 EMS pre-CMOS process. Adapted from Smith et al. [1995]

(This figure also appears in the color-plate section.)

MOS device

Insulation plug poly nitride

Moving structure

Handle wafer

Insulator (SiO2 )

Figure 4.24 Analog Devices’ SOIMEMS process. Adapted from Lewis et al. [2003]. (This figure also appears in the color-plate section.)

interconnects to the polysilicon plugs, the nitride cap over the MEMS structure is removed using one additional mask. The release process is then an ordinary sacrificial silicon oxide etch step. Analog Devices (AD), the pioneer in monolithic integration of inertial MEMS, significantly improved their initial technology by perfecting an alternative preCMOS approach – the SOIMEMS technology – which had originally been developed by UC Berkeley [Lemkin et al. 1999]. It employs SOI and CMOS processes in close proximity. A simplified schematic diagram of the SOIMEMS cross-section is shown in Fig. 4.24. The fabrication starts with DRIE etching of trenches (insulation plugs in Fig. 4.24) in the SOI wafer, which is aimed at isolating MEMS substructures from each other and from the CMOS part. The etching stops at the buried insulator oxide. The trenches are refilled with isolating nitride and subsequently planarized. Now the standard CMOS process takes place; in the case of Analog Device’s SOIMEMS it is a 0.6-µm BiCMOS process, which is much more powerful than the pioneering, old 3-µm “iMEMS” process, in particular by virtue of allowing the integration of dense digital structures for signal conditioning, calibration, and self-testing. After (Bi)CMOS completion all dielectrics are removed from the structural region, and DRIE-etch MEMS structuring is performed. The microstructures are then released by etching the buried oxide using HF. Care

4.3 Integrated processes

PECVD nitride

PECVD oxide

LPCVD nitride

metal

BPSG

thermal oxide

sensor polysilicon

ground plane

n+ -runner

p-substrate

195

p-doped

Figure 4.25 Analog Devices’ “iMEMS” process (the CMOS part is not shown). Adapted from Core et al. [1993]. (This figure also appears in the color-plate section.)

is taken to prevent collapse of the moving structure during drying, by using temporary photoresist pedestals. The typical thickness of the structural layer is 10 µm, which significantly improves the potential performance of inertial MEMS in comparison with the old “iMEMS” process of AD that features a 2–4 µm structural poly layer. Further approaches to pre-CMOS processes are based on the combination of silicon fusion bonding and DRIE [Parameswaran et al. 1995], and on the implementation of CMOS circuitry within anisotropically etched recessed cavities, which then, together with the moving structure, are anodically bonded face to face to constraint wafers, forming capacitive interfaces [Matsumoto and Eshai 1993], among others.

Intra-CMOS MEMS The first monolithic integrated accelerometer was produced by Analog Devices using the repeatedly cited “iMEMS” process [Core et al. 1993, Senturia 2001, Chapter 19], which is a typical intra-CMOS process. The polysilicon MEMS structures are inserted by interrupting the CMOS flow. Usually this is done before the back-end metallization. Figure 4.25 illustrates the process structure. Using a slightly doped p-type wafer, the process starts with CMOS front-end fabrication: implementation of n-wells for PMOS devices, sources, drains, and polysilicon gates for the CMOS transistors, and bases and emitters for the bipolar transistors. Here also the interconnects between the sensor area and the circuit region, as well as ground planes under the future moving structure, are inserted in the form of n+ -doped regions. A p-implant between the n+ -regions is used in order to improve their mutual isolation. The front-end processing is completed by overcoating the transistor region with an LPCVD nitride layer and BPSG deposition. LPCVD deposition over the transistor region is added to provide an etch stop for the final release etch of the sensor structure. The sensor process begins with removing all dielectric layers from the sensor region. A 1.6-µm-thick sacrifical oxide layer is deposited and opened for the

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n+ -interconnects. Sensor polysilicon deposition (2 µm) and annealing follows. The polysilicon processing should guarantee a final tensile stress of 40–75 MPa. After annealing, a PECVD oxide is deposited over the wafer and removed, together with the sacrificial oxide, from the CMOS area, where contact openings are patterned and contacts to the n+ -runners are formed. The BiCMOS process is resumed with back-end interconnect formation. Finally, an additional passivation of the transistor region with two PECVD oxide and nitride layers follows. These layers serve as the etch stop for the final sacrificial oxide etching. During release and drying, temporary photoresist pedestals prevent collapse of the movable structure. Additionally, the entire structure is covered with an anti-sticking agent to prevent sticking in operation. Dicing is performed from the back side of the wafer protecting the fragile front die by two layers of tape with recessed cavities. For final packaging AD uses a hermetic ceramic package. Despite the complicated nature of this expensive 24-mask process it was used successfully for high-volume production of accelerometers and gyroscopes (series ADXL and ADXRS). The structural thickness was extended later from 2 µm to 4 µm. Other companies such as Infineon and Freescale (Motorola) also followed the intra-CMOS approach for pressure-sensor fabrication. Not surprisingly, industrial intra-CMOS MEMS are usually based on in-house CMOS hostprocesses, because the fine tuning requires a close interaction with process engineers to adapt the recipes for the necessary process steps.

Post-CMOS MEMS Post-CMOS MEMS processing is very popular, in particular within the scientific community, which usually has broad access to micromachining equipment. Almost any idea regarding how to form structures in the existing material or to add them can be realized, provided that the thermal budget is limited by the strong requirements of the prior aluminum metallization. Thus, high-temperature deposition and annealing steps, as required for polysilicon deposition, are excluded. However, the whole repertoire of low-temperature processes including sputtering, electroplating, wet and dry etching, and PECVD is available. The most effective means for structuring post-CMOS MEMS is the use of bulk silicon or of dedicated layers deposited during the CMOS process. In Fig. 4.26 four basic approaches to post-CMOS MEMS processing are shown [Baltes 1997]:

r sacrificial oxide etching for polysilicon structure (Fig. 4.26(a)): the oxide layers are selectively removed by anisotropic etching in order to release a polysilicon beam r sacrificial metal etching for cavity structure (Fig. 4.26(b)): local removal of the first metallization layer via pad-like structures, while protecting the openings to operating pads by photoresist

4.3 Integrated processes

(a)

(c)

polysilicon

197

passivation nitride

silicon oxide

(b)

(d)

photoresist

silicon substrate

metal

polysilicon

Figure 4.26 Four basic approaches to post-CMOS micromachining: (a) sacrificial oxide

etching for polysilicon structure, (b) sacrificial metal etching for cavity structure, (c) front-side bulk micromachining for beam forming, and (d) back-side wafer etching for membrane creation. Adapted from Baltes [1997].

r front-side bulk micromachining for beam forming as discussed in Section 4.1.2 (Fig. 4.26(c))

r back-side wafer etching for membranes or movable masses, whereby the etch stop can be a dielectric layer or an additional etch stop in order to retain active (with implemented CMOS structures) or passive parts of the substrate (Fig. 4.26(d)). Back-side bulk micromachining of membrane-based pressure sensors with CMOS-integrated piezoresistors has been the state of the art since the end of the 1970s and is most likely the most often used type of post-CMOS MEMS. The basic approach (Fig. 4.26(d)) is often modified by using not a dielectric etch stop and a corresponding dielectric membrane, but an electrochemical or other etch stop, retaining part of the bulk silicon as indicated in Fig. 4.9. Large companies such as Freescale (Motorola), Bosch, NEC, and others produce piezoresistive pressure sensors in high volumes, slightly modifying the standard CMOS process in order to optimize the performance of the piezoresistors. As mentioned, the ETH Z¨ urich and Austria Microsystems AG developed an industrial post-CMOS bulk micromachining process with an electrochemical etch stop [Mueller et al. 2000]. Similar developments were achieved by the Carnegie Mellon University (CMU), the Tohoku University, the Fraunhofer Society, and others. Post-CMOS processing was enhanced significantly by the development of a front-side anisotropic dry-etch process by the Carnegie Mellon University [Fedder et al. 1996, Xie et al. 2002] as shown in Fig. 4.27(a). The dielectric layers are anisotropically etched using CHF3 –O2 chemistry. The metal layers serve as etch masks. Thus, minimum feature sizes of beams are defined by the

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(a)

(b) dielectric layers

silicon substrate

silicon substrate gate polysilicon

gate polysilicon a)

a)

silicon substrate substrate silicon

metal

metal

polysilicon

dielectric layers

anchored stator

anchored stator

movable structures

Figure 4.27 CMU’s post-CMOS process. (a) Release of dielectric structures. (b) Release of combined dielectric–silicon structures. Adapted from Fedder et al. [1996] and Xie et al. [2002].

CMOS design rules and can be scaled down with progress in CMOS technology. After etching down to the substrate, an isotropic etch step based on SF6 –O2 is added to under-etch the oxide structures. Despite the fact that the process has been used successfully for MEMS prototyping, including various inertial sensors, large structures with embedded polysilicon and metal lines are endangered by curling due to the thermal mismatch and the residual stress or stress gradients. This effect is mitigated by using a dominant crystalline silicon layer as shown in Fig. 4.27(b). In a first step the wafer is anisotropically back-side etched, leaving a membrane with desired microstructure thickness between 10 and 100 µm. Then, as in the older CMU process, an anisotropic front-side etch follows (see the middle picture in Fig. 4.27(b)); however, the former isotropic etch release of the microstructure is substituted by an anisotropic back-side etch step (bottom picture in Fig. 4.27(b)). If the silicon under small beams has to be removed, an additional lateral etch step can be used. In this way selected silicon areas can be electrically isolated from each other (bottom picture in Fig. 4.27(b)) . The process was also used for the design of some inertial sensors [Xie et al. 2002, Xie and Fedder 2003].

Add-on structures An alternative to structuring existing layers or substrate is to build the MEMS structures completely on top of the finished CMOS wafer not touching the CMOS layers. The add-on structures are usually incorporated using surface micromachining techniques. As noted, polysilicon deposition and annealing would require a modification of the basic CMOS process using a more temperature-robust tungsten metallization. To avoid similar modifications, alternative materials such as

References

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polycrystalline Si–Ge and pure poly-Ge films have been developed. They can be deposited by LPCVD or PECVD systems at temperatures below 500 ◦ C. Metal deposition is also post-CMOS compatible. An electroplated nickel ring gyroscope on top of a CMOS substrate was developed and demonstrated by Delphi-Delco, Motorola, and the University of Ann-Arbor, Michigan, at the end of the 1990s [Sparks et al. 1997]. Additive stacking of structural nickel layers based on a standard electroforming process was used by Alper et al. [2007] in order to build vertical comb capacitances in which the sensing areas of the fingers are horizontal, rather than being vertical as in SMM implementations. Such an arrangement allows one to design gyroscopes with in-plane driven proof masses that are sensitive with respect to in-plane rate signals, i.e. to out-of-plane deflections (see Chapter 8). Sometimes the addition of MEMS structures by wafer bonding is beneficial and offers a cost-effective solution for the combination of zero-level packaging and MEMS structuring. An example is a simple beam accelerometer bonded on top of a standard CMOS substrate [Brandl and Kempe 2001].

Remark A general remark is necessary with respect to CMOS–MEMS co-integration. The development and industrialization of modern deep-sub-micrometer technologies is a costly and time-consuming process. The insertion of micromachining process steps and corresponding modifications are complicated and seldom justified. Usually, a clear separation of CMOS and micromachining process modules is preferable. The CMOS process should not be interrupted. Post-CMOS MEMS and pre-CMOS MEMS on SOI are the most promising candidates. Leading CMOS fabs such as TSMC have opened their doors to similar approaches supporting the growth of fab-less MEMS companies.

References Abe, T., Messner, W., and Reed, M. (1995). Effective methods to prevent stiction during post-release-etch processing, in Proceedings of IEEE Micro Electro Mechanical Systems (MEMS ’95), Amsterdam, pp. 94–99. Allen, J., Kinney, R., Sarsfield, J. et al. (1998). Integrated micro-electromechanical sensor development for inertial applications. IEEE AES Systems Magazine, 36–40. Alper, S. E., Ocak, I. E., and Akin, T. (2007). Ultrathick and high-aspect-ratio nickel gyroscope using EFAB multilayer additive electroforming. Journal of Microelectromechanical Systems, 16(5):1025–1035. Andersson, G. I., Hedenstierna, N., Svensson, P., and Pettersson, H. (1999). A novel silicon bulk gyroscope, in Transducers ’99, Extended Abstracts, paper no. 3D1.1.

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5

First-level packaging

First-level packaging (FLP) of MEMS is a decisive step within the fabrication chain of the final product. During FLP the MEMS die(s) is (are) integrated into a package that has to protect the MEMS structure, including the signal-processing elements. The package has to create an electrical interface to the higher-level components and a well-controlled coupling of the physical measurand or the manipulated variable to the sensor or actuator. The FLP process includes all assembly steps between wafer finishing and final encapsulation such as dicing, die separation, die attach, interconnection between dies (if any), connection to the outside connectors, and encapsulation. FLP of MEMS uses many techniques developed for IC packaging within the microelectronics industry [Tummala et al. 1993]. Ceramic, metallic, and plastic packages have been adapted for FLP of MEMS. However, many MEMS devices, in particular those with optical, chemical or mechanical access channels to the outside world, require specialized solutions, which often are manufactured using dedicated packaging lines. Nevertheless, the infrastructure of the IC-packaging industry is an important basis for a cost-efficient packaging especially of types of MEMS devices for which no mechanical contact to the environment is required, as is the case for some field measurements (e.g. magnetic, temperature, and inertial field measurements). Leading packaging companies such as ASE (Korea), Amkor, Carsem, and Kyocera have extended their portfolios, often in close cooperation with the final producer, towards such solutions. As stressed in many publications, MEMS packaging carries high costs, is considered to be under-represented within the scientific literature, and is often decisive for reaching a given target specification in time [Baltes et al. 2004, Hsu 2004, Jung et al. 2004]. A common root cause is the huge variety of combinations of sensors/actuators and application conditions which up to now has prevented a higher level of standardization. Standardization means the availability of proven and fully characterized product types. Characterization includes reliability certification within specified environmental conditions over a device’s lifetime, which strongly depends on the interaction between the MEMS structure and packaging. It is not standardized MEMS packages that are needed, but standardized packaged MEMS solutions. In contrast to the IC industry, the flexibility of packaging is limited by the very high tooling cost for nearly any change in geometry of a package, and by

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insufficient material characterization, which forces one to reuse the few wellcharacterized materials known to the designer. Another problem is related to the impossibility of direct testing of the package performance with respect to required sensor functions. Usually the ultimate impact of package parameters on function and reliability can be revealed only by exhaustive testing and qualification of the final device. Thus, package modifications are time-consuming and costly. FLP is strongly interwoven with sensor design and testing/qualification. An interdisciplinary effort involving system designers, process and packaging experts, and specialists in testing and qualification seems to be the best precondition for keeping a complex MEMS design on track. Inertial MEMS impose special requirements on the first-level package, which, of course, depend on the application. Since accelerometers and gyroscopes do not need access to the surrounding medium, the need for protection against outside chemical attacks is alleviated. The most important requirements are related to achieving

r r r r r

low size and cost good to excellent decoupling from package-induced stress high reliability and low performance degradation over a device’s lifetime a low thermal gradient adequate accuracy of geometric positioning and orientation of the MEMS structure, even under the impact of shock and vibration.

In some cases, despite the low masses of MEMS carriers, the package has to contribute to a low-pass filtering of external high-frequency vibrations, as, for instance, in the case of vibratory gyroscopes under stochastic shock pulses with a very high-frequency acceleration spectrum (stone impingement in cars). The construction of such suspension-like packages is beyond the scope of this book. The weighting of the different factors depends on the application area. For instance, automotive companies specify the requirement of operation within a temperature range of −40 to +125 ◦ C under harsh humidity conditions. In contrast, biomedical applications have very moderate temperature requirements but may need biological compatibility with human systems.

5.1

FLP packages Most of the FLP technologies of inertial MEMS are borrowed from IC packaging and, where needed, extended towards stress decoupling. There are more than 100 types of standardized (compliant with JEDEC or EIAJ standards) geometric packages in use by the microelectronics industry. Transistor-outline packages (TO), which usually are based on metallic or ceramic carriers, are popular for military applications due to their high robustness. For small pin-counts cheaper plastic small-outline versions (PSOT

5.1 FLP packages

(a)

(b)

cap ceramics (lid) ASIC die

207

seal glass

lead frame

lead frame

substrate ceramics

lid

substrate ceramics

MEMS die

Figure 5.1 Examples of first-level packaging of hybrid MEMS. (a) A ceramic-leaded

DIL. (b) A leadless package.

packages) exist. A dual-inline package (DIP) is a rectangular ceramic or plastic package with two rows of through-hole pins. SOIC (small-outline IC with a pinpitch of 1.27 mm) and SOP (small-outline package with reduced pitch) packages are small and two-side-leaded for surface mounting; therefore the bent pins are configured sitting up on the board. There exist subfamilies such as PSOP (plastic SOP) with a reduced pin-pitch of 0.65 mm, shrunk versions of SOP (SSOP) with up to 70 pins, thin SSOPs called TSSOP with reduced height, and so on. Quadratic flat packages (QFP – quad flat pack) with pin rows on any of the sides are used with various strategies for their connection to the board (quad flat non-leaded – QFN; and quad flat with J-leaded pins – QFJ). They allow the realization of very high pin-counts. Pin-grid arrays with connecting balls distributed over the bottom side of the package are used to further enhance the number of connections. MEMS packages usually do not require a high number of pins. Hence, DILand SOP-like packages are preferred candidates for encasing inertial MEMS. However, inertial MEMS need a hermetic package, which in the case of open MEMS structures is typically made from a metal, a ceramic or a combination thereof, since these materials provide a very high barrier to gas diffusion. In general, the packages can be divided into cavity packages and overmolded packages. In cavity packages the dies are in mechanical contact with the carrier at their ground plane only. Thus, the sensitive MEMS structures may remain open, provided that they are temporarily protected during dicing, and the assembly is performed under high-cleanroom-class conditions in order to avoid particle contamination. Overmolded packages create material contact around the entire MEMS die, hence requiring appropriate robustness. A typical arrangement of a hybrid MEMS assembly in a ceramic cavity package is sketched in Fig. 5.1. The substrate and cap ceramic are manufactured, for instance, by co-firing laminated ceramic layers with accordingly punched rectangular holes. External leads are created by brazing a lead frame to the carrier (Fig. 5.1(a)) or by metallization on top and through the ceramic layers (Fig. 5.1(b)). The MEMS and ASIC dies are attached to the substrate ceramics by the use of adhesives, interconnected, and bonded to the lead frame. In a final step a ceramic cap or lid is sealed to the substrate, and a hermetic cavity is created.

208

First-level packaging

(a)

MEMS die

adhesive (glue)

ASIC die

lead frame

(b)

mold

Figure 5.2 The cross-section of an SOIC overmolded plastic package. (a) Side view. (b) Front view.

This simple example reveals the complicated nature of FLP. What footprint of the substrate ceramic should be chosen in order to get a small package? How does the die attach transfer the stress induced by the thermal mismatch between ceramics (CTE ∼ 7 ppm/◦ C) and the silicon die (CTE ∼ 1.7 to 3.2 ppm/◦ C within the temperature range between −50 ◦ C and 125 ◦ C)? What minimum height is possible considering the dimensions of the sensor die and the acceptable height of the bond loops? What kind of lid should be chosen, a cup-shaped lid or a flat lid? What lid material should be used, epoxyresin, metal, glass, or ceramic? What kind of sealing should be used, a eutectic bond, a solder bond, or adhesive sealing? Is the MEMS die appropriately protected from contamination during assembly? The list of questions could be extended. If instead of ceramic a plastic package is considered, additional questions arise. First, plastic packages have to host some conductors in order to electrically connect the MEMS to the external world. The common way of achieving this is to add conductors by insert-molding a patterned metal lead frame. The lead frame provides also the external package terminations. The present standard for lead frames is plated copper, which has to be finished by plating with gold, silver, or palladium on a nickel base in order to support wire bonding. The next question is related to the package type: overmolded or cavity-like. The cheapest package solution is an overmolded package as shown in Fig. 5.2. Whereas with a ceramic package an unprotected MEMS structure is feasible, with an overmolded package some zero-level packaging is absolutely mandatory. The MEMS and ASIC dies are mounted by adhesives on a copper lead frame, connected with each other, and bonded to the inner lead-frame fingers. Since the zero-level-packaged MEMS die is usually wafer bonded, its height is greater then the ASIC die’s thickness. To reduce the overall height of the package a lead frame down-set is often introduced, as shown in Fig. 5.2(a). This down-set is limited to the mounting area of the MEMS die, because a general down-set would further imbalance the entire package, making it very sensitive to temperaturedependent warping. It is intuitively clear that the package-induced stress within the overmolded package is considerable larger and less predictable than that within a ceramic or not-overmolded plastic package. However, the cost pressure, particularly for high-volume products, forces system designers to accept this challenge [Darveaux and Munukutla 2005]. The choice of the mold and

5.2 FLP technologies

209

Figure 5.3 A bird’s-eye view of a 24-pin open-cavity package. With permission of

SensorDynamics AG.

lead-frame material, of the adhesives and of the actual geometry, should give answers regarding the additional questions related to the plastic encapsulation of inertial MEMS. At first glance use of a ceramic-like cavity package made from plastic such as a so-called open-cavity plastic package should avoid at least part of the problems typical of overmolded packages. The open-cavity or pre-molded plastic package has a similarity to ceramic packages in featuring only a bottom contact area between die and package and being sealed by a top lid. Figure 5.3 may impart an impression of this relatively new type of package, which was developed predominantly for moisture-resistant IC applications and has been used from the beginning also for the needs of MEMS packaging. The preform material is normally a thermoplastic liquid-crystal polymer (LCP) that can be injection-molded. It exhibits pronounced shrinking and anisotropic behavior and is not suited for hermetic packaging. A more careful comparison with overmolded packages shows that the induced stress may be even larger than in SOIC, provided that additional coatings around the MEMS die are used. Thus, a careful selection of materials and assembly geometry is needed in order to reduce the stress impact to below the level associated with overmolded packages.

5.2

FLP technologies To carry out FLP design for inertial MEMS it is insufficient to consider only the package type and geometries. The final performance and cost depend mainly on material parameters, on material interactions, and on the complexity of the fabrication steps. To select an appropriate solution all materials and fabrication steps must be evaluated in order to get an estimate of their reliability, performance, and cost.

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First-level packaging

5.2.1

Dicing and die separation Microelectronic ICs are protected by nitride passivation layers and easily withstand the stress associated with diamond sawing, including exposure to vibrations, cooling water, and debris. The wafer usually is mounted on a dicing tape with sticky backing to keep the individual dies in a fixed position during sawing. The adhesive forces are small enough to allow a subsequent pick-up procedure. The ultrathin diamond blades of dicing saws rotate at speeds between a few thousand and about 60 000 rpm and may generate high-frequency vibrations, especially in the case of fast sawing. Sawing speeds are between 0.1 and some hundreds of mm/s. The spacing between the separated dies is typically 50 µm, provided that a sawing blade of thickness ∼20 µm is used. A directed stream of water cools the blade. Generally speaking, the microelectronic dicing procedures are not directly suited for MEMS dies with unprotected fragile structures. Vibrations and the cooling-water stream may destroy sensitive MEMS structures, moisture may cause sticking, and debris may damage or block substructures. Wafer-level cap-sealing as described in Chapter 4 eliminates the problems listed above. Back-side sawing using front-side-protecting foils (or vice versa) is another way, which has been exploited for instance by Analog Devices. Both approaches involve considerable effort. Temporary protection by spin-coated polymers, which can be stripped off after dicing and before separation, is sometimes used. However, the removal of a deposited polymer without leaving residues is difficult and often not reliable enough. In contrast, fully protected MEMS structures made using wafer-level bonding are robust and their use creates ideal preconditions for a first-level packaging using cheap overmolded packages. Alternative dicing methods have been developed in order to try to avoid dangerous vibrations as well as exposure to water or moisture during separation. These include wafer breaking, laser cutting, diamond-wire cutting, and abrasivejet machining. Wafer breaking requires pre-implemented grooves, which may, for instance, be formed by a bulk-micromachining etch step on the back side of the wafer. It can be applied only if no other break-sensitive structures are present. Laser cutting, despite being a dry process, generates considerable amounts of dangerous debris. Diamond wafer cutting is based on thin wire loops impregnated with diamond dust, which have the tendency to abrade quickly or even to break and, thus, are less robust than diamond blades. Owing to the low efficiency of this method, it is not well suited for high-volume manufacturing. The same is valid for abrasive-jet machining, where fine particles are accelerated in a gas stream and strike out particles on the surface under the nozzle that are then collected by a dust collector. The packaging steps after dicing strongly determine the compatibility of separation techniques with a given MEMS process. If hermetic encapsulation of an inertial MEMS without mechanical contact between the package and the sensitive MEMS structure is intended, the structure may be open, and, hence, the

5.2 FLP technologies

211

dicing process requires the use of additional means of protection. This is the case for hermetic packaging within ceramic cases. If cheap overmolded or nonhermetic plastic packages are sought, hermetic encapsulation must be created before FLP and then automatically serves as protection during dicing.

5.2.2

Die attachment The separated die must be attached to a substrate material. The FLP substrate may be a ceramic carrier or a metallic lead frame used within plastic packages. The substrate may be also the bottom of a plastic carrier, as in the case of pre-molded cavities, or a PCB material for chip-on-board (COB) assembly etc. The interaction between chip and carrier is dominated by the thermal expansion coefficients (CTEs) and the elastic properties of the two materials and the adhesive between them. Die attachment has to provide mechanical, electrical, and thermal support to the die. It is one of the most important locations where stress coupling between the surrounding package and the MEMS die occurs. The selection of bond type, of substrate material or lead frame, and of the adhesive material and its geometry (thickness) is often decisive for the final performance of the packaged device and its long-term stability. Both anodic bonding on glass substrates and eutectic and solder bonds to metallic or ceramic substrates are reliable but sensitive to small process variations. They are more and more commonly being replaced by cheaper polymeric adhesives such as epoxy resins or silicones that are more robust against variations of the process parameters. They may be isolating or conductive; the latter are usually manufactured by adding conductive particles, for instance, in the form of grainy silver. A typical die-attachment process sequence based on polymeric adhesives includes first dispensing an adhesive onto the substrate using micro-dispensers, which work like inkjet printers. Dispensing is performed preferably by dropping minute amounts of adhesive onto the substrate at well-defined locations. The dispensement pattern, amount of adhesive, and its viscosity-caused difluence are decisive for an efficient bond connection. In a second step a good die is picked up from the wafer by the use of vacuum tools. In the case of unprotected MEMS structures care must be taken to contact the die in safe pick-up areas only, such as the edge regions. The die is placed onto the adhesive and pressed down with an exactly defined pressure. The third step is curing of the adhesive – a thermally induced cross-linking procedure, which hardens and stabilizes the adhesive. A comprehensive overview on adhesive bonding can be found in Habenicht [2009]. Die attachment for hybrid MEMS is even more difficult, because two dies have to be placed, which entails modification of the pick-and-place equipment. Often a tracing of the individual MEMS and ASIC dies without or with internal markings is required.

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First-level packaging

Table 5.1. Coefficients of thermal expansion and elasticity moduli for some packaging materials at room temperature

Material Silicon Copper Alloy 42 Al2 O3 ceramics Alumina (∼99%) Pyrex 7740 Glass Plastic Solder-like adhesive Epoxy-resin-like adhesive Silicone gel

CTE (ppm/◦ C) 2.5 17 4.5–5 6.5–8.5 6.7 3.1 9–10 13–20 52 25–125 300 (above Tg )

Young modulus (GPa) 130–190 (orientation-dependent) 110–128 144 270–415 350 68 50–90 10–25 24 2.5–4 0.001–0.05

If the overall height can be reduced to an acceptable value by using thinned ASIC wafers and MEMS dies, stacked solutions with a bonded MEMS die on top of an ASIC are sometimes considered for hybrid MEMS. Since the temperature coefficients of the two dies are identical no stresses emerge at their interface. The ASIC is attached to the substrate. The advantage is a considerably reduced footprint; however, the assembly flow is more complicated and requires special MEMS-to-ASIC connection technologies. However, this approach is becoming ever more popular.

Packaging materials In Table 5.1 the CTEs and Young moduli of some packaging materials are collected. Most of the data have been extracted from the Internet and various handbooks (e.g. Shackelford and Alexander [2001]) and are presented to impart an impression of the orders of magnitude involved. For instance, the elasticity of a silicone gel is five orders of magnitude larger then that of a ceramic. Silicon has the smallest temperature coefficient of all packaging materials, this being as much as 100 times smaller than that for silicone gel. The exact values of the parameters presented in Table 5.1 must be considered with caution, because the variety of different material types is exploding, and exact values are meaningful only for a given product. Materials with the same market name offered by different companies usually do not have the same properties, and, more seriously, the parameter spread within samples of a given product is sometimes comparable to or even larger than the data-sheet differences between different products. Data necessary for stress evaluation, such as Young’s modulus or the temperature dependencies of viscosity and elasticity, are normally missing because the materials were primarily developed for IC packaging, for which these parameters are not so relevant. For an ongoing MEMS design it is therefore recommended

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that one should carry out one’s own temperature characterization of the mold materials, plastic adhesives, and coating materials to be used, despite the quite considerable cost. Many adhesives and mold compounds are epoxy-resin-based. The more elastic silicone-based adhesives are used for high-elasticity interconnect layers and die coating. Nearly all polymeric materials exhibit temperature-dependent properties, which are associated with changes in the material’s cross-linking. The temperature, at which the compound exhibits the largest changes of its deformation ability is called the glass temperature, Tg . Above Tg both the elasticity and the CTE increase rapidly and plastic deformation takes place. Typical changes of mold-compound parameters are by factors on the order of three to five for the CTE and five to twenty for Young’s modulus. For mold compounds the glass temperature should be above the maximal operating range, i.e. above 125 ◦ C for automotive applications. In contrast, the glass temperature of silicones and soft adhesives should be below the operational temperature range in order to avoid their hardening at low temperatures. The elastic and plastic properties of many mold compounds and adhesives are often accompanied by some viscoelasticity and viscoplasticity, i.e. by timedependent elastic and plastic deformations under stress. If an adhesive layer is loaded with a constant stress, a spontaneous shear strain results, which will increase with time if the load is subsequently held constant. Removing the loading stress will lead to a spontaneous decrease of strain, which will decrease further with time. This reversible change corresponds to viscoelasticity (inelastic relaxation). If the cycle of loading and unloading is repeated, the maximum strain increases slightly from cycle to cycle. This irreversible deformation or viscoplastic strain (creep) may cause unpredictable long-time changes of the stress-dependent MEMS performance. It is mainly caused by internal diffusion processes. Creep becomes non-negligible at elevated temperatures above approximately half of the melting temperature. Since the melting temperature of polymers may be on the same order as the glass temperature, creep may become a critical process, endangering the long-time reliability of the package. Thus, hysteresis upon temperature cycling and long-time changes under load are main concerns in plastic packages of stress-sensitive inertial MEMS. For FEM simulations of die-attachment stress and its dynamic behavior exact models of material properties including viscoelasticity and viscoplasticity are needed. However, the relevant model parameters are usually missing from data sheets and must be extracted from one’s own measurements.

Die-attachment-induced stress The die-attachment area can be considered as a three-layer composite as drawn in Fig. 5.4. The top layer, of height h1 , is for instance the MEMS die; the bottom layer, of height h2 , the substrate; and the thin, intermediate layer, of height ha , the adhesive: ha ≪ h1 , h2 . The small working cavity inside the MEMS die is neglected for stress calculation. At a certain temperature, which is normally the

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First-level packaging

(a)

(b)

L

MEMS die

h1

cavity

ha

d1

Ea ;

adhesive

h2

neutral line

E1 ;

substrate

R

E2 ;

x

da

neutral line

neutral line

d2

z1 za z2

R1

R2

Figure 5.4 (a) The three-layer model of die attachment. (b) Stress-induced bending.

curing temperature of the adhesive, at which the bonding happens, all three layers are free of stress. A temperature change makes the layers contract or expand according to their individual temperature coefficients αi . To a first approximation the entire multilayer composite bends with a neutral line as shown in the top part of Fig. 5.4(b). The corresponding curvature is κ(∆T ) = 1/R(∆T ). The coordinate systems of the different layers zi and the distances di of their axes from the neutral bending line are explained in the lower part of Fig. 5.4(a). The temperature change induces stresses in the three-layer composite. Interfacial shear stresses τ at the boundaries of the adhesive film and the adherent layers as well as peeling stresses σ appear. The peeling stress has its maximum close to the end of the adhesive layer, which stretches the adhesive layer as illustrated exaggeratedly in the bottom picture of Fig. 5.4(b). Strictly speaking, the MEMS die and the substrate have different curvatures 1/R1 and 1/R2 . However, since the vertical dimension of the assembly is much smaller than Ri , the difference can be neglected. Interfacial stresses may cause delamination between the adhesive and one of the adherents. Cracking stresses may occur within the top or bottom layer. The interfacial stresses as well as the cracking stresses determine possible failures of the assembly and should be estimated during package design. The cracking stress is the normal stress on top of the top layer and can cause microcracks to propagate through the layers. In order to mitigate the interfacial stresses the materials of the top and bottom layers should have CTEs as similar as possible to each other, and – if possible – the edges of the assembly components should be slanted. The most relevant parameter for the impact of die-attachment stresses on the MEMS function is the curvature of the MEMS die. Ground-plane warping occurs where the ground plane represents a physical reference plane within the MEMS

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cavity. Ground-plane warping may induce stresses in the MEMS structure if the latter is anchored at different points to the ground plane. If only one anchor point fixes the structure, the distances of MEMS subcomponents from the ground plane change, possibly causing changes in capacitance, air-film thickness, or other parameters. If the structure is fixed not by point-like anchors but by extended areas, large deformations may take place. For the package design it is important to have at one’s disposal some analytical expressions. There are many publications dealing with stress in three-layer or bilayer structures with thin layers of adhesive between them (e.g. Dunn et al. [2002], Lu et al. [1992], Suhir [2006], Suhir et al. [2007], Sujan et al. [2006], and Wang et al. [2000]). Most authors model the layers as beams. The results differ quantitatively, depending on the model assumptions. However, for a rough and ready estimate of the curvature change of a three-material composite with a thin and elastic adhesive layer (low elasticity modulus in comparison with those of the adherent layers) the following formula derived by Suhir [1986] has been proven to be accurate enough [Lu et al. 1992]: ∆κ =

[6/(h1 + h2 )](1 + η)2 (α2 − α1 )∆T µ ¶µ ¶ f (x), h2 E20 h1 E10 2 2 3(1 + η) + 1 + η + h1 E10 h2 E20

f (x) = 1 − 

(5.1)

cosh(2x/Lc ) , cosh(L/Lc )

3 Lc = 2  2

− 21 1 1 3(h1 + h2 )2 + + 3 0  h1 E10 h2 E20 h1 E1 + h32 E20  , h1 (1 + ν1 ) h2 (1 + ν2 ) ha (1 + νa )  + +2 E1 E2 Ea

(5.2)

where η = h2 /h1 and Ei0 = Ei /(1 − νi ), with Ei and νi Young’s modulus and Poisson’s ratio of the ith layer. Here ∆κ =

1 1 − R(T + ∆T ) R(T )

is the curvature change under temperature load. If T is the equilibrium temperature, R(T ) = ∞. f (x) is a correction factor accounting for the dependency of R on x. Clearly, at the end of the plate (x = ±L/2) the curvature does not change, or, in other words, at the edges the plates remain planar if they were not bent before. Since for soft adhesives and typical bond geometries Lc ≪ L, the correction factor for not-too-small x behaves like f (x) ≈ 1 − exp[−(L − 2x)/Lc ], and, for x → 0, f (x) ≈ 1 − 2 exp(−L/Lc ). If the limit ha → 0 for h2 ≪ h1 is calculated, which corresponds to a thin layer on top of a thick film, and the stress in the thin layer according to

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σ2 = E2 (α2 − α1 )∆T is inserted, Stoney’s classical expression [Stoney 1909] follows: σ2 =

E1 h21 , 6Rh2

(5.3)

which emphasizes the consistency of Suhir’s approach. It should be noted that the adhesive layer has no impact on the curvature as long as the correction factor can be neglected. The thickness and elasticity of the adhesive are mainly responsible for the internal interfacial shear and peeling stresses. An example should help to give the reader a feeling for the deformations induced. A capped silicon MEMS die of height 1 mm and length 3 mm is mounted on a 270-µm-thick copper lead frame using a 10-µm-thick layer of epoxy-resin-like adhesive with Ea = 2.7 GPa and νa = 0.33 (αa = 50 ppm). For the calculation silicon is approximated by an isotropic material with E1 = 168 GPa, ν1 = 0.28, and α1 = 2.5 ppm; and the copper has E = 128 GPa, ν2 = 0.3, and α2 = 17 ppm. If the curing temperature is 140 ◦ C and the system is cooled down to room temperature, 25 ◦ C, the radius of curvature is 850 mm. This corresponds to a warping of a central 1-mm2 ground plane of the cavity adumbrated in Fig. 5.4(a), which causes 300-nm deflections of the points located on the x- or y-axis at a distance of 0.5 mm from the center. The maximal interfacial shear and peeling stresses τm ax and σm ax at the end of the layers (x = L/2) can be determined straightforwardly using the expression derived by Wang et al. [2000]: Ea [(1 + ν1 )α1 − (1 + ν2 )α2 ]∆T, 2λ(1 + νa )ha µ ¶ λ2 2χ = 1− 2 − βτm ax . 2χ λ

τm ax =

(5.4)

σm ax

(5.5)

The coefficients λ, χ, and β are defined as (Ei0 = Ei /(1 − νi ) as in (5.1)) s µ ¶ Ea 1 + ν1 1 + ν2 λ=2 + 0 , 2(1 + νa )ha E10 h1 E2 h 2 · µ ¶¸ 41 Ea0 1 + ν1 1 + ν2 χ= 3 + 0 3 , (5.6) (1 + νa )ha E10 h31 E2 h 2 µ ¶ 1 + ν1 1 + ν2 2ha (1 + νa ) − λ E10 h21 E20 h22 Ea β=3 . µ ¶2 µ ¶ 1 + ν1 1 + ν2 1 + ν1 1 + ν2 2(1 + νa )ha 4(1 − νa ) + 0 +6 + 0 3 E10 h1 E2 h 2 E10 h31 E2 h 2 Ea For the example above the calculated maximal shear stress is 120 MPa and the peeling stress is 27 MPa. Both values have to be related to the adhesion forces at the adhesive-to-adherent interfaces.

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(a)

(b)

217

(c)

Figure 5.5 Three bonding types. (a) Wire bonding. (b) Tape automated bonding. (c) Flip-chip assembly.

Tape

lead

die

Figure 5.6 Tape automated bonding.

For an accurate evaluation of the internal stresses the temperature- and moisture-dependent properties of the materials, the elastoplastic properties around the yield point, the impact of an elongated substrate layer exceeding the dimensions of the MEMS die, the anisotropy of silicon, and other factors should be included. FEM simulation may help to improve the results; however, the models can only be as good as their input parameters. Many of them are not available and should be extracted by additional measurements, for instance of the mentioned viscoelasticity and viscoplasticity parameters or stress values at delamination. A final reliability verification of package performance by standardized qualification procedures is necessary.

5.2.3

Electrical interconnection After die attachment the electrical connection between the densely pitched MEMS and/or ASIC pins and the package pins is established. Gold or aluminum wires are most common for wire bonding or tape automated bonding (TAB). Wire bonding connects thin wires of diameters between about 15 and 70 µm to the bond pads of the die and to the corresponding lead-frame fingers as illustrated by Fig. 5.5(a). TAB (Fig. 5.5(b)) is based on a multilayer polymer tape with embedded metallic interconnects. The interconnects protrude through windows left for the dies and into the outer bond areas, as is shown schematically in Fig. 5.6. The tape is accurately positioned above the die(s) so that the tape-embedded leads – the spider – meet the contact pads of the die. Bonding is usually performed in parallel for all contact pads, and connects the spider with the outer leads, which will be placed into contact with the final substrate (e.g. PCB) after tape separation. TAB is a high-density interconnect technique and, thus, not really central to MEMS packaging. The resulting package is flat and reliable, but requires the

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wire clamps

piezoelectric stack

wire

horn electrical flame-off electrode

tool

ultrasonic vibrations MEMS/IC die substrate

heater stage

Figure 5.7 A bonding head.

design and fabrication of a product-specific tape. High-volume products with larger pin-counts are candidates for a broader penetration of TAB into the world of MEMS. Similar statements are valid for flip-chip packages. As indicated in Fig. 5.5(c) the chip is bonded face-down (flipped) to the substrate. Conductive solder bumps are formed on bond pads of the die, which can be distributed over the entire chip area. Bonding is performed by heat; also de-bonding is possible, which is an interesting feature for rapid prototyping. Flip-chip bonding completely eliminates leads, thereby increasing the reliability of the bond connection. Again, high pincount at lowest cost is possible. Usually the space between the bumps is filled with carefully designed underfillers in order to strengthen the mechanical contact and to reduce the considerable local stress at the bumps which may occur within the substrate–die interface. Flip-chip packaging of inertial MEMS requires stressrobust sensing structures. The complicated stress field at the interface to the substrate, with potentially large stress gradients, may cause difficult-to-predict behavior, which endangers the performance particularly of sensors working over a wide temperature range. The most important bonding technique for inertial MEMS is still wire bonding. As in all bonding processes heat, pressure, or ultrasonic energy is applied to the wire once it has been brought into intimate contact with the bond pad. The corresponding bonding techniques are thermocompression, wedge–wedge ultrasonic, and thermosonic wire bonding, which are often used in combination. Thermocompression includes intensive heating of a wire tip by an electrical flame-off electrode (EFO), which is part of the bonding head. The principal bonding-head arrangement is shown in Fig. 5.7. If a high DC voltage is applied between the wire and the electrode, the arc generated causes the tail to melt, producing, after solidification, the ball. The capillary guides the ball-terminated wire towards the bond pad, where it is deformed and bonded by the heat from the heater stage and/or by the ultrasonic energy generated by the piezoelectric stack and transformed by the horn to the bond location. The first three pictures of the bonding cycle, presented in Fig. 5.8, illustrate the ball bonding described above. Ball bonding requires pad pitches of no less then 100 µm, but is the most robust bonding process.

5.2 FLP technologies

ball bond

wedge bond

219

Loop formation

tail formation

Figure 5.8 A ball-bonding–wedge-bonding cycle.

Ultrasonic wedge bonding uses a carrying tool with a flat tip-end. The tool guides the wire to the bond pad, where it presses the wire to the electrical interconnect. The bonding energy streams from the ultrasonic transducer. Wedge (or stitch or fishtail) bonding is often the second interconnect bond process, as shown in the last four pictures of the bonding cycle presented in Fig. 5.8. In this case a bond loop is formed, when the wire is guided from the first ballbond location to the second wedge-bond interconnect. The wire is considerably deformed by the bond tool to create a well-defined weak spot, where it breaks when the clamped wire is torn off after bonding. Thermosonic bonding proceeds similarly to wedge bonding, using the same tool. The only difference is the source of the bond energy: the heat is supplied by the heater stage. The combination of ultrasonic and thermosonic methods enhances the bonding strength. In all cases the wire-bonding process is a solid-phase weld where electrons are shared or diffusion of atoms takes place. Any contamination or insufficiently intimate bonding contact may cause failures. Thermosonic bonding has the highest reliability, purely ultrasonic the lowest. Since the bonding processes depend on the combination of material parameters, pressure, temperature, and ultrasonic energy dosage, it is difficult to find the optimal recipe. The bonds are endangered in operation, especially after the bond interface is overmolded or coated by a protective film. Typical failures are wedge-bond lifting, ball-bond neck breakage as shown in Fig. 5.9, wedge-bond heel breakage, and bonding-wire breakage along the wire. They may take place, in particular, as the result of variable stress applied to the bond interface and to the wires by the mold or coating material during temperature loads. Bond-pad cracks up to cratering (tearing of material together with the attached silicon) as a result of internal bond-pad stresses are also not unusual if a multilayer bond pad of an ASIC is not properly designed. It must be noted that the bonding process requires a rigid support in order for the necessary pressure to be developed during bond pulses. This is not always given, in particular if the MEMS die is bonded to the substrate by a soft and thick adhesive and may retreat under mechanical pulses. In the case of a hybrid MEMS–ASIC solution two dies have to be attached on a common lead frame and bonded together as shown in Fig. 5.2. Usually the

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First-level packaging

20 µm

20 µm

Figure 5.9 An example of an unharmed ball bond and of ball-bond neck breakage.

ASIC is bonded to the lead-frame fingers in a second bonding step. The choice of bond types and configurations of the bond loops is important for the reliability of high-performance overmolded or coated inertial MEMS and must be made as early as possible within the design cycle in order to leave enough time for experimental verification.

5.2.4

Encapsulation The ultimate goal of first-level packaging is encapsulation of the MEMS die or, in the case of hybrid MEMS, of both MEMS and ASIC dies. In accordance with the package type, encapsulation is divided into cavity forming or overmolding. The dominant package materials are metal, ceramic, and plastic. As mentioned, metal and ceramic cavity packages are well suited for reliable hermetic packages and can be used to host open MEMS structures, because a mechanical contact is required only between the carrier and the ground plane of the die(s). The lid can be soldered, brazed, eutectic, or glass-frit-bonded, or attached by adhesives. For cost reasons metal and ceramic materials are more and more commonly being replaced by polymer plastics: either thermoplastic (e.g. polyamide, polyester, polyphenylene sulfide (PPS), polyvinyl chloride, polyurethane) or thermosetting plastics (e.g. epoxy resin, polyimide).

r Thermoplastic polymers are liquid when heated to high enough temperatures and freeze to a glassy state when cooled sufficiently. Above a glass-transition temperature their elasticity rises. r In contrast to thermoplastics, thermosets irreversibly change their structure during curing by cross-linking into a rigid 3D structure, and cannot be remelted or re-shaped after cure. Thermoplastic materials can be injection-molded, whereas thermosets are usually transfer-molded, because they are generally low-viscosity materials when entering the mold and need high molding pressures. The mold is heated and closed during

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the curing process. The molding tool incorporates ejector pins that eject the part from the mold tool after it has hardened. The usage of a closed mold leads to smaller geometric tolerances than for injection-molding, whereby the part is cooled down after injection and mold removal.

Overmolded plastic packages Overmolded packages as shown in Fig. 5.2 are exposed to large stress at the silicon surfaces, which is caused mainly by the mismatch of temperature coefficients between silicon on the one side and the lead frame and mold material on the other. The CTE of an epoxy-resin mold can be very similar to the CTE of copper (∼17 ppm); however, both differ considerably from the CTE of silicon (∼2.5 ppm). For temperatures above the glass temperature Tg , which is usually around 130 ◦ C, the CTE mismatch increases dramatically, since the mold expansion grows by a factor of four to five. This increase is, of course, somewhat mitigated by the growing elasticity of the mold compound. The impact of the “elastic” mold material (E ∼ 10–25 GPa) with large CTE can be viewed in a simplified manner as the action of a “rubber strap” compressing the silicon (E ∼ 168 GPa) when it is cooled down from curing to room temperature. Neglecting the adhesive, at the curing temperature the assembly can be considered stress-free. The stress developed at lower temperatures depends on the geometry of the entire construction and especially on the silicon–mold interface. Normally it is largest at the edges of the silicon dies. Clearly, a symmetric embedding of a silicon die in a mold mitigates large stress gradients. A common challenge for all plastic packages is the reflow solder shock that occurs during soldering of the final package onto a PCB. For lead-free soldering the solder temperature follows a defined profile and attains values above 245 ◦ C for 10–40 s (JEDEC Standard 22-A113D). The package has not only to withstand suchlike temperature loads, but also should retain the pre-reflow stress conditions within the operating range in order to avoid irreversible stress-dependent parameter changes of the sensor.

Coating Stress reduction at the silicon–mold interfaces can be achieved by die-coating. The preferred coating materials are high-elasticity silicones such as roomtemperature-vulcanizing (RTV) silicone rubbers or elastomers. An example of an overmolded package with die-coating is presented in Fig. 5.10, where the general FEM model is shown in the top-left picture, and the details of the coat coverage in the top-right and bottom-left embodiments. The MEMS die is a wafer-bonded composite of a 2.5-mm × 3-mm-bottomed sensor die and a cap die; thus featuring a relatively large height. Only the MEMS die is coated. Since the coating elastomers with elasticity in the MPa range are highly viscous, the dispensing procedure is difficult, especially with respect to achieving uniform gel coverage. Edge phasing of the MEMS die facilitates the coverage of the sharp corners and edges.

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First-level packaging

FEM model: general view

FEM model: side view

FEM model: detail

Lead-frame deformation during cooling down from 170 °C to 20 °C

Figure 5.10 The FEM model and lead-frame deformation during temperature change. With permission of SensorDynamics AG. (This figure also appears in the color-plate section.)

In the bottom-right picture the calculated lead-frame deformation at room temperature is presented in a highly exaggerated manner. The simulation is based on a temperature-dependent description of the CTE and elasticity of all of the materials involved. The ASIC is positioned in the middle of the mold compound and has only a small impact on the lead-frame deformation under the MEMS die. If a relatively hard (E ≃ 100 MPa) and thick (∼50 µm) adhesive is used between the MEMS die and the lead frame, the maximal deformation change of the 2-mm × 2-mm ground plane on the surface of the sensor die is less than 100 nm over an operating temperature range of −40 to +125 ◦ C. Optimization of the adhesive, e.g. use of soft adhesives such as RTV silicone rubbers with E of a few MPa, can further reduce the impact of stress. An elastic and thick adhesive, however, begins to act like a spring between the lead frame and the MEMS die, which may cause non-negligible relative movements. They are often not tolerable for inertial MEMS because they disturb the mechanical transfer function of the sensitive inertial proof mass. An important effect in die-coated overmolded packages is the emergence of a gap between the mold and the coating material as shown in Fig. 5.11. This occurs

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Mold material

Cap die

gap bottom sensor die gel-coat

leadframe

Figure 5.11 The FEM model of a coat–mold interface within an overmolded package.

With permission SensorDynamics AG. (This figure also appears in the color-plate section.)

as a result of the high CTE of the gel coat material, which shrinks considerably during its cooling down, and of the limited silicone-to-mold adhesion, which is not sufficient to maintain the contact between the two materials. The changing width of the gap is a potential source for interconnect-wire breakage, especially at low temperatures, for which the adhesion of the wires to the gel material becomes very strong. Wires crossing the gap should have as much room as possible for free movement to adsorb the emerging stresses (see the different wire lengths illustrated in Fig. 5.11). The design of the wire loop as well as of the gel shape has to take this fact into consideration, and one must provide a flat crossing angle with maximum wire length inside the gap. The amount of gel used for MEMS-die coating may also be a critical issue. On the one hand, a sufficient edge coverage is required; on the other, a sufficient thickness of the encasing mold must be guaranteed in order to keep the necessary dimensional stability. The selection of the silicone with respect to dispensability and curing, glass temperature, and CTE is of utmost importance for a robust and reliable packaging.

Pre-molded plastic packages Alongside overmolded plastic packages, open-cavity plastic packages have gained popularity. They are made from thermoplastics like the mentioned LCP or from PPS (polyphenylene sulfide), PPA (polyphthalamide), or PEEK (polyetheretherketone), but LCP is the preferred material. They are resistant to lead-free soldering temperatures and exhibit low moisture absorption, preventing susceptibility to the formidable pop-corn effect. Since thermoplastics, unlike thermosets, can be remelted and thus reused, there is no waste such as epoxy-resin mold runners remaining after the moldtransfer process. Open-cavity packaging is environmentally friendly.

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First-level packaging

mold die

v

lead frame – exposed pad

Figure 5.12 An exposed pad package.

The open-cavity package provides a cavity, albeit not a hermetic one, and therefore requires additional die protection against corrosion. Since, due to the large diffusion channels along the feedthroughs, hermeticity cannot be guaranteed anyway, package closure sometimes is performed by deploying a metallic or plastic lid, which is simply pressed into a recessed frame at the top of the cavity instead of using the more expensive adhesive bonding, melting, or infrared sealing. The inserted dies and interconnects within pre-molded cavity packages are coated, or the cavities are filled with an encapsulant such as a fluid or gel to delay diffusion of moisture and gas and subsequent corrosion of the interconnects. In some cases one may abstain from the use of a closing lid. The high CTE of the elastic coating or fill material may cause a large stress for the embedded wires and wire bonds, which should be estimated by FEM simulation. Pre-molded cavities as shown in Fig. 5.3 are well suited for hybrid MEMS. Nearly all package types, such as SOIC, TSSOP, and QFP, among others, have been realized. However, die attachment might be not trivial due to the low adhesion of glues to the glassy surface of a standard liquid-crystal polymer. Laser roughening is a possible countermeasure. The use of other thermoplastics such as PPS is another way out. If a reliable bond between the die and the plastic is formed, the induced stress no longer follows the simple layer model as used in the section on “Dieattachment-induced stress.” Temperature-induced deformations of the bottom plate as part of a bathtub-like construction must be added. Particularly, in the case of anisotropic LCP they may increase the strain level significantly and require a careful design of the adhesive interface layer.

Summary Despite the many problems associated with encapsulating inertial MEMS in plastic housings, plastic packaging most likely will continue to dominate for long high-volume applications. Chip stacking for multichip hybrid MEMS in pre-molded as well as in overmolded packages will help to reduce package sizes. In order to keep the overall height small, thinned wafers are used instead of ∼500- or 300-µm-thick standard wafers. Also exposed pad solutions as shown in Fig. 5.12 are suited for height reduction despite the fact that the main

References

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motivation for using this package is often the desirability of better heat transfer. The handling of the strain is difficult in both approaches. Usually such solutions are exploited first for less sensitive inertial sensors with reduced temperature operating ranges such as, for instance, for consumer applications. Flip-chip-based ball grid arrays offer further advantages in yield and electrical reliability but are not yet well prepared for inertial MEMS applications with high accuracy requirements.

References Baltes, H., Brand, O., Fedder, G. et al. (2004). Enabling Technology for MEMS and Nanodevices. Weinheim: Wiley-VCH. Darveaux, R. and Munukutla, L. (2005). Critical challenges in packaging MEMS devices, in Advanced Semiconductor Manufacturing Conference and Workshop 11–12 April 2005, 2005 IEEE/SEMI, pp. 210–216. Dunn, M., Zhang, Y., and Bright, V. (2002). Deformation and structural stability of layered plate microstructures subjected to thermal loading. Journal of Microelectromechanical Systems, 11(4):372–384. Habenicht, G. (2009). Applied Adhesive Bonding. Weinheim Wiley-VCH. Hsu, T. R. (2004). MEMS Packaging. London: INSPEC, The Institution of Electrical Engineers. Jung, E., Wiemer, M., Becker, K. F., and Aschenbrenner, R. (2004). Impact of packaging on MEMS devices, in Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS (DTIP 2004), Montreux, pp. 7–11. Lu, G. Q., Kromann, G. B., Mogilevsky, B., and Gupta, T. K. (1992). Evaluation of die-attach adhesives by curvature measurements, in InterSociety Conference on Thermal Phenomena 1992, pp. 155–158. Shackelford, J. F. and Alexander, W. (2001). CRC Materials Science and Engineering Handbook. Boca Raton, FL: CRC Press, 3rd edn. Stoney, G. (1909). The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society, London, A, 82:172. Suhir, E. (1986). Calculated thermally induced stresses in adhesively bonded and soldered assemblies, in Proceedings of International Symposium on Microelectronics, Atlanta, GA, pp. 383–392. Suhir, E. (2006). Interfacial thermal stresses in a bi-material assembly with a low-yield-stress bonding layer. Modelling and Simulation in Material Science Engineering, 14:1421–1432. Suhir, E., Lee, Y. C., and Wong, C. P. (2007). Micro- and Opto-Electronic Materials and Structures: Physics, Mechanics, Design, Reliability, Packaging. Berlin: Springer. Sujan, D., Sridhar, T., Seetharamu, K., Murthy, M. V. V., and Hassan, A. Y. (2006). An accurate solution for thermo-mechanical stresses in tri-material

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assembly in electronic packages, in International Conference on Electronic Materials and Packaging 2006, EMAP 2006, pp. 1–8. Tummala, R., Rymaszewski, E., and Klopfenstein, A. (1993). Microelectronics Packaging Handbook, Part II. New York: Chapman and Hall. Wang, K. P., Huang, Y. Y., Chandra, A., and Hu, K. X. (2000). Interfacial shear stress, peeling stress, and die cracking stress in trilayer electronic assemblies. IEEE Transactions on Components and Packaging Technologies, 23(2):309– 316.

6

Electrical interfaces

Electronic interfaces between MEMS and higher-level measurement and control systems may include not only drive and read-out electronics but also comprehensive built-in test blocks and communication interfaces such as SPI (synchronous serial peripheral interface), CAN-bus (asynchronous controller area network), I2 C (inter-integrated circuit, a two-wire serial interface developed by Philips), and others. Decisive for efficient use of a sensing element is the front-end electronics, which dominantly determines the signal-to-noise ratio, linearity, and power consumption of the whole sensor system [Hagleitner 2005]. The front-end electronics must be selected and matched with the specific sensor properties. Consequently, the design of the front-end electronics is an inseparable part of the system design. In this chapter some principles and basic blocks for forming the front-end electronics are presented. In contrast, the implementation of high-level signal processing and control blocks according to the system concept is usually the task of IC-design specialists and follows the road of ASIC-design methodologies. It is not considered here. However, some interrelations between front-end and high-level electronics must be evaluated before starting a system design. Front-end electronics is analog, whereas high-level signal processing is digital. Low-noise input amplifiers occupy large areas while fast digital signal processing requires high integration densities. Hence, analog front-ends can be realized advantageously in moderate technologies such as 0.25–3-µm CMOS. Cost savings in digital processing, however, call for deep-sub-micrometer technologies. The technological compromise depends on the relation between the areas occupied by the two types of circuitry. The technology choice is especially important in the case of MEMS–ASIC cointegration. Combination of deep-sub-micrometer IC technologies with MEMS processes is not only very complicated but, most importantly, also very expensive during design and production ramp up. Any new iteration or correction requires a new mask set with costs easily exceeding the 0.5 million dollar limit. This is particularly distressing because the digital part occupies a negligible portion of the much larger MEMS and front-end area. As long as cost sharing for mask sets by MEMS-multi-project-wafer (MPW) approaches remains a dream, the better choice is mostly a co-integration of MEMS and moderate analog processes, leaving highly complex digital processing – if needed – to a separated sub-micrometer

228

Electrical interfaces

chip. The first example of such an approach was the iMEMS process of Analog Devices described in Chapter 4. Another conflict between technologies for analog front-ends and digital blocks stems from the different voltage levels. The analog front-end communicates with the MEMS device, which usually needs driving or biasing voltages significantly larger than the digital core. A large voltage is beneficial also with respect to the electromagnetic-compatibility (EMC) robustness of non-shielded devices or of wiring connections. The digital core in modern sub-micrometer processes is driven by voltages typically between 0.45 and 1.2 V. It is most likely that supply voltages for digital circuitry will decrease further, to below 0.2 V. Of course, at the price of higher costs and complexity, any technology can be extended towards higher voltages by introducing additional masks for thicker gate oxides, for dedicated high-voltage transistors, and for enforced isolation of different devices. The present standard for input–output devices of mixedsignal sub-micrometer technologies is 5 V and 3.3 V. This is sometimes not sufficient, and in these cases special high-voltage processes are needed for front-end blocks. Since the transducers in inertial MEMS have to serve for a broad variety of sensitivity ranges as well as for different accuracy and drift requirements, individual solutions for any concrete implementation must be considered. This is even more necessary because the sensor environment, which exhibits parasitic capacitances, shielding, and shunt and wiring resistances, may be quite different. The impact of unwanted pull-in forces in nonlinear capacitances also depends strongly on the operating-voltage selection. Altogether, the system designer has to know the merits of different approaches for building a front-end and has to evaluate the pros and cons of alternative solutions. Many details of implementation are not so important at this stage. It is the aim of this chapter to give a very concise introduction to some basic principles of front-end design and the key properties of some selected solutions. This should serve as a guideline for specifying electronic blocks and for targetorientated searching of improved solutions.

6.1

Sensing electronics – building blocks The first and most sensitive connection between the sensor and the front-end electronics usually takes place at the gate of a MOSFET (or MOS) transistor. Alternative bipolar transistors would feature excellent noise characteristics compared with those of a MOS transistor of the same size and power consumption. They are available as isolated devices within BiCMOS technologies and also as byproducts within high-voltage BCD technologies. However, they are especially advantageous for high-frequency applications, which are not typical for inertial MEMS, and, therefore, in light of the significantly higher cost they are used rather seldom in inertial MEMS.

229

6.1 Sensing electronics

(a)

(b) oxide B

G

gate

n+

B CSB

S

D

n-channel

p-channel

drain gate oxide

Contact polysilicon

S

D G

source

channel

L

n+ CDB

p-type substrate body

Figure 6.1 Unipolar MOSFET transistors. (a) Circuit symbols for n- and p-channel

MOSFETs. (b) A schematic view of an n-channel MOSFET.

6.1.1

The MOS transistor Since the MOSFET transistor is the basic device in CMOS, it should be discussed briefly, without going too far into the physical basics that are presented, for instance, in Sze [1981] and Naemen [2003]. Figure 6.1(a) shows the commonly used circuit symbols of n-channel and p-channel enhancement transistors, and Fig. 6.1(b) illustrates the structure of an n-channel enhancement MOSFET. The source and drain regions are heavily doped n+ -regions within a p-type substrate. They are connected by polysilicon or equivalent contacts to the electrical wires of the surrounding circuitry. The gate contact supplies the positive gate voltage VGS to the gate oxide. The generated electrical field attracts the majority carriers – the electrons – into the region underneath the gate oxide, forming in this way an inverted conductivity region: the channel. It should be noted that the potential of the source region is not necessarily equal to the substrate potential, so there is a difference between the potential of the body (substrate) and that of the source. This is accounted for by the introduction of the body voltage, which is defined with respect to the source and becomes particularly important for serial connections of MOSFETs with different source-to-substrate voltages.

Drain current Let a positive drain-to-source voltage VDS be applied. If the gate voltage is less than a certain value, called the threshold voltage VT , the channel is cut off, causing a very low drain-to-source current. In this mode the n-doped wells form reverse-biased diodes to the p-doped substrate. For gate voltages above VT the device starts to conduct, and the drain current ID increases with increasing gate voltage. If the threshold voltage is positive, the device is called an enhancement-mode MOSFET; otherwise it is a depletion transistor. The last conduct a drain current to the source even if the gate-tosource voltage VGS is zero. If for a given gate voltage VGS the drain voltage VDS is increased, the gate voltage drop along the channel changes the profile of the channel as adumbrated

230

Electrical interfaces

in Fig. 6.1(b). The channel height decreases towards the drain terminal, where only the difference between the gate and drain voltages is effective. In this socalled linear or triode mode the drain current depends on the effective gate voltage, VGS − VT , as well as on the drain voltage. On increasing the drain voltage further the channel becomes pinched off and the carriers must drift under the influence of the positive drain voltage towards the drain. This is the saturated region. On cutting off and switching on the current through the MOSFETs by use of a gate voltage, a nearly ideal switch results, which is the basis for building nonlinear digital devices such as gates, (N)ANDs, (N)ORs, and other Boolean functions. For the design of analog front-ends, however, the small-signal behavior is of interest. The drain current within the three regions described can be modeled by the following equations [Weste and Eshraghian 1988]:    0, VGS − VT ≤ 0, cut-off, 2 IDS = β[(VGS − VT )VDS − VDS /2](1 + λn VDS ), 0 < VDS < VGS − VT , linear,   (β/2)(V − V )2 (1 + λ V ), 0 < V − V < V , saturation, GS

T

n DS

GS

T

DS

(6.1)

where β = µn Cox

W L

(6.2)

is the MOS-transistor gain, depending on the effective surface mobility of the electrons within the channel µn, on Cox = εox /tox (the oxide capacitance per unit area; tox is the oxide thickness), on L (the channel length), and on W (the transistor width). λn is a constant accounting for the slight increase in drain current with growing VDS in the saturation region. This increase is caused by a shift of the channel’s pinch-off point with changing drain voltage (channel-length modulation) and, in the end, determines the output resistance of the transistor. The order of magnitude for λn is λn ∼ 10−7/L V−1 . Its inverse 1/λn = VA is called the Early voltage. For continuity reasons the factor (1 + λn VDS ) is introduced also into the linear region. In Fig. 6.2 a typical drain-current dependency on the drain–source voltage is shown for various gate-to-source voltages. The impact of different voltages between body and source, VBS 6= 0, is not directly reflected in Eq. (6.1). As is intuitively clear, different source-to-body voltages change the effective gate field and, therefore, the threshold voltage. The dependency is given by VT = VT0 +

³p ´ p tox p 2eεSi NA −VBS + 2ΦF − 2ΦF εox

(6.3)

6.1 Sensing electronics

x 10

231

-3

3.5

3

V

GS

=4V

2.5

IDS

saturation region

linear region

2

VGS= 3 V

1.5

1 V

=2V

V

= 1.5 V

3.5

4

GS

0.5

GS

0

0.5

1

1.5

2

2.5

3

4.5

5

VDS

Figure 6.2 The dependency of the drain current on the drain–source voltage.

with e the electron charge, NA the doping concentration of the substrate, εSi the dielectric constant of the silicon substrate, and ΦF = (kT /e)ln(NA /ni ) the substrate potential (ni is the intrinsic carrier concentration, which at room temperature is about 1016 /m3 ). A p-channel transistor exhibits the same characteristics with correspondingly changed material constants. Most importantly, the hole mobility, µp , is roughly a factor of two smaller than µn, leading for the same transistor geometry to a transistor gain that is smaller by a factor of two. Of course, the different nature of the majority carrier changes the voltage and current polarities, leading to a source voltage higher than the drain voltage, and a drain current flowing out of the drain rather than flowing in.

The small-signal model Amplifiers, switches, and comparators are built using MOS transistors. Switches and comparators are modeled using large-signal characteristics, whereas amplifiers are modeled by using the small-signal model.

The low-frequency small-signal model A MOSFET in amplifiers is usually operated within the saturation region, where the drain current at low frequencies can be represented approximately as a gateto-source voltage-driven current source. The transistor is biased by constant VDS and VGS , which determine the operating point, so that VGS → VGS + vGS ;

VDS → vDS + vDS ;

IDS → IDS + iD .

(6.4)

The relation among the small input and output signals vGS , vDS , and iDS is according to Eq. (6.1) approximately · ¸ vGS iD = gm vGS 1 + (6.5) + g0 vDS ∼ = gm vGS + g0 vDS , 2(VGS − VT ) where the transconductance gm is given by gm = β(VGS − VT )(1 + λn VDS ) = 2

p IDS ∼ = 2βIDS VGS − VT

(6.6)

232

Electrical interfaces

(a)

v1/f

(b)

vR

iD

gate

v1/f

vR

drain

drain

vin

iD

CGD

gate

vin

vGS

g0

vGS gm

source

CGS

g0

vGS

CGB

vGS gm

source body

vBS

vBS gmB

CDB

CSB

Figure 6.3 Equivalent circuits of a MOSFET transistor. (a) Low-frequency equivalent

circuit. (b) High-frequency equivalent circuit.

and the output conductance g0 by g0 =

β 1 ∼ (VGS − VT )2 = λn IDS = λn IDS . 2 1 + λn VDS

(6.7)

Here the last expression in Eq. (6.1) for a drain current in a given operating point was used. The MOSFET acts as a transconductance amplifier generating output current, which is controlled by the applied gate voltage. For small gate voltages, vGS , the term vGS /[2(VGS − VT )] can be neglected and the output current linearly depends on the input voltage. The output resistance r0 = 1/g0 is to a good approximation independent of the applied voltage, depending only on λn . The resulting small-signal equivalent circuit is presented in Fig. 6.3(a).

Noise in MOS transistors According to the derived approximation, the DC-input resistance in the smallsignal, low-frequency model is infinite; however, the noise sources must be added at the input. The dominant noise in MOSFETs is the channel noise, which can be divided into two parts: the first is the thermal noise, vR , of the carriers within the channel and is reflected by its equivalent resistance. If the channel profile is taken into consideration, the equivalent resistance is 2/(3gm ). Thus, the spectral density of this noise component, which is related to the gate input, is ¯ hvR2 i ¯¯ 8kT SR = = . (6.8) ¯ ∆f ¯ 3gm ∆f →0

The mean square thermal input noise hvR2 i within the band ∆f is inversely proportional to gm . That is, the signal-to-thermal-noise ratio related to the input increases with growing gm . According to Eqs. (6.6) and (6.2) a large transconductance means a large operating drain current and a large transistor width. This is why a low-noise CMOS design is closely associated with an area- and power-consuming circuitry. There is a second noise source, v1/f , related to low-frequency drain-current fluctuations. They are caused by imperfections at the oxide–semiconductor interface that constitute trapping centers for charges within the inversion layers. The drain-current fluctuations fall with increasing frequency and represent the

6.1 Sensing electronics

233

so-called flicker or “one-over-f ” noise, which in a simplified manner can be described by the following equation: ¯ 2 A F −1 hv1/f i ¯¯ KF × IDS S1/f = = , (6.9) ¯ 2 WL ∆f ¯ f Cox ∆f →0

where KF and AF are semi-empirical constants that are often derived from theoretical or empirical models. For instance, according to Tsividis [1999] the coefficient AF can be set to unity, neglecting the weak dependency on the operating point. However, as in the case of thermal noise, the inverse proportionality to the transistor size (width W ) is again present. Here the exact expression for the noise density is less important than the fact that the noise may be very high at low frequencies. Only at the corner frequency, fC , does the flicker noise become equal to the thermal noise. For higher frequencies flicker noise can be neglected. Therefore, sensing at low frequencies using MOSFETs requires special care to suppress or even bypass the impact of this noise, for instance, by transferring the signal spectrum to higher frequencies by modulation techniques.

The high-frequency small-signal model The simple low-frequency model according to Fig. 6.3(a) must be extended towards high frequencies by adding different capacitances. In fact, for analog small-signal operation the gate-to-source, CGS , and gate-to-drain, CGD , capacitances are the most important, the latter due to the so-called Miller effect.1 It should be expected that CGS = W LCox ; however, such a model would imply that the channel extends fully from source to gain, connecting all parts of the capacitance to the source. But in saturation mode the channel is pinched off, and only a part is effectively connected to the source. Besides that, usually there is an overlap region with length Lov between the polysilicon-gate oxide structure and the source well, which forms an overlap capacitance CGSov = W Lo c Cox . The following approximation for the total gate-to-source capacitance in the saturation mode is commonly accepted: µ ¶ 2 CGS = (6.10) L + Lov W Cox . 3 The gate-to-drain capacitance, CGD , is significantly smaller, including only the overlap effect between the drain well and the polysilicon-gate oxide stack; however, due to the Miller effect its effective contribution may be much bigger. Additionally, the capacitances between gate and body, CGB , between source and body, CSB , and between drain and body, CDB , must be included. Eventually, the total equivalent circuit of a MOSFET becomes the structure shown in Fig. 6.3(b). 1

The Miller effect describes the increase of the effective input capacitance within an inverting amplifier caused by the feedback capacitance between the input and output terminals.

234

Electrical interfaces

VDD

1:1

M3

M4 I0/2

i

I0/2

i

I0/2

i

CLoad

-

+ v+

M1

I0/2

M2

i

i

I0

RLoad

v-

IB

I0/2 M5

vout

iout =2 i

AGND

M6

VSS

Figure 6.4 An operational transconductance amplifier (OTA) with active load.

6.1.2

Operational and transconductance amplifiers Signal sensing and amplification may be related to internal (inside the electronic block) and external (connected to external input sources or output loads) interfaces. If a sensor signal in the form of a voltage should be transformed to a load with high impedance, transconductance amplifiers are needed. They have a large output resistance and are able to drive small capacitances or quite large input resistances, as are typical for an internal signal transfer on chip. If, however, external pads with connections to low-ohmic loads needs to be driven, small output resistances of the driving stages are beneficial. This is the typical application of operational amplifiers (Op Amps). An Op Amp can be considered as a voltage-controlled voltage source, whereas a transconductance amplifier can be thought of as a voltage-controlled current source. However, there is no border between operational transconductance amplifiers (OTAs) and Op Amps, because an OTA can be extended towards an Op Amp by adding corresponding input–output transformers such as source followers or buffer amplifiers. Moreover, a transconductance amplifier with differential input is usually used as the input stage of an Op Amps.

A simple transconductance amplifier In Fig. 6.4 one of the simplest configurations is presented, to demonstrate typical design principles and imperfections of Op Amps. First of all, the input and output signals, v+ , v− , and vout , are all related to a chip-internal voltage – the analog ground AGND. The analog ground is an internal circuit node, where ideally a perfect voltage generator sets the desired voltage level with respect to the substrate. In CMOS circuits, which are supplied by a positive voltage, VDD − VSS , this internal ground is often set to 21 (VDD − VSS ); however, other input settings are possible.

6.1 Sensing electronics

235

The signal v+ − v− in the differential OTA pair according to Fig. 6.4 should generate an output current or voltage. Three transistor pairs are used: the intrinsic differential stage M1 –M2 , consisting of two identical n-channel MOS transistors; the pair M3 –M4 ; and the pair M5 –M6 . The last two pairs constitute current mirrors, which are extensively used in Op Amp design for setting operating points (biasing) and transforming voltages into currents. A current mirror consists of two source- and gate-connected transistors. Since VGS2 = VGS1 , a simple mirror has according to Eqs. (6.1) and (6.2) the following ratio between the drain currents, provided that channel-length modulation is neglected: IDS2 Iout β2 W2 L1 = = = . IDS1 Iin β1 W1 L2

(6.11)

The current mirror M5 –M6 in Fig. 6.4 generates a constant current I0 , which is the sink for the transistors M1 and M2 of the differential pair. Transistors M1 and M2 as well as transistors M3 and M4 of the upper mirror are identical. Any small-signal current generated by either one of the transistors M1 and M2 must flow in the opposite direction through the other one because the current through M5 is set to a fixed DC value I0 . Since VDS = VGS the upper mirror works with diode-connected MOSFETs, generating the same DC and small-signal current in both transistors. The upper mirror constitutes the load for the differential pair. If the transistors are well matched, the direct current through M3 –M1 and through M4 –M2 is I0 /2. The small-signal current i is generated by the differential signals at the inputs v+ and v− . If v+ = −v− the common source voltage is vm = 21 (v+ + v− ) = 0 and the currents i1 and i2 from M1 and M2 circle through both transistors M1 and M2 as shown in Fig. 6.4. Together with the current i1 = i mirrored via M3 –M4 this current flows into the load: iout = i1 − i2 = 2i1 = 2i. With v+ = 21 (v+ − v− ) and i = i1 = gm 1 v+ = 21 gm 1 (v+ − v− ) one gets the output current iout = gm (v+ − v− ). The maximal voltage gain A for Rload → ∞ is vload gm gm ∼ A= = = v+ − v− g02 + g04 2g0

(6.12)

(6.13)

because the load current flows into the parallel-connected output resistances of M2 and M4 . The load capacitance Cload leads to a 3-dB drop of the gain at the frequency at which 2πf1 Cload = 2g0 . The gain–bandwidth product GBW= Af1 is therefore given by GBW =

gm βI0 = . 2πCload 2πCload

(6.14)

If the input signals are not symmetric, the gain drops. If, for instance, v− = 0, the common source voltage vm changes to vm = 21 v+ , shifting the input voltages

236

Electrical interfaces

VS+

v+

+ vout

v-

VS-

vsat

vout

v+ - v- vsat

Figure 6.5 The Op Amp symbol and the input–output characteristic.

by the same value, so that the actual input voltages with respect to the common ′ ′ = 21 v+ and v− source are v+ = − 21 v+ . Therefore the gain is half of the value valid for symmetric inputs. If the input signals move further away from the differential condition, the gain drops to zero for v+ = v− , because no current can flow into the DC-fixed M5 . The structure presented is far from perfect. First, the output-voltage swing is strongly limited. If the output voltage becomes lower than the voltage at the inverting input, v− , one of the transistors is pushed into the linear region because VDS < (VGS − VT ). Another disadvantage of this structure is the large offset voltage, which can be mitigated by an increasing amplifier gain. It is also clear that disturbances on the power line (on VDD ), which are coupled via parasitic capacitances to the drains mainly of M1 and M4 , will be transferred to the output. The list could be continued. Many improvements exist. A myriad of operational and transconductance amplifier types for wide application range is available to the system designer [Carter and Mancini 2003, Grey et al. 2009, Howe and Sodini 1997, Jung 1997, 2004, Laker and Sansen 1994]. They are characterized by their performance parameters, and for proper system evaluation one requires a basic understanding of the principles behind them, but not of all the details of their implementation.

Models of operational and transconductance amplifiers Op Amps are amplifier circuits with two inputs and one (single-ended) or two outputs (differential output). The well-known symbol of a single-ended Op Amp is shown at the top in Fig. 6.5, together with the supply connections VS+ and VS− which are related to the system ground AGND. Usually they are omitted. The output signal ideally follows the difference of the input signals with very high gain A, vout = A(v+ − v− ),

(6.15)

6.1 Sensing electronics

237

zic i+

v+

+

+ -

zout

zid i-

v-

+ -

+ -

vout

A ( voff)

-

zic

Figure 6.6 A simplified model of a differential-input, single-ended-output Op Amp.

albeit only within a small input range due to the limitation of the output voltage by the supply voltages (Fig. 6.5 bottom). The Op Amp consists of a differential input stage like the transconductance amplifier described in the previous section, a high-gain amplifier stage, and an output amplifier with very low output resistance. A simplified model is presented in Fig. 6.6. The input circuitry is substituted by differential and common-mode impedances zic and zid , the output by a voltage generator A(v+ − v− + voff ) in series with an output impedance zout . For an ideal Op Amp all input impedances are infinite, zid = zic = ∞, the offset voltage is zero voff = 0, the output impedance is also zero, zout = 0, and the gain A becomes very large.

Golden rules for Op Amp analysis In practice the gain is not infinite but on the order of 103 to 105 (at low frequencies). This is large enough for one to neglect the difference signal ε = v+ − v− in comparison with the input and output signals themselves. Thus, one can set v+ ∼ = v− . As shown in the previous section, the input currents i+ and i− are the currents flowing into the gates of the input transistors and practically are caused only by leakage effects. Therefore, at low frequencies they are negligible. Under such conditions the analysis of Op Amp applications in stable circuit configurations becomes very simple on applying two rules, sometimes called “the golden rules,” namely (1) There is no input current. (2) The two input voltages are nearly equal.

Non-inverting amplifiers Applying these rules to the example shown in Fig. 6.7(a), the current i through the feedback impedance z2 is, according to rule 1, equal to the current through the source impedance z1 . Therefore the following simple relations

238

Electrical interfaces

(a)

(b) z2

z1 i

vin

v-

-

i

vout

z2

z1

vin

v-

+

i

-

vout

i

+

Figure 6.7 Op Amps in (a) non-inverting and (b) inverting configurations.

hold: vout = i(z1 + z2 ),

v− = iz1 ,

vout = A(vin − v− ),

(6.16)

where the last equation is the “inner” OpAmp relation. The voltage vin is applied to the non-inverting input of the Op Amp, while v− is defined by the feedback divider z2 and z1 . Increasing vin increases the output voltage vout , which leads to a reduction of the acting input difference vin − v− and, thus, to a negative feedback. The negative feedback, which reduces the input signal difference, increases the closed-loop input impedance even if the open-loop input impedance may be low. A negative-feedback condition is mandatory for stable operation and, hence, for the applicability of the two rules mentioned above. On solving Eqs. (6.16) with respect to vout (vin ) one gets vout z1 + z2 z2 ∼ = =1+ . vin z1 + (z1 + z2 )/A z1

(6.17)

Usually the configuration is used with resistive impedances z1 = R1 and z2 = R2 . The gain is then positive, i.e. the input signal is not inverted. The configuration is known as a non-inverting Op Amp and becomes a voltage follower (unity-gain buffer) for R1 = ∞. Such buffers are used if the load has to be driven with large currents, but the previous stage has weak driving capability and can drive only high-load impedances.

The inverting amplifier The inverting amplifier, presented in Fig. 6.7(b), can be analyzed literally in the same way as the non-inverting one, starting from the obvious equations vout = i(z1 + z2 ) + vin ,

v− = iz1 + vin ,

vout = A(v+ − v− ) = −Av− , (6.18)

solution of which delivers the closed-loop gain vout z2 z2 ∼ =− =− . vin z1 + (z1 + z2 )/A z1

(6.19)

239

6.1 Sensing electronics

(a)

(b) R

vin

i

CFB v-

-

RFB i

vin

vout

i

i

v-

-

vout

C

+

+

Figure 6.8 An inverting Op Amp used as (a) an integrator and (b) as a differentiator.

Integrating and differentiating amplifiers The inverting Op Amp is a very useful configuration because it may serve as an integrating or differentiating amplifier as shown in Figs. 6.8(a) and (b).2 Indeed, if the feedback impedance is z2 = 1/(jωCFB ), the input–output relation becomes, according to Eq. (6.19), vout 1 =− , vin jωRCFB

(6.20)

which in the time domain is equivalent to vout (t) = −

1 RCFB

Z

dt vin (t).

(6.21)

The feedback capacitor CFB integrates the current i flowing through R or CFB into a charge. R The output Rvoltage is proportional to the charge: Vout = Q/CFB with Q = dt i = −(1/R) dt vin (t). The input impedance seen from the side of the voltage source is pre-dominantly a capacitive reactance. Therefore, this configuration is often called a “charge amplifier.” Analogously, the gain for a differentiating Op Amp according to Fig. 6.8(b) is vout /vin = −jωRFB C, which in the time domain corresponds to the relation vout (t) = −RFB C

dvin (t) . dt

(6.22)

The inverting amplifier is often used in order to sum input signals by connecting voltage sources vk − via their impedances zk 1 to the inverting input. Combination of amplification and integration can easily be realized by proper choice of the impedances.

Transimpedance amplifiers It is worth noting that, if an input current iin is fed into an Op Amp with feedback resistance RFB as shown in Fig. 6.9(a), this current flows via RFB into 2

Of course, non-inverting Op Amps can also be connected with external impedances in order to form non-inverting integrators and differentiators.

240

Electrical interfaces

(a)

(b) RFB v-

iin

iin

-

vout

v+

+

v-

-

gm (v+-v- )

iout

+

Figure 6.9 (a) Transimpedance and (b) transconductance amplifiers.

the ground, generating a voltage v− at the inverting input according to v− = iin RFB + vout = iin RFB − Av− ,

→

vout = −

A RFB iin ∼ = −RFB iin . 1+A (6.23)

The gain is then given by vout = −

A RFB iin ∼ = −RFB iin . 1+A

(6.24)

Therefore, such a configuration transforms currents to voltages and is called a transimpedance amplifier. It is the counterpart to the transconductance amplifier (OTA), which transforms voltages into currents. For completeness the symbol of a transconductance amplifier operating according to Eq. (6.12) is presented in Fig. 6.9(b). All of the configurations considered here are acting under negative feedback, i.e. the output voltage or some fraction of it is transferred back to the inverting input. There are also many applications using a positive-feedback configuration relative to the non-inverting input. Of course, such circuits are unstable within the linear operating range, but may feature very interesting behavior with stable points within the nonlinear region. A well-known example is the standard comparator with hysteresis (Schmitt trigger). However, the discussion of such applications would take us too far outside the scope of this book.

The real Op Amp The idealized Op Amp model is useful for a first, quick analysis of a given application. The second step is the inclusion of some imperfections like frequencydependent gain, offset, etc.

Open-loop gain and the gain–bandwidth product The dynamic behavior of Op Amps is most dominantly affected by the frequencydependent open-loop gain A(jω) =

A0 . 1 + jω/ω0

(6.25)

The gain–bandwidth product, f = GBW = A0 f0 , indicates (of course, for large A0 ) at which frequency, A0 f0 , the Op Amp gain becomes unity.

Figure 4.17 A released epi-poly structure within vacuum cavity. Courtesy of ISIT.

Litho 1 + back-side RIE wet oxidation

Litho 2 + front-side RIE wafer turnover

Oxide removal KOH cavity etch

Litho 3 + gold electroplating plating base deposition wet oxidation

getter deposition

Figure 4.19 PSM-X2 process flow 2. Courtesy of ISIT.

LPCVD buried poly-Si thick isolation oxide Si substrate deposition protection layer deposition sacrificial oxide 1 Litho 1 + RIE buried poly deposition sacrificial oxide 2 Litho 2 + RIE protection layer

Litho 4 + RIE isolation oxide Litho 3 + RIE sacrificial oxide

Litho 5 + dimple wet etch

thick epi-poly deposition

epi-poly polishing

Litho 6 + AlCu wet etch deposition AlCu

Litho 7 + DRIE epi-poly

HF vapor-phase etch

Figure 4.18 PSM-X2 process flow 1. Courtesy of ISIT.

CMOS area

CMOS-poly 1

CMOS-poly 2

MEMS area

PECVD TEOS

PECVD nitride

metal

pad

sac oxide

n-well

p-well

nitride seal

poly stud

mechanical poly

field oxide/TEOS/BPSG

epitaxial silicon layer

nitride

poly 0

n-type silicon substrate

Figure 4.23 Sandia’s M3 EMS pre-CMOS process. Adapted from Smith et al. [1995].

Moving structure

MOS device

P

Insulation plug poly nitride

Handle wafer

Insulator (SiO2 )

Figure 4.24 Analog Devices’ SOIMEMS process. Adapted from Lewis et al. [2003].

PECVD nitride

PECVD oxide

LPCVD nitride

metal

BPSG

thermal oxide

sensor polysilicon

ground plane

n+ runner

p-substrate

p-doped

Figure 4.25 Analog Devices’ “iMEMS” process (the CMOS part is not shown). Adapted from Core et al. [1993].

FEM model: general view

FEM model: detail

Lead-frame deformation during cooling down from 170 °C to 20 °C

FEM model: side view

Figure 5.10 The FEM model and lead-frame deformation during temperature change. With permission of SensorDynamics AG.

Mold material

Cap die

gap bottom sensor die gel-coat

leadframe

Figure 5.11 The FEM model of a coat–mold interface within an overmolded package. With permission of SensorDynamics AG.

6.1 Sensing electronics

241

The pole at ω = ω0 is primarily not a result of unavoidable capacitances like gate capacitances, which limit the gain at higher frequencies, but is usually deliberately built-in in order to stabilize the circuit. This dominant pole should neutralize high-frequency poles and related possible instabilities. In practice the pole usually is controlled by a compensating capacitance Cc between the high-gain stages of the Op Amp, which shifts the unwanted poles to very high frequencies. Typical GBW products are on the order of a few MHz and can be extended up to 100 MHz or so. The single-pole gain model according to Eq. (6.25) is helpful to determine the approximate step responses of different Op Amp applications. It also is related to the maximum speed, max dvout /dt (or slew rate), with which the output signal may change. If the slew-rate limitation is caused by the compensation capacitance Cc related to the dominant pole, s = −jω0 , then the slew rate is limited by the maximum current available to load this capacitance. Indeed, since i = Cc dvc /dt and vout = A′ vc , the slew rate SR is given by µ ¶ µ ¶ dvout 1 dvc 1 im ax SR = max = ′ max = ′ , (6.26) dt A dt A Cc where A′ is the gain between the voltage along the compensating capacitance and the output, and im ax is the maximum (saturation) output current of the previous gain stage. The slew rate corresponds to the slope of the front or back edges of an output pulse if a square wave is applied to the input. It is on the order of V /µs.

The power-supply rejection ratio Two very important Op Amp parameters are related to the suppression of disturbances that are coupled in via the power-supply lines and via the input connections. The output-related power-supply rejection ratios (PSRRs) characterize the change of the output signal caused by a change of the corresponding supply voltage: PSRRDD =

∆vout , ∆VDD

PSRRSS =

∆vout . ∆VSS

(6.27)

Sometimes the input-related PSRR is used, which is the ratio between the disturbance ∆VDD on the power line VDD that generates the same output change as a given input signal vin = v+ − v− . It is simply the output-related PSRR divided by the open-loop Op Amp gain A. Variations of the supply voltages are mainly caused by spikes generated within the digital part or within a switched capacitor block of a chip, and coupled to the power lines (and ground). They can be created by clock-distribution lines, by adjacent signal lines, or by substrate coupling. The PSRR is frequencydependent, decreasing unfortunately with ω, since at high frequencies the

242

Electrical interfaces

(a)

(b)

vout1

- -

v-

v+

-

+

vout

(1) +

v-

+ - A(v+-v-)

+ -

R

A(v+-v-)

-vout

v+

+

--

R

++ (2) vout2

Figure 6.10 Op Amps with differential input and balanced differential output. (a) A

fully differential Op Amp. (b) Two single-ended Op Amps forming a balanced Op Amp.

coupling between disturbance sources and power lines as well as between power lines and output increases. A high PSRR does not suffice to guarantee a quiet output signal, because the levels of the disturbing signals may be different. An analytical evaluation of PSRRs is difficult. Normally a comprehensive circuit simulation based on models including the coupling capacitances and resistance extracted from the layout must be performed. At low frequencies PSRRs on the order of 100 to 120 dB are typical, which may decrease quickly to 0 dB on approaching the 10-MHz range.

Fully differential Op Amps A powerful countermeasure against power-supply coupling is a balanced design. Assuming that the structure can be split into two signal paths that are subject to nearly the same parasitic supply and ground coupling, one can form two differential outputs signals with opposite signs that – after summing – have a considerably reduced sensitivity to coupled-in disturbances. The corresponding structure is a differential-input-balanced, differential-output Op Amp as presented in Fig. 6.10(a). The two output voltages are balanced, i.e. symmetric with respect to the analog ground. Fully differential amplifiers with non-symmetric output voltages with respect to ground are called non-balanced and are also used in many applications. In Fig. 6.10(b) the idea of a very early implementation of a fully differential balanced Op Amp according to Gregorian and Temes [1986] is shown, which, however, is not optimal with respect to the PSRR, because it does not fully exploit structural symmetry. As can easily be seen, the voltage of the inverting input of the second amplifier is the common-mode output v−2 = 21 (vout1 + vout2 ). Since it is very close to ground, it follows that vout2 = −vout1 . Assuming a very careful design of a fully differential analog front-end, appropriate shielding of sensitive analog nodes, guard rings around analog circuitry connected to the “quiet” potential, separated supply lines and connections for analog and digital circuits, guard rings around digital parts connected to a digital supply, and, last but not least, the complete electrical separation of analog

6.1 Sensing electronics

243

and digital parts, the PSRR-related output disturbances can be brought well below the thermal noise level within the frequency range 0 to 1 MHz, which is the typical range for inertial sensors.

The common-mode rejection ratio The common-mode rejection ratio (CMRR) characterizes the suppression of signals common to both inputs. The Op Amp should react to input differences only; however, in many applications large common-mode signals vm = 21 (v+ + v− ) are superimposed upon them. The total output signal of an Op Amp is 1 vout = A(v+ − v− ) + ACM (v+ + v− ) = Avin + ACM vm . 2

(6.28)

The CMRR is defined as the ratio between the differential input gain and the common-mode gain: ¯ ¯ ¯ ¯ ¯ A ¯ ¯ A ¯ ¯ ¯ ¯ ¯ CMRR = ¯ or in dB CMRR = 20 log ¯ (6.29) ¯; ¯. ¯ ACM ¯ ¯ ACM ¯

The CMRR is also frequency-dependent, just as the PSRR is. A high CMRR is mandatory in order to suppress noise on the transmission lines to both inputs. It is also needed in order to reduce the impact of large DC common-mode signals, which may drive the output into saturation. Most inertial sensors create minute changes of capacitance or resistance pairs in opposite directions. The much larger common-mode part may cause an offset at the output, which in principle is not critical as long as it is small enough and stable over temperature and supply variations. However, even this condition is usually not fulfilled, which makes compensation difficult if not impossible. Typical values of the CMRR are between 65 and 120 dB. Remarkably, inverting configurations like that in Fig. 6.7(b) are much less sensitive to CMRR problems. The non-inverting input v+ is at virtual ground. The inverting input v− differs very little from ground, so the common-mode input signal is very small, at least as long as the Op Amp remains within the linear region.

Offset, chopping, and auto-zeroing The last Op Amp parameter is the offset voltage. The input-related offset is typically between some tens of microvolts and some tens of millivolts. It may be caused by tolerance-related mismatches between resistors or transistor parameters such as the threshold voltage and transistor gain. There may be also a systematic offset caused by an asymmetry in the circuit configuration. Any stable offset can be compensated at the amplifier’s output, provided that the output offset is small enough and does not drive the output signal into saturation. Therefore, the most dangerous effect is the temperature dependency and even more the drift of the offset over time, which may be affected by radiation, stress, and other factors.

244

Electrical interfaces

(b) v-

-

vin

(a)

+

vin-

-

vin+

vout

vout

AOP

S

BOP

SW1

vCOP

v-

+

Z

COP

-

+

S

SW2 AN

vNout -BN

CN

Z vN

Figure 6.11 Op Amps with improved offset. (a) A chopper amplifier. (b) The principle

of auto-zeroing.

In principle, a stable but temperature-dependent offset can be eliminated by temperature-dependent calibration, but a more accurate, drift-independent way is the introduction of a continuously running offset calibration (auto-zeroing or correlated double sampling (CDS)), or the elimination of the offset by carrier modulation, which is often called chopping or chopper stabilization [Enz and Temes 1996]. The classical chopper amplifier chops the input signal as shown in Fig. 6.11(a) by synchronously switching with sufficiently high frequency fCh both input signals between the two Op Amp terminals. In this way a differential signal vin = (vin+ − vin− )z(t) = (2/π)(vin+ − vin− )Σk = 0 sin((2k + 1)2πfCh t) (with z(t) the chopping ± 1 pulse sequence) is generated. The resulting spectrum corresponds to an amplitude-modulated carrier with carrier frequencies at (2k + 1)fCh and adjacent signal sidebands. If the carrier frequency is high enough, the whole signal spectrum lies outside of the 1/f -noise region and can be amplified, bandpass filtered, and synchronously demodulated. The offset and – very importantly – the 1/f noise are eliminated. Contrary to the common understanding, this approach can be extended to quite high chopper frequencies (carriers) and signal bandwidths. The problems with intermodulation products and limited switching time can be largely eliminated by using not a rectangular, switch-generated carrier, but genuine single-carrier amplitude modulation as in Section 6.2.3. The auto-zeroing technique is a sampling or semi-sampling technique. The offset and slowly changing disturbances (e.g. 1/f noise) are sampled and then subtracted from the disturbed signal. The correlated double sampling is a special case of auto-zeroing in which the disturbing signal is sampled twice in each clock period: first in the absence of signal and second in its presence. Notice that, in contrast to the chopper technique, the auto-zeroing process modulates not only the signal but also the broad-band noise of the amplifier and surrounding noise sources. Since the switching frequency is usually much below the bandwidth frequency of the amplifier, noise-aliasing effects result.

6.1 Sensing electronics

245

The auto-zeroing method is commonly demonstrated using the example shown in Fig. 6.11(b) (e.g. Kugelstadt [2005]). The main amplifier AOP is complemented by a calibrating or “nulling” amplifier AN . Additionally to the input control, both amplifiers can be controlled by offset-compensating voltages vCO P and vN using the corresponding built-in open-loop gains, BOP and (−BN ), for these control paths: vout = AOP (vin + voffO P ) + BO P vCO P ,

vNout = AN (vin + voffN ) − BN vN . (6.30)

voffOP and voffN are the offsets to be compensated. The offset-control signals are not fed into the input terminals, in order not to corrupt the input resistances, but into a later stage. This is why the loop gains A and B are different. The offset compensation control gives one the possibility of feeding an offsetreduced signal sample into the main amplifier and from time to time updating the error correction of the nulling amplifier. The operating cycle is divided into two steps: (1) autocalibration of the nulling amplifier (AN ) and (2) operation of the main amplifier (AO P ) with added, offset-reduced signal. During autocalibration the input of the nulling amplifier is shorted and the output is connected with the storage capacitance CN , so that after a short settling time the sampled output and the voltage across the capacitance are given by vN = vCN = AN voffN − BN vCN ;

→

vCN = voffN

AN . 1 + BN

(6.31)

If the compensator gain BN were equal to AN , the stored voltage would be simply the offset itself (since AN ≫ 1). During the operating phase the switches of the nulling amplifier establish a connection to the input signal, and an output connection to the offsetcompensating path as shown in Fig. 6.11(b). Therefore, the output of the nulling amplifier generates a voltage across the capacitance CO P according to µ ¶ voffN vNout = vCOP = AN (vin + voffN ) − BN vCN = AN vin + , (6.32) 1 + BN which for large BN corresponds to a relative reduction of the offset by the compensating gain BN. Since the output signal of the main Op Amp is described by the first equation in (6.30), it becomes, after substitution of vCO P , vout = vin (AOP + BO P AN ) + AO P voffO P + AN BO P

voffN . 1 + BN

(6.33)

During the following nulling phase the main amplifier is supported by the voltage vCOB which was sampled and stored in the previous cycle, the change of which during one cycle is negligible. Summarizing, the relative weight of the signal has increased by the factor 1 + BO P (AN /AO P ) whilst the additional offset from

246

Electrical interfaces

vin+

+ -

vout+ R2

R3

R1

RG

-vin-

+ +

vout

+ + R1

voutR2

R3

Figure 6.12 A typical instrumentation amplifier.

the nulling amplifier is proportional only to AN (BO P /BN )voffN . The efficiency of this compensation method becomes obvious when large gains are considered, and the gains of the amplifiers are set equal to each other, AO P = AN = A, BO P = BN = B: µ ¶ voffO P + voffN vout = BA vin + . (6.34) B The offset impact is reduced by the open-loop gain of the offset-control path (more exactly by B/2, because the additional offset of the nulling amplifier has to be accounted for). Auto-zeroing configurations are able to achieve offset values on the order of microvolts. If the cycle time is short, the fluctuations caused by the 1/f noise are also correspondingly reduced. However, the high-frequency noise may increase significantly, mainly due to aliasing effects but also due to clock-feedthrough and other parasitic couplings.

Instrumentation amplifiers There are classical solutions for how to build excellent Op Amps with very high open-loop gain, high input impedance, and small input noise. They exploit the symmetry principle as far as possible, leading to very low PSRR, CMRR, and offset as well as offset drift. They are called instrumentation amplifiers, because they are well suited for usage in measurement and test equipment. In Fig. 6.12 a typical representative of this amplifier type is presented. It is easy to see that the first two Op Amps create an output voltage difference µ ¶ 2R1 vout+ − vout− = 1 + (vin+ − vin− ), (6.35) RG which can be transformed into a single-ended output voltage by the third Op Amp, µ ¶ 2R1 R3 vout = 1 + (vin+ − vin− ). (6.36) RG R2

6.2 Sensor interfaces

247

The first two Op Amps act basically as input buffers. The total common-mode gain is small, resulting only from the mismatches between the corresponding resistors and between the common-mode gains of the first two Op Amps. Instrumentation amplifiers are often used for direct measurement of resistances, for instance, of piezoresistors.

6.2

Sensor interfaces

6.2.1

Resistive interfaces Piezoresistors and thermoresistors are transducers appropriate for use in accelerometers. The stress-induced resistance changes are diminutive. Thus, the large temperature dependency makes a differential measurement mandatory. The difference between two (or, even better, four) nearly equal resistors with opposite stress-created changes is nearly temperature-independent. If transducers with opposite signal changes are not available, constant reference resistors can be used, though, of course, this entails a loss of sensitivity. Thermoresistors as used in convective or bubble accelerometers are arranged per se in differential configurations. In principle, a resistance measurement can be performed by applying a voltage to the resistor and measuring the current generated or, vice versa, feeding a known current into the resistor and measuring the voltage across it. Both solutions require voltage or current sources that are highly stable over the temperature and operating conditions. At least ratiometric behavior is necessary in order to correct the supply dependency during analog–digital conversion by changing the threshold levels synchronously with the supply voltage. Further tasks include handling of the common-mode voltage, suppression of the offset drift, compensation of nonlinearity and of the temperature dependency of the sensitivity, and, last but not least, attaining a good performance with respect to noise. The typical arrangement for a differential resistance measurements is the oldfashioned Wheatstone bridge as used in Chapter 2 and shown once more in Fig. 6.13. If all four resistors are transducers and the opposing resistors in the bridge change with opposite sign (R1 = R4 = R − ∆R; R2 = R3 = R + ∆R), then the bridge is called a full bridge, which features four active resistors. If only two resistors are active, the bridge is called a half-bridge. The two remaining resistors act as a reference. The differential output voltage vout = vout1 − vout2 is simply µ ¶ R3 R4 R2 R3 − R1 R4 vout = (vs1 − vs2 ) − = vs , (6.37) R1 + R3 R2 + R4 (R1 + R3 )(R2 + R4 ) where vs = vs1 − vs2 is the supply voltage of the bridge. The common-mode voltage is accordingly µ ¶ 1 vs R3 R4 vCM = (vout1 + vout2 ) = + . (6.38) 2 2 R1 + R3 R2 + R4

248

Electrical interfaces

vS1

R1

R2 vout1

vout2

R3

R4

vS2

Figure 6.13 A Wheatstone bridge.

In the case of an ideal symmetric full bridge the output and common-mode voltages are vout = vs

∆R = ∆R I0 , R

vCM =

vs ; 2

(6.39)

and in the case of an ideal half-bridge (R1 and R3 active and R2 = R4 = R) µ ¶ ∆R I0 vs ∆R vout = vs = ∆R , vCM = 1+ . (6.40) 2R 2 2 2R I0 is the current through the bridge. If just one transducer is available, the output voltage becomes a nonlinear function of ∆R, vout = vs

∆R . 4R(1 + ∆R/(2R))

(6.41)

Therefore, despite the greater effort involved, it is desirable to realize at least a half-bridge transducer. In both cases – half- and full-bridge implementation – the mismatch between resistors causes an offset R20 R30 − R10 R40 vout−off = vs . (6.42) (R10 + R30 )(R20 + R40 ) Remarkably, since resistors within the branches R1 –R3 and R2 –R4 have nearly identical temperature coefficients, the bridge offset is to a first approximation temperature-independent. The output noise of the bridge has a spectral density hvout−n i2 /∆f = 4kT R. Indeed, any resistor contributes a noise power 4kT R ∆f /4 = kT R ∆f , because the noise voltage of any resistor is halved by the resistor divider. Therefore, the signal-to-noise ratio at the output of a full bridge is within the bandwidth ∆f , ∆R I0 SNR = √ . 2 kT R ∆f

(6.43)

Thus, large transducer sensitivities expressed in terms of resistance change per input value (stress or temperature), large bridge currents, and small bandwidths are decisive for a high resolution.

6.2 Sensor interfaces

+ QP

RP

CP

CC

Voltage mode

RB

249

vout

RFB

R1

CFB

Sensor Rin

QP

RP

CP

RFB

CC Charge mode

-

CFB vout

+

Figure 6.14 Voltage-mode (top) and charge-mode amplifiers (bottom) for piezoelectric

sensors.

As an example a piezoresistive stress measurement is considered. According to Eqs. (2.87) and (2.88) of Chapter 2 the relative change of piezoresistors orientated along the [110] direction and perpendicular to it is given by ∆R/R = (π44 /2)σ1 , provided that σ2 = 0. The last condition holds for the stress measurement of a bent beam orientated along the [110] direction. σ1 is the lateral stress along the beam at its surface. The SNR becomes in this case s π44 R SNR = I0 σ1 . (6.44) 2 kT ∆f Unfortunately, bridge noise is not the only noise source within the system. If, for instance, the bridge output signal is directly connected with an instrumentation-like Op Amp as presented in Fig. 6.12, the thermal and flicker noise of the input amplifier add, which may lead to intolerable performance degradation. Since the calculation of the output noise is very simple, it will be omitted here. Auto-zeroing, chopping, or input modulation must be used in order to eliminate at least the 1/f noise together with the offset. For resonant accelerometers with piezoresistive read-out, described in Section 7.3 of Chapter 7, stress-related changes in resistance only at the resonance frequency are of interest. Since the frequency of the typical resonant beam is much higher than the corner frequency of flicker noise, the latter does not disturb a correct measurement by a standard, low-noise Op Amp.

6.2.2

Piezoelectric interfaces Voltage-mode amplifiers Piezoelectric transducers, as explained in the section on “Piezoelectric sensors in MEMS” (Chapter 2), usually exploit standard structures as shown in Fig. 6.14.

250

Electrical interfaces

The top picture presents the so-called voltage-mode amplifier configuration, for which the output voltage is proportional to the voltage between v+ and analog ground: v+ = iQ zP′ . iQ = sQ is the current created by the charge generator and flowing through the parallel connection, zP′ , of shunting sensor impedance, external biasing resistor RB , and parasitic capacitance CC . The parasitic capacitance CC stems mainly from coupling capacitances of wiring connections between the sensor and the input stage. The latter are especially important if the sensor and electronics are located on different chips or even at different locations on a PCB. If the sensor and input amplifier can be placed in close proximity, namely with a small CC , the voltage-mode amplifier is a preferred choice. It should be noted that the intrinsic shunt resistance RP of a piezoelectric sensor is very large – in the range of GΩ and higher. Since the current through the feedback impedance is i = vout /(zFB + R1 ), the voltage at the inverting input is v− = iR1 . Together with the inner Op Amp equation vout = A(v+ − v− ), this system of equations leads to the following input–output relation: vout =

QP s CP + CC s + ωP

µ

1+

RFB R1 (1 + s/ωFB )

¶

.

(6.45)

The transfer function is a bandpass with upper and lower cut-off frequencies 1 , RFB CFB µ ¶ 1 1 1 1 ωP = + ≃ , RP RB CP + CC RB (CP + CC )

ωFB =

(6.46)

respectively. For frequencies ωP ≪ ω ≪ ωFB the gain is vout /QP = [1/(CP + CC )](1 + RFB /R1 ) ∼ [1/(CP + CC )]RFB /R1 (RFB ≫ R1 ). Large RP and RB are mandatory in order to achieve a small lower cut-off frequency and, thus, to transfer low frequencies of the signal. A large RB is also necessary in order to avoid having a large offset caused by the biasing current. Increasing the parasitic capacitance CC would decrease the lower cut-off frequency; however, there would then be a direct penalty to be paid in the form of a decreased gain. In contrast, the upper cut-off frequency can be adjusted according to the required bandwidth of the signal by proper choice of CFB .

Charge-mode amplifiers The transfer characteristic of a charge-mode amplifier according to the bottom picture in Fig. 6.14 can be derived analogously: vout = −

QP RP s 1 QP s 1 ∼ . (6.47) =− CFB RP + Rin s + ωFB 1 + s/ωP CFB s + ωFB 1 + s/ωP

6.2 Sensor interfaces

251

Here the feedback and sensor impedances have exchanged their places. The lower and upper cut-off frequencies are 1 , RFB CFB µ ¶ 1 1 1 1 ∼ ωP = + , = RP Rin CP + CC Rin (CP + CC )

ωFB =

(6.48)

respectively. The feedback impedance is now responsible for the lower cut-off frequency and the sensor impedance for the upper one. The senor injects charges into the feedback capacitance, hence the naming convention. The maximum value of the feedback resistance should be limited by the tolerable charge on the feedback capacitance, which in the case of very slow QP can drive the Op Amp into saturation. An alternative to the feedback resistance can be a reset network that from time to time removes accumulated charges from the feedback capacitor. An advantage of a charge-mode amplifier is the lack of any dependency of the gain on the sensor and wiring capacitances. The gain is determined by the feedback capacitance. However, on decreasing the feedback capacitance, the lower cut-off frequency increases, thereby corrupting low-frequency signal transfer. This can be mitigated by increasing the feedback resistance RFB . Unfortunately, reasonable feedback resistances are limited to some 100 kΩ if highly resistive polysilicon is used, and to some 100 MΩ if accordingly biased MOS transistors, which emulate resistors, are used. Also switched capacitors (see Section 6.2.3) allow one to realize large resistor values. For a feedback capacitance of 1 nF and a resistance of 100 MΩ the lower cut-off frequency is 1.6 Hz, which in some cases may still be too large. This example shows the difficulties of a full CMOS implementation of piezoelectric interfaces. Sometimes expensive off-chip feedback resistors and/or capacitors are used. Nevertheless, the charge-mode amplifier gives enough freedom in adjusting the desired bandpass characteristic and is usually preferred if the coupling capacitance CC has a significant impact in comparison with CP .

Noise in piezoelectric interfaces As an example, the noise performance of a charge-mode amplifier will be analyzed. The analysis is based on the equivalent circuit in Fig. 6.15, where the noise sources are shown explicitly. vnP , vnR , and vnFB are thermal noise sources of the corresponding resistors. vnin is the input-related Op Amp noise consisting of thermal and flicker components. CP′ is the total capacitance CP′ = CP + CC . The input capacitance of the Op Amp, Cin , is taken into consideration. The signal transfer function of a piezoelectric transducer plus electronic input stage

252

Electrical interfaces

RFB vnR

vnFB

CFB

Rin

RP

QP

vnin

vout C’P

vnP

Cin

+

Figure 6.15 Noise sources in a piezoelectric charge amplifier.

is, according to Eq. (6.47),3 vout 1 s 1 HS = =− . QP CFB s + ωFB 1 + s/ωP

(6.49)

The noise contributions at the output can be characterized by their spectral densities, whereby the relation RP ≫ Rin is taken into account: ¯2 µ ¶2 ¯¯ ¯ RFB 1 ¯ ¯ SR P (ω) = (6.50) ¯ ¯ 4kT RP , ¯ (1 + jω/ωFB )(1 + jω/ωP ) ¯ RP ¯2 µ ¶2 ¯¯ ¯ RFB 1 ¯ ¯ SR i n (ω) = (6.51) ¯ ¯ 4kT Rin , ¯ 1 + jω/ωFB ¯ RP ¯ ¯2 ¯ ¯ 1 ¯ ¯ SR F B (ω) = ¯ (6.52) ¯ 4kT RFB , ¯ 1 + jω/ωFB ¯ ¯ ¯2 ¯ jωCin RFB RFB 1 + jωRin Cin ¯¯ ¯ SAm pl-in (ω) = ¯1 + + ¯ Sn-in . ¯ (1 + jω/ωFB )(1 + jω/ωP ) RP 1 + jω/ωFB ¯ (6.53)

Sn-in = hvnin i2 /∆f |∆f →0 is the input-related spectral noise density of the amplifier. According to Eqs. (6.50)–(6.52) thermal noise of all external sources is drastically suppressed by the low-pass character of HS /(jω), which becomes active above the low cut-off frequency. For frequencies higher than the low cut-off frequency this noise quickly decreases like a 1/f noise. The amplifier-generated noise can be estimated for ωFB ≪ ω ≪ ωP as ¯ µ · ¶ µ ¸ ¯¯2 ¶2 ¯ 1 Rin 1 Cin ¯ ¯ SAm pl-in (ω) = ¯1 + Cin 1 + + Sn-in . ¯ Sn-in ≃ 1 + ¯ CFB RP jωRP ¯ CFB (6.54)

3

Sometimes it is more convenient to work with voltage gain instead of with charge gain. The transfer between them is defined by the sensor equation V P = Q P /C P .

6.2 Sensor interfaces

253

A small ratio Cin /CFB improves the noise performance. On summing all noise contributions within the range ωFB ≪ ω ≪ ωP and considering Rin ≪ RP , the total spectral density Sn-out becomes µ ¶ ω2 RFB Cin + CFB Sn-out = 4kT RFB FB 1 + + Sn-in . (6.55) 2 ω RP CFB The output noise at low frequencies is dominated by the noise of the external resistors RP and RFB , which, according to the first term of Eq. (6.55), exhibits a characteristic 1/f behavior. For increasing frequencies the dominant term is [(Cin + CFB )/CFB ]Sn-in . One may raise the question of whether it is possible to suppress the 1/f term by auto-zeroing or chopping techniques. The answer is no, because the output term of the Op Amp is correlated, but the resistor noise at the input is not. Chopping, auto-zeroing, or correlated double sampling therefore will deal with independent noise portions, the powers of which will add. As an example a beam-strain measurement according to Fig. 2.16 in Chapter 2 (see also Eq. (2.104)) is considered. Since the sensor signal is given by the generated charge QP = d31 EP LBε1 , the SNR at the output within the small bandwidth ∆f is SNR|⊂∆f =

CFB

d31 LBEP p ε1 , 2 /ω 2 )]∆f [(1 + Cin /CFB )Sn-in + 4kT RFB ωFB

(6.56)

∼ QP /CFB . It depends implicitly on the sensor capacitance because vout-signal = because CP = ǫ33 BL/H, where LB is the sensor area and H its thickness. Assuming a PZT sensor with area LB = 10−9 m2 , piezoelectric sensitivity d31 = −275 pC/N, √ Young’s modulus EP = 70 GPa, an input-related noise density of 20 nV/ Hz, and a feedback capacitance √ of 100 pF, the asymptotic resolution at high frequencies is equal to ε1 / Hz = 10−10 . This is an excellent value that unfortunately deteriorates for lower frequencies. For lower frequencies alternative sensing methods such as the use of capacitive transducers become superior.

6.2.3

Capacitive interfaces In inertial MEMS capacitances are arranged in differential or single-ended configurations. Differential structures prevail. The only relevant exception is the measurement of single-sided plate capacitances as used in some accelerometers. What has to be measured? Usually only differences within a pair of capacitances are of interest, not absolute values. The reason is that the capacitances are used to measure position changes with respect to a reference position, which usually can be determined by a reference capacitance. This makes the measurement simpler, because all measurements can be performed with respect to this properly chosen reference capacitance in the form of differential measurements. If both capacitances of a pair change their values with opposite signs, one of

254

Electrical interfaces

them can be considered as a time-varying “reference” with the same rest value as the first capacitance. To save real estate the measurement of single-ended capacitances is often performed using a reference that is implemented not as a MEMS structure, but as an on-chip capacitance. Such references are difficult to match exactly with a sensing cap. Additionally, the temperature coefficients of the two capacitors may be different, leading to a temperature-dependent offset. For high-accuracy sensing this approach is not exploited. Another important attribute of the measurement task is the spectrum of capacitance changes to be captured. For accelerometers the position changes and corresponding capacitance changes usually span from DC to some kHz. No regions of the spectrum are accentuated. This case will be called “low-pass sensing.” In contrast, in resonant accelerometers the sensing capacitances are subject to harmonic excitations with fairly high frequencies in ranges above some tens or hundreds of kHz. Low-frequency changes are not of interest. The capacitance changes are captured within a bandpass region. Therefore, this case will be denoted as “bandpass sensing.” In gyroscopes most of the movements are close to harmonic oscillations; however, the frequencies are usually in a range between some kHz and 50 kHz. For low-frequency gyroscopes it is possible to capture the whole spectrum between DC and some maximum frequency, and to separate the frequency region of interest in a postprocessing step. However, bandpass sensing is preferably used mainly in connection with an initial transfer of the capacitance changes into a highfrequency region by chopping. One of the reasons for making this differentiation is the role of 1/f noise. For low-pass sensing special measures have to be taken in order to suppress the impact of 1/f noise. For bandpass sensing 1/f noise can be neglected, provided that there is a corner frequency below the lowest frequency of the signal spectrum.

Principles of capacitive sensing Baxter [1997] distinguishes among DC, AC with synchro-demodulation (impedance measurement), and oscillatory capacitance measurements. In the context of inertial MEMS it is more convenient to classify the acquisition of capacitance changes according to the following four principles:

r r r r

sensing the current through a capacitance measuring the voltage across a capacitance determining the stored charge determining the resonance frequency of an oscillator depending on a capacitance.

The first three principles will be discussed briefly in the next few paragraphs. They all require voltage sources in order to generate current, charge, and voltage across the capacitor.

255

6.2 Sensor interfaces

(a)

(b) vn1

RFB

v1

CPs1

v1

i

i1

Cs1 Cs2

i2 RP

vnP

vn-in vCP

iP

v2

CPs2

vnFB

-

Cs1

iFB

v0

+

Cs2

CP

v2

+

v0

-

vn2

Figure 6.16 Principles of capacitance sensing: (a) current and (b) voltage sensing.

The basic configuration is a capacitance pair, either capacitance of which is excited by an applied voltage as shown in Figs. 6.16 and 6.19. CS1 and CS2 are the capacitances to be sensed, whereby one of them may be also a constant reference capacitance. If fully differential sensing is assumed, the capacitances change with opposite signs: Cs1 = Cs + ∆Cs1 ,

Cs2 = Cs − ∆Cs2 ,

∆Cs1 ∼ = ∆Cs2 ;

(6.57a)

Cs ≈ Cs′ .

(6.57b)

otherwise Cs1 = Cs + ∆Cs1 ,

Cs2 = Cs′ ,

The capacitance CP summarizes all parasitic capacitances of the common plates and their connecting wires to ground. v1 and v2 are the applied voltages, which in general may be time-varying. In Fig. 6.16(a) the details of the circuitry are presented. The parasitic capacitances CPs1 and CPs2 of opposite sensing plates are connected in parallel to the low-ohmic outputs of the voltage sources. Therefore they can be neglected for a first-order analysis. RP is a wiring resistance between the common plate and the input, which generates thermal noise vnP . A more correct model should insert also wiring resistances between partitioned parasitic capacitances. However, for a first-order analysis the chosen model is sufficient. The noise, created by source – and interconnect – resistances of the voltage generators, together with disturbances, coupled to the output of the voltage sources, is summarized into output noise sources vn1 and vn2 . The dotted elements are relevant for noise analysis and can be neglected during signal-flow analysis.

Current sensing The circuit presented in Fig. 6.16(a) corresponds to a transimpedance amplifier according to Fig. 6.9(a) that senses the input current. The currents i1 and i2 through the capacitors are (neglecting RP ) i1 =

d [(v− − v1 )Cs1 (t)], dt

i2 =

d [(v− − v2 )Cs2 (t)] dt

(6.58)

256

Electrical interfaces

v1 =V cos( t) Cs1 Cs2

-

v2 =-v1

-VSD sin( t )

RFB

v0

GBP

+

Low pass filter

vout

Figure 6.17 The principle of charge modulation–demodulation.

and iFB = i1 + i2 + CP

d v0 − v− v− = , dt RFB

v0 = −Av− .

(6.59)

On solving as usual for large Op Amp gain A, the output voltage becomes v0 = −RFB i = −RFB

d [v1 Cs1 + v2 Cs2 ]. dt

(6.60)

Since the voltage at the inverting input v− is negligible in comparison with the driving voltages vi , the term (vi Csi ), i = 1, 2, represents practically the stored charge Qi : ii = (d/dt)Qi = (d/dt)(vi Csi (t)). In more detail, the total current through the two capacitors is given by i=

d dv1 dv2 dCs1 dCs2 (Q1 + Q2 ) = Cs1 + Cs2 + v1 + v2 . dt dt dt dt dt

Signal modulation If the applied voltages are constant and, for simplicity, with opposite signs, v2 = −v1 = V , the output voltage is µ ¶ dCs1 dCs2 v0 = V − . (6.61) dt dt The derivatives are proportional to the velocity, not to the position. Thus, current sensing with a constant voltage does not deliver the desired information. Integrating the output is possible, but introduces unavoidable offset and randomwalk errors. However, if – as in resonant accelerometers or for sensing the drive movement of gyroscopes (e.g. Geen et al. [2002]) – bandpass sensing of a harmonic capacitance-change is required, the output signal provides the sought information on frequency and amplitude. For low-pass sensing correct position-dependent sensing is likewise possible, as illustrated by Fig. 6.17. For that purpose the applied voltages are chosen as signal-modulating harmonic functions v1 = V cos(ωt) = −v2 that transfer the low-pass spectrum into the high-frequency region. The output becomes · µ ¶¸ dCs1 dCs2 v0 = RFB V ω(Cs1 − Cs2 ) sin(ωt) − cos(ωt) − . (6.62) dt dt For large frequencies ω and slow capacitance changes the second term in Eq. (6.62) is negligible: the output represents the sought-after capacitance

6.2 Sensor interfaces

257

v1(t) v2 (t)

2 V

T

Figure 6.18 Excitation voltages for charge modulation.

changes. After bandpass amplification with gain GBP and synchro-demodulation (analog multiplication of the output signal by VSD cos(ωt + π/2)) the final output is proportional to Cs1 − Cs2 , while the emerging second harmonic can be filtered away: 1 vout-SD =∼ = − GBP ω(Cs1 − Cs2 )RFB V VSD . 2

(6.63)

Often harmonic voltages are substituted by periodic sequences with rectangular pulse shapes of amplitude ∆V as presented in Fig. 6.18: v1 (t) = P ∆V (−1)k z(t − kT /2) with z(t) = 1 for 0 ≤ t < T /2 and z(t) = 0 otherwise. In this case one often speaks of chopper-stabilization technique. An example of a fully differential chopper-stabilized amplifier with DC-offset cancellation to avoid saturation is given, for instance, in Wu et al. [2002]. The rectangular pulse sequence consists of all odd harmonics and the corresponding spectral components of the chopped input signal. Such sequences can be generated easily with small real estate by digital circuitry. The advantage is that the generated force acting on the capacitance is constant, because it features a quadratic dependency on the applied voltage (see Chapter 2). Harmonic excitations, in contrast, create forces containing the second harmonic. In order to avoid disturbing signals, the mechanical system must be able to suppress this harmonic. Usually such a condition can be maintained by use of a proper mechanical design. A disadvantage of rectangular pulse modulation is the emergence of sharp pulses that may be coupled via parasitic capacitances into the signal paths. The high-frequency modulation technique requires a bandwidth of the transimpedance amplifier that is several times that of the modulation frequency. Correspondingly large unity-gain frequencies of the basic Op Amp are needed. Remarkably, the described AC modulation eliminates the 1/f noise, because bandpass amplification after a transimpedance amplifier suppresses lowfrequency contributions of the Op Amp’s output, including 1/f noise. Even components that have not fully been eliminated are transformed by the synchrodemodulator into a frequency band around f = ω/(2π), where they cause no disturbance, while the components of interest are demodulated down into the DC region.

Noise The output noise of a current-sensing amplifier is mainly determined by the input-related Op Amp noise vn-in , the thermal noise of the feedback resistor

258

Electrical interfaces

vnFB , the noise of the wiring resistor vnP , and the output noise of the driving voltage sources vni . The output noise of driving stages is transferred in the same way as are the driving voltages themselves. Assuming for the noise analysis constant sensing capacitances Cs1 = Cs2 = Cs and uncorrelated identical noise of the voltage sources with densities Sn1 = Sn2 , the output spectrum can be easily derived: Sn-out = 4kT RFB + (ωRFB CS )2 (4kT RP + Sn1 ) + [1 + (ωRFB Ctot )2 ]Sn-in (6.64) with Ctot = Cs1 + Cs2 + CP . The Op Amp noise Sn-in consists of 1/f -noise components, which in the case of low-pass sensing require countermeasures like the amplitude-modulation technique (chopping) described above or similar. Further, since the signal output is proportional to RFB , the SNR at high frequencies 2 is may be corrupted if the second term with power density proportional to RFB badly controlled. This component includes the noise from wiring resistors and from voltage sources. Thus, the driving voltages have to be “noise-free” and the wiring resistor must be as small as possible. This is a general requirement for all types of capacitance sensing. For current sensing the source noise with Sn1 must be well controlled, especially at high frequencies.

Voltage sensing Voltage sensing means that the voltage at the common node of both capacitances is transferred by a buffer amplifier directly to the output as shown in Fig. 6.16(b). The current balance at the common node delivers (d/dt)(CP + Cs1 + Cs2 )v+ = (d/dt)(v1 Cs1 + v2 Cs2 ) or, since the integration constant (the charge in the absence of applied voltages) is zero, v0 =

v1 Cs1 + v2 Cs2 v1 Cs1 + v2 Cs2 ∼ Q1 + Q2 = . = CP + Cs1 + Cs2 Ctot Ctot

(6.65)

Input-related noise and the noise of driving sources create the following output noise: µ ¶2 Cs Sn-out = 4kT RP + 2 Sn1 + Sn-in , (6.66) Ctot which has to be complemented by the noise of the subsequent amplifier, because the first stage has only unity gain. The output is proportional to the sensing capacitances and with v1 = −v2 = v provides the difference v0 = v(Cs1 − Cs2 )/(CP + Cs1 + Cs2 ). In contrast to current sensing, parasitic capacitances may have a significant impact on the signal gain. This disadvantage is partially balanced by the advantage of direct capacitance sensing. The low-pass and bandpass sensing methods presented in the previous section as well as chopping and auto-zeroing can be used also for voltage sensing.

259

6.2 Sensor interfaces

(a)

(b) RFB1

Cs1

CFB1

-

RFB

v1

v0+

CFB

Cs1 -

Cs2

v0

CFB2

Cs2

+

v2

v0-

+

v1

CP

RFB1

Figure 6.19 Charge sensing with (a) single-ended and (b) differential output with symmetric charge sensing.

Charge sensing Charge sensing is a powerful method for parasitic-insensitive capacitance measurement. The principle is shown in Fig. 6.19(a). Ideally, a charge amplifier would not include a feedback resistor RFB as introduced in the picture, but would include only an integrating capacitor CFB . However, RFB is necessary in order to provide a DC feedback to the Op Amp input. In this case the DC value at the inverting input is approximately zero, avoiding a drift of the input node voltage, which may drive the amplifier into saturation. Taking into consideration a limited open-loop gain, the output signal obeys the equation v0 = −

1 1 + (1/A)(1 + Ctot /CFB )

µ

v1 Cs1 + v2 Cs2 CFB + 1/(sRFB )

¶

v1 Cs1 + v2 Cs2 ∼ . =− CFB (6.67)

The factor 1/[1 + (1/A)(1 + Ctot /CFB )] reflects the frequency-dependent chargeamplifier gain. The bandwidth is, from Eq. (6.25), approximately fB =

GBW . 1 + Ctot /CFB

(6.68)

Large total capacitances require large unity-gain frequencies and GBW, and, therefore, expensive Op Amp implementations. As stated, the feedback resistance assures that for s = jω → 0 the output voltage approaches zero, avoiding nonlinear saturation effects. The approximation of Eq. (6.67) by the equation on the right is valid for 2πfB > 1/(RFB CFB ). The charge gain v0 /(Q1 + Q2 ) ∼ = −1/CFB is uniform for this frequency range. Hence, charge sensing is especially useful for bandpass sensing and input chopping with f ≫ 1/(2πRFB CFB ). For instance, for a 2pF-feedback capacitance and a 20-MΩ-feedback resistor the operating-frequency range should be f ≫ 4 kHz.

260

Electrical interfaces

With ωRFB CFB ≫ 1 the output noise spectral density is µ ¶ µ ¶ µ ¶ 4kT RFB 2CS 2 CS 2 Ctot 2 Sn-out = + 4kT RP + 2 Sn 1 + 1 + Sn-in . (ωRFB CFB )2 CFB Ctot CFB (6.69)

Comparison and improvements It is helpful to compare the SNR of all three sensing methods. Assuming that the input-related Op Amp noise dominates the SNR and that in the case of current sensing the feedback resistances and frequencies are sufficiently large (ωRFB Ctot ≫ 1), the signal-to-noise ratios can be roughly estimated, ( 1/Ctot for current and voltage sensing, Q1 + Q2 SNR = √ (6.70) Sn-in ∆f 1/(CFB + Ctot ) for charge sensing. However, in the case of voltage sensing the noise of the following amplifier must be added, whereas for current and charge sensing it is usually negligible (large gain of the input stage). For all three sensing methods the impact of parasitic capacitances between common plates and ground is very important. Thus, the design of capacitances has to consider the different fringing effects as well as parasitic capacitances to ground shields. For all three sensing methods single-ended solutions were presented. They suffer from bad PSRR and external CMR. Another disadvantage is the usage of two voltage sources with possible mismatch. These problems can be solved by using fully differential amplifiers with single voltage sources as for the case of charge sensing illustrated by Fig. 6.19(b). The output signal is v0+ − v0− = −

v1 (Cs1 − Cs2 ) . CFB

(6.71)

The structure is powerful but has to handle a large internal common-mode swing v1 at the inputs of the Op Amp. If the Op Amp cannot master the voltage swing, a supporting input common-mode feedback (ICMFB) must be implemented.

Switched-capacitor sensing Switched-capacitor amplifiers are appropriate front-end circuits for capacitance sensing [Allen and Sanchez-Sinencio 1984]. They are gladly used especially in mixed-signal circuits, where anyway a digital part provides clock and digital control signals, which can be adapted for the control of the switches. Initially, switched-capacitor circuits were introduced in analog design because they allow one to emulate precise and large resistances. Capacitances and clock cycles can be realized with high accuracy. A capacitor ratio is implemented with tolerances on the order of 0.1%, compared with an absolute accuracy of passive components in the range of 30%. Thus, equivalent resistors in switchedcapacitor circuits exhibit an accuracy one to two orders of magnitude higher

261

6.2 Sensor interfaces

(a)

1

S1

2

i1

(b)

i2

v1

C

(a1)

2

1

S2

v2

v1

C

v2

1

i2

(a2)

1

v1 2

(a3)

2

i1

t T/2

T

2

1

v2

C

3T/2

Figure 6.20 (a) The principle of switched-capacitor operation. (b) A bilinear switched

capacitor.

than that for implemented resistors. The silicon area occupied by a switchedcapacitor resistor is several orders of magnitude smaller than the area of resistors built from polysilicon or some other material. Both factors are decisive for the broad application of switched-capacitor techniques in the design of filters, DC– DC and AD converters, sigma–delta converters etc. In capacitive interfaces the possibility of seamlessly integrating switched-capacitor amplifiers and correlated double-sampling techniques is a great advantage.

The principle of switched-capacitor operation Switched-capacitor circuits are based on rapidly switched capacitances for which the signal can be considered constant during a full switching cycle. The continuous flow of charges is substituted by the transport of charge portions. The principle is illustrated in Fig. 6.20(a). Two switches S1 and S2 are controlled by two non-overlapping control clocks shown in the lower picture (a3). S1 is closed when the clock signal φ1 is high, and S2 is closed when φ2 is in the high position. The analysis of switched-capacitor circuits always follows the same methodology. First analyze the time-domain relations between desired voltage variables. Second, convert this equation into the z-domain. Third, find the desired transfer function within the z-domain. Fourth, replace the argument of the ztransformation by exp(jωT ). There is no room for including these steps here. The presentation will be limited to the first step only, which describes the basic processes within the switched-capacitor network. For readers interested in gaining a deeper understanding the books by Liu [2006], Quinn and Roermund [2007], and Unbehauen and Cichocki [1989] can be recommended. To explain the switched-capacitor principle the example in Fig. 6.20(a1) is usually adopted. If the clock cycle T is small in comparison with the speed of variation of applied voltages, the current i can be substituted by its average ¯i. Since i1 is zero during the second half-period and since the current is the derivative of the charge on the capacitor C, the following equation holds: µ µ ¶ ¶ Z T Z T 2 1 1 T ¯i1 = 1 i1 (t)dt = i1 (t)dt = Q − Q(0) . (6.72) T 0 T 0 T 2

262

Electrical interfaces

The accumulated charge at the end of the first half-cycle is Q(T /2) = Cv1 (T /2), whereas the charge at the beginning of the cycle is determined by Cv2 (0), which is the accumulated charge at the end of the previous half-cycle. Therefore µ µ ¶ ¶ C ¯i1 = C v1 T − v2 (0) ∼ v1 (t) − v¯2 (t)). (6.73) = (¯ T 2 T The current is proportional to the applied voltage and obviously emulates Ohm’s law ¯i1 = (¯ v1 (t) − v¯2 (t))/R with R=

T . C

(6.74)

On performing the same analysis for a switched capacitor as in Fig. 6.20(a2) and for a bilinear configuration according to Fig. 6.20(b) one gets for the first an equivalent resistance R = T /C and for the bilinear capacitor R = T /(4C). Many other capacitor configurations, including serial-parallel connections, cascodes etc., are used. In all cases, similar relations result. One may expect that the usage of noiseless capacitances will avoid the problem of thermal noise. This is not the case, because the switches always have an on-resistance, Ron , that actively produces thermal noise during their on-phase: SR on = 4kT Ron . This noise is filtered by the low pass formed by the resistance Ron and capacitance C. At the capacitance node the noise spectrum is SnC = The total output noise voltage calculated as

4kT Ron . 1 + (ωCRon )2

(6.75)

p R∞ hvnC i2 = [1/(2π)] 0 SnC dω can readily be

p hvnC i2 =

r

kT . C

(6.76)

It depends not on the on-resistance, but only on the switched capacitance C and is called kT /C noise (“kT over C noise”). The lack of a dependency on Ron is not surprising, because a noise power growing with Ron is compensated for by a reduced noise bandwidth ∆fnC = 1/(4Ron C). It should be noted that the kT /C noise is typically wide-band noise, which often can be approximated by white noise. Indeed, typical on-resistances are on the order of 0.1 to 10 kΩ, which for typical capacitances on the order of 2 to 20 pF result in noise bandwidths between 1.25 and 1250 MHz.

Switched-capacitor amplifiers Switched-capacitor operation assumes correct sampling. That means that all settling processes are finished within a short time after a switch is closed or opened. The settling time interval should be about one order of magnitude smaller than half of the cycle time itself. Therefore, the condition for proper functioning of switched-capacitor amplifiers is a large enough Op Amp bandwidth. It should be ten to twenty times higher than the switching frequency. Under this assumption

263

6.2 Sensor interfaces

(a)

(b)

1

CFB

2

CFB

-

Cs

v0

-

Cs

CP

CP

+ 2

1

v1

2

1

2

+ 1

v0 CL

1

v1

Figure 6.21 Switched-capacitor amplifiers with different switching schemes: (a) With

input and feedback switches and (b) with input and output switches.

a first analysis of switched-capacitor amplifiers can be performed, neglecting the settling processes and considering only charges and voltages after the settling processes have finished. There are many possibilities to realize switched-capacitor amplifiers with an identical transfer characteristic. Two basic configurations are shown in Fig. 6.21. For the amplifier in Fig. 6.21(a) the sensing and feedback capacitances are embedded within a switch environment. If φ1 is high, the output is connected with the inverting input, resulting in v− = v0 = 0. Therefore the voltage vCS across the sensing capacitance Cs (T /2) is vCS (T /2) = −v1 . The total charge within the system is concentrated within Cs and equals Qs (T /2) = −v1 (T /2)Cs (T /2). Within the next half-cycle φ2 is high, and therefore the charged capacitance Cs is connected via the feedback capacitance to the output. The charge redistribution is governed by the charge-conservation law, µ ¶ T QFB (T ) + Qs (T ) + QP (T ) = Qs , QFB (T ) = (v0 (T ) − v− (T ))CFB , 2 µ ¶ µ ¶ µ ¶ T T T Qs (T ) = v− (T ))Cs (T ), QP (T ) = v− (T )CP , Qs = −v1 Cs . 2 2 2 (6.77) On applying the Op Amp equation v0 = −v− /A, for large open-loop gain one gets µ ¶ T Cs (T /2) C¯s v0 (T ) = −v1 or v0 (T ) = −¯ v1 . (6.78) 2 CFB CFB The output voltage changes between zero in the first half-cycle and the value given by Eq. (6.78) during the second. If the transfer processes are fast, then the sequence represents a series of rectangular pulses, the amplitudes of which are the sampled at t2k + 1 = (2k + 1)T /2 + ε values and the average of which is − 21 v1 (T /2)C¯s /CFB . The second amplifier according to Fig. 6.21(b) is basically identical with Fig. 6.21(a). The output voltage during the second half-cycle is equal to the

264

Electrical interfaces

v1 1+2

3

CFB

1

vin

Cs1

CH

Cs2 1+2

CP 3

-v1

1+2

3

+ 1

v0

1

vH

1+2

t T1

T2

T

Figure 6.22 A switched-capacitor amplifier with correlated double sampling.

voltage given by Eq. (6.78). However, the insertion of a buffering load capacitance CL makes sure that the output voltage does not go back to zero after the loading cycle, but remains equal to the output voltage from the previous half-cycle, leading to a smooth output signal. The principal difference of switched-capacitor amplifiers from the charge amplifiers considered here is their consistent performance down to DC. No feedback resistors are required, because drift into saturation is avoided by the first half-cycle, which represents a reset phase for all capacitances and input voltages. Low-pass sensing becomes possible with a constant exciting voltage v1 = constant. If instead of a single-ended capacitor Cs the differential arrangement according to Fig. 6.16(b) or Fig. 6.19(a) with v2 = −v1 is used, the output voltage from Eq. (6.78) has to be modified, v0 (T ) = v1

Cs2 (T /2) − Cs1 (T /2) C¯s2 − C¯s1 = v1 . CFB CFB

(6.79)

Correlated double sampling in switched-capacitor amplifiers Correlated double sampling (CDS) is a powerful method for eliminating correlated disturbances like offset and 1/f noise. A CDS procedure performs sampling and storing of disturbances in a first half-cycle and sampling and storing of a signal plus disturbance in a second half-cycle, additionally subtracting the disturbance which was stored in the first half-cycle. As long as the disturbances between the first and the second half-cycle are strongly correlated they are cancelled out during the second half-cycle. CDS can be implemented in all of the versions of sensing discussed before. In switched-capacitor amplifiers an elegant integration of CDS by introducing a third subcycle as explained in Fig. 6.22 is possible [Wongkomet and Boser 1998]. In a first cycle φ1 is high during nT ≤ t < nT + T1 , and all capacitors are reset to zero. In a second subcycle nT + T1 ≤ t < nT + T2 the sensing capacitances and the error-storing capacitance CH remain connected to ground, but the input-related noise and offset source vin now injects charges into all capacitances

6.2 Sensor interfaces

according to µ ¶ ¯¯ 1 1 ¯ v0 = Q + ¯ CFB Ctot ¯

, t=T 2

¯ Q ¯¯ v− = vin + ¯ Ctot ¯

, t=T 2

265

QH = v0 CH |t=T 2 . (6.80)

Q is the charge on the feedback capacitance that is equal to the charge on the total capacitance Ctot = Cs1 + Cs2 + CP , because both are the product of integrating the current created by v0 . Using the inner Op Amp equation one gets for large open-loop gains µ ¶ Ctot v0 (T2 ) = −vin (T2 ) 1 + . (6.81) CFB During the third subcycle nT + T2 ≤ t < nT + T3 the sensing capacitances are connected to the voltage sources, and all other switches are open. This corresponds to the standard measurement cycle of a charge amplifier, where here the disturbances vin are included. No current is flowing into the error-storing capacitance. Thus, the Op Amp output voltage is, after the corresponding calculations, µ ¶ Cs2 (T3 ) − Cs2 (T3 ) Ctot v0 (T3 ) = v1 − vin (T3 ) 1 + . (6.82) CFB CFB The voltage at the output node of the error-storing capacitor CH is vout = v0 (T3 ) − vH (T2 ) = v0 (T3 ) − v0 (T2 ) and therefore given by µ ¶ Cs2 (T3 ) − Cs2 (T3 ) Ctot vout = v1 − (vin (T3 ) − vin (T2 )) 1 + . (6.83) CFB CFB One can see that slowly changing input-related disturbances, vin (Tk ), are cancelled out. The subcycles may have different durations so that the loss of signal measurement time (T3 − T2 )/T can be reduced. A fully differential implementation of the principle reduces the PSRR and CMRR further. The advantages of switched-capacitor amplifiers must be considered in the context of their noise performance. Noise aliasing increases the role of Op Amp noise in comparison with other noise sources. The analysis is quite complicated, because due to undersampling the broad-band noise of the amplifier is folded down into the pass band and transformed to the output, whereas the transfer characteristics during the corresponding subcycles may be different. As a rule of thumb one may imagine the spectrum of the Op Amp noise as uniform up to the unity-gain frequency fu = GBW and assume that the noise of all N = 2fu /fs spectral intervals is folded down in the pass band. Thus, the noise spectrum will increase by the factor N [Hsien et al. 1981, Petkov and Boser 2004, Enz and Temes 1996]. However, for a more accurate estimate the noise transfer during all periodically repeating subcycles must be calculated for any concrete structure and compensation method. In doing this the non-stationary character of the switched noise has to be taken into account [Gerosa et al. 2001, Goette and Gobet

266

Electrical interfaces

1989, Vasudevan and Ramakrishna 2003]. Since an analytical representation is cumbersome, appropriate simulation models embedded into CAE design tools are usually used.

6.3

Data converters Analog-to-digital (AD) converters (ADCs) and digital-to-analog (DA) converters (DACs) are the most commonly used interfaces between the analog and digital worlds. ADCs transform a sampled signal into a digital representation, whereas DACs first re-transform a sequence of digital words into signal values and secondly reconstruct a time-continuous signal by a hold procedure during the sampling interval and by subsequent low-pass filtering.

6.3.1

Sampling and hold Usually AD conversion is performed in two steps: sampling and hold. The sampling is accomplished in equidistant steps v(t) ⇒ {v(nTS )}, n = . . . −1, 0, 1, . . ., and the hold interval is needed in order to realize the actual AD transformation v(nTS ) ⇒ [vn ], where [vn ] is a digital word of length N . The number N determines the digital resolution. If the signal remains constant within the hold interval, the system is called a zeroth-order hold. Higher-order polynomial extrapolations may be used. As is well known, the ratio between the sampling frequency fS = 1/TS and the upper frequency of the signal spectrum fv determines the possibility of an accurate signal reconstruction from the (exact) sampled values. According to Nyquist’s theorem the relation fS ≥ 2fv

(6.84)

must hold in order for one to reconstruct the original function from its sampled values v(t) =

∞ X

v(nTS )

−∞

sin(ωS (t − nTS )/2) . ωS (t − nTS )/2)

(6.85)

fS = 2fv is the so-called Nyquist frequency. Guaranteeing that the Nyquist condition holds may require the insertion of appropriate anti-aliasing filters into the signal path before sampling. If the Nyquist condition is not fulfilled, undersampling takes place. Namely, the sampled signal v {s} (t) can be represented as v {s} (t) =

X n

v(t)p(t − nTS ) = v(t)

1 X sin(kωS p0 /2) −j k ω S p 0 /2 j k ω S t e e , TS kωS p0 /2 k

(6.86)

6.3 Data converters

267

(a) -fS

-fS/2

fv < fS/2

fS

f

-fS

-fS/2

fv < fS/2

fS

f

2fS

f

(b)

(c) -2fS

-3fS/2

-fS

-fS/2

fS/2 < fv fS

3fS/2

Figure 6.23 Nyquist sampling (b) and undersampling (c) of the original spectrum (a).

where p(t) is the rectangular sampling pulse of length p0 and height 1/p0 , ( 1/p0 for 0 ≤ t < p0 , (6.87) p(t) = 0 for p0 ≤ t < TS . For p0 → 0 the sampling pulse converges to the delta function. The expression on the right in Eq. (6.86) is simply the Fourier-series representation of the periodic pulse sequence. Therefore, the Laplace transform of the sampled signal is given by Z ∞ {s} v {s} (t)e−j st dt V (s) = −∞ Z ∞ X sin(kωS p0 /2) 1 = dt v(t) e−j k ω S p 0 /2 e−(s−j k ω S )t . TS −∞ kωS p0 /2 k

Correspondingly, the spectra of the sampled and original signals obey the relation 1 X sin(kωS p0 /2) −j k ω S p 0 /2 V {s} (jω) = e V (j(ω − kωS )), (6.88) TS kωS p0 /2 k

where V (jω) is the spectrum of the input signal. For the sake of clarity the case of delta-pulse sampling p0 → 0 is accentuated: 1 X V {s} (jω) = V (j(ω − kωS )). (6.89) TS k

Equation (6.89) is presented here in order to clarify the aliasing effect stressed in the last section. If the input spectrum extends to frequencies limited by ω ≤ 1 2 ωS as illustrated in Figs. 6.23(a) and (b), the baseband spectrum and its replicas V (j(ω − kωS )), k 6= 0, do not overlap. Hence, the components with k 6= 0 do not disturb the frequency band |ω| ≤ 21 ωS . This is the case of Nyquist sampling. The violation of the Nyquist criterion as shown in Fig. 6.23(c) folds parts of the replicas V (j(ω − kωS )) into the band ω ≤ 21 ωS , preventing correct reconstruction of the signal.

268

Electrical interfaces

If sampling is performed by a rectangular pulse sequence according to Eq. (6.87), the structure of the sampled spectrum remains intact. Only the coefficients at the replicas are less than unity and decrease with increasing k.

6.3.2

Single-sample conversion in the amplitude domain If the Nyquist criterion is fulfilled, the usual means of AD conversion is a singlesample conversion. Single-sample conversion means that during the hold interval the sampled signal is quantized by parallel or successive comparison with a set of thresholds vth-i . Usually the thresholds vth-i are given by 2N values in the range vm in ≤ v(nTS ) ≤ vm ax : vth-i = vm in + ∆v i/2N , i = 0, 1, . . . , 2N , with ∆v = vm ax − vm in . The intervals between the thresholds are numbered in the same way. The digital number of the interval within which the signal falls represents the quantized digital output of the ADC. The most common conversion methods are as follows.

r Serial conversion by consecutive comparison of v(nTS ) with a digitally controlled ramp vRam p (tk −1 ≤ t < tk ) = constant × k; k = 1, . . . , 2N ; up to 2N steps are required. r Resistive or capacitive division of v(nTS ) using an R- or C-division tree, and consecutive comparison between two possible states. The division tree is designed as a stepwise-accessible set of thresholds {vth-k } = {vth-(k −1) } ± ∆v/2k , k = 1, . . . , N with {vth-1 } = vm in + ∆v/2. In the first step it is decided whether the sample is located in the upper or lower half-range v(nTS ) > < {vth-1 }. Assume v(nTS ) > vm in + ∆v/2. In the second step the sampled value is then compared with the corresponding thresholds vm in + ∆v/2 ≤ v(nTS ) < vm in + 3 ∆v/4 and vm in + 3 ∆v/2 ≤ v(nTS ) < vm ax of the two quarter-ranges within the selected half-range. In the next branch it is decided where the tree will be activated. At most N steps are needed. r Successive approximation by comparison with thresholds, which are generated by a DAC according to the same scheme as for the division tree r Single or two-step flash conversion with 2N − 1 comparators working in parallel with all thresholds. One step is needed for full conversion. The difficulties of such an approach (offset, feedthrough) limit the resolution to about 6 to 8 bits. The difficulties can be partially mitigated by dividing the conversion cycle into two parts: the first for the most significant bits and the second for the remaining bits. All of the methods outlined are related to domain-conserving approaches, i.e. any conversion step transforms amplitudes into amplitudes or into their quantized counterparts. There exists an impressive number of architectures and publications describing different methods and properties of such Nyquist-type ADCs [Maloberti 2006, Rasavi 1995, Plasche 2003]. Astonishing sampling speeds of up to 10 gigasample/s and high accuracies are reported. However, inertial sensors

6.3 Data converters

269

impose very moderate requirements, at least with respect to the sampling rates, which allows one to concentrate on resolution, nonlinearity, and offset errors.

6.3.3

Time-domain conversion For many applications including inertial MEMS a conversion of sampled values into time parameters of some standard signals is beneficial. The sampled values {v(nTS )} should determine the value of the governing time parameter linearly. The time parameter is often a result of combining many subsequent samples, calculating, for instance, an average pulse density over a certain time interval. One of the time parameters that could be used is the frequency of a harmonic or periodic function with constant amplitude. Since an assignment of a time parameter should happen periodically within any sampling interval, difficult matching problems emerge at the interval limits. More convenient and well established is the conversion of sampled signal values into pulse widths and pulse densities with regular or stochastic distributions of the pulse location. The width or the number of pulses per sampling interval carries the information on the signal.

Pulse-width and pulse-density modulation One of the oldest time-parameter conversion techniques is pulse-width modulation (PWM). An analog pulse-width-modulated signal with duty cycles ϑk = ∆k /TS (0 ≤ ϑ < 1, ∆k is the length of the kth pulse) is described by X Z(t − kTS − ∆k ) (6.90) vPW M (t) = Vm in + (Vm ax − Vm in ) k

with Z(t − kTS − ∆k ) = Zk = h(t − kTS ) − h(t − kTS − ∆k ); h(t) is the unit step function. Since the pulses Zk are not overlapping, Zk Zl = δk ,l Zk , it immediately follows that X 2 2 2 2 Z(t − kTS − ∆k ). (6.91) vPW M (t) = Vm in + (Vm ax − Vm in ) k

The beautiful property of pulse-modulated signals is that they preserve the linear (or nonlinear) dependency on the modulation parameter ∆k even after squaring. The traditional analog PWM sequence is generated using a sawtooth generator and a comparator. The principle is illustrated in Fig. 6.24(a). As soon as the signal exceeds the sawtooth level, the starting pulse is set back to zero. For ADC and DAC applications the analog sawtooth is replaced by a clock-controlled digital sawtooth, shown in Fig. 6.24(b), that can easily be generated using simple counters and comparators. For any discrete level a cycle time TC has to be spent. Therefore, the accuracy of data conversion is limited by the number of available time slots within one sampling period R = TS /TC = fC /fS . R is sometimes called the oversampling ratio despite the fact that it has nothing to do with any oversampling. The necessary number of representing bits is given by log2 R.

270

Electrical interfaces

(a)

(b) signal

k

TC

kTS

(k+1)TS

(k+2)T

kTS

(k+1)TS

(k+2)T

Figure 6.24 Analog and digital generation of PWM modulated signals. (a) Analog generation of a PWM signal using a sawtooth and a comparator. (b) Digital sawtooth generation.

signal

(a)

(b)

TP kTS

(k+1)TS

kTS

(k+1)TS

Figure 6.25 Comparison of PWM and PDM signals. (a) A typical PWM signal. (b) A pulse-density-modulated signal.

An important feature of the output function, vPW M , is the distribution of its spectral components and the related question of whether or not it is easy to separate the different components created by the sampling process. Consider the Fourier series of a pulse-modulated signal for a constant pulse length ∆k = ∆. In correspondence with Eq. (6.86) the Fourier representation is vPW M (t) =

∆ X sin(kωS ∆/2) −j k ω S ∆/2 j k ω S t e e . TS kωS ∆/2

(6.92)

k

The baseband component has the amplitude C0 = ∆/TS . The other spectral components are located at kfS and have the following relative amplitudes: |Ck | sin(kπ∆/TS ) = . C0 kπ∆/TS

(6.93)

For a duty cycle ∆/TS = 0.25 the spectral component at fS features 90% of the baseband component, the component at 2fS features 64% etc. The example demonstrates the difficulties of baseband separation in the presence of higherharmonic replicas. Very sharp filters are needed. This task simplifies considerably, however, if the compact packaged time slots are evenly distributed over the sampling interval. This kind of modulation is called pulse-density modulation (PDM). In Fig. 6.25(a) a PWM signal is shown, and in 6.25(b) its equivalent PDM representation. Here for simplicity it is assumed that the PWM signal is coded with four time slots, which in the case of PDM are distributed equally over the whole sampling interval. The PDM signal has now a basic period of TP = TS /4 – four times shorter than the sampling time.

6.3 Data converters

271

Thus, the spectral lines are arranged around kfP = 4kfS . They can be more easily separated from their higher-harmonic replicas than in the case of PWM. In the given example the frequency components at kfS = 4kfP are all zero, because they fall on the zero-crossing points of the sine function. Generally speaking, the kfS replicas are more strongly attenuated than in the case of PWM. The example shows that the spectral content of a PDM signal with time-varying ∆k is distributed over the basic frequency interval [0, fC = N fS ]; however, on average the separation of the baseband signal is much less disturbed by replicas than it is in the case of PWM. It can be shown that the situation improves further if the time slots are distributed stochastically over the sampling interval.

Σ∆ Converters It is now self-evident how to generate a stochastic pulse distribution using the sampling process itself. The sampling process with time steps TS , where the signal fulfills the Nyquist criteria fv < 21 fS , can be substituted by an oversampling process, where the signal is sampled in every time slot TC = 1/fC ≪ TS . A comparator generates a pulse of length TC , the amplitude of which depends on its belonging to a given quantization interval. The large number of comparison procedures provides the needed accuracy of sampling. For instance, for a one-bit quantizer the output is one or zero (or plus one and minus one), and the ratio between the number of one-pulses and the total number of pulses within an appropriately chosen time interval is proportional to the average input signal. The output counter of the produced bit-stream is the Σ part of a Σ∆ converter. Output-counting corresponds to a process of digital filtering, which is an inevitable final part of a Σ∆ converter. In contrast to PWM/PDM, the ratio R = fC /(2fv ) is now a true oversampling rate. Clearly, the quantization of the sampled signal is much more difficult during the short time interval TC than that during the long Nyquist-sampling interval TS = 1/(2fv ). However, there is no need and no intention to realize the same sampling accuracy as in the case of baseband sampling, because the now much larger discretization inaccuracy can be assigned to a distinguishable error signal – the so-called quantization noise. The quantization noise is responsible for a stochastic assignment of pulse locations. Its spectrum can accordingly be formed by combining, for instance, two or more consecutive sampling and quantization steps. This process is called noise shaping and is aimed at shifting the quantization-noise spectrum as much as possible into a region outside the signal spectrum. Using the spectral properties of this noise, it can be suppressed by appropriate output filters, achieving at the end astonishing sampling accuracies. Oversampling reduces drastically the requirement on the selectivity of costly anti-aliasing filters for AD converters, because the higher sampling rate allows one to accept signal and distortion residuals outside the intended bandwidth, which will be filtered away after quantization as shown in Fig. 6.26(b). The output filter of a Σ∆ converter is thus a digital filter (decimation filter) and can

272

Electrical interfaces

(a)

v

Low Pass filter

y

ADC

(b)

v

Low Pass filter

Low Pass filter

R

y

Modulator

Figure 6.26 A comparison of (a) AD and (b) Σ∆ converters.

(a)

(b)

yn

y= KQ v Q

n

v vn

KQ

y +

Figure 6.27 (a) The quantization characteristic and quantization error. (b) The

equivalent circuit of the quantizer.

be implemented using cheap digital circuitry. In general, Σ∆ conversion transfers the dominant analog signal processing of conventional ADC and DAC into the digital domain. The result is considerable savings in CMOS area and power, and a higher accuracy. In Fig. 6.26 the differences between the structures of classical ADCs and of Σ∆ converters are illustrated. The advantages of Σ∆ converters are their large dynamic range, small nonlinearities, high resolution, digital output, and simplicity of implementation.

Quantization noise The key terminus in oversampling converters is the quantization error εn = ε(nTC ), which is given by the difference between the quantized signal yn = y(nTC ) and the linearly scaled input signal vn = v(nTC ), y n = K Q vn + ε n .

(6.94)

The quantization process is illustrated in Fig. 6.27(a). ∆Q is the quantization interval and KQ the quantizer gain. The input–output relation of a quantizer can be represented by the equivalent circuit shown in Fig. 6.27(b). The crucial step in the theory of Σ∆ converters is the approximation of the quantization error, ε, by an input-independent stochastic process. It is clear

6.3 Data converters

273

that the quantization error is not a true stochastic process and always depends on the input signal. However, if one assumes that the signal is changing quite stochastically and is equally distributed within one quantization interval (the “busy” condition), then the error can be decoupled from the input signal and also considered as uncorrelated (white) noise with uniform probability distribution pε (ε) = 1/∆Q . In this case the noise variance can readily be determined: σε2

=

Z

−

∆Q 2 ∆Q 2

ε2 pε (ε)dε =

∆2Q . 12

(6.95)

The quantization error is assumed to be constant during a sample interval and uncorrelated between samples. Thus, the quantization noise folds in the frequency range [−fC /2, fC /2]. Using the assumption of white quantization noise, the spectral density is therefore Sε (ω) =

∆2Q TC 12

(6.96)

and decreases with increasing sampling frequency fC = 1/TC . Despite the obvious deficiencies the error model described above is used to gain insight into the qualitative behavior of different Σ∆ converters. The discomfort experienced certainly by many readers is eliminated by exact supporting simulations and additional considerations of different signal classes, which definitely violate the “busy” condition. Among them are DC signals with certain amplitudes, which may create so-called “pattern noise” or death zones. Special measures such as dithering are usually implemented in order to bring the reality closer to the model and, for instance, to eliminate the periodic quantization noise which is created in the case of certain DC inputs (idle tones). One way to create dither is, for instance, the addition of a pseudo-random sequence with spectral components outside the signal band to the input. Countermeasures are required, in particular, for modulators based on quantizers with a low number of discretization levels, for instance, for one-bit quantizers. In the case of oversampling the error variance σε′ = (Sε (ω)2fv )1/2 after separation (after filtering) of the baseband [−fv , fv ] decreases according to Eq. (6.96) by the oversampling rate R = fc /(2fv ), ′

σε2 = Sε 2fv =

σε2 . R

(6.97)

This is the basic equation of oversampling converters. After removal of all spectral components above fv by appropriate output filtering, the impact of the √ quantization error has been reduced by a factor of R. That is, any doubling of the oversampling rate decreases the power of the quantization noise by 3 dB and therefore improves the digital resolution by half a bit. Consideration of the one-bit quantizer should demonstrate the picture. The signal has a range −vm ax ≤ v ≤ vm ax , i.e. ∆Q = 2vm ax . In any step the quantizer decides whether v > 0 or v < 0 and correspondingly assigns the output plus or

274

Electrical interfaces

(a) One-step delay

v(nTC) +

+ -

z-1 +

u

(b)

Quantizer: ± VRef

y(nTC)

± VRef

+ v(nTC)

+ -

+

+

+

-

z-1 +

y(nTC)

z-1

Figure 6.28 (a) First- and (b) second-order Σ∆ modulators (with the digital output filter omitted).

minus VRef . This corresponds to one-bit resolution. If a resolution of N bits is required, an oversampling rate of R = 22(N −1) is needed; 5-bit resolution corresponds to an oversampling rate of 256, but 10-bit resolution would already need the giant number of R = 262 144. Thus, further steps are needed in order to suppress the impact of the quantization noise.

Noise shaping According to the assumptions made above, the quantizer creates uncorrelated error samples that have a white spectrum. By noise shaping one shifts the error spectrum from the low-frequency baseband f < fv into regions above fv , while keeping the noise power constant. For instance, on substituting the quantizer noise εn by the difference εn − εn −1 , y n + 1 = vn + ε n + 1 − ε n ,

(6.98)

one clearly decreases the low-frequency components while at the same time increasing the high-frequency content. Equation (6.98) suggests the structure of the system: the error signal must be fed back and subtracted from the input. Figure 6.28(a) shows the corresponding structure. It is easy to see that this structure realizes exactly the relation according to Eq. (6.98). Using the z-transforms P −k with appropriate complex z, Y (z), V (z), and E(z) (e.g. Y (z) = ∞ −∞ y(kT )z guaranteeing convergence) the output signal can be represented as Y = z −1 V + (1 − z −1 )E.

(6.99)

The input signal is delayed by one time step; however, the noise spectrum has been shaped. Since for a sampled signal the Fourier transformation follows from the z-transform on substituting z = ej k ω T , the noise spectrum can be directly written as µ ¶ ωTC Sε-out (ω) = |1 − e−j ω T C |2 Sε (ω) = 4 sin2 Sε (ω) (6.100) 2 2 with Sε = (VRef /12)TC according to Eq. (6.95). The term 4 sin2 (ωTC /2) is the noise transfer function, which amplifies the noise spectrum at high frequencies f > 2fv and reduces it within the low-frequency range. At frequencies f > fC /2 the spectrum is mirrored about f = fC /2. For large oversampling ratios (fv ≪ fC 2 and thus sin(ωT /2) ∼ within the signal = ωT /2) the total output variance σε-out

6.3 Data converters

275

bandwidth is 2 σε-out

V2 = Ref TC 12

Z

fv

2

4 sin

−f v

µ

ωTC 2

¶

2 dω ∼ VRef π2 . = 2π 12 3R3

(6.101)

Doubling the oversampling√rate R results now in an increase of the signal-tonoise ratio by a factor of 8 or, equivalently, in an increase of the resolution by one and a half bits. Using as in the example above a one-bit quantizer, an oversampling rate of R = 256 leads to 12-bit resolution. Higher-order modulators improve the resolution further. For instance, a second-order modulator features an input–output relation according to Y = z −1 V + (1 − z −1 )2 E

→

V 2 π4 2 ∼ σε-out = Ref 12 5R5

(6.102)

and can be realized by a sequence of two structures as illustrated in Fig. 6.28(b) on feeding back the quantized signal to the input of any of the first-order subsystems. √ √The resolution of a second-order modulator is improved by a factor 32 = 4 2 on doubling the oversampling rate. To generalize, Σ∆ modulators of Nth order follow the scheme Y = z −1 V + (1 − z −1 )N E

→

2 σε-out = σε2

π 2N 1 , 2N + 1 R2N + 1

(6.103)

which allows one to estimate easily the output noise variance given on the right of the previous equation [Norsworthy and Schreier 1997]. Of course, there is a limit on increasing the order of the modulator set by additional wide-band noise sources such as thermal noise and switching noise (jitter), which at high frequencies start to dominate the total noise, exceeding the contribution of the quantization noise. The implementation of higher-order modulators is based on a combination of first- and/or second-order modulators and on cancelling out of lower-order quantization noise generated by the substructures. They require a very accurate matching of the different stages while higher-order loop filters suffer from an inherent tendency to exhibit signal-dependent instabilities. Any design of Σ∆ modulators needs a careful analysis of possible unwanted limit cycles. Some analytical estimates can be derived using stochastic linearization of the quantizer characteristic as proposed in Ardalan and Paulos [1987] and applied to embedded Σ∆ converters in Handtmann [2005].

Discrete-time modulators In discrete-time (DT) Σ∆ modulators sample-and-hold structures are used to operate synchronously with the sampled signals within all substages of the modulator. Sample-and-hold (S&H) stages include switched capacitors. Therefore, the dominant noise source in DT modulators is usually the kT /C noise. Figure 6.29 shows a simulation model of a typical first-order DT Σ∆ modulator, where the noise sources represent the kT /C noise generated in the S&H

Electrical interfaces

BS

In1

-K-

1

++

++

Zero-Order Hold

++

++

1/z

Zero-Order Hold

Unit delay

1

Comparator

-K-

276

Random Source

Random Source 1

Random Source 2

++

x

VRef 2

-1

Zero-Order Hold

Figure 6.29 A simulink model of a first-order modulator.

(a) 10

(b)

0

10

10

-2

10

resolution : 40 microV/sqrt Hz dynamic range: 83 dB

-4

10

10

dynamic range: 124 dB -6

-8

10

-12

10

resolution: 0.33 microV/sqrt Hz -4

10

-10

10

-2

10

-8

10

10

10

-6

10

0

-10

10

-14

-12

10

-16

10

3

10

4

Frequency

10

5

10

-14

-16

10

3

10

4

10

5

Frequency

Figure 6.30 (a) The spectrum of a first-order DT Σ∆ modulator. (b) The spectrum of a fifth-order DT Σ∆ modulator.

stages. Their noise power is simply Pn = kT /CS& H with CS& H the switched capacitance of the S&H stage. The central block highlighted in gray is the heart of the modulator. The function z −1 is realized by a delay line within the feedback structure. Using this structure, higher-order modulators can be built, for instance, a fifth-order modulator [Strle 2009], the simulation results of which are used here in order to demonstrate the impact of the noise shaping. Figures 6.30(a) and (b) illustrate different spectra of the first- and fifth-order modulators for a sampling frequency of 1 MHz and a reference voltage of 0.6 V. Dither is part of the system. The impressive improvement achieved by a fifthorder system in comparison with a √first-order modulator is characterized by a resolution increasing from 40 µV/ Hz for the first-order modulator √ √ to 0.33 µV/√ Hz. Correspondingly, the dynamic range grows from 83 dB/ Hz to 124 dB/ Hz at a frequency of 4000 Hz. The simulation results for the fifth-order system were experimentally verified for an operating temperature of 130 ◦ C without significant deviation.

277

6.3 Data converters

(a)

(b) y

v(t) +

D

DAC:

Q

± VRef

S/H

bitstream

v(t)

+

+

-

S/H -

y

clock fC

± VRef

Figure 6.31 (a) First- and (b) second-order continuous time Σ∆ modulators.

Interestingly, without kT /C noise the resolution of the fifth-order system improves by only 5% in comparison with the “noisy” case, indicating that the quantization noise is still limiting the final resolution.

Continuous-time modulators For convenience up to now the modulators have been presented by invoking their discrete-time implementations, where the input is the sampled signal. Many practical implementations, however, are continuous-time modulators, as illustrated by Fig. 6.31(a) for a first-order modulator. This is especially the case when the Σ∆ modulator is embedded into an electromechanical feedback loop as described in the next chapter. The time-continuous input signal and the fed-back bitstream are integrated and applied to the quantizer. The comparator(s) switches the output as soon as one of the thresholds is crossed. This may happen at any time. However, the subsequent D-flip-flop synchronizes any of these changes with the next rising edge of the clock pulse. Thus, the output is a temporally discrete pulse sequence, that is synchronous with the clock. In order to analyze the structure it is convenient to introduce fictitious switches and sample-and-hold stages as shown in Fig. 6.31(b) for a second-order modulator. Since the signal can be considered nearly constant during one clock cycle, such an approximation does not significantly distort the real picture. However, the small time uncertainties due to the clock jitter in the feedback DAC generally increase the noise floor and make time-continuous converters more sensitive to jitter. Since the loop filter operates like an anti-aliasing filter, it becomes possible to eliminate the pre-filter which is needed for discrete converters. The continuous-time and discrete-time modulators can be considered mathematically equivalent. Indeed, on approximating an integration step by intout (n) = intout (n − 1) + TC intin (n), the transfer function in the z-domain is TC /(z − 1). For an integrator gain α = 1/TC one can immediately see that, for instance, the model of the time-continuous modulator according to Fig. 6.31(a) coincides with Eq. (6.99), which was derived for the time-discrete version.

Bandpass Σ∆ converters In communication systems analog-to-digital conversion is often required not for low-pass signals, but for bandpass signals. In inertial MEMS it may also be beneficial sometimes to use bandpass converters, in particular for the demodulation

278

Electrical interfaces

of nearly harmonic sensing-signals in gyroscopes. The Σ∆ bandpass converters follow the same noise-shaping path as that derived for low-pass converters (see Eq. (6.103)) with the difference that the integrators are replaced by resonators. This is equivalent to the substitution z −1 → −z −2 . The zeros of the noise-transfer function are now at fC /4 and the quantization noise is shaped away from this frequency. For arbitrary notch frequencies f 6= fC /4 the general transformation z ⇒ −z

z+a , az + 1

−1 < a < 1,

(6.104)

is applied, which allows one to select the desired frequency by variation of a. In any case, high-order converters are required. Since the method is still rather seldom used in inertial MEMS, it is not considered further here. Excellent overviews on bandpass Σ∆ converters can be found in de La Rosa et al. [2002], Salo [2003], and Engelen and van de Plassche [1999].

Decimation Σ∆ converters can be used as embedded systems, as will be discussed in Chapter 7, or as AD converters. In the last case the data generated, which are encoded at high sampling rates, are not well suited for processing. They must be decimated to a low word rate. A direct sub-sampling by taking for instance every Mth sample would increase the noise in the baseband and lead the oversampling ad absurdum. Noise contributions from all frequency intervals (k/M )fC ± fv are the reason for this (k is an integer). Even in the self-evident case of counting the pulses within R time slots (running averaging) and sub-sampling the result with frequency f = 2fv the procedure would generate unacceptable noise. The high-frequency noise must be filtered away efficiently first, before the sample rate is reduced. If the sampling rate should be reduced to fC /M , the noise and possible spurious signals must be removed from the region above fC /(2M ) in order to avoid aliasing effects within the sub-sampled signal. The filters should have good stopband attenuation and small passband variations. A one-step realization of a filter with a small transition band (i.e. transfer region between passband and stopband) requires a high-order filter with a large number of filter coefficients. Such a filter is area- and power-consuming. Fortunately, filtering by a cascade of several simple stages considerably reduces the complexity of the circuitry. The most common filter structure is a cascade of integrator combs followed – if necessary – by finite-impulse-response (FIR) filters. An integrator comb realizes a simple running averaging procedure, zn =

M −1 1 X yn −k M k=0

or

Z=

M −1 1 X −k 1 1 − z −M z Y = Y. M M 1 − z −1

(6.105)

k=0

It is often called (not very correctly) a “sinc” filter, despite the fact that a true “sinc” filter describes a hypothetical low-pass filter with brick-wall frequency response.

279

6.3 Data converters

0

-10

n=1

Attenuation (dB)

-20

-30 n=2

-40

-50 n=3

-60

-70

-80 0

0.2

0.4

0.6

0.8

1

f/fC

Figure 6.32 The filter transfer function of cascaded comb filters with eight input samples (n is the number of cascades).

The transfer function of the comb filter in the frequency domain G(jω) = Z(jω)/Y (jω) follows from Eq. (6.105) with z = ej ω T C : G(jω) =

sin(M ωTC /2) −j (M −1)ω T C /2 sinc(M ωTC /2) −j (M −1)ω T C /2 e = e , M sin(ωTC /2) sinc(ωTC /2) (6.106)

with sinc(x) = sin(x)/x. Figure 6.32 illustrates the transfer behavior of cascaded comb filters with one to three cascades and an averaging length of eight samples. The passband interval is below fC /8, and the attenuation in the stopband is more then −45 dB for a threefold cascade. The noise density after filtering, S ′ (ω), is the product of the noise density of the Σ∆ output Sε-out (ω) and the transfer function |G(jω)|2n of the comb filter cascade, S ′ (ω) =

1 sin2n (M ωTC /2) Sε (ω). M 2 sin2(n −N ) (ωTC /2)

(6.107)

Here the output noise density of an N th-order Σ∆ modulator was used, so that Sε (ω) is the intrinsic spectral density of the quantizer according to Eq. (6.96). One sees that for n < N the result becomes useless. If after filtering a sub-sampling with fC /M – the decimation – is performed, the necessary word length, w, is, in the case of a one-bit quantizer, w = 1 + n log2 (M ), where n is the number of cascades. This follows from the fact that an n-stage system with averaging length M of the individual stages spans a time interval of nM samples. For a multi-bit quantizer with Q quantization levels (log2 (Q) − 1) bits have to be added. Increasing n leads to larger word lengths. However, it is important to use always the minimal necessary word length in order to avoid white noise introduced by rounding off surplus bits. So the question of how many stages n should be implemented for a first anti-aliasing filter arises.

280

Electrical interfaces

According to Candy [1986] and Ghosh [2006], for a given order N of a Σ∆ modulator n = N + 1 cascades are sufficient. Remarkably, cascading leads to very simple structures. The threefold cascade, for instance, can be represented as G3 =

1 1 1 1 × × × (1 − z −M ) × (1 − z −M ) × (1 − z −M ) × 3 . 1 − z −1 1 − z −1 1 − z −1 M (6.108)

Viewing the first three stages as negative-feedback structures with delay TC in the feedback loop, so that yout-n = yout-(n −1) + y(in-n ) , the next three stages represent feedforward structures that subtract the delayed input from the nondelayed one yout-n = yin-n − yin-(n −M ) . The delay in these stages is M TT and can be realized by clocking these stages with M TC . The result is a very efficient structure based on subtractors, adders, and registers (delay stages). There is no need to store any filter coefficients. Further filtering stages are commonly used for the next decimation steps. The requirement for a small transient band leads to more complicated filter structures like the FIR filters mentioned above, with a transfer characteristic in P −k the z-domain M (hk = 1 results in a comb filter). A good introduction k = 0 hk z to the design of FIR filters can be found, for instance, in Shenoi [2006].

References Allen, P. E. and Sanchez-Sinencio, E. (1984). Switched Capacitor Circuits. New York: Van Nostrand Reinhold. Ardalan, S. H. and Paulos, J. J. (1987). An analysis of nonlinear behavior in delta–sigma modulators. IEEE Transactions on Circuits and Systems, 34(6):593–603. Baxter, L. K. (1997). Capacitive Sensors Design and Applications. New York: IEEE Press. Candy, J. C. (1986). Decimation for sigma delta modulation. IEEE Transactions on Communications, 34(1):72–76. Carter, B. and Mancini, R., eds. (2003). Op Amps for Everyone. Oxford: Newnes, 3rd edn. de La Rosa, J. M., Perez-Verdu, B., and Vazquez, A. R. (2002). Systematic Design of CMOS Switched-Current Bandpass SIGMA-Delta Modulators for Digital Communication Chips. Dordrecht: Kluwer Academic Publications. Engelen, J. V. and van de Plassche, R. (1999). Bandpass Sigma Delta Modulators. Dordrecht: Kluwer Academic Publications. Enz, C. C. and Temes, G. C. (1996). Reducing the effects of op-amp imperfections: autozeroing, correlated double sampling, and chopper stabilization. Proceedings of the IEEE, 84:1584–1614.

References

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Geen, J. A., Sherman, S. J., Chang, J. F., and Lewis, S. R. (2002). Micromachined integrated gyroscope with 50◦ /h Allan deviation. IEEE Journal of Solid-State Circuits, 37(12):1860–1866. Gerosa, A., Rubin, R., and Neviani, A. (2001). A simplified analysis of noise in switched capacitor networks from a circuit design perspective. ECCTD01 – European Conference on Circuit Theory and Design. Espoo, pp. 261–264. Ghosh, A. (2006). Decimation filtering for complex sigma delta analog to digital conversion in a low-IF receiver. www.serc.iisc.ernet.in/cadl/pub/ ghoshdef05.ppt. Goette, J. and Gobet, C. A. (1989). Exact noise analysis of SC circuits and an approximate computer implementation. IEEE Transactions on Circuits and Systems, 36(4):508–521. Gregorian, R. and Temes, G. C. (1986). Analog MOS Circuits and Systems. New York: Wiley and Sons. Grey, P. R., Hurst, P. J., Lewis, S. H., and Meyer, R. G. (2009). Analysis and Design of Analog Integrated Circuits. New York: John Wiley & Sons, 5th edn. Hagleitner, C. (2005). Circuit and system integration, in CMOS-MEMS, ed. H. Baltes, O. Brand, G. K. Fedder et al., Weinheim: Wiley-VCH, pp 513– 577. Handtmann, M. (2005). Dynamische Regelung mikroelektromechanischer Systeme (MEMS) mit Hilfe kapazitiver Signalwandlung und Kraftr¨ uckkopplung. Ph.D. thesis, Munich University of Technology; Aachen: Shaker Verlag. Howe, R. T. and Sodini, C. G. (1997). Microelectronics: An Integrated Approach. New York: Prentice Hall. Hsien, K. C., Gray, P. R., and Messerschmitt, D. G. (1981). A low-noise chopperstabilized differential switched-capacitor filtering technique. IEEE Journal of Solid-State Circuits, 16(6):708–715. Jung, W. G., ed. (1997). IC Op-Amp Cookbook. New York: Prentice Hall, 3rd edn. (2004). Op Amp Applications Handbook, Analog Devices. Oxford: Newnes. Kugelstadt, T. (2005). Auto-zero amplifiers ease the design of high-precision circuits. Analog Applications Journal, Texas Instruments, 19–27. Laker, K. R. and Sansen, W. M. C. (1994). Design of Analog Integrated Circuits and Systems. New York: McGraw-Hill, Inc., 5th edn. Liu, M. (2006). Demystifying Switched Capacitor Circuits. Amsterdam: Elsevier. Maloberti, F. (2006). Data Converters. Berlin: Springer. Naemen, D. A. (2003). Semiconductor Physics and Devices. New York: McGrawHill, 3rd edn. Norsworthy, S. R. and Schreier, R. (1997). Delta–Sigma Data Converters: Theory, Design and Simulation. New York: IEEE Press. Petkov, V. P. and Boser, B. E. (2004). Capacitive interfaces for MEMS, in Enabling Technology for MEMS and Nanodevices, Vol. 1, ed. H. Baltes, O. Brand, G. K. Fedder et al., Weinheim: Wiley-VCH, pp. 49–92.

282

Electrical interfaces

Plasche, R. V. D. (2003). CMOS Integrated Analog-to-Digital and Digital-toAnalog Converters. Boston: Kluwer. Quinn, P. J. and Roermund, A. H. M. V. (2007). Switched-Capacitor Techniques for High-Accuracy Filter and ADC Design. Berlin: Springer. Rasavi, B. (1995). Principles of Data Conversion System Design. New York: Wiley–IEEE Press. Salo, T. (2003). Bandpass delta–sigma modulators for radio receivers. Ph.D. thesis, Helsinki University of Technology, Electronic Circuit Design Laboratory. Shenoi, B. A. (2006). Introduction to Digital Signal Processing and Filter Design. New York: Wiley. Strle, D. (2009). Personal communication. University of Ljubljana, EE Department. Sze, S. M. (1981). Physics of Semiconductor Devices. New York: John Wiley & Sons, Inc., 2nd edn. Tsividis, Y. (1999). Operation and Modeling of the MOS Transistor. Oxford: Oxford University Press. Unbehauen, R. and Cichocki, A. (1989). MOS Switched-capacitor and Continuous-time Integrated Circuits and Systems: Analysis and Design. Berlin: Springer. Vasudevan, V. and Ramakrishna, M. (2003). Computation of noise spectral density in switched capacitor circuits using the mixed-frequency-time technique, in DAC 2003, Anaheim, pp. 538–541. Weste, N. and Eshraghian, K. (1988). Principles of CMOS VLSI Design – A System Perspective. New York: Addison-Wesley Publishing Company. Wongkomet, N. and Boser, B. E. (1998). Correlated double sampling in capacitive position sensing circuits for micromachined applications, in IEEE APCCAS 1998: Proceedings of 1998 IEEE Asia-Pacific Conference on Circuits and Systems. Microelectronics and Integrating Systems, pp. 723–726. Wu, J., Fedder, G. K., and Carley, L. R. (2002). A low-noise low-offset chopperstabilized capacitive-readout amplifier for CMOS MEMS accelerometers. Proceedings of IEEE International Solid-State Circuits Conference, 2002, pp. 428– 430.

7

Accelerometers

7.1

General measurement objectives Accelerometers convert accelerations into deflections or stress deviations. The deflections or stresses are transformed into an electrical vector signal Y¯out that ˆ should represent a more or less optimal estimate of the acceleration vector a ¯. ˆ (Estimates of a function f (t) are denoted by f (t).) The most common estimation algorithms are simple linear filtering procedures that smooth the noise of the reactions captured. From a more general viewpoint, an inertial sensor is a system, the output signal of which depends on six inertial forces caused by three linear accelerations ¯ T = (ax , ay , az , Ωx , Ωy , Ωz ). Even and three rate signals or their derivations: X in for sensors, which usually are approximated by 1D systems, the impact of all inertial forces is virtually present in the form of cross-coupling effects. In a 1D system the response of the five undesired outputs is suppressed to negligible levels in comparison with the reaction of the main component. Ideally, a one- to three-dimensional accelerometer should deliver an output signal Y¯out that satisfies the following criteria.

r Within a given measurement range X˜ ∈ [Xm in , Xm ax ; i = 1, 2, 3] it represents ¯ in , a linearly scaled estimate of the input signal, X ¯ ˆ in − X¯0 ), Y¯out = S(X

(7.1)

where S is the scale-factor matrix. X¯0 is the offset, which often is called bias.

r It constitutes a “good” estimate of the input component (for instance in terms of the minimum squared error or of the maximum a-posteriori probability), i.e. resolves it with high accuracy within the specified measurement range and bandwidth. The resolution √ is usually characterized by the spectral density of the output signal in g/ Hz that is equal to the noise density and, thus, that can be differentiated from it. r In the case of there being no input signal it remains below a predefined limit close to the zero position (small bias), ¯TX ¯ 0 ≤ ε. X 0

(7.2)

284

Accelerometers

Anchor Spring

Mass

Acceleration

}

y

Deflection x x

Figure 7.1 The principle of a linear spring–mass accelerometer.

r It should exhibit negligible cross-coupling. That is, the output signal is not disturbed by other linear or angular accelerations. The matrix S should be diagonal and for 1D systems should collapse into a matrix with one non-zero coefficient only. r It is stable over temperature variations and over time. r It has no drop-out and no malfunction during operation. These requirements are normally described in detail by the corresponding specifications and reflected in the data sheet of a commercially available sensor. The main specification parameters are sensitivity, linearity, resolution, bias, bias drift, cross-axis sensitivity, shock robustness, vibration sensitivity against high-frequency accelerations (outside the measurement frequency band), self-test capability, and safety (probability of undetected errors). They are relevant also for gyroscope systems. Parameters such as safety require an overall probabilistic evaluation of all error sources and implemented failure limits which, unfortunately, goes beyond the scope of this book. In the next sections basic principles and models of “MEMS-friendly” accelerometers that provide the basis for evaluating most of the specification parameters are considered.

7.2

The spring–mass system One of the most successful arrangements of a 1D acceleration sensor is a simple spring–mass system with one degree of freedom as shown in Figs. 7.1–7.3. The spring–mass system with one DOF is best suited for discussing the typical behavior and models of most accelerometer types. As illustrated in Fig. 7.1 the mass m of a linear accelerometer1 is suspended by springs, which, in the case of a 1D accelerometer, should yield only in the x-direction and feature infinite stiffness in all other directions. Thus, the 1

The attribute “linearity” is here related to the acting acceleration, not to the properties of the accelerometer.

7.2 The spring–mass system

R

285

Center of Gravity

Acceleration

z

θ x

Figure 7.2 The principle of a torsional spring–mass accelerometer.

Fixed comb fingers

Torsional springs

Inertial mass

Figure 7.3 The principle of an angular accelerometer.

sensor responds only to accelerations ax in the x-direction. Any deviation from such ideal behavior will just determine the unwanted “parasitic” effects such as cross-coupling of orthogonal acceleration and rate components into the output signal. Therefore, the construction of the corresponding suspensions as treated in Chapter 3, Section 3.1, is of utmost importance for the performance of any spring–mass accelerometer. A torsional (or pendulous) accelerometer according to Fig. 7.2 [Fricke and Obermeier 1995, Selvakumar et al. 1996, Rose et al. 2003] consists of an asymmetric mass, suspended by flexural hinges, that may rotate under the impact of external linear and angular accelerations about the y-axis. A special type of pendulous accelerometer is completely asymmetric, exhibiting a proof mass on one side of the hinge only, as in the repeatedly mentioned first batch micromachined accelerometer of Roylance and Angell [1979]. The angular accelerometer as in Fig. 7.3, in contrast, has a symmetric mass distribution and, thus, is insensitive to linear acceleration. Ideally it would react only to angular accelerations about the z-axis. All three systems show identical dynamic behavior.

286

Accelerometers

7.2.1

The transfer functions The basic behavior of an idealized spring–mass system is characterized by its transfer function.

The common dynamics of linear, torsional, and angular accelerometers If the total spring constant of all suspending springs in a linear deflecting accelerometer is denoted by k and the damping coefficient by c, the dynamic behavior of the mass relative to the substrate x(t) is described according to Newton’s law by the relation m(¨ xsub + x) + cx˙ + kx = 0, where xsub is the corresponding movement of the substrate, carrying the accelerometer, within an inertial system of coordinates. Since the substrate acceleration is the entity to be measured, −¨ xsub = ax , the internal mass deflection x obeys the equation m¨ x + cx˙ + kx = F = max + NB .

(7.3)

Analogously, for the torsional accelerometers the balance of torques requires (7.4) Iy θ¨ + cθ θ˙ + kθ θ = Rm cos θ az + Rm sin θ ax − Iy θ¨sub + NB . R Iy is the axial moment of inertia, Iy = V ρ(x2 + z 2 )dV , ρ the mass density, V the volume of the body, and R the distance of the center of gravity (COG) from the y-axis. Rm cos θ az represents the torque around the y-axis created by the acceleration az , and Rm sin θ ax is the torque created by the acceleration ax , acting in the x-direction. The dominant term Rm cos θ az should be much larger than the torque generated by the substrate rotation |Rmaz | ≫ I|θ¨sub | and the cross-torque Rm sin θ ax . In contrast, angular accelerometers measure θ¨sub . Thus, the terms |Rmaz | and |Rmax | must be suppressed. Therefore, they feature a symmetric mass distribution with R = 0 and are usually arranged as an in-plane rotating circularsymmetric disc with attached radial combs for capturing the capacitance changes between moving and fixed fingers as shown in Fig. 7.3. Since for small angles cos θ ∼ = 1 and since for a negligible or constant2 crosscoupling term ax equations (7.3) and (7.4) have the same structure, the following analysis, which is related to the linear spring–mass system, is valid also for torsional and angular accelerometers.

Brownian noise The average value of the damping force cx˙ in Eq. (7.3) is proportional to the velocity of the moving object and predominantly determined by the friction with the ambient medium. For MEMS this is usually air or fill gases such as nitrogen. The damping force has a stochastic part NB stemming from the probabilistic nature of the particle collisions with the moving body. The subscript B stands 2

In the case of constant cross-acceleration the spring constant k θ must be substituted by k θ − Rma x .

7.2 The spring–mass system

287

for “Brownian.” The very fast fluctuations of the gas particles can be represented by Brownian motion. The model is equivalent to the thermal noise of a resistor described by the Nyquist equation. Thus, the equivalent expression for the spectral density of the mechanical Brownian noise is [Gabrielson 1993] SB = 4kT c,

(7.5)

where k is the Boltzmann constant k = 1.38066 × 1023 J/K and T is the absolute temperature in degrees Kelvin. The damping coefficient c can be determined using the results of Chapter 3, Section 3.2.

The step and impulse responses of accelerometers First, the spring–mass system is considered for a steady acceleration ax = constant. Neglecting the noise, there exists an equilibrium point, kx = max , where the spring force compensates the external acceleration force. The Brownian noise causes small fluctuations around this point. For arbitrary input accelerations the dynamic behavior of the system can be characterized by the impulse response, xδ (t), relating the output to the input by the convolution Z t xδ (t − t′ )a′x (t′ )dt′ (7.6) x(t) = −∞

or by the corresponding frequency-transfer function. The impulse response xδ (t) is the reaction on an applied delta-impulse. Introducing the natural resonance frequency and the damping ratio, 1 p ω02 = k/m, δ = c/ k m, (7.7) 2 respectively, the normalized dynamic equation becomes NB = a′x . m For a′x = δ(x) the inhomogeneous equation has the following  ¡√ ¢ √ −δτ 2 2   e sin 1 − δ τ / 1 − δ , xδ (τ ) · ω0 = e−δτ τ, ³ √ ´ √  √  e−δτ e δ 2 −1 τ − e− δ 2 −1 τ / ¡2 δ 2 − 1¢ , x ¨ + 2δω0 x˙ + ω02 x = ax +

(7.8) solution: for δ < 1, for δ = 1,

(7.9)

for δ > 1,

where the normalized time τ = ω0 · t has been introduced. In Fig. 7.4 the normalized impulse-transfer function xδ (τ ) · ω0 is shown for critical damping (δ = 1), for under-damping (δ < 1) and for over-damping (δ > 1). Often, instead of the impulse response, the unit-step response is used. The unit-step response is the reaction on a unit step: a′x = 1 for t > 0 and a′x = 0 before t = 0. Impulse-transfer and unit-step representations are equivalent to each other (h(t) = xδ (t) = dxstep (t)/dt).

Accelerometers

0.6

0.5

δ = 0.5

0.4

δ=1 x ω0

0.3

δ = 1.5 0.2

0.1

0

-0.1

0

1

2

3

4

5

6

7

8

τ

Figure 7.4 An impulse-transfer function.

1.5

δ = 0.25 δ = 0.707 1

δ=1

x ω20

288

δ=2 0.5

0

0

1

2

3

4

5

6

7

8

9

10

τ

Figure 7.5 A unit-step response.

Solving Eq. (7.3) for a unit-step input and for zero initial conditions yields  √ ¡ ¢ ¡√ ¢  1 − e−δτ / 1 − δ 2 sin 1 − δ 2 τ + ψ0 , for δ < 1,    1 − e−τ (1 + τ ), for δ = 1, √ xstep (τ ) = ω0−2 √ £ −δτ ¡ √ ¢¤ 2 − 1 [(δ + 2 − 1)e δ 2 −1 τ  1 − e / 2 δ δ   √  √ 2  −(δ − δ 2 − 1)e− δ −1 τ ], for δ > 1, (7.10) where p ψ0 = arcsin 1 − δ 2 .

Figure 7.5 shows the unit-step reaction for various levels of damping. An ideal system should follow the unit step with small delay and small deviations from the input step. A real system shows different behavior. For under-critical damping, δ < 1, an overshoot is present, as well as oscillations that attenuate for longer the smaller the damping. In the case of over-damping, δ > 1, the response has too small a rise time and is sluggish. Critical damping with δ = 1 avoids overshoots

289

7.2 The spring–mass system

(a)

10

1

(b) δ = 0.05

δ = 0.25

140

δ = 0.5

0

phase in °

(X/Ax) ω0

2

120

δ = 0.5 10

δ = 0.05

160

δ = 0.25 10

180

-1

δ = 0.707 δ = 1.0

δ = 0.707

100

80

δ = 1.0

60

δ = 2.0

δ = 2.0

40

20 -2

10 -1 10

10

0

10

0 -1 10

1

10

f/f0

0

10

f/f0

Figure 7.6 Frequency-transfer functions of a linear spring–mass system. (a) The amplitude-transfer function. (b) The phase-transfer function.

and oscillations. For δ not too small the response time tresp is approximately tresp = 4δ/ω0 , where tresp is the time at which 90% of the acceleration level is reached for the first time.

Under-damped, critically damped, and over-damped systems The damping ratio δ is a key design parameter. It determines the time behavior or, alternatively, the shape of the frequency response. The frequency-transfer function, which is the Fourier-transformed impulse response, is illustrated by Figs. 7.6(a) (amplitude-transfer function) and (b) (phase-transfer function). Ax (jω) and X(jω) are the spectra of a′x (t) and x(t). At the bandwidth frequency fB (ω = 2πf ) the transfer function

with

ej ψ X(jω)/Ax (jω) = p 2 (ω0 − ω 2 )2 + 4δ 2 ω02 ω 2 ψ = − arctan

µ

2δω0 ω ω02 − ω 2

¶

decreases its amplitude in comparison with the value at f = 0 by 3 dB, qp fB /f0 = (2δ 2 − 1)2 + 1 − (2δ 2 − 1).

(7.11)

(7.12)

(7.13)

It is evident that

1≤

fB ≤ f0

q √ 1 + 2 = 1.55.

(7.14)

As for all second-order systems, at the bandwidth frequency the phase of the output signal lags by 90◦ .

1

290

Accelerometers

√ For damping ratios δ > 1/ √ 2 the amplitude-transfer function has only one maximum at f = 0. For δ > 1/ 2 an additional maximum emerges at p fres = f0 1 − 2δ 2 . (7.15)

This maximum represents the resonance of the system. For systems without such additional resonance the maximal bandwidth is achieved at a damping rate √ of δ = 1/ 2. According to Eq. (7.13) the maximal bandwidth without a resonance maximum is thus fB = √ f0 . Together with an acceptably small overshoot, this is why the value δ = 1/ 2 is considered the optimal damping rate. For under-damped sensors high-frequency input components (caused for instance by environmental vibrations) tend to create distortions of the output signal. Thus, large mechanical deflections and subsequent instabilities may occur. Large deflections may entail nonlinear distortions or, even worse, mechanical contacts of the proof mass with its environment. Instead, for over-damped systems the output signal does not correctly follow the high-frequency input signals: the system suppresses them. An optimal damping close to the critical value δ = 1 is obviously the best compromise between suppression of high frequencies and a fast step-transfer response. The damping rate δ and quality factor Q are equivalent quantities. Mathematically the Q-factor is defined as the relative amplitude at the resonance frequency: Q = X(fres )/X(0) =

1 √ . 2δ 1 − δ 2

(7.16)

For small damping rates Q ∼ = 1/(2δ). If the resonance bandwidth, ∆f (not the low-pass bandwidth which was considered up to now), is defined as the difference between the frequencies f2 and f1 at which the resonance peak is reduced by 3 dB, then these frequencies satisfy the following equation: √ 1 2 2 Q/ 2 = [(ω02 − ω1,2 ω02 ]− 2 )2 + 4δ 2 ω1,2 (7.17) or, on solving the equation, 2δ (f2 − f1 )/f0 = ∆f /f0 = √ . 1 − 2δ 2

(7.18)

In view of Eq. (7.15) the final relation is ∆f /fres =

2δ ∼ = 2δ = Q−1 . 1 − 2δ 2

(7.19)

This is the well-known relation between the quality factor and the resonance bandwidth which is used for electrical resonant circuits. The quality factor is well adapted to mechanical resonators and also used for systems with quite small Q of around 2 to 5. Such sensors are sometimes highly desirable for applications in harsh environments in which it is advantageous to suppress intrinsically mechanical distortions at higher frequencies.

7.2 The spring–mass system

291

Table 7.1. Mechanical sensitivity versus bandwidth for optimal damping

Bandwidth (Hz) Sensitivity (µm/g)

1 2600

10 26

100 0.26

1000 0.0026

The trade-off between sensitivity and bandwidth The intrinsic limitations of the mechanical sensor are set by its sensitivity and its bandwidth or its resonance frequency. If the mechanical DC sensitivity, i.e. the ratio between the asymptotic deflections after the impact of a step-like acceleration, is denoted by Sm ech , Sm ech = x(∞)/ax = 1/ω02 , the product of the mechanical sensitivity and the bandwidth is qp 1 Sm ech · fB = (2δ 2 − 1)2 + 1 − (2δ 2 − 1), 2π · ω0

(7.20)

(7.21)

which for critical damping becomes Sm ech · fB |δ=1 = 0.64/(2π · ω0 ) and for optimal damping, δ = 0.707, Sm ech · fB |δ= 0.707 = 1/(2π · ω0 ). Since fB |δ= 0.707 = f0 the last expression can be rewritten as Sm ech · f02 = Sm ech · fB2 |δ=0.707 = 1/(4π 2 ).

(7.22)

This equation states a strong limitation for sensitivity and bandwidth: the larger the requested bandwidth the smaller the sensitivity of a spring–mass accelerometer and vice versa. It serves as a good illustration of the order of magnitude of the expected mechanical proof-mass deflections. The expected deflections for 1 g acceleration and various bandwidths are summarized in Table 7.1. For instance, for a bandwidth of 1000 Hz the deflection is very small and lies in the nanometer range. Since with Eq. (7.14) the bandwidth and the natural resonance frequency differ at most by 55%, the requirement for high sensitivity is tantamount to the need for low resonance frequencies. However, low resonance frequencies require large masses and well-yielding springs. MEMS devices can be realized with characteristic dimensions between some millimeters in the wafer-plane direction and some 10 to 100 µm in the out-of-plane direction. Masses on the order of 10 ng to 300 µg result. The spring rates cannot be reduced arbitrarily because this would lead to very thin springs with extreme sensitivity to damage by shock and vibration. Therefore, especially for the range of low-mass accelerometers, it is impossible to match the sensitivity–bandwidth target. Small proof masses are typical for surface-micromachined (SMM) accelerometers with very limited vertical dimensions. Resonance frequencies may exceed 10 kHz. The corresponding sensitivities decrease to sub-nanometer/g.

292

Accelerometers

If the SMM accelerometer is in-plane sensitive, slide damping dominates, which is small. Undercritical damping and bandwidths that are far too large result. Thus, noise and acceleration components above the target measurement frequency range must be filtered out. Often electronic damping by feedback action has to be introduced. This is particularly the case for combined accelerometers and gyroscopes within a common cavity, the low vacuum level of which is determined by the drive power of the vibrating gyroscope and, hence, leads to small accelerometer damping. To achieve sufficiently large sensitivities high-performance, low-noise transducers are needed anyway. Surface-micromachined inertial MEMS entail particularly high challenges to electronic-circuitry and transducer technique. Fortunately, the existing sensitive transducers and electronic input amplifiers allow one to resolve displacements within the sub-nanometer range, which was the precondition for the broad introduction of SMM MEMS accelerometers. Accelerations down to the micro-g range have become measurable by such devices. Larger masses and correspondingly small resonance frequencies as well as optimal damping can be more easily achieved in bulk micromachined accelerometers with dominant squeeze damping. Here, in contrast to the case with SMM accelerometers, the damping is sometimes undesirably high due to the squeeze damping within the small gaps between the proof mass and the counterelectrodes (e.g. Peeters et al. [1991]). Such over-damped systems are the typical candidates for resolving the sensitivity–bandwidth trade-off by appropriate feedback control. For higher sensitivities exceeding even the possibilities of bulk technologies, classical accelerometers will retain their role for a long time to come, for instance in seismology. Limitations caused by relations between sensitivity and bandwidth similar to those for the basic spring–mass system are characteristic of all inertial MEMS. They largely determine the possible areas of application. However, such limits are not fixed: higher-order DOF systems with coupled masses as well as new MEMS technologies are expanding the borders and should open the way to ever more sensitive inertial MEMS.

7.2.2

Accelerometer imperfections Imperfections of the mechanical sensor corrupt the total system parameters such as cross-coupling, nonlinearity, shock and vibration sensitivity etc. A schematic view of a typical linear MEMS accelerometer is shown in Fig. 7.7 (e.g. Mukherjee et al. [1999]). Surface micromachining is assumed. The proof mass is suspended by a system of elastic beams arranged in such a way that they are compliant in the x-direction and stiff in the y- and z-directions. Thus, the mass is substantially sensitive only to accelerations in the x-direction. The deflection in the x-direction is measured by a capacitive comb transducer

7.2 The spring–mass system

x

x

293

Spring suspension

Sensing Combs

y

Stopper

Anchor Top cap

Sensing Mass

Substrate Cut along the x-axis

Figure 7.7 A typical accelerometer structure realized in surface-micromachining

technology.

as described in Section 2.4.4. The changes in capacitance of each moving finger relative to the two adjacent, substrate-fixed counter-electrodes are captured differentially. Sensor imperfections can be roughly divided into structural and manufactural imperfections, spring and damping nonlinearities, and misalignment errors.

Structural imperfections Structural imperfections are related to the measurement principle rather than to imperfections of design or fabrication. For instance, the cross-coupling of a torsional accelerometer as on the right-hand side of Eq. (7.4) is a generic effect and can be reduced only by properly decreasing the operational deflection angle θ, e.g. by applying feedback control. Another kind of structural imperfection is the limited stiffness of any ideally fabricated suspension against bending and torsion about all three axes. Design measures can increase the torsional stiffness about the y-axis in comparison with that of a single beam by using a multi-beam suspension with two distant anchors as has been done in the design in Fig. 7.7. The same principle can be used to balance torques about the x-axis with a four-anchor suspension. Importantly, internal forces, such as electrostatic excitations causing torques, can be avoided by using symmetric fields and a symmetric mass distribution. For suchlike symmetric proof masses external rotations of the substrate,

294

Accelerometers

however, may excite torques anyway and consequently rotate the proof mass around its center of gravity. Fortunately, in most application areas, such as the automotive and consumer markets, the maximal substrate rotation and the moments of inertia are small, and the rotational stiffness of the suspension system is able to suppress the induced proof-mass rotation sufficiently. Hence, to a first approximation the torques acting on the proof mass can be neglected. For accelerometers on rapidly rotating platforms special analysis is needed. Under such assumptions structural imperfection of a spring suspension means that accelerations in any of the six DOF cause corresponding deflections, but not cross-coupling. Therefore, an external acceleration in the z- and y-directions would impact the performance of an x-accelerometer only indirectly by disturbing the transducer function, changing the overlapping area between the fingers but not causing an x-deflection. In the example discussed here the dominant parasitic deflection is in the z-direction, while the mass is suspended much more stiffly in the y-direction.

Manufacturing imperfections of springs Probably the most important error sources are manufacturing imperfections. They can cause skewing of beam’s cross-sections and fraying of sidewalls, which usually are not constant over the wafer, but increase with growing distance from the center of the wafer. Often the distribution is anisotropic, causing orientationdependent geometric differences. If the suspension is deflected along one of the main axes, a displacement along the orthogonal axes and corresponding crosscoupling results. The effect can be calculated using Eq. (3.44) in Chapter 3.

Residual stress within springs Depending on the manufacturing process, some residual stress may remain within the polysilicon layer, which will have an impact on the spring constant (see Chapter 3, the section on “Residual stress in bending beams”). Instability and temperature dependency of the residual stress may cause hysteresis effects and tiny changes of the spring constants. Bulk micromachined springs made from undoped Si do not exhibit such effects.

Spring nonlinearity and hysteresis In order to achieve requested performance parameters within given size limitations, often quite large ratios between the mechanical deviation and the beam length have to be tolerated. In this case the emerging large stress will nonlinearly change the spring constant of both Si and polysilicon beams. Usually, a cubic force-deflection dependency is a good approximation. The nonlinearity can be controlled very well during design and manufacturing. In contrast, spring hysteresis is difficult to control because it depends on the inner stress, material properties, structural defects etc. Hysteresis is similar to a sort of memory. It reflects the attempt of the springs to retain some of the material changes which occurred during the distortion after the distortion has been

7.2 The spring–mass system

295

removed. The hysteresis causes a residual deflection/strain after the acceleration impact that depends on the applied dynamic load. Thus, the zero position is not repeatable. This is one of the reasons for the dangerous bias instabilities. Even a small hysteresis should be avoided if at all possible.

Anchor-induced stress If the spring system is anchored at more than one point, the different temperature coefficients of the substrate, isolating layers, and substrate-counterparts (top cap) may cause displacements of the anchors that lead to additional, orientationdependent strain. The strain will be transferred to the springs. Consequently, if a force along one of the orthogonal axes is applied to the spring, the deflection will consist of components along the other two axes. Cross-coupling of the acceleration components is the result. Additionally, the spring constants may change slightly, which produces sensitivity errors. To avoid anchor-induced stress, stressrelief structures have to be incorporated into the suspension.

Damping nonlinearity As insinuated in Chapter 3, Section 3.2.3, the different damping mechanism may cause nonlinear effects. Especially for squeeze damping within small gaps as is typical for z-accelerometers, the damping ratio δ depends nonlinearly on the displacement z. For the example shown in Fig. 7.7 this kind of damping may also occur between moving and fixed fingers of the comb transducer.

Misalignment of the center of gravity Less critical for accelerometers are design and manufacturing tolerances related to pattern generation and transfer. Deviations from the target geometry of the proof mass may appear particularly with respect to the etch-hole pattern used in surface micromachining in order to facilitate the removal of the underlying sacrificial layer by gas-phase etching. Such deviations may lead to a small shift of the center of gravity (COG). Usually this effect is negligible, provided that the etch holes are fine-grained.

Orientation misalignment More critical is the non-identicalness of the substrate orientation and the orientation of the mounting platform of the packaged sensor. To account for such an orientation error the platform-assigned acceleration vector has to be transformed into the substrate-assigned acceleration vector. The orientation error mainly depends on the accuracy of retention of the orientation during first-level packaging. For well-controlled packaging technologies typical orientation errors are on the order of 1◦ and less. For a 3D accelerometer the measured acceleration components can be recalculated back into the platform vector provided that the disorientation is known. For a 1D sensor this is possible only under the assumption that the orthogonal acceleration components do not exert a significant impact on ax .

296

Accelerometers

However, usually the individual “substrate-to-platform” orientation error cannot be determined without destructive analysis. Thus, an acceleration in the yor z-direction in platform coordinates features a component in the x-direction in substrate coordinates that is proportional to the sine of the orientation error. Hence, the maximal error depends on the maximal acceleration applied to the orthogonal axes.

Summary Manufacturing tolerances and material properties of the substrate, isolating layers, and top cap as well as of the building materials such as silicon or polysilicon may create stress, deformations, and geometric imperfections that will lead to cross-coupling, shifting of the rest position, nonlinearity, and hysteresis.

A simplified accelerometer model with imperfections The effects described above can be collected in an extended model of the accelerometer. First, the effect of cross-coupling via springs should be accounted for. According to Chapter 3, Eq. (3.5), for negligible torsional terms the relation between forces and deflections can be written as Fx = kx x + kxy y + kxz z,

Fy = ky y + kxy x + ky z z,

Fz = kz z + kxz x + ky z y. (7.23)

Since a careful spring design is aimed at avoiding cross-coupling terms it can be assumed that the stiffness in the y- and z-directions is much larger than that in the x-direction and that the imperfections lead to small cross-coupling terms only: ky ≫ kx ,

kz ≫ kx ,

kxy ≪ ky ,

kxz ≪ kz ,

ky z ≪ ky , kz . (7.24)

The dynamic equations of the spring–mass system become m¨ x + cx x˙ + kx x = max − kxy y − kxz z, m¨ y + cy y˙ + ky y = may − kxy x − ky z z,

(7.25)

m¨ z + cz z˙ + kz z = maz − kxz x − ky z y.

Considering that ωz2 = kz /m and ωy2 = ky /m are much higher than the resonance frequency of the x-movement, the second and third equations can be approximated in the light of (7.24) by the static relations ky y = may and kz z = maz . On substituting into the first equation, the disturbed relation for the x-displacement results: x ¨ + 2δω0 x˙ + ω02 x = ax −

kxy kxz ay − 2 az . 2 ωy ωz

(7.26)

7.2 The spring–mass system

297

This equation has to be modified by incorporating the shift of the rest position as well as the nonlinearity and the hysteresis of the spring system: x ¨ + 2δ(x, x)ω ˙ 0 x˙ + γ(x − x0 )ω02 (x − x0 ) = ax −

kxy kxz ay − 2 az + nB . ωy2 ωz

(7.27)

The factor γ(x − x0 ) =

kx (x − x0 ) kx (0)

(7.28)

accounts for spring nonlinearity, hysteresis, and a position-dependent elastic damping term that possibly arises in squeeze-damped systems. The damping nonlinearity is included by substituting the constant δ by a nonlinear function δ(x). Its character can be estimated from the following considerations. The proof mass normally is arranged symmetrically between fixed counter-walls constituted for instance by the opposing fixed fingers or by bottom and top electrodes over and below the proof mass. Then the damping forces on both sides of the corresponding faces have to be added. For free-molecule viscosity Peeters et al. [1991] quote the damping force with "µ ¶3 µ ¶3 # 1 A2 1 1 FS = µ + x. ˙ 2 2 D0 − x D0 + x This expression can be made plausible for low Knudsen numbers and refined for rarefied gases. Indeed, as shown in Chapter 3 in the section “Low-frequency squeeze damping,” the squeeze-damping coefficients for relatively slow deflections are proportional to 1/D03 for low Knudsen numbers and proportional to 1/D0 for rarefied gases (see Eq. (3.243)), where D0 is the gap between the moving and fixed structures. If the low-speed damping coefficient is envisioned as the result of an iterative solution process whereby the gap is stepwise substituted by the actual gap D0 ± x, nonlinearities close to the types δ(x) ∼ 1/(D0 − x)3 + 1/(D0 + x)3 for sufficiently low Knudsen numbers and close to δ(x) ∼ 1/(D0 − x) + 1/(D0 + x) for rarefied gases can be expected. This simple consideration likewise makes it clear that the damping coefficient of accelerometers predominantly depends on x rather than on x. ˙ For completeness the normalized Brownian noise nB is added, nB = NB /m.

(7.29)

For pendulous accelerometers the structure of Eq. (7.27) remains invariant; however, the right-hand side must be extended by addition of the structural cross-coupling terms according to Eq. (7.4).

Cross-coupling The qualitative evaluation of the cross-coupling terms for steady input accelerations is quite simple. If the cross-coupling sensitivity Sxx i |a x =a i for i = 2, 3 is

298

Accelerometers

defined as the ratio of the x-deflection caused by an orthogonal component ai and the deflection caused by the same acceleration applied in the x-direction then Sxx i |a x =a i =

kxx i ω02 , kx ωi2

i = 2, 3.

(7.30)

For a given ratio kxx i /kx the cross-sensitivity is smaller the larger the resonance frequencies in the orthogonal directions. The cross-coupling terms kxx i /kx can be estimated by finite-element methods (FEMs) using different models for the impact of the manufacturing imperfections on the cross-sections of the spring beams. The corresponding values are usually below 1%. In general, an important design target is resonance frequencies of all resonance modes as high as possible with respect to the deployed mode. This reduces not only the impact of cross-coupling effects, but also parasitic excitations by highfrequency vibrations. The cross-coupling distortion ∆θ = θ(az , ax ) − θ(az , ax )|a x = 0 of a torsional accelerometer is a multiplicative effect depending on the acceleration in the z-direction. For steady accelerations and small θ a simple calculation using Eq. (7.4) yields ∆θ Rm Rm = ax ≃ ax . θ(az , ax )|a x = 0 kθ − Rmax kθ

(7.31)

This shows that the measurement error is proportional to the desired measurand. Hence, the cross-coupling error of a pendulous accelerometer should be specified not in terms of absolute values but as the relative error. Unlike in the case of linear accelerometers, for pendulous accelerometers the relative error depends on the spring constant kθ . Consequently, implementing spring stiffening by feedback control as explained in the next section will decrease the pendulous error of a torsional accelerometer. Equation (7.27) represents a generic collection of imperfection effects. The model must be adapted for any practical application. Even for well-behaved, smooth nonlinearities a solution is often difficult and requires numerical methods. However, for many types of nonlinearities there exists a lot of accumulated topological knowledge such as locations of stability points, limit cycles or bifurcation points. For practical applications it is important to avoid or reduce the impact of inevitable nonlinearities on the desired ideal behavior of the sensor. One approach to this end is the introduction of feedback control.

7.2.3

Accelerometer feedback control The application of feedback control requires an actuator that is able to create compensating forces acting on the proof mass. Ideally, the actuator should be linear. This can be, for instance, a comb actuator. Such force-feedback systems

7.2 The spring–mass system

299

have been used successfully for many years in order to suppress large deflections and to improve the dynamic behavior of the accelerometer (e.g. Kampen et al. [1994], Matsumoto and Esaki [1993], Stuart-Watson [2006], and Yun et al. [1992]). Introducing a feedback loop should reduce the impact of nonlinearities and, more generally, reduce the dynamic deflections. Linear feedback control can decrease or increase damping and bandwidth, respectively, and somehow decouple the dynamics of the loop output yout and the sensor deflection x, giving room for optimization. The general target of an acceleration force-feedback system is to compensate for any deviation from the rest position without introducing additional noise. Thus, the system remains substantially around its zero position, avoiding large deviations and resulting nonlinear effects. Other possible feedback strategies may try to compensate not for the position but for the velocity, or for both. Velocity compensation requires larger feedback forces the larger the velocity, and is difficult to realize. Such methods have not achieved broad relevance. Closed-loop accelerometers are generally superior to open-loop systems. The strong reduction of proof-mass deflection alleviates the operational stress of the sustaining springs and, thus, reduces wear and fatigue. Since in closed-loop systems the springs experience much smaller deflections, they can be designed with higher compliance, which reduces possible hysteresis effects. In open-loop accelerometers the transducers have to handle large displacements, which may lead to large nonlinearity errors. In closed-loop sensors the displacement is very small, therefore closed-loop accelerometers are much less reliant on transducer linearity. To a first approximation the sensitivity of a closed-loop linear accelerometer to cross-coupling is not reduced since the ratio between ax and the cross-coupling terms does not change. However, an improvement may take place because the cross-coupling spring constants kxx i may indeed be reduced in comparison with an open-loop system due to the implementation of softer springs. In contrast, the cross-coupling of a torsional accelerometer according to Eq. (7.31) reduces, because the cross-acceleration ax is multiplied by the reduced deflection angle sin θ, changing the ratio between the two components az cos θ and ax sin θ in Eq. (7.4). The price to be paid is the introduction of an actuator. There are three ways to generate actuating forces:

r introduction of additional electrodes into the MEMS structure, separating sense and actuation in space

r time division of actuation and sensing in different time slots r voltage division by superposition of high-frequency excitation voltages onto the relatively slow actuating force and separation of the corresponding highfrequency sense signal.

300

Accelerometers

NB a’x m

+ + -

proof mass

x0

m -1 s2+ 2

2

s+

+ + x

transducer kT

nE

electronics yout

+

k E (s)

+

x

feedback k F (s)

error

force

-+

x set

Figure 7.8 A principal-block diagram of accelerometer force-feedback control.

The disadvantages of each method are obvious: spatial separation requires sensor area, or for constant area leads to lower sensitivities; time division entails also lower sensitivities, requiring more complicated electronic circuitry; and voltage division would reduce the available voltage levels for sensing and actuation or require high-voltage processes. All three methods are used in practice. The following considerations, however, are, for simplicity, based on the assumption of space division.

The linearized feedback model In Fig. 7.8 the linearized model of a force-feedback accelerometer is shown. To emphasize the nature of the feedback control, according to the terminology of control theory the proof mass is treated as “plant” to be controlled. The “plant” is disturbed by the input signal xin = ma′x + NB , where (see Eq. (7.27)) a′x = ax −

kxy kxz ay − 2 az , ωy2 ωz

(7.32)

and consists of a small bias x0 . The set point for the proof mass is normally xset = 0; however, if the bias is known, the set point can also be chosen to be xset = x0 . To induce a control action the actual deflection must be estimated (measured). The estimator block consists of the transducer and a (first) electronic stage with transfer characteristics kT and kE (s) correspondingly. Owing to the linearity of the transducer and electronic stages the output signal yout represents a scaled estimate of the deflection yout = αˆ x. The feedback control block generates the compensating forces as a linear functional of the estimated deflection x ˆ or, more precisely, of yout . The feedback block thus consists of the electronic drive stages and the actuator. The dominant noise contributions are the electronic noise nE and the Brownian noise nB , with spectral density functions SE (ω 2 ) and SB = 4kT c correspondingly.

7.2 The spring–mass system

x0 + n’

NB a’x m

+ + -

301

proof mass F(s)

+

x

+

x set

feedback H(s)

Figure 7.9 A principal-block diagram of accelerometer force-feedback control.

The output signal yout plays a double role: it serves as an estimate of the position of the proof mass and as a first (scaled) estimate of the measurant, a′x . The quality of the estimation with respect to the acceleration can be further improved by additional out-of-loop signal processing (post-loop filters) not shown here. Such improvements may be related to additional linear filtering procedures or – in the case of more sophisticated models including sensor and transducer nonlinearities – to nonlinear estimation algorithms. As is known from estimation theory, an optimal estimate for systems working within the linear range can be achieved by linear filtering.

Closed-loop transfer functions It is important to understand how the feedback control changes the behavior of the proof mass and the quality of the acceleration measurement. Under the assumptions made above, the model of the force-feedback control can be presented a little bit more clearly as illustrated in Fig. 7.9. The transfer function of the proof mass with constant damping and γ = 1 s = s/ω0 ) follows from Eq. (7.27) and can be written (˜ F (s) =

1 1 1 1 = m s2 + 2δω0 s + ω02 mω02 s˜2 + 2δ˜ s+1

(7.33)

The electronic noise is transformed to the output of the sensor block: n′ = (1/kT )nE . Since the set point α · xset is assumed to be constant it can also be transformed to the same location at the sensor output: xset = [1/(kT kE0 )]α · xset . kE0 is the DC gain of the electronic block. The three stages – transducer, electronics, and feedback kF (s) – form a common feedback block with transfer characteristic H(s) = kT kE (s)kF (s).

(7.34)

The output signal according to Fig. 7.8 is related to the variables in Fig. 7.9 by the equation yout = kT kE (s)x + kE (s)nE + kT kE0 (x0 − xset ).

(7.35)

The closed-loop equation establishes the dependency between the output deflection and the input set point as well as the disturbing acceleration. It follows

302

Accelerometers

immediately from Fig. 7.9 that · µ ¶¸ 1 H(s) ′ x= F (s)H(s)(xset − x0 ) + F (s)max + F (s) NB − , nE 1 + F (s)H(s) kT (7.36) yout =

kT kE (s) [−(xset − x0 ) + F (s)ma′x + n], 1 + F (s)H(s)

(7.37)

where the total noise n is defined by n = F (s)NB +

nE . kT

(7.38)

Clearly, kE · n is the summary output noise of the open-loop system (kF = 0) composed of the electronic noise of the input stage and of the filtered and transferred Brownian noise. Within control theory the set point, xset , represents the input variable, and the status of the “plant,” x, represents the output. The difference between them is the error signal. In deviation from that, for the accelerometer feedback system the distortion a′x is considered to be the input variable. The feedback control should scale down the acceleration-induced deflections but should follow the acceleration changes with high accuracy in order to extract an undistorted estimate. The deflection must be scaled up again by out-of-loop amplification and filtering. However, the signal-to-noise ratio (SNR) of the closed-loop output should be at least no worse than that for the open-loop output, provided that the same bandwidth is considered. The deflection, x, and output, yout , calculated for a constant acceleration a′x = a0 illustrate the basic properties of the feedback control. Neglecting the noise, the corresponding values are · ¸ 1 a0 x= K0 (xset − x0 ) + 2 , (7.39) 1 + K0 ω0 · ¸ kT kE0 a0 yout = −(xset − x0 ) + 2 , (7.40) 1 + K0 ω0 where K0 = F (s = 0)H(s = 0) =

1 kT kE0 kF0 . mω02

(7.41)

K0 is the loop gain (see Eqs. (7.33) and (7.34)). In order to include the classical proportional-integral-differential (PID) controller, for which H(s) = kP +

kI + kD s, s

(7.42)

the case H(s = 0) = kT kE0 kF0 = ∞ is allowed. Let us now put xset = x0 (compensated bias). Then the closed-loop operation reduces the output (7.40) as well as the proof-mass deflection (7.39) by a factor

7.2 The spring–mass system

303

1 + K0 in comparison with an open-loop system. Usually the loop gain is large, K0 ≫ 1, in order to achieve good deflection suppression. If the controller consists of an integral part, a constant input acceleration generates a zero-output signal, and, hence, cannot be resolved. Therefore, PID controllers with ki 6= 0 are not suited for acceleration measurements including DC values. More appropriate controller functions are given by H(s) = H(0)

1 + α1 s + α2 s2 + · · · + αm sm , 1 + β1 s + β2 s2 + · · · + βn sn

m ≤ n.

(7.43)

As could be expected, for xset = 0 the impact of a given bias x0 on the output signal is not reduced because in a closed-loop system the ratio between the input acceleration and the bias term ω02 x0 does not change. According to Eq. (7.39) the bias deflection x0 is also not reduced in the case of large loop gain. This shows that there is no reason to include the bias compensation in the feedback loop. An out-of-loop compensation should be preferred. Thus, for future considerations the bias and set point can be assumed to be zero.

Dynamic behavior The dynamic behavior under feedback control can be characterized by the changing transfer function. For large loop gains |F (s)H(s)| = |F (s)kT kE (s)kF (s)| ≫ 1 the output is approximated by F (s) 1 ma′x ≈ ma′x , 1 + F (s)H(s) H(s) kT kE (s)F (s) 1 = ma′x ≈ ma′x . 1 + F (s)H(s) kF (s)

x= yout

(7.44) (7.45)

Therefore, as long as the frequency-dependent loop gain is significantly larger than unity, the output, yout , follows the inverse feedback filter. If kF (s) is constant, the input–output transfer is flattened and the bandwidth is increased in comparison with an open-loop system. Resonance peaks of the the sensor characteristic are largely suppressed. Similarly, the deflection x becomes also independent of the proof-mass transfer function F (s); however, it is shaped by the inverse loop transfer H −1 (s = jω), the frequency dependency of which is normally determined by the electronic filter. Since |F (jω)H(jω)| features a low-pass characteristic, the condition for large loop gains |F (jω)H(jω)| ≫ 1 can be substituted by the inequality ω ≪ ωCO , where ωCO is the crossover frequency for which the loop gain becomes unity: |F (jωCO )H(jωCO )| = 1.

Spring stiffening Looking more into the details, if the transfer characteristic of the feedback loop (7.34) is constant, the feedback control acts like a spring stiffening, increasing the resonance frequency and thus the bandwidth of the system. Indeed,

304

Accelerometers

if H(s) = constant, the closed-loop transfer function becomes X(s) 1 1 = 2 = 2 (7.46) 2 2 . A′x (s) s + 2δω0 s + ω0 (1 + K0 ) s + 2δCL ωCL s + ωCL √ Thus, the closed-loop resonance ωCL = ω0 1 + K0 has increased in compari√ son with the open-loop resonance by a factor of 1 + K0 , ωCL = ω0

p 1 + K0 ,

δ δCL = √ . 1 + K0

(7.47)

Since the damping of the closed-loop system δCL is given by the identity δCL ωCL = δω0 , the closed-loop damping has reduced by the same factor. Therefore, the closed-loop transfer function features a larger bandwidth and smaller damping than those of the open loop. Increasing loop gain may transform the signal response from an over-damped, via a critically damped, into a heavily under-damped mode with long-lasting oscillations as adumbrated in Fig. 7.5. The introduction of the loop gain K0 > 0 is equivalent to a stiffening of the sensor spring, leading to deviations being reduced by the factor 1/(1 + K0 ). In view of Eq. (7.41) and kx = mω02 , this factor describes an additional term of the spring constant, 1 kx = . 1 + K0 kx + kT kE0 kF0

(7.48)

If the feedback characteristic H(jω) is a higher-order filter, increasing the feedback gain K0 may lead to instabilities. Therefore, the stability reserve has to be checked. An appropriate parameter is the phase margin of the frequencydependent loop gain F (jω)H(jω). It states how many degrees away from −180◦ the phase characteristic is at the frequency at which |F (jω)H(jω)| = 1. When the phase margin equals zero, the loop gain has become −1, achieving the instability point.

Shock testing in operation Spring stiffening significantly improves the shock robustness of a force-feedback accelerometer. Let’s consider, for instance, a drop-like shock test, in which the system has to survive the fall from a height of 1.2 m onto a concrete floor. Usually this test is relevant for non-operating systems; however, some drop-detection accelerometers have to work throughout the whole of the dropping event. Provided that there is no damage to the springs, the most dangerous effect is sticking. Thus, the deflection should not bridge any gap to the neighboring fingers or electrodes even under worst-case conditions. A drop-test acceleration can be modeled as half a sine wave, aDrop sin(πt/TDrop ), 0 ≤ t ≤ TDrop , with amplitude around 2000g and a contact-pulse time on the order of hundreds of microseconds. If the feedback control is slow, the pulse can be approximated by a deltafunction a(t) = (2TDrop /π) aDrop δ(t), and the maximal deflection in the most

7.2 The spring–mass system

305

sensitive direction corresponds to the maximum of the impulse transfer function (Eq. (7.9)). For δ = 0 this maximum is xδm ax = 1/ω0 ; for δ = 1 it reduces to xδm ax = (1/ω0 )e−1 , decreasing further for increasing damping. The improved √ shock robustness is caused by the increase from ω0 to 1 + K0 ω0 of the resonance frequency and can be estimated as xDrop =

2 2 TDrop aDrop xδm ax TDrop aDrop ≤ . π π ωCL

(7.49)

For a closed-loop resonance frequency of 5 kHz and a drop pulse duration TDrop = 100 µs the worst-case deflection is around 40 µm, which, of course, is still far above the tolerable values. In the alternative case of a very fast feedback control the output response can be estimated by calculating the response to a steady input according to Eq. (7.39): aDrop xDrop = 2 , (7.50) ωCL indicating a much better drop-pulse suppression. For instance, for a closed-loop bandwidth of 20 kHz the resulting worst-case deflection has reduced to 1.2 µm. Of course, packaging and assembly significantly change the actual pulse, which makes a more detailed analysis necessary. However, the drop test is a representative example for shock impacts and fast vibrations, demonstrating the urgent need for a fast and overshoot-free system reaction. Despite the fact that only a frequency-independent feedback was considered, it can be expected that the nature of the closed-loop response (7.46) remains valid also for weakly frequency-dependent control blocks with finite zero gain H(0). Therefore, the dynamics can be allayed or speeded up by a proper choice of the filter kE (s) or kF (s).

The signal-to-noise ratio The SNR is the ratio between the signal and the noise power, SNR = 2 2 Psignal /Pnoise . With Psignal = yeff and Pnoise = ynRM S it can be related also to the effective amplitudes of the signal yeff and noise ynRM S : SNR = (yeff /ynRM S )2 . To determine the amplitude SNR, SNR′ = yeff /ynRM S , the output signal, yout , is divided into a deterministic part, ya , and a stochastic part, yn : |ya | , ynRM S ½ ¾1/2 Z fo u t 1 = Sy (ω 2 )dω . 2π 0

SNR′ = ynRM S

(7.51) (7.52)

For an applied constant acceleration the effective amplitude is the acceleration √ value itself; for a harmonic input signal the effective amplitude is a0 / 2. For simplicity the SNR is usually compared for constant input acceleration, thus, ya =

kT kE0 a0 . (1 + K0 )ω02

306

Accelerometers

For signals with arbitrary spectral distribution it is more convenient to consider the noise floor, i.e. the frequency-dependent RMS value of the output noise within 1-Hz bandwidth. This allows one to calculate the SNR for arbitrary input signals. The spectral density Sy (ω) of the output noise is obviously Sy (ω) =

¤ £ |kE (jω)|2 SE (ω) + kT2 |F (jω)|2 SB . 2 |1 + F (jω)H(jω)|

(7.53)

SE and SB = 8kT δω0 m are the spectral densities of the electronic and the Brownian noise, respectively (see Eq. (7.5) with c/m = 2δω0 ). As long as the loop gain |F (jω)H(jω)| ≫ 1, the output spectrum can be approximated by "¯ # ¯ ¯ kE (jω) ¯2 1 ¯ ¯ Sy (ω) = SE (ω) + SB . |kF (jω)|2 ¯ F (jω)kT ¯

(7.54)

In an open-loop system the output spectrum Sy op enlo op

¯ ¯ ¯ kE (jω) ¯2 ¯ ¯ SE (ω) + SB =¯ F (jω)kT ¯

is |kF (jω)|2 larger. Since in this approximation the closed-loop output is given by yout = ma′x /kF0 , the SNR within an output bandwidth fout ≪ fCO does not change with respect to the open-loop SNR. For very small output bandwidths fout ≪ f0 a crude estimate of the SNR can be derived from (7.51)–(7.53) and (7.33) by substituting all transfer functions by their values at ω = 0 and assuming an out-of-loop brick-wall filter with bandwidth fout : SNR′ =

ya ynRM S

=

kT 1 a0 p . 2 ω0 fout (SE (0) + [kT2 /(m2 ω04 )] SB )

(7.55)

Clearly, for increasing frequency the feedback force decreases, and the high-frequency components of the noise in Eq. (7.53) are no longer compensated. Hence, with increasing output bandwidth a reduction of the SNR follows. At high frequencies (|F (jω)| → 0) the nearly frequency-independent “white” electronic noise is, according to (7.53), transferred to the output shaped only by kE (jω). Therefore, it is here the most dangerous factor. Even in the case of open-loop operation a low-pass output filter is necessary in order to limit the RMS. In contrast, the Brownian noise acts only within a limited frequency range determined by the transfer characteristic of the proof mass. Without any further in-loop and out-of-loop filters the contribution of the Brownian noise to the total

7.2 The spring–mass system

307

RMS value is limited to ynBrownRM S = kT kE0

½

1 2π

Z

0

∞

|F (jω)|2 SB dω |1 + F (jω)H(jω)|2

¾1/2

=

½

ω0 SB 8δ(1 + K0 )

¾1/2

.

(7.56)

Here we used the known relation Z ∞ 0

dx π = . (x2 − 1)2 + 4b2 x2 4b

Generally speaking, the SNR of a feedback accelerometer must be designed very carefully. The price for larger bandwidth and improved dynamic behavior may be a smaller SNR than for an open-loop system. The combination of inloop and out-of-loop filters helps one to find an acceptable compromise between dedicated feedback dynamics and the desired output SNR.

Closed-loop dynamics As demonstrated, the dynamic behavior of a closed-loop accelerometer with frequency-independent feedback H(s) = constant is characterized by an increased resonance frequency and reduced damping. If the open-loop response is too sluggish, it should be speeded up. If it features large overshoots, increased bandwidth will impair the situation. Additionally, outside the desired output-frequency band, shock and acceleration suppression has to be guaranteed in order to eliminate large proof-mass deflections. This may be in conflict with the required behavior for in-bandwidth accelerations. Feedback compensation is used in order to achieve an acceptable compromise. Thereby it has to be considered that the dynamic behavior of the final output signal kPL (s)yout is strongly affected by the post-loop filter kPL (s). The larger the ratio between the feedback bandwidth and the output frequency fCO /fout , the stronger the impact of kPL (s). A proper combination of feedback and postloop filters supports a compromise between disturbance suppression and proper tracing of the input signal within the desired bandwidth. Feedback compensation is determined by the frequency characteristic of the loop filter H(s) = kT kE (s)kF (s). In the simplest case H(s) is a first-order filter H(s) = H(0)(s + sz )/(s + sp ), where (−sz ) is the zero and (−sp ) the pole of the transfer function. If |sz | < |sp | the filter is a so-called lead compensator, otherwise it is a lag compensator [Franklin et al. 1998, Nise 2008]. The idea behind lead or lag filters can be most easily illustrated for a transfer function F (s) = F (0)[1/[(s + s1 )(s + s2 )]] with real poles and zeros, where |s1 | < |s2 |. To speed up the loop the task of a lead filter is, for instance, the compensation of the pole s1 , which is located nearest to the imaginary axis, by a corresponding zero of the compensator: H(s) = H(0)(s + s1 )/(s + sp ) with |sp | > |s1 |. The feedback-dominating

Accelerometers

1.6

1.4

normalized x and yout

308

fp /f0 =3.33 ; fz /f0 =2.5

K0 = 8

Closed Loop

δ = 0.5

1.2

1

0.8

yout

0.6

x

fp/f0=5 ; fz/f0=1 0.4

0.2

0 0

Lead Compensation

Open Loop

1

2

3

4

5

τ

Figure 7.10 Closed-loop lead compensation.

product is then given by F (s)H(s) = F (0)H(0)(s + sz )/[(s + sp )(s + s2 )] and leads to a faster transfer function F H/(1 + F H). Lead–lag controllers are popular for stabilization of many feedback systems. Lead compensation is equivalent to the addition of a derivative term, whereas lag compensation emphasizes the integral behavior.3 In the following example the first-order transfer characteristic is realized by the electronic block: kE (s) = kE0

1 + s/(2πfz ) fp s + 2πfz = kE0 . 1 + s/(2πfp ) fz s + 2πfp

(7.57)

fp and fz are the bandwidths of the low-pass and high-pass factors, respectively. According to Eq. (7.37) the output then behaves proportionally to F H/(1 + F H) and the x-deflection proportionally to F/(1 + F H). The electronic filter (7.57) can be represented as the sum of an all-pass component and a low-pass component. It allows one to demonstrate qualitatively the different types of feedback behavior. Typical output step responses yout (τ ), τ = ω0 t, for a system with open-loop damping δ = 0.5 and loop gain K0 = 8 are shown in Figs. 7.10 and 7.11. For different pole and zero frequencies fp /f0 and fz /f0 the lead compensation is illustrated by Fig. 7.10. The unit step responses of open- and closed-loop systems without compensators (kE = kE0 ) are inserted for comparison. The zero and pole frequencies of the lead compensator are chosen to lie below and above the crossover frequency, respectively. For the example presented here the crossover frequency is fCO = 2.9f0 . The resulting step response for the loop output signal yout (dotted lines) exhibits a shorter response time than that for the uncompensated closed loop, and a smaller overshoot. The sensor deflection x (dot–dashed lines) follows even faster than yout . For increasing pole frequencies the overshoot in the x-deflection can be completely eliminated. 3

A pure integrating component 1/s was excluded because it does not allow one to measure a DC acceleration.

7.2 The spring–mass system

309

3

K =8

y

δ = 0.5

x

out

0

normalized x and yout

2.5

fp/f0=0.5 ; fz/f0=1 2

fp/f0=0.1 ; fz/f0=0.2 1.5

1

Closed Loop 0.5

Open Loop 0 0

1

Lag Compensation

2

3

4

5

τ

Figure 7.11 Closed-loop lag compensation.

Figure 7.11 demonstrates the output behavior for lag compensation. The pole and zero frequencies are well below the crossover frequency. Decreasing values of the pole and zero frequencies tend to lead to slight improvement in the asymptotic behavior (dotted lines), reducing the steady-state error. The latter is generally worse than for a lead controller. For the chosen parameters the overshoot of the loop output is also slightly reduced in comparison with that for closedloop operation without a controller. However, the sensor deflection (dot–dashed lines) tends to exhibit large overshoots and long-lasting oscillations, clearly indicating the preference for lead compensation if the dynamic behavior should be improved. If the compensator is realized by the feedback block in Fig. 7.8, kF (s) = kF (0)[1 + s/(2πfz )]/[1 + s/(2πfp )], rather than by the first set of electronics, the deflection and output signals have identical behaviors proportional to F/(1 + F H), which corresponds to the deflection derived in the previous example. In general, parameter variations of a lead or lag controller within a forcefeedback accelerometer have less impact on the dynamic behavior than does the variation of the loop gain. Higher-order controllers such as cascaded lead–lag compensators can extend the range. However, for most practical applications the first blueprint of the feedback loop can be designed without any sophisticated loop filters.

7.2.4

Feedback control with nonlinear actuators Among other things, feedback control should improve robustness against external shocks and vibration. As has been shown, this is really the case as long as the feedback loop is linear. In practice, however, feedback actuators are normally nonlinear capacitive devices, and care must be taken to avoid nonlinearity errors and instabilities.

310

Accelerometers

Sensing

Beam

Forcing electrodes

Figure 7.12 A capacitive feedback accelerometer.

In order to analyze feedback nonlinearities the system according to Fig. 7.8 is represented by an ordinary differential equation x ¨ + 2δω0 x˙ + ω02 x = a′x +

Fact NB + m m

(7.58)

with a nonlinear actuating force Fact . Depending on the actuator type, the force exhibits different degrees of nonlinearity.

Bidirectional capacitive actuators The z-sensitive accelerometer as illustrated in Fig. 7.12 uses two plate capacitances that are driven in anti-phase and, thus, create a bidirectional force according to · ¸ 1 ∂C1 (D − z) ∂C2 (D + z) Fact = (V0 − VFB )2 + (V0 + VFB )2 2 ∂z ∂z · ¸ 2 2ε0 AV0 V02 + VFB 2 = 2 −(1 + ξ )V + ξ , (7.59) FB D (1 − ξ 2 )2 V0 where ξ = z/D is the normalized deflection, V0 the constant bias voltage, VFB the feedback voltage, and A the overlap area. Remarkably, £the bidirectional actuator consists of an intrinsic positive¤ feedback term 2ε0 AV02 /[D2 (1 − ξ 2 )2 ] ξ, which in the end is responsible for possible pull-in effects. This normally small contribution entails also a correction of the feedback gain. Indeed, for small deflections the linearized relation Fact = − (2ε0 AV0 /D0 ) (VFB − ξV0 ) holds. The gain of the feedback actuator is generally µ ¶¯ ∂Fact ∂Fact ∂ξ ¯ kFB = − + . (7.60) ¯ ∂VFB ∂ξ ∂VFB V F B =0

For larger deflections the feedback force may change sign, causing fatal positive feedback with subsequent pull-in. For the analysis the simplest case is that in which the feedback voltage is directly proportional to the sensed deflection VFB = kT kE z, so that ∂ξ/∂VFB = 1/(kT kE D). Since the electronics limits the feedback voltage to VFB ≤ V0 the

7.2 The spring–mass system

311

maximal processable deflection, zM , is zM =

V0 kT kE

or

ξM =

zM . D

(7.61)

Therefore, after substitution into Eq. (7.59) one gets the actuator force Fact = −

2ε0 AV02 1 − ξM 1 − ξ 2 /ξM 1 − ξ 2 /ξM ξ = −k k k D ξ T E FB D0 ξM (1 − ξ 2 )2 (1 − ξ 2 )2

(7.62)

with feedback gain

2ε0 AV0 (1 − ξM). (7.63) D2 The characteristic ξ = 0 and p is cubic in ξ and changes sign at the points √ √ ξp = ± ξ = ± V / (k k D). For deflections larger then |ξ| ≥ ξ 1,2 M 0 T E M (or z ≥ V0 D/(kT kE )) the feedback becomes positive and the operation of the system becomes unstable. Large bias voltages V0 help to prevent inversion of the feedback direction. Small gaps are the most dangerous factors for such instabilities. With the loop gain K0 = kT kE kF /(mω02 ) the dynamic equation (7.58) can be rewritten kFB =

z¨ + 2δω0 z˙ + ω02 z = a′z − K0 ω02

1 − ξ 2 /ξM z (1 − ξ 2 )2

(7.64)

and used for an estimate of the steady nonlinearity error. This is defined for a constant acceleration a0 , or as the deviation, ǫ, of the step response for t → ∞ from its ideal value (equal to unity) ǫ=

(1 + K0 )z∞ ω02 − 1. a′z

(7.65)

Performing simple calculations assuming ξ < ξM ≪ 1, the error can be expressed as µ ¶ a0 ξ2 ∼ z = 2 − K0 1 − z ⇒ ω0z ξM ǫ≈

1 K0 a20 mD K02 a20 ≈ , ξM D2 (1 + K0 )3 ω04 2ε0 AV02 (1 + K0 )3 ω02

(7.66)

where the relation 1/ξM = Dmω 2 K0 /(V0 kFB) has been used. Despite the fact that the error increases quadratically with applied acceleration, it remains negligibly small even for growing loop gains. If, for instance, an accelerometer with a gap of 2 µm, resonance frequency of 5 kHz, thickness of the moving mass h = 100 µm, A = (200 µm)2 , and supply voltage 2 V is used, the error is ǫ ≈ 6.6 × 10−6 [K02 /(1 + K0 )](a0 /g)2 . This excellent performance is the cutting edge for alternative actuator implementations. The considerations can be easily extended towards torsional accelerometers as per Fig. 7.2 with tilting-plate capacitors. The formulas are a little bit more complicated but reveal the same parametric tendencies as those in the case of parallel-plate actuators.

312

Accelerometers

Levitation combs Folded beams

Figure 7.13 A surface-micromachined capacitive-feedback accelerometer. Adapted from Lu et al. [1995].

Single-sided actuators Often the technological difficulties do not allow one to implement double-sided actuators without significantly great effort. Not only bulk-micromachined but also surface-micromachined accelerometers as, for instance, presented in Fig. 7.13 lack the balancing electrode on the top side [Lu et al. 1995]. Owing to the four folded beams a high stiffness in the x- and y-directions is achieved here; however the accelerometer is quite sensitive to moments about the diagonals. Remarkably, this one of the first designs of z-accelerometers in surfacemicromachining technology exhibits a counter-force created by the surrounding combs that is directed away from the underlying electrode. It is generated by the levitation effect and not only reduces the pull-in tendency, but also constitutes a restoring force. Nonetheless, in all such cases an inherently more nonlinear unidirectional actuator, consisting of a single-plate capacitor only, has to be accepted. The following shows that analog feedback compensation is not suited to cope with the nonlinearity of a single-sided accelerometer. First, in order to generate a bidirectional feedback force a prestressing of the springs by a constant bias voltage V0 must be introduced.4 Therefore the dynamic equation is · ¸ NB ε0 A (V0 + VFB )2 V02 z¨ + 2δω0 z˙ + ω02 (z − z0 ) = a′z + − − . (7.67) m 2m (D + z)2 (D + z0 )2 In the absence of a feedback signal and of external accelerations the voltage V0 creates the prestressed position (bias shift), z0 , according to z0 = − 4

1 ε0 A V 2. 2 2mω0 (D + z0 )2 0

(7.68)

The usage of levitation forces as counter-forces can be considered exceptional. Such a counterforce can easily be added to the following model and is not treated here.

7.2 The spring–mass system

313

The deviation z ′ = z − z0 from z0 represents the reaction to an external acceleration. To avoid pull-in the deflection must, according to Chapter 2, fulfill the condition z0 < D/3. Secondly, assuming again a direct proportionality VFB = kT kE (z − z0 ) between the feedback voltage and z − z0 , one gets with the normalized variables ξ=

z − z0 , D

VFB =

kT kE ξ , D

ξ0 =

z0 , D

ξM =

zM V0 = D kT kE D

(7.69)

the following dynamic equation for the noiseless case: · µ ¶¸ a′ (1 + ξ0 )2 ξ ξM ξ¨ + 2δω0 ξ˙ + ω02 ξ = z − ω02 K0 ξ 1 + 1 + (7.70) D (1 + ξ + ξ0 )2 2ξM 1 + ξ0 with feedback gain kFB =

ε0 AV0 1 − ξM /(1 + ξ0 ) D2 (1 + ξ0 )2

and loop gain K0 =

kT kE kFB . mω02

The steady-state error can be estimated for small ξ using a standard iterative approximation, µ ¶2 K0 mD2 ǫ = −a0 . (7.71) 1 + K0 2ε0 AV02 The error remains nearly constant with growing loop gain and is far too much for precise measurements. For the example in the previous section the error becomes ǫ ≈ −0.127 [K0 /(1 + K0 )]2 a0 /g, which emphasizes the aggravating impact of single-sided-actuator nonlinearities on the steady-state error. The dynamic behavior is also strongly affected by the large-signal effects as illustrated in Fig. 7.14 for two damping levels of δ = 1 and δ = 5. Here the simulated (Simulink) step responses for large and small signals are shown for an accelerometer with the same parameters as those used above and with closed-loop gain K0 = 10. The time is normalized, τ = ω0 t. For large positive accelerations the response is speeded up, whereas for negative accelerations the response is delayed. Summarizing, the analog feedback technique applied to a single-sided accelerometer is not an acceptable solution if reasonable linearity is required.

Linearization and embedded Σ∆ converters Linearization The dominant reason for the feedback nonlinearity in single-sided actuators is the quadratic dependency of the acting force on the control signal VFB . The inversely quadratic dependency on the deflection is negligibly smaller within the operating range.

Accelerometers

2

1.8

a = –2 g

1.6 2 (z-z0) ω0(1+K0)/a 0

314

1.4

0

δ=1

1.2

a0= 0.1 g

1

0.8

a =2g 0

0.6

δ=5

0.4

K0=10

0.2

0 0

2

4

6

8

10

τ

Figure 7.14 Small- and large-signal step responses of a single-sided feedback accelerometer.

The nonlinearity can be completely eliminated using charge control, for which, according to Chapter 2, Eq. (2.120), the force does not depend on the gap. Charge control is used rather seldom due to the considerable implementation difficulties. The best way to eliminate the dominant quadratic nonlinearity is to make V 2 proportional to the sensed deflection z or z − z0 . The standard approach is to use embedded Σ∆ modulators that provide high-frequency rectangular pulse sequences with two amplitudes as discussed in Section 6.3.3.5 The bit-stream BS generated by a one-bit Σ∆ quantizer is a pulse sequence X θk Z(t − kTC ) (7.72) BS = k

with θk = 0 or θk = 1. The average pulse density of the bit-stream represents the capacitance changes or the measured deflection z. Z(t) is the unit-pulse function of length TC . A one-bit DAC with output levels Vm in and Vm ax with respect to the common ground converts the bit-stream into a sequence of voltage pulses v(t) = Vm in + (Vm ax − Vm in ) BS. The orthogonality of the non-overlapping pulses guarantees the linear dependency of the squared function on the information carriers θk : X θk Z(t − kTC ). (7.73) v 2 (t) = Vm2 in + (Vm2 ax − Vm2 in ) k

If the output of the quantizer is ±1-pulses: θk′ = 2(θk − 21 ) = ±1, the equivalent representation of the DAC output is v(t) = 21 (Vm ax + Vm in ) + 21 (Vm ax − Vm in ) 5

Multi-bit quantizers combined with multi-bit DACs in the feedback loop as, for instance, in Lang and Tielert [1999] can also be used.

7.2 The spring–mass system

NB

a’z m

+

+

-

Sensing element

fC

nE + +

FS(s)

kTE(s)

S&H Quantizer

DAC

Vmax

kF +

+ +

yout

Compensator

feedback

kQ

315

Vmin

kF

+ DAC

Figure 7.15 An electromechanical Σ∆ feedback loop.

BS′ , while the squared voltage becomes: v 2 (t) =

X 1 2 1 (Vm ax + Vm2 in ) + (Vm2 ax − Vm2 in ) θk′ Z(t − kTC ). 2 2

(7.74)

k

The fast pulse sequence v 2 replaces the continuous force by a sequence of fast strikes. The solution of the governing system equation (7.58) can be written using the impulse transfer function zδ (t), · ¸ Z t NB Fact (τ ) dτ zδ (t − τ ) a′z (τ ) + z(t) = + . (7.75) m m −∞ Obviously, if the reaction time of the accelerometer is long, the acting force can be substituted by its average over a time interval of length RTC , where R is the oversampling rate. Hence, the system reacts like under the impact of a force derived from the linearly measured deflection. Such a model approximates the reality better the higher the Nyquist frequency of the sensed signal in comparison with the bandwidth of the accelerometer transfer function.

Embedded Σ∆ converters The advantages of Σ∆ converters such as high dynamic range and small nonlinearities together with the benefits of their simple implementation destine them for usage not just in feedback-controlled single-sided accelerometers but in all accelerometers and gyroscopes. The possibility of deriving immediately a digital output signal is another reason for their broad acceptance. Embedded Σ∆ converters are very popular, particularly in accelerometer feedback systems [Dong et al. 2006, Henrion et al. 1990, Kraft et al. 1998, Lemkin and Boser 1999, Lu et al. 1995, Petkov and Boser 2005, Smith et al. 1994, Yun et al. 1992]. Figure 7.15 shows a typical arrangement of an embedded electromechanical Σ∆ modulator. In the case of accelerometers the sensing element, F (s),

316

Accelerometers

represents a second-order filter that shapes the total quantization noise at the output of the actuator kF . Ideally, it should approximate a double integration as in second-order Σ∆ converters. This is the case for critically damped or overdamped systems. However, the poles of the mechanical filter cannot be freely chosen. Therefore, in the case of large mechanical bandwidths where the sampling frequency of the Σ∆ modulator is no longer some orders of magnitude larger than the resonance frequency, the frequency behavior must be corrected by a compensating filter. In the simplest case a low-pass filter can be added, reducing the overall loop bandwidth below the resonance frequency of the sensor (e.g. AnalogDevices [1993]). Some sensitivity loss results. A better solution is the insertion of a lead compensator aimed at guaranteeing the necessary phase margin for stability. Boser and Howe [1996] suggest a transfer characteristic of the compensator HC (z) = α − z −1 with α ∼ 2. In Fig. 7.15 the capacitive transducer together with the front-end electronics is represented by the block kTE (s). Since the input amplifier resides in the forward branch of the feedback loop before the quantizer, linearity, gain tolerances, and slew rate are much less critical then in an open-loop implementation. Hence, the design focus can be put on noise performance in exchange for a lower signal performance. Sampling and hold is performed by using a zero-order-hold stage. It can be placed also before the compensator block. The bit-stream output BS′ = P ′ ′ k θk Z(t − kTC ) with θk = ±1 is fed back into a 1-bit DAC with output levels Vm ax and Vm in . The voltage squared v 2 , generating the acting force, obeys the relation (7.74). For the sake of compactness the following notation will be used: v 2 = V02 + V12 × BS′ ,

V02 =

1 2 1 (V + Vm2 in ), V12 = (Vm2 ax − Vm2 in ). 2 m ax 2

(7.76)

kF is the force-forming block, the structure of which depends on whether a single- or double-sided capacitive actuator is used. In symmetric actuators the bit-stream-generated force is applied with appropriate sign to one electrode, and the force generated by the inverted bit-stream is applied to the other. In the absence of additional phase lag this is equivalent to applying the acting voltage to only the electrode furthest away from the moving structure. The benefit is a larger robustness with respect to pull-in effects, because the actual pulse is always trying to push the moving mass closer to the rest position. Not shown in Fig. 7.15 are the decimation filters at the output, which transform the bit-stream into the output code. The noise performance of a Σ∆ feedback accelerometer should not be worse than in an open-loop system. That is, the contribution of the quantization noise to the total output noise should be as small as possible. Using the equivalent-circuit representation of the quantizer, shown in Fig. 7.15 as gray blocks in the right upper corner and below the DAC, the output signal

317

7.2 The spring–mass system

can be represented by Yout (s) =

1 [FS (s)kTE (s)Hcom p (s)kQ (ma′z (s) + NB + εDAC ) 1 + FS (s)H(s) + kTE (s)Hcom p (s)kQ nE + ε],

(7.77)

where H(s) = kTE (s)Hcom p (s)kQ kF and Hcom p (s) is the transfer function of the compensator circuit plus possible additional noise-shaping electronics. kF represents the gain of the bit-stream-to-force transformation and, strictly speaking, depends on the position z. This more symbolic than exact relation allows one to draw some conclusions. First, assuming a large loop gain K0 = FS (0)kTE (0)Hcom p (0)kQ kF ≫ 1 within the signal bandwidth 2fa ≪ fC , the following simplification holds: Yout (s) =

1 nE ε (ma′z + NB + εDAC ) + + . kF kF FS (s)Hcom p (s) FS (s)H(s)

(7.78)

The quantizer noise is shaped most significantly by the closed-loop transfer characteristic (FS (s)H(s))−1 . The input-related electronic noise nE is also slightly shaped, while the signal plus Brownian noise together with the electromechanical DAC error are transferred without any shaping. Therefore, the most critical factor is the equivalent electromechanical DAC error εDAC , which is the result of the transformation of the voltage into a force. It is modeled and simulated in the following example.

A single-sided accelerometer with Σ∆ feedback According to Eq. (7.58) the dynamics of the single-sided accelerometer under the impact of the bit-stream-generated force follows, according to (7.76), the equation z¨ + 2δω0 z˙ + ω02 z = a′z +

NB ε0 A V02 + V12 × BS′ − . m 2m (D + z)2

(7.79)

Here the bit-stream BS′ is a ±1 pulse sequence. If the accelerometer is in the rest position z0 , the average of the bit-stream is zero. This corresponds to a prestressing by the voltage V02 , ω02 z0 = −

ε0 A V02 . 2m (D + z0 )2

(7.80)

The deviation from the prestressed position ∆z is therefore · 2 ¸ ′ 2 V02 ˙ + ω02 ∆z = a′z + NB − ε0 A V0 + V1 × BS − ¨ + 2δω0 ∆z ∆z . m 2m (D + z0 + ∆z)2 (D + z0 )2 (7.81) If the deviation from the prestressed position is positive, the force attracts the mass towards the position z = z0 . For negative deviations no electromechanical force is applied, restricting the movement to the action of the restoring force of the prestressed spring.

318

Accelerometers

Electronic noise

Brownian noise

Sensing element

m

+

+

-

+-

1/s

+-

1/m

1/s

kCV

z dC

+

+

kCE

Sine-wave mass

Z to dC

2*

dC to V

Electronic gain

Transducer & front-end electronics

-g1

+-

1/s

g1

+-

Out 1

1/s

Zero-Order Relay Hold g2

g2

z FFB

Additional noise shaping

S’ Force-feedback

Figure 7.16 A simulation model of a fourth-order continuous-time Σ∆ feedback accelerometer.

The following example is aimed at demonstrating typical properties of Σ∆ feedback accelerometers. The mechanical system represents a critically damped bulk-micromachined accelerometer as in the example of the sections “Bidirectional capacitive actuators” and “Single-sided actuators” with a 200 µm × 200 µm × 100 µm heavy proof mass (m = 9.2 µg). The resonance frequency is located at f0 = 5000 Hz. High-frequency charge sensing with an excitation amplitude of 0.5 V performs capacitive position sensing. The corresponding capacitance-to-voltage gain amounts to 0.25 × 1012 V/F. Correlated double sampling suppresses the 1/f noise. The input-related noise of the charge amplifier is √ 10 nV/ Hz. The clock and sampling frequency is 500 kHz. A target bandwidth of 250 Hz is realized. Figure 7.16 illustrates the simulation model of a fourth-order continuous-time feedback loop. The sensing element delivers two orders of the Σ∆ modulator; the remaining two orders are introduced by additional noise-shaping circuitry. The integrator gains g1 and g2 are chosen in order to guarantee stability (g1 = g2 = πfC /10). Since the clock frequency is small enough, time delays in the quantizer can be neglected. Because of the critical damping the system works without an additional compensator. In Fig. 7.17(b) the output spectrum of the bit-stream is presented for a 200-Hz input signal with amplitude 1g. The nonlinearity of the one-sided actuation due to the proof-mass motion creates a second harmonic, but with an acceptably small amplitude. The corresponding nonlinearity error for applied steady acceleration is less then 6% for 100g input. It increases quadratically with the applied steady acceleration.

319

7.3 Resonant accelerometers

(a)

10

(b)

0

First harmonic for 1 g input

First harmonic for 1 g input 10

10

-2

δ = 1; f0=5 kHz

δ = 1 f0 = 5 kHz 10

-4

10

-6

noise floor 90 µg /sqrt Hz

noise floor 1.1 mg/sqrt Hz

-4

10

-8

Second harmonic (HD= -55.9 dB)

HD= -41.0 dB

10

-6

10

10

10

-8

10

10

-10

10

-10

1

10

2

3

10 Frequency

10

4

10

5

-12

-14

10

1

10

2

3

10 Frequency

10

4

10

5

Figure 7.17 Spectra of (a) second- and (b) fourth-order Σ∆ force-feedback

accelerometers.

Thus, the Σ∆ feedback has reduced the nonlinearity of the analog feedback single-sided accelerometer according to (7.71) by orders of magnitude. If necessary the nonlinearity can be further reduced by controlling the DAC voltages according to the sign of the bit-stream pulse as proposed in Dong et al. [2006]. For the example presented here the output noise is formed by a balanced contribution of electronic and Brownian noise. The quantization noise within the signal√band is well suppressed, resulting in a theoretical overall noise floor of 90 µg/ Hz, which in practice, of course, may be difficult to achieve due to parasitic effects. For comparison, in Fig. 7.17(a) the bit-stream spectrum for the second-order force-feedback accelerometer without additional noise shaping is shown. Here the noise shaping is performed exclusively by the sensing element itself. The resolution is more then ten times lower than for the fourth-order system, demonstrating the usefulness of additional noise-shaping circuitry. The large noise floor masks also the nonlinear distortion which is revealed in the fourth order system.

7.3

Resonant accelerometers Resonant sensors exploit changes of resonance frequencies, phases, or resonance amplitudes of the mechanical transducer caused by the applied accelerations. Frequency-modulated sensors are especially attractive due to their high sensitivity and large dynamic range. The frequency output provides a smooth interface to digital signal processing, and high signal-to-noise ratios, provided that optimal frequency demodulation techniques are used.

320

Accelerometers

FN

L

Clamp

Clamp

Figure 7.18 A resonant beam.

Sensing comb Drive combs

Clamp

Sensing comb

Figure 7.19 A double-ended tuning fork (DETF).

7.3.1

Resonant beams One of the most often used mechanical resonance transducers is the axially loaded clamped-clamped beam, which like a violin string features a set of natural resonance frequencies. The resonant beam as shown in Fig. 7.18 represents a forceto-frequency converter. The clamped-clamped beam has a natural frequency f0 that is shifted under the impact of an axial force FN . To measure the resonance frequencies the beam must be excited, for instance by comb drives as shown in Fig. 7.19. Here, one uses not a single beam, but two parallel beams that share the applied axial force. Two harmonic drive forces in anti-phase excite the beams, and the deflections are capacitively picked up by sensing combs. With the driving and sensing amplifiers embedded in an oscillator loop, the system oscillates at the actual resonance frequency. The arrangement according to Fig. 7.19 is known as a double-ended tuning-fork (DETF) resonator (e.g. Nguyen and Howe [1993], Roessig et al. [1997], Seshia et al. [2002], and Su et al. [2005] and works – with respect to any of the beams – with one-sided excitation. For a single beam the arrangement of the driving and sensing combs can be designed to be fully symmetric.

Resonance vibration – exact solution The deflection of a single beam under axial force follows from Eq. (3.47) after differentiating twice and considering that σ0 S → FN and δ 2 Mσy /δx2 = −q, · ¸ d2 d2 w(x) d2 w(x) EI (x) − FN = q(x), y 2 2 dx dx dx2

(7.82)

7.3 Resonant accelerometers

321

where the residual stress force σ0 S was re-substituted by an arbitrary axial force FN . To extend this static equation towards time-varying deflections, the inertial forces have to be added. A beam slice with length dx has the mass ρS dx, so the distributed load has to be substituted by subtracting the inertial force: q(x, t) ⇒ q(x, t) − ρS ∂ 2 w(x, t)/∂t2 , where ρ is the mass density, · ¸ ∂2 ∂ 2 w(x, t) ∂ 2 w(x, t) ∂ 2 w(x, t) EI (x) − F + ρS = q(x, t). (7.83) y N ∂x2 ∂x2 ∂x2 ∂t2 Formula (7.83) represents the general dynamic beam equation without damping. The resonance frequencies of the homogeneous doubly clamped beam (Iy (x) = constant) with length L can be found by searching for the periodic solutions of the corresponding homogeneous equation. The solution is presented as w(x, t) =

∞ X

Wn (x)cn cos(ωn t + ψn )

(7.84)

0

with ωn natural frequencies, and Wn (x) the shape function of the nth vibrational mode. After substitution of (7.84) into (7.83) (q(x, t) = 0) and normalization of the variable x according to ξ = x/L, the shape function obeys the following equation: d4 Wn (ξ) d2 Wn (ξ) − 2δF + kn4 Wn = 0, 4 dξ dξ 2

(7.85)

where δF =

FN L2 , 2 EIy

kn4 =

ρSL4 2 ω . EIy n

(7.86)

Any of the shape functions has to satisfy the boundary conditions at both clamped ends, Wn (0) = Wn′ (0) = Wn (1) = Wn′ (1) = 0.

(7.87)

According to the theory of linear ordinary differential equations W (ξ) can be presented as a superposition of the four fundamental solutions Wn =

4 X

Ai ep i ξ

(7.88)

i= 1

with pi the roots of the characteristic equation p4 − 2δF p2 + kn4 = 0.

(7.89)

On solving (7.89) the characteristic roots are p1,2 = ±kn 1 ,

p3,4 = ±kn 2

with kn 1 =

qp kn4 + δF2 − δF ,

kn 2 =

qp kn4 + δF2 + δF .

(7.90)

322

Accelerometers

On inserting (7.88) with roots from (7.90) into the boundary conditions (7.87) one gets the following algebraic equations: A1

+ A2

+ A3

+ A4 = 0,

jk1n A1

− jk1n A2

+ kn 2 A3

− kn 2 A4 = 0,

j k1 n

e

j k1 n

jk1n e

A1

−j k 1 n

+e

−j k 1 n

A1 − jk1n e

kn 2

A2

+e

kn 2

A2 + kn 2 e

A3

+ e−k n 2 A4 = 0, −k n 2

A3 − kn 2 e

(7.91)

A4 = 0,

which have a non-trivial solution only if the determinant is zero. This condition delivers the conditional equation for the natural frequencies: cosh kn 2 cos kn 1 −

δF sinh kn 2 sin kn 1 − 1 = 0, kn2

(7.92)

whereas according to Eq. (7.86) the resonance frequencies obey the relation s EIy 2 ωn (δF ) = k (δF ). (7.93) ρSL4 n In the absence of an axial load (δF = 0), the equation contracts to cosh kn2 (0) cos kn2 (0) = 1. For large kn (0) (n ≥ 3) the solution can be approximated by kn (0) =

(2n + 1)π for n ≥ 3 2

and

k1 (0) = 4.730,

k2 (0) = 7.8532. (7.94)

The numerical solutions k1 (0) and k2 (0) are quite close to the approximation for large n. By expanding the solution of Eq. (7.92) around the force-free point and assuming small axial forces (δF < 6), the following approximation can be derived (e.g. Bouwstra and Geijselears [1991] and Bao [2000]): r 1 ωn (δF ) = ωn (0) 1 + γn δF , (7.95) 6 where the coefficient γn is given by γn =

12(kn − 2) for n ≥ 3 kn3

and

γ1 = 0.2949,

γ2 = 0.1453.

(7.96)

With (7.86) and (7.93) the frequency change ∆ωn (FN ) = ωn (δF ) − ωn (0) is then γn L2 ∆ωn ∼ FN . = ωn 24 EIy

(7.97)

Thus, for a homogeneous rectangular beam the frequency shift of the first natural frequency is ∆ω1 (FN ) ∼ 0.147L2 FN . = ω1 (FN ) Ebh3

(7.98)

7.3 Resonant accelerometers

323

The longer the beam and the smaller the cross-sectional dimensions, the larger the sensitivity to axial loads, which explains some of the preference for surfacemicromachined resonators. An example may impart an impression regarding the orders of magnitude. A 3-µm thick, 20-µm wide, and 200-µm long silicon beam (ρ = 2300 kg/m3, E = 168 GPa) has a first natural frequency of 659 kHz. To get a sensitivity of 30 Hz/g a mass of 71 µg is needed, provided that the axial force is directly generated by the mass acceleration FN = mg. This corresponds to an attached proof mass of area 3 mm2 with a layer thickness of 10 µm. Consequently, leverage of the acting acceleration force is highly desirable in order to reduce the dimensions of the proof mass and to increase sensitivity. The example shows further that the first vibration frequency may be on the order of some hundreds of kHz. Thus, large forces F are needed in order to get a reasonable excitation. Since the excited amplitude x0 is given by x0 ∼ mQF0 /ω12 , large quality factors Q are beneficial for high-frequency resonators. Therefore, high-frequency resonant accelerometers usually are driven at low pressures and feature quality factors well above 1000.

Resonance frequencies by the energy method The exact solution of the beam equation (7.83) is difficult, especially if the beam is inhomogeneous and transverse loads are attached to the beam. Sometimes a simpler resonance estimate can be derived using energy methods. The energyconservation principle requires that at resonance the maximum potential energy Up ot = Uelastic of a dynamic system equals its maximum kinetic energy Ukin . The potential strain energy according to Eq. (2.38) in Chapter 2 can easily be calculated for an Euler beam (σy = σz = σxy = σxz = σy z = 0, σx = Eεx = −Ezw′′ (x, t)): Uelastic

1 = 2

1 dV σx εx = 2 V

Z

Z

L

dx EIy w′′2 (x, t).

(7.99)

0

The kinetic energy is the sum of the motion energy of all of the volume elements included, dUkin = 21 dm(dw(x, t)/dt)2 = 21 ρ dV (dw(x, t)/dt)2 ,

Ukin =

1 2

Z

L

0

dx ρbh

µ

dw(x, t) dt

¶2

.

(7.100)

Since the deflection represents the superposition of modes,

w(x, t) =

∞ X

n=0

Wn (x) cos(ωn t + ψn ),

(7.101)

324

Accelerometers

according to the Rayleigh–Ritz method the energy-conservation principle is applicable to any of the vibration modes, Z 1 L Uelastic,n = dx EIy Wn′′2 (x) cos2 (ωn t + ψn ), 2 0 (7.102) Z L 21 2 2 Ukin,n = ωn dx ρbhWn (x) sin (ωn t + ψn ), 2 0

so that from max Uelastic,n = max Ukin,n the Rayleigh quotient follows: RL dx EIy Wn′′2 (x) 2 ωn = R0 L . (7.103) 2 0 dx ρbhWn (x)

Rayleigh’s quotient can be estimated by approximating the exact shape function Wn (x) by a static deflection for an appropriate load, for instance, by the solution of Eq. (7.82) with constant distributed load q. On applying the procedures described in Section 3.1.4 (Eq. (3.29)) with boundary conditions for a clamped-clamped beam w(0) = w′ (0) = w(L) = w′ (L) = 0 in the absence of an axial load, FN = 0, one gets the shape function q w(x) = x2 (x − L)2 (7.104) 24EIy and the corresponding resonance frequency s s √ EIy h E ω1 = 504 = 6.46 2 . 4 ρbhL L ρ The exact solution according to (7.93) and (7.94) delivers ω1 = p EIy /(ρbhL4 ). Here the error is less than 0.5%.

7.3.2

(7.105) √ 500.5×

Resonant accelerometer systems Many embodiments of resonant beams are known. For instance, the beam can be excited by local heaters embedded in the beam, which in the case of good thermal isolation allows one to achieve quite high excitation frequencies. In this case sensing is usually performed by piezoresistors [Aikele et al. 2001, Ohlckers et al. 1998, Burrer and Esteve 1995, Ferrari et al. 2005]. Piezoresistors are used also in combination with electrostatic drives as reported by Burns et al. [1996]. Resonant accelerometers typically include a leverage mechanism that transforms the inertial force into an increased axial load that is applied to the resonant beam. In Fig. 7.20 a common principle of mass–beam coupling in bulk-micromachined accelerometers is shown [Burrer and Esteve 1995, Ferrari et al. 2005]. The proof mass is suspended by relatively thick hinges, which dominate the deflection. The significantly thinner and longer beam has a different neutral axis with respect to the hinges. Thus, the beam deflection generates stretching or compression of the beam. The leverage is determined by the ratios between beam and hinge lengths

7.3 Resonant accelerometers

325

resonant beam

FN

hinge

hinge

resonant beam

seismic mass

Figure 7.20 A bulk-micromachined resonant accelerometer with hinge-beam

suspension.

Torsional beam

Anchor

resonant Feedback electrode beam

FN

Sense

Drive

Figure 7.21 A bulk-micromachined resonant accelerometer with a torsion bar.

and thicknesses. Usually an identical isolated reference beam is implemented within the bulk silicon in order to eliminate first-order temperature dependencies. The piezoresistors should be implemented at the end of the beam where the oscillating beam features the maximal stress. Surprisingly low resonance frequencies of around 100 kHz are reported for this kind of accelerometers (according to Burrer and Esteve [1995], 90 kHz with sensitivity 200 Hz/g). A different transfer principle is demonstrated in Fig. 7.21 [Esashi 1996]. Here the proof-mass deflection twists a torsion bar, to the upper edge of which the resonant beam is attached. Hence, the twisting of the bar creates an axial load on the beam. Wide torsion bars are required, which is feasible in bulk micromachining. Sensing and driving as well as feedback actuation can be realized by electrostatic excitation. In surface-micromachined resonant accelerometers lateral deflections are transferred into lateral beam loads. One of the first principles was that of the asymmetric microlever shown in Fig. 7.22 ([Roessig et al. 1995, 1997] with 68 kHz and 45 Hz/g; Seshia et al. [2002] with 145 kHz and 30 Hz/g). It was used for implementations using Analog-Devices’ and Sandia’s monolithic integrated MEMS

326

Accelerometers

anchor 1

DETF

2

proof mass

acceleration

Figure 7.22 A surface-micromachined resonant accelerometer with asymmetric

leverage. cantilever hinge proof mass

acceleration

fixed

resonant beam

drive/sense block

Figure 7.23 A resonant accelerometer with hinge-beam coupling.

processes. A lateral proof-mass deflection causes stretching and compressing forces at the attached DETFs via the anchored lever. The small horizontal beam to the anchor in Fig. 7.22 can be considered stiff in the x-direction, providing a lever point at the end for the vertical bar with its attached proof mass. Both DETFs are loaded, one with a positive and one with a negative load-force, which allows a differential frequency sensing, eliminating first-order temperature and common-mode stress effects. Double-DETF sensing was extended later towards cascaded symmetric microlevers as described in Su et al. [2005] and implemented in SOI technology. A similar principle using a single resonator beam is presented in Aikele et al. [2001] and shown in Fig. 7.23. The seismic mass is suspended by a hinge via a stiff intermediate bar. Deflections generate axial beam forces, whereas the leverage effect can be designed by changing the distance of the beam’s attachment to the hinge. The second end of the resonator beam is anchored via a special part of the shown drive/sense block consisting of a resistive heater and a sensing piezoresistor. The special shaping of the drive/sense block (details are omitted here) with a thermally isolating opening allows one to drive the beam at frequencies above 400 kHz and to get a sensitivity of around 70 Hz/g.

7.4

Beam accelerometers On the system level all of the accelerometers considered up to now could be well described by spring–mass models. Such a lumped-element approximation

7.4 Beam accelerometers

Bond interface

327

Bond wire

Cantilever

Actuator electrode

Sensing electrode

Sensor ASIC

Figure 7.24 A schematic view of the beam accelerometer.

Figure 7.25 The polysilicon beam – top view.

is no longer applicable for pronounced distributed-mass movements such as, for instance, in the case of mass-loaded beams. Such beams are of interest not only for direct acceleration measurements [Brandl and Kempe 2001] but also as a means for stress measurement in embedded piezoresistors, and as feedback actuators in tunneling accelerometers, which will be described in the next section. The use of simple cantilever beams without any additional proof masses has the advantage of technological elementariness and adaptability to low-, medium-, and high-g applications. Brandl and Kempe [2001] presented an accelerometer concept based on the unification of the monolithic and hybrid approaches and using only low-cost add-ons to a standard CMOS process. Two silicon dies are joined together as shown in Fig. 7.24. The top die forms the mechanical sensing component shown separately in Fig. 7.25, and the bottom die carries the actuating and sensing electrodes together with the ASIC. The sensing element is a singly clamped polysilicon cantilever, i.e. a mass-loaded spring. It is fabricated by photolithography and reactive-ion etching followed by a subsequent release step using anisotropic wet etching in KOH solution. This yields a very robust, cost-efficient sensing element with a most simple shape and practically no crossaxis sensitivity. The sensing-element die is joined face to face to a CMOS ASIC using a eutectic bonding procedure on the wafer level. Thus, the cantilever is hermetically sealed and electrically shielded. In this way, a zero-level package for the movable cantilever is simultaneously provided, allowing standard plastic injection molding of the composite structure. An air gap of a few micrometers is formed between the cantilever and the ASIC. The ASIC contains the counter-electrodes for the actuation and capacitive position sensing of the cantilever, and the electronics for signal processing and

328

Accelerometers

(b)

2

1.5

(a)

first eigenmode

z Cantilever

L

B

w(x)

D1

Actuator electrode x1

x

D2

Eigenmode

1

0.5

0

third eigenmode

-0.5

xD

Sensing electrode x2

second eigenmode

-1

-1.5 0

0.2

0.4

0.6

0.8

1

x/L

Figure 7.26 (a) An embedded cantilever beam and (b) its first three spatial

eigenmodes.

trimming. The integrated capacitive distance-sensing circuit is able to measure √ changes down to < 0.5 aF/ Hz. For excellent linearity and large bandwidth up to some kHz (depending on the configuration of the cantilever for the different full-scale ranges), the system is operated in closed-loop mode. The actuating force is a PWM modulated signal in order to generate a force that depends linearly on the capacitance change. This electrostatic actuation keeps the mechanical cantilever close to its initially adjusted position. Loop stability is ensured by the uncritical modal behavior of the cantilever and overcritical squeezed-film damping in the air gap. The bandwidth of the output signal is limited by outside loop filtering (typically 100 Hz for low-g and 1 kHz for high-g application). The typical sensor performance is characterized, for instance, for a 7-g z-axis sensor in a PLCC28 package by a sensitivity of 250 mV/g and a peak-to-peak noise at 140 Hz bandwidth of < 0.01g √ (resolution typically 130 µg/ Hz).

7.4.1

Beam dynamics The dynamics of the beam under acceleration and actuating/sensing forces is a well-suited example in order to demonstrate the application of energy methods to the analysis of elastic members in MEMS. It was used for dimensioning the sensor and feedback parameters of the beam accelerometer. The principal configuration of the embedded beam is shown in Fig. 7.26(a). The beam is prebent by residual stress, which can be very well controlled and used to establish a negative tip deflection, B, away from the bottom ASIC. This gives the possibility of introducing a larger electrical prebending in the opposite direction without a pull-in effect and, thus, a larger restoring force, which is necessary in order to run the feedback loop. The shape function of the prestressed beam for small residual stresses is proportional to x2 : w0 (x) = Bx2 /L2 . For further analysis it will be assumed that the deformation caused by the residual stress and the deformation created by the applied external forces add, because

7.4 Beam accelerometers

329

for small displacements the principle of superposition holds. Thus, it is possible to analyze the beam deflection assuming zero prebending, and add to the result the static prebending value.

The principle of virtual work The principle of virtual work states that for a system that is initially in equilibrium without external loads, which deforms and displaces quasi-statically under applied external forces, the work done by the external forces through the virtual displacements equals the internal work or energy change characterizing the deformed and displaced state. Virtual displacements are displacements for which such constraints as boundary conditions remain. According to d’Alembert’s principle the external forces may include inertial forces. For the cantilever beam under consideration the virtual work done by external forces in moving through the virtual displacement δw(x) around w(x) is δWext =

Z

L

[ρS(a − w(x)) ¨ + qD (x) +

0

X

Fi δ(x − xi )]dx δw,

i= 1,2

(7.106)

where the inertial force is taken into consideration by the term ρS w(x)dx, ¨ the damping force per unit length by qD (x), and the applied acceleration force per unit length by ρSa. The Fi are the approximating point loads induced by the actuator and sensing electrodes at positions xi (see Fig. 7.26(a)). The virtual internal work δWint corresponds to the change of the elastic energy according to Eq. (2.38). In the case of the Euler beam only the stress σxx and the strain εxx are different from zero. Thus δWint =

Z

dV δUelastic =

Z

L

dx

0

V

Z

dS δ S

µ

1 σxx εxx 2

¶

=

Z

0

L

dx

Z

dS σxx δεxx .

S

(7.107)

In Section 3.1.3 it was shown that for the Euler beam σxx (z) = Ezw′′ (x) whereas here the sign “−” has changed to “+” because the z-axis is directed inversely to the orientation in Section 3.1.3. Therefore, the internal virtual work for the same virtual displacements as in (7.106) can be expressed as δWint =

Z

0

L

∂ 2 w ∂ 2 δw dx E dS z = ∂x2 ∂x2 S Z

2

Z

L

dx EIy w′′ δw′′ .

(7.108)

0

According to the principle of virtual work δW = δWint − δWext = 0, and the final expression is, with ξ = x/L, Z

0

1

dξ

·

¸ Z 1 EIy ′′ ′′ 1 X dξ[ ρSa + qD ]δw + w δw + ρS w ¨ δw = Fi δw(ξi ). 4 L L i=1,2 0

(7.109)

330

Accelerometers

Integrating by parts yields ¶ ¸ · ¸1 Z 1 · 2 µ ∂ EIy ′′ ∂(EIy w′′ ) ′′ ′ dξ ¨ δw + EIy w δw − w + ρS w δw ∂ξ 2 L4 ∂ξ 0 0 Z 1 1 X dξ[ ρSa + qD ]δw + = Fi δw(ξi ). (7.110) L i=1,2 0 Since the equation is valid for all compliant virtual displacements the following equation results: µ ¶ ∂ 2 EIy ′′ 1 X ¨ = ρSa + qD + w + ρS w Fi δ(ξ − ξi ), (7.111) 2 4 ∂ξ L L i= 1,2 with boundary conditions EIy w′′ (1) = 0,

Eigenmode expansion

∂(EIy w′′ ) ¯¯ = 0. ¯ ξ= 1 ∂ξ

(7.112)

The solution of (7.111) for homogeneous beams, Iy (x) = constant, will be sought as an eigenmode expansion. For this purpose the eigenmodes Wn cos(ωn + ψn ) of the homogeneous equation µ ¶ ∂ 2 EIy ′′ ¨=0 (7.113) w + ρS w ∂ξ 2 L4 with boundary conditions w(0) = w′ (0) = w′′ (1) = w′′′ (1) = 0 are determined. As in Section (7.3.1), a non-trivial solution exists only if the characteristic equation is fulfilled: cos kn cosh kn + 1 = 0; ⇒ k1 = 1.875, k2 = 4.694, k3 = 7.855, . . .

(7.114)

The corresponding eigenfrequencies are k2 ωn = n2 L

s

EIy . ρS

(7.115)

On comparing the characteristic roots, kn , with those of a clamped-clamped beam according to Eq. (7.93), it is found that the eigenfrequencies of the spatial cantilever modes are significantly lower, as could be expected intuitively. In view of the boundary conditions the eigenmodes obey the relation Wn (ξ) = cn [(sin kn + sinh kn )(cos(kn ξ) − cosh(kn ξ))

+ (cos kn + cosh kn )(sinh(kn ξ) − sin(kn ξ))].

(7.116)

The first three spatial eigenmodes are presented in Fig. 7.26(b). They are normalized by 50% of the deflection at the free end of the beam: 1 = cos kn sinh kn − sin kn cosh kn . cn

(7.117)

7.4 Beam accelerometers

331

In this case the eigenfunctions and their second derivatives are not only orthogonal, Z 1 Z 1 dξ Wn′′ Wm′′ = 0, for n 6= m, (7.118) dξ Wn Wm = 0 and 0

0

but also orthonormal, 1

Z

0

dξ Wj2 = 1.

(7.119)

An approximative solution of the cantilever equation can be found by assuming that the real as well as the virtual displacements can be represented (in the method of assumed modes) as w(x, t) =

N 1X Zn (t)Wn (x), 2 n =1

δw(x, t) =

N X

Wn (x)δZn (t).

(7.120)

n=1

Since Wn (1) = 2, the Zn (t) describe the relative mode deflection at the tip of the beam. In light of the eigenmodes’ orthonormality and the lack of dependency of the virtual displacements δZn (t), the substitution of (7.120) into (7.109) leads to the resulting modal differential equations Z 1 Z 1 EIy 2 X ′′ 2 ¨ ρS Zn + 4 Zn [ρSa + qD ]Wn (ξ)dξ + (Wn ) dξ = 2 Fi Wn (ξi ). L L i= 1,2 0 0 (7.121)

Since the series (7.120) converges very fast, it is sufficient to consider only the first two members, ρS Z¨1 + Q1 + α11

EIy 2 X Z1 = β1 ρSa + Fi W1 (ξi ), 4 L L i=1,2

EIy 2 X ρS Z¨2 + Q2 + α22 4 Z2 = β2 ρSa + Fi W2 (ξi ), L L i= 1,2

(7.122)

with α11 =

Z

1

0

β1 = 2

Z

(W1′′ )2

dξ = 12.362,

α22 =

Z

1

0

1

W1 dξ = 1.565 89,

β2 = 2

0

Z

(W2′′ )2 dξ = 485.52,

(7.123)

1 0

W2 dξ = −0.867 86

(7.124)

and the damping terms Qn = −2

Z

1

qD Wn dξ, 0

n = 1, 2.

(7.125)

332

Accelerometers

Damping and electrostatic forces In contrast to the case in the section “Low-frequency squeeze damping” in Chapter 3, the gap under the plate-like beam is not constant along x. Therefore, the results for a rectangular beam are not directly applicable. The squeeze damping force per unit length, qD (ξ), for a bent beam can most easily be found by solving the Navier–Stokes equation (Eq. (3.118)), µ△¯ v = µ ∂ 2 v¯/∂z 2 = ∇p, with homogeneous boundary conditions at the beam’s R b/2 tip under the assumption 12L2 /b2 ≫ 1. It can be shown that qD = −b/2 p dy is approximately given by Ã ! √ cosh[ 12(L/b)ξ] D˙ ′ 3 √ qD (ξ) = µ b −1 , (7.126) D3 cosh[ 12(L/b)] where µ′ is the effective viscosity according to Eq. (3.160) or according to Veijola’s approximation (3.103). In order to optimize the capacitive gaps for sensing and actuation, the distance between beam and substrate is designed to be position-dependent and, from Fig. 7.26(a), is given by D(ξ) =

1 (Z1 (t)W1 (ξ) + Z2 (t)W2 (ξ)) + Bξ 2 + D0 (ξ) 2

(7.127)

with ( D1

D0 (ξ) =

for ξ < xD /L,

D2

for ξ > xD /L.

(7.128)

Substitution into Eq. (7.125) delivers the mode-damping forces Q1 = c1,1 Z˙ 1 + c1,2 Z˙ 2 ,

Q2 = c1,2 Z˙ 1 + c2,2 Z˙ 2

(7.129)

with cn ,m = − 2µ′ b3 ×

Z

0

1

dξ

Ã

[ 21 (Z1 W1 (ξ)

! √ cosh[ 12(L/b)ξ] £√ ¤ −1 cosh 12(L/b)

Wn (ξ)Wm (ξ) + Z2 W2 (ξ)) + Bξ 2 + D0 (ξ)]3

(m = 1, 2). They are proportional to Z˙ 1 and Z˙ 2 ; however, the damping coefficients cn ,m depend nonlinearly on the amplitudes Z1 and Z2 . Considering the deflections along the lengths of the electrodes L1 and L2 as constant, the electrostatic forces Fi at positions x1 and x2 can be approximated by Fi = −

ε0 bLi Vi2 . 2 [ 21 (Z1 W1 (ξi ) + Z2 W2 (ξi )) + Bξi2 + D0 (ξi )]2

(7.130)

Vi is the voltage at the ith electrode. The electrostatic forces depend nonlinearly on Z1 and Z2 and may cause a pull-in effect of the beam within the capacitor

7.4 Beam accelerometers

333

regions. The analysis follows the methodology presented in Chapter 2. Indeed, the equilibrium points for applied constant voltages V1 and V2 and acceleration a are given by EIy 2 X Fj = αj j 4 Zj − βj ρSa − Fi Wj (ξi ) = 0, j = 1, 2. (7.131) L L i= 1,2 Similarly to Eq. (2.131), the equilibrium point becomes unstable if  ∂F ∂F  1

1

 ∂Z1 ∂Z2   det   ∂F ∂F  < 0. 2 2 ∂Z1 ∂Z2

(7.132)

For given voltages Eqs. (7.130)–(7.132) have to be resolved with respect to Z1 and Z2 in order to find the bifurcation points.

Static deflection With Z1 (t) = constant and Z2 (t) = constant the static equilibrium for a uniformly distributed load q = ρSa obeys, according to Eq. (7.122), the equation EIy EIy Z1 = β1 q, α22 4 Z2 = β2 q. 4 L L Therefore, the total tip deflection is given by µ ¶ 4 β1 β2 L L4 w(L) = + q = 0.1249 q. α11 α22 EIy EIy α11

(7.133)

(7.134)

Comparison with the exact solution according to the first equation in Table 3.1, w(L) = [L4 /(8EIy )]q, reveals that the error is less than 0.1%. The same can easily be proven for point loads. The example confirms the rapid convergence of the eigenmode expansion. It should be noted that even the first-eigenmode approximation gives very reasonable results. For instance, for the given example the error at the tip would be less than 1.5%. This gives rise to limit dynamic simulations by the firsteigenmode approximation.

7.4.2

Model implementation Dynamic modeling is carried out using the first eigenmode only, which, on the basis of Eq. (7.122), leads to the following equation: g(Z1 ) ˙ Z¨1 + 2δω1 Z1 + ω12 Z1 g0 η1 V12 η2 V22 = β1 a − µ ¶2 − µ ¶2 , B 2 Z1 W1 (ξ1 ) D2 B 2 Z1 W1 (ξ2 ) 1+ ξ + + ξ + D1 1 D1 2 D1 D1 2 D1 2 (7.135)

334

Accelerometers

where ω12 = α11

EIy EIy = k14 , ρSL4 ρSL4

δ = g0

µ′ b3 , ω1 ρSD13

ηi =

ε0 bW1 (ξi ) Li , ρSD12 L

i = 1, 2. (7.136)

The coefficient g(Z1 ) represents the nonlinear damping characteristic. The integral representation of the damping coefficient c11 according to (7.129) can thus be substituted by a very-well-fitting analytical function g(Z1 ): c1,1 = 2

µ′ b3 g(Z1 ), D13

(7.137)

with g0 (1 + γZ1 /D1 )3 ! √ Z 1 Ã cosh[ 12(L/b)ξ] W12 (ξ) ∼ √ dξ 1 − = · ¸3 . cosh[ 12(L/b)] 0 D0 (ξ) Bξ 2 Z1 W1 (ξ) + + D1 D1 D1 2

g(Z1 ) =

γ is a fitting parameter close to 32 , and g0 = g(0). Remarkably, the prebending reduces the squeeze damping more than could be expected from substituting the gap by an average value. The eigenfunction W1 (ξ) generally assigns more weight to the forces closer to the tip than to the cantilever-support end. Equation (7.135) represents a lumped-element approximation of the cantilever and is an representative example for spring–mass accelerometers with nonlinear damping and nonlinear electrostatic forces. To simplify the presentation it is assumed that the sensing voltage V2 can be neglected. Starting from Eq. (7.135), it is convenient to introduce the normalized deflection y=

Z1 ω12 − y0 , D1 β1 a0

(7.138)

where the acceleration is represented by a = a0 h(t). h(t) may be the unit-step function or a sinusoidal excitation etc. y0 is the static deflection caused by the prebending voltage and obeying the condition y0 = −k

2 V10 . (1 + b1 + ry0 )2

(7.139)

V10 is the properly chosen prestressing voltage. The following parameters are introduced: r=

β1 a0 W1 (ξ1 ) , D1 ω12 2

k=

ε0 W1 (ξ1 )L1 , β1 a0 ρhD12 L

b1 =

B 2 ξ . D1 1

(7.140)

335

7.4 Beam accelerometers

(a) 1.6

(b) 300 Pa

1.8

1.6

0

1.4

500 Pa

a0=+/-500g

300 Pa

a =1 g

1.4

500 Pa

1.2

lower curves for negative accelerations (towards the substrate)

1.2

0

y/a

y/a

0

1

0.8

5000 Pa

1000 Pa 0.6

5000 Pa

0.6

atmospheric pressure

0.4

0.4

0.2

0 0

1

0.8

0.2

5

10

15

20

25

30

τ

0 0

atmospheric pressure 5

10

15

20

25

30

τ

Figure 7.27 The impact of nonlinear damping on (a) small and (b) large signal step

responses.

Now, for the closed-loop beam accelerometer with τ = ω1 t one has the following equation: · ¸ 2 d2 y g dy V12 V10 + 2δ + y = h(t) − k − . dτ 2 g0 dτ (1 + b1 + r(y + y0 ))2 (1 + b1 + ry0 )2 (7.141)

The impact of nonlinear damping The impact of the nonlinear damping term g/g0 = 1/(1 + α(y + y0 ))3 in Eq. (7.141) – with α = γβ1 a0 /ω12 derived from Eq. (7.137) using (7.138) – is of general interest for all spring–mass accelerometers that are subject to squeeze damping. The qualitative response can be demonstrated most simply for an open-loop accelerometer without prebending (V10 = 0, y0 = 0). For the chosen example of a beam with L = 570 µm, b = 170 µm, h = 2.4 µm, D1 = 2.4 µm, D2 = 1.6 µm, B = 1 µm, and xD = 350 µm the damping at atmospheric pressure (µ′ = 17.2 × 10−6 Pa s) is δ ∼ = 28, indicating heavily over-damped operation. This is the working regime of the beam accelerometer according to Brandl and Kempe [2001]. As seen in Figs. 7.27(a) and (b) the impact of the damping nonlinearity on the step response is negligible for large pressure, > 1000 Pa. However, for lower pressures the effective viscosity µ′ decreases. The damping may become undercritical, causing overshoots and oscillations. Assuming the validity of Veijola’s approximation (Eq. (3.109)), the corresponding step responses for various pressure levels and accelerations are presented in Figs. 7.27(a) and (b). The damping nonlinearity becomes especially relevant at low pressures and large accelerations as shown in Fig. 7.27(b). For instance, at 300 Pa the maximal deviation for ±500g accelerations differs by around 40%. For step excitations of the beam towards the substrate the step response behaves as in a more strongly damping environment. In contrast, for excitations forcing the beam to move away from the substrate the behavior approaches

Accelerometers

1.5

1.4

Loop gain K0=10

1000 Pa

1.3

a0= 500 g

a0= 1 g

a = –500 g 0

1.2

1.1

y/a0

336

1

0.9

0.8 atmospheric pressure

0.7

0.6

0.5 0

5

10

15

20

25

τ Figure 7.28 The closed-loop step response of a prebent cantilever.

features of less damped beams. Such behavior corresponds to the intuitive expectations.

Feedback control To eliminate the quadratic dependency between the feedback force and the applied voltage, the acting voltage should be a pulse-width-modulated (PWM) or pulse-density-modulated (sigma–delta) signal with constant amplitude. In this case V12 and the correspondingly generated average force depend linearly on the measured deflection, 2 + kTE y, V12 = V10

(7.142)

where kTE is the equivalent gain of the electronic block of the transducer. Since the loop gain is defined as K0 = F (0)H(0) with H(0), the gain of the transducer, electronics, and actuator, the factor kTE , is according to Eq. (7.141), kTE =

K0 (1 + b1 + ry0 )2 . k

(7.143)

The control loop is slightly nonlinear, because, for a steady deflection, y∞ , the 2 2 terms V10 /(1 + b1 + r(y∞ + y0 ))2 and V10 /(1 + b1 + r + y0 )2 on the right-hand side of Eq. (7.141) do not completely compensate each other. The operating point of the pre-bent beam shifts. Fortunately, the impact of this nonlinearity is small, as is demonstrated in Fig. 7.28, where the step response for small and large acceleration inputs is shown for the same example as in Fig. 7.27(b) and for V10 = 2 V. Even for accelerations on the order of 500g the residual error is only around 2%. The impact of the damping nonlinearity is significantly reduced in comparison with open-loop operation.

7.5 Various other accelerometer principles

Nitride cantilever

337

Electrodes

Hinge

Proof mass Tunneling tip

Squeeze-damping perforation

Figure 7.29 A tunneling accelerometer with the proof mass embedded in the feedback

loop. Adapted from Liu and Kenny [2001].

7.5

Various other accelerometer principles

7.5.1

Tunneling accelerometers Tunneling accelerometers emerged in 1988 [Baski et al. 1988]. Since then prototypes of tunneling accelerometers and applications have repeatedly been reported [Dong et al. 2005, Hartwell et al. 1998, Liu and Kenny 2001, Rockstad et al. 1996], despite the fact that a commercial breakthrough is still pending. Tunneling accelerometers are usually spring–mass systems with one DOF. They exploit electron-tunneling transducers. A tunneling transducer consists of a very small gap between a conducting tip and a plane electrode, where the electrons can tunnel through under an applied potential difference. Since the gap is on the order of atomic distances, sharp tips or at least cotters are needed in order to have well-defined distances and large enough electrical fields between the plane electrode and the tip. The current generated by the applied voltage is gap-dependent, It = I0 e−α i

√ Φz

≃ I0 e−0.725×10

10

z

,

(7.144)

−1 where αi = 1.025 ˚ A eV−0.5 is a constant, Φ ≃ 0.5 eV is the tunneling barrier, which is typically around 0.5 eV, and I0 is proportional to the tunneling bias, VB , across the tunneling electrode gap z. If the initial tip separation is, for instance, z0 = 10 ˚ A, the change of the tunneling current is approximately ∆It ≃ −0.725×10 1 0 ∆z e , which represents a huge sensitivity of around 70%/˚ A. The tunneling transducer is strongly nonlinear. Therefore, the distance measurement is performed in closed-loop operation. Different approaches exist. Probably the simplest approach is that demonstrated in Fig. 7.29, where the proof mass is slightly biased in order to set an initial separation z0 between the tunneling tip and the counter-electrode on the proof mass. The corresponding tunneling current (on the order of nA) defines the reference position. Since the actuating force is unidirectional, the feedback loop stabilizes the deflection around this position, using the restoring force induced √ by the position setting. Because variations and setting inaccuracies of I0 e−α i Φz 0 are difficult to handle, usually the low-frequency components are not measured. The tunneling tip is carried by a stiff nitride beam that is covered in gold [Liu and Kenny 2001]. The usage of a noble metal for the tunneling tip and

338

Accelerometers

Tunneling tip Set electrodes

Proof mass

Fast beam

Feedback electrodes

Figure 7.30 A tunneling accelerometer with a tracing fast beam. Adapted from

Rockstad et al. [1996].

counter-electrodes is mandatory in order to avoid oxidation and subsequent surface changes. Generally, the erosion of the tunneling tip by the large tip fields is the most important factor still limiting the longevity of tunneling accelerometers. Tip-shaping of the nitride beam is performed by coating a low-stress nitride onto a pyramidal silicon tip, which is created in the recess of the bottom wafer by anisotropic under-etching of a small masked area, and which subsequently is etched from the back side, releasing the cantilever and hollowing out the tip. Deposition of gold onto the electrodes is done by liftoff of Cr/Pt/Au layers. The back-side etching is used also for forming lead-throughs, reducing squeeze√ −9 damping. A resolution of 20 × 10 g/ Hz and bandwidth between 5 Hz and 1.5 kHz have been achieved. A similar feedback system was used for a symmetrically suspended proof mass with four springs on a quadratic plate (e.g. Dong et al. [2005]). A more sophisticated feedback concept is illustrated in Fig. 7.30. According to Rockstad et al. [1996] the proof-mass electrodes are used for setting the initial distance and the corresponding tunneling current. However, the tracing of the proof-mass deflections is performed by a small, wide-bandwidth beam shown to the right of the proof mass. This beam allows a very fast tracing without corrupting the intrinsic sensitivity of the proof mass. The proof-mass electrodes can be √ used to add a slow feedback component to the system. A resolution of 10−7 g/ Hz or better between 10 and 200 Hz has been demonstrated. The tunneling tip is directly etched within the recess of the proof-mass wafer and coated with gold on intermediate Ti/Pt adhesion/barrier layers. Lateral tunneling tips are necessary in order to design lateral tunneling accelerometers. Hartwell et al. [1998] used a bulk-micromachined lateral silicon tip, which is Al-coated and, after oxidation, covered by an evaporated gold layer, as are also the coated combs. The generation of typical Al–Au intermetallics √ is avoided by the Al-oxidation step. The reported resolution is 20 µg/ Hz at 100 Hz. The big advantage of tunneling accelerometers is the potentially high signal level of the transducer, which reduces the impact of electronic noise. At higher frequencies, outside of the 1/f noise of the tunneling effect, resolutions close to those determined by the thermo-mechanical Brownian noise are achievable. This destines the devices for high-precision measurements such as underwater acoustic measurements. The longevity performance of present-day accelerometers

339

7.5 Other accelerometer principles

(a) (b)

Temperature sensor 2

T

H ea

te r

acceleration

Cavity

Temperature sensor 1

Bond pads

Silicon temperature sensor 2

heater

temperature sensor 1

x

Figure 7.31 (a) The design of a convective accelerometer (adapted from Luo et al. [2001]) and (b) the temperature distribution along x without (full line) and with (dashed line) acceleration.

has still to be improved in order to meet the needs of commercial high-volume applications.

7.5.2

Convective and bubble accelerometers It is a dream of many design-engineers to create an ultra-robust inertial sensor able to withstand extreme shocks and vibration without damage. This would not only eliminate many reliability problems in harsh environments but also extend the application into new areas like blast supervision. One way to solve this problem is by the elimination of acceleration proof masses and substituting them by liquid or gaseous bubbles, which change their thermal flow under the impact of inertial forces. The well-known phenomenon of convection, i.e. the transport of particles carrying thermal energy, was used in convective accelerometers emerging in the middle of the 1990s [Dao et al. 1996]. The principle of convective accelerometers is illustrated in Fig. 7.31(a) (e.g. Luo et al. [2001] or Leung et al. [1997]). A heater is placed in the center of a cavity and at least two symmetrically arranged temperature sensors are positioned on both sides of the heater. The cavity is sealed within a hermetic package – ceramic or metal – that isolates the enclosed gas from the environment and creates a more or less isothermal boundary. In the absence of lateral accelerations the heater generates a symmetric temperature profile as shown in Fig. 7.31(b), because the thermally induced convection is equal on both sides. If a lateral acceleration is applied, the inertial forces create an asymmetric convection flow, which shifts the temperature profile as indicated and, hence, creates a temperature difference between the two sensors. The same happens if the sensor is tilted and a skewed acceleration field acts on the gas. Modelling of the convection heat transfer is based on the equation of continuity, ∇¯ v = 0, the equation of momentum, ρ¯ v · ∇¯ v = −∇p + µ ∇2 v¯ + ρ¯ a, the equation of energy, ρcp v¯ · ∇T = k ∇2 T , and the equation of state, ρ = p/(RT), which

340

Accelerometers

have to be solved with corresponding boundary conditions. Here these are v¯, the velocity vector of the flow, ρ, the density of gas, p, the pressure, and a ¯, the acceleration vector. The constants are correspondingly µ, the viscosity, cp , the specific heat, k, the thermal conductivity, and R, the gas constant. Numerical solutions have been analyzed for instance by Luo et al. [2003] and have confirmed the validity of a simpler model suggested by Leung et al. [1997]. Leung et al. [1997] argued that, since the convection heat-transfer process is governed by the Grashof number Gr [Holman 1972], the latter is linearly proportional to the sensitivity, S, of the accelerometer, S = constant × Gr = constant ×

ax ρ2 l3 β ∆T , µ2

(7.145)

where l is a characteristic linear dimension, which is usually associated with the width of the heater, and β is the coefficient of expansion. As can be seen, the device sensitivity increases linearly with the temperature difference ∆T between the heater and the cavity wall, which is proportional to the heater power. Linearity of the device applies for Grashof numbers 10−2 < Gr < 103 [Luo et al. 2003]. As can be expected, increasing the gas density rapidly improves the sensitivity. The actual sensitivity depends on the gas filling, housing, and packaging volume [Billat et al. 2002], and also on the location of the temperature sensor etc., and may vary by orders of magnitude. The temperature sensors used in convective accelerometers are thermistors (thermo-resistors) or thermopiles and are usually embedded into Wheatstone bridges. It is important to guarantee a good thermal isolation between the heater and the temperature sensors as well as between the heater and cavity and the environment. The usage of porous silicon between the heater/sensors and the silicon substrate may be a promising approach [Goustouridis et al. 2007]. Since thermal convection is a relatively slow process, the reaction times of convective accelerometers are large and the corresponding bandwidths are low – on the order of some 10 Hz (e.g. Milanovic et al. [1998] and Mailly et al. [2003]). An interesting idea to extend the convection principle towards gyroscopes was demonstrated by Zhu et al. [2006]. It is based on the fact that a constant acceleration caused, for instance, by the Earth’s gravity generates two contrariwise circular convection flows within the right and left half-chambers, which, under the impact of a perpendicular rate signal, create Coriolis forces. The resulting flow asymmetries are measured by four thermo-sensors and are proportional to the applied rate signal.

Bubble accelerometers One way to speed up the reaction of convection accelerometers is the usage of gaseous bubbles created by vaporizing a liquid contained within the microchamber. In Fig. 7.32 the bubble formation within the liquid is shown. The bubble-nucleation process under applied heat is described, for instance, in Lin

7.6 From 1D to 6D accelerometers

Thermal bubble

Isothermal lines

341

Acceleration

Housing Heater

T-sensor

Liquid

Si-substrate

Unpowered – no bubble

Powered – no acceleration

Bubble under acceleration

Figure 7.32 A thermal bubble accelerometer.

et al. [1998]. The bubble is a stable and well-separated gas formation within the working liquid that appears at temperatures higher than the “activation temperature.” Under acceleration the heated bubble is forced to move in the acceleration direction because the center of gravity of the higher-density liquid is shifted against the acceleration. Therefore the sensor located in the acceleration direction is heated up more than the opposite sensor. The exact process description of this two-phase-flow problem with dominant surface tension and thermocapillary effects follows basically the same approach as for convective accelerometers and is briefly delineated in Liao et al. [2005]. Sensitivities of about 1 ◦ C/g and response times of 60 ms could be achieved. The bubble accelerometer was successfully commercialized by the company MEMSIC, which offers 2D accelerometers for 0.25g to 35g ranges and for reasonable bandwidths.

7.6

From 1D to 6D accelerometers There are two principles that can be applied in order to create multi-dimensional accelerometers:

r parallel on-chip implementation of N one-DOF-sensitive accelerometers r creation of N -DOF sensing structures consisting of fewer than N proof masses (most commonly only one proof mass). While the first principle requires a low cross-sensitivity with respect to accelerations that are not in the main sensing direction, the second needs a well-balanced and controlled cross-sensitivity. Ideally, each of the output signals should respond to the acceleration in one direction only. However, this requirement is not mandatory, because, subject to certain limitations, the components of the acceleration vector can be derived from the vector of the output signals if the cross-coupling is known.

342

Accelerometers

cantilever

centrally symmetric beams

central anchorsurrounding mass

torsion bars

quad-beam/bridge

Figure 7.33 Types of proof-mass suspensions in bulk-micromachined accelerometers.

7.6.1

1D accelerometers The broad advent of micromachined accelerometers is closely connected with the introduction of micromachined 50g airbag sensors into safety systems of cars. Substituting for the former bulky and expensive discrete sensors, they paved the way for numerous new applications like ESP, gesture recognition, and shock protection in consumer products etc. One of the first airbag sensors was the SA20 of SensoNor (Norway), which was launched after many obstacles and a lengthy industrialization period in 1992. In the middle of the 1990s it dominated the market, at least in Europe. The senor was based on a micromachined cantilever beam loaded with a glued ceramic proof mass and equipped with piezoresistors. An advanced thermoplastic package filled with silicone oil housed the mounted sensor and provided the necessary damping [Ohlckers and Jakobsen 1998]. Later, the prepared resonance accelerometer, SA30, which was intended to substitute for the SA20, could not repeat the success of the first product. Be that as it may, the most common 1D accelerometers are single-proof-mass structures with capacitive or piezoresistive/piezoelectric transducers. Some solutions were presented in Figs. 7.1 and 7.7 for in-plane capacitive acceleration measurement, in Figs. 7.2, 7.12, 7.13, and 7.25 for out-of-plane capacitive accelerometers, and in Fig. 7.3 for an angular in-plane capacitive accelerometer. More systematically, the different mass suspensions can be classified into types as is done, for instance, in Fig. 7.33 for bulk-micromachined accelerometers. Similar structures are the starting point for the creation of multidimensional accelerometers. Resonant, tunneling, and convective accelerometers are not well suited for extension into the world of multi-dimensional sensing. However, due to the simple integrability of small transducers onto the surface of the sensing structure, piezoresistive and piezoelectric accelerometers are particularly promising candidates for multi-dimensional sensing, offering the freedom to create appropriately shaped sensing structures without the severe limitations of necessary addition

343

7.6 From 1D to 6D accelerometers

(a)

(b)

Lm

LB R2 bB

R1

R3

Beam hB

R4

FS

Proof mass hm

My LB

az

x

ax

Lm

z

Figure 7.34 A 1D piezoresistive accelerometer with beam-mass structure. (a) The

piezoresistive beam-mass accelerometer. (b) The beam-mass accelerometer model.

of form-limiting and area-consuming capacitive transducers. They are especially annoying for in-plan measurement when comb-like electrodes attached to or integrated into the proof mass are needed.

Piezoresistive accelerometers The pursuit of high sensitivities and correspondingly of large proof masses has lead to some preference of bulk-micromachined acclerometers. This is especially true for the early days of MEMS accelerometers. Silicon proof masses and suspensions are particularly well suited for the integration of piezoresistors. However, the lack of suited piezoresistive or piezoelectric actuators makes it difficult to realize a seamless integrated feedback control. Hence, in contrast to capacitive accelerometers, most piezoresistive/piezoelectric accelerometers are operated in open-loop mode. Thus, reduction of a possible pendulous effect is not possible.

Cross-sensitivity One-side suspended accelerometers feature an intrinsic 3D effect, causing a significant cross-sensitivity. This effect is characteristic for BMM sensors and can be completely avoided in surface-micromachined structures with identical vertical dimensions for suspensions and proof mass. In Fig. 7.34 the model of the the BMM cantilever-beam piezoresistive accelerometer as described in Section 4.3.1 is presented [Roylance and Angell 1979]. Assuming that the distributed proof mass can be approximated by a point mass located at the center of gravity xCO G = LB + 21 Lm , zCO G = (hm − hB )/2, the loaded beam satisfies according to Chapter 3 the equation EIy wIV (x) = 0 with boundary conditions EIy w′′′ (LB ) = −FS , EIy w′′ (LB ) = −My , and w(0) = w′ (0) = 0. The acceleration in the z-direction causes a force at the beam tip FS = maz , while the moment My is generated by accelerations in the z- as well as in the x-direction: My = −maz

Lm hm − hB − max . 2 2

(7.146)

The acceleration in the x-direction creates namely a force max applied at the COG, which is transformed into a momentum by the lever arm (hm − hB )/2.

344

Accelerometers

On solving the beam equation one gets ½ · µ ¶ ¸ ¾ 2mx2 Lm hm − hB w= 3 L + a − x + a , B z x Ebh3B 2 2

(7.147)

where the inertial moment of the beam Iy = bh3B /12 was used (b is the beam width). Since the lateral stress is given by σ(z, x) = −zEw′′ (x) the stress at the upper beam surface (z = −hB /2) of the beam is µ ¶ · µ ¶ ¸ hB LB 6m Lm hm − hB σ − , = L + a − x + a . (7.148) B z x 2 2 Ebh2B 2 2 Assuming for simplicity that the Wheatstone bridge with the four p-type piezoresistors is located at the middle of the beam, the output voltage is, according to Section 2.2.2, vout =

π44 3m[az (LB + Lm ) + ax (hm − hB )] V. 2 bh2B

(7.149)

V is the supply voltage of the bridge. The sensitivity, Sz = vout /az , strongly depends on the thickness of the beam, and is corrupted by possible axial accelerations. Since the cross-coupling sensitivity Sx,z is the ratio of the signals caused by vertical and axial accelerations of the same value, one gets Sz ,x =

hm − hB . LB + Lm

(7.150)

For a beam of thickness 25 µm, and for a proof mass with Lm = 500 µm ≫ LB and thickness hm = 100 µm the cross-sensitivity of 15% is far too high for precision measurements. The cross-sensitivity Sy ,z with respect to transverse accelerations in the y-direction is Sy ,z = hB /b, because the output is, as before, determined by Eq. (7.149), provided that hB is substituted by b and vice versa. With a beam width of 150 µm the cross-sensitivity for the example considered here is 16%, which is also very high, but it can easily be reduced by splitting the beam into two beams each of width b/2. Such a splitting does not change the sensitivities with respect to vertical and axial accelerations, but reduces the response to transverse forces.

Quad-beam accelerometers In order to reduce the cross-sensitivity caused by the axial and lateral acceleration, a quad-beam accelerometer can be used [Sandmeier et al. 1987]. Instead of a single beam, two split beams carry the proof mass as illustrated in Fig. 7.35(a). The stress measurement is performed at the two ends of each beam. Under the impact of an axial acceleration, ax , the beams deform as shown in Fig. 7.35(b). The stresses at the roots of the beams at R1 and R4 exhibit the opposite sign. For instance, in the figure the stress at R1 is compressive, while at R4 it is tensile. In contrast, on applying a vertical acceleration az , the beams deform symmetrically with respect to the vertical symmetry axis of the proof mass. Thus, the correspondingly generated stress values at R1 and R4 have the

345

7.6 From 1D to 6D accelerometers

(a)

LB R1

(b)

Lm R2

R3

R1

R4 hB

ax

R2

My1

x

R3

R4

My 2

R5 bB R6

R7

R8

z

FS1 az

R1

R4

V

FS2 R2

R3

vou t

R5

R8

R6

R7

Figure 7.35 (a) The principle of the quad-beam accelerometer. (b) Beam deformation

for axial acceleration and the principle of stress compensation.

same sign. Hence, the sum R1 + R4 is sensitive with respect to az and insensitive with respect to ax . The same consideration is valid for the resistor pairs R2 –R3 , R5 –R8 , and R6 –R7 . Therefore, on adding the resistors belonging to the roots to the tips of the corresponding beams, the cross-coupling effect to a large extent cancels out. The corresponding arrangement of the piezoresistors within a Wheatstone bridge is delineated in Fig. 7.35(b) at the bottom. Typical improvements of the cross-sensitivity in comparison with single-beam accelerometers are by one order of magnitude, which for many applications is still too large. In such cases a twin-mass accelerometer offers the potential for further reduction of the axial cross-sensitivity by about one order of magnitude down to the 10−3 level [Shen et al. 1992]. Two double-beam suspended masses are coupled by a central beam carrying a Wheatstone bridge. In the case of axial acceleration the resistors at the two ends of the bridge beam experience opposite stresses, which are cancelled out by arranging the two resistors in the diametral legs of the Wheatstone bridge. Unfortunately, the elimination of the bending moment caused by a vertical acceleration at the beam tip, and the usage of four parallel beams in quadbeam and twin-mass accelerometers, reduces the sensitivity considerably. In comparison with a cantilever-beam accelerometer the bandwidth is correspondingly increased. The quad-beam accelerometer can be used also for measurement of the lateral acceleration ax . In this case the resistors with opposite stresses do not have to be added, but must be put in opposite legs of a Wheatstone bridge as illustrated in Fig. 7.36 for the resistors at the beam roots R1 , R4 , R5 , and R8 . R1 and R4 feature changes with opposite signs in the case of axial acceleration, and with the same sign in the case of vertical acceleration. The synchronous changes are cancelled out within the bridge. Of course, also the resistors at the beam tips can be used instead of the root resistors. By increasing the height of the proof mass and adapting the lateral dimensions the sensitivity with respect to axial accelerations can be optimized.

346

Accelerometers

R1

R4

V

vout

R5

R8

Figure 7.36 A Wheatstone bridge for lateral acceleration sensing.

C1

(a)

D+zC

hB

hm

x

tg

LB

C2

Lm

w’(LB) ax

z

az

(b)

C1L hB

z

hm

C1R

x

C2L

C2R

ax

az

Figure 7.37 Capacitive bulk-micromachined accelerometers. (a) A capacitive cantilever-beam accelerometer. (b) A capacitive cantilever-beam accelerometer.

The example reveals one of the basic approaches towards multi-dimensional accelerometers: balancing of mechanical sensitivities and corresponding combination of the resistor output signals in order to separate the reactions for the different sensing directions.

Capacitive accelerometers Most capacitive accelerometers are manufactured by surface micromachining. They use comb capacitors as shown in Fig. 7.7 or plate capacitances as in the case of z-accelerometers according to Fig. 7.13. The cross-sensitivity is low and depends on careful suspension design. Bulk-micromachined capacitive accelerometers feature basically the same proof-mass structures as piezoresistive acceleration sensors. In Fig. 7.37(a) the model of a capacitive cantilever-beam accelerometer is delineated. The capacitive sensing structure is less sensitive with respect to axial accelerations than the stress-driven piezoresistor. The sensing capacitances C1 and C2 are given by Z xC O G + L m 2 dx C1(2) = ε0 bm . ′ D ± [w(LB ) + (Lm /2)w (LB ) + (x − xCO G )w′ (LB )] x C O G − L 2m (7.151)

7.6 From 1D to 6D accelerometers

347

As before, xCOG = LB + Lm /2 is the position of the COG on the x-axis. Since the proof mass is assumed to be stiff, ∆z = w(LB ) + (Lm /2)w′ (LB ) is the COG shift in the z-direction under acceleration, and (x − xCO G )w′ (LB ) is the additional gap change at position x due to the tilting of the proof mass. If the length of the proof mass is not too large, so that (Lm /2)w′ (LB ) ≪ D, the tilt does not contribute to a capacitance change and the capacitance is C1(2) = ε0 bm

Lm . D ± [w(LB ) + (Lm /2)w′ (L)]

(7.152)

With the usual approximation ∆z ≪ D the capacitance changes due to vertical and axial accelerations can be easily calculated using Eq. (7.147). The corresponding cross-sensitivity is Sxz =

∆Ca x 1 (hm − hB )(LB + Lm ) 1 hm − hB = ≃ . ∆Ca z 3 (LB + Lm )2 − L2B /3 3 LB + Lm

(7.153)

This is three times less than for a piezoresistive transducer. However, most importantly, on extending the structure towards a quadbeam accelerometer as shown in Fig. 7.37(b), the cross-sensitivity theoretically becomes zero, because the COG does not shift under the impact of axial acceleration but just tilts. This suggests the split of the measurement capacitances as indicated in the figure in order to measure the difference between these subcapacitances, C1R − C1L , which is proportional to the tilt angle. By solving the two-beam equation by using the symmetry property of the quad-beam structure, it can be shown that the tip deflection and the slope at the tip of the left beam in Fig. 7.37(b) are equal to w(LB ) =

maz L3B , 2EbB h3B

w′ (LB ) = −

L3B (hm − hB ) max . Lm (LB + Lm ) EbB h3B

(7.154)

Since the slope is the tangent of the tilt angle of the stiff proof mass, the capacitance change can be derived, with the simplification 1/(D + xw′ ) ∼ = (1/D)(1 − xw′ /D), as · ¸ Z Lm 2 1 1 C1R − C1L = ε0 bm dx − D − xw′ D + xw′ 0 3 bm Lm LB (hm − hB ) ∼ . (7.155) = ε0 max 2ED2 bB h3B (LB + Lm ) On substituting the proof-mass deflection under vertical acceleration into the expression for the capacitance change, one gets the cross-sensitivity of the capacitive quad-beam accelerometer, Sxz =

∆(C1R + C1L ) hm − hB = . C1R − C1L Lm + LB

(7.156)

The cross-sensitivity is the same as for a piezoresitive cantilever-beam accelerometer (see Eq. (7.150)) and therefore very high. Thus, a capacitive quad-beam

348

Accelerometers

(a) 2

(b)

y

1 A

x

3 A

4

1 1

x z

3 3

Figure 7.38 A 3D-accelerometer with four masses according to Roedjegard et al. [2005]: (a) top view and (b) cut along A–A.

accelerometer with split capacitances is a well-suited candidate for a 2D accelerometer.

3D accelerometers with four masses If a cantilever-suspended proof mass similar to that in Fig. 7.37(a) is capacitively sensed by only the top electrode, the capacitance change is proportional to the shift of the center of gravity, bm Lm Lm ′ ∆C ∼ ∆z, ∆z = w(LB ) + w (LB ). = ε0 D2 2 The deflection depends on ax and az , ∆z = αx ax + αz az ,

(7.157)

(7.158)

where 2mLB αx = Ebh3B

µ

2L2B

3 + 3LB Lm + L2m 2

¶

,

αz =

mLB (hm − hB )(Lm + LB ). Ebh3B

If now a second proof mass is arranged along the x-axis but with the opposite direction as shown in Fig. 7.38(a) (sensing element number 3) the deflection obeys the relation ∆z3 = −αx ax + αz az with opposite sign for the ax -acceleration. Thus, from the measurement of both capacitance changes the accelerations ax and az can be extracted. Locating now altogether four identical sensing elements in a cloverleaf position, the following four equations hold: ∆z1 = αx ax + αz az , ∆z2 = αy ay + αz az ,

∆z3 = −αx ax + αz az ,

∆z4 = −αy ay + αz az .

(7.159)

They can be resolved with respect to the three acceleration components. Roedjegard et al. [2005] have shown that the coefficients αi can be chosen in order to get uniform sensitivity. Their prototype was fabricated in SOI technology with 600-µm-high proof masses and 6-µm-high beams and confirmed the directionindependent resolution.

Piezoelectric accelerometers Piezoelectric accelerometers are built similarly to piezoresistive devices. The piezoelectric transducers are located on the supporting beams measuring the induced stress. A significant drawback of piezoelectric accelerometers is their

7.6 From 1D to 6D accelerometers

349

PZT elememt

electrodes PZT elememt

proof mass

beam

A

A

top electrode

A

Cut along A-A

frame

Figure 7.39 A quad-beam accelerometer with piezoelectric sensing. Adapted from Beeby et al. [2001].

bandpass character, which does not allow them to sense DC or very-lowfrequency accelerations (see Chapter 2). Piezoelectric layers can be sputtered or sol–gel deposited. Thick-film screenprinting of PZT with layers of thickness up to 60 µm was employed by Beeby et al. [2001], who combined this technique with bulk-micromachining. In Fig. 7.39 the arrangement used by Beeby et al. is shown, which is quite typical for piezoelectric accelerometers. The four suspending beams and the pads are covered by the bottom electrode, followed by the PZT layer, and then covered by the top electrode as delineated on the right-hand side of Fig. 7.39. A very large proof mass of area 2 mm2 and height ∼ 500 µm, and the long beams LB ∼ 1 mm with height about 40 µm result in high sensitivity at a considerable bandwidth, on the order of 15–20 kHz. The reported cross-sensitivity of about 4% is on the same order as that for piezoresistive accelerometers. However, the simulated value was more than one order of magnitude better. Beeby et al. [2001] tried to demonstrate the possibility of piezoelectric actuation for generating self-test signals that are mandatory in safety critical applications. On exciting three of the four beams at resonance a discernible change of the sensor output at the fourth beam could be observed. However, in contrast to the simple self-testing in capacitive accelerometers the costs and robustness of the method are questionable. Other piezoelectric accelerometers have been realized with impressive performance parameters, but they tend to be large and bulky. Nemirovsky et al. [1996] realized an accelerometer in which the piezoelectric layer is divided into pads that are clamped between the bottom electrode (on the substrate surface) and the proof mass, which acts as the second electrode. Acceleration in the z-direction creates strain normal to the surface of the pads. Excellent bandwidths of up to 200 kHz could be achieved using, inter alia, damping optimization by filling in the spacing between the pads with polymers. The relatively large dimensions of piezoelectric accelerometers become more and more acceptable, if the proof mass is used for multiple-dimensional sensing.

350

Accelerometers

piezoresistors – axially stressed tiny beams

y-lateral accelerometer x-lateral accelerometer z-accelerometer

reference piezoresistive beams

piezoresistors

Figure 7.40 A 3D piezoresistive shock accelerometer. Adapted from Dong et al. [2008].

7.6.2

2D and 3D accelerometers Parallel implementation The parallel implementation of 1D accelerometers allows one to combine highperformance sensing structures that can be optimized for low cross-sensitivity. The cross-sensitivity can even be improved by correcting, for instance, a possible impact of the 3D effect on one accelerometer by using the measurement results of the other.

Piezoresistive 3D accelerometers A typical example of a 3D piezoresistive acclerometer with an inventive transducer design is presented in Fig. 7.40 [Dong et al. 2008]. The 4.4 mm × 2.3 mm × 1 mm sensor chip is aimed at sensing shock accelerations in the range up to 100 000g. Therefore, the 440-µm-thick proof masses for lateral sensing are suspended by strong beams of width 280 µm, leading to resonance frequencies of about 300 kHz. Sensing is performed by two tiny beams 4 µm wide and 50 µm high, which are p-doped and act as piezoresistors. On properly designing the geometry of the beams and their location, one of them is stretched under lateral acceleration, while the other one is compressed. Together with the external reference beam resistors they form a half-Wheatstone bridge, which is quite insensitive to accelerations in the z-direction. The z-accelerometer is designed as a twin-mass sensor. The deflection is sensed by four piezoresistors located at the central bridge as shown in Fig. 7.40 (below right). They have the same thickness as the tiny beams. The three sensors have similar sensitivities of about 2.2 µV/g per 5 V in the x- and y-directions and 2.6 µV/g per 5 V in the z-direction. The measured crossaxis sensitivity is less then 2.1%.

3D capacitive accelerometers As an example of a 3D capacitive accelerometer the surface-micromachined, monolithically integrated, low-g accelerometer described in Lemkin and Boser

7.6 From 1D to 6D accelerometers

351

y-axis

z-axis z-axis reference

x-axis

500 µm

© 1997 IEEE

Figure 7.41 A 3D capacitive accelerometer. Slightly adapted from Lemkin et al. [1997], with permission.

[1999] and Lemkin et al. [1997] may serve. It is manufactured using Sandia’s M3 EMS process (see Chapter 4) and consists of three on-chip sensing structures as shown in Fig. 7.41 The lateral sensing elements carry differential combs in the form of frame-based capacitors, while the z-sensor is a simple plate capacitor between the proof mass and the ground plane. In order to create a differential output, an identical, stiffly anchored reference sensor is added as shown in the middle of the figure. All sensors are embedded in Σ∆-force-feedback loops and √ show a good resolution. The noise floor of the lateral elements is 110 µg/ Hz for the x-axis, √ √ or 160 µg/ Hz for the y-direction. The z-sensor, with a noise floor of 990 µg/ Hz, is roughly one order of magnitude less accurate. The sensors and electronics occupy an area of 4 mm × 4 mm and are designed for bandwidths of about 100 Hz. The paradigm of combined capacitive single-axis accelerometers was extended towards monolithic integration using post-CMOS processing of a standard 0.35-µm 2P4M TSMC process [Tsai et al. 2008]. The structure is formed by metal wet etching and dielectric dry etching in a manner similar to the CMU process described in Chapter 4. Three metal layers are left within the etched combs and form the comb capacitances while the bottom and top metal layers constitute the electrodes of a movable plate and the fixed counter-electrode. Similar structures were commercialized and found broad application within the consumer market. Instead of a symmetrically suspended plate for sensing the z-acceleration, torsional accelerometers are used, as for instance within BoschSensortec’s SMB380 surface-micromachined, capacitive, low-g sensor for consumer applications. √ It features a small size (3 mm × 3 mm × 0.9 mm) and a noise floor of 500 µg/ Hz [von Janacek 2007]. Often companies like Bosch, Kionix, and Freescale still use two-chip solutions. Combined single-axis torsional accelerometers for 2D or 3D sensing fabricated in BMM were investigated, for instance, in Rose et al. [2003] and Lapadatu et al.

352

Accelerometers

(a)

(b)

Figure 7.42 A high-performance 3D accelerometer manufactured in a combination of

BMM and SMM. (a) The z-sensing element. (b) The lateral sensing element. From Chae et al. [2005], with permission.

[2001]. They feature good robustness, destining them for application within the automotive area. Among the large number of single-sensor-per-axis accelerometers the design of Chae et al. [2005] (based on Yazdi and Najafi [2000]) should be mentioned, because it describes one of the highest-performance 3D accelerometers. It is based on a combined SMM–BMM process that allows one to create large masses and counter-electrodes on a single wafer without the need to bond different wafers together. The structure of the z-axis sensing element is shown in Fig. 7.42(a); the x- or y-sensitive elements are presented in Fig. 7.42(b). The proof masses, together with the suspending springs for the z-accelerometer, are manufactured using BMM, while the electrodes and the suspension beams for the lateral sensing elements are produced using polysilicon SMM technology. The large but thin electrodes (2–3 µm) for the z-axis accelerometers are stiffened by lateral ribs, which are created by refilling the prepared trenches in the BMM sensing body. The sensing electrodes for the lateral sensing elements are fabricated in a similar way. The perforation of the z-sensing electrodes allows one to match the damping to the required mechanical noise. The fabricated large √ structure (7 mm × 9 mm) features an excellent noise floor on the order of 1 µg/ Hz for all three directions.

7.6 From 1D to 6D accelerometers

y

353

x

anchor

Rx4

Ry1

(attached to surrounding frame)

Ry2 cross beams

Rx2

Rx3 Rz4 Rz3 Ry3

Rx1

Ry4

Rz2

Rz1

proof mass

Figure 7.43 A single-mass 3D accelerometer with piezoresistive read out. Adapted

from Amarasinghe et al. [2006].

2D and 3D accelerometers with multi-DOF sensing elements The general approach is to use two or three different weak bending modes of the same spring–mass system which responds to acceleration in two or three directions.

Single-mass piezoresistive 3D accelerometers Piezoresistive 3D accelerometers usually employ the 3D effect described in the section “Piezoresistive accelerometers.” Therefore, thick proof-masses suspended by thin beams are preferred. In Fig. 7.43 a schematic representation of a crossbeam-suspended proof mass according to Amarasinghe et al. [2006] is shown (earlier work was carried out by Plaza et al. [1998]). The structure represents a combination of centrally symmetric beams and a surrounding mass and is fabricated in bulk-micromachining technology. The outer dimensions of the four proof-mass segments are 3 mm × 3 mm × 0.48 mm while one beam element is 10 µm thick, 80 µm wide, and 700 µm long. The beam axes in the h100i plane are orientated along the h110i and h1¯ 10i directions. For any sensing direction four piezoresistors Rx,(y ),(z );i are implemented near the beam ends as shown in Fig. 7.43. On considering the dominant stress components caused by the deformation shapes under lateral and vertical accelerations, it becomes obvious that the dominant stress component within the beams is the axial stress σ1 . Thus, the resistances of the p-type diffused piezoresistors in all beams change according

354

Accelerometers

(a)

(b) fixed comb fingers

x-springs

y-springs

anchor

anchor

x–y–z– suspension

proof mass

left leftupper upper corner of of corner proof-mass proof mass

fixed comb fingers

Figure 7.44 Schematic representations of 2D and 3D capacitive accelerometers. (a) Detail of a capacitive 2D accelerometer, adapted from Weinberg [1999]. (b) A capacitive 3D accelerometer (with the z-reference capacitor omitted).

to ∆R/R ≈ (π44 /2)σ1 . Under x-acceleration Rx,1 and Rx,4 vary with opposite signs, as do also Rx,3 and Rx,2 . Therefore, full Wheatstone bridges for both lateral sensing directions can be formed. Under the impact of z-acceleration Rz ,1 and Rz ,4 change, however, with identical sign, that is oppositely to the change of Rz ,2 and Rz ,3 . Hence, two active legs of the Wheatstone half-bridge are composed of the sums Rz ,1 + Rz ,4 and Rz ,2 + Rz ,3 , which are insensitive to deformations caused by lateral accelerations. The remaining two resistors are reference resistors outside the beam area. The 10g sensor, targeting biomedical applications, shows a cross-axis sensitivity of less than 4%, which is less than that for a similar system employing the piezoresistive effect of MOSFETs instead of piezoresistors [Takao et al. 1998]. The cross-beam concept with surrounding mass can be used also for piezoelectric sensing as proposed by Huang et al. [2003]. Eight active PZT cells are located at the ends of the beams – two on each beam, and one outside in an unstrained status. The cross-axis sensitivity is theoretically very good. Indeed, due to the large area occupied by the PZT elements, layout mismatch, which is one of the reasons for crosstalk in case of piezoresistive transducers, plays a considerably smaller role.

2D and 3D capacitive accelerometers with a single sensing element The world’s first commercial 2D, single-mass accelerometer, ADXL202, was introduced by Analog-Devices in 1999 [Weinberg 1999, Analog-Devices 1999]. It was manufactured using the proprietary iMEMS SMM process with 2-µm-thick polysilicon. The proof mass is anchored via a two-dimensional spring suspension system as shown in Fig. 7.44(a). Here only the left upper corner of the entire structure is highlighted. The suspension system is designed in order to suppress cross-axis signals by using four symmetric suspensions with multiply

7.6 From 1D to 6D accelerometers

355

folded beams. Comb-sensing is performed by the embedded comb fingers along the sides of the proof mass (only the first finger per side is shown). √ The 2g accelerometer has a noise floor of about 500 µg/ Hz, a bandwidth typically equal to 60 Hz (up to 5 kHz) and a cross-axis sensitivity of about 2%. Two-dimensional torsional capacitive accelerometers based on asymmetric mass distribution rotating about the z-axis and about one of the in-plane axes are another type of smart accelerometer (e.g. Schwarzelbach et al. [2008]). A logical step towards a 3D capacitive accelerometer is the inclusion of zmeasurement, reducing the stiffness of the single-mass suspension in the zdirection. A corresponding arrangement is shown in Fig. 7.44(b). It follows the indications given in Boser [1996]. A single proof mass is suspended by a symmetric spring system that is compliant in the x-, y-, and z-directions. The comb fingers along the sides of the outer part sense differentially the lateral deflections, while the z-deflection is sensed by the electrode deposited onto the substrate under the central part of the proof mass. Differential sensing of the z-deflection can be achieved by implementing a reference plate outside the structure as in the case of Fig. 7.41. The principle of this structure is used by Analog-Devices in its ADXL330 exploiting the same technology as for the ADXL202/210. The packaged device occupies a space of 4 mm × 4 mm × 1.45 mm and features an extremely low power consumption of 180 µA at a supply voltage of 1.8 V. For a dynamic range of ± 3g and a bandwidth less then 550 Hz (in the z-direction) the noise floor is less than √ 350 µg/ Hz. The typical cross-sensitivity amounts to 1% [Analog-Devices 2007]. In order to further decouple the lateral and vertical deflections, the inner zsensing part can be embedded in the outer, frame-like surroundings by using a z-compliant suspension system as in Fig. 7.13. A corresponding approach was demonstrated by Qu et al. [2008], who nested a torsional accelerometer operating according to the principle shown in Fig. 7.2 into a frame similar to that shown in Fig. 7.44(a). The prototype was fabricated and Xie using an improved post-CMOS process with back-side silicon etch [Qu √ was demonstrated: 12 and 14 µg/ 2007]. A surprisingly good noise floor Hz for √ lateral accelerations and 110 µg/ Hz for vertical accelerations.

7.6.3

6D accelerometers A 6D accelerometer senses three linear and three angular accelerations. In contrast, an inertial measurement unit (IMU) is commonly understood as a device measuring three linear accelerations and three angular velocities (rates). Owing to the necessity to create Coriolis forces, IMUs are more complicated and costly than multi-dimensional accelerometers. For low-to-mediumperformance applications like in biomechanics and computer games, accelerometers sensing more than three acceleration components are affordable because they have the potential for small size and low cost.

Accelerometers

(a)

(b) R’x1

R’x2

R’x3

y

R’x4

about x

x

R’z1

R’z2

y2

R ’z

3

,R

’x

3, R

’z

4

proof mass

Rx

R ’x

1

R’z3

1

R z ’ y1 R

anchor

about z

2

Rz

’ y2

4

R

R

356

R’z4

cross beams

Figure 7.45 A piezoresistive 6D accelerometer. (a) A schematic drawing of the piezoresistive 6D accelerometer. (b) Displacement of y-beams for angular acceleration about the x-axis (top) and about the z-axis (bottom). Adapted from Amarasinghe et al. [2007].

The design of piezoresistive or piezoelectric multi-dimensional accelerometers basically turns out to be the creation of a stress pattern, particular characteristics of which correspond to reactions of a singular acceleration component and can be well sensed and distinguished. The stress pattern is usually located at the surface of the suspending structure and has to be accessible in the sense that placement of piezo-sensors and electrical wiring is possible. One of the few representatives of a 6D accelerometer was developed by Amarasinghe et al. [2007]. The sensing element is shown schematically in Fig. 7.45(a), where for the sake of clarity half of the proof mass is removed. The mass is suspended by a cross-beam with four elastic members along the x- and y-axes. It is fabricated by bonding together the top and bottom parts. The 450-µm-thick, n-type, (100) bottom SOI wafer with a 10-µm-thick silicon device layer serves as the starting material for the bottom part of the sensing element. This consists of the cross-beam, carrying 20 boron-diffused piezoresistors. The top part was bonded manually using silicon resin. The piezoresistors are placed accordingly on the surface of the cross-beam in order to form six Wheatstone bridges – one for each acceleration component. The beam axes are orientated along the h110i and h1¯ 10i directions of the silicon. As in the case of the 3D accelerometer, the dominant stress is again the axial stress σ1 . In contrast to the 3D accelerometer shown in Fig. 7.43, the proof mass is symmetric with respect to all three axes and, therefore, does not feature the

7.6 From 1D to 6D accelerometers

357

3D effect. Thus, lateral acceleration no longer causes twisting, but only creates tension and compression of the two beams in the axial direction. The corresponding stress can be sensed by piezoresistors in the middle of two opposite beams; for instance, by Rx1 on the left beam and by Rx2 on the right (not denoted in Fig. 7.45(a)). Both resistances change under x-axial stress, but with opposite signs, and constitute the active legs of a Wheatstone half-bridge. The sensitivity is low, because the stress is inversely proportional to the area of the cross-section rather than to the larger ratio of the beam length divided by bh2B as is the case for bent beams. As for the 3D accelerometer, the acceleration in the z-direction is sensed with high sensitivity by the resistor pairs Rz 1 –Rz 4 and Rz 2 –Rz 3 , which constitute the z-sensitive half-bridge. The angular accelerations about the lateral axes – ϕ¨ and ψ¨ – are captured on the basis of the deformation pictures shown in Fig. 7.45(b) at the top. Under ′ ′ change and Rx2 angular acceleration about the x-axis the sensing resistances Rx1 their values in opposite directions and form, together with the resistor pair on the opposite beam, a full Wheatstone bridge. The positions of the corresponding resistors Ry′ 1 and Ry′ 2 for angular accelerations about the y-axis are shown in Fig. 7.45(a). Angular acceleration about the z-axis causes a deformation state, as is schematically highlighted in Fig. 7.45(b) at the bottom. The beams are bent in-plane. The transverse beam coordinate (x or y) exchanges place with the zcoordinate for vertical bending. Hence, on locating two piezoresistors Rz′ 1 and Rz′ 2 (hidden in the picture) or Rz′ 3 and Rz′ 4 at the same axial coordinate, they will change their values with opposite signs. A corresponding full Wheatstone bridge can be built. The experimental data confirmed the concept. The lateral sensitivities are roughly one order of magnitude less than the angular and z-sensitivities. Remarkably, the measured crosstalk was about 4%. Chapsky et al. [2007] demonstrated a macroscopic prototype of a 6D, singlemass accelerometer consisting of a cube. The cube is suspended at all eight vertexes with three springs at any time (with damping inserts) that are sensitive along the x-, y- and z-axes. Different displacement sensing principles were investigated. The macroscopic model was still far from miniaturization, but it offers a slightly different approach to a 6D accelerometer. The prototype 6D accelerometers described here may be considered as starting points for the design of single-mass capacitive accelerometers. The capacitances have to capture deflections and tilt angles.

Summary At present multi-dimensional accelerometers are still at the beginning of their development and need improvements regarding uniformity of sensitivity, offset stability over temperature, batch-fabrication, and cost. Principles of magnetic sensing and actuation, proposed early in 1994 by Abbaspour-Sani et al.

358

Accelerometers

[1994], exploitation of acceleration-induced movement of encased mercury drops and corresponding capacitance changes through the walls of the encasing cube [Shuangfeng et al. 2008] as well as usage of inclined cantilever beams, which are intrinsically sensitive in two acceleration directions (e.g. Andersson [1995]), are candidates for deeper investigation. Also optical sensing may become important as soon as the integration of cost-efficient optical sources can be solved.

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Accelerometers

Henrion, W., DiSanza, L., Ip, M., Terry, S., and Jerman, H. (1990). Wide dynamic range direct digital accelerometer, in IEEE Solid-State Sensor and Actuator Workshop, Hilton Head, SC, pp. 153–157. Holman, J. (1972). Heat Transfer. New York: McGraw Hill, Inc., 3rd edn. Huang, P. S., Ren, T. L., Lou, Q. W. et al. (2003). Design of a triaxial piezoelectric accelerometer. Integrated Ferroelectrics, 56:1115–1122. Kampen, R. V., Vellekoop, M., Sarro, P., and Wolfenbuttel, R. (1994). Application of electrostatic feedback to critical damping of an integrated silicon capacitive accelerometer. Sensors and Actuators A, 43:100–106. Kraft, M., Lewis, C., Hesketh, T., and Szymkowiak, S. (1998). A novel micromachined accelerometer capacitive interface. Sensors and Actuators A, 68:466– 473. Lang, C. and Tielert, R. (1999). A low noise accelerometer with digital PID-type controller and multibit force feedback, in Proceedings of the 25th European Solid-State Circuits Conference, Duisburg, pp. 250–253. Lapadatu, D., Habibi, S., Reppen, B., Salomonsen, G., and Kvisteroy, T. (2001). Dual-axes capacitive inclinimeter/low-g accelerometer for automotive applications. Proceedings of the IEEE, 39(4):34–37. Lemkin, M. and Boser, B. E. (1999). A three-axis micromachined accelerometer with a CMOS position-sense interface and digital offset-trim electronics. IEEE Journal of Solid-State Circuits, 34(4):456–468. Lemkin, M. A., Ortiz, M. A., Wongkomet, N., Boser, B. E., and Smith, J. H. (1997). A 3-axis surface micromachined SD accelerometer. International SolidState Circuit Conference, pp. 202–204. Leung, A. M., Jones, J., Czyzewska, E., Chen, J., and Pascal, M. (1997). Micromachined accelerometer with no proof mass, in Technical Digest of International Electron Device Meeting, pp. 899–902. Liao, K. M., Chen, R., and Chou, B. C. (2005). A novel thermal-bubble-based micromachined accelerometer, in The 13th International Conference on SolidState Sensors, Actuators and Microsystems, 2005, Transducers ’05; Digest of Technical Papers, volume 1, pp. 519–522. Lin, L., Pisano, A., and Carey, V. (1998). Thermal bubble formation on polysilicon micro resistors. ASME Journal of Heat Transfer, 120:735–742. Liu, C. H. and Kenny, T. W. (2001). A high-precision, wide-bandwidth micromachined tunneling accelerometer. Journal of Microelectromechanical Systems, 3:425–433. Lu, C., Lemkin, M., and Boser, B. E. (1995). A monolithic surface micromachined accelerometer with digital output. IEEE Journal of Solid-State Circuits, 30(12):1367–1373. Luo, X., Li, Z., Guo, Z., and Yang, Y. (2003). Study on linearity of a micromachined convective accelerometer. Microelectronic Engineering, 65:87–101. Luo, X. B., Yang, Y. J., Zheng, F., Li, Z. X., and Guo, Z. Y. (2001). An optimized micromachined convective accelerometer with no proof mass. Journal of Micromechanics and Microengineering, 11(5):504–508.

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8

Gyroscopes

The need to capture the relative rotation of a platform like a ship or an airplane led to the development of classical gyroscopes. Gyroscopes acquire angular orientations or their changes (rates) using Coriolis forces emerging within a rotating, non-inertial system of coordinates. Since the Earth rotates with a quite significant angular speed, any reference system connected to the Earth’s surface is at best only an approximation of an inertial coordinate system. Therefore, navigation-orientated applications in aerospace and marine systems need correction for the Earth’s rotation vector. However, for automotive, consumer, and biomedical applications the Earth’s surface is a sufficiently good approximation of an inertial system.

8.1

Some basic principles A classical gyroscope consists of a flying wheel, the large buoyancy of which, together with friction-compensating mechanisms, allows one to approximate its rotation as a force-free movement within an inertial space. If the platform rotates, the principle of conservation of angular momentum keeps the wheel’s motion unchanged within the inertial system, causing the angular velocity vector to change orientation within the platform system. By observing this change, the platform’s rotation angle can be derived. The same mechanism forces Foucault’s pendulum to “draw” rosettes on the floor – the platform of the non-inertial, rotating system of the Earth’s surface. The relative motion of a body within the platform’s system of coordinates can equivalently be described by using virtual entities like Coriolis forces or moments. The measurement principle of so-called “rate” gyroscopes exploits the Coriolis force as a coupling mechanism between the platform’s rotation and mass movements with respect to the platform. The Coriolis force emerges as a vector ¯ and the relative velocity of a mass product of the platform’s angular rate, Ω, ¯ ¯ × v¯rel ). It is the output of element, dm, within the platform, FCor = −2 dm(Ω a right-handed coordinate triple x–y–z with an angular rate about the x-axis, a relative velocity along the y-axis and the Coriolis force in the direction of the z-axis. Correspondingly, a Coriolis moment of a mass element is created by ¯ ×ω the relation −2(dJD Ω ¯ rel ), where JD is a matrix of dyadic inertial moments

8.1 Some basic principles

(a)

(b)

input

Corio li mome s nt

(c) Co ri for olis ce

e1 drive

365

vdrive

e3 output

e2

e1

Figure 8.1 Principles of rate measurements based on application of the Coriolis force: (a) a gyrostat, (b) a vibrating string, and (c) a vibrating bar.

explained below, and ω ¯ is the relative angular velocity of the elementary mass. In rate gyroscopes the relative velocity is kept constant and thus a Coriolis force proportional to the platform’s angular velocity is created. By measuring deflections caused by the Coriolis force, the angular velocity can be derived. Correspondingly, such gyroscopes are called “rate” gyroscopes. In contrast, with “angular” or “rate-integrating” gyroscopes one tries to determine the platform’s orientation by keeping the body in “free” motion by feedback control, not disturbing the Coriolis forces, and measuring the average (integrated) orientation of the body within the platform. Since the orientation of the body within the inertial space remains unchanged, the orientation of the body with respect to the platform reflects the platform’s orientation within the inertial space. Basically, angular and rate gyroscopes differ in terms of the underlying control and measurement paradigm, using the same mechanical construction of an oscillating or rotating mass. Most likely, the traditional spinning-wheel gyroscope as shown in Fig. 8.1(a) is one of the oldest representatives of macroscopic rate gyroscopes. A flywheel mounted within a gimbal is rotated about the drive axis, e¯1 , with a constant speed. An orthogonal angular platform rotation, Ω2 , creates a Coriolis moment about the sensing axis, e¯3 , and rotates the gimbal. The sensing moment is measured by applying a compensating torque, the value of which provides the desired information on the platform’s rotation. The solution of numerous problems like that of a “frictionless” bearing, the supply of energy to the wheel over rotating contacts, exact calibration etc. have made the creation of high-precision gyroscopes a beacon of engineering excellence. However, miniaturization met growing barriers in the form of the increasing impact of friction and corresponding difficulties in creating suitable bearings. Alternatives such as the vibrating string were investigated, which paved the way into the world of vibrating gyroscopes. No longer is a constant rotation used to generate the Coriolis force, but oscillatory movements in the form of linear or rotational proof-mass deflections. Figure 8.1(b) illustrates the Coriolis deflection of a mass oscillating in the e¯1 –¯ e3 -plane. Under the impact of a platform rotation Ω the vibrating body deviates into the orthogonal e¯2 –¯ e3 -plane. The amplitude

366

Gyroscopes

(a)

(b)

Coriolis force

FC

e1

input

sense drive mode

FC

sense mode

FC

oscillating torque

e3 drive

FC

Figure 8.2 Principles of rate measurements based on vibrating forks and shells: (a) a tuning fork and (b) a vibrating ring.

of the orthogonal mode is a measure of the applied rate Ω. Similarly, the tip of a vibrating beam is deflected in the orthogonal direction under the impact of an axial rotation as shown in Fig. 8.1(c). Neither of these principles achieved commercial success, in contrast to the tuning fork shown in Fig. 8.2(a) (e.g. Morrow [1955]). Here two tips of a “prong” vibrating in anti-phase are deflected in opposite directions in the e¯1 –¯ e2 -plane. Thus, at the common carrier beam a moment that can be measured is created. Of course, also the deflections within the sensing plane reflect the information on an applied angular rate, Ω2 , and can be captured.1 Balanced, H-shaped quartz tuning forks based on two rigidly connected forks – one mirrored with respect to the e¯1−¯ e3 -plane – were commercially very successful within the first generation of MEMS-based automotive clusters (e.g. Madni et al. [1996], Madni and Costlow [2001], Madni et al. [2003]). Shell-based micromachined gyroscopes – often called wine-glass resonator gyroscopes – could take a large part of the automotive market share as well. The principle of such ring or cylinder rate gyroscopes is demonstrated in Fig. 8.2(b) (see e.g. Fox [1988]). The ring has two in-plane resonances that are ideally at the same frequency and separated in space by 45◦ as indicated by the two drawings. On exciting one mode an out-of-plane rate signal generates Coriolis forces that transfer energy into the second mode. The corresponding deflections are measured. Ring gyroscopes are among the potential candidates for multi-axis rate sensing (see Eley et al. [2000]). Also bulk acoustic-wave gyroscopes exploiting similar mode-coupling mechanisms and resonating in the MHz region are contending with flexural gyroscopes promising high robustness and sensitivity. In the the last decade the most successful structures for building MEMS gyroscopes have turned out to be simple flat, rigid proof masses that are particularly well suited for implementation in standard surface- or bulk-micromachining technologies. Figure 8.3 illustrates two main methods of operation. The proof mass 1

In general, all gyroscopes based on two masses oscillating with same amplitude in opposite directions are now called tuning-fork gyroscopes, irrespective of whether they are fork-like or not.

8.2 Kinematics of gyroscopes

(a)

367

(b)

se

drive vibrat io

e3

ns ev

ibr a

tio n

e3

e1

n

e2

drive vibrat ion

e2

e1 sense

vibrat io

n

Figure 8.3 The principle of rigid-body MEMS gyros: (a) linear and (b) rotatory.

can be either vibrated in a translational movement as in Fig. 8.3(a), or excited into rotational oscillations as in Fig. 8.3(b), forming a vibrating-wheel gyroscope. In the first case (e.g. Bernstein et al. [1993]) the Coriolis force created by an e¯3 angular-rate component is directed along the e¯1 -axis lying in the plane of the platform. It causes the plate to vibrate additionally in the e¯1 -direction, provided that a suitable platform suspension exists. By capturing this sensing motion the angular-rate component can be calculated. A tuning-fork arrangement with two such coupled masses is most likely the commercially most successful representative of modern MEMS gyroscopes. In the case of vibrating-wheel gyroscopes a disk as shown in Fig. 8.3(b) oscillating in-plane about e¯3 is subject to a tilting Coriolis torque about the e¯1 -axis. The torque is caused by an angular-rate component in the e¯2 -direction. If an appropriate suspension supports both rotations – about the drive and sensing axes – the sensing motion can be picked out and the corresponding rate signal derived. There are a great many constructions according to the rate-sensing principle described above. Different suspensions and multiple-mass constructions may support a one- to multiple-DOF motion of the proof masses. The structures may be embedded in gimbal-like frames limiting the motion of different parts and thereby separating the driving and sensing motions. Before going into more details, an introduction to the kinematics of rigid-body movement within a non-inertial platform will be given for readers who are not skilled in this area.2

8.2

Kinematics of gyroscopes In what follows the coordinate-system model according to Fig. 8.4(a) will be used. It consists of three coordinate frames (notation as in Chapter 2) with their 2

The author has observed that faulty models of rotatory MEMS gyroscopes appear regularly in the literature, emphasizing the need for a sound understanding of the underlying methods.

368

Gyroscopes

(a)

(b)

e3=E3

E3

E2

E1

3

dm

e3 E2

E3

1

P

r R

2

r0

R0

E2 e1

e2

E1

Figure 8.4 A rigid body in different coordinate systems and rotation of a coordinate

system. (a) Radius vectors of a mass particle of a rigid body in non-inertial and inertial systems of coordinates. (b) An illustration of Bryan angles achieved by real or virtual gimbal suspension.

unit base vectors T

¯ = [E ¯1 , E ¯2 , E ¯3 ]), Σ: E

Σe : e¯T = [¯ e1 , e¯2 , e¯3 ],

Σε : ε¯T = [¯ ε1 , ε¯2 , ε¯3 ]. (8.1)

¯1 , E ¯2 , and E ¯3 represents the inertial reference The frame Σ with base vectors E system in which Newton’s law P˙ = F¯

principle of linear momentum,

¯ L˙ = M

principle of conversation of angular momentum,

(8.2)

R R ˙ ¯ ¯˙ ¯ dm and L = is valid. P = V R V R × R dm are the total momentum and the ¯ total angular momentum of the body, respectively, and F¯ and R Rexternal R R M are the force and momentum, respectively, acting on the rigid body. V dm = V dm denotes the volume integral over all mass elements of the body. Σe constitutes the platform frame, which in inertial MEMS usually is assigned to the silicon substrate. Σε is a co-moving coordinate system strapped to an arbitrary point P of a body that moves with respect to the platform. The body is the proof mass of a gyroscope or one of the moving members if the gyroscope consists of more than one. Since the gyroscope is linked with the platform, the measurement system captures the dynamic state of the constitutive parts of the gyroscope with respect to the frame Σe . ¯ can be represented in any of the coordinate frames: An arbitrary vector R ¯= R

X i

¯i = RE ,i E

X i

Re,i e¯i =

X i

Rε,i ε¯i ,

(8.3)

8.2 Kinematics of gyroscopes

369

¯=E ¯ T RE = e¯T R = ε¯T R . In what follows the or in a more compact form as R e ε subscript ε, indicating the affiliation of a coordinate vector to the body frame, will be omitted in order to simplify the presentation of the final dynamic equation. Within the different frames the three radius vectors of a selected infinitesimal body element with mass dm are, according to Fig. 8.4(a), related to each other by ¯=R ¯ 0 + r¯, R

r¯ = r¯0 + ρ¯.

(8.4)

¯ 0 and r¯0 describe the translations between the frame origins. r¯ and R ¯−R ¯ 0 are R subject to the same rotations as if ΣE and Σe had the same origin. The same is valid for ρ¯ and r¯ − r¯0 for the relative rotation of the body with respect to the platform.

8.2.1

Platform rotation and angular velocity The handling of rigid-body rotations requires a clear definition of the rotation model used in a given application. As stated in Chapter 2, the rotation of base vectors e¯ pertaining to Σe with respect to ΣE is described by the rotation matrix S: X ¯ ¯j . e¯ = SE (8.5) Sˆij E or, for the singular basis vectors, e¯i = j

The rotation matrix was derived using Bryan angles ψ1 , ψ2 , ψ3 . The first coun¯1 by ψ1 , the second about the E ¯ {2} -axis of the terclockwise rotation is about E 2 ¯ {2} -axis of the secondly rotated rotated frame by ψ2 , and the third about the E 3 coordinate system by ψ3 = ψz . Figure 8.4(b) illustrates this rotation using a real or virtual Cardan suspension with three gimbals, the first rotation being ¯1 , the second within the first about E ¯ {1} , and correspondingly the third about E 2 ¯ {2} . The sequence (formed by the drawn body itself) within the second about E 3 of transformations yields ¯ {1} = S1 (ψ1 )E ¯, E

¯ {2} = S2 (ψ2 )E ¯ {1} , E

¯ {3} = S3 (ψ3 )E ¯ {2} = e, E

(8.6)

where the partial and total transformations were given by Eqs. (2.60)–(2.62). A continuous rotation changes the basis vectors of the rotated frame in time. From Eq. (8.5) one gets the following change of Σe with respect to ΣE : ˙ T e¯ = Ω¯ ¯ = SS ˜ e. e¯˙ = S˙ E

(8.7)

˙ T + SS˙ T = 0 holds. Therefore, the elements of the Since SST = I, the relation SS T ˜ = SS ˙ ˜ ij + Ω ˜ j i = 0. Thus, Ω ˜ is a skew-symmetric matrix Ω satisfy the relation Ω matrix with zero diagonal elements and can be represented by   0 −Ωe,3 Ωe,2 ˜ = SS ˙ T =  Ωe,3 (8.8) Ω 0 −Ωe,1  . −Ωe,2 Ωe,1 0

370

Gyroscopes

Substitution into Eq. (8.7) yields e¯˙ 1 = Ωe,2 e¯3 − Ωe,3 e¯2 ,

e¯˙ 2 = Ωe,3 e¯1 − Ωe,1 e¯3 ,

e¯˙ 3 = Ωe,1 e¯2 − Ωe,2 e¯1 . (8.9)

¯ in terms of the vector Ωe On defining the (pseudo-)vector of angular velocity Ω assigned in the Σe -coordinate by ¯ = e¯T Ωe Ω

with

ΩeT = [Ωe,1 , Ωe,2 , Ωe,3 ]

(8.10)

it is easy to see (note that e¯1 × e¯2 = e¯3 etc.) that the three equations (8.9) are equivalent to ¯ × e¯i , e¯˙ i = Ω

i = 1, 2, 3,

(8.11)

which implies the relation ˙ T = Ω×, ¯ SS

(8.12)

˙ TA ¯ =Ω ¯ The extraction of an angular velocity ¯ × A. i.e. for an arbitrary vector SS T ˙ vector Ωe from SS may be a little bit lengthy. More conveniently, the coordinate vector can be derived using the physical meaning. Indeed, the angular velocity ¯ follows from the relation for infinitesimal increments, Ω ¯ {2} ¯ {1} + dψ3 E ¯1 + dψ2 E ¯ dt = dψ1 E Ω 3 2

or

¯ {2} + ψ˙ 3 e¯3 . ¯ {1} + ψ˙ 2 E ¯ = ψ˙ 1 E Ω 2 1 (8.13)

¯1 = E ¯ {1} , E ¯ {1} = In the last equation the Bryan-angle scheme was considered: E 1 2 T ¯ {2} = e¯3 . On introducing the coordinate vectors ψ˙ = [ψ˙ 1 , 0, 0], ψ˙ T = ¯ {2} , E E 3 2 1 2 T [0, ψ˙ 1 , 0], ψ˙ = [0, 0, ψ˙ 1 ] Eq. (8.13) can be transformed into a component-vector 3

representation ¯ =E ¯ {1}T ψ˙ + E ¯ {2}T ψ˙ + e¯T ψ˙ Ω 1 2 3

(8.14)

that after performing the base transformation takes the form Ωe = S3 S2 ψ˙ 1 + S3 ψ˙ 2 + ψ˙ 3 . On doing the calculation, one gets    ˙   ψ1 c2 c3 s3 0 Ωe,1 Ωe =  Ωe,2  =  −c2 s3 c3 0   ψ˙ 2  , s2 0 1 Ωe,3 ψ˙ 3

(8.15)

(8.16)

where, as in Chapter 2, the notations ci = cos ψi and si = sin ψi have been used.

Relative velocity and acceleration within the platform In order to analyze the movement of a mass element dm within the non-inertial frame Σe , the absolute movement within the inertial frame must be decomposed into a movement of the non-inertial frame and a movement of the mass element relative to this frame.

371

8.2 Kinematics of gyroscopes

¯˙ = R ¯˙ 0 + r¯˙ . The absolute velocity is the time derivative of the radius vector R P P ¯ T ¯ × r¯ the time derivaSince with Eq. (8.11) e¯˙ re = i e¯˙ i re,i = i (Ω × e¯i )re,i = Ω tive can be represented as ¯˙ = R ¯˙ 0 + e¯˙ T re + e¯T r˙ = R ¯˙ 0 + Ω ¯ × r¯ + r¯◦ , R e

(8.17)

or, considering only r¯, ◦

r¯˙ = Ω × r¯ + r¯ .

(8.18)

The first two terms in Eq. (8.17) describe the velocity of the non-inertial frame ◦ while the component r¯= e¯T r˙ e is the relative velocity of the mass element with respect to the non-inertial frame Σe as if the platform were frozen in space. The acceleration of the mass element is obtained by differentiating Eq. (8.17) using (8.18), ¨¯ = R ¨¯ + Ω ¯˙ × r¯ + Ω ¯ × (Ω ¯ × r¯) + 2Ω ¯ × r¯◦ + ◦◦r¯ . R 0

(8.19)

The first three terms compose the acceleration of the non-inertial frame a ¯Σ e , the last term is the acceleration of the mass element a ¯rel with respect to the non¯ × r¯◦ is the Coriolis acceleration force – a mixed inertial frame, and a ¯Cor = 2Ω term of the relative velocity and the angular velocity of the non-inertial frame: ¨¯ = a ¯Cor + a ¯rel , R ¯=a ¯Σ e + a

(8.20)

where ¨¯ + Ω ¯˙ × r¯ + Ω ¯ × (Ω ¯ ×¯ r), a ¯Σ e = R 0

¯ × r¯◦ , a ¯Cor = 2Ω

◦◦

a ¯rel = r¯ .

¯ × (Ω ¯ × r¯) describes the centrifugal forces acting on mass elements in the frame Ω Σe .

8.2.2

Body rotation in a non-inertial system A body rotation is most easily represented by rotation of a co-moving coordinate system Σε pertaining to an arbitrary body point P. Within this coordinate frame all mass elements are fixed so that the radius vector to any other point of the body does not depend on time, ρ¯ = constant. The relative movement of Σε with respect to the platform frame Σe is determined by a time-dependent rotation matrix T, ε¯ = T¯ e

⇒

ρε = Tρe ,

(8.21)

which defines the relative movement ˙ e = TT ˙ T ε¯ = ω ε¯˙ = T¯ ¯ × ε¯.

(8.22)

ω ¯ is the relative angular velocity given by ω ¯ × = TTT .

(8.23)

372

Gyroscopes

With D = TS the absolute movement of a body point with respect to the inertial frame is correspondingly ¯ = DE ¯ ε¯ = TSE

⇒

ρε = TSρE = DρE .

(8.24)

The total rotation D is associated with an absolute angular velocity of Σε with respect to the inertial frame: ˙ T =ω DD ˜ abs = ω ¯ abs ×

(8.25)

Since for subsequent infinitesimal rotations of Σe with respect to ΣE and of Σε with respect to Σe the rotation angles add, one can write ¯ +ω ω ¯ abs = Ω ¯.

(8.26)

The coordinate vector of the absolute angular velocity, ω abs , within the body frame is obviously the sum of the relative velocity coordinate vector ω plus the relative movement projected into Σε of the non-inertial frame with respect to ΣE : ω abs = T Ωe + ω .

(8.27)

Some authors prefer a stepwise calculation of the absolute angular velocity ω abs according to Bryan’s rotation scheme: {1}

ω abs = T1 Ω + θ˙ 1 ,

{2}

{1}

ω abs = T2 ω abs + θ˙ 2 ,

{2}

ω abs = T3 ω abs + θ˙ 3 ,

(8.28)

T T T where θ¯ T = [θ1 , θ2 , θ3 ] and θ˙ 1 = [θ˙1 , 0, 0], θ˙ 2 = [0, θ˙2 , 0], θ˙ 3 = [0, 0, θ˙1 ] are the components of the relative body rotation with respect to the platform according to Bryan’s scheme. The components of the absolute angular velocity become (using the notation ci = cos θi and si = sin θi )        ω1 Ωe,1 c3 c2 c3 s2 s1 + s3 c1 −c3 s2 c1 + s3 s1 ωabs,1  ωabs,2  =  −s3 c2 −s3 s2 s1 + c3 c1 s3 s2 c1 + c3 s1   Ωe,2  +  ω2 , ω3 Ωe,3 s2 −c2 s1 c2 c1 ωabs,3  ˙     θ1 c2 c3 s3 0 ω1  ω2  =  −c2 s3 c3 0   θ˙2  . (8.29) s2 0 1 ω3 θ˙3

For deriving the equations of motion the angular acceleration ω ¯˙ abs is needed. In light of (8.26) it becomes ◦

or

P ◦ ¯˙ + ω ˙ ¯ × (Ω ¯ +ω ¯ +ω Ω ¯˙ = i {[e¯˙ i (Ωe,i + ω e,i )] + [¯ eT ˙ e,i )]} = Ω ¯ )+ Ω ¯ i (Ωe,i + ω ◦

◦ ¯˙ + ω ¯ ×ω ¯ +ω Ω ¯˙ = Ω ¯+Ω ¯.

(8.30)

373

8.2 Kinematics of gyroscopes

◦

◦ ¯ and ω Ω ¯ are the relative angular accelerations with respect to the platform frame. The relative platform acceleration is ◦

˙ e. Ωe = TΩ

(8.31)

It describes the acceleration of the non-inertial frame with respect to ΣE .

8.2.3

The angular-momentum theorem ¯˙ × R ¯˙ = 0, ¯=R ¯ ′ + ρ¯ and R Now Newton’s law Eq. (8.2) can be applied. Since R 0 the angular-momentum conservation law is transformed by means of the first equation (8.2) into Z Z ¨ ¯ ¨ ¯ =R ¯ 0′ × F¯ + dm(¯ ρ × R), (8.32) dm R and M F¯ = V

V

¯ 0 + r¯0 is the radius vector from the ΣE -origin to the origin of the ¯′ = R where R 0 R R body frame. With the COG radius m¯ ρ = V dm ρ¯ (m = V dm) this results in Z ¨ ¯C. ¯ −R ¯ 0′ × F¯ − m¯ ¯ 0′ = dm(¯ ρ × ρ¨ ¯) = M (8.33) M ρC × R V

On locating the center of gravity at the origin of the body frame the third term ¯ C presents the binding torque to on the left-hand side vanishes. In this case M the platform plus possible external torques like these created by damping effects, both of which are related to the center of gravity of the body. ¯= ω ¯˙ abs × ρ¯ + ω ¯ abs × On applying formula (8.17) for a co-moving frame, i.e. ρ¨ (¯ ωabs × ρ¯), the equation transforms into Z ¯C = dm[¯ ρ × (ω ¯˙ abs × ρ¯) + ρ¯ × ω ¯ abs × (¯ ωabs × ρ¯)]. (8.34) M V

With the help of the following transformation rules for a double vector product, ρ¯ × (ω ¯˙ × ρ¯) = ω ¯˙ ρ¯T ρ¯ − ρ¯ρ¯T ω ¯˙ = (¯ ρT ρ¯ I − ρ¯ρ¯T )ω ¯˙ ,

ρ¯ × ω ¯ × (¯ ω × ρ¯) = ω ¯ [¯ ρT (¯ ω × ρ¯)] − (¯ ω × ρ¯)¯ ρT ω ¯=ω ¯ × (¯ ρT ρ¯ I − ρ¯ρ¯T )¯ ω, the expressions within Eq. (8.34) can be rebuilt. On introducing the tensor of inertia Z dm(¯ ρT ρ¯ I − ρ¯ ρ¯T ) (8.35) J= V

the angular-momentum equation takes the form

¯ C = Jω M ¯˙ abs + ω ¯ abs × J¯ ωabs .

(8.36)

This is the well-known Euler equation that should be adapted for the case of body movement with respect to a non-inertial platform being considered here. Into this Eqs. (8.26) and (8.30) should be substituted, yielding ◦

◦ ¯ ×ω ¯ +Ω ¯ × J¯ ¯ × JΩ ¯ + J Ω= ¯ M ¯C. Jω ¯ + [J(Ω ¯) + ω ¯ × JΩ ω] + ω ¯ × J¯ ω+Ω

(8.37)

374

Gyroscopes

The equation describes the relative motion ω ¯ (t) under the impact of platform ¯ ¯ and the applied moments are rotation and applied moments MC . The vector Ω considered as given functions of time. The term in square brackets represents the Coriolis moment and is the main factor responsible for transforming the platform rotation into an intended reaction of the body. On performing some lengthy calculations the Coriolis term can be rearranged: ◦

◦

¯ ×ω ¯ × JΩ ¯ + J Ω= ¯ M ¯C, ¯) + ω ¯ × J¯ ω+Ω Jω ¯ + 2(JD Ω

(8.38)

where the tensor JD =

Z

dm ρ¯ρ¯T

(8.39)

V

will be called the “tensor of dyadic inertial moments” because it is constituted by the dyadic product ρ¯ρ¯T = ρ¯ ⊗ ρ¯. Equation (8.38) reflects the governing forces more clearly than does the compact Euler equation (8.36) or alternative derivations based on the Lagrangian approach, both of which require quite cumbersome transformations for the absolute angular velocity. The equation has to be solved with respect to the angles [θ1 , θ2 , θ3 ] = θT , ¯. which, according to Eq. (8.29), are hidden in ω

Coordinate-vector representation It is convenient to express the tensor equation (8.38) in the coordinate frame Σε : ◦

Jω + 2(JD Ω × ω ) + ω × Jω + Ω × JΩ + JΩ= M C . ◦

The matrix of inertia within the body frame has the coefficients   2 Z ρ2 + ρ23 −ρ1 ρ2 −ρ1 ρ3 dm  −ρ1 ρ2 ρ21 + ρ23 −ρ2 ρ3  J = {Jij } = V −ρ1 ρ3 −ρ2 ρ3 ρ21 + ρ22

(8.40)

(8.41)

and the matrix of dyadic moments JD is   2 Z ρ1 ρ1 ρ2 ρ1 ρ3 (8.42) dm  ρ1 ρ2 ρ22 ρ2 ρ3  . JD = {JijD } = V 2 ρ1 ρ3 ρ2 ρ3 ρ3 R P It is easy to see that JiiD = V dm ρ2i = 21 ( j Jj j − 2Jii ) and JijD = −Jij , i 6= j. In practical gyroscopes the moving bodies are usually symmetric with respect to body axes. In this case the non-diagonal elements of the matrix of inertia and of the dyadic moment’s matrix vanish, and only the principal moments of inertia remain: Jij = Ji δij and JijD = JiD δij . Possible non-diagonal elements may be brought to zero also for non-symmetric bodies by rotating the body until the axes match with the so-called principal axes. In the following it is assumed that the frame Σε fits with the principal body axes if not explicitly stated otherwise.

8.2 Kinematics of gyroscopes

375

Equation (8.40) becomes under these assumptions ◦

J1 ω˙ 1 − 2J3D Ω3 ω2 + 2J2D Ω2 ω3 + (J2D − J3D )(ω2 ω3 + Ω2 Ω3 ) + J1 Ω1 = MC,1 , ◦

J2 ω˙ 2 + 2J3D Ω3 ω1 − 2J1D Ω1 ω3 − (J1D − J3D )(ω1 ω3 + Ω1 Ω3 ) + J2 Ω2 = MC,2 , ◦

J3 ω˙ 3 − 2J2D Ω2 ω1 + 2J1D Ω1 ω2 + (J1D − J2D )(ω1 ω2 + Ω1 Ω2 ) + J3 Ω3 = MC,3 , (8.43) where in the interest of having a catchy representation inertial moments Ji as well as dyadic moments JiD have been used. Any degree of freedom i is governed by two Coriolis moments created by the corresponding two orthogonal pairs Ωj ωk and Ωk ωj and “amplified” by the dyadic moments JjD and JkD , respectively. Any gyroscope needs at least two DOF in order to generate and to deploy an action of Coriolis forces, which makes them more complicated and challenging than simple one-DOF accelerometers.

8.2.4

The momentum equation The angular-momentum equation (8.38) or (8.40) has to be completed by the momentum equation, which follows immediately from Eqs. (8.2) and (8.19). For ρ¯c = 0 one gets ◦◦ ¯ × r¯◦ 0 + Ω ¯ × (Ω ¯ × r¯0 ) + Ω ¯˙ × r¯0 ] = F¯ + m¯ m[ r¯0 + 2Ω a.

(8.44)

¨ 0 is the acceleration of the platform within the inertial frame. The decisive a ¯ = −R ¯ × r¯◦ 0 . term for Coriolis gyroscopes is, of course, the Coriolis force F¯C = −2mΩ Transformation of Eq. (8.44) into coordinate-vector form and resolving the double vector product yields ◦ ◦◦ m[ r 0,e + 2Ωe × r¯0,e − J(Ωe )r0,e + Ω˙ e × r0,e ] = F¯e − mae ,

(8.45)

where the matrix J(Ωe ) is defined by (8.42) with Ωe,i instead of ρi . Here ae = SaE is the platform acceleration, aE , projected into the platform coordinates. F e are binding forces of the body to the platform complemented by external forces such as damping. Binding forces and torques, in principle, are gradient components of a potential U along the coordinate axes e¯ and about the {1} {2} rotation axes e¯1 , e¯2 , and e¯3 , respectively, e.g. Fi,e = −∂U /∂ri,e . In practice it is quite common to derive the forces and torques by considering the interaction between the body and the platform without constructing an exact potential function. Omitting the subscript “e” and assuming binding forces in the form of decoupled spring forces −ki ri and external damping forces −ci r˙i + NB,i including their stochastic fraction in the form of Brownian noise, and supposing further the

376

Gyroscopes

presence of applied driving forces FD,i , the components of r obey the relations m¨ r1 + c1 r˙1 + [k1 − m(Ω22 + Ω23 )]r1 = − 2mΩ2 r˙3 + 2mΩ3 r˙2

− m[Ω1 (r2 Ω2 + r3 Ω3 ) + Ω˙ 2 r3 − Ω˙ 3 r2 ]

+ FD,1 + mae,1 + NB,1 , m¨ r2 + c2 r˙2 + [k2 −

m(Ω21

m¨ r3 + c3 r˙3 + [k3 −

m(Ω21

+

Ω23 )]r2

= 2mΩ1 r˙3 − 2mΩ3 r˙1

− m[Ω2 (r1 Ω1 + r3 Ω3 ) − Ω˙ 1 r3 + Ω˙ 3 r1 + FD,2 + mae,2 + NB,2 ,

+

Ω22 )]r3

= − 2mΩ1 r˙2 + 2mΩ2 r˙1

− m[Ω3 (r1 Ω1 + r2 Ω2 ) + Ω˙ 1 r2 − Ω˙ 2 r1

+ FD,3 + mae,3 + NB,3 .

(8.46)

Since confusion with other interpretations is unlikely, for simplicity relative velocities and accelerations, hitherto labeled by circles above the symbols, are denoted using overdots. Equation (8.46) is linear in ri . The Coriolis forces are faster the larger the oscillation frequency. Among other effects, the centrifugal forces reduce the spring constants of any of the DOF, and hence shift their resonance frequencies. This effect can be used for designing frequency-dependent vibrating gyroscopes (forks), in which not a Coriolis force is used, but centrifugal forces (e.g. Jeuken et al. [1971] and Moussa and Bourquin [2006]). However, normally for Coriolis forks this effect can be 2 neglected. Indeed, the term ki /m = ωres,i represents the resonance frequency of P the ith DOF. It is shifted by the value j Ω2j − 2Ω2i , which even for high-rate signals around 2π rad/s is on the order of 1 Hz.

8.2.5

The small-angle approximation On comparing the two momentum equations (8.45) and (8.38) one sees that their structures are similar, but, in contrast to (8.45), Eq. (8.38) for rotary vibration is substantially nonlinear. For vibratory MEMS gyroscopes, however, the equation can be reduced also to a linear equation. The justification comes from the fact that rotation-based vibratory gyros usually are driven with small driving angles and even smaller sensing angles |θi | ≪ 1. This allows one to overcome the inherent nonlinearities of the governing equation (8.38). Furthermore, the ◦ vibration is set much faster than possible rate changes, Ω, which, thus, to a first approximation can be considered very small. Typical values are vibrations in the kHz region that must be compared with rates in the 1-Hz region. In a linear approximation, i.e. assuming according to Eq. (8.29) that ωi = θ˙i and Ω1 = Ωe,1 + θ3 Ωe,2 − θ2 Ωe,3 , Ω2 = −θ3 Ωe,1 + Ωe,2 + θ1 Ωe,3 , Ω3 = θ2 Ωe,1 − θ1 Ωe,2 + Ωe,3 , one gets a relation that is structurally equivalent to Eq. (8.46) and allows one to estimate the nonlinear distortion caused by large input

8.2 Kinematics of gyroscopes

377

¯ rates Ω: J1 θ¨1 + cθ,1 θ˙1 + [kθ,1 − (J2D − J3D )(Ω2e,2 − Ω2e,3 )]θ1 = − 2J D Ωe,2 θ˙3 + 2J D Ωe,3 θ˙2 −

3 D J3 )[Ωe,2 Ωe,3

2 D (J2

− + θ2 Ωe,1 Ωe,2 − θ3 Ωe,1 Ωe,3 ] ˙ ˙ − J1 (Ωe,1 + θ3 Ωe,2 − θ2 Ω˙ 3 ) + MD,1 + NB,1 ,

J2 θ¨2 + cθ,2 θ˙2 + [kθ,2 − (J1D − J3D )(Ω2e,1 − Ω2e,3 )]θ2 = 2J D Ωe,1 θ˙3 − 2J D Ωe,3 θ˙1 1

+

(J1D

−

3 D J3 )[Ωe,1 Ωe,3

− θ1 Ωe,1 Ωe,2 + θ3 Ωe,2 Ωe,3 ] ˙ ˙ − J2 (−θ3 Ωe,1 + Ωe,2 + θ1 Ω˙ 3 ) + MD,2 + NB,2 ,

J3 θ¨3 + cθ,3 θ˙3 + [kθ,3 − (J2D − J1D )(Ω2e,2 − Ω2e,1 )]θ3 = − 2J1D Ωe,1 θ˙2 + 2J2D Ωe,2 θ˙1

+ (J2D − J1D )[Ωe,1 Ωe,2 + θ1 Ωe,1 Ωe,3 − θ2 Ωe,2 Ωe,3 ] (8.47) − J3 (θ2 Ω˙ e,1 − θ1 Ω˙ e,2 + Ω˙ 3 ) + MD,3 + NB,3 .

Here the angular momentum M C was substituted by the sum of applied torques MD,i plus the decoupled spring torques (−kθ,i θi ) plus the corresponding damping moments including the Brownian noise (−cθ,i θ˙i + NB,i ). The formulas reveal some basic properties of oscillating masses that correspondingly are also valid for translation-based gyroscopes.

r Any degree of freedom is under impact of two Coriolis forces: one is usually the desired mode; the other one creates a cross-sensitivity against one of the orthogonal components of the angular velocity. r In order to eliminate small-signal cross-couplings the proof mass must be driven about one axis, say ε¯3 (drive mode), and the suspension should be very stiff in one of the orthogonal directions, say about ε¯2 , i.e. MD1 = MD2 = 0 and k2 → ∞ or θ2 = 0. In this case the sensing mode θ1 is exposed only to the Coriolis force 2J2D Ωe,2 θ˙3 , and there is no small-signal cross-coupling from Ωe,3 , but there exists a back-coupling into the drive mode equal to (−2J2D Ωe,2 θ˙1 ). r A planar structure with J3D ≪ J1D , J2D is not suitable for measuring angular rate signals in the z-direction (direction e¯3 ), because the associated Coriolis torque 2J3D Ωe,3 ω1(2) is very small. Rotation-based planar MEMS gyroscopes are therefore mainly utilized for in-plane rate measurement. A flat structure for in-plane measurements also reduces cross-coupling into the sensing mode even in the case of residual oscillation about the ε¯2 -axis. r Nonlinear distortions of the sense signal θ1 are caused by the term given as (J2D − J3D )[Ωe,2 Ωe,3 − θ1 (Ω2e,2 − Ω2e,3 ) + θ2 Ωe,1 Ωe,2 + θ3 Ωe,1 Ωe,3 ], which in practice can be estimated for a given operating range of angular velocity components Ωe,i and for maximal sensing and driving angles. If all angular components have the same operating range, the Coriolis forces have to be compared with the disturbing mixed terms (JiD − JkD )θi Ωe,1 Ωe,j , which for a low

378

Gyroscopes

Table 8.1. Performance classes of gyroscopes

Performance class Rate grade

Resolution (◦ /s) 0.1–1

Bias stability (◦ /hr) 10–1000

Tactical grade

0.01–0.1

0.1–10

Inertial grade

<0.001

0.0001–0.01

Application examples Automotive (anti-skid, rollover), consumer (camera stabilization, game control), medicine Robotics, military (munition guidance) Inertial navigation, space research

driving frequency may become non-negligible. The seemingly dominant term (J2D − J3D )Ωe,2 Ωe,3 can be neglected, because it is not within the frequency range of high-frequency vibration. Notably, large angular platform velocities change the stiffness of all DOF and, thus, may slightly shift the resonance frequencies of the modes as discussed above for the translational case. r The drive mode also experiences distortions from large angular velocities. Assuming slow changes, a feedback control can efficiently stabilize the amplitude of θ˙3 (t) (but not the frequency). Summarizing, the small-angle assumption and the practical application conditions of inertial MEMS with |Ωi | ≤ 10 give rise to a commonly used simplification for translation- and rotation-based Coriolis gyroscopes: ◦◦ ¯ × r¯◦ 0,e ] = F e + N B − mae , m[ r¯0 + 2Ω

J ω + 2(JD Ω × ω ) = M C + N B . ◦

(8.48)

If the gyroscope is composed of more than one body, the corresponding equations or their more exact counterparts, (8.46) and (8.43), respectively, have to be written for each body, whereby the external forces F e and torques M C include the interaction between the bodies.

8.3

The performance of gyroscopes In Table 8.1 rate gyroscopes are divided roughly into three performance classes that determine their possible applications. The overwhelming majority of present-day MEMS gyroscopes belongs to the rate-grade class, with steadily falling prices at present on the order of $10 for a one-axis gyroscope. Only a few companies produce tactical-grade MEMS gyroscopes. Inertial-grade gyroscopes are today still bulky constructions with prices on the order of one million dollars. However, the race to enter the tactical-grade and, ultimately, the inertial-grade performance class with MEMS gyroscopes is on.

8.3 Performance

379

For commercial products the most important performance parameters are specified in detail in data sheets. Among them are the following for the area of automotive applications, for example.

r Measurement range: typically ±(75–100)◦ /s and ±300◦ /s. r Operating and storage temperature range: the most stringent requirements are

r r

r

r r

r

r

r

r

set by automotive applications with an operating range from −40 ◦ C to 125 ◦ C and storage temperatures between −60 ◦ C and 150 ◦ C. For commercial and medical applications operating ranges with variations of ±20 ◦ C to at most ±60 ◦ C around room temperature are common. Additionally the device has to survive without degradation (without failures and without stress-changing plastic deformation of the package) the soldering process, which includes periods with peak temperatures above 250 ◦ C. Lifetime: more than 17 years and guaranteeing more than 50 000 ignition cycles. Shock survivability: in nearly all mass applications the sensor must survive a drop from about 1.25 m height onto a concrete floor (tested with a ∼(1500– 2000)g half-sine pulses of duration about 0.5 ms, in all three directions). Sensitivity/scale factor and sensitivity error. This parameter characterizes the output voltage per ◦ /s input. It is strongly determined by the electronic gain. More information on the intrinsic sensitivity is revealed by the mechanical scale factor which characterizes the transducer input per ◦ /s (for instance the mechanical deflection or the generated stress). It is often not disclosed. The sensitivity error in automotive applications is typically less than 1%–3%. Nonlinearity: global and local nonlinearity for rate-grade gyroscopes are on the order of 0.1%–0.3% and 2%–4%, respectively. Cross-axis sensitivity. This characterizes the output generated by a rate signal applied about an orthogonal axis. It should be on the order of less than 1%– 2%. Acceleration sensitivity. This is one of the most difficult requirements for automotive applications. It limits the rate output generated by 10-ms-long acceleration test pulses with amplitudes <(20–30)g to fractions of ◦ /s. It is sometimes specified as 0.1–0.2◦ /s per g. An additional, very hard requirement for automotive sensors to satisfy is often set by the stone-impinging test, which requires the distortions of the rate output to be limited to some ◦ /s while the acting acceleration spectrum may propagate by as much as more than 50 kHz. Vibration sensitivity: a vibration spectrum specified up to some thousand Hz with total acceleration around 10g should not generate an output exceeding a given bias shift and noise contribution. Resolution. This is usually defined for the specified output bandwidth. More representative is the spectral resolution, which for automotive applications is √ about 0.01–0.04◦ /s per Hz. Bandwidth: normally set by the electronics and on the order of 10–100 Hz.

380

Gyroscopes

r Angular random walk (ARW). This characterizes the standard deviation of √

r

r

r

r

the integrated output signal and is given in ◦ / Hz. Since the rate output noise is predominantly filtered white noise, the integrated output is a Wiener (random-walk or √ Brownian) process with standard deviation σ increasing with time T as σ = T × ARW. Bias. The zero-rate output (ZRO), also called offset or bias, is nominally set to a small value at room temperature. Over the temperature range of operation it should not exceed typically 0.5–2◦ /s. Including aging and other drift factors, the total bias usually may be 50%–75% larger. Bias drift. This includes the so-called run-in drift caused by the self-heating of the device after it is switched on, and all other variations over the device’s lifetime. Bias (in)stability: a fundamental measure that characterizes the best bias-drift performance under optimal averaging conditions. It is derived as the minimum of the Allen-variance curve and usually given in ◦ /hr. Standard parameters. Parameters and exploitation conditions such as the power supply, power consumption, package type, size, especially footprint, alignment accuracy, weight etc. must be added to complete a full specification.

If the gyroscope senses about more then one axis, the requirements have to include all sensing axes. The reliability requirements dictate to a large extent the system architecture, especially in safety-critical applications. The implementation of continuous self-testing and failure monitoring may become a must. Double checks are necessary in order to exclude false alarms. A typical requirement for automotive safety-critical applications is a residual error probability3 of less than 10−9 /hr. Rate-integrating gyroscopes should be specified by a similar performance parameter list with angles instead of rates to be measured. However, representative target parameters for volume applications do not yet exist.

8.4

Rate-integrating gyroscopes

8.4.1

Two-DOF gyroscopes The simplest gyro structures are suspended plates that afford a two-DOF movement as shown in Fig. 8.5. Drive actuators and sensing transducers such as driving and sensing combs (left), or driving combs and sensing plates under the disk (right), are omitted. Both structures can be operated as angular gyroscopes; 3

This is the probability of undetected failures, whereas a failure is defined on the basis of slightly relaxed performance parameters.

8.4 Rate-integrating gyroscopes

(a)

(b)

y

z

x

drive

s en se m otio

n

z

381

y sens e

drive motion

x

Figure 8.5 Principal arrangements of single-body MEMS vibratory gyroscopes with

two-DOF suspensions. (a) A translation-based gyroscope. (b) A rotation-based gyroscope.

however, for the sake of clearness the linear gyro according to Fig. 8.5(a) will be treated in detail. The COG vibrates in the x–y-plane; the elastic flexure suspension suppresses out-of-plane motions. If the platform acceleration has a frequency spectrum far below the resonance frequencies, it causes a slowly changing offset that can be filtered out. Thus, for any in-plane linear, vibrating gyroscope the governing equation becomes4 m¨ x + cx x˙ + [kx − m(Ω2y + Ω2z )] = 2mΩz y˙ − my[Ωx Ωy − Ω˙ z ] + Fx + NB ,x , m¨ y + cy y˙ + [ky − m(Ω2 + Ω2 )] = −2mΩz x˙ − mx[Ωy Ωx + Ω˙ z ] + Fy + NB ,y . x

z

(8.49)

If the z-motion of an in-plane gyroscope is not completely suppressed, according to Eq. (8.46) additional Coriolis terms of −2mΩy z˙ in the first equation and Ωx z˙ in the second must be added. They become relevant only for very-large-rate signals Ωx(y ) and bad decoupling between the in-plane and out-of-plane motion. Assuming resonance frequencies ωx2 = kx /m and ωy2 = ky /m much larger than ¯ all inertial forces the applied rate signals and supposing only slow changes of Ω, except for the Coriolis force can be neglected, resulting in Fx + nB,x , m Fy y¨ + 2δy y˙ + ωy2 y = −2Ωz x˙ + + nB,y , m

x ¨ + 2δx x˙ + ωx2 x = 2Ωz y˙ +

(8.50)

with damping coefficients 2δx = cx /m and 2δy = cy /m.

8.4.2

The principle of angular gyroscopes Gyroscopes are based on a weak coupling force – the Coriolis force. The Coriolis force in MEMS gyroscopes ranges from a few to some hundred pN. It is 4

Neglecting the impact of acceleration within the governing dynamic equations does not mean that there is no impact at all. Acceleration may change driving and sensing conditions and in this way disturb signal pick-up and feedback control.

382

Gyroscopes

only one of the numerous coupling forces existing in a real MEMS gyroscope with typical imperfections caused by the limited manufacturing accuracies. The angular gyroscope is especially well suited to demonstrating the impact of the typical cross-coupling effects. It is sensitive to nearly all imperfections, be they MEMS-technology-related effects or concerning the stability and accuracy of the electronics and control. The same effects take place in rate gyroscopes, but their weight in the final performance may be different and will be commented on separately. Turning to angular gyroscopes, it should be noted first that – to the best of the author’s knowledge – despite there having been many publications, there are not yet stable working prototypes of angular MEMS gyroscopes, let alone commercial devices. However, the interest is formidable because angular gyroscopes allow the direct determination of a rotation angle, bypassing the rate integration that is necessary in rate gyroscopes and that leads to an ARW error that increases with time. Angular gyroscopes are potential candidates for tactical- and inertial-grade devices. The idea behind the direct angle determination is to elucidate features of the body motion that can be distinguished within the platform frame and that depend on this angle. One such characteristic is the orientation axis of movement patterns generated, for instance, by the center of gravity (COG) of a plate oscillating in the x–y-plane (e.g. Fig. 8.5(a)). In order to make the pattern stable over many cycles, the x- and y-oscillations have to be performed with the same frequency. This leads to the desired equations of motion x ¨ + ω02 x = 2Ωz y, ˙ y¨ +

ω02 y

= −2Ωz x, ˙

(8.51) (8.52)

which describe coupled motions with the same properties as Foucault’s pendulum. Within the inertial coordinate system the same equation holds with Ωz = 0. Therefore, the pattern of motion is fixed within the inertial system. However, within the non-inertial frame, with the Coriolis forces acting here a platform rotation shifts the pattern in the opposite direction. The COG motion for Ωz = 0, or equivalently in the inertial frame, is given by x = x0 cos(ω0 t) + (x˙ 0 /ω0 ) sin(ω0 t) and y = y0 cos(ω0 t) + (y˙ 0 /ω0 ) sin(ω0 t). By eliminating trigonometric functions, one gets with α2 x2 + β 2 y 2 − 2γxy = 1 a pattern of COG movement in the form of an ellipse. α, β, and γ depend on the initial conditions x0 , y0 , x˙ 0 , and y˙ 0 . The main axis is rotated by an angle that is zero for γ = 0. If the line of oscillation is aligned with the Ex -axis of the inertial system and if the ellipticity is zero, then x = a cos(ω0 t + ψ0 ) and y = 0. Within the platform rotated by ϕ the transformed coordinates take the form x = a cos(ω0 t + ψ0 ) cos ϕ and y = −a cos(ω0 t + ψ0 ) sin ϕ. The zero-ellipticity mode represents the ideal angular gyroscope.

383

8.4 Rate-integrating gyroscopes

(a)

0.2

(b) 0.8

0.15

0.6

Orbit without rotation

0.1

0.4

0.05

0.2

0

0

-0.05

-0.2

-0.1

-0.4

-0.15

-0.6

-0.2 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.8 -1

Orbit with platform rotation

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 8.6 The orbital pattern of a freely vibrating COG. (a) The orbit of COG

vibration for Ωz = 0. (b) Orbital precession for Ωz 6= 0.

Orbital parameters More generally speaking, the x- and y-oscillations within the platform frame can be expressed by (see also the basic paper by Friedland and Hutton [1978]) x = aC cos ϕ − bS sin ϕ,

x˙ = −ω0 (aS cos ϕ + bC sin ϕ),

C = cos ψ,

ψ = ω0 t + ψ 0 ,

y = aC sin ϕ + bS cos ϕ,

y˙ = ω0 (−aS sin ϕ + bC cos ϕ),

(8.53)

with S = sin ψ,

which satisfy Eq. (8.52) with Ωz = 0 and explicitly reflect the orientation of the ellipse in space by the angle ϕ. With K = C cos ϕ; L = C sin ϕ; M = S cos ϕ; N = S sin ϕ two relations hold: K 2 + L2 + M 2 + N 2 = 1 and tan ϕ = L/K = N/M . On solving them, the orbital parameters become p p E + E 2 − ω02 H 2 E − E 2 − ω02 H 2 2 2 a = , b = , ω02 ω02 ω 2 xy + x˙ y˙ ω0 (xx˙ + y y) ˙ tan(2ϕ) = 2 2 0 2 , tan(2ψ) = − 2 2 . 2 2 2 ω0 (x − y ) + (x˙ − y˙ ) ω0 (x + y ) − (x˙ 2 + y˙ 2 ) (8.54) Here E = 21 [ω02 (x2 + y 2 ) + x˙ 2 + y˙ 2 ] is the normalized total energy (m = 1) of the plate, and H = xy˙ − y x˙ = (¯ r × r¯˙ )3 is the normalized angular momentum. It characterizes the degree of parallelism of the state vector and the velocity vector. The main axis b becomes zero if the angular momentum vanishes. Such a condition is highly desirable for easy detection of the orientation angle ϕ. ϕ is constant as long as Ω3 = 0. An example of an undisturbed orbit is shown in Fig. 8.6(a). If the platform starts to rotate, all four orbital parameters a, b, ϕ, and ψ become time-dependent. On substituting Eq. (8.53) into (8.52) and averaging the parameters over one oscillation period [Friedland and Hutton 1978], one gets

1

384

Gyroscopes

(a) 0.8

Orbit for frequency mismatch of 2 %

(b)

Orbit with platform rotation

0.3 0.6

0.2 0.4

0.1

0.2

0

0

-0.2

0.1

-0.4

0.2

-0.6

0.3 -0.8 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 8.7 Line precession for matching resonances and orbit in the case of frequency

mismatch. (a) Line precession for Ωz = 0.005ω0 and ideal frequency matching. (b) Orbital changes caused by frequency mismatch without any rotation.

after some transformations a simple and intuitively clear relation, a˙ = 0,

b˙ = 0,

ψ˙ = ω0 ,

ϕ˙ = −Ωz .

(8.55)

According to this approximation the long-term orbital behavior is described by a constant ellipse precessing clockwise according to the last equation in (8.55) with speed equal to angular velocity Ωz . A pattern evolution and a corresponding line precession for zero ellipticity are illustrated by Figs. 8.6(b) and 8.7(a), respectively. The basic assumption is a slow change Ωz ≪ ω0 of the angular position of the platform. In light of Eq. (8.53) the extraction of the orientation angle can be performed by low-pass filtering the synchronously demodulated signal and calculating µ ¶ LP(y sin(ω0 t + ψ0 )) ϕ = −arctan − ϕ0 LP(x sin(ω0 t + ψ0 )) µ ¶ LP(y cos(ω0 t + ψ0 )) = arctan − ϕ0 , LP(x cos(ω0 t + ψ0 ))

(8.56)

where LP means the low-pass-filtered part and ϕ0 is the initial orientation. In the case of zero ellipticity according to Eq. (8.53) with b = 0 the dependency on ψ0 cancels out. In the general case of non-zero ellipticity ψ0 is unknown and has to be derived during an initial calibration procedure whereby the phase-locked loop (PLL), generating sin (ω0 t + ψ0 ) and cos(ω0 t + ψ0 ), is synchronized. However, a quickly increasing phase difference between a voltage-controlled oscillator (VCO) and a MEMS oscillator makes such an approach rather impractical. Alternatively four signals x sin(ω0 t), x cos(ω0 t), y sin(ω0 t), and y cos(ω0 t) can be formed, and the corresponding four trigonometric equations for the low-pass parts of Eq. (8.53) can be solved – which amounts to a difficult signal-processing task. Therefore, even for signal-detection purposes zero ellipticity should be established.

8.4 Rate-integrating gyroscopes

8.4.3

385

An imperfection model Angular gyroscopes are based on freely oscillating masses, which, of course, are an idealization. Moreover, the resonance frequencies in the two DOF have to be identical. In practice, unavoidable damping, asymmetries, and cross-couplings between the two deflections cause an unstable and disturbed motion.

A linear model of a realistic gyroscope When it is applied to rate gyroscopes Eq. (8.50) is an idealization. A model of a realistic angular or rate gyroscope with two DOF must at least reflect crosscouplings between the two modes. Yet, neglecting nonlinearities in the damping and spring forces, a more comprehensive model is described by ˜ r˙ + F + N B , M¨ r + Cr˙ + Kr = 2ΩM

(8.57)

with µ ¶ µ ¶ ¶ µ x Fx NB,x , NB = , F= , y Fy NBy ¶ µ ¶ ¶ µ ¶ µ µ 0 Ωz kx kx,y cx cx,y mx mxy ˜ , Ω= . , K= , C= M= −Ωz 0 ky ,x ky cy ,x cy mxy my

r=

With respect to a higher-dimensional, multi-body extension it is written in matrix form.

r The generally non-diagonal modal mass matrix M = {mi,j = mj,i } is the mass

matrix post-multiplied by the mode-shape matrix and pre-multiplied by its transpose. It may account for asymmetries in the mode shapes of the vibrating plate, or, generally, for the mode-shape matrix of a multi-DOF system. In the case of a two-DOF structure, mx and my may also be intentionally different if the body is formed from coupled members that are differently active in the two vibration coordinates. r C = {cx,y } represents the damping matrix, including possible nonproportional damping terms, and K = {kx,y } is the stiffness matrix. Nonproportional damping means that the damping matrix is not formed by the weighted sum of the mass and spring matrix – C 6= αM + βK – as in the case of Rayleigh or “classical” damping. On the contrary, it consists of crosscoupling terms that may be caused by localized, discrete sources of damping such as asymmetric energy dissipation within support regions (anchors) or by interactions between orthogonal flow components around the moving body, especially close to the cavity walls (e.g. “up-swimming” effects). r Non-diagonal stiffness coefficients characterize anisoelasticity. They emerge if the principal axes do not perfectly coincide with the measurement coordinates x and y. Thus, with Eq. (3.44) (Chapter 3) the cross-coupling terms of the

386

Gyroscopes

spring matrix can be expressed by kx,y = ky ,x =

kx ′ − ky ′ sin(2ϕ). 2

(8.58)

kx ′ and ky ′ are the stiffness coefficients along the principal axes that are rotated by the angle ϕ with respect to the coordinate axes. As mentioned in Chapter 3, manufacturing imperfections such as imparities between the different beams, beam-width variations, and cross-sectional skewing of individual beams caused by anisotropy of the etching process are among the dominant effects. Lateral comb offset caused, for instance, by levitation forces may also contribute to coupling effects. Since the matrices M and K are symmetric, the solution of the eigenvalue problem (ω 2 I − M−1 K)¯ x = 0 delivers orthogonal modes (eigenvectors) – a property that will be extensively used in the following considerations. It must be noted that the linear two-DOF model represents even for a stiff body a reduction of a three-dimensional structure that is supported by flexures that are compliant in all three directions. Especially for in-plane suspensions with relatively long beams, the compliance in the z-direction is comparable to the in-plane compliance. The principal axes of the suspension may be orientated in space quite arbitrarily around the main axes, depending on the anisotropic character of the manufacturing tolerances and material properties. Approximation of the real movement by a two-DOF motion using, for instance, reduction techniques known from the modal analysis, is a mathematical approach that is not well suited to revealing the nature of the underlying physical effects. This is even more true for the non-proportional damping terms. Nevertheless, the model is appropriate for investigating the different gyroscopic effects caused by the cross-coupling terms. The parameters of the model can be identified by exciting the drive motion at various frequencies and measuring the sensing response or vice versa [Phani and Seshia 2004, Phani et al. 2006]. In the case of angular gyroscopes, static deflection measurements or the analysis of dynamic Lissajous patterns have been proposed as well [Painter and Shkel 2002].

A standard gyroscope model with imperfections The external forces on the right-hand side of Eq. (8.57) may include applied forces as well as unintentionally existing forces such as acceleration, centrifugal forces or feedthrough of a possible out-of-plane motion. N B is the Brownian noise. Assuming symmetric anisoelasicity, kx,y = ky ,x , and symmetric nonproportional damping, cx,y = cy ,x , as well as mx = my , the following notation is introduced: ω02 = (ωx2 + ωy2 )/2, ∆ = (ωx2 − ωy2 )/2, kx,y /m = ky ,x /m = γy ,x ωx2 = γx,y ωy2 , γx,y = kx,y /ky 6= γy ,x = kx,y /kx , Dx = cx /m, Dy = cy /m, Dx,y = cx,y /m, fx = Fx /m, fy = Fy /m, nB,x = NB,x /m, and nB,y = NB,y /m.

8.4 Rate-integrating gyroscopes

387

Now Eq. (8.57) can be rewritten in standard form: x ¨ + ω02 x − 2Ωy˙ = fx − Dx x˙ − Dx,y y˙ − ∆ x − γy ,x ωx2 y + nB,x , y¨ + ω02 y + 2Ωx˙ = fy − Dy y˙ − Dx,y x˙ + ∆ y − γx,y ωy2 x + nB,y .

(8.59)

The model is valid in equal measure for any kind of two-DOF gyroscope with singular suspension. It can be easily transformed for the case of rotatory movements by substituting the mass and the Coriolis force by the corresponding inertial moment and by the Coriolis torque, respectively.

8.4.4

Imperfection in angular gyroscopes For angular gyroscopes the right-hand side of Eq. (8.59) describes perturbations from the desired free motion. The equations can be solved using perturbation methods, leading, for fx = fy = 0 among other cases, to an equation for the averaged phase evolution [Friedland and Hutton 1978], ab (∆ cos(2ϕ) + γx,y ωy2 sin(2ϕ)) ω0 (a2 − b2 ) a2 + b2 − [(−Dx + Dy ) sin(2ϕ) + Dx,y cos(2ϕ)]. 2(a2 − b2 )

ϕ˙ = −

(8.60)

A non-ideal spring matrix The first effect to be considered is a non-ideal spring matrix. If kx 6= ky , the free oscillations feature slightly different resonance frequencies ∆ 6= 0. Such systems are called anisotropic. The frequency mismatch simulates the presence of an angular motion as shown in Fig. 8.7(b) and has to be eliminated. It prevents a stable straight-line oscillation with zero ellipticity. On the other hand, from Eq. (8.60) it follows that the impact of frequency mismatch vanishes for zero ellipticity (b = 0). Therefore, reducing the ellipticity to zero includes the compensation of frequency mismatch, which underlines the necessity of guaranteeing a line-precession mode for angular gyroscopes. The same is true for the impact of anisoelasticity terms, γx,y ωy2 x, which also prevent the system oscillating in the line-precession mode. However, since the natural frequencies for an anisoelastic but isotropic system are equal, the orientation angle of the ellipse remains stable and can be measured. According to Eq. (8.54), a reduction of the ellipticity is equivalent to an angular-momentum reduction: H = xy˙ − y x˙ ⇒ 0. Instead of the ratio b/a a convenient measure for ellipticity or quadrature can be defined as [Shkel et al. 1999] Z 1 T P = (xy˙ − y x)dt. ˙ (8.61) 2 0 For two signals x = cos(ω0 t) and y = cos(ω0 t + φ) the angular momentum is H = ˙ ˙ −(ω0 + φ/2) sin φ and accordingly the ellipticity P = −π(1 + φ/(2ω 0 )) sin φ. If the signals are in phase, the ellipticity is zero; if they are in quadrature, the

388

Gyroscopes

ellipticity is ±π for constant φ. Therefore, to have a stable zero ellipticity, φ˙ must be zero (elimination of anisotropy), and for constant φ the cross-couplinggenerated ellipticity must be brought to zero.

Non-proportional damping The second effect stems from non-proportional damping. Clearly, according to Eq. (8.59) the effect of off-diagonal damping elements δx,y 6= 0 is indistinguishable from a Coriolis-force action and, hence, cannot be eliminated by any control procedure. It has to be excluded by design. However, the dominant damping terms δx and δy have to be compensated despite the fact that, if they are equal, they are not disturbing the orientation angle (see Eq. (8.60)) but simply attenuate the signal level to zero.

8.4.5

Gyroscope control In angular gyroscopes the feedback control has to ramp the gyroscopes up to a certain energy level, and must also compensate for damping, adjust stiffness in order to match the two resonance frequencies, and eliminate anisoleasticity terms – all without disturbing the action of Coriolis forces. The control criteria are a constant energy level and minimal ellipticity – or, in other words, minimal quadrature in order to guarantee a straight-line oscillation. Different approaches have been proposed [Painter and Shkel 2002, 2003a, 2003b; Piyabongkarn and Rajamani 2002, Shkel et al. 1999], but all employ similar underlying compensation strategies. They can be described as follows. 1. Damping compensation. The feedback force FC must, of course, be proportional to (x, ˙ y) ˙ T . The proportionality factor to compensate for dissipative losses and to reach accordingly a given energy level, E0 , is the weighted difference between the actual and the desired energy level · ¸ 1 FC,x = αC E0 − (mx˙ 2 + my˙ 2 + kx x2 + ky y 2 ) x, ˙ 2 (8.62) · ¸ 1 2 2 2 2 FC,y = αC E0 − (mx˙ + my˙ + kx x + ky y ) y, ˙ 2 with αC > 0. Piyabongkarn and Rajamani [2002] have shown that such control is stable, and – with an appropriate choice of αC – allows one to match the actual steady-state energy with the desired energy level. 2. Resonance frequency tuning. A frequency-mismatch compensation for known spring constants can be performed by adding the following compensation forces: Fω ,x = αω (kx − k0 )x,

Fω ,y = αω (ky − k0 )y;

or, alternatively, Fω ,x = αω′ ∆ω x,

Fω ,y = αω′ ∆ω y,

8.5 Rate gyroscopes

389

where

p √ √ kx + ky − 2 k0 ωx + ωy ∆ω = − ω0 = . (8.63) 2 2m k0 is the target spring constant. However, for unknown spring constants, their estimation introduces noise and bias, which lead to control errors up to instabilities. The PI control proposed in Piyabongkarn and Rajamani [2002], namely ˆ ˆ F˙ω ,x = [kI (kˆx − kˆy ) + kP (k˙ x − k˙ y )]x,

ˆ ˆ F˙ ω ,y = −[kI (kˆx − kˆy ) + kP (k˙ x − k˙ y )]y,

(8.64)

resolves this problem by integrating the differences between estimated spring constants and tuning them so that they become equal to each other. kˆx(y ) and ˆ k˙ x(y ) are calculated estimates that are based on a consecutive zero-crossing measurement of x and y, for instance, kˆx = mπ 2 /∆T 2 . ∆Ti is the measured i

time interval between two zero crossings. Averaging reduces the estimation error. 3. Ellipticity or quadrature compensation. In order to compensate for the ellipticity caused by spring anisoelasticity, the control law should suppress the cross-coupling terms and, therefore, should have the form FQ,x = αQ P y,

FQ,y = −αQ P x.

(8.65)

On substituting into Eq. (8.59) one sees that steady state is reached for P = 0, i.e. for zero ellipticity. The control strategies described here can be extended towards full adaptive control (e.g. Shkel et al. [1999]). Usually measurability of all state variables x, y, x, ˙ and y˙ is assumed – a precondition that is hard to guarantee in the presence of noise and disturbances. Sometimes the angle measurement is combined with rate measurement by adding a harmonic excitation far away from resonance and measuring the Coriolis deflection at this frequency. Owing to the large distance to resonance, the rate sensitivity is intrinsically low. Despite the big advantage of direct angular measurement, rate-integrating gyroscopes have still to demonstrate their robustness and sensitivity in practical applications.

8.5

Rate gyroscopes MEMS-based rate gyroscopes are very successful on the market. They consist of two functional blocks: an actuator creating a velocity field that is necessary in order to generate a Coriolis force, and an accelerometer that is sensing the generated deflection. Both are realized by resonators oscillating at or

Gyroscopes

drive

drive spring

mS y

drive comb

390

mD

sense spring

x motor-sense combs

sense direction

Figure 8.8 The typical structure of a z-gyroscope.

near the corresponding resonance frequencies. An applied rate signal transfers energy from the drive mode into the sensing mode, where the transfer effect is measured. If only one component of the angular rate vector has to be captured, structures with two DOF can be used. Instead of single-mass architectures with two-DOF suspensions that, due to the anisoelasticity effects, are intrinsically prone to coupling between the modes, decoupled two-mass systems are preferred. An example of a z-gyroscope is shown in Fig. 8.8. The usage of two 1D suspensions that are isolated from each other instead of a common 2D suspension demonstrates how driving and sensing motions can be decoupled within a two-body arrangement. Here the suspensions are separated by an outer, stiff frame with mass mD that comprises the outer drive and the inner sensing springs. The two 1D suspensions are compliant in orthogonal directions. The drive suspension is stiff in the y-direction, inhibiting movement of the drive frame in the sensing direction. The sensing springs are stiff in the drive direction, forcing the two bodies commonly to deflect in the drive direction. Thus, the drive motion is decoupled from the sensing deflection and, more importantly, orthogonal drive-force components in the y-direction, caused by mismatch or cross-coupling, are significantly absorbed by the drive suspension, which is stiff in this direction. Also the action of the Coriolis force on the drive frame is suppressed. Properly designed, such frame systems allow one to implement a desired mode decoupling. They will be discussed in a bit more detail in the next section. Owing to the drive deflection being orders of magnitude larger, the diminutive changes of the sensing capacitances may be subject to varying fringe fields at the end of the sensing boxes. Their impact can be suppressed by side shields as presented in Chapter 2, in the section “Frame-based capacitors” (the shields are not shown in Fig. 8.8). The chosen example is well suited to analyze the impact of the drive motion on the sensing output. It represents also a basic structure for some commercially successful products.

391

8.5 Rate gyroscopes

The following analysis remains valid for single-mass structures without separating frames (mD = 0). In all cases, comb-drive fingers are attached to the frame. A fraction of the comb drive – the motor-sensing combs – is normally used for a separate measurement of the driving deflections, which is necessary in order to build up a feedback control.

Dynamic equations and transfer functions of a two-DOF gyroscope The dynamic equations of the two-mass system are derived assuming that the drive and sensing masses have identical x-coordinates – xD = xS = x – and the drive frame cannot move in the y-direction – yD = 0. Assuming small enough rate signals, Eq. (8.46) becomes, analogously to Eq. (8.49), mS FD ΩZ y˙ + + nB,x , mD + mS mD + mS FS y¨ + 2δS ωS y˙ + ωS2 y = −2ΩZ x˙ + + nB,y , mS

x ¨ + 2δD ωD x˙ + ωD2 x = 2

(8.66)

with p ωD2 = kD /(mD + mS ) and ωS2 = kS /mS the resonance frequencies, and δD = √ cx /[2 kD (mD + mS )] and δS = cy /(2 kS mS ) the damping coefficients. The first equation in (8.66) describes the common deflection of the drive frame and the sensing mass; the second equation is for the Coriolis deflection of the sensing mass that can be compensated for by a possible feedback force FS . The y-movement in a gyroscope according to Fig. 8.8 is measured using sensing boxes or, alternatively, by comb fingers attached to the right and left of the inner proof mass. The decoupling reduces the small backward action of the Coriolis force generated by the sensing deflection by a factor mS /(mS + mD ). For convenience, the transfer functions of the drive and sensing resonators are introduced: 1 FD (jω) = 2 = |FD (jω)|e−j ϕ D , 2 ωD − ω + 2jδD ωD ω (8.67) 1 −j ϕ S FS (jω) = 2 , = |F (jω)|e S ωS − ω 2 + 2jδS ωS ω where 1

|FD (jω)| = p 2 , (ωD − ω 2 )2 + 4δD2 ωD2 ω 2 1 |FS (jω)| = p 2 , 2 (ωS − ω )2 + 4δS2 ωS2 ω 2

8.5.1

µ

¶ 2δD ωD ω ϕD = arctan , ωD2 − ω 2 µ ¶ 2δS ωS ω ϕS = arctan . ωS2 − ω 2

System architecture In a vibratory rate gyroscope the structure is driven by a harmonic force FD = F0 cos(ωF t). Matching of ωF with the drive resonance and a stable drive amplitude can be achieved by various types of feedback-control system. The drive resonator can be excited just like any mechanical oscillator by feeding back

392

Gyroscopes

amplitude URef control

D(s)

+

x0GD cos x0 cos(

Ft –

Ft –

D–

D)

– cos(

Ft –

sin(

D

H(s) Ft –

’Ref ’Ref

F(s)

F0cos

Ft

VCO

PLL

Drive & sense resonators

y0sin(

UAGC

Sense interface

Ft –

y0G0sin

D–

Ft –

sin(

rate demodulator

S)

D

–

Ft – Ref)

S–

cos(

HI(s)

rate output

Ft – Ref)

HQ(s)

HFB(s)

feedback Figure 8.9 The standard architecture of a rate-gyroscope system.

the amplified output. The amplitude of such a self-oscillating structure will saturate at a level defined by the inherent nonlinearities. Thus, a well-controlled amplitude requires an amplitude-control loop that sets the feedback gain in accordance with the difference between the actual and the desired amplitude. The demodulation of the sensing signal can be carried out by a simple amplitude detection. Other approaches to drive control have been proposed, for instance, on the basic of a nonlinear loop with force feedback that is proportional not to the deflection but to the velocity x(t) ˙ [M’Closkey et al. 2001, Sung et al. 2009]. Since the Coriolis force is proportional to x˙ rather than to x, the stabilization of x˙ seems to be quite natural. Most of the existing systems use sophisticated control systems based on a phase-locked loop (PLL) embedded in an amplitude-control loop in order to perform a powerful, noise-optimal synchro-demodulation. The standard architecture of such a rate-gyroscope system is sketched in Fig. 8.9. The sensor-interface block represents different versions of interfaces described in Chapter 6, for instance, capacitive interfaces with unmodulated or modulated charge sensing and with asymmetric or symmetric input according to Figs. 6.19(a) and 6.19(b). The PLL and amplitude-control blocks, drive resonator, associated “motor”drive/sensing capacitances, and the electrical interface form the primary loop. The latter includes also the electronic driving stages and the drive actuators denoted in Fig. 8.9 by D(s). The generation of all necessary carrier signals cos(ωF t − φi ) is assigned to the voltage-controlled oscillator (VCO). In

8.5 Rate gyroscopes

393

practice that operation is often carried out as a digital signal generator controlled by a clock that is synchronized by the incoming drive resonator signal. All electrical blocks such as the linear filters F (s), FD (s), HI (s) etc. may exhibit temperature- and long-time-dependent gain and phase shifts. In the case of digital implementations the corresponding time delays must be accounted for. The amplitude reference URef may be well stabilized or intentionally controlled by temperature changes in order to compensate, for instance, for a temperaturedependent transducer gain. The rate-signal output is derived within the rate-demodulator block. Optionally, a feedback to the sensing resonator may be implemented, forming the base of a secondary loop. The block HFB (s) may contain up-modulators if a modulated feedback force at the driving frequency is used. It may also transform the measured drive signal into a contribution to the feedback force (this connection is not shown in Fig. 8.9). Different approaches exist (e.g. Batur et al. [2006], Dong et al. [2007], Fei and Batur [2009], and Park and Horowitz [2003, 2004]), most of them on a theoretical level. In the overwhelming majority of commercial products open-loop demodulation is used, sometimes complemented by a quadrature-compensation loop as described later.

The drive resonator Since according to Eq. (8.66) y˙ is proportional to ΩZ , the Coriolis term in the first equation is proportional to Ω2Z and – for a first analysis – can be omitted. In the case of negligible Brownian noise the corresponding deflection is then given by x = x0 cos(ωF t − ϕDF ),

ϕDF = ϕD (ωF )

(8.68)

with x0 =

F0 |FD (jωF )|. mD + mS

The maximal deflection is reached at resonance, where ϕDF =

π 2

and

x0,m ax =

F0 QD . mD + mS ωD2

(8.69)

QD ∼ = 1/(2δD ) is the quality factor of the drive resonator. Low damping helps to save driving force and thus area and mass that are needed for implementing driving capacitances and/or voltages. In the light of the spare actuating forces, vibratory gyros are usually driven at resonance with well-controlled amplitude in order to exclude the impact of variations in QD and ωD . This task is performed by the primary loop according to Fig. 8.9. The typical drive deflections are on the order of a few micrometers to 20 µm.

Sensing The Coriolis movement y(t) represents the reaction of the sensing resonator to an amplitude-modulated carrier −2Ωz x˙ = 2Ωz x0 ωF sin(ωF t − ϕDF ). According

Gyroscopes

(a)

(b)

Resonator transfer characteristic 1

0.9

F sin(

Ft –

0.8

sin(

2 x bandwidth

0.7

Ft – –

0.6

0.5

Sense resonator

Amplifier

Low-pass filter HI(s)

z(t)

Frequency split

0.4

0.3

Rate-modulated carrier

0.2

2x0

394

FS(j t)

GB(j t)

0.1

0 0.6

0.7

0.8

0.9

1

1.1

1.2

1. 4

1.3

Figure 8.10 Rate-signal detection in vibratory gyroscopes. (a) The typical signal path for a vibratory gyroscope. (b) Non-resonant sensing detection.

to Eq. (8.66), one gets for a constant-rate signal and FS = 0 (no feedback) y = 2x0 ωF Ωz |FS (jωF )| sin(ωF t − ϕDF − ϕSF ),

with ϕSF = ϕS (jωF ). (8.70)

It is well known from communication theory that in the presence of Gaussian noise an optimal amplitude demodulation has to be carried out by synchrodemodulation, i.e. by multiplication by sin(ωF t − ϕDF − ϕSF ), suppression of the second harmonic, and subsequent Wiener filtering. In Fig. 8.9 the ratedemodulator block carries out these operations. Of course, in the case of moderate resolution requirements, suboptimal, bruteforce demodulation techniques such as signal squaring and low-pass filtering can be used. Also quite complicated adaptive-input estimation methods can be applied [Feng et al. 2007]. On estimating the low-pass part of the demodulated signal, one gets simply LP y = x0 ωF Ωz |FS (jωF )|.

(8.71)

The proportionality factor x0 ωF |FS (jωF )| is called the mechanical sensitivity or mechanical scale factor. If the difference between the driving frequency and the sensing resonance is denoted by △ = ωS − ωF , where (|△| ≪ ωS ), the rate deflection in degrees per second can be expressed as π x0 p yΩ /(◦ /s) = . (8.72) 360 △2 + δS2 ωS2

yΩ /(◦ /s) characterizes the deflection for a rate signal of 1◦ /s.

The output rate spectrum For a time-varying rate signal the signal path should be considered in more detail. The typical input processing for the rate-modulated carrier is once more shown in Fig. 8.10(a). The Coriolis signal passes the sensing resonator followed by the transducer and input amplifier stages with total transfer characteristic GB (jω). The amplifier stages are sometimes designed with a bandpass characteristic in order to exploit the the narrow-band character of the modulated carrier. The

395

8.5 Rate gyroscopes

synchro-demodulator is implemented as a multiplier fed with the reference carrier sin(ωF t − γref ). The phase angle γref compensates for the total phase shift ϕDF + ϕSF plus the additional phase shift arising within the transducer and amplifier stages. Subsequent stages HI (s) eliminate the second harmonic and a possibly present residual carrier. The low-pass-filter stages are designed to implement the desired overall transfer characteristic. R If the rate-signal spectrum is denoted5 by SΩ Z (jΩ) = dt ΩZ (t)e−j Ωt and the total transfer function FS (jω)GB (jω) by FS′ (jω), then the transfer characteristic H(jΩ) between the rate signal and the low-pass fraction of the synchrodemodulator output spectrum, ZΩ Z (jΩ), can be derived as ZΩ Z (jΩ) = H(jΩ)SΩ Z (jΩ)

(8.73)

with H(jΩ) =

1 x0 ωF [e−j (γ r e f −ϕ D F ) FS′ (j(Ω − ωF )) + ej (γ r e f −ϕ D F ) FS′ (j(Ω + ωF ))]. 2

Considering GB (jω) = G∗B (−jω) and taking into account the narrow-band character of the signal, the transfer function of the amplifier stages within the frequency intervals around ω = ±ωF can be approximated by GB (j(Ω − ωF )) = G0 ej γ B ,

GB (j(Ω + ωF )) = G0 e−j γ B

for ω ∈ [±ωF ± |Ωm ax |],

(8.74)

where Ωm ax is the maximum frequency of the rate-signal spectrum. This condition has to be guaranteed by design with high accuracy. Substitution into Eq. (8.73) yields H(jΩ) =

8.5.2

1 x0 ωF G0 [e−j (γ r e f −ϕ D F −γ B ) FS (j(Ω − ωF )) 2 + ej (γ r e f −ϕ D F −γ B ) FS (j(Ω + ωF ))].

(8.75)

Resonance sensing In the case of matched modes the two resonance frequencies should be identical and equal to the driving frequency, ωF = ωD = ωS .

(8.76)

Therefore, with ∆ωS = δS ωS the bandwidth of the sensing resonator, yΩ /(◦ /s) =

π x0 QS π x0 = , 90 ωS 180 ∆ωS

ϕS,F = ϕS,res =

π . 2

(8.77)

It should be noted that, with ϕS,res = π/2, ϕDF = ϕD,res = π/2 and Eqs. (8.68)–(8.70) the x- and y-deflections are in phase: x = x0 cos(ωF t − π/2) and 5

In order to emphasize the low-frequency character of the rate signal, the frequencies are denoted not by ω but by Ω. They should not be confused with the rate signal itself.

396

Gyroscopes

y = 2x0 ωF Ωz |FS (jωF )| cos(ωF t − π/2). Noteworthy, in the case of resonance sensing, is that the driving force is subject to three −90◦ phase shifts: the first by the resonating drive shifting the oscillating force by −90◦ ; the second by the Coriolis-force generation shifting the drive oscillation again by −90◦ ; and the third by the resonating sensing element creating the last phase shift of −90◦ . Looking at Eq. (8.77) reveals that, in order to resolve rate signals of 0.1◦ /s, deflections of ∼28 pm must be measured if a bandwidth of 50 Hz and a drive deflection of 5 µm are assumed. The sensitivity is larger the higher QS is. However, unavoidable variations of the resonator’s quality factor, QS , and, more importantly, of the matching conditions, caused by temperature changes and aging, affect the sensitivity. Consequently, mode-matched gyroscopes are often designed with relative large resonator bandwidths. For instance, for a 10-kHz gyroscope with Q-factors of 100 (bandwidth ∼ 50 Hz) the relative frequency tolerances should be on the order of less than a dozen Hz (less than ∼0.15%). This is still within the limits of modern MEMS technologies. Automatic mode matching by electrostatic trimming of the sensing resonator using the spring-softening effect can mitigate the technological matching requirements. It allows one to realize high-Q resonant gyroscopes as demonstrated, for instance, for prototypes of high-Q (tuning-fork) gyroscopes (e.g. Sharma et al. [2007]). Another problem associated with matched-mode gyroscopes with high Qfactors is the small transfer bandwidth for the rate signal. Assuming a rate spectrum with frequencies much smaller than the resonance frequencies, |Ωm ax | ≪ ωF , the transfer characteristic can be approximated according to Eqs. (8.75) and (8.67) by µ ¶ e−j ψ Ω Hres (jΩ) = x0 G0 sin(γref − ϕ − γB ) p 2 , ψ = arctan . ∆ωS ∆ωS + Ω2 (8.78) With optimal phase setting γref = ϕD,res + ϕS,res + γB the equation transforms into Hres (jΩ) =

x0 G0 QS e−j ψ p . ωS 1 + (Ω/∆ωS )2

(8.79)

Thus, the open-loop bandwidth is given by the bandwidth of the sensing resonator. The gain–bandwidth product does not depend on the bandwidth of the sensing resonator, thereby opening a gap between the sensitivity and the bandwidth as was also observed in the case of accelerometers. Theoretically, the bandwidth of high-Q, mode-matched gyroscopes can be enlarged by closed-loop operation. However, closed-loop operation requires meticulous phase control of all of the blocks involved, which is difficult to realize. Commercially available z-gyroscopes are based on the low-Q approach. Impressive performance parameters of a mass-producible, commercially available zgyroscope were reported in Geen et al. [2002] (zero-rate output stability characterized by 50◦ /hr Allen variance, resonance frequencies of 15 KHz, a Q-factor

397

8.5 Rate gyroscopes

(a)

(b)

Sense frame

1

Total Coriolis force

QD,i= 400;

sense

......

Drive N

Drive 2

Drive 1

0.8

Frequency spacing: 30% of sense bandwidth

Sense resonator

0.6

0.4

Drive resonators 0.2

0 0.99

0.992

0.994

0.996

drive

0.998

1

1.002

1.004

1.006

1.008

1.01

Drive frequency fF/fS

Figure 8.11 The principle of multiple drives for resonant sensing. (a) The principle of a multi-DOF drive. (b) The total Coriolis force of a multi-DOF drive.

of 45, operation at atmospheric pressure, large driving voltages requiring charge pumps with off-chip, external components).

Multi-DOF drives for resonant sensing The requirements for exact mode matching can be slightly reduced by using multiple drives with somewhat different resonance frequencies [Ancar and Shkel 2003]. A possible arrangement is shown in Fig. 8.11(a). N drive masses are attached to the sensing frame that is fixed in the drive direction. The resonance frequencies ωD,i are spaced equidistantly and chosen in order to overlap the expected sensing resonance. Drive-resonance setting with high relative accuracy is possible using identical drive suspensions and slightly different drive masses. The local variation of spring constants is very small despite any global manufacturing tolerances. Relative variation of masses can be performed also with high accuracy. With Eq. (8.68) the Coriolis force generated by the ith drive is FC,i = 2Ωz ωF mi x0,i sin(ωF t − ϕi ), where the individual deflections and phase angles are given by x0,i = (F0 /mi )|FD,i (jωF )| and ϕi = ϕD,i (jωF ). The total Coriolis P force FC,Σ = FC,i is therefore FC,Σ = FC,Σ,0 sin(ωF t − ϕΣ ),

(8.80)

with p FC,Σ,0 = 2ΩZ ωF F0 α2 + β 2 , X X α= cos ϕi |FD,i (jωF )|, β= sin ϕi |FD,i (jωF ),

µ ¶ β ϕΣ = arctan . α

In Fig. 8.11(b) an example of the normalized total Coriolis force is presented. The resonance frequencies of the eight driving resonators are spaced by an interval equal to 30% of their bandwidth. A larger spacing leads to an unacceptable ripple in the frequency dependency of the total Coriolis force. Their normalized individual transfer characteristics are shown in the figure. The effective bandwidth of the equivalent total drive is roughly doubled in comparison with a single-drive system. The price to be paid is high: one must use eight individually driven resonators that occupy significantly more area than a single drive with

398

Gyroscopes

the same amplitude of the Coriolis force. Within a given footprint the sensing area is significantly reduced, leading to an inferior sensitivity. Where possible, it would seem to be more reasonable to increase the damping by a factor of two to accomplish the same effect.

8.5.3

Non-resonant sensing Stable sensitivity even for high-Q gyroscopes is achieved by introducing a mode mismatch expressed by a frequency split, △, between the two resonance frequencies. The split is usually on the order of 2%–10% of the resonance frequencies, but significantly larger than the bandwidth of both resonators, △ = ωS − ωD ,

|△| ≫ ∆ωS ≈ ∆ωD ,

ωF = ω D .

(8.81)

The location of the two rate-signal sidebands with respect to the sensing resonance is schematically illustrated in Fig. 8.10(b). Clearly, since the signal is far outside the resonator bandwidth, the gain is reduced. In this case the term 2δS ωS (Ω ± ωF ) in the expression for the resonator’s transfer function, Eq. (8.67), can be neglected, and one gets for constant-rate signals π x0 ∼ 0. yΩ /(◦ /s) = (8.82) cos(γref − ϕDF − γB ), ϕSF = 360 △ The sensitivity ratio between resonant and non-resonant sensing is yΩ,res /yΩ,non-res =

△ . ∆ωS

(8.83)

The sensitivity loss is not as dramatic as expected if a realistic split of say 300– 500 Hz is compared with a low-Q bandwidth of say 30–100 Hz for resonance sensing. According to Eqs. (8.75) and (8.67) the rate-transfer function for non-resonant sensing becomes Hsplit (jΩ) =

x0 G0 [△′ cos(γref − ϕDF − γB ) + jΩ sin(γref − ϕDF − γB )], 2(△′ 2 − Ω2 ) (8.84)

where △′ = △ − Ω2 /(2ωF ). Thus, in contrast to the resonance case, the drive and sensing motions are, with ϕFS ∼ = 0, phase shifted by 90◦ (in quadrature). This is a fundamental fact that plays a decisive role in determining the nature of cross-coupling effects. For a correct phase setting, γref = ϕDF + γB , it yields Hsplit (jΩ) =

x0 G0 1 x0 G0 1 ∼ . = ′ ′ 2 2△ 1 − (Ω/△ ) 2△ 1 − Ω2 /△2

(8.85)

Not surprisingly, the gain increases with growing modulation frequency, Ω, because according to Fig. 8.10(b) the increase in gain of the left sideband overcompensates for the decrease of the right one. Only for very large modulation frequencies does the total gain reduce.

8.5 Rate gyroscopes

399

Bandwidth reduction has to be performed by the low-pass filter HI (jω) following the synchro-demodulator. This may require high-order filters because the transfer characteristic should have a large attenuation for frequencies above the stopband frequency Ω ≥ ΩStop ≪ △′ in order to avoid distortions that can be caused by rate-signal components active near the sensing resonance peak. Remarkably, within the validity of the approximation there is no phase shift within the whole input path, which eases signal handling up to modulation frequencies close to the frequency split. The frequency split is stable over time. The temperature dependency is small because the drive and sensing resonances are governed by similar temperature dependencies of the flexures. If necessary, it can be corrected by calibration. The robustness of the frequency-split mode is intrinsically superior with respect to the resonance mode. The price to be paid is a reduced sensitivity. To resolve a rate signal of 0.1◦ /s according to Eq. (8.82) a mechanical deflection of ∼5 pm must be captured, providing a frequency split of 300 Hz and the same drive deflection as in the example for resonant sensing (x0 = 5 µm). This corresponds to one twentieth of the radius of an atom. It is worth to note, that without remarkable loss of sensitivity the damping of the sensing resonator can be increased in order to improve the robustness against mechanical disturbances such as vibrations at frequencies close to the sensing resonance. I.e. the resonance gain can be decreased by increased mechanical damping or by electronic feedback that is proportional to the deflection velocity.

Three-DOF non-resonant sensing An idea to improve the robustness against variations of the matching conditions in mode-matched gyroscopes was described in the paragraph on “Multi-DOF drives for resonant sensing.” The one-DOF drive was substituted by a multiDOF drive. The same approach can be applied for improving the sensitivity and robustness of non-resonant gyroscopes by implementing more than one coupled sensing resonator with different resonance frequencies [Acar and Shkel 2006, Schofield et al. 2008]. There is no necessity to use a large number of sensing resonators, because the sensing resonances can be set on the right- and left-hand sides of the drive resonance, requiring only two sensing resonators. An example of a two-DOF sensing resonator is shown in Fig. 8.12(a). The first nested proof mass, mS1 , surrounds a second embedded mass, mS2 , which consists of the sensing boxes. The two sensing masses are coupled with each other by the suspension with stiffness kS2 . The first sensing mass is connected with the driving frame by a suspension with y-stiffness kS1 . The dynamic equations for the three-DOF gyroscope are, in the linear-rate approximation, FD , mD + mS1 + mS2 2 2 y¨1 + 2δS1 ωS2 y˙ 1 + ωS1 y1 = γωS2 y2 − 2ΩZ x, ˙ x ¨ + 2δD ωD x˙ + ωD2 x =

2 2 y¨2 + 2δS2 ωS2 y˙ 2 + ωS2 y2 = ωS2 y1 − 2ΩZ x, ˙

(8.86) (8.87)

400

Gyroscopes

kD

drive

(a)

(b)

x

1.8

x 10

-6

1.6

y mS1

1.2

mS2

Sensing mode transfer characteristic

1.4 Drive-mode transfer characteristic

1

0.8

0.6

kS2

0.4

0.2

mD

kS1

0 2500

3000

3500

Figure 8.12 A non-resonant gyroscope with two-DOF sensing. (a) A gyroscope with

two sensing masses. (b) The sensing and drive responses. 2 2 = kS2 /mS2 , = (kS1 + kS2 )/mS1 , ωS2 where ωD2 = kD /(mD + mS1 + mS2 ), ωS1 and γ = mS2 /mS1 . In Fig. 8.12(b) the drive and sensing responses are calculated for fD = 3000 Hz, fS1 = 2850 Hz, fS2 = 3150 Hz, QD = QS1 = 100, and γ = mS2 /mS1 = 0.1. One sees that the sensing response lies within the flat region between the two sensing resonances that is insensitive to damping and, to some extent, also to mismatch changes. The principle works only for relative small ratios mS2 /mS1 , which limits the area available for sensing boxes, and – for a given frequency split – may overcompensate for the increase of sensitivity achieved by having a larger plateau value than the value of the sensing gain in the case of one sensing resonator. Schofield et al. [2008] reported an experimentally verified improvement of the temperature dependency of sensitivity by about one order of magnitude, provided that the drive-deflection amplitude is kept constant. However, the achieved sensitivity change of around 1.5%/50 ◦ C is still too much for many applications. Thus, electronic calibration using an internal temperature sensor may be necessary anyway. This limits the application of the method to cases where the saving of electronic compensation constitutes a sufficient advantage (provided that the sensitivity loss in comparison with a two-DOF gyroscope within the same footprint area is acceptable). Theoretically, the same approach can be used for the coupling of two drive resonators and one sensing resonator (see Acar [2004] and Acar and Shkel [2009]). In this case, in Fig. 8.12(b) the drive response corresponds to the sensing response shown, and the sensing response to the drive characteristic shown. However, since again small mass ratios between the two driving masses are required, a large area within a given footprint is occupied by the drive system, reducing the sensitivity drastically.

8.5.4

Noise The total output noise after synchro-demodulation is composed of the “electronic” noise, which is determined by the transducer and the subsequent amplifier

8.5 Rate gyroscopes

401

stages, and by the Brownian noise. The Brownian noise defines the ultimate limit for resolving small-rate signals. The decisive noise component is the Brownian noise of the sensing resonator, nB,y , with spectral density SB,y = 4kT cy /m2S = 8kT δS ωS /mS , while the impact of Brownian noise of the drive resonator is incomparably smaller. According to (8.66) the sensing resonator noise causes a stochastic deflection, yn , that has the spectral density Sn ,y (ω) Sy n (ω) = |FS (jω)|2 SB,y .

(8.88)

The resonator noise clearly has a narrow-band character and, thus, can be represented in the form (e.g. Kempe [1976]) yn = ηy cos(ωS t + φ) − νy cos(ωS t + φ)

(8.89)

with arbitrary initial phase angle φ. Since the spectrum Sy n (ω) is assumed to be highly symmetric around the resonance peaks, the components ηy (t) and νy (t) are statistically independent and feature identical spectral density, Sη y (Ω) = Sν y (Ω) = Sy n (Ω − ωS ) + Sy n (Ω + ωS ) ∼ =

SB,y . + δS2 ωS2 )

2ωS2 (Ω2

(8.90)

After passing the transducer and amplifier stages, the synchro-demodulated output noise becomes zy n = G0 [ηy sin(△t) + νy cos(△t)].

(8.91)

It is obvious that the correlation function is hzy n (t)zy n (t + τ )i = G20 /2hηy (t)ηy (t + τ )i cos(△τ ). Thus, the corresponding total spectral density yields6 ½ · ¾ ¸ SB,y 1 1 2 Sz n (Ω) = G0 + + SE , 8ωS2 (△ + Ω)2 + δS2 ωS2 (△ − Ω)2 + δS2 ωS2

(8.92)

where the spectral density SE of the input-related electronic noise has been added. The spectral density of the electronic noise is flat and, within the bandwidth of interest, can be approximated by white noise.7 To a first approximation the signal-to-noise ratios for resonant and nonresonant sensing are equal. Indeed, assuming a constant-rate signal, the Brownian-noise output is given by s G0 2kT δS ωS ∆f zBnoise,s = , (8.93) ωS mS (△2 + δS2 ωS2 ) where ∆f is the output bandwidth. By comparison with the rate-output signal according to Eq. (8.71), the noise-equivalent rate signal ΩBrownian , where the 6 7

R∞

Here the formula 0 [cos(αx)/(β 2 + x2 )]dx = [π/(2β)]e −αβ for α ≥ 0 was used. If the reference signal of the synchro-demodulator is a rectangular pulse sequence with frequency components (2k + 1)ω F, k = 1, . . . , care has to be taken first to filter out electronic noise components above 2ω F in order to avoid demodulated cross-products.

402

Gyroscopes

SNR is unity, is given by ΩBrownian

1 = x0 ωS

r

8kT δS ωS ∆f mS

(8.94)

and does not depend on the frequency split. A closer analysis reveals that the spectral signal-to-noise ratios differ only for large modulation frequencies comparable to the frequency split. For the most relevant case of |Ω| ≪ |△| the noise performance of the non-resonant sensing mode does not deteriorate in comparison with that of the resonant case. However, the impact of the electronic noise may be incomparably higher because the Brownian-noise fraction is roughly by the factor ∆ωS /△ smaller than that in the resonant case. As long as the electronic noise is dominant the challenge of non-resonant sensing is to reduce the electronic input noise in accordance with the sensitivity loss on going from resonant to non-resonant sensing. If, as an example, a gyroscope of mass 5 µg, with its sensing resonance at 10 kHz, with a bandwidth of 5 Hz (QS = 1000) and with a drive excitation of 5 µm is considered, the equivalent noise within a 25-Hz output bandwidth is equal to ΩBrownian ∼ = 0.01◦ /s – a value that is not so far away from the resolution requirements for automotive applications (≃0.1◦ /s).

8.5.5

The zero-rate output The zero-rate output (ZRO) has to be stable in order for it to be eliminated by calibration. However, bias drift further corrupts an accurate measurement. The uncompensated or residual (after compensation) bias of a gyroscope is one of the most difficult phenomena. It depends on the sensor itself, on the packaging stress, on imperfections of the transducer, on an exact phase setting for demodulation, on the frequency stability of the VCO, and, last but not least, on the electronic offset. All of these particular fractions are temperature-dependent, some of them are subject to aging, and nearly all are stress-dependent. As mentioned, within the automotive market typical requirements are a total bias of less than 2–3◦ /s over the whole temperature range −40 to 125 ◦ C and during a lifetime of 17 years for the completely packaged device with all inherent stress sources. Consumer products are confronted with less-challenging features; in contrast, navigation-grade products are under the pressure of continuously decreasing bias limits. In vibratory gyroscopes there are two types of bias, the so-called real bias and the quadrature bias, or, in short, the R-bias and the Q-bias. The term Q-bias is a little bit misleading, because it denotes a vibratory component at the input of the synchro-demodulator that is 90◦ phase-shifted with respect to the rate-modulated carrier of the Coriolis force x(t). ˙ Such a component does not necessarily create a bias at the output. Only in the case of incorrect phase setting of the reference signal does a bias emerge.

8.5 Rate gyroscopes

403

y

sensing axis sensing mode axis

zero-quadrature axis

drive axis

driven mode axis x

Figure 8.13 Alignment errors of a capacitive z-gyroscope.

Mechanical bias sources The mechanical sources of the ZRO are geometric and/or mode-misalignment errors. The suspended masses will typically oscillate not perfectly parallel to the substrate. The drive and sensing motions will be not ideally orthogonal. The COG of the body does not fit perfectly with the center of the suspension, leading to a slight wobble. By reducing the model to a two-DOF description by projecting out-of-plane deviations onto the in-plane movement one gets only a first-order model. Within this approximation Fig. 8.13 illustrates different imperfections. First of all, the x-axis is defined as the orientation axis of the drive force, thus being parallel to the fixed drive-comb fingers shown in Fig. 8.8. The sensing axis is the axis of maximal sensitivity and is orientated not exactly along the y-axis but, due to manufacturing flaws, rotated by a small angle α. Owing to anisoelasticity or drive-axis misalignment, the driven mode, i.e. the movement of the COG in the absence of an external rate signal, takes place along the driven mode axis that may be rotated by the angle β with respect to the drive axis x. The projection of this movement onto the sensing axis is generally not zero, but consists of components that are coupled to the sensing movement. The coupling creates accelerations in the sensing direction that contaminate the output. Vice versa, the position-sensed output does not contain contaminations stemming from the drive motion if the driven axis is orthogonal to the sensing axis. This position is called the “zero-quadrature axis.” First the impact of β 6= 0 is analyzed assuming ideal alignment of the sensing axis (α = 0). For this the gyroscope model with imperfections represented by Eq. (8.59) is adapted to the case of the two-mass gyroscope being considered here: F0 cos(ωF t) mS + 2ΩZ y˙ − 2λS,D δD ωD y˙ mS + mD mS + mD − γS,D ωD2 y + nB,x , FS y¨ + 2δS ωS y˙ + ωS2 y = −2ΩZ x˙ + − 2λD,S δS ωS x˙ − γD;S ωS2 x + nB,y , mS

x ¨ + 2δD ωD x˙ + ωD2 x =

(8.95)

404

Gyroscopes

where ωD2 = kD /(mS + mD ), ωS2 = kS /mS and λS,D = cx,y /cx and λD,S = cx,y /cy are the coefficients of non-proportional damping, and γS,D = kS,D /kD and γD,S = kD,S /kS are the cross-coupling coefficients. Since the rate deflection is orders of magnitude smaller than the drive motion, it can be neglected, and the drive motion is approximated by Eq. (8.68): x = x0 cos(ωF t − ϕDF ). Therefore, the sensing deflection obeys the relation · ¸ ωS2 y = 2x0 ωF |FS (jωF )| (ΩZ + λD,S ∆ωS ) sin ψ − γD;S cos ψ , (8.96) 2ωF where ψ = ωF t − ϕDF − ϕSF . Thus, the COG moves along an ellipse, the parameters of which depend now on the additional parameters λD,S and γD,S . It is illustrative to consider the movement in the absence of a rate signal and of non-proportional damping. Using Eq. (8.54), the corresponding rotation angle can be derived: tan(2β) = −

γD,S ωS2 |FS | 2 ω 4 |F |2 cos(ϕSF ). 1 − γD,S S S

(8.97)

For non-resonant sensing (ϕSF = 0) the COG oscillates along a straight line, rotated by the angle β, along the driven axis. However, for resonant sensing the rotation angle converges according to Eq. (8.96) to zero and an ellipticity develops. To conclude, the general expression for the relative output signal of a positionsensing transducer (after synchro-demodulation with reference signal sin(ωF t − γref )) is derived from Eq. (8.96) (ωS ∼ = ωF ): ·µ ¶ ¸ z ΩZ 1 = 360 cos φ − + λ ∆f γ f sin φ , (8.98) D,S S D,S S zΩ Z /(◦ /s) 2π 2 with φ = ϕDF + ϕSF + γB − γref . The two emerging additional components are main sources of the gyroscope offset. Thereby, low damping and large sensing masses (small resonator bandwidth ∆fS ) decrease the contribution from non-proportional damping. Analogously, high resonance frequencies increase the impact of anisoelasticity on the quadrature error, provided in both cases that the coupling rates λD,S and γD,S are constant. Similar extensions of mode-coupling analysis for doubly decoupled systems have been presented, e.g. by Braxmeier et al. [2003]. Analogously, Weinberg and Kourepenis [2006] considered later the ZRO of a tuning-fork arrangement.

8.5 Rate gyroscopes

405

Q-bias The second component in Eq. (8.96) is in quadrature with the Coriolis term – hence the denotation “Q-bias” for the demodulated fraction. If the anisoelasticity coefficient γD,S equals 1%, the equivalent input amplitude, γS,D ωS2 /(2ωF ), becomes equal to 18 000◦ /s for a 10-kHz gyroscope. This is at least two orders of magnitude larger then a maximal rate signal of 180◦ /s. In practice, quadrature components exceeding the maximal rate signal by up to three orders of magnitude have been reported. They require an excessive dynamic range of the transducer and front-end electronics that may cause additional noise and significant nonlinear distortions. A well-designed and stable-rate gyroscope should exhibit quadrature components no more than five to ten times larger than the maximal-rate signal. To achieve this, an efficient mode decoupling and careful control of the manufacturing conditions, including – if necessary – post-manufacturing screening, are common. Q-bias compensation by a feedback loop can be applied where this is not sufficient. In the case of correct phase setting – γref = φ – the quadrature component is suppressed. However, a phase error of 1◦ leads to a quadrature error leakage of 1.7% into the output. Exact phase generation for the reference signal is decisive for good Q-bias suppression. Accurate phase setting over the whole temperature range in the presence of noise is not a trivial task. It includes not only the correction of the temperature-dependent resonator phases but also the elimination of temperature-dependent electronic phase shifts.

Q-bias compensation Since according to Eq. (8.95) the Q-bias is created by a force (−γD,S ωS2 mS x) that represents a certain small fraction of the drive movement, it can be compensated for by adding the same force in anti-phase. There are two ways of doing this. The first corresponds to the classical feedback approach, where the force is generated by the measured drive motion and applied to feedback electrodes. The feedback electrodes should be separated from the sensing electrodes in order to avoid electrical coupling into the sensing channel. The difficulties in realizing such an approach again arise from the different phase shifts emerging within the electronic blocks. For instance, what is available at the output of the front-end electronics is not x(t), but a phase-shifted replica. Feeding it back with some controlled gain introduces additional phase shifts, mainly within the driver stages. In the end, the difficulties of stable phase control make such an approach quite impractical. The second way, going back to Clark [1999] and Clark et al. [1996], is based on parametric amplification of DC-feedback voltages within the sensing electrodes and limited to capacitive or other force-creating interfaces. The principle is most easily demonstrated for a gyroscope with sensing combs arranged as shown in Fig. 8.14. The combs, moving in the drive direction, modulate the DC voltages

406

Gyroscopes

y

DS-y(t)

V3 =V0- V

V1 =V0+ V

V2 =-(V0- V)

X0-x(t)

sensing mass

x

X0+x(t) V4 =-(V0+ V)

Figure 8.14 Parametric modulation of feedback combs.

V1 to V4 applied to the stationary combs. Thus, a drive-motion-dependent force is created. More precisely, the instantaneous forces between the stationary and moving fingers in the y-direction are · ¸ 1 V12 V22 Fy ,left = ε0 H(L − x) − , 2 (D − y)2 (D + y)2 (8.99) · ¸ 1 V32 V42 Fy ,right = ε0 H(L + x) − . 2 (D − y)2 (D + y)2

L is the overlap of the comb fingers in the rest position, and H is their height. The total force is then, for V1 = V0 + ∆V , V2 = ±(V0 − ∆V ), V3 = ±(V0 − ∆V ), and V4 = ±(V0 + ∆V ), exactly proportional to the drive motion: FΣ,y = −2ε0

HV0 ∆V x(t). DS2

(8.100)

By controlling ∆V by the demodulated Q-bias the force can be chosen to null the quadrature component. This can be done off-line, for instance, during trimming, or on-line – in operation. An electronic implementation of the whole feedback loop with continuous time and digital controllers is discussed in Saukosi et al. [2007]. Notably, there emerges at the sensing mass a small torque about the z-axis that is created by the stationary forces Fy ,left = −Fy ,right = 2ε0 HLV0 ∆V /DS2 acting in opposite directions. If it leads to a rotation, an additional output signal proportional to the drive motion emerges and may corrupt the whole undertaking. A corresponding large rotational stiffness must be implemented by design. The sensed output depends on the different possible signs of the asymmetric voltages. If the standard sensing combination as shown in Fig. 8.14 is chosen, the following charge flow results (see Chapter 6): QS = 2ε0

H (LV0 y + DS ∆V x). DS2

(8.101)

It contains the desired rate output, but also a non-negligible additional component proportional to ∆V x. Depending on the kind of operation, it adds a quadrature signal in the case of non-resonant sensing and an in-phase signal in the case of resonant sensing. Therefore, it is meaningful to separate the feedback capacitances from the remaining part of the sensing capacitances and in this way avoid the coupling to the input of the front-end electronics. Another way of decoupling is AC sensing as described in Chapter 6 (V0 ⇒ V0 + VAC cos(ωM o d t)).

8.5 Rate gyroscopes

DS

y

V3= V0 - V

x

V2= -(V0 + V)

drive

sense

V1=V0 + V

407

V4=-(V0 - V)

Figure 8.15 An example of force creation by mechanical modulation.

For different geometric gyroscope designs different modulating structures exist [Johnson 2009, Wyse 2005]. If, for instance, sensing boxes are used, a possible modulating structure may take the shape shown in Fig. 8.15. The stationary plates of some or all sensing boxes are divided into two parts with slightly asymmetric excitation voltages. The walls of the moving structure are no longer parallel with respect to the stationary plates, but inclined by the angle α on the left-hand side, and by the angle −α on the right-hand side (α ≪ 1). The total force in the y-direction is calculated as above by integrating the infinitesimal forces along the x-axis. In the linear approximation one gets FΣ,y =

4ε0 LH 8ε0 LH [y(V02 + ∆V 2 ) + 2V0 ∆V tan α x] ≈ x V0 ∆V tan α. DS3 DS3 (8.102)

L is the length of the stationary plates. The charges generated are proportional only to y and are not disturbed by the x-motion: QS = (4ε0 LHV0 /DS2 )y. This allows one to use the whole transducer for sensing even with DC excitation voltages and one does not need separate dedicated sensing boxes for the force generation. Other geometric shapes are possible. For instance, in the example given, by inclining the stationary plates additionally to the walls by the same angle, the force-generation gain can be doubled. Sometimes it may be preferable to use a fraction of, or all of, the sensing combs (or boxes) with a common DC voltage, but with non-centered positions of the movable fingers or walls. If, for instance, the rest positions of the movable fingers are with distances D1 to the upper fixed finger and D2 to the lower one, where D1 6= D2 (for instance, D1 = 2D2 ), an x-dependent force emerges even for identical DC voltages applied to the fixed fingers. This principle can be used for combining comb fractions with positive and negative shifts of the rest position of the movable fingers [Lemkin et al. 2006] to generate the feedback force. Since the zero-position shift may be large, the force generated may easily exceed the values achievable for voltage asymmetries. Another approach for reducing the Q-bias targets the roots of cross-coupling. Since the cross-coupling effects in flexures are proportional to the stress generated by the deflections, stress-compensating structures may be implemented.

408

Gyroscopes

Geen and Carow [2000] were the first to apply this principle by introducing two pivoting beams into a drive flexure in order to alleviate the tensile stress. It is also possible to try to find out the effects of geometric misalignment in a given design and to reduce them by tuning the mechanical positions of the sensor elements. Such an approach is highly dependent on the actual geometry and manufacturing imperfections and each case has to be considered individually.

Q-bias and self-testing For safety-critical applications a continuous self-testing of the whole mechanical structure is required in order to exclude failures caused by mechanical damage. The described method of Q-bias compensation using asymmetrically applied voltages is well suited for the generation of Q-bias changes that can be monitored. Usually a slowly changing voltage ∆V (t) is used. Its frequency is higher than the maximum rate frequency but significantly lower then the resonator frequencies. It may be superimposed on the compensating DC value. In this case its frequency must be higher than the bandwidth of the compensation loop. Since the Q-bias is subject to long-term changes caused by the aging crosscoupling effects, care must be taken to remain within the expected limits, or to adapt the self-test excitation level over the lifetime of the device.

The impact of transducer imperfections The sensing structure

r The considerations in the last section have shown that voltage asymmetries as well as tilting of the moving structure may be additional sources of bias and should be avoided within a capacitive sensing interface. Tilting of the moving structure with respect to stationary combs generates forces as well as electrical output signals that are proportional to the drive motion. Misalignment tilt as indicated in Fig. 8.13 may emerge when the substrate is exposed to stress and the anchor points are dislocated. For sensing boxes one gets after integrating the elementary capacitances of a sensing box tilted by the misalignment angle α along the x-axis, ∆C = ε0 H0

Z

L 2

− L2

dξ[(DS1 + (ξ − x) tan α)−1 − (DS2 − (ξ − x)) tan α)−1 ],

ε0 H0 L ∆C ∼ (y + x tan α). =2 DS2

(8.103)

The rate signal is represented by the first term. The distortion is proportional to x (and α). Hence, for non-resonant sensing the sensing deflection is in quadrature with the drive signal, and the distortion corresponds to a Q-bias signal. More critical is the situation for resonant sensing, where the drive and sensing signals are in phase and the distortion is coupled directly in the real part of the bias. Therefore, particularly for resonant sensing, the sensing-axis misalignment must be controlled during manufacturing and testing.

8.5 Rate gyroscopes

409

r Another source of quadrature signals may be stray capacitances. They are present in combs at the ends of the moving fingers and in sensing boxes at the ends of the fixed plates. Both are dependent on the positions of the moving counterparts and create a capacitance change that is modulated by the drive movement. In the case of sensing boxes they can be effectively eliminated by introducing small shields with the same potential as the moving structure, as discussed in Chapter 2, in the section “Frame-based capacitors.” Such countermeasures make sensing boxes superior to comb solutions. r Anisoelastic cross-couplings of the drive motion into out-of-plane deflections can also generate a quad bias. Using the equation describing the z-deflection analogously to the second equation in (8.66), one gets for the parasitic outof-plane motion z = −γDz

ωz2 x, ωz2 − ωF2

(8.104)

p where γDz = kxz /kz is the out-of-plane coupling coefficient and ωz = kz /mS is the out-of-plane resonance of the sensing mass. Assuming not an ideal ysensitive transducer but a real capacitive transducer, the capacitance changes caused by a z-modulation are, to a very crude approximation (C ∝ (H − 2z)), ∆Cz = 4ε0 LzV /D, while the y-sensitivity is ∆Cy /y = 2ε0 V HL/D2 . Therefore, the ratio between the quadrature components caused by in-plane and out-of-plane cross-coupling zQ /yQ is, using Eq. (8.96), p p △2 + ∆ωS2 zQ D γDz D γDz △2 + ∆ωS2 =4 ≈ 4 normally ≪ 1. yQ H γDS ωS (1 − ωS2 /ωz2 ) H γDS ωS (8.105) Considering the decreasing z-sensitivity around the z = 0 position (see Chapter 2, in the section “Linear combs”), the ratio becomes even smaller. Only in the case of very large γDz or of z-resonances close to the sensing resonance do the contributions become comparable.

Driving structure The effect of a misaligned drive structure, where, for instance, the moving fingers of the comb drive are inclined with respect to the stationary fingers by an angle β as indicated in Fig. 8.13, is the same as in the case of anisoelasticity. An additional force, FD,y , in the y-direction is generated, which directly or via the drive frame in Fig. 8.8 is transferred to the sensing element. In the case of non-resonant sensing the effect is indistinguishable from the effect of anisoelasticity and can be compensated for by the same means as for the standard quadrature correction.

R-bias According to Eq. (8.98) the R-bias corresponds to a value of zR-bias = 360λD,S ∆fS [◦ /s]. zΩ /(◦ /s)

(8.106)

410

Gyroscopes

If the non-proportional damping coefficient accounts for only 0.1%, the R-bias for a sensing bandwidth of 5 Hz becomes 1.8◦ /s. This example demonstrates the urgent necessity to control all possible sources of non-proportional damping, especially in the regions where the suspensions are attached to the substrate and where orthogonal flow coupling may arise. Unfortunately, there are only a few models reflecting the interactions among structural damping, material and form inhomogeneity, internal friction, and stress. Most of the support-loss models are based on the theory of thermoelastic damping and assume symmetric and regular supports that do not reflect possible cross-coupling terms. An efficient way to minimize anisotropic support losses is by the reduction of the stress concentration within all supports and flexures. Also orthogonal flow coupling has still not been well investigated. From a pragmatic point of view, care should be taken to avoid such effects completely, by minimizing flow-coupling effects. Thus, large distances between moving structures and surrounding walls are preferable, which is usually in contradiction with the requirements for small gaps within capacitive interfaces and other arealimiting criteria.

Other bias sources The ZRO may be caused by other sources than mechanical misalignment (see also Saukosi et al. [2008]). They are mainly related to electrical cross-coupling effects such as feedthrough of the driving voltages to the sensing channel, or coupling of the driving voltages directly to the sensing electrodes. In the last case not only is the sense input corrupted but also a direct excitation of the sensing resonator may take place. Another root cause may be feedthrough of the driving voltages to different reference signals disturbing a correct phase setting of the synchro-demodulator reference signal. Last but not least, an electronic offset is present in all electronic blocks and may generate additional components at the demodulator’s output or directly add to the output signal. Some of the countermeasures are as follows.

r The impact of direct coupling of drive signals into the sensing channel can be mitigated by using high-frequency excitation voltages for the detection of capacitance changes. However, since this requires additional effort, simpler solutions are often welcome. r In the case of differential sensing a low impedance of the commonly biasing circuits for both capacitances, and symmetric (common-mode) coupling to both inputs of the input amplifier can reject the coupled-in voltages to a considerable extent. r Most importantly, the geometry of the sensor design as well as the electronic layout must be very well controlled in order to minimize dangerous stray capacitances.

8.5 Rate gyroscopes

8.5.6

411

Bias stability The bias drift depends on changing environmental conditions and lifetime. It is mainly caused by low-frequency changes. Since the sources of bias drift are hard to localize, usually a total bias that includes the nominal bias plus its drift is specified. The characterization of bias stability is difficult because long-term changes are masked by the different noise sources. Short-term fluctuations are related to the measurement time, which is inversely proportional to the output bandwidth. They are estimated within the framework of the standard noise analysis of a gyroscope. Therefore, for stability analysis a technique that allows one to identify different additional sources of fluctuation with a longer time horizon is needed. The ZRO is a frequency- and phase-sensitive object onto which the output noise is superimposed. However, the additive (“white” within the measurement bandwidth) noise characterizes only the mentioned short-time part of the bias fluctuations. Noise is further generated by the fluctuations of the resonance frequencies of the resonators and of the VCO-controlled drive frequency. For a large quad bias also phase fluctuations may contribute via the synchro-demodulator. Thus, the ZRO output behaves similarly to the frequency of a mechanical or electrical oscillator. The frequency of an oscillator is subject to modulations with random-walk noise of spectral density ∝ 1/f 2 , with flicker noise ∝ 1/f , and with white noise. The phase is modulated by flicker noise and white noise, resulting in frequency-related spectral contributions proportional to f and f 2 . Further types of stationary and non-stationary noise sources may exist. Therefore, one can expect that the random bias fluctuations within the measurement interval f < fm eas can be represented by a linear combination of at least five independent noises with spectral density [Yafei et al. 2007] SΩ (f ) =

2 X

2 N(0,α) fα,

(8.107)

−2

where α = −2, −1, 0, 1, 2. In order to capture instability phenomena the observed ZRO has to be analyzed with respect to the contributions of the different noise sources. A spectral analysis is very time-consuming, considering, inter alia, that some of the noise sources belong to non-stationary (e.g. flicker) processes.

The Allan variance A more powerful, well-established tool is the Allan variance, which was originally developed for the analysis of instabilities in oscillators and later adapted to the characterization of ring laser gyroscopes (IEEE Standard 647-1995). It substitutes the frequency analysis by a time-domain estimate that is based on the combination of successive better noise averaging and sharper separation of longterm changes. The Allan variance simply represents the variance of consecutive,

Gyroscopes

Root Allan Variance

10

3

10

2

Measured Allan variance [°/h] Allan Variance

412

1/2

Rate random walk N 0,-2T , N0,-2=0.14315(°/h)*sqrt(Hz)

10

Bias instability N0,-1=2.5834°/h

1

Angle random walk N 0,0 T N0,0=17.0617(°/h)/sqrt(Hz)

-1/2

Confidence bounds

,

0

10 -2 10

10

-1

10

0

1

2

10 10 Averaging interval T (sec)

10

3

10

4

10

5

Figure 8.16 An example of a measured Allan variance plot. With permission of

SensorDynamics AG.

averaged differences: N −2 X 1 1 h(Ωi+1 (T ) − Ωi (T ))2 i = (Ωi+ 1 (T ) − Ωi (T ))2 , 2 2(n − 1) 0 Z 1 (i+ 1)T Ωi (T ) = hΩ(t)|T i = Ω(t)dt. T iT

σ 2 (T ) =

(8.108)

Ω(t) is here the ZRO, given usually in ◦ /hr. The basic relation between the Allan variance and spectral densities follows from this definition, Z ∞ sin4 (πf T ) df SΩ (f ) σ 2 (T ) = 4 , (8.109) (πf T )2 0 where for non-stationary spectral components the time-averaged spectra must be used. This relation allows one to identify the contributions of the different noise sources in the Allan plot – the dependency of the Allan variance on the averaging interval T in a log–log representation. In Fig. 8.16 an example of a measured Allan plot for a torsional gyroscope is given. The different noise contributions can be separated. For instance, the angle random walk is the noise of the integrated rate output and caused mainly by 2 the white-noise contribution SΩ,ARW (f ) = N(0,0) , which is also subject to the standard noise analysis: Z ∞ 2 N0,0 sin4 (πf T ) 2 2 df σARW (T ) = 4N(0,0) = . (8.110) (πf T )2 T 0

8.5 Rate gyroscopes

413

Thus, the slope of the Allan plot in the T-interval dominated by the angle random walk contribution is − 21 and the value of N(0,0) can be simply read out as the ordinate of the plot at T = 1. The ARW of a gyroscope is therefore nothing other than the square root of the spectral density of the output noise. Similarly, the bias instability is dominated by a 1/f flicker-noise contribution 2 f −1 with cut-off frequency f0 , above which the spectrum can be set to zero. N0,−1 The sources are components of the device that are susceptible to flicker noise. On performing the calculations involved in deriving the power spectral density and calculating the Allan variance, one gets a plateau-like plot that extends from approximately T0 = 1/f0 to infinity, but falls quickly for time intervals decreasing below T0 . Therefore, for T < T0 it does not significantly disturb the angle walk contribution. The bias instability N0,−1 can be read out as the ordinate of the plateau which is approximately equal to the minimum of the Allan plot. The last main contribution is the rate random walk – a fluctuation the origins of which are widely unknown, but which determines the long-term drift of a 2 f −2 and leads to a rate signal. It is caused by the spectral component N0,−2 1 branch growing with slope 2 that dominates the plot for time intervals noticeably exceeding the plateau-onset interval T0 . 2 f α with α > 0 are filtered out after The neglected spectral contributions N0,α a very short averaging time. Summarizing, understanding and improving bias stability is the precondition for entering the inertial-grade range and is obviously inseparably linked with further investigations in material science and engineering in order better to localize the individual root causes of flicker and rate random walk noise.

8.5.7

Acceleration suppression and tuning forks An inherent drawback of all linear gyroscopes is their large sensitivity to linear accelerations. Torsional gyroscopes are superior with respect to this kind of external disturbance because they are designed to be as stiff as possible against linear deflections. In contrast, a linear gyroscope is always compliant at least in the drive and sensing directions. The most dangerous accelerations are, of course, those acting in the sensing direction. In the case of the linear z-gyroscope according to Fig. 8.8 considered as an example the sensing resonator is subject to linear accelerations, ay , in the ydirection that must be compared with the Coriolis acceleration 2Ωz x. ˙ For sufficiently high resonance frequencies the acceleration can be considered as a lowpass filtered process. An orientation on the expected deflections, ya , is given by the action of a constant acceleration: ya =

ay ,0 . ωS2

(8.111)

An acceleration of only one g shifts the sense electrodes of a 10-kHz gyroscope by 2.5 nm – a value that is comparable to the amplitude of a Coriolis

414

Gyroscopes

mD1

drive 1

mS1

coupling springs KCD, KCS kCS12

kCD11

sense 1

S1P

mD2

drive 2

mS2

kCS22

kCD21

S1N

S2P

S2N

sense 2

Figure 8.17 A anti-phase-driven tuning fork with acceleration suppression.

deflection generated by a full-scale rate signal of 100◦ /s. In automotive applications the gyroscope must work without significant disturbances under environmental shocks and vibrations – i.e. in the presence of accelerations exceeding the 100g limit. The acceleration sensitivity increases with growing excitation frequencies and becomes especially critical around the sensing resonance. Considering that modern audio equipment in cars can generate sounds up to 16 kHz with considerably loud pressure, the sensing resonance for critical linear automotive gyroscopes should be above this region. The dominant direction of possible spectral contributions of yet higher frequency, for instance, caused by stone impingement, must be at least orthogonal to the sensing direction. In some application areas a good separation between an acting low-pass acceleration spectrum and the sensing resonance frequency is guaranteed. In such cases the gyroscope can be used for a combined angular-rate and acceleration measurement separating the low-pass and high-pass parts of the sensed signals. Unacceptable nonlinear distortions should be excluded by limiting the peak acceleration using, for instance, motion “stoppers.” However, the primary target of the gyroscope design is to suppress accelerations within the angular-rate channel.

Anti-phase-driven identical gyroscopes Since the orientation of the Coriolis force depends on the orientation of the drive velocity vector, it is obvious that two identical gyroscopes, for instance, according to Fig. 8.8, having parallel drive and sensing axes and driven in anti-phase, will exhibit sensing motions in opposite directions. Assuming ideal anti-phase drive motion x1 (t) = −x2 (t) in such an arrangement – corresponding to Fig. 8.17 but without the central coupling part – the lower electrodes S1P feature capacitances

8.5 Rate gyroscopes

415

C1P = ε0 A/(DS − ya − yΩ ) and C2P = ε0 A/(DS − ya + yΩ ); the upper ones have correspondingly C1N = ε0 A/(DS + ya + yΩ ) and C2N = ε0 A/(DS + ya − yΩ ). On connecting the left lower electrodes S1P with the right upper ones S2N and performing differential sensing the Coriolis contributions add and the acceleration contributions subtract from each other (yΩ ≪ DS ): ∆C = n(C1P − C1N + C2N − C2P ) = 4nε0 AyΩ

DS2 + ya2 A ≃ 4nε0 2 yΩ . 2 2 2 (DS − ya ) DS (8.112)

n is the number of boxes within one part; A is the area of overlap between the fixed fingers and the box wall. One sees that the large acceleration response is significantly suppressed and no longer overloads the front-end electronics. Since the capacitive-plate-like transducer is nonlinear, the usual right-hand approximation holds only for ya ≪ DS . However, a 100g static acceleration with 250 nm deflection causes already a sensitivity error of ya2 /DS2 ≃ 6% assuming a gap of 1 µm. This example illustrates the practical limits of anti-phase acceleration suppression. The sensitivity error can be corrected, at least theoretically, by measuring ya in a second transducer channel with a differential arrangement of parallel-connected electrodes S1P and S2P and correspondingly S1N and S2N .

Tuning-fork gyroscopes The described idea of cross-sensing of identical gyroscopes driven in anti-phase was proposed, for instance, by Geen and Carow [2000]. However, a practical application needs a very careful matching of the two drive resonances. The two drive resonances and with them the force-to-drive transfer functions never match completely. This makes the frequency stabilization and phase detection of both sensors with summed motor-sense signals ±(x1 + x2 ) difficult. Thus, a mechanical synchronization of both drive resonators is beneficial and can be realized in the tuning-fork gyroscopes. Anti-phase synchronization is achieved by introducing a mechanical coupling of the drive motions, for instance, in the form of a coupling spring KCD as shown in Fig. 8.17. It is advantageous to synchronize also the sensing deflections by using coupling springs KCS . This not only increases the robustness but also mitigates the impact of torques about the central coupling point. Neglecting, as usual, terms nonlinear in ΩZ and the impact of the Coriolis force on the drive motion, the dynamic equations become FD1 , mD1 + mS1 ′ FD2 FD1 2 2 x¨2 + 2δD2 ωD2 x˙2 + ωD2 x2 − ωCD2 x1 = =− , mD2 + mS2 mD2 + mS2 2 2 y¨1 + 2δS1 ωS1 y˙1 + ωS1 y1 − ωCS1 y2 = −2ΩZ x˙1 , 2 2 x¨1 + 2δD1 ωD1 x˙1 + ωD1 x1 − ωCD1 x2 =

2 2 y¨2 + 2δS2 ωS2 y˙2 + ωS2 y2 − ωCS2 y1 = −2ΩZ x˙2 ,

(8.113)

416

Gyroscopes

where the resonance frequencies are determined by 2 ωD1 = 2 ωCD1 =

KD1 + KCD , mD1 + mS1

2 ωD2 =

KD2 + KCD , mD2 + mS2

KCD , mD1 + mS1

2 ωCD2 =

KCD . mD1 + mS1

The spring constants KD(S)i are given by the parallel connection of the corresponding four supporting folded beams while the coupling rate is formed by the serial connection of two parallel-connected springs attached to the left and right drive frames, KCD =

kCD1 kCD2 , kCD1 + kCD2

kCDi = kCDi1 + kCDi2 .

(8.114)

Analogously for the sensing parameters: 2 = ωS1

KS1 + KCS , mS1

2 = ωS2

KS2 + KCS , mS2

2 = ωCS1

KCS , mS1

2 = ωCS2

KCS . mS1

The coupling springs may be centralized as in Fig. 8.17 or split into two or more coupling members, usually at the “inner” top and bottom corners of the drive frame. They may be cross-shaped or have distributed crab-leg-like shapes etc. In Eq. (8.113) small differences in the applied forces are taken into consider′ ation by setting FD2 = −FD1 . Thus, the force-to-drive transfer functions satisfy the relations 1 ′ 2 + Q2 (s)FD1 ], FD1 [−ωC2 mD1 + mS1 1 2 2 ′ [Q1 (s)Q2 (s) − ωC1 ωC2 ]x2 = [ω 2 FD1 − Q1 (s)FD1 ], mD2 + mS2 C1

[Q1 (s)Q2 (s) − ωC2 1 ωC2 2 ]x1 =

(8.115)

where 2 Q1 (s) = s2 + 2δD1 ωD1 s + ωD1 ,

2 Q2 (s) = s2 + 2δD2 ωD2 s + ωD2 .

The most important feature of the coupled system is revealed by these equations: the homogeneous equations for both movements are identical irrespective of any matching inaccuracy of springs, masses, and damping. Consequently, all of the resonance frequencies are identical, as are the transfer processes. For negligible damping the resonance frequencies are given by r 1 2 1 2 2 2 2 )2 − ω 2 ω 2 , ωH(L) = (ωD1 + ωD2 ) ± (8.116) (ω − ωD2 C1 C2 2 4 D1 where “H” denotes the high and “L” the low resonance frequency of the coupled system. With increasing coupling the resonance frequencies migrate from ωD1 and ωD2 upwards to ωH and downwards to ωL . Since the characteristic equations for both partial subsystems remain identical for any mismatch, the case of ideal matching, ωD1 = ωD2 = ωD , ωC1 = ωC2 =

8.5 Rate gyroscopes

417

ωC1 , is now representative also for small parametric differences, ωH2 = ωD2 + ωC2 =

KD + 2KC , mD + mS

ωL2 = ωD2 − ωC2 =

KC . mD + mS

(8.117)

′ = FD1 = FD , the In the case of equal forces applied in anti-phase, FD2 = −FD1 system degenerates into two independent subsystems with the same resonance frequency:

[s2 + 2δD ωD s + ωH2 ]x1(2) = +(−)

FD . mD + mS

(8.118)

However, if the two forces are applied in phase (FD2 = FD1 = FD ), the transfer function becomes FD [s2 + 2δD ωD s + ωL2 ]x1(2) = (8.119) mD + mS and features a resonance at the lower resonance frequency fL . Here the two subsystems move in the same direction, like a hula hoop around a beautiful girl. This mode is therefore often called the hula mode. External vibrations with frequencies around fL may excite this mode. Consequently, it is meaningful to set the lower frequency as high as possible outside the frequency range of possible external vibrations. Since with Eq. (8.117) the difference is ωH2 − ωL2 = 2ωC2 , the coupling frequency fC should be chosen to be large enough to separate clearly the higher and lower resonances, but small enough not to drive the higher frequency into frequency regions that are difficult to realize. Literally the same conclusions can be derived for the sensing motion on substituting the anti-phase driving forces by the two Coriolis forces. The two high frequencies ωH and ωS,H have to be designed in order to meet the target frequency split. The practical design of a tuning fork should consider also the acceleration sensitivity in the z-direction. A first-order calculation can be based on the zcompliance of the suspensions of the individual, uncoupled subsystems according to Fig. 8.17.8 The corresponding model describes the coupled motion of the drive and sensing frames, both in the z-direction. Additionally, the two coupled subsystems may feature a slight tilting mode whereby both inner sensing plates tilt in anti-phase around two axes that are parallel to the y-axis and shifted from the center of the sensing frame towards the common center of symmetry. A relative dislocation of the COGs of sensing and driving frames within one subsystem is one of the reasons for this. The superposition of the two mechanisms constitutes a butterfly mode. Distortions caused by this mode may become especially dangerous if they are within the frequency range of the driving or sensing mode. If two sensing channels are used – one for differential sensing and one for common-mode sensing – angular rates about the z-axis and accelerations in the y-direction can be measured, as was realized in Nagao et al. [2004]. The Coriolis 8

For common z-deflections the coupling between the two subsystems is inactive.

418

Gyroscopes

oscillations and the spectrum of acting accelerations are separated in space and not necessarily in the frequency region. Tuning-fork gyroscopes and their derivatives are extending the application of z-gyroscopes. They feature good sensitivity and efficient footprint usage. Some representative commercially available gyroscopes will be presented in Section 8.6.2.

8.5.8

Drive-motion control and spring nonlinearities A good stabilization of drive motion is one of the preconditions for a stable and temperature-independent sensitivity. Drive-mode stabilization based on an amplitude-control loop with a nested phase detection of the resonators output by a phase-locked loop (PLL) as shown in Fig. 8.9 is well established. Figure 8.9 serves as a reference for the following notations and relations.

The phase-locked loop For the discussion of the PLL let the normalized driving force fD (t) = FD (t)/(mS + mD ) be given by fD = f0 cos(ωF t − ϕ0 ) = f˜D ej ω F t + f˜D∗ e−j ω F t with some initial phase to be determined below (in Fig. 8.9 the initial phase is set to zero). The complex amplitude is f˜D = 21 f0 e−j ϕ 0 . In contrast to the past, the amplitude and phase of the driving force within the control loop should be considered not as constant but as slowly changing functions (in comparison with ωF ). Interpreting the resonator transfer function FD (s) = 1/(s2 + 2δD ωδD s + ωD2 ) as an operator with s = d/dt and separating the fast and slow motions, the output can be represented symbolically as x(t) = FD (s)f (t) = FD (s)(f˜0 ej ω F t + f˜0∗ e−j ω F t ) ∼ = ej ω F t FD (jωF + s)f˜0 + e−j ω F t FD (−jωF + s)f˜0∗ .

(8.120)

Under the assumption of a frequency-independent transducer/front-end with phase shift −γD , the input voltage to the PLL is then equal to GD (ej (ω F t−γ D ) x˜0 + e−j (ω F t−γ D ) x˜∗0 ) with x˜0 = FD (jωF + s)f˜0 . ′ ) (for simplicity with unit The VCO output voltage is sin Ψ = sin(ωF t − γref amplitude). The PLL equation can be derived by performing the synchrodemodulator multiplication, ignoring the higher harmonics, and using the well˙ − ω0 = −KVCO vin : known VCO equation Ψ ′ =− ωF − ω0 − sγref

KVCO GD ′ F (s)Im[FD (jωF + s)f0 ej (γ r e f −γ D −ϕ 0 ) ]. 2

(8.121)

ω0 is the VCO frequency at zero input, which at room temperature is set normally in the vicinity of the resonator resonance. The loop filter F (s) has to be chosen in such a way as to guarantee a good noise filtering. Often a PID controller is used. In any case, the bandwidth of the PLL is small. After start-up, during normal operation with very slow temperature and environmental changes, this allows one to neglect the operator s within the

8.5 Rate gyroscopes

419

expression for the resonator’s transfer characteristic: FD (jωF + s) ≃ FD (jωF ) = |FD (jωF )|e−j ϕ D (ω F ) . Therefore, the PLL equation can be rewritten s ∆ϕ = ωF − ω0 −

dϕD 1 F0 − KVCO GD |FD (jωF )|F (s) sin ∆ϕ (8.122) dt 2 mS + mD

with ′ ∆ϕ = γref − ϕ0 − ϕD − γD .

During the fast start-up phase or during recovery cycles after locking has been lost the full equation (8.121) should be used, for instance, for simulations using MatLab/Simulink.9 A stationary case (“s = 0”) is possible only if the resonator drift dϕD /dt is constant or zero. However, even in this situation the VCO phase ′ does not completely compensate for the phase shifts ϕ0 + ϕD + γD but only γref with a small phase error ∆ϕ = 2(ωF − ω0 − dϕD /dt)/(KVCO F (0)|FD (0)|GD ). In the light of an implemented large loop gain 21 KVCO F0 |FD (jωF )|GD ≫ 1 this error is usually not critical. It can be eliminated completely using a PI controller with infinite gain at DC.

The amplitude loop The amplitude loop is controlled by the voltage Uref that determines the amplitude of the driving force via the voltage UAGC . The subscript “AGC” stands for automatic gain control. On performing the same steps as for the PLL, from the drawings it follows that UAGC is given by ′

UAGC = H(s){Uref − GD Re[FD (jωF + s)f0 ej (γ r e f −γ D −ϕ 0 ) ]}.

(8.123)

The applied force F is the result of the modulation of UAGC with the VCO output ′ + γcorr ) and of subsequent amplification by some gain stages with cos(ωF − γref −j δ D0 e followed by the capacitive actuator amplification. For instance, in the case of linear combs (see Chapter 2, Eq. (2.184)) with N fingers, with DC voltage VDC and with a gap D one gets ′ F = SD0 UAGC cos(ωF t − γref + γcorr − δ),

S = 4ε0

N HVDC D0 . D

(8.124)

′ − γcorr + δ. The additional phase γcorr has Thus F0 = SD0 UAGC and ϕ0 = γref been introduced in order to be able to compensate for the different phase shifts within the loop. On substituting these expressions into Eq. (8.123) the amplitude-control equation becomes ½ ¾ GD SD0 ′ 1+ H(s)Re[FD (jωF + s)ej (γ r e f −γ D −ϕ 0 ) ] UAGC = H(0)Uref . mS + mD (8.125)

9

A semi-analytic approach based on an approximative description of FD (jω F + s) by a series expansion in s often delivers sufficiently good results.

420

Gyroscopes

If again the case of slow changes is considered (FD (jωF + s) ≃ FD (jωF )), the equation acquires a clearer shape, (1 + K0 |hD (jωF )|h(s))UAGC = H(0)Uref

(8.126)

with K0 =

GD SD0 H(0) cos ∆ϕ, 4(mS + mD )δD ωD2

h(s) =

H(s) , H(0)

hD (jωF ) = 2δD ωD2 FD (jωF ). The normalized transfer functions, hD (jωD ) = 1 and h(0) = 1, are introduced for convenience. The loop gain K0 is then the maximal resonance gain for constant amplitudes and phases. With x0 = |FD (jωF )|[SD0 /(mS + mD )]UAGC the equation can be rewritten with respect to the driving amplitude, · ¸ 1 H(0)SD0 + K0 h(s) x0 = K1 Uref , K1 = . (8.127) |hD (jωF )| 2(mD + mS )δD ωD2

Correct phase setting After substituting for ϕ0 , the phase error ∆ϕ is, according to (8.122), ∆ϕ = γcorr − γD − ϕD − δ.

(8.128)

In order to drive the resonator at resonance, ϕD must be set to π/2. Thus, the correcting phase shift must be kept equal to γcorr =

π + γD + δ 2

(8.129)

over the whole temperature range. Since the PLL holds the phase error close to zero, a violation of this condition leads to a resonator phase different from π/2 and, thus, to a driving frequency ωF 6= ωD . Therefore, the challenges in electronic design are the temperature- and agingindependent phase shifts γD , ϕD , and δ, or, at least, well-controlled temperature dependencies that can be compensated for by a corresponding γcorr (T ). For high-precision applications the choice of γcorr (T ) is usually made during an initial trimming procedure of the gyroscope at various T. Here γcorr (T ) of the VCO/signal generator is set to the value at which the driving excitation approaches its maximum. The large effort necessary often limits such an approach to a single temperature setting.

Sensitivity and amplitude control The AGC voltage is aimed at setting a certain excitation amplitude x0 . Since x0 = |FD (jωF )|[SD0 /(mS + mD )]UAGC , the necessary reference voltage can be calculated. For sufficiently large loop gains K0 > 1 an estimate is immediately

8.5 Rate gyroscopes

421

derived from Eq. (8.126):10 2Uref x0 ∼ . = GD

(8.130)

According to Eq. (8.84) and Fig. 8.9 the sensitivity SΩ of a non-resonant rate gyroscope with frequency split is SΩ =

HI (0)G0 x0 cos(γref − ϕ0 − ϕD − γB ). 2△

(8.131)

HI (0) is the gain of the electronic output stages. ′ − ϕ0 − ϕD − γD close to zero, but The PLL keeps the phase error ∆ϕ = γref ′ ad hoc not the difference ∆ϕ = γref − ϕ0 − ϕD − γB . If the equality γB = γD is guaranteed by a proper design of the motor-sense and rate front-end stages, both ′ and γref as well as both phase errors become identical, which is VCO phases γref the normally implemented operating condition for both in-phase and quadrature synchro-demodulators. In order to guarantee with high accuracy a temperature-independent sensitivity, the T-dependency of the product G0 x0 /△ must be eliminated or compensated by the output stages HI (0, T ). Since x0 ∼ = 2Uref /GD , the sensitivitydominating product G0 x0 /△ = G0 Uref /(GD △) will have a reduced T-variation if identical temperature dependencies of the two transducer channels are implemented. The remaining drift is lowered towards the impact of the frequency-split variation. A convenient way to eliminate the residual temperature dependencies is to program the reference voltage Uref (T ) so that the corresponding sensitivity changes are compensated for via a temperature-dependent drive excitation x0 (T ). The precondition is an accurate enough temperature sensor. The trimming procedure requires then a rate measurement at at least three different temperatures and is a costly procedure, having significant impact on the final price.

Spring nonlinearities and the resonator transfer function Spring nonlinearities are often discussed as disturbing effects. However, moderate nonlinearities may play a positive role during the trimming and set-up procedures of the gyroscope. The dynamic equation of a nonlinear drive resonator according to Fig. 8.8 is µ ¶ x2 FD x ¨ + 2δD ωD x˙ + ωD2 1 + 2 x = . (8.132) xnl mS + mD It corresponds to a cubic deflection-to-force relation, Fspring = kD (x)x = kD0 x + kD3 x3 , which is typicalp for most of the MEMS springs. The nonlinearity amplitude is given by xnl = kD0 /kD3 and approaches infinity for a linear spring. 10

Note that the synchro-demodulator gain was set to 21 . For practical calculations the real gain has to be added to the transducer and front-end gain.

422

Gyroscopes

For a selective system like a harmonic oscillator that operates close to the resonance frequency the first harmonic dominates the oscillation. Therefore, it is possible to substitute the contribution of the nonlinear term by the impact of its first harmonic [Landau and Lifshitz 2003]. More precisely, with x = x0 cos ψ, ψ = ωF t + ϕ0 , the term x3 /x2nl is substituted by its harmonic linearization11 Z π x3 1 jψ ∗ −j ψ = c e + c e with c = dψ x30 cos3 (ψ)e−j ψ . (8.133) 1 1 1 x2nl 2πx2nl −π On inserting the result into Eq. (8.132) one gets the nonlinear transfer function FD (jωF , x0 ) = |FD (jωF , x0 )|e−j ϕ D (j ω F ,x 0 ) , 1 |FD (jωF , x0 )| = p 2 , 2 2 [ωD (1 + 3x0 /(4xnl )) − ωF2 ]2 + 4δD2 ωD2 ωF2 µ ¶ 2δD ωD ωF ϕD (jωF , x0 ) = arctan . ωD2 (1 + 3x20 /(4x2nl )) − ωF2

(8.134)

The formula reveals that the resonance frequency of the nonlinear resonator shifts with growing deflection amplitude: s 3x2 fD,res = fD 1 + 20 . (8.135) 4xnl For moderate nonlinearities the relation may be very helpful for an initial setting of a desired drive amplitude: the drive force is increased until an implemented mechanical stop-point is reached. In parallel the corresponding resonance-frequency shifts are measured. At the stop-point with known x0,Stop the nonlinearity xnl can be identified and used for the final setting of a desired drive amplitude or of a corresponding frequency shift. However, the nonlinear spring may cause instabilities. In operation they have been safely excluded. Let’s consider the amplitude loop with an embedded nonlinear resonator. For simplicity in Eq. (8.127) the stationary case in which h(0) = 1 and ∆ϕ ∼ = 0 is assumed: "p # [ωD2 (1 + 3x20 /(4x2nl )) − ωF2 ]2 + 4δD2 ωD2 ωF2 (8.136) + K0 x0 = K1 Uref . 2δD ωD ωF For a given amplitude of the driving force, determined by Uref , the deflection x0 reaches its maximum at the frequency at which ωD2 (1 + 3x20,m ax /(4x2nl )) − ωF2 = 0. Hence, x0,m ax = K1 Uref /(1 + K0 ). Inserting this into Eq. (8.136) and squaring 11

A more accurate estimate of higher harmonics (2k + 1)ψ and sub-harmonics ψ/(2k + 1) can be found in corresponding handbooks on nonlinear oscillations or in MEMS-related publications such as Braghin et al. [2007].

423

8.5 Rate gyroscopes

Open-loop transfer function for different QD

(a)

(b)

1

Nonlinear transfer function for different loop gain 1

fD = 10 kHz; xnl = 50 µm;

K0=100 0.8

x0max = 5 µm;

x0 /x 0max

x0 /x 0max

0.8

QD = 1000

0.6

0.4

0.2

0

fD = 10 kHz;

K0=10

QD= 5000;

0.6

xnl= 50 µm; 0.4

QD = 100000

-10

x0max = 5 µm

QD = 10000

K0=0

0.2

0

10

20

30

40

0

-10

0

fF - fD (Hz)

10

20

30

40

50

ff - fD (Hz)

Figure 8.18 Transfer characteristic of a drive resonator with nonlinear springs. (a) The

open-loop drive resonator response. (b) The closed-loop resonator response.

delivers the conditional equation for the x0 (fF − fD ) dependency (fF ≈ fD ): (· ) · µ ¸2 ¶ ¸2 3x20 (K0 + 1)x0,m ax 2 2 4 2 ωD 1 + 2 − ωF = 4δD ωD − K0 − 1 . 4xnl x0

(8.137)

Near resonance the equation can be easily solved with respect to (fF − fD )(x0 ). Figure 8.18(a) shows the inverse function x0 (fF − fD ) for various Q-factors. With increasing quality factor the dependency features a more and more pronounced hysteresis. The curves consist of three branches: the upper and lower stable stationary states and the middle, unstable state between the points where the slope dx0 /df becomes infinite. On increasing the frequency fF the drive deflection grows until it approaches its maximal value, before jumping down during a further infinitesimal step. Vice versa, on decreasing the frequency from large values the stationary state approaches the point at which the slope dx0 /d(fF − fD ) becomes infinite. Here the amplitude jumps up to the corresponding stable status on the upper branch and decreases further with falling frequency. In Fig. 8.18(a) this behavior is labeled for the resonator with Q = 1000. Fortunately, on embedding the resonator into the control loop the increasing loop gain mitigates the tendency towards forming a hysteresis as demonstrated in Fig. 8.18(b). In order to exclude the emergence of the hysteresis it is sufficient to fulfill the following condition: 3 4

µ

x0,m ax xnl

¶2

QD ≤

p K0 + 1.

(8.138)

This inequality results from investigating the upper bound where the precondition for an infinite slope vanishes. The equation that determines the slope is derived by differentiating Eq. (8.137). The infinity-slope points can then be analyzed.

424

Gyroscopes

8.6

Gyroscope architectures The following discussion is focused on capacitive gyroscopes, although the suspension-related considerations can be extended to other types of gyroscopes.

8.6.1

Mode-decoupling architectures In 1D gyroscopes at least two modes exist: the primary or drive mode and the secondary or sensing mode. They represent the two main modes. Intentionally introduced additional modes (DOFs) may exist. The design of continuously improving gyroscopes with high sensitivity, low bias, and good bias stability leads to the question of how single or multiple vibrating masses with two orthogonal main modes should be arranged and suspended in order to be robust against imperfections in geometry and material parameters. Minimization of cross-coupling effects between the drive and sensing modes is the key element. The main modes should interact by the Coriolis force or torque only. Thus, “mode decoupling” means that the parasitic interactions between the main modes caused by the cross-coupling effects are reduced in comparison with a more or less arbitrarily chosen reference. Unfortunately, the methods of sensitivity analysis are not well suited to derive a general approach for solving the mode-decoupling problem. As a result a great variety of different “solutions” exist in the literature, many of them protected by patents. It is often difficult to quantify them. One criterion for the efficiency of mode decoupling should be the resulting uncompensated ZRO, especially the Q-bias, which is related to the full-scale rate signal. It should be accompanied by the bias-stability performance – the figure of merit for the “goodness” of a gyroscope.

8.6.2

z-Gyroscopes A general statement can be made with respect to the mode-decoupling capability of a z-gyroscope architecture: a single-mass system requires a 2D suspension that must per se be compliant in two orthogonal, in-plane directions. For a zgyroscope this may be crab-leg-, serpentine- (or similar) or H-type suspensions as described in Chapter 3, in the section “Suspensions.” As mentioned earlier, small asymmetries within such suspensions cause large cross-coupling effects and, thus, a large Q-bias.

Mode decoupling by frame-based architectures This is why modern capacitive z-gyroscopes use mode-decoupling measures, for instance, in the form of frame-based architectures. The bodies and frames are linked with each other or with the substrate by 1D suspensions that are much less prone to cross-coupling effects.

8.6 Gyroscope architectures

(a)

425

(b)

sense mass

sense mass

drive frame

drive frame

Figure 8.19 Drive-frame architectures. (a) Type I: drive frame suspended from the

substrate. (b) Type II: drive frame with sensing mass suspended from the substrate.

(a)

(b)

drive mass

drive mass

sensing frame

sensing frame

Figure 8.20 Sensing-frame architectures. (a) Type III: sensing frame suspended from

the substrate. (b) Type IV: sensing frame with drive mass suspended from the substrate.

There are two variants: drive-frame-based (with outer drive frame) and sensing-frame-based (with outer sensing frame) as illustrated by Figs. 8.19 and 8.20. However, as will be shown, what is more important is the kind of suspension of the drive mass. The outer frames can be suspended directly from the substrate as shown in Figs. 8.19(a) and 8.20(a) (see also Palaniapan et al. [2003]), or vice versa, the inner masses can be suspended from the substrate anchors as in Figs. 8.19(b) and 8.20(b). The outer frames are then carried by the inner masses. The advantages and disadvantages can be summarized as follows.

r The architectures of Types I and IV with direct suspension of the drive frame and drive mass, respectively, from the substrate (Figs. 8.19(a) and 8.20(b)) have the advantage that forces that are orthogonal to the drive direction and caused by the rotation of the driven-mode axis with respect to the drive axes meet a large resistance in the y-direction. Hence, the forces are balanced by the counter-forces of the drive springs that are stiff in the

426

Gyroscopes

y-direction. The coupling energy is transferred to the substrate rather than to the sensing mass. Of course, the x-motion of the sensing mass may be still misaligned with respect to the zero-quadrature axis. However, since this misalignment is mainly caused by the disorientation of the sense springs, which are very stiff in the x-direction, it is normally much less than the drive-axis misalignment. For a given drive resonance the drive springs are stiffer than in the cases in which only one of the masses is performing a drive movement: kD = ωD2 (mD + mS ). The Coriolis force is proportional to the sensing mass only; however, the damping of the sensing element is smaller than that in the cases of co-moving masses. Thus, the sensing gain for resonant sensing is increased. Owing to the possible impact of changing fringe fields, the large xamplitude of the inner sensing mass may be a disadvantage if proper countermeasures are not implemented (side shields in the case of sensing boxes). The available space for drive combs in Type IV gyroscopes is usually less than that in the Type I architecture, because the inner mass is geometrically significantly smaller than an external drive frame. More sophisticated topologies can resolve such problems. r If the drive frame or the drive mass is suspended from the sensing mass/frame as in Figs. 8.19(b) and 8.20(a) (Types II and III; for Type III see also Palaniapan et al. [2002]) the misalignment of the driven axis with respect to the drive axis is directly transferred into a y-motion of the inner sensing mass or outer sensing frame, respectively. The drive conditions are better than for Types I and IV, because the sensing frames do not co-move with the drive frame, and the drive forces needed are smaller. However, drive-force generation is usually not a critical point. The Coriolis force is now proportional to the drive mass. The sense damping is determined by both bodies, because the drive mass follows also the Coriolisforce-induced sensing movement. An advantage may be the suppressed drive motion in the sensing frame, which allows one to neglect electrostatic fringe-field changes. Architectures I and IV are superior with respect to mode decoupling, provided that the impact of fringe-field changes is eliminated. However, sophisticated elastic couplings between the drive and sensing frames may mitigate the drive-axis misalignment projection as well. The structures discussed above are normally used within a tuning-fork arrangement in order to suppress external accelerations. The external coupling springs impose geometric limitations on the arrangement of frame and combs so that the preference for one of the two types is dictated by geometric considerations (optimal usage of a given footprint). A successful implementation of the Type III architecture within two anti-phase-driven, mechanically independent cells was performed by AnalogDevices in their ADXRS gyroscopes [Geen and Krakauer 2003]. The gyroscope

8.6 Gyroscope architectures

(a)

sensing frame

427

(b)

Coriolis frame

drive frame

Coriolis frame

drive frame

sensing frame

Figure 8.21 Double-frame decoupling. (a) The closed double-frame architecture.

(b) The open double-frame architecture.

operates under atmospheric pressure, which, according to Geen and Krakauer [2003], should lead to a good insensitivity to shocks and vibration.

Doubly decoupled z-gyroscopes The decoupling mechanism described above can be extended towards fixing not only the orientations of the driven axis but also the alignment of the sensing axis. This would further reduce the misalignment between the driven-mode and the zero-quadrature axes and completely eliminate the impact of changing electrical fringe fields.

Doubly decoupled frame architectures One approach is based on a double-frame architecture as shown in Fig. 8.21(a) (according to Willig et al. [2004]). The system is separated into three frames: the first is the drive frame suspended from the substrate, and thus with restricted guidance for the driven-mode axis. The second is the inner nested sensing frame, also suspended directly from the substrate, preventing motions in the x-direction. Only the third, intermediate frame is able to perform deflections in the xand y-directions. It is the only frame generating a Coriolis force, and is often called the Coriolis frame. The Coriolis frame decouples drive-induced orthogonal forces from the sensing frame and sensing-axis misalignments from the drive frame. Such a cell embedded within a tuning-fork architecture is used by the company Bosch in its product MM3. Figure 8.22 shows a SEM √ photograph of the MEMS structure. The noise floor is about < 0.003◦ /s per Hz and the typical bias instability is less then 3◦ /hr [Ernst 2007, Gomez et al. 2005, Neul et al. 2007, Willig and Moerbe 2003]. It must be noted that in order to embed two such doubly decoupled cells into a tuning fork the Coriolis frames must be coupled with each other. Thus, slitting of the outer drive frame is necessary in order to insert connecting members. This disturbs the stiffness and causes additional modes, especially with out-of-plane oscillating free ends of the slitted frame. Hence, the

428

Gyroscopes

Figure 8.22 Bosch’s doubly decoupled z-gyroscope. Courtesy of Bosch GmbH.

corresponding FEM design requires careful placement of the most dangerous modes outside the operating frequencies. As an alternative to the closed-frame architecture, an open-frame arrangement was proposed by Geiger and Lang [2002]. The topology is illustrated in Fig. 8.21(b). The Coriolis frame transfers the drive motion plus the Coriolis deflection to the outer sensing frame, where the drive motion is absorbed by the x-stiff sensing springs. The conjunction of two such structures in a tuning fork can be achieved, for instance, by encasing the sensing frame nearly completely by the Coriolis frame and coupling the outer right edge of the enhanced cell with its mirrored counterpart.

Doubly decoupled frame-free topologies The principle of restricting the drive motion and the sensing motion to axial motions by suppressing the orthogonal components is not necessarily bound to frame structures. For instance, Alper et al. [2007] realized prototypes of SOI gyroscopes according to the topology shown in Fig. 8.23. Drive and sensing members are fixed in their spatial orientation by suspensions from the substrate that are stiff in the respective orthogonal directions. The proof mass is suspended symmetrically with respect to the drive and the sensing components. The misalignment of the proof mass is therefore not transformed to the sensing axis. Alper et al. [2007] reported an uncompensated Q-bias of less then 70◦ /s, demonstrating the decoupling capabilities of such a design. However, the integration of the cell described above into a tuning-fork gyroscope seems to be difficult, because one-sided coupling structures seriously disturb the intended symmetry of the design.

8.6 Gyroscope architectures

429

Proof mass

drive sense

Figure 8.23 A doubly decoupled proof mass. Adapted from Alper et al. [2007].

8.6.3

In-plane-sensitive gyroscopes For 1D gyroscopes there are five possible approaches to measure in-plane rate signals using flat MEMS structures:

r lateral drive oscillation of the (flat) proof mass along one of the in-plane axes

r r r

r

and sensing the Coriolis deflection that is orthogonal to the drive and to the in-plane rate motion, i.e. – out-of-plane out-of-plane drive of the proof mass and sensing the lateral Coriolis deflection anti-phase linear drive oscillation (tuning fork) and out-of-plane rotation of the embedding frame structure rotatory drive oscillation of the proof mass about the z-axis and sensing of the out-of-plane tilt about the orthogonal in-plane axis (in-plane drive – out-of-plane sensing) rotary drive about the in-plane axis that is orthogonal to the rate axis, and sensing of the in-plane rotation about the z-axis (out-of-plane drive – in-plane sensing).

The first two approaches represent linear gyroscopes, the last two correspond to torsional gyroscopes. The third approach consists of a combination of linear and rotatory oscillations.

In-plane-sensitive linear gyroscopes A possible representative of a linear in-plane-sensitive gyroscope is shown in Fig. 8.24 (see Bhave et al. [2003] and also Xie and Fedder [2001]). The proof mass is suspended by relatively long and wide springs to the sensing frame. They are compliant mainly in the z-direction. The proof mass is excited by a voltage applied to the underlying electrode. The Coriolis deflection, caused by a rate signal in the y-direction, is transferred to the sensing frame and detected by the x-sensitive boxes. In comparison with the alternative version of a linear in-plane gyroscope, where the drive motion is performed in-plane and the sensing deflection happens along the z-axis (see Mochida et al. [1999]), the z-drive version has the advantage

Gyroscopes

drive mass

sensing frame

430

Ωy

x

Figure 8.24 An out-of-plane-driven and in-plane-sensitive gyroscope. Adapted from Bhave et al. [2003].

that the dominant squeeze damping of a sensing plate would be overwritten by the drive forces. The resulting possibility of atmospheric-pressure operation is beneficial for cost-efficient package solutions that require hermeticity but not a vacuum environment. Since the proof mass can be excited by the underlying electrode only, both versions need Σ∆ techniques in order to implement a linear force feedback as in the case of single-sided accelerometers. Both versions are also sensitive to external accelerations in the x-direction as well as in the z-direction. In comparison with torsional gyroscopes they have been able to gain remarkable commercial success.

Linear–rotatory gyroscopes Tuning forks give the unique possibility of creating architectures that combine a linear drive with a rotatory sensing motion. One of the first MEMS gyroscopes designed by the Draper Laboratory and manufactured in a silicon-on-glass technology was such a tuning-fork system with two masses driven in anti-phase, where the out-of-plane tilting was sensed [Bernstein et al. 1993]. The measured rate was, thus, in-plane. However, architectures for in-plane-driven and in-plane sensing systems are also possible. The basic idea is the combination of linear drive motion and rotatory sensing motion. The principle is illustrated in Fig. 8.25 for a z-sensitive gyroscope [Cadarelly 2007, Ichinose 2004] (possible drive-coupling springs are not shown). Two masses, m1 and m2 , vibrate in anti-phase along the x-axis. Here, an out-of-plane rate signal ΩZ creates anti-phase Coriolis forces FCor1 and FCor2 . Whereas in a standard tuning fork for in-plane measurements such torque is absorbed by the drive springs, here the masses are nested in a frame that can rotate around the z-axis. The resulting torque about the z-axis rotates the whole frame, and the rotation can be sensed by radially arranged sensing boxes. The governing equations for in-plane as well as out-of-plane rate signals are derived from the tuning-fork drive mechanism according to Eq. (8.118), and from

8.6 Gyroscope architectures

FCor1

proof mass 2

proof mass 1

FCor2

431

sensing boxes

drive combs

Figure 8.25 A linear–rotatory z-gyroscope.

the moment equation for the sense deflection: FD , mD ˙ Z, + cθ Z θ˙Z + kθ Z θZ = 4LmD xΩ ˙ ˙ Y. + cθ y θY + kθ Y θY = 4LmD xΩ

x ¨ + 2δD ωD x˙ + ωH2 x = Jθ Z θ¨Z Jθ θ¨Y Y

(8.139)

The Jθ Y ( Z ) = J2(3) are the moments of inertia of the whole structure, and L is the distance between the center of the system and the COG of one of the masses. If kθ Y → ∞, the system reacts to out-of-plane rates ΩZ . If, vice versa, kθ Z → ∞, a y-gyroscope is formed, in which the central ring must be suspended in order to tilt about the y-axis and be stiff about the x- and z-axes. Differential sensing of the Coriolis oscillation by underlying plate electrodes is then the method of choice. Hence, using very similar frame structures, it becomes possible to sense outof-plane as well as in-plane rate signals.

8.6.4

Torsional gyroscopes Torsional MEMS gyroscopes based on planar proof masses are sensitive with respect to in-plane rate signals and are not well suited for z-gyroscopes. Indeed, for slowly changing, small-rate signals Eq. (8.47) reduces to J1 θ¨1 + cθ,1 θ˙1 + kθ,1 θ1 = −2J2D Ωe,2 θ˙3 + 2J3D Ωe,3 θ˙2 + MD,1 , J2 θ¨2 + cθ,2 θ˙2 + kθ,2 θ2 = 2J1D Ωe,1 θ˙3 − 2J3D Ωe,3 θ˙1 + MD,2 , J3 θ¨3 + cθ,3 θ˙3 + kθ,3 θ3 = −2J1D Ωe,1 θ˙2 + 2J2D Ωe,2 θ˙1 + MD,3 .

(8.140)

Thus, a rate signal about the z-axis generates the Coriolis torques R2J3D Ωe,3 θ˙2 and 2J3D Ωe,3 θ˙1 that are proportional to the dyadic moment J3D = ρ V z 2 dV = ρAH03 /12 = mH02 /12. Since the vertical dimension of plate-like structures is

432

Gyroscopes

much smaller then the lateral dimensions, the moment J3D is orders of magnitude smaller then the in-plane dyadic moments: J3D ≪ max(J1D , J2D ). Hence, torsional z-gyroscopes require technologies allowing one to extend the z-dimension of the structure accordingly, as, for instance, in the first batchmanufactured micromachined gyroscope described by Greiff et al. [1991]. At least thick SOI or bulk-micromachined layers as in Kuisma et al. [1997] or Saukoski et al. [2005, 2006] should be exploited. Therefore, as acknowledged by the authors (C. Acer, personal communication, 2010) the concept of a flat, rotational zgyroscope (see Acar [2004] and Acar and Shkel [2009]) is very questionable. It features a near-zero sensitivity and seems to be the result of a wrong calculation of the Coriolis moment.12 A similar trial was undertaken by Alper and Akin [2000] using a very flat but large plate. The plate oscillates about an in-plane axis and is nested within a driving gimbal that oscillates about the orthogonal in-plane axis. However, the authors neither included models nor reported any achieved sensitivity values.

1D torsional gyroscopes A flat, in-plane-sensitive torsional gyroscope exploits the Coriolis torques 2J2D Ωe,2 θ˙3 , 2J1D Ωe,1 θ˙3 , 2J1D Ωe,1 θ˙2 , and/or 2J2D Ωe,2 θ˙1 . As mentioned in Section 8.6.3, the two possibilities regarding how to build a 1D gyroscope are

r in-plane drive and out-of-plane sensing using one of the first two Coriolis torques

r out-of-plane drive and in-plane sensing using one of the two last Coriolis torques. Both concepts have been successfully used for commercial products, the last in a mode-decoupled arrangement to be considered in the section “Decoupled torsional gyroscopes”.

Torsional gyroscopes with in-plane drive The simplest variant of a 1D, z-driven gyroscope is shown in Fig. 8.26 and goes back to Funk et al. [1999]. The butterfly-like proof mass is driven by radial combs into oscillations about the z-axis. The action of the Coriolis torques is limited to rotations about the y-axis. Tilt about the x-axis is suppressed by virtue of the design of the central suspension. Therefore, with θ1 = 0 and the notation Ωe,1 = Ωx , the governing equations become J2 θ¨2 + cθ,2 θ˙2 + kθ,2 θ2 = 2J1D Ωx θ˙3 , J3 θ¨3 + cθ,3 θ˙3 + kθ,3 θ3 = −2J1D Ωx θ˙2 + MD,3 . 12

(8.141)

Equation (6.8) in Acar [2004] and that on page 24 plus Chapter 7 in Acar and Shkel [2009] are based on a Coriolis torque calculated with a factor (−Jx + Jy + Jz ) instead of (Jx + Jy − Jz ) = 2JzD .

8.6 Gyroscope architectures

433

y drive combs

sensing plate

z z-

compliant suspension

x

Y

y

sensing electrodes

Figure 8.26 An in-plane-driven and out-of-plane-sensitive gyroscope.

In analogy to Eq. (8.72) the mechanical sensitivity yields θ2,0 JD θ3,0 θ3,0 ∼ = 1 p . =p 2 2 2 2 Ωx J2 △ + δS ωS △ + δS2 ωS2

(8.142)

R For flat structures, |zm ax | ≪ |xm ax |, the inertial moment J2 = ρ V dV (x2 + z 2 ) R is nearly identical with the dyadic moment J1D = ρ V dV x2 , so to a good approximation the last relation in Eq. (8.142) holds. The long butterfly structure in conjunction with the underlying fixed electrodes contributes to a large sensing capacitance, leading to a remarkable sensitivity. The company Bosch manufactured and sold millions of parts fabricated according to such a design under the name SMGXXX (MM2-type) [Ernst 2007, Thomae et al. 1999]. The structure of Eq. (8.141) is identical with that of linear gyroscopes (Eq. (8.49)). Clearly, all of the parasitic effects discussed for linear gyroscopes are present within torsional gyros. Since the central 2D suspension used is prone to misalignment errors and anisoelasticity effects, stringent requirements with regard to manufacturing tolerances must be applied. Otherwise large Q-bias may result. This leads to the need to seek decoupled structures of torsional gyroscopes.

Decoupled torsional gyroscopes The principles of decoupling are the same as for linear z-gyroscopes and as described in the section “Mode decoupling by frame-based architectures.”

434

Gyroscopes

sensing plate X

torsion spring

Figure 8.27 A decoupled gyroscope with inner drive. Adapted from Geiger et al. [1998].

In-plane-drive–out-of-plane-sensing torsional gyroscopes For instance, the analogy to the Type IV architecture is reflected by the approach of Geiger et al. [1998] (see also Geiger et al. [1999, 2002]) and illustrated in Fig. 8.27. The inner drive is suspended by springs that are stiff with respect to rotations about both in-plane axes (1D suspension). The 1D torsional springs between the outer frame and the drive mass force both bodies to co-rotate about the z-axis. Under the impact of a Coriolis force generated by a rate signal, ΩX , the outer frame seesaws about the lateral axis of the torsional springs (∼ y-axis). Underlying electrodes allow for differential sensing. They should feature sufficiently large surroundings to eliminate the impact of changing fringe fields that may emerge due to the co-movement with the drive. Since the outer frame is extended in the x-direction, the structure is quite sensitive and allows one to achieve noise levels of about 0.025◦ /s within a 50-Hz bandwidth, as demonstrated using a thick SOI technology (Geiger et al. [2002]). If the outer frame is suspended by torsional springs from the substrate and the inner drive disk by radial springs not from the substrate but from the outer sensing frame, a Type III configuration results. Type I and Type II topologies may be derived in the same way. This exercise is left to the reader. Also doubly decoupled configurations with an intermediate ring-like Coriolis frame can be realized.

Decoupled torsional gyroscopes with three DOF It is not easy to realize a 1D z-torsional suspension with long radial springs. Such springs are needed, for instance, for a Type II architecture with inner sensing plate and attached outer drive frame. The topology is shown in Fig. 8.28(a). For SMM layers the limited aspect ratio of the springs leads to a 3D suspension that is compliant for rotations about the z-axis as well as about the x- and yaxes. Compliance about the x-axis could be suppressed by changing the angles between the springs. However, since the sensing plate is suspended by a nearly ideal 1D torsional y-suspension, oscillations of the outer ring about the x-axis are not transferred to the plate anyway and can be neglected. Nonetheless, at least three DOF are present within this arrangement described by Reeds and Hsu

8.6 Gyroscope architectures

(a)

(b) 10

435

0

frequency split

ring

10

RZ

-1

10

-2

sensing response

X

plate

10

-3

drive response 10 PY

-4

10

4

frequency (Hz)

10

5

Figure 8.28 Three-DOF torsional gyroscopes with outer drive. (a) A decoupled

torsional gyroscope with outer drive and two-DOF sensing. Adapted from Reeds and Hsu [2002]. (b) The drive and sensing transfer functions.

[2002] and Reeds et al. [2003]. If the outer ring is excited, the Coriolis torque D 2JR,1 ΩX θ˙R,Z causes a common rocking of ring and plate about the y-axis that can be sensed differentially by the two electrodes under the right and left sides of the plate. Assuming an ideal 1D suspension of the plate and small rate signals, the relevant dynamic equations for ring and plate can be derived using, for instance, Eq. (8.140): JR,Z θ¨RZ + cRZ θ˙RZ + kRZ θRZ = MZ , JR,Y θ¨RY + cRY θ˙RY + kRY (θRY − θPY ) = 2JXD ΩX θ˙RZ , JP,Y θ¨PY + cPY θ˙PY + kPY θPY = kRY (θRY − θPY ).

(8.143)

The indices “R” and “P” stand for “ring” and “plate.” θRZ , θRY , and θPY are the rotation angles of the ring about the z-axis and of the ring and the plate about the y-axis, respectively. X, Y , and Z substitute for the former axis notation e¯1 , e¯2 , 2 2 2 and e¯3 . With ωRZ = kRZ /JR,Z , ωRY = kRY /JR,Y , ωPY = (kRY + kPY )/JP,Y , and the corresponding damping values the mechanical transfer function for constant-rate signals is θPY 2J D ωθRZ 0 KRY =j X ΩX JPY ×

1 4 , 2 2 − ω 2 + 2jδPY ωPY ω) − βωRY − ω 2 + 2jδRY ωRY ω)(ωPY (ωRY (8.144)

where β = JRY /JPY is the ratio of the inertial moments of the ring and the plate. In contrast to the two-DOF sensing considered in Section 8.5.3, under the heading “Three-DOF non-resonant sensing,” the two resonances of the scale factor are positioned far away from each other. For a typical design with fRZ = 10 kHz, fRY = 15 kHz, fPY = 45 kHz, and a ratio of the inertial moments of

436

Gyroscopes

Y

C1

C3 sensing axis

drive ring

C2

sensing cap C4

Figure 8.29 An example of a torsional tuning-fork gyroscope. Adapted from Geen

[1999].

β = 4.5 the drive and sensing transfer functions are as shown in Fig. 8.28(b) (QRZ = 1000, QRY = QPY = 300). The third DOF leads to a high-frequency resonance, here around 45 kHz, and to a shift of the common sensing resonance from 15 kHz for the ring alone down to a value close to the drive resonance. It allows one to adapt the spring values freely in order to set a predefined frequency split between the drive and the common sensing resonances that is, in the example given, around 300 Hz. As for all architectures of Type II, the misalignment between the drive axis z and the driven axis is directly projected into a quadrature oscillation of the plate. However, use of an appropriate multiple-spring suspension of the ring mitigates manufacturing imperfections, especially sidewall tilts of the springs, to some extent. The inner sensing springs are intrinsically rigid and, thus, need careful design in order to optimize an appropriate stress relief over the operating temperature range.

Torsional tuning forks The extension of the tuning-fork principle to a torsional gyro is obvious (see e.g. Geen [1999]). The opposite linear drive motions have to be substituted by antiphase torsional oscillations as shown in Fig. 8.29. The two rings with butterflyshaped trusses are excited by the drive combs attached to the blade-edges. The oscillation is mechanically synchronized by the central coupling springs, which are stiff in the tangential direction but compliant for deflections along and twisting about the sensing axis (x). Motor-sense combs may be located inside the rings or arranged as part of the outer drive combs. An applied rate signal ΩY creates anti-phase tilting moments about the output axis (x). The capacitances C1 and C4 change in a direction that is opposite to the changes of C2 and C3 . The rate can be extracted by differential measurement of the pairs C1 + C4 and C2 + C3 . As in the case of linear tuning forks, external accelerations in the

8.6 Gyroscope architectures

y- torsional drive spring

drive ring

sense ring

437

drive combs

sensing capacitors

X

motor sensing capacitors

Figure 8.30 A decoupled gyroscope with out-of-plane drive and in-plane sensing.

Adapted from Adams et al. [2003].

z-direction create a common-mode change of the two capacitance pairs, which can be detected additionally. Notably, a rate signal about the x-axis, ΩX, generates a rocking moment about the two axes parallel to the y-axis and passing through the COGs of both rings. Thus, by adapting the geometry to centrally symmetric disks and separating the two ring capacitances into a left pair and a right pair, an x–y-sensitive gyroscope can be created (e.g. Geen [2004]).

Out-of-plane-drive–in-plane-sensing torsional gyroscopes In order to achieve sufficiently large drive angles out-of-plane drive architectures require a considerable gap under the rocking drive. If, for instance, a drive angle of 1◦ is to be realized using a drive structure that is 0.75 mm wide, a minimum gap of D ≥ 13 µm is needed in order to avoid any mechanical contact. Clearly, using extended underlying electrodes and sufficiently large drive voltages as, for instance, in Cardarelli [2006], such ample gaps make capacitive actuation quite difficult. An efficient way out is the exploitation of levitating comb drives as used in Kionix’s gyroscope. The patented configuration is shown in Fig. 8.30 [Adams et al. 2003]. It consists of an outer drive and an inner sensing ring. The sensing ring is here suspended from the substrate by eight beams while the two torsional springs between the drive and sensing frames allow the drive ring to perform torsional movements about the y-axis. To hold the sensing ring in-plane, a high aspect ratio of the suspending beams is needed. Thus, a thick structural layer in conjunction with a large gap underneath are the technological preconditions for such a type of gyroscope.

Gyroscopes

(a)

(b) X

piezoactuator

(c)

Z

z

z

X

piezoactuator

drive motion

piezosensor

sense motion

x

Vertical supporting beam R1 R2

drive motion piezo- sense motion sensor (torsion)

ZnO-silicon driving beam

y drive R4 R3

se ns e

438

x X

tiny sensing beams

x

Figure 8.31 Different sensing modes of a beam gyroscope. (a) A beam gyroscope with

linear drive and sensing. (b) A beam gyroscope with linear drive and torsional sensing. (c) A beam gyroscope according to Zhang et al. [2006].

The sensing capacitances are implemented as interdigitated fingers located between stationary fingers. Outer ring-shaped capacitances sense the rocking motion of the drive ring and serve as input transducers for the drive-stabilization feedback loop. The levitating comb drives that are attached on one side to the inner ring and on the other to the drive frame must be driven by applying different voltages to each of the rings. Hence, an electrical-isolation technique is needed in order to create different potentials on the two rings. The technology used allows the creation of electrically isolated regions of the moving structures by introducing trenches filled with dielectric material. Drive and ring are decoupled not only mechanically as in all previous architectures but also electrically. A previous version of such a gyroscope but with inner out-of-plane drive and outer in-plane sensing was described earlier in Adams et al. [1999].

8.7

Non-planar MEMS gyroscopes In the early phase of MEMS gyroscopes many undertakings were not yet focused on planar structures and technologies close to standard CMOS equipment. Also miniaturization was not yet on the present level, with today’s overall sizes of inertial MEMS less then 2 mm × 2 mm. Batch-fabricated gyroscopes using not only established microelectronic materials but also glass, quartz etc. have been successfully developed and commercialized. However, they are increasingly being replaced by smaller and cheaper planar structures. Despite the fact that most of the early developments ended up in the cul-de-sac of limited miniaturization capabilities, some of the underlying principles have experienced a renaissance in the last few years. Among them may be promising candidates for multi-DOF sensing of rate signals and linear accelerations.

8.7.1

Beam gyroscopes The principal arrangement of a beam gyroscope is shown in Figs. 8.31(a) and (b). The customarily used configuration corresponds to Fig. 8.31(a), where a beam

8.7 Non-planar MEMS gyroscopes

439

with or without an additional mass at the tip (not shown) is vibrated in the z-direction, and the sensing deflection in the y-direction caused by the Coriolis force is captured by piezoelectric, piezoresistive, or capacitive transducers. Drive excitation is performed by piezoelectric actuators as shown in Fig. 8.31, or by electrostatic forces generated by an electrode under the beam (or additionally under the plate attached to the beam). In one of the first batch-fabricated prototypes of Maenaka and Shiozawa [1994] and Maenaka et al. [1996] the oscillating bar was embedded between two parallel electrodes on the left and the right of the beam. The bar was bonded to a glass carrier grooved under the bar and with an electrode within the groove. The bar was vertically [Maenaka and Shiozawa 1994] or in-plane [Maenaka et al. 1996] excited at resonance by a piezoelectric actuator. The small gas damping allows operation under atmospheric pressure. Drive motion and Coriolis-deflection sensing were performed capacitively. The sensor size was about 20 mm × 5 mm × 2 mm. Up to now such dimensions have been typical for beam-like gyroscopes and represent a serious obstacle to their broad use. Electrostatic drive and piezoresistive sensing were used by Li et al. [1999] and Yang et al. [2002] within a modern, bulk-micromachined two-beam gyroscope. Frequency mismatch between the two bars did not allow an experimental verification of the double-mass operation. Only single-mass operation was measured. The squeeze damping acts against the drive force and is reduced by grooves within the electrodes and the underlying silicon in order to operate the gyroscopes at atmospheric pressure. Monitoring of the vertical drive motion as well as capturing of the Coriolis deflection is performed by two piezoresistive halfbridges. The footprint of 8.5 mm × 7.5 mm is still large and is determined by the 7-mm-long beam-mass structure. Proof masses of about 3000 µm × 1000 (and 3000) µm × 400 µm have been exploited by Zhang et al. [2006] (see also Madni et al. [2008]) for a prototype of a robust, frequency-mismatch-tolerant gyroscope. The principle is illustrated schematically in Fig. 8.31(c). The proof mass is suspended by two orthogonal beams. A piezoelectric actuator on the drive beam with peak-to-peak voltages of 32 V excites the proof mass vertically to amplitudes larger than 100 µm. The sensing motion is determined by the supporting sensing beam that is surrounded by two tiny, stress-enforcing beams carrying the piezoresistors or piezoelectric sensors. The slide damping across the attached mass is small, allowing one to realize quite large Q-factors for the sensing mode at atmospheric pressure. The large drive motion together with the use of two tiny sensing beams like in the arrangement described in Chapter 7, Fig. 7.40, allows one to achieve a high resolution while keeping a reasonable footprint. In order to suppress the large acceleration dependency in the z-direction, a two-mass, anti-phase-driven arrangement can be exploited. The models of beam gyroscopes represent often-simplified two-DOF approximations following Eqs. (8.50). However, a deeper analysis, especially of the resonance frequencies, is usually based on the application of Lagrange’s or

440

Gyroscopes

Hamilton’s principles. In Esmaeili et al. [2005] the case of linear drive and sensing under axial rate signals is treated, while in Bhadbhade et al. [2008] the results for flexural–torsional vibrations of a mass-loaded beam can be found. Beam gyroscopes based on proprietary ceramics with piezoelectric readout found their way into the volume market (Murata) mainly for platform stabilization of cameras and games.

Frequency-output beam gyroscopes In Section 8.2.4, Eq. (8.46), it was shown that the stiffness and, hence, resonance frequencies of suspended bodies depend on the rate signal applied. This fact has led to the concept of beam gyroscopes with direct frequency output that is noise-robust and independent of exact amplitude settings. If a beam is excited in the z- and y-directions, the difference between the measured changes of the two resonance frequencies is, according to Moussa and Bourquin [2006], proportional to the rate signal ΩX for large enough rate inputs. Thus, frequency measurement may replace amplitude demodulation. However, the unavoidable mismatch between the two resonance frequencies leads, for smallrate signals, to a nonlinearly increasing frequency offset. Additionally, for smallrate signals the inherent parasitic mode couplings within the beam cause a lockin effect between the two excited modes. At present, it seems to be not yet clear whether refined concepts can bring this alluring approach to a practical application.

8.7.2

Quartz tuning forks Quartz-based tuning forks as well as ring gyroscopes are an inseparable part of the history of inertial MEMS. The unique advantage of quartz is its piezoelectricity, which allows one to combine high-quality vibrating structures efficiently with space-saving piezoelectric actuation and sensing. A simple tuning fork has four main modes: in-plane and out-of-plane oscillations of both tines, which may occur in phase or in anti-phase. On exciting the two tines within the fork plane to oscillate in opposite directions (anti-phase in-plane mode), a rate signal along the fork axis generates out-of-plane Coriolis forces in opposite directions, and, thus, an oscillating moment about the support (see Fig. 8.2(a)). In order to decouple the drive and sensing modes an H-shaped arrangement of two tuning forks according to Fig. 8.32(a) was the preferred configuration of the designers at BEI Sensors/Systron Donner, PanasonicMathsuishita, Fujitsu, Toyota, and other firms. Here the oscillating moment is transferred via the tuned suspension to the pick-up tines. The two resonance frequencies are positioned with a small frequency split of about 3%. Actuation and sensing can be performed by adding electrodes to the drive and sensing tines, respectively. Since for quartz the coefficients of the direct piezoelectric matrix are almost zero but d21 = −d11 and d41 = −d52 = − 21 d62 , it can be shown that

441

8.7 Non-planar MEMS gyroscopes

(a)

(b)

sensing electrodes

e ns se

can lid

X

silicon

upper pole pedestal glass

magnet

drive electrodes

lower pole

support glass

can base

tuned suspension drive

3

2

z

4

a pl

1

rm tfo

y 8

x

5 7

6

Figure 8.32 Quartz tuning-fork and ring gyroscopes. (a) A quartz tuning fork.

Adapted from Madni et al. [1996]. (b) A ring gyroscope according to Hopkin [1997].

the push–pull forces should be generated by electrical fields perpendicular to the y-surfaces of the tines but with opposite sign. The necessary field configuration is shown in Fig. 8.32(a). Therefore, electrodes are needed on all four planes of the tine. They are connected in order to form an electrical field that is symmetric with respect to the x–z-plane. Accordingly, the sensing electrodes must be located at the y-plane surfaces, in the form of double electrodes capturing the polarization difference of the upper and lower halves (in the z-direction) of a sensing tine. Notably, there are no analytical models describing the double tuning fork. Approximative two-DOF models with empirical parameter fitting may serve as guidelines for understanding the basic principle; however, for a detailed design a FEM analysis is indispensable. The quartz tuning fork of BAE Sensors √ is a high-performance gyroscope with typical noise floor down to 0.0017◦ /s per Hz (according to the Internet data sheets of the company) and very good bias stability. It started its broad market penetration in 1996 and for a long time held considerable market shares within the defense and automotive industries. The quartz forks are batch manufactured, but many complicated technology steps follow. The length of the sensing element was reduced from about 25 mm to around 8 mm. The difficulties in further scaling down the sensing element in the end led to a step-by-step replacement by other products.

8.7.3

Ring gyroscopes The principle of ring gyroscopes is based on the existence of two fundamental modes (more precisely, a mode pair) as shown in Fig. 8.2(b). The modes are

442

Gyroscopes

rotated by 45◦ with respect to each other. Under the impact of Coriolis forces the nodes shift, and energy is coupled from the first into the second resonance mode – the sensing mode (see e.g. Hopkin [1997]). British Aerospace Systems and Equipment together with Sumitomo Precision Products Company Ltd. have developed a ring gyroscope using a batchmanufactured silicon ring suspended by eight spider-leg springs as shown in Fig. 8.32(b), bottom. The about 100-µm-high ring is bonded to a glass pedestal and permeated by a constant, vertical magnetic field (Fig. 8.32(b), top). The upper plane of springs and the ring are covered by an appropriate metal-wiring pattern that connects always two neighboring springs and the intermediate ring segment. Thus, there are eight isolated, conducting segments of the ring, which are rotated by 45◦ with respect to each other. Lorentz forces in the radial direction Frad,k /dl = ik B (B is the magnetic induction) generated by currents ik through two opposite segments “1” and “5” can be used to drive the ring into resonance. The 90◦ -shifted segments “3” and “7” are aimed at sensing the drive motion and in this way one can stabilize it by feedback control. Sensing detection is performed using the induced voltages within segments “2” and “6” that are angularly separated from the drive segments by 45◦ . Sensing feedback is implemented using the remaining two ring segments “4” and “8”. The system has a good noise performance (typically 0.15◦ /s in 30-Hz bandwidth) and excellent shock robustness. As long as the ring remains within the homogeneous magnetic field, vibrations do not disturb the operation. However, the silicon ring’s elasticity modulus is anisotropic and, besides this, the main modes are prone to diminutive asymmetries. They split into modes with a small frequency mismatch (on the order of Hz). Therefore, highly automated laser trimming is needed in order to guarantee a small frequency split that in the end is responsible for a quite complicated bias behavior. Not surprisingly, the system features a great number of modes. Using a simple FEM model as shown in Fig. 8.33, bottom, the author counts for a 10-kHz ring more then 50 different modes within a frequency interval of up to 100 kHz. Most of them are not critical, but singular modes must be controlled in order to exclude parasitic effects. The ring gyroscope described above was successfully sold mainly to the automotive market by BAE and Silicon Sensing Systems. The external magnet makes further miniaturization difficult. Logically, the principle has been transformed into capacitively driven and sensed rings. Despite the fact that the ring modes themselves are inherently robust against shock and vibration, capacitive instead of magnetic transducers introduce a significant sensitivity to linear accelerations. The outstanding vibration robustness of the BAE sensor is obviously lost. Nevertheless, the first attempt was made by Putty and Najafi [1994] and Chang et al. [1998]. They used electroforming of a nickel ring into a thick polyimide mold within a post-CMOS process and achieved resolutions of around 0.5◦ /s in 10-Hz bandwidth. A further trial was undertaken by Ayazi and Najafi [2001], who

8.7 Non-planar MEMS gyroscopes

Bulk Si disk (~800 µm)

443

nominal gap (~200 nm)

sense ive dr

30o

secondorder elliptical modes

Figure 8.33 The principle of a capacitive acoustic-wave disk gyroscope. Adapted from

Johari [2008].

transformed the concept into a high-aspect-ratio polysilicon ring. Experimentally a resolution of 1◦ /s in 1-Hz bandwidth has been demonstrated; however, √ the improvement potential was estimated as resolution down to 0.01◦ /s per Hz. Regrettably, after these prototypes the concept obviously was not further developed and did not gain market relevance.

8.7.4

Bulk acoustic-wave gyroscopes However, the emergence of acoustic-wave silicon disk gyroscopes can be considered as a kind of fundamental renaissance of the ring-shell-mode coupling principle (Johari and Ayazi [2006], Johari [2008]). Two second-order elliptical bulk acoustic modes of a centrally suspended single-crystalline silicon disk are excited by drive electrodes arranged around the disk. Figure 8.33 illustrates the function: the two degenerate modes are rotated by ◦ 30 with respect to each other. To a good approximation they can be imagined as corner-smoothed equilateral triangles. Ideally, they would feature the same resonance frequency. A Z-rate creates Coriolis forces acting on the bulk particles mainly in the tangential direction. Therefore, energy will be transferred from the driving mode to the sensing mode, where it is picked up by the sensing electrodes. The alternating drive and sensing electrodes around the disk of hight 50 µm are displaced by 30◦ and feature typical gaps of about 200 nm between them and the disk. The great potential of the approach is caused by the very high resonance frequencies on the order of some MHz, as well as by the diminutive particle displacements of the acoustic-wave modes that are incomparably smaller than in the case of flexural modes (about some tens of nanometers). This reduces not only the thermoelastic damping but also the impact of the surrounding gas,

444

Gyroscopes

altogether the impact of the Brownian noise is reduced. Superior Q-factors and good bias stability result. Since no flexures are needed, shock and vibration have only a minor impact. The concept is in the phase of commercialization by the company Qualtre (iSuppli [2010]). Extensions towards in-plane rate measurements are under development.

8.8

2D and 3D gyroscopes and ways towards a 6D IMU The broad introduction of 1D gyroscopes in electronic stability control (ESC) systems of cars (Mercedes A Class) represented the real start of MEMS gyroscopes. Many applications such as rollover detection, suspension control of cars, navigation aid in the automotive area; or camcorder and mobile-phone stabilization, GPS-aided navigation, human-motion analysis, transportation (Segway) in the consumer and industrial markets; or platform stabilization and missile navigation within the military area followed. Functional and accuracy requirements differ by orders of magnitude (e.g. accuracy from 10◦ /s to 0.1◦ /s). Some applications are satisfied by 1D gyroscopes, many need two 1D acceleration sensors and one 1D gyroscope (ECS), others require even more sensed signals up to the full 6D inertial measurement unit (IMU) that allows one to acquire all three acceleration components plus three rate or angular-acceleration signals and to perform autonomous navigation at least for limited time intervals. Despite the IMU being a common target, dedicated inertial MEMS with fewer than six components to be measured will dominate the inertial MEMS market for a long time due to cost reasons and due to the difficulties involved in meeting completely different performance requirements for different application classes. However, the integration of more than one sensed component in a common system has become one of the formative trends in developing new inertial MEMS. Multi-component inertial MEMS can be composed out of 1D sensors. This is costly, energy-consuming, and space-consuming. As in the case of accelerometers, merging 1D sensors is possible at different levels:

r common signal processing for N parallel sensing elements: the mechanical structures remain separated – often within the same package or even on chip – but the signal processing is common, using the benefits of joint power supplies, busses, BIST etc. up to shared front-ends with time or frequency multiplexing r creation of sensing elements for N components with fewer than N 1D subsystems or with reduced signal-processing complexity in comparison with N 1D subsystems. In contrast to merging 1D accelerometers, the fusion of 1D gyroscopes need not primarily target a smaller number of proof masses but may be done to reduce the overall number of functional members to a minimum. A second target is the reduction of necessary drive and sensing channels in comparison with N 1D sensors.

8.8 2D and 3D gyroscopes

445

Remarkably, many of the vibratory gyroscopes exhibit a parasitic acceleration sensitivity that can be used to form combined rate and acceleration sensors. Appending an acceleration measurement to a rate sensor without significant change of the mechanical structure is often a cheap add-on possibility. For instance, all in-plane-driven and -sensed linear gyroscopes are inherently sensitive with respect to accelerations in the direction of the sensing deflection. The tuning fork suppresses the acceleration impact by anti-phase drive and sensing; however, as mentioned repeatedly, additional common-mode sensing allows detection of one of the in-plane acceleration components. A 2D inertial sensor results. Or, some single-mass architectures with out-of-plane excitation and means for in-plane sensing can be optimized for a combined rate and acceleration sensor as done by Roszhart [2000], where the oscillations of a vertically driven proof mass are in-plane sensed by at least two opposing electromechanical resonators. Furthermore, the limited stiffness in the z-direction causes torsional gyroscopes that are in-plane driven and out-of-plane sensed to become sensitive with respect to z-accelerations. These can be captured by common-mode sensing of both sensing capacitances (see e.g. Fig. 8.27, where the z-acceleration sensitivity is mainly caused by the z-compliance of the external sensing frame). Suchlike DOFs extending the number of orthogonal linear and angular deflections are the means for creating N D inertial sensors.

8.8.1

Single-mass multiple-DOF inertial sensors Gyroscope-free, single-mass IMUs There are different concepts for N -DOF inertial sensing. One may start with a single-mass system, linear and angular accelerations of which are measured as described, for instance, in Chapter 7, Section 7.6.3. This is a classical approach of a gyroscope-free IMU. It needs at least six accelerometers with a common proof mass. In the proposal of Chen et al. [1994] the six accelerometers are more advantageously located at the centers of the six faces of a cube. They measure accelerations along the diagonals in such a way that the six sensing directions form a regular tetrahedron. The principal disadvantage of gyroscope-free IMUs is the quick divergence of the measurement error with integration time. Park et al. [2005] proposed the use of an additional 3D accelerometer that allows one to derive the angular rate without integration of the angular accelerations. In this way the growth of the angular-rate error is limited. Nevertheless, gyroscope-free IMUs remain inferior to gyroscope-based IMUs and, thus, will not be the subject of further consideration.

Single-mass, gyroscope-based IMUs A vibrating mass with three pairs of orthogonal displacement sensors may not only detect three angular velocity components but also acquire and separate

Gyroscopes

(a)

(b)

z

D FCY

y

FCZ

x

y

proof mass

dr iv e co &s m en bs s e

446

bottom electrode

C

x

FCX

FCZ

x

FCX

A

vA

FCZ x

FCZ

FCY

B

Figure 8.34 Single-mass gyroscopes. (a) A single-mass gyroscope/IMU according to John and Vinay [2006]. (b) The trajectory of the single-mass COG and corresponding Coriolis forces.

low-frequency accelerations. The principle is illustrated by Fig. 8.36. A suspended mass is vibrated in two orthogonal directions x and y. Originally, a suspended cube was proposed in order to verify the principle [Luinge 2002]; however, here a proposal according to John and Vinay [2006] is cited because it is closer to a MEMS realization. The vibration is excited by forces that are in quadrature so that the COG draws a circular trajectory. The corresponding Coriolis forces emerging in one cycle along the trajectory are shown in Fig. 8.34(b), where FC,X ,(Y ),(Z ) denote the Coriolis forces generated by the angular rate components ΩX ,(Y ),(Z ) , respectively. Assuming that the velocities x˙ = x˙ 0 sin(ωt) and y˙ = y˙ 0 cos(ωt) are ¯ × v¯] exactly in quadrature, the components of the Coriolis force F¯C = −2m[Ω become     −ΩZ y˙ 0 cos(ωt) FC,X .  FC,Y  = −2m  (8.145) ΩZ x˙ 0 sin(ωt) ΩX y˙ 0 cos(ωt) − Ωy x˙ 0 sin(ωt) FC,Z

The Z-component ΩZ can be derived by in-plane sensing, while in-phase and inquadrature synchro-demodulation of the z-deflection allows one to extract both of the in-plane angular rates ΩX and ΩY . Since the acceleration spectrum is assumed to be negligible at the drive frequency, it can be separated by low-pass filtering the transducer signals. Thus, in principle, all of the functions of a IMU are present. However, the practical implementation meets many difficulties. The main problem is the separation of the small Coriolis-force-induced deflections from the background of large vibratory oscillations. The reference trajectory has to be subtracted from the measured total deflection. This suggests that one should use adaptive control techniques whereby the system is compared with an ideal reference system that is free from Coriolis forces, damping, and cross-coupling effects. By estimating all unknown parameters including the angular-rate vector, the desired signals can be extracted (see John and Vinay [2006]). However, many questions such as noise impact,

8.8 2D and 3D gyroscopes

C3

C2

drive

Cdrive

C1 pr oo f

m as s

C4

447

C5

suspending beam

C6

Figure 8.35 A 5D inertial sensor. Adapted from Watanabe et al. [2006].

disturbances by parasitic modes, efficient separation of drive and sensing combs etc. are still awaiting answers. In view of what has been stated above, the way towards multi-component inertial sensors is predominantly via bottom-up approaches whereby more and more simple rate- and acceleration-sensing functions are merged into a multicomponent system.

5D inertial sensors The difficulties with controlling a circular trajectory can be bypassed by being satisfied with just five inertial components. Out-of-plane rate-signal measurement is skipped. In this case the proof mass must be vibrated in the z-direction. This creates Coriolis torques originating from both of the in-plane rate components. The three acceleration components should be extracted as low-frequency fractions of the corresponding deflections. Watanabe et al. [2006] presented a prototype of a SOI bulk-micromachined sensor that is excited in the z-direction by two electrodes above and under the proof mass in resonance. Capacitive detection allows one to extract the accelerations and the two in-plane rates. The principle is demonstrated in Fig. 8.35. The proof mass is suspended by four beams. They are 20 µm high and formed by an active Si layer isolated by a buried oxide layer from the 675-µm thick SOI wafer. Anodic bonding of the SOI wafer to upper and lower glass wafers forms a vacuum-encapsulated cavity with electrodes on both glass wafers as indicated in Fig. 8.35. Eight sensing electrodes, C1 to C8 , capture all tilting movements plus a common-mode z-deflection emerging as a result of a z-acceleration. Under the impact of Coriolis torques the proof mass oscillates about the corresponding inplane axis and causes capacitance changes. The change of the capacitance pairs (C1 + C4 ) − (C2 + C3 ) is, for instance, a measure for the tilt angle about the x-axis. Accordingly, the rotation about the y-axis is captured by the difference of the capacitance pairs under the proof mass. A large frequency split of about 30% was used, whereby the matching of the two sensing modes was only around 5%, indicating quite a good robustness against fabrication tolerances at reasonable sensitivities.

448

Gyroscopes

(a)

(b) drive combs X

drive

sense plate

C3

C1 C4 Y

X

Y -sensing electrodes

X -sensing electrodes

Figure 8.36 2D torsional gyroscopes. (a) The 2D torsional gyroscope according to Juneau et al. [1997]. (b) The input-decoupled 2D torsional gyroscope according to Tsai and Fang [2006].

The acceleration measurement uses the same principle as that explained in Chapter 7, in the section “6D accelerometers.” However, a non-negligible crosscoupling between drive motion and acceleration measurement was observed.

8.8.2

2D gyroscopes Single-plate 2D gyroscopes The simplest extension of a 1D in-plane torsional gyroscope to a 2D gyroscope exploits the rotation of the proof mass about two in-plane axes. A system is designed according to Fig. 8.26 or Fig. 8.28(a) with in-plane symmetry about the origin, with identical torsional flexures arranged along both in-plane axes, so that the plate may rock about the x- and y-axes in response to the corresponding angular velocities Ωy and Ωy . In order to separate the two rate components, four sensing electrodes like slices of a cake can be placed underneath the plate. One of the first implementations by the Berkeley Sensors & Actuators Center [Juneau et al. 1997] is shown schematically in Fig. 8.36(a). The torsionally excited disk is suspended by two pairs of torsional flexures that limit the tilt motions to rotations about the x- and y-axes. On measuring differentially the capacitance changes between the pair C1 –C3 and the pair C2 –C4 , both of the angular-rate components ΩX and ΩY can be extracted. The design features an inherent cross-axis sensitivity that increases with growing sensing deflections. Thus, target resolution and cross-axis sensitivity have to be compromised within such a design. Since the structure is suspended by long flexures, z-accelerations may change all four sensing capacitances. Hence, an extension of the 2D gyroscope to a 3D inertial sensor is obvious.

Decoupled 2D gyroscopes In order to reduce the cross-coupling between the axes, a decoupled architecture should be used. An example proposed by Tsai and Fang [2006] is given in

449

8.8 2D and 3D gyroscopes

(a)

(b) Y

FCZ e

FCZ

riv

X

Vd

FCZ

Y

m3

e

V

dr iv e

m3

m4

MCX MCY

MCY MCX

m2

m1

e

FCZ

e riv

Vd

riv

FCZ

FCZ

Vd

V

dr

iv

e

FCZ

m2

V

m1

dr iv e

m4

V

FCZ

dr iv

e riv

Vd

X

Figure 8.37 Design principles of 3D gyroscopes. (a) The principle of a radially driven,

four-mass 3D gyroscope according to Wood et al. [1996]. (b) A tangentially driven, four-mass 3D gyroscope.

Fig. 8.36(b). A suspended ring is vibrated about the z-axis. It carries an inner plate and an outer ring. The inner plate may rotate with respect to the annular drive frame about the x-axis, whereas the outer ring rotates about the y-axis. An applied rate ΩX forces the outer ring to rock about the y-axis, ideally without exciting the inner plate. Vice versa, the rate ΩY causes a tilt of the inner plate about the x-axis, not impacting the outer frame. The drive motion is a common rotation of all three members. The four drive-suspending springs have non-negligible torsional compliance about the x- and y-axes. That leads to a system with five DOF: one for the drive motion, two for the rotation of the drive ring and the outer frame about the y-axis, and two for the rotations of the drive ring and the inner plate about the x-axis. The analysis is similar to that for the three-DOF torsional gyroscope considered under the heading “Decoupled torsional gyroscopes with three DOF.” To adjust the resonance frequencies Tsai and Fang [2006] used a triple-beam torsional spring for the link between the outer and drive rings that is stiff enough with respect to linear forces. The fine tuning can be performed by electrostatic tuning using separate parts of the sensing electrodes.

8.8.3

3D gyroscopes Anti-phase excitation of at least two proof-mass pairs and consecutive transfer of the opposing Coriolis forces into torques is a basic principle for building 3D gyroscopes. As early as in 1996 Wood et al. [1996] sketched an arrangement of four masses, m1 to m4 , that are synchronously excited in the radial direction as illustrated by Fig. 8.37(a). The suspensions are sufficiently stiff against out-of-plane deflections that any of the four masses may be deflected only in the radial direction with respect to the four frame-corners. The Coriolis reactions are denoted by FCZ and MCX (Y ) , where the axis index belongs to the

Gyroscopes

synchronizing springs

ra d co ial d m ri bs v e

X -sensing

ns Z in g

Y -sensing

se

450

y

x

Figure 8.38 A fully decoupled 3D gyroscope.

originating rate component. Under the impact of an angular-velocity component in the z-direction all four masses experience tangential Coriolis forces, FCZ , which rotate the whole structure about the z-axis. An in-plane rate component, say ΩX , generates opposite Coriolis forces at masses m2 and m4 that tilt the whole structure about the x-axis. Accordingly, the component ΩY forces the structure to rotate about the y-axis. It is obvious that the principle described above can be applied also for tangential excitations as shown in Fig. 8.37(b). Here, a rate component ΩZ forces the masses to respond with radial oscillations, while the components ΩX and ΩY are responsible for the torques about the y- and x-axes respectively. It must be noted that the four-mass principle has an inherent cross-axis sensitivity mainly between the in-plane and the out-of-plane rate components. If, for instance, a large z-signal component forces the radially driven system to oscillate tangentially, in-plane rate signals are coupled into the orthogonal sensing axes. There are many ways to implement architectures according to Fig. 8.37. Often ring-shaped frames are used instead of square carriers. Decoupling of the sensing modes reduces cross-axis sensitivities. For instance, Tsai et al. [2008] (see also Tsai and Sue [2008]) analyzed a tangentially oscillating structure with an outer ring and an inner disk. Four proof masses are nested between the disk and the outer ring and are tangentially co-moving with the oscillating ring. The ring and disk capture Coriolis rotations that are induced by the in-plane rate components, while the four proof masses detect the z-signal.

A fully decoupled 3D gyroscope and extension towards an IMU A fully decoupled 3D gyroscope is shown schematically in Fig. 8.38 [Kempe 2010b]. Implementation details are presented in Fig. 8.39. The gyroscope

8.8 2D and 3D gyroscopes

451

Figure 8.39 Implementation detail of the fully decoupled 3D gyroscope.

consists of eight radially driven segments angularly spaced by 45◦ . Four segments that are angularly separated by 90◦ are not paired by the central suspension but suspended directly from the substrate. The high z-stiffness of the suspension neutralizes out-of-plane Coriolis forces. Two opposite segments of the other four plates are linked by the Cardan suspension (or equivalent) in two pairs, either of which may tilt about the y- or x-axis in response to the corresponding components ΩX and ΩY . Electrodes placed underneath allow independent differential capacitive measurement for any of the in-plane rate signals. The four ΩZ -sensitive frames nest sub-frames that can be deflected orthogonally to the radial movement. The sensing boxes within the sub-frames capture the Coriolis deflections stemming from the ΩZ component. All segments perform a common drive motion that is enforced by the eight radially arranged synchronization springs. The system is implemented in the thick polysilicon process PSM-X2 described in Chapter 4. The resonance frequencies are located close to 20 kHz. The sensor features excellent cross-axis insensitivity and practically equal resolution in all three axes. The specified noise level is less then 0.1◦ /s within 10-Hz bandwidth. A slightly simplified variant of the inertial sensor described here (see Kempe [2010a]) with somewhat increased cross-axis sensitivity consists of only four segments that are pair-wise linked with the central suspension as in Fig. 8.38. All segments carry suspended sub-frames with sensing boxes that can deflect orthogonally to the radial orientation and, hence, can capture the ΩZ component. The out-of-plane tilt of a pair of opposite segments is the response to the corresponding in-plane rate component. The principle described above is easily transformed into a system with tangentially driven segments.

452

Gyroscopes

Operation as IMUs Both mechanical architectures can be operated as a full IMU: the in-plane accelerations are captured by the common-mode deflections of the sensing boxes within two opposite ΩZ -sensitive frames. The z-acceleration is more difficult to measure with the same linearity and stability, because of the lack of a reference capacitance for the four out-of-plane capacitance-segments. With enlarged z-stiffness of the whole structure, a 5D gyroscope/ accelerometer (three rate and two acceleration components) results. It should be emphasized that the possibility in principle of sensing all six inertial components is usually not sufficient to build an IMU. For instance, the requirements on the damping behavior of the acceleration-sensing components and the rate-signal sensors may be conflicting. Such practical limitations often lead to solutions with separated accelerometers and gyroscopes within different cavities.

Some principles of designing multi-body 3D gyroscopes The approaches sketched above reveal some principles that can be applied in designing 3D multi-body gyroscopes.

r A common, mechanically synchronized drive motion of the participating proof

r r

r r

masses reduces the signal-processing effort decisively. Radial or tangential drive motion of proof-mass segments are the alternative variants. Grouping in-plane-symmetric segments in tilting pairs allows one to capture the in-plane rate-signal components. In the case of radial drive the z-rate can be measured either by sensing the tangential motion of sub-frames embedded into the segments, or by summing the tangential Coriolis forces by transferring them to an outer or inner frame carrying the segments. In the last case the frame rotation is a measure of the applied z-rate. In the case of tangentially driven segments the z-rate can be captured by sensing the radial deflection of embedded sub-frames. For tangential drive the paired segments can be substituted by rings that are electrically segmented into two halves in order to detect their out-of-plane tilting motion. However, the detection of the z-rate requires here additional, radially movable segments (as, for instance, were realized by Tsai et al. [2008]).

References Acar, C. (2004). Robust micromachined vibratory gyroscopes, Ph.D.thesis, Section 6.3 Gyroscope dynamics. University of California, Irvine. Acar, C. and Shkel, A. (2009). MEMS Vibratory Gyroscopes. Sections 2.1.2 Torsional gyroscopes and 4.3.1 Torsional suspension systems. Acar, C. and Shkel, A. M. (2006). Inherently robust micromachined gyroscopes with 2-DOF sense-mode oscillator. Journal of Microelectromechanical Systems, 15(2):380–387.

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Kuisma, H., Ryhaenen, P. T., Lahdenperae, J. et al. (1997). A bulk micromachined silicon angular rate sensor, in 1997 International Conference on SolidState Sensors and Actuators, Transducers ’97, Chicago, pp. 875–877. Landau, L. D. and Lifshitz, E. M. (2003). Mechanics. London: ButterworthHeinemann, 3rd edn. Lemkin, M. A., Clark, W. A., Juneau, T. N., and Roessing, A. W. (2006). Structure for attenuation or cancellation of quadrature error. US Patent 7,051,590 B1. Li, X., Bao, M., Yang, H., and Lu, D. (1999). A micromachined piezoresistive angular rate sensor with a composite beam structure. Sensors and Actuators A, 72:217–223. Luinge, H. J. (2002). Inertial sensing of human movement. Ph.D. thesis University of Twente, Twente University Press; Also available in print: http:// www.tup.utwente.nl/tupress/catalogue/book/index.jsp?isbn=9036518237. Madni, A. M. and Costlow, L. E. (2001). A third generation, highly monitored, micromachined quartz yaw rate sensor for safety-critical automotive stability control, in Proceedings of the IEEE Aerospace Conference, pp. 2523–2534. Madni, A. M., Costlow, L. E., and Knowles, S. J. (2003). Common design techniques for gyrochip quartz rate sensors for both automotive and aerospace/defense markets. IEEE Sensors Journal, 3:2569–578. Madni, A. M., Wan, L. A., and Hammons, S. (1996). A micro-electromechanical quartz rotational rate sensor for inertial applications, in Proceedings of the IEEE Aerospace Applications Conference, Aspen, volume 2, pp. 315–332. Madni, A., Zou, Q., Kim, E. S. et al. (2008). Silicon inertial sensors formed using MEMS. US Patent 7,360,422 B2. Maenaka, K., Fujita, T., Konishi, Y., and Maeda, M. (1996). Analysis of a highly sensitive silicon gyroscope with cantilever beam as vibrating mass. Sensors and Actuators A, 54:568–573. Maenaka, K. and Shiozawa, T. (1994). A study of silicon angular rate sensors using anisotropic etching technology. Sensors and Actuators A, 43:72–77. M’Closkey, R. T., Vakakis, A., and Gutierrez, R. (2001). Mode localization induced by a nonlinear control loop. Journal of Nonlinear Mechanics, 25(1– 3):221–236. Mochida, Y., Tamura, M., and Ohwada, K. (1999). A micromachined vibrating rate gyroscope with independent beams for the drive and detection modes, in 12th IEEE International Conference on Micro Electro Mechanical Systems, MEMS ’99, Orlando, pp. 618–623. Morrow, C. T. (1955). Zero signals in the Sperry tuning fork gyrotron. Journal of the Acoustic Society of America, 27(3):581–584. Moussa, H. and Bourquin, R. (2006). Theory of direct frequency output vibrating gyroscopes. IEEE Sensors Journal, 6(2):310–315. Nagao, M., Watanabe, H., Shirai, E. N. K., Aoyama, K., and Hasimoto, M. (2004). A silicon micromachine gyroscope and accelerometer for vehicle

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stability control system, in 2004 SAE World Congress, Detroit; SAE Technical Paper Series 2004-01-1113. Neul, R., Gomez, U. M., Kehr, K. et al. (2007). Micromachined angular rate sensors for automotive applications. IEEE Sensors Journal, 7(2):302–309. Painter, C. C. and Shkel, A. M. (2002). Identification of anisoelasticity for electrostatic “trimming” of rate integrating gyroscopes, in 2002 SPIE Annual International Symposium on Smart Structures and Materials, San Diego, pp. 157– 168. (2003a). Detection of orientation and predicted performance of a MEMS absolute angle measuring gyroscope, in Proceedings of the 4th International Workshop on Structural Health Monitoring (SHM2003), Stanford, pp. 1011– 1018. (2003b). Active structural error suppression in MEMS vibratory rate integrating gyroscopes. IEEE Sensors Journal, 3(5):595–606. Palaniapan, M., Howe, R. T., and Yasaitis, J. (2002). A surface-micromachined z-axis frame microgyroscope, in Proceedings of International Electron Devices Meeting, San Francisco, pp. 203–206. (2003). Performance comparison of integrated z-axis frame microgyroscopes, in Proceedings of IEEE – MEMS 2003, pp. 482–485. Park, S. and Horowitz, R. (2003). Adaptive control for the conventional mode of operation of MEMS gyroscopes. Journal of Microelectromechanical Systems, 12(1):489–494. (2004). New adaptive mode of operation for MEMS gyroscopes. Journal of Dynamic Systems, Measurement, and Control, 126(4):800–810. Park, S., Tan, C. W., and Park, J. (2005). A scheme for improving the performance of gyroscope-free inertial measurement unit. Sensors and Actuators A, 121:410–420. Phani, A. S. and Seshia, A. A. (2004). Identification of anisoelasticity and nonproportional damping in MEMS gyroscopes, in Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, volume 2, pp. 343– 346. Phani, A. S., Seshia, A. A., Palaniapan, M., Howe, R. T., and Yasaitis, J. A. (2006). Modal coupling in micromechanical vibratory rate gyroscopes. IEEE Sensors Journal, 6(5):1144–1152. Piyabongkarn, D. and Rajamani, R. (2002). The development of a MEMS gyroscope for absolute angle measurement, in Proceedings of the American Control Conference, Anchorage, pp. 1960–1965. Putty, M. W. and Najafi, K. (1994). A micromachined vibrating ring gyroscope, in Proceedings of the Solid-State Sensor and Actuator Workshop, Hilton Head, pp. 213–220. Reeds, J. W. and Hsu, Y. W. (2002). Angular rate sensor having a sense element constrained to motion about a single axis and flexibly attached to a rotary drive mass. International Patent Application WO 02/103364 A2.

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Reeds, J. W., Hsu, Y. W., and Dao, P. C. (2003). MEMS sensor with single central anchor and motion-limiting connection geometry. US Patent 6,513,380 B2. Roszhart, T. V. (2000). Micromachined acceleration and Coriolis sensor. US Patent 6,032,531. Saukosi, M., Aaltonen, L., and Halonen, K. A. I. (2007). Zero-rate output and quadrature compensation in vibratory MEMS gyroscopes. IEEE Sensors Journal, 7(12):1693–1652. (2008). Effect of synchronous demodulation in vibratory MEMS gyyroscopes: a theoretical study. IEEE Sensors Journal, 8(10):1722–1733. Saukoski, M., Aaltonen, L., Salo, T., and Halonen, K. (2005). Fully integrated charge sensitive amplifier for readout of micromechanical capacitive sensors, in Proceedings of the IEEE International Symposium on Circuits and Systems, Kobe, pp. 5377–5380. (2006). Integrated readout and control electronics for a microelectromechanical angular velocity sensor, in Proceedings of the European Solid-State Circuits Conference, Montreux, pp. 243–246. Schofield, A. R., Trusov, A. A., and Shkel, A. M. (2008). Effects of operational frequency scaling in multi-degree of freedom MEMS gyroscopes. IEEE Sensors Journal, 8(10):1672–1680. Sharma, A., Zaman, M. F., and Ayazi, F. (2007). A 0.2◦ /hr micro-gyroscope with automatic CMOS mode matching, in IEEE International Solid-State Circuits Conference, 2007, ISSCC2007, pp. 368–388. Shkel, A. M., Horowitz, R., Seshia, A. A., Park, S., and Howe, R. T. (1999). Dynamics and control of micromachined gyroscopes, in Proceedings of the American Control Conference, San Diego, pp. 2119–2124. Sung, W. T., Sung, S., Lee, J. Y. et al. (2009). Development of a lateral velocitycontrolled MEMS vibratory gyroscope and its performance test. Journal of Micromechanics and Microengineering, 18:1–13. Thomae, A., Schellin, R., Lang, M. et al. (1999). A low cost angular rate sensor in Si-surface micromachining technology for automotive application, in SAE 1999 World Congress, Detroit, Technical Paper 1999-01-0931. Tsai, D. H. and Fang, W. (2006). Design and simulation of a dual-axis sensing decoupled vibratory wheel gyroscope. Sensors and Actuators A, 126: 33–40. Tsai, N. C. and Sue, C. Y. (2008). Fabrication and analysis of a micro-machined tri-axis gyroscope. Journal of Micromechanics and Microengineering, 18:1–14. Tsai, N. C., Sue, C. Y., and Lin, C. C. (2008). Design and fabrication of a novel tri-axis micro-gyroscope, in Proceedings of the 3rd IEEE International Conference on Nano/Micro Engineered and Molecular Systems, Sanya, pp. 457–461. Watanabe, Y., Mitsui, T., Mineta, T., Matsu, Y., and Okada, K. (2006). SOI micromachined 5-axis motion sensor using resonant electrostatic drive and non-resonant capacitive detection mode. Sensors and Actuators A, 130– 131:116–123.

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Weinberg, M. S. and Kourepenis, A. (2006). Error sources in in-plane silicon tuning-fork MEMS gyroscopes. Journal of Microelectromechanical Systems, 15(3):479–491. Willig, R. and Moerbe, M. (2003). New generation of inertial sensor clusters for ESP- and future vehicle stabilizing systems in automotive applications, in 2003 SAE World Congress, Detroit; SAE Technical Paper Series 2003-010199AMAA. Willig, R., Thomae, A., Kuhlmann, B. (2004). Rotation rate sensor. US Patent 6,705,164 B2. Wood, D., Cooper, G., Burden, J. S., Harris, A. J., and Cruickshank, J. L. (1996). A monolithic silicon gyroscope capable of sensing about three axes simultaneously, in IEE Colloquium on Silicon Fabricated Inertial Instruments, London, pp. 2.1–2.3. Wyse, S. F. (2005). Quadrature compensation technique for vibrating gyroscopes. US Patent 6,883,361 B2. Xie, H. and Fedder, G. F. (2001). A CMOS-MEMS lateral-axis gyroscope, in 14th IEEE International Conference on Micro Electro Mechanical Systems, MEMS ’01, Interlaken, pp. 162–165. Yafei, R., Xizheng, K., and Yijie, L. (2007). MEMS gyroscope performance estimate based on Allan variance, in The Eighth International Conference on Electronic Measurements and Instruments, ICEMI ’2007, vol. 1, pp. 260–262. Yang, H., Bao, M., Yin, H., and Shen, S. (2002). A novel bulk micromachined gyroscope based on a rectangular beam-mass structure. Sensors and Actuators A, 96:145–151. Zhang, H., Zou, Q., Kim, E. S. et al. (2006). Frequency-mismatch-tolerant silicon vibratory gyroscope without vacuum package for automotive applications, in World Automotive Congress WAC 2006, Budapest.

9

Test and calibration

Testing of inertial MEMS includes 1. 2. 3. 4. 5.

device characterization tests product qualification tests fabrication tests product tests in-field tests especially self-tests.

Characterization tests Characterization tests are aimed at investigating the performance limits of a device under development. The knowledge of their dependency on fabrication tolerances, environmental conditions, and lifetime is necessary in order to determine pass–fail criteria for production and qualification. They include, for instance, performance characterization over temperature and temperature cycling, determination of shock robustness, especially the impact of different types of vibration, Q-bias tests for gyroscopes for various fabrication parameter settings etc.

Qualification tests Qualification tests are the basis of qualification procedures for different stages of the fabrication process (wafer production and packaging) and for the final product and product components. Process qualification procedures end with a qualified production process. Product qualification tests are product-specific and are performed in accordance with standard requirements for a given application area of the product. On the basis of representative test ensembles they should guarantee that a given product or product component fabricated on a given fabrication line and tested in accordance with a given test sequence will function in the field under the specified environmental conditions over the specified lifetime with a failure rate less than a requested limit. For automotive applications they include, for instance, around 50 standardized tests such as preconditioning (survival of reflow soldering without degradation), temperature humidity bias or the HAST test (highly accelerated temperature/humidity stress test) with a specified humidity load at given temperatures, temperature cycling tests, for instance, with 1000 cycles from −50 to 150 ◦ C, high-temperature storage tests and high-temperature

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461

operating-lifetime tests with electrical biasing e.g. for 1000 hours at 150 ◦ C, interconnect tests, electro-migration and hot-carrier-injection tests, various ESD and latch-up tests, EMC (electromagnetic compatibility) tests, hermeticity tests, drop tests, various acceleration-to-acceleration and acceleration-to-rate-signal cross-sensitivity tests etc.

Fabrication tests Fabrication tests are aimed at monitoring the whole production chain. They include a dense set of parametric tests for all production steps that ideally should be independent of the nature of the fabricated product but guarantee the repeatability of all manufacturing procedures within predefined tolerances. They are fixed during the process-qualification procedure. Different test devices such as sheet resistances, contact resistance structures (Kelvin bridges), beam resonators, and beam arrays for stress monitoring and so on are tested during the fabrication flow in order to detect parametric deviations as early as possible. Continuity tests should discover failing interconnects. Since the fabrication process for inertial MEMS includes zero- and first-level packaging, outgoing and incoming inspections for different fabrication stages are usual. For instance, before the common first-level packaging of an ASIC with a SMM gyroscope as described in Chapter 4, Section 4.3.2, the gyroscope-die is tested on wafer level and the failing zero-level-packaged parts are labeled in order to eliminate them during first-level packaging. Here fabrication and product testing become interlinked.

Product testing and calibration Product testing and calibration is the heart of the final production test. It should be largely independent of the parametric fabrication tests. Using a qualified production line, product testing and calibration can be performed outside the production facilities. This gives fabless design houses the possibility of developing and selling products that are manufactured on foreign production lines. The exact coordination of the different test set points and sub-tests within the fabrication lines, including the packaging facility, is a complicated task and requires a lot of experience in fabricating MEMS devices. Usually the final product testing and calibration are set up in close proximity to the development team. Besides visual control, electrical tests of the signalprocessing unit, and standard continuity and geometric-conformity tests, they focus on calibrating the device and measuring the performance parameters at various temperatures and supply voltages (usually at the default and at the extremal values). Functionality checks of self-testing procedures are part of the performance testing as well. Calibration and performance testing are key procedures within the whole test system. They are needed many times, for instance, conducting different qualification tests where the performance parameters such as sensitivity and bias have to be compared before and after different load cycles.

462

Test and calibration

Figure 9.1 A rate table with climate chamber and loaded multi-test board. Courtesy of SensorDynamics AG.

Inertial MEMS require special test equipment for calibration and performance testing. The entity to be measured – acceleration or angular velocities – must be applied to the DUT (device under test) and is not available within the broad repertoire of microelectronic test equipment. Shakers and rate tables have been developed especially for the needs of inertial MEMS testing. For low-g accelerometers it is sometimes sufficient to perform a static calibration by using the Earth’s gravity and simply turning the device to two positions with ±g load. However, dynamic characterization demands vibrating platforms such as shakers that are usually based on the loudspeaker-membrane principle. Rate tables are rotating tables with precision rate control. Usually they carry a carefully mounted multi-test PCB with a large number of sockets/DUTs as shown in Fig. 9.1. Here the gyroscopes are sensitive in-plane around the vertical axis of the rate table. One sees that deformations or orientation mismatches of the board and the individual sockets cause orthogonal rate components, which must be avoided, especially for the characterization of cross-sensitivity. Rate tables are best suited for characterization and qualification purposes. However, due to the long soak times for setting low or high test temperatures, the throughput for three-point temperature testing is low. Flip tables have been developed especially to overcome this drawback. In Fig. 9.2 two mounting heads of a flip table from Multitest Elektronische Systeme GmbH, Germany, each carrying a DUT, are shown. Both heads rotate around their axial orientation. Therefore, an out-of-plane gyroscope with its sensitive

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Figure 9.2 Heads of a flip table for production testing of gyroscopes. Courtesy of SensorDynamics AG.

axis parallel to the head axis can be tested. The heads can be turned by 90◦ . In this position they can test gyroscopes with in-plane-sensitive axes as well. The flip table is normally coupled to a standard microelectronic handler allowing fully automated loading. For high or low temperatures a local mini-climate chamber guarantees the desired temperature, whereas in the interest of small setting times a pre-heating or pre-cooling is performed during the loading process. Test times in the region of seconds per head result.

Self-testing Electrical self-tests are widely used in integrated-circuit testing. Such tests should activate gates of logical devices or building blocks of analog devices as much as possible in order to check their functionality. They can be carried out in-field, for instance, after any switching on of a device. In particular, in safety-critical applications suchlike self-tests consisting of monitoring of predefined limits of analog blocks or correct functioning of digital logic are mandatory in order to achieve a low probability of undetected failures. For inertial MEMS the self-testing of mechanical structures is accompanied by the creation of actuating stimuli. The use of Q-bias signals for self-testing of gyros was mentioned in the section “Q-bias.” Other possibilities include the actuation of sensing structures outside of the operational frequency region – in the case of gyroscopes, by slow actuation signals superimposed on the rapidly oscillating Coriolis forces. Auxiliary actuation by modulated oscillations with modulation frequencies outside the frequency region of the rate signals is possible as well.

Burn-in tests Burn-in tests belong to the category of reliability tests. Usually they are performed as part of the qualification and final product tests. In contrast, most of the other reliability tests are included in the generic material characterization as well

464

Test and calibration

as the product-characterization and qualification tests. For instance, a vibrating gyroscope’s fatigue testing is done during characterization of materials by investigating the dependency of fracture failures on applied periodic stress with different amplitudes and frequencies (e.g. Bagdahn and Sharpe [2003], Boroch et al. [2004], Tabata and Tsuchiya [2008], and Muhlstein et al. [2001]). The same is true with respect to the determination of fracture limits under the impact of isolated large forces. The results are usually load limits that must be put into any design of inertial MEMS. The aim of production burn-in is to exclude in-field infant mortality. Generally, the dependency of the failure rate of microelectronic and MEMS devices on time follows a typical bathtub curve. Accelerated testing during burn-in at elevated temperatures or humidity levels allows one to force the emergence of such failures, which follow the Arrhenius model and are responsible for the initial falling branch of the bathtub curve. The failure rate, λ, at temperature T of suchlike failures follows the function λ = λ0 exp[−(Ea /k)(1/T − 1/T0 )], where Ea is the activation energy that must be exceeded to cause the failure mode, and λ0 is the failure rate at temperature T0 . The factor exp[−(Ea /k)(1/T − 1/T0 )] represents the acceleration between the failure rates at the two temperatures. Permeation and leakage within packages, threshold shifts, oxidation processes, and other effects can be modeled by the Arrhenius formula. Burn-in is a cost-adding factor and can be skipped if after a sufficient duration of field-failure observation the infant mortality level is small enough.

Failure analysis Any producer of inertial MEMS has to be able to react immediately to failure events in-field. This is especially crucial for safety-critical automotive applications, for which unidentified failures, especially without clear risk evaluation, lead immediately to a stopping of production and possibly to recall actions. A well-defined RMA (return merchandise authorization or return material authorization) process backed by appropriate methods and equipment for failure analysis has to be established at the supplier. A highly skilled failure-analysis team with immediate access to functional and parametric testers, micro-probers, X-ray, light, SEM, photon, thermal-emission, and acoustic microscopes, focusedion-beam equipment, and chemical re-engineering is a necessary precondition for a state-of-the art product accompaniment for high-volume inertial MEMS.

References Bagdahn, J. and Sharpe, W. N. Jr. (2003). Fatigue of polycrystalline silicon under long-term cyclic loading. Sensors and Actuators A, 103:9–15. Boroch, R. E., Bernhard, W., Kehr, K., Hauer, J., and Mueller-Fiedler, R. (2004). Fracture strength and the fatigue of polycrystalline silicon under static and

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long term high frequency cyclic load, in DTIP 2004, Design, Test, Integration and Packaging of MEMS and MOEMS, Montreux, pp. 389–394. Muhlstein, C. L., Brown, S. B., and Ritchie, R. O. (2001). High-cycle fatigue of single-crystal silicon thin films. Journal of Microelectromechanical Systems, 10(4):593–600. Tabata, O. and Tsuchiya, T., eds. (2008). Reliability of MEMS. Weinheim: WileyVCH.

Concluding remarks

MEMS accelerometers and gyroscopes exploit a wide variety of means for excitation and sensing. Actuation is performed by electrostatic, electromagnetic (e.g. using a magnetic z-gyro; see Maenaka et al. [2005]), piezoelectric, and thermal actuation effects. For sensing electrostatic, electromotive-force, piezoresistive, piezoelectric, and thermal effects are used. The excitation may be linear or angular. It may be in-plane or out-of-plane as well as about an in-plane or an out-of-plane axis. Skewed excitation of gyroscopes using the anisotropy of bulk silicon also has been demonstrated [Andersson et al. 1999]. The integration of coherent optical sources, resonators, and photo-detectors into MEMS opens the way for optical acceleration sensing. It is not just rigid-body systems that can serve as a base for inertial sensors. Any velocity field in gases or fluids is disturbed by the Coriolis effect which creates orthogonal forces on the moving particles. 2D and 3D micro-fluidic or convective gas gyroscopes can be built (e.g. Zhou et al. [2005] and Dau et al. [2008]), exploiting, for instance, velocity-dependent thermal transport processes. There is still a broad field for innovations within the area of inertial MEMS. However, the future of inertial MEMS will depend mainly on new and expanding applications. Inertial MEMS are increasingly merging with higher-level measurement systems such as GPS for navigation purposes. They are becoming basic components of complex user interfaces, as, for instance, in games and smartphones. They are merging with other sensors such as magnetosensors, ultrasonic and optical sensors etc. Combined accelerometers and Hall sensors for orientation (compass-like) or position measurement have been developed. This co-integration is proceeding hand-in-hand with a continuous miniaturization that is decisive for high yield and low cost. The pressure for monolithic integration of sensors and signal processing increases with advancing miniaturization, since otherwise the interconnects become a bottleneck. Packaging remains an important reliabilityand cost-defining factor. On-wafer encapsulation will help to meet the challenges of first-level packaging. The growing complexity of merged sensor systems forces one to reduce the test effort. Built-in self-testing is becoming standard not just for safety-critical application areas. Summarizing, the future of inertial MEMS seems to be inextricably linked with their further miniaturization and penetration into new applications and

References

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multi-domain sensing systems. These unique midgets will become elementary building blocks for many thousands of future technical systems. It is to be hoped that they will be used for the collective good only and not for war and destruction.

References Andersson, G. I., Hedenstierna, N., Svensson, P., and Petterson, H. (1999). A novel silicon bulk gyroscope, in Transducers ’99, International Conference on Solid State Sensors and Actuators 1999, Sendai, pp. 902–905. Dau, V. T., Dao, D. V., Shiozawa, T., and Sugiyama, S. (2008). Simulation and fabrication of a convective gyroscope. IEEE Sensors Journal, 8(9):1530–1538. Maenaka, K., Ioku, S., Sawai, N., Fujita, T., and Takayama, Y. (2005). Design, fabrication and operation of MEMS gimbal gyroscope. Sensors and Actuators A, 121:6–15. Zhou, J., Yan, G., Zhu, Y., Xiao, Z., and Fan, J. (2005). Design and fabrication of a microfluid angular rate sensor, in Proceedings of the 18th IEEE International Conference on Micro Electro Mechanical Systems (MEMS 2005), pp. 363–366.

Index

Σ∆ converter feedback accelerometer, 318 output spectrum, 318 single-sided accelerometer, 317 transfer characteristic, 317 1D gyroscope, 443 2D accelerometer capacitive, 354 2D gyroscope, 444, 448 decoupled, 448 with single plate, 448 3D gyroscope fully decoupled, 450 3D accelerometer, 348 capacitive, 350, 355 cross-beam suspended proof mass, 353 multi-DOF sensing, 353 piezoelectric, 348 piezoresistive, 350 with four masses, 348 3D gyroscope, 444, 449 multi-body, 452 5D inertial sensor, 447 6D accelerometer, 355 piezoresistive, 356 abrasive-jet machining, 210 acceleration sensitivity, 379 accelerometer, 283 1D, 284, 342 angular, 284 bias, 283 capacitive, 346 cross-sensitivity, 346 closed loop, 299 closed-loop dynamics, 307 closed-loop transfer function, 301 critically damped, 289, 291 cross-coupling, 284, 297 cross-sensitivity, 298 dynamic equation, 286, 287 feedback control, 298

468

impulse response, 287 impulse transfer function, 287 loop gain, 303 model with imperfections, 296 nonlinear damping, 335 offset, 283 over-damped, 289, 292 pendulous, 285 piezoresistive, 343 beam-mass, 343 cross sensitivity, 343 quad beam, 344 resonant, 319, 324 sensitivity, 284 sensitivity–bandwidth trade-off, 291 spring stiffening, 303 step response, 287 torsional, 285, 289 transfer function, 286 twin mass, 345 under-damped, 289 accelerometer imperfections, 292 anchor-induced stress, 295 damping nonlinearity, 295 manufacturing imperfections, 294 misalignment, 295 spring nonlinearity, 294 spring hysteresis, 294 structural, 293 adhesive, 211 curing, 211 dispensing, 211 epoxy, 211, 213 polymeric, 211 silicone, 211, 213 Allan variance, 411 Allan plot, 412 aluminum interconnect, 157 analog ground, 69 angular gyroscope, 365, 381 frequency mismatch, 387 imperfections, 387 line precession, 384

Index

angular gyroscope (cont.) orbital parameters, 383 zero-ellipticity, 387 angular-momentum equation small-angle approximation, 376 angular-momentum theorem, 373 angular random walk (ARW), 380 angular velocity absolute, 372 platform rotation, 369 vector, 370 anisoelasticity, 385, 387, 388, 389, 403, 405, 409 annealing, 155, 184 Arrhenius model, 464 ashing, 153 beam, 80 area moment of, 85 bending equation, 82 bending moment, 83 bending torque, 83 biaxial moment, 85 cantilever, 82 cantilever beam, 87 clamped, 82 clamped at both ends, 82 compressive stress, 94 cross-coupling stiffness, 92 displacement equation for, 86 geometric moment of inertia, 85 guided, 82 main axes, 85 maximal stress in, 101 point load on bridge, 90 rectangular, 88 residual stress, 93 residual stress gradients in, 93 skewed bending, 85, 90 spring rate under residual stress, 95 stress distribution, 84, 85 tensile stress, 94 trapezoidal, 90 virtual cut, 83 with applied moment, 90 with distributed load, 87, 90 with force at free end, 90 with force at guided end, 90 with inclined sidewall, 90 beam accelerometer, 326 beam dynamics, 328, 334 damping, 332 eigenmodes, 330 instability, 333 modal differential equations, 331 nonlinear damping in closed loop, 336

469

prebending, 328 single clamped cantilever, 327 beam chain, 104 crab-leg spring, 105 folded beam, 104 serpentine, 104 U-shaped beam, 104 bending stiffness, 86 Bernoulli equation, 85 Bernoulli’s hypothesis, 85 bias, 380 drift, 380 instability, 380 stability, 411 bidirectional actuation, 65 bottom coverage, 158 Brownian noise, 286, 307, 375, 377, 393, 401 spectral density, 287 Bryan angles, 369, 370 Cardan suspension, 369 bubble accelerometer, 339, 340 bubble formation, 341 buckling, 95, 102 Euler buckling limit, 103 onset, 102 bulk micromachining, 153, 163, 170, 181, 183 back-side, 197 calibration, 460, 461 capacitance force balance, 53 frame-based, 67 induced charges, 50 instability of tilting plate, 58 negative electrostatic spring rate, 54 nonlinear distortion, 56 parallel plate, 51 plate, 48 rectangular plate, 57 stability condition in plate, 54 tilting plate, 48, 55 vector of electrostatic forces, 50 capacitive actuator, 48 capacitive interface, 253 bandpass sensing, 254 charge sensing, 259 correlated double sampling, 264 current sensing, 255 differential sensing, 51, 57 low-pass sensing, 254 noise, 257 noise in, 52 signal modulation, 256 switched-capacitor sensing, 260 voltage sensing, 258

470

Index

capacitive transducers, 47 capacitive sensing, 51 Castigliano’s theorem, 104 cavity sealing, 188 charge amplifier, 51 chemical mechanical polishing, 153 CMOS-MEMS, 192 add-on structures, 198 intra-CMOS MEMS, 182, 195 post-CMOS MEMS, 182, 196 post-CMOS-MEMS CMU process, 197 pre-CMOS MEMS, 182, 193 co-integration, 181 coating, 213, 221 comb capacitor, 62 half-comb, 62 linear comb, 48, 62 radial comb, 48, 66 spring-rate matrix of linear, 66 compliance matrix, 22 continuity equation, 115 continuous-flow equation, 114 convective accelerometer, 339 asymmetric convection, 339 Grashof number, 340 coordinate system, 25 coordinate frame, 367 coordinate frames, 26 inertial, 364, 382 inertial frame, 372 non-inertial, 364 body rotation, 371 partial rotations of, 28 platform frame, 368 rotation of, 25 rotation of coordinate frames, 26 Coriolis force, 364, 371, 377, 381, 390, 397 Coriolis moment, 364, 374 creep, 213 curing temperature, 214 current mirror, 235 damping, 108 damping force, 79 data converter, 266 analog-to-digital converter, 266 continuous-time modulator, 277 decimation, 278 digital-to-analog converter, 266 discrete-time modulator, 275 noise shaping, 274 one-over-f noise, 233 oversampling, 271 pulse-density modulation, 269 pulse-width modulation, 269

quantization noise, 272 sigma-delta converter, 271 transconductance, 231 delamination, 214 deposition, 153, 155 APCVD, 155 chemical vapor, 153, 155 CVD, 155 electrodeposition, 153 epitaxial, 156 LPCVD, 155 PECVD, 155 physical vapor, 153 printing, 161 silicon, 155 sol–gel, 153 spin casting, 153 diamond wire cutting, 210 dicing, 173, 205, 210 die attach, 205, 211 die separation, 205 differential measurement, 253 doping, 153, 161 annealing, 161 diffusion, 161 ion implantation, 161 drag forces, 143 drive-in diffusion, 153 drop test, 304, 379 dyadic product, 27 elasticity matrix, 21 electrostatic forces, 49 electrostatic trimming effect, 58 embedded Σ∆ converter, 315 encapsulation, 220 etching, 153, 162 anisotropic, 153, 164 back-side, 166 deep reactive-ion etching – DRIE, 170 DRIE, 190 dry etching, 168 electrochemical etch stop, 3, 167 etch stop, 167 isotropic, 153, 162 notch effect, 171 plasma etching, 168 reactive-ion etching – RIE, 168 selectivity, 165 under-etching, 166 undercut, 162 underetching, 164 vapor, 168 wet, 153 wet etching, 162

Index

Euler angles, 28 non-standard, 29 standard, 28 Euler equation, 373, 374 coordinate-vector representation, 374 Euler–Bernoulli equation, 86 failure analysis, 464 return material authorization (RMA), 464 feedback control feedback compensation, 307 lag controller, 308 lead controller, 308 feedthrough, 174 first-level packaging (FLP), 172, 205 flip table, 462 flow models, 110 fluid flow, 108 bulk viscosity, 115 compressible, 115 continuous flow, 111 Couette flow, 111, 114 free molecular flow, 145 incompressible, 115 Maxwellian boundary condition, 113 nature of, 109 Poiseuille flow, 125 slip condition, 113 Stokes flow, 111 frequency split, 399 fringe capacitances, 69 fringe field, 390 front-end electronics, 227 gas density, 121 gauge factor, 33 getter deposition, 185, 187 glass temperature, 213 grounded shield, 70 gyroscope, 364 2-DOF model, 386 beam, 438 with frequency output, 440 doubly decoupled, 427 frame-free, 428 imperfections, 385 model, 385, 386 measurement range, 379 mode decoupling, 424 frame-based architectures, 424 performance classes, 378 polysilicon ring, 443 quartz tuning fork, 440 ring, 441 ring gyroscope, 366 rotation-based, 378

471

shell-based, 366 spinning wheel, 365 translation-based, 378 tuning fork, 366 vibrating, 365, 381 vibrating beam, 366 vibrating disk, 367 vibrating string, 365 wine-glass resonator, 366 gyroscope architectures, 424 gyroscope control, 388 damping compensation, 388 ellipticity compensation, 389 frequency-mismatch compensation, 388 Helmholtz equation, 131 hermeticity, 172 Hooke’s law, 14 ideal-gas-state equation, 116 induced stress, 213 cracking, 214 interfacial shear, 214, 216 peeling, 214, 216 inertial measurement unit (IMU), 355, 444, 452 ion-beam milling, 169 ion implantation, 153, 159 isothermal speed of sound, 116 kinematics of gyroscopes, 367 rigid body, 368 Knudsen number, 109 kT /C noise, 262 laser cutting, 210 lead-frame, 207 lead-frame deformation, 222 levitation, 69 anti-levitation, 73 in drive combs, 71 of combs, 69 reduction of, 73 lifetime, 379 line of oscillation, 382 liquid-crystal polymer LCP, 209 lithography, 153 electron beam, 153, 161 optical proximity, 153 projection, 153 matrix of dyadic moments, 374 matrix of inertial moments, 374 matrix of spring rates, 50

472

Index

mean free path, 109 MEMS prototyping processes, 192 MUMPs, 192 SUMMiT V, 192 metal deposition, 157 electroless, 157 electroplating, 157 sputtering, 157 microfabrication, 153 momentum equation, 375 monolithic integration, 154 MOS transistor, 228, 229 depletion-mode, 229 drain, 229 drain current, 229, 230 early voltage, 230 enhancement mode, 229 flicker noise, 233 gain, 230 gate, 228 high-frequency small-signal model, 233 linear region, 230 MOSFET, 228, 229 n-channel, 229 noise, 232 output conductance, 232 p-channel, 229 saturation region, 230 small-signal equivalent circuit, 232 small-signal model, 231 source, 229 molding, 158 LIGA, 159 Navier–Stokes equation, 110, 115, 332 Newtonian liquids, 112 nonlinear actuator, 309 bidirectional actuator, 310 linearization, 313 nonlinearity error, 311 single-sided, 312 step response, 313 non-proportional damping, 385, 388 Nyquist’s theorem, 266 offset, 380 op amp, 234 analog ground, 234 auto-zeroing, 243 carrier modulation, 244 charge amplifier, 239 chopping, 243 common-mode rejection ratio, 243 correlated double sampling, 244

differentiating, 239 fully differential, 242 gain, 236 gain–bandwidth product, 240 golden design rules, 237 instrumentation amplifier, 246 integrating, 239 inverting amplifier, 238 model, 236 non-inverting, 237 offset, 243 open-loop gain, 240 power-supply rejection ratio, 241 real, 240 slew rate, 241 switched-capacitor amplifier, 262 transconductance amplifier, 234 transimpedance amplifier, 239 unity-gain buffer, 238 voltage follower, 238 operating temperature, 379 operational amplifiers, 234 orthogonal modes, 386 OTA, 234 voltage gain, 235 oxidation, 153 package, 206 ceramic, 207 dual inline (DIP), 207 exposed pad, 224 flip-chip, 218 lead-free soldering, 221 open cavity, 209 overmolded, 208, 221 plastic, 208 pre-molded, 209, 223 PSOP, 207 PSOT, 207 QFP, 207 reflow solder shock, 221 SOIC, 207 SOP, 207 SSOP, 207 transistor outline (TO), 206 TSSOP, 207 package materials, 212 silicone, 221 epoxy resin, 221 room-temperature-vulcanizing silicone (RTV), 221 thermoplastic, 220, 223 thermosetting plastic, 220 parasitic capacitances, 52 passivation, 154 pattern transfer, 153

Index

patterning, 154 mask, 154 photoresist, 154 pendulous effect, 106 phase-lock loop (PLL), 392 photoresist, 159 piezoelectric effect, 39 piezoelectric equations, 42 piezoelectric sensors, 43 piezoelectric transducers, 39 piezoelectric interface, 249 charge mode, 250 noise, 251 PZT sensor, 253 voltage mode, 249 piezoresistivity, 13, 30 matrix of piezo-coefficients, 31 piezoresistive materials, 34 resistivity tensor, 31 piezoresistor, 33 Hall-like, 38 on polysilicon, 38 on silicon, 34 temperature compensation of, 36 planarization, 160 chemical–mechanical polishing (CMP), 160 plate suspension, 105 by crab-legs, 106 by folded beams, 106 by hairpins, 107 by serpentine, 107 four-beam, 106 H-type, 107 torsional, 107 platform rotation, 369 Poisson equation, 99 Poisson’s ratio, 15, 23 polysilicon, 13 potential energy, 49 principal axes, 91 principal stiffness, 92 principle of virtual work, 329 method of assumed modes, 331 pull-in effect, 53 pull-in voltage, 54 stability region, 55 Q-bias, 402, 405 anisoelastic cross-coupling, 409 compensation, 405 parametric amplification, 405 for self-test, 408 phase error, 405 quadrature bias, 402 stray capacitances, 409 transducer imperfections, 408

Q-factor, 290 quality factor, 108 R-bias, 402, 409 real bias, 402 rate gyroscope, 364, 365, 382, 389 2-DOF, 391 3-DOF sensing, 399 amplitude control, 392, 419, 420 anti-phase-driven, 414 decoupled, 390 decoupled torsional, 433 drive resonator, 393 in-plane sensitive, 429 linear, 429 linear-rotatory, 430 mode-matching, 396 noise, 400 non-resonant sensing, 398 output rate spectrum, 394 phase setting, 420 PLL control, 418 rate demodulator block, 393 rate transfer function, 398 resonance sensing, 395 sensing, 393 spring nonlinearities, 421 instabilities, 422 resonance shift, 422 torsional, 431 in-plane driven, 432 out-of-plane driven, 437 torsional with 3 DOF, 434 transfer functions, 391 rate gyrosope multi-DOF drive, 397 rate-integrating gyroscope, 365, 380 rate table, 462 residual error probability, 380 resist deposition, 157 ashing, 159 polymer, 157 resist stripping, 159 spin coating, 157 resistive interface, 247 full bridge, 248 half bridge, 248 resonant beam, 320 double-ended tuning fork, 320 energy method, 323 force leverage, 324 microlever, 326 Rayleigh–Ritz method, 324 resonance frequencies, 321 resonance frequencies of loaded beam, 322

473

474

Index

Reynolds’ equation, 125, 126 linearized, 127 modified, 138 Reynolds’ number, 117 sampling and hold, 266 sensitivity error, 379 sensor interface, 247 shaker, 462 shear modulus, 23 shock survivability, 379 signal-to-noise ratio, 305 closed loop, 306 silicon, 13 diamond structure, 22 mechanical properties, 13, 23 piezoresistance of, 24 silicon dioxide, 156 boron phosphosilicate glass (BPSG), 156 CVD oxide, 156 phosphosilicate glass (PSG), 156 TEOS, 156 silicon nitride, 157 LPCVD nitride, 157 silicon on insulator (SOI), 172, 181 customized SOI, 191 SOI-MEMS process, 189 single-mass inertial sensors, 445 gyroscope-based, 445 gyroscope-free, 445 slide damping, 117 Couette flow, 118 pure viscous flow, 122 quality factor, 119 slip equation, 118 Stokes flow, 120, 123 spectral density, 306 spring, 79 compliance, 80 parallel connection of, 103 serial connection of, 103 spring rate, 80 stiffness, 80 spring connection, 103 spring matrix principal axes, 386 spring nonlinearities, 418 spring softening, 50, 59 spring–mass system, 284 squeeze damping, 124 at high frequencies, 131 at low frequencies, 127 for annular plate, 129 for cylindrical plate, 128 for rectangular plate, 128 for torsion plate, 129

of fast circular plate, 135 of fast rectangular plate, 132, 134 of torsion plate, 136 step response of plate, 133 squeeze damping with perforation, 137 at atmospheric pressure, 138 at low pressure and high frequency, 140 penetration rate, 138 step coverage, 158, 160 stiction, 164 anti-sticking agent, 164 stiffness matrix, 81 Stokes equation, 117 Stoney’s formula, 216 storage temperature, 379 strain, 14 strain tensor, 19 shear strain, 20 strain gauge, 33 strain rate, 111, 114 stress, 14 shear stress, 18 stress tensor, 17 transfer of, 38 stress concentration, 101 stress–strain in isotropic materials, 23 stress–strain relation, 21 structural damping, 147 support loss, 148 surface micromachining, 13 surface micromachining, 153, 163, 181, 184, 192 PSM-X2 process, 185 sacrificial layer, 153 surface roughness, 166 suspension, 103 Sutherland equation, 112 tape automated bonding (TAB), 217 tensor rotation of, 27 transformation of, 30 tensor of dyadic moments, 374 tensor of inertia, 373 test, 460 burn-in, 463 device characterization, 460 fabrication, 460, 461 infant mortality, 464 product, 460, 461 product qualification, 460 self-test, 380, 460, 463 thermal expansion coefficient, 21 thermoelastic damping, 147, 410 torsion bar, 97

Index

with arbitrary cross-section, 97 annular, 101 cylindrical, 96, 101 polar moment, 97 rectangular, 100 torsion spring, 96 cylindrical bar, 97 spring rate, 97 stiffness, 97 th eo ry o f S a int Ven a nt, 9 7 total angular momentum, 368 total momentum, 368 transducer, 33 tuning fork, 413, 427, 430 acceleration sensitivity, 417 acceleration suppression, 413, 415 Hula mode, 417 torsional, 436 tunneling accelerometer, 337 with fast tracing beam, 337 ultra-fine leak test, 185 underfiller, 218 velocity absolute, 371 angular, 374 of non-inertial frame, 371 relative, 370 vibration sensitivity, 379 vibratory gyroscope, 376 planar structure, 377 viscoelasticity, 213 viscoplasticity, 213

viscosity, 111, 112 Voigt’s notation, 19 wafer bonding, 172 anodic bonding, 176 bond strength, 178 by adhesive, 180 eutectic, 179 fusion bonding, 175 glass-frit bonding, 177 polymer adhesive, 180 polymer bonding, 180 thermocompression, 180 wafer breaking, 210 Wheatstone bridge, 37, 247 Wheatstone half-bridge, 38 wire bonding, 217, 218 ball, 218 ball-bond neck break, 219 bond lifting, 219 bond pad crack, 219 thermocompression, 218 thermosonic, 218 ultrasonic, 218 wedge, 219 wedge-bond heel break, 219 wire break, 219 Young’s modulus, 14, 23 z-gyroscope, 390, 396, 413, 424 zero-rate output (ZRO), 380, 402 bias sources, 403 electrical cross-coupling, 410 zero-level packaging, 172

475