The field of microsystems has an interesting history. In the late 1950s, the physicist Richard P. Feynman [1] delivered a presentation during which he posed two technological challenges designed to stimulate development of new ways to design and build miniature machines. The first challenge was to write a page of a book on a surface 25 000 times smaller in linear dimensions than the original, to be read by an electron microscope. The second prize was to build a fully functional, miniature, electric motor with dimensions less than 1∕64 of an inch. Surprisingly, the second prize was claimed first in 1960, using conventional fabrication methods. On the other hand, the technology required to accomplish the first challenge did not emerge until 1987. One early MEMS device, the resonant gate transistor shown in Fig. C.1, was demonstrated in 1967 [2], but the field did not really begin to emerge till the 1980s and 1990s. Today there is a vast array of methods and processes for fabricating micromechanical devices. This appendix offers a concise introduction to the fundamental concepts of microfabrication and to the ways it is used to create MEMS devices.

Most of the fabrication processes emerged from the revolution in the microelectronics industry. Those adapted or specifically developed for MEMS are referred to collectively as micromachining. These batch processes have the considerable advantage that they can deliver large numbers of device replicas on a wafer simultaneously.

Basic assumptions and concepts

In Chapter 1, we introduced the notion of the lossless electromechanical coupling. Then, in Chapter 2, the principles and approximations of circuit-based device modeling were reviewed. The primary goal of Chapter 3 is to introduce the energy-based technique for determining the electrical force operative in a capacitive transducer. This force effects the electromechanical transduction of energy between electrical and mechanical forms, and we cannot predict the behavior of a MEMS device without it. Electromechanical interactions in capacitive devices arise from either of two physical origins. First and far more familiar is the Coulombic interaction of electric charges at a distance. The force exerted on an electrostatic charge q is qE̅, where E̅, the vector electric field, is the superposition of the force fields created by all the other charges. The other, less well-known force mechanism originates from the interactions of an electric field with the dipoles that constitute liquids and solids. These dipoles can be either induced or permanent. The essential requirement for an observable force is a non-uniform electric field. All dipoles have zero net charge, but if the positive and negative charge centers experience slightly different electric field vectors, there will be a net force. For a small dipole having moment p̅, this force may be approximately expressed by p̅·∇E̅. An ensemble of dipoles in any solid (or liquid) can experience a net body force, called the ponderomotive effect. The classic book by Landau and Lifshitz presents a general electroquasistatic formulation for the volume density of the ponderomotive force [1].

Fundamentals of circuit theory

This chapter summarizes basic circuit theory concepts, shows how to model capacitive MEMS actuator structures as simple circuit devices, and introduces elements of multiport network theory for later use in modeling electromechanical systems. The approach is a very practical one, based on reasonable approximations and easy-to-use inspection methodologies. Students seeking more background on the theoretical underpinnings of circuit modeling should refer to Appendix A, which summarizes the electroquasistatic and magnetoquasistatic approximations, reveals the origins of the models for capacitors and inductors, and relates them to basic circuit theory.

Our emphasis on circuit-based representations for MEMS devices is motivated by the fact that they can be embedded directly into the electronic system models for the control and sensing circuitry. With this groundwork, we will later investigate conventional implementations of capacitive sensors and actuators, including inverting operational-amplifier circuits, two-plate and three-plate topologies, and the half-bridge differential scheme. For sufficiently complex systems, software tools such as PSPICE or CADENCE might be used, but in this text on fundamentals, we restrict the focus to systems that can be treated analytically.

Introduction

The piezoelectric effect is widely exploited in actuators and sensors larger than about a millimeter. A familiar example is the crystal oscillator, which is heavily relied on as a stable frequency standard in electronics. Migrating piezomaterials into smaller scale devices has been stymied until fairly recently by serious fabrication challenges. The main problem is that the common piezoelectric solids are either ceramics or crystals, neither of which is amenable to the surface and bulk microfabrication processes used for MEMS. This situation is now starting to change. New materials and the associated microfabrication processes needed to incorporate them into submillimeter structures are being developed. Examples include thin aluminum nitride films sputtered on such substrates as Pt and crystalline Si [1]. Figure 8.1 shows some interesting and novel structures that have now been fabricated. Piezoelectric-based MEMS product lines are on the market and further entries may be anticipated.

For use as a MEMS material, piezoelectrics have compelling advantages. First, their response is linear over a large dynamic range. This attribute simplifies the requirements placed on the signal-conditioning electronics. Perhaps more importantly, piezoelectrics possess high energy densities. Because of their favorable scaling for thin film-based structures, MEMS-scale actuators and sensors using the piezoelectric effect deliver, respectively, large forces and strong signals. Furthermore, they have the practical advantage of being self-biasing, obviating the need for a DC voltage source. Finally, piezoelectric materials are generally inexpensive.

Introduction

Previous chapters of this text have relied exclusively on lumped parameter capacitance models with mechanical motion represented by one or perhaps a few discrete mechanical variables. In these models, the capacitive electrodes have been assumed to be rigid structures. In MEMS, however, many of the common designs do not fit such a description. For example, one of the most widely exploited structures is the cantilevered beam, a 1-D continuum. Also, pressure sensors and microphonic transducers are usually based on deformable 2-D continua, typically circular diaphragms. Other applications of MEMS continua exist and are growing in numbers. For example, Fig. 6.1 shows a deformable mirror, developed by Boston Micromachines Corporation, for use in laser-pulse-shaping applications where the deformable mirror modifies the phase of the spectral components of the laser pulse to achieve desired temporal pulse characteristics.

This chapter develops modeling approaches for such continua and demonstrates that it is usually possible to devise reduced-order, lumped-parameter models that capture the essential behavior and reveal important trade-offs amongst the system parameters. Section 6.2 employs this approach, using the simple example of a cantilevered beam operated as a capacitive transducer. After introducing some basics from the mechanics of continua, a familiar-looking lumped parameter model is extracted and then tested for accuracy by comparing the predicted resonant frequency with the well-known analytical solution for the cantilevered beam. In the initial exercise, certain assumptions and approximations are presented without much explanation but these are tested and justified in Section 6.3 in a detailed revisit to the problem. The effort leads naturally to the electromechanical two-port transducer representations of Chapter 4. In subsequent sections, the same treatment is extended to the important geometry of the circular diaphragm.

Introduction

MEMS devices are usually exploited in practical measurement and actuation technologies by integrating them on-chip with the necessary sensing circuitry and electronic drives. While the electromechanical transducer itself is the heart of any MEMS system, its capabilities can be realized only with appropriate amplification, regulation, and signal conditioning. Electronic design has always been central to actuator and sensor technologies, but the development of microfabricated devices has presented new challenges for circuit designers. Devices with dimensions of the order of tens to hundreds of microns have very small capacitances – C < 1 pF – and circuits must be reliably sensitive down to ΔC ~ 10 fF. Further, typical devices fabricated on chips suffer significant parasitics, requiring that serious attention be paid to electrostatic shielding and to the issues of signal strength and noise.

In this chapter, we introduce and analyze some of the basic operational-amplifier-based circuit topologies for capacitive sensing. The presentation focuses on how to integrate the two-port electromechanical models developed in Chapter 4 with simple amplifier circuits. The emphasis is on basic principles at the systems level. The starting point is a brief review of the ideal operational amplifier and its most important circuit implementation, namely, the inverting amplifier configuration. We then apply this very robust and adaptable circuit to the basic DC biased two-plate capacitive sensor, showing that such systems provide good sensitivity but entail the serious problem of large DC voltage offsets. Three-plate sensing schemes are then introduced as a way to avoid DC offsets and to take advantage of the inherent sensitivity of differential measurements.

Background

In this chapter, we introduce the linear electromechanical two-port, its multiport generalization, and equivalent circuit models for them. This MEMS representation makes it possible to invoke the so-called cascade design paradigm. The resulting electromechanical multiport networks and equivalent circuits broadly describe the critical behavior of microelectromechanical devices and systems. For the cascaded sensor depicted in Fig. 4.1a, the mechanical input might be a pressure perturbation acting on a diaphragm or an acceleration acting on a proof mass. In whatever form taken, this input acts on a mechanical element that possesses mass and is constrained by a restoring spring, mechanical damping, and possibly other influences. The electromechanical coupling converts this motion into an electrical signal, which is then appropriately conditioned and electronically amplified by a detector. For the actuator in Fig. 4.1b, an input electrical signal drives the electromechanical transducer, which in turn produces a force acting on the mechanical system. The system then responds with the desired motion. These schemes are typical examples of cascaded systems consisting of sequentially connected two-port networks.

The crucial assumption of the electromechanical two-port construct is that the amplitudes of the electrical and mechanical perturbations are small enough to justify linearization of the coupling mechanism. Linearization leads directly to transmission matrices similar to those introduced in Chapter 2. More importantly, the full power of linear systems theory, starting with AC phasors and transfer functions, becomes available. Initially, students with electrical engineering backgrounds will have the advantage because they are already adept at using these methods to design and analyze electronic amplifiers and filters.

Background

The trend toward device miniaturization and large-scale integration, which has already revolutionized electronics, now promises a profound transformation of engineered mechanical systems, reducing their size by orders of magnitude while vastly increasing their capabilities. Microelectromechanical systems (MEMS) are now found in automotive airbag systems, computer projectors, digital cameras, gyroscopic sensors, and many other devices. Their small size invites a high degree of on-chip integration with essential drive, detection, and signal conditioning circuitry. These collective advances have spawned another new term, the system-on-a-chip, with its own inevitable acronym, SOC. Consider as an example the digital camera. Nowadays, even rather inexpensive models have MEMS chips installed to sense the camera's orientation with respect to gravity and to detect and compensate for the inadvertent jolts and motions of the picture taker. Such features would have been well beyond the expectations of the owner of even the most expensive SLR camera of 20 years ago. The capabilities mentioned above are made possible by mechanical devices with dimensions less than a millimeter or so, fabricated on a chip side by side with all the required control and drive electronics.

Additional evidence for the vitality of this new technology is that the microfabrication industry has appropriated the term foundry to describe their facilities. This word dates from sixteenth-century French. One of the authors (TBJ) has a vivid childhood recollection from the 1950s of the nightly spectacle of fumes and fire belching impressively from the venting chimneys atop a metal casting foundry in the small Midwestern city where he grew up. This plant produced manhole covers and other essential yet mundane components of the urban infrastructure.

The growing interest in microsystems, and particularly in MEMS technology, has reasserted electromechanics as a key discipline. This book fills the need for a textbook that presents the fundamentals of electromechanics, classifies structures according to their functional capabilities, develops systematic modeling methods for the design of MEMS devices integrated into electronic systems, and provides practical examples derived from selected microdevice technologies. It is written for engineering students and physical science majors who want to learn about such systems. A further ambition is that the book will find its way slowly onto the shelves of practicing engineers involved in MEMS design and development.

Organization

The organization proceeds from basics to systems-oriented applications. The first three chapters focus on fundamentals of circuits and lumped parameter electromechanics. Chapter 1 provides some historical context, introducing key terminology and then offering a general description of electromechanical transducers based on power and energy considerations. Chapter 2 introduces the crucial concept of circuit-based modeling. Because the vast majority of MEMS devices are capacitive, this chapter focuses on circuits with capacitors and resistors. Chapter 3, drawing heavily on Part 1 of H. H. Woodson and J. R. Melcher's text, Electromechanical Dynamics, presents the classic, energy-based formulation for electromechanical interactions. The treatment here differs from their text by concentrating on capacitive microelectromechanical devices and introducing the geometries and dimensions characteristic of MEMS technology.

Introduction

The general equations of motion, developed in Chapter 10, are, without question, the complete and proper model of our dynamic system. However, because of their nonlinear character, they are not directly amenable to the use of the many mathematical tools that are available for linear systems. For example, many vibration and automatic control techniques are directly valid only for linear systems. Indeed, even the electronic instrumentation that is available for measurement of the dynamics of multibody systems is often designed to operate in the frequency domain and, thus, inherently assumes that the system treated is linear.

If there is any hope for a general solution technique for these equations of motion, it is probably through numeric integration by digital computer. We will investigate such an approach in Chapter 14. However, before looking at the general case, let us first study the dynamics of our system in the local vicinity of its current posture.

Linearization Assumptions

At its current posture, whatever posture this might be, we assert that the system in question exists in accordance with our general dynamic equations of motion. If it is not in equilibrium in the sense of being stationary or operating at constant velocity, then it is in dynamic equilibrium, meaning that its accelerations are consistent with these same equations of motion.

Mechanical Design: Synthesis versus Analysis

There are two completely different aspects of the study of mechanical systems: design and analysis. The concept embodied in the word design might be more properly termed synthesis, the process of contriving a scheme or a device for accomplishing a given purpose. Design is the process of developing the sizes, shapes, material compositions, types and arrangements of parts, and manufacturing processes so that the final system will perform a prescribed task. Although there are many phases of the design process that can be approached in a well-ordered scientific manner, the process is, by its very nature, as much an art as a science. It calls for imagination, intuition, creativity, judgment, and experience. The role of science in the design process can be viewed as providing tools to be used as the designer practices this art. Computer programs and computations that allow a designer to simulate a system and evaluate its potential performance play an important role in helping the designer practice the art. This is why scientific techniques such as the matrix methods discussed in this text play such an important role in dealing with the design of three-dimensional mechanisms and multibody systems.

Introduction

Through simulation of multibody systems as explained in the preceeding chapters, we can solve a variety of useful problems with no further enhancement. However, with the methods explained so far, we still lack the capability to simulate collisions, either between moving bodies or between a single moving body and its fixed surroundings. Simulation software developed strictly with the formulae presented so far assumes that a moving body may simply pass through others with no interference or impact. Clearly, this can benefit from enhancement.

Collision or contact between bodies cannot be detected unless it is through computations relating the geometries of the bodies’ surfaces. Therefore, we must have accurate geometric shapes for all bodies for which collisions are to be considered, and in as much detail and accuracy as we wish to monitor their possible contact. We need data for vertices, edges, and surfaces, and we need to distinguish the material from the exterior sides of such surfaces. Therefore, we need solid models of the bodies to be considered. Either constructive solid geometry (CSG) or boundary representation (B-Rep) or hybrid combinations may be considered, but wire-frame data are not sufficient.

Introduction

In order to make a systematic study of mechanisms and multibody systems and to develop general methods for their analysis by digital computer, we must be able to recognize and precisely describe certain basic information that governs their operation. For example, it is clear that, at some point, we must explicitly identify certain dimensional information, such as part shapes and dimensions, in order to perform the analysis. However, before we reach this stage, another even more fundamental problem confronts us. We must first study each system enough to determine how its various parts are interrelated – that is, which part is connected to which, and what is the nature of each connection. In other words, we need to understand the kinematic architecture of the multibody mechanical device. In the kinematics literature, the term “structural analysis” has sometimes been used for this type of analysis. Here, however, we use the term “kinematic architecture” to avoid confusion with the statics use of structural analysis.

In the classic methods of analysis, both graphic and analytic, the task of recognizing the architecture of a mechanism or multibody system did not require reduction to a step-by-step procedure. No real difficulties arose because the analyst, through experience, developed an intuitive feeling for analyzing problems of a given type. As the analysis progressed, he or she could continually make decisions based on experience as to what steps should be taken in what order and what techniques might be applied to accomplish each step.