Introduction

In the previous chapters, we have investigated lumped parameter electromechanical conversion, linear multiport representations for actuators and sensors, and the effects of external constraints on transducer response. We used a powerful electromechanical analog to synthesize circuit models for transducers, examined the signal conditioning and amplification stages needed to turn MEMS devices into practical systems, and finally developed a methodology to represent mechanical continua, viz., beams and plates, as lumped parameter systems. We are now ready to put these modeling tools to use in the analysis of some practical MEMS devices.

This chapter provides concise system-level technical presentations of four important MEMS applications: pressure sensors, accelerometers, gyroscopes, and energy harvesters. These applications have been selected according to diverse criteria. For example, pressure sensors, relatively uncomplicated as MEMS go, were among the first types of sensors and actuators to be miniaturized. MEMS accelerometers, more complicated than pressure sensors, are very widely used in automotive airbag systems. Micro-mechanical gyroscopes, now available commercially, are considerably more complex than accelerometers, with respect to both the dynamics and the system-level electromechanical drive–sense scheme needed to make them work. Finally, the MEMS energy harvester is included because it is a concept that shows great promise but needs further development before real market penetration can occur.

This appendix reviews the essentials of lumped parameter mechanical systems relevant to microelectromechanical sensors and actuators. Our focus here is mechanical resonators. This choice is based on the reality that, in the MEMS world, stictive phenomena such as van der Waals forces or capillarity are always lurking. Like a tree snagging a kite from the air, these forces can fasten upon a moving element and affix it permanently to an adjacent surface. For this reason, resonance is usually the best way to exploit large motions reliably in a MEMS device. We start with a simple spring-mass system with one degree of freedom and Newton's second law in scalar form. All the main concepts are derived from the single-degree-of-freedom linear model, and then generalized to multiple-degree-of-freedom models and continuous systems using modal analysis.

The approach taken in this appendix is tailored to students having some familiarity with Laplace transforms, from the perspective of either resonant electric circuits or mechanical resonators. The starting point is the differential equation of motion; simple Laplace techniques are introduced so that solutions to the equation can be obtained and the important properties of this solution can be studied.

Preliminaries

For the first century of its existence, the engineering discipline of electromechanics focused almost exclusively on high-current relays, magnetic actuators, and, of course, rotating machines, the latter ranging from fractional horsepower induction motors all the way to gigawatt-rated synchronous alternators [1, 2]. Capacitive transducers never really figured in the discipline because of the focus maintained by the electric power industry on electrical ↔ mechanical power conversion and control. It might seem ironic then that the first eight chapters of this text exclusively concern electrostatic transducers. This emphasis, warranted by the dominance of such devices in the technology of MEMS, testifies to the stern rule of physical scaling in the engineering of useful devices. On the scale of millimeters and below, electrostatic forces hold a considerable advantage.

It would be a mistake, however, to foster the impression that magnetic MEMS will have no role in the future of microsystems technology. They already fill some significant niches in micromechanics and many believe their full potential remains to be tapped. The area-normalized scaling analysis presented in Section A.6 of Appendix A reveals that, based on the present state of materials science and process technology for magnetic materials, the dimensional “break-even” point between capacitive and magnetic MEMS devices is near one millimeter, that is, ~103 microns. This result might seem to suggest that magnetic devices face insurmountable challenges in overtaking capacitive devices in the size range of ~100 microns; however, scaling analyses are prone to the biases built into the assumptions used to formulate them, and as a result, may misrepresent the situation.

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.