Analysis and Design Principles of MEMS Devices (eBook)

Minhang Bao (Autor)

eBook Download: PDF

2005 | 1. Auflage
328 Seiten
Elsevier Science (Verlag)
978-0-08-045562-4 (ISBN)

Infos

Sensors and actuators are now part of our everyday life and appear in many appliances, such as cars, vending machines and washing machines. MEMS (Micro Electro Mechanical Systems) are micro systems consisting of micro mechanical sensors, actuators and micro electronic circuits. A variety of MEMS devices have been developed and many mass produced, but the information on these is widely dispersed in the literature. This book presents the analysis and design principles of MEMS devices. The information is comprehensive, focusing on microdynamics, such as the mechanics of beam and diaphragm structures, air damping and its effect on the motion of mechanical structures. Using practical examples, the author examines problems associated with analysis and design, and solutions are included at the back of the book. The ideal advanced level textbook for graduates, Analysis and Design Principles of MEMS Devices is a suitable source of reference for researchers and engineers in the field.

* Presents the analysis and design principles of MEMS devices more systematically than ever before.

* Includes the theories essential for the analysis and design of MEMS includes the dynamics of micro mechanical structures

* A problem section is included at the end of each chapter with answers provided at the end of the book.

Inhalt

Front Cover 1
Analysis and Design Principles of MEMS Devices 4
Copyright Page 5
Contents 10
Preface 8
Summary 14
Chapter 1. Introduction to MEMS devices 16
§1.1. Piezoresistive pressure sensor 16
§1.2. Piezoresistive Accelerometer 22
§1.3. Capacitive Pressure Sensor, Accelerometer and Microphone 25
§1.4. Resonant Sensor and Vibratory Gyroscope 29
§1.5. Micro Mechanical Electric and Optical Switches 33
§1.6. Micro Mechanical Motors 35
§ 1.7. Micro Electro Mechanical Systems 39
§1.8. Analysis and Design principles of MEMS Devices 43
References 45
Chapter 2. Mechanics of beam and diaphragm structures 48
§2.1. Stress and Strain 48
§2.2. Stress and Strain of Beam Structures 59
§2.3. Vibration Frequency by Energy Method 78
§2.4. Vibration Modes and the Buckling of a Beam 88
§2.5. Damped and forced vibration 99
§2.6. Basic Mechanics of Diaphragms 112
§2.7. Problems 123
References 128
Chapter 3. Air Damping 130
§3.1. Drag Effect of a Fluid 130
§3.2. Squeeze-film Air Damping 139
§3.3. Damping of Perforated Thick Plates 159
§3.4. Slide-film Air Damping 165
§3.5. Damping in Rarefied Air 174
§3.6. Problems 185
References 188
Chapter 4. Electrostatic Actuation 190
§4.1. Electrostatic Forces 190
§4.2. Electrostatic Driving of Mechanical Actuators 196
§4.3. Step and Alternative Voltage Driving 213
§4.4. Problems 222
References 227
Chapter 5. Capacitive Sensing and Effects of Electrical Excitation 228
§5.1. Capacitive Sensing Schemes 228
§5.2. Effects of Electrical Excitation — Static Signal 241
§5.3. Effects of Electrical Excitation — Step Signal 250
§5.4. Effects of Electrical Excitation — Pulse Signal 254
§5.5. Problems 257
References 259
Chapter 6. Piezoresistive Sensing 262
§6.1. Piezoresistance Effect in Silicon 262
§6.2. Coordinate Transformation of Second Rank Tensors 269
§6.3.Coordinate Transformation of Piezoresistive Coefficient 276
§6.4. Piezoresistive Sensing Elements 280
§6.5. Polysilicon Piezoresistive Sensing Elements 289
§6.6. Analyzing Piezoresistive Bridge 298
§6.7. Problems 316
References 319
Answers to the Problems 320
Subject Index 324

Textauszug

Chapter 2

Mechanics of Beam and Diaphragm Structures

Crystalline silicon is an excellent mechanical material as well as an excellent electronic material. The mechanical properties of bulk silicon are quite ideally governed by the theory of elasticity in a large temperature range. It has been speculated that the mechanical properties of silicon may change when the geometries of the mechanical structure are scaled down. Fortunately, however, no significant changes in the mechanical properties have been observed so far for silicon mechanical structures in micrometer scale. Therefore, it will be assumed throughout this book that the mechanical properties of a silicon micro structure are ideally elastic. This assumption implies that, if the deformation produced by external forces does not exceed a certain limit, it disappears once the forces are removed (i.e., Hooke’s law, see §2.1.3).

As a crystalline material is anisotropic, the mechanical properties of silicon are orientation dependent and the relations among mechanical parameters are tensor equations, which are quite complicated as shown in §2.1. Thus, for the simplicity of analysis, homogeneous assumption is used in most part of this book. The homogeneous assumption simplifies analytical analysis greatly without causing significant errors in the results.

Even with these assumptions, approximations have to be made for analytical analysis of most practical problems. However, the results are generally accurate enough for design optimization, as in most cases the performance of mechanical sensors is more significantly affected by the variations of geometric parameters determined by process control rather than by the assumptions and approximations. If a more precise result is required, a numerical analysis has to be made using a computer aided design (CAD) tool in addition to the analytical analyses. This will be beyond the scope of this book.

As silicon mechanical structures with beams and diaphragms are the most important parts for MEMS devices, mechanics of beam and diaphragm structures will be studied in this chapter according to the theory of elasticity for homogeneous material [1,2].

§2.1 Stress and Strain

§2.1.1 Stress

According to the theory of elasticity, external forces acting on a solid-state body produce internal forces between the portions of the body and cause deformation. If the external forces do not exceed a certain limit, the deformation disappears once the forces are removed. To describe the internal forces, the stress tensor is introduced. Mathematically, stress is a second rank tensor, which has nine components as shown by the matrix

(2.1.1)

where the three diagonal components are referred to as normal stresses and the six off-diagonal components are called shearing stresses.

To illustrate the definition of the components of stress tensor, let us examine an elemental cube inside the body as shown in Fig. 2.1.1. The six faces of the cube are denoted as x, , y, , z, , according to the normal of the faces. (Note: a bar over a letter indicates a negative sign for the letter.)

Fig. 2.1.1 Components of the stress tensor

In the figure, Tij, the component of a stress tensor, is defined as the force in a specific direction j (the second subscript) on a unit area on a specific surface i (the first subscript) of the elemental cube. For examples, TXX in Fig. 2.1.1 is the normal force per unit area of the x-plane, TXY is the force in the y-direction applied on a unit area of the x-plane and TXZ is the force in the z-direction per unit area of x-plane, and so forth.

The signs of the tensor components are determined by the direction of the force relative to the normal of the plane. For example, for the x-plane, the normal stress component caused by a force in the x-direction is defined as positive but that caused by a force in the -direction is defined as negative. The stress component caused by the tangential forces on the -plane and in the y- and z-directions are defined as positive while those by tangential forces in the - and -directions are defined as negative. Furthermore, for the plane, the stress component caused by a force in the -direction is positive but that caused by a force in the y-direction is negative. Similarly, the stress components caused by the tangential forces in the - and -directions on the -plane are defined as positive and those by the forces in the x- and z- directions on the -plane are defined as negative.

According to the condition of force balance, the TXX in two opposite parallel planes (x- and x -planes) should be equal in quantity and sign, and the same is true for the TYY and the TZZ. Also from the condition of torque balance, we have

(2.1.2)

This means that the matrix of stress tensor T is symmetric and has only six independent components. Therefore, they are often denoted by a simplified notation system

(2.1.3)

Therefore, equation (2.1.1) can be written as

§2.1.2 Strain

With stresses, deformation is produced inside the material. For easy to understand, let us first look at the deformation of a one-dimensional material as shown in fig. 2.1.2. If the displacement of an original position x is u(x) and the displacement of an original position x+Δx is u(x+Δx), the strain (the relative elongation of the material) in the one-dimensional material has only one component,

Fig. 2.1.2 Deformation of a one-dimensional material

For a three dimensional material, the deformation of the material is schematically shown in Fig. 2.1.3. The displacement components for the point r (x,y,z) are u(x,y,z), v(x,y,z) and w(x,y,z) in the x-, y- and z-directions, respectively, and the displacement components for the point r′ (x+Δx, y+Δy, z+Δz) are u+Δu, v+Δv and w+Δw, respectively. The three components are all functions of x, y and z.

Fig. 2.1.3 Deformation in a three-dimensional material

Therefore, the incremental displacement between point r (x, y, z) and point r′ (x+Δx, y+Δy, z+Δz) can be expressed as

(2.1.4)

For a solid-state body rotating around the origin O of a coordinate system with an angular velocity , the velocity of the end of the vector is . According to equations , we have .

Therefore, an angular displacement can be expressed as

(2.1.5)

where

Therefore, the last term on the right side of Eq. (2.1.4) is

If no rotational movement for the solid-state material is allowed, the last term on the right side of Eq. (2.1.4) is zero and the equation can be written as

(2.1.6)

The 3 by 3 matrix in the equation is referred to as a strain tensor in solid-state physics. The three diagonal components in the matrix are called the normal strain components of the strain tensor

(2.1.7)

It is quite clear that the three normal components of strain tensor shown in Eq. (2.1.7) are the relative elongations along the three coordinate axes. The six off-diagonal components are referred to as the shearing strain components of the strain tensor

(2.1.8)

Therefore, Equation (2.1.6) is written as

(2.1.9)

where (e) represents the strain tensor, a tensor of the second rank. As the strain tensor (e) is a symmetrical tensor with only six independent components, simplified notations can be used

Thus, we have

It has been found that the three shearing strain components are related to the angular distortion of the material. To verified these results, we consider the distortion of angle ∠APB, a right angle included by the elemental sections of PA=dx and PB=dy in the XY plane as shown in Fig. 2.1.4. Due to a deformation, the original positions A, P and B move to A′, P′, and B′, respectively. If u(x, y) and v(x, y) are the displacements in the x- and y-directions for point P(x,y), respectively, the displacement of point A in the y-direction is

Fig. 2.1.4 Angular deformation by shearing stress

And the displacement of point B in the x-direction is

The...

Produkt-Details

• Erscheint lt. Verlag: 12.4.2005
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie ► Analytische Chemie
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• Themenwelt: Wirtschaft
• ISBN-10: 0-08-045562-X / 008045562X
• ISBN-13: 978-0-08-045562-4 / 9780080455624