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MECHANICALLY AMPLIFIED CAPACITIVE
STRAIN SENSOR

by
JUN GUO

Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy

Dissertation Adviser: Dr. Wen H. Ko
Department of Electrical Engineering and Computer Science
CASE WESTERN RESERVE UNIVERSITY

May, 2007

CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

Jun Guo

______________________________________________________ candidate for the Ph.D. degree *.

Wen H. Ko

(signed)_______________________________________________
(chair of the committee)

Alexis R. Abramson

________________________________________________

Thomas P. Kicher

________________________________________________

Steven Garverick

________________________________________________

Darrin J. Young

________________________________________________

Christian A. Zorman

________________________________________________

03/23/2007

(date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein.

Dedication

To my parents.

TABLE OF CONTENTS

Mechanical Amplified Capacitive Strain Sensor

Table of Contents ......................................................................................................................... 1 List of Tables ................................................................................................................................ 7 List of Figures .............................................................................................................................. 9 Acknowledgements .................................................................................................................. 15

  • Abstract
  • ................................................................................................................................. 16

  • Chapter 1
  • Introduction to Strain Measurement ................................................. 18

1.1. 1.2.
Introduction and motivation............................................................................................ 18 Thesis organization ......................................................................................................... 23

  • Chapter 2
  • A Review of Micromachined Strain Gages ........................................ 25

2.1. 2.2.
Introduction..................................................................................................................... 25 Metallic-foil strain gage.................................................................................................. 25

2.2.1 Principle........................................................................................................ 26 2.2.2 Gage factor and sensitivity............................................................................ 28 2.2.3 Instrumentation ............................................................................................. 29 2.2.4 Resolution of a typical metallic-foil strain gage........................................... 30

Piezoresistive strain gage................................................................................................ 32

2.3.1 Piezoresistive effect ...................................................................................... 32

2.3.

1
2.3.2 Longitudinal and transverse piezoresistive coefficients ............................... 33 2.3.3 Impurity concentration (doping level) and temperature effect on piezoresistance coefficients....................................................................................... 35

2.3.4 Resolution and issues.................................................................................... 36

Resonant silicon micromachined strain gage.................................................................. 37

2.4.1 Principle........................................................................................................ 38 2.4.2 Excitation and detection scheme................................................................... 39 2.4.3 Resolution: minimum detectable frequency ................................................. 40

Capacitive strain gage..................................................................................................... 42

2.5.1 Principle and configurations ......................................................................... 42 2.5.2 Performance of capacitive strain gage.......................................................... 44

Summary......................................................................................................................... 46
2.4. 2.5. 2.6.

  • Chapter 3
  • Capacitive Strain Sensor Employing a Buckled Beam Mechanical

Amplifier ........................................................................................................................ 48

3.1. 3.2.
Introduction..................................................................................................................... 48 Mechanical amplifier ...................................................................................................... 49

3.2.1 Background................................................................................................... 50 3.2.2 Principle........................................................................................................ 51

Analytical modeling........................................................................................................ 52

3.3.1 Derivation of the curvature function under an applied force........................ 53

3.3.

2
3.3.2 Calculate the mechanical gain ...................................................................... 56 3.3.3 Small input force simplification and the structure nominal mechanical gain 59 3.3.4 Large deflection compensation..................................................................... 60 3.3.5 Calculate the buckling beam equivalent stiffness in x-direction .................. 60

Finite-Element-Analysis verification.............................................................................. 61

3.4.1 FEA verification of nominal mechanical...................................................... 61 3.4.2 FEA verification of equivalent stiffness ....................................................... 62

3.4.

3.4.3 FEA verification of mechanical gain corresponding to large displacement input
63

3.5. 3.6.
Nonlinearity of the buckled beam amplifier ................................................................... 65 Fundamental noise analysis ............................................................................................ 67

3.6.1 Resistive thermal noise estimation................................................................ 68 3.6.2 Mechanical thermal noise (Brownian motion noise) estimation .................. 70

The maximum stress of the structure and locations........................................................ 73 Device fabrication........................................................................................................... 75 Fabricated device ............................................................................................................ 77
3.7. 3.8. 3.9. 3.10. Testing............................................................................................................................. 80

3.10.1 Test fixture .................................................................................................... 80 3.10.2 Device functionality test under microscope.................................................. 83 3.10.3 Capacitive readout circuitry.......................................................................... 84

3
3.10.3.1 C/V characteristics using MS3110®.................................................. 84 3.10.3.2 C/V characteristics using developed interface electronics................ 88

3.11. Conclusions..................................................................................................................... 93

  • Chapter 4
  • Sensor Backing Design ........................................................................ 95

  • 4.1.
  • Strain transmission ratio of a solid backing......................................................................95

4.1.1 Transmission ratio definition .......................................................................... 95 4.1.2 Strain transmission ratio of a solid backing.................................................... 96

Transmission ratio of a folded-spring backing................................................................102

4.2.1 Folded-spring backing structure analysis...................................................... 103 4.2.2 Anchor location and package gain................................................................ 106 4.2.3 Package gain discussions.............................................................................. 109

4.2.

4.2.3.1 4.2.3.2
The maximum package gain for piezoresistive sensor design.......... 109 The maximum displacement for capacitive sensor design ................111

4.3. 4.4.
Overall sensor module backing design ........................................................................... 116 Summary and conclusions .............................................................................................. 119

  • Chapter 5
  • Sensing Structure Design .................................................................... 120

5.1. 5.2. 5.3. 5.4.
Structure modification ....................................................................................................120 Capacitive output sensitivity calculation ........................................................................122 Fringe capacitance effect on comb drive finger design ..................................................124 Nominal capacitance and parasitic capacitance..............................................................126

4

5.5. 5.6.
Series resistance consideration .......................................................................................127 Structure mechanical property ........................................................................................128

5.6.1 Nonlinearity of the buckled beam amplifier................................................. 128 5.6.2 Resonance modes and frequencies................................................................ 128 5.6.3 Mechanical thermal noise analysis ............................................................... 130

Fabrication process .........................................................................................................131 Fabricated device ............................................................................................................135 Summary and conclusions ..............................................................................................136
5.7. 5.8. 5.9.

  • Chapter 6
  • Testing Fixtures and Device Testing ................................................. 137

  • 6.1.
  • Test fixtures for capacitance output characteristics ........................................................137

6.1.1 Four-point strain testing fixture .................................................................... 137 6.1.2 Bent cantilever beam testing fixture ............................................................. 141 6.1.3 Capacitive strain sensor applying procedure ................................................ 144

Package gain & mechanical gain measurement under microscope ................................148 Capacitive output characteristics ....................................................................................151

6.3.1 Interfaced to an off-the-shelf C/V converter................................................. 151

6.2. 6.3.

6.3.2 Interfaced to a developed low noise C/V converter...................................... 153

Temperature behavior .....................................................................................................155 Step response ..................................................................................................................159 Frequency response testing.............................................................................................160
6.4. 6.5. 6.6.

5
6.6.1 Test fixture .................................................................................................... 161 6.6.2 Resonant cantilever beam resonant frequency calculation........................... 161 6.6.3 Actuation displacement versus generated strain........................................... 163 6.6.4 Test fixture behavior ..................................................................................... 163 6.6.5 Sensor Dynamic behavior............................................................................. 165

  • Conclusions.....................................................................................................................166
  • 6.7.

Bibliography.................................................................................................................. 168

6
LIST OF TABLES
Table 1.1. Major requirements for the MEMS strain gage module used for a

  • rolling-element bearing.
  • 22

  • 29
  • Table 2.1. Properties of strain gage conductors.

Table 2.2. Longitudinal and transverse piezoresistance coefficients for various

  • combination of direction in cubic crystals.
  • 35

35 47 59
Table 2.3. Adiabatic piezoresistance coefficients at room temperature. Table 2.4. Summary of four types of micromachined strain gages. Table 3.1. Properties of hyperbolic cosine and sine function. Table 3.2. Comparison of nominal mechanical gains obtained analytically and from FEA.
62

Table 3.3. Comparison of structure stiffness obtained analytically and from FEA. Table 3.4. Design values of the buckled beam strain sensor. Table 3.5. Parameters of the test fixture.
63 79 81
Table 4.1. Calculated transmission ratio with various bonding adhesive thickness. The structure is with thickness of 480 µm and geometrical dimensions is illustrated in

123 129 141
Table 5.1. Device parameters. Table 5.2. Resonant frequency and shape of each part of the sensing structure. Table 6.1. Parameters of the developed four-point test fixture.

7
Table 6.2. Parameters of the developed bent cantilever beam test fixture. Table 6.3. Beam parameters.
143 162

8

LIST OF FIGURES

  • Figure 1.1. Strain definition.
  • 18

19 25 26 30
Figure 1.2. Strain-stress relationship for common materials. Figure 2.1. Metallic-foil strain gage. Figure 2.2. The resistance change of a cylindrical conductive wire under strain. Figure 2.3. Wheatstone bridge configuration. Figure 2.4. Piezoresistor under a) longitudinal and b) transverse stress. The small red

  • arrow on the resistor points in the direction of the exciting current.
  • 33

Figure 2.5. Variation of the coefficient π11 in n-type layers with temperature and surface

  • concentration.
  • 36

38 39 40 42 44 45 50 52 52
Figure 2.6. A resonant beam subjected to a strain load, ε. Figure 2.7. Resonator excitation and detection diagram. Figure 2.8. Electrostatic excitation and capacitive detection. Figure 2.9. The diagram of a capacitive strain sensor. Figure 2.10. Capacitive strain sensor configuration. Figure 2.11. High sensitivity of a capacitive strain sensor. Figure 3.1. The diagram of a capacitive strain sensor. Figure 3.2. Principle of the buckled beam amplification scheme. Figure 3.3. A buckled beam strain sensor with differential output.

9

  • Figure 3.4. (a) Simplified structure. (b) Forces acting on the beam.
  • 54

57
Figure 3.5. Beam modeling for calculating Δx. Figure 3.6. Comparison of mechanical gain obtained analytically and by FEA with l=300 μm , w=1 μm , and h=30 μm with the maximum applied strain of ± 2000 με .

65
Figure 3.7. Finite Element Analysis result of relationship between nonlinearity, mechanical

  • gain, and the maximum strain.
  • 67

68 69 72
Figure 3.8. Device electrical model. Figure 3.9. The diagram of the front amplifier architecture of the interface electronics. Figure 3.10. Device resonant behavior simulation. Figure 3.11. FEA analysis of the structure under the maximum strain indicates that the

  • location of maximum stress is near the anchor.
  • 74

75 78
Figure 3.12. Major fabrication steps. Figure 3.13. An SEM picture of a buckled beam capacitive strain sensor. Figure 3.14. Description of the test strip and bent-beam test fixture. (a) Top view of the
4-inch device wafer and a test strip. (b) Cross-section view a test strip mounted on

  • a bent-beam test fixture.
  • 81

  • 82
  • Figure 3.15. A picture of the bent-beam test fixture.

Figure 3.16. Microphotographs of sensor comb drive fingers before and after an applied

  • tensile strain.
  • 83

10
Figure 3.17. A computer illustration of the bent-beam test fixture. Figure 3.18. Coupled-field finite-element analysis of the structure under applied strain. Figure 3.19. Measured capacitance change as a function of applied strain. Figure 3.20. Strain sensor electronic interface architecture. Figure 3.21. Sensing electronics die photo [40].
85 86 87 89 90 92 92 93 96 98 100
Figure 3.22. System testing board. Figure 3.23. Output voltage vs. input strain. Figure 3.24. Output noise spectral density. Figure 4.1. Strain transfer mechanism. Figure 4.2. Sensor substrate geometric dimensions. Figure 4.3. FEA results on bonding adhesive thickness effect on transmission ratio. Figure 4.4. Measured capacitance change as function of the applied strain using the

  • four-point test fixture
  • 101

  • 102
  • Figure 4.5.The structure of a folded-spring backing for the capacitive strain sensor.

Figure 4.6. Top view of a folded-spring backing and the geometrical dimensions used.
The structure has a thickness of 480 μm .
103

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    Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA DETC2007-35426 MECHANICAL DOMAIN PARAMETRIC AMPLIFICATION Jeffrey F. Rhoads∗ Nicholas J. Miller Department of Mechanical Engineering Department of Mechanical Engineering Michigan State University Michigan State University East Lansing, Michigan 48824 East Lansing, Michigan 48824 Email: [email protected] Email: [email protected] Steven W. Shaw Brian F. Feeny Department of Mechanical Engineering Department of Mechanical Engineering Michigan State University Michigan State University East Lansing, Michigan 48824 East Lansing, Michigan 48824 Email: [email protected] Email: [email protected] ABSTRACT plications of parametric amplification in macroscale mechanical Though utilized for more than fifty years in a variety of systems. power and communication systems, parametric amplification, the process of amplifying a harmonic signal with a parametric pump, has received very little attention in the mechanical engi- INTRODUCTION neering community. In fact, only within the past fifteen to twenty years has the technique been implemented in micromechanical Parametric amplification, the process of amplifying an ex- systems as a means of amplifying the output of resonant micro- ternal, harmonic signal with a parametric pump, is a well es- transducers. While the vast potential of parametric amplifica- tablished concept in the field of electrical engineering, where tion has been demonstrated, to date, in a number of micro- and it has been employed for more than fifty years in applications nano-mechanical systems (as well as a number electrical sys- ranging from power and communication systems to Josephson tems), few, if any, macroscale mechanical amplifiers have been junctions [1, 2].
  • A History of Engineering & Science in the Bell System

    A History of Engineering & Science in the Bell System

    A History of Engineering and Science in the Bell System The Early Years (1875-1925) A History of Engineering and Science in the deal with developments in specific areas of Bell System: The Early Years (1875-1925) offers technology that led towards its realization. a detailed view of the first fifty years of tele­ The evolution of station apparatus, wire and phone technology. The narrative goes well wireless (radio) transmission, switching, and beyond a simple statement of events to deal various non-voice services is described in de­ with the "how" and "why" of technological tail. Other chapters cover the development innovation. It examines the underlying of materials and components for these new motivations and evaluates the long-term im­ communications systems, the origins of portance of the engineering, scientific, and quality control techniques, and the begin­ organizational achievements that were the nings of scientific research in the Bell System. substance of the first five decades of re­ An early chapter is devoted to the corporate markable progress in telecommunications. structures of the period since these organiza­ While this technical history is primarily an tional arrangements provided a continuing account of Bell System achievements, the synergy of business planning and advancing treatment has not been constrained by cor­ technological skills. porate boundaries and thus recognizes fundamental contributions originating out­ side of the System. A History of Engineering and Science in the During the half-century covered by The Bell System: The Early Years (1875-1925) is Early Years, a whole new field of technology illustrated with over 500 photographs and evolved.