A MICROSCALE DIFFERENTIAL CAPACITIVE DIRECT WALL …
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A MICROSCALE DIFFERENTIAL CAPACITIVE DIRECT WALL SHEAR STRESS SENSOR By VIJAY CHANDRASEKHARAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1
Transcript of A MICROSCALE DIFFERENTIAL CAPACITIVE DIRECT WALL …
A MICROSCALE DIFFERENTIAL CAPACITIVE DIRECT WALL SHEAR STRESSSENSOR
By
VIJAY CHANDRASEKHARAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2009
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c© 2009 Vijay Chandrasekharan
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To my parents and grandparents, who have been my inspiration throughout
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ACKNOWLEDGMENTS
Financial support for this work was provided by NASA Langley Research Center and
the University of Florida. I thank my advisor Prof. Mark Sheplak for his role as a mentor,
both technically and otherwise. I would specially like to thank him for providing the right
kind of guidance and the intellectual freedom to realize my research interests. I am also
grateful to my committee members David Arnold, Lou Cattafesta, and Nam Ho Kim for
their technical inputs, helping me succeed in this project.
I am especially grateful to many of my former and present colleagues at the Inter-
disciplinary Microsystems Group (IMG). David Martin and Karthik Kadirvel helped me
understand interface circuits for capacitive sensors through insightful discussions, even when
they were pressed for time. I probably do not have better words to thank Jeremy Sells and
Jessica Meloy for their help with this project at the design implementation and testing stage.
The numerous hours of discussions and the efforts of Jeremy and Jessica really brought out
the best in us as a team and not just my efforts. I would like to thank Benjamin Griffin and
Brian Homeijer for all those hours of brainstorming and technical discussions. It was the
best time in graduate school in terms of the learning process that we started together back
in 2003. My special thanks to Matt Williams for his willingness to help numerous times,
may it be on mechanics or proof reading my dissertation. I am grateful to John Griffin for
his help with the experimental setup. I thank Brandon Bertolucci for his technical, as well
as his photographic assistance. I thank Sara Homeijer, Sheetal Shetye, Naigang Wang, and
Janhavi Agashe for their help with SEM images. I am also grateful to all of the students at
IMG. Honestly, I learnt a great deal from discussions with fellow graduate students at IMG
than from discussions with faculty. Learning process was never as easy and effective before
and perhaps may not be the same after IMG. I would like to thank the faculty at IMG for
facilitating such a research environment in the group.
I am thankful to Ken Reed at TMR engineering for his excellent and timely machining
work for sensor packages. Other technical assistance was provided at the University of
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Florida by Prof. Ho-Bun Chan and his student Konstantinos Ninios with the supercritical
release and wire bonding. I would also like to thank Al Ogden for his willingness to help in
the cleanroom.
Last but foremost of all, I thank my parents, G.Chandrasekharan and Ranjani Chan-
drasekharan, and my sister, Sudha Chandrasekharan, for their support and encouragement
at every stage of graduate school. My parents are my role models and have taught me by
example, good work ethic, diligence, and practical thinking, which is directly reflected in the
successful outcome of this project.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1 Wall Shear Stress and Boundary Layers . . . . . . . . . . . . . . . . . . . . 181.1.1 Laminar Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . 211.1.2 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . 221.1.3 Small Scales in Turbulence and Sensor Requirements . . . . . . . . 26
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1 Shear Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . 352.1.1 Indirect Measurement Techniques . . . . . . . . . . . . . . . . . . . 352.1.2 Direct Measurement Techniques . . . . . . . . . . . . . . . . . . . . 37
2.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 DEVICE MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 Quasi-Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.1 Small Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . 543.1.2 Large Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . 573.1.3 Electrostatic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2.1 Lumped Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.2.2 Electromechanical Transduction . . . . . . . . . . . . . . . . . . . . 843.2.3 Equivalent Sensor Circuit . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Higher Order Effects and Noise . . . . . . . . . . . . . . . . . . . . . . . . 953.3.1 Fringing Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.3.2 Parasitic Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . 973.3.3 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 DESIGN OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Sequential Quadratic Programming-SQP . . . . . . . . . . . . . . . . . . . 1014.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1 Sensor Performance Requirements . . . . . . . . . . . . . . . . . . . 103
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4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3.1 Objective and Design Variables . . . . . . . . . . . . . . . . . . . . 1044.3.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.3.3 Design Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 DEVICE FABRICATION AND PACKAGING . . . . . . . . . . . . . . . . . . . 121
5.1 Fabrication Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.1.1 Floating Element Trench . . . . . . . . . . . . . . . . . . . . . . . . 1235.1.2 Seedless Electroplating of Nickel . . . . . . . . . . . . . . . . . . . . 1245.1.3 Backside Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.4 Die Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.5 The MEMS Shear Stress Sensor . . . . . . . . . . . . . . . . . . . . 128
5.2 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 EXPERIMENTAL CHARACTERIZATION . . . . . . . . . . . . . . . . . . . . 131
6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.1.1 Impedance Measurements . . . . . . . . . . . . . . . . . . . . . . . . 1316.1.2 Mean Shear Stress Measurement . . . . . . . . . . . . . . . . . . . . 1336.1.3 Dynamic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 1376.1.4 Noise Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.2.1 Impedance Measurements . . . . . . . . . . . . . . . . . . . . . . . . 1426.2.2 Mean Shear Stress Characteristics . . . . . . . . . . . . . . . . . . . 1446.2.3 Dynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 1486.2.4 Noise Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.2.5 Experimental Uncertainty Estimation . . . . . . . . . . . . . . . . . 159
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . 162
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.2 Non-Idealities in Sensor Design and Characterization . . . . . . . . . . . . 164
7.2.1 Design Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.2.2 Characterization Aspects . . . . . . . . . . . . . . . . . . . . . . . . 168
7.3 Recommendations for Future Sensor Designs . . . . . . . . . . . . . . . . . 175
APPENDIX
A MECHANICAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.1 Beam Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178A.1.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 178A.1.2 Small Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . 181
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A.1.3 Large Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . 183A.2 Lumped Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.2.1 Lumped Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193A.2.2 Lumped Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . 195
B TWO PORT ELEMENT MODELING . . . . . . . . . . . . . . . . . . . . . . . 198
B.1 Two Port Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198B.2 Linear Conservative Transducers . . . . . . . . . . . . . . . . . . . . . . . . 199
C SHEAR STRESS IN PWT WITH REFLECTIONS . . . . . . . . . . . . . . . . 205
D PROCESS TRAVELER AND PACKAGING DETAILS . . . . . . . . . . . . . . 209
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
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LIST OF TABLES
Table page
1-1 Laminar boundary layer parameters [8]. . . . . . . . . . . . . . . . . . . . . . . 22
3-1 Sensor geometry for effective tether length calculation. . . . . . . . . . . . . . . 66
4-1 Lower and upper bounds for design variables and flow specifications. . . . . . . 110
4-2 Constant parameters for optimization. . . . . . . . . . . . . . . . . . . . . . . . 110
4-3 Capacitive shear stress optimization results for Cp + Ci = 2.2 + 0.3 pF . . . . . . 112
4-4 Tolerance chart for variables and constants. . . . . . . . . . . . . . . . . . . . . 119
4-5 Design sensitivity based on Monte Carlo (MC) simulation. . . . . . . . . . . . . 119
6-1 Measurement settings for static calibration in the flow cell. . . . . . . . . . . . . 145
6-2 Sensitivity estimates using Monte Carlo technique on the mean shear stress mea-surement data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6-3 Measurement settings of B&K PULSE Multi-Analyzer System (Type 3109) fordynamic calibration in the PWT. . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6-4 Spectrum analyzer settings for noise measurement in a double Faraday cage. . . 156
6-5 Integrated voltage noise floor of the sensor at different frequency ranges. . . . . 158
6-6 Comparison of predicted and measured sensor performance parameters. . . . . . 161
7-1 Comparison of measured sensor performance with previous work. . . . . . . . . 163
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LIST OF FIGURES
Figure page
1-1 Velocity profile of viscous fluid flow over a flat plate. . . . . . . . . . . . . . . . 19
1-2 Schematic of a 2D boundary layer for flow over a flat plate showing both thelaminar and turbulent regions (adapted from [8]). . . . . . . . . . . . . . . . . 22
1-3 Near wall, mean velocity in a turbulent boundary layer or law of the wall (adaptedfrom [11]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1-4 Schematic of energy distribution in turbulence as a function of wavenumber atsufficiently high Re (adapted from [13]). . . . . . . . . . . . . . . . . . . . . . . 28
1-5 Reynolds number for different flight regimes at sea level (adapted from [15]). . 29
1-6 Variation of Kolmogorov length and time scales with flow Reynolds number (x =1m,U∞ = 50m/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1-7 Variation of shear stress and corresponding shear force on sensor area (Asensor =η2) with flow Reynolds number (x = 1m,U∞ = 50m/s). . . . . . . . . . . . . . . 31
2-1 Simplified 2-d schematic showing the plan view (top) and cross-section (bottom)of a typical floating element shear stress sensor structure (adapted from reviewby Naughton and Sheplak [6]). . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2-2 Schematic of cross section of polyimide capacitive shear stress sensor with differ-ential capacitive readout scheme using integrated depletion mode P-MOS tran-sistors (adapted from Schmidt et al. [37]). . . . . . . . . . . . . . . . . . . . . . 40
2-3 Schematic of a) folded beam with lateral capacitive change and b) force feedbackcapacitive structure (adapted from Pan et al. [38]). . . . . . . . . . . . . . . . . 41
2-4 Top view of cantilever based floating element capacitive shear sensor (adaptedfrom Zhe et al. [44]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2-5 Cross sectional schematic of a)two diode and b)’split’ diode (3 diodes) opticalshutter based micromachined shear stress sensor (adapted from Padmanabhan’swork [21]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2-6 Conceptual schematic showing bottom and side view of fabry-perot floating ele-ment shear stress sensor (adapted from [52]). . . . . . . . . . . . . . . . . . . . . 46
2-7 Schematic of the optical Moire shear stress sensor. The shift of the amplifiedMoire pattern is imaged using a microscope attached to a CCD camera (withpermission from author [54]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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2-8 3D schematic of a piezoresistive sensor with top side tether implants oriented inthe direction of flow for high shear stress measurements (adapted from Ng et al.and Goldeberg et al. [56, 57]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2-9 3D schematic of a side implanted piezoresistive shear stress sensor. The implantsare such that two tethers are in tension while two are in compression for highersensitivity (with permission from author [60]). . . . . . . . . . . . . . . . . . . 49
3-1 Schematic of geometry of the differential, capacitive shear stress sensor. . . . . 53
3-2 Simplified mechanical model of the floating element structure. . . . . . . . . . . 55
3-3 Schematic of parallel plate capacitor analogous to overlapping comb fingers. . . 59
3-4 Simplified schematic of individual a)tether, b)comb finger, and c)floating elementcapacitances, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3-5 Schematic of non-uniform tether capacitors with primary and secondary gaps. . 64
3-6 Differential capacitance sensing model for sensing capacitors with the asymmetricgaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3-7 Simplified classification of capacitive interface electronic circuits. . . . . . . . . 69
3-8 Simplified sensor circuitry with a charge amplifier along with noise sources. . . 72
3-9 Simplified sensor circuitry with a voltage amplifier along with noise sources. . . 74
3-10 Simplified differential capacitance sense circuitry using synchronous modulationand demodulation technique with a voltage amplifier. . . . . . . . . . . . . . . 75
3-11 Schematic indicating spectra of the sensor output at each stage of the sensorcircuitry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3-12 Schematic of capacitive transduction scheme using a single sense capacitance. . . 86
3-13 Schematic of capacitive transduction scheme using differential capacitive sensingtechnique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3-14 Schematic of equivalent circuit for the differential capacitive shear stress sensor. 89
3-15 Resonant modes of floating element structure. . . . . . . . . . . . . . . . . . . . 91
3-16 Asymmetric comb finger structure to estimate effect of fringing electric fields. . 96
3-17 Simplified schematic of sensor circuitry with noise sources. . . . . . . . . . . . . 98
4-1 Schematic of operating space of the sensor. . . . . . . . . . . . . . . . . . . . . . 103
4-2 Sensitivity of optimized MDS to design variables for Design 1. . . . . . . . . . . 115
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4-3 Sensitivity of optimized MDS and constraints due to change in optimized variablevalues of Design 1 (cont· · · ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4-4 Sensitivity of optimized MDS, Sτ,overall, and fmin to tolerances in variables inconstants of Design 1 using Monte Carlo simulation. . . . . . . . . . . . . . . . 120
5-1 Schematic showing the plan view of a single die of the proposed shear stress sensorand its section view indicating various layers. . . . . . . . . . . . . . . . . . . . 121
5-2 Step by step fabrication process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5-3 SEM images of comb finger structure etched using DRIE, giving 3D perpective. 124
5-4 SEM image of a cleaved wafer sample depiciting a) vertical sidewalls and b)uniformly plated nickel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5-5 Microscopic image of a released floating element sensor structure from Design 3(Table 4.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5-6 A 2 mm× 2 mm sense element on a 5 mm× 5 mm sensor die. . . . . . . . . . . 128
5-7 Schematic of sensor package for shear stress characterization. . . . . . . . . . . . 129
5-8 Photographs of sensor packaged on a 30 mm× 30 mm PCB. . . . . . . . . . . . 130
6-1 Schematic showing the die-level impedance measurement setup for the sensor. . 132
6-2 Schematic showing the mean shear stress/static calibration setup using Poiseuilleflow in a 2-D channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6-3 Schematic with optical image of sensor die (5 mm × 5 mm) indicating float-ing element, contact pads, and interface circuit (voltage buffer) for mean shearcharacterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6-4 Schematic of the biasing circuit scheme to control phase and amplitude of biasingsignals to null out output from a potential mismatch in sensor capacitances. . . 137
6-5 A schematic of the dynamic calibration setup for measuring shear sensitivity usingrigid termination with the sensor located at pressure node and velocity maximum.139
6-6 A schematic of the dynamic calibration setup for measuring pressure sensitivityusing normal incidence acoustic waves. . . . . . . . . . . . . . . . . . . . . . . . 141
6-7 A schematic of the dynamic calibration setup for shear stress measurement withplane progressive acoustic waves. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6-8 A schematic of the noise measurement setup for the packaged shear stress sensorusing a double Faraday cage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6-9 Capacitance from impedance measurements on the pre-packaged sensor die. . . 144
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6-10 Linear sensor output voltage as a function of mean shear stress at bias voltageamplitude of 2.5 V at 10 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6-11 Distribution of the measured shear stress sensitivity due to uncertainty in themeasured shear stress and sensor output voltage. . . . . . . . . . . . . . . . . . 147
6-12 Sensor output voltage at 1.128 kHz at a bias voltage of 10 V . Sensor is placedat a quarter wavelength (velocity maxima) from the rigid termination. . . . . . 150
6-13 Voltage output as a function of pressure at 4.2 kHz at different bias voltages. . 151
6-14 Linear sensor output voltage as a function of shear stress at 4.2kHz at 3 differentbias voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6-15 Sensor output voltage normalized by bias voltage as a function of shear stress at4.2 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6-16 Schematic showing a set of overlapping comb fingers deflection due to both shear(δs) and due to pressure (δp). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6-17 Frequency response of sensor at Vb = 10 V using τin = 0.5 Pa as the referencesignal up to the testing limit of 6.7 kHz. . . . . . . . . . . . . . . . . . . . . . 155
6-18 Measured output referred noise floor of the packaged sensor in Vrms/√
Hz atdifferent bias voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6-19 Zoomed in plot of the output referred noise floor of the packaged sensor near1 kHz at different bias voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7-1 Schematic with the top view and section of a sensor bond pad showing the asso-ciated electrical impedances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7-2 Schematic with the top view and section of a sensor bond pad showing the asso-ciated electrical impedances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7-3 Effective sense capacitance variation with frequency as a function of substrateresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7-4 Capacitance measurement drift indicated via the first and last measurement priorto ensemble averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7-5 Drift in mean capacitance with subsequent dc bias sweeps. . . . . . . . . . . . . 170
7-6 Magnitude of sensor to microphone transfer function as a function of acousticpressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7-7 Phase of sensor to microphone transfer function as a function of acoustic pressure.171
7-8 Transfer function between the sensor and the reference microphone for normalacoustic incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
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7-9 Comparison of combined (shear and pressure) and pressure transfer functionsmeasured between sensor and reference microphone. . . . . . . . . . . . . . . . 173
7-10 Comparison mean shear stress measurements at Vb = 2.5 V at different channelwidths, h, in the flow cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7-11 Mechanical LEM for pressure sensitivity of the sensor. . . . . . . . . . . . . . . 176
A-1 A deflected beam with arbitrary curvature. . . . . . . . . . . . . . . . . . . . . . 178
A-2 Simplified mechanical model of the floating element structure. . . . . . . . . . . 180
A-3 Schematic of one half of a clamped-clamped beam under large deflection. . . . . 189
B-1 General representation of an ideal two-port element. . . . . . . . . . . . . . . . 198
B-2 Circuit representation of the transformer and gyrator. . . . . . . . . . . . . . . 199
B-3 Impedance to impedance analogy representation of a two port element. . . . . 199
B-4 Equivalent circuit representation of a transducer using impedance analogy. . . . 202
B-5 Schematic of a parallel plate capacitive transducer. . . . . . . . . . . . . . . . . 202
B-6 Circuit representation of a capacitive transducer for constant charge biasing. . 204
C-1 Setup for shear stress in PWT for a general impedance termination. . . . . . . 205
D-1 PCB Layout and dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
D-2 Drawing of Lucite plug. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
14
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
A MICROSCALE DIFFERENTIAL CAPACITIVE DIRECT WALL SHEAR STRESSSENSOR
By
Vijay Chandrasekharan
May 2009
Chair: Mark SheplakMajor: Mechanical Engineering
A shear stress sensor measures the frictional force due to fluid flow adjacent to a surface.
Shear stress sensors have a wide variety of applications which include the fundamental study
of turbulence, aerodynamic drag measurement devices, feedback sensors for flow control
applications, and non-intrusive industrial flow measurement devices.
Several research efforts in the past have been directed towards developing a shear shear
stress sensor for time resolved wall shear stress measurements. Conventional techniques
rely on velocity measurements and heat/mass transfer techniques to infer the shear stress.
While these measurement techniques are suitable to determine mean shear stress in well-
known two dimensional flows, they are unsuitable for complex three dimensional flows as
they significantly alter the flow or rely on semi-empirical theory for a measurement. Shear
stress sensors require flush mounted sensors on the surface to avoid flow disturbance. It
is also necessary that these sensors measure the shear force directly and be small enough
to have sufficient resolution both in space and time. These sensors when assembled in the
form of an array on a surface, may be used to understand the basic nature of turbulent flow
and flow separation. Preventing flow separation via flow control techniques helps to reduce
separation drag and improve energy efficiency of moving vehicles.
A microelectromechanical systems (MEMS) shear stress sensor may serve as an instru-
ment grade sensor for many applications. A lack of such a sensor on a laboratory or a
commercial level, despite several research undertakings indicates the challenging nature of
15
this problem. Challenges and requirements for shear stress sensors are explained. The previ-
ous work on MEMS scale shear stress sensors are reviewed in detail with emphasis on scope
of improvement.
A differential capacitive shear stress sensor design is presented for application in bound-
ary layer flows. The sensor is the first in its kind to use an asymmetric comb finger structure.
This enables a simple two mask fabrication process using bulk micromachining, in which a
single etching process defines the sensor structure, comb fingers, and electrical contact pads.
A thorough analysis of the mechanical, electrical, electromechanical transduction, and inter-
face circuit models is presented. A system level optimization is performed, which includes
the sensor and the associated interface electronics.
The characterized sensor has the highest dynamic range (> 102 dB) and the lowest noise
floor/MDS (14.9 µPa) reported to date for a micromachined direct shear stress sensor. The
sensor has a bandwidth of 6.2 kHz, which is defined by its resonance. It is the first sensor to
demonstrate a pressure rejection of 64 dB for comparable forcing in shear and pressure. In
comparison to previous work on direct MEMS shear stress sensors, this sensor outperforms
its best predecessors by roughly two orders of magnitude in minimum detectable shear stress
and at least an order of magnitude in dynamic range.
16
CHAPTER 1INTRODUCTION
Viscous or skin friction drag due to relative velocity between a body and the surrounding
fluid medium is a major source of energy dissipation in aircraft and automobiles. Drag has
an adverse effect on the fuel efficiency of the vehicle. Annual consumption of hydrocarbon
based fuels runs into billions of gallons [1]. Furthermore, depleting petroleum reserves in the
world emphasize the importance of drag reduction to improve fuel efficiency.
In 2000, the annual inflation-adjusted aviation fuel cost in the United States was about
$10 billion [2]. The cost per barrel of crude oil has since risen from an average value of
$27.53 in 2000 to $147 in July 2008 [1,3]. While it has subsequently dropped to $40.26, the
long term prospects indicate market volatility and project prices as high as $186 per barrel
in 2030 [1, 4]. Increasing world automobile and air traffic together with ever increasing fuel
prices thus necessitate effective techniques to reduce drag. During landing, take off and
cruise conditions, skin friction drag is approximately 50% of the total drag [2, 5]. A 20%
reduction would translate into approximately $1 billion in annual fuel savings worldwide [2].
In underwater vehicles almost 90% of the drag is associated with skin friction. A 20%
reduction in drag in such vehicles can help increase their speeds by as much as 6.8% [2].
Viscous drag also determines the characteristics of different wall bounded shear flows
and is also important for characterizing the state of turbulent boundary layers. This helps
in understanding different flows from a fundamental fluids perspective and aids in their
control [6]. Quantifying drag reduction requires accurate measurement of wall shear stress,
which is drag force per unit area.
An ideal shear stress sensor should be able to accurately capture both the static and
dynamic characteristics of the flow. To capture the turbulent flow physics, a shear stress
sensor requires sufficiently high bandwidth to provide acceptable temporal resolution and
small dimensions for good spatial resolution. A microscale sensor possesses the reduced
physical size and low inertance required to satisfy both spatial resolution and bandwidth
17
requirements [6]. But, none exist that provide a reliable shear stress measurement due to
sensitivity drift and insufficient dynamic range, bandwidth, and/or shear stress resolution.
The goal of this research is the design, fabrication, and characterization of a microscale
sensor for direct measurement of time resolved wall shear stress on a surface adjacent to a
flow.
In the sections that follow, the mechanism of wall shear stress generation and its rele-
vance is briefly reviewed. Design requirements for the sensor are determined from relevant
length and time scales of the flow physics. The objectives and expected contributions of
this research are outlined, and at the conclusion of the chapter, the overall dissertation
organization is presented.
1.1 Wall Shear Stress and Boundary Layers
A fluid in motion transmits both tangential and normal stress between adjacent fluid
particles. Tangential stress is a result of viscosity, a fluid property that quantifies the inter-
molecular forces inherent to a moving fluid. In the vicinity of a solid surface, there exists a
velocity gradient between the fluid and the solid. At the fluid-solid interface in most situa-
tions, the viscous forces cause the fluid molecules to adhere to the surface, commonly known
as the no-slip boundary condition. This gives rise to a velocity gradient normal to the wall,
resulting in a shear stress [7].
For a Newtonian fluid in a 2D boundary layer, the shear stress is linearly related to the
velocity gradient via the viscosity of the fluid. Mathematically, the fundamental relation is
τwall = µdu
dy
∣∣∣∣y=0
, (1–1)
where τwall is the shear stress, µ is the dynamic viscosity, u is the streamwise velocity of the
fluid, and y is the coordinate normal to the wall. Figure 1-1 depicts the velocity gradient
due to flow over a flat plate. The velocity is zero at the stationary wall due to the no-slip
boundary condition, and reaches the free stream velocity, U∞, at a finite distance from the
wall. This viscous dominated region close to the wall is known as a boundary layer. Usually,
18
x
y
Direction of Flow
u = 0
u = 0.99 U∞
0
w
y
du
dyτ µ
=
=
δ
Figure 1-1. Velocity profile of viscous fluid flow over a flat plate.
the boundary layer thickness, δ, is defined as the distance from the surface where the local
fluid velocity is 99% of the free stream velocity i.e., u = 0.99U∞ [8].
Skin friction measurement has been of interest since the early 1870s [9]. The reduction
of skin friction is one of the important reasons behind wall shear stress measurement. In
addition to skin friction, drag is also caused due to flow separation. Flow separation occurs
when the adverse pressure gradient overcomes the momentum of the flow. The region of
adverse pressure gradient is also known as the pressure recovery region. When the flow
separates, it effectively encounters a deformed body after the point of separation. This
results in a net integrated pressure force in the direction of flow, causing pressure drag.
From a flow control perspective, separation drag is detrimental for an aircraft as it reduces
lift and its fuel efficiency. Thus, wall shear stress measurement is extremely important for
flow control where flow separation is delayed using a variety of control techniques [8]. From
the discussion so far, it is evident that the measurement of wall shear stress has practical
implications and also gives insight into the flow physics. As stated by Haritonidis, “The mean
stress is indicative of the overall state of the flow over a given surface while the fluctuating
stress is a footprint of the individual processes that transfer momentum to the wall” [10].
In general, the viscous diffusion of flow momentum from the wall due to shearing effects
results in a boundary layer [7]. The non-dimensional Reynolds number, Re, which is the
ratio of inertial to viscous forces, is used to describe the extent of viscous diffusion, resulting
19
in a boundary layer. Mathematically, the Reynolds number is expressed as,
Rel =ρUl
µ, (1–2)
where ρ is the density of the fluid, U is a typical velocity scale of the flow, and l is a typical
length scale of interest in the flow. The Reynolds number may also be interpreted as the
ratio of viscous diffusion to convective time scales in the flow [8]. Thus, high values of Re
result in thin boundary layers while low values of Re result in thicker boundary layers in
both laminar and turbulent flows. In an incompressible flow over a thin body, the boundary
layer flow is usually laminar for 1000 < Re < 106 and turbulent at higher Re [8]. Before
delving into different kinds of boundary layers, it is informative to introduce some important
parameters. First is the displacement thickness, δ∗, which is a measure of the mass deficit
in the boundary layer due to viscous retardation of the fluid. For an incompressible flow, it
is expressed as [8],
δ∗ =
∞∫
0
(1− u
U∞
)dy. (1–3)
Second is the momentum thickness, θ, which represents the flow momentum lost due to shear-
ing effects transmitted to the bounding surface. The momentum thickness is mathematically
expressed as [8],
θ =
∞∫
0
u
U∞
(1− u
U∞
)dy. (1–4)
The wall shear stress and momentum thickness are related to the friction and drag coeffi-
cients, Cf and CD, for flow over flat surfaces with zero pressure gradient through [8],
Cf =τw
12ρU∞
2 = 2dθ
dx(1–5)
and CD =D
12ρU∞
2L=
2θ(L)
L, (1–6)
20
where D is the drag force per unit width for a flat plate given by
D = θ(ρU∞
2)
=
L∫
0
τw(x)dx. (1–7)
Equations 1–5 and 1–6 are general in nature and are valid for both laminar and turbulent
boundary layers for incompressible flow over flat plates [8].
Section 1.1.1 describes a laminar boundary layer, which is formed at relatively low
Reynolds numbers. General characteristics of a steady, two-dimensional boundary layer and
their relevance to wall shear stress measurements are provided. In Section 1.1.2, turbulent
boundary layers formed at high Reynolds numbers are described. Section 1.1.3 provides
a discussion on relevant length scales and time scales essential for the understanding of
turbulent flow physics. These length and time scales are translated later into sensor design
requirements.
1.1.1 Laminar Boundary Layer
In general, laminar boundary layers are less complex than turbulent boundary layers.
A laminar boundary layer is characterized by smooth, orderly flow and is generally observed
at lower Reynolds numbers. The vorticity generated at the wall due to tangential pressure
gradients is transported away through viscous diffusion. The extent of discernable viscous
diffusion away from the bounding surface determines the boundary layer thickness, δ. Usu-
ally, δ is defined as the distance from the surface where the local fluid velocity is 99% of the
free stream velocity [8].
Blasius’ solution is used for laminar boundary layers with zero pressure gradient (ZPG).
This method assumes large values of Re but not enough for transition into turbulence.
Table 1-1 provides the relevant parameter values for a laminar boundary layer using the
Blasius solution. These relations may be used to theoretically validate the measurement
accuracy of the shear stress sensor in a ZPG laminar boundary layer.
21
Table 1-1. Laminar boundary layer parameters [8].
Parameter Blasius Solution
θx
√Rex 0.664
δ∗x
√Rex 1.721
δ99%x
√Rex 5.0
Cf
√Rex 0.664
CD
√Rex 1.328
Stable
laminar
flow
x0
U
Recrit
T/S
waves
Spanwise
vorticity
Three
dimensional
vortex
breakdown
Turbulent
spots
Fully
turbulent
flow
Edge
contamination
LaminarTransition length
Turbulent
Retr
U
Top View
Side View
Figure 1-2. Schematic of a 2D boundary layer for flow over a flat plate showing both thelaminar and turbulent regions (adapted from [8]).
1.1.2 Turbulent Boundary Layer
Turbulent boundary layers are formed approximately at Rex > 106 for flow over a flat,
smooth, plate. Turbulent flows are generally characterized by random fluctuations in the
flow, diffusivity, large Re, three dimensional vorticity fluctuations, and dissipation [11]. A
wall-bounded turbulent flow results in a turbulent boundary layer. Figure 1-2 is a schematic
22
of flow over a flat plate that shows the laminar, transition, and turbulent regions of the
boundary layer.
Unlike laminar boundary layers, turbulent momentum transport occurs via both viscous
diffusion and macroscopic mixing where the latter dominates. A non-dimensional length,
y+, known as the viscous wall unit, is used to express the extent of the influence of viscous
diffusion. The characteristic viscous length scale used for non-dimensionalization is
l+ =ν
u∗, (1–8)
where u∗ is the friction velocity and ν is the kinematic viscosity of the fluid. A wall unit is
therefore defined as,
y+ = y/l+ = u∗y/ν. (1–9)
The friction velocity is defined as
u∗ =
√|τw|ρ
. (1–10)
It noteworthy that the fundamental turbulent boundary layer non-dimensional parameters
depend on u∗, which is estimated from τw, further emphasizing the importance of wall shear
stress measurements. As opposed to a laminar boundary layer, the shear stress in turbulent
boundary layers has an additional component referred to as the Reynolds stress [11],
τtr = −ρu′v′, (1–11)
where u′ and v′ are the fluctuating components of the velocities. The bar in Equation 1–11
represents the time average at fixed locations, assuming a statistically steady but inhomo-
geneous flow field [11]. The velocities decomposed in terms of a time averaged mean and
fluctuating components, are represented as
u = u + u′ (1–12)
and v = v + v′. (1–13)
23
The stress represented by Equation 1–11 is a term in the Reynolds-averaged momentum equa-
tion derived from the Navier Stokes equation by decomposing velocity, pressure and stresses
into their respective mean and fluctuating components. Physically, this term serves as the
mechanism for momentum transport due to macroscopic mixing in a turbulent boundary
layer, similar to viscous diffusion in a laminar boundary layer. Note that this is true for wall
units, y+ > 5, whereas for y+ ≤ 5, viscous diffusion continues to dominate.
In the past, von Karman’s integral approach has been used to obtain flow parameters
for a turbulent boundary layer just as it has been for laminar boundary layers. In this
dissertation, it will be used to provide rough scaling information. An assumed velocity
profile suggested by Prandtl based on pipe flow theory is [8]
u
U∞≈
(y
δ
)1/7
. (1–14)
Using Equation 1–5 and Equations 1–6 with the assumed velocity of Equation 1–14 results
in [8]
Reδ ≈ 0.16Re6/7x ,
δ
x≈ 0.16
Re1/7x
, and Cf ≈ 0.027
Re1/7x
. (1–15)
The expressions in Equation 1–15 agree well with published flat plate data [8] and serve as
rough estimates for validating mean shear stress measurements in turbulent flows.
A typical near wall non-dimensional mean velocity profile of a turbulent boundary layer
is shown in Figure 1-3. The near wall turbulent boundary layer consists of three different
regions: the viscous sublayer, the buffer layer, and the inertial sublayer. The portion of the
boundary layer for which y+ ≤ 5 is called the viscous sublayer. The mean flow in this layer
scales as [12, 13]
u
u+= f(y+) ∼ y+. (1–16)
Equation 1–16 indicates that the mean velocity scales linearly with y+. Similar to a laminar
boundary layer, the momentum transport in this region is dominated by viscous diffusion.
However, unlike the laminar boundary layer, the viscous sublayer still possesses fluctuations
24
10−1
100
101
102
103
104
0
5
10
15
20
25
30
y+
u+
=u/u
∗
Viscous Sublayer Inertial Sublayer
u+ = y+
u+ = 1/κ log(y+) + B
BufferLayer
Figure 1-3. Near wall, mean velocity in a turbulent boundary layer or law of the wall(adapted from [11]).
due to turbulence in the flow [11]. As previously mentioned, the wall shear stress mea-
sured in a turbulent boundary layer thus possesses both mean and fluctuating components.
The viscous sublayer is also referred to as the laminar sublayer due to the similarity of its
properties and those of laminar boundary layers [12]. At y+ > 50, it is found that [12]
u
u+= f(y+) ∼ 1
κlog(y+) + B, (1–17)
where κ is known as the von Karman constant and B is a universal constant. Both are
determined empirically. Typically, κ = 0.4 and B ≈ 4 − 5. In this region, known as the
inertial sublayer, the shear stress is dominated by the turbulent exchange with negligible
contribution from viscous effects. Equation 1–17 shows that in the inertial sublayer, the
mean velocity has a logarithmic variation with y+.
The transition region between the viscous and the inertial sublayers is known as the
buffer layer. The buffer layer usually exists between 5 ≤ y+ ≤ 30. In this region the shear
effects are transferred by a combination of both the viscous effects and turbulent exchanges.
25
The velocity profile in this region is thus a smooth combination of the linear and logarithmic
variations with y+. Spalding and Chui [14] provide a composite formula valid from y+ = 0
to y+ > 100 expressed as
y+ = u+ + eκB
[eκu+ − 1− κu+ − (κu+)2
2− (κu+)3
6
]. (1–18)
Since this dissertation focusses on developing a wall shear stress sensor, the present discussion
on turbulence is limited to the near wall region and excludes wakes. Furthermore, the analysis
is based on 2-D turbulent flow assumption.
1.1.3 Small Scales in Turbulence and Sensor Requirements
As stated previously, viscous forces tend to be negligibly small relative to inertial forces
at large values of Re. Negligible viscous effects would imply that there is no conceivable
energy dissipation in the flow, which is contrary to physical intuition. The non-linear terms in
the Navier Stokes equations drive an energy cascade. This maintains a finite dissipation level
in the flow by generating small scale motions that can be influenced by viscous effects [11].
A sensor is therefore necessary to measure these small scale motions. The wall shear stress,
if measured accurately, can aid in extracting valuable information about these small scale
fluctuations.
A shear stress sensor has to be non-intrusive in nature to ensure that the flow remains
unaltered. The sensor is therefore ideally placed flush with the wall. If the sensor roughness
is within the viscous sublayer, it does not disturb the flow and is considered hydraulically
smooth. The sensor measures shear stress contributions from both the small scale fluctua-
tions and the mean flow. For a reliable measurement while avoiding spatial averaging effects,
the sensor dimensions should be smaller than the length scale of these small scale fluctua-
tions of interest. Similarly, the sensor should have sufficient bandwidth to resolve the small
time scales (high frequencies) associated with small scale fluctuations. Kolmogorov derived
various scales of motion using an elegant energy based theory called Kolmogorov’s universal
equilibrium theory [11]. The respective “Kolmogorov scales” for length, time, and velocity
26
are
η ≡ (ν3/ε
)1/4,T ≡ (ν/ε)1/2 , and υ ≡ (νε)1/4 , (1–19)
where ε is the dissipation rate per unit mass. The small scale motions are statistically
independent of large scale motions and the mean flow. However, they rely on the energy
supply from the large scale motions and the viscous dissipation to sustain their motion [11].
According to Kolmogorov’s theory, near equilibrium the rate of energy supplied should be
approximately the same as the rate of energy dissipated. The dissipation rate is therefore
expressed in terms of the large scale motions as
ε ∼ u3/`, (1–20)
where u is the characteristic velocity scale of large eddies (u ≈ 0.01U∞) and ` is the length
scale related to the size of those eddies, also referred to as the integral scale [11]. In a
turbulent boundary layer the size of large eddies are approximately equal to the boundary
layer thickness δ. Therefore the inviscid estimate for ε in Equation 1–19, is used with δ as
the length scale instead of ` to relate the large scale motions to the Kolmogorov scales as
η
δ∼
(uδ
ν
)−3/4
= Re−3/4δ , (1–21)
Tu
δ∼
(uδ
ν
)−1/2
= Re−1/2δ , (1–22)
andυ
u∼
(uδ
ν
)−1/4
= Re−1/4δ . (1–23)
Combining Equations 1–15, 1–21, 1–22, and 1–23, the Kolmogorov scales in terms of the
distance from the leading edge, x, of a flat plate and the corresponding Reynolds number,
Rex, are estimated via the 1/7th power law as follows:
η ∼ 0.632xRe−11/14x ,
T ∼ 40x
U∞Re−4/7
x , (1–24)
and υ ∼ 0.016U∞Re−3/14x .
27
log k
log E
k = O(δ-1) k = O(η-1)
k-5/3
Figure 1-4. Schematic of energy distribution in turbulence as a function of wavenumber atsufficiently high Re (adapted from [13]).
A typical 1-D distribution of turbulent energy, E(k), as a function of the wavenumber,
k, is shown in Figure 1-4. The turbulent wave number k is defined as
k = ω/v ≡ 2π
δ, (1–25)
where ω is the frequency, v is the turbulent eddy velocity, and δ is the corresponding eddy
length scale. In general, turbulent energy in a boundary layer is concentrated in the large
turbulent eddies, which dominate momentum transfer via macroscopic mixing. These eddies
have length scales on the order of the boundary layer thickness δ. Therefore, from Equa-
tion 1–25, the frequencies corresponding to these structures scale as, O(δ−1). This energy
gets redistributed from the large scale structures to the small scale turbulent structures on
the order of the Kolmogorov scales (η) with frequencies that scale as O(η−1). As Rex in-
creases, these scales get smaller, resulting in challenging sensor design even with modern
micromachining technology. Thus, shear stress sensor length scales and bandwidths are
application specific with stringent design requirements with increasing values of Re. The
different flight regimes as a function of Reynolds number are shown in Figure 1-5.
28
Figure 1-5. Reynolds number for different flight regimes at sea level (adapted from [15]).
The challenges in sensor design for resolving Kolmogorov scales may be further explained
via the following study. Consider flow over a flat plate with U∞ = 50m/s and a shear
stress sensor of sensing area, Asensor = η2, placed at x = 1 m from the leading edge. For
this example, any variation in Re is due to change in thermodynamic parameters. From
Equations 1–5 and 1–15,
τw =1
2ρU2
∞Cf =1
2ρU2
∞0.027
Re1/7x
. (1–26)
29
106
107
108
109
10−2
10−1
100
101
102
η(µ
m)
Kolmogorov Scales
Reynolds Number Rex
106
107
108
10910
3
104
105
106
1/T
(Hz)
Figure 1-6. Variation of Kolmogorov length and time scales with flow Reynolds number(x = 1m, U∞ = 50m/s).
The corresponding shear force, Fs, on the sensor is
Fs = τwAsensor = τwη2. (1–27)
The variation of η and T at different flight Reynolds numbers for x = 1 m is shown in
Figure 1-6. Figure 1-7 shows the variation of τw and Fs with Rex. While the shear stress levels
remain at moderate values, the sensing area (η2) and the corresponding Fs drop dramatically.
For example, at Rex ∼ 1 × 107, η ∼ 2 µm and 1/T ∼ 12 kHz. Furthermore, for Asensor =
η2 ∼ 4 µm2, Fs = 16 pN and τw = 4 Pa, which is the maximum measurable shear stress. For
a given range of Rex the variation in η is O(102) higher than the variation in T. Similarly,
the variation of Fs is also much higher than the change in τw, making shear stress sensing
more challenging at higher Rex. Now consider a measurement where the turbulent shear
stress is roughly 40 dB re 1 Pa below the mean shear stress of 4 Pa with a preferred signal
to noise ratio of 20 dB. Thus, the sensor needs to have a minimum detectable signal (MDS)
of 4 mPa or a dynamic range of 60 dB. The shear force corresponding to this MDS is 16 fN .
30
The analysis clearly elucidates the stringent requirements and the challenging nature of shear
stress sensor design to resolve Kolmogorov scales in both space and time at high Rex.
106
107
108
109
2
2.5
3
3.5
4
4.5
5
5.5
6
τ w(P
a)
Reynolds Number Rex
106
107
108
10910
−9
10−8
10−7
10−6
10−5
10−4
10−3
Fs
(µN
)
Figure 1-7. Variation of shear stress and corresponding shear force on sensor area (Asensor =η2) with flow Reynolds number (x = 1m, U∞ = 50m/s).
It is important to note that the Kolmogorov scales represent the smallest fluid structures
in turbulence, but not necessarily the smallest scales of interest from the sensing perspective.
For instance, in flow control applications, shear stress sensors may be used as feedback
sensors for drag reduction. Numerous research articles on flow control for turbulent drag
reduction sight turbulent streaks as a major cause of skin friction drag. Turbulent streaks
are alternating spanwise regions of low and high speed fluid oriented in the streamwise
direction and are quiescent most of the time [16]. At low Re values, the streaks are spaced
100l+ apart [16] and are roughly 40l+ wide [17], circumventing the need for the sensor
to always resolve the Kolmogorov scales. For example, for measuring shear stress from
turbulent streaks that are 40l+ wide, let us assume the sensor may be at most 20l+ in its
maximum sensing dimension. Using the example in the previous paragraph, for τw = 4 Pa,
20l+ ≈ 200 µm, which is two orders of magnitude greater than the length scales required
31
to resolve Kolmogorov scales. Thus, similar to the previous case, for a dynamic range of
60 dB for shear stress, the corresponding maximum and minimum shear forces acting on the
sensor are 160 nN and 160 pN , respectively. These numbers indicate that the stringency
of design in considerably lower for applications such as flow control as against resolution of
Kolmogorov scales.
Micromachined sensors offer the potential to satisfy both the spatial and temporal res-
olution requirements for different applications. No direct/indirect shear stress sensor exists
that satisfies both spatial and temporal resolution requirements while providing accurate
shear stress measurements at the Kolmogorov scales. This will be shown in the review of
previous research presented in Chapter 2. Padmanabhan [18] used a floating element shear
stress sensor while both Lofdahl et al. [19] and Alfredsson et al. [20] use thermal shear
stress sensors. Their research efforts have sensor length scales of 4l+ [21], 5l+ [19], and
10 − 20l+ [20]. As stated earlier, to accurately measure the Kolmogorov fluctuations, the
sensor dimensions must be equal to or smaller than these scales. Sensor resolution at the
Kolmogorov scales essentially allows one to use the sensor for a wide variety of applications.
However, previous efforts indicate that the sensor dimensions are restricted by signal resolu-
tion issues and fabrication limitations. Numerous direct and indirect micromachined sensors
have been previously developed and will be reviewed in Chapter 2 with emphasis on their
strengths and limitations.
Floating element sensors used for direct measurement of wall shear stress may also
be sensitive to vibrations and pressure fluctuations. If these undesirable sensitivities are
not eliminated, the shear stress measurements may be erroneous. The magnitude of the
fluctuating pressure forces is approximately two orders of magnitude higher than for the
shear forces. Direct numerical simulations to study the wall pressure effects show that, based
on frequency, the wall pressure fluctuations are 7 − 20 dB higher than the streamwise wall
shear stress and 15 − 20 dB higher than the spanwise component [22]. The sensor design
should also ensure minimum errors due to misalignment, pressure gradients and gaps [9].
32
The gaps must be less than a few viscous length scales (< 5l+) to ensure that there is no
flow disturbance (hydraulically smooth) [7]. With micromachined floating element sensors
gaps as small as 4 µm are achievable. Detailed discussions on floating element sensors and
implementation of these design requirements are provided in Chapter 2 and 3, respectively.
1.2 Research Objectives
This dissertation provides details on the development of a MEMS based capacitive
floating element sensor for direct, time resolved wall shear stress measurements. The sen-
sor employs a differential capacitance measurement scheme, which helps to eliminate the
undesirable cross axis sensitivity of the sensor. A capacitive sensor is also insensitive to tem-
perature variations. The sensor employs simple fabrication technology and is suitable for
stand alone measurements of wall shear stress. The aim is to design, optimize, and fabricate
a benchmark shear stress sensor to reliably measure turbulent variations within its spatial
and temporal resolution limitations. The goal is to measure up to 10 Pa of shear stress at
frequencies up to 10 kHz and achieve a spatial resolution of ≈ O(100l+) to demonstrate the
proof of concept of the proposed sensor.
1.3 Dissertation Organization
The dissertation is organized into seven chapters. Chapter 1 introduced wall shear
stress and motivation for its measurement. Scaling studies and corresponding sensor re-
quirements were presented. Chapter 2 is a review of previous research efforts to develop
microscale, direct wall shear stress sensors. Existing sensing techniques, including capacitive
transduction using comb fingers, are discussed. Chapter 3 presents the sensor’s mechanical
and electrostatic model, electromechanical transduction, interface circuitry, and the noise
model. Mathematical representations for noise and sensitivity are derived to use in a design
optimization strategy. Chapter 4 contains the formulation of the optimization problem and
an explanation of the optimization scheme to be implemented. Optimization results are
presented together with sensitivity analysis for the design. Chapter 5 provides details of the
fabrication steps and the packaging scheme for the sensor. Chapter 6 explains the sensor
33
characterization procedure and discusses the results. Chapter 7 provides conclusions and
gives recommendations for the next generation of capacitive shear stress sensors.
34
CHAPTER 2BACKGROUND
In this chapter, previous research efforts directed towards developing shear stress sensors
are described. Existing measurement techniques are briefly explained along with key figures
of merit. Transduction schemes commonly used in direct MEMS wall shear stress sensors
are described. Finally, a review of direct MEMS wall shear stress sensors is presented. The
chapter summarizes the various techniques for wall shear stress measurement drawn mainly
from previous review articles [6, 9, 10,23].
Different techniques exist for shear stress measurement. As stated in Chapter 1, shear
stress may be directly determined from the measurement of shear force on a known area.
However, other methodologies also exist and are used in practice to estimate the shear stress
from measured flow parameters such as velocity, joulean heating rate etc. Thus based on
the quantity measured, the techniques are broadly classified as direct (measures shear force)
or indirect (measures other quantitites) measurement techniques. Shear stress sensors may
also be categorized as conventional (macroscale) or microscale sensors. Several research
efforts in the past have been directed towards developing both direct and indirect microscale
sensors. Microscale direct sensors, due to their direct nature and favorable scaling to capture
turbulent flow physics (see Chapter 1), are better suited for quantitative time-resolved shear
stress measurement. The discussion in this chapter mostly concentrates on previous research
on direct micromachined sensors.
2.1 Shear Measurement Techniques
2.1.1 Indirect Measurement Techniques
Indirect techniques rely on a known correlation between the measured parameter and
shear stress to estimate the latter. Heat/mass transfer based devices, surface/flow obstacle
devices and velocity profile measurement techniques are used to indirectly estimate wall shear
stress. Reviews by Winter [9] and Haritonidis [10] describe the benefits and limitations of
these measurement techniques.
35
Heat/mass transfer based devices rely on the exchange of heat and mass flux between
the device and the flow. Heated films/wires are examples of devices based on heat transfer
techniques. The benefits of these techniques include high sensitivity, reasonable dynamic
response, small size, simple structure, and non-intrusive mounting. However, temperature
sensitivity, tedious and non-repeatable calibration due to heat loss to substrate/air limit the
use of these sensors for reliable shear stress measurement. Naughton and Sheplak [6], and
Sheplak et al. [23] discuss the developments and limitations of micromachined thermal shear
stress sensors.
Preston tubes, stanton tubes, razor blades, steps and fences are surface obstacle based
shear stress measurement devices. Since these devices obstruct the flow they are useful
in thick boundary layer flows. The measurement is sensitive to the size and geometry of
these devices. These devices are also limited to mean measurements and lack time resolved
shear stress measurement capability. The shear stress estimation is based on empirical
correlations relating 2-D turbulent boundary layer profiles to the measured property. Lastly,
the calibration is sensitive to the location of the probe in the boundary layer. Therefore,
these techniques are of little use in complex three-dimensional flows.
Velocity profile based techniques use invasive probes like pitot tubes or optical (non-
invasive) techniques like laser doppler velocitmetry (LDV) or particle image velocimetry
(PIV). Optical techniques are limited to thick boundary layers because it is very difficult to
seed the near-wall region of the flow. The velocity profile measurements are then matched
with the logarithmic law of the wall also referred to as the Clauser plot [24]. This method
relies on the validity of the Clauser plot and the ability to identify the logarithmic region of
the near wall boundary layer.
The reported micromachined indirect wall shear stress sensors include thermal sensors
[25–28], micro-fences [29], micro-pillars [30–33], and laser based sensors [34, 35]. Micro-
fences consist of a cantilever structure whose deflection due to the flow is measured using
piezoresistve transduction. Micro-pillars use an array of pillars on the wall within the viscous
36
sublayer. The tip deflections of the pillars are optically measured and correlated to the shear
stress. Laser Doppler based sensors are based on the measurement of the doppler shift of
light scattered from the particles passing through a diverging fringe pattern in the viscous
sublayer of a turbulent boundary layer.
All these techniques rely on the invariance of the near-wall structure of a two dimensional
turbulent boundary layer and are thus not suited for three-dimensional flows.
2.1.2 Direct Measurement Techniques
Direct measurement sensors as stated previously are most suitable since they measure
the shear force directly and do not require assumptions about the flow field [10]. Direct shear
stress measurement techniques include thin-oil film interferometry, liquid crystal coatings and
floating element based sensors. The recent review by Naughton and Sheplak [6] discusses the
operating principle, developments, uncertainties and limitations of these techniques. Both oil
film and liquid crystal coatings need optical access which limit their widespread use. Both
techniques involve complex post processing to analyze the data and are limited to slowly
varying mean measurements [6]. These two techniques are therefore not discussed here.
Most of the direct measurement shear stress sensors employ a sensing element with a
known area attached to a compliant structure. The compliant structure provides a restoring
force when the sensing element is moved by shear force. A floating element sensor is one of
the most widely used direct measurement sensor structure. The sensing element is suspended
over an open cavity by means of tethers that also act as restoring springs. Figure 2-1 shows a
schematic of a floating element sensor. The motion of the floating element is transduced into
a proportional electrical/optical signal to measure the shear stress. These devices, however,
have issues of their own as first explained by Winter [9]. The following problems are generally
associated with conventional floating element sensors [9, 10]:
1. Minimum detectable shear stress (MDSS) increases when the sensor size is decreased
to improve spatial resolution. A small sensor size results in smaller integrated shear
force (Figure 1-7) and thus a lower stress sensitivity and correspondingly higher MDSS.
37
Le
Flow
X
Floating ElementTether
X
t
gk k
Velocity
Lt Lt
Wt
Le
We
Figure 2-1. Simplified 2-d schematic showing the plan view (top) and cross-section (bottom)of a typical floating element shear stress sensor structure (adapted from reviewby Naughton and Sheplak [6]).
2. The effect of essential gaps around the floating element
3. The effect of misalignment of the floating element with respect to the surrounding
surface
4. Forces due to pressure gradients
5. Effects of gravity and acceleration when placed on a moving object or on a non-level
surface
6. Massive devices result in poor temporal resolution
Naughton and Sheplak [6], based on previous research efforts, provide a detailed dis-
cussion showing the favorable scaling of micromachined shear stress sensors for quantitative
time resolved shear stress measurements. Their findings show that micromachined sensors
offer the potential for high temporal and spatial resolution. They theoretically offer at least
five-orders of magnitude improvement in the sensitivity bandwidth product and about two
38
orders of magnitude improvement in spatial resolution [6]. In a more recent review, Sheplak
et al. summarize the benefits of micro-machined sensors as follows: [23]
1. MEMS sensors can measure shear forces of the order 0.1 nN . This corresponds to a
shear stress of order 10 mPa for a floating element size of 100 µm× 100 µm.
2. Sensors fabricated using standard micromachining technology do not need assembly of
sensor components, eliminating some misalignment issues although sensor packaging
and installation on the test setup is still prone to misalignment errors.
3. Micromachining allows gaps of O(1 µm), rendering the surface hydraulically smooth
except at very high Re [7].
4. Three orders of magnitude reduction in scale compared to conventional sensors greatly
reduces pressure gradient errors in micromachined sensors.
5. The cross-axis sensitivity of these sensors, with respect to acceleration, is three orders
of magnitude smaller than conventional sensors because of reduced sensor mass.
6. Thermal expansion errors are mitigated due to monolithic fabrication techniques but
careful packaging is also important to avoid these errors.
To highlight the benefits and challenges faced by MEMS based shear stress sensors,
this section will focus on previous research, specifically on micromachined direct shear stress
sensors. Based on the transduction scheme used, the sensors are categorized as capacitive,
optical, and piezoresistive, respectively.
2.1.2.1 Capacitive Technique
Schmidt et al. were the first to develop a micromachined shear stress sensor with a
capacitive transduction scheme for measuring shear stress [36,37]. This was the first proof of
concept device indicating feasibility of micromachined floating element shear stress sensors.
The sensor was surface micromachined with polyimide using aluminum as the sacrificial
layer. The sensor used a differential capacitance read out scheme with integrated depletion
mode P-MOS transistors to sense the change in capacitance. A schematic of this device is
shown in Figure 2-2. The floating element was 500 µm× 500 µm× 30 µm and the tethers
39
Floating-ElementEmbedded
Conducter
Cps1 Cdp Cps2
Silicon
Passivated
Electrodes
VDS
Csb2Csb1
On Chip
Off Chip
+
-Vd
Figure 2-2. Schematic of cross section of polyimide capacitive shear stress sensor with differ-ential capacitive readout scheme using integrated depletion mode P-MOS tran-sistors (adapted from Schmidt et al. [37]).
were 1000 µm × 5 µm × 30 µm. The sensor demonstrated a sensitivity of 52 µV/Pa [37].
It however, suffered from drift due to moisture on the order of several µV/min. Moisture
has two different effects, 1) Moisture induces hydroscopic in-plane stresses that affects the
mechanical sensitivity and 2) It changes the dielectric properties of the polyimide resulting
in drift [36]. In general, any charge accumulation at the air dielectric interface will result in
drift [6].
Pan et al. and Hyman et al. developed comb finger-based capacitive shear stress
sensors [38–40]. They developed two sensor designs, one based on differential capacitive
measurement, and the other based on force balancing of moving capacitive plates using
a feedback signal. Figure 2-3 shows a schematic of both sensors. In the first design a
40
Comb finger
Drive
Folded
Beam
V+
Tether
Floating
Element Anchor
Release
Holes
V-
Expanded View of Comb Finger
Structures
V-V+
Flow
C2
C1
b)
Release
Hole
Floating
Element/Electrode
Bottom
electrode
Flow
a)
Figure 2-3. Schematic of a) folded beam with lateral capacitive change and b) force feedbackcapacitive structure (adapted from Pan et al. [38]).
folded beam structure support a shear stress sensing element. The folded beams provided
the requisite restoring force. In this sensor, the output due to capacitance change was
undetectable due to high parasitic attenuation. Hyman et al. attribute the parasitics to
the direct electrical contact being made to the device [39]. Optically measured deflection in
a known flow resulted in a mechanical sensitivity of 9Pa/µm or 0.11 µm/Pa. The second
41
generation sensors used a structure similar to a commercial accelerometer where a shear stress
sensing element is used instead of a proof mass. Using the reciprocal and conservative nature
of electrostatic transducers [41, 42], the mechanical force sensed using the sense electrodes
is countered by applying electrostatic force on the actuation electrodes. This force feedback
method maintains the sensor at its nominal position i.e., zero displacement. On chip circuitry
with sufficiently high loop gain also mitigate parasitic effects [40]. These sensors had a high
sensitivity of 1.03 V/Pa. At high shear stresses, the electrical actuation force was not enough
to maintain null deflection of the sensor element, potentially limiting its dynamic range. The
dynamic and noise characteristics of these sensors were not reported.
Zhe et al. developed a cantilever based shear stress sensor, consisting of a sensing
element attached to the end of a cantilever beam, that provided the restoring force [43, 44].
The sensing element is 200 µm × 500 µm × 50 µm and the beam is 3000 µm × 10 µm ×50 µm. Two capacitors are formed across the gaps between the floating element and the
surrounding substrate as shown in Figure 2-4. The differential capacitance change due to
element deflection is measured using an off the shelf circuit component MS3110 from Irvine
Sensors. Flow calibration of the sensor resulted in a high noise floor of 0.04 Pa and a
large sensitivity of 337 mV/Pa due to compliance of the long beam [44]. The authors
expressed concern over sensor misalignment in the channel leading to errors. There was 13%
uncertainty in the measurement, which was attributed to the potential misalignment errors
and channel height uncertainty. There were other sources of uncertainty and scatter in the
data, which are still being investigated [44]. The dynamic behavior of the sensor was not
reported. A charge amplification scheme is used in the interface circuitry. Using this scheme
renders the sensor sensitivity independent of the parasitics in the system but, the noise floor
is sensitive to parasitics (see Section 3.1.3.3). This explains the high noise floor of the sensor.
Furthermore, all capacitive sensors inherently are high impedance devices and are therefore
susceptible to electromagnetic interference (EMI).
42
Sense Electrode
Actuation Electrode
Pad
Flow
t, C2t, C1
nl
n2
El g
b
L
SS’
E’ E
x
y
z
n2
n1
h
b
Figure 2-4. Top view of cantilever based floating element capacitive shear sensor (adaptedfrom Zhe et al. [44]).
McCarthy et al. and Tiliakos et al. have attempted to develop a similar sensor using
differential capacitance measurement and force feedback for high shear stress applications
(10 Pa−10 kPa). They however did not perform an experimental characterization and their
work is therefore not discussed any further [45–47].
Similarly, Desai et al. designed and fabricated a novel MEMS structure for 2-D shear
stress measurement using the wafer thickness for the floating element [48]. The sensor
proposed to use a differential capacitive transduction. Though the sensor was fabricated, no
experimental characterization has been reported.
2.1.2.2 Optical Techniques
Padmanabhan et al. designed the first micro-scale optical shear stress sensor with photo
diodes as detection elements using an optical shutter technique [18, 21, 49, 50]. This sensor
also had integrated electronics on the sensor die for electronic signal read-out. Two differ-
ent generations of this device were built. The sensor was comprised of a floating element
(120 µm× 120 µm and 500 µm× 500 µm) acting as the optical shutter for two photo diodes
located under the element at the leading and trailing edges as shown in Figure 2-5. Four
43
Photodiodes
n+ n+
n-Type Si
Floating-element shutterFlow
Incident Light from a Laser Source
p-Type Si Substrate
Photodiodes
n+ n+
n-Type Si
Floating-element shutterFlow
Incident Light from a Laser Source
p-Type Si Substrate
(a)
(b)
Figure 2-5. Cross sectional schematic of a)two diode and b)’split’ diode (3 diodes) opticalshutter based micromachined shear stress sensor (adapted from Padmanabhan’swork [21]).
tethers 500 µm long and 10 µm wide suspended the floating element 1.2 µm over a silicon
substrate. The tethers and floating element were 7 µm thick. A coherent uniform light
source located above the sensor illuminates the exposed area of the two photodiodes, result-
ing in a differential photocurrent. The photocurrent is ideally zero when there is no motion
as the illuminated areas of the two photodiodes are the same by design. In the presence of
a force due to shear stress, the lateral sensor motion increases the illuminated area of one
photodiode while the area for the other decreases by the same amount, resulting in a dif-
ferential photocurrent. The differential photocurrent is proportional to both the magnitude
and direction of the shear stress. Intensity gradients across the floating element resulted in
44
erroneous outputs with the first generation sensors. The second generation of this sensor
implemented a split diode configuration and resulted in 94% reduction in sensitivity to in-
tensity variations [21]. Static calibration results indicated maximum nonlinearity of 1% over
shear stress ranging from 1.4 mPa to 10Pa. The sensor was also characterized dynamically
via Stokes’ layer excitation. This is was the first sensor use this dynamic calibration tech-
nique [51]. The dynamic response exceeded 10 kHz and the sensitivity for 120 µm sensor
was shown to be 0.097%/Pa (percentage in photocurrent/shear stress) [50]. The sensor was
designed to allow off-chip read-out of the output signal. These sensors were also relatively in-
sensitive to EMI and stray capacitance, unlike capacitive sensors [21]. The detection scheme
was effectively designed to minimize common mode signals due to pressure fluctuations or
out of plane acceleration and vibration with promising results [50]. The shortcomings of
this sensor were front side electrical contacts and remote location of the light source. The
remote location of the light source with respect to the sensor may make the measurement
sensitive to mechanical motion of the light source, such as vibration and thermal expansion
of its mounting [23].
Tseng and Lin used an optical fiber based micro-Fabry-Perot interferometry technique
[52]. The floating element is built on a flexible membrane using two layers of SU-8 resist
as shown in Figure 2-6. The 1.5 mm× 1.5 mm× 20 µm membrane provides the requisite
restoring force. The floating element was 200 µm × 200 µm in length and width. The
reflecting portion of the floating element was 400 µm high. With no gaps, this sensor
circumvented flow disturbance issues, was truly flush mounted, and could also be used in
different fluids. Two optical fibers aligned orthogonal to each other allow 2-D shear stress
measurements, though this ability was not demonstrated experimentally. The detection
technique was also immune to EMI. The static calibration results indicate a sensitivity of
0.65 Pa/nm or 1.54 nm/Pa and temperature sensitivity of 3.4 nm/K [52]. The temperature
sensitivity was thus higher than the shear stress sensitivity. The sensor also had a high noise
floor of ≈ 2 ∼ 3 nm, which corresponds to a shear stress of 1.3 Pa. The dynamic response
45
Floating Element
Glue Groove
A A'
Input (Output) Fiber
MRTV1 Membrane Air or Liquid Flow
Reflection Mirror
(Floating Element)d
IR1R2
Figure 2-6. Conceptual schematic showing bottom and side view of fabry-perot floating ele-ment shear stress sensor (adapted from [52]).
of the sensor was not reported, but the large height of the floating element suggests that
the sensor would have a large mass and therefore a lower bandwidth. The dynamic range
of the sensor was also not reported. The sensor may also have been sensitive to pressure
fluctuations and vibrations of the membrane, supporting the floating element.
Horowitz et al. designed an optical floating element shear stress sensor that used an
aligned wafer bonding/thin back process [53, 54]. Figure 2-7 shows a schematic of the sen-
sor. The sensor has optical gratings on the backside of the floating element and on the
top side of a transparent pyrex wafer. The gratings form a geometric Moire pattern that
amplifies the mechanical motion of the floating element. The Moire pattern displacement
46
Reflected Moiré
FringeIncident
Incoherent Light
Floating ElementLaminar Flow Cell
Pyrex
Silicon
Aluminum Gratings
(Floating Element &
Base Gratings)
Tethers
Figure 2-7. Schematic of the optical Moire shear stress sensor. The shift of the amplifiedMoire pattern is imaged using a microscope attached to a CCD camera (withpermission from author [54]).
was measured using a linescan CCD camera and imaged using a microscope. By design, the
sensor is insensitive to out of plane motion due to pressure, vibration etc. since the motion
produces no change in the Moire fringe pattern. Like most optical sensors, this sensor is
also insensitive to EMI. Drawbacks of this sensor include a bulky optical setup and the as-
sociated packaging requirements, restricting its use to an optical bench. The sensor had a
static mechanical sensitivity of 0.26 µm/Pa with a linear response up to the testing limit of
1.3 Pa. The dynamic testing indicated a resonant frequency of 1.7 kHz, and a noise floor
of 6.2 mPa/√
Hz [54].
2.1.2.3 Piezoresistive Technique
Ng and Goldberg et al. were the first to use piezoresistive implants on a floating element
sensor to measure large shear stress (1 kPa − 100 kPa) for polymer extrusion processes
[55–57], using axial loading on the tethers. They used a wafer bonding and etch back
47
technology with top side tether implants to form piezoresistors. The piezo resistors were
connected in a half bridge configuration to obtain an electrical output. A simplified schematic
of the sensor is shown in Figure 2-8. The tethers were aligned along the direction of the flow
to withstand the high shear stresses. The floating element was 120 µm × 140 µm × 5 µm
while the tethers were 30 µm × 10 µm × 5 µm, respectively. The sensor had back side
electrical contacts which represented an important landmark in terms of achieving truly
flush mounted non-optical sensors for shear stress measurement. Sheplak et al. noted that
use of anisotropic wet etching process for the backside contact resulted in large die sizes [23].
Flow
X
Y
180 m
120 m
120 m
10 m
Plate Thickness = 5 m
Figure 2-8. 3D schematic of a piezoresistive sensor with top side tether implants oriented inthe direction of flow for high shear stress measurements (adapted from Ng et al.and Goldeberg et al. [56, 57]).
Barlian et al. studied side tether implants and additional top-side implants to also
sense out of plane motion in a floating element shear stress sensor [58, 59]. They presented
mechanical characterization, temperature and doping effects, and the effects of annealing on
noise for a single implanted cantilever beam (6000 µm × 400 µm × 15 µm) [59]. However,
Barlian’s initial flow characterization results were preliminary and had significant scatter in
their measurements, warranting further investigation [58].
48
Al-Si(1%)
p++interconnect
Silicon tether
Sidewall
implanted
piezoresistor
Floating
element
Bond
pad
n-well
Reverse
bias contact
tether
centerline
Figure 2-9. 3D schematic of a side implanted piezoresistive shear stress sensor. The implantsare such that two tethers are in tension while two are in compression for highersensitivity (with permission from author [60]).
Li et al. used side implanted piezoresistors on the tethers of the floating element sensor.
A schematic of a side implanted floating element sensor is shown in Figure 2-9 [60]. They
presented preliminary shear stress characterization results for a 1 mm2 floating element with
tethers 1 mm long and 30 µm wide. The floating element and tethers were 50 µm thick.
The sensor was calibrated using a Stokes layer excitation to generate shear stress in a plane
wave tube [51]. The sensor was linear up to a measured shear stress of 2 Pa and possessed
a normalized sensitivity of 2.83 µV/V/Pa. The noise floor was 11.4 mPa at 1 kHz and
the temperature sensitivity was 0.42 mV/C [61]. The upper end of the dynamic range was
limited by the testing limit of 2 Pa [61]. The dynamic response of the sensor indicated
a flat frequency response up to the testing limit of 6.7 kHz. This sensor demonstrated
promising results and offers a robust sensor for shear stress measurements. High sensitivity
to in-plane motion due to sidewall implants, low sensitivity to out of plane motion both
due to geometry and a simple read out scheme, are the benefits of this sensor design. In a
49
balanced Wheatstone bridge, out of plane motion acts as a common change in all resistors
resulting in no output. The sensor however had an unbalanced bridge due to poor uniformity
in doping for side wall piezoresistive implants, resulting in higher than expected out of plane
sensitivity. The drawbacks of this sensor include a high temperature coefficient of resistance,
limiting applied bias voltages and thus lowering sensitivity, and drift due convective heat
transfer to the flow similar to hot wires.
2.2 Conclusion
A review of various shear stress sensor techniques was provided. Basic principle, merits,
and issues with existing non-direct measurement techniques were presented based on pre-
vious reviews. The benefits of micromachined direct shear stress sensors were highlighted,
followed by a detailed review of previous work on these sensors organized according to their
transduction mechanisms. Each transduction mechanism has strengths and limitations that
are yet to be resolved, leaving ample room for further development of micromachined shear
stress sensors.
Optical transduction has increased levels of complexity when the whole measurement
system is scaled down. At small scales optical alignment and noise issues can be significant.
Floating element piezoresistive shear stress sensors may drift due to Joulean heating and
require active temperature compensation. Spencer’s research in the area of pressure sensors
indicates that the raw sensitivity of capacitive pressure sensors is 10-20 times the sensitivity
of piezoresistive sensors with similar geometric shape and dimensions [62]. This is amply
supported by the fact that a large number of silicon microphones are mostly capacitive in
nature [63]. Spencer’s observation may be valid if capacitive and piezoresistive transducers
are compared in general. Further, bulk micromachined capacitive sensors are simple to
fabricate, inherently noiseless, and are insensitive to temperature changes. Susceptibility to
EMI, effects of circuit noise and parasitics (see Chapter 3) remain a concern for capacitive
sensors. However, the several benefits of capacitive sensors offer opportunities to design
around their limitations.
50
In this dissertation, attempts are made to overcome some limitations of capacitive
sensors. The drifts due to charge accumulation at the air-dielectric interface observed by
Schmidt et al. [36] is avoided using metal coated electrodes for the capacitors. An unpack-
aged (die form) low noise voltage amplifier is used with very low input capacitance and is
directly wire bonded to the sensor to mitigate the adverse effect of parasitics. An overall
system optimization is performed which includes the sensor and the interface circuitry to
obtain the best design performance from the sensor. The next chapter discusses the design
and modeling of a floating element based differential capacitive shear stress sensor.
51
CHAPTER 3DEVICE MODELING
Design and modeling of a micromachined direct capacitive wall shear stress sensor is
presented in this chapter. An electromechanical model is developed using a differential ca-
pacitive transduction scheme. Detailed quasi-static models are developed and used to predict
sensitivity, minimum detectable signal and frequency response of the electromechanical sen-
sor. The proposed shear stress sensor is required to meet stringent requirements on spatial
and temporal resolution while accurately measuring both the mean and the fluctuating shear
stress. In order to maximize overall performance and meet sensor requirements, a design
optimization is required (Chapter 4). Mechanical and electrical models developed in this
chapter are essential components for the optimization study.
This chapter is organized into three major sections. In Section 3.1, an existing quasi-
static deflection model presented in [36,64] is extended to include the effect of comb fingers.
Expressions for mechanical deflections are presented using both linear and nonlinear theories.
A novel asymmetric differential capacitive sensing scheme is proposed. Electrostatic behavior
of the sensor is explained. A short comparison of capacitive interface circuits is given,
followed by integration of the chosen circuitry with the device model. In Section 3.2, the
dynamic characteristics of the shear stress sensor are analytically studied. Lumped element
modeling is employed to estimate the dynamic characteristics of the sensor. In Section 3.3,
higher order effects such as the fringing fields and parasitic capacitance effects are explained.
The noise of the system is also discussed.
3.1 Quasi-Static Model
The proposed shear stress sensor model developed in this chapter consists of a floating
element structure similar to the ones discussed in Section 2.1.2. A rectangular proof mass,
suspended over a small cavity by four compliant tethers, forms the floating element. A flow
across the floating element surface exerts a shear force, which produces a proportional lateral
deflection. The tethers act as springs that restore the floating element to its mean/reference
52
position in the absence of the flow. One end of each tether is attached to the floating element
and the other end is attached to a fixed substrate. Interdigitated comb fingers between
the fixed substrate and the floating element serve as parallel plate capacitors and provide
electrical transduction. The capacitance due to the additional gaps between the stationary
substrate and both the floating element and tethers is also utilized. The schematic of the
sensor, shown in Figure 3-1, depicts the comb fingers, tethers, and the floating element.
Direction of
flow
y
x
Anchors
Tethers
Wt
Lt
Le
We
Comb
Fingers
We
Tt
z
y
LoWf
TtTt
g01
g02
Lt
Wf
TtTt
g01
g02 Wt
g01
Tt
A
A
Side view
Section A-ATop View
Floating
Element
Figure 3-1. Schematic of geometry of the differential, capacitive shear stress sensor.
A deflection changes the capacitance between the surrounding substrate and the tethers,
floating element and between comb fingers, resulting in a proportional change in voltage
(Section 3.1.3.1). The resulting change in voltage is detected using a voltage buffer (see
Section 3.1.3.3). For analysis purposes, the tethers are modeled as clamped beams and
53
the floating element is a rigid mass. Nonlinear deflection of the floating element structure
introduces undesirable harmonic distortion in the spectral content of the sensor output.
The nonlinear deflection may be restricted to a small percentage of the linear deflection to
preserve spectral fidelity of the time resolved measurement. Hence, both small deflection
(linear) and large deflection (non-linear) analyses are needed in the design process in order
to predict the linear range of a particular geometry.
3.1.1 Small Deflection Analysis
In this section, the mechanical model of the differential capacitive shear stress sensor
for small deflection is presented. The following assumptions enable the derivation of the
mechanical model using beam theory [36,64].
• The length of the tether, Lt, is much larger than its width, Wt, and tether thickness,
Tt. The tethers may therefore be treated as beams.
• The floating element and the attached comb fingers move as a rigid mass under the
applied shear stress and is modeled for in-plane motions.
• The material is homogeneous, isotropic and linearly elastic.
• Euler Bernoulli beam theory is used for both small and large deflection analyses, re-
spectively [65].
In the simplified mechanical model as shown in Figure 3-2, each pair of tethers form
a clamped-clamped beam, with a point load P applied at center by the shear stress, τw,
on the floating element. A uniformly distributed load, Q, along the length of the tethers
accounts for the shear force on the tethers [37]. Two such beams share the load applied on
the floating element because of the geometric symmetry. Thus a single pair of tethers that
form the beam experience a force due to the shear stress given as
P =τwWeLe
2+
τw (NWfLf )
2[N ] (3–1)
and Q = τwWt, [N/m] (3–2)
54
Direction
of Flow
Le
Clamped
Boundary
MB
RA
P Q
2Lt
P Q
RA
V
Wt
Q
xx=0
Tt
xx=0
x=Lt
MA
MA Mx
RB
x
z
y
Rigid body motion
P
Figure 3-2. Simplified mechanical model of the floating element structure.
where We is the width of the floating element, Le is the length of the floating element, N
is the number of comb fingers on the floating element, Wf is the width of each comb finger,
and Lf is the length of each comb finger. The governing differential equation is given by
Mx
EI=
d2w (x)/dx2
(1 +
(dw(x)
dx
)2) 3
2
. (3–3)
where E is the Young’s modulus of elasticity, Mx is the moment along the length of the
beam, I is the moment of inertia about the y-axis, and w is the in-plane deflection in the
direction of the applied load. For both small and large deflections, the slope dw(x)dx
¿ 1 and
the simplified governing equation becomes
Mx
EI= d2w (x)
/dx2. (3–4)
Substituting for the the moment, Mx, in terms of the forces P and Q results in a third order
differential equation, expressed in terms of internal moments and reaction forces, requiring
55
three boundary conditions. Since the beam is symmetric, the solution is obtained by solving
for half the length of the beam i.e., a single tether. Furthermore, the length of the floating
element is not accounted for the moment computations because it does not affect the bound-
ary condition at the tethers where they connect to the rigid floating element and hence the
beam deflection characteristics. The boundary conditions for a clamped-clamped beam are
mathematically stated as,
w(x = 0) = 0, (no deflection)
dw
dx
∣∣∣∣x=0
= 0, (zero slope)
anddw
dx
∣∣∣∣x=Lt
= 0. (symmetry) (3–5)
The Euler Bernoulli theory and small beam deflections in response to an applied force are
assumed while solving Equation 3–4. The theory hypothesizes that a straight line transverse
to the neutral axis remains straight, inextensible, and normal before and after deformation
[65]. Assuming pure bending and small deflections allows the nonlinear extensional strain
along the length of the beam to be neglected. The detailed derivation of the model is
presented in Appendix A. The linear deflection using this theory is,
w (x) = − τw
4ETtW 3t
(3 (LeWeLt + NWfLfLt) + 8WtL2t ) x2
− (2LeWe + 2NWfLf + 8WtLt) x3 + 2Wtx4
. (3–6)
The center (maximum) deflection represents the movement of the floating element. Since
most of the transduction takes place via comb fingers and the floating element, the center
(maximum) deflection significantly contributes to the transduction process. Furthermore, in
the lumped element model developed in Section 3.2, this center deflection is used to estimate
the lumped parameters. The maximum deflection at the center of the beam (x = Lt) is
δ = −w (Lt) =τwWeLe
4ETt
(1 +
NWfLf
WeLe
+ 2WtLt
WeLe
)(L3
t
W 3t
). (3–7)
56
In the summation term in Equation 3–7, there are three terms contributing to the deflection:
the first term is due to the floating element, the second term is due to the comb fingers and
the final one is due to the tethers.
3.1.2 Large Deflection Analysis
Beam bending results in rotation of the transverse normal. For large beam deflections
(rotation of 10− 15), not all of the nonlinear terms in the Green-Lagrange strain displace-
ment relationships are negligible [65]. Solutions are obtained using two different solution
techniques; the Rayleigh-Ritz method (Section 3.1.2.1) and an analytical solution technique
(Section 3.1.2.2). The validity of both these solution techniques have been verified previously
using finite element analysis and compared with the linear quasi-static solution in [64].
3.1.2.1 Energy Method
The energy method solution utilizes the principle of virtual work to arrive at a solution
using the Rayleigh-Ritz method [65]. Appendix A provides the detailed derivation of the
solution using this method. The expression for the floating element deflection based on this
method is
δ
(1 +
3
4
(δ
Wt
)2)
=τwWeLe
4ETt
(1 +
NWfLf
WeLe
+ 2WtLt
WeLe
)(L3
t
W 3t
). (3–8)
Equation 3–8 is nonlinear and is solved using numerical solution techniques. Unlike the
previous analytical technique, which yielded a solution for deflection as a function of the
position along the beam, the energy method is used only to solve for the central deflection
with an assumed shape function or mode shape. The center deflection is the quantity of
interest because the permissible nonlinear deflection is defined with reference to the linear
central deflection in Equation 3–7 (see Section 4.3.2.2). The term,
(1 +
3
4
(δ
Wt
)2)
, (3–9)
is a consequence of beam stiffening due to deformation along the neutral axis. The solution
approaches the linear solution in Equation 3–7 for small deflections i.e., δ/Wt << 1.
57
3.1.2.2 Analytical Solution
In this method, an axial force, Fa, responsible for the nonlinear strain along the length
of the beam is estimated. The axial force is obtained by integrating the stress along the
length of the beam. The expression for the central beam deflection (Appendix A) using this
solution technique is,
δ =
− P
2Faλsinh (λLt) +
Q
2Fa
L2t +
P
2Fa
Lt
−(cosh (λLt)− 1)
Faλ sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
)
,(3–10)
where
λ =
√12Fa
ETtW 3t
, (3–11)
and Fa =EWtTt
2Lt
Lt∫
0
(∂w
∂x
)2
dx. (3–12)
An iterative step-wise method for solving Equation 3–10 is provided in Appendix A.
3.1.3 Electrostatic Behavior
The previous section presented the expressions for the mechanical deflection, δ, of the
floating element sensor structure in response to an applied shear stress, τw. In this section,
the electrical response to a mechanical motion is analyzed. Fundamental relationships for
an electrostatic transducer are provided. A detailed analysis of the interdigitated capacitive
comb fingers and the interface circuitry implemented in the current sensor is developed.
3.1.3.1 Electrostatic Transducers
The capacitive shear stress sensor developed in this thesis is an example of an electro-
static transducer. An accelerometer with comb fingers is another common example of an
electrostatic transducer. An electrostatic transducer consists of at least two electrodes, sep-
arated by a finite distance by a dielectric medium to form one or more variable capacitors.
58
One of the electrodes remains fixed while the other electrode is free to move in response to
a physical input signal such as pressure, shear stress, acceleration, etc. For a more detailed
discussion on electrostatic transducers the reader is referred to Rossi [41] and Hunt [66].
Tt
g0
L0
Wf
+VB
Ground (0 V)
+VB
C
Direction of
motion
g0
Tt
Figure 3-3. Schematic of parallel plate capacitor analogous to overlapping comb fingers.
Background. A model of a parallel plate electrostatic transducer with variable gap is
shown in Figure 3-3. The transducer consists of two conducting parallel plates separated by
a dielectric medium such as air. One plate is fixed and the other is free to move, such that
the gap between the plates may change. The capacitance between the plates is
C = εA/g, (3–13)
where ε is the permittivity of the medium, A is the area of overlap, and g is the plate
separation distance. For the current configuration, the movable plate is assumed to have
one dimensional motion such that only the gap between the plates changes while the area of
overlap stays constant. For illustration, consider a pair of overlapping comb fingers as the
parallel plates, the initial gap between the plates is g0 and the area is A = L0Tt, where L0 is
the overlapping length. Consequently, when the system is at rest, the capacitance between
the plates is
C0 =εL0Tt
g0
. (3–14)
The following assumptions must be true for Equations 3–13 and 3–14 to hold:
1. g ¿ √A, i.e., g ¿ L0 and g ¿ Tt.
59
2. The electric field, EE is normal to the plates.
An applied force results in a change in gap δ(t); the time varying capacitance is then given
by
C(t) = C0
[1− δ(t)
g0
]−1
. (3–15)
The voltage between the capacitor plates is V (t) = Q(t)/C(t), where Q(t) is the charge on
the plates. Substituting Equation 3–15 into this expression gives the voltage,
V (t) =Q(t)
C0
[1− δ(t)
g0
]. (3–16)
The change in electrical potential energy and co-energy(*) stored in the capacitor are ex-
pressed as [42]
dEp = Fedδ + V dQ (charge variation), (3–17)
and dE∗p = −Fedδ + QdV (voltage variation), (3–18)
respectively. If no additional charge or voltage is applied to the capacitor electrodes, a change
in stored potential energy is effected by the motion, δ of the movable plate. Consequently,
the electrostatic force is given as,
Fe =dEp
dδ
∣∣∣∣Q
= −dE∗p
dδ
∣∣∣∣V
. (3–19)
The stored electrical potential energies in the capacitor are
Ep =
Q(t)∫
0
V (t)dQ =
Q(t)∫
0
Q(t)
C(t)dQ =
1
2
Q(t)2
C(t). (3–20)
and E∗p =
V (t)∫
0
Q(t)dV =
V (t)∫
0
C(t)V (t)dV =1
2C(t)V (t)2. (3–21)
Substituting Equation 3–21 and 3–20 into Equation 3–19 results in
Fe =d
dδ
(1
2
Q(t)2
C(t)
)= − d
dδ
(1
2C(t)V (t)2
). (3–22)
60
Substituting the expression for the time dependent capacitance from Equation 3–15 into the
above expression results in,
Fe = −Q(t)2
2C0g0
(3–23)
or Fe = − V (t)2C0
2g0
(1− δ
g0
)2 . (3–24)
Note that the force can be expressed either in terms of the voltage V (t) or in terms of charge
Q(t). No prior assumptions were made regarding the method of applying charge or voltage
to the capacitor plates. Equation 3–23 indicates that when there is a constant charge across
the capacitor, Fe is independent of the plate motion δ. This avoids the electrostatic pull-in
instability for the constant voltage case in Equation 3–24 as δ approaches g0 [66].
A restoring force due to the mechanical compliance of the structure opposes the deflec-
tion of the movable plate. The mechanical force Fm(t) in terms of the compliance, Cme, is
expressed as
Fm =δ(t)
Cme
. (3–25)
The electrical force and the mechanical force are opposite in direction. Using Equations 3–
16, 3–23, 3–24, and 3–25, the characteristic electrostatic equations for the system are written
as
V (t) =Q(t)
C0
[1− δ(t)
g0
], (3–26)
and F (t) =δ(t)
Cme
− Q(t)2
2C0g0︸ ︷︷ ︸Charge Control
=δ(t)
Cme
− V (t)2C0
2g0
(1− δ
g0
)2
︸ ︷︷ ︸V oltage Control
. (3–27)
Inspection of Equations 3–26 and 3–27 indicates that voltage, V (t), and force, F (t), are
coupled. In addition, the voltage and the force on the capacitor are nonlinearly related. The
system therefore needs to be linearized. The linearization of the system and the approach
for two port modeling is explained in Appendix B. In the next section, a two port model of
the entire sensor structure is illustrated based on the derivations in Appendix B.
61
3.1.3.2 Electrostatics of the sensor structure
In this section, the electrostatic behavior of the developed shear stress sensor is dis-
cussed. A two port model based on the approach explained in Appendix B is derived to
illustrate the electrostatic transduction mechanism. Consider Figure 3-1 and Figure 3-4,
where there are pairs of capacitances formed by different gaps on either side of the sensor
structure as follows:
• C1a , C2a , between interdigitated comb fingers with primary gap g01
• C1b, C2b
, between interdigitated comb fingers with secondary gap g02
• C1c , C2c , between the floating element edges normal to the flow and the surrounding
substrate with primary gap g01
• C1d, C2d
, between the tether and the surrounding substrate with primary gap g01
• C1e , C2e , between the tether and the surrounding substrate with secondary gap g02
The comb finger capacitance formed by N interdigitated comb fingers results in N − 1
capacitors connected in parallel. Half the comb fingers form capacitors with the gap g01 and
the other half form capacitors with the gap g02. The nominal comb finger capacitances for
both the primary and secondary gaps are therefore written as
C1a = C2a =(N − 1)
2
εTtL0
g01
(3–28)
and C1b= C2b
=(N − 1)
2
εTtL0
g02
. (3–29)
The capacitances C1c and C2c are,
C1c = C2c =εTtLe
g01
. (3–30)
While the comb finger and floating element capacitances are easily modeled as parallel
plate capacitors, the tethers acting as cantilevers form a non-uniform gap requiring further
model simplification. Consider the tether capacitor with non-uniform gap as shown in Fig-
ure 3-5. Since the electrode surfaces are metalized, a uniform surface charge density, σ, is
62
L0 Wf
TtTt
g01
g02
Interdigitated
comb finger(C1a,C2a)
(C1b,C2b)
a)
g01
Floating Element
g01 = Primary gap
g02 = Secondary gap
(C1c,C2c)
Surrounding silicon
substrate
b)
Lt
Wf
TtTt
g01
g02 Wt
Surrounding silicon
substrate
Tether(C1d,C2d)
(C1e,C2e)
Surrounding
silicon substrate
c)
Figure 3-4. Simplified schematic of individual a)tether, b)comb finger, and c)floating elementcapacitances, respectively.
assumed on the tethers and the surrounding substrate. By definition, the capacitance is [67],
−∫
+
EE · ds = Q/C, (3–31)
where EE is the electric field, ds is the differential vector distance/gap between the two
electrodes (positive [+] and negative [−]). For a parallel plate capacitor with a dielectric
63
g20
g10
Tether
y
Lf
Surrounding
Substrate
Surrounding
Substrate
(C1d,C2d)
(C1e,C2e)
Figure 3-5. Schematic of non-uniform tether capacitors with primary and secondary gaps.
with constant permittivity, ε, EE is normal at both conductor surfaces, which simplifies
Equation 3–31 to
−∫
+
EE · ds =σ
ε
−∫
+
ds = Q/C. (3–32)
Assuming small deflection of the tethers, it is reasonable to assume that the electric field is
normal to the conducting surfaces, forming the tether capacitors. Now consider an elemental
area dA of the tether; using Equation 3–32, the capacitance formed by this element is
dC =dQ
σε
g(x)∫0
dg
. (3–33)
Expressing charge as dQ = σdA, Equation 3–33 is rewritten as
dC =εdA
g(x). (3–34)
Since the thickness stays constant over the length of the tether, dA = Ttdx. The gap
reduction/increase due to the deflection of the tethers is
g(x) = g0 ± w(x). (3–35)
64
For decreasing gap, the capacitance is evaluated by using Equation 3–6 for w(x) and solving
the integral,
Cng =
Lt∫
0
εTt
g0 − τw
4ETtW 3t
(3 (LeWeLt + NWfLfLt) + 8WtL2t ) x2
− (2LeWe + 2NWfLf + 8WtLt) x3 + 2Wtx4
dx, (3–36)
where ng indicates non-uniform gap. Equation 3–36 is numerically evaluated in MAPLE.
The change in capacitance due to the non-uniform gap variation is computed using
∆Cng = Cng − Ctether, (3–37)
where Ctether =εTtLt
g0
. (3–38)
The capacitance change is also computed using a simple parallel plate model with tether
length, Lt and using the center deflection δ for the gap change. The capacitance change for
this case is
∆Cparallel = Ctetherδ
g0
. (3–39)
Equations 3–37 and 3–39 are numerically compared for a few geometries for small deflections
using MAPLE. Table 3-1 shows the geometry and parameters for this study. The results in
Table 3-1 indicate that for small deflections, ∆Cparallel is approximately twice the value of
∆Cng. For ease of modeling, this difference is accounted as an effective tether length, Lteff ,
allowing the tethers to be modeled as parallel plate capacitors. The effective tether length
is therefore,
Lteff ≈ Lt
2. (3–40)
65
Table 3-1. Sensor geometry for effective tether lengthcalculation.
Parameter Value
Shear stress τw 10 Pa
Tether Length Lt 1000 µm
Tether Width Wt 10 µm
Floating Element Length Le 1000 µm
Floating Element Width We 1000 µm
Tether Thickness Tt 45 µm
Number of comb fingers N 100
Comb Finger Width Wf 4 µm
Comb Finger length Lf 150 µm
Initial gap for tethercapacitance g0
3.5 µm
Dielectric Permittivity ε 8.85× 10−12 F
Floating Element Young’sModulus E †
160 MPa
∆Cparallel 6.637 fF
∆Cng 13.65 fF
† Average value in the 100 plane of silicon.
Using the parallel plate model, the tether capacitances for primary and secondary gaps are
C1d= C2d
=εTtLteff
g01
(3–41)
and C1e = C2e =εTtLteff
g02
, (3–42)
respectively.
All the capacitances with the same subscript (1 and 2) are in parallel by design. The
net capacitance is therefore the sum of the individual capacitances expressed as
C1 = C1a + C1b + C1c + C1d + C1e (3–43)
and C2 = C2a + C2b + C2c + C2d + C2e. (3–44)
66
Each component capacitor of C1 forms a differential capacitance with the corresponding
component of C2. Thus, C1 and C2 together form the differential capacitance pair for the
sensor. The nominal capacitances obtained by substituting Equations 3–28 - 3–30, 3–41,
and 3–42 in Equations 3–43 and 3–44 are
C0 = C10 = C20 =(N − 1)
2
εTtL0
g01
+(N − 1)
2
εTtL0
g02
+εTtLe
g01
+εTtLteff
g01
+εTtLteff
g02
or C0 = εTt
[3Lteff + Le + (N−1)
2Lf
g01
+Lteff + (N−1)
2Lf
g02
]. (3–45)
The differential capacitance model thus formed using the asymmetric gaps is graphically
represented using two capacitors for each of the sensing capacitors C1 and C2 (Figure 3-6).
C1
g01
g02
C2
g01
g02
Figure 3-6. Differential capacitance sensing model for sensing capacitors with the asymmetricgaps.
67
For ease of representation an overall effective length is defined based on Equation 3–45 and
is given as
Leff =
(3Lteff + Le +
(N − 1)
2Lf
)+
(Lteff +
(N − 1)
2Lf
)g01
g02
. (3–46)
Substituting Equations 3–46 into 3–45, the nominal capacitance is simplified and rewritten
as
C10 = C20 =εTtLeff
g01
. (3–47)
In this section, the electrostatic equations were derived for a parallel plate transducer in
general and for a single pair of comb fingers in particular. Capacitances formed by different
geometric elements of the sensor structure, tethers, comb fingers, and floating element, were
described. An effective tether length was determined for modeling tether capacitance change
using the parallel plate assumption. Lastly, the nominal capacitances for the differential
capacitor scheme to be implemented for the proposed sensor were presented. The next
section describes details of the sensing circuitry.
3.1.3.3 Interface Circuits
In a capacitive sensor the change in capacitance usually manifests itself in terms of a
proportional charge or voltage change. Either the voltage is measured while holding the
charge constant or vice-versa. A sensing circuitry is always required to measure charge
or voltage change in capacitive sensors due to the large sensor impedance. This section
concentrates on the circuit requirements to measure the voltage/charge output obtained from
the capacitive floating element sensor with interdigitated comb fingers shown in Figure 3-
1. The measurement requirements for the sensor include, measuring dc and ac change in
capacitance due to shear stress. Typical sensor parameters that influence the interface circuit
design are C0 ≈ 0.4− 1.5 pF , ∆C ≈ 3− 25 fF . The sensor designs target a maximum shear
stress of 10 Pa. A high sensor resolution enables measurement of weak turbulent motion
and maximizes dynamic range. This requires the interface circuit to have a low noise floor.
68
Furthermore, parasitic capacitances in the system which effect sensor performance have to be
considered while designing the interface circuit. With these requirements in mind, existing
analog interface circuits for capacitive transducers are briefly reviewed. Next, equations
for the synchronous modulation and demodulation (MOD-DMOD) technique, used for the
capacitive shear stress sensor, are derived.
Interface circuits for capacitive sensors can be classified into two basic categories, open-
loop and closed-loop. Each method can be further sub-categorized into analog (no clock
signals) and digital (uses clock signals) based on the type of electronic components. Figure 3-
7 shows the block diagram illustrating the classification of interface circuitry. This work
focuses on open-loop techniques with analog electronic components.
Interface Electronic
Circuits
Open Loop Closed Loop
Continuous
Time
Discrete
Time
Switched
Capacitor
Methods
Charge
Amplifier
Voltage
Amplifier
Synchronous
MOD/
DEMOD
Oscillators
(LC and RC)
Figure 3-7. Simplified classification of capacitive interface electronic circuits.
Before describing the details of the proposed sensor circuitry, existing open-loop, analog
interface circuit configurations will be discussed. The most commonly used analog, open-loop
69
interface circuits include charge amplifiers, voltage amplifiers, oscillators (RC and LC), and
synchronous modulation/demodulation (MOD-DMOD) techniques [68–71]. The principle of
operation of oscillators (RC and LC) formed using sense capacitances is different from the
other three techniques. Oscillators require complicated interface circuitry and are beyond the
scope of this work. The primary goal of the current work is to obtain reliable sensor output
voltage, while limiting the complexity of the sensor fabrication and interface electronics.
At a constant capacitance, charge and voltage were established in Equation 3–16 to be
directly proportional. Any change in capacitance with a constant charge or voltage applied
to the capacitor produces a proportional voltage or charge output, respectively. An interface
circuit can therefore be designed to sense either the charge or the voltage across the capacitor.
Constant voltage biasing with a charge amplifier and constant charge scheme in conjunction
with a voltage amplifier are the two techniques generally used in capacitive sensors, resulting
in an output voltage.
Comparative Study. The floating element sensor has two sets of capacitors C1 and
C2 as discussed previously in Section 3.1.3.1 and also shown in Figure 3-1. This sensor geom-
etry enables measurement of a differential capacitance change. Both single and differential
capacitance sensing schemes have been implemented for sensors previously. Martin [72] pro-
vides a comparison of single and differential capacitance sensing schemes using both charge
(constant voltage bias) and voltage (constant charge bias) amplifiers. An appropriate bias-
ing scheme for an electrostatic transducer is important from the interface circuit and overall
performance perspective. It is therefore relevant to perform a comparative study of charge vs
voltage amplifiers and single vs differential capacitance measurement schemes. Kadirvel [73]
and Martin [72] provide insight into the aspects of different biasing techniques. In the past
several other researchers have also studied the interface circuit schemes using charge and
voltage amplifiers [69–71]. In this section, a comparative study of the constant charge and
constant voltage biasing schemes is presented.
70
A common input signal to the capacitances, such as hydrostatic pressure fluctuations,
ideally results in no net differential capacitance change. Thus, output due to any common
mode signal is suppressed by the differential measurement scheme. An added benefit of using
the differential capacitance measurement scheme, in conjunction with a charge amplifier, is
the doubling of the sensitivity compared to the single capacitance measurement [72]. For
a variable gap capacitive sensor, such as a microphone, a constant voltage biasing with a
voltage amplifier makes the sensor electrodes susceptible to electrostatic pull-in [42,66]. Pull-
in occurs when the mechanical restoring force is unable to balance the electrostatic force,
causing instability. In this biasing scheme, the reduction of the gap between the electrodes
increases the electrostatic force (refer Equation 3–24), eventually leading to pull in. The use
of constant charge biasing in combination with a voltage amplifier avoids this problem. The
constant charge biasing creates a constant electric field across the gap independent of the
plate motion (see Equation 3–23), preventing electrostatic pull-in [66].
The study conducted by Kadirvel [73] reveals several details about sensitivity and noise
in charge and voltage amplifiers. The sensitivity of charge amplifiers is independent of the
parasitic capacitance, while parasitics have a pronounced effect on the sensitivity of voltage
amplifiers. The adverse effect of parasitics in voltage amplifiers increases significantly with
a drop in sensor capacitance. However, with an increase in parasitic capacitance, the noise
increases in charge amplifiers. The study also revealed that noise performance in both charge
and voltage amplifiers is better when they are optimized for low current noise.
Charge Amplifiers. A charge amplifier circuit typically has an integrating capacitor,
Cf , in the feedback path that is inversely proportional to sensitivity, which is expressed as
Schg−amp = −2Vb
g0
C0
Cf
δ
τw
. (3–48)
Figure 3-8 shows a typical charge amplifier circuit with noise sources. While decreasing Cf
increases the sensitivity, it also increases the cut-on frequency of the circuit. The cut-on
71
B
B
1
2
f
outin
f
iRf
Va
ia
p i
Figure 3-8. Simplified sensor circuitry with a charge amplifier along with noise sources.
frequency is
fcut−on =1
2πRfCf
, (3–49)
where Rf is the feedback resistor that sets the dc operating point. Lowering Cf also lowers
the bandwidth and increases the noise in charge amplifiers [73]. This can be seen from
the expression for the total noise power spectral density (PSD) at the output of a charge
amplifier, given as
Sv0chg−amp= Sva
(1 +
Ctot
Cf
)2
+
∣∣∣∣1
jωCf
∣∣∣∣2 [
Sia + SiRf
], (3–50)
where Ctot = C1 +C2 +Cp +Ci. Cp and Ci represent the parasitic and the input capacitances
in the system, respectively. SiRf= 4kT/Rf is the noise PSD of Rf represented as a current
source and Sva and Sia are the amplifier voltage and current noise PSDs, respectively. The
noise due to Rf is 1/f 2 shaped and increases significantly at low frequencies [73]. A high
value of Rf can help mitigate this effect by lowering the current noise, 4kT/Rf . This also
lowers the cut-on frequency, thereby increasing the bandwidth. The highest value of Rf that
72
may be used is limited by the availability of off-the-shelf components (1 GΩ) for a board
level circuit implementation. The limiting values of Rf for a CMOS based interface circuit
implementation can be much smaller (O(100 KΩ)). A study of the specific circuit under
consideration is needed to truly understand the noise performance tradeoffs.
Voltage Amplifiers. A voltage amplifier typically has a bias resistor, Rb, that sets
the dc operating point for the amplifier. Figure 3-9 shows a typical voltage amplifier circuit
including dominant noise sources. Any capacitance at the input of the amplifier, except
the sensor capacitance, forms a potential divider reducing the voltage at the input of the
amplifier. The expressions for sensitivity and noise for the voltage amplifier circuit are
Svamp =C1 − C2
C1 + C2 + Cp + Ci
(Vb
g0
)(δ
τw
)(3–51)
and Sv0v−amp= Sva + Sia
(RbZi
Rb + Zi
)2
+ Svb
(Zi
Rb + Zi
)2
, (3–52)
where Zi = 1/(jωCtot) and Svb = 4kTRb is the voltage noise PSD of Rb. Higher values of
parasitics in comparison to the sensor capacitance lower the sensitivity (see Section 3.2.3). A
larger sensor capacitance should therefore help mitigate the ill effects of parasitics. However,
the ratio, ∆CC0
, should not drop significantly because it results in the loss of sensitivity.
This will be illustrated mathematically while explaining the expression for sensitivity of
the proposed sensor in Section 3.2.3. The combination of the bias resistor Rb and the total
capacitance Ctot (including parasitics), form a high pass filter. The cut-on frequency, 12πRbCtot
can be lowered using a large bias resistor. However, this will increase the noise contribution
at the output. The simplicity of the voltage amplifier circuit makes it an attractive option
for capacitive sensors. The reader is referred to Kadirvel’s work [73] on interface circuits for
a detailed understanding of the tradeoffs involved in using charge and voltage amplifiers.
Martin [74] reported better results using a voltage amplifier compared to a charge ampli-
fier with off the shelf components for a dual back plate capacitive microphone with nominal
capacitances similar to the sensor developed in this work. The same voltage amplifier will
be used for the proposed sensor. Desirable traits of shear stress sensors include their ability
73
-
Rb
Cp Ci
Svb=
4kTRb
Sva
Sia
Vout
VB
-VB
C1
C2
Figure 3-9. Simplified sensor circuitry with a voltage amplifier along with noise sources.
to measure both the mean and fluctuating shear stress components. In the proposed sensor,
mean and fluctuating shear stress components would correspond to mean and fluctuating
changes in capacitance respectively. As stated previously, both charge and voltage amplifiers
possess a characteristic cut-on frequency, 1/2πRC, making them unsuitable for measuring
dc (mean) capacitance changes.
The synchronous MOD-DMOD scheme enables the measurement of static capacitance
changes or making mean measurements. This technique has been used widely in the past for
capacitive inertial sensors [69, 70]. The following section explains the MOD-DMOD scheme
for differential capacitance measurement using a voltage amplifier. The interface circuit
is explained and derivations for the output voltage of the system are provided. For the
derivations presented, each capacitor is assumed to be a parallel plate capacitor.
MOD-DMOD with Voltage Amplifier. A MOD-DMOD circuit for differential
capacitance measurement using a voltage amplifier is as shown in Figure 3-10. Each sense
capacitor is represented as a parallel combination of two capacitors which change in the
opposite sense. The capacitors in parallel correspond to the gaps, g01 and g02 that change in
a opposite sense with the sensor motion. A unity gain voltage follower is used to represent
the voltage amplifier. The non-inverting input is connected to the common electrode of the
capacitor (the floating element in this case). The other two electrodes of the capacitors
74
+
-
vc=vacsin(ct)
-vc=-vacsin(ct)
C1
C2 vout1
RbCp Ci
vc=vrefsin(ct)
vout
g02
g02
g01
g01
Multiplier/
Demodulator
low-pass
filter
Figure 3-10. Simplified differential capacitance sense circuitry using synchronous modulationand demodulation technique with a voltage amplifier.
are biased directly using sinusoidal voltage signals, +vc and −vc respectively. For similar
implementations for inertial sensors, square waves are generally used [75]. However, square
waves inherently have harmonic content which may interfere with the measurement signal.
Square waves also possess less power in the fundamental frequency compared to sine waves
at a given amplitude, lowering the sensitivity (Section 3.2.3). Cp is the parasitic capacitance
due to on board connection lines and wire bonds. Ci is the input capacitance of the amplifier.
The bias resistor Rb sets the dc operating point of the amplifier. The combination of the
total capacitance at the input of the amplifier Ctot and Rb form a high-pass filter with
a cut-on frequency, 1/2πRbCtot. The charge at the common node remains approximately
constant provided the ac bias voltage frequency is greater than the cut-on frequency, i.e.,
fcut−on > 1/2πRbCtot [66]. The ac bias voltages on the two capacitors are equal in amplitude
and frequency, but opposite in phase. Thus, in the absence of a physical excitation, i.e., no
change in capacitance, the voltage at the common node is ideally zero (C1 ≈ C2). In case
of mismatched capacitors, the nominal output voltage will have an amplitude proportional
to the mismatch at the biasing frequency, restricting the dynamic range of the sensor. An
appropriate phase adjustment circuitry may be used to null the output resulting from an
initial mismatch in nominal capacitances (see Chapter 6). A shear stress input causes a
change in capacitance resulting in a voltage change at the middle node. The amplifier
75
provides a voltage output corresponding to the capacitance change. Note, the mean wall
shear stress acting on the sensor produces a dc change in capacitance. In the absence of
fluctuating bias voltages, an output due to a dc change in capacitance is high-pass filtered.
The electromechanical coupling is not considered in the present analysis. The expressions
derived in this section are purely based on a change in capacitance due to a finite change in
gap between capacitive sensor elements i.e., free electrical impedance (refer Appendix B).
For ease of understanding, Figure 3-11 shows a simple schematic showing the expected
sensor output spectrum at every stage of the circuitry presented. The biasing results in a
modulated output at the amplifier output, where the mean change in capacitance is at the
carrier frequency and the dynamic change is observed at the sidebands of the modulated
output. There is an additional noise component from the amplifier. The modulated output
is then demodulated and low pass filtered to extract the original modulating/input signal.
There will be additional dc offset voltages and noise contribution from the demodulator
and the low pass filter, which are not shown in the figure. Especially, the net 1/f noise
component at the output, at low frequencies, needs consideration to ensure sufficient signal
resolution. The derivations that follow provide a better understanding of the benefits of the
modulation technique to measure static capacitance change and hence the mean shear stress.
The instantaneous capacitances for the sensor are
C1 = εTt
[3Lteff + Le + (N−1)
2Lf
g11
+Lteff + (N−1)
2Lf
g12
](3–53)
and C2 = εTt
[3Lteff + Le + (N−1)
2Lf
g21
+Lteff + (N−1)
2Lf
g22
], , (3–54)
where g11 and g12 are instantaneous gaps for C1 and g21 and g22 are for C2. The instantaneous
charge on the capacitors C1 and C2 are
q1 = vcC1 (3–55)
and q2 = −vcC2. (3–56)
76
dc change
ac change ()
frequency
PS
D
Amplifier Noise PSD
frequency
~ dc change x
carrier amplitudeamplifier
PSD
frequency dc change multiplier/
demodulator
low-pass filter
PS
D
frequency dc changeFigure 3-11. Schematic indicating spectra of the sensor output at each stage of the sensor
circuitry.
where, ±vc = ±Vac sin (ωct) are the sinusoidal bias voltages that act as the carrier signals
for the modulation process.
The bias voltage frequency is chosen to be higher than the bandwidth of the sensor while
ensuring that no charge dissipates through Rb. Consequently, the bias resistor is essentially
treated as an open circuit for the voltage analysis. The combination of sense and parasitic
capacitances form a potential divider, with bias voltage sources connected to one electrode
of each of the sense capacitors. The analysis assumes one sense capacitor is biased at a time,
followed by superposition to obtain the net output voltage.
77
Consider the case when capacitor C1 is biased. The total charge, Qin1 , is the same as
the charge across the sense capacitance, C1. Therefore,
Qin1 = Q1 (3–57)
and Ctot1vc = C1(vc − vinC1), (3–58)
where Ctot1 = C1(C2+Cp+C1)
C1+C2+Cp+Ci. Substituting for Ctot1 and rearranging the equation results in,
vinC1= vc
C1
C1 + C2 + Cp + Ci
. (3–59)
Similarly when the capacitor C2 is considered,
Qin2 = Q2 (3–60)
and Ctot2(−vc) = C2
(−vc − vinC2
), (3–61)
where Ctot2 = C2(C1+Cp+C1)
C1+C2+Cp+Ci. Substituting for Ctot2 and rearranging the equation results in
vinC2= −vc
C2
C1 + C2 + Cp + Ci
. (3–62)
The total input voltage is due to the bias voltages applied to both C1 and C2, obtained by
adding Equations 3–59 and 3–62. The result is,
vin = vinC1+ vinC1
and vin = vcC1 − C2
C1 + C2 + Cp + Ci
. (3–63)
The output voltage of the amplifier is
vout1 = Gvin,
and vout1 = GvcC1 − C2
C1 + C2 + Cp + Ci
, (3–64)
where G is the closed loop gain of the amplifier. The voltage follower shown in Figure 3-10
has a unity gain; i.e., G = 1. The equations however, are derived for an amplifier with
arbitrary gain. Equations 3–63 and 3–64 verify that the input, and hence the output of the
78
amplifier, is proportional to the differential capacitance. Substituting for the capacitances
from Equations 3–53 and 3–54, Equation 3–64 can be simplified and rewritten as,
vout1 = Gvc
(3Lteff + Le + (N−1)
2Lf
) [g21−g11
g11g21
]+
(Lteff + (N−1)
2Lf
) [g22−g12
g12g22
](3Lteff + Le + (N−1)
2Lf
) [g21+g11
g11g21
]+
(Lteff + (N−1)
2Lf
) [g22+g12
g12g22
]+ Cp+Ci
εTt
.
(3–65)
By design, the nominal capacitances C10 and C20 are assumed to be equal as shown in
Equation 3–45. Thus, the changes in capacitances are also equal in magnitude but opposite
in direction. Consider a floating element motion such that C1 increases and C2 decreases,
resulting in
g11 = g01 − δ, g12 = g02 + δ, g21 = g01 + δ, and g22 = g02 − δ. (3–66)
Note here the deflection δ is only used to analyze the change in capacitance. The effect of
electromechanical coupling is not included and will be accounted for later (see Section 3.2.3).
Substituting Equations 3–66 and 3–45 into Equation 3–65 gives
vout1 = Gvc
1−
(Lteff +
(N − 1)
2Lf
)
(3Lteff + Le + (N−1)
2Lf
)[g201
g202
]
1 +
(Lteff + (N−1)
2Lf
)(3Lteff + Le + (N−1)
2Lf
)[g01
g02
]
(2C0
2C0 + Cp + Ci
)δ
g01
. (3–67)
The term,
Hgap =
1−
(Lteff + (N−1)
2Lf
)(3Lteff + Le + (N−1)
2Lf
)[g201
g202
]
1 +
(Lteff + (N−1)
2Lf
)(3Lteff + Le + (N−1)
2Lf
)[g01
g02
] , (3–68)
in Equation 3–67 represents an attenuation term due to the presence of the secondary gap
g02 which partially cancels out the capacitance change due to the primary gap g01. This is
79
represented graphically using two capacitors for C1 and C2 in Figure 3-6 and Figure 3-10.
As g01/g02 → 0, Hgap → 1 i.e., no attenuation. On the contrary, as g01/g02 → 1, Hgap → 0
i.e., 100% attenuation. If g01/g02 > 0, the differential scheme still holds, but the two gaps
switch functionalities, where g01 acts as the secondary gap and g02 acts as the primary gap.
The term,
Hc1 =2C0
2C0 + Cp + Ci
, (3–69)
in Equation 3–67 represents the attenuation due to parasitics [63]. However, the effect of
electromechanical transduction is not accounted yet and will be incorporated in the next
section. The effects of attenuation and the design considerations that are required to mini-
mize them are also discussed later. Substituting Equations 3–68 and 3–69 in Equation 3–67,
and setting bias voltage, vc = Vac sin (ωct), gives
vout1 = GHgapHc1
(δ
g01
)Vac sin (ωct) . (3–70)
Equation 3–70 represents the sensor output voltage amplitude that is modulated by the
carrier signal. The modulated output signal is subsequently demodulated using a multiplier
(demodulator) to retrieve the original signal.
During the demodulation process, the output from the voltage amplifier is multiplied
with a reference signal having the same frequency, ωc, as the sinusoidal bias voltages. As-
suming a reference voltage, vref = Vref sin (ωct), the output of the multiplier is
vout =1
U(vout1vref ) , (3–71)
where U is the scaling factor of the multiplier with units in volts (V). Combining Equations 3–
70 and 3–71 and substituting for vref results in
vout = HgapHc1
(δ(t)
g01
)(G
U
)VacVref sin2 (ωct)
or vout = HgapHc1
(δ(t)
g01
)(G
2U
)VacVref (1− cos (2ωct)) . (3–72)
80
In a turbulent flow, the shear stress exerted by the flow has two components; shear stress due
to the mean flow and the fluctuating shear stress due to turbulent structures. A sinusoidal
fluctuation is assumed as a simple case for analysis purposes. The resulting deflection is
written as
δ(t) = δm + δt sin (ωt + φ) , (3–73)
where δm is the deflection due to the mean flow, and δt sin (ωt + φ) is the deflection due to
turbulent flow. Combining Equation 3–73 for deflection with Equation 3–72 results in
vout = VacVrefHgapHc1
(G
2U
)(1
g01
)(δm + δt sin (ωt + φ)) (1− cos (2ωct)) (3–74)
or vout = VacVrefHgapHc1
(G
2U
)(1
g01
)
δm + δt sin(ωt + φ)
−δm cos(2ωct)
−δt sin(ωt + φ) cos(2ωct)
. (3–75)
Equation 3–75 gives the net demodulated voltage output from the multiplier. This
signal has several frequency components, a dc component, δm, a low frequency component
from the flow, δt sin(ωt + φ), and high frequency components due to the carrier signal,
δm cos(2ωct) − δt sin(ωt + φ) cos(2ωct). The components of interest are the dc and the low
frequency signals, corresponding to the mean and fluctuating parts of the flow, respectively.
The higher frequency components are therefore low-pass filtered after the demodulation
process. The low-pass filtered output voltage is then given as
vout = VacVrefHgapHc1
(G
2U
)(1
g01
)(δm + δt sin(ωt + φ)) . (3–76)
In this section, the expression for output voltage using the proposed interface circuitry
using capacitors C1 and C2 were presented. Voltage output was derived purely based on
capacitance change resulting from electrode motion, disregarding the effect of electrome-
chanical coupling. An attenuation factor, Hgap due to the secondary gap g02 was identified.
So far, a mechanical model and an electrical model have been developed for the proposed
shear stress sensor. In the next section, simplified passive elements (lumped elements) for
81
the sensor are obtained from the mechanical model. The mechanical and electrical domain
are then related using an electromechanical transduction scheme.
3.2 Dynamic Modeling
The dynamic model of the sensor is developed using the principle of lumped element
modeling. A brief description of the concept of lumped element modeling is provided here
before delving into the particular details for the sensor. In a lumped element model (LEM),
the distributed properties of the system such as mass, compliance and dissipation are lumped
about a single point of interest. This simplifies the mathematical modeling of the sensor
structure, which is necessary for predicting the sensor response to a given input. However,
the validity of LEM requires that the wavelength of physical input signal be much larger
than the length scale of the device. This allows for the decoupling of spatial and temporal
variations in the governing system equations and application of the quasi-static solution to
solve for the dynamic system response [41].
Lumping about a single point allows distributed system representation using discrete cir-
cuit/mechanical elements (inductor/intertance, capacitor/compliance and resistor/damper).
There are three kinds of passive circuit elements: the generalized resistor, the generalized
inductor and the generalized capacitor. The generalized resistor represents the dissipation,
the generalized capacitor represents the storage of potential energy, and the generalized in-
ductor represents the storage of kinetic energy in the system for the respective domains. In
the mechanical domain the deflected spring stores the potential energy, the moving mass
possesses kinetic energy, and the damper acts as the dissipator.
To track power transfer during electromechanical transduction, the conjugate power
variables of effort and flow are generally used. The product of the conjugate power variables
in each energy domain results in power. For example, in the mechanical domain, force is
the effort variable and velocity is the flow variable. The dynamic mechanical system equa-
tions are later studied using equivalent Kirchoff’s laws for velocity (analogous to current),
and shear force (analogous to voltage). An equivalent circuit is formed based on common
82
flow or common effort shared by individual lumped elements. The analysis of the circuit
yields information such as bandwidth and sensitivity of the sensor. A noise analysis of the
equivalent circuit helps to estimate the minimum detectable signal (MDS).
3.2.1 Lumped Parameters
In this section, the lumped mechanical elements for the floating element structure are
presented. Since, the transduction is effected by the mechanical shear force acting on the rec-
tilinearly translating floating element structure, it is intuitive to use the mechanical lumped
elements to model the sensor. First, the lumped mass, Mme is estimated followed by the
lumped compliance, Cme. The derivations for Mme and Cme are presented in Appendix A.
When a deflection occurs due to shear force on the sensor, the tethers act as the springs
(compliance), storing potential energy. The distributed mass of the tethers and the rigid
mass of the floating element with comb fingers, lumped to the center, possess kinetic energy.
Since, majority of the transduction occurs at the floating element (x = Lt), the center de-
flection, δ = w(Lt), seems to be the natural choice for lumped parameter estimation. There
are two different sources of dissipation: elastic damping due to internal friction and fluidic
damping due to viscous effects under the suspended floating element structure [42].
A linear system is required to ensure spectral fidelity of the sensor. The energy and the
co-energy of a linear system are equal [41]. Therefore, the linear mechanical (Equation 3–6)
is used to compute the lumped mass and compliance of the floating element structure instead
of the nonlinear model.
3.2.1.1 Lumped Mass
To compute the lumped mass, Mme, the total kinetic co-energy due to the motion of
the beam is equated to the equivalent kinetic co-energy of the lumped system. For a linear
system,
W ∗KE =
∫dW ∗
KE =
∫ f0
0
pdf =1
2Mmef0
2, (3–77)
where p represents momentum and f0 represents flow. The total lumped mass is the sum
of the effective mass of the tethers lumped about the center, and the mass of the floating
83
element. As shown in Appendix A, the lumped mechanical mass of the sensor is
Mme = ρTtLeWe
1 + 19235
(WtLt
LeWe
)+ 3
(NWf Lf
LeWe
)+ 384
35
(WtLt
LeWe
)(NWf Lf
LeWe
)
+1072105
(WtLt
LeWe
)2
+ 3(
NWf Lf
LeWe
)2
+ 1072105
(NWf Lf
LeWe
)(WtLt
LeWe
)2
+3(
NWf Lf
LeWe
)2 (WtLt
LeWe
)+ 2048
315
(WtLt
LeWe
)3
+(
NWf Lf
LeWe
)3
(1 +
NWf Lf
LeWe+ 2 WtLt
LeWe
)2 . (3–78)
3.2.1.2 Lumped Compliance
The total potential energy is equated to the equivalent potential energy of the lumped
system to estimate the lumped compliance. For a linear system,
W ∗PE =
∫dW ∗
PE =
∫ q0
0
edq =1
2
δ2
Cme
, (3–79)
where e and q are effort and displacement respectively. As shown in Appendix A, the
compliance of the tethers is given as
Cme =1
4ETt
(Lt
Wt
)3
(1 +
NWf Lf
WeLe+ 2
(WtLt
WeLe
))2
1 + 2(
NWf Lf
WeLe
)+ 4
(WtLt
WeLe
)+ 4
(NWf Lf WtLt
(WeLe)2
)
+6415
(WtLt
WeLe
)2
+(
NWf Lf
WeLe
)2
. (3–80)
The packaging and the flow around the sensor decide the damping in the system. An
accurate model for the damping is not available at the moment. The damping in the system
is therefore neglected for modeling purposes and will be experimentally deduced.
3.2.2 Electromechanical Transduction
This section presents the approach used to model energy transduction from the me-
chanical domain to the electrical domain. This energy transfer is modeled using idealized
two-port elements which permit energy transduction sans losses. A general approach for
modeling linear conservative transducers and an example of a capacitive transducer is pre-
sented in Appendix B. The two port model, transduction factor, and the electromechanical
84
coupling factor for the shear stress sensor are provided. First, the transduction process us-
ing a single capacitor transducer is explained followed by one for the differential capacitance
scheme.
An impedance analogy is used for modeling the transduction process. The approach
enables transduction of force to voltage (both effort variables). For the capacitive shear
stress sensor, a transformer is used to model the transduction of the mechanical energy into
electrical energy. The electromechanical transduction factor, ϕ′, also known as the turns
ratio in a transformer, accounts for the coupling between two different energy domains.
The actual change in capacitance due to the effective shear force, fe, that is transduced, is
included via the fictitious transduction voltage, v0, to maintain a constant charge. Using
the equations developed in Appendix B, the sensor two-port model using a single sensor
capacitance, C10, is expressed as,
Vout
Fshear
=
1
jωC10
− Vb
jωgeff
− Vb
jωgeff
1
jωCme
I
U
, (3–81)
where Vb is the bias voltage (ac or dc) across the electrodes. In Equation 3–81, CMO and
CEB in Equation B–30 have been replaced with the Cme and C10, respectively. Therefore,
CMO = Cme (open mechanical compliance) (3–82)
and CEB = C10 = C20. (blocked electrical capacitance) (3–83)
Similarly, the electromechanical transduction factor is
ϕ′ =Vb
geff
Cme, (3–84)
where
geff =g01
Hgap
. (3–85)
85
me fe o
10inC1
p i
φ’
Figure 3-12. Schematic of capacitive transduction scheme using a single sense capacitance.
Equation 3–85 represents the effective gap, which accounts for the loss in sensitivity due
to gap attenuation discussed earlier using Equation 3–68. The electromechanical coupling
factor is
κ2 =V 2
b
g2eff
C10Cme. (3–86)
For illustration purposes, Figure 3-12 shows the transduction mechanism for a capacitive
transducer in general, and the proposed shear stress sensor in particular with a single ended
capacitance (C10). The free electrical capacitance C ′10 is defined as
C ′10 =
C10
(1− κ2). (3–87)
In Equation 3–87, the effect of the electromechanical coupling has been incorporated. The
voltage, v0 is related to the effective shear force, fe, acting on the floating element sensor
through the turns ratio as follows,
v0 = ϕ′fe. (3–88)
Substituting Equation 3–84 in the expression above, the voltage, vin at the input of the
interface circuitry is
vin =C ′
10
C ′10 + Cp + Ci
v0. (3–89)
86
me fe
φ’
φ’
in1
in2
10
20
in
fe
fe
Figure 3-13. Schematic of capacitive transduction scheme using differential capacitive sens-ing technique.
Substituting Equations 3–88 and 3–84 into the equation above gives
vin =
(C ′
10
C ′10 + Cp + Ci
)Vb
geff
Cmefe. (3–90)
Equation 3–90 represents the voltage available at the input of the interface circuitry i.e.,
amplifier (ideal amplifier).
So far, a single sense capacitance was used to illustrate the transduction process and an
expression for the amplifier input voltage was obtained. Next, the amplifier input voltage
using both the sense capacitors for the differential capacitance sensing scheme is described.
Figure 3-13 shows the differential transduction scheme for the floating element sensor.
The same principle used in deriving Equation 3–90 is used to obtain the net output
voltage for the differential capacitance scheme. One sense capacitor is considered at a time
while the other acts as parasitic. Subsequently, superposition is used to obtain the total
87
amplifier input voltage. Using Equation 3–90, the individual amplifier input voltages are
vinC1=
(C ′
10
C ′10 + C ′
10 + Cp + Ci
)Vb
geff
Cmefe (3–91)
and vinC2=
(C ′
20
C ′10 + C ′
10 + Cp + Ci
)Vb
geff
Cmefe. (3–92)
The net voltage available to the interface circuit is given by the sum of Equations 3–91 and
3–92 as
vin =
(C ′
10 + C ′20
C ′10 + C ′
10 + Cp + Ci
)Vb
geff
Cmefe. (3–93)
Substituting for the electromechanical transduction factor of Equation 3–87 and the effective
gap of Equation 3–85 into Equation 3–93, gives
vin =
(C10 + C20
C10 + C10 + (Cp + Ci)(1− κ2)
)Hgap
Vb
g01
Cmefe. (3–94)
The first term in Equation 3–93 includes the effect of electromechanical coupling in the
parasitic attenuation term (see Equation 3–69). Recognizing that the nominal capacitances
are the same from Equation 3–45 the new parasitic attenuation term is
Hc =
(2C0
2C0 + (Cp + Ci)(1− κ2)
). (3–95)
Thus, expression for the voltage at the interface circuit input is
vin = HgapHcVb
g01
Cmefe. (3–96)
A typical electromechanical coupling factor for the sensor structure is about 3.6%. As cou-
pling gets weaker the attenuation Hc ≈ Hc1 in Equation 3–69 i.e., the free electrical capaci-
tance (C ′10) approaches the blocked electrical capacitance (C10). In the next section, an
equivalent circuit is provided using lumped elements from Section 3.2.1 and the transduc-
tion scheme developed in this section. The derived transduced voltage is compared with the
expression in Equation 3–67 to further explain the effect of electromechanical transduction
via the electromechanical coupling factor, κ2 (see Equation 3–86).
88
fshear
e
me
me
me fe
φ’
φ’
in1
in2
10
20
in
fe
fe
To interface
circuit
Figure 3-14. Schematic of equivalent circuit for the differential capacitive shear stress sensor.
3.2.3 Equivalent Sensor Circuit
The electromechanical transduction model derived in Section 3.2.2 links the mechanical
and the electrical system of the sensor. In this section, the complete equivalent circuit of the
sensor is presented as shown in Figure 3-14. The resonant frequency (bandwidth) and the
static sensitivity of the sensor is estimated from the frequency response of this equivalent
circuit.
As discussed in Section 3.1.3.3, the output voltage is buffered using a high impedance
voltage follower. Ideally, the infinite input impedance of the voltage follower allows to assume
an open circuit condition after the transduction. Furthermore, weak electromechanical cou-
pling (κ2 ¿ 1) strengthens this assumption. Thus, loading of the system in Figure 3-14 due
to the interface circuitry and parasitics, is neglected. The frequency response is therefore de-
termined using the open circuit configuration and is dominated by the mechanical response
of the system. Although, the loading due to parasitics is neglected for the transduction,
subsequent parasitic attenuation (Hc) of the transduced signal is important to determine
the overall sensor sensitivity. It is noteworthy that the open circuit assumption leads to a
89
stiffer/less sensitive device, resulting in a conservative estimate of the sensor performance.
However, this also slightly overpredicts the bandwidth estimated from the resonance, which
is usually between the open circuit and short circuit resonance frequencies. The mechanical
system in Figure 3-14 is a simple second order system (spring mass damper or LCR) and is
governed by the differential equation,
fshear(t) = md2δ
dt2+ R
dδ
dt+ kδ, (3–97)
where fshear represents the shear force, Mme represents the lumped mass, k represents the
effective stiffness of the tethers, and R represents the effective damping in the system. The
stiffness, k is
k =1
Cme
. (3–98)
The differential equation in 3–97 is solved using Fourier transform, resulting in the following
frequency response function:
H(ω) =δ(jω)
fshear(jω)=
Cme
MmeCme(jω)2 + RCme(jω) + 1. (3–99)
The undamped natural frequency (open circuit resonant frequency), of the system is
fr =1
2π
√1
CmeMme
. (3–100)
For modeling purposes, the damping is assumed to be negligibly small. In that case, the
damped natural frequency is approximately equal to the undamped natural frequency of the
system. The validity of this assumption will be verified during experimental characterization
of the sensor.
The accuracy of the resonant frequency prediction is verified via finite element analy-
sis (FEA) in COMSOL using modal analysis. The simulation involves the moving floating
element (1 µm × 1000 µm × 45µm), tethers (1000µm × 15µm × 45µm), and comb fingers
(170µm× 4µm× 45µm) with a clamped boundary condition at the free end of the tethers.
90
The structure has a total of 102 comb fingers on either side of the floating element. An eigen
frequency analysis is performed with Lagrange quadratic elements, allowing large deforma-
tions. For the first mode or the shear/in-plane mode, the LEM predicts a resonant frequency
of 5.23 kHz compared to 5.18 kHz from FEA, which are in agreement to within 1.2%. The
first six mode shapes for the simulated sensor geometry and the corresponding frequencies
are shown in Figure 3-15.
(a) Mode 1 (fr = 5.177 kHz)
(b) Mode 2 (fr = 14.39 kHz)
(c) Mode 3 (fr = 25.08 kHz)
(d) Mode 4 (fr = 46.91 kHz)
(e) Mode 5 (fr = 140.9 kHz)
(f) Mode 6 (fr = 141.0 kHz)
Figure 3-15. Resonant modes of floating element structure.
Next, the expression for the output voltage and static sensitivity of the sensor is derived.
Consider the effective force, fe used in the the transduction process. This force is same as
the last term in Equation 3–97. This is also evident from a comparison of Equation 3–25,
91
3–99, and the transfer function or gain factor, G(ω), between fe and fshear,
G(ω) =fe
fshear
=1
MmeCme(jω)2 + RCme(jω) + 1. (3–101)
Therefore,
fe = kδ =δ
Cme
, (3–102)
Thus, the input voltage in Equation 3–96 given to the interface circuitry is obtained by
substituting Equation 3–102 to give
vin = HgapHcδ
g01
Vb. (3–103)
Considering a general case, a voltage amplifier with a gain G, an input vin, and a sinusoidal
bias voltage, Vac sin(ωct) is considered for the interface circuitry. The output voltage of this
amplifier is
vout1 = Gvin = GHgapHc
(δ
g01
)Vac sin(ωct). (3–104)
The modulated sensor output was previously presented in Equation 3–70, which did
not account for the electromechanical coupling between electrical and mechanical domains.
Comparison with Equation 3–70 shows that all terms are the same except the term Hc in
Equation 3–104, which accounts for the electromechanical coupling in the system. Similarly,
the effect of electromechanical coupling is incorporated in the demodulated output signal in
Equation 3–76, rewritten with Hc instead of Hc1 as
vout = VacVrefHgapHc
(G
2U
)(1
g01
)(δm + δt sin(ωt + φ)) . (3–105)
A steady laminar flow results only in a mean wall shear stress with no fluctuations (ω = 0).
A steady laminar flow is therefore used to estimate the static sensitivity of the sensor to wall
92
shear stress. The static sensitivity is defined as,
Sτ, static =Output voltage
(vout
∣∣ω=0
)
Mean wall shear stress (τw). (3–106)
Using the above definition, the expression for the overall static sensitivity, including atten-
uation, is
Sτ, overall = VacVref︸ ︷︷ ︸1
HgapHc︸ ︷︷ ︸2
(G
2U
)
︸ ︷︷ ︸3
(δm/τw
g01
)
︸ ︷︷ ︸4
. (3–107)
The static sensitivity of the sensor alone, neglecting the attenuation terms, is
Sτ, sensor = Vac
(δm/τw
g01
). (3–108)
It is instructive to study the effect of each term in Equation 3–107 on the sensitivity. Pro-
ceeding term by term, term 1 indicates that the static sensitivity is directly proportional to
the amplitude product of the bias voltage (Vac) and the reference voltage (Vref ) supplied to
the multiplier. The upper limit of Vac is determined from the pull-in limit or the dielectric
breakdown limit of air, with a built in factor of safety. Usually the pull-in constraint domi-
nates (see Chapter 4). Vref is limited by the maximum input voltage that can be applied to
the multiplier. While the sensitivity may be improved by increasing Vac and Vref , the source
noise contribution may also increase and should be accounted for while choosing the values
of Vac and Vref .
Term 2 captures two attenuating effects on the sensitivity: the effect of the secondary
gap g02 via Hgap and the sensitivity scaling in relation to the nominal sensor capacitance, C0,
and parasitic capacitances, Cp + Ci via Hc. Equation 3–68 indicates that Hgap < 1, lowering
the sensitivity. Hgap increases as g02 increases in comparison to g01. It is evident that the
condition g01 ¿ g02 needs to be satisfied to minimize attenuation from this term. Higher g02,
however, lowers the number of comb fingers and thus the total capacitance, which is needed
to overcome parasitics. Thus, there is a trade-off between the number of comb fingers and
g02.
93
The condition Hc ¿ 1 or ((2C0)/(2C0 + Cp + Ci)) ¿ 1 or 2C0 À Cp + Ci is desirable
to make the sensitivity independent of the adverse effects of parasitics. However, it becomes
increasingly difficult to satisfy this condition as the sensor size is scaled down. Guarding,
a shielding technique in which the terminals of the parasitic capacitance are tied to the
common-mode potential of the sensor system, is often used to minimize the attenuation due
to parasitics [42, 68]. However, the finite input capacitance of the amplifier still remains,
which cannot be guarded. Using an amplifier with very low or negligible input capacitance
may therefore improve sensitivity. Term 2 is always less than unity if the conditions g01 ¿ g02
and ((2C0)/(2C0 + Cp + Ci)) ¿ 1 are not satisfied. A good estimate of parasitic capacitances
and an appropriate choice of g02 is essential during the design process to ensure this term is
as close to unity as possible i.e., minimum attenuation.
Term 3 in Equation 3–107 is the ratio of the closed loop gain, G, of the amplifier to
twice the scaling factor, U , of the multiplier. Thus, to increase sensitivity, a high gain
amplifying stage and a multiplier with a low scaling factor are favorable. Furthermore, a
high gain amplifying stage helps to boost the signal to noise ratio (SNR), minimizing the
effect of noise added in the subsequent circuitry. The value of gain G is again limited by the
maximum allowable voltage input to the multiplier.
Term 4 in Equation 3–107 represents the contribution from the actual sensitivity of the
sensor via the transduction itself (see Equation 3–108). The numerator (δm/τw) represents
the mechanical sensitivity of the sensor. The ratio δm/g01 represents a simplified form of
the change in capacitance to the original capacitance or the change in gap to the original
gap. The combined ratio thus represents the simplest form of sensitivity, which is percentage
change in capacitance per unit shear stress. Ideally, this value should be as large as possible
for the sensor design; however, bandwidth and linearity requirements constrain the upper
limit of this ratio.
In summary, the dynamic modeling section discussed lumped element modeling and
this concept was used to compute lumped elements from the distributed mechanical system.
94
Electromechanical transduction for the sensor system was mathematically illustrated. An
equivalent circuit representation of the sensor was provided. Expressions for the frequency
response and resonant frequency were given. Expressions for the demodulated sensor output
voltage and static sensitivity were given. Lastly, the relevance and contribution of different
terms in the sensitivity expression were explained.
3.3 Higher Order Effects and Noise
In this section, some of the foreseeable higher order effects like fringing fields and par-
asitic capacitances are discussed. The fringing fields are edge effects that add additional
parasitic capacitance, increasing attenuation. A reasonable model for fringing fields aids in
better design and prediction of sensor performance. A model to estimate the noise from the
interface circuitry is also presented. This noise model is used to estimate the minimum de-
tectable signal of the sensor, which is used as the objective function for design optimization
in Chapter 4.
3.3.1 Fringing Fields
The capacitance equations discussed up to this point assume an ideal parallel plate
capacitor. In an ideal parallel plate capacitor, the electric field lines are normal to the parallel
planes of the capacitor. However, due to the finite dimensions of the plate, some curved
electric field lines exist at their edges. These electric fields are referred to as fringing electric
fields. These fringing electric fields manifest themselves as a fringing field capacitance,
Cfr. Any capacitance which is unaccounted for when using the parallel plate assumption
is treated as fringing field capacitance. It is noteworthy that any additional capacitance
that does not change with the physical input to the sensor is parasitic in nature. Thus, the
fringing field capacitance also adds to the parasitic capacitance of the system. The fringing
field capacitance is assumed to be invariant with the change in nominal capacitance for small
input signals i.e., small change in gaps. This is a reasonable assumption because a small
change in gap does not change the fringing field capacitances. This is also verified via FEA
simulations, which are performed for both an initial gap size and that of a typical full scale
95
deflection of the sensor. Figure 3-16 shows the electric field lines in the gaps g01 and g02 for
a single comb finger simulated in 3D, using FEA in COMSOL.
Lf
g02
Wf
g01
g02
Insulating
Boundary
1 V
0 V (ground)
Figure 3-16. Asymmetric comb finger structure to estimate effect of fringing electric fields.
The comb fingers are simulated because their contribution dominates the sensor nominal
capacitance in most designs (Table 4.4) and due to their asymmetric gaps. Comb fingers
made of perfect electrical conductors are assumed for this simulation and the dielectric
medium is air (εr = 1). The various geometric dimensions for this analysis are, g01 = 3.5 µm,
g02 = 20 µm, Wf = 4 µm, and Lf = 170 µm. The color gradient of the field lines indicates the
variation of electric potential in the gaps. Typically for the comb finger, the sensor fringing
field capacitance is ≈ 10% of the nominal capacitance i.e.,Cf r
C0≈ 0.1. The ratio of change in
fringing field capacitance to the total change in capacitance is ≈ 13% i.e.,∆Cf r
∆C0≈ 0.13 and
hence neglected. Such an approximation only leads to a conservative estimate of the sensor
performance while maintaining modeling simplicity.
96
3.3.2 Parasitic Capacitance
As stated earlier in Section 3.1.3.3, parasitic capacitance has a detrimental effect on
the sensitivity of the sensor. An accurate estimate of the parasitic capacitances is thus
essential to accurately predict device performance characteristics. Theoretical models are
not presently available to obtain such an estimate.
Some sources of parasitics in a typical MEMS sensor are polysilicon/metal lines, wire
bonds, PCB traces and BNC cables. The contribution of these sources may vary from
sensor to sensor, based on length of metal lines, quality of wire bonds etc. Martin [72]
compares sensitivities of a dual backplate microphone, using both a charge amplifier and
a voltage amplifier. The sensitivity measured using a charge amplifier is independent of
parasitic capacitances, but is influenced by parasitics when measured with a voltage amplifier.
Thus, a comparison of the sensitivities measured using the two biasing schemes helps in
isolating the associated parasitic capacitance of the sensor. Similar to Martin’s work, the
sensor developed in this dissertation also uses the SiSonicTM microphone amplifier , courtesy
Knowles Acoustics [76]. It uses the same differential capacitance measurement scheme as
used in Martin’s sensor. Martin’s work shows experimental value of parasitic capacitances
vary from 0.92 pF to 2.23 pF from tests conducted on seven different microphones. The
proposed sensor has similar interface circuitry/sensor packaging and the same amplifier as
Martin’s microphone. The higher value of parasitic capacitance, 2.23 pF , should therefore
provide a conservative estimate of parasitics for design purposes.
3.3.3 Noise Model
In this section the noise model of the sensor is described. The ratio of the noise floor
[V ] to the sensitivity [V/Pa] of the sensor determines the minimum detectable shear stress
(MDS). The MDS is also the lower end of the dynamic range of the sensor and is discussed
in greater detail in Chapter 4. For a given bandwidth, the MDS is mathematically defined
97
as
MDS =Noise[V ]
Sensitivity[V/Pa]. (3–109)
As a first order estimate, the noise from the mechanical and fluidic system is considered
negligible compared to the noise contribution from the interface circuitry. Thus, a noise
model for the interface circuitry is developed to obtain the noise floor for the sensor. A
modulation-demodulation scheme is used for the sensor interface circuitry as previously
shown in Figure 3-10. A voltage amplifier amplifies the modulated signal, the output of which
is supplied to the input of the multiplier (demodulator). If an amplifier with a sufficient closed
loop gain is used, the noise contribution from the later stages may be neglected provided
they have a comparable or lower noise floor. Therefore, in the noise model the input referred
noise of the amplifier is analyzed without the contribution of the subsequent stages.
-
Rb
Cp Ci
Svb=
4kTRb
Sva
SiaVout
C1
C2
Figure 3-17. Simplified schematic of sensor circuitry with noise sources.
In Figure 3-17 three noise sources are considered: thermal noise of the bias resistor Rb,
the input referred voltage noise Sva, and the current noise Sia of the amplifier. The noise
sources are assumed to be uncorrelated, allowing superposition to compute the total noise
at the output. The noise sources are presented in terms of their spectral density and must
be integrated over the noise bandwidth to obtain the total rms noise voltage at the amplifier
output. A single noise source is considered at a time and the corresponding output noise is
98
determined. The total impedance at the amplifier input is
Zi =1
jω(C1 + C2 + Cp + Ci)=
1
jωCT
, (3–110)
where CT = C1 + C2 + Cp + Ci. (3–111)
First, consider the noise due to the source Sva. No current flows into the impedance Zi and
all the voltage is amplified and appears at the output. Thus,
S1 = G2Sva. (3–112)
Now, consider output noise due to the current source Sia. The impedance for this current
source is (Rb||Zi). The noise contribution to the output is therefore,
S2 = G2Sia
(RbZi
Rb + Zi
)2
. (3–113)
Finally, the noise contribution due to the thermal noise source Svb is considered. The noise
contribution from the resistor is
S3 = G2Svb
(Zi
Rb + Zi
)2
. (3–114)
The total input referred noise in the circuit is obtained by summing the output noise con-
tribution from each individual source and dividing it by the square of the closed loop gain.
The total input noise is given as,
Snoise,in =S1 + S2 + S3
G2,
and =1
G2
[Sva + Sia
(RbZi
Rb + Zi
)2
+ Svb
(Zi
Rb + Zi
)2]
. (3–115)
Combining Equation 3–107 and Equation 3–115, the MDS for the sensor is
MDS =Snoise,in
Sτ,overall
. (3–116)
99
The thermal noise of the resistor is usually much smaller than the input referred noise
of the amplifier. The amplifier noise usually dominates the noise floor. Equation 3–115
and Equations 3–111 show that the contribution of the current noise increases as the total
capacitance, CT decreases due to small device capacitance. Hence it is desirable to have
minimum contribution from this term for microscale electrostatic sensors. Moreover, no
theoretical/empirical models exist for the accurate estimation of parasitics that contribute
to CT . Thus to have a predictable noise estimate, it is desirable to use an amplifier designed
for a very low current noise.
100
CHAPTER 4DESIGN OPTIMIZATION
Optimization is a widely used design tool both in industry and academia. It is a method-
ical approach to making design choices that lead to the best possible outcome. Optimization
helps to explore and exploit the performance of a given sensor structure as a function of
geometry, materials, etc. It also gives physical insight into the design problem itself and
helps understanding of design tradeoffs.
This chapter explains the methodology used for the optimized design of the capaci-
tive shear stress sensor described in the previous chapter. An appropriate cost function
representing sensor performance is identified and optimized. Constraint functions are also
formulated based on performance requirements of the sensor and limitations of the fabrica-
tion process. The expressions and relations developed in Chapter 3 are used to formulate
mathematical functions and constraints of the resulting nonlinear constrained optimization
problem. Sequential quadratic programming (SQP), which is one of the several available
optimization techniques, is used for optimization. The constrained optimization problem
is implemented using the optimization toolbox in MATLAB. Specifically, the SQP func-
tion fmincon in MATLAB uses a gradient descent based method to optimize nonlinearly
constrained problems.
Section 4.1 gives a brief overview of the optimization technique used, its benefits, and
its shortcomings. Section 4.2 explains the design requirements, identifies the design vari-
ables, objective function, and the constraints. Section 4.4 provides optimization results and
sensitivity analysis.
4.1 Sequential Quadratic Programming-SQP
Sequential quadratic programming uses a direction seeking algorithm to arrive at the
minimum/maximum of a given function. SQP implemented in the current optimization
problem uses a function known as the Lagrangian formed using both the objective and
101
constraint functions [77]. The Lagrangian is represented as,
L(x, λ) = f −n∑
i=1
λiGi. (4–1)
where f is the objective function, Gi represents the ith inequality constraint, gi, transformed
into an equality constraint, and λi is a parameter called the Lagrangian multiplier. The
Lagrangian in Equation 4–1 accounts for the constraints in a given optimization problem
and λi represents the relation between the objective and constraint functions when the
constraints are active. For instance, if no constraints are present or are inactive (λi = 0),
the Lagrangian simply reduces to the objective function. Optimality is ensured by solving
for λi such that the Kuhn-Tucker conditions are satisfied [77]. The Kuhn-Tucker conditions
are necessary conditions for optimality and are stated as
∇L (x, t, λ) = ∇f −n∑
i=1
λi · ∇Gi = 0,
λi ·Gi = 0 for i = 1, · · · , n
λi ≥ 0 for i = 1, · · · , n.
(4–2)
The first condition in Equation 4–2 shows that the optimization algorithm relies on the
reducing gradients of both the objective and constraint functions and solves for λi such
that ∇L = 0. Consequently, based on the initial values of design variables, the solver may
arrive at a local minimum instead of the global minimum of the objective in the design
space. Repeating the optimization for several different initial conditions provides greater
possibility of arriving at the global minimum [77]. The second and third conditions are for
the constraints, which indicate if the constraints are satisfied while being active or inactive
for optimality i.e., Gi = 0 and λi 6= 0 or Gi < 0 and λi = 0.
The benefits of this optimization technique include faster convergence, easy implementa-
tion, and high accuracy. Drawbacks of SQP are that it requires smooth analytical functions
to compute gradients and is not a global optimization algorithm.
102
4.2 Optimization
4.2.1 Sensor Performance Requirements
Sensors in general need to satisfy several target specifications like large dynamic range
and bandwidth, in addition to high sensitivity and low noise floor. These terms and their
desirable values are briefly explained in this section. The operating space for a sensor may be
represented by a plot as shown in Figure 4-1. The operating space is bound by the limits of
the sensor performance. Ideally, a good design is aimed at maximizing the sensor operating
space. The extents of the operating space are bound by the dynamic range and bandwidth
of the sensor.
Frequency (Hz)
Shear Stress (Pa)
Bandwidth (Hz)
Dynamic Range (Pa)
Non-linearity
(Mechanical
structure, Pull-in)
±3dB point
(flat band
response)
Minimum detectable signal
(Sensor structure, electronics)
Figure 4-1. Schematic of operating space of the sensor.
The dynamic range of a sensor is defined as the maximum range of the physical input
that the sensor can reliably measure. The upper end of the dynamic range is determined
103
by the maximum input signal (shear stress) the sensor can measure without significant non-
linearity/harmonic distortion. The lower end of the dynamic range is limited by the noise
floor/minimum detectable signal (MDS) of the sensor. For an input within its dynamic
range, the sensor bandwidth is defined by the range of frequencies over which the sensor has
a constant response (sensitivity).
In an optimization problem, the lower and upper end of the bandwidth are set via
constraints based on the frequency range of interest. The lower end of the bandwidth is at
dc since it is required to measure mean shear stress. The cut-on frequency of the interface
circuit is considered negligible in this case, allowing dc measurements. The upper end of the
dynamic range is constrained by the maximum permissible non-linearity in the output that
is ideally set by the target physical input to be measured. Thus, with the three sides bound
via constraints, the operating space is maximized by minimizing the MDS (fourth side) of
the sensor. This is consistent with Spencer’s study, which indicates that MDS is useful as
true figure of merit for studying sensor performance [62]. Minimizing MDS in Equation 3–
109 thus translates into maximizing the overall sensitivity while minimizing the noise floor
of the sensor.
4.3 Problem Formulation
4.3.1 Objective and Design Variables
The goal of this optimization is to minimize the minimum detectable signal (shear stress)
(MDS) for the differential capacitive shear stress sensor. In the expression for the output
given by Equation 3–105, the termVref
2U, represents a contribution from the demodulation
circuitry. In the current circuit implementation, this term is not variable since off-the-shelf
components are used for demodulation. A unity gain voltage follower is used as the amplifier
i.e., G = 1. Thus, the simplified expression for the overall sensitivity is
Sτ,overall = VacHgapHc
(δm/τw
g01
). (4–3)
104
As discussed in Section 3.3.3, the mechanical thermal damping in the system is assumed
to be negligible. As a result, the electronics dominate the overall noise floor. The SiSonic
amplifier used in this implementation has a measured input referred voltage noise power
spectral density of 4×10−16 V 2
Hzfor thermal noise with a corner frequency, fce = 395 Hz [72].
The current noise power spectral density of this amplifier was negligibly small, 5×10−31 ( A2
Hz)
and it has an internal bias resistor. If an amplifier with a higher current noise is used, a change
in the nominal sensor capacitance alters the input impedance, Zi, affecting the overall noise
floor (Equation 3–115). Also, it may need an external bias resistor, whose noise contribution
varies with the sensor capacitance. With the present noise specifications for the amplifier,
the noise floor reduces to its voltage noise and is represented using a curve fit as
So = 4× 10−16 (1 + 395/f) . (4–4)
The optimization is performed for a overall static sensitivity, Sτ,overall, at 1 kHz, with
1 Hz bin-width. Since the amplifier has a fixed noise floor at 1 kHz, the objective function
only maximizes Sτ,overall, assuming fixed parasitic capacitance. However, to maintain gen-
erality, the MDS is retained as the objective function. The objective function is therefore
given as
F (−→X ) = MDS (Pa) =
√S0
∣∣f=1 kHz
Soverall
or F (−→X ) =
√S0
∣∣f=1 kHz
VacHgapHc
(δm/τw
g01
) , (4–5)
where−→X is the design variable vector as shown in Equation 4–6. Note, the overall sensitivity
includes the non-idealities and attenuation factors (Hc and Hgap) due to parasitics and
asymmetric gaps. The LEM developed in Chapter 3 is used to estimate the frequency
response of the coupled electromechanical system. The floating element, tethers, and comb
fingers have the same thickness, Tt, which is decided by the material layer thickness during
105
fabrication. A constant charge biasing scheme is used in conjunction with a voltage buffer.
The voltage change corresponding to a change in capacitance is sensed using the voltage
buffer. Assuming silicon as the sensor material with known properties, the variables of
interest for this optimization problem are as follows:
1. Bias voltage amplitude Vac
2. Tether length Lt
3. Tether width Wt
4. Element length Le
5. Element width We
6. Tether thickness Tt
7. Comb finger overlap L0
8. Comb finger width Wf
9. Primary gap g01
10. Secondary gap g02
This results in 10 design variables represented as a vector,
−→X = (Vac, Lt,Wt, Le,We, Tt, L0,Wf , g01, g02) . (4–6)
The comb finger length, Lf , is longer than the overlap, L0, to accommodate the gap at the
end of each comb finger, given as
Lf = L0 + 20 µm. (4–7)
The number of comb fingers is computed using the the design variables as
N =2(We − 2Wt)
(2Wf + g01 + g02). (4–8)
Using this expression, the maximum number of comb fingers that can be accommodated
along the width of the floating element are obtained. The larger the number of comb fingers,
the higher the nominal capacitance, which is essential to overcome the parasitic attenuation,
106
Hc. This may marginally increase the mass of the sensor and correspondingly lower the
resonant frequency. However, the optimizer has more than one variable to vary to offset this
increase in mass.
4.3.2 Constraints
Constraints for the optimization are set based on performance criteria, design specifi-
cations, and fabrication limitations. The bounds on the design variables are set based on
application and fabrication limitations. As stated in Chapter 1 the goal is to design a sensor
to measure τmax ≤ 10 Pa. The characteristic viscous length scale, l+, for this shear stress
is approximately 5.3 µm. The spatial resolution is set by the maximum sensor size which
is restricted to ≤ 400l+ via upper bounds (UB) for a conservative design to demonstrate
proof of concept. Three different scales are considered 100l+, 200l+, and 400l+, respectively.
The lower bounds (LB) are set by the sensing area needed to have a reasonable signal to
noise ratio sans attenuation. Nominal capacitance, fabrication and hydraulic smoothness
requirements are used to determine the bounds on the gaps around the sensor. The largest
gap is roughly 3.8l+, thus ensuring that hydraulic smoothness is maintained. The upper
(UB) and lower (LB) bounds on the design variables are (Table 4-1)
LB ≤ (Vac, Lt, Wt, Le,We, Tt, L0,Wf , g01, g02) ≤ UB. (4–9)
4.3.2.1 Bandwidth
This constraint ensures that the sensor possesses the bandwidth specified by the design
requirements. The bandwidth constraint is given as
fhigh − dc ≥ fmin, (4–10)
fmin is the minimum acceptable bandwidth set by the design specification. A ±3 dB fre-
quency is chosen which is defined as the ±3 dB change in the value of the frequency response
function from its flat band value (sensitivity). The lower end of the bandwidth is at dc since
the sensor needs to measure the mean flow. There is no mechanical cut-on frequency for the
107
sensor (second order system) and the electrical cut-on frequency is assumed to be negligible.
fhigh is ±3 dB frequency at high frequencies, set via the damping in the system. As stated
in Section 3.2, the mechanical thermal damping in the system is neglected. Therefore, fhigh
is defined as the undamped natural frequency, fres of the system for estimation purposes.
Thus, the bandwidth constraint may be simply written as
fmin ≤ fres. (4–11)
4.3.2.2 Mechanical Nonlinearity
Any non-linearity in the system results in harmonics at the output. To maintain spec-
tral fidelity of time resolved measurements, the nonlinearities in the system have to be con-
strained. The linearity of the estimated MDP under maximum incident shear stress τmax, is
ensured using the nonlinearity constraints mathematically stated as follows,
δNL − δ
δ≤ 3%, (4–12)
where δNL is the nonlinear deflection computed using Equation 3–8 and δ is the linear
deflection obtained from Equation 3–7.
4.3.2.3 Electrostatic Pull-in
Ideally the capacitances that form the differential scheme remain at their mean position
under increasing bias voltages due to equal and opposite electric fields balancing each other.
However, when the pull-in voltage is reached, even a small perturbation causes the diaphragm
to jump to one of the stationary electrodes. Pull-in occurs when the mechanical force is no
longer sufficient to counter the electrostatic force causing the movable electrode to collapse to
the stationary electrode in an electrostatic transducer. A discussion on the pull-in instability
can be found in [42, 66]. For a differential capacitance scheme, the pull-in voltage is given
as [78]
VPI =
√g30
2CmeεA. (4–13)
108
Assuming g02 >> g01 i.e., pull-in in dominated by the primary gap, and using the small-
est possible electrode separation at maximum forcing, the conservative estimate for pull-in
voltage for the proposed sensor is written as
VPI =
√(g01 − δ)3
2Cmeε(3Lteff + Le + (N − 1)L0). (4–14)
The amplitude of the bias voltage amplitude Vac is constrained to 85% of the pull-in voltage
given as
Vac ≤ 0.85VPI . (4–15)
There is an additional nonlinearity due to nonlinear output voltages at large deflections
and is illustrated here mathematically. Assuming g02 À g01, the nonlinear output voltage,
VNL, for a differential capacitive sensing scheme with matched capacitors is
VNL = Vacδ
g01
C0
C0 + (Cp+Ci)
2
(1− δ2
g201
) . (4–16)
Typically, the full scale deflection δ ≈ 0.1 µm and g01 ≈ 3 µm. Thus, 1 − δ2/g201 ≈ 0.999,
indicating that a nonlinear output voltage due to large defections, i.e., δ ∼ g01 is unlikely. A
constraint to limit VNL is therefore not formulated to simplify the optimization problem.
Mathematically, the problem is to minimize the MDS subject to constraints or
Minimize
F (−→X ) = MDS,
subject to
g1 = fmin/fres − 1 ≤ 0,
g2 =δL − δNL
0.03δL
− 1 ≤ 0,
g3 =Vac
0.85VPI
− 1 ≤ 0,
109
and gi which are to be determined from
LB ≤ (Vac, Lt, Wt, Le,We, Tt, L0,Wf , g01, g02) ≤ UB.
4.3.3 Design Specifications
This section provides the target design specifications and the constants used in the
optimization. The maximum input shear stress and the required bandwidth of the sen-
sor determine the target designs. Upper and lower bounds are based on spatial resolution
requirements and micromachining limitations. The target design specifications and the vari-
able bounds are shown in Table 4-1. The constants and material properties used for the
optimization are shown in Table 4-2.
Table 4-1. Lower and upper bounds for design variables and flow specifications.
Vac Lt Wt Le We Tt L0 Wf g01 g02 fmin τmax
(V ) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (µm) (kHz) (Pa)
5−25 100−1000
10−40
100−2000
100−2000
45−50
5−150
4−20 3.5−15
3.5−20
10 5&10
Table 4-2. Constant parameters for optimization.
Parameter Value
Electrical Permittivity of Free Space ε0 8.854× 10−12 F/m
Relative Permittivity of Air εr 1.005
Floating Element Young’s Modulus E † 168 GPa
Density of Floating Element ρ 2330 kg/m3
Amplifier Input Capacitance Ci 0.3 pF
Parasitic Capacitance Cp 2.2 pF
† Average value in the 100 plane of silicon.
4.4 Results and Discussion
Optimization results for five different designs are presented in this section. Two target
shear stresses are used, τmax = 5 Pa and τmax = 10 Pa. The bandwidth target is fmin =
5 kHz and 10 kHz. The upper bounds on the floating element length and width are also
110
varied to study the effect of different sensing areas. The results of the optimization are
summarized in Table 4-3.
The bias voltage, Vac is restricted either by its upper bound or by the active pull-in
constraint for all the designs. The tether length, Lt always hits its UB tending to maximize
the sensitivity by increasing the tether compliance. However, the tether width is not always
at its LB because of the bandwidth and the active pull-in constraint. Thus, the bandwidth
is maximized and the pull-in instability is avoided, but, at the cost of sensitivity. This is
better understood from the relations
Sensitivity ∼ Cme, (4–17)
, (4–18)
and VPI ∼ 1√Cme
. (4–19)
As stated already, the active pull-in constraint also restricts Vac which in turn lowers the
sensitivity (refer Equation 3–107). Thus, the detrimental effect of the pull-in voltage is
two-fold, it limits Vac and Cme, lowering the sensitivity.
The optimizer always maximizes the sensor sensing area via Le and We to increase the
available shear force, improving sensitivity. The tether thickness is always pushed to its
lower bound. Equation 3–80 indicates,
Cme ∝ 1
Tt
, (4–20)
which means a lower tether thickness increases Cme, improving the sensitivity. Ideally, a thin
floating element structure would greatly improve sensitivity but it would lower the resonance
frequency and hence the sensor bandwidth. A thinner element also reduces the overlapping
area (L0Tt) of the capacitors, lowering nominal capacitance. A high nominal capacitance is
essential to minimize attenuation due to parasitics. Also, smaller Tt makes the mechanical
structure more sensitive to undesirable pressure fluctuations.
111
The initial finger overlap, L0 and the primary gap, g01 are always at their UB and LB
respectively. Both are essential to achieve a high nominal capacitance to minimize parasitic
attenuation. The secondary comb separation g02 is also maximized because it has a change
opposite to g01. As a result of g02, the overall capacitance change,∆C is lower than the
achievable value with only g01. A large value of g02 is therefore desirable to maximize ∆C
and hence the sensitivity. This is also explained in Chapter 3 via the gap attenuation factor,
Hgap in Equation 3–68. Large g02 also decreases the number of comb fingers N , lowering
nominal capacitance. Thus, a tradeoff exists between nominal capacitance needed to mitigate
parasitic attenuation, and larger g02 required to lower Hgap.
In some cases, the comb finger width, Wf is also maximized. Increased width con-
tributes to increase in sensing area and decrease in number of comb fingers N . Therefore,
thicker fingers result in lower nominal capacitance. Similar to the discussion in the previous
paragraph, there is tradeoff between improved sensitivity and lower C0 i.e., higher parasitic
attenuation, resulting from wider fingers.
Table 4-3. Capacitive shear stress optimization results for Cp + Ci = 2.2 + 0.3 pF .
DesignParameter
Design 1 Design 2 Design 3 Design 4 Design 5
τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 10 Pa
fmin = 5 kHz fmin = 5 kHz fmin = 5 kHz fmin = 10 kHz fmin = 5 kHz
UB Le = 2000 µm Le = 1000 µm Le = 500 µm Le = 1000 µm Le = 1000 µm
UB We = 2000 µm We = 1000 µm We = 500 µm We = 1000 µm We = 1000 µm
Bias VoltageVac (V )
25.00 22.78 16.46 25.00 22.21
TetherLengthLt (µm)
1000 1000 1000 1000 1000
Tether WidthWt (µm)
23 15 10 23 15
ElementLengthLe (µm)
2000 1000 500 1000 1000
ElementWidthWe (µm)
2000 1000 500 1000 1000
112
Table 4-3. Continued
DesignParameter
Design 1 Design 2 Design 3 Design 4 Design 5
τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 10 Pa
fmin = 5 kHz fmin = 5 kHz fmin = 5 kHz fmin = 10 kHz fmin = 5 kHz
UB Le = 2000 µm Le = 1000 µm Le = 500 µm Le = 1000 µm Le = 1000 µm
UB We = 2000 µm We = 1000 µm We = 500 µm We = 1000 µm We = 1000 µm
Tether/ElementThicknessTt (µm)
45 45 45 45 45
Comb FingerOverlapL0 (µm)
150 150 150 150 150
Comb FingerLengthLf (µm)
170 170 170 170 170
Comb FingerWidthWf (µm)
5.1 20 20 4.0 20
PrimaryCombSeparationg01 (µm)
3.5 3.5 3.5 3.5 3.5
SecondaryCombSeparationg02 (µm)
20 20 20 15.4 20
Number ofComb FingersN
116 31 15 71 31
ResonantFrequency/Bandwidthf (kHz)
5 5 5.2 10 5
Design ShearStressτmax (Pa)
5 5 5 5 10
SensitivitySoverall
(dB re 1 V/Pa)
−28.98 −34.33 −40.57 −43.05 −34.50
SensitivitySoverall
(mV/Pa)
35.60 19.20 9.360 7.040 18.80
MDS at1 kHz (µPa)
0.562 1.040 2.140 2.840 1.060
Attenuationdue to Cp +Ci, Hc (dB)
−5.10 −9.86 −12.65 −6.90 −9.86
113
Table 4-3. Continued
DesignParameter
Design 1 Design 2 Design 3 Design 4 Design 5
τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 5 Pa τmax = 10 Pa
fmin = 5 kHz fmin = 5 kHz fmin = 5 kHz fmin = 10 kHz fmin = 5 kHz
UB Le = 2000 µm Le = 1000 µm Le = 500 µm Le = 1000 µm Le = 1000 µm
UB We = 2000 µm We = 1000 µm We = 500 µm We = 1000 µm We = 1000 µm
TetherCapacitanceCtet (pF )
0.18 0.18 0.18 0.18 0.18
FloatingElementCapacitanceCE (pF )
0.23 0.11 0.06 0.11 0.11
Comb FingerCapacitanceCf (pF )
1.16 0.30 0.14 0.73 0.30
CapacitanceC0 (pF )
1.56 0.59 0.38 1.03 0.59
Full ScaleDeflectionδmax (nm)
60 60 50 10 120
CapacitanceChange∆C (fF )
22.6 8.73 5.22 3.55 17.5
PercentageChange(∆C/C0) %
1.45 1.48 1.37 0.35 2.95
Pull inVoltageVPI (V )
29.5 26.9 19.4 39.1 24.2
4.4.1 Sensitivity Analysis
The sensitivity of the optimized MDS to variation of the design variables is studied in
this section. The sensitivity analysis of Design 1 in Table 4.4 is presented. The MDS and
the variables are both normalized by their respective optimum values. The variables are
perturbed 20% on either side of their optimum value. Figure 4-2 shows the sensitivity of
normalized MDS to the ten normalized design variables. However, some of these solutions
may be invalid because the constraints may be violated and are studied separately.
The results indicate the MDS is highly sensitive to variation in tether dimensions, Lt
and Wt. The design has minimum sensitivity to the secondary gap, g02 and the comb finger
114
0.8 0.9 1 1.1 1.20.5
1
1.5
2
X/Xopt
f obj(X
)/f ob
j(Xop
t)
Vac
Lt
Wt
Le
We
Tt
L0
Wf
g01
g02
Figure 4-2. Sensitivity of optimized MDS to design variables for Design 1.
width Wf . The MDS variation with each individual variables and constraints is also studied.
Only one design variable is perturbed at a time to understand the effect on the MDS. The
region of the plot that is not shaded or hatched indicates the feasible region. The shaded
portion indicates the region where the constraint is violated whereas the hatched portion
indicates the infeasible region set by the bounds. Figure 4-3 shows the results for each design
variable.
Consider the result in Figure 4-3(b) for Lt, for which the optimal MDS has the highest
sensitivity. The optimal MDS is limited by the upper bound of Lt. Any further increase
would have lowered the MDS but again restricted simultaneously by the pull-in and resonant
frequency constraints. The presented analysis provides insight into the expected variations
in the objective and constraint functions resulting from change in the optimized variables.
However, this study is limited to change in one variable at a time and as stated earlier may
result in violation of constraints for other variables.
The expected change in sensor performance (Design 1) due to simultaneous variations
in the design variables and parameters of interest is also studied statistically, using a Monte
115
(a) Vac
(b) Lt
(c) Wt
(d) Le
Figure 4-3. Sensitivity of optimized MDS and constraints due to change in optimized variablevalues of Design 1.
116
(e) We
(f) Tt
(g) L0
(h) Wf
Figure 4-3. Sensitivity of optimized MDS and constraints due to change in optimized variablevalues of Design 1 (cont· · · ).
117
(i) g01 (j) g02
Figure 4-3. Sensitivity of optimized MDS and constraints due to change in optimized variablevalues of Design 1 (cont· · · ).
Carlo (MC) simulation. In this simulation, a Gaussian distribution is assumed for each
parameter of interest. The approximate tolerances and uncertainties, that may be expected
in the variables and constants, are used to build the Gaussian profile. Table 4-4 shows the
tolerance chart for the various parameters used in this study.
The performance parameters of interest for this analysis are MDS, overall sensitivity
(Sτ,overall), and bandwidth (fmin). The simulation uses 50000 samples for each input/design
parameter of interest. The variations in MDS, Sτ,overall, and fmin are represented in the form
of histograms after outlier rejection in Figure 4-4. The distribution is slightly skewed to the
nonlinearity in the system and due to correlation between the variables. For instance, when
gap g01 increases g02 proportionally decreases and vice versa. However, for this analysis
all variables were assumed to be uncorrelated. For small perturbations these effects are not
pronounced, resulting in a gaussian distribution. The vertical red line in the figures indicates
the optimized value from Table 4.4. The mean and standard deviation for each of the design
parameters are summarized in Table 4-5. The table shows that despite finite tolerances
118
and uncertainties, the predicted performance of the sensor is likely to be within the 95%
confidence intervals for MDS, Sτ,overall, and fmin.
Table 4-4. Tolerance chart for variables and constants.
Parameter Tolerances (σ)
Bias Voltage Vac ±0.2 V
Tether Length Lt ±5 µm
Tether Width Wt ±0.5 µm
Floating Element Length Le ±10 µm
Floating Element Width We ±10 µm
Tether Thickness Tt ±1 µm
Initial Finger Overlap L0 ±5%
Comb Finger Width Wf ±5%
Primary Comb Separation g01 ±15%
Secondary Comb Separation g02 ±1 µm
Dielectric Permittivity ε ±10%
Density of Floating Element (Si) ρ † ±10%
Floating Element Young’s Modulus E ±10%
Amplifier Input Capacitance Ci ±10%
Parasitic Capacitance Cp ±10%† Average value in the 100 plane of silicon.
Table 4-5. Design sensitivity based on Monte Carlo (MC) sim-ulation.
DesignParameter
OptimizedValue
Mean Value(µMC)
95% ConfidenceInterval (2σ)
MDS (µPa) 0.562 0.528 0.292− 0.765
Sτ,overall
(mV/Pa)35.60 34.92 20.70− 49.13
fmin (kHz) 5.00 4.96 4.28− 5.65
4.5 Conclusion
This chapter provided details of the optimization for the proposed differential capacitive
shear stress sensor. The optimization principle was briefly discussed followed by the problem
119
0.2 0.3 0.4 0.5 0.6 0.70
200
400
600
800
1000
MDS (µPa)
(a) MDS
15 20 25 30 35 40 450
200
400
600
800
Sτ,overall (mV/Pa)
(b) Sτ,overall
4 4.5 5 5.50
200
400
600
800
1000
1200
Resonant Frequency (kHz)
(c) Bandwidth (fmin)
Figure 4-4. Sensitivity of optimized MDS, Sτ,overall, and fmin to tolerances in variables inconstants of Design 1 using Monte Carlo simulation.
formulation i.e., design variables, objective function, constraints, and variable bounds. The
optimized results for five different designs were presented. Lastly, a sensitivity analysis was
performed to study the effect of variation of the optimized design variables on the MDS and
constraints for Design 1. A Monte Carlo simulation was performed to investigate the effect
on sensor performance due to simultaneous variation in design variables and constants. The
optimization provided better insight into the design physics. Dominant design variables that
influence the design most are identified easily via the sensitivity analysis.
120
CHAPTER 5DEVICE FABRICATION AND PACKAGING
This chapter describes a detailed fabrication process for the metal coated capacitive
floating element shear stress sensor. The fabrication process uses a 2-mask Si bulk microma-
chining and metal plating processes to form a released, metal passivated, floating element
sensor. The fabrication process starts with a 100 mm (100) p-type silicon on insulator (SOI)
wafer with a 45 µm highly doped silicon layer above a 2.0 µm buried silicon dioxide (BOX)
layer. The highly doped device layer has a low resistivity of 0.001 − 0.005 Ω − cm. The
top silicon (device layer) and BOX are backed by a 500 µm thick, float zone, bulk silicon
substrate needed to provide mechanical support. The wafers are also double-side polished to
allow patterning on both sides of the wafer. Deep reactive ion etching (DRIE), an anisotropic
silicon etch process, defines the floating element structure on the device layer. The fabrica-
tion process is described followed by the sensor packaging. Figure 5-1 shows the schematic
of a single die of the shear stress sensor.
A A
A-A
p++ Si (45 )SiO2 (2.0
)Float Zone Si
(500 )v+ v-
Electroplated
Ni (~0.2 )Plan View
Section
View
Figure 5-1. Schematic showing the plan view of a single die of the proposed shear stresssensor and its section view indicating various layers.
121
5.1 Fabrication Process
The sensor fabrication is a simple process involving two lithography steps, three etching
steps, and an electroplating step shown in Figure 5-2. First, the floating element structure is
defined using DRIE. Next, a thin layer of nickel is electroplated on the device layer. A front
to back alignment step defines the mask for the backside etch. This is followed by backside
DRIE with the BOX layer serving as an etch stop. Lastly, the floating element structure is
released by wet etching the BOX layer.
p++
p++
p++
Highly doped
device layer
Float Zone
bulk substrate
SOI Wafer
Photoresist
AZ1512
Pattern using
EVG 620
Mask Aligner
DRIE on Si
Device layer
(STS)
Pattern Back
Side with AZ
9260
DRIE on back
side.
BOX etch
stop
EVG 620
Mask Aligner
BOE dip to
remove BOX
Directly
electroplate
Ni onto highly
doped Si
Device layer
(0.2 )
Photoresist
SiO2 (BOX)
Silicon
Nickel
(a)
(b)
(c)
(d)
(d)
(e)
(f)
(g)
(h)
Figure 5-2. Step by step fabrication process.
The motivation for electroplating on the silicon is briefly explained here before illustrat-
ing the fabrication process. The highly doped micromachined device layer enables seedless
electroplating on the entire exposed sensor silicon surface. Seedless electroplating eliminates
several steps associated with seeded electroplating, significantly simplifying the fabrication
process. Electroplating eliminates unwanted charge accumulation on the dielectric native
122
oxide formed on the device surface by providing a conductive metal coating. This allevi-
ates drifts due to trapped charge at dielectric interfaces. Electroplating also eliminates an
additional alignment step for metallization to make electrical contacts to the device silicon.
The details of the fabrication process are described in the sections that follow. A detailed
process flow describing the process parameters, equipment and the labs used for fabrication
are provided in Appendix D.
5.1.1 Floating Element Trench
An RCA clean is performed to clean the wafer surface before beginning the sensor
fabrication process. Photoresist is patterned on the device surface of the SOI wafer to define
the floating element. Standard anisotropic DRIE technique is used to form the floating
element, tethers, and comb fingers on the device layer with the BOX layer as an etch stop.
In addition to having an etch stop for DRIE, using an the SOI also enables precise device
layer thickness, enabling closer match with the design value. Electrical isolation of three
terminals for the differential capacitance measurement scheme, is achieved during this DRIE
step. Both alignment marks needed for front to back alignment for the sensor release and
the dicing marks for die separation, are etched during this DRIE step.
The challenge during this fabrication step is the ability to etch high aspect ratio, narrow,
parallel-walled trenches defining the comb fingers. Narrow trenches increase capacitance and
also allow more comb fingers in the available space increasing the overall static capacitance.
Large static capacitance is critical to mitigate detrimental effects of parasitic capacitance,
discussed in Chapter 3. Parallel walls enable a uniform electric field between the comb
fingers, which is the theoretical basis of the sensing scheme analyzed in Chapter 3. Figure 5-
3 shows scanning electron microscopy (SEM) images indicating the quality of the etch and
the asymmetric gaps (Chapter 3) of the floating element structure.
123
TetherComb
Fingers
Floating
Element
Stationary
Substrate
Asymmetric
Gaps
200
20
Figure 5-3. SEM images of comb finger structure etched using DRIE, giving 3D perpective.
5.1.2 Seedless Electroplating of Nickel
Seedless electroplating simplifies the fabrication process by eliminating the need to de-
posit a conductive metal seed layer [79]. The use of a seed layer also adds additional fabri-
cation steps. The interdigitated comb fingers have a separation distance of 3.5 µm resulting
in deep - narrow trenches. For sufficiently thin metal layers, only evaporating or sputtering
may suffice, eliminating the electroplating step. However, evaporating or sputtering metal
uniformly onto the high aspect ratio trench sidewalls poses a significant fabrication chal-
lenge due to poor side wall coverage. If used as a seed layer, such a deposition will result in
non-uniform metal plating. This may result in non-uniform gaps causing difference in the
capacitance between adjacent comb fingers, promoting mismatch between C10 and C20. Also
124
the possibility of charge trapping on the exposed silicon surface raises concerns regarding
drift. Electroplating being a pre-release step, a seed layer will also result in metal deposition
at the bottom of the trenches. This adds a metal removal step to isolate the capacitive
electrodes of the sensor. Nickel electroplating is performed on the previously micromachined
highly doped device layer during this step. Nickel also provides good corrosion resistance
that allows the use of the sensor in harsh environments [80].
A commercially available high speed nickel sulfamate bath from Technic Inc is used
for the electroplating process [81]. The solution is maintained at a constant temperature of
90 F during the plating process with constant stirring. The electroplating process is based
on the work presented in [79]. A nickel electrode is used as the anode and the highly doped
conducting device layer of the SOI wafer serves as the cathode. Previous studies claim that
from an electroplating standpoint, highly doped silicon has conductivities similar to that of
normally doped silicon with a conductive layer [79].
Silicon surface preparation prior to electroplating is essential to ensure good plating
quality. Organic residue left from previous processing is removed using a standard piranha
(3 : 1, H2SO4 : H2O2) clean procedure. The wafers are immersed in the piranha solution
at 120C for 10 min. A thin layer of native oxide is formed on the silicon surface in the
piranha and if exposed to air. The oxide creates an insulating non-conducting surface in-
hibiting the electroplating process. Buffered oxide etch (6 : 1) (BOE) is used to remove
this oxide. The wafer is immersed in the BOE solution for 5 min. The BOE immersion
makes the silicon surface hydrophobic [79]. This is immediately followed by a 5 min dip in
2-propanol. The propanol treatment wets the silicon surface and makes it hydrophillic. The
propanol treatment provides a more uniform growth of Ni during the plating process [79].
Following the propanol treatment the wafer is immersed into the plating bath with the power
supply switched on. Immersion into the aqueous plating bath, without current switched on,
facilitates oxidation inhibiting the plating process [79].
125
The plating is done in two phases, the rapid nucleation phase called striking and the
fine plating phase for uniformity. A high current density for a short time interval in the
nucleation phase called striking, helps rapid nucleation of Ni on the silicon surface. This
creates locally metalized regions facilitating further electroplating. A lower current density
and longer plating time in the fine plating phase gives finer grain size and a more uniform
surface. This electroplating step creates a 0.2 µm thick coating of Ni on the micromachined
structure created in the previous step.
Vertical sidewalls resulting from the etching process and the uniformity of the elec-
troplating process is verified by looking at SEM images of a cleaved sample of the wafer.
Figure 5-4 shows both vertical sidewalls and uniformity of the plated metal.
5.1.3 Backside Release
This is the final and the most important phase of fabrication. Most of the yield depends
on this release phase of sensor fabrication. A “front to back alignment” enables back side
patterning of the wafer followed by the backside etch.
The trenches defining the floating element in the device layer are filled with photoresist
to provide additional mechanical strength and prevent breaking of tethers during release.
Photoresist 10 µm in thickness is spun on the device layer to serve as a protective coating.
The back side of the wafer is now spun with photoresist and patterned using the front to
back mask aligner.
The wafer is attached with its face down on to a carrier wafer, pre-spun with photoresist.
The carrier wafer provides mechanical support and heat dissipation path during the back
side etch. DRIE is used to etch away the bulk silicon using the BOX as the etch stop.
5.1.4 Die Separation
The processed wafer is separated from the carrier wafer using photoresist stripper, and
cleaned for the die separation step. Heat sensitive, thermal release tapes (REVALPHA) by
Nitto Denko Corporation are used in this step. The front side of the wafer is covered with
a single sided tape for protection. The wafer is mounted on another carrier wafer using a
126
(a)
(b)
SiliconNickel
100 20
0.143 Figure 5-4. SEM image of a cleaved wafer sample depiciting a) vertical sidewalls and b)
uniformly plated nickel.
double sided heat sensitive tape. Using a dicing saw, the wafer is diced along the dicing lines
etched on to the wafer in the initial DRIE step. The wafer is then placed on a hot plate
at 80C to remove the tape, resulting in individual die. Each individual die is immersed in
BOE for 20 min to release the sensors. The sensors are immediately immersed in methanol
and then supercritically dried in CO2 to avoid stiction. Figure 5-5 shows an optical image
of a fully processed, released floating element sensor die, etched on the SOI wafer.
127
5.1.5 The MEMS Shear Stress Sensor
Prior to dicing, the wafer consists of 200 die. Only one wafer out of the four processed
wafers is released. The released die measured 5 mm×5 mm in size. The seemingly large size
of the die for a MEMS sensor is not a limitation, but is for better handling and to withstand
repeated tests, thus ensuring first-run success of the proof-of-concept sensor. Figure 5-6
shows the sensor die placed on a Florida quarter to get a perspective of the size of the
sensor.
500 500
Surrounding
Substrate
Bond
PadsBond
Pads
Figure 5-5. Microscopic image of a released floating element sensor structure from Design 3(Table 4.4).
Figure 5-6. A 2 mm× 2 mm sense element on a 5 mm× 5 mm sensor die.
128
5.2 Packaging
The fabricated sensor die needs to be packaged to characterize the sensor performance.
A recess is cut into a printed circuit board (PCB) such that the sensor is flush with the PCB
surface which again is mounted flush in a Lucite plug that fits with the calibration setup.
Figure 5-7 shows the schematic showing the overall packaging scheme for the sensor. The
contacts to the board are made using gold wire bonds. The Knowles SiSonic amplifier is
placed on the back side of the PCB in close proximity to the sensor to minimize parasitic
attenuation arising from the connection traces on the PCB (Figure 5-8). SMB connectors on
the backside of the PCB allow electrical contact to the sensor for biasing, input and output
signals, also shown in Figure 5-8. The detailed drawings of the PCB and Lucite plug are
provided in Appendix D along with vendor details.
Sensor
Die
Printed
Circuit Board
(PCB)Lucite
Plug
Figure 5-7. Schematic of sensor package for shear stress characterization.
129
Flush mounted
sensorSiSonic
TMvoltage
follower
SMB
connectors
Figure 5-8. Photographs of sensor packaged on a 30 mm× 30 mm PCB.
130
CHAPTER 6EXPERIMENTAL CHARACTERIZATION
In this chapter, the differential capacitive shear stress sensor, designed and realized in
Chapter 3 and Chapter 5, is experimentally characterized. Design 1 in Table 4-3 is experi-
mentally studied for impedance, static, dynamic, and noise characteristics. The experimental
setup for these measurements is described. Experimental results and discussions that follow
give physical insight into the sensor performance characteristics.
6.1 Experimental Setup
This section describes the setup for experimental characterization of the shear stress
sensor. The setup for die-level impedance measurements on the sensor for estimating its
nominal capacitance is presented. The laminar flow cell used for mean shear stress mea-
surement is illustrated. The dynamic characterization using a plane wave tube (PWT) is
explained, in which dynamic shear stress is generated using acoustic plane waves. Lastly,
the setup for noise floor measurements on the post-packaged sensor in a Faraday cage, is
explained.
6.1.1 Impedance Measurements
Impedance measurements on the pre-packaged sensor are performed in a probe station
using a four point measurement technique. Since the packaged sensor has the amplifier con-
nected to it, which alters the overall impedance of the system, the impedance measurement
is made at the die level. The schematic for this measurement is shown in Figure 6-1. The
measurement setup consists of four probes, two of which (Hp and Hc) are connected to one
bond pad / electrode and the other two (Lp and Lc) are connected to the other bond pad
/ electrode. Hp and Hc stand for high potential and high current and Lp and Lc stand for
low current and low potential, respectively. The impedance of the sensor is measured using
131
Voltage
Current
Lc Hc
Lp HpProbe
Station
HP 4294 A
Impedance
Analyzer
Hp – High Potential
Hc – High Current
Lp – Low Potential
Lc – Low Current
Figure 6-1. Schematic showing the die-level impedance measurement setup for the sensor.
an HP 4294A impedance analyzer, which uses an I-V method where
Zsensor =V
I=
VHp−Lp
IHc−Lc
(6–1)
and Zsensor = R− jXc = R +1
jωC1,2
. (6–2)
The impedance measurement helps to estimate the nominal sensor capacitances, inclusive of
the parasitics i.e., C10+p1 and C20+p2. The subscripts p1 and p2 represent parasitic capaci-
tances in addition to the nominal values C10 and C20, respectively. In case of a mismatch
i.e., C10 6= C20, the capacitor electrodes experience different electrical forces when biased.
As a result, the floating element settles at a new mean position and therefore a new nom-
inal capacitance value at each bias voltage. The sensor nominal capacitance is therefore
estimated as a function of the applied bias voltage. This test also helps to quantify the
mismatch between C1 and C2.
132
Biasing
Circuit
Heise Pressure
Gauge
PC
(LabView)
Mass Flow
Controller
Source
Meter
P1 P2
xh
P
Bandpass
Filter
DAQ
Sensor
Figure 6-2. Schematic showing the mean shear stress/static calibration setup using Poiseuilleflow in a 2-D channel.
6.1.2 Mean Shear Stress Measurement
Mean shear stress measurements are performed in a laminar flow cell, which consists of
two aluminum plates separated by a known distance to form a channel. Figure 6-2 shows
the schematic of the sensor calibration setup for static/mean shear stress inputs. Two
dimensional Poiseuille flow is assumed in the channel, 330 mm long, 100 mm wide, and
1 mm in height, which may be varied if necessary. The expression of shear stress for steady,
fully developed, 2-D flow in a channel is [82]
τw =h
2
dP
dx, (6–3)
where dP/dx is the pressure gradient driving the flow and h is the channel height. As long as
the pressure drop along the length of the channel is linear, measuring the pressure differential
between two pressure taps that are separated by a known distance, ∆x, results in
τw =h
2
P2 − P1
∆x= −h
2
∆P
∆x. (6–4)
The linear pressure drop was previously verified by [83] and [84]. The Poiseuille flow as-
sumption is confirmed by checking for compressibility effects via the Mach number, M , and
ensuring laminar flow via the Reynolds number, Re. The average velocity in terms of the
133
pressure gradient and channel height is [82]
V = − 1
12µ
dP
dxh2, (6–5)
The maximum velocity is given as
Vmax =3
2V . (6–6)
Based on the velocities, M and Re are expressed as,
M =Vmax
c(6–7)
and Re =V h
ν, (6–8)
where c is the isentropic speed of sound in air and ν is the kinematic viscosity of air. For
the maximum tested shear stress based on Equation 6–4 is 2.9 Pa, M ≈ 0.06 and Re ≈ 442.
This validates the Poiseuille flow assumption, which is true if Re < 1400 and M < 0.3 [82].
+
-RbCp+Ci
Vout
Vac sin(ωt)
-Vac sin(ωt)
C1
C2
-Vcc
+Vcc
Sensing Element
Bond pad for
sense element
through tethers
Substrate on
either side of bond
pad form
electrodes
Figure 6-3. Schematic with optical image of sensor die (5 mm × 5 mm) indicating float-ing element, contact pads, and interface circuit (voltage buffer) for mean shearcharacterization.
134
The sensor plug is mounted on the flow cell and biased with sinusoidal bias sources for
modulating the sensor output voltage. Figure 6-3 shows a picture of a fabricated sensor and
a schematic of the interface circuit connections for this measurement. For static shear stress
characterization, the flow rate through the channel is varied and sensor output voltage is
measured. A change in flow rate alters the wall shear stress via a change in the pressure
gradient along the length of the channel. The flow rate is varied electronically using an
AALBORG GFC47 mass flow controller, and the corresponding pressure differential and
sensor output voltage are recorded. A Keithley 2400 sourcemeter serves as the voltage source
to control the flow controller. The pressure differential is measured using the Heise ST-2H
pressure gauge using a differential pressure module (50”H2O). An SR560 low noise amplifier
with unity gain bandpass filters the sensor output (around the biasing frequency) for noise
reduction. This output is then fed to a DAQ card. Pressure drop from the Heise pressure
gauge and the sensor output voltage are acquired using LabView on a PC. Equation 6–4 and
the recorded data are used to estimate the sensor’s mean shear stress sensitivity.
6.1.2.1 Biasing Scheme Implementation
The initial plan to implement the MOD-DMOD scheme discussed in Section 3.1.3.3,
is not implemented due to issues like dc offsets and drift associated with the demodulation
circuitry. Only the modulation scheme is implemented for mean shear stress measurements.
This section explains the reasons and the practical implementation of the current biasing
scheme. The configuration shown in Figure 6-3 results in a modulated voltage at the amplifier
input. A shear induced mean capacitance change produces a corresponding change in this
voltage amplitude. The maximum voltage that may be applied to the input of the SiSonic
voltage buffer is limited to 100 mV . Without an applied shear stress, ideally matched
sense capacitors and out-of-phase (180) bias voltages result in no voltage at the input node
(Section 3.1.3.3) [42]. Mathematically, this may be expressed as (Equation 3–64)
vin = Vacsin(ωt)C1 − C2
C1 + C2 + Cp + Ci
= 0, for C1 = C2. (6–9)
135
In case of a mismatch between the capacitors, C2 = λC1, a proportional sinusoidal voltage
exists at the input node given as
vinmismatch= Vacsin(ωt)
C1 − C2
C1 + C2 + Cp + Ci
(6–10)
or vinmismatch= Vacsin(ωt)
(1− λ)C1
C1 + C2 + Cp + Ci
, for C2 < C1 and λ < 1. (6–11)
This mismatch voltage limits the dynamic range for the static shear stress measurement since
its upper end is limited at 100 mV . The mismatch voltage also increases with increase in
the bias voltage amplitude, Vac, limiting Vac to smaller values and thus lowering sensitivity.
The mismatch voltage may be nulled by using bias voltages of different amplitudes. For
illustration, let Vac1 = Vacsin(ωt) and Vac2 = −βVacsin(ωt) be the bias voltages applied to
C1 and C2, then Equation 6–11 may be rewritten as
vinmismatch=
C1Vac1 − C2Vac2
C1 + C2 + Cp + Ci
= Vacsin(ωt)(1− λβ)C1
C1 + C2 + Cp + Ci
. (6–12)
Thus, if λβ = 1 in Equation 6–12, then vin = 0 i.e., with no shear stress input, the nominal
output voltage of the biased sensor, with mismatched capacitors, is zero. To achieve λβ = 1,
a biasing circuitry is required that can independently control both amplitude and phase of
the two bias voltage signals. Note, that the mismatch is a practical aspect which depends
on fabrication process and is usually quantified via electrical characterization.
A custom made biasing circuit is designed and implemented, which consists of four
AD4898 operational amplifiers, shown in Figure 6-4, with the required phase adjustment
and biasing capabilities. A sinusoidal signal from a function generator is input to two
op-amps, which serve as phase adjust circuits. They are connected in an all pass filter
configuration [85], having variable capacitors, Cvar, to independently adjust the phase of
their outputs. The outputs of the two filter op-amps are input to the voltage buffer and the
inverting op-amp with an adjustable gain (β = R2
R1). The output from the non-inverting and
inverting amplifiers serve as bias voltages for the sensor capacitances, C1 and C2 (assuming
C2 = λC1). Thus, this biasing scheme implementation helps to null out the initial sinusoidal
136
+
‒
+
+
‒
‒
Vac
+
‒
R1
R1
R1
Cva
r
R1
R1
R1
Cva
r
R1
R2
AD
4898
AD
4898
AD
4898
AD
4898
Phase Adjust
Stage
Inverter
Voltage
Buffer
Vac sin(ωt)
-Vac sin(ωt)
C1
C2
Figure 6-4. Schematic of the biasing circuit scheme to control phase and amplitude of biasingsignals to null out output from a potential mismatch in sensor capacitances.
voltage at the middle node via phase and amplitude adjustment. This helps to use higher
bias voltages, improving sensitivity and dynamic range.
6.1.3 Dynamic Measurements
The sensor is characterized for dynamic shear sensitivity, pressure sensitivity, and fre-
quency response in the PWT. Stokes’ layer excitation from propagating or standing acoustic
plane waves is used to estimate the dynamic sensitivity, linearity and frequency response
of the sensor. The sensor configuration for this measurement is the same as shown in Fig-
ure 6-3, but with dc biasing for the sensing capacitors. A known oscillating shear stress
input is generated using acoustic plane waves in a duct [51]. The oscillating acoustic field
in conjunction with the no-slip boundary condition at the duct wall results in an oscillating
velocity gradient, generating a frequency-dependent shear stress. This enables a theoretical
estimate of the wall shear stress if the acoustic pressure is known at a given axial location
137
in the duct. For plane progressive waves, the input shear stress, τin, corresponding to the
amplitude of the total acoustic pressure, p′, at a given frequency, ω, is theoretically given
as [86]
τinprog = −p′√
jων
cej(ωt−kx0)tanh
(a
√jω
ν
), (6–13)
where k = ω/c is the acoustic wave number, a is the duct width and x0 is the axial location
along the length of the duct. The term η =√
a2ω/ν is the non-dimensional Stokes number.
Equation 6–13 is a 1-D shear stress solution and may be used for 2-D acoustic fields for
η > 2 [86]. The 1-D solution in Equation 6–13 is used in this dissertation because the lowest
frequency for measurements in the PWT is 1 kHz, which corresponds to η = 520.
There are three different measurement setups, each of which vary in terms of the termi-
nation used in the PWT and the sensor location. The first setup uses a rigid termination at
the end of the PWT for shear stress measurements, the second setup has the sensor mounted
at the end for normal acoustic incidence for pressure measurement, and the last setup uses an
anechoic termination for combined shear and pressure measurements. There may be acoustic
reflections, which vary as a function of the PWT termination. For standing acoustic plane
waves, which has reflected waves, the input shear stress is
τinstanding=
(−1
c
√jων tanh
(a
√jω
ν
)ejkds −Re−jkds
ejk(ds−δ) + Re−jk(ds−δ)
)p′ejωt, (6–14)
where R is the complex acoustic reflection coefficient. The sensor and a reference microphone,
which measures acoustic pressure amplitude, are at distances, ds and dps from the termination
(Figure C-1). Appendix C provides the theoretical derivation for the dynamic wall shear
stress based on acoustic reflections from the termination.
6.1.3.1 Dynamic Shear Stress Measurement Setup
This section describes the calibration of the sensor for dynamic shear stress inputs. The
measurement setup and the resulting standing acoustic wave pattern in the PWT is shown
in Figure 6-5. Plane waves are generated by a BMS 4590P compression driver (speaker),
138
Techron 7540 Power
Supply Amplifier
B & K Pulse
Analyzer System
PC
Acoustic Plane
Wave
Shear Stress
SensorSpeaker
Microphones
Rigid
Termination
R=1
PressureVelocity
Standing Wave Pattern
Figure 6-5. A schematic of the dynamic calibration setup for measuring shear sensitivityusing rigid termination with the sensor located at pressure node and velocitymaximum.
mounted at one end of the PWT, while the other end is fitted with a rigid termination. The
PWT consists of a 1” × 1” cross section square duct, with a cut-on frequency of 6.7 kHz
for higher order modes in air, above which Equation 6–13 is no longer valid. The sensor
being tested and a reference microphone (B&K 4138, 1/8”) are flush mounted at known
positions along the length of the tube and the acoustic frequency is chosen such that the
sensor is at a velocity maxima and the microphone is at a pressure maxima. A B&K PULSE
Multi-Analyzer System (Type 3109) acts as the microphone power supply, data acquisition
unit, and signal generator for the compression driver.
For a perfectly rigid termination (R = 1), the acoustic wave is reflected in phase with
the incident wave, resulting in a pressure maxima (doubling) and a velocity minima at the
termination [87]. The reference microphone is placed adjacent (< 3 mm away) to the rigid
termination (pressure maxima) without appreciable phase errors at the testing frequencies
(6.6 kHz maximum). At a pressure node the sensor response is dominated by shear stress,
which is a function of the velocity gradient. Another microphone, located at the same
139
axial location as the sensor, monitors the pressure at its node to confirm minimum pressure
contribution to the sensor output.
6.1.3.2 Dynamic Pressure Calibration Setup
When used in a real application, the sensor output may have additional components due
to the out-of-plane motion of the sensor caused by forces such as pressure (Section 1.1.3). The
out-of-plane sensitivity of the sensor is a result of a mismatch (non-ideal case) in the sensor
nominal capacitance. For out-of-plane motion both sensor capacitors change in the same
sense i.e., increase or decrease simultaneously. Thus the differential output voltage is zero
for matched capacitors. However, a capacitance mismatch results in a proportional output
voltage. This may be simply illustrated using the example in Section 6.1.2.1. Consider
C2 = λC1, where λ < 1, and with dc bias voltages, ±Vb. Consider the dynamic out-of-
plane motion is sinusoidal i.e., sin(ω1t). The resulting input voltage to the amplifier due to
sinusoidal variation in the capacitance and based on Equation 6–11 is
vinmismatchp= Vb
(1− λ)C1sin(ω1t)
C1 + C2 + Cp + Ci
. (6–15)
Thus, if λ = 1 (no mismatch) the output is zero while if 0 < λ < 1, the sensor has a finite
sensitivity to out-of-plane motion due to forces such as pressure.
To study the out-of-plane sensitivity, if any, the sensor is subjected to normal forces
(pressure). The pressure response of the sensor is directly measured by mounting the pack-
aged sensor at the end of the PWT to impart normal acoustic incidence (Figure 6-6).
6.1.3.3 Combined Shear and Pressure Measurement Setup
As discussed in the previous section, with mismatched capacitors, the sensor output
voltage may be a composite result of the possible multi-axis (in-plane and out-of-plane)
motion of the floating element. In this section the response of the sensor under simultaneous
shear stress and pressure inputs is investigated. To emphasize the importance of this study,
consider a shear stress of 0.5 Pa at 4.2 kHz. The corresponding acoustic pressure (based
on Equation 6–13) is approximately 142.7 dB or 273 Pa. This pressure is over two orders of
140
Techron 7540 Power
Supply Amplifier
B & K Pulse
Analyzer System
PC
Acoustic
Plane Wave
Shear Stress
SensorSpeaker
Microphone
Figure 6-6. A schematic of the dynamic calibration setup for measuring pressure sensitivityusing normal incidence acoustic waves.
magnitude higher than the shear stress value. Let Sshear and Spressure be the respective shear
and pressure sensitivities and let Vshear = Sshearτw and Vpressure = Spressurep be the respective
voltages due to shear and pressure inputs. Now, if Sshear ≈ 100 Spressure, then for the current
example, Vshear ≈ Vpressure. Thus the pressure sensitivity may be two orders of magnitude
smaller than shear sensitivity but, if the pressure input is two orders of magnitude higher
than the shear input, the pressure contribution to the total output is the same as that from
shear.
To investigate this, the PWT is fitted with an anechoic termination (a 30.7” long fiber-
glass wedge), which minimizes acoustic reflections to ensure plane propagating waves in the
duct. The composite sensitivity (shear and pressure) and the frequency response of the
sensor are studied with this setup, shown in Figure 6-7. The setup in Section 6.1.3.1 is not
used to study the frequency response of the sensor because the fixed sensor location (nodal
position) for pressure minima in the standing wave pattern limits the number of acoustic
frequencies.
6.1.4 Noise Measurement
A noise measurement of the packaged sensor provides an estimate of its shear stress res-
olution. It is most relevant to measure the noise floor in the actual shear stress measurement
setup. However, the noise floor of the data acquisition system for the dynamic measurements
141
Techron 7540 Power
Supply Amplifier
B & K Pulse
Analyzer System
PC
Acoustic Plane
Wave
Anechoic
Termination
Shear Stress
Sensor
Speaker
Microphone
Figure 6-7. A schematic of the dynamic calibration setup for shear stress measurement withplane progressive acoustic waves.
is at 1× 10−6 V/√
Hz, which is much higher than that of the sensor, limiting its utility for
noise measurements. Also, the acquisition system may be different based on a measurement
setup. The measurement in the Faraday cage helps to determine the absolute noise floor
of the sensor itself independent of the acquisition system used in a real application. The
packaged sensor, biased with known voltages, is placed in a double Faraday cage without
a physical input, as shown in Figure 6-8. The output of the SiSonic buffer is measured
using a SR785 spectrum analyzer. The details of the sources of noise and the experimental
setup for this measurement can be found in [88]. This experiment is repeated both with
(total noise) and without the sensor (setup noise) to isolate the sensor noise from that of the
measurement system.
6.2 Results and Discussion
This section provides the results of the experiments described in the previous sections.
The settings for each characterization setup is provided. The impedance, static, dynamic,
and noise characteristics of the sensor are explained via experimental plots.
6.2.1 Impedance Measurements
This section presents the impedance measurement results performed on the pre-packaged
sensor die using the probe station and the impedance analyzer. The impedance is measured
142
Spectrum
Analyzer
(SR 785)
Shielded Box
Shear Stress
Sensor
Shielded Box
+
-Rb Cp+i
Vout
Vb
-Vb -Vcc
+Vcc
Computer
(LabView)
Figure 6-8. A schematic of the noise measurement setup for the packaged shear stress sensorusing a double Faraday cage.
across C1 and C2 to estimate their nominal capacitances. The dc bias voltage across the
capacitors is swept from 0 − 18 V with a 400 mV (peak) source signal at 50 kHz. The
bandwidth for this measurement is set to 5, which precisely controls the sweep parameter
(dc bias voltage). The averaging in the system is turned off and the measurement is repeated
31 times with 100 points for each measurement. Ensemble averages are used to estimate the
mean and standard deviation of the measured capacitance as a function of bias voltage. Fig-
ure 6-9 shows the results from the impedance measurements. The results are significantly
different from predicted capacitance value in Chapter 4 due to large parasitic capacitances.
The predicted capacitance is 1.56 pF while the measured values are approximately 14.8 pF
and 16.3 pF for C10+p1 and C20+p2, respectively. The parasitic capacitance is formed be-
tween the highly doped device layer and the float zone bulk silicon substrate separated by
the dielectric BOX layer. The difference between C10+p1 and C20+p2 is roughly 1.5 pF , which
143
0 5 10 15
16.3
16.31
16.32
16.33
Bias (V)
C20
(pF
)
0 5 10 15
14.8
14.805
14.81
14.815
Bias (V)
C10
(pF
)
Figure 6-9. Capacitance from impedance measurements on the pre-packaged sensor die.
is approximately same as the predicted sensor capacitance. Based on the fabrication pro-
cess and the sensor geometry such a difference is unlikely in the sensor capacitance itself,
suggesting that the difference must be due to the unexpectedly large parasitic capacitance.
Furthermore, a suspected drift due to the probe station was a limitation in obtaining an
accurate quantitative measurement of C10+p1 and C20+p2. This restricted further character-
ization to estimate C10 and C20 and also to quantify their mismatch. The details of the
measurement drift, unexpected parasitic capacitance, and the consequent non-idealities in
the sensor performance are discussed in Section 7.2.2.1 and Section 7.2.1.1.
6.2.2 Mean Shear Stress Characteristics
This section presents the results from the experiments performed in the flow cell for
static calibration of the sensor. The sensor is biased with sinusoidal voltages having am-
plitudes ±2.5 V and frequencies of 10 kHz, respectively. Even with fine tuning using the
biasing circuitry, an ≈ 40 mV amplitude signal remains at the sensor output, while the
permissible input voltage to the SiSonic buffer is 100 mV . If a good capacitance match is
144
achieved during fabrication, higher bias voltages as per design may be applied. During the
measurement, a time delay of 20s is set between every change in state of the flow controller
and data acquisition to enable the flow to reach a steady state. The data acquisition from
the Heise is limited due to its slow updating speed while the sensor signal is at 10 kHz,
requiring higher sampling rates. Therefore, the pressure and voltage were sampled inde-
pendently, with the pressure sampled at the highest read rate allowed by the serial port
and the sensor signal sampled at 40 kHz and a uniform window is applied with no overlap.
Although simultaneous signal acquisition is ideal, because the measurement is static/mean,
enough averages in the data yield reasonable results. The acquisition time is limited to
the time taken for the pressure measurement due to the slow data acquisition. It is ideal
to have long acquisition times, since it is a mean measurement. However, due to drift in
the biasing voltage, resulting from potential charge build up on the on-the-board variable
capacitors, Cvar, the data acquisition times are kept relatively small. This measurement
uses autospectral estimates of the sensor output voltage at 10 kHz. The equipments and
settings for this measurement are summarized in Table 6-1. The number of averages and
the sampling times are chosen such that, for each flow rate setting, the pressure and sensor
voltage measurements approximately end at the same time.
Table 6-1. Measurement settings for static calibration in the flow cell.
Pressure differential data Number of averages 200
Sensor voltage data Sampling Frequency (kHz) 40
Number of Samples 2000
Number of linear spectral averages 500
Keithley 2400 sourcemeter compliance (mA) 100
SR560 Coupling AC
Filter frequency band (kHz) 3− 30
Gain 1
The shear stress input varies from 0.18 Pa to 2.9 Pa. Figure 6-10 shows the sensitivity
plot for this mean shear measurement. The initial voltage due to the capacitive mismatch is
145
subtracted from the measured values. The normalized random uncertainty in the measured
voltage is 4.5% (Section 6.2.5). A linear curve fit on the data set is used to estimate the
sensor sensitivity (R2 = 0.9958). The sensor has a linear sensitivity of 0.94 mV/Pa up to
the testing limit of 2.9 Pa.
0.5 1 1.5 2 2.5 3
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
Shear Stress (Pa)
Vol
tage
(m
V)
Figure 6-10. Linear sensor output voltage as a function of mean shear stress at bias voltageamplitude of 2.5 V at 10 kHz.
The plot indicates a large uncertainty in the shear stress value compared to the output
voltages but the fit is based only on the mean values of shear stress and sensor output
voltage. Such large error values warrant further analysis to deduce the range of sensitivities
within the measured uncertainty limits. This is achieved via a Monte Carlo simulation
assuming a Gaussian profile for the uncertainty at each point. Each measurement point is
randomly perturbed from its mean value within its error bounds in both shear stress and
output voltage. A linear fit on the resulting curve is used to estimate the sensitivity. This
procedure is repeated 50000 times to build a distribution of the overall shear stress sensitivity
represented in the form of a histogram. The corresponding distribution of the normalized
residuals values (R2) for each fit are also plotted. Figure 6-11 shows the histograms for the
sensitivity and R2. The sensitivity and R2 values and the 95% confidence intervals from this
146
study are summarized in Table 6-2. The 2D scatter of the sensitivity and R2 indicates that
the sensitivity from the measurement is towards an extreme of the distribution where the
fit is better with R2 = 0.9958. However, the perturbations about the mean measured values
result in a sensitivity that is much different (≈ 1.02 mV/Pa) but with lower mean quality
of fit (R2 = 0.8288). The ellipsoidal shape of the scatter plot provides some intuition into
this effect. An important assumption for this analysis is that the perturbations for each
data point are uncorrelated, which is reasonable since each measured data point should be
a statistically independent value.
0.98 1 1.02 1.04 1.060
500
1000
1500
Sensitivity (mV/Pa)
(a) Sensitivity
0.75 0.8 0.85 0.9 0.950
500
1000
1500
Normalized Residual (R2)
(b) R2
(c) Joint Histogram
0.98 1 1.02 1.04 1.06
0.75
0.8
0.85
0.9
0.95
Sensitivity(mV/Pa)
Nor
mal
ized
Res
idua
l (R
2 )
(d) 2D Scatter
Figure 6-11. Distribution of the measured shear stress sensitivity due to uncertainty in themeasured shear stress and sensor output voltage.
147
Table 6-2. Sensitivity estimates using Monte Carlo technique onthe mean shear stress measurement data.
Parameter Mean Value (µMC) 95% ConfidenceInterval (2σ)
Sensitivity (mV/Pa) 1.017 0.997− 1.037
NormalizedSensitivity(mV/V/Pa)
0.407 0.399− 0.415
Normalized Residual(R2)
0.8288 0.7751− 0.8825
6.2.3 Dynamic Characteristics
As previously described (Section 6.1.3), the sensor characterization in the PWT involves
three different setups. Dynamic shear sensitivity results are obtained using the rigid termi-
nation in the PWT. Pressure sensitivity results are illustrated with the sensor mounted at
the end of the PWT for normal acoustic incidence. The frequency response and combined
sensitivity (shear + pressure) results, using the anechoic termination, are provided. During
each measurement, the recorded quantities are as follows: the respective auto power spectral
densities of the sensor and the reference microphone outputs and the transfer function and
coherence between the sensor and the reference microphone signal. For all sensitivity esti-
mates, the measured autospectra of the respective sensor and reference microphone signals
are used. For the frequency response measurement, the computed sensitivities and the trans-
fer function (sensor to microphone) are used. The measurement settings were kept the same
for all dynamic measurements and are summarized in Table 6-3. The normalized random
error for the autospectrum estimates is 3.2%. The normalized random errors for the mag-
nitude and phase of the frequency response function are each 0.04% (Refer Section 6.2.5).
6.2.3.1 Dynamic Shear Sensitivity
The sensor dynamic sensitivity for shear is measured using a rigid termination (Fig-
ure 6-5), where the sensor is placed at a quarter wavelength (velocity maxima) away from
148
Table 6-3. Measurement settings of B&K PULSE Multi-Analyzer System (Type 3109) for dynamic calibra-tion in the PWT.
FFT mode zoom
Windowing function uniform
Percentage Overlap 0
Frequency span 200 Hz − 6.4 kHz
Number of samples/FFT lines 400
Sampling time 62.5 ms
Time steps 122.1 µs
Total acquisition time 62.5 s
Frequency bin width 16 Hz
Number of linear spectral averages 1000
the termination. The chosen frequency based on the sensor location is 1.128 kHz. The input
shear stress is increased by raising the input sound pressure level (SPL) in steps of 5 dB
from 80 dB to 160 dB. The upper and lower ends of the SPL range are set by the perfor-
mance limits of the compression driver. The shear stress corresponding to the input acoustic
pressure varies from 0.1 mPa to 1.9 Pa. Figure 6-12 shows the dynamic shear sensitivity of
the sensor. The sensitivity of the sensor is deduced using a linear curve fit on the data set
(R2 = 0.99995). The estimated sensor sensitivity is 7.66 mV/Pa at dc bias of Vb = 10 V .
During this measurement, the SPL at the sensor axial location was ≈ 40 dB lower than the
maximum pressure measured by the reference microphone at the termination.
The normalized sensor sensitivities from both the mean and dynamic shear stress mea-
surements are 0.407 mV/V/Pa and 0.766 mV/V/Pa, respectively. The difference between
the two values is roughly 47%, which may be attributed to non-idealities in the sensor and
in the measurement setup several of which are discussed in Chapter 7.
149
10−3
10−2
10−1
100
10−2
10−1
100
101
Shear Stress (Pa)
Vol
tage
(m
V)
Figure 6-12. Sensor output voltage at 1.128 kHz at a bias voltage of 10 V . Sensor is placedat a quarter wavelength (velocity maxima) from the rigid termination.
6.2.3.2 Pressure Sensitivity
The sensor mounted at the end of the PWT for normal acoustic incidence, is experi-
mentally characterized for pressure sensitivity at 4.2 kHz and dc bias voltages of Vb = 5 V ,
8 V , and 10 V , respectively (Figure 6-13).
The frequency (4.2 kHz) is chosen for comparison with the combined sensitivity mea-
surements that are presented subsequently. The SPL range for the experiment is 80−150 dB
varied in steps of 5 dB, to be consistent with the shear sensitivity tests. The minimal varia-
tion of the output at low SPL is because the sensor output voltage is below the noise floor of
the measurement system. The pressure sensitivity of the sensor at Vb = 10 V is 4.8 µV/Pa
(R2 = 0.9976), which is about 1000 times smaller than the shear sensitivity of the sen-
sor (Figure 6-5 and Figure 6-10). A comparison of the dynamic and shear sensitivities at
Vb = 10 V results in a pressure rejection, Hp, of approximately 64 dB for comparable forcing
in shear and pressure, which is computed as
Hp = 20log(Sshear/Spressure). (6–16)
150
100
101
102
103
10−3
10−2
10−1
100
101
Pressure (Pa)
Vol
tage
(m
V)
Vb = 10 V
Vb = 8 V
Vb = 5 V
Figure 6-13. Voltage output as a function of pressure at 4.2 kHz at different bias voltages.
6.2.3.3 Combined Sensitivity and Frequency Response
The sensor sensitivity due to a possible multi-axis motion resulting from both shear
and pressure inputs is tested in this study. The anechoic termination is used, which ideally
results in plane progressive acoustic waves. Weak reflections if any are accounted for via
Equation 6–14. Since there are no pressure nodes with progressive acoustic waves (with
weak reflections), the sensor encounters both shear and pressure. The experiments are
conducted at three different dc bias voltages, 5 V , 8 V , and 10 V , at a frequency of 4.2 kHz.
This frequency is chosen to ensure high signal to noise ratio (for shear stress) even at low
SPLs, which is limited by the driving capability of the compression driver (From Equation 6–
13, τ ∼ p′√
ω). At the same time, the frequency is kept well below the estimated sensor
resonance (5 kHz). The input SPL is varied in steps of 5 dB from 80 dB to 150 dB. The
shear stress corresponding to the measured pressure input varies from 0.4 mPa to 1.16 mPa.
The sensitivity plots corresponding to this measurement are shown in Figure 6-14.
Note that although the voltage measurement is plotted as a function of shear stress, cross
axis (out-of-plane) sensor sensitivity due to pressure also contributes to the output voltage.
In all three cases, the sensor exhibits a linear sensitivity (R2min = 0.9995) up to the testing
151
10−3
10−2
10−1
100
10−2
10−1
100
101
Shear Stress (Pa)
Vol
tage
(m
V)
Vb = 10 V
Vb = 8 V
Vb = 5 V
Figure 6-14. Linear sensor output voltage as a function of shear stress at 4.2kHz at 3 differentbias voltages.
limit of 1.1 Pa. The measured sensitivities are 23 mV/Pa, 19 mV/Pa, and 11 mV/Pa,
respectively at Vb = 10 V , 8 V , and 5 V . The combined sensor sensitivity is directly
proportional to the applied bias voltage as expected (Equation 4–3). A plot of the normalized
sensitivity (output voltage per voltage bias) indicates the same via the collapse in the data
for different bias voltages (Figure 6-15). The normalized sensitivities agree closely to
within 6% and are 2.268 mV/V/Pa, 2.316 mV/V/Pa, and 2.196 mV/V/Pa, at Vb = 10 V ,
8 V , and 5 V , respectively. Comparison of the normalized sensitivity estimates from the
rigid (2.268 mV/V/Pa) and anechoic termination (0.766 mV/V/Pa) experiments indicates
significant difference (66%) in the sensor performance in the presence of large pressure signals.
Note, unlike the anechoic case, for the rigid termination the sensor is primarily forced
by shear stress given that it is at a pressure node and velocity maximum. The monitored
pressure at this node is ≈ 40 dB or two orders of magnitude below the peak acoustic pressure.
Also, with a combined input having large pressure signals in addition to shear, the sensor
shear sensitivity may drop from its nominal value. For example, consider the sensor has
a finite out-of-plane deflection. This reduces the overlapping area of the capacitors, which
152
10−3
10−2
10−1
100
10−5
10−4
10−3
Shear Stress (Pa)
Nor
mal
ized
Sen
sitiv
ity (
V/V
)
Vb = 10 V
Vb = 8 V
Vb = 5 V
Figure 6-15. Sensor output voltage normalized by bias voltage as a function of shear stressat 4.2 kHz.
lowers the sensitivity to in-plane motion, shown graphically in Figure 6-16. Furthermore, the
effective in-plane compliance may be lower since the deflected (out-of-plane) structure offers
a higher stiffness to in-plane motion. The stiffening of the device may not be an issue for
small out-of-plane deflections, but for large deflections (large pressures) this may significantly
alter device performance. This may need further theoretical and experimental analysis for
better understanding of the sensor response to combined shear and pressure inputs. The
frequency response of the sensor is measured for Vb = 10 V , with the anechoic termination
in place (Figure 6-7). As previously stated a rigid termination may not be used for this
experiment since the pressure nodes in the resulting standing wave pattern are limited. This
restricts the different discreet frequencies available for testing the frequency response. The
sensor and the reference microphone are placed at the same axial location along the length
of the tube. The expression of the frequency response normalized by the input shear stress
is [86]
H(f) =τout
τin
=V (f)
τin
∂τ
∂V, (6–17)
153
Original
Position
Displaced
PositionStationary
Comb Fingers
Figure 6-16. Schematic showing a set of overlapping comb fingers deflection due to both
shear (δs) and due to pressure (δp).
where V (f) is the sensor output voltage corresponding to the known shear stress input, τin,
which is theoretically estimated using Equation 6–13. The term ∂τ/∂V is the inverse of
the flat-band sensitivity magnitude of the sensor. There is an additional phase component
in the transfer function between the sensor and microphone. This component changes as
a function of the termination used, SPL, and the respective locations of the sensor and
reference microphone in the acoustic field (Section 7.2.2.2). Correcting for this phase, θ,
Equation 6–17 may be rewritten as
H(f) =τout
τin
=V (f)
τin
∂τ
∂Ve−jθ. (6–18)
154
To get a more accurate estimate of the normalized frequency response, the results are com-
puted using the sensitivity magnitude from the rigid termination experiments at Vb = 10 V .
To be consistent with the frequency response measurements, the phase value from the com-
bined sensitivity experiment, with the anechoic termination, is utilized. Figure 6-17 shows
the magnitude and phase of the measured frequency response function (FRF) conducted at
Vb = 10 V for a theoretical shear stress input magnitude of 0.5 Pa. The SPL is approxi-
mately 140 dB± 2 dB over the range of testing frequencies. A single frequency is excited at
a time and the SPL is adjusted to maintain a constant shear stress (theoretical).
0 1000 2000 3000 4000 5000 6000 70000
20
40
60
Frequency (Hz)
|H(f
)| (
τ out/τ
in)
0 1000 2000 3000 4000 5000 6000 7000−150
−100
−50
0
50
Frequency (Hz)
Pha
se(D
eg)
Figure 6-17. Frequency response of sensor at Vb = 10 V using τin = 0.5 Pa as the referencesignal up to the testing limit of 6.7 kHz.
As expected from the design, the FRF shows a clear second-order system response with
a flat band region and a resonance frequency of ≈ 6.2 kHz. The quality factor, Q, is
approximately 79. Assuming a second order system, this translates into a damping ratio,
155
ζ = 0.0063, which is computed using
Q =1
ζ√
1− ζ2. (6–19)
The phase stays constant at zero and shifts to ≈ 90 at resonance. However, there are certain
aspects of the data, which need closer attention and further investigation. For instance, the
normalized magnitude, |H(f)| is only qualitative since only the shear sensitivity is used for
the estimation while the measurement is a combination of both shear and pressure. Fur-
thermore, beyond resonance the phase starts to flatten out well before reaching 180. Both
these effects may be attributed to the pressure sensitivity of the sensor and its effect on the
combined measurement. Preliminary investigation along those lines and recommendations
to further improve the FRF estimate of the sensor, both during design and characterization
are discussed in Section 7.2.2.2.
6.2.4 Noise Characteristics
The noise of the packaged sensor is measured at three dc bias voltages: Vb = 10 V , 8 V ,
and 5 V . The spectrum analyzer settings for the noise measurement, performed piecewise
over a span of 12.8 kHz, are summarized in Table 6-4.
Table 6-4. Spectrum analyzer settings for noise measurement in a double Fara-day cage.
Parameter Settings
Span (Hz) 100 400 1600 12800
Number of samples/FFTlines
800 800 800 800
Bin width (Hz) 0.125 0.5 2 16
Number of linear spectralaverages
30 240 1200 4800
Channel coupling AC AC AC AC
Windowing function Hanning Hanning Hanning Hanning
Window overlap (%) 75 75 75 75
156
100
102
104
10−7
10−6
10−5
10−4
10−3
Frequency [Hz]
PS
D [V
rms/r
tHz]
Vb = 10 V
Vb = 8 V
Vb = 5 V
Figure 6-18. Measured output referred noise floor of the packaged sensor in Vrms/√
Hz atdifferent bias voltages.
800 900 1000 1100 1200
0.8
1
1.2
1.4
1.6
1.8
x 10−7
PS
D [V
rms/r
tHz]
Frequency [Hz]
V
b = 10 V
Vb = 8 V
Vb = 5 V
Figure 6-19. Zoomed in plot of the output referred noise floor of the packaged sensor near1 kHz at different bias voltages.
157
The measured noise power spectral density of the sensor is as shown in Figure 6-18.
The highest measured thermal noise power spectral density is 114 nV/√
Hz at f = 1 kHz
with 1 Hz frequency bin for Vb = 10 V . Figure 6-19 shows a zoomed in plot of the sensor
noise floor around 1 kHz. The measured noise power spectral density increases with the
applied bias voltage. Two potential reasons exist for such a dependence on Vb. First, the
noise contribution from the bias source itself may increase with its output voltage, raising
the sensor noise floor. Second, the nominal capacitances C1 and C2 may change with the
applied bias. At each bias voltage the floating element settles at a new mean position, which
is a result of electromechanical coupling and the initial nominal capacitance mismatch. The
noise at the amplifier input is thus a function of the resulting impedance divider formed
by the sensing capacitors. Table 6-5 shows the integrated noise floor of the sensor over the
entire range of measurement and over specific frequency ranges of interest. This is relevant
for time domain shear stress measurements for real time flow control applications.
Table 6-5. Integrated voltage noise floor of the sensor at differ-ent frequency ranges.
Frequency range Integrated noise floor
15.6 mHz − 12.8 kHz (totalmeasured noise)
1.5 mV
10 Hz − 10 kHz 27.3 µV
100 Hz − 10 kHz 18.2 µV
1 kHz − 10 kHz 10.2 µV
The MDS of the sensor, calculated from Equation 4–5 using the dynamic sensitivity at
Vb = 10 V , is 14.9 µPa (f = 1 kHz @ 1 Hz bin). This corresponds to a mechanical deflection
δ = 0.17 pm and a capacitance change ∆C = 75 zF , which are computed using the design
values for the geometry and assuming parallel plate capacitances. As discussed previously,
the quantitative accuracy of |H(f)| (shear output per unit shear input as a function of
frequency) is questionable due to the combined shear and pressure response. However,
an accurate estimate of |H(f)|, if available, may be used to obtain an MDS spectrum in
158
µPa/√
Hz. Such a spectrum may then be used with a weighting function, represented by
the shape of the turbulent energy spectrum in a given application, to estimate the integrated
sensor noise floor (MDS) in Pa.
Using the maximum tested linear range of shear stress as the upper limit and the noise
floor at 1 kHz as the lower limit, the dynamic range of the sensor is at least 102 dB. Since
the sensor output is still linear at the maximum available shear stress input, its dynamic
range is expected to be higher than 102 dB. The sensor was designed for τmax = 5 Pa, and
if this design limit were achieved, the dynamic range would be 110.5 dB.
6.2.5 Experimental Uncertainty Estimation
The random uncertainty for the presented measurements is computed for both spec-
tral density and frequency response estimates used in this chapter. For the autospectral
estimates, which are used for computing static, dynamic, and pressure sensitivities, the
normalized random error is computed using [89]
εr =
√V ariance[Gss(f)]
Gss
=1√
ndeff
, (6–20)
where Gss is the estimate of the auto powerspectral density of the sensor output voltage, Gss
is the mean power spectral density from the measurement. The symbol, [∧], stands for the
estimator of a given quantity. The effective number of averages, ndeff, rounded to an integer
value, is given as
ndeff= 1 +
(N − 1)
1− r, (6–21)
where N , is the number of averages during the measurement and r is the percentage over-
lap of the windowing function used. All the measurements in this chapter except noise
measurements are performed with a uniform window, for which r = 0.
159
The normalized random error for the magnitude,∣∣∣Hms(f)
∣∣∣, and phase, |φms(f)|, of the
frequency response function estimate are [89]
εr||Hms(f)|= |Gms(f)|
Gmm(f)
=
√1− γ2
ms(f)
|γms(f)|√2ndeff
(6–22)
and εr|φms(f)=
√1− γ2
ms(f)
|γms(f)|√2ndeff
. (6–23)
Here Gmm and Gms represent autos and cross power spectral densities, respectively, and γms
represents the coherence function between the sensor and the reference microphone signal.
6.3 Summary
The sensor functionality and performance is demonstrated for both mean and dynamic
shear stress inputs. The results show great promise in the conceptual design of the proof
of concept sensor. At a bias of 10 V , the sensor possesses a linear dynamic sensitivity
of 7.66 mV/Pa up to the testing limit of 1.9 Pa and a noise floor of 114 nV/√
Hz (f =
1 kHz @ 1 Hz bin). This translates into an equivalent MDS of 14.9 µPa (f = 1 kHz @ 1 Hz bin)
and a dynamic range of 102 dB. The sensor has a relatively flat frequency response with
bandwidth defined by its resonance at 6.2 kHz. The sensor functionality is demonstrated
for both static and dynamic shear stress input. The normalized mean and dynamic shear
sensitivities are 0.407 mV/V/Pa (mean shear) and 0.766 mV/V/Pa (dynamic shear), re-
spectively. The sensor has a pressure sensitivity of 4.8µV/Pa which translates to a pressure
rejection of 64 dB. Table 6-6 provides a comparison of the predicted and measured values
for the sensor.
The bandwidth and dynamic range estimates are in close agreement while the sensitivity,
noise floor, and MDS estimates are significantly different. The difference in predicted and
measured sensor performance has several possible explanations. A large parasitic capacitance
exists between the sensor electrodes due to the electrical path through the float zone silicon
substrate, lowering sensitivity. The capacitance mismatch and the noise contribution of the
biasing sources increases the overall noise floor of the system. These two effects are directly
160
Table 6-6. Comparison of predicted and measured sensor performance param-eters.
Parameter Predicted Measured
Normalized SensitivitySoverall (mV/V/Pa)
1.424 0.407 (mean)/0.766(dynamic)
Bandwidth (kHz) 5 6.2
Noise Floor(nVrms/
√Hz
∣∣f=1 kHz @ 1 Hz bin
)20 114 @ Vb = 10 V
MDS(µPa
∣∣f=1 kHz @ 1 Hz bin
)0.562 14.9
Dynamic Range atVb = 10 V andτmax = 1.9 Pa (dB)
107 102
reflected in the higher MDS and lower dynamic range of the sensor. The difference in resonant
frequencies may be due to a different flexural rigidity (EI). The assumed Young’s modulus
of Si may be different from that of the actual wafer. Also, geometric variations during
fabrication may result in a different moment of inertia I. Lastly, the pressure gradient
errors, and forces from complex flow around the sensor may alter the effective force on the
sensor leading to errors (flow different during static and dynamic shear). Recommended
changes to the sensor design and fabrication and an approach for improved characterization
are presented in Chapter 7.
161
CHAPTER 7CONCLUSIONS AND FUTURE WORK
This chapter summarizes the main research contributions of this work. Recommenda-
tions are provided for future work with suggestions for improvement while highlighting some
of the non-idealities in the current design.
7.1 Conclusions
Direct wall shear stress measurement continues to be of importance for the fundamental
study of turbulence, flow control, and for non-intrusive flow measurement applications. A
measurement standard for wall shear stress and its lack thereof, especially for high Reynold’s
number applications, limits the improvement of existing aerodynamic systems and testing
facilities. This continues to drive the development of micromachined direct wall shear stress
sensors since they are non-intrusive and scale favorably for high speed applications. Micro-
machined floating element sensors have been widely investigated for direct wall shear stress
measurements. The previous research efforts were reviewed in Chapter 2, which highlighted
their capabilities and limitations.
Using some lessons learnt from previous research, three key areas were identified and
focusing on these contributed to the overall improvement in sensor performance. These
key areas, which also give the main contributions of this work, are a simple and novel
sensor structure, an effective and potentially inexpensive fabrication process, and a system
level optimization of the design. The sensor was designed to have an asymmetric comb
finger structure to form capacitors. While achieving a relatively flush surface, this design
allows the use of a single material layer to define the sensor structure and electrical contact
pads. Additional capacitance between the stationary device substrate and both the floating
element and tethers, was also utilized to improve sensitivity. The geometry of the sensor
and a differential sensing scheme help in the attenuation of any output due to out-of-plane
motion resulting from pressure, vibration, etc.
162
The sensor structure was fabricated on a highly conductive substrate (highly doped Si).
A potentially cost effective technique for mass production of such sensors was developed using
a two mask fabrication process. A seedless electroplating on the highly doped Si helped to
passivate the sensing electrodes with metal to prevent drift arising from charge accumulation
on the sensing electrodes.
In the system level design and optimization, the electrical and mechanical systems and
the interface circuit were included. This resulted in an overall improvement in the predicted
and realized sensor performance to give a high sensitivity and low noise floor. This work
presented the mechanical and electrical modeling, a system level optimization, fabrication,
and characterization of the first differential capacitive sensor to make both mean and dynamic
shear stress measurements with sufficient pressure rejection.
The sensor has the highest dynamic range (> 102 dB) and the lowest noise floor/MDS
(14.9 µPa) reported to date for a micromachined direct shear stress sensor. It is also the first
MEMS sensor to have quantified the fluctuating pressure sensitivity for a floating element
shear stress sensor. The sensor possesses a pressure rejection of 64 dB. The dynamic
characterization revealed a flat frequency response, capturing the resonance of the sensor
structure in both magnitude and phase at ≈ 6.2 kHz, which has never been done before.
Table 7-1 provides a comparison of the sensor performance with some previous work. The
comparison indicates that the present work outperforms the previous best by three orders
of magnitude in MDS and at least an order of magnitude in dynamic range.
Table 7-1. Comparison of measured sensor performance with previous work.
Shear StressSensor
Element Size(mm×mm)
τw Range (Pa) fmax
(kHz)Sensitivity(mV/Pa)
Present Work 2.0× 2.0 1.9−14.9×10−6 6.2 7.66
Zhe [44] 3.2× 3.2 0.16− 0.04 0.531 337
Padmanabhan [50] 0.5× 0.5 10− 1.4× 10−3 16 † 320
Schmidt [37] 0.5× 0.5 13− 0.01† 10 0.47† Theoretically predicted values.
163
These encouraging results represent a step towards development of instrument grade
direct shear stress sensors. The benefits of the comprehensive design, fabrication, and pack-
aging approach, based on physical insight, are reflected in the demonstrated sensor per-
formance. The ability to make successful measurements with such a sensor also presents
the opportunity to identify some of the non-idealities with the sensor design and in the
characterization. The next section discusses some of these non-idealities.
7.2 Non-Idealities in Sensor Design and Characterization
Despite very encouraging sensor performance results, some issues exist both in the
sensor design and the characterization setup. These need to be addressed for better design
and accurate calibration of the sensor for real world applications. Experimental evidence or
simulations that helped identify some of these issues are presented in this section.
7.2.1 Design Aspects
This section provides a discussion on the non-idealities, which pertain to the sensor
design and may be addressed in subsequent designs. Based on the characterization results,
parasitic capacitance and pressure sensitivity are the two main design aspects that may be
improved upon.
7.2.1.1 Parasitic Capacitance
The first issue is the parasitic capacitance, which results from having a float zone bulk
substrate, instead of an insulator, underneath the device layer. Figure 7-1 shows a schematic
of a single bondpad to explain the sources of parasitic capacitance. During fabrication,
only the parallel plate capacitance, CP , between the bond pad and the stationary substrate
around it was considered. Float zone (FZ) bulk silicon was used in the SOI, assuming that
the electrical path through this silicon would be negligibly small. However, this resulted in
a finite resistive path through the silicon. As a result, the capacitance due to the dielectric
SiO2 between the highly conductive device layer and the weakly conductive bulk substrate
adds to the parasitic capacitance. This capacitance due to the stationary substrate is CoxS
and that from the bond pad is CoxBP .
164
RFZ RFZ
CoxBP
CoxSCoxS
CP CP
A A
Section View
Top View
Bond Pad
Stationary
Substrate
Bulk
Silicon
Device
Silicon
Oxide
(BOX)
-Vb+Vb
Figure 7-1. Schematic with the top view and section of a sensor bond pad showing theassociated electrical impedances.
The circuit model for the entire sensor, including the additional parasitic impedances,
is shown in Figure 7-2. As an example, a single ended sense capacitance value between
points 1 and 2 is simulated as a function of frequency for substrate resistance, RFZ = 10 KΩ
and RFZ = 1 MΩ. The parallel plate capacitance model is used for computing the various
capacitance values. The substrate and bond pad areas serve as parallel plates separated
by the dielectric BOX thickness. The simulated results are shown in Figure 7-3. The
plot indicates that as the substrate resistance or the frequency increases, the additional
parasitic capacitance from the substrate vanishes and the capacitance reaches a much lower
value (CP ). Thus, by choosing a high resistivity substrate and a high frequency bias signal
the adverse effects of parasitic substrate capacitance may be mitigated, improving overall
sensitivity.
This analysis was based on constant values of CoxS and CoxBP . In reality, these capaci-
tance values are time dependent and for mean shear stress measurements, they may vary
165
CP CP
CoxS CoxS
CoxBP
RFZRFZ
C2C1
+Vb -Vb
1 2 3
Figure 7-2. Schematic with the top view and section of a sensor bond pad showing theassociated electrical impedances.
102
104
106
108
0
2
4
6
8
10
12
Frequency (Hz)
Cef
f (pF
)
RFZ
=10 KΩ
RFZ
=1 MΩ
Figure 7-3. Effective sense capacitance variation with frequency as a function of substrateresistance.
with the applied sinusoidal bias voltages [90]. These capacitors are similar to metal oxide
semiconductor (MOS) gate capacitors. In the present device, the highly conductive device
layer, the BOX, and the float zone bulk substrate form the MOS capacitors. The capaci-
tance value depends on the depletion layer thickness which varies with the bias voltage value.
A delta-depletion model is used to explain the voltage dependence of the MOS capacitor.
166
Detailed description of this model can be found in [91]. Based on this model, the MOS
capacitance value as a function of the applied bias voltage is given as [91]
C =Cox√
1 + Vbsin(ωt)Vδ
, for π < ωt < 2π. (7–1)
Here, Cox is the substrate capacitance due to the BOX, excluding the depletion width. The
term
Vδ ≡ q
2
Ksx20
K20ε0
NA, (7–2)
where q is the unit charge on an electron, x0 is the BOX thickness, NA is the total number
of acceptors in the p-substrate (FZ), and Ks and Ko are the relative permittivities of silicon
and BOX, respectively. Equation 7–1 indicates that the parasitic MOS capacitance varies
nonlinearly with an applied sinusoidal bias voltage. Thus the attenuation factor Hc (Equa-
tion 3–95), which is a function of the parasitic capacitance, also induces a nonlinear variation
in the sensor sensitivity. Therefore, the overall sensor sensitivity, Sτ,overall, also varies with
an applied fluctuating bias voltage. This may inhibit the use of the sensor for simultaneous
mean and dynamic shear stress measurements with sinusoidal bias voltages. Given all these
different issues, using an insulating substrate instead of float zone silicon will eliminate the
parasitic substrate capacitance (MOS) issues, improving overall sensor performance.
7.2.1.2 Pressure Sensitivity
The output due to out of plane sensitivity of the sensor was mitigated in the design
by adjusting the structural geometry of the design and using the differential capacitance
scheme. The back cavity underneath the sensor, formed by the backside release process, has
an associated compliance/stiffness, Cmecav , due to the compressibility of the fluid in it. In a
lumped sense, the cavity compliance opposes the out of plane motion of the sensor referred to
as the cavity stiffening. If the out of plane compliance of the sensor is Cmeop , the attenuation
167
of the pressure induced sensor output due to cavity stiffening is
Hcavity =Cmecav
Cmecav + Cmeop
. (7–3)
In addition to this, the back cavity is vented to the atmosphere through the gaps around
the sensor, which are necessary for a released floating element structure. The vent offers a
resistance to fluidic flow across the sensor thickness between the front and back side. The
cavity compliance and vent resistance, Rv, form a high pass filter with a corner frequency of
f =1
2π√
RvCmecav
. (7–4)
Microphone designers commonly design the compliance and the vent resistor to have a low
cut-on frequency. In contrast, in a shear stress sensor, these should be designed to move this
cut-on frequency beyond the operating bandwidth for shear stress. This modeling aspect has
not been explored previously and could greatly benefit floating element shear stress sensor
design.
7.2.2 Characterization Aspects
The characterization of the sensor revealed some interesting results which need fur-
ther investigation for better comparison with results from the predictive tools developed in
this dissertation. This section concentrates on three different aspects of characterization:
impedance measurements, dynamic sensitivity, pressure frequency response, and errors from
pressure gradient effects.
7.2.2.1 Impedance Measurement Drift
As mentioned in Section 6.2.1, an accurate estimate of the difference between C10 and
C20 was not achieved due to drift associated with the probe station. Figure 7-4 shows the
drift in the measurement of one of the sensor capacitors. The plot shows the first and the last
measurement sweep out of the 31 sweeps of the dc bias voltage for ensemble averaging. The
trend in the the drift is also indicated in Figure 7-5, which is a plot of the mean capacitance
168
for each dc bias sweep vs the sweep/measurement number N . The figure indicates that
initially, the drift in mean capacitance is high but tends to decrease as time progresses.
0 5 10 15 2014.79
14.795
14.8
14.805
14.81
14.815
14.82
Bias (V)
C10
(pF
)
FirstLast
Figure 7-4. Capacitance measurement drift indicated via the first and last measurement priorto ensemble averaging.
However, longer measurements indicate that the drift continues over time. The drift is about
5 fF over the course of 31 bias voltage sweeps, which is roughly 22% of the predicted full scale
capacitance change. Such a high drift is therefore a limitation in effectively quantifying the
nominal capacitance and the change in capacitance as a function of bias voltage. Potential
causes of drift in the probe station include, charge build up on the probes and poor contact
of the probe tips with the sensor bond pads. A lack of a standard substrate was also a
limitation in the complete understanding of the nature and source of the drift.
7.2.2.2 Dynamic Shear Measurements
The dynamic shear stress measurements on the sensor were performed with two differ-
ent kinds of terminations at the end of the PWT, the anechoic termination and the rigid
termination. Note that, for the anechoic termination the reference microphone is at the same
axial location as the sensor, whereas for the rigid termination, it is located at the termination
(Section 6.1.3). The dynamic sensitivity was determined based on the autospectral estimates
169
0 10 20 30 4014.802
14.804
14.806
14.808
14.81
Measurement Number (N)
C10
mea
n (pF
)
Figure 7-5. Drift in mean capacitance with subsequent dc bias sweeps.
of the microphone and sensor signals while maintaining high coherence values between these
signals. A constant sensitivity implies that the transfer function, Hms, between the sensor
and the microphone signal should be constant. However, contrary to expectation, Hms, varies
with the SPL. Figure 7-6 and Figure 7-7 show the transfer function as a function of SPL for
the calibrations with the rigid and anechoic terminations, respectively, at Vb = 10 V .
The maximum variation in the magnitude, |Hms|, over the range of SPLs is ≈ 221 nV
for the rigid termination while it is ≈ 2.99 µV for the anechoic termination. Similarly, the
maximum variation in the phase, φms, is ≈ 2.5 for a rigid termination and it is ≈ 11
for the anechoic termination. Since the same microphone was used for the reference SPL
measurements, this variation is potentially a function of the termination at the end of the
tube and the SPL at the sensor location. In contrast to the anechoic termination case, for the
rigid termination, the SPL is ≈ 40 dB lower than the peak SPL at the reference microphone.
This intuitively suggests that local SPL at the sensor location significantly affects the sensor
output. While the high SPL variation may be attributed to the acoustic nonlinearity, the
low SPL behavior needs further investigation.
170
100
102
104
106
6.9
7
7.1
7.2x 10
−6
|Hm
s|(V
/Pa)
Rigid
100
102
104
106
3.9
4
4.1
4.2x 10
−5
Pressure (Pa)
|Hm
s|(V
/Pa)
Anechoic
Figure 7-6. Magnitude of sensor to microphone transfer function as a function of acousticpressure.
100
102
104
106
−24
−23
−22
−21
φm
s(D
eg)
Rigid
100
102
104
106
70
75
80
Pressure (Pa)
φm
s(D
eg)
Anechoic
Figure 7-7. Phase of sensor to microphone transfer function as a function of acoustic pressure.
171
Clearly, the comparison indicates that, using a rigid termination with the sensor placed
at an odd multiple of the quarter wavelength (nλ/4 where n is odd) is a better technique to
measure the dynamic behavior of the sensor. Therefore, for future dynamic measurements
for both sensitivity and frequency response, it is recommended to use a rigid termination
with variable location of the rigid plane for tuning. This will ensure that the sensor may
always be located at a pressure minima and a velocity maxima.
7.2.2.3 Dynamic Pressure Measurements
The sensor demonstrated a high pressure rejection, which has never been quantified in
the MEMS shear stress literature. The study of the pressure frequency response is also im-
portant for quantitative shear measurements, since the physical phenomenon usually consists
of shear and pressure inputs. Similar to the dynamic shear measurement studies, the trans-
fer function between the sensor and the reference microphone was investigated for variation
with the input SPL. Figure 7-8 shows the variation of the transfer function with the input
SPL at Vb = 10 V .
100
102
104
106
3
4
5
x 10−6
|Hm
sp
ressu
re
|(V
/Pa)
100
102
104
106
−150
−140
−130
−120
−110
Pressure (Pa)
|φm
sp
ressu
re
|
Figure 7-8. Transfer function between the sensor and the reference microphone for normalacoustic incidence.
172
This exercise gives results similar to dynamic shear measurements, where both magni-
tude and phase of the transfer function vary with SPL. This requires further investigation
from both the sensor design and acoustics point of view for a better understanding of these
results.
1000 2000 3000 4000 5000 600010
−6
10−5
10−4
|Hm
s|(V
/Pa)
Shear+pressurepressure
1000 2000 3000 4000 5000 6000
−50
0
50
100
Frequency (Hz)
φm
s(D
eg)
Shear+pressurepressure
Figure 7-9. Comparison of combined (shear and pressure) and pressure transfer functionsmeasured between sensor and reference microphone.
Next, the combined pressure and shear transfer function measured with the anechoic
termination is compared with the pressure transfer function measured using normal acoustic
incidence shown in Figure 7-9. A higher, combined pressure-shear transfer function indicates
that despite similar and higher pressure forces, the sensor remains more sensitive to shear in
the frequency range of measurement than it is to pure pressure. The scatter in the pressure
data is due to the poor performance of the sensor for pressure inputs. Note, that the sensor
was not designed for pressure measurements and relies purely on the capacitance mismatch
(ideally zero) for this measurement. Similar qualitative nature of the two magnitude re-
sponses may be due to the shaping of the overall transfer function by the pressure response
173
of the sensor. Also, for out-of-plane motion of the sensor, close to its in-plane resonance fre-
quency, a small perturbation in the in-plane direction may be sufficient to excite its in-plane
resonant motion. An FEA simulation is performed to ensure that the pressure and shear
resonances of mechanical structure are not in close proximity. The results reveal that the
first mode, which is the in-plane mode, is at 5.2 kHz and the second mode, which is the
out-of-plane mode, is at 14.4 kHz. These values are to within 4% of the LEM predictions.
Thus, for better understanding of the sensor performance a LEM with combined shear and
pressure inputs is necessary, including the effects of cavity stiffening and the cut-on frequency
for pressure.
7.2.2.4 Mean Shear Stress Measurements
The mean shear stress measurements showed sensitivities that were considerably lower
than the dynamic sensitivity estimates. This may be attributed to the nonlinear variation in
the parasitic substrate capacitance discussed previously. Pressure gradient effects may also
contribute to the errors in this measurement. A theoretical estimate of the effective shear
stress is [37]
τeff = τw
(1 +
2Tt
h+
g
h
), (7–5)
where g is the gap underneath the sensor. The additional terms are due to the force acting
on the side of the sensor and the shearing force underneath the sensor structure. These are
simplified estimates and may be different for each sensor and measurement setup. An attempt
was made to quantify these experimentally. For this the mean shear stress measurements are
repeated at Vb = ±2.5 V , with a different channel height, h (Figure 7-10). The measurement
is done twice for h = 0.5 mm and compared with the results for h = 1 mm. The respective
sensitivities are 0.932 mV/Pa and 0.870 mV/Pa for h = 0.5 mm and 0.879 mV/Pa for
h = 1 mm. The variability of estimated sensitivities for h = 0.5 mm is higher than the
difference between sensitivities for h = 1 mm and h = 0.5 mm. These measurements are
therefore not enough to quantify the errors due to pressure gradient effects. This variability
174
may also be due to drift in the biasing circuitry due to charging of the variable capacitor on
the PCB (Figure 6-4). This problem also needs to be addressed from the circuits perspective.
0.5 1 1.5 2 2.50.5
1
1.5
2
2.5
3
Shear Stress (Pa)
Vol
tage
(m
V)
h=1 mmh=0.5 mmh=0.5 mm
Figure 7-10. Comparison mean shear stress measurements at Vb = 2.5 V at different channelwidths, h, in the flow cell.
7.3 Recommendations for Future Sensor Designs
Future designs of the current sensor may be significantly improved based on experiences
drawn from the design, fabrication and characterization aspects of this work. Several sugges-
tions were included in Section 7.2 from both design and characterization stand points. This
section combines them and recommends an approach for the next generation of the direct
differential capacitive shear stress sensors.
For the sensor design, the pressure response needs to be accounted for in the model.
This will include the effects of the cavity compliance and the damping/vent resistance to vent
the cavity to the ambient pressure. This will considerably improve the pressure sensitivity
issues. A simple LEM for the sensor for pressure forcing is represented in Figure 7-11,
with the various mechanical lumped elements in pressure are indicated with the subscript
175
‘p’. For the design optimization, the maximum pressure force may be set to be orders
fpressure
ue
Rmep
Mmep
Cmep fp
+
-
Rvent
Ccav
Transduction
Figure 7-11. Mechanical LEM for pressure sensitivity of the sensor.
of magnitude higher than the shear forcing. This may yield some interesting results. The
design with combination of cavity stiffening and cut-on frequency may enable thinner tethers
improving in-plane sensitivity while resulting in sufficient pressure rejection. However, the
trade-off would be lower nominal capacitance. From the circuits standpoint, the optimization
may include the current noise of the amplifier and the thermal noise of the bias resistor.
This allows to understand the trade-off involved in using different amplifiers instead of the
SiSonic. Furthermore, the demodulation and biasing circuits may be included as part of
the optimization to develop a predictive tool for the overall system performance, instead of
piecewise analysis.
The sensor may be fabricated by wafer bonding a silicon substrate to a pyrex wafer.
This will eliminate the parasitic substrate capacitance issues altogether. Using a thin-back
of the floating element prior to bonding to the pyrex may enable higher bandwidths by
lowering the mechanical mass. Backside electrical contacts using electronic through wafer
176
internconnects (ETWI) in the pyrex will help eliminate wire bonds, which disturb the flow.
A compliant, dielectric, hydrophobic layer on the floating element will eliminate errors due
to forces on the side of the floating element and forces due to flow underneath the sensor
structure.
The packaging used in this dissertation was developed based on packaging and testing
capabilities available in a laboratory setup. However, the package may be significantly
improved using shielded cables, which can carry multiple signals, providing power supply to
the amplifier, biasing signals for the sensor, and the sensor output voltage. This will further
lower the size of the PCB providing a more compact package for using them in shear stress
sensor arrays.
177
APPENDIX AMECHANICAL ANALYSIS
In this appendix the mechanical analysis for the floating element structure is presented.
First, the expression for linear deflection is derived from the governing differential equations.
Next, the non-linear deflection of the sensor structure is presented using two different solution
techniques, the energy method and the analytical technique. Lastly, the expressions for
lumped mass and compliance are derived based on the principle of conservation of energy.
A.1 Beam Deflection
A.1.1 Governing Equation
The governing differential equation is derived before explaining the solution procedure.
The curvature of a beam is first determined to formulate the governing equations. Consider
an arbitrary curve as shown in Figure A-1.
x
y
φs
Figure A-1. A deflected beam with arbitrary curvature.
The length of the arc for such a curve is
L =n∑
i=1
√(x− xi)
2 + (y − yi)2
=n∑
i=1
√(∆x)2 + (∆y)2
=n∑
i=1
√1 +
(∆y∆x
)2(∆x)
=b∫
a
√1 +
(dydx
)2dx.
(A–1)
178
The curvature, κ or1
ρfor a curve is defined as,
1
ρ= κ =
∣∣∣∣dϕ
ds
∣∣∣∣ . (A–2)
Considering a positive curvature Equation A–2 is rewritten as
1
ρ= κ =
dϕ
ds. (A–3)
Using the definition of the slope, the angle, ϕ is
ϕ = tan−1
(dy
dx
). (A–4)
Using Equation A–1 the differential length, ds along the curve is
ds =
√1 +
(dy
dx
)2
dx. (A–5)
The curvature is rewritten in terms of the variation along the x-axis as
1
ρ= κ =
dϕ
ds=
dϕ
dx
dx
ds. (A–6)
Substituting Equation A–5 and A–4 into Equation A–6 results in
1
ρ= κ =
dϕ
ds=
d
dx
(tan−1
(dy
dx
)) 1√
1 +(
dydx
)2
=d2y/dx2
(1 +
(dydx
)2)3/2
.
(A–7)
Now consider a beam under pure bending. The curvature of such a beam is [92]
1
ρ=
M
EI, (A–8)
179
where M is the resisting moment, E is the Young’s modulus of elasticity, and I moment of
inertia of the beam. Combining Equation A–7 and A–8 gives
Mx
EI=
d2w (x)/dx2
(1 +
(dw(x)
dx
)2)3/2
, (A–9)
where Mx is the moment along the length of the beam, and w (instead of y) is the deflection
in the transverse direction due to the applied load. Equation A–9 is the governing equation
for the deflection of a beam due to pure bending. For both small and large deflections, the
slope dw(x)dx
¿ 1. Therefore, the simplified governing equation is
Mx
EI= d2w (x)
/dx2. (A–10)
Direction
of Flow
Le
Clamped
Boundary
MB
RA
P Q
2Lt
P Q
RA
V
Wt
Q
xx=0
Tt
xx=0
x=Lt
MA
MA Mx
RB
x
z
y
Rigid body motion
P
Figure A-2. Simplified mechanical model of the floating element structure.
180
Consider a beam as shown in Figure A-2. This is same as Figure 3-2, which is repeated
here for convenience. The assumptions for beam deflection were previously stated in Sec-
tion 3.1.1 and therefore not repeated here. In the simplified mechanical model, for the shear
stress sensor, each pair of tethers form a clamped-clamped beam, with a point load P applied
at center by the shear stress, τw, on the floating element. A uniformly distributed load, Q,
along the length of the tethers accounts for the shear force on the tethers. Two such beams
share the load applied on the floating element because of the geometric symmetry as shown
in Figure A-2. Thus, the shear force each tether pair that form beams is
P =τwWeLe
2+
τw (NWfLf )
2, (A–11)
and Q = τwWt. (A–12)
(A–13)
Due to symmetry the solution is obtained by solving for half the beam length. The boundary
conditions for a clamped-clamped beam are mathematically stated as
w(x = 0) = 0, (no deflection)
dw
dx
∣∣∣∣x=0
= 0, (zero slope)
anddw
dx
∣∣∣∣x=Lt
= 0. (symmetry) (A–14)
A.1.2 Small Deflection Analysis
The Euler Bernoulli theory and small beam deflections in response to an applied force are
assumed while solving Equation 3–4. The theory hypothesizes that a straight line transverse
to the neutral axis remains straight, inextensible, and normal before and after deformation
[65]. With these assumptions, the linear solution for small beam deflection is obtained.
181
The reaction forces, RA and RB are determined first in terms of the applied loads, and
are given as
RA = RB =P
2+ QLt. (A–15)
The resisting moment Mx is
Mx = −MA + RAx− Qx2
2. (A–16)
Substituting Equation A–16 in A–10 gives
d2w (x)
dx2=
1
EI
(−MA + RAx− Qx2
2
). (A–17)
Solving the above equation gives the deflection and its for the beam as
dw (x)
dx=
1
EI
(−MAx + RA
x2
2− Qx3
6
)+ c1, (A–18)
and w (x) =1
EI
(−MA
x2
2+ RA
x3
6− Qx4
24
)+ c1x + c2. (A–19)
Now, the boundary conditions are applied to solve for the constants as follows:
w (0) =c2 = 0, (A–20)
dw (0)
dx=c1 = 0, (A–21)
dw (Lt)
dx=
1
EI
(−MALt + RA
L2t
2− QL3
t
6
)= 0. (A–22)
Simplifying Equation A–22 results in
MA =RALt
2− QL2
t
6. (A–23)
Substituting Equation A–15 into A–23 and simplifying it gives
MA =PLt
4+
QL2t
3. (A–24)
182
The deflection in terms of the applied forces is obtained by substituting Equation A–15,
A–20, A–21, and A–23 into Equation A–19 and given as
w (x) =1
EI
(−
(PLt
4+
QL2t
3
)x2
2+
(P
2+ QLt
)x3
6− Qx4
24
). (A–25)
The above equation is rewritten in terms of the applied shear stress. Substituting Equa-
tion A–11 and A–12, the simplified expression for the deflection is
w (x) = − τw
EI
((3 (LeWeLt + NWfLfLt) + 8WtL
2t ) x2
48
−(LeWe + NWfLf + 4WtLt) x3
24+
Wtx4
24
).
(A–26)
Using the expression for moment of inertia,
I =TtW
3t
12, (A–27)
Equation A–26 is further simplified using Equation A–27 to give
w (x) = − τw
4ETtW 3t
(3 (LeWeLt + NWfLfLt) + 8WtL2t ) x2
− (2LeWe + 2NWfLf + 8WtLt) x3 + 2Wtx4
. (A–28)
The -ve sign indicates that the deflection is downwards as per the chosen sign convention.
The maximum deflection occurs at the centre i.e., at x = Lt. The maximum deflection is
expressed as,
δ = −w (Lt) =1
ETtW 3t
(τwLeWeL
3t
4+
τwNWfLfL3t
4+
τwWtL4t
2
). (A–29)
A.1.3 Large Deflection Analysis
Large beam deflection causes extensional strain along the neutral axis, previously ig-
nored in the small deflection analysis. Some nonlinear terms in the Green Lagrange strain
displacement relationships, are not negligible for large deformations. Detailed solutions using
two different techniques, the energy method (Section 3.1.2.1) and an analytical solution tech-
nique (Section 3.1.2.2) are illustrated. First, an expression for the total strain in the beam
183
due to both bending and stretching is derived. The total strain in the beam is expressed as
εt = εbending + εstetching. (A–30)
The bending and stretching strains are [65],
εbending = ε1xx = −z
∂2w
∂x2(A–31)
and εstretching = ε0xx =
∂u
∂x+
1
2
(∂w
∂x
)2
. (A–32)
The total change in length of the beam due to stretching is
δLts =
2Lt∫
0
ε0xxdx
=
2Lt∫
0
(∂u
∂x+
1
2
(∂w
∂x
)2)
dx
= [u (x)]2Lt
0︸ ︷︷ ︸0
(clamped boundary)
+1
2
2Lt∫
0
(∂w
∂x
)2
dx
=1
2
2Lt∫
0
(∂w
∂x
)2
dx.
(A–33)
Thus the strain in terms of the transverse deflection due to stretching of the beam is
ε0xx =
δLts
2Lt
=1
4Lt
2Lt∫
0
(∂w
∂x
)2
dx. (A–34)
Combining Equation A–30, A–31, and A–34, the total strain in the beam is rewritten as
εt = −z∂2w
∂x2+
1
4Lt
2Lt∫
0
(∂w
∂x
)2
dx. (A–35)
184
A.1.3.1 Energy Method
The energy method solution utilizes the principle of virtual work to arrive at a solution
using the Rayleigh-Ritz method [65]. The principle is represented as
U = Stored energy −Work done, (A–36)
where U is the total energy.
Stored energy. The energy stored in any elastic structure due to displacement is
W (q1) =
q1∫
0
e (q) dq, (A–37)
where e represents effort (Force, Pressure etc.) and f represents a flow variable (displace-
ment, velocity etc.). For a distributed system energy density is a parameter of interest rather
than the energy i.e.,
W = W∆∀. (A–38)
Now in terms of units
W =J
m3=
N −m
m3=
N
m2
m
m, (A–39)
which indicates the energy density may be represented as the product of stress and strain.
Thus, energy density for normal stress may be written as,
W =
ε∫
0
σ (ε) dε, (A–40)
where σ is the stress and ε is the strain. As long as the deflection is within the linear elastic
limit of the material, Equation A–40 may be rewritten as
Wnormal =
ε∫
0
Eεdε =1
2Eε2 =
1
2σε. (A–41)
185
Similarly, the energy density due to shear stress may be written as
Wshear =1
2τγ. (A–42)
The total energy is thus
W =
∫∫∫(Wnormal + Wshear)dxdydz. (A–43)
A consequence of Euler Bernoulli beam theory is, there is no shear in the beam. Thus
considering only the normal energy density and substituting Equation A–41 into A–43 gives,
W =
Tt∫
0
Wt/2∫
−Wt/2
2Lt∫
0
1
2σεtdxdydz. (A–44)
Work Done. The work done by the external forces acting on the beam is
Work = Pδ + QLtδ = (P + QLt) δ. (A–45)
Substituting Equation A–44 and A–45 into Equation A–36 results in
U =
Tt∫
0
Wt/2∫
−Wt/2
2Lt∫
0
1
2Eε2
t dxdydz − (P + QLt) δ. (A–46)
Substituting for P and Q from Equation A–11 and A–12, the above equation is rewritten as
U =
Tt∫
0
Wt/2∫
−Wt/2
2Lt∫
0
1
2Eε2
t dxdydz
︸ ︷︷ ︸I1
−(τw
2(WeLe + NWfLf ) + τwWtLt
)δ
︸ ︷︷ ︸I2
. (A–47)
An estimate of the strain εt is needed to evaluate the integral I1 in Equation A–47. In the
solution procedure using the Rayleigh-Ritz method, a trial function for the deflection is used
to evaluate I1, which is
w =δ
2
(1 + cos
(π (Lt − x)
Lt
)). (A–48)
186
Therefore,
dw
dx=
δπ
2Lt
(sin
(π (Lt − x)
Lt
)), (A–49)
andd2w
dx2= − δπ2
2L2t
(cos
(π (Lt − x)
Lt
)). (A–50)
The strain ε is computed by combining Equation A–35, A–49, and A–50 to give
εt = z
(δπ2
2L2t
(cos
(π (Lt − x)
Lt
)))+
1
4Lt
2Lt∫
0
(δ
2
π
Lt
(sin
(π (Lt − x)
Lt
)))2
dx
=
(zδπ2
2L2t
(cos
(π (Lt − x)
Lt
)))+
δ2π2
16L3t
2Lt∫
0
(1− cos
(2π(Lt−x)
Lt
))
2dx
=
(zδπ2
2L2t
(cos
(π (Lt − x)
Lt
)))+
δ2π2
32L3t
[x +
Lt
2πsin
(2π (Lt − x)
Lt
)]2Lt
0
= zδπ2
2L2t
cos
(π (Lt − x)
Lt
)+
δ2π2
16L2t
.
(A–51)
The integral I1 in Equation A–47 is evaluated, using the expression for ε from Equation A–51
as follows,
I1 =1
2
Tt∫
0
Wt2∫
−Wt2
2Lt∫
0
E
(zδπ2
2L2t
cos
(π (Lt − x)
Lt
)+
δ2π2
16L2t
)2
dxdydz
=Tt
2
Wt2∫
−Wt2
2Lt∫
0
E
(zδπ2
2L2t
cos
(π (Lt − x)
Lt
)+
δ2π2
16L2t
)2
dxdz
=ETt
2
Wt2∫
−Wt2
(z2δ2π4
8L4t
[2Lt] +δ4pi4
256L4t
[2Lt]
)dz
= ETt
(δ2π4W 3
t
96L3t
+δ4π4Wt
256L3t
).
(A–52)
187
Substituting Equation A–52 into A–47, the total potential energy of the system is expressed
as
U = ETt
(δ2π4W 3
t
96L3t
+δ4π4Wt
256L3t
)
︸ ︷︷ ︸I1
−(τw
2(WeLe + NWfLf ) + τwWtLt
)δ
︸ ︷︷ ︸I2
. (A–53)
According to the principle of virtual work,“the total potential energy is stationary with
respect to any virtual displacement” [42]. This implies that the change in the total potential
energy due to the change in any virtual parameter should be minimum (ideally zero) to
maintain the stationary conditions of U i.e.,
dU
dδ= 0. (A–54)
Substituting Equation A–53 in the above equation and knowing that π4
96≈ 1 and π4
128≈ 3
4
results in
δ
(1 +
3
4
(δ
Wt
)2)
=τwWeLe
4ETt
(1 +
NWfLf
WeLe
+ 2WtLt
WeLe
)(L3
t
W 3t
). (A–55)
Equation A–55 gives the expression for the deflection using energy method under large
deflection. The accuracy of the solution is however limited by the choice of trial function.
A.1.3.2 Analytical Method
As discussed already under large deflections for a beam in bending, there is extensional
strain along the length of the beam. An axial force, Fa responsible for this strain is computed
in this method. The solution presented here is not a closed form analytical solution and
requires an iterative procedure to solve for the deflection of the beam. The governing equation
is same as Equation A–9 and still retains the assumption d2wdx2 ¿ 1. The schematic to the
right in Figure A-3 represents the free body diagram of the body of one half of the clamped-
clamped beam shown in the figure to the left.
Balancing moments at a given section at a distance x from the free end gives,
M (x) =P
2(x) +
Qx2
2+ Fa (w (x)− w (0))−M0, (A–56)
188
Fa
P/2Q
Lt
M0
Fa
M0
P/2M(x) Q
x
Figure A-3. Schematic of one half of a clamped-clamped beam under large deflection.
where M0 is the resisting moment at x = 0. The governing equation for this method obtained
by substituting Equation A–56 into A–9, resulting in
EId2w (x)
dx2− Faw (x) =
P
2(x) +
Qx2
2− Faw (0)−M0,
d2w (x)
dx2−
(Fa
EI
)
︸ ︷︷ ︸λ2
w (x) =Q
2EI︸︷︷︸X
x2 +P
2EI︸︷︷︸Y
x− (Faw (0) + M0)
EI︸ ︷︷ ︸Z
, (A–57)
ord2w (x)
dx2− λ2w (x) = Xx2 + Y x− Z. (A–58)
To solve Equation A–57, the axial force, Fa, due to the large deflection is determined.
The axial strain under large deflection is same as Equation A–34 with limits changed for one
half of the beam given as
ε0xx =
δLts
Lt
=1
2Lt
Lt∫
0
(∂w
∂x
)2
dx. (A–59)
For the tether the longitudinal stress is expressed as
σxx =Fa
Area= Eε0
xx, (A–60)
where the area of the tether, Area = TtWt. Therefore, the axial force,
Fa = Eε0xxWtTt. (A–61)
Combining Equation A–61 and Equation A–59 gives
Fa = EWtTt
2Lt
Lt∫
0
(∂w
∂x
)2
dx. (A–62)
189
Using this estimate, Equation A–57 is solved, which has a homogenous and a particular
solution written as
w = wg + wp. (A–63)
The homogeneous differential equation is
d2wg (x)
dx2− λ2wg (x) = 0, (A–64)
which has a solution given by
wg (x) = C1 cosh (λx) + C2 sinh (λx) . (A–65)
The inhomogeneous equation to obtain particular solution is
d2wp (x)
dx2− λ2wp (x) = Xx2 + Y x− Z. (A–66)
Assume a particular solution of the form
wp (x) = ax2 + bx + c. (A–67)
Substituting Equation A–67 into A–66 gives
2a− λ2(ax2 + bx + c
)= Xx2 + Y x− Z. (A–68)
Matching coefficients of different powers of x,
a = −X
λ2,
b = − Y
λ2, (A–69)
and c =λ2Z − 2X
λ4(A–70)
Substituting coefficients from Equation A–69 into Equation A–67 gives
wp (x) = −X
λ2x2 − Y
λ2x +
λ2Z − 2X
λ4. (A–71)
190
The total deflection obtained by combining Equation A–63, A–65, and A–71 is
w (x) = C1 cosh (λx) + C2 sinh (λx)− X
λ2x2 − Y
λ2x +
λ2Z − 2X
λ4, (A–72)
and the deflection gradient is
dw (x)
dx= C1λ sinh (λx) + C2λ cosh (λx)− 2X
λ2x− Y
λ2. (A–73)
Applying the boundary conditions in Equation A–14 gives,
dw (0)
dx= C2λ− Y
λ2= 0 ⇒ C2 =
Y
λ3, (A–74)
anddw (Lt)
dx= C1λ sinh (λLt) +
Y
λ2cosh (λLt)− 2X
λ2Lt − Y
λ2= 0
⇒ C1 =Y + 2XLt − Y cosh (λLt)
λ3 sinh (λLt). (A–75)
Substituting Equation A–74 and A–75 into Equation A–72 the deflection is rewritten as
w (x) =
[Y + 2XLt − Y cosh (λLt)
λ3 sinh (λLt)cosh (λx)
+Y
λ3sinh (λx)− X
λ2x2 − Y
λ2x +
λ2Z − 2X
λ4
].
(A–76)
Now, using values of X, Y , Z, and λ from Equation A–57 the above equation becomes,
w (x) =
[1
Faλ sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
)cosh (λx)
+P
2Faλsinh (λx)− Q
2Fa
x2 − P
2Fa
x + w (0) +M0
Fa
− Q
λ2Fa
].
(A–77)
Recognizing that w (x = 0) = w (0), the resisting moment M0 is determined and is expressed
as
M0 =Q
λ2− 1
λ sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
). (A–78)
191
Substituting Equation A–78 into A–79 gives,
w (x) =
[P
2Faλsinh (λx) +
(cosh (λx)− 1)
Faλ sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
)
− Q
2Fa
x2 − P
2Fa
x + w (0)
].
(A–79)
The last boundary condition i.e., w(Lt) = 0 results in
w (0) =
[− P
2Faλsinh (λLt)− (cosh (λLt)− 1)
Faλ sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
)
+Q
2Fa
L2t +
P
2Fa
Lt
].
(A–80)
Equation A–80 is the expression for deflection of the beam under large deflections, which
includes the effect of strain in the neutral axis. The gradient of the deflection is written as
dw
dx=
[P
2Fa
cosh (λx) +sinh (λx)
Fa sinh (λLt)
(QLt +
P
2− P
2cosh (λLt)
)
− Q
Fa
x− P
2Fa
].
(A–81)
Using the expressions derived so far the following steps are followed to determine the deflec-
tion in an iterative method.
1. Start with an initial value of axial force Fa
2. Obtain the eigen value given by λ =√
Fa
EI
3. Estimate the deflection w (x) and the gradient dwdx
4. Calculate Fa again using the relation Fa = E WtTt
2Lt
Lt∫0
(∂w∂x
)2dx
5. Repeat step 1 to 4 until |F n+1a − F n
a |/F n+1a ≤ 1e− 8
6. Use Fa to obtain the maximum center deflection w (0)
A.2 Lumped Parameters
The basic concepts of lumped element modeling were explained in Section 3.2 in Chap-
ter 3. In this section, the derivations for lumped mass and lumped compliance of the floating
element sensor structure are presented.
192
A.2.1 Lumped Mass
The lumped mass is computed by using the total kinetic energy due to the motion of
the beam and equating it to the equivalent kinetic energy of the lumped system. For a linear
system,
W ∗KE =
∫dW ∗
KE =
f0∫
0
pdf =1
2Mlumpedf
20 , (A–82)
where, p and f represent momentum and flow, respectively. The expression for linear de-
flection in Equation A–28 is used. The velocity or flow variable represented in terms of the
deflection is
v (x) = jωw (x) . (A–83)
The velocity at x = Lt is
v (Lt) = jωw (Lt) . (A–84)
Combining Equation A–83 and A–84,
v (x) =v (Lt)
w (Lt)w (x) . (A–85)
Consider a differential element of the tether of mass dm at a location x. The momentum of
the element is then given by,
p = dmv (x) = ρTtWtv (x) dx, (A–86)
where ρ is the mass density of the beam. By definition of co-energy, the differential kinetic
co-energy is expressed as
dW ∗KE = ρTtWtdx︸ ︷︷ ︸
dm
v (x) dv (x) . (A–87)
193
Therefore,
W ∗KE =
∫dW ∗
KE
W ∗KE =
Lt∫
0
v(x)∫
0
ρNiTtWtv (x) dv (x)dx
W ∗KE =
Lt∫
0
ρNiTtWtv2 (x)
2dx.
(A–88)
Combining Equation A–88 and A–85 gives
W ∗KE =
Lt∫
0
ρNiTtWtv2 (Lt)
w2 (Lt)
w2 (x)
2dx
=ρNiTtWt
2
v2 (Lt)
w2 (Lt)
Lt∫
0
w2 (x) dx.
(A–89)
In the above computation, the distributed tether mass is lumped at the center with a dis-
placement equal to the tip deflection. Combining, Equation A–82 and A–89 gives
Mtme = 4ρNiTtWt
w2 (Lt)
Lt∫
0
w2 (x) dx. (A–90)
Note the additional factor 4 in Equation A–90, which accounts for mass of four tethers
suspending the floating element. The total lumped mass of the entire floating element is the
sum of the tether and the floating element mass given as
Mme = Mtme + Melement = Mtme + ρNi (WeLe + NWfLf ) Tt. (A–91)
194
Combining Equation A–28, A–29, A–90, and A–91 and evaluating the integral results in
(solve in MAPLE)
Mme = ρTtLeWe
1 + 19235
(WtLt
LeWe
)+ 3
(NWf Lf
LeWe
)+ 384
35
(WtLt
LeWe
) (NWf Lf
LeWe
)
+1072105
(WtLt
LeWe
)2
+ 3(
NWf Lf
LeWe
)2
+ 1072105
(NWf Lf
LeWe
)(WtLt
LeWe
)2
+3(
NWf Lf
LeWe
)2 (WtLt
LeWe
)+ 2048
315
(WtLt
LeWe
)3
+(
NWf Lf
LeWe
)3
(1 +
NWf Lf
LeWe+ 2 WtLt
LeWe
)2 . (A–92)
A.2.2 Lumped Compliance
The lumped compliance is evaluated by computing the total potential energy of the
system and equating it to the equivalent potential energy of the lumped system. For a linear
system,
WPE =
∫dW ∗
PE =
q0∫
0
edq =1
2
w2 (Lt)
Clumped
, (A–93)
where e and q are effort and displacement, respectively. Consider an element of the beam of
length dx. The total distributed force acting on a half section of the beam is
F =
Lt∫
0
(Q +
P
2δ (x− Lt)
)dx, (A–94)
where P and Q are defined in Equation A–11 and A–12 and δ represents a dirac delta
function. Thus the potential energy in Equation A–93 is rewritten as
WPE =
Lt∫
0
w(x)∫
0
(Q +
P
2δ (x− Lt)
)dw (x) dx. (A–95)
Substituting Equation A–11 and A–12 into A–95 gives the potential energy expressed in
terms of the shear stress, τw, to give
WPE =
Lt∫
0
w(x)∫
0
τw
(Wt +
(WeLe
4+
NWfLf
4
)δ (x− Lt)
)dw (x) dx. (A–96)
195
Equation A–96 requires the forcing shear stress to be expressed in terms of the distributed
system to solve for the total potential energy. Substituting for τw from Equation A–28,
Equation A–96 is expressed as
WPE =
Lt∫
0
w(x)∫
0
4ETtW3t
(Wt +
(WeLe
4+
NWf Lf
4
)δ (x− Lt)
)(
(3(LeWeLt+NWf Lf Lt)+8WtL2t)x2
−(2LeWe+2NWf Lf+8WtLt)x3+2Wtx4
)
w (x)dw (x) dx
=
Lt∫
0
4ETtW3t
(Wt +
(WeLe
4+
NWf Lf
4
)δ (x− Lt)
)(
(3(LeWeLt+NWf Lf Lt)+8WtL2t)x2
−(2LeWe+2NWf Lf+8WtLt)x3+2Wtx4
)
w2 (x)
2dx.
(A–97)
The distributed deflection w (x) is expressed in terms of the central deflection w(Lt) by
eliminating the shear stress τw from Equation A–28 and A–29, to give
w (x) = w (Lt)
((3LeWeLt+3NWf Lf Lt+8WtL2
t)x2
−(2LeWe+2NWf Lf+8WtLt)x3+2Wtx4
)
WeLeL3t
[1 +
NWf Lf
WeLe+ 2 WtLt
WeLe
] . (A–98)
Combining Equation A–97 and Equation A–98 results in
WPE =
Lt∫
0
w2 (Lt)
[2ETtW3
t
(Wt+
(WeLe
4 +NWf Lf
4
)δ(x−Lt)
)
W2e L2
eL6t
[1+
NWf LfWeLe
+2WtLtWeLe
]2
] ((3LeWeLt+3NWf Lf Lt+8WtL2
t)x2
−(2LeWe+2NWf Lf+8WtLt)x3
+2Wtx4
)dx
= w2 (Lt)
(ETtW
4t
L2t WeLe
)
1 +NWf Lf
WeLe+ 32
15
(WtLt
WeLe
)
(1 +
NWf Lf
WeLe+ 2
(WtLt
WeLe
))2
+
ETtW3t
(WeLe
2+
NWf Lf
2
)w2 (Lt)
WeLeL3t
[1 +
NWf Lf
WeLe+ 2 WtLt
WeLe
]
= ETt
(Wt
Lt
)3
1 + 2(
NWf Lf
WeLe
)+ 4
(WtLt
WeLe
)+ 4
(NWf Lf WtLt
(WeLe)2
)
+6415
(WtLt
WeLe
)2
+(
NWf Lf
WeLe
)2
2(1 +
NWf Lf
WeLe+ 2
(WtLt
WeLe
))2 w2 (Lt) .
(A–99)
Since the computation above is only for a single tether, the total potential energy of the
floating element suspended by four tethers is four times the value in Equation A–99, written
196
as,
WPE = 2ETt
(Wt
Lt
)3
1 + 2(
NWf Lf
WeLe
)+ 4
(WtLt
WeLe
)+ 4
(NWf Lf WtLt
(WeLe)2
)
+6415
(WtLt
WeLe
)2
+(
NWf Lf
WeLe
)2
(1 +
NWf Lf
WeLe+ 2
(WtLt
WeLe
))2 w2 (Lt) . (A–100)
Combining Equation A–93 and A–100 and solving for Cme gives the lumped compliance of
the floating element sensor, expressed as
Cme =1
4ETt
(Lt
Wt
)3
(1 +
NWf Lf
WeLe+ 2
(WtLt
WeLe
))2
1 + 2(
NWf Lf
WeLe
)+ 4
(WtLt
WeLe
)+ 4
(NWf Lf WtLt
(WeLe)2
)
+6415
(WtLt
WeLe
)2
+(
NWf Lf
WeLe
)2
. (A–101)
197
APPENDIX BTWO PORT ELEMENT MODELING
B.1 Two Port Models
A two-port element is represented in general as shown in Figure B-1, where A & B and
C & D are conjugate power variables (effort and flow). By definition, the power transfer is
always into the two-port element and hence must be conserved [42],
PowerNET = AB − CD = 0 or AB = CD. (B–1)
Two-Port
Element
+ +
− −
A D
B C
Figure B-1. General representation of an ideal two-port element.
Two types of linear elements satisfy the condition in Equation B–1. One is a transformer
and the other is a gyrator, represented as follows:
Transformer
D
C
=
n 0
0 1/n
A
B
(B–2)
Gyrator
D
C
=
0 n
1/n 0
A
B
(B–3)
The transformer and gyrator as circuit elements are shown in Figure B-2. The transformer
model in Equation B–2 represents the transduction of effort/flow in one domain to corre-
sponding effort/flow in a different energy domain via the transduction factor, n. The gyrator
model in Equation B–3 relates effort in one energy domain to flow in a different domain.
For example, electrodynamic speakers are modeled using a gyrator where electrical current
198
+ +
--
A
B
D
Cn
Gyrator
+ +
--
A
B
D
C1:n
Transformer
Figure B-2. Circuit representation of the transformer and gyrator.
(B) through a coil in a magnetic field produces a proportional actuation force (D) via elec-
trodynamic coupling. A transformer model is applicable for a capacitive microphone where
pressure (A) is transduced into a proportional voltage (D) due to diaphragm deflection.
Two port networks also allow representation using an impedance analogy or an admit-
tance analogy based on definition of A,B,C, andD in Equation B–1. For an impedance
analogy the across variables, A & D are effort variables. For admittance analogy the across
variables A & D are flow variables. This definition usually depends on the specific transduc-
tion problem. As a convention, in the impedance analogy an effort (eg: pressure, voltage,
force) is an across variable and a flow (eg: current, volumetric flow rate, velocity) is a through
variable. An example of an impedance to impedance analogy for a two-port element is shown
in Figure B-3.
Transducer
Element
+
−
I U
V
_
+
F
Figure B-3. Impedance to impedance analogy representation of a two port element.
B.2 Linear Conservative Transducers
This section presents models for linear conservative transducers (LCT) based on the
discussion for two port models in the previous section. These transducers linearly transduce
energy from one energy domain to the other. The linear transduction maintains spectral
199
fidelity of the input signal. For an electromechanical transducer, variables in the electrical
domain are expressed in terms of those in the mechanical domain and vice versa. Consider
the transducer element shown in Figure B-3 with the instantaneous values of effort variables
expressed in terms of flow variables as,
v = v (i, u) andf = f (i, u) , (B–4)
where v & i are voltage and current for the electrical domain and f & u are force and
velocity for the mechanical domain, respectively. In the Fourier domain these variables are
represented as,
V = V (I, U) andF = F (I, U) . (B–5)
Applying chain rule to Equation B–4,
dv =∂v
∂i
∣∣∣∣u=0
di +∂v
∂u
∣∣∣∣i=0
du, (B–6)
and df =∂f
∂i
∣∣∣∣u=0
di +∂f
∂u
∣∣∣∣i=0
du. (B–7)
Linearizing about a mean condition and replacing differentials with fluctuations in the Fourier
domain results in,
V =V
I
∣∣∣∣U=0
I +V
U
∣∣∣∣I=0
U = ZEBI + TEMU, (B–8)
and F =F
I
∣∣∣∣U=0
I +F
U
∣∣∣∣I=0
U = TMEI + ZMOU. (B–9)
Thus the characteristic equation of a linear transducer is,
V
F
=
ZEB TEM
TME ZMO
I
U
, (B–10)
where the impedances are defined as follows:
ZEB ≡ V
I
∣∣∣∣U=0
, (B–11)
200
the blocked electrical impedance,
ZEF ≡ V
I
∣∣∣∣F=0
, (B–12)
the free electrical impedance,
ZMO ≡ F
U
∣∣∣∣I=0
, (B–13)
the open-circuit mechanical impedance,
ZMS ≡ F
U
∣∣∣∣V =0
, (B–14)
the short-circuit mechanical impedance,
TEM ≡ V
U
∣∣∣∣I=0
, (B–15)
the open-circuit electromechanical transduction factor, and
TME ≡ F
I
∣∣∣∣U=0
, (B–16)
the blocked electromechanical transduction factor.
For a reciprocal transducer, TME = TEM = T , which implies that the same velocity, U
can be obtained using both a force and a voltage, and in addition the same current, I can
be obtained using a force and a voltage [66]. If a transducer is reciprocal, the impedance
transformation factor is,
ϕ′ =T
ZMO
. (B–17)
The equivalent circuit of a LCT is shown in Figure 31 where the free electrical impedance is
ZEF = ZEB − T 2
ZMO
=(1− κ2
)ZEB, (B–18)
201
where κ is the electromechanical coupling factor expressed as [42],
κ2 =T 2
ZEBZMO
. (B–19)
+
−
F MSZ
U
I
EFZ
+
−
V
1:φ ′
MOZ
Figure B-4. Equivalent circuit representation of a transducer using impedance analogy.
x′
x
0x x=
0x =
fixed
plates
movable
plates
A
Figure B-5. Schematic of a parallel plate capacitive transducer.
Example:. Consider a parallel plate capacitor with a fixed plate and movable plate as
shown in Figure B-5.
The nominal gap between the plates is x0 and the change in gap due to a sens-
ing/actuation force is x′ (t) such that,
x = x0 − x′ (t) . (B–20)
202
The capacitance is computed as,
CE(t) =ε0A
x=
ε0A
x0 − x′(t)=
ε0A
x0
(1− x′(t)
x0
)−1
= CEB
(1− x′(t)
x0
)−1
, (B–21)
where CEB is the blocked capacitance when x′ (t) = 0. The voltage is related to the capaci-
tance and stored charge between the parallel plates by,
V (t) =Q(t)
CE(t)=
Q(t)
CEB
(1− x′(t)
x0
). (B–22)
Equation B–22 is the sum of the voltage at rest and that due to electromechanical coupling.
The electrostatic force for a constant charge between the plates is [42],
FE = −1
2
Q2
CEBx0
= −1
2
Q2
ε0A. (B–23)
The compliance/stiffness of the diaphragm results in a mechanical restoring force,
FM = kx′ (t) =x′ (t)CMO
, (B–24)
where k is the mechanical stiffness and CMO is the open circuit compliance of the diaphragm.
Adding Equation B–23 and B–24, the total force on the diaphragm is,
F = FM + FE =x′(t)CMO
− 1
2
Q2
ε0A=
x′(t)CMO
− 1
2
Q2
CEBx0
. (B–25)
Assume Q(t) = Q0 + Q′(t) and V (t) = V0 + V ′(t) with the constraint that the perturbations
are small compared to the mean, i.e., x′(t)x0
<< 1;Q′(t)Q0
<< 1; and V ′(t)V0
<< 1. Applying these
assumptions, the linearized form of Equation B–22 and B–25 are,
V (t) =Q′(t)CEB
− V0x′(t)
x0
, (B–26)
and F (t) =x′(t)CMO
− V0Q′(t)
x0
. (B–27)
The two-port model representation is generally applicable for transducers that measure per-
turbations about a mean. Thus dropping the ”primes” and representing Equation B–26 and
203
B–27 in terms of their Fourier components results in,
V =
(1
jωCEB
)I +
(− V0
jωx0
)U (B–28)
and F =
(− V0
jωx0
)I +
(1
jωCMO
)U. (B–29)
The matrix form of the governing equations for a capacitive transducer is,
V
F
=
ZEB TEM
TME ZMO
I
U
=
1jω CEB
−V0
jω x0
−V0
jω x0
1jω CMO
I
U
. (B–30)
Equation B–30 indicates the transducer is reciprocal but indirect in nature. Comparing
Equation B–30 and B–17,
ϕ′ =T
ZMO
= − V
x0
CMO. (B–31)
The equivalent circuit representation for a capacitive transducer is shown in Figure B-6.
This is consistent with the general modeling approach of a linear conservative transducer
discussed in the previous section. Using Equation B–19, the compliance CEF is,
1
jωCEF
=(1− κ2
) 1
jωCEB
, (B–32)
where
κ2 =T 2
ZEBZMO
=
(V
jωx0
)2/
1
(jω)2 CEBCMO
=V 2
x20
CEBCMO (B–33)
.
+
−
F
U
I
+
VMO
C
EFC
0
0
1:
M
x
C V
−
0
1:MO
VC
x−
_
Figure B-6. Circuit representation of a capacitive transducer for constant charge biasing.
204
APPENDIX CSHEAR STRESS IN PWT WITH REFLECTIONS
This chapter shows the solution procedure to correct for effects of reflection for fluctu-
ating wall shear stress measurements in the plane wave tube. The derivation is for a general
impedance termination at the end of the tube having a reflection coefficient, R as shown in
Figure C-1.
Techron 7540 Power
Supply Amplifier
B & K Pulse
Analyzer System
PC
Acoustic Plane
Wave
Shear Stress
SensorSpeaker
Microphone
General
Impedance
Termination
R=Z
PressureVelocity
Standing Wave Pattern
Left Running
Wave (Incident)
Right Running
Wave (Reflected)
dps
ds
Figure C-1. Setup for shear stress in PWT for a general impedance termination.
Assume a fully developed flow in a duct, driven by an oscillating pressure gradient. The
governing differential equation is
∂u
∂t= −1
ρ
∂p
∂xejωt + ν
∂2u
∂y2. (C–1)
205
Based on no-slip boundary condition at the duct wall and the finite velocity at the center of
the duct the expression for velocity by solving Equation C–1 is [93]
u (y, t) =jejωt
ρω
∂p
∂x
1−
cosh
(y√
jων
)
cosh
(a√
jων
)
. (C–2)
Now assume a left running (incident wave) plane acoustic wave along the length of the duct
given by [87],
p1 = p1(x)ejωt = p′1e(jωt−kx)
∴ ∂p1
∂x= −jω
cp′1e
(jωt−kx) (C–3)
where x is the position along the length of the duct. The corresponding velocity due to this
pressure is,
u1 (x, y, t) =jejωt
ρω
∂p1
∂x
1−
cosh
(y√
jων
)
cosh
(a√
jων
)
. (C–4)
Similarly, for a right running wave (reflected wave) the pressure is
p2 = p2(x)ejωt = p′2e(jωt+kx)
∴ ∂p2
∂x= −jω
cp′2e
(jωt+kx) (C–5)
The corresponding reflected velocity is
u2 (x, y, t) =jejωt
ρω
∂p2
∂x
1−
cosh
(y√
jων
)
cosh
(a√
jων
)
. (C–6)
Using linear superposition, the total velocity at a given location, x, along the length of the
duct is
u (x, y, t) = u1 (x, y, t) + u2 (x, y, t) . (C–7)
206
Note: u1&u2 account for the direction of the wave via the sign associated with the pressure
gradient in Equations C–3 and C–5, respectively. Substituting Equations C–4 and C–6 in
Equation C–7 results in
u (x, y, t) =jejωt
ρω
1−
cosh
(y√
jων
)
cosh
(a√
jων
)
(∂p1
∂x+
∂p2
∂x
). (C–8)
The wall shear stress due to this velocity is,
τwall = µdu
dy
∣∣∣∣y=a
. (C–9)
Substituting the expression for velocity from Equation C–8,
τwall =ejωt
j
√jµ
ρωtanh
(a
√jω
ν
)(∂p1
∂x+
∂p2
∂x
). (C–10)
Now, substituting for the pressure gradients from Equations C–3 and C–5 gives
τwall = −1
c
√jωµ
ρtanh
(a
√jω
ν
)(p′1e
j(ωt−kx) − p′2ej(ωt+kx)
). (C–11)
Defining distances from the termination x = l− d, where l is the length of the tube and d is
the distance from the termination, the τwall in the above expression may be rewritten as
τwall = −1
c
√jωµ
ρtanh
(a
√jω
ν
)(p′1e
j(ωt−kl+kd) − p′2ej(ωt+kl−kd)
), (C–12)
or τwall = −1
c
√jωµ
ρtanh
(a
√jω
ν
)p′1e
−jkl
︸ ︷︷ ︸P+
ej(ωt+kd) − p′2ejkl
︸ ︷︷ ︸P−
ej(ωt−kd)
, (C–13)
or τwall = −1
c
√jωµ
ρtanh
(a
√jω
ν
)P+
(ejkd − P−
P+e−jkd
)ejωt. (C–14)
The reflection coefficient at a given termination based on incident and reflected pressures is
defined as,
R =p′2e
jkl
p′1e−jkl=
P−
P+. (C–15)
207
Substituting R from Equation C–15 in Equation C–14 results in,
τwall = −1
c
√jων tanh
(a
√jω
ν
)P+
(ejkd −Re−jkd
)ejωt. (C–16)
Assuming that the shear stress sensor is located at a position ds from the termination,
τwall = −1
c
√jωv tanh
(a
√jω
ν
)P+
(ejkds −Re−jkds
)ejωt. (C–17)
The pressure P+ejωt has to be expressed in terms of total pressure measured by the micro-
phone at a distance dps from the termination. The total pressure at a distance dps is,
p (dps) ejωt = p1 + p2 =(p′1e
j(ωt−kl+kdps) + p′2ej(ωt+kl−kdps)
)
p (dps) ejωt = p′1e−jklej(ωt+kdps) + p′2e
jklej(ωt−kdps)
p (dps) ejωt =(P+ejkdps + P−e−jkdps
)ejωt
pmeasuredejωt = p (dps) ejωt = P+
(ejkdps + Re−jkdps
)ejωt.
(C–18)
Substituting for P+ from Equation C–18 in terms of pmeasured, into Equation C–17,
τwall =
(−1
c
√jων tanh
(a
√jω
ν
)ejkds −Re−jkds
ejkdps + Re−jkdps
)pmeasurede
jωt. (C–19)
Let the separation between the sensor and microphone be δ = ds − dps, then the wall shear
stress may be expressed as
τwall =
(−1
c
√jων tanh
(a
√jω
ν
)ejkds −Re−jkds
ejk(ds−δ) + Re−jk(ds−δ)
)pmeasurede
jωt. (C–20)
208
APPENDIX DPROCESS TRAVELER AND PACKAGING DETAILS
Wafer:. 100±0.1 mm” p-type (100) SOI wafer of thickness 545±10 µm, 2 µm±5% thick
buried oxide (BOX) layer, 45±1 µm thick device layer with resistivity 0.001−0.005 ω− cm,
and 500± 10 µm thick handle wafer with resistivity > 1000 ω − cm.
Masks:.
• Floating element DRIE mask (FEM)
• Backside/Cavity etch mask (CEM)
Process Steps:.
1. Wafer cleaning
(a) SC-1 (Ion Strip) - 15 : 3 : 2 H2O : H2O2 : NH4OH at 75C for 10 min
(b) SC-2 (Organic Strip) - 16 : 3 : 1 H2O : H2SO4 : H2O2 at 75C for 10 min
(c) Oxi-clean - 50 : 1 H2O : HF at room temperature for 30s
(d) Acetone/Methanol/DI wash
2. Etch on device layer to define sensor structure
(a) Coat with HMDS for 5 min
(b) Spin resist (1 µm, AZ 1512) - spin at 500 rpm 100 rpm/s 5s−4000 rpm 100 rpm/s 50s
(c) Soft bake 60s (hot plate) at 95 C
(d) Expose using mask FEM - EVG620 Ch-A 29.5 mW/cm2 for 1 s (hard contact)
(e) Develop using AZ 300MIF for 60 s
(f) Post exposure bake (oven) - 95C for 60 min
(g) Etch using STS-DRIE - recipe VJ pol50, 42 cycles and recipe BAO2 VJ (avoids
footing), 20 cycles
(h) Acetone/Methanol/DI wash to clean photoresist
3. Nickel electroplating
(a) Piranha clean to remove organic CHF3 passivation - 3 : 1 H2SO4 : H2O2 at
120 C for 10 min
209
(b) Oxide removal using BOE (6 : 1) for 5 min
(c) Immediate 2-propanol immersion for surface wetting and to prevent oxidation
(d) Immediate immersion in Technic Nickel sulfamate solution at 90 F (Terminals -
Ni electrode positive, SOI wafer negative)
(e) Begin plating simultaneously with immersion - 30s strike with 125 mA (nucle-
ation), 5 min plating with 30 mA (finer grains), 5 min plating with 7 mA (very
fine grains)
(f) Rinse with DI water
4. Backside/Cavity etch
(a) Coat backside with HMDS for 5 min
(b) Spin resist on backside (10 µm, AZ 9260) - spin at 500 rpm 100 rpm/s 5s −2000 rpm 100 rpm/s 50s
(c) Spin AZ 9260 on handle silicon wafer - 2000 rpm 1000 rpm/s ramp rate
(d) Soft bake 30min (oven) at 95 C
(e) Front to back align using EVG620
(f) Expose using mask CEM - EVG Ch-B 62 mW/cm2 for 23 s (hard contact)
(g) Develop using AZ 300MIF for 3 min
(h) Spin resist (5 µm, AZ 9260) - spin at 500 rpm 100 rpm/s 5s−4000 rpm 100 rpm/s 50s
(i) Place SOI wafer face down on handle wafer and apply gentle pressure for uniform
contact
(j) Post exposure bake (Oven) - 95C for 60 min
(k) Etch back side cavity using STS-DRIE - recipe VJ pol50, 470 cycles and recipe
BAO2 VJ (avoids footing), 130 cycles
(l) Remove PR mask - Acetone/Methanol/DI wash
(m) Separate from handle wafer using S-46 PR stripper - heat 80C for ≈ 15 min
(n) Rinse Acetone/Methanol/DI
5. Die separation
210
(a) Lay down double sided heat sensitive tape REVALPHA (120C) on back side of
wafer (without air bubbles)
(b) Stick wafer, face up on a handle silicon wafer for support during dicing
(c) Lay down single sided heat sensitive tape REVALPHA (80C) on front side of
wafer (without air bubbles)
(d) Dice wafer using dicing saw - Tresser 620 - Kulicke and Soffa blade S1235 - thick-
ness (1− 1.2 mm)
(e) Heat on hot-plate at 80C to peel off tape from front side of wafer
(f) Heat on hot-plate at 120C to separate die from tape and handle wafer
6. Die release
(a) Etch BOX in BOE (6:1) for 20 min
(b) Immediate immersion in methanol
(c) Super critical dry in CO2 (Prof. Ho-Bun Chan’s lab)
Packaging:.
1. Sensor packaging in PCB
(a) Cut recess in PCB for sensor mounting (CNC recipe name) - PCB layout shown
in Figure D-1 - made at Sierra Proto Express
(b) Glue SiSonic to back side of PCB using epoxy (Dualbond 707) - cure at 35C for
10 min
(c) Wire bond to bond pads using ball size 4. (Prof. Ho-Bun Chan’s lab)
(d) Cover bonds with epoxy (Dualbond 707) - cure at 35C for 10 min
(e) Solder SMB connectors for electrical contact and fill through vias with solder
(f) Glue sensor in place with epoxy (Dualbond 707) - cure at 35C for 10 min
(g) Wirebond using ball size 6 and force at 100. (Prof. Ho-Bun Chan’s lab)
2. Sensor PCB packaging in plug
(a) Sensor PCB is press fit in Lucite plug - drawing and dimensions shown in Figure D-
2 - made at TMR engineering (Ken Reed)
211
Figure D-1. PCB Layout and dimensions.
212
6
10
60
40
27
,1
R46,12
12,7
10,16
10
6
5
10,16
17,86
1,62
5
SC
AL
E1
,00
0
SC
AL
E1
,00
0
Ce
nte
r of th
rou
gh
ho
le is
R=
75
SC
AL
E1
,00
0
SC
AL
E1
,00
0
Figure D-2. Drawing of Lucite plug.
213
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220
BIOGRAPHICAL SKETCH
Vijay Chandrasekharan grew up in Nagpur, a city located right in the center of India.
After high school, he went to the National Institute of Technology Karnataka (NITK), India
where he received his bachelor’s in mechanical engineering in 2002. After working for a year,
he started graduate school in the Department of Mechanical and Aerospace Engineering at
the University of Florida in 2003. He received his MS in 2006 and is currently completing his
doctoral degree at the Interdisciplinary Microsystems Group working under the guidance of
Prof. Mark Sheplak. Stemming from his inclination towards product development, Vijay’s
research interests include micromachined (MEMS) sensor and actuator design, modeling,
fabrication and characterization.
221
