DESIGN STRATEGIES AND EXPERIMENTAL VALIDATION OF …
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The Pennsylvania State University The Graduate School College of Engineering DESIGN STRATEGIES AND EXPERIMENTAL VALIDATION OF HIGH-PERFORMANCE LOGGING-WHILE-DRILLING PIEZOCOMPOSITE TRANSDUCERS A Dissertation in Electrical Engineering by Runkun Jiang © 2017 Runkun Jiang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2017
Transcript of DESIGN STRATEGIES AND EXPERIMENTAL VALIDATION OF …
The Pennsylvania State University
The Graduate School
College of Engineering
DESIGN STRATEGIES AND EXPERIMENTAL VALIDATION
OF HIGH-PERFORMANCE LOGGING-WHILE-DRILLING
PIEZOCOMPOSITE TRANSDUCERS
A Dissertation in
Electrical Engineering
by
Runkun Jiang
© 2017 Runkun Jiang
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2017
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The dissertation of Runkun Jiang was reviewed and approved* by the following:
Qiming Zhang
Distinguished Professor of Electrical Engineering
Dissertation Adviser
Chair of Committee
Bernhard R. Tittmann
Schell Professor of Engineering Science and Mechanics
Zhiwen Liu
Professor of Electrical Engineering
Sumeet Kumar Gupta
Monkowski Assistant Professor of Electrical Engineering
Kultegin Aydin
Professor of Electrical Engineering
Head of the Department of Electrical Engineering
*Signatures are on file in the Graduate School.
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ABSTRACT
In the oil and gas industry, logging-while-drilling (LWD) acoustic transducers have been found
to provide valuable sonic information of the borehole rock formation such as compressional
wave velocity and shear wave velocity. These acoustic transducers consist of transmitters and
receivers. Transmitters send out acoustic waves through the borehole. Waves then get received
by an array of receivers. Through the phase delay of the arriving signals, one can calculate the
mechanical and acoustic properties of the borehole rock formation and this information can be
used further to indicate lithology, determine porosity, detect over-pressured zones, check well-to-
well correlation, etc. This dissertation covers exhaustively on original research work on LWD
acoustic transmitters and receivers, including design and optimization, fabrication, and testing.
Some necessary theoretical background was given in the Theories chapter. The fundamentals of
elasticity, wave motion and wave equations were introduced. Wave theories on acoustic
impedance matching (reflection and refraction), and attenuation were also covered briefly.
Piezoelectricity constitutive equations and piezoelectrically stiffened wave equations were
outlined. And theories on different piezoelectric vibration modes were presented in details, as
well as the three-port network, equivalent circuit, and electrical impedance matching analysis of
piezoelectric transducers. The compositing effect on piezocomposite transducers was explained
and verified experimentally also in this chapter. These theories are central for understanding and
optimizing LWD transducers.
One of the research objectives is to design LWD transmitters that meet acoustic requirements
such as transmitting power, transmitting voltage response (TVR), directivity, etc. In the
Optimization chapter, focus was given to a detailed methodology for applying COMSOL
Multiphysics® to achieve this goal. Material properties, meshing techniques, and physics
coupling were presented in details. Displacement frequency responses of two piezocomposite
transducer designs were compared and general design strategies were come up with. Targeted
studies confirmed these design strategies. Two important strategies is disintegrating the height
direction to reduce the height mode resonance around 8-10 kHz, and slanting the piezoelectric
ceramic pieces to broaden response. A comparison of acoustic performance parameters including
acoustic field spatial distribution, absolute acoustic pressure, TVR and directivity was made
between the two designs. An extensive comparison between d33 and d31 configurations revealed
the advantage and disadvantage of each.
Usually LWD transducers work under extreme environment such as high temperature, high
pressure, corrosive chemicals, and strong vibrations. This requires rigorous packaging for the
piezocomposite transducers. In the Fabrication chapter, first some fabrication topics were
discussed, including epoxy selection, solder selection, thermal expansion coefficient
consideration, and in-vacuum bonding setup and method. These discussions are a summary of
trial and error along the project progress. It might seem concise but it is equivalent to an
immense amount of work. In-vacuum epoxy bonding and uniform thickness spacer are proven to
achieve ultrahigh bonding strength in combination. Once the techniques were discussed, the
fabrication of a successful high-performance piezocomposite transducer prototype was presented
step-by-step. Typical steps were piezoelectric ceramic cutting, packaging material machining,
epoxy bonding, and impedance analysis. Their performances matched the computation
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simulations. The preliminary prototypes leading to the final successful prototype were explained
in the Appendix A. The first high-performance piezocomposite transducer featured slant-cut
ceramics, resulting in broadband response at the expense of reduced resonance peaks. The
second high-performance piezocomposite transducer featured non-slant-cut ceramics, bringing
about strong resonance peaks but less broad response. One of the contributions of this
dissertation work is to have successfully developed fabrication processes to use high-temperature
polymer polyether ether ketone (PEEK) which is also corrosion resistant. The application of this
polymer simplified the transducer design and fabrication significantly. Challenges conquered
include ultrahigh bonding strength for large PEEK pieces, especially with in-vacuum bonding
which leaves no trapped air bubbles suitable for high pressure applications.
Extensive testing is needed before the transducers can be used in the field. In the Testing chapter,
protocols for multiple tests were established. These tests are anticorrosion testing, to make sure
transducers can withstand corrosive drilling fluids; thermal cycle testing, to make sure
transducers can withstand high working temperature repeatedly without deteriorating in quality;
high voltage testing, to make sure transducers can withstand high driving voltage without
dielectric breakdown; high hydrostatic pressure testing, to make sure transducers can withstand
high working pressure in the oil well; vibration testing, to make sure transducers can withstand
strong vibration in the drilling practice; and acoustic testing, to make sure transmitters can
transmit enough power at designated driving frequency for the logging application, and have
desired TVR and directivity. The prototypes we fabricated passed all the tests.
Receivers were discussed separately and presented in the Receiver Considerations chapter. Some
receiver design strategies were looked into first. Structure-stress interaction studies by COMSOL
Multiphysics® compared different piezoelectric ceramic configurations to find the receiver with
the highest receiving sensitivity (RS) and signal-to-noise ratio (SNR). Different packaging
materials were studied also aiming to improve receiver performance. In terms of fabrication and
testing, there were more similarities between transmitters and receivers than differences.
Therefore, this chapter focused on the deviations rather than repeating the same processes. Using
the same fabrication techniques, receiver prototypes were manufactured and their impedance
analysis was presented. They featured a flat response between 11 kHz and 15 kHz, which is a
desired performance for LWD receivers so that the receiver is equally sensitive to all frequencies
and less prone to excitation variations.
Future work can be in four directions. The first one is to fabricate d33 mode transmitters, which
will improve transmitting power and reduce material cost. The second one is to expand design
and fabrication from the current monopole to dipole and quadrupole. Multipole transmitters will
obtain certain data not available for monopole transmitters, especially shear data in slow rock
formation. The third one is on 3D time-transient COMSOL Multiphysics® simulations of sonic
well logging. It will enable transmitter and receiver design optimization in a virtual logging
environment. Last but not least, guided wave simulations can be done on drill collar periodic
groove design to create broader stopbands, which will then facilitate transmitter designs.
Keywords: LWD acoustic transducer, COMSOL design and optimization, extreme-environment
fabrication, transducer testing protocol
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TABLE OF CONTENTS
LIST OF FIGURES ....................................................................................................................... ix
LIST OF TABLES ........................................................................................................................ xii
LIST OF ABBREVIATIONS ...................................................................................................... xiii
ACKNOWLEDGMENTS ........................................................................................................... xiv
CHAPTER 1 INTRODUCTION .................................................................................................... 1
1.1 Background ........................................................................................................................... 1
1.1.1 Applications of Piezoelectric Devices ............................................................................ 1
1.1.2 Geophysics of Oil Drilling ............................................................................................. 1
1.1.3 Working Principles of Acoustic LWD ........................................................................... 2
1.1.4 History of Acoustic LWD............................................................................................... 4
1.2 Motivations............................................................................................................................ 4
1.2.1 Advantages of the Approach ........................................................................................... 4
1.2.2 Uniqueness of the Work .................................................................................................. 5
1.2.3 Advancing State of the Art.............................................................................................. 5
1.3 Research Objectives .............................................................................................................. 6
1.3.1 Optimize Piezocomposite Transmitters with COMSOL Multiphysics® ........................ 6
1.3.2 Investigate Longitudinal Mode Compared to Transverse Mode .................................... 6
1.3.3 Improve Fabrication Techniques for Piezocomposite Transducers ................................ 7
1.3.4 Establish Testing Protocols for LWD Transducers ......................................................... 7
1.3.5 Optimize Piezocomposite Receivers with COMSOL Multiphysics® ............................. 8
1.4 Outline ................................................................................................................................... 8
CHAPTER 2 THEORIES ............................................................................................................. 10
2.1 Elasticity and Acoustics ...................................................................................................... 10
2.1.1 Wave Equations ............................................................................................................ 10
2.1.2 Reflection, Transmission, Absorption .......................................................................... 11
2.2 Piezoelectricity .................................................................................................................... 12
2.2.1 Material Structure ......................................................................................................... 12
2.2.2 Temperature Effects ..................................................................................................... 13
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2.2.3 Stress Effects ................................................................................................................ 14
2.2.4 Constitutive Equations .................................................................................................. 14
2.2.5 Piezoelectrically Stiffened Elastic Constants ............................................................... 15
2.3 Vibration Modes .................................................................................................................. 15
2.3.1 Transverse Mode .......................................................................................................... 15
2.3.2 Longitudinal Mode ....................................................................................................... 16
2.3.3 Thickness Mode ............................................................................................................ 17
2.3.4 Radial Mode ................................................................................................................. 17
2.3.5 Hoop Mode ................................................................................................................... 18
2.4 Transducer Electrical Considerations.................................................................................. 19
2.4.1 Three-Port Network ...................................................................................................... 19
2.4.2 Equivalent Circuits ....................................................................................................... 20
2.4.3 Electrical Impedance Matching .................................................................................... 21
2.4.4 Bandwidth Discussion .................................................................................................. 21
2.5 Compositing Effect ............................................................................................................. 22
2.5.1 Analytical Solutions ..................................................................................................... 22
2.5.2 Experimental Validation ............................................................................................... 24
2.5.3 Computational Validation............................................................................................. 25
2.6 FEM Discussion .................................................................................................................. 25
2.6.1 Elements and Nodes ..................................................................................................... 26
2.6.2 Integral Form of Partial Differential Equations ............................................................ 27
CHAPTER 3 OPTIMIZATION .................................................................................................... 28
3.1 COMSOL Multiphysics® .................................................................................................... 28
3.1.1 Materials ....................................................................................................................... 28
3.1.2 Geometry and Meshing ................................................................................................ 31
3.1.3 Physics Coupling .......................................................................................................... 33
3.2 Displacement Analysis ........................................................................................................ 34
3.3 Targeted Design Studies ...................................................................................................... 37
3.3.1 Slant Angle Broadening Effect ..................................................................................... 37
3.3.2 Height Mode Reducing Effect ...................................................................................... 38
3.4 Design Strategies ................................................................................................................. 39
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3.5 Acoustic Analysis ................................................................................................................ 40
3.5.1 Acoustic Pressure Spatial Distribution ......................................................................... 40
3.5.2 Acoustic Pressure Frequency Response ....................................................................... 41
3.5.3 Transmitting Voltage Response.................................................................................... 43
3.5.4 Directivity ..................................................................................................................... 44
3.6 d33 Transmitters ................................................................................................................... 45
3.6.1 Comparison between d31 and d33 Transmitters ............................................................. 45
3.6.2 d33 Transmitter Design Considerations ......................................................................... 48
CHAPTER 4 FABRICATION ..................................................................................................... 52
4.1 Fabrication Topics ............................................................................................................... 52
4.1.1 Epoxy Selection ............................................................................................................ 52
4.1.2 Coefficient of Thermal Expansion Mismatch .............................................................. 54
4.1.3 Soldering Consideration ............................................................................................... 55
4.1.4 Trapped Air Issue ......................................................................................................... 56
4.1.5 In-Vacuum Bonding ..................................................................................................... 57
4.1.6 Uniform Thickness Spacer ........................................................................................... 58
4.2 Successful Prototype ........................................................................................................... 59
4.2.1 PZT Ceramic Grouping ................................................................................................ 59
4.2.2 PEEK Tube Fitting ....................................................................................................... 61
4.2.3 Impedance Analysis ...................................................................................................... 62
4.2.4 Drill Collar Fitting ........................................................................................................ 67
CHAPTER 5 TESTING ................................................................................................................ 68
5.1 Anticorrosion Test ............................................................................................................... 68
5.2 Thermal Cycle Test ............................................................................................................. 69
5.3 High Voltage Test ............................................................................................................... 70
5.4 High Pressure Test .............................................................................................................. 72
5.5 Vibration Test ...................................................................................................................... 72
5.5 Acoustic Test ....................................................................................................................... 73
5.5.1 Excitation Signal........................................................................................................... 73
5.5.2 Transmitting Voltage Response.................................................................................... 74
5.5.3 Directivity ..................................................................................................................... 75
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CHAPTER 6 RECEIVER CONSIDERATIONS ......................................................................... 77
6.1 Receiver Simulations........................................................................................................... 77
6.1.1 Structure-Stress Interaction .......................................................................................... 77
6.1.2 PEEK vs. Rubber Packaging ........................................................................................ 80
6.2 Receiver Fabrication and Testing ........................................................................................ 82
6.2.1 Receiver Fabrication ..................................................................................................... 82
6.2.2 Receiver Impedance Analysis ...................................................................................... 85
6.2.3 Receiver Drill Collar Fitting ......................................................................................... 87
6.2.4 Receiver Testing ........................................................................................................... 87
CHAPTER 7 SUMMARY AND FUTURE WORK .................................................................... 88
7.1 Summary ............................................................................................................................. 88
7.2 Future Work ........................................................................................................................ 89
7.2.1 Cost Reduction with d33 Transmitters .......................................................................... 89
7.2.2 Dipole and Quadrupole Transmitters ........................................................................... 90
7.2.3 Transmitter-Receiver System Simulations ................................................................... 91
7.2.4 Stopband Design with Guided Waves Studies ............................................................. 93
REFERENCES ............................................................................................................................. 96
APPENDIX A ............................................................................................................................. 100
A1.1 Slant Broadband Transmitter Prototype ......................................................................... 100
A1.2 Non-Slant High-Resonance Transmitter Prototype ........................................................ 104
APPENDIX B ............................................................................................................................. 108
A2.1 COMSOL Simulations on Magnetoelectric Sensors ...................................................... 108
A2.2 Future Simulations on Magnetoelectric Sensors ............................................................ 115
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LIST OF FIGURES
Number Name Page
1.1 Examples of piezoelectric devices in daily life 1
1.2 Schematic logging techniques in oil drilling 2
1.3 Working principles of LWD acoustic transducers 3
1.4 Segmented transmitter multipole source 4
1.5 Longitudinal mode compared to transverse mode transducer configurations 7
2.1 The x - directed stresses acting on a volume element in homogeneous bulk 10
2.2 PZT ceramic crystal structure 12
2.3 Temperature dependence of electromechanical coupling coefficients 13
2.4 Temperature dependence of piezoelectric charge constants 13
2.5 Effect of stress on piezoelectric properties 14
2.6 Configuration of a transverse mode sample 15
2.7 Configuration of a longitudinal mode sample 16
2.8 Configuration of a thickness mode sample 17
2.9 Configuration of a radial mode sample 17
2.10 Configuration of a hoop mode sample 18
2.11 Transducer regarded as a three-port black box 19
2.12 The Mason equivalent circuit 20
2.13 The Krimholtz, Leedom, and Matthaei (KLM) model 20
2.14 Matching circuit for an acoustic transducer 21
2.15 Fabrication steps of a low-power piezocomposite transducer 22
2.16 Wave velocity/resonance frequency dependence on PZT volume fraction 23
2.17 Admittance and resistance curves for low-power piezocomposite transducer 24
2.18 Computational and experimental data comparison for low-power transducer 25
2.19 Nodal placements in a quadratic triangular element and its resulting mesh 26
2.20 The domain with an element e , boundary and element boundary e 27
3.1 Physics coupling in COMSOL Multiphysics® programming environment 28
3.2 Geometry of a two-tier non-slant and a three-tier slant transmitter 32
3.3 COMSOL model of displacement and acoustic analysis of transmitters 33
3.4 Displacement frequency response of the two-tier non-slant transmitter 34
3.5 Displacement frequency response of the three-tier slant transmitter 36
3.6 Geometry in COMSOL to study slant angle broadening effect 37
3.7 Displacement frequency response showing slant angle broadening effect 38
3.8 Geometry in COMSOL to study height mode reducing effect 38
3.9 Displacement frequency response showing height mode reducing effect 39
3.10 Acoustic pressure spatial distribution of two-tier non-slant design 40
3.11 Acoustic pressure spatial distribution of three-tier slant design 41
3.12 Acoustic pressure frequency response of two-tier non-slant design 42
3.13 Acoustic pressure frequency response of three-tier slant design 42
3.14 TVR frequency spectrum of two-tier non-slant design 43
3.15 TVR frequency spectrum of three-tier slant design 43
3.16 Directivity of two-tier non-slant design 44
x
3.17 Directivity of the three-tier slant design 45
3.18 Geometry of the d31 mode piezocomposite transmitter 45
3.19 Geometry of the d33 mode piezocomposite transmitter 46
3.20 The comparison of radial displacement between d31 mode and d33 mode 47
3.21 d33 mode model featuring 8 PZT ceramic pieces in each quarter 48
3.22 Comparison of radial displacement between d31 mode and 8-piece d33 mode 49
3.23 Displacement color graph for d31 mode 50
3.24 Displacement color graph for 4-piece d33 mode 50
3.25 Displacement color graph for 8-piece d33 mode 51
4.1 Testing specimens, lap-shear test equipment, and shear stress curves 53
4.2 Transmitter breakdown caused by epoxy silver paste wire bonding 55
4.3 Grooves were made on the PZT ceramic to facilitate wire soldering 56
4.4 Failed transmitter prototype after high pressure test 56
4.5 Trapped air studies with different ambient pressure bonding techniques 57
4.6 In-vacuum bonding system and different actuation holders 58
4.7 Optic fiber and glass fiber fabric as uniform thickness spacer 58
4.8 PZT pieces with no cutting needed 59
4.9 Capacitance data of 24 PZT pieces 60
4.10 Solidworks drawing of center PEEK tubes 61
4.11 Three sets of PEEK tubes, and one set in details 62
4.12 PEEK center frame showing wire grooves and PZT fitted in PEEK frame 62
4.13 Finished successful prototypes and impedance analyzer 63
4.14 Capacitance data for the 6 individual transmitters before/after annealing 63
4.15 Admittance data for the 6 individual transmitters before/after annealing 64
4.16 Conductance data for the 6 individual transmitters before/after annealing 64
4.17 Impedance data for the 6 individual transmitters before/after annealing 65
4.18 Resistance data for the 6 individual transmitters before/after annealing 65
5.1 BORE-GEL® drilling mud solution and anticorrosion test 68
5.2 Thermal cycle test heating profile and programmable furnaces 69
5.3 Admittance data before and after thermal cycle test for the prototype 70
5.4 Setup for high voltage testing and charging and discharging currents 71
5.5 Capacitance curve for transmitters before and after high voltage test 72
5.6 13 kHz transmitting and receiving acoustic signals for transmitters 74
5.7 Measured transmitting voltage response 74
5.8 TVR simulated by COMSOL Multiphysics® 75
5.9 Measured directivity 76
5.10 Directivity simulated by COMSOL Multiphysics® 76
6.1 Structure-stress interaction studies of receivers 77
6.2 Electric potential distribution of receivers with different PZT configurations 78
6.3 Receiving sensitivity curves for the three PZT configurations 79
6.4 Signal to noise ratio curves for the three PZT configurations 80
6.5 Electric potential distribution of the two rubber-packaged receivers 81
6.6 RS curves for the two rubber packaged receivers vs. PEEK packaging 81
6.7 SNR curves for the two rubber packaged receivers vs. PEEK packaging 82
6.8 24 PZT pieces in each half of receivers, 6 in series, 4 groups in parallel 83
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6.9 Holders to hold center PEEK tube during milling 83
6.10 PZT pieces were soldered for desired electrical connections in PEEK frame 84
6.11 In-vacuum bonding system 84
6.12 Two halves of finished receiver number one 85
6.13 Capacitance curves for four parts of the receivers before annealing 85
6.14 Resistance curves for four parts of the two receivers before annealing 86
6.15 Conductance curves for four parts of the two receivers before annealing 86
7.1 8-piece d33 transmitter compared to d31 transmitter for cost reduction 89
7.2 Dipole and quadrupole sources and their velocity dispersion curves 90
7.3 Transmitter-receiver system simulations 91
7.4 Wave propagation in the transmitter-receiver system simulations 92
7.5 Terminal voltages in the transmitter-receiver system simulations 93
7.6 Periodic groove configurations as stopband designs 94
7.7 Dispersion curves for SH modes on an isotropic plate with free boundaries 94
A1.1 Slant broadband transmitter prototype geometry and packaging 100
A1.2 Capacitance with dielectric loss data for 15 ° slanted PZT ceramic piece 101
A1.3 Resistance curves of two half-tube slant transmitters 102
A1.4 Impedance curves of two half-tube slant transmitters 102
A1.5 Admittance curves of two half-tube slant transmitters 103
A1.6 Conductance curves of two half-tube slant transmitters 103
A1.7 Capacitance curves of two half-tube slant transmitters 104
A1.8 Non-slant high-resonance transmitter prototype geometry and packaging 105
A1.9 Capacitance curves for non-slant high-resonance transmitter one and two 106
A1.10 Conductance curves for non-slant high-resonance transmitter one and two 106
A1.11 Resistance curves for non-slant high-resonance transmitter one and two 107
A2.1 Geometric construction of the magnetoelectric sensor model 109
A2.2 Permanent magnet magnetizes the entire system 110
A2.3 Biomagnetic field in the liver phantom now the magnetizing part 110
A2.4 Biomagnetic field in a pixelated liver model at different lateral positions 111
A2.5 Physics coupling of the pixelated liver model 112
A2.6 Magnetic flux density versus vertical distance 113
A2.7 Magnetic flux density versus lateral distance 114
A2.8 Magnetic flux density versus liver iron concentration 114
A2.9 Computer generated pixelated ellipsoid complete 115
A2.10 Computer generated pixelated ellipsoid at layer 52 116
A2.11 Computer generated pixelated ellipsoid at layer 42 116
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LIST OF TABLES
Number Name Page
3.1 Key properties of PZT materials 29
3.2 Radial poling base vectors 30
3.3 Tangential poling base vectors 30
3.4 Properties of linear elastic materials 31
3.5 Properties of acoustic domain materials 31
4.1 A list of epoxy properties compared with PEEK properties 53
4.2 Pass/fail results of epoxies under different conditions based on shear test 54
4.3 PZT pieces divided into groups during mass production 60
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LIST OF ABBREVIATIONS
Number Term Abbreviation
1 Microelectromechanical Systems MEMS
2 Radio Frequency RF
3 Surface Acoustic Wave SAW
4 Bulk Acoustic Wave BAW
5 Sound Navigation and Ranging SONAR
6 Logging-While-Drilling LWD
7 Finite Element Method FEM
8 Transmitting Voltage Response TVR
9 Sound Pressure Level SPL
10 Receiving Sensitivity RS
11 Signal-to-Noise Ratio SNR
12 Lead Zirconate Titanante PZT
13 Polyether Ether Ketone PEEK
14 American Society for Testing and Materials ASTM
15 Coefficient of Thermal Expansion CTE
16 Computer Numerical Control CNC
17 Root-Mean-Square RMS
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ACKNOWLEDGMENTS
I wish to express the most gratitude to my adviser, Dr. Qiming Zhang. Without him, this
dissertation would be impossible. Being my mentor, he equipped me with an important skill set
and led me in the right direction; being my role model, he spurred me to achieve more goals and
envision bigger dreams. I would like to recognize all other committee members, Dr. Bernhard R.
Tittmann, Dr. Zhiwen Liu, and Dr. Sumeet Kumar Gupta. Thank you for the advising at various
stages.
Special thanks go to all the professors who have taught and enlightened me. It was a profound
part of my doctoral education. I would like to thank my labmates, Lei Mei, among all, for their
assistance and advice. I learned exceedingly from them. I would also like to acknowledge the
graduate assistant staff for their support in departmental issues.
I wish to express my gratitude and love to my best friend and partner, Justin Moore, who has
been treating me with respect, care, and love. I want to recognize all my family members and
friends, who brought laughter and fun into the several years of my graduate schooling. Thank
you very much.
I am deeply indebted to Monix Energy Solutions, Inc. for sponsoring the PhD project that I
worked on with immense interest and great passion. I would like to thank Pennsylvania State
University for giving me an opportunity to further my education six years ago. In these six years,
I got immersed in such an exciting learning and living climate and met so many wonderful
people. These are the most valuable resources that I will treasure for the rest of my life.
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DEDICATION
This dissertation is dedicated to my nieces
Kinley Bender, Yanxi Zhang, and my nephews
Ky Bender, Jacob Lucas, Zishuo Zhang.
1
CHAPTER 1 INTRODUCTION
1.1 Background
1.1.1 Applications of Piezoelectric Devices
The general field of this dissertation is in piezoelectric devices. One will be surprised by how
widely piezoelectric devices are used in all aspects of our lives. Industry-wise, they are used in
aerospace, automotive, acoustics, biomedical, microelectromechanical systems (MEMS), and in
this case, oil and gas. Piezoelectric devices are normally utilized in one of the following four
categories, actuators and sensors including linear and rotary motors, position controllers, energy
harvesters, pressure sensors, accelerometers, gyroscopes, load cells; radio-frequency (RF)
MEMS including RF MEMS switches, surface acoustic wave (SAW) filters, bulk acoustic wave
(BAW) filters, tunable cavity filters; flow control including synthetic jet, inkjet printers, active
valves, micro pumps, flow sensors; and acoustic devices including sound navigation and ranging
(SONAR), hydrophones, speakers, microphones, and hearing aids. For this dissertation, it falls
into acoustic devices domain. Some examples of piezoelectric devices in daily life is shown in
Figure 1.1.
Fig. 1.1 Examples of piezoelectric devices in daily life (Internet sources).
1.1.2 Geophysics of Oil Drilling
The world will consume 1,100 billion barrels of oil and 4,475 trillion cubic feet of natural gas
between 2010 and 2040 [1]. The increasing demand calls on geophysicists, geologists, drilling
engineers, reservoir engineers, and production engineers to work together and utilize a multitude
of measurement tools to achieve improved formation and completion evaluation. Essential
questions to be asked include: Are there any hydrocarbons, and if so are they oil or gas? Where
are the hydrocarbons? How much hydrocarbon is contained in the formation? How producible
are the hydrocarbons [2]?
2
The complicated nature of properties of reservoir rocks, such as porosity, permeability,
consolidation, homogeneity, fractures, etc., makes it difficult to answer these questions. The
partitioning of the hydrocarbon and the brine, the “saturation”, is also of considerable importance
not only for the ultimate production procedure but also for the interpretation of seismic
measurements [2]. Furthermore, hydrocarbon extraction belongs to an extremely costly industry
with a high occupational injury rate. A deep water well of duration of 100 days costs around
US$100 million. The fatality rate among oil and gas workers is eight times higher than the all-
industry rate of 3.2 deaths for every 100,000 workers [1].
To obtain relevant information to reduce drilling cost, ensure safe completion and production,
and understand the complicated reservoir rock formation, an array of measurement tools have
been developed in the past years. Based on the operation, they can be roughly grouped into
wireline logging and logging while drilling (LWD). Based on broad disciplines, there are three
major measurement techniques: electrical, nuclear, and acoustic [3, 4]. The electrical technique
developed was a measurement of resistivity or conductivity. The nuclear technique explored
neutron, gamma ray, and nuclear magnetic resonance. The acoustic technique relied on acoustic
wave propagation along the borehole [4-6]. This is the major field of investigation in this
dissertation work. A schematic drawing of different logging techniques being used in oil drilling
is given in Figure 1.2.
Fig. 1.2 Schematic logging techniques in oil drilling [7].
1.1.3 Working Principles of Acoustic LWD
The propagation of waves in the borehole can be abstracted as in Figure 1.3 (Left). Since stone
formations always have larger compressional wave velocity than the drilling fluids, from Snell’s
law (Figure 1.3 (top right)), we know there is always an incident angle (critical angle) where the
refracted compressional wave is along the interface, travelling at the speed of the compressional
wave velocity in the stone formations. This is called the compressional head wave. In fast
formation, the shear wave velocity in the stone formation is larger than the compressional wave
velocity in the drilling fluids, so there will be a shear head wave as well. However, in slow
formation, the shear wave velocity in the stone formation is slower than the compressional wave
velocity in the drilling fluids, so there is no shear head wave in this kind of formation [4]. In
terms of penetration depth, it has the magnitude of one wavelength. For shale with a
3
compressional wave velocity of 1500 m/s (same with in water) and assuming a driving frequency
of 13 kHz, it results in a wavelength of about 115 mm. This give us an idea on the penetration
depth.
Fig. 1.3 (Left) Schematic drawing showing transmitter sends out acoustic waves, the waves get critically
refracted at the drilling mud-rock formation interface, and then get received by the receivers [2]; (top
right) wave reflection and transmission at an interface, which gives rise to the critically refracted waves
based on Snell’s law; (bottom right) waves arrive at different receivers at delayed times. There are
compressional, shear, and Stoneley waves for fast formation. There are no shear waves for slow formation
[3].
As shown in Figure 1.3 (bottom right), acoustic LWD tools have multiple receivers, and the
sonic signal arrives later as the transmitter-receiver distance increases. The same is for the shear
wave and Stoneley wave. Since in the slow formation there is no shear wave, there is only
compressional and Stoneley wave. From the phase delay one can calculate the velocity of the
sound waves, or for petrophysicists, the reciprocal termed slowness is more frequently used.
Mechanical properties such as Young’s modulus, shear modulus, bulk modulus, etc. can be
obtained from the slowness data using the following equations [3-6]:
2 2
4 9 3 2
3 3 6 2
b b
c s
a a G KG K GM G K M E v
K G K Gt t
The compressional modulus, M , is computed from the compressional slowness time ( ct ) and
bulk density, b . The shear modulus, G , is computed from the shear slowness time ( st ) and
bulk density. The a in both equations is a unit conversion constant. In turn, these two moduli are
used to compute the bulk modulus, K , Young’s modulus, E , and Poisson’s ratio, v .
4
1.1.4 History of Acoustic LWD
The absence of LWD acoustic tools has only been recognized by the oil drilling community since
1989 when preliminary feasibility study of sonic while drilling was initiated within
Schlumberger to address some common problems: suppressing collar arrivals, transmitter and
receiver mounting on drill collars, interference of drilling noise, and processing sonic waveforms
downhole [8]. During early field tests, some ground rules were come up with. For maximum
sonic output, it is desirable to have a resonant acoustic transmitter. And the transmitter design
takes advantage of the collar geometry to achieve an acoustic resonator that is excited with a
moderate-duration voltage pulse around the resonant frequency.
Over the next few years, a few companies including Halliburton and Baker Hughes also
subsequently carried out research and design of tools for measuring formation compressional
wave slowness while drilling. There were some continued efforts to overcome drilling noise
issues, drill collar isolator designs, and mechanical strength challenges [9]. LWD acoustic
logging was becoming increasingly common, serving real-time petrophysical, geophysical, and
drilling applications, and as an economical substitute for wireline sonic logging in many high-
angle or high-cost applications [10].
The earlier focus was mainly on compressional wave velocity measurements. Therefore,
monopole source was sufficient. However, as the needs to drill formations with different
characteristics increased, it was desirable to measure formation shear wave velocity. In that
context, companies raced to develop dipole and quadrupole tools, and it eventually led to
multipole tools that can operate in different modes based on the electrical excitations [11-13].
Figure 1.4 shows such a LWD acoustic transducer.
Fig. 1.4 Segmented transmitter multipole source [12, 13].
1.2 Motivations
1.2.1 Advantages of the Approach
LWD acoustic transducers are still supplemental tools in the oil drilling industry. Seismic data
give an evaluation of a large area in a cost-effective and time-efficient fashion. Wireline logging
uses numerous other tools including resistivity and nuclear to provide comprehensive and more
5
specific information. However, the advantages of the LWD acoustic transducers are irreplaceable
by other means. First, it gives a well-specific data that help to correlate to existing wireline
logging and seismic data. It corrects any discrepancies and is more accurate to each specific well
than any other approaches. With each drilling practice, more data is obtained only via LWD tools
that can be used to improve information on the entire drilling area. Secondly, LWD acoustic
tools provide real-time data that are needed to make on-site decisions. Examples include
adjusting drilling mud weight to balance the formation pressure. If there is a drastic change in
formation pressure, drilling mud weight needs to be adjusted right away to ensure safe operations.
Only LWD data can be used to achieve that goal. Another example is deciding the drilling
directions. Nowadays, not only is there vertical drilling, there are also horizontal drilling and
directional drilling. As long as there is petroleum reserve, modern technology enables drilling in
any directions. For each well, the direction can be different. LWD acoustic transducers “see” the
wide angle of drilling as they are installed on the head of the drill collar. And that “sight” can
help determine the drilling direction to maximize production.
In comparison with other techniques, acoustic transducers can get mechanical data unobtainable
by resistivity or nuclear tools. Needless to say, all these tools are unique in their own ways.
Resistivity tool measures the resistance or conductance of earth strata. Nuclear sources emit fast
neutrons that interact with rock formation, which further emits gamma ray, and then gets
detected by sensors. Acoustic transducers send out sonic waves through the formation before
they get received by the receivers. Whether it is resistance, or gamma ray spectrum, or slowness,
there are empirical data affected by rock properties such as lithology, porosity, saturation, etc.
All three measurements can be compared with empirical data and these rock properties can then
be interpreted. However, acoustic tools can measure compressional and shear wave velocities
which are determined by mechanical properties of the rocks. The mechanical parameters cannot
be obtained from other techniques than acoustic transducers.
1.2.2 Uniqueness of the Work
The originality of this dissertation work is sometimes questioned because there are existing
products from companies like Schlumberger, Baker Hughes, and Halliburton. Though the field of
LWD acoustic transducers led by major oil service companies continues to grow strong with a
number of multipole tools available commercially, their design and fabrication processes are
highly confidential and protected by numerous patents. For smaller oil service companies, they
are still in the pathfinding stage. This dissertation work is aimed to make breakthrough in an
extremely secretive field where there is very limited information and references. The
breakthrough will shine light on the design and fabrication processes of the LWD transducers,
which will be available to smaller oil service companies and the public. This is the uniqueness of
the work. In a way, it is no less than making complete new findings in some fields.
1.2.3 Advancing State of the Art
Even for leading oil service companies, they are still facing challenges to use new materials as
packaging materials so that the transducers can meet the mechanical strength requirements.
Commercially available products still use multiple layers of materials, rubber for anticorrosion,
epoxy for high temperature and thermal expansion protection, and fiber glass for mechanical
toughness. It makes the acoustic response less clear with so many acoustically mismatched
6
layers. In this dissertation work, we chose polyether ether ketone (PEEK) as single packaging
material. It is corrosion resistant, mechanically tough, and retains the superior properties under
high temperature. With all these advantages in one material, we can use this one layer to protect
the ceramic pieces. It reduced the acoustically mismatched layers, and improved the acoustic
response. This is a breakthrough even among leading oil service companies, and without doubt is
advancing the state of the art.
Another benefit of using PEEK as packaging material is that it simplified the fabrication process.
Because PEEK is in solid form, the frame can be machined into the exact shape to house ceramic
elements. Once the epoxy bonding obstacle was conquered as was done in this dissertation work,
the fabrication process was greatly simplified. In comparison, it was extremely difficult to
maneuver ceramic elements when trying to mold them into fluidic epoxy. It was also very
technically challenging to cover the transducers with the rubber layer. The space between the
rubber layer and the transducers needed to be vacuumed for high hydrostatic pressure purposes.
However, if it was not done properly, the degassing process could break the ceramic elements
inside the transducers. There were more technical pitfalls than our new fabrication process.
1.3 Research Objectives
1.3.1 Optimize Piezocomposite Transmitters with COMSOL Multiphysics®
For piezocomposite LWD transducers with complicated geometry and composition, design by
analytical solutions or Mason equivalent circuit is in disadvantage. Finite element method (FEM)
as a numerical technique for finding approximate solutions to boundary value problems for
partial differential equations stands out in this problem [14]. Among those, COMSOL
Multiphysics® was chosen for the design process because of its ease to couple multiple physics
together [15, 16]. Overall, FEM is time-efficient and cost-effective.
The design optimization of LWD piezocomposite transmitters involves a number of
requirements:
(1) Clean, broadband response from 10 to 15 kHz to meet drill collar stopband design [17-21];
(2) Peak resonance at 13 kHz to meet main driving frequency requirement;
(2) High power tolerance with a driving voltage of over 1000 V;
(3) High acoustic transmitting power over 10,000 Pa at 1 m;
(4) High transmitting voltage response (TVR) over 130 dB at 1 m;
(5) Excellent directivity with ± 10 dB sound pressure level (SPL) at 1 m except at connecting
parts.
1.3.2 Investigate Longitudinal Mode Compared to Transverse Mode
In the scope of this dissertation research, a more common vibration mode for oil drilling
transmitters was adopted, the transverse vibration mode. However, investigating longitudinal
mode (Figure 1.5) piezocomposite transmitters has some merits. One of the most outstanding one
is that d33 coefficient is greater than d31 coefficient by a factor of 2.5 approximately [22]. This
will increase the transmitting power significantly.
7
Fig. 1.5 Longitudinal mode (left) compared to transverse mode (right) piezoelectric ceramic
configuration.
1.3.3 Improve Fabrication Techniques for Piezocomposite Transducers
Fabrication of piezocomposite LWD transducers for oil drilling applications has always been
challenging because of the following harsh working conditions [23]:
(1) Maximum working temperature is at 200 °C;
(2) Maximum working pressure is at 20,000 psi (140 MPa);
(3) Strong vibrations of 20 GRMS between 2 Hz and 1 kHz from the drilling practice;
(4) High impact of 500 GRMS for duration of 2 ms from the drilling practice;
(5) Corrosive gas and drilling fluids.
Research in this dissertation scope aims to fabricate successful prototypes as optimized and
summarize a system of fabrication techniques ranging from material selection, machining, to
epoxy bonding. With the selection of high-temperature mechanically-tough corrosion-resistant
PEEK polymer as the packaging material, the fabrication techniques are especially important
because this material will simplify the transducer design with its excellent all-in-one properties.
Instead of using multiple acoustically-mismatched layers, one PEEK layer is enough. Challenges
for bonding large PEEK pieces will be resolved in this dissertation work.
1.3.4 Establish Testing Protocols for LWD Transducers
Complying with the design requirements and the harsh working conditions, the following testing
protocols need to be established.
(1) Anticorrosion testing, to make sure transducers can withstand corrosive drilling fluids;
(2) Thermal cycle testing, to make sure transducers can withstand high working temperature
repeatedly without deteriorating in quality;
(3) High voltage testing, to make sure transducers can withstand high driving voltage without
dielectric breakdown;
(4) High hydrostatic pressure testing, to make sure transducers can withstand high working
pressure in the oil well;
(5) Vibration testing, to make sure transducers can withstand strong vibration in the drilling
practice;
8
(6) Acoustic testing, to make sure transmitters can transmit enough power at designated driving
frequency for the logging application, and meet the TVR and directivity requirements.
1.3.5 Optimize Piezocomposite Receivers with COMSOL Multiphysics®
The design optimization of LWD piezocomposite receivers has its own uniqueness:
(1) Flat, broadband response from 2 to 25 kHz to receive wide range of signals; unlike the
transmitters which are favored for a resonance peak, the receivers are preferred to have a flat
response;
(2) High receiving sensitivity (RS) over -210 dB with signal source at 1 m;
(3) High signal-to-noise ratio (SNR);
(4) Excellent directivity to receive signals from different directions.
Beyond the design requirements, the receivers also have to sustain extreme working environment,
including high temperature to 200 °C, high hydrostatic pressure at 20,000 psi (140 MPa), strong
vibrations of 20 GRMS between 2 Hz and 1 kHz, high impact of 500 GRMS for duration of 2
ms, and corrosive chemicals.
1.4 Outline
The structure of the dissertation is such that there are five chapters following the Introduction
and prior to the Summary and Future Work: Theories, Optimization, Fabrication, Testing, and
Receiver Considerations.
In the Theories chapter, the fundamentals of elasticity, wave motion and wave equations were
introduced. Wave theories on acoustic impedance matching (reflection and refraction), and
attenuation were also covered briefly. Piezoelectricity constitutive equations and
piezoelectrically stiffened wave equations were outlined. And theories on different piezoelectric
vibration modes were presented in details, as well as the three-port network, equivalent circuit,
electrical impedance matching analysis, and bandwidth discussion of piezoelectric transducers.
The compositing effect on piezocomposite transducers was explained and verified
experimentally also in this chapter. These theories are central for understanding and optimizing
LWD transducers.
In the Optimization chapter, focus was given to a detailed methodology for applying COMSOL
Multiphysics® to optimizing LWD transducer design parameters. Material properties, meshing
techniques, and physics coupling were presented in details. Displacement frequency responses of
two piezocomposite transducer designs were compared and general design strategies were come
up with. Targeted studies confirmed these design strategies. A comparison of acoustic
performance parameters including acoustic field spatial distribution, absolute acoustic pressure,
TVR and directivity was made between the two designs. An extensive comparison between d33
and d31 configurations revealed the advantage and disadvantage of each.
In the Fabrication chapter, first some fabrication topics were discussed, including epoxy
selection, solder selection, thermal expansion coefficient consideration, and in-vacuum bonding
setup and method. These discussions are a summary of trial and error along the project progress.
It might seem concise but it is equivalent to an immense amount of work. Once the techniques
9
were discussed, the fabrication of a successful high-performance piezocomposite transducer
prototype was presented step-by-step. Typical steps were piezoelectric ceramic cutting,
packaging material machining, epoxy bonding, and impedance analysis. Of course, the success
was not attained in one trial. Earlier prototypes that gave important insights and knowledge on
the successful prototype were explained in Appendix A. The first high-performance
piezocomposite transducer featured slant-cut ceramics, resulting in broadband response at the
expense of reduced resonance peaks. The second high-performance piezocomposite transducer
featured non-slant-cut ceramics, bringing about strong resonance peaks but less broad response.
The innovative way of bonding PEEK in vacuum resulted in ultrahigh bonding strength for this
material, which is a breakthrough that resulted in simplified transducer design. For LWD
transducers to work in oil drilling conditions such as high temperature, high pressure, corrosive
chemicals, etc., multiple materials such as epoxy, fiber glass, rubber were used, which made the
fabrication process tedious. With PEEK being mechanically tough, corrosion resistant, and
remaining stable in properties at elevated temperatures, it combines all the needs in one material
and greatly simplifies the design and fabrication once the bonding strength issue was resolved in
this dissertation work.
In the Testing chapter, protocols for multiple tests were established. These tests are anticorrosion
testing, to make sure transducers can withstand corrosive drilling fluids; thermal cycle testing, to
make sure transducers can withstand high working temperature repeatedly without deteriorating
in quality; high voltage testing, to make sure transducers can withstand high driving voltage
without dielectric breakdown; high hydrostatic pressure testing, to make sure transducers can
withstand high working pressure in the oil well; vibration testing, to make sure transducers can
withstand strong vibration in the drilling practice; and acoustic testing, to make sure transmitters
can transmit enough power at designated driving frequency for the logging application, and have
desired TVR and directivity.
In the Receiver Considerations chapter, some receiver design strategies were looked into.
Structure-stress interaction studies by COMSOL Multiphysics® compared different piezoelectric
ceramic configurations to find the receiver with the highest RS and SNR. Different packaging
materials were studied also aiming to improve receiver performance. Using the same fabrication
techniques, receiver prototypes were manufactured and their impedance analysis was presented.
10
CHAPTER 2 THEORIES
2.1 Elasticity and Acoustics
2.1.1 Wave Equations
A wave is “a disturbance [of a medium] from a neutral or equilibrium condition that propagates
without the transport of matter”. Here, we distinguish between two types, or modes, of plane
waves: (a) transverse and (b) longitudinal [24]. Wave motion is deeply rooted in elasticity and
acoustics theories.
Fig. 2.1 The x - directed stresses acting on a volume element in homogeneous bulk (unbounded) medium
[24].
Figure 2.1 depicts the x - directed stresses acting on a volume element within the bulk material.
Derived from Newton’s second law, the x , y , and z vector components of the three-
dimensional wave equation (in any medium) are 2
2
2
2
2
2 (2.1.1)
xyxx xz x
yx yy yz y
zyzx zz z
u
x y z t
u
x y z t
u
x y z t
In solid mechanics, the stress-strain relationship is succinctly written in tensor form as
2 (2.1.2)ij ij ij
where xx yy zz and the indices , ,i j k equal , , or x y z .
Using that relationship and the definition of strain for an isotropic, homogeneous material, one
can obtain 2
2
2( ) (2.1.3)x
x
uu
t x
11
for the x direction, where: 2 2 2
2
2 2 2 (2.1.4)
x y z
Similarly, for the y and z directions: 2
2
2( ) (2.1.5)
y
y
uu
t y
22
2( ) (2.1.6)z
z
uu
t z
Now, assume the energy of the wave is propagating with phase velocity v in the x direction.
Equation (2.1.3) yields a phase velocity of 1/2
2 (2.1.7)lv
and both equations (2.1.5) and (2.1.6) yield the same phase velocity of 1/2
(2.1.8)tv
The coefficients (comprised of material properties only) are called the Lamé constants:
2 (2.1.9)(1 )
E
(2.1.10)(1 )(1 2 )
E
where is the shear constant and is also considered a Lamé constant.
2.1.2 Reflection, Transmission, Absorption
As wave propagates, it can impinge onto interfaces separating two different materials. In normal
incidence, the reflection and transmission coefficients can be written in terms of acoustic
impedances:
2 1
2 1
(2.1.11)r
i
P Z Zr
P Z Z
2
2 1
2 (2.1.12)t
i
P Zt
P Z Z
Acoustic impedance is defined as
(2.1.13)aZ v
where is the density, and av is the acoustic velocity.
As we can see from the above section, acoustic velocity is determined by density and elastic
moduli of the matter. In generalized cases,
= (2.1.14)a
Yv
where Y is the Young’s modulus.
Thus
12
(2.1.15)Z Y
In the oblique incidence case, a convenient trigonometric relation known as Snell’s law gives the
propagation directions of the scattered waves.
sin sin sin sin sin (2.1.16)i rs rl ts tl
i rs rl ts tlv v v v v
In application, the amplitude of any propagating wave diminishes with distance traveled, by
absorption, scattering, beam spreading, dispersion, etc. Mathematically, attenuation can be
represented as a decaying exponential.
0 (2.1.17)I xI I e
where 0I is initially measured acoustic intensity, I is measured acoustic intensity at a distance
x from the initial measurement point, I is attenuation coefficient for acoustic intensity.
Another method is to measure attenuation in decibels (dB). 2
dB/10
2
0 0 0
10 or dB 10log 20log (2.1.18)I P P
I P P
2.2 Piezoelectricity
2.2.1 Material Structure
A piezoelectric crystal has an asymmetric atomic lattice. Since the center of positive charge does
not coincide with its center of negative charge, when an electric field is applied to such a
material, two neighboring atoms will not move the same distance and it changes its mechanical
dimensions. Conversely, an electric field is generated in a piezoelectric material that is strained.
It is illustrated in Figure 2.2 [25]. One example of naturally occurring piezoelectric crystals is
quarts. Synthetic piezoelectric ceramics such as barium titanate (BaTiO3), lithium niobate
(LiNbO3), lead zirconate titanate (PZT), and more were discovered for research and applications
[26]. Some polymers like polyvinylidene fluoride (PVDF) [27-32] also exhibit piezoelectric
effect by attracting and repelling the intertwined long-chain molecules when an electric field is
applied.
Fig. 2.2 Piezoelectricity is rooted in the ceramic crystal structure [25].
13
2.2.2 Temperature Effects
Effect of temperature on the main piezoelectric parameters of soft PZT ceramic has been
investigated by Miclea, et al [33]. Figure 2.3 (left) shows the variation of the electromechanical
coupling factor pk with temperature, from room temperature up to the Curie point, situated
around 350 °C. One observes that up to 150 °C pk remains constant after which it decreases
relatively slowly up to about between 250 °C, with a decreasing rate of about 0.2%/°C and above
250 °C, the decreasing is rather sudden. Figure 2.3 (right) illustrates the behavior of mechanical
quality factor mQ with temperature. One can also observe that between the room temperature
and 250 °C it grows up slowly and steadily after which it remains nearly constant up to about
300 °C and then suddenly falls off due to the quick depoling of the sample [26].
Fig. 2.3 The temperature dependence of (left) the electromechanical planar coupling coefficient pk and
(right) the mechanical quality factor mQ between room temperature and the Curie point [26].
The behavior of the loss tangent tan is shown in Figure 2.4 (left). One observes an
insignificant increase up to 250 °C, after which it suddenly goes up. The behavior of the charge
constants 33d and 31d are presented in Figure 2.4 (right) [26].
Fig. 2.4 The temperature dependence of (left) the loss tangent tan and (right) the charge constants 33d
and 31d between room temperature and the Curie point [26].
14
2.2.3 Stress Effects
The reason why we study the effect of temperature on the basic piezoelectric properties is that
the LWD transducers operate at elevated temperatures. The trend in the change of the properties
give us a better idea how the devices will perform under high-temperature conditions. Same
thing is for high pressure. The transducers also operate at very high hydrostatic pressure.
Therefore, the effect of stress on basic piezoelectric properties were also studied by Zhao, et al
[34]. Figure 2.5 (a) to (f) shows the effect of T3 stress on dielectric constant and dielectric loss,
s33, s13, k33, d33, and d31, respectively. In general, high pressure decreases all the properties,
resulting in a reduced performance.
Fig. 2.5 Effect of T3 stress on (a) dielectric constant and dielectric loss, (b) s33, (c) s13, (d) k33, (e) d33, and
(f) d31 [34].
2.2.4 Constitutive Equations
Relationship between mechanical, electrical, and piezoelectric considerations is summarized as
piezoelectric constitutive equations. There are four forms of constitutive equations.
Strain-Charge Form:
(2.2.1)
E
ij ijkl kl kij k
T
i ikl kl ik k
S s T d E
D d T E
Stress-Charge Form:
(2.2.2)
E
ij ijkl kl kij k
S
i ikl kl ij k
T c S e E
D e S E
Strain-Field Form:
(2.2.3)
D
ij ijkl kl kij k
T
i ikl kl ik k
S s T g D
E g T D
15
Stress-Field Form:
(2.2.4)
D
ij ijkl kl kij k
S
i ikl kl ik k
T c S h D
E h S D
where S is the strain tensor, T is the stress tensor, E is the electric field intensity, D is the
electric displacement, s is elastic compliance constant, c is elastic stiffness constant, the
superscript E stands for at constant electric field, the superscript D stands for at constant
electric displacement, d is piezoelectric charge constant, g is piezoelectric voltage constant, e
is piezoelectric stress constant, h is piezoelectric deformation constant, is permittivity, and
is impermittivity [35].
2.2.5 Piezoelectrically Stiffened Elastic Constants
For plane-wave propagation in the infinite piezoelectric medium, the following equation stands: ( )
, (2.2.5)v
jk k vv ju ii
( ) / (2.2.6)v E S
jk ijkl i l mij m i lnk l n rs r s vjkvc n n e n n e n n n n c
denotes the piezoelectrically stiffened elastic constants for plane-wave propagation in the
direction in , where i iv n x denotes the magnitude of length in the propagation direction and in
the components of the unit wave normal relative to the crystal axes, and the convention that a
repeated Greek index is not to be summed has been adopted. When the second term in equation
(2.2.6) vanishes, the elastic constant is said to be unstiffened [35].
Steady-state plane-wave solutions of equation (2.2.5) may be written ˆ( )
(2.2.7)vi x v
j ju A e
where jA represents the amplitude of each displacement component. From equations (2.2.5) and
(2.2.7) the three wave velocities for the direction in may be found from the three assumed-
positive eigen-values(n) of ( ) ( ) 0 (2.2.8)v n
jk jkc
by means of the relation ( ) ( ) 2( ) , 1,2,3 (2.2.9)n nc v n
2.3 Vibration Modes
2.3.1 Transverse Mode
Fig. 2.6 Configuration of a transverse mode sample.
16
The configuration of a transverse vibration mode sample is shown in Figure 2.6. It satisfies
, , 3l b l w w b such that the elastic conditions are [35-37]:
0; 0; 0; 0; 0x y z x y zT T T S S S
For this vibration mode, the series resonance frequency is
11
1 1 (2.3.1)
2s E
fl s
The electromechanical coupling factor is
3131
33 11
(2.3.2)T E
dk
s
And the parallel resonance frequency is related to the two by
2
31
( )tan (2.3.3)
2 2
p p s
s s
f f fk
f f
2.3.2 Longitudinal Mode
Fig. 2.7 Configuration of a longitudinal mode sample.
The configuration of a longitudinal vibration mode sample is shown in Figure 2.7. It satisfies the
same conditions as the transverse mode, , , 3l b l w w b , and the elastic conditions are [35-
37]:
0; 0; 0; 0x y z x y zT T T S S S
For this vibration mode, the parallel resonance frequency is
33
1 1 (2.3.4)
2p D
fl s
The electromechanical coupling factor is
3333
33 33
(2.3.5)T E
dk
s
17
And the series resonance frequency is related to the two by
2
33
( )tan (2.3.6)
2 2
p ss
p p
f ffk
f f
2.3.3 Thickness Mode
Fig. 2.8 Configuration of a thickness mode sample.
The configuration of a thickness vibration mode sample is shown in Figure 2.8. It satisfies
, l t w t such that the elastic conditions are [35-37]:
0; 0; 0; 0; 0x y z x y zT T T S S S
For this vibration mode, the parallel resonance frequency is
331 (2.3.7)
2
D
p
cf
t
The electromechanical coupling factor is
33
33 33
(2.3.8)tS D
ek
c
And the series resonance frequency is related to the two by
2( )
tan (2.3.9)2 2
p sst
p p
f ffk
f f
2.3.4 Radial Mode
Fig. 2.9 Configuration of a radial mode sample.
18
The configuration of a radial vibration mode sample is shown in Figure 2.9. The elastic
conditions are [35-37]:
0; 0; 0; 0x y z x y zT T T S S S
For this vibration mode, the parallel resonance frequency is
111 (2.3.10)
4
p
p
cf
r
where 1111 2 2
11 12( ) ( )
Ep
E E
sc
s s
The electromechanical coupling factor is
31
33 11
(2.3.11)p
pp p
ek
c
where 2
31 3131 33 33
11 12 11 12
2,p p T
E E E E
d de
s s s s
pk is related to 31k , by the well-known relation
31
2 (2.3.12)
1p p
k k
where 12 11
p E Es s
And the series resonance frequency is related to the coupling factor and the parallel resonance
frequency by an empirical relation [38] 2
22.51( )
(2.3.13)p s p s
p
p p
f f f fk
f f
2.3.5 Hoop Mode
The most common vibration modes, transverse, longitudinal, thickness, and radial, have been
covered in details. However, these pure vibration modes have strict dimensional requirements so
that the corresponding elastic conditions can be met. In real applications, the diverse geometry
types rarely result in one predominant mode but several modes coupled together. For example,
for the sample poled in the radial direction depicted in Figure 2.10, both hoop (circumferential)
mode and height mode exist. This sometimes can be a problem because in LWD application the
hoop mode is more ideally matched to the liquid load and is the one of greater interest [39]. The
height mode vibrations then become noise and need to be rid of by all means.
Fig. 2.10 Configuration of a hoop mode sample.
19
In analysis, different apparent vibration modes can be traced back to the fundamental modes. The
two modes here, for instance, are both essentially transverse mode. Therefore, one can use the
formulas derived for the transverse mode to analyze resonance frequency, electromechanical
coupling factor, etc. For the hoop mode, the series resonance frequency is
11
1 1 (2.3.14)
2s E
fa s
Since there are two modes coupling for this case, to obtain the electromechanical coupling factor,
one can use the effective coupling factor formula
2 2 2 2= / (2.3.15)eff p s pk f f f
2.4 Transducer Electrical Considerations
2.4.1 Three-Port Network
The piezoelectric transducer can be regarded as a “black box” having one or more mechanical
ports and one electrical port, as illustrated in Figure 2.11.
Fig. 2.11 Transducer regarded as a three-port black box.
The parameters satisfy the following matrix manipulation [40]:
1 1
2 2
3 3
0
cot cosec
cosec cot (2.4.1)
1
C a C a
C a C a
hZ l Z l
F vh
F j Z l Z l v
V Ih h
C
where 0
S AC
l
,
0CZ Z A , 1/2
0 0
D
mZ c ,
1/2
0ma Dc
,
S
eh
,
2(1 )D Ec c K ,
and where A is surface area of ceramic element, l is initial length of ceramic element, 0m is
density, Ec is elastic stiffness under constant electric field, S is relative dielectric constant under
constant strain, K is electromechanical coupling factor, e is piezoelectric constant, and is
angular frequency.
20
2.4.2 Equivalent Circuits
Fig. 2.12 The Mason equivalent circuit.
The matrix formula (2.4.1) results in the Mason equivalent circuit of Figure 2.12. In this circuit,
the transformer ratio is 0 0 / /SN hC eC eA l . All other parameters are the same as those in
the three-port network.
It is fitting to obtain electrical impedance of a stand-alone piezoelectric transducer using the
Mason equivalent circuit. However, when we consider the operation of a transducer exciting a
wave in an acoustic medium, it is more convenient to use yet another equivalent circuit, the
KLM model [41, 42]. It is convenient because it expresses the acoustic parameters in terms of an
equivalent transmission line. This makes it easier to design multiple matching layers. On the
other hand, the electrical terminal parameters are expressed in terms of an equivalent lumped
circuit, which is convenient for the design of electrical matching networks at commonly-used
frequencies. The disadvantage of the KLM model is that the ratio of the required transformer
varies with frequency. As shown in Figure 2.13, the transformer ratio is 1/2
0 0 0( / ) sinc( / 2 )T Ck C Z , and 2
0 0' / [ sinc( / 2 )]TC C k .
Fig. 2.13 The Krimholtz, Leedom, and Matthaei (KLM) model.
Other equivalent circuits include the Van Dyke model and Redwood model [43, 44]. Different
models can be chosen depending on the specific applications.
21
2.4.3 Electrical Impedance Matching
Electrical impedance matching is important in designing piezoelectric transducers, especially
concerning bandwidth and efficiency. Here we consider placing the transducer in an inductive
circuit that can tune out its series capacity at the center frequency. In addition, we consider the
situation when a transformer is employed for impedance matching, that is, to match 22
0
0 0 1 2
4 CTa
ZkR
C Z Z
to the impedance of the input or output circuit. The circuit is shown in
Figure 2.14.
Fig. 2.14 Matching circuit for an acoustic transducer.
Under these circumstances, the ratio of the electrical power transferred to the transducer to the
available input power is [40]:
0
2 2
0 0
4 (2.4.2)
( ) ( 1/ )
aT
a a
R R
R R X X C
where X is the reactance of the inductor, 0R is the transformed value of the input resistance, and
a a aZ R jX is the motional impedance of the transducer at an arbitrary frequency.
2.4.4 Bandwidth Discussion
For the bandwidth discussion, it will be limited for two reasons. First, the value of aR , the
radiation resistance, varies with frequency due to the acoustic properties of the transducer. This
gives rise to an effective acoustic Q , defined as 0Q f f , where 0f is the center frequency of
the response and f is the 3-dB bandwidth. A transducer with poor acoustic matching at each
end has a high value of aQ ; one that is well matched at each end has a low aQ . Second, the
circuit used to tune out the transducer capacity, usually a series or parallel inductance, has an
effective circuit Q called eQ (the electrical Q ) of value 0 0 01/e aQ C R . eQ tends to be small if
the coupling coefficient Tk is large.
Using KLM model, we can derive:
1 2
(2.4.3)2
Ca
ZQ
Z Z
22
1 2
2 (2.4.4)
8e
T C
Z ZQ
k Z
The optimum bandwidth of the transducer is obtained with e aQ Q . It follows that
0
4(3 ) (2.4.5)Tkf dB
f
These considerations make the PZT ceramics good choices for broadband, high-efficiency, low-
frequency transducers. As a class they have a very high 2
Tk , which makes 0aR relatively high.
They also have high dielectric constants, thus transducers can have capacitive reactances and
input resistances of the order of 50 . As this impedance is typical for most power sources, such
a design leads to a system that can be reasonably well matched [40].
2.5 Compositing Effect
2.5.1 Analytical Solutions
Fig. 2.15 Fabrication steps of a low-power piezocomposite transducer.
Understanding how compositing changes the transducer performance is very important in
designing high-performance piezocomposite transducers. We fabricated a low-power prototype
aiming to understand the compositing effect, as shown in Figure 2.15. One PZT piece was
packaged in polyether ether ketone (PEEK). The entire frame had the dimensions of 105 mm×67
mm×6.6 mm. The dimensions of the PZT were 95 mm×57 mm×6.35 mm. Here we focus on the
compositing effect rather than details of fabrication and material properties.
Using a similar method described in [45], we derived the composite transverse wave velocity for
this case. One can generalize it as piezoelectric ceramic embedded in elastic matrix.
For the PEEK polymer phase, in the transverse mode, it satisfies:
1 11 1 12 2 12 3 2 3 1, (2.5.1)p p p p p p p pT c S c S c S S S S
For the ceramic phase, using stress-charge constitutive equations, one can obtain:
1 11 1 12 2 13 3 31 3 (2.5.2)c E c E c E c cT c S c S c S e E
In describing the coupling of the phases, a key approximation embodies that the ceramic and
polymer move together in a uniform transverse oscillation, thus
1 1 1 (2.5.3)p cS S S
In relating the electric fields in the two phases, we shall neglect all fringing fields and take the
electric fields to be the same in both phases, namely
3 3 3 (2.5.4)p cE E E
23
For the other two uniform strain components, we assume it is a collective effort of the polymer
and the ceramic:
2 2 2 1
3 3 3 1
(1 )
(1 ) (2.5.5)
p c
p c
S q S qS S
S q S qS S
where q is the volume fraction of the piezoelectric ceramic and (1 ) p cq q .
We can then represent both 1
pT and 1
cT with only uniform terms:
1 11 12 1
1 11 12 13 1 31 3
( 2 )
( ) (2.5.6)
p p
c E E E c
T c c S
T c c c S e E
Our final approximation is that the effective total stress is by averaging over the contributions of
the constituents, namely
1 1 1 11 1 31 3(1 ) (2.5.7)p cT q T qT c S e E
where 11 11 12 11 12 13(1 )( 2 ) ( )p E E E cc q c c q c c c and 31 31e qe .
To get the composite transverse wave velocity, these equations must be supplemented by the
expression for the composite density, namely
(1 ) (2.5.8)p cq q
And then the transverse wave velocity is obtained by
11 (2.5.9)c
v
And the transverse mode resonance frequency is
31 (2.5.10)2
vf
l
For the material properties in this case, one can plot the transverse wave velocity and transverse
mode resonance frequency with respect to the volume fraction of the piezoelectric ceramic, as
shown in Figure 2.16.
Fig. 2.16 Composite transverse wave velocity and resonance frequency dependence on the piezoelectric
ceramic volume fraction.
24
2.5.2 Experimental Validation
Using the derivations, we calculated the analytical solutions of the series resonance frequency
(maximum admittance Y ) and the parallel resonance frequency (maximum resistance R ) for
pure PZT, PZT with PEEK frame, and finalized piezocomposite transducer. Next we compared
them to their experimental data as shown in Figure 2.17.
Fig. 2.17 Experimental admittance and resistance curves for pure PZT, PZT with PEEK frame, and
finalized piezocomposite transducer.
For pure PZT, its volume fraction can be treated as 1. Its analytical resonance frequency is 14.6
kHz, closely matching the experimental series resonance frequency 14.5 kHz of the unbonded
sample. Its experimental parallel resonance frequency was at 15.0 kHz. Therefore the
electromechanical coupling factor for the unbonded sample is:
,31 ,31
31
,31
3.14 15 kHz 14.5 kHz0.3
2 2 14.5 kHz
p s
s
f fk
f
For PZT with bonded wires, its series resonance frequency was 14.2 kHz and its parallel
resonance frequency was 14.7 kHz. They are lower than the analytical solutions respectively, due
to the mass loading effect of the wires. The experimental electromechanical coupling factor for
the PZT with bonded wires is also 0.3. From the admittance curve, we can also calculate the
resonance quality factors using the method described in [36]. The quality factors are similar and
about 80 for both before and after the wires are bonded to the PZT.
For the PZT framed with PEEK, the volume fraction of piezoelectric ceramic is 0.77,
corresponding to a calculated resonance frequency of 13.8 kHz. The experimental series
resonance frequency for the framed PZT was 13.7 kHz. The experimental parallel resonance
frequency was 14.2 kHz. The electromechanical coupling factor for the framed sample is thus
0.3. The quality factor for the composite is about 60. It means there is added damping caused by
the PEEK frame.
25
For the finalized piezocomposite transducer, the volume fraction of PZT is 0.26, corresponding
to a resonance frequency of 12.0 kHz. The experimental series resonance frequency for the
finalized composite piezo-transducer was 12.2 kHz. The experimental parallel resonance
frequency was 12.5 kHz. The electromechanical coupling factor for the finalized sample is 0.25.
The efficiency of the finalized transducer converting electrical energy to mechanical energy is
smaller than the pure PZT or the PZT with the frame. The PEEK frame reduces the efficiency.
The quality factor for the composite transducer is about 60, similar to the one with the frame, but
smaller than the pure PZT one. It is also caused by PEEK damping.
The matching between experimental data to the analytical solutions confirmed that the transverse
wave velocity abides by the compositing effect. And the damping of elastic materials bring down
the peak value.
2.5.3 Computational Validation
Fig. 2.18 Computational and experimental admittance data comparison for pure PZT, PZT with PEEK
frame, and finalized piezocomposite transducer.
Using COMSOL Multiphysics®, a model resembling the low-power piezocomposite transducer
was built and admittance simulations results were obtained and compared with the experimental
data. As shown in Figure 2.18, the computational and experimental results correlated superbly.
2.6 FEM Discussion
The behavior of a physical system is usually governed by some boundary conditions and a set of
equations, some of them partial differential (PDE). Direct analysis of such a system can be very
difficult, especially if the structure is complicated. However, FEM can solve physical systems of
immense complexity through approximations of the PDEs and discretization of the system
geometry. The method is generally performed as in the following steps:
1. Governing equations of the system are identified, and an approximation is derived;
2. The system is discretized into a finite number of elements;
26
3. An approximation of the dependent variable is introduced in each element;
4. The integral form equation is evaluated for each element;
5. The solution in each element is assembled as a global matrix equation;
6. Said matrix equation is solved;
7. Values of interest can be calculated from the solution of the matrix [46].
2.6.1 Elements and Nodes
In finite element method, each domain in a physics model is divided into elements. These
elements are usually polygons with three or four corners. There is always a node at each corner
of the element, and often one or more nodes equally spaced along the sides. An example of a
quadratic triangular element and its resulting mesh is shown in Figure 2.19.
Fig. 2.19 Nodal placements in a quadratic triangular element and its resulting mesh.
For each of the six modes in this element, a shape function iN is defined such that it is unity at
node i , but zero at all other nodes and outside of the element. In addition, the sum of all the
shape functions is unity anywhere within an element. These functions are then, along with the
nodal values of the dependent variable au , used to obtain the approximate value of the dependent
variable, u , anywhere within the element:
1
ˆ (2.6.1)n
a a
a
u u N u
with n being the number of nodes in the element and u being the exact value of the dependent
variable.
It must be noted that iu is a vector only if there is more than one dependent variable. For
example, in solid mechanics, the dependent variables are the displacements in x, y, and z
direction. This interpolation is a good approximation for variables as long as said variables are
constantly increasing or decreasing inside an element, which implies that the sinusoidal
variations associated with a wave will require several elements per wavelength to obtain a good
approximation [47-49].
27
2.6.2 Integral Form of Partial Differential Equations
Fig. 2. 20 The domain with an element e , boundary and element boundary e .
In a physics problem, the variable u satisfies a set of differential equations, ( )A u , anywhere
within the system or domain, , and a set of conditions, ( )B u , on the boundary, , of said
domain. The equations and conditions can be expressed as:
1 1
2 2
( ) ( )
( ) 0 ( ) 0 (2.6.2)( ) ( )
A B
A B
u u
A u B uu u
In the finite element method, the system is discrete, so the approximate value of u is only
defined in a finite number of nodes. Some of the nodal values, au , might be known through the
boundary conditions, but the absolute majority is unknown. The objective is then to find
functions or operators bG and bg so that:
ˆ ˆ( ) ( ) 0, 1,2,3,..., (2.6.3)b bd d b n
G u g u
where n is set as the total number of nodes.
Due to the nature of finite integral, this equation can also be written as:
1
ˆ ˆ( ) ( ) 0, 1,2,3,..., (2.6.4)e e
m
b b
e
d d e m
G u g u
where m is set as the total number of elements.
This implies that the integral can be calculated in every element, and then summed up to yield an
approximate solution for the entire domain. Thus giving the value of u in every node.
Given the definition of ( )A u in equation (2.6.2), it is clear that:
( ) 0 (2.6.5)T d
v A u
where v is a set of n arbitrary test function. The same conclusion can be made for the boundary
condition with the test functions v that are defined on the boundary. This equation is however,
not valid when u is approximated. Still, if the test functions are replaced by weighting functions,
then equation (2.6.5) and its boundary counterpart become valid once more. In the Galerkin
method, which is the one used by COMSOL Multiphysics®, these weighting functions are
identical to the shape functions, thus yielding:
ˆ ˆ( ) ( ) 0, 1,2,3,..., (2.6.6)T T
b bd d b n
N A u N B u
This equation is in the same form as equation (2.6.3), and can be applied for evaluation in each
element [47-49].
28
CHAPTER 3 OPTIMIZATION
3.1 COMSOL Multiphysics®
Fig. 3.1 Piezoelectric transducer physics coupling in COMSOL Multiphysics® programming environment.
COMSOL Multiphysics® is an FEM tool widely used in many physics problems. The modeling
is generally performed in the following steps:
(1) Define parameters and variables;
(2) Build geometry;
(3) Define material properties and assign to domains;
(4) Set boundary conditions and physics coupling;
(5) Customize mesh size and mesh;
(6) Set steps and range and compute;
(7) Plot results.
For piezocomposite transmitters, the displacement analysis was completed in the Piezoelectric
Devices (pzd) module under Structural Mechanics. The acoustic response analysis was done in
the Acoustic-Piezoelectric Interaction (acpz) module under Acoustics, Acoustic-Structure
Interaction. In the Receiver Considerations chapter, for piezocomposite receivers, the receiving
sensitivity analysis was also done in Acoustic-Piezoelectric Interaction (acpz) module. All of
them were studied in the frequency domain. For a complete virtual sonic well logging
simulation, it integrated Electrical Circuit (cir), Solid Mechanics (solid), and Acoustic-
Piezoelectric Interaction, Time-Transient (acpztd) modules. Figure 3.1 shows the common
physics coupling in piezocomposite transducer problems. The following sections describe
materials, geometry and meshing, and physics in details [15, 16, 50].
3.1.1 Materials
The most important material in the piezocomposite transducers is the piezoelectric material. It is
the so-called functional material. From the Theories chapter, we have determined that PZT is a
suitable group of piezoelectric ceramic materials for high-efficiency, broadband transducer
application. In selecting a PZT material for transmitters, a “rule of thumb” is to choose one that
has high mechanical quality factor and low loss, essentially a PZT-4 material. In selecting a PZT
material for receivers, a general rule is to choose one that has large piezoelectric charge constant
and high dielectric constant, essentially a PZT-5A material. For both, especially for high-
temperature transducers, we also need to choose one that has large electromechanical coupling
factor and high Curie temperature [51-56]. Table 3.1 shows the key properties of some PZT
29
materials from two sources [57, 58]. For transmitters, we decided on Shanghai S33, and for
receivers, we decided on APC 850.
Table 3.1 Key properties of PZT materials from two sources [57, 58].
PZT Materials Q tanδ d31 d33 K k31 k33 Tc
APC 840 500 0.6 125 290 1275 0.35 0.72 325
APC 850 80 2 175 400 1900 0.36 0.72 360
Shanghai F15-6 1200 0.5 145 360 1280 0.33 0.71 310
Shanghai S33 73 1.6 200 530 1920 0.4 0.68 360
Based on the two materials, their material properties are entered into COMSOL Multiphysics®
materials library. The material properties are as below.
Density 37750 kg/m
Mechanical isotropic damping loss factor 0.01, dielectric loss factor 0.02
Strain-Charge Form E
ij ijkl kl kij k
T
i ikl kl ik k
S s T d E
D d T E
12 2
16.4 5.74 7.22 0 0 0
5.74 16.4 7.22 0 0 0
7.22 7.22 18.8 0 0 010 m /N
0 0 0 47.5 0 0
0 0 0 0 47.5 0
0 0 0 0 0 44.3
Es
12
0 0 0 0 584 0
0 0 0 584 0 0 10 C/N
171 171 374 0 0 0
d
12
0
1730 0 0
0 1730 0 (8.85 10 F/m)
0 0 1700
T
Stress-Charge Form E
ij ijkl kl kij k
S
i ikl kl ij k
T c S e E
D e S E
1( )E Ec s
Using MATLAB to find the inverse matrix ofEs
30
12
0.1203 0.0752 0.0751 0 0 0
0.0752 0.1203 0.0751 0 0 0
0.0751 0.0751 0.1109 0 0 010 Pa
0 0 0 0.0211 0 0
0 0 0 0 0.0211 0
0 0 0 0 0 0.0226
Ec
E
ip iq qpe d c
To extend it,
31 31 11 12 33 13
33 31 13 33 33
15 15 44
( )
2
E E E
E E
E
e d c c d c
e d c d c
e d c
2
0 0 0 0 12.3435 0
0 0 0 12.3435 0 0 C/m
5.3431 5.3431 15.7924 0 0 0
e
12
0
916 0 0
0 916 0 (8.85 10 F/m)
0 0 830
S
Table 3.2 Radial poling base vectors.
Radial x y z
x1 -sin(atan2(Y,X)) cos(atan2(Y,X)) 0
x2 0 0 1
x3 cos(atan2(Y,X)) sin(atan2(Y,X)) 0
Table 3.3 Tangential poling base vectors.
Tangential x y z
x1 cos(atan2(Y,X)) sin(atan2(Y,X)) 0
x2 0 0 1
x3 -sin(atan2(Y,X)) cos(atan2(Y,X)) 0
There are two possible vibration modes used in the piezocomposite transducers. For the
transmitters with fabricated prototypes, it was d31. For the receivers with fabricated prototypes, it
was d33. We also discussed another transmitter design, which operates at d33 as well. Based on
31
the vibration modes, two poling directions, radial and tangential, are defined according to a base
vector system given in Table 3.2 and 3.3.
Properties of other materials including linear elastic materials and acoustic domain materials are
listed here. As a special case, rubber is sometimes treated as linear elastic material (transmitting),
but sometimes treated as acoustic domain material (receiving).
Table 3.4 Properties of linear elastic materials [59-63].
Materials Density
(kg/m3)
Young’s
Modulus
(GPa)
Poisson’s
Ratio
Isotropic
Damping
Loss Factor
PEEK 1300 3.5 0.4 0.02
Epoxy 1270 3 0.4 0.02
Rubber (Small Strain) 1300 0.01 0.45 0.07
Steel 7850 205 0.28 0.01
Table 3.5 Properties of acoustic domain materials [64-66].
Materials Density
(kg/m3)
Wave
Velocity
(m/s)
Acoustic
Attenuation
Coefficient (dB/m)
Hard Rubber 1300 1680 130
Silicone Rubber 1100 1060 8
Water 1000 1500 1×10-5 (near 0)
Granite 2600 5500 1
3.1.2 Geometry and Meshing
Figure 3.2 shows the geometries built in COMSOL Multiphysics® for the piezocomposite
transmitter studies. Top left is the PZT configuration for a two-tier non-slant piezocomposite
transmitter. It contains 8 pieces of PZT ceramics. Each piece is 5 mm thick, 50.8 mm tall, and it
has an inner radius of 75 mm and an outer radius of 80 mm. Its center arc length is 102 mm. Top
right is the PZT configuration for a three-tier slant piezocomposite transmitter. It contains 12
PZT ceramic pieces that have a slant angle of 15°. To have the slant angle of 15°, the PZT
ceramic pieces have a short outer arc length of 104 mm, and a long outer arc length of 118 mm.
The height, thickness, and radii of the PZT pieces are the same as the two-tier non-slant ones.
Bottom left shows the packaging for the two-tier non-slant piezocomposite transmitter. The
height of the entire transmitter is 175 mm. The PZT ceramics are wrapped with epoxy/fiber glass
composite that is 2.5 mm thick inside and outside. Then it is wrapped with anti-corrosive rubber
32
that is 2.5 mm thick inside and outside. In the arc length direction, there is about 20 mm spacing
between two PZT pieces filled with epoxy/fiber glass composite. In the height direction, there is
about 5 mm spacing between two PZT pieces filled with epoxy/fiber glass composite as well. At
the top and bottom of the transmitter, there is about 21.7 mm tall epoxy, and 12.5 mm tall rubber.
Not shown here is the packaging for the three-tier slant piezocomposite transmitter, which is
very different in nature. The PZT pieces are inlaid in a center PEEK frame, then sandwiched in
an inner PEEK tube and an outer PEEK tube that are 2.5 mm thick each. In between each tier of
PZT pieces is a 10 mm tall rubber ring to disintegrate the height direction.
Bottom right shows the piezocomposite transmitter installed on a steel drill collar to imitate the
LWD tool. The whole structure is put in a cylindrical and half-spherical water domain to study
the acoustic response of the transmitter. In real LWD application, the drill collar is surrounded
by a thin layer of drilling fluid and the waves propagate in rock formations. The water domain
has a 0.4 m radius, and a 0.1 m perfectly matched layer. The entire height of the water domain is
1.2 m.
Fig. 3.2 (Top left) PZT configuration for a two-tier non-slant piezocomposite transmitter; (top right) PZT
configuration for a three-tier slant piezocomposite transmitter; (bottom left) geometry of a packaged two-
tier non-slant transmitter; (bottom right) geometry of a piezocomposite transmitter installed on steel drill
collar and placed in water domains for acoustics simulation.
33
Different parts require different mesh sizes. For water domains, mesh elements should be no
larger than 1/6 of wavelength. Therefore, the maximum element size is 10 mm. The minimum
element sizes can be decided based on how fast one wants the model to run. Here, the minimum
element size is 5 mm. For the transmitters, the maximum element size should be no larger than
1/3 of the shortest edge and is chosen to be 1.5 mm. The minimum element size is set to be 0.275
mm. The mesh is free tetragonal.
3.1.3 Physics Coupling
Fig. 3.3 (Left) Physics coupling for displacement analysis of the piezocomposite transmitters; (right)
physics coupling for acoustic analysis of the piezocomposite transmitters.
The physics coupling in COMSOL Multiphysics® for displacement analysis of the
piezocomposite transmitters is shown in Figure 3.3 (Left). Ground and voltage boundaries are
applied on the inside and outside surface of the curved PZT ceramic pieces. The poling is in the
radial direction. Therefore, the main operating mode is d31. Fixed boundaries are on the inner
most surface of the transmitter. For the displacement analysis, the radial displacement RD, the
height displacement ZD, and the total displacement TD, are defined as: 2 2
2 2
RD = sqrt ((pzd.uAmpX) + (pzd.uAmpY) )
ZD = pzd.uAmpZ
TD = sqrt (RD + ZD ) (3.1.1)
with pzd.uAmpX , pzd.uAmpY , pzd.uAmpZ being COMSOL Multiphysics® parameters for
amplitude of three components of displacement at any point.
The physics coupling in COMSOL Multiphysics® for acoustic analysis of the piezocomposite
transmitters is shown in Figure 3.3 (right). The water domains are computed as the pressure
acoustics model. Perfectly matched layers are defined in definitions, and far-field calculations
34
are applied to the outer surfaces of perfectly matched layers. The acoustic-structure boundary is
the boundary between the water domain and the steel drill collar. The top of the steel drill collar
is defined as fixed constraints. For the transmitter, the boundary conditions including the poling,
the ground, and the voltage boundaries are the same as in the displacement analysis. The
COMSOL Multiphysics® parameter for absolute acoustic pressure is acpz.absp, which is used in
acoustic pressure spatial distribution graphing and acoustic pressure frequency response plotting.
The transmitting voltage response (TVR) definition for 1 m down is:
rms
rms rms
= sqrt (0.5*pfar(0,0,-1)*conj(pfar(0,0,-1))) [Pa]
TVR = 20*log10( /V /1[ Pa/V]) (3.1.2)
p
p
where pfar is a COMSOL Multiphysics® parameter for far field acoustic pressure. The strict
definition of TVR is used for plotting TVR frequency spectrum.
The sound pressure level (SPL) used in plotting directivity is defined as:
10SPL 20log ( /1 ) (3.1.3)p Pa
3.2 Displacement Analysis
2500 5000 7500 10000 12500 15000 17500 20000
-1.0x10-5
0.0
1.0x10-5
2.0x10-5
3.0x10-5
4.0x10-5
5.0x10-5
6.0x10-5
7.0x10-5
Dis
pla
cem
en
t (m
)
Frequency (Hz)
Total Displacement
Radial Displacement
Z Displacement
Figure 3.4 Displacement frequency response of the two-tier non-slant piezocomposite transmitter.
The displacement frequency response of a representative domain point on the two-tier non-slant
piezocomposite transmitter is plotted in Figure 3.4. Using transverse mode series resonance
frequency equation, we can estimate several basic resonance peaks.
35
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the arc length of the half ring, it’s about 243 mm. The corresponding resonance frequency is:
12
1 15.7 kHz
2 0.243 7750 16.4 10sf
For the height length of the transmitter, it’s about 175 mm. The corresponding resonance
frequency is:
12
1 18 kHz
2 0.175 7750 16.4 10sf
For the arc length of the ceramic pieces, it’s about 102 mm. The corresponding resonance
frequency is:
12
1 113.7 kHz
2 0.102 7750 16.4 10sf
Also to take note is that the above calculations used an equation and material data for a pure
ceramic piece. With composite transmitters, there is compositing effect as described and proven
experimentally in the Theories chapter. So taking that into consideration while using the above
numbers as reference, the analysis of each resonance peak is as below:
(1) 2.4 kHz, ring resonance frequency; 4.8 kHz dipole ring mode; 9.6 kHz quadrupole ring mode
(not visible due to another 10 kHz mode);
(2) 4.9 kHz, 5.1 kHz, 5.3 kHz, broadened transverse resonance frequencies with the main one
caused by the arc length of the half ring;
(3) 8 kHz, 10 kHz, both are transverse resonance frequencies in the height direction; 8 kHz
caused by the height of the entire cylinder, and 10 kHz a major acoustic impedance mismatch
length (bottom of the cylinder to the top of the upper PZT piece, or top of the cylinder to the
bottom of the lower PZT piece); they feature strong Z displacement and weak radial
displacement;
(4) 11.5 kHz, transverse resonance frequency caused by the PZT piece arch length (lower than
13 kHz because of the composite); it features strong radial displacement and weak Z
displacement;
(5) 15 kHz, third harmonic by the 4.9 kHz, 5.1 kHz, and 5.3 kHz; because three resonances come
in together, it is a very strong peak. They feature strong radial displacement and weak Z
displacement. The three Z displacement components are still visible.
The displacement frequency response of a representative domain point on the three-tier slant
piezocomposite transmitter is plotted in Figure 3.5. There are definitely some differences caused
by different designs such as disintegrated height direction and less acoustically-mismatched
layers. However, the basic modes can still be calculated in a similar fashion.
36
4000 6000 8000 10000 12000 14000 16000 18000
0.0
2.0x10-5
4.0x10-5
6.0x10-5
8.0x10-5
1.0x10-4
1.2x10-4
1.4x10-4
1.6x10-4
Total Displacement
Radial Displacement
Z Displacement
Dis
pla
cem
ent (m
)
Frequency (Hz)
Fig. 3.5 Displacement frequency response of the three-tier slant piezocomposite transmitter.
In comparing the displacement frequency responses of two piezocomposite transducer designs
shown in Figure 3.4 and 3.5, we can easily come up with the following distinctions:
(1) Both have resonance peaks at around 5 kHz and 13-15 kHz. These are the first and third
harmonics of the hoop mode and they feature strong radial displacement but weak height
displacement. This is the desired vibration mode. Three-tier slant design has a broader peak
around 15 kHz caused by the slanting ceramic pieces.
(2) For two-tier non-slant design, there is another resonance peak around 8-10 kHz, which
features strong height displacement but weak radial displacement. This one is the height
vibration mode, and is not desired. Three-tier slant design eliminates this peak by having rubber
rings in between each two tiers of PZT pieces as absorber and buffer.
(3) Three-tier slant design has cleaner responses than two-tier non-slant design. This is due to the
fact that three-tier slant design employs fewer layers of packaging materials. Each interface will
cause reflection and creates minor resonance peaks.
(4) The amplitude of three-tier slant design displacement is much larger since it uses 12 pieces of
PZT instead of 8 pieces as in two-tier non-slant design. The transmitting power is increased
significantly.
37
3.3 Targeted Design Studies
To obtain broadband transmitters that feature strong 13-15 kHz circumferential mode but
reduced 8-10 kHz height mode, it seems it is particularly useful to focus on two arguments based
on the displacement analysis. The first one is to use slant angle to control the peak broadening.
The second one is to use the inhomogeneous rubber rings to disintegrate the height direction so
that a reduced height mode is achieved. To study these two hypotheses, we did targeted design
studies to focus on one variable at a time.
3.3.1 Slant Angle Broadening Effect
Fig. 3.6 Geometry in COMSOL Multiphysics® to study slant angle broadening effect. A rectangular shape
is easier to study than a cylindrical one. The composition and dimensions of the entire union do not
change. The only thing that changes is the slant angle.
As shown in Figure 3.7, the slant angle changes from 0 degree (which represents no slant) to 20
degrees. As the slant angle increases, the resonance peak caused by the longer end of the PZT
ceramic element moves left gradually, broadening the overall resonance response. 15 degrees
was chosen for a slant-cut transmitter prototype because when it goes up to 20 degrees, the peak
caused by the shorter end of the PZT ceramic element starts to get reduced as well.
To estimate the theoretical central resonance frequency, we used the same equation:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the length of the ceramic piece in the middle, it’s about 102 mm. The corresponding
resonance frequency is:
12
1 113.7 kHz
2 0.102 7750 16.4 10sf
With a slant cut, the shorter edge and longer edge move the resonance peak up and down
respectively, resulting in broadened response.
38
11000 12000 13000 14000 15000 16000
0.0
5.0x10-9
1.0x10-8
1.5x10-8
2.0x10-8
2.5x10-8
3.0x10-8
3.5x10-8
No Slant
5 Degrees
10 Degrees
15 Degrees
20 Degrees
Dis
pla
cem
ent (m
)
Frequency (Hz)
Fig. 3.7 Displacement frequency response showing slant angle broadening effect.
3.3.2 Height Mode Reducing Effect
Fig. 3.8 Geometry in COMSOL Multiphysics® to study height mode reducing effect. A two-tier slant
rectangular shape is used to simplify the cylindrical complications. The only thing that changes is the
material filling the gap between the two tiers. One option is to have nothing at all (completely
disintegrated). Another option is to have PEEK (completely integrated). And the last option is to have
rubber as absorber and buffer.
39
8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000
0.0
1.0x10-8
2.0x10-8
3.0x10-8
4.0x10-8
5.0x10-8
PEEK Connection
No Connection
Rubber Connection
Dis
pla
cem
ent (m
)
Frequency (Hz)
Fig. 3.9 Displacement frequency response showing height mode reducing effect.
As shown in Figure 3.9, nothing-filled gap and rubber-filled gap designs both has the height
mode much reduced compared to PEEK-filled gap design. The difference is since rubber is a
very absorbing material, the rubber-filled gap design also damped the circumferential resonance
peak. Based on this simulation, a gap disintegrating the upper and lower parts of the transducer
has the reasonable reducing effect for the height mode yet not affecting the circumferential mode.
Therefore, in fabricating prototypes, the gap design is used. The theoretical estimation of the two
major peaks, height mode peak and ceramic piece peak, is the same as before and it matches the
FEM simulations.
3.4 Design Strategies
Based on the displacement analysis of the two-tier non-slant design and the three-tier slant
design, as well as targeted studies, the following design strategies should be instituted:
(1) Slant cut PZT ceramic pieces can broaden the response around 13-15 kHz;
(2) More pieces of PZT in total can increase the transmitting power; so to achieve the same
effect, the driving voltage can be lowered. This will make it easier for the driving circuit design;
(3) The change of slant directions between the PZT ceramic tiers is to ensure a uniform
directivity;
(4) The rubber ring or a gap in between two neighboring tiers decouples the resonance in the
height direction, removing the undesired resonance around 8-10 kHz;
40
(5) With improved packaging technique, we only use one capsuling material (high-temperature,
anti-corrosive PEEK polymer), which eliminates the multi-layer structures (epoxy/glass fiber and
rubber), thus reducing acoustically mismatched interfaces. This should make the response
cleaner.
The two more important design strategies are slanting the PZT elements and disintegrating the
tiers of the PZT elements. Slanting the PZT elements could broaden the displacement frequency
response directly. Disintegrating the tiers of the PZT elements can reduce the height mode
vibration, making the circumferential vibration the dominant mode. Otherwise, the height mode
signal would propagate along the drill collar and also be received and become noise.
3.5 Acoustic Analysis
3.5.1 Acoustic Pressure Spatial Distribution
Fig. 3.10 Acoustic pressure spatial distribution of two-tier non-slant design.
41
Fig. 3.11 Acoustic pressure spatial distribution of three-tier slant design.
One of the most important design parameters is the acoustic pressure. In the design requirements,
the acoustic pressure should be no smaller than 10,000 Pa. The acoustic pressure spatial
distribution at 13 kHz for the two designs is shown in Figure 3.10 and 3.11. In the figure, the
acoustic pressure in the yellow and red regions is over 10,000 Pa. It shows clearly that at 13 kHz,
the yellow and red regions in the three-tier slant design are expanded drastically compared with
the two-tier non-slant design.
3.5.2 Acoustic Pressure Frequency Response
Presented in Figure 3.12 and 3.13 is the frequency response of the acoustic pressure at several
representative water domain points for the two piezocomposite transmitter designs. Generally
speaking, the acoustic pressure frequency response matches the displacement frequency response.
For two-tier non-slant design, it shows that the overall acoustic pressure cannot meet the
requirement of 10,000 Pa at 1 m, with an average of around 8,000 Pa. There is another acoustic
peak region around 8 kHz, which is the undesirable height mode. Also, because there are so
many acoustically mismatched interfaces, the overall acoustic frequency response is not clean.
For three-tier slant design, from 10 kHz on, the acoustic pressure exceeds 10,000 Pa, resulting in
an average of 20,000 Pa. Also the response is much cleaner due to reduced acoustic mismatch
layers.
42
Fig. 3.12 Acoustic pressure frequency response of two-tier non-slant design.
Fig. 3.13 Acoustic pressure frequency response of three-tier slant design.
43
3.5.3 Transmitting Voltage Response
Fig. 3.14 Transmitting voltage response frequency spectrum of two-tier non-slant design.
Fig. 3.15 Transmitting voltage response frequency spectrum of three-tier slant design.
44
Figure 3.14 and 3.15 presents the TVR plot of one representative point (1 m from the
piezocomposite transmitter) featured in the acoustic pressure frequency response figure for the
two designs. For the two-tier non-slant design, the figure shows TVR fluctuations corresponding
to resonance peaks around 5 kHz, 8-10 kHz, 13-15 kHz, etc. In between 10 and 15 kHz, the
average TVR is 125 dB, which is below the requirement of 130 dB. For three-tier slant design,
there is a broadband TVR from 10 to 15 kHz. It is proven to be a broadband device. In between
10 kHz and 15 kHz, the TVR values are between 135 dB and 145 dB, much larger than the
requirement of 130 dB.
3.5.4 Directivity
Another quantitative parameter is directivity. Here we plot the sound pressure level (SPL) in the
360° azimuthal view in the middle plane of the piezocomposite transmitter to represent
directivity. Normally 10 m radius is taken to represent far field directivity. It is required to be
less than +/- 10 dB fluctuations. Frequencies we are interested in are between 10 and 15 kHz. For
both designs, as we can see in Figure 3.16 and 3.17, directivity normally has fluctuations at the
discontinuities with the PZT pieces, a.k.a., at the spacing. However, at the PZT regions, since
three-tier slant design features less reflecting interfaces, it has much better directivity than two-
tier non-slant design. The central SPL increased from 160 dB to 180 dB because of added
ceramic pieces and increased transmitting power.
Fig. 3.16 Directivity of two-tier non-slant design.
45
Fig. 3.17 Directivity of the three-tier slant design.
3.6 d33 Transmitters
3.6.1 Comparison between d31 and d33 Transmitters
Fig. 3.18 Geometry of the d31 mode piezocomposite transmitter.
The geometry of the d31 mode piezocomposite transmitter is shown in Figure 3.18. It utilizes
PEEK as packaging material. The structure is three-layer. The center PEEK frame houses PZT
46
ceramic pieces, and is sandwiched by the inner and outer PEEK tubes. The middle part is cut out
to eliminate resonance in the height direction. The left figure shows nine representative points,
the radial displacements of which are used in comparison with other models. The right figure
shows d31 mode PZT configuration. The electrical boundary is on the inside surface of the PZT
ceramic piece, and the ground boundary is on the outside surface of the PZT ceramic piece. The
green arrow shows the poling direction, which is radial and outward. The applied voltage is 1000
Volts.
Fig. 3.19 Geometry of the d33 mode piezocomposite transmitter.
The geometry of the d33 mode piezocomposite transmitter is shown in Figure 3.19. The only
change in the transmitters with two different modes is in the PZT configurations. The PEEK
frame, and inner and outer PEEK tubes remain the same. For d31 mode, the PZT in each quarter
is one cylindrical piece. For d33 mode in this model, there are four pieces of PZT (one cylindrical
piece divided vertically). The adjacent ones have opposite poling directions, as shown with green
arrows. The electrical boundaries are shown in red, and the ground boundaries are shown in blue.
And they are on the sides, not on the inside and outside surfaces as in d31 mode.
To compare these two modes, at first, we set the electric field intensity as the same. The applied
voltage for d31 mode is 1000 Volts, and the thickness of the PZT ceramic piece is 5 mm, so the
electric field intensity is 1000 V/5 mm. For d33 mode, the distance between the electrical and
ground boundaries is much larger (approximately 27 mm). To have the same electric field
intensity, the applied voltage is set to 5400 Volts.
For the PZT-5A material used in the model, d31 coefficient is -171 pC/N, and d33 coefficient is
374 pC/N. d33 is about 2.2 times larger than d31. S3 = d33 E, S1 = d31 E. Since the electric field
intensity is set the same, it is expected that the strain and the displacement of the d33 mode is 2.2
times larger than the d31 mode. The comparison of radial displacement between d31 mode and d33
mode is shown in Figure 3.20.
47
3000 4500 6000 7500 9000 10500 12000 13500 15000 16500 18000
0.0
1.0x10-5
2.0x10-5
3.0x10-5
4.0x10-5
5.0x10-5
6.0x10-5
7.0x10-5
8.0x10-5
Radia
l D
ispla
cem
ent (m
)
Frequency (Hz)
d31
mode
d33
mode 4 pieces in each quarter
Same 1000 V/5 mm electric field intensity
Fig. 3.20 The comparison of radial displacement between d31 mode and d33 mode.
The black lines represent d31 mode. The blue lines represent d33 mode. The multiple lines show
the radial displacement of the nine representative points. The resonance peak around 4-6 kHz is
caused by the half-hoop resonance of the entire structure. The resonance peak around 10-15 kHz
is caused by the main PZT circumferential mode resonance in each quarter. The height resonance
peak around 8 kHz is suppressed, as expected from the structural design. The applied voltage for
d31 mode is 1000 Volts, and for d33 mode is 5400 Volts. Therefore, they both operate at the same
electric field intensity. As predicted by theoretical analysis, d33 coefficient is 2.2 times larger
than d31 coefficient, and it is shown that the radial displacement for d33 mode is approximately 2-
2.5 times larger than the radial displacement for d31 mode. The estimation of the theoretical
resonance peaks is as follows.
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the length of the ceramic piece in the middle, it’s about 102 mm. The corresponding
resonance frequency is:
12
1 113.7 kHz
2 0.102 7750 16.4 10sf
48
For the length of the half hoop, it’s about 259 mm. The corresponding resonance frequency is:
12
1 15.4 kHz
2 0.259 7750 16.4 10sf
That corresponds to the simulation results. As for the height mode, it is eliminated through
height mode disintegration.
3.6.2 d33 Transmitter Design Considerations
It is simply not practical to supply 5400 Volts to drive the transmitter. One can increase the
number of PZT pieces in the d33 mode design in each quarter, so that the distance between the
electrical and ground boundaries is decreased. And to maintain the same electrical field intensity,
a lower applied voltage can be achieved. But this will increase the complexity of transmitter
fabrication. After compromising between lowering voltage supply and increasing fabrication
complexity, a d33 mode model with eight PZT pieces in each quarter is built. The geometry is
shown in Figure 3.21.
Fig. 3.21 The geometry of a d33 mode model featuring eight PZT ceramic pieces in each quarter.
Compared to the d33 model featuring four PZT pieces in each quarter, the overall structure remains the
same, except that the number of PZT pieces is increased from four to eight. The red and blue boundaries
show the electrical and ground boundaries, respectively. The green arrows show the poling directions.
The distance between the electrical and ground boundaries is reduced by half. And the d33
coefficient is 2.2 times larger than d31 coefficient. So jointly, they bring the applied voltage down
by 4.4 times, if the radial displacement between the two modes is to be the same. The original d33
mode voltage is 5400 Volts, so the reduced one is 1200 Volts.
We compared the d31 mode and the eight-piece d33 mode under the same applied voltage of 1000
Volts. It is expected that the eight-piece d33 mode radial displacement is only slightly lower than
the d31 mode. It is verified in Figure 3.22.
For designing considerations, the eight-piece d33 mode is the break-even design. It possesses the
same level of transmitting power with the d31 mode. However, fabricating a one-piece cylindrical
PZT ceramic faces more technical challenges and higher cost. In comparison, dividing one-piece
cylindrical PZT vertically into eight pieces basically results in eight rectangular PZT pieces.
Fabricating rectangular PZT pieces are much easier and cheaper.
49
Also, if a better performance is desired of the transmitter, one can simply increase the number of
PZT pieces in the d33 mode from eight to any number beyond, like nine, ten, etc. Since the eight-
piece d33 mode has the same displacement with d31 mode, any more pieces will result in a higher
displacement and transmitting power, and better performance.
3000 4500 6000 7500 9000 10500 12000 13500 15000 16500 18000
0.0
5.0x10-6
1.0x10-5
1.5x10-5
2.0x10-5
2.5x10-5
3.0x10-5
3.5x10-5
Radia
l D
ispla
cem
ent (m
)
Frequency (Hz)
d31
mode
d33
mode 8 pieces in each quarter
Same 1000 Volts applied
Fig. 3.22 Comparison of radial displacement between d31 mode and eight-piece d33 mode. The same 1000
Volts is applied to both designs and they show equal performance. Compared to four-piece d33 mode, for
eight-piece d33 mode, the distance between electrical and ground boundaries decreases by half. Also, d33
coefficient is inherently 2.2 times larger than d31 coefficient. Combing these two factors, one can bring
down the voltage supply from 5400 Volts to 1200 Volts. Supplying 1000 Volts, it shows similar
displacement curves with d31 mode. Therefore, eight-piece d33 mode is the break-even design. Dividing
one cylindrical PZT piece into eight pieces vertically basically results in rectangular PZT pieces. It is
much easier and cheaper to fabricate rectangular PZT pieces than cylindrical PZT pieces. So there is more
merit to fabricate eight-piece d33 mode than d31 mode. Also, to have a better performance, one can simply
increase the number to nine, ten, etc. However, the more pieces there are, the more complex it is to
fabricate the transmitter. So there is a tradeoff between better performance and fabrication complexity.
The ideal transmitter should have PZT in one quarter between eight and twelves pieces.
Figure 3.23, 3.24, 3.25 shows the displacement color graph for d31 mode, four-piece d33 mode,
and eight-piece d33 mode, respectively, at 12.5 kHz. They all show similar resonance patterns,
which assures the exchangeability between different designs for same applications. The only
difference is in the amplitude of displacement.
50
Fig. 3.23 Displacement color graph for d31 mode.
Fig. 3.24 Displacement color graph for four-piece d33 mode.
51
Fig. 3.25 Displacement color graph for eight-piece d33 mode.
52
CHAPTER 4 FABRICATION
Piezocomposite transducers work in extreme environment featuring high temperature (up to
200 °C), high pressure (up to 140 MPa), strong vibration, corrosive chemicals, etc. A common
practice to protect them is to shield them in materials that are mechanically and chemically
resistant to these harsh conditions. Existing products use multiple materials for different
shielding purposes, rubber for anticorrosion, epoxy for high temperature, and fiber glass for
mechanical toughness. The disadvantage of using multiple layers of different materials is that it
adds to the number of acoustically mismatched layers, which results in messy acoustic responses
due to reflections. Polyether ether ketone (PEEK) is a material that combines all the shielding
properties. It is chemically resistant, mechanically tough, as well as stable at high temperature.
So being able to use this one material will make the transducer fabrication process easier, on top
of reducing the number of acoustic mismatch layers. The challenge is that it is difficult to bond
large pieces of PEEK with high bonding strength. This dissertation work offers a series of
fabrication techniques that resolved this issue. In the packaging process, epoxy bonding is
preferred. Epoxy selection is of ultimate importance in ensuring high bonding strength. Besides,
matching coefficients of thermal expansion among different materials is desired. Consideration
on soldering choice is also key to avoiding electrical break down. With all these respects, a failed
prototype, however, revealed a problem on the microscopic level. Epoxy bonding in ambient
resulted in trapped air unseen by naked eye. It’s harmless for ambient pressure applications but
detrimental for high pressure applications. Bonding in vacuum left no trapped air. Therefore, an
in-vacuum epoxy bonding setup and method was developed. Also, a uniform thickness spacer
such as optic fiber and glass fiber cloth proved to improve the bonding strength as well. The
following sections will present these fabrication topics in details.
4.1 Fabrication Topics
4.1.1 Epoxy Selection
Epoxy bonding is the most crucial step that guarantees the piezocomposite transducers to work in
high-temperature, high-pressure, and strong-vibration working environment. Some general
guidelines to ensure a strong bonding are as follows.
(1) Surface should be clean of all grease, oil, dirt, etc. Acetone and then deionized water are used
as the solvent here.
(2) Entrapped air in the epoxy mixture should be removed by either letting it stand for several
minutes or vacuum degassing.
(3) The bond lines should be between 0.005ꞌꞌ - 0.010ꞌꞌ (0.127 mm to 0.254 mm). To achieve such
a thin layer, applying forces by clamping devices is useful. In addition, a smooth surface finish
such as smaller than 1 micron roughness can make sure the bonding layer is of uniform thickness.
Besides these guidelines, different epoxies work differently on different materials and under
different conditions. Therefore, empirical testing was imperative to select the most suitable
epoxy. Apparent shear strength of single-lap-joint adhesively bonded specimens (PEEK-PEEK,
and PEEK-Ceramic-PEEK) by tension loading was measured. The standard specimens were
made based on the American Society of Testing and Materials (ASTM) D1002-10
recommendation [67] shown in Figure 4.1 (top). The Instron lap-shear test equipment was
available in the Mechanical Testing Lab. The failed specimens are shown in Figure 4.1 (bottom
53
left). And the tensile stress curves are shown in Figure 4.1 (bottom right). Three specimens for
each epoxy under each condition was prepared to rule out randomness. The failure point on the
stress curve gave a quantitative indication on the strength of epoxy bonding for the
corresponding epoxy type and bonding conditions. The properties of a list of epoxies are shown
in Table 4.1. These epoxies were tried under different annealing conditions especially, and were
assigned pass or fail in Table 4.2.
Fig. 4.1 Standard testing specimens, lap-shear test equipment, failed specimens, and shear stress curves.
Table 4.1 A list of epoxy properties compared with PEEK properties [68-70].
Epoxy Duralco 4538
(Flexible)
Duralco 4538
(Rigid)
Bond-It
7050 EP 5340
Aremco-
Bond 2150 PEEK
Maximum
Temperature (°F) 450 450 450 400 400 600
Thermal Expansion
(×10-5/°C) N/A 4.0 4.8 5.4 1.8 5.9
Tensile Strength
(psi) 6000 8000 5000 9000 2350 14000
Chemical Resistance Outstanding Outstanding Excellent Good Good Outstanding
54
Table 4.2 Pass or fail results of epoxies under different annealing conditions based on lap-shear test.
Adhesive Manufacturer
PEEK-
PEEK
150°C
PEEK-
PEEK
200°C
1st
PEEK-
PEEK
200°C
2nd
PEEK-
PEEK
200°C
3rd
PEEK-
PZT
150°C
PEEK-
PZT
200°C
1st
PEEK-
PZT
200°C
2nd
PEEK-
PZT
200°C
3rd
Duralco
4538
(Flexible)
Duralco √ √ √ √ √ √ √ √
Duralco
4538
(Rigid)
Duralco √ √ √ √ √ √ ×
Bond-It
7050 Duralco √ √ √ √ × ×
EP 5340 Eagle
Polymers √ √ √ √ ×
Aremco-
Bond
2150
Aremco √ √ ×
×
4.1.2 Coefficient of Thermal Expansion Mismatch
Based on the lap-shear test, Duralco 4538 Flexible epoxy was chosen for piezocomposite
transducer fabrication. The flexibility of this epoxy gives rises to an elongation factor instead of
a coefficient of thermal expansion (CTE), which compensates the mismatch between PEEK and
PZT CTE. PEEK CTE is 59.4 µm/m-ºC and PZT CTE is 7 µm/m-ºC. The difference is
approximately 50 µm/m-ºC. 6 2
2
thermal strain = CTE difference temperature increase = 50 10 m/m- C 200 C = 1 10
displacement = strain length = 10 100 mm = 1 mm
We can compare this displacement with that of PZT from electrical excitation. 12 4
4
strain = piezoelectric charge constant electric field = 200 10 m/V 2000 V / 5 mm = 0.8 10
displacement = strain length = 0.8 10 100 mm = 8 m
This displacement can reach 20 times larger (0.2 mm) at resonance. Still, compared to the
resonance displacement, the thermal strain displacement is 5 times larger, much more significant
and needs to be treated carefully. Otherwise, it would disassemble the piezocomposite
transducers easily. Epoxy Duralco 4538 has an elongation factor of 8%, which will compensate
the thermal strain displacement in the shear direction. Through thermal cycle testing, it was also
verified that the transducers fabricated with this epoxy could withstand thermal strain. adhesive layer elongation = 0.08 100 mm = 8 mm
55
4.1.3 Soldering Consideration
To begin with, high temperature conducting epoxy silver paste (Duralco 120, 260 °C [71]) was
used to attach wires to the PZT pieces. Unfortunately after some initial testing, there appeared to
be an electrical break down problem to one of the first transmitter prototypes. Figure 4.2 (Top)
shows the pulse excitation responses for the two transmitter prototypes in questions. The left is
the correct response, but the right shows the response after the electrical break down. Figure 4.2
(bottom) shows the capacitance and resistance analysis of the two prototypes. The left is again
the correct curve, but the right shows the incorrect curve.
Fig. 4.2 (Top) Correct (left) and incorrect (right) pulse excitation response and (bottom) correct (left) and
incorrect (right) capacitance and resistance analysis for two transmitter prototypes.
56
Therefore, high temperature solder was used for the remaining fabrication works. Besides, as
shown in Figure 4.3 (Left), one groove was made on each side of the PZT piece. Silver electrode
was fired on again after the groove was made. This groove enables easier soldering of wire onto
the ceramic piece, as shown in Figure 4.3 (right). Also, it leaves a flatter surface, which
facilitates the bonding process.
Fig. 4.3 (Left) grooves were made on each side of the PZT piece and (right) it facilitated wire soldering.
4.1.4 Trapped Air Issue
With the above considerations in mind, a flat slant transmitter prototype was fabricated (Figure
4.4 left and middle). The PEEK frame cutting was done with a water jet cutter available in the
Learning Factory. The pattern was designed with Solidworks and a computer controlled the high-
speed water jet stream to move and cut the flat PEEK piece. The slant cut PZT piece had a long-
side length of 95 mm, and a short-side length of 80 mm, making the slant angle 15°. The height
was still 6.35 mm, and the width remained 57 mm. The entire frame had dimensions 210
mm×150 mm×6.35 mm. The grooves were 1 mm wide, 1 mm deep. The wire outlet mouth was 8
mm tall, 12 mm wide. It was bonded with Duralco 4538. After high pressure test, the prototype
was totally disassembled though, as shown in Figure 4.4 (right).
Fig. 4.4 Flat slant transmitter prototype (left and middle) totally disassembled after high pressure test
(right).
After careful scrutiny, it was found out that the problem lay on the microscopic level. Since the
bonding took place in ambient, it appeared that trapped air in the bonding process contributed
majorly to the deteriorating bonding strength under high hydrostatic pressure. To prove this
hypothesis, a glass slide on PZT bonding test was carried out, one in ambient (Figure 4.5 Left),
one in ambient but with a sliding movement (Figure 4.5 Middle), and one in 1% vacuum (Figure
4.5 Right). It was shown that although all bonding appeared uniform by naked eye, under
microscopy (magnification 1,000×), the one bonded in vacuum had no trapped air at all.
57
However, the two bonded in ambient has numerous minuscule trapped air bubbles, with sliding
movement one slight better. A camera was attached to the optic microscopy to capture these
images. These trapped air bubbles are harmless for applications in ambient, but under high
pressure, they can deteriorate the epoxy bonding strength.
Fig. 4.5 (Left) Glass on PZT bonded in ambient. (Middle) Glass with PZT bonded with sliding method.
(Right) Glass on PZT bonded in vacuum. All magnification is 1,000×.
4.1.5 In-Vacuum Bonding
To remove trapped air, a system that allowed in-vacuum epoxy bonding was developed. Epoxy
was first applied to the parts to be bonded. These parts were placed on a motor-controlled holder
in a vacuum box. The motor could be activated by a computer outside the vacuum box. Once the
whole system was pumped to 1% vacuum, the motor was activated to push the parts into contact,
achieving bonding. Since the entire process took place in vacuum, it left no trapped air.
Figure 4.6 (left) shows the in-vacuum bonding system. The motor controller was wired outside
the vacuum box to a voltage generator. The connection parts were carefully sealed with vacuum
grease to ensure vacuum. The actuation could be controlled by a computer program. For liner
movement, it consisted of retraction and extension. For rotary movement, it consisted of
clockwise and counterclockwise rotation. To bond, the surfaces of two parts were covered with
epoxy separately and put inside the vacuum box. Then the pump was switched on to pump the
vacuum box to 1% vacuum. The two parts stayed in vacuum for 30 minutes so that the epoxy
could be degassed. After that, the computer controlled the motor to move the two parts into
contact. For flat parts, as shown in Figure 4.6 (top right), it used a linear motor
retraction/extension apparatus. Sandwiching PEEK piece was lowered to the PZT piece at the
bottom. For curved parts, as shown in Figure 4.6 (bottom right), it used a rotary motor and the
rotation was transferred into a linear motion of the half-tubular holders. Since the entire
operation took place in vacuum, there was no air trapped in the bonding process. This is
consistent with Bascom and Cottington’s findings [72].
58
Fig. 4.6 In-vacuum bonding system and actuation holders for bonding parts of different geometries.
4.1.6 Uniform Thickness Spacer
Literature research revealed that the optimum epoxy thickness for bonding is between 50 µm and
150 µm [73]. When pushed too hard, epoxy flew out of the bonding interface between two parts
and the layer became too thin and non-uniform. To counter this problem, spacers of uniform
thickness such as optic fiber and glass fiber fabric were utilized. They ensured that the epoxy
bonding layer had a uniform thickness as decided by the spacer thickness. In Figure 4.7 (left), it
was 125 µm optic fiber on a one-piece transducer prototype; in Figure 4.7 (right), it was 150 µm
glass fiber woven fabric on a real-size transducer prototype. By adding the glass fibers, the
epoxy-glass system constituted a composite material, and the gain in the mechanical strength was
further due to fiber reinforcement and load distribution across the joint [74, 75]. Transducer
prototypes fabricated with combination of in-vacuum bonding and uniform thickness spacer
techniques had ultrahigh bonding strength and sustained the high hydrostatic pressure tests.
Fig. 4.7 Optic fiber (left) and glass fiber fabric (right) as uniform thickness spacer.
59
4.2 Successful Prototype
Several earlier prototypes led to the final successful prototype. The earlier prototypes were
presented in Appendix A. In this section, fabrication details of the successful prototype were
explained step-by-step. They were non-slant high-resonance transmitter prototype and they were
reproduced for three sets. In this process, suppliers were finalized for possible scaling up of
transmitter production. The transmitter had a major working resonance peak between 10-15 kHz.
The peak around 8 kHz was basically eliminated. The fabrication process comprised of PZT
ceramic grouping, PEEK tube fitting, and in-vacuum bonding. The testing consisted of
impedance analysis at different stages. The in-vacuum bonding with spacer fabrication technique
was used. This section focuses on finalizing PZT and PEEK designs so that suppliers can
manufacture in large quantities.
4.2.1 PZT Ceramic Grouping
Fig. 4.8 PZT pieces with no cutting needed.
To facilitate mass production, the PZT ceramic pieces were designed to fit PEEK frames directly
out of factory so no additional cutting was required. The PZT ceramic pieces had inner radius 75
mm, outer radius 80 mm, height 50.8 mm, and angle at center 80 °. All the PZT pieces are shown
in Figure 4.8. There were 24 pieces of PZT ceramic pieces for 3 sets of transmitters, and their
properties are in great consistency, as shown in the capacitance data in Figure 4.9.
The peak estimation is based on this equation:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the arc length of the ceramic piece, it’s about 108 mm. The corresponding resonance
frequency is:
60
12
1 113 kHz
2 0.108 7750 16.4 10sf
The calculation corresponds to the first peak, which is the arc length resonance and the main
working resonance peak.
For the height of the ceramic piece, it’s about 50.8 mm. The corresponding resonance frequency
is:
12
1 127.5 kHz
2 0.0508 7750 16.4 10sf
It corresponds to the second peak, which is the height direction resonance. There is the third peak
in the measurement, which is the third harmonic resonance of the arc length.
0 5000 10000 15000 20000 25000 30000 35000
-1.3x10-7
-1.0x10-7
-7.5x10-8
-5.0x10-8
-2.5x10-8
0.0
2.5x10-8
5.0x10-8
7.5x10-8
1.0x10-7
1.3x10-7
1.5x10-7
Ca
pa
cita
nce
(F
)
Frequency (Hz)
Fig. 4.9 Capacitance data of 24 PZT pieces.
Based on the capacitance data at 40 Hz, the 24 PZT pieces were grouped into 6 groups so that the
sums of the capacitance in each group were equivalent. Apparently, there could be different ways
of grouping them. Table 4.3 is just one example.
Table 4.3 PZT pieces divided into 6 groups.
Groups Numbered PZT Pieces Corresponding Capacitances and Sum (All in nF)
Group 1 2, 3, 12, 18 19.4+19.5+19.7+19.5=78.1
Group 2 10, 11, 13, 21 19.6+19.6+19.6+19.2=78
Group 3 1, 6, 14, 16 19.3+19.5+19.6+19.6=78
Group 4 4, 5, 9, 24 19.2+19.6+19.6+19.6=78
Group 5 7, 15, 17, 19 19.7+19.4+19.5+19.4=78
Group 6 8, 20, 22, 23 19.6+19.4+19.5+19.5=78
61
All PZT ceramic pieces had grooves made on them, each groove being 18.5 mm away from the
edge. After the grooves were made, silver electrode was again fired in the grooves. The grooves
facilitated wire soldering so that the wire was even with the ceramic surface. This will help
seamless bonding in the later fabrication stages.
In previous prototypes, if any PZT pieces needed to be cut to change dimensions or to create
slants, a diamond saw was used. It is available in the Materials Preparation Lab.
4.2.2 PEEK Tube Fitting
To facilitate mass production, Solidworks drawings of PEEK tubes were provided to a factory
with Computer Numerical Control (CNC) machining ability and exact replicas of PEEK frames
were manufactured with efficiency and ease. The drawing of the frame is shown in Figure 4.10.
Fig. 4.10 Solidworks drawing of center PEEK tubes.
The Solidworks designs were translated into machine language and used to control a 5-axis CNC
machine for PEEK tube cutting. After the tubes were machined, they were cut into half tubes for
further use. Each individual transmitter consisted of three PEEK tubes, inner, center, and outer.
The inner PEEK tube had an inner diameter of 145 mm, an outer diameter of 150 mm, and a
height of 152 mm. The center PEEK tube had an inner diameter of 150 mm, an outer diameter of
160 mm, and a height of 152 mm. The outer PEEK tube had an inner diameter of 160 mm, an
outer diameter of 165 mm, and a height of 152 mm. The wire outlet mouth was also
manufactured, with a height of 12 mm, and a width of 8 mm. For the center tube, 4 windows
were machined out to house the non-slant PZT ceramic pieces. These windows had the same
dimensions with those of the PZT ceramic pieces, therefore the fitting was made easy. The
machined PEEK tubes and the fitting are shown in Figure 4.11 and Figure 4.12, respectively.
62
Fig. 4.11 Three sets of PEEK tubes, and one set in details. The wire mouth was already machined as part
of the complete tube. This will prevent the mud from entering from the connection part.
Fig. 4.12 PEEK center frame showing the wire grooves and PZT fitted in the PEEK frame.
In previous prototypes, a lot of machining was also done in the Learning Factory on the milling
and lathing machines. Some was also completed by machinists on CNC machines in the Machine
Shop.
4.2.3 Impedance Analysis
The finished prototypes are shown in Figure 4.13. They were analyzed with the Agilient 4294A
impedance analyzer in the Electrical Characterization Lab. The impedance analysis gave
indication on the resonance performance of the transducers, and also provided information for
the electrical matching in the circuit design.
63
Fig. 4.13 Finished successful prototypes and impedance analyzer in the Electrical Characterization Lab.
Figure 4.14 through Figure 4.18 shows the capacitance, admittance, conductance, impedance,
and resistance data for the 6 individual transmitters before and after annealing.
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
2.0x10-8
3.0x10-8
4.0x10-8
5.0x10-8
6.0x10-8
7.0x10-8
8.0x10-8
9.0x10-8
1.0x10-7
1.1x10-7
1.2x10-7
1.3x10-7
Ca
pa
cita
nce
(F
ara
ds)
Frequency (Hz)
Before Annealing
After Annealing
Fig. 4.14 Capacitance data for the 6 individual transmitters before and after annealing.
64
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0.0
1.0x10-3
2.0x10-3
3.0x10-3
4.0x10-3
5.0x10-3
6.0x10-3
7.0x10-3
8.0x10-3
9.0x10-3
1.0x10-2
1.1x10-2
1.2x10-2
1.3x10-2
Ad
mitta
nce
(S
iem
en
s)
Frequency (Hz)
Before Annealing
After Annealing
Fig. 4.15 Admittance data for the 6 individual transmitters before and after annealing.
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0.0
1.0x10-3
2.0x10-3
3.0x10-3
4.0x10-3
5.0x10-3
6.0x10-3
7.0x10-3
8.0x10-3
Co
nd
ucta
nce
(S
iem
en
s)
Frequency (Hz)
Before Annealing
After Annealing
Fig. 4.16 Conductance data for the 6 individual transmitters before and after annealing.
65
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
Imp
ed
an
ce
(O
hm
s)
Frequency (Hz)
Before Annealing
After Annealing
Fig. 4.17 Impedance data for the 6 individual transmitters before and after annealing.
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0
20
40
60
80
100
120
140
160
180
200
Re
sis
tan
ce
(O
hm
s)
Frequency (Hz)
Before Annealing
After Annealing
Fig. 4.18 Resistance data for the 6 individual transmitters before and after annealing.
66
The black lines show the data for the 6 individual transmitters before annealing, and the red lines
show the data for the 6 individual transmitters after annealing. First, these figures showed the
consistency of the data before and after thermal cycle test, which means that the transmitters
have stable performance at high temperatures. Second, the overlapping of the 12 lines
demonstrated the consistency between each individual transmitter.
To look at the resonance performances, the peak around 4-5 kHz was the resonance in the arc
length of the half tube. The peak around 10-15 kHz was the major working resonance peak,
which was expected and it is the circumferential mode. There was no resonance around 8-10 kHz,
which was reduced by disintegrating height direction. These impedance analysis data are
consistent with those of computational simulations. They can also be verified by the same
theoretical calculations of the series resonance frequencies:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the arc length of the ceramic piece, it’s about 108 mm in this case. The corresponding
resonance frequency is:
12
1 113 kHz
2 0.108 7750 16.4 10sf
It matches exactly with the main working resonance frequency.
The finished transmitter has an arc length of the half hoop of about 259 mm. The corresponding
resonance frequency is:
12
1 15.4 kHz
2 0.259 7750 16.4 10sf
This corresponds to the lowest resonance mode and it is the half ring hoop mode.
The finished transmitter has height of about 152 mm. The corresponding resonance frequency is:
12
1 19.2 kHz
2 0.152 7750 16.4 10sf
If the height direction were not disintegrated, this would match to the height mode vibration. But
since the height direction was disintegrated, this mode was eliminated almost completely, which
is consistent with the design strategy.
Another important parameter in the transducer performance is bandwidth. For sure, it is ideal that
the transducer operates at resonance frequency, so that the transmitting power is the maximum.
Therefore, inherently, it requires a high quality factor Q. However, the higher the quality factor
is, the narrower the bandwidth is. Because the excitation signal is a pulse, there is inevitably a
bandwidth associated with that pulse. So on the other hand, we also want certain bandwidth for
the transducer response. It is a tradeoff that we need to consider. As discussed in Theories
chapter, the optimum combination of bandwidth and quality factor requires a high
electromechanical coupling factor, which is a material property and by choosing PZT, we are
guaranteeing an optimum transducer. Furthermore, by matching PZT with well-chosen
packaging materials, it will also bring about a good bandwidth. Since we are talking about the
quality factor for stand-alone packaged transducers without electrical matching, this quality
factor is an effective acoustic quality factor.
67
We can use admittance curves to calculate this quality factor and bandwidth. 3dB bandwidth
corresponds to the value of admittance when it is 0.707 or 1/ 2 of the maximum admittance at
resonance peak. And the quality factor is then defined as:
3
r
dB
fQ
f
where Q is the quality factor, rf is the central resonance frequency, and 3dBf is the 3dB
bandwidth.
By scrutinizing the raw admittance data for one of the transmitters around 13 kHz resonance, the
exact central resonance frequency is 13.34 kHz, corresponding to a maximum value of
admittance of 1.0528E-02. The two frequencies that approximately correspond to the 0.707 of
the maximum admittance value are 12.62 kHz and 13.52 kHz. That gives a 3dB bandwidth of 0.9
kHz. And the quality factor is 14.8.
4.2.4 Drill Collar Fitting
The transducers need to be fit on a test drill collar for testing purposes. For this set of
transmitters, a test drill collar closely matching a real-application drill collar was manufactured
by our industry collaborator. The outside diameter of the inner steel tube was 138 mm. In
between the test drill collar and the transmitters was a 1.6 mm thick rubber layer. The purpose of
the rubber layer was to ensure a tight fitting and also to function as a backing material. This
rubber layer was not meant to protect the transmitters from corrosions because the PEEK and
epoxy used in the transmitters could achieve this purpose already. By eliminating a rubber layer
inside and outside of the transmitters, it would reduce acoustically mismatched layers and result
in cleaner and stronger responses. The inside diameter of the outer steel clamping half tubes was
167 mm. The thickness of the metal clamps was 5 mm.
Wires of the transmitters were linked to black push-pin wire connectors. These connectors has
two functions. One is to prevent mud from entering the transmitters, and the other is to facilitate
easy connection to testing instruments or other drill collar electronics. One simply pushes the
matching pin connector into the black socket. Before the transmitters were installed onto the drill
collar, they were covered with vacuum grease throughout and any gaps were filled with vacuum
grease as well. Inside the transmitters was a layer of compressible rubber right on the test drill
collar. It functions as a backing materials and also ensures a tight fitting. Outside, metal clamps
were tightened onto the transmitters with a set of screws. The fitted transducers will go through
acoustic and hydrostatic testing in the Testing chapter.
68
CHAPTER 5 TESTING
5.1 Anticorrosion Test
Fig. 5.1 BORE-GEL® drilling mud solution and anticorrosion test.
The piezocomposite transducers work in corrosive drilling mud environment. It is necessary to
conduct anticorrosion test to make sure the transducers maintain the same performance level
after being exposed to the drilling mud for an extended period of time. Here, a BORE-GEL® [76]
drilling mud solution was prepared according to the following procedures:
1. Pre-treat make-up water with soda ash at a concentration of 1.2 - 2.4 kg/m3. Slowly add soda
ash through high shear mixer to facilitate uniform treatment of active fluid system (target pH
range: 8.5 - 9.5).
2. Using a high shear mixer, add BORE-GEL® slowly through hopper at a rate not to exceed 75
g/sec.
3. Mix BORE-GEL® fluid with water for 15 to 20 minutes (normal conditions: 30 - 42 kg/m3) for
effective hydration. The typical pH of the BORE-GEL® mud system is 10.2 - 11.5.
Flat low-power transmitter prototype shown in the inset of Figure 5.1 (right) was immersed in
the drilling mud solution for 18 hours. The admittance data were taken before and after the mud
immersion. The comparison shown in Figure 5.1 (right) revealed no change and assured stability
of the transducers in the corrosive environment.
The theoretical verification of the series resonance frequency for this flat low-power prototype is:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
The transmitter has a length of 105 mm in this case. The corresponding resonance frequency is:
12
1 113.3 kHz
2 0.105 7750 16.4 10sf
The experimental data is lower because of the compositing effect.
69
5.2 Thermal Cycle Test
Piezocomposite transducers work at high temperature and a thermal cycle test is crucial to check
their thermal stability. To do the thermal cycle test, transmitter prototypes were placed in
programmable furnaces (Figure 5.2 right) in the Shared Furnace Lab, a heating profile (Figure
5.2 left) was programmed, and a thermal cycle was performed, typically taking 24 hours or more.
Multiple cycles might have been used in certain cases.
Details of the heating profile is as follows:
1. Heat up the furnace from room temperature (20 °C) at the rate of 1 °C/min to 50 °C, and then
hold at that temperature for 30 minutes.
2. Continue the heating-holding process for every 30 °C until it reaches 170 °C.
3. Heat up the furnace from 170 °C at the rate of 1 °C/min to 200 °C, and then hold at that
temperature for 10 hours.
4. Cool down the furnace from 200 °C to room temperature at a rate no more than 1 °C/min.
Fig. 5.2 Thermal cycle test heating profile and programmable furnaces.
For the successful transmitter prototypes as shown in Figure 5.3 (inset), they went through such
thermal cycle tests. Figure 5.3 shows the admittance data for the 6 individual transmitters before
and after annealing. It showed great consistency before and after, verifying their thermal stability
and reliability.
In terms of resonance frequencies, there are the major working frequency caused by arc length of
ceramic pieces and the minor hoop mode frequency of the half ring. Based on the calculations of
the series resonance frequencies:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
70
For the arc length of the ceramic piece, it’s about 108 mm in this case. The corresponding
resonance frequency is:
12
1 113 kHz
2 0.108 7750 16.4 10sf
It matches exactly with the main working resonance frequency.
The finished transmitter has an arc length of the half hoop of about 259 mm. The corresponding
resonance frequency is:
12
1 15.4 kHz
2 0.259 7750 16.4 10sf
This corresponds to the lowest resonance mode and it is the half ring hoop mode.
Again, the height mode vibration is eliminated here.
Fig. 5.3 Admittance data before and after thermal cycle test for the prototype.
5.3 High Voltage Test
To make sure the transmitters can withstand high driving voltage without breaking down, a high
voltage test was carried out. The setup is shown in Figure 5.4 (top). It includes a function
generator, a high voltage amplifier and an oscilloscope.
71
Fig. 5.4 Setup for high voltage testing includes a function generator, a high voltage amplifier, and an
oscilloscope. The charging and discharging currents are shown in the bottom.
For this test, a square wave was generated with a frequency of 0.1 Hz. Three levels of voltages
were tested. The first one had a low voltage of 650 volts and a high voltage of 1050 volts,
making it a 400 volts step up that lasted for 5 seconds at trigger. The second one had a low
voltage of 500 volts and a high voltage of 1200 volts, making it a 700 volts step up that lasted for
5 seconds. The third level had a low voltage of 0 volts and a high voltage of 1500 volts, making
it a 1500 volts step up that lasted for 5 seconds. The third level is mostly approximate to the
actual driving voltage pulse.
The charging and discharging currents of the transmitters that went through high voltage testing
is shown in Figure 5.4 (bottom). These are normal charging and discharging currents and not
leaking currents. This means the transmitter can withstand 1500 volts without breaking down.
Figure 5.5 shows the comparison of capacitance curves of both transmitter half-tubes before and
after the high voltage test. It showed no noticeable change at all. Therefore, there was no
breakdown and the transmitter prototypes can endure high voltage excitation. The resonance
frequencies are exactly the same with the prototypes that went through thermal cycle test, so the
theoretical calculations are skipped here. It has two resonance peaks, one at 4 kHz for the half
ring hoop mode, and another one at 13 kHz caused by the ceramics pieces, which is the main
resonance mode. Because of the disintegrated height, height mode resonance is completely
annihilated.
72
Fig. 5.5 Capacitance curve for the transmitter half-tubes before and after high voltage test.
5.4 High Pressure Test
The high hydrostatic pressure testing facility was provided by our industry collaborator. It is an
underground facility to reduce risks of high pressure testing if done above the ground. The
pressure is hydrostatic compressive load supplied by testing oil or water. The liquids can be
heated to generate high-temperature high-pressure conditions at the same time.
For the high pressure test, the below protocol was used. High pressure checkpoint at every 40
MPa helped determine the failure pressure should that happen. For instance, unheated high
pressure testing for peak pressure of 20, 60, 100, and 140 MPa were carried out first. For each
peak pressure testing, a reasonable pressure profile was adopted. 30 minutes to one hour was
taken to pump the pressure to peak pressure. Prototypes stayed at peak pressure for 30 minutes. 2
hours were taken to reduce the pressure to normal.
The same process was repeated for high-temperature high-pressure testing. In this scenario,
different temperatures such as 100 ºC, 150 ºC, and 200 ºC were tested along with all
combinations of high pressures.
The method to check whether a transducer endured the high pressure was twofold. First, check
their physical intactness and appearance and see if there are visible cracks. Second, redo the
impedance analysis and acoustic tests to see if they still work the same. In fact, even for the final
products, each transducer will go through high pressure testing before being used in oil wells.
5.5 Vibration Test
Vibration testing facility was also provided by our industry collaborator. Transmitters were
installed on test drill collars and test drill collars were installed on the vibration testing base. The
73
vibration testing base was programmed to undergo shaking, bumping, revolving, etc. similar to
the vibrations commonly experienced in real drilling process. The method to check whether a
transducer passed the vibration test was also by looking at their physical appearances and by
redoing the impedance analysis and comparing before and after data. Like the high hydrostatic
test, each transducer will go through high pressure testing before being used in oil wells.
The prototypes we developed using PEEK as packaging material passed all these tests, including
anticorrosion test, thermal cycle test, high hydrostatic pressure test, and vibration test. The
techniques we used such as selecting the correct epoxy, in-vacuum bonding, and uniform
thickness spacer helped ensure an ultrahigh bonding strength. This breakthrough of bonding
large pieces of PEEK for the LWD transducer application greatly simplified the transducer
design and fabrication.
5.5 Acoustic Test
5.5.1 Excitation Signal
The electrical excitation setup includes a computer with software-controlled signal output; a
matching circuit with transformers, converters, etc.; and an oscilloscope. The transducers were
installed on the test drill collar, as outlined in Chapter Four. Signals with different frequencies
and different numbers of wavelengths were supplied by the computer and its software. The
matching circuit makes sure the maximum power was supplied to the transmitter. And the
oscilloscope reads the electrical transmitting signals. This electrical excitation testing can help to
check the impedance matching and give a rough idea of whether the transmitter works before it
is put to acoustic excitation testing.
The acoustic excitation setup includes a hydrophone, a water tank, the transmitter and its
matching circuit, the signal input from the computer, the oscilloscope, and an amplifier.
Hydrophone was on the left end of the water tank to receive acoustic signal. Transmitter test
collar was hang on the right end of the water tank transmitting acoustic signal. Electrical
excitation was to the immediate right of the transmitter test collar. An amplifier to the left of the
oscilloscope was used to amplify the hydrophone signal. This test checks the acoustic signal
strength and gives a relevant assessment of the transmitters for the oil drilling applications.
The transmitting and receiving acoustic signals for the transmitters are shown in Figure 5.6. The
first signal was the excitation signal. The signal after the gap was the receiving signal by the
hydrophone. The frequency was 13 kHz, which is a common frequency for the LWD purposes.
The signals show sufficient amplitude for LWD applications.
74
Fig. 5.6 13 kHz transmitting and receiving acoustic signals for the transmitters.
5.5.2 Transmitting Voltage Response
The transmitting voltage response test was carried out by our industry collaborator in the
Institute of Acoustics, Chinese Academy of Sciences. The transmitter prototype was tested with
a frequency range of 3 kHz to 25 kHz. The TVR plot is shown in Figure 5.7. TVR is a good
representation of the efficiency of the transducer. Higher TVR means more electrical power is
converted into acoustic power by the transducer.
Fig. 5.7 Measured transmitting voltage response.
75
Figure 5.8 shows the TVR plot simulated by COMSOL Multiphysics®. It corresponds to the 5
kHz to 16 kHz range of the measured TVR. They matched closely and exhibited two resonance
peaks, one at 4 kHz for the half-ring mode, and the other at 13 kHz for the major PZT arc length
mode. The peak at 8 kHz was eliminated because of the disintegrated height direction. The
design strategy was verified by the TVR plot. The desired TVR value for the transmitters is 130
dB, which was achieved for a broad band from 11 kHz to 15 kHz. The TVR plot demonstrated a
high-performance transducer with broad band, desired resonance frequencies, and high TVR
value.
Fig. 5.8 TVR simulated by COMSOL Multiphysics®.
5.5.3 Directivity
The directivity test was also carried out by our industry collaborator in the Institute of Acoustics,
Chinese Academy of Sciences. Figure 5.9 shows the directivity plot at 14 kHz frequency.
Directivity demonstrates the uniformity of energy dissipation by the transducer into the aqueous
environment. For monopole transducer, the ideal directivity is a circular pattern. However,
because of the positioning of the active piezoelectric elements, there is less power in the gaps.
Therefore, this transducer displays a four-lobe pattern. Since this pattern is innate, normally a 20
dB variation is acceptable.
Figure 5.10 shows the directivity plot simulated by COMSOL Multiphysics®. It matched well to
the measured directivity. They both showed four-lobe pattern, which was caused by the
geometrical distribution of piezoelectric ceramic pieces in the transducers. The variations in SPL
were both about 20 dB, which met the requirements for such LWD transducers. For absolute
values, the measurement had a central SPL of 140 dB, which is smaller than the simulated value
of 160 dB.
76
Fig. 5.9 Measured directivity.
Fig. 5.10 Directivity simulated by COMSOL Multiphysics®.
77
CHAPTER 6 RECEIVER CONSIDERATIONS
6.1 Receiver Simulations
6.1.1 Structure-Stress Interaction
Fig. 6.1 Receiver is placed in a cubic water domain to study its receiving from acoustic wave to electrical
signals (top left). Three PZT configurations in the PEEK frame were simulated to study the structure-
stress interaction. The first one (top right) has a linear pattern and has six PZT pieces in a quarter of the
half ring, which is the configuration we used in our fabricated receiver prototype. These six pieces are in
series, and the four groups of six pieces are in parallel. The second one (bottom left) has a linear pattern
and has four PZT pieces in a quarter of the half ring, which is the configuration used in a previous
receiver design. Similarly, four pieces are in series, and four groups of four pieces are in parallel. The
third one (bottom right) has a chessboard pattern, with two linear patterns slightly shifted with respect to
each other. Twelve pieces are in series, and four groups of twelve pieces are in parallel. Fabrication
complexity increases from four-piece design, to six-piece design, to chessboard design. But the contact
area also increases from four-piece design, to six-piece design, to chessboard design.
To study the effect of structure-stress interaction on the receiving sensitivity and signal to noise
ratio, three different PZT configurations in the PEEK frame were simulated with COMSOL
Multiphysics®. For the three designs, the percentage of PZT volume is the same, which ensures
the same composite density and elastic moduli. But the contact area between PEEK and PZT are
78
different for the three designs, resulting in three different structure-stress interactions, as shown
in Figure 6.1.
The dimensions of these receivers are such that the inner radius of the inner tube is 72.5 mm; the
inner radius of the center tube is 75 mm, the outer radius of the center tube is 80 mm; the outer
radius of the outer tube is 82.5 mm. This makes the thickness of the PZT pieces 5 mm. The
height of the entire receiver is 40 mm. For the six-piece design, the height of the PZT pieces is
20 mm, and the width is 6.5 mm. For the four-piece design, the height of the PZT pieces remains
20 mm, but the width is expanded to 9.75 mm, so that the PZT volume is the same. For the
chessboard design, the height of the PZT pieces is 10 mm, since each PZT piece in the six-piece
design is divided into two to form the chessboard pattern. The width remains unchanged so that
the PZT volume is the same.
Fig. 6.2 Incident plane wave and its resultant acoustic wave pattern in the cubic water domain (top left),
and electric potential distribution of the three different PZT configurations (top right six piece in a quarter,
bottom left four piece in a quarter, and bottom right chessboard design).
As a receiver, it senses an acoustic wave signal and converts it into an electrical signal. To better
represent its function, the receiver is placed in a cubic water domain (Figure 6.1 top left). An
incident plane wave of 10,000 Pa is applied on the back side of the cubic water domain (Figure
6.2 top left). All other sides are applied with plane wave radiation boundary condition. When
waves arrive at the back of the receiver, it functions on the PEEK package as boundary load
defined by acoustic pressure. As linear elastic material, PEEK frame passes the force to the
active piezoelectric PZT ceramics, and charge signals are summed as receiving signals. The
79
inner boundary of the receiver is set fixed boundary condition. These are the physics principles
of the simulation. The electric potential distribution of the three different designs are shown in
Figure 6.2. They demonstrate a similar pattern which is expected because they have a similar
composite structure.
Quantitatively, we are interested in receiving sensitivity and signal to noise ratio of the receiver.
These are two important parameters to check the performance of a receiver.
Receiving sensitivity (RS) is the ratio of the receiving voltage to the acoustic pressure measured
at 1 m away from the receiver, in comparison with a reference sensitivity 1 /V Pa in water. In
decibel representation, RS is defined as
Receiver Acoustic10
/( ) 20log ( ) (6.1.1)
1 /
V PRS dB
V Pa
Signal to noise ratio (SNR) is self-explanatorily the ratio of the output signal to the noise signal.
Here only the thermal noise is considered. For a given bandwidth, the root mean square (RMS)
of the noise voltage is given by
Noise 4 (6.1.2)BV K TR f
where BK is Boltzmann’s constant, T is temperature in Kelvin, R is impedance, and f is
bandwidth in hertz over which the noise is measured.
Therefore, in decibel representation SNR is defined as
Receiver10
Noise
( ) 20log ( ) (6.1.3)V
SNR dBV
2500 5000 7500 10000 12500 15000 17500 20000 22500 25000
-250
-240
-230
-220
-210
-200
-190
-180
-170
-160
Re
ce
ivin
g S
en
sitiv
ity (
dB
)
Frequency (Hz)
6 Piece in a Quarter
4 Piece in a Quarter
Chessboard
Fig. 6.3 Receiving sensitivity curves for the three PZT configurations.
80
2500 5000 7500 10000 12500 15000 17500 20000 22500 25000
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
Sig
na
l to
No
ise
Ra
tio
(d
B)
Frequency (Hz)
6 Piece in a Quarter
4 Piece in a Quarter
Chessboard
Fig. 6.4 Signal to noise ratio curves for the three PZT configurations.
The receiving sensitivity and signal to noise ratio curves for the three PZT configurations are
shown in Figure 6.3 and Figure 6.4. They resemble each other in shapes except in their values,
which is understandable, since both of them root in a ratio of receiving voltage to somewhat a
constant. For each figure, the chessboard design has the highest receiving sensitivity and signal
to noise ratio, followed by six-piece design, and the four-piece design has the lowest receiving
sensitivity and signal to noise ratio. This shows that the chessboard design has the best structure-
stress interaction, since it has the most contact area between elastic material PEEK and
piezoelectric material PZT. Four-piece design has the worst structure-stress interaction, while the
six-piece design is in the middle. Fabrication-wise, it’s most complex to fabricate chessboard
though, so six-piece is the best in a comprehensive way.
The receiving sensitivity of the three designs is all over -210 dB, and the signal to noise ratio of
the three designs is all over 90 dB. These are very good receiver performances. All three designs
exhibit a resonance peak around 20 kHz due to the composite structure. But between 10 kHz and
15 kHz working range, they all show a flat response, which is desirable.
6.1.2 PEEK vs. Rubber Packaging
Another important goal here is to compare PEEK packaging with rubber packaging. While
PEEK is treated as linear elastic material with damping, rubber is normally treated as another
acoustic domain where waves propagate with attenuation, a.k.a. acoustic lens. So to compare
these two packaging designs requires different physics coupling in COMSOL Multiphysics®.
Two different rubbers were studied here, hard rubber (attenuation factor 130 dB/m) and silicone
rubber (attenuation factor 8 dB/m). For this study, the PZT configuration is set as the six piece in
a quarter. The electric potential distribution of the PZT configuration for both hard rubber
81
(Figure 6.5 left) and silicone rubber (Figure 6.5 right) is shown. They demonstrate a similar
pattern.
Fig. 6.5 Electric potential distribution of the two rubber-packaged receivers (left hard rubber, and right
silicone rubber). Both are in six piece PZT configurations.
Figure 6.6 and Figure 6.7 show the receiving sensitivity, and signal to noise ratio curves for the
two rubber packaging, in comparison with PEEK packaging.
2500 5000 7500 10000 12500 15000 17500 20000 22500 25000
-250
-245
-240
-235
-230
-225
-220
-215
-210
-205
-200
-195
-190
-185
-180
-175
-170
-165
-160
Re
ce
ivin
g S
en
sitiv
ity (
dB
)
Frequency (Hz)
PEEK
Silicone Rubber
Hard Rubber
Fig. 6.6 Receiving sensitivity curves for the two rubber packaged receivers in comparison with PEEK
packaging.
82
2500 5000 7500 10000 12500 15000 17500 20000 22500 25000
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
Sig
na
l to
No
ise
Ra
tio
(d
B)
Frequency (Hz)
PEEK
Silicone Rubber
Hard Rubber
Fig. 6.7 Signal to noise ratio curves for the two rubber packaged receivers in comparison with PEEK
packaging.
For each figure, the PEEK packaging has the highest receiving sensitivity and signal to noise
ratio, followed by hard rubber packaging, and silicone rubber packaging. This is mostly due to
the fact that silicone rubber has the highest attenuation of waves, PEEK has the lowest damping,
and hard rubber is in the middle. While PEEK packaging exhibits a resonance peak around 20
kHz due to the composite structure, both hard rubber and silicone rubber packaging are exempt
of that resonance, mostly because of the high damping. Overall, PEEK packaging has a better
performance than rubber packaging.
6.2 Receiver Fabrication and Testing
Receiver fabrication and testing are fundamentally similar to transmitter fabrication and testing.
Therefore, this section focuses on the difference rather than repeating the same processes.
Fabrication of low-noise high-sensitivity piezocomposite receivers included PZT ceramic pieces
electrical connections, PEEK frame preparation, PZT pieces soldering into the PEEK frame, and
in-vacuum bonding. Impedance analysis was carried out and it showed that the receivers had a
flat frequency response in between 11 kHz and 15 kHz, which is a desired performance for LWD
receivers. There was a half ring resonance around 10.5 kHz. The receivers were fitted on a test
drill collar. Limited testing was conducted including thermal cycle testing. It verified the high-
temperature stability of the receivers.
6.2.1 Receiver Fabrication
As shown in Figure 6.8, there were 24 pieces of PZT-5A (APC International, APC 850) in each
half of the receiver. Every 6 pieces were in series, and 4 such groups were in parallel. Each piece
83
had the dimension of 20 mm × 6.5 mm × 5mm. The poling was in 6.5 mm direction, making it
operate at d33 mode.
Fig. 6.8 Every 6 pieces of PZT were in series. 4 such groups were in parallel in the receiver connection.
3 3
12
0 3
20 10 5 101900 8.85 10 / 260
6.5 10r
A m mC F m pF
d m
For 6 pieces in series, the capacitance was 260 6 43pF .
For 4 such groups in parallel, the capacitance was 43 4 172 pF .
The slots and wire grooves on PEEK frames were manufactured with a milling machine. To
facilitate milling, first a holder was machined, as shown in Figure 6.9 (Left). The finished PEEK
center frames are shown in Figure 6.9 (Right). The center piece had an inner diameter of 150 mm,
outer diameter of 160 mm, and a thickness of 5 mm. The height was 40 mm.
Fig. 6.9 (Left) Holders to hold center PEEK tube during milling. (Right) Center PEEK tubes after milling.
After all PZT pieces were fitted in the PEEK frame, they were soldered to each other and to the
main wire to achieve the designed electrical connection configuration. For each 6 pieces soldered,
a capacitance measurement was done to make sure they were connected. High temperature solder
was used to make sure the device can operate in extreme environment seen in the oil drilling
conditions. PZT pieces soldered in the PEEK frame is shown in Figure 6.10.
84
Fig. 6.10 PZT pieces were soldered for desired electrical connections in the PEEK frame.
The same bonding technique used in the transmitter fabrication was conducted to bond these two
receivers. It was bonded in vacuum to make sure there was no trapped air in between layers.
Fiber fabric spacer was used to ensure a uniform bonding thickness. Both measures are to
improve the bonding strength. The in-vacuum epoxy bonding setup is shown again in Figure
6.11 and one bonded receiver is shown in Figure 6.12.
Fig. 6.11 The entire set up included a vacuum box, a vacuum pump, a rotary motor with controller, a
voltage generator, and a computer. The fixture was specially made for the half-tube geometry. Center
PEEK tube was placed on the center plate. Inner and outer sandwiching tubes were placed on the fixture.
The rotary motor moved the fixture closer or farther apart. After the tubes were set up and the system was
pumped to vacuum, the rotary motor was controlled by the computer to move the fixture close up. The
inner and outer PEEK tubes were then pushed onto the center PEEK tube, achieving the in-vacuum
bonding. The advantage is that it left no trapped air to improve bonding strength.
85
Fig. 6.12 Two halves of finished receiver number one. They were bonded with the same technique used to
bond the transmitters.
6.2.2 Receiver Impedance Analysis
Impedance analysis was done on both receivers with Agilent 4294A Impedance Analyzer. Here
are the results for capacitance (Figure 6.13), resistance (Figure 6.14) and conductance (Figure
6.15) data. Four parts of the two receivers showed a flat frequency response curve from 11 kHz
to 15 kHz, which is desired for receiving signals so that the receiver is equally sensitive to all
frequencies and less prone to excitation variations. The peak around 10.5 kHz was caused by the
half ring resonance. For each figure, the values and the positions of the peaks varied due to PZT
material difference and fabrication deviations. But in general, the curves were consistent. And
for the capacitance, the measurement matched the theoretical calculations.
8000 9000 10000 11000 12000 13000 14000 15000 16000
0.0
5.0x10-11
1.0x10-10
1.5x10-10
2.0x10-10
2.5x10-10
3.0x10-10
Ca
pa
cita
nce
(F)
Frequency (Hz)
Receiver 1 Part 1
Receiver 1 Part 2
Receiver 2 Part 1
Receiver 2 Part 2
Fig.6.13 Capacitance curves for four parts of the receivers before annealing.
86
8000 9000 10000 11000 12000 13000 14000 15000 16000
0
5x103
1x104
2x104
2x104
3x104
Re
sis
tan
ce
(Oh
ms)
Frequency (Hz)
Receiver 1 Part 1
Receiver 1 Part 2
Receiver 2 Part 1
Receiver 2 Part 2
Fig. 6.14 Resistance curves for four parts of the two receivers before annealing.
8000 9000 10000 11000 12000 13000 14000 15000 16000
0.0
5.0x10-7
1.0x10-6
1.5x10-6
2.0x10-6
2.5x10-6
3.0x10-6
3.5x10-6
Co
nd
ucta
nce
(Sie
me
ns)
Frequency (Hz)
Receiver 1 Part 1
Receiver 1 Part 2
Receiver 2 Part 1
Receiver 2 Part 2
Fig. 6.15 Conductance curves for four parts of the two receivers before annealing.
87
To calculate the theoretical resonance frequency, the compositing effect needs to be revisited.
This is the difference between the transmitter and the receiver. The major working resonance of
the transmitter is caused by the arc length of ceramic pieces. Therefore, for the transmitters, the
direct use of the series resonance frequency is correct. But for the receivers, it is so composited
that it needs to be calculated in the way outlined in Theories chapter.
11 11 12 11 12 13(1 )( 2 ) ( )p E E E cc q c c q c c c
(1 ) p cq q
And then the transverse wave velocity is obtained by
11cv
And the transverse mode resonance frequency is
312
vf
l
6.2.3 Receiver Drill Collar Fitting
The finished receivers had an inner diameter of 148 mm, and a thickness of 14 mm in the radial
direction. The height was 40 mm. The mouth for the wire outlet had a height of 13 mm, and a
width of 13 mm.
To make sure the two parts of the receivers would fit onto the drill collar, an Acetal plastic tube
imitating the drill collar of the same size was machined. The outside diameter of the imitation
drill collar was 144 mm. In between the imitation drill collar and the receivers was a 1.6 mm
thick rubber layer. The purpose of the rubber layer was to ensure a tight fitting and also to
function as a backing material. This rubber layer was not meant to protect the receivers from
corrosions because the epoxy used on the receivers could achieve this purpose already. Outside
the receivers should be clamps which were not made here.
6.2.4 Receiver Testing
All the mechanical testing for the receivers were the same with the testing for the transmitters,
including anticorrosion test, thermal cycle test, high hydrostatic pressure test, and vibration test.
88
CHAPTER 7 SUMMARY AND FUTURE WORK
7.1 Summary
In summary, this dissertation conveys an extensive collection of original research work on the
optimization, fabrication, and testing of LWD acoustic transmitters and receivers. In the
Optimization chapter, focus was given to a detailed methodology for applying COMSOL
Multiphysics® to optimizing LWD transducer design parameters. Material properties, meshing
techniques, and physics coupling were presented in details. Displacement frequency responses of
two piezocomposite transducer designs were compared and general design strategies were come
up with. Targeted studies confirmed these design strategies. A comparison of acoustic
performance parameters including acoustic field spatial distribution, absolute acoustic pressure,
TVR and directivity was made between the two designs. An extensive comparison between d33
and d31 configurations revealed the advantage and disadvantage of each.
In the Fabrication chapter, first some fabrication topics were discussed, including epoxy
selection, solder selection, thermal expansion coefficient consideration, in-vacuum bonding setup
and method, and uniform thickness spacer. These discussions are a summary of trial and error
along the project progress. It might seem concise but it is equivalent to an immense amount of
work. Once the techniques were discussed, the fabrication of the high-performance
piezocomposite transducers was presented step-by-step. Typical steps were piezoelectric ceramic
cutting, packaging material machining, epoxy bonding, and impedance analysis. The earlier
prototypes leading to the final successful prototype as presented are explained in Appendix A.
The first one is high-performance piezocomposite transducer featuring slant-cut ceramics,
resulting in broadband response at the expense of reduced resonance peaks. The second one is
high-performance piezocomposite transducer featuring non-slant-cut ceramics, bringing about
strong resonance peaks but less broad response. This dissertation work made breakthrough on
bonding large piece of PEEK with ultrahigh bonding strength. With PEEK having excellent
mechanical toughness, thermal and chemical stability, it can withstand all the drilling conditions.
It then replaces multiple materials and can be used singularly in the LWD transducer application.
This simplifies both the design and the fabrication for future LWD transducers. And it resulted in
LWD transducers with very high performance, both the transmitters, and the receivers.
In the Testing chapter, protocols for multiple tests were established. These tests are anticorrosion
testing, to make sure transducers can withstand corrosive drilling fluids; thermal cycle testing, to
make sure transducers can withstand high working temperature repeatedly without deteriorating
in quality; high voltage testing, to make sure transducers can withstand high driving voltage
without dielectric breakdown; high hydrostatic pressure testing, to make sure transducers can
withstand high working pressure in the oil well; vibration testing, to make sure transducers can
withstand strong vibration in the drilling practice; and acoustic testing, to make sure transmitters
can transmit enough power at designated driving frequency for the logging application, and have
desired TVR and directivity.
In the Receiver Considerations chapter, some receiver design strategies were looked into.
Structure-stress interaction studies by COMSOL Multiphysics® compared different piezoelectric
ceramic configurations to find the receiver with the highest RS and SNR. Different packaging
89
materials were studied also aiming to improve receiver performance. Using the same fabrication
techniques, receiver prototypes were manufactured and their impedance analysis was presented.
7.2 Future Work
Future work can be in four directions. The first one is to fabricate d33 mode transmitters, which
will improve transmitting power and reduce material cost. The second one is to expand design
and fabrication from the current monopole to dipole and quadrupole. Multipole transmitters will
obtain certain data not available for monopole transmitters, especially shear data in slow rock
formation. The third one is on 3D time-transient COMSOL Multiphysics® simulations of sonic
well logging. It will enable transmitter and receiver design optimization in a virtual logging
environment, which will have more reference values. Last but not least, guided wave simulations
can be done on drill collar periodic groove design to create broader stopbands, which will then
facilitate transmitter designs.
7.2.1 Cost Reduction with d33 Transmitters
Fig. 7.1 8-piece d33 transmitter compared to d31 transmitter.
The detailed simulations for comparison between d33 and d31 transmitters have been given in
Optimization chapter. As reminded in Figure 7.1, the 8-piece d33 transmitter has the same
transmitting power and other performances with d31 transmitter under the same applied voltage.
However, since the 8-piece design practically divided the quarter-circular PZT pieces into 8
rectangular pieces that approximately resembled the circular form, it would reduce the material
cost drastically. Quarter-circular PZT pieces are much more expensive than small rectangular
90
pieces. Nonetheless, it will require some work to overcome the fabrication challenges associated
with 8-piece d33 transmitters. But once the technical difficulties are tackled, the cost reduction
will be very rewarding and make this product very competitive to those from competitors.
Furthermore, one can use 12-piece or 16-piece designs to improve the transmitting power
significantly which will also bring an edge over other products.
7.2.2 Dipole and Quadrupole Transmitters
Fig. 7.2 Dipole and quadrupole sources and their velocity dispersion curves [13].
Some logging data cannot be obtained by monopole transducers but dipole and quadrupole
transducers. Especially for slow formation, there is no shear data by monopole sources. Because
both formation dipole and quadrupole dispersive waves approach shear wave velocity, these
sources can obtain shear data. It is definitely rewarding to research into these sources, however,
there are a lot of technical challenges in this realm because of the dispersive nature of the waves
(Figure 7.2). Computationally, COMSOL Multiphysics® makes it straightforward to extend
transducer designs beyond monopole case. By defining a phase shift on the electric field exerted
on each quarter part of the transducer, we can achieve dipole and quadrupole transducer
simulations. Fabrication-wise, it is important to have 4 or 8 quadrants mechanically separated
(cut or using dampening materials) and electrically isolated, for both transmitters and receivers
for the dipole and quadrupole cases.
91
7.2.3 Transmitter-Receiver System Simulations
After studying transmitters and receivers separately, it is imperative to simulate the transmitter-
receiver system. It is more accurate to the working principles of LWD acoustic transducers. It
will also allow transmitter and receiver optimization on a codependent level. I conducted a
simple 2D COMSOL Multiphysics® simulation on such a system. However, only 3D simulation
can take into considerations the geometries of the transmitter and receivers. Figure 7.3 (Left)
shows the geometry imitating an oil well borehole. The materials of the transmitter and the
receivers are also shown.
Fig. 7.3 (Left) Geometry, materials, and meshing of the 2D virtual sonic logging model. (Right)
COMSOL programming environment showing the details of physics coupling.
Figure 7.3 (Right) shows the physics coupling of this model. It includes acoustic-piezoelectric
interaction, solid mechanics, and electrical circuit modules. For details of this model, please refer
to [77]. The electrical circuit module supplies an excitation signal to the transmitter (terminal 1).
The function is: 100 exp( (( ) / ( / 2)) ^ 2) (7.2.1)t period period
where the period is the reciprocal of frequency 10 kHz.
The acoustic-piezoelectric interaction module couples the transducers and the water domain
together. The transmitter terminal 1 is voltage and supplied by electrical circuit. The receiver
terminal 2 and 3 are both charge. After waves propagate through granite domain, they get back to
water domain by normal acceleration. Top and bottom of the water domain is set cylindrical
wave radiation. The left edges of the transducers are set fixed boundaries. This module is time
transient so that we can observe how waves propagate and electric signals arrive along time.
The solid mechanics module couples the water domain and the granite domain together. The
waves propagate from the water domain to the granite domain by boundary load. The top,
bottom, and right edges of the granite domain are set low-reflecting boundaries to simulate an
92
infinite rock formation.
A first look at the model results is how the waves propagate through the water domain and the
granite domain. Here the sound pressure level (Figure 7.4) is used to reveal acoustic propagation.
For the water domain, the sound pressure level is: 20 log10( /1 6) (7.2.2)p e
where p is a COMSOL parameter for pressure in the water domain.
For the granite domain, the sound pressure level is:
20 log10( . /1 6) (7.2.3)solid pm e
where solid.pm is a COMSOL parameter for pressure in the granite domain.
Fig. 7.4 Sound pressure level in the water domain and granite domain along time. One can see the wave
propagation. It starts from the transmitter, propagates, and gets reflected from the boundary as time goes
on. It keeps propagating and soon fills the entire space. Since there is only a one-time excitation, the
acoustic pressure decreases as it fills more space.
When the transmitter is excited at 0.0001 s, waves start in that region. As time goes on, it
propagates to a larger area at 0.0002 s. At 0.0004 s, waves hit the boundary and reflect. It keeps
propagating down towards the receivers and eventually all space is filled with acoustic pressure
(0.001 s). Since there is only a one-time excitation, the sound pressure level decreases as the
waves fill more space.
Another aspect of interest is how the signals are received by the receivers. These received signals
are the basis for further analysis for the acoustic and mechanical properties of the rock formation.
At 0.0001 s, the transmitter is excited and the transmitter voltage is shown as terminal 1 voltage.
The transmitting power propagates as acoustic waves. After some time of travel, around 0.0015
s, the first receiver detects the signal as terminal 2 voltage. This voltage is multiplied by 2000
and offset by 25 Volts. Around 0.0021 s, the second receiver detects the signal as terminal 3
voltage. This voltage is also multiplied by 2000 and offset by 50 Volts. The amplification of the
receiving signals is common in the field. The offsetting is so that all three terminal voltages can
be shown in one figure. Since the transmitter-receiver distance increases, there is a time
difference as when the signal arrives. Based on this time difference, wave velocity can be
calculated. Since the granite is treated as linear elastic material with only Young’s modulus, there
is only the compressive wave. In reality, there are also shear wave and Stoneley wave. Figure 7.5
basically shows the working principles of oil well sonic logging.
93
Fig. 7.5 Terminal voltages. Terminal 1 is the transmitting signal. Terminal 2 and terminal 3 are the
receiving signals. The receiving voltages are amplified by 2000 and offset so that they can be shown on
the same graph with the transmitting voltage. After the transmitter is excited at 0.0001 s, the first
receiving signal gets detected around 0.0015 s, and the second around 0.0021 s. The time difference is due
to the increase of the transmitter-receiver distance. Wave velocity can be calculated from this time
difference. Since granite is treated as linear elastic material with just Young’s modulus considered, there
is only compressive wave. In reality, there is also shear wave and Stoneley wave. This figure shows the
working principles of oil well sonic logging.
7.2.4 Stopband Design with Guided Waves Studies
In oil drilling, guided waves propagating along the drill collar is least desired. Critically refracted
waves at the drilling mud and rock formation interface is the source of useful information for the
borehole logging. Guided waves along the drill collar only add noise to the useful information
and should be reduced by all means. In this dissertation work, disintegrating the height direction
of the transducer and eliminating the height mode vibrations that can more easily propagate
along the drill collar was achieved.
Another common practice to eliminate guided waves is to design periodic groove configurations
on the drill collar as shown in Figure 7.6 (Top) [14]. By combining different lengths of periodic
grooves, a stopband can be created to prevent waves at certain frequencies to propagate (Figure
7.6 (Bottom)). This is why it is important to also study stopband designs along with transmitter
optimization. Because only with both studies one can match transmitters to have frequency
responses that fall into the stopband. This way, at these frequencies, transmitting waves will not
propagate along the drill but just around the borehole.
94
Fig. 7.6 (Top) A combination of periodic groove configurations to act as acoustic isolator. (Bottom) A
stopband created by periodic groove configurations [14].
The theoretical background behind the stopband design is given in Figure 7.7. Guided waves can
only propagate at strictly defined modes. They also have a property called low-frequency cutoff
because of their dispersive nature. Therefore, if we design a series of stopbands that cut off
several modes of guided waves, we have a stopband that cut off a range of frequencies.
Fig. 7.7 Dispersion curves for SH modes on an isotropic plate with free boundaries [21].
95
Values of propagation constant for different waveguide modes n are: 22 2
2 2 (7.2.4)s
s
n nk
b V b
For to be real for the thn mode, /sn V b (i.e., there is a low-frequency cutoff for all but
the 0n mode).
96
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100
APPENDIX A
A1.1 Slant Broadband Transmitter Prototype
This section depicts the development of a slant broadband transmitter prototype, focusing mainly
on its geometry and impedance performance. Figure A1.1 shows the ceramic pieces (Top left),
PEEK frame fit with ceramic pieces (top right), packaged transmitter (bottom left), and the
finished transmitter (bottom right). The development of this prototype helped accumulate
knowledge and experience in developing the final successful prototype.
Fig. A1.1 (Top left) slant ceramic pieces; (top right) PEEK frame fit with slant ceramic pieces; (bottom
left) packaged transmitter; (bottom right) finished transmitter prototype.
The original curved PZT ceramic pieces had inner radius 75 mm, outer radius 80 mm, height
50.8 mm, and angle at center 85 °. To have the slant angle of 15 °, the PZT ceramic pieces were
cut with a diamond saw. The final curved PZT ceramic pieces had a short outer arc length of 104
mm, and a long outer arc length of 118 mm. The slant directions were different so that the
maximum directivity could be ensured. Grooves for wires were cut by a sonic milling tool. The
slant ceramic piece demonstrated broadened resonance response, as exemplified by two split
101
resonance peak at the same position. The singular peak at 13 kHz was now split into two peaks,
which spanned 13 kHz to 15 kHz. The cause of the two peaks is two edges with different lengths.
The phase data corresponded to the broadening as well. This is consistent with simulation data.
Capacitance with dielectric loss curves is given in Figure A1.2.
Fig. A1.2 Capacitance with dielectric loss data for 15 ° slanted PZT ceramic piece. The x axis is in Hz.
The capacitance Cp data is in Farads. The dielectric loss D data is percentage.
There were three PEEK tubes, inner, center, and outer. The inner PEEK tube had an inner
diameter of 145 mm, an outer diameter of 150 mm, and a height of 152 mm. The center PEEK
tube had an inner diameter of 150 mm, an outer diameter of 160 mm, and a height of 152 mm.
The outer PEEK tube had an inner diameter of 160 mm, an outer diameter of 165 mm, and a
height of 152 mm. The windows for ceramic pieces were cut out by CNC milling machines.
For each transmitter, from the impedance measurements, it showed properties as predicted by
COMSOL simulations. Take the resistance curves in Figure A1.3 for example, the peak around
8-9 kHz was reduced drastically. The peak around 13 kHz was split into two peaks and
broadened significantly, caused by the slanted PZT ceramic pieces. The long and short edges of
PZT corresponded to low and high peaks around 13 kHz, broadening the entire peak from 10 to
15 kHz. The slant-to-broaden design strategies were verified experimentally.
Other measurement curves including impedance, admittance, conductance, and capacitance, are
also shown in Figure A1.4 through Figure A1.7.
0.0000E+00
1.0000E+00
2.0000E+00
3.0000E+00
4.0000E+00
5.0000E+00
6.0000E+00
-1.0000E-08
-5.0000E-09
0.0000E+00
5.0000E-09
1.0000E-08
1.5000E-08
2.0000E-08
2.5000E-08
3.0000E-08
3.5000E-08
0 5000 10000 15000 20000 25000 30000
Cp - D
DataTraceA Real
DataTraceB Real
102
7500 10000 12500 15000 17500-25
0
25
50
75
100
125
150
175
Resis
tance (
Ohm
s)
Frequency (Hz)
Half-Tube Transmitter One
Half-Tube Transmitter Two
Fig. A1.3 Resistance curves of two half-tube transmitters.
7500 10000 12500 15000 17500100
200
300
400
500
600
Imp
eda
nce
(O
hm
s)
Frequency (Hz)
Half-Tube Transmitter One
Half-Tube Transmitter Two
Fig. A1.4 Impedance curves of two half-tube transmitters.
103
7500 10000 12500 15000 175001.0x10
-3
2.0x10-3
3.0x10-3
4.0x10-3
5.0x10-3
6.0x10-3
7.0x10-3
8.0x10-3
Adm
itta
nce (
Sie
mens)
Frequency (Hz)
Half-Tube Transmitter One
Half-Tube Transmitter Two
Fig. A1.5 Admittance curves of two half-tube transmitters.
7500 10000 12500 15000 17500-5.0x10
-4
0.0
5.0x10-4
1.0x10-3
1.5x10-3
2.0x10-3
2.5x10-3
3.0x10-3
3.5x10-3
Co
ndu
cta
nce
(S
iem
en
s)
Frequency (Hz)
Half-Tube Transmitter One
Half-Tube Transmitter Two
Fig. A1.6 Conductance curves of two half-tube transmitters.
104
7500 10000 12500 15000 175002.5x10
-8
3.0x10-8
3.5x10-8
4.0x10-8
4.5x10-8
5.0x10-8
5.5x10-8
6.0x10-8
6.5x10-8
Ca
pacita
nce
(F
ara
ds)
Frequency (Hz)
Half-Tube Transmitter One
Half-Tube Transmitter Two
Fig. A1.7 Capacitance curves of two half-tube transmitters.
Below is the theoretical calculation of the series resonance frequency for the slant transmitter:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the arc length of the ceramic piece in the middle, it’s about 108 mm in this case. The
corresponding resonance frequency is:
12
1 113 kHz
2 0.108 7750 16.4 10sf
Because of the cut, the shorter edge brings one of the split resonance peaks higher towards 15
kHz. The longer edge brings one of the split resonance peaks lower towards 10 kHz. The center
as calculated stays around 13 kHz. The splitting because of the slanting is one design strategy
that can be used to make broadband transmitters. It is verified in the prototype.
A1.2 Non-Slant High-Resonance Transmitter Prototype
Where there is an advantage, there comes a disadvantage. The slant transmitter prototype
featured broadband performance, but the magnitude of the resonance peaks were reduced
consequently. Therefore, this section depicts the development of an alternative, a set of curved
high-power non-slant high-resonance piezocomposite transmitter prototype. It featured non-slant
PZT pieces to increase resonance peaks. One can choose either prototype for the best application
in specific situations.
105
The original curved PZT ceramic pieces had inner radius 75 mm, outer radius 80 mm, height
50.8 mm, and angle at center 85 °. The non-slant PZT pieces were simply cut down from the
original sized PZT pieces in angle from 85 ° to 80 ° with a diamond saw.
There were three PEEK tubes, inner, center, and outer. The inner PEEK tube had an inner
diameter of 145 mm, an outer diameter of 150 mm, and a height of 152 mm. The center PEEK
tube had an inner diameter of 150 mm, an outer diameter of 160 mm, and a height of 152 mm.
The outer PEEK tube had an inner diameter of 160 mm, an outer diameter of 165 mm, and a
height of 152 mm. Because the PEEK tubes were to house non-slant PZT pieces, they were easy
to machine manually using a 3-axis milling machine. Then they were cut into halves with a
vertical band saw.
Figure A1.8 shows the non-slant ceramic pieces (Top left), PEEK frame with non-slant windows
(top right), PEEK frame fit with ceramic pieces (bottom left), and the finished transmitter
(bottom right).
Fig. A1.8 (Top left) non-slant ceramic pieces; (top right) PEEK frame with non-slant windows; (bottom
left) PEEK frame fit with ceramic pieces; (bottom right) finished transmitter prototype.
106
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0.0
1.0x10-8
2.0x10-8
3.0x10-8
4.0x10-8
5.0x10-8
6.0x10-8
7.0x10-8
8.0x10-8
9.0x10-8
1.0x10-7
Ca
pa
cita
nce
(F
)
Frequency (Hz)
Transmitter 1 Before High Voltage Test
Transmitter 2 Before High Voltage Test
Fig. A1.9 Capacitance curves for non-slant high-resonance transmitter one and two.
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0.0
5.0x10-4
1.0x10-3
1.5x10-3
2.0x10-3
2.5x10-3
3.0x10-3
Co
nd
ucta
nce
(S
iem
en
s)
Frequency (Hz)
Transmitter 1 Before High Voltage Test
Transmitter 2 Before High Voltage Test
Fig. A1.10 Conductance curves for non-slant high-resonance transmitter one and two.
107
2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
0
20
40
60
80
100
120
140
Re
sis
tan
ce
(O
hm
s)
Frequency (Hz)
Transmitter 1 Before High Voltage Test
Transmitter 2 Before High Voltage Test
Fig. A1.11 Resistance curves for non-slant high-resonance transmitter one and two.
Figure A1.9 to A1.11 shows the capacitance, conductance, and resistance data for the two half-
tube transmitters. Based on the theoretical resonance frequencies:
11
1 1
2s E
fl s
For the PZT material, 37750 kg/m , 12 2
11 16.4 10 m /NEs .
For the arc length of the ceramic piece, it’s about 108 mm in this case because there is no slant.
The corresponding resonance frequency is:
12
1 113 kHz
2 0.108 7750 16.4 10sf
Ideally there should be no splitting at 13 kHz. But because this was an earlier prototype, the
ceramic pieces were cut manually and there could be some inconsistency in the arc length. That
is why there is some broadening. In the final prototype showed in Fabrication chapter, the
ceramic pieces were manufactured in the same mold with well-designed dimensions, and
therefore, there was no band broadening at all.
For the arc length of the half ring, it’s about 259 mm. The corresponding resonance frequency is:
12
1 15.4 kHz
2 0.259 7750 16.4 10sf
This is the peak around 4-5 kHz caused by the resonance in the arc length of the half tube. There
was no resonance around 8-10 kHz again due to the disintegrated height direction. The peak
around 10-15 kHz was the major working resonance peak.
108
APPENDIX B
A2.1 COMSOL Simulations on Magnetoelectric Sensors
Piezoelectric materials have numerous applications including transducers, transformers, actuators,
sensors, etc. Therefore, beyond my transducer work, I also worked on some magnetoelectric
sensor application. The sensor is a laminate of two materials, magnetostrictive material that
strains in a changing magnetic field, and piezoelectric material that outputs an electrical signal
from the strain. The application for the magnetoelectric sensor suits any situation where the
changing magnetic field is the source of the signal. Specifically in this part of work, it detects
biomagnetic field changes in human organs such as the liver and the heart. The biomagnetic field
depends on the iron concentration and this can be used to detect iron overload.
Iron is a trace mineral that plays a vital role in the human body. However, absorbing and
accumulating excessive iron in body organs (iron overload) can damage or even destroy an organ
such as liver. Conventional liver biopsy is disadvantaged due to its invasiveness and discomfort.
Biomagnetic Liver Susceptometry (BLS) such as Superconducting Quantum Interference Device
(SQUID) can determine the liver iron concentration in vivo, but it requires high cost and operates
with complicated procedures. There is a great need for noninvasive, low-cost tissue iron
detection methods. Dr. Zhang’s group is leading advances in the magnetoelectric composite
sensors that exhibit an ultrahigh AC magnetic sensitivity under the presence of a strong DC
magnetic field at room temperature. These advances open up innovative possibilities for
compact-size, portable, affordable, and room-temperature medical instruments for quantitative
determination of tissue iron concentrations.
To supplement the experimental advances, it is very important to look at the same problem using
simulation tools such as COMSOL. To have a medical device that can interpret measurement
data quickly and accurately, a database for livers of different sizes as well as when measured at
different distances (both lateral and vertical) need to be built. Since FEM is cost-effective and
time-efficient, one can build a detailed database with an accurate model. The goal in this
Appendix is to build such accurate COMSOL model so that enough data can be extracted for the
database. A future working magnetoelectric sensor based liver iron concentration detection
device can correlate measurement data to this database so that it can interpret the measurement
quickly and accurately.
In a typical biomagnetic susceptometer, a high magnetic field ( ')B r applied to the tissue
generates a magnetization,
0 ( ') ( ') ( ') (A2.1)M r r B r
where ( ')r is the magnetic susceptibility of the tissue at 'r , 0 is the magnetic permeability of
the free space. Since the induced magnetic field due to tissues is significantly weaker in
magnitude (~10-6) compared with the applied field, the applied field ( ')B r is used in Equation
(A2.1). The biomagnetic field at the sensor location r , due to magnetization ( ')M r of the tissue
in volume element d at 'r , is then
109
5 30( ) 3[ ( ') ] / ( ') / (A2.2)4
dB r M r R R R M r R d
where - 'R r r . Substituting Equation (A2.1) into Equation (A2.2), and integrating over the
whole liver volume , the induced biomagnetic field from the liver (or tissue) at the sensor
location, ( )B r , can be obtained as
5 31( ) ( ') 3[ ( ') ] / ( ') / (A2.3)
4B r r B r R R R B r R d
For magnetoelectric sensors under a strong DC magnetic field bias generated by a permanent
magnet, the physics can be broken down into two steps. The first step is the magnetization of the
permanent magnet, generating a magnetic field near the liver. The second step is the induced
magnetization of the liver (sort of functioning as a permanent magnet), generating a magnetic
field near the sensor. Based on Equation (A2.3), the induced magnetization of the liver depends
on the sensor-liver distance. Therefore, more accurate simulation model should use a pixelated
liver geometry and do an element-by-element calculation. However, an explanation of the
physics can still be made by an averaged liver.
Fig. A2.1 Geometric construction of the permanent magnet, magnetic sensor, and liver phantom.
As shown in Figure A2.1, the permanent magnet has a rectangular shape with length 5.08 cm,
width 2.54 cm, and height 2.54 cm. The magnetic sensor is also rectangular shape with length 13
mm, width 6 mm, and height 1 mm. Only magnetostrictive layer is considered here as we are
only studying the induced biomagnetic field in this part of simulation. Later on, the model can
have solid mechanics and piezoelectricity modules added to convert the magnetic field to strain
by magnetostriction and from strain to electrical signal by piezoelectricity. And at that stage, a
complete sensor structure will be used. The liver is represented with an ellipsoid, with three
constants being a = 6 cm, b = 7 cm, and c = 3.5 cm. The magnetic sensor is 2.02 cm below the
permanent magnet (centroid to centroid distance). The liver phantom can have ranging sizes by
changing the ellipsoidal constants. It can also have ranging vertical and lateral positions which
can be easily realized in the geometry part of the model.
110
The permanent has an M = 1.03 × 106 A/m, and µr = 1.05. For air, µr = 1; for sensor, µr = 6. For
liver phantom, µr = [ferratin concentration × 1600× 10-6 – (1-ferratin concentration) × 9.32× 10-6]
+ 1. Normal level for ferratin concentration in liver is 0.005 mg/g. But different concentrations
can be used to obtain relevant data as well. As aforementioned, there are two steps to obtain the
induced biomagnetic field from the liver at the sensor. The first step is shown in Figure A2.2.
This step is where the permanent magnet is magnetizing the entire system, inducing a DC bias
field in the magnetic sensor, and also more importantly, inducing a biomagnetic field in the liver
phantom. Compared to the permanent magnet, the induced biomagnetic flux density is much
smaller in scale.
Fig. A2.2 Step one: permanent magnet magnetizes the entire system, inducing a DC bias field in the
magnetic sensor, and more importantly, inducing a biomagnetic field in the liver phantom.
Fig. A2.3 Step two: the biomagnetic field in the liver phantom is now the magnetizing part, and it induces
a biomagnetic flux density at the sensor region.
111
In the second step, the permanent magnet is disabled, and we are focusing on the effect of the
biomagnetic field on the magnetic sensor, as shown in Figure A2.3. In step two, the biomagnetic
field in the liver phantom is now the magnetizing part, and it induces a biomagnetic flux density
at the sensor region. This biomagnetic flux density will cause the magnetostrictive material to
strain, and this strain will convert into electrical signal by piezoelectricity. If there are different
concentrations of iron in the liver, or different positions and sizes of the liver, the biomagnetic
field will be different.
Fig. A2.4 Biomagnetic field in a pixelated liver model at different lateral positions.
The averaged liver model was only used to explain the physics of the magnetization processes. It
is not accurate based on Equation (A2.1) through Equation (A2.3), or by common sense. Because
the farther away one part of the liver is from the permanent magnet, the weaker the induced
magnetic field is, thus the weaker the magnetization of that liver part on the sensor. It is shown in
Figure A2.4.
Based on this observation, a pixelated liver model was adopted. The model had the same sizes
with the averaged liver model, but the liver was divided into 400 elements. It was constructed
layer-by-layer. At each layer away from the middle layer, certain elements were rid of to
maintain an ellipsoid shape. This construction was rather arbitrary. In the future, a computer
112
generated pixelated ellipsoid can be used to represent the liver phantom accurately. Because the
number of the elements is quite large, it can be tedious to use the model. Therefore for certain
studies, only a portion of the elements are needed based on symmetry. For example, in the
vertical distance study, only a quarter of the total number of the elements are needed. In the
lateral distance study, only half of the total number of the elements are needed.
Figure A2.5 shows the physics coupling in COMSOL. Material properties were as described in
previous paragraphs. Geometry utilized the pixelated liver model. Probes were defined in
Definitions and a number of constants and variables were given in Parameters for easier
alteration of the model in the future. The main physics is Magnetic Fields, No Currents (mfnc).
Figure A2.5 (Left) shows the first step where permanent magnet is magnetizing the entire system.
In this one, all the liver elements as magnetic entities are disabled. Figure A2.5 (right) shows the
second step where the permanent magnet is disabled and all the liver elements function as the
magnetizing entities.
Fig. A2.5 (Left) step one physics coupling of the pixelated liver model; (right) step two physics coupling
of the pixelated liver model.
113
The first study we did is the effect of vertical distance between the liver phantom and the sensor
on the magnetic flux density. The distance studied was between 2 cm and 14 cm, which is a
normal vertical range. The distance is defined between the top of the ellipsoid phantom with the
bottom of the sensor. The results are shown in Figure A2.6. It highly matched the results from
competitors’ groups. More data fitting needs to be done but it seems like it’s fitting to ~1/R3
trend, which is correct based on theoretical derivations. The vertical distance has a very drastic
effect on the magnetic flux density and it decreases very quickly as the distance increases.
Results of pixelated model versus averaged model are compared. Besides at 2 cm, at all other
distances, results by parts are lower than results by average.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
1.0x10-8
2.0x10-8
3.0x10-8
4.0x10-8
5.0x10-8
Ma
gn
etic F
lux D
en
sity (
T)
Vertical Distance (cm)
By Parts
Average
Fig. A2.6 Magnetic flux density versus vertical distance between the ellipsoid phantom and the magnetic
sensor both by parts and by average.
For the effect of lateral position (x-direction for this part) on magnetic flux density, the liver
phantom was placed from center (0 cm) to 15 cm right to the sensor (+15 cm). At each distance,
it was measured using the x-axis coordinates of the ellipsoid center and the sensor center. It is
expected that the results should be symmetrical to the left side at the same distance. Therefore,
only one-sided lateral study was done. The vertical distance was kept at 2 cm down. The results
are shown in Figure A2.7. In general, it is a less drastic effect compared with vertical distance.
Still, as lateral distance increases, the magnetic flux density decreases. Also, in terms of by parts
and by average, it has the same results with the vertical study. Except for at center, results by
average are higher than results by parts. It further proves the merits of using pixelated liver
model.
114
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.0
5.0x10-9
1.0x10-8
1.5x10-8
2.0x10-8
2.5x10-8
3.0x10-8
3.5x10-8
4.0x10-8
4.5x10-8
5.0x10-8
Ma
gn
etic F
lux D
en
sity (
T)
Lateral Distance (cm)
By Parts
Average
Fig. A2.7 Magnetic flux density versus lateral distance between the phantom and the magnetic sensor (x-
direction) both by parts and by average.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0
1.0x10-8
2.0x10-8
3.0x10-8
4.0x10-8
5.0x10-8
6.0x10-8
Ma
gn
etic F
lux D
en
sity (
T)
Concentration (mg/g)
By Parts
Average
Fig. A2.8 Magnetic flux density versus liver iron concentration both by parts and by average.
115
Figure A2.8 gives the dependence of the magnetic flux density on the ferratin (iron)
concentration the liver ellipsoid. It ranges from normal concentration to extreme overload. It
mostly showcases a linear correlation. But because of the existence of water and water being
paramagnetic, it is negatively linear first until the paramagnetism of water was compensated by
the ferromagnetism of the ferratin. Then it starts to be positively linear. These results were
verified by experiments in the group. For same concentration, results by parts are always larger
than those by average.
A2.2 Future Simulations on Magnetoelectric Sensors
The shortcoming of the current COMSOL model is that the pixelated liver phantom is arbitrarily
determined. Future direction for a more accurate model is to use computer generated pixelated
objects to represent ellipsoid or any shape of liver or tissue. As shown in Figure A2.9 to Figure
A2.11, one of such methods gives a best representation of an ellipsoid with elements. Height,
width, depth can be used to determine the size of the ellipsoid. Precision can be used to adjust the
elemental layout. It also gives layer-by-layer details so that one can build a geometry by stacking
up elements in the same way. One only needs to note the relative positions of each element. With
this more accurate representation of the liver ellipsoid, more accurate data can be obtained for
the database taking into consideration of liver sizes, distances, iron concentrations, etc.
Fig. A2.9 Computer generated pixelated ellipsoid.
116
Fig. A2.10 Computer generated pixelated ellipsoid at layer 52.
Fig. A2.11 Computer generated pixelated ellipsoid at layer 42.
VITA
Runkun Jiang was born in Xishangping Village, Huangshanpu Town, Yishui County, Linyi City,
Shandong Province, China, on April 2, 1989, the son of Xiubiao Jiang and Jihui Wu. He has two
older sisters, Runyan Jiang, and Runye Jiang. After completing his high school diploma at
Yishui No. 1 High School, Linyi City, Shandong Province, in 2006, he entered Northwestern
Polytechnical University at Xi’an City, Shaanxi Province, receiving the degree of Bachelor of
Engineering in Materials Science and Engineering in July, 2010. With a prestigious Chiang Chen
Overseas Graduate Student Fellowship, he entered the Graduate School in the Department of
Engineering Science and Mechanics at The Pennsylvania State University at University Park, PA
in August, 2010, receiving a Master of Engineering degree in May 2012. He then continued in
the Department of Electrical Engineering in August 2012, receiving a Master of Science degree
in December 2013. During his stay at the University, he was elected President of Engineering
Graduate Student Council (2013-2015), organized and participated in a quantity of
extracurricular activities, outreach, and services. He was part of several professional associations,
and was presented a number of awards throughout the Graduate School.
