Characterization and Modeling of Piezoelectric Devices
An Independent Study
Presented to the Faculty of
The Department of Electrical and Computer Engineering
Villanova University
In Partial Fulfillment
Of the Requirements for the Degree of
Master of Science in Electrical Engineering
By
Sean Pearson
February 20th, 2006
Under the Direction of Dr. Pritpal Singh
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Villanova University Department of Electrical and Computer Engineering
Graduate Program
ECE 9030: Independent Study
Approval Form
Student’s Name: Sean Pearson Department: Electrical Engineering Full Title of Independent Study: Examination and Modelling of Piezoelectric Energy Harvesters Date Submitted: Faculty Advisor: Pritpal Singh Date: Chairperson: Pritpal Singh Date:
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Table of Contents • Introduction 1 • Project aims and goals 1 • History of Piezoelectricity 1 • Piezoelectric Theory 3
o Physics of Piezoelectricity 3 • Piezoelectric Materials 12
o Crystalline Materials 13 o Piezoelectric Ceramics 14 o Piezoelectric Material Comparison 15
• Experimental Test Materials 18 o Advanced Cerametrics Incorporated 18 o Omnitek Incorporated 21
• Experimental Test Procedures 22 Loading Situations 22 Steady State Tests 22 Drop Tests 25 Impedance Tests 27
• Experimental Results and Discussion 29 o PZT Material 29
Hard PZT 5a 29 • Drop Tests 29 • Steady State tests 30 • Impedance Tests 31
Soft PZT 5a 32 • Drop Tests 32 • Steady State tests 33 • Impedance Tests 34 • Soft PZT Linearity Confirmation 34
Bi-Morph Materials 35 • Drop Tests 35 • Steady State tests 36 • Impedance Tests 36
Resonators 37 • Drop Tests 38 • Steady State tests 39 • Impedance Tests 39
Calculations and Final Results 40 • Conclusions 45 • References 46 • Appendix 1 – Young’s Modulus Calculations 48 • Appendix 2 – CMA-R Type 3 Datasheet 52 • Appendix 3 – PZT type 5a Datasheet 54 • Appendix 4 – PZT type 8 Datasheet 55
1
Introduction
This project is being conducted with the intention of developing a system
that can be used to custom design, simulate and test piezoelectric materials for
use in an energy harvesting system. This system is going to be used to
supplement other energy harvesting systems that are being used in certain
applications, where batteries are not the optimal power source. These
applications include remote sensing and munitions fuzing, amongst others. The
ability to develop a power source to a specific requirement will be of great
advantage in powering electronics for these applications.
Project aims and goals
The specific goal of this project is to design, simulate and test a
piezoelectric energy harvester for powering on-board electronics in a munitions
application. This application is characterized by high G forces and a short
operational life. Knowing this, tools were needed to be able to predict the output
from a device when subjected to a mechanical load. This is necessary since
testing these devices is expensive, and to have an accurate simulation tool would
save a lot of time and money. Once this is accomplished, a power converter
needs to be designed so that the power which is harvested can be stored and
used by the electronics.
History of Piezoelectricity
Curiosity about the piezoelectric effect dates back thousands of years. It
was first noticed in rocks which would repel other rocks when they were heated.
These rocks, which were actually Tourmaline crystals, eventually found their way
2
into Europe. Once the crystals arrived in Europe, they were scrutinized by the
scientists of the day. In the mid 1700’s, this effect was given the name of
Pyroelectricity, which means electricity by heat. [1]
The Curie brothers were the first to discover the direct piezoelectric effect.
This title means the correlation between input mechanical force and output
electrical energy. They first published their research results on August 2, 1880
[1]. The converse piezoelectric effect, which means mechanical deformation by
application of an electric field, was predicted in 1881. The first applications of
piezoelectricity were in the area of sonar, where quartz plates were used to emit
high frequency waves. These waves would bounce off an object and return to a
receiver, indicating to the operator the presence of an object below. Today,
major applications of piezoelectric materials are in sensors, where their linear
response makes them ideal for making mechanical measurements. A growing
field for these devices is in actuators, where piezoelectrics are used to cause a
mechanical movement [1].
3
Piezoelectric Theory
Physics of Piezoelectricity
Piezoelectricity is known as a linear phenomenon. Its name literally
means electricity by pressure. Electricity is generated when the material is
mechanically deformed. When the material is deformed, it polarizes, creating an
electric field, which allows electricity to be harvested from the material. The
converse effect works in much the same way. When a potential is applied across
the material, it will cause the polarization, which in turn will pull the material and
deform it.
Piezoelectric materials can express both an isotropic and anisotropic
characteristic. They are isotropic when they are unloaded, and therefore, their
properties are not dependent on which axis of the material is being examined.
When the material is loaded, however, it will exhibit isotropic properties.
Therefore, it is important in which direction one examines the material.
The piezoelectric constants are defined as Xab, where X is the constant
symbol, a is the axis where one is examining the electrical properties, and b is
the axis where one is examining the mechanical properties. These are shown in
figure 1. Here, all axes are labeled, and shown are the different linear directions,
1, 2, and 3, and the radial directions, 4, 5, and 6. An example of this axis
nomenclature is K13 , the electro-mechanical coupling coefficient, where the
electrical characteristic is on the X axis, and the mechanical characteristic is on
the Z axis. [2] Therefore, if the material is being mechanically excited on the Z
axis, the electrical output is being measured on the X axis.
4
There is one characteristic equation which governs all piezoelectric
devices. It is called the piezoelectric equation, and it is given in equation 1 [2].
This equation relates the compressive force per unit area to the charge density
on the piezoelectric electrodes. This is the basic equation used in analysis of
piezoelectric devices.
AFddD ijjiji ⋅=⋅= σ (1)
Where:
Di = Electric Displacement (or Charge Density)
dij = Piezoelectric Constant
σj = Mechanical Stress
F = Force
A = Area
The Curie brothers noticed that the piezoelectric effect is linear, so that the
charge produced is directly proportional to the stress to which the material is
subjected. These two properties are linked by the piezoelectric strain coefficient.
This same coefficient is used for the converse piezoelectric effect. Figure 2
Figure 1 – Axis Nomenclature [2]
5
shows different loading situations of piezoelectric materials, including both the
direct effect and the converse effect. In both cases it is seen how the material
reacts to the given excitation, either mechanical or electrical.
Figure 3 shows different loading situations, and it also shows how these
materials can be cut and oriented. The a column shows plate shaped elements
for the longitudinal effect. The b column shows plate shaped elements for the
sheet effect. The c columns shows rod shaped elements for the transverse
effect. The d column shows elements in the shape of a hollow cylinder or a
truncated cone. Such elements can only be made of piezoelectric ceramics.
They can be polarized either radially for the longitudinal effect or in axial direction
for the sheer effect. The e column shows bimorph elements as bending beams
Figure 2 – Piezoelectric Loading [1]
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(exploiting the transverse effect). The f column shows torsion sensitive elements
(exploiting the shear effect). [1]
Piezoelectric materials also exhibit electrical properties. They have a
defined capacitance, resistance, and inductance, and therefore exhibit an
electrical resonance, where the electro-mechanical coupling peaks. These
characteristics are directly related to the area and the piezoelectric modulus e. [3]
Table 1 shows all of the common piezoelectric constants, with descriptions
of the constant and its units. These constants are used by manufacturers to
characterize their devices. In addition, these constants are used in models
governing the behavior of the piezoelectric devices. Using these models, the
output from piezoelectric devices can be accurately predicted, allowing
simulations to be completed. The most important parameters are the
piezoelectric constant “dij”, the coupling coefficient “kij”, and Young’s modulus “Ya
ij”.
Figure 3 – Various Piezoelectric Elements and their possible loadings [1]
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Table 1 – Piezoelectric Material Constants Piezoelectric
Constant Description
εr Relative Dielectric Constant • Used in calculating the capacitance of the material
tan δ Loss Tangent • A frequency dependent ratio between the real and parts
of the impedance of a capacitor. • A large dielectric constant implies a lot of dielectric
absorption [4] kij Coupling Coefficient
• A measure of the coupling between the mechanical energy converted to electrical charge, and the mechanical energy input
Di Electric Displacement (C/m2) • Charge Density
dij Piezoelectric Constant (C/N) • The piezoelectric charge coefficient is the ratio of electric
charge generated per unit area to an applied force [2] gij Piezoelectric Voltage Constant (Vm/N)
• The voltage constant is equal to the open circuit field developed per unit of applied stress, or as the strain developed per unit of applied charge density or electric displacement [5]
eij Piezoelectric Modulus (C/m2) • The ratio of strain to applied field, or charge density to
applied mechanical stress [5] Ni Frequency Constant (m/s)
• The frequency constant is the product of the resonance frequency and the linear dimension governing the resonance. [6]
Qm Mechanical Quality Factor • Measure of how well a system will resonate at or close to
its resonance frequency [7] ρ Density (kg/m3)
• Ratio of mass to volume σE Poisson’s Ratio
• A measure of how, when a material is stretched in one direction, it becomes thinner in the other two [8]
saij Elastic Compliance (m2/N)
• The inverse to Young’s Modulus • Ratio of mechanical strain to stress [6]
Ya ij Young’s Modulus (Pascals) • Ratio of mechanical stress to strain
*See fig 4 for explanation of “ a ”
8
Figure 5 shows a piezoelectric material loaded compressively through its
thickness. It is assumed that the material is cylindrical, and that it is being
compressed between two rigid masses. Equation 2 shows the stress component
exerted on the material [1]. Using this stress, and assuming the terminals are
short-circuited the electric flux density can be calculated using equation 3 [1].
Finally, the charge placed on the terminals can be calculated from equation 4 [1].
All of these calculations are based on the fundamental piezoelectric equation.
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KrFT⋅
=π
(2)
211
1111KrFdTdD
⋅=⋅=
π (3)
All strains in the material are constant or mechanical deformation is blocked in any direction.
All stresses on material are constant or no external forces.
Electrodes are perpendicular to 3 axes. Relative dielectric constant ($3s/$0).
Electrodes are perpendicular to 1 axis. Relative dielectric constant ($1T/$0).
Stress or strain is equal in all directions perpendicular to 3 axis
Stress or strain is in shear from around 2 axis.
Electromechanical coupling factor
Electrodes are perpendicular to 1 axis. Electromechanical coupling factor.
Hydrostatic stress or stress is applied equally in all directions. Electrodes are perpendicular to 3 axis (Ceramics).
Applied stress, or piezoelectrically induces strain is in 3 direction.
Piezoelectric charge coefficient.
Electrodes are perpendicular to 3 axis. Piezoelectric charge coefficient.
Applied stress, or the piezoelectrically induced strain in shear form around 2 axis.
Applied stress, or the piezoelectrically induced strain is in the 1 direction.
Electrodes are perpendicular to 1 axis. Piezoelectric voltage coefficient.
Electrodes are perpendicular to 3 axis. Piezoelectric voltage coefficient.
Compliance is measured with closed circuit.
Compliance is measured with open circuit.
Stress or strain is shear around 3 direction. Strain or stress is in 3 direction Elastic compliance.
Stress or strain is in 1 direction. Strain or stress is in 1 direction Elastic compliance.
Fig 4 – Piezoelectric Symbol Terminology [9]
9
FdQ ⋅= 11 (4)
Where:
T1 = Mechanical Stress
F = The force exerted on the material
rk = the radius of the material
D1 = Electric Displacement (or Charge Density)
d11 = Piezoelectric Constant
Q = Charge
Figure 6 shows a piezoelectric material which is loaded compressively
through its length. Again, the same equations apply, although instead of the
material being cylindrical, it is instead a rectangular bar. Equation 5 is used for
the stress component of the material [1]. Equation 6 is used for the flux density,
and equation 7 calculates the charge on the terminals [1]. Again, these
Figure 5 – Piezoelectric Material Loaded Longitudinally [1]
10
equations are a more specific application of the fundamental piezoelectric
equation.
baFT⋅
=1 (5)
2121 TdD ⋅= (6)
FaldF
abbldblDQ 12121 === (7)
Where:
T1, T2 = Mechanical Stress
F = The force exerted on the material
a = Width of the material
b = Height of the material
D1 = Electric Displacement (or Charge Density)
d12 = Piezoelectric Constant
Q = Charge
l = Length of the material
Figure 6 - Piezoelectric Material Loaded Transversely [1]
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All of these formulas are summarized by the diagram shown in figure 7.
As shown, one can see how all of the properties relate to one another. It is seen
how the electric field is derived by the stress T, or the strain S. From these
measurable quantities, it is also seen how all of the piezoelectric constants,
illustrated in Table 1, are interrelated. Using this information, a piezoelectric
material can be fully characterized for use under any situation.
Fig 7. – Relation of Piezoelectric Constants [10]
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Piezoelectric Materials
There are two main types of piezoelectric materials, crystalline materials
and ceramic materials. Crystalline materials, such as quartz, occur naturally.
They were found to exhibit piezoelectric properties as long as 100 years ago.
Recent advancements have yielded man-made materials that also exhibit
piezoelectric properties. These materials have begun to be used in many
applications, from sensor applications to powering remote electronics in areas
where other power sources are unavailable [1]. Figure 8 shows a
comprehensive list of general piezoelectric material applications.
An example of such an application is the use of sensors on bridges. For
older bridges, monitoring of modern loads on the bridge has become an
Fig 8 – Piezoelectric Material Application [1]
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important area of research. For existing structures, having to retrofit the structure
with wiring for a monitoring system is expensive and time-consuming [10]. Using
sensors powered by piezoelectric materials, which transmit their data using a RF
link in a burst at periodic intervals, the real time load and stresses on a bridge
can be determined. This capability enables an easy retrofit, and is a very cost-
effective way to monitor physical structures. [11]
Figure 9 shows a basic topology of this type of system. The mechanical
vibrational energy of the bridge is harvested, and used to power the onboard
electronics. These electronics include the sensors, an A/D converter, a
microcontroller for data processing, and the necessary RF devices to transmit the
data to a remote receiver.
Crystalline Materials
Crystalline materials were the first materials identified to exhibit
piezoelectric properties. These materials, particularly quartz, are found naturally,
Fig 9 – Piezoelectric Powered Wireless Sensor Array [11]
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especially in areas of the South Pacific [1]. Since these materials are crystalline,
they are especially sensitive to their cut and orientation, and they will exhibit
different piezoelectric properties depending on the crystal orientation [1]
Since the advent of piezoelectric sensors, the demand for quartz crystals
has outstripped the natural supply. Therefore it became necessary to develop
ways to artificially create the crystals. Methods were developed and many
Quartz piezoelectric materials today are grown artificially in autoclaves. It was
found that with a pressure between 1 and 2 kilo Bars and at a temperature of
between 350 to 450 OC, Quartz can be grown [1].
There are, however, problems with artificially creating quartz. One such
problem is the effect of twinning. This occurs when Quartz of two different
orientations intergrow. Twins can also form under loading, affecting the
piezoelectric coefficient. It is best that this occurrence be avoided and must be
considered when designing, or designing with, piezoelectric materials [1].
Piezoelectric Ceramics
Another common group of piezoelectric materials other than quartz is a
ceramic material which has been developed more recently. Piezoceramic
materials are man-made, and come in many different types. These materials
exhibit high coupling coefficients, and are very flexible, so they are very suited to
custom applications. Another advantage of ceramic piezoelectric materials is
that since they are man made, they do not suffer from the problems which natural
materials have regarding scarcity, and crystal orientation. The ceramic material
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examined through the course of this research is lead-zitronite-titanate
(PbZrO3,PbTiO3), commonly referred to as PZT material.
The PZT materials can be manufactured into various subtypes. These
subtypes can be custom designed for different uses and environments. The sub-
type materials are created by doping, or introducing impurities into the materials.
This will change the physical properties, and by controlling the amount and types
of impurities introduced, the physical properties can be modified to the designer’s
needs. The subtypes are given a standard designation, such as PZT-5A. For
the research conducted at Villanova, the materials examined were PZT-5A and
PZT-8 [6].
Piezoelectric Material Comparison
There are many options for the use of piezoelectric materials. As
discussed, the main materials are the crystalline materials and the ceramic
materials. When designing a system, the engineer must choose between these
two options. Further, the designer must choose which material subtype to use.
Some general considerations of use are presented next.
Ceramic materials are much cheaper than crystalline materials to use.
You do not have to grow them, nor do you have to cut them properly. They are
widely available. Crystals can be rare, and finding the proper one for a specific
application can be difficult. Finally, the ceramic materials are usually much more
sensitive than the crystalline materials. [1]
Table 2 shows some of the typical piezoelectric constant values for some
common materials. The materials shown, from left to right are Tourmaline, a
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type of quartz material, Ceramic Multilayer Actuator – Ring (CMAR), which is a
piezoceramic manufactured by Noliac, and PZT type 5a and 8 materials, which
are manufactured by several companies.
Symbol Unit Tourmaline
(Quartz) Noliac CMAR PZT 5A PZT 8 ε3,r 7.5 1325.63 1875 1000 tan δ (3
X) 0.003 0.02 0.004 TC > ºC 330 370 300 kp 0.568 0.62 0.51 kt 0.471 0.45 0.45 k31 0.327 0.34 0.3 k33 0.684 0.67 0.64 k15 0.553 0.69 0.55 d31 C/N 3.40E-13 -1.28E-10 -1.76E-10 -9.70E-11 d33 C/N 1.83E-12 3.28E-10 4.09E-10 2.25E-10 d15 C/N 3.63E-12 3.27E-10 5.85E-10 3.30E-10 dh C/N 2.51E-12 7.24E-11 5.80E-11 3.10E-11 g31 V m/N -1.09E-02 -1.10E-02 -1.09E-02 g33 V m/N 2.80E-02 2.57E-02 2.54E-02 g15 V m/N 3.89E-02 3.82E-02 2.89E-02 Np m/s 2209.94 2000 2340 Nt m/s 2038.09 1940 2060 Nc m/s 1015.41 930 1070 Qm,t 372.71 60 1000 ρ kg/m3 3100 7700 7750 7600 s11
E m2/N 3.85E-12 1.30E-11 1.67E-11 1.15E-11 s12
E m2/N -4.80E-13 -4.35E-12 -5.20E-12 -3.60E-12 s33
E m2/N 6.36E-12 1.96E-11 1.72E-11 1.35E-11 s66 m2/N 8.66E-12 3.47E-11 4.37E-11 2.83E-11 s11
D m2/N 1.16E-11 1.50E-11 1.01E-11 s12
D m2/N -5.74E-12 -7.10E-12 -4.80E-12 s33
D m2/N 1.05E-11 9.40E-12 8.50E-12 Y11
E GPa 76.93 61 87 Y33
E GPa 50.92 53 74 Y11
D GPa 86.16 69 99 Y33
D GPa 95.61 106 118
Table 2 – Common PZT Material Properties
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There are drawbacks to the ceramic materials. For some PZT materials,
their sensitivity can degrade over time, an effect called “aging”. For applications
where consistent and reproducible measurements are necessary, such as
sensors, this is a most undesirable trait [1].
These materials usually have very high temperature sensitivity, making
their thermal operating range very limited. This makes these materials
unsuitable for more extreme environments, especially high temperature ones. At
high temperatures the properties of these materials change and tend to degrade
as temperature increases. This change becomes complete when the ambient
temperature increases to the Curie temperature of the material [8]. At this point,
the material will lose all of its polarization, losing its piezoelectric properties.
Typical Curie temperatures for PZT materials are on the order of 200 oC.
Finally, these materials are pyroelectric, so when being used in sensors,
noise will increase as their temperature increases. These materials exhibit a
lower resistivity than the quartz materials, which can be a potential problem for
designers. In sensor applications, a high resistance is needed in applications
where the measurand is quasistatic. [1]
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Experimental Test Materials
Initial testing of the piezoelectric devices consisted of tests conducted on
various sample materials supplied to the project. These materials came from
Advanced Cerametrics Incorporated (ACI), and Omnitek.
Advanced Cerametrics Incorporated
The materials from ACI were in the form of bare materials, i.e. the
piezoelectric ceramic materials themselves. They came in several varieties, and
several material subtypes. These varieties included the “soft” material, which is
loaded transversely, and also a “hard material”, which is loaded longitudinally.
For this project, the subtypes examined were the PZT 5a and PZT 8.
The materials examined were all ceramic materials, created artificially.
They came in two main types. ACI manufactures the actual piezoelectric fibers.
These are string-like materials that when subjected to mechanical stresses,
generate electricity. These string-like materials are then embedded into ceramic
materials that allow them to be used under various conditions.
The first type, referred to as the “hard” material, is a hard piece of material
which generates electricity when it is compressed. This material is shown in
figure 10. Figure 11 shows a technical drawing of the material, with the physical
Figure 10 – Hard PZT 5a Material
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dimensions of the material. The term hard does not refer to the piezoelectric
type, but merely to its physical characteristics. This material is one in which the
piezoelectric fibers are embedded along the vertical axis of the material. When
the material is compressed, the fibers are also compressed vertically, causing
electricity to be generated at the ends of the fibers, or the top and bottom plates
of the material.
The second material is referred to as the “soft” material. Figure 12 shows
a photograph of the soft material. Figure 13 shows a dimensioned technical
drawing of the soft material. It is called the soft material not because it is
sponge-like, but rather because it is flexible. Again, the term “soft material” does
Fig 11 – “Hard” Material Diagram and Loading; Units are in Inches
20
not refer to the piezoelectric type, but its physical characteristics. In this material,
the piezoelectric strands are oriented along the length of the material, so that
when the material is bent along its long axis, the strands are stretched, and
placed under tension. This action causes electricity to be generated.
The soft materials come in two different varieties, the regular material, and
also a bi-morph material. The bi-morph material is one in which two of the
regular soft test materials are placed in a sandwich, with a hard piece of material
in between. Essentially the device is two “soft” materials connected in parallel.
The middle material is much harder than the regular test materials, and since the
piezoelectric elements are bonded to it, a much higher output voltage is seen
from this device.
Figure 12 – Soft PZT Type 5a Material
Fig 13 – “Soft” PZT Loading Diagram
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Omnitek Incorporated
Omnitek is another company which manufactures piezoelectric devices.
However, they have taken a slightly different approach to the problem. Instead of
simply using a bare material, they have decided to construct a more complex
system with the hope of harvesting more energy. They designed and
constructed a mechanical resonator that, when subjected to acceleration, will
absorb and store the energy in a mechanical system. As that energy is released,
it is absorbed within the piezoelectric material and thus allows the generation of
electricity. A cut-through schematic diagram of the type 3000 resonator is shown
in figure 14.
Fig 14: Resonator Drawing
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Experimental Test Procedures
Loading Situations
There are many different ways in which these materials can be loaded.
Most of these devices work best when the piezoelectric material is loaded in
compression. Depending on how the material is physically constructed, when it
is subjected to compressive forces, it will generate electricity. The usual loading
is to somehow apply a force to the material, which compresses it. This is
accomplished by either placing a mass upon the top of the material and having
the system oscillate, or, another option is to have a moving mass impact the
material, yielding a higher force, albeit for a shorter time period.
The tests conducted on the piezoelectric materials were completed to
attempt to determine certain model parameters of the materials. It was
determined that the two best ways to test these materials were simple steady
state accelerations and high-G impact accelerations. By using a mass, placed
upon the top of the bare materials, any acceleration will compress the material,
and thus produce an output electrical current and voltage. The resonators,
however, do not have this problem, as they are already pre-loaded, and need no
external mass, only an input acceleration.
Steady State Tests
A vibration table manufactured by Bruel and Kjaer was used for the steady
state tests. The model number of the table is 4809. The table is driven by an
amplifier, also manufactured by Bruel and Kjaer, type 2706. The amplifier is
driven by an Agilent 33120A 15 MHz Function/Arbitrary Waveform Generator.
23
Measurements were taken with an Agilent 34401A 6.5 Digit Multimeter, an
Agilent 54622D Mixed Signal Oscilloscope, a B&K Charge Accelerometer, Type
4371 which was connected to a B&K Charge Amplifier Type 2635. This test
setup produces a maximum acceleration of approximately 15 g’s, depending on
the oscillating frequency.
The Agilent multimeter was used to take single readings, such as the DC
voltage on a capacitor, the amplitude of an RMS voltage signal, etc. The majority
of the measurements were made with the oscilloscope. The voltage from the
material was fed into the oscilloscope on channel 2, and the charge amplifier
output was fed into channel 1. This allows the user to correlate the output, and
extract the exact real-time acceleration, and the output from the material. The
oscilloscope is connected to a computer through a GPIB link, which allows all the
data from the oscilloscope to be downloaded straight into Microsoft Excel. This
allows for much easier data processing and analysis. The basic setup for the
shaker table is shown in figure 15.
Figure 15 – Steady State Vibration Table Setup
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The shaker table is a variable amplitude and frequency shaker table. It
allows the user to vary its amplitude from 0 to 15 g’s, and frequency from
approximately 1 Hz to 50 kHz. This is useful, because it allowed the testing of
the linearity of the material, along with facilitating easy examination of
mechanical resonances. The table is very similar to that of the core of a
loudspeaker. It consists of a metal magnetic core which has a coil of wire wound
around it. When AC current from an amplifier is passed through it, it causes the
magnetic core, and thus the table to vibrate. The source signal is provided from
a conventional signal generator. A close up of the table itself, with the soft
material mounted, is shown in figure 16.
The way that a material is mounted on the table depends on the way in
which it is to be excited. A compressive material, such as the hard PZT should
be mounted with a mass above and below it so that as it oscillates, the masses
will compress the material, thus yielding better output. Materials such as the soft
PZT and the Bi-morph material are mounted in a cantilever position, so that they
Figure 16 – Soft PZT Material mounted on table
25
may oscillate freely. Materials such as the resonator will oscillate freely as long
as they are placed under acceleration, therefore no masses are needed.
Drop Tests
The drop test is used to provide very high impact accelerations, greater
than 15,000 g’s. The drop platform itself is approximately 2 meters tall, and is
constructed of two steel plates, one which acts as the base plate, and one which
acts as the drop plate. From the base plate, two polished metal pipes extend,
and are held in parallel by a wooden clamp at the top. The second steel plate
rides on these pipes using two ball bearing sleeves.
Depending on what material is being examined, the mounting and use of
the drop test can vary. For a material such as the hard PZT type, for optimal
output, the materials should be compressed. To facilitate this, the material is
sandwiched between the drop plate and the bottom plate.
The impact force is applied through a cushion, which is designed to
extend the force of the impact to better resemble the acceleration curve
experienced in the gun environment. To do this, some of the acceleration
amplitude must be sacrificed, but since it is known that piezoelectric devices are
linear, it is easy to extrapolate the lab output to that of the gun environment. The
cushion is positioned between the base plate and the drop plate. If the test
material must also be placed between the base plate and the drop plate, the
cushion is placed on top of the material.
The cushion is made of rubber, layered, and held together with electrical
tape. Many materials such as wood, foam, and wax were examined to be used
26
as cushions. These materials were found to dampen the acceleration to a point
where it became useless. The rubber material acted more like a spring, storing
and real easing the energy of the drop, effectively reproducing the acceleration
curve.
Measurements were taken with an Agilent 54622D Mixed Signal
Oscilloscope and a B&K Charge Accelerometer, Type 4371 which was
connected to a B&K Charge Amplifier Type 2635. The data was collected from
the oscilloscope using the GPIB interface, and downloaded into Microsoft Excel
for processing. Channel one from the oscilloscope was the acceleration data,
and channel two was the voltage data from the device. This method of data
retrieval allows for easier processing of the data.
Figure 17 shows the accelerometer, drop platform, and measurement
equipment. It should be noted that the test material in this photo is mounted on
top of the drop plate, although during testing, it resided in between the plates.
Shown on the lab bench from right to left are the:
• Personal computer used to collect the data
• Agilent 54622D Oscilloscope
• Equipment rack containing from top to bottom the:
o Agilent 34401A Multimeter
o Agilent 33120A Function Generator
o Agilent E3631A DC Supply
• Type 2635 Charge Amplifier
27
When drop testing the resonator, the optimal method to excite the device
is to apply acceleration to it. This action will set the resonator into oscillation,
thus producing a voltage output. This is easy to do by simply mounting the
resonator to the top of the drop plate. When dropping it, the impact of the plate
will provide an acceleration to excite the resonator. This will set the resonator
oscillating, and thus generate a voltage output. The cushion is still used to
provide the same acceleration curve as with the hard material.
For the soft materials, such as the soft PZT and the Bi-morph material, the
material is mounted in a cantilever position off of the edge of the drop plate, and
is allowed to oscillate freely. This was found to provide the best output from the
device. Again, the cushion is used to generate an acceleration curve similar to
the gun environment.
Figure 17 – Drop Test Setup
28
Impedance Tests
One final test completed is an impedance test. Peak power transfer is a
basic principle of electrical engineering. It states that when the load impedance
is equal to the source impedance, the maximum power will be transferred from
the source to the load. Since this is an energy harvester device, the goal is to
get the most energy out of the materials. To test the impedance of the source
material, the material was first excited at its mechanical resonance. The
mechanical resonance was determined using the steady state test procedure.
Then, for a steady state input at resonance, the load impedance was changed,
and the power was calculated form the measured voltage and the load
impedance. The plots were examined, and the impedance where the highest
power output occurred was used for further tests.
The impedance tests were conducted on the same test apparatus as the
steady state test. A variable resistor box, in addition to a breadboard setup was
used to vary the resistive load in an easy increment. The breadboard was
attached to the piezoelectric material. The variable resistor box was connected
in series with a fixed resistor on the breadboard. This allowed the user to move
the sweep range of the resistor box to anything they chose by placing any
through-hole resistor in series with the variable resistance box.
The results were recorded in Microsoft Excel by inputting the load voltage
readings taken on the Agilent 34401A multimeter, and the power was
automatically calculated and plotted. The best power transfer occurred at 45 kilo
ohms. This is the load that was used in the steady state and drop tests. Figure
29
18 shows a simple schematic diagram showing all of the components used in the
tests, and how they are all interconnected.
Fig 18: Impedance Test Schematic Diagram
30
Experimental Results and Discussion
PZT Material
Hard PZT 5a
The hard material is comprised of strands of PZT fiber, which are
embedded into a ceramic material. The fibers are oriented along the short
dimension of the material, which is therefore called the height. To excite the
fibers, the material must be compressed, which is difficult to do under certain
loading situations. One of these samples was encased in a low temperature
jeweler’s wax to better enhance the survivability of the material.
The raw wax was placed into a beaker and melted using an oven. A mold
was created from wood, lined in plastic, and used to cast the material. The mold
was .2032 meters long, .0508 meters wide, and .0195 meters deep. The mold
was placed on its end, the material was suspended inside it, and the molten wax
was poured in around it. After allowing it to properly cure, the mold was cracked
open, and ready to be used.
Drop Tests
The material used in the drop test was the one which had been encased in
jeweler’s wax. It was placed in between the drop plate and the bottom plate, so
that when the plate falls, the entire mold is compressed, and thus the material is
also compressed. For these tests, the rubber cushion was placed on top of the
mold. The results are shown in figure 19. An acceleration plot, recorded from
the Bruel & Kjaer accelerometer is shown. It is easy to see the direct correlation
between acceleration and output voltage. As the acceleration increases, the
31
output voltage is seen to correspondingly increase. Once the acceleration
reaches a certain point, the output voltage plateaus. This phenomenon is
probably due to a maximum compression of the material.
Steady State tests
The steady state tests for this device demonstrate the resonance of the
device. The material was sandwiched in between two thin metal plates which
were fastened to the top of the shaker table using two screws. It was then
excited and the output was recorded. The mechanical resonance for this
material was identified using the shaker table. By completing a frequency
sweep, it was found that it resonated at approximately 240 Hz. The load when
completing these tests was approximately 30 kilo ohms. This load was chosen
Gun Barrel Comparison
-2000
200400
600800
10001200
1400
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Time (S)
Acce
lera
tion
(G's
)
Cushioned Drop TestNormalized Gun Barrel Data
Ha rd PZT 5a Volta ge vs Tim e
0255075
100125150175200
0 0.005 0.01 0.015 0.02
Tim e (s )
Volta
ge (V
)
Figure 19 – Acceleration and Rectified Voltage Plots for Hard PZT Drop Test
32
arbitralally, and was done so with the consideration of not to load the material too
heavily. The results of the steady state tests are shown in figure 20.
Impedance Tests
The impedance tests were conducted on the same test apparatus as the
steady state test. The plot shows the power transfer curve below, and the
maximum power transfer is observed to occur at 800 kilo ohms. This is the load
that was used in the drop tests conducted previously. The results of the
impedance tests are shown in figure 21.
Power v. Frequency DC
0.000E+002.000E-104.000E-106.000E-108.000E-101.000E-091.200E-091.400E-091.600E-09
50.00 150.00 250.00 350.00 450.00
Frequency (Hz)
Pow
er (W
)
Figure 20 – Resonance Frequency Sweep, Hard PZT Material
Im p e d a n c e M a tc h in g y = -6 E -3 3 x 4 + 9 E -2 7 x 3 - 6 E -2 1 x 2 + 5 E -1 5 x + 7 E -1 1R 2 = 0 .9 9 7 4
0 .0 0 0 E + 0 0
5 . 0 0 0 E -1 0
1 . 0 0 0 E -0 9
1 . 5 0 0 E -0 9
2 . 0 0 0 E -0 9
2 . 5 0 0 E -0 9
0 2 0 0 0 0 0 4 0 0 0 0 0 6 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0
R e sista n c e (O h m s)
Pow
er (W
atts
)
Figure 21 - Hard PZT 5a Power Transfer Curve
33
Soft PZT 5a
The soft PZT material is the same basic PZT material which is used in the
hard PZT material, but instead of being embedded in a ceramic, it is woven into a
strip which is very flexible. The whole weave is then embedded into plastic for
durability reasons. Since again the material is a piezoelectric fiber, it must be
excited along its length, mainly by placing it in either tension or compression. To
accomplish this, the material must be stretched. This can be done by bending
the material. Because of the soft plastic casing, it is very easy to stretch and
bend the material.
Drop Tests
When this material was tested, it was mounted in a cantilever position. In
this position, it can freely oscillate, and therefore generate electricity. It was
attached to the drop plate, and the end of the material was hung off of the plate.
It was attached with the contacts on the mounted side, so that the wires and clips
would not tear free of the material on impact. The cushion was again used,
being placed in between the drop plate and the base plate, and the test were
completed. The output from the test was recorded and is shown in figure 22.
34
Steady State tests
The soft PZT material was also tested under steady state conditions. The
material was mounted in much the same way as it was for the drop tests. It was
fastened near the terminaled end on the shaker table, and the majority of the
material was allowed to oscillate freely. It was mounted in between two thin
metal plates, which were attached to the table with screws. The advantage to
this test is to examine mechanical resonances. For the soft PZT material, it was
found that the material had a mechanical resonance of approximately 34 HZ.
Gun Barrel Comparison
-50
0
50
100
150
200
250
300
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005
Time (S)
Acce
lera
tion
(G's
)
Cushioned Drop TestNormalized Gun Barrel Data
Soft PZT Voltage vs. Time
-0.5
0
0.5
1
1.5
2
0 0.002 0.004 0.006 0.008 0.01
Time (s)
Out
put (
V)
Figure 22 – Soft PZT Drop Test Output
35
This value is logical due to the lack of mechanical rigidity in the material. The
results of these tests are shown in figure 23.
Impedance Tests
The impedance tests were conducted on the same test apparatus as the
steady state test. Using the procedure outlined in the test procedure section, the
maximum power transfer was observed to occur at 45 kilo ohms. This is the load
that was used in the steady state and drop tests.
Linearity Confirmation
One other test series conducted was to validate the linearity of the
piezoelectric materials. Since testing could only be completed in a low g
environment, the entire set of test data must be drastically extrapolated to
determine material output in the gun environment. Therefore the linearity of
these devices is extremely important.
In order to validate the linearity of the material’s voltage output versus
acceleration, the amplitude of the acceleration applied to the shaker table was
Power v. Frequency
2E-11 4E-11 6E-11 8E-11 1E-10
1.2E-10 1.4E-10
0 100 200 300 400 500 Frequency
Power
Figure 23 – Resonance Frequency Sweep, Soft PZT Material
36
varied and the output voltage was recorded. The test results are shown in figure
24 and it is clear that there is a linear trend to the data.
Bi-Morph Materials
The bi-morph material is two soft PZT materials which are sandwiched
together with a shim in between. The two paralleled PZT materials are similar to
the soft PZT material discussed previously. The shim can be anything, but
usually is some kind of bendable metal. The PZT materials are bonded to the
shim, so that when the sandwich is bent, the PZT materials are actually
stretched, and thus produce a higher output voltage. These materials usually
have a much higher output than a single PZT material due to their construction.
For this reason, they were chosen for examination for this project.
Drop Tests
Shown in figure 25 are the results for the bi-morph drop tests. The
material was attached to the drop platform in a cantilever fashion and the tests
Soft PZT Acceleration vs. Voltage Output
y = 0.0116xR2 = 0.9448
0
2040
60
80
100120
140
0 2000 4000 6000 8000 10000 12000
Acceleration (g's)
Ope
n Ci
rcui
t Vol
tage
(V)
Figure 24 - Linearity Confirmation
37
were completed using the drop test procedure. The acceleration curve closely
matches the normalized gun barrel data. Characteristic of the bi-morph material,
the output voltage is very high, greater than 150 volts.
Steady State tests
This material was tested in steady state acceleration, and was found to
have a very high output voltage, greater than 50 volts open circuit. It was
mounted in the same cantilever fashion as the soft PZT material.
Impedance Tests
The impedance tests were conducted on the same test apparatus as the
steady state test. Figure 26 shows the power transfer curve and the maximum
Gun Barrel Comparison
-5000
50010001500200025003000350040004500
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005Time (S)
Acce
lera
tion
(G's
)
Cushioned Drop TestNormalized Gun Barrel Data
Voltage v. Time
-50
0
50
100
150
200
0 0.001 0.002 0.003 0.004 0.005
Time (S)
Vol
tage
(V)
Figure 25 – Bi-Morph Drop Test Results
38
power transfer is seen to occur at 28 kilo ohms. This is the load that was used in
the drop tests conducted above.
Resonators
The resonators examined were manufactured by Omnitek. They use the
CMA-R type 3 PZT material manufactured by Noliac (see table 2 or appendix 2).
This material is machined and cut into a thin ring shape. Internally it is
comprised of 24 thin stacked piezoelectric rings, each connected in parallel. It is
mounted in between a mechanical mass spring resonator, and the housing for
the resonator. When excited, the mass-spring will start to oscillate. While
oscillating, it will compress the PZT material, causing it to generate electricity.
Drop Tests
The resonator was mounted to the top of the drop plate. When dropped,
the mass inside experiences the same acceleration that is measured by the
accelerometer on the plate. This measurement can then be used to model the
Impedance Matching
y = -5E-26x4 + 1E-20x3 - 1E-15x2 + 3E-11x + 4E-07R2 = 0.9869
6.000E-07
6.200E-07
6.400E-07
6.600E-076.800E-07
7.000E-07
7.200E-07
7.400E-07
0 20000 40000 60000 80000 100000
Resistance (Ohms)
Pow
er (W
atts
)
Figure 26 – Bi-Morph Impedance Test
39
operation of the device, and predict its voltage output for various loading
scenarios. The results for the drop tests, the acceleration curve, and the voltage
output, are shown in figure 27.
Steady State tests
Steady state tests were also conducted on these materials. These tests
consisted of mounting the resonator and an accelerometer onto the top plate of
the shaker table. This table then excited the resonator into oscillation, and the
output voltage was measured. Since the resonator was designed to oscillate, the
peak output frequency was known, and was seen to be about 1300 Hz, as
Resonator Acceleration vs. Time
-200
-100
0
100
200
300
400
500
0.3 0.32 0.34 0.36 0.38 0.4
Time (s)
Acce
lera
tion
(g's
)
Resonator Open Circuit Voltage vs. Time
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.3 0.32 0.34 0.36 0.38 0.4
Time (s)
OC
Vol
tage
(V)
Figure 27 – Resonator Drop Test Output
40
expected. At this frequency the impedance tests were conducted. The
mechanical resonator’s resonance frequency was also examined and verified.
This was done by placing the resonator on the table, and scanning the
frequencies until the peak response was found.
Impedance Tests
The impedance tests were conducted on the same test apparatus as the
steady state test. A variable resistor box, in addition to a breadboard setup, was
used to vary the resistive load in a constant increment. The results were
recorded in Microsoft Excel, and the power was automatically calculated and
plotted. The plot shows the power transfer curve below, and as shown, the
maximum power transfer occurred at 3.5 Mega-ohms. This is the load that was
used in the tests conducted above. The results from these tests are shown in
figure 28.
Resonator Impedance Matchingy = -5E-36x4 + 7E-29x3 - 4E-22x2 + 1E-15x + 3E-11
R2 = 0.9971
0.000E+00
2.000E-10
4.000E-10
6.000E-10
8.000E-10
1.000E-09
1.200E-09
1.400E-09
0 1000000 2000000 3000000 4000000 5000000 6000000
Resistance (Ohms)
Pow
er (W
atts
)
Fig 28 – Resonator Impedance Matching at 1300 Hz
41
Calculations and Final Results
Table 3 shows the summary of the piezoelectric values calculated for the
different materials studied. These values were calculated by using the measured
acceleration, the mass of the mass spring resonator and the maximum output
voltage measured. The electric field was calculated by using the voltage
measured and the thickness of the piezoelectric material assuming a uniform
field.
The first step was to calculate the stress on the material. This was done
by calculating the area of the material, and then calculating the compressive
force. In the case of the resonator, the compressive force was calculated by
multiplying the mass of the resonator by the peak acceleration measured. Then
the total area of the material was calculated either using the inner and outer radii
of the material for the ring materials, or the length and width for the rectangular
materials. The stress was calculated by dividing the force by the area.
For the hard PZT material, the material was compressed between the
drop plate and the bottom of the drop platform. When calculating the
compressive force, the mass of the plate was multiplied by the acceleration of the
plate. For calculating the area, the way in which the material was tested must be
accounted for. There were two different tests completed, one in which the
material was bare, and another in which the material was encapsulated in
jeweler’s wax. For the bare test, the area is simply the length of the material
multiplied by its width. For the encapsulated test, the area is the length of the
encapsulated material, multiplied by its width. This is different from the bare test
42
because in the cast material, the force is spread over the total area of the
casting, not just the material itself. The stress was then calculated by dividing
the force by the area.
The next step was to calculate the strain on the material. This was done
using Young’s modulus of elasticity. The value of Young’s modulus was taken
from the material data sheets. Since Young’s modulus is the ratio of stress to
strain for a particular material, it is easy to calculate the strain on a material when
the stress is known. This was completed for all the materials examined, and
used further in the calculations. Please see appendix 1 for these calculations.
With the stress and strain known, it is easy to calculate the rest of the
parameters. As shown, all the parameters have the “33” designation on them,
meaning that the mechanical excitation is in the vertical axis, as is the
measurement of the electrical response.
The first parameter calculated was the piezoelectric modulus. This was
calculated by using the formula shown in eq 8. The next step is to calculate the
electric field, this is shown in eq. 9. Next calculated was the voltage constant.
This was calculated as shown in eq 10. The next constant calculated was the
piezoelectric constant, which is shown in eq 11. Finally, the elastic compliance
was calculated as shown in eq. 12.
33
3333 E
Te = (8)
hVE 3
33 = (9)
43
3333
3333
1eT
Eg == (10)
33
3333 E
Sd = (11)
33
3333 T
Ss = (12)
Table 3 shows the final results calculated for the materials examined. As
shown, the first row of values is for the Noliac material. The percent errors for
this material are substantially higher than those for the other materials. This is
attributed to how the compressive force for this material had to be calculated
using the equivalent mass for the material, rather than making a more direct
measurement. The best result is the d33 result. It is shown to have a 44 %
error. However the calculated result is on the same order of magnitude as the
expected result.
The second and third rows are for the PZT 5a material. The second row
gives the results for the bare material, while the third row gives the results for the
encapsulated material. The results for the bare material are seen to be accurate,
with the piezoelectric constant showing less than a 3 % error. The results for the
encapsulated material are less accurate, but are still on the same order of
magnitude as the expected results.
44
Overall, these tests confirmed the expected values of the measured
parameters. Accuracy could have been further improved with better
measurement equipment, more controlled tests, or a better test procedure.
Another major factor in these tests were simply the results being measured. The
high impact accelerations and high voltages allowed for great variation in the test
data, mainly in high frequency oscillation seen on the measured acceleration
curves. In spite of these obstacles, the expected values were still confirmed, as
shown.
CMAR Measured Expected %Error e33 10.21 16.0 36.27%g33 0.0979 0.0267 267.22%d33 2.38E-10 4.25E-10 44.11%s33 2.33E-11 2.32E-11 0.24% Bare PZT 5a Measured Expected %Error e33 42.06 g33 2.38E-02 2.57E-02 7.49%d33 3.97E-10 4.09E-10 2.98%s33 9.43E-12 9.40E-12 0.36% Cast PZT 5a Measured Expected %Error e33 43.86 g33 2.28E-02 2.57E-02 11.29%d33 4.14E-10 4.09E-10 1.17%s33 9.43E-12 9.40E-12 0.36%
Table 3 – Calculated Piezoelectric
Parameters
45
Conclusions
In conclusion, this report summarizes the work completed thus far on the
project. The calculations carried out, particularly those on the PZT 5a materials
conclusively showed that the mathematics behind these materials is very well
understood, and these materials respond to mechanical excitation in a
predictable way. One major goal, verifying the linearity of the materials, was
completed. This is critical in developing ways to design these materials to a
specific environment. The mechanical resonances were identified and explored.
These resonances were found to depend greatly on the material itself, and also
on how it is mechanically loaded. Finally, the peak power transfer of these
materials was identified and verified. This information will allow designers to
develop circuits which can harness the most power from these devices. Further
research of these devices will concentrate on the goal of developing models for
the materials, particularly the Noliac CMAR materials, and the mechanical
resonators.
46
REFERENCES
[1] Gautschi, Gustav. “Piezoelectric Sensorics”. Berlin: Springer – Verlag,
2004
[2] Phillips, James R. "Piezoelectric Technology Primer." CTS Wireless
Components: 17 pgs. 6 Dec 2005
<www.ctscorp.com/components/Datasheets/piezotechprimer.pdf>.
[3] Arnau, Antonio., ed. “Piezoelectric Transducers and Applications”. Berlin:
Springer – Verlag, 2004
[4] Dielectric Loss Tangents. Signal Consulting Inc. 6 Dec 2005
<http://www.sigcon.com/Pubs/news/4_5.htm>
[5] Limitations - Piezoelectric Tutorial 11 Morgan Electro Ceramics
Piezoelectric Tutorial 11. Morgan ElectroCeramics. 6 Dec 2005.
<http://www.morganelectroceramics.com/piezoguide18.html>
[6] Noliac.com. Noliac Inc. 6 Dec 2005
<http://www.noliac.com/index.asp?id=98>.
[7] Quality Factor. Ruye Wang. 7 Feb 2006
<http://fourier.eng.hmc.edu/e84/lectures/ch3/node9.html >
[8] What is Possions Ratio?. Rod Lakes. 7 Feb 2006
<http://silver.neep.wisc.edu/~lakes/PoissonIntro.html >
[9] Seacor Piezoelectric Ceramics – Definitions and Terminology. Seacor
Piezoelectric Ceramics. 8 Dec 2005
<http://www.seacorpiezo.com/defin_terms/symbol.html>
47
[10] Lin, C.H. et al. "Preliminary Attempt to Create a Smart Bridge Design and
Implimentation" National Taiwan University: 12 pgs. 6 Dec 2005
< http://www.engineering.ucsb.edu/~lin/publications/bridge.pdf>.
[11] Riedel, Tod. "Power Considerations for Wireless Sensor Networks."
Sensors Online Magazine (March 2004): 12 pars. 6 Dec 2005
< http://www.sensorsmag.com/articles/0304/38/pf_main.shtml>.
48
Appendix 1 – Young’s Modulus
AFT =
Where T is the stress, F the force, and A the area
StrainStressY =
Where Y = Young’s Modulus
Therefore:
YTStress =
49
For CMAR
OR 0.006 m IR 0.002 m Area 0.000101 m^2 Thickness 0.000067 m
Y31 43.00 GPa Compressive Force 15.32 Newtons
mass 0.00375 Kg Stress T 152438.11 Newtons/m^2 spring constant 2.00E+06 N/m Strain S 3.545E-06 Peak Acceleration 417 g's Peak Voltage 1 volts Peak electric Field 14925.37 V/M e33 1.02E+01 g33 9.79E-02 d33 2.38E-10 s33 2.33E-11
50
For the Bare PZT 5a
L 0.0762 m W 0.009525 m Area 0.000726 m^2 Thickness 0.00635 m
Y 106 GPa Compressive Force 807.6572 Newtons
Mass of plate 9.26 Kg Stress T 1112774 Newtons/m^2 Strain S 1.05E-05 Peak Acceleration 8.9 g's Peak Voltage 168 volts Peak electric Field 26456.69 V/M e33 4.21E+01 g33 2.38E-02 d33 3.97E-10 s33 9.43E-12
51
For the Cast PZT 5a
L 0.2032 m W 0.0508 m Area 0.010323 m^2 Thickness 0.00635 m
Y 106 GPa Compressive Force 11978.74 Newtons
Mass of plate 9.26 Kg Stress T 1160442 Newtons/m^2 Strain S 1.09E-05 Peak Acceleration 132 g's Peak Voltage 168 volts Peak electric Field 26456.69 V/M e33 4.39E+01 g33 2.28E-02 d33 4.14E-10 s33 9.43E-12
*please note, the thickness is not that of the casting, but that of the material, since it is not affected by the stress on the material
52
Appendix 2 – CMAR 3, S1 Material
Symbol Unit H1 S1 S2 ε1,r
X 1193.04 1795.99 2438.00 ε3,r
X 1325.63 1802.77 2874.16 ε1,r
S 828.30 1129.69 1341.00 ε3,r
S 699.67 913.73 1221.61 tan δ (3
X) 0.003 0.017 0.016 TC > ºC 330 350 235 kp 0.568 0.592 0.643 kt 0.471 0.469 0.524 k31 0.327 0.327 0.370 k33 0.684 0.699 0.752 k15 0.553 0.609 0.671 d31 C/N -1.28E-10 -1.70E-10 -2.43E-10 d33 C/N 3.28E-10 4.25E-10 5.74E-10 d15 C/N 3.27E-10 5.06E-10 7.24E-10 dh C/N 7.24E-11 8.50E-11 8.82E-11
g31 V m/N -0.0109 -0.0107 -0.0096
g33 V m/N 0.0280 0.0267 0.0226
g15 V m/N 0.0389 0.0373 0.0321
e31 C/m2 -2.80 -3.09 -5.06 e33 C/m2 14.7 16.0 21.2 e15 C/m2 9.86 11.64 13.40
h31 V/m -
4.52E+08-
3.82E+08-
4.68E+08 h33 V/m 2.37E+09 1.98E+09 1.96E+09 h15 V/m 1.34E+09 1.16E+09 1.13E+09 Np m/s 2209.94 2011.08 1970.47 Nt m/s 2038 1953 1966 N31 m/s 1500 1400 1410 N33 m/s 1800 1500 1500 N15 m/s 1018 896 822 Qm,p 776 89 76 Qm,t 373 74 195 ρ kg/m3 7700 7700 7460 σE 0.334 0.389 0.340 s11
E m2/N 1.30E-11 1.70E-11 1.70E-11 s12
E m2/N -4.35E-12 -6.60E-12 -5.78E-12 s13
E m2/N -7.05E-12 -8.61E-12 -8.79E-12
53
s33E m2/N 1.96E-11 2.32E-11 2.29E-11
s44E =
s55E m2/N 3.32E-11 4.35E-11 5.41E-11
s66 m2/N 3.47E-11 4.71E-11 4.56E-11 s11
D m2/N 1.16E-11 1.51E-11 1.47E-11 s12
D m2/N -5.74E-12 -8.41E-12 -8.10E-12 s13
D m2/N -3.47E-12 -4.08E-12 -3.30E-12 s33
D m2/N 1.05E-11 1.19E-11 9.94E-12 s44
D = s55
D m2/N 2.31E-11 2.73E-11 2.98E-11 c11
E N/m2 1.68E+11 1.47E+11 1.34E+11 c12
E N/m2 1.10E+11 1.05E+11 8.97E+10 c13
E N/m2 9.99E+10 9.37E+10 8.57E+10 c33
E N/m2 1.23E+11 1.13E+11 1.09E+11 c44
E = c55
E N/m2 3.01E+10 2.30E+10 1.85E+10 c66 N/m2 2.88E+10 2.12E+10 2.20E+10 c11
D N/m2 1.69E+11 1.49E+11 1.36E+11 c12
D N/m2 1.12E+11 1.06E+11 9.21E+10 c13
D N/m2 9.33E+10 8.75E+10 7.58E+10 c33
D N/m2 1.58E+11 1.44E+11 1.51E+11 c44
D = c55
D N/m2 4.34E+10 3.66E+10 3.36E+10 Y11
E GPa 76.93 58.98 58.82 Y33
E GPa 50.92 43.10 43.65 Y11
D GPa 86.16 66.04 68.13 Y33
D GPa 95.61 84.25 100.57
54
Appendix 3 – PZT Type 5a Material
55
Appendix 4 – PZT Type 8 Material
