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Ultrasonic pulse generation by apiezoelectric transducer at cryogenic
temperatures
By
MARTA GRANADOS JIMÉNEZ
SUPERVISED BY:JOSE EDUARDO GARCÍA GARCÍA
FRANCESC SUÑOL GALOFRE
Department of PhysicsUNIVERSITAT POLITÈCNICA DE CATALUNYA
A dissertation submitted to the Universitat Politècnica deCatalunya in accordance with the requirements of the degreeof ENGINEERING PHYSICS in the Escola Tècnica Superiord’Enginyeria de Telecomunacions de Barcelona.
JUNE 2018
ABSTRACT
The use of cryogenic liquid propellants such as liquid hydrogen or liquid oxygen has arisena problem for future space missions. As a result of long storage periods bubbles are formed inpropellant tanks due to boil-off. In order to eliminate these bubbles that can be harmful to theengines, it has been proposed an innovative cryogenic technique: the control of bubbles by meansof ultrasonic waves generated by piezoelectric ceramic transducers.
In this project, a simple approach for studying this issue has been proposed. A so calledtransducer-rod system is built to test the possibility of using piezoelectric ceramics to generateultrasonic pulses at cryogenic temperatures. This simple system consists in an Aluminum rodwith two transducers, made with piezoelectric ceramics, at each end of the rod. Once the systemis built, a pulse is sent to the source transducer, which will generate the ultrasonic signal sensedat the receiver transducer. From this, the propagation through the medium of the pulse generatedby the transducer has been studied. The measurements were obtained in a closed-loop cryogenicsystem. Liquid Helium is recirculated through all the system, cooling down a so called cold-finger.Through thermal contact the cold-finger is able to cool the cryogenic chamber to temperaturesaround T = 20 K. The results show that none of the tested piezoelectric ceramics is able to work atcryogenic temperatures of around T = 20 K. However, one of the piezoelectric materials availablewas able to preserve its properties until T = 200 K and to keep working until T = 130 K.
It can be concluded that our chosen piezoelectric ceramic is not able to work at cryogenictemperatures. However, other interesting results were achieved. The first result is that theresonance frequency of the studied ceramics does not depend on temperature. That means thereis no need of changing the initial pulse perturbation throughout the experimental process. Thesecond result is that the speed of sound in Aluminum slightly increases as temperature decreases,thereby confirming that Aluminum is a good material for ultrasound propagation at cryogenictemperatures. Lastly, the amplitude of the received pulse decreases as temperature decreases.With that last result, we can determine the lower working temperature for each piezoelectricceramic transducers.
i
DEDICATION AND ACKNOWLEDGEMENTS
I would like to express my gratitude to my project’s director, Jose Eduardo Garcia. He offeredme to join his group and introduced me to the world of piezoelectric materials and cryogenics.I also want to thank my co-director, Francesc Suñol, for always helping me to understand thephysics behind the experiments.
I would like to acknowledge all the group in the Department of Physics at Escola TècnicaSuperior d’Enginyeria de Telecomunicacions de Barcelona for welcoming me and treating melike one of the group. In particular, I would like to thank Diego Ochoa for all his help during thecryogenic part of the project.
Last but not least, I would like to thank my family and my friends for encouraging andsupporting me during this four years of the Engineering Physics Degree.
iii
TABLE OF CONTENTS
Page
List of Tables vii
List of Figures ix
1 Introduction 1
2 Theoretical background 32.1 Piezoelectric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Properties of piezoelectric ceramics . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Types of piezoelectric ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Piezoelectric transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Acoustic impedance, reflectivity and attenuation . . . . . . . . . . . . . . . . 8
2.3 Acoustic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Transducer waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Experimental signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Experimental setup 153.1 Shape and material selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Piezoelectric material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Material of the propagation medium . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.3 Shape of the medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Transducer-rod system build-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Conditioning of the sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Cryogenic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Results 254.1 Piezoelectric properties as a function of temperature . . . . . . . . . . . . . . . . . . 25
4.2 Resonance frequency study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Pulse at cryogenic temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Pulse amplitude as a function of temperature . . . . . . . . . . . . . . . . . . . . . . 31
v
Table of Contents
4.5 Resonance frequency as a function of temperature . . . . . . . . . . . . . . . . . . . 32
4.6 Pulse arrival time as a function of temperature . . . . . . . . . . . . . . . . . . . . . 33
4.7 Sound velocity in the medium as a function of temperature. . . . . . . . . . . . . . . 35
5 Conclusions 37
Bibliography 39
Appendix A 41Analytical solution of the acoustically-forced underdamped oscillator model . . . . . . . 41
Appendix B 43Measurements of the different configurations of the transducer-cylinder systems. . . . . 43
vi
LIST OF TABLES
TABLE Page
3.1 Piezoelectric materials properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Sound velocity, density and acoustic impedance of the different candidate materials. . 17
3.3 Times of pulse arrival at different lengths. Being tarrival the time from source to
receptor and t1st echo the time to travel back and forth. . . . . . . . . . . . . . . . . . . . 18
vii
LIST OF FIGURES
FIGURE Page
2.1 Crystal structure of a traditional piezoelectric ceramic . . . . . . . . . . . . . . . . . . . 4
2.2 Schematic representation of the piezoelectric effect. a) Direct piezoelectric effect.
Where σ is the applied stress that creates a current. b) Converse piezoelectric effect.
Where E is the applied electric field that creates an expansion or a contraction
depending on the sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Direction of forces affecting a piezoelectric material. . . . . . . . . . . . . . . . . . . . . 5
2.4 Schematic representation of the main parts of an ultrasonic transducer. . . . . . . . . 8
2.5 Propagation of an ultrasonic pulse through different mediums. . . . . . . . . . . . . . . 9
2.6 Comparison between the theoretical prediction (eq 2.15 and the experimental signal
obtained at the receiver transducer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Experimental signal obtained. At time ∆t arrives at the receiver transducer (blue) the
pulse and at time 2∆t the first echo arrives to the source transducer (red). . . . . . . . 14
3.1 Transducer-rod system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Chosen piezoelectric ceramics. Their dimensions are 6 mm × 6 mm × 1 mm for the
square one and D= 4.5 mm × 0.5 mm for the disc. . . . . . . . . . . . . . . . . . . . . . 17
3.3 Variety of cylinders tested. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Final form of the transducer-rod system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Sample holder final system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 a) Liquid Helium compressor. b) Cold-finger from below. . . . . . . . . . . . . . . . . . . 22
3.7 Cryogenic temperature cooling system. a) Chamber of the cryostat. b) Vacuum pump
system. c) Temperature controller. d) Pressure controller. . . . . . . . . . . . . . . . . . 23
4.1 Piezoelectric properties of the 855 piezoelectric ceramic as a function of temperature. 26
4.2 Resonance study of the different piezoelectric ceramics. In blue there is the electric
impedance and in red the phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Input pulses for the different piezoelectric ceramic systems. . . . . . . . . . . . . . . . . 29
4.4 Evolution of the pulse response of the piezoelectric transducers at different tempera-
tures. The arrows mark the time of the pulse arrival at each temperature. . . . . . . . 30
ix
List of Figures
4.5 Comparison of the pulse amplitude for the different piezoelectric ceramics. . . . . . . 31
4.6 Comparison of the resonance frequency as a function of temperature for the different
piezoelectric ceramics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 Comparison of the pulse arrival time as a function of temperature for the different
piezoelectric ceramics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.8 Comparison of the sound velocity in Aluminum as a function of temperature for the
different piezoelectric ceramics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1 Two Aluminum cylinders of L = 15 mm were tested with different glues. . . . . . . . . 44
2 This three measurement correspond to three Aluminum cylinders of L = 15 mm,
L = 100 mm, L = 200 mm. The transducers are attached to the cylinder with superglue
in the three cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 These three measurements correspond to three Aluminum cylinders of L = 75 mm
and diameters D = 10 mm, D = 15 mm and D = 20 mm. The transducers are attached
to the cylinder with superglue in the three cases. . . . . . . . . . . . . . . . . . . . . . . 46
x
CH
AP
TE
R
1INTRODUCTION
The future of space exploration missions resides in understanding the behaviour of cryogenic
liquids over long periods of storage. Cryogenic propellants such as liquid hydrogen and liquid
oxygen play a fundamental role. The feasibility of using liquid hydrogen or liquid oxygen engines
depends on the success of storing these propellants under microgravity conditions for extended
periods of time [1].
Cryogenic fluid management in microgravity provides several physical challenges such as
locating the liquid in the tank, heat and mass transfer at the tank walls, boiling, natural
convection, the dilution of gas into liquid phase and many more [2–5]. One of the biggest issues is
the loss of cryogenic propellant due to boil-off. However, loosing propellant is not the only problem,
the bubbles formed as a result of it are an issue too. These bubbles can generate foam structures
that can be hazardous in different stages of the flight phase such as operations in orbit.
Since the current heat insulation technologies are not able to prevent or even control boil-off
over long periods of times, other techniques are required to minimize the effects of boiling in
fuel tanks. A recently proposed strategy to overcome the boil-off drawbacks consists in the use
of acoustic fields for the control and elimination of bubbles. The force due to an acoustic wave
generated by a piezoelectric transducer should be able to detach the bubbles from the tanks walls
and move them to a subcooled liquid where they would collapse. Nowadays, this technique is
being studied in microgravity conditions and non-cryogenic temperatures. However, to be able
to function at space it first has to be tested at cryogenic temperatures similar to those at outer
space. So, the aim of this project is to begin with this challenging problematic faced at long space
1
Chapter 1. Introduction
missions and test the functionality of piezoelectric transducers at such temperatures.
As detailed later on, there are several types of piezoelectric materials. In the last years it has
been discovered that single-crystal piezoelectric materials exhibit good functionality at cryogenic
temperatures, which could mean a good behaviour as a transducer [6]. However, this project
is centered in the test of piezoelectric ceramics because single-crystal piezoelectric are a cost
prohibitive solution to this problematic. If it is proved that piezoelectric ceramics are able to work
at cryogenic temperatures it would mean a much affordable solution to the boil-off issue.
Despite the fact that there is a lack in experimental works focusing on the behaviour of
piezoelectric ceramics at cryogenic temperatures, several attempts have been performed with
the aim to find a material with high piezoelectric properties in a cryogenic environment. A
piezoelectric system was engineered to increase the intrinsic and extrinsic contributions to the
piezoelectric response at cryogenic temperatures by adjusting the Curie temperature [7]. For
that reason, this project will be centered in materials with low Curie temperature. It has to be
noted that a great piezoelectric effect was discovered in SrTiO3 (STO) single crystals at cryogenic
temperatures, but this effect only emerges for temperatures below 10 K [8]. The applications
studied require working at temperatures of liquid hydrogen (around 20 K), which make the STO
not adequate.
Due to the lack of potential industrial applications, a deep research to find an appropriate
piezoelectric material working at cryogenic temperatures has seldom been addressed. It has
been shown that a realistic application for space missions can justify the challenge of finding
such material. Theoretical and experimental studies that provide the key piezoelectric device to
perform an acoustic approach to control bubbles at cryogenic temperatures will be carried out in
this study. Another aspect that has to be taken into account in a near future is the content of toxic
elements (e.g Pb) that can become an environmental issue. This may require the replacement of
Pb-based materials in electronic components [9]. The Pb-free piezoelectric materials seem to be a
promising alternative since they have a low Curie temperature, however it has been shown in
previous projects that their properties decay considerably at cryogenic temperatures [10].
To sum it up, long term space missions will require an efficient control of boil-off in fuel tanks
at cryogenic temperatures. The use of the acoustic waves that piezoelectric materials can generate
is a promising technology system to prevent this issue. Although there is very little understanding
on materials showing a good piezoelectric response at cryogenic temperatures, the search of the
appropriate material for this application implies a challenge with the risk of not finding any
suitable material. This study is structured as follows: in Chapter 2, the theoretical background
needed to develop and understand the project is described; in Chapter 3 the experimental set-up
details are presented; in Chapter 4 the results obtained from different tests are exposed and
finally in Chapter 5 the conclusions of the project are reported.
2
CH
AP
TE
R
2THEORETICAL BACKGROUND
In this chapter, the key role of acoustic fields in the control of boil-off bubbles and the working
principle and parts of a piezoelectric transducer are discussed. Piezoelectric materials are the
basis of the transducers that generate the acoustic fields needed, therefore, the properties and
types of this materials are explained. In addition to that, the form of the resulting acoustic signal
is detailed.
2.1 Piezoelectric materials
Certain materials become electrically polarized when they are strained. This effect is called the
direct piezoelectric effect. It is experimentally observable by the appearance of bound electrical
charges at the surfaces of a strained medium. It is a linear phenomenon and consequently the
polarization changes sign when the sign of the strain is reversed.
Piezoelectricity is related to the microscopic structure of solids. The atoms of a solids and
the electrons within the atoms are displaced when the material is deformed. This displacement
produces microscopic electrical dipoles within the medium as we can see in figure 2.1. In certain
materials, these dipole moments combine to give an average macroscopic moment producing
electrical displacement [11].
3
Chapter 2. Theoretical background 2.1. Piezoelectric materials
Figure 2.1: Crystal structure of a traditional piezoelectric ceramic
The constitutive equations of piezoelectricity describe the lineal interaction between the elec-
tric and mechanic systems. Generally, the magnetic effects are not considered in the piezoelectric
phenomenon. Consequently, it is considered a piezoelectric material where the influence of the
magnetic field is negligible (H = 0). The equations that describe the piezoelectric effect can be
written as [12]:
(2.1) Dm = dTmiTi j +εT
mkEk ,
(2.2) Si = sE,Ti j Ti j +dT
mEm .
Where Dm is the electric displacement and Si is the deformation. Ti j corresponds to mechan-
ical strain that depends on the direction in which it is being applied and Em is the electric field
in the direction m. The indices take values i, j = 1..6 and m = 1,2,3.
From equation 2.1 it can be observed that for E = 0 a displacement appears when applying a
mechanical strain. This is known as the direct piezoelectric effect (2.3)
(2.3) Dm = dTmiT j .
From equation 2.2 it is observed that for Ti j = 0 deformations can be obtained by applying an
electric field. That is known as the converse piezoelectric effect (2.4)
(2.4) Si = dTi jm Em .
4
Chapter 2. Theoretical background 2.1. Piezoelectric materials
The direct piezoelectric effect is accompanied by the converse piezoelectric effect. A schematic
of the explained above can be seen in figure 2.2.
A A
DIRECT PIEZOELECTRIC EFFECT
σ
σ
σ
σ
EE
CONVERSE PIEZOELECTRIC EFFECT
EXPANSION CONTRACTION
a)
b)
Figure 2.2: Schematic representation of the piezoelectric effect. a) Direct piezoelectric effect.Where σ is the applied stress that creates a current. b) Converse piezoelectric effect. Where E isthe applied electric field that creates an expansion or a contraction depending on the sign.
2.1.1 Properties of piezoelectric ceramics
Piezoelectric ceramics are anisotropic, so their physical properties depend on the direction of
the applied mechanical or electric force. Usually, each constant has two subscripts that refer to
the direction of those quantities. Generally, the direction of the positive polarization is made to
coincide with the z-axis as can be seen in figure 2.3.
3
1
2
Polarization
Z
XY
6
5
4
Figure 2.3: Direction of forces affecting a piezoelectric material.
There are several physical constants that define the characteristics of a piezoelectric ceramic.
The most used ones are the following:
• Piezoelectric charge constant (d). It is the polarization generated per unit of mechani-
5
Chapter 2. Theoretical background 2.1. Piezoelectric materials
cal stress (T) applied to the ceramic or consequently, the mechanical strain (S) experienced
by the ceramic per unit of electric field applied. The most used is the d33 charge constant.
It represents the induced polarization in direction 3 per unit of stress applied in the polar-
ization direction. Conversely, is the induced strain in direction 3 per unit of electric field
applied in direction 3.
• Electromechanical coupling factor (k). It is an indicator of the effectiveness with
which a piezoelectric material converts electrical energy into mechanical energy or vice
versa. Even though a high electromechanical coupling factor would mean an efficient
energy conversion it does not take into account the dielectric or mechanical losses of the
piezoelectric ceramic.
• Piezoelectric voltage constant (g). It is the electric field generated by a piezoelectric
material per unit of mechanical stress applied or the mechanical strain experienced per
unit of electric displacement applied. The most common piezoelectric voltage constant is
d33 which measures the induced electric field in the polarization direction per unit of stress
applied in direction 3, or the induced strain in direction 3 per unit of electric displacement
applied in direction 3.
• Curie Temperature (Tc). It is the temperature at which piezoelectric materials lose their
permanent polarization. Hence, above the Curie temperature piezoelectric ceramics lose
their piezoelectric properties.
2.1.2 Types of piezoelectric ceramics
The vast majority of the piezoelectric ceramics used in industrial applications crystallize in
a perovskite structure. The most important materials due to its multiple applications have
been widely studied, they are: the barium titanate (BaTiO3) and the lead zirconate titanate
(PbZr1−xTixO3), known as PZT. The schematic crystalline structure of a traditional perovskite
piezoelectric ceramic is presented in figure 2.1. Barium titanate was the first piezoelectric ceramic
developed commercially for acoustic applications. Nowadays, it has been replaced by the PZT due
to its excellent properties.
In this study, we focus on PZT piezoelectric ceramics, since they are the most widely used
piezoelectric ceramic materials. The PZT is an oxide based piezoelectric material that in compari-
son to the other known oxide piezoelectric materials exhibits greater sensitivity. The PZT are
chemically inert and have a great piezoelectric coefficient, so it has a greater electric sensitivity
than other ceramics. Also, one of its attractive is that it can be manufactured at a very low price
compared to other types of piezoelectric materials such as single crystals.
6
Chapter 2. Theoretical background 2.1. Piezoelectric materials
It is possible to modify the physical properties of piezoelectric ceramics by adding dopants
to their crystalline structure. With this addition the dielectric, piezoelectric and mechanical
properties can be optimized for different applications. Depending on the valence of the ions the
dopants are classified in donors or acceptors (in analogy with the nomenclature used in semicon-
ductors). The principal changes produced in the PZT when adding dopants are a modification in
the dielectric constant, the electromechanical coupling factor and the piezoelectric coefficient.
When small amounts of donor dopants are added to a ceramic formulation, the effects of
extrinsic contribution on the piezoelectric properties are enhaced. As a result of donor dopant,
soft piezoelectric ceramics are obtained. They are characterized by large electromechanical
coupling factors, large piezoelectric constants, high permitivity, large dielectric constants and
low mechanical quality factors. Soft ceramics can produce larger displacements and therefore
wider signal bandwidths. Soft ceramics present lower Curie points, generally below 300°C and
are suitable for sensing applications. However, their drawback is that they present high electrical
and mechanical losses.
On the other side, when adding acceptor dopants in a ceramic formulation a decrease of the
dielectric constant, the piezoelectric coefficient and the mechanical coupling factor is observed.
The result of acceptor dopants are the hard piezoelectric ceramics that present opposite properties
to the ones of their soft counterparts. Their Curie point is usually above 300°C. Hard ceramics
are suitable for high mechanical loads and high voltages since they exhibit low mechanical and
dielectric losses.
PZT are classified as Navy types depending on their performance characteristics. That means
that a piezoelectric ceramic qualifies as a certain type not because of its chemical composition
but because of its physical and electrical properties. The Navy types are the following:
• Navy type I: Hard. Able to generate high amplitudes maintaining low mechanical and
dielectric loses.
• Navy type II: Soft. High sensitivity, has mechanical and dielectric loses at high voltages.
• Navy type III: Hard. Similar to Navy type I but has less sensitivity. Good at high voltages.
• Navy type IV: Soft. Low Curie temperature, good for medium voltage applications.
• Navy type V: Soft. An intermediate between Navy type II an Navy type VI. Good for high
energy applications.
• Navy type VI: Soft. Good for applications that require big mechanical deformations.
7
Chapter 2. Theoretical background 2.2. Piezoelectric transducer
2.2 Piezoelectric transducer
A transducer is a device that converts one form of energy into another. The case of our study is
centered in ultrasonic transducers which convert electrical energy into mechanical energy in
the form of an ultrasonic wave and vice versa. The main components are the active element, the
backing, the wear plate and the matching layer. In figure 2.4, a sketch of a typical ultrasonic
transducer is shown.
MATCHING LAY ER
ACTIV E ELEMENT
CONNECTORSBAKING
Figure 2.4: Schematic representation of the main parts of an ultrasonic transducer.
• Active element: is a piezoelectric material. It converts electrical energy into ultrasonic
energy. The most commonly used are polarized ceramics.
• Backing: is a highly attenuative material that is used to control the vibration of the
transducer by absorbing the energy radiating from the back face of the active element.
• Wear plate: is a layer used to protect the transducer element from the testing environment.
• Matching layer: a layer of a determined material to maximize the transition of energy to
the propagation medium.
2.2.1 Acoustic impedance, reflectivity and attenuation
In order to get a proper functional transducer, there are several factors that have to be taken into
account such as the acoustic impedance of the material, the reflectivity of the ultrasonic waves in
the medium and the attenuation due to the medium material.
The acoustic impedance is a property that measures the opposition of the material to the
acoustic flow. It is defined as the product of the material density and the speed of sound
(2.5) Z = ρc ,
8
Chapter 2. Theoretical background 2.2. Piezoelectric transducer
which units are Rayls ( 1 Rayl = 1 kgsm2 ).
The boundary between two materials of different acoustic impedance is called an acoustic
interface. When sound arrives at an acoustic interface at normal incidence, some amount of
energy is reflected and some is transmitted across the boundary.
The simplest situation, and the one discussed in this project, is the reflection and transmission
when waves are impinging normal to the surface. The situation can be described mathematically
in terms of three propagating waves: the incident wave, the reflected wave and the transmitted
wave (figure 2.5).
Z1
MEDIUM 1
Z2
MEDIUM 2
Incident wave (Ii)
Relfected wave (Ir)
Transmitted wave (It)
Figure 2.5: Propagation of an ultrasonic pulse through different mediums.
The acoustic wave equation describes the propagation of sound and is given by
(2.6)δ2 pδx2 − 1
c2δ2 pδt2 = 0 ,
where p is the sound pressure, x the particle displacement, c the speed of sound at the
medium and t is the time. Solving equation 2.6 for a travelling wave in two mediums one can
obtain the reflection and transmission coefficients in intensity [13]
(2.7) R =(
Z1 −Z2
Z1 +Z2
)2,
(2.8) T = 4Z1Z2
(Z1 +Z2)2 .
9
Chapter 2. Theoretical background 2.3. Acoustic fields
Here, Z1 is the acoustic impedance of the first medium and Z2 of the second. Therefore, when
there is no absorption of energy on the interface (the case of Z1 = Z2), R+T = 1. In the case in
which Z2 << Z1 then R → 1, T → 0. Hence, almost all of the energy of the wave is reflected and
therefore, there is no transmission.
The dB loss of energy in an acoustic interface for an acoustic wave travelling from medium 1
to medium 2 is given by
(2.9) dBloss = 10log10
[4Z1Z2
(Z1 +Z2)2
].
If the first and second medium had the same or a similar value of the acoustic impedance the
loss of energy would be negligible. Unfortunately, the piezoelectric material and the propagation
medium used in the experiments do not have the same acoustic impedance. This can become a
problem if the differences are large enough, since it will affect the transmission of the acoustic
wave. The acoustic coupling layer is used between the two materials to minimize the effect of the
change in the acoustic impedance.
To get a maximum transmission, a matching layer with a specific acoustic impedance is
needed
(2.10) Zmatchinglayer =√
ZpiezoZmaterial .
This value of the acoustic impedance for the matching layer, maximizes the energy transmis-
sion to the propagation medium.
Finally, an ultrasound pulse attenuates as it progresses through a medium. There are three
causes of attenuation: diffraction, scattering and absorption. Attenuation will play an important
role in choosing the medium material length.
2.3 Acoustic fields
Acoustics is the study of time-varying deformations or vibrations in material media. All material
substances are composed of atoms which can be forced into vibrational motion around its equi-
librium positions. Acoustic theory deals only with the macroscopic phenomena, the structure at
microscopic level only matters as it affects the medium’s macroscopic behaviour.
When particles of a medium are displaced from their equilibrium positions, internal restoring
forces arise. These elastic restoring forces between particles, combined with the inertia of the
10
Chapter 2. Theoretical background 2.3. Acoustic fields
particles lead to oscillatory motions of the medium. These vibrations on the medium can either
be traveling waves or localized vibrations and both are due particle displacement, material
deformation and internal restoring forces.
Acoustic waves are longitudinal waves that can exhibit phenomena such as diffraction,
reflection and interference. Ultrasonic waves do not have any polarization since they oscillate
along the same direction as they move. In this study, we are dealing with ultrasonic travelling
waves produced at the source transducer that travel through our chosen material and are received
at the receiver transducer.
The travelling wave produced by the piezoelectric transducer will depend on the material’s
speed of sound as stated in equation 2.6. Therefore, the material used to test our cryogenic
technique has to accomplish several conditions that will be later discussed in chapter 3.
2.3.1 Transducer waveform
Assuming that the piezoelectric material behaves as an underdamped oscillator, the signal
received at the receiver transducer is defined as the solution of a forced oscillator, in which the
external force is the acoustic wave travelling through the medium. The signal sent to the source
transducer is a rectangular pulse Π(t) with an amplitude Apulse and width tw. The form of the
acoustic signal is given by the solution of the following non-homogeneous differential equation.
(2.11) X +2ξωn X +ω2nX =Π(t),
where X is the form of the acoustic signal travelling through the medium, ωn = 2π fn with fn
being the natural frequency of the piezoelectric material and ξ is the damping coefficient. The
pulse sent can be written as a combination of Heaviside functions: Π(t)= Apulse [Θ(t)−Θ(t− tw)],
where Θ(t) is the Heaviside step function.
A solution of equation 2.11 can be written as [14]
(2.12) X (t)= Apulse [g(t)− g(t− tw)] ,
where
(2.13) g(t)=[1− e−at
(cos(bt)+ a
bsin(bt)
)]Θ(t) ,
11
Chapter 2. Theoretical background 2.3. Acoustic fields
with a = ξωn and b = ωn√
1−ξ2 . The equation 2.12 represents the acoustic signal that
travels through the medium and is received by the receiver transducer. The source and receiver
transducers are identical. Therefore, the form of the electric signal induced in the receiver will be
the solution of an underdamped oscillator under an external force driven by the acoustic signal
in equation 2.12. Then the equation describing the receiver transducer becomes
(2.14) Xr +2ξωn Xr +ω2nXr = X (t) ,
where Xr is the signal generated at the receiver transducer. The analytical solution of
equation 2.14 is detailed in Appendix A and can be written as
(2.15) Xr(t)= Apulse
b[T1(t)−T2(t)]Θ(t) ,
where T1(t) and T2(t) are given by
(2.16) T1(t)= e−at[
beat −asin(bt)−bcos(bt)a2 +b2
]− e−at
[tsin(bt)
2− a [sin(bt)−btcos(bt)]
2b2
],
(2.17) T2(t)= e−at [beat −asin(bt)−bcos(bt)]
a2 +b2 −
− e−a(t−tw)[
2btsin[b (t− tw)]+cos[b (tw − t)]−cos[b (tw + t)]4b
]−
− ae−a(t−tw) [cos(btw)sin(bt)−btcos[b (t− tw)]]2b2 .
In order to determine the validity of this approach, the solution (Equation (2.15)) has been
plotted and compared with the experimental received electric signal. The presented model takes
four independent parameters as input variables, set according to the transducer and electric
pulse characteristics. It has been studied the response signal to a rectangular pulse of width
tw = 140 ns and an amplitude of 2V in which the transducer resonance frequency is fn = 4.3
MHz. The damping coefficient has been set to ξ= 0.08. The comparison between the experimental
pulse and the theoretical one can be seen in figure 2.6. Good agreement is achieved despite the
simplicity of the model.
12
Chapter 2. Theoretical background 2.3. Acoustic fields
Figure 2.6: Comparison between the theoretical prediction (eq 2.15 and the experimental signalobtained at the receiver transducer.
2.3.2 Experimental signal
When exciting the source transducer with an electric pulse this will resonate and generate an
ultrasonic signal as seen on figure 2.7 that will travel through the medium. This signal will arrive
at the end of the medium and will be transmitted to the receiver transducer at a time ∆t = Lcmaterial
.
Since there is a an acoustic interface (medium-transducer), part of the received signal will be
reflected backwards towards the source transducer. This reflected signal will travel back to the
source transducer and will arrive at a time 2∆t.
This phenomenon was experimentally observed both at room temperature and also in a
cryogenic environment. In figure 2.7, the pulse sent by the source transducer, the signal at the
receiver and the echo (back to the source transducer) can be clearly distinguished.
13
Chapter 2. Theoretical background 2.3. Acoustic fields
∆t
2∆t
Figure 2.7: Experimental signal obtained. At time ∆t arrives at the receiver transducer (blue)the pulse and at time 2∆t the first echo arrives to the source transducer (red).
14
CH
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3EXPERIMENTAL SETUP
Once all the elements needed to carry out the different tests of our project are identified, it is
important to build the appropriate experimental set-up. In this Chapter all the decisions made
regarding the different types of materials, shapes and conditioning of the experimental system
are discussed.
3.1 Shape and material selection
In order to test the piezoelectric transducers at cryogenic temperatures a medium was needed
for the propagation of the ultrasonic pulse. The aim of the experiment is to test the system with
liquid hydrogen or liquid oxygen. However, working with cryogenic liquids can be challenging
and considerably expensive. Our first approach focuses on the propagation of ultrasonic waves
through the fuel tank (which is a solid medium), independently of the nature of the liquid fuel to
be used in a space mission. The schematic of our system can be seen in figure 3.1.
3.1.1 Piezoelectric material
As it has been said, the chosen type of piezoelectric material are soft piezoelectric ceramics. Even
though this type of material has the most promising characteristics for cryogenic applications
they provide large dielectric losses. Hence, soft piezoelectric materials cannot deal with high
currents. For that reason and as it will be detailed later, the initial perturbation sent to the
system is a rectangular pulse.
15
Chapter 3. Experimental setup 3.1. Shape and material selection
L
Figure 3.1: Transducer-rod system.
Table 3.1 shows the different properties of the three American Piezo (APC) materials available.
The three of them are soft piezoelectric ceramics being the 855 and 851 a Navy type VI and the
850 a Navy type II.
Physical property 855 850 851
YE11 (1010 N/m2) 5.9 6.3 6.3
YE33(1010 N/m2) 5.1 5.4 5.4ρ(g/cm3) 7.6 7.6 7.6c (m/s) 2786 2879 2879
Z (MRa) 21.1 21.8 21.8Tc (°C) 200 360 320
d33(10−12 m/V) 630 400 300
Table 3.1: Piezoelectric materials properties.
The chosen material is the 855 ceramic, as in figure 3.2, because it has the lowest Curie
temperature and the higher piezoelectric constant. However, the 851 ceramic will be tested
in order to check its properties at cryogenic temperatures, taking into account that it is more
affordable and common.
3.1.2 Material of the propagation medium
In order to choose the propagation medium material several factors had to be taken into account.
For getting an accurate result of our test, it is important that the characteristic of the medium
material do not change or slightly change as a function of temperature. If the material deterio-
rates as the temperature decreases, we could be seeing changes in the signal produced by the
degradation of material and not from the own deterioration of the piezoelectric transducer.
First of all, a study of several material properties at room temperature was carried out. In
table 3.2 is shown a comparison between them [15]:
16
Chapter 3. Experimental setup 3.1. Shape and material selection
6 mm4:5 mm
Figure 3.2: Chosen piezoelectric ceramics. Their dimensions are 6 mm × 6 mm × 1 mm for thesquare one and D= 4.5 mm × 0.5 mm for the disc.
Longitudinal velocity (m/s) Shear velocity (m/s) ρ (g/cm3) Z (MRa)
Al 6320 3130 2.7 17.06Steel 5890 3240 7.9 45.63Cu 4660 2260 8.96 41.61
Brass 4430 2120 8.64 37.3
Table 3.2: Sound velocity, density and acoustic impedance of the different candidate materials.
The chosen material was Aluminum for several reasons:
• It has an acoustic impedance more similar to the soft piezoelectric ceramics than any other
material considered.
• It has the lowest density, meaning that it weights less than any other material and will be
easier to cool.
• It is the most affordable material and easier to get than the others.
3.1.3 Shape of the medium
At first, the considered shape was a cylinder since any other shape such as rods or hexagons
could present unwanted geometrical edge effects. Cylinders of different diameters and lengths
were tested as it can be see in figure 3.3.
After running some tests at room temperature, the results were in accordance with the
predictions, even though the amplitude of the sent pulse was quite small. The diameter did not
change much the signal, so the chosen one was D = 15 mm since the majority of our piezoelectric
17
Chapter 3. Experimental setup 3.1. Shape and material selection
Figure 3.3: Variety of cylinders tested.
ceramics have a diameter of D = 10 mm. To choose the length of the cylinder it had to be taken
into account the size of the sample-holder, finally the chosen length was 75 mm. Moreover, this
length provides a good balance between the two arrival times; if the rod is too short, tarrival and
t1st echo are coupled, then the response signals could overlap each other (see figure 3.3). Even
though in comparison between the shortest and largest cylinder there is no much difference in
the pulse received, having a shorter length means making sure less attenuation will appear. On
the other side, if the rod is too large, it would not fit into the chamber of the cryostat and there
would be too much mass to cool down.
Length (mm) tarrival(µs) t1st echo(µs)
50 7.9 15.875 11.8 23.7
100 15.8 31.6150 23.7 47.4200 31.6 63.2
Table 3.3: Times of pulse arrival at different lengths. Being tarrival the time from source toreceptor and t1st echo the time to travel back and forth.
The difficulties in cooling down the system is another important factor. Arriving at cryogenic
temperatures with big systems is quite challenging since it could take up to 4-6h. It is crucial to
have a good thermal contact. Apart from that, reaching temperatures lower than 20 K is pretty
difficult. Between the cold-finger and the cryogenic chamber exists a temperature gradient that
18
Chapter 3. Experimental setup 3.2. Transducer-rod system build-up
cannot be eliminated. Moreover, there is always going to be heat radiation to the sample-holder.
That is why a good thermal contact with the platform of the sample-holder is so important. With
the cylinder, only a straight line was in contact with the sample holder. However, if using a rod, a
whole rectangular surface would be in contact with the sample holder, meaning a better heat/cold
transmission.
Although a rod could mean some geometrical edge effects, it provides a better thermal contact
than a cylinder, so allowing the system to thermalise better and faster. For all the stated reasons,
the chosen shape of the medium material was a rod of length L = 75 mm and width h = 15 mm.
At the end the chosen geometry for our medium is a rod, however, a lot of tests were made
with the cylinders. These tests helped to make the decisions regarding of the medium. These
results can be seen in appendix B.
3.2 Transducer-rod system build-up
As explained in Chapter 2, the transducer is formed with the active element (in our application
the 855 piezoelectric ceramic), the baking material and the matching layer. Knowing that the
acoustic impedance of the ceramic is Zpiezo = 21.1 MRa, it is important for a good transmittance
of the pulse that the baking has a different Z than the piezoelectric. The chosen baking is air, not
only for the simplicity but also for its acoustic impedance Zair = 4.2 ·10−4 MRa. Air meets the
requirement to avoid the transmittance of energy backwards, which is Zpiezo >> Zbaking.
The last parameter to be defined is the matching layer. In the system the matching layer
is not only going to be the layer protecting the piezoelectric ceramic but also the way to keep
the transducer and rod together. After testing different glue options, the resulting system was
figure 3.4. The reason of this build-up is due to the difficulties of finding a glue that can resist
cryogenic temperatures. Neither Superglue, conductive epoxies nor General electric glue can
preserve its properties at cryogenic temperatures thus altering our tests. The only way to fully
keep the transducers and the medium together throughout all the test was to mechanically attach
them. Also, between the transducer and the medium a semiconductor paste was placed in order
to avoid having air between them because that would mean that a big part of the energy would
be reflected to the transducer.
19
Chapter 3. Experimental setup 3.3. Conditioning of the sample holder
Figure 3.4: Final form of the transducer-rod system.
As it has been said, our system needs an air baking to prevent the transmittance of the wave
backwards. The fact that a mechanical system is needed to attach the transducers to the rod
may seem that there is no air left as baking. There is no substance between the mechanical
system and the transducer to avoid air, like the semiconductor paste between the transducers
and the rod. Consequently, there will always be a thin layer of air as baking enough to prevent
the backwards transmission.
3.3 Conditioning of the sample holder
Once the transducer-rod system is assembled, the next step is to find a way to fix it in the sample
holder of the cryostat. The sample holder is just a platform that can be fixed in the cryostat. Due
to the vibrations of the cryogenic system when its cooling down, the transducer-rod system could
fall off from it if not properly fixed to it. The chosen set-up for the sample holder is shown in
figure 3.5.
(a) Sample holder from above. (b) Sample holder from the side.
Figure 3.5: Sample holder final system.
Apart from fixing the transducer-rod system the sample holder has other functions:
20
Chapter 3. Experimental setup 3.4. Cryogenic system
• Fixing the connector wires from which the measures are going to be read.
• Fixing the temperature sensor of the cryostat to the platform to know the actual tempera-
ture of the system.
• Isolating the transducer-rod system from the radiation coming from the cryogenic chamber,
helping it to get cooler easily.
Working at cryogenic temperatures is quite challenging, not all materials are able to resist
such temperatures and something so simple as fixing a rod in a platform can become an issue.
The platform is made of a tough material that is why we were unable to drill a hole in it. The
first approach proposed was to fix the transducer-rod system directly at the platform with some
bridle. Apart from that, the radiation had to be taken into account. In the cryogenic chamber, the
platform is in direct contact with the cold-finger that cools down the whole system, therefore the
rest of the chamber is not at such temperatures. That explains why it is important to our system
to be in good thermal contact with the platform since the cold will be transmitted by conduction.
The complexity of our final system is able to firmly fix the samples to the platform without
the need of puncturing it. Not only is that important but it also isolates the system from the heat
radiation of the chamber.
3.4 Cryogenic system
The cryogenic cooling device is composed by different subsystems, as it can be seen in figure 3.7.
Our aim is to test the transducer-rod system at cryogenic temperatures of around 20 K. The
cryogenic chamber has a volume of around 8L, reaching cryogenic temperatures in such a large
chamber can be quite challenging and the procedure follows different steps. First of all, it is
crucial to create a good vacuum of around 10−6 mbar in the chamber. The aim of the vacuum
pump is to eliminate all particles in the chamber, including air particles since air is composed
of water vapour and it condenses at TH2O = 273 K. If some amount of water were found in the
chamber when decreasing the temperature, it would condense at TH2O and it would be really
hard to reach temperatures lower than that. Once the vacuum is the adequate, the next step is to
connect the compressor and its refrigerator.
The compressor will pump Helium gas, which is at a temperature of THe = 4 K, to the
refrigeration system of the chamber that can be seen in figure 3.6. It consists in a closed cycle
system in which the helium gas is recirculated in all of the circuit decreasing the temperature
in the so called cold-finger. Through thermal contact the decrease of temperature is going to be
transmitted to the bigger chamber (a) on figure 3.7, and therefore to the sample holder that is in
direct contact with the refrigeration system.
21
Chapter 3. Experimental setup 3.4. Cryogenic system
a)
b)
Figure 3.6: a) Liquid Helium compressor. b) Cold-finger from below.
22
Chapter 3. Experimental setup 3.4. Cryogenic system
a)
b)
c)d)
Figure 3.7: Cryogenic temperature cooling system. a) Chamber of the cryostat. b) Vacuum pumpsystem. c) Temperature controller. d) Pressure controller.
23
CH
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4RESULTS
After choosing and building the appropriate experimental setup, the different tests of the
transducer-rod systems were carried out. From that tests, several characteristics of the piezoelec-
tric materials and the medium can be calculated, such as the resonance frequency of the ceramic
material and the sound velocity of the propagating medium as a function of temperature. Apart
from studying the transducer-rod system, the characteristics of the piezoelectric material 855
were studied as a function of temperature.
4.1 Piezoelectric properties as a function of temperature
The first approach taken was to study the properties of the chosen 855 piezoelectric ceramic as a
function of temperature. In figure 4.1, it can be seen two of the most relevant piezoelectric char-
acteristics, the d31 piezoelectric constant and the kp piezoelectric coefficient. Both magnitudes
decrease as temperature decreases, a clear sign of the deterioration of the piezoelectric properties
as it was expected. These measurements were obtained by introducing the piezoelectric ceramic
into the cryogenic chamber. The data in figure 4.1 was measured automatically following the
procedure described in previous works [16].
The electromechanical coupling factor is one of the most important parameters in piezoelec-
tricity, defined as
25
Chapter 4. Results 4.1. Piezoelectric properties as a function of temperature
(4.1) k2 = mechanical energy storedelectric energy supplied
≤ 1 .
The kp is the factor for a thin disk for the electric field in direction 3 (parallel to direction
in which ceramic element is polarized) and radial vibrations in direction 1 and direction 2 (both
perpendicular to direction in which ceramic element is polarized). Consequently, if this coefficient
decreases means the conversion of mechanical to electric energy (or vice versa) is less efficient.
Figure 4.1: Piezoelectric properties of the 855 piezoelectric ceramic as a function of temperature.
Moreover, looking at the piezoelectric constant, we recall that d31 is the induced polarization
in direction 3 (parallel to direction in which ceramic element is polarized) per unit stress applied
in direction 1 (perpendicular to direction in which ceramic element is polarized) or vice versa. If
this magnitude decreases, that means that for the same electric field applied (the pulse sent will
not change over temperature) it induces less strain. Meaning that the signal received/sent by the
piezoelectric transducer will have less amplitude as temperature decreases.
26
Chapter 4. Results 4.2. Resonance frequency study
4.2 Resonance frequency study
As it has been explained before, the study consists on sending a rectangular pulse of a certain
width and amplitude to the source transducer. The pulse will be transmitted by the source
transducer through the material medium and will be sensed at the receiver transducer. The
adequate width of the pulse is given by the resonance frequency of the piezoelectric material.
First of all, the resonance properties of the different piezoelectric ceramics were studied with an
impedance analyzer.
In figure 4.2, the different resonances of the three piezoelectric ceramic tested can be seen. In
all three of them, it can be observed a set of resonance frequencies between 0 MHz and 1.5 MHz
in the case of the square 855 and the 851 disc and between 0 MHz and 3.5 MHz in the case of
the 855 disc. That set of resonance frequencies corresponds to the axial resonance mode, the one
depending on the radius or length (square ceramic) of the ceramic. The first peak of impedance
corresponds to the fundamental resonance frequency of the axial mode. The rest of the peaks
that can be seen correspond to the harmonics of the axial mode.
The resonance at which the experiment takes place is the frequency corresponding to the
thickness mode. That resonance is found at around 2 MHz for the square 855 and the 851
piezoelectric ceramics, while for the round 855 is at around 4 MHz. That difference is due to the
fact that the disc ceramic has half width of the other ones. The resonance frequency of a cylinder
with radius R and thickness h is given by equation 4.2.
(4.2) fr = c2
√(mh
)2 +( qnp
πR
)2,
where n = 0,1,2, ... and m and p = 1,2,3, ... qnp are the values that cancel the nth order Bessel
function, Jn(qnp) = 0. The part consisting on the Bessel function of equation 4.2 refers to the
radial resonance frequencies, in our case, it only matters the fundamental (m = 1) thickness
resonance frequency given by fr = c2h .
Therefore, if the thickness of a piezoelectric is doubled, the resonance frequency is divided by
a factor 2. The 855 square piezoelectric ceramic has a thickness of hsquare = 1 mm and the disc of
hdisc = 0.5 mm, that is why the thickness resonance frequency is multiplied by a factor 2.
Then, the appropriate pulse width is given by equation 4.3
(4.3) tw = Tr
2= 1
2 fr.
27
Chapter 4. Results 4.2. Resonance frequency study
(a) Resonance study of the disc 855 ceramic
(b) Resonance study of the square 855 ceramic
(c) Resonance study of the 851 ceramic
Figure 4.2: Resonance study of the different piezoelectric ceramics. In blue there is the electricimpedance and in red the phase.
28
Chapter 4. Results 4.3. Pulse at cryogenic temperatures
In the case of the square 855 and 851 piezoelectric ceramics the pulse width is of tw = 260
ns and for the round 855 ceramic is of tw = 130 ns. It is important to send a pulse with a
width corresponding to the resonance frequency of the piezoelectric material since it maximizes
the signal generated at the transducer. If the piezoelectric transducer is not excited with the
appropriate pulse, the amplitude of the signal generated would decrease. This is because the
ceramic is not able to resonate at its resonance frequency. The amplitude of the sent pulse is
decided to be of 2V. Experimentally it could be seen that increasing that value saturates the
signal at the source transducer.
(a) Input pulse for the disc 855 ceramic. Pulseof width tw=130 ns.
(b) Input pulse for the square 855 ceramic andthe disc 851 ceramic. Pulse of width tw=260 ns.
Figure 4.3: Input pulses for the different piezoelectric ceramic systems.
4.3 Pulse at cryogenic temperatures
Once the pulse characteristics were defined, the cryogenic systems were tested. After exciting the
source piezoelectric transducer with its appropriate pulse (figure 4.3), a signal is generated. In
the obtained measurements, the received pulse at the receiver transducer and the first echo at the
source transducer can be seen , just as in figure 2.7. However, it will only be studied the received
signal at the receiver transducer. In figure 4.4, the sensed signals at different temperatures for
the three piezoelectric materials configurations can be seen.
One of the main differences between the three set of pulses is their form. It is clear that for
each piezoelectric ceramic the form of the pulse changes. Not only is the material important but
also the geometry of it. Since the 855 material changes the form of the pulses depending if it is
square or a disc. In the three of them it can be seen a decrease in the amplitude of the signal as
temperature decreases. The lowest temperature signal is the last that can be sensed, therefore is
the lowest temperature at which the transducer is able to keep working. From that last signal, it
is clear that the properties of the piezoelectric ceramic deteriorate with temperature as it does
not have the same form of the rest.
29
Chapter 4. Results 4.3. Pulse at cryogenic temperatures
(a) Pulses at different temperatures of the disc 855 ceramic
(b) Pulses at different temperatures of the square 855 ceramic
(c) Pulses at different temperatures of the 851 ceramic
Figure 4.4: Evolution of the pulse response of the piezoelectric transducers at different tempera-tures. The arrows mark the time of the pulse arrival at each temperature.
30
Chapter 4. Results 4.4. Pulse amplitude as a function of temperature
From this signals, we will obtain some interesting information of the piezoelectric materials
we are testing. The most important is the amplitude of these signals as a function of temperature.
From the frequency of the signals, one can obtain the resonance frequency of the piezoelectric
transducer as a function of temperature. And finally, the time arrival of the signals and the
velocity of sound in the propagation medium as a function of temperature can also be known.
4.4 Pulse amplitude as a function of temperature
The amplitude and the following measures of the received signal were calculated with a Matlab
code. In this case, the amplitude is calculated with the peak value of the signal. As it can be seen
in figure 4.5, the pulse amplitude remains the same or slightly decreases in the first 220 K to 300
K. However, at around 200 K there is a peak in the amplitude of the three ceramics. That is quite
unusual, since it can be clearly seen the peak even though the materials are different. After some
research, it was concluded that this phenomenom was due to the semiconductive paste between
the piezoelectric ceramic and the medium material. That paste is used avoid having air between
the medium material and the piezoelectric. To confirm this, another test was made without the
semiconductive paste and the peak at 200 K was not seen. That could be explained as a change
in the physical properties of the paste, probably changing its acoustic impedance, which provides
a better transmission of the signal.
Figure 4.5: Comparison of the pulse amplitude for the different piezoelectric ceramics.
31
Chapter 4. Results 4.5. Resonance frequency as a function of temperature
From figure 4.5 and figure 4.4 it can be concluded that neither of the three configurations
work at cryogenic temperatures. The one that is able to work at the lower temperature is the
855 disc piezoelectric ceramic, as expected. The lowest working temperature for the 855 disc is
T = 130 K. The square 855 fails to work at around T = 160 K and the 851 disc at around T = 180
K. It could be interesting to find another material that is able to maximize the transmission of
the signal just as it happens with the semiconductive paste but at lower temperatures, since that
could extend the functionality of the ceramic.
Two types of the 855 ceramic material have been tested, the disc and the square (figure 3.2).
Even though they are made from the same material, the square ceramic fails to work at a
temperature 30 K higher than the disc one. Since all the experiments were carried out with the
same conditions, this discordance is probably due to the geometry of the material. Arriving to the
conclusion that it is more efficient to work with thinner discs than with squares in the thickness
resonance mode.
Finally, although the results show that there is no piezoelectric ceramic able to work at
cryogenic temperatures there is a positive result. Both 855 ceramic materials maintain its
properties as temperature decreases. They both can work until 200 K maintaining the amplitude
of its signals. As in contrary with the 851 ceramic that slowly decreases its amplitude, meaning a
decrease of its properties. Therefore, the 855 materials is a good candidate for applications that
require piezoelectric materials at temperatures from 200 K.
4.5 Resonance frequency as a function of temperature
Another form of knowing the resonance frequency of our piezoelectric materials, apart from using
the impedance analyzer, is to calculate the frequency of the received signal. Once the rectangular
pulse is sent to the source transducer, it will generate a signal that will have the resonance
frequency of the piezoelectric material. The frequency of the signals is calculated with an average
of two periods, the first three peaks (t1, t2, and t3) are taken, therefore, two periods of the signal
are considered. Then, the frequency is calculated with equation 4.4
(4.4) fpiezo = 2t3 − t1
.
32
Chapter 4. Results 4.6. Pulse arrival time as a function of temperature
Figure 4.6: Comparison of the resonance frequency as a function of temperature for the differentpiezoelectric ceramics.
In figure 4.6, it is clear that their resonance frequency hardly changes over temperature.
Which means that as temperature decreases it is not necessary to adjust the width of the pulse to
the new resonance frequency. This result rules out the possibility that a change in the resonance
frequency of the ceramic could affect the transmission of the signal, generating signals of lower
amplitude. It can be concluded that the resonance frequency of the 855 and 851 piezoelectric
ceramics do not depend on temperature.
4.6 Pulse arrival time as a function of temperature
To calculate the arrival time of the pulse, the cross-correlation is used. The cross-correlation
function between two signals X (t) and Y (t) provides a quantitative value of how similar are
these two signals. This method has been widely used during the last decades [17, 18]. The
cross-correlation function for continuous signals is
(4.5) RXY (τ)=∫ ∞
−∞X (t)Y (t+τ)dt .
The maximum of the cross-correlation between two signals indicates the point in time where
the signals are best aligned. Consequently, the time delay between two signals is determined by
33
Chapter 4. Results 4.6. Pulse arrival time as a function of temperature
the argument of the maximum of the cross-correlation
(4.6) τdelay = arg maxt
(RXY (τ)) .
Therefore, to calculate the arrival time of each temperature, the signal received is always
compared to the received signal at room temperature (reference signal). Once the τdelay of the
signal at a certain temperature with the reference signal is known the arrival time of the signal
can be calculated as
(4.7) tarrival = tre f erence +τdelay ,
where the tre f erence is set with the matlab code.
2.5 %
Figure 4.7: Comparison of the pulse arrival time as a function of temperature for the differentpiezoelectric ceramics.
In figure 4.7 it is clear that the signal is faster as temperature decreases, since the arrival
time decreases. There is a difference of around 2.5% in the values of the piezoelectric materials
that can be due to the geometry of the signal received. As seen in figure 4.4, that can condition
the way the cross-correlation calculates the exact arrival time.
34
Chapter 4. Results 4.7. Sound velocity in the medium as a function of temperature.
4.7 Sound velocity in the medium as a function of temperature.
Knowing the pulse arrival time, it is easy to obtain the sound velocity in Aluminum from
equation 4.8. The results are shown in figure 4.8.
(4.8) cAl =L
tarrival
2 %
Figure 4.8: Comparison of the sound velocity in Aluminum as a function of temperature for thedifferent piezoelectric ceramics.
The speed of sound is almost the same in the three cases, since it only matters the propagation
medium and not the piezoelectric material, as it can be seen in figure 4.8. The slightly differences
between the values of the velocity come from the determination of the arrival time of the pulse
(figure 4.7), which corresponds to a difference of 2.5%.
The speed of sound in Aluminum hardly varies with the decrease of temperature, which
indicates that the acoustic properties of our propagation medium do not change. It can be
appreciated in figure 4.8 that there is a change of around 100 m/s between the velocity at T = 300
K and T = 150 K, which is a change of 1%. A total difference of 2% between the sound velocity of
the 855 disc and the 851 disc has been measured.
35
CH
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5CONCLUSIONS
The aim of this project was to study the feasibility of using piezoelectric ceramic transducers
to generate ultrasonic pulses at cryogenic temperatures. The future use of these piezoelectric
technology would consist in the control of bubbles formed in the cryogenic liquid propellants of
rockets in space missions. In order to accomplish our objective three main goals were reached: a
piezoelectric ceramic transducer was developed, a transducer-rod system for cryogenic tempera-
tures was built and the properties and functionality of this cryogenic technique were tested at
cryogenic temperatures.
First of all, the active element of the transducer, the piezoelectric ceramic, was studied. Among
all the different piezoelectric materials the chosen one were ceramics, more specifically a soft
piezoelectric ceramic. Even though there are other types of piezoelectric materials, such as single-
crystals, that in theory have better properties to work at cryogenic temperatures, the challenge
was to test a more affordable option: piezoelectric ceramics. The chosen soft piezoelectric ceramic,
a 855 American Piezo, had promising characteristics for functioning at cryogenic temperatures,
such as a low Curie temperature. On the other side, different materials available to be the medium
of propagation of the ultrasonic signal were considered. The material chosen was Aluminum, due
to several factors such as its price and its stable properties as a function of temperature.
The second part of the theoretical study was to understand the transducer-rod system we
were working on. The form and characteristics of the ultrasonic signal generated at the source
transducer had to be understood. At the end, the system is simplified as if the piezoelectric
ceramic behaved as an underdamped oscillator and the received signal can be estimated as the
solution of the equation of the underdamped oscillator excited by an ultrasonic pulse. The results
37
Chapter 5. Conclusions
showed that the theoretical model reproduced quite well the experimental values obtained.
Once the cryogenic ultrasonic pulse generating system was defined, the second step of the
project was to set the cryogenic cooler, the cryostat. Working at such low temperatures (of around
20 K) is such a challenge since not everything is able to resist cryogenic conditions. This part took
most of our time as we were always improving the sample-holder that went into the cryogenic
chamber. In spite of all the drawbacks a final adequate set-up was built up and the different tests
could be carried out.
Although the 855 American Piezo was the chosen piezoelectric ceramic, another one, the 851
soft ceramic, with not the same promising characteristics as the 885, was tested too. The aim of
this test was to confirm that our chosen ceramic could work better at cryogenic temperatures.
Finally, after the data obtained, it can be concluded that none of the soft piezoelectric ceramics
tested were able to resist cryogenic temperatures as low as 20 K. It has to be noted that the 855
piezoelectric ceramic was able to work at around T = 130 K whereas the 851 ceramic stopped
working at around T = 190 K. The 855 material is able to preserve its properties until T = 200
K, meaning that it can work at around T = 200 K without deteriorating its response. Although
it has been discovered that the 855 piezoelectric material could not work for our cryogenic
application, it is an interesting option for applications that require temperatures ranging from
room temperature to around 200 K.
Another conclusions from the data obtained are that the resonance frequency of the different
piezoelectric ceramics tested is temperature independent. The results showed that their frequency
did not change as temperature decreased. On another note, through the pulse time arrival at
the receiver transducer it was possible to estimate the sound velocity in Aluminum. It could be
seen that the speed of sound increased as temperature decreased, probably due to a change in
the acoustic properties of aluminum as a function of temperature.
In conclusion, we proved that the actual piezoelectric ceramics are not able to fulfill the
challenge that is generating ultrasounds at cryogenic temperatures. The future of ultrasound
generation at outer space temperatures resides in the manufacturing of a special piezoelectric ce-
ramic. A material with the appropriate properties and characteristics to resist such temperatures
without deteriorating.
38
BIBLIOGRAPHY
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40
APPENDIX A
Analytical solution of the acoustically-forced underdampedoscillator model
Here is detailed the solution of equation 2.14, following the procedure proposed in [22].
The electric signal provided by the receiver transducer is the solution of the non-homogeneous
differential equation
(1) Xr +2ξωn Xr +ω2nXr = X (t) ,
where X (t) is the form of the acoustic signal travelling through the flowing medium,
(2) X (t)= Apulse [g(t)− g(t− tw)] ,
with
(3) g(t)=[1− e−at
(cos(bt)+ a
bsin(bt)
)]Θ(t) ,
being a = ξωn, b =ωn√
1−ξ2 , and Θ(t) the Heaviside step function.
Fortunately, an analytical solution of (1) exists and can be obtained with the aid of the Green
function,
(4) Xr(t)= Apulse
b
∫ t
−∞[g(τ)− g(τ− tw)] e−a(t−τ) sin[b(t−τ)]dτ .
41
Appendix A Analytical solution of the acoustically-forced underdamped oscillator model
Expanding the integral in two parts,
(5) Xr(t)= Apulse
b
∫ t
−∞g(τ)e−a(t−τ) sin[b(t−τ)]dτ︸ ︷︷ ︸
T1(t)
−Apulse
b
∫ t
−∞g(τ− tw)e−a(t−τ) sin[b(t−τ)]dτ︸ ︷︷ ︸
T2(t)
.
The first term on the right-hand side of equation (5) is designated as T1(t), whereas the
second term is denoted by T2(t). The Heaviside step function contained in g(t) transforms the
integration limits from (−∞, t] to [0, t]. Therefore, the term T1(t) can be rewritten as
(6) T1(t)=∫ t
0
[1− e−aτ
(cos(bτ)+ a
bsin(bτ)
)]· e−a(t−τ) sin[b (t−τ)]dτ .
Expanding the three terms in the parenthesis,
(7) T1(t)=∫ t
0e−a(t−τ) sin[b (t−τ)]dτ− e−at
∫ t
0cos(bτ)sin[b (t−τ)]dτ−
− ae−at
b
∫ t
0sin(bτ)sin[b (t−τ)]dτ .
Integrating the three terms separately and rearranging, one obtains
(8) T1(t)= e−at[
beat −asin(bt)−bcos(bt)a2 +b2
]− e−at
[tsin(bt)
2− a [sin(bt)−btcos(bt)]
2b2
].
Following a similar procedure for the term T2(t), one arrives at
(9) T2(t)= e−at [beat −asin(bt)−bcos(bt)]
a2 +b2 −
− e−a(t−tw)[
2btsin[b (t− tw)]+cos[b (tw − t)]−cos[b (tw + t)]4b
]−
− ae−a(t−tw) [cos(btw)sin(bt)−btcos[b (t− tw)]]2b2 .
Rearranging terms, one can write the final form of the solution to (1), which is given by
(10) Xr(t)= Apulse
b[T1(t)−T2(t)]Θ(t) .
42
APPENDIX B
Measurements of the different configurations of thetransducer-cylinder systems.
Several measures had to be done before building up the final transducer-rod system. With the
cylinders we tested different glues for the transducers, different lengths and different diameters.
All these measurements helped us decide which was the optimum system.
First, we compared a short cylinder of L = 15 mm with different types of glue. We tested
Superglue and a conductive epoxy (see figure 1). The main difference between the Superglue and
the conductive epoxy is the amplitude of the received signal. In this case the Superglue showed
less amplitude, mainly due to its low acoustic impedance. On the other hand, the conductive
epoxy, had a bigger amplitude, transmitting better the signal from the medium to the transducer.
At the end, none of those glues were used due to the complications caused by the cryogenic
conditions. Neither of them was able to resist such low temperatures.
Then, we tested some cylinders with different lengths (see figure 2). Although the conductive
epoxy showed a better amplitude of the pulses, it was decided to test these configurations with
superglue, just because superglue is easier and faster to manipulate. The only case in which the
1st echo of the source transducer can be clearly seen is the L = 100 mm case. As for the receptor
transducer signal, the main difference is seen in the shortest cylinder in comparison with the
other two. As explained in chapter 3, the final chosen length was L = 75 mm for several reasons.
Finally, three diameters were considered: D = 10 mm, D = 15 mm and D = 20 mm (see
figure 3). In the three cases a repetition of the received signal can be seen after it. The main
difference between them is that as the diameter of the cylinder increases, the separation between
the repetition of the signal increases too. That can also be seen in the source transducer first
echo. Therefore, the second signal must be due to some unknown radial effects. At the end, since
we are only studying the first arrived signal, the chosen diameter is D = 15 mm as explained in
chapter 3.
43
(a) Source transducer, L = 15 mm. Attached withsuperglue.
(b) Receptor transducer, L = 15 mm. Attached withsuperglue.
(c) Source transducer, L = 15 mm. Attached with aconductive epoxy.
(d) Receptor transducer, L = 15 mm. Attached witha conductive epoxy.
Figure 1: Two Aluminum cylinders of L = 15 mm were tested with different glues.
Appendix B Measurements of the different configurations of the transducer-cylinder systems.
(a) Source transducer, L = 15 mm. (b) Receptor transducer, L = 15 mm.
(c) Source transducer, L = 100 mm. (d) Receptor transducer, L = 100 mm.
(e) Source transducer, L = 200 mm. (f) Receptor transducer, L = 200 mm.
Figure 2: This three measurement correspond to three Aluminum cylinders of L = 15 mm, L = 100mm, L = 200 mm. The transducers are attached to the cylinder with superglue in the three cases.
45
Appendix B Measurements of the different configurations of the transducer-cylinder systems.
(a) Source transducer, D = 10 mm. (b) Receptor transducer, D = 10 mm.
(c) Source transducer, D = 15 mm. (d) Receptor transducer, D = 15 mm.
(e) Source transducer, D = 20 mm. (f) Receptor transducer, D = 20 mm.
Figure 3: These three measurements correspond to three Aluminum cylinders of L = 75 mm anddiameters D = 10 mm, D = 15 mm and D = 20 mm. The transducers are attached to the cylinderwith superglue in the three cases.
46
