**Meet the editor**

Dr. Farzad Ebrahimi was born in Qazvin, Iran, in 1979. He graduated in Mechanical engineering, from the University of Tehran, Iran, in 2002. He received his Msc and PhD in Mechanical engineering, with a specialization in applied design from the University of Tehran, Iran, in 2009. Since 2002, he has been working at the "Smart Materials and Structures Lab" Research Center of the

Faculty of Mechanical Engineering at the University of Tehran, where he is a researcher of smart functionally graded materials and structures. He is involved in several international journals as editor and reviewer. He serves on the editorial board of "Far East Journal of Mechanical Engineering and Physics". He is the author of books "Smart Functionally Graded Plates" and "Progress in Analysis of Functionally Graded Structures", Nova Science Publishers, NY. He also served as the editor of the book "Advances in Vibration Analysis Research", InTech – Open Access Publisher, Croatia. His research interests focus on areas of smart materials and structures, plate and shell theory, vibration analysis of continuous systems, composite materials and structures, functionally graded materials and structures, finite element analysis and fracture mechanics and he has published several researches in these fields. His research in these areas has been presented at international conferences and appeared in academic journals such as "Journal of Mechanical Science and Technology", "Smart Materials and Structures", "European Journal of Mechanics" and "Archive of Applied Mechanics and Thin Walled Structures". Farzad has also strong collaboration with Iranian industries on gas and oil projects, as well as serves as ad-hoc referee in several top academic journals.

Contents

**Preface VII**

Chapter 2 **Modeling and Investigation**

Chapter 3 **Distributed-Parameter Modeling**

Adam Wickenheiser

**Part 1 Modeling of Piezoelectric Transducers 1**

Chapter 1 **Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 3**  Tao Li, Jan Ma and Adrian F. Low

**of One-Dimensional Flexural Vibrating**

Andrzej Buchacz and Marek Płaczek

**Part 2 Applications of Piezoelectric Transducers in Structural Health Monitoring 85**

**in Structural Health Monitoring:**

Chapter 5 **Application of Piezoelectric Transducers**

Chapter 6 **Piezoelectric Transducers Applied**

**Mechatronic Systems with Piezoelectric Transducers 27**

**of Energy Harvesting Structures with Discontinuities 53**

**Distortion in Wideband Piezoelectric Transducer Systems 75**  Said Assous, John Rees, Mike Lovell, Laurie Linnett and David Gunn

Chapter 4 **Bandwidth Enhancement: Correcting Magnitude and Phase**

**in Structural Health Monitoring Techniques 87**  Najib Abou Leyla, Emmanuel Moulin, Jamal Assaad,

**Data Acquisition and Virtual Instrumentation for Electromechanical Impedance Technique 105**  Fabricio Guimarães Baptista and Jozue Vieira Filho

Farouk Benmeddour, Sébastien Grondel and Youssef Zaatar

### Contents

#### **Preface XI**

	- **Part 2 Applications of Piezoelectric Transducers in Structural Health Monitoring 85**

Preface

A transducer is a device that converts energy from one form to another. Presently, piezoelectric material is commonly used as a basic component of transducers. Piezoelectric devices are a very reliable and inexpensive means of converting electrical energy into physical motion and exhibit a high tolerance to environmental factors such as electromagnetic fields and humidity. The piezoelectric transducer converts electric signals into mechanical vibrations or vice versa by utilizing the morphological change of a crystal which occurs on voltage application, or conversely by monitoring the

Piezoelectric transducers have many applications. In particular, piezoelectric diaphragms have been employed as pressure sensors, in speakers for audio equipment, fluid ejection, fluid pumping and printing applications. Piezoelectric ceramic transducers have also found many important applications in adaptive structures for vibration control and acoustic noise suppression in modern space, civilian and military systems, such as launch vehicles, space platforms, aircraft,

This book reports on the state of the art research and development findings on this very broad matter through original and innovative research studies exhibiting various

In particular, it introduces recent research results on important issues at the field of

The book has been organized in two parts. The first part deals with the modeling approaches of the piezoelectric transducers e.g. modeling of the horn-type piezoelectric ultrasonic transducers in chapter 1 and the modeling and investigation of one-dimensional flexural vibrating mechatronic systems with piezoelectric transducers in chapter 2. Also, the distributed-parameter modeling of energy harvesting structures with discontinuities and the correcting magnitude and phase distortion in wideband piezoelectric transducer systems is introduced in chapter 3 and 4. In the second part, the applications of piezoelectric transducers in structural health monitoring and the data acquisition and virtual instrumentation for electromechanical impedance

voltage generated by a pressure applied on a crystal.

piezoelectric transducers modeling and applications.

technique are introduced in chapters 5 and 6, respectively.

submarines and helicopters.

investigation directions.

### Preface

A transducer is a device that converts energy from one form to another. Presently, piezoelectric material is commonly used as a basic component of transducers. Piezoelectric devices are a very reliable and inexpensive means of converting electrical energy into physical motion and exhibit a high tolerance to environmental factors such as electromagnetic fields and humidity. The piezoelectric transducer converts electric signals into mechanical vibrations or vice versa by utilizing the morphological change of a crystal which occurs on voltage application, or conversely by monitoring the voltage generated by a pressure applied on a crystal.

Piezoelectric transducers have many applications. In particular, piezoelectric diaphragms have been employed as pressure sensors, in speakers for audio equipment, fluid ejection, fluid pumping and printing applications. Piezoelectric ceramic transducers have also found many important applications in adaptive structures for vibration control and acoustic noise suppression in modern space, civilian and military systems, such as launch vehicles, space platforms, aircraft, submarines and helicopters.

This book reports on the state of the art research and development findings on this very broad matter through original and innovative research studies exhibiting various investigation directions.

In particular, it introduces recent research results on important issues at the field of piezoelectric transducers modeling and applications.

The book has been organized in two parts. The first part deals with the modeling approaches of the piezoelectric transducers e.g. modeling of the horn-type piezoelectric ultrasonic transducers in chapter 1 and the modeling and investigation of one-dimensional flexural vibrating mechatronic systems with piezoelectric transducers in chapter 2. Also, the distributed-parameter modeling of energy harvesting structures with discontinuities and the correcting magnitude and phase distortion in wideband piezoelectric transducer systems is introduced in chapter 3 and 4. In the second part, the applications of piezoelectric transducers in structural health monitoring and the data acquisition and virtual instrumentation for electromechanical impedance technique are introduced in chapters 5 and 6, respectively.

#### VIII Preface

The advantage of the book "**Advances in Piezoelectric Transducers"** is its open access, fully searchable by anyone anywhere, and in this way it provides the forum for dissemination and exchange of the latest scientific information on theoretical as well as applied areas of knowledge in the field of piezoelectric transducers.

The present book is a result of contributions of experts from international scientific community working in different aspects of piezoelectric transducers. The introductions, data, and references in this book will help the readers know more about this topic and help them explore this exciting and fast-evolving field.

The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area.

I hope that readers will find the book useful and inspiring by examining the recent developments in piezoelectric transducers.

> **Dr. Farzad Ebrahimi**  Mechanical Engineering Department, University of Tehran – Tehran, I. R. Iran

VIII Preface

The advantage of the book "**Advances in Piezoelectric Transducers"** is its open access, fully searchable by anyone anywhere, and in this way it provides the forum for dissemination and exchange of the latest scientific information on theoretical as well as

The present book is a result of contributions of experts from international scientific community working in different aspects of piezoelectric transducers. The introductions, data, and references in this book will help the readers know more about

The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and

I hope that readers will find the book useful and inspiring by examining the recent

**Dr. Farzad Ebrahimi**

I. R. Iran

Mechanical Engineering Department, University of Tehran – Tehran,

applied areas of knowledge in the field of piezoelectric transducers.

this topic and help them explore this exciting and fast-evolving field.

what kind of open problems are in this area.

developments in piezoelectric transducers.

**Part 1**

**Modeling of Piezoelectric Transducers**

## **Part 1**

## **Modeling of Piezoelectric Transducers**

**1**

*Singapore*

**Horn-Type Piezoelectric Ultrasonic Transducer:**

The piezoelectric transducers can be categorized into sonic, ultrasonic and megasonic transducers based on the operating frequency. In each category, the design and function of the transducers vary significantly. The sonic transducers work at an audible frequency range, typically less than 20 kHz. In this frequency range, the transducers could be designed in the bending mode. Examples of the sonic transducers are the bimorph cantilever and buzzer (siren) unimorph transducer (APC International, Ltd., 2002; Yeo et al., 2008). The buzzer transducer is made of a piezoelectric disk attached to a metallic substrate. It is able to produce a sound level more than 100 dB at resonance in the frequency range of kHz (Bai et al., 2007; Miclea et al., 2008). At off-resonance conditions, the buzzer transducer can also be used as an actuator for piezoelectric diaphragm pump (Accoto et al., 2000; Woias, 2005). The ultrasonic transducers usually work at a frequency range from ~20 kHz to ~200 kHz (Abramov, 1998; Atar et al., 1999; Chen & Wu, 2002; Chu et al., 2002; Hongoh et al., 2004; Lin, 1995, 2004a; Mattiat, 1971; Medis & Henderson, 2005; Prokic, 2004; Radmanovic & Mancic, 2004; Sherrit et al., 1999a, 1999b; Tsuda et al., 1983; Wiksell et al., 2000). Transducers in this range could also be designed in the bending mode. Some examples are the tube transducer for cylindrical ultrasonic motor and disk/ring transducer for travelling wave motor (Li et al., 2007a; Uchino, 2003). But in this frequency range, a more common transducer design is the longitudinal mode. Langevin (sandwich, converter, Tonpiltz, etc.) transducer represents a typical structure (Abramov, 1998; Atar et al., 1999; Chen & Wu, 2002; Chu et al., 2002; Hongoh et al., 2004; Lin, 1995, 2004a, 2004b; Mattiat, 1971; Medis & Henderson, 2005; Prokic, 2004; Radmanovic & Mancic, 2004; Sherrit et al., 1999a, 1999b; Tsuda et al., 1983; Wiksell et al., 2000). It is sometimes connected with a horn (wave guide, booster, sonotrode, etc.) to focus or transmit energy. This type of transducer provides a wide range of applications, including welding, machining, sonochemistry, cleaning, underwater communication, ultrasonic surgery, etc. The megasonic transducers work in the frequency range of MHz. The most widely used vibration mode is the thickness mode for this range. One of the typical applications of the transducers in this category is the megasonic cleaning (Kapila et al., 2006). It provides advantages such as gentle and controllable cavitation, which will incur less damage on the cleaned parts compared with

its traditional ultrasonic counterpart. Hence it is more suitable for precision cleaning.

**1. Introduction**

 \*

Corresponding Author

*1School of Materials Science and Engineering, Nanyang Technological University*

*2National University Heart Centre; National University Health System,*

**Modelling and Applications**

Tao Li1\*, Jan Ma1 and Adrian F. Low2\*

### **Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications**

Tao Li1\*, Jan Ma1 and Adrian F. Low2\*

*1School of Materials Science and Engineering, Nanyang Technological University 2National University Heart Centre; National University Health System, Singapore*

#### **1. Introduction**

The piezoelectric transducers can be categorized into sonic, ultrasonic and megasonic transducers based on the operating frequency. In each category, the design and function of the transducers vary significantly. The sonic transducers work at an audible frequency range, typically less than 20 kHz. In this frequency range, the transducers could be designed in the bending mode. Examples of the sonic transducers are the bimorph cantilever and buzzer (siren) unimorph transducer (APC International, Ltd., 2002; Yeo et al., 2008). The buzzer transducer is made of a piezoelectric disk attached to a metallic substrate. It is able to produce a sound level more than 100 dB at resonance in the frequency range of kHz (Bai et al., 2007; Miclea et al., 2008). At off-resonance conditions, the buzzer transducer can also be used as an actuator for piezoelectric diaphragm pump (Accoto et al., 2000; Woias, 2005). The ultrasonic transducers usually work at a frequency range from ~20 kHz to ~200 kHz (Abramov, 1998; Atar et al., 1999; Chen & Wu, 2002; Chu et al., 2002; Hongoh et al., 2004; Lin, 1995, 2004a; Mattiat, 1971; Medis & Henderson, 2005; Prokic, 2004; Radmanovic & Mancic, 2004; Sherrit et al., 1999a, 1999b; Tsuda et al., 1983; Wiksell et al., 2000). Transducers in this range could also be designed in the bending mode. Some examples are the tube transducer for cylindrical ultrasonic motor and disk/ring transducer for travelling wave motor (Li et al., 2007a; Uchino, 2003). But in this frequency range, a more common transducer design is the longitudinal mode. Langevin (sandwich, converter, Tonpiltz, etc.) transducer represents a typical structure (Abramov, 1998; Atar et al., 1999; Chen & Wu, 2002; Chu et al., 2002; Hongoh et al., 2004; Lin, 1995, 2004a, 2004b; Mattiat, 1971; Medis & Henderson, 2005; Prokic, 2004; Radmanovic & Mancic, 2004; Sherrit et al., 1999a, 1999b; Tsuda et al., 1983; Wiksell et al., 2000). It is sometimes connected with a horn (wave guide, booster, sonotrode, etc.) to focus or transmit energy. This type of transducer provides a wide range of applications, including welding, machining, sonochemistry, cleaning, underwater communication, ultrasonic surgery, etc. The megasonic transducers work in the frequency range of MHz. The most widely used vibration mode is the thickness mode for this range. One of the typical applications of the transducers in this category is the megasonic cleaning (Kapila et al., 2006). It provides advantages such as gentle and controllable cavitation, which will incur less damage on the cleaned parts compared with its traditional ultrasonic counterpart. Hence it is more suitable for precision cleaning.

<sup>\*</sup> Corresponding Author

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 5

a casing. Due to the vibration of the transducer, an acoustic field will be established inside the casing. As a result, the liquid will be forced to flow due to the nonlinear acoustic effects, such as streaming and cavitation. The unique features of the pump are high pressure head, continuous flow, no body oscillation and insensitive to the tubing length. The potential

Fig. 1 shows the typical structure of the horn-type piezoelectric transducer. Generally, the transducer comprises four parts, i.e., the back mass, the piezoelectric stack, the front mass and the horn. The piezoelectric stack is clamped between the front and back mass. The horn is coupled to the front mass through the method such as bolt. In this configuration, the function of the piezo stack is to generate vibration, which is then amplified by the horn. The horn together with the front and back mass also has function of impedance match to maximize the energy transfer from the piezoelectric stack to the loadings at the tip of the transducer (Chen et al., 2004). The horn can have various designs based on the applications, such as conical profiled, step profiled, exponential profiled, etc (Hongoh et al., 2004; Horita, 1967; Khmelev et al., 2005; Minchenko, 1969; Muhlen, 1990). To maximize the vibration amplitude, the horn can be designed with multiple sections. Each section has the same or different profiles. The total length of the transducer is usually multiples of the half

Equivalent circuit network theory is a well-accepted method in piezoelectric material study and transducer design (Li et al., 2008, 2009, 2010; Mason,1964; Prokic, 2004; Radmanovic & Mancic, 2004). In this method, the transducer is divided into different sections. Each section will be represented using a two- or three-port electrical network. The networks are then connected to represent the whole transducer. For example, in the case of Fig. 1, four networks can be used to represent the back mass, piezo stack, front mass and the horn. The behaviour of the transducer can then be analyzed from the obtained electrical model. Conventionally, the method is efficient in the transducer design, such as calculating the dimension based on the given frequency. But in the case of performance simulation or analysis, such as calculation of stress distribution, predication of power consumption, estimation of quality factor, etc, the application of this theory is limited. In the present work, we proposed an improved equivalent circuit theory, called Finite Electric Circuit Element Modelling (FECEM), which are convenient and efficient in both transducer design and

applications of the pump are sonochemistry and microfluidics.

Fig. 1. General structure of a horn-type piezoelectric transducer

Back mass Piezo stack Front mass Horn

wavelength.

**3. FECEM theory**

**2. General structure of the horn-type piezoelectric transducer**

The capability of the transducer is closely associated with the frequency, vibration mode and wavelength. As a result, different transducers will achieve different performances in a wide range of applications. In comparison, the ultrasonic transducer with longitudinal vibration mode is the only one that is able to achieve both large vibration amplitude and high power (density) simultaneously. The vibration velocity of the transducer is limited by the mechanical, thermal and electrical properties of the materials. Currently, the PZT material is only able to achieve a vibration velocity less than 1 m/s as compared to 20 m/s for the titanium material in the same ultrasonic frequency range (Li et al., 2007b; Mason & Wehr, 1970; Muhlen, 1990). Thus, by using the titanium horn, the transducer can achieve larger vibration amplitude. In addition, the transducer at the axial direction has a relative long wave length compared to the thickness mode. A large volume of the PZT material could be used to increase the power capacity of the transducer (Mason, 1964). Furthermore, the power can be focused to a small area by the horn, which further improves the power density of the transducer. Due to the superiorities of high amplitude and high power (density), the horn type transducer has been found to have many unique applications.

The shape of the horn-type transducer varies significantly depending on the applications. Conventionally, the modelling of the transducer can be based on electrical circuit theory and FEA theory. Typically, the electrical circuit theory is used to design the transducer while FEA theory is used to analyze the performance of the transducer. It is convenient to use the electrical theory to design the dimension. However, for analysis of parameters, such as mode shape, stress distribution, power consumption, etc, the FEA theory is more advantageous. In the present work, we proposed an improved electrical circuit theory, called Finite Electrical Circuit Element Modelling (FECEM), which combines functions of both design and analysis. In this method, the transducer can be represented using a network, which are the connections of electrical circuit elements. The great advantage of the proposed method is the high efficiency in the development of piezoelectric transducers. We also proposed a method to estimate the mechanical quality factor *Qm* which is important to estimate parameters like velocity and power consumption. Conventionally, *Qm* parameter was either assumed or measured. In the present work, the FECEM was also compared with ANSYS FEA 2D analysis. Consistent results have been found.

The ANSYS 2D fluid-structure acoustic modelling was also introduced in the present work. In this modelling, the system response can be simulated when the transducer is loaded in the acoustic field. This model is useful to analyze the loading effect on the transducer. It is also essential to study the distributions and magnitude of sound pressure under different boundary conditions.

Lastly, in this chapter two applications of the horn-type transducer are introduced. They are thrombolysis transducer and acoustic pump. Both can be designed and analyzed using the above mentioned FECEM and ANSYS 2D theories. The thrombolysis transducer is a horntype transducer connected with a long and thin transmission wire. It can be used to emulsify the thrombus in the blood vessel. The transducer was modelled and characterized in the present work. Efficient and effective emulsification results have been observed which proves the design and analysis were successful. The acoustic pump is another application of the horn-type transducer. The pump possesses a simple structure with just a transducer and a casing. Due to the vibration of the transducer, an acoustic field will be established inside the casing. As a result, the liquid will be forced to flow due to the nonlinear acoustic effects, such as streaming and cavitation. The unique features of the pump are high pressure head, continuous flow, no body oscillation and insensitive to the tubing length. The potential applications of the pump are sonochemistry and microfluidics.

#### **2. General structure of the horn-type piezoelectric transducer**

Fig. 1 shows the typical structure of the horn-type piezoelectric transducer. Generally, the transducer comprises four parts, i.e., the back mass, the piezoelectric stack, the front mass and the horn. The piezoelectric stack is clamped between the front and back mass. The horn is coupled to the front mass through the method such as bolt. In this configuration, the function of the piezo stack is to generate vibration, which is then amplified by the horn. The horn together with the front and back mass also has function of impedance match to maximize the energy transfer from the piezoelectric stack to the loadings at the tip of the transducer (Chen et al., 2004). The horn can have various designs based on the applications, such as conical profiled, step profiled, exponential profiled, etc (Hongoh et al., 2004; Horita, 1967; Khmelev et al., 2005; Minchenko, 1969; Muhlen, 1990). To maximize the vibration amplitude, the horn can be designed with multiple sections. Each section has the same or different profiles. The total length of the transducer is usually multiples of the half wavelength.

Fig. 1. General structure of a horn-type piezoelectric transducer

#### **3. FECEM theory**

4 Advances in Piezoelectric Transducers

The capability of the transducer is closely associated with the frequency, vibration mode and wavelength. As a result, different transducers will achieve different performances in a wide range of applications. In comparison, the ultrasonic transducer with longitudinal vibration mode is the only one that is able to achieve both large vibration amplitude and high power (density) simultaneously. The vibration velocity of the transducer is limited by the mechanical, thermal and electrical properties of the materials. Currently, the PZT material is only able to achieve a vibration velocity less than 1 m/s as compared to 20 m/s for the titanium material in the same ultrasonic frequency range (Li et al., 2007b; Mason & Wehr, 1970; Muhlen, 1990). Thus, by using the titanium horn, the transducer can achieve larger vibration amplitude. In addition, the transducer at the axial direction has a relative long wave length compared to the thickness mode. A large volume of the PZT material could be used to increase the power capacity of the transducer (Mason, 1964). Furthermore, the power can be focused to a small area by the horn, which further improves the power density of the transducer. Due to the superiorities of high amplitude and high power (density), the horn type transducer has been found to have many unique

The shape of the horn-type transducer varies significantly depending on the applications. Conventionally, the modelling of the transducer can be based on electrical circuit theory and FEA theory. Typically, the electrical circuit theory is used to design the transducer while FEA theory is used to analyze the performance of the transducer. It is convenient to use the electrical theory to design the dimension. However, for analysis of parameters, such as mode shape, stress distribution, power consumption, etc, the FEA theory is more advantageous. In the present work, we proposed an improved electrical circuit theory, called Finite Electrical Circuit Element Modelling (FECEM), which combines functions of both design and analysis. In this method, the transducer can be represented using a network, which are the connections of electrical circuit elements. The great advantage of the proposed method is the high efficiency in the development of piezoelectric transducers. We also proposed a method to estimate the mechanical quality factor *Qm* which is important to estimate parameters like velocity and power consumption. Conventionally, *Qm* parameter was either assumed or measured. In the present work, the FECEM was also compared with

The ANSYS 2D fluid-structure acoustic modelling was also introduced in the present work. In this modelling, the system response can be simulated when the transducer is loaded in the acoustic field. This model is useful to analyze the loading effect on the transducer. It is also essential to study the distributions and magnitude of sound pressure under different

Lastly, in this chapter two applications of the horn-type transducer are introduced. They are thrombolysis transducer and acoustic pump. Both can be designed and analyzed using the above mentioned FECEM and ANSYS 2D theories. The thrombolysis transducer is a horntype transducer connected with a long and thin transmission wire. It can be used to emulsify the thrombus in the blood vessel. The transducer was modelled and characterized in the present work. Efficient and effective emulsification results have been observed which proves the design and analysis were successful. The acoustic pump is another application of the horn-type transducer. The pump possesses a simple structure with just a transducer and

ANSYS FEA 2D analysis. Consistent results have been found.

applications.

boundary conditions.

Equivalent circuit network theory is a well-accepted method in piezoelectric material study and transducer design (Li et al., 2008, 2009, 2010; Mason,1964; Prokic, 2004; Radmanovic & Mancic, 2004). In this method, the transducer is divided into different sections. Each section will be represented using a two- or three-port electrical network. The networks are then connected to represent the whole transducer. For example, in the case of Fig. 1, four networks can be used to represent the back mass, piezo stack, front mass and the horn. The behaviour of the transducer can then be analyzed from the obtained electrical model. Conventionally, the method is efficient in the transducer design, such as calculating the dimension based on the given frequency. But in the case of performance simulation or analysis, such as calculation of stress distribution, predication of power consumption, estimation of quality factor, etc, the application of this theory is limited. In the present work, we proposed an improved equivalent circuit theory, called Finite Electric Circuit Element Modelling (FECEM), which are convenient and efficient in both transducer design and

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 7

(4)

(5)

(6)

(7)

(8)

*Z ZZ Z Z Z Z Z ZZ*

This is actually the constitutive relations between the forces and velocities for the elements. This relation can also be represented using the following "chain" or "ABCD" matrix

> 1 2 1 2 *U AB U I CD I*

> >

*AZZZ BZZ Z Z*

122 11 2 2

2

11 21 11 22 12 21 21

*Z*1bolt+ *Z*1pzt

*Z*2bolt+ *Z*2pzt

The *Z* matrix and *ABCD* matrix can be converted to each other based on the following Eqs. 7 and 8. This provides much convenience for different parameter calculations (Balabanian &

*DZZZ*

21 22 21 1 *A B Z Z ZZ ZZ Z*

11 12

*Z Z A C AD BC C*

The piezoelectric stack comprises two parts, PZT and bolt. It can be represented using a three-port network element as shown in Fig. 3, in which *V* is the input electrical voltage, *C*0

*U*<sup>1</sup> *U*<sup>2</sup>

*1:*

*I*<sup>1</sup> *I*<sup>2</sup>

*C D Z ZZ*

21 22 1

*Z*1bolt+ *Z*1pzt

*V C*<sup>0</sup>

*Z Z C DC*

2 122

1 /

*C Z*

(Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993)

Bickart, 1981; Hayt & Kemmerly, 1993).

Fig. 3. Piezoelectric stack element (Lin, 2004b)

where

where

analysis. In this method, the transducer will also be divided into multiple sections. But the section will be further divided into a number of subsections, called element. An element is represented using an electric circuit network. The connections of the networks constitute the electrical circuit model of the transducer. The number of elements can be flexibly selected based on the complexity of the structure and parameters to be solved. The advantages of this method are parameters such as mode shape, stress distribution and mechanical quality factor can be easily solved. In the following sections, the FECEM will be introduced using Langevin transducer as an example. For simplicity, the structure of horn is ignored. The transducer is considered to contain front mass, piezoelectric stack and back mass. The front and back masses are supposed to be made of the same material of aluminium. And the piezoelectric material is PZT.

#### **3.1 Electrical circuit element**

Naturally, the Langevin transducer mentioned above can be divided into front mass section, piezoelectric stack section and back mass section. The front and back mass sections are uniform cylinders. Therefore, they can be represented using the same element as shown in Fig. 2 (Lin, 1987).

Fig. 2. Front and back mass elements

This is a two-port network, in which *U*1, *U*2 and *I*1, *I*2 are the voltages and currents at the input and output terminals. Physically, they mean the forces and velocities at the two ends of the vibrating element. Z1 and Z2 are impedances in the network. They can be expressed as

$$Z\_1 = j\rho c S \tan \frac{kl}{2} \tag{1}$$

$$Z\_2 = \frac{-j\rho cS}{\sin kl} \tag{2}$$

where is density, *c* is sound speed, *S* is cross sectional area of the element, *k* is wave number and *l* is the length of the element. Based on Fig. 2, the two-port voltages can be expressed in terms of the two-port currents using the "Z" matrix (Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993)

$$
\begin{bmatrix} \mathcal{U}\_1 \\ \mathcal{U}\_2 \end{bmatrix} = \begin{bmatrix} Z\_{11} & Z\_{12} \\ Z\_{21} & Z\_{22} \end{bmatrix} \begin{bmatrix} I\_1 \\ I\_2 \end{bmatrix} \tag{3}
$$

where

6 Advances in Piezoelectric Transducers

analysis. In this method, the transducer will also be divided into multiple sections. But the section will be further divided into a number of subsections, called element. An element is represented using an electric circuit network. The connections of the networks constitute the electrical circuit model of the transducer. The number of elements can be flexibly selected based on the complexity of the structure and parameters to be solved. The advantages of this method are parameters such as mode shape, stress distribution and mechanical quality factor can be easily solved. In the following sections, the FECEM will be introduced using Langevin transducer as an example. For simplicity, the structure of horn is ignored. The transducer is considered to contain front mass, piezoelectric stack and back mass. The front and back masses are supposed to be made of the same material of aluminium. And the

Naturally, the Langevin transducer mentioned above can be divided into front mass section, piezoelectric stack section and back mass section. The front and back mass sections are uniform cylinders. Therefore, they can be represented using the same element as shown in

This is a two-port network, in which *U*1, *U*2 and *I*1, *I*2 are the voltages and currents at the input and output terminals. Physically, they mean the forces and velocities at the two ends of the vibrating element. Z1 and Z2 are impedances in the network. They can be expressed as

*U*<sup>1</sup> *U*<sup>2</sup>

*Z*2

*Z*<sup>1</sup> *Z*<sup>1</sup>

*I*<sup>1</sup> *I*<sup>2</sup>

<sup>1</sup> tan

<sup>2</sup> sin *j cS <sup>Z</sup>*

number and *l* is the length of the element. Based on Fig. 2, the two-port voltages can be expressed in terms of the two-port currents using the "Z" matrix (Balabanian & Bickart,

> 1 11 12 1 2 21 22 2 *U ZZ I U ZZ I*

*kl Z j cS*

2

(2)

*kl*

is density, *c* is sound speed, *S* is cross sectional area of the element, *k* is wave

(1)

(3)

piezoelectric material is PZT.

**3.1 Electrical circuit element**

Fig. 2. Front and back mass elements

1981; Hayt & Kemmerly, 1993)

Fig. 2 (Lin, 1987).

where

$$\begin{cases} \mathbf{Z}\_{11} = \mathbf{Z}\_{1} + \mathbf{Z}\_{2} \\ \mathbf{Z}\_{12} = \mathbf{Z}\_{2} \\ \mathbf{Z}\_{21} = \mathbf{Z}\_{2} \\ \mathbf{Z}\_{22} = \mathbf{Z}\_{1} + \mathbf{Z}\_{2} \end{cases} \tag{4}$$

This is actually the constitutive relations between the forces and velocities for the elements. This relation can also be represented using the following "chain" or "ABCD" matrix (Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993)

$$
\begin{bmatrix} U\_1 \\ I\_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} U\_2 \\ -I\_2 \end{bmatrix} \tag{5}
$$

where

$$\begin{cases} A = \left(\mathbf{Z}\_1 + \mathbf{Z}\_2\right) / \mathbf{Z}\_2\\ B = \mathbf{Z}\_1 \left(\mathbf{Z}\_1 + 2\mathbf{Z}\_2\right) / \mathbf{Z}\_2\\ \mathbf{C} = \mathbf{1} / \mathbf{Z}\_2\\ D = \left(\mathbf{Z}\_1 + \mathbf{Z}\_2\right) / \mathbf{Z}\_2 \end{cases} \tag{6}$$

The *Z* matrix and *ABCD* matrix can be converted to each other based on the following Eqs. 7 and 8. This provides much convenience for different parameter calculations (Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993).

$$
\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} \mathbf{Z}\_{11}/\mathbf{Z}\_{21} & (\mathbf{Z}\_{11}\mathbf{Z}\_{22} - \mathbf{Z}\_{12}\mathbf{Z}\_{21})/\mathbf{Z}\_{21} \\ \mathbf{1}/\mathbf{Z}\_{21} & \mathbf{Z}\_{22}/\mathbf{Z}\_{21} \end{bmatrix} \tag{7}
$$

$$
\begin{bmatrix} Z\_{11} & Z\_{12} \\ Z\_{21} & Z\_{22} \end{bmatrix} = \begin{bmatrix} A/\mathbb{C} & (AD - BC)/\mathbb{C} \\ 1/\mathbb{C} & D/\mathbb{C} \end{bmatrix} \tag{8}
$$

The piezoelectric stack comprises two parts, PZT and bolt. It can be represented using a three-port network element as shown in Fig. 3, in which *V* is the input electrical voltage, *C*0

Fig. 3. Piezoelectric stack element (Lin, 2004b)

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 9

11 1 1 2 2

*ZZ Z Z Z*

*bolt pzt bolt pzt*

can be built as shown in Fig. 4 (Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993).

*I*<sup>1</sup> *I*<sup>11</sup> *I*<sup>12</sup> *I*i1 *I*i2 *I*n1 *I*n2

*U*<sup>1</sup> *U*<sup>11</sup> *U*<sup>12</sup> *U*i1 *U*i2 *U*n1 *U*n2

*bolt pzt bolt pzt*

(21)

*I*n

*U*n

*bolt pzt bolt pzt*

22 1 1 2 2

The vibrating elements are mechanically connected in series. Therefore the electrical model

*A*<sup>i</sup> *B*<sup>i</sup>

*C*<sup>i</sup> *D*<sup>i</sup> i

*U*(i-1)2=*U*i1

*-I*(i-1)2=*I*i1

The output of one element is the input of the adjacent one. Therefore, the following relation

*U A B A B A B U AB U I C D C D C D CD I I*

> *U U A B ICD I*

For the piezoelectric stack, assuming there are *m* elements, the following relation can be

*n pzt pzt n m n pzt pzt n m U V A B U V*

*UU V AB A B*

*I CD I*

*n n pzt pzt m pzt pzt n pzt pzt n m pzt pzt*

*I CD I C D*

Similarly, for the back mass, supposing the number of elements is *q*, so there is

*front front n front front n*

For the front mass, assuming there are *n* elements, the constitutive relation will be

1 1

*ii nn n n ii nn n n*

1 0

1

(22)

*U*n=*U*n2 -*I*n=-*I*n2

*A*<sup>n</sup> *B*<sup>n</sup>

*C*<sup>n</sup> *D*<sup>n</sup> n

(23)

(24)

(25)

*ZZ Z Z Z*

12 2 2 21 2 2

*ZZ Z ZZ Z*

**3.2 Finite electrical circuit element model**

*A*<sup>1</sup> *B*<sup>1</sup>

*C*<sup>1</sup> *D*<sup>1</sup> 1

Fig. 4. Finite electrical circuit element model

1 11 1 11

Rearranging, Eq. 24 turns to be

exists

*U*1=*U*<sup>11</sup>

*I*1=*I*<sup>11</sup>

obtained

is the static capacitance, is the transformer ratio, Z1bolt, Z2bolt, Z1pzt and Z2pzt are the impedances from the bolt and PZT, respectively. *Z*1bolt and *Z*2bolt can also be expressed using Eqs. 1 and 2, respectively. The expressions for the rest parameters are shown below (Lin, 2004b; Sherrit et al., 1999a, 1999b).

$$Z\_{1\text{p2}t} = j\rho c^{D}S\tan\frac{k^{D}l}{2} \tag{9}$$

$$Z\_{2\text{pxt}} = \frac{-j\,\rho\text{c}^{D}S}{\sin k^{D}l} - \frac{\rho^{2}}{j\rho\text{C}\_{0}}\tag{10}$$

$$\mathbf{C}\_0 = \frac{\mathbf{S}}{l \overline{\beta}\_{33}} \tag{11}$$

$$\varphi = \frac{\mathcal{S}\_{33}\mathcal{S}}{1s\_{33}^D \overline{\mathcal{J}}\_{33}} \tag{12}$$

$$\overline{\beta}\_{33} = \beta\_{33}^T \left( 1 + \frac{g\_{33}^2}{s\_{33}^D \beta\_{33}^T} \right) \tag{13}$$

$$\mathbf{s}\_{33}^{D} = \mathbf{s}\_{33}^{E} \left(\mathbf{1} - k\_{33}\right) \tag{14}$$

$$\mathbf{c}^{D} = \sqrt{\frac{1}{\rho \mathbf{s}\_{33}^{D}}} \tag{15}$$

$$k^D = \frac{\alpha \nu}{\mathcal{L}^D} \tag{16}$$

$$\mathcal{J}\_{33}^T = 1 \;/\; \varepsilon\_{33}^T \tag{17}$$

$$d\_{33} = d\_{33} \mathcal{J}\_{33}^T \tag{18}$$

$$k\_{33} = \frac{d\_{33}}{\sqrt{s\_{33}^{E} \varepsilon\_{33}^{T}}} \tag{19}$$

In the above expressions, *d*33 is piezoelectric coefficient, 33 *T* is dielectric constant, 33 *E s* is compliance coefficient. The constitutive relation for the piezoelectric stack element will hence be

$$
\begin{bmatrix} \mathcal{U}\_1 - \phi V \\ \mathcal{U}\_2 - \phi V \end{bmatrix} = \begin{bmatrix} Z\_{11} & Z\_{12} \\ Z\_{21} & Z\_{22} \end{bmatrix} \begin{bmatrix} I\_1 \\ I\_2 \end{bmatrix} \tag{20}
$$

where

$$\begin{cases} \mathbf{Z}\_{11} = \mathbf{Z}\_{1bult} + \mathbf{Z}\_{1pzt} + \mathbf{Z}\_{2bult} + \mathbf{Z}\_{2pzt} \\ \mathbf{Z}\_{12} = \mathbf{Z}\_{2bult} + \mathbf{Z}\_{2pzt} \\ \mathbf{Z}\_{21} = \mathbf{Z}\_{2bult} + \mathbf{Z}\_{2pzt} \\ \mathbf{Z}\_{22} = \mathbf{Z}\_{1bult} + \mathbf{Z}\_{1pzt} + \mathbf{Z}\_{2bult} + \mathbf{Z}\_{2pzt} \end{cases} \tag{21}$$

#### **3.2 Finite electrical circuit element model**

8 Advances in Piezoelectric Transducers

impedances from the bolt and PZT, respectively. *Z*1bolt and *Z*2bolt can also be expressed using Eqs. 1 and 2, respectively. The expressions for the rest parameters are shown below (Lin,

1p tan

*zt*

*jcS <sup>Z</sup>*

0

33 33

1 *<sup>T</sup>*

*<sup>D</sup>* 1 *<sup>D</sup> c*

> *D <sup>D</sup> k c*

33

In the above expressions, *d*33 is piezoelectric coefficient, 33

hence be

where

*<sup>d</sup> <sup>k</sup> s*

33 33 1 / *T T*

33 33 33 *<sup>T</sup> g d*

compliance coefficient.

The vibrating elements are mechanically connected in series. Therefore the electrical model can be built as shown in Fig. 4 (Balabanian & Bickart, 1981; Hayt & Kemmerly, 1993).

$$\begin{array}{c c c c c c c c c} I\_1 & I\_{11} & I\_{12} & I\_{12} & \dots & I\_{1n} & I\_{1n} & I\_{12} & \dots & I\_{1n} & I\_{1n} \\ \hline U\_1 & U\_{11} & & I\_{11} & & U\_{12} & & U\_{1n} & & \dots & & I\_{1n} & I\_{1n} \\ \hline & & & & & & & & & & & & & \\ \hline & & & & & & & & & & & & & \\ U\_1 = U\_{11} & & & & & & & & & & & & \\ I\_1 = I\_{11} & & & & & & & & & & & \\ I\_1 = I\_{11} & & & & & & & & & & & \end{array}$$

Fig. 4. Finite electrical circuit element model

The output of one element is the input of the adjacent one. Therefore, the following relation exists

$$
\begin{bmatrix} \boldsymbol{U}\_{1} \\ \boldsymbol{I}\_{1} \end{bmatrix} = \begin{bmatrix} \boldsymbol{A}\_{1} & \boldsymbol{B}\_{1} \\ \boldsymbol{C}\_{1} & \boldsymbol{D}\_{1} \end{bmatrix} \bullet \cdots \bullet \begin{bmatrix} \boldsymbol{A}\_{i} & \boldsymbol{B}\_{i} \\ \boldsymbol{C}\_{i} & \boldsymbol{D}\_{i} \end{bmatrix} \bullet \cdots \bullet \begin{bmatrix} \boldsymbol{A}\_{n} & \boldsymbol{B}\_{n} \\ \boldsymbol{C}\_{n} & \boldsymbol{D}\_{n} \end{bmatrix} \begin{bmatrix} \boldsymbol{U}\_{n} \\ -\boldsymbol{I}\_{n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{U}\_{n} \\ -\boldsymbol{I}\_{n} \end{bmatrix} \tag{22}
$$

For the front mass, assuming there are *n* elements, the constitutive relation will be

$$
\begin{bmatrix} \mathcal{U}\_1 \\ I\_1 \end{bmatrix} = \begin{bmatrix} A\_{front} & B\_{ffront} \\ \mathcal{C}\_{front} & D\_{ffront} \end{bmatrix} \begin{bmatrix} \mathcal{U}\_n \\ -I\_n \end{bmatrix} \tag{23}
$$

For the piezoelectric stack, assuming there are *m* elements, the following relation can be obtained

$$
\begin{bmatrix} \mathcal{U}\_n - \varphi V \\ -I\_n \end{bmatrix} = \begin{bmatrix} A\_{pzt} & B\_{pzt} \\ C\_{pzt} & D\_{pzt} \end{bmatrix} \begin{bmatrix} \mathcal{U}\_{n+m} - \varphi V \\ -I\_{n+m} \end{bmatrix} \tag{24}
$$

Rearranging, Eq. 24 turns to be

$$
\begin{bmatrix} \mathbf{U}\_n \\ -I\_n \end{bmatrix} = \begin{bmatrix} A\_{pzt} & B\_{pzt} \\ C\_{pzt} & D\_{pzt} \end{bmatrix} \begin{bmatrix} \mathbf{U}\_{n+m} \\ -I\_{n+m} \end{bmatrix} + \begin{bmatrix} A\_{pzt} - \mathbf{1} & B\_{pzt} \\ C\_{pzt} & D\_{pzt} - \mathbf{1} \end{bmatrix} \begin{bmatrix} -\rho V \\ 0 \end{bmatrix} \tag{25}
$$

Similarly, for the back mass, supposing the number of elements is *q*, so there is

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 11

*E d T E*

where *E* is Young's modus, *v* is velocity and *x* is position. The stress is an important factor

The input current to the transducer is solved using the following relation (Lin, 2004b)

*I V j*

where *I*1 and *I*2 are velocities at the surface of the PZT. Solving the current, the electrical properties such as impedance and power consumption can hence be estimated accordingly.

As seen in Section 3.2.2, before the harmonic analysis, an equivalent load *R* has to be assumed. Without it, the vibration magnitude of the transducer will go to infinity. In practice, the load can be the results of the external loadings or the internal losses. In the case of free vibration, the load is due to the internal mechanical loss, which is the function of the

> *m <sup>M</sup> <sup>R</sup> Q*

The mechanical loss per unit volume is estimated using the following equation (Li et al.,

1 11 1 2 2 tan

tangent, and <sup>2</sup> *w sT* 1 2 is the elastic energy density. In Eq. 35, *s* and *Qm* is solely material dependant. But the stress *T* not only depends on the materials, but also relies on the position *x*. Therefore, to calculate the total loss for a single material *i*, the following integration should

> *l l* 11 1 *i i*

*mi mi mi*

(36)

1

 (37)

1 *<sup>n</sup>*

*i mi*

*Q*

*QQ Q*

where *M* is the equivalent mass. To find *M* and *Qm*, the following method is applied.

*W sT sT w Q Q*

0 0

1 2

*W w Adx wAdx W*

where *A* is the cross sectional area of the material, *Wi* is the elastic energy for material *i*. Assuming the transducer is made of *n* different materials, total mechanical loss of the

*total n i*

*W WW W W*

2 2 *<sup>m</sup>*

in which *s* is the compliance coefficient, *T* is the stress, tan 1

that affects heat generation and vibration amplitude (Li et al., 2007b, 2007c).

**3.2.3 Resistive load and mechanical quality factor**

mechanical quality factor *Qm* (Chen & Wu, 2002)

**(a) Mechanical quality factor**

transducer can hence be calculated as

2007b, 2007c)

be applied

*dx*

(32)

*C II* 0 12 (33)

(34)

*m m Q* is the mechanical loss

*m m*

 (35)

$$
\begin{bmatrix} \boldsymbol{\mathcal{U}}\_{n+m} \\ -\boldsymbol{I}\_{n+m} \end{bmatrix} = \begin{bmatrix} \boldsymbol{A}\_{back} & \boldsymbol{B}\_{back} \\ \boldsymbol{\mathcal{C}}\_{back} & \boldsymbol{D}\_{back} \end{bmatrix} \begin{bmatrix} \boldsymbol{\mathcal{U}}\_{n+m+q} \\ -\boldsymbol{I}\_{n+m+q} \end{bmatrix} \tag{26}
$$

Combining Eqs. 23, 25 and 26, the relation for the whole system can be obtained as

$$
\begin{bmatrix} \mathbf{U}\_1 \\ I\_1 \end{bmatrix} = T\_1 \begin{bmatrix} \mathbf{U}\_{n+m+q} \\ -I\_{n+m+q} \end{bmatrix} + T\_2 \begin{bmatrix} -\rho V \\ \mathbf{0} \end{bmatrix} \tag{27}
$$

where

$$
\begin{bmatrix} T\_1 = \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} A\_{front} & B\_{front} \\ C\_{front} & D\_{front} \end{bmatrix} \begin{bmatrix} A\_{pzt} & B\_{pzt} \\ C\_{pzt} & D\_{pzt} \end{bmatrix} \begin{bmatrix} A\_{buck} & B\_{back} \\ C\_{buck} & D\_{back} \end{bmatrix} \tag{28}
$$

$$\begin{bmatrix} T\_2 = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} = \begin{bmatrix} A\_{frunt} & B\_{front} \\ C\_{front} & D\_{front} \end{bmatrix} \begin{bmatrix} A\_{pzt} - \mathbf{1} & B\_{pzt} \\ C\_{pzt} & D\_{pzt} - \mathbf{1} \end{bmatrix} \tag{29}$$

#### **3.2.1 Modal analysis**

Modal analysis is to find the resonant frequencies and the corresponding mode shapes of the transducer. The resonant frequencies can be found using the system Z matrix. Assuming the short circuit condition, i.e.,*V*=0, transform the *ABCD* matrix *T*1 into the Z matrix Z1 based on Eq. 7, and then set the determinant to zero

$$\left| Z\_1 \right| = 0 \tag{30}$$

The solutions of the equation are the resonant frequencies. The mode shape can be obtained based on the relation in Eq. 22. Assuming *I*1=1, *U*1=0, i.e., unit velocity and zero force at the free end, the velocity at each point can hence be solved. In another word, the mode shape can be found.

#### **3.2.2 Harmonic analysis**

The harmonic analysis is to find the performance of the transducer as a function of frequency. The parameters interested might include magnitude of velocity, magnitude of stress, power consumption, impedance characteristics, etc. This can be done based on the system relation Eq. 27. Assuming one mechanical port is loaded with a resistor *R*, the other mechanical port is short circuited, i.e., no force, and the electrical port is applied with 1 V, Eq. 27 will turn to

$$
\begin{bmatrix} I\_1 R \\ I\_1 \end{bmatrix} = T\_1 \begin{bmatrix} 0 \\ -I\_{n+m+q} \end{bmatrix} + T\_2 \begin{bmatrix} -qV \\ 0 \end{bmatrix} \tag{31}
$$

Solving the above equation, the velocity at the two ends (*I*1 and *I*n+m+q) can hence be obtained. The stress can then be estimated based on the mode shape, velocity obtained and the following relation (Silva, 1999)

$$T = E\varepsilon = \frac{E}{\alpha} \frac{d\nu}{d\mathbf{x}}\tag{32}$$

where *E* is Young's modus, *v* is velocity and *x* is position. The stress is an important factor that affects heat generation and vibration amplitude (Li et al., 2007b, 2007c).

The input current to the transducer is solved using the following relation (Lin, 2004b)

$$I = V \cdot j \alpha \mathbb{C}\_0 - \varphi \left( I\_1 + I\_2 \right) \tag{33}$$

where *I*1 and *I*2 are velocities at the surface of the PZT. Solving the current, the electrical properties such as impedance and power consumption can hence be estimated accordingly.

#### **3.2.3 Resistive load and mechanical quality factor**

As seen in Section 3.2.2, before the harmonic analysis, an equivalent load *R* has to be assumed. Without it, the vibration magnitude of the transducer will go to infinity. In practice, the load can be the results of the external loadings or the internal losses. In the case of free vibration, the load is due to the internal mechanical loss, which is the function of the mechanical quality factor *Qm* (Chen & Wu, 2002)

$$R = \frac{\alpha M}{Q\_m} \tag{34}$$

where *M* is the equivalent mass. To find *M* and *Qm*, the following method is applied.

#### **(a) Mechanical quality factor**

10 Advances in Piezoelectric Transducers

*n m back back nmq n m back back n m q*

1 2 <sup>1</sup> 0 *nmq nmq U V U T T*

*A B A B AB A B*

*CD C D C D C D*

*A B AB A B*

*CD C D C D*

Modal analysis is to find the resonant frequencies and the corresponding mode shapes of the transducer. The resonant frequencies can be found using the system Z matrix. Assuming the short circuit condition, i.e.,*V*=0, transform the *ABCD* matrix *T*1 into the Z matrix Z1

The solutions of the equation are the resonant frequencies. The mode shape can be obtained based on the relation in Eq. 22. Assuming *I*1=1, *U*1=0, i.e., unit velocity and zero force at the free end, the velocity at each point can hence be solved. In another word, the mode shape

The harmonic analysis is to find the performance of the transducer as a function of frequency. The parameters interested might include magnitude of velocity, magnitude of stress, power consumption, impedance characteristics, etc. This can be done based on the system relation Eq. 27. Assuming one mechanical port is loaded with a resistor *R*, the other mechanical port is short circuited, i.e., no force, and the electrical port is applied with 1 V,

1 2

Solving the above equation, the velocity at the two ends (*I*1 and *I*n+m+q) can hence be obtained. The stress can then be estimated based on the mode shape, velocity obtained and

*I R V T T*

*nmq* 0

0

1

1

*I I*

1

1

<sup>1</sup> *Z* 0 (30)

*front front pzt pzt back back front front pzt pzt back back*

> *front front pzt pzt front front pzt pzt*

(26)

(27)

(28)

(29)

(31)

*U AB U I CD I*

Combining Eqs. 23, 25 and 26, the relation for the whole system can be obtained as

1

1

2

based on Eq. 7, and then set the determinant to zero

*T*

*T*

**3.2.1 Modal analysis**

can be found.

Eq. 27 will turn to

**3.2.2 Harmonic analysis**

the following relation (Silva, 1999)

where

*I I*

The mechanical loss per unit volume is estimated using the following equation (Li et al., 2007b, 2007c)

$$\mathcal{W} = \frac{1}{2} \rho \text{s} \, T^2 \, \tan \delta\_m = \frac{1}{2} \, \text{s} \, T^2 \, \cdot \, \text{o} \cdot \frac{1}{Q\_m} = \text{w} \cdot \text{o} \cdot \frac{1}{Q\_m} \tag{35}$$

in which *s* is the compliance coefficient, *T* is the stress, tan 1 *m m Q* is the mechanical loss tangent, and <sup>2</sup> *w sT* 1 2 is the elastic energy density. In Eq. 35, *s* and *Qm* is solely material dependant. But the stress *T* not only depends on the materials, but also relies on the position *x*. Therefore, to calculate the total loss for a single material *i*, the following integration should be applied

$$\text{Cov}\_{i}^{\prime} = \int\_{0}^{l} \text{cov} \frac{1}{Q\_{mi}} A \, d\mathbf{x} = \frac{1}{Q\_{mi}} \, o \int\_{0}^{l} w A \, d\mathbf{x} = o \mathcal{W}\_{i} \frac{1}{Q\_{mi}} \tag{36}$$

where *A* is the cross sectional area of the material, *Wi* is the elastic energy for material *i*. Assuming the transducer is made of *n* different materials, total mechanical loss of the transducer can hence be calculated as

$$\mathcal{W}'\_{total} = \mathcal{W}'\_1 + \mathcal{W}'\_2 + \dots + \mathcal{W}'\_n = o\sum\_{i=1}^n \mathcal{W}\_i \frac{1}{Q\_{mi}} \tag{37}$$

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 13

circuit model. The mathematical respresention of the circuit model was also estibilished. Accordingly, the modal analysis and harmonic analysis can be achieved by solving the network equations. The mechanical quality factor and equivalent mass were derived, too. The FECEM was also compared with ANSYS 2D modelling. The results show reasonable

The piezoelectric transducer can also be analyzed using ANSYS 2D simulation (ANSYS, Inc., 2010; Li et al., 2008, 2009, 2010). One of the advantages of this method is that the acoustic field with various boundary conditions can be coupled to the piezoelectric transducer. Therefore, the model is useful to analyze the response of the transducer under different loading conditions. It is also essential for the study of the sound pressure distributions at

Fig. 6 shows the transducer-fluid coupled model, in which the 2D model revolving about the axial of rotation will produce the 3D structure of the system. The liquid medium is water which directly contacts the mechanical structure of the piezoelectric transducer at the structure-liquid interface. For the piezoelectric transducers, the axisymmetric structure element Plane42 for metallic parts and couple field element Plane13 for PZT materials were applied. The Plane42 element has degrees of freedom UX and UY, and Plane13 element has degrees of freedom VOLT, UX and UY. For liquid medium, acoustic element Fluid29 was applied. In the acoustic domain, it has degrees of freedom PRES. And at the boundary and interface, it possesses degrees of freedom UX, UY and PRES. The boundary condition can be controlled by the absorption coefficient MU. In this case, the horizontal boundary was set "hard" and curved boundary is set "absorbing", corresponding to MU=0 and MU=1,

Length (mm)

 ANSYS 2D FECEM 1D

Fig. 5. Comparison of mode shape of Langevin transducer

0.0

Relative displacement

accuracy.

**4. ANSYS 2D modelling**

different boundary conditions.

0.5

1.0

On the other hand, the mechanical loss can also be expressed as

$$\mathcal{W}'\_{\text{total}} = o\mathcal{W}\mathcal{V}\_{\text{total}}\frac{1}{\mathcal{Q}\_m} \tag{38}$$

where 1 *n W W total i* is the total elastic energy, and *Qm* is the equivalent mechanical quality

factor of the transducer. Equalizing Eq. 37 and 38, and rearranging, the following relation can finally be obtained

$$\frac{\mathcal{W}\_{total}}{Q\_m} = \sum\_{i=1}^{n} \frac{\mathcal{W}\_i}{Q\_{mi}} \tag{39}$$

This equation means the equivalent quality factor *Qm* can be estimated using the mechanical elastic energy and the individual material quality factor.

#### **(b) Equivalent mass M**

Because the piezoelectric transducer can be modelled using both lumped parameters and distributed parameters, the following relation exists (Li, 2004)

$$\frac{1}{2}\frac{1}{2}MV\_t^2 = \int\_0^l \frac{1}{2}v\_\chi^{-2}\rho A d\mathbf{x} \tag{40}$$

where *v*x is the velocity at position *x*, is density, *A* is cross sectional area and *Vt* is the velocity at the tip. In Eq. 40, the left part is the kinetic energy of the transducer expressed in lumped parameters and right part in distributed parameters. From the derived mode shape mentioned earlier, *v*x in Eq. 40 can be easily known. Therefore the equivalent mass can be conveniently solved.

#### **3.2.4 Results for Langevin transducer**

Table 1 and Fig. 5 compare the results between FECEM modelling and ANSYS 2D modelling (details in next section). It can be seen that the results are reasonably consistent. Therefore the FECEM theory is effective.

Table 1. Comparison of simulation results between ANSYS and FECEM theories

#### **3.3 Concluding remarks**

The FECEM theory is introduced in this section. This theory is useful for both transducer design and performance analysis. Based on this theory, the transducer can be divided into finite numbers of electrical cricuit elements, which are connected to form the electrical

Fig. 5. Comparison of mode shape of Langevin transducer

circuit model. The mathematical respresention of the circuit model was also estibilished. Accordingly, the modal analysis and harmonic analysis can be achieved by solving the network equations. The mechanical quality factor and equivalent mass were derived, too. The FECEM was also compared with ANSYS 2D modelling. The results show reasonable accuracy.

#### **4. ANSYS 2D modelling**

12 Advances in Piezoelectric Transducers

*total total*

factor of the transducer. Equalizing Eq. 37 and 38, and rearranging, the following relation

1 *<sup>n</sup> total <sup>i</sup> m i mi W W Q Q*

This equation means the equivalent quality factor *Qm* can be estimated using the mechanical

Because the piezoelectric transducer can be modelled using both lumped parameters and

2 2 0

velocity at the tip. In Eq. 40, the left part is the kinetic energy of the transducer expressed in lumped parameters and right part in distributed parameters. From the derived mode shape mentioned earlier, *v*x in Eq. 40 can be easily known. Therefore the equivalent mass can be

Table 1 and Fig. 5 compare the results between FECEM modelling and ANSYS 2D modelling (details in next section). It can be seen that the results are reasonably consistent. Therefore

Model Resonant frequency Qm velocity Conductance G

The FECEM theory is introduced in this section. This theory is useful for both transducer design and performance analysis. Based on this theory, the transducer can be divided into finite numbers of electrical cricuit elements, which are connected to form the electrical

ANSYS (2D) 65845 849 0.3 0.2 FECEM 67471 816 0.32 0.24 Difference 2% 4% 6% 20%

Table 1. Comparison of simulation results between ANSYS and FECEM theories

*l MV v Adx t x*

1 1 2 2

*W W <sup>Q</sup>*

1

*m*

(39)

(40)

is density, *A* is cross sectional area and *Vt* is the

*W W total i* is the total elastic energy, and *Qm* is the equivalent mechanical quality

(38)

On the other hand, the mechanical loss can also be expressed as

elastic energy and the individual material quality factor.

distributed parameters, the following relation exists (Li, 2004)

where

1

can finally be obtained

**(b) Equivalent mass M**

conveniently solved.

where *v*x is the velocity at position *x*,

**3.2.4 Results for Langevin transducer**

the FECEM theory is effective.

**3.3 Concluding remarks**

*n*

The piezoelectric transducer can also be analyzed using ANSYS 2D simulation (ANSYS, Inc., 2010; Li et al., 2008, 2009, 2010). One of the advantages of this method is that the acoustic field with various boundary conditions can be coupled to the piezoelectric transducer. Therefore, the model is useful to analyze the response of the transducer under different loading conditions. It is also essential for the study of the sound pressure distributions at different boundary conditions.

Fig. 6 shows the transducer-fluid coupled model, in which the 2D model revolving about the axial of rotation will produce the 3D structure of the system. The liquid medium is water which directly contacts the mechanical structure of the piezoelectric transducer at the structure-liquid interface. For the piezoelectric transducers, the axisymmetric structure element Plane42 for metallic parts and couple field element Plane13 for PZT materials were applied. The Plane42 element has degrees of freedom UX and UY, and Plane13 element has degrees of freedom VOLT, UX and UY. For liquid medium, acoustic element Fluid29 was applied. In the acoustic domain, it has degrees of freedom PRES. And at the boundary and interface, it possesses degrees of freedom UX, UY and PRES. The boundary condition can be controlled by the absorption coefficient MU. In this case, the horizontal boundary was set "hard" and curved boundary is set "absorbing", corresponding to MU=0 and MU=1,

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 15

curved absorbing boundary. The maximum pressure is located at the centre of the transducer surface. It decreases as the distance from the transducer surface increases. The magnitude and distribution of the pressure wave are sensitive to the boundary conditions. Therefore the

Thrombolysis transducer is a device used to deal with the vascular thrombotic occlusive disease, which is a major cause of morbidity and mortality in the developed world. The development of a blood clot or thrombus in a blood vessel compromises distal blood flow and is the usual cause of a heart attack or stroke. Established treatment is the urgent removal or dissipation of the occluding thrombus. This is achieved with the use of a simple aspiration catheter, mechanical thrombectomy, or pharmacological agents such as thrombolytic drugs (Atar et al., 1999; Brosh et al., 1998; Janis et al., 2000; Ma et al., 2008; Siegel & Luo, 2008). Ultrasonic emulsification of the blood clot is another technique for thrombolysis. This is achieved by acoustic cavitation and mechanical fragmentation (Bond & Cimino, 1996; Chan et al., 1986). Compared with conventional mechanical thrombectomy techniques ultrasonic thrombolysis exhibits the advantage of inherent tissue selectivity (Rosenschein et al., 1994; Tschepe et al., 1994). This is because thrombus is highly susceptible to ultrasonic cavitational emulsification, while the arterial walls, which are lined with cavitation-resistant matrix of collagen and elastin, are not. Ultrasound energy has also been shown to improve myocardial reperfusion in the presence of coronary occlusion (Siegel et al., 2004). The system of ultrasonic thrombolysis generally contains a generator, a transducer and a catheter. The tip of the transducer will go through the catheter lumen to the blood clot. Then the generator will provide control signal and power to the transducer. The vibration of the tip will emulsify the blood clot. In the following sections, the thrombolysis transducer developed in NTU (Nanyang Technological University) will be introduced.

Fig. 9 shows the structure of the piezoelectric thrombolysis transducer designed in the present work. The transducer consists of five parts. The first part is the end cap. It serves to prestress the adjacent PZT stack and also adjusts the mechanical impedance applied to the stack. The second part is the PZT stack, which is clamped between the end cap and the horn. It is the most crucial part of the transducer, where the vibration is generated. The third part

boundary conditions should be optimized accordingly based on the applications.

Fig. 8. Sound pressure pattern in the liquid domain

**5. Thrombolysis transducer**

**5.1 Structure**

respectively. Fig. 6 (b) shows the meshed model of the system. After meshing, an electrical load of 1 V was applied to the PZT stack. To be noted, the transducer analysis and the acoustic analysis can be modelled separately, which might be more efficient for certain problems.

Fig. 6. 2D modelling of the transducer-fluid coupled system, (a) model and (b) meshed model

Fig. 7 shows the harmonic analysis results of the transducer displacement at the resonant frequency. The colour represents the magnitude of the displacement in the unit of meter. ANSYS provides the harmonic displacement with the real and imaginary part. Because at resonance, the electrical excitation and displacement response has 90 deg phase difference, the imaginary part dominates. It is also noticed that the resonant frequency is 64510 Hz in the figure, which is different from 65845 Hz in Table 1. This is because the liquid loading changes the resonant frequency of the transducer.

Fig. 7. Real and imaginary part of the transducer displacement, (a) real, (b) imaginary

Fig. 8 shows the acoustic pressure distribution in the acoustic field. The colour represents the magnitude of the pressure wave in the unit of dB. In this case, the transducer-fluid interface is the pressure wave radiation surface. The horizontal hard boundary resembles an infinite baffle. A travelling pressure wave is therefore generated at the transducer radiation surface. It then propagates in the half hemisphere liquid domain.  The maximum pressure is located at the centre of the transducer surface. It decreases as the distance from the transducer surface increases. The magnitude and distribution of the pressure wave are sensitive to the boundary conditions. Therefore the boundary conditions should be optimized accordingly based on the applications.

Fig. 8. Sound pressure pattern in the liquid domain

#### **5. Thrombolysis transducer**

14 Advances in Piezoelectric Transducers

respectively. Fig. 6 (b) shows the meshed model of the system. After meshing, an electrical load of 1 V was applied to the PZT stack. To be noted, the transducer analysis and the acoustic analysis can be modelled separately, which might be more efficient for certain

(a) (b)

Fig. 6. 2D modelling of the transducer-fluid coupled system, (a) model and (b) meshed

Boundaries

Liquid

Fig. 7. Real and imaginary part of the transducer displacement, (a) real, (b) imaginary

Fig. 8 shows the acoustic pressure distribution in the acoustic field. The colour represents the magnitude of the pressure wave in the unit of dB. In this case, the transducer-fluid interface is the pressure wave radiation surface. The horizontal hard boundary resembles an infinite baffle. A travelling pressure wave is therefore generated at the transducer radiation surface. It then propagates in the half hemisphere liquid domain. And finally it is fully absorbed at the

changes the resonant frequency of the transducer.

(a) (b)

Fig. 7 shows the harmonic analysis results of the transducer displacement at the resonant frequency. The colour represents the magnitude of the displacement in the unit of meter. ANSYS provides the harmonic displacement with the real and imaginary part. Because at resonance, the electrical excitation and displacement response has 90 deg phase difference, the imaginary part dominates. It is also noticed that the resonant frequency is 64510 Hz in the figure, which is different from 65845 Hz in Table 1. This is because the liquid loading

problems.

Axis of rotation Interface

Transducer PZT

model

Thrombolysis transducer is a device used to deal with the vascular thrombotic occlusive disease, which is a major cause of morbidity and mortality in the developed world. The development of a blood clot or thrombus in a blood vessel compromises distal blood flow and is the usual cause of a heart attack or stroke. Established treatment is the urgent removal or dissipation of the occluding thrombus. This is achieved with the use of a simple aspiration catheter, mechanical thrombectomy, or pharmacological agents such as thrombolytic drugs (Atar et al., 1999; Brosh et al., 1998; Janis et al., 2000; Ma et al., 2008; Siegel & Luo, 2008). Ultrasonic emulsification of the blood clot is another technique for thrombolysis. This is achieved by acoustic cavitation and mechanical fragmentation (Bond & Cimino, 1996; Chan et al., 1986). Compared with conventional mechanical thrombectomy techniques ultrasonic thrombolysis exhibits the advantage of inherent tissue selectivity (Rosenschein et al., 1994; Tschepe et al., 1994). This is because thrombus is highly susceptible to ultrasonic cavitational emulsification, while the arterial walls, which are lined with cavitation-resistant matrix of collagen and elastin, are not. Ultrasound energy has also been shown to improve myocardial reperfusion in the presence of coronary occlusion (Siegel et al., 2004). The system of ultrasonic thrombolysis generally contains a generator, a transducer and a catheter. The tip of the transducer will go through the catheter lumen to the blood clot. Then the generator will provide control signal and power to the transducer. The vibration of the tip will emulsify the blood clot. In the following sections, the thrombolysis transducer developed in NTU (Nanyang Technological University) will be introduced.

#### **5.1 Structure**

Fig. 9 shows the structure of the piezoelectric thrombolysis transducer designed in the present work. The transducer consists of five parts. The first part is the end cap. It serves to prestress the adjacent PZT stack and also adjusts the mechanical impedance applied to the stack. The second part is the PZT stack, which is clamped between the end cap and the horn. It is the most crucial part of the transducer, where the vibration is generated. The third part

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 17

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Length (m)

proportional to the source strength (International standard, IEC 1998; Li et al., 2009). Therefore, in the present work, a horn is applied to amplify the vibration velocity from the PZT crystal. And a ball or a short cylinder tip is attached at the distal end of the

The acoustic properties of the transducer were characterized qualitatively. As the acoustic pressure becomes larger and larger, two phenomena might be observed around the tip of the transducer, i.e., cavitation and streaming (Abramov, 1994, 1998; Young, 1999). Cavitation is the generation and burst of bubbles in the liquid due to the high amplitude of the acoustic

Fig. 10. Vibration amplitude distribution along the length of the transducer

Fig. 11. Acoustic pressure pattern generated by the tip in the liquid

**5.4 Acoustic characterization**

Transmission wire

Tip

transmission wire to enhance the radiation area.

0.0

Relative amplitude

0.5

1.0

Horn

PZT stack + End cap

1.5

of the transducer is the horn, which functions to magnify the displacement produced by the PZT stack. The fourth part is a long and thin transmission wire, which should be flexible but sufficiently stiff for energy transmission. The last part is a distal vibration tip that consists of a ball or a short cylinder with an enlarged diameter (~ 1.5 mm) compared to the connecting transmission wire (~ 0.5 mm). The enlarged diameter increases acoustic power emission to the surrounding liquid and blood clot. The vibration produced by the PZT stack is transmitted through the horn and then the transmission wire to the distal vibration tip. The acoustic energy emitted from the tip is finally used to emulsify the clot.

Fig. 9. Schematic illustration of the piezoelectric thrombolysis device

The transducer operates at ~26.7 kHz longitudinal vibration mode. This is a low ultrasonic frequency aiming at reducing the heat generation (Francis, 2001; Li et al., 2007b; Siegel et al., 2000). The diameter of the PZT stack is 10 mm. Maximum input power is 20 W. The length of transmission wire is 1 m, which is made of a high strength material, Ti-6Al-4V, for achieving a high vibration velocity (Li et al., 2007b; Mason & Wehr, 1970; Muhlen, 1990). For practical operations, due to the long length of the transmission wire, the bending mode could be excited, which would increase loss and decrease the efficiency of the device. To avoid this, the transmission wire should be coaxial with the horn.

#### **5.2 FECEM modeling**

Because of the high aspect ratio of the device (length/diameter), it is very advantageous to model the device using the FECEM theory. Fig. 10 shows the mode shape of the transducer. It can be seen that along the axial direction, the displacement generated by the PZT stack is first amplified by the horn, then further amplified by the transmission wire. As a result, the tip has a much larger displacement than the PZT stack. Also considering the smaller area of the vibration tip to the PZT stack, it can be said that the energy produced by the PZT stack has been focused to the tip through this design. Therefore, the tip will work effectively for the blood clot emulsification.

#### **5.3 ANSYS 2D acoustic modeling**

During practical operation, the vibration tip will be surrounded by liquid and produces an acoustic field. Fig. 11 shows the simulation of the acoustic field generated by an OD 1.5 mm and length 3 mm vibration tip. The tip is connected to an OD 0.5 mm transmission wire, and the vibration frequency of the tip is 30 kHz. The simulation was carried out using ANSYS 2D acoustic analysis. Fig. 11 shows that the maximum ultrasonic pressure is located at the top and bottom surfaces of the vibration tip, which is normal to the displacement direction. This indicates that emulsification should be most effective at these locations. The radiation area multiplied by the normal surface velocity of the tip is known as the source strength (International standard, IEC 1998; Li et al., 2009). The ultrasonic pressure amplitude is 16 Advances in Piezoelectric Transducers

of the transducer is the horn, which functions to magnify the displacement produced by the PZT stack. The fourth part is a long and thin transmission wire, which should be flexible but sufficiently stiff for energy transmission. The last part is a distal vibration tip that consists of a ball or a short cylinder with an enlarged diameter (~ 1.5 mm) compared to the connecting transmission wire (~ 0.5 mm). The enlarged diameter increases acoustic power emission to the surrounding liquid and blood clot. The vibration produced by the PZT stack is transmitted through the horn and then the transmission wire to the distal vibration tip. The

The transducer operates at ~26.7 kHz longitudinal vibration mode. This is a low ultrasonic frequency aiming at reducing the heat generation (Francis, 2001; Li et al., 2007b; Siegel et al., 2000). The diameter of the PZT stack is 10 mm. Maximum input power is 20 W. The length of transmission wire is 1 m, which is made of a high strength material, Ti-6Al-4V, for achieving a high vibration velocity (Li et al., 2007b; Mason & Wehr, 1970; Muhlen, 1990). For practical operations, due to the long length of the transmission wire, the bending mode could be excited, which would increase loss and decrease the efficiency of the device. To

End cap PZT stack Horn Transmission wire Tip

Because of the high aspect ratio of the device (length/diameter), it is very advantageous to model the device using the FECEM theory. Fig. 10 shows the mode shape of the transducer. It can be seen that along the axial direction, the displacement generated by the PZT stack is first amplified by the horn, then further amplified by the transmission wire. As a result, the tip has a much larger displacement than the PZT stack. Also considering the smaller area of the vibration tip to the PZT stack, it can be said that the energy produced by the PZT stack has been focused to the tip through this design. Therefore, the tip will work effectively for

During practical operation, the vibration tip will be surrounded by liquid and produces an acoustic field. Fig. 11 shows the simulation of the acoustic field generated by an OD 1.5 mm and length 3 mm vibration tip. The tip is connected to an OD 0.5 mm transmission wire, and the vibration frequency of the tip is 30 kHz. The simulation was carried out using ANSYS 2D acoustic analysis. Fig. 11 shows that the maximum ultrasonic pressure is located at the top and bottom surfaces of the vibration tip, which is normal to the displacement direction. This indicates that emulsification should be most effective at these locations. The radiation area multiplied by the normal surface velocity of the tip is known as the source strength (International standard, IEC 1998; Li et al., 2009). The ultrasonic pressure amplitude is

acoustic energy emitted from the tip is finally used to emulsify the clot.

Fig. 9. Schematic illustration of the piezoelectric thrombolysis device

avoid this, the transmission wire should be coaxial with the horn.

**5.2 FECEM modeling**

the blood clot emulsification.

**5.3 ANSYS 2D acoustic modeling**

Fig. 10. Vibration amplitude distribution along the length of the transducer

proportional to the source strength (International standard, IEC 1998; Li et al., 2009). Therefore, in the present work, a horn is applied to amplify the vibration velocity from the PZT crystal. And a ball or a short cylinder tip is attached at the distal end of the transmission wire to enhance the radiation area.

Fig. 11. Acoustic pressure pattern generated by the tip in the liquid

#### **5.4 Acoustic characterization**

The acoustic properties of the transducer were characterized qualitatively. As the acoustic pressure becomes larger and larger, two phenomena might be observed around the tip of the transducer, i.e., cavitation and streaming (Abramov, 1994, 1998; Young, 1999). Cavitation is the generation and burst of bubbles in the liquid due to the high amplitude of the acoustic

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 19

The piezoelectric thrombolysis transducer consists of an end cap, a PZT stack, a horn, a transmission wire and a vibration tip. The transducers vibrate longitudinally and generate maximum acoustic pressure at the tip. The acoustic pressure induces effects such as cavitation and streaming at the tip. Blood clot can be effectively emulsified using the

The piezoelectric pump has various designs (Laser & Santiago, 2004; Luong & Nguyen, 2010; Yeo et al., 2008). According to frequency, the pumping mechanism will be different. At the sonic low frequency range, the diaphragm pump is a popular design, which utilizes the reciprocating movement of piezo transducer to displace the liquid. However, as frequency increases, the diaphragm mechanism becomes more and more difficult to achieve. One of the important reasons is the response mismatch between the diaphragm and valves (Hu et al., 2004; Zhang et al., 2003). However, there are other pumping mechanisms. At the ultrasonic and megasonic frequency range, the acoustic effect could be used (Chen & Lal, 2006; Frampton et al., 2004; Hasegawa et al., 2008; Koyama, et al., 2010; Li et al., 2010; Zhang et al., 2010). It has been well known that the intense and nonlinear ultrasound could induce effect such as cavitation and streaming, which are able to generate the flow of the liquid (Abramov, 1994; Li et al., 2010; Raton et al., 2007). This work will report a piezoelectric

Fig. 14 shows the structure of the piezoelectric pump designed in NTU. The pump comprises a piezoelectric transducer, a reflecting condenser and a casing. The structure of the transducer is similar as the thrombolysis transducer, but without transmission wire. There is a through hole inside the transducer working as the flow path. The transducer works approximately at 30 kHz. The reflecting condenser has a "" shape, which focused the acoustic energy at its corner. The casing has a cylindrical shape with inlet on the wall. The casing, condenser and horn surface form the boundaries of the acoustic field. Because the pump mechanism is based on the acoustic effect, the pressure magnitude and pattern inside the field will affect the performance of the pump significantly. The distribution of

sound pressure level inside the casing is therefore simulated in the next section.

Fig. 13. Emulsification of the blood clot by the transducer tip

**5.6 Concluding remarks**

designed transducer.

**6. Acoustic pump**

acoustic pump based on these acoustic effects.

**6.1 Structure of the pump**

pressure. Along with the burst of the bubbles is the high intensity shock wave and impinging of the liquid, which is normally very strong and even sufficient to break a hard surface. Fig. 12 (a) shows the cavitation bubble clusters generated at the vibration tip in the silicon oil. The cluster is usually generated at the center of the surface and then flows away along the acoustic axis. When the distance becomes larger from the surface, the bubbles might be agglomerating and floating upwards due to the buoyancy force. The cavitation threshold of the water is larger than silicon oil and is also frequency, temperature and static pressure dependant (Abramov, 1994, 1998; Young, 1999). Generally in water, only when the acoustic field is very strong, visible bubble clusters can be observed. The generation of the bubble cluster at the tip surface also proves that the highest intensity of acoustic energy is around the tip. This is consistent with the earlier theoretical analysis.

The second important phenomenon is acoustic streaming, which is the flow of the liquid as the result of high and nonlinear acoustic pressure field and the generation of bubble clusters (Abramov, 1994). The flow pattern is usually that the flow is along the acoustic axis and outwards, the surrounding liquid will flow to the tip and compensate the outflows. The streaming effects have both positive and negative effect in application of ultrasonic thrombolysis. For the positive aspect, the streaming will expedite the ablation of the blood clot, especially when the fibrinolytic agents are present (Atar et al., 1999). The negative effect is that the streaming is inclined to push away the blood clot, which increases the difficulty to control, especially, the floating clot. Fig. 12 (b) is the demonstration of acoustic streaming, which pushes the water away from the tip surface. Because the demonstration is near the water surface, the droplets jumped into the air, resulting in the phenomenon so called atomization (NII et al., 2006).

Fig. 12. Acoustic cavitation (a) and acoustic streaming (b)

#### **5.5 Blood clot emulsification**

The transducer was tested to emulsify the blood clot in an anechoic tank. The tank was filled with water and lined with sound absorption materials both at the walls and the bottom. The dimensions of the tank are 0.6 × 0.6 × 1.3 m3. A holder made of natural latex of 30 µm thick was used to contain the blood clot. The clot was prepared by naturally coagulating fresh rabbit blood overnight at 6 oC. During operation, the tip of the transducer was pointed at the clot surface as shown in Fig. 13. The blood clot was immediately emulsified when the power was provided. The whole procedure documents rapid clot lysis (~ 750 mg/min) and confirm the effectiveness of the transducer in thrombolysis.

Fig. 13. Emulsification of the blood clot by the transducer tip

#### **5.6 Concluding remarks**

18 Advances in Piezoelectric Transducers

pressure. Along with the burst of the bubbles is the high intensity shock wave and impinging of the liquid, which is normally very strong and even sufficient to break a hard surface. Fig. 12 (a) shows the cavitation bubble clusters generated at the vibration tip in the silicon oil. The cluster is usually generated at the center of the surface and then flows away along the acoustic axis. When the distance becomes larger from the surface, the bubbles might be agglomerating and floating upwards due to the buoyancy force. The cavitation threshold of the water is larger than silicon oil and is also frequency, temperature and static pressure dependant (Abramov, 1994, 1998; Young, 1999). Generally in water, only when the acoustic field is very strong, visible bubble clusters can be observed. The generation of the bubble cluster at the tip surface also proves that the highest intensity of acoustic energy is around the tip. This is consistent with the earlier

The second important phenomenon is acoustic streaming, which is the flow of the liquid as the result of high and nonlinear acoustic pressure field and the generation of bubble clusters (Abramov, 1994). The flow pattern is usually that the flow is along the acoustic axis and outwards, the surrounding liquid will flow to the tip and compensate the outflows. The streaming effects have both positive and negative effect in application of ultrasonic thrombolysis. For the positive aspect, the streaming will expedite the ablation of the blood clot, especially when the fibrinolytic agents are present (Atar et al., 1999). The negative effect is that the streaming is inclined to push away the blood clot, which increases the difficulty to control, especially, the floating clot. Fig. 12 (b) is the demonstration of acoustic streaming, which pushes the water away from the tip surface. Because the demonstration is near the water surface, the droplets jumped into the air, resulting in the phenomenon so called

The transducer was tested to emulsify the blood clot in an anechoic tank. The tank was filled with water and lined with sound absorption materials both at the walls and the bottom. The dimensions of the tank are 0.6 × 0.6 × 1.3 m3. A holder made of natural latex of 30 µm thick was used to contain the blood clot. The clot was prepared by naturally coagulating fresh rabbit blood overnight at 6 oC. During operation, the tip of the transducer was pointed at the clot surface as shown in Fig. 13. The blood clot was immediately emulsified when the power was provided. The whole procedure documents rapid clot lysis (~ 750 mg/min) and confirm

theoretical analysis.

atomization (NII et al., 2006).

**5.5 Blood clot emulsification**

Fig. 12. Acoustic cavitation (a) and acoustic streaming (b)

(a) (b)

the effectiveness of the transducer in thrombolysis.

The piezoelectric thrombolysis transducer consists of an end cap, a PZT stack, a horn, a transmission wire and a vibration tip. The transducers vibrate longitudinally and generate maximum acoustic pressure at the tip. The acoustic pressure induces effects such as cavitation and streaming at the tip. Blood clot can be effectively emulsified using the designed transducer.

#### **6. Acoustic pump**

The piezoelectric pump has various designs (Laser & Santiago, 2004; Luong & Nguyen, 2010; Yeo et al., 2008). According to frequency, the pumping mechanism will be different. At the sonic low frequency range, the diaphragm pump is a popular design, which utilizes the reciprocating movement of piezo transducer to displace the liquid. However, as frequency increases, the diaphragm mechanism becomes more and more difficult to achieve. One of the important reasons is the response mismatch between the diaphragm and valves (Hu et al., 2004; Zhang et al., 2003). However, there are other pumping mechanisms. At the ultrasonic and megasonic frequency range, the acoustic effect could be used (Chen & Lal, 2006; Frampton et al., 2004; Hasegawa et al., 2008; Koyama, et al., 2010; Li et al., 2010; Zhang et al., 2010). It has been well known that the intense and nonlinear ultrasound could induce effect such as cavitation and streaming, which are able to generate the flow of the liquid (Abramov, 1994; Li et al., 2010; Raton et al., 2007). This work will report a piezoelectric acoustic pump based on these acoustic effects.

#### **6.1 Structure of the pump**

Fig. 14 shows the structure of the piezoelectric pump designed in NTU. The pump comprises a piezoelectric transducer, a reflecting condenser and a casing. The structure of the transducer is similar as the thrombolysis transducer, but without transmission wire. There is a through hole inside the transducer working as the flow path. The transducer works approximately at 30 kHz. The reflecting condenser has a "" shape, which focused the acoustic energy at its corner. The casing has a cylindrical shape with inlet on the wall. The casing, condenser and horn surface form the boundaries of the acoustic field. Because the pump mechanism is based on the acoustic effect, the pressure magnitude and pattern inside the field will affect the performance of the pump significantly. The distribution of sound pressure level inside the casing is therefore simulated in the next section.

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 21

Fig. 16 shows the performance of the pump. From Fig. 16 (a), it can be seen that the maximum flow rate is about 10 mL/min. And maximum pressure head is about 250 kPa. The maximum pressure head is voltage dependant. As voltage increases, the pressure head can be further increased as shown in Fig. 16 (b). Compared with the regular piezoelectric diaphragm pump whose pressure head is mostly in the magnitude of ~50 kPa (Laser & Santiago, 2004; Luong & Nguyen, 2010), the advantage of the acoustic pump is the high pressure head. The reason for this is that the acoustic cavitational effect contains large

Pressure head (kPa)

(a) (b)

Fig. 16. Performance of the acoustic pump, (a) flow rate *vs* back pressure, (b) pressure head

The acoustic pump is based on the nonlinear acoustic effect such as cavitation and streaming. The cavitation effect contributes significantly on the high pressure feature of the pump. The advantage of the piezoelectric acoustic pump designed in NTU is simple

The general structure of the horn-type transducer is the Langevin transducer with a horn. The advantage of such device is that both large amplitude and high power (density) can be obtained simultaneously. FECEM theory can be used for the development of the piezoelectric transducer. This method is based on two-port network theory. Parameters like dimension, frequency, stress, velocity, quality factor, impedance, etc, can all be solved using this method. It also shows consistent results with ANSYS 2D analysis. The ANSYS 2D analysis is effective for the piezoelectric structure, acoustic field, or structure-acoustic coupled field analysis. It is also important in the development of piezoelectric devices. Two applications of horn-type transducer were also introduced in this work. The thrombolysis transducer is used for blood clot emulsification. Due to high aspect ratio of the structure, it is also a good example to be analyzed using the FECEM theory. The acoustic pump, on the other hand, possesses a simple structure for pumping applications. The working principle is nonlinear acoustic effects of streaming and cavitation. The major advantage of the pump is

the high pressure head. It might be used for sonochemistry or microfluidics.

20 40 60 80 100 120 140

Voltage (Vpp)

energy, which is able to generate a high force to overcome the back pressure.

0 50 100 150 200 250

structure, less oscillation and high pressure head.

Back pressure (kPa)

**6.3 Performance test**

0.0 2.0 4.0 6.0 8.0 10.0 12.0

**6.4 Concluding remarks**

**7. Conclusions**

Flow rate (mL/min)

*vs* voltage

#### **6.2 Modeling and simulation**

Fig. 15 is the simulation of the pressure pattern inside the pump casing. The casing was simplified as a cylinder in this 2D axisymmetric simulation. The length of the cylinder is one wavelength. The graph shows the right half piece of the model. The horn tip generates 1 µm displacement at 30 kHz. The colour represents the pressure magnitude in the unit of dB. Two boundary conditions are compared. Fig. 15 (a) is the absorbing condition. As mentioned earlier, a travelling wave will be generated at the tip. It propagates inside the casing and finally is absorbed at the casing boundary. The maximum pressure is located at the horn tip. The magnitude is approximately 187 dB. Fig. 15 (b) shows the hard or reflecting boundary condition. In this case, a standing wave will be established inside the casing. The maximum pressure is still at the horn tip. But the magnitude is increased to 255 dB, which is much larger than that of the absorbing boundary condition. This indicates that the reflecting boundary will be more effective than the absorbing boundary to focus the energy or increase the pressure magnitude. In practice, the reflecting boundary could be achieved by using materials with high acoustic characteristic impedance such as metals (Feng, 1999).

Fig. 14. The schematic structure of the piezoelectric acoustic pump designed in NTU

Fig. 15. The pressure pattern inside casing, (a) Absorbing boundary, (b) Reflecting boundary

#### **6.3 Performance test**

20 Advances in Piezoelectric Transducers

Fig. 15 is the simulation of the pressure pattern inside the pump casing. The casing was simplified as a cylinder in this 2D axisymmetric simulation. The length of the cylinder is one wavelength. The graph shows the right half piece of the model. The horn tip generates 1 µm displacement at 30 kHz. The colour represents the pressure magnitude in the unit of dB. Two boundary conditions are compared. Fig. 15 (a) is the absorbing condition. As mentioned earlier, a travelling wave will be generated at the tip. It propagates inside the casing and finally is absorbed at the casing boundary. The maximum pressure is located at the horn tip. The magnitude is approximately 187 dB. Fig. 15 (b) shows the hard or reflecting boundary condition. In this case, a standing wave will be established inside the casing. The maximum pressure is still at the horn tip. But the magnitude is increased to 255 dB, which is much larger than that of the absorbing boundary condition. This indicates that the reflecting boundary will be more effective than the absorbing boundary to focus the energy or increase the pressure magnitude. In practice, the reflecting boundary could be achieved by using

materials with high acoustic characteristic impedance such as metals (Feng, 1999).

Outlet

Fig. 14. The schematic structure of the piezoelectric acoustic pump designed in NTU

Casing (Absorbing)

Condenser (Reflecting)

(a) (b)

Reflecting condenser

Reflecting

Reflecting

Reflecting

Piezoelectric transducer

Casing

Fig. 15. The pressure pattern inside casing, (a) Absorbing boundary, (b) Reflecting boundary

**6.2 Modeling and simulation**

Inlet

Horn Surface (reflecting)

> Horn tip

Fig. 16 shows the performance of the pump. From Fig. 16 (a), it can be seen that the maximum flow rate is about 10 mL/min. And maximum pressure head is about 250 kPa. The maximum pressure head is voltage dependant. As voltage increases, the pressure head can be further increased as shown in Fig. 16 (b). Compared with the regular piezoelectric diaphragm pump whose pressure head is mostly in the magnitude of ~50 kPa (Laser & Santiago, 2004; Luong & Nguyen, 2010), the advantage of the acoustic pump is the high pressure head. The reason for this is that the acoustic cavitational effect contains large energy, which is able to generate a high force to overcome the back pressure.

Fig. 16. Performance of the acoustic pump, (a) flow rate *vs* back pressure, (b) pressure head *vs* voltage

#### **6.4 Concluding remarks**

The acoustic pump is based on the nonlinear acoustic effect such as cavitation and streaming. The cavitation effect contributes significantly on the high pressure feature of the pump. The advantage of the piezoelectric acoustic pump designed in NTU is simple structure, less oscillation and high pressure head.

#### **7. Conclusions**

The general structure of the horn-type transducer is the Langevin transducer with a horn. The advantage of such device is that both large amplitude and high power (density) can be obtained simultaneously. FECEM theory can be used for the development of the piezoelectric transducer. This method is based on two-port network theory. Parameters like dimension, frequency, stress, velocity, quality factor, impedance, etc, can all be solved using this method. It also shows consistent results with ANSYS 2D analysis. The ANSYS 2D analysis is effective for the piezoelectric structure, acoustic field, or structure-acoustic coupled field analysis. It is also important in the development of piezoelectric devices. Two applications of horn-type transducer were also introduced in this work. The thrombolysis transducer is used for blood clot emulsification. Due to high aspect ratio of the structure, it is also a good example to be analyzed using the FECEM theory. The acoustic pump, on the other hand, possesses a simple structure for pumping applications. The working principle is nonlinear acoustic effects of streaming and cavitation. The major advantage of the pump is the high pressure head. It might be used for sonochemistry or microfluidics.

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 23

Hasegawa, T.; Koyama D.; Nakamura, K. & Ueha, S. (2008). Modeling and Performance

Hayt, W.H. & Kemmerly J.E. (1993). *Engineering Circuit Analysis* (Sixth edition), McGRAW-

Hongoh, M.; Yoshikuni, M.; Miura, H.; Miyamoto, R.; Ueoka T. & Tsujino, J. (2004)

Horita, R.E. (1967). Free-Flooding Unidirectional Resonators for Deep-

Hu, M.; Du, H.J.; Ling, S.F.; Fu, Y.Q.; Chen, Q.F.; Chow, L. & Li, B. (2004). A Silicon-on-

International standard, IEC 61847, (1998-01). *Ultrasonics-Surgical Systems-Measurement and*

Janis, A.D.; Buckely, L.A. & Gregory, K.W. (2000). Laser Thrombolysis in an In-Vitro Model,

Kapila, V.; Deymier, P.A.; Shende, H.; Pandit, V.; Raghavan S. & Eschbach, F.O. (2006).

Khmelev, V.N.; Tchyganok,S.N.; Barsukov, R.V. & Lebedev, A.N. (2005). Design and

Koyama, D.; Wada, Y.: Nakamura, K.; Nishikawa, M.; Nakagawa, T. & Kihara, H. (2010). An

Laser, D.J. & Santiago J.G. (2004). A Review of Micropumps, *J. Micromech. Microeng.*, Vol.14,

Lee, D.R. & Loh, B.G. (2007). Smart Cooling Technology Utilizing Acoustic Streaming, *IEEE*

Li, Tao (2004). *Development of Piezoelectric Tubes for Micromotor*, PhD thesis, Nanyang

Li, Tao; Chen, Y.H.; Ma, J. & Boey F.Y.C. (2007a). Metal-PZT Composite Piezoelectric

Li, Tao; Chen, Y.H. & Ma, J. (2007b). Frequency Dependence of Piezoelectric Vibration

Vol.47, No. 5, (May 2008), pp.4248-4252, ISSN 00214922

HILL, Inc., ISBN 0-07-228364-5, New York

2004), pp.2896-2900, ISSN 0021-4922

2004) pp.382-387, ISSN 0960-1317

5778204914, Erlagol, July 1-5, 2005

2007), pp.691-699, ISSN 1521-3331

Technological University

0924-4247

pp. 1073-1076, ISSN 1013-9826

(January 2010), pp.253-261, ISSN 0885-3010

No.6, (June 2004), pp.R35-R64, ISSN 0960-1317

*Declaration of the Basic Output Characteristics*

ISSN 0277-786X, San Jose, CA, USA, January 22-25, 2000

1- 12, ISSN 0277-786X, Yokohama, Japan, April 18, 2006

(January 1967), pp.158-166

Evaluation of an Ultrasonic Suction Pump, *Japanese Journal of Applied Physics*,

Configuration of a 30-mm-Diameter 94 kHz Ultrasonic Longitudinal Vibration System for Plastic Welding, *Japanese Journal of Applied Physics*, Vol.43, No.5b, (May

Ocean Transducers, *The Journal of the Acoustical Society of America*, Vol.41, No.1,

Insulator Based Micro Check Vavle, *J. Micromech. Microeng.*, Vol.14, No.3, (March

*Proc. SPIE - The International Society for Optical Engineering*, Vol.3907, pp.582-599,

Megasonic Cleaning, Cavitation, and Substrate Damage: an Atomistic Approach, *Proc. Of SPIE - The International Society for Optical Engineering*, Vol.6283, pp.628324-

Efficiency Analysis of Half-Wave Piezoelectric Ultrasonic Oscillatory Systems, *6th International Siberian workshop and tutorial EDM'2005, Session II*, pp.82-85, ISBN

Ultrasonic Air Pump Using an Acoustic Travelling Wave Along a Small Air Gap, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*, Vol.57, No.1,

*Transactions on Components and Packaging Technologies*, Vol.30, No.4, (December

Transducers and Ultrasonic Motors, *Key Engineering Materials*, Vol.334-335, (2007)

Velocity, *Sensors and Actuators A*, Vol.138, No.2, (August 2007) pp.404-410, ISSN

#### **8. References**

22 Advances in Piezoelectric Transducers

Abramov, O.V. (1994). *Ultrasound in Liquid and Solid Metals*, CRC Press, ISBN<PHONE_NUMBER>,

Abramov, O.V. (1998). *High-Intensity Ultrasonics Theory and Industrial Applications*, Cordon

Accoto, D.; Carrozza M.C. & Dario P. (2000). Modelling of Micropumps Using Unimorph

ANSYS, Inc. (2010) *ANSYS Fluid Analysis Guide and Coupled-Field Analysis Guide*, Release 13 APC international, Ltd. (2002). *Piezoelectric Ceramics: Principles and Applications*, APC

Atar, S.; Luo, H.; Nagai, T. and Siegel, R.J. (1999). Ultrasonic Thrombolysis: Catheter-

Bai, M.R.; Chen R.L.; Chuang C.Y.; Yu C.S. & Hsieh H.L. (2007). Optimal Design of Resonant

*Soc. Am.*, Vol.122, No.3, (September 2007), pp. 1568-1580, ISSN 0001-4966 Balabanian, N. & Bickart, T.A. (1981). *Linear Network Theory Analysis, Properties, Design and*

Bond, L.J. & Cimino W.W. (1996). Physics of Ultrasonic Surgery Using Tissue

Brosh, D.; Miller, H.I.; Herz, I.; Laniado, S. & Rosenschein, U. (1998) Ultrasound

Chan, K.; Watmough, D.J.; Hope D.T. & MOIR, K. (1986). A New Motor-Driven Surgical

Chen, X. & Lal A. (2006). Theoretical and Experimental Study of a High Flow Rate

Chen, Y.C. & Wu, S. (2002). A Design Approach of Tonpiltz Transducer, *Jpn. J. Appl. Phys.,*

Chen, Y.C.; Wu, S. & Chen P.C. (2004). The Impedance-Matching Design and Simulation on

Chu, W.P.; Li, H.L.; Chan, L.W.; Ng, M.W. & Liu C.K. (2002). Smart Ultrasonic Transducer

Francis, C.W. (2001). Ultrasound-Enhanced Thrombolysis. *Echocardiography*, Vol. 18, No. 3,

Feng, R. (1999). Ultrasonics Handbook, Nanjing University, ISBN 7-305-03354-5, Nanjing Frampton, K.D.; Minor, K. & Martin, S. (2004). Acoustic Streaming in Micro-Scale

*Synthesis*, Matrix publisher, Inc., ISBN 0-916460-10-X, Beaverton

Vol.1, No. 1, (January 1998), pp.11-18, ISSN 1462-8848

4244-0201-8, Vancouver, BC, October 2-6, 2006

(September 2004), pp.38-45, ISSN 0924-4247

(April 2002), pp.95-100, ISSN 0254-0584

(April 2001), pp. 239-246, ISSN 0742-2822

1129, ISSN 0003-682X

Vol. 41, No.6A, (June 2002), pp. 3866-3877, ISSN 00214922

Piezoelectric Actuator and Ball Valves, *J. Micromech. Microeng.*, Vol.10, No. 2, (June

Delivered and Transcutaneous Applications. *European Journal of Ultrasound*, Vol.9,

Piezoelectric Buzzer from a Perspective of Vibration-Absorber Theory, *J. Acoust.*

Fragmentation, *Ultrasonics*, Vol. 34, No. 2-5, (June 1996), pp. 579-585, ISSN 0041-

Angioplasty: an Update Review, *International Journal of Cardiovascular Interventions*,

Probe and its In Vitro Comparison with the Cavitron Ultrasonic Surgical Aspirator, *Ultrasound in Medicine & Biology*, Vol.12, No.4, (April 1986), pp.279-283, ISSN 0301-

Ultrasonic Horn Pump, *2006 IEEE Ultrasonics Symposium*, pp.2409-2412, ISBN 1-

High Power Electro-acoustical Transducer, *Sensors and Actuators A,* Vol.115, No. 1,

for Wire-Bonding Applications, *Materials Chemistry and Physics*, Vol. 75, No.1-3

Cylindrical Channels, *Applied Acoustics*, Vol. 65, No.11, (November 2004), pp.1121-

and Breach Science Publishers, ISBN<PHONE_NUMBER>, Russia

International, Ltd., ISBN<PHONE_NUMBER>, Mackeyville

No.1 (March 1999), pp. 39–54, ISSN 09298266

2000), pp. 277-281, ISSN 0960-1317

**8. References**

Boca Raton

624X

5629

Horn-Type Piezoelectric Ultrasonic Transducer: Modelling and Applications 25

Prokic, M. (2004). *Piezoelectric Transducers Modeling and Characterization*, MPI, Switzerland Radmanovic, M.D. & Mancic D.D. (2004). *Design and Modelling of the Power Ultrasonic*

Vol.74, No.12, (December 1994), pp.1263-1266, ISSN 0002-9149

Rosenschein, U.; Frimerman, A.; Laniado S. & Miller, H.I. (1994). Study of the Mechanism of

Sherrit, S.; Dolgin, B.P.; Bar-Cohen, Y.; Pal, D.; Kroh J. & Peterson, T. (1999a). Modeling of

Siegel, J.; Atar, S.; Fishbein, M.C.; Brasch, A.V.; Peterson, T.M.; Nagai, T.; Pal, D.; Nishioka,

Siegel, J. & Luo, H. (2008). Ultrasound Thrombolysis, *Ultrasonics*, Vol.48, No.4, (August

Siegel, R.J., Suchkova, V.N.; Miyamoto, T.; Luo, H.; Baggs, R.B.; Neuman, Y.; Horzewski, M.;

Tschepe, J.; Aspidov, A.A.; Helfmann J. & Herrig, M. (1994). Acoustical Waves via Optical

Tsuda, Y.; Mori E. & Ueha, S. (1983). Experimental Study of Ultrasonic Surgical Knife, *Jpn. J. Appl. Phys.*, Vol.22, Supplement 22-3 (1983) pp.105-107, ISSN 0021-4922 Uchino, K. & Giniewicz J.R. (2003). *Micromechatronics*, CRC Press, ISBN<PHONE_NUMBER>, New

Wiksell, H.; Martin, H.; Coakham, H.; Berggren A. & Westermark, S. (2000). Miniaturized

Woias, P.; Micropumps - Past, Progress and Future Prospects, *Sensors and Actuators B,*

Yeo, C.Y.; Shim, W.K.; Wouterson, E.; Li, Tao & Ma, J. (2008). Piezoelectric Materials for

Vol. 11, No.1, (March 2000), pp.41-46, ISSN 09298266

Vol.105, No.1, (February 2005), pp.28-38, ISSN 0925-4005

Young, F.R. (1999). *Cavitation*, Imperial College Press, ISBN<PHONE_NUMBER>, Singapore

651, ISBN 0-7803-5722-1, Caesars Tahoe, NV, USA, October 17-20, 1999 Sherrit, S.; Leary, S.P.; Dolgin B.P. & Bar-Cohen Y. (1999b). Comparison of the Mason and

*Transducers*, MPI, ISBN 86-80135-87-9, Switzerland

02638762

USA, October 17-20, 1999.

pp.2026-2029, ISSN 1524-4539

York

York

September 01, 1993

(2008), pp.225-228

2008) pp.312-320, ISSN 0041-624X

*Engineering Research and Design*, Vol.84, No.5A, (May 2006), pp.412-41, ISSN

Ultrasound Angioplasty from Human Thrombi and Bovine Aorta, *Am. J. Cardiol.*,

Horns for Sonic/Ultrasonic Applications, *1999 IEEE Ultrasonics Symposium*, pp.647-

KLM Equivalent Circuits for Piezoelectric Resonators in the Thickness Mode, *1999 IEEE Ultrasonics Symposium*, pp.921-926, ISBN 0-7803-5722-1, Caesars Tahoe, NV,

T.; Chae, J.S.; Birnbaum, Y.; Zanelli, C. & Luo, H. (2000). Noninvasive, Transthoracic, Low-frequency Ultrasound Augments Thrombolysis in a Canine Model of Acute Myocardial Infarction, *Circulation*, Vol.101, No.17 (May 2000)

Suorsa, V.; Kobal, S.; Thompson, T.; Echt D.; & Francis, C.W. (2004) Ultrasound Energy Improves Myocardial Perfusion in the Presence of Coronary Occlusion, *J. Am. Coll. Cardiol.*, Vol.44, No.7, (October 2004), pp.1454-1458, ISSN 0735-1097 Silva, C.W.D. (1999). *Vibration Fundamentals and Practice*, CRC Press, ISBN<PHONE_NUMBER>, New

Fibers for Biomedical Applications, *Proc. SPIE- Biomedical Optoelectronic Devices and Systems*, Vol.2084, pp.133-143, ISBN<PHONE_NUMBER>512, Budapest, Hungary, 01

Ultrasonic Aspiration Handpiece for Increased Applicability, *Eur. J. Ultrasound*,

Impedance Driven Micro-Channel Flow, *Functional Materials Letters*, Vo.1, No.3,

24 Advances in Piezoelectric Transducers

Li, Tao; Chen, Y.H.; Boey, F.Y.C. & Ma, J. (2007c). High Amplitude Vibration of Piezoelectric

Li, Tao; Chen, Y.H. & Ma, J. (2008). Development of Miniaturized Piezoelectric Multimode

Li, Tao; Chen, Y.H. & Ma, J. (2009). Development of a Miniaturized Piezoelectric Ultrasonic

Li, Tao; Chen, Y.H; Lew F.L. & Ma J. (2010). Design, Characterization, and Analysis of a

Lin, S.Y. (1995). Study on the Multifrequency Langevin Ultrasonic Transducer, *Ultrasonics,*

Lin, S.Y. (2004a). Effect of Electric Load Impedances on the Performance of Sandwich

Lin, S.Y. (2004b). *Ultrasonic Transducer Principle and Design*, Science Press, ISBN 7-03-013419-

Lin, Z.M. (1987). *Ultrasonic Horn Principle and Design*, Science press, ISBN 7-03-000008-0,

Luong, T.D. & Nguyen N.T. (2010). Surface Acoustic Wave Driven Microfluidics - A Review, *Micro and Nanosystems*, Vol.2, No.3, (2010) pp.217-225, ISSN 18764029 Ma, J., Low F.H.A. & Boey Y.C.F. (2010) *Micro-Emulsifier for Arterial Thrombus Removal*,

Mason, W.P. (1964). *Physical Acoustics Principles and Methods*, *Volume I - Part A*, Academic

Mason, W.P. & Wehr J. (1970). Internal Friction and Ultrasonic Yield Stress of the Alloy 90 Ti

Mattiat, O.E. (1971), *Ultrasonic Transducer Materials*, Plenum Press, ISBN<PHONE_NUMBER>, New

Medis, P.S. & Henderson H.T. (2005). Micromachining Using Ultrasonic Impact Grinding, *J. Micromech. Microeng.*, Vol.15, No. 8, (August 2005), pp.1556-1559, ISSN 0960-1317 Miclea, C.; Tanasoiu, C.; Iuga, A.; Spanulescu, I.; Miclea, C.F.; Plavitu, C.; Amarande, L.;

Minchenko, H. (1969). High-Power Piezoelectric Transducer Design, *IEEE Transactions on Sonics and Ultrasonics*, Vol. SU-16, No. 3, (July 1969), pp.126-131, ISSN 0018-9537 Muhlen, S.S. (1990). Design of an Optimized High-Power Ultrasonic Transducer, *1990 IEEE Ultrasonics Symposium*, pp.1631-1634, Honolulu, HI , USA, December 04-07, 1990 NII, S.; Matsuura, K.; Fukazu, T.; Toki, M. & Kawaizumi F. (2006). A Novel Method to

6 Al 4 V, *J. Phys. Chem. Solids*, Vol.31, No.8, (August 1970), pp.1925-1933, ISSN 0022-

Cioangher, M.; Trupina, L.; Miclea, C.T. & Tanasoiu, T. (2008). A High Performance PZT Type Material Used as Sensor for an Audio High Frequency Piezoelectric Siren, *2008 International Semiconductor Conference*, pp.185-188, ISBN 978-1-4244-

Separate Organic Compounds through Ultrasonic Atomization, *Chemical*

Vol.56, No.3, (March 2009), pp. 649-659,ISSN 0885-3010

Vol.25, No.4, (April 2010), pp.221-226, ISSN 10426914

(October 2004), pp.1280-1286, ISSN 0885-3010

PCT/SG2008/000323, WO 2010/027325 A1

2004-9, Sinaia, Romania, October 13-15, 2008

Press, ISBN<PHONE_NUMBER>, New York

Vol. 33, No.6, (November 1995), pp.445-448, ISSN 0041-624X

13853449

2, Beijing

Beijing

3697

York

pp.61-64, ISSN 10226680

Bending Actuators*, J. Electroceram*, Vol.18, No.3-4, (August 2007), pp.231-242, ISSN

Transducer for Projection Purpose, *Advanced Materials Research*, Vol.47-50, (2008)

Transducer, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*,

Miniaturized Piezoelectric Transducer, *Materials and Manufacturing Processes*,

Piezoelectric Transducers, *IEEE Trans. Ultrason., Ferroelec., Freq. Contr.*, Vol.5, No.10

*Engineering Research and Design*, Vol.84, No.5A, (May 2006), pp.412-41, ISSN 02638762

**2**

*Poland*

**Modeling and Investigation of One-Dimensional**

Piezoelectricity has found a lot of applications since it was discovered in 1880 by Pierre and Jacques Curie. There are many applications of the direct piezoelectric effect - the production of an electric potential when stress is applied to the piezoelectric material, as well as the reverse piezoelectric effect - the production of strain when an electric field is applied (Moheimani & Fleming, 2006). In this chapter analysis of mechatronic systems with both direct and reverse piezoelectric effects applications in mechatronic systems are presented. In considered systems piezoelectric transducers are used as actuators – the reverse piezoelectric effect application, or as vibration dampers with the external shunting electric circuit – the direct piezoelectric effect. In the first case piezoelectric transducers can be used as actuators glued on the surface of a mechanical subsystem in order to generate desired vibrations or also to control and damp vibrations in active damping applications (Kurnik et al., 1995; Gao & Liao, 2005). In this case electric voltage is generated by external control system and applied to the transducer. In the second case piezoelectric transducers are used as passive vibration dampers. A passive electric network is adjoined to transducer's clamps. The possibility of dissipating mechanical energy with piezoelectric transducers shunted with passive electric circuits was experimentally investigated and described in many publications (Buchacz & Płaczek, 2009a; Fein, 2008; Hagood & von Flotow, 1991; Kurnik, 2004). There are two basic applications of this idea. In the first method only a resistor is used as a shunting circuit and in the second method it is a passive electric circuit composed of a resistor and inductor. Many authors have worked to improve this idea. For example multimode piezoelectric shunt damping systems were described (Fleming et al., 2002). What

is more there are many commercial applications of this idea (Yoshikawa et al., 1998).

Mechatronic systems with piezoelectric sensors or actuators are widely used because piezoelectric transducers can be applied in order to obtain required dynamic characteristic of designed system. It is very important to use very precise mathematical model and method of the system's analysis to design it correctly. It was proved that it is very important to take into consideration influence of all analyzed system's elements including a glue layer between piezoelectric transducer and mechanical subsystem (Pietrzakowski, 2001; Buchacz & Płaczek, 2010b). It is indispensable to take into account geometrical and material

**1. Introduction**

**Flexural Vibrating Mechatronic Systems with**

**Piezoelectric Transducers**

Andrzej Buchacz and Marek Płaczek

*Silesian University of Technology,*

### **Modeling and Investigation of One-Dimensional Flexural Vibrating Mechatronic Systems with Piezoelectric Transducers**

Andrzej Buchacz and Marek Płaczek *Silesian University of Technology, Poland*

#### **1. Introduction**

26 Advances in Piezoelectric Transducers

Zhang, A.L & Wei, Y.Q. (2010). Generation of Droplets for Lab-on-a-Piezoelectric-Substrate

Zhang, J.H.; Wang, D.K.; Wang, S.Y. & Qnuki, A. (2003). Research on Piezoelectric Pump-

978-1-4244-9822-2 ,Xiamen, China, December 10-13, 2010

2003), pp.107-110, ISSN 05776686

Utilizing Surface Acoustic Wave, *Proceedings of the 2010 symposium on piezoelectricity, acoustic waves, and device applications (SPAWDA 2010)*, pp.68-71, ISBN

Lagging of Valve, *Chinese Journal of Mechanical Engineering*, Vol.39, No.5, (May

Piezoelectricity has found a lot of applications since it was discovered in 1880 by Pierre and Jacques Curie. There are many applications of the direct piezoelectric effect - the production of an electric potential when stress is applied to the piezoelectric material, as well as the reverse piezoelectric effect - the production of strain when an electric field is applied (Moheimani & Fleming, 2006). In this chapter analysis of mechatronic systems with both direct and reverse piezoelectric effects applications in mechatronic systems are presented. In considered systems piezoelectric transducers are used as actuators – the reverse piezoelectric effect application, or as vibration dampers with the external shunting electric circuit – the direct piezoelectric effect. In the first case piezoelectric transducers can be used as actuators glued on the surface of a mechanical subsystem in order to generate desired vibrations or also to control and damp vibrations in active damping applications (Kurnik et al., 1995; Gao & Liao, 2005). In this case electric voltage is generated by external control system and applied to the transducer. In the second case piezoelectric transducers are used as passive vibration dampers. A passive electric network is adjoined to transducer's clamps. The possibility of dissipating mechanical energy with piezoelectric transducers shunted with passive electric circuits was experimentally investigated and described in many publications (Buchacz & Płaczek, 2009a; Fein, 2008; Hagood & von Flotow, 1991; Kurnik, 2004). There are two basic applications of this idea. In the first method only a resistor is used as a shunting circuit and in the second method it is a passive electric circuit composed of a resistor and inductor. Many authors have worked to improve this idea. For example multimode piezoelectric shunt damping systems were described (Fleming et al., 2002). What is more there are many commercial applications of this idea (Yoshikawa et al., 1998).

Mechatronic systems with piezoelectric sensors or actuators are widely used because piezoelectric transducers can be applied in order to obtain required dynamic characteristic of designed system. It is very important to use very precise mathematical model and method of the system's analysis to design it correctly. It was proved that it is very important to take into consideration influence of all analyzed system's elements including a glue layer between piezoelectric transducer and mechanical subsystem (Pietrzakowski, 2001; Buchacz & Płaczek, 2010b). It is indispensable to take into account geometrical and material

Modeling and Investigation

*b)* with piezoelectric actuator

have time unit (Pietrzakowski, 2001).

by equations:

of One-Dimensional Flexural Vibrating Mechatronic Systems with Piezoelectric Transducers 29

Fig. 1. Shape of the considered system: *a)* with piezoelectric passive vibration damper,

Structural damping of the beam and glue layer was taken into account in mathematical models of considered systems using Kelvin-Voigt model of material. It was introduced by replacing Young's modulus of the beam and modulus of elasticity in shear of the glue layer

> 1 , *E E bb b <sup>t</sup>*

1 , *G G <sup>k</sup> <sup>t</sup>*

where *η<sup>b</sup>* and *ηk* denote structural damping coefficients of the beam and the glue layer that

It was assumed that the beam is made of steel and piezoelectric transducer is a PZT transducer. Geometric and material parameters of the system's elements: mechanical subsystem – the beam,

the glue layer and the piezoelectric transducer are presented in tables 1, 2 and 3.

*b m* 0,04 <sup>3</sup> <sup>7850</sup> *<sup>b</sup>*

*h m <sup>b</sup>* 0,002 <sup>5</sup> 8 10 *<sup>b</sup>*

Table 1. Parameters of the mechanical subsystem

Geometric parameters Material parameters *l m* 0,24 *E MPa <sup>b</sup>* 210000

(1)

(2)

*kg m*

*s*

parameters of all system's components because the omission of the influence of one of them results in inaccuracy in the analysis of the system.

This work presents the issues of modeling and testing of flexural vibrating mechatronic systems with piezoelectric transducers used as actuators or vibration dampers. Analysis method of the considered system will be presented, started from development of the mathematical model, by setting its characteristics, to determine the influence of the system's properties on these characteristics.

The discussed subject is important due to increasing number of applications, both simple and reverse piezoelectric phenomena in various modern technical devices. The process of modeling of technical devices with piezoelectric materials is complex and requires large amounts of time because of the complexity of the phenomena occurring in these systems. The correct description of the system by its mathematical model during the design phase is fundamental condition for proper operation of designed system. Therefore, in the work the processes of modeling, testing and verification of used mathematical models of onedimensional vibrating mechatronic systems will be presented. A series of discrete – continuous mathematical models with different simplifying assumptions will be created. Using created models and corrected approximate Galerkin method dynamic characteristics of considered systems will be designated. An analysis of influence of some geometrical and material parameters of system's components on obtained characteristics will be conducted. Mathematical model that provides the most accurate analysis of the system and maximum simplification of used mathematical tools and minimize required amount of time will be indicated. Identification of the optimal mathematical model that meets the assumed criteria is the main purpose of this work, which is an introduction to the task of synthesis of onedimensional vibrating continuous systems.

#### **2. Considered system with piezoelectric transducer and assumptions**

The main aim of this work is to designate dynamic characteristics of a mechatronic system with piezoelectric transducer used as an actuator or passive vibration damper. It is a cantilever beam which has a rectangular constant cross-section, length *l ,* width *b* and thickness *hb*. Young's modulus of the beam is denoted *Eb*. A piezoelectric transducer of length *lp* is bonded to the beam's surface within the distance of *x1* from a clamped end of the beam. The transducer is bonded by a glue layer of thickness *hk* and Kirchhoff's modulus *G*. The glue layer has homogeneous properties in overall length. The system under consideration in both cases (with piezoelectric actuator or vibration damper) is presented in Fig. 1.

In order to analyze vibration of the systems following assumptions were made:

28 Advances in Piezoelectric Transducers

parameters of all system's components because the omission of the influence of one of them

This work presents the issues of modeling and testing of flexural vibrating mechatronic systems with piezoelectric transducers used as actuators or vibration dampers. Analysis method of the considered system will be presented, started from development of the mathematical model, by setting its characteristics, to determine the influence of the system's

The discussed subject is important due to increasing number of applications, both simple and reverse piezoelectric phenomena in various modern technical devices. The process of modeling of technical devices with piezoelectric materials is complex and requires large amounts of time because of the complexity of the phenomena occurring in these systems. The correct description of the system by its mathematical model during the design phase is fundamental condition for proper operation of designed system. Therefore, in the work the processes of modeling, testing and verification of used mathematical models of onedimensional vibrating mechatronic systems will be presented. A series of discrete – continuous mathematical models with different simplifying assumptions will be created. Using created models and corrected approximate Galerkin method dynamic characteristics of considered systems will be designated. An analysis of influence of some geometrical and material parameters of system's components on obtained characteristics will be conducted. Mathematical model that provides the most accurate analysis of the system and maximum simplification of used mathematical tools and minimize required amount of time will be indicated. Identification of the optimal mathematical model that meets the assumed criteria is the main purpose of this work, which is an introduction to the task of synthesis of one-

**2. Considered system with piezoelectric transducer and assumptions**

In order to analyze vibration of the systems following assumptions were made:

material of which the system is made is subjected to Hooke's law,

displacements are small compared with the dimensions of the system.

the system has a continuous, linear mass distribution,

the system's vibration is harmonic,

flat cross-sections,

The main aim of this work is to designate dynamic characteristics of a mechatronic system with piezoelectric transducer used as an actuator or passive vibration damper. It is a cantilever beam which has a rectangular constant cross-section, length *l ,* width *b* and thickness *hb*. Young's modulus of the beam is denoted *Eb*. A piezoelectric transducer of length *lp* is bonded to the beam's surface within the distance of *x1* from a clamped end of the beam. The transducer is bonded by a glue layer of thickness *hk* and Kirchhoff's modulus *G*. The glue layer has homogeneous properties in overall length. The system under consideration in both cases (with piezoelectric actuator or vibration damper) is presented in

 planes of cross-sections that are perpendicular to the axis of the beam remain flat during deformation of the beam – an analysis is based on the Bernoulli's hypothesis of

results in inaccuracy in the analysis of the system.

properties on these characteristics.

dimensional vibrating continuous systems.

Fig. 1.

Fig. 1. Shape of the considered system: *a)* with piezoelectric passive vibration damper, *b)* with piezoelectric actuator

Structural damping of the beam and glue layer was taken into account in mathematical models of considered systems using Kelvin-Voigt model of material. It was introduced by replacing Young's modulus of the beam and modulus of elasticity in shear of the glue layer by equations:

$$E\_b \, ^\star = E\_b \left( \mathbf{1} + \eta\_b \, \frac{\partial}{\partial t} \right) \, \tag{1}$$

$$\mathbf{G}^\* = \mathbf{G} \left( \mathbf{1} + \eta\_k \frac{\partial}{\partial t} \right) \tag{2}$$

where *η<sup>b</sup>* and *ηk* denote structural damping coefficients of the beam and the glue layer that have time unit (Pietrzakowski, 2001).

It was assumed that the beam is made of steel and piezoelectric transducer is a PZT transducer. Geometric and material parameters of the system's elements: mechanical subsystem – the beam, the glue layer and the piezoelectric transducer are presented in tables 1, 2 and 3.

Table 1. Parameters of the mechanical subsystem

Modeling and Investigation

**subsystem**

where:

obtained:

of One-Dimensional Flexural Vibrating Mechatronic Systems with Piezoelectric Transducers 31

In order to designate dynamic characteristics of considered systems correctly it is important to use very precise mathematical model. Very precise method of the system's analysis is very important too. It is impossible to use exact Fourier method of separation of variables in analysis of mechatronic systems, this is why the approximate method must be used. To analyze considered systems approximate Galerkin method was chosen but verification of this method was the first step (Buchacz & Płaczek, 2010c). To check accuracy and verify if the Galerkin method can be used to analyze mechatronic systems the mechanical subsystem was analyzed twice. First, the exact method was used to designate dynamic flexibility of the mechanical subsystem. Then, the approximate method was used and obtained results were

The equation of free vibration of the mechanical subsystem was derived in agreement with d'Alembert's principle. The external force *F*(*t*) was neglected. Taking into account equilibrium of forces and bending moments acting on the beam's element, after

> 2 4 4 2 4 , , , *<sup>y</sup> xt y xt <sup>a</sup> t x*

> > <sup>4</sup> . *b b b b*

*Ab* and *Jb* are the area and moment of inertia of the beam's cross-section. In order to determine the solution of the differential equation of motion (7) Fourier method of separation of variables was used. Taking into account the system's boundary conditions, after transformations the characteristic equation of the mechanical subsystem was

> cos . cosh *kl*

Graphic solution of the equation (9) is presented in Fig. 3. The solution of the system's

characteristic equation approach to limit described by equation:

1

*E J <sup>a</sup>*

(7)

*<sup>A</sup>* (8)

*kl* (9)

**3. Approximate Galerkin method verification – Analysis of the mechanical**

juxtaposed. The mechanical subsystem is presented in Fig.2.

Fig. 2. Shape of the mechanical subsystem

transformations a well known equation was obtained:

Table 2. Parameters of the piezoelectric transducer

Table 3. Parameters of the glue layer

Symbols *ρb* and *ρp* denote density of the beam and transducer. *d31* is a piezoelectric constant, *e33T* is a permittivity at zero or constant stress, *s11E* is flexibility and *c11E* is a Young's modulus at zero or constant electric field.

Dynamic characteristics of considered systems are described by equations: