ANALYSIS AND MODELING OF

PIEZOELECTRIC TRANSFORMERS

Ehson Muhammad Syed

A thesis submitted in conformity with the requirements

for the degree of Masters of Applied Sciences

Graduate Department of Electrical and Cornputer Engineering

University of Toronto

O Copyright by Ehson Muhammad Syed 2001

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ANALYSIS AND MODELING OF PIEZOELECTRIC TRANSFORMERS

Master of Applied Science 200 1

Ehson Muhammad Syed

Graduate Department of Elecûical and Compter Engineering

University of Toronto

Abstract

This thesis presents an equivalent circuit rnodel for a Rosen type piezoelectric

transformer. Transmission line equations for various vibration modes with a clamped

boundary condition on one end of each section are developed. The partial expansion

theorem is used to convert the transmission line model into an equivalent circuit model.

Losses are introduced in a pst-processing step using the experimental value of

mechanical quality factor. L'Hospitals mle is used to simplifi the circuit near a desired

resonance fiequency. Experimental results confirm the existence of multimode resonance

and anti-resonance behavior. The modeling methodology is then extended to multi-

layered rectangular and circular geometries.

The applicability of two DC-DC converter topologies in connection with a

piezoelectric transformer i.e. Class-E and an Asymrnetrical converter are discussed.

Also, experiments on a three winding multi-layered device connected to an asymmetrical

converter are performed. Results indicate that the third winding cannot be used to

provide a feedback voltage since the voltage does not track the secondary output voltage.

DEDICATION

This work is dedicated to.. . .. . . .. . .. .. .

my beloved parents and siblings

and

al1 of my hardworking teachers and professors

(iii)

Acknowledgements

The author would like to thank the following individuals:

Professor Dr. Francis P. Dawson for his intensive guidance, encouragement,

instructive suggestions and full support throughout the work and for providing

necessary information for pursuing this work efficiently.

a Mr. Brian Whitnell and Mr. Michel Gagne of Agilent Technologies for helping

out with the measurements with the HP's Network Analyzer at the University of

Toronto and at their place.

Mr. Jack Goldstein fiom the Power Labs at the University of Toronto for his help

in ordering the parts and cornponents.

- Philippe Blanchard, a Ph.D. student and Edward Chen, an undergraduate student

for their help with the converter design.

Finally, 1 would like to thank my wonderful and ever loving parents for their continuous

support, encouragement and confidence.

Table of Contents

List of Symbols .................................................................................................................. ix

List of Figures .................................................................................................................... xi

List of Tables ................................................................................................................... xvi

Chapter: 1 INTRODUCTION ............................................................................................. 1

.............................................................................................................. 1 -1 Bac kground 2

1.2 ThesisObjectives ..................................................................................................... 5

.......................................................................................................... 1.3 Thesis O u t h e 6

.............................................................................................. Chapter: 2 BASIC THEORY 7

............................................................................. 2 . 1 Poling of a Piezoelectric Matenal 7

.............................................................. 2 . 2 Definition of FieId and Material Properties 8

................................................................................................. 2 . 3 Piezoelectric Effect IO

................................................................... 2.3.1 Polarity of Piezoelectric Effect 11

2 . 4 Shapes and Vibrational Modes ............................................................................... 12

2.4.1 Transverse (length) Vibration Mode ............................................. 13

...................................................... 2.4.2 Thickness Vibration Mode 14

.......................................................... 2.4.3 Shear Thickness Mode 14

.......................................................................................... 2 . 5 Velocity of Propagation 15

............................................................................................. 2 . 6 Piezoelectnc Material 16

......................................................................................... 2 . 7 Multi-layered Structures 19

.................................................................................... 2 . 8 Piezoelectric Transfomers -20 . . .................................................................... 2.8.1 Applications -20

...................................................................... 2.8.2 Properties ..2O

............................. 2.8.3 Structure of a Single Rosen-type Transformer 2 1

....................................... 2.8.4 Stress and Displacement distributions 22

Chapter: 3 MODELING ................................................................................................. -24

3.1 Basic Piezoelectric (Tensor) Equations .................................................................. 24

3.2 Piezoelectric Constants and Coefficients .............................................................. -29

. . 3.3 Equivalent Circuit Modeling ................................................................................... 31

3.3.1 Equivalent circuit of a transducer in Thickness vibration mode ........... 32

........................................ 3.3.2 Equivalent Circuit Under Mechanical Stress 41

... 3.3.3 Equivalent Circuit of a Transducer in Longitudinal vibration Mode 54

3.3.4 Equivalent Circuit of a Transformer ........................................ 57

3.3.4.1 Equivalent Circuit of a Transformer ................................. 59

3.3.4.2 Process Flow Chart ...................................................... 60

3.4 Extension Of The Mode1 to a Three Section Rosen-type Piezoelectric ....................................................................................................................... Transformer 60

.............................................................................................. 3.4.1 Modeling 61

3 .4.2 Lumped Circuit Mode1 ......................................................................... 63

Chapter : 4 EXPERIMENT AND RESULTS ............................................................... -64

.................................................................................................. 4.1 Testing Procedure -64

4.2 Testing of 3-layered Rosen-type Transformer ........................................................ 66

4.3 Testing of multi-layered Circutar Transformer ....................................................... 71

............................................................................................................. 4.4 Conclusion 76

Chapter: 5 CONVERTER DESIGN ................................................................................. 77

5.1 A DC-DC Class E ZVS Converter Design .............................................................. 77

........................................................................ 5.1.1 Description 78

................................................... 5.1.2 Equations and Mathematical Modeling 79

.............. 5.1.3 Class E converter Design using the Piezoelectric Transformer -86

................................................................................... 5.1.4 Simulation Results 89

............................................................................. 5 -2 Asymmetrical Converter Design -9 1

5.2.1 Asyrnmetrical Converter Design using a Piezoelectric Transformer ...... 92

................................................................................... 5.2.2 Simulation Results 94

............................................................................. 5.2.3 Expenmental Analysis 96

........................................................................................... 5.2.4 Conclusion 1 0 3

.......................................................................... Chapter: 6 THESIS CONCLUSION 1 0 4

..................................................................... Proposed Future Work 105

APPENDIX: A 1 Piezoelectric Equations in Cartesian Coordinate System ....... A l . 1

......... APPENDIX: A2 Piezoelectric Equations in Cy lindrical Coordinate System A2-1

APPENDIX: A3 Equivalent circuit of a transducer in Longitudinal vibration modeA3- 1

APPENDIX: A4 Determination of off-resonance equivalent impedance using .......................................................................................................... L'Hospital's Rule A4-1

.............................................................................................. A4.1 For tanh fùnction A4-1

...................................................... ................................. A4.2 For coth fùnction ....... A 4 4

.......................................................................................... A4.3 For cosech h c t i o n A4-8

APPENDIX: A5 Flow Chart ................................................................................... A5-1

........................................................................................ A5.1 Pre-processing Stage A5-1

.............................................................................................. A5.2 Processing Stage AS-2

A5.3 Post Processing Stage ...................................................................................... AS-4

APPENDIX: A6 Equivalent Circuit Modeling of a Circular Disc type Piezoelectric ...................................................................... Transformer assuming a radial vibration A6-1

A6.1 Equivalent Circuit Modeling of a Circulas Disc in a radial vibration ............. A64

A6.2 Equivalent Circuit Modeling of a Circular Piezoelectric Transformer in a radial vibration ........................................................................................................................ A6-9

APPENDIX: A7 Equivalent Circuit Modeling of a multi-layered Circular Disc type Piezoelectric Transformer under thickness vibration .................................................. A7-1

.......................................................................................................... A7.1 Modeling A7-3

(vii)

APPENDIX: A8 Asymmetrical Converter design and circuit.. . . . . . . . .. . . . . . . . . . . . . . . . ... A8- 1

APPENDIX: A9 Simulation Programs. . . . . . .. . . . . . . . . . . . . . . . . . ...... .... . ......... ....... ........ ... A9- 1

List of References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . .RI

(viii)

List of Symbols

Radius (m)

Thickness (m)

Width (m)

Area (m;)

Surface Area (m'l)

Dens ity (kg/mL)

Frequency (Hz)

Angular frequency (radians/'.)

Impedance (R)

Resistance (a)

Capacitance Vmads)

Current (A)

Voltage (0

Resonant frequency (Hz)

Anti-resonant fiequency (Hz)

Stress (i= 1-6) reduced form of 7, ( j = I - 6 ) ( ~ / r n ~ )

Dielectric Displacement (i=I-3) (C/m2)

Electric field ( i=l-3) (V/m)

Strain (i=l-3) reduced form of S, (j=1-6) ( d m )

Displacement (i=I-3) (m)

Displacement Velocity (i= 1-3) ( d s )

Position (m)

J

Piezoelectric charge coefficient (i= 1 -3), reduced form of duk ÿ. k= 1-6) (Cm - - - -

Piezoelectric voltage coefficient (i= 1-3), reduced form of g ,, ÿ. k=f -6) ( Y ,

Thickness electromechanical coupling constant

Planar electromechanical coupling constant

Thickness Shear electromechanical coupling constant

Length Thickness electromechanical coupling constant

Length electromechanical coupling constant

Radial electromechanical coupling constant

Free permittivity (constant stress) ( i j=f -3) (F/m)

Clamped permittivity (constant strain) (iJ= 1-3) (F/m)

Permittivity, a combination as defined in radial mode.

Piezoelectric coeffiecient as defined in radial mode.

Elastic Sti ffness (~/m')

Elastic Cornpliance (constant field), reduced form of SC (ij.k=I-6) (rn2mc)

Poisson's ratio (radial mode)

Static Capacitance (F)

Mechanical Quality factor

List of Figures

Figure Description Polarization of Ceramic material to generate Piezoelectric Effect

Examples of Piezoelectric Effect

Different Shapes of a PZT

Transverse Vibration Mode

Thickness Vibration Mode

Shear Vibration Mode

Admittance versus frequency of a PZT ceramic

Admittance versus frequency of a PbTiO, ceramic

X- and Y-Poled Multilayered Stacking

A simplified representation of a Rosen Type Piezoelectric Transformer

Dimensions (in mm) of a Piezoelectric Transformer

Displacernent and Stress Utilization

Axis Representation

A simpl ified representation of a Piezoelectric Transformer

Rosen type Resonator in the Thickness Vibration mode

Equivalent Circuit Representation of a Piezoelectric Resonator

Piezoelectric Resonator under Stress

Electromechanical equivalent circuit of a resonator under thickness vibration

Equivalent circuit with terminal short circuited

Redrawn equivalent circuit with tenninal short circuit

Equivalence of an L-Type network plus a transformer to an L-Type network reversed

in direction

3.10 Equivalent circuit represenbtion after ernploying the simplification technique L

3.11 Equivalent circuit representation after referring the middle branch to the secondary

side in figure 3.1 1

3.12 Simplified equivalent circuit without losses

3.13 Equivalent circuit of a resonator for thickness vibration with al1 the circuit parameters

3.14

1 3.16 1 A transversely vibrated piezoelectric resonator under stress

Representation of Strain and Strain Velocity

3.15 Cornplete equivalent circuit model with losses

3.1 7 Electromechanical equivalent circuit for the longitudinal vibration section

3.18

I

3-20 1 Simplified equivalent circuit of a Rosen type piezoelectric transformer

Simplified equivalent circuit without losses

3.19 Equivalent circuit of a resonator assuming thickness vibration with al1 the circuit

parameters

1 3.22 I

1 Simplification of off resonance branches

3.21 Detailed equivalent circuit of a Rosen type piezoelectric transformer

3.23 Outline of a three section altemately poled piezoelectric transformer

3.24 Equivalent circuit representation of the middle section

3.25 Complete equivalent circuit model of a piezoelectric transformer shown in figure 3.24

3.26 Equivalent circuit representation of a three section alternately poled piezoelectric

transformer

3.27 Detai led equivalent circuit representation of a three section a fternately poled

piezoelectric transformer in terms of lumped parameters

4.1 Setup for Measuring Zin

4.2

(xii)

Setup for measuring V2'NI

4.3 Measured Input Admittance vs. Frequency Characteristics of the 3-layered Rosen-type

transformer

Measured and simulated input admittance Input Admittance vs. Frequency

Characteristics of the 3-layered Rosen-type transformer

Measured and simulated Input Admittance vs. Frequency Characteristics of the 3-

layered Rosen-type transformer, afier correction in material parameters is made

Best fit equivalent circuit model near the 3" resonance mode

Comparison of Detailed and Simplified (near 3" resonance) Input Admittance vs.

Frequency Simulation of the 3-layered Rosen-type transforrner

Measured Voltage Ratio vs. Frequency of the 3-layered Rosen-type transformer

Measured Input Admittance vs. Frequency Characteristics of multi-layered Circular

Disc transformer with an open circuit at the auxiliary winding

Input Admittance vs. Frequency Characteristics of multi-layered Circular Disc

transformer. Comparison of simuiated and measured results

Best fit equivalent circuit model of a multi-layered PT near the resonance frequency

Input Admittance vs. Frequency Characteristics of multi-layered Circular Disc

transformer. Comparison of simplified and detailed simulation

Measured Output and Input Voltage Ratio vs. Frequency of the multi-layered Circular

transformer

Basic Class E Converter

A parallel RC network

Class E converter with the an equivalent R and C

Class E converter with the a parallel capacitance across the load resistance

Approximate Equivalent Circuit

Class E Converter based on Piezoelectric Transformer

Waveforms for a Class-E converter using a multi-layered Piezoelectric Transformer

(xiii)

5.8 Class-E Converter using a multi-layered Piezoelectric Transformer with different

values of Choke Inductance

5.9 Asymmetrical Converter with an input inductance

5.10 Simulated waveforms of an Asymmetrical Converter with different input inductance

values

5.11

1 5.13 1 Voltage Transformation ratio (output/input) versus Frequency at different loads

S imulated waveforms of an Asymmetrical Converter (Li=250uH)

L

S. 12 Measured voltage and current waveforms at the input terminais of a PT, when used

with an asymmetrical converter (Li4OOuH)

5.14 Voltage Transformation ratio (auxiliary/input) vs. Frequency at different loads

5.15 Temperature rise (AT) vs. Frequency versus the at different loads

5.16 The Voltage Transformation ratio (output/input) versus Frequency at 100 ohms for

di fferent sam ples

3.1 7 The Voltage Transformation ratio (auxiliary/input) versus Frequency at 100 ohms for

different samples.

5.18

1 A6.1 I

1 Equivalent Circuit of a Piezoelectric Resonator (radial)

The temperature rise (AT) versus Frequency at 100 ohms for different samples

A3. I I

Electromechanical equivalent circuit for the longitudinal vibration section

A6.2 Equivalent circuit of a radial resonator

A6.3 A radially vibrated Piezoelectric Transformer

A6.4 Equivalent circuit of a Piezoelectric Transformer under radial vibration

A7.1

(xiv)

Outline of a three section alternately poled piezoelectric transformer

A7.2 Details of the Input Section of a three section alternately poled piezoelectric

A7.3 - Details of the Output Section of a three section alternately poled piezoelectric

transformer

A7.4 Equivalent circuit mode1 of a circular disc multi-layered piezoeiectric transformer

A7.5 Sirnplified equivalent circuit modef of a circular disc multi-layered piezoelectric

transformer

L

A7.6 Simplifieci Model with Star-Delta Transformations

A8. I Power Stage of a PT based Asymmetrical Converter L

A8.2 Control Stage of a PT based Asymrnetrical Converter

Table 2.1

C

3.2 Design parameters-3 layered PT

Description Definition o f Symbols in this chapter

3.1 Equivaient Matrix notation for Tensor Representation

5.1 Comparison between the results for simulation and calculation

5.2 Summary of Results @220VAC/300 VDC

5.3 Characteristics of Auxiliary Winding (sample#5)

5.4 Characteristics of Auxiliary Winding (different samples)

A8.1

(xvi)

- Component List-Power Stage

A8.2 Com ponent List-Control Stage

Chapter: 1

INTRODUCTION

There is an increasing interest in finding components or circuit architectures that will lead

to higher power densities in switched mode power supplies. Higher power densities are

achieved by (i) reducing the losses in components through the use of optirnized circuit

architectures, (ii) utilizing materials with improved electricai and thermal properties, (iii)

constmcting planar structures to increase the surface area for heat transfer, (iv)

integrating components (v) increasing the switching fkequencies (vi) reducing the number

and size of components Il]. The minimum possible size is ultimately constrained by

losses and electromagnetic compatibility issues.

In an effort to achieve the goal of miniaturization, various technologies, which integrate

the fùnctionality of a reactance and transformer in one device, are being investigated.

Piezoelectric transformers (PT) have recently received attention as a possible alternative

to magnetic based transformers since they are planar, have low losses (96% efficiency),

incorporate a transformer and resonant structure in one package and transmit the energy

fiom prirnary to secondary by acoustic means [2]. Leakage fields are constrained to the

primary and secondary side and these stray fields can be minimized [3].

Piezo-transformers offer a potential advantage of a thimer profile compared to a

magnetic transformer and a simpler manufacturing process due to the lack of windings.

However there are other complexities such as geometries, polarization, mounting, which

need to be considered.

Page 1

At this time, the model for these devices is not well understood specifically from the

point of view of how a transformer can be designed to meet a specific application

requirement. Considering the lack of information available on modeling, this thesis

focuses on advancing our understanding of the model for a piezoelectric transformer. In

addition, converter topologies suitable for use with piezoelectric transfomers are also

discussed.

1.1 Background

Modeling of a simple resonator in terms of an electrical equivalent circuit has always

been of interest to scientists. S. Butterworth gave a description of capacitive

electrornechanical systerns and he concluded that any physical system codd be presented

as an electrical network consisting of a capacitance in parallel with a series LCR network

[4]. It was W.P. Mason, who was credited with developing an electromechanical model

for a loss-less rectangular piezoelectric resonator [5, 63. After Mason, many scientists

worked on the modeling of single segmented or multiple segmented systems. A detailed

circuit analysis for a segmented electromechanical system that extends the approach

presented in [5] was docurnented in [7]. The behavior of al1 segments was described in

terms of general boundary conditions at either end of the system.

Due to the smaller size and high resonant Iiequency, xnulti-layered transducers, i.e.

stacking of ceramic filters on top of each other, have always been attractive to scientists.

The modeling of these devices is more complex. The theoretical treatrnent of stacked

resonators has been reported in [8] and a simplified electromechanical circuit model has

been developed for a ferro-electric cylindrical tube consisting of any nurnber of

Page 2

longitudinally polarized coaxial segments. This has made it possible to analyze

composite resonators in a transducer design.

In 1956, T. Tanaka manufactured various types of mechanical filters to prove the

usefulness of piezoelectric devices as actuators. He revealed the importance of specific

vibration modes and coupling methods for resonators. The discovery of multiple

segmented transducers and their applications gave birth to piezoelectric transfomers

(PTs).

Studies of Piezoelectric transfomers began in 1957 in the US and in 1961 in Japan [9].

PTs did not become comrnercially successful due to the use of poor materials and the

existence of a competing magnetic technology. Researchers in Japan have made

additional efforts to produce efficient and compact PTs for portable applications e.g.

laptop displays since the 1990s. Effkiencies from 85% [IO] to 92% [I l ] have been

reported. It was not until the development of piezoelectric ceramics with large

electromechanical coupling coefficients, that the opportunity for their practical use was

real ized.

C.A. Rosen proposed [12] a PT operating in the length vibration mode. The device was

rectangular in shape and was named after him (Rosen-type PT). This invention opened

the doors for M e r investigations. In [13]-[15], a PT operating in the thickness

extensional vibration mode was proposed. This discovery lead to m e r reductions in

the size of the device.

Fundamental limits of energy density and power throughput for a PT have been discussed

in [16]. The fundamental limitations are imposed by a maximum electric field strength,

Page 3

maximum surface charge density, maximum stress and maximum strain for the

piezoelectric material.

Recently a Rosen type PT was analyzed [17] using a finite element method. An electrical

equivalent circuit mode1 was developed fiom this analysis and was used to validate the

analysis. The calcuiated and experimental results were in good agreement but some

differences in resonant fiequency, transformation ratio and power were observed. In al1

the cases the models were assumed to be linear but in reality the device is far fiom being

iinear.

PTs are advantageous in some DC-DC converter applications for instance cold cathode

florescent lamp divers. This device is cheaper to produce compared to an

electromagnetic transformer due to the PT'S inherent high voltage isolation charactenstic.

In [ 141, a multi-layered piezoelectric cerarnic transformer for switching power supplies is

described. This PT operates in the second rcsonant mode (at IMHz) and al1 segments

operate in the thickness vibration mode. In [18], development of a PT converter is

combined with a zero-voltage switching converter. This PT operates in the thickness

extensional mode, and it is claimed that this design eliminates capacitive turn-on losses.

The losses are due to the low input impedance of large input capacitance in a Rosen-type

design. A inodel of a PT designed for high voltage step-up applications is presented in

1191. The combination of a PT and a DC-DC class-E converter for a low profile

application is described in [20]. The paper shows that the loss of the PT depends on the

load and switching frequency. A matching network is presented and designed to provide

maximum output power. It is also shown that the impedance mismatch in the matching

networks for a PT and the conduction losses of the matching components cause a

significant loss. A PT based converter with pulse width modulation (PWM) is presented

in [10]. RecentIy many applications of PTs in power converters for cold cathode

fluorescent larnp and miniature battery charger have been reported in [21-251, and an

asymmetncal converter design with series input capacitance is investigated in 1261. A

new control PWM with PT is presented in [27] and it is shown that stable voltage

regulation and ZVS over a wide range of input power is achievable by selecting an

appropriate value of input inductor. However, a new modeling approach is reported in

1281. This approach is towards understanding a multi-layered PT, but it is not based on

material and geometrical parameters.

Unfortunately, the design of these devices for a specific application is problematic for the

following reasons: an equivalent circuit based model derived fiom physical principles

does not exist in the literature; no research to date has demonstrated how losses can be

incorporated into the equivalent circuit model; existing circuit models are complex and

do not predict multi resonance or anti-resonance behavior.

1.2 Thesis Objectives

This thesis advances the state of the art by developing a circuit -based model that can

predict multimode resonance and anti-resonance behavior of different types of

piezoelectric transformers. Moreover, the proposed design approach makes a direct link

between the equivalent circuit components and the mechanical dimensions and material

properties. The approach is systematic and relies on the following assumptions: the

system is considered to be linear and loss-less, although the non-linearity issue is beyond

the scope of this thesis. Non-linearities can be considered as a perturbation to the circuit

Page 5

parameters of the first order and can be incorporated as part of a post-processing step. A

detailed analysis of a selected number of devices is presented. The simulation results are

validated for a number of devices using an HP Network Analyzer (4395A). Finally,

converters suitable for use with a piezoelectric transformer are identified and described in

the context of the application environment.

1.2 Thesis Outline

Chapter: 2 reviews the fundarnentals of piezoelectricity. Different types of vibration

modes and geometries are discussed along with the stress and displacement distribution

in a PT. A brief description of piezoelectnc material properties is also given.

Chapter: 3 presents the methodology for modeling a PT. A general introduction to the

basic piezoelectric equations is given. The chapter begins by considering a single

resonator under no mechanical stress. The analysis is later extended to include the case of

a single resonator operating in the thickness mode with a mechanical stress applied.

Then a complete transformer mode1 is presented and a detailed description of the

simplification techniques employed are described. Finally. the application of these

equations for a specific device is discussed.

Chapter: 4 consists of results and the analysis for different geometnes. Simulation results

are analyzed and compared with experimental data.

Chapter: 5 briefly discusses the application of a PT in a DC-DC converter based on a

Class-E converter topology. Experimentd results for an asyrnmetrical bridge

configuration are also given.

Page 6

Chapter: 2 BASIC THEORY

Piezoelectricity is the property held by some materials in which a mechanical

deformation of the material results in an interna1 electric field or vice versa. The

piezoelectric effect is found in crystals that have no center of symmetry like quartz,

Rochelle salt and many synthetic crystalline ceramics [29]. The most widely used

piezoelectric materials are polycrystalline composite ceramics of lead zicronate titanate

b g h coupling factors, high piezoelectric and dielectric constants and wide operating

range for temperature and stress) and barium titanate (widely used for tramducers with

moderate power levels and sensitivity). Specific additives are included to give each

composition specific dielectric, piezoelectric and physical properties.

2.1 Poling of a Piezoelectric Material

Any polycrystalline ceramic is composed of a multitude of randomly oriented crystals

(dipoles) and the bulk properties are the sum of the properties of these crystallites.

Figure: 2. la illustrates this concept, where arrows indicate the direction of polarization.

In the manufacturing of piezoelectric ceramics, a suitable ferroelectric material is first

fabricated into a desired shape and electrodes are applied. The piezoelectric element is

then heated to a temperature referred to as the Curie temperature: the temperature above,

which the spontaneous polarization and piezoelectric effect cease to exist. The heating is

performed in the presence of a strong DC field. This polarizes the ceramic (Le. aligns the

molecular dipoles of the ceramic in the direction of an applied field), as shown in figure:

2. lb. The polarization field remains fkozen in place when the temperature is reduced

Page 7

below the Curie point and the field is removed (figure: 2. lc). The greater the number of

domains aligned, the greater the piezoeiectric effect [29,30].

Figure: 2.1 Polarization of Ceramic material to generate Piezoelectric Effect [29)

The polarization (poling) of piezoelectric materials is permanent. However when

working with the materiais, the temperature of the material should be kept well below the

Curie point. Therefore, the material should not be exposed to very strong alternating

current, as this will give nse to an increase in temperature and hence depoling will result

[30]. Also in order to avoid cracking in the materiai, the stress imposed should not

exceed specific limits. This stress could be caused by temperature gradients, excessive

mechanical stress, or fabrication flaws.

2.2 Definition of Field and Material Properties

Before any M e r discussion, it is necessary to have some basic knowledge about the

material and field properties of the chosen material. These are summarized below:

Stress (T): applied force per cross-sectional area.

Strain (S): the ratio of change in dimension to the actual dimension.

Page 8

Electric Field (E): the ratio of the voltage applied or generated to the distance between

the electrodes.

Electric Disphcement (D): the product of electric field intensity (E) and the

permittivity.

Dielectrie Permittivity ( E ) : the proportionality factor that relates electrical displacement

(D) to an electric field (E) under a constant stress (T). It is given as follows:

Piezoelectric Distortion Constant (d): it relates the mechanical strain (S) developed in

response to an applied electric field (E) with no stress (T) applied. In a general form it is

given as:

Piezoelectric Elasticity Constant or Cornpliance (SE): it relates strain due to an applied

stress, in the presence of a constant electric field. In a general form it is given as:

Where ,E is the compliance given a constant electric field (E).

The inverse of compliance is referred to as Young's Modulus.

Electromechanical Coupling Coefficient (k): it is defined as the ability of a

piezoelectric material to transform electrical energy into mechanical energy, and vice

versa. It is also referred to as the piezoelectric efficiency of a piezoelectric ceramic and

is given by:

Page 9

EIectriculStored Energy Mechanical Input energy

The value of a coupling coefficient is unique for each vibration mode. It is expressed as a

number less than unity. It can also be related to the other piezoelectric coefficients in the

following form:

Deositv: is related tc the mass and volume of a piezoelectric material by the following

expression:

muss P = volume

Mechanical Qualitv Factor: It determines the sharpness of the resonant peak.

2.3 Piezoelectric Effect

A piezoelectric substance is one that produces or develops a surface electnc charge

(distributed) when a mechanical stress is applied i.e. the material is squeezed or stretched

(direct effect). Conversely, a mechanical deformation (substance shnnks or expands) is

produced when an electric field is applied (converse effect).

Figure: 2.2 describes the pictorial f o m of the piezoelectric effect. Figure: 2.2a shows the

piezoelectric matenal without any stress or charge. If the matenal is compressed, a

voltage of the same polarity will appear between the electrodes (figure: 2.2b).

Page 10

Figure: 2.2 Examples of Piezoelectric Effect (291

If the material is stretched then an opposite polarity of voltage will appear (figure: 2 . 2 ~ ) .

Conversely if a voltage is applied, the material will deform. Voltage of opposite polarity

to the polarity setup by the polarization field will cause matenal expansion (figure: 2.26).

Applying a voltage wiIl cause the material to cornpress (figure: 2.2e). If an AC signal is

applied to the piezo device Üxn the device will vibrate at the same fiequency as the

signal assuming the electrical fiequency coincides with the mechanicd resonant

fiequency of the piezoelectric material (Figure: 2.2n.

2.3.1 Polarity of Piezoelectric Effect

An electric field of the sarne polarity as the polarization field will cause an elongation

along the direction of polarization and a contraction in al1 directions perpendicular to the

poiing axis. In contrast, a reverse field will cause contraction dong the poling axis and

Page 11

expansion in the transverse directions. The deformation remains as long as a field is

maintained.

Similarly, a compressive force applied perpendicular to the poling axis produces an

electric field of the sarne polarity as the poling axis. In contrast, an application of a

reversed applied force would reverse the polarity of the generated electric field. The

positive electrode on the finished ceramic is usuaily identified by a polarity mark. This is

the electrode to which the positive voltage is applied during the poling operation [3 11.

2.4 Shapes and Vibrational Modes

The multi-resonant behavior of piezoelectric ceramics depends on their shape, orientation

of polarization and the direction of the electric field. The displacement pattern within a

piezo-device depends on the mechanical frequency excited. The type of displacement

pattern or bending is referred to as the vibration mode. The piezo-device c m be made

into various shapes to achieve different vibration modes or alternatively the vibration

mode required will dictate the basic shape of the resonator.

The vibration mode used is dictated by the target frequency of the resonator and the

desired stress distribution. Resonators have been designed for fiequencies fiom several

kHz to several MHz. Figure: 2.3 shows the various shapes of piezoelectric ceramics.

Page 12

l 1

Figure: 2.3 Different Shapes of a PZT (30)

The modes are described as k i n g transverse, thickness or shear thickness. A description

of each of these modes follows:

2.4.1 Transverse (length) Vibration Mode

For this mode the direction of vibration is oriented orthogonal to the direction of

polarization. Figure: 2.4 shows a rectangular plate shape vibrating in this mode. The

length in the direction of propagation for this mode is much greater than its thickness and

width. nie resonant frequency depends on the length, hence a large length irnplies a low

resonant fiequency drive, and a large surface area of the electrodes irnplies a higher input

impedance. Since, the electromechanical coupling factors associated with the other

modes are very small as compared to that of the transverse mode, this mode generates

single resonant frequency, and/or the other resonant points are very far apart.

Figure: 2.4 Transverse Vibration Mode

Page 13

Where P= Direction of Polarization E= Direction of Electric Field Arrows indicate direction of vibration

2.4.2 Thickness Mode

For this mode, the vibration is oriented along the direction of polarization. Figure: 2.5

shows that disk and rectangular plate shapes are employed in this mode, where the

thickness (length of propagation) is much smaller as compared to its width, length, or

diameter. The resonant fiequency depends on the thickness of the device hence a thin

device implies a high fiequency drive and a small surface area of the electrodes implies a

low intemal impedance. This structure exhibits multiple points of resonance, as the

electromechanical coupling factors associated with the other modes are also present.

Wav Motion 1 Wav Motion 4

Figure: 2.5 Thickness Vibration Mode

2.4.3 Shear Thickness Mode

In this mode the piezoelectric ceramic expands in thickness as well as diagonaily.

Figure: 2.6 shows that rectangular plate shapes are employed in this mode and the length

in the direction of propagation is much smaller than the surface area of the electrodes.

The electric field is orthogonal to the direction of polarization, causing a shear vibration

along the surface. The resonant fiequency is determined by the thickness of the device.

Page 14

Figure: 2.6 Shear Vibration Mode

2.5 Velocity of Propagation

The velocity of propagation through a piezoelectric ceramic has a specific value for each

vibration mode. For a piezoelectric ceramic with a certain shape and vibration mode, the

relationship between wavelength h of a vibration and the propagation length I at a

resonant point is given by:

The sound velocity is given by:

s = f,.A

wheref, is the resonant fiequency. Therefore

Where N, the fiequency constant (see also Table: 2. l ) , depends on the vibration mode

and material properties [29].

Page 15

TABLE 2.1 Definition of Symbols Used in this cbapter

Frequency constants, when the; poling axis is perpendicular to that of stress or strain;

poling axis is in the same direction as stress and strain; poling axis is perpendicular I to that of stress or strain and to electric field; and poling as well as strain or stress are

al 1 in the same direction

Electromechanical Coupling Coeficient indicates that the poling is in the direction

of the 3-mis, and the stress or strain is along the 1-axis.

Electromechanical Coupling Coefficient indicates that the poling as well as strain or

stress is al1 in the direction of the 3-ais. (also referred as a planar mode)

Electromechanical Coupling CoefTacient indicates that the poling is in the direction

of the 3-axis, and the stress or strain is along the f -axis (also referred as a thickness

mode)

The fiequencies for the specific modes are listed below:

Transverse mode: f, = - N33 ; I = length of the resonator 1

N3 1 Thickness mode: f , = -; t = thickness of the resonator f

NI 5 Shear mode: f, = -; t = thickness of the resonator t

NP Radial mode: f, = - ; d = diameter of the resonator d

2.6 Piezoelectric Material

Many applications use the resonance point, since the cerarnic has a very hi&

electrornechanical transfonning eficiency at this point. When piezoelectnc ceramics are

molded in different shapes they can possess multi resonance behavior depending on their

operating vibration modes. For exarnple, for the piezoelectric device operating in the

Page 16

thickness vibration mode, a piezoelectric materiai, with a large electromechanical

coupling factor kt (see also Table: 2 4 , is advantageous. In general, lead zicronate-

titanate solid solution (PZT) family ceramics not only have a large coupling factor kt, but

also a large electromechanical coupling factor k3/ and k,. Therefore, when an AC voltage

is applied to a resonator made fiom PZT ceramics that is desired to support the thickness

extensional vibration mode, the high order modes of length andor width extensional

vibrations could cause undesirable vibrations. For this reason it is extremely difficult to

suppress these undesired vibrations caused by the other modes (Le. k3/) . Thus, to reaiize

a resonator, which has a resonant response due only to the thickness extensional vibration

mode without spurious vibrations, piezoelectric materials with large anisotropy between

k& are required [13,14].

The lead titanate (PbTi03) family of ceramics consists of materials with large

piezoelectric anisotropy, where k, is larger than 50% and k, is less than 5%. In

consequence, it is possible to suppress the undesired vibrations fiom the resonator. The

admittance versus fiequency characteristic of a PZT cerarnic plate and a PbTi03 plate,

whose dimensions are 5Omm long, 25mm wide and Imm thick, are shown in figures: 2.7

and 2.8, respectively. The PbTi03 cerarnic plate operating in the thickness extension

vibration mode has a clearer resonant response compared with the PZT ceramic plate.

Page 17

Frequency (MHz)

Figure: 2.7 Admittance versus frequency of a PZT ceramic (141

This demonstrates why PbTi03 ceramics are used when the piezoelectric device is

operated in the thickness extensional vibration mode. In figure: 2.7, the planar coupling

factor k,, denotes the coupling between the elecûic field in the thickness direction (in

direction 3, as shown in figure: 2.5) and the simuItaneous mechanical actions (in the I

and/or 2 directions). The thickness coupling factor kt denotes the coupling between the

electric field in the thickness direction and the mechanical vibration in the same direction.

1 0 ~ ~ ~ ~ ~ ' ~ ~ ~ ~ ' " ' ~ ' ~ ~ 0.5 1 1.5

Frequency (MHz)

Figure: 2.8 Admittance versus frequency of a PbTi03 ceramic [14)

The requirement of suppressing the undesired vibrations is very critical, as this will cause

loss of power due to the excitation of unwanted resonant modes.

Page 18

2.7 Multi-layered Structures

Any number of piezoelectric layers may be stacked on top of one another. Increasing the

volume of piezo-ceramic increases the energy that may be delivered to a load. As the

nurnber of layers grows, so does the diaculty of accessing and wiring al1 the layers.

Typically, more than 3 layers become impractical [32].

The co-fired stack is a practical way to assemble and wire a large nurnber of piezoelectric

layers into one monolithic structure. The stack, which comprises a large number of

piezoelectric iayers, is a very stiff structure. It also has a high capacitance since the

plates making the structure are connected electricaliy in parallel. The device is suitable

for handling high force and collecting a large volume of charge. The tiny motions of

each layer contribute to the overall displacement. Stack motion on the order of microns to

tens of microns, and a force from hundreds to thousands of Newtons is typical[32].

X-Poled refers to the case where the polarization vectors for each of the m o layers point

in opposite directions (figure: 2 . 9 ~ ) ~ specifically, towards each other. In contrast, Y-

Poled refers to the case where the polarization vectors for each of the rwo layers point in

the sarne direction (figure: 2. Pb).

(a) (b)

Figure: 2.9 X- and Y-Poled Multilayered Stacking (321

Page 19

2.8 Piezoelectric Transformers

The direct and converse piezoelectric effects are used in a piezoelectric transformer,

where power fiom one level to another is transfomed through a vibrating structure

(acoustically). Depending on the geometry and material parameters of the piezoelectric

resonators/crystais, desired voltage transformation(s) c m be obtained.

2.8.1 Applications

Piezoelectric Transformers are widely used in many industrial/commercial applications,

such as power supplies for the back lighting of LCDs in notebooks or laptops and for

high voltage power supplies for ring laser gyros, ozone generators, deflectors in Cathode

Ray Tubes (CRT), copy/fax machines, air cleaners, image intensifiers, munitions fuses

[33] and power supplies that provide power to adaptive wing structures for helicopters.

At present, piezoelectric transformers are used for low power (up to 10 watt), large step

up transformation applications. Devices with power levels of 80W are being developed

WI -

2.8.2 Properties

Piezoelectric transformers have the following properties:

Higher power density compared to magnetic transformers [13].

Higher efficiency and a lighter weight compared to magnetic transfomiers.

No stray leakage field between primary and secondary since energy is coupled

fiom primary to secondary by acoustic means.

Page 20

No fire hazard.

Large step up voltages, since voltage isolation is provided between the primary

and secondary, by a high dielectric constant material.

2.8.3 Structure of a Simple Rosen-type Transformer

The transformer consists of two piezoelectric cerarnic plates of equal cross-sections

rigidly bonded together or made fiom one piece. The voltage ratio depends on the

geometrical dimensions and the piezoelectric properties of the piezoelectric ceramic.

Figure: 2.10 shows the layout of a specific piezoelectric transformer manufactured by

Philips and figure: 2.11 shows the dimensions of the device.

Figure: 2.10 A simplified Rosen Type Piezoelectric Transformer (33 J

Figure: 2.1 1 Dimensions (in mm) of a Piezoelectric Transformer 1331 (Shaded area is an electrode)

The transformer is operated by applying an AC signal between the electrodes of the first

half (the input half). If the frequency matches that of the n a W vibrating frequency of

Page 2 1

the length vibration of the bar, then a voltage of signifiant amplitude will appear at the

secondary (output) electrode.

2.8.4 Stress and Displacement distributions

The resonant behavior of a transformer is less obvious than a simple resonator because of

the lack of symrnetry and the electrical independence of the two haives. As a result of a

high mechanical quality factor, the particle displacement is nearly sinusoidal. The

displacement and stress distributions for a piezoelectric transformer are shown in figure:

2.12, where u (x) = Displacement, s (x) = stress.

The first mode (A=2L) is the half wavelength mode or the fundamental mode, as the

transformer length is equal to a half wavelength and the input and output layers expand

and contract simultaneously. Since the generated charge quantity is in proportion to the

stress value in the piezoelectric matenal, the part, to which the maximum stress is

applied, should be used effectively i.e. the stress should be maximized between the end

plates.

For the first mode, the interface between the input and output parts is exposed to the

greatest stress. This area is typicalIy a region in which the poling of the material changes

and cracks could develop here. So, the vibration energy between the input and the output

parts cannot be utilized effectively. On the other hand in the second mode (A=L) or one

wavelength mode, the input and output layers expand and contract alternatively. The

second mode is superior to the first mode since the stress at the interface between the

Page 22

primary and secondary section is zero. The third mode (A=2/3L) results in the strain in

each section being bi-directional and the vibrational node is not in the center causing

maximum stress at the interface between the primary and secondary sections 113,351.

Figure: 2.12 Displacement and Stress Utilùation P5J

Page 23

Chapter: 3 MODELING

3.1 Basic Piezoelectric (Tensor) Equations

In their original state, ceramic materials are composed of a multitude of crystallites with

random domain orientation (isotropic) and hence no piezoelectric properties [1,3,6]. By

the application of a temporary high electric field (as explained earlier) they becorne

anisotropic, retain the polarization and become piezoelectric. This is referred to as the

poling operation, and the poling direction, by convention, defmes the 2-axis of a three-

dimensional orthogonal axis system [6]. In figure 3.1, the X, Y and Z-axes are

represented as 1,2 and 3 respectively.

Figure 3.1 : Axis Representation

The property of piezoelectricity is rnathematically described by a phenomenological

model, derived fiom thermodynamic potentials. The derivations are not unique and the

set of equations describing the piezoelectric effect depends on the choice of potential and

the independent variables used [6].

When a piezoelectric substance has an electric field E applied across its electmdes and is

maintained at a constant temperature at al1 points, it produces distortion (elongation) S

that is a linear fùnction of the electric field, if the field strength is not too large [l]. The

Page 24

tensor representation of this phenornenon of piezoelectricity under a constant temperature

condition and small strain is:

where:

E, represents the electric field, first order tensor

qk represents the strain, second order tensor, and

duk represents the piezoelectric coefficient, third order tensor.

The dqk tensor is composed of the following 3 layers of syrnmetrical matrices:

Ist.layer(i = 1)

A general third rank tensor has 33=27 independent components. Each matnx in the duk

tensor is syrnmetrical in j and k i.e. d123=d13t, d213=d231 etc. Therefore, some of these

coefficients cm be simplified to a reduced form by noting that there is a redundancy in

the stress and strain variables, leaving 18 independent dQk coefficients [6]. This reduction

Page 25

in coefficients is advantageous for matrix notation and can be represented in a new form

as shown in table: 3.1.

TABLE: 3.1 Equivalent Mat& Notations for Tensor Representations

1 Tensor Representation Matrix Notation 1

Therefore:

The subscripts

respectively. The

Sj = c d , ~ , , i =1,2,3and j =1,2 ,....., 6

and 6 refer to shear directions about the 1, 2 and

(3 4

3 directions

first subscript (i) refers to the electric field direction and the second

subscript (j) gives the direction of mechanical stress or strain.

From Hooke's law we have strain proportional to stress and this proportionality is given

as follows:

Page 26

where sE is the cornpliance or elasticity constant expressing the proportionality between

the strain and stress, and the superscript E refers to the fact that the value of s is obtained

under the condition of a constant electric field. In tensor form:

S, = s;~, ; where j = k = 1,2, ..., 6 (3-8)

Hence, for a piezoelectric cerarnic, these relationships can be combined to give a

complete relationship of strain S depending on the stress T with an electric field E applied

across its electrodes as follows [6]:

S j = s ~ ~ + s ~ , ~ , + s ~ 3 ~ 3 + ~ f i ~ 4 + s f , ~ , + s f , ~ , + d , j E , + d 2 j E 2 + d 3 j E 3 J - - (3.9)

Where i = 1,2,3 and j = k = 1,2 ,....., 6

Similarly, a relationship exists for the electric displacement D as a iünction of E and T,

which is given as follows:

LI, =dilT, +d, ,T, - - +d,,T, +d,,T, +d,,T, +di6T6 + E ~ E ~ +&LE? + E ~ E ~ (3.1 1)

or

Where i, I = 1,2,3; j =1,2, ....., 6

E; is the dielectric constant, and the superscript T infers that the dielectric constant

is obtained under the condition of a constant stress.

Equations 3.10 and 3.12 are called the basic piezoelectric equations. Electric field E and

Page 27

electric displacement D are represented in vector form, while stress T and distortion S are

represented in tensor form.

Three vibration modes are exploited in practice and are commonly referred to as the

transverse mode, the thickness mode and the shear mode. The basic piezoelectric

equations for the three basic vibration modes are given as follows:

Transverse Length vibration

Consider figure: 2.5. The polarization is in the x-plane [l] therefore:

T2=T3=0

Also no shear stress exists, therefore

T5=T6=0

This implies from equations 3.9 and 3.11 that:

SI = s,lE Tl + d31 E3

0 3 = d31 TI + E 3

Thickness vibration

Consider figure: 2.4. The polarization is in the z-plane [1] therefore:

Tl =T2=0

Also no shear stress exists, therefore:

T5=Ta=0

This implies fiom equations 3.9 and 3.11 that:

S3 = ~ 3 3 ~ Tj + d33 E3

D3 = d 3 3 T3 + ~ 3 3 ~ E3

Page 28

Thickness shear vibration

Consider figure: 2.6. n i e polarization is in the shear-plane [l] therefore:

Tl=T2 =T3=0

Also shear stresses Tg (12 of 21, are not present but T5 ( I J , 311 exists, therefore fiom equations

3.9 and 3.1 1:

Ss = sssE T5 + dI5 El (3.15 a)

DI = dis Ts + &llT El (3.15 b)

3.2 Piezoelectric Constants and Coefficients

After deriving the basic piezoelectric equations and before modeling an equivalent circuit

it is necessary to define certain constants. Let us consider the transverse length mode

given by expressions 3.13a and 3.136. Expressing E3 as a function of D3 in equation

3.136 and substituting in equation 3. I3a we amve at the following expression:

Where g31, also called the voltage output coefficient, is the ratio of the piezoelectric strain

constant dJ1 and the dielectric constant E& . The voltage output coefficient refers to the

field strength, when a uniform stress is applied under the condition of no electrical field

[29,3 11. Its value for the different types of vibration modes is given as follows:

d3 1 Transverse: g3, = - 4

Page 29

d33 Thickness: g,, = - 4

Shear: g,, = - 4

Equation 3.16 can be written as:

Where 41 (sometimes referred to as k) is the electromechanical coupling coefticient, as

defined in chapter: 2. Its value for the different types of the vibration modes is given as

follows:

Transverse: k,, = d33

,/m Thickness: k3, = 4 1

J s X

Shear: k15= 4 5 ,/m The Young's modulus, as defined in chapter: 2, for the different vibration modes is given

as:

E 1 Transverse: 5, = - E SI 1

1 Thickness: Y,," = - E

s33

E 1 Shear: Y,, = - E s55

Page 30

The piezoelectric distortion constant d , for different vibration modes is given as follows:

Transverse: d3, = k,, /% Thiekness: = k33 ,/%

Equivalent Circuit Modeling

The basic piezoelectric equations derived in section 3.1 will be used to develop an

equivalent circuit model. It is also required to express these equations in diflerent

coordinate systems, so that they c m be employed for different geometries. Appendices

A l and A 2 present these equations in the Cartesian and Cylindrical coordinate systems

respective1 y.

As explained earlier, a simple piezoelectric transformer is comprised of two piezoelectric

resonatorslfilters as shown in figure: 3.2. In order to develop an equivalent circuit of a

transformer, it is necessary to first develop the models of individual resonators.

Page 3 1

input half output half Figure: 3.2 A simplified representation of a Piezoelectric Transformer

3.3.1 Equivalent circuit of a transducer in Thickness vibration mode

Consider first the input half of a Rosen type resonator in the thickness vibration mode, as

shown in figure: 3.3.

Figure: 3.3 Rosen type Resonator in the Thickness Vibration mode

The bar is long and thin with dimensions designated as I for length, w for width and r for

thickness. I is much greater than w and r, and r-W. The bar is polarized in the thickness

(z-direction) and the x and y planes coincide with the planes of the electrodes [1,3,5].

The extensional vibration in the z-direction is given by Newton's law as:

Page 32

Where ul is the displacement in the cerarnic plate in the x-direction. The relations

between stress, electric field (only E3 exists) and the induced strain is given by:

SI = S ~ T , + S E T , +&T, +d3,E3 (3.19)

The assurnption in deriving the simplified equation of motion is a plane wave

propagating along the length axis and a zero stress in the lateral direction. Thus the

lateral inertia does not have to be considered and consequently [3,5]:

T2=T3=0.

Hence fiom equations 3.18 and 3.19, the equation of motion reduces to the following

fonn:

Therefore equation 3.19 can be written as;

SI = S ~ T , +d3,E3

where p is the density of the crystal and u, is the displacement of the crystal in the x-

direction.

Expressing Tl as a fùnction of E3 and SI, we obtain

Differentiating equation 3.22, we obtain

Also fiom the basic piezoelectric equations we have:

D, =d3,T, +&LE,

Page 33

Substituting equation 3.22 into 3.24 gives us

where

It is assumed that the state of the body is uniform dong its Iength and the electric field is

constant i.e. independent of x [ I l , therefore:

and thus

Hence equation 3.20 can be rewritten as:

a2u, - 1 as, PT--- .Y; ax

Since

Hence equation 3.29 c m be restated in the following way:

Page 34

Let

1 = - (3.32) p s i

Where v is the velocity of the propagating wave in the piezoelectric medium. Then for a

loss-less system:

For simple hannonic motion, the variation of u, with time can be written in phasor foxm

as [3,5]:

Substituting this in equation 3.33 results in the following expression

The solution of equation 3.35 with two arbitrary boundary conditions is:

- O X W X u, = ACOS-+ Bsin-

v U

To determine A and B, we differentiate equation 3.36 with respect to x:

From equation 3.21, we have:

At the boundaries, x=O and x=I (the crystal length), the stress is TI=O [l]. Under these

conditions:

Page 35

Let

d E 4 1 Asinyl = - a + c o s y l - E ,

and

1 1 --+-] Y sinyl tanyl

sinyx sinyx --- + cos yx sin yl tan yl

The current in the piezoelectric device is the rate of change of the surface charge with

respect to time and for a single harmonic voltage [3,6] it is given by:

Where S is the surface area and dS = drdy

Hence, equation 3.33 implies;

C

~ = j o J ~ , d x = j o w O O

Introducing SI from equation 3.42 and integrating fiom O to t , we have:

Page 36

The admittance of the crystal is therefore:

Low frequency characteristics

At very low frequencies the admittance in (3.46) reduces to the following capacitance [6]:

where we have made use of equation 3.26. Therefore:

Resonance frequencies

A resonant fiequency is the fiequency at which the admittance is infinite or impedance is

t zero (short circuit case). With reference to equation 3.16, if tan y - = m or altematively

2

4' 1.L y - = - then resonance will occur [5 ] .

2 2

Equation 3.36 exhibits a multiple resonant behavior, defined as follows:

where:

n = h - I and m=I, 2, ...

The resonant fiequency fi) is deiemined by the electrical cornpliance sllE, density p and

the length of the crystal. Therefore the resonant frequencies are:

Page 37

nie anti-resonance fiequency is the fiequency at which admittance is zero or impedance

is infinite (open circuit case). With reference to equation 3.16, it occurs when:

or altematively

y t Y[ 4-1 -çot - = -- 2 2 .;;s1;

We defined the electromechanical coupling coefficient in section: 3.2 as:

Now substituting the value of EL fiom equation 3.26 into equation 3.496 and simplieing

gives :

Therefore, the anti-resonance fiequencies can be determined using equation 3 . 4 9 ~ .

Determination of circuit parameters

From equation 3.46,

Therefore:

Page 38

Let

Equation 3.5 1 then becomes:

where

and

k:, tan n ypie:o = G [- 1 - k:, .]

Now the electrical behavior can be investigated under a constant voltage condition, with

varying w .

Resonance occurs when a is equal to a,, defined as follows:

me 1 a,, = - = - n l r for n = 2 m - 1 , m=1,2 ,.....

2v 2

tan a The fùnction - in equation 3.53 can be expanded in the following way using the

a

partial fraction expansion [34] :

tan a -=8[ 1 + 1 + .......-.. ] = 2 Pn

a zZ -40' 9z2 -4aZ nais1 - (ala. )2

Where p, is given by:

Page 39

8 1 P, = -- for n=2rn-1 n2 n2

This means that the piezoelecvic impedance is expressed by a number of LC, series

circuits in parallel [3,6]. Hence:

u,,,=

From the equivalent circuit point of view, a series resonance of L and C,, in the

piezoelectric branch gives rise to this resonance. On the other hand anti-resonance

corresponds to a combined effect of al1 branches of LC, near the resonant fiequency and

Y,l,C,riCar represented by the capacitor Co, as shown in figure 3.4 [3].

4

Figure: 3.4 Equivalent Circuit Representation of a Piezoelectric Resonator

Using the expressions 3.55, 3.56 and 3.54 we obtain the expression for capacitance and

using

, we obtain:

Page 40

Hence :

From equation 3.55 and 3.57 we have:

This implies:

After simplification and substitution, we O btain:

3.3.2 EQUIVALENT CIRCUIT UNDER MECHANICAL STRESS

The above representation (section: 3.3.1) of the thickness vibration equivalent circuit is

derived under a stress-fiee mechanical condition at the end terminals. This is therefore

not useful for treating extemal mechanical loads or cascaded sections. For this purpose,

the inclusion of arbitrary mechanical boundaries is required [3].

The pieu>-device in figure: 3.5 is considered around the resonance point. Under fiee

conditions, when no force is applied to the end surfaces, the mechanical terminals are

Page 4 1

considered short-circuited. In fact the mechanical action is delivered through both ends

of the transducer [3]. This equivalent circuit with six terminais (2 electrical and 4

mechanical) was developed by Mason [5,6].

Figure: 3.5 Piezoelectric Resonator under Stress

A compressive force is taken as an external mechanical variable (i.e. F = -T, wr ). Later

subscripts 1 and 2 are used to represent the mechanical ends at x= - 412 and x= + 472

respectively.

From equation 3.22 we have:

Then fiom 3.22, we have:

In general form, we also have:

u = ueJN'

and the displacement velocity is

Page 42

There fore:

ry W X W X u = ACOS-+ Bsin-

O U

In terms of velocity we have fiom equation 3.65:

It is desired to utilize the particle velocities at the two ends of the resonator as boundary

conditions [8] which are defined by:

(at x=-112)

4 -- - U2 (at x=-1/2) iw

Constants A and B in equation 3.65 can be determined by differentiating equation 3.65

with respect to x, and evaluating the constants A and B for the given boundary conditions,

we obtain:

u2 = A C O S ( ~ C / ~ ) + ~ s i n ( r U 2 )

Solving for A and B, we obtain:

U , sin y(.f?/2) - U2 sin y (-t/2) A =

jw sin yC

B = u2 cos y (- !/2) - LI, cos y ( q 2 )

jwsin yC

Page 43

The strain may be written in terms of the denvative of the particle displûcement, so by

using equations 3.61, 3.71 and 3.72, we obtain the stress expressions for each mechanical

end as:

4, T(x=-t /2)+?E3 = 1 [u, - Ul cos yt]

SI I SE jvsin

d3 l T , ( x = t / 2 ) + , E 3 = 1

s i jvs in y! [CI, cosye - ul ]

4 1

At x=-V2, the compressive force is FI=-wtT,. Hence multiplying both sides of equation

3.73 by wr gives:

d3 l - wt -4 + wEV = - 1 [LI, cos yP - LI, ]

SI I sfi s f i jv s iny t

Let:

Using

and letting

We obtain:

4 - @ V = - O0 [ ~ , - u ~ c o s ~ ~ ] j sin y4

Page 44

Where:

Similarly equation 3.73 would imply, F2=-wrT, at x=+1/2 therefore by multiplying

equation 3.74 by ivt and M e r simplifying we get:

- F ~ + # V = ' O [CI, COS y e - CI, 1 jsin y [

Fz - # V = [- U, + U , cos y t ] jsin y t

Equations 3.76 and 3.77 c m be written in the following matrix form:

Since cos A = cosh jA and jsin A = sinh jA [35], we c m rewrite (3.78) as follows:

Z= - 1 ]= Z . [cosjhyt cosyt sinjhyt - 1 cosjhy! dl 1

Also

coth jye - sinh jy4 21, + 4 2

-"12 1 (3.79) coth j y t 2 Z21+z22

sinh j y l

w here :

Page 45

Based on the above equations, the equivalent circuit c m be drawn as shown in figure:

3.6:

Figure 3.6:~lectromechaniea~ equivalcnt circuit of s resoantor und& thickness vibration

In figure 3.6:

Fil & F2/ are the forces at the ends UI & U2 are the velocities ut the two ends 7

t = thickness p = density s 1 1 " = Cornpliance

YJ z,, = Z0Tmdy-- 2

d3 1 = Piezoelectric Struin Constant (= MJ d3i/s,lE

With reference to figure: 3.6, we can derive the equation for current as follows:

The above equivalent circuit is valid for al1 frequencies.

Simplification

The resonator is clamped on one end (clamped drive) and fiee on the other end (inertia

Page 46

drive). The exact network under these conditions can be obtained by shorting the

terminal at x=-I/2 Le. F = O and leaving the other end open, as shown in figure: 3.7 [SI.

Figure: 3.7 Equivalent circuit with terminal short circuited

Figure: 3.7 can be redrawn as shown in figure: 3.8:

Figure: 3.8 Redrawn equivalent circuit with terminal short circuit

Figure: 3.8 c m be simplified using the procedure described in 1371. The process is

explained in figure: 3.9.

Page 47

Figure: 3.9 Equivalence of an L-Type network plus a transformer with an L-Type network reversed in direction

where:

< and Zb are the new equivalent impedances and a is the result of this transformation.

Using the technique of figure: 3.9, figure: 3.8 can be redrawn and is shown in figure:

Figure: 3.10 Equivalent circuit represeotation after employing the simplification technique

Now referring the middle side to the secondary on the right, we obtain a circuit as shown in figure: 3.11.

Page 48

Figure: 3.1 1 Equivalent circuit representation after refemng the middle branch to the secondary side in figure 3.11

In figure: 3.11, let

- j4Z0 3 Z A = y t Y + 2 jZo tan- = -2ZoC0~hj--

sin y[ 2 2

Also, let

ire Let ,=- CI

Page 49

Considering the hyperbolic fùnctions of equations 3.87 and 3.89, we have from the partial

expansion theorern [3 81 :

I cotha = -+ 2a 2a ......... 7 + * + a z 2 + a - 4n2+a

Equation 3.90 can be modeled as parallel combination of an inductor L I , and an infinite

number of inductors L, and capacitors C,, in series.

Similady for the parallel branch we have:

Equation 3.91 can be modeled as a parailel combination of an infinite number of

inductors Ln and capacitors C, in series. Thus considering both branches, the equivalent

circuit can be formulated as shown in figure: 3. I2.

Figure: 3.12 Simplified equivalent circuit without losses

Page 50

Determination of circuit parameters

The use of the partial fraction expansion theorem for the parallel and series branches

enables us to determine the equivalent circuit parameters in tenns of Ls and Cs.

For the Series Branch

For the Parallel Branch

Hence figure: 3.12 c m be redrawn in ternis of equivalent circuit parameters, as shown in

figure: 3.13.

Modeling losses

Piezoelectric cerarnics suffer energy losses, which are attributable to mechanical,

dielectric and piezoelectnc effects. Mechanical losses are the dominant of the 3 types

near resonance and the latter two are not of significant importance [39].

Page 5 1

Figure: 3.13 Equivalent circuit of a resonator for thickness vibration with al1 the circuit parameters

Mechanical Iosses are due to the delay between the strain and the force applied i.e. a

hysteresis curve is traced over one cycle. The mechanical quality factor Q,,, is the figure

of merit most ofien utilized in regards to the mechanical losses. The quality factor is

defined as the ratio of stress in phase with strain velocity to the stress out of phase with

strain velocity 1351, (figure: 3.14).

a, = tan +c

Q,,, = cot @c = tan(90 - 4=)

I Strain Figure: 3.14 Vector Representation of Strain and Velocity

The electrical analogies of velocity and stress are current and voltage. The quality factor

for an equivalent circuit is the ratio of reactance to resistance [35]. Hence, in a series

K R circuit, we can define:

Electrical Quality factor= Qe =

Page 52

In a mechanical system we use the nomenclature Qm instead of Q. to denote the quality

factor [35]. Thus:

Qm = Qe (3.97)

Most rigorous denvations of the Iumped circuit parameters for elecûical equivalent

circuits of piezoelectric resonators show that the above-expfained method for determining

the quality factor gives good accuracy near resonance. It may not be as accurate for other

fiequencies including the anti-resonant fiequencies [3 51.

Since the value of mechanical quality factor Q, is given by the manufacturer, it is used to

introduce mechanical losses in ternis of an equivalent R, using the following relationship;

where Qmn is Qm for each resonant fiequency. Hence the circuit s h ~ w n in figure: 3.13

can be redrawn with losses incorporated as show in figure: 3.15:

Dielectric Losses

J I

Figure: 3.15 Compkte equivaknt circuit mode! with losses

Page 53

It is also possible to incorporate the dielectric losses by placing a resistor in parallel with

the input capacitance. One interesting question is whether or not the branch containing

the inductance should include a series resistance. This may be the piezoelectric loss that

authors have referred to in a previous publication [39]. Apparently the authors came to

the conclusion that it would be impossible to separate the mechanical and piezoelectric

losses. Perhaps our circuit makes the positioning of this additional loss element obvious.

3.3.3 EQUIVALENT CIRCUIT OF A TRANSDUCER FOR THE LONGITUDINAL VIBRATION MODE

The analysis of the longitudinal vibration mode is similar to the thickness vibration mode

with the exception of the electrical boundary conditions. The wave is assurned to

propagate along the length axis with zero stress in the lateral direction [3]. Figure: 3.16

shows a transducer operating in the transverse (length) vibration mode.

Figure: 3.16 A longitudinally vibrated piezoelectric resonator under stress

The equivalent circuit modeling approach follows dong the same lines as the description

given in section: 3.3.2. Details are explained in Appendix A3. The equivalent circuit of a

transducer under longitudinal vibration is shown in figure: 3.17.

Page 54

Figure: 3.17: Electromecbanical equivalent circuit for the longitudinal vibration section

In figure: 3.17:

F2 & F2* are the forces ut the ends Uf & U' are the velocities ut the huo ends

7

I = length w = widrh t = rhickness p = density ~ 3 3 ~ = Cornpliance w = Angular velocity dJ3 = Piezoelectric Strain Constant for the longitudinal mode Steps for simplifiing the circuit in figure: 3.17 are the same as those given in section:

3.3.2 and explained in appendix A3. The simplification process would give us an

equivalent circuit as shown in figure: 3.18.

Figure: 3.18 Simplified equivalent circuit witbout losses

Page 55

In figure: 3.18, the expressions for the impedances or admittances are the same as derived

in section: 3.2, i.e.

Determination of circuit parameters

These parameters are determined in the sarne way as was determined for the thickness

vibration mode, namely to use the partial tiaction expansion method to represent the

hyperbolic functions.

For the Series Branch

For the Parallel Branch

There fore :

Hence figure: 3.18 can be redrawn in terms of equivalent circuit parameters, as shown in

figure: 3.19.

Page 56

Figure: 3.19 Equivaknt circuit of a resoaator assuming longitudinal vibration with al1 the circuit parameters

An equivalent circuit with losses for a transducer under longitudinal vibration conditions

can be obtained in a similar way as outlined in section: 3.3.2.

3.3.4 EQUIVALENT CIRCUIT OF A TRANSFORMER

The equivalent circuit of a piezoelectric transformer s h o w in figure: 3.2 is just a

combination of the open mechanical ends of the equivalent circuits of individual

transducers or sections. The simplified representation of the circuit is obtained by

applying transformations and circuit simplifications that result in an equivalent

piezoelectric transformer model. This can either be shown in a detailed form (in terms of

hyperbolic fùnctions), or with the lumped parameters with losses induded. Figure: 3.20

shows the results after combining and simplieing:

Page 57

Figure: 3.20: Simplified equivalent circuit of a Rosen type piezoelectric transformer

In figure: 3.20:

Z , , = 22, coth j- ( 3

A detailed transformer mode1 wîth al1 the lumped parameters is shown in figure: 3.21.

Figure 3.21: Detailed equivalent circuit of a Rosen type piezoelectric transformer

Page 58

3.3.4.1 Further Simplification to the Model

The model shown in figure: 3.21 is quite complicated. Usuaily, we are operating close to

a particular resonant frequency. In which case the details of the remaining resonances is

not important, unless the device is subjected to a voltage that generates harmonies that

coincides with the other resonant fiequencies. A simplified equivalent transformer model

either side of a resonance frequency c m be obtained by applying L'Hospital's Rule to

determine the equivalent impedance of parallel off resonance branches for the series and

parallel sections. This process of determining the equivaient circuit is explained in

Appendix: A 4 This action reduces the order of the rnodel as s h o w in figure: 3.22.

(a) series

(b) parallel

Figure 3.22: Simplification of off resonance branches

where for the series branch:

and for the parallel branch:

Page 59

J Y, =- 1iI-n a-.", =,7

220

cot a

a

3.3.4.1 Process Flow Chart

This whole process of equivalent circuit modeling c m be summarized in a flow chart

fom. We have divided the whole process into 3 steps i.e. pre-processing, processing,

and post-processing. These steps and the flow chart are given in Appendix: AS. A

transformer mode1 is also s h o w for a circular disc operating in the radial vibration mode.

It is given in Appendix: A6.

3.4 Extension of the Model to a Three Section Rosen- type Piezoelectric Transformer

As explained in Chapter: 2, a multi-layered structure is capable of handling high force,

and an increase in device volume increases the energy that may be transferred to a load.

Due to the layered structure, a higher resonant frequency (specific mode) can be

achieved, depending on the stress distribution. We used a three section altematel y poled

piezoelectric transformer for testing and analysis purposes. Figure 3.23 shows the

general outline of the device, while Table: 3.2 contains an estimate of the material and

geometry data.

Page 60

- "out

Figure 3.23: Outline of a three section alternately poled piezoelectric transformer

TABLE 3.2 DESIGN PARAMETERS

1

Piezoelectric Charge Coefficient (dJl) 1 -14i~lB'~ m N

Density (p)

1

1 1 Piezoelectric Charge Coefficient (4) 1 3 10 ~ 1 0 " ~ m N

7.97~ 10' kg/m'

1

Elastic Compliance (s,,") 1 11.5 xlO-" m L M I

Elastic Cornpliance (sUE) 1 15.9 xlO-IL mZM 1

Free Perrnittivity ( E ~ ] ) 1 1380 F/m I

Length of each section ( f ) ) 13.2 mm I

W idth (w ) 1 7.6 mm

3.4.1 Modeling

Thickness ( t )

The modeling of this device follows dong the same lines as the device modeled earlier.

1.0 mm

The first and last sections have the sarne equivalent circuit as given in the previous

sections i-e. 3.3.1 and 3.3.2 for thickness and longitudnal vibration modes respectively.

In contrast, the middle section must be handled differently. The middle section is poled

in an opposite direction to the first section and is comected electrkaiiy in parallei.

Figure: 3.24 shows the equivalent circuit representation of the middle section. If we

combine al1 the three sections we will have a circuit as shown in figure: 3.25.

Page 6 1

- -

Figure 3.24: Equivalent circuit representation of the middle section

ia input Section

2nd Input Section

Ouput Section

Figure: 3.25: Complete equivalent circuit model of a piezoelectric transformer sbown in figure: 3.24

Upon simplification and application of boundary conditions, the circuit in figure 3.24 c m

be redrawn as shown in figure 3.26. This figure gives a complete equivalent circuit model

of an alternately poled piezoelectric transformer.

~ k . 3.26 Equivalent circuit representation of a tbree section altemately poled piezoelectric transformer

Page 62

In figure: 3.26:

~t i 2, = z,, coth j-;.

3.4.2 Lumped Circuit Model

We ernploy the methodology described in the section: 3.3, to determine the lumped

parameters associated with each branch using the Partial Fraction Expansion of

hyperbolic tùnctions. Figure: 3.27 shows a detailed mode1 in terms of lumped circuit

parameters in which the middle resonator is transformed fiom a Y configuration to a A

configuration.

Fig. 3.27 Detailed equivalent circuit representation of a three section altemately poled piezoelectric transformer in terms of lumped parameters

In figure: 3.27:

L A I g and CAl,,s represent the expansion of ZH. L B ~ ~ and Celd represent the expansion

of &, LA2d and CAld represent the expansion of &, LB~,$ and Ceins represent the

expansion of ZB2, and CA& represent the expansion of ZA, L B ~ and C~f i represent

the expansion of ZB

Page 63

Chapter: 4

EXPERIMENT AND RESULTS

A 3-layered piezoelectric transformer (described in Chapter: 3) and a multi-layered

circular disc type structure (described in Appendix: A7) are considered. The

experimental plan, test results, and also the derivation of a simple circuit mode1 near the

resonant frequency based on the techniques described in Chapter: 3 are described in this

chapter.

4.1 Testing Procedure

Test A: Yi, Characterktic:

The proposed admittance arrangement is shown in figure: 4.1.

1. The V-I impedance method was used to veri@ the simulation results. This

method of measurement requires a value for the input current 1. This is achieved

by determining the voltage across the resistor Ri,, and calculating Ih using the

following relationship:

2. A HP Network Analyzer i.e. HP4395A was used to perform the experiments. The

comection setup is shown in figure: 4.1. The resonant fiequencies, anti-resonant

fiequencies, and admittance characteristics as a function of fiequency were

extracted fiom these results.

3. Only small signal analysis was performed considering the limitations of the

maximum voltage levels available fiom the Network Analyzer. A fkequenc y

Page 64

range of 10-200 kHz was used. Frequencies lower than IkHz were not practical

due to the low signai to noise ratio.

See Note 1

DUT

Note 2

Note 1: Rh is adjusted such that the maximum current Iin under short circuit

conditions does not exceed the rating of the analyzer, which for our case is 100 R

Note 2: The loads considered for RI. were 1 OOY IOOOY I ka, I OOkO and open circuit

i.e. infinite resistance.

Note 3: Piezoelectric Transformer(s) with rated input voltage of 1 OV, 1 W step-up and

300V, IOW-15W step-down were tested. The estimated Qs of these devices are 1800,

and 180 respectively. The voltage transformation ratio for these devices is

approximately 1: l O (step up), 1 O: I (step down) respectively.

Test B: Ourpu to Input Vo/tage Ratio (Y.,)

The proposed voltage divider scheme for a step up configuration is shown in figure:

Page 65

See Note 1

I DUT

Nehivork Analyzer

4 I

Piezoelectric - L I

R I

Transformer I

1 6 Note 2 See Note 3

Figure: 4.2 Setup for measuring V2'Ni

Note 4: RDjvider is adjusted such that the maximum voltage Y2 ' under open circuit or

low load conditions does not exceed the rating of the analyzer/probe.

The voltage division is achieved using HP's active probe i.e. HP1 6338A with input signal

divider option. We used a divider ratio of 100.9.

4.2 Testing of 3-layered Rosen-ty pe Transformer

The piezoelectric transformer discussed in section: 3.4 was tested first. Two transformers

were tested. These transformers are distributed comrnercially by Mitsui Chemicals,

Japan and the device nurnber is PT130A02. The material and geometrical information is

shown in table: 3.1.

Admittance Measurement

Figure: 1.3 shows the results obtained at different Ioads for one of the samples. The Y-I

characteristics were measured first by recording the input voltage and the output voltage

across the terminais of the device over a fiequency sweep. The input admittance

characteristics as a function of fiequency were calculated aflerwards.

Page 66

-100 .

3rd Resonant Mode

-250 .

- 300 O 0 2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

F requency (Hz> 10'

Fig,ure 4.3: Measured Input Admittance vs. Frequency characteristics of the 3- layered Rosen-type transformer

From figure: 4.3 it is clear that there are different resonant modes present. For our device

it is advisable to operate the device near the 3rd resonance mode as it ailows the device to

operate with zero stress at the section interfaces, as described in Chapter: 2. This

corresponds to a fiequency of 120-130kH.z (the operating fiequency specified by the

manufacturer).

We validated Our model of figure: 3.25 i.e. detailed circuit model of a 3-layered Rosen-

type PT by simulating the circuit model. Figure: 4.1 shows a comparison of our

simulated and measured results for a load of Ik 0

It can be observed fiom figure: 4.4 that Our model predicts the multimode resonance and

anti-resonance behavior very well. The shifts in the peaks are due to the uncertainties in

material and geometrical tolerances and a close response between the experimental and

simulation results were obtained by changing the value of thickness extensional

compliance ( s i ) by 80% (up). Fine adjustments to the offsets between the two curves

Page 67

were obtained by adjusting the longitudinal compliance (s:) to 90% of its specified

value. The adjustments allowed us to obtain an exact relation between the experimental

and simulated resonant fiequency. In order to provide near exact correlation between the

simulation and measured results in terms of the amplitude, the dielecûic constant (6; )

was increased up to 5% of its given value. This variation between estimated and actuai

data is possible since these parameters can change during the cofiring stage. Figure: 4.5

shows the results after these adjustments are made and the simulation program is given in

appendix: Ag. It is also noted that very slight changes in the density ( p ) and the

piezoelectric constants (d,, or d,, ) affect the distance between different resonant modes,

and hence equidistance between the resonant peaks is not achieved. The values of the

mechanical quality factor (Q) will affect the peak value and also the width of a peak. -50 .- I

-350 '---- 1 O 0.5 1 1.5 2 2.5

Frequency (Hz) w 1 O 5

Figure 4.4: Measured and sirnulated input admittance vs. frequency characteristics of the 3-layered Rosen-type transformer

In Chapter: 3 we discussed the method of simplifying the equivalent circuit mode1 near

the desired frequency and resonant mode using L'Hospital's d e . Hence, by employing

Page 68

the methodology, we simplified the detailed equivalent circuit representation of the three

segment altemately poled piezoelectric transformer, figure: 3.28. The simplified model

for the transformer operating near the 3'd resonance mode is shown in figure: 4.6, while

the simulation results of the model are shown in figure: 4.7.

-6 O

-100 -

- -

-300 3

-35 0 O 0.2 0.4 0.6 0.8 1 1.2 1.4 lb 1.8 2

Frequency (Hz) r 1 2

Figure 4.5: Measured and simulated input admittance vs. frequency characteristics of the 3-layered Rosen-type transformer, after correction in material parameters is

made

Figure 4.6: Best fit equivalent circuit model near the 3rd resonance mode

It can be seen in figure: 1.6 that the main resonant branch is retained in the model and al1

other off resonant branches in parallel are replaced by an equivalent impedance. This

equivalent impedance can be determined by applying L'Hospital's rule (Appendix: A4).

Page 69

1 Detailed Simulation

3rd resonant mode Point of interest for

1 Simplification

Figure 4.7: Cornparison of detailed and sirnplifîed (near 3rd resonance) input admittance vs. frequency simulation of the 3-layered Rosen-type transformer

Losses are incorporated by placing resistances in each resonant branch, as explained in

Chapter: 3. The negative branches have negative resistances inserted. The addition of

resistances in each resonant branch Ieads to a decrease in the resonant and anti-resonant

peaks as well as broadening of the peaks. This is in agreement with the operation.

Output to Input Voltage Ratio

The test procedure given in figure: 4.2 was used to determine the transformation ratio of

the transformer under varying load conditions. Figure: 4.8 shows the results obtained. A

transformation ratio of about 30 was observed under open circuit conditions.

Page 70

Figure 4.8: Measured voltage Ratio vs. Frequency of the 3-layered Rosen-type transformer

4.3 Testing of multi-layered Circular Transformer

The mode1 of a piezoelectric transformer given in Appendix: A7 was tested. 3 out of the

5 proto-type sarnples were characterized. The results are discussed below. Material data

for these devices is listed in Table: A7.1.

Admittance Measurement

Figure: 4.9 shows the results obtained at different loads for one of the samples when the

auxiliary is open circuited.

From figure: 4.9, it is clear that there are different resonant modes present. The

fiequency of interest, as given by the manufacturer, is near 99-IlOkHz. The responses

obtained with the remaining sarnples are identical; hence, they are not shown separately.

Upon loading the awtiliay winding (i.e. R-=IO, IOOk), it is noted that the output

remained unaffected and the ratio of auxiliary to input voltage levels remained at

0.3-0.32.

Page 71

-350 O 0 .2 0.4 0 .6 0.8 1 1 .2 1.4 1.6 1.8 2

Fraquancy (Hz) l o s

Figure 4.9: Measured input admittance vs. frequency characteristics of multi- layered circular dise transformer with an open circuit rt the auxiliary winding

We validated our model of figure: A7.5 (i.e. detailed circuit mode1 of a multi-layered

Circular PT) by simulating the circuit model. Figure: 4.10 shows a cornparison of our

simulated and measured results for a I O 0 load, given an auxiliary load of 1 OOk.

Our mode1 predicts many peaks including the measured peaks observed through the

measurements. The other peaks are not observed during the measurement due to the fact

that the excitation levels are very low (mV). There are also some small undulations.

These represent modes that are only weakly excited and which are heavily darnped.

Unforhmately due to the driving limitations of the equiprnent, we were unable to generate

a large enough signal to excite these modes. This low excitation was not a problem for

the Rosen-type as it is designed to operate with a IO V input.

Page 72

-50 I

Srmulation

-1 00 -

Mode of imporîance

s E 9 -250 ,

Frequency (Hz) x l o s

Figure 4.10: Input admittance vs. frequency characteristics of multblayered circular disc transformer. Comparison of simulated and measured results

Now having the experimentd results, we employed the simplification methodology using

L'Hospital's rule, and simplified the detailed equivalent circuit representation of the

multi-layered circular piezoelectric transformer, figure: A 7.5. The simulation results of

the simplified model are shown in figure: 4.11, while the model for the transformer

operating near the desired resonant fiequency is shown in figure: 4.12.

Simplified

i -100 , . ,

S I

M 08s ursd . . 1 o . . . . - i I .. . . . . - I

. . . . , '

I 1

! I

-250 !

7

f i

-300 L.

O 0 . 2 0 .4 0 .6 0 . 8 1 1.2 1.4 1.6 1.8 2 Frequsncy (Hz ) 1 o5

Figure 4.1 1: Input admittance vs. frequency characteristics of multi-layered circular disc transformer. Comparison of simplified and measured simulation

Page 73

to output terminal 2

Input terminal

- - -

(b)ln put Section

Output terminal

Figure 4.12: Best fit equivalent circuit mode1 of a multi-layered PT near the resonance frequency

Figure: 4.1 1 should be viewed with caution, as the simulation is based on the circuit

given in figure: 4.10 and is used to investigate the response near the fiequency range of

99-IOlkiiz. Therefore, al1 the results outside of this fiequency range have no meaning.

Page 74

Output to Input Voltape Ratio

The test procedure given in figure: 3.2 is used to determine the transformation ratio of the

transformer under different loading conditions. Figure: 4.13 shows the results obtained

with the auxiliary terminais lefl unloaded.

The voltage levels at higher loads i.e. >500 ohms are found to be greater than 1.0.

Results obtained when the auxiliary branch was not loaded did not affect the output

significantly and the ratios remained the sarne. The auxiliary is f o n d to trace the input.

The awiiliary to input ratio was observed to be 0.3 in contrast to the value of 0.28

determined through the mode1 (Appendix: A 7).

O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) x 1 0 !5

Figure 4.13: Measured output and input Voltage ratio vs. frequency of the multi- layered circular transformer

Page 75

4.4 CONCLUSION

Our models for both of the test transformer cases predict the multi-mode resonance and

anti-resonance peaks. The partial fiaction expansion theorem was used to generate the

equivalent circuit mode1 for each resonant branch. Near resonance, the equivalent circuit

can be represented by a series L and C, and a paraliel equivalent reactance obtained by

applying L'Hospital's rule to the hyperbolic fünction for the admittance minus the

admittance of the remaining parallei branches evaluated at the resonant fiequency

(Appendix: A*. Losses can be incorporated through a pst-processing step. The

addition of resistances in each resonant branch leads to a decrease in the resonant and

anti-resonant peaks as well as broadening of the peaks.

Page 76

Chapter: 5

Converter Design

In this chapter, two zero voltage switching (ZVS) DC-DC converter topologies, a Class-E

and Asymmetrical Converter suitable for use with a piezoelectric transformer are

proposed. These topologies have the lowest component count and are investigated for a

given application. For the Class E converter, a simplified analysis and Spice simulation

are performed using the conventional piezoelectric transformer mode1 used by many

other researchers. This topology is found to be suitable for step-up applications. The

asymmetrical converter topology is investigated using Spice simulations and

experirnents. It is found that the asymmetrical topology is suitable for step-down

applications. The asymmevical topology was also used to investigate the suitability of

using the voltage across an auxiliary winding as a feedback control signal. Experiments

showed that the auxiliary voltage does not track the secondary voltage hence the voltage

across this arrangement is not suitable as a control feedback signal. The proposed piezo

transformer was also found to be unsuitable for a 220 V input voltage.

5.1 A DC-DC Class E ZVS Converter Design

The advantage of the Class-E converter is the single switch implementation and the

elimination of switching losses if operated under a ZVS condition [40]. ZVS switching

condition is satisfied, when the switching fiequency is higher than the series resonant

fiequency .

Page 77

The piezoelectric transformer mode1 is ideally suited for a Class-E converter since its

equivalent circuit represents the circuit typically associated with a Class-E converter load.

In terms of component count, PTs provide an appropriate resonant structure in one

package.

5.1 .1 Description

The basic circuit of the Class-E ZVS inverter is shown in figure: 5.1. It consists O

power MOSFET operating as a switch, an LrCrRo series resonant circuit, a shunt capacitor

Cl, and a choke inductor Li. Resistor R, is an ac load and inductor L, is assumed high

enough so that the ac ripple on the dc supply current Ii can be neglected [40].

When the switch is ON, the resonant circuit consists of LrCrRo, since capacitor Cl is

shorted by the switch. The resonant fiequency is given by:

Figure: 5.1 Basic Class E Converter

However, when the switch is OFF, the resonant circuit consists of L,CrCIR, connected in

series. The current through Li is assumed to remain constant. The resonant fiequency is

given by:

Page 78

The switch is turned ON when the voltage Y, across the switch and capacitor CI is zero.

The energy stored in the shunt capacitance is zero when the switch tums ON therefore

zero turn-on switching loss is achieved. To achieve ZVS tum-on of the switch, the

operating fiequency f should be greater than the resonant fieq~encyf.~ [40].

5.1 .2 Equations and Mathematical Modeling

Consider the circuit s h o w in figure: 5.1. The current through the series resonant circuit

is assumed sinusoidal and the current 4 is assumed to be almost DC. These assumptions

are not exactly valid, but will allow us to develop a circuit that will give us some

guidance on component ratings. We begin by letting:

i = I , sin(wt + 8) (5-3)

where:

L represents the amplitude, and 0 represents the phase of the current I with respect to the

voltage Vcoi.

For the time interval O 5 wf 5 2 x 0 , (where D is the duty cycle) the switch is ON and

therefore: i,, = 0, and the switch voltage V, = O. Consequently, the current through the

MOSFET is given by:

i, = Ii - In, sin(wt + O) O < m r S S n D (5.4)

For the time interval 2 x 0 < u t I 2n, the switch is OFF, hence:

i, = O

Page 79

The current through the capacitor is given by:

During this interval, the voltage across the capacitor Cl and the switch is given by:

In order to achieve ZVS, the voltage across switch S and shunt capacitance Cl must be

zero when the switch is turned ON. Thus the ZVS condition is expressed as:

i.e. when the switch tums ON, Y, is zero. Using equation 5.6 and applying an ampsec

balance to capacitor Cl we obtain:

Substituting equation 5.10 into 5.5 gives:

1 - 2z(1- D)sin(wt + 8) O c w t I 2 n D

cos(2z D + 8) - cos 0 O 21tDcwt 1 2 ~

Likewise, substituting 5.10 into 5.6 gives:

and, substituting 5.10 into 5.8 gives:

Page 80

Under optimal operating conditions, there is no current through the diode and the switch

operates under a ZVS condition. In this case, both the switch voltage Y' and its slope (i.e.

dVs - ) are zero when the switch tums ON (at ot = Sn). Using this condition, a

d ( W

relationship between phase 0 and duty cycle D c m be obtained by differentiating

dVs equation 5.13 and setting - - - O at w t = 21t . Therefore:

d ( W

Substituting w t = 2 z and simplieing results in:

COS~ZD-1 tan 8 =

27r(l- D) + sin 2nD

cos 2 z D - 1 9 = tan"

27r(l- D) + sin 27r D

Using equation 5.13 and appIying a volt-sec balance across the inductor, we obtain an

expression for dc input voltage 6 given by the following expression:

Remangement of equation 5.16 gives:

(1 - D)[x(l - D) cos z D + sin ZD = R,

' i oC, tan(x D + 0) sin X D

Page 8 1

which is the equivalent dc input resistance of the Class E converter as see fiom the dc

source.

From equations 5.13 and 5.17 we can obtain a relationship between the input voltage and

the switch voltage:

The current through the series-resonant circuit is considered sinusoidai, therefore using

equation 5.13 and the Fourier formula, the fùndamental component of the voltage across

resistor Ro is given by:

= - 2 sin ZD sin(lrD + 8) n(1- D) Y

Hence, the output power fiom equation: 5.19 is

v 2 sin' ZD sin' (dl + 8 ) p, =-- - v,' (5.20) 2 4 , rr2(1- D ) ' R ~

This expression can be used to determine the input dc current given an output power and

load resistance as a fûnction of the duty cycle and input voltage.

In order to determine the peak values of the switch current and the voltage across the

switch, we differentiate 5.11 and 5.13 with respect to m t to determine the maximum

value. The result is:

Page 82

Substituting 3.21 into 5.11, gives:

L < m m > -- ~ ( 1 - D) - 1 -

4 sin z D sin(z D + 8)

5. f 3 can also be rewritten as:

Substitution of equation 5.23 into 5.13 gives the maximum voltage across the switch.

To design the value of the choke inductance Li, let us consider the operation at a

switching fiequency equal to the resonant fiequency (optimal case). In such a case, when

the switch is ON the voltage across the choke inductor is V,, hence the peak to peak value

of the tipple cwrent in the choke inductor is:

where n is the percent ripple factor.

The minimum value of inductance for a given maximum allowable ripple is:

Let us assume a npple current of less than 10% with respect to the dc current li ut D4.5.

Using equation 5.17 we obtain:

( 1 - D)[lr(l - D)coszD + sinnD] L i ( m i n ) = 2 f (1 0% or less) wC, tan(z D + 8) sin lr D

Page 83

L. . = I

~(rnin) 2 f (1 0% or less) R,

In the case of a PT, we have an output capacitance across the load resistor in the

equivalent circuit model (Chapter: 3). In order to develop a PT based Class-E converter

model; we must modifi the expression slightly. Let us consider a parallei RC circuit, as

shown in figure: 5.2.

Figure: 5.2 A paraltel RC network

In figure: 5.2, the equivalent impedance as seen fiom the input can be expressed as

follows:

Equation 5.28 can also be written as:

The reactance factor for R, and C2 is defined as [40]:

Therefore fiom equations 5.29 and 5.30, let:

Page 84

Hence, fiom equations 5.30 and 5.31, we obtain:

Rearrangement of 5.32 gives:

We can also write:

Using 5.33 and 5.34, equation 5.29 c m be rewritten as:

This is a series RC network with the above-derived values of Rs and Cs. Using this

relation, we obtain a circuit as shown in figure 5.3. Cr in figure: 5.1 can now be

considered as a series combination of C and Cs.

Figure: 5.3 Class E converter with a an equivalent R and C

Page 85

Refemng to figure 5.3 and using 5-31 and 5.35, we obtain:

Using the above-derived expressions and relationships, figure: 5.3 cm now be s h o w to

be equivalent to figure: 5.4 at least to first order. The circuit in figure: 5.4 is the

equivalent circuit of a simplified PT. Li

---. R, -.

Figure: 5.4 Class E converter with a parallel capacitance across the load resistance

5.1.3 Class E converter Design using the Piezoelectric Transformer

The circuit given in figure: 5.5 has been widely used by other researchers and is based on

expenmental measurements but has not been rigorously derived fiom first principles.

Values for the circuit components can be obtained using a system identification tool with

expenmentally derived impedance characteristics.

Figure: 5.5 Approximate Equivalent Circuit

Page 86

The following values were provided by the supplier based on measurements taken using

an impedance analyzer.

L,=O. 00 7924 H

c,=o. 32107 x 1 0 - ~

RL=13 ohms A2=j. 67

Using figure: 5.3, the complete design with simplified circuit parameters c m be drawn as

shown in figure: 5.6. The components in figure: 5.6 are values referred to the primary

side of the PT shown in figure: 5.5. Li Le

Figure: 5.6 Class E Converter based on Piezoelectric Transformer

In figure: 5.6:

L'=0.0391 H &"=2.2 k ohms

~ ' = 6 . 5 x IQ"F CO2 =O. 768 nF

Col=I.55 nF

Using equations 5.2 and 5.3, we have:

fol=99.83 kHz

fo2=1o1.90 kHz

Page 87

For our analysis, we considered an operating fiequency of 100 kHz. For the given step

down (IO: I ) transformer under investigation, Y, is 320V and the output Power is 15W.

For D=OS and an operating fiequency of 100 kHz, we obtain fiom equation 5.17, 5.18,

and 5.2 7 :

R , =326.84R, Li =0.01634 H , =1139.4 V

These results show that the value of peak voltage across the switch is about 4 times the

input DC voltage. We therefore conclude that the Class-E topology is not suited for step-

down (high voltage) applications, assuming 1 1 V or 22O/j12O V are considered as high

voltage levels, and 5, 10, or l 5 V are considered as low voltage levels. It is best suited for

step-up applications fiom low to high voltage levels, where practical values of inductance

and switch voltages are small due to a low input voltage. In order to ver@ our

conclusion we considered a Rosen-type step-up (1:IO) transformer, where the input

voltage is IOV, and the component values are estimated as follows [17]:

L,=0.2193 H Al=l

Cr= 14.556 pF A2=81. 8509

RL= 700 kohms fol ~ 8 9 . 0 79 kHz

Coz=3. 9 1 74 pF fO2=89. 998 kHz

Col= 7O1.96pF

Now using the above-derived expressions for a Class-E converter and these numerical

values, we obtain the following data:

R , =806.37a, Li =0.045 H , V,(,,,,, 2 5 0 V

These results v e n v our initial statement and the stress across the switch is just 5OV.

Page 88

The given simplified circular disc transformer mode1 given in figure: 5.5 has been

simulated using PSpice at an operating fiequency of I00kH z (Appendix: Ag) . The

reason for performing the simulation was to verify the validity of the empirical method in

ternis of component values and waveforms. The results were found to be in good

agreement with the calculations. Table: 5.1 shows the comparison between the simulated

and calculated results in terms of the maximum values of switch voltage Y,, current

through Col, and current through Li.

TABLE: 5.1 Cornparison between the results from a simulation and calculation

@L=IS otrms,pIOo kHz

It can be concluded that the fùndamental component analysis provides a good

approximation for switch voltage, and switch current, while the inductor current observed

through simulation is lower than expected. The calculated value of input inductance is

found to be on the high side, and ZVS could not be achieved with this value for the

chosen operating conditions, as shown in figure: 5.7. However, simulation verifies

minimum ripple content with this value. We cannot place great confidence in the exact

location of the optimal ZVS location since the waveform for analysis purposes was

simplified. Thetefore, to evaluate our circuit for ZVS, a different operating fiequency or

a lower value of inductor should be used. Since operating fiequency is dictated by the

Switch Voltage Inductor Current Switch Current Suggested Input Inductor for ZVS

Page 89

Max. Values obtained througb calculations

1139.84 V 0.9515 A 2.723 A

0.01634 H

Max. Values obtained through simulation

1104.44 V 0.636 A 2.046 A

0.0008 H

equivalent circuit of PT, simulations with different values of input inductance were

performed to determine a lower value of inductance that would lead to ZVS. An

inductance of about 0.8mH was able to achieve this requirement. In figure: 5.8, the

wavefoms of current through the inductor and the voltage across the input capacitor for

different values of inductance are shown. The impact of the choice of inductance on ZVS

is evident. Figure: 5.8 shows a ZVS operation with a lower input inductance, but the cost

is a higher ripple current. Although, initially we assumed a relatively large choke

inductance in order to achieve a low ac ripple on the dc supply current for Class-E

operation, it is suggested in 1421 that a large ripple cment is dso possible.

l 1 1 I 1 1 1 I

4 0 ~ A - - - - - - - - - - - - - - - - - - - - - - - - - - . . . . . . . . . . . . . . . . . . . . . . -----..------- - - - - - - - -- - - - - - - - - - - - - - - J

CI Cmcnt îhrough the Mput Inchrctor. Ii 2.OKV - - - - - - - - - - - c-- - - - - - - - - - - - a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure: 5.7 Wnveforms for a C1ass-E coav&%b usiiig a multi-lnyered Piezoelectric Transformer (L4.016342H)

Page 90

1 I 1 I 1 I 1 I 1 I

SEm> j I -1.OW - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A

e Voiîage across the switch (Vs), Cm order of spmbois. M. 0 l634H, 0.8mH)

Figure: 5.8 Class-E Converter using a multi-layered Piezoelectric Transformer with different values of Choke Inductance

5.2 Asymmetrical Converter Design

The purpose of using this topology was threefold: to investigate the permissible range of

input voltage for a PT provided by a manufacturer; to study the behavior of an auxiliary

winding and to look into the possibility of using the auxiliary voltage for feedback

control; to consider the applicability of an asymmetrical bridge converter.

An asyrnmetrical converter is a combination of a Class-D series-parallel and a Class-D

ZVS inverter. In a Class-D Series-ParalIel Inverter, there is no parallel capacitor (Coi).

In contrast, a Class-D ZVS inverter has no parallel capacitance (Cor).

Page 91

5.2.1 Asymmetrical Converter Design using a Piezoelectric Transformer

Figure: 5.9 shows an Asymmetrical Converter with the sarne piezoelectric transformer

used for the Class-E converter analysis. It is noted that an inductor Li is inserted in series

with the PT. The function of Li is similar to that of a low pass filter. It attenuates higher

fiequency components of the quasi-square wave voltage and thus supplies the PT with a

sinusoidal waveform [4 1 1. The inductor prevents rnultimode resonances fiom k ing

excited and allows ZVS to occur. It also reduces the circulating current through the input

shunt capacitor [42,43] and helps to maintain a continuous flow of current through the

resonant circuit, even during the dead time interval (diodes conducting, switches do not

conduct). A suitable value for this inductor is determined using a PSpice simulation.

Figure: 5.9 Asymmetrical Converter with an input inductance

The asymrnetncal converter of figure: 5.9 consists of two bi-directional switches S/ and

S2 and a resonant circuit L, '-Cr '-Cm ' 'ORL ' ', where RL ' ' represents an ac load and Co2 ' '

represents the shunt capacitance. The switches SI and S2 consist of power MOSFET QI,

/diode Dl, and MOSFET Q2 /diode D2 respectively. The switches are dnven altemately

by a rectangular pulse at the switching fiequency with a suficiently long dead time i.e.

the ON duty cycle of each drive voltage is less than 0.5 (or 50%)[40].

Page 92

A fundamentai component analysis can be used to analyze the salient features of the

circuit in figure 5.9. The resonant circuit (when the switch S2 is OFF) is similar to a third

order resonant low p a s filter circuit. The corner fiequency of the tank circuit of figure

5.9 is given by:

The behavior of the circuit in figure 5.9 is slightly different fiom that of a Class-E

converter, as the input capacitor is always part of the resonant circuit. The input

impedance of the resonant circuit is given by:

where

and

Page 93

From the above equations (5.40-5.42), it is apparent that at f = f,, 8 > 0 and the

resonant circuit represents an inductive load. For f c fr , 6 is less than zero and the

resonant circuit represents a capacitive load. Setting 0 in equation 5.40 to zero would

give:

I

Substituting the component values provided by the supplier (section 5.1.3) into the above

expressions (equation 5.3 7 and 5.43) would give:

fo = 103.972 kHz

f r - = 0.894 fo a f, = 92.97 kHz

Since a fkquency greater thanf, is required for ZVS, an operating fiequency of 95kHz is

used for M e r analysis.

5.2.2 Simulation Results

As proposed earlier (section: 5.2.2), the value of input inductor was determined using the

simulation (Appendix: A9). Circuit behavior was investigated for different inductance

values and results are shown in figure: 5.10.

Page 94

-80- O. 99- O. 99- O. 994u 0.996- O. 99- 1. O O O u

O . Input Inâuctor Cuntnt ('ni order of syrnbols Li0.40.500. and 700uH) Ti or

Figure: 5.10 Simulated waveforms for an Asymmetrical Converter witb different input inductance values

It is observed fiom figure: 5-16 that in the absence of an inductor, there is a discontinuity

in current during the dead time i.e. when both switches are OFF. The voltage at the input

terrninals is not sinusoidal. This may cause higher order harmonies to influence the

circuit behavior and thus oscillations at the other frequencies may occur. This problem

can be resolved by a proper selection of the inductor value, as shown in figure: 5.10.

These plots help designers determine the value of inductor. It is clear that in case of no

inductor we observe a discontinuous behavior and high spikes (up to 8A) during the

switching transition. In contrast, a larger inductor guarantees a smooth sinusoidal input

current to the transformer. The condition of ZVS must also be sotisfied, but it was found

that at lower inductor values ZVS could not be achieved. As far as ZVS is concemed, the

Page 95

higher the inductor value, the greater the range of loading conditions under which ZVS is

achieved. However, the use of a very large inductor is not a good choice due to its size

and cost. Notice in figure: 5.10 that inductor values of 500 and 700 uH results in an

approximately sinusoidal voltage and curent at the input.

5.2.3 Experimental Analysis

For experimental purposes, we tested two PTs. PT #I is a five terminal device, while #2

is a 6-terminal device i.e. Multi-layered Circular disc transformer (Appendix: A7).

Simulated and actud waveforms for the cunent through the input inductor and for the

voltage across the input terminals PTW are shown in figures: 5.11 and 5.12 respectively.

T1.e

Figure: 5.1 1 Simulated Waveforms of an Asymmetrical Converter (Li=2SOuH)

An inductor of 25OuH is chosen for simulation. This value corresponds approximately

with the value used in the experiment. The affect of the MOSFET drain to source

Page 96

capacitance is considered during simulation. The experimental results are in good

agreement with the simulation results. Thus, the inclusion of MOSFET capacitances and

the simplified piezoelectric transformer model, provided by the manufacturer and used by

the researchers, appear to be adequate for design purposes.

-200 l

O 1 2 3 4 5 6 7 8 9 10 11 Time (u

400 '- I

O 1 2 3 4 5 6 7 8 9 I O 11 Time (u

Figure: 5.12 Measured voltage and current waveforms at the input terminais of a PT, when used with an asymmetrical converter (Li=400uH)

Further experiments were performed with the circular disc multi-layered PT device. Five

circular Piezoelectric Transformer samples were used to experimentally verifj the

operating characteristics. The converter has been modified so that the device could be

tested for both 110 and 220 VAC, with an optional auxiliary output. The converter

design and circuit is shown in Appendix: A8. A duty cycle ratio of 0.42 is used during

testing, while ZVS is achieved if the circuit is operated at a switching fiequency in excess

Page 97

of 98.5 kHz. However, an operating fiequency of 99.5 kHz was used in order to maintain

a temperature nse within an acceptable value (figure: 5.14). One of the five samples

(sample #2) was found to be faulty, white sample #4 was f o n d tu be defective due to an

abnormal temperature rise (as much as 90 OC below 99 kHz (figure: 5-14)). Voltage

transformation ratios with sample #I were found to be very different fiom the other

sarnples (figure: 5.14).

Test results for 300VDC Operation

The complete characteristics of the converter for a 300 VDC input were not determined.

Testing was constrained due to an unacceptable temperature nse (as much as 104

degrees), which resulted in the destruction of the device. Only a few readings were

recorded at a load of 100 l2 and a frequency of 98.5400 kHz, and the results are iisted in

Table 5.2. From these results, it appears that the device is not suitable to meet the

specifications. We thus chose to use I6OVDC for which the previous devices were

known to work.

Test results for 16OVDC Operation

Testing given an ac input voltage of 110 VAC or 160 VDC (after rectification) was

perfonned for most of the analysis. A temperature rise of as much as 71 OC at

fiequencies around 96-98 kHz was observed. Figures: 5.13, 5.14, and 5.15 show the

outputhnput voltage transformation ratio, auxiliarylinput voltage transformation ratio and

temperature charactenstics of the converter for varying loading conditions. These plots

are based on steady state values recorded during the expenment.

Page 98

The auxiliary winding was found to be tracking the input voltage at the transformer

terminals and was unaffected by the loading conditions at the output. This behavior was

expected as explained in Chapter: 4 and Appendix: A 7. The auxiliary voltage for most of

the cases was O. 4-0.3 times the input voltage, except for R,=l2.5 and 25 ohms where the

ratio varied between 0-35-0.46 and 0.24-0.35 respective1 y. This behavior was not

observed with the other loads.

TABLE: 5.2 SummaV of Results @ 220VAC/300 W C

I - - * 10 ohms

Frequency kHz 97.5 98.5 99.5 1 00

Frequency k ~ z 97.5 98.5 99.5 1 O0

OutpuVInput Voltage transformation ratio

OutpuVInput Voltage transformation ratio

1 1

- - y 100 ohms

Page 99

Sample 1 0.0 154 0.0 149 0.0098 0.008 1

Sample 1 0.0498 0.0425 0.0412 0.0383

Freq uency kHz 97.5 98.5 99.5 100

:age ratio

Sample 5 NIA

0.362 0.343 0.349

- -- -

Aux.Anput Voltage transformation ratio

Aux.Anput Voltage transformation ratio

1 1

- - -

AT CC) Sample 3

0.025 0.022 0.0 1 7

0.0 157

AT CC) 1

Sample 3 0.077 1 0.070 1 0.0698 0.055

AT ml

Sample 5 0.0245 0.024 0.0193 0.0 164

Sample 5 0.33 1 0.372 0.365 0.36 1

Sample 1 0.292 0.3 1 1 0.30 1 0.299

Outputfinput Voltage transformation ratio

Sample 5 73.4 73.2 77.9 75.1

Sample 3 0.324 0.365 0.362 0.359

Sample I 76.2 74.9 76

75 -6

Sample 5 0.0757 0.0639

" 0.053 0.049

Sample 5 N/ A 77.6 74.7 70.1

Sample 1 N/A 79.2 75.2 74

Aux.Anput Vol transformation

Sample 3 75 74 79

75.3

Sample 5 N/A

0.102 0.101 0.09 1

Sample 3 N/A 78.9 75.1 72.7

Sample 1 NIA

0.074 0.073 0.068

Sample 1 NIA

0.359 0.362 0.359

Sample 1 0.324 0.365 0.362 0.359

Sample 3 N/A

0.101 0.952

0.0874

Sample 3 NIA 0.349 0.35 1 0.359

Sample 3 0.388 0.359 0.35 0.35

Sample 5 0.39 0.367 0.342 0.348

Sample 1 80.1 79

74.9 71.3

Sample 3 78.5 75.9 74.9 72

Sample 5 79.2 76.4 74 7 1

Figure 5.13: Voltage Transformation ratio (output/input) vs. Frequency for

different loads, keeping auxiliary unloaded

Figure 5.14: Voltage Transformation ratio auxiliary/input) vs. Frequency for

different loads, keeping auxiliary unloaded

Figure 5.15: Temperature rise (AT) vs. Frequency at different loads, keeping auxiliary unloaded

High temperatures (AT=10-50) were recorded over an operating frequency range of

97-98.5 KHr while low temperature values were recorded (AT= 19-25) around 99400.5

kHz. An input to output voltage transformation ratio of 10:f was achieved with a load of

100 ohms at an operating frequency of 99.5400 kHz. A temperature nse AT of 25

degrees was recorded.

Page 100

Other sarnples were also tested with the same loading conditions (100 ohms). Figures:

5.16, 5.1 7, and 5.18 show the outputhput voltage transformation ratio, awiliary/input

voltage transformation ratio and temperature charactenstics for different piezoelectric

transformer samples.

, For sample 4. ~99kHz. I

25 L T rise is 60+

Figure 5.16: Voltage Transformation ratio (outpuüinput) versus Frequency for a 100 ohm load and for different samples

1

20 : 95 96 97 98 99 100 101 102 103

Fmquency (kHz)

Figure 5.17: Voltage Transformation ratio (auxiliary/input) versus Frequency for a 100 ohm load and for different samples

Figure 5.18: The temperature rise (AT) versus Frequency for a 100 ohm load and for different sam~les

The other task was to detennine how the auxiliary winding tracked the load voltage given

different auxiliary loading conditions. A load of 100 ohms at the output was fixed as the

base load, and a fiequency of 99.5 kHz was considered. Sarnple #5 was tested first and

Page 101

the results are shown in Table: 5.3. The procedure was then repeated for the other

samples and the observations are surnmarized in Table: 5.3.

Table: 5.3 shows the behavior of auxiliary winding for various auxiliary loading

conditions. It is clear that although the load at the auxiliary changed fiom very light to

very heavy, the output voltage level remained unaffected. Changes in the voltage

available at the auxiliary were obsewed with the varying load. The same voltage levels

were observed at the output and the auxiliary for a load of 1 .Z k-ohms. In table: 5.4 it is

shown that a similar response was observed for sample #3 and #5, while the other

samples show results that deviate significantly fiom those of sample #3 and #5. Table:

5.4 revalidates that concIusion. From this, we conclude that the auxiliary voltage cannot

be used as a feedback signal for controlling the converter.

TABLE: 5.3 Characteristics of Auxiliary Windings (Sample #5)

@IIOVAC, R&OOaP99.5 kHz

t Open I 82 1 5.34

Page 102

TABLE: 5.4 Characteristics of Auxiliary Windings

5.2.4 Conclusion

SampIe

# 1

The design of an asymrnetrical converter in conjunction with a piezoelectric transformer

has been investigated. The differences between the experimental data and simulation are

due to the approximate mode1 provided by the manufacturer. The circular disc type PT

provided by the manufacturer was found suitable for a s!ep-down type DC-DC converter

design given that the voltage was IIOVAC/160VDC. These samples were designed to

deliver 10 W at 220VAC/300VDC, but fiom observations of temperature rise this cannot

be the case. The behavior of the auxiliary winding is also investigated. It is found that

the auxiliary output tracks the input signal and is not useful as a control feedback voltage.

Page 103

V~ur i l i a ry W C )

15.2

Voutpue W C )

12.5 # 3 1 t 5.3 1 15.3

Chapter: 6

Thesis Conclusion

The characteristics of piezoelectric transformers are investigated in this thesis. A

simplified approach for developing a steady state model of a simple Rosen type

piezoelectric transformer has been presented. A flow chart has also been developed to

assist the reader in understanding the link between the physical and circuit-based model.

The model has been successfùlly extended to different geometries and complex

structures, and expenmental results are found to be consistent with the theoretical

predictions. Differences between theory and experiment can be attributed to uncertainties

in the materid parameters. It has been shown that the material data (cornpliance, in

particular) could be off as much as 80%. Some caution should therefore be exercised in

believing the data provided by the manufacturer.

Two topologies (Le. Class-E and an asymrnetrical bridge converter) have been identified

as suitable for piezoelectric transformers. The class-E converter design is recommended

for low voltage step-up applications as the switch is exposed to a potential, which is four

times higher than the input bus. In addition, the value of the input inductance is large.

Through our findings, it c m also be concluded that an asymmetrical converter is suitable

for high voltage step-down applications as the voltage across the driver switches is the

sarne as the input bus. It is also found that for an asymmetrical converter the input

inductor guarantees soft switching and acts as a low-pass filter together with the input

Page 104

capacitance of the transformer, thus providing a sinusoidal voltage to the piezoelectric

transformer. This prevents multimode resonances fiom king excited.

The major contribution of this thesis is a methodology for linking electricai circuit

parameters with the mechanical geometry and material constants. The Partial Expansion

Theorem was used to generate an exact representation to the circuit model. L'Hospital

Rule was used to simplify the model about the resonance fiequency without losing

information relating the material and geometrical constants to the electrical circuit

parameters.

It has also k e n shown that losses cm be introduced into single resonant branches, using

the mechanical quality factor and related data. It is known that there are other losses

present, but the literature indicated that these losses are unimportant in devices that are

operated near the main resonance fiequency. It is possible to incorporate other losses, but

at this point in time, there is no means of decoupling piezoelectric losses fkom dielectric

and mechanical iosses. Moreover, the values of these parameters are detennined by the

level of excitation. Our model gives guidance regarding suitable location for introducing

lossy components (resistors).

Proposed Future Work

The following topics are proposed for future work:

1. Incorporating temperature characteristics and dependencies in the model.

2. Considering the effects of extemal stress, strain, and hence mounting.

Page 105

3. Using a system identification technique to determine the equivalent circuit

parameters after characterizhg the devices using a network anaiyzer.

4. Recommending that manufacturers provide tolerance data for the material and

physical design.

5. Using a simple resonator to develop a complete mode1 that accounts for ail the

physical, material, mechanical properties, and other non linearities i.e. losses and

temperature.

6. Developing an integrated circuit with al1 the control and switch drivers on a single

chip for a complete converter design.

Page 106

APPENDIX: A1

Piezoelectric Equations in a Cartesian Coordinate System

The fundamental piezoelectnc equations in the Cartesian coordinate system [6] are given

by :

where j = 42 ,... .,6

The linear strain in the x, y and z directions is given by [6];

The remaining strain coefficients are usually defined by:

du2 a u 3 du, du, S, =- +- ' S5 =- au h l +-, S6 =- a2 dx

+- ?Y & ? Y

In vector form, Newton's law can be written as:

where ui are the displacements of the crystal in x, y and z directions, Fi are the

components of force in x, y and z directions exerted on the crystal.

The resultant force in any direction is obtained by summing al1 the forces with

components in that direction. Hence the total resultant force along the x direction is the

partial derivative of the stresses [6] or

Similady, for the other two directions

1 kdyd7 (x direction)

These c m be expressed in the general tensor form [6];

where i,1= I,2,3, ... 6

Hence, using equations A1 -4 and A1.9, we can obtain [6];

(y direction) (A 1.7)

(r direction) (A 1.8)

For the longitudinal bar wîth its length along x, the only stress different fiom zero is Tl,

and hence the only equation of motion for this bar is:

APPENDIX: A2

Piezoelectric Equations in a Cylindrical Coordinate System

The fundamental piezoelectric equations in the cylindrical coordinate system are 131:

S , = SET, +SET, + S ~ T . - + d,, Et

S, = SET, + S ~ T , + SLT- - + d,, E,

S= =s,";(T, + T , ) + s , ~ T = +d, ,E=

Sc = s z c +dl&

se = S ~ T , + d , , ~ ,

S , = sLTr,

Dr = &:Er + d 1 5 ~ ~

D, = &&Ee +dIsT'

D: = EL E. + d 3 , ( T , + T m ) + d,,T'

The strain

1 du, u , s, =-- au:

+Z-, S- =- r dB r - &

au, u, 1 % Sr@ = - +-+-- 3 S E = ; +- au au; d e r r d e dz dr

1 au: au, se = -- +- r d 8 d e

The equations of motions are

Appendix: A3

Equivalent circuit of a transducer in Longitudinal vibration mode

Consider the transducer shown in figure: 3.17. From Newton's law we have [3,5]:

The wave is assumed to propagate dong the length axis with a zero stress in the lateml

direction. Hence no lateral inertia exists [l] i.e.

T1=O, T2=0

but

Where u~ is the displacement velocity in z-direction. From the basic piezoelectric

equations for this mode we have:

Rearranging equation A3.3 gives:

Substituting equation A3.5 into A3.4 gives us the following expression:

Solving for E3 and using it in equation A3.5 results in the following expression:

Substituting equation A3.7 in A3.2 results in:

Since for this mode

A 3 . 8 gives:

Since the electrical field is along the length of the bar and the displacement does not vary

along the length of the bar [3], we have:

Differentiating A3.4 with respect to z gives:

Now substituting equation A3.7 in for T3 and using equation A3.11, we obtain the

following expression:

T- + g33 T g

4 3

Using A3.9 and substituting it into A3.13, gives:

Rearranging gives:

Substitution of A3.15 into A3.10 gives:

Simplification gives:

For simple harmonic motion, the variation of u3 with time can be written in the phasor

form as [1,5,6]:

C1 u3 = u3e i d (A3.18)

Hence A3.17 becomes:

A solution of this equation is

where:

The displacement velocity is:

In terms of velocity we have from A3.18:

It is desired to utilize the particle velocities at the two ends of the resonator for boundary

conditions [8] defined as follows:

Hence we obtain:

- C I , sin y ( - l / 2 ) + LI, sin y ( t / 2 ) B =

jw sin y l

Therefore the strain S3 is given by:

au 3= = u, cos y { z - (- 42/2)) - U, cos y { ( l / 2 ) - z) dx s 3

jw sin y l Y

- - U, cos y{z - (- P/2)f - U, cos y ( ( l / 2 ) - z) VPS& (A3.24) jw sin y [

From A3.4 we have:

D3 = d3,T, + E; E)

Substituting T' h m equation A3.9 and simplieing gives:

Rearranging gives:

Substituting A3.25 into equation A3.5, gives:

Hence:

Substituting A3.21 into A3.25 gives:

and

I =LI3 =-

j o w t

Therefore A3.28 can be written as:

1 E;= 1 d33 1 --- P ~ V

d:3 j o w t s$ [ U ~ C O S @ - ( - ~ ~ ) } - U ~ C O S Y { ( P / ~ ) - Z ] ] (A3.30)

$3 -F s 3 3

The voltage potential between the two plates can be obtained by integrating the electric

field over the distance between the plates [3,8], i.e.

Substituting A3.30 gives:

It is clear from A3.32 that:

Hence:

v = I 4 3 1 +-- iwco, s: oc, J[U, -u1 1

where the electromechanical transformation ratio is given by:

4, = jwCo2V + wt(U2 - U, ) s33

Similarly the stress is (fiom A3.3):

Substituting for E3 gives:

d 3 3 1 E r3 = --- I + - 1 PS33 [u, cos(z - (- P/z)} - U, COS y ( ( q 2 ) - z } ] S E juwt S; j sin y t

For the two sides of the resonator:

4 3 1 T,(z = 412) =--- E I + Pv [ U , C O S ~ ~ - U , ] s33 jowt j sin y.4'

4 3 1 T3(z = 4 1 2 ) = --- I + Pv [u2 - U , C O S ~ P ] SE jwwt j s i n y l

At 2=1/2. the compressive force =-wtT.. Therefore multiplying both sides of equations

A3.40 by wt gives:

d33 wt pvwt -F,+--I= [LI, cos y! - U , ] SU jwwf jsinyP

d3, ~ : : 4 wr -F2 + - - P I = [LI, cos y4 - CI, ] s: E:: t jowr j sin ye

4 3 Co* I = - F , +- z0 [u, cos y t - LI,] j@Co2 j sin y!

- F2 + Y' I = zo [LI, COS ye - u, ] jwC,, js inyt

At z-=Z/2, the compressive force =-wtT' , similarly for A3.39:

4- I = - z0 [u2 - U, c0syc] j ~ c 0 2 j sin y t

Combining A3.4 1 and A3.42 into a matrix form results in the following expression:

Hence, the equivalent circuit representation for equation A3.43 is shown infigure 3.18.

Now by employing the same simpliming procedure as explained in section: 3.3.2 for the

thickness vibration mode, we can obtain a circuit as shown in figure: A3.1.

Figure A3.1: Electromechanical equivalent circuit for the longitudinal vibration section

Appendix: A4

Determination of off-resonance equivalent impedance using L'Hospital's Rule

We have to consider other means of representing the hyperbolic functions used in our

model, i.e. tanh, cosech, and coth.

A4.1 For tanh function

From equation 3.87 we have:

1 YA =- Y[ tanh j -

2 2 , 2

Let

Equation 3.99 can be rewritten as:

Using the partial fraction expansion [38] we have:

Therefore:

From equations A4.2 and Ad. 3 we obtain:

tan a y=-- a

Let

Therefore

tan a y=- - P a

L'Hospital's rule is used to determine the value of Y when:

CI + a,

L'Hospital's rule is used to determine the value of Y when:

- da

Hence:

- U 3 U I

- 2a2 cota

a:

for

Since the application of limits gives an indeterminate quantity, we have to apply the rule

a second time

Therefore:

This admittance can be easily represented as a capacitive conductance that represents al1

the off resonance branches placed in parallel with the resonant branch evaluated at the

resonant fiequency.

A4.2 For coth function

From equation 3.89 we have:

Let

coth ju = - j cot a

Equation 3.98 can be rewritten as:

YB =-- ~e j coth- 22, 2

Using the partial fraction expansion method [38] we have:

1 2a cotha =-+- 1 1

+ 2

+ . . . . . . . . .

Let

Therefore for al

cot a a

Admittance c m be found by using equations A4.15 and A4.18 fkom the RHS, which

gives:

cot a

Let

Therefore

cota y=---- P a 2

l - [;)

L 2

L'Hospital's rule is used to determine the value of Y when:

But since:

2 sec a=l+tan2a

Equation A422 can be written as:

Since the application of limits gives an indetenninate quantity, we have to apply the rule

a second time

L'Hospital's rule is used to determine the value of Y when:

a + a , = z

and we obtain:

hence:

for

Therefore:

This admittance can be easily represented as a capacitive conductance representing al1 the

off resonance branches placed in paralle1 with the resonant branch evaluated at the

resonant fiequency.

A4.3 For cosech function

Frorn equation 3.112 we have:

2, = Z,, sinh j y l ,

Therefore:

Let

csc h ja = - j csc a

Equation A4.29 can be rewritten as:

yA = -- ' cscho 20,

Using the partial fraction expansion method [38] we have:

Let

Therefore for a=al

Admittance can be found by using equations A4.29 and A4.34 and we obtain:

Let

Therefore

csca 2 a r

L'Hospital's rule is used to determine the value of Y when:

Since the application of limits gives an indeterminate quantity, we have to apply the rule

- da

a second time

- -

"+Ut s ina~- ( l )>] '+acosa~ -[:)>]- 2a sin a a:

L'Hospital's rule is used to determine the value of Y when:

$Y - &2

this results in

2 - - + P cosa + Pasina - Pcosu - - a1

""1

4a - ( ~ ] ] - 2 ~ i - a S i n -($~]+co{-(i)i]-$acosa-- ci' sina - - 2$ 4

Hence:

for

Therefore:

This admittance cm be easily represented as a capacitive or an inductive conductance

(depending on the sign), representing al1 the off resonance branches placed in parallel

with the resonant branch evaluated at the resonant fiequency.

Appendix: A5

Flow Chart

ASA Pre-processing Stage

/ Determine the / /'

number of \ ; sections (1.2, ....) /'

1 Dimensions and Material 1 Properties

For every section, determine: dimensions of the section 1 dimensions of the electrodes vibration mode determined by Y the direction of polarization \ with respect to the electris

\ field 1

Determine Piezoklectric Material Constants and Coefficients, i.e.

density p), charge coefficient (dij), free permittivity (E&

elastic compliance (sijE) mechanical Q

i

Dimensions: Rectangular geometry (length, height and width). Circular structure(height, inner and outer diameter)

Vibration Mode: thickness transverse shear

Processing Stage s

A5.2 Processing Stage

Piezoelectric equations, as a function of stress 1, strain S, electric field E, displacernent D and piezoelectric coefficients

Equation of Motion a 2 u i a T , a t 2

= - a x ,

Set the boundary conditions a Write and solve the wave equation

Determine equations for velocity (u,)

CJu Determine strain S: S = -

ax

Rearrange equations for D to determine electric field E, as a function of D and S

v Determine voltage V: E, = -

t

Detemine the current I: 1 = jwQ = jw J J D ~ ~ s Calculate:

- Input capacitance as a function of dimensions and permitivity - Transformation ratio as a function of piezo coefficients

Post Processing Stage

Applying boundary conditions leads to equivalent circuit:

Impedances repmsented as a function of coth, tanh, csch and sech. l

Apply Partial Fraction Expansion to hyperbolic functions for each branch.

This gives a description of the multiresonant \ behavior

Equivalent circuit can be redrawn in tems of Ls, Cs and Rs (parallel or series combinations)

Use Q to detemine appropriate resistance Rs for each branch

Detailed Circuit

Circuit is in detailed form

Circuit I

Determine the equivalent

i

parallel branches using L'Hospital's Rule

Go to Post- \ Processing II ,

Repeat Processing and Post -Processing 1 for

every section of the transformer

Combine mechanical (open)ends of every section Apply transformations and

circuit simplifications

In case of: Detailed circuit- complete transforrner model for multi- resonant behavior Simplified circuit- model for single resonant behavior w e L'Hospital's rule

\ process

Appendix: A6

Equivalent Circuit Modeling of a Circular Disc type Piezoelectric Transformer assuming a radial vibration

A6.1 Equivalent Circuit Modeling of a Circular Disc in a radial vibration

For a radialy vibrating disc, we assume the thickness to be small enough that the change

of stress is negligible along the z-direction [5,6]. Since the stresses are zero on the

surface [ I l , we can set:

T= =Tc =Ta = O

4 = O

Since the motion is entirely radial [3];

Uo = O

and

Tr, = O

Furthemore, since the field is applied only along the z-direction i.e.:

D , = D , = O

Hence the basic equations in Appendix: A 2 become [3]:

S , = S ~ T , + s ~ T , +d3i E:

Sm = SLT, + S ~ T , +d3,EZ

@ =&E; +d,,(~' +Tm)

Also:

w here

Superscript P is used for radial mode [43].

Dr, Do, Dz are the electrical displacements along the r , 8, z directions respectively.

Er, E, , E, are the electric fields along the r , 8, z directions respectively.

u,, u, , u_ are the displacements dong the r , 8, z directions respectively.

S,, S , , S= , S, , S,, S , are the strain tensors dong the r , 8, z directions

T', TM, T= ,Tc, T,,, Teare the stresses' tensors along the r , 6, z directions

p is the density of the matenal -

Note that, since the plating on the surface is an equi-potential surf'ace, Ez is not a function

of r, therefore:

The equation of motion, for a simple harmonic motion, becomes:

The general solution of the wave equation for a steady state forced vibration is of the

forrn:

where

and JI is the Bessel fünction of first kind.

Taking the derivative of Ur with respect to r in equation A6.13 gives the following

relationship:

Hence

At the boundary r=a (the radius of the disc), the stress

Using the recurrence relationship for Bessel functions, equation Ad. 13 gives:

Simplifiing we get:

and

Using equation A6.1 O, we have:

Since the value of Dz at the surface is equal to the surface charge density [6], then the

total charge Q on the plate is:

Q = E & ~ ' E = + EZ (A6.22)

:JO (E) +

-

(y)] v a

The impedance of the plate is given by:

Let

Also, let

Let

where

The resonant fiequency occurs when [6]:

Denoting its roots by Rn, and using A6.15, the resonant angular fiequency of the nth order

is given by [6,44]:

The piezoelectric admittance Y, is expanded wit respect to the k Rn, the poles for Y,

[3,6]. The elements are:

Using equations A6.29, A637 and A6.38, an equivalent circuit is drawn as shown in

figure Ad. I .

* Figure A6.1 Equivalent Circuit of a Piezoelectric Resonator

Equation A6.24 can be rewritten as:

(radial)

where

Ur = jwu,

Let

Also

Multiplying both sides of equation (A6.44) by xat , gives:

Let

where

Let

Hence, the equivalent circuit of this radial resonator is s h o w in figure A6.2:

Figure: A6.2 Equivalent circuit of a radial resonator

A6.2 Equivalent Circuit Modeling of a Circular Piezoelectric Transformer for a radial vibration

Lets consider a radial mode transformer of the shape shown in figure: A6.3. In this type,

one resonator is a circular disc of radius a, while the other is a cylindrical plate attached

to the radial disc and has a radius e.

Figure: A63 A radially vibrnted Piezoelectric Transformer

The equations for the radial mode cylinder will be derived the same way as we did for the

disc.

From (6.13)

where

For this case, equation (6.19) becomes

Let

Let

= Y,, = jaco 2(e,q l2 4 3 4(y) [ 4 (y) +(cL-c.I

Let

Hence the equivalent circuit and transformer mode1 is shown in figure: A6.4.

Figure: A6.4 Equivalent circuit of a Piezoelectric Transformer under radial vibration

Appendix: A7

Modeling of a Multi-layered Circular Disc Piezoelectric Transformer

Another device we investigated was a multi-layered circular disc piezoelectric

transformer (a prototype). Al1 layers operate in the thickness vibration mode. The outline

of the device is shown in figure: A 7.1. The device contains 3 layers for the input section,

al1 of them mechanically in parallel but electrically in series as shown in figure: A7.2.

The middle layer in the input section is designated as an auxiliary as shown in figure:

A7.2. The input layer is sandwiched between two output sections, each section itself has

5 layers, al1 aitemately poled and mechanically in parallel. Their electrical connections

are shown in figure: A7.3.

Output part

Input part

Output part

Figure A7.1: Outline of a three section alternately poled piezoelectric transformer

I I

I I

Auxiliary i Connection !

a I

I a

Figure A7.2: Details of the Input Section of a three section alternately poled piezoelectric transformer

Figure A7.3: Details of the Output Section of a three section altemately poled piezoelectric transformer

A7.1 Modeling

Modeling of this device does not difler fiom that of a very simple piemelecaic structure.

It is just a matter of cementing individual circuits for each layer. Al1 the layea are in

thickness mode. The layers for the input section are poled in the same direction while for

the output sections they are altemately poled. For modeling, we used the technique

presented in sections: 3.3.2, 3.3.3 and 3.4. Figure: A 7.4 shows a detailed circuit mode1

of this device.

For al1 output layers:

For input layer 1 :

For input layer 2:

L 0 2

'12 = Sinhjy d

Y 4, 2, ,, = Z,,, Tunhj - 2

For input layer 3:

L o i 3 ' I Z i 3 = Sinhjy d,

#3 = 4 3 d 3 d ~ 3 3 ~ (A7.12)

The voltage across the middle input layer or the auxiliary output depends on the ratio of

the thickness of layers i.e. the voltage transmission is achieved by the thickness ratio

between the individual layers in the input section.

From equation A7.13, it is clear that the voltage at the auxiliary depends on the voltage at

the input terminal. The equivalent circuit shown in figure: 3.30 c m be simplified to a

circuit shown in figure: A7.5.

ZA& is the input capacitance of the second input or the auxiliary layer referred to the primary of the input layer no. 2 or amiliary

Zco3 ' is the input sapacitance of the 3" input layer referred to primary of the input layer no. 3.

Figure: A7.6 cm be simplified further to figure: A7.7 where a star to delta transformation is performed on each branch at the output

and input sections. This is required in order to use the partial expansion theorem for the hyperbolic functions, as we are limited only

to cosech, coth, sech and tanh.

w - Figure: A7.7 Simplified Model with Star-Delta Transformations

Appendix: A8

Asymmetrical Converter design and circuit

The complete circuit design is shown in figures: A8.1 (Power Stage) and A8.2 (Control

Stage). The list of components for each stage is given in Table: A8. I (Power Stage) and

A8.2 (Control Stage).

Note that the fiequency is determined using equations A8.2 and A8.2. These expressions

are provided by the manufacturer. Refer to the data sheets of UC386 1 for more details.

TABLE A8.1 Components List-Power Stage

Reference # Cl

Description Capacitor IOOuF, 250V

C2 C3 C4 CS C6 C7 DI D2 FI LI QI 42 R1 R2 R3 R4 R5 R6 Raux Rload U1 U2 U3

Capacitor 1 OOnF, 250V Capacitor 1200uF 63V Capaci tor 1200uF 63V Capacitor IOnF, 630V Zapac itor 1 OnF, 630V Capacitor 220pF, 1OOV Zener Diode 14V, 500m W Zener Diode 14V, 500m W Fuse 0.25A, 220V, SLO BLO Inductor SOOuH NFET 400V NFET 400V Resistor 45k (1 20V) or 75k (220V), 0.5U Resistor 10 ohms, 0.5 W Resistor 10 ohms, 0.5 W Resistor 10 ohms, 0.5 W Resistor I O ohms, 0.5 W Resistor 1 ohm, 3W Resistor Ik, 2k, 3k ..... 1M ohms Non-Inductive Resistor 1,2,5,10,15,20,25,50 ohms Bridge Rectifier 400V, 1.1 A Bridge Rectifier 400V, 1.1 A Bridge Rectifier 400V, 1.1 A

N2 OlCT2

AUX w1

AUX w 2

Figure: A8.1 Power Stage of a PT based Asymmetrical Converter

TABLE A8.1 Components List-Control Stage

Capaci tor 100nF, SOV Capacitor 100nF, SOV Capacitor 1 nF, 50V Capacitor 220pF, 100V Capacitor 1 nF, IOOV Capacitor IOnF, 50V Capacitor 1 OOnF, 50V Capacitor IOuF, SOV Capacitor 1 OOnF, 50V Capaci tor lOuF, 50V Capacitor 20pF Capacitor 1 nF, lOOV Capacitor InF, IOOV Capacitor lOnF, 50V Diode 100V, lOOmA Zener Diode Optional Zener Diode Optional Trim pot 2k, O.SW Trimpot 2k, 0.5 W Trimpot 2k, 0.5 W Trimpot Sk, 0.5W Resistor 10 ohms, 0.5 W Resistor 3.3 ohms, 0.5 W Resistor 1 Ok, 0.5W Resistor 1 Ok, O.5W Resistor 1 Ok, 0.5 W Resistor 22k, 05 W Resistor 4.7k, 0.5 W Resistor 220k, 0.5 W Resistor Slectable for frequency requiremes Resistor 10k, 0.5W Resistor IOk, 0.5 W Resistor 100 ohms, 0.5 W Resistor 100 ohms, 0.5 W Resistor 100 ohms, 0.5 W Resistor 10 ohms, 0.5 W Resistor 1 ohm, 0.5W IC Resonant Mode UC3861N IC High and Low Side DriverIR2 1 1 O IC Opto Isolator LTV713V

Reference # Cl

IIC Shunt Re~ulator Ti43ACLP

Description Capacitor 47uF, 5OV

C2

Aux Suppiy (14-16VDc)

con

USED La

=F DRIVE 1 RTN

Appendix: A9

Simulation Programs

A9.1 Detailed Admittance Analysis of a 3-layered Rosen-type Piezoelectric Transformer

%This code is for Rosen Type Piezoelectric Transformer - Admittance Characteristics*/

%Cornmon f = [10:260:200e+3] ;%Hz sigma=7.97e+3; %Density kg/mA2 epsilon0=8.854e-12 ;%dielectric constant F/m epsilon33T=1220*epsilonO; %free permitivitty F/m omega=2*pi*f; %angualr

%Input part-Thickness .ll=13.2e-3 ; %m wl=7.6e-3 ; %m tl=le-3 ;%m sllE=11.5e-12; %elastic compliance mA2/N d31=-141e-12; gammal=omega*sqrt(sigrna*s11E); % ZO1=wl*tl*sqrt(sigma/sllE);%Characteristic Impedance

%Output half -Longitudinal 12=13.2e-3 ;%m w2=7.6e-3; %m t2=le-3; %m d33=310e-l2;%g33*epsilon33T s33E=15.9e-12 ;%elastic compliance mA2/N gamma2=omega*sqrt (sigma*s33E) ; % Z02=w2*t2*sqrt(sigma/s33E);%Characteristic Impedance

%load resistance RL=l.O0e3; %ohms

%output capacitance COS C02=~2*t2*(epsilon33T-d33~2/~33E)/12; %output capacitance F

si=d33*~2*tZ/(lS*s33E); %output half transformation ratio phi=wl*d3l/sllE; %input half transformation ratio

k31=d3l/sqrt(epsilon33T*sllE);%electromechanical coupling coeffiecient Thickness mode CO1=2*(wl*ll*epsilon33T*(1-k31A2))./tl;%nput capacitance F

ZB2=2.*ZO2.*tanh(j.*gamma2.*12/2)%simplified circuit parameters ZB1=2.*ZOl.*tanh(j.*gamma1.*Il/2);%simpliied circuit parameters ZB=ZBl. *ZB2. / (ZBl+ZB2)

ZA2=+2.*Z02.*coth(j.*gamma2.*l2./2); Z A I = + ~ . *ZOl. *coth (j . *gammal*ll. /2) ; % s i m p i f i e d circuit parameters Zll=ZOl. *tanh ( j . *gammal. *12/2) ; Z12=201./sinh(j.*gamma1.*12);

%RL parallel CO2 ZL= (RL. / (l+j*ornega*C02*RL) ) ; ZLT=4*siA2.*ZL;Referred ZNC02T=4*siA2.*ZNC02;Referred

A=ZNC02T+ZLT B=A+ZA2 C=B.*ZB2./ (B+ZB2) D=C+Z11 E=Z12. *D. / (Z12+D) F=E+Z 11 G=F. *ZBl. / (F+ZBl) H=G+ZAl I=H/ (4*phiA2) J=I, *ZCOl. / (I+ZCOl) K=l*/J plot ( f , (abs (KI

%End of Program

A9.2 Simplified Analysis of a 3-layered Rosen-type Piezoelectric Transformer near 3'<' Resonant Mode

%This code is for Rosen Type Piezoelectric Transformer - Impedance Characteristics*/

%Corrunon f=[lO:260:25Oe3] % [10Oe3:260:200e+3] ;%Hz sigma=7.97e+3; %Density kg/mA2 epsilon0=8.854e-12 ;%dielectric constant F/m epsilon33T=1380*epsilonO; %free permitivitty F / m omega=2 *pi*£; %angualr

%Input part-Thickness 11=13.2e-3 ;%m wl=7.6e-3 ; %m t l=le-3 ; %m sllE=11,5e-12; %elastic compliance mA2/N d31=-14 le-12; garnma1=omega*sqrt(sigma*s11E); % ZO1=wl*tl*sqrt(sigma/s11E);%Characteristic Impedance

%Output half -Longitudinal 12=13-2e-3 ;%m w2=7.6e-3; %m t2=le-3; %m d33=310e-I2;%g33*epsilon33T s33E=15.9e-12 ;%elastic compliance mA2/N gamma2=omega*sqrt(sigma*s33E); % Z02=w2*t2*sqrt(sigma/s33E);%Characteristic Impedance

%load resistance RL=1000;%ohms Qm=1800; Mechanical Q

%output capacitance CO2 C02=~2*t2*(epsilon33T-d33~2/~33E)/12; %output capacitance F ZNCOS=l. / (-3 *omega*C02) ;

si=d33*~2*t2/(12*~33E); %output half transformation ratio phi=wl*d3l/sllE; %input half transformation ratio

k3l=d3l/sqrt(epsilon33T*s11E;)%electromechanicl coupling coeffiecient Thickness mode CO1=2*(wl*ll*epsilon33~*(1-k31~2))./tl;%input capacitance F

%circuit parameters

ZAlO=ZOl*pi/ (2*Qm) +j *omega*sigma*wl*t1*11/2+1. / (j *omega*2*1 l*sllE./(wl*tl*piA2)); ZAll=-j*piA2*Z01*4/14;%Lhospital ZAl2=+j*omega*sigma*wl*tI*11/2+1./(j*omega*2*l1*sllE./(25*w l*tl*piA2));%simplified circuit parameters ZAl=l. / (1. /ZAlO+l. /ZAll) ;

ZA2O=ZO2*pi/(2*Qm)+j*omega*sigma*w2*t2*12/2+1./(j*omega*2*~ 2*s33E./(w2*t2*piA2))%simplified circuit parameters ZA21=-j*piA2*Z02*4/14;%LHospital Z~22=+j*omega*sigma*w2*t2*12/2+1./(j*omega*2*12*s33E./(25*w 2*t2*piA2));%simplified circuit parameters ~~23=+j*omega*sigma*w2*t2*12/2+1./(j*omega*2*12*s33E./(49*w 2*t2*piA2));%simplified circuit parameters

ZB10=j*omega*wl*tl*ll*sigma%simp1ified circuit pararneters ZB11=-j*8*ZOl*piA2*1/1%LHospital ZB12=+j*omega*sigma*wl*tl*11/2+l./(j*omega*ll*sllE./(2*wl*t 1*4*piA2));%simplified circuit parameters ZB13=-j*8*ZOl*piA2*9/17%LHospital ZB14=+j*omega*sigma*~1*t1*ll/4+l~/(j*0mega*ll*sllE./(wl*tl* 16*piA2));%simplified circuit parameters ~ ~ 1 5 = + j *ornega*sigma*wl*tl*ll/4+l. / (j *omega*ll*sllE. / (wl*tl* 25*piA2));%sirnplified circuit parameters ZBl=l. / (1. /ZBlO+l. /ZBll

ZB20=j*ornega*w2*t2*12*sigrna%simplified circuit parameters ZB21=pi*Z02/Qm+j*omega*~igma*~2*t2*12/2+l./(j*omega*l2*s33E ./(2*w2*t2*piA2))%simplified circuit parameters ZB22=-j*8*Z02*piA2*4/7%LHospital ZB23=+j*omega*sigma*~2*t2*12/2+1./(j*omega*l2*s33E~/(2*9*w2 *t2*piA2));%simplified circuit parameters ZB24=+j*omega*sigma*~2*t2*12/2+1./(j*omega*12*s33E~/(2*16*w 2*t2*piA2));%simplifled circuit parameters

ZAAO=pi*ZO1/(4*Qm)+j*omega*~igma*w1*t1*11/4+1./(j*omega*4*1 l*sllE./(wl*tl*piA2))%simp1ified circuit parameters

ZAAl=pi*Z01*3/(4*Qm)+j*omega*sigma*wl*tl*ll/4+1./(j*omega*4 *ll*sllE./(9*wl*tl*piA2))%simplified circuit parameters ~~~2=-j*pi~2*ZOl*4/7%LHospital ~~~3=+j*omega*sigma*wl*t~*l~/~+1./(j*0mega*4*11*s11E./(49*w l*tl*pi"2));%simplified circuit parameters

%RL p a r a l l e l CO2 ZL= (RL. / ( l + j *omega*CO2*RL) ) ; ZLT=4*siA2.*ZL ZNC02T=4*siA2.*ZNC02

%End of Program

A9.3 Analysis of the material uncertainities of a 3- layered Rosen-type Piezoelectric Transformer

%This code is for Rosen Type Piezoelectric Transformer - Admittance Characteristics*/

%Cornmon f=[10:260:350e+3];%Hz sigma=7.97e+3; epsilon0=8.854e-12 ;%dielectric constant F/m epsilon33T=1380*epsilonO;%free permitivitty F/m epsilon33T=epsilon33T*l.O5 omega=2*pi*f; %angualr

sllE=11.5e-12 sllE=l.848*sllE%elastic compliance mA2/N

ZO1=wl*tl*sqrt(sigma/sl1E);%Characteristic Impedance

%Output half -Transverse l2=13.2e-3 ; %m w2=7.6e-3; %m t2=le-3; %m d33=31Oe-l2;%g33*epsilon33T s33E=15.9e-12 ;%elastic compliance mA2/N s33E=0.90fs33E;

gamma2=omega*sqrt(sigma*s33E); % Z02=w2*t2*sqrt(sigma/s33E);%Characteriçtic Impedance

%load resistance RL=1000; %ohms Qm=1800;Mechanical Q

%output capacitance CO2 C02=~2*t2*(epsilon33~-d33~2/~33E)/12; %output capacitance F ZNCOZ=l. / ( - j *omega*C02) ;

si=d33*~2*t2/(12*~33E); %output half transformation ratio phi=wl*d3l/sllE; %input half transformation ratio

k31=d3l/sqrt(epsilon33T*sllE)%electromechanical coupling coeffiecient Thickness mode; CO1=2* (wl*ll*epsilon33T* ( 1-k31A2) ) Jtl; % p u capacitance F

ZB2=2. * Z 0 2 . *tanh ( j . +gamrna2. *12/2) %simplied circuit parameters ZB1=2. *-201. *tanh ( j . *gamma1 * l l / 2 ) ; Bsimplified circuit parameters ZB=ZBl.*ZB2./(ZBl+ZB2)

ZA2=+2. *ZO2. *coth ( j . *gamrna2. "12. /2) ; ZAl=+2.*ZO1.*coth(j.*gammal*ll./2);%simpied circuit parameters Zll=ZOl. *tanh ( j . *garnrnal. *12/2) ; Z12=ZOl. /sinh (j . *gamma1 . *l2) ;

%RL parallel CO2 ZL= (RL* / (l+j*omega*CO2*RL) ) ; ZLT=4*siA2. *ZL; Referred ZNC02T=4*siA2.*ZNC02;Referred

A=ZNC02T+ZLT B=A+ZAZ C=B. *ZB2. / (B+ZB2) D=C+Z11 E=Z12. *D. / (Z12+D) F=E+Z11 G=F. *ZB1. / (F+ZBl) H = G + Z A l I = H / (4*phiA2) J=I. *ZCOi. / (I+ZCOl) K=1. /J plot (f, 20*1og (abs (K) ) )

%End of Progxam

A9.4 Analysis of a Class-E converter

vdl 50 O pulse(0V IV fO.O*per} {0.004perJ {0.004per) {OS*per) {per}) .param per= { 1 /(f?eq)) fieq=95kHz *.step panun fieq list 95kHz *+ 80kHz *+ 105kHz * vbus 1 4 320v *visense2 1 2 dc 0.0 11 1 2 0.01634H * r100 2 3 0.2 S1 3 4 50 O hfswitch dsl 4 2 dpwr cossl 2 4 10pf *

visense 2 5 dc 0.0 cd1 5 4 1.55nf 1 5 6 0.0391H r 6 7 110 c 7 8 65pf * rgroundl 4 0 1 G *

cd2 8 4 768.45pf rload 8 4 2200 * .mode1 hfswitch VSWITCH(rofFle6 ron=2 vofF0.0 von=l .O) .mode1 dpwr D(is= 1 e- 1 5 rs=O.O 1 ) .probe .options RELTOL=O. 1 ABSTOL=l UA VNTOL=l mV .tran O. 1 us 1 ooous ous o.osus .end

A9.5 Analysis of an Asymmetrical Converter

vdl 50 O pulse(0V 1 V (O.O*per) {O.Ol *per) (0.0 1 *per) {0.38*per) (per)) vd2 5 1 O pulse(0V 1 V {OS*per) (0.01 *per) (0.0 1 *per) {0.38*per) {per)) . pararn per= ( 1 /(fie@ ) f?eq=95 kHz .step pararn fieq list 9 5 W +96kHz +99kHz + 1 OOkHz +101kHz * vbus 1 3 l5OV rlOO 1 80 0.2 S 1 80 2 50 O hfswitch dsl 2 1 dpwr cossl 1 2 lOpf * rlOl 2 81 0.2 S2 81 3 51 O hfswitch ds2 3 2 dpwr coss2 2 3 SOpf * visense 60 4 dc 0.0 *

rground 1 3 0 1 G *

cd2 7 3 768.4Spf rload 7 3 2000 * .mode1 hfswitch VSWTCH(rofP1 e6 ron=2 vofF0.0 von= 1 .O) .mode1 dpwr D(is= 1 e- 1 5 rs=O.O 1 ) .probe .options RELTOL=O. 1 ABSTOL=l UA VNTOL=l mV &an O. 1 us 1 OOOus Ous O. 1 us

List of References

[1] K. Uchino, Piezoelectric Actuators and Ultrasonic Motors, Boston: Kluwer Academic

Publishers, 1997

[2] J.B. Ferreira, J.D. Van Wyk, Electromagnetic Energy Propagation in Power

Electronic Converters: Toward Future Electromagnetic Integration, Proceedings of

the IEEE, Vol. 89, No. 6, June 2001, pp.876-879

133 T. Ikeda, Fundamentais of Piezoelectricity. Oxford University Press, 1996.

[4] S. Butterworth, On Electrically-Maintained Vibrations, Proceedings of Physics

Society, Vol. 27, 191 5, pp. 4 10-424

[SI W. P. Mason, Electromechanical Trançducers and Wave Filters, 2" Edition, Toronto:

D. Van Nostrand Company Inc, 1948.

[6] W. P. Mason, Piezoelectric Crystals and their Application to Ultrasonics, 2nd Edition,

Princeton: D. Van Nostrand Company Inc, 1964

[7] G.E. Martin, On the Theory of Segmented Electromechanical System, The Journal of

the Acoustical Society of America, Vol. 36, No. 2, July 1964, pp. NIA

[8] G.E. Martin, Vibrations of Coaxially Segmented, Longitudinally Polarized

Ferroelectric Tubes, The Journal of the Acoustical Society of Amenca, Vol. 36, No.

8, Aug. 1964, pp. 1496-1 506

[9] T. Tanaka, Piezoelectric Devices in Japan, Ferroelectrics, Vol. 40, 1982, pp. 167- 1 87

[IO] T. Zaitsu, T. Shigehisa, M. Shoyama, T. Ninomiya; Piezoelectric Transformer

Converter with PWM Control; APEC796 Conference Proceedinns, Vol. 1, M m h

1996, pp. 279-283

[II] S. Kawashima, O. Ohinishi, H. Hakamata, S. Tagarni, A. Fukouka, T. Inuoue, S.

Hirose; Third Order Longitudinal Mode Piezoelectric Ceramic Transformer and its

Applications to High Voltage Power Inverter; IEEE Ultrasonics Symposium 1994

Proceedings, Volume: 1, 1994, pp. 525-530

[12] C.A. Rosen, Ceramic Transformers and Filters, General Electric Company,

Syracuse, N.Y. and Proceedings of the Electronic Components Society, 1956, pp. 205

[13] T. Zaitsu, T. Inuoue, O. Ohnishi, I. Iwarnoto, 2 MHz Power Converters with

Piezocerarnic Transformer, IEEE MTELEC '92 Proceedings, 1992, pp. 430-437

[14] O. Ohnishi, H. Kishie, 1. Iwarnoto, Y. Sasaki, T. Zaitsu, T. Inuoue, Piezoelectric

Ceramic Transformers Operating in Thickness Extensional Vibration Mode for Power

Supply, Ultasonic Symposium 1992 Proceedings, 1992, pp. 483-488

[15] T. Inuoue, O. Ohnishi, N. Ohde, Thickness Mode Vibration Piezoelectric

Transformer, US Patent 5 1 1 8982, 1992

[16] A.M. Flynn, S.R. Sanders, Fundamental Limits on Energy Transfer and Circuit

Considerations for Piezoelectric Transformers, IEEE PESC 98 Record, Vol. 2, 1998,

pp. 1463- 1471

[17] S. Hallaret, E. Sarruete, B. LePloufie, Numerical and Anaiytical Modeling of a

Piezoelectric Transformer and Experimental Verifkation, Laboratoire d'Electriciteq

Signaux et Robotique, France and the University of Tokyo, Tokyo

1181 T. Zaitsu, O. Ohnishi, T. Inuoue, T. Ninomiya, F.C. Lee, G.C. Hua; Piezoelectric

Transformer Operating in Thickness Extensional Vibration and its Application to

Switching Converter; IEEE PESC Record'94, Vol. 1, pp. 585-589

1191 G. Ivensky, M. Shvarstas, S.B. Yaakov; Analysis and Modeling of a Piezoelectric

Transformer in High Output Voltage Applications; IEEE M E C 2000, Vo1.2, 2000,

1081-1087

[20] C.Y. Lin, F.C. Lee; Design of a Piezoelectric Transformer and its Matching

Network; IEEE PESC '94 Record, Vol. 1, 1994, pp.607-612

[2 11 J.S. Lee, Y.H. Lee, H.I. Chai, M.S. Yoon, K.J. Lim, The Characteristics of New

Piezoelectric Ballast for Fluorescent T8 Larnp, IEEE ISIE 2001 Proceedings, 2001,

Vol. 2, pp. 947-95 1

[22] D.Jiaz, F. Nuno, M.A. Prieto, J.A. Martin, A New Control for an AC/DC

Converter Based on a Piezoelectric Transformer, IEEE M E C 2001, Vol. 1, pp. 497-

503

[23] P. Alou, J.A. Cobos, M. Sanz, J. Uceda, M. Rivas, J. Navas, Subharmonic

Driving: A New Concept to Drive Piezoelectric Transformers in Power Converters,

IEEE APEC 200 2, Vol. 1,200 1, pp. 487-49 1

[24] J. Navas, T. Bove, J.A. Cobos, F. Nmo, K. Brebol, Miniaturised Battery Charger

using Piezoelectric Transformers, IEEE APEC 200 1, Vol. 1,200 1, pp. 492-496

[25] C. S. Moo, W. M. Chen and H. K. Hsieh, An Electronic Ballast With

Piezoelectric Transformer for Cold Cathode Fluorescent Larnps, IEEE ISIE 2001,

Vol. 1,2001, pp.36-4 1

[26] M.J. Prito, J. Diaz, and J.A. Martin, A Very Simple Dc/DC Converter Using

Piezoelectric Transformer, IEEE PESC 200 1 Record, Vo1.4,200 1, pp. 1 755- 1760

[27] S. Hamamura, D. Kurose, and T. Ninomiya; Piezoelectric Transformer AC/DC

Converter Over a Worldwide range of Input Voltage by Combined PWM and PFM

Control, IEEE PESC 2001 Record, Vol. 1,2001, pp.416-42 1

[28] J.A. Oliver, R. Prieto, J.A. Cobos, M. Sanz and J. Uceda, ID Modeling of Multi-

Layer Piezoelectric Transformer, IEEE PESC 2001 Record, Vo1.4, 2001, pp.2097-

2102

[29] Murata Electronics, PZT Application Manual, June 1999

1301 APC International Limited, [Online document], [cited 1999-2001 July 271,

Available HTTP: http://www.americanpiezo.com

[3 1 ] Channel Industries Inc., Piezoelectric Ceramics Catalogue, 2000

[32] Piezo Systerns Inc, "Piezo Education", [Oniine document], [cited 1999-200 1 July

271, Available HTTP: http://www.piezo.com

[33] Philips Components, Piezoelectric Transformers Catalogue, 1997- 1 999

[34] The Face Companies, [Online document], [cited

www. faceco.com

[35] Mike Smith, Piezoelectric Transformers, Draft

200 1 July 271, Available HTTP:

thesis, Kavanagh Labs, UCSD,

[Online document], [cited 1999 Oct. 141, Available HTTP: www.ucsd.edu

[36] N.M. Moms, Essential Formulae, 2nd Ed., London: Macmillan Press Ltd., 1993

1371 T.E. Shea, Transmission Networks and Wave Filters, D. Van Nostrand

[38] J. Harris, H. Stocker, Handbook of Mathematics and Computational Science, New

York: Springer-Verilog, 1 998

1391 K. Uchino, S. Herose, Loss Mechanisms in Piezoelectrics-How to Measure

Different losses Separately, IEEE Transactions on Ultrasonics, Ferroelectrics and

Frequency Control, Vol. 48, No. 1, Jan- 2001, pp. 307-32 1

1401 M. Kazimierczdc, D. Czarkowski, Resonant Power Converters, New York: Jon

Wiley and Sons Inc., 1995

[41] S. Hamamura, T. Zaitsu, T. Ninomiya, M. Shoyama, Noise Charactenstics of

Piezoelectric Transformer DC-DC Converter, IEEE PESC '98 Record, Vol. 2, 1998,

pp. 1 262- 1267

[42] M.K. Kazimierczuk, K. Puczko, Exact Analysis of Class E Tuned Power

Amplifier at any Q and Awitch Duty Cycle, IEEE Transactions on Circuits Systems,

Vol. CAS-34, No. 2, Feb. 1987, pp. 149-1 59

[43] T. Yamane, S. Hamamura, T. Zaitsu, T. Ninomiya, M. Shoyama, Y. Fuda,

EfEciency Improvement of Piezoelectric Transformer DC-DC Converter, IEEE PESC

'98 Record, Vol. 2, 1998, pp. 1255-1 26 1

[44] S. Shemt, Losses, Dispersion, and Field Dependance of Piezoelectric Materials,

Ph.D. Thesis, Queens University, ON, Canada, 1997, pp. 23-24,A17