Rémy, Romain, Sébastien, Ghislain, Sylvain, Thierry, Charley Vincent Othman

Léo

Éric Antoine Benjamin

Yan

Marco Marc

Andres Thibault

Jérôme

Remerciement

Durant mon projet de thèse, j'ai eu beaucoup de soutient de la part de nombreuse personne et je tiens à les remercier énormément, néanmoins ces quelques mots ne suffisent pas à exprimer ma reconnaissance! Je tiens à remercier énormément Ghislain Despesse, mon directeur de thèse, pour tout son temps consacré à nos discussions interminables, ses précieux conseils techniques et de m'avoir partager sa patient pour la science.

Je remercie également François Costa, mon codirecteur de thèse, pour m'avoir prodigué de précieux conseils, pour son temps consacré et son soutien.

A new mechanism has been developed recently based on piezoelectric resonators. The mechanism consists of transforming an input electrical energy into a mechanical energy before restoring it to an electrical load with the right voltage. The piezoelectric materials are widely used for sensing (accelerometer, force sensor …), harvesting energy (vibration, deformation), or actuating (sonar, ultrasonic cleaner, scanning microscope). These materials offers several advantages, such as can be integrated while achieving small scale, and offering high quality factor and electromechanical coupling factor, enabling high power density and high efficiency operation. Also, they have an LC-resonator behavior, which enables their use in resonant converters. Nevertheless, their usage in power converters should be investigated for a better utilization.

Therefore, in the first part of this thesis, we studied the conversion mechanism, that alternates between connected voltage stage for charge transfer and open circuit stage for soft changing of the voltage level across the piezoelectric resonator (PR) and performing softswitching, reducing losses and EMI interference. A generic model of six-stages based switching sequence is developed, this model estimates the timing of each stage and the amplitude of the current related to the amplitude of the vibration. Furthermore, we propose a dc-dc converter based on PR targeting high-to-low voltage and low power applications (up to 100 W), suitable for grid to load power supply. Indeed, the proposed topology enables high step-down voltage conversion and minimizes the stored energy in the PR.

In the second part, we propose a design of the DC-DC converter based on PR, with an actual circuit for an input voltage up to 250 V and an output voltage up to 125 V. This converter achieved an output power up to 50 W with an efficiency of 93 % for radial mode vibration at frequency near 90 kHz, and an output power up to 175 W for thickness mode vibration at a T frequency of 1 MHz. Experimental waveforms are exhibited, which show how the conversion cycle evolves through the various operating conditions.

After that, we have focused on the development of a first control strategy for closed-loop operation for such a conversion principle. The control strategy enables load regulation, while achieving the conversion cycle, the soft switching operation, soft-charging and phase/frequency control for the synchronization with the mechanical vibration of the PR. The control strategy is then implemented in a Field Programmable Gate Arrays (FPGA) with the development of a suited control circuit, for a high step-down DC-DC converter. An actual prototype was designed for input voltage up to 120 V and output voltage < 48 V. The implemented control demonstrates good performances in terms of response time (less than 2 ms) and the voltage overshoot not exceeds 10 % of the output voltage.

At the end of this manuscript, we investigate on piezoelectric materials suited for power converters based on piezoelectric resonators. The investigation aims the increasing of the operating frequency, reducing the spurious modes interference, achieving high quality and electromechanical factors and mechanical mounting by using lithium niobate (LNO). Then, a fabricated LNO based PR is presented, with an actual circuit for an operating frequency up to 7 MHz. The proposed circuit covers a power range of 2-32 W with an input to output voltage of 60 to 20 V. This part is presented in APPENDIX A.

Then, we finish by presenting our perspectives and main challenges that can be investigated in the future, and the potential opportunities of power conversion based on piezoelectric resonators. Résumé ujourd'hui, la demande de dispositifs électroniques continue d'augmenter avec des exigences de miniaturisation et de réduction de la taille des alimentations.

La majeure partie du volume des alimentations est généralement occupée par les composants passifs. Par conséquent, l'un des principaux défis consiste à réduire la taille de ces composants passifs. Les solutions conventionnelles basées sur les composants magnétiques ont tendance à être limitées par les propriétés physiques des matériaux ferromagnétiques. Leur miniaturisation est donc freinée par ces limitations intrinsèques, ce qui rend difficile la réalisation d'alimentations à haut rendement et à haute densité de puissance.

Un nouveau mécanisme a été développé récemment, basé sur des résonateurs piézoélectriques (ou piézo-résonateurs PR). Ce mécanisme consiste à transformer l'énergie électrique prélevée à une source en énergie mécanique stockée dans un résonateur mécanique avant de la restituer sous forme électrique à une charge sous la tension souhaitée. Les matériaux piézoélectriques sont largement utilisés pour les applications telles que la mesure (accéléromètre, capteur de pression…), des systèmes de récupération d'énergie (vibratoire, déformation mécanique…), des actionneurs électromécaniques (sonar, nettoyeur ultra-son, injecteur…). Ces matériaux offrent plusieurs avantages, tels que la réduction de volume ou encore la possibilité d'intégration. Par ailleurs, leur facteur de qualité élevé et leur facteur de couplage électromécanique élevé permettent d'atteindre une densité de puissance élevée et un fonctionnement à haut rendement. Ces matériaux peuvent se comporter comme des résonateurs LC, permettant ainsi leur utilisation dans les convertisseurs résonants. Néanmoins, leur utilisation dans les convertisseurs de puissance doit donc être investiguée pour une meilleure utilisation.

Dans la première partie de cette thèse, nous avons étudié le mécanisme de conversion, qui alterne entre une phase à tension constante pour échanger des charges électriques et une phase à circuit ouvert pour changer le niveau de tension du PR de manière douce et ainsi effectuer des commutations à zéro de tension. Ce fonctionnement permet la conversion de puissance avec de faibles pertes et de faibles interférences électromagnétiques. Un modèle générique pour un cycle de conversion contenant six phases est développé, ce modèle permet d'estimer la durée de chaque phase ainsi que l'amplitude du courant qui est l'image de l'amplitude des vibrations du résonateur. Nous proposons également un convertisseur continu-continu basé sur le PR visant une application pour une conversion de haute vers basse tension et une puissance faible A Nous terminons ce manuscrit en présentant nos perspective et les principaux défis qui peuvent être étudiés, ainsi que les opportunités potentielles de la conversion d'énergie basée sur les résonateurs piézoélectriques.

Table of Figures

List of Tables

Chapter I: Introduction

Abstract: This chapter is the introduction of this dissertation, where the scope and the objectives of this thesis are presented.

Content

I.1 Scope

The scope of this dissertation is to expose the research that has been conducted during this PhD project. This project was carried out at CEA-Leti (Carnot's Institutes) and SATIE laboratory, and supported by French Agence Nationale de la Recherche (ANR) and via Carnot Funding. This research has been mostly published as journal/conference papers and patents, [APPENDIX B].

I.2 Background and motivations

Nowadays, all electronics need to be powered by grid-connected supplies or batteries. These electronic devices operate at various voltage amplitude, frequency and power. Therefore, power converters are required to change the power source's waveform and to adapt it to the electronic devices need, as illustrated in Figure I.1. Thus, these converters become unavoidable in our daily life. Certainly, the demand of power converters is constantly growing, which is providing a serious economic challenge. Indeed, according to the report of Market Sand Markets in Global Power Converters in 2022, the market of only dc-dc converters is estimated to grow by USD 7.7 billion between 2021 and 2026 all over the world [START_REF]DC-DC converter Market Size Share Global forecast to[END_REF]. This growth is due to the increase of the power consumption in industrial automation (industry 4.0), telecommunication, aerospace industry, electric vehicle …etc., and the constant demand in cleaner energy as renewable energy.

Among the large family of power converters, we can classify two categories: the direct transfer converters and the indirect ones. The basic operating of these lasts, which are considered in this

Power Converter

Power source Power Load work, consists of taking electrical energy from the source and to store it temporally in a passive component. Then, this stored energy is transferred to the load. The electrical energy can be stored in electromagnetic, electrostatic or mechanical energy form. As the source and the load can be found in different forms (AC current, DC current), designing power converter is a serious process. Indeed, the design process consists of well selecting the topology of the converter, the semiconductors and the passive components…, as illustrated by Figure I.2. Moreover, that should fit the requirement in terms of power, efficiency, size reduction and cost. Considering that, the design process becomes complex and tough as it depends on multiple parameters, as illustrated in Figure I.2. Therefore, the major challenge of the design is to provide a compact converter that ensures high efficiency (i.e., low heat sink) and load constraints (i.e., power, voltage, frequency).

Commercially, the most used power converters are based on electromagnetic energy conversion. These converters use inductors or transformers in order to operate as step-up or Power step-down converters, or as isolated converters. They can operate with high efficiency and high power. However, the magnetic components have parasitic effects increasing losses like copper losses, hysteresis and Eddy Current losses in magnetics, limiting the operating frequency and the energy stored by cycle due to materials and provide poor efficiency with size reduction [START_REF] Perreault | Opportunities and challenges in very high frequency power conversion[END_REF]- [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF]. Also, inductors radiate since the electromagnetic field is not completely enclosed in the material. Therefore, miniaturization of the converter becomes limited by the magnetic components, as they take a significant part of the volume of the converter and their performances decline with size reduction [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF]. Some researchers have investigated on size reduction of magnetic components by increasing the switching frequency and/or on using improved magnetic materials [START_REF] Perreault | Opportunities and challenges in very high frequency power conversion[END_REF], [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF], but the power density of magnetic components based converters remains limited by their materials' fundamental physical limits [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF].

The capacitors used as energy storage in power electronics can easily be integrated into monolithic structures, which can be assembled in parallel or series from small cells to fit the requirements in terms of power and voltage [START_REF] Seeman | Analysis and optimization of switched capacitor dc-dc converters[END_REF], [START_REF] Makowski | Performance limits of switched capacitor dc-dc converters[END_REF]. Moreover, the switched capacitance converters (SCCs) can achieve high-power density for low-power applications (less than 1W), which could be 100-1000 times greater than magnetic converters [START_REF] Seeman | Analysis and optimization of switched capacitor dc-dc converters[END_REF]. However, their efficiency drops significantly for variable gain. Furthermore, it needs to operate at variable switching frequency for maintaining high efficiency with load variations (i.e., variable output power) [START_REF] Makowski | Performance limits of switched capacitor dc-dc converters[END_REF].

Piezoelectric transformers (PTs) were used to replace the magnetic transformers of high step-up power converters and low power, especially for cold cathode fluorescent back light (CCFL) to overcome the drawbacks of the magnetic solutions, [START_REF] Lin | Piezoelectric Transformer Characterization and Application of Electronic Ballast[END_REF]- [START_REF] Lin | Inductor-less piezoelectric transformer electronic ballast for linear fluorescent lamp[END_REF]. PTs-based power converters are compact and low profile suited for integration in thin devices like cell phone. However, their efficiency, without additional inductor in order to achieve ZVS operation, is quite poor, or it needs a specific condition for ZVS operation [START_REF] Chen | ZVS considerations for a phase-lock control dc/dc converter with piezoelectric transformer[END_REF]- [START_REF] Horsley | Analysis of inductorless zero-voltage-switching piezoelectric transformer-based converters[END_REF]. Ultimately, the market of the backlight was widely replaced by led lightning. Thus, the commercial breakthrough has still to show up with the current market demand. Hence, the investigation on alternative mechanism of conversion should be conducted to offer new perspectives in power electronics.

Therefore, this dissertation deals with indirect power conversion based on piezoelectric resonators, since these materials can achieve high power density and efficiency compared to magnetic and electrostatic devices [START_REF] Kyaw | Fundamental examination of multiple potential passive component technologies for future power electronics[END_REF]. Indeed, the advantage of the piezoelectric components are listed below:

• Reducing EMI problems: o no magnetic field generation by the storage element as long as it is mechanical. o it is immune to electromagnetic interference EMI.

o Soft switching operation avoiding fast electric field variations.

• High quality factor (low losses): potential high efficiency.

• High mechanical energy storage capability: potential of high power density conversion.

• Low cost, thanks to the simple build-up and manufacturing.

• Low profile and planar shape.

• Integration on silicon with controlled process such as quartz oscillator.

• Inherent electrical insulation.

• Possibility to achieve complex geometries such as Tubes, disks, plates, focal bowls… As reveled by the title, our main objective is to develop an uninsulated power converter for AC to DC voltage conversion and low power applications (between 10 to100 W) by using piezoelectric components in order to miniaturize the power supply. The structure of AC to DC converter can be composed of multiple conversion stages (i.e., rectifier stage, isolated dc-dc, PFC, etc.…) which depends on the power level, voltage, insulation requirement, etc.… [START_REF] Hernandez | Etude et réalisation d'un convertisseur AC/DC Buck Boost réversible à haut rendement pour alimentation de secours[END_REF].

Still, most of AC-DC converters contains a regulated DC-DC power stage in order to deliver the energy at a constant voltage level to the output load. In domestic and housing applications, this DC-DC stage is a high-to-low voltage converter; hence, we focus on a high ratio DC-DC power converter in this dissertation. This converter will use a piezoelectric resonator as the main transient energy storage.

I.3 Thesis objectives

The main objective of this thesis is to investigate a new mechanism of power conversion that uses an alternative to the conventional magnetic power storage. The conversion mechanism that we focus on is based on mechanical energy storage. Indeed, the piezoelectric properties enable to store electrical energy as mechanical one through the reverse effect of the piezoelectricity, and to restitute this energy as electrical form thanks to the direct effect. Therefore, the objectives of thesis can be summarized by:

• Identifying the topology of the piezo DC-DC converter and the related conversion cycle.

• Providing an analytical model.

• Developing an actual prototype for experimental verification.

• Providing and developing a control strategy for regulating the output voltage.

• Implanting the control strategy.

I.4 Thesis structure and content

This dissertation is structured as follow:

Chapter II:

In this chapter, we present an overview of the state of the art of the piezoelectricity and power converters based on piezoelectric components. The recent progress are also presented.

Chapter III:

This chapter presents the power conversion using a piezoelectric resonator as storage element. A generic representation of the power conversion is established, and a relative generic model is provided. We also present the most relevant topologies and conversion cycles. A comparison between the conversion cycles are made considering step-down converters.

Therefore, we identify the relevant converter topology and cycle that we will use for our applications.

Chapter IV:

In this chapter, we propose a design and gives the constraints of our converter. Then, we present our experimental setup. Afterward, experimental results are exhibited over various operating points. The influence of the voltage, frequency and power are stated.

Chapter V:

This chapter shows a generic control strategy of conversion cycle based on piezoelectric resonator. Therefore, this strategy is applied on our topology. Simulation under MATLAB-Simulink is carried out, which validates the operation of the developed control strategy. An experimental prototype is built in order to validate the control. Therefore, we present the experimental results to fulfil the validation process.

Chapter VI:

This chapter concludes our work and present the major challenges and perspectives of piezobased power converters.

Chapter II: Overview and State of Art

Abstract: This chapter gives the state of the art and overview of the piezo-based power converters. Also, this chapter illustrates the recent improvements and progress of such converters, and the major challenges that we should deal with.

Content

II.1 Introduction

The goal of this chapter is to give the state of art of the piezoelectricity through its history and applications with a particular attention on the power conversion using piezoelectric components, in which we explain the mechanism of the conversion through an example of implementation. In addition, a synthesis of the main papers in the literature dealing with our topic are cited and classified by subjects.

II.2 The piezoelectricity: history and applications

In 1880, the French physicists Curie Brothers discover the piezoelectricity [START_REF]IEEE standard on piezoelectricity[END_REF]. The piezoelectricity consists of generating electricity from a pressure. Therefore, the piezoelectricity is the interaction between the mechanical and electrical states in the piezoelectric material providing thus an electromechanical coupling. Resulting from that, two effects of the piezoelectricity are defined:

The direct effect: it is the result of a displacement of electrical charge when a mechanical force is applied.

The reverse effect: Lippman in 1881 demonstrated the reverse effect. The reverse effect is the result of a mechanical strain when an electrical field is applied. In literature, the mechanical strain is very small, for instance: the lead zirconate titane (PZT) material provides a mechanical strain around 0.1 %. The low mechanical strain can be very useful for achieving The usage of the piezoelectric materials in power converters comes in the late of 1950s, when C. Rosen has invented the first piezoelectric transformer (PT) (called the Rosen-type after the inventor) [START_REF] Rosen | Electromechanical transducer[END_REF], and then PT-based power converter [START_REF] Rosen | Ceramic transformers and filters[END_REF]. Therefore, with the progress in the piezoelectric material, many researches have been conducted in this domain. To replace the magnetics based converters, the piezoelectric based converters were a commercial achievement but only for high step-up voltage conversion and low power (< 5 W) applications. Indeed, the usage of PT-based converters to supply LCD display with CCFL back-light has succeeded.

However, the Back-lighting was rapidly replaced by Led lighting which is supplied with low voltage. Even if the PT-based converters are not limited by the same drawbacks comparing to electromagnetic transformer converters, a major challenge remains in developing them for high-to-low voltage or low-voltage conversion and power up to 100 W, in order to meet the growing demand in the switched-mode power supply (SMPS) and low-voltage converters.

In the last decade, the piezoelectric resonators (PRs) were firstly used in switched capacitor converters as replacement for the capacitors [START_REF] Park | A resonant-type stage-up dc/dc converters with piezoelectric transducers[END_REF]- [START_REF] Seo | A magnetic component-less series resonant converter using a piezoelectric transducer for low profile application[END_REF], the PRs can provide a higher quality factor and also get benefit from the natural inductive behavior in order the converter to operate with ZVS conditions. However, the voltage gain remains to be fixed to 2 (in step-up conversion), which requires using cascade converters for having another conversion gain. In addition, these converters inherit the same drawbacks as the switched capacitor converters in terms of voltage regulation and low efficiency with load variation [START_REF] Seeman | Analysis and optimization of switched capacitor dc-dc converters[END_REF]- [START_REF] Makowski | Performance limits of switched capacitor dc-dc converters[END_REF]. With the recent developments in PR-based converters [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF], a new potential field has emerged that consists of using a single PR as storage component in power converters without requiring transformer or additional inductors in the conversion systems. Moreover, it enables employing the PR component in step-up and step-down topologies. Subsequently, several research studies have emerged [START_REF] Pollet | A New Non-Isolated Low-Power Inductorless Piezoelectric DC-DC Converter[END_REF]- [START_REF] Braun | Optimized Resonators for Piezoelectric Power Conversion[END_REF] that investigate on power converters based on PR. They exhibit the performances of the PR-based converters in several use cases. Until now, we have identified three major research teams on using PRs in power electronics: CEA-Leti, MIT and Sandford University.

II.3 Piezo-based power converters

II.3.1 Mechanical Energy Conversion

The first use of the mechanical energy in order to store transiently the electric energy as replacement for the magnetic was made by J. M. Noworolski et S. R. Sanders [START_REF] Noworolski | Microresonant devices for power conversion[END_REF]. Basically, J. M. Noworolski was motivated by the fact that twice energy can be stored in the Silicon as mechanical energy compared to magnetic energy in the same volume. In their research, they have used a variable capacitance (VC) to transfer electric energy to a mechanical form. The operating principle of their converter consists in using multiple cycles of pumping energy from the input source to the mechanical resonator at the resonance frequency of the VC, then, once the energy stored in the VC reaches the desired value, a boosting cycle is applied to transfer the energy stored to the load. Even if the integration of the converter was achieved, its efficiency was low, because during the pumping stage (i.e., the pumping stage consists of charging and discharging the capacitance to increase the displacement of the free electrode), hard switching was applied to discharge the capacitance when the energy of the capacitance is at its maximum value. This leads to high power losses. In addition, the converter is limited only to boost conversion.

In 2017, G. Despesse and B. Pollet have submitted the first patent on power converter based on piezoelectric resonator that describes a new mechanism of energy conversion [START_REF] Despesse | Power converter[END_REF].

The power converter based on piezoelectric resonator (PR) was firstly inspired from the research made by S. Ghandour, G. Despesse and S. Basrour during their research work in integration of power converters as MEMS using a VC [START_REF] Ghandour | Design of a new mems dc/dc voltage step-down converter[END_REF]. The initially developed power converter is based on using a variable capacitance where by construction one electrode is to close-on the switches with ZVS operation. The presented converter in [START_REF] Ghandour | Design of a new mems dc/dc voltage step-down converter[END_REF] illustrates an efficiency up to 80 % in simulation for a step-up converter from 10 V to 20 V with a power of the order of mW for a VC having an area of 1 cm².

The advantage of such a converter is that the fabrication process of the VC is compatible with the current process in silicon industry enabling to integrate the power converters, even though some difficulties had been noted for the fabrication of the free electrode requiring some developments. However, the VC should be permanently polarized, and if the zero voltage crossing is occurred, the VC should be re-polarized which provides a major issue. Also, the power density (W/cm²) shown of this converter remains low, less than 10 mW/cm² (considering only the area of VC).

Therefore, to overcome these limitations, B. Pollet &.al. [START_REF] Despesse | Power converter[END_REF] have started the investigation on the use of the piezoelectric components instead of the electrostatic devices. In both cases, energy is stored in a mechanical form, but in the piezoelectric case, the energy transfer between electrical and mechanical domains is done by using the piezoelectric properties and not any more by using electrostatic actuation/regeneration. This enables a significant increase in power level that can be exchanged between the mechanical and electrical domain at each cycle thanks to high electromechanical coupling factor of the recent piezoelectric materials (k²>0.3).

Furthermore, the storage and transduction mechanism are merged, removing the need of adding air-gap, electrodes, and guidance/spring. Finally, the power density is significantly increased, about a factor 1000 (Watt range scale obtained for piezoelectric materials compared to mW range per centimeter scale for the VC).

In addition, the piezoelectric materials are extensively developed as MEMS components in actuator or in sensing applications [START_REF] Lu | Design, fabrication, and integration of piezoelectric mems devices for applications in wireless sensor network[END_REF]- [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF], which make them suitable for integration and miniaturization. Where, Qin is the charge taken from the input source, while Qout is the charge transferred to the load. Q3 is the charge evacuated during the stage 3 (short-circuit), used for balancing the total charge of the PR while energy exchanged during stages 1 and 5 are balanced.

II.3.2 Example of implementation of PR-based DC-DC converter

Therefore, the conversion gain can be deduced by (II.3.2). As stated in (II.3.2), the conversion gain is just the ratio between the charge stored in the PR and the one transferred to load. This operating principle enables to operate with energy balance, making as consequence a sustainable oscillation. The charge balance clamps the amplitude of the voltage across VP.

𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛 = 𝑄 𝑖𝑛 |𝑄 𝑜𝑢𝑡 | = 1 - 𝑄 3 |𝑄 𝑜𝑢𝑡 | ≤ 1 (II.
Therefore, the losses are mainly depending on the quality factor of the materials (damping effect) that may be low as the PR may achieve a high quality factor (> 1000 for PZT), and also on conduction losses of the switches (no switching losses). Assuming this operation, the converter may achieve a high efficiency and a high power density, as demonstrated in [START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF].

Thus, this operation can be extended in numerous conversion cycles and topologies, depending on the power conversion requirements and implementation.

II.4 Piezo-based power converters: projects and applications

In this part, we have identified the recent scientific articles or patents on the development of power converters based PR that give an overview of the state of the art illustrating the various topics and application targets. Therefore, through the recent research projects, we can summarized the 3 main research related topics as follow:

II.4.1 Topology of power converters based on PR

The integration of the PR as a storage component in DC-DC converter leads to question about the best topology, the conversion cycle, the number of switches…etc., based on how the PR is used. In [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF]- [START_REF] Touhami | A New Topology of DC-DC Converter Based on Piezoelectric Resonator[END_REF], the authors enumerate the main topologies for step-up, step-down, high step-down/high step down converters. Also, isolated PRs-based converters have been developed in [START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF].

In [START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF], the DC-DC converter presented operates as step-up (0< Vout < Vin) or step-down (0< Vin < Vout) conversion. In this topology (Figure II.5), the PR stores the energy taken from the input source, and then transfers it to the output load. motivates our researches in exploring the numerous topologies and conversion cycles that get the best advantage of the PR.

II.4.2 Close-loop Control

The control of a such dc-dc converters that employs PR reveals to be more complex than the classical solutions in magnetics based converters. Indeed, the control should take into consideration the conversion cycle, variable frequency, soft switching/charging and load regulation. In [START_REF] Pollet | A New Non-Isolated Low-Power Inductorless Piezoelectric DC-DC Converter[END_REF], authors have developed a control strategy that takes into consideration the previous issues for a step-up converter with low voltage conversion 10-20 V and power of 0.5 W for the switching sequence {Vin, 0, Vout}. Also, during my PhD, the MIT [START_REF] Piel | Feedback Control for a Piezoelectric-Resonator-Based DC-DC Power Converter[END_REF] has implemented a control law in a microcontroller for a step-down converter for a voltage conversion 30 -10 V and a power of < 1 W, the control shows good performances in terms of transient response and load regulation (response time is about 14.8 ms to reach 2 % of the output voltage set point). In [START_REF] Touhami | Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezoelectric Resonator[END_REF], we have developed and implemented a first control law for such a conversion mechanism. It has been used for high-tolow voltage converter for a 120 V input voltage and an output voltage of < 48 V and power up to 16 W. This control strategy shows a response time for load regulation less than 2 ms, it can be used for any switching sequence, and will be detailed chapter V.

Also, it reveals that spurious modes can show-up at high frequency operation which limits the operating power range (i.e., variable frequency), [START_REF] Pollet | A New Non-Isolated Low-Power Inductorless Piezoelectric DC-DC Converter[END_REF]- [START_REF] Braun | Optimized Resonators for Piezoelectric Power Conversion[END_REF]. Therefore, fixed frequency control can be used by using burst mode as proposed by Stanford University [START_REF] Stolt | Fixed-Frequency Control of Piezoelectric Resonator DC-DC Converters for Spurious Mode Avoidance[END_REF]. It consists of using power conversion cycles to transfer the energy to load followed by Zero Voltage cycles to cut-off the power transfer. The ratio between the number of conversion cycles and zero voltage cycles gives the mean power. Since, the PR can achieve a high quality factor (i.e., the operating frequency is near the resonance), the losses in the PR remains lower compared to magnetic converters. Nevertheless, the efficiency is lower than the optimal control developed in [START_REF] Pollet | A New Non-Isolated Low-Power Inductorless Piezoelectric DC-DC Converter[END_REF], [START_REF] Piel | Feedback Control for a Piezoelectric-Resonator-Based DC-DC Power Converter[END_REF]- [START_REF] Touhami | Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezoelectric Resonator[END_REF]. In addition, frequency should be, in any cases, regulated as the resonant frequency of the material depends on the temperature.

II.4.3 Piezoelectric materials

Piezoelectric components are anisotropic materials such as ceramics produced by powder metallurgy process (PZT), and crystals (Lithium Niobate) fabricated by crystal growth process.

Therefore, their performances are highly depending on their physical properties (i.e., strain, Elastic and dielectric constants), the shape, the mutual orientation of driving electric field and acoustic wave propagation direction [START_REF] Erhart | Piezoelectric Ceramic Resonators[END_REF]. All these considerations should be taken into account for choosing the best material enabling high power density and high efficiency. In harvesting applications, the figure of merit (FOM) is given by the product of the quality factor Q (inverse of loss rate), and the electromechanical coupling factor k² (see its definition in (II.4.1)). This same FOM is also applicable for PR based power converters to maximize the efficiency (cf.

Pollet's thesis [START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF]). It is a simple approach for classifying the piezoelectric materials in terms of efficiency.

𝑘 2 = 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑐𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑑 𝑡𝑜 𝑚𝑒𝑐ℎ𝑛𝑖𝑐𝑎𝑙 𝑓𝑜𝑟𝑚 𝑖𝑛𝑝𝑢𝑡 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 (II.4.1)
In Figure II.12 [START_REF] Touhami | Piezoelectric Materials for the DC-DC Converters Based on Piezoelectric Resonators[END_REF], we have plotted the efficiency as a function of the product Qk² for the conversion cycle presented in [START_REF] Touhami | A New Topology of DC-DC Converter Based On Piezoelectric Resonator[END_REF], considering hard PZT and Lithium Niobate from several manufacturers. This figure illustrates disparities of the materials, and show the efficiency that may be achieved. However, this FOM is not sufficient, as it does not give an indication on the material that can achieve the highest power density. In [START_REF] Boles | Evaluating Piezoelectric Materials and Vibration Modes for Power Conversion[END_REF], a FOM (FOMADP) based on calculation of the power density (W/cm²) is proposed. The main PZT materials are then placed on a 2D graph showing the area power density as a function of loss ratio (i.e., k²Q), giving the physical limits of the materials, as illustrated in Figure II.13. But overall, all these FOMs give only indications for choosing the best material. To determine the power density limit, it lacks of data verified and the measurement setups or technics to determine the properties of the materials are not specified and/or normalized for all the manufacturers.

The paper [START_REF] Braun | Optimized Resonators for Piezoelectric Power Conversion[END_REF] and our paper [START_REF] Touhami | Piezoelectric Materials for the DC-DC Converters Based on Piezoelectric Resonators[END_REF] introduced the use of LNO materials for power converters, with a specific focus on the sizing of the piezoelectric resonator. These articles show the impact of the geometry, size and mounting on the performances of the resonator and the spurious modes disturbances. They well illustrate the influence of the geometry, the dimensions and the mechanical mounting on the reduction of the spurious modes through experimental results and actual prototypes.

Figure II.12.Efficiency Vs k²Q (thickness mode vibration), [START_REF] Touhami | Piezoelectric Materials for the DC-DC Converters Based on Piezoelectric Resonators[END_REF], APPENDIX A.

II.5 State of the art synthesis

We have summarized the relevant references in TABLE II.1, and sorted by subject and by research team. Some references deal with multiple subjects, so, they are cited in multiple cases.

The references in red color are prior to this thesis work, while all other researches appeared simultaneously to this work.

From this table we can deduce the effort that was invested by each team. For the Stanford team, they were the first to use the LNO in dc-dc converter based on PR in order to operate at high frequency. Also, to deal spurious modes, burst modulation was implemented in open loop.

In addition, they have put their interest in very high frequency converters by using the LNOresonator as filter for cancelation of the second harmonic in Φ2 inverter and output filter for the class E structure.

For the MIT team, they were the second team working on piezo-based converter after CEA- materials that can provide a high efficiency and a high power density. Also, design and optimization of piezo components are subject of their work.

As illustrated in the table, in CEA-leti, we have put our interest on exploring the possibilities of the piezo-based converters. Numerous patents and papers have been submitted/accepted dealing with DC-DC converters based on PRs for isolated and non-isolated structures before and during this thesis. Also, developing control strategy is investigated with the scope of developing actual porotypes for validation. In addition, we investigate on design and fabrication of piezoelectric resonators through internal collaboration with the different departments. To sum up, several subjects are aimed to investigate on the opportunities of piezoelectric in power conversion.

II.6 Conclusion

So far, energy conversion based on using mechanical storage components such as piezoelectric resonators remains new and poorly studied at the beginning of this thesis project.

Up to now, few cases are explored and investigated, while numerous variants of power converters based on piezoelectric resonators are still to be investigated and developed for a better use of the piezoelectric materials.

In this thesis project, we aim to develop a power converter that uses a piezoelectric resonator as a storage component for high-to-low voltage conversion and low power applications, while minimizing the energy stored in the material, which allows the miniaturization of the power converter. In addition, the conversion mechanism is not well developed and clarified in the literature, thus, during the thesis we have made efforts to properly detail the operating principle and provide a generic model to build a conversion cycle based on six switching stages.

The control of such a converter is not yet addressed in the scientific literature. Conventional control is not sufficient for a PR-based converter, as it requires frequency control, soft charging, soft switching, and load regulation at the same time while keeping the control synchronized with the motion of the PR, for high efficiency operation. Furthermore, there are no papers (to our knowledge) demonstrating the feasible implementation of control law on a PR-based converter prior to this thesis [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF]. Therefore, during this thesis project, we aim to develop a control strategy and design a control circuit for a step-down DC-DC converter for high-to-low voltage conversion. The implementation will be developed with a real control circuit for the experimental proof.

In addition, we have conducted research on increasing the operating frequency using alternative materials such as Lithium Niobate which was not explored at the beginning of the thesis. This material can provide a high quality factor and a high electromechanical coupling factor, and have low permittivity, which makes it suitable for high frequency operation.

Therefore, we have provided a design on LNO-based PR for high frequency operation with achieving high quality and coupling factors. Moreover, the proposed design takes into consideration the reduction of the spurious modes' disruption.

Chapter III: Piezoelectric resonator based power converter Abstract: In this chapter, we start by presenting the operating principle of piezo-based dcdc converters. Then, we propose a generic analytical model for steady-state operation. Also, we enumerate the conversion cycles and the topologies. Furthermore, a simulation is presented to verify the operating principle. Finally, we propose a method to compare the conversion cycles. We conclude the chapter with a discussion and analysis. The piezoelectric resonator (PR) in power conversion operates by using the both piezoelectricity effects. Over a single conversion period, the PR works as a piezoelectric actuator to store the electrical energy in mechanical form, and as a piezoelectric transducer to transfer the mechanical energy in electrical energy to supply the load. A PR is composed of a piezoelectric ceramic plate or crystal with two electrodes on each side; it can operate in thickness vibration mode or longitudinal vibration mode.

Content

To replace the inductive behavior of inductor in power converter, the PR operates in the inductive region of its impedance (i.e., between the resonant and anti-resonant frequencies).

Compare to a L-C resonator (Q < 50), PR benefits of a high quality factor of the ceramic, such as PZT (𝑄 > 1000). However, due to the physical behavior of the PR and the need for operating near the resonance frequency, the conversion mechanism and the approach should be adapted to the PR based DC-DC energy converter. This chapter first presents the equivalent circuit model of PR, the main operating principle and the conversion cycle of piezo-based power conversion. Then, the conversion cycle is derived for any switching sequences. Therefore, this chapter presents an analytical model based on energy approach by taking as example a switching sequence adapted for high-to-low voltage applications. The electrical waveforms and the power conversion efficiency are performed by simulation in order to illustrate and to assess the operating principle. Yet, the PR operates between the resonant and the anti-resonant frequency, the current i in L-C branch can be approximated by a sinusoidal AC current source thanks to the high quality factor of the ceramics like PZT, then the current i is given by:

𝑖(𝑡) = 𝐼 • sin 𝜔𝑡 (III.2.2)
Where, I is the amplitude of current i in LCR branch¸ and 𝜔 = 2𝜋𝑓 is the pulsation and f the operating frequency.

III.2.2 Operational principle

As in a conventional energy converter, the energy is taken from the input source, stored temporarily in a passive component before being transferred to the load, with changing the type of the energy. Therefore, by using a PR as a storage element, it should at least being connected • {Vin-Vout, Vout} : where Vin=2Vout.

• {Vin, Vout-Vin} : where Vin=½Vout.

• {Vin-Vout, 0}: where Vin=Vout.

• {Vin, Vout}: where Vin=Vout.

Therefore, a third connected stage named Vc is required to operate with a variable voltage conversion gain. This stage allows balancing the PR's charge and energy independently. So, we can construct a conversion cycle where we define the voltages {Va, Vb, Vc} as follow:

• Each voltage level is defined during only a single polarity of the current i which means that no reactive energy is exchanged with the PR. t3-t3': βi becomes positive and the capacitance CP is being discharged. The voltage βVP goes from βVzvst3 to βVa. This stage is needed in some cases to turn-on switches with zero voltage switching-on. However, if βVa=βVzvst3, then this stage in not required. As, we consider the general case, this stage is taken into account for the following analysis.

{ 𝑉 𝛽 = median{V 1 , V 2 , V 3 } 𝛽 = -sign (𝑄 𝑉 𝛽 ) (III.2.8) { 𝛽𝑉 𝑎 = max{𝛽V 1 , 𝛽V 2 , 𝛽V 3 } 𝛽𝑉 𝑏 = median{𝛽V 1 , 𝛽V 2 , 𝛽V 3 } 𝛽𝑉 𝑐 = min{𝛽V 1 , 𝛽V 2 , 𝛽V 3 } (III.2.
t3'-t4: PR is connected to βVa, and exchanges a charge Qa.

t4-t5:

The PR is isolated, and the current βi discharges the capacitance CP, so, the voltage βVP decreases from βVa to reach the voltage βVc at t=t5.

t5-t5':

The PR is connected to βVc, and Qc is the electric charge exchanged between the PR and βVc. This stage ends to let the voltage go from βVc to βVzvst6.

t5'-t6: During this stage, the PR is isolated, and the voltage βVP decreases from βVc to reach βVzvst6 at t=t6. This stage ends when the current i changes of sign at the end of period. This cycle is repeated at the PR's oscillation frequency. In some cases, as proposed by [START_REF] Stolt | Fixed-Frequency Control of Piezoelectric Resonator DC-DC Converters for Spurious Mode Avoidance[END_REF],

the cycle can be applied only on a few periods, during the other periods the piezoelectric being short-circuited. This enables to reduce the output power level without reducing too much the power per cycle, reducing the operating frequency shift. The high quality factor being sufficient to keep the mechanical oscillation amplitude between two effective cycles.

III.2.3 Energy approach

The conversion principle of the piezo-based power converter is based on a balance of energy and charge over each period of conversion in steady-state operation. Therefore, for the generic This representation makes it easy to visualize and to understand the energy and charge transfer over a period of conversion in steady state operation. Icirculating is the amplitude of the current required to charge and discharge the capacitance CP, this amplitude is required even with no load. Iuseful is the amplitude of the current required to provide the load current Iout, its amplitude is proportional to Iout and depends on the switching sequences. Regarding to the charge transfer utilization K-factor 1 defined in [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF], we can express the K-factor as a function of our parameters, as given by (III.3.21).

Once the amplitude of the current I is determined as a function of the operating conditions, manage the output current through t4, this equation can be used for any switching sequence. 1 The K-factor in [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF], quantifies the charge transfer delivered to the output load. Also, the K-factor indicates how efficiently is used the PR. This factor is mainly depending on the switching sequences and Vin and Vout. Furthermore, the relation between t4 and Iout helps to determine the loop to regulate the output voltage.

III.4 Converter topologies and conversion cycles III.4.1 Switching Sequences Implementation

As mentioned before, the voltages Va, Vb and Vc can be zero (a short-circuit), the input voltage, the output voltage, a combination of them, with a positive or negative polarity (Vi=+/-Vin+/-Vout or Zero). Therefore, starting from these various combinations, we can enumerate the conversion cycles, where some of them are mentioned and/or explored in [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF]- [START_REF] Touhami | A New Topology of DC-DC Converter Based on Piezoelectric Resonator[END_REF]. For example, for a given 3-voltage levels triplet {Vin-Vout, Vout, 0V} [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF], which operates as stepdown converter, we can have Vout> Vin-Vout or Vout< Vin-Vout. In case of Vout> Vin-Vout, the voltage level Vin-Vout is the median voltage, so, a positive power flow is provided from this voltage leading to a factor β equals to -1. Thus, we can identify the voltages {Va, Vb, Vc} by using (III.2.9) as below: Where,

𝛾 3 ′ =
𝛼 𝑎 = 0, 𝛼 𝑏 = -1, 𝛼 𝑐 = +1, 𝛽 = -1, 𝐾 = 𝑉 𝑖𝑛 2𝑉 𝑜𝑢𝑡 .
In case of Vin-Vout >Vout, Vout is then the median voltage, so, a positive power flow is taken under this connected stage leading to β equals to 1. Then, the voltages {Va, Vb, Vc} are determined as:

The amplitude of current I is then deduced:

Where,

𝛼 𝑎 = -1, 𝛼 𝑏 = +1, 𝛼 𝑐 = 0, 𝛽 = +1, 𝐾 = 𝑉 𝑖𝑛 2(𝑉 𝑖𝑛 -𝑉 𝑜𝑢𝑡 )
. In addition, the expression of the current I is given for each conversion cycle, which is separated into the circulating current and the current needed to achieve the output current. After choosing Actually, the current amplitude I gives the key element of physical constraints for the PR and the switches. Indeed, at a specific frequency in its inductive behavioral, the PR has a current limitation per square centimeter, and then, for minimizing the piezoelectric material volume, we have to minimize the current amplitude I for a given output current value. Also, the amplitude I can be used to compare the converter topologies and conversion cycles. Indeed, at the same current amplitude I, the output current can be higher by maximizing K-factor. The maximum value for the K-factor is 1. K=1 means that 𝐼 = operates with a gain ratio lower than 0.5. At the opposite, the K-factor is lower (i.e., high amplitude of the current I) for a switching sequence like {Vin, -Vout, 0}, where a half of the period is connected to zero without any power transfer. However, this switching sequence can operate at any gain ratio. Therefore, regarding the operating conditions and the need of the power/voltage and the conversion ratio, the switching sequence should be carefully selected to minimize I, but also to minimize the switch number and the voltages they have to sustain. topologies with an unidirectional switch, a diode in series with unidirectional switch can be used as a replacement. Also, for some switching sequences, two active switches can be replaced by diodes, as their conduction stage depends on the polarity of the PR's current i.

{ -𝑉 𝑎 = max{-(V in -V out ), -V out , 0} = 0 -𝑉 𝑏 = median{-(V in -V out ), -V out , 0} = -(V in -V out ) -𝑉 𝑐 = min{-(V in -V out ), -V out , 0} = -V out (III.
{ 𝑉 𝑎 = max{V in -V out , -V out , 0} = V in -V out 𝑉 𝑏 = median{V in -V out , V out , 0} = V out 𝑉 𝑐 = min{V in -V out ,
Considering the number of switches employed, the possible switching sequences are summarized in nine-switching sequences. TABLE III. [START_REF] Perreault | Opportunities and challenges in very high frequency power conversion[END_REF] shows the switching sequences related to the converter topologies. By reducing the number of switches, the implementation of the switching sequences is easier [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF]. However, besides the number of switches, this approach does not take into consideration how the PR is used, but only minimizing the number of the switches. damping effect, in order to select the appropriate switching sequence and the converter topology in order to minimize the energy stored in the material as first approach (while omitting the losses of the switches, number of the switches and their electrical constraints). Therefore, as we are considering a step-down converter AC-Grid (230/110 VAC) to load (<100V) applications, we have selected a six switching sequences from those listed in TABLE III.2 enabling a gain ratio conversion lower than 0.5. Some switching sequences are modified to consider the soft switching operation by considering the reached extremums Vzvst3 and Vzvst6, for a fair comparison. • avoiding the Zero-Voltage stage.

• using multiple connected stages with the output.

These criteria can be used to select the switching sequence that holds a better utilization of the PR, enabling to reduce the constraint in terms of size of the material while maintaining a high efficiency and a high power density.

In the Figure III.15, we have plotted the efficiency as a function of the gain ratio. This figure illustrates that the conversion cycle {Vin-Vout, Vout t, -Vout} offers the highest efficiency compared to the other switching sequences, while at the opposite {Vin, -Vout, 0} gives the worst case. The switching sequence {Vin-Vout, -Vout, 0} gives the same efficiency evolution as {Vin, 0, Vout }, since this switching sequence is just offsetting by Vout regarding to {Vin, 0, Vout }. The difference between the two previous cycles is that {Vin-Vout, -Vout, 0} has a peak-to-peak voltage of Vin+Vout to ensure the ZVS condition while {Vin, 0, Vout} has just Vin, leading to a higher Icirculating.

III.7 Analysis and discussion

A large number of conversion cycles are possible for six-stages operations. However, few of them are interesting for a better use of the PR in terms of efficiency and power density.

Indeed, using the estimation of the efficiency and the current amplitude of the resonator gives a guideline to determine the appropriate switching sequence and then deducing the converter topology. In addition, the presented analysis provides an evaluation of the operation of the stepdown dc-dc converters based on PR, which leads to select the converter with high efficiency and power density.

The current amplitude determined previously can be found by computing the derivative equations of the current/voltage {i, Vp, Vc} numerically as demonstrated in [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF], with additional assumptions to obtain the exact steady state. This method is tedious, as it should be computed by using 18 variables and required appropriate initial conditions to reach the same behavior of the efficiency/current as previously with slightly higher accuracy [START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF]. A numerical model is presented in [START_REF] Pereira | Operating Frequency Prediction of Piezoelectric DC-DC Converters[END_REF], this model firstly considers a sinusoidal current waveform i, and then uses Fourier transforms of VP to evaluate an induced equivalent capacitance placed in parallel to CP.

It enables to find the operating frequency and the current amplitude I and the angular transition times. The main advantage of this model is to find the operating frequency, to reduce transient

IV.1 Introduction

The switching sequences based on dc-dc converters that employs PR as storage element were validated in a prior work [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF] for various operating conditions in terms of voltage and power applications. As we are aiming for high-to-low voltage conversion, we have selected the conversion cycle by respecting the following criteria:

➢
V P + V P -
The equivalent electrical circuit model has been modeled in prior work [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF], [START_REF] Van Dyke | The piezo-electric resonator and its equivalent network[END_REF], BVD circuit. The equivalent model is composed of L-C-RP series branch that represents the motional behavior of the resonator for a vibration mode, in parallel with the capacitance named CP.

The equivalent circuits are shown in For the radial mode, the model parameters are determined by the equations (IV.2.1) to (IV.2.5), [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF]. These equations result from solving the fundamental equations of the piezoelectricity, which are well detailed in literature [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF]. The resonant frequency can be determined by the diameter D and the resonant constant frequency in radial mode NP (Hz.m) which depends on the physical proprieties of the material, and given by the manufacturers.

𝐶 𝑃 =
𝑅 𝑃 = 2𝜋𝑓 𝑟 𝐿 𝑄 (IV.2.5)
Where, 𝜀 33 𝑆 is the clamped primitivity, as it is determined by assuming blocked-electrodes and can be measured at high frequency above 15 MHz as suggested by [START_REF] Touhami | Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezoelectric Resonator[END_REF].

In the literature [START_REF]IEEE standard on piezoelectricity[END_REF], [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF], the capacitance C is also defined by 𝑘 𝑒𝑓𝑓 2 𝐶 𝑃 for this vibration mode, where 𝑘 𝑒𝑓𝑓 2 is the effective coupling factor and determined by:

𝑘 𝑒𝑓𝑓 2 = 𝑓 𝑎𝑟 2 -𝑓 𝑟 2 𝑓 𝑎𝑟 2 (IV.2.6)
Thus, using this simplified model, one can determine the electrical model by using the geometry parameters of the PR, the diameter D, the thickness th, and the physical proprieties of the material Q, kp, 𝜀 33 𝑆 and NP which are generally given by the manufacturers.

(b) Thickness mode: (𝐸 ⃗ ∥ 𝐷 ⃗ ⃗ )

In thickness vibration mode, the equivalent circuit parameters are determined by (IV.2.7)-(IV.2.11), [START_REF] Bhugra | Piezoelectric MEMS Resonators[END_REF]. Also, the resonant frequency in this mode is determined as a function of the thickness th of the resonator and the constant frequency in thickness mode of the material named Nt (Hz.m). The previous models described only the fundamental resonant mode regarding to the driving electric field. So, to fully describe the resonator behavior, an extra LCR branch should be added in parallel with CP for each vibration modes. However, for a first approximation, we will use only the fundamental mode to design the resonator.

𝐶

As shown in the previous part, the BVDM model is mainly linked to the physical proprieties and the geometry of the PR. For a simplified design, we have designed a resonator by ensuring the following conditions:

• High k²Q

In the prior work [START_REF]IEEE standard on piezoelectricity[END_REF], the PR is selected for its high figure of merit k²Q. Indeed, having high quality factor means low power loss due to low damping effect in the material, while a high coupling factor ensures a high electromechanical energy conversion ratio and then a high part of the energy mechanically stored can be exchanged at each cycle. Therefore, it is interesting to choose a ceramic with high k²Q. Regarding the datasheet of the supplier, we have selected the PZT C213 from Fuji ceramics which exhibits the highest k²Q (ex: Q=2500, kP =58%) value [START_REF] Touhami | Piezoelectric Materials for the DC-DC Converters Based on Piezoelectric Resonators[END_REF].

• Insulation

As we are aiming to use the PR for high-to-low voltage power converter, the PR should withstand the maximum voltage applied. For most PZT based ceramics, the maximal electric field is between 1 kV/mm and 1.5 kV/mm (assumed), so, by considering the worst case of 1 kV/mm and a maximum voltage applied of 750 V (2Vacmax=2 x 350 V), the thickness of the PR is then equal to 0.75 mm. The thickness could be higher but it leads to a lower power density [START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF].

• Operating frequency

As mentioned previously, the PR operates between its series and parallel frequency (inductive region). Also, for the radial mode operation, the series resonant frequency is determined by its diameter. So, for operating frequency around 90 kHz, the diameter is calculated by using (IV.2.3), and equal to 25 mm considering C-213 material.

Therefore, considering the previous assumptions, the PR is sized and the equivalent circuit parameters are summarized in

IV.4 Experimental results

In this part, we illustrate the performances of the piezo based dc-dc converter through various voltage and power conditions. Also, parallelization of PRs is investigated for increasing the power. In addition, we employ the thickness mode operation for high frequency operation 1MHz, while the current density is measured over 3-PRs (three different diameters) at the same frequency operation (thickness = 2 mm). The setup used for the measurement is shown in However, by decreasing the gain ratio, the stage Vin-Vout becomes higher than the stage Vout, so, in this case, the third stage Vp= -Vout is needed for balancing the charge. In addition, the third stage's time duration increased by decreasing the gain ratio, as illustrated throughout the Therefore, regarding how the waveforms evolved at various loads, as it is depending on time duration of the connected stage and the open stage, a control law can be deduced for ensuring the energy and charge balance. In addition, we note that the frequency decreases from 100.08 kHz to 96.56 kHz, while the output power increases. This frequency variation should be taken into account in the control law; for low power, the frequency operation is approaching the anti-resonant frequency of the PR, and by increasing the output power toward the full power, the operating frequency decreased up to the series resonant frequency.

(a): V in =200 V, V out =66.7 V, Pout =6 W, f=100.08 kHz, 𝜂=92.6%

(b): V in =200 V, V out =66.7 V, Pout =10 W, f=98.54 kHz, 𝜂=94.08%.
(c): V in =200 V, V out =66.7 V, Pout =16 W, f=96.56 kHz, 𝜂=92.5%. The experimental waveforms are in good agreement with the theoretical waveforms, also, the influence of the timing as function of the power and voltage on the conversion cycle enables to determine thereafter the control strategy for a close-loop control. values. This figure illustrates how frequency evolves with the power. We note that the tendency is the same for each gain ratio, and the curves are shifted down for lower gain. While, the PR operates in free oscillation, the operating frequency evolves regarding the operating condition in terms of voltage and power.

IV.4.2 Frequency operation

As mentionned in the previous section, the amplitude of the oscillation increases when the frequency is approaching the resonant frequency, leading to high power transfer. At the opposite, when the frequency is near the anti-resonant frequency, the amplitude of the oscillation decreases and the power decreases as well. In this part, the efficiency of the converter is evaluated experimentally. The measurements take into account the losses of the PR and of the switches; the power supply of the gate drivers is omitted since the power supply is not optimized. However, we have measured its value to 1.1W@5Vdc. ), higher power and efficiency are obtained by increasing the output voltage, since the power losses of the switches and the PR remains constant (Ploss α I²). Also, at the same efficiency, higher output power is obtained which also validates that the efficiency is mainly depending on the current (omitting dielectric losses). For the analytical model, the conduction losses of the switches S1 and S2 are obtained by computing 0.5RdsonISeff² with ISeff the RMS current flowing through the switch. Regarding to diodes, their conduction losses are determined by computing IDVf, where ID is the mean current flowing through the diode and Vf the forward voltage. Furthermore, for the experimental results, the losses are calculated by using the experimental waveforms.

However, the PR loss are deduced by subtracting the losses of semiconductors from the total losses, since we do not have fully access to the current i (as we can measure i only during conduction stage).

For the simulation results, the losses are obtained directly from Spice model for each power components.

As shown in Figure IV.17, the estimated losses obtained from analytical model and simulation match with the experimental losses with a good accuracy for the semiconductor components. However, the PR provides higher losses in experimental results due to spurious modes excitation, which induces a significant amplitude increase of current I in the PR. Also, the effect of the spurious modes increases the losses in the semiconductors, however, as the losses in the switches are low the effect of spurious modes is barely noticeable.

IV.4.5 Parallelization of PRs

In this part, we have used 3-PRs connected in parallel in order to increase the output power and to illustrate the scaling of power with the global piezoelectric area. The PRs have the same geometry: a disk shape with a diameter of 25 mm and a thickness of 0.75mm. TABLE IV. [START_REF] Yang | A low-loss inductor structure and design guidelines for high-frequency applications[END_REF] shows the equivalent model parameters of each PR. A deviation of the resonant frequencies can be noted. However, at the same output power, the efficiency falls since there is a current exchange between PRs to balance the voltages between resonators leading to increase the losses by I².

Also, as we have 3-PRs, the equivalent parallel capacitance becomes three times higher 3CP, so, the circulating current increases as well. However, due to spurious modes, the inductive region is highly disturbed, which limits the operating frequency range, as shown in Figure IV.22. Therefore, the experimental results are provided near the resonant frequency where the output power is higher. given by the datasheet C-213 is 2500.

Futhermore, at this operating frequency, the circulating current becomes significant ( 𝑉 𝑖𝑛 +𝑉 𝑜𝑢𝑡 2 𝐶 𝑃 𝜔), which increases considerably the current amplitude I and then the losses (0.5Rp.I²). For instance, the circulationg current to charge/discharge the capacitance CP of PR3 is of 6.9 A for input-output voltage of 200-60 V. Nevertheless, a higher ouput power is reacheable at higher frequency, an output power of 175 W was reached, with an efficiency of 80 %. An improvement of the electrical contacts should be carried out to increase the quality factor of the setup.

In conclusion, in thickness mode, the thickness defines the operating frequency while the surface defines the maximal current amplitude value I, the maximal current density being about This comparison aims to determine the maximum reachable current density for each resonant mode, at same operating conditions Vin and Vout. Also, for achieving these current densities, the operating frequency for each vibration mode is selected to be near the resonant frequency. Indeed, Figure IV. [START_REF] Touhami | A New Topology of DC-DC Converter Based On Piezoelectric Resonator[END_REF] shows the measured waveforms for the radial and thickness modes, for an input-output voltage of 200-60 V.

For the radial mode at f = 91 kHz, we have measured an output power of 22 W, and an estimated current density of 0.2 A/cm² in the PR. For thickness mode that operates at 1 MHz, the output power measured is 100 W and a current density about 1 A/cm². From these results, increasing the frequency by a factor 10, the power and current density are increased by a factor 5.

It can be noted that for the waveforms measured in radial mode, the time-duration of the connected stages is dominant compared to the open circuit stages used for charging/discharging CP. One can deduce that the useful current amplitude is higher than the circulating current. In In this part, we propose a simple comparison between an inductor L (magnetic energy) and a PR having the same inductive value L between its resonant and anti-resonant frequencies.

Therefore, we compare the energy stored per unit volume between the mechanical (EPR) and magnetic storage (EL).

To do so, we have taken as an example a PR with diameter of 25 mm and thickness of 0.75 mm. In addition, we take the amplitude of the ac current for PR (I=1 A determined experimentally) and a resonant frequency around 100 kHz. This PR provides:

𝐿 = 1.1 mH, 𝐸 𝑃𝑅 = 1 2
𝐿 𝐼 𝑃𝑅 2 = 0.55 mJ and Volume = 0.368 cm 3 , 𝑅 𝑃 = 0.5 Ω Hence, we come up with a commercial inductor referenced by MSS1210-105KED (we try our best to keep the volume of L as low as possible while having a high Q factor) that gives the same energy stored and same inductance value L:

duration of the open stages becomes preponderant at low power rate. This reliance appears again in the output power vs. operating frequency relation.

The presented power converter performs a high efficiency, which illustrates the potential of this PR based converter. Indeed, this power converter exhibits an efficiency higher than 85 % over a wide range of power and voltage. Also, a peak output power of 50 W is reached for input-output voltage of 250-117 V with an efficiency up to 94 %. Moreover, a higher efficiency is expected by using a synchronous rectifier, as the main losses of the semiconductors are located in the diodes. However, the challenging part of synchronous rectifier is the control, which should be optimized to ensure ZVS operation.

A potential of increasing the output power is investigated by employing several parallel PRs or by increasing the operating frequency. Indeed, the use of parallel resonators is an easy way to increase the output power. However, a small dispersion between resonators parameters induces a circulating current between resonators to balance the voltages; this circulating current increases the losses, limiting thus the scaling of the output power and reducing the efficiency.

Operating at a higher frequency of 1 MHz has been demonstrated, it exhibits a higher output power, and a higher current and power density.

Indeed, at 1MHz operation, an output power exceeding 100 W was reached for a PR volume of 0.98 cm 3 , with an efficiency up to 80 %. However, a cooling system is required for longterm operation, since a temperature of 100°C is reached. Furthermore, the mechanical and electrical contact should be considered to reduce the electrical resistance of the connection and to reduce the impact on the mechanical vibrations (damping). Furthermore, we have shown the capability to increase the output power by increasing the area of the resonator.

Avoiding spurious modes remains a serious challenge for designing the PR. As illustrated, the spurious modes provide additional losses: additional high frequency current flows through the switches and some of energy is stored in another mechanical mode and is ultimately dissipated.

V.1 Introduction

As demonstrated in previous chapter, DC-DC converter based on piezoelectric resonator enable high efficiency and high power density without using any magnetic components.

Therefore, one challenge of such converter is the control, while using simple and low cost components. To our knowledge, the control of such converter remains poorly developed and little studied [START_REF] Pollet | A new inductorless DC-DC piezoelectric flyback converter[END_REF]. In order to be attractive for commercial applications by using a low-cost electronics solution increases the challenge.

Indeed, the conversion cycle using a PR as a storage element operates with six-switching

stages, alternating open-circuit stages and constant voltage stages, which is simple but requires a control to ensure:

• Zero voltage switching and soft charging,

• Energy and charge balance,

• Output voltage regulation.

Those criteria make the operation efficient, as it reduces the switching losses while it ensures the output voltage regulation. Therefore, considering these constraints and the need of managing 6 stages, the control becomes difficult as a whole.

Generally, the objective of the control when designing a dc-dc power converter is to regulate the output voltage to a desired constant value. Nevertheless, for a high efficiency operation, the control must ensure the zero-voltage switching of the semiconductors and, in addition in our case, the energy and charge balance, which is more difficult.

In this chapter, we present the design and the implementation of a cost effective control of a PR based DC-DC converter. This control is valid for any switching sequences. The control strategy is validated experimentally.

V.2 Issues of control

. [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF] shows the generic conversion cycle, while we identify the key elements of the control. In this figure, we described the optimal control that minimizes the amplitude of the current i. This control ensures:

• ZVS,

• Soft charging,

• Load regulation,

• Synchronization with the mechanical movement (zero current crossing detection).

V.2.1 ZVS operation

The switching losses are proportional to the frequency, the square of the voltage ∆V across the switches before closing, the capacitance of the transistor Coss, as given by (V.2.1). The frequency f can be selected in order to reduce the volume of passive components. Indeed, increasing the frequency allows to reduce the stored energy per cycle (i.e., Ecycle=P/f), that reduces the volume of the passive components. The Coss value depends on the technology of transistor (IGBT, MOSFET, HEMT), which is designed regarding the operating conditions V, f and P and the cost. Therefore, minimizing the switching losses can be done by minimizing the voltage across the switches named ΔV just before closing the switch (i.e., Psw α Δ𝑉 2 ). In both cases, the operating conversion is not optimal, since it induces a higher amplitude of current I if t2 < t2opt leading to higher losses and implying to oversize the PR to achieve the requirements of the conversion, or higher switching losses if t2 > t2opt. Therefore, we need to control t2 in order to let VP reaching Vzvst3 in t3 maximizing by that the efficiency and the power density.

β V b β V a β V c t 𝑇/2 𝑇 0 βV p β V zvst3

V.2.3 Synchronization with the current i

As mentioned previously, the operating frequency depends on the load: at high output power, the operating frequency gets closer to the resonant frequency; at lower power, the operating frequency approaches the anti-resonant frequency, as illustrated in Figure V.5. Thus, the control should take into consideration the fact that the frequency moves. In addition, the phase of current i should be known, as the switching sequences are implemented regarding the polarity of i as shown previously. Thus, the phase and the frequency should be determined/measured to implement the control, which acts like a vector control of a rotational electrical machine. In the above section, we present a few methods to measure the phase and frequency of the resonator. The following design of the converter and the feedback regulation are made for an input voltage of 120 V and an output voltage < 48 V, and output power up to 20 W.

V.3.1 Synchronization with PR's deformation

As the conversion cycle is composed of connected stages and open-circuit stages, current i can be measured directly only during the connected stages configurations. Therefore, current i cannot be sensed all over the resonant period, which makes it difficult to measure its phase and frequency. So, to determine these quantities, we can use one of the methods shown in TABLE

V.1. The advantages and drawbacks of each method are enumerated. Note: the phase and In the case of the switching sequence illustrated in Figure IV.6, we note that VP-is equal to Vout when i is positive (i.e., the diode D3 is turning-off), and negative when VP-is equal to zero (i.e., the diode D3 is turning-on). So, we can use the conduction state of D3 to determine the current zero crossing and then the phase of current i. Therefore, a single threshold

+ - R 𝑖 𝑒𝑎𝑠 = 𝑘 * 𝑖 𝑉 𝑒𝑎𝑠 = 𝑅 * 𝑖 𝑒𝑎𝑠 R 1 R 2 𝑉 𝑃 𝑑 𝑑𝑡 𝑠𝑖𝑔𝑛( 𝑑𝑉 𝑃 𝑑𝑡 )
Open stage :

V.3.4 Output voltage regulation

As illustrated in equation (V.2.2), the output power is mainly linked to the stored energy in the PR during the stage VP=Va (=Vin-Vout in this case), which is the time duration of [t3',t4].

From the generic representation, the instant t4 for the switching sequence {Va=Vin-Vout, Vb=Vout, Vc=-Vout } is written as a function of the electrical parameters (for lossless resonator: RP=0):

The output load is modeled as a resistor RL in parallel with an output capacitance Cout as illustrated in Figure V.12. We consider that Cout is sufficiently high for filtering the high frequency current ripple on the output voltage, and RL is depending on the desirable output power. Therefore, by using (V. synchronize the PWM of S1 and S2 with the current, at each period. Figure V. [START_REF] Kyaw | Fundamental examination of multiple potential passive component technologies for future power electronics[END_REF] shows the implemented blocks in Simulink, where each control loop is identified by its inputs and outputs.

V.5 Simulation Results

Once each block has been designed as illustrated in the previous section, we have implemented the control strategy for an input-output voltage of 120 -48 V and an output power of 10 W. Therefore, to increase the current amplitude i, and then the mechanical oscillation amplitude, the PR takes higher energy from the input source than the energy transferred to the output load.

So, the conversion cycle operates with a positive energy balance during this regime to increase the mechanical energy and so the current i amplitude and at the end the output power. Once, the amplitude of the current becomes significant, the resonator transfers energy to the load and the output voltage increases. Thus, the output voltage reaches the set value 48 V, and the converter operates with energy and charge balance, leading to a constant amplitude of the current in the LCR branch. to reduce the sensing noise. Also, buffer amplifiers are used for impedance matching. In addition, we used digital isolator ADUM1201 and isolated ADC AMC1204 in order to isolate the control circuit from the power converter. This reduces the EMI noise in the control circuit and it protects the circuit and the users. The sampling of the output voltage is achieved at 78 kHz with a resolution on 14 bits. The PWMs S1 and S2 are generated by the FPGA after calculating the duty cycles, and then transmitted to the isolated gate drivers SI8271. We measured on this board a power consumption of 1 W at 5 V voltage supply that includes also the power supply of the gate drivers operating at 100 kHz. This circuit is designed for a frequency operation around 100 kHz and an output dc-current of 1 A, the selected transistors are STD5NM50T4 for S1-S2 and the diodes are US1G for D1 to D4 in this prototype.

V.3 Experimental verification

The objective of this part is to validate the control strategy of the proposed dc-dc power converter: the transient response and the steady-state operation. Also, the robustness of the control to load variations is also tested.

𝑖 𝑜𝑢𝑡

In Figure V.27, we have instantly changed the load RL from 329 Ω to 177 Ω. Thus, the output power increases from 7 W to 13 W. As shown, before the load variation, the power converter operates under energy balance Ein1 = Eout1(+Eloss) at an output power about 7 W, which is noted by a constant amplitude of the current (sustainable regime). When a variation on the load is applied, the energy balance is not conserved, as the amplitude of the current i is not sufficient to maintain the output voltage to its set point, i.e. to reach the output power requirement.

Therefore, during this transient regime, the input energy is higher than the output one Ein1 > Eout1(+Eloss) at each resonant period, which increases the current i amplitude. Once, it reaches a sufficiently high value, the resonator can store sufficient energy to transfer the right power level to the output load. Therefore, a new energy balance Ein2 = Eout2(+Eloss) is established for the output power of 13 W, which leads to a constant amplitude of i (sustainable regime). This transient takes about 2 ms, enabling to regulate the output voltage with an overshoot not exceeding 10 V. This test validates the operating of the control strategy while the output power changes.

𝑖 𝑃

In Figure V.28, we have performed a load variation by increasing RL from 177 Ω to 329 Ω, in order to check if the control strategy is able to regulate the output voltage while the output power decreases. As shown, the output voltage is regulated after almost 3.5 ms. However, we can observe a current overshoots on iP, since S1 is not turned with ZVS. Indeed, these overshoots are due to the regulation of t2 that needs to be adjusted to ensure the soft charging of VP to reach Vin at t3, making it possible to turn-on S1 with ZVS. As shown, the ZVS of S1 is regulated after almost 7 ms.

Overall, the implemented control strategy enables to regulate the output voltage even when load variations occur.

V.3.4 Efficiency

𝑖 𝑃

the converter exhibits a good efficiency over the various operating points, higher than 90 % for most of the cases. In addition, it demonstrates the capability of the control strategy to operate at various gain levels.

V.4 Analyze and discussion

The proposed control strategy was able to regulate the output voltage at various operating conditions as demonstrated previously. Nevertheless, it should be noted that this control is sensitive to voltage measurement accuracy, which is needed for the control loop. Indeed, the sensing interface of the voltage is depending on the voltage divider and the pre-amplification and amplification stages, which should be well designed and calibrated. Also, the control of dcdc converter based on PR reveals to be more complex than the classical ones which operate at fixed frequency and with only two stages for most of them (one PWM having only two values and dead time). Indeed, to control a PR based converter, it requires frequency and phase tracking, two ZVS detection, a soft-charging regulation, and a load regulation. However, the implementation shows the feasibility of controlling such a converter by using comparators and simple control loops and low-cost controller while exhibiting a quite high performance in terms of response time and overshoot. Moreover, the experimental behavior was observed very close to the simulated one. High efficiency (higher than 90%) was obtained at various gain levels and power operating points for high to low voltage conversions. These satisfactory results give encouragements for further works on this new conversion principle.

V.5 Summary of the chapter

In this chapter, we have developed and designed a control strategy of a step-down dc-dc converter based on PR that operates on a six-stages conversion cycle. Energy balance, soft charging and ZVS were all considered in the development of the control law. The control strategy was firstly verified by simulation in transient and steady-state regimes. Then, we have designed the power and the control circuits for experimental checking for an input voltage of 120 V, an output voltage < 48V and an output power up to 16 W. The control strategy was synthetized by using VHDL language and then implemented in a FPGA. The experimental results have demonstrated the feasibility to regulate the output voltage of the dc-dc converter while ensuring the six-stages conversion cycle. Indeed, the control loop is able to regulate the converter in order to follow the output voltage set point at the starting of operation (transient regime), steady-state operation and when load variations are applied. This chapter was published in the EPE2020 conference, [START_REF] Touhami | Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezoelectric Resonator[END_REF]

VI.1 Conclusions

In this thesis, we focused on the development of novel PR-based power converters for high to low voltage and low power applications. Replacing the L-C resonant circuit in quasi-resonant or resonant DC-DC converters with PRs reduces the number of components. In addition, PRs offer a low-profile design and a high power density, which facilitates their miniaturization and integration. They can also offer low losses and high electromechanical coupling factor. However, due to the electrical model and peculiarities of piezoelectric resonators, we need to adapt the power conversion mechanism and the conversion cycle, in order to exploit the performances of these materials.

The interest of using piezoelectric material is demonstrated. The mechanism of the conversion is explained with a method of construction for any conversion cycle using sixswitching stages alternating between constant voltage stage and open circuit stage. The conversion cycle consists of taking energy from the input source, storing it as mechanical energy in a piezo material, then, transferring that energy in electrical form to the load. Therefore, during this cycle, the energy and charge are balanced, resulting in a sustained oscillation and low power loss. A generic analytical model was also presented to predict the timing of each stage and the amplitude of the current in the motional branch. The expression of the current enables a better understanding of the impact of the switching sequence, the properties of the material and the electric parameters of the converter like the input voltage, the output voltage and the load. Also, it enables a fast sizing of the power switches. A comparison between the various conversion switching sequences was conducted in order to identify the appropriate one for our application. This comparison was achieved over various operating conditions, and stated which switching sequence can minimize the constraints on the material and maximize efficiency as well. An actual prototype was built for experimental verification.

The experimental measurements exhibit a high efficiency and high power density, as for instance: a 100 W output power for a volume of PR of 1 cm 3 and an input to output voltage of 200 V-60 V was achieved in steady state (180 W in transient pic power).

A control strategy was developed which can be used for any switching sequence for any DC-DC converter based on PR, enabling energy/charge balance, soft-switching and softcharging. A control strategy was designed and implemented by considering our selected switching sequence and topology. The control was validated in simulation (SIMULINK). Then, we have designed an actual circuit and the control laws were implemented in FPGA. The control strategy exhibits a good performance in terms of transient time response (less 2 ms) and load perturbation rejection.

To sum up, the power conversion based on piezoelectric components gives a good promise in power electronics which aiming for the miniaturization, the integration on silicon, the high efficiency and the high power density. Nevertheless, several research subjects require more investigations and explorations for maturing this conversion technique.

VI.2 Perspectives and future challenges

The recent advancements have shown serious opportunities for further research in PR based power converters and in potential applications, as well as high voltage conversion or low voltage for integrated power supply. It requires supplementary investigation and optimization, which are presented below: Design and optimization of the PR materials for discrete as well as integrated applications is needed, for a better exploitation of their performances. Indeed, as the PRs are not extensively employed in power converters, the guideline of designing the PR is lacking. This guideline design should take into consideration the mechanical mounting, the fabrication process, reducing the effect of spurious modes, electrical connection (bonding, silver glue…) and the heat sink. Also, optimization considering an additional mass connected mechanically to the PR for a higher power density should be investigated as it may offer a bigger energy density by just adding an inert mass [START_REF] Bonavia | Augmented Piezoelectric Resonators for Power Conversion[END_REF]. Also, this guideline must be combined with the circuit design as the IC.

Further explorations in power converters, conversion cycles and control is needed, in order to cover all the possibilities of using PRs as storage components. In the end, we should be able to meet all the needs of power converters.. It is also possible to design multi-port converters (several inputs and/or outputs) that share the same piezoelectric resonator [START_REF] Despesse | Power converter[END_REF]. Therefore, the control and the power management can be developed as consequence. Also, using more than six-switching sequence enables to store additional energy in the PR or in mass (by exciting the material at a higher amplitude of oscillation than needful) and then transfer it immediately to the load when necessary [Patent 2], reducing by that way the time response of the system. Moreover, isolated magnetic-(and less) power converters were investigated in this thesis but not presented as it is subject of patents.

Figure I.

  Figure I.1. Power converter Schematic. .............................................................................

1 .

 1 Figure I.1. Power converter Schematic. .............................................................................

Figure I. 2 .

 2 Figure I.2. Power converter: design requirements. ............................................................

Figure II. 1 .

 1 Figure II.1. (a) Direct effect, (b) Reverse effect of piezoelectricity. ..................................

Figure II. 2 .

 2 Figure II.2. Field of application of the piezoelectric devices. ............................................

Figure II. 3 .

 3 Figure II.3. (a) Mechanical structure of the VC, (b) Resonant electrostatic model of the VC [32]. ..........................................................................................................................................

Figure II. 4 .

 4 Figure II.4. The topology of the converter based on VC [32]. ...........................................

Figure II. 5 .

 5 Figure II.5. Topology of a dc-dc converter using PR as storage component. ....................

Figure II. 6 .

 6 Figure II.6. Waveforms of the step-down conversion cycle 20-10 V [36]. .......................

Figure II. 7 .

 7 Figure II.7. Picture of circuit, [36]. ....................................................................................

Figure II. 8 .

 8 Figure II.8. Efficiency Vs Output Power, for Vin=10 V, and Vout=15, 20, 25, 30 V, [36].

Figure II. 9 .

 9 Figure II.9. (a) Converter topology, (b) picture of the prototype, [27]. .............................

Figure II. 10 .

 10 Figure II.10. Efficiency Vs Output power, for Vout=40 V and Vout/Vin=0.2 to 0.47[START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF]. .

Figure II. 11 .

 11 Figure II.11. Structure of control step-up converter based on PR for 10-20 V, [36]. ........

Figure II. 12 .

 12 Figure II.12.Efficiency Vs k²Q (thickness mode vibration), [41], APPENDIX A. ............

Figure II. 13 .

 13 Figure II.13. Hard PZT materials comparison: FOMAPD (Areal Power Density) Vs FOMM (loss ratio Pout/Ploss), [42]. ........................................................................................................

Figure III. 1 .

 1 Figure III.1. PR's: (a) electrical model based on Butterworth-Van Dyke (BVD) model [50], (b) impedance plot. ...................................................................................................................

Figure III. 2 .

 2 Figure III.2. PR's electrical connections: (a) Connected stage, (b) open-circuit stage. .....

Figure III. 3 .

 3 Figure III.3. Conversion cycle for: (a) Vc<Vb<Va, (b) Va<Vb<Vc .....................................

Figure III. 4 .

 4 Figure III.4. Generic conversion cycle. ..............................................................................

Figure III. 5 .

 5 Figure III.5. (a) Connected stage, (b) Open-circuit stage. ..................................................

Figure III. 6 .

 6 Figure III.6. VP vs Q diagram. ............................................................................................

Figure III. 7 .

 7 Figure III.7. Examples of switching sequences. .................................................................

Figure III. 8 .

 8 Figure III.8. Converter Topologies based on four switches [27], [36]. .............................. 67

Figure V. 8 .

 8 Figure V.8. Theoretical waveforms of the comparator used for phase/frequency tracking. ................................................................................................................................................

Figure V. 9 .

 9 Figure V.9. Theoretical waveforms of the comparator used for phase/frequency tracking. ................................................................................................................................................

Figure V. 10 .

 10 Figure V.10. Schematic diagram for the compactor at t=t3. .............................................

Figure V. 11 .

 11 Figure V.11. Operating principle of the regulation of t2 : (a) t2 < t2opt, (b) t2 > t2opt. .........

Figure V. 12 .

 12 Figure V.12. Output load model. ......................................................................................

Figure V. 13 .

 13 Figure V.13. Converter topology by using toolbox SimScape. .......................................

Figure V. 14 .

 14 Figure V.14. Loop control of the output voltage: Vout_ref = 48 V. ....................................

Figure V. 15 .

 15 Figure V.15. Soft-charging control for t2. ........................................................................

Figure V. 16 .

 16 Figure V.16. Close-loop control in Simulink: green-block for Vout control, yellow-block for soft-charging t=t2 and bleu-block for the synchronization with the current i. ........................

Figure V. 17 .

 17 Figure V.17. Simulated waveforms during transient-response for : Vin=120 V, Vout= 48 V and Pout=10 W. .......................................................................................................................

Figure V. 18 .

 18 Figure V.18. Steady-state waveforms for : Vin=120 V, Vout=48 V and Pout=10 W..........

Figure V. 19 .

 19 Figure V.19. Conception flow. .........................................................................................

Figure V. 20 .

 20 Figure V.20. Schematic block of the feedback control in ISE-Xilinx. ............................

Figure V. 21 :

 21 Figure V.21: Simplified synoptic of the control circuit. ..................................................

Figure V. 22 :

 22 Figure V.22: PR-based DC-DC circuit with the proposed control. .................................

Figure V. 23 :

 23 Figure V.23: Picture of the prototype circuits. .................................................................

Figure V. 24 :

 24 Figure V.24: Experimental Setup. ....................................................................................

Figure V. 25 :

 25 Figure V.25: Transient response for: Vin=120 V, Vout = 48 V and P =10 W. ..................

Figure V. 26 :

 26 Figure V.26: Steady-state regime for: Vin=120 V, Vout=48 V and P=10 W. ...................

Figure V. 27 :

 27 Figure V.27: Dynamic response for a power step from 7 W to 13 W. ............................

Figure V. 28 :

 28 Figure V.28: Dynamic response for a power step from 13 W to 7 W. ............................

Figure V. 29 :

 29 Figure V.29: Efficiency vs Output Power. .......................................................................

I. 1 I. 2 I. 3

 123 Scope ..................................................................................................... Background and motivations ............................................................. Thesis objectives .................................................................................. I.4 Thesis structure and content ..............................................................

Figure I. 1 .

 1 Figure I.1. Power converter Schematic.

Figure I. 2 .

 2 Figure I.2. Power converter: design requirements.

II. 1

 1 Introduction ......................................................................................... II.2 The piezoelectricity: history and applications ................................. II.3 Piezo-based power converters ............................................................ II.3.1 Mechanical Energy Conversion ........................................................................ II.3.2 Example of implementation of PR-based DC-DC converter ............................ II.4 Piezo-based power converters: projects and applications .............. II.4.1 Topology of power converters based on PR ..................................................... II.4.2 Close-loop Control ............................................................................................ II.4.3 Piezoelectric materials ...................................................................................... II.5 State of the art synthesis ..................................................................... II.6 Conclusions ..........................................................................................

FigureFigure II. 2 .

 2 Figure II.1. (a) Direct effect, (b) Reverse effect of piezoelectricity.

3 .Figure

 3 Figure II.3. (a) Mechanical structure of the VC, (b) Resonant electrostatic model of the VC [32].

Figure II. 4 .

 4 Figure II.4. The topology of the converter based on VC [32].

Figure II. 5

 5 Figure II.[START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF] shows the topology of the DC-DC converter that has been used for validation in[START_REF] Pollet | Convertisseurs DC-DC piézoélectrique avec stockage provisoire d'énergie sous forme mécanique[END_REF]. The circuit is composed of three switches {kin, kout, k3} and a PR that has been modeled for its motional part by a sinusoidal current source (assumed near the resonant frequency) in parallel with the intrinsic capacitance of CP. This circuit enables connecting the PR with the input source Vin, the output load Vout and Zero voltage (short-circuit).

Figure II. 5 .

 5 Figure II.5. Topology of a dc-dc converter using PR as storage component.

3 . 2 )Figure II. 6 .

 326 Figure II.6. Waveforms of the step-down conversion cycle 20-10 V [36].

Figure II. 7

 7 shows the prototype that has been used for experimental verification of power conversion based on PR. Also, it has been the first prototype used for experimental verification. This prototype exhibits a high efficiency (>90 %) for an input voltage 10 V, an output voltage range 15-30 V, and an output power up to 1.5 W. This topology operates a wide voltage range conversion, but all the energy transferred to the load is stored in the PR, which increases the constraints on the PR. The authors of[START_REF] Boles | Enumeration and Analysis of DC-DC Converter Implementations Based on Piezoelectric Resonators[END_REF] presented a step-down dc-dc converter based on a PR. This topology is composed of four switches. The switching sequence of the proposed converter uses {Vin-Vout, 0V, Vout}, therefore, a part of the energy is transferred directly to the load during the stage Vin-Vout, while a second part is stored in the PR and then transferred during the stage Vout.

Figure II. 7 .

 7 Figure II.7. Picture of circuit, [36].

Figure II. 8 .Figure II. 10

 810 Figure II.8. Efficiency Vs Output Power, for Vin=10 V, and Vout=15, 20, 25, 30 V, [36].

Figure II. 9 .

 9 Figure II.9. (a) Converter topology, (b) picture of the prototype, [27].

Figure II. 10 .

 10 Figure II.10. Efficiency Vs Output power, for Vout=40 V and Vout/Vin=0.2 to 0.47 [27].

Figure II. 11 .

 11 Figure II.11. Structure of control step-up converter based on PR for 10-20 V, [36].

  Figure II.13. Hard PZT materials comparison: FOMAPD (Areal Power Density) Vs FOMM (loss ratio Pout/Ploss), [42].

III. 1

 1 Introduction ......................................................................................... 47 III.2 Operating Principle ............................................................................ 47 III.2.1 Equivalent circuit model of PR ......................................................................... 47 III.2.2 Operational principle ........................................................................................ 49 III.2.3 Energy approach ............................................................................................... 54 III.3 Steady state analytical model ............................................................. 56 III.3.1 Charge transfer .................................................................................................. 56 III.3.2 Energy transfer .................................................................................................. 57 III.3.3 Load current ...................................................................................................... 58 III.3.4 Estimation of the amplitude of the current i ..................................................... 59 III.4 Converter topologies and conversion cycles ..................................... 61 III.4.1 Switching Sequences Implementation .............................................................. 61 III.4.2 Piezo-Based dc-dc Converters: converter sequences ........................................ 66 III.4.3 Example of implementation of a switching sequence ....................................... 69 III.5 Simulation to validate the operating principle ................................. 71 III.6 Efficiency Estimation .......................................................................... 73 III.7 Analysis and Discussion ...................................................................... 77 III.8 Summary of the chapter ..................................................................... 78 III.1 Introduction

III. 2

 2 Operating Principle III.2.1 Equivalent circuit model of PR Piezo-materials behave like dielectrics but also like active materials able to convert electrical energy into mechanical energy and vice-versa. Therefore, an equivalent circuit can represent both effects. The two electrodes, in relation with the dielectric property of the plate acts as a natural capacitance (or called blocked capacitance) named CP while an L-C-RP series electrical circuit models the electro-mechanical property of the PR close to the resonance.

Figure III. 2 .

 2 Figure III.2.a shows the equivalent circuit based on Butterworth-Van Dyke (BVD) model.

  The Figure II.1b shows the impedance of the PR between the resonant frequency fr (L-C) (and the anti-resonant frequency far (L-(C//CP)) Once the PR is placed in the converter, we define two operating configurations: one where the PR is connected to a constant voltage source/load named connected stage and a second configuration where the PR is placed in open-circuit (isolated) named open-circuit stage.

Figure III. 2

 2 Figure III.2 shows these both configurations of the PR, where:

Figure III. 2 .

 2 Figure III.2. PR's electrical connections: (a) Connected stage, (b) open-circuit stage.

Figure III. 3a .

 3a Figure III.3a. In the second case with Va < Vb, the classification becomes Va<Vb<Vc. Since Va<Vc in this case, the current i should be negative in the half period where Va and Vc are located to charge the capacitance CP and thus going from Va to Vc during the open-stage between Va and Vc, as illustrated Figure III.3b. These two cases are represented in Figure III.4.

9 )

 9 The electrical connections to the constant voltage values are illustrated in Figure III.5, where the LC branch is modeled by a sinusoidal current source. Thus, the operating cycle is divided in six-stages as follows: t0-t1: All switches are OFF during this stage, the resonant current β i charges the capacitance CP (considering the sign of current in Figure III.4), so, the voltage βVP goes from βVc to βVb.t1-t2:The PR is connected to βVb . So, the PR exchanges electric charge under βVb : Qb. t2-t3: Again, all switches are OFF and PR is isolated. The voltage βVP goes from βVb to βVzvst3 by discharging the capacitance CP with current βi, since βi is negative.

Figure III. 4 .

 4 Figure III.4. Generic conversion cycle.

Figure III. 5 .

 5 Figure III.5. (a) Connected stage, (b) Open-circuit stage.

  conversion cycle, Equation (III.2.10) gives the charge balance and equation (III.2.10) gives the energy balance without considering the losses in the PR and the switches. Thus, the conversion cycle can be represented in a diagram VP vs Q, where the open-circuit stage operates with constant charge (Q=constant), while connected stage operates at constant voltage (VP=constant).

Figure III. 6

 6 Figure III.6 illustrates VP vs Q for the switching sequence illustrated in Figure III.4. As shown, the diagram alternates between constant voltage (horizontal displacement) and constant charge stage (vertical displacement). From this diagram, the charge exchanged during each connected stage are deduced by (III.2.12)-(III.2.14), and the energy exchanged during each constant voltage stage is represented by the area under the curve. The cycle is closed, which means that the charge are balanced. The global energy exchanged during one cycle is the area surrounded, the area surrounded in one direction is positive and the area surrounded in the opposite direction is negative. Here, the positive area in orange is equal to the negative area in blue and purple, meaning a zero global energy exchange respecting thus the energy balance.

Figure III. 6 .

 6 Figure III.6. VP vs Q diagram.

3

 3 Steady state analytical modelIn this part, we propose an analytical model to calculate the time-duration of each stage and to find the amplitude of current I for any six-switching sequences to fit the requirements in terms of the output power, and also to establish the control laws by identifying the key parameters to control. The current amplitude I is a main sizing parameter for the PR. That amplitude depends on the switching sequences, so, having I as a function of the switching sequence and the operating conditions (Vin, Vout and Pout) allows to design the PR. To do so, we propose a method to evaluate I over the various switching sequences and operating points.III.3.1 Charge transferFor simplification, i is considered as a sinusoidal waveform. This approximation will be validated in the simulation part. The charge exchanged during each constant voltage stage is estimated by integrating the current i (III.2.2), as given in (III.3.1)-(III.3.3), where 𝛾 𝑖 = cos 𝜔𝑡 𝑖 . Thus, the charge depends only on the time duration of the connected stage and the amplitude of I.

4 𝜔)

 4 the conversion cycle can be plotted by estimating the transition time between each stage using (III.3.22)-(III.3.30). These equations show that the instants are mainly linked to the operating conditions {Vin, Vout, Pout, f and Cp}. For cycles defined under charge and energy balance conditions, and for conversion cycle based on six stages, the output load is regulated by acting on the instant t4= arccos ( 𝛾 because this instant controls the charge Qa while the others instants are determined to ensure the ZVS operation and soft charging of CP. Therefore, the equation (III.3.27) can be used as a transfer function to

𝜋 2 𝐼

 2 𝑜𝑢𝑡 (omitting the circulating current) which corresponds to the average value of a rectified sine wave. It is obtained while the switching sequence ensures multiple connections with the output load during each constant voltage stages. It is for example the sequence {Vin-Vout, Vout, -Vout}. This switching sequence

Figure III. 7 • 3 -

 73 Figure III.7 shows some examples of switching sequences for step-down converters and step-up converters. For instance, the conversion cycle (A) shown in Figure III.7 is suitable for low gain ratio conversion 2Vout < Vin, and it enables a direct energy transfer thanks to the stage Va=Vin-Vout allowing by that to reduce the energy stored in the PR.

Figure III. 9

 9 Figure III.9 shows the topology derived from the topology shown in Figure III.8h, which ensures the switching sequence {Vin, 0, Vout} with g=Vout/Vin < 1. The derived topology allows to modify the previous switching sequence into {Vin-Vout, -Vout, Vout}, but it needs two additional switches S4 and S5. By increasing the number of connections with the load, this switching sequence reduces the current I and then the energy stored in the PR and it maximize the charge transfer.

Figure III. 8 .

 8 Figure III.8. Converter Topologies based on four switches [27], [36].

FigureFigure III. 10 .

 10 Figure III.10a shows the steady-state operation of the dc-dc converter based on the PR for the topology shown in Figure III.9 and for the switching sequence {Vin-Vout, Vout, -Vout}. As noted before, the current in the LCR branch is assumed sinusoidal, since the PR operates in the inductive region and has a high quality factor.

Figure III. 11 .III. 4 .

 114 Figure III.11. Circuit configurations.

Figure III. 13

 13 Figure III.13 shows the voltage waveforms of the PR (Vp and its terminals Vp+ and Vp-) and the current i in the LCR branch, the simulation is done for a 120 V to 40 V step-down conversion and for an output power of 5 W. The steady state is reached after 10 periods. The computed instants {t0…t6} are firstly calculated from equations (III.3.22)-(III.3.30) and for an operating frequency of 98 kHz, which are tuned in the simulation since the proposed model did not consider the parasitic capacitances of switches in the calculation of the instants. As shown in Figure III.13, the illustrated waveforms matches the theoretical waveforms presented in Figure III.10b. Indeed, the simulation waveforms exhibit the connected stages and the opencircuit stages, and the switching sequence transition times {t0…t6} are depicted. Also, the sinusoidal waveform approximation of the current i established previously is verified by the simulation. The waveforms reveal that the ZVS of S2 is not achieved due to the parasitic capacitance of S3 (or D2), however, we propose a solution to overcome that limitation detailed in APPENDIX C.

Figure III. 12 .

 12 Figure III.12. Simulation circuit.

Figure III. 13 .

 13 Figure III.13. Simulated waveforms in the time domain for the switching sequence : {Vin-Vout, Vout, -Vout}: Vin = 120 V, Vout = 40 V, Pout= 5 W and 𝜂 = 99.1 %.

Figure III. 14 5 Figure III. 14 .

 14514 Figure III.[START_REF] Chen | ZVS considerations for a phase-lock control dc/dc converter with piezoelectric transformer[END_REF] shows the evolution of the efficiency as a function of the output power for g = 0.083, 0.25 and 0.5 for input voltage of 120 V and for topologies adapted to these gain values enumerated in TABLE III.2. In this figure, we note that for low gain, the switching sequence {Vin-Vout, Vout, -Vout} offers the highest efficiency over the output power range compared to the other switching sequences, since this stage sequence yields a maximum of connected stage to the load, reducing the amplitude of current I and then reducing the losses.

Figure III. 15 .

 15 Figure III.15. Efficiency Vs Gain for Pout = 15 W and Vin = 120 V.

➢➢

  Figure IV.1. (a) The conversion cycle {Vin-Vout, Vout, -Vout}, (b) the proposed dc-dc topology.

Figure IV. 3 .Figure IV. 3 .

 33 Figure IV.3. Equivalent BVD circuit for: (a) Radial mode vibration (b) thickness mode vibration, [35].

FigureFigure IV. 4 .

 4 Figure IV.4b shows the impedance of the PR measured by using the impedance analyzer Wayne Kerr 6500B. As we operate in radial mode, the electrical contact may disturb the resonance. So, to reduce the impact of the electric connections, we have used the setup shown in Figure IV.4a. This setup does not require the soldering (avoiding heat soldering) and the connections are taken at the center of the resonator, at the neutral axe. As shown, we have used a spring contact from the top with a low pressure applied to not disturb the electric model.

IV. 3

 3 Circuit Design To validate the operating principle and to quantify the performances of the dc-dc converter based on PR, an actual circuit is proposed in this part. The switches S3 to S6 shown in Figure IV.1b operate as a synchronous rectifier which are suited for reducing the losses. So, these switches can be replaced by diodes. Figure IV.5 shows the converter topology where the switches S3 to S6 are replaced by D1 to D4. This topology enables reducing the number of controlled switches, which simplifies the experimental verification, as only two command signals are required but that degrades the efficiency.

Figure IV. 5 .

 5 Figure IV.5. The converter topology.

Figure IV. 7 .

 7 Figure IV.7. Photograph of the prototype.

  1 A. Vf = 485mV @ T = 25 °C Gate driver SI8721 VGS = +6 V/-3 V Rgon/off = 10 Ω

Figure IV. 8 .

 8 Figure IV.8. The duty cycles of S1 and S2 are generated by using the signal generator AFG 3011C. These duty cycles are pre-calculated (by using the analytical model shown in the previous chapter) at the central frequency, and then tuned while varying the frequency between the resonant and anti-resonant frequency. As illustrated in Figure IV.8, the efficiency is measured between the output and the input by using two-power analyzers HMC8015, without taking into consideration the power consumption of the gate drivers that was estimated at 0.5 W @ 100kHz.

Figure IV. 8 .

 8 Figure IV.8. Setup used for the experimental validation.

Figure IV. 9

 9 Figure IV.9 validates the operating principle of the dc-dc converter based on PR presented in the previous chapter. Indeed, Figure IV.9a-c illustrates the waveforms evolution of the voltage {Vp, Vp+, Vp-} and the current iP, for an input voltage Vin = 100, 150 and 200 V, a constant output voltage Vout = 48 V and a power Pout =10 W.

Figure

  Figure IV.9b to Figure IV.9c. Furthermore, we note that the time duration of the connected stage decreases by increasing the input voltage, while the open-stage duration increases. Indeed, the voltage needs to go from Vin to -Vout before going back to Vout. So, by increasing Vin, the time duration of open-stages increases to let VP evolving by charging and discharging the capacitance CP.

  Figure IV.10 shows the evolution of Vp, Vp+, Vp-and ip while the input voltage Vin = 200 V and the output voltage Vout = 66.7 V (G = 0.3). The output power varies from 6 to 16 W (Figure IV.10a to Figure IV.10c).

Figure IV. 10 .

 10 Figure IV.10. Experimental waveforms in time-domain for Vin = 200 V, Vout = 66.7 V and for output power: (a) Pout =6 W, (b) Pout =10 W, (c) Pout =16 W.

Figure IV. 11

 11 Figure IV.11 shows the output power as a function of the frequency over various gain

  Indeed, for low power operation, the time duration of open stages is preponderant, which means that the frequency is defined by L-(C//CP) for major part of conversion cycle. Although, by increasing the power conversion, the time duration of connected stage becomes preponderant leading to operating frequency near the resonant frequency defined by L-C.

Figure IV. 11 .

 11 Figure IV.11. Output Power vs Frequency, for Vin = 150 V and G=0.4, 0.33, 0.25 and 0.1667.

Figure IV. 12 2 (

 122 Figure IV.12 illustrates the evolution of frequency as a function of the voltage gain for output voltage Vout = 48 V. As shown, by increasing the gain, the operating frequency decreases leading to high amplitude of I for a constant load (Pout and Vout are constants). Indeed, by increasing the gap between the input and output voltages, the amplitude of the current Icirculating required to charge/discharge CP increases as well (IV.4.2). So, the proportion of open-circuit time increases and making the frequency approaching the anti-resonant frequency.

Figure IV. 12 .

 12 Figure IV.12. Gain Vs Frequency for Pout = 2 to 10 W.

Figure IV. 13 to

 13 Figure IV.13 to Figure IV.15 show the efficiency as a function of the output power over various gain levels for g < 0.5. The efficiency of the converter increases with the gain ratio and the input voltage (IV.4.3), even though the circulating current increases because it remains very low compared to the useful current. This leads to an efficiency which is mainly related to the current.

Figure IV. 13 .

 13 Figure IV.13. Efficiency vs output power for gain G = 0.48 and for different input voltages.

Figure IV. 14 .

 14 Figure IV.14. Efficiency vs output power for gain G = 0.33 and for different input voltages.

Figure IV. 15 .

 15 Figure IV.15. Efficiency vs output power for gain G = 0.167 and for different input voltages.

Figure IV. 16

 16 Figure IV.[START_REF] Kyaw | Fundamental examination of multiple potential passive component technologies for future power electronics[END_REF] illustrates the evolution of the efficiency as a function of the power, for an output voltage of 48 V, for various gain ratios. As shown, by decreasing the gain (i.e., increasing the input voltage in this case) while maintaining the output voltage constant, the efficiency decreases, as the part of the circulating current Icirculating increases with (Vin+Vout) leading to higher amplitude of the current I at same output power, leading to increase the loss by I².

Figure IV. 17 .

 17 Figure IV.17. Loss distribution for: Vin = 250 V, g = 0.33 and Pout = 25 W ± 10 %.

Figure IV. 18

 18 Figure IV.[START_REF]IEEE standard on piezoelectricity[END_REF] shows the thermal image of the prototype at full amplitude of current I (I ≈ 1 A), where the hottest areas show the components having the highest losses. This thermal image is obtained after we have blacked the circuit for having emissivity near 1, and then inserted in box at room temperature and without cooling.

Figure IV. 18 .

 18 Figure IV.18. Thermal image of the prototype at Vin=250 V, g=0.33 and Pout= 25 W ± 10 % and I ≈1 A, Δ𝑇 = 28 °𝐶.

Figure IV. 19

 19 Figure IV.[START_REF] Rosen | Electromechanical transducer[END_REF] shows the waveforms of the voltage across PRs and the currents flowing through the PRs. As expected, higher output power is achieved by increasing the number of PR and then the global piezoelectric area of the PR. A power of 25 W is reached by using 2 PRs, since by using a single PR only 16 W is reached.

Figure IV. 19 .

 19 Figure IV.19. Experimental waveforms of 2-PR for: Vin=120V, Vout=48V and Pout=25 W.

Figure IV. 20 .

 20 Figure IV.20. Experimental waveforms of 3-PR for: Vin=120V, Vout=48V and Pout=38 W.

Figure IV. 21 .IV. 4 . 6

 2146 Figure IV.21. Efficiency Vs. Output power for Vin=120 V and Vout= 48 V.

Figure IV. 22 .

 22 Figure IV.22. Impedance plot in inductive region for the resonator with D=25mm and Th=2mm.

Figure IV. 23 Figure IV. 23 . 5 AFigure IV. 24 . 5 . 7 .

 232352457 Figure IV.23 shows the waveforms of the PR1's voltage and current in the thickness mode vibration. As illustrated, the switching sequence used is {Vin-Vout, Vout, -Vout}, the soft charging stages and the connected stages at 1 MHz operation appear clearly. The current amplitude is estimated to 1.41 A by using (IV.4.4), where the useful current is 0.785 A and the circulating current part is 0.633 A. This current amplitude of I=1.41 A reached near the resonant frequency, is close to the maximum current reachable with this material (C-213) @ 1MHz, it corresponds to a current density of 1.2 A/cm². The circulating current Icirculating increases linearly with frequency, as shown in I.4.2. At high frequency operation, this part becomes significant, limiting the useful current Iuseful part and then the output current.

Figure IV. 25 .

 25 Figure IV.25. Efficiency Vs. Output power for Vin=200 V and Vout=60 V.

  Figure IV.26.Waveform of voltage and current of PR near resonant frequency for: (a) radial mode, (b) thickness mode.

Figure V. 1 .

 1 Figure V.1. Generic conversion cycle.

In Figure V. 2 ,Figure V. 2 .

 22 Figure V.2. No ZVS operation: (a) t1> t1opt, (b) t1< t1opt.

Δ𝑉Figure V. 3 Figure V. 3 .

 33 Figure V.3 shows the waveform of voltage VP during the open stage between Vb and Va,where the soft charging is not ensured leading to two cases:

Figure V. 4 .

 4 Figure V.4. Reverse power flow during the stage Vin+Vf for t2 < t2opt.

Figure V. 5 .•

 5 Figure V.5. Power Vs frequency. .

Figure V. 7 .

 7 Figure V.7. Waveforms in time-domain for Vin =120 V and Vout = 48 V.

  𝑃+ > (𝑉 𝑖𝑛 -𝑉 𝑜𝑓𝑓𝑠𝑒𝑡 ) 𝑎𝑡 𝑡 = 𝑡 3 → 𝑡 2 ((𝑛 + 1)𝑇) = 𝑡 2 (𝑛𝑇) + Δ𝑡 𝑖𝑓 𝑉 𝑃+ < (𝑉 𝑖𝑛 -𝑉 𝑜𝑓𝑓𝑠𝑒𝑡 ) 𝑎𝑡 𝑡 = 𝑡 3 → 𝑡 2 ((𝑛 + 1)𝑇) = 𝑡 2 (𝑛𝑇) -Δ𝑡 (V.3.1) 𝑉 𝑜𝑓𝑓𝑠𝑒𝑡 is the precision of the controller, and should be as small as possible since the switching losses of S1 are depending on this value (i.e., Psw(S1)=1/2CissVoffset²f), as illustrated in Figure V.11. Δ𝑡 is the incremental step of t2. The implementation of this regulation is simple but it can take several periods to converge depending on Δ𝑡 value.

Figure V. 10 .

 10 Figure V.10. Schematic diagram for the comparator at t=t3.

)

  Figure V.13 shows the dc-dc converter and the PR's model by using the toolbox of Simulink SimScape, the output load used is modeled by a (RL//Cout) circuit. The key electrical parameters are shown inTABLE V.2.

Figure V. 13 .

 13 Figure V.13. Converter topology by using toolbox SimScape.

Figure V. 14 .

 14 Figure V.14. Loop control of the output voltage: Vout_ref = 48 V.

Figure V. 16 .

 16 Figure V.16. Close-loop control in Simulink: green-block for Vout control, yellow-block for soft-charging t=t2 and bleu-block for the synchronization with the current i.

Figure V. 16

 16 shows the close-loop control implemented in Simulink. The pink block contains the DC-DC converter shown Figure V.13, the blue block enables the synchronization with the current I, the yellow-block contains the regulation of the soft charging by controlling t2 and which generates the pulse of S2, the green-block is the block for regulation of the output voltage to it set point through t4.

Figure V. 17

 17 Figure V.[START_REF] Hernandez | Etude et réalisation d'un convertisseur AC/DC Buck Boost réversible à haut rendement pour alimentation de secours[END_REF] shows the simulated waveforms during the transient mode. It shows a good regulation of the output voltage to 48 V, with an overshoot not exceeding 7 V. At the beginning, we can observe a delay of 0.58 ms during which the output voltage is close to 6 V. This delay is due to the saturation of the control's parameters. This delay can be avoided by using initial conditions close to the steady state operation. During the transient mode, the amplitude of current i is not sufficiently high to charge and discharge the capacitance CP, since the extrema of VP (low value -Vout) is not reached, no stage VP = -Vout is visible as observed in Figure V.17.

Figure V. 18

 18 Figure V.18 displays the waveforms of the converter during the steady-state operation.Clearly, the VP voltage waveform shows that the conversion cycle of our topology fits to the theoretical prediction, where the connected and the open-circuit stages can be observed. The mean value of the output voltage is 48 V, the peak-to-peak value of the output voltage ripple is 180 mV.

Figure V. 17 .

 17 Figure V.17. Simulated waveforms during transient-response for : Vin=120 V, Vout= 48 V and Pout=10 W.

V. 1

 1 Figure V.18. Steady-state waveforms for : Vin=120 V, Vout=48 V and Pout=10 W.

Figure V. 21 :

 21 Figure V.21: Simplified synoptic of the control circuit.

Figure V. 22

 22 Figure V.22 shows the circuit with the proposed control schematic. It illustrates an overview of the control strategy. The various measurements used for generating the PWMs are illustrated. A state machine implemented in FPGA is used for gererating the PWMs regarding to the timing of the conversion cyle.

Figure V. 23

 23 Figure V.23 shows the designed circuits to validate the control feedback. The voltages are sensed by using a voltage divider. These measurements are filtered through first order RC filters

Figure V. 22 :

 22 Figure V.22: PR-based DC-DC circuit with the proposed control.

Figure V. 23 :

 23 Figure V.23: Picture of the prototype circuits.

Figure V. 24

 24 shows the setup used to validate the control strategy. The input voltage is generated by a dc source, while the output voltage is emulated by the load resistor RL of 230 Ω for an output power of 10 W at Vout = 48 V. Moreover, parallel resistors are added to emulate load variations by connecting and disconnecting them. The input and the output powers are measured to determine the efficiency of the converter. The FPGA is programmed by using ISE 14.1 with JTAG connector.

Figure V. 24 :•

 24 Figure V.24: Experimental Setup.

Figure V. 25 :Figure V. 26

 2526 Figure V.25: Transient response for: Vin=120 V, Vout = 48 V and P =10 W.

Figure V. 26 :

 26 Figure V.26: Steady-state regime for: Vin=120 V, Vout=48 V and P=10 W.

Figure V. 27 :

 27 Figure V.27: Dynamic response for a power step from 7 W to 13 W.

Figure V. 29

 29 Figure V.[START_REF] Braun | Optimized Resonators for Piezoelectric Power Conversion[END_REF] illustrates the efficiency of the converter as a function of the output power over various gain levels. It should be noted that the driving power (estimated to 1 W) is not considered in the efficiency calculation since the control board is overdesigned. As illustrated,

Figure V. 29 :

 29 Figure V.29: Efficiency vs Output Power.

  entitled: Implementation of Control Strategy for Step-down DC-DC Converter Based on Piezoelectric Resonator, and presented in a talk. This published paper discusses the closed-loop control of a DC-DC converter based on a piezoelectric resonator, which demonstrates experimentally the first implementation of a control for this type of converter.

TABLE II .

 II 1.Relevant references arranged by subjects and research teams ...................... TABLE III.1. Electrical circuit and the corresponding mechanical circuit........................ TABLE III.2. Switching Sequences and Current Expressions. .......................................... TABLE III.3. Examples of Switching Sequences Used with the Proposed Topology. ..... TABLE III.4. PR's circuit C-213 from Fuji .......................................................................

TABLE IV .

 IV 1. BVDM model parameters of the PR in radial mode from Fuji Ceramics C-213. ...........................................................................................................................................TABLE IV.2. Electrical constraints of switches. ............................................................... TABLE IV.3. Key part components. .................................................................................. TABLE IV.4. Equivalent model of 3-PRs in radial mode vibration. ............................... TABLE IV.5. BVD circuit of 3-PRs in thickness mode vibration for C-213. ................. TABLE IV.6. BVD circuit of a PR with D =25 mm and th =2 mm. ...............................

TABLE IV .

 IV [START_REF] Makowski | Performance limits of switched capacitor dc-dc converters[END_REF]. Current Vs Frequency: Vin=200 V, Vout= 60 V and CP=3.3nF. ................

TABLE V .

 V [START_REF]DC-DC converter Market Size Share Global forecast to[END_REF]. Techniques of sensing the current for the phase/frequency tracking. ........

TABLE V .

 V 2. Key parameters. ..........................................................................................

TABLE V .

 V 3. Control parameters. ....................................................................................

	Y D List of symbols Elastic modulus at constant displacement. Y E Elastic modulus at constant electric field.
	C 𝜌	Motional capacitance in the electrical equivalent circuit. Density of the material kg/m 3 .
	CP 𝜀 0	Static capacitance in the equivalent circuit. Electric constant.
	D d33 𝜀 33 𝑠	Diameter. Piezoelectric coefficient where the polarization face the electrodes Permittivity at constant strain where the polarization is longitudinal to the
	far	The ani resonant frequency in the equivalent circuit. electrode.
	fr 𝜔	The resonant frequency in the equivalent circuit. Radial frequency rad/s.
	G	The conversion gain between the input and the output voltage.
	I	The amplitude of the current in the motional branch in the equivalent circuit.
	Iout	The average value of the output current.
	i	The current of the equivalent circuit.
	iin	The input current.
	iout	The input current
	iP	The current out of the resonator.
	k	Electromechanical coupling factor.
	k31	Electromechanical coupling factor, where the polarization is normal to the
		electrode.
	k33	Electromechanical coupling factor, where the polarization is orthogonal to the
		electrode.
	L	Dynamic inductance in the equivalent circuit.
	NP	Constant frequency for a disk in radial mode vibration.
	Nt	Constant frequency for a disk in thickness mode vibration.
	m	Mass of the material in kg.
	𝑃 ⃗	Polarization vector.
	Pout	The mean output power.
	PR	Piezoelectric Resonator.
	PT	Piezoelectric Transformer.
	Q	The quality factor.
	RP	The motional resistance illustrating the dumping losses.
	th	Thickness of the material.
	VC	Variable capacitance.
	Vin	The input voltage of an electrical source.
	Vout	The output voltage.
	VP	The voltage across the piezoelectric resonator.

TABLE II

 II

			[Patent 6]
		PT-based converters		[43]
		Variable frequency control (optimal control)	[36], [38], [Patent 2]	[37]
	Control Methods	Fixed frequency control: reactive energy	[Patent 2]
		Burst modulation	[Patent 2]		[39]
		Design and fabrication	[41]	[48]	[29]
	Materials	Materials and limits	[41]	[42]	[29]
		"Augmented Piezo''	[Patent 2]	[48]
		Φ2 inverter: L-Piezo Class
		(suppression of the second	[45], [49]		[46]
	VHF-filter	harmonic)
		Output Filter : Class E			[47]

.1.Relevant references arranged by subjects and research teams Subject CEA-Leti MIT Stanford Topology PR-based converters: Non-Isolated [24], [25], [

26], [28], [31] [27]

PRs-based converters: Isolated [31], [Patent 3-

TABLE III .

 III [START_REF]DC-DC converter Market Size Share Global forecast to[END_REF]. Electrical circuit and the corresponding mechanical circuit.

	Electrical system	Voltage	Current	Charge	Electrical resistance	Inductance Capacitance
		V	I	q	RP	L	C
	Mechanical system	Force	Velocity	Displacement Mechanical resistance	Mass	Compliance

  to two voltage levels named Va and Vb; where Va and Vb can be the input or the output voltages or composed of input and output voltages combinations. In addition, open-circuit is used to let the voltage level of VP evolves (Va➔ Vb or Vb➔ Va) in order to turn-on switches in ZVS mode.

By ensuring the energy balance, the amplitude of i is constant, leading to a sustained oscillation.

With charge balance, the voltage VP is capped between a minimum and maximum value throughout of a period of conversion. Equations (III.2.3) and (III.2.4) give the energy and

•

  Zero voltage Switching thanks to soft-charging.

	𝑄 𝑎 = 𝑄 𝑏	(III.2.3)
	𝑉 𝑎 𝑄 𝑎 = 𝑉 𝑏 𝑄 𝑏	(III.2.4)

Considering (III.2.3) and (III.2.4), it induces to Va= Vb. In this case, the possible values for the voltages couples {Va, Vb} are limited to:

•

  The constant voltage stages Va and Vb are defined to occur on opposite polarities of the current i. • The voltage level Vc is defined after Va occur and at the same current i polarity as Va, as shown in Figure III.3. It results thus that Qa and Qc have the same polarity. states that if Va > Vb then Vb > Vc, and if Va < Vb then Vb< Vc, because Qa and Qc have the same polarity by definition. In case of Va > Vb, we obtain the following classification: Vc<Vb<Va. Since Va>Vc, the current i should be positive in the half period where Va and Vc are located, because we need to discharge the capacitance CP (according to (III.2.1)) for going from Va to Vc during the open-stage between Va and Vc, as shown in

	Then, the energy and charge balance become:
	𝑄 𝑎 + 𝑄 𝑏 + 𝑄 𝑐 = 0	(III.2.5)
	𝑉

𝑎 𝑄 𝑎 + 𝑉 𝑏 𝑄 𝑏 + 𝑄 𝑐 𝑉 𝑐 = 0 (III.2.6)

We can remove Qb in (III.2.6) by using (III.2.5), the energy balance can be re-written by:

The equation (III.2.7)

  𝑄 𝑎 + 𝑄 𝑏 + 𝑄 𝑐 = 0 (III.2.10) 𝛽 𝑉 𝑎 𝑄 𝑎 + 𝛽𝑉 𝑏 𝑄 𝑏 + 𝛽𝑄 𝑐 𝑉 𝑐 = 0 (III.2.11)

	t 3	𝑉 𝑣𝑠𝑡3
	t 3 '	𝛽𝑉 𝑎	t 4
	V p (V)	𝛽𝑉 𝑎 𝑄 𝑎
	t 2			𝛽𝑉 𝑏	t 1
				𝛽𝑉 𝑏 𝑄 𝑏
				𝛽𝑉 𝑐 𝑄 𝑐	Q (C)
			t 5	𝑉 𝑐	t 6
		𝑄 𝑎 = 𝑞 4 -𝑞 3′	(III.2.12)

  In addition, during open-stage, as the voltage VP is evolving by charging and discharging the capacitance CP, the electric charge required for each open stage is determined by (CP ΔU), as Vc} can be the input voltage/the output voltage or a linear combination of the input and the output voltage, or 0V. Thus, by considering a general case, we introduce three parameters named (𝛼 𝑎 , 𝛼 𝑏 , 𝛼 𝑐 ) for each voltage level respectively {Va, Vb, Vc}, so, 𝛼 𝑎 (respectively 𝛼 𝑏 , 𝛼 𝑐 ) is defined regarding to Va (respectively Vb, Vc) by: 𝛽 𝑉 𝑎 𝑄 𝑎 + 𝛽𝑉 𝑏 𝑄 𝑏 + 𝛽𝑉 𝑐 𝑄 𝑐 = Vout, -Vout} charges are transferred to the load during all the three connected stage minimizing the amplitude of I compared to Iout.For the switching sequence (Vin, Zero, Vout), the charge transfer to the load occurs only during one stage which inducing a higher current amplitude I at same output current compared to the previous switching sequence.This expression can be reworded as a function of Qa, as shown in (III.3.18). The expression (III.3.18) states that the current 𝐼 𝑜𝑢𝑡 is mainly depending on Qa which itself depends on the time duration of [t3', t4]. The transition time t3' is the instant where Vp is equal to Va and it cannot be changed, the stage duration is then only controlled by t4 which is free to move. Equation This expression is applicable to all switching sequences, and takes into account the PR's proprieties and the operating points (Vin, Vout, Iout).

	𝑄 𝑏 = 𝑞 2 -𝑞 1 𝑄 𝑐 = 𝑞 5′ -𝑞 5 𝛽(𝑉 𝑏 -𝑉 𝑣𝑠𝑡 )𝐶 𝑃 = 𝛽 𝐼 𝜔 𝛽(𝑉 𝑣𝑠𝑡3 -𝑉 𝑏 )𝐶 𝑃 = 𝛽 𝐼 𝜔 𝛽(𝑉 𝑎 -𝑉 𝑣𝑠𝑡3 )𝐶 𝑃 = 𝛽 𝜔 𝐼 By using charge balance, we deduce a relation between each charge stage: (𝛾 1 -𝛾 0 ) (𝛾 3 -𝛾 2 ) (𝛾 3′ -𝛾 3 ) Where A and B are: 1 2 𝑅 𝑃 𝐼 2 𝑇 𝑄 𝑏 = 𝐴𝑄 𝑎 + 𝐵 𝑄 𝑐 = -(1 + 𝐴)𝑄 𝑎 -𝐵 𝑃 𝑜𝑢𝑡 𝑉 𝑜𝑢𝑡 = -(𝛼 𝑎 𝑄 𝑎 + 𝛼 𝑏 𝑄 𝑏 + 𝛼 𝑐 𝑄 𝑐 ) 𝑇 for the switching sequence {Vin-Vout, 𝐼 𝑜𝑢𝑡 = where 𝐴 0 = 𝑉 𝑐 -𝑉 𝑏 𝑉 𝑎 -𝑉 𝑐 , 𝐼 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔 = 𝛽(𝑉 𝑍𝑉𝑆𝑡3 -𝑉 𝑍𝑉𝑆𝑡6 ) 2 𝐶 𝑃 𝜔.	(III.2.13) (III.2.14) (III.3.5) (III.3.6) (III.3.7) (III.3.11) (III.3.12) (III.3.13) (III.3.17)
	𝛽(𝑉 𝑐 -𝑉 𝑎 )𝐶 𝑃 = 𝛽 𝐴 = 𝑉 𝑐 -𝑉 𝑎 𝑉 𝑏 -𝑉 𝑐 , 𝐵 =	𝐼 𝜔 2𝛽(𝑉 𝑏 -𝑉 𝑐 ) (𝛾 5 -𝛾 4 ) 𝑅 𝑃 𝐼 2 𝑇	(III.3.8) (III.3.14)
	𝛽(𝑉 𝑣𝑠𝑡 -𝑉 𝑐 )𝐶 𝑃 = 𝛽	𝐼 𝜔	(𝛾 5′ -𝛾 )	(III.3.9)
	𝑉 𝑎 (𝛾 4 -𝛾 3 ′ ) + 𝑉 𝑏 (𝛾 2 -𝛾 1 ) + 𝑉 𝑐 (𝛾 5 ′ -𝛾 5 ) = 𝐼 𝑜𝑢𝑡 = -((𝛼 𝑎 -𝛼 𝑐 ) + (𝛼 𝑏 -𝛼 𝑐 )𝐴 ) 𝑄 𝑎 𝑇 -𝐵 𝑇 (𝛼 𝑏 -𝛼 𝑐 ) 𝜋𝑅 𝑃 𝐼 𝛽	(III.3.15) (III.3.18)
	These expressions give the charge exchanged during each connected stage. So, assuming the charge balance during steady state operation (no accumulation of charge), one can find a relation between the instants {t0..., t6}, as is given by (III.3.4). 𝑄 𝑎 = 𝛽 𝐼 𝜔 (𝛾 4 -𝛾 3′ ) (III.3.1) 𝑄 𝑏 = 𝛽 𝐼 𝜔 (𝛾 2 -𝛾 1 ) (III.3.2) 𝑄 𝑐 = 𝛽 𝐼 𝜔 (𝛾 5′ -𝛾 5 ) (III.3.3) III.3.2 Energy transfer As mentioned before, the PR operates with energy balance leading to sustained oscillation. Equation (III.2.11) illustrates the energy balance without including the losses of PR. By considering the electric model of PR, its losses (i.e., estimated by 1/2RPI²) can be integrated in the energy balance equation as shown in (III.3.11). 𝐼 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔 = 𝛽(𝑉 𝑣𝑠𝑡3 -𝑉 𝑣𝑠𝑡 ) 2 𝐶 𝑃 𝜔 (III.3.10) { 𝛼 𝑎 = -1 𝑖𝑓 𝑉 𝑎 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (-𝑉 𝑜𝑢𝑡 ) 𝛼 𝑎 = +1 𝑖𝑓 𝑉 𝑎 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (+𝑉 𝑜𝑢𝑡 ) 𝛼 𝑎 = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (III.3.16) (𝛾 4 -𝛾 3 voltage excursion. 𝑇 𝐼 𝑜𝑢𝑡 = -((𝛼 𝑎 -𝛼 𝑐 ) + (𝛼 𝑏 -𝛼 𝑐 )𝐴 ) 𝑄 𝑎 (III.3.19)

′ ) + (𝛾 2 -𝛾 1 ) + (𝛾 5′ -𝛾 5 ) = 0 (III.

3.4)

given in (III.

[START_REF] Hanson | Measurements and performance factor comparisons of magnetic materials at high frequency[END_REF]

.5)-(III.3.9).

Assuming that the voltage βVP is capped between its minimum (i.e., βVzvst6) and maximum (i.e., βVzvst3) values, the sum of charge (III.3.5) to (III.3.6) gives the charge required to go from minimum to maximum amplitude of VP: 𝛽(𝑉 𝑣𝑠𝑡3 -𝑉 𝑣𝑠𝑡 )𝐶 𝑃 , while the sum of (III.3.7) to (III.3.9) gives the charge required to go from maximum to minimum amplitude of VP: 𝛽(𝑉 𝑣𝑠𝑡 -𝑉 𝑣𝑠𝑡3 )𝐶 𝑃 . Consequently, the amplitude of the circulating current Icirculating required to charge and discharge the capacitance CP is then deduced by (III.3.10). This part of the current is required only for charging and discharging the capacitance CP without any power transfer to load, and it is directly proportional to the CP𝜔 (i.e., inverse of the impedance CP) value and the By including equations (III.3.1)-(III.3.3) in (III.3.11), one can obtain a second relation between the instants {t0..., t6} as shown in (III.3.15). The energy balance is mainly linked to time-duration of each constant voltage stage and to the voltage levels. This expression is simple, and can be simpler if one voltage level is equal to 0, and moreover simplified by removing losses.

III.3.3 Load current

To determine the current flowing to the output load, we consider a constant output voltage without ripple. Therefore, the mean output current Iout is determined by summing the charges exchanged during each constant voltage stage where Vout is involved. As said before, the voltage levels {Va, Vb, Then, the load current is calculated by using (III.3.17). This expression shows that the average current transferred to the load is highly depending on the switching sequences. Indeed, (III.3.18) reveals that the degree of freedom is reduced to one parameter (in general case, t4 is chosen to manage the output power), and the other instants {t0, t1, t2, t3, t3', t5, t5', t6} are deduced directly to ensure ZVS, soft charging, energy and charge balance. Therefore, Equation (III.3.18) gives a way to control the output (current or voltage). For a lossless PR (i.e., RP=0, so, B=0), the equation (III.3.18) can be simplified by: III.3.4 Estimation of the amplitude of the current i As mentioned in charge transfer section, the current I is determined by the one needed to charge and discharge the capacitance CP (III.3.10) and by the one required to supply the output load (III.3.18) which depends on the switching sequence and the load. By combining the previous equations (III.3.4)-(III.3.9) with (III.3.19) and considering lossless PR (i.e., B=0) for reaching a readable equation, one deduces the amplitude of the current I which is given by (III.3.20).

  𝐼 = 𝛽𝜋 (𝛼 𝑏 -𝛼 𝑐 ) + (𝛼 𝑎 -𝛼 𝑐 )𝐴 0 𝐼 𝑜𝑢𝑡 + 𝐼 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔 = 𝐼 𝑢𝑠𝑒𝑓𝑢𝑙 + 𝐼 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔 (III.3.20)

		𝐾 =	(𝛼 𝑎 +𝛼 𝑏 )-(𝛼 𝑐 -𝛼 𝑎 )𝐴 0 2𝛽	(III.3.21)
	Where 𝐵 0 =	𝑎 𝑐 -𝛼 𝑎 𝛼 𝑎 +𝛼 𝑏
			𝛾 0 = 1		(III.3.22)
		𝛾 1 = 𝛽(𝑉 𝑏 -𝑉 𝑍𝑉𝑆𝑡 )	𝛽𝐼 𝐶 𝑃 𝜔	+ 𝛾 0	(III.3.23)
		𝛾 2 = 𝛽(𝑉 𝑏 -𝑉 𝑍𝑉𝑆𝑡3 )	𝛽𝐼 𝐶 𝑃 𝜔	+ 𝛾 3	(III.3.24)
			𝛾 3 = -1		(III.3.25)

. From the proposed model, one can first determine the current I from (III.3.20), and then deduces the instants t0 to t6 on the cycle thanks to equations (III.3.22) to (III.3.30).

TABLE III .

 III 2 shows the various possible switching sequences {Va, Vb, Vc} with a valid classification sequence {Va, Vb, Vc}, with the corresponding parameters {𝛽, 𝛼 𝑎 , 𝛼 𝑏 , 𝛼 𝑐 } for each switching sequence. The following switching sequences shown in TABLE III.2 do not take into account the number of required switches or the need or not of bidirectional switches.

TABLE III . 2 .

 III2 Therefore, regarding the operating conditions shown in TABLE III.2, these switching

	Figure III.9. The proposed topology [28].
	sequences can be applied in consequence. Also, for operation as step-up dc-dc converter g>1,
	S1 should be replaced by a bidirectional switch, which allows that the converter can operate as
	buck-boost converter.

TABLE III .

 III 3. Examples of Switching Sequences Used with the Proposed Topology.

		III.4.3 Example of implementation of a switching sequence
		V P +				V P -
		Conditions	Vin	Vin-Vout	-Vout	Zero	Vout
	Mode 1	Any	i>0		i >0	i <0
	Mode 2	2Vout <Vin		i >0	i >0		i <0
	Mode 3	Vout<Vin	i>0			i >0	i <0
	Mode 4	Vout<Vin	i>0		i >0		i <0
	Mode 5	2Vout <Vin		i>0		i >0	i <0
	Mode 6	2Vout >Vin		i>0		i <0	i <0
	Mode 7	Vout <Vin<2Vout	i>0	i >0			i <0

TABLE III

 III

			.4. PR's circuit C-213 from Fuji
	Parameter	RP	CP	C	L	fr	far
	Value	0.4Ω	8.4nF	2.9nF	1.1mH	89kHz	104kHz

TABLE IV .

 IV 1. BVDM model parameters of the PR in radial mode from Fuji Ceramics C-213. : the capacitance CP determined is not the "clamped" capacitance. As the capacitance CP is measured by using an impedance analyzer. So, this capacitance is obtained by applying a constant amplitude AC electric field that induces a displacement arising. The "clamped" capacitance CP can be measured at high frequency out of the resonance >15 MHz, but the measurement becomes highly sensitive to the setup.

	d	th	CP *	C	L	R	Q	kP	fr
	calculated 25mm 0.75mm 8.6nF 2.85nF 1.1mH 0.242Ω 2500 58% 90kHz
	measured 25mm 0.75mm 8.4nF 2.9nF 1.1mH 0.5Ω** 1500 52% 89.5kHz
	CP

*

TABLE IV

 IV

			.3. Key part components.
	Component	Ref	Voltage/ current	Electrical characteristics
	S1 and S2	GaN 66508B	650 V, 30 A	Rdson = 50 mΩ @T=25 °C

TABLE IV .

 IV [START_REF] Yang | A low-loss inductor structure and design guidelines for high-frequency applications[END_REF]. Equivalent model of 3-PRs in radial mode vibration.

	Parameters	L	C	RP	CP	fr
	PR1	1.1 mH	2.9 nF	0.548 Ω	8.4nF±10%	89.1kHz
	PR2	1.1 mH	2.824 nF	0.506 Ω	8.35nF±10% 90.01KHz
	PR3	1.1 mH	2.925 nF	0.501 Ω	8.38nF±10% 88.72kHz

TABLE IV .

 IV [START_REF] Sullivan | On size and magnetics: Why small efficient power inductors are rare[END_REF] shows the BVD equivalent circuit in thickness mode vibration of three PRs having various diameters(12, 25 and 40 mm). The PRs are selected in order to operate at the same resonant frequency, which is 1 MHz. Therefore, the thickness is deduced by using (IV.2.11) and it is equal to 2 mm. The various diameters are to evaluate the dependency of the power with the piezoelectric material area.

TABLE IV

 IV

			.5. BVD circuit of 3-PRs in thickness mode vibration for C-213.
		D (mm)	Th (mm)	L (µH) C (nF) CP (nF) RP (Ω)	fr (MHz)	Q	kt (%)
	PR1	12	2	133	0.17	0.775	7.4	1.04	200	42.7
	PR2	25	2	34	0.735	3.3	0.39	1.049	500	43
	PR3	40	2	11.7	2.01	8.44	0.5	1.038	300	45

TABLE IV

 IV

					.6.
		TABLE IV.6. BVD circuit of a PR with D =25 mm and th =2 mm.
	Vibration mode	L	C	RP	CP	fr	Q	k
	Radial	2.7mH	1.19nF	1.087Ω	3.3nF	88.8kHz	1387	0.51
	Thickness	32µH	0.72nF	0.2049Ω	3.3nF	1.051MHz	1026	0.42

  TABLE IV.7, we have estimated Icirculating, Iuseful and I by using (IV.4.3) for the radial and thickness mode. As shown, by increasing the frequency, the circulating current becomes significant, which is increasing the amplitude I. Also, as Icirculating is only used for charging/discharging CP, this amplitude induces losses even without load. So, having a high Icirculating reduces the efficiency, especially at low power operations. Thus, the design of the PR should consider the ratio between Icirculating and Iuseful. A simple solution for keeping a high efficiency operation over wide power range consists of parallelizing the PRs as a function of power segments.

TABLE IV

 IV

			.7. Current Vs Frequency: Vin=200 V, Vout= 60 V and CP=3.3nF.
		Value	Radial mode F = 91 kHz	Thickness mode F = 1.08 MHz
	𝐼 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑛𝑔 =	(𝑉 𝑖𝑛 + 𝑉 𝑜𝑢𝑡 )2𝜋𝑓𝐶 𝑝 2	0.273 A	2.73 A
	𝐼 𝑢𝑠𝑒𝑓𝑢𝑙 =	𝜋 2	𝐼 𝑜𝑢𝑡	0.6 A	2.62 A
		𝐼			0.9 A	5.4 A
	IV.5 Inductor Vs. PR

  operating point for small variations around initial point 0 (by using Taylor series development at first order), where 𝑡 4 = 𝑡 4 0 + Δ𝑡 4 and 𝑣 𝑜𝑢𝑡 = 𝑉 𝑜𝑢𝑡 0 + ΔV 𝑜𝑢𝑡 .

	Therefore, by linearizing the equation (V.3.4), one can obtain a relation between Δ𝑡 4 and
	ΔV 𝑜𝑢𝑡 (using Taylor development 1 st order):
	cos 𝜔𝑡 4 = 1 -Then, we deduce the static transfer function H(s) by 4𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛 + 𝑉 𝑜𝑢𝑡 𝐶 𝑃 𝜔 𝐼 (1 + ΔV 𝑜𝑢𝑡 2𝑉 𝑜𝑢𝑡 for a desired operating point. This ) (V.3.2) Δ𝑡 4 𝑉 𝑖𝑛 gain gives a slew rate of the variation of the Vout when a small change of t4 occurs, and it is
	expressed in V/s.
	Δ𝑉 𝑜𝑢𝑡 Δ𝑡 4	= 𝜔 sin 𝜔𝑡 4 0 (1 +	𝜋𝐶 𝑃 𝜔 𝑅 𝐿	1 +	𝑉 𝑖𝑛 𝑉 𝑜𝑢𝑡 0
	𝐼 =	𝜋 2	𝑉 𝑜𝑢𝑡 𝑅 𝐿	+	𝐶 𝑃 𝜔𝑉 𝑜𝑢𝑡 2	(1 +	𝑉 𝑖𝑛 𝑉 𝑜𝑢𝑡	)	(V.3.3)
		Figure V.12. Output load model.
	cos 𝜔𝑡 4 = 1 -	4𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛	+	2 (1 + 2 1 + 𝜋𝐶 𝑃 𝜔/𝑅 𝐿 + 𝑉 𝑖𝑛 /𝑉 𝑜𝑢𝑡 𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛 )	(V.3.4)

3.2), equation (V.3.3) can be rewritten as a function of Vout by removing I. Thus, equation (V.3.4) gives the relation between t4 and Vout: this expression can be linearized around an

TABLE V .2.

 V

	-𝜔 Δ𝑡 4 * sin 𝜔𝑡 4 0 = -4	Δ𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛	+	4 1 + 𝜋𝐶 𝑃 𝜔/𝑅 𝐿 + 𝑉 𝑖𝑛 /𝑉 𝑜𝑢𝑡0 Δ𝑉 𝑜𝑢𝑡 𝑉 𝑖𝑛	(V.3.5)
	TABLE V.2. Key parameters.
	Parameter		Value
	L			1.1 mH
	RP			0.6 Ω
	C			2.9 nF
	CP			8.9 nF
	Rdson			100 mΩ
	Vf			0.7 V
	Cout			10 µF
	f			[fr, far]=[90 kHz, 105 kHz]

(jusqu'à 100 W), cette structure est adaptée pour les appareils s'alimentant à partir du réseau domestique. En effet, la topologie proposée permet une conversion à fort abaissement de tension et une minimisation de l'énergie stockée dans le PR. Dans la deuxième partie, nous proposons la conception d'un convertisseur continu-continu basé sur le PR, conçu et réalisé pour une tension d'entrée allant jusqu'à 250 V et une tension de sortie allant jusqu'à 125 V. Ce convertisseur a atteint une puissance de sortie de 50 W avec un rendement de 93 % pour un mode vibration radial à une fréquence proche de 90 kHz, et une puissance de sortie de 175 W en mode de vibration d'épaisseur à une fréquence de 1 MHz. Des formes d'onde expérimentales sont présentées, qui montrent comment le cycle de conversion évolue en fonction des différentes conditions de fonctionnement. Par la suite, nous nous sommes concentrés sur le développement d'une première stratégie de contrôle pour un fonctionnement en boucle fermée pour ce nouveau principe de conversion. La stratégie de contrôle permet la régulation de la tension de sortie, tout en réalisant le cycle de conversion, des commutations douces et le contrôle de phase/fréquence pour la synchronisation du cycle de pilotage avec la vibration mécanique du PR. La stratégie de contrôle est implémentée dans un Field Programmable Gate Arrays (FPGA), constituant le circuit de contrôle adapté pour un convertisseur DC-DC fortement abaisseur. Un prototype a été conçu pour une tension d'entrée allant jusqu'à 120 V et une tension de sortie < 48 V. Le contrôle implémenté a montré de bonnes performances en termes de temps de réponse (moins de 2 ms) et de dépassement de tension n'excédant pas 10 % de la tension de sortie. Dans une dernière étape de la thèse, nous avons investigué les matériaux piézoélectriques dédiés aux convertisseurs de puissance basés sur des résonateurs piézoélectriques. L'étude visait l'augmentation de la fréquence en utilisant du Niobate de Lithium (LNO) afin de : -réduire les modes parasites perturbant le bon fonctionnement du résonateur, -atteindre un facteur qualité et un fort couplage électromécanique, -améliorer le montage mécanique et la connexion électrique avec le circuit électronique. Par la suite, nous avons fabriqué un PR à base de LNO en salle blanche, avec un circuit dédié pour une fréquence de fonctionnement allant jusqu'à 7 MHz. Le circuit proposé a pu être validé sur une plage de puissance de 2 -32 W avec une tension d'entrée de 60 V et une tension de sortie de 20 V. Cette partie est présentée dans l'APPENDIX A.

To conclude, the piezo-based power conversion may become one of the most important breakthroughs in power electronics for miniaturization and integration of power converters.

Normalized current i

High Step-down converter: 2Vout <Vin Va= Vin-Vout Vb= Vout Vc =-Vout Vzvst3=Vin

Step-down converter:

Vout <Vin Va= Vin Vb= Vout Vc =-Vout or Zero

Step-up converter:

Vin time simulation and to evaluate harmonic spectrum generated by the switching sequence.

However, this approach requires a complex model and calculations, and the convergence time/accuracy depends on the calculation step number. Therefore, the approach here gives a fast solution to evaluate the efficiency, with simplified and readable equations. In addition, the proposed generic analytical model makes it easier to select the cycle, the topology, to identify the key parameters for the PR and to define the general structure of the closed-loop to regulate the output power/voltage.

Hence, a generic implementation of the control for six switching sequence can be built with the same method and can be used for any switching sequence regardless of the topology of the converter.

III.8 Summary of the chapter

In this chapter the operating principle of the dc-dc converter based on PR is fully detailed and explained, as well as the most relevant conversion cycles and topologies. A generic model is described, enabling to determine the current amplitude, the time duration of each stage and the efficiency for each switching sequence. A simulation based on a Spice model validates the operating principle while exhibiting a high efficiency. In addition, an efficient conversion cycle was identified for a high-to-low voltage conversion ratio. This one will be used for the rest of this thesis.

Chapter IV: The piezoelectric-based DC-DC step-down converter: experimental validation Abstract: This chapter is aimed to demonstrate experimentally the operating principle of the DC-DC converter based on a piezoelectric resonator. An actual prototype is designed to operate at various voltage and power levels demonstrating high efficiency operation. Parallelization of the PRs is investigated for a scalable power conversion. In addition, high frequency operation is shown using the thickness mode, which enables a high power level conversion up to 175 W. We conclude with discussion and the presentation of some challenges to overcome. The PR volume is about four times smaller than that of the inductor and power losses are lower thanks to its higher quality factor. In addition, PR offers a planar shape easier for implementation in thin devices (th = 0.75 mm). Still, at this frequency, the inductor remains competitive compared to PR.

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V.3 Design of the control strategy

The main idea to implement the control is based on sensing the voltage of the PR, Vin and Vout at specific instants of the conversion cycle, then, we use these measurements for achieving the soft-switching, ZVS and load regulation. The information tied to these conditions are processed digitally and implemented in a state machine in order to generate the PWMs synchronized with the mechanical movement thanks to a zero current crossing detection of ip.

In this section, we propose to design a control strategy by considering the switching sequence {Vin-Vout, Vout, -Vout} with the topology proposed in the previous chapter. In the control design, we consider -the soft charging, -ZVS conditions, -synchronization with i -and load regulation. The frequency is then determined by counting the number of clock cycles (f=fclk/ncycle).

Where, fclk is the clock reference of the control circuit.

V.3.2 ZVS

The turning-on of S1 is achieved at the beginning of the period, when the voltage VP+ is approaching Vin (i.

The time duration of the stage VP =Vout is controlled by turning off S2 at t =t2. This instant t2 is used to regulate the charge balance. This means that the instant t = t2 should be controlled by a closed loop in order to let VP reach the voltage level Vin at the half of the resonant period.

Indeed, if the instant t2 > t2opt, then VP cannot reach Vin at t3 and S1 cannot be turned-on with ZVS condition. In other hand, if t2 < t2opt, then VP reaches Vin before t3, so S1 can be turned-on with ZVS at t = t3. However, the reverse diode of S1 is turned-on and a part of the stored energy in the resonator is restored to the input source through the reverse diode of S1 and D4. In both cases, the efficiency of the converter is reduced. Therefore, to regulate the t2 value, we can use a simple comparator to compare voltage VP+ with Vin. In this case, we use VP+ as it is the image of VP, since VP+ is referenced to the ground, avoiding a differential measurement of VP. The result of the comparison at t = t3 is then used to increase or decrease t2 for the next resonant period. Design and Implementation of the control and functional verification.

Logical Synthesis

Instantiation of the inputs/and outputs of each IP core block by using Xilinx software.

VHDL/Verilog Synthesis

Low level programming of each IPblock.

Placing and routing

Compiling the programming regarding the FPGA target.

Generating the .bin file File program to configure the FPGA.

V.2 Circuit design and considerations

In this section, we describe the circuit that we designed for experiments. Beforehand, we list the control parameters with the measurements awaited in order to identify the resources as illustrated in

Chapter VI: Conclusion

Abstract: This chapter summarizes the various investigation subjects that have been conducted in this thesis, and the main results are presented. In addition, we present the future challenges and opportunities for the piezo-based power conversion in aims of miniaturization and integration of power converters.

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