Use of Superconductors in the Excitation System of ...€¦ · Use of Superconductors in the...

Use of Superconductors in the Excitation System of

Electric Generators Adapted to Renewable Energy Sources

YBCO superconducting magnets for low speed synchronous generators

João André Guerreiro Arnaud

Thesis to obtain the Master of Science Degree in

Electrical and Computer Engineering

Supervisor: Prof. Paulo José da Costa Branco

Examination Committee

Chairperson: Prof. Rui Manuel Gameiro de Castro

Supervisor: Prof. Paulo José da Costa Branco

Members of the Committee: Prof. Pedro Domingos Santos do Sacramento

October 2015

iii

Dedicado a

Toda a família e amigos

v

ACKNOWLEDGEMENTS

First of all, I’d like to thank Prof. Paulo Branco for the opportunity to work on this thesis and all

the help provided. Secondly, I’d like to thank my colleague Ivo Prada for all the help he gave me. Two

brains think better than one, and his contribution was very helpful, particularly in the initial stage of this

work. I also would like to thank all the students that have shared my supervisor for providing a good

work environment.

Finally, I would like to thank all my friends and family who have supported me through all these

years, and who have filled this very important part of my life with significant memories and good lessons,

which have made me grow immensely.

vi

ABSTRACT

This thesis studies the general use of high-temperature superconducting materials in the

transverse flux excitation system for low speed electrical generators, in particular the use of YBCO

superconductors. First, an electro-thermal coupled model for simulation of bulk superconductors and its

hysteretic magnetization was developed, tested and experimentally verified. The characteristics of high

temperature superconductors under the influence of time-varying magnetic fields have been studied,

specifically for the material YBCO, and their implication on joule losses, namely taking advantage of the

electro-thermal model implemented in a finite element software. It was found that joule losses increase

with the applied magnetic field and, in an almost linear dependency, with its frequency. Similarly, for

better internal characteristics of the material, which provide better performance, higher losses are

obtained. Regarding the two cooling techniques, and due to the characteristic hysteresis cycle of the

material, it has been shown that Field Cooling has lower losses than Zero Field Cooling. Some

temperature analyzes are done to support the results obtained.

Keywords

High temperature superconductors; YBCO; HTS modeling; Superconducting magnets; Bulk

superconductors

viii

RESUMO

Esta tese estuda o uso de materiais supercondutores de alta temperatura no sistema de

excitação de fluxo transverso para geradores eléctricos de baixa velocidade, em particular o uso do

material YBCO. Foi desenvolvido, testado, e experimentalmente verificado um modelo eléctrico e

térmico acoplado para simulação de supercondutores maciços e a sua magnetização de histerese.

Foram estudadas as características de supercondutores de alta temperatura sob a influência de

campos magnéticos variáveis no tempo, especificamente para o material YBCO, e a influência destas

nas perdas de joule, tirando partido do modelo implementado num software de elementos finitos. Os

resultados mostram que as perdas de joule aumentam com o campo magnético aplicado, e de maneira

quase linear com a sua frequência. Similarmente, para melhores características instrínsecas do

material, que proporcionam melhor rendimento, são obtidas maiores perdas. Em relação às duas

técnicas de arrefecimento, e devido ao ciclo de histerese característico do material, foi demonstrado

que a técnica de Field Coling tem menores perdas que a técnica de Zero Field Cooling. São feitas

também algumas análises de temperatura que suportam os resultados obtidos.

Palavras-chave

Supercondutores de alta temperatura; YBCO; Modelação de supercondutores de alta

temperatura; Magnetos supercondutores; Supercondutores maciços

x

TABLE OF CONTENTS

Acknowledgements ....................................................................................................................... v

Abstract......................................................................................................................................... vi

Resumo ....................................................................................................................................... viii

List of Figures ............................................................................................................................... xii

Nomenclature .............................................................................................................................. xvi

List of Tables ............................................................................................................................. xviii

List of Acronyms ........................................................................................................................ xviii

1 Introduction .......................................................................................................................... 1

1.1 Structure and Contents .................................................................................................. 2

1.2 State of the Art ............................................................................................................... 3

1.3 Motivation and Goals ..................................................................................................... 4

2 High Temperature Superconductors: Basic Concepts ........................................................ 5

2.1 Maxwell’s Equations ....................................................................................................... 6

2.2 The Superconducting State ........................................................................................... 7

2.3 Critical Region ................................................................................................................ 8

2.4 Constitutive Laws ........................................................................................................... 9

2.5 Type-II Superconductors .............................................................................................. 11

2.5.1 Flux Pinning ....................................................................................................... 12

2.6 Cooling Techniques ..................................................................................................... 13

2.6.1 Zero Field Cooling.............................................................................................. 13

2.6.2 Field Cooling ...................................................................................................... 14

3 Electromagnetic and Thermal Modeling............................................................................ 15

3.1 Electromagnetic Modeling ............................................................................................ 16

3.2 Thermal Modeling ........................................................................................................ 21

4 Results Analysis ................................................................................................................ 25

4.1 Sinusoidal Magnetic Field Analysis .............................................................................. 26

4.1.1 Magnetic field amplitude analysis ...................................................................... 34

4.1.2 Frequency analysis ............................................................................................ 37

xi

4.1.3 JC0 analysis ........................................................................................................ 40

4.1.4 B0 analysis ......................................................................................................... 43

4.1.5 Geometric dimensions analysis ......................................................................... 46

4.2 Field Cooling Transient Analysis for Trapped Field ..................................................... 49

4.2.1 Initial magnetic field analysis ............................................................................. 51

4.2.2 Applied field derivative analysis ......................................................................... 54

4.2.3 JC0 analysis ........................................................................................................ 57

4.2.4 B0 analysis ......................................................................................................... 60

4.2.5 Geometric dimensions analysis ......................................................................... 63

4.2.6 Long term study ................................................................................................. 66

4.3 Experimental Results ................................................................................................... 69

4.4 Thermal Analysis .......................................................................................................... 72

4.5 Electromagnetic and Thermal Analysis ........................................................................ 78

4.5.1 Heating analysis ................................................................................................. 79

4.5.2 Cooling analysis ................................................................................................. 86

5 Conclusions ....................................................................................................................... 89

5.1 Achievements ............................................................................................................... 90

5.2 Further Developments .................................................................................................. 91

5.3 Final Remarks .............................................................................................................. 91

References .................................................................................................................................. 93

xii

LIST OF FIGURES

Figure 2-1 – Volume defining the superconducting region ...................................................................... 8

Figure 2-2 – Characteristic states of type-I superconductors [16]........................................................... 8

Figure 2-3 – Characteristic states of type-II superconductors [16].......................................................... 8

Figure 2-4 – Dependency of the electric field with relative current density and n parameter ................. 9

Figure 2-5 – Critical current density dependency of the applied magnetic flux density for different values

of 𝐵0 ...................................................................................................................................................... 10

Figure 2-6 – Critical temperature dependency with magnetic flux density ............................................ 10

Figure 2-7 – Magnetization curve of type-I and type-II superconductors [16] ....................................... 11

Figure 2-8 – Flux pinning schematic [20]. (a) – Example of flux tubes distribution in the superconductor

(b) – Detail of a flux tube with superconducting current vortices........................................................... 12

Figure 2-9 – Example of the Meissner effect in the superconductor ..................................................... 13

Figure 2-10 – Faraday’s Law ................................................................................................................. 14

Figure 3-1 – 2D schematic and respective axis for the HTS electromagnetic model ........................... 16

Figure 3-2 – Schematic of the model used in the electromagnetic simulations .................................... 17

Figure 3-3 – Schematic of the magnetization of the superconductor .................................................... 18

Figure 3-4 – Example of a magnetization curve .................................................................................... 19

Figure 3-5 – Example of average eddy current losses inside the superconductor ............................... 20

Figure 3-6 – Superconductor thermal conductivity as a function of temperature .................................. 22

Figure 3-7 – Superconductor specific heat capacity as a function of temperature ............................... 22

Figure 3-8 – Nitrogen density as a function of temperature .................................................................. 23

Figure 3-9 – Nitrogen thermal conductivity as a function of temperature ............................................. 23

Figure 4-1 – Example of magnetic flux densities applied in the case of ZFC and FC .......................... 27

Figure 4-2 – Evolution of the current density inside the superconductor for a sinusoidal magnetic field

applied in ZFC and FC .......................................................................................................................... 33

Figure 4-3 – Hysteresis cycle dependency of magnetic flux density 𝐵𝑚 .............................................. 34

Figure 4-4 – Instantaneous power density losses dependency of magnetic flux density 𝐵𝑚 ............... 35

Figure 4-5 – Power density losses dependency of magnetic flux density 𝐵𝑚 ...................................... 36

Figure 4-6 – Hysteresis cycle dependency of applied frequency .......................................................... 37

Figure 4-7 – Instantaneous power density losses dependency of applied frequency .......................... 38

xiii

Figure 4-8 – Power density losses dependency of applied frequency .................................................. 38

Figure 4-9 – Hysteresis cycle dependency of maximum critical current density ................................... 40

Figure 4-10 – Instantaneous power density losses dependency of maximum critical current density . 41

Figure 4-11 – Power density losses dependency of maximum critical current density ......................... 41

Figure 4-12 – Hysteresis cycle dependency of 𝐵0 parameter .............................................................. 43

Figure 4-13 – Instantaneous power density losses dependency of 𝐵0 parameter ............................... 44

Figure 4-14 – Power density losses dependency of 𝐵0 parameter....................................................... 45

Figure 4-15 – Hysteresis cycle dependency of superconductor width .................................................. 46

Figure 4-16 – Instantaneous power density losses dependency of superconductor width ................... 47

Figure 4-17 – Power density losses dependency of superconductor width .......................................... 47

Figure 4-18 – Example of magnetic flux density applied in the case of Field Cooling .......................... 49

Figure 4-19 – Evolution of the current density inside the superconductor for a linearly decreasing

magnetic field applied in a process of Field Cooling ............................................................................. 50

Figure 4-20 – Current distribution in the superconductor after FC in trapped field ............................... 50

Figure 4-21 – Evolution of the maximum trapped magnetic flux density for different initial magnetic flux

densities................................................................................................................................................. 51

Figure 4-22 – Comparison of the evolution of the current density distribution inside the superconductor

for different initial magnetic flux densities .............................................................................................. 52

Figure 4-23 – Comparison of the final magnetic flux density distribution for different initial magnetic flux

densities................................................................................................................................................. 53

Figure 4-24 – Evolution of the maximum trapped magnetic flux density for different derivatives ......... 54


for different applied field derivatives ...................................................................................................... 55

Figure 4-26 – Comparison of the final magnetic flux density distribution for different applied field

derivatives.............................................................................................................................................. 56

Figure 4-27 – Evolution of the maximum trapped magnetic flux density in the superconductor for different

values of 𝐽𝐶0 .......................................................................................................................................... 57


for different maximum critical current densities ..................................................................................... 58

Figure 4-29 – Comparison of the final magnetic flux density distribution for different maximum critical

current densities .................................................................................................................................... 59

Figure 4-30 – Evolution of the maximum trapped magnetic flux density for different values of 𝐵0 ...... 60

xiv


for different values of 𝐵0 ........................................................................................................................ 61

Figure 4-32 – Comparison of the evolution of the magnetic flux density distribution for different values

of 𝐵0 ...................................................................................................................................................... 62

Figure 4-33 – Evolution of the maximum trapped magnetic flux density for different values of

superconductor width ............................................................................................................................ 63


for different superconductor widths ....................................................................................................... 64

Figure 4-35 – Comparison of the final magnetic flux density distribution for different superconductor

widths ..................................................................................................................................................... 65

Figure 4-36 – Long term study of trapped magnetic flux density after FC study .................................. 66

Figure 4-37 – Maximum magnetic flux density for different initial derivatives of a sinusoidal field applied

after magnetization using Field Cooling ................................................................................................ 67

Figure 4-38 – Maximum magnetic flux density for different frequencies of a sinusoidal field applied after

magnetization using Field Cooling ........................................................................................................ 67

Figure 4-39 – Maximum magnetic flux density for different amplitudes of a sinusoidal field applied after

magnetization using Field Cooling ........................................................................................................ 68

Figure 4-40 – Schematic of the lab experiment .................................................................................... 69

Figure 4-41 – a) Early stages of the coil setup b) Example of the magnet and superconductor setup 70

Figure 4-42 – Experimental results for the time it takes the YBCO bulk superconductor to lose

superconductivity ................................................................................................................................... 71

Figure 4-43 – Schematic of the thermal simulations ............................................................................. 72

Figure 4-44 – Evolution of the superconductor temperature ................................................................. 73

Figure 4-45 – Final temperature distribution after 2 minutes of cooling using default liquid nitrogen

parameters ............................................................................................................................................ 73

Figure 4-46 – Cooling process for different values of thermal conductivity .......................................... 74

Figure 4-47 – Cooling process for different values of specific heat capacity ........................................ 74

Figure 4-48 – Cooling process for different values of specific heat capacity and thermal conductivity 75

Figure 4-49 – Detail of the cooling process for different values of specific heat capacity and thermal

conductivity ............................................................................................................................................ 75

Figure 4-50 – Final temperature of the cooling process ........................................................................ 76

Figure 4-51 – Evolution of the superconductor temperature during the cooling and heating process.. 76

Figure 4-52 – Temperature after 3 minutes of removing the liquid nitrogen ......................................... 77

xv

Figure 4-53 – Setup of the electromagnetic and thermal simulations ................................................... 78

Figure 4-54 – Evolution of the superconductor maximum and average temperatures ......................... 79

Figure 4-55 – Temperature distribution after 3 minutes ........................................................................ 79

Figure 4-56 – Evolution of the maximum internal and critical current densities .................................... 80

Figure 4-57 – Evolution of the maximum and minimum critical current densities ................................. 80

Figure 4-58 – Comparison between magnetic flux density and critical current density ........................ 81

Figure 4-59 – Evolution of the ratio 𝐽/𝐽𝐶 inside the superconductor ..................................................... 82

Figure 4-60 – Evolution of the ratio 𝐽/𝐽𝐶 for two points inside the superconductor .............................. 82

Figure 4-61 – Evolution of the average power loss in the superconductor ........................................... 83

Figure 4-62 – Evolution of absolute value of the magnetization of the superconductor ....................... 83

Figure 4-63 – Comparison between different two different maximum magnetic flux density values .... 84

Figure 4-64 – Comparison for different frequency values ..................................................................... 84

Figure 4-65 – Comparison of the heating process between ZFC and FC conditions ........................... 85

Figure 4-66 – Evolution of the superconductor temperature after removing the applied field .............. 86

Figure 4-67 – Evolution of the maximum and minimum critical current density and maximum internal

current density after removing the applied field .................................................................................... 87

Figure 4-68 – Evolution of the ratio 𝐽/𝐽𝐶 in the superconductor after removing the applied field ......... 87

Figure 4-69 – Evolution of the average power losses in the superconductor after removing the applied

field ........................................................................................................................................................ 88

Figure 4-70 – Evolution of the magnetization in the superconductor after removing the applied field . 88

xvi

NOMENCLATURE

𝑬 Electric field vector

𝑫 Displacement field

𝑱 Current density vector

𝑯 Magnetic field

𝐻 Norm of the magnetic field vector

𝑩 Magnetic flux density vector

𝐵 Norm of the magnetic flux density vector

Φ Magnetic flux

𝐻𝐶 Critical magnetic field

𝐻𝐶1 Lower critical magnetic field

𝐻𝐶2 Higher critical magnetic field

𝐻0 Maximum applied magnetic field

𝑇𝐶 Critical temperature

𝑇𝐶0 Maximum critical temperature (with no magnetic field applied)

𝐽𝐶 Critical current density

𝐽𝑐0 Maximum critical current density (with no magnetic field applied)

𝛼 Maximum critical current density (with no magnetic field applied, at 0 K)

𝜇 Magnetic permeability

𝜇𝑟 Relative magnetic permeability

𝜀 Electrical permittivity

𝜀𝑟 Relative electrical permittivity

𝜌𝑞 total charge density

𝜎 Electrical conductivity

𝜌 Electrical resistivity

𝐵0 Parameter of the equation describing the dependency 𝐽𝑐(𝐵)

𝐸0 Parameter of E-J power-law

xvii

𝑛 Parameter of E-J power-law

𝑁 Number of magnetic moments

𝑛𝑚 Number density of magnetic moments

𝑚0 Magnetic moment vector

𝑀 Magnetization

𝑝 Atmospheric pressure

𝑇 Temperature

𝑊ℎ Hysteresis energy losses per cycle due to magnetization

𝑃ℎ Hysteresis power losses due to magnetization

𝑄 Heat Source

𝐽𝑠𝑐_𝑧 Current density of the superconductor in the z-direction

𝐸𝑠𝑐_𝑧 Electric field of the superconductor in the z-direction

xviii

LIST OF TABLES

Table 3-1 – Parameters used for the thermal simulations .................................................................... 21

Table 4-1 – Experimental results for the time it takes the YBCO bulk superconductor to lose

superconductivity ................................................................................................................................... 71

LIST OF ACRONYMS

FC Field Cooling

HTS High Temperature Superconductor

LTS Low Temperature Superconductor

YBCO Yttrium Barium Copper Oxide - YBa2Cu3O7

ZFC Zero Field Cooling

1 INTRODUCTION

This chapter describes how this dissertation is structured. A brief current state of the art regarding

the use of bulk superconductors in electrical machines is also presented. To conclude, the motivation

and goals of the dissertation are stated, as well as the expected results.

2

1.1 STRUCTURE AND CONTENTS

This thesis is divided in five chapters, each with a specific purpose. Chapter 1 is only an

introductory one, where the state of the art, motivation and goals are presented. Chapter 2 is dedicated

to explaining some basic concepts of high temperature superconductors necessary to the understanding

of this work. Chapter 3 describes the parameters and development of the electromagnetic and thermal

coupled FEM model of bulk YBCO superconductor used along this thesis. Chapter 4 presents the results

of the simulations done using the model described and also the experiments done to verify the model.

Chapter 5 completes this work by presenting the main conclusions of this work.

3

1.2 STATE OF THE ART

Superconducting electromechanical power converters have been researched since the 1950s for

applications where a high efficiency and power density is required. The first approach was replacing

copper coils by superconducting ones because they are able to carry electric current without significant

energy dissipation, due to negligible electrical resistance. Initially, the low critical temperature coils were

used to generate the excitation field of synchronous machines, but did not support high variations of

magnetic field. AC superconducting wire allowed fully superconducting synchronous machines [1].

Another approach has been using conventional machines with the addition of superconductors.

In synchronous reluctance machines, the rotor moves towards a position favoring a value of maximum

magnetic flux. The torque produced by these machines is proportional to the difference of d and q axis

inductances. It is possible to amplify this difference by placing a ferromagnetic material in one axis and

a bulk superconductor operating as a magnetic shield in the other axis, therefore increasing the level of

electromagnetic torque [2] [3].

There are many innovative structures using high-temperature superconductors (HTS) for their

property of high current density transport or magnetic levitation. An example which exploits the magnetic

behavior of bulk HTS is the hysteresis machine, which works using the hysteresis of high temperature

superconductors [4]. Another approach is using HTS coils which have the main advantage of higher

current densities ranging from 10 to 1000 times that of copper, and also allow higher air-gap fields

without any active iron, allowing higher armature loading and, when replacing all or a part of copper

coils, generally increases machine efficiency.

Electromechanical power converters based only on HTS bulk materials have reached a new

progress, improving machine efficiency [5] [6]. By exploiting the flux trapping and flux shielding

properties of HTS, different machine designs can be achieved. Trapping higher fields in HTS bulks using

the novel pulsed-field magnetization technique [4] [7] [8] [9] [10] has opened the appearing of HTS bulk

magnets that can trap nearly 2 T at 77K. This allows to surpass the neodymium magnets (NdFeB) field

in about 40% and the use of HTS magnets as the excitation system of electromechanical converters.

However, the liquid nitrogen cooling system (including the cryostat) can minimize the advantages

of using an HTS machine. Problems such as transporting the cryogenic coolant to the superconductors,

determining the cryostat location (either in the rotor and/or in a fixed stator, or even surrounding the

whole machine), and determining the core material (air-core vs iron-core machines or hybrid topologies)

are all open topics that show it is not yet evident which design is best [11].

HTS machines are still in development. In 2006, Siemens started the development of a 4MVA

HTS generator [12]. Recently, a 1 MW HTS synchronous motor has been developed in China [5]. Also,

some projects for synchronous generators for wind power generation have been conducted, such as a

12 MW LTS generator [11] [13] [14] and a 10 MW salient-pole HTS generator [1]. Superconducting

linear generators have also been constructed in the last few years using not only HTS coils but also

HTS bulks [2] [3].

4

1.3 MOTIVATION AND GOALS

The motivation for this work comes from previous thesis regarding the use of superconductors in

electric machines. A better understanding of these materials and their properties is needed to allow

innovation towards more efficient machines.

The goals include the study, modeling and analysis of the process of magnetization in high

temperature superconducting materials, the verification of the ability to maintain the level of

magnetization of YBCO superconductor material when subjected to a time-variant magnetic field, and

an analysis of the advantages and disadvantages of this solution.

Expected accomplishments include an electromagnetic macroscopic model for simulation of bulk

superconductors and its hysteretic magnetization, the determination of a set of parameters and their

characteristics to describe the behavior of high-temperature superconductors in magnetic field

confinement and an explanation of the advantages and disadvantages of this solution for the realization

of excitation systems.

5

2 HIGH TEMPERATURE SUPERCONDUCTORS: BASIC CONCEPTS

This chapter provides a summary of high temperature superconductors’ characteristics,

describing their behavior and unique phenomena.

6

2.1 MAXWELL’S EQUATIONS

First of all, Maxwell’s equations and its parameters and variables are revised. The four equations

are:

Gauss’s law

∇ ∙ 𝑬 =𝜌𝑞

𝜀0

∯ 𝑬 ∙ 𝑑𝑺 =1

𝜀0∭ 𝜌𝑞𝑑𝑉

Ω𝜕Ω

(2.1)

Gauss’s law for magnetism

∇ ∙ 𝑩 = 0

∯ 𝑩 ∙ d𝑺 = 0𝜕Ω

(2.2)

Faraday’s law of induction

∇ × 𝑬 = −𝜕𝑩

𝜕𝑡

∯ 𝑬 ∙ d𝓵 = −𝑑

𝑑𝑡∬ 𝑩 ∙ 𝑑𝑺Σ𝜕Σ

(2.3)

Ampère’s law with Maxwell’s addition

∇ × 𝑩 = 𝜇0 (𝑱 + 𝜀0𝜕𝑬

𝜕𝑡)

∮ 𝑩 ∙ 𝑑𝓵𝜕𝛴

= 𝜇0∬𝑱 ∙ 𝑑𝑺𝛴

+ 𝜇0𝜀0𝑑

𝑑𝑡∬𝑬 ∙ 𝑑𝑺𝛴

(2.4)

In the equations, 𝑬 is the electric field, 𝑩 is the magnetic flux density, 𝑱 is the current density, 𝜌𝑞

is the total charge density, 𝜀0 is the permittivity of free space, 𝜇0 is the permeability of free space.

Gauss’s law (2.1) states that the net outward normal electric flux through any closed surface is

proportional to the total electric charge enclosed within that closed surface [15]. It shows that the static

electric field points away from positive charges and towards negative charges. In the field line

description, electric field lines begin only at positive electric charges and end only at negative electric

charges.

Gauss’s law for magnetism (2.2) states that magnetic field lines only exist in closed loops. They

neither begin nor end.

7

Faraday’s law (2.3) describes how a time varying magnetic field induces an electric field. This

dynamically induced electric field has closed field lines just as the magnetic field, if not superposed by

a static (charge induced) electric field.

Similarly, Ampère's law with Maxwell's addition (2.4) states that magnetic fields can be generated

in two ways: by electrical current (the original "Ampère's law") and by changing electric fields ("Maxwell's

addition").

Besides those four equations, three isotropic constitutive relations, (2.5), (2.6), and (2.7), can be

established, where 𝑯 is the magnetic field, 𝑫 is the displacement field, 𝜎 is the conductivity of the

material (equal to the inverse of its resistivity: 𝜎 = 1/𝜌), 𝜀 is the permittivity of the material (𝜀 = 𝜀𝑟𝜀0), 𝜇

is the permeability of the material (𝜇 = 𝜇0𝜇𝑟).

𝑩 = 𝜇 𝑯 (2.5)

𝑫 = 𝜀 𝑬 (2.6)

𝑱 = 𝜎 𝑬 (2.7)

2.2 THE SUPERCONDUCTING STATE

In order to fully grasp the concepts shown in this thesis, it is vital to understand what exactly the

superconducting state is and what high temperature superconductivity is. Superconductivity is not only

characterized by a negligible resistivity, but also by magnetic field “expulsion” from the superconductor’s

interior.

High temperature superconductivity is associated with Type-II superconductors. These have a

property called flux pinning in which magnetic field can be trapped by the superconductor, thus acting

like a permanent magnet. This phenomenon is explained in detail ahead.

The transition to the state of superconductivity can be done in two different ways, regarding the

application of magnetic field at the moment of transition: with no applied magnetic field, called Zero Field

Cooling (ZFC), or with an applied field, called Field Cooling (FC).

8

2.3 CRITICAL REGION

There are three different quantities that restrict the superconducting state of a certain material:

the critical temperature 𝑇𝐶 , the critical magnetic field 𝐻𝐶 or critical magnetic flux density 𝐵𝐶 , and the

critical current density 𝐽𝐶.

Figure 2-1 – Volume defining the superconducting region

If any of these critical quantities is exceeded, the superconducting state is lost. However, it can

happen that these quantities are exceeded only locally, turning some parts of the superconductor to a

normal state. These parts will turn superconductive again once the quantities in question fall back below

the respective critical value. In the case of type-II superconductors, two critical magnetic flux densities

can be defined: the first, 𝐵𝐶1, where the superconductor loses the property of magnetic field “expulsion”

and enter a state of flux pinning. The second, 𝐵𝐶2, when the field is high enough to disable the magnetic

properties of the superconductor, causing the loss of superconductivity:

Figure 2-2 – Characteristic states of type-I superconductors [16]

Figure 2-3 – Characteristic states of type-II superconductors [16]

9

2.4 CONSTITUTIVE LAWS

In superconductors, as there is no proportionality between E and J, resistivity cannot be defined

as in equation (2.7). The relation between the electric field E and the current density J is given by a non-

linear E-J power-law [17]. In this relation, 𝐸0 and 𝑛 are constants depending on the superconductor type,

and 𝐽𝐶 is the critical current density of the superconductor.

𝑬 = 𝐸0 (𝑱

𝐽𝐶)𝑛

(2.8)

Below is shown the evolution of the electric field with the ratio between the current density and its

critical value for the YBCO material and for different n values.

Figure 2-4 – Dependency of the electric field with relative current density and n parameter

The critical current density is function of magnetic flux density applied in the superconductor,

being formulated by equation (2.9). Here, 𝐽𝑐0 and 𝐵0 are constants dependent of the material, and 𝐵 is

the norm of the magnetic flux density vector (𝐵 = |𝑩|) [17]:

𝐽𝐶(𝐵) =𝐽𝐶0𝐵0𝐵0 + 𝐵

(2.9)

The term 𝐽𝑐0 in (2.9) represents the maximum critical current density of the material (for 𝐵 = 0),

while 𝐵0 represents the value of the applied magnetic flux density that reduces the value of the critical

current density to half of its maximum value: 𝐽𝐶(𝐵0) = 𝐽𝐶0/2. Below is shown the evolution of the critical

current density with the applied magnetic flux density for different 𝐵0 values.

10

Figure 2-5 – Critical current density dependency of the applied magnetic flux density for different values of 𝐵0

The critical current density can also be considered dependent on the temperature, given by

equation (2.10), as stated by [18]. In this equation, 𝛼 represents the critical current density at 0 K with

no applied field.

𝐽𝑐(𝐵, 𝑇) = 𝐽𝑐0(𝑇, 𝑇𝐶) [𝐵0

𝐵0 + 𝐵] = 𝛼 (1 − (

𝑇

𝑇𝑐)2

)

3 2⁄

[𝐵0

𝐵0 + 𝐵] (2.10)

The critical temperature is also affected by the magnetic flux density over the superconductor

[19]. In this case, 𝑇𝐶0 is the critical temperature when 𝐵 = 0.

𝑇𝐶(𝐵) = 𝑇𝐶0𝑒−𝐵/30 (2.11)

Figure 2-6 – Critical temperature dependency with magnetic flux density

11

2.5 TYPE-II SUPERCONDUCTORS

In this thesis, only Type-II superconductors are studied in detail. However, it’s necessary to

distinguish between the two types of superconductors, in order to better understand the work developed.

Type-I superconductors are known to expel all magnetic field from its interior. This type of

superconductors can only sustain magnetic fields lower than 𝐵𝐶, for temperatures lower than 𝑇𝐶, as

illustrated in Figure 2-2. Generally, Type-I superconductors are also low temperature superconductors

(LTS), only able to hold their superconducting state at extremely low temperatures (usually below 10K)

and most of them are elemental metals.

Figure 2-3 indicates that Type-II superconductors not only repel magnetic fields, but can also trap

it inside. These superconductors have two critical magnetic fields, 𝐵𝐶1and 𝐵𝐶2. Figure 2-7 shows that,

below 𝐵𝐶1, all magnetic field is expelled (fully superconducting state). Between 𝐵𝐶1and 𝐵𝐶2, the magnetic

field partially penetrates the superconductor, in what it is defined as mixed state. Above 𝐵𝐶2 , the

superconductor loses its superconductivity (normal state).

Figure 2-7 – Magnetization curve of type-I and type-II superconductors [16]

Type-I superconductor Type-II superconductor

Superconducting

state

Normal state

𝐵𝐶 𝐵𝐶2 𝐵𝐶1

Magnetic Flux Density (𝐴/𝑚2)

Ma

gn

etiza

tion

(−𝐴/𝑚

) Mixed state

12

2.5.1 FLUX PINNING

In the mixed state, the magnetic field is able to penetrate inside the bulk of the superconductor.

This penetration only exists in tiny tubes known as flux tubes (Figure 2-8 (a)), which are surrounded by

superconducting current vortices, as shown in Figure 2-8 (b).

Figure 2-8 – Flux pinning schematic [20]. (a) – Example of flux tubes distribution in the superconductor (b) – Detail of a flux tube with superconducting current vortices

The flux tubes are enclosed in areas of non-superconducting region, surrounded by

superconducting current vortices. Note that this continues to satisfy the condition of magnetic repulsion

by the superconductor. Flux pinning is thus the phenomenon where the flux tubes are pinned in place

by the superconducting vortices. As this happens in the mixed state, it is exclusive to Type-II

superconductors. These vortices can move around the superconductor, as they tend to repel each other

due to the Lorentz force interaction, but are usually held in place by impurities of the superconducting

material, such as grain boundaries and lattice defects, with a pinning force [21].

However, if the Lorentz force is greater than the pinning force, the flux tubes move. This can

happen because of two reasons. One is the reduction of the pinning force caused by superconductor

heating (flux creep). The other is the increase of the Lorentz force due to current densities higher than

the critical current density (flux flow).

Examples of superconductors in levitation often use this phenomenon to pin a permanent magnet

or a superconducting in place while levitating.

(a) (b)

13

2.6 COOLING TECHNIQUES

As discussed before, and considering the magnetic properties of high temperature

superconductors, there are two ways that the transition to superconductivity can occur: with or without

an applied field.

2.6.1 ZERO FIELD COOLING

The first technique is called Zero Field Cooling (ZFC), in which the temperature transition to

superconductivity is made without any magnetic field applied on the material. This causes the

superconductor to expel all magnetic field applied on it, up until the first critical magnetic flux density

𝐵𝐶1. For higher magnetic fields it enters the mixed state, where some magnetic field can penetrate inside

the material.

2.6.1.1 MEISSNER EFFECT

The Meissner effect is the phenomenon of magnetic field expulsion occurring in the

superconductor. This happens in the case of Zero Field Cooling and holds as long as the magnetic flux

density applied is not higher than the critical magnetic flux density 𝐵𝐶1, above which the superconductor

enters the mixed state. However, the magnetic field is not completely expelled, penetrating in some

depth at the surface of the superconductor (as the magnetic field is able to penetrate in the region where

the currents circulate), as illustrated in Figure 2-9 where the currents cancel the magnetic field inside

the superconductor.

Figure 2-9 – Example of the Meissner effect in the superconductor

14

In Figure 2-9 it is possible to visualize Faraday’s Law: the applied field is increasing with time in

the y-direction. The electric field induced inside the superconductor produces a current density that

generates an opposite magnetic field with time, according to the right hand rule, cancelling the magnetic

field inside the superconductor. Figure 2-10 illustrates Faraday’s Law, given by equation (2.3):

Figure 2-10 – Faraday’s Law

2.6.2 FIELD COOLING

The second technique is called Field Cooling (FC), in which the transition is made with a magnetic

field applied on the superconductor before the transition to the superconducting state. In this case, if the

source of the field is slowly removed (to avoid high peaks of current which can cause the superconductor

to lose superconductivity), the superconductor will tend to maintain its previous internal field, becoming

magnetized, and can be used like a permanent magnet.

As in the case of Zero Field Cooling, what is observed is that the superconductor tends to maintain

its internal magnetic field. In the case of Field Cooling, the internal currents will try to maintain the

magnetic field applied at the moment of transition into the superconducting state.

𝑱 = 𝐽𝐶 (𝑬

𝐸0)1/𝑛

∇ × 𝑬 = −𝝁𝟎𝜕𝑯

𝜕𝑡

𝑯

15

3 ELECTROMAGNETIC AND THERMAL MODELING

In this chapter, the parameters, conditions, and setup for the model used for the simulations are

described.

16

3.1 ELECTROMAGNETIC MODELING

The model to be presented was based in the 2D H-formulation model presented in [17]. As shown

in Figure 3-1, the magnetic field is applied along the x-y plane, so the current density and electric field

will only have a component in the z-direction (it is admitted that the rod has infinite length in the z-

direction). A schematic of the simulation is shown below.

Figure 3-1 – 2D schematic and respective axis for the HTS electromagnetic model

By applying Ampère’s Law (2.4), and assuming that Maxwell’s addition (the electric field

derivative) is in this case negligible (quasi-static regime), the current density and the electric field in the

superconductor, in the z-direction, can be obtained as given by (3.1) and (3.2), respectively. Notice that

equation (3.2) is the result of using (3.1) in (2.8).

𝐽𝑠𝑐_𝑧 = (

𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

) 𝑒𝑧 (3.1)

𝐸𝑠𝑐_𝑧 = 𝐸0(

𝜕𝐻𝑦𝜕𝑥

−𝜕𝐻𝑥𝜕𝑦

𝐽𝑐(𝐵))

𝑛

𝑒𝑧 (3.2)

Substituting the previous equations in Faraday’s Law (2.3) results in two coupled equations, given

by (3.3), that relates the magnetic field evolution in time with the electric field inside the superconductor.

The relative magnetic permeability 𝜇𝑟 was considered to be equal to 1.

𝜕(𝐸𝑠𝑐_𝑧) 𝜕𝑦⁄ = −𝜇0𝜇𝑟

𝜕𝐻𝑥𝜕𝑡

−𝜕(𝐸𝑠𝑐_𝑧) 𝜕𝑥⁄ = −𝜇0𝜇𝑟𝜕𝐻𝑦

𝜕𝑡

(3.3)

SUPERCONDUCTOR 𝑯(𝑡)

𝑥

𝑦

𝑧

17

The goal is to verify the effects of a time variable magnetic field when applied to an YBCO bulk

material. In this example [17], the YBCO superconductor is a 12x4cm rod inserted in an air domain as

shown in Figure 3-2.

Figure 3-2 – Schematic of the model used in the electromagnetic simulations

For the air domain, the relation between 𝑬 and 𝑱 is linear, 𝑬 = 𝜌𝑱, where 𝜌 is the resistivity of air

(𝜌 = 1 × 106 Ω ∙ 𝑚 [17]). Using this equation to define the electric field in (3.3) and also making use of

equation (3.1) and a 𝜇𝑟 = 1, yields the coupled equations (3.4).

𝜕 (𝜌 (

𝜕𝐻𝑦


)) 𝜕𝑦⁄ = −𝜇0𝜇𝑟𝜕𝐻𝑥𝜕𝑡

−𝜕 (𝜌 (𝜕𝐻𝑦


)) 𝜕𝑥⁄ = −𝜇0𝜇𝑟𝜕𝐻𝑦

𝜕𝑡

(3.4)

On the boundary between the superconductor and the air, the boundary condition is set up as

(2.2). On the outer boundary of the air domain a boundary condition is set in time as shown in (3.5):

𝐻𝑥 = 𝑓𝑥(𝑡)

𝐻𝑦 = 𝑓𝑦(𝑡) (3.5)

The magnetization of a certain material can be defined as equation (3.6) where 𝑁 is the number

of magnetic moments in the sample, 𝑉 is the total volume of the material, 𝑛𝑚 is the number density of

magnetic moments and 𝒎𝟎 is the vector that defines the magnetic moment.

SUPERCONDUCTOR

𝑥

𝑦

𝑧

AIR

18

𝑴 =

𝑁

𝑉𝒎𝟎 = 𝑛𝑚𝒎𝟎 [𝐴/𝑚] (3.6)

The magnetic moment vector is given by equation (3.7) where 𝒓 is the position vector pointing

from the origin to the location of the volume element (𝒓 = (𝑥, 𝑦, 𝑧)), 𝑱 is the current density vector.

𝒎𝟎 = 𝒓 × 𝑱 (3.7)

Figure 3-3 – Schematic of the magnetization of the superconductor

Because the simulation is in 2D, and the magnetic field is applied only in the y-direction, it is useful

to see the magnetization in that same direction. 𝑱 in this case becomes 𝑱 = (0,0, 𝐽𝑠𝑐_𝑧) . The

magnetization is given by expression (3.8), where 𝑆 is the cross section of the superconductor domain.

𝑴 = (∫ (−𝑥 ∙ 𝐽𝑠𝑐_𝑧) 𝑑𝑆𝑆

𝑆) 𝑒𝑦 (3.8)

It is then possible to compute the YBCO hysteresis losses when subjected to a time-variant

magnetic field, which are given by the area defined by its magnetization curve and computed using

equation (3.9).

𝑊𝐻 = ∮𝐵 𝑑𝑀 [𝐽/𝑚3/𝑐𝑦𝑐𝑙𝑒] (3.9)

SUPERCONDUCTOR

𝑥

𝑦

𝑧

𝑟 = (𝑥, 𝑦)

𝑱 𝑒𝑧

𝑴 𝑒𝑥 𝑒𝑦

19

Figure 3-4 – Example of a magnetization curve

Figure 3-4 shows the magnetization curve for a sinusoidal magnetic field applied to the

superconductor after Zero Field Cooling. The magnetization starts at zero and as the magnetic field is

increasing in the y-direction, it will acquire negative values in that same direction in order to counter the

applied magnetic field (region 1). After a certain value of magnetic field applied, it enters in the mixed

state, as described before (regions 2 and 3). Because multiple frequencies are generally used, it helps

to see the results of (3.9) in power density units, as in (3.10):

𝑃ℎ = 𝑓∮𝐵 𝑑𝑀 [𝑊/𝑚3] (3.10)

For Figure 3-4, a magnetic field with a frequency of 5 Hz was used, and equation (3.10) yielded

a power density loss of 𝑃ℎ = 4.2106 𝑊/𝑐𝑚3.

It is also possible to obtain the YBCO hysteresis losses density by multiplying the internal current

density by the associated electric field, as given by (3.11):

𝑄 = 𝐸𝑠𝑐_𝑧 𝐽𝑠𝑐_𝑧 [𝑊/𝑚3] (3.11)

The average YBCO losses density has a typical curve as shown by Figure 3-5:

1

2 2

3 3

20

Figure 3-5 – Example of average eddy current losses inside the superconductor

The losses can be computed averaging (3.11) in the superconductor cross section (the values

shown in Figure 3-5) and averaging in the time period related with the frequency of the applied magnetic

field:

𝑃𝑄 =1

𝑇∫

∫ 𝑄𝑆

𝑑𝑆

𝑆

𝑡+𝑇

𝑡

[𝑊/𝑚3] (3.12)

In order to obtain a steady state result, the time average needs to be done for a full period of the

mixed state (regions 2 and 3), after the initial fully superconducting state (region 1). In the case of Figure

3-5, the frequency of the applied magnetic field was 5 Hz, so the period was 0.2 seconds. An average

in time was done from 0.05 to 0.25 seconds to account for a full period, while ignoring the initial fully

superconducting state. The yielded value of power density losses was 𝑃𝑄 = 4.2118 𝑊/𝑐𝑚3 . As

demonstrated, both methods are valid in computing the losses from the superconductor, with minimum

error.

Eddy current losses exist when there is flux pinning, in the non-superconducting parts of the

material. For frequencies usually used in power applications (below 200 Hz) these losses are usually

considered to be negligible in comparison to hysteresis losses [22].

1 2 3

21

3.2 THERMAL MODELING

The critical current density 𝐽𝐶 changes with temperature according to equation (2.10). Knowing

the temperature distribution in the HTS bulk material allows estimating with more accuracy the local

electromagnetic variables. The thermal phenomena, which appear as soon as an electric current or a

magnetic field are applied to the superconductor, are characterized by the heat diffusion equation (3.13).

∇ ∙ (𝜆(𝑇)∇𝑇) − 𝜌𝑚𝐶𝑃(𝑇)

𝜕𝑇

𝜕𝑡+ 𝑝𝑄 = 0 (3.13)

In (3.13), 𝜆 is the thermal conductivity (in 𝑊/(𝑚 ∙ 𝐾) ), 𝜌𝑚 is the mass density (in

𝑊/(𝑚 ∙ 𝐾)), 𝐶𝑃 is the specific heat capacity (in 𝐽/(𝐾𝑔 ∙ 𝐾)). Variable 𝑝𝑄 is a volumetric power loss (in

𝑊/𝑚3) representing the heat source.

For equation (2.10), the parameter 𝛼 was calculated to be 𝛼 = 1.134 × 108 𝐴/𝑚2 to fit with the

assumption of a maximum critical current density 𝐽𝐶0 defined to be 2 × 107 𝐴/𝑚2 at 77 K (2.9).

Table 3-1 shows the thermal parameters used for each material: YBCO, liquid nitrogen, styrofoam

and air. The last two materials are used in exclusively thermal simulations.

Table 3-1 – Parameters used for the thermal simulations

Parameters

Material

YBCO superconductor Liquid Nitrogen Styrofoam Air

Mass density

𝐾𝑔/𝑚3

5.9 × 103 [18] Temperature

dependent 1 200 [23] 𝑝

0.02897

8.314

1

𝑇 1

Thermal conductivity

𝑊/(𝑚 ∙ 𝐾)

6.91846 + (0.35147)𝑇 +

(3.31 × 10−3)𝑇2 −

(1.67834 × 10−4)𝑇3 +

(1.04056 × 10−6)𝑇4 [24]

Temperature

dependent 1 0.05 [23] ~0.0225 1

Specific heat capacity

𝐽/(𝑘𝑔 ∙ 𝐾)

38.762 + (1.4428)𝑇 [24] 2.042 × 103 [25] 1300 [23] ~1000 1

Ratio of specific

heats 𝐶𝑃/𝐶𝑉 Not applicable 1.4013 [26] Not applicable 14 1

1 Parameter provided in simulation software

22

Figure 3-6 illustrates how the YBCO thermal conductivity changes with temperature up to 300 K.

It is has a non-linear characteristic, and it shows a maximum value around 45 K [24].

Figure 3-6 – Superconductor thermal conductivity as a function of temperature

Figure 3-7 illustrates how the YBCO superconductor specific heat changes with temperature,

showing that the specific heat capacity increases linearly with a temperature increase for the interval

considered [24].

Figure 3-7 – Superconductor specific heat capacity as a function of temperature

23

Table 3-1 also lists some thermal parameters of liquid nitrogen, used for cooling in experimental

studies and also considered in the simulations involving thermal effects. Figure 3-8 and Figure 3-9 plot

how the liquid nitrogen mass density and thermal conductivity are affected by temperature, respectively.

The presented temperature interval is where the nitrogen is defined as a liquid, for a range of pressures

up to 33 atm (graphs provided in simulation software).

Figure 3-8 – Nitrogen density as a function of temperature

Figure 3-9 – Nitrogen thermal conductivity as a function of temperature

25

4 RESULTS ANALYSIS

This chapter presents the results obtained according to the description provided in the previous

chapter, and describes the consequences of these results. A more detailed explanation of each study

is presented.

26

The replacement of permanent magnets by YBCO bulk superconductors in low-speed

synchronous generators requires prior testing of their behavior for some conditions appearing during

the generator operation.

The most important one is how the superconductor behaves when subjected to time-variant

magnetic fields. This chapter aims to answer some questions such as what are the losses dependency

of the magnitude and frequency of the perturbing magnetic field, and if ZFC and FC superconductors

show similar results.

4.1 SINUSOIDAL MAGNETIC FIELD ANALYSIS

The magnetization curve taken by the superconductor depends on multiple factors. Here it is

demonstrated the dependency of external factors like the amplitude and frequency of the applied

magnetic field, and internal factors like maximum critical current density 𝐽𝐶0, magnetic flux density 𝐵0

and width of the superconductor. The results presented were compared with [27], and the values fit the

same order of magnitude. In the YBCO model presented here, the parameters used were taken by [17].

For equation (2.8), the parameters are 𝐸0 = 1 × 10−4 𝑉/𝑚 and 𝑛 = 21, and for equation (2.9), the

parameters used are 𝐽𝐶0 = 2 × 107 𝐴/𝑚2 and 𝐵0 = 0.1 𝑇, which was determined by simulation to best fit

with the results presented in [17]. The simulations in this subchapter admit a constant temperature of

77 K with an already superconducting material.

First results will illustrate the magnetization techniques of Zero Field Cooling and Field Cooling.

In (4.1) and (4.2) are shown the x- and y- magnetic field components for ZFC and FC boundary

conditions, respectively. The amplitude of the sinusoidal magnetic field 𝐻𝑚 is defined by 𝐵𝑚/𝜇0, and the

trapped magnetic 𝐻𝑀 is defined by 𝐵𝑀/𝜇0 . The default frequency for these illustrative studies was

defined to be 𝑓 = 5 𝐻𝑧, with 𝜔 = 2𝜋𝑓, and magnetic flux densities were set to be 𝐵𝑚 = 1.26 𝑇 and 𝐵𝑀 =

1.26 𝑇.

𝑍𝐹𝐶:

𝐻𝑥 = 0𝐻𝑦 = 𝐻𝑚sin (𝜔𝑡)

(4.1)

𝐹𝐶:

𝐻𝑥 = 0𝐻𝑦 = 𝐻𝑀 +𝐻𝑚sin (𝜔𝑡)

(4.2)

Figure 4-1 shows an example of the magnetic flux densities applied to the superconductor,

defined by (4.1) and (4.2).

27

Figure 4-1 – Example of magnetic flux densities applied in the case of ZFC and FC

Using the examples of Figure 4-1, it’s possible to describe what happens in the superconductor

in terms of its internal currents:

In the case of ZFC, when the magnetic field is applied after the transition to superconductivity,

the Meissner effect is observed at first, when the superconductor is in the fully superconducting state

and expels the applied magnetic field. As the applied field is still low, comparing to the parameter 𝐵0

defined in equation (2.9), the critical current density takes high values. As the magnetic field increases,

the superconductor will reach the mixed state and the magnetic field will penetrate in its interior. As this

happens, the current density will decrease, according to equation (2.9), and the internal currents will

gradually fill the superconductor, as it still tries to counter the applied field, but with lower current

densities. When the magnetic field decreases, currents in the opposite direction will appear on the edges

of the superconductor, in order to oppose the changing in magnetic field. These currents gradually fill

the material, like described before, and its value will increase according to (2.9). These currents are

responsible for the magnetization of the superconductor, causing the phenomenon of hysteresis

observed during the cycle. An illustration of the changing currents in the superconductor is shown in

Figure 4-2.

For the FC case, the principle is the same. The differences are the lack of initial Meissner effect

and the value of the internal current density, which will be lower, as the magnetic field takes higher

values. This disparity in the values of the internal current density also affects the magnetization and

consequently the hysteresis cycle will also be affected.

28

Time (s)

ZFC FC

0.000

0.001

0.005

0.010

29

Time (s)

ZFC FC

0.025

0.050

0.055

0.060

30

Time (s)

ZFC FC

0.075

0.095

0.100

0.105

31

Time (s)

ZFC FC

0.110

0.125

0.150

0.155

32

Time (s)

ZFC FC

0.160

0.175

0.195

0.200

33

Time (s)

ZFC FC

0.205

0.210

0.225

0.250

Figure 4-2 – Evolution of the current density inside the superconductor for a sinusoidal magnetic field applied in ZFC and FC

Regarding the current densities higher than the critical value, it causes the resistivity of the

material and, consequently, the losses, to increase. It still can, however, hold the magnetic properties

of the superconducting state.

34

4.1.1 MAGNETIC FIELD AMPLITUDE ANALYSIS

As shown in the previous example, the magnetic field applied to the superconductor affects its

internal current density distribution, and consequently, its magnetization. The values analyzed range

from 0.01 to 1.26 T, this last value corresponding to the knee or working point regarding the saturation

of silicon steel used in electric machines. The dashed curves correspond to the ZFC simulations, while

the solid curves correspond to the FC simulations, which were done using 𝐵𝑀 = 1.26 𝑇. The other

parameters used were 𝑓 = 5𝐻𝑧, 𝐽𝐶0 = 2 × 107 𝐴/𝑚2 and 𝐵0 = 0.1 𝑇.

Figure 4-3 shows the hysteresis cycle for various values of magnetic flux density 𝐵𝑚.

Figure 4-3 – Hysteresis cycle dependency of magnetic flux density 𝐵𝑚

It is shown that ZFC yields a symmetric hysteresis cycle, which fits with the symmetric applied

magnetic field. For FC, this does not happen due to the asymmetry of the applied field. The figure also

clearly shows that the sign of the magnetization depends on the derivative of the applied magnetic field:

positive derivatives yield negative magnetizations, and vice versa, which fits with the tendency of the

superconductor to counteract any changes in its internal field.

Due to the characteristic shape of the hysteresis cycle, higher losses are associated with ZFC or

low values of 𝐵𝑀 and high magnetic field amplitudes (𝐵𝑚), and lower losses are associated with FC with

higher values of 𝐵𝑀 and lower magnetic field amplitudes.

35

Figure 4-4 shows the instantaneous power density losses for the same values of magnetic flux

density 𝐵𝑚. The values is this figure are taken from equation (3.12), which cannot be separate from

equation (3.11) in this analysis, given its dependence with the internal current density distribution.

Figure 4-4 – Instantaneous power density losses dependency of magnetic flux density 𝐵𝑚

Given equation (2.9), it is expected that for lower magnetic fields, the currents will be higher, thus

increasing the losses. It is seen that the power losses peaks coincide with the instants of higher

magnetization, and consequently higher current densities.

Figure 4-5 shows the power density losses for different values of magnetic flux density 𝐵𝑚.

36

Figure 4-5 – Power density losses dependency of magnetic flux density 𝐵𝑚

It is clear that the losses have a strong dependence with the magnetic field amplitude. In the ZFC

case, particularly until 0.4 T, and in a less strong manner after that. The difference in the ZFC and FC

cases will depend on the trapped field 𝐵𝑀. Higher trapped fields will yield lower losses for the same

magnetic field amplitude. The values studied fall within practical values used in electric machines, thus

it’s important to take this into account regarding the use of superconducting elements in practical

applications.

37

4.1.2 FREQUENCY ANALYSIS

Considering the use of HTS magnets in low speed synchronous generators, the presence of

electric currents circulating in the stator windings create magnetic fields which will disturb the HTS

magnets. In this context, it is important to study how these perturbations affect the operation

characteristics of HTS magnets and their associated power losses. As in previous sections, ZFC and

FC cases were considered, and the magnitude of the applied magnetic flux density was kept constant,

with 𝐵𝑚 = 1.26 𝑇 . Other parameters used were 𝐽𝐶0 = 2 × 107 𝐴/𝑚2 and 𝐵0 = 0.1 𝑇 .The frequencies

studied ranged from 0.25 to 100 Hz. Figure 4-6 shows the hysteresis cycle for the various frequencies.

Figure 4-6 – Hysteresis cycle dependency of applied frequency

It can be seen that a change in frequency does not alter the hysteresis curve significantly.

However, this curves do not correspond to the same time period, and we need to look at the losses

expressed in power units. Looking at equation (3.11) and taking in consideration the slight increase in

the area of the hysteresis cycle for an increase in frequency, it is expected that the power density losses

dependency with frequency to be higher than linear (the losses dependency would be linear if the

hysteresis cycle area did not change with frequency).

38

Figure 4-7 shows the instantaneous power density losses for different frequencies of the applied

magnetic field.

Figure 4-7 – Instantaneous power density losses dependency of applied frequency

It is shown that higher frequencies have higher spikes, but for smaller time periods, which means

that the difference in height is almost balanced (due to the non-linearity shown above) by the width of

the spike.

Figure 4-8 shows the power density losses for different values of frequency.

Figure 4-8 – Power density losses dependency of applied frequency

39

Frequency is a very important parameter to take into account using HTS bulks in electric

machines. As can be seen, these materials have higher efficiencies for machines that work at low

speeds.

40

4.1.3 JC0 ANALYSIS

The parameter 𝐽C0 represents the maximum critical current density in the superconductor. As

such, it is a good indicator of the expected orders of magnitude, according to equation (2.9). This is a

parameter worth studying due to the influence of the manufacturing process of bulk HTS in the definition

of this parameter. The parameters used were 𝑓 = 5𝐻𝑧, 𝐵𝑀 = 1.26 𝑇, 𝐵𝑚 = 1.26 𝑇 and 𝐵0 = 0.1 𝑇.

Figure 4-9 and Figure 4-10 show the hysteresis cycle and instantaneous power losses,

respectively, for different values of maximum critical current density 𝐽C0.

Figure 4-9 – Hysteresis cycle dependency of maximum critical current density

A higher critical current density means that for the same applied magnetic field, there will be a

higher current density, which means a higher magnetization, as shown above. The difference is more

evident when the magnetic field is minimum inside the superconductor, and the critical current density

reaches its maximum value. Higher current densities also mean higher power losses, according to

equation (3.11).

41

Figure 4-10 – Instantaneous power density losses dependency of maximum critical current density

Figure 4-11 shows the power density losses for different values of 𝐽𝐶0.

Figure 4-11 – Power density losses dependency of maximum critical current density

42

It is shown that the 𝐽𝐶0 parameter is critical regarding the power losses in the superconductor.

The previous figures show an important dependency with the 𝐽𝐶0 parameter. This parameter is very

important since it specifies the expected maximum order of magnitude of the internal currents in the

superconductor. It is easy to relate the higher current densities with higher losses, as can be observed

in Figure 4-10. However, higher current densities also yield higher magnetizations, as shown in Figure

4-9. The non-linearity shown in Figure 4-11 could be attributed to equation (2.9).

43

4.1.4 B0 ANALYSIS

The 𝐵0 parameter defines how much the critical current density will decrease in the presence of

a magnetic field, more specifically the value of magnetic flux density at which the critical current density

drops to half its maximum value. As such, it is also a very important parameter in all applications

involving magnetic fields. The parameters used were 𝑓 = 5𝐻𝑧, 𝐵𝑀 = 1.26 𝑇, 𝐵𝑚 = 1.26 𝑇 and 𝐽𝐶0 = 2 ×

107 𝐴/𝑚2. Figure 4-12 and Figure 4-13 show the hysteresis cycle and instantaneous power losses,

respectively, for different values of the 𝐵0 parameter.

Figure 4-12 – Hysteresis cycle dependency of 𝐵0 parameter

According to equation (2.9), higher values of 𝐵0 yield higher critical current densities for the same

value of magnetic flux density (higher than zero), so consequently it will yield higher magnetizations.

Note that the difference between the values studied in now more pronounced for higher fields due to the

maximum critical current density 𝐽𝐶0 being the same.

44

Figure 4-13 – Instantaneous power density losses dependency of 𝐵0 parameter

As described before, the higher critical current densities yield by higher values of 𝐵0 will yield

higher power losses according to equation (3.11).

Figure 4-14 shows the power density losses for different values of 𝐵0.

45

Figure 4-14 – Power density losses dependency of 𝐵0 parameter

The results show that the 𝐵0 parameter is also determining regarding the power losses of the

superconductor. As higher values of this parameter are associated with higher critical current densities,

they also come with higher power losses. The non-linearity shown in Figure 4-14, similar to Figure 4-11,

could also be explained by equation (2.9). Still regarding this equation, it is seen that the 𝐵0 parameter

preferably takes the highest value possible in order to decrease the critical current density as least as

possible, which will yield higher magnetizations, higher trapped magnetic fields and better magnetic

shielding for higher magnetic fields. However, the higher losses for higher values of this parameter need

to be taken into account.

46

4.1.5 GEOMETRIC DIMENSIONS ANALYSIS

Another study was done regarding the geometric dimensions of the superconductor, specifically

its width, for the same height. The superconductor dimensions have implications in the distribution of

the internal currents, specifically by affecting the area where these currents can circulate. By extension,

this study also applies to the area of the cross section of the superconductor. The parameters used were

𝑓 = 5𝐻𝑧, 𝐵𝑀 = 1.26 𝑇, 𝐵𝑚 = 1.26 𝑇, 𝐽𝐶0 = 2 × 107𝐴/𝑚2 and 𝐵0 = 0.1 𝑇.

Figure 4-15 and Figure 4-16 show the hysteresis cycle and instantaneous power losses,

respectively, for ZFC (dashed lines) and FC (solid lines), for values of the superconductor width ranging

from 4 to 20 cm.

Figure 4-15 – Hysteresis cycle dependency of superconductor width

Intuitively, one can theorize that bigger dimensions allow for a larger distribution of internal

currents inside the superconductor. As described before, higher currents create higher magnetizations.

This is demonstrated in Figure 4-15.

.

47

Figure 4-16 – Instantaneous power density losses dependency of superconductor width

Likewise, higher currents mean higher power density losses, also demonstrated in Figure 4-16.

Figure 4-17 – Power density losses dependency of superconductor width

48

From the results obtained, a linear dependency of the power density losses can be seen regarding

the width of the superconductor. This leads to problems of optimization regarding the dimensions of the

superconductors in electric machines. However, since a 2D FEM analysis was used, this result will have

to be confirmed and studied further by 3D simulations.

49

4.2 FIELD COOLING TRANSIENT ANALYSIS FOR TRAPPED FIELD

This section studies some transient tests with the objective of magnetizing the superconductor

using the Field Cooling technique, by looking at the value of magnetic flux density trapped in in the

superconductor after the initial field is removed and studying which parameters will be more important

to increase this value. The simulations in this subchapter admit a constant temperature of 77 K with an

already superconducting material.

An external field is applied, decreasing linearly from an initial value 𝐻𝑀0, defined by 𝐵𝑀0/𝜇0, to

zero for a specified time interval 𝑃. The boundary condition is set up as (4.3):

𝐻𝑥 = 0

𝐻𝑦 = 𝐻𝑀0 −𝐻𝑀0𝑃𝑡, (0 ≤ 𝑡 ≤ 𝑃)

(4.3)

Initially, the results of [17] were replicated. Furthermore, some conditions of this simulation were

studied, including, similarly as in the previous section, the initial applied magnetic flux density 𝐵𝑀0, the

derivative of the applied field (by varying 𝑃), the maximum critical current density 𝐽𝐶0, the parameter 𝐵0,

the geometric dimensions of the superconductor (specifically its width), and a long term analysis is also

done where it is studied the influence of a small amplitude sinusoidal field applied after removing the

initial field. A description of this process is made, regarding the internal currents in the superconductor.

Figure 4-18 – Example of magnetic flux density applied in the case of Field Cooling

Initially, the applied field limits the current density according to its relation with the 𝐵0 parameter,

as shown in equation (2.9). Like described before, the currents emerge from the edges and gradually

fill the superconductor. Due to (2.9), the current density also increases, and after the applied magnetic

field reaches zero, it suffers a slight decay. The internal currents then produce a magnetic field which

resembles the one of a permanent magnet. The described process is illustrated in Figure 4-19.

50

0.00 s

0.01 s

0.05 s

0.10 s

0.15 s

0.20 s

0.30 s

0.40 s

Figure 4-19 – Evolution of the current density inside the superconductor for a linearly decreasing magnetic field applied in a process of Field Cooling

The final current distribution in the center of the superconductor (y=0) is shown in Figure 4-20 –

Current distribution in the superconductor after FC in trapped field.

Figure 4-20 – Current distribution in the superconductor after FC in trapped field

51

4.2.1 INITIAL MAGNETIC FIELD ANALYSIS

In this study, the initial applied is varied while the instant when the applied field reaches zero is

kept constant (𝑃 = 0.05 𝑠). The parameters used were 𝐽𝐶0 = 2 × 107𝐴/𝑚2 and 𝐵0 = 0.1 𝑇. The magnetic

flux density below is the maximum value in the upper boundary of the superconductor.

Figure 4-21 – Evolution of the maximum trapped magnetic flux density for different initial magnetic flux densities

Given the results in Figure 4-21, it can be seen that the maximum initial field does not significantly

affect the maximum trapped field for values equal or higher than 0.25 T, assuming the same values of

𝐽𝐶0 and 𝐵0. For lower values, the initial field is approximately kept constant.

Figure 4-22 shows the evolution of the current density for two values of 𝐵𝑀0. For the lowest value,

the maximum critical current density is closer to 𝐽𝐶0 and, as the change in magnetic field in small, the

currents expand only slightly. In the case of the highest value, the maximum current density is lower and

need more space to counter the change in magnetic field. As such, they fill the entire volume of the

superconductor. As the initial field is progressively removed, the current density increases. It can be

seen that the maximum trapped magnetic field is limited by the current distribution, reaching its limit

when the currents fill the superconductor.

52

Time (s)

𝐵𝑀0 = 0.1 𝑇 𝐵𝑀0 = 1 𝑇

0.010

0.025

0.035

0.050

Figure 4-22 – Comparison of the evolution of the current density distribution inside the superconductor for different initial magnetic flux densities

53

Figure 4-23 shows a comparison of the magnetic field of the final state for the two values of 𝐵𝑀0

shown in Figure 4-22. A relation between the magnetic field and current density is clearly visible: the

points with higher magnetic field are the ones with lower current density.

𝐵𝑀0 = 0.1 𝑇 𝐵𝑀0 = 1 𝑇

Figure 4-23 – Comparison of the final magnetic flux density distribution for different initial magnetic flux densities

The illustration of the distribution of the magnetic flux density on the superconductor helps to

understand the differences shown before and can help with the design of electric machines using this

technology.

54

4.2.2 APPLIED FIELD DERIVATIVE ANALYSIS

In this case, the maximum applied magnetic flux density is of 0.5 T. The parameters used were

𝐽𝐶0 = 2 × 107𝐴/𝑚2 and 𝐵0 = 0.1 𝑇. The goal was to attest if the trapped field value could be influenced

by the derivative of the magnetic field when removing it from the superconductor vicinity.

Figure 4-24 – Evolution of the maximum trapped magnetic flux density for different derivatives

According to the results obtained and shown in Figure 4-24, the derivative of the applied field

does not affect the maximum trapped field in steady state. For lower derivatives, the trapped field is

slightly lower at the moment when the external field is completely removed, although the differences are

not significant in the long term.

Figure 4-25 shows the evolution of the current density for two values of 𝑃, which is approximately

identical, with only slight differences. For the lowest value of 𝑃, a higher value of current density can be

seen, which accounts for what is seen in Figure 4-24.

55

Time (s)

𝑃 = 0.005 𝑠 𝑃 = 1 𝑠

0.2𝑃

0.5𝑃

0.7𝑃

𝑃

Figure 4-25 – Comparison of the evolution of the current density distribution inside the superconductor for different applied field derivatives

56

Figure 4-26 shows a comparison of the magnetic field of the final state for the two values of 𝑃

shown in Figure 4-25. Again, there is a slightly higher magnetic field for the lower 𝑃, which will have no

impact in the long term, as seen in Figure 4-24.

𝑃 = 0.005 𝑠 𝑃 = 1 𝑠

Figure 4-26 – Comparison of the final magnetic flux density distribution for different applied field derivatives

The results show that the derivative of the external field has no influence on the trapped field in

steady state.

57

4.2.3 JC0 ANALYSIS

As stated before, higher current densities yield higher magnetic fields, and the goal in this section

was to evaluate how 𝐽𝐶0 influences the final value of the trapped field. The parameters used were 𝐵𝑀0 =

0.5 𝑇, 𝑃 = 0.05 𝑠, and 𝐵0 = 0.1 𝑇.

Figure 4-27 – Evolution of the maximum trapped magnetic flux density in the superconductor for different values of 𝐽𝐶0

Given the results achieved and shown in Figure 4-27, it is evident that for higher maximum current

densities, higher magnetic fields can be trapped.

Figure 4-28 shows the evolution of the current density for two values of 𝐽𝐶0. A lower critical current

density means that it will expand more because lower current densities need more volume to counter

the change in magnetic field. The opposite happens for higher critical current densities. For the higher

value of 𝐽𝐶0, currents don’t fill the superconductor, but these currents are much higher that the case for

the lower value of 𝐽𝐶0, which in fact yield a higher trapped field.

58

Time

(s)

𝐽𝐶0 = 5 × 106 𝐴/𝑚2 𝐽𝐶0 = 1 × 10

8 𝐴/𝑚2

0.010

0.025

0.035

0.050

Figure 4-28 – Comparison of the evolution of the current density distribution inside the superconductor for different maximum critical current densities

59

Figure 4-29 show the final state of magnetic flux distribution, which supports what was described

before. For the same conditions, higher maximum current densities allow higher trapped fields.

𝐽𝐶0 = 5 × 106 𝐴/𝑚2 𝐽𝐶0 = 1 × 108 𝐴/𝑚2

Figure 4-29 – Comparison of the final magnetic flux density distribution for different maximum critical current densities

60

4.2.4 B0 ANALYSIS

This parameter is also very important in magnetic field trapping. Higher values of 𝐵0 allow higher

current densities, which allow higher trapped magnetic fields. The parameters used were 𝐵𝑀0 = 0.5 𝑇,

𝑃 = 0.05 𝑠, and 𝐽𝐶0 = 2 × 107 𝐴/𝑚2.

Figure 4-30 – Evolution of the maximum trapped magnetic flux density for different values of 𝐵0

The results in Figure 4-30 predictably show that the higher the values of 𝐵0, the higher is the

magnetic field that can be trapped in the superconductor.

Figure 4-31 shows the evolution of the current density for two values of 𝐵0. As described before,

for the same applied magnetic field, higher values of 𝐵0 allow for higher current densities, which in turn

allow higher trapped fields.

61

Time (s)

𝐵0 = 0.02 𝑇 𝐵0 = 2 𝑇

0.010

0.025

0.035

0.050

Figure 4-31 – Comparison of the evolution of the current density distribution inside the superconductor for different values of 𝐵0

62

Figure 4-32 show the final state of the magnetic flux distribution. As in the previous section, it

shows the importance of the current density (defined by the parameter 𝐵0) in the value of the trapped

field .

𝐵0 = 0.02 𝑇 𝐵0 = 2 𝑇

Figure 4-32 – Comparison of the evolution of the magnetic flux density distribution for different values of 𝐵0

63

4.2.5 GEOMETRIC DIMENSIONS ANALYSIS

As before, an analysis was made in order to study the influence of the superconductor width, now

in terms of trapped field. The parameters used were 𝐵𝑀0 = 0.5 𝑇, 𝑃 = 0.05 𝑠, 𝐽𝐶0 = 2 × 107𝐴/𝑚2 and

𝐵0 = 0.1 𝑇.

Figure 4-33 – Evolution of the maximum trapped magnetic flux density for different values of superconductor width

The results in Figure 4-33 show an ability of hold higher magnetic fields for wider

superconductors. This can be useful when projecting electric machines.

Figure 4-34 shows the evolution of the current density for two values of superconductor width.

For larger widths, there is more volume for the currents to occupy, which means that there is room for

higher currents. However, contrastingly with the losses analysis in section 4.1.5, the trapped field does

not show a linear dependency with this parameter. That might be to the non-linearity of the current

distribution inside the superconductor (the “U” shape) and the limit that is the filling of the superconductor

by these currents.

64

Time (s)

𝑐 = 4 𝑐𝑚 𝑐 = 20 𝑐𝑚

0.010

0.025

0.035

0.050

Figure 4-34 – Comparison of the evolution of the current density distribution inside the superconductor for different superconductor widths

65

Figure 4-35 shows the final state of the distribution of the magnetic flux density in the

superconductor, which illustrates a substantial difference.

𝑐 = 4 𝑐𝑚 𝑐 = 20 𝑐𝑚

Figure 4-35 – Comparison of the final magnetic flux density distribution for different superconductor widths

The difference in magnetic flux density distribution for different geometries is something that can

also be useful in the design of electric machines using this technology.

66

4.2.6 LONG TERM STUDY

A long term study was done in order to analyze the value of the trapped field after the applied

field had been removed. In this study, the parameters were set as 𝐵𝑀0 = 1 𝑇 and 𝑃 = 1 𝑠, 𝐽𝐶0 = 2 ×

107𝐴/𝑚2 and 𝐵0 = 0.1 𝑇.

Figure 4-36 – Long term study of trapped magnetic flux density after FC study

The results in Figure 4-36 show some decay in the maximum trapped value. However, in

experiments done in the lab, this decay was not observed. In practical applications, the decay is very

slow and negligible.

Another study was done to see the influence of a sinusoidal magnetic field applied after the initial

field was removed. Using 𝐵𝑀0 = 0.5 𝑇 and 𝑃 = 1 𝑠, a sinusoidal field with an amplitude 𝐵𝑚 = 0.1 𝑇 and

a frequency 𝑓 = 0.25 𝐻𝑧 was applied. Other parameters used were 𝐽𝐶0 = 2 × 107𝐴/𝑚2 and 𝐵0 = 0.1 𝑇.

Two situations were considered: with positive and negative initial derivative. The results are presented

in Figure 4-37.

67

Figure 4-37 – Maximum magnetic flux density for different initial derivatives of a sinusoidal field applied after magnetization using Field Cooling

Given the results obtained, it is observed that a sinusoidal magnetic field applied after a process

of Field Cooling decreases the magnetization of the superconductor. The initial derivative of the applied

field does not influence the magnetization of the superconductor in the long term.

Another study was done to attest the influence of the frequency of the applied field in the

demagnetization of the superconductor, shown in Figure 4-38.

Figure 4-38 – Maximum magnetic flux density for different frequencies of a sinusoidal field applied after magnetization using Field Cooling

The results shown no difference regarding the use of different frequencies.

68

Another study was done to attest the influence of the amplitude of the applied field in the

demagnetization of the superconductor, shown in Figure 4-39. The dashed lines follow the case where

no field was applied minus the RMS value of the amplitude of the applied field.

Figure 4-39 – Maximum magnetic flux density for different amplitudes of a sinusoidal field applied after magnetization using Field Cooling

Results show that the demagnetization depends on the amplitude of the applied field and that the

maximum magnetic field density tends to be reduced by the RMS value of the applied field.

69

4.3 EXPERIMENTAL RESULTS

An experiment was done to verify how much time a superconductor magnetized using FC takes

to lose its superconducting state when a periodic external magnetic field is applied. This was done by

applying a sinusoidal magnetic field with different amplitudes.

Figure 4-40 – Schematic of the lab experiment

A coil of 1mm diameter wire was used, with 1000 turns. A magnet was used in order to see the

moment of loss of superconductivity (as the magnetic properties of the superconductor are lost, the

levitation by flux pinning ends, and the magnet falls). The frequency applied was 50 Hz. The magnetic

flux density was produced using currents of 1, 2, 3 and 4 A. With the resources available, a current of 5

A was impossible to reach. The magnetic field was measured by placing a secondary coil, with 6 turns

and 6 cm of diameter, in the position of the superconductor and calculated through the induced voltage.

The values of magnetic flux density obtained were 46.9, 93.8, 140.1, 187.6 mT, respectively.

AIR

MAGNET

YBCO

STYROFOAM

AIR

CO

IL C

OIL

LIQUID NITROGEN

70

a)

b)

Figure 4-41 – a) Early stages of the coil setup b) Example of the magnet and superconductor setup

It was verified that, while submerged in liquid nitrogen, the superconductor never lost the

superconducting state. As such, the tests were done after cooling the superconductor and then removing

the liquid nitrogen. The results are shown in Table 4-1 and Figure 4-42, with the case of no applied field

for comparison.

These results fit well with the expectations from the theoretical model and simulations. The

decrease in superconducting time is associated with higher losses for higher magnetic fields.

71

Table 4-1 – Experimental results for the time it takes the YBCO bulk superconductor to lose superconductivity

𝑩𝒎𝒂𝒙 (𝒎𝑻) 𝟎 𝟒𝟔. 𝟗 𝟗𝟑. 𝟖 𝟏𝟒𝟎. 𝟏 𝟏𝟖𝟕. 𝟔

𝒕𝒎𝒂𝒙

1:40 1:35 1:06 1:16 1:32

1:55 1:46 2:06 1:20 1:30

2:05 1:35 1:56 2:31 1:30

2:05 1:38 1:58 1:20 1:22

1:57 1:38 1:52 1:21 1:23

2:30 1:50 1:38 1:43 1:20

2:12 1:54 1:53 1:34 1:35

2:12 1:51 1:37 1:56 1:10

2:00 1:40 1:51 1:35 1:43

2:07 1:49 1:51 1:39 1:50

2:44 2:07 2:18 1:58 1:22

2:47 2:19 1:56 1:30 1:44

2:47 2:19 2:02 1:21 1:19

𝒕𝒂𝒗 2:14 2:04 1:51 1:37 1:29

Figure 4-42 – Experimental results for the time it takes the YBCO bulk superconductor to lose superconductivity

72

4.4 THERMAL ANALYSIS

The process of decreasing the YCBCO bulk temperature by submerging it in liquid nitrogen is first

studied. The cooling process observed in the lab revealed a cooling time (from room temperature until

superconductivity) around 1 minute for a bulk piece with dimensions 4x4x1.5 cm, approximately.

A FEM simulation of the cooling process was done. In the simulation, a bulk piece with a diameter

of 4 cm and a height of 1.5cm was used, and an initial temperature of 293.15 K (20 ºC) for the

superconductor, styrofoam, and air and 77 K (-196.15 ºC) for the liquid nitrogen were used. The thermal

parameters used are shown in Table 3-1 and the simulation setup is shown in Figure 4-43.

Figure 4-43 – Schematic of the thermal simulations

Figure 4-44 shows the evolution of the average temperature of the superconductor after

immersing it in liquid nitrogen. To help in the sense of scale, the temperatures of 93 K, corresponding

to the critical temperature of the superconductor, and 77 K, corresponding to the boiling point of liquid

nitrogen, are marked.

AIR

YBCO STYROFOAM

LIQUID NITROGEN

73

Figure 4-44 – Evolution of the superconductor temperature

Figure 4-45 shows the temperature distribution after 2 minutes of cooling, which better illustrates

the temperature differences between the superconductor and the liquid nitrogen.

Figure 4-45 – Final temperature distribution after 2 minutes of cooling using default liquid nitrogen parameters

As this did not fit with the experimental observations, an adjustment of the parameters of the liquid

nitrogen was done, adjusting its thermal conductivity and heat capacity, by multiplying these by factors

K and CP, respectively.

At first, an analysis of the thermal conductivity was done, as seen in Figure 4-46, which shows a

larger initial temperature drop for higher thermal conductivities, which does not hold in the long term and

is not enough to bring the superconductor to superconductivity.

74

Figure 4-46 – Cooling process for different values of thermal conductivity

It was then done an analysis regarding the specific heat capacity, shown in Figure 4-47. The

cooling process proved less sensible to this parameter, and superconductivity was still not reached.

Figure 4-47 – Cooling process for different values of specific heat capacity

As the fitting of one parameter alone was insufficient, an analysis of both parameters was done.

Figure 4-48 shows how the evolution of the average temperature in the superconductor changes for

different pairs of K and CP. By increasing both the thermal conductivity and the specific heat capacity

simultaneously, in this case by the same factor (K=CP), results resembling the experimental ones were

obtained.

75

Figure 4-48 – Cooling process for different values of specific heat capacity and thermal conductivity

Figure 4-49 shows a detail of Figure 4-48, where can be seen that for K and CP equal to 150, a

cooling time of around 1 minute can be obtained.

Figure 4-49 – Detail of the cooling process for different values of specific heat capacity and thermal conductivity

Figure 4-50 presents the temperature distribution after 2 minutes, which shows the

superconductor and the liquid nitrogen with about the same temperature, due to the color scale.

However, notice that the minimum temperature (77.5 K) corresponds to the liquid nitrogen, while the

maximum temperature (293 K) corresponds to the air and exterior part of the styrofoam.

76

Figure 4-50 – Final temperature of the cooling process

The time that the superconductor takes to lose the superconducting state was also studied by

removing the liquid nitrogen. In experimental observations, by simply removing the liquid nitrogen,

superconductivity was lost in approximately 2 minutes and 14 seconds, as shown in the previous

section. Figure 4-51 shows the evolution of the average temperature during a cooling process of 5

minutes and for 3 minutes after removing the liquid nitrogen:

Figure 4-51 – Evolution of the superconductor temperature during the cooling and heating process

The results show that the after removing the liquid nitrogen, the superconductor loses the

superconducting state after 2 minutes and 9 seconds, which is close to the experimental results.

77

Figure 4-52 shows the temperature distribution after 3 minutes of removing the liquid nitrogen.

Figure 4-52 – Temperature after 3 minutes of removing the liquid nitrogen

These results fit well with the experimental observations. However, it is worth mentioning that

higher cooling periods yield longer times until the loss of superconductivity in these simulations.

Experimentally, this was not studied, but the reality is far more complex that these simulations show, as

they do not contemplate phase changes nor fluid flow.

78

4.5 ELECTROMAGNETIC AND THERMAL ANALYSIS

The superconductor can present significant power losses causing thermal effects that can

degrade its performance, leading to an eventual loss of superconductivity in certain parts or even in the

entire superconductor. Hence, it is important to derive a model describing the coupled electromagnetic

and thermal phenomena in the superconductor.

Figure 4-53 illustrates the setup considered for these simulations: only the liquid nitrogen and the

YBCO bulk are considered. The geometrical, electromagnetic and thermal parameters used are the

same as used in the previous simulations.

Figure 4-53 – Setup of the electromagnetic and thermal simulations

Note that these simulations do not take into account any phase change from liquid nitrogen to

nitrogen gas, as they follow the model described in Chapter 3 and also the conditions described in

section 4.1.

YBCO

LIQUID NITROGEN

79

4.5.1 HEATING ANALYSIS

Figure 4-54 shows the evolution of the superconductor maximum and average temperatures

when subjected to an external sinusoidal field after ZFC. The applied field had a maximum magnetic

flux density 𝐵𝑚 = 1.26 𝑇 and a frequency 𝑓 = 0.25 𝐻𝑧.

Figure 4-54 – Evolution of the superconductor maximum and average temperatures

Results show that the temperature evolution presents two distinct thermal constants: a shorter

one of about 10 seconds, and a longer one in the order of minutes.

Figure 4-55 – Temperature distribution after 3 minutes

80

The final temperature of the previous simulation is shown in Figure 4-55. As has been shown in

previous sections, the internal currents originate from the edges, and have a higher value in these areas

due to the trapped field in the middle. As such, the superconductor will tend to be hotter around these

parts, but as they are also in contact in the liquid nitrogen, they are also the parts that are better cooled,

making the hottest parts being between the middle and the edges of the piece.

This simulation was also useful to study the evolution of other variables and how they evolve over

time, like the critical current density, the average power losses inside the superconductor, and the

magnetization. The evolution of the maximum internal and critical current densities is shown in Figure

4-56.

Figure 4-56 – Evolution of the maximum internal and critical current densities

Figure 4-57 – Evolution of the maximum and minimum critical current densities

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The maximum and minimum critical current densities is shown in Figure 4-57. In the instants of

lower magnetic flux density inside the superconductor, the results show a small decrease in the

maximum internal current density, which contrasts with an increase, similar to the temperature increase,

for the maximum critical current density, after an initial decrease. Note that these values of maximum

critical current density will correspond to the middle area, shown in Figure 4-58, in the instant that the

trapped field is minimized because of the increasing field in the opposite direction. This increase in

critical current density might be explained by the decrease in the internal current density, causing a

lower trapped field, which results in a lower magnetic flux density for that same instant.

Figure 4-58 – Comparison between magnetic flux density and critical current density

Although the previous results are important in the sense that they give a better understanding of

the values of the internal and critical current densities, the fact that Figure 4-56 shows values of critical

current density increasingly higher than that of the internal current density suggests that the losses

should decrease. However, the values are evaluated for the entire superconductor, which means it does

not hold necessarily for every point. As has been shown before, these variables take a whole range of

values inside the superconductor. As such, a better way to “see” the losses is to look at the ratio between

the internal current density and the critical current density, which is shown in Figure 4-59.

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Figure 4-59 – Evolution of the ratio 𝐽/𝐽𝐶 inside the superconductor

The fact that the internal currents surpass the critical current density does not mean a loss of

superconductivity. Instead, what this means is that the electrical resistivity of the superconductor

becomes not negligible. The fact that the ratio 𝐽/𝐽𝐶 can reach values of 2.7 does not necessarily mean

high power losses either, as it is only valid for a small volume. On average, the overall ratio remains in

a constant range around 0.7 and 1.2, approximately. As an example, the ratio 𝐽/𝐽𝐶 is shown for 2 points

inside the superconductor, shown in Figure 4-60.

Figure 4-60 – Evolution of the ratio 𝐽/𝐽𝐶 for two points inside the superconductor

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Note that these are examples taken on single points and do not necessarily reflect the behavior

of the superconductor in the surrounding region. Point 1 is worth considering, given that it is in the middle

of one of the hot spots of the superconductor. The maximum ratio is approximately constant, while the

minimum shows an increase after an initial decrease, which suggests increasing losses in that point.

Point 2 shows significant losses throughout the analysis, but it is also close to the edge, where it is better

cooled, which explains its lower temperature. Remember that the average ratio remains in a constant

range, as shown in Figure 4-59, implying lower power losses than what might be suggest by Figure

4-60.

Figure 4-61 – Evolution of the average power loss in the superconductor

Figure 4-61 shows the evolution of the average power losses. It can be seen that, as the internal

current density decreases, the power losses decrease accordingly, as has also been shown in previous

sections.

Figure 4-62 – Evolution of absolute value of the magnetization of the superconductor

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Figure 4-62 shows the magnetization of the superconductor in absolute values. The results show

a decrease similar to the previous variables. As described before, the magnetization depends on the

internal currents, so as the currents decrease, so does the magnetization.

Figure 4-63 – Comparison between different two different maximum magnetic flux density values

A comparison of different conditions was also done, in order to validate the previous decoupled

simulations. Figure 4-63 shows a comparison between different values of magnetic field amplitude. As

can be seen, these results are expected taking into account the losses calculated in the previous

simulations. However, the relation between the temperature evolution between the 1.26 T and the 0.6

T field is lower than what the electromagnetic study suggests. This might be due to the lower critical

current densities at higher temperatures in the case of the 1.26 T field, which would imply lower losses.

Figure 4-64 – Comparison for different frequency values

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An analysis of the frequency of the magnetic field was also done, shown in Figure 4-64. With the

results obtained, the influence of the frequency in the superconductor losses and its almost linear

characteristic is evident, as for twice the frequency there is almost a double in temperature rise.

However, in the case of higher frequency, the temperature evolution appears to be slightly lower than

expected, suggesting the same decrease in power loss for higher temperatures. Higher frequencies

were studied, for example 50 Hz for the same conditions, but it was found that the superconductor lost

superconductivity too quickly for comparison with the previous examples. This loss of superconductivity

was localized, not global, but it stopped the simulation and no solution was found to solve this problem.

This does not mean that in real applications the same result will happen. As has been stated, the thermal

part of the model cannot be seen as an accurate representation of real experiments.

Figure 4-65 – Comparison of the heating process between ZFC and FC conditions

Furthermore, a comparison between ZFC and FC conditions was done, with 𝐵𝑀 = 1.26 𝑇 for the

FC process, shown in Figure 4-65. The results support the differences seen in the electromagnetic

study, which for 1.26 T shows a difference of about twice the losses for ZFC, in comparison with the FC

case.

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4.5.2 COOLING ANALYSIS

This study was done taking as initial conditions the final temperature distribution from the previous

heating analysis that ended after 3 minutes, and now the external applied field is removed, leaving the

process of cooling by liquid nitrogen.

Figure 4-66 shows the evolution of the maximum and average temperatures.

Figure 4-66 – Evolution of the superconductor temperature after removing the applied field

It can be seen that in less than 30 seconds the maximum temperature drops below 77.5 K, which

represents a drop larger than 1.5 K. From there, the temperature drop is slower. The first thermal

constant can be attributed to the transient interval of the electromagnetic variables, as will be shown in

subsequent figures, and the second thermal constant is related with the stabilization of the same

variables.

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Figure 4-67 – Evolution of the maximum and minimum critical current density and maximum internal current density after removing the applied field

Figure 4-67 shows the evolution of the maximum internal and critical current densities. As

described before, this result is just illustrative of the maximum values in question. The temperature drop

allows for an increase in the critical current density, as expected. As the initial time step was a zero in

the waveform of the applied field, the maximum critical current density takes a high value. However, not

as high as it can go, because of the trapped field, which, by being a zero in the applied field, is not

minimized with the opposing magnetic field. The internal current density will decrease as has been

observed in the trapped field analysis, so these results fit well with the previous simulations.

Figure 4-68 shows the evolution of the ratio between the internal and critical current densities.

Figure 4-68 – Evolution of the ratio 𝐽/𝐽𝐶 in the superconductor after removing the applied field

88

In Figure 4-68 it can be seen a steep decrease in the ratio between the internal and critical current

densities, falling quickly below 1, on average, and rendering the resistivity and the losses quickly

negligible, as can be seen in Figure 4-69.

Figure 4-69 – Evolution of the average power losses in the superconductor after removing the applied field

Figure 4-69 shows the evolution of the average power losses in the superconductor. It is shown

that the power losses quickly become negligible, which fits with the steep decrease in temperature seen

in Figure 4-66.

Figure 4-70 – Evolution of the magnetization in the superconductor after removing the applied field

Figure 4-70 shows the evolution of the magnetization in the superconductor. As seen before in

the previous studies, with a large enough field, there will be a trapped field in the superconductor. This

result shows an evolution of the magnetization similar to the decay of trapped field that was obtained in

the section 4.2.6, which, in addition to the results from the previous figures, adds to the consistency of

the results.

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5 CONCLUSIONS

This chapter finalizes this work, summarizing conclusions and pointing out aspects to be

developed in future work.

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5.1 ACHIEVEMENTS

The goals stated in the first chapter included the study, modeling and analysis of the process of

magnetization in high-temperature superconducting material, in this case YBCO. Additionally, the

verification of the ability to maintain the level of magnetization of the material when subjected to a time-

variant magnetic field was also covered and also an analysis of the advantages and disadvantages of

this solution. The expected achievements were the development of an electromagnetic macroscopic

model for simulation of bulk superconductors and its hysteretic magnetization, which included the

determination of a set of parameters and their characteristics to describe the behavior of high-

temperature superconductors in magnetic field confinement. Finally, a description and explanation of

the advantages and disadvantages of this solution was expected.

The electromagnetic analysis allowed for an extensive study regarding the behavior of high

temperature superconductors subjected to magnetic fields. The magnetization and hysteresis cycle of

an YBCO bulk superconductor was studied, and the dependency with the applied magnetic field and

internal characteristics was characterized. In the intrinsic parameters study, the results show that these

parameters can be a determining factor regarding the expected capabilities and losses of a bulk

superconductor. The behavior of HTS regarding trapped fields was also studied, since the objective is

to use them to replace permanent magnets. The dependency of the trapped field with the applied field

and various parameters was studied. Experimentally, the conditions for which superconductivity is lost

were studied. Results show that superconductivity is lost more rapidly for higher applied magnetic fields,

as expected by the model. However, the loss of superconductivity was not observed experimentally

while the superconductor was submerged in liquid nitrogen, and as such was not studied in the model.

The thermal and electromagnetic analysis allowed the understanding of the dynamics of HTS in

a more practical environment when considering the liquid nitrogen. It was possible to observe the effect

of the temperature rise on the superconductor and how it influences all the variables in the

superconductor. However, the thermal model needs to be improved, since the liquid nitrogen has a

phase change (from liquid to vapor), which has not been considered in the model. However, the model

pointed out the thermal effects in the superconductor material and its characteristic time constants.

Regarding the advantages and disadvantages of using bulk HTS in the excitation systems of

electric machines, they come down to the intrinsic characteristics of the superconductor and the cooling

system used. As has been demonstrated, higher values of the intrinsic parameters 𝐵0 and 𝐽𝐶0 give better

results, in the sense that they allow for higher currents, which in turn improves performance in both

magnetic shielding and magnetic field trapping. The dependence of the critical current density with

temperature also provides better results for lower temperatures. These advantages come with

disadvantages of their own. Higher values of the parameters mentioned also come with higher power

losses due to the higher currents. Likewise, better cooling systems are more expensive. As has been

mentioned, no loss of superconductivity was obtained with the superconductors submerged in liquid

nitrogen. However, for more practical applications, mainly in the use in electric machines, a more specific

analysis has to be made, specially an analysis with the superconductors inserted in the machine.

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5.2 FURTHER DEVELOPMENTS

In the analysis of the behavior of HTS with trapped fields, the fact that the model yielded the same

trapped field for different initial applied fields and different derivatives is worth looking into, as well as

the decay observed in the long term study. The decay observed in the long term study was not observed

in experimental results. Also, the loss in trapped field regarding the influence of an external applied

magnetic field is worth considering because of the cogging torque in electric machines.

In the thermal analysis experiments no loss of superconductivity was observed when the

superconductor was quenched in liquid nitrogen. A study with higher magnetic fields is needed to

analyze this in more detail.

The purpose of the thermal simulations was only to study how the temperature affects the

properties of the superconductor and their relations, not to simulate a cooling system. Thus, a more

realistic model of the thermal simulation is needed.

Finally, an analysis of the HTS pulse magnetization process has to be planned and studied, both

simulated and experimental. Also, a comparison with conventional excitation systems, taking into

account the electromagnetic, thermal and economic costs of the different solutions, is needed.

5.3 FINAL REMARKS

Generally, the advantages of using superconductors are the reduction of resistive losses, due to

the very low resistivity, and an increased power density, due to the possibility of higher currents and

higher magnetic fields.

However, the cost of the cooling system needed to achieve the low temperatures required needs

to be considered in practical applications. With the cheap access to liquid nitrogen, this becomes an

easy solution to the cooling problem regarding HTS, in particular the material studied, YBCO. The loss

in trapped field due to external magnetic fields is also a disadvantage in terms of performance.

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REFERENCES

[1] P. Tixador, F. Simon, H. Daffix and M. Deleglise, "150-kW Experimental Superconducting," IEEE Transactions on Applied Superconductivity, vol. 9, no. 2, pp. 1205-1208, June 1999.

[2] B. Oswald, M. Krone, M. Söll, T. Strasser, J. Oswald, K. J. Best, W. Gawalek and L. Kovalev, "Superconducting Reluctance Motors with YBCO Bulk Material," IEEE Transactions on Applied Superconductivity, vol. 9, no. 2, pp. 1201-1204, 1999.

[3] G. J. Barnes, M. McCulloch and D. Dew-Hugues, "Computer Modelling of Type II Superconductors in Applications," Supercond. Sci. Technol., vol. 12, pp. 518-522, 1999.

[4] H. W. Kim, Y. S. Jo, J. H. Joo, S. Park, H. M. Kim and J. Hur, "Study of Hybrid-Type Field Coil for Superconducting Machines," IEEE Transactions on Applied Superconductivity, vol. 24, no. 3, 2014.

[5] M. Zhang, W. Wang, Y. R. Chen and T. Coombs, "Design Methodology for HTS Bulk Machine for Direct-DrivenWind Generation," IEEE Transactions on Applied Superconductivity, vol. 22, no. 3, 2012.

[6] Z. Huang, M. Zhang, W. Wang and T. A. Coombs, "Trial Test of a Bulk-Type Fully HTS Synchronous Motor," IEEE Transactions on Applied Superconductivity, vol. 24, no. 3, 2014.

[7] Z. Deng, M. Miki, B. Felder, K. Tsuzuki, Shinohara, N, R. Taguchi, K. Suzuki and M. Izumi, "The Effectiveness of Pulsed-Field Magnetization with Respect to Different Performance Bulk Superconductors," Journal of Superconductivity and Novel Magnetism, vol. 25, no. 1, pp. 61-66, 2012.

[8] Z. Deng, M. Miki, K. Tsuzuki, B. Felder, R. Taguchi, N. Shinohara and M. Izumi, "Pulsed Field Magnetization Properties of Bulk RE-Ba-Cu-O as Pole-Field Magnets for HTS Rotating Machines," IEEE Transactions on Applied Superconductivity, vol. 21, no. 3, 2011.

[9] T. Oka, Y. Yamada, T. Horiuchi, J. Ogawa, S. Fukui, T. Sato, K. Yokoyama and M. Langer, "Magnetic Flux-Trapping of Anisotropic-Grown Y-Ba-Cu-O Bulk Superconductors during and after Pulsed-Field Magnetizing Processes," Journal of Physics: Conference Series, vol. 507, no. 1, 2014.

[10] T. Oka, H. Seki, J. Ogawa, S. Fukui, T. Sato and K. Yokoyama, "Performance of Trapped Magnetic Field in Superconducting Bulk Magnets Activated by Pulsed Field Magnetization," IEEE Transactions on Applied SUperconductivity, vol. 21, no. 3, 2011.

[11] M. Cavique, "Study and characterization of restrictive thermal phenomena in low-speed electric generators for utilization of renewable energy," Master Thesis - Instituto Superior Técnico, Lisbon, 2014.

[12] Siemens, "Superconducting Generators - Siemens Global Website," Siemens, 2006. [Online]. Available: http://www.siemens.com/innovation/en/publikationen/publications_pof/pof_spring_2006/motors_articles/superconducting_generators.htm. [Accessed 11 September 2014].

[13] D. Oliveira, "Electromagnetic Design of a PM Synchronous Generator for a 20 kW Ultralow Head Hydro Turbine," Master Thesis - Instituto Superior Técnico, Lisbon, 2015.

94

[14] R. Qu, Y. Liu and J. Wang, "Review of Superconducting Generator Topologies for Direct-Drive Wind Turbines," IEEE Transactions on Applied Superconductivity, vol. 23, no. 3, 2013.

[15] R. A. Serway, “Superconductivity,” in Physics for Scientists and Engineers with Modern Physics, 4th edition, 1996, pp. 367-384.

[16] F. Bouquet and J. Bobroff, "Category:Superconductors - Wikimedia Commons," LPS - Laboratoire de physique des solides, 7 December 2011. [Online]. Available: http://commons.wikimedia.org/wiki/Category:Superconductors. [Accessed 8 September 2014].

[17] Z. Hong, H. M. Campbell and T. A. Coombs, "Computer Modeling of Magnetisation in High Temperature Superconductors," IEEE Transactions on applied superconductivity, vol. 17, no. 2, pp. 3761-3764, 2007.

[18] H. Fujishiro and T. Naito, "Simulation of temperature and magnetic field distribution in superconducting bulk during pulsed field magnetization," Superconductor Science and Technology, vol. 23, 2010.

[19] J. H. Koo and G. Cho, "Magnetic field dependence on transition temperature Tc in cuprate superconductors," Solid State Communications, vol. 129, pp. 191-193, 2004.

[20] F. G. Blatter, "Vortices in High Temperature superconductors," Reviews of Modern Physics, 1994.

[21] M. Sjöström, "Hysteresis Modelling of High Temperature Superconductors," Lausanne, 2001.

[22] F. Grilli, E. Pardo, A. Stenvall, D. Nguyen, W. Yuan and Gömöry, "Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents," IEEE Transactions on Applied Superconductivity, 2013.

[23] COMSOL, "A Warm Sunny Day on the Beach under a Parasol - Comsol 4.4".

[24] J. Feng, "Thermohidraulic-quenching simulation for superconducting magnets made of YBCO HTS tape," Plasma Science and Fusion Center - Massachusetts Institute of Technology, Cambridge, 2010.

[25] S. Barros, "POWERLABS Cryogenic Demonstrations," Power Labs, 2010. [Online]. Available: http://www.powerlabs.org/ln2demo.htm. [Accessed 13 September 2014].

[26] Air Liquide, "Nitrogen, N2, Physical properties, safety, MSDS, enthalpy, material compatibility, gas liquid equilibrium, density, viscosity, flammability, transport properties," Air Liquide, 2013. [Online]. Available: http://encyclopedia.airliquide.com/Encyclopedia.asp?LanguageID=11&CountryID=19&Formula=&GasID=5&UNNumber=&EquivGasID=5&PressionBox=1&btnPression=Calculate&VolLiquideBox=&MasseLiquideBox=&VolGasBox=&MasseGasBox=&RD20=29&RD9=8&RD6=64&RD4=2&RD3=22&RD8=27&RD2. [Accessed 13 September 2014].

[27] M. Zhang and T. A. Coombs, "3D modeling of high-Tc superconductors by finite element software," Superconductor Science and Technology, vol. 25, no. 1, 2012.

[28] M. K. Wu and others, “Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Compound System at Ambient Pressure,” Physical Review Letters, vol. 58, no. 9, p. 908–910, 1987.

[29] L. K. Kovalev, K. V. Ilushin, V. T. Penkin, K. L. Kovalev, A. E. Larionoff, S. M.-A. Koneev, K. A. Modestov, S. A. Larionoff, V. N. Poltavets, I. I. Akimov, V. V. Alexandrov, W. Gawalek, B. Oswald

95

and G. Krabbes, "High output power reluctance electric motors with bulk high-temperature superconductor elements," Superconductor Science and Technology, no. 15, pp. 817-822, 2002.

[30] P. Hoffman, "Superconductivity," 12 March 2009. [Online]. Available: http://users-phys.au.dk/philip/pictures/physicsfigures/node12.html. [Accessed 8 September 2014].

[31] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, 2004, pp. 273-278.

[32] L. D. Landau and E. M. Lifschitz, Course of Theoretical Physics: Electrodynamics of Continuous Media, vol. 8, Oxford: Butterworth-Heinemann, 1984.

[33] G.-J. Lee, "Superconductivity Applications in Power Systems," in Applications of High-Tc Superconductivity, Intech, 2011, pp. 45-74.

[34] M. Tinkham, Introduction to Superconductivity, Dover Publications, 2004.

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