Pedro Nuno Barracha de Freitas Ribeiro - ULisboa · Magnetic levitation train systems are being...

“Scale-up” of the Portuguese Superconductor type-ZFC

Magnetic Levitation System Fulfilling the Functional

Criteria of the Maglev-Cobra

Pedro Nuno Barracha de Freitas Ribeiro

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

Member of the Committee: Prof. António Eusébio Velho Roque

October 2015

iii

ACKNOWLEDGEMENTS

I would like to thank my supervisor professor Paulo Branco for assigning me to this thesis, and for his

support throughout its development. I would also like to thank João Arnaud for the help he has given

me, and which was crucial in the beginning stages of development of the FEM program.

v

ABSTRACT

Magnetic levitation train systems are being developed as part of a new generation of mass

transportation systems. This thesis will deal with the latest trend of these type systems, the

superconductor magnetic levitation system using zero-field cooling, with a track made of permanent

magnets, the ZFC-Maglev. This thesis will try to propose a competitive solution, using data retrieve

from a similar system, the Maglev-Cobra train, which uses field cooling. To this end, a study of the

levitation and guidance forces is made for the ZFC-Maglev system so that fulfills the functional criteria

of the Maglev-Cobra.

Keywords

Maglev; Superconductor; Zero Field Cooling; ZFC; Permanent Magnets

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RESUMO

Comboios com sistemas de levitação magnética têm sido desenvolvidos como parte de uma nova

geração de sistemas de transporte de massas. Esta tese trata a tendência mais recente deste tipo de

sistemas, o sistema de levitação magnética supercondutora que usa zero-field cooling, com um trilho

de magnetos permanentes, o ZFC-Maglev. Esta tese vai tentar apresentar uma solução competitiva,

usando dados retirados de um sistema semelhante, o do comboio Maglev-Cobra, que usa field

cooling. Para este fim, é realizado um estudo das forças de levitação e guiamento para o ZFC-Maglev

de modo a desenvolver uma solução que cumpra os critérios funcionais do Maglev-Cobra.

Palavras-Chave

Maglev; Supercondutor; Zero Field Cooling; ZFC; Magnetos Permanentes

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TABLE OF CONTENTS

Acknowledgements ....................................................................................................................... iii

Abstract.......................................................................................................................................... v

Resumo ........................................................................................................................................ vi

List of Figures ................................................................................................................................ x

List of Tables ............................................................................................................................... xiii

List of Acronyms .......................................................................................................................... xv

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

1 - Introduction ....................................................................................................................... 1

1.1 - State of the art ............................................................................................................. 2

1.1.1 - Advantages of magnetic levitation systems ...................................................... 2

1.1.2 - Different passive layouts for superconducting magnetic levitation systems ..... 2

1.1.3 – The Vehicle Maglev-Cobra ............................................................................... 6

1.2 – Fundamental aspects about superconductivity ........................................................... 8

1.2.1 - Type I superconductors ..................................................................................... 9

1.2.2 - Type II superconductors .................................................................................. 10

1.3 – Superconductor Cooling techniques ......................................................................... 11

1.3.1 - Zero-field cooling (ZFC) ................................................................................... 11

1.3.2 - Field-cooling (FC) ............................................................................................ 11

1.4 - Motivation and goals .................................................................................................. 12

1.5 - Thesis organization .................................................................................................... 14

2 - FEM model and its validation .......................................................................................... 15

2.1 – Model regions and the type II superconductor equations ......................................... 16

2.1.1 - Introduction to the distributed parameters simulation method ........................ 16

2.1.2 - Common field equations to all model regions ................................................. 17

2.1.3 - Air constitutive equation .................................................................................. 17

2.1.4 – Type II Superconductor electromagnetic equations ....................................... 17

2.2 – Electromagnetic Force computation ......................................................................... 18

2.2.1 - Description of the Maxwell Stress Tensor ....................................................... 18

viii

2.2.2 – Example: Application of the Maxwell Stress tensor to a cubic volume ........... 20

2.2.3 Simplified analysis of the interaction between a permanent magnet and a

superconductor: expected results ..................................................................................... 21

2.3 – The YBCO bulk supercondcutor: its FEM Model validation ...................................... 23

2.3.1 – Finite element modelling (FEM) and analysis of the superconductors and magnet

structures ........................................................................................................................... 24

2.3.2 - First attempt at validation: 2D model ............................................................... 26

2.3.3 - Second attempt at validation: 3D model .......................................................... 28

2.4 – ZFC-Maglev topology: levitation force ...................................................................... 29

2.4.1 - Considered superconductor and magnet geometry ........................................ 29

2.4.2 - Simulation results ............................................................................................ 30

2.4.3 -The experimental set-up for the zfc-maglev geometry ..................................... 32

2.4.4 – Experimental levitation force results ............................................................... 35

3 - ZFC-Maglev scale-up to the Maglev-cobra .................................................................... 38

3.1 - Influence of track and superconductor disposition in the levitation and guidance forces

39

3.1.1 - Study 1 and 2: Track layout evaluation ........................................................... 40

3.1.2 - Study 3: Distance between the superconductors ............................................ 42

3.1.3 - Pre-study conclusions ..................................................................................... 43

3.2 - Establishing the final track and superconductor geometry ........................................ 45

3.2.1 - Search trial-and-error method ......................................................................... 45

3.2.2 - Geometry A ...................................................................................................... 47

3.2.3 - Geometry B ...................................................................................................... 48

3.2.4 - Geometry C ..................................................................................................... 50

3.2.5 - Geometry D ..................................................................................................... 51

3.2.6 - Geometry selection for the ZFC-Maglev ......................................................... 52

3.2.7 - Guidance study for geometry C ....................................................................... 54

4 – Technical and econimic analysis Comparison between the ZFC-Maglev and Maglev-Cobra

55

4.1 – Levitation and guidance Force comparison .............................................................. 56

4.2 - Implementation cost analysis ..................................................................................... 57

ix

4.3 - Operational cost analysis ........................................................................................... 60

5 – Final Conclusions and future work ................................................................................. 64

5.1 – Future work ............................................................................................................... 65

References .................................................................................................................................. 66

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LIST OF FIGURES

Figure 1.1 - Basic superconductor to permanent magnet geometry for HTS magnetic levitation .......... 3

Figure 1.2 - Most common superconductor based magnetic levitation system [4] ................................. 3

Figure 1.3 - Layout where two magnets are placed opposing each other to increase field strength ...... 4

Figure 1.4 - Field distribution of the two opposing magnets layout ......................................................... 4

Figure 1.5 -Halbach permanent magnet layout ....................................................................................... 4

Figure 1.6 - Field distribution of the Halbach array ................................................................................. 5

Figure 1.7 - T shaped permanent magnet track. (1) Vehicle body, (2) Bed plate, (3) T-shaped magnet

layout, (4) Iron, (5)(7)(9) Liquid Nitrogen Cryostat, (6)(10) high temperature superconductors [7] ........ 6

Figure 1.8 - Illustration of the spring like restitution force of a HTS block in field cooling ....................... 7

Figure 1.9 – One module of the Maglev-Cobra vehicle [9] ...................................................................... 7

Figure 1.10 -Linear motor used by the Maglev-Cobra train [10] ............................................................. 8

Figure 1.11 - Superconductivity region [1] ............................................................................................... 9

Figure 1.12- Illustration of the Meissner effect ...................................................................................... 10

Figure 1.13 - The three possible states for a type II superconductor .................................................... 10

Figure 1.14 - Illustration of the formation of vortices in type II superconductors ................................... 11

Figure 1.15 - Illustration of how a superconductor block creates and maintains an image of the original

magnetic field within itself when in field cooling [8] ............................................................................... 12

Figure 1.16 – Use o an iron core to increase field strength in the Maglev-Cobra track geometry ........ 12

Figure 1.17 - Proposed track topology: the superconductors are locked in place by the permanent

magnets’ field ........................................................................................................................................ 13

Figure 2.1 - Steps for the electromagnetic force computation .............................................................. 16

Figure 2.2 – Stress tensor vectors applied to a surface of a cubic volumetric element ........................ 20

Figure 2.3 - Illustration of the resulting magnetic flux of the interaction between a permanent magnet

and a perfect superconductor ................................................................................................................ 22

Figure 2.4- Simulation of a superconductor over a permanent magnet ................................................ 24

Figure 2.5 - Example of the magnitude of the magnetic field for the same surface for x, y e z ............ 25

Figure 2.6 - Elements’ dimensions ........................................................................................................ 26

Figure 2.7 -Graph of the experimental force per superconductor ......................................................... 26

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Figure 2.8- 2D field distribution for an air gap of 1(cm) ......................................................................... 27

Figure 2.9 - Graph comparing the experimental force with the simulated ones .................................... 28

Figure 2.10 - ZFC track and superconductor placement, a mesh of 0.5(cm) was used ....................... 29

Figure 2.11 - Permanent magnet orientation in the ZFC track .............................................................. 30

Figure 2.12 - Cuts made in the 𝑦𝑧 plane along the 𝑥 direction ............................................................. 30

Figure 2.13 - Magnetic field distribution of both cuts ............................................................................. 31

Figure 2.14 - Visualization of the electric current desnity and the magnetic flux distribution lines within

the superconductors .............................................................................................................................. 32

Figure 2.15- Photo of the piezoelectric force sensor Scaime K12+LMVu ............................................. 33

Figure 2.16 - Triple magnet trail disposition for the ZFC-Maglev .......................................................... 33

Figure 2.17 - Disposition of the superconductors inside the foam box following the same geometry

used in the simulation ............................................................................................................................ 34

Figure 2.18 - Total assembly for the experiment ................................................................................... 34

Figure 2.19 - Permanent magnet field for two cuts in the absence of the superconductors ................. 35

Figure 2.20 - Permanent magnet field for two cuts in the presence of the superconductors ................ 36

Figure 2.21 - Graphic comparing the simulated force increased by the average error with the

experimental force for the ZFC-Maglev track ........................................................................................ 37

Figure 3.1 – (a) Schematic of one Maglev-Cobra module and its cryostats; (b) Photo of the cryostat

utilized by the levitation system of the Maglev-Cobra train [1] .............................................................. 38

Figure 3.2 - Illustration of the superconductor block disposition ........................................................... 40

Figure 3. 3 - Magnet placement in study 1. (a) Standard case; (b) Increase in 𝑥; (c) Decrease in 𝑥 ... 40

Figure 3. 4 - Magnet placement in study 1. (a) Standard case; (b) Increase in 𝑦; (c) Decrease in 𝑦 ... 41

Figure 3. 5 - Superconductor placement in study 3. (a) Standard case; (b) Increase in the gap; (c)

Decrease in the gap .............................................................................................................................. 42

Figure 3.6 - Distribution of the magnetic field for a displacement of -5(mm): (a) left surface of the left

block; (b) right surface of the left block; (c) left surface of the right block; right surface of the right block

............................................................................................................................................................... 44

Figure 3.7 – Flowchart of the search trail-and-error method ................................................................. 46

Figure 3.8 - Geometry A achieved for a ZFC-Maglev module .............................................................. 47

Figure 3.9- Mesh of 5(mm) used for the simulation of geometry A ....................................................... 47

Figure 3.10 - Geometry B achieved for a ZFC-Maglev module ............................................................ 48

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Figure 3.11 - Mesh of 5(mm) used for the simulation of geometry B .................................................... 49

Figure 3.12 - Geometry C achieved for a ZFC-Maglev module ............................................................ 50

Figure 3.13 - Mesh of 5(mm) used for the simulation of geometry C .................................................... 50

Figure 3.14 - Geometry D achieved for a ZFC-Maglev module ............................................................ 51

Figure 3.15 - Mesh of 5(mm) used for the simulation of geometry D .................................................... 52

Figure 3.16 - Graph of the force in function of the air gap for all geometries; Points A, B, C and D

indicate where each curve passes the desired levitation force value of 2500(N) ................................. 53

Figure 3.17 - Graph of the guidance force in function of the air gap for a lateral displacement of -

5(mm) .................................................................................................................................................... 53

Figure 3.18- Graph of the guidance force in function of the lateral displacement for geometry C ........ 54

Figure 4.1 - Levitation force comparison between the Maglev-Cobra and the ZFC-Maglev scale-up .. 56

Figure 4.2 – Total guidance force comparison; (a) Maglev-Cobra guidance force, and (b) ZFC-Maglev

guidance force; Red line indicates the minimum guidance force of 900(N) .......................................... 57

Figure 4.3 - Dimensions (cm) of the cross section of the Maglev-Cobra rail ........................................ 58

Figure 4.4 - Photo of the rail topology used by Maglev-Cobra track [1] ................................................ 58

Figure 4.5 - Rail topology of the ZFC-Maglev solution .......................................................................... 59

Figure 4.6 - Difference in quantity of permanent magnets used per meter (figure scale in centimeters)

............................................................................................................................................................... 59

Figure 4.7 - Graph of the power losses in function of the magnitude of the applied magnetic field over

the superconductor for 5(Hz) [16] .......................................................................................................... 60

Figure 4.8 - Symbolic illustration of the distance ∆x, for which a time period is defined for the magnetic

field acting on the superconductors ....................................................................................................... 61

Figure 4.9 - Power losses density in function of the Maglev speed ...................................................... 62

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LIST OF TABLES

Table 2.1 - Generic YBCO superconductor parameters used in the FEM program ............................. 18

Table 2.2 - 2D simulated and experimental force per superconductor, in function of the distance

between the permanent magnet and superconductor ........................................................................... 27

Table 2.3 - 3D simulated and experimental force per superconductor, in function of the distance

between the permanent magnet and superconductor ........................................................................... 28

Table 2.4 – ZFC-Maglev topology simulated levitation force per superconductor ................................ 32

Table 2.5 - Comparison between the experimental force and the simulated force per superconductor

............................................................................................................................................................... 37

Table 3.1 - Variation of the distance between the magnets in the x direction, for an air gap of 15(mm).

In the case of the guidance force the displacement is of -5(mm) from its central position ................... 41

Table 3.2 - Variation of the distance between the magnets in the y direction, for an air gap of 15(mm).

In the case of the guidance force the displacement is of -5(mm) from its central position ................... 42

Table 3.3 - Variation of the gap between the superconductors, for an air gap of 15(mm). In the case of

the guidance force the displacement is of -5(mm) from its central position .......................................... 43

Table 3.4 - Contribution of each facet for the guidance force, for a displacement is of -5(mm) from its

central position ...................................................................................................................................... 43

Table 3.5 - Dimensions of the elements in geometry A ........................................................................ 47

Table 3.6 - Levitation and guidance force in function of the air gap, with the guidance force computed

for a displacement of –5(mm) from its central position, for geometry A ......................................... 48


for a displacement of -5(mm) from its central position, for geometry B ......................................... 49

Table 3.8 - Dimensions of the elements in geometry C ........................................................................ 50


for a displacement of –5(mm) from its central position, for geometry C ........................................ 51


for a displacement of -5(mm) from its central position, for geometry D ......................................... 52

Table 4.1 - Implementation cost per 100(km) for the Maglev-Cobra track, in millions of US dollars, in

November of 2011 ................................................................................................................................. 57

Table 4.2 - Implementation cost per 100(km) for the ZFC-Maglev track layout, in millions of US dollars,

using the values of November of 2011 .................................................................................................. 60

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Table 4.3 - Consumption of liquid nitrogen by the Maglev-Cobra in a 12 hour window ........................ 63

Table 4.4 - Consumption of liquid nitrogen by the ZFC-Maglev in a 12 hour window........................... 63

xv

LIST OF ACRONYMS

FC Field Cooling

FEM Finite Element Program

HTS High Temperature Superconductor

YBCO Yttrium Barium Copper Oxide

ZFC Zero-Field Cooling

xvi

NOMENCLATURE

𝑬 Electric field

𝑱 Current density

𝑯 Magnetic field

𝐁 Magnetic flux density

𝑬𝒂𝒊𝒓 Electrical field of the air

𝜌 Electrical resistivity

𝑬𝑺𝑪 Electrical field within the superconductor

𝐸0 Critical electrical field of the superconductor

𝐽𝑆𝐶 Current density within the superconductor

𝐽𝑐 Critical current density of the superconductor

𝑛 State of superconductivity of the superconductor

𝐽𝑐0 Parameter of that varies from superconductor to superconductor

𝐵0 Parameter of that varies from superconductor to superconductor

𝑇𝑐 Critical temperature of the superconductor

𝐻𝑐 Critical magnetic field for type I superconductors

𝐻𝑐1 First value of critical magnetic field for type II superconductors

𝐻𝑐2 Second value of critical magnetic field for type II superconductors

𝑓𝑑 Force density in the computation of the force by the Lorentz method

𝑓𝑖 Density of force in one direction

𝐹𝑖 Total force in one direction

𝜇 Magnetic permeability

𝑛𝑓𝑡 Normal to the front facet of a cube

𝑛𝑏𝑘 Normal to the back facet of a cube

𝑛𝑟𝑡 Normal to the right facet of a cube

𝑛𝑙𝑡 Normal to the left facet of a cube

𝑛𝑖𝑛𝑓 Normal to the inferior facet of a cube

xvii

𝑛𝑠𝑢𝑝 Normal to the superior facet of a cube

𝑆 Area of a surface

𝐹𝑥 Force in the 𝑥 direction

𝐹𝑦 Force in the 𝑦 direction

𝐹𝑧 Force in the z direction

𝑙 Length of a path

Ψ Linked flux

𝑊𝑚 Magnetic co-energy

𝑣 Velocity

𝑓 Frequency

∆𝑥 Distance between two coupled lateral and central permanent magnets

∆𝑡 Period of distance ∆𝑥

𝑄 Heat energy

𝑚 Mass of liquid nitrogen

𝐶𝑝 Specific heat at constant pressure

∆𝑇 Temperature difference

𝑡 Time duration of a liquid nitrogen deposit

𝑃𝑙𝑜𝑠𝑠 Loss potency

𝑉𝑛𝑖𝑡𝑟𝑜𝑔𝑒𝑛 Volume of liquid nitrogen

𝑉𝑐𝑟𝑦𝑜𝑠𝑡𝑎𝑡 Volume of the cryostat

𝑉𝑠𝑢𝑝𝑒𝑟𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟𝑠 Total volume of the superconductors in one cryostat

1

1 - INTRODUCTION

The industrial revolution of the XVIII and XIX centuries, led to an exponential growth in population in

the XX century within urban centres. Mass transport became a problem, as the population had the

need to commute between home and work place. Motorized transports became imperative, due to the

unstoppable increase in city size. Thus began the collective mass transport system: bus, train and

metro.

The necessity of an efficient and non-polluting public transportation system, with competitive

implementation and maintenance costs, is nowadays a vital part of a society evermore focused in

living in big city centres.

However, these means of transportation are characterized by a high implementation and operational

cost. Furthermore they may have their own specifications, which may not be compatible with the

environment where they need to be placed. Examples are the curvature ray need for a train to turn,

the inclination of the tracks [1], maximum train speed and also how power is supplied to it.

In recent years a new type of transportation is being researched: the superconducting levitation train,

also known as Maglev. This system uses magnetic fields to make the carriage of the train levitate,

usually leaving propulsion to linear motors.

The scientific area of energy (DEEC) from Instituto Superior Técnico (IST), has been developing this

type of technology since 2006, in conjunction with the Superconductivity Laboratory from the federal

university of Rio de Janeiro (UFRJ).

The Maglev offers a set of advantages in comparison to the traditional train system. As it does not rely

in attrition in order to move, the Maglev systems requires less power, while at the same time

producing less sound pollution. It can also be inserted in overpasses for easier city integration.

This thesis is inserted in the theme “Superconducting levitation vehicles in zero-field cooling”. It aims

at developing a new system that fulfils the functional criteria of the Maglev-Cobra developed by the

UFRJ, which uses field-cooling for its superconducting levitation system. Thus, this thesis will consist

in a study of the levitation and guidance force required for a functional prototype.

2

1.1 - STATE OF THE ART

Nowadays there are two levitation systems being employed, each using active elements, namely

electromagnets. The Transrapid, of German design, has an experimental track in Germany, and has

been built in China, connecting Shanghai to its airport. In Japan it is used the JR-Maglev, which has

been approved for construction until 2027, and recently has achieved a new velocity record of

603(Km/h) for magnetic levitation systems [2].

However, the latest trend has been the development of a levitation based on passive elements, such

as permanent magnets and superconductors. Depending on how superconductors are cooled,

different type of levitation systems are possible to be made. Cooling strategies will be treated in point

– Superconductor Cooling techniques, and being that the state of the art will present the benefits of

magnetic levitation systems and different layout configurations.

1.1.1 - ADVANTAGES OF MAGNETIC LEVITATION SYSTEMS

Trains using magnetic levitation systems, or Maglev trains, offer certain advantages in relation to

conventional trains which operate with wheels on a track [1] [3].

1 - The inexistence of contact between the train and the rail of the track, leads to a decrease in

attrition losses. As a consequence the energetic requirements for the propulsion system are lessen;

2 - It allows simpler designs, with fewer parts, due to the absence of the wheels. This translates into

fewer maintenance costs;

3 - The weight of the train is distributed, instead of concentrated on its axis. This allows for less robust

and lighter structures;

4 - A smaller curvature ray for turning, allowing the tracks to fit better in city limits, and allows for

higher slopes, due to the absence of slipping and sliding.

Superconductor based magnetic levitation systems offer additional benefits. As they only use passive

elements, it allows for lower operational costs, since there is only the need to cool down the

superconductors with liquid nitrogen.

1.1.2 - DIFFERENT PASSIVE LAYOUTS FOR SUPERCONDUCTING MAGNETIC

LEVITATION SYSTEMS

The basic superconductor to permanent magnet placement and layout is based on overlapping the

superconductors over the permanent magnets as shown in Figure 1.1, being that they will maintain a

safe distance between them. The main struggle with this type of system has been generating a

3

tangential magnetic field strong enough to support the levitation force required, while limiting the

number of superconductors used, as they are the most expensive component of the two.

Figure 1.1 - Basic superconductor to permanent magnet geometry for HTS magnetic levitation

This system will substitute the traditional iron rail in the tracks and the train’s wheels, as shown in

figure Figure 1.2, taken from the paper [4]. In this paper a permanent magnet guideway is used, and it

was concluded that a hybrid levitation system composed of both superconductors and permanent

magnets is superior to just the usage of superconductors.

Figure 1.2 - Most common superconductor based magnetic levitation system [4]

4

Other attempts at increasing the levitation force have being centred on the tracks layout. Papers [5] [6]

compares two possible magnet layout. The first one consists in aligning two magnets with opposite

poles, and bringing them together, using an iron core to guide the magnetic field. This solution will

increase the field’s density per unit of volume.

Figure 1.3 - Layout where two magnets are placed opposing each other to increase field strength

The field distribution is shown in Figure 1.4. It can be seen that the magnetic field is much stronger in

the middle iron core, having an upwards direction.

Figure 1.4 - Field distribution of the two opposing magnets layout

The other solution is based on an Halbach array. This array positions the permanent magnets in such

a way it creates a strong periodic field on upper side, while having a weak field on the lower side.

Figure 1.5 -Halbach permanent magnet layout

5

The field distribution can be seen in Figure 1.6.

Figure 1.6 - Field distribution of the Halbach array

The Halbach array will enhance the levitation system, by allowing a more efficient usage of the

magnetic field, since it remains practically in the upper side of the layout, thus leading to stronger

levitation forces.

One drawback of these two solutions is the significant magnetic forces between the permanent

magnets. Due to this fact, special mechanical systems need to be employed to effectively bring the

magnets together in the required order, but also extra structures need to be used to avoid any

deformations.

Another problem with a superconductor based magnetic levitation system is its guidance force. This

force is required to stabilize the vehicle on its tracks, and it is of major concern when the t vehicle has

to make a curve at high speeds. A topology for this issue was proposed in the paper [7]. There a T-

shaped geometry is proposed, using both an Halbach array from Figure 1.5 for levitation and the

opposing permanent magnet poles geometry for guidance from Figure 1.3.

6

Figure 1.7 - T shaped permanent magnet track. (1) Vehicle body, (2) Bed plate, (3) T-shaped magnet layout, (4) Iron, (5)(7)(9) Liquid Nitrogen Cryostat, (6)(10) high temperature superconductors [7]

This topology does increase guidance and stability forces, while generating enough levitation force.

The obvious cost of this solution is the high usage of permanent magnets and superconductors, as

well as the problem with the high magnetic forces between the magnets that can make this solution

not viable.

One feature common for all previous topologies is the use of not only ferromagnetic pieces, but also

bulk superconductors that are cooled in the presence of a magnetic field (field cooling). The magnetic

circuit topology of Figure 1.3 was adopted by the Brazilian group that is implementing a low-speed

vehicle called the Maglev-Cobra.

The solution that will be proposed on this thesis will be based on the requirements of the Maglev-

Cobra. In the following point, it will be presented the main features of this system.

1.1.3 – THE VEHICLE MAGLEV-COBRA

The Maglev-Cobra is a vehicle which uses a levitation system based on field cooled superconductors,

and tracks that are made of permanent magnets and ferromagnetic pieces as shown in Figure 1.3. As

stated before, the cooling process will be presented afterwards, however a simplistic explanation can

be given of this phenomenon. Consider a block connected by springs to four walls, in such a way that

all springs are in their lowest level of potential energy. Any movement that the block may suffer will

lead to the actuation of the springs, in order to restore the block to its original position.

7

Figure 1.8 - Illustration of the spring like restitution force of a HTS block in field cooling

A photo of the Maglev-Cobra vehicle can be seen in Figure 1.9.

Figure 1.9 – One module of the Maglev-Cobra vehicle [8]

On the photo of the module it is possible to see the cryostats, in which the superconductors are

lodged, along its borders. There are six cryostats per each module, having each 24 superconductors.

Each segment can hold up to ten passengers, thus leading to a normalized total weight of 1500(Kg).

It is also possible to see the linear motor in its central part for its propulsion system. In Figure 1.10 it is

shown a picture of the primary and secondary of the linear motor. The Maglev-Cobra vehicle was

designed to operate at low speed, usually around 8(m/s) [9] [10].

8

Figure 1.10 -Linear motor used by the Maglev-Cobra train [9]

1.2 – FUNDAMENTAL ASPECTS ABOUT SUPERCONDUCTIVITY

The superconductive state is characterized by a set of electrical and magnetic properties, which are

revealed when certain materials are refrigerated below a certain temperature, usually very low ones.

The main indication that a material is in a superconductive state is a negligible electrical resistance.

This effect was discovered during the first decade of the XX century soon after Heike Kamerlingh

Onnes [3] was able to liquefy helium at 4,2(K). During his experiments to determine the electrical

resistivity of mercury in function of the temperature, he took notice that the resistance value was not

measurable when the mercury was below a given temperature.

Afterwards, it was verified that the superconductive state would also disappear when certain critical

values of the magnetic field or the current were achieved. Therefore, a superconductor is

characterized by three critical parameters: critical temperature (𝑇𝑐), critical density current (𝐽𝑐) and

critical magnetic field (𝐻𝑐), above which the superconductor loses its superconductive state.

Later it was discovered that almost all elements possess a superconductive state, if submitted to very

low temperatures. In 1986 Bednorz and Müller [11] created a composite made of 𝐿𝑎𝐵𝑎𝐶𝑢𝑂, whose

critical temperature was of 35(K). Thus began the development of high temperature superconductors

(HTS), being that today the critical temperatures are of around 100(K). These superconductors are

made of a composite of Yttrium, Barium, Copper and Oxygen (𝑌𝐵𝑎𝐶𝑢𝑂). With the critical temperature

of these composites, the low cost liquid nitrogen can now be used in the refrigeration instead of liquid

Helium.

A new development in superconductor technology was the discovery that not all superconductor

behave as the same, giving birth to type I and type II superconductors. Their similarities and

differences will be present in the next two points.

9

1.2.1 - TYPE I SUPERCONDUCTORS

Type I superconductors are the ones that are either in a superconductive state, or in its normal state,

not having a mixed state as indicated in Figure 1.11. This type of materials is characterized by a

perfect diamagnetism, being the superconductive state limited by the three critical parameters already

mentioned.

Figure 1.11 - Superconductivity region [1]

When this material is in its conductive state, Meissner effect occurs, where the magnetic field is

repelled by the superconductor, surrounding it as illustrated in Figure 1.12. The magnetic field

repulsion has it genesis on the surface currents generated in the superconductor by the same field,

effectively shielding the superconductor from the field. However, type I superconductors are not viable

for levitation systems, as their critical magnetic field value is too low.

10

Figure 1.12- Illustration of the Meissner effect

1.2.2 - TYPE II SUPERCONDUCTORS

The type II superconductors share the same limitation of type I superconductors, represented in

Figure 1.13, with the addition of a new state, the mixed state, which depends on the magnitude of the

applied external magnetic field.

Figure 1.13 - The three possible states for a type II superconductor

The mixed state exists between two field values (𝐻𝑐1 and 𝐻𝑐2) as indicated in Figure 1.13. This state

allows for field penetration within the superconductor, generating vortices, which are regions of non-

superconductivity, surrounded by current in the superconductive region. The creation of these vortices

is due to impurities within the material that constitutes the superconductor.

11

Figure 1.14 - Illustration of the formation of vortices in type II superconductors

The superconductor also exhibits increasing values of electrical resistivity, until it reaches its normal

state at 𝐻𝑐2. Their most common critical temperature is around 100(K), so they are usually cooled with

liquid nitrogen, which is usually at 77(K).

1.3 – SUPERCONDUCTOR COOLING TECHNIQUES

In this topic it will be presented the two cooling methods. Each method has its unique functional

characteristics, having advantages in regard to one another. This will lead to different system

problems and requirements.

1.3.1 - ZERO-FIELD COOLING (ZFC)

Zero-field cooling is realized by cooling the superconductors in the absence of a magnetic field. By

using this type of cooling, Meissner effect will occur until one of its critical parameters (𝐽𝑐, 𝑇𝑐, 𝐻𝑐2) is

reached. Being that in type II superconductors after the field value surpasses its first critical field value,

there will be field penetration.

1.3.2 - FIELD-COOLING (FC)

By contrast, field-cooling is made by cooling the superconductors now in the presence of a magnetic

field. The superconductor is magnetized with an image of the field that coursed through him at the

initial time of cooling. The induced internal currents, unlike the ZFC case that allow the Meissner

effect, will try to maintain the initial field distribution in the superconductor.

12

The superconductor can now be considered a type of permanent magnet, if it is placed elsewhere

outside the original magnet field, as shown in Figure 1.15. However, if it remains within the field, the

superconductor will be pinned down acting as a spring fixing it in the initial position.

Figure 1.15 - Illustration of how a superconductor block creates and maintains an image of the original magnetic field within itself when in field cooling [11]

1.4 - MOTIVATION AND GOALS

The present thesis falls within scope of the development of superconductor based magnetic levitation

systems. More specifically, it will search for a solution that has lower implementation and operation

costs than the Maglev-Cobra, while having its functional criteria.

As stated before, the Maglev-Cobra vehicle uses track layout from Figure 1.3, where two permanent

magnets are aligned with their opposite poles, and an iron core in the middle of them. This layout is

used in order to create a field powerful enough to allow a field-cooling levitation system, allowing the

required levitation and guidance forces.

Figure 1.16 – Use o an iron core to increase field strength in the Maglev-Cobra track geometry

13

The main advantage of field-cooling is the guarantee of lateral stability. When cooled by this method,

superconductors try to maintain their initial position, which translates into a restitution force into its

central position. By opposition, zero-field cooling by itself does not guarantee vertical nor lateral

stability. Despite this, it has been shown in [3] that it generates a higher levitation force, by realizing

strong field repletion. This phenomenon is similar to placing a leaf over the water coming out of a

fountain, the leaf will stay on top over the water, as long as it is pouted uniformly, but any perturbation,

will make it fall.

A ZFC-Maglev alternative has been proposed by Professor António Dente and Professor Paulo

Branco [12] consists in a new track topology shown in Figure 1.17. The track takes advantage of ZFC

Meissner effect to levitate and lock the superconductors in place, by making the field lines close

around the superconductors, and thus producing a guidance force. This is achieved by placing two

columns of superconductors, thus allowing the magnetic field to close through the gap between the

two columns.

Figure 1.17 - Proposed track topology: the superconductors are locked in place by the permanent magnets’ field

This thesis will analyse this novel topology called ZFC-Maglev, in order to verify if it is possible to do a

scale-up of that ZFC topology into a full size system that will have to fulfil the Maglev-Cobra’s

functional criteria, while trying to have a lower implementation and maintenance cost. The final result

of this thesis will be the dimensions of required components for the new system.

14

1.5 - THESIS ORGANIZATION

This thesis is organized in five chapters. The first chapter is an introductory one, informing the reader

of what already has been done in the field. In order to do a scale-up of a zero-field cooling levitation

system, it is necessary to build a simulator from which trustworthy data can be taken.

Thus, Chapter 2 will deal with the elaboration of a FEM model, presenting the mathematical models

necessary to compute the force generated by the pairing of the superconductor with a permanent

magnet. Then steps are made to validate the model.

Chapter 3 will present the process by which the final scale-up geometry was attained. It will point out

the functional system requirements, and display the levitation and guidance force values of the arrived

solution.

The final ZFC-Maglev geometry will be compared with the one used by Maglev-Cobra in Chapter 4,

being presented the strong points from each solution, and the advantages that one solution has over

the other.

Chapter 5 will be a resume of all that was done, and will conclude on how favourable the elaborated

solution is, while providing insight into new ways of optimizing it.

15

2 - FEM MODEL AND ITS VALIDATION

The levitation system is composed by two crucial elements: the permanent magnets and the type II

superconductors.

With this chapter, it is intended to determine, from a FEM simulation, the influence of the presence of

superconductors in zero field cooling (ZFC) over a magnetic field, produced by a set of permanent

magnets forming a certain array geometry. In particular it is intended to determine the strength of the

force generated by the superconductors for levitation proposes (force over the z axis). To this end

several steps will be described to develop a FEM program and valid its results.

16

2.1 – MODEL REGIONS AND THE TYPE II SUPERCONDUCTOR EQUATIONS

2.1.1 - INTRODUCTION TO THE DISTRIBUTED PARAMETERS SIMULATION METHOD

The approximation of a type II superconductor in ZFC condition to a magnet, leads to the circulation of

currents in its interior, which are most significant in its surfaces, so that the external field is repelled.

From these currents it is possible to determine the electrical and magnetic field distributions, both for

the superconductors as for their surroundings.

Figure 2.1 - Steps for the electromagnetic force computation

There can be considered three distinct model regions: the air, the superconductors and the permanent

magnets. In the air and superconductors regions the density current (𝑱) and the magnetic field (𝑯) can

be determined by the same set of equations (equations (1) and (2)), being that the only difference

between the two regions is the function by which the electrical field (𝑬) is defined, equations (3) or (4),

respectively. The region defined by the magnets is considered only as a source of field, and is

modelled so that it has a remnant magnetic flux density.

17

2.1.2 - COMMON FIELD EQUATIONS TO ALL MODEL REGIONS

As stated in the previous section, the computation of density current (𝑱 = [𝐽𝑥, 𝐽𝑦 , 𝐽𝑧]) and the magnetic

field (𝑯 = [𝐻𝑥 , 𝐻𝑦 , 𝐻𝑧]) are common to all regions, being taken directly from Maxwell’s equations for

slow varying phenomena in time (quasi-stationary regime).

𝑱 = ∇ × 𝑯 →

{

𝐽𝑥 =

𝜕𝐻𝑧𝜕𝑦

−𝜕𝐻𝑦

𝜕𝑧

𝐽𝑦 =𝜕𝐻𝑥𝜕𝑧

−𝜕𝐻𝑧𝜕𝑥

𝐽𝑧 =𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

, ∇. 𝑱 = 0 (1)

∇ × 𝑬 = −𝜕𝑩

𝜕𝑡→

{

𝜕𝐸𝑧𝜕𝑦

−𝜕𝐸𝑦

𝜕𝑧= −𝜇0𝜇𝑟

𝜕𝐻𝑥𝜕𝑡

𝜕𝐸𝑥𝜕𝑧

−𝜕𝐸𝑧𝜕𝑥

= −𝜇0𝜇𝑟𝜕𝐻𝑦

𝜕𝑡𝜕𝐸𝑦

𝜕𝑥−𝜕𝐸𝑥𝜕𝑦

= −𝜇0𝜇𝑟𝜕𝐻𝑧𝜕𝑡

, ∇. 𝑩 = 0 (2)

2.1.3 - AIR CONSTITUTIVE EQUATION

The air is considered a linear region [13], where Ohm’s Law can be applied. Thus the electrical field is

given by relation (3), where 𝜌 is the air electrical resistivity.

𝑬𝒂𝒊𝒓 = 𝜌𝑱 (3)

2.1.4 – TYPE II SUPERCONDUCTOR ELECTROMAGNETIC EQUATIONS

This thesis uses a macroscopic modelling approach of type II superconductors. The key departure

comes in the form of a non-linear E-J relationship. For the simulated model, it was considered the

model presented in the papers [13] [14], where the electrical field and current are given by an E-J

characteristic law, that takes the form of equation (4). The electrical field ESC within the superconductor

is given by a function of the superconductor parameters: E0 , Jc(𝐵), n and B0 . The critical electric

current density is field dependent and given by relation (5).

𝐄𝐒𝐂 = E0 (JSCJc(B)

)n

(4)

18

Jc(B) =Jc0 × B0B0 + ‖𝑩‖

, ‖𝑩‖ = 𝜇𝑜𝜇𝑟√𝐻𝑥2 + 𝐻𝑦

2 + 𝐻𝑧2 (5)

The n parameter represents the possible states of conductivity by the superconductor [3], being that

when it has the value of 1, it is in its resistive state, and when it tends to infinite it is in its ideal

superconductive state, that is, ESC = 0 in all superconductor volume . As for E0 parameter it is the

value of the critical electrical field.

For the computation of the critical current of the superconductor it is considered Jc0 and B0 which are

parameters that depend on the type of superconductor. These parameters essentially regulate the

density current in function of the norm of the magnetic flux density applied to the superconductor.

Table 2.1 presents the generic values of a YBCO superconductor, which were used in the FEM

program.

Table 2.1 - Generic YBCO superconductor parameters used in the FEM program

𝐵0(T) 0,1

Jc0(A/m2) 2 × 107

E0 (V/m) 1 × 10−4

n 21

2.2 – ELECTROMAGNETIC FORCE COMPUTATION

To determine the strength of the electromagnetic forces generated in the superconductor’s surface by

a magnetic field, it will be used the Maxwell Stress Tensor. This technique allows the computation of

forces, knowing only the distribution of the magnetic field over a closed surface.

2.2.1 - DESCRIPTION OF THE MAXWELL STRESS TENSOR

The Maxwell stress tensor [15] is a derivation of the computation of the force by the Lorentz method,

for a unit of volume where the distribution of the charge is unknown.

𝑓𝑑 = 𝜌𝑬 + 𝑱 × 𝑩 (6)

The Maxwell tensor allows the computing of the force, recurring only to the distribution of the magnetic

field (since this is the dominant field in the region, the electrical field contribution can be considered

null), and thus with the perk of not being needed to know any electric current density.

For a Cartesian coordinate system (𝑥, 𝑦, 𝑧) → (1,2,3) the Maxwell tensor takes the form:

𝑇𝑖𝑗 = 𝜇𝐻𝑖𝐻𝑗 −𝜇

2𝛿𝑖𝑗(𝐻1

2 + 𝐻22 + 𝐻3

2) , 𝛿𝑖𝑗 = {1 , 𝑖 = 𝑗0 , 𝑖 ≠ 𝑗

(7)

19

Where 𝑇𝑖𝑗 is the Maxwell stress tensor, and 𝛿𝑖𝑗 is the Kronecker delta. Writing (7) in the form of a

stress tensor matrix yields (8), which gives the stress tensor components.

𝑻 =

[ 𝜇

2(𝐻𝑥

2 − 𝐻𝑦2 − 𝐻𝑧

2) 𝜇𝐻𝑥𝐻𝑦 𝜇𝐻𝑥𝐻𝑧

𝜇𝐻𝑥𝐻𝑦𝜇

2(𝐻𝑦

2 − 𝐻𝑥2 −𝐻𝑧

2) 𝜇𝐻𝑦𝐻𝑧

𝜇𝐻𝑥𝐻𝑧 𝜇𝐻𝑦𝐻𝑧𝜇

2(𝐻𝑧

2 − 𝐻𝑥2 − 𝐻𝑦

2)]

(8)

For a direction 𝑖 , it is possible to determine the force density (𝑓𝑖 ) by computing the associated

divergence of stress components 𝑇𝑖𝑗 as in (9).

𝑓𝑖 = ∑𝜕𝑇𝑖𝑗

𝜕𝑥𝑛

3

𝑛=1

, 𝑗 = {1,2,3} (9)

To obtain the total force over a direction 𝑖 (𝐹𝑖), it is computed the integration of 𝑓𝑖 for the volume 𝑉 of

the region enclosed by the chosen surface. Since the force density is obtained from a divergence, the

Gauss Theorem can be used to compute the force over a closed surface 𝑆𝑉.

𝐹𝑖 =∭∑𝜕𝑇𝑖𝑗

𝜕𝑥𝑛

3

𝑛=1

𝑑𝑉

𝑉

= ∯𝑇𝑖𝑗 . 𝑛𝑆𝑉 𝑑𝑆

𝑆𝑉

(10)

The total force is now computed using a force per unit of area 𝑇𝑖𝑗 . 𝑛𝑆𝑉, where 𝑛𝑆𝑉 is the norm to the

surface 𝑆𝑉.The force per unit of area in each direction takes the form of (11).

{

𝑓𝑥 = 𝑇𝑥𝑥𝑛𝑥 + 𝑇𝑥𝑦𝑛𝑦 + 𝑇𝑥𝑧𝑛𝑧𝑓𝑦 = 𝑇𝑦𝑥𝑛𝑥 + 𝑇𝑦𝑦𝑛𝑦 + 𝑇𝑦𝑧𝑛𝑧𝑓𝑧 = 𝑇𝑧𝑥𝑛𝑥 + 𝑇𝑧𝑦𝑛𝑦 + 𝑇𝑧𝑧𝑛𝑧

(11)

20

2.2.2 – EXAMPLE: APPLICATION OF THE MAXWELL STRESS TENSOR TO A CUBIC

VOLUME

Considering a cube delimited region as that shown in Figure 2.2, equation (11) allows the visualization

that the total stress, over 𝑥, 𝑦 or 𝑧 directions, has components in all facets of the cube.

Figure 2.2 – Stress tensor vectors applied to a surface of a cubic volumetric element

The geometry of the cube consists in six 𝑆𝑉 faces: front (ft), back (bk), left (lt), right (rt), inferior (inf)

and superior (sup). For the referential(𝑥, 𝑦, 𝑧), the surface norms take the values:

𝑥 𝑎𝑥𝑖𝑠: 𝑛𝑓𝑡 = 1 , 𝑛𝑏𝑘 = −1

𝑦 𝑎𝑥𝑖𝑠: 𝑛𝑟𝑡 = 1 , 𝑛𝑙𝑡 = −1

𝑧 𝑎𝑥𝑖𝑠: 𝑛𝑠𝑢𝑝 = 1 , 𝑛𝑖𝑛𝑓 = −1

21

Using the force per unit of area it is possible to reach the total force 𝐹𝑥 over the 𝑥 direction as in (12).

𝐹𝑥 =∬𝑇𝑥𝑥 . 𝑛𝑥𝑑𝑆

𝑆𝑥

+∬𝑇𝑥𝑦 . 𝑛𝑦𝑑𝑆

𝑆𝑦

+∬𝑇𝑥𝑧 . 𝑛𝑧𝑑𝑆

𝑆𝑧

=

=∭𝜇

2(𝐻𝑥

2 − 𝐻𝑦2 − 𝐻𝑧

2). 𝑛𝑓𝑡𝑑𝑆

𝑆𝑓𝑡

+∭𝜇

2(𝐻𝑥

2 −𝐻𝑦2 −𝐻𝑧

2). 𝑛𝑏𝑘𝑑𝑆

𝑆𝑏𝑘

+

+∭𝜇𝐻𝑥𝐻𝑦 . 𝑛𝑟𝑡𝑑𝑆

𝑆𝑟𝑡

+∭𝜇𝐻𝑥𝐻𝑦 . 𝑛𝑙𝑡𝑑𝑆

𝑆𝑙𝑡

+ ∭𝜇𝐻𝑥𝐻𝑧 . 𝑛𝑠𝑢𝑝𝑑𝑆

𝑆𝑠𝑢𝑝

+∭𝜇𝐻𝑥𝐻𝑧 . 𝑛𝑖𝑛𝑓𝑑𝑆

𝑆𝑖𝑛𝑓

=

=𝜇

2𝑆𝑓𝑡_𝑏𝑘 [(𝐻𝑥

2 − 𝐻𝑦2 − 𝐻𝑧

2)𝑓𝑡− (𝐻𝑥

2 − 𝐻𝑦2 − 𝐻𝑧

2)𝑏𝑘] +

+𝜇𝑆𝑟𝑡_𝑓𝑡 [(𝐻𝑥𝐻𝑦)𝑟𝑡 − (𝐻𝑥𝐻𝑦)𝑙𝑡] + 𝜇𝑆sup _𝑖𝑛𝑓[(𝐻𝑥𝐻𝑧)𝑠𝑢𝑝 − (𝐻𝑥𝐻𝑧)𝑖𝑛𝑓]

(12)

The computation for the 𝑦 and 𝑧 directions is made in a similar way. Therefore, it is only shown the

final result for 𝐹𝑦 in (13) and 𝐹𝑧 in (14).

𝐹𝑦 =𝜇

2𝑆𝑟𝑡_𝑙𝑡 [(𝐻𝑦

2 − 𝐻𝑥2 − 𝐻𝑧

2)𝑟𝑡− (𝐻𝑦

2 −𝐻𝑥2 − 𝐻𝑧

2)𝑙𝑡] +

+𝜇𝑆𝑓𝑡_𝑏𝑘 [(𝐻𝑥𝐻𝑦)𝑓𝑡 − (𝐻𝑥𝐻𝑦)𝑏𝑘] + 𝜇𝑆sup _𝑖𝑛𝑓 [(𝐻𝑦𝐻𝑧)𝑠𝑢𝑝 − (𝐻𝑦𝐻𝑧)𝑖𝑛𝑓]

(13)

𝐹𝑧 =𝜇

2𝑆sup _𝑖𝑛𝑓 [(𝐻𝑧

2 − 𝐻𝑥2 − 𝐻𝑦

2)𝑠𝑢𝑝

− (𝐻𝑧2 −𝐻𝑥

2 − 𝐻𝑦2)𝑖𝑛𝑓] +

+𝜇𝑆𝑟𝑡_𝑙𝑡 [(𝐻𝑦𝐻𝑧)𝑟𝑡 − (𝐻𝑦𝐻𝑧)𝑙𝑡] + 𝜇𝑆𝑓𝑡_𝑏𝑘[(𝐻𝑥𝐻𝑧)𝑓𝑡 − (𝐻𝑥𝐻𝑧)𝑏𝑘]

(14)

2.2.3 SIMPLIFIED ANALYSIS OF THE INTERACTION BETWEEN A PERMANENT

MAGNET AND A SUPERCONDUCTOR: EXPECTED RESULTS

In order to study the levitation force generated by a set of superconductors, it is needed to consider

that the generated force depends on four fundamental factors:

- The field strength of the magnet;

- The size of the magnet;

- The size of the superconductor;

- The distance between the superconductor and the magnet.

22

Consider the system represented in Figure 2.3 composed by a permanent magnet and a

superconductor above it. To evaluate the electromagnetic forces between the two, a magnetic

stationary regime study will be made. In this regime, equation set (15) is considered.

{∇ × 𝑯 = 𝑱 ∇. 𝑩 = 0 𝑩 = 𝜇𝑯

(15)

Since the superconductor is in zero-field cooling, it can be assumed that there shall not be any

significant magnetic field penetration within it. As such, the ideal superconductor behaves as a region

of null magnetic permeability ( 𝜇𝑆𝐶 = 0 ), and the superconductor region behaves as a perfect

diamagnetic region. Thus, the magnetic field lines are “repelled” from the superconductor, enveloping

it until they close, as shown in figure Figure 2.3.

Figure 2.3 - Illustration of the resulting magnetic flux of the interaction between a permanent magnet and a perfect superconductor

Beginning by applying Ampère’s Law to the path 𝑆1 , using the integral form of ∇ × 𝑯 = 𝑱 . By

application of Stokes’ Theorem, equation (16) is obtained.

∮ 𝑯 𝑑𝑙𝑠1

= ∬𝑱. 𝑛𝑆𝑠1 𝑑𝑆

𝑆𝑠1

→ 𝐻𝑃𝑀1𝑙𝑃𝑀1 + 𝐻𝑎𝑖𝑟1𝑙𝑎𝑖𝑟1 = 0 (16)

The same procedure is used for path 𝑆2 yielding equation (17).

𝐻𝑃𝑀2𝑙𝑃𝑀2 + 𝐻𝑎𝑖𝑟2𝑙𝑎𝑖𝑟2 = 0 (17)

23

Applying ∇.𝑩 = 0 to the surface volume 𝑆𝑣, which englobes a frontier of the permanent magnet along

the 𝑧 axis, and using Gauss’s Theorem, equation (18) is obtained.

∬ 𝑩.𝒏𝒆𝒙𝒕𝑑𝑆𝑆𝑣

= 0 ↔ 𝐵𝑃𝑀1𝑆𝑃𝑀2+ 𝐵𝑃𝑀2

𝑆𝑃𝑀2− 𝐵𝑎𝑖𝑟1

𝑆𝑃𝑀2− 𝐵𝑎𝑖𝑟2

𝑆𝑃𝑀2= 0

↔ 𝐵𝑃𝑀1 + 𝐵𝑃𝑀2 = 𝐵𝑎𝑖𝑟1 + 𝐵𝑎𝑖𝑟2

(18)

As the field within the permanent magnet is equal to it reminiscent field 𝐵𝑟, the equalities in equation

(19) can be stated.

𝐵𝑃𝑀1 = 𝐵𝑃𝑀1 = 𝐵𝑎𝑖𝑟1 = 𝐵𝑎𝑖𝑟2 =𝐵𝑟2

(19)

The linked flux between the superconductor and the permanent magnet can be considered as follows

in equation (20).

Ψ = 𝐵𝑟𝑆𝑃𝑀 (20)

Since the generated force is of magnetic origin, some conclusions can be taken from the magnetic co-

energy through equation (21).

𝐹 = (𝜕𝑊𝑚

𝜕𝑥)Ψ=𝑐𝑜𝑛𝑠𝑡

→ 𝐹(B𝑟) ∝ 1

2𝐵𝑟

2𝑆𝑃𝑀 (1

𝑥2) (21)

It can be expected that when the air gap between the permanent magnet and the superconductor is

smaller, the higher the force will be, and that this force will have a pace of 1

𝑥2.

2.3 – THE YBCO BULK SUPERCONDCUTOR: ITS FEM MODEL VALIDATION

In order to do a successful scale-up for a ZFC based levitation system, there is the need to compute

the differential model formed by equations (1) to (5) that can determine, with enough accuracy, the

magnetic field distribution resulted from the interaction of the superconductor with a permanent

magnet. Knowing the field distribution, one can compute all electromagnetic force components using

Maxwell’s Stress Tensor as described previously.

To validate the model two steps are made. In the first step, it will be simulated the superconductor and

magnet geometry whose experimental results were presented in Painho’s thesis [3]. The experiment

presented in this thesis is ideal for a 2D simulation, so it is a good starting point for model validation.

After the 2D simulation holds up, a 3D version is developed. At this point, a new experimental activity

will be made, now using the ZFC-Maglev track topology considered for the future scale up model. The

24

3D model results will be confronted with the results obtained from the experiment effectuated at the

laboratory.

2.3.1 – FINITE ELEMENT MODELLING (FEM) AND ANALYSIS OF THE

SUPERCONDUCTORS AND MAGNET STRUCTURES

A numerical software capable to compute the FEM model was used. This program allows the

simulation of various physics problems, both using its libraries, as by imputing the differential

equations of the problem.

Making note of equations (12)(13)(14) from section 2.2.2, one only needs to know the surface values

of the magnetic field over the superconductor. Thus, for each superconductor surface, the values of

the magnetic field are exported to a file, with a resolution given by the utilized finite element mesh,

meaning that each surface is characterized by a number 𝑛 of points (the number of meshes in the

surface).

As the field distribution is not necessarily uniform over each surface, the Maxwell Stress Tensor

equations are applied to each mesh element. As a consequence, instead of using the total area (𝑆) it

is used 𝑆/𝑛 , followed by the sum of each mesh element contribution, in order to know the total

electromagnetic force over a given direction, as given by equation (22).

𝐹𝑡𝑜𝑡𝑎𝑙 =∑𝑆

𝑛𝑓𝑖

𝑛

𝑖=1

(22)

To exemplify the need for the application of equation (22), a simulation of a superconductor over a

permanent magnet was made, as shown in Figure 2.4.

Figure 2.4- Simulation of a superconductor over a permanent magnet

25

The magnetic field components can be visualized for each Cartesian direction on the bottom surface

of the superconductor can be visualized in Figure 2.5. As the field is repelled from its surface, it has a

smooth alignment in 𝑥 and 𝑦 directions, the directions parallel to the surface, being strongly pixelated

in 𝑧, as in this direction the magnetic field has a weaker component due to the repulsion.

Figure 2.5 - Example of the magnitude of the magnetic field for the same surface for x, y e z

26

2.3.2 - FIRST ATTEMPT AT VALIDATION: 2D MODEL

As stated before, the first attempt at verifying the proposed model for the computation of forces is

made based on the results of the thesis [3]. The geometry is presented in Figure 2.6. It consists of

eight permanent magnets with a remnant field of 1,25(T) and six 𝑌𝐵𝑎2𝐶𝑢3𝑂7−𝑥 superconductors. This

geometry is ideal for a 2D cut in the 𝑦𝑧 plane, as it is quite symmetrical considering any plane, mainly

in the 𝑥 direction.

Figure 2.6 - Elements’ dimensions

The ZFC force per superconductor is shown in Figure 2.7 for an air gap between 0,5(cm) and

2,25(cm). By analysing the results, the 1

𝑥2 pace of the force in function of the distance is verified, as

deduced by the approximation in section 2.2.3.

Figure 2.7 -Graph of the experimental force per superconductor

The 2D plane is defined in a region where both the superconductor and the permanent magnets exist,

as shown in Figure 2.8. Hence, based on the magnetic field distribution at the superconductor surface,

all stress tensors were computed to determine the levitation force at each superconductor.

27

Figure 2.8- 2D field distribution for an air gap of 1(cm)

Comparing Figure 2.8 and Figure 2.3, it is verified that the simplification proposed in 2.2.3 was correct,

as observing the figures the field lines do in fact go around the superconductor.

By exporting the values of the magnitude of the 𝑥, 𝑦 and 𝑧 fields on the superconductor surfaces, and

computing the levitation force by the proposed model, Table 2.2 is obtained, where the force is given

by superconductor. It shows the simulated force and also the correspondent experimental one,

previously measured in Painho’s thesis [3].

Table 2.2 - 2D simulated and experimental force per superconductor, in function of the distance between the permanent magnet and superconductor

Air gap (cm) Simulated force (N) Experimental force (N) [3] Error (%)

0,75 9,72 6,67 46

1 6,23 5,17 21

1,25 4,14 4,00 3

1,5 2,87 2,83 1

1,75 2,08 2,33 11

2 1,56 1,67 7

2,25 1,20 1,17 3

Despite the results for the 0,75(cm) air gap showing a significant error of 46%, the 2D simulated

values show low errors against the experimental values. As such, the next step is creating the 3D

version of the FEM program.

28

2.3.3 - SECOND ATTEMPT AT VALIDATION: 3D MODEL

Using the geometry of Figure 2.6, and the same set of equations used for the 2D experiment, a 3D

version of the FEM program was developed. The results are listed in table Table 2.3.

Table 2.3 - 3D simulated and experimental force per superconductor, in function of the distance between the permanent magnet and superconductor

Air gap (cm) Simulated force (N) Experimental force (N) [3] Error (%)

0,75 15,37 6,67 131

1 9,19 5,17 78

1,25 5,80 4,00 45

1,5 3,87 2,83 37

1,75 2,69 2,33 15

2 1,98 1,67 19

2,25 1,48 1,17 27

For a simpler visualization, all levitation forces (2D, 3D and experimental values) are shown in Figure

2.9.

Figure 2.9 - Graph comparing the experimental force with the simulated ones

Though the error for an air gap inferior to 1(cm) is quite high, both the 2D and 3D simulated values do

tend to stick to the experimental values. The high error may be caused by the FEM program, or might

be an experimental error. Since there is not a way to be certain, a final experimental validation will be

made by doing an experiment now with the topology required for the ZFC-Maglev levitation system,

comparing it after with the values given by the FEM model.

29

2.4 – ZFC-MAGLEV TOPOLOGY: LEVITATION FORCE

The most relevant forces for a Maglev system are levitation and guidance. This means that the

levitation force is computed using equation (14) and the guidance force by equation (13). Though all

surfaces contribute to the generated force in one direction, their major component comes from the

surfaces that are perpendicular to intended direction.

Due to the Meissner effect, the levitation force main contribution will come from the inferior surface, as

the field component according to 𝑧 is very weak, leading to a positive force contribution from this

surface.

The same can be said for the guidance force. The left and right lateral facets will determine if the

vehicle remains on track or not.

2.4.1 - CONSIDERED SUPERCONDUCTOR AND MAGNET GEOMETRY

The proposed configuration consists of four YBCO superconductors of type 2, overlapped over twelve

permanent magnets, whose remnant magnetic flux density is of 1,25(T). The dimension of the

components is shown in Figure 2.10. There is an air gap of 1(cm) between the superconductors, and

the magnets have a spacing of 1,5(cm) over 𝑥 direction and of 2(cm) over the 𝑦 direction.

Figure 2.10 - ZFC track and superconductor placement, a mesh of 0.5(cm) was used

The magnets are oriented differently in each row of magnets as shown in Figure 2.11. The lateral

magnets have their polarization according to the positive direction of the 𝑧 axis, while the central row

magnets are in the opposite direction.

30

Figure 2.11 - Permanent magnet orientation in the ZFC track

2.4.2 - SIMULATION RESULTS

By making a cut along the 𝑦𝑧 plane in the geometry of Figure 2.10, in such a way that it intersects the

magnets and the superconductors, the distribution of the magnetic field in space can be observed.

Due the offset between the lateral magnets and the central magnets, two cuts were made for two

different points along the 𝑥 direction, as presented in Figure 2.12.

Figure 2.12 - Cuts made in the 𝑦𝑧 plane along the 𝑥 direction

31

In Figure 2.13 the superior graph shows the plane that has cut the two lateral magnets. These have

the same magnetic polarity and are responsible for the tangential magnetic field below the

superconductors, which ends in the magnet located in the middle. However, this magnet and

associated magnetic field is best seen in the inferior graph of Figure 2.13.

In both graphs of Figure 2.13, streamlines were used to compose a map of the magnetic field

distribution in the ZFC-Maglev system. Both graphs also show that the magnetic flux closes around

the superconductor blocks borders, as indicated by the red arrows.

Figure 2.13 - Magnetic field distribution of both cuts

Within the superconductors there are flux lines closed in a spiracle shape, which point to the passage

of current. The current density is stronger near the borders of the superconductor blocks, having near

to null values within its core, as shown in Figure 2.14.

32

Figure 2.14 - Visualization of the electric current desnity and the magnetic flux distribution lines within the superconductors

Table 2.4 presents the simulated results, where the force is given by superconductor as function of the

air gap to the permanent magnet track.

Table 2.4 – ZFC-Maglev topology simulated levitation force per superconductor

Air gap (cm) Simulated force (N)

0,75 7,40

1 5,40

1,25 3,90

1,5 2,91

1,75 2,19

2 1,65

2,25 1,22

2,5 0,93

2.4.3 -THE EXPERIMENTAL SET-UP FOR THE ZFC-MAGLEV GEOMETRY

In order to determine the levitation force, the sensor Scaime K12+LMVu shown in Figure 2.15 was

used. It is a compression force piezoelectric sensor with an error of 0,1%, capable of registering forces

up to 150(N). Before the start of the experiment, the sensor was calibrated. For this procedure, it was

registered the voltage given by the sensor to four different weights of known mass, in order to get a

regression line. The regression line is then used to compute the force values, using the voltage values

given by the sensor during the experimental activity.

33

Figure 2.15- Photo of the piezoelectric force sensor Scaime K12+LMVu

At first the superconductors where cooled by liquid nitrogen in the absence of the field of the

permanent magnets, so that they were in ZFC mode.

Figure 2.16 shows a photo of a track module for the ZFC-Maglev composed by twelve 𝑁𝑑𝐹𝑒𝐵

permanent magnets with a remnant field of 1,25(T), following the disposition proposed for the Maglev

vehicle with levitation based in zero field cooling (ZFC). The lateral permanent magnets have the

same polarity, while the central magnets have opposite polarity and are unaligned relative to the two

other magnet lanes.

Figure 2.16 - Triple magnet trail disposition for the ZFC-Maglev

34

Over the magnets, there are four superconductors of 𝑌𝐵𝑎2𝐶𝑢3𝑂7−𝑥 within a foam box with the same

dimensions used for the simulation, as shown in Figure 2.17. The superconductors are arranged in

columns of two. The box is sealed by a foam top for better thermal isolation of the YBCO blocks. The

foam includes a hole used for filling the box with liquid nitrogen.

Figure 2.17 - Disposition of the superconductors inside the foam box following the same geometry used in the simulation

The total assembly is presented in Figure 2.18. This assembly allows the for an air gap between

0,75(cm) and 2,5(cm).

Figure 2.18 - Total assembly for the experiment

35

2.4.4 – EXPERIMENTAL LEVITATION FORCE RESULTS

Before the presentation and analysis of the levitation force results obtained using the experimental set

up introduced in the previous section, one will show how the magnetic field is distributed along the

ZFC-Maglev track, before and after the placement of the ZFC-Maglev superconductor module above

the permanent magnet track.

Figure 2.19 - Permanent magnet field for two cuts in the absence of the superconductors

36

Figure 2.20 - Permanent magnet field for two cuts in the presence of the superconductors

As explained in section 1.3.1 the Meissner effect of type II superconductors repels the magnetic field

of permanent magnets. For this topology this translates into a magnetic flux compression, with the

magnetic lines under the superconductors being brought closer to each other, thus intensifying the

magnetic field under the superconductor.

37

The experimental results are expressed in Table 2.5 in force per superconductor.

Table 2.5 - Comparison between the experimental force and the simulated force per superconductor

Air gap (cm) Simulated force (N) Experimental force (N) Error (%)

0,75 7,40 8,54 13

1 5,40 6,33 15

1,25 3,90 4,90 21

1,5 2,91 3,54 18

1,75 2,19 2,88 24

2 1,65 2,19 25

2,25 1,22 1,63 25

2,5 0,93 1,21 23

There is a consistent error between the experimental and simulated values on average of 21%. This

comes from the numerical precision associated with the mesh used. This could be removed using a

finer mesh in FEM model. However, the computational effort was too much for our current hardware.

Since, the order of the error values is almost the same, independent of the air gap, the FEM model

can be considered validated. Hence, Figure 2.21 shows the simulated values increased by the

average error and the experimental values.

Figure 2.21 - Graphic comparing the simulated force increased by the average error with the experimental force for the ZFC-Maglev track

38

3 - ZFC-MAGLEV SCALE-UP TO THE MAGLEV-COBRA

This chapter introduces the methodology by which the final ZFC superconducting magnetic circuit

geometry for the magnetic levitation system scale-up was attained, taking into account the

requirements of the Maglev-Cobra vehicle. All data relative to the Maglev-Cobra vehicle was obtained

from thesis that dealt with its conceptual design [1] [9].

The superconducting levitation system has the following characteristics:

1 - There are six cryostats per module of the vehicle as shown in Figure 3.1(a), each containing 24

YBa2Cu3OX superconductors, whose dimensions are of 63x63x13(mm);

Figure 3.1 – (a) Schematic of one Maglev-Cobra module and its cryostats; (b) Photo of the cryostat utilized by the levitation system of the Maglev-Cobra train [1]

2 - The bottom wall of the cryostat has a thickness of 2(mm);

3 - Each module has a mass of 500(kg), and it is capable of sustaining up to ten passengers of

100(kg);

4 - The field cooling position is made at a distance of 25(mm) from the tracks;

5 - The guidance force must be such that it should allow the train to do curves at a speed of 3,6(m/s),

having a maximum displacement from its movement centre of 10(mm)

39

Considering the collected characteristics, this chapter will treat the development of a ZFC-Maglev

system with the following characteristics:

- Each module has to support a maximum weight of 15000(N) distributed by six cryostats, which

corresponds to 2500(N) per cryostat;

- The module should levitate between 5(mm) and 25(mm), therefore the levitation system must

generate enough force within this set of height values;

- Since the cryostat’s bottom wall has a thickness of 2(mm), the minimum height assumed was of

10(mm);

- The maximum guidance force is of 5400(N) distributed by the six cryostats, which corresponds to

900(N) per cryostat.

In order to understand key elements of the Maglev system for the generation of levitation and

guidance force, a pre-study is made using the geometry of the experimental activity (Figure 2.10).

3.1 - INFLUENCE OF TRACK AND SUPERCONDUCTOR DISPOSITION IN THE LEVITATION AND GUIDANCE FORCES

From the theoretical study in Chapter 2, the points listed below are known to lead to an increase in the

generated levitation force by the Maglev system composed by the coupling of ZFC superconductors

with permanent magnets. The points are:

- Increase of the superconductor’s inferior area;

- Increase of the magnet’s superior area;

- Increase of the magnetic field density;

- Reduction of the air gap between the superconductor and the magnet.

In an attempt to improve the generated levitation and guidance forces, three studies will be made. The

first two studies are about the layout of the magnet track, and the third about the superconductor

disposition. For these studies it is used the geometry presented in the last chapter (2.4.1), and the

following convention is used: the left block is the block in the negative side of the 𝑦𝑦 axis, and the right

block is the block on the positive side of the 𝑦𝑦 axis.

It is of worth remembering the polarization of the magnets. The lateral magnets have their polarization

according to the direction of the 𝑧 axis, while the central row magnets are in the opposite direction as

shown in Figure 2.11.

A reminder should be made about how the levitation and guidance forces are generated. As indicated

in point 2.2.2, the main contributions to the force generated over a direction are from facets

perpendicular to that direction. This means that for the guidance force, the most fundamental facets of

40

the superconductors will be their left and right laterals, and for the levitation force it will be the superior

and inferior facets.

Figure 3.2 - Illustration of the superconductor block disposition

3.1.1 - STUDY 1 AND 2: TRACK LAYOUT EVALUATION

These two studies aim at understanding how the magnet positioning affects the generated levitation

and guidance forces. To access the influence in the generated forces, a total variation of 5(mm) is

made. The standard distance in 𝑥 between the magnets is of 15(mm), and in 𝑦 is of 20(mm).

Figure 3. 3 - Magnet placement in study 1. (a) Standard case; (b) Increase in 𝑥; (c) Decrease in 𝑥

41

Table 3.1 - Variation of the distance between the magnets in the x direction, for an air gap of 15(mm). In the case of the guidance force the displacement is of -5(mm) from its central position

Study 1 Levitation force (N) Guidance Force (N)

Left block Right block Resulting force

Standard distance 10,8 -0,08 2,54 2,46

Increase of 5(mm) 9,0 0,07 1,93 2,00

Decrease of 5(mm) 13,8 -0,26 3,48 3,23

Figure 3. 4 - Magnet placement in study 1. (a) Standard case; (b) Increase in 𝑦; (c) Decrease in 𝑦

42

Table 3.2 - Variation of the distance between the magnets in the y direction, for an air gap of 15(mm). In the case of the guidance force the displacement is of -5(mm) from its central position




Increase of 5(mm) 9,6 -0,93 2,81 1,87

Decrease of 5(mm) 12,0 0,69 1,91 2,60

The immediate conclusion is that the approximation of the magnets, either in the 𝑥 or 𝑦 directions,

leads to a higher magnetic flux density, which results in an increase in both levitation and guidance

forces.

It can also be verified that for a negative displacement of 5(mm) of the superconductor blocks in

relation to the then central position, the major component of the restitution force comes from the HTS

block on the opposite side of the displacement, the right block. Thus, it can be concluded that the

magnetic field distribution is more intense between the superconductors than in their outer lateral

surfaces.

3.1.2 - STUDY 3: DISTANCE BETWEEN THE SUPERCONDUCTORS

This study was conducted to determine how the gap between the superconductors interacts with the

field distribution, in order to find out how it affects the levitation and guidance force generation. As

stated in the introduction, it is the gap between the columns of superconductor blocks that allows the

magnetic field to close through it. For elements of the considered size, between 2 and 3 centimetres

long, a gap of 10(mm) was originally considered. To evaluate how the gap affects the generated

levitation and guidance forces, a total variation of 5(mm) is made to this gap.

Figure 3. 5 - Superconductor placement in study 3. (a) Standard case; (b) Increase in the gap; (c) Decrease in the gap

43

Table 3.3 - Variation of the gap between the superconductors, for an air gap of 15(mm). In the case of the guidance force the displacement is of -5(mm) from its central position




Increase of the gap 10,1 0,89 2,00 2,89

Decrease of the gap 12,2 -1,62 3,47 1,86

The approximation of the superconductors leads to an increase in levitation force, since it increases

the magnetic energy stored per unit of volume due to a smaller gap. This means that the magnetic

field of the central magnet is more effectively used.

However, the same result is not obvious for the guidance force. An increase in the gap benefits the

guidance force. Despite this, it is of note that though the restitution force of the left block does

increase, there is a reduction on the restitution force of the right block. Therefor the benefit of

increasing the gap between the superconductors may not be directly applied in all cases, being

dependent on the interaction between the superconductor’s and permanent magnet’s layout.

3.1.3 - PRE-STUDY CONCLUSIONS

From the results obtained from studies 1 and 2, which dealt with the permanent magnet track, it is

verified that an increase in the number of permanent magnets per unit of area, leads to stronger

forces. The results from study 3 have shown that there is a better usage of the central magnet field,

when the superconductors are closer to each other, at a cost of the system’s stability. This is due to a

smaller guidance force when the superconductor blocks are brought together.

A thourough study of how the guidance force is generated was made. It was determined that the block

opposite to the dispalcemente is the main responsible for the restitution force. Using the simulation

data, the force contributions of the left and right lateral surfaces of each superconductor block was

retrived for a displcement of -5(mm), and is presented in Table 3.4.

Table 3.4 - Contribution of each facet for the guidance force, for a displacement is of -5(mm) from its central position

Guidance force (N)

Left block Right block Total force

Left facet Right facet Total Left facet Right facet Total

1,33 -1,55 -0,21 2,70 -0,31 2,38 2,17

Both blocks contribute to the guidance force. However, the magnetic field is so strong in the inner gap

between the superconductors, that it off balances the force generated by the left superconductor

block. The strength of the field is presented in Figure 3.6. Since the magnetic field is mostly tangential

in the gap, it was chosen to show the norm of the magnetic field. The surfaces with stronger field

values are (b) and (c), which correspond to the inner surfaces of the gap.

44

Figure 3.6 - Distribution of the magnetic field for a displacement of -5(mm): (a) left surface of the left block; (b) right surface of the left block; (c) left surface of the right block; right surface of the right block

45

3.2 - ESTABLISHING THE FINAL TRACK AND SUPERCONDUCTOR GEOMETRY

3.2.1 - SEARCH TRIAL-AND-ERROR METHOD

The levitation height dictates how much force the levitation system can produce. Despite this fact,

when planning a new potential solution it should not be considered the most relevant parameter, and

priority should be given to increasing the size of the elements of system: permanent magnet track and

superconductor blocks. The search method for a solution that satisfies all the functional criteria

established in the introduction of the chapter is shown in the flowchart of Figure 3.7.

All simulations were made using the finite element model developed and validated in Chapter 2, by

using six superconductors (two columns of three) and enough magnets to cover the space occupied

by the superconductors. The usage of so few elements, both magnets and superconductors, is due to

the computational demands of a larger model 3D model. The model uses a mesh of 5 (mm) on all

superconductor surfaces.

From the Maglev-Cobra specifications shown before, it is established that each cryostat must support

at least 2500(N) of levitation force. In order to get a competitive solution, regarding the number of

superconductors used in the Maglev-Cobra, each superconductor must develop a minimum of 104(N)

of levitation strength. This means that the proposed solution must develop a levitation force of around

600(N), with a minimum air gap of 10(mm), as the simulated model uses six superconductors.

For each iteration of a potential solution, in a first instance it is verified if the minimum levitation force

is guaranteed. If this criterion is met, it can proceed to guidance force verification, where the force

must be of restitution, which means it must be opposed to the lateral displacement direction. If this

point fails, it can be opted to improve the solution, or discard it and search for a new one.

46

Figure 3.7 – Flowchart of the search trail-and-error method

47

3.2.2 - GEOMETRY A

After successive attempts following the flowchart of figure Figure 3.7, by increasing elements’ size, the

first ZFC-Maglev geometry showing force values within the established criteria is obtained. The

permanent magnets and superconductors dimensions are listed in Table 3.5 and Figure 3.8 shows the

layout of the magnets (relative distance between magnets was kept the same), being that the gap

between the superconductors is of 20(mm).

Table 3.5 - Dimensions of the elements in geometry A

Dimensions(mm) Permanent Magnets Superconductors

Length 50 60

Width 50 60

Height 20 15

Figure 3.8 - Geometry A achieved for a ZFC-Maglev module

Figure 3.9- Mesh of 5(mm) used for the simulation of geometry A

48

The simulated forces are shown in Table 3.6. This table lists how levitation and guidance forces

evolve in function of the air gap from 5(mm) to 20(mm). The guidance forces were measured for a

displacement of -5(mm) from the superconductor’s central position.

By inspecting the table, it can be verified that the necessary levitation force can be obtained for an air

gap between 7,5(mm) and 10(mm). However the minimum air gap is 10(mm). Hence, in an attempt to

increase the levitation force, geometry B is proposed, where the gap between the superconductors is

reduced, in order to increase the levitation force.

Table 3.6 - Levitation and guidance force in function of the air gap, with the guidance force computed for a displacement of –5(mm) from its central position, for geometry A

Air gap (mm) Levitation force (N) Guidance force (N)

5 902 69

7,5 657 60

10 495 50

12,5 379 41

15 292 34

17,5 231 27

20 182 23

3.2.3 - GEOMETRY B

Geometry B uses the components with the same dimensions as in Table 3.5, and with a gap between

superconductors of 10(mm). The simulated forces are presented in the Table 3.7.

Figure 3.10 - Geometry B achieved for a ZFC-Maglev module

49

Figure 3.11 - Mesh of 5(mm) used for the simulation of geometry B

Table 3.7 - Levitation and guidance force in function of the air gap, with the guidance force computed for a displacement of -5(mm) from its central position, for geometry B


5 986 54

7,5 719 50

10 541 43

12,5 418 36

15 324 30

17,5 254 24

20 201 20

The reduction in the superconductor’s gap led to slight stronger force values. However, levitation

forces of 600(N) were still not verified for an air gap superior to 10(mm). Despite the fact the

necessary force is not yet obtained, it is in its vicinity. Thus, for getting a stronger magnetic field, the

height of the permanent magnets will be increased, resulting in geometry C analysed next.

50

3.2.4 - GEOMETRY C

This geometry is based on geometry A, being the only difference the height of the permanent magnets

that changed from 15(mm) to 30(mm). Table 3.8 lists the dimensions of each element and Figure 3.12,

shows the dimensions indicated for the geometry.

Table 3.8 - Dimensions of the elements in geometry C

Dimensions(mm) Permanent Magnets Superconductors

Length 50 60

Width 50 60

Height 30 15

Figure 3.12 - Geometry C achieved for a ZFC-Maglev module

Figure 3.13 - Mesh of 5(mm) used for the simulation of geometry C

51

The force results for this geometry are listed in the Table 3.9. For an air gap of 12,5(mm), the levitation

force criterion is almost met. Thus, by making a linear approximation, it can be established that for an

height of 11,25(mm) the levitation force will be about 638N.

Table 3.9 - Levitation and guidance force in function of the air gap, with the guidance force computed for a displacement of –5(mm) from its central position, for geometry C


5 1277 107

7,5 928 95

10 705 77

12,5 577 62

15 447 53

17,5 353 43

20 278 36

In an attempt to get the levitation force for an height of 12,5(mm), geometry D is proposed in the next

section, where the distance between superconductors is decreased to 1(cm).

3.2.5 - GEOMETRY D

Figure 3.14 shows geometry D where the gap between superconductors has been reduced to

10(mm). All other dimensions remained the same as in geometry C. Levitation and guidance forces for

different heights are listed in table Table 3.10.

Figure 3.14 - Geometry D achieved for a ZFC-Maglev module

52

Figure 3.15 - Mesh of 5(mm) used for the simulation of geometry D

Table 3.10 - Levitation and guidance force in function of the air gap, with the guidance force computed for a displacement of -5(mm) from its central position, for geometry D


5 1401 85

7,5 1044 81

10 802 67

12,5 614 53

15 492 47

17,5 390 38

20 309 32

Considering this geometry, Table 3.10 shows that the levitation force for an air gap of 12,5(mm)

surpasses the needed 600(N), despite showing a lower guidance force than geometry C, from 62(N)

to 53(N).

3.2.6 - GEOMETRY SELECTION FOR THE ZFC-MAGLEV

In order to achieve the same levitation force of 2500(N) per cryostat of the Maglev-Cobra for a

minimum height of 10(mm), the results in Figure 3.16 indicate that only two geometries can be

selected, geometry C or geometry D. These allow having respectively 2499(N) and 2658(N) for an air

gap of about 12,5(mm) in the ZFC-Maglev

53

Considering that each superconductor has a cross section of 60x60(mm2), the ZFC-Maglev would

need 26 superconductors per cryostat, instead of the 24 superconductors used in the Maglev-Cobra.

Figure 3.16 - Graph of the force in function of the air gap for all geometries; Points A, B, C and D indicate where each curve passes the desired levitation force value of 2500(N)

Figure 3.17 shows that for the guidance force, the best geometry is C, since it is the one that develops

the highest restitution force. Thus, it is with this geometry that a complete guidance force study will be

made, where the guidance force will be computed for larger lateral displacements than 5(mm).

Figure 3.17 - Graph of the guidance force in function of the air gap for a lateral displacement of -5(mm)

54

3.2.7 - GUIDANCE STUDY FOR GEOMETRY C

Using the FEM program, the total guidance force generated by the superconductors is computed for a

set of lateral displacements within the interval [-4,4](cm). The result for 26 superconductors, that is 13

superconductors in two columns in parallel, is shown in Figure 3.18.

Figure 3.18- Graph of the guidance force in function of the lateral displacement for geometry C

The maximum guidance force in Figure 3.18 is of 680(N) for a displacement of 2(cm), which is still

below the required 900(N) for a maximum displacement of 1(cm) established by the criteria of the

Maglev-Cobra. However, one has to limit the allowed interval for lateral displacement into a stable and

nearly linear region situated between [-1,1](cm), as indicated in Figure 3.18. In this case, the

maximum guidance forces stayed at 500(N).

Having attained the required levitation force, it can be established at this time that the guidance force

is a weakness of the ZFC-Maglev system topology.

55

4 – TECHNICAL AND ECONIMIC ANALYSIS COMPARISON BETWEEN THE ZFC-MAGLEV AND

MAGLEV-COBRA

In this chapter, a comparative technical and economic analysis will be made, between the final ZFC-

Maglev solution of Chapter 3, geometry C, and the actual Maglev-Cobra system. It will start by

comparing the generated levitation and guidance forces per cryostat, and then the implementation and

operational system costs are analysed and compared.

56

4.1 – LEVITATION AND GUIDANCE FORCE COMPARISON

The forces developed by the Maglev-Cobra, both levitation and guidance, was retrieve from the thesis

[1]. This thesis concerns the optimization of the permanent magnet track, and as such includes

information about the original geometry. The forces presented here are in force per meter, since the

total force will be higher, the longer the vehicle. With proper spacing, each cryostat fits in one meter,

and there are two parallel cryostats per meter. As such, the force data is considered to be the

developed force by two cryostats, and thus it is divided by two, for comparison with the force values

previously obtained in Chapter 3 for geometry C, by scaling proportionally the force values from 6

superconductors to 26.

Figure 4.1 - Levitation force comparison between the Maglev-Cobra and the ZFC-Maglev scale-up

The developed solution for the ZFC-Maglev is capable of generating more force than the one used by

the Maglev-Cobra, as shown in Figure 4.1. This means that for levitation proposes the Portuguese

track layout is better, as it allows for a higher developed levitation force.

The main drawback however, as stated in Chapter 3, is the guidance force. The main problem with

zero-field cooling is system stability in terms of keeping the vehicle on the tracks. Figure 4.2 compares

the guidance force developed by each Maglev. The Portuguese layout does implement a solution

which has a natural stability point in the middle of the track. However, Figure 4.2 (b) shows that the

“Scale-up” solution is not capable of reaching the minimum guidance force value off 900(N) for a

displacement of 1(cm). By contrast the restitution force of the Maglev-Cobra is much higher at 1(cm),

having the value of 4000(N), thus having a good safety margin for the required displacement.

57

Figure 4.2 – Total guidance force comparison; (a) Maglev-Cobra guidance force, and (b) ZFC-Maglev guidance force; Red line indicates the minimum guidance force of 900(N)

4.2 - IMPLEMENTATION COST ANALYSIS

Maglev-Cobra’s thesis [1] has estimated the cost of the Maglev-Cobra implementation for a 100(km)

track. Table 4.1 lists the cost per 100(km) in the unit of millions of US dollars for steel, permanent

magnets, aluminium and superconductors.

Table 4.1 - Implementation cost per 100(km) for the Maglev-Cobra track, in millions of US dollars, in November of 2011

Steel cost Permanent Magnet cost Aluminium cost Superconductor cost Total cost

2,07 1320 0,87 1,87 1335

For the cost analysis, each element type has been considered individually, being that the total cost

was shown at the end of the analysis. About 96% of the implementation cost comes from the

permanent magnets found in the tracks. The Maglev-Cobra track is composed of two rails whose

cross section dimensions are indicated in Figure 4.3. Each rail is always made of modules such as the

58

one in the photo of Figure 4.4. A special reference is given to the steel core between the magnets and

on their sides.

Figure 4.3 - Dimensions (cm) of the cross section of the Maglev-Cobra rail

Figure 4.4 - Photo of the rail topology used by Maglev-Cobra track [1]

The track used by the ZFC-Maglev solution would consist in two rails like the one presented in Figure

4.5. The dimensions of the permanent magnets are the same as the ones previously indicated in

Table 3.8 of Chapter 3.

59

Figure 4.5 - Rail topology of the ZFC-Maglev solution

This type of track consists only of permanent magnets and aluminium as a fitting to keep the magnets

fixed in place. This way, the rail has a direct economic advantage over the one used by the Maglev-

Cobra, as it does not need a steel core. A second point in its favour the total volume of permanent

magnets used. In Figure 4.6 a visual comparison is made, concerning the volume of magnets used

per meter. In it, one meter of the ZFC-Maglev rail is sorted in parallel with the two permanent magnets

that form one rail of the Maglev-Cobra.

Figure 4.6 - Difference in quantity of permanent magnets used per meter (figure scale in centimeters)

60

The ZFC-Maglev rail uses 53% less volume of magnets. This translates in a reduction of 702 million

US dollars. As both Maglev tracks use aluminium, this cost will not be evaluated, and it will be

assumed that it has the same value.

In terms of superconductors needed, there is an expected cost increase of 108% for the ZFC-Maglev,

since it uses 26 superconductors per cryostat, instead of 24.

Thus the total cost estimated for the ZFC-Maglev is of 623 million US dollars, which represents a

reduction of 54% of the implementation cost compared with that of the Maglev-Cobra. The cost per

material for the ZFC-Maglev is revised in Table 4.2.

Table 4.2 - Implementation cost per 100(km) for the ZFC-Maglev track layout, in millions of US dollars, using the values of November of 2011

Steel cost Permanent Magnet cost Aluminium cost Superconductor cost Total cost

0 618 0,87 2,02 621

4.3 - OPERATIONAL COST ANALYSIS

As the superconducting levitation system is composed of passive elements, the only operational cost

will be the cooling system for the superconductors, in order for them to be in their superconductive

state at temperatures below the 90(K).This makes the required quantity of liquid nitrogen to cool the

superconductors, the only operational cost. To obtain a quantitative value of the needed liquid

nitrogen, results from the master thesis [16], were used to evaluate the superconductor power losses

in function of the magnitude of a periodic applied magnetic field over a superconductor and for a

frequency value that will depend of the vehicle speed and track dimensions as explained next.

Figure 4.7 - Graph of the power losses in function of the magnitude of the applied magnetic field over the superconductor for 5(Hz) [16]

61

Figure 4.8 is a schematic representation of the magnetic field distribution on the ZFC-Maglev track.

The illustration indicates the distance ∆𝑥 for which a time period is defined according to the Maglev

speed. From it, it becomes possible to define a frequency for the magnetic field on the

superconductors, as the field between the centres of two consecutive lateral permanent magnets. This

distance is of 6,5(cm).

Figure 4.8 - Symbolic illustration of the distance ∆x, for which a time period is defined for the magnetic field acting on the superconductors

From distance ∆𝑥, it was possible to determine the magnetic field frequency using equations (23) and

(24), by imposing the velocity at which the vehicle moves.

𝑣 =∆𝑥

∆𝑡→ ∆𝑡 =

∆𝑥

𝑣 (23)

𝑓 =1

∆𝑡→ 𝑓 =

𝑣

∆𝑥 (24)

Using the results from the simulation of geometry C for an air gap of 12,5(mm), it was possible to

determine the average of the magnetic field which crosses the inferior surface of the superconductor

blocks (𝐵𝑧), which is equal to 1(mT). This field was chosen as it is the one that will induce currents

within the superconductors, in the first few millimetres. Using data from the graph from Figure 4.7, for

a frequency of 5(Hz) the power loss density can be estimated to be about 5,5 × 10−5 (W/cm3). As the

power losses are proportional to the frequency, the losses can be computed for other frequencies

proportionally. Therefore, the power losses density as function of the Maglev speed is given in Figure

4.9.

62

Figure 4.9 - Power losses density in function of the Maglev speed

Recalling Figure 2.14 the current density was concentrated near the surfaces of the superconductor

blocks. Thus, the considered volume to compute the total power losses is restricted to a 1(mm) shell

near the superconductor surface.

Knowing the total power losses, it is possible to estimate the duration of the liquid nitrogen deposit.

Since it has a pressure valve, the heating process of the cooled superconductors is made at a

constant pressure, allowing the use of Heat Law equation (25).

𝑄 = 𝑚𝐶𝑝∆𝑇 (25)

In (26), the term 𝑚 is the mass of liquid nitrogen (whose mass density is of 800(kg/m3) [16]), and 𝐶𝑝 its

specific heat at constant pressure (2,042 × 103(Jkg-1

K-1

) [16]). The liquid nitrogen is at a temperature

of 77(K), and evaporates at a temperature of 78(K). Thus, it was possible to determine the duration of

the deposit using equation (26) with ∆𝑇 = 1 and 𝑄 = 𝑃𝑙𝑜𝑠𝑠 × 𝑡.

𝑡 =𝑚𝐶𝑝

𝑃𝑙𝑜𝑠𝑠 (26)

The required liquid nitrogen for the Maglev-Cobra is assumed to be 5(l) per day [17], having each

cryostat a storage capacity of 2,5(l) [18]. Assuming that one Maglev-Cobra vehicle operates for 12

hours, the power losses density can be computed using the proposed simplified thermal model,

knowing that at the time the superconductors had the dimensions of 64x32x12(mm), though also using

24 superconductors. The results are shown in Table 4.3.

63

Table 4.3 - Consumption of liquid nitrogen by the Maglev-Cobra in a 12 hour window

Volume of liquid

nitrogen (l)

Power losses of the 24

superconductors (W)

Power losses density

(W/cm3)

5 0,19 3,1 × 10−4

In order to compare the efficiency of the Maglev-Cobra and ZFC-Maglev in terms of liquid nitrogen

consumption, the volume of liquid nitrogen is computed for the same 12 hour operation time, using the

information gathered from Figure 4.7 simplified thermal model is applied to generate Table 4.4.

Table 4.4 - Consumption of liquid nitrogen by the ZFC-Maglev in a 12 hour window

Volume of liquid

nitrogen (l)

Power losses of the 26

superconductors (W)

Power losses density

(W/cm3)

3,68 0,14 5,5 × 10−5

Comparing the results for an operation time of 12 hours, shows that ZFC-Maglev consumes about

27% less liquid nitrogen than the Maglev-Cobra. This means that the ZFC-Maglev will have a much

lower operational cost.

64

5 – FINAL CONCLUSIONS AND FUTURE WORK

The work developed in this thesis was focused on developing a competitive solution for a Maglev

system, using the Portuguese solution proposed by professors António Dente and Paulo Branco, so

that it would fulfil the functional criteria of the Maglev-Cobra.

The studies concerning the track and superconductor placement, found that an increase in number of

permanent magnets per unit of area in the track, will lead to stronger levitation forces. As for the

guidance force, it was postulated that its major portion comes from the facets present in the gap

between the superconductors. The size of this gap should be dimensioned taking into account the

layout of the track’s permanent magnets, though, in some cases, a bigger gap should lead to higher

guidance forces.

The final geometry developed during the “scale-up” process has superconductors whose size is

almost identical to those used by the Maglev-Cobra’s levitation system. This allows the train to switch

between track types, as only the cooling method must be changed.

The proposed solution offers an outstanding economical advantage. With a near identical investment

cost in superconductors, the track cost was almost halved, allowing the saving of 702 million US

dollars, as it does not employ a steel core to increase the magnetic field density. It also fairs better for

the generation of a levitation force, due to clever magnet placement.

The operational costs were computed using a rather simplistic thermal model, which was not

validated. However, if it holds true, the ZFC-Maglev consumes 27% less liquid nitrogen than the

Maglev-Cobra. This adds to the economical advantage of the ZFC-Maglev, as it is expected that it will

consume less liquid nitrogen.

The main disadvantage of the Portuguese solution is not being able to generate enough guidance

force to reach the minimum value. Though it does reach two thirds of the minimum value, it does so at

double the maximum lateral displacement distance.

65

5.1 – FUTURE WORK

Further studies need to be done concerning the consumption of liquid nitrogen by the levitation

system, in order to obtain a better understanding of the operation costs involved in this type of

magnetic levitation system. Using the data collected in this thesis and the thesis [16] a new FEM

program may be developed to analyse the power losses of the system.

In regards to the guidance force, a possible solution might be the introduction of more or bigger

permanent magnets in curves. Lateral permanent magnets will establish a new field over the lateral

facets of the superconductors, allowing for better guidance force values. Alternatively, increasing the

height of superconductors may also be done, as an increase in lateral area will also bring higher force

values.

A second possible solution to the guidance force, is including a second type of cryostat whose

superconductors are used only for guidance force generation, similarly to the T-shaped guideway

presented in the State of the Art.

66

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