AN INVESTIGATION INTO ELECTRODYNAMIC MAGNETIC LEVITATION USING THE FINITE ELEMENT METHOD

Transcription

1 GRADUADO EN INGENIERIA ELECTROMECANICA AN INVESTIGATION INTO ELECTRODYNAMIC MAGNETIC LEVITATION USING THE FINITE ELEMENT METHOD Autor:Mario Ginel Pérez Director: Craig Baguley Madrid Agosto 2014

2 Table of Contents Table of Contents Section I Report... 8 Chapter 1 Introduction Significance and background Electromagnetic Suspension Electrodynamic Suspension Objectives and Organisation Tools Chapter 2 Development of Magnetic Levitator Design E Shaped Model U Shaped Model Theory Analysis of the Process Critical Frequencies Chapter 3 Analysis of the Primary Magnetic Field Flux Lines Chapter 4 Analysis of the Secondary Force Impedance Frequency I

3 Table of Contents 4.4 Structures and shapes Chapter 5 Results and Conclusions Further Research References 61 Appendix A 64 εaxwell s Equations Eddy Current Theory Deriving the Eddy Current Equation Matrix Inductance Matrix Resistance Virtual Forces (Eddy Current) Section II Economic Study Economic Study Section III Datasheet Datasheet II

4 List of figures List of figures [21]Figure 1. Application for attractive lifting force. 10 [15]Figure 2. Linear Induction Motor 11 [20]Figure 3. Levitation techniques in train industry 12 [19]Figure 4. Current loops to calculate impedance matrix 15 [9]Figure 14. Different values of magnetic susceptibility. 25 [10] Figure 6. Magnetic field shielded. 14 [12]Figure 5. Iron-wire core and energised coil at the bottom. 17 [12]Figure 6. Transformation of axial flux motor 18 [12]Figure 7. Axial flux motor 19 [9]Figure 8. Forces produced when the disk levitate. 20 [12]Figure 9. Two concentric coils and radial laminated core. 20 [9]Figure 10.. Stability characteristics of two concentric coils model. 21 [13]Figure 11. Steps from circular plate levitator to rectangular one. 22 [12]Figure 12. E shaped model 22 Figure 13. U shaped model. 23 Figure 16. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 0 mm. E design. 30 Figure 17. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 30 mm. E design. 30 Figure 18. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 60 mm. E design. 31 III

5 List of figures Figure 19. B evaluated at 1500 Amp-turns and 50 Hz. E design. 31 Figure 20. B evaluated at 3000 Amp-turns and 50 Hz. E design. 32 Figure 21. Vector B evaluated at 2000 Amp-turns and 50 Hz. E design. 32 Figure 22. B evaluated at 1500 Amp-turns and 50 Hz. U design. 33 Figure 23. B evaluated at 3000 Amp-turns and 50 Hz. U design. 33 Figure 24. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 0 mm. E design. 34 Figure 25. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 30 mm. E design. 34 Figure 26. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 60 mm. E design. 35 Figure 27. Flux lines evaluated at 1500 Amp-turns and 50 Hz. E design. 35 Figure 28. Flux lines evaluated at 3000 Amp-turns and 50 Hz. E design. 36 Figure 29. Flux lines evaluated at 3000 Amp-turns and 50 Hz. 'U' design. 36 Figure 30. Possible equivalent circuit. 37 Figure 31. Possible equivalent circuit. 37 Figure 32. Simulation of levitation at 50 Hz, 1500 Amps-turns and 5 mm gap. E design. 38 Figure 33. Simulation of levitation at 50 Hz, 1500 Amps-turns and 5 mm gap. U design. 38 Figure 34. Force versus Gap at different values of current. U design. 39 Figure 35. Force versus Gap at different values of current. E design. 40 Figure 36. Inductance versus gap at 1500 A and 50 Hz. 'E' design. 41 Figure 36. Inductance versus gap at 1500 A and 50 Hz. 'U' design. 41 Figure 38. Resistance versus gap at 1500 A and 50 Hz. 'E' design. 42 IV

6 List of figures Figure 39. Resistance versus gap at 1500 A and 50 Hz. 'U' design. 42 Figure 40. Voltage versus Gap evaluated at different currents. 'U' design. 44 Figure 41.. Voltage versus Gap evaluated at different currents. 'E' design.44 Figure 42. Comparison of forces versus gap at different currents and frequencies. 'U' design. 46 Figure 43. Comparison of forces versus gap at different currents and frequencies. 'E' design. 46 Figure 44. Resistance versus frequency evaluated at different 5 mm gap and different currents. 'U' design. 47 Figure 45. Resistance versus frequency evaluated at different 5 mm gap and different currents. 'E' design. 47 Figure 46. Inductance versus frequency evaluated at 5 mm gap and different frequencies. 'U' design. 48 Figure 47. Inductance versus frequency evaluated at 5 mm gap and different frequencies. 'U' design. 48 Figure 48. Simulation of levitation at 5mm gap, 1500 Amp-turns and 50 Hz 50 Figure 49. Force versus separation between cores at 5 mm gap, 1500 Amps-turns and 50 Hz. 51 Figure 50. Resistance versus separation between cores at 5 mm gap, 1500 Ampsturns and 50 Hz. 51 Figure 51. Inductance versus separation between cores at 5 mm gap, 1500 Ampsturns and 50 Hz. 52 Figure 52. Lifting force versus size of the coil at 5 mm gap and different currents and 50 Hz. 53 Figure 53. Simulation of levitation at 5 mm gap, 50 Hz, 1500 Amps-turns and 34 mm size coils. 53 Figure 54. Voltage versus size of coils at different currents and 50 Hz. 54 V

7 List of figures Figure 55. Secondary shaped to achieve better lateral stabilization force. 54 Figure 56. Force versus gap evaluated at different thicknesses of secondary and 1500 Amp-turns and 50 Hz. U design. 55 Figure 57. Force versus gap evaluated at different thicknesses of secondary and 1500 Amp-turns and 50 Hz. E design. 56 Figure 58. Levitation experiment. 58 VI

8 Introduction List of Tables Table 1 Variation of Force

9 Introduction Section I REPORT 8

10 Introduction Chapter 1 INTRODUCTION Along this section, an introduction to this project is made. First of all, previous work and related technology is analysed. Then objectives and organisation is explained and finally, tools used are shown. 1.1 SIGNIFICANCE AND BACKGROUND Magnetic levitation is the physical phenomenon of levitation objects using electromagnetic forces, basically, overcoming the gravitational force with a lift force. But this lift force can be either attractive or repulsive, generating two main sorts of magnetic levitation. In addition, adding propulsion on the perpendicular axis can achieved linear movement. Historically, lift force was attempted by using permanent magnets until Samuel Earnshaw formulated his theorem in Then Maxwell applied this theorem to the electrostatic field and Braunbek developed the theory to show that this instability was due to the values of μ and being greater than their free-space values.but in 1939, Walther Meissner and Robert Ochsenfeld discovered the superdiamagnetism (The Meissner effect) which states that some conductors below their critical temperature, become superconductors, with an internal resistance of zero. Braunbeck extended it stating that a system of permanent magnets must also contain diamagnetic material or a superconductor in order to obtain stable, static magnetic levitation or suspension. 9

11 Introduction ELECTROMAGNETIC SUSPENSION So the first example of an application for an attractive lifting force, also known as electromagnetic suspension was done by Emile Bachelet who stabilized magnetic force by controlling current intensity and switching on and off power to the electromagnets at desired frequencies. He was awarded a patent in March 1912 for his levitating transmitting apparatus (patent no. 1,020,942). Later on, In 1934 Hermann Kemper applied Bachelet s concept to the large scale, creating the first design of a maglev. Then suspension with controlled dc electromagnets became feasible with the appearance of high-power electronics. Although, not as elegant as the other levitation techniques, this method is reliable and provided the first levitation system for ground transport [1]. Finally in 1979 the Transrapid electromagnetically suspended train carried passengers for a few months as a demonstration on a 908 m track in Hamburg for the first International Transportation Exhibition.[2] [21]Figure 1. Application for attractive lifting force. 10

12 Introduction ELECTRODYNAMIC SUSPENSION In repulsive levitation or also known as electrodynamic levitation, one of the first designs was done by Bedford, Peer, and Tonks in Bedford levitator consisted in an aluminium plate placed on two concentric cylindrical coils, and driven with an AC current showing 6-axis stable levitation. Then, In the 1950s, a technique was developed where small quantities of metal were levitated and melted by a magnetic field of a few tens of khz and finally Eric Laithwaite developed the first lineal induction motor, combining levitation and thrust [3]. [15]Figure 2. Linear Induction Motor Basically, there are three tested types of electrodynamic levitation [1]: Levitation with superconductors may be based on the perfect diamagnetism in the Meissner state that allows stable levitation of a superconductor in magnetic fields generated, for example, by a superconducting coil. Alternatively, the levitation may rely on the repulsion between a superconducting magnet and a conducting plate or guideway on which it moves. This repulsion originates from the eddy currents in the plate that cause both lift and drag. Interestingly, the drag (caused by the ohmic loss in the plate) decreases as the velocity of the vehicle increases. This is so because at high velocities almost all the energy in the eddy currents generated by the front end of the magnet is regained 11

13 Introduction when the rear end of the magnet passes by; at low velocities part of the wake of the eddy current has decayed by then. Levitation may use the eddy currents generated in a conducting plate by an ac coil. Such ac levitation systems may also provide the propulsion of the vehicle by a linear induction motor. Actually, there is a forth type of tested levitation, investigated in train industry, but in this case is made of unpowered loops of wire in the track and permanent magnets in the train. It is the Inductrack and it was invented by Richard F. Post in California. The permanent magnets are arranged in configurations called Halbach arrays, which is a special arrangement of permanent magnets that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side [4]. As the Halbach magnet array passes over the loops of wire, the sinusoidal variations in the field induce a voltage in the track coils. So The track thus creates its own magnetic field which lines up with and repels the permanent magnets, creating the levitation effect [5,6]. Thus the movement is necessary to achieve levitation. [20]Figure 3. Levitation techniques in train industry 12

14 Introduction So applications for magnetic levitation mainly involve situations where friction should be reduced or eliminated such as transportation (high and low speed trains), low friction bearings, wind turbines or flywheel energy storage. But there are other applications which use similar physical processes such as induction heating or eddy-current breaking even though levitation is not the ultimate aim. 13

15 Introduction 1.2 OBJECTIVES AND ORGANISATION The aim of this project is to investigate magnetic levitation, focused on electrodynamical levitation to design and build a magnetic levitation machine for demonstration purposes, capable of levitating an aluminium disc from above. Basically, a magnetic levitator machine operates like a transformer, one primary carries current which generates a magnetic field and this magnetic field induces a voltage through the secondary. In addition, two possible designs will be developed and analysed, including different parameters such as force, current, frequency or inductance. But in most practical levitation systems, equations either cannot be formulated, or they are so complicated as to make the solution impracticable. So numeric methods are used for simulations and calculations. 14

16 Introduction 1.3 TOOLS As it was said before, in most levitation systems, equations usually make the solution impracticable. So numeric methods are used for simulations and calculations, in this case the program used will be ANSYS Maxwell. In all mathematical models of the machines used in this project, the current flow is in the z direction. This assumption would be correct for an enough long magnetic levitator machine. The program is able to calculate results like force or impedance matrix. But it is necessary to explain how this is results are interpreted. An impedance matrix summarizes the relationship between AC voltages and AC currents in multiconductor systems. Given the two current loops below, the relationships between voltages and currents in each loop is as follows: [19]Figure 4. Current loops to calculate impedance matrix 15

17 Introduction Vi and Ii are phasors. Z11 = R 11 + j L 11 (self-impedance of Loop 1). Z12 = R 12 + j L 12 (mutual impedance between Loops 1 and 2). Z22 = R 22 + j L 22 (self-impedance of Loop 2). A further explanation of the procedures carried out by this program is included in Appendix 1, obtained from the contents of the program [19]. 16

18 Development of Magnetic Levitator Design. Chapter 2 DEVELOPMENT OF MAGNETIC LEVITATOR DESIGN. To explain the designs discussed in this project, firstly a simple demonstration is carried out. An iron-wire core is set up with the core vertical and an energised coil at the bottom. If the core is long enough and the coil is energised, the ring will take a position of equilibrium like shown in the next figure. [12]Figure 5. Iron-wire core and energised coil at the bottom. But it will need physical contact with the iron core to stay suspended in that position, thus removing the iron core will make unstable the system and impossible for the disc to keep levitation. However this experiment doesn t explain levitation in the complete sense, but it is in itself a good explanation of the generation of vertical forces to come over gravity. 17

19 Development of Magnetic Levitator Design. In addition, it doesn t imply a stable levitation without mechanical contact. In fact, as it was explained according to Earnshaw theorem, a magnet or charge cannot, at ordinary temperatures, stably levitate by means of any system of other permanent magnet or D.C. energised magnets. 2.1 E SHAPED MODEL But comparing levitation with how shaded-pole induction motors work, it is easy to realize that there is a link between them [11].For example, in the reluctancestart motor, the phase changes induced by a shading-ring generate force on the ring itself. In figure 8, the development from the conventional machine to the levitated ring is shown. [12]Figure 6. Transformation of axial flux motor Basically, at the first step, the system is an axial flux motor which is a conventional motor unrolled about one axis and rolled again about another. So the 18

20 Development of Magnetic Levitator Design. conducting cylinder will have induced currents generated by the flux of the 3- phase voltages coils [12]Figure 7. Axial flux motor Now, at 8(b), the airgap between the cylinder and the stator coils can be identified as the one in a conventional shaded pole machine. But in 8(c), the number of shaded-poles is reduced to the minimum. Finally, the shading ring will produce force on itself by the induced current generated by the coil. But with this configuration, it is easy to see that levitation will be unstable. Because if the disc is not exactly situated in the centre of the coil, the originated forces on one side will be bigger than the ones on the other side, so these forces will push the disc off. 19

21 Development of Magnetic Levitator Design. [9]Figure 8. Forces produced when the disk levitate. So to generate stability, it is necessary to produce inward-travelling fields to compensate the outward forces. This can be achieved placing another concentric coil and a radial laminated core as it is shown in figure 11. [12]Figure 9. Two concentric coils and radial laminated core. In this model, the current I2 carried by the inner coil generates lift force while current I1 carried by the outer coil provides stability against the lateral displacement. According to [9, 11, 12], and as it can be seen in figure 12, this model requires a bigger inner current than the outer one and also I2 lagging I1 in phase by approximately 0-45º. 20

22 Development of Magnetic Levitator Design. [9]Figure 10.. Stability characteristics of two concentric coils model. But there are practical difficulties when the configuration before indicated is set up, the assembly of radial laminations in the primary seems to be infrequent and complicated. But there is a smooth transition between cylindrical symmetry and square configuration as it is shown in the next figure. Firstly, the stator iron is divided into a number of rectangular blocks arranged as shown in figure 13a, and the behaviour of this arrangement is very little different from that of a truly cylindrical system. Then, if the number of blocks is reduced to four as in figure 13b, the primary is found to be able to levitate a plate of the appropriate size [13]. If, however, the currents in each slot were returned beneath the individual blocks, each of them would be independent from each other. Furthermore, analysing one special case where Iy1 = Iy2 = Iy3 = Iy4 = 0, and the block is elongated in the XX' direction, stability is obtained with Ix1 = Ix2 = Ix3 = Ix4, as can be realised in practice [13]. 21

23 Development of Magnetic Levitator Design. a b c d [13]Figure 11. Steps from circular plate levitator to rectangular one. In addition, if the two coils shown before are connected as in the next figure, a more flexible structure can be made which allows the two blocks to be displaced from each other by any distance. For this model, two coils with equal and opposite currents and same number of turns are enough to make the plate levitate [12]. [12]Figure 12. E shaped model 22

24 Development of Magnetic Levitator Design. 2.2 U SHAPED MODEL The other analysed model is a U shaped iron yoke which is tested in [14] and which produce a force of repulsion and lateral stabilisation between the secondary plate and the yoke. This model is also used by Eric Laithwaite in [15] as a linear induction motor separating the coils into small groups to generate a travelling field, also known as electromagnetic river, producing propulsion lift and guidance. As in the previous design, it has two primary excitation coils around its limbs and both coils carry the same single phase current and same number of turns. Figure 13. U shaped model. 23

25 Development of Magnetic Levitator Design. 2.3 THEORY According to Earnshaw s theorem; no charge can be in stable equilibrium in an electrostatic field under the influence of electric forces alone [8]. Even though this theorem is referred to electrostatic field, it was lately extended to the magnetic field too. In this case, this might be understood as magnetic stable levitation cannot be attempted but as it was said in 1.1, this levitation is not attempted with permanent magnets, its electrodynamic levitation inducing an EMF with ac current. In classical electromagnetism, magnetic field generated can be affected by the presence of different materials. In a vacuum, B and H are proportional to each other but inside a material this relation changes and is governed by the next equation: where o is the permeability of free space, and B is the internal magnetic flux density (T), produced by a magnetic field H [A/m] and the magnetisation, M [A/m]. M is a measure of the tendency of the magnetic moments resulting from the electron spin and orbital motion within a material to align and enhance, or counteract and diminish the applied field [9]. Magnetisation can be defined as: where χ is the magnetic susceptibility, so the permeability of any material related with free space is: 24

26 Development of Magnetic Levitator Design. Within ferromagnetic materials magnetic moments large numbers of magnetic moments can be aligned through the application of an applied magnetic field. This means a very strong B-field can be generated with the application of a relatively small H-field to a material with a high level of magnetisation [9]. [9]Figure 14. Different values of magnetic susceptibility. Within materials that are conductive voltages can be induced according to Faraday s δaw: Thus the free electrons in the conductors are influenced by Faraday induction Law, causing them to circulate around the applied magnetic field lines. This movement will form eddy current loops and will define EMF as the energy available from a unit charge that has travelled once around the wire loop. So these eddy currents produce an inducted magnetic field that opposes the applied field and the force generated by the interaction of these two fields is governed by the Lorentz Law. However, a magnetic field can change with the time either using ac current or employing relative motion between the generator of the magnetic field and the 25

27 Development of Magnetic Levitator Design. induced material. This last point explains the difference between inductrack and electrodynamic suspension shown in figure 3. So, assuming current flow through a conductor consists of electrons in motion, then its charge is qe and its velocity v(r) depends on the radius. Thus at a specific radius, a radial Lorentz Force is given by: ANALYSIS OF THE PROCESS As it was said in 1.1, there are two kinds of levitation magnetic force: attractive and repulsive. Attractive levitation is unstable without a feedback loop control, but nowadays either analogue or digital control techniques are available. It consists in a piece of ferromagnetic material attracted to a magnetic flux, which can be created either by permanent magnets or DC currents. However, in electrodynamic levitation, eddy currents are generated in a conducting body when the body is subjected to a time-varying magnetic flux. The interaction of the eddy currents with the magnetic flux generates forces and levitates the body. For example, in electrodynamic system maglev, the changing is produced by a superconducting magnet on the moving train. This changing magnetic flux generates circulating currents in stationary conducting loops over which the train levitates. The interaction of the induced currents with the magnetic field creates the forces [10]. 26

28 Development of Magnetic Levitator Design. So a simplified explanation of the process that the levitator carries out would be: Firstly, by Amperes s δaw, the ac current circulating through the coils generates a time-varying magnetic flux. This flux has both axial (z) and radial (r) components and part of this flux cross the conducting plate above the levitator. Then, by Faraday s δaw of induction explained before, the changing magnetic flux going through the plate induces an EMF and hence, current flow in the plate. Finally, by δorentz s δaw, the current induced in the plates is in the direction of the flux. So the motion of the electrons interacts with the radial component of the magnetic field to generate lift force CRITICAL FREQUENCIES According to [10], for an applied magnetic field tangential to the surface of a wide flat plate, the characteristic length over which the field decays in the plate is the so-called skin depth. This value of skin depth is given by: Where f is the operating frequency, is the magnetic permeability of the plate and is the electrical conductivity of the plate. The magnetic field inside the plate decays with this characteristic length. So the circulating current in the plate create a magnetic field that opposes the incident field. Hence the field is shielded from the inside of the plate. In the next figure, is shown the simulation results for flux lines and magnetic flux density 27

29 Development of Magnetic Levitator Design. from finite element analysis. In the plot the red areas are higher magnetic field, and the field intensity decreases the further away from the coil you are. [10]Figure 15a. Magnetic field shielded. [10] Figure 15b. Magnetic Field not shielded In the figure 6b, the magnetic flux lines for the case of low-frequency means that the operating frequency is sufficiently low so that the induced magnetic (due to induced currents) is small compared to the incident field. Thus, the field passes through the plate as if it weren t there at all. Since there is minimal induced current, there is minimal lift force. However, figure 6a, is high frequency excitation where the incident magnetic field is shielded from passing through the plate. This is due to large induced circulating currents in the plate. Note that the flux lines are squished beneath the coil. The resultant induced currents may be used to generate a lift force (as in Maglev) or may be used to heat the conducting plate (as in induction heating). So the simple calculation shown above, find this minimum plate thickness at a given operating frequency to reach the high frequency limit [10]. So from this point, it is understood that the frequency of the current used for the coils will affect both, lift force and temperature of the plate. 28

30 Analysis of the Primary. Chapter 3 ANALYSIS OF THE PRIMARY. Firstly, it is important to establish the size of each machine. According to experiments developed in [12],[14] and [16], the size for the iron core used in these cases is around 90 millimetres height and 100 millimetres width, so the characteristics of the iron core used are specified in the Datasheets. Coils will have 290 turns and the core used for both design will be the same but for the U design, the limb in the middle will be removed. Furthermore, as it was pointed before, there will be two important elements that will affect to the process of levitation, magnetic field (related with the Lorentz force) and flux lines (related with Eddy currents induced in the secondary). But also other factors such as frequency or characteristics of secondary will be related and analysed then. 3.1 MAGNETIC FIELD Clearly, once the E design is completely developed, first question that comes up is the distance between both iron cores. It will of course depend on the size of the secondary, but it is also necessary to see how affects to the magnitude of the magnetic field. So since this value will depend on the current through the coils, it is necessary to evaluate this distance fixing every parameter of the simulation except the distance between the cores. 29

31 Analysis of the Primary. Figure 16. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 0 mm. E design. Figure 17. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 30 mm. E design. 30

32 Analysis of the Primary. Figure 18. B evaluated at 2000 Amp-turns, 50 Hz and cores separated by 60 mm. E design. So, as it is shown in the figures above, doing a parametric setup, and according to the values of the scale, it is needless to say that as far as it is known now, the distance between the cores barely affects to the magnitude of the magnetic field. But, as it was said before, the magnetic field will be directly related to the magnitude of the current through the coils as it is shown below. Figure 19. B evaluated at 1500 Amp-turns and 50 Hz. E design. 31

33 Analysis of the Primary. Figure 20. B evaluated at 3000 Amp-turns and 50 Hz. E design. Finally, according to εaxwell s equations, magnetic field is a vector which will be related with the direction of the current and the structure of the coils and cores. So the direction of this vector is at it is shown in figure 21: Figure 21. Vector B evaluated at 2000 Amp-turns and 50 Hz. E design. However, in the case of th U design, for the time being, any parametric analysis can t be analysed, just the size of the coils but it will be studied later in presence of the secondary. But, the magnetic field of the E design seems to be stronger than the U design at both, low and high values of current. 32

34 Analysis of the Primary. Figure 22. B evaluated at 1500 Amp-turns and 50 Hz. U design. Figure 23. B evaluated at 3000 Amp-turns and 50 Hz. U design. 33

35 Analysis of the Primary. 3.2 FLUX LINES Once again, what looks like to be the most important parameter is the distance between the cores for the E design, in this case, it seems to affect more than in the case of magnetic field. When both cores are separated, flux lines in the middle tent to rise, while ones at the ends reduce its route. Figure 24. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 0 mm. E design. Figure 25. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 30 mm. E design. 34

36 Analysis of the Primary. Figure 26. Flux lines evaluated at 2000 Amp-turns, 50 Hz and cores separated by 60 mm. E design. Not surprisingly, the flux lines also are strongly affected by the value of the current. Figure 27. Flux lines evaluated at 1500 Amp-turns and 50 Hz. E design. 35

37 Analysis of the Primary. Figure 28. Flux lines evaluated at 3000 Amp-turns and 50 Hz. E design. However, the primary of the U design creates more slightly powerful flux lines than the U design using the same current, thus bigger eddy current loops will be generated through the plate but these ones will interact with a lower magnetic field. Figure 29. Flux lines evaluated at 3000 Amp-turns and 50 Hz. 'U' design. 36

38 Analysis of the Secondary Chapter 4 ANALYSIS OF THE SECONDARY So the system developed has two parts, the primary, analysed in the previous chapter, and the secondary. This secondary is what it is aimed to levitate and the interaction of these two components, similar to the process in a transformer, lead to an equivalent circuit [17]. Figure 30. Possible equivalent circuit. Where and are referred to the primary (coils) and and to the secondary (plate). It is easy to see in (b) similarities with the transformer, so in (c), secondary values are referred to the primary to get an equivalent circuit. In addition, according to [10], another simple possible electrical model for this experiment could be the next one, dividing the resistance due to levitation process: R coil L c R W Figure 31. Possible equivalent circuit. 37

39 Analysis of the Secondary Where Rcoil is the resistance of the coil in free-space due to the finite resistance of the wire, L is the inductance seen at the coil terminals and RW is the resistance due to eddy-current losses in the conducting plate. Being the plate the secondary, and the coil the primary. So knowing an equivalent circuit, different parameters are analysed through several simulations. Next figures show an example of the simulation of levitation, showing the secondary used in the simulations according to the references indicated in chapter two. Figure 32. Simulation of levitation at 50 Hz, 1500 Amps-turns and 5 mm gap. E design. Figure 33. Simulation of levitation at 50 Hz, 1500 Amps-turns and 5 mm gap. U design. 38

40 Analysis of the Secondary 4.1 FORCE Firstly, according to levitation principles, force must be the most important parameter for this experiment. So the simulation has been run with a 10 mm thickness plate and different currents. These currents are expressed in Amp-turns, depending on the value of turns of the coils. This first figure shows the behaviour of the U system under these conditions. Force_y [newton] XY Plot 1 Curve Info Force_y Current='1500A' Freq='50Hz' Xsize='44.3mm' Force_y Current='2250A' Freq='50Hz' Xsize='44.3mm' Force_y Current='3000A' Freq='50Hz' Xsize='44.3mm' Gap [mm] Figure 34. Force versus Gap at different values of current. U design. As it was expected, with constant current, the force induced in the plate decreases with the size of the gap between the primary and the secondary. Now it is shown the same situation but applied to the E design. 39

41 Analysis of the Secondary Force_y [newton] XY Plot 5 Curve Info Force_y Current='1500A' Freq='50Hz' Force_y Current='2250A' Freq='50Hz' Force_y Current='3000A' Freq='50Hz' Gap [mm] Figure 35. Force versus Gap at different values of current. E design. Comparing the last two figures, it is shown that E design is able to supply almost twice force using the same value of current-turns. Also, it can be appreciated that the relation between the force and the current applied is not completely lineal. Analysing the ratio between increments of current over force, different values are obtained: U = = % = = % E = = % = = % Table 1 Variation of Force 1 So the same increment in the current means a bigger increment of force if the simulation is run at higher values of current through the coils. 40

42 Analysis of the Secondary 4.2 IMPEDANCE When the plate is impulse upwards, RW decreases and the terminal inductance of the coil increases. This is because a bigger airgap increases the value of the inductance because the magnetic flux needs to travel through a longer lane and also this magnetic field is modified due to induced currents [10]. So if we analysed these values using keeping constant current and frequency: Induc Maxwell2DDesign1 Curve Info Total_Inductance Current='1500A' Freq='50Hz' Total_Inductance [mh] Gap [mm] Figure 36. Inductance versus gap at 1500 A and 50 Hz. 'U' design XY Plot 6 Curve Info Total_inductance Current='1500A' Freq='50Hz' Maxwell2DDesign Total_inductance [mh] Gap [mm] Figure 37. Inductance versus gap at 1500 A and 50 Hz. 'E' design. 41

43 Analysis of the Secondary Therefore, the inductance is a function of the height of the plate above the coils and also, as it was expected, this value increases with the size of the airgap. But in the case of the resistance, the result is different: Resis Curve Info Total_resistance Current='1500A' Freq='50Hz' Total_resistance [ohm] Gap [mm] Figure 38. Resistance versus gap at 1500 A and 50 Hz. 'E' design XY Plot 10 Curve Info Total_reistance Current='1500A' Freq='50Hz' Total_reistance [ohm] Gap [mm] Figure 39. Resistance versus gap at 1500 A and 50 Hz. 'U' design. 42

44 Analysis of the Secondary Apparently, the resistance decreases until an approximated value of height, but then it increases so the conclusions indicated in [10] are not completely true. This could be understood as a maximum of efficiency depending on the height due to cooper loses, being this maximum around 4 mm height. But anyway, this variation in the resistance is minimum so it won t be translate as a big improvement of the efficiency in terms of percentage. Finally, as it was explain in the first chapter, the impedance matrix generated by the program as a solution is: Where n is the number of turns. So to calculate the total voltage necessary in each case: But simulations are run with a constant current: So, as the amount of voltage dropped the wires is the total voltage necessary: 43

45 Analysis of the Secondary So applying this equation to the matrix given by the program used, we obtain the next necessary voltage for each case: mag(voltage) [A] XY Plot 2 Curve Info mag(voltage) Current='1500A' Freq='50Hz' Xsize='44.3mm' mag(voltage) Current='2250A' Freq='50Hz' Xsize='44.3mm' mag(voltage) Current='3000A' Freq='50Hz' Xsize='44.3mm' Gap [mm] Figure 40. Voltage versus Gap evaluated at different currents. 'U' design. mag(voltage) [A] XY Plot 4 Curve Info mag(voltage) Current='1500A' Freq='50Hz' mag(voltage) Current='2250A' Freq='50Hz' mag(voltage) Current='3000A' Freq='50Hz' Gap [mm] Figure 41.. Voltage versus Gap evaluated at different currents. 'E' design. 44

46 Analysis of the Secondary But according to these two last figures, although the E design supplies almost twice more force with the same current, it also needs around five times more voltage. Moreover, the evolution of the voltage according to current increments seems to be lineal, around 450 V for the E design and 90 V for the U design per 750 Amps-turns. 45

47 Analysis of the Secondary 4.3 FREQUENCY So first of all, to analyse the frequency, a study about the behaviour of the force is again carried out. In this case, a comparison between two different currents with lower and higher frequencies and a constant airgap is done. Force_y [newton] XY Plot 11 Curve Info Force_y Current='1500A' Freq='50Hz' Force_y Current='1500A' Freq='200Hz' Force_y Current='2250A' Freq='50Hz' Force_y Current='2250A' Freq='200Hz' Gap [mm] Figure 42. Comparison of forces versus gap at different currents and frequencies. 'U' design. Force_y [newton] XY Plot 6 Curve Info Force_y Current='1500A' Freq='50' Force_y Current='1500A' Freq='200' Force_y Current='2250A' Freq='50' Force_y Current='2250A' Freq='200' Gap [mm] Figure 43. Comparison of forces versus gap at different currents and frequencies. 'E' design. 46

48 Analysis of the Secondary In both models, four times more frequency means an improvement in the force between 20% and 25%. So, apparently, this seems to be a good way to obtain better lift force. However, a deeper analysis, including values of resistance and inductance shows the next results XY Plot 8 Curve Info Total_reistance Current='1500A' Gap='5mm' Total_reistance Current='2250A' Gap='5mm' Maxwell2DDesign1 Total_reistance [ohm] Freq [Hz] Figure 44. Resistance versus frequency evaluated at different 5 mm gap and different currents. 'U' design. Total_resistance [ohm] Name X Y m m XY Plot 6 Curve Info Total_resistance Current='1500A' Gap='5mm' Total_resistance Current='2250A' Gap='5mm' m1 m Freq [Hz] Figure 45. Resistance versus frequency evaluated at different 5 mm gap and different currents. 'E' design. 47

49 Analysis of the Secondary On one hand, it is shown that the total resistance increases almost three times for the U design and four times for the E design. So, despite of obvious consequences in necessary voltage and input power, it can be understood that as a result of this increment of frequency, much more power is dedicated to heat up the plate. Total_inductance [mh] XY Plot 9 Curve Info Total_inductance Current='1500A' Gap='5mm' Total_inductance Current='2250A' Gap='5mm' Freq [Hz] Figure 46. Inductance versus frequency evaluated at 5 mm gap and different frequencies. 'U' design. Total_Inductance [mh] Name X Y m m XY Plot 6 Curve Info Total_Inductance Current='1500A' Gap='5mm' Total_Inductance Current='2250A' Gap='5mm' Maxwell2DDesign m1 m Freq [Hz] Figure 47. Inductance versus frequency evaluated at 5 mm gap and different frequencies. 'U' design. 48

50 Analysis of the Secondary On the other hand, even though the inductance is reduced 10% and 25% respectively in U and E designs, the resultant impedance will be multiplied by four times more frequency. So again in this case, as a result of this increment of frequency, it is necessary an excessive improvement of voltage. Moreover, although only two currents are analysed, it can be seen in figures 42 and 44 that at using higher currents, the resistance is slightly reduced at high frequencies. But in the case of the inductance, it is not affected by current values in the U design, however for the E structure, there is a small change in this value. 49

51 Analysis of the Secondary 4.4 STRUCTURES AND SHAPES During the last parameter analysis of the E design, in all cases it was considered that both cores were together. So now, force is tested depending on the distance between the cores which will affect the size of the plate. This is because to achieve stable levitation, according to [12] and [13], the edges of the plate must been approximately in the centre of the coils. So first, a case with both cores separated is shown. Figure 48. Simulation of levitation at 5mm gap, 1500 Amp-turns and 50 Hz Then, running a simulation comparing the force versus this distance between cores, results are as shown. 50

52 Analysis of the Secondary Point1 Curve Info mag(force_y) Current='1500A' Freq='50' Gap='5mm' mag(force_y) [newton] Name X Y Point Point Point Lateral_Displacement [mm] Figure 49. Force versus separation between cores at 5 mm gap, 1500 Amps-turns and 50 Hz. It seems that this lateral separation between the cores won t strongly affect to the lifting force, in fact, resistance and inductance are hardly affected either XY Plot 3 Curve Info Total_resistance Current='1500A' Freq='50' Gap='5mm' Total_resistance [ohm] Lateral_Displacement [mm] Figure 50. Resistance versus separation between cores at 5 mm gap, 1500 Amps-turns and 50 Hz. 51

53 Analysis of the Secondary XY Plot 4 Curve Info Total_Inductance Current='1500A' Freq='50' Gap='5mm' Total_Inductance [mh] Lateral_Displacement [mm] Figure 51. Inductance versus separation between cores at 5 mm gap, 1500 Amps-turns and 50 Hz. However, according to the explanations expressed in the beginning of this chapter, related with the relation between inductance and airgap, this fact could not have any sense at all. But, actually, even though the separation between cores is much bigger, the flux lines generated by the inner part of the coils, which induce eddy currents in the plate, travel through the same size of airgap. While the flux generated by the outer coils is used to control the lateral stabilization completes the same path. Finally, the interaction between the magnetic field and eddy currents is almost the same, because the magnetic field decreases very fast with the distance as it is shown in previous simulations. So despite of the fact that lifting force is not almost affected, higher values are obtained when both cores are together. Now looking at the U design, there is only one modification possible, varying the size of the inner coils apparently should affect to the flux lines. So it will mean a different flux through the plate, changing also eddy currents. Next simulation shows these results. 52

54 Analysis of the Secondary m m m Force_y [newton] Name X Y Curve Info XY Plot 14 Force_y Current='1500A' Freq='50Hz' Gap='5mm' Force_y Current='2250A' Freq='50Hz' Gap='5mm' Force_y Current='3000A' Freq='50Hz' Gap='5mm' m1 m2 Maxwell2DDesign m Xsize [mm] Figure 52. Lifting force versus size of the coil at 5 mm gap and different currents and 50 Hz. Figure 53. Simulation of levitation at 5 mm gap, 50 Hz, 1500 Amps-turns and 34 mm size coils. As shown in figure 49, there is maximum lifting force depending on the size of the coils. This difference is bigger at higher currents but now is necessary to know how this new structure affects to other parameters such as total impedance, affecting directly to the voltage. 53

55 Analysis of the Secondary Name X Y m m m mag(voltage) [A] XY Plot 14 Curve Info mag(voltage) Current='1500A' Freq='50Hz' Gap='5mm' mag(voltage) Current='2250A' Freq='50Hz' Gap='5mm' mag(voltage) Current='3000A' Freq='50Hz' Gap='5mm' m1 m2 m Xsize [mm] Figure 54. Voltage versus size of coils at different currents and 50 Hz. As it can be seen in figure 51, this change in the structure also affects to the voltage. It increases the necessary voltage at the maximum registered force as it is indicated. But although during these analysis lateral stabilization forces haven t been tested, lateral stabilization is assumed as it is explained in different references used. In addition, according to [12] and [13] in the case of the U design and [14] for the E design, the shape of the plate levitated in the simulations is enough to generate lateral stability forces (smaller than lift force). But as it is explained in [16], it is possible to increase these lateral stability forces modifying the shape of the plate with sides inclined 65, as it is shown in the next figure. Figure 55. Secondary shaped to achieve better lateral stabilization force. 54

56 Analysis of the Secondary The interaction of the magnetic field and the eddy currents generated in the plate will increase both, lateral stabilisation and lift force but also, the weight of the secondary is higher due to a bigger plate. As it was explained in first chapter, the critical frequency establishes a relation between the frequency used and the characteristic length over which the field decays in the plate, so it is expected this thickness of the plate will affect to levitation process. Force_y [newton] XY Plot 12 Curve Info Force_y Current='1500A' Freq='50Hz' Thickness='5mm' Force_y Current='1500A' Freq='50Hz' Thickness='7.5mm' Force_y Current='1500A' Freq='50Hz' Thickness='10mm' Force_y Current='1500A' Freq='50Hz' Thickness='12.5mm' Force_y Current='1500A' Freq='50Hz' Thickness='15mm' Gap [mm] Figure 56. Force versus gap evaluated at different thicknesses of secondary and 1500 Amp-turns and 50 Hz. U design. 55

57 Analysis of the Secondary Force_y [newton] XY Plot 14 Curve Info Force_y Current='1500A' Freq='50Hz' Thickness='5mm' Force_y Current='1500A' Freq='50Hz' Thickness='7.5mm' Force_y Current='1500A' Freq='50Hz' Thickness='10mm' Force_y Current='1500A' Freq='50Hz' Thickness='12.5mm' Force_y Current='1500A' Freq='50Hz' Thickness='15mm' Gap [mm] Figure 57. Force versus gap evaluated at different thicknesses of secondary and 1500 Amp-turns and 50 Hz. E design. According to these results, a thicker plate means a bigger lifting force but this increment is not lineal, this improvement of the force is also reduced with thicker plates. In the case of aluminum, at 50 Hz, this skin depth is approximately 12 mm. So as it can be seen in the analysis, after 12.5 mm thickness, lifting force is hardly affected by this parameter. 56

58 Results and Conclusions Chapter 5 RESULTS AND CONCLUSIONS To sum up, it seems that the E design is more powerful, obtaining stronger lifting force using same current than the U design. However, it uses more material (another iron core) and also needs more voltage to achieve same current. This extra necessary voltage is due to the structure of the system, it affects to the values of resistance and inductance. Inductance is increased almost five times in the E design, because the flux lines generated by outer parts of the coils are used to generate lateral stability instead of using all the flux to induce eddy current loops in the plate. So depending on the application, each model is more suitable, achieving more lateral stabilization or levitation using less power. However, frequency variation doesn t seem to be a useful method to improve levitation. It has been proven that levitation force is increased, but resulting in much higher values of impedance of the system. So this improvement of the lift force is better achieved increasing the current. There is another factor, which is not studied in this project, but thermal effects strongly affect to electrical properties of materials. In this case, for the aluminum, the resistance increases approximately 0.4% per degree Celsius. So the induced current will be smaller as the plate is heat up. 57

59 Results and Conclusions Finally, in the experiment, the E design was chosen to be test and some characteristics were observed. However this experiment couldn t be completed in the university facilities due to limited equipment, because of high values of current and voltage. So these results are limited: It was verified that different plate thicknesses affect to the lifting force as in the simulations. But also has an inverse relationship with the lateral stability. The plate reaches an equilibrium height from the beginning; my theory is that this system is unstable the first millimeters until it reaches this equilibrium point. Due to a short longitudinal size of the levitator, equilibrium wasn t achieved in the longitudinal axis. Levitation was achieved with approximately 400 Volts. Figure 58. Levitation experiment. 58

60 Results and Conclusions In addition, magnetic scaling laws show that large magnetic, elements are more efficient in energy conversion than smaller ones. Conversely, small-scale levitation experiments are likely to be very power hungry (or unable to levitate at all before they burn up) [10]. Thus with materials nowadays available, it is necessary to use cooling systems to improve the efficiency. So there mainly two uses developed nowadays using this kind of levitation, trains, as the linear induction motor developed by Eric Laithwaite or the Inductrack and magnetic bearings. This last one, doesn t require any control electronics to operate and it works by the electrical currents generated by motion causing a restoring force. But there other new possible feature applications for this levitation such as magnetic wind turbines which are able to increase generation capacity by 20% over conventional wind turbines and decrease operational costs by 50% [22] and flywheel energy storage. 59

61 Results and Conclusions 5.1 FURTHER RESEARCH According to this experiment, further work might include test the actual behaviour of the U design which doesn t need so high voltage as it was shown during simulations. Also, a deeper analysis of the actual behaviour of the design used in the experiment may be carried out. On one hand, it might be also possible to check how actually frequency changes affect the levitation during the experiment. But on the other hand, to improve levitation, it also possible to develop a magnetic river, creating a linear induction motor. The lifting force will be greater under action of the travelling magnetic field, due to induced eddy currents in the conductive plate by both transformer action and linear motion effects. But there is one last possible modification; new feature materials such as graphene could develop much efficient systems. Graphene is characterized by being a greater conductor of electricity and heat (much more efficient than aluminium), similar to a superconductor. In addition, besides this obvious characteristic, its weight is also almost negligible. 60

62 References REFERENCES [1] E H Brandt δevitation in Physics January [2] [3] [4] K. Halbach (1980). "Design of permanent multipole magnets with oriented rare earth cobalt material". Nuclear Instruments and Methods 169 (1): [5] Laithwaite, Eric R. (February 1975). "Linear electric machines A personal view". Proceedings of the IEEE 63 (2): [6] "A New Approach for Magnetically Levitating Trains and Rockets". llnl.gov. Retrieved by Arnie Heller. [7] R. F. Post, ìthe Inductrack: a simpler approach to magnetic levitation, Proceedings of the Sixteenth International Conference on Magnet Technology, IEEE Transactions on Applied Superconductivity, Vol. 10, ,2000. [8] Introduction to Electromagnetic Theory: A Modern Perspective [9] Sangster, Alan J., 2012 Jan Fundamentals of Electromagnetic Levitation: Engineering Sustainability Through Efficiency. [10] Eddy current magnetic levitation. Models and experiments Marc T. Thompson

63 References [11] Cashmore, D.H. Electromagnetic Levitation. M.Sc. Thesis, Manchester University (1963). [12] Eric δaithwaite Induction machine for special purposes. [13] Prof. Eric δaithwaite Electromagnetic Levitation Vol. 112, No. 12, DECEMBER [14] The performance of induction levitators by J F Eastham and D Roger. September [15] Eastham, J F and δaithwaite: δinear induction motors as electromagnetic rivers Oct [16] Jacek F. Gieras, Fellow, IEEE, Jacek εews, and Pawel Splawski. Analytical Calculation of electrodynamic Levitation Forces in a Special-Purpose Linear Induction εotor January/February [17] Barry, N; Hudgins, J. δevitation of an aluminium disc in a magnetic flux well Jul 2007 [18] J.F. Gieras, Influence of structure and material of secondary suspended electrodynamically on steady performance characteristics of LIM with transverse flux. [19] Contents of ANSYS Maxwell. [20] The Monorail Society Website Technical Pages 62

64 References [21] U.S. Departament of Transportation Final Report on the National εaglev Initiative. [22] Wind Power Plant Using Magnetic Levitation Wind Turbine Dinesh N Nagarkar, Dr. Z. J. Khan July

65 Appendix A APPENDIX A Maxwell s Equations The eddy current field simulator solves for time harmonic electromagnetic fields governed by Maxwell s equations: E is the electric field. B is the magnetic flux density. D is the electric displacement, E. H is the magnetic field intensity. J is the conduction current density, E. is the charge density. The eddy current solver assumes that all time-varying electromagnetic quantities in the problem have the form: Using Euler s formula: If = t+, F(t) equals the real portion of e j( t+ ) : 64

66 Appendix A Now, because each time-varying quantity has the form and are equal to j D and j B. Therefore, with this simplification and the relations H= B, D= E, and J= E, εaxwell s equations reduce to: Eddy Current Theory Time-varying currents flowing in a conductor produce a time-varying magnetic field in planes perpendicular to the conductor. In turn, this magnetic field induces eddy currents in the source conductor and in any other conductor parallel to it. The eddy current field solver calculates the eddy currents by solving for A and in the field equation: Where: A is the magnetic vector potential. is the absolute magnetic permeability. is the conductivity. is the electric scalar potential. is the angular frequency at which all quantities are oscillating. is the absolute permittivity. 65

67 Appendix A Deriving the Eddy Current Equation The eddy current field solver uses the finite element method to compute A and using these two relationships: Where: A is the magnetic vector potential. is the electric scalar potential. is the magnetic permeability. is the conductivity. is the angular frequency at which all quantities are oscillating. is the permittivity. I T is the total current flowing in conductors. Matrix Inductance To compute the inductance of the current loop, the simulator calculates the average energy, WAV, of the system after a field solution is computed: Since the instantaneous energy of the system is equal to: 66

68 Appendix A Where the instantaneous value of the current is related to the peak value of the current by i = I Peak* cos( t+ ). The average value for the energy can then be found by integrating the instantaneous energy: From this, the average energy of the system is equal to: The inductance, therefore is: The software assumes that the object for which impedance is being computed has a peak current of one ampere per coil turn flowing through it. Thus, the inductance is simply 4W AV. Matrix Resistance To compute the resistance, the simulator calculates the ohmic loss, P, after a field solution has been computed: The ohmic loss is related to the resistance by: The resistance is therefore: 67

69 Appendix A The system assumes that the object for which impedance is being computed has a peak current of one ampere per coil turn flowing through it. Therefore, the resistance is simply 2P. Note that the resistance for an eddy current problem will be higher than the equivalent DC resistance, due to the skin concentration of currents. Virtual Forces (Eddy Current) Virtual force in an eddy current problem is computed pretty similar to this way: To compute the force on an object, the system uses the principle of virtual work. In the structure shown below, the force on the plate in the direction of the displacement, s, is given by the following relationship: Where W(s,i) is the magnetic coenergy of the system. The current, i, is held constant. Unlike the classical virtual work method, the plate is not actually moved during the force computation. Instead, only the triangles that lie along the outside surface of the object are virtually distorted. Thus, the force computation only requires one field solution. But in the case of Eddy Current, the difference is that the value computed is the average value of force over time, not the instantaneous force at a given time. 68

70 Appendix A The difference between the time-averaged (or DC) force, AC force, and instantaneous force is shown below: Force oscillates at twice the frequency of the source current and magnetic field: Where: f F is the frequency of the force. f S is the frequency of the source current and magnetic field. T F is the period of the force. 69

71 Appendix A The time-averaged (or DC) force, AC force, and instantaneous force can be determined by: The AC force, FAC must be evaluated at a particular phase (= t) in order to determine its magnitude at an instant in time. However, the peak value of the AC force is reported as the AC εagnitude in the force and torque solution panel for Maxwell 2D. 70

72 Appendix A Section II ECONOMIC STUDY 71

73 Economic Study ECONOMIC STUDY As this Project is for demonstration purposes, it is complicated to establish if the levitation machine is profitable or not. There isn t any return rate, but since this technology came out, it has been quite complicated to develop this technology as fast as others due to its high costs. Just with special materials, such as superconductors is possible to improve the efficiency of this technology. 72

74 Economic Study Section III DATASHEET 73

75 Datasheet 1 DATASHEET 1 74