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1 ELECTROMAGNETIC LEVITATION OF A DISC Rodrigo Valle, Fábio Neves, Rubens de Andrade Jr., Richard M. Stephan LASUP/UFRJ Abstract: This paper presents a teaching experiment that explores the levitation of a disc of ferromagnetic material in the presence of the magnetic field produced by a single electromagnet. In comparison with the classical experiment of the levitation of a sphere, the main advantage of the proposed laboratory bench is due to the uniform magnetic field distribution in the air gap that allows analytical calculations. The work illustrates the important connection between theory, mathematical modeling, design, simulation and experimental verification, emphasizing the opportunities that this study can bring to education in subjects like control, magnetic circuits, power electronics and electro mechanic energy conversion. The proposal can be seen as an introduction of the main issues of mechatronics and is being used as example to raise the interest of undergraduate electrical engineering students. Keywords: magnetic levitation, electromagnetic forces, stability, mechatronics. 1 - Introduction Laboratory experiments with electromagnetic levitation of spheres have been the subject of a series of technical papers and studies [1-23]. The excellent visual effect provided by this assembly makes levitating spheres to be found in several museums. They can also be purchased for decorative purposes as a quick internet search shows [24-27]. Also, learning kits of sphere levitation systems for hobbyists and undergraduate students can be found at [28-31]. However, analytical calculations for this experiment

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ELECTROMAGNETIC LEVITATION OF A DISC

Rodrigo Valle, Fábio Neves, Rubens de Andrade Jr., Richard M. Stephan

LASUP/UFRJ

Abstract: This paper presents a teaching experiment that explores the levitation of a

disc of ferromagnetic material in the presence of the magnetic field produced by a single

electromagnet. In comparison with the classical experiment of the levitation of a sphere,

the main advantage of the proposed laboratory bench is due to the uniform magnetic

field distribution in the air gap that allows analytical calculations. The work illustrates the

important connection between theory, mathematical modeling, design, simulation and

experimental verification, emphasizing the opportunities that this study can bring to

education in subjects like control, magnetic circuits, power electronics and electro

mechanic energy conversion. The proposal can be seen as an introduction of the main

issues of mechatronics and is being used as example to raise the interest of

undergraduate electrical engineering students.

Keywords: magnetic levitation, electromagnetic forces, stability, mechatronics. 1 - Introduction

Laboratory experiments with electromagnetic levitation of spheres have been the

subject of a series of technical papers and studies [1-23]. The excellent visual effect

provided by this assembly makes levitating spheres to be found in several museums.

They can also be purchased for decorative purposes as a quick internet search shows

[24-27]. Also, learning kits of sphere levitation systems for hobbyists and undergraduate

students can be found at [28-31]. However, analytical calculations for this experiment

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are practically impossible due to the non-linear distribution of the magnetic field, which

leads to solutions supported on finite elements simulations [12]-[13] or purely empirical

ones.

This paper shows that the levitation of a disc of ferromagnetic material is better

suited for educational purposes since the verification of theoretical knowledge is

straightforward. Furthermore, the experiment considers not only forces, as in the

classical case of spheres, but also mechanical and electromagnetic moments.

Nevertheless, the construction is simple enough to be reproduced and executed.

This paper is organized in the following way. Section 2 describes the experiment

and the analytical calculations are presented in Section 3. The design and

implementation of the control system and power electronic circuits are part of Section 4.

Experimental results are presented in Section 5. Section 6 shows the pedagogical

importance of this work and presents suggestions for future work. Finally, conclusions

are drawn in Section 7.

2 – Experimental bench

The basic idea of this assembly is to establish a simple system that allows the

experimental verification of analytical equations of magnetic [32] and mechanical forces.

The determination of magnetic forces can be obtained easily in the case of

constant magnetic fields. A simple way to establish a magnetic field under these

conditions is illustrated in figure 1, which shows the cross section of a modified version

of the disc near a cylindrical electromagnet. This modification introduces cut outs in the

disc in order to direct the path of magnetic flux increasing the uniformity of the magnetic

field in the air gap region and standardizing the distribution of electromagnetic force.

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This change allows the analytic calculation that will be presented in section 3.1. Such

solution was not possible in the case of the sphere because of the non-linear distribution

of the magnetic field in the air gap. The dimensions of the disc and the electromagnet

can be seen in figure 1. The electromagnet was fabricated with N=3126 turns of copper

wire, with resistance equal to 5.2 'ohm.

The magnetic field in the region of air between the faces of the disc and the

electromagnet, for small gaps (x<<R in figure 1), can be considered having constant

magnitude. Assuming the system in a vertical position with the electromagnet fixed and

the disc free in space, the maintenance of a constant air gap requires the presence of a

control system that measures this distance and properly imposes the current needed to

support the weight of the disc. To maintain the face of the disc parallel to the face of the

electromagnet, the gravity center should be lowered as illustrated in figure 2. In other

words, the angle θ between the face of the disc and the horizontal line, measured from

the center of the disc, must be equal to zero.

R = 3.50 10-2m

R1 = 0.02 m

R2 = 0.03 m

Cutout = 0.01 m

H= 0.11 m

Hd = 1.50 10-2m

(a) (b) Figure 1 - Cross section of the levitating disc experiment of the (a) front view and (b) top

view.

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L = 1.25 10-1m

R`= 1.60 10-1m

Figure 2 - Displacement of the Gravity Center (GC) of the experiment.

3 - Analytical Calculations

3.1 - Magnetic Circuit

Calling “A” the active area of the electromagnet, R1 was chosen so that:

(1)

For x << R, the air reluctance is composed of two equal parts given by:

(2)

where "x" represents the length of the gap. Considering the magnetic permeability of

iron much greater than that of air, the flux φ established in the air gap is determined by

the magnetic circuit of figure 3, where μ0 is 4π10-7 NA−2.

.2

2

2

2

1

ARRR

,21

A

x

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Figure 3 - Magnetic circuit.

(3)

The magnetic flux density B is given by:

(4)

3.2 - Electromagnetic Force

The electromagnetic force f can be determined by the derivative of stored

energy aE in relation to displacement. The stored energy is:

(5)

then:

(6)

The negative sign that appears in the equation above indicates an attraction

force. To facilitate algebraic manipulations, the term was replaced by a constant

x

iNANi

42

.22/ x

iN

AB

,222

1

2

122

x

ANiAx

BBHVEa

.8

22

x

iAN

dx

dEf a

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K. Therefore, the expression of electromagnetic force, considering the coordinate axis of

figure 2, is given by:

(7)

The intensity of force per unit area is:

(8)

At this point, it is interesting to note that for a density of magnetic field of 1T in the

air gap, equation (8) leads to a value of 4.00 105 Nm-2, which is quite significant.

Calling the weight of the disc P, the current io necessary to support it in

equilibrium at a distance xo of the electromagnet, according to equation (7), is given by:

(9)

Linearizing the force around the equilibrium position given by (xo, io), results in:

(10)

According to the references adopted in figure 2, the rate of variation of force with

positive displacements must be positive, which is confirmed when it is realized that x0 is

negative. Calling:

(11)

and

(12)

the electromagnetic force can be rewritten as:

(13)

.2

1

22

22

B

x

Ni

A

f

.

2

x

iKf

.),(

2

0

x

iKixfP o

.),(),(),( 000000 iixi

fxix

x

fixff

,2),(2

0

0

00x

iKKix

i

fi

.iKxKPf ix

3

0

2

0

3

0

2

000

||22),(

x

iK

x

iKKix

x

fx

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3.3 - Dynamics of the Vertical Displacement

The dynamic equation governing the vertical displacement of the disc,

represented in figure 4, is given by:

Figure 4 - Free Body Diagram.

(14)

where "m" is the mass of the disc. Substituting (13) in equation (14), it follows:

(15)

After simple algebraic manipulations on this equation and applying Laplace, the

transfer function of the system is obtained:

(16)

(17)

Considering the current variation in the electromagnet as the input variable, the

system has two real poles positioned atm

K x .

.2

2

iKxKdt

xdm ix

),()(2 sm

Ks

m

Ks ix

.)(

)(

2

m

Ks

m

K

s

s

x

i

f

PMomentum

Mechanical

Momentum

neticElectromag

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3.4 - Mechanical Momentum

Any small deviation angle θ from the center of the disc, as suggested in figure 2,

will destroy the equilibrium point. This condition will be compensated by a mechanical

momentum given by:

Mechanical Momentum = (18)

For small θ:

Mechanical Momentum = (19)

3.5 - Electromagnetic Momentum

This calculation will be simplified neglecting the region of the cutouts. The radius

of the dome added at the bottom of the disc was designed in such a way that angular

displacements around the equilibrium point (x0=-0.01m) would not affect the position

measurement given by an ultrasonic sensor located below the dome, and consequently

the circulating current in the electromagnet.

The material of the dome and the rod do not play an important hole from the

electromagnetic point of view since the flux lines will go preferably through the

ferromagnetic disc. In the present experiment, the dome is made of lead and the rod of

stainless steel.

Based on equations (9) and (11), the force variation per area (pressure) is given

by:

(20)

The electromagnetic momentum, for a small angular displacement θ, results from

the surface integral of the electromagnetic moment element (dp) seen in equation (21).

Figure 5 shows the area element used in the integration.

.

||

22

xkxRx

P

A

f

.sinPL

.PL

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(21)

Figure 5 - Determination of the momentum produced by electromagnetic forces.

Where:

(22)

Integrating 1:

(23)

3.6 - Dynamics of the Angular Displacement

To maintain the disc surface parallel to the electromagnet surface, the

mechanical momentum must be higher than the momentum produced by the

electromagnetic forces, according to figure 4. Therefore, using equations (19) and (23):

(24)

(25)

1

R

rsen

RrRRr

rdrrRr 1

42222222

82

8

.40

4

R

r

RkdpMomentumeticEletromagn

,

4||

2 4

2

R

Rx

PPL

.||2

2

x

RL

r

R

dr

2 (R2-r2)1/2

.rx

Area Equation (20) Lever

arm

.22Momentum ofElement 22 rrRdrxkdp

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In the proposed laboratory experiment [33], R = 3.50 10-2 m and xo = -0.01 m,

implying L > 6.10 10-2 m.

4 - Control system design and implementation

4.1 - Control System

The control system was implemented in real time Simulink toolbox, a tool for real-

time simulations of Matlab. The sensor signals are transmitted to the computer and

processed, generating the command signal to the power system. The sampling time of

the ultrasonic sensor signal (U-Gage S18UUAR from Banner) is 1.00 10-3 s, while the

sampling time of the current sensor (LA 25-NP from LEM) and the command signal are

66.67 10-6 s.

Figure 6 shows the experiment under study, where xo is the reference position

relative to the ultrasonic position sensor.

Figure 6 - Representation of the experiment.

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For this experimental bench, the gain of the ultrasonic position sensor is

Kpos=248.36Vm-1 and the gain of the current sensor (Hall effect sensor) Kc=0.51VA-1.

The calibration curve and linearization around region of interest are given in [33].

4.2 - Controller design

Based on equation (17), the control system is represented in figure 7.

Figure 7 - Control System.

The gain of the plant, as well as the position of its poles, depends on the

equilibrium position (xo) as equations (11), (12) and (17) demonstrate. Therefore, the

correct approach to control this system is to design an adaptative controller. However, in

any case, a PD controller can be robust enough to stabilize the system as shown in

figure 8. For a matter of simplicity, the controller was synthesized [33] considering the

plant parameters Kx and Ki constant with the values for x0 at - 0.01 m. Based on

equation (9), for m= 1.38 kg and g= 9.81m/s2, the current at the equilibrium point is

given by equation (26).

(26)

Therefore, according to equations (11) and (12), Kx = 2.70 103 Nm-1 and

Ki = 4.45 101 NA-1. In this work, a lead compensator, that stabilizes the system for small

C(s) Δx Δi +

-

m

Ks

m

K

x

i

2

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variations (Δx<<R) around this equilibrium position, was designed following classical

approaches [34].

Figure 8 – (a) Root Locus with Lead compensator and (b) the unit step response of

1.00 10-3 m of the closed loop system with a Lead compensator.

The parameters of the lead compensator were chosen with the help of the Root

Locus, outlined in figure 8a, just to stabilize the system at x0. The command in Matlab to

use this tool is rltool [35].

In order to eliminate the steady state error, which is about 50% in figure 8b, an

integral term, with gain KI, was adjusted experimentally. Thus:

(27)

The Root Locus of the complete system is illustrated in figure 9a, where the

natural resonance frequency is 53 rad s-1 (obtained through the rltool), meaning that

sinusoidal inputs with this frequency will have the largest amplitude in the output. The

step response of this system is in figure 9b, where the zero steady-state error occurs

because of the integral term:

.1

2.0280

5694.6

1)(

ss

s

sK

Ps

ZsKsC Ip

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Figure 9 – (a) Root Locus with complete C(s) and (b) the unit step response of

1.00 10-3 m on the closed loop system with the same C(s).

The simulation block diagram of the system is illustrated in figure 10 and the step

response is equal to figure 9b obtained with the rltool.

Figure 10 – Simulink block diagram.

4.3 - Current Regulated Circuit

The power circuit consists of two mosfets (IRF640N), two diodes (BYT79) and a

source of 30V DC forming a bridge showed in figure 11.

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Figure 11 – Power electronics circuit.

When the current reference (the output of the position controller in figure 6) signal

is positive, the two mosfets are driven and a positive voltage is applied to the

electromagnet (+ VCC), i.e. the current through the electromagnet increases. When the

current reference signal is negative, the mosfets are blocked and a negative voltage is

applied to the electromagnet (-VCC), i.e. the current through it decreases.

Therefore, the mosfets will switch on and off in such a way that the average

current will be equal to the reference current io required for the equilibrium position xo.

5 – Experimental Results

5.1 – Closed Loop Step Response

Starting from the equilibrium point, two step variations in the reference position

were imposed. Figure 12 shows the disc levitating at the reference position and

figure 13 shows the experimental response of the system to positive and negative steps.

VCC Electromagnet

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Figure 12 - Disc levitating.

Figure 13 – Position step response.

Considering the measurement noise, these results are already expected by the

simulation presented in figure 9.

The results showed also that the controller managed to stabilize the system with

approximately zero steady-state error, disregarding the noise of sensor. However, there

is a small asymmetry in relation to positive and negative steps, indicating the

nonlinearity of the system and emphasizing the limits of the linearization carried out in

equations (10), (11) and (12).

5.2 – Closed Loop Frequency Response

To obtain the frequency response of the system, sinusoidal variations in the

reference were imposed. The responses are shown in figure 14:

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(a) (b)

(c) (d)

(e) (f)

Figure 14 – Output measured position and reference sinusoidal signals for: a) 0,1Hz; b) 0,5Hz; c) 1,0Hz; d) 10,0Hz; e) 15,0Hz; f) 20,0Hz.

For frequencies of 0.1 Hz, 0.5 Hz and 1 Hz the amplitude of the output increases

with the frequency and the phase shift is close to zero, indicating that these frequencies

are below the natural resonance frequency of the system, which is 53 rads-1 (8.43 Hz)

according to Section 4.2. For frequencies of 10 Hz, 15 Hz and 20 Hz the amplitude of

0 5 10 15 20 25 30 3557.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

0 5 10 1557.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

0 1 2 3 4 5 6 7 8 9 1057.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 157.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.557.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.557.5

58

58.5

59

59.5

60

60.5

61

Time (s)

Po

siti

on

(m

m)

Measure

Reference

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the output decreases with -180° phase shift, indicating that these frequencies are

greater than the natural resonance frequency of the system.

6 – Pedagogical Value

This work aimed initially to challenge two undergraduate electrical engineering

students from different training emphases, who have to develop a project as part of the

requirements to graduate in engineering. One of them came from automation and

control, the other from power electronics. A healthy and creative interdisciplinary living,

that also involved the faculty advisers, resulted.

The importance of mathematics, electromagnetic theory [32], mechanics, power

electronics and control [34], with an integrated and holistic vision, was emphasized

during the course of this project. The value of this experiment as a laboratory

demonstration for first semester electrical engineering students was then recognized

and it is now part of the course EEE200 Introduction to Electrical Engineering [36]. A

survey carried out at the end of each semester with the students of EEE200 shows circa

90% approval of this demonstration. The students say that the demonstration makes

clear the importance to study mathematics and physics, which is usually not very

attractive.

The work bench is also being used for new projects involving graduates in

electrical engineering. Presently, there is a student developing an adaptive controller to

compensate for the variation in parameters Kx and Ki. Another student is in charge of the

development of a position observer to substitute the expensive ultrasonic position

sensor. As taught by equation (4), the position measurement may be replaced by

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measuring the flux density in the air gap (B) and the current in the electromagnet (i) with

two low cost Hall-effect sensors. Moreover, some of these students are now pursuing a

Master of Science (M.Sc.) degree on the research area of Magnetic Bearings, which is

better understood after acquiring knowledge in electromagnetic levitation systems such

as the one in this manuscript.

Controllers to reject sinusoidal perturbations or to achieve predetermined

performance indexes by application of classical control theory as in [2, 4-8, 11, 18, 20,

22, 23] and of modern and advanced control theory as in [10, 14, 19, 21] can also be

tested with the proposed hardware. Issues of noise reduction, electromagnetic

interference (EMI) and magnetic field could also be subjects of study.

Interested readers in reproducing the assembly of this work can find a list of

materials, hardware information, components data-sheets, detailed drawings, Matlab

codes, Simulink diagrams and students improvements to this work in [36]. The operation

of this experiment can be seen in Youtube:

http://www.youtube.com/watch?v=BWYCe1PBoW8&feature=player_embedded.

7 - Conclusion

This paper has described a simple laboratory experiment combining the teaching

of control, electromagnetism, mechanics, power electronics, instrumentation and signal

processing. The students can see the importance of mathematics, electromagnetic

theory, mechanics and control with an integrated and holistic vision. Analytical

expressions could be used to establish a mathematical model of the system and

experimental results confirmed the theoretical approach. The experiment motivates first

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semester students and serves also as research subject for graduating and M.Sc.

students.

8 – Acknowledgement

The authors would like to thank FAPERJ and CNPq for the financial support and

to Mr. G.F. Santana and Mr. O.J. Machado for the mechanical assembly.

9 – References

1. Jayawant, B.V.; Sinha, P.K.; Wheeler, A.R.; Whorlow, R.J.; Willsher, J.; "Development of 1-ton magnetically suspended vehicle using controlled d.c. electromagnets", Proceedings IEEE , vol.123, pp. 941 - 948, 1976. 2. Wong, T. H.; "Design of a Magnetic Levitation Control System - An Undergraduate Project", IEEE Transactions on Education, vol.E-29, no.4, pp.196-200, Nov 1986. 3. Sinha, P.K.; Electromagnetic suspension: dynamics and control, IEE Control Engineering Series, England 1987. 4. Charara, A.; de Miras, J.; Caron, B.; "Nonlinear control of a magnetic levitation system without premagnetization", IEEE Transactions on Control Systems Technology, vol.4, no.5, pp.513-523, Sep 1996. 5. Hurley, W.G.; Wolfle, W.H.; "Electromagnetic design of a magnetic suspension system", IEEE Transactions on Education, vol.40, no.2, pp.124-130, May 1997. 6. Oliveira, V.A.; Costa, E.F.; Vargas, J.B.; "Digital implementation of a magnetic suspension control system for laboratory experiments", IEEE Transactions on Education, vol.42, no.4, pp.315-322, Nov 1999. 7. El Hajjaji, A.; Ouladsine, M.; "Modeling and nonlinear control of magnetic levitation systems", IEEE Transactions on Industrial Electronics, vol.48, no.4, pp.831-838, Aug 2001. 8. Galvao, R.K.H.; Yoneyama, T.; de Araujo, F.M.U.; Machado, R.G.; "A simple technique for identifying a linearized model for a didactic magnetic levitation system", IEEE Transactions on Education, vol.46, no.1, pp. 22- 25, Feb 2003. 9. Naumovic, M.B.; "Modeling of a didactic magnetic levitation system for control education", 6th International Conference on Telecommunications in modern satellite, cable and broadcasting service, vol.2, no., pp. 783- 786 vol.2, Serbia and Montenegro, Oct 2003.

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10. Yang, Z.-J.; Miyazaki, K.; Kanae, S.; Wada, K.; "Robust position control of a magnetic levitation system via dynamic surface control technique", IEEE Transactions on Industrial Electronics , vol.51, no.1, pp. 26- 34, Feb 2004. 11. William, G.H.; Hynes, M.; Wölfle, W.H.; "PWM control of a magnetic suspension system", IEEE Transactions on Education, vol. 47, no. 2, pp. 165 - 173, May 2004. 12. Gomes, R.R.; Sotelo, G.G.; Stephan, R. M.; "Comparação de configurações para um levitador eletromagnético pelo método dos elementos finitos", Congresso Brasileiro de Eletromagnetismo, São Paulo, 2004. 13. Gomes, R.R.; Sotelo, G.G.; Stephan, R. M.; "Desenvolvimento de um sistema didático para levitação eletromagnética com o auxílio do método dos elementos finitos", Congresso Brasileiro de Automática, Gramado, 2004. 14. Oliveira, V.A.; Tognetti, E.S.; Siqueira, D.; "Robust controllers enhanced with design and implementation processes", IEEE Transactions on Education, vol.49, no.3, pp.370-382, Aug. 2006 15. Baranowski, J.; Piatek, P.; "Nonlinear dynamical feedback for motion control of magnetic levitation system", 13th Power Electronics and Motion Control Conference, EPE-PEMC, pp. 1446 - 1453, Poland, 2008. 16. Dragos, C.-A.; Preitl, S.; Precup, R.-E.; Petriu, E.M.; "Magnetic Levitation System laboratory-based education in control engineering", 3rd Conference on Human System Interactions (HSI), pp.496-501, May 2010. 17. Bandal, V.S.; Vernekar, P.N.; "Design of a discrete-time sliding mode controller for a magnetic levitation system using multirate output feedback", American Control Conference (ACC), pp.4289-4294, June/July 2010. 18. Shiakolas, P.S.; Piyabongkarn, D.; “Development of a real-time digital control system with a hardware-in-the-loop magnetic levitation device for reinforcement of controls education”, IEEE Transactions on Education, vol.46, no.1, pp.79-87, Feb 2003. 19. Shiakolas, P.S.; Van Schenck, S.R.; Piyabongkarn, D.; Frangeskou, I.; “Magnetic levitation hardware-in-the-loop and MATLAB-based experiments for reinforcement of neural network control concepts”, IEEE Transactions on Education, vol.47, no.1, pp.33-41, 2004. 20. Davey, K.; Klimpke, B.; “Computing forces on conductors in the presence of dielectric materials”, IEEE Transactions on Education, vol.45, no.1, pp.95-97, 2002. 21. Bittar, A.; Moura Sales, R.; “H2 and H∞ Control for MagLev Vehicles”, IEEE Control Systems Magazine, vol. 18, no.4, pp.18–25, 1998.

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22. Ellis, J.; “K-12 teachers provide meaningful technical projects for teams of first-year engineering students”, Frontiers in Education Conference, vol.2, 1995. 23. Jayawant, B.V.; “Electromagnetic suspension and levitation”, Physical Science, Measurement and Instrumentation, Management and Education - Reviews, IEE Proceedings A, vol.129, no.8, pp.549-581, 1982. 24. XUMP.com & Innovation Frontier Inc. (2011, Jan 1) Science supplies, Toys and Gifts [Online]. Available at: http://www.xump.com/Science/Floating-Magnetic-Globes.cfm

25. Fascinations and XyNexT (2011, Jan 1) [Online]. Available at: http://www.fascinations.com/unique-toys-gifts/space-mission.htm 26. Edmund Scientifics (2011, Jan 1) [Online]. Available at: http://www.scientificsonline.com/floating-ideas-cosmic-series-levitating-display.html 27. National Geographic Store (2011, Jan 1) [Online]. Available at: http://shop.nationalgeographic.com/ngs/product/maps/globes/levitating-globe 28. Lilienkamp, K.A.; Lundberg, K.; “Low-cost magnetic levitation project kits for teaching feedback system design”, American Control Conference (ACC), vol.2, pp.1308-1313, 2004. 29. Arc Tec - Guy Marsden (2011, Jun) Magnetic levitation kit [Online]. Available at: http://www.arttec.net/Levitation/index.html 30. LNS Technologies Levitator Kit (2011, 13 Jun) [Online]. Available at: http://www.techkits.com/#lev 31. Zeltom Educational and Industrial Control Systems (2011, 13 Jun) Electromagnetic levitation system [Online]: Available at: http://zeltom.com/emls.aspx 32. Hayt, W.H.; Buck, J.A.; “Engineering Electromagnetics”, Ed. Mc Graw Hill, 6th edition, 2001, New York; 33. Valle, R.L.S.; Levitação eletromagnética de um disco, B.Sc. Project, DEE/UFRJ, Rio de Janeiro, 2010. 34. Franklin, G.F.; Powell, J.D.; Naeini, A.E.; “Feedback Control of Dynamic Systems”, Ed. Prentice Hall, 4th edition, 2002, New Jersey; 35. MATLAB (2011, Jun 16) SISO Design Tool Details [Online]. Available at: http://www.mathworks.com/help/toolbox/control/ug/bsupvb0.html 36. DEE (2011, Jan 1) [Online]. Available at: http://www.dee.ufrj.br/IntroEng/index.htm

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Author’s Biographical Information

Rodrigo Valle graduated in Electrical Engineering from the Federal University of

Rio de Janeiro (UFRJ) in 2010. He is currently working at ELETROBRAS. His research

interests include electrical machines, power system and power electronics.

Fábio Neves is studying Control and Automation Engineering at the Federal

University of Rio de Janeiro (UFRJ). He is currently working at the Laboratory of Applied

Superconductivity/UFRJ. His research interests include magnetic levitation, educational

technology and industrial automation.

Rubens de Andrade Jr. received.the B.Sc., M.Sc and D.Sc. degrees in Physics

from Universidade Estadual de Campinas (UNICAMP), in 1985, 1989 and 1995

respectively. Since 1999, he has been with the Department of Electrical Engineering,

UFRJ. He has worked with selective surfaces for solar heaters, electrochemical alloy

deposition, vortex dynamics of type II superconductors, HTS preparation and

characterization (Hg-1212) and vortex dynamics of Hg based superconductors (Hg-1212

and Hg-1223). At moment, his main interests is in the applications of superconducting

materials in power electrical systems and transportation, he has also interest in the

simulation of superconducting devices.

Richard M. Stephan received the B.Sc. degree in Electrical Engineering from

Instituto Militar de Engenharia (IME), Rio de Janeiro, in 1976; the M.Sc. degree in

Electrical Engineering from Universidade Federal do Rio de Janeiro (UFRJ) in 1980, and

the Dr.-Ing. degree in Electrical Engineering from Ruhr Universität Bochum, Germany, in

1985. He has an MBA degree (2005) from the Center for Scientific Enterprise, London

(CSEL), on Technology Enterprise Development. During 1977, he worked as an

engineer at Furnas Centrais Elétricas, Rio de Janeiro. Since 1978, he has been with the

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Department of Electrical Engineering, UFRJ. He spent a sabbatical leave at CEPEL, the

Research Center of ELETROBRAS in 1993. His main interests are in the fields of

applications of superconductivity, control of electrical drives and power electronics.

Dr. Stephan is member of SOBRAEP and IEEE.