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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
2
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.
3
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.
4
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
5
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
6
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
7
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
8
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
9
(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
10
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.
11
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
12
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
13
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.
14
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
15
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:
16
(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
17
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
23
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.
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