Download Design and Implementation of a Magnetic Levitation System Using

Design and Implementation of a Magnetic Levitation
System Using Phase Lead Compensation Technique
Nabila Shawki
Shahana Alam
Ashoke Kumar Sen Gupta
Department of Electrical and Electronic
Engineering
CUET
Chittagong, Bangladesh.
[email protected]
Department of Electrical and Electronic
Engineering
CUET
Chittagong, Bangladesh.
[email protected]
Faculty Member
Faculty of Electrical and Computer
Engineering, CUET
Chittagong, Bangladesh.
[email protected]
Abstract—The fundamental objective of this paper is to describe
the design and implementation of a magnetic levitation system
using phase lead compensation technique. It also verifies the
possibility of implementing a system with equipments available in
Bangladesh with nominal expenditure. Magnetic levitation is the
phenomenon of suspending an object in space using magnetic
force. By overcoming the pull of gravitational force, a maglev
system can levitate an object into space using a force actuator
and proper controlling methods. The theoretical background of
the magnetic levitation is studied from mathematical perspective
that leads to deriving the model of the main system and the
associated controller. The design is carried out using MATLAB,
following Root Locus method, with a primary target of levitating
a steel ball of 21.6 g at a distance of 1 cm below the coil tip. The
system was practically implemented and tested with actual mass.
It is observed that the system has achieved the goal by levitating
the object at predetermined distance. Moreover, it has been
tested for several other masses and the system was capable of
levitating all of them which makes it a robust system.
Keywords—magnetic levitation; electromagnetic force; phase-lead
compensation; system stability; root locus method.
I.
INTRODUCTION
Magnetic levitation is the phenomenon of suspending an
object in space using magnetic force. The natural event of an
object getting pulled towards the center of the earth due to the
gravitational force is quite strenuous to overcome. A maglev
system upholds an object against the natural force of gravity
by using a dynamic system involving an electromagnet with
control over the current flow through it, thus adjusting the
electromagnet's energy. To achieve magnetic levitation, a
dynamic system must have to be implemented with a proper
control system. In practical systems, the usual techniques to
implement magnetic levitation are controlled DC
electromagnets, tuned LCR circuits, superconductors, induced
eddy currents etc. [1], [2]. Magnetic levitation presents
enormous advantages where there is presence of friction as it
is capable of dampening vibration, providing more speed,
minimizing mechanical strains without causing environmental
hazards and pollution. Magnetic trains are the most familiar
example of implementation of magnetic levitation [3].
This paper primarily focuses on studying the principle of
978-1-4799-6062-0/14/$31.00©2014 IEEE
magnetic levitation technique with proper controlling method
leading towards design and implementation of a practical
system which suspends a steel object into space at
predetermined distance using an electromagnet and phase lead
compensation technique. The secondary objective of this
paper is devising an optimal functioning structure with low
cost. The implemented system uses cheap items collected from
local market. The final system is an excellent tool to study
control system and its applications [3].
II.
SYSTEM AND MATHEMATICAL MODELING
A magnetic levitation system is inherently an unstable
system. If an object is put too close to a magnet, it will be
attracted to the magnet and attached to it. If it is placed too far,
the magnet will not be able to exert any force upon the object
[1]. Therefore, an electro-dynamic system with proper control
mechanism has to be sketched to apply precise amount of
force to hold the object suspended into the space at a definite
distance. In this paper, the levitation system consists of an
electromagnet, a controller, a position sensor and the object to
be levitated. The block diagram of the stable closed loop
system is given in Fig. 1.
A. The Subsystems
The electromagnet works as the force actuating device.
When the object, a steel ball, is placed under the
electromagnet, it encounters two forces, one is the
gravitational force and other one is force from the
electromagnet. The goal is to balance between the two forces
in order to suspend the ball in a desired position from the tip
of the electromagnet [4]. This position is also called the steady
state position. In ideal case, these are the only forces to
Fig. 1. Block Diagram of Magnetic Levitation System
The inductance value of the coil varies with the insertion of
the object [6]. The variation occurs due to the movement of
the ball vertically. This can be depicted in a curve shown in
Fig. 3. In this calculation the direction towards the
gravitational force is taken positive i.e. downward direction.
The concept of co-energy is necessary to understand the
fundamental force mechanism of the electromagnet on the
object. Co-energy is basically the energy stored in the
electromagnet due to the generated magnetic field with current
flow. It can be defined mathematically as, [1], [5]-[7], [11]
1
W' i,y = i2 L(y)
(1)
2
L(y) = L1 +
Fig. 2. Schematic Diagram of Magnetic Levitation System
consider [5]. There are other factors such as air flow,
surrounding light, fluctuation in line voltage etc. but these are
neglected while designing the system [1], [5]. The schematic
diagram is shown in Fig. 2.
There is a position sensor which provides the information
respective to the current location of the object. In this
arrangement, the ball is only stabilized in vertical plane. The
rotation around its own axis is not considered. Suspending the
ball in the desired direction at precise position is the heart of
this system design. A light dependent resistor (LDR) and a
light emitting diode (LED) construct the position sensor.
B. Mathematical Modeling
The parameters of the electromagnet essential for the
mathematical analysis are the resistance, the inductance and
the current flow within it. The mass of the steel ball, the
vertical displacement are the other parameters in addition to
those. A list of necessary parameters is given below:
i
Current flow through electromagnet (A)
i′
Perturbation current (A)
Io
Steady-state position current (A)
y
Vertical displacement (m)
y′
Perturbation displacement (m)
Y0
Steady state position (m)
L(y) Total inductance of the electromagnet (H)
L0
Additional inductance caused by the levitated
object (H)
L1
Coils inductance in the absence of the object
(H)
m
Mass of the steel object (kg)
L0 1 2
i L(y)
y
2
1+
Substituting the value of L(y) in Equation (1) gives,
1
Lo Yo
2
y
W' i, y = i2 (L1 +
)
(3)
According to the equation of motion, the forces acting on the
levitated object and the summation of these forces are,
m
d2 y
dt2
=mg + F
(4)
Differentiating (4) with respect to y, the force of the
electromagnet (F) is,
F=
δW'(i, y)
δy
L0 Y0 i2
=
(5)
2y2
L0 Y0
Let, C =
(Nm2 A-2 ), [8]
2
Here, C is the electromagnetic constant or the electromagnetic
strength. The (5) gives,
F=-C
i 2
(6)
y
Equations (4) and (6) can be simplified as,
C=mg
Y0 2
I0
Using Taylor series expansion, the non-linear equation of
force F can be linearized as, [10],
δF(I, y)
δF(I, Y)
y'+
i'
(7)
F (i′, y′) = F (I0 , Y0 ) +
δy
Here, i' i
δi
I and y'= y –Y. Equation (6) gives,
F=m
d2 y
dt 2
2C
I0 2
Y0 3
y' - 2C
I
Y0 2
i'
(8)
The electrical equivalent circuit of the electromagnet can be
shown as Fig. 4. [1], [5].
The circuit equation is
δi
V= Ri + L (y)
(9)
δt
Fig. 3. Variation of Inductance of Coil with Position
(2)
Y0
Fig. 4. Equivalent Circuit of Coil
Equation (9) is non-linear because the inductance of the coil is
dependent on the object position. If it is assumed that the
system is properly designed and the object would always stay
or near to the equilibrium position then y tends to be equal to
Y0 or y = Y0 which makes L(y) = L1+L0.
Again assume that the electromagnet’s inductance, L1, is very
large and the inductance caused by the object’s presence, L0,
is negligible compared to L0.
That gives equation (9) as
δi
V= Ri + L1
(10)
δt
The motion of the object has to be considered too. This can be
expressed using Newton’s Law of Motion (F = ma),
d2 y
(11)
F=m 2
dt
The sensor must be working in the liner region according to
the equation, i = Kx or i = − kx+b, [13],
This is modeled as a gain element.
V = αy
(12)
Here, α = the gain of the sensor (V/m) and V= voltage across
the sensor.
This whole system is described by (8), (10), (11), (12). The
system’s transfer function can be given by the ratio of the
position of the steel ball below the magnet, Y(s), to the current
through the magnet I(s).
Y(s)
G(s) =
(13)
I(s)
As the input voltage of the magnet is proportional to its
current at constant reactance and the output voltage across the
sensor is directly proportional to the position of the steel ball
below the electromagnet, (13) can be expressed as,
V (s)
G(s) = s
(14)
Vm (s)
Taking Laplace transforms of the system (9), (10), (11), and
(12), and simplifying them gives the total open loop transfer
function, [1], [2],[4],
2I C
- α 20
Y0 L1 m
G s =
s+
III.
(15)
2CI0 2
R
(s2 )
L1
mY30
DESIGN APPROACH
The parameters of the basic system are presented in Table
1. Resistance and Inductance of the coil are measured using
LCR meter. Most of the parameter are arbitrarily assumed but
in realistic manner for the sake of theoretical and behavioral
study.
TABLE I.
PARAMETERS OF MAGNETIC LEVITATION SYSTEM
System Parameters
Equilibrium Distance, Y0
Value
0.01 m
Equilibrium Current, I0
0.5 A
Mass of the object, m
0.0216 kg
Force Constant, C
8.47×10-5 Nm2A-2
Coil Resistance, R
6.4 ohms
Coil Inductance, L1
0.0237H
Sensor Gain, α
511.4 V/m
Fig. 5.
Uncompensated Root Locus
By putting the value of system parameter in equation (15) we
have,
G s =
-846700
s3 +270s2 -1962s-529800
(16)
It can be expressed as,
G s =
-846700
s+269.992 (s2 -1962.22)
(17)
From the denominator of the system’s transfer function, the
open loop poles of the system are calculated as 269.992,
44.296, −44.296.
In this system, the quantity that can be directly controlled
with a DC amplifier driving the magnet is its voltage output
L
and the term 1 represents the lag in the resulting current due
R
to largely to the magnetic inductance. The term (s2−1962.22)
represents the force-distance characteristics. It is a comparable
to the spring stiffness constant but here it is negative. The
other poles obtained here are 44.296 and −44.296. The pole
44.296 gives rise to instability due to its location in the right
half plane.
Equation (17) gives the open loop characteristics of the
system. A root locus plot of the transfer function is drawn
using MATLAB. This is the uncompensated root locus of the
magnetic levitation system [1], [5].
Fig. 5 shows that the root locus of this system lies in right
half plane thus indicating the system’s instability. As in the
plot, the pole 44.296 can be seen in the right half plane and no
value of system gain can nullify the effect of this pole to
stabilize the system. Therefore, insertion of a compensator is a
must to pull the root locus into the left half plane. A zero has
to be added between the first left hand plane and the origin
i.e., between -44.296 and origin. The pole of the compensator
should be introduced as such that it is in deeper location than
the deepest left hand pole of the system. This ensures the
stabilization of the system by pulling the root locus into the
left half plane [1].
Fig. 7. Step Response at Gain 16.083
Fig. 6. Compensated Root Locus
IV.
COMPENSATED ROOT LOCUS
For compensation, a phase lead controller is chosen
because it is simplest method to achieve stability of a
magnetic levitation system. A root locus method is followed
for the fact that it offers the clear advantage of giving the
designer the ability of choosing the pole and zero location thus
to impose control over the transient response. The equation of
the compensator is given by, [1], [5], [8],
s+z
C(s) = K
(18)
s+p
It is called lag compensator when z > p and lead
compensator when z < p while K denotes the gain. The lead
compensator is an approximate derivative. It works to increase
the bandwidth and speed of the response while minimizing
overshoot. It lessens the rise time and settling time by
increasing damping which improves the overall transient
response. The characteristic of the phase lead system is that it
improves the magnitude of the phase. The maximum phase
increase can be gained, [4],
δ =sin-1
1- α
1+ α
; α=
z
p
According to rules of thumb, p is 10 times of z, (18) can be
expressed as,
s + 100c
C(s) = K
(19)
s + 1000c
Constant c and K are chosen to achieve the required
performance. c is assumed to be 0.35 rads-1. It gives a zero
between the first pole and origin as well as a compensation
pole deeper than the deepest pole of the system.
s + 35
C(s) = K
(20)
s + 350
The value of K is iterated from 0 to 25. Not all of the gain
points give a satisfactory stable system. A sgrid is taken and a
stable gain point is chosen considering rise time, settling time
and overshoot [5].
The selected gain and its related parameters are obtained
as:
K = 16.083
p = 1.0e+002 *
−4.7803, −0.5680 + 1.3496i, −0.5680 −1.3496i, −0.2841
Transfer function:
1.362×107 s+4.766×108
s4 +620 s3 + 9.255×104 s2 + 1.24×107 s +2.912×108
The step response of the system is given in Fig. 7. It has a rise
time of 0.0129 s, settling time 0.0689 s, overshoot of 27%,
damping ratio of 0.39 at a natural frequency 146.5 rad s−1.
V.
SYSTEM REALIZATION AND EQUIPMENTS
A. Electromagnet Circuit:
In this circuit, the electromagnet that is used has a core of
steel bolt. A 25AWG magnet wire is wounded around the core
with 1440 numbers of turns in 24 layers. The resistance and
inductances are 6.4Ω and 0.0237H. A power transistor is used
to supply the driving current to the magnetic coil. A medium
power transistor 2N3055 is used in this circuit. Generally the
transistor’s gain varies in the range of 10 to 250. A variable
resistance is used with the transistor. It has to be set
experimentally to adjust the gain to achieve a successful
levitation. The circuit is shown in Fig. 8.
B. Controller Circuit:
The transfer function of the controller is given by:
s + 35
C(s) = 16.083
(21)
s + 350
In the circuit diagram shown in Fig. (9), R1, R2 and C form the
compensator network and R3 and R4 provides the system gain.
The transfer function between the R1, R2 and C is ratio of the
input voltage to the controller V1 to the voltage V2 is
Fig. 8.
Electromagnet Coil Driver Circuit
C. Position Sensor circuit:
The position sensor circuit made using a LED and a LDR
provides the position of the object. A variable resistance is
added to adjust the feedback to the system. The related circuit
is given in Fig. 10.
The whole system is connected with an UM741 op-amp. The
circuit design is given in Fig. 11 [5].
Fig. 9.
Controller Circuit
Fig. 10.
Position Sensor Circuit
V2
V1
=
R2 (R1 sC+1)
R2 R1 sC+1 +R1
R
V2
V1
=
s+ R R2
1 2C
s+
R1 + R2
R1 R2 C
In (21) R1 and C are selected to be 280 KΩ and 0.1µF
respectively and then the R2 is calculated to be 31.8KΩ, [4].
The gain of the compensator circuit is
R
K = 1+ f = 16.083
R0
Or, Rf = 15.083 R0
If R0 = 10KΩ, Rf = 150 KΩ
The compensator circuit is shown in Fig. 9.
A wooden frame [5], [14] is used to hold the
electromagnet, position sensor and the circuit. It is constructed
by wood in such a way that the electromagnet is adjusted at
the top of the frame. The Light emitting diode (LED) and the
photo sensor (LDR) are adjusted horizontally in the same axis
at a level where the steel ball is to be levitated. Two pipes
made of rubber of diameter 3cm are used to provide shielding
so that the light emitted by the LED is directed to the LDR [1],
[4]-[6].
The working mechanism of this circuit can be described as
follows: The controller exercises power over the current flow
through the electromagnet. The position sensor provides
location of the top of the ball. When the ball is attracted too
near to the electromagnet, the amount of light received by the
LDR is low which increases its resistance. This in turn lessens
the current flow in the electromagnet and the force in it
weaker. Hence, the ball moves away from the electromagnet.
When the ball is too far from the electromagnet, the process
reverses making the field stronger. In this method, the
controller balances the magnetic force and the weight of the
object with feedback from the position sensor and the ball is
set to the precise position with exact amount of force acting
upon it.
VI.
SYSTEM OBSERVATIONS:
The implemented magnetic levitation circuit is shown in
Fig. 12. After the implementation of the system, it was found
that the system fulfilled the desired target. The design aimed
to levitate a steel ball of 21.6g at a distance of 1 cm below the
magnetic coil. Practical observation showed that the system
succeeded to obtain the desired result. The system was tested
for various objects of different masses. It could levitate all of
these objects subjected to supply voltage. The data is shown in
Table II:
Fig. 11. Circuit Arrangement of Magnetic Levitation System with Phase-lead Controller
controller equipment etc. Magnetic Levitation system is
difficult to implement because of its intrinsic nonlinearity.
Implemented design only imposes control in the vertical
direction leaving the rotation of the ball around its own axis
untouched. The force actuator and the sensor can be
ameliorated using panoply of coils and sensors instead of
single elements. Experimenting with the propulsion technique,
the core technology of magnetic train, which works with
repulsion force instead of attraction force, can bring a whole
new level in transport and communication sector. If more
study and research can be introduced involving magnetic
levitation, a revolutionary advancement is possible in
transportation and industrial sectors.
ACKNOWLEDGMENT
Fig. 12. Implemented Magnetic Levitation Circuit
TABLE II.
Mass (g)
21.6
23.5
18.7
11.1
10
EXPERIMENTAL DATA
Distance (cm)
1
0.9
1.1
1.5
1.5
Supply Voltage (V)
15
15
15
8
8
The system is strongly subjected to environmental effects
as the slightest change of ambient light can change the system
behavior. Electrical interference from any measurement
appliances, such as a multi-meter, can alter the system
response. High frequency noise is supply and thermal noise in
amplifier can cause disruption too. The supply voltage for the
system should be a pure dc source.
Total implementation cost is remarkably reasonable from
the perspective of economy in Bangladesh. The coil has been
manufactured by a steel bolt, which can be collected easily
from the mechanical shops, wounded around with 25 AWG
magnet wire. The circuit equipments are quite accessible in
local electronic shops and can be purchased at low cost. The
wooden frame can be constructed in any woodshop at a
feasible price. The power sources for providing the supply
voltage are available at university laboratories.. Compared to
the system designs that are regularly used with expensive
equipment such as dSPACE DSP Controller Board [2], costly
DC power supplies, metal test-bed structure etc., the design
proposed in here uses locally available inexpensive parts and
circuit appliances. As magnetic levitation is a new arena of
study from the perspective of Bangladesh which also provides
a tremendous opportunity to be familiar with Control System
and its applications, this inexpensive design characteristic
qualifies it suitable for general students of Bangladesh.
VII. CONCLUSION
Levitation of an object is the basic form of magnetic
levitation creating an opportunity to be familiarized with all
the fundamental aspects of control system and electromagnet
theory. The controller can be upgraded using lead lag
controller, PWM technique, state-space controller, dedicated
First of all we are thankful to the Almighty for his
blessings and help. We are grateful to our supervisor for his
constant support, genuine motivation and invaluable
directions. We would also like to thank the teachers of the
Department of Electrical and Electronic Engineering,
especially Dr. Muhammad Ahsan Ullah for his indispensable
guidance to organize the final research paper. We also would
like to pay gratitude to our families as they aided us for the
collection of the supplies as well as for their relentless
encouragement and ardent belief in us. And finally, our
wonderful friends deserve thanks for always being there for us
whenever we needed them, for their consistent helping hands
and precious advices.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
Sintayehu Challa, “Magnetic levitation on a steel ball,” Addis Ababa
University, Addis Ababa, Ethiopia, April 2007.
Stephen C. Paschall II, “Design, fabrication and control of a single
actuator magnetic levitation system,” Texas A&M University, Texas,
USA, August 2002.
Magnetic Levitation. [Online].
Available: http://en.wikipedia.org/wiki/Magnetic_levitation
Milica B. Naumović, Boban R. Veselić, ‘‘Magnetic Levitation System
in Control Engineering Education,’’ FACTA UNIVERSITATIS Series:
Automatic Control and Robotics, Vol. 7, No 1, 2008, pp. 151 – 160
Dahiru Sani Shu’aibu and Sanusi Sani Adamu, ‘‘Design and
implementation of a phase-lead compensator for magnetic levitation,’’
Global journal of Engineering and Technology, Vol. 2 No. 1 March
2009 pp 35-49.
H.H. Woodson and J.R. Melcher, Electromechanical Dynamics, John
Wiley & Sons, 1968.
James W. Nilsson and Susan A. Riedel, Electric Circuits, Prentice Hall,
2000.
Jen-Hsing Li, “DSP-based control of a PWM-driven magnetic
levitation system,” in World Transactions on Engineering and
Technology Education, Vol.5, No.1, 2006.
N.S. Nise, Control System Engineering, Wiley India Pvt. Limited, 1989.
H.K. Dass, Advanced Engineering Mathematics, 20th Edition, S. Chand
and Company Limited, 2005.
U.A. Bakshi and M.V. Bakshi, Electrical Machines - I, Technical
Publications, 2010.
William Gerard Hurley, ‘‘PWM control of a magnetic suspension
system,’’ IEEE Transactions on Education, VOL. 47, NO. 2, May 2004.
Valer Dolga and Lia Dolga, ‘‘Modeling and simulation of a magnetic
levitation system,’’ Fascicle of Management and Technological
Engineering, Volume VI (XVI), 2007.
Chai Pei Keon, “Magnetic levitation - driver and controller,” B.Sc.
Thesis, Universiti Teknologi Malaysia, Malaysia, April 2010.