Optimization of Eight Pole Radial
Active Magnetic Bearing
K. P. Lijesh
Mechanical Department,
Indian Institute Technology Delhi,
New Delhi 110016, India
e-mail: [email protected]
Harish Hirani
Associate Professor
Mechanical Department,
Indian Institute Technology Delhi,
New Delhi 110016, India
e-mail: [email protected]
In the current paper, studies carried out to design an eight pole
electromagnetic bearing have been presented. The magnetic
levitation force, accounting the copper and iron losses, was
maximized for the given geometric constraints. Derivation of
winding constraint equation in terms of wire diameter, number of
turns, and dimensions of pole has been presented. Experiments
were conducted to establish the constraints related to temperature
rise. Finally, the dimensions of the electromagnet for maximizing
the force obtained using numerical optimization have presented.
[DOI: 10.1115/1.4029073]
Keywords: electromagnet, optimization, losses in AMB
Introduction
Active magnetic bearings (AMBs), made of eight poles as
shown in the Fig. 1(a), find their wider usage in flywheel energy
storage systems [1], turbomolecular vacuum pump [2], artificial
heart blood pump [3], etc., due to their built-in fault diagnostics,
vibration free operation, and friction and wear [2] characteristics.
Each pair of pole makes an electromagnet consists of iron-core
and copper-winding. On passing the current through the copperwire of electromagnet, it attracts the rotor. Using four pairs of
electromagnets, the motion of rotor in horizontal and vertical (x
and y) directions can be controlled. The detailed description of
working of electromagnet has been provided by Schweitzer and
Maslen [4].
An electromagnet have been shown in the Fig. 1(b), where Ws
is the thickness of the outer rim of the stator; dp is the inner diameter of the stator at the poles; L is the width of the bearing along
the shaft axis; l is length of the pole; Wp is the width of the pole;
and a is the half angle made by the two consecutive poles at the
centre of the bearing. The electromagnet is made of N turns of
copper wire having cross-sectional diameter dw. The A shaft of
diameter Dshaft is made of paramagnetic material and a ferromagnetic sleeve of thickness Wrotor is mounted on the shaft. In the
present work, Wrotor ¼ Wp ¼ Ws has been considered to maintain
the same saturation limit for the magnetic flux density [4]. Using
these parameters, the attraction force, exerted by an electromagnet
on the rotor (as illustrated in Fig. 1(b)), is given as [5]
F¼g
l0 AG
ðNIÞ2 cos a
G2
(1)
Contributed by the Tribology Division of ASME for publication in the JOURNAL OF
TRIBOLOGY. Manuscript received August 24, 2014; final manuscript received
November 9, 2014; published online December 18, 2014. Assoc. Editor: Bugra Ertas.
Journal of Tribology
Here, N is the number of turns in single coil, I is the current, AG
is the area of air gap for one pole, G is the air gap, and g is the
loss due to magnetic flux leakage and fringe effect. The value of g
can vary from 0.6 (for very closely spaced poles with leakage) to
0.9 (for very well spaced with low leakage).
From Eq. (1), it can be inferred that designing a magnetic
bearing with more number of turns, higher current, and lesser air
gap increases the magnetic force. But with more number of
turns, the volume of the electromagnet increases. Increase in the
current is limited by wire diameter and material saturation.
Reducing the air gap reduces the reaction time for the controller
to react and increases the complexity of controller. Therefore,
there is a need to follow a systematic procedure [6] for designing
the AMB.
Hsiao et al. [6] optimized the load carrying capacity and force
slew rate by constraining the bearing geometry. Hsiao et al. [6]
did not consider the losses, such as copper loss and iron loss,
which constraint the performance of magnetic bearing. Further,
Hsiao et al. [6] put a constraint on the minimum distance between
the tips of the poles, which does not ensure the complete separation between the tips of winding. In the present study, both the
aspects (separation between winding, copper, and iron losses)
have been accounted for maximization of the magnetic force.
Shelke and Chalam [7] optimized the weight of the electromagnet and analyzed the losses (copper and iron loss) considering
different number of poles and concluded that eight pole radial
magnetic bearing reduces the total losses. As per Shelke and
Chalam [7], on reducing outer diameter of magnetic bearing
copper loss increases. This may be due to higher current to be provided in shorter stator radius compared larger radius stator to
achieve same magnetic force. On the other hand, increase in outer
diameter of the stator increases the core losses because the core
loss proportionally increases with the increase in volume. Based
on these considerations Shelke and Chalam [7] suggested a wire
of cross-sectional area of 1.48 cm2 (1.38 cm diameter) for 30 A.
Bakay et al. [8] explored the effect of angular velocity of rotor on
copper and iron losses of AMB. As per Yanhua and Lie [9], the
iron loss is proportional to the thickness of lamination and it is a
higher-order function of flux density (B). For eddy currents, it is
B2, and for hysteresis it is B1.6. It is noteworthy that various
researchers [8–10] analyzed the losses in electromagnet but did
not consider losses as a constraint. In the present study, the losses
as constraints have been accounted.
After reviewing the literature, it appears that researchers have
either considered geometrical constraints or analyzed the losses
occurring in the electromagnet. Designing an efficient AMB
requires considerations of all the geometrical constraints and
losses. In the present work, constraint due to losses have been
included by equating the heat generation and heat liberated and
an appropriate winding interference constraints have been
considered. In the present research, an effort have been made to
maximize load carrying capacity using numerical method, for the
given outer diameter of stator, shaft diameter, axial length, and
the maximum permissible current.
Formulations
In AMB, the constraints can be classified as: (1) geometrical
and (2) losses. The limit on outer diameter of stator, limit on the
shaft diameter, winding space limitation, etc., belong to geometrical constraints. Limit on the maximum current to be supplied to
electromagnet, as a function of temperature rise, rotating speed,
etc., is categorized as loss. Explanation of each constraint is provided below.
Geometrical Constraints. Geometric constraints ensure the
physical nonintersection of the pole tips and sufficient gap
between the pole tips so that the wire can be wound onto the pole.
The winding space constraint limits the number of coils of any
particular diameter. The excessive current in the coil is limited by
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Fig. 1 (a) Active magnetic bearing and (b) two orthographic views of the top electromagnet
the pole magnetic saturation limits. The excessive current causes
excessive losses.
The geometrical constraints include: (1) wire diameter due to
current, (2) area of pole, (3) maximum diameter of stator, (4)
maximum diameter of rotor, (5) winding space, and (6) interference of winding. The geometrical notations of the variables have
been shown in Fig. 1(b).
Pole Diameter. Shaft is made of paramagnetic material to allow
minimum field (minimize loss) to pass through. A ferromagnetic
sleeve is used to respond the magnetic field generated by electromagnet. With this arrangement, there will be very little flux in the
shaft and the position sensors would not be affected by it. The
radial thickness of this ferromagnetic ring (Wrotor) should be equal
to the pole width (Wp) so that the shaft get saturated only when
the poles get saturated; thus, shaft is surrounded by ring having
thickness Wrotor of ferromagnetic material. The air gap (G) is kept
very less and so all the field is assumed to pass through the shaft
path. Thus, the inner diameter of the stator becomes
(2)
dp ¼ Dshaft þ 2 Wp þ G
Interference of Winding. Bakay et al. [8] considered the interference of the pole as constraint. As per authors of the present
paper, the minimum gap between of tips of winding, as shown in
Fig. 2, must be greater than two times of wire diameter. Therefore,
in the present work, the interference of winding has been
accounted.
The distance between the tips of winding must be at least twice
the thickness of wire
2dw 2w
Fig. 2 Winding constraint
(3)
where dw is the diameter of the winding, Lw is the length of the
winding, w is the thickness of winding, If dw is the wire diameter,
and “n” is the number of layer of winding then “w ¼ ndw.” Equation (3) is modified as
dw ndw
Ndw
1
Lw
(4)
Equation (4) makes sure that the number of layers of winding
should be more than or equal to one but cannot be less than one.
In order to prevent interference of the winding tips and to
ensure that there is a minimum gap of 2dw between the tips, the
following condition expressed by Eq. (5) must be satisfied:
8 Wp þ 2w þ 2 cos adw 2p 0:5dp þ t
(5)
The use of Eq. (5) ensures that the sum of the sum of the width
of the eight poles (along with the winding width) and minimum
gap on both sides of the pole is always lesser than the circumference of circle of radius 0.5 dp þ t.
Wire Diameter Constraint Due to Current. Maximum current
in the wire is limited by the area of the current carrying wire and
its current density. The equation is given below [7]
rffiffiffiffiffiffiffiffiffiffiffi
4Imax
(6)
dw np
where n is the maximum current density of copper wire which
depends on the cooling mechanism, shape of winding and insulation type, etc. In the present study, the current density is limited to
6 A/mm2 [6].
Area of the Pole. Ferromagnetic materials are made of randomly distributed microscopic regions called as magnetic
domains (small magnets). The random distribution of the domains
results in zero net magnetic field. In the presence of externally
applied magnetic field (H), the magnetic domains align
themselves and increase in the magnetic flux density (B) of the
ferromagnetic material. The maximum value of flux density (B) is
limited by the onset of material saturation (B ¼ Bsat). Saturation
occurs when all of the domains have been aligned. Once the effective saturation limit has been reached, further increase in applied
field (i.e., amp-turns) will result in minimal gain of force output.
The flux generated in the air gap of the magnetic bearing due to
the current (I) and number of turns (N) must be less than the saturation limit of the material hence the constraint due to saturation
is given by
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stator back iron. This model considers that the windings themselves have a fairly poor thermal conductivity in the direction of
interest. Even if the windings are potted in a thermal compound,
this material tends to have conductivities on the order of 1–5 W/m
K, well lower than the copper or steel; hence, the value of K in the
present work is assumed to be 2.5 W/m K. L is the thickness perpendicular to the heat flow from inside to outside, T1 is the inside
temperature, and Ta is the outside ambient temperature.
Heat transfer due to convection between solid and fluid (i.e.,
air) is given as
Q_ conv ¼ hSðT1 Ta Þ
Fig. 3 Winding space and air gap
NI Bsat l0
G
(7)
Maximum Diameter of Stator. Maximum diameter of stator
constraint is given as
Dstator ¼ Dshaft þ 2 Lw þ Wp þ t þ G
(8)
where t is gap given at the bottom of the pole for avoiding loosening of winding during operating condition
Winding Space Constraint. Maximum wire turns that can be
accommodated around the pole are approximated by the area
between the two poles (Fig. 1(b)). This analysis assumes that 80%
of the coil slot area is usable based on close packed windings [6]
as shown in Fig. 3.
2ndw Lp 0:8 dp þ 2t þ Lw sinðaÞ Wp cosðaÞ Lp cosðaÞ (9)
where Lp is the length of the pole.
Constraints Due to Losses. The losses (i.e., heat generation)
constraints will drive the winding operational temperature that
must be maintained below some limit to ensure long life. Losses
negatively impact the working efficiency of the machine, which
could be very important in some applications like flywheels.
The losses in the electromagnet can be divided into the categories: (1) copper losses and (2) iron losses. The copper losses are
due to the current flowing in the coils. The iron losses occur due
to ferromagnetic material used in both the stator and the rotor.
Copper Losses. According to Joule’s law, the current (I) supplied to wire having resistance (R) generates heat proportional to
I2R, where I is the current flowing in the wire and R is the resistance of the wire
where S is the total surface area of contact between the fluid and
the solid, natural convection coefficient (h) of 10 W/m2 K was
selected [10] for the analysis, and Ta represents the surface
temperature of the core exposed to the ambient, i.e., outer
diameter of the bearing.
Due to the limited temperature range expected, that radiation
influences will be negligible hence heat dissipation through radiation not considered in the optimization. The net heat dissipated
from the magnetic bearing is given below.
Iron Losses. Based on iron loss models presented earlier
[10,11],
eddy current and hysteresis
losses
may be approximated
as 8:4tWp f 2 B2sat =qnpðV Þ þ Kh B1:6
sat fV
ðT1 Ta Þ
Q_ dis Q_ conv þ Q_ cond ¼ hSðT1 Ta Þ þ KA
L
(10)
Pironloss ¼ Peddy þ Physt
qlw
pdw2 =4
(11)
Pironloss ¼
where qis the resistivity of current carrying wire and lw is the total
length of the wire.
The generated heat is transferred through conduction, convection, and radiation methods. Heat transfer due to conduction by
solid material is given as [9]
Q_ cond
T1 Ta
¼ KA
L
(12)
where K is the coefficient of thermal conductivity of the material.
For a free convection application (no forced air flow), much of the
heat is conducted from the windings into the poles and into the
(14)
The core of an AMB is made from soft iron, which is a conducting material with desirable magnetic characteristics. Any conductor will have currents induced in it when it is rotated in a
magnetic field. These currents, induced in the AMB armature
core, are called eddy currents. The power loss due to eddy current
must be minimized to reduce the heating of the core. The magnetic core of electromagnets is made of ferromagnetic material
which consists of numbers of domains. These domains are
arranged in such a random manner, that net resultant magnetic
field is zero. Whenever external magnetic field is applied to that
substance, these randomly directed domains get arranged parallel
to the axis of applied field. After removing the external field, a
number of domains again come to random positions, but some of
them still remain in their charged position. Because of these
unchanged domains, the substance becomes slightly magnetized
for longer duration. Some energy is required to neutralize those
domains, which is termed as hysteresis loss. Hysteresis loss is proportional to volume of the conductor, supplied magnetic field, and
frequency of domain reversal.
To reduce eddy currents losses in stator material lamina are
used. However, in the presence of unbalance force, iron losses
occur in the stator and rotor
Q_ gen ¼ I 2 R
R¼
(13)
8:4tWp f 1:5 B2sat
0:8845
V
V þ 0:06253B1:6
sat f
qp
(15)
where “t” is thickness of laminated core, length of the pole “l,”
“f” is the frequency of the rotation, and V is the volume of the
conductor.
The total losses in the electromagnet is
Ploss ¼ Peddy þ Physt þ Q_ gen
(16)
For the electromagnet to work for long time the heat generated
must be less than the heat dissipated, i.e.,
Q_ dis Ploss
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Fig. 6 Change in temperature of winding with respect to time
Fig. 4 Experimental setup with electromagnet core
Current Constraint Due to Temperature Rise. The maximum
value of temperature (T1) is limited based on the reduction of
magnetic flux density estimated by experiments. The rise in temperature of winding reduces the magnetic property and it is necessary to consider the degradation of the magnetic property with
temperature. To understand this behavior, a core (Fig. 4(a)) and
an experimental setup (Fig. 4(b)) were manufactured. 110 number
of turns of 18 gauge wire was wound around each pole of electromagnet (as shown in Fig. 5(a)) and 4 A current was passed.
The temperatures of the winding for different currents were
measured using the four thermocouples (two thermocouples to
measure the outside winding temperature, one thermocouple to
measure the inside temperature, and remaining one thermocouple
to measure the core temperature) at different locations on electromagnet as shown in Fig. 5. The variation in temperature inside the
winding (position “1” in Fig. 5(b)) with respect to time at various
currents is shown in Fig. 6. From this figure, it can be inferred that
the temperature rise is substantial for currents 4 A and 3.5 A, while
the temperature remains almost constant for current of 3.2 A.
The change in surface magnetic flux density was measured
using the Gauss meter by attaching its probe to the setup as shown
in Fig. 7. Three currents 4 A, 3.5 A, and 3.2 A were supplied to the
winding. The magnetic flux densities with rise in temperature
were recorded. Each experiment was conducted twice. The results
are plotted in Fig. 8. From this figure, it can be concluded that the
supply current to be restricted to 0.8 times of the rated current.
The possible reasons for the lower magnetic field are given
below:
•
Space Constraint: Lower number of turns (110) used in the
electromagnet due to the space constraints. For these
Fig. 5
•
•
turns (110) and 4 A current, the value of H comes out to be
50 A/m. The value of B for H ¼ 50 is 0. 2 T for silicon steel
[12].
Dimensions of Gauss-Meter Probe: The width of the Gaussmeter probe, used for measuring the magnetic field, is 4 mm.
It is very difficult to keep the Gauss probe at the corners of
the pole, so the probe was placed at the center of the pole.
The magnetic field will be the maximum at the corners of
the core and reduces toward the center. This is another
reason for further reduction in the magnetic field.
Quality of the Electrical Steel: The core was fabricated
using low grade silicon steel due to nonavailability of pure
silicon steel. Since the magnetic flux density depends on the
core material; therefore, a reduced magnetic field was
achieved with the use of this core material.
Results and Discussion
In the present research, an electromagnetic bearing is developed
for a typical application, in which limit on axial length and the
outer diameter of the stator have been defined. The maximum current that can be supplied to the purchased controller is limited to
4 A. Hence, in the present work, the objective is to maximize the
force, i.e., (Eq. (1)) for the given set of Eqs. (2), (4)–(9), and (17)
and inputs as listed in Table 1. The parameters have been illustrated in Fig. 9.
Formulation of Problems. To use numerical optimization, the
constraints equations (2), (4)–(9), and (17) and the objective function (1) have to be simplified. The steps, followed for simplification of each constraints equation, are given below:
Location of thermocouple: (a) experimental measurement and (b) schematic figure
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Fig. 7 Measurement of magnetic flux density using Gauss
meter
Fig. 9 Dimensions of electromagnet
(4) By substitution the values of a ¼ 22.5deg along with the
values of “Lw ¼ 0:0355 0:91Wp ” obtained from
Eq. (18) and values of dp ¼ 0:021 þ 2Wp from Eq. (2) in
Eq. (8)
2N
Fig. 8 Reduction in magnetic flux density with rise in
temperature
pdw2
0:8 0:021 þ 2Wp þ 1:1 0:0355 0:91Wp
4
sinð22:5 degÞ Wp cosð22:5 degÞ
0:0355 0:91Wp cosð22:5 degÞ
pdw2
0:8 Wp þ 0:0601 sinð22:5 degÞ
4
Wp cosð22:5 degÞ 0:0355 Wp
) 2N
Table 1 Parameter considered for the AMB
Outer diameter, Dstator
Radial air gap, G
Diameter of the shaft, Dshaft
Length of the AMB, L
Saturation of the material (Bsat)
Copper current density (n)
Resistivity (q)
Heat transfer coefficient (h)
Maximum current (Imax)
g
Ambient temperature (Ta)
Winding temperature (T1)
100 mm
0.5 mm
20 mm
35 mm
1.2 T
6 A/mm2
1.68 108 (X m)
10 W/(m2 K)
4A
0.6
35 C
65 C
(1) On substituting Dshaft ¼ 20 mm, Dstator ¼ 100 mm, and
G ¼0.5 mm from Table 1 in Eq. (7), 0:1 ¼ 0:02
þ 2 Lw þ t þ Wp þ 0:0005 . This can be rewritten as
Lw ¼ 0:0359 0:909Wp
(18)
(2) On substituting the values of current density of copper wire
(n ¼ 6 A/mm2) from Table 1 in Eq. (5)
dw2 4Imax
1:88 107
(19)
(3) By substituting the values of Bsat ¼ 1.2T, l0 ¼ 4p107, and
G ¼ 0.5 mm in Eq. (7)
IN On rearranging
0:0005 1:2
4p 107
NI 477
cosð22:5 degÞ
pðdw Þ2
0:8 0:0555 0:5421Wp
) 2N
4
0:0355 Wp cosð22:5 degÞ
) Ndw2 0:475 0:002 0:0748Wp þ 0:5421Wp2
Modified final equation is given by
) Ndw2 0:475 0:002 0:0748Wp þ 0:5421Wp2 0
(21)
(5) The expression “dp ¼ Dshaft þ 2 Wp þ G ” of Eq. (2) can
be used in Eq. (4), and substituting the values of variables
(Dshaft ¼ 20 mm and G ¼ 0.5 mm) from Table 1
8 Wp þ 2w þ 2 cos adw 2p 0:5dp þ t
Ndw
8 Wp þ 2
þ 2 cosð22:5Þdw 2p 0:5dp þ 0:1Lw
Lw
8Wp Lw þ 2Ndw þ 1:847dw Lw 3:14dp þ 0:628L2w
0:287Wp 0:728Wp2 þ 2Ndw þ 0:066dw 1:67dw Wp
0:11dp þ 2:857dp Wp þ 8:1 104 0:04Wp þ 0:5Wp2
0:327Wp 1:28Wp2 þ 2Ndw þ 0:066dw 1:67dw Wp
(20)
0:11dp 2:857dp Wp 8:1 104 0
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Substituting Eqs. (23a)–(23e) in Eq. (17), we get
(6) Similarly
(a) Power loss due to eddy current
Peddy
8:4tWp f 1:5 B2sat
¼
V
qp
Peddy
8:4tWp f 1:5 B2sat ¼
lLWp
qp
5:51 106 Wp3 þ 1:11 105 Wp2 þ 0:0039Wp
þ 2:13 108 Wp þ 7:48 1010 I 2 N=dw2
6 104 Wp2 þ 1170Wp þ 1:06 104
5:51 106 Wp3 þ 1:71 105 Wp2 1170Wp
þ 2:13 108 Wp þ 7:48 1010 I 2 N=dw2
Peddy ¼ 8:1 107 Wp2 ðLLw Þ
1:06 104 0
Peddy ¼ 8:1 107 Wp2 ð0:035Þ 0:039 2Wp
(7) On substituting g ¼ 0.6, l0 ¼ 4p107,
G ¼ 0.0005 m, and AG ¼ 0.035Wp in Eq. (1)
Peddy ¼ 1:11 105 Wp2 5:665 106 Wp3
Final modified equation for power loss due to eddy current is given by
Peddy ¼ 1:11 105 Wp2 5:665 106 Wp3
a ¼ 22.5 deg,
4p 107 0:035Wp
ðIN Þ2 cosð22:5 degÞ
0:000752
maxðFÞ ¼ 0:0433Wp ðIN Þ2
F ¼ 0:6
(25)
(23a)
Numerical Optimization. For numerical optimization, in the
present work, MATLAB has been used for obtaining the optimum solution. In MATLAB, the optimization has been carried out using constrained minimization function (fmin) and “Interior Trust” method
for the considering all the specified constraints. Equation (26) was
converted to a minimization objective function and given as
(b) Power loss due to hysteresis
Physt ¼ Kh B1:6
sat fV
Physt ¼ 0:06253f 0:1155 B2sat f Lw LWp
Physt ¼ 0:1Wp 0:039 2Wp
minðFÞ ¼ 0:0433Wp ðIN Þ2
Physt ¼ 0:0039Wp 0:2Wp2
Final modified equation for power loss due to hysteresis
is given by
Physt ¼ 0:0039Wp 0:2Wp2
(24)
(23b)
(c) Heat generation in current carrying wire
(26)
The nonlinear constraints is given as
14:78Ndw2 < 2:0718Wp2 0:16061Wp þ 0:0031
Imax
4:7 106
dw2
0:327Wp 1:28Wp2 þ 2Ndw þ 0:066dw 1:67dw Wp 0:11dp
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
qLw
pdw2 =4
2 qN Wp þ L þ 0:001
¼I
pdw2 =4
¼ 2:13 108 Wp þ 7:48 1010 I 2 N=dw2
>
2:857dp Wp 8:1 104 0
>
>
>
>
>
NI 477 0
>
>
>
>
6 3
5 2
>
5:51 10 Wp þ 1:71 10 Wp 1170Wp
>
>
>
;
þ 2:13 108 Wp þ 7:48 1010 I 2 N=dw2 1:06 104 0
Final heat generation equation is given by
Qgen ¼ 2:13 108 Wp þ 7:48 1010 I 2 N=dw2
The problem has been solved with the aid of MATLAB toolbox.
The functional and nonlinear constraint tolerance was considered
to be 106. The values of objective function with respect to number of iteration have been shown in Fig. 10. The number of
Qgen ¼ I 2
Qgen
Qgen
(23c)
(27)
(d) Heat transfer due to convection
Q_ conv ¼ 0:2hLWp ðT1 Ta Þ
Q_ conv ¼ 2:1Wp
Final heat transfer due to convection is given by
Q_ conv ¼ 2:1Wp
(23d)
(e) Heat transfer due to conduction
ðTi Ta Þ
Q_ cond ¼ KLp Wp
L
Ti Ta
¼ 2:5 0:039 2Wp Wp
L
¼ 83Wp 14512Wp2
(23e)
Fig. 10 Objective function with respect to the number of
iteration
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Table 2
Results using numerical optimization
Parameter
Value
Wp (m)
dw (m)
Imax (A)
N
Force (N)
0.015
0.001
2.9
113
157
winding constraint equations in terms of wire diameter, number of
turns, and dimensions of pole have been derived and constraints
related to current by measuring magnetic flux density with
increase temperature have been established. The maximization of
levitation force has been achieved, for the given constraints, using
numerical optimization method.
Acknowledgment
iterations required for the finding the optimum solution is 28 and
the function value is 157 N and the values of variables are tabulated in Table 2.
The results obtained by the optimization method were substituted in objective function (26) and constrained equations
(19)–(21) to determine the feasibility of the optimum solution.
The optimum force of 157 N was found to be almost equal to
157.15 N which was obtained by Eq. (1) after substitution of the
values of number of turns (N), width of pole (Wp), and current (I).
The observation for different constraint equations is provided
below:
•
•
•
•
The wire diameter of 1 mm estimated by optimization was
found to be much lesser than 0.78 mm obtained by using
substituting the value of current in Eq. (6)), satisfying the
constraint equation (19).
The optimum value of product of number of turns and current, 327, were also found to be much smaller than 477, satisfying the constraint equation (20).
Similarly, the LHS value obtained by substituting the values
of width of the pole (Wp), number of turns (N), and wire diameter (dw) in constraint equation (21) is greater than zero,
which satisfies the constraint condition.
Similarly, the pole interference (Eq. (22)) values and loss
constants (Eq. (24)) were found to found to be stratifying
the conditions.
Conclusions
In the present paper, a systematic procedure to design an electromagnet for magnetic radial bearing has been detailed. The
This research was supported by Council of Scientific and
Industrial Research, New Delhi, India [Grant No. 70(0073)/2013/
EMR-II].
References
[1] Sahinkaya, M. N., Abulrub, A. H. G., Keogh, P. S., and Burrows, C. R., 2007,
“Multiple Sliding and Rolling Contact Dynamics for a Flexible Rotor/Magnetic
Bearing System,” IEEE/ASME Trans. Mechatronics, 12(2), pp. 179–189.
[2] Noh, M. D., Cho, S. R., Kyung, J. H., Ro, S. K., and Park, J. K., 2005, “Design
and Implementation of a Fault-Tolerant Magnetic Bearing System for Turbomolecular Vacuum Pump,” IEEE/ASME Trans. Mechatronics, 10(6), pp.
626–631.
[3] Williams, R. D., Keith, F. J., and Allaire, P. E., 1990, “Digital Control of Active
Magnetic Bearings,” IEEE Trans. Ind. Electron., 37(1), pp. 19–27.
[4] Schweitzer, G., and Maslen, H., 2009, Magnetic Bearings: Theory, Design, and
Application to Rotating Machinery, Springer-Verlag, Berlin, Germany.
[5] Chang, H., and Chung, S. C., 2002, “Integrated Design of Radial Active
Magnetic Bearing Systems Using Genetic Algorithms,” Mechatronics, 12(1),
pp. 19–36.
[6] Hsiao, F. Z., Fan, C. C., Chieng, W. H., and Lee, A. C., 1996, “Optimum
Magnetic Bearing Design Considering Performance Limitations,” JSME Int. J.
Ser. C, 39(3), pp. 586–596.
[7] Shelke, S., and Chalam, R. V., 2011, “Optimum Energy Loss in Electromagnetic Bearing,” 3rd International Conference on Electronics Computer Technology (ICECT), Kanyakumari, India, Apr. 8–10, pp. 374–379.
[8] Bakay, L., Dubois, M., Viarouge, P., and Ruel, J., 2009, “Losses in an
Optimized 8-Pole Radial Magnetic Bearing for Long Term Flywheel Energy
Storage,” International Conference on Electrical Machines and Systems,”
Tokyo, Japan, Nov. 15–18, pp. 1–6.
[9] Yunus, C. A., 2011, Heat and Mass Transfer: Fundamentals and Applications,
Tata McGraw-Hill, New Delhi, India.
[10] Yanhua, S., and Lie, Y., 2002, “Analytical Method for Eddy Current Loss in
Laminated Rotors With Magnetic Bearing,” IEEE Trans. Magn., 38(2), pp.
1341–1347.
[11] Saito, T., Takemoto, S., and Iriyama, T., 2005, “Resistivity and Core Size
Dependencies of Eddy Current Loss for Fe–Si Compressed Cores,” IEEE Trans.
Magn., 41(10), pp. 3301–3303.
[12] Boldea, I., and Nasar, A. S., 2002, The Induction Machine Handbook, CRC,
Boca Raton, FL.
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