Magnetic thrust bearing is a device which is used to support the object by controlling the magnetic field. Permanent magnets or electromagnets or both are used to produce magnetic field. The type of magnetic bearing discussed in this paper is a single acting active magnetic thrust bearing. A prototype magnetic thrust bearing is made to study the thrust capability. The measured values of force are compared with theoretical values. A leakage factor is estimated. The experiments are done at different air gaps from 1mm to 5 mm in steps of 0.5 mm. The variation of leakage factor is plotted at different air gaps. An attempt is made to find the optimum air gap between the stator and rotor of AMTB

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 1880-1896, Article ID: IJMET_10_01_186
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=1
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
ESTIMATION OF LEAKAGE FACTOR FOR
ACTIVE MAGNETIC THRUST BEARING
V. V. Kondaiah
Center for Design and Analysis, Department of Mechanical Engineering, CMR College of
Engineering & Technology, Kandlakoya, Hyderabad, Telangana-501401
Jagu S. Rao
Department of Mechanical Engineering, RGUKT, Nuzvid, Andhra Pradesh
V. V. Subba Rao
Department of Mechanical Engineering, Jawaharlal Nehru Technological University
Kakinada, Kakinada, Andhra Pradesh
ABSTRACT
Magnetic thrust bearing is a device which is used to support the object by
controlling the magnetic field. Permanent magnets or electromagnets or both are used
to produce magnetic field. The type of magnetic bearing discussed in this paper is a
single acting active magnetic thrust bearing. A prototype magnetic thrust bearing is
made to study the thrust capability. The measured values of force are compared with
theoretical values. A leakage factor is estimated. The experiments are done at different
air gaps from 1mm to 5 mm in steps of 0.5 mm. The variation of leakage factor is
plotted at different air gaps. An attempt is made to find the optimum air gap between
the stator and rotor of AMTB.
Keywords: Active magnetic thrust bearings (AMTB), leakage factor, Dynamic
stiffness.
Cite this Article: V. V. Kondaiah, Jagu S. Rao and V. V. Subba Rao, Estimation of
Leakage Factor for Active Magnetic Thrust Bearing, International Journal of
Mechanical Engineering and Technology, 10(1), 2019, pp. 1880-1896.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=1
1. INTRODUCTION
An active magnetic bearing works on the principle of electromagnetic suspension. It consists
of an electromagnet assembly, a set of power amplifiers which supply current to the
electromagnets, a controller, and gap sensors with associated electronics to provide the
feedback required to control the position of the rotor within the gap.
A Magnetic thrust bearing has an electromagnetic stator and a rotor. Allaire et al. [1]
presented the design of a prototype of thrust magnetic bearing for the high load-to-weight

2. V. V. Kondaiah, Jagu S. Rao and V. V. Subba Rao
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ratio. Groom and Bloodgood [2] proposed a model by adding the loss and leakage factors to
ideal models with and without bias permanent magnets. Subsequently, Bloodgood et al. [3]
applied the theory for the optimal design of a thrust magnetic bearing with bias permanent
magnets. Rao and Tiwari [4] implemented multi-objective genetic algorithms (MOGAs) for
the optimization of active magnetic thrust bearings (AMTB) with pure electro magnets
considering the power-loss and the weight as minimization type objective functions. David et
al. [5] explained the leakage, fringing and eddy current effects in design of magnetic bearings.
Bekinal et al. [6] Experimented on permanent magnet thrust bearing and compared the
theoretical and practical force generated in their test setup. Kondaiah et al. [7] experimented
on active magnetic thrust bearing on a universal testing machine for an air gap of 3mm and
measured the actual force between stator and rotor of bearing. In this work the bearing is
made with laminated CRGO sheets
In the present work, a prototype of single acting active magnetic thrust bearing has been
made and tested for thrust capability on the own fabricated test setup. The stator and rotor of
the bearing is made with the solid metal of mild steel. The solid metal is used because the
eddy current losses in the thrust applications are negligible [1]. The attractive force between
stator and rotor has been measured on the test setup. The measured values of force are validated with
theoretical values computed using Lorentz principle. The experiments are done at different
current inputs and varying the gap between stator and rotor from 1 mm to 5 mm in steps of
0.5 mm. The average leakage factor is calculated at different air gaps. The variation of
leakage factor is plotted with respect to air gap. A comparison of theoretical, actual and
predicted forces has been discussed at different air gaps.
2. NOMENCLATURE
gA Area of air gap
B Magnetic flux density
aF Actual Force developed
pF Force on one pole face
preF Predicted force
thF Theoretical force
R Total reluctance of magnetic circuit
gR Reluctance of air gap
fR Reluctance of iron path
ch Axial length of stator
th Axial length of stator pole with base
i Current in coil
k Leakage factor
avk Average leakage factor
l Effective magnetic circuit gap
bl Axial length of stator base
dl Axial length of thrust rotor
fl Length of iron path

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gl Height of Air gap
n Number of turns
cir Inner radius of coil gap
cor Outer radius of coil gap
ir Inner radius of stator
or Outer diameter of stator
v Voltage supplied
%th Percentage of error between theoretical force and actual force
%pre Percentage of error between predicted force and actual force
,ni Magneto motive force
r Relative permeability of silicon steel
 Magnetic flux
g Permeability of free space
4𝜋 × 10−7
Web/amp-turn-meter
3. DESCRIPTION OF MAGNETIC THRUST BEARING
A magnetic thrust bearing has an electromagnetic stator and a rotor, which are separated by an air gap, as
illustrated in Fig. 1. In its simplest form, the electromagnetic stator is formed by an inner and outer pole
connected by a common base. The exploded view, in Fig. 1, clearly shows the stator, shaft, winding, and thrust
collar of a single acting bearing. The stator and rotor disc are made up of mild steel. The mild steel discs are
turned into the desired shapes on the lathe machine. The winding, which occupies the space between the inner
and outer poles of the stator, produces the magnetic flux in the bearing. Magnetic flux paths are flowing through
the inner and outer poles through the rotor disc. It is important to provide a good flux path to avoid leakage from
the magnetic components [1].
Figure 1 Components of AMTB

4. V. V. Kondaiah, Jagu S. Rao and V. V. Subba Rao
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4. THEORETICAL MODELLING
The model of the magnetic thrust bearing used is based on one dimensional electromagnetic
theory. Several assumptions are made in this derivation for the sake of simplicity as given
below.
1. No leakage of flux takes place between the inner and outer poles.
2. The intensity of flux is always below saturation level.
3. Changes in the current input are small compared to the steady state level.
4. Axial shaft motions are small compared to the steady state air gap.
5. One dimensional model of the magnetic path was used.
Figure 2 Geometry of AMTB
The geometry of AMTB is shown in Fig.2, pole face area of air gap gA is given by
   2 2 2 2
g ci i o coA r r r r    
(1)
These areas of the outer and inner poles are made equal so that the magnetic flux density
has the same level in each pole. The pole face area then equals the air gap area gA . Thus the
thickness of the base could be evaluated from
2g ci bA r t
(2)
The rotor disc thickness equals to back wall thickness. The reluctance of each air gap is given
by
0
g
g
g
l
R
A

(3)
And the reluctance of the iron path is
0
f
f
r g
l
R
A 

(4)
Let the length of air gap magnetically be equal to the iron path with value

5. Estimation of Leakage Factor for Active Magnetic Thrust Bearing
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f
e
r
l
l


(5)
The total reluctance of the magnetic circuit is
   02 2g f g e gR R R l l A   
(6)
Thus the effective magnetic gap l is
2 g el l l 
(7)
Magneto Motive Force (ϑ) is equal to the number of turns in the coil times the current
ni  (8)
The magnetic flux could be found from
0 gA ni
R l

  
(9)
And the flux density in the path is given by
0 g
g
A ni
B
A l

 
(10)
The magnetic flux density must not exceed the saturation level for the particular magnetic
material involved. Typical values for silicon iron are 1.2 to 1.6 Tesla and for rare earth
materials up to 2.0 Tesla. The attractive force developed at each pole face is
22
0
2
2 2
g
p
g
A
F
A l
 

 
(11)
The total force developed is
22
0
2
2
g
th p
g
A
F F
A l
 

  
(12)
The actual force is reduced some extent due to leakage effects. A leakage parameter k ,
can be calculated for the thrust bearing geometry as
th
a
F
k
F

(13)
Having some practical values of aF at a particular air gap, an average leakage factor, avk
could be predicted that fits thF with the practical results. And the thrust bearing load capacity
is then modified to include as
2
0
2 2
g
pre
av
A
F
l k
 

(14)
Where preF is predicted force.

6. V. V. Kondaiah, Jagu S. Rao and V. V. Subba Rao
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5. PROTOTYPE OF MAGNETIC THRUST BEARING
A prototype of single acting thrust bearing was constructed for testing of load capacity.
The rotor and stator parts separately are shown Fig. 3. The lead wires for the coil come out of
holes in the stator base for connection to the control circuit. The dimensions of the prototype
have been given in Table 1.
Table 1 Dimensions of prototype manufactured
Parameter Symbol
Value
(mm)
Inner diameter of stator ir 20
Inner diameter of coil gap,
cir
30
Outer diameter of coil gap,
cor
45
Outer diameter of stator, or 52.5
Depth of coil gap, d 3.8
Axial length of thrust
runner, th
10
Air gap, gl 1: 0.5: 5
(a) AMTB stator (b) AMTB rotor
Figure 3 Prototype of AMTB
6. DESCRIPTION OF TEST SETUP
The test setup used in this work, shown in Fig.4. is fabricated to measure the magnetic force
between stator and rotor parts of the bearing. The frame holds the components of AMTB and
load. The stator part of bearing is hold by upper horizontal bar of the frame and it has nut and
bolt arrangement to move up and down. The rotor and load are hold by the middle horizontal
bar. The rotor shaft is free to move in the middle horizontal bar. The shaft of rotor passes
through the middle bar and dead weights are attached to it.

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Figure 4 Test setup for AMTB
A variator is used to change the current input. An ammeter and a voltmeter are used to
measure the input current and voltage respectively.
7. TEST PROCEDURE
As the difference between the relative permeability of air and aluminum is negligible, the air
gap between stator and rotor is maintained by keeping aluminum plates of desired thickness.
The thickness of aluminum plates ranging from 1mm to 5mm in the steps of 0.5mm. The
voltage is varied from 50 V to 120 V. The input current is varied from 0 Amp to 6 Amp. At
each value of current the attractive force is measured by applying equivalent force in the
opposite direction. The experiment is repeated three times for each air gap and the average
values have been taken as final values.
8. RESULTS AND DISCUSSIONS
Though the experimentation has been done at different air gaps 1 mm to 5 mm in steps of 0.5
mm, the results of the AMTB at an air gap 1 mm have been shown in Table 2 for observation.
The actual force, theoretical force, predicted force, the percentage of error between actual
force and theoretical force and percentage of error between predicted and actual force have
been tabulated.

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Table 2 Results of AMTB when the air gap is 1 mm
V i Fa Fth k Fpre
90 2.16 65.95 199.09 1.74 58.14 66.87 -13.44
95 2.31 72.81 227.62 1.77 66.47 68.01 -9.53
100 2.45 78.22 254.58 1.81 74.34 69.28 -5.21
105 2.56 86.72 279.39 1.79 81.59 68.96 -6.29
110 2.73 94.65 316.86 1.83 92.53 70.13 -2.29
115 2.86 101.34 348.53 1.86 101.78 70.92 0.43
120 2.96 107.04 373.27 1.86 109.01 71.32 1.79
kav = 1.85
Theoretical force and Predicted force has been calculated at each air gap between stator
and rotor of AMTB, and these are compared with actual force measured.
Figure 5 Actual, Theoretical and Predicted force for 1mm airgap
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
400.00
2.17 2.32 2.45 2.57 2.73 2.87 2.97
Force(N)
current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)

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Figure 6 Actual, Theoretical and Predicted force for 1.5 mm airgap
Figure 7 Actual, Theoretical and Predicted forces for air gap 2mm
0.00
50.00
100.00
150.00
200.00
250.00
300.00
1.87 2.08 2.40 2.67 2.98 3.28 3.58 3.88
Force(N)
current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
1.88 2.13 2.50 2.78 3.15 3.37 3.67 3.85
Thrustforce(N)
current (A)
Actual force (N)
Theoreticalforce (N)
Predicted force (N)

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Figure 8 Actual, Theoretical and Predicted force for 2.5mm air gap
Figure 9 Actual, Theoretical and Predicted forces for 3mm air gap
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
2.32
2.48
2.70
2.90
3.12
3.30
3.48
3.68
3.85
4.03
4.18
4.28
4.45
4.63
Force(N)
current (A)
Actual force
Theoretical force
Predicted force
0.00
20.00
40.00
60.00
80.00
100.00
120.00
2.38
2.58
2.82
3.02
3.22
3.42
3.58
3.77
3.93
4.08
4.23
4.37
4.50
4.68
Force(N)
Current(A)
Actual force(N)
Theoretical force (N)
Predicted force(N)

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Figure 10 Actual, Theoretical and Predicted force for 3.5mm
Figure 11 Actual, Theoretical and Predicted force for air gap 4mm
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
3.13 3.33 3.53 3.72 3.87 4.05 4.22 4.35 4.55
Force(N)
Current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
3.77
3.92
4.03
4.22
4.42
4.57
4.75
4.93
5.10
5.20
5.32
Force(N)
current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)

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Figure 12 Actual, Theoretical and Predicted force for air gap 4.5mm
Figure 13 Actual, Theoretical and Predicted force for air gap 5mm
In Fig. 5 to Fig. 13 the actual force, theoretical force and predicted force have been shown
at different air gaps from 1 mm to 5mm in steps of 0.5 mm. From these plots it can be
observed that there is large differences between theoretical force and actual force, however
after introducing leakage factor, the difference between the predicted force and actual force
drastically reduced for all the air gaps. The percentage of error between theoretical force and
actual force is ranging from 77 to 266 and the percentage of error between predicted force and
actual force is ranging from 13.25 to 0.16
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
4.70 4.93 5.13 5.30 5.42 5.57 5.68 5.78
Force(N)
current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)
0.00
10.00
20.00
30.00
40.00
50.00
60.00
5.23 5.38 5.52 5.65
Force(N)
current(A)
Actual force(N)
Theoretical force(N)
Predicted force(N)

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Table 3 Change of leakage factor and max % of error with respect to air gap
Air
Gap
(mm)
Average
Leakage factor
(kav)
Max % of
error of
predicted force
1 1.8505 11.85
1.5 1.829 10.84
2 1.5475 10.89
2.5 1.5229 4.99
3 1.4149 11.59
3.5 1.4306 8.06
4 1.4232 12.2
4.5 1.655 13.26
5 1.6765 6.2
The average value of leakage factor and maximum percentage of error between actual and
predicted forces has been calculated at all air gaps from 1 mm to 5 mm in the steps of 0.5 mm.
There are shown in Table 3. The variation of leakage factor with air gap has been plotted in
Fig.14. It is observed that the average value of leakage factor is minimum and almost equal
from the air gaps 2 mm to 4 mm. The average leakage factor value is more when the distance
between stator and rotor of AMTB is too close and too far.
The variation of leakage factor with air gap has been plotted in Fig 14. It is observed that
it is almost equal from the air gaps 2 mm to 4 mm.
Figure 14 Average leakage factor vs air gap
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1 1.5 2 2.5 3 3.5 4 4.5 5
Averageleakagefactor
Air Gap (mm)
Average leakage factor

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Figure 15 change of max % of error with air gap
The maximum percentage of error between predicted force and actual force with air gap
has plotted in Fig 15. It is observed that the maximum percentage of error is minimum
between the air gaps of 2 mm to 3.5 mm. At an air gap of 4.5 mm percentage of error is
maximum and is 13.26. The minimum percentage of error at an air gap of 2.5 mm and is 4.99.
Table 4 Variation of current with air gap (voltage constant)
Voltage
(V)
Current (A)
95 2.323.132.323.853.934.224.575.135.23
100 2.453.282.454.034.084.354.755.305.38
105 2.573.422.574.184.234.554.935.425.52
110 2.733.582.734.284.374.805.105.575.65
115 2.873.732.874.454.504.955.205.68 --
120 2.973.882.974.634.685.105.325.78 --
air gap
(mm)
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0
2
4
6
8
10
12
14
1 1.5 2 2.5 3 3.5 4 4.5 5
%oferror
Air gap(mm)
max % error

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Figure 16 Variation of current with air gap (voltage constant)
Next the results have been analyzed at constant voltage. The variation of current with air
gap has been shown in Table 4. The variation of current with air gap at constant voltage has
been plotted in Fig. 16. The plots are made at constant voltages from 95 V to 120 V in steps
of 5 V. It is observed that current is increasing with increase of air gap.
Table 5 variation of force with air gap (voltage constant)
Voltage
(V)
actual force (N)
95 72.81 53.6945.9843.3936.4330.6129.1718.8415.57
100 78.22 56.2449.2148.1339.9833.6530.9820.8017.00
105 86.72 61.4951.0652.4541.7735.0934.5622.9119.05
110 94.65 66.1253.6654.0545.1141.9436.6124.7220.34
115 101.3473.1759.8257.8546.5043.4137.2027.62 --
120 107.0478.9762.7859.8249.1246.1138.6729.48 --
air gap
(mm)
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
1 1.5 2 2.5 3 3.5 4 4.5 5
current(amp)
Air gap(mm)
voltage 95
voltage 100
voltage 105
voltage 110
voltage 115

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Figure 17 Variation of force with air gap (voltage constant)
The variation of force with air gap has been shown in Table 5. The variation of force with
air gap at constant voltage has been plotted in Fig. 17. It is observed that force is decreasing
with increase of air gap.
Figure 18 Average power loss/force/mm gap
The average power loss per unit force per unit mm air gap calculated at each air gap has
been calculated. The variation of average power loss per unit force per unit mm air gap with
air gap has been plotted in Fig. 18. It is observed that average power loss per unit force per
unit mm air gap is minimum and almost equal between the air gaps 2 mm and 4 mm.
9. CONCLUSIONS
In this paper one dimensional magnetic flux theory is used to find the theoretical force
between stator and rotor parts of AMTB. A test setup is designed and fabricated to find actual
force between stator and rotor. A leakage factor is introduced to find the predicted force. The
experiments are carried out at different air gaps from 1 mm to 5 mm in the steps of 0.5 mm.
0.00
20.00
40.00
60.00
80.00
100.00
120.00
1 2 3 4 5 6 7 8 9
Force(N)
Air gap(mm)
voltage 95
voltage 100
voltage 105
voltage 110
voltage 115
voltage 120
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
1.5 2 2.5 3 3.5 4 4.5 5
averagepowerloss/force/mm
air gap (mm)

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The percentage of error between theoretical force and actual force is ranging from 77 to 266
and the percentage of error between predicted force and actual force is ranging from 0.16 to
13.25. It shows that the predicted force is more closure to the actual force and is helpful in
design of AMTB. The variation of average leakage factor with air gap is minimum and almost
equal between the air gaps 2 mm to 4 mm and the average power loss per unit force per unit
mm air gap is also minimum and almost equal between the air gaps 2 mm and 4 mm. With
these results it may be concluded that the optimum air gap between stator and rotor from 2
mm to 4 mm.
REFERENCES
[1] Allaire PE, Mikula A, Banerjee BB, Lewis DW, ImlachJ.Design and test of magnetic
thrust bearing. J Franklin Inst 1989;326(6):831–847
[2] Groom NJ, Bloodgood VD. A comparison of analytical and experimental data for a
magnetic actuator, NASA-2000-tm210328; 2000.
[3] BloodgoodJr VD, Groom NJ, Britcher CP. Further development of an optimal design
approach applied to axial magnetic bearings. NASA-2000-7ismb-vdb, 2000.
[4] Rao. J. S., Tiwari. R., Optimum Design and Analysis of thrust Magnetic Bearings using
Multi objective Genetic Algorithms, International Journal for Computational methods in
Engineering Sciences and Mechanics, 9;223-245,2008
[5] David. C., Meeker, Eric. H., Myounggyu. D., Noh. An Augmented circuit Model for
Magnetic Bearings including Eddy currents, Fringing, and leakage. IEEE Transactions
32(4) July 1996.
[6] Bekinal et al, Permanent magnet thrust bearing: Theoretical and practical results, Progress
in Electromagnetic Research B,56,269-287,2013
[7] Rao J. S., Kondaiah V. V. and Rao V. V. S., Validation of thrust capacity of active
magnetic bearing, International Journal of Engineering Research ,vol. 3, 108-112,2014
[8] Mr.Vijay Shankar A Finite Difference Approach To Pressure Distribution On Fixed Pad
Thrust Bearing Under Isothermal Condition. International Journal of Mechanical
Engineering and Technology, 8(7), 2017, pp. 1837–1843.
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