design and analysis the magnetic levitation of the ball using J-Mag designer and optimize the system importing result to the Matlab

1. DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
FACULTY OF ENGINEERING
UNIVERSITY OF RUHUNA
ME 6302 Automatic Control Engineering
ASSIGNMENT NO 01
(DESIGN AND ANALYSIS A MAGNETIC LEVITATION SYSTEM
USING J-MAG DESIGNER AND MATLAB)
Diameter of the ball : 22mm
Selected permanent magnet : N50 (Neomax reversible)
NAME : WEERASINGHA P.K.C.P.
REG NO : EG/2014/2566
DATE : 03/09/2017

2. ABSTRACT
Overcoming the grip of the Earth’s gravity has been a major challenge for years. However, the
work of scientists and engineers who have found many ways to levitate a variety of objects is
being applied in the field of transportation in several countries. Maglev is the means of floating
one magnet over another. According to a theorem attributed to Earnshaw, it is impossible to
achieve static levitation using any combination of fixed magnets and electric charges. There are,
however, ways to levitate by getting around the assumptions of the theorem. Magnetic levitation
employs diamagnetism. In this report consist with magnetic levitation experiment for 350g ball
from ss400 material. From −3𝐴 to +3𝐴 current with 0.25𝐴 step and design for 25 cases. Using
magnetic levitation result from J-Mag Designer, analysis with MATLAB.

3. ii
ACKNOWLEDGEMENT
Under the module ME6302 automatic control engineering, we follow the control system
behavior magnetic lavation is one of this control system. As an assignment completed this report.
It is very pleasure to have the experiences and guidance during completing of assignment task
Dr. Buddika Annasiwaththa, lecturer at Department of Mechanical and Manufacturing
Engineering, Faculty of Engineering Ruhuna. We would like to give our gratitude for leading us
by teaching how to do a project according to the procedure. I have taken efforts in this assignment
with the help of many persons. However, it would not have been possible without the kind
support and help of them. I would like to extend our sincere thanks to Dr. Buddika Annasiwaththa
and all others who helped us.
Thank you,
Weerasingha P.K.C.P.
EG/2014/2566
Faculty of Engineering
University of Ruhuna.

4. iii
TABLE OF CONTENTS
Abstract..........................................................................................................................................i
Acknowledgement........................................................................................................................ii
Table of Contents ....................................................................................................................... iii
List of tables .................................................................................................................................v
List of figures ..............................................................................................................................vi
1 Magnetic levitation system in control engineering ...............................................................1
1.1 Magnetic Levitation Defined .........................................................................................1
1.2 Introduction....................................................................................................................1
1.3 Assignment Task............................................................................................................2
1.4 Manufacturer details for component in magnetic levitation system..............................2
2 Design a simple zero power controlled magnetic levitation system .....................................4
2.1 2D model........................................................................................................................4
2.2 3D model........................................................................................................................5
3 Simulation .............................................................................................................................5
3.1 J-mag Designer ..............................................................................................................5
3.2 Simulation process.........................................................................................................6
3.3 Plotting graph...............................................................................................................10
3.4 The zero-power levitation air gap (𝒛𝟎) of the SS400 ball ...........................................12
4 Curve fitting tool in MATLAB R2016a..............................................................................18
4.1 Force equation..............................................................................................................18
4.2 Calculating constant and Root Mean Square Error (RMSE).......................................20
4.3 Calculation of theoretical force....................................................................................21
4.4 Difference between calculated force and J-Mag Designer force.................................27
5 Magnetic flux density plot...................................................................................................33
5.1 −𝟑 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap ...........................................................................................33
5.2 +𝟑 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap ...........................................................................................34
5.3 𝟎 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap ..............................................................................................34
5.4 −𝟑 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap ...........................................................................................35

5. iv
5.5 +𝟑 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap ...........................................................................................35
5.6 𝟎 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap (zero power levitation air gap (𝒛𝟎)) ....................................36
6 Simulation details................................................................................................................36
6.1 Mesh.............................................................................................................................36
7 Computer characteristics.....................................................................................................38
8 Linearize the equation .........................................................................................................38
9 Discussion ...........................................................................................................................40
References ..................................................................................................................................42

6. v
LIST OF TABLES
Table 1. AWG copper wire selection ...........................................................................................3
Table 2. Calculate the total resistance and length of copper coil .................................................8
Table 3. Created cases for the design .........................................................................................10
Table 4. Relationship between force and airgap length..............................................................13
Table 5. J-Mag data ....................................................................................................................14
Table 6. Calculated theoritical force...........................................................................................21
Table 7. Time taken to complete each case................................................................................37

7. vi
LIST OF FIGURES
Figure 1. magnetic levitation system model.................................................................................1
Figure 2. N50 1/2"x1" Neodymium Rare Earth Cylindrical Magnet...........................................2
Figure 3. 0.65 mm copper wire.....................................................................................................3
Figure 4. 2D drawing of system ...................................................................................................4
Figure 5. 3D model.......................................................................................................................5
Figure 6. 1st
round of coil .............................................................................................................6
Figure 7. 2nd
round of coil ............................................................................................................7
Figure 8. Electromagnetic circuit .................................................................................................8
Figure 9. After creating mesh of model........................................................................................9
Figure 10. (a) Quantity of mesh given by J-Mag Designer, (b) Quality of mesh of model .......10
Figure 11. The graph of Length Vs. Force acting on Y axis......................................................11
Figure 12. The graph of Length Vs. Force acting on Y axis with reference line as weight of ball
....................................................................................................................................................12
Figure 13. Surface fitting graph for force equation....................................................................19
Figure 14. Surface fitting graph for calculated force..................................................................26
Figure 15. Difference between absolute force and calculated force...........................................32
Figure 16. Flux density in the model −3 𝐴 with 1.0 𝑚𝑚 air gap ..............................................33
Figure 17. Flux density in the model +3 𝐴 with 1.0 𝑚𝑚 air gap ..............................................34
Figure 18. Flux density in the model 0 𝐴 with 1.0 𝑚𝑚 air gap .................................................34
Figure 19. Flux density in the model −3 𝐴 with 4.3 𝑚𝑚 air gap ..............................................35
Figure 20. Flux density in the model +3 𝐴 with 4.3 𝑚𝑚 air gap ..............................................35
Figure 21. Flux density in the model 0 𝐴 with 4.3 𝑚𝑚 air gap .................................................36
Figure 22. Linearization of equation (C)....................................................................................39
Figure 23. cure fitting for J-Mag data with polynomial function...............................................40

8. 1
1 MAGNETIC LEVITATION SYSTEM IN CONTROL ENGINEERING
1.1 Magnetic Levitation Defined
Magnetic levitation is cited on Microsoft Academic Search as follows: “magnetic levitation is a
technique to suspend an object without any support other than that of a magnetic field”. The
definition goes on to state, “In a simple configuration, an electromagnet is used to lift up a
ferromagnetic object and the gravity pulls down the object”. Stated another way, current passed
through electromagnets creates an attractive magnetic levitation force. Magnetic levitation is also
referred to as magnetic suspension[1].
1.2 Introduction
Magnetic levitation is becoming widely applicable in magnetic bearing, high-speed ground
transportation (bullet trains), vibration isolation, flywheels and levitation melting, household
fixtures and decorations, magnetic levitation in universities, etc. magnetic bearing support radial
and thrust loads in rotating machinery. Also, magnetic suspension generates levitation action is
rectilinear motion devices such as high-speed ground transportation system[1].
The portable magnetic levitation system (MagLev), shown in figure 1, design and analysis using
J-mag Designer and MATLAB software and plot the graph between the current flowing in
copper coil, air gap between ss400 steel ball and the force action on ball when the ball levitates
in air by the magnetic force generated by the
• Electromagnet
• Permanent magnet
• Or both permanent and electromagnet
Figure 1. magnetic levitation system model

9. 2
1.3 Assignment Task
Under the module ME 6302 automatic control engineering, use computer aided design and
analysis tools (J‐mag Designer and MATLAB) to identify the relationship between force, current
and air gap of a simple zero power controlled magnetic levitation system.
1.4 Manufacturer details for component in magnetic levitation system
In this designing process first calculate the radius of the ball and dimension of the permanent
magnet which is available in the market.
1. ss400 ball
• Radius of the ss400 ball =
70+150
10
= 22 𝑚𝑚
2. Magnet;
• Radius = 1/2 𝑖𝑛𝑐ℎ
• Height = 1 𝑖𝑛𝑐ℎ
Figure 2. N50 1/2"x1" Neodymium Rare Earth Cylindrical Magnet[2]
Custom details
• Material : Sintered NdFeB. Grade N50
• Gauss Ratings : 14,200 Gauss

10. 3
• Pulling Force : 51 lbs.
• Pole Orientation : Axially magnetized. Poles on the flat surfaces.
• Coating : Ni + Cu + Ni triple layer plated, the best coating available
• Tolerance : The tolerances of all the dimensions are +/- 0.002in with coating.
Magnet available in CMS magnets
3. Copper coil
In this design need to flow maximum 5A current to the its coil. But the duration of the current
flowing is very small due to that reason the heat dissipation from the coil is less. Then consider
American Wire Gauge (AWG) to get suitable diameter for the copper coil wire[3].
Table 1. AWG copper wire selection
AWG gauge Conductor
diameter (mm)
Ohms per
Km
Maximum amps
for chassis
wiring
Maximum amps
per power
transmission
22 0.64516 52.9392 7 0.92
23 0.57404 66.7808 4.7 0.729
The 5A between AWG gauge number 22 and 23, therefore used gauge number 22 to the design.
Figure 3. 0.65 mm copper wire[4]
Custom details
• Metric Wire Size : 0.65𝑚𝑚

11. 4
• Net Wire Weight : 50 𝑔 𝑝𝑒𝑟 𝐶𝑜𝑖𝑙
• Length on spool : 16.5 𝑚
• Cross Sectional Area : 0.33187 𝑚𝑚2
• Temperature Rating : 212 °𝐶 (Class 'H')
• Voltage Break Through : 11000 𝑉 Minimum Dielectric Strength
• Grade : Dual Coat
• Base Coating : Polyester Imide
• Top Coating : Polyamide Imide
• IEC Standards : IEC 60317-13 / DIN 46416-7
Copper coil available in eBay
2 DESIGN A SIMPLE ZERO POWER CONTROLLED MAGNETIC LEVITATION
SYSTEM
2.1 2D model
Based on the above calculation create a 2D sketch for the magnetic levitation system.
Figure 4. 2D drawing of system

12. 5
✓ The plastic bobbin is design for 3D printing and its vertical axis thickness of 0.50 𝑚𝑚
✓ For the bolt, it is design for custom design and it can create from machining by ss400
bar.
✓ The initial air gap between magnet lower surface and top surface of ss400 ball is
10.0 𝑚𝑚.
2.2 3D model
Then using 2D sketch created 3D solid model using SolidWorks R2016a
✓ Take axis of the bolt is Y (then ball levitate through the Y axis).
✓ Take origin as reference point of model.
✓ For the levitate propose ss400 ball set to motion region from magnet lower surface
1.0 𝑚𝑚 to 10.0 𝑚𝑚.
✓ Create ACIS.sat file format for import to J-mag designer.
Figure 5. 3D model
3 SIMULATION
Basically, used J-mag Design 16.0 to magnetic simulation and create graph between force acting
on the ball and the airgap length.
3.1 J-mag Designer
Special simulation tool used for simulate the
✓ Magnetic
✓ Thermal

13. 6
✓ Structural
✓ Electric
✓ Thermal stress
✓ Transformer
In this deign we used magnetic transient analysis, because it is need to calculate force and air
gap of the model. It can directly import solid works file to the J-mag designer.
3.2 Simulation process
• Step 1
After importing the solid model to the J-mag designer, define material for each part of the design;
• Ball and bolt : ss400
• Bobbin : Plastic
• Coil : Copper
• Magnet : Neomax reversible N50
for the ball and bolt should be allowed the eddy current because when flux is acting on the metal
surface its create eddy current inside it. But coil surface is coated then it’s no need to allow eddy
current.
• Step 2
In the system has electromagnet, for create electromagnet used copper coil. The coil resistance
should be calculated. The coil has 700 turn each one round has 50 turns, then it has 14 rounds.
• 1st
round of copper wire
the diameter between 1st
round is sum of the bobbin
outer diameter and two times of radius of the coil, it is
shown figure 6.
Diameter in one coil turn = 𝜋𝑑
= 𝜋 𝑥 22.65 𝑚𝑚
Figure 6. 1st
round of coil

14. 7
= 71.1570736 𝑚𝑚
Bobbin has 50 turns in one round. Then,
Total length of one round = 50 𝑥 71.1570736 𝑚𝑚
= 3557.85368 𝑚𝑚
• 2nd
round of copper wire
The diameter between 2nd
round is sum of the
bobbin outer diameter and two times of radius
of the coil, it is shown figure 7.
Diameter in one coil turn = 𝜋𝑑
= 𝜋 𝑥 (22 𝑥 1 + 1) 𝑥 0.65 𝑚𝑚
= 59.53318079 𝑚𝑚
Bobbin has 50 turns in one round. Then,
Total length of one round = 50 𝑥 59.53318079 𝑚𝑚
= 2976.659039 𝑚𝑚
According to the figure 7, the total length of each one round can be calculated 2nd
round to 20th
round by taking n = 1, 2, …, 19.
The resistance of the unit lent of the copper wire of 22 AWG gauge in metric standard is 52.9392
Ohms per Km.
Then further calculation is shown in table 2 and it gives
Total length of coil = 94.24777961 𝑚
Total resistance of coil = 4.989402054 𝛺
Figure 7. 2nd
round of coil

15. 8
Table 2. Calculate the total resistance and length of copper coil
Number of
round
Diameter Length of one
turn
Number of turn in
each round
Total length of one
round
1 22.65 71.15707360 50 2772.455517
2 23.95 75.24114405 50 2976.659039
3 25.25 79.32521450 50 3180.862562
4 26.55 83.40928495 50 3385.066084
5 27.85 87.49335540 50 3589.269607
6 29.15 91.57742585 50 3793.473129
7 30.45 95.66149630 50 3997.676652
8 31.75 99.74556675 50 4201.880174
9 33.05 103.8296372 50 4406.083697
10 34.35 107.9137077 50 4610.287219
11 35.65 111.9977781 50 4814.490742
12 36.95 116.0818486 50 5018.694264
13 38.25 120.1659190 50 5222.897787
14 39.55 124.2499894 50 5427.101309
Total length(mm) 68392.47207
Total length(m) 68.39247207
Ohms per Km 52.9392
Total resistance (𝜴) 3.620642757
The coil should be contained the number of turn and resistance of the coil.
Figure 8. Electromagnetic circuit
• Step 3
The initial condition should be applied on the model. There are three basic conditions,
1) FEM coil
2) Translation motion of ball
3) Nodal force action on the ball

16. 9
When applied the FEM coil it should be link to the created circuit in step 2 and also give to the
initial starting point of current flowing through the coil.
When the translation motion applied on the ball is tend to move through the Y axis. The motion
is taken by 0.1 𝑚𝑚 step taking throughout the whole moving distance of ball. As an example, in
this model we take 1𝑚𝑚 to 5 𝑚𝑚 for motion region. If it has 0.1 𝑚𝑚 step total steps of 51 with
first step. In this model applied downward motion and therefore the graph of force on Y Axis
Vs. length (mm) are shown in figure 11.
Nodal fore link to the translation motion.
• Step 4
The accuracy of the calculation depends on the mesh of the model, when it has small mesh
element the accuracy of the output is higher. For that model used 1.0 𝑚𝑚 and air element which
contain in control volume to the calculation is 2.0 𝑚𝑚.
Figure 9. After creating mesh of model
This model has
✓ 268 754 elements
✓ 52 676 nodes
✓ Total element of 321 430

17. 10
(a) (b)
Figure 10. (a) Quantity of mesh given by J-Mag Designer, (b) Quality of mesh of model
• Step 5
In this model, used different current value and it is starting −3𝐴 to +3𝐴 with 0.25𝐴 difference
then it has 25 different current value. Each one take as case, then it has 25 all cases. 13th
case
with 0 𝐴 current.
Table 3. Created cases for the design
case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
I(A)
-3.00
-2.75
-2.50
-2.25
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.3 Plotting graph
One of main output of the simulation is the graph of force action on ball through Y Axis Vs.
length through Y axis from initial point which has 1.00 𝑚𝑚 away from magnet. First consider
case 13th
, and it should be cut within 3 𝑚𝑚 to 6 𝑚𝑚 region. And after 1st
and 2nd
cases are run
because of the get clear idea about the saturation of the model with the magnetic flux. If it is
saturated, should be change the physical parameter of the model. These parameter is.
✓ Number of turn of coil
✓ Magnet diameter
✓ Diameter of bolt, etc.

18. 11
Figure 11. The graph of Length Vs. Force acting on Y axis
The weight of the ball =
4
3
𝑥 𝜋 𝑥 𝑟3
𝑥 𝜌 𝑥 𝑔
=
4
3
𝑥 𝜋 𝑥 (0.022)3
𝑥 7860 𝑥 9.81
= 3.439126932 𝑁

19. 12
3.4 The zero-power levitation air gap (𝒛 𝟎) of the SS400 ball
It can apply reference line to the graph to get length when force is 3.43913 𝑁.
Figure 12. The graph of Length Vs. Force acting on Y axis with reference line as weight of ball
It also has these types of graph for all 25 cases. When ball have 3.43913 𝑁 and length of X axis
of reference value for the Y is given by table number 4.
From 1st case to 22nd
case when force is 3.43913 N, curve intersect between 3.00 mm to 6.00mm
region according to the above table. But after 22nd
case intersecting point goes to right hand side
to X axis.
Using these 25 graphs get the reading of airgap length (mm) where 0.1mm each step from 1.00
mm to 6.00 mm and force of each step. These step is previously defined as 51 steps for total
motion region.

20. 13
Table 4. Relationship between force and airgap length
Case number Length of graph (mm) Airgap length (mm)
1 2.86073 3.86073
2 2.82835 3.82835
3 2.80681 3.80681
4 2.79684 3.79684
5 2.79862 3.79862
6 2.81229 3.81229
7 2.83790 3.83790
8 2.87467 3.87467
9 2.92925 3.92925
10 3.00363 4.00363
11 3.07115 4.07115
12 3.14929 4.14929
13 3.24947 4.24947
14 3.36787 4.36787
15 3.49870 4.49870
16 3.65361 4.65361
17 3.82482 4.82482
18 4.00651 5.00651
19 4.20457 5.20457
20 4.43227 5.43227
21 4.67430 5.67430
22 4.93335 5.93335
23 - -
24 - -
25 - -
After the simulation process, it created the 25 graphs. The graph has
✓ X axis as length (it gives 0 𝑚𝑚 to 5 𝑚𝑚 in positive direction; because initially set the
motion direction as the −𝑌 direction).
✓ Y axis gives the Force in Y direction
Using that graph get force value in each 0.1 𝑚𝑚 step of each one of graph. Then for every 25
graphs contain this force value and it can create 51 𝑥 25 matrix
✓ 25 - −3𝐴 to +3𝐴 with 0.25𝐴 time step.
✓ 51 – 1.0 𝑚𝑚 to 6.0 𝑚𝑚 with 0.1 𝑚𝑚 time step includes with 1.0 𝑚𝑚.

21. 14
Table 5. J-Mag data
Length(mm)
AirGap(mm)
Current (A)
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
0
1
26.3815772
8
26.1721058
4
26.0097762
25.8977232
8
25.8394772
6
25.8365702
7
25.8899916
6
26.0003596
5
26.1681329
9
26.3935718
6
26.6768276
5
27.0181432
4
27.4177514
9
27.8757278
5
28.3921707
7
28.9672057
4
29.6008564
9
30.2931404
31.0441124
31.8535710
2
32.7215943
2
33.6482333
7
34.6330563
4
35.6761735
36.7778001
9
0.1
1.1
24.4528786
5
24.2612225
7
24.1154810
1
24.0186861
7
23.9740027
5
23.9828555
4
24.0462922
6
24.1647356
8
24.3386843
2
24.5683966
7
24.8539803
25.1956899
5
25.5937258
1
26.0481635
9
26.5591119
1
27.1266866
7
27.7508866
4
28.4317394
4
29.1692953
6
29.9633775
30.8140490
4
31.7213599
3
32.6849164
9
33.7047866
4
34.7811698
9
0.2
1.2
22.7322899
1
22.5540402
4
22.4205777
4
22.3347272
22.2994619
8
22.3160025
4
22.3857067
1
22.5086126
6
22.6852673
6
22.9159450
4
23.2007408
9
23.5398822
7
23.9335900
8
24.3819053
3
24.8849360
8
25.4427907
3
26.0554692
3
26.7229935
4
27.4454065
7
28.2225502
3
29.0544840
8
29.9412555
1
30.8824983
7
31.8782449
3
32.9287020
6
0.3
1.3
21.0704327
2
20.9053315
9
20.7838568
3
20.7087903
8
20.6829047
3
20.7072041
1
20.7832125
6
20.9107977
4
21.0904469
9
21.3224402
8
21.6069003
3
21.9440063
5
22.3339761
1
22.7768629
7
23.2727721
5
23.8217962
2
24.4239267
7
25.0791919
7
25.7876213
3
26.5490849
1
27.3636278
28.2313010
5
29.1517569
5
30.1250010
2
31.1512505
0.4
1.4
19.5015141
1
19.3479945
4
19.2370340
1
19.1712534
19.1532422
3
19.1839869
3
19.2649268
9
19.3958907
6
19.5773968
1
19.8096781
9
20.0928366
6
20.4270619
3
20.8125567
7
21.2493569
4
21.7375701
5
22.2772849
8
22.8684892
5
23.5111999
3
24.2054478
3
24.9511213
7
25.7482569
26.5968974
7
27.4967505
4
28.4477472
1
29.4501400
4
0.5
1.5
17.9888190
2
17.8467411
6
17.7461735
17.6896052
9
17.6795003
17.7167956
1
17.8027848
8
17.9373568
7
18.1209819
5
18.3538640
5
18.6361182
1
18.9679248
4
19.3494691
2
19.7807955
20.2620120
1
20.7931934
21.3743216
1
22.005424
22.6865214
9
23.4175187
9
24.1984385
25.0293264
3
25.9099164
5
26.8401089
1
27.8201652
2
0.6
1.6
16.9124518
7
16.7767282
4
16.6814098
3
16.6289244
7
16.6216513
9
16.6603621
2
16.7465242
16.8798446
6
17.0608270
3
17.2896855
17.5665141
9
17.8914678
9
18.2647580
8
18.6864089
6
19.1565197
2
19.6751705
8
20.2423361
9
20.8580479
8
21.5223233
7
22.2350818
9
22.9963381
2
23.8061344
9
24.6642379
5
25.5705120
6
26.5252238
3
0.7
1.7
15.7635129
8
15.6351761
5
15.5462827
7
15.4991252
15.4959457
6
15.5374769
5
15.6252068
4
15.7587413
6
15.9385948
2
16.1649595
3
16.4379401
7
16.7576928
8
17.1243869
2
17.5380640
8
17.9988320
5
18.5067414
3
19.0617757
5
19.6639645
6
20.3133259
6
21.0097905
21.7533657
1
22.5440933
5
23.3817656
2
24.2662172
4
25.1977194
4
0.8
1.8
14.6546848
6
14.5327767
2
14.4493801
1
14.4066874
14.4068156
9
14.4504394
3
14.5391207
2
14.6723368
7
14.8506214
6
15.0741539
3
15.3430334
8
15.6574255
7
16.0174758
1
16.4232207
9
16.8747846
8
17.3721912
5
17.9154376
1
18.5045448
1
19.1395366
4
19.8203503
1
20.5469862
9
21.3194917
6
22.1376689
7
23.0013398
8
23.9107726
4
0.9
1.9
13.6061874
4
13.4910864
6
13.4135854
4
13.3757614
8
13.3796666
13.4259497
6
13.5160381
7
13.6494755
4
13.8267767
7
14.0481145
7
14.3135509
3
14.6232493
4
14.9773735
15.3759337
1
15.8190526
6
16.3067735
4
16.8390624
17.4159528
2
18.0374759
18.7035773
3
19.4142389
9
20.1695189
20.9692317
9
21.8131892
3
22.7016506
5
1
2
12.4488549
8
12.3411888
9
12.2702303
3
12.2380045
7
12.2464040
1
12.2960082
3
12.3883843
1
12.5228776
2
12.7000510
6
12.9200653
3
13.1829911
4
13.4889773
3
13.8381796
6
14.2306250
4
14.6664143
6
15.1455832
2
15.6681103
8
16.2340212
1
16.8433409
5
17.4960236
4
18.1920559
18.9314758
8
19.7141404
8
20.5398207
6
21.4087849
1

22. 15
1.1
2.1
11.6160995
9
11.5129006
1
11.4455294
7
11.4159411
7
11.4259683
9
11.4761764
3
11.5680058
8
11.7009078
9
11.8753870
4
12.0915999
9
12.349619
12.6495852
7
12.9916482
2
13.3758286
6
13.8022373
5
14.2708941
3
14.7817832
15.3349314
6
15.9303631
2
16.5680381
6
17.2479391
1
17.9701017
6
18.7344031
19.5405933
8
20.3889458
4
1.2
2.2
10.5624307
10.4664680
1
10.4055053
8
10.3814395
9
10.3960076
7
10.4496417
3
10.5439376
2
10.6781936
8
10.8529406
3
11.0683078
1
11.3243816
4
11.6213209
3
11.9592369
2
12.3381531
7
12.7581854
9
13.2193572
2
13.7216426
6
14.2650648
4
14.8496564
5
15.4753820
1
16.1422143
6
16.8501955
5
17.5992098
6
18.3890099
7
19.2198402
8
1.3
2.3
9.82951422
3
9.73679055
2
9.67826670
5
9.65574089
4
9.67088542
2
9.72416472
9
9.81707122
6
9.94892829
2
10.1202699
8
10.3312085
7
10.5818224
6
10.8722807
6
11.2026806
9
11.5730567
4
11.9835150
1
12.4340736
6
12.9247045
6
13.4554371
6
14.0263027
14.6372715
2
15.2883180
7
15.9794723
1
16.710644
17.4815671
3
18.2924852
8
1.4
2.4
9.10309032
1
9.01405011
1
8.95842208
6
8.93796296
4
8.95428308
5
9.00781713
1
9.10001055
4
9.23020377
2
9.39891599
2
9.60625137
9.85228500
8
10.1371784
5
10.4610244
3
10.8238519
11.2257771
3
11.6668056
3
12.1469110
7
12.6661300
8
13.2244954
6
13.8219776
4
14.4585495
3
15.1342386
15.8489701
7
16.6024681
7
17.3949709
6
1.5
2.5
8.4815802
8.39556778
5
8.34217068
8.32308737
9
8.33987212
8.39293572
4
8.48373068
3
8.61154036
4
8.77690566
8
8.97993630
5
9.22068656
6
9.49930608
7
9.81590908
3
10.1705280
5
10.5632525
4
10.9940891
7
11.4630199
4
11.9700689
3
12.5152739
7
13.0986113
8
13.7200486
1
14.3796140
8
15.0772451
7
15.8126595
6
16.5860875
8
1.6
2.6
7.85846475
6
7.77517566
2
7.72378919
7.70596003
4
7.72319858
8
7.77587779
5
7.86538969
3
7.99106477
6
8.15341591
2
8.35254618
8.58851479
9
8.86145159
2
9.17147513
5
9.51861884
4
9.90296575
10.3245187
9
10.7832612
2
11.2792213
11.8124346
12.3828829
9
12.9905307
5
13.6354052
8
14.3174574
1
15.0363968
9
15.7924486
4
1.7
2.7
7.42260414
7.34004879
9
7.28867126
7
7.27006447
4
7.28567071
6
7.33587835
1
7.42208260
4
7.54360679
4
7.70092966
5
7.89415779
6
8.12334812
9
8.38865623
4
8.69016947
7
9.02791532
1
9.40198934
9
9.81240088
1
10.2591284
10.7421968
3
11.2616431
5
11.8174544
1
12.4095948
13.0380862
7
13.7028895
4
14.4037105
15.1407681
1
1.8
2.8
6.95735277
5
6.87630340
7
6.82572913
8
6.80717386
6
6.82205073
7
6.87073534
5
6.95455545
7.07288889
8
7.22618883
7
7.41455872
7
7.63805876
1
7.89685119
4
8.19098625
3
8.52050640
4
8.88550559
5
9.28599047
9
9.72192363
4
10.1933446
5
10.7002881
9
11.2427486
9
11.8206826
8
12.4341189
8
13.0830249
3
13.7671047
2
14.4865676
5
1.9
2.9
6.46651685
6
6.38767991
5
6.33863961
2
6.32087694
4
6.33577490
2
6.38372765
2
6.46596715
8
6.58191243
8
6.73200705
4
6.91636342
7.13501936
6
7.38812129
1
7.67572745
7.99790037
1
8.35472814
5
8.74619036
9.17225652
1
9.63298955
8
10.1284086
9
10.6585081
2
11.2232447
2
11.8226424
6
12.4566833
4
13.1250627
4
13.8279871
2
2
3
6.05563618
9
5.97758853
5.92867909
5
5.91035265
7
5.92395119
2
5.96985370
6
6.04927186
3
6.16163944
8
6.30740172
6.48663765
9
6.69939918
3
6.94583271
3
7.22601255
2
7.53996354
9
7.88778916
6
8.26947405
2
8.68499487
5
9.13439061
9
9.61769250
5
10.1348947
8
10.6859572
6
11.2708988
9
11.8897048
12.5420755
7
13.2282114
7
2.1
3.1
5.64794578
5.57088880
1
5.52234319
2
5.50370849
5.51629669
6
5.56044405
6
5.63741807
3
5.74658932
5
5.88843198
8
6.06299811
7
6.27034981
9
6.51063249
1
6.78392910
2
7.09023975
2
7.42966655
8
7.80220599
1
8.20784443
2
8.64660037
4
9.11850690
2
9.62356399
7
10.1617363
9
10.7330393
5
11.3374616
8
11.9747126
1
12.6449739
1
2.2
3.2
5.31975473
1
5.24309484
5.19430767
8
5.17474963
7
5.18573991
3
5.22763080
4
5.30155051
3
5.40697642
8
5.54434951
4
5.71374585
4
5.91519713
7
6.14884094
8
6.41476019
6.71298350
9
7.04359112
7
7.40656667
6
7.80190578
7
8.22963967
1
8.68978605
5
9.18234647
1
9.70728557
7
10.2646230
7
10.8543505
8
11.4761729
4
12.1302738

23. 16
2.3
3.3
4.94288452
2
4.86729749
2
4.81899201
8
4.79929391
6
4.80947232
4.84986165
8
4.92163584
7
5.02421095
1
5.15803050
2
5.32317169
3
5.51968040
8
5.74768315
4
6.00725186
2
6.29841736
7
6.62125985
6
6.97575258
8
7.36189848
8
7.77973640
9
8.22927854
8
8.71052713
8
9.22344527
3
9.76804428
2
10.3443395
1
10.9520260
8
11.5912813
4
2.4
3.4
4.64721012
5
4.57215092
6
4.52378505
3
4.50340286
4.51224393
7
4.55064508
5
4.61972742
1
4.71895577
3
4.84875016
5.00918563
2
5.20030451
6
5.42223514
9
5.67504476
7
5.95877724
4
6.27350831
9
6.61920104
3
6.99585694
6
7.40351775
8
7.84219964
3
8.31190100
1
8.81258981
5
9.34426753
1
9.90695545
2
10.5003639
11.1246498
7
2.5
3.5
4.35722548
3
4.28257792
7
4.23404464
6
4.21287401
4.22031023
3
4.25669359
9
4.32305871
8
4.41892984
5
4.54472319
5
4.70051304
9
4.88633175
6
5.10230088
3
5.34848543
9
5.62492840
8
5.93170621
8
6.26878976
7
6.63617043
8
7.03389608
5
7.46196721
7.92040520
7
8.40916262
3
8.92825115
3
9.47768550
5
10.0571916
4
10.6669091
6
2.6
3.6
4.06639946
9
3.99198723
3.94312127
7
3.92102632
4
3.92691517
7
3.96112229
4
4.02466754
5
4.11708409
1
4.23877687
7
4.38981414
8
4.57024103
4.78017392
7
5.01967788
6
5.28879435
3
5.58760187
1
5.91606806
2
6.27419398
7
6.66201761
8
7.07954352
4
7.52679341
7
8.00372618
2
8.51033915
1
9.04664924
2
9.61239654
7
10.2077016
2
2.7
3.7
3.81119030
5
3.73728024
3.68838469
3
3.66568700
1
3.67038001
5
3.70279035
1
3.76393999
3.85334805
5
3.97141711
5
4.11821667
7
4.29379044
4
4.49825652
5
4.73166452
7
4.99405316
2
5.28550822
7
5.60598807
7
5.95549376
8
6.33406093
1
6.74170181
7.17843299
3
7.64421125
6
8.13904391
5
8.66294578
5
9.21566821
9
9.79731654
5
2.8
3.8
3.57555952
4
3.50272899
4
3.45440433
6
3.43173406
8
3.43587967
1
3.46718145
3
3.52656667
2
3.61364366
9
3.72877933
3
3.87203354
4.04344708
5
4.24313726
1
4.47116134
2
4.72755313
5
5.01238993
3
5.32563077
7
5.66728382
9
6.03739119
1
6.43595237
7
6.86298734
7
7.31845439
5
7.80236115
7
8.31472318
5
8.85530351
4
9.42418615
9
2.9
3.9
3.35091135
4
3.27843319
3
3.22994579
3
3.20656654
3.20944619
5
3.23889413
5
3.29585563
7
3.37991348
1
3.49144932
8
3.63051725
3.79714611
6
3.99145715
5
4.21351332
6
4.46334970
2
4.74103400
1
5.04653336
9
5.37984837
9
5.74102496
3
6.13006460
1
6.54698454
7
6.99174593
7
7.46434777
3
7.96480834
4
8.49290718
1
9.04871539
3
3
4
3.18383871
5
3.11168758
4
3.06304655
1
3.03900603
3.04069446
9
3.06842863
3.12311862
9
3.20436936
5
3.31254057
7
3.44769138
1
3.60986738
1
3.7991804
4.01568047
7
4.25939366
8
4.53039753
8
4.82865694
4
5.15417958
8
5.50700870
8
5.88713758
2
6.29459683
6
6.72933654
8
7.19135564
7
7.68068074
2
8.19709395
2
8.74066113
9
3.1
4.1
2.96073796
1
2.88928187
2.84085483
6
2.81650967
2
2.81736013
2.84372244
1
2.89646233
9
2.97522653
7
3.08035092
6
3.21189430
1
3.36989061
2
3.55446655
4
3.76566644
2
4.00351428
8
4.26808351
5
4.55933302
2
4.87727647
5
5.22195750
4
5.59336202
4
5.99152061
1
6.41638689
7
6.86796307
6
7.34627092
7.85110830
4
8.38251884
3
3.2
4.2
2.75353888
3
2.68221484
8
2.63344722
6
2.60826487
1
2.60775938
9
2.63225409
2.68255926
1
2.75837249
8
2.86001742
4
2.98754346
4
3.14098421
3.32046459
3.5260202
3.75768856
3
4.01554312
3
4.29954569
1
4.60970613
4
4.94605820
6
5.30859709
8
5.69735644
6
6.11228825
8
6.55339287
1
7.02069125
9
7.51398899
4
8.03332005
1
3.3
4.3
2.60873379
9
2.53753009
7
2.48843171
2
2.46243924
1
2.46063766
5
2.48335245
6
2.53135611
8
2.60437004
4
2.70270589
2.82641447
5
2.97553298
8
3.15017421
4
3.35037675
4
3.57617757
9
3.82764708
9
4.10474479
4
4.40747195
3
4.73587802
3
5.08995610
9
5.46973912
5.87517372
4
6.30626225
1
6.76302643
1
7.24528602
7
7.75305988
6
3.4
4.4
2.43668970
4
2.36614806
1
2.31726642
1
2.29101691
7
2.28846448
6
2.30992341
6
2.35616546
5
2.42690539
5
2.52245388
6
2.64286000
2
2.78815619
2
2.95845724
9
3.15380591
4
3.37423378
3
3.61980901
2
3.89048687
3
4.18627730
4
4.50722537
1
4.85332397
5
5.22460696
2
5.62101630
2
6.04255469
6
6.48924630
5
6.96091933
1
7.45757999
9

24. 17
3.5
4.5
2.29853609
1
2.22785454
3
2.17839937
5
2.15111288
2
2.14705053
4
2.16653570
3
2.21030350
7
2.27809715
2
2.37021113
7
2.48669420
2
2.62758656
1
2.79300188
7
2.98297348
6
3.19753030
7
3.43673834
9
3.70056430
4
3.98902161
5
4.30215322
4
4.63993981
8
5.00242355
1
5.38955034
7
5.80132357
2
6.23776471
7
6.69870634
4
7.18414962
6
3.6
4.6
2.17431847
4
2.10397516
7
2.05444265
9
2.02664443
4
2.02161237
8
2.03967816
6
2.08153877
8
2.14696098
8
2.23623610
5
2.34940148
6
2.48649451
1
2.64763631
1
2.83285198
6
3.04217833
8
3.27567449
6
3.53330057
7
3.81507969
8
4.12104268
8
4.45117849
4
4.80553299
6
5.18404760
6
5.58672034
8
6.01357418
9
6.46444926
6
6.93934192
6
3.7
4.7
2.04260869
6
1.97251706
1
1.92281671
1
1.89440815
4
1.88832131
9
1.90486582
4
1.94475461
6
2.00773217
7
2.09408736
7
2.20387393
2.33712267
6
2.49394952
2
2.67437798
7
2.87843633
9
3.10619722
6
3.3576233
3.63272671
8
3.93153921
2
4.25404748
9
4.60030117
3
4.97024379
1
5.36387298
1
5.78121419
5
6.22211592
1
6.68656115
8
3.8
4.8
1.93766154
6
1.86771133
2
1.81775319
3
1.78866494
2
1.78146494
8
1.79647109
1.83434149
9
1.89487509
1
1.97833739
2.08477452
2
2.21421988
8
2.36678757
9
2.54251087
9
2.74141625
7
2.96356787
3
3.20892690
9
3.47750396
4
3.76934368
2
4.08442906
9
4.42280604
4.78441345
2
5.16925230
4
5.57735055
2
6.00856564
8
6.46286736
3.9
4.9
1.82783756
2
1.75831461
9
1.70839927
2
1.67894464
7
1.67094923
6
1.68472690
8
1.72092638
1.77934853
1
1.86025511
2
1.96369320
6
2.08969685
2
2.23837817
3
2.40976406
6
2.60387809
5
2.82078884
5
3.06045975
3.32290670
5
3.60816529
3.91621783
6
4.24710876
4
4.60078384
7
4.97724127
8
5.37650800
3
5.79844430
5
6.24301631
7
4
5
1.71909701
3
1.64993755
7
1.60000595
2
1.57013292
2
1.56130889
9
1.57385121
3
1.60836543
8
1.66468095
2
1.74305155
5
1.84352284
4
1.96613618
8
2.11099449
1
2.27811751
2.46753644
3
2.67931515
6
2.91341419
3.16985896
1
3.44867420
5
3.74984335
2
4.07341520
6
4.41932943
8
4.78759312
4
5.17822596
8
5.59109150
8
6.02615613
2
4.1
5.1
1.62646408
2
1.55737880
3
1.50715337
6
1.47659752
6
1.46669103
7
1.47774026
4
1.51034299
4
1.56433951
9
1.63997737
9
1.73730068
2
1.85634034
2
1.99719980
1
2.15990363
4
2.34449019
4
2.55101657
7
2.77944207
1
3.02978325
2
3.30206840
7
3.59629282
8
3.91250415
3
4.25063926
7
4.61069772
3
4.99270299
9
5.39653011
3
5.82213978
3
4.2
5.2
1.53375667
6
1.46487589
4
1.41448728
2
1.38337690
8
1.37251469
4
1.38221230
3
1.41303086
1.46484362
1
1.53788377
6
1.63219112
6
1.74780292
6
1.88481707
7
2.04325439
5
2.22315435
2.42456786
5
2.64745947
9
2.89185041
8
3.15776590
1
3.44519676
6
3.75419173
4.08468582
4.43668627
2
4.81021255
1
5.20514607
5
5.62144011
8
4.3
5.3
1.45415515
6
1.38559260
6
1.33517314
8
1.30366503
7
1.29202641
2
1.30056301
9
1.32981984
1
1.37967771
7
1.45036432
6
1.541923
1.65438677
2
1.78785417
3
1.94234884
4
2.11790265
1
2.31457268
2.53231957
1
2.77116677
7
3.03113975
1
3.31223284
1
3.61448658
6
3.93784228
4
4.28230061
4
4.64788450
8
5.03448569
7
5.44204395
4.4
5.4
1.37955073
1.31127637
3
1.26080863
1
1.22889503
1
1.21647710
4
1.22386279
2
1.25158991
7
1.29953624
3
1.36792875
1.45680815
8
1.56620621
6
1.69622549
1.84688278
3
2.01821129
3
2.21026663
6
2.42301323
2.65647226
2
2.91067229
5
3.18559931
7
3.48129901
4
3.79771538
5
4.13484775
4
4.49272144
6
4.87122108
5.27029631
4.5
5.5
1.30811914
2
1.24008795
1
1.18952192
7
1.15715076
4
1.14391178
3
1.15011847
4
1.17624992
9
1.22224123
3
1.28829859
5
1.3744558
1.48075439
1.60728852
5
1.75407578
9
1.92115169
3
2.10856863
1
2.31628853
5
2.54434221
4
2.79274559
8
3.06149143
1
3.35062540
5
3.66009139
5
3.98988934
6
4.34004018
8
4.71043973
6
5.10103097
4.6
5.6
1.24021167
8
1.17230705
8
1.12153387
8
1.08860688
7
1.07445443
4
1.07939463
7
1.10387616
2
1.1478566
1.21152887
6
1.29493623
3
1.39811565
7
1.52116059
4
1.66408604
1.82692332
9
2.00972394
5
2.21246250
1
2.43516128
8
2.67783765
9
2.94048070
6
3.22313829
6
3.52575563
9
3.84833437
7
4.19089346
1
4.55333642
9
4.93559364
3

25. 18
4.7
5.7
1.18576350
7
1.11809594
1
1.06724243
8
1.03389925
1
1.01898343
8
1.02279875
8
1.04581355
1.08796474
4
1.14945009
9
1.23031177
7
1.33058264
5
1.45035481
5
1.58964832
2
1.74848689
3
1.92693129
6
2.12494595
7
2.34255731
4
2.57978297
1
2.83660966
3
3.11308497
3.40915884
4
3.72482718
9
4.06011136
4
4.41491608
8
4.78916592
3
4.8
5.8
1.12698535
7
1.05992028
1
1.00935686
7
0.97597326
1
0.96067491
1
0.96376007
3
0.98568294
4
1.02638729
2
1.08606697
1.16476296
9
1.26250630
9
1.37938885
7
1.51543020
3
1.67065676
1.84512285
4
2.03879669
7
2.25169912
7
2.48384885
3
2.73524052
3
3.00591696
9
3.29582524
6
3.60496468
8
3.93335155
6
4.28090502
4
4.64754829
9
4.9
5.9
1.06586929
7
0.99900571
9
0.94834162
3
0.91453763
1
0.89848775
2
0.90050649
9
0.92097906
4
0.95991204
1.01747924
6
1.09372067
1.18866794
6
1.30240750
8
1.43495493
9
1.58634447
1.75662818
4
1.94577298
5
2.15380462
7
2.38073399
6
2.62655668
5
2.89132006
3.17497083
5
3.47750572
4
3.79894202
7
4.13919844
7
4.49819619
1
5
6
1.01758380
6
0.95084810
1
0.90001173
4
0.86571540
1
0.84884558
1
0.84970853
9
0.86870292
9
0.90582194
2
0.96123591
3
1.03498623
4
1.12710441
5
1.23767536
3
1.36671985
7
1.51426892
9
1.68036720
9
1.86498880
9
2.06815366
2
2.28987784
6
2.53015983
2
2.78904125
5
3.06646559
3.36243421
4
3.67696512
4.00997809
8
4.36139693
9
4 CURVE FITTING TOOL IN MATLAB R2016A
The values which represented in table 5 export in MATLAB R2016a and using curve fitting tool
create 𝑋𝑌𝑍 axis graph[5],
✓ X – Current, I (A)
✓ Y – Air Gap Length z (mm)
✓ Z – Force F (N)
4.1 Force equation
For that calculation used force equation of,
𝐹 = 𝑎
(𝑖+𝑏)2
(𝑧+𝑐)2
(A)
and, for MATLAB R2016a used,
𝐹 = 𝑎
(𝑥+𝑏)2
(𝑦+𝑐)2
(B)
Because of the above X, Y, Z.

26. 19
Figure 13. Surface fitting graph for force equation

27. 20
4.2 Calculating constant and Root Mean Square Error (RMSE)
After surface fitting, it gives the details for above equation (B)
General model:
f(x,y) = a*((x + b)^2)/((y + c)^2)
Coefficients (with 95% confidence bounds):
a = 0.3185 (0.2947, 0.3423)
b = 20.9 (20.15, 21.65)
c = 1.149 (1.114, 1.184)
Goodness of fit:
SSE: 2029
R-square: 0.9732
Adjusted R-square: 0.9732
RMSE: 1.263
in this experiment Root Mean Square Error (RMSE) is 1.263, but this value is higher than which
excepted value. Because if the physical model which used in the simulation. If this value can be
lower changing,
✓ physical dimension of bolt
✓ turn of the copper wire, etc.

28. 21
4.3 Calculation of theoretical force
In above step is helped to the fine a, b, c constant foe the equation A and using that constant
calculate the fore according to the equation A. The table 6 is shown calculated force every current
and length step.
Table 6. Calculated theoritical force
Length
(mm)
AirGap
(mm)
Current (A)
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
0
1
22.097475
84
22.719034
15
23.349213
25
23.988013
14
24.635433
8
25.291475
24
25.956137
47
26.629420
47
27.311324
26
28.001848
83
28.700994
18
29.408760
31
30.125147
22
30.850154
91
31.583783
38
32.326032
64
33.076902
67
33.836393
49
34.604505
09
35.381237
47
36.166590
63
36.960564
57
37.763159
29
38.574374
79
39.394211
08
0.1
1.1
20.176070
55
20.743583
53
21.318967
71
21.902223
08
22.493349
65
23.092347
4
23.699216
35
24.313956
49
24.936567
83
25.567050
35
26.205404
07
26.851628
98
27.505725
09
28.167692
38
28.837530
87
29.515240
56
30.200821
43
30.894273
5
31.595596
76
32.304791
21
33.021856
86
33.746793
69
34.479601
72
35.220280
95
35.968831
36
0.2
1.2
18.494792
58
19.015014
54
19.542451
78
20.077104
31
20.618972
12
21.168055
22
21.724353
6
22.287867
26
22.858596
21
23.436540
44
24.021699
95
24.614074
75
25.213664
83
25.820470
19
26.434490
84
27.055726
77
27.684177
99
28.319844
49
28.962726
27
29.612823
34
30.270135
69
30.934663
33
31.606406
25
32.285364
45
32.971537
94
0.3
1.3
17.015234
09
17.493838
99
17.979081
97
18.470963
02
18.969482
13
19.474639
32
19.986434
58
20.504867
9
21.029939
3
21.561648
77
22.099996
32
22.644981
93
23.196605
61
23.754867
36
24.319767
19
24.891305
08
25.469481
05
26.054295
08
26.645747
19
27.243837
37
27.848565
62
28.459931
94
29.077936
33
29.702578
79
30.333859
32
0.4
1.4
15.706370
13
16.148159
28
16.596075
88
17.050119
92
17.510291
42
17.976590
37
18.449016
77
18.927570
62
19.412251
91
19.903060
66
20.399996
86
20.903060
51
21.412251
61
21.927570
15
22.449016
15
22.976589
6
23.510290
5
24.050118
85
24.596074
65
25.148157
89
25.706368
59
26.270706
74
26.841172
34
27.417765
39
28.000485
89
0.5
1.5
14.542918
89
14.951982
46
15.366719
58
15.787130
26
16.213214
5
16.644972
3
17.082403
65
17.525508
57
17.974287
04
18.428739
07
18.888864
66
19.354663
81
19.826136
52
20.303282
78
20.786102
61
21.274595
99
21.768762
93
22.268603
43
22.774117
49
23.285305
1
23.802166
28
24.324701
01
24.852909
3
25.386791
15
25.926346
56
0.6
1.6
13.504111
62
13.883955
59
14.269067
85
14.659448
4
15.055097
25
15.456014
4
15.862199
83
16.273653
56
16.690375
59
17.112365
9
17.539624
52
17.972151
42
18.409946
62
18.853010
11
19.301341
89
19.754941
97
20.213810
35
20.677947
01
21.147351
97
21.622025
23
22.101966
77
22.587176
61
23.077654
75
23.573401
17
24.074415
9
0.7
1.7
12.572759
27
12.926406
14
13.284957
95
13.648414
72
14.016776
44
14.390043
1
14.768214
72
15.151291
28
15.539272
8
15.932159
26
16.329950
68
16.732647
04
17.140248
36
17.552754
62
17.970165
83
18.392482
18.819703
11
19.251829
17
19.688860
18
20.130796
14
20.577637
05
21.029382
91
21.486033
72
21.947589
48
22.414050
19
0.8
1.8
11.734536
86
12.064606
19
12.399253
46
12.738478
66
13.082281
8
13.430662
88
13.783621
9
14.141158
86
14.503273
75
14.869966
58
15.241237
35
15.617086
06
15.997512
71
16.382517
29
16.772099
81
17.166260
27
17.564998
67
17.968315
01
18.376209
28
18.788681
49
19.205731
64
19.627359
73
20.053565
76
20.484349
72
20.919711
62

29. 22
0.9
1.9
10.977429
33
11.286202
72
11.599258
68
11.916597
21
12.238218
32
12.564121
99
12.894308
24
13.228777
06
13.567528
44
13.910562
4
14.257878
94
14.609478
04
14.965359
71
15.325523
96
15.689970
77
16.058700
16
16.431712
12
16.809006
65
17.190583
75
17.576443
43
17.966585
67
18.361010
49
18.759717
87
19.162707
83
19.569980
36
1
2
10.291298
55
10.580772
44
10.874261
22
11.171764
9
11.473283
47
11.778816
93
12.088365
29
12.401928
55
12.719506
69
13.041099
74
13.366707
67
13.696330
5
14.029968
23
14.367620
85
14.709288
37
15.054970
77
15.404668
08
15.758380
28
16.116107
37
16.477849
35
16.843606
24
17.213378
01
17.587164
68
17.964966
25
18.346782
7
1.1
2.1
9.6675421
88
9.9394710
41
10.215171
45
10.494643
4
10.777886
91
11.064901
97
11.355688
58
11.650246
74
11.948576
45
12.250677
72
12.556550
53
12.866194
9
13.179610
82
13.496798
29
13.817757
31
14.142487
88
14.470990
01
14.803263
68
15.139308
91
15.479125
69
15.822714
02
16.170073
9
16.521205
33
16.876108
32
17.234782
85
1.2
2.2
9.0988227
23
9.3547546
23
9.6142362
01
9.8772674
6
10.143848
4
10.413979
01
10.687659
31
10.964889
29
11.245668
95
11.529998
28
11.817877
3
12.109305
99
12.404284
37
12.702812
42
13.004890
15
13.310517
57
13.619694
66
13.932421
43
14.248697
89
14.568524
02
14.891899
83
15.218825
32
15.549300
49
15.883325
34
16.220899
87
1.3
2.3
8.5788507
03
8.8201568
17
9.0648097
56
9.3128095
21
9.5641561
11
9.8188495
27
10.076889
77
10.338276
83
10.603010
73
10.871091
44
11.142518
99
11.417293
35
11.695414
55
11.976882
57
12.261697
41
12.549859
08
12.841367
58
13.136222
9
13.434425
05
13.735974
02
14.040869
81
14.349112
44
14.660701
88
14.975638
16
15.293921
26
1.4
2.4
8.1022100
84
8.3301092
4
8.5611692
71
8.7953901
79
9.0327719
62
9.2733146
21
9.5170181
56
9.7638825
67
10.013907
85
10.267094
02
10.523441
06
10.782948
97
11.045617
76
11.311447
43
11.580437
97
11.852589
39
12.127901
68
12.406374
85
12.688008
9
12.972803
82
13.260759
62
13.551876
3
13.846153
85
14.143592
27
14.444191
57
1.5
2.5
7.6642166
35
7.8797959
8.0983651
69
8.3199244
42
8.5444737
18
8.7720129
99
9.0025422
82
9.2360615
7
9.4725708
61
9.7120701
56
9.9545594
54
10.200038
76
10.448508
06
10.699967
37
10.954416
69
11.211856
11.472285
32
11.735704
65
12.002113
97
12.271513
31
12.543902
64
12.819281
98
13.097651
32
13.379010
67
13.663360
02
1.6
2.6
7.2608024
01
7.4650344
21
7.6720990
63
7.8819963
26
8.0947262
12
8.3102887
19
8.5286838
47
8.7499115
97
8.9739719
69
9.2008649
63
9.4305905
78
9.6631488
14
9.8985396
73
10.136763
15
10.377819
25
10.621707
98
10.868429
32
11.117983
29
11.370369
88
11.625589
09
11.883640
92
12.144525
37
12.408242
45
12.674792
14
12.944174
46
1.7
2.7
6.8884209
11
7.0821785
76
7.2786235
87
7.4777559
45
7.6795756
49
7.8840826
99
8.0912770
95
8.3011588
38
8.5137279
27
8.7289843
62
8.9469281
43
9.1675592
71
9.3908777
44
9.6168835
65
9.8455767
31
10.076957
24
10.311025
1
10.547780
31
10.787222
86
11.029352
76
11.27417
11.521674
59
11.771866
53
12.024745
81
12.280312
44
1.8
2.8
6.5439689
67
6.7280378
8
6.9146597
59
7.1038346
06
7.2955624
19
7.4898432
7.6866769
47
7.8860636
61
8.0880033
42
8.2924959
89
8.4995416
04
8.7091401
86
8.9212917
34
9.1359962
5
9.3532537
32
9.5730641
81
9.7954275
97
10.020343
98
10.247813
33
10.477835
65
10.710410
93
10.945539
18
11.183220
4
11.423454
58
11.666241
73
1.9
2.9
6.2247217
82
6.3998109
02
6.5773284
43
6.7572744
04
6.9396487
86
7.1244515
89
7.3116828
12
7.5013424
55
7.6934305
19
7.8879470
04
8.0848919
09
8.2842652
35
8.4860669
81
8.6902971
48
8.8969557
35
9.1060427
43
9.3175581
71
9.5315020
2
9.7478742
9
9.9666749
79
10.187904
09
10.411561
62
10.637647
57
10.866161
94
11.097104
74
2
3
5.9282789
25
6.0950296
94
6.2640932
33
6.4354695
43
6.6091586
24
6.7851604
76
6.9634750
98
7.1441024
91
7.3270426
55
7.5122955
9
7.6998612
95
7.8897397
71
8.0819310
17
8.2764350
35
8.4732518
23
8.6723813
82
8.8738237
11
9.0775788
11
9.2836466
82
9.4920273
24
9.7027207
36
9.9157269
19
10.131045
87
10.348677
6
10.568622
09

30. 23
2.1
3.1
5.6525190
73
5.8115132
62
5.9727126
41
6.1361172
1
6.3017269
69
6.4695419
18
6.6395620
56
6.8117873
84
6.9862179
03
7.1628536
11
7.3416945
09
7.5227405
96
7.7059918
74
7.8914483
42
8.0791099
99
8.2689768
46
8.4610488
83
8.6553261
1
8.8518085
27
9.0504961
34
9.2513889
3
9.4544869
17
9.6597900
93
9.8672984
59
10.077012
02
2.2
3.2
5.3955619
5
5.5473284
43
5.7011998
8
5.8571762
62
6.0152575
89
6.1754438
6
6.3377350
75
6.5021312
35
6.6686323
39
6.8372383
87
7.0079493
8
7.1807653
18
7.3556862
7.5327120
26
7.7118427
97
7.8930785
12
8.0764191
71
8.2618647
75
8.4494153
24
8.6390708
17
8.8308312
54
9.0246966
36
9.2206669
62
9.4187422
32
9.6189224
47
2.3
3.3
5.1557361
9
5.3007568
58
5.4477889
09
5.5968323
42
5.7478871
58
5.9009533
56
6.0560309
36
6.2131198
99
6.3722202
44
6.5333319
72
6.6964550
82
6.8615895
74
7.0287354
48
7.1978927
05
7.3690613
45
7.5422413
66
7.7174327
7
7.8946355
57
8.0738497
25
8.2550752
77
8.4383122
1
8.6235605
26
8.8108202
24
9.0000913
05
9.1913737
68
2.4
3.4
4.9315520
92
5.0702669
05
5.2109056
41
5.3534682
99
5.4979548
79
5.6443653
82
5.7926998
08
5.9429581
56
6.0951404
27
6.2492466
2
6.4052767
35
6.5632307
73
6.7231087
34
6.8849106
17
7.0486364
23
7.2142861
51
7.3818598
02
7.5513573
75
7.7227788
7
7.8961242
89
8.0713936
29
8.2485868
93
8.4277040
78
8.6087451
86
8.7917102
17
2.5
3.5
4.7216784
32
4.8544899
13
4.9891434
41
5.1256390
13
5.2639766
32
5.4041562
95
5.5461780
05
5.6900417
6
5.8357475
6
5.9832954
06
6.1326852
97
6.2839172
34
6.4369912
17
6.5919072
45
6.7486653
18
6.9072654
37
7.0677076
01
7.2299918
11
7.3941180
67
7.5600863
68
7.7278967
15
7.8975491
07
8.0690435
44
8.2423800
27
8.4175585
56
2.6
3.6
4.5249226
48
4.6521997
78
4.7812421
95
4.9120498
97
5.0446228
86
5.1789611
61
5.3150647
22
5.4529335
7
5.5925677
03
5.7339671
23
5.8771318
28
6.0220618
2
6.1687570
98
6.3172176
62
6.4674435
12
6.6194346
49
6.7731910
71
6.9287127
8
7.0859997
74
7.2450520
55
7.4058696
22
7.5684524
76
7.7328006
15
7.8989140
4
8.0667927
52
2.7
3.7
4.3402138
69
4.4622955
07
4.5860703
71
4.7115384
62
4.8386997
79
4.9675543
23
5.0981020
93
5.2303430
91
5.3642773
14
5.4999047
65
5.6372254
42
5.7762393
45
5.9169464
75
6.0593468
32
6.2034404
15
6.3492272
25
6.4967072
62
6.6458805
25
6.7967470
15
6.9493067
31
7.1035596
74
7.2595058
43
7.4171452
39
7.5764778
62
7.7375037
11
2.8
3.8
4.1665883
1
4.2837862
03
4.4026095
88
4.5230584
64
4.6451328
3
4.7688326
87
4.8941580
36
5.0211088
75
5.1496852
05
5.2798870
26
5.4117143
38
5.5451671
4
5.6802454
34
5.8169492
19
5.9552784
94
6.0952332
6
6.2368135
18
6.3800192
66
6.5248505
05
6.6713072
35
6.8193894
56
6.9690971
67
7.1204303
7
7.2733890
63
7.4279732
48
2.9
3.9
4.0031766
72
4.1157781
2
4.2299413
07
4.3456662
34
4.4629529
01
4.5818013
08
4.7022114
55
4.8241833
42
4.9477169
69
5.0728123
35
5.1994694
42
5.3276882
88
5.4574688
75
5.5888112
01
5.7217152
67
5.8561810
73
5.9922086
19
6.1297979
05
6.2689489
31
6.4096616
97
6.5519362
02
6.6957724
48
6.8411704
33
6.9881301
59
7.1366516
24
3
4
3.8491932
45
3.9574634
43
4.0672353
08
4.1785088
4
4.2912840
39
4.4055609
06
4.5213394
39
4.6386196
4
4.7574015
07
4.8776850
42
4.9994702
44
5.1227571
13
5.2475456
49
5.3738358
52
5.5016277
22
5.6309212
6
5.7617164
64
5.8940133
36
6.0278118
74
6.1631120
8
6.2999139
53
6.4382174
93
6.5780227
6.7193295
74
6.8621381
15
3.1
4.1
3.7039264
41
3.8081105
71
3.9137396
95
4.0208138
15
4.1293329
29
4.2392970
39
4.3507061
43
4.4635602
42
4.5778593
36
4.6936034
25
4.8107925
08
4.9294265
87
5.0495056
6
5.1710297
28
5.2939987
92
5.4184128
5
5.5442719
02
5.6715759
5
5.8003249
93
5.9305190
3
6.0621580
62
6.1952420
9
6.3297711
12
6.4657451
29
6.6031641
4
3.2
4.2
3.5667305
6
3.6670556
41
3.7687721
93
3.8718802
16
3.9763797
11
4.0822706
77
4.1895531
15
4.2982270
24
4.4082924
04
4.5197492
55
4.6325975
77
4.7468373
71
4.8624686
37
4.9794913
73
5.0979055
81
5.2177112
6
5.3389084
11
5.4614970
32
5.5854771
25
5.7108486
9
5.8376117
25
5.9657662
32
6.0953122
11
6.2262496
6
6.3585785
81

31. 24
3.3
4.3
3.4370186
03
3.5336951
43
3.6317125
51
3.7310708
25
3.8317699
68
3.9338099
77
4.0371908
54
4.1419125
98
4.2479752
1
4.3553786
89
4.4641230
36
4.5742082
5
4.6856343
31
4.7984012
8
4.9125090
96
5.0279577
8
5.1447473
31
5.2628777
5
5.3823490
35
5.5031611
89
5.6253142
09
5.7488080
97
5.8736428
53
5.9998184
76
6.1273349
66
3.4
4.4
3.3142559
83
3.4074794
53
3.5019958
98
3.5978053
17
3.6949077
11
3.7933030
8
3.8929914
23
3.9939727
41
4.0962470
33
4.1998143
4.3046745
42
4.4108277
58
4.5182739
49
4.6270131
15
4.7370452
55
4.8483703
7
4.9609884
59
5.0748995
23
5.1901035
62
5.3066005
75
5.4243905
63
5.5434735
25
5.6638494
62
5.7855183
74
5.9084802
6
3.5
4.5
3.1979550
06
3.2879071
6
3.3791069
16
3.4715542
75
3.5652492
36
3.6601918
01
3.7563819
68
3.8538197
37
3.9525051
09
4.0524380
84
4.1536186
62
4.2560468
42
4.3597226
25
4.4646460
11
4.5708169
99
4.6782355
9
4.7869017
84
4.8968155
81
5.0079769
8
5.1203859
81
5.2340425
86
5.3489467
93
5.4650986
03
5.5824980
15
5.7011450
31
3.6
4.6
3.0876700
22
3.1745200
77
3.2625747
1
3.3518339
2
3.4422977
08
3.5339660
74
3.6268390
18
3.7209165
39
3.8161986
38
3.9126853
15
4.0103765
69
4.1092724
01
4.2093728
11
4.3106777
99
4.4131873
65
4.5169015
08
4.6218202
29
4.7279435
27
4.8352714
04
4.9438038
58
5.0535408
9
5.1644824
99
5.2766286
87
5.3899794
52
5.5045347
95
3.7
4.7
2.9829931
49
3.0668988
5
3.1519682
92
3.2382014
75
3.3255983
98
3.4141590
62
3.5038834
67
3.5947716
12
3.6868234
98
3.7800391
24
3.8744184
91
3.9699615
99
4.0666684
48
4.1645390
37
4.2635733
67
4.3637714
37
4.4651332
48
4.5676588
4.6713480
93
4.7762011
26
4.8822179
4.9893984
14
5.0977426
69
5.2072506
65
5.3179224
01
3.8
4.8
2.8835504
94
2.9646590
7
3.0468925
92
3.1302510
59
3.2147344
71
3.3003428
3
3.3870761
34
3.4749343
83
3.5639175
78
3.6540257
19
3.7452588
05
3.8376168
36
3.9310998
14
4.0257077
37
4.1214406
05
4.2182984
19
4.3162811
79
4.4153888
84
4.5156215
35
4.6169791
31
4.7194616
73
4.8230691
6
4.9278015
93
5.0336589
72
5.1406412
96
3.9
4.9
2.7889988
14
2.8674478
38
2.9469849
21
3.0276100
62
3.1093232
62
3.1921245
21
3.2760138
38
3.3609912
13
3.4470566
47
3.5342101
4
3.6224516
92
3.7117813
02
3.8021989
7
3.8937046
97
3.9862984
83
4.0799803
27
4.1747502
3
4.2706081
92
4.3675542
12
4.4655882
9
4.5647104
28
4.6649206
24
4.7662188
78
4.8686051
91
4.9720795
63
4
5
2.6990225
47
2.7749407
16
2.8519118
43
2.9299359
25
3.0090129
65
3.0891429
6
3.1703259
13
3.2525618
22
3.3358506
87
3.4201925
1
3.5055872
88
3.5920350
24
3.6795357
16
3.7680893
64
3.8576959
69
3.9483555
31
4.0400680
49
4.1328335
24
4.2266519
56
4.3215233
44
4.4174476
88
4.5144249
89
4.6124552
47
4.7115384
62
4.8116746
32
4.1
5.1
2.6133311
75
2.6868390
16
2.7613663
83
2.8369132
76
2.9134796
95
2.9910656
4
3.0696711
12
3.1492961
1
3.2299406
34
3.3116046
85
3.3942882
61
3.4779913
64
3.5627139
93
3.6484561
49
3.7352178
3
3.8229990
38
3.9117997
72
4.0016200
32
4.0924598
18
4.1843191
31
4.2771979
7
4.3710963
35
4.4660142
26
4.5619516
44
4.6589085
87
4.2
5.2
2.5316568
79
2.6028673
83
2.6750655
5
2.7482513
81
2.8224248
74
2.8975860
3
2.9737348
49
3.0508713
32
3.1289954
77
3.2081072
85
3.2882067
56
3.3692938
91
3.4513686
88
3.5344311
49
3.6184812
72
3.7035190
59
3.7895445
08
3.8765576
21
3.9645583
96
4.0535468
35
4.1435229
36
4.2344867
01
4.3264381
29
4.4193772
19
4.5133039
73
4.3
5.3
2.4537524
42
2.5227716
48
2.5927481
25
2.6636818
72
2.7355728
9
2.8084211
78
2.8822267
37
2.9569895
67
3.0327096
67
3.1093870
38
3.1870216
79
3.2656135
91
3.3451627
73
3.4256692
26
3.5071329
49
3.5895539
43
3.6729322
07
3.7572677
42
3.8425605
48
3.9288106
24
4.0160179
7
4.1041825
88
4.1933044
75
4.2833836
34
4.3744200
62
4.4
5.4
2.3793893
74
2.4463168
94
2.5141726
74
2.5829567
13
2.6526690
13
2.7233095
71
2.7948783
9
2.8673754
68
2.9408008
05
3.0151544
03
3.0904362
6
3.1666463
76
3.2437847
52
3.3218513
88
3.4008462
84
3.4807694
39
3.5616208
54
3.6434005
28
3.7261084
62
3.8097446
56
3.8943091
09
3.9798018
22
4.0662227
95
4.1535720
27
4.2418495
19

32. 25
4.5
5.5
2.3083562
4
2.3732857
39
2.4391157
85
2.5058463
79
2.5734775
21
2.6420092
11
2.7114414
49
2.7817742
34
2.8530075
67
2.9251414
48
2.9981758
77
3.0721108
54
3.1469463
79
3.2226824
51
3.2993190
72
3.3768562
4
3.4552939
56
3.5346322
19
3.6148710
31
3.6960103
9
3.7780502
98
3.8609907
53
3.9448317
56
4.0295733
06
4.1152154
05
4.6
5.6
2.2404571
51
2.3034767
82
2.3673704
72
2.4321382
21
2.4977800
28
2.5642958
94
2.6316858
19
2.6999498
02
2.7690878
45
2.8390999
45
2.9099861
05
2.9817463
23
3.0543806
3.1278889
35
3.2022713
3
3.2775277
83
3.3536582
94
3.4306628
65
3.5085414
94
3.5872941
81
3.6669209
28
3.7474217
33
3.8287965
96
3.9110455
19
3.9941685
4.7
5.7
2.1755104
12
2.2367032
2
2.2987447
49
2.361635
2.4253739
72
2.4899616
65
2.5553980
8
2.6216832
16
2.6888170
73
2.7567996
52
2.8256309
51
2.8953109
73
2.9658397
15
3.0372171
79
3.1094433
64
3.1825182
71
3.2564418
99
3.3312142
48
3.4068353
19
3.4833051
1
3.5606236
24
3.6387908
58
3.7178068
14
3.7976714
91
3.8783848
9
4.8
5.8
2.1133473
09
2.1727915
92
2.2330603
45
2.2941535
67
2.3560712
6
2.4188134
22
2.4823800
55
2.5467711
57
2.6119867
3
2.6780267
72
2.7448912
84
2.8125802
66
2.8810937
18
2.9504316
4
3.0205940
32
3.0915808
94
3.1633922
26
3.2360280
28
3.3094882
99
3.3837730
41
3.4588822
53
3.5348159
34
3.6115740
86
3.6891567
07
3.7675637
98
4.9
5.9
2.0538110
09
2.1115806
53
2.1701515
41
2.2295236
72
2.2896970
46
2.3506716
64
2.4124475
25
2.4750246
29
2.5384029
77
2.6025825
68
2.6675634
02
2.7333454
79
2.7999288
2.8673133
65
2.9354991
72
3.0044862
23
3.0742745
17
3.1448640
55
3.2162548
36
3.2884468
6
3.3614401
28
3.4352346
39
3.5098303
93
3.5852273
9
3.6614256
31
5
6
1.9967555
7
2.0529203
57
2.1098641
29
2.1675868
86
2.2260886
27
2.2853693
53
2.3454290
63
2.4062677
58
2.4678854
38
2.5302821
02
2.5934577
51
2.6574123
84
2.7221460
02
2.7876586
04
2.8539501
91
2.9210207
62
2.9888703
18
3.0574988
59
3.1269063
84
3.1970928
94
3.2680583
89
3.3398028
67
3.4123263
31
3.4856287
79
3.5597102
12
Similarly, previous step these result import to the excel and using curve fitting tool crested the
graph for Force in 𝑁, Airgap length in 𝑚𝑚 and Current in 𝐴.
This step gives,
Linear interpolant:
f(x,y) = piecewise linear surface computed from p, where x is normalized by mean 0 and std
1.803 and where y is normalized by mean 3.5 and std 1.473
Coefficients:
p = coefficient structure
Goodness of fit:
SSE: 7.591e-28
R-square: 1
Adjusted R-square: NaN
RMSE: NaN

33. 26
Figure 14. Surface fitting graph for calculated force

34. 27
4.4 Difference between calculated force and J-Mag Designer force
Difference between calculated force and J-Mag Designer provided force is shown in table 7.Length(mm)
AirGap(mm)
Current (A)
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
0
1
-4.284101443
-3.453071684
-2.660562942
-1.909710146
-1.204043462
-0.545095026
0.066145802
0.62906082
1.143191273
1.608276963
2.024166528
2.390617069
2.707395727
2.974427062
3.191612608
3.358826893
3.476046187
3.54325309
3.560392683
3.527666447
3.444996312
3.312331198
3.130102955
2.898201298
2.616410891
0.1
1.1
-4.276808107
-3.517639035
-2.796513298
-2.116463087
-1.480653108
-0.890508139
-0.347075907
0.149220816
0.597883509
0.998653682
1.351423768
1.655939034
1.911999274
2.119528795
2.278418968
2.388553888
2.449934791
2.462534062
2.426301396
2.341413712
2.207807814
2.025433764
1.794685231
1.515494305
1.187661476
0.2
1.2
-4.237497328
-3.539025699
-2.878125952
-2.257622887
-1.680489853
-1.147947317
-0.661353112
-0.220745402
0.173328846
0.5205954
0.820959056
1.074192479
1.280074751
1.438564862
1.549554761
1.612936047
1.628708764
1.596850955
1.517319707
1.390273114
1.21565161
0.993407816
0.72390788
0.407119524
0.042835874
0.3
1.3
-4.055198624
-3.411492593
-2.804774863
-2.237827362
-1.713422601
-1.232564789
-0.796777983
-0.405929838
-0.060507687
0.239208497
0.493095988
0.700975579
0.862629497
0.978004395
1.046995033
1.069508859
1.045554278
0.975103117
0.858125863
0.69475246
0.48493782
0.22863089
-0.07382062
-0.422422232
-0.817391175
0.4
1.4
-3.795143976
-3.199835256
-2.640958132
-2.121133476
-1.64295081
-1.207396559
-0.815910125
-0.468320146
-0.165144894
0.093382475
0.307160199
0.475998581
0.599694833
0.678213211
0.711446001
0.699304616
0.641801252
0.538918922
0.390626815
0.197036519
-0.041888303
-0.326190724
-0.655578199
-1.029981827
-1.449654154
0.5
1.5
-3.445900131
-2.894758701
-2.379453919
-1.902475033
-1.466285799
-1.071823313
-0.720381222
-0.411848305
-0.146694911
0.074875025
0.252746452
0.386738973
0.476667394
0.522487279
0.524090596
0.481402585
0.394441318
0.263179429
0.087595999
-0.132213688
-0.396272221
-0.704625419
-1.057007148
-1.453317764
-1.893818663
0.6
1.6
-3.408340247
-2.892772653
-2.412341983
-1.969476063
-1.56655414
-1.204347719
-0.884324363
-0.606191092
-0.370451439
-0.177319594
-0.026889678
0.080683525
0.145188534
0.166601149
0.144822179
0.079771395
-0.028525844
-0.180100963
-0.374971398
-0.613056664
-0.89437135
-1.218957872
-1.586583203
-1.99711089
-2.450807932
0.7
1.7
-3.190753708
-2.708770017
-2.261324821
-1.85071048
-1.479169323
-1.147433847
-0.856992117
-0.607450078
-0.399322023
-0.232800267
-0.107989489
-0.025045839
0.015861435
0.014690539
-0.028666214
-0.114259438
-0.242072644
-0.412135387
-0.62446578
-0.878994359
-1.175728654
-1.514710434
-1.895731893
-2.318627754
-2.783669249
0.8
1.8
-2.920147998
-2.468170529
-2.050126654
-1.668208738
-1.324533892
-1.019776545
-0.755498816
-0.531178011
-0.347347706
-0.204187349
-0.101796124
-0.040339509
-0.0199631
-0.040703496
-0.102684866
-0.205930979
-0.350438945
-0.536229804
-0.763327357
-1.031668818
-1.341254644
-1.692132034
-2.084103218
-2.51699016
-2.991061022
0.9
1.9
-2.628758111
-2.204883738
-1.814326758
-1.459164267
-1.141448282
-0.861827773
-0.621729927
-0.42069848
-0.259248329
-0.137552163
-0.055671994
-0.013771303
-0.012013791
-0.050409755
-0.129081887
-0.24807338
-0.407350281
-0.606946164
-0.846892144
-1.127133902
-1.447653322
-1.808508416
-2.209513913
-2.650481401
-3.131670285

35. 28
1
2
-2.157556423
-1.760416447
-1.395969104
-1.066239675
-0.773120539
-0.517191299
-0.300019019
-0.120949078
0.019455631
0.121034408
0.183716529
0.207353177
0.191788569
0.136995816
0.042874001
-0.090612444
-0.263442302
-0.475640935
-0.72723358
-1.018174284
-1.348449667
-1.718097871
-2.126975801
-2.574854513
-3.06200221
1.1
2.1
-1.948557398
-1.573429564
-1.230358024
-0.921297767
-0.648081484
-0.411274459
-0.2123173
-0.050661145
0.073189415
0.15907773
0.206931538
0.216609633
0.187962598
0.120969634
0.015519963
-0.128406247
-0.31079319
-0.531667772
-0.791054209
-1.088912475
-1.425225096
-1.800027863
-2.213197773
-2.664485068
-3.15416299
1.2
2.2
-1.463607978
-1.111713389
-0.791269176
-0.504172128
-0.252159274
-0.035662713
0.143721693
0.286695606
0.39272832
0.461690475
0.493495654
0.487985064
0.445047451
0.364659252
0.246704662
0.091160345
-0.101947997
-0.332643411
-0.600958567
-0.90685799
-1.250314526
-1.631370229
-2.049909369
-2.505684624
-2.998940412
1.3
2.3
-1.25066352
-0.916633735
-0.613456949
-0.342931373
-0.106729311
0.094684797
0.259818542
0.389348543
0.482740746
0.539882873
0.560696524
0.5450126
0.492733862
0.403825829
0.278182404
0.115785421
-0.083336987
-0.319214257
-0.591877659
-0.901297502
-1.247448255
-1.630359876
-2.049942116
-2.505928974
-2.998564025
1.4
2.4
-1.000880236
-0.683940871
-0.397252814
-0.142572785
0.078488877
0.26549749
0.417007602
0.533678795
0.614991862
0.660842647
0.671156048
0.645770521
0.584593336
0.487595531
0.354660845
0.185783759
-0.01900939
-0.259755226
-0.536486561
-0.849173812
-1.197789907
-1.582362306
-2.002816328
-2.458875896
-2.950779387
1.5
2.5
-0.817363565
-0.515771885
-0.243805511
-0.003162937
0.204601598
0.379077274
0.5188116
0.624521205
0.695665193
0.732133851
0.733872888
0.700732669
0.63259898
0.529439318
0.391164147
0.217766835
0.009265381
-0.234364284
-0.513159994
-0.827098073
-1.176145967
-1.560332102
-1.979593845
-2.433648888
-2.922727558
1.6
2.6
-0.597662356
-0.310141241
-0.051690127
0.176036293
0.371527623
0.534410923
0.663294154
0.758846821
0.820556057
0.848318783
0.842075778
0.801697223
0.727064538
0.618144309
0.474853504
0.297189186
0.085168101
-0.161238013
-0.442064722
-0.757293903
-1.10688983
-1.490879904
-1.909214959
-2.361604749
-2.848274181
1.7
2.7
-0.534183229
-0.257870223
-0.010047679
0.207691471
0.393904933
0.548204348
0.669194491
0.757552044
0.812798261
0.834826566
0.823580014
0.778903037
0.700708268
0.588968244
0.443587382
0.264556363
0.051896705
-0.194416519
-0.474420287
-0.788101653
-1.1354248
-1.516411676
-1.931023015
-2.37896469
-2.860455667
1.8
2.8
-0.413383807
-0.148265527
0.088930621
0.29666074
0.473511682
0.619107855
0.732121497
0.813174763
0.861814505
0.877937262
0.861482843
0.812288992
0.730305481
0.615489845
0.467748137
0.287073702
0.073503963
-0.173000667
-0.452474863
-0.764913045
-1.110271746
-1.488579797
-1.899804532
-2.343650134
-2.820325918
1.9
2.9
-0.241795075
0.012130987
0.238688831
0.436397461
0.603873884
0.740723936
0.845715654
0.919430018
0.961423465
0.971583584
0.949872543
0.896143944
0.810339531
0.692396777
0.54222759
0.359852383
0.145301651
-0.101487537
-0.380534404
-0.691833139
-1.035340633
-1.411080842
-1.819035769
-2.258900797
-2.730882381
2
3
-0.127357264
0.117441163
0.335414138
0.525116887
0.685207432
0.81530677
0.914203235
0.982463043
1.019640935
1.02565793
1.000462111
0.943907057
0.855918466
0.736471486
0.585462657
0.402907329
0.188828836
-0.056811807
-0.334045823
-0.642867456
-0.983236527
-1.355171966
-1.758658923
-2.193397972
-2.659589376
2.1
3.1
0.004573293
0.240624461
0.450369449
0.63240872
0.785430273
0.909097862
1.002143983
1.065198059
1.097785914
1.099855494
1.071344689
1.012108106
0.922062772
0.801208589
0.649443441
0.466770855
0.253204451
0.008725736
-0.266698375
-0.573067863
-0.910347458
-1.278552429
-1.677671588
-2.107414154
-2.5679619

36. 29
2.2
3.2
0.075807219
0.304233603
0.506892202
0.682426625
0.829517676
0.947813056
1.036184562
1.095154807
1.124282825
1.123492533
1.092752244
1.03192437
0.94092601
0.819728517
0.668251669
0.486511836
0.274513384
0.032225104
-0.240370731
-0.543275654
-0.876454323
-1.239926435
-1.633683619
-2.057430704
-2.511351355
2.3
3.3
0.212851668
0.433459366
0.628796892
0.797538427
0.938414838
1.051091698
1.13439509
1.188908948
1.214189742
1.210160279
1.176774674
1.113906419
1.021483587
0.899475338
0.747801488
0.566488778
0.355534282
0.114899148
-0.155428822
-0.455451861
-0.785133063
-1.144483756
-1.533519286
-1.95193477
-2.399907567
2.4
3.4
0.284341967
0.498115979
0.687120588
0.850065439
0.985710942
1.093720297
1.172972387
1.224002383
1.246390266
1.240060988
1.20497222
1.140995625
1.048063967
0.926133373
0.775128104
0.595085108
0.386002855
0.147839617
-0.119420772
-0.415776712
-0.741196185
-1.095680638
-1.479251374
-1.891618718
-2.332939651
2.5
3.5
0.364452948
0.571911987
0.755098795
0.912765003
1.043666399
1.147462696
1.223119287
1.271111914
1.291024365
1.282782357
1.246353541
1.181616352
1.088505777
0.966978837
0.8169591
0.63847567
0.431537164
0.196095726
-0.067849143
-0.360318839
-0.681265909
-1.030702047
-1.408641961
-1.814811611
-2.249350605
2.6
3.6
0.458523179
0.660212548
0.838120918
0.991023573
1.11770771
1.217838868
1.290397177
1.335849479
1.353790826
1.344152974
1.306890799
1.241887893
1.149079212
1.028423309
0.879841642
0.703366586
0.498997084
0.266695162
0.006456251
-0.281741361
-0.597856559
-0.941886676
-1.313848627
-1.713482507
-2.140908864
2.7
3.7
0.529023564
0.725015267
0.897685677
1.045851461
1.168319764
1.264763972
1.334162104
1.376995036
1.392860199
1.381688087
1.343434998
1.277982821
1.185281949
1.06529367
0.917932188
0.743239148
0.541213494
0.311819593
0.055045204
-0.229126262
-0.540651582
-0.879538071
-1.245800546
-1.639190357
-2.059812834
2.8
3.8
0.591028785
0.781057209
0.948205252
1.091324395
1.209253159
1.301651235
1.367591364
1.407465205
1.420905872
1.407853486
1.368267253
1.30202988
1.209084092
1.089396084
0.942888561
0.769602483
0.569529688
0.342628074
0.088898128
-0.191680112
-0.49906494
-0.833263989
-1.194292815
-1.58191445
-1.996212911
2.9
3.9
0.652265318
0.837344927
0.999995514
1.139099694
1.253506706
1.342907173
1.406355818
1.44426986
1.45626764
1.442295085
1.402323326
1.336231134
1.243955548
1.125461499
0.980681267
0.809647705
0.612360241
0.388772943
0.138884331
-0.13732285
-0.439809735
-0.768575325
-1.123637911
-1.504777022
-1.912063768
3
4
0.66535453
0.845775859
1.004188757
1.13950281
1.25058957
1.337132276
1.39822081
1.434250275
1.444860931
1.429993661
1.389602863
1.323576713
1.231865172
1.114442184
0.971230184
0.802264316
0.607536876
0.387004628
0.140674292
-0.131484756
-0.429422595
-0.753138154
-1.102658042
-1.477764378
-1.878523024
3.1
4.1
0.74318848
0.918828701
1.072884859
1.204304143
1.311972799
1.395574598
1.454243803
1.488333705
1.49750841
1.481709123
1.440901896
1.374960033
1.283839218
1.16751544
1.025915277
0.859079828
0.666995427
0.449618446
0.206962969
-0.061001581
-0.354228835
-0.672720987
-1.016499808
-1.385363175
-1.779354703
3.2
4.2
0.813191677
0.984840793
1.135324967
1.263615346
1.368620322
1.450016588
1.506993853
1.539854526
1.54827498
1.532205791
1.491613368
1.426372782
1.336448436
1.221802811
1.082362458
0.918165569
0.729202277
0.515438826
0.276880027
0.013492244
-0.274676533
-0.587626639
-0.925379049
-1.287739334
-1.67474147
3.3
4.3
0.828284804
0.996165047
1.143280839
1.268631584
1.371132302
1.450457521
1.505834736
1.537542555
1.54526932
1.528964215
1.488590048
1.424034036
1.335257577
1.222223701
1.084862007
0.923212986
0.737275378
0.526999727
0.292392926
0.033422069
-0.249859515
-0.557454154
-0.889383579
-1.245467552
-1.625724921

37. 30
3.4
4.4
0.877566279
1.041331393
1.184729477
1.306788401
1.406443226
1.483379664
1.536825958
1.567067346
1.573793147
1.556954299
1.51651835
1.45237051
1.364468035
1.252779332
1.117236243
0.957883496
0.774711156
0.567674152
0.336779587
0.081993613
-0.196625739
-0.499081171
-0.825396842
-1.175400957
-1.549099738
3.5
4.5
0.899418915
1.060052617
1.200707541
1.320441393
1.418198703
1.493656097
1.546078461
1.575722585
1.582293972
1.565743882
1.526032101
1.463044955
1.37674914
1.267115704
1.134078651
0.977671286
0.797880169
0.594662356
0.368037162
0.117962431
-0.155507761
-0.452376779
-0.772666114
-1.116208328
-1.483004595
3.6
4.6
0.913351548
1.07054491
1.208132051
1.325189486
1.420685331
1.494287908
1.54530024
1.573955551
1.579962533
1.563283828
1.523882058
1.46163609
1.376520826
1.268499461
1.137512868
0.983600931
0.806740531
0.606900839
0.384092909
0.138270862
-0.130506717
-0.422237849
-0.736945502
-1.074469814
-1.434807132
3.7
4.7
0.940384453
1.094381789
1.229151581
1.343793321
1.437277079
1.509293238
1.559128851
1.587039434
1.59273613
1.576165194
1.537295815
1.476012077
1.39229046
1.286102698
1.157376141
1.006148137
0.83240653
0.636119589
0.417300604
0.175899953
-0.088025892
-0.374474567
-0.683471526
-1.014865257
-1.368638757
3.8
4.8
0.945888948
1.096947738
1.229139399
1.341586116
1.433269523
1.50387174
1.552734635
1.580059292
1.585580188
1.569251196
1.531038917
1.470829257
1.388588934
1.284291479
1.157872732
1.00937151
0.838777214
0.646045202
0.431192466
0.194173091
-0.06495178
-0.346183144
-0.649548958
-0.974906676
-1.322226064
3.9
4.9
0.961161252
1.109133219
1.238585649
1.348665415
1.438374026
1.507397612
1.555087458
1.581642682
1.586801536
1.570516934
1.532754839
1.473403128
1.392434904
1.289826602
1.165509638
1.019520577
0.851843526
0.662442902
0.451336376
0.218479526
-0.036073419
-0.312320654
-0.610289125
-0.929839114
-1.270936755
4
5
0.979925534
1.12500316
1.25190589
1.359803003
1.447704065
1.515291747
1.561960475
1.58788087
1.592799132
1.576669666
1.5394511
1.481040533
1.401418206
1.300552921
1.178380814
1.034941341
0.870209089
0.68415932
0.476808604
0.248108138
-0.00188175
-0.273168134
-0.565770721
-0.879553047
-1.2144815
4.1
5.1
0.986867093
1.129460212
1.254213007
1.36031575
1.446788658
1.513325377
1.559328118
1.584956591
1.589963255
1.574304003
1.537947919
1.480791563
1.402810359
1.303965954
1.184201253
1.043556966
0.88201652
0.699551625
0.496166991
0.271814978
0.026558703
-0.239601388
-0.526688772
-0.834578469
-1.163231196
4.2
5.2
0.997900204
1.137991489
1.260578269
1.364874473
1.44991018
1.515373727
1.560703989
1.586027711
1.591111701
1.57591616
1.540403831
1.484476814
1.408114293
1.311276798
1.193913407
1.056059579
0.89769409
0.71879172
0.51936163
0.299355105
0.058837117
-0.202199572
-0.483774423
-0.785768855
-1.108136145
4.3
5.3
0.999597286
1.137179042
1.257574977
1.360016835
1.443546478
1.507858159
1.552406896
1.57731185
1.582345341
1.567464037
1.532634907
1.477759418
1.402813929
1.307766575
1.192560269
1.057234371
0.90176543
0.726127991
0.530327707
0.314324038
0.078175687
-0.178118026
-0.454580032
-0.751102063
-1.067623887
4.4
5.4
0.999838644
1.135040521
1.253364043
1.354061682
1.436191908
1.49944678
1.543288473
1.567839225
1.572872055
1.558346245
1.524230044
1.470420886
1.396901969
1.303640095
1.190579647
1.057756209
0.905148591
0.732728233
0.540509145
0.328445642
0.096593725
-0.155045932
-0.426498651
-0.717649053
-1.028446791
4.5
5.5
1.000237098
1.133197788
1.249593858
1.348695615
1.429565738
1.491890736
1.53519152
1.559533001
1.564708972
1.550685648
1.517421487
1.46482233
1.39287059
1.301530758
1.190750441
1.060567705
0.910951741
0.741886622
0.5533796
0.345384985
0.117958902
-0.128898593
-0.395208432
-0.680866429
-0.985815565

38. 31
4.6
5.6
1.000245472
1.131169724
1.245836594
1.343531333
1.423325594
1.484901257
1.527809657
1.552093203
1.557558969
1.544163713
1.511870447
1.460585729
1.39029456
1.300965607
1.192547385
1.065065281
0.918497007
0.752825206
0.568060788
0.364155885
0.141165289
-0.100912644
-0.362096865
-0.64229091
-0.941425143
4.7
5.7
0.989746906
1.118607279
1.231502311
1.327735749
1.406390534
1.467162908
1.50958453
1.533718471
1.539366974
1.526487875
1.495048306
1.444956158
1.376191393
1.288730286
1.182512068
1.057572314
0.913884585
0.751431277
0.570225656
0.370220141
0.15146478
-0.086036331
-0.34230455
-0.617244597
-0.910781033
4.8
5.8
0.986361953
1.112871312
1.223703478
1.318180307
1.395396349
1.45505335
1.496697111
1.520383865
1.525919759
1.513263803
1.482384975
1.433191409
1.365663515
1.27977488
1.175471178
1.052784197
0.911693099
0.752179174
0.574247776
0.377856072
0.163057006
-0.070148754
-0.321777471
-0.591748317
-0.8799845
4.9
5.9
0.987941712
1.112574934
1.221809918
1.314986041
1.391209294
1.450165164
1.491468461
1.515112589
1.52092373
1.508861897
1.478895456
1.430937971
1.364973861
1.280968894
1.178870988
1.058713238
0.92046989
0.764130059
0.589698151
0.3971268
0.186469292
-0.042271085
-0.289111634
-0.553971056
-0.83677056
5
6
0.979171764
1.102072257
1.209852395
1.301871485
1.377243046
1.435660814
1.476726134
1.500445816
1.506649525
1.495295868
1.466353336
1.419737021
1.355426145
1.273389675
1.173582982
1.056031953
0.920716657
0.767621014
0.596746553
0.408051639
0.201592799
-0.022631347
-0.264638789
-0.524349319
-0.801686727
This difference can be plot in MATLAB in similar way.
the variance of the absolute force from calculated force is polluted from the MATLAB and it is
represented in figure

39. 32
Figure 15. Difference between absolute force and calculated force

40. 33
5 MAGNETIC FLUX DENSITY PLOT
Considering the all 25 cases it takes -3A to +3𝐴 for 0.25𝐴 current step with 1.0 𝑚𝑚 which 1st
step in length create the magnetic flux density shown in bellow figures. In this figure is shown
the maximum and minimum flux density point in the model. Also, all the model has maintained
maximum 1.5 T magnetic flux density in the its all area.
This model has zero power levitation air gap (𝑧0) is 4.24947 𝑚𝑚 and assume it is 4.3 𝑚𝑚.
5.1 −𝟑 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap
Figure 16. Flux density in the model −3 𝐴 with 1.0 𝑚𝑚 air gap
In this figure shows the color bar and its show the magnetic flux density color values for different
regions.
Top left corner shows the,
✓ Case number
✓ Step number and
✓ Time taken to start given step
Also, down left corner shows the maximum and minimum values of the model

41. 34
5.2 +𝟑 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap
Figure 17. Flux density in the model +3 𝐴 with 1.0 𝑚𝑚 air gap
5.3 𝟎 𝑨 and 𝟏. 𝟎 𝒎𝒎 airgap
Figure 18. Flux density in the model 0 𝐴 with 1.0 𝑚𝑚 air gap

42. 35
5.4 −𝟑 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap
Figure 19. Flux density in the model −3 𝐴 with 4.3 𝑚𝑚 air gap
5.5 +𝟑 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap
Figure 20. Flux density in the model +3 𝐴 with 4.3 𝑚𝑚 air gap

43. 36
5.6 𝟎 𝑨 and 𝟒. 𝟑 𝒎𝒎 airgap (zero power levitation air gap (𝒛 𝟎))
Figure 21. Flux density in the model 0 𝐴 with 4.3 𝑚𝑚 air gap
6 SIMULATION DETAILS
6.1 Mesh
This model has only part mesh elements
✓ Mesh size = 1.0 𝑚𝑚
✓ Air element size = 2.0 𝑚𝑚
Geometry details
✓ Number of elements = 1 330 070
✓ Number of nodes = 235 702

44. 37
Table 7. Time taken to complete each case
Case
number
Date Starting
time
Time taken to complete simulation
Days Hours Mins Sec
1 2017/09/12 01:25 0 06 30 34
2 2017/09/12 01.34 0 06 21 36
3 2017/09/12 07.56 0 06 31 31
4 2017/09/12 07.57 0 06 30 47
5 2017/09/12 14.29 0 06 28 37
6 2017/09/12 21.32 0 05 44 34
7 2017/09/12 15.08 0 06 22 35
8 2017/09/12 21.19 0 06 14 03
9 2017/09/13 03.17 0 0 09 54
10 2017/09/13 03.34 0 06 08 03
11 2017/09/13 09.29 0 06 18 45
12 2017/09/13 09.43 0 06 18 08
13 2017/09/13 19.04 0 06 19 20
14 2017/09/13 15.49 0 06 36 02
15 2017/09/13 16.03 0 06 37 27
16 2017/09/13 22.26 0 06 22 37
17 2017/09/13 22.42 0 06 21 41
18 2017/09/14 04.50 0 06 29 32
19 2017/09/14 05.04 0 06 31 42
20 2017/09/14 11.20 0 06 32 49
21 2017/09/14 11.37 0 06 31 46
22 2017/09/14 17.54 0 06 20 56
23 2017/09/14 18.10 0 06 20 53
24 2017/09/15 10.15 0 05 04 27
25 2017/09/15 19.05 0 06 26 44
Total time taken to simulate 3 23 40 44

45. 38
7 COMPUTER CHARACTERISTICS
Processor model : Intel(R) coreTM
i5-7200U CPU @ 2.50 GHz
Processer speed
Processer base frequency : 2.50 GHz
Maximum turbo frequency : 3.10 GHz
Number of cores : 2
RAM amount : 8.00 GB
Total space taken to simulation
Size : 63.6 GB (68,290,856,898 bytes)
Size on disk : 63.6 GB (68,291,661,824 bytes)
Contains : 434 Files, 105 Folders
8 LINEARIZE THE EQUATION
For this case, nonlinear equation of maglev model is,
𝐹 = 𝑎
(𝑖+𝑏)2
(𝑧+𝑐)2
and,
Constant of a, b, c is
✓ a = 0.3185
✓ b = 20.9
✓ c = 1.149.
In the simulation process, got the 𝑧0 is 0𝐴 current with 4.24947 𝑚𝑚 airgap. Then linearize
above equation for
𝒊 = 𝟎 and 𝒛 = 𝟒. 𝟐𝟒𝟗𝟒𝟕
Rewrite above equation,
𝐹 = 0.3185
(𝑖+20.9)2
(𝑧+1.149)2
(C)
𝑓(𝑖, 𝑧)│0, 4.24947 = 𝑓(𝑖0, 𝑧0) +
𝜕𝑓
𝜕𝑖
(𝑖0, 𝑧0)(𝑖 − 𝑖0) +
𝜕𝑓
𝜕𝑧
(𝑖0, 𝑧0)(𝑧 − 𝑧0)
𝜕𝑓
𝜕𝑖
= 2 𝑥 0.3185
(0+20.9)
(4.24947+1.149)2
and
𝜕𝑓
𝜕𝑧
= −2 𝑥 0.3185
(0+20.9)2)
(4.24947+1.149)3

46. 39
Therefore,
𝑓(𝑖, 𝑧)│0, 4.24947 = 0.3185
(0+20.9)2
(4.24947+1.149)2
+ 2 𝑥 0.3185
(0+20.9)
(4.24947+1.149)2
(𝑖 − 0) −
2 𝑥 0.3185
(0+20.9)2)
(4.24947+1.149)3
(𝑧 − 4.24947)
𝑓(𝑖, 𝑧)│0, 4.24947 = 0.3185
(20.9)2
(4.24947+1.149)2
+ 2 𝑥 0.3185
(20.9)
(4.24947+1.149)2
(𝑖) −
2 𝑥 0.3185
(20.9)2)
(4.24947+1.149)3
(𝑧 − 4.24947)
𝑓(𝑖, 𝑧)│0, 4.24947 = 4.773760475 + 1.768560527(𝑧 − 4.24947)
𝑓(𝑖, 𝑧)│0, 4.24947 = 1.768560527 𝑧 + 12.2892053784029
This is the linearize equation for the equation (C). using MATLAB, it can be linearized as shown
in below figure
Figure 22. Linearization of equation (C)

47. 40
9 DISCUSSION
This process had taken more time to finish because of it has simulation process and it took large
time period. in this model has more than 6 hours for each case. Also, other works which related
to the assignment has taken more time. During the finite element analysis (FEA) of the model
time depend on the simulation based on size of model and element size of model which given.
As an example, in here used element size is 1.0 𝑚𝑚, if used 4.0 𝑚𝑚 it will reduce to 45 mins
per each case. Then time can be reduce increasing the size of the element but it mainly affected
to the output result of the FEA process. Even the shape of graph which we excepted can be
different when the size of elements is higher.
The physical model mainly affects to the result and changing the shape, dimensions of the model
can be design more accurate system. Then we design model from Solid works and test 13, 1,25
cases respectively, to get result which we excepted.
Other thing is saturation of ss400 material. For get accurate result from this model it should be
avoid the saturation. In this report shown in figure 16 and 17 there is no any saturation of the
entire process of this model.
After plotting the 3D curve using MATLAB curve fitting tool it gets 1.263 RMSE value it more
then higher value of excepted one. it can be reduced if we use polynomial function with degree
of 3 for both current and airgap instead given function of equation A, it can be get very closer
value of the excepted value.
Figure 23. cure fitting for J-Mag data with polynomial function

48. 41
Then it given coefficients, RMSE and other details of curve of,
Linear model Poly33:
f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y
+ p12*x*y^2 + p03*y^3
Coefficients (with 95% confidence bounds):
p00 = 48.17 (48.02, 48.32)
p10 = 2.119 (2.079, 2.159)
p01 = -25.74 (-25.9, -25.59)
p20 = 0.466 (0.4559, 0.4762)
p11 = -0.36 (-0.3834, -0.3366)
p02 = 5.026 (4.977, 5.074)
p30 = 0.0009047 (-0.001605, 0.003415)
p21 = -0.05856 (-0.06124, -0.05588)
p12 = 0.01587 (0.01259, 0.01915)
p03 = -0.3407 (-0.3452, -0.3361)
Goodness of fit:
SSE: 54.96
R-square: 0.9993
Adjusted R-square: 0.9993
RMSE: 0.2084
As shown RMSE value is reduced from 1.263 to 0.2084.

49. 42
REFERENCES
[1] “Applications And Future Of Magnetic Levitation - High Tech Point.” [Online].
Available: http://htpoint.com/news/applications-and-future-of-magnetic-levitation/.
[Accessed: 23-Aug-2017].
[2] “N50 1/2"x1" Neodymium Rare Earth Cylindrical Magnet - CMS
Magnetics, Inc.” [Online]. Available: http://www.magnet4sale.com/n50-1-2x1-
neodymium-rare-earth-cylindrical-magnet/. [Accessed: 20-Aug-2017].
[3] “American Wire Gauge Chart and AWG Electrical Current Load Limits table with skin
depth frequencies and wire breaking strength.” [Online]. Available:
https://www.powerstream.com/Wire_Size.htm. [Accessed: 10-Aug-2017].
[4] “0.65mm ENAMELLED COPPER WIRE - COIL WIRE, HIGH TEMPERATURE
MAGNET WIRE - 50g | eBay.” [Online]. Available: http://www.ebay.co.uk/itm/0-
65mm-ENAMELLED-COPPER-WIRE-COIL-WIRE-HIGH-TEMPERATURE-
MAGNET-WIRE-50g-/122022171564. [Accessed: 20-Aug-2017].
[5] “Curve Fitting Toolbox - MATLAB.” [Online]. Available:
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