28
CHAPTER 4
DESIGN OF THE ELECTROMAGNETIC MODULE
After a qualitative description about EES requirements and constraints, they will be quantified in details in this chapter. However before that, it is necessary to:
- describe the necessary formulas to make magnetic field analysis;
- describe IPM characteristics, that is necessary to define magnetic torque and force.
4.1 - Magnetic field background
In order to fulfill the working configurations, C1 and C2, EES should generate a magnetic field to have enough force to lift the WCE against its own weight gravity, to move it along the colonic lumen and to produce WCE orientations.
For these reasons, the equations, that allow to evaluate the magnetic field produced by EES and, as consequence, to define EES parameters, are now explained.
Within an external induced magnetic field with flux density B, IPM will exert a magnetic torque and force.
The magnetic torque expressed in cylindrical reference system, measured in N*m, is [14]:
29
=
×
=
×
=
θ θB
M
B
M
B
M
V
Z Z R RB
M
B
m
τ
m(1)
where:
m (A*m2) = magnetic dipole moment;
V (m3) = IPM volume;
M(A/m) = IPM magnetization;
B (T) = applied magnetic field’s flux density.
When a permanent magnet, with its dipole moment m, is placed somewhere in the space, a torque tends to align the magnetization of IPM along the direction of magnetic field, generated by EES.
The magnetic force, measured in N, is:
F
m= V
(
M
⋅
∇
)
B
(2)
where:
V (m3) = IPM volume;
M(A/m) = IPM magnetization;
∇
B (T/m) = magnetic field gradient.The equation (2) can be simplified considering that the magnetic capsule is small compared to the spatial changes of B. Under this assumption, B can be considered fairly uniform within the capsule and therefore, assuming that the capsule has spatially homogeneous magnetic properties, B represents the value of the applied magnetic field’s flux density in the center of the IPM. Considering that, the force can be expressed as:
30
F
m= V
M
∇
B
(3)
The magnetic field gradient induces a magnetic force that allows capsule’s translation movements. The magnetic force components in the cylindrical coordinates of space are (4):
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
∂
∂
+
∂
∂
=
θ
θ
θ
θ θ θ θ θ θ θB
M
B
M
B
M
z
B
M
z
B
M
z
B
M
r
B
M
r
B
M
r
B
M
V
F
F
F
Z Z R R Z Z R R Z Z R R m mZ mR(4)
The magnetic torque and force intensity are directly proportional to the magnitude and gradient of the applied magnetic field, IPM volume and magnetization.
4.2 - IPM characteristics
The magnetic force and torque between EES and WCE is strongly influenced by IPM.
Considering the anatomical constraints, in SUPCAM platform it was decided to use a spherical capsule with a diameter equal to 2.3 cm and, a cylindrical shape of the permanent magnet was chosen in order to optimize the WCE space and to obtain the required degrees of freedom.
A first issue to consider is the dimensioning of IPM, which should be in according with the WCE available space and compatible with the space
31
occupied by others WCE components, electronics and mechanics. For this reason, IPM has to have a volume, approximately the 10% of WCE room.
Knowing that WCE volume is (5),
3
5575
33
4
mm
r
V
WCE=
π
=
(5)
IPM volume should be around 400 mm3 and therefore a suitable solution is to have an IPM with a height of 9.5 mm and a diameter of 7.5 mm.
Another aspect is IPM material. It was chosen a Neodymium-Iron-Boron (NdFeB), because these magnets are the strongest types available and allow to have a high strength in a small package.
The type of NdFeB that was chosen is N52 because it maximizes the magnetic interaction. This grade, which indicates the material performances, means that the material can be reached a maximum temperature of 80°C (indicated by the letter N) and that the maximum energy produced in the magnet volume is 52 MGOe. This value is obtained from the hysteresis curve (Fig.20) and it represents the point on this curve where the value of flux density B (T) has to be multiplied by the value of field strength H in correspondence of its maximum.
So, a magnet with a bigger maximum energy product will have greater strength (allowing to obtain IPM dimension reduction and/or the applied currents inside EES).
32
Fig.21. Hysteresis curve
Residual flux density Br (T) is the magnetic induction remaining in a
saturated magnetic material after the magnetizing field H has been removed (Fig.21). For the chosen magnet, this values is equal to 1.48 T and it can be used to define a residual magnetization Mr as:
A
m
B
M
r=
r=
1
.
177
*
10
6/
µ
(6)
where µ is the magnetic permeability of free space and it is equal to 4π∗10−7 Α/m.
Finally, it is necessary to decide what is the direction of the magnetization, which indicates how the magnetic dipoles are oriented.
To perform an accurate diagnosis, the camera has to center the colon lumen. To achieve this issue, the internal permanent magnet has to be disposed with its magnetization parallel with to camera axis vision, and so IPM magnetization direction was chosen axial in order to achieve the required DoFs and also to increase the interaction of magnetic flux lines when the electromagnet is in C2 position.
33
When the magnetization is axial, IPM dipoles are oriented as shows in Fig.22(a). Fig.22(b) shows how IPM is positioned with respect the camera.
Fig.22. (a) Axial magnetization; (b) Position of IPM in the capsule
To summarize, IPM features are listed in Tab.1.
Tab.1. IPM features
4.3 – EES technical requirements
In this paragraph, the calculation of magnetic force and torque targets that EES has to satisfy in order to obtain WCE control are illustrated.
SHAPE Cylindrical
VOLUME ≈ 400 mm3
DIMENSIONS 9.5x7.5 mm
MATERIAL NdFeBo
GRADE OF MAGNETIZATION N52
RESIDUAL FLUX DENSITY 1.48 T
34
4.3.1 Magnetic gradient target
WCE translation along X, Y and Z axis depend on the magnetic force Fm. In particular, the force is directly proportional to the magnetic field gradient
∇
B (see equation 3).EES has to be designed to produce an appropriate magnetic field B, that allows to have a suitable magnetic field gradient to pull and to lift the WCE along precise directions.
For the design of EES, it was necessary to known what was the required magnetic field gradient value.
In reference to the equation (3), the magnetic field gradient, measured in (T/m), is:
M
F
B
mV
=
∇
(6)
This magnetic field gradient, necessary to move the capsule, is evaluated in a specific direction in the space and relevant to the two working conditions, C1 and C2.
For this reason, the magnetic force Fm is imposed equal to the capsule
weight force Fw, as indicated in equation (7). A rough yet conservative weight
prediction for a capsule may be two times the PillCam weight, that is around 4 g, that is considered a gold standard for capsule endoscopy. Therefore the capsule weight is equal to 8 g.
F
m=
F
w=
mg
=
78
.
4
mN
(7)
where:
m (kg) = WCE total weight
35
After the definition of IPM characteristics, the magnetic field gradient target is calculated in equation (8):
T
m
VM
mg
VM
F
B
=
w=
=
0
.
166
/
∇
(8)
where: m (kg) = 8*10-3 g (m/s2) = 9,81 V (m3) = 400*10-7M (A/m) = 1,177*106 (see equation 6)
Now it will be explained, through body diagrams, in what directions the value in equation (8) has to be guaranteed and what are the magnetic force components involved.
Since the configuration C1 is used to find and capture WCE, the condition to satisfy is that the magnetic force Fm must be equal or higher than the WCE
weight force Fw (Fig.23).
In particular the magnet (and consequently the capsule) are oriented as shown in Fig. 22 and the WCE is lifted along the Z axis.
36
Fig.23. Body diagram in C1
Based on equation (4), the magnetic force component involved in the lift is FmZ (9):
∂
∂
+
∂
∂
+
∂
∂
=
z
B
M
z
B
M
z
B
M
V
F
Z Z R R mZ θ θ(9)
This equation can be simplified because IPM magnetization is only along Z axis due to a torques as expressed in (N*m) that aligns the permanent magnet with the magnetic field of the EES and places it in the center of EES, and so the force is:
z
B
VM
F
Z Z mZ∂
∂
=
(10)
37
Therefore, EES needs to be able to generate a magnetic field, so that the magnetic field gradient
z BZ
∂ ∂
is equal to the target value in the (8).
Also in configuration C2, to have the WCE attraction, the condition to be satisfied is that the magnetic force Fm must be equal or higher than the WCE
weight force Fw. In this case, the attraction has to be guaranteed for all length
of the coil and the lift of capsule is along R axis. As consequence, the magnetic force component is (11):
∂
∂
+
∂
∂
+
∂
∂
=
r
B
M
r
B
M
r
B
M
V
F
Z Z R R mR θ θ (11)Fig.24. Body diagram in C2
According with the WCE position, the magnetic force, expressed in (11) is defined as follows:
- on the coil edge, EES magnetic field is mainly direct along R axis; as consequence, IPM is positioned along this axis and its magnetization
38
vector is composed only by the R component (Fig.24a). The calculation of the magnetic force corresponds to:
r
B
VM
F
R R mR∂
∂
=
(12)
- in the coil center, EES magnetic field is mainly direct along Z axis; as consequence, IPM is positioned along this axis and its magnetization vector is composed only by the Z component (Fig.24c). The calculation of the magnetic force corresponds to:
r
B
VM
F
Z Z mR∂
∂
=
(13)
In this condition, the interaction between EES and IPM magnetic fields allows to maintain the capsule with a camera centered with the colon lumen.
- from the coil edge to center, IPM assumes intermediate positions, in which its magnetization is composed by R and Z components (Fig.24b) and the force is equal to:
∂
∂
+
∂
∂
=
r
B
M
r
B
M
V
F
Z Z R R mR(14)
Therefore, the magnetic field gradient, expressed in equation (12), (13) and (14) has to be equal or greater than the target value in the (8).
39
After the attraction condition is satisfied, the other issue to define is WCE locomotion, that occurs in configuration C2, in order to guarantee the correct examination of all colon. In this case, the translation is along Z axis and the magnetic force is equal to:
z
B
VM
F
Z Z mZ∂
∂
=
(15)
To achieve locomotion, it is necessary that this magnetic force, named drag force Fd (as indicated in Fig.25), is higher than the friction force Fa.
However, the condition of attraction has to be respected too.
Fig.25. Body diagram in C2 locomotion
4.3.2 Magnetic torque target
The torque tends to align the magnetic moment of IPM with the applied magnetic field, generated by EES. Therefore, the torque allows the orientation
40
of the capsule. The inertial momentum measures the resistance of a body to rotate. The capsule, that has a spherical shape, has an inertial moment equal to:
2
4.23
3
*
10
7 25
2
m
kg
mr
I
WCE=
=
−⋅
(16)
where:m (kg) = WCE total weight;
r (m) = WCE radius.
Therefore, it is necessary to verify if magnetic field produced by EES, that is capable to generate the required magnetic gradient expressed in equation 8, is also able to generate torques for pitch and yaw orientation, that are greater than the product between WCE inertial momentum and required angular speed:
τ
≥
I
WCEω
(17)
where:
τ(N*m) = magnetic torque;
I (kg*m2) = WCE inertial momentum; ω(rad/s) = angular speed.
If the condition in (17) is satisfied, WCE should align with the EES magnetic field.
In particular, in the pitch and yaw orientation, respectively in (18) and (19), the magnetic torque is expressed as:
41
τ
pitch=
VM
ZB
Zsin(
α
)
(18)
τ
yaw=
VM
YB
Ysin(
α
)
(19)
where α is the desired angle between the IPM magnetization and the EEC magnetic field direction.
In the paragraph “5.3 – Analysis of magnetic forces and torques generated by final EES” will be detailed the relevant calculation.
4.4 – Methods of magnetic field simulation
To have the correct dimensioning of the electromagnet and to guarantee the generation of the required torques and forces, some methods are necessary to calculate and simulate in advance which is the magnetic field that EES can generate in the space.
In this thesis, it was chosen to simulate the generated magnetic field using two different approaches:
- finite element method (FEM), using a software simulation named COMSOL (Comsol Multiphysics 4.3a, Inc., Sweden);
42
4.4.1. Comsol
COMSOL Multiphysics is a powerful interactive environment for modeling and solving scientific and engineering problems.
This software simulation offers many modules for analyze mechanical, electrical, magnetic and chemical phenomena.
For this project, the interested module is represented by AC/DC module (Fig.26), which allows to find solutions relevant to electromagnetic field problems.
Inside this module, the user interface chosen is “Magnetic Field”, in which Ampere’s law is solved for a magnetic vector potential.
Fig.26. Comsol modules
The study of EES magnetic flux density is considered in a static condition where the current and the field have no time variation. In this case, magnetostatic Maxwell’s equations are:
43
∇
×
H
=
J
(20)
∇ B
⋅
=
0
(21)
where:
H (A/m) = magnetic field intensity;
B (T) = magnetic flux density;
J (A/m2) = current density.
The equation (20) is the Ampere’s law for static cases. The magnetic flux density and the magnetic field intensity are united in a linear constitutive relation:
B
=
µ
H
(22)
where µ is the magnetic permeability of free space (4π∗10−7 Α/m).
For the resolution of the equations (20) and (21), it is more convenient to use a vector potential A.
With the introduction of this vector potential A, it is possible to express the flux density B in the following way:
∇
⋅
B
=
0
⇒
B
=
∇
×
A
(23)
Substituting the equation (23) in the (21), the result is:
44
In this equation the vector potential A is represented by an approximation function and the current density is the input parameter for the simulation of the magnetic flux density.
For a coil, the current density is expressed as:
S
I
N
J
=
tot*
(25)
where:Ntot = number of windings of the EES;
I(A) = current intensity;
S(m2) = EES section.
This software computes the solution of these differential equations using a numerical method, which is the finite element method (FEM). In this approach, the space is divided into a finite number of subregions, called elements, and the nodes define each element. These elements can betriangles and quadrilaterals for 2D domains (Fig.27), hexahedral and tetrahedral for 3D domains.
Fig.27. Comsol mesh
ELEMENT NODE
45
The approximating function of the vector potential A is calculated for each node of the mesh in the space and combining the value of A with the equation (23), it is possible to compute the value of B.
4.4.1.1 Comsol simulation
After a short description of the principle on how Comsol software works, in this paragraph it will be explained the steps to have the value of the magnetic flux density along a direction [15].
Since the electromagnet has a cylindrical shape, it is possible to take advantage from its symmetry and as consequence, the model is built using the 2D interface.
The required steps are:
1) definition of geometries; 2) definition of materials; 3) definition of magnetic field; 4) definition of mesh;
5) definition of results.
1) Since the simulation is in 2D, the coil is represented as a rectangle and it is positioned respect to the symmetry axis (Fig.28). In particular, “R1” is rectangle, that represents the space around the electromagnet; R2” represents the difference between the external and internal radius and “R3” represents the internal radius;
46
Fig.28. Definition of geometry
2)
The selected materials are (Fig.29): for the rectangle “R2”, the chosen material is copper, because it represent the space that contains the wire windings; for the rectangle “R3”, the chosen material is air or iron, depending on what is inserted in the electromagnet core; for the rectangle “R1”, the chosen material is air, because it used to simulate the space around the electromagnet;
47
3) As defined in equation (24), to have the magnetic field value, the input parameter is the external current density J and its value depends from the current intensity and the wire diameter. In this case, the interested J component is along θ axis and this density is applied to rectangle “R2”, because it represents the coil windings (Fig. 30);
Fig.30. Definition of external current density
4) The selected elements, in which the space is divided, are triangles and their size has been chosen as “extra fine”, in order to obtain a high number of nodes in which the magnetic field is calculated (see Fig.27) and to minimize the error even if there is a higher computational time;
5) At this point, the simulation allows to have the distribution of magnetic flux density, which intensity is represented as a color map (Fig.31). Also it is possible to have the magnetic field direction (the red arrows in Fig.31) inside and outside the electromagnet.
48
Fig.31. Result of simulation
To know what is the trend of the magnetic field in a precise direction, it is possible to use a line (named CUTLINE) and the output is a series of discrete values, that are located in the intersection between the cut line and the meshed values (Fig. 32).
49
4.4.2 Analytical model
The calculation of coil magnetic field can be obtained also with an analytical model [16]. It was selected this model, because it allows the computation of the magnetic field in points outside the electromagnet.
In fact, when an electric current is passed inside the solenoid windings, it produces a uniform magnetic field also in a volume of space outside of the solenoid itself.
Applying the Biot-Savart’s law to a single loop, it is possible to have the magnetic field generated by just a winding (Fig.33):
3
4
r
d
I
l
ρ
B
=
∫
×
π
µ
(26)
The equation (26) represents the sum of magnetic fields generated by each element
Idl
of which the loop is composed and this calculation regards the magnetic field in the internal point P of the turn.50
The magnetic field generated over the length of a solenoid [-L/2,L/2], Fig.34, is expressed in the equation (27):
−
=
∫ ∫
∫ ∫
∫ ∫
− − − 2 / 2 / 2 0 2 / 2 / 2 0 3 3 2 / 2 / 2 0 3cos
0
cos
4
L L L L L Ldz
d
R
dz
d
b
dz
d
L
IbN
π π πθ
ρ
θ
ρ
θ
θ
ρ
θ
ξ
π
µ
B
(27)
Fig.34. Solenoid where:
B (T) = magnetic field intensity;
µ
= magnetic permeability of the free space;I (A) = current intensity;
N = number of wire turns;
b (m) = solenoid radius;
L (m) = solenoid height;
R (m) = coordinate along the r axis;
51 ζ1 (m) = Z L − − 2 ; ξ2 (m) = Z L − 2 .
For the aim of this thesis, it is necessary to know what is the magnetic field created in the space outside the electromagnet and an approximation of the equation (27) allows to have the magnetic fieldcomponents in an external point Q (Fig.35), as expressed in equation (28):
Fig.35. Magnetic field at an external point Q
(
)
(
)
[
]
(
)
(
)
[
]
) 28 ( 0 4 2 1 2 / 1 2 2 2 / 3 2 2 3 2 2 / 3 2 1 2 2 / 3 2 2 2 2 + − + + − + = = = − − − − − ζ ζ ζ ζ θζ
ζ
ζ
ζ
ζ
ζ
µ
R R R R R R L N Ib B B B z rTherefore, this equation is valid only when b << |R|, that means that the solenoid radius has to be lesser than Q radial distance from the axis of the solenoid.
52
EES will have an internal core, in which it is possible to have air or iron. Since the model computes the magnetic field only for air core, it will be necessary to find a proportional coefficient, that allows to compare the result between FEM simulation and the analytical calculation, when the first will be computed with an iron core
.
4.5 – Validation of magnetic field simulation models
The aforementioned methods, described in section 4.4, have to be validated for the design of the EES in terms of accuracy.
The results of FEM simulations and of analytical model are here analyzed, for evaluating the magnetic field strength along the interested direction for C1 and C2 configurations.
For the calculation of magnetic flux density, it is necessary to define some parameters, such as: external and internal diameter, height, number of windings, and current density. For that purpose, the coil used to validate these method has the following characteristics:
- height = 120 mm;
- external diameter = 150 mm; - internal diameter = 20 mm; - number of windings = 740
and the coil is supplied with a current of 5 A, that corresponds to a current density of 0.48 A/mm2.
53
1) Magnetic field in configuration C1:
as already explained, in this configuration the magnetic field on interest is along the Z axis component and its trend is observed from a distance of 5 cm from the EES edge.
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 1 2 3 4 5 6 7 8 x 10-3 Distance [m] M a g n e ti c f ie ld [ T ] FEM Model
Fig.36. Magnetic fields in configuration C1
The blue curve, in Fig.36, shows the magnetic field trend calculated with FEM simulation, in which the internal diameter of EES is filled with an iron core. Instead, the red curve, still in Fig.36, shows the magnetic field trend calculated with the modeland in this case the EES internal diameter is an air core, in according with the analytical expressions.
The ratio between the two curves allows to find a mean proportional coefficient equal to 1,8, that applied to the analytical model provides to simulate the iron core in analytical model. Fig.37 shows a “like for like” comparison between the two models highlighting a good correspondence among them.
54 8 . 1 _ = = Model FEM t coefficien Iron 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 1 2 3 4 5 6 7 8 x 10-3 Distance [m] M a g n e ti c f ie ld [ T ] FEM Model
Fig.37. Magnetic fields graph in C1 with iron coefficient
2)
Magnetic field for the attraction in configuration C2:
the magnetic field is evaluated at the edge of coil and, the component is along R axis. Also in this condition, the trend is observed from a distance of 5 cm from the EES edge
.
In the Fig.38 it is represented the trend of FEM, with iron, and model, with air.
In the Fig.39, applying the same iron coefficient, still equal to 1.8, it is possible to have a correspondence between the two methods.
55
Fig.38 Magnetic fields for the attraction in configuration C2
0.050 0.1 0.15 0.2 0.5 1 1.5 2 2.5x 10 -3 Distance [m] M a g n e ti c f ie ld [ T ] FEM Model
Fig.39. Magnetic fields graph in C2 with iron coefficient
3) Magnetic field for the locomotion in configuration C2 :
for the locomotion, the magnetic field is evaluated along Z axis and its trend is considered for the entire length of coil (Fig. 40).
56
Fig.40. Magnetic fields for the attraction in configuration C2
Also in this case, it is necessary to consider the derived iron coefficient (Fig.41) to have the comparison with the two magnetic field simulations.
0 0.02 0.04 0.06 0.08 0.1 0.12 5 6 7 8 9 10x 10 -4 Distance [m] M a g n e ti c f ie ld [ T ] FEM Model
Fig.41. Magnetic fields graph in C2 locomotion with iron coefficient
Finally, the mean error and the standard deviation are calculated for each configuration (Tab.2) and the obtained results demonstrate a good matching between two methods.
57
MEAN ERROR
±
STANDARD DEVIATION [T]FEM-Model in C1 attraction
(1.46 ± 1.40)*10
-4 FEM-Model in C2 attraction(0.50 ± 0.15)*10
-4 FEM-Model in C2 locomotion(0.30 ± 0.26)*10
-4Tab.2. Mean error and standard deviation
In order to derive the above graphs and the reported errors, some commands have been implemented in Matlab. In appendix A it is reported the followed steps for the configuration C1.
After, the FEM results are compared with a real condition; for this reason, a hall-effect probe (Koshava 5, Wuntronic GmbH, Germany), supported by a robotic arm (Mitsubishi Melfa RV-6SL), has been used to measure the magnetic field generated by an EES (specifications are the same presented before). The probe has been positioned in the appropriate directions to have the same conditions used in the simulations.
The following figures show the comparison between the FEM simulation outputs and the experimental data. In this case, both curves are obtained with the presence of iron core.
In the Fig.42, the blue curve represents the magnetic field calculated with FEM method and the green curve represents the magnetic field measured experimentally, when the configuration is C1.
58 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 1 2 3 4 5 6 7 8 x 10-3 Distance [m] M a g n e ti c f ie ld [ T ] FEM Measured
Fig.42. Magnetic field in C1 for FEM simulation and measured with probe
In the Fig.43 the magnetic field is considered in C2 configuration in the phase of attraction (C2).
0.050 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10 -3 Distance [m] M a g n e ti c f ie ld [ T ] FEM Measured
Fig.43. Magnetic field in C2 for FEM simulation and measured with probe
In the Fig.44 the magnetic field is considered in C2 configuration in the phase of locomotion.
59
0 0.02 0.04 0.06 0.08 0.1 0.12 5 6 7 8 9 10x 10 -4 Distance [m] M a g n e ti c f ie ld [ T ] FEM Measured
Fig.44. Magnetic field in C2 for FEM simulation and measured with probe
In Tab.3 the mean errors and the standard deviation are reported for each case. Also this comparison between obtained values with FEM simulations and obtained values with a real measures allows to have a good matching between the relevant output.
The difference between FEM and measured data is mainly due to a manual error in positioning the probe.
MEAN ERROR
±
STANDARD DEVIATION [T]FEM-Measured in C1 attraction
(1.88 ± 1.65)*10
-4 FEM-Measured in C2 attraction(0.15 ± 0.08)*10
-4FEM-Measured in C2 locomotion
(0.11 ± 0.06)*10
-460
To decide what is the better method to use for calculating the magnetic field generated by EES, it is necessary to compare the experimental data respectively with FEM simulations and with the analytical model outputs.
In the following figures (Fig.45, Fig.46 and Fig.47) the magnetic field calculated with FEM, with analytical model and measured with the probe are shown all together.
Fig.45. Comparison of magnetic fields between three methods in C1 attraction
Fig.46. Comparison of magnetic fields between three methods in C2 attraction
61
Fig.47. Comparison of magnetic fields between three methods in C2 locomotion
Considering the three conditions (C1-attraction, attraction and C2-locomotion) has been evaluated:
- the average error between analytical model outputs and experimental data which is equal to (0.74 ± 0.58)*10-4 T;
- the average error between FEM outputs and experimental data which is equal to (0.71 ± 0.6)*10-4 T
it is possible to conclude that FEM simulationscan be adopted for computing the magnetic field, which value will be used for the determination of magnetic force and torque during EES design.
