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Modelling of magnetic field distributions of elliptical cylinder permanent magnets with diametrical magnetization
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
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Research articles
Modelling of magnetic field distributions of elliptical cylinder permanent magnets with diametrical magnetization
T
Van Tai Nguyena,b, , Tien-Fu Lub ⁎
a b
Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam School of Mechanical Engineering, University of Adelaide, SA 5005, Australia
ARTICLE INFO
ABSTRACT
Keywords: Elliptical cylinder Modelling of magnetic field Permanent magnet Diametrical magnetization
This research presents semi-analytical and closed-form models to calculate the magnetic field distribution of elliptical cylinder permanent magnets with uniform diametrical magnetization at any point in three dimensional (3D) space. Using the magnetic charge approach, an accurate and fast-computed model is derived. The semianalytical model yields results in excellent agreement with those of Finite Element Analysis. Moreover, it took less than 0.65 ms to compute each component of the magnetic field of the cylinder on a modern personal computer, which demonstrates its efficiency over the well-known Finite Element Analysis method, in terms of computation time. The accuracy and efficiency of the closed-form expressions are analysed and compared with the semi-analytical model. Two and three dimensional analyses of the magnetic field distribution of diametrically magnetised cylinders with different elliptical profiles are also conducted in this study, using the derived model. The analytical model can be used to calculate the magnetic field of an annular elliptical cylinder using the principle of superposition. In cases where the major and minor semi-axes of the elliptical cylinder are equal, it becomes a circular cylinder; therefore, the derived model can be used to compute the magnetic field of a circular cylinder with diametrical magnetization, which can be shown to outperform the existing analytical model in terms of computational cost.
1. Introduction In electrical machines, permanent magnets have been widely used for many years. Recently, they have been implemented extensively in non-contact sensing applications [1–3], permanent magnetic gears [4–6], permanent magnetic couplings [7,8], permanent magnet-bearings [9], non-contact cam mechanisms [10], energy harvesters [11,12] and magnetic guns [13]. In practical applications, the computation of the magnetic field, interaction forces and torques between permanent magnets is of great importance [14–18]. While the expression of these forces can be derived analytically and semi-analytically using the interaction energy [14,15], they and the interaction torques can be expressed based on the Coulombian approach [16,17] if the magnetic field is modelled. Moreover, understanding the magnetic field distribution of a permanent magnet is necessary for many other purposes, such as non-contact sensing and magnetic resonance imaging applications [18]. Therefore, in order to facilitate the design and dynamical modelling of a system using permanent magnets, many methods have been devised to compute and study their magnetic field distribution. The most common one is the well-known Finite Element Analysis (FEA)
⁎
based method. However, this method is time-consuming when high precision is required, due to the smaller mesh that needs to be applied. To address the disadvantages of the FEA, semi-analytical and analytical approaches have been developed to calculate the magnetic field of permanent magnets in different shapes [17]. Nguyen et al. [19] developed exact analytical expressions of the magnetic field distribution of circular cylinder permanent magnets with diametrical magnetization. Rakotoarison et al. [20] derived the fast-computed semi-analytical model to compute the magnetic field of permanent magnets with radial magnetization. Ravaud et al. [21] studied the magnetic field created by a parallelepiped magnet with various and uniform magnetization. Kim et al. [22] studied the polarization of a permanent magnet to produce the predefined magnetic field distribution. Acevedo et al. [23] studied the magnetic field of multipole magnetic configurations analytically. Ravaud et al. [24] developed the analytical model to calculate the magnetic field distribution of permanent-magnet rings with axial and radial magnetizations. However, the magnetic field distribution of elliptical cylinder permanent magnets with uniformly diametrical magnetization (the magnetization is uniformly parallel and perpendicular to the central axis of the cylinder) has not yet been modelled analytically
Corresponding author at: Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam. E-mail addresses: [email protected], [email protected] (V.T. Nguyen).
https://doi.org/10.1016/j.jmmm.2019.165569 Received 15 March 2019; Received in revised form 11 June 2019; Accepted 13 July 2019 Available online 16 July 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
and studied in the literature. This could be because of the highly nonlinear profile, which increases the difficulty of modelling the field distribution of a permanent magnet of this kind. It is the fact that elliptical profiles can be found both in nature and literature, for example, the orbit of each planet in the solar system is approximately an ellipse [25] and elliptical gears have been found to acquire great efficiency in motion and power transmission solutions [26–28]. This could lead to the curiosity of understanding the magnetic field distribution of objects with elliptical profiles [29] as well as their potential application. Therefore, it is of great interest to study the magnetic field distribution of cylinder permanent magnets with an elliptical profile. It can also be agreed that proper understanding and having a fast-computed model of the magnetic field distribution of permanent magnet of this kind are important for its application. For instance, an accurate and fast-computed analytical expression of the magnetic field would assist in the design and dynamic modelling of systems using a permanent magnet of this kind, such as the design and dynamical modelling of an elliptical permanent magnet used in surface magnetic resonance imaging [30], the design and dynamic modelling of the system of a novel magnetic coupler [31], and the design and analysis of permanent magnet gears [4–6] and couplings [7,8] with elliptical profiles. Furthermore, in education, teachers would have a novel collection of a fast-computed analytical model of the magnetic field distribution to demonstrate the physics behind the elliptical cylinder permanent magnet to their students, beyond those commonly found in textbooks like ring-shaped, circular cylinder, parallelepiped and cuboidal permanent magnets [17]. Therefore, this study aims to model the magnetic field distribution of an elliptical cylinder permanent magnet with diametrical magnetization analytically. As a result, an accurate and fast-computed analytical model of the magnetic field has been developed. Using the novel model, two and three dimensional analyses of the magnetic field distribution of an elliptical cylinder have been conducted. The derived model is validated against the well-known Finite Element Method. The efficiency of the analytical model is analysed. The rest of this paper is organized as follows. Section 2 describes in detail the mathematical modelling of the magnetic field distribution. Section 3 presents the Finite Element Analysis and results. Section 4 analyses the magnetic field distribution and Section 5 draws the conclusion.
Fig. 2. Top view of the elliptical cylinder.
Fig. 3. Schematic of the elliptical profile of the cylinder [32].
2. Mathematical modelling
magnetization vector J of the cylinder is uniformly parallel and perpendicular to the central axis of the cylinder, and Ξ is the angle between J and the azimuthal axis X (Fig. 2). The cylinder has a finite length of h (Fig. 1); the parameters of the elliptical profile are depicted in Fig. 3, where F1 and F2 are the foci; a and b are the lengths of the major and minor semi axes, respectively. Hence, the ellipse has the function represented by Eq. (1). It also can be described by the parametric representation using the sine and cosine functions as V = (x, y) = (acosτ, bsinτ), 0 ≤ τ < 2π; here τ can be defined as shown in Fig. 3, based on de la Hire [32], where A is on a circle with a radius of a and B is on a circle with a radius of b.
The geometry of a computed cylinder with an elliptical profile is presented in Fig. 1. Figure 2 presents the top view of the cylinder and illustrates the surface charge distribution s . It is assumed that the
y2 x2 + 2 =1 a2 b
(1)
Based on the magnetic charge or Coulombian approach [17], the magnetic field intensity HK at any point K (r, α, z) in a cylindrical coordinate system, which lies on plane O′X′Y′ (Fig. 1), can be calculated using Eq. (2)
HK =
1 4 µ0
s s
|PK |3
v
PK ds + v
|PK |3
PK dv
(2)
where µ 0 is the permeability of the vacuum, P is a point on the surface of the cylinder (Fig. 1), s is the surface charge density, and v is the
Fig. 1. Geometry of a computed elliptical cylinder. 2
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
volume charge density. The volume charge density v in Eq. (2) can be calculated as the divergence of the magnetization vector: v
=
(3)
·J
Since the magnetisation vector J is uniformly parallel, its divergence is equal to zero, that means v = 0. In other words, the volume charge has no contribution to the magnetic field; hence, the magnetic field intensity HK at point K can be calculated using only the surface charge:
HK =
1 4 µ0
s
|PK |3
s
PK ds
where the surface charge density s
(4) s
can be calculated as follows:
Fig. 5. Parameters used in the mathematical formulation.
(5)
=J. n
Here n is the unit vector normal to the surface of the cylinder. On the top and bottom surfaces of the cylinder, the angles between the unit vectors n of these surfaces and vector J are 90°. Therefore, the surface charge density of these surfaces s = 0 . This means only the cylindrical surface of the cylinder contributes to the computed magnetic field. Fig. 4 depicts the tangent and normal vectors of the elliptical profile of the cylinder as its projection on the O′X′Y′ plane. A tangent vector of the ellipse at point P′(x, y) can be defined as follows:
From Fig. 3, we have
acos = r0 cos ¸
(10)
bsin = r0 sin ¸
(11)
P0 is a point on the elliptical profile. Inserting Eqs. (10) and (11) into Eq. (9) yields: s
t =
dV = ( asin , bcos ) d
1 b2cos2 + a2sin2
= J. n =
a 2 b
2
r02
()
a
+ b JY sin +
( ) sin a 2 b
2
(14)
Inserting Eq. (14) into Eq. (12) with some simplifications, we achieve:
r0 =
(8)
ab (15)
a2sin2 + b2cos2
The infinitesimal area ds in Eq. (4) can be computed as follows: 2
dx d
ds = dldzP =
bJXcos + aJYsin b2cos2 + a2sin2
( ) sin
b J cos a X b 2 cos2 a
Dividing Eq. (10) by Eq. (11), we achieve:
Since the magnetization vector J can be presented as the sum of projected vector JX and JY on axes O′X′ and O′Y′ (Fig. 5) J = JX + JY , the surface charge density s is derived as follows: s
+
=
a cot = cot b
(7)
(bcos , asin )
()
a
+ b JY sin r0
(13)
Therefore, the unit vector n can be expressed by dividing the above normal vector by its magnitude to achieve:
n =
b J cos r0 a X b 2 cos2 r0 2 a
= J. n =
(6)
Rotating the tangent vector t by 90° in a clockwise direction, a normal vector N of the ellipse at the given point is achieved:
N = (bcos , asin )
(12)
a2cos2 + b2sin2
r0 = O'P0 =
+
dy d
2
d dzP =
b2cos2 + a2sin2 d dzP
(16)
Here, dl is the infinitesimal length of the elliptical profile. Inserting Eqs. (13) and (16) into Eq. (4) produces:
(9)
HKs =
1 4 µ0
=2 =0
zP = h PK zP = 0 |PK|3
b a JX cos + JY sin a b b 2 a 2 2 cos2 + sin a b
()
()
(17)
b 2cos2 + a2sin2 dzP d
×
Taking the derivative of Eq. (14) with some simplifications gives:
d =
d b cos2 a
a
+ b sin2
(18)
The component using Eq. (14) as follows:
b2cos2
b2cos2 + a2sin2 =
+
a2sin2
can be expressed through
b 4cos2 + a4sin2 b2cos2 + a2sin2
by
(19)
Inserting Eqs. (18) and (19) into Eq. (17) with some simplifications produces:
HKs =
Fig. 4. Tangent and normal vectors of the elliptical profile. 3
1 4 µ0
=2 =0
zP = h zP = 0
PK ab3JX cos + ba3JY sin d dzP 2 2 2 2 3/2 |PK|3 (b cos + a sin )
(20)
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
From Figs. 1 and 5, the vector PK projected on the radial, azimuthal and axial ( ur, u and uz are the unit vectors respectively) as follows:
can be directions
PK = PP' + P'P" + P"K = (z zP ) uz + r0sin( - ) u + (r r0cos(
)) ur
(21)
Here zP is the axial coordinate of point P, and z is the axial co-
ordinate of points K and P'; is the angle between vector O'P' and axis O′X′ (Fig. 5). Inserting Eq. (21) into Eq. (20) gives:
HKs =
=2 =0
1 4 µ0
zP = h (z zP ) uz + r0sin( - ) u + (r r0cos( - )) ur 3 zP = 0 (r02+r 2 2r0rcos( - )+(z zP )2) 2 ab3JX cos + ba3JY sin
×
d (b 2cos2 + a2sin2 )3/2
dzP
(22)
After integrating Eq. (22) based on z, the axial, azimuthal and radial components can be expressed as follows (Appendix A). 2.1. The axial component H(3D) K(z) (r, , z) The axial component H(3D) K(z) (r, , z) was derived as follows:
H(3D) K(z) (r, , z) =
1 4 µ0
=2 =0
1 h)2 + 2
(z
ab3JX cos + ba3JY sin
×
(b2cos2 + a2sin2 )3/2
1 z2 + 2
d
(23)
). where 2 = r02 + r 2 2r0 r cos( Eq. (23) can be further approximated using Simpson’s method (Appendix B) to achieve the closed-form expression, as follows:
H(3D) K(z) (r, , z) = ×
1 2N µ 0
N n=0
1 (z
× where
S (n )
Fig. 6. An annular cylinder with an elliptical profile: (a) three dimensional (3D) geometry; (b) top view of the cylinder.
1
h)2 + 2 ( (n))
z2 + 2 ( (n))
ab3JX cos (n) + ba3JY sin (n)
2(
(n)) = r0 2 + r2
(Appendix B) to obtain the closed-form expression, as follows:
(23a)
(b2cos2 (n) + a2sin2 (n))3/2
H (3D) K(r) (r, , z) =
2r0rcos( - (n)) .
×
2.2. The tangential (Azimuthal) component H(3D) K( ) (r, , z) The azimuthal component H(3D) K( ) (r, , z) was obtained, as follows:
H(3D) K( ) (r, , z) = ×
1 4 µ0
r0sin(
=2 =0
h z 2
2 + (h
z)2
×
z
+
2
2+ z2
)(ab3JX cos + ba3JY sin ) d (b 2cos2 + a2sin2 )3/2
×
×
1 2N µ 0
h z 2 ( (n))
2 ( (n))+ (h
z)2
+
N n=0
(24)
S (n ) z
2 ( (n))
The radial component H(3D) K(r) (r, , z) was obtained, as follows:
×
=2 =0
h z 2
2 + (h z)2
+
(r r0cos( ))(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
+
z 2 ( (n))
z 2
2 ( (n))+ z2
(r-r0cos( - (n)))(ab3JX cos (n) + ba3JY sin (n)) (b2cos2 (n) + a2sin2 (n))3/2
(25a)
HK (ai, b i)
(26)
where HK (a o, bo) is the magnetic field created by an elliptical cylinder with the outer parameters of major semi-axis a o and minor semi-axis bo . On the other hand, HK (ai, bi) is the magnetic field created by an elliptical cylinder with the inner parameters of major semi-axis ai and minor semi-axis bi . Both cylinders have the same thickness of h and magnetization J as illustrated in Fig. 6(a) and (b). From the magnetic field intensity HK , the magnetic flux density BK can be computed with
(24a)
2.3. The radial component H(3D) K(r) (r, , z)
1 4 µ0
z)2
HK (annular) = HK (a o, bo)
2 ( (n))+ z2
r0sin( - (n))(ab3JX cos (n) + ba3JY sin (n)) (b 2cos2 (n) + a2sin2 (n))3/2
H(3D) K(r) (r, , z) =
2 ( (n))+ (h
S (n )
In the case of a = b, the elliptical cylinder simplifies to a circular cylinder; hence the derived expressions could also be implemented to calculate the magnetic field created by a circular cylinder permanent magnet with uniformly diametrical magnetization. In the case of an annular cylinder with an elliptical profile (Fig. 6), the magnetic field of this cylinder HK (annular) at any point K can be computed using the principle of superposition, as follows:
Once again, Eq. (24) can be further approximated using Simpson’s method (Appendix B) to obtain the closed-form expression, as follows:
H (3D) K(z) (r, , z) =
h z 2 ( (n))
N n=0
1 2N µ 0
2+ z2
(25)
Eq. (25) can be further approximated using Simpson’s method 4
BK=µ 0 HK (in the air space)
(27)
BK=µ 0 HK + J (inside the magnet)
(28)
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
3. Finite Element Analysis (FEA) and Comparison results 3.1. Comparison of semi-analytical expressions with FEA The semi-analytical expressions of the axial (Eq. (23)), the azimuthal (Eq. (24)) and the radial (Eq. (25)) components were implemented in MATLAB R2016b (MathWorks, Natick, MA, USA) to compute the magnetic field of an elliptical cylinder with diametrical magnetization made of rare earth material with the finite length h = 5 mm; major semi-axis a = 6 mm; minor semi-axis b = 3 mm; the remanence J = 1 Tesla and the angle between J and the azimuthal axis X is π/6 rad (Ξ = π/6). The global adaptive quadrature and default error tolerances [33] were used to solve the semi-analytical expressions. In order to verify the results of the analytical model, Finite Element Analysis (FEA) was conducted using Electromagnetic Simulation Software® (EMS) (EMWorks, Inc., Montreal, Quebec, Canada) for the elliptical cylinder with the same parameters. The FEA aims at solving the following governing equations Eqs. (29)–(31) (Maxwell’s equations) of magnetic field for the magnetostatic analysis [34]:
× H = Js
(29)
·B = 0
(30)
And the constitutive relation connects B and H (31)
B =µ (H + H c)
Here H (A/m) is the magnetic field intensity; B (T) is the magnetic flux density; Js (A/m2) is the source current density; µ is the magnetic
permeability (H/m) and Hc (A/m) is the coercive force or coercivity. The coercivity of the magnetic material used in this study is µ and the magnetic permeability is 800,000 A·m−1 1.20536 × 10−6 (H/m). Plots of the computed magnetic field distribution on the plane z = 7 mm and along the radial distance with an azimuthal angle α = π/ 3 (rad) and r from −20 mm to 20 mm are depicted in Fig. 7((a) Axial component; (b) Radial component; (c) Azimuthal component). The values of the magnetic field of 12 random points extracted from Fig. 7 are listed in Table 1. The errors between the results obtained by the analytical model and by FEA, which are estimated using Eq. (32) are also defined in Table 1.
Error =
BAnalytical
BFE model
BFE model
× 100%
(32)
Here BAnalytical – the magnetic flux density computed using the semianalytical model; BFE model – the magnetic flux density computed using the FEA. It is demonstrated based on Fig. 7 and Table 1 that the results of the analytical model derived in this research are in excellent agreement with those computed using the FEA (the average errors are 0.23% for the axial component; 0.38% for the azimuthal component and 0.16% for the radial component). It is noted that, when evaluated in MATLAB R2016b on a personal computer (Intel Core i7-6700 CPU, 3.40 GHz), it took an average of 0.6 ms to compute the radial component, 0.4 ms to compute the azimuthal component and 0.35 ms to compute the axial component, using the derived expressions at a single location (4000 sample points were used in this evaluation). On the other hand, it could take up to several hours with a small scale of mesh (the smaller the mesh is, the more computation time it consumes) to complete the simulation using FEA on the same personal computer.
Fig. 7. FEA verification results (the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m: (a) Axial component; (b) Radial component; (c) Azimuthal component.
simplifies to a circular cylinder. In the work of Nguyen et al. [19], an exact analytical model based on complete and incomplete elliptic integrals was developed to compute the three components of the magnetic field of a circular cylinder with diametrical magnetization. In order to demonstrate the efficiency of the developed model in this study, the semi-analytical expressions Eqs. (23)–(25) and the exact analytical model were used to calculate the magnetic field generated by a circular cylinder with the geometrical parameters as follow: the radius is R = a = b = 0.005 m; the thickness h = 0.005 m; the remanence J = 1 Tesla and vector J is along Y axis; Plots of the computed magnetic field distribution on the plane z = 7 mm; the radial distance r = 0.01 m and the azimuthal angle α = [0, 2π] (rad) are depicted in Fig. 8. It is observed that the results obtained from the semi-analytical
3.2. Comparison of a semi-analytical model and an existing analytical model In a case where the two semi-axes are equal, an elliptical cylinder 5
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Table 1 Semi-Analytical results vs FEA results for 12 points with random r (mm) (Max – Maximum; Min – Minimum; Aver – Average). Points (r mm, 60° , 7 mm)
Axial component (Gauss) Semi-Analytical results
−19.59 −18.59 −16.78 −12.96 −9.75 −6.13 0.10 5.13 9.75 13.17 15.78 18.19
−15.4837 −18.7911 −27.2918 −67.1742 −164.5458 −507.2350 31.2412 672.4053 164.5458 63.6438 34.0078 20.3533 Max 0.45
Aver 0.23
Azimuthal component (Gauss)
FEA results
Semi-Analytical results
−15.4523 −18.7644 −27.2893 −67.1613 −165.0748 −504.9246 31.3573 675.3980 164.7702 63.5470 33.9344 20.2955 Min 0.009
15.1991 17.7630 24.0408 50.4986 106.9623 264.7460 215.6923 318.1446 106.9623 48.2798 28.7752 18.9469
Max 0.82
Aver 0.38
FEA results 15.1212 17.6924 23.9857 50.2791 107.0150 262.5984 216.7766 318.4012 106.4388 48.0761 28.7032 18.8860 Min 0.05
Radial component (Gauss) Semi-Analytical results 42.4949 48.6420 62.8861 113.5919 189.1612 229.0508 −885.6404 142.7812 189.1612 109.8084 72.9705 51.4120 Max 0.29
Aver 0.16
FEA results 42.3682 48.5431 62.8546 113.2647 189.0493 228.7799 −884.6202 143.1480 189.1907 109.5542 72.8962 51.3024 Min 0.05
Fig. 10. Accuracy of the closed-form expression against the semi-analytical model of the azimuthal component ((the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m).
Fig. 8. Comparison of semi-analytical model and existing analytical model [19] (the parameters of the used elliptical cylinder are: R = a = b = 0.005 m, h = 0.005 m, J = 1 T along Y axis); plots along the azimuthal angle α = [0, 2π] (rad) with z = 0.007 m, and the radial distance r = 0.01 m: Bax, Baz and Br are the axial, azimuthal and radial components, respectively; indexes E and S indicate the magnetic field computed using the exact [19] and semi-analytical models.
Fig. 11. Accuracy of the closed-form expression against the semi-analytical model of the radial component ((the parameters of the elliptical cylinder used are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m): singular regions of radial distance (r) are [−0.0049 m, −0.0039 m] and [0.0042 m, 0.0046 m].
Fig. 9. Accuracy of the closed-form expression against the semi-analytical model of the axial component ((the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m): Singular region of radial distance (r) is [−0.0058 m; 0.0072 m]. 6
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 12. Average computational times using closed-form and semi-analytical expressions: Closed-Axial (Semi-Axial), Closed-Azimuthal (Semi-Azimuthal) and ClosedRadial (Semi-Radial) denote the computational costs of the closed-form (semi-analytical) expressions of the axial, azimuthal and radial components, respectively.
Table 2 Accuracy and efficiency analysis of the closed-form expressions (Max – Maximum error; Min – Minimum error; Aver – Average error; Time – Average execution time; * denotes the average computational costs using the semi-analytical expressions (see Section 3.1)). Axial component
Azimuthal component
Radial component
Nθ
Max (%)
Min (%)
Aver (%)
Time (ms) 0.35*
Max (%)
Min (%)
Aver (%)
Time (ms) 0.4*
Max (%)
Min (%)
Aver (%)
Time (ms) 0.6*
100 400 700
2.15 0.53 0.31
0.23 0.056 0.032
0.52 0.16 0.09
0.09 0.35 0.62
2.87 0.72 0.36
0.56 0.14 0.08
1.48 0.35 0.17
0.12 0.48 0.84
2.95 0.74 0.42
5e−12 7e−14 4e−14
0.96 0.25 0.12
0.12 0.49 0.84
model are in excellent agreement with those of the exact analytical model; the maximum, minimum and average errors between these two models are lower than 0.008% for all three components (Bax, Baz and Br (Fig. 8) are the axial, azimuthal and radial components, respectively; indexes E and S indicate the magnetic field computed using the exact [19] and the semi-analytical models, respectively). However, the computational costs of the semi-analytical expressions are 14 times faster for the axial component, 240 times faster for the azimuthal component and 360 times faster for the radial component than those of the exact analytical model. (The simulation was carried out on the same personal computer with the configuration parameters as presented in Section 3.1). Therefore, the currently developed semi-analytical model has been proved to be more efficient than the analytical model derived by Nguyen et al. [19] in terms of the computational time.
number of meshing points Nθ (written as Nθ in the Figures to be compact) ranges from 20 to 700. Figs. 9–11 depict the errors between these two models for the axial, azimuthal and radial components, respectively. It can be seen from Figs. 9–11 that the errors decrease following the increase of the meshing points. Singular regions are observed for the axial (Fig. 9) and radial (Fig. 11) components. In these singular regions, the maximum error is greater than 200%, the minimum error is greater than the maximum outside the regions and the average error is greater than 100%. Therefore, it is not recommended to apply the closed-form expressions to compute the magnetic field inside the singular regions. The semi-analytical expressions should be used in these cases. As a trade-off between accuracy and efficiency, along with a decrease in the error rate, the average computational times of the three components (with the same conditions used for the semi-analytical model described in Section 3.1, where it is demonstrated that the average computational times using the semi-analytical expressions are 0.35 ms, 0.4 ms and 0.6 ms to compute the axial, azimuthal and radial components, respectively) increase (Fig. 12). Table 2 shows that the closed-form expressions can be utilised to compute the magnetic field efficiently when the number of meshing points is Nθ = 100 with the error rate at less than 3%. When Nθ = 700, the errors are less than 0.5%, however, the computational times are greater than those of the semi-analytical expressions (see Section 3.1). This suggests that the semi-analytical model should be implemented when high accuracy (where the errors are equal to or less than 0.5%) is required. On the other hand, the closed-form expressions could be used when low accuracy (the errors are greater than 0.5%, in this case those points in the singular regions should be excluded) is allowed. It is also worth noting
3.3. Accuracy and efficiency of the closed-form expressions The semi-analytical expressions have been demonstrated to predict the magnetic field of a diametrically magnetized elliptical cylinder correctly with a low computational cost. However, it is still interesting to investigate the possibility of solving these expressions analytically. As presented above, the closed-form expressions of the magnetic field can be obtained approximately using the Simpson’s method with the meshing points Nθ in the θ variable (Appendix B). The closed-form expressions were implemented to calculate the three components of the magnetic field created by an elliptical cylinder. The geometrical parameters and the coordinates of the computed points are the same as those used in Section 3.1. The achieved results are compared with those obtained using the semi-analytical expressions (see Section 3.1). The
7
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 13. Distribution of the axial component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (alpha) with a radial distance (b) the plot along the azimuthal angle r = 10 mm.
Fig. 14. Distribution of the azimuthal component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (alpha) with a radial distance (b) the plot along the azimuthal angle r = 10 mm.
that the suitable number of meshing points should be determined individually, according to the changes in the geometrical parameters of an elliptical cylinder, to achieve the desired accuracy.
demonstration. It is noted that although the different axis ratios c are used, all the ellipses have the same area; in other words, all the cylinders have the same volume. Figs. 13–15 depict the magnetic field distribution of the axial component, the azimuthal component and the radial component, respectively. Figs. 13(a) and 15(a) show that the magnetic field distributions of the axial and radial components of the cylinder with different elliptical profiles along the radial distance have the same form and the magnetic field distribution of the circular cylinder (c3) lies between the those of c1, c2, c4 and c5. Fig. 14(a) shows that the distribution of the azimuthal component is in a different form, depending on the axis ratio c. Figs. 13(b), 14(b) and 15(b) demonstrate that the magnetic field distributions of the axial, azimuthal and radial components of the circular cylinder along the azimuthal angle are in the sinusoidal waveform, whereas the magnetic field distributions of those components of the cylinder with elliptical profiles seem to be in a multiorder, function-like waveform. In order to provide more insight into the magnetic field distribution of an elliptical cylinder, three dimensional (3D) plots are generated for
4. Analysis of the magnetic field distribution The fast-computed semi-analytical model derived in this study enables the generation of the magnetic field distribution of a cylinder with different elliptical profiles. The axis ratio of the major semi-axis and the minor semi-axis c = a/b is implemented to represent the different elliptical profiles. The thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6 (Figs. 1 and 2); the axial coordinate of the computed profile is z = 7 mm. For each magnetic field component, plots are generated along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3 and along the azimuthal angle (alpha) with a radial distance r = 10 mm. There are five elliptical profiles with axis ratios c1 = 8/4; c2 = 16/ 2; c3 = 32 / 32 (circular cylinder); c4 = 2/16; c5 = 4/8 used in this
8
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 15. Distribution of the radial component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (b) the plot along the azimuthal angle (alpha) with a radial distance r = 10 mm.
the axial, azimuthal and radial components, as illustrated in Fig. 16. The used geometry parameters of the cylinder (Figs. 1 and 2) are a = 0.006 m, b = 0.003 m, J = 1 T with Ξ = π/6; plots are generated on the plane z = 0.007 m and in the interval r = [−0.02 m, 0.02 m], alpha = [0 rad, 2π rad]. Fig. 16(a)–(c) demonstrate that the maxima and minima of the magnetic field on the plot region on the plane are multiple and do not necessarily lie on the central axis z of the cylinder. 5. Conclusion This study presents accurate and fast-computed semi-analytical expressions of the magnetic field distribution generated by an elliptical cylinder with uniformly diametrical magnetization at any point in 3D space. The results computed by the semi-analytical model are in excellent agreement with those of FEA. Moreover, the closed-form expressions of the magnetic field are derived and compared with the semianalytical ones. These closed-form model could be used when low accuracy is allowed. On the other hand, the semi-analytical model could be applied when high accuracy is required. The derived models can be used to calculate the magnetic field of an annular cylinder with an elliptical profile and a circular cylinder. The plots of the magnetic field distribution of a cylinder with different elliptical profiles along the azimuthal angle demonstrate that, whilst the circular cylinder has a sinusoidal waveform, the elliptical cylinders have multi-order function-
Fig. 16. 3D plots of the magnetic field distribution of an elliptical cylinder with a = 0.006 m, b = 0.003 m, J = 1 T with Ξ = π/6; the plots are generated on the plane z = 0.007 m and in the interval r = [−0.02 m, 0.02 m], alpha = [0 rad, 2π rad]: (a) axial component; (b) azimuthal component; (c) radial component.
like waveforms. The 3D plots of the magnetic field distribution of an elliptical cylinder on a plane shows that there are multiple maxima and minima of the distribution on the plotted region. Those extrema do not necessarily lie on the central axis of the cylinder. 9
V.T. Nguyen and T.-F. Lu
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
Acknowledgments
Analyses. The authors would like to thank the School of Mechanical Engineering, University of Adelaide, Australia for providing the equipment, facilities and assistances for this research. We are also grateful to Alison-Jane Hunter for her time helping to edit the English.
The authors are grateful to the EMWORKS Company for providing the license for the EMS software to conduct the Finite Element Appendix A
From Eq. (22), the axial component, H(3D) K(z) (r, , z) can be represented as follows:
H(3D) K(z) (r, , z) =
1 4 µ0
×
=2 =0
zP = h zP = 0
z zP (r02+r 2
ab3JX cos + ba3JY sin d (b 2cos2 + a2sin2 )3/2
2r0rcos( - )+(z
3
z P ) 2) 2
dzP
(A.1)
Integrating Eq. (A.1) based on zP yields:
H(3D) K(z) (r, , z) =
1 4 µ0
=2 =0
zP = h
1 (z
z P) 2 + 2
zP = 0
ab3JX cos + ba3JY sin
×
(b 2cos2 + a2sin2 )
d 3/2
(A.2)
Expanding Eq. (A.2) to achieve:
H(3D) K(z) (r, , z) =
1 4 µ0
×
=2 =0
1 h)2 + 2
(z
1 z2 + 2
ab3JX cos + ba3JY sin
d (b2cos2 + a2sin2 )3/2
(A.3)
From Eq. (22), the azimuthal component
H(3D) K( ) (r,
, z) =
1 4 µ0
×
=2 =0
zP = h zP = 0
H(3D) K( ) (r,
r0sin( - ) (r02+r 2
ab3JX cos + ba3JY sin
d (b 2cos2 + a2sin2 )3/2
2r0rcos( - )+(z
, z) can be expressed as follows: 3
z P ) 2) 2
dzP
(A.4)
Integrating Eq. (A.4) based on zP produces:
H(3D) K( ) (r, , z) = ×
1 4 µ0
=2 =0
zP = h
zP z 2
2 + (z
2
zP)
zP = 0
r0sin( - )(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
(A.5)
Expanding Eq. (A.5) gives:
H(3D) K( ) (r, , z) = ×
1 4 µ0
=2 =0
h z 2
2 + (h
z)2
r0sin( - )(ab3JX cos + ba3JY sin ) (b 2cos2 + a2sin2 )3/2
+
H(3D) K(r) (r, , z) =
×
=2 =0
2+ z2
d
(A.6)
From Eq. (22), the radial component 1 4 µ0
z 2
zP = h zP = 0
H(3D) K(r) (r, r
, z) can be expressed as follows:
r0cos( - )
(r02+r 2 2r0rcos( - )+(z ab3JX cos + ba3JY sin d dzP (b 2cos2 + a2sin2 )3/2
3
z P ) 2) 2
(A.7)
Integrating Eq. (A.7) over zP yields:
H(3D) K(r) (r, , z) = ×
1 4 µ0
=2 =0
zP = h
zP z 2
2 + (z
2
zP)
zP = 0
(r-r0cos( - ))(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
(A.8)
Expanding Eq. (A.8) produces:
H(3D) K(r) (r, , z) = ×
1 4 µ0
=2 =0
h z 2
2 + (h
z)2
(r-r0cos( - ))(ab3JX cos + ba3JY sin ) (b 2cos2 + a2sin2 )3/2
+
z 2
2+ z2
d
(A.9)
Appendix B The parameters used in the Simpson’s method mentioned in this research can be obtained as those presented in the work of Furlani et al. [35]. They are described as follows: 10
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Nθ denotes the number of mesh points in the θ variable, and θ(n) indicates the values at which the integrand is evaluated. This yields:
=
2 ; N
(n ) =
2 n (n = 0, 1, 2, …, N ) N
(B.1)
The integration coefficients Sθ(n) are defined as follow,
S (n) =
1 3 4 3 2 3 1 3
(n=0), (n=1, 3, 5…), (n=2, 4, 6…), (n=N ).
(B.2)
2009. [19] V.T. Nguyen, T.-F. Lu, Analytical expression of the magnetic field created by a permanent magnet with diametrical magnetization, Progr. Electromagn. Res. C 87 (2018) 163–174. [20] H.L. Rakotoarison, J.P. Yonnet, B. Delinchant, Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization, IEEE Trans. Magn. 43 (4) (2007) 1261–1264. [21] R. Ravaud, G. Lemarquand, Magnetic field produced by a parallelepipedic magnet of various and uniform polarization, Progr. Electromagn. Res. 98 (2009) 207–219. [22] W. Kim, M. Kim, Y.Y. Kim, Polarization of a permanent magnet to yield specific magnetic field Distribution, J. Appl. Phys. 104 (2008) pp. 064915-1 – 064915-6. [23] M.T.M. Acevedo, V.D.D. Zhabon, O. Otero, Analytical study of the magnetic field generated by multipolar magnetic configuration, J. Phys. Conf. Ser. 687 (2016) 1–6. [24] R. Ravaud, G. Lemarquand, V. Lemarquand, C. Depollier, Analytical calculation of the magnetic field created by permanent-magnet rings, IEEE Trans. Magn. 44 (8) (2008) 1982–1989. [25] H. Karttenen, P. Kroger, H. Oja, M. Poutanen, K.J. Donner, Fundamental Astronomy, fifth ed., Edition, Springer, 2007, p. 507. [26] K. Hideo, The design, manufacture and application of non-circular gears, first ed., Nikkan Industrial Press, Tokyo, 2001. [27] B.W. Bair, Tooth profile generation and analysis of crowned elliptical gears, ASME J. Mech. Des. 131 (2009) 0745031–074503-6. [28] C.F. Chen, C.B. Tsay, Computerized tooth profile generation and analysis of characteristics of elliptical gears with circular-arc teeth, J. Mater. Process. Technol. 148 (2004) 226–234. [29] G.O. McClurg, Magnetic field distributions for a sphere and for an ellipsoid, Am. J. Phys. 24 (1956) 496–499. [30] M. Ciarrocchi, V.D. Miccoli, M. Alecci, A. Sotgiu, A. Galante, Design of an elliptical permanent magnet for surface magnetic resonance imaging, Meas. Sci. Technol. 20 (1) (2008) 1–4. [31] F. Filsoufi, R.S. Farivar, L. Aklog, C.A. Anderson, R.H. Chen, S. Lichtenstein, J. Zhang, D.H. Adams, Automated distal coronary bypass with a novel magnetic coupler (MVP system), J. Thoracic Cardiovasc. Surg. 127 (1) (2004) 185–192. [32] K. Strubecker, Vorlesungen über Darstellende Geometrie, Vandenhoeck & Ruprecht, Gottingen, 1967, p. 26. [33] https://www.mathworks.com/help/matlab/ref/integral.html (accessed 03/2019). [34] EMS’s User Guide, ElectroMagneticWorks Inc. – Available at: www.emworks.com (accessed 03/2019). [35] E.P. Furlani, S. Reznik, W. Jason, A three-dimensional field solution for bipolar cylinders, IEEE Trans. Magnet. 30 (1994) 2916–2919.
References [1] Z. Wu, L. Bian, S. Wang, An angle sensor based on magnetoelectric effect, Sens. Actuat. A-Phys. 262 (2017) 108–113. [2] B. Lenzo, M. Fontana, S. Marcheschi, Trackhold: a novel passive arm-support device, J. Mech. Robot. Trans. ASME 8 (2016) 021007-9. [3] S.-T. Wu, J.-Y. Chen, S.-H. Wu, A rotary encoder with an eccentrically mounted ring magnet, IEEE Trans. Instrum. Meas. 63 (2014) 1907–1915. [4] Y.-C. Wu, C.-T. Chan, A novel bicycle rear hub transmission with a magnetic gear mechanism, Adv. Mech. Eng. 10 (2018) 1–9. [5] W. Zhao, J. Liu, J. Ji, G. Liu, Z. Ling, Comparison of coaxial magnetic gears with and without magnetic conducting ring, IEEE Trans. Appl. Superconduct. 26 (4) (2016). [6] M. Chen, K.-T. Chau, C.H.T. Lee, C. Liu, Design and analysis of a new axial-field magnetic variable gear using pole-changing permanent magnets, Progr. Electromagn. Res. 153 (2015) 23–32. [7] E.P. Furlani, Analytical analysis of magnetically coupled multipole cylinders, J. Phys. D Appl. Phys. 33 (2000) 28–33. [8] R. Ravaud, V. Lemarquand, G. Lemarquand, Analytical design of permanent magnet radial couplings, IEEE Trans. Magn. 46 (11) (2010). [9] B. Paden, N. Groom, J.F. Antaki, Design formulas for permanent-magnet bearings, J. Mech. Des. 125 (2003) 734–738. [10] T. Takazakura, T. Morita, T. Sugiura, Dynamical behaviour of nocontact driving cam mechanism using permanent magnets, Int. J. Appl. Electromagn. Mech. 52 (2016) 169–174. [11] L.-Q. Chen, W.-A. Jiang, Internal resonance energy harvesting, J. Appl. Mech. 82 (2015) 031004-1–031004-11. [12] W. Yang, S. Towfighian, A hybrid nonlinear vibration energy harvester, Mech. Syst. Sig. Process. 90 (2017) 317–333. [13] D. Vokoun, M. Beleggia, L. Heller, Magnetic guns with cylindrical permanent magnets, J. Magn. Magn. Mater. 324 (2012) 1715–1719. [14] O.J. Suarez, P. Vargas, E.E. Vogel, Energy and force between two magnetic nanotubes, J. Magn. Magn. Mater. 321 (2009) 3658–3664. [15] D. Vokoun, M. Beleggia, L. Heller, P. Sittner, Magnetostatic interactions and forces between cylindrical permanent magnets, J. Magn. Magn. Mater. 321 (2009) 3758–3763. [16] J. Li, E.S. Barjuei, G. Ciuti, Y. Hao, P. Zhang, A. Menciassi, Q. Huang, P. Dario, Magnetically-driven medical robots: an analytical magnetic model for endoscopic capsules design, J. Magn. Magn. Mater. 452 (2018) 278–287. [17] E.P. Furlani, Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications, Academic Press, 2001. [18] J.M.D. Coey, Magnetism and magnetic materials 617 Cambridge University Press,
11
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Research articles
Modelling of magnetic field distributions of elliptical cylinder permanent magnets with diametrical magnetization
T
Van Tai Nguyena,b, , Tien-Fu Lub ⁎
a b
Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam School of Mechanical Engineering, University of Adelaide, SA 5005, Australia
ARTICLE INFO
ABSTRACT
Keywords: Elliptical cylinder Modelling of magnetic field Permanent magnet Diametrical magnetization
This research presents semi-analytical and closed-form models to calculate the magnetic field distribution of elliptical cylinder permanent magnets with uniform diametrical magnetization at any point in three dimensional (3D) space. Using the magnetic charge approach, an accurate and fast-computed model is derived. The semianalytical model yields results in excellent agreement with those of Finite Element Analysis. Moreover, it took less than 0.65 ms to compute each component of the magnetic field of the cylinder on a modern personal computer, which demonstrates its efficiency over the well-known Finite Element Analysis method, in terms of computation time. The accuracy and efficiency of the closed-form expressions are analysed and compared with the semi-analytical model. Two and three dimensional analyses of the magnetic field distribution of diametrically magnetised cylinders with different elliptical profiles are also conducted in this study, using the derived model. The analytical model can be used to calculate the magnetic field of an annular elliptical cylinder using the principle of superposition. In cases where the major and minor semi-axes of the elliptical cylinder are equal, it becomes a circular cylinder; therefore, the derived model can be used to compute the magnetic field of a circular cylinder with diametrical magnetization, which can be shown to outperform the existing analytical model in terms of computational cost.
1. Introduction In electrical machines, permanent magnets have been widely used for many years. Recently, they have been implemented extensively in non-contact sensing applications [1–3], permanent magnetic gears [4–6], permanent magnetic couplings [7,8], permanent magnet-bearings [9], non-contact cam mechanisms [10], energy harvesters [11,12] and magnetic guns [13]. In practical applications, the computation of the magnetic field, interaction forces and torques between permanent magnets is of great importance [14–18]. While the expression of these forces can be derived analytically and semi-analytically using the interaction energy [14,15], they and the interaction torques can be expressed based on the Coulombian approach [16,17] if the magnetic field is modelled. Moreover, understanding the magnetic field distribution of a permanent magnet is necessary for many other purposes, such as non-contact sensing and magnetic resonance imaging applications [18]. Therefore, in order to facilitate the design and dynamical modelling of a system using permanent magnets, many methods have been devised to compute and study their magnetic field distribution. The most common one is the well-known Finite Element Analysis (FEA)
⁎
based method. However, this method is time-consuming when high precision is required, due to the smaller mesh that needs to be applied. To address the disadvantages of the FEA, semi-analytical and analytical approaches have been developed to calculate the magnetic field of permanent magnets in different shapes [17]. Nguyen et al. [19] developed exact analytical expressions of the magnetic field distribution of circular cylinder permanent magnets with diametrical magnetization. Rakotoarison et al. [20] derived the fast-computed semi-analytical model to compute the magnetic field of permanent magnets with radial magnetization. Ravaud et al. [21] studied the magnetic field created by a parallelepiped magnet with various and uniform magnetization. Kim et al. [22] studied the polarization of a permanent magnet to produce the predefined magnetic field distribution. Acevedo et al. [23] studied the magnetic field of multipole magnetic configurations analytically. Ravaud et al. [24] developed the analytical model to calculate the magnetic field distribution of permanent-magnet rings with axial and radial magnetizations. However, the magnetic field distribution of elliptical cylinder permanent magnets with uniformly diametrical magnetization (the magnetization is uniformly parallel and perpendicular to the central axis of the cylinder) has not yet been modelled analytically
Corresponding author at: Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam. E-mail addresses: [email protected], [email protected] (V.T. Nguyen).
https://doi.org/10.1016/j.jmmm.2019.165569 Received 15 March 2019; Received in revised form 11 June 2019; Accepted 13 July 2019 Available online 16 July 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
and studied in the literature. This could be because of the highly nonlinear profile, which increases the difficulty of modelling the field distribution of a permanent magnet of this kind. It is the fact that elliptical profiles can be found both in nature and literature, for example, the orbit of each planet in the solar system is approximately an ellipse [25] and elliptical gears have been found to acquire great efficiency in motion and power transmission solutions [26–28]. This could lead to the curiosity of understanding the magnetic field distribution of objects with elliptical profiles [29] as well as their potential application. Therefore, it is of great interest to study the magnetic field distribution of cylinder permanent magnets with an elliptical profile. It can also be agreed that proper understanding and having a fast-computed model of the magnetic field distribution of permanent magnet of this kind are important for its application. For instance, an accurate and fast-computed analytical expression of the magnetic field would assist in the design and dynamic modelling of systems using a permanent magnet of this kind, such as the design and dynamical modelling of an elliptical permanent magnet used in surface magnetic resonance imaging [30], the design and dynamic modelling of the system of a novel magnetic coupler [31], and the design and analysis of permanent magnet gears [4–6] and couplings [7,8] with elliptical profiles. Furthermore, in education, teachers would have a novel collection of a fast-computed analytical model of the magnetic field distribution to demonstrate the physics behind the elliptical cylinder permanent magnet to their students, beyond those commonly found in textbooks like ring-shaped, circular cylinder, parallelepiped and cuboidal permanent magnets [17]. Therefore, this study aims to model the magnetic field distribution of an elliptical cylinder permanent magnet with diametrical magnetization analytically. As a result, an accurate and fast-computed analytical model of the magnetic field has been developed. Using the novel model, two and three dimensional analyses of the magnetic field distribution of an elliptical cylinder have been conducted. The derived model is validated against the well-known Finite Element Method. The efficiency of the analytical model is analysed. The rest of this paper is organized as follows. Section 2 describes in detail the mathematical modelling of the magnetic field distribution. Section 3 presents the Finite Element Analysis and results. Section 4 analyses the magnetic field distribution and Section 5 draws the conclusion.
Fig. 2. Top view of the elliptical cylinder.
Fig. 3. Schematic of the elliptical profile of the cylinder [32].
2. Mathematical modelling
magnetization vector J of the cylinder is uniformly parallel and perpendicular to the central axis of the cylinder, and Ξ is the angle between J and the azimuthal axis X (Fig. 2). The cylinder has a finite length of h (Fig. 1); the parameters of the elliptical profile are depicted in Fig. 3, where F1 and F2 are the foci; a and b are the lengths of the major and minor semi axes, respectively. Hence, the ellipse has the function represented by Eq. (1). It also can be described by the parametric representation using the sine and cosine functions as V = (x, y) = (acosτ, bsinτ), 0 ≤ τ < 2π; here τ can be defined as shown in Fig. 3, based on de la Hire [32], where A is on a circle with a radius of a and B is on a circle with a radius of b.
The geometry of a computed cylinder with an elliptical profile is presented in Fig. 1. Figure 2 presents the top view of the cylinder and illustrates the surface charge distribution s . It is assumed that the
y2 x2 + 2 =1 a2 b
(1)
Based on the magnetic charge or Coulombian approach [17], the magnetic field intensity HK at any point K (r, α, z) in a cylindrical coordinate system, which lies on plane O′X′Y′ (Fig. 1), can be calculated using Eq. (2)
HK =
1 4 µ0
s s
|PK |3
v
PK ds + v
|PK |3
PK dv
(2)
where µ 0 is the permeability of the vacuum, P is a point on the surface of the cylinder (Fig. 1), s is the surface charge density, and v is the
Fig. 1. Geometry of a computed elliptical cylinder. 2
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
volume charge density. The volume charge density v in Eq. (2) can be calculated as the divergence of the magnetization vector: v
=
(3)
·J
Since the magnetisation vector J is uniformly parallel, its divergence is equal to zero, that means v = 0. In other words, the volume charge has no contribution to the magnetic field; hence, the magnetic field intensity HK at point K can be calculated using only the surface charge:
HK =
1 4 µ0
s
|PK |3
s
PK ds
where the surface charge density s
(4) s
can be calculated as follows:
Fig. 5. Parameters used in the mathematical formulation.
(5)
=J. n
Here n is the unit vector normal to the surface of the cylinder. On the top and bottom surfaces of the cylinder, the angles between the unit vectors n of these surfaces and vector J are 90°. Therefore, the surface charge density of these surfaces s = 0 . This means only the cylindrical surface of the cylinder contributes to the computed magnetic field. Fig. 4 depicts the tangent and normal vectors of the elliptical profile of the cylinder as its projection on the O′X′Y′ plane. A tangent vector of the ellipse at point P′(x, y) can be defined as follows:
From Fig. 3, we have
acos = r0 cos ¸
(10)
bsin = r0 sin ¸
(11)
P0 is a point on the elliptical profile. Inserting Eqs. (10) and (11) into Eq. (9) yields: s
t =
dV = ( asin , bcos ) d
1 b2cos2 + a2sin2
= J. n =
a 2 b
2
r02
()
a
+ b JY sin +
( ) sin a 2 b
2
(14)
Inserting Eq. (14) into Eq. (12) with some simplifications, we achieve:
r0 =
(8)
ab (15)
a2sin2 + b2cos2
The infinitesimal area ds in Eq. (4) can be computed as follows: 2
dx d
ds = dldzP =
bJXcos + aJYsin b2cos2 + a2sin2
( ) sin
b J cos a X b 2 cos2 a
Dividing Eq. (10) by Eq. (11), we achieve:
Since the magnetization vector J can be presented as the sum of projected vector JX and JY on axes O′X′ and O′Y′ (Fig. 5) J = JX + JY , the surface charge density s is derived as follows: s
+
=
a cot = cot b
(7)
(bcos , asin )
()
a
+ b JY sin r0
(13)
Therefore, the unit vector n can be expressed by dividing the above normal vector by its magnitude to achieve:
n =
b J cos r0 a X b 2 cos2 r0 2 a
= J. n =
(6)
Rotating the tangent vector t by 90° in a clockwise direction, a normal vector N of the ellipse at the given point is achieved:
N = (bcos , asin )
(12)
a2cos2 + b2sin2
r0 = O'P0 =
+
dy d
2
d dzP =
b2cos2 + a2sin2 d dzP
(16)
Here, dl is the infinitesimal length of the elliptical profile. Inserting Eqs. (13) and (16) into Eq. (4) produces:
(9)
HKs =
1 4 µ0
=2 =0
zP = h PK zP = 0 |PK|3
b a JX cos + JY sin a b b 2 a 2 2 cos2 + sin a b
()
()
(17)
b 2cos2 + a2sin2 dzP d
×
Taking the derivative of Eq. (14) with some simplifications gives:
d =
d b cos2 a
a
+ b sin2
(18)
The component using Eq. (14) as follows:
b2cos2
b2cos2 + a2sin2 =
+
a2sin2
can be expressed through
b 4cos2 + a4sin2 b2cos2 + a2sin2
by
(19)
Inserting Eqs. (18) and (19) into Eq. (17) with some simplifications produces:
HKs =
Fig. 4. Tangent and normal vectors of the elliptical profile. 3
1 4 µ0
=2 =0
zP = h zP = 0
PK ab3JX cos + ba3JY sin d dzP 2 2 2 2 3/2 |PK|3 (b cos + a sin )
(20)
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
From Figs. 1 and 5, the vector PK projected on the radial, azimuthal and axial ( ur, u and uz are the unit vectors respectively) as follows:
can be directions
PK = PP' + P'P" + P"K = (z zP ) uz + r0sin( - ) u + (r r0cos(
)) ur
(21)
Here zP is the axial coordinate of point P, and z is the axial co-
ordinate of points K and P'; is the angle between vector O'P' and axis O′X′ (Fig. 5). Inserting Eq. (21) into Eq. (20) gives:
HKs =
=2 =0
1 4 µ0
zP = h (z zP ) uz + r0sin( - ) u + (r r0cos( - )) ur 3 zP = 0 (r02+r 2 2r0rcos( - )+(z zP )2) 2 ab3JX cos + ba3JY sin
×
d (b 2cos2 + a2sin2 )3/2
dzP
(22)
After integrating Eq. (22) based on z, the axial, azimuthal and radial components can be expressed as follows (Appendix A). 2.1. The axial component H(3D) K(z) (r, , z) The axial component H(3D) K(z) (r, , z) was derived as follows:
H(3D) K(z) (r, , z) =
1 4 µ0
=2 =0
1 h)2 + 2
(z
ab3JX cos + ba3JY sin
×
(b2cos2 + a2sin2 )3/2
1 z2 + 2
d
(23)
). where 2 = r02 + r 2 2r0 r cos( Eq. (23) can be further approximated using Simpson’s method (Appendix B) to achieve the closed-form expression, as follows:
H(3D) K(z) (r, , z) = ×
1 2N µ 0
N n=0
1 (z
× where
S (n )
Fig. 6. An annular cylinder with an elliptical profile: (a) three dimensional (3D) geometry; (b) top view of the cylinder.
1
h)2 + 2 ( (n))
z2 + 2 ( (n))
ab3JX cos (n) + ba3JY sin (n)
2(
(n)) = r0 2 + r2
(Appendix B) to obtain the closed-form expression, as follows:
(23a)
(b2cos2 (n) + a2sin2 (n))3/2
H (3D) K(r) (r, , z) =
2r0rcos( - (n)) .
×
2.2. The tangential (Azimuthal) component H(3D) K( ) (r, , z) The azimuthal component H(3D) K( ) (r, , z) was obtained, as follows:
H(3D) K( ) (r, , z) = ×
1 4 µ0
r0sin(
=2 =0
h z 2
2 + (h
z)2
×
z
+
2
2+ z2
)(ab3JX cos + ba3JY sin ) d (b 2cos2 + a2sin2 )3/2
×
×
1 2N µ 0
h z 2 ( (n))
2 ( (n))+ (h
z)2
+
N n=0
(24)
S (n ) z
2 ( (n))
The radial component H(3D) K(r) (r, , z) was obtained, as follows:
×
=2 =0
h z 2
2 + (h z)2
+
(r r0cos( ))(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
+
z 2 ( (n))
z 2
2 ( (n))+ z2
(r-r0cos( - (n)))(ab3JX cos (n) + ba3JY sin (n)) (b2cos2 (n) + a2sin2 (n))3/2
(25a)
HK (ai, b i)
(26)
where HK (a o, bo) is the magnetic field created by an elliptical cylinder with the outer parameters of major semi-axis a o and minor semi-axis bo . On the other hand, HK (ai, bi) is the magnetic field created by an elliptical cylinder with the inner parameters of major semi-axis ai and minor semi-axis bi . Both cylinders have the same thickness of h and magnetization J as illustrated in Fig. 6(a) and (b). From the magnetic field intensity HK , the magnetic flux density BK can be computed with
(24a)
2.3. The radial component H(3D) K(r) (r, , z)
1 4 µ0
z)2
HK (annular) = HK (a o, bo)
2 ( (n))+ z2
r0sin( - (n))(ab3JX cos (n) + ba3JY sin (n)) (b 2cos2 (n) + a2sin2 (n))3/2
H(3D) K(r) (r, , z) =
2 ( (n))+ (h
S (n )
In the case of a = b, the elliptical cylinder simplifies to a circular cylinder; hence the derived expressions could also be implemented to calculate the magnetic field created by a circular cylinder permanent magnet with uniformly diametrical magnetization. In the case of an annular cylinder with an elliptical profile (Fig. 6), the magnetic field of this cylinder HK (annular) at any point K can be computed using the principle of superposition, as follows:
Once again, Eq. (24) can be further approximated using Simpson’s method (Appendix B) to obtain the closed-form expression, as follows:
H (3D) K(z) (r, , z) =
h z 2 ( (n))
N n=0
1 2N µ 0
2+ z2
(25)
Eq. (25) can be further approximated using Simpson’s method 4
BK=µ 0 HK (in the air space)
(27)
BK=µ 0 HK + J (inside the magnet)
(28)
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
3. Finite Element Analysis (FEA) and Comparison results 3.1. Comparison of semi-analytical expressions with FEA The semi-analytical expressions of the axial (Eq. (23)), the azimuthal (Eq. (24)) and the radial (Eq. (25)) components were implemented in MATLAB R2016b (MathWorks, Natick, MA, USA) to compute the magnetic field of an elliptical cylinder with diametrical magnetization made of rare earth material with the finite length h = 5 mm; major semi-axis a = 6 mm; minor semi-axis b = 3 mm; the remanence J = 1 Tesla and the angle between J and the azimuthal axis X is π/6 rad (Ξ = π/6). The global adaptive quadrature and default error tolerances [33] were used to solve the semi-analytical expressions. In order to verify the results of the analytical model, Finite Element Analysis (FEA) was conducted using Electromagnetic Simulation Software® (EMS) (EMWorks, Inc., Montreal, Quebec, Canada) for the elliptical cylinder with the same parameters. The FEA aims at solving the following governing equations Eqs. (29)–(31) (Maxwell’s equations) of magnetic field for the magnetostatic analysis [34]:
× H = Js
(29)
·B = 0
(30)
And the constitutive relation connects B and H (31)
B =µ (H + H c)
Here H (A/m) is the magnetic field intensity; B (T) is the magnetic flux density; Js (A/m2) is the source current density; µ is the magnetic
permeability (H/m) and Hc (A/m) is the coercive force or coercivity. The coercivity of the magnetic material used in this study is µ and the magnetic permeability is 800,000 A·m−1 1.20536 × 10−6 (H/m). Plots of the computed magnetic field distribution on the plane z = 7 mm and along the radial distance with an azimuthal angle α = π/ 3 (rad) and r from −20 mm to 20 mm are depicted in Fig. 7((a) Axial component; (b) Radial component; (c) Azimuthal component). The values of the magnetic field of 12 random points extracted from Fig. 7 are listed in Table 1. The errors between the results obtained by the analytical model and by FEA, which are estimated using Eq. (32) are also defined in Table 1.
Error =
BAnalytical
BFE model
BFE model
× 100%
(32)
Here BAnalytical – the magnetic flux density computed using the semianalytical model; BFE model – the magnetic flux density computed using the FEA. It is demonstrated based on Fig. 7 and Table 1 that the results of the analytical model derived in this research are in excellent agreement with those computed using the FEA (the average errors are 0.23% for the axial component; 0.38% for the azimuthal component and 0.16% for the radial component). It is noted that, when evaluated in MATLAB R2016b on a personal computer (Intel Core i7-6700 CPU, 3.40 GHz), it took an average of 0.6 ms to compute the radial component, 0.4 ms to compute the azimuthal component and 0.35 ms to compute the axial component, using the derived expressions at a single location (4000 sample points were used in this evaluation). On the other hand, it could take up to several hours with a small scale of mesh (the smaller the mesh is, the more computation time it consumes) to complete the simulation using FEA on the same personal computer.
Fig. 7. FEA verification results (the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m: (a) Axial component; (b) Radial component; (c) Azimuthal component.
simplifies to a circular cylinder. In the work of Nguyen et al. [19], an exact analytical model based on complete and incomplete elliptic integrals was developed to compute the three components of the magnetic field of a circular cylinder with diametrical magnetization. In order to demonstrate the efficiency of the developed model in this study, the semi-analytical expressions Eqs. (23)–(25) and the exact analytical model were used to calculate the magnetic field generated by a circular cylinder with the geometrical parameters as follow: the radius is R = a = b = 0.005 m; the thickness h = 0.005 m; the remanence J = 1 Tesla and vector J is along Y axis; Plots of the computed magnetic field distribution on the plane z = 7 mm; the radial distance r = 0.01 m and the azimuthal angle α = [0, 2π] (rad) are depicted in Fig. 8. It is observed that the results obtained from the semi-analytical
3.2. Comparison of a semi-analytical model and an existing analytical model In a case where the two semi-axes are equal, an elliptical cylinder 5
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Table 1 Semi-Analytical results vs FEA results for 12 points with random r (mm) (Max – Maximum; Min – Minimum; Aver – Average). Points (r mm, 60° , 7 mm)
Axial component (Gauss) Semi-Analytical results
−19.59 −18.59 −16.78 −12.96 −9.75 −6.13 0.10 5.13 9.75 13.17 15.78 18.19
−15.4837 −18.7911 −27.2918 −67.1742 −164.5458 −507.2350 31.2412 672.4053 164.5458 63.6438 34.0078 20.3533 Max 0.45
Aver 0.23
Azimuthal component (Gauss)
FEA results
Semi-Analytical results
−15.4523 −18.7644 −27.2893 −67.1613 −165.0748 −504.9246 31.3573 675.3980 164.7702 63.5470 33.9344 20.2955 Min 0.009
15.1991 17.7630 24.0408 50.4986 106.9623 264.7460 215.6923 318.1446 106.9623 48.2798 28.7752 18.9469
Max 0.82
Aver 0.38
FEA results 15.1212 17.6924 23.9857 50.2791 107.0150 262.5984 216.7766 318.4012 106.4388 48.0761 28.7032 18.8860 Min 0.05
Radial component (Gauss) Semi-Analytical results 42.4949 48.6420 62.8861 113.5919 189.1612 229.0508 −885.6404 142.7812 189.1612 109.8084 72.9705 51.4120 Max 0.29
Aver 0.16
FEA results 42.3682 48.5431 62.8546 113.2647 189.0493 228.7799 −884.6202 143.1480 189.1907 109.5542 72.8962 51.3024 Min 0.05
Fig. 10. Accuracy of the closed-form expression against the semi-analytical model of the azimuthal component ((the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m).
Fig. 8. Comparison of semi-analytical model and existing analytical model [19] (the parameters of the used elliptical cylinder are: R = a = b = 0.005 m, h = 0.005 m, J = 1 T along Y axis); plots along the azimuthal angle α = [0, 2π] (rad) with z = 0.007 m, and the radial distance r = 0.01 m: Bax, Baz and Br are the axial, azimuthal and radial components, respectively; indexes E and S indicate the magnetic field computed using the exact [19] and semi-analytical models.
Fig. 11. Accuracy of the closed-form expression against the semi-analytical model of the radial component ((the parameters of the elliptical cylinder used are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m): singular regions of radial distance (r) are [−0.0049 m, −0.0039 m] and [0.0042 m, 0.0046 m].
Fig. 9. Accuracy of the closed-form expression against the semi-analytical model of the axial component ((the parameters of the used elliptical cylinder are: a = 0.006 m, b = 0.003 m, h = 0.005 m, J = 1 T, Ξ = π/6); plots along the radial distance with z = 0.007 m, azimuthal angle α = π/3 (rad) and r from −0.020 m to 0.020 m): Singular region of radial distance (r) is [−0.0058 m; 0.0072 m]. 6
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 12. Average computational times using closed-form and semi-analytical expressions: Closed-Axial (Semi-Axial), Closed-Azimuthal (Semi-Azimuthal) and ClosedRadial (Semi-Radial) denote the computational costs of the closed-form (semi-analytical) expressions of the axial, azimuthal and radial components, respectively.
Table 2 Accuracy and efficiency analysis of the closed-form expressions (Max – Maximum error; Min – Minimum error; Aver – Average error; Time – Average execution time; * denotes the average computational costs using the semi-analytical expressions (see Section 3.1)). Axial component
Azimuthal component
Radial component
Nθ
Max (%)
Min (%)
Aver (%)
Time (ms) 0.35*
Max (%)
Min (%)
Aver (%)
Time (ms) 0.4*
Max (%)
Min (%)
Aver (%)
Time (ms) 0.6*
100 400 700
2.15 0.53 0.31
0.23 0.056 0.032
0.52 0.16 0.09
0.09 0.35 0.62
2.87 0.72 0.36
0.56 0.14 0.08
1.48 0.35 0.17
0.12 0.48 0.84
2.95 0.74 0.42
5e−12 7e−14 4e−14
0.96 0.25 0.12
0.12 0.49 0.84
model are in excellent agreement with those of the exact analytical model; the maximum, minimum and average errors between these two models are lower than 0.008% for all three components (Bax, Baz and Br (Fig. 8) are the axial, azimuthal and radial components, respectively; indexes E and S indicate the magnetic field computed using the exact [19] and the semi-analytical models, respectively). However, the computational costs of the semi-analytical expressions are 14 times faster for the axial component, 240 times faster for the azimuthal component and 360 times faster for the radial component than those of the exact analytical model. (The simulation was carried out on the same personal computer with the configuration parameters as presented in Section 3.1). Therefore, the currently developed semi-analytical model has been proved to be more efficient than the analytical model derived by Nguyen et al. [19] in terms of the computational time.
number of meshing points Nθ (written as Nθ in the Figures to be compact) ranges from 20 to 700. Figs. 9–11 depict the errors between these two models for the axial, azimuthal and radial components, respectively. It can be seen from Figs. 9–11 that the errors decrease following the increase of the meshing points. Singular regions are observed for the axial (Fig. 9) and radial (Fig. 11) components. In these singular regions, the maximum error is greater than 200%, the minimum error is greater than the maximum outside the regions and the average error is greater than 100%. Therefore, it is not recommended to apply the closed-form expressions to compute the magnetic field inside the singular regions. The semi-analytical expressions should be used in these cases. As a trade-off between accuracy and efficiency, along with a decrease in the error rate, the average computational times of the three components (with the same conditions used for the semi-analytical model described in Section 3.1, where it is demonstrated that the average computational times using the semi-analytical expressions are 0.35 ms, 0.4 ms and 0.6 ms to compute the axial, azimuthal and radial components, respectively) increase (Fig. 12). Table 2 shows that the closed-form expressions can be utilised to compute the magnetic field efficiently when the number of meshing points is Nθ = 100 with the error rate at less than 3%. When Nθ = 700, the errors are less than 0.5%, however, the computational times are greater than those of the semi-analytical expressions (see Section 3.1). This suggests that the semi-analytical model should be implemented when high accuracy (where the errors are equal to or less than 0.5%) is required. On the other hand, the closed-form expressions could be used when low accuracy (the errors are greater than 0.5%, in this case those points in the singular regions should be excluded) is allowed. It is also worth noting
3.3. Accuracy and efficiency of the closed-form expressions The semi-analytical expressions have been demonstrated to predict the magnetic field of a diametrically magnetized elliptical cylinder correctly with a low computational cost. However, it is still interesting to investigate the possibility of solving these expressions analytically. As presented above, the closed-form expressions of the magnetic field can be obtained approximately using the Simpson’s method with the meshing points Nθ in the θ variable (Appendix B). The closed-form expressions were implemented to calculate the three components of the magnetic field created by an elliptical cylinder. The geometrical parameters and the coordinates of the computed points are the same as those used in Section 3.1. The achieved results are compared with those obtained using the semi-analytical expressions (see Section 3.1). The
7
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 13. Distribution of the axial component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (alpha) with a radial distance (b) the plot along the azimuthal angle r = 10 mm.
Fig. 14. Distribution of the azimuthal component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (alpha) with a radial distance (b) the plot along the azimuthal angle r = 10 mm.
that the suitable number of meshing points should be determined individually, according to the changes in the geometrical parameters of an elliptical cylinder, to achieve the desired accuracy.
demonstration. It is noted that although the different axis ratios c are used, all the ellipses have the same area; in other words, all the cylinders have the same volume. Figs. 13–15 depict the magnetic field distribution of the axial component, the azimuthal component and the radial component, respectively. Figs. 13(a) and 15(a) show that the magnetic field distributions of the axial and radial components of the cylinder with different elliptical profiles along the radial distance have the same form and the magnetic field distribution of the circular cylinder (c3) lies between the those of c1, c2, c4 and c5. Fig. 14(a) shows that the distribution of the azimuthal component is in a different form, depending on the axis ratio c. Figs. 13(b), 14(b) and 15(b) demonstrate that the magnetic field distributions of the axial, azimuthal and radial components of the circular cylinder along the azimuthal angle are in the sinusoidal waveform, whereas the magnetic field distributions of those components of the cylinder with elliptical profiles seem to be in a multiorder, function-like waveform. In order to provide more insight into the magnetic field distribution of an elliptical cylinder, three dimensional (3D) plots are generated for
4. Analysis of the magnetic field distribution The fast-computed semi-analytical model derived in this study enables the generation of the magnetic field distribution of a cylinder with different elliptical profiles. The axis ratio of the major semi-axis and the minor semi-axis c = a/b is implemented to represent the different elliptical profiles. The thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6 (Figs. 1 and 2); the axial coordinate of the computed profile is z = 7 mm. For each magnetic field component, plots are generated along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3 and along the azimuthal angle (alpha) with a radial distance r = 10 mm. There are five elliptical profiles with axis ratios c1 = 8/4; c2 = 16/ 2; c3 = 32 / 32 (circular cylinder); c4 = 2/16; c5 = 4/8 used in this
8
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Fig. 15. Distribution of the radial component (the thickness of the used cylinder is h = 5 mm; the magnetic remanence J = 1 T with Ξ = π/6; the axial coordinate of the computed profile is z = 7 mm): (a) the plot along the radial distance r in the interval [−20 mm, 20 mm] with an azimuthal angle = /3; (b) the plot along the azimuthal angle (alpha) with a radial distance r = 10 mm.
the axial, azimuthal and radial components, as illustrated in Fig. 16. The used geometry parameters of the cylinder (Figs. 1 and 2) are a = 0.006 m, b = 0.003 m, J = 1 T with Ξ = π/6; plots are generated on the plane z = 0.007 m and in the interval r = [−0.02 m, 0.02 m], alpha = [0 rad, 2π rad]. Fig. 16(a)–(c) demonstrate that the maxima and minima of the magnetic field on the plot region on the plane are multiple and do not necessarily lie on the central axis z of the cylinder. 5. Conclusion This study presents accurate and fast-computed semi-analytical expressions of the magnetic field distribution generated by an elliptical cylinder with uniformly diametrical magnetization at any point in 3D space. The results computed by the semi-analytical model are in excellent agreement with those of FEA. Moreover, the closed-form expressions of the magnetic field are derived and compared with the semianalytical ones. These closed-form model could be used when low accuracy is allowed. On the other hand, the semi-analytical model could be applied when high accuracy is required. The derived models can be used to calculate the magnetic field of an annular cylinder with an elliptical profile and a circular cylinder. The plots of the magnetic field distribution of a cylinder with different elliptical profiles along the azimuthal angle demonstrate that, whilst the circular cylinder has a sinusoidal waveform, the elliptical cylinders have multi-order function-
Fig. 16. 3D plots of the magnetic field distribution of an elliptical cylinder with a = 0.006 m, b = 0.003 m, J = 1 T with Ξ = π/6; the plots are generated on the plane z = 0.007 m and in the interval r = [−0.02 m, 0.02 m], alpha = [0 rad, 2π rad]: (a) axial component; (b) azimuthal component; (c) radial component.
like waveforms. The 3D plots of the magnetic field distribution of an elliptical cylinder on a plane shows that there are multiple maxima and minima of the distribution on the plotted region. Those extrema do not necessarily lie on the central axis of the cylinder. 9
V.T. Nguyen and T.-F. Lu
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
Acknowledgments
Analyses. The authors would like to thank the School of Mechanical Engineering, University of Adelaide, Australia for providing the equipment, facilities and assistances for this research. We are also grateful to Alison-Jane Hunter for her time helping to edit the English.
The authors are grateful to the EMWORKS Company for providing the license for the EMS software to conduct the Finite Element Appendix A
From Eq. (22), the axial component, H(3D) K(z) (r, , z) can be represented as follows:
H(3D) K(z) (r, , z) =
1 4 µ0
×
=2 =0
zP = h zP = 0
z zP (r02+r 2
ab3JX cos + ba3JY sin d (b 2cos2 + a2sin2 )3/2
2r0rcos( - )+(z
3
z P ) 2) 2
dzP
(A.1)
Integrating Eq. (A.1) based on zP yields:
H(3D) K(z) (r, , z) =
1 4 µ0
=2 =0
zP = h
1 (z
z P) 2 + 2
zP = 0
ab3JX cos + ba3JY sin
×
(b 2cos2 + a2sin2 )
d 3/2
(A.2)
Expanding Eq. (A.2) to achieve:
H(3D) K(z) (r, , z) =
1 4 µ0
×
=2 =0
1 h)2 + 2
(z
1 z2 + 2
ab3JX cos + ba3JY sin
d (b2cos2 + a2sin2 )3/2
(A.3)
From Eq. (22), the azimuthal component
H(3D) K( ) (r,
, z) =
1 4 µ0
×
=2 =0
zP = h zP = 0
H(3D) K( ) (r,
r0sin( - ) (r02+r 2
ab3JX cos + ba3JY sin
d (b 2cos2 + a2sin2 )3/2
2r0rcos( - )+(z
, z) can be expressed as follows: 3
z P ) 2) 2
dzP
(A.4)
Integrating Eq. (A.4) based on zP produces:
H(3D) K( ) (r, , z) = ×
1 4 µ0
=2 =0
zP = h
zP z 2
2 + (z
2
zP)
zP = 0
r0sin( - )(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
(A.5)
Expanding Eq. (A.5) gives:
H(3D) K( ) (r, , z) = ×
1 4 µ0
=2 =0
h z 2
2 + (h
z)2
r0sin( - )(ab3JX cos + ba3JY sin ) (b 2cos2 + a2sin2 )3/2
+
H(3D) K(r) (r, , z) =
×
=2 =0
2+ z2
d
(A.6)
From Eq. (22), the radial component 1 4 µ0
z 2
zP = h zP = 0
H(3D) K(r) (r, r
, z) can be expressed as follows:
r0cos( - )
(r02+r 2 2r0rcos( - )+(z ab3JX cos + ba3JY sin d dzP (b 2cos2 + a2sin2 )3/2
3
z P ) 2) 2
(A.7)
Integrating Eq. (A.7) over zP yields:
H(3D) K(r) (r, , z) = ×
1 4 µ0
=2 =0
zP = h
zP z 2
2 + (z
2
zP)
zP = 0
(r-r0cos( - ))(ab3JX cos + ba3JY sin ) d (b2cos2 + a2sin2 )3/2
(A.8)
Expanding Eq. (A.8) produces:
H(3D) K(r) (r, , z) = ×
1 4 µ0
=2 =0
h z 2
2 + (h
z)2
(r-r0cos( - ))(ab3JX cos + ba3JY sin ) (b 2cos2 + a2sin2 )3/2
+
z 2
2+ z2
d
(A.9)
Appendix B The parameters used in the Simpson’s method mentioned in this research can be obtained as those presented in the work of Furlani et al. [35]. They are described as follows: 10
Journal of Magnetism and Magnetic Materials 491 (2019) 165569
V.T. Nguyen and T.-F. Lu
Nθ denotes the number of mesh points in the θ variable, and θ(n) indicates the values at which the integrand is evaluated. This yields:
=
2 ; N
(n ) =
2 n (n = 0, 1, 2, …, N ) N
(B.1)
The integration coefficients Sθ(n) are defined as follow,
S (n) =
1 3 4 3 2 3 1 3
(n=0), (n=1, 3, 5…), (n=2, 4, 6…), (n=N ).
(B.2)
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