This study presents a fast-computed analytical model to compute the magnetic field generated by a cone with arbitrarily uniform magnetization. Based on the charge model, algebraic and geometrical analyses, the analytical expressions of the axial, azimuthal, and radial components of the magnetic field produced by the cone at any given point in space are derived. The model was in excellent agreement with the Finite Element Model (FEM), as is proved in this research. It is demonstrated that it took an average of less than 1 ms to execute each expression at a single point on the modern personal computer, whereas the FEM simulation consumed 779 s. Moreover, using the developed model, the magnetic field distribution of a cone used in magnetic resonance imaging with varying magnetization is analyzed. As a result, the distributions of the axial and radial components are a cosine-like wave with opposite directions, except for the case where the slope angle of the magnetization is equal to π/2; or, in other words, the magnetization is axial. On the other hand, the distribution of the azimuthal component is a sine-like wave, except for the noted case where the magnetization is axial.

In order to facilitate the design and optimization of permanent magnet devices,1–5 a fast-computed (semi-)analytical model of the magnetic field distribution of basic-shaped (such as ellipsoid, sphere, tile, ring, cone, circular and elliptical cylinders, and parallelepiped) permanent magnets have been developed.6–16 Moreover, due to the superposition principle,6,15 the magnetic field distribution of a complex-shaped permanent magnet, which is formed by the summation or substitution of basic-shaped ones, can be computed using the models of the field distribution created by these permanent magnets of elemental shapes.11,14 This implies that it is of great importance to obtain a fast-computed expression of the magnetic field generated by any given permanent magnets in their basic shapes. In the current literature, the magnetic field distribution of some permanent magnets with basic shapes has been developed (semi-)analytically. For instance, Ravaud et al.8 derived an analytical model to compute the magnetic field produced by a parallelepipedic magnet of various and uniform polarizations; McClurg10 studied the magnetic field distribution of a sphere and an ellipsoid analytically. The analytical expressions of the magnetic field distribution of elliptical cylinder permanent magnets with diametrical11 and axial magnetizations14 have also been developed. However, the derivation of mathematical models of the magnetic fields produced by conical permanent magnets is limited. Most recently, Nguyen15 has derived analytical expressions of three components of the magnetic field created by a conical magnet with axial magnetization. These expressions are more fast-computed than can be achieved by the Finite Element Analysis (FEA), and they are useful for the analysis of the magnetic field distribution of a conical magnet used in magnetic resonance imaging. Nonetheless, the magnetic field distribution of a conical permanent magnet with arbitrarily uniform magnetization has not yet been analytically modeled and analyzed. Although the magnetic field distribution of permanent magnets of any geometries can be solved using the FEA, this method could be time-consuming as being demonstrated in the current literature.11,14,15 Moreover, having in hand fast-computed expressions and understanding the magnetic field distribution could facilitate the design and optimization of devices where this type of permanent magnet could be applied, such as in magnetic resonance imaging.3,4,15

To address this gap in the current literature, this research aims to derive analytical expressions of the magnetic field distribution of a permanent magnet in a cone shape with arbitrarily uniform magnetization. The analytical model is developed based on the charge model,6 algebraic and geometrical analyses. Based on this model, the analysis of the magnetic field distribution is also carried out to provide readers with an insight into the distribution when the magnetization varies.

The rest of this paper is organized as follows: Section II presents the step derivation of the magnetic field distribution. The results obtained from the derived analytical model are validated by those of the Finite Element Model (FEM) and described in more detail in Sec. III. Section IV analyzes the magnetic field distribution in different magnetization directions for an application in magnetic resonance imaging. Section V draws a conclusion.

Due to the superposition principle, the magnetic field HJ generated by a conical permanent magnet with arbitrarily uniform magnetization J can be represented as the summation of the magnetic field produced by two cones with the same geometrical parameters as those of the original cone [the cone height is h (m) and the vertex angle is 2φ (rad) (Fig. 1)]; one is with the axial magnetization JZ, which creates the magnetic field HJZ, and the another one is with the diametrical magnetization JY, which creates the magnetic field HJY (it is noted that diametrical magnetization means that the direction of the magnetization is perpendicular to the symmetry axis OZ of the cone).

FIG. 1.

Construction of a cone with arbitrarily uniform magnetization.

FIG. 1.

Construction of a cone with arbitrarily uniform magnetization.

Close modal

From a mathematical perspective, this can be described as Eqs. (1) and (2),

HJ=HJZ+HJY.
(1)

Denoting the slope angle as Ω (Fig. 1) between J and the horizontal plane (OXY), we have

JZ=JsinΩ and JY=JcosΩ.
(2)

In the author’s previous article,15 the magnetic field distribution HJZ of a conical permanent magnet with the axial magnetization JZ has been expressed analytically. The author recommends readers to refer to that publication for more details on the analytical expressions. For the sake of brevity, but without any loss of generality, they are not presented repeatedly in this research. This also means that, in order to compute the magnetic field distribution of a cone with arbitrarily uniform magnetization analytically, it is necessary to develop an analytical model to express the magnetic field generated by a cone with diametrical magnetization. This task is addressed in the rest of this section.

The geometrical parameters of the cone used in the derivation are depicted in Fig. 2. The height of the cone is denoted as h (m); the vertex angle is 2φ (rad); the radius of the base circle is R (m); vb and v are the unit vectors normal to the base and lateral surfaces, respectively; vh and vv are projections of vector v on the horizontal and vertical directions; and S and N represent the south and north poles. Without the loss of generality, it is assumed that the magnetization JY is along the OY axis.

FIG. 2.

Geometrical parameters of a cone.

FIG. 2.

Geometrical parameters of a cone.

Close modal

Based on the charge model,6 the magnetic field produced by a cone at point N in free space (Fig. 3) can be computed using Eq. (3),

HN=14πμ0sσsMN3MNds+vσvMN3MNdv,
(3)

where MN is the vector linking the source point M belonging to the cone at the computed point N; σs and σv are the surface and volume charge densities, respectively; and μ01.2566370×106 H/m is the vacuum permeability.

FIG. 3.

Source and computed points.

FIG. 3.

Source and computed points.

Close modal

Since the magnetic field is uniformly distributed, the volume charge density σv is equal to zero.12,15 Thus, only the surface charge density contributes to the magnetic field. As a result, Eq. (3) can be simplified as follows:

HN=14πμ0sσsMN3MNds.
(4)

Now, let us consider the sectional plane O′X′Y′ parallel to plane OXY, where the source point M lies. The projection of point N on this plane is denoted as G (Fig. 3).

Figure 4 depicts the section of the cone on plane O′X′Y′ and vectors MM,MG used for the following geometrical analysis.

FIG. 4.

Projection plane O′X′Y′ and vectors for the geometrical analysis.

FIG. 4.

Projection plane O′X′Y′ and vectors for the geometrical analysis.

Close modal

The cylindrical coordinate system with three unit vectors ir, the radial direction, iα, the azimuthal direction, and iz, the axial direction is applied in the further derivation. In this coordinate system, source point M has coordinates (rM, α, and zM) and computed point N has coordinates (r, β, and z). It is worth noting that rM and r are the radial distances; α and β are the azimuthal angles; and zM and z are the coordinates of the axial axis.

From Figs. 3 and 4, vector MN can be presented in the form of other vector components as follows:

MN=MM+MG+GN=rMsinβαiα+rrMcosβαir+zzMiz.
(5)

The surface charge density, which can be calculated as the dot product of the unit normal vector and the magnetization vector6 generated by the base surface, is equal to zero due to the perpendicularity between the unit normal vector vb of this surface and the magnetization vector JY (JYvb=0). Therefore, only the surface charge density of the lateral surface of the cone contributes to the magnetic field intensity. Based on Figs. 2 and 4, this density can be computed using the following equation:

σs=JYv=JYvh=JYcosαcosφ.
(6)

Based on Fig. 5, some geometrical parameters, such as rM and zM, and the infinitesimal area of the lateral surface ds can be further derived using the cone’s slant height ξ as

rM=ξsinφ,
(7)
zM=ξcosφ,
(8)

and

ds=rMdαdξ=ξsinφdαdξ.
(9)

Inserting Eqs. (5) and (6) into Eq. (4), including Eqs. (7)–(9), we have the following equation:

HN=JYsin2φ8πμ0α=0α=2πξ=0ξ=h2+R2rMsinβαiα+rrMcosβαir+zξcosφizrM2+r22rMrcosβα+zξcosφ232cosαξdαdξ.
(10)

After integrating Eq. (10) based on ξ, the three components of the magnetic field can be analytically expressed as follows:

FIG. 5.

Cone’s geometrical parameters with its slant height.

FIG. 5.

Cone’s geometrical parameters with its slant height.

Close modal

The axial component HNAxial is obtained as follows:

HNAxial=JYsin2φ8πμ0α=0α=2π2zΨ+2ΘcosφΨ2cosφΞ4Θz+2ΨΘcosφ4ΘΨ2ΞΞΨ+Θ4Θz+2ΨΘcosφ4ΘΨ2Θcosφln2ΞΞΨ+Θ+ΞΨ2ΘΨcosαdα,
(11)

where Ξ=h2+R2; Ψ = 2r sin φ cos(β − α) + 2z cos φ; Θ = r2 + z2; without much difficulty, it can be algebraically proved that 4Θ > Ψ2, except for the singular point where r = z = 0.

For points on the axial axis (OZ; r = 0, δ, z), the magnetic field intensity is equal to zero as a result of the integration of the odd function of cos(α) over the range [0, 2π] in Eq. (11).

The axial component of the magnetic field intensity HNAxial(Abitr) is produced by a cone with arbitrarily uniform magnetization, and can be computed using the following equation:

HNAxial(Abitr)=HNAxial+HNAxial[15],
(12)

where HNAxial[15] is the axial component of a magnetic field created by a cone with the same geometrical parameters, but the magnetization has an axial orientation15 with the magnitude JZ = J sin Ω (Fig. 1). HNAxial[15] can be calculated using the analytical expression of the axial component, which was developed in the author’s previous article.15 For the purpose of brevity, this expression is not repeated here.

The azimuthal component HNAzim is derived as follows:

HNAzim=JYsinφsin2φ8πμ0α=0α=2πln2ΞΞΨ+Θ+ΞΨ2ΘΨ22ΘΨ2Ξ+2ΨΘ4ΘΨ2ΞΞΨ+Θ+2ΨΘ4ΘΨ2Θ×sinβαcosαdα.
(13)

For points on the axial axis (OZ; r = 0, δ, z), the magnetic intensity HNAzim(OZ) can be expressed as follows:

HNAzim(OZ)=JYsinβsinφsin2φ8μ0×ln2ΞΞΨ0+Θ0+ΞΨ02Θ0Ψ022Θ0Ψ02Ξ+2Ψ0Θ04Θ0Ψ02ΞΞΨ0+Θ0+2Ψ0Θ04Θ0Ψ02Θ0,
(14)

where Ξ=h2+R2; Ψ0 = 2z cos φ; Θ0=z2; without much effort, it can be proved that 4Θ0>Ψ02, except for the singular point where z = 0.

The azimuthal component of magnetic field intensity HNAzimuthal(Abitr) produced by a cone with arbitrarily uniform magnetization can be computed using the following equation:

HNAzimuthal(Abitr)=HNAzimuthal+HNAzimuthal[15],
(15)

where HNAzimuthal[15] is the azimuthal component of the magnetic field created by a cone with the same geometrical parameters, but the magnetization has the axial orientation15 with a magnitude of JZ = J sin Ω (Fig. 1). HNAzimuthal[15] can be calculated using the analytical expression of the azimuthal component, which was developed in Ref. 15. For the purpose of brevity, this expression is not repeated here.

The radial component HNRadi is represented as follows:

HNRadi=JYsin2φ8πμ0α=0α=2π22ΘΛΨ2Λ+ΨrΞ+2ΨΘΛ4Θr4ΘΨ2ΞΞΨ+ΘΛln2ΞΞΨ+Θ+ΞΨ2ΘΨ2ΨΘΛ4Θr4ΘΨ2Θ×cosαdα,
(16)

where Ξ=h2+R2; Ψ = 2r sin φ cos(β − α) + 2z cos φ; Θ = r2 + z2; and Λ = sin φ cos(β − α), and it is mentioned again that 4Θ > Ψ2, except for the singular point where r = z = 0.

For the points on the axial axis (OZ; r = 0, δ, z), the radial component HNRadi(OZ) can be expressed as follows:

HNRadi(OZ)=JYsinφsin2φcosβ8μ022Θ0Ψ02Ξ+2Ψ0Θ04Θ0Ψ02ΞΞΨ0+Θ0ln2ΞΞΨ0+Θ0+ΞΨ02Θ0Ψ02Ψ0Θ04Θ0Ψ02Θ0,
(17)

where Ξ=h2+R2; Ψ0 = 2z cos φ; Θ0 = z2; and it can be realized that 4Θ0>Ψ02, except for the singular point where z = 0.

The radial component of magnetic field intensity HNRadial(Abitr) produced by a cone with arbitrarily uniform magnetization can be computed using the following equation:

HNRadial(Abitr)=HNRadial+HNRadial[15],
(18)

where HNRadial[15] is the radial component of the magnetic field created by a cone with the same geometrical parameters, but the magnetization has the axial orientation15 with the magnitude JZ = J sin Ω (Fig. 1). HNRadial[15] can be calculated using the analytical expression of the radial component, which was developed in Ref. 15. For the purpose of brevity, this expression is also not repeated here.

From the magnetic field intensity HN, the magnetic flux density BN can be computed with the following equations:11,15

BN=μ0HN(in the air space),
(19)
BN=μ0HN+J(inside the magnet).
(20)

In order to validate the analytical model developed in this study to compute the magnetic field generated by a diametrically magnetized cone, a Finite Element Model (FEM) was built in the Electromagnetic Simulation Software® (EMS) (EMWorks, Inc., Montreal, QC, Canada), integrated with 3D CAD INVENTOR® (AUTODESK, Inc., CA). The geometrical parameters of the cone used in this simulation are height h = 0.010 m; vertex angle 2φ = π/3 (rad); and the magnetization, which is along the axis OY, as depicted in Fig. 2, with magnitude 1 (T).

The analytical results were obtained by coding the developed expressions of the three components of the magnetic field, which were created by a cone with the same geometrical parameters and magnetization used in the FEM in MATLAB R2016b (MathWorks, Natick, MA).

The comparison results between the two models are shown in Fig. 6 [the plot was generated along the axial direction z = [−0.0015, 0.012] (m) with the radial distance r = 0.00825 (m) and azimuthal angle β = 4.957 (rad)]. Moreover, the results of 14 points with random axial coordinate z (mm) are extracted and listed in Table I. Figure 6 and Table I demonstrate that the results of the developed analytical expressions are in excellent agreement with those of the FEM (the maximum, average, and minimum errors between the two models are 0.51%, 0.14%, and 0.0057%, respectively, for the axial component; 0.17%, 0.05%, and 4.45 × 10−5%, respectively, for the azimuthal component; and 0.99%, 0.26%, and 0.0027%, respectively, for the radial component).

FIG. 6.

Comparison between FEM and analytical expressions. The height of the cone is h = 0.010 m; the vertex angle is 2φ = π/3 rad; the magnetization is J = 1 T. Bax, Baz, and Bra indicate the axial, azimuthal, and radial components; indexes Ana and FE denote the results of the analytical FE models.

FIG. 6.

Comparison between FEM and analytical expressions. The height of the cone is h = 0.010 m; the vertex angle is 2φ = π/3 rad; the magnetization is J = 1 T. Bax, Baz, and Bra indicate the axial, azimuthal, and radial components; indexes Ana and FE denote the results of the analytical FE models.

Close modal
TABLE I.

Comparison of the analytical results and those obtained by FEM for 14 random points.

Points (r = 0.00825 m,Axial component (gauss)Azimuthal component (gauss)Radial component (gauss)
z m, β = 4.957 rad)Analytical modelFEAAnalytical modelFEAAnalytical modelFEA
−0.8181 −59.8687 −59.8653 −165.7066 −165.7111 21.6651 21.5873 
−0.2727 −65.3803 −65.4131 −182.2708 −182.2264 28.0656 28.0159 
1.7727 −85.8743 −85.8113 −257.5002 −257.4356 63.5014 63.4997 
9.8182 92.1019 92.5757 −462.4400 −462.4561 232.8043 232.7093 
3.2727 −96.3648 −96.2581 −324.2330 −324.1736 102.8055 102.6393 
4.0909 −98.0242 −98.0037 −362.9352 −363.2549 129.0379 128.7251 
10.5 118.0807 117.7244 −427.2725 −426.9482 194.8541 194.5199 
5.4545 −89.7257 −89.5719 −425.9476 −425.6513 178.4385 178.1625 
6.1364 −78.2050 −78.2665 −453.8830 −454.6911 204.1356 203.8762 
7.7727 −22.2021 −22.2167 −496.6415 −496.2163 255.2363 255.6377 
2.5909 −92.4938 −92.4305 −292.9522 −292.9801 83.4821 83.2599 
0.6818 −75.2204 −75.2368 −214.8197 −214.7843 42.1795 42.1099 
10.091 104.2757 103.9000 −449.4231 −449.4233 218.8197 217.9128 
−1.3636 −54.5664 −54.4806 −150.5720 −150.6286 16.3184 16.1575 
Points (r = 0.00825 m,Axial component (gauss)Azimuthal component (gauss)Radial component (gauss)
z m, β = 4.957 rad)Analytical modelFEAAnalytical modelFEAAnalytical modelFEA
−0.8181 −59.8687 −59.8653 −165.7066 −165.7111 21.6651 21.5873 
−0.2727 −65.3803 −65.4131 −182.2708 −182.2264 28.0656 28.0159 
1.7727 −85.8743 −85.8113 −257.5002 −257.4356 63.5014 63.4997 
9.8182 92.1019 92.5757 −462.4400 −462.4561 232.8043 232.7093 
3.2727 −96.3648 −96.2581 −324.2330 −324.1736 102.8055 102.6393 
4.0909 −98.0242 −98.0037 −362.9352 −363.2549 129.0379 128.7251 
10.5 118.0807 117.7244 −427.2725 −426.9482 194.8541 194.5199 
5.4545 −89.7257 −89.5719 −425.9476 −425.6513 178.4385 178.1625 
6.1364 −78.2050 −78.2665 −453.8830 −454.6911 204.1356 203.8762 
7.7727 −22.2021 −22.2167 −496.6415 −496.2163 255.2363 255.6377 
2.5909 −92.4938 −92.4305 −292.9522 −292.9801 83.4821 83.2599 
0.6818 −75.2204 −75.2368 −214.8197 −214.7843 42.1795 42.1099 
10.091 104.2757 103.9000 −449.4231 −449.4233 218.8197 217.9128 
−1.3636 −54.5664 −54.4806 −150.5720 −150.6286 16.3184 16.1575 

As mentioned in Sec. II, based on the analytical model developed in this current study and that of the author’s previous work,15 the magnetic field generated by a cone with arbitrarily uniform magnetization can be computed analytically. To validate this, a model of a cone with the same geometrical parameters as those of the one implemented in Sec. III A was built in EMS integrated with INVENTOR. The magnetization of the cone at this time had a slope angle Ω = π/4, and its magnitude is J = 2 T. Moreover, the analytical model, which can be represented as the algebraic summation of those developed in this study and those developed in Ref. 15 [Eqs. (12), (15), and (18)], was coded in MATLAB R2016b. The results of the comparison are plotted in Fig. 7 (the plot was generated along the axial direction z = [−0.0015, 0.012] m with the radial distance r = 0.008 25 m and azimuthal angle β = 4.957 rad). Furthermore, the results of the 14 random points that are used in Table I were extracted and are listed in Table II. From Fig. 7 and Table II, it is concluded that the results of the analytical model are in excellent agreement with those of the FEM (the maximum, average, and minimum errors between the two models are 0.62%, 0.24%, and 0.05%, respectively, for the axial component; 0.14%, 0.06%, and 0.008%, respectively, for the azimuthal component; and 0.69%, 0.27%, and 0.036%, respectively, for the radial component).

FIG. 7.

Comparison between the FEM and analytical expressions for the arbitrarily uniform magnetized cone. The height of the cone is h = 0.010 m; the vertex angle is 2φ = π/3 rad; the magnetization has the slope angle Ω = π/4 rad, and its magnitude is J = 2 T. aBax, aBaz, and aBra indicate the axial, azimuthal, and radial components; indexes Ana and FE denote the results of the analytical and FE models, respectively.

FIG. 7.

Comparison between the FEM and analytical expressions for the arbitrarily uniform magnetized cone. The height of the cone is h = 0.010 m; the vertex angle is 2φ = π/3 rad; the magnetization has the slope angle Ω = π/4 rad, and its magnitude is J = 2 T. aBax, aBaz, and aBra indicate the axial, azimuthal, and radial components; indexes Ana and FE denote the results of the analytical and FE models, respectively.

Close modal
TABLE II.

Comparison of the analytical results of an arbitrarily uniform magnetized cone and those obtained by the FEM for 14 points.

Points (r = 0.00825 m,Axial component (gauss)Azimuthal component (gauss)Radial component (gauss)
z m, β = 4.957 rad)Analytical modelFEMAnalytical modelFEMAnalytical modelFEM
−0.8181 21.6102 21.7200 −165.7066 −165.7938 −225.2348 −224.7978 
−0.2727 6.7828 6.7561 −182.2708 −182.2500 −241.5753 −240.9652 
1.7727 −82.2723 −81.7691 −257.5002 −257.4637 −290.7283 −290.3790 
9.8182 −391.1025 −390.7076 −462.4400 −462.6739 611.9609 611.7414 
3.2727 −186.0311 −185.5419 −324.2330 −324.0046 −294.7771 −294.1961 
4.0909 −255.9562 −255.6254 −362.9352 −363.1533 −275.4525 −274.8153 
10.5 −244.9007 −244.1744 −427.2725 −427.4012 681.2204 679.7142 
5.4545 −386.3896 −385.4932 −425.9476 −425.3470 −191.9615 −191.4261 
6.1364 −452.0258 −451.1190 −453.8830 −454.3709 −118.8286 −118.0120 
7.7727 −562.6418 −562.2303 −496.6415 −496.1794 163.0489 163.9408 
2.5909 −134.7311 −134.5163 −292.9522 −292.9299 −298.0908 −297.6009 
0.68182 −27.7009 −27.7197 −214.8204 −214.9406 −268.0691 −267.5137 
10.091 −334.6861 −332.9267 −449.4231 −449.8453 648.2057 645.7909 
−1.3636 33.3573 33.3737 −150.5720 −150.7034 −208.7058 −208.2549 
Points (r = 0.00825 m,Axial component (gauss)Azimuthal component (gauss)Radial component (gauss)
z m, β = 4.957 rad)Analytical modelFEMAnalytical modelFEMAnalytical modelFEM
−0.8181 21.6102 21.7200 −165.7066 −165.7938 −225.2348 −224.7978 
−0.2727 6.7828 6.7561 −182.2708 −182.2500 −241.5753 −240.9652 
1.7727 −82.2723 −81.7691 −257.5002 −257.4637 −290.7283 −290.3790 
9.8182 −391.1025 −390.7076 −462.4400 −462.6739 611.9609 611.7414 
3.2727 −186.0311 −185.5419 −324.2330 −324.0046 −294.7771 −294.1961 
4.0909 −255.9562 −255.6254 −362.9352 −363.1533 −275.4525 −274.8153 
10.5 −244.9007 −244.1744 −427.2725 −427.4012 681.2204 679.7142 
5.4545 −386.3896 −385.4932 −425.9476 −425.3470 −191.9615 −191.4261 
6.1364 −452.0258 −451.1190 −453.8830 −454.3709 −118.8286 −118.0120 
7.7727 −562.6418 −562.2303 −496.6415 −496.1794 163.0489 163.9408 
2.5909 −134.7311 −134.5163 −292.9522 −292.9299 −298.0908 −297.6009 
0.68182 −27.7009 −27.7197 −214.8204 −214.9406 −268.0691 −267.5137 
10.091 −334.6861 −332.9267 −449.4231 −449.8453 648.2057 645.7909 
−1.3636 33.3573 33.3737 −150.5720 −150.7034 −208.7058 −208.2549 

In addition, it is worth mentioning that simulated on a personal computer (Intel Core i7-6700 CPU, 3.40 GHz), using MATLAB R2016b, it took an average of less than 0.5 (ms) to execute each analytical expression of the magnetic field components at a single point (40 000 random points were implemented in this simulation) created by a cone with diametrical magnetization. Moreover, it took an average of less than 1 ms to execute each analytical expression of the magnetic field components generated by a cone with arbitrarily uniform magnetization. In contrast, the computation costs of simulation of the FEM on the same personal computer for both cases were 779 s.

In this study, the magnetic field distribution, which is produced by an annular cone with arbitrarily uniform magnetization, is analyzed using the developed analytical expressions. The cone is assumed to be applied in magnetic resonance imaging for a human head.15Figure 8 depicts both the schematic representation of this application and the geometrical parameters of the cone.

FIG. 8.

A permanent magnet cone used in magnetic resonance imaging.

FIG. 8.

A permanent magnet cone used in magnetic resonance imaging.

Close modal

The cone has the height h (m); the inner vertex angle is 2φ2 (rad); the outer vertex angle is 2φ1 (rad); the inner and outer base radii are Ri (m) and RO (m), respectively; the slope angle of its magnetization is Ω (rad) with the magnitude J (T); the human head has the average radius of Rh (m); the considered plane of the human head is the center plane Δ with a radius of Rh (m); this plane lies inside the cone, with a distance a (m) from the cone’s base surface. The values of these parameters are listed in Table III.

TABLE III.

Annular cone’s parameters.

ParametersValues
h (m) 0.3 
φ1 (rad) 0.6174 
φ2 (rad) 0.4636 
Ri (m) 0.15 
RO (m) 0.213 
Ω (rad) [0, π/6, π/3, π/2] 
J (T) 
Rh (m) 0.094 
a (m) 0.02 
ParametersValues
h (m) 0.3 
φ1 (rad) 0.6174 
φ2 (rad) 0.4636 
Ri (m) 0.15 
RO (m) 0.213 
Ω (rad) [0, π/6, π/3, π/2] 
J (T) 
Rh (m) 0.094 
a (m) 0.02 

Due to the superposition principle, the magnetic field HN(annular) created by the annular cone at any given point N in space can be computed using the following equation:

HN(annular)=HN2φ1HN2φ2,
(21)

where HN2φ1 is the magnetic field created by a solid cone that has the same geometrical parameters as those of the outer cone (h, 2φ1) and the same magnetization as that of the annular cone. Similarly, HN2φ2 is the magnetic field produced by a solid cone with the same geometrical parameters of the inner cone (h, 2φ2) and the same magnetization as that of the annular cone.

The plots of the magnetic field distribution of the axial, azimuthal, and radial components with varying slope angle Ω are depicted in Figs. 9, 10, and 11, respectively. Figures 9(a), 10(a), and 11(a) show the three-dimensional (3D) plot of the magnetic field distribution of these components. On the other hand, Figs. 9(b), 10(b), and 11(b) illustrate the distribution along a two-dimensional (2D) plot along the radial distance r = [0, 0.094] m and the azimuthal angle β = π/6; Figs. 9(c), 10(c), and 11(c) depict the distribution along a two-dimensional plot along the azimuthal angle β = [0, 2π] rad and the radial distance r = 0.015 m.

FIG. 9.

Distribution of the axial component: (a) 3D plot of the axial component according to different slope angles Ω; (b) 2D plot of the axial component according to different slope angles Ω along the radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the axial component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

FIG. 9.

Distribution of the axial component: (a) 3D plot of the axial component according to different slope angles Ω; (b) 2D plot of the axial component according to different slope angles Ω along the radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the axial component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

Close modal
FIG. 10.

Distribution of the azimuthal component: (a) 3D plot of the azimuthal component according to different slope angles Ω; (b) 2D plot of the azimuthal component according to different slope angles Ω, along radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the azimuthal component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

FIG. 10.

Distribution of the azimuthal component: (a) 3D plot of the azimuthal component according to different slope angles Ω; (b) 2D plot of the azimuthal component according to different slope angles Ω, along radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the azimuthal component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

Close modal
FIG. 11.

Distribution of the radial component: (a) 3D plot of the radial component according to different slope angles Ω; (b) 2D plot of the azimuthal component according to different slope angles Ω, along radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the azimuthal component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

FIG. 11.

Distribution of the radial component: (a) 3D plot of the radial component according to different slope angles Ω; (b) 2D plot of the azimuthal component according to different slope angles Ω, along radial distance r = [0, 0.094] m and β = π/6 rad; (c) 2D plot of the azimuthal component according to different slope angles Ω, along angle β = [0, 2π] rad and r = 0.015 m.

Close modal

For the axial component, Figs. 9(a) and 9(c) show that the distribution of the axial component is a cosine-like wave along the azimuthal angle β when the slope angle Ω is not equal to π/2. If Ω = π/2, the value of the magnetic flux density is constant along the azimuthal direction (Baxial = −500 G). Figure 9(b) demonstrates that the magnitude of the axial component decreases with the increase of the slope angle, and it is more uniformly distributed when Ω = π/2.15 For the azimuthal component, Figs. 10(a) and 10(c) show that the distribution of the azimuthal component is a sine-like wave along β when Ω is not equal to π/2, and the component is equal to zero when Ω = π/2.15Figure 10(b) illustrates that this component tends to be more uniformly distributed along r when Ω increases. For the radial component, Figs. 11(a) and 11(c) show that the distribution of this component along β is a cosine-like wave and in the opposite direction from the axial component when Ω is not equal to π/2. When Ω = π/2, this component is a constant long β (Bradial = −75.29 G). Figure 11(b) demonstrates that along r, the distribution of the radial component becomes less uniformly distributed with the increase in Ω.

In this research, fast-computed analytical expressions of the magnetic field generated by a cone with arbitrarily uniform magnetization are developed. First, the analytical model of a magnetic field generated by a cone with diametrical magnetization is developed based on the charge model, algebraic and geometrical analyses. Using this model, combining with the analytical model developed by the author in the previous publication for a cone with axial magnetization, the magnetic field created by a cone with arbitrarily uniform magnetization can be computed analytically due to the principle of superposition. The expressions were validated with a Finite Element Model (FEM). Moreover, as is demonstrated, the analytical model is much faster in terms of the computational cost than the FEM. Using the developed expressions, the magnetic field distribution of a cone with arbitrarily uniform magnetization, which is assumed to be applied in magnetic resonance imaging, is analyzed. As shown in this study, the distribution of the magnetic field produced by a cone with arbitrarily uniform magnetization is sinusoidal, except for the case where the slope angle is equal to π/2 (axial magnetization). Furthermore, with the changes in the slope angle Ω, the magnitude and uniformity of the magnetic field vary accordingly.

The author is grateful to the EMWORKS Company for providing the license for the EMS software to conduct the finite element analyses. The author would like to thank the School of Mechanical Engineering, University of Adelaide, Australia, for providing the equipment, facilities, and assistance for this research. The author acknowledges the financial support from Thuyloi University according to the Decision No. 989/QD-DHTL. The author is also grateful to Dr. Alison-Jane Hunter for her time helping to edit the English.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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