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Figure 5. Decomposition of the circular surface in rectangular element. Figure 4. Approximating area under a curve using rectangles. To calculate the magnetic. force magnetic between cylindrical magnets, we should first calculat the scalar magnetic potential resulting from one of the circular surfaces (Fig. 5) and any given point | (arbitrary observation point) [23-25]. The potential can be calculated by the formula:

Figure 5 Decomposition of the circular surface in rectangular element. Figure 4. Approximating area under a curve using rectangles. To calculate the magnetic. force magnetic between cylindrical magnets, we should first calculat the scalar magnetic potential resulting from one of the circular surfaces (Fig. 5) and any given point | (arbitrary observation point) [23-25]. The potential can be calculated by the formula:

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Figure 1. A scheme of two sets of cylindrical magnets. The benefit for these applications is that magnetic forces act without physical contact over larger distances than electrostatic, piezoelectric or other schemes [5, 8, 9]. In the literature related to permanent magnets, calculations of forces between magnets of various shapes and geometries have their relevance in the context of several applications. For investigating the magnetic force between two sets of cylindrical permanent magnets (Fig. 1), the force engendered between two parallel magnets should be calculated firstly.
Figure 1. A scheme of two sets of cylindrical magnets. The benefit for these applications is that magnetic forces act without physical contact over larger distances than electrostatic, piezoelectric or other schemes [5, 8, 9]. In the literature related to permanent magnets, calculations of forces between magnets of various shapes and geometries have their relevance in the context of several applications. For investigating the magnetic force between two sets of cylindrical permanent magnets (Fig. 1), the force engendered between two parallel magnets should be calculated firstly.
Figure 2. Two cylindrical magnets with parallel magnetization. Figure 2 shows that the magnetic configuration consists of two magnets with axial polarization. The dimensions of the system are shown in Fig. 2. 2a and 2A are the radii of magnet 1 and magnet 2. 2C’ and 2c are the thicknesses of the magnets. The cylinder shown in Fig. 2 is uniformly magnetized in the z direction.
Figure 2. Two cylindrical magnets with parallel magnetization. Figure 2 shows that the magnetic configuration consists of two magnets with axial polarization. The dimensions of the system are shown in Fig. 2. 2a and 2A are the radii of magnet 1 and magnet 2. 2C’ and 2c are the thicknesses of the magnets. The cylinder shown in Fig. 2 is uniformly magnetized in the z direction.
Figure 3. Model of the permanent magnet by the Coulombian approach. Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 3. Model of the permanent magnet by the Coulombian approach. Progress In Electromagnetics Research C, Vol. 91, 2019
By adding the field created by the two surfaces, we obtain the field created by an elementary cuboida magnet (Fig. 6). The system of magnet is shown in Fig. 6, in which the cylindrical magnet is replacec by cuboidal shape composed of N cuboidal magnets.
By adding the field created by the two surfaces, we obtain the field created by an elementary cuboida magnet (Fig. 6). The system of magnet is shown in Fig. 6, in which the cylindrical magnet is replacec by cuboidal shape composed of N cuboidal magnets.
Figure 6. Elementary decomposition of cylindrical magnet in cubic elements. The total magnetic field is always calculated from the gradient:
Figure 6. Elementary decomposition of cylindrical magnet in cubic elements. The total magnetic field is always calculated from the gradient:
In Fig. 7, we present the magnetic flux density created by cylindrical magnet for the number o: subdivisions: N = 4 and N = 40.
In Fig. 7, we present the magnetic flux density created by cylindrical magnet for the number o: subdivisions: N = 4 and N = 40.
Figure 7. The magnetic flux density in XY plane for (a) N = 4 elements, (b) N = 40 elements. Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 7. The magnetic flux density in XY plane for (a) N = 4 elements, (b) N = 40 elements. Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 8. Magnetic induction B, obtained from measurement and our results. The comparison between the magnetic field calculation and the measured values when the sensors move along the z-direction is shown in Fig. 8.
Figure 8. Magnetic induction B, obtained from measurement and our results. The comparison between the magnetic field calculation and the measured values when the sensors move along the z-direction is shown in Fig. 8.
For the calculation of interaction forces between two cylindrical magnets, we divided the first and second into (N, M) elementary cuboidal magnets. The interaction energy between two elementary magnets t and s, as shown in Fig. 11, is given by
For the calculation of interaction forces between two cylindrical magnets, we divided the first and second into (N, M) elementary cuboidal magnets. The interaction energy between two elementary magnets t and s, as shown in Fig. 11, is given by
Figure 9. Modeling of permanent magnet annular Superposition principle.
Figure 9. Modeling of permanent magnet annular Superposition principle.
Figure 10. Field radial component versus the radial distance r of the observation point; D = 3mm, R; = 25mm, Ra = 28mm.
Figure 10. Field radial component versus the radial distance r of the observation point; D = 3mm, R; = 25mm, Ra = 28mm.
Figure 11. System of two rectangular elements. As shown in Fig. 11, the dimensions of the first elementary cuboidal magnet are a, b, and c, and its polarization is J. For the second elementary magnet, the dimensions are A, B and C. Its polarization is J’, and the coordinate of its center is a, 3, and y. Avl, and Ax2; are the widths of the first and second elementary magnets. The upper magnet moves in translation along the y-axis above the lower fixed magnet. The total aenetic force is given by:
Figure 11. System of two rectangular elements. As shown in Fig. 11, the dimensions of the first elementary cuboidal magnet are a, b, and c, and its polarization is J. For the second elementary magnet, the dimensions are A, B and C. Its polarization is J’, and the coordinate of its center is a, 3, and y. Avl, and Ax2; are the widths of the first and second elementary magnets. The upper magnet moves in translation along the y-axis above the lower fixed magnet. The total aenetic force is given by:
Figure 12. Force components for ¢ = 10mm, r = 10mm and x = 10mm. where N and M are the number of first and second elementary magnets, respectively. To verify the validity of the analytical results, the force between two magnets was calculated Fig. 12 shows the force distribution, and we notice that the force in the direction is in a sinusoidal forr
Figure 12. Force components for ¢ = 10mm, r = 10mm and x = 10mm. where N and M are the number of first and second elementary magnets, respectively. To verify the validity of the analytical results, the force between two magnets was calculated Fig. 12 shows the force distribution, and we notice that the force in the direction is in a sinusoidal forr
Table 1. Comparison of magnetic force (magnet set 4 x 4).
Table 1. Comparison of magnetic force (magnet set 4 x 4).
Figure 13. Force between the two cylindrical magnets versus the axial axis. Figure 13 shows variation of the force versus axial distance between the two cylindrical magnets. We can see that the calculation describes the experimental behavior fairly well.
Figure 13. Force between the two cylindrical magnets versus the axial axis. Figure 13 shows variation of the force versus axial distance between the two cylindrical magnets. We can see that the calculation describes the experimental behavior fairly well.
Figure 14. A Scheme of three sets of cylindrical magnets; R = 2mm and ¢ = 8mm.
Figure 14. A Scheme of three sets of cylindrical magnets; R = 2mm and ¢ = 8mm.
Figure 15. Calculated data for attraction force between three magnet set (2 x 2), (4 x 4) and (6 x 6) Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 15. Calculated data for attraction force between three magnet set (2 x 2), (4 x 4) and (6 x 6) Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 16. (a) System composed of a permanent magnet and thin coil. (b) The coil is replaced by four bars). (c) Bar-shaped volume current density. The force formula between cylindrical magnets can also be used to evaluate the force between thin coils and magnet. The fastest way to calculate forces in the cylindrical linear single-axis actuator [21, 26— 28] is to use the Lorenz force calculations, as the estimation of the magnetic field of the coils is not necessary [21,22]. The structure is shown in Fig. 16(a). The coil is modeled by four straight bars, as shown in Fig. 16(b), where each bar is defined by dimensions 2a x 2b x 2c. The total magnetic force acting on the thin coil is given by
Figure 16. (a) System composed of a permanent magnet and thin coil. (b) The coil is replaced by four bars). (c) Bar-shaped volume current density. The force formula between cylindrical magnets can also be used to evaluate the force between thin coils and magnet. The fastest way to calculate forces in the cylindrical linear single-axis actuator [21, 26— 28] is to use the Lorenz force calculations, as the estimation of the magnetic field of the coils is not necessary [21,22]. The structure is shown in Fig. 16(a). The coil is modeled by four straight bars, as shown in Fig. 16(b), where each bar is defined by dimensions 2a x 2b x 2c. The total magnetic force acting on the thin coil is given by
Figure 17. PM and massive conductor in (a) x-direction, (b) y-direction. Progress In Electromagnetics Research C, Vol. 91, 2019
Figure 17. PM and massive conductor in (a) x-direction, (b) y-direction. Progress In Electromagnetics Research C, Vol. 91, 2019
Table 2. Design parameters for magnet coil system. Figure 18. The magnetic induction intensity of force magnetic on the z-axis.
Table 2. Design parameters for magnet coil system. Figure 18. The magnetic induction intensity of force magnetic on the z-axis.
Figure 19. Axial force ‘Fz’ obtained from measurement and our approximation in different positions.
Figure 19. Axial force ‘Fz’ obtained from measurement and our approximation in different positions.
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