A method for constructing a magnetic field distribution model of a triangular permanent magnet

Abstract

The application discloses a kind of construction methods of triangle permanent magnet magnetic field distribution model, the method is: the three-dimensional molecular ring current model of triangle permanent magnet is converted into two-dimensional analytical model, permanent magnet is uniformly magnetized, its body current is zero, the magnetic field at any field point in space is only generated by the closed current loop of permanent magnet surface, based on ampere molecular ring current hypothesis and molecular ring current model, application Biot-Savart law and magnetic field superposition principle, construct the external space magnetic field distribution model of triangle permanent magnet, the external space magnetic field distribution model of this two-dimensional permanent magnet array considers the distribution waveform and amplitude of the normal component of its external space magnetic flux density, optimize the permanent magnet monomer gap value of two-dimensional rectangular permanent magnet array, and the reliability and effectiveness of analytical formula are verified using finite element analysis results, and the distribution characteristics that external space magnetic flux density of triangle permanent magnet varies with this important motor design parameter, space air gap value are given.

Description

Technical Field

[0001] This invention belongs to the technical field of magnetic field distribution models of triangular permanent magnets, and relates to a method for constructing a magnetic field distribution model of a triangular permanent magnet. Background Technology

[0002] The distribution of the external magnetic field of permanent magnets is fundamental to their engineering applications. In planar moving-magnet linear motors, according to Ampere's left-hand rule, the normal component Bz of the external magnetic flux density of the mover permanent magnet array determines the horizontal thrust it experiences. The intuitive analytical expression for magnetic flux density is of great significance for the design and optimization of the mover structure in planar linear motors. Therefore, it is necessary to conduct analytical studies on the external magnetic field of permanent magnets. Among these, rectangular permanent magnets are the most mature and have the most widespread engineering applications. However, research has been limited to single rectangular permanent magnets and one-dimensional rectangular permanent magnet arrays.

[0003] With the development of precision machining and other technologies, and the increasing demands of consumers and the market for product performance, permanent magnets with other symmetrical shapes are gradually being researched and applied. Furthermore, the literature “JLG Janssen, JJH Paulides, EA Lomonova. Influence of magnet shape on the performance of coreless axial flux permanent magnet synchronous generators. Electrical Engineering, 104, 959–968 (2022)” points out that permanent magnets with triangular, hexagonal, and octagonal shapes may contribute to the improvement of magnetic machine performance. The literature “S. Amin, S. Madanzadeh, S. Khan, et al. Three-dimensional analysis field calculation of triangular magnet segments applied to inclined linear permanent magnet actuators [J]. Compel, 2010, 29(4): 984-993” mentions that in coreless axial flux permanent magnet synchronous generators, compared to trapezoidal or oblique magnets, the use of triangular permanent magnets achieves higher induced electromotive force and output power. The literature “N. Majernik, J. Brosenzweig. Halbach undulator using right-angled triangular magnets. Physics Review. Accelerators and the Speed ​​of Light. 2019, 22(9): 092401-092401” mentions that the Halbach array based on right-angled triangular magnets has performance comparable to that of isosceles triangular magnets and superior to that of upper and lower lattice arrays. The literature “TBIBRAHIM, AHMEMON, F. MEMON, P. NALLAGOWDEN and NAMOHD ZAMRI. Modeling and verification of a triangular magnetic array linear synchronous permanent magnet generator for wave energy conversion. International Conference on Intelligent and Advanced Systems (ICIAS) 2018. 2018, pp. 1-6” mentions that in a direct-drive permanent magnet linear generator for wave energy conversion, isosceles triangular permanent magnets have achieved better electromagnetic performance than traditional rectangular permanent magnets, and the open-circuit magnetic flux density has been studied using the Fourier series method. The paper "JLG Janssen, JJH Paulides, EA Lomonova. Influence of magnet shape on the performance of coreless axial flux permanent magnet synchronous generator. Electrical Engineering, 104, 959–968 (2022)" proposes a method for calculating the magnetic field of prismatic permanent magnets based on the magnetic charge method, but the analytical formula is complex and not easy to learn from and promote. The paper "A. Deshmukh, L. Petit, MU Khan, F. Lamarque and C. Prelle. Development of a six-digit digital electromagnetic actuator. 2017 IEEE Advanced Intelligent Mechatronics (AIM) International Conference. 2017: 975-980" mentions the use of hexagonal permanent magnets as movers in three-dimensional micro digital electromagnetic actuators.The literature “A. Deshmukh, L. Petit, MUKhan, F. Lamarque and C. Prelle. A novel 12 discrete position three-dimensional electromagnetic digital actuator. IEEE / ASME Mechatronics Bulletin. 2018, 23(4): 1653-1661” equates a hexagonal permanent magnet to a rectangular permanent magnet to analyze its external spatial magnetic field distribution. Currently, there is no analytical formula describing the external spatial magnetic field distribution of a hexagonal permanent magnet. In magnetocooling systems, using an octagonal array can improve the cooling capacity of the refrigerator while maintaining a simple and compact system structure. In magnetic particle imaging technology, considering both application and field-line-free performance, the octagon is the most suitable magnet shape. However, none of these studies have analyzed the external spatial magnetic field of an octagonal permanent magnet. Currently, there is also no analytical formula describing the external spatial magnetic field distribution of an octagonal permanent magnet. In fact, the literature “A.Deshmukh, L.Petit, MUKhan, F.Lamarque and C.Prelle. Development of a six-digit electromagnetic actuator. 2017 IEEE Advanced Intelligent Mechatronics (AIM) International Conference. 2017:975-980” all use regular hexagonal permanent magnets, and the literature “CELIK, SERDAR Kural, MEHMET, HAMDI. Design of an octagonal Heilbeck magnetic array for a magnetic refrigerator. Heat Transfer Engineering, 2018, 39(4):391-397” all use regular octagonal permanent magnets.

[0004] In summary, existing technologies do not record numerical calculation methods for the magnetic field distribution of triangular permanent magnets. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a method for constructing a magnetic field distribution model of a triangular permanent magnet, so as to solve the technical problems existing in the prior art.

[0006] The technical solution adopted in this invention is as follows: a method for constructing a magnetic field distribution model of a triangular permanent magnet. The method is as follows: the three-dimensional molecular circulation model of the triangular permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its volume current is zero. Then, the magnetic field at any field point P(x,y,z) in space is generated only by the closed current loop on the surface of the permanent magnet. Let the surface magnetization current of the permanent magnet be J, and its relationship with the magnetization vector M is given by equation (1), where n is the unit outward normal direction of the magnetic medium surface. The magnitude of the surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, i.e., equation (2). In equation (2), M = B r / μ0, B r The remanent magnetic flux density of the permanent magnet is expressed in Tesla (T); μ0 = 4π × 10⁻⁶ -7H / m is the vacuum permeability. The current intensity I of a thin current loop l with thickness dz0 is given by equation (3). Equation (4) is the vector expression of the Biot-Savart law, used to calculate the magnetic induction intensity generated by the line current element Idl at any point in space. The integral of equation (4) over the closed current loop l yields the magnetic induction intensity B generated at the field point P(x,y,z). l Equation (5) is given by integrating equation (5) over the thickness h of the permanent magnet, resulting in the magnetic induction intensity B generated by the entire permanent magnet at that field point, which is given by equation (6):

[0007]

[0008] In the above formulas, r represents the radius vector from the origin to the field point P(x,y,z); r' represents the radius vector from the origin to the source point (x0,y0,z0); r-r' represents the radius vector from the source point to the field point; M is the magnetization vector of the permanent magnet, and i, j, and k represent unit vectors;

[0009] Calculate the normal component of the magnetic induction intensity produced at field point P(x,y,z) by all current-carrying conductors parallel to current-carrying conductor 1 (l1) in the two-dimensional analytical model within the triangular permanent magnet. For example, let k1, k2, and k3 be the slopes of the triangles in the two-dimensional analytical model. Substituting equation (3) into equation (5), we obtain the magnetic induction intensity dB1 generated at point P(x,y,z) as equation (7). Integrating equation (7) over thickness h and substituting it into equation (8), we obtain the magnetic induction intensity B1 generated at point P(x,y,z) by all current-carrying conductors parallel to l1 in the permanent magnet, as shown in equation (9). Its component B x B y B z Equations (10)-(12) are respectively. Relations (13)-(15) exist in l1. Substituting them into equation (12), we get equation (16). Equation (16) simplifies to standard integral form, and finally yields the result shown in equation (17).

[0010] Define functions F, F1, and F2 with coordinate values ​​x, y, and z as independent variables, as shown in equations (18)-(20), and with coefficients as shown in equations (21)-(30). As shown in equation (31); similarly, calculate the magnetic induction intensity generated at P(x,y,z) by all current-carrying conductors parallel to current-carrying conductors 2 and 3 in the permanent magnet, and obtain B according to the principle of magnetic field superposition. z B x B y The parsing follows B zThe steps are complete; then the external spatial magnetic field of the three-dimensional molecular circulation model of the triangular permanent magnet is described by equations (32)-(34):

[0011] y0=k1x0+b (13)

[0012] dz0=0 (14)

[0013] dy0=k1dx0 (15)

[0014]

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] In the above equations (21)-(30), T ij H i These are coefficients derived from the aforementioned parameters, i = 1 to 3, j = 1 to 3;

[0022]

[0023]

[0024]

[0025]

[0026] In the above equations (32)-(34), is a coefficient, k1, k2 and k3 are the slopes of the diagonal lines of the triangle in the two-dimensional analytical model; b is the offset value of the first diagonal line.

[0027] The beneficial effects of this invention are as follows: Compared with the prior art, this invention is based on the Ampere molecular circulation hypothesis and molecular circulation model, and applies the Biot-Savart law and the principle of magnetic field superposition to derive a triangular external space magnetic field distribution model. The reliability and effectiveness of the analytical formula are verified by finite element analysis results. Furthermore, the distribution characteristics of the external space magnetic flux density of triangular, regular hexagonal, and regular octagonal permanent magnets as a function of the important motor design parameter of the air gap value are given. Attached Figure Description

[0028] Figure 1 It is a three-dimensional molecular circulation model diagram;

[0029] Figure 2 It is a two-dimensional analytical model;

[0030] Figure 3 It is a three-dimensional distribution diagram of the magnetic field in the external space of an isosceles triangular permanent magnet;

[0031] Figure 4 It is a diagram showing the distribution characteristics of the magnetic field in the external space of an isosceles triangular permanent magnet as the field point moves away from the surface of the permanent magnet. Detailed Implementation

[0032] The present invention will be further described below with reference to specific embodiments.

[0033] Example 1: A method for constructing a magnetic field distribution model of a triangular permanent magnet. The method is as follows: the three-dimensional molecular circulation model of the triangular permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its volume current is zero. Then, the magnetic field at any field point P(x,y,z) in space is generated only by the closed current loop on the surface of the permanent magnet. Let the surface magnetization current of the permanent magnet be J, and its relationship with the magnetization vector M is given by equation (1), where n is the unit outward normal direction of the magnetic medium surface. The magnitude of the surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, i.e., equation (2). Equation (2) M=B r / μ0, Br is the remanent magnetic flux density of the permanent magnet, in Tesla (T); μ0=4π×10 -7 H / m is the vacuum permeability. The current intensity I of a thin current loop l with thickness dz0 is given by equation (3). Equation (4) is the vector expression of the Biot-Savart law, used to calculate the magnetic induction intensity generated by the line current element Idl at any point in space. The integral of equation (4) over the closed current loop l yields the magnetic induction intensity B generated at the field point P(x,y,z). l Equation (5) is given by integrating equation (5) over the thickness h of the permanent magnet, resulting in the magnetic induction intensity B generated by the entire permanent magnet at that field point, which is given by equation (6):

[0034]

[0035] In the above formulas, r represents the radius vector from the origin to the field point P(x,y,z); r' represents the radius vector from the origin to the source point (x0,y0,z0); r-r' represents the radius vector from the source point to the field point; and M is the magnetization vector of the permanent magnet.

[0036] The triangular permanent magnet is divided into current-carrying conductors l1, l2, and l3. First, the normal component of the magnetic induction intensity produced by all current-carrying conductors parallel to current-carrying conductor l1 in the two-dimensional analytical model at the field point P(x,y,z) is calculated. For example, let k1, k2, and k3 be the slopes of the triangles in the two-dimensional analytical model. Substituting equation (3) into equation (5), we obtain the magnetic induction intensity dB1 generated at point P(x,y,z) as equation (7). Integrating equation (7) over thickness h and substituting it into equation (8), we obtain the magnetic induction intensity B1 generated at point P(x,y,z) by all current-carrying conductors parallel to l1 in the permanent magnet, as shown in equation (9). Its component B x B y B z Equations (10)-(12) are respectively. Relations (13)-(15) exist in l1. Substituting them into equation (12), we get equation (16). Equation (16) simplifies to standard integral form, and finally yields the result shown in equation (17).

[0037] Define functions F, F1, and F2 with coordinate values ​​x, y, and z as independent variables, as shown in equations (18)-(20), and with coefficients as shown in equations (21)-(30). As shown in equation (31); similarly, calculate the magnetic induction intensity generated at P(x,y,z) by all current-carrying conductors parallel to current-carrying conductors 2 and 3 in the permanent magnet, and obtain B according to the principle of magnetic field superposition. z B x B y The parsing follows B z The steps are complete; then the external spatial magnetic field of the three-dimensional molecular circulation model of the triangular permanent magnet is described by equations (32)-(34):

[0038] y0=k1x0+b (13)

[0039] dz0=0 (14)

[0040] dy0=k1dx0 (15)

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048] In the above equations (21)-(30), T ij H i These are coefficients derived from the aforementioned parameters, i = 1 to 3, j = 1 to 3;

[0049]

[0050]

[0051]

[0052]

[0053] In the above equations (32)-(34), is a coefficient, k1, k2, and k3 are the slopes of the diagonal lines of the triangle in the two-dimensional analytical model, and b is the offset value of the first diagonal line.

[0054] Simulation Verification: In the analytical study of the magnetic field of a triangular permanent magnet based on the magnetic charge method, the isosceles triangular permanent magnet has a base length of 40 mm, a vertex distance of 34 mm from the base, a thickness of 7 mm, and a remanent magnetic flux density Br = 1.23 T. Its structure is established as follows... Figure 1 Using the coordinate system shown, and applying equation (22), the normal component Bz of the magnetic flux density in the external space 1 mm above it is obtained, as follows: Figure 3 As shown. The results of the analytical expression are consistent with those in the literature "JLG Janssen, JJH Paulides, EA Lomonova. Influence of magnet shape on the performance of coreless axial flux permanent magnet synchronous generator. Electrical Engineering, 104, 959–968 (2022)", proving that the analytical expression for the external spatial magnetic field distribution of the triangular permanent magnet derived in this invention is correct.

[0055] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of protection of the claims.

Claims

1. A method for constructing a magnetic field distribution model of a triangular permanent magnet, characterized in that: The method is as follows: the three-dimensional molecular circulation model of the triangular permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its volume current is zero. Then, the magnetic field at any field point P(x,y,z) in space is generated only by the closed current loop on the surface of the permanent magnet. Let the surface magnetization current of the permanent magnet be J, and its relationship with the magnetization vector M is given by equation (1), where n is the unit outward normal direction of the magnetic medium surface. The magnitude of the surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, i.e., equation (2). In equation (2), M = B r / μ0, B r The remanent magnetic flux density of the permanent magnet is expressed in Tesla; μ0 = 4π × 10⁻⁶ -7 H / m is the vacuum permeability. The current intensity I of a thin current loop l with thickness dz0 is given by equation (3). Equation (4) is the vector expression of the Biot-Savart law, used to calculate the magnetic induction intensity generated by the line current element Idl at any point in space. The integral of equation (4) over the closed current loop l yields the magnetic induction intensity B generated at the field point P(x,y,z). l Equation (5) is given by integrating equation (5) over the thickness h of the permanent magnet, resulting in the magnetic induction intensity B generated by the entire permanent magnet at that field point, which is given by equation (6): In the above formulas, r represents the radius vector from the origin to the field point P(x,y,z); r' represents the radius vector from the origin to the source point (x0,y0,z0); r-r' represents the radius vector from the source point to the field point; M is the magnetization vector of the permanent magnet, and i, j, and k represent unit vectors; First, calculate the normal component of the magnetic induction intensity produced at the field point P(x,y,z) by all current-carrying conductors parallel to the first current-carrying conductor l1 in the two-dimensional analytical model within the triangular permanent magnet. Let k1, k2, and k3 be the slopes of the triangles in the two-dimensional analytical model. Substituting equation (3) into equation (5), we obtain the magnetic induction intensity dB1 generated at point P(x,y,z) as equation (7). Integrating equation (7) over thickness h and substituting it into equation (8), we obtain the magnetic induction intensity B1 generated at point P(x,y,z) by all current-carrying conductors parallel to l1 in the permanent magnet, as shown in equation (9). Its magnetic induction intensity component B x B y B z Equations (10)-(12) are respectively. Relations (13)-(15) exist in l1. Substituting them into equation (12), we get equation (16). Equation (16) simplifies to standard integral form, and finally yields the result shown in equation (17). Define functions F, F1, and F2 with coordinate values ​​x, y, and z as independent variables, as shown in equations (18)-(20), and with coefficients as shown in equations (21)-(30). As shown in equation (31); similarly, calculate the magnetic induction intensity at P(x,y,z) of all current-carrying conductors parallel to current-carrying conductors 2 and 3 in the permanent magnet, and obtain B according to the principle of magnetic field superposition. z B x B y The parsing follows B z The steps are complete; then the external spatial magnetic field of the three-dimensional molecular circulation model of the triangular permanent magnet is described by equations (32)-(34): y0=k1x0+b (13) dz0=0 (14) dy0=k1dx0 (15) In the above equations (21)-(30), T ij and H i These are coefficients derived from the aforementioned parameters, i = 1 to 3, j = 1 to 3; In the above equations (32)-(34), is a coefficient, k1, k2 and k3 are the slopes of the diagonal lines of the triangle in the two-dimensional analytical model; b is the offset value of the first diagonal line.

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