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

Abstract

The application discloses a kind of octagonal permanent magnet magnetic field distribution model construction methods, this method is: the three-dimensional molecular ring current model of octagonal permanent magnet is converted into two-dimensional analytical model, permanent magnet is uniformly magnetized, its bulk 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 octagonal permanent magnet, 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 octagonal permanent magnet varies with this important motor design parameter, air gap value.

Description

Technical Field

[0001] This invention belongs to the technical field of magnetic field distribution model of regular octagonal permanent magnet, and relates to a method for constructing a magnetic field distribution model of regular octagonal 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 octagonal 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 octagonal permanent magnets to rectangular permanent magnets to analyze their external spatial magnetic field distribution. Currently, there is no analytical formula describing the external spatial magnetic field distribution of octagonal permanent magnets. In magnetocooling systems, using octagonal arrays can improve the cooling capacity of refrigerators 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 octagonal permanent magnets. Currently, there is also no analytical formula describing the external spatial magnetic field distribution of octagonal permanent magnets. In fact, the literature “A.Deshmukh, L.Petit, MUKhan, F.Lamarque and C.Prelle. Development of a six-digit digital electromagnetic actuator. 2017 IEEE Advanced Intelligent Mechatronics (AIM) International Conference. 2017:975-980” all use octagonal 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” also uses octagonal permanent magnets.

[0004] In summary, no numerical analytical model of the magnetic field distribution of a regular octagonal permanent magnet has been provided in the existing literature. 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 regular octagonal permanent magnet, so as to solve the technical problems existing in the prior art.

[0006] The technical solution adopted in this invention is: a method for constructing a magnetic field distribution model of a regular octagonal permanent magnet, the method being as follows:

[0007] The three-dimensional molecular circulation model of a regular octagonal permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its volume current is zero. Therefore, 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. s The relationship between the magnetization vector M and the magnetic medium 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. The magnetic flux density B generated by the entire permanent magnet at this field point is given by equation (6). Substituting this into equation (7), we get the magnetic flux density expression (8) and its component equations (9)-(11):

[0008]

[0009] 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;

[0010] For the dz0 plane in the two-dimensional analytical model, the two-dimensional analytical model consists of eight current-carrying conductors distributed in the two-dimensional plane. Based on the derivation of the expression B2 for the magnetic induction intensity produced by the second current-carrying conductor (labeled 2) at the spatial field point P(x, y, z), the other seven current-carrying conductors are derived as follows:

[0011] In the second current-carrying conductor, the slope of the current-carrying conductor in the two-dimensional plane is k, because:

[0012] y0=-kx0+b

[0013] so,

[0014] dy0 = -kx0

[0015] and

[0016] dz0=0

[0017]

[0018]

[0019] In the formula, x4, x3 represent the x-coordinate values ​​of the current-carrying conductor labeled 2; y3, y4 represent the y-coordinate values ​​of the current-carrying conductor labeled 2, and the current-carrying conductor points from (x4, y3) to (x3, y4);

[0020] And because

[0021]

[0022] In the formula, This refers to the line element on the second current-carrying conductor; This represents the vector from the source point (x0, y0, z0) on the second current-carrying conductor to the field point P(x, y, z); and Represents the direction vectors of the three coordinate axes in a three-dimensional coordinate system;

[0023] so

[0024]

[0025]

[0026]

[0027] but

[0028]

[0029]

[0030]

[0031] Define functions F, F1, and F2, each with coordinates (φ1, φ2, φ3), respectively. The function notation for (ξ1,ξ2,ξ3) as independent variables is:

[0032]

[0033]

[0034]

[0035] but

[0036]

[0037]

[0038]

[0039] Similarly, the magnetic induction intensity produced by the other seven current-carrying conductors at the field point P(x, y, z) can be obtained:

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058]

[0059]

[0060]

[0061] make:

[0062]

[0063]

[0064] Then, according to the principle of superposition of magnetic fields, we get:

[0065]

[0066]

[0067]

[0068] In the above formula, It is a coefficient, where x1, x2, x3, x4, y1, y2, y3, and y4 represent the component values ​​of the coordinates of the eight vertices of the regular octagon in the two-dimensional analytical model; the vertex coordinates in the counterclockwise direction from the top left corner are (x1, y2), (x2, y1), (x3, y1), (x4, y2), (x4, y3), (x3, y4), (x2, y4), and (x1, y3); k and b are the slope and bias value of the oblique line of the regular octagon in the two-dimensional analytical model, respectively; x0, y0, and z0 are the components of the source point coordinates; and x, y, and z are the components of the field point coordinates. T i G j The coefficients derived from the aforementioned parameters are i = 1 and 2, j = 2, 4, 6 and 8; h is the thickness of the permanent magnet.

[0069] The beneficial effects of this invention are as follows: Compared with the prior art, this invention is based on Ampere's molecular circulation hypothesis and molecular circulation model, and applies Biot-Savart law and the principle of magnetic field superposition to derive the external space magnetic field distribution model of a regular octagonal permanent magnet. The reliability and effectiveness of the analytical formula are verified by finite element analysis results, and the distribution characteristics of the external space magnetic flux density of the regular octagonal permanent magnet as a function of the important motor design parameter of the air gap value are given. Attached Figure Description

[0070] Figure 1 It is a three-dimensional molecular circulation model diagram;

[0071] Figure 2 It is a two-dimensional analytical model;

[0072] Figure 3 This is a comparison chart of finite element analysis and analytical results of the magnetic field distribution in the external space of a regular octagonal permanent magnet. Detailed Implementation

[0073] The present invention will be further described below with reference to specific embodiments.

[0074] Example 1: As Figure 1-2 As shown, a method for constructing a magnetic field distribution model of a regular octagonal permanent magnet is as follows:

[0075] The three-dimensional molecular circulation model of the regular octagonal permanent magnet is converted into a two-dimensional analytical model. In the two-dimensional analytical model, the origin is at the center, the upper and lower sides are horizontal current-carrying conductors, the left and right sides are numerical current-carrying conductors, and the remaining current-carrying conductors are centrally symmetrical. Starting from the lower horizontal current-carrying conductor, the conductors are numbered counterclockwise: 1, 2, 3, 4, 5, 6, 7, and 8. After the permanent magnet is uniformly magnetized, its volume current is zero. Therefore, the magnetic field at any field point P(x,y,z) in space is generated solely by the closed current loop on the surface of the permanent magnet. Let the surface magnetization current of the permanent magnet be J.s The relationship between the magnetization vector M and the magnetic medium 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 -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. The magnetic flux density B generated by the entire permanent magnet at this field point is given by equation (6). Substituting this into equation (7), we get the magnetic flux density expression (8) and its component equations (9)-(11):

[0076]

[0077] 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;

[0078] For the dz0 plane in the two-dimensional analytical model, the two-dimensional analytical model consists of eight current-carrying conductors distributed in the two-dimensional plane. Based on the derivation of the expression B2 for the magnetic induction intensity produced by the second current-carrying conductor (labeled 2) at the spatial field point P(x, y, z), the other seven current-carrying conductors are derived as follows:

[0079] In the second current-carrying conductor, the slope of the current-carrying conductor in the two-dimensional plane is k, because:

[0080] y0=-kx0+b

[0081] so,

[0082] dy0 = -kx0

[0083] and

[0084] dz0=0

[0085]

[0086]

[0087] In the formula, x4, x3 represent the x-coordinate values ​​of the current-carrying conductor labeled 2; y3, y4 represent the y-coordinate values ​​of the current-carrying conductor labeled 2, and the current-carrying conductor points from (x4, y3) to (x3, y4);

[0088] And because

[0089]

[0090] In the formula, This refers to the line element on the second current-carrying conductor; This represents the vector from the source point (x0, y0, z0) on the second current-carrying conductor to the field point P(x, y, z); and Represents the direction vectors of the three coordinate axes in a three-dimensional coordinate system;

[0091] so,

[0092]

[0093]

[0094]

[0095] but

[0096]

[0097]

[0098]

[0099] Define functions F, F1, and F2, each with coordinates (φ1, φ2, φ3), respectively. The function notation for (ξ1,ξ2,ξ3) as independent variables is:

[0100]

[0101]

[0102]

[0103] but

[0104]

[0105]

[0106]

[0107] Similarly, the magnetic induction intensity generated by the other seven current-carrying conductors at the field point P(x,y,z) can be obtained.

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122]

[0123]

[0124]

[0125]

[0126]

[0127]

[0128]

[0129] make:

[0130]

[0131]

[0132] Then, according to the principle of superposition of magnetic fields, we get:

[0133]

[0134]

[0135]

[0136] In the above formula, It is a coefficient, where x1, x2, x3, y1, y2, y3, and y4 represent the component values ​​of the coordinates of the eight vertices of the regular octagon in the two-dimensional analytical model. The coordinates of the vertices in the counterclockwise direction from the top left corner are (x1, y2), (x2, y1), (x3, y1), (x4, y2), (x4, y3), (x3, y4), (x2, y4), and (x1, y3), respectively. k and b are the slope and bias value of the oblique line of the regular octagon in the two-dimensional analytical model, respectively. x0, y0, and z0 are the components of the source point coordinates, and x, y, and z are the components of the field point coordinates. T i G i The coefficients are derived from the aforementioned parameters. h is the thickness of the permanent magnet.

[0137] Simulation Verification: In the finite element analysis software ANSYS, a regular octagonal permanent magnet model with a side length of 2mm and a thickness of 2mm was established, with a residual magnetic flux density Br = 1.31T, to verify the normal component B of its external space magnetic flux density. z The analytical expression. Figure 3 To measure B at a point 0.1 mm above a 1.2 mm diameter octagonal permanent magnet. z Distribution. In the figure, solid lines represent the results of finite element analysis; dashed lines represent the results of analytical expressions. Figure 3 In the process, due to the selection of boundary conditions and mesh division, the amplitude error at the curve ends is relatively large; peak noise causes burrs and poor smoothness at the curve top. However, overall, the analytical expression can describe the normal component B of the magnetic flux density of the external magnetic field of the regular octagonal permanent magnet. z .

[0138] 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 scope of the technology 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 regular octagonal permanent magnet, characterized in that: The method is as follows: The three-dimensional molecular circulation model of a regular octagonal permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its volume current is zero. Therefore, 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. s The relationship between it and 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. Integrating equation (5) over the thickness h of the permanent magnet, we obtain the magnetic induction intensity B generated by the entire permanent magnet at the field point as equation (6). Substituting this into equation (7), we get the magnetic flux density expression (8) and its component equations (9)-(11). 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; For the dz0 plane in the two-dimensional analytical model, the two-dimensional analytical model consists of eight current-carrying conductors distributed in the two-dimensional plane. Based on the derivation of the expression B2 of the magnetic induction intensity generated by the second current-carrying conductor with the lower right corner number 2 at the spatial field point P(x,y,z), the other seven current-carrying conductors are derived. In the second current-carrying conductor, the slope of the current-carrying conductor in the two-dimensional plane is k, because: y0=-kx0+b so, dy0 = -kx0 and dz0=0 In the formula, x4, x3 represent the x-coordinate values ​​of the current-carrying conductor labeled 2; y3, y4 represent the y-coordinate values ​​of the current-carrying conductor labeled 2, and the current-carrying conductor points from (x4, y3) to (x3, y4); And because In the formula, This refers to the line element on the second current-carrying conductor; This represents the vector from the source point (x0, y0, z0) on the second current-carrying conductor to the field point P(x, y, z); and Represents the direction vectors of the three coordinate axes in a three-dimensional coordinate system; so, but Define functions F, F1, and F2, each with coordinates (φ1, φ2, φ3), respectively. The function notation for (ξ1,ξ2,ξ3) as independent variables is: but Similarly, the magnetic induction intensity produced by the other seven current-carrying conductors at the field point P(x, y, z) is: make: Then, according to the principle of superposition of magnetic fields, we get: In the above formula, x1, x2, x3, x4, y1, y2, y3, y4 represent the component values ​​of the coordinates of the eight vertices of the regular octagon in the two-dimensional analytical model; the coordinates of the vertices counterclockwise from the top left corner are (x1, y2), (x2, y1), (x3, y1), (x4, y2), (x4, y3), (x3, y4), (x2, y4), and (x1, y3); k and b are the slope and bias value of the oblique line of the regular octagon in the two-dimensional analytical model, x0, y0, z0 are the components of the source point coordinates, and x, y, z are the components of the field point coordinates; T i G j The coefficients derived from the aforementioned parameters are i = 1 and 2, j = 2, 4, 6 and 8, and h is the thickness of the permanent magnet.

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