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

Abstract

The application discloses a kind of regular hexagon permanent magnet magnetic field distribution model construction methods, this method is: the three-dimensional molecular ring current model of regular hexagon 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, the external space magnetic field distribution model of regular hexagon permanent magnet is constructed, and the reliability and effectiveness of analytical formula are verified using finite element analysis results, and the distribution characteristics that regular hexagon permanent magnet external space magnetic flux density changes with this important motor design parameter, air gap value.

Description

Technical Field

[0001] The invention belongs to the technical field of regular hexagonal permanent magnet magnetic field distribution models and relates to a method for constructing a regular hexagonal permanent magnet magnetic field distribution model. Background Art

[0002] The external magnetic field distribution of permanent magnets is the basis of their engineering applications. In a planar moving magnet linear motor, according to Ampere's left-hand rule, the normal component B of the external magnetic flux density of the permanent magnet array is z The horizontal thrust exerted on the magnet determines the magnetic flux density. A straightforward analytical formula for magnetic flux density is crucial for the design and optimization of planar linear motor rotor structures. Therefore, it is necessary to analytically study the external magnetic field of permanent magnets. Rectangular permanent magnets are the most mature and widely used in engineering applications. However, research has been limited to single rectangular permanent magnets and one-dimensional rectangular permanent magnet arrays.

[0003] With the development of technologies such as precision machining and the improvement of product performance requirements of consumers and the market, other permanent magnets with symmetrical shapes have gradually been studied and applied. In addition, the document "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)" points out that permanent magnets with triangular, hexagonal, octagonal and other shapes may help improve the performance of magnetic machines. The document "S. Amin, S. Madanzadeh, S. Khan, et al. Three-dimensional analytical field calculation of triangular magnet segments used in inclined linear permanent magnet actuators [J]. Compel, 2010, 29 (4): 984-993" mentions that in coreless axial flux permanent magnet synchronous generators, the use of triangular permanent magnets achieves higher induced electromotive force and output power than trapezoidal or oblique magnets. The literature "N. Majernik, J. B. Rosenzweig. Halbach undulator using right-angled triangle magnets. Physics Review. Accelerators and Light Speed. 2019, 22(9): 092401-092401" mentions that the performance of the Halbach array based on right-angled triangle magnets is comparable to that of isosceles triangle magnets and better than that of the upper and lower lattice arrays. The literature "TBIBRAHIM, AHMEMON, F. MEMON, P. NALLAGOWDEN and NAMOHD ZAMRI. Modeling and verification of triangular magnetic array linear synchronous permanent magnet generator for wave energy conversion. 2018 International Conference on Intelligent and Advanced Systems (ICIAS). 2018, pp. 1-6" mentions that in a direct-drive permanent magnet linear generator for wave energy conversion, the use of isosceles triangle permanent magnets achieved better electromagnetic performance than traditional rectangular permanent magnets, and the open-circuit magnetic flux density was 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 Generators. 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 difficult to apply and generalize. 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 a three-dimensional micro-digital electromagnetic actuator.The document "A. Deshmukh, L. Petit, MUKhan, F. Lamarque and C. Prelle. A novel 12-discrete position three-dimensional electromagnetic digital actuator. IEEE / ASME Transactions on Mechatronics. 2018, 23(4): 1653-1661" equates a hexagonal permanent magnet to a rectangular permanent magnet to analyze its external magnetic field distribution. Currently, there is no analytical formula to describe the external magnetic field distribution of a hexagonal permanent magnet. In magnetic refrigeration systems, the use of octagonal arrays can not only improve the cooling capacity of refrigerators, but also keep the system structure simple and compact. In magnetic particle imaging technology, the octagon is the most suitable magnet shape in terms of both application and field line-free performance. However, the external magnetic field of an octagonal permanent magnet has not been analyzed. Currently, there is no analytical formula to describe the external magnetic field distribution of an octagonal permanent magnet. In fact, the literature "A. Deshmukh, L. Petit, MUKhan, F. Lamarque and C. Prelle. Development of 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 octagonal Halbach magnetic array for magnetic refrigerator. Heat Transfer Engineering, 2018, 39(4): 391-397" all use regular octagonal permanent magnets.

[0004] In summary, no numerical analytical model for regular hexagons is given 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 hexagonal permanent magnet, so as to solve the technical problems existing in the prior art.

[0006] The technical solution adopted by the present invention is: a method for constructing a magnetic field distribution model of a regular hexagonal permanent magnet, the method comprising:

[0007] The three-dimensional molecular circulation model of a regular hexagonal permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its body 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 s , its relationship with the magnetization vector M is as follows: (1), where n is the unit outer normal direction of the magnetic medium surface. The surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, that is, (2), where M = B r / μ0,B r is the residual magnetic flux density of the permanent magnet, in Tesla (T); μ0=4π×10 -7H / m, is the vacuum permeability, then the current intensity I of the thin layer current loop l with a thickness of dz0 is formula (3), formula (4) shows the vector expression of the Biot-Savart law, which is used to calculate the magnetic induction intensity generated by the line current element Idl at any field point in space. The integration of formula (4) on the closed current loop l gives the magnetic induction intensity B generated at the field point P (x, y, z) l The equation (5) is integrated over the thickness h of the permanent magnet to obtain the magnetic induction intensity B generated by the entire permanent magnet at this field point, which is equation (6). Substituting it into equation (7), we get the magnetic flux density expression (8) and its components (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 is a distribution of six current-carrying conductors in the two-dimensional plane. Based on the derivation of the magnetic induction intensity expression B5 generated by the fifth current-carrying conductor numbered 5 at the spatial field point P(x, y, z), the other five current-carrying conductors are derived:

[0011] In the fifth 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=-kdx0

[0015] and

[0016] dz0=0

[0017] x0∈[x1,x2]

[0018] y0∈[y2,y1]

[0019] Where x1, x2 represent the x-coordinate values ​​of the current-carrying conductor numbered 5; y2, y1 represent the y-coordinate values ​​of the current-carrying conductor numbered 5, and the current-carrying conductor points from (x1, y2) to (x2, y1);

[0020] Also because:

[0021]

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

[0023] so,

[0024]

[0025]

[0026]

[0027] but

[0028]

[0029]

[0030]

[0031] Define functions F, F1 and F2, each with coordinate values ​​(φ1, φ2, φ3), The function notation with (ξ1,ξ2,ξ3) as the independent variable is:

[0032]

[0033]

[0034]

[0035] Then we get:

[0036]

[0037]

[0038]

[0039] Similarly, the magnetic induction intensity generated by the other five current-carrying conductors at the field point P(x, y, z) is

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] make

[0056]

[0057]

[0058] According to the principle of magnetic field superposition, we can get:

[0059]

[0060]

[0061]

[0062] In the above formula, is a coefficient, x1, x2, x3, y1, y2, y3, y4 represent the component values ​​of the six vertex coordinates of the regular hexagon in the two-dimensional analytical model, respectively. The vertex coordinates in the counterclockwise direction from the upper left corner are (x1, y2), (x2, y1), (x3, y2), (x3, y2), (x2, y4) and (x1, y3); k and b are the slope and bias value of the oblique line of the regular hexagon in the two-dimensional analytical model, x0, y0, z0 are the components of the source point coordinates, x, y, z are the components of the field point coordinates, T i , G j The coefficients derived from the above parameters are i=1 and 2, j=2, 3, 5 and 6, and h is the thickness of the permanent magnet.

[0063] Beneficial effects of the present invention: Compared with the existing technology, the present invention is based on Ampere's molecular circulation hypothesis and molecular circulation model, applies the Biot-Savart law and the magnetic field superposition principle, derives the external space magnetic field distribution model of a regular hexagonal permanent magnet, uses finite element analysis results to verify the reliability and effectiveness of the analytical expression, and gives the distribution characteristics of the external space magnetic flux density of the regular hexagonal permanent magnet as a function of the spatial air gap value, an important motor design parameter. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 It is a diagram of a three-dimensional molecular circulation model;

[0065] Figure 2 It is a two-dimensional analytical model;

[0066] Figure 3 This is a comparison chart of the finite element analysis and analytical results of the magnetic field distribution in the external space of a regular hexagonal permanent magnet. DETAILED DESCRIPTION

[0067] The present invention will be further described below with reference to specific embodiments.

[0068] Example 1: A method for constructing a magnetic field distribution model of a regular hexagonal permanent magnet, the method comprising:

[0069] The three-dimensional molecular circulation model of the regular hexagonal permanent magnet is converted into a two-dimensional analytical model. The coordinate origin of the two-dimensional analytical model is located at the center, the upper and lower sides are horizontal current-carrying conductors, and the remaining current-carrying conductors are symmetrical. The conductors are numbered counterclockwise starting from the horizontal current-carrying conductor on the lower side: 1, 2, 3, 4, 5 and 6. After the regular hexagonal permanent magnet is uniformly magnetized, its body 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 s , its relationship with the magnetization vector M is as follows: (1), where n is the unit outer normal direction of the magnetic medium surface. The surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, that is, (2), where M = B r / μ0,B r is the residual magnetic flux density of the permanent magnet, in Tesla (T); μ0=4π×10 -7 H / m, is the vacuum permeability, then the current intensity I of the thin layer current loop l with a thickness of dz0 is formula (3), formula (4) shows the vector expression of the Biot-Savart law, which is used to calculate the magnetic induction intensity generated by the line current element Idl at any field point in space. The integration of formula (4) on the closed current loop l gives the magnetic induction intensity B generated at the field point P (x, y, z) lThe equation (5) is integrated over the thickness h of the permanent magnet to obtain the magnetic induction intensity B generated by the entire permanent magnet at this field point, which is equation (6). Substituting it into equation (7), we get the magnetic flux density expression (8) and its components (9)-(11):

[0070]

[0071] 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.

[0072] For the dz0 plane in the two-dimensional analytical model, the two-dimensional analytical model is a distribution of six current-carrying conductors in the two-dimensional plane. Based on the derivation of the magnetic induction intensity expression B5 generated by the fifth current-carrying conductor numbered 5 at the spatial field point P(x, y, z), the other five current-carrying conductors are derived:

[0073] In the fifth current-carrying conductor, the slope of the current-carrying conductor in the two-dimensional plane is k, because:

[0074] y0=-kx0-b

[0075] so,

[0076] dy0=-kdx0

[0077] and

[0078] dz0=0

[0079] x0∈[x1,x2]

[0080] y0∈[y2,y1]

[0081] Also because:

[0082]

[0083] so,

[0084]

[0085]

[0086]

[0087] but

[0088]

[0089]

[0090]

[0091] Define functions F, F1 and F2, each with coordinate values ​​(φ1, φ2, φ3), The function notation with (ξ1,ξ2,ξ3) as the independent variable is:

[0092]

[0093]

[0094]

[0095] Then we get:

[0096]

[0097]

[0098]

[0099] Similarly, the magnetic induction intensity generated by the other five current-carrying conductors at the field point P(x, y, z) is

[0100]

[0101]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107]

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115] make

[0116]

[0117]

[0118] According to the principle of magnetic field superposition, we can get

[0119]

[0120]

[0121]

[0122] In the above formula, is a coefficient, x1, x2, x3, y1, y2, y3, y4 represent the component values ​​of the six vertex coordinates of the regular hexagon in the two-dimensional analytical model, respectively. The vertex coordinates in the counterclockwise direction from the upper left corner are (x1, y2), (x2, y1), (x3, y2), (x3, y3), (x2, y4) and (x1, y3); k and b are the slope and bias value of the oblique line of the regular hexagon in the two-dimensional analytical model, x0, y0, z0 are the components of the source point coordinates, x, y, z are the components of the field point coordinates, T i , G j The coefficients derived from the above parameters are i=1 and 2, j=2, 3, 5 and 6, and h is the thickness of the permanent magnet.

[0123] Simulation verification: In the finite element analysis software ANSYS, a regular hexagonal permanent magnet model with a side length of 3mm and a thickness of 2mm was established. The residual magnetic flux density B r =1.31T, used to verify the normal component B of the magnetic flux density in the external space z The analytical expression of . Figure 3 Take x as 1.8 mm, and B at 0.5 mm above the regular hexagonal permanent magnet z The solid line in the figure is the result of finite element analysis; the dotted line is the result of analytical expression. Figure 3 It shows that the results of the analytical formula are highly consistent with the results of the finite element analysis, that is, it can accurately describe the normal component B of the magnetic flux density of the external space magnetic field of the regular hexagonal permanent magnet. z , the analytical expression is correct.

[0124] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of changes or replacements within the technical scope disclosed by the present invention, which should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A method for constructing a magnetic field distribution model of a regular hexagonal permanent magnet, characterized by: The method is: The three-dimensional molecular circulation model of a regular hexagonal permanent magnet is converted into a two-dimensional analytical model. After the permanent magnet is uniformly magnetized, its body 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 s , its relationship with the magnetization vector M is as follows: (1), where n is the unit outer normal direction of the magnetic medium surface. The surface magnetization current of the radially magnetized permanent magnet is equal to the magnetization intensity of the permanent magnet, that is, (2), where M = B r / μ0,B r is the residual magnetic flux density of the permanent magnet, in Tesla; μ0=4π×10 -7 H / m, is the vacuum permeability, then the current intensity I of the thin layer current loop l with a thickness of dz0 is formula (3), formula (4) shows the vector expression of the Biot-Savart law, which is used to calculate the magnetic induction intensity generated by the line current element Idl at any field point in space. The integration of formula (4) on the closed current loop l gives the magnetic induction intensity B generated at the field point P (x, y, z) l The equation (5) is integrated over the thickness h of the permanent magnet to obtain the magnetic induction intensity B generated by the entire permanent magnet at this field point, which is equation (6). Substituting it into equation (7), we get the magnetic flux density expression (8) and its components (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 is a distribution of six current-carrying conductors in the two-dimensional plane. Based on the derivation of the magnetic induction intensity expression B5 generated by the fifth current-carrying conductor numbered 5 at the spatial field point P(x, y, z), the other five current-carrying conductors are derived: In the fifth current-carrying conductor, the slope of the current-carrying conductor in the two-dimensional plane is k, because: y0=-kx0-b so, dy0=-kdx0 and dz0=0 x0∈[x1,x2] y0∈[y2,y1] Where x1, x2 represent the x-coordinate values ​​of the current-carrying conductor numbered 5; y2, y1 represent the y-coordinate values ​​of the current-carrying conductor numbered 5, and the current-carrying conductor points from (x1, y2) to (x2, y1); Also because: Where, It represents the line element on the fifth current-carrying conductor; It represents the vector from the source point (x0, y0, z0) on the fifth current-carrying conductor to the field point P(x, y, z); and Represents the direction vectors of the three coordinate axes in the three-dimensional coordinate system; so, but Define functions F, F1 and F2, each with coordinate values ​​(φ1, φ2, φ3), The function notation with (ξ1,ξ2,ξ3) as the independent variable is: Then we get: Similarly, the magnetic induction intensity generated by the other five current-carrying conductors at the field point P(x, y, z) is make According to the principle of magnetic field superposition, we can get: In the above formula, is a coefficient, x1, x2, x3, y1, y2, y3, y4 represent the component values ​​of the six vertex coordinates of the regular hexagon in the two-dimensional analytical model, respectively. The vertex coordinates in the counterclockwise direction from the upper left corner are (x1, y2), (x2, y1), (x3, y2), (x3, y3), (x2, y4) and (x1, y3); k and b are the slope and bias value of the oblique line of the regular hexagon in the two-dimensional analytical model, x0, y0, z0 are the components of the source point coordinates, x, y, z are the components of the field point coordinates, T i , G j The coefficients derived from the above parameters are i=1 and 2, j=2, 3, 5 and 6, and h is the thickness of the permanent magnet.

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