Modeling method for three-dimensional magnetic field of permanent magnet coupling based on molecular current theory

Abstract

The invention belongs to the technical field of mechanical transmission, and provides a permanent magnet coupling three-dimensional magnetic field modeling method based on a molecular current theory, which comprises the following steps: firstly, according to a molecular current viewpoint, enabling a uniformly magnetized permanent magnet to be equivalent to a form that a plurality of current loops are tightly wound on the surface of the uniformly magnetized permanent magnet; secondly, correcting the Biosaval law, and solving the magnetic field intensity of an equivalent close winding current ring on the surface of the permanent magnet in vacuum; then, considering the influence of a soft magnetic medium on the magnetic induction intensity, and solving the magnetic induction intensity contribution of a single permanent magnet to a geometric space according to a magnetic medium constitutive equation; and finally, solving the total magnetic induction intensity distribution of the magnet array. According to the permanent magnet coupling three-dimensional magnetic field modeling method based on the molecular current theory, a simple and efficient magnetic field calculation method is provided for the field of optimization design and application of permanent magnet couplings.

Description

Technical Field

[0001] The invention belongs to the technical field of mechanical transmission, and in particular relates to a three-dimensional magnetic field modeling method of a permanent magnetic coupling based on molecular current theory. Background Art

[0002] As a new type of high-performance contactless magnetic bearing, permanent magnetic coupling has the advantages of good seismic isolation performance, simple structure, low centering accuracy requirements, and no friction loss compared to traditional mechanical bearings. It is widely used in important engineering fields such as petroleum, shipping, and new energy. In practical engineering applications, it is of great significance to model and calculate the magnetic field distribution of permanent magnetic coupling. Due to the complex magnetic field structure of permanent magnetic coupling, it is difficult to analyze and calculate the magnetic field. At present, finite element analysis software is mostly used to model the magnetic field distribution to achieve the purpose of auxiliary design. However, finite element calculation consumes a lot of resources, takes a long time to calculate, and has a slow response, which is not conducive to adaptive control of the transmission system. Analytical modeling solves this problem. The existing analytical modeling methods mainly include the equivalent magnetic charge method, the equivalent magnetic circuit method and the equivalent current method. Among them, the equivalent magnetic charge method is complete under the theoretical background of classical electromagnetism, but in the theoretical calculation process, this method has the problem of difficulty in describing the magnetic charge distribution of non-prism-like / cylindrical geometric bodies, which makes it difficult to perform physical modeling when solving the magnetic field distribution problem of permanent magnets of other configurations, and thus it is impossible to use the equivalent magnetic charge method for modeling; the equivalent magnetic circuit method is based on the viewpoint of abstracting the influence of magnetic medium on magnetic flux into a "path" to solve the problem that magnetic medium of complex geometric configuration is difficult to analyze and model. In the context of practical engineering applications, this method has the problem of inaccurate physical modeling and unclear physical images, which affects the modeling accuracy. The equivalent current method based on molecular current theory is the essential model of classical electromagnetic theory to explain the magnetic source of permanent magnets. Therefore, this method can model the magnetic field of permanent magnets of relatively complex geometric configurations while maintaining a certain calculation accuracy, which can effectively solve the problems existing in the above two traditional methods. However, the molecular current method is rarely used in engineering at present, and the research objects are mostly permanent magnets with excellent geometric symmetry. There is a lack of engineering application scenarios, and theoretical research is the main focus.

[0003] Li Ruohan et al.'s patent "Magnetic Field Calculation Method and System for Local Elliptical Cylindrical Magnets Based on Equivalent Magnetic Charge Method", patent number CN202310587121.X, introduces a magnetic field calculation method for local elliptical cylindrical magnets based on the equivalent magnetic charge method. This method equates the magnetic charge that contributes to the magnetic field of an axially uniformly magnetized elliptical cylindrical magnet to the upper and lower parallel elliptical surfaces, and then models the magnetic field based on the magnetic charge theory. Since this method is only for uniformly axially magnetized cylindrical magnets, the magnetic charge distribution calculated by the equivalent magnetic charge method is only concentrated on the upper and lower surfaces, which is convenient for mathematical calculations. If non-cylindrical magnets or non-axially magnetized cylindrical magnets are considered, this method will find it difficult to calculate the charge distribution function, and thus cannot solve the magnetic field distribution problem.

[0004] Li Bin et al.'s patent "Permanent Magnet Spherical Motor Cogging Torque Analysis Method Based on Equivalent Magnetic Circuit Method", patent number CN201911097266.1, uses the equivalent magnetic circuit method to calculate the air gap magnetic flux density function in the permanent magnet spherical motor. This method abstracts the direction of the magnetic flux into four magnetic circuits to complete the magnetic field modeling of the permanent magnet engineering structure with relatively complex geometric configuration. However, this method makes a simple assumption that the magnetic lines of force of the permanent magnet are uniformly distributed, and does not consider the impact of leakage magnetic flux on the system. Although the analysis of the four magnetic circuits is correct, it cannot guarantee uniqueness, and can only roughly model the direction of the magnetic flux. In addition, this method is limited to the analytical modeling of the air gap magnetic field. By two-dimensionalizing the air gap magnetic field to simplify the calculation, it lacks application scenarios for three-dimensional magnetic field modeling. In summary, this method can solve the magnetic field modeling problem of complex structures, but it cannot guarantee accuracy.

[0005] In the article "Magnetic Field Calculation of Ring Permanent Magnets in Magnetic Suspension Twisting Mechanisms", Tang Yu et al. obtained the magnetic field distribution function of solid ring permanent magnets based on molecular current theory. However, since the calculation process is performed in a vacuum by default, and in actual engineering applications, magnetic steel is mostly adsorbed on ferromagnetic materials to work, and only the magnetic field distribution of a well-symmetrical ring magnetic steel is studied, and there is a lack of calculation of the magnetic field distribution of special-shaped magnetic steels, so it is not applicable in the field of engineering applications.

[0006] Therefore, a new three-dimensional magnetic field modeling method for permanent magnet coupling based on molecular current theory is proposed. The magnetic field distribution is visualized through analytical model, which is of great significance for guiding the design of permanent magnet coupling and expanding the application scenarios of permanent magnet coupling. Summary of the invention

[0007] The purpose of the present invention is to provide a three-dimensional magnetic field modeling method for a permanent magnet coupling based on molecular current theory, to overcome the deficiencies of the prior art, to solve the problems of long calculation time of the finite element method and difficulty in modeling the magnetic field of a permanent magnet structure with poor symmetry, and to accurately and quickly perform magnetic field modeling calculations for a permanent magnet coupling with a magnetic tile structure.

[0008] The technical solution adopted by the present invention is:

[0009] A three-dimensional magnetic field modeling method for permanent magnet coupling based on molecular current theory. First, according to the molecular current viewpoint, the uniformly magnetized permanent magnet is equivalent to several current loops tightly wound on its surface. Secondly, the Biot-Savart law is corrected to solve the magnetic field intensity of the equivalent tightly wound current loops on the surface of the permanent magnet in a vacuum. Then, considering the influence of soft magnetic medium on the magnetic induction intensity, the magnetic induction intensity contribution of a single permanent magnet to the geometric space is solved according to the magnetic medium constitutive equation. Finally, the total magnetic induction intensity distribution of the magnet array is solved.

[0010] The method specifically comprises the following steps:

[0011] The first step is to establish a mathematical model of a single permanent magnet equivalent closely wound current loop

[0012] According to the molecular current theory, the magnetizing current density of the permanent magnet is

[0013]

[0014] Where: is the current density vector, is the magnetization intensity vector, is a partial differential operator;

[0015] Assume that the permanent magnet is uniformly magnetized inside, that is is a constant, so the current density inside the permanent magnet is 0.

[0016] Considering the magnetizing current linear density on the permanent magnet surface

[0017]

[0018] Where: is the magnetizing current linear density, is the surface normal vector;

[0019] The above content establishes the relationship between magnetizing current and magnetizing intensity. Next, it is necessary to analyze the configuration of permanent magnet coupling to solve practical problems.

[0020] Taking a certain type of permanent magnet coupling structure as an example, it includes a permanent magnet coupling outer rotor 1 and a permanent magnet coupling inner rotor 2; the inner and outer rotor permanent magnets 4 and 3 are adsorbed on the surfaces of the inner and outer rotor cylinder walls 6 and 5, both of which are tile-shaped configurations and arranged in a circular array. The polarization direction is NS, and the two poles are arranged alternately, and it is assumed that the inner and outer magnets are installed completely aligned.

[0021] For a tile-shaped permanent magnet 3 or 4 in this type of permanent magnet coupling, its spatial configuration is: concentric arcs with the same angle at both ends form the bottom surface, and then vertically stretch to a certain height in the longitudinal direction to obtain a three-dimensional configuration. Take the common center of the two arcs of the bottom surface as the origin, take the bottom surface as the xy plane, take the vertical z axis, and construct a three-dimensional rectangular coordinate system. The positive direction of the y axis is taken as the direction of the line connecting the origin and the midpoint of the bottom arc, and the positive direction of the z axis is taken as the direction from the bottom surface to the upper surface. After taking the positive direction of the y axis, since the projection of the three-dimensional configuration on the xy plane is symmetrical about the y axis, the positive direction of the x axis is not constrained. The vertical rectangular surface along the positive direction of the x axis is defined as the left side surface, and the arc surface corresponding to the bottom arc with a smaller radius is defined as the inner arc surface. Similarly, the right side surface and the outer arc surface are defined.

[0022] In the above coordinate system, the current line density of the six surfaces of the permanent magnet to be determined can be obtained as follows:

[0023]

[0024] Right side:

[0025]

[0026] Outer arc surface:

[0027]

[0028] Inner arc surface:

[0029]

[0030] Upper surface:

[0031]

[0032] Lower surface:

[0033]

[0034] Where: M is the magnetization intensity of the permanent magnet, are the unit vectors in rectangular coordinates, θ0 is the arc angle of the tile-type permanent magnet;

[0035] At this point, it is not difficult to find that the magnetization current distribution on the surface of this type of permanent magnet is mainly composed of six intervals on each surface, each interval can be equivalent to a group of closely wound current-carrying wires, and the six intervals can form four groups of closely wound current loops.

[0036] At this point, the mathematical model of the equivalent closely wound current loop of a single permanent magnet has been established.

[0037] Step 2: Calculation of the three-dimensional magnetic field strength of a single permanent magnet

[0038] Since the subsequent calculation steps need to consider the influence of magnetic media, the equivalent closely wound current loop mathematical model obtained in the first step obtains the vacuum magnetic induction intensity according to the standard Biot-Savart law. For magnetic media, it is more convenient to calculate to obtain the magnetic induction. Therefore, the Biot-Savart law in vacuum is mathematically transformed to obtain

[0039]

[0040] Where: is the magnetic field intensity, I is the current intensity, is the direction vector of the current element, is the direction vector of the action point, t and l are the corresponding vector modulus lengths;

[0041] The relationship between current line density and current element vector is as follows:

[0042]

[0043] Where: dτ is the infinitesimal element corresponding to the current line density flux;

[0044] Then formula (9) can be written as

[0045]

[0046] Consider the magnetization current elements of the six surfaces for the next point in the cylindrical coordinates (r x ,θ x ,z x )'s magnetic field strength contribution, taking the current element coordinates as (r,θ,z):

[0047]

[0048] Where: r1 is the radius of the inner arc surface, r2 is the radius of the outer arc surface, and the magnetic field intensity of equations (13) to (18) is The magnetizing current element corresponding to each surface in equations (3) to (8) is Contribution of magnetic field strength;

[0049] Considering the simplicity of calculation, the first four equations above are integrated in the cylindrical coordinate system, and then the rectangular coordinate transformation is performed after calculation; the last two equations are integrated in the rectangular coordinate system corresponding to the cylindrical coordinates, and the integration area is the upper and lower surfaces of the permanent magnet.

[0050] Then the magnetic field strength of a single permanent magnet at a point in space is

[0051]

[0052] Step 3: Consider the magnetic medium and calculate the magnetic induction intensity

[0053] In magnetic media, the magnetic field constitutive equation is:

[0054]

[0055] Where: μ r is the relative magnetic permeability of the magnetic medium, and μ0 is the magnetic permeability of vacuum.

[0056] Step 4: Analytical calculation of the full magnetic field distribution of the permanent magnet array

[0057] The above content has obtained the magnetic induction intensity distribution function in the eigen coordinate system of a single permanent magnet. Next, it is necessary to transform the eigen coordinates into the coordinates shown in the figure for solution to obtain the new coordinates (r x * ,θ x * ,z x * ). For the permanent magnet coupling configuration considered in the first step, the inner and outer rotor permanent magnets 4 and 3 are evenly distributed on the inner and outer rotor cylinder walls 6 and 5 to form a permanent magnet array. For any permanent magnet in the permanent magnet array, the following coordinate transformation form is available:

[0058]

[0059] Where: is the unit vector of the rectangular coordinate system with the axis as the origin, β ij is the coordinate transformation tensor, B j is the magnetic induction intensity component in the original coordinate system;

[0060] Then, through the coordinate transformation relationship, the coordinate (r x ,θ x ,z x ) is converted to the total coordinate system (r x * ,θ x * ,z x * )get

[0061]

[0062] Finally, the magnetic induction intensity vector of each permanent magnet in the permanent magnet array is summed up to obtain

[0063]

[0064] At this point, the calculation of the magnetic field distribution of the permanent magnet coupling is completed.

[0065] The beneficial effect of the present invention is that a novel three-dimensional magnetic field modeling method for permanent magnetic coupling is proposed, and the method realizes the rapid calculation of the three-dimensional magnetic field of permanent magnetic coupling based on molecular current theory. In the air gap of the permanent magnetic coupling and inside the magnetic medium, the method can ensure a certain calculation accuracy, and can provide a theoretical basis for the calculation of transmission performance of the permanent magnetic coupling in the design stage. At the same time, the method is expected to combine computer technology, optimize the magnetic field visualization modeling software algorithm, and promote the development of digital design technology. Compared with traditional finite element analysis and numerical algorithms, the advantage of using the molecular current viewpoint is that the mathematical modeling is more accurate, the calculation consumes less resources, and the physical image is clearer. At the same time, the method takes into account the influence of magnetic medium on magnetic field modeling, and has wider applicability. In summary, the three-dimensional magnetic field modeling method of permanent magnetic coupling based on molecular current theory proposed in the present invention provides a simple and efficient magnetic field calculation method for the optimization design and application field of permanent magnetic coupling. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 It is a schematic diagram of a cylindrical permanent magnetic coupling;

[0067] Figure 2 It is a cross-sectional view of the permanent magnet array distribution of a cylindrical permanent magnet coupling;

[0068] Figure 3 It is a cross-sectional view of the axial structure of a cylindrical permanent magnetic coupling;

[0069] Figure 4 It is a schematic diagram of the permanent magnet of a cylindrical permanent magnet coupling;

[0070] Figure 5 It is a flow chart of the method for calculating the magnetic field of a permanent magnetic coupling based on the molecular current theory;

[0071] In the figure: 1-permanent magnetic coupling outer rotor, 2-permanent magnetic coupling inner rotor; 3-permanent magnet of outer rotor, 4-permanent magnet of inner rotor; 5-outer rotor cylinder wall, 6-inner rotor cylinder wall. DETAILED DESCRIPTION

[0072] The embodiments of the present invention are further described below in conjunction with the accompanying drawings and technical solutions.

[0073] This embodiment uses a cylindrical permanent magnetic coupling with 4 pairs of magnetic poles for modeling and calculation.

[0074] Among them, the inner diameter of the outer rotor permanent magnet 3 is 130mm, the outer diameter is 160mm, the height is 200mm, and the opening angle is 30 degrees; the inner diameter of the inner rotor permanent magnet is 65mm, the outer diameter is 95mm, the height is 200mm, and the opening angle is 30 degrees; the outer diameter of the outer rotor cylinder wall 5 is 350mm, the inner diameter is 320mm, and the internal cavity length is 250mm; the outer diameter of the inner rotor cylinder wall 6 is 200mm, the diameter of the permanent magnet mounting groove is 130mm, and the total length is 250mm.

[0075] A three-dimensional magnetic field modeling method for permanent magnetic coupling based on molecular current theory Figure 5 As shown, the specific calculation steps are as follows:

[0076] The first step is to determine the magnetization intensity and size data of the permanent magnet and the equivalent close-wound current loop mathematical model

[0077] The magnetization intensity is set to M = 3.02 × 10 5 A / m, radial parallel uniform magnetization. Substitute into equations (3)(4)(5)(6)(7)(8) to establish the mathematical model of equivalent closely wound current loop.

[0078] Step 2: Calculation of vacuum magnetic field strength

[0079] Substituting the equivalent current model obtained in the first step into equation (19) yields the original coordinate function relationship of the magnetic field intensity. Figure 2 Taking the permanent magnet 4 as an example, Figure 4 In the cylindrical coordinate system shown in the figure, the reference point (r x ,θ x ,z x )=(0.33,0,0.1) as an example to calculate, and the magnetic field strength at this point is

[0080]

[0081] Step 3: Perform magnetic media correction

[0082] The inner and outer cylinder walls are made of #45 steel, which can be regarded as a soft magnetic material. The relative magnetic permeability μ r =623, substituting into formula (20), we can get the permanent magnet 1 at the reference point (r x ,θ x ,z x )=(0.33,0,0.1) is

[0083]

[0084] Step 4: Calculate the magnetic induction intensity of the full permanent magnet array

[0085] Place each permanent magnet relative to the reference point (r x ,θx ,z x )=(0.33,0,0.1) are superimposed and substituted into formula (23) to obtain the total magnetic induction intensity

[0086]

[0087] Since the selected reference point is in the magnetic medium, the initial absolute magnetic induction intensity of this point is relatively large. If this point is a vacuum, the total magnetic induction intensity is

[0088]

[0089] According to the above modeling method, the coordinate value of the reference point can be changed to obtain the magnetic induction intensity at any position in the space, and the three-dimensional coordinates can be discretized to obtain the three-dimensional magnetic field distribution in the entire space.

[0090] At this point, the magnetic field distribution modeling and calculation of this type of permanent magnet coupling has been completed.

[0091] This three-dimensional magnetic field modeling method for permanent magnetic coupling based on molecular current theory is mainly aimed at a permanent magnetic coupling of a certain configuration, considering the radially magnetized tile-shaped permanent magnet with non-traditional geometric symmetry, and solving the annular array arrangement. Based on the molecular current theory, an equivalent closely wound current ring mathematical model is constructed, and the magnetic field distribution function of a single permanent magnet is obtained by using the modified Biot-Savart law and the electromagnetic field constitutive equation. The full magnetic field distribution is solved by combining the permanent magnet array and the spatial configuration of the coupling.

[0092] This method is not limited to the permanent magnet and permanent magnet coupling configuration to be solved, but is applicable to the solution of the magnetic field distribution of permanent magnets and permanent magnet couplings with any geometric shape and any spatial configuration. This calculation method solves the problems of long finite element calculation time, poor accuracy and narrow adaptability of traditional calculation methods. It expands the solution method for the calculation problem of magnetic field modeling of permanent magnet couplings, laying a foundation for the structural optimization design of permanent magnet couplings, adaptive control and other application fields.

Claims

1. A three-dimensional magnetic field modeling method for permanent magnetic coupling based on molecular current theory, characterized in that: The following steps are involved: The first step is to establish a mathematical model of a single permanent magnet equivalent closely wound current loop According to the molecular current theory, the magnetizing current density of the permanent magnet is: Where: is the current density vector, is the magnetization intensity vector, is a partial differential operator; Assume that the permanent magnet is uniformly magnetized inside, that is is a constant, so the current density inside the permanent magnet is 0; Consider the magnetizing current line density on the permanent magnet surface: Where: is the magnetizing current linear density, is the surface normal vector; The permanent magnetic coupling comprises a permanent magnetic coupling outer rotor (1) and a permanent magnetic coupling inner rotor (2); the outer rotor permanent magnet (3) and the inner rotor permanent magnet (4) are respectively adsorbed on the surface of the outer rotor cylinder wall (5) and the inner rotor cylinder wall (6), and are both tile-shaped and arranged in a ring array, with polarization directions, i.e., NS poles arranged alternately, and the inner and outer magnetic steels are installed in full alignment; The spatial configuration of any permanent magnet in the permanent magnet coupling is: the concentric arcs with the same angle at both ends form the bottom surface, and then the three-dimensional configuration is obtained by vertically stretching a certain height in the longitudinal direction; the common center of the two arcs of the bottom surface of the three-dimensional configuration is taken as the origin, the bottom surface is taken as the xy plane, and the vertical z axis is taken to construct a three-dimensional rectangular coordinate system; wherein, the positive direction of the y-axis is taken as the direction of the line connecting the origin and the midpoint of the bottom arc, and the positive direction of the z-axis is taken as the direction from the bottom surface to the upper surface. After taking the positive direction of the y-axis, since the projection of the three-dimensional configuration in the xy plane is symmetrical about the y-axis, the positive direction of the x-axis is not constrained; the vertical rectangular surface along the positive direction of the x-axis is defined as the left side surface, and the arc surface corresponding to the bottom arc with a smaller radius is defined as the inner arc surface, and the right side surface and the outer arc surface are defined similarly; in the three-dimensional rectangular coordinate system, the current line densities of the six surfaces of the permanent magnet to be determined are obtained as follows: Left side: Right side: Outer arc surface: Inner arc surface: Upper surface: Lower surface: Where: M is the magnetization intensity of the permanent magnet, are the unit vectors in rectangular coordinates, θ0 is the arc angle of the tile-type permanent magnet; The magnetization current distribution on the surface of this type of permanent magnet is mainly composed of six sections on each surface, each section can be equivalent to a group of closely wound current-carrying wires, and the six sections constitute four groups of closely wound current loops; At this point, the mathematical model of the equivalent closely wound current loop of a single permanent magnet has been established; Step 2: Calculation of the three-dimensional magnetic field strength of a single permanent magnet By mathematically transforming Biotsafar's law in vacuum, we can obtain Where: is the magnetic field intensity, I is the current intensity, is the direction vector of the current element, is the direction vector of the action point, t and l are the corresponding vector modulus lengths; The relationship between current line density and current element vector is as follows: Where: dτ is the infinitesimal element corresponding to the current line density flux; Then formula (9) can be written as: Consider the magnetization current elements of the six surfaces for the next point in cylindrical coordinates (r x ,θ x ,z x )'s magnetic field strength contribution, taking the current element coordinates as (r,θ,z): Where: r1 is the radius of the inner arc surface, r2 is the radius of the outer arc surface, and the magnetic field strength of equations (13) to (18) is The magnetizing current element corresponding to each surface in equations (3) to (8) is Contribution of magnetic field strength; Then the magnetic field strength of a single permanent magnet at a point in space is: The third step is to calculate the magnetic induction intensity based on the known spatial configuration of the permanent magnetic coupling and considering the magnetic medium; The constitutive equation of spatial magnetic medium is: Where: μ r is the relative magnetic permeability of the magnetic medium, μ0 is the magnetic permeability of vacuum; Step 4: Analytical calculation of the full magnetic field distribution of the permanent magnet array The above-mentioned magnetic induction intensity distribution function in the eigen coordinate system of a single permanent magnet is obtained. It is necessary to transform the eigen coordinates into the coordinates shown in the figure for solution to obtain the new coordinates (r x * ,θ x * ,z x * ); For the permanent magnet coupling configuration in the first step, the outer rotor permanent magnet (3) and the inner rotor permanent magnet (4) are evenly distributed in a ring shape on the surface of the outer rotor cylinder wall (5) and the inner rotor cylinder wall (6) to form a permanent magnet array; for any permanent magnet in the permanent magnet array, the following coordinate transformation form is available: Where: is the unit vector of the rectangular coordinate system with the axis as the origin, β ij is the coordinate transformation tensor, B j is the magnetic induction intensity component in the original coordinate system; Then, through the coordinate transformation relationship, the coordinate (r x ,θ x ,z x ) is converted to the total coordinate system (r x * ,θ x * ,z x * )get: Finally, the magnetic induction intensity vector of each permanent magnet in the permanent magnet array is summed to obtain: At this point, the calculation of the magnetic field distribution of the permanent magnet coupling is completed.

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