A method for calculating magnetic force of radial magnetized trapezoidal magnetic steel considering full surface magnetic charge distribution

Abstract

This invention belongs to the field of permanent magnet drive and spatial magnetic field analytical calculation technology. It discloses a method for calculating the magnetic force of radially magnetized trapezoidal magnets considering the magnetic charge distribution on the entire surface. The steps are as follows: First, establish a three-dimensional spatial geometric coordinate system and solve for the surface equivalent magnetic charge density perpendicular to the magnetic polarization intensity vector direction in each trapezoidal magnet; Second, derive the analytical expression of the normal vector of the inclined side of the trapezoidal magnet and solve for the magnetic charge distribution on its edge equivalent surface; Third, construct a discrete integral equation for the entire surface and calculate the total radial magnetic force between the first trapezoidal magnet and the second trapezoidal magnet. This method makes up for the theoretical truncation error caused by ignoring the side magnetic charge in the traditional analytical model. Based on this high-fidelity theoretical analytical model, the true edge magnetic field distortion and spatial coupling characteristics at the magnetic pole junction can be accurately obtained, providing strong theoretical support for the precision design of various high-end permanent magnet drive equipment. It is a calculation method with universality and practical application value.

Description

Technical Field

[0001] This invention belongs to the field of permanent magnet transmission and spatial magnetic field analytical calculation technology, and relates to a method for calculating the magnetic force of radially magnetized trapezoidal magnets that considers the magnetic charge distribution across the entire surface. Background Technology

[0002] With the rapid development of modern industry and high-end equipment manufacturing technology, more stringent requirements have been placed on the reliability, stability, and power density of high-performance transmission systems. Permanent magnet transmission mechanisms (including permanent magnet couplings, high-performance permanent magnet motors, magnetic gears, and magnetic bearings) are gradually replacing traditional mechanical transmissions and becoming core power components in aerospace, deep-sea exploration, new energy vehicles, and precision manufacturing due to their wear-free operation, non-contact transmission, overload protection, and excellent vibration isolation characteristics. These magnetic transmission mechanisms use a rotor as the core carrier for power transmission, and its magnetic poles are typically composed of multiple independent permanent magnets arranged in a circumferential array. To perfectly fit the annular topology of the rotating body and further improve the internal space filling rate, the rotor often uses trapezoidal or fan-shaped cross-section magnetic steel arrays. However, for radially magnetized trapezoidal magnets, due to the geometric tilt angle on their sides, the magnetization vector is not perpendicular to the side normal. This inherent geometric feature results in equivalent magnetic charges distributed on the tilted sides, leading to a strong edge magnetic field distortion effect in the magnetic pole junction region, resulting in an extremely complex spatial distribution of the air gap magnetic field. Ignoring the influence of the magnetic charge distribution on the inclined side on the overall spatial magnetic field will result in a significant deviation from the true physical laws, making it impossible to accurately obtain the high-precision interaction force between magnets and its spatial evolution. This leads to substantial theoretical calculation errors and severely affects the accurate study of the mechanical properties and transmission performance of high-performance permanent magnet equipment. Therefore, developing a magnetic force calculation method for radially magnetized trapezoidal magnets that considers the magnetic charge distribution across the entire surface is of great significance for the performance analysis and optimization of permanent magnet equipment.

[0003] To address the analytical calculation of magnetic fields and forces in trapezoidal magnets, Zhang Yang et al., in their patent "A Self-Actuating Assembled Permanent Magnet Coupling and a Torque Characterization Method" (CN 115276368 B), proposed an analytical model based on the equivalent magnetic charge method to solve for the interaction force and torque between magnets. However, this model has significant limitations in boundary treatment; its integral solution only covers the surface perpendicular to the magnetic polarization vector direction, completely ignoring the magnetic charge distribution on the inclined sides. Because the side magnetic fields are not included in the superposition calculation, this method cannot accurately characterize the distortion effect of the edge magnetic field, resulting in a large theoretical truncation error in the final torque value. Regarding the analytical calculation of axial magnetic force, Wang Peng et al., in their article "Axial Force Analysis of Cage Rotor Asynchronous Magnetic Coupler" published in *Mechanical Design and Manufacturing*, constructed an analytical model of axial force based on the equivalent magnetic charge method and the current image method. However, this method also has theoretical deficiencies in defining the three-dimensional geometric integration domain; its integration range is still limited to the surface perpendicular to the magnetic polarization vector direction, implicitly ignoring the equivalent magnetic charge generated by the inclined sides. This neglect of the contribution of the lateral magnetic field makes it impossible to accurately reproduce the spatial magnetic field coupling characteristics at the magnetic pole junction, resulting in the calculated analytical value of the axial force being difficult to fully match the actual physical distribution.

[0004] Therefore, proposing a method for calculating the magnetic force of radially magnetized trapezoidal magnets that considers the magnetic charge distribution across the entire surface has important guiding significance for the research of permanent magnet magnetic transmission technology. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention provides a method for calculating the magnetic force of radially magnetized trapezoidal magnets that considers the distribution of magnetic charge across the entire surface. The aim is to fully consider the equivalent surface magnetic charge generated by the geometric inclination of the side surfaces of the radially magnetized trapezoidal magnets, and to accurately reconstruct the edge magnetic field distortion effect at the pole junction using the superposition of magnetic charge across the entire surface. This provides strong theoretical support for the transmission performance and optimized design of permanent magnet equipment. Based on the theory of equivalent magnetic charge, this method precisely derives the equivalent surface magnetic charge density across the entire surface, including the surface perpendicular to the magnetic polarization vector and the inclined side surfaces, by strictly defining the relationship between the three-dimensional geometric normal vector and the radial magnetization vector of the trapezoidal magnet. Furthermore, it incorporates the contribution of the side edge magnetic field into the analytical framework of the overall spatial magnetic field, ultimately achieving high-fidelity analytical calculation of the magnetic force of the trapezoidal magnets.

[0006] The technical solution of the present invention: A method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the magnetic charge distribution across the entire surface, comprising the following steps: The first step is to establish a three-dimensional spatial geometric coordinate system and solve for the surface equivalent magnetic charge density perpendicular to the magnetic polarization vector direction in each trapezoidal magnet. First, two trapezoidal magnets with radial air gaps are defined as the first trapezoidal magnet and the second trapezoidal magnet; a three-dimensional spatial geometric coordinate system is then established. OXYZ The origin of the coordinate system is the geometric center of the end face of the first trapezoidal magnet on the side furthest from the radial air gap. O The axial length of each trapezoidal magnet is set as follows: L ,for Z Axial direction; radial height of each trapezoidal magnet is H mag ,for Y Axial direction; tangential direction of each trapezoidal magnet is X Axial direction; the first trapezoidal magnet is located in Y = H mag The width of the end face of the plane is W top ,lie in Y = The width of the end face of the plane is W bot The second trapezoidal magnet and the first trapezoidal magnet are along... Y The shafts are arranged radially opposite each other, and the width of their end faces is... W top The surfaces of the first and second trapezoidal magnets face each other directly, and the minimum distance between them is the radial air gap distance. G The second trapezoidal magnet is located at Y = H mag + G The width of the end face of the plane is W top ,lie in Y = 2 H mag + G The width of the end face of the plane is W bot ; Each trapezoidal magnet is set to use along Y Radial parallel magnetization in the positive axis direction introduces a magnetic polarization vector. J Its size is equivalent to the remanence of a trapezoidal magnet. B r , is represented as: (1) Secondly, calculate the magnetic polarization vector perpendicular to the first and second trapezoidal magnets. J Equivalent magnetic charge density of the four surfaces in the direction: For the first trapezoidal magnet, the surface closest to the air gap is located at Y = H mag A plane, the unit vector of which is the outward normal to the surface is denoted as . n 1_top= [0, 1, 0] T Its surface furthest from the air gap is located Y = 0 plane, the outward normal unit vector of this surface is denoted as n 1_bot = [0, -1, 0] T For the second trapezoidal magnet, its surface near the air gap is located... Y = H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_bot = [0, -1, 0] T Its surface furthest from the air gap is located Y = 2 H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_top = [0, 1, 0] T According to the basic principle of the equivalent magnetic charge method, the equivalent magnetic charge densities of the four surfaces mentioned above are expressed as follows: (2) (3) (4) (5) The second step is to derive the analytical expression of the normal vector of the inclined side of the trapezoidal magnet and solve for the magnetic charge distribution on its edge equivalent surface. First, define the inclined side surfaces of each trapezoidal magnet. Each trapezoidal magnet has a symmetrically distributed... X There are four inclined sides in total, consisting of two inclined sides on each side of the positive and negative half-axis; the inclined sides are the connecting ends of the trapezoidal magnets with a width of... W top The width of the top surface and the end face is W bot The inclined boundary plane of the bottom surface; calculate the length of the hypotenuse of the inclined side based on the geometric dimensions of the trapezoidal magnet cross-section. L side for: (6) Based on the established three-dimensional spatial geometric coordinate system OXYZ The outward unit normal vector of each inclined side is derived: For the first trapezoidal magnet, it is located at... X The normal vector of the right side of the positive half-axis is denoted as... n 1_R ,lie in X The normal vector of the left side of the negative half-axis is denoted as...n 1_L , is represented as: (7) (8) For the second trapezoidal magnet, since it is radially and vertically opposite to the first trapezoidal magnet, it is located in... X The normal vector of the right side of the positive half-axis is denoted as... n 2_R ,lie in X The normal vector of the left side of the negative half-axis is denoted as... n 2_L , is represented as: (9) (10) Secondly, calculate the equivalent surface magnetic charge density at the edges of the four inclined sides: since the side normal vector contains Y Axial components, which are related to the magnetic polarization vector J The dot product is not zero; by introducing the unit normal vector of the side surface, the equivalent surface magnetic charge density of the edge, which is actually distributed on the four inclined side surfaces, is calculated. s side Its calculation formula is expressed as: (11) in, n side Take the corresponding n 1_R , n 1_L , n 2_R or n 2_L , s side They correspond to as s 1_R , s 1_L , s 2_R , s 2_L ; The third step is to construct a discrete integral equation for the entire surface and calculate the total radial magnetic force between the first trapezoidal magnet and the second trapezoidal magnet. First, spatial mesh generation and derivation of discrete parameters are performed: Based on the complete symmetry of the spatial geometry of the first and second trapezoidal magnets, the tangential and axial interaction magnetic forces between the first and second trapezoidal magnets cancel each other out to zero. Therefore, only the total radial magnetic force between the first and second trapezoidal magnets is analytically solved; the entire surface of the first trapezoidal magnet is assumed to be the integration domain. S1. The entire surface of the second trapezoidal magnet is the integral domain. S 2; Integral domain S 1 and the integral domain S The geometric surfaces on 2 are respectively along X Axial direction division N x portion, along Z Axial direction division N z share; For perpendicular to the magnetic polarization intensity vector J The geometric area of ​​discrete surface elements of a surface in a certain direction. A main Represented as: (12) in, W Take the corresponding W top or W bot ; For the inclined side, its discrete surface element geometric area A side Represented as: (13) Let the first trapezoidal magnet be discretely distributed across its entire surface. i The spatial center coordinates of each face element are ( x 1,i , y 1,i , z 1,i The second trapezoidal magnet has a full surface discrete pattern of the first... j The spatial center coordinates of each face element are ( x 2,j , y 2,j , z 2,j Then the radial coordinate difference between the two face elements is... y i,j Represented as: (14) The spatial linear distance between the center points of two facet elements | r ij The geometric formula for | is: (15) Secondly, construct the discrete integral equation for the entire surface: the total radial magnetic force between the two trapezoidal magnets F y The discretization solution formula is as follows: (16) in, m 0 is the permeability of free space. N total = 4× N x × N z The total number of discrete surface elements participating in the integration of a single trapezoidal magnet, including two surfaces perpendicular to the direction of the magnetic polarization vector and two inclined side surfaces; s 1,i and s 2,j The value of is determined as follows: if the discrete surface element is located on a surface perpendicular to the direction of the magnetic polarization vector, then s 1_top , s 1_bot , s 2_bot or s 2_top The corresponding magnetic charge density is assigned to the respective positions. s 1,i or s 2,j If the discrete surface element is located on an inclined side, then the edge equivalent surface magnetic charge density will be... s side The corresponding magnetic charge density is assigned to the corresponding position. s 1,i or s 2,j ; A 1,i and A 2,j The geometric area is determined as follows: if the discrete surface element is located on a surface perpendicular to the direction of the magnetic polarization vector, then the corresponding geometric area is taken as... A main The calculation results; if the discrete surface element is located on the inclined side, then the corresponding geometric area is taken as... A side The calculation results.

[0007] The beneficial effect of this invention is that it proposes a method for calculating the magnetic force of radially magnetized trapezoidal magnets that considers the magnetic charge distribution across the entire surface. This method overcomes the limitations of traditional analytical models that only consider the surface perpendicular to the magnetic polarization vector while neglecting the magnetic charge on the inclined sides. By rigorously deriving the geometric normal vector of the inclined sides of the trapezoidal magnet, the contribution of the equivalent magnetic charge at the side edges is accurately incorporated into the analytical framework of the overall spatial magnetic field. Through the construction of a discrete integral equation across the entire surface, the high-precision interaction magnetic force between the radially magnetized trapezoidal magnets is accurately calculated. This method compensates for the theoretical truncation error caused by neglecting edge effects in traditional simplified models, greatly improving the accuracy of analytical magnetic force calculation. It helps to comprehensively reveal the true spatial magnetic field distortion and coupling characteristics at the magnetic pole junction, providing strong theoretical support for the precision design research of various high-end permanent magnet transmission equipment. It is a high-fidelity and universally applicable analytical method. Attached Figure Description

[0008] Figure 1 This is a flowchart of a method for calculating the magnetic force of a radially magnetized trapezoidal magnet that considers the magnetic charge distribution across the entire surface.

[0009] Figure 2 It is a schematic diagram of a three-dimensional spatial structure of a pair of radially magnetized trapezoidal magnets. Detailed Implementation

[0010] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0011] Example As shown in Figure 1, a method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the magnetic charge distribution across the entire surface is as follows: The first step is to establish a three-dimensional spatial geometric coordinate system and solve for the surface equivalent magnetic charge density perpendicular to the magnetic polarization vector direction in each trapezoidal magnet. First, determine the key parameters of the trapezoidal magnet: Figure 2 This is a schematic diagram of a three-dimensional spatial structure of a pair of radially magnetized trapezoidal magnets, showing the axial length of the trapezoidal magnets in a permanent magnet transmission mechanism. L It is 40mm, located in Y = H mag plane and Y = H mag + G End face width of the plane W top It is 10mm, located in Y = 0 plane and Y = 2 H mag + G End face width of the plane W botThe radial height of the trapezoidal magnet is 15mm. H mag The radial air gap distance between the first and second trapezoidal magnets is 10mm. G The remanence of the magnet material is 5mm. B r It is 1.38T.

[0012] Secondly, the surface equivalent magnetic charge density perpendicular to the magnetic polarization vector direction in the trapezoidal magnet is calculated: The trapezoidal magnet is assumed to be radially parallel magnetized, and the magnetic polarization vector... J The equivalent scalar magnitude is 1.38T; according to the basic principle of the equivalent magnetic charge method, from equations (2), (3), (4) and (5), it can be seen that the equivalent magnetic charge densities of the four surfaces perpendicular to the direction of the magnetic polarization intensity vector are respectively s 1_top = 1.38T, s 1_bot = -1.38T, s 2_bot = -1.38T, s 2_top = 1.38T.

[0013] The second step is to derive the analytical expression of the normal vector of the inclined side of the trapezoidal magnet and solve for the magnetic charge distribution on its edge equivalent surface. First, the length of the hypotenuse of the inclined side of the trapezoidal magnet is calculated: based on the cross-sectional geometry of the trapezoidal magnet, the length of the hypotenuse is calculated using equation (6). L side =10.31mm; Based on the obtained hypotenuse length, the outward unit normal vector of each inclined side is derived by equation (7), equation (8), equation (9) and equation (10); Secondly, calculate the equivalent magnetic charge density at the edges of the four inclined sides: since the side normal vector contains Y The axial component, whose dot product with the magnetic polarization vector is not zero. Substituting the magnetic polarization vector and the outward unit normal vector of each inclined side into equation (11), the effective surface magnetic charge density of the actual edge distribution on the left and right inclined sides of the first trapezoidal magnet is calculated. s 1_R = s 1_L =0.33T; Due to the radial and vertical relative arrangement of the second trapezoidal magnets, the equivalent surface magnetic charge density of the actual distribution on its left and right inclined sides is calculated. s 2_R = s 2_L = -0.33T.

[0014] The third step is to construct a discrete integral equation for the entire surface and calculate the total radial magnetic force between the first trapezoidal magnet and the second trapezoidal magnet. First, spatial grid generation and discrete parameter calculation are performed: the integration domain is defined. S 1 and the integral domain S The geometric surfaces on 2 are respectively along X Axial direction division N x = 30 copies, along Z Axial direction division N z = 30 parts, then the total number of discrete surface elements participating in the integration for a single trapezoidal magnet is N total = 3600. (This likely refers to a number of items.) W top , W bot , L , N x and N z Substituting the parameter values ​​into equation (12), the width of the end face in the surface perpendicular to the magnetic polarization vector direction is calculated. W top The geometric area of ​​the discrete surface element is approximately 0.44 mm². 2 The width of the end face is W bot The geometric area of ​​the discrete surface element is approximately 0.67 mm². 2 ;Will L side , L , N x and N z Substituting the parameter values ​​into equation (13), the geometric area of ​​the discrete surface element on the inclined side is calculated to be approximately 0.46 mm². 2 The radial coordinate difference between the discrete surface elements of the first trapezoidal magnet and the second trapezoidal magnet is calculated using equations (14) and (15), respectively. y i,j and spatial straight-line distance | r ij |

[0015] Secondly, a full-surface discrete integral equation is constructed to calculate the total radial magnetic force. To achieve double discrete summation, the local magnetic charge density and geometric area of ​​the discrete surface elements divided on the surface of each trapezoidal magnet need to be assigned values. When the discrete surface element is located on the surface perpendicular to the direction of magnetic polarization intensity vector, the equivalent magnetic charge density ±1.38T obtained in the first step is substituted, and the geometric area calculation result of the corresponding surface perpendicular to the direction of magnetic polarization intensity vector is taken. When the discrete surface element is located on the inclined side, the edge equivalent surface magnetic charge density obtained in the second step is substituted (where the left and right inclined sides of the first trapezoidal magnet are assigned 0.33T, and the left and right inclined sides of the second trapezoidal magnet are assigned -0.33T), and the geometric area calculation result of the corresponding inclined side is taken. By combining the data containing the spatial coordinates, geometric area and magnetic charge density of the above discrete surface elements, the double discretization superposition solution is performed by equation (16) to accurately calculate the total radial magnetic force of a pair of trapezoidal magnets at an air gap of 5mm. F y = 68.59 N.

[0016] This method utilizes the three-dimensional geometry and equivalent magnetic charge model of trapezoidal magnets to derive the full-surface equivalent surface magnetic charge density, including the surface perpendicular to the magnetic polarization vector direction and the inclined side surface. It constructs a full-surface discrete integral equation and performs spatial cross-coupling solution to accurately calculate the total radial magnetic force between trapezoidal magnets. This compensates for the theoretical truncation error caused by neglecting the side magnetic charge in traditional analytical models. Based on this high-fidelity theoretical analytical model, the true edge magnetic field distortion and spatial coupling characteristics at the magnetic pole junction can be accurately obtained, providing strong theoretical support for the precision design of various high-end permanent magnet transmission equipment. It is a calculation method with universality and practical application value.

Claims

1. A method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the full surface magnetic charge distribution, characterized in that, The steps are as follows: The first step is to establish a three-dimensional spatial geometric coordinate system and solve for the surface equivalent magnetic charge density perpendicular to the magnetic polarization vector direction in each trapezoidal magnet. First, two trapezoidal magnets with radial air gaps are defined as the first trapezoidal magnet and the second trapezoidal magnet; a three-dimensional spatial geometric coordinate system is then established. OXYZ The origin of the coordinate system is the geometric center of the end face of the first trapezoidal magnet on the side furthest from the radial air gap. O The axial length of each trapezoidal magnet is set as follows: L ,for Z Axial direction; radial height of each trapezoidal magnet is H mag ,for Y Axial direction; tangential direction of each trapezoidal magnet is X Axial direction; the first trapezoidal magnet is located in Y = H mag The width of the end face of the plane is W top ,lie in Y = The width of the end face of the plane is W bot The second trapezoidal magnet and the first trapezoidal magnet are along... Y The shafts are arranged radially opposite each other, and the width of their end faces is... W top The surfaces of the first and second trapezoidal magnets face each other directly, and the minimum distance between them is the radial air gap distance. G The second trapezoidal magnet is located at Y = H mag + G The width of the end face of the plane is W top ,lie in Y = 2 H mag + G The width of the end face of the plane is W bot ; Each trapezoidal magnet is set to use along Y Radial parallel magnetization in the positive axis direction introduces a magnetic polarization vector. J Its size is equivalent to the remanence of a trapezoidal magnet. B r , is represented as: Secondly, calculate the magnetic polarization vector perpendicular to the first and second trapezoidal magnets. J Equivalent magnetic charge density of the four surfaces in the direction: For the first trapezoidal magnet, the surface closest to the air gap is located at Y = H mag A plane, the unit vector of which is the outward normal to the surface is denoted as . n 1_top = [0, 1, 0] T ; Its surface away from the air gap is located Y = 0 plane, the outward normal unit vector of this surface is denoted as n 1_bot = [0, -1, 0] T ; For the second trapezoidal magnet, its surface near the air gap is located Y = H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_bot = [0, -1, 0] T ; Its surface away from the air gap is located Y = 2 H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_top = [0, 1, 0] T According to the basic principle of the equivalent magnetic charge method, the equivalent magnetic charge densities of the four surfaces mentioned above are expressed as follows: ； The second step is to derive the analytical expression of the normal vector of the inclined side of the trapezoidal magnet and solve for the magnetic charge distribution on its edge equivalent surface. The third step is to construct a discrete integral equation for the entire surface and calculate the total radial magnetic force between the first trapezoidal magnet and the second trapezoidal magnet.

2. The method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the full surface magnetic charge distribution according to claim 1, characterized in that, The specific implementation process of the first step is as follows: First, two trapezoidal magnets with radial air gaps are defined as the first trapezoidal magnet and the second trapezoidal magnet; a three-dimensional spatial geometric coordinate system is then established. OXYZ The origin of the coordinate system is the geometric center of the end face of the first trapezoidal magnet on the side furthest from the radial air gap. O The axial length of each trapezoidal magnet is set as follows: L ,for Z Axial direction; radial height of each trapezoidal magnet is H mag ,for Y Axial direction; tangential direction of each trapezoidal magnet is X Axial direction; the first trapezoidal magnet is located in Y = H mag The width of the end face of the plane is W top ,lie in Y = The width of the end face of the plane is W bot The second trapezoidal magnet and the first trapezoidal magnet are along... Y The shafts are arranged radially opposite each other, and the width of their end faces is... W top The surfaces of the first and second trapezoidal magnets face each other directly, and the minimum distance between them is the radial air gap distance. G The second trapezoidal magnet is located at Y = H mag + G The width of the end face of the plane is W top ,lie in Y = 2 H mag + G The width of the end face of the plane is W bot ; Each trapezoidal magnet is set to use along Y Radial parallel magnetization in the positive axis direction introduces a magnetic polarization vector. J Its size is equivalent to the remanence of a trapezoidal magnet. B r , is represented as: Secondly, calculate the magnetic polarization vector perpendicular to the first and second trapezoidal magnets. J Equivalent magnetic charge density of the four surfaces in the direction: For the first trapezoidal magnet, the surface closest to the air gap is located at Y = H mag A plane, the unit vector of which is the outward normal to the surface is denoted as . n 1_top = [0, 1, 0] T ; Its surface away from the air gap is located Y = 0 plane, the outward normal unit vector of this surface is denoted as n 1_bot = [0, -1, 0] T ; For the second trapezoidal magnet, its surface near the air gap is located Y = H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_bot = [0, -1, 0] T ; Its surface away from the air gap is located Y = 2 H mag + G A plane, the unit vector of which is the outward normal to the surface is denoted as . n 2_top = [0, 1, 0] T According to the basic principle of the equivalent magnetic charge method, the equivalent magnetic charge densities of the four surfaces mentioned above are expressed as follows: 。 3. The method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the full surface magnetic charge distribution according to claim 2, characterized in that, The specific implementation process of the second step is as follows: First, define the inclined side surfaces of each trapezoidal magnet. Each trapezoidal magnet has a symmetrically distributed... X There are four inclined sides in total, consisting of two inclined sides on each side of the positive and negative half-axis; the inclined sides are the connecting ends of the trapezoidal magnets with a width of... W top The width of the top surface and the end face is W bot The inclined boundary plane of the bottom surface; calculate the length of the hypotenuse of the inclined side based on the geometric dimensions of the trapezoidal magnet cross-section. L side for: Based on the established three-dimensional spatial geometric coordinate system OXYZ The outward unit normal vector of each inclined side is derived: For the first trapezoidal magnet, it is located at... X The normal vector of the right side of the positive half-axis is denoted as... n 1_R ,lie in X The normal vector of the left side of the negative half-axis is denoted as... n 1_L , is represented as: For the second trapezoidal magnet, since it is radially and vertically opposite to the first trapezoidal magnet, it is located in... X The normal vector of the right side of the positive half-axis is denoted as... n 2_R ,lie in X The normal vector of the left side of the negative half-axis is denoted as... n 2_L , is represented as: Secondly, calculate the equivalent surface magnetic charge density at the edges of the four inclined sides: since the side normal vector contains Y Axial components, which are related to the magnetic polarization vector J The dot product is not zero; by introducing the unit normal vector of the side surface, the equivalent surface magnetic charge density of the edge, which is actually distributed on the four inclined side surfaces, is calculated. σ side Its calculation formula is expressed as: in, n side Take the corresponding n 1_R , n 1_L , n 2_R or n 2_L , σ side They correspond to as σ 1_R , σ 1_L , σ 2_R , σ 2_L .

4. The method for calculating the magnetic force of a radially magnetized trapezoidal magnet considering the full surface magnetic charge distribution according to claim 3, characterized in that, The specific implementation process of the third step is as follows: First, spatial mesh generation and derivation of discrete parameters are performed: Based on the complete symmetry of the first trapezoidal magnet and the second trapezoidal magnet in spatial geometry, the interaction magnetic forces generated between the first trapezoidal magnet and the second trapezoidal magnet in the tangential and axial directions cancel each other out to zero. Therefore, only the total radial magnetic force between the first trapezoidal magnet and the second trapezoidal magnet is analytically solved. Let the entire surface of the first trapezoidal magnet be the integration domain. S 1. The entire surface of the second trapezoidal magnet is the integral domain. S 2; Integral domain S 1 and the integral domain S The geometric surfaces on 2 are respectively along X Axial direction division N x portion, along Z Axial direction division N z share; For perpendicular to the magnetic polarization intensity vector J The geometric area of ​​discrete surface elements of a surface in a certain direction. A main Represented as: in, W Take the corresponding W top or W bot ; For the inclined side, its discrete surface element geometric area A side Represented as: Let the first trapezoidal magnet be discretely distributed across its entire surface. i The spatial center coordinates of each face element are ( x 1,i , y 1,i , z 1,i The second trapezoidal magnet has a full surface discrete pattern of the first... j The spatial center coordinates of each face element are ( x 2,j , y 2,j , z 2,j Then the radial coordinate difference between the two face elements is... y i,j Represented as: The spatial linear distance between the center points of two facet elements | r ij The geometric calculation formula for | is: Secondly, construct the discrete integral equation for the entire surface: the total radial magnetic force between the two trapezoidal magnets F y The discretization solution formula is as follows: in, μ 0 is the permeability of free space. N total = 4× N x × N z The total number of discrete surface elements participating in the integration of a single trapezoidal magnet, including two surfaces perpendicular to the direction of the magnetic polarization vector and two inclined side surfaces; σ 1,i and σ 2,j The value of is determined as follows: if the discrete surface element is located on a surface perpendicular to the direction of the magnetic polarization vector, then σ 1_top , σ 1_bot , σ 2_bot or σ 2_top The corresponding magnetic charge density is assigned to the respective position. σ 1,i or σ 2,j If the discrete surface element is located on an inclined side, then the edge equivalent surface magnetic charge density will be... σ side The corresponding magnetic charge density is assigned to the corresponding position. σ 1,i or σ 2,j ; A 1,i and A 2,j The geometric area is determined as follows: if the discrete surface element is located on a surface perpendicular to the direction of the magnetic polarization vector, then the corresponding geometric area is taken as... A main The calculation results; if the discrete surface element is located on the inclined side, then the corresponding geometric area is taken as... A side The calculation results.

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