An equivalent surface current method for solving the torque of a cylindrical Halbach synchronous magnetic coupling transmission device

Abstract

The application relates to the field of permanent magnet transmission, and discloses an equivalent surface current method for solving the torque of a cylindrical Halbach synchronous magnetic coupling transmission device. The method is based on the Biot-Savart closed solution of a standard rectangular surface current unit, and divides the 90-degree discrete Halbach array permanent magnet of the cylindrical permanent magnet coupling into two types of radial magnetization and tangential magnetization. Different equivalent conversion schemes are adopted for the cylindrical surface of the permanent magnet, the fan ring end surface and the inclined side surface, and the permanent magnet is uniformly converted into a standard surface current unit which can be analytically solved according to the Ampere molecular ring current theory. After multi-coordinate system translation and rotation transformation, each surface magnetic field component is mapped to a global cylindrical coordinate system and vector synthesis is completed to obtain the magnetic field of a single permanent magnet at any point in space. Finally, the analytical solution of the coupling transmission torque is completed by the Maxwell tensor method. It has been verified that, under the common working condition of 8-10 magnetic pole pairs, the torque calculation result of the method has a deviation of less than 2% from the finite element simulation value, and the solving speed of a single group of parameters is greatly improved compared with the traditional finite element method.

Description

Technical Field

[0001] This invention belongs to the technical field of transmission performance analysis of synchronous magnetic coupling transmission devices, specifically relating to a high-precision torque calculation method for a cylindrical Halbach array synchronous magnetic coupling transmission device. Background Technology

[0002] Cylindrical synchronous magnetic coupling drives rely on a ring array of permanent magnets arranged radially inside and outside to generate electromagnetic torque, thereby achieving non-contact transmission of rotational mechanical energy. Traditional cylindrical synchronous magnetic coupling drives mostly employ a permanent magnet array structure with alternating positive and negative magnetization along the axial direction, while the Halbach permanent magnet array structure is now being increasingly widely used in cylindrical synchronous magnetic coupling drive equipment. The Halbach array is a special arrangement of permanent magnets. By gradually deflecting the magnetization direction of each permanent magnet unit at fixed angles, the magnetization direction inside the structure exhibits an approximately sinusoidal distribution along the circumference, possessing excellent unilateral magnetization characteristics, which can effectively improve the air gap magnetic induction intensity and permanent magnet energy utilization rate.

[0003] Rated transmitted torque is a core performance indicator of the cylindrical Halbach synchronous magnetic coupling drive. Achieving high-precision and rapid torque calculation is crucial for the structural design and parameter iterative optimization of this type of equipment. Currently, torque calculation methods for synchronous magnetic coupling drives are mainly divided into two categories: numerical calculation methods and theoretical analytical methods. Finite element numerical simulation methods offer high accuracy but are time-consuming, with torque calculations for a single set of structural parameters taking several minutes to hours. This cannot support the tens of thousands of iterative calculations required by intelligent optimization algorithms, resulting in extremely low parameter optimization efficiency. Traditional theoretical analytical methods, including subdomain analytical methods and equivalent magnetic circuit methods, are mostly applicable only to two-dimensional idealized structural models and are difficult to perform efficient and accurate torque analytical calculations for cylindrical coupling structures composed of fan-shaped permanent magnets.

[0004] The surface current method, based on Ampere's molecular circulation hypothesis, treats permanent magnets as equivalent to macroscopic current models on the outer surface for magnetic field calculation, offering advantages such as high computational accuracy and strong scalability. However, current research largely focuses on planar permanent magnet structures, and a torque analysis method for cylindrical synchronous magnetic coupling transmission devices adapted to discretized Halbach arrays has not yet been developed. This significantly restricts the structural optimization and engineering application of Halbach arrays in high-power cylindrical permanent magnet transmission equipment. Summary of the Invention

[0005] To address the problems of low computational efficiency and insufficient analytical accuracy in existing technologies, this invention proposes an equivalent surface current solution method for the torque of a cylindrical Halbach synchronous magnetic coupling transmission device. Specifically designed for a cylindrical discrete Halbach synchronous magnetic coupling transmission device with four magnets forming a cycle and each magnet's magnetization direction rotating sequentially by 90°, a high-precision torque analytical model is constructed. The specific technical solution is as follows:

[0006] Step 1: Establish a baseline solution element, defining a standard rectangular surface current with dimensions and normal directions parallel to the coordinate axes, and defining the surface current density as M. Based on the Biot-Savart law, perform an integral derivation to solve for the x, y, and z components of the magnetic induction intensity at any coordinate point (l, m, n) in space, providing a basic formula for subsequent calculation of the equivalent magnetic field on the surface of the permanent magnet.

[0007] Step two involves classifying and establishing a coordinate system for the permanent magnet blocks of the cylindrical Halbach synchronous magnetic coupling drive. The permanent magnets in this drive can be categorized into radial and tangential magnetization based on their magnetization method. Each single sector-ring permanent magnet comprises six structural surfaces: upper and lower cylindrical curved surfaces, front and rear sector ring planes, and left and right inclined surfaces. According to Ampere's molecular circulation theory, a uniformly magnetized permanent magnet can be represented as a surface current, with the surface current density equal to the magnetization intensity M0 of the permanent magnet. The current direction is tangential to the magnetization direction of the permanent magnet. Based on these characteristics, for radially magnetized magnets, only the current-magnetic field of the front and rear sector ring curved surfaces and the left and right inclined surfaces needs to be calculated. For tangentially magnetized magnets, due to the side tilt angle structural characteristics, the magnetic field contribution of all six surfaces needs to be calculated. A dedicated coordinate system is established based on the structural characteristics of the cylindrical coupling. The center of the coupling is taken as the origin, the axis perpendicular to the plane of the permanent magnet array is taken as the Z-axis, the symmetry line of any permanent magnet's center is taken as the Y-axis, and the circumferential tangential direction is taken as the X-axis, thus completing the spatial coordinate system construction.

[0008] Step 3: For the curved surface and spatial planar structure of the permanent magnet in the cylindrical coupling, a differentiated equivalent approximation strategy is adopted: when the number of magnetic pole pairs is not less than 6, the upper and lower cylindrical curved surfaces of the permanent magnet can be directly equivalent to standard rectangular planes; the front and rear fan ring planes can be equivalent to rectangles with a width of Rtanα / rtanα based on the inner rotor / outer rotor magnetic blocks; for the left and right inclined planes, through spatial coordinate rotation transformation, they are converted into standard rectangular planes in the reference coordinate system to satisfy the solution conditions of the analytical formula.

[0009] Step four: Based on the two magnetization methods of the permanent magnet, radial and tangential, calculate the spatial magnetic field generated by the current on each equivalent surface. Through coordinate transformation, convert the magnetic induction intensity components in all rectangular coordinate systems into radial, tangential, and axial components in the global cylindrical coordinate system. Vector superposition of the magnetic field components on all working surfaces of a single permanent magnet is performed to finally obtain the three-dimensional magnetic induction intensity distribution of a single Halbach permanent magnet at any position in space.

[0010] Step 5: Based on Maxwell's tensor method, select the annular plane in the middle of the air gap as the torque integration calculation surface, divide it into regions according to the period, calculate the torque of the permanent magnet generated in each region within a single period, and combine the number of periods to obtain the steady-state transmission torque of the cylindrical Halbach synchronous magnetic coupling transmission device under different relative rotation angles, thus completing the entire torque analytical solution process.

[0011] Furthermore, the beneficial effects of the present invention are as follows:

[0012] 1) High analytical calculation accuracy: For the irregular structure of cylindrical fan-shaped permanent magnets, differentiated equivalent treatment and micro-element subdivision strategy are adopted for different surfaces to fully preserve the geometric structural features of permanent magnets. The error between analytical results and finite element simulation results is less than 2%.

[0013] 2) Significantly improved computational efficiency: The entire solution system is built based on pure theoretical analytical formulas, avoiding the operational steps of finite element analysis. The torque calculation efficiency of a single set of structural parameters is greatly improved compared with the finite element numerical method, significantly shortening the structural design cycle of the cylindrical Halbach synchronous magnetic coupling transmission device. Attached Figure Description

[0014] Figure 1 The diagram shows a flowchart of an equivalent surface current method for calculating the torque of a cylindrical Halbach synchronous magnetic coupling drive device provided by the present invention.

[0015] Figure 2 The diagram shows a standard rectangular surface current.

[0016] Figure 3 The diagram shown is a schematic of the magnetization of the magnetic blocks in a cylindrical Halbach synchronous magnetic coupling drive device.

[0017] Figure 4 The figure shown is a coordinate diagram of the permanent magnet in a cylindrical Halbach synchronous magnetic coupling drive device.

[0018] Figure 5 The diagram shows the equivalent treatment method for the upper and lower cylindrical surfaces;

[0019] Figure 6 The diagram shows the equivalent treatment method for the front and rear fan toroidal surfaces;

[0020] Figure 7 The diagram shows the equivalent treatment method for a spatial inclined plane.

[0021] Figure 8 The diagram shown is a reference section for torque calculation of a cylindrical synchronous magnetic coupling transmission device.

[0022] Figure 9The figure shows the finite element model of the cylindrical Halbach synchronous magnetic coupling drive device.

[0023] Figure 10 The figure shown is a comparison between the analytical solution and the finite element method for the cylindrical torque. Detailed Implementation

[0024] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can fully understand and implement the present invention. This embodiment provides a torque analysis method for a cylindrical Halbach synchronous magnetic coupling transmission device based on the surface current method. The overall solution process is as follows: Figure 1 As shown, the specific implementation steps are as follows:

[0025] S1, as Figure 2 As shown, a standard rectangular surface current model is constructed, and the surface current density is set to M. According to the Biot-Savart law, the magnetic field generated by this surface current at any point in space can be characterized by equation (1):

[0026] (1)

[0027] In the formula, S represents the region where the surface current is located, ds represents the infinitesimal element of the current plane, and r represents the radius vector pointing from the infinitesimal element to the spatial point. The magnetic flux density at a point (l, m, n) in space can be calculated using equation (1):

[0028] (2)

[0029] In the formula, dl is a linear infinitesimal element along the current direction in the plane, and a, b, g, and h are respectively the spatial boundary dimension parameters of the standard rectangular plane. The above formula constitutes the basic analytical unit for solving all equivalent surface current and magnetic field problems.

[0030] Taking the analytical solution of the above equation, we get:

[0031] (3)

[0032] (4)

[0033] (5)

[0034] S2, as Figure 3As shown, the cylindrical Halbach permanent magnet array contains two types of permanent magnet units: radially magnetized and tangentially magnetized. The equivalent surface current of both types of magnets is tangential to their magnetization direction. Specifically, the equivalent current of the radially magnetized magnets is distributed on the upper and lower cylindrical surfaces and the left and right inclined surfaces, while the equivalent current of the tangentially magnetized magnets is distributed on the upper and lower cylindrical surfaces and the front and rear fan-ring planes. The magnetization intensity of the permanent magnets is set to M0, and the current density on each working surface of the radially magnetized magnets is M0; the current density on the left and right inclined surfaces of the tangentially magnetized magnets is M0cosα, and the current density on the other working surfaces is M0. A dedicated spatial coordinate system is established as follows: Figure 4 As shown, with the center of the coupling's central axis as the origin, the axial extension direction as the Z-axis, the circumferential line of the center of the single permanent magnet as the Y-axis, and the circumferential tangential direction as the X-axis, the key geometric parameters of the permanent magnet are defined as: inner diameter r, outer diameter R, central angle 2α of the single magnet, and axial thickness H, providing a reference for subsequent magnetic field calculation and coordinate transformation.

[0035] S3, applying different equivalent treatments to the three surfaces of the permanent magnet block in the cylindrical synchronous magnetic coupling transmission device:

[0036] Upper and lower cylindrical surfaces: such as Figure 5 As shown, when the central angle of the magnetic pole is small, the upper and lower surfaces of the magnetic block can be regarded as standard rectangular planes. At this time, the analytical solution of the magnetic field of the standard rectangular plane S1 can be used to solve the problem.

[0037] Front and rear fan ring surfaces: such as Figure 6 As shown, the front and rear fan-shaped annular surfaces are equivalent to different rectangles depending on whether they are located on the outer rotor or inner rotor magnet. If it is the outer rotor, the lower arc is the equivalent rectangle with a width of rtanα as the bottom edge; if it is the inner rotor, the upper arc is the equivalent rectangle with a width of Rtanα as the side edge.

[0038] Left and right sloping surfaces: such as Figure 7 As shown, the original coordinate system xOy is rotated by an angle α along the positive and negative z-axis to construct two auxiliary coordinate systems x1Oy1 and x2Oy2, so that the left and right tilting planes of the permanent magnet are respectively attached to the reference plane of the auxiliary coordinate system. The tilting plane is transformed into a standard rectangular solution model through spatial coordinate transformation.

[0039] S4. In the cylindrical coordinate system of the cylindrical synchronous magnetic coupling transmission device, take any point with coordinates P(ρ,θ,n). When analyzing the magnetic field at this point on the front, back, upper and lower surfaces, transform the coordinates of P into the Cartesian coordinate system (ρcosθ, ρsinθ,n), and its x, y, and z magnetic induction components are [B x0 B y0 B z0For the magnetic field at this point on the left and right surfaces, P is successively transformed into points (ρcos(θ-α), ρsin(θ-α),n) in the x1Oy1 coordinate system and (ρcos(θ+α), ρsin(θ+α),n) in the x2Oy2 coordinate system, with their x, y, and z magnetic induction intensity components being [Bx1,By1,Bz1], [B... ... x2 B y2 B z2 ].

[0040] The analytical solution of the magnetic flux density components of the standard rectangular current plane obtained by S1 is expressed in parametric form to facilitate the representation of complex formulas:

[0041] (6)

[0042] The tangential and radial components of the magnetic field strength at point P in cylindrical coordinates are represented by the three-coordinate components of the magnetic field strength in different coordinate systems:

[0043] (7)

[0044] (8)

[0045] (9)

[0046] The sum of the magnetic induction intensities produced by each plane of the permanent magnet at point P is:

[0047] (10)

[0048] Using the standard rectangular current-plane magnetic induction intensity parametric formula for coordinate transformation, the magnetic field of various magnetic blocks at point P in cylindrical coordinates is solved, where the magnetic field of the radially magnetized permanent magnet is:

[0049] (11)

[0050] (12)

[0051] The magnetic field distribution of the tangentially magnetized permanent magnet is as follows:

[0052] (13)

[0053] (14)

[0054] (15)

[0055] In the formula, ε = Rtanα is used when calculating the magnetic field generated by the inner rotor, while ε = rtanα is used when calculating the magnetic field of the outer rotor. g and h represent the positions of the front and rear sectors of the permanent magnet, |gh| = H, and:

[0056] (16)

[0057] (17)

[0058] (18)

[0059] By superimposing the magnetic field distributions of each plane of the magnetic block using vectors, the three-dimensional magnetic field distribution of a single Halbach array permanent magnet can be obtained.

[0060] S5, such as Figure 8 As shown, the annular plane in the middle of the air gap of the cylindrical Halbach synchronous magnetic coupling drive is selected as the torque integration reference plane, and it is divided into regions corresponding one-to-one with the permanent magnet blocks. The permanent magnet torque generated in each region is calculated based on Maxwell's tensor method. Taking region ② as an example, the following is defined:

[0061] (19)

[0062] (20)

[0063] Then the torque exerted by the magnetic field in region ② on the entire coupling is:

[0064] (twenty one)

[0065] If the number of permanent magnets in a single cycle is 4 and the number of magnetic pole cycles is N, then the total magnetic torque generated between the master and slave rotors of the coupling is:

[0066] (twenty two)

[0067] Where T ① T ② T ③ T ④ This refers to the torque exerted on the coupling system by each permanent magnet in the region directly opposite the pole during a single magnetic pole cycle.

[0068] like Figure 9 As shown, simulations were performed in AYSYS Maxwell on multiple sets of cylindrical Halbach synchronous magnetic coupling transmission devices with a permanent magnet central angle of π / 12 and an air gap thickness of 12.5 mm. The finite element torque transmission results within the rotation angle β ∈ [0, π / 6] were obtained and compared with the torque obtained by this method. Figure 10As shown, the analytical torque and the finite element simulation torque of the cylindrical synchronous magnetic coupling transmission device have an error of no more than 2% within the range of the required rotation angle, indicating high solution accuracy.

Claims

1. A method for calculating the equivalent surface current of a cylindrical Halbach synchronous magnetic coupling transmission device, characterized in that, Includes the following steps: S1. Construct a standard rectangular surface current model with surface current density M in space, and obtain the three-coordinate components of the magnetic induction intensity generated by the surface current at any point in space according to the Biot-Savart law. S2. A classification and coordinate system is established for the permanent magnet blocks of the cylindrical Halbach synchronous magnetic coupling drive. The permanent magnets in the cylindrical Halbach synchronous magnetic coupling drive can be divided into two categories according to their magnetization method: radial magnetization and tangential magnetization. Both include six surfaces: upper and lower cylindrical curved surfaces, front and rear fan-ring planes, and left and right inclined surfaces. Based on Ampere's molecular circulation theory, for radially magnetized magnets, only the current-magnetic field of the front and rear fan-ring curved surfaces and the left and right inclined surfaces needs to be calculated. For tangentially magnetized magnets, the magnetic field contribution of all six surfaces needs to be calculated. A coordinate system is established based on the structural characteristics of the cylindrical coupling's permanent magnets. The center of the coupling is taken as the origin, the axis perpendicular to the plane of the permanent magnet array is taken as the Z-axis, the symmetry line of any permanent magnet's center is taken as the Y-axis, and the circumferential tangential direction is taken as the X-axis, thus completing the construction of the spatial coordinate system. S3 uses different methods to transform the upper and lower cylindrical curved surfaces, front and rear fan ring planes, and left and right inclined surfaces of the two permanent magnets in the cylindrical synchronous magnetic coupling transmission device into standard rectangular surface current units, and solves the magnetic field of each surface through the analytical expression of magnetic induction intensity obtained in S1. S4, superimpose the magnetic field component vectors of each equivalent surface and map them to the global cylindrical coordinate system to obtain the radial, tangential and axial magnetic induction intensity of a single permanent magnet at any point in space; S5. Select the annular plane in the middle of the air gap as the torque integration surface and divide the region. Combine the permanent magnet magnetic field obtained in S4 and complete the analytical solution of torque based on Maxwell's stress tensor method.

2. The torque analysis method for a cylindrical Halbach synchronous magnetic coupling transmission device according to claim 1, characterized in that, In step S3, the upper and lower cylindrical surfaces can be regarded as standard rectangular planes when the central angle is small; the front and rear fan ring surfaces are equivalent to rectangles with width rtanα / Rtanα depending on the location of the outer / inner rotor; for the left and right inclined planes, the original coordinate system xOy is rotated by an angle α along the positive and negative directions of the z axis to construct two auxiliary coordinate systems x1Oy1 and x2Oy2 so that the left and right inclined planes of the permanent magnet are respectively attached to the reference plane of the auxiliary coordinate system, and the inclined planes are transformed into standard rectangular planes through spatial coordinate transformation.

3. The torque analysis method for a cylindrical Halbach synchronous magnetic coupling transmission device according to claim 1, characterized in that, In step S4, an arbitrary point P(ρ,θ,n) is selected in the cylindrical coordinate system of the cylindrical synchronous magnetic coupling transmission device. When analyzing the magnetic field at this point on the front, back, upper, and lower surfaces, the coordinates of P are transformed into a point (ρcosθ, ρsinθ,n) in the rectangular coordinate system, and its x, y, and z magnetic induction intensity components are [B x0 B y0 B z0 For the magnetic field at this point on the left and right surfaces, P is successively transformed into points (ρcos(θ-α), ρsin(θ-α),n) in the x1Oy1 coordinate system and (ρcos(θ+α), ρsin(θ+α),n) in the x2Oy2 coordinate system, with their x, y, and z magnetic induction components being [B], respectively. x1 B y1 B z1 ]、[B x2 B y2 B z2 ].

4. Based on the analytical solution of the magnetic flux density components of the standard rectangular current plane obtained in S1, the calculation formula for the magnetic flux density corresponding to P(l,m,n) is transformed into parametric form: （1） The tangential and radial components of the magnetic field strength at point P in cylindrical coordinates are represented by the three-coordinate components of the magnetic field strength in different coordinate systems: （2） （3） （4） Since the three coordinate systems have the same z-axis, according to the properties of the cylindrical coordinate system, the sum of the magnetic induction intensities produced by each plane of the permanent magnet at point P is: （5） Using the standard rectangular current-plane magnetic induction intensity parametric formula for coordinate transformation, the magnetic field of various magnetic blocks at point P in cylindrical coordinates is solved, where the magnetic field of the radially magnetized permanent magnet is: （6） （7） The magnetic field distribution of the tangentially magnetized permanent magnet is as follows: （8） （9） （10） In the formula, ε = Rtanα is used when calculating the magnetic field generated by the inner rotor, while ε = rtanα is used when calculating the magnetic field of the outer rotor. g and h represent the positions of the front and rear sectors of the permanent magnet, |gh| = H, and: （11） （12） （13） The three-dimensional magnetic field distribution of a single Halbach array permanent magnet can be obtained by superimposing the plane magnetic field distribution vectors corresponding to the magnet blocks in each magnetization direction.

5. The torque analysis method for a cylindrical Halbach synchronous magnetic coupling transmission device according to claim 1, characterized in that, In step S5, the annular plane in the middle of the air gap is selected as the torque integration reference plane, and it is divided into small regions corresponding to each permanent magnet, i.e., the number of regions equals the number of permanent magnets. Every four small regions constitute one cycle. The torque exerted on the rotor system by the magnetic induction intensity in a single region is calculated using the Maxwell tensor method. Taking region ② as an example, the following is defined: （14） （15） In the cylindrical synchronous magnetic coupling transmission device, the torque applied by the magnetic field in region ② is: （16） The total magnetic torque is: （17） Where T ① T ② T ③ T ④ N represents the torque exerted on the coupling system by each permanent magnet in the region directly opposite the pole during a single magnetic pole cycle, where N is the number of cycles.

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