Analytical method of air-gap magnetic field for eccentric harmonic magnetic gear with halbach array

Abstract

The application provides a Halbach array eccentric harmonic magnetic gear air gap magnetic field analytical method, which has the following characteristics and comprises the following steps: in step S1, an eccentricity is calculated according to an eccentricity ratio; in step S2, a radial air gap relative permeance function is obtained according to a hyperbolic cotangent transformation; in step S3, a concentric analytical model is established to obtain air gap magnetic fields when the stator permanent magnet acts alone and when the low-speed inner rotor permanent magnet acts alone; in step S4, the air gap magnetic field is corrected according to the radial air gap relative permeance function, the eccentricity is combined, and based on the linear superposition principle, a radial component and a tangential component of the air gap magnetic flux density of the eccentric harmonic magnetic gear are obtained; and in step S5, the electromagnetic torque is obtained based on the Maxwell stress tensor method according to the radial component and the tangential component. In summary, the method can establish an accurate analytical model, and then the electromagnetic torque can be calculated from the design parameters.

Description

Technical Field

[0001] This invention relates to a method for analyzing the air gap magnetic field of a magnetic gear, specifically a method for analyzing the air gap magnetic field of a Halbach array eccentric harmonic magnetic gear. Background Technology

[0002] Magnetic gears possess advantages such as low noise, no lubrication required, maintenance-free operation, and inherent overload protection, making them promising candidates for applications in wind power generation and electric vehicles. Compared to concentric harmonic magnetic gears, eccentric harmonic magnetic gears can achieve higher torque density and higher transmission ratios. In eccentric harmonic magnetic gears, the Halbach permanent magnet array exhibits excellent magnetization, resulting in a better sinusoidal magnetic field in the air gap, reduced harmonic content, and increased motor output torque. Therefore, Halbach array eccentric harmonic magnetic gears achieve even higher torque density.

[0003] The Halbach array eccentric harmonic magnetic gear consists of the following key components: a stator core, a low-speed inner rotor, Halbach array permanent magnets, a non-uniform air gap, bearings, and a high-speed inner rotor. The high-speed inner rotor slides in contact with the low-speed inner rotor via bearings. The stator permanent magnet structure and the inner rotor permanent magnet structure are non-concentric, forming a non-uniform air gap. When the high-speed inner rotor rotates, causing the non-uniform air gap to rotate accordingly, the air gap distribution exhibits a sinusoidal periodic change. The low-speed permanent magnet rotor will rotate slowly accordingly to maintain the original magnetic field condition and torque output. The high-speed inner rotor is equivalent to the harmonic generator in a mechanical harmonic gear.

[0004] The key to the design of eccentric magnetic gears lies in electromagnetic analysis, but existing technologies have failed to establish an accurate analytical model to calculate the electromagnetic torque from the design parameters. Summary of the Invention

[0005] This invention is made to solve the above-mentioned problems, and aims to provide a method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear with a Halbach array.

[0006] This invention provides an analytical method for the air gap magnetic field of an eccentric harmonic magnetic gear with a Halbach array, characterized by the following steps: Step S1, calculating the eccentricity ε based on the eccentricity e of the eccentric harmonic magnetic gear with a Halbach array; Step S2, obtaining the radial air gap relative permeability function f based on the hyperbolic cotangent transform. r Step S3: Establish a concentric analytical model to obtain the first air gap magnetic field when the stator permanent magnet acts alone and the second air gap magnetic field when the low-speed inner rotor permanent magnet acts alone; Step S4: Based on the radial air gap relative permeability function f rBy correcting the first and second air gap magnetic fields and combining them with the eccentricity ε, the radial component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained based on the principle of linear superposition. r (r,θ) and tangential component B θ (r,θ); Step S5, based on radial component B r (r,θ) and tangential component B θ (r,θ) is obtained based on Maxwell's stress tensor method to obtain the electromagnetic torque.

[0007] The Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method provided by this invention may also have the following features: Step S3 includes the following sub-steps: Step S3-1, setting the premise assumptions of the concentric analytical model, dividing the analytical region of the concentric analytical model into three regions, namely, the low-speed inner rotor permanent magnet region I, the air gap region II, and the stator permanent magnet region III; Step S3-2, establishing the Laplace equation or Poisson equation satisfied by the vector magnetic potential of the three regions. When the stator permanent magnet acts alone, the formula of the Laplace equation or Poisson equation satisfied by the vector magnetic potential is as follows: In the formula A II A represents the vector magnetic potential of air gap region II when the stator permanent magnet acts alone. III M represents the vector magnetic potential of region III of the stator permanent magnet. r M represents the radial component of the magnetization of the permanent magnet. θ Let r and θ be the tangential component of the permanent magnet's magnetization, r and θ be the axes of the polar coordinate system, and μ0 be the permeability of free space. When the permanent magnet of the low-speed internal rotor acts alone, the formula for the Laplace equation or Poisson equation satisfied by the vector magnetic potential is as follows: In the formula A′ II A represents the vector magnetic potential of air gap region II when the permanent magnet of the low-speed inner rotor acts alone. I To determine the vector magnetic potential of the permanent magnet region I of the low-speed inner rotor, the permanent magnet Halbach array is magnetized, and the formula for the magnetization intensity M is as follows: M = M r r+M θ θ(3), where: In the formula, n represents the calculated harmonic order of the air gap magnetic field and the permanent magnet magnetic field, θ0 represents the offset degree between the magnet and the initially set angle, and M... rn (n) and M θn (n) is the Fourier expansion of the magnetization of the Halbach permanent magnet, B rLet q be the relative permeability, q be the number of blocks per pole of the Halbach array, l be the l-th block in the q-block array, and r and θ be the axes of the polar coordinate system. In step S3-3, based on the boundary conditions of the three regions, solve formulas (1) and (2) to obtain the vector magnetic potential expressions for the three regions. Based on the vector magnetic potential expressions, obtain the radial component B of the magnetic flux density generated by the stator permanent magnet acting alone in the air gap region II under the concentric condition. rII and tangential component B θII That is, the first air gap magnetic field, and the radial component B′ of the magnetic flux density generated by the low-speed inner rotor acting alone in air gap region II under concentric conditions. II and tangential component B′ θII That is, the boundary condition of the interface when the second air gap magnetic field is acting alone, i.e., when the stator permanent magnet acts alone, is: In the formula R ms R is the internal radius of the stator permanent magnet. s R is the inner radius of the stator. r Given the outer radius of the low-speed inner rotor, and the boundary conditions at the interface when the permanent magnet of the low-speed inner rotor acts alone: In the formula R mr The outer radius of the permanent magnet of the low-speed inner rotor.

[0008] The analytical method for the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear provided by this invention can also have the following characteristics: The underlying assumptions are that the calculation is performed in a two-dimensional field, ignoring end effects; the core permeability is infinite, saturation effects are ignored; the permanent magnet's BH curve is linear, and its relative permeability is μ. r =1.

[0009] The Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method provided by this invention may also have the following features: Step S4 includes the following sub-steps: Step S4-1, establish an r-θ coordinate system with the rotor center as the origin, and according to the radial air gap relative permeability function f r For radial component B rII and tangential component B θII After correction, the radial component B of the magnetic flux density in the air gap region II in the r-θ coordinate system is obtained. rII_ecc and tangential component B θII_ecc radial component B rII_ecc and tangential component B θII_ecc The formula is as follows: B rII_ecc (r,θ)=B rII (r,θ)f r -B θII (r,θ)sinα(10),B θII_ecc (r,θ)=B θII (r,θ)cosα(11), Step S4-2: Establish a ξ-ψ coordinate system with the stator center as the origin, and calculate the radial air gap relative permeability function f. r For the radial component B′ rII and tangential component B′ θII After correction, the radial component Br′ of the magnetic flux density in air gap region II in the ξ-ψ coordinate system is obtained. II_ecc and tangential component B′ θII_ecc radial component Br′ II_ecc and tangential component B′ θII_ecc The formula is as follows: Br′ II_ecc (ψ,ξ)=Br′ II (ψ,ξ)fr-B′ θII (ψ,ξ)sinβ(13),B′ θII_ecc (ψ,ξ)=B′ θII (ψ,ξ)cosβ (14), Step S4-3, based on the radial component Br II_ecc Tangential component Bθ II_ecc Radial component B′r′ II_ecc and tangential component B′ θII_ecc Based on the principle of linear superposition, the radial component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained. r (r,θ) and the tangential component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear θ (r,θ), radial component B r (r,θ) and tangential component B θ The formula for (r,θ) is as follows: B r (r,θ)=B rII_ecc (r,θ)+B′ rII_ecc (ψ,ξ)cosκ-B′ θII_ecc (ψ,ξ)sink(16), B θ (r,θ)=B θII_ecc (r,θ)+B′ rII_ecc (ψ,ξ)sink+B′ θII_ecc (ψ,ξ)cosκ(17), ξ=r-εcosθ(18), κ=ψ-θ (20).

[0010] The Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method provided by the present invention may also have the following features: in step S1, the formula for calculating the eccentricity ε is: ε=e·g(21), where g is the measured air gap length of the Halbach array eccentric harmonic magnetic gear.

[0011] The Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method provided by this invention may also have the following feature: wherein step S2 includes the following sub-steps: step S2-1, according to the hyperbolic cotangent transformation, the z-plane xy coordinate system is transformed into the w-plane uv orthogonal coordinate system, which is expressed as: In the formula, λ is a positive constant related to the rotor, stator radius, and relative offset distance between the stator and rotor centers, and e is a natural constant. Substituting formula (23) into formula (22) and rearranging, we obtain another expression for z. Dividing z into real and imaginary parts, we can obtain the following by simultaneous elimination: According to formula (24), two sets of orthogonal eccentric circular clusters are obtained, representing the equipotential lines of the irrotational field and the magnetic field lines in the electromagnetic field, respectively. Two circles are selected from the two sets of orthogonal eccentric circular clusters, and the radii of the two circles are u. s with u r These correspond to the rotor outer radius Rr and stator inner radius R of the Halbach array eccentric harmonic magnetic gear, respectively. s The centers of the two circles are (x, y) and (x, y). s ,0) and (x r ,0); Step S2-2, in the uv orthogonal coordinate system, define the air gap magnetic potential as Ω ecc Given (u,v), where the rotor boundary magnetic potential is 1 and the stator boundary magnetic potential is 0, then Ω ecc The formula for (u,v) is as follows: Combining formula (23) and electromagnetic field theory, formula (25) is transformed in polar coordinates to obtain the following formula: In the formula, μ0 is the vacuum permeability, and r and θ are the axes of the polar coordinate system. When the rotor is not eccentric, according to the Laplace equation and boundary conditions in the air gap region, the formula for the radial magnetic flux density in the air gap is as follows: Based on formulas (26) and (27), the radial air gap relative permeability function f is obtained. r The formula is as follows:

[0012] The Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method provided by this invention may also have the following feature: wherein, in step S5, the formula for the electromagnetic torque is as follows: In the formula L ef r is the axial length of the motor with Halbach array eccentric harmonic magnetic gears. g Let T be the radius of integration. cog It is electromagnetic torque.

[0013] The role and effect of invention

[0014] According to the analytical method for the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear of the present invention, the air gap magnetic field when the stator permanent magnet and the low-speed inner rotor permanent magnet act independently is obtained through a concentric analytical model. The air gap magnetic field is corrected by the radial air gap relative permeability function, and the radial and tangential components of the air gap magnetic flux density of the eccentric harmonic magnetic gear are obtained based on the principle of linear superposition. Thus, the electromagnetic torque is obtained according to Maxwell's stress tensor method. Therefore, the analytical method for the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear of the present invention can establish an accurate analytical model, and then calculate the electromagnetic torque from the design parameters. Attached Figure Description

[0015] Figure 1 This is a schematic diagram of the Halbach array eccentric harmonic magnetic gear in an embodiment of the present invention;

[0016] Figure 2 This is a flowchart illustrating the method for analyzing the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear in an embodiment of the present invention.

[0017] Figure 3 This is a schematic diagram illustrating the relationship between curve coordinates u and v in the z-plane in an embodiment of the present invention;

[0018] Figure 4 This is a schematic diagram of the analytical region of the concentric analytical model in an embodiment of the present invention;

[0019] Figure 5 This is a schematic diagram showing the relationship between the rotor coordinate system r-θ and the stator coordinate system ξ-ψ in an embodiment of the present invention. Detailed Implementation

[0020] To make the technical means, creative features, objectives and effects of this invention easier to understand, the following embodiments, in conjunction with the accompanying drawings, specifically illustrate the Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method of this invention.

[0021] Figure 1 This is a schematic diagram of the Halbach array eccentric harmonic magnetic gear in an embodiment of the present invention.

[0022] like Figure 1 As shown, the Halbach array eccentric harmonic magnetic gear includes: a stator core, a low-speed inner rotor, a Halbach array permanent magnet, a non-uniform air gap, bearings, and a high-speed inner rotor. s O is the center of the stator. r R is the center of the rotor. r R is the outer radius of the rotor. s R is the inner radius of the stator. mr R is the outer radius of the permanent magnet of the low-speed inner rotor.ms The radius of the stator permanent magnet is given. The arrow on the Halbach array permanent magnet indicates the magnetization direction of each permanent magnet. The white and gray blocks on the Halbach array permanent magnet represent different magnetic poles.

[0023] In this embodiment, the eccentricity of the Halbach array eccentric harmonic magnetic gear is 0.5, and the maximum torque is 42 N·m.

[0024] Figure 2 This is a schematic flowchart of the Halbach array eccentric harmonic magnetic gear air gap magnetic field analysis method in an embodiment of the present invention.

[0025] like Figure 2 As shown, the analytical method for the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear includes the following steps:

[0026] Step S1: Calculate the eccentricity ε based on the eccentricity e of the Halbach array eccentric harmonic magnetic gear.

[0027] The formula for calculating the eccentricity ε is as follows:

[0028] ε=e·g (21),

[0029] In the formula, g is the measured air gap length of the Halbach array eccentric harmonic magnetic gear.

[0030] Step S2: Obtain the radial air gap relative permeability function fr according to the hyperbolic cotangent transform.

[0031] Step S2 includes the following sub-steps:

[0032] Step S2-1: According to the hyperbolic cotangent transformation, the xy coordinate system in the z-plane is transformed into the uv orthogonal coordinate system in the w-plane, which can be expressed as:

[0033]

[0034]

[0035] In the formula, λ is a positive constant related to the rotor and stator radii and the relative offset distance between the stator and rotor centers, and e is a natural constant.

[0036] Substituting formula (23) into formula (22) and rearranging, we obtain another expression for z. Dividing z into real and imaginary parts, we can obtain the following by simultaneous elimination:

[0037]

[0038] According to formula (24), two sets of orthogonal eccentric circular clusters are obtained, which represent the equipotential lines of the irrotational field and the magnetic field lines in the electromagnetic field, respectively.

[0039] Figure 3 This is a schematic diagram illustrating the relationship between curve coordinates u and v in the z-plane in an embodiment of the present invention.

[0040] like Figure 3 As shown, in the z-plane, when u and v are constants, they present as two sets of orthogonal eccentric circles. From these, two circles can be obtained by using appropriate values ​​of u, such that the radii of the two circles are u and v respectively. s with u r These correspond to the rotor outer radius R of the Halbach array eccentric harmonic magnetic gear. r and stator inner radius R s The centers of the two circles are (x, y) and (x, y). s ,0) and (x r ,0).

[0041] Step S2-2, in the uv orthogonal coordinate system, define the air gap magnetic potential as Ω. ecc Given (u,v), where the rotor boundary magnetic potential is 1 and the stator boundary magnetic potential is 0, then Ω ecc The formula for (u,v) is as follows:

[0042]

[0043] Combining formula (23) and electromagnetic field theory, formula (25) is transformed in polar coordinates to obtain the following formula:

[0044]

[0045] In the formula, μ0 is the permeability of free space, and r and θ are the axes of the polar coordinate system.

[0046] When the rotor is not eccentric, the formula for the radial magnetic flux density in the air gap is obtained according to the Laplace equation and boundary conditions in the air gap region as follows:

[0047]

[0048] Based on formulas (26) and (27), the radial air gap relative permeability function f is obtained. r The formula is as follows:

[0049]

[0050] Step S3: Establish a concentric analytical model to obtain the first air gap magnetic field when the stator permanent magnet acts alone and the second air gap magnetic field when the low-speed inner rotor permanent magnet acts alone.

[0051] Step S3 includes the following sub-steps:

[0052] Step S3-1: Set the premise assumptions of the concentric analytical model and divide the analytical region of the concentric analytical model into three regions.

[0053] The underlying assumptions are as follows: calculations are performed in a two-dimensional field, and end effects are ignored; the core permeability is infinite, and saturation effects are ignored; the permanent magnet's BH curve is linear, and its relative permeability is μ. r =1.

[0054] Figure 4 This is a schematic diagram of the analytical region of the concentric analytical model in an embodiment of the present invention.

[0055] like Figure 4 As shown, the analytical region is divided into three regions: the low-speed inner rotor permanent magnet region I, the air gap region II, and the stator permanent magnet region III. The white and gray blocks in the low-speed inner rotor permanent magnet region I and the stator permanent magnet region III represent different magnetic poles.

[0056] Step S3-2: Establish the Laplace equation or Poisson equation satisfied by the vector magnetic potential in the three regions. When the stator permanent magnet acts alone, the formulas for the Laplace equation or Poisson equation satisfied by the vector magnetic potential are as follows:

[0057]

[0058] In the formula A II A represents the vector magnetic potential of air gap region II when the stator permanent magnet acts alone. III M represents the vector magnetic potential of region III of the stator permanent magnet. r M represents the radial component of the magnetization of the permanent magnet. θ Let r and θ be the tangential components of the magnetization of the permanent magnet, r and θ be the axes of the polar coordinate system, and μ0 be the permeability of free space.

[0059] When the permanent magnet of the low-speed internal rotor acts alone, the formula for the Laplace equation or Poisson equation satisfied by the vector magnetic potential is as follows:

[0060]

[0061] In the formula A′ II A represents the vector magnetic potential of air gap region II when the permanent magnet of the low-speed inner rotor acts alone. I The vector magnetic potential is the region I of the low-speed internal rotor permanent magnet.

[0062] The formula for magnetization M of a permanent magnet Halbach array is as follows:

[0063] M = M r r+M θ θ (3),

[0064] in:

[0065]

[0066]

[0067]

[0068]

[0069] In the formula, n represents the calculated harmonic order of the air gap magnetic field and the permanent magnet magnetic field, θ0 represents the offset degree between the magnet and the initially set angle, and M... rn (n) and M θn (n) is the Fourier expansion of the magnetization of the Halbach permanent magnet, B r θ represents the relative permeability, q represents the number of blocks per pole of the Halbach array, l represents the l-th block in the q-block array, and r and θ represent the axes of the polar coordinate system.

[0070] Step S3-3: Based on the boundary conditions of the three regions, solve formulas (1) and (2) to obtain the vector magnetic potential expressions for the three regions. Based on the vector magnetic potential expressions, obtain the radial component B of the magnetic flux density generated by the stator permanent magnet acting alone in the air gap region II under the concentric condition. rII and tangential component B θII That is, the first air gap magnetic field, and the radial component B′ of the magnetic flux density generated by the low-speed inner rotor acting alone in air gap region II under concentric conditions. rII and tangential component B′ θII That is, the second air gap magnetic field.

[0071] When the stator permanent magnet acts alone, the boundary conditions at the interface are:

[0072]

[0073] In the formula R ms R is the internal radius of the stator permanent magnet. s R is the inner radius of the stator. r The outer radius of the low-speed inner rotor.

[0074] When the low-speed internal rotor permanent magnet acts alone, the interface boundary conditions are as follows:

[0075]

[0076] In the formula R mr The outer radius of the permanent magnet of the low-speed inner rotor.

[0077] Step S4, based on the radial air gap relative permeability function f rBy correcting the first and second air gap magnetic fields and combining them with the eccentricity ε, the radial component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained based on the principle of linear superposition. r (r,θ) and tangential component B θ (r,θ).

[0078] Step S4 includes the following sub-steps:

[0079] Step S4-1: Establish an r-θ coordinate system with the rotor center as the origin, and calculate the radial air gap relative permeability function f. r For radial component B rII and tangential component B θII After correction, the radial component B of the magnetic flux density in the air gap region II in the r-θ coordinate system is obtained. rII_ecc and tangential component B θII_ecc .

[0080] Step S4-2: Establish a ξ-ψ coordinate system with the stator center as the origin, and calculate the radial air gap relative permeability function f. r For the radial component B′ rII and tangential component B′ θII After correction, the radial component B′ of the magnetic flux density in air gap region II in the ξ-ψ coordinate system is obtained. rII_ecc and tangential component B′ θII_ecc .

[0081] Figure 5 This is a schematic diagram showing the relationship between the rotor coordinate system r-θ and the stator coordinate system ξ-ψ in an embodiment of the present invention.

[0082] like Figure 5 As shown, radial component B rII_ecc and tangential component B θII_ecc The formula is as follows:

[0083] B rII_ecc (r,θ)=B rII (r,θ)f r -B θII (r,θ)sinα (10),

[0084] B θII_ecc (r,θ)=B θII (r,θ)cosα (11),

[0085]

[0086] Radial component B′ rII_ecc and tangential component B′ θII_ecc The formula is as follows:

[0087] B′ rII_ecc(ψ,ξ)=B′ rII (ψ,ξ)f r -B′ θII (ψ,ξ)sinβ (13),

[0088] B′ θII_ecc (ψ,ξ)=B′ θII (ψ,ξ)cosβ (14),

[0089]

[0090] Step S4-3, based on the radial component R rII_ecc Tangential component R θII_ecc Radial component B′ rII_ecc and tangential component B′ θII_ecc Based on the principle of linear superposition, the radial component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained. r (r,θ) and the tangential component B of the air gap magnetic flux density of the eccentric harmonic magnetic gear θ (r,θ), radial component B r (r,θ) and tangential component B θ The formula for (r,θ) is as follows:

[0091] B r (r,θ)=B rII_ecc (r,θ)+B′ rII_ecc (ψ,ξ)cosκ-B′ θII_ecc (ψ,ξ)sinκ (16),

[0092] B θ (r,θ)=B θII_ecc (r,θ)+B′ rII_ecc (ψ,ξ)sinκ+B′ θII_ecc (ψ,ξ)cosκ (17),

[0093] ξ=r-εcosθ (18),

[0094]

[0095] κ=ψ-θ (20).

[0096] Step S5, based on radial component B r (r,θ) and tangential component B θ (r,θ) is obtained based on Maxwell's stress tensor method to obtain the electromagnetic torque.

[0097] The formula for electromagnetic torque is as follows:

[0098]

[0099] In the formula L ef r is the axial length of the motor with Halbach array eccentric harmonic magnetic gears. g Let T be the radius of integration. cog It is electromagnetic torque.

[0100] The role and effect of the embodiments

[0101] According to the analytical method for the air gap magnetic field of the Halbach array eccentric harmonic magnetic gear involved in this embodiment, the air gap magnetic field when the stator permanent magnet and the low-speed inner rotor permanent magnet act independently is obtained through a concentric analytical model. The air gap magnetic field is corrected by the radial air gap relative permeability function, and the radial and tangential components of the air gap magnetic flux density of the eccentric harmonic magnetic gear are obtained based on the principle of linear superposition. Thus, the electromagnetic torque is obtained according to Maxwell's stress tensor method. In summary, this method can establish an accurate analytical model, and then calculate the electromagnetic torque from the design parameters.

[0102] The above embodiments are preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention.

Claims

1. A method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear using a Halbach array, characterized in that, Includes the following steps: Step S1: Calculate the eccentricity e based on the eccentricity e of the Halbach array eccentric harmonic magnetic gear. ; Step S2: Obtain the radial air gap relative permeability function based on the hyperbolic cotangent transform. ; Step S3: Establish a concentric analytical model to obtain the first air gap magnetic field when the stator permanent magnet acts alone and the second air gap magnetic field when the low-speed inner rotor permanent magnet acts alone. Step S3 includes the following sub-steps: Step S3-1: Set the premise assumptions of the concentric analytical model and divide the analytical region of the concentric analytical model into three regions, namely, the low-speed inner rotor permanent magnet region I, the air gap region II, and the stator permanent magnet region III. Step S3-2: Establish the Laplace equation or Poisson equation satisfied by the vector magnetic potential of the three regions. When the stator permanent magnet acts alone, the formula for the Laplace equation or Poisson equation satisfied by the vector magnetic potential is as follows: （1）， In the formula This refers to the vector magnetic potential of the air gap region II when the stator permanent magnet acts alone. This refers to the vector magnetic potential of the stator permanent magnet region III. The radial component of the magnetization of the permanent magnet. This represents the tangential component of the magnetization intensity of the permanent magnet. , The axes are in the polar coordinate system. The permeability of free space, When the low-speed internal rotor permanent magnet acts alone, the formula for the Laplace equation or Poisson equation satisfied by the vector magnetic potential is as follows: （2）， In the formula This refers to the vector magnetic potential of the air gap region II when the low-speed inner rotor permanent magnet acts alone. Let be the vector magnetic potential of the low-speed internal rotor permanent magnet region I. The formula for magnetization M in a permanent magnet Halbach array is as follows: （3）， in: （4）， （5）， （6）， （7）， In the formula, n represents the calculated harmonic order of the air gap magnetic field and the permanent magnet magnetic field. This refers to the offset in degrees between the magnet and the initially set angle. and Here is the Fourier expansion of the magnetization of the Halbach permanent magnet. Let q be the relative permeability, q be the number of blocks per pole of the Halbach array, and l be the l-th block in the q-block array. , These are the axes of the polar coordinate system; Step S3-3: Based on the boundary conditions of the three regions, solve formulas (1) and (2) to obtain the vector magnetic potential expressions for the three regions. Based on the vector magnetic potential expressions, obtain the radial component of the magnetic flux density generated by the stator permanent magnet acting alone in the air gap region II under concentric conditions. and tangential components That is, the first air gap magnetic field, and the radial component of the magnetic flux density generated by the low-speed inner rotor acting alone in the air gap region II under concentric conditions. and tangential components That is, the second air gap magnetic field. When the stator permanent magnet acts alone, the interface boundary conditions are as follows: （8）， In the formula Let be the radius of the stator permanent magnet. The inner radius of the stator. The outer radius of the low-speed inner rotor. When the low-speed inner rotor permanent magnet acts alone, the interface boundary conditions are as follows: （9）， In the formula The outer radius of the permanent magnet of the low-speed inner rotor; Step S4, based on the radial air gap relative permeability function The first air gap magnetic field and the second air gap magnetic field are corrected, combined with the eccentricity. Based on the principle of linear superposition, the radial component of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained. and tangential components ; Step S5, based on the radial component and the tangential component Based on Maxwell's stress tensor method, the electromagnetic torque is obtained, wherein in step S5, the formula for the electromagnetic torque is as follows: （29）， In the formula The axial length of the motor with Halbach array eccentric harmonic magnetic gears. Let be the radius of integration. It is electromagnetic torque.

2. The method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear using a Halbach array according to claim 1, characterized in that: in, The underlying assumptions are that the calculations are performed in a two-dimensional field, ignoring end effects; the core permeability is infinite, ignoring saturation effects; the permanent magnet's BH curve is linear, and its relative permeability is [missing information]. .

3. The method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear using a Halbach array according to claim 1, characterized in that: in, Step S4 includes the following sub-steps: Step S4-1: Establish an r-θ coordinate system with the rotor center as the origin, based on the radial air gap relative permeability function. For the radial component and the tangential component After correction, the radial component of the magnetic flux density of the air gap region II in the r-θ coordinate system is obtained. and tangential components The radial component and the tangential component The formula is as follows: （10）, （11）, （12）； Step S4-2: Establish a ξ-ψ coordinate system with the stator center as the origin, based on the radial air gap relative permeability function. For the radial component and the tangential component After correction, the radial component of the magnetic flux density of the air gap region II in the ξ-ψ coordinate system is obtained. and tangential components The radial component and the tangential component The formula is as follows: （13）, （14）, （15）； Step S4-3, based on the radial component The tangential component The radial component and the tangential component Based on the principle of linear superposition, the radial component of the air gap magnetic flux density of the eccentric harmonic magnetic gear is obtained. and the tangential component of the air gap magnetic flux density of the eccentric harmonic magnetic gear The radial component and the tangential component The formula is as follows: （16）， （17）， （18）, （19）, （20）。 4. The method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear using a Halbach array according to claim 1, characterized in that: in, In step S1, the eccentricity The calculation formula is: （21）, In the formula, g is the measured air gap length of the Halbach array eccentric harmonic magnetic gear.

5. The method for analyzing the air gap magnetic field of an eccentric harmonic magnetic gear using a Halbach array according to claim 1, characterized in that: in, Step S2 includes the following sub-steps: Step S2-1: According to the hyperbolic cotangent transformation, the z-plane xy coordinate system is transformed into the w-plane uv orthogonal coordinate system. The transformation is expressed as: （22）, （23）, In the formula Let e ​​be a positive constant related to the rotor and stator radii and the relative offset distance between the stator and rotor centers, where e is the natural constant. Substituting formula (23) into formula (22) and rearranging, we obtain another expression for z. Dividing z into real and imaginary parts, we can obtain the following by simultaneous elimination: （24）, According to formula (24), two sets of orthogonal eccentric circular clusters are obtained, representing the equipotential lines of the irrotational field and the magnetic field lines in the electromagnetic field, respectively. Two circles are selected from the two sets of orthogonal eccentric circular clusters, and the radii of the two circles are respectively... and These correspond to the rotor outer radius of the Halbach array eccentric harmonic magnetic gear. and stator inner radius The centers of the two circles are respectively and ; Step S2-2, in the uv orthogonal coordinate system, define the air gap magnetic potential as... If the rotor boundary magnetic potential is 1 and the stator boundary magnetic potential is 0, then The formula is as follows: （25）， Combining formula (23) and electromagnetic field theory, in polar coordinates, formula (25) is transformed to obtain the following formula: （26）, In the formula The permeability of free space, , The axes are in the polar coordinate system. When the rotor is not eccentric, the formula for the radial magnetic flux density of the air gap is obtained according to the Laplace equation and boundary conditions in the air gap region as follows: （27）, According to formulas (26) and (27), the radial air gap relative permeability function is obtained. The formula is as follows: （28）。

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