A three-dimensional analytical method for calculating the air-gap magnetic field of a stator without magnetic yoke modular axial flux motor at no-load

Abstract

A kind of three-dimensional analytical method of no-load air-gap magnetic field of stator non-magnetic yoke modular axial flux motor, it is related to the calculation method of axial flux motor no-load air-gap magnetic field, comprising the following steps: S1: by solving Laplace equation under column coordinate system to calculate motor slotless no-load air-gap magnetic field;S2: complex relative permeance function is solved using the conformal mapping method, based on slotless no-load air-gap magnetic field and complex relative permeance function calculation slot no-load air-gap magnetic field;S3: according to the magnetic circuit characteristics of stator non-magnetic yoke modular axial flux motor, establish its equivalent magnetic circuit model;S4: according to the slotless air-gap magnetic field, complex relative permeance function and equivalent magnetic circuit model establish the analytical model of axial flux motor no-load air-gap magnetic field.The present application has the advantages of high solution accuracy and fast calculation efficiency.

Description

Technical Field

[0001] This invention relates to a method for calculating the no-load air gap magnetic field of an axial flux motor, specifically a three-dimensional analytical method for calculating the no-load air gap magnetic field of a stator-less modular axial flux motor. Background Technology

[0002] Stator-free modular axial flux motors possess advantages such as high torque density, high efficiency, and short axial length, making them suitable for applications in electric vehicles, underwater vehicles, wind turbines, and related fields. Air gap magnetic field calculation is fundamental to motor performance analysis, including electromagnetic characteristics, vibration, and noise. Its accuracy and efficiency are crucial for motor design and optimization. Due to the structural characteristics of stator-free modular axial flux motors, their magnetic field exhibits complex three-dimensional properties, increasing the difficulty of magnetic field calculation. Although the three-dimensional finite element method can accurately calculate the magnetic field distribution and electromagnetic performance parameters of axial flux motors, it requires a large number of meshes, places high demands on hardware, and is time-consuming. Therefore, during the development phase of stator-free modular axial flux motors, a precise and rapid three-dimensional analytical method for the air gap magnetic field is needed.

[0003] Currently, analytical methods for the no-load air gap magnetic field of axial flux motors mainly include the subdomain method, the magnetomotive force and magnetic permeability method, and the equivalent magnetic circuit method. The subdomain method typically divides the axial flux motor into subdomains such as stator teeth, stator slots, air gap, and permanent magnets, solves Maxwell's equations within each subdomain, and then substitutes boundary conditions to obtain the scalar or vector magnetic potential within each subdomain. While this method can analyze the magnetic field distribution of different parts of the motor, the solution process is relatively complex. Patent CN 112347627A derives the expression for the air gap magnetic field based on Laplace's equations or Poisson's equations and boundary conditions in different solution domains, but it cannot consider the curvature effect, leakage flux effect, and magnetic saturation effect of the axial flux motor. Patent CN 114006559A proposes an analytical method for the magnetic field of an axial switched reluctance motor that considers magnetic saturation and edge effects through an adaptive convergent iterative algorithm and a radial correction function, but this method neglects the curvature effect and leakage flux effect of the motor. Meanwhile, when motor parameters change, the corresponding radial correction function needs to be recalculated using the finite element method, resulting in poor versatility. The magnetomotive force and permeability method utilizes complex relative permeability functions to consider the slotting effect of the motor. Patent CN 113868929 A considers the influence of stator slotting and edge leakage flux effects of axial flux motors using complex relative permeability functions and radial correction functions, respectively. This method is fast and can reveal the general laws of the air gap magnetic field, but it ignores the curvature effect, leakage flux effect, and magnetic saturation effect of the axial motor, reducing the calculation accuracy of the analytical model. Nonlinear iteration based on equivalent magnetic circuit models and material parameters can analyze the leakage flux and magnetic saturation of axial flux motors. Patent CN 111327170A establishes a dynamic equivalent magnetic circuit model for an axial permanent magnet levitation flywheel motor. This model can analyze the influence of leakage flux effects, but different magnetic circuit models need to be established according to the relative positions of the stator and rotor. To ensure the solution accuracy of the equivalent magnetic circuit model, it is usually necessary to increase the number of nodes in the model, which drastically increases the complexity of the solution model and the calculation time. Furthermore, the equivalent magnetic circuit method cannot reveal the general laws governing the unloaded air gap magnetic field.

[0004] In summary, the following problems exist when using the above method to calculate the no-load air gap magnetic field of a stator-less modular axial flux motor:

[0005] (1) The subdomain method and the magnetomotive force and magnetic permeability method cannot comprehensively consider the curvature effect, leakage magnetic effect and magnetic saturation effect of the stator yoke-less modular axial flux motor, which will affect the accuracy of the analytical results.

[0006] (2) The equivalent magnetic circuit method requires different equivalent magnetic circuit models to be established according to the relative positions of the stator and rotor. The solution equation is complex and cannot reveal the general law of the unloaded air gap magnetic field. Summary of the Invention

[0007] To address the problem that existing technologies cannot fully consider various nonlinear factors in stator-yoke-less modular axial flux motors, this paper proposes a three-dimensional analytical method for the no-load air gap magnetic field of stator-yoke-less modular axial flux motors, which can be used to improve the efficiency of motor design and optimization.

[0008] The technical solution adopted by this invention to overcome the shortcomings of the prior art is as follows:

[0009] A three-dimensional analytical method for the no-load air gap magnetic field of a stator-yoke-free modular axial flux motor includes the following steps:

[0010] S1: Calculate the slotless air gap magnetic field of the motor by solving the Laplace equation in cylindrical coordinates;

[0011] S2: Solve the complex relative permeability function using the conformal transformation method, and calculate the slotted empty air gap magnetic field based on the slotless empty air gap magnetic field and the complex relative permeability function.

[0012] S3: Based on the magnetic circuit characteristics of the stator-less modular axial flux motor, establish its equivalent magnetic circuit model;

[0013] S4: An analytical model of the no-load air gap magnetic field of the axial flux motor is established based on the slotless air gap magnetic field, the complex relative permeability function, and the equivalent magnetic circuit model.

[0014] S5: Calculate motor performance parameters based on unloaded air gap magnetic field.

[0015] S1 includes the following steps:

[0016] S1.1 Based on the structural characteristics of the stator-less modular axial flux motor, a cylindrical coordinate system is established with the center of the lower rotor back iron as the origin. The motor axial, radial, and circumferential directions are respectively taken as the z-direction, r-direction, and θ-direction. The stator-less modular axial flux motor model is established in the cylindrical coordinate system.

[0017] S1.2 In the stator-yoke-less modular axial flux motor model, the permanent magnet domain and the air gap domain are selected as the solution domains for the motor. The permanent magnet domain is defined as 0 ≤ z ≤ h. m The region has an air gap of h. m ≤z≤h m The region of +g. Where, h m denoted as the thickness of the permanent magnet, and g is the length of the air gap.

[0018] The permanent magnets of the S1.3 stator-less modular axial flux motor are sector-shaped and axially magnetized, with alternating N and S poles. An expression for magnetization intensity is established based on the permanent magnet pole parameters:

[0019]

[0020] In the formula, B r θ is the remanence of the permanent magnet, μ0 is the permeability of free space, and θ is the magnetic remanence of the permanent magnet. p θ is the angle occupied by the magnetic poles. o Let T be the angle between the magnetic poles, and T be the period of the magnetization, T = 2(θ). p +θ o ).

[0021] S1.4 In both the permanent magnet domain and the air gap domain, the scalar magnetic potential satisfies the Laplace equation:

[0022]

[0023] In the formula, For scalar magnetic potential, For the scalar magnetic potential of the permanent magnet domain. This represents the scalar magnetic potential of the air gap domain.

[0024] S1.5 Solving the general solution of the Laplace equation using the method of separation of variables to calculate the scalar magnetic potential:

[0025]

[0026] In the formula, J n (i n r) is an nth-order Bessel function of the first kind, Y n (i n r) is an nth-order Bessel function of the second kind, i n a1, a2, b1, b2, c1, and c2 are terms with undetermined coefficients.

[0027] S1.6 Determine the boundary conditions of the solution domain:

[0028] (1) At z = 0 and z = h m The scalar magnetic potential at the +g plane is zero, as shown in the following expression:

[0029]

[0030] (2) At the interface between the permanent magnet domain and the air gap domain, z = h m In the plane, the axial magnetic flux density is continuous with the scalar magnetic potential, and the specific expression is as follows:

[0031]

[0032] (3) When r approaches 0, the scalar magnetic potential of the motor is a finite value. When r approaches the boundary outer diameter r p At that time, the scalar magnetic potential is zero. The specific expression is as follows:

[0033]

[0034] In the formula, k finite It is a finite value.

[0035] S1.7 expands the magnetization into a Fourier-Bessel series, as shown in the following expansion:

[0036]

[0037] In the formula, M nk Let i be the Fourier-Bessel coefficients. nk The coefficient term can be calculated based on boundary conditions.

[0038] Based on the general solution and boundary conditions of the equation, the scalar magnetic potential expressions for the permanent magnet domain and the air gap domain can be obtained in S1.8:

[0039]

[0040]

[0041] In the formula, and The constants are calculated based on the boundary conditions, where p is the number of pole pairs of the motor, and J is the constant. np (i nk r) is a Bessel function of the first kind with order np.

[0042] S1.9 Based on the scalar magnetic potential of the air gap domain, the slotless no-load air gap magnetic flux density of the motor in the axial, tangential, and radial directions can be calculated:

[0043]

[0044]

[0045]

[0046] In the formula, J np-1 (i nk r) is a Bessel function of the first kind with order np-1, J np+1 (i nk r) is a Bessel function of the first kind with order np+1.

[0047] S2 includes the following steps:

[0048] S2.1 Using the conformal mapping method to calculate the complex relative permeability function, it can be expressed as:

[0049] λ(r,θ,z)=λ a (r,θ,z)+iλ b (r,θ,z)

[0050] In the formula, λa (r,θ,z) is the real part of the complex relative permeability, λ b (r,θ,z) represents the imaginary part of the relative magnetic permeability of the complex number, and i is the imaginary symbol.

[0051] S2.2 Expand the complex relative permeability function into a Fourier series form

[0052]

[0053]

[0054] In the formula, λ aro (r,z) represents the constant term in the real part expansion of the relative permeability function, λ arm (r,z) are the coefficients of the pole number term in the real part expansion of the relative permeability function, λ brm (r,z) are the coefficients of the pole number terms in the imaginary part expansion of the relative permeability function.

[0055] S2.4 Calculate the slotted air gap magnetic flux density of the motor based on the obtained slotless no-load air gap magnetic flux density and complex relative permeability function. The slotted no-load air gap magnetic flux densities of the motor in the axial and radial directions are:

[0056] B gzs (r,θ,z)=B gzw (r,θ,z)×λ a (r,θ,z)+B gθw (r,θ,z)×λ b (r,θ,z)

[0057] B gθs (r,θ,z)=B gθw (r,θ,z)×λ a (r,θ,z)-B gzw (r,θ,z)×λ b (r,θ,z)

[0058] S3 includes the following steps:

[0059] S3.1 Because the axial flux motor has additional leakage magnetic paths at the inner and outer diameters, the motor is divided into inner diameter domain, center domain, and outer diameter domain, and magnetic circuit models are established separately for each domain. The solution domain is divided according to the motor radius, r. i ≤r≤r ri The region is the inner diameter region of the motor, r ri ≤r≤r ro The region is the central area of ​​the motor, r ro ≤r≤r o The region is the outer diameter region of the motor. Where r ri For the inner diameter and outer diameter, r ro For the outer diameter domain inner diameter, r ir is the inner diameter of the motor. o This refers to the outer diameter of the motor.

[0060] S3.2 Analyzes the magnetic circuit characteristics of the stator-yoke-less modular axial flux motor. In the motor's central region, the magnetic circuit mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnets, and the self-leakage magnetic circuit of the permanent magnets. In the motor's inner diameter region, the magnetic circuit mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnets, the self-leakage magnetic circuit of the permanent magnets, and the leakage magnetic circuit at the inner diameter. In the motor's outer diameter region, the magnetic circuit mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnets, the self-leakage magnetic circuit of the permanent magnets, and the leakage magnetic circuit at the outer diameter.

[0061] S3.3 Based on the magnetic circuit characteristics of the central domain of the stator-less modular axial flux motor, an equivalent magnetic circuit model is established.

[0062] S3.4 Based on the magnetic circuit characteristics of the inner and outer diameter domains of the stator-less modular axial flux motor, an equivalent magnetic circuit model is established.

[0063] S3.5 In the main magnetic circuit of a stator-less modular axial flux motor, the permanent magnet reluctance R mi Equivalent air gap magnetic reluctance R g Equivalent stator reluctance R s and equivalent rotor reluctance R r The calculation formula is as follows:

[0064]

[0065]

[0066]

[0067]

[0068] In the formula, μ m h is the relative permeability of the permanent magnet. r and h s These are the rotor back iron thickness and stator thickness, respectively. m S g and S s These represent the cross-sectional areas of the permanent magnet, air gap, and stator in the region being sought, respectively. a For the polar distance, l r Let μ be the radial length of the region in question. r (B r ) and μ s (B s These are the relative permeability of the rotor and the relative permeability of the stator, respectively.

[0069] S3.6 In the permanent magnet self-leakage magnetic circuit of a stator-less modular axial flux motor, the equivalent air gap reluctance R gmand equivalent rotor reluctance R rm The calculation formula is as follows:

[0070]

[0071]

[0072] S3.7 In the leakage magnetic circuit between permanent magnet poles, the equivalent air gap magnetic reluctance R gmm and equivalent rotor reluctance R rmm The calculation formula is as follows:

[0073]

[0074]

[0075] In the formula, l o The distance between the two magnetic poles is denoted as .

[0076] In the leakage magnetic circuit at the inner diameter of S3.8, the equivalent air gap magnetic reluctance R of the inner diameter is... bgi Equivalent rotor reluctance R of inner diameter bri The calculation formula is as follows:

[0077]

[0078]

[0079] In the formula, l ri This is the radial length of the rotor within the inner diameter region.

[0080] In the leakage magnetic circuit at the outer diameter, the equivalent air gap magnetic reluctance R at the outer diameter is... bgo Equivalent rotor reluctance R to outer diameter bro The calculation formula is as follows:

[0081]

[0082]

[0083] In the formula, l ro The radial length of the rotor in the outer diameter region.

[0084] S3.9 Within the motor's central domain, the formulas for calculating the unloaded axial air gap magnetic flux density and the unloaded tangential air gap magnetic flux density, after considering leakage flux and magnetic saturation effects, based on the equivalent magnetic circuit model of the central domain, are as follows:

[0085]

[0086]

[0087] In the formula, α in The saturation leakage flux coefficient of the central domain is...

[0088] S3.10 Within the inner diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the inner diameter region:

[0089]

[0090]

[0091] In the formula, α ri The saturation leakage flux coefficient of the inner diameter domain is...

[0092]

[0093] Within the outer diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the outer diameter region:

[0094]

[0095]

[0096] In the formula, α ro The saturation leakage flux coefficient in the outer diameter domain is...

[0097]

[0098] S3.11: Based on the no-load air gap magnetic flux density and the BH curves of the stator and rotor, the saturation leakage flux coefficient is calculated iteratively, and the influence of the motor magnetic saturation effect and leakage flux effect on the no-load air gap magnetic flux density is analyzed.

[0099] S4 includes the following steps:

[0100] S4.1 Based on the above steps, the mathematical expressions for the axial no-load air gap magnetic flux density and tangential no-load air gap magnetic flux density of the stator yoke-less modular axial flux motor, considering curvature effect, slotting effect, leakage magnetic flux effect and magnetic saturation effect, are as follows:

[0101]

[0102]

[0103] In the formula, α b S is the saturation leakage flux coefficient of the motor. The saturation leakage flux coefficient of different regions of the motor is calculated by S3.

[0104] S4.2: Analyze the main spatial orders of the axial no-load air gap magnetic flux density and the tangential no-load air gap magnetic flux density of the motor based on mathematical expressions.

[0105] S5 includes the following steps:

[0106] S5.1 Based on the obtained no-load air gap magnetic flux density, an analytical model of the motor's no-load back electromotive force is established, and its expression is:

[0107]

[0108] In the formula, ψ phase The corresponding air gap flux linkage, α phase_j Let α be the starting angle of the j-th winding in the desired phase. τ n is the span angle of a single winding. phase N represents the number of windings per phase. c This represents the number of turns in the winding.

[0109] S5.2 Establishing an analytical model of the motor's no-load cogging torque using Maxwell's stress tensor method:

[0110]

[0111] In the formula, l is the radial length of the motor.

[0112] Compared with the prior art, the advantages of the present invention are as follows:

[0113] 1) This method can comprehensively consider the curvature effect, slotting effect, leakage magnetic effect and magnetic saturation effect of the stator-less modular axial flux motor, and has high solution accuracy.

[0114] 2) Using this method, the mathematical expression of the air gap magnetic field distribution of the stator-yoke-less modular axial flux motor can be obtained, and based on this, the general law of air gap magnetic field distribution can be derived.

[0115] 3) The proposed method has high computational efficiency and low memory usage. Under the same conditions, its computation time is less than 2% of that of the three-dimensional finite element method, and its memory usage is less than 0.5% of that of the three-dimensional finite element method. Attached Figure Description

[0116] Figure 1 This is a model diagram of a modular axial flux motor with a stator and no magnetic yoke.

[0117] Figure 2 This is a schematic diagram of the solution domain for the motor.

[0118] Figure 3 This is a schematic diagram of the equivalent magnetic circuit region division of the motor.

[0119] Figure 4 and Figure 5 This is a schematic diagram of the main magnetic flux circuit of the motor, in which Figure 4 This is a schematic diagram of the magnetic circuit of the motor in the central region. Figure 5This is a schematic diagram of the magnetic circuits in the inner and outer diameter regions of the motor, excluding the magnetic circuits shown in section 4.

[0120] Figure 6 , Figure 7 and Figure 8 This is the equivalent magnetic circuit model diagram of the motor, in which Figure 6 This is an equivalent magnetic circuit model established based on the magnetic circuit characteristics of the motor's central region. Figure 7 This is an equivalent magnetic circuit model established based on the magnetic circuit characteristics of the inner diameter domain of the motor. Figure 8 An equivalent magnetic circuit model is established based on the magnetic circuit characteristics of the outer diameter domain of the motor.

[0121] Figure 9 This is the flowchart of the magnetic circuit model iteration.

[0122] Figure 10 This is a schematic diagram of the three-dimensional distribution of the axial air gap magnetic field obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the finite element method.

[0123] Figure 11 This is a schematic diagram showing the three-dimensional distribution of the axial air gap magnetic field obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the method of the present invention.

[0124] Figure 12 This is a schematic diagram of the three-dimensional distribution of the tangential air gap magnetic field obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the finite element method.

[0125] Figure 13 This is a schematic diagram showing the three-dimensional distribution of the tangential air gap magnetic field obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the method of the present invention.

[0126] Figure 14 This is a comparison chart of the back electromotive force obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the finite element method and the method of this invention.

[0127] Figure 15 This is a comparison chart of the cogging torques obtained by calculating the stator yoke-less modular axial flux motor as shown in Table 1 using the finite element method and the method of this invention.

[0128] Figure 16 This is a comparison chart of the axial air gap magnetic field at the average radius obtained by calculating the stator yoke-less modular axial flux motor as shown in Table 1 using the method of this invention and the finite element method, and the experimental results.

[0129] Figure 17This is a comparison chart of the tangential air gap magnetic field at the average radius obtained by calculating the stator yoke-less modular axial flux motor as shown in Table 1 using the method of this invention and the finite element method, and the experimental results.

[0130] Figure 18 This is a comparison chart of the axial air gap magnetic field order characteristics at the average radius obtained by calculating the stator yoke-less modular axial flux motor as shown in Table 1 using the method of this invention and the finite element method, and the experimental results.

[0131] Figure 19 This is a comparison chart of the tangential air gap magnetic field order characteristics at the average radius obtained by calculating the stator yoke-less modular axial flux motor as shown in Table 1 using the method of this invention and the finite element method, and the experimental results.

[0132] Figure 20 This is a comparison chart of the axial air gap magnetic field distribution along the radius obtained by calculating the stator-yoke-less modular axial flux motor as shown in Table 1 using the method of this invention and the finite element method, and the experimental results. Detailed Implementation

[0133] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0134] This invention uses a 20-pole, 18-slot stator-less, modular axial flux motor as an example. This motor has a dual-rotor, single-stator structure. The main parameters of the motor are shown in Table 1:

[0135] Table 1 Main parameters of the motor

[0136]

[0137]

[0138] S1: Calculation of the slotless air gap magnetic field of the motor by solving the three-dimensional Laplace equation:

[0139] S1.1 Based on the structural characteristics of the motor, a cylindrical coordinate system is established with the center of the rotor back iron as the origin. This coordinate system uses the motor's axial, radial, and circumferential directions as the z-axis, r-axis, and θ-axis, respectively. A modular axial flux motor model without a stator yoke is established in the cylindrical coordinate system, as follows: Figure 1 As shown, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron.

[0140] S1.2 Selects the permanent magnet domain and air gap domain in the stator-yoke-less modular axial flux motor model as the solution domain for the motor, such as Figure 2As shown. The region 0 ≤ z ≤ 4 mm is the lower permanent magnet region, and the region 4 mm ≤ z ≤ 5.5 mm is the lower air gap region. Among them, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron.

[0141] The permanent magnets of the S1.3 stator-less modular axial flux motor are sector-shaped and axially magnetized, with alternating N and S poles. The remanence intensity B of the selected permanent magnets... r The magnetization period is 1.2T, the period T is π / 10 rad, and the free permeability is 4π×10⁻⁶. -7 Wb / (A·m). The expression for the magnetization of a permanent magnet is as follows:

[0142]

[0143] In the formula, μ0 is the free permeability, and θ p θ is the angle occupied by the magnetic poles. o This represents the angle occupied by the magnetic pole spacing.

[0144] S1.4 Due to the symmetrical structure of the motor, the solution is taken using the lower permanent magnet domain and the lower air gap domain as examples. Within the lower permanent magnet domain and the lower air gap domain, the scalar magnetic potential satisfies the Laplace equation:

[0145]

[0146] In the formula, For scalar magnetic potential, For the scalar magnetic potential of the permanent magnet domain. This represents the scalar magnetic potential of the air gap domain.

[0147] S1.5 The general solution for the scalar magnetic potential is obtained by solving the Laplace equation using the method of separation of variables:

[0148]

[0149] In the formula, J n (i n r) is an nth-order Bessel function of the first kind, Y n (i n r) is an nth-order Bessel function of the second kind, i n a1, a2, b1, b2, c1, and c2 are terms with undetermined coefficients.

[0150] S1.6 Determine the boundary conditions of the solution domain:

[0151] (1) When r approaches 0, the scalar magnetic potential of the motor is a finite value. When r approaches the outer diameter of the boundary r p At that time, the scalar magnetic potential is zero. The specific expression is as follows:

[0152]

[0153] In the formula, k finite It is a finite value.

[0154] (2) At z = 0 and z = h m The scalar magnetic potential at the +g plane is zero, as shown in the following expression:

[0155]

[0156] (3) At the interface between the permanent magnet domain and the air gap domain, the axial magnetic flux density and scalar magnetic potential are continuous, and the specific expression is as follows:

[0157]

[0158] S1.7 Since the magnetization of a permanent magnet is an even function, the Fourier series expansion of the magnetization is a cosine series. The Fourier-Bessel series expansion of the magnetization is:

[0159]

[0160] In the formula, M nk Let i be the Fourier-Bessel coefficients. nk The coefficient term can be calculated based on boundary conditions.

[0161] S1.8 Substituting the general solution of the scalar magnetic potential in the permanent magnet domain and the air gap domain with the magnetization of the permanent magnet into the boundary conditions, we can obtain a2 = 0, b1 = 0, c2 = 0, i nk It should satisfy its relationship with r p The product of these terms forms the zeros of the Bessel function of the first kind. To simplify the expression, the product of a1, b2, and c2 in the general solution can be reduced to a single coefficient.

[0162] Based on the analysis, the scalar magnetic potential expressions for the permanent magnet domain and the air gap domain can be simplified as follows:

[0163]

[0164]

[0165] In the formula,

[0166] p is the number of pole pairs of the motor, J np (i nk r) is a Bessel function of the first kind with order np.

[0167] Based on the scalar magnetic potential of the air gap region obtained from S1.8, the axial magnetic flux density, tangential magnetic flux density, and radial magnetic flux density in the air gap region without slots can be calculated using S1.9.

[0168]

[0169]

[0170]

[0171] In the formula, J np-1 (i nk r) is a Bessel function of the first kind with order np-1, J np+1 (i nk r) is a Bessel function of the first kind with order np+1.

[0172] S2: Solve for the complex relative permeability function using the conformal mapping method, and multiply the slotless air gap magnetic field by the complex relative permeability function to obtain the slotted air gap magnetic field:

[0173] S2.1 Using the conformal mapping method to calculate the complex relative permeability function, it can be expressed as:

[0174] λ(r,θ,z)=λ a (r,θ,z)+iλ b (r,θ,z)

[0175] In the formula, λ a (r,θ,z) is the real part of the complex relative permeability, λ b (r,θ,z) represents the imaginary part of the relative magnetic permeability of the complex number, and i is the imaginary symbol.

[0176] S2.2 Expand the complex relative permeability function into a Fourier series form, as shown in the following expression:

[0177]

[0178]

[0179] In the formula, λ aro (r,z) represents the constant term in the real part expansion of the relative permeability function, λ arm (r,z) are the coefficients of the pole number term in the real part expansion of the relative permeability function, λ brm (r,z) are the coefficients of the pole number terms in the imaginary part expansion of the relative permeability function.

[0180] S2.4 Calculate the slotted axial and tangential air gap magnetic flux density of the motor based on the slotless air gap magnetic flux density obtained in S1.9 and the complex relative permeability function obtained in S2.2:

[0181] B gzs (r,θ,z)=B gzw (r,θ,z)×λ a (r,θ,z)+Bgθw (r,θ,z)×λ b (r,θ,z)

[0182] B gθs (r,θ,z)=B gθw (r,θ,z)×λ a (r,θ,z)-B gzw (r,θ,z)×λ b (r,θ,z)

[0183] S3: Based on the magnetic circuit characteristics of the stator-less modular axial flux motor, its equivalent magnetic circuit model is established:

[0184] S3.1 Because axial flux motors have additional leakage magnetic paths at the inner and outer diameters, the motor is divided into inner diameter domain, center domain, and outer diameter domain, and magnetic circuit models are established separately for each domain, such as... Figure 3 As shown, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron. In this example, 65mm≤r≤70mm is the inner diameter range of the motor, 70mm≤r≤90mm is the center diameter range of the motor, and 90mm≤r≤95mm is the outer diameter range of the motor.

[0185] S3.2 analyzes the magnetic circuit characteristics of a stator-less, yoke-less modular axial flux motor. In the motor's central domain, the magnetic circuit mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnets, and the self-leakage magnetic circuit of the permanent magnets, such as... Figure 4 As shown. 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron. In addition to the main magnetic circuits mentioned above, there are also additional leakage magnetic circuits at the inner and outer diameters of the motor, such as... Figure 5 As shown. 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, 5 is the lower rotor back iron, and 6 is the motor shaft.

[0186] S3.3 Based on the magnetic circuit characteristics of the central region of the stator-less modular axial flux motor, an equivalent magnetic circuit model is established, such as... Figure 6 As shown in the figure. Among them, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron.

[0187] S3.4 Based on the magnetic circuit characteristics of the inner diameter domain of the stator-less modular axial flux motor, an equivalent magnetic circuit model is established, such as... Figure 7 As shown in the figure. Among them, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron.

[0188] Based on the magnetic circuit characteristics of the outer diameter domain of a stator-less modular axial flux motor, an equivalent magnetic circuit model is established, such as... Figure 8As shown in the figure. Among them, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 4 is the lower permanent magnet, and 5 is the lower rotor back iron.

[0189] S3.5 In the main magnetic circuit of a stator-less modular axial flux motor, the permanent magnet reluctance R mi Equivalent air gap magnetic reluctance R g Equivalent stator reluctance R s and equivalent rotor reluctance R r The calculation formula is as follows:

[0190]

[0191]

[0192]

[0193]

[0194] In the formula, μ m h is the relative permeability of the permanent magnet. r and h s These are the rotor back iron thickness and stator thickness, respectively. m S g and S s These represent the cross-sectional areas of the permanent magnet, air gap, and stator in the region being sought, respectively. a For the polar distance, l r Let μ be the radial length of the region in question. r (B r ) and μ s (B s These are the relative permeability of the rotor and the relative permeability of the stator, respectively.

[0195] S3.6 In the permanent magnet self-leakage magnetic circuit of a stator-less modular axial flux motor, the equivalent air gap reluctance R gm and equivalent rotor reluctance R rm The calculation formula is as follows:

[0196]

[0197]

[0198] S3.7 In the leakage magnetic circuit between permanent magnet poles, the equivalent air gap magnetic reluctance R gmm and equivalent rotor reluctance R rmm The calculation formula is as follows:

[0199]

[0200]

[0201] In the formula, l o The distance between the two magnetic poles is denoted as .

[0202] In the leakage magnetic circuit at the inner diameter of S3.8, the equivalent air gap magnetic reluctance R of the inner diameter is... bgi Equivalent rotor reluctance R of inner diameter bri The calculation formula is as follows:

[0203]

[0204]

[0205] In the formula, l ri This is the radial length of the rotor within the inner diameter region.

[0206] In the leakage magnetic circuit at the outer diameter, the equivalent air gap magnetic reluctance R at the outer diameter is... bgo Equivalent rotor reluctance R to outer diameter bro The calculation formula is as follows:

[0207]

[0208] In the formula, l ro The radial length of the rotor in the outer diameter region.

[0209] S3.9 Within the motor's central domain, the formulas for calculating the unloaded axial air gap magnetic flux density and the unloaded tangential air gap magnetic flux density, after considering leakage flux and magnetic saturation effects, based on the equivalent magnetic circuit model of the central domain, are as follows:

[0210]

[0211]

[0212] In the formula, α in The saturation leakage flux coefficient of the central domain is...

[0213] S3.10 Within the inner diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the inner diameter region:

[0214]

[0215]

[0216] In the formula, α ri The saturation leakage flux coefficient of the inner diameter domain is...

[0217]

[0218] Within the outer diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the outer diameter region:

[0219]

[0220]

[0221] In the formula, α ro The saturation leakage flux coefficient in the outer diameter domain is...

[0222]

[0223] S3.11: To analyze the impact of motor magnetic saturation on air gap flux density, the saturation coefficient is calculated iteratively. First, the air gap flux density is used as the system input to calculate the stator and rotor flux densities. Then, based on the stator and rotor flux density values ​​and the BH curve, the corresponding permeability of each part of the magnetic circuit is calculated, and the saturation coefficient is solved. Finally, the iteration is complete when the error between two adjacent calculations is less than 1.0%. The iterative flowchart of the magnetic circuit model is shown below. Figure 9 As shown.

[0224] S4: An analytical model of the unloaded air gap magnetic field of the axial flux motor is established based on the slotless unloaded air gap magnetic field, the complex relative permeability function, and the equivalent magnetic circuit model.

[0225] S4.1 Based on the above steps, the mathematical expressions for the axial no-load air gap magnetic flux density and tangential no-load air gap magnetic flux density of the stator yoke-less modular axial flux motor, considering curvature effect, slotting effect, leakage magnetic flux effect and magnetic saturation effect, are as follows:

[0226]

[0227]

[0228] In the formula, α b S is the saturation leakage flux coefficient of the motor. The saturation leakage flux coefficient of different regions of the motor is calculated by S3.

[0229] S4.2: Based on the mathematical expressions, the main spatial orders of the axial unloaded air gap magnetic flux density and the tangential unloaded air gap magnetic flux density are shown in Table 2. The main spatial orders of the axial unloaded air gap magnetic flux density and the tangential unloaded air gap magnetic flux density are both np and np±mq.

[0230] Table 2 Spatial order composition of air gap magnetic field

[0231]

[0232] S5: Calculation of no-load back electromotive force and cogging torque of motor based on air gap magnetic field:

[0233] S5.1 Number of windings per phase n of the motor in this paper phase The number of turns is 6, and the number of winding turns is N. c The value is 54. An analytical model of the motor's no-load back electromotive force is established based on the air gap magnetic field obtained in S4.1.

[0234]

[0235] In the formula, ψ phase The corresponding air gap flux linkage, α phase_j Let α be the starting angle of the j-th winding in the desired phase. τ This refers to the span angle of a single winding.

[0236] S5.2 An analytical model of the motor's no-load cogging torque is established using the Maxwell stress tensor method:

[0237]

[0238] In the formula, l is the radial length of the motor.

[0239] The stator-less modular axial flux motor shown in Table 1 was calculated using the finite element method and the method of this invention (analytical method). The magnetic field test experiment of the stator-less modular axial flux motor shown in Table 1 was carried out using a teslameter.

[0240] (1) The three-dimensional spatial distribution of the unloaded axial air gap magnetic field and the unloaded tangential magnetic field at the center of the 1 / 4 circumferential air gap as a function of the circumferential angle was calculated using the finite element method and analytical method. The calculation results are as follows: Figure 10 , Figure 11 , Figure 12 and Figure 13 As shown in the figure, the period and amplitude of the analytical results and the finite element results are in good agreement.

[0241] (2) The no-load back electromotive force and cogging torque of the motor were calculated using the finite element method and analytical method, and the results are as follows: Figure 14 and Figure 15 As shown, compared with the finite element method, the average errors of the analytical method are 5.88% and 7.55%, respectively.

[0242] (3) Direct test experiments were conducted on the unloaded air gap magnetic field of the stator-yoke-less modular axial flux motor to verify the correctness of the analytical model. The analytical calculation results, three-dimensional finite element simulation results, and experimental results for the unloaded axial and tangential air gap magnetic fields at the average radius are as follows: Figure 16 and Figure 17 As shown. The analytical calculation results, three-dimensional finite element simulation results, and experimental results for the order characteristics of the axial and tangential air gap magnetic fields under no-load conditions at the average radius are as follows. Figure 18 and Figure 19As shown in the figure. The analytical calculation results of the unloaded axial air gap magnetic field and the tangential air gap magnetic field are in good agreement with the three-dimensional finite element results and experimental results. The comparison of the errors, efficiency and resource consumption of the analytical method and the finite element method is shown in Table 3. The analytical results, finite element results and experimental results of the radial distribution of the axial air gap magnetic field of the motor are shown in the figure. Figure 20 As shown in the figure. Calculations show that the average errors between the analytical results and the finite element method and experimental results are 4.65% and 3.07%, respectively, with root mean square errors of 0.0513 and 0.0365, respectively. The calculation data indicates that the proposed method has high solution accuracy and fast computational efficiency. Under the same conditions, its computation time is less than 2% of that of the three-dimensional finite element method, and its memory usage is less than 0.5% of that of the three-dimensional finite element method.

[0243] Table 3 Comparison of error, efficiency, and resource consumption between analytical methods and finite element methods.

[0244]

Claims

1. A three-dimensional analytical method for the no-load air gap magnetic field of a stator-yoke-less modular axial flux motor, characterized in that, Includes the following steps: S1: Calculate the slotless air gap magnetic field of the motor by solving the Laplace equation in cylindrical coordinates; S2: Solve the complex relative permeability function using the conformal transformation method, and calculate the slotted empty air gap magnetic field based on the slotless empty air gap magnetic field and the complex relative permeability function. S3: Based on the magnetic circuit characteristics of the stator-less modular axial flux motor, establish its equivalent magnetic circuit model; S4: An analytical model of the no-load air gap magnetic field of the axial flux motor is established based on the slotless air gap magnetic field, the complex relative permeability function, and the equivalent magnetic circuit model. S3 includes the following steps: S3.1 Because the axial flux motor has additional leakage magnetic paths at the inner and outer diameters, the motor is divided into inner diameter domain, center domain, and outer diameter domain, and magnetic circuit models are established separately for each domain; the solution region is divided according to the motor radius, r i ≤r≤r ri The region is the inner diameter region of the motor, r ri ≤r≤r ro The region is the central area of ​​the motor, r ro The region ≤r≤r0 is the outer diameter region of the motor; where r ri For the inner diameter and outer diameter, r ro For the outer diameter domain inner diameter, r i r is the inner diameter of the motor. o The outer diameter of the motor; S3.2 Analysis of the magnetic circuit characteristics of the stator-yoke-less modular axial flux motor: In the motor's central domain, the motor's magnetic circuit mainly includes the main magnetic circuit, the permanent magnet inter-pole leakage magnetic circuit, and the permanent magnet self-leakage magnetic circuit; in the motor's inner diameter domain, the motor's magnetic circuit mainly includes the main magnetic circuit, the permanent magnet inter-pole leakage magnetic circuit, the permanent magnet self-leakage magnetic circuit, and the leakage magnetic circuit at the inner diameter; in the motor's outer diameter domain, the motor's magnetic circuit mainly includes the main magnetic circuit, the permanent magnet inter-pole leakage magnetic circuit, the permanent magnet self-leakage magnetic circuit, and the leakage magnetic circuit at the outer diameter. S3.3 Based on the magnetic circuit characteristics of the central domain of the stator-free modular axial flux motor, an equivalent magnetic circuit model is established. S3.4 Based on the magnetic circuit characteristics of the inner and outer diameter domains of the stator-free modular axial flux motor, an equivalent magnetic circuit model is established. S3.5 In the main magnetic circuit of a stator-less modular axial flux motor, the permanent magnet reluctance R mi Equivalent air gap magnetic reluctance R g Equivalent stator reluctance R s Equivalent rotor reluctance R r The calculation formula is as follows: In the formula, μ m h is the relative permeability of the permanent magnet. r and h s These are the rotor back iron thickness and stator thickness, respectively. m S g and S s These represent the cross-sectional areas of the permanent magnet, air gap, and stator in the region being sought, respectively. a For the polar distance, l r Let μ be the radial length of the region in question. r (B r ) and μ s (B s These are the relative permeability of the rotor and the relative permeability of the stator, respectively. S3.6 In the permanent magnet self-leakage magnetic circuit of a stator-less modular axial flux motor, the equivalent air gap reluctance R gm Equivalent rotor reluctance R rm The calculation formula is as follows: S3.7 In the leakage magnetic circuit between permanent magnet poles, the equivalent air gap magnetic reluctance R gmm Equivalent rotor reluctance R rmm The calculation formula is as follows: In the formula, l o The distance between the two magnetic poles; In the leakage magnetic circuit at the inner diameter of S3.8, the equivalent air gap magnetic reluctance R of the inner diameter is... bgi Equivalent rotor reluctance R of inner diameter bri The calculation formula is as follows: In the formula, l ri The radial length of the rotor within the inner diameter region; In the leakage magnetic circuit at the outer diameter, the equivalent air gap magnetic reluctance R at the outer diameter is... bgo Equivalent rotor reluctance R of outer diameter bro The calculation formula is as follows: In the formula, l ro The radial length of the rotor in the outer diameter region; S3.9 Within the motor's central domain, the formulas for calculating the unloaded axial air gap magnetic flux density and the unloaded tangential air gap magnetic flux density, after considering leakage flux and magnetic saturation effects, based on the equivalent magnetic circuit model of the central domain, are as follows: In the formula, α in The saturation leakage flux coefficient of the central domain is... ; S3.10 Within the inner diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the inner diameter region: In the formula, α ri The saturation leakage flux coefficient of the inner diameter domain is... Within the outer diameter region of the motor, the calculation formulas for the no-load axial air gap magnetic flux density and the no-load tangential air gap magnetic flux density, after considering leakage flux effect and magnetic saturation effect, are as follows, based on the equivalent magnetic circuit model of the outer diameter region: In the formula, α ro The saturation leakage flux coefficient in the outer diameter domain is... ； S3.11: Based on the no-load air gap magnetic flux density and the stator and rotor BH curves, the saturation leakage flux coefficient is calculated iteratively, and the influence of motor magnetic saturation effect and leakage flux effect on no-load air gap magnetic flux density is analyzed.

2. The three-dimensional analytical method for the no-load air gap magnetic field of a stator-less modular axial flux motor according to claim 1, characterized in that, S1 includes the following steps: S1.1 Based on the structural characteristics of the stator-yoke-free modular axial flux motor, a cylindrical coordinate system is established with the center of the lower rotor back iron as the coordinate origin; the motor axial, radial, and circumferential directions are respectively taken as the z-direction, r-direction, and θ-direction; a model of the stator-yoke-free modular axial flux motor is established in the cylindrical coordinate system. S1.2 The permanent magnet domain and air gap domain in the stator-yoke-less modular axial flux motor model are selected as the solution domains for the motor; the permanent magnet domain is 0≤z≤h. m The region has an air gap of h. m ≤z≤h m The region of +g; where h m Where is the thickness of the permanent magnet, and g is the air gap length; The permanent magnets of the S1.3 stator-less modular axial flux motor are sector-shaped and axially magnetized, with alternating N and S poles. An expression for magnetization intensity is established based on the permanent magnet pole parameters: In the formula, B r θ is the remanence of the permanent magnet, μ0 is the permeability of free space, and θ is the magnetic remanence of the permanent magnet. p θ is the angle occupied by the magnetic poles. o Let T be the angle between the magnetic poles, and T be the period of the magnetization, T = 2(θ). p +θ o ); S1.4 In both the permanent magnet domain and the air gap domain, the scalar magnetic potential satisfies the Laplace equation: In the formula, φ is the scalar magnetic potential. pm φ is the scalar magnetic potential of the permanent magnet domain. g For the scalar magnetic potential of the air gap domain; S1.5 Solving the general solution of the Laplace equation using the method of separation of variables to calculate the scalar magnetic potential: In the formula, J n (ir) is the nth-order Bessel function of the first kind, Y n (ir) is an nth-order Bessel function of the second kind, i n a1, a2, b1, b2, c1, and c2 are terms with undetermined coefficients; S1.6 Determine the boundary conditions of the solution domain: (1) At z=0 and z=h m The scalar magnetic potential at the +g plane is zero, as shown in the following expression: (2) At the interface between the permanent magnet domain and the air gap domain, z=h m In the plane, the axial magnetic flux density is continuous with the scalar magnetic potential, and the specific expression is as follows: (3) When r approaches 0, the scalar magnetic potential of the motor is a finite value; when r approaches the boundary outer diameter r p At that time, the scalar magnetic potential is zero; the specific expression is as follows: In the formula, k finite It is a finite value; S1.7 expands the magnetization into a Fourier-Bessel series, as shown in the following expansion: In the formula, M nk For Fourier and Bessel coefficients; i nk The coefficient term can be calculated based on boundary conditions; Based on the general solution and boundary conditions of the equation, the scalar magnetic potential expressions for the permanent magnet domain and the air gap domain can be obtained in S1.8: In the formula, and The constants are calculated based on the boundary conditions, where p is the number of pole pairs of the motor, and J is the constant. np (i nk r) is a Bessel function of the first kind with order np; S1.9 Based on the scalar magnetic potential of the air gap domain, the slotless no-load air gap magnetic flux density of the motor in the axial, tangential, and radial directions can be calculated: In the formula, J np-1 (i nk r) is a Bessel function of the first kind with order np-1, J np+1 (i nk r) is a Bessel function of the first kind with order np+1.

3. The three-dimensional analytical method for the no-load air gap magnetic field of a stator-less modular axial flux motor according to claim 1, characterized in that, S2 includes the following steps: S2.1 Using the conformal mapping method to calculate the complex relative permeability function, it can be expressed as: In the formula, λ a (r,θ,z) is the real part of the complex relative permeability, λ b (r,θ,z) represents the imaginary part of the complex relative permeability, where i is the imaginary sign; S2.2 Expand the complex relative permeability function into a Fourier series form In the formula, λ aro (r,z) represents the constant term in the real part expansion of the relative permeability function, λ arm (r,z) are the coefficients of the pole number term in the real part expansion of the relative permeability function; λ brm (r,z) are the coefficients of the pole number terms in the imaginary part expansion of the relative permeability function; S2.4 Calculate the slotted air gap magnetic flux density of the motor based on the obtained slotless no-load air gap magnetic flux density and complex relative permeability function. The slotted no-load air gap magnetic flux densities of the motor in the axial and radial directions are: 。 4. The three-dimensional analytical method for the no-load air gap magnetic field of a stator-less modular axial flux motor according to claim 1, characterized in that, S4 includes the following steps: S4.1 Based on the above steps, the mathematical expressions for the axial no-load air gap magnetic flux density and tangential no-load air gap magnetic flux density of the stator yoke-less modular axial flux motor, considering curvature effect, slotting effect, leakage magnetic flux effect and magnetic saturation effect, are as follows: In the formula, α b The saturation leakage flux coefficient of the motor is calculated from S3. S4.2: Analyze the main spatial orders of the axial no-load air gap magnetic flux density and the tangential no-load air gap magnetic flux density of the motor based on mathematical expressions.

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