Three-dimensional analytical method of load air-gap magnetic field for stator without magnetic yoke modular axial flux motor

Abstract

The application discloses a kind of three-dimensional analytical method of stator non-magnetic yoke modular axial flux motor load air gap magnetic field, it is related to the calculation method of axial flux motor load air gap magnetic field, comprising the following steps: S1: division armature reaction solving region, and establish the equivalent current surface model of armature winding current;S2: Laplace equation is solved in armature reaction solving domain, obtain the expression of armature reaction three-dimensional 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, calculate the motor load air gap magnetic field after considering magnetic saturation effect and leakage effect;The method comprehensively considers the slotting effect, curvature effect, leakage effect and magnetic saturation effect of stator non-magnetic yoke modular axial flux motor, and the accuracy is higher;Using the method can solve the armature reaction axial, tangential and radial air gap magnetic field of stator non-magnetic yoke modular axial flux motor;High calculation efficiency and less memory occupation.

Description

TECHNICAL FIELD

[0001] The present application relates to a kind of axial flux motor load air gap magnetic field calculation method, specifically a kind of three-dimensional analytical method of stator non-magnetic yoke modular axial flux motor load air gap magnetic field considering curvature effect, leakage and magnetic saturation. BACKGROUND

[0002] Stator non-magnetic yoke modular axial flux motor has compact structure, high power density and torque density, and has broad application prospect in new energy and other related fields. Motor armature reaction air gap magnetic field is one of the important indicators for evaluating motor performance, and based on armature reaction air gap magnetic field, key parameters such as motor output torque, efficiency and loss can be analyzed. Due to the geometric structural characteristics of axial flux motor, the magnetic field distribution is uneven in three-dimensional space, which increases the difficulty of its magnetic field calculation. In order to accurately evaluate the running state of stator non-magnetic yoke modular axial flux motor, it is crucial to accurately calculate its armature reaction air gap magnetic field.

[0003] The main methods for calculating the air-gap magnetic field of the armature reaction of axial flux motors include the finite element method and the analytical method. The three-dimensional finite element method calculates the magnetic field distribution of the axial flux motor by discretizing the model into several units and solving Maxwell's equations for each unit. To ensure the accuracy of the three-dimensional finite element model, precise meshing is required, which significantly increases the calculation time of the model, leading to a sharp increase in the time cost of the design and optimization of axial flux motors. The analytical methods for the magnetic field of axial flux motors mainly include the sub-domain method, the relative permeance method, and the equivalent magnetic circuit method. The relative permeance method uses the conformal transformation method to solve the complex relative permeance function and analyzes the effect of stator slotting on the air-gap magnetic field. Patent CN113868929A calculates the air-gap magnetic field of an external rotor axial flux motor using the relative permeance function and solves the electromagnetic force wave based on the air-gap magnetic field to analyze the vibration and noise of the motor. The sub-domain method divides the motor into different regions and solves the Poisson equation in each sub-domain to obtain the air-gap magnetic field of the motor. Patents CN117477882A and CN117477881A divide the area between the inner and outer radii of the rotor, the area between the upper and lower surfaces of the rotor in the axial direction, and the area between the air-gap and the stator interface into three annular cylindrical regions: the internal sub-domain, the middle sub-domain, and the external sub-domain. In the three-dimensional cylindrical coordinate system, the Poisson equation and the Laplace equation are solved in the above sub-domains, and the equivalent air-gap is calculated by introducing the Carter coefficient to analyze the effect of stator slotting on the magnetic flux density. However, the Carter coefficient cannot analyze the effect of slotting on the magnetic field distribution of the motor, cannot accurately consider the influence of complex slotting, and may introduce large errors. The equivalent magnetic circuit method converts the complex magnetic field problem into a magnetic circuit problem by equivalent the motor components to corresponding magnetic sources or magnetic resistances to calculate the magnetic field distribution of the motor. Patent CN116796675A considers the leakage magnetic phenomenon of the axial motor and establishes an equivalent magnetic circuit model of the stator without a magnetic yoke. According to the model, the motor is optimized for multiple targets. This model can analyze the leakage magnetic phenomenon of the motor, but different magnetic circuit models need to be established according to the relative position of the stator and the rotor. With the increase in the number of nodes of the equivalent magnetic circuit model, the matrix dimension and the solving time of the model increase sharply. Patent CN116244847A proposes a three-dimensional analytical method for calculating the no-load air-gap magnetic field of a stator without a magnetic yoke modular axial flux motor. It discloses a method for calculating the no-load axial air-gap magnetic field B pmz (r,θ,z) and the tangential air-gap magnetic field B pmθ (r,θ,z) generated by the permanent magnet of the motor, but it has not solved the three-dimensional air-gap magnetic field generated by the armature reaction of the motor.

[0004] In summary, the existing analytical methods have the following problems when calculating the load air-gap magnetic field of the stator without a magnetic yoke modular axial flux motor:

[0005] (1) The slotting effect, curvature effect, leakage magnetic effect, and magnetic saturation effect of the stator without a magnetic yoke modular axial flux motor are not fully considered, which may affect the accuracy of the analytical results.

[0006] (2) The above analysis methods cannot calculate the three-dimensional air gap magnetic field generated by the stator yokeless modular axial flux motor armature reaction. SUMMARY

[0007] The purpose of the present application is to solve the above-mentioned deficiencies of the prior art, and provide a three-dimensional analysis method for the load air gap magnetic field of a stator yokeless modular axial flux motor, which can be used to improve the design and optimization efficiency of the motor.

[0008] The technical solution adopted by the present application to solve the above-mentioned deficiencies of the prior art is:

[0009] A three-dimensional analysis method for the load air gap magnetic field of a stator yokeless modular axial flux motor, characterized in that it comprises the following steps:

[0010] S1: dividing the armature reaction solving region and establishing an equivalent current surface model of the armature winding current;

[0011] S2: solving Laplace equation in the armature reaction solving domain to obtain the expression of the three-dimensional air gap magnetic field of the armature reaction;

[0012] S3: establishing an equivalent magnetic circuit model according to the magnetic circuit characteristics of the stator yokeless modular axial flux motor, and calculating the motor load air gap magnetic field considering the magnetic saturation effect and the leakage magnetic effect.

[0013] The specific steps of S1 are as follows:

[0014] S1.1: Establish a cylindrical coordinate system according to the structural characteristics of the axial flux motor, with the center of the rotor back iron as the coordinate origin, and the motor axial direction, radial direction and circumferential direction as the z-axis, r-axis and θ-axis respectively.

[0015] S1.2: In the calculation of the armature reaction magnetic field, the permanent magnet is equivalent to air; the area between the interface of the rotor and the permanent magnet and the interface of the air gap and the stator is defined as the armature reaction solving domain, which is a circular annular region of 0≤z≤h m +g, r i ≤r≤r o , wherein h m is the thickness of the permanent magnet, g is the air gap length, r i is the inner diameter of the motor, r o is the outer diameter of the motor; z is the z-axis coordinate, and r is the r-axis coordinate.

[0016] S1.3: Determine the direction of the three-phase winding current; (the direction of the three-phase winding current can be determined according to the winding method of the motor winding).

[0017] S1.4: Establish a current surface model generated by the armature winding:

[0018] ,

[0019] where N coil is the number of turns, ± denotes the current direction, i is the current of the slot corresponding phase, l so is the slot opening width.

[0020] S1.5: Fourier-Bessel decomposition is made to the function f(x, y, z) as follows:

[0021] ,

[0022] where A nk and B nk are Fourier-Bessel coefficients, J n is the first kind Bessel function of order n, i nk is the undetermined coefficient.

[0023] The specific steps of S2 are as follows:

[0024] S2.1: In the armature solution domain, the scalar magnetic potential φ a satisfies the Laplace equation;

[0025] .

[0026] S2.2: The scalar magnetic potential general solution of the armature domain is calculated by using the separation of variables method,

[0027] ,

[0028] where a1, a2, b1, b2, c1 and c2 are undetermined coefficient terms, Y n is the second kind Bessel function of order n;

[0029] The undetermined coefficients are calculated according to the boundary conditions of the armature solution domain:

[0030] (1) When r→0, the scalar magnetic potential tends to 0, because when r→0 , then a2=0;

[0031] (2) The scalar magnetic potential is zero at the z=0 plane, , and c2=0 is calculated;

[0032] (3) The scalar magnetic potential at the z=h m +g plane is equal to the current density J, , and the following is obtained:

[0033] ,

[0034] , ​

[0035] (4) To meet the needs of solving, the scalar magnetic potential is 0 at the outer radius:

[0036] ,

[0037] where, is the kth zero point of J n (r, θ, z), r b is the outer radius;

[0038] 5) According to the above calculation process, the expression of the scalar magnetic potential general solution of the armature domain is:

[0039] .

[0040] S2.3 The axial air gap magnetic flux density, tangential air gap magnetic flux density and radial air gap magnetic flux density of the armature reaction are respectively:

[0041] ,

[0042] ,

[0043] ,

[0044] where, μ0 is the vacuum permeability, J n-1 is the first kind of Bessel function of order n-1, J n+1 is the first kind of Bessel function of order n+1.

[0045] The specific steps of S3 are as follows:

[0046] S3.1: Calculate the no-load axial air gap magnetic field B pmz (r, θ, z), tangential air gap magnetic field B pmθ (r, θ, z) and radial air gap magnetic field B pmr (r, θ, z) generated by the permanent magnet of the motor.

[0047] S3.2: Since the stator without magnetic yoke modular axial flux motor has edge leakage, the motor is divided into inner diameter domain, middle domain and outer diameter domain when establishing the equivalent magnetic circuit model; define the outer diameter of the inner diameter domain and the inner diameter of the outer diameter domain as r ri and r ro ; according to the radius division of the motor, the region of r i ≤r≤r ri is the inner diameter domain of the motor, the region of r ri ≤r≤r ro is the center domain of the motor, and the region of r ro ≤r≤r o is the outer diameter domain of the motor.

[0048] S3.3: Analyze the magnetic flux path characteristics of the motor, the motor magnetic circuit mainly includes the main magnetic circuit, the permanent magnet inter-pole leakage magnetic circuit, the permanent magnet self-leakage magnetic circuit, the inner diameter leakage magnetic circuit (inner diameter domain) and the outer diameter leakage magnetic circuit (outer diameter domain); according to the magnetic circuit of the above-mentioned stator non-magnetic yoke modular axial flux motor, an equivalent magnetic circuit model is established.

[0049] S3.4: Based on the air gap magnetic field, the permanent magnet magnetic flux source in the equivalent magnetic circuit model is calculated m and the armature magnetic flux source a :

[0050] ,

[0051] .

[0052] S3.5: Calculate the magnetic resistance in each magnetic circuit in the rotor equivalent magnetic circuit model. The calculation formulas of the stator non-magnetic yoke modular axial flux permanent magnet magnetic resistance R mi , the equivalent air gap magnetic resistance R g , the equivalent stator magnetic resistance R s , the equivalent rotor magnetic resistance R r , the permanent magnet self-leakage magnetic equivalent air gap magnetic resistance R gm , the permanent magnet self-leakage magnetic equivalent rotor magnetic resistance R rm , the permanent magnet inter-pole leakage magnetic equivalent air gap magnetic resistance R gmm , the permanent magnet inter-pole leakage magnetic circuit equivalent rotor magnetic resistance R rmm , the inner diameter equivalent air gap magnetic resistance R bgi , the inner diameter equivalent rotor magnetic resistance R bri , the outer diameter equivalent air gap magnetic resistance R bgo and the outer diameter equivalent rotor magnetic resistance R bro are as follows:

[0053] ,

[0054] ,

[0055] ,

[0056] ,

[0057] ,

[0058] ,

[0059] ,

[0060] ,

[0061] ,

[0062] ,

[0063] ,

[0064] .

[0065] where μ m is the relative permeability of the permanent magnet, h r and h s are the rotor back iron thickness and the stator thickness, S m , S g and S s are the cross-sectional area of the permanent magnet, the air gap and the stator in the region of interest, l a is the pole pitch, l r is the radial length of the region of interest, μ r (B r ) and μ s (B s ) are the relative permeability of the rotor and the stator, l o is the distance between two magnetic poles, l ri is the rotor radial length of the inner diameter region, l ro is the rotor radial length of the outer diameter region.

[0066] S3.6: Calculate the saturation leakage coefficient by iteration method. First, input the air gap flux density, calculate the permanent magnet flux source and armature flux source, then calculate the corresponding permeability of each part of the magnetic circuit according to the stator flux density, rotor flux density and B-H curve, solve the saturation leakage coefficient, finally, when the error of two adjacent calculations is less than 1.0%, the iteration is completed, and the values of saturation leakage coefficients a and b are obtained.

[0067] The calculation formula of saturation leakage coefficient a and b parameters is as follows:

[0068] ,

[0069] ,

[0070] where: , , , , , , , .

[0071] S3.7: Based on the equivalent magnetic circuit model, the load air gap flux density expression after considering leakage and magnetic saturation is as follows:

[0072] ,

[0073] ,

[0074] ,

[0075] According to the load air gap magnetic flux obtained above, the electromagnetic torque is calculated by using Maxwell stress tensor method:

[0076] ,

[0077] In the formula, l is the radial length of the motor.

[0078] Compared with the prior art, the present application has the following advantages:

[0079] 1) The method can comprehensively consider the slotting effect, curvature effect, leakage effect and magnetic saturation effect of the stator yokeless modular axial flux motor, and has high accuracy.

[0080] 2) The method can be used to solve the armature reaction axial, tangential and radial air gap magnetic field of the stator yokeless modular axial flux motor.

[0081] 3) The method has high calculation efficiency and occupies less memory. Under the same conditions, the calculation time is about 1.5% of that of three-dimensional finite element. BRIEF DESCRIPTION OF DRAWINGS

[0082] Figure 1 Stator yokeless modular axial flux motor cylindrical coordinate system.

[0083] Figure 2 is a schematic diagram of the current direction of the stator yokeless modular axial flux motor.

[0084] Figure 3 is an equivalent magnetic circuit model diagram of the stator yokeless modular axial flux motor.

[0085] Figure 4 is a magnetic saturation iteration flowchart.

[0086] Figure 5 is a comparison diagram of the axial air gap magnetic field obtained by using the present application and the finite element method.

[0087] Figure 6 is a comparison diagram of the tangential air gap magnetic field obtained by using the present application and the finite element method.

[0088] Figure 7 is a comparison diagram of the radial air gap magnetic field obtained by using the present application and the finite element method.

[0089] Figure 8 is a comparison diagram of the electromagnetic torque obtained by using the present application and the finite element method.

[0090] Reference: 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator core, 4 is the armature winding, 5 is the lower permanent magnet, and 6 is the lower rotor back iron. DETAILED DESCRIPTION

[0091] The application will be described in detail below in combination with the drawings and specific examples.

[0092] The application takes a stator non-magnetic yoke modular axial flux motor as an example. The motor adopts a block stator structure. The main parameters of the motor are shown in Table 1:

[0093] Table 1 Main parameters of the motor

[0094] .

[0095] S1: Divide the solution domain of the motor magnetic field, and establish the current surface model of the motor;

[0096] S1.1: Establish a cylindrical coordinate system in combination with the structure of the axial flux motor, set the center of the rotor back iron as the coordinate origin, and set the axial, radial and circumferential directions of the motor as the z-axis, r-axis and theta-axis respectively. The mathematical model of the stator non-magnetic yoke modular axial flux motor is established in the cylindrical coordinate system, as shown in Figure 1 , wherein 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator core, 4 is the armature winding, 5 is the lower permanent magnet, and 6 is the lower rotor back iron.

[0097] S1.2: When calculating the armature reaction magnetic field, the permanent magnet needs to be equivalent to air. The armature winding is equivalent to the current surface of the motor slot opening, and the area between the rotor and the permanent magnet interface and the air gap and the stator interface is defined as the armature reaction solution domain. The circular annular area of 0≤z≤6.5mm, 185mm≤r≤275mm is the armature reaction solution domain.

[0098] S1.3: Determine the winding current direction, the current direction inside the three-phase winding of the example axial flux motor of the application is as shown in Figure 2 .

[0099] S1.4: Establish the current surface model generated by the armature winding:

[0100] ,

[0101] In the formula, N coil is the number of turns, ± indicates the current direction, i a is the A-phase winding current, i is the current corresponding to the phase of the slot opening, and l so is the stator slot opening width.

[0102] S1.5: Fourier-Bessel decomposition is made on the current surface model for solving the equation:

[0103] ,

[0104] In the formula, A nk and B nk J represents the Fourier-Bessel coefficients. n Let i be a Bessel function of the first kind with order n. nk The coefficient is denoted by and can be calculated based on boundary conditions.

[0105] S2: Solve the Laplace equation within the armature reaction solution domain to obtain expressions for the axial, tangential, and radial air gap magnetic fields of the armature reaction;

[0106] S2.1: Within the armature solution domain, the scalar magnetic potential φ a Satisfies the Laplace equation:

[0107] .

[0108] S2.2: The scalar magnetic flux solution of the armature domain is calculated using the method of separation of variables:

[0109] ,

[0110] In the formula, a1, a2, b1, b2, c1, and c2 are terms with undetermined coefficients, Y n It is a Bessel function of the second kind with order n.

[0111] S2.3: Calculate the undetermined coefficients based on the boundary conditions of the armature solution domain determined by physics.

[0112] (6) When r→0, the scalar magnetic potential tends to 0. Because when r→0 If a2 = 0, then a2 = 0;

[0113] (7) The scalar magnetic potential at the z=0 plane is zero. The calculation yields c2=0;

[0114] (8) z=h m The scalar magnetic potential at the +g plane is equal to the current density J. Substituting, we get:

[0115] ,

[0116] ,

[0117] (1) To meet the solution requirements, the scalar magnetic potential at the calculated radius outside the motor is 0: (Supplement the ink calculation formula)

[0118] ,

[0119] In the formula, For J n The kth zero, r b For the outer radius calculation, r in this paper b =1.5r o ;

[0120] (2) Based on the above calculation process, the expression for the scalar magnetic potential flux solution of the armature domain is:

[0121] .

[0122] S2.4: The axial air gap magnetic flux density, tangential air gap magnetic flux density, and radial air gap magnetic flux density of the armature reaction are respectively:

[0123] ,

[0124] ,

[0125] ,

[0126] In the formula, μ0 is the free permeability, J n-1 It is a Bessel function of the first kind with order n-1.

[0127] S3: Analyze the flux path of the stator-less modular axial flux motor, establish its equivalent magnetic circuit model, and analyze the effects of magnetic saturation effect and leakage flux effect on the load air gap magnetic field of the axial flux motor.

[0128] S3.1: Calculate the no-load axial air gap magnetic field B generated by the permanent magnet of the motor. pmz (r,θ,z) and tangential air gap magnetic field B pmθ (r,θ,z): The calculation process is described in detail in "Patent CN116244847A, A Three-Dimensional Analytical Method for the No-Load Air Gap Magnetic Field of a Stator-less Modular Axial Flux Motor".

[0129] S3.2: Due to edge leakage flux in the stator-less modular axial flux motor, the motor is divided into an inner diameter domain, an intermediate domain, and an outer diameter domain when establishing the equivalent magnetic circuit model. The outer diameter of the inner diameter domain and the inner diameter of the outer diameter domain are defined as r, respectively. ri and r ro The solution region is divided according to the motor radius. The region of 185mm≤r≤190mm is the inner diameter region of the motor, the region of 190mm≤r≤270mm is the center region of the motor, and the region of 270mm≤r≤275mm is the outer diameter region of the motor.

[0130] S3.3: Analyze the characteristics of the magnetic flux path of the motor. The motor magnetic circuit mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnet, the self-leakage magnetic circuit of the permanent magnet, the leakage magnetic circuit at the inner diameter, and the leakage magnetic circuit at the outer diameter. Based on the magnetic circuit characteristics of the stator-yoke-less modular axial flux motor, the equivalent magnetic circuit method is used to equate the permanent magnet and armature winding as magnetic flux sources, and the stator, rotor, permanent magnet, and air gap in different magnetic circuits are equated as magnetic reluctance, establishing their equivalent magnetic circuit models, such as... Figure 3 As shown in the figure. Among them, 1 is the upper rotor back iron, 2 is the upper permanent magnet, 3 is the stator, 5 is the lower permanent magnet, and 6 is the lower rotor back iron.

[0131] S3.4: Calculate the permanent magnet flux source φ in the equivalent magnetic circuit model based on the air gap magnetic field. m and armature flux source φ a :

[0132] ,

[0133] .

[0134] S3.5: Calculate the magnetic reluctance in each magnetic circuit of the rotor equivalent magnetic circuit model; the permanent magnet reluctance R of the stator without yoke modular axial flux. mi Equivalent air gap magnetic reluctance R g Equivalent stator reluctance R s Equivalent rotor reluctance R r The equivalent air gap reluctance R of the permanent magnet's self-leakage magnetic field gm The equivalent rotor reluctance R of the permanent magnet self-leakage magnetic field rm Equivalent air gap reluctance R between permanent magnet poles gmm Equivalent rotor reluctance R of permanent magnet inter-pole leakage magnetic circuit rmm Inner diameter equivalent air gap magnetoresistance R bgi Equivalent rotor reluctance R of inner diameter bri Outer diameter equivalent air gap magnetoresistance R bgo Equivalent rotor reluctance R to outer diameter bro The calculation formula is as follows:

[0135] ,

[0136] ,

[0137] ,

[0138] ,

[0139] ,

[0140] ,

[0141] ,

[0142] ,

[0143] ,

[0144] ,

[0145] ,

[0146] ,

[0147] 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 ) represent the relative permeability of the rotor and the relative permeability of the stator, respectively. o l is the distance between the two magnetic poles. ri l is the radial length of the rotor in the inner diameter region. ro This is the radial length of the rotor in the outer diameter region.

[0148] S3.6: Based on the equivalent magnetic circuit model, the expression for the load air gap magnetic flux density considering leakage flux and magnetic saturation is as follows:

[0149] ,

[0150] ,

[0151] The formulas for calculating parameters a and b are as follows:

[0152] ,

[0153] ,

[0154] In the formula: , , , , , , , .

[0155] S3.7: Iterative analysis of the influence of motor magnetic saturation and leakage flux on the load air gap magnetic field through saturation leakage flux coefficient, such as... Figure 4 As shown. First, the air gap magnetic flux density is used as the permanent magnet flux source and armature flux source for system input. Then, the stator and rotor magnetic flux density values ​​are calculated. Based on the BH curve, the corresponding permeability of each part of the magnetic circuit is solved, and the saturation leakage coefficients a and b are solved. Finally, the iteration is completed when the error between two adjacent calculations is less than 1.0%.

[0156] Calculation of electromagnetic torque using Maxwell's stress tensor method:

[0157] In the formula, l is the radial length of the motor.

[0158] Method Validation

[0159] The three-dimensional simulation results and analytical results of the stator-yoke-less modular axial flux motor shown in Table 1 were obtained by using the three-dimensional finite element method and the method of the present invention.

[0160] (1) The axial, tangential, and radial air gap magnetic fields of the armature reaction at the center of the 1 / 2 circumferential air gap were calculated using the finite element method and analytical method, and the calculation results are as follows: Figure 5 , Figure 6 and Figure 7 As shown, compared with the finite element method results, the root mean square errors of the armature reaction axial, tangential, and radial air gap magnetic flux density obtained by the analytical method are 0.0742, 0.0345, and 0.002245, respectively. The results show that the period and amplitude of the analytical results agree well with those of the finite element method.

[0161] (2) The electromagnetic torque of the motor was calculated using the finite element method and analytical method. The calculation results are as follows: Figure 8 As shown in the figure. According to the calculation results, the period and amplitude of the analytical results and the finite element results are in good agreement.

[0162] (3) The time taken to calculate the electromagnetic torque of the motor by the finite element method and the analytical method is 28h47min5s and 25min38s respectively. By comparison, the analytical calculation time is only about 1.5% of the finite element calculation time.

Claims

1. A three-dimensional analytical method for the air gap magnetic field of a stator-yoke-less modular axial flux motor load, characterized in that, Includes the following steps: S1: Divide the armature reaction solution region and establish an equivalent current surface model of the armature winding current; S2: Solve the Laplace equation within the armature reaction solution domain to obtain the expression for the three-dimensional air gap magnetic field of the armature reaction; S3: Based on the magnetic circuit characteristics of the stator-free modular axial flux motor, establish its equivalent magnetic circuit model and calculate the air gap magnetic field of the motor load after considering magnetic saturation effect and leakage magnetic effect. The specific steps of S1 are as follows: S1.1: A cylindrical coordinate system is established based on the structural characteristics of the axial flux motor. The origin of the coordinate system is set at the center of the rotor back iron, and the axial, radial and circumferential directions of the motor are respectively used as the z-axis, r-axis and θ-axis. S1.2: When calculating the armature reaction magnetic field, the permanent magnet is treated as equivalent to air; the region between the rotor-permanent magnet interface and the air gap-stator interface is defined as the armature reaction solution domain, which is 0≤z≤h. m +g, r i ≤r≤r o A circular region, where h m where g is the thickness of the permanent magnet, g is the air gap length, and r is the thickness of the permanent magnet. i r is the inner diameter of the motor. o The outer diameter of the motor is represented by z; z is the z-axis coordinate and r is the r-axis coordinate. S1.3: Determine the direction of the three-phase winding current; S1.4: Establish the current surface model generated by the armature winding: ， In the formula, N coil The number of turns in the winding is given by , ± indicates the current direction, i is the current in the phase corresponding to the slot, and l is the current in the phase corresponding to the slot. so This refers to the width of the stator slot. S1.5: Will Perform Fourier-Bessel decomposition: ， In the formula, A nk and B nk J represents the Fourier-Bessel coefficients. n Let i be a Bessel function of the first kind with order n. nk These are coefficients to be determined.

2. The three-dimensional analytical method for the air gap magnetic field of a stator-less modular axial flux motor load according to claim 1, characterized in that, The specific steps of S2 are as follows: S2.1: Within the armature solution domain, the scalar magnetic potential φ a Satisfies the Laplace equation; ， S2.2: The scalar magnetic flux solution of the armature domain is calculated using the method of separation of variables. ， In the formula, a1, a2, b1, b2, c1, and c2 are terms with undetermined coefficients, Y n It is a Bessel function of the second kind with order n; Calculate the undetermined coefficients based on the boundary conditions of the armature solution domain: (1) When r→0, the scalar magnetic potential tends to 0, because when r→0 If a2 = 0, then a2 = 0; (2) The scalar magnetic potential at the z=0 plane is zero. The calculation yields c2=0; (3) z=h m The scalar magnetic potential at the +g plane is equal to the current density J. Substituting, we get: ， ， (4) To meet the solution requirements, the scalar magnetic potential at the calculation radius outside the motor is 0: ， In the formula, For J n The kth zero, r b The outer radius is used for calculation. (5) The expression for the scalar magnetic potential flux solution in the armature domain is: ； S2.3 Armature reaction axial air gap magnetic flux density, tangential air gap magnetic flux density and radial air gap magnetic flux density are respectively: ， ， ， In the formula, μ0 is the free permeability, J n-1 J is a Bessel function of the first kind with order n-1. n+1 It is a Bessel function of the first kind with order n+1.

3. The three-dimensional analytical method for the air gap magnetic field of a stator-less modular axial flux motor load according to claim 1, characterized in that, The specific steps of S3 are as follows: S3.1: Calculate the no-load axial air gap magnetic field B generated by the permanent magnet of the motor. pmz (r,θ,z), tangential air gap magnetic field B pmθ (r,θ,z) and radial air gap magnetic field B pmr (r,θ,z); S3.2: Due to edge leakage flux in the stator-less modular axial flux motor, the motor is divided into an inner diameter domain, an intermediate domain, and an outer diameter domain when establishing the equivalent magnetic circuit model; the outer diameter of the inner diameter domain and the inner diameter of the outer diameter domain are defined as r, respectively. ri and r ro 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 ≤r≤r o The region is the outer diameter area of ​​the motor; S3.3: Analyze the characteristics of the magnetic flux path of the motor. The magnetic circuit of the motor mainly includes the main magnetic circuit, the inter-pole leakage magnetic circuit of the permanent magnet, the self-leakage magnetic circuit of the permanent magnet, the leakage magnetic circuit at the inner diameter, and the leakage magnetic circuit at the outer diameter. Based on the magnetic circuit of the above-mentioned modular axial flux motor without stator yoke, establish its equivalent magnetic circuit model. S3.4: Calculate the permanent magnet flux source ϕ in the equivalent magnetic circuit model based on the air gap magnetic field. m and armature flux source ϕ a : ， ； S3.5: Calculate the magnetic reluctance in each magnetic circuit of the rotor equivalent magnetic circuit model: permanent magnet reluctance R of the axial magnetic flux of the stator without yoke modularization. mi Equivalent air gap magnetic reluctance R g Equivalent stator reluctance R s Equivalent rotor reluctance R r The equivalent air gap reluctance R of the permanent magnet's self-leakage magnetic field gm The equivalent rotor reluctance R of the permanent magnet self-leakage magnetic field rm Equivalent air gap reluctance R between permanent magnet poles gmm Equivalent rotor reluctance R of permanent magnet inter-pole leakage magnetic circuit rmm Inner diameter equivalent air gap magnetoresistance R bgi Equivalent rotor reluctance R of inner diameter bri Outer diameter equivalent air gap magnetoresistance R bgo Equivalent rotor reluctance R to outer diameter bro 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 ) represent the relative permeability of the rotor and the relative permeability of the stator, respectively. o l is the distance between the two magnetic poles. ri l is the radial length of the rotor in the inner diameter region. ro The radial length of the rotor in the outer diameter region; S3.6: Calculate the saturation leakage coefficient by iterative method: First, input the air gap magnetic flux density, calculate the permanent magnet flux source and armature flux source, then calculate the corresponding magnetic permeability of each part of the magnetic circuit based on the stator magnetic flux density, rotor magnetic flux density and BH curve, solve for the saturation leakage coefficient, and finally, when the error between two adjacent calculations is less than 1.0%, the iteration is completed, and the values ​​of the saturation leakage coefficients a and b are obtained. The formulas for calculating the saturation leakage coefficients a and b are as follows: ， ； In the formula: , , , , , , , ; S3.7: Based on the equivalent magnetic circuit model, the expression for the load air gap magnetic flux density considering leakage flux and magnetic saturation is as follows: ， ， ； The electromagnetic torque is calculated using Maxwell's stress tensor method based on the load air gap magnetic flux density. ， In the formula, l is the radial length of the motor.

Images