Calculation and optimization methods for winding coefficients of coreless motors

Abstract

This application discloses a method for calculating and optimizing the winding coefficient of a coreless motor, comprising: constructing a radial magnetic induction intensity model of the permanent magnet considering three-dimensional effects / axial magnetic field attenuation based on the geometric and material parameters of the permanent magnet; calculating the axial efficiency factor of the winding based on the winding geometric parameters and the radial magnetic induction intensity model of the permanent magnet, the axial efficiency factor including the straight-side axial efficiency factor and the inclined-side axial efficiency factor; modifying the traditional pitch factor formula based on the axial efficiency factor to obtain the pitch factor considering the three-dimensional effect of the magnetic field of the coreless motor, and calculating the total winding coefficient of the winding based on the pitch factor. The accuracy of the three-dimensional distribution of magnetic induction intensity and the winding coefficient of the coreless motor calculated by this method is close to the three-dimensional finite element simulation results, while reducing the calculation time from several hours to milliseconds, achieving high precision and high efficiency.

Description

Technical Field

[0001] This application relates to the field of motor design technology, and in particular to a method for calculating and optimizing the winding coefficients of a coreless motor. Background Technology

[0002] In implantable medical devices, especially axial-flow blood pumps, the drive motor needs to provide stable power within an extremely compact space while avoiding damage to blood components. Slotless permanent magnet motors have become the mainstream choice for these applications due to their inherent zero cogging torque and simple structure. Their hollow structure naturally provides the physical space for the integration of the blood flow channel and impeller.

[0003] However, to create unobstructed blood flow channels, these motors must be designed with large air gaps. This leads to a key technical challenge: while traditional radial flux motors typically have a uniform magnetic field distribution along the axial direction, the magnetic field inside these motors exhibits a strong three-dimensional effect. The most significant feature is the severe attenuation of radial magnetic flux along the axial direction. This attenuation causes significant errors in design calculations based on two-dimensional models, making it impossible to accurately predict motor performance. Furthermore, the windings of these slotless motors typically employ cup-shaped self-supporting coils. These cup-shaped coils have long ends and complex geometry, and the coupling effects in this non-uniform magnetic field lack accurate analytical models.

[0004] Currently, the design and analysis of such motors mainly rely on empirical estimations or approximate two-dimensional models, which have very low accuracy and cannot provide a reliable basis for the design of motors used in precision instruments. Another design tool is three-dimensional finite element simulation based on electromagnetic field simulation software such as Ansys Maxwell. Although three-dimensional finite element analysis has high accuracy, its computational cost is extremely high. A single simulation takes several hours, and the total time for multi-objective optimization can reach tens of days or even longer, making it impossible to perform rapid parameter iteration and multi-objective optimization during the design phase. Summary of the Invention

[0005] In a first aspect, embodiments of this application provide a method for calculating the winding coefficient of a hollow cup motor, the method comprising the following steps: Obtain the geometric and material parameters of the permanent magnet and windings of the motor; Based on the geometric and material parameters of the permanent magnet in the motor, a radial magnetic flux density model of the permanent magnet is constructed. ; Based on winding geometry parameters and radial magnetic induction model Calculate the axial efficiency factor, which includes the straight-edge axial efficiency factor. and the axial efficiency factor of the hypotenuse Straight-edge axial efficiency factor The ratio of the average radial magnetic flux density of the straight side of the winding to the radial magnetic flux density of the center plane, and the axial efficiency factor of the hypotenuse. It is the ratio of the average radial magnetic flux density of the inclined section of the winding to the radial magnetic flux density of the central plane; Based on the axial efficiency factor, the traditional pitch coefficient formula is modified to obtain the pitch coefficient, and the total winding coefficient of the winding is calculated based on the pitch coefficient. The total winding coefficient is the product of the pitch coefficient and the distribution coefficient.

[0006] In one embodiment, a radial magnetic flux density model of the permanent magnet is constructed based on the geometric and material parameters of the permanent magnet. include: Based on the geometric parameters of the permanent magnet, a scalar magnetic potential description of the magnetic field of the diameter-magnetized permanent magnet is constructed; Based on the scalar magnetic potential description, axial harmonic decomposition is performed on the magnetic field of the permanent magnet to obtain the radial-axial magnetic density function. radial-axial magnetic density function Expressed as radial magnetic flux density function using the method of separation of variables With axial harmonic components The combination; Solving for the radial magnetic flux density function Regarding winding radius r The ordinary differential equation is used to obtain the radial magnetic flux density function. The general solution, radial magnetic flux density function The general solution is a linear combination of the first and second kind of first-order modified Bessel functions; By utilizing the derivative recurrence relation and boundary conditions of the modified Bessel function, the coefficients of the first-order modified Bessel function of the second kind can be obtained. With the axial impulse function characterizing the finite length of the permanent magnet The relationship between them; Axial impulse function Based on the Fourier cosine series expansion, the axial impulse function after the series expansion is... The coefficients can be obtained by using the orthogonality of the cosine function. ; coefficients Substitute back to radial magnetic flux density function The general solution yields the expression for the scalar magnetic potential; Based on the expression for scalar magnetic potential, the annular permanent magnet of the hollow cup motor is considered as the difference set of two concentric solid cylindrical permanent magnets. According to the superposition principle, the radial magnetic induction intensity model of the permanent magnet is obtained. .

[0007] In one embodiment, calculating the axial efficiency factor includes: Radial magnetic flux density model Axial projection is performed on the plane of maximum magnetic flux density to obtain the average radius of the winding. At the point of maximum magnetic flux density plane The radial magnetic flux density along the axial direction z Distributed magnetic flux density function ; Based on magnetic flux density function Calculate the integral of the straight-side segment of the winding along the axial direction, then divide by the length of the straight-side segment projected along the axial direction to obtain the average radial magnetic flux density of the straight-side segment. Calculate the ratio of the average radial magnetic flux density of the straight-side segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the straight-side segment. ; Based on magnetic flux density function Calculate the integral of the hypotenuse segment along the axial direction of the winding, then divide by the length of the hypotenuse segment projected along the axial direction to obtain the average radial magnetic flux density of the hypotenuse segment. Calculate the ratio of the average radial magnetic flux density of the hypotenuse segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the hypotenuse. .

[0008] In one embodiment, the axial harmonic components are projected onto the axial projection region of the straight edge segment. After performing a definite integral and dividing by the length of the straight side segment, the geometric mapping coefficients of the straight side segment are obtained. ; By projecting the axial harmonic components onto the axial projection range of the hypotenuse segment After performing the definite integral, divide by the axial projection length of the hypotenuse segment, and then multiply by the cosine of the inclination angle of the hypotenuse segment. Obtain the geometric mapping coefficients of the hypotenuse segment. ; By calculating the harmonic coefficients of each order At the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients with straight edge segments The sum of the products is used to obtain the average radial magnetic flux density of the straight-side segment; Calculate the harmonic coefficients of each order At the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients of hypotenuse segment The sum of the products is used to obtain the average radial magnetic flux density of the hypotenuse segment; Among them, harmonic coefficient and average radius of winding Radial attenuation factor at the location These are two coefficients in the radial magnetic induction intensity model.

[0009] In one embodiment, the harmonic coefficient ; In the formula The permeability of free space, This represents the magnitude of the remanence of the permanent magnet. Axial pulse function The basis function coefficients according to the Fourier cosine series expansion.

[0010] In one embodiment, the radial attenuation factor Calculated using the following formula: ; In the formula, , The average radius of the winding. The outer radius of the permanent magnet is [missing information]. Let be the radius of the permanent magnet. Let be the wave number of the m-th axial harmonic component. (*) represents the zeroth-order modified Bessel function of the second kind. (*) represents the first-order modified Bessel function of the second kind.

[0011] In one embodiment, the coefficients of the second-order first-order modified Bessel function This can be expressed by the following formula: ; In the formula, The outer radius of the permanent magnet is [missing information]. Let be the wave number of the m-th axial harmonic component. (*) represents the zeroth-order modified Bessel function of the second kind. (*) represents the first-order modified Bessel function of the second kind. This represents the magnitude of the remanence of the permanent magnet. Axial pulse function The basis function coefficients according to the Fourier cosine series expansion.

[0012] In one embodiment, the traditional pitch coefficient formula is modified based on the axial efficiency factor using the following formula: ; In the formula, This represents the total length of a single coil of the winding projected axially. This indicates the length of the straight side segment of a single coil. Indicates the straight-side pitch of a single coil. This represents the axial efficiency factor of the straight edge. This represents the axial efficiency factor of the hypotenuse.

[0013] Secondly, this application also provides an optimization method for a hollow cup motor. The optimization method uses the winding coefficient calculation method of the first aspect to calculate the axial efficiency factor, thereby obtaining the straight-side axial efficiency factor. and the axial efficiency factor of the hypotenuse Straight-edge axial efficiency factor Explicitly dependent on the straight-side length of a single coil in the winding axial efficiency factor of hypotenuse Explicitly dependent on the straight-side length of a single coil in the winding The total length of a single coil projected axially. and the axial projection length of each side of the inclined segment of a single coil. ; Define axial length-to-diameter ratio axial major diameter The ratio is the total length of the axial projection of a single coil of the winding. With permanent magnet size h The ratio; Define straight edge ratio straight edge ratio The length of the straight side segment of a single coil in the winding. The total length of the individual coils of the winding projected in the axial direction The ratio; Based on axial length-to-diameter ratio Compared to straight edges Based on the coil geometry of the winding, the axial efficiency factor of the straight side is... and the axial efficiency factor of the hypotenuse Perform variable substitution to change the axial efficiency factor of the straight edge. and the axial efficiency factor of the hypotenuse Reconstruct them separately for axial length-to-diameter ratio Compared to straight edges The function; Based on the straight-edge axial efficiency factor after variable substitution and the axial efficiency factor of the hypotenuse The total winding coefficient is calculated. Based on the total winding coefficient and winding geometry parameters, the effective flux linkage amplitude of each phase of the winding is obtained; The optimization objectives are defined as maximizing the output potential energy and minimizing the conductor volume, with the output potential energy being proportional to... Axial length-to-diameter ratio The conductor volume is proportional to the product of the total winding coefficient and the total winding coefficient. In the formula, The axial length-to-diameter ratio, For straight sides, The axial efficiency factor of the straight edge after variable substitution and the axial efficiency factor of the hypotenuse The calculated total winding coefficient; Torque constraints, thermal constraints, and manufacturing constraints are set to maximize the output potential energy and minimize the conductor volume. A multi-objective optimization algorithm is used to search for the Pareto front, and the final design scheme is obtained based on the knee point criterion.

[0014] Thirdly, this application also provides a computer-readable storage medium storing a processing program configured to execute, at runtime, a method for calculating the winding coefficients of a coreless motor as described in any of the first aspects, or an optimization method for a coreless motor as described in the second aspect.

[0015] The above-described method for calculating the winding coefficient of a hollow cup motor obtains the geometric parameters of the permanent magnet, the winding geometric parameters, and the material parameters of the motor. Based on the geometric and material parameters of the permanent magnet, a radial magnetic induction intensity model of the permanent magnet's magnetic field considering three-dimensional effects / axial magnetic field attenuation is constructed. By combining the winding geometry parameters and the magnetic induction intensity model, the axial efficiency factor of the straight side, which quantitatively characterizes the magnetic field utilization rate of the straight and inclined sides of the winding, is calculated. and the axial efficiency factor of the hypotenuse ;Utilizing the straight-edge axial efficiency factor and the axial efficiency factor of the hypotenuse A modified formula for the traditional winding pitch coefficient is used to obtain the pitch coefficient considering the three-dimensional effect of the magnetic field in a coreless motor. The total winding coefficient is then calculated based on this pitch coefficient with high accuracy. This method considers the axial magnetic field inhomogeneity in the three-dimensional effect, solving the problem of large prediction errors caused by neglecting axial magnetic field attenuation in traditional two-dimensional models. The accuracy of the three-dimensional magnetic induction intensity distribution and winding coefficients of the coreless motor calculated by this method is close to that of three-dimensional finite element simulation results. Simultaneously, the calculation time is reduced from several hours to milliseconds, achieving both high accuracy and high efficiency. Attached Figure Description

[0016] Figure 1 This is a flowchart illustrating a method for calculating the winding coefficients of a hollow cup motor in one embodiment; Figure 2 This is a schematic diagram of the hollow cup motor in one embodiment; Figure 3 This is a schematic diagram of the geometric parameters of the motor in one embodiment; Figure 4 This is a schematic diagram of the winding geometry parameters in one embodiment; Figure 5 A flowchart illustrating the radial magnetic flux density model of a permanent magnet in one embodiment; Figure 6 This is a flowchart illustrating the calculation of the axial efficiency factor in one embodiment; Figure 7A comparison diagram of the radial magnetic flux density model calculated by the method of this application and the simulation of the three-dimensional finite element method is shown in one embodiment; Figure 8 This is a comparison chart of the back EMF waveforms obtained by the method of this application and different analytical models at a speed of 20000 RPM in one embodiment, based on no-load back EMF prediction and finite element simulation. Detailed Implementation

[0017] The present application will be described in detail below with reference to the specific embodiments shown in the accompanying drawings. However, these embodiments do not limit the present application. Any structural, methodological, or functional modifications made by those skilled in the art based on these embodiments are included within the protection scope of the present application.

[0018] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes the element.

[0019] In one embodiment, such as Figure 1 As shown, this application provides a method for calculating the winding coefficient of a coreless motor, which includes the following steps: Step 101: Obtain the permanent magnet geometric parameters, winding geometric parameters, and material parameters of the motor; Specifically, a hollow cup motor, such as Figure 2 As shown, the coreless motor has a rotor at its center, surrounded by permanent magnets, and an outermost stator core. Unlike ordinary motors, the stator of a coreless motor lacks slots; instead, it features cup-shaped windings (i.e., cup-shaped coils) directly embedded in the core, creating a large air gap for cooling or fluid passage. The coreless motor also includes a complete model, rotor components, and winding wires.

[0020] like Figure 3 As shown, the geometric parameters of the permanent magnet in the motor include the inner radius of the permanent magnet. Outer radius Axial height of permanent magnet The winding geometry of the motor includes the average radius of the winding. . Figure 3 middle, This refers to the inner diameter of the stator core. The axial direction refers to the direction parallel to the axis of rotation of the motor rotor (see reference). Figure 2 (In the z-axis direction), the magnetic field exhibits a non-uniform distribution (attenuation) along this direction.

[0021] Winding geometric parameters such as Figure 4 As shown, the winding has a cup-shaped structure. The winding geometry parameters include the total length. (That is, the total length of the coil projected axially), length of the coil's straight side Length of the hypotenuse on each side of the coil (The length of the projection of the hypotenuse of each coil onto the coil axis), and the pitch of the straight side segment. (That is, the distance between the two straight sides), the angle of inclination of the hypotenuse. (That is, the angle between the extension direction of the inclined segment of the coil and the radial direction of the winding).

[0022] The material parameters of a hollow cup motor include the remanence amplitude of the permanent magnet. .

[0023] Step 102: Based on the geometric and material parameters of the permanent magnet of the motor, construct a radial magnetic flux density model of the permanent magnet. ; Specifically, for a diameter-magnetized permanent magnet, based on its geometric and material parameters, in cylindrical coordinates, the diameter-magnetized permanent magnet is equivalent to a magnetization source with a specific orientation and spatial distribution. Here, radial direction can refer to the direction perpendicular to the motor axis (see reference). Figure 2 (in the r-axis direction), is the radial magnetic field. and average radius of winding The direction of measurement.

[0024] Furthermore, by employing the equivalent magnetic charge method, the magnetic field problem is transformed into the solution of a scalar magnetic potential in the passive region outside the permanent magnet, which satisfies the Laplace equation.

[0025] At the interface between the permanent magnet and air, the normal discontinuity of magnetization leads to the transformation into a Neumann boundary condition. Since the permanent magnet is diameter-magnetized, the excitation source has… Dependency relationship: Using the method of separation of variables, and based on the orthogonality of trigonometric functions, the scalar magnetic potential can be simplified into an angular fundamental mode, i.e., expressed as... With a radial coordinate and axial coordinates function The product of.

[0026] To solve Furthermore, to accurately address the axial magnetic field attenuation caused by finite-length magnets, a harmonic decomposition method is introduced to... Expanding this into a sum of Fourier cosine series of axial harmonic components, the radial function can then be solved. Radial function It satisfies the first-order modified Bessel equation. By applying boundary conditions and utilizing the derivative recursion of the modified Bessel function, specific coefficients in the first-order modified Bessel equation can be determined.

[0027] Based on the superposition principle, the magnetic field of a ring-shaped permanent magnet can be obtained by subtracting the fields of two concentric solid cylindrical magnets, thus yielding a radial magnetic induction model. .

[0028] Step 103: Model based on winding geometry parameters and radial magnetic induction intensity Calculate the axial efficiency factor, which includes the straight-edge axial efficiency factor. and the axial efficiency factor of the hypotenuse Straight-edge axial efficiency factor The ratio of the average radial magnetic flux density of the straight side of the winding to the radial magnetic flux density of the center plane, and the axial efficiency factor of the hypotenuse. It is the ratio of the average radial magnetic flux density of the inclined section of the winding to the radial magnetic flux density of the central plane; Specifically, using the obtained radial magnetic flux density model At the average radius of the winding Maximum magnetic flux density plane By projecting the data onto the axial coordinate z, we obtain the radial magnetic flux density distribution function that varies only with the axial coordinate z. .

[0029] Furthermore, regarding the winding structure, the winding is decomposed into straight-side segments (axial intervals). ) and the hypotenuse segment (axial interval) The winding consists of two parts. The straight side segment can refer to the coil side of the winding parallel to the motor axis, and its axial length is denoted as... This is the main region where effective electromagnetic energy conversion occurs. The inclined segment can be the sloping part connecting the two straight segments in the winding, forming a certain angle with the axis (inclination angle). β Its projected length on each side along the coil axis is denoted as , which constitute the ends of the winding.

[0030] Calculate the average radial magnetic flux density of the straight-side segment and the hypotenuse segment in the non-uniform magnetic field, respectively: For the straight edge segment, in the axial interval Integrating the radial magnetic flux density distribution function and dividing by the length of the straight-side segment yields the average radial magnetic flux density of the straight-side segment. For the hypotenuse segment, the geometric projection effect of its tilt angle β needs to be considered (i.e., multiplied by cos...). β ), and then in the axial range Integrating the radial magnetic flux density distribution function and dividing by the axial projection length of the hypotenuse segment yields the average radial magnetic flux density of the hypotenuse segment.

[0031] The calculated average radial magnetic flux density of the straight side segment is compared with the radial magnetic flux density of the center plane of the winding center plane (z=0). Divide to obtain the axial efficiency factor of the straight side. Similarly, the average radial magnetic flux density of the hypotenuse segment and the radial magnetic flux density of the central plane are compared. Divide to obtain the axial efficiency factor of the hypotenuse. .

[0032] Step 104: Based on the axial efficiency factor, modify the traditional pitch coefficient formula to obtain the pitch coefficient, and calculate the total winding coefficient of the winding according to the pitch coefficient. The total winding coefficient is the product of the pitch coefficient and the distribution coefficient.

[0033] Specifically, using the calculated straight-edge axial efficiency factor and the axial efficiency factor of the hypotenuse The traditional formula for calculating winding pitch coefficient is modified.

[0034] Based on the straight-edge axial efficiency factor and the axial efficiency factor of the hypotenuse The traditional pitch coefficient is corrected using the following formula. : ; In the formula, This represents the total length of a single coil of the winding projected axially. This indicates the length of the straight side segment of a single coil. Indicates the straight-side pitch of a single coil. This represents the axial efficiency factor of the straight edge. This represents the axial efficiency factor of the hypotenuse.

[0035] Furthermore, the distribution coefficient is determined based on the phase band type of the winding. :

[0036] Ultimately, the total winding coefficient of the winding From the corrected pitch coefficient With distribution coefficient Multiplying them together yields the result, i.e. .

[0037] In this embodiment, the radial magnetic flux density model constructed by this method This method quantifies magnetic field non-uniformity, replacing the uniformity assumption of traditional two-dimensional models. It accurately quantifies coil flux linkage and quantitatively describes the impact of three-dimensional magnetic field attenuation on different parts of the coil. While maintaining near-finite element accuracy of three-dimensional models, it reduces computation time from hours to milliseconds, significantly improving efficiency. Furthermore, by introducing geometric mapping coefficients and axial efficiency factors, it achieves rapid coupling between the magnetic field and winding geometry parameters, as well as accurate calculation of winding coefficients, thereby enhancing the reliability of equipment performance prediction.

[0038] In one embodiment, such as Figure 5 As shown, a radial magnetic flux density model of the permanent magnet is constructed based on the geometric and material parameters of the permanent magnet in the motor. This includes the following steps: Step 501: Based on the geometric parameters of the permanent magnet, construct a scalar magnetic potential description of the magnetic field of the diameter-magnetized permanent magnet; Specifically, in cylindrical coordinate system Below, based on material parameters (the amplitude of remanence of the permanent magnet material) ) and permanent magnet geometric parameters (permanent magnet axial height) The magnetization vector is expressed as: .

[0039] in, This represents the amplitude of the remanence of the permanent magnet material. It is an axial impulse function characterizing the finite length of the magnetic field of a permanent magnet. , This refers to the axial height of the permanent magnet.

[0040] Furthermore, using the equivalent magnetic charge method, in the passive region outside the permanent magnet ( In this context, the magnetic field is composed of scalar magnetic potential. Describe it and satisfy the Laplace equation: .

[0041] Because the permanent magnet is magnetized by diameter, the excitation source has explicit... Dependency relationship: Based on the method of separation of variables and utilizing the orthogonality of trigonometric functions, all higher-order angular harmonic terms in the general solution are set to 0. Therefore, the scalar magnetic potential description can be simplified to: .

[0042] Step 502: Based on the scalar magnetic potential description, perform axial harmonic decomposition on the permanent magnet magnetic field to obtain the radial-axial magnetic density function. radial-axial magnetic density function Expressed as radial magnetic flux density function using the method of separation of variables With axial harmonic components The combination; Specifically, the harmonic decomposition method is adopted, which introduces a field encompassing the entire effective field region, taken as the axial length of the permanent magnet. h non-integer multiples of artificial cycles L Harmonic order m ( m (e.g., =1,2,3,...,M), construct a complete orthogonal eigenfunction system, thereby transforming the finite-length problem into a periodic problem for solution.

[0043] Based on the even symmetry of the scalar magnetic potential with respect to the plane (z=0) in the motor, the first m Wavenumber of axial harmonics It can be defined as: .

[0044] This definition ensures the harmonic function It satisfies the symmetry condition of the midplane.

[0045] Furthermore, it is possible to convert the radial-axial magnetic density function Expanded into the Fourier cosine series of the aforementioned axial harmonic components, the radial-axial magnetic density function Expressed as radial magnetic flux density function using the method of separation of variables With axial harmonic components Linear combination: .

[0046] Step 503: Solve for the radial magnetic flux density function Regarding winding radius r The ordinary differential equation is used to obtain the radial magnetic flux density function. The general solution, radial magnetic flux density function The general solution is a linear combination of the first and second kind of first-order modified Bessel functions; Specifically, the above radial-axial magnetic density function Substituting the expression into the Laplace equation yields the following partial differential equation: .

[0047] By utilizing the orthogonality of harmonic functions, the radial magnetic flux density function is obtained. The satisfied ordinary differential equation is a first-order modified Bessel equation: .

[0048] The general solution to the first-order modified Bessel equation is a linear combination of the first-type and first-order second-type modified Bessel functions, i.e. .

[0049] in, This represents the first-order modified Bessel function of the first kind. exist r It exhibits an exponential growth trend when it increases; This represents the second kind of first-order modified Bessel function. exist r It exhibits an exponential decreasing trend as it increases; and These are undetermined coefficients, which will be determined by specific boundary conditions; It is the first m Axial harmonic wavenumber.

[0050] Step 504: Using the derivative recurrence relation and boundary conditions of the modified Bessel function, obtain the coefficients of the first-order modified Bessel function of the second kind. With the axial impulse function characterizing the finite length of the permanent magnet The relationship between them; Specifically, due to the first kind of modified Bessel function It grows exponentially with increasing r, in order to ensure that the solution is within the physical domain ( If the value is bounded, its coefficient must be set to... Therefore, the radial magnetic flux density function It can be simplified to: .

[0051] At the interface between the permanent magnet and air, the scalar magnetic potential satisfies the Neumann boundary condition: .

[0052] Will Substitute the Neumann boundary conditions. When calculating the derivative, the recurrence relation for the derivative of the modified Bessel function of the second kind must be used. After substituting and simplifying the recurrence relation, the coefficients are obtained. With the axial impulse function characterizing the finite length of the permanent magnet Relationship: .

[0053] in, It is a zero-order modified Bessel function of the second kind.

[0054] Step 505: Convert the axial pulse function Based on the Fourier cosine series expansion, the axial impulse function after the series expansion is... The coefficients can be obtained by using the orthogonality of the cosine function. ; Specifically, coefficient With the axial impulse function characterizing the finite length of the permanent magnet The relationship is as follows: .

[0055] Axial impulse function The expansion is performed according to the Fourier cosine basis function system (based on the midplane). (The even symmetry of ) is shown in the expansion as follows: .

[0056] Axial impulse function based on series expansion The coefficients can be obtained by using the orthogonality of the cosine function. : .

[0057] Step 506: Adjust the coefficients Substitute back to radial magnetic flux density function The general solution yields the expression for the scalar magnetic potential; Step 507: Based on the expression for scalar magnetic potential, the annular permanent magnet of the hollow cup motor is considered as the difference set of two concentric solid cylindrical permanent magnets. According to the superposition principle, the radial magnetic induction intensity model of the permanent magnet is obtained. .

[0058] Specifically, in obtaining the coefficient Afterwards, Substitute back the boundary condition determination coefficients general solution of radial magnetic flux density function In the middle, we obtain an outer radius of The radial function corresponding to a solid cylindrical magnet.

[0059] Substituting the radial function back into the radial-axial magnetic density function and scalar magnetic potential description This yields an intermediate expression for the scalar magnetic potential of the solid cylindrical magnet.

[0060] Based on the principle of superposition, a ring-shaped permanent magnet can be considered as two magnets whose inner radii are respectively... and outer radius The difference between the radii of two concentric solid cylindrical magnets is calculated by subtracting their corresponding scalar magnetic potential expressions and introducing harmonic coefficients. sum function After simplification, we obtain the final expression for the scalar magnetic potential: .

[0061] Furthermore, a radial magnetic flux density model for permanent magnets can be obtained. The expression: .

[0062] In the formula, The permeability of free space, The outer radius of the permanent magnet is [missing information]. Let be the radius of the permanent magnet. For the first m The wave number of the first-order axial harmonic component. (*) represents the zeroth-order modified Bessel function of the second kind. (*) represents the first-order modified Bessel function of the second kind.

[0063] In one embodiment, such as Figure 6 As shown, the calculation of the axial efficiency factor includes the following steps: Step 601: Convert the radial magnetic induction intensity model Axial projection is performed on the plane of maximum magnetic flux density to obtain the average radius of the winding. At the point of maximum magnetic flux density plane The radial magnetic flux density along the axial direction z Distributed magnetic flux density function ; Specifically, the radial magnetic induction intensity model In the plane of maximum magnetic flux density ( By performing axial projection, the average radius of the winding is obtained. At the point of maximum magnetic flux density plane The radial magnetic flux density along the axial direction z Distributed magnetic flux density function : ; Wherein, the radial attenuation factor of the m-th harmonic at the winding radius r .

[0064] Step 602: Based on magnetic flux density function Calculate the integral of the straight-side segment of the winding along the axial direction, then divide by the length of the straight-side segment projected along the axial direction to obtain the average radial magnetic flux density of the straight-side segment. Calculate the ratio of the average radial magnetic flux density of the straight-side segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the straight-side segment. ; Specifically, the straight edge segment is located in the axial interval. Magnetic flux density function within this interval Perform a definite integral and divide by the length of the straight side segment projected along the axis. That is, the average radial magnetic flux density of the straight edge segment is obtained, and the expression is: .

[0065] Furthermore, the radial magnetic flux density of the central plane (z=0) is calculated. The radial magnetic flux density in the central plane is .

[0066] Straight-edge axial efficiency factor The average radial magnetic flux density of the straight-side segment and the radial magnetic flux density of the central plane The ratio, expressed as: .

[0067] To improve computational efficiency, a geometric mapping coefficient for the straight edge segment can be introduced. The geometric mapping coefficient for the straight edge segment is: ,in Therefore, it can be quickly calculated using the following formula. : .

[0068] Step 603: Based on magnetic flux density function Calculate the integral of the hypotenuse segment along the axial direction of the winding, then divide by the length of the hypotenuse segment projected along the axial direction to obtain the average radial magnetic flux density of the hypotenuse segment. Calculate the ratio of the average radial magnetic flux density of the hypotenuse segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the hypotenuse. .

[0069] Specifically, the hypotenuse segment is located in the axial interval. Calculate the magnetic flux density function on each side of this interval. When integrating, the geometric projection of the inclination angle β of the hypotenuse must be considered (multiplied by cos( β Then divide by the length of the axial projection of each hypotenuse segment. The average radial magnetic flux density of the hypotenuse segment is obtained: .

[0070] Hypotenuse axial efficiency factor Defined as the average radial magnetic flux density of the hypotenuse segment and the magnetic flux density of the central plane The ratio, expressed as: .

[0071] The geometric mapping coefficient of the hypotenuse segment is .in

[0072] The following formula can be used for quick calculation: .

[0073] in, Harmonic coefficients Radial attenuation factor, and These are the geometric mapping coefficients. This represents the total length of a single coil of the winding projected axially. This indicates the length of the straight side segment of a single coil.

[0074] In one embodiment, the axial harmonic components are projected onto the axial projection range of the straight edge segment. After performing a definite integral and dividing by the length of the straight side segment, the geometric mapping coefficients of the straight side segment are obtained. ; By analyzing the axial harmonic components on each side of the hypotenuse segment's axial projection interval. After performing the definite integral, divide by the axial projected length of each hypotenuse segment, and then multiply by the cosine of the inclination angle of the hypotenuse segment. Obtain the geometric mapping coefficients of the hypotenuse segment. ; By calculating the harmonic coefficients of each order At the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients with straight edge segments The sum of the products is used to obtain the average radial magnetic flux density of the straight-side segment; Calculate the harmonic coefficients of each order At the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients of hypotenuse segment The sum of the products is used to obtain the average radial magnetic flux density of the hypotenuse segment; Among them, harmonic coefficient and average radius of winding Radial attenuation factor at the location These are two coefficients in the radial magnetic induction intensity model.

[0075] Specifically, through the axial projection interval of the straight edge segment For axial harmonic components Perform a definite integral and divide by the length of the straight side. Obtain the geometric mapping coefficients of the straight edge segment. : .

[0076] in, Geometric mapping coefficients of straight edge segments It can quantify the average effect of the magnetic field over the length of a straight side segment.

[0077] By the axial projection interval of the hypotenuse segment Above, for each side of the axial harmonic components Perform a definite integral and divide by the axial projection length of each hypotenuse segment. Then multiply by the cosine of the angle of inclination of the hypotenuse segment. Obtain the geometric mapping coefficients of the hypotenuse segment. : .

[0078] Geometric mapping coefficients of hypotenuse segment The average effect of the quantized magnetic field on the inclined segment and its projection attenuation due to the inclination.

[0079] By calculating the harmonic coefficients of each order At the average radius of the winding Radial attenuation factor at the location Geometric mapping with straight edge segments The sum of the products of the three factors is used to obtain the average radial magnetic flux density of the straight-side segment: .

[0080] By calculating the harmonic coefficients of each order of axial harmonics At the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients of hypotenuse segment The sum of the products of the three factors yields the average radial magnetic flux density of the hypotenuse segment: .

[0081] In the formula, the harmonic coefficient ; Permeability of free space; This represents the magnitude of the remanence of the permanent magnet. Axial pulse function The basis function coefficients according to the Fourier cosine series expansion.

[0082] Straight-edge axial efficiency factor With the axial efficiency factor of the hypotenuse That is, the summation of the average radial magnetic flux density and the magnetic flux density in the central plane mentioned above. The ratio, that is: ; .

[0083] Based on the same concept, this application also provides an optimization method for a hollow cup motor. The optimization method uses the above-mentioned winding coefficient calculation method to calculate the axial efficiency factor and obtain the straight-side axial efficiency factor. and the axial efficiency factor of the hypotenuse Straight-edge axial efficiency factor Explicitly dependent on the straight-side length of a single coil in the winding axial efficiency factor of hypotenuse Explicitly dependent on the straight-side length of a single coil in the winding The total length of a single coil projected axially. And the axial projection length of the hypotenuse of a single coil. ; Define axial length-to-diameter ratio axial major diameter The ratio is the total length of the axial projection of a single coil of the winding. With permanent magnet size h The ratio; Define straight edge ratio straight edge ratio The length of the straight side segment of a single coil in the winding. The total length of the individual coils of the winding projected in the axial direction The ratio; Based on axial length-to-diameter ratio Compared to straight edges Based on the coil geometry of the winding, the axial efficiency factor of the straight side is... and the axial efficiency factor of the hypotenuse Perform variable substitution to change the axial efficiency factor of the straight edge. and the axial efficiency factor of the hypotenuse Reconstruct them separately for axial length-to-diameter ratio Compared to straight edges The function; Based on the straight-edge axial efficiency factor after variable substitution and the axial efficiency factor of the hypotenuse The total winding coefficient is calculated. Based on the total winding coefficient and winding geometry parameters, the effective flux linkage amplitude of each phase of the winding is obtained; The optimization objectives are defined as maximizing the output potential energy and minimizing the conductor volume, with the output potential energy being proportional to... Axial length-to-diameter ratio The conductor volume is proportional to the product of the total winding coefficient and the total winding coefficient. In the formula, The axial length-to-diameter ratio, For straight sides, The axial efficiency factor of the straight edge after variable substitution and the axial efficiency factor of the hypotenuse The calculated total winding coefficient; Torque constraints, thermal constraints, and manufacturing constraints are set to maximize the output potential energy and minimize the conductor volume. A multi-objective optimization algorithm is used to search for the Pareto front, and the final design scheme is obtained based on the knee point criterion.

[0084] Specifically, based on the defined axial length-to-diameter ratio Compared to straight edges Based on the coil geometry of the winding, the axial efficiency factor of the obtained straight side is... and the axial efficiency factor of the hypotenuse Perform variable substitution to reconstruct them into functions only of ξ and α, i.e. and .

[0085] Furthermore, the reconstructed efficiency factor is substituted into the corrected pitch coefficient. formula: .

[0086] Pitch coefficient Multiply by the distribution coefficient determined by the phase band type. The total winding coefficient is obtained. = Based on this total winding coefficient, the total winding coefficient... By combining the winding geometry parameters, the effective flux linkage amplitude per phase of the winding can be obtained. The flux linkage amplitude is used to characterize the net flux linkage after the non-uniform three-dimensional magnetic field is coupled with the winding geometry.

[0087] The amplitude of the back potential per pole is The electromagnetic torque is Copper loss is .

[0088] in, The number of turns in series per phase. The rotor's mechanical angular velocity (unit: radians / second). m For phase number, For power factor, Peak phase current (unit: amperes).

[0089] Furthermore, the optimization objectives include maximizing the output potential energy and minimizing the conductor volume. Maximizing output potential energy: The amplitude of the motor's back electromotive force and electromagnetic torque are both proportional to the amplitude of the flux linkage per phase. . ,therefore As a dimensionless output capacity index, it comprehensively reflects that the longer the winding and the more conductors, the greater the output potential; at the same time, the longer the winding, the weaker the end magnetic field and the lower the utilization rate.

[0090] Minimize copper loss and volume: And the total volume of the conductor is: ),in This represents the total length of the coil.

[0091] Therefore, a clear correspondence can be established between two objective functions and design variables: Output target: Volume target: .

[0092] The objective function can be set as: maximizing the output potential capability. Minimize conductor volume The constraints include: the electromagnetic torque T is not less than the rated load requirement. Copper loss should not exceed the thermal safety limit. The tilt angle of the cup-shaped coil meets the minimum manufacturing process requirements. .

[0093] To maximize output potential capability and minimizing conductor volume With the objective of automatically searching for the Pareto optimal frontier of the design parameters (ξ, α) under constraints, a multi-objective optimization algorithm (such as NSGA-II) is used. Based on the knee point criterion, the compromise point that achieves the best balance between output and volume on this frontier is selected as the final design scheme.

[0094] Based on the above-mentioned method for calculating the winding coefficient of the hollow cup motor and the optimization method for the hollow cup motor, the advantages of this application can be verified through the following comparative cases and data: 1. Corrected three-dimensional magnetic field distribution for high prediction accuracy: Existing technologies assume that the magnetic field around the winding is uniformly distributed along the axial direction. The radial magnetic induction intensity model established in this application accurately quantifies the axial magnetic field attenuation. To verify the accuracy, ten different sizes of motor permanent magnet configurations were selected:

[0095] The error in predicting the radial magnetic flux density at the average radius of the winding is calculated using the following formula: ; .

[0096] like Figure 7 As shown, the radial magnetic induction intensity model of this application is used respectively. and 3D finite element simulation results Comparative calculations show that the proposed method and the results of 3D finite element simulation are in high agreement under different size configurations. The root mean square error (RMSE) is below 4.15 mT for all ten configurations, and the normalized RMS error is below 1.36%. Figure 7 Points of different shapes represent data obtained from three-dimensional finite element simulation, and lines represent the radial magnetic induction intensity distribution calculated by the method of this application.

[0097] 2. Advantages in winding factor calculation accuracy: By introducing an axial efficiency factor to quantitatively describe the different effects of magnetic field attenuation on the straight and slanted sections of the winding, this invention significantly improves the accuracy of winding factor calculation. Verification was conducted using four different motor configurations. Four motor configurations with different winding structures and permanent magnet sizes:

[0098] The parameters and results are compared as follows:

[0099] The total winding coefficient predicted by the model in this application is in high agreement with the results of the three-dimensional finite element simulation, and the error is still less than 5.0% even under extreme dimensions. In contrast, the existing model, which ignores axial attenuation, has a significantly higher predicted value and a significant error.

[0100] 3. Advantages of Motor Back EMF Prediction: Based on the verification of the accuracy of the magnetic field and winding coefficients, the predictive capability of the motor's back EMF was further verified through physical experiments. Prototype parameters and results are as follows:

[0101] like Figure 8 As shown, the no-load back EMF test was conducted at a speed of 20,000 RPM. The peak error between the predicted value and the experimental value of the model in this application was 4.22%, which is a high degree of agreement.

[0102] 5. Compared to high-precision 3D finite element simulation, the model provided in this application reduces time and computational costs. All simulations were performed on a computer equipped with an AMD Ryzen 9 7950X processor and 32GB of memory. The model in this application ran on MATLAB R2024a, and the finite element simulation used ANSYS Electromagnetics Suite 2022 R1. While maintaining accuracy comparable to 3D finite element simulation, this application achieves a computational speed improvement of over 80,000 times, enabling large-scale parameter scanning and rapid multi-objective optimization.

[0103] Furthermore, the winding coefficient calculation method for the hollow cup motor proposed in this application has flexible scalability and broad application prospects. For different winding topologies, the geometric mapping vector methodology constructed in this application can be extended to other winding shapes, such as windings with rhomboid, rectangular, or sinusoidal ends. Only specific geometric parameters need to be substituted and the corresponding geometric mapping vector constructed to continue the entire modeling and optimization process. For different optimization objectives, the multi-objective optimization framework of this application can flexibly replace optimization objectives and constraints. For example, maximizing efficiency, minimizing torque ripple, minimizing volume, or suppressing specific harmonic content can be used as new optimization objectives to adapt to different application scenarios.

[0104] In other applications, the technical solution of this application can also be applied to the following fields: other micro medical devices, such as ventricular assist devices, cochlear implant drivers, etc.; high-precision industrial servo systems, used in robot joints, precision machine tool spindles, and other applications where high motion control precision is required and cogging torque needs to be eliminated; and in the aerospace and electric vehicle fields, used in flywheel energy storage systems, high-speed compressors, electric fuel pumps, etc.

[0105] Based on the same concept, this application also provides a computer-readable storage medium storing a processing program configured to execute the above-described method for calculating the winding coefficients of a coreless motor, or to execute the above-described method for optimizing a coreless motor.

[0106] Computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means.

[0107] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. A computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the flow or function according to the embodiments of this application is generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device.

[0108] The various embodiments in this specification are described in a related manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the system embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions in the method embodiments. The above are merely preferred embodiments of this application and are not intended to limit the scope of protection of this application. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application are included within the scope of protection of this application.

Claims

1. A method for calculating the winding coefficient of a hollow cup motor, characterized in that, The method for calculating the winding coefficient includes the following steps: Obtain the geometric and material parameters of the permanent magnet and windings of the motor; Based on the geometric and material parameters of the permanent magnet in the motor, a radial magnetic flux density model of the permanent magnet is constructed. ; Based on winding geometry parameters and the radial magnetic induction model Calculate the axial efficiency factor, which includes the straight-edge axial efficiency factor. and the axial efficiency factor of the hypotenuse The straight-edge axial efficiency factor The axial efficiency factor of the inclined side is the ratio of the average radial magnetic flux density of the straight side segment of the winding to the radial magnetic flux density of the center plane. It is the ratio of the average radial magnetic flux density of the inclined section of the winding to the radial magnetic flux density of the central plane; Based on the axial efficiency factor, the traditional pitch coefficient formula is modified to obtain the pitch coefficient, and the total winding coefficient of the winding is calculated based on the pitch coefficient. The total winding coefficient is the product of the pitch coefficient and the distribution coefficient.

2. The method for calculating the winding coefficient of a hollow cup motor according to claim 1, characterized in that, Based on the geometric and material parameters of the permanent magnet of the motor, a radial magnetic flux density model of the permanent magnet is constructed. include: Based on the geometric parameters of the permanent magnet, a scalar magnetic potential description of the magnetic field of the diameter-magnetized permanent magnet is constructed; Based on the scalar magnetic potential description, axial harmonic decomposition is performed on the permanent magnet's magnetic field to obtain the radial-axial magnetic density function. The radial-axial magnetic density function Expressed as radial magnetic flux density function using the method of separation of variables With axial harmonic components The combination; Solve for the radial magnetic flux density function Regarding winding radius r The ordinary differential equation is used to obtain the radial magnetic flux density function. The general solution, the radial magnetic flux density function The general solution is a linear combination of the first and second kind of first-order modified Bessel functions; By utilizing the derivative recurrence relation and boundary conditions of the modified Bessel function, the coefficients of the second kind of first-order modified Bessel function are obtained. With the axial impulse function characterizing the finite length of the permanent magnet The relationship between them; The axial pulse function Based on the Fourier cosine series expansion, the axial impulse function after the series expansion... The coefficients are obtained by using the orthogonality of the cosine function. ; The coefficient Substitute back to the radial magnetic flux density function The general solution yields the expression for the scalar magnetic potential; Based on the expression for the scalar magnetic potential, the annular permanent magnet of the hollow cup motor is considered as the difference set of two concentric solid cylindrical permanent magnets. According to the superposition principle, the radial magnetic induction intensity model of the permanent magnet is obtained. .

3. The method for calculating the winding coefficient of a hollow cup motor according to claim 2, characterized in that, The calculation of the axial efficiency factor includes: The radial magnetic induction intensity model Axial projection is performed on the plane of maximum magnetic flux density to obtain the average radius of the winding. At the point of maximum magnetic flux density plane The radial magnetic flux density along the axial direction z Distributed magnetic flux density function ; Based on the magnetic flux density function Calculate the integral of the straight-side segment of the winding along the axial direction, and then divide by the length of the straight-side segment projected along the axial direction to obtain the average radial magnetic flux density of the straight-side segment. Calculate the ratio of the average radial magnetic flux density of the straight-side segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the straight-side segment. ; Based on the magnetic flux density function Calculate the integral of the hypotenuse segment along the axial direction of the winding, then divide by the length of the hypotenuse segment projected along the axial direction to obtain the average radial magnetic flux density of the hypotenuse segment. Calculate the ratio of the average radial magnetic flux density of the hypotenuse segment to the radial magnetic flux density of the center plane to obtain the axial efficiency factor of the hypotenuse. .

4. The method for calculating the winding coefficient of a hollow cup motor according to claim 3, characterized in that, By projecting the axial harmonic components onto the axial projection range of the straight edge segment After performing a definite integral and dividing by the length of the straight side segment, the geometric mapping coefficients of the straight side segment are obtained. ; By projecting the axial harmonic components onto the axial projection range of the hypotenuse segment... After performing the definite integral, divide by the axial projection length of the hypotenuse segment, and then multiply by the cosine of the inclination angle of the hypotenuse segment. Obtain the geometric mapping coefficients of the hypotenuse segment. ; By calculating the harmonic coefficients of each order , at the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients of the straight edge segment The sum of the products is used to obtain the average radial magnetic flux density of the straight edge segment; Calculate the harmonic coefficients of each order , at the average radius of the winding Radial attenuation factor at the location Geometric mapping coefficients of the hypotenuse segment The sum of the products is used to obtain the average radial magnetic flux density of the hypotenuse segment; Wherein, the harmonic coefficient and the average radius of the winding Radial attenuation factor at the location These are two coefficients in the radial magnetic induction intensity model.

5. The method for calculating the winding coefficient of a hollow cup motor according to claim 4, characterized in that, The harmonic coefficient ; In the formula The permeability of free space, This represents the magnitude of the remanence of the permanent magnet. The axial pulse function The basis function coefficients according to the Fourier cosine series expansion.

6. The method for calculating the winding coefficient of a hollow cup motor according to claim 4, characterized in that, Radial attenuation factor Calculated using the following formula: ； In the formula, , The average radius of the winding. The outer radius of the permanent magnet is... Let be the radius of the permanent magnet. Let be the wave number of the m-th axial harmonic component. (*) represents the zeroth-order modified Bessel function of the second kind. (*) represents the first-order modified Bessel function of the second kind.

7. The method for calculating the winding coefficient of a hollow cup motor according to claim 2, characterized in that, The coefficients of the second kind of first-order modified Bessel function This can be expressed by the following formula: ； In the formula, The outer radius of the permanent magnet is... For the first m The wave number of the first-order axial harmonic component. (*) represents the zeroth-order modified Bessel function of the second kind. (*) represents the first-order modified Bessel function of the second kind. This represents the magnitude of the remanence of the permanent magnet. The axial pulse function The basis function coefficients according to the Fourier cosine series expansion.

8. The method for calculating the winding coefficient of a hollow cup motor according to claim 1, characterized in that, Based on the aforementioned axial efficiency factor, the traditional pitch coefficient formula is corrected using the following formula: ； In the formula, This represents the total length of a single coil of the winding projected axially. This indicates the length of the straight side segment of a single coil. Indicates the straight-side pitch of a single coil. This represents the axial efficiency factor of the straight edge. This represents the axial efficiency factor of the hypotenuse.

9. An optimization method for a hollow cup motor, characterized in that, The optimization method uses the winding coefficient calculation method described in any one of claims 1 to 8 to calculate the axial efficiency factor, obtaining the straight-side axial efficiency factor. and the axial efficiency factor of the hypotenuse The straight-edge axial efficiency factor Explicitly dependent on the straight-side length of a single coil of the winding The axial efficiency factor of the hypotenuse Explicitly dependent on the straight-side length of a single coil of the winding The total length of a single coil projected axially. and the axial projection length of each side of the inclined segment of a single coil. ; Define axial length-to-diameter ratio The axial major diameter The ratio is the total length of the axial projection of a single coil of the winding. With permanent magnet size h The ratio; Define straight edge ratio The straight edge ratio The length of the straight side segment of a single coil in the winding. The total length of the individual coils of the winding projected in the axial direction The ratio; Based on the axial length-to-diameter ratio and the ratio of the straight edge Based on the coil geometry of the winding, the axial efficiency factor of the straight side is... and the axial efficiency factor of the hypotenuse Perform variable substitution to change the straight-edge axial efficiency factor. and the axial efficiency factor of the hypotenuse Reconstructed respectively with respect to the axial length-to-diameter ratio and the ratio of the straight edge The function; Based on the straight-edge axial efficiency factor after variable substitution and the axial efficiency factor of the hypotenuse The total winding coefficient is calculated. Based on the total winding coefficient and the winding geometric parameters, the effective flux linkage amplitude of each phase of the winding is obtained; Define optimization objectives, which include maximizing the output potential energy and minimizing the conductor volume, wherein the output potential energy is proportional to... Axial length-to-diameter ratio The conductor volume is proportional to the product of the total winding coefficient. In the formula, The axial length-to-diameter ratio is... The straight edge ratio is... The axial efficiency factor of the straight edge after variable substitution and the axial efficiency factor of the hypotenuse The calculated total winding coefficient; Torque constraints, thermal constraints, and manufacturing constraints are set up with the goal of maximizing output potential energy and minimizing conductor volume. A multi-objective optimization algorithm is used to search for the Pareto front, and the final design scheme is obtained based on the knee point criterion.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a processing program configured to execute, when running, the winding coefficient calculation method for a coreless motor as described in any one of claims 1 to 8, or the optimization method for a coreless motor as described in claim 9.

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