Optimal design method for electromagnetic parameters of micro-servo motor

Abstract

This invention discloses a method for optimizing the electromagnetic parameters of a micro servo motor, belonging to the field of motor design technology. The method includes: determining a one-dimensional analytical initial flux linkage value based on pre-inputted motor geometric dimensions and material property data; generating initial parameters containing air gap flux and cogging torque harmonic amplitudes through a surface mapping function; importing the initial parameters into a finite volume method differential solver and introducing a transposition method to construct boundary conditions, obtaining a corrected air gap flux density waveform; establishing a target optimization function based on the corrected air gap flux density waveform and the initial parameters, this function containing torque density and electromagnetic loss components; optimizing the winding turns, pole arc coefficient, and magnet thickness using a sequential quadratic programming method to obtain an optimized parameter range; and determining the final electromagnetic parameter set within the optimized parameter range using numerical extrapolation.

Description

Technical Field

[0001] This invention relates to the field of motor design technology, specifically to a method for optimizing the electromagnetic parameters of a micro servo motor. Background Technology

[0002] The electromagnetic parameter optimization design of micro servo motors directly determines their key performance characteristics such as torque density, efficiency, and torque ripple. In existing technologies, determining the electromagnetic parameters of such motors often relies on multiple iterative calculations using the finite element method (FEM), obtaining a better solution by scanning combinations of design variables such as the number of winding turns, pole arc coefficient, and magnet thickness. This method requires repeatedly solving the two-dimensional or three-dimensional electromagnetic field of the entire motor model in each iteration step, and the computational burden increases dramatically with the increase in parameter dimensions. To reduce computational costs, some schemes use analytical models or equivalent magnetic circuit methods for initial value estimation. However, these simplified models struggle to accurately capture the cogging effect and harmonic components in the air gap magnetic field, leading to significant deviations between the initial parameters and the actual model, thus affecting the efficiency and convergence stability of subsequent optimization paths.

[0003] In the optimization process, existing optimization objectives typically focus only on one aspect: torque magnitude or electromagnetic losses. A weighted summation method is used to transform multiple objectives into a single objective, but the selection of weights is highly subjective, making it difficult to achieve an autonomous balance between torque output capability and electromagnetic losses. Furthermore, for the candidate parameter range obtained through optimization, there is a lack of effective means to establish a reliable verification mechanism between numerical accuracy and the stability of the optimization results. This often leads to performance degradation of candidate parameters under actual nonlinear field coupling conditions. Micro servo motors have stringent requirements for response speed and control accuracy. These shortcomings create a gap between design results and actual needs, urgently requiring a parameter determination method that can balance computational efficiency and model accuracy, and automatically seek the optimal trade-off between torque density and losses. Summary of the Invention

[0004] The technical problem this invention aims to solve is to address the shortcomings of existing micro servo motor electromagnetic parameter optimization design, such as the difficulty in balancing the accuracy and computational efficiency of air gap magnetic field harmonic capture during the initial parameter generation stage, the inability of the optimization target to autonomously balance torque density and electromagnetic loss, and the insufficient numerical stability in the parameter range confirmation stage. This invention provides an optimization design method that can generate high-precision initial magnetic field waveform correction values, construct a product-type target optimization function, and stably determine the parameter set through numerical extrapolation.

[0005] To achieve the above objectives, the present invention provides the following technical solution: The present invention provides a method for optimizing the electromagnetic parameters of a micro servo motor, which combines analytical calculation, finite volume method, and numerical optimization to achieve multi-objective precise optimization of the motor's electromagnetic parameters. The method includes: Based on the pre-input geometric dimensions and material properties of the motor, a one-dimensional analytical initial flux linkage value is determined for the motor in its initial operating state. This initial flux linkage value is used as input to generate initial parameters through a surface mapping function. These initial parameters include the air gap flux and the amplitude of the cogging torque harmonics. The initial parameters are then imported into a finite volume method differential solver, and a transposition method is introduced to construct the boundary conditions of the differential solver, resulting in a corrected air gap flux density waveform. Based on the corrected air gap flux density waveform and the initial parameters, a target optimization function is established, which incorporates the motor's torque density component and electromagnetic loss component. Based on this target optimization function, a sequential quadratic programming method is used to optimize the number of winding turns, pole arc coefficient, and magnet thickness of the motor, obtaining an optimized parameter range. Within this optimized parameter range, a numerical extrapolation method is used to determine the final electromagnetic parameter set.

[0006] As a preferred embodiment of the present invention, when determining the one-dimensional analytical initial flux linkage value, the stator inner diameter, rotor outer diameter, and core stacking coefficient of the motor are extracted as the geometric dimension data, and the remanent magnetization and coercivity of the permanent magnet are extracted as the material property data. A flux linkage analytical function is established with the rotor position angle as the independent variable. The remanent magnetization and coercivity are substituted into the flux linkage analytical function to obtain the one-dimensional analytical initial flux linkage value. This allows for the rapid acquisition of a flux linkage reference value reflecting the initial magnetic field state of the motor from basic electromagnetic parameters, providing an accurate starting point for subsequent parameter generation.

[0007] As another preferred technical solution of the present invention, the process of generating initial parameters through a surface mapping function specifically involves: mapping the one-dimensional analytical initial flux linkage value to a two-dimensional planar coordinate system to form a flux linkage contour distribution; performing Fourier series decomposition on the flux linkage contour distribution to obtain the fundamental component and multiple harmonic components; defining the amplitude of the fundamental component as the air gap flux; and performing modulo operation on the harmonic components whose order is close to the number of tooth slots to obtain the amplitude of the tooth slot torque harmonic. The above processing can reconstruct two-dimensional field characteristics from a single flux linkage curve, accurately extracting initial design parameters sensitive to torque output and torque ripple.

[0008] As a further preferred technical solution of the present invention, when obtaining the corrected air gap magnetic flux density waveform, a differential solver based on the finite volume method divides the solution region of the motor into multiple control volume units. On each control volume unit, a discrete equation for the magnetic vector potential is established according to Ampere's circuital law. The stator slot structure of the motor is equivalent to a periodic permeability function using a substitution method, and this is used as the boundary condition to solve the discrete equation, thereby obtaining the corrected air gap magnetic flux density waveform. Preferably, when constructing the boundary conditions using the substitution method, the permeability function of the stator slot is expressed as a Fourier series, and periodic boundary conditions are used to connect adjacent slot regions. This method significantly reduces the solution scale while preserving the slot effect, balancing the accuracy and efficiency of air gap magnetic field calculation, and making the magnetic flux density waveform closer to the actual physical distribution.

[0009] As a preferred embodiment of establishing the objective optimization function in this invention, the torque density component is calculated based on Maxwell's tensor method and the modified air gap magnetic flux density waveform; the electromagnetic loss component is calculated based on Joule's law and the phase resistance of the motor; the square of the difference between the torque density component and the preset torque value is multiplied by the electromagnetic loss component to obtain the objective optimization function. This optimization function incorporates both torque output capability and energy loss into the evaluation system, ensuring that the optimization direction pursues both high torque density and loss suppression, which is beneficial to improving the overall performance of the servo motor.

[0010] As a preferred embodiment of the present invention for optimizing motor parameters, initial values ​​and optimization boundaries are set for the number of winding turns, the pole arc coefficient, and the magnet thickness, respectively; the first-order partial derivatives of the objective optimization function with respect to the above three parameters are calculated to construct a gradient vector; based on the gradient vector and the Hessian matrix of the objective optimization function, a quadratic programming subproblem is solved to obtain the search direction; a one-dimensional line search is performed on the search direction to update the values ​​of the number of winding turns, the pole arc coefficient, and the magnet thickness until the magnitude of the gradient vector is less than a preset threshold. Specifically, when solving the quadratic programming subproblem, the BFGS formula is used to approximate and update the Hessian matrix of the objective optimization function to obtain an approximate Hessian matrix; the optimization boundary is converted into equality constraints and inequality constraints using the Lagrange multiplier method, and the search direction is obtained by solving the Kuhn-Tak condition based on the approximate Hessian matrix. Furthermore, the multiplier update in the Lagrange multiplier method adopts an augmented Lagrange form, introducing a penalty factor. This optimization process uses the quasi-Newton method to approximate second-order information and ensures that the optimization variables are updated within the range allowed by the technology and performance through constraint processing. It can stably converge to the optimal solution region that satisfies multiple constraints, effectively shortening the motor design cycle.

[0011] As a preferred technical solution for determining the final electromagnetic parameter set in this invention, multiple verification points are selected within the optimized parameter range. The Runge-Kutta method is used to solve the nonlinear field-circuit coupling equation of the motor to obtain the output torque value corresponding to each verification point. Richardson extrapolation is performed on the output torque values ​​of all verification points to obtain the extrapolation limit value. The verification point parameters corresponding to the extrapolation limit value are used as the final electromagnetic parameter set. By eliminating discrete errors through numerical extrapolation, the final determined electromagnetic parameter set has a higher confidence level in the continuous parameter space, effectively improving the torque density of the micro servo motor and reducing electromagnetic losses, thereby achieving accurate prediction of motor performance and system optimization.

[0012] The technical effects and advantages provided by the present invention in the above technical solution are as follows: The one-dimensional analytical initial flux linkage values ​​are mapped to a two-dimensional planar coordinate system using a surface mapping function, forming a flux linkage contour distribution. This distribution is then subjected to Fourier series decomposition to obtain the fundamental component and multiple harmonic components. The fundamental amplitude is used as the air gap flux, and the amplitudes of the harmonic components with orders close to the number of tooth slots are obtained by modulo operation, thus generating initial parameters containing harmonic characteristics. These initial parameters are directly imported into the finite volume method's differential solver. Simultaneously, a substitution method is introduced to equate the stator tooth structure to a periodic permeability function as boundary conditions. The corrected air gap flux linkage waveform is obtained by discretizing the solution on the control volume element based on Ampere's circuital law. This combination method directly provides initial quantities containing cogging torque characteristics by mapping the analytical model, avoiding iterative divergence or slow convergence caused by initial value deviations in numerical solvers without prior information. The substitution method accurately equivalences the cogging region with periodic boundary conditions, and can obtain high-precision air gap magnetic field waveform correction results with low grid density under a finite volume discrete framework, significantly shortening the calculation time from initial parameters to accurate magnetic field distribution.

[0013] In the process of establishing the objective optimization function, the torque density component is calculated using the Maxwell tensor method based on the modified air gap magnetic flux density waveform, and the electromagnetic loss component is calculated based on Joule's law. The square of the difference between the torque density component and the preset torque value is multiplied by the electromagnetic loss component to form the objective function expressed as a product. The winding turns, pole arc coefficient, and magnet thickness are optimized using the sequential quadratic programming method. Constraints are handled using the BFGS approximation of the Hessian matrix and the augmented Lagrange multipliers. After obtaining the candidate parameter set within the optimization interval, the Runge-Kutta method is used to solve the nonlinear field-circuit coupling equation to calculate the output torque value at the verification point. Then, Richardson extrapolation is performed on these torque values, and the parameters corresponding to the extrapolation limit values ​​are used as the final electromagnetic parameter set. The product-type objective function creates a nonlinear constraint relationship between torque deviation and loss, avoiding the need for weighting coefficient selection and automatically achieving coordinated adjustment of torque density and loss during the optimization process. Sequential quadratic programming, guided by gradient information, quickly approximates the extreme values ​​within the constraints, while numerical extrapolation extracts asymptotic limit values ​​based on the calculation results of multiple verification points, eliminating fluctuations caused by numerical discretization or nonlinear coupling of individual calculation nodes. The final parameter set can still stably maintain the optimization objective level under actual field-circuit coupling conditions. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0015] Figure 1 This is a flowchart of the electromagnetic parameter optimization design method for micro servo motors; Figure 2 This is a flowchart of the one-dimensional analytical process for determining the initial flux linkage value; Figure 3 This is a flowchart of the initial parameter generation method based on the surface mapping function; Figure 4 These are one-dimensional analytical initial flux linkage curves corresponding to rotor position angles under different pole pair numbers. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] See Figure 1This invention provides a method for optimizing the electromagnetic parameters of a micro servo motor, comprising the following steps: determining a one-dimensional analytical initial flux linkage value of the motor in its initial operating state based on pre-inputted geometric dimension data and material property data; using the one-dimensional analytical initial flux linkage value as input, generating initial parameters through a surface mapping function, the initial parameters including the air gap flux and the amplitude of the cogging torque harmonics of the motor; importing the initial parameters into a finite volume method differential solver, and simultaneously introducing a transposition method to construct the boundary conditions of the differential solver to obtain a corrected air gap flux density waveform; establishing a target optimization function based on the corrected air gap flux density waveform and the initial parameters, the target optimization function including the torque density component and the electromagnetic loss component of the motor; optimizing the number of winding turns, pole arc coefficient, and magnet thickness of the motor using a sequential quadratic programming method to obtain an optimized parameter range; and determining the final electromagnetic parameter set within the optimized parameter range using a numerical extrapolation method. Example 1:

[0018] In specific implementation, please refer to Figure 2 The pre-input motor geometry data includes the stator inner diameter. Rotor outer diameter axial length of iron core Core stacking coefficient Extreme logarithms Initial value of the number of turns in series per phase Fundamental winding coefficient Etc.; Pre-input material property data includes the remanence density of the permanent magnet. and coercivity Extract the stator inner diameter of the motor. Rotor outer diameter and core stacking coefficient As geometric dimension data; extracting the remanence density of the permanent magnet. and coercivity As material property data. Established based on rotor position angle. Let be the flux linkage analytical function with respect to the independent variable. When establishing the flux linkage analytical function, first determine the initial magnet thickness. Initial magnet thickness Based on stator inner diameter With rotor outer diameter The difference setting is as follows: The ratio of 0.8 is set because in the initial design of surface-mount micro servo motors, the ratio of the permanent magnet thickness to the total gap on one side is usually selected as 0.8 to establish a sufficiently large magnetic flux density in the air gap. (Single-side air gap length) Depend on Confirmed. The restoring permeability of the permanent magnet at the operating point. Through coercivity and remanence calculate, ,in Vacuum permeability. Air gap magnetic flux density amplitude. According to Ohm's law for magnetic circuits, it can be expressed as: .Will Substituting the expression, we get Effective length of the iron core Based on the pre-input core axial length and core stacking coefficient get, Magnetic flux per pole for Amplitude of unloaded flux linkage in one phase for The final system established is based on the rotor position angle. The flux linkage analytic function with respect to the independent variable is: in, This represents the initial flux linkage value in one-dimensional analysis. Indicates the rotor position angle. This indicates the initial number of series turns per phase as input. Indicates the fundamental winding coefficient. Indicates the stator inner diameter. Indicates the core stacking factor. This indicates the pre-input axial length of the iron core. This represents the remanence density of a permanent magnet. Represents the extreme logarithm. Indicates the rotor outer diameter, Indicates the initial magnet thickness. The value is , Represents the permeability of free space. The value is H / m, This represents the coercivity of the permanent magnet. The extracted remanence density... and coercivity And by substituting the geometric dimensions into the above flux linkage analytical function, for any given rotor position angle... The corresponding one-dimensional analytical initial flux linkage value can be calculated. This allows for the determination of the initial flux linkage value in one-dimensional analysis.

[0019] See Figure 4 In the figure, the horizontal axis represents the rotor position angle θ, ranging from 0 to 2π radians, and the vertical axis represents the one-dimensional analytical initial flux linkage value ψ, in Weber (Wb). The legend shows the flux linkage waveform variation trends when the pole pair number p takes 2, 4, and 6, respectively.

[0020] When p=2, the period of the flux linkage waveform is the longest, and it is a single cosine waveform. The flux linkage amplitude is the largest, about ±0.16Wb. The peak and trough appear at θ=0 and θ=π respectively, which is consistent with the characteristics of the fundamental period of the cosine function in the formula (analytical function of flux linkage in Example 1).

[0021] When p=4, the period of the flux linkage waveform is halved, exhibiting two complete cosine periods. The amplitude drops significantly to about ±0.08Wb, which is about half of that when p=2. The frequency of the waveform doubles, reflecting the characteristic that the increase in the number of pole pairs leads to a doubling of the frequency of the flux linkage waveform.

[0022] When p=6, the period of the flux linkage waveform is further shortened, and three complete cosine periods appear. The amplitude is further reduced to about ±0.05Wb, and the waveform frequency is three times that when p=2. This shows that the increase in the number of pole pairs makes the flux linkage frequency and amplitude change inversely. Example 2:

[0023] In specific implementation, please refer to Figure 3 The surface mapping function is used to generate the amplitude of the air gap flux and cogging torque harmonics of the motor based on the one-dimensional analytical initial flux linkage values. It maps the one-dimensional analytical initial flux linkage values ​​to a two-dimensional plane coordinate system, forming a flux linkage contour distribution. Specifically, in the one-dimensional analytical initial flux linkage values... In the middle, rotor position angle For angle variables, For the corresponding flux linkage value; set the number of rotor position angle sampling points. and the number of initial phase offset sampling points , The value is 1024. The value is 1024, chosen because the two-dimensional discrete Fourier transform requires a sampling number that is an integer power of 2 to facilitate the use of the fast Fourier transform algorithm. 1024 provides sufficient frequency resolution. Define a two-dimensional array. row index The range of values ​​is Column index The range of values ​​is Row index The corresponding rotor position angle sampling value is Column index The corresponding initial phase offset sample value is Two-dimensional array The element values ​​are determined according to the following formula: in, This represents the value of the magnetic flux linkage contour lines distributed across discrete sampling points. This represents the one-dimensional analytical initial flux linkage value function obtained from Example 1. Indicates the first Each rotor position angle sample value Indicates the first One initial phase offset sample value, This represents a modulo operation, which keeps the angle at a certain value. Within the range. By combining the flux linkage waveform of one rotor rotation with different initial phase offsets in the above manner, a two-dimensional matrix is ​​formed, which is the discrete representation of the flux linkage contour distribution.

[0024] In practical implementation, the magnetic flux linkage contour distribution is decomposed using Fourier series to obtain the fundamental component and multiple harmonic components. A two-dimensional discrete Fourier transform is then applied to the two-dimensional array. Decomposing the data, the formula for the two-dimensional discrete Fourier transform is as follows: in, Represents the coefficients of the two-dimensional discrete Fourier transform. Indicates the harmonic order in the row direction. The range of values ​​is , Indicates the harmonic order in the column direction. The range of values ​​is , and These are row indexes and column indexes, respectively. and These represent the number of sampling points in the row direction and the number of sampling points in the column direction, respectively. Represents the imaginary unit. This represents the base of the natural logarithm. The fundamental component obtained from Fourier decomposition corresponds to... coefficient of time The multiple harmonic components correspond to and Coefficients that are not both zero .

[0025] In practical implementation, the amplitude of the fundamental component is defined as the air gap magnetic flux. Air gap magnetic flux We obtain it according to the following formula: in, This represents the air gap magnetic flux. This represents the magnitude of the fundamental component coefficient.

[0026] In practical implementation, the harmonic components whose order is close to the number of stator slots are modulo-calculated to obtain the amplitude of the cogging torque harmonic. Specifically, the stator slot number of the motor is obtained. , It is the integer part of the ratio of the number of stator slots to the number of pole pairs in the motor. Searching for multiple harmonic components, the harmonic order in the row direction... absolute value and closest harmonic pair Select the harmonic order in the column direction The components. The amplitude of the cogging torque harmonic. The following can be obtained by taking the modulus of the harmonic component coefficient: in ,Right now Take and The harmonic order with the smallest absolute value of the difference is then determined. This completes the generation of the initial parameters, including the air gap magnetic flux. Amplitude of cogging torque harmonics . Example 3:

[0027] In practical implementation, the finite volume method's differential solver is used to calculate the two-dimensional electromagnetic field distribution of the motor. Based on the finite volume method's differential solver, the solution domain of the motor is divided into multiple control volume elements. The two-dimensional cross-sectional solution domain of the motor includes the rotor core, permanent magnets, air gap, stator teeth, and stator yoke. A cylindrical coordinate system is used for meshing, and the radial coordinate system uses... Indicates that the circumferential angle coordinates are... Indicated. The radial direction from the center of the shaft to the outer diameter of the stator is divided into... layer, The value is 60; in the circumferential direction, a complete pole pitch or a slot pitch is divided into... part, The value is set to 120. The radial mesh is refined within the air gap region, with the number of radial layers in the air gap being [number missing]. The value is set to 12 to ensure the accuracy of the air gap magnetic field calculation. Each control volume element is a fan-shaped annular region, with the center point of the control volume element serving as a node, and the magnetic vector potential at the node... The variable to be solved.

[0028] In practical implementation, a discrete equation for the magnetic vector potential is established for each control volume element based on Ampere's circuital law. For any control volume element, the integral form of Ampere's circuital law is expressed as: Magnetic Vector Potential The line integral along the boundary of the control volume element is associated with the current flowing through the interior of that control volume element. In a two-dimensional static magnetic field, the governing equations simplify to... ,in The magnetic permeability of the material, Let be the axial current density. Integrating the governing equation over the control volume element, and using Gauss's theorem, transforming the volume integral into a boundary flux integral, yields the result for the node-specific equation. Discrete equations: ,in , , , , Representing nodes respectively The magnetic vector potential of its adjacent nodes in the east, west, south, and north directions. , , , , For discrete coefficients, The source term contributes to the current density. The dispersion coefficients are determined by the permeability and interface length at the control volume element interface. The permanent magnet region is treated using an equivalent surface current sheet model, incorporating the excitation effect generated by the permanent magnet into the source term. .

[0029] In practical implementation, based on the transposition method, the stator slot structure of the motor is equivalent to a periodic permeability function, and this is used as the boundary condition. When constructing the boundary condition using the transposition method, the permeability function of the stator slot is expressed in Fourier series form: in, This indicates that the stator tooth structure varies with angle in the circumferential direction. The changing equivalent permeability function; This represents the average component of the stator cogging permeability. The permeability distribution within a single slot spacing is obtained by spatial averaging, with units of . ; Indicates the first Permeability amplitude of the second harmonic component It is determined by the ratio of the slot width to the tooth pitch, and is extracted through static magnetic field calculation; Indicates the harmonic order; This represents the truncation order of a Fourier series. The value of 9 is based on the fact that after Fourier decomposition of the permeability distribution of the actual stator slot, the relative root mean square error of the permeability waveform reconstructed by the first 9 harmonic components and the original permeability waveform is reduced to less than 0.3%, which can fully characterize the cogging effect. This represents the number of stator slots, specifically the integer part of the ratio of the number of stator slots to the number of pole pairs. During the transposition method, the actual geometry of the stator teeth and slots is no longer meshed. Instead, on the surface of the first layer of control volume elements on the air gap side at the air gap-stator interface, the material permeability is set to an equivalent permeability function that varies according to the aforementioned Fourier series. This allows the magnetic permeability modulation effect of the tooth groove to be introduced into the finite volume method calculation.

[0030] In practical implementation, periodic boundary conditions are used to connect adjacent slot regions. The period of the stator slot structure in the circumferential direction is... Along a circular axis at a certain midpoint in the radial direction of the air gap region, the magnetic field distribution of adjacent slot regions exhibits repetition. At the circumferential boundaries of the finite volume method computational domain, i.e., at an angle of... and Periodic boundary conditions are set on the two radii: Meanwhile, the normal derivative of the magnetic vector potential along the radial direction on the boundary also satisfies the continuity condition. By using the periodic boundary condition, the magnetic vector potential distribution on the entire circumference of the motor can be obtained by periodic extension by discretizing and solving only a single slot pitch region.

[0031] In practical implementation, the corrected air gap magnetic flux density waveform is obtained by solving the discrete equations. The discrete equations established on each control volume element are combined with the periodic boundary conditions and the parallelism condition of the outer boundary magnetic field lines to construct a system of linear algebraic equations. This system of linear algebraic equations is solved using the conjugate gradient method with incomplete Cholesky decomposition of the preconditions. The convergence criterion for the iterative solution is set as the residual vector. The norm decreases to the initial norm. The magnetic vector potential at all nodes is obtained after solving the problem. Along the circumference of the air gap center, i.e., the radial coordinate... The location, among which The outer diameter of the rotor. For the inner diameter of the stator, extract the nodes on the circumference. The radial component of the air gap magnetic flux density was calculated using a central difference scheme. and tangential components Radial component tangential component The calculated distribution sequence of air gap magnetic flux along the circumference is the corrected air gap magnetic flux waveform. Example 4:

[0032] In practice, the process of establishing the objective optimization function consists of three steps. The first step involves calculating the torque density component based on Maxwell's tensor method and the corrected air gap magnetic flux density waveform. A closed integration path is constructed along the circumference of the air gap centerline, with the radius of this closed integration path being the radius of the air gap centerline. , ,in The outer diameter of the rotor. Let be the stator inner diameter. On a closed integral path, Maxwell's tensor method expresses the electromagnetic torque as a circular integral of the tangential magnetic stress. The discrete form of the torque calculation formula is: ,in Indicates electromagnetic torque. Indicates the effective length of the iron core. Based on the pre-input core axial length With core overlap coefficient Multiplying them together, we get Represents the permeability of free space. H / m, This represents the number of circumferential discrete segments on a closed integral path. The value is 720. Indicates the first The radial component of the corrected air gap magnetic flux density waveform on each discrete segment. Indicates the first Tangential components of the corrected air gap magnetic flux density waveform on discrete segments Indicates the first The angular intervals corresponding to each discrete segment. The cylindrical volume of the outer surface of the motor rotor. for Torque density component Defined as the ratio of electromagnetic torque to the volume of the cylinder on the outer surface of the rotor. The unit is .

[0033] In the practical implementation, the second step involves calculating the electromagnetic loss components based on Joule's law and the phase resistance of the motor. The phase resistance of the motor... The calculation involves the number of winding turns. Average length of single-turn coil in stator slot pre-input Pre-input conductor cross-sectional area and the conductivity of copper wires electrical conductivity The value is taken at a reference temperature of 20℃. S / m. Phase resistance The calculation formula is: Phase current amplitude of the motor Based on the pre-input rated current density and conductor cross-sectional area Sure, Electromagnetic loss component This represents the total copper loss of the three-phase winding. Phase current amplitude and phase resistance Substituting the expression, we obtain the electromagnetic loss component. expansion .

[0034] In the specific implementation, the third step involves squaring the difference between the torque density component and the preset torque value, then multiplying this squared value by the electromagnetic loss component to obtain the target optimization function. (Preset torque value) Given the motor design requirements, the objective optimization function is... The expression is Objective optimization function The construction makes the torque density component Approaching the preset torque value At the same time, electromagnetic loss component Minimize the objective function The optimal solution is the design variable that yields the minimum value when both factors are weighed.

[0035] In practical implementation, sequential quadratic programming is used to optimize the number of winding turns, pole arc coefficient, and magnet thickness of the motor. Initial values ​​and optimization boundaries are set for each of these parameters. (Number of winding turns...) initial value Set as the initial value of the number of series turns per phase as pre-input Finding the optimal lower boundary Set as 0.5 times, seeking the upper boundary Set as The 2.0 times, and the basis for choosing 0.5 times and 2.0 times, is that the adjustment range of the number of turns of the micro servo motor is constrained by the slot fill factor and winding inductance matching conditions. A number of turns exceeding this range will cause the slot fill factor to exceed the limit or the electrical time constant to fail to meet the response bandwidth requirements. (Polar arc coefficient) initial value Set to 0.78, optimize the lower boundary. Set to 0.60, optimize the upper boundary. The initial value of 0.78 is set to 0.95, and the basis for choosing 0.78 is that the theoretical value of the polar arc coefficient corresponding to the maximum amplitude of the fundamental wave under the ideal square wave air gap magnetic flux density distribution is... The criteria for choosing 0.60 and 0.95 as boundaries are as follows: when the pole arc coefficient is below 0.60, the magnetic flux per pole decreases significantly, leading to insufficient torque capacity; when the pole arc coefficient is above 0.95, the inter-pole leakage flux increases sharply, and the structural strength is difficult to guarantee. Magnet thickness. initial value Set as the initial magnet thickness Finding the optimal lower boundary Set as 0.6 times, seeking the upper boundary Set as The basis for choosing 1.5 times, 0.6 times and 1.5 times is that too thin magnets will cause irreversible demagnetization risk, while too thick magnets will excessively compress the air gap space and increase costs.

[0036] In practical implementation, the first-order partial derivatives of the objective optimization function with respect to the number of winding turns, pole arc coefficient, and magnet thickness are calculated to construct the gradient vector. This is done in the sequential quadratic programming method... In the next iteration, design the variable vector. Based on the number of winding turns Polar arc coefficient and magnet thickness constitute, Gradient vector The three components are the objective optimization function. For the number of winding turns Polar arc coefficient and magnet thickness The partial derivatives. Due to the objective optimization function... Since there is no analytical expression, the first-order partial derivatives are calculated using the finite difference method. (For the number of winding turns...) Fixed polar arc coefficient and magnet thickness Keep the number of winding turns unchanged. Add a small perturbation , Values The finite volume method's differential solver is re-executed to calculate the new corrected air gap magnetic flux density waveform, and then the objective function is recalculated based on the new corrected air gap magnetic flux density waveform. The value of the partial derivative Approximately central difference Polar arc coefficient and magnet thickness Calculate the partial derivatives in the same way and The disturbance of the polar arc coefficient The value is 0.0001, representing the disturbance amount of the magnet thickness. Values Therefore, the gradient vector is constructed. .

[0037] In practical implementation, the quadratic programming subproblem is solved based on the gradient vector and the Hessian matrix of the objective function to obtain the search direction. The BFGS formula is used to approximately update the Hessian matrix of the objective function, resulting in an approximate Hessian matrix. In the... In the next iteration, the approximate Hessian matrix is ​​denoted as... , dimension , representing the objective optimization function The second-order partial derivatives approximate the matrix with respect to the three design variables. In the first iteration, Let it be the identity matrix. After obtaining the... Search direction in the next iteration and step size factor Then, the update amount of the design variables. The change in the gradient vector .

[0038] Using the Lagrange multiplier method, the optimization boundary is transformed into equality constraints and inequality constraints. The constraints of the optimization boundary are: , , , , , The multiplier update in the Lagrange multiplier method uses an augmented Lagrange form, introducing a penalty factor. Augmented Lagrange function. Defined as ,in Denotes the Lagrange multiplier vector. Index representing the constraint condition. This represents the penalty factor vector. Penalty factor The initial value is set to The basis is within the augmented Lagrange framework. An initial penalty factor of magnitude can prevent numerical ill-conditioning while ensuring that constraint violations are sufficiently small. Lagrange multipliers The initial value is set to zero. After each iteration of the quadratic programming sequence, the Lagrange multipliers are updated: Punishment factor In the amount of constraint violation Compared to the constraint violation amount in the previous iteration If the descent is insufficient, amplification is applied, with an amplification factor of 2.0. This is done in the approximate Hessian matrix. Based on this, the subproblem of quadratic programming is expressed as: minimizing Satisfying constraints , The search direction is obtained by solving the Kuhn-Tak condition. The Kuhn-Tak conditions include the stationarity condition that the linear combination of the constraint gradient and the Lagrange multipliers equals zero gradient, the complementary relaxation condition, and the original constraint condition. The effective set method is used to solve this quadratic programming subproblem, yielding the search direction that satisfies the Kuhn-Tak conditions. And the corresponding Lagrange multiplier estimates.

[0039] In practice, a one-dimensional line search is performed along the search direction to update the winding turns, pole arc coefficient, and magnet thickness. The one-dimensional line search employs an imprecise line search method with strong Wolfe conditions, starting from an initial trial value of the step factor. Starting from this point, gradually reduce the step size factor until the sufficient descent condition and curvature condition are met. The sufficient descent condition requires... , where the coefficient Values Curvature requirements , where the coefficient Values .Pick and The basis is the standard strong Wolfe condition parameter values ​​in sequential quadratic programming, which can ensure both sufficiency of the linear search and convergence efficiency. After finding the step size factor that satisfies the strong Wolfe condition, the design variables are updated. The approximate Hessian matrix is ​​updated using the BFGS formula, which is as follows: The convergence criterion for the sequential quadratic programming method is the gradient vector. Length of the module Less than a preset threshold, preset threshold Values ,Pick The basis is that the order of magnitude of the design variables is 1 to... When the gradient norm is less than This indicates that the objective function has reached a flat region with respect to changes in the design variables, and the optimality condition is approximately satisfied. When the magnitude of the gradient vector is less than a preset threshold... When the iteration stops, the point where the current design variable is located is taken as the optimization result of the sequential quadratic programming method, and the number of winding turns at this point is recorded. Polar arc coefficient and magnet thickness . Example 5:

[0040] In practical implementation, after the sequential quadratic programming method completes the optimization, an optimized parameter interval containing the optimal design variable points is obtained. This interval consists of the winding turns interval, the pole arc coefficient interval, and the magnet thickness interval. ,in This represents the number of winding turns when the sequential quadratic programming method stops iterating. Indicates the half-width of the winding turns interval. Values , The initial value for the number of turns per phase in series is used as the pre-input value. The coefficient 0.05 is chosen to cover the global optimum that might be missed due to gradient threshold truncation, within a 5% variation in the number of turns near the optimal solution. The polar arc coefficient range is... ,in This represents the polar arc coefficient value when the sequential quadratic programming method stops iterating. This represents the half-width of the polar arc coefficient range. The value is set to 0.015. This value is chosen because the engineering manufacturing precision of the polar arc coefficient is typically on the order of 0.01 to 0.02; variations within this range will not render the structure unfeasible. The magnet thickness range is... ,in This represents the magnet thickness value at which the sequential quadratic programming method stops iterating. This represents the half-width of the magnet thickness range. Values , The initial magnet thickness is set to 0.03 because the change in electromagnetic performance tends to be gradual when the magnet thickness is adjusted within 3% of the optimal solution.

[0041] In practical implementation, multiple verification points are selected within the optimized parameter range. A three-factor, three-level full factorial experimental design is used to select verification points: three levels are selected within the winding turns range, namely... , , Three levels are taken within the polar arc coefficient range, namely: , , Three levels were selected within the thickness range of the magnet, namely... , , The three factors, combined at three levels, form a total of 27 verification points. Each verification point corresponds to a set of motor design parameters, denoted as... ,in For the checkpoint index, .

[0042] In practical implementation, the Runge-Kutta method is used to solve the nonlinear field-circuit coupling equations of the motor, obtaining the output torque value corresponding to each verification point. The nonlinear field-circuit coupling equations of the motor are established in a rotating coordinate system, with the phase winding current as the reference. , Rotor mechanical angular velocity and rotor position angle Let the variables be state variables, and the equation be in the form of: , , , ,in Indicates direct-axis inductance. Indicates quadrature axis inductance. Indicates phase resistance. Represents electric angular velocity. , For extreme logarithms, and For direct-axis and quadrature-axis voltages, It is a permanent magnet flux chain. For rotational inertia, The damping coefficient is... This refers to the load torque. Electromagnetic torque. The modified air gap magnetic flux density waveform is calculated in real time using the Maxwell tensor method. In each integration time step, the above field-circuit coupling equations are first updated using the current winding current and rotor position as inputs, and then the finite volume method differential solver is invoked to update the air gap magnetic flux density distribution and calculate the electromagnetic torque. Then the electromagnetic torque Substitute the equations into the mechanical equations and perform the integration at the next time step.

[0043] In its implementation, the Runge-Kutta method uses a fourth-order classical form. The state vector... From time Advance to The recursive format is: in, express The state vector at time t, express The state vector at any given time; Indicates the time step. The value is the electric cycle of the motor. of , , The step size of 1 / 200 of the electric period is chosen as the rated electric angular velocity, based on the local truncation error of the Runge-Kutta fourth-order method. Proportional to the step size, taking a step size of 1 / 200 of the electrical cycle can reduce the single-step integration error to less than [a certain value]. Magnitude; , , , The four intermediate increment vectors representing the Runge-Kutta method are calculated as follows: , , , ,function The right-hand side of the field-circuit coupling equation is determined by the voltage equation, the mechanical equation, and the electromagnetic torque calculation.

[0044] In practice, the solution is started from a zero initial state, and the same working condition simulation is performed for each verification point: rated voltage excitation is applied, and load torque is... Set to preset torque value 0.8 times that, simulation continued One electrical cycle, take the last one The average electromagnetic torque over one electrical cycle is taken as the output torque value. For each verification point... With time step Perform a complete simulation to obtain the output torque value. Then, using time steps Repeat the full simulation once to obtain the output torque value. Double the simulation time steps when the step size is halved to cover the same physical duration.

[0045] In practice, Richardson extrapolation is performed on the output torque values ​​at all verification points to obtain the extrapolated limit values. For the verification points... Richardson extrapolation is performed using the theoretical accuracy order of the Runge-Kutta fourth-order method. The extrapolation formula is as follows: Global error order of Runge-Kutta's fourth-order method The value is 4. The extrapolation formula is specified as follows: Eliminating the time step through Richardson extrapolation. The main truncation error term is used to obtain the check point. Approximate accurate output torque value After extrapolation was performed at all 27 verification points, 27 extrapolation limit values ​​were obtained. .

[0046] In practice, the parameters of the verification points corresponding to the extrapolated limit values ​​are used as the final set of electromagnetic parameters. From all verification points, the extrapolated limit values ​​are selected. With preset torque value The checkpoint with the smallest absolute difference between the two is denoted by its index. ,Right now . Index Corresponding verification point parameters As the final set of electromagnetic parameters, the final set of electromagnetic parameters includes the number of winding turns. Polar arc coefficient and magnet thickness .

[0047] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A method for optimizing the electromagnetic parameters of a micro servo motor, characterized in that, Includes the following steps: Based on the pre-input geometric dimensions and material properties of the motor, the one-dimensional analytical initial flux linkage value of the motor in its initial operating state is determined. The one-dimensional analytical initial flux linkage value is used as input, and initial parameters are generated through a surface mapping function. The initial parameters include the air gap flux of the motor and the amplitude of the cogging torque harmonic. The initial parameters are imported into the finite volume method's differential solver, and the substitution method is introduced to construct the boundary conditions of the differential solver, thus obtaining the corrected air gap magnetic flux density waveform. Based on the corrected air gap magnetic flux density waveform and the initial parameters, a target optimization function is established, which includes the torque density component and electromagnetic loss component of the motor. Based on the objective optimization function, the number of winding turns, pole arc coefficient, and magnet thickness of the motor are optimized using the sequential quadratic programming method to obtain the optimal parameter range; Within the optimized parameter range, the final electromagnetic parameter set is determined by numerical extrapolation.

2. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The one-dimensional analytical initial flux linkage value of the motor in its initial operating state is determined based on the pre-input geometric dimension data and material property data of the motor, specifically as follows: The stator inner diameter, rotor outer diameter, and core stacking coefficient of the motor are extracted as the geometric dimension data; The remanence and coercivity of the permanent magnet are extracted as material property data. Establish an analytical function for flux linkage with rotor position angle as the independent variable; Substituting the remanent magnetization and the coercivity into the magnetic flux linkage analytical function yields the one-dimensional analytical initial magnetic flux linkage value.

3. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The initial parameters are generated using the surface mapping function, specifically as follows: The one-dimensional analytical initial flux linkage value is mapped to a two-dimensional plane coordinate system to form a flux linkage contour distribution; The magnetic flux linkage contour distribution is decomposed into Fourier series to obtain the fundamental component and multiple harmonic components. The amplitude of the fundamental component is defined as the air gap magnetic flux; The amplitude of the cogging torque harmonic is obtained by taking the modulus of the harmonic components whose order is close to the number of tooth slots.

4. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The process involves importing the initial parameters into the finite volume method's differential solver, and simultaneously introducing the substitution method to construct the boundary conditions of the differential solver, resulting in the corrected air gap magnetic flux density waveform. Specifically: Based on the differential solver of the finite volume method, the solution domain of the motor is divided into multiple control volume units; For each of the control volume units, a discrete equation for the magnetic vector potential is established based on Ampere's circuital law; Based on the aforementioned transposition method, the stator cogging structure of the motor is equivalent to a periodic permeability function, and this is used as the boundary condition. The corrected air gap magnetic flux density waveform is obtained by solving the discrete equation.

5. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The establishment of the objective optimization function is specifically as follows: The torque density component is calculated based on Maxwell's tensor method and the modified air gap magnetic flux density waveform. The electromagnetic loss components are calculated based on Joule's law and the phase resistance of the motor. The target optimization function is obtained by squaring the difference between the torque density component and the preset torque value and multiplying it by the electromagnetic loss component.

6. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The optimization of the motor winding turns, pole arc coefficient, and magnet thickness using sequential quadratic programming is as follows: Initial values ​​and optimization boundaries are set for the number of turns of the winding, the pole arc coefficient, and the thickness of the magnet, respectively; Calculate the first partial derivatives of the objective optimization function with respect to the number of winding turns, the pole arc coefficient, and the magnet thickness, and construct the gradient vector; Based on the gradient vector and the Hessian matrix of the objective optimization function, solve the quadratic programming subproblem to obtain the search direction; A one-dimensional line search is performed in the search direction to update the values ​​of the number of winding turns, the polar arc coefficient, and the magnet thickness until the magnitude of the gradient vector is less than a preset threshold.

7. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 6, characterized in that, The step of solving the quadratic programming subproblem based on the gradient vector and the Hessian matrix of the objective optimization function is as follows: The Hess matrix of the objective optimization function is updated approximately using the BFGS formula to obtain an approximate Hess matrix; Using the Lagrange multiplier method, the optimization boundary is transformed into equality constraints and inequality constraints; Based on the approximate Hessian matrix, the search direction is obtained by solving the Kuntak condition.

8. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 1, characterized in that, The determination of the final electromagnetic parameter set using numerical extrapolation is specifically as follows: Multiple verification points are selected within the optimized parameter range; The nonlinear field-circuit coupling equation of the motor is solved using the Runge-Kutta method to obtain the output torque value corresponding to each of the verification points; Richardson extrapolation was performed on the output torque values ​​at all the aforementioned verification points to obtain the extrapolation limit values; The verification point parameters corresponding to the extrapolated limit value are used as the final electromagnetic parameter set.

9. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 4, characterized in that, When constructing boundary conditions using the transposition method, the permeability function of the stator slot is expressed as a Fourier series, and periodic boundary conditions are used to connect adjacent slot regions.

10. The method for optimizing the electromagnetic parameters of a micro servo motor according to claim 7, characterized in that, The multiplier update in the Lagrange multiplier method adopts the augmented Lagrange form and introduces a penalty factor.

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