Torque quality analysis and design method of an alternate-pole permanent magnet Vernier motor

Abstract

This invention discloses a torque quality analysis and design method for an alternating pole permanent magnet vernier motor, belonging to the technical field of electromagnetic performance analysis and optimization design of permanent magnet motors. First, after obtaining the amplitude and phase information of each order of spacetime harmonics in the air gap magnetic flux density, an analytical expression for electromagnetic torque is constructed using Maxwell's stress tensor method. This yields the conditional relationship between the average torque and torque ripple generated by spacetime harmonics, enabling quantitative calculation and separation analysis of the contributions of different orders of spacetime harmonics to the average torque and torque ripple. Second, based on the stator and rotor magnetic permeability modulation functions, the function expressions for the permanent magnet magnetic field and armature magnetic field of the alternating pole permanent magnet vernier motor are constructed. The key design parameters of the motor associated with spacetime harmonics are clarified, and a multi-objective optimization model is established to improve the torque quality of the motor. This invention can not only accurately analyze the output torque performance of an alternating pole permanent magnet vernier motor, but also achieve stator and rotor structural parameter optimization, torque ripple suppression design, and torque quality improvement design.

Description

Technical Field

[0001] This invention relates to electromagnetic performance analysis and optimization design technology for permanent magnet motors, specifically to a torque quality analysis and design method for an alternating pole permanent magnet vernier motor. Background Technology

[0002] Permanent magnet vernier motors are motor structures that achieve low-speed, high-torque output based on the principle of magnetic field modulation. They offer advantages such as high torque density and small size, and have promising application prospects in fields such as electric vehicles, aerospace, and robotics. Traditional surface-mount permanent magnet vernier motors typically employ an alternating polarity permanent magnet structure to create a high pole-log magnetic field, but this results in a large amount of permanent magnets and some magnetic leakage. Alternating pole permanent magnet vernier motors, by using a unipolar excitation method with permanent magnets, utilize a rotor salient-pole iron core as an equivalent magnetic pole of opposite polarity, forming a pair of magnetic poles together with the permanent magnets. This reduces the amount of permanent magnets used while effectively reducing magnetic leakage, enhancing the main magnetic flux, and improving magnetic flux utilization. Thus, while reducing material costs, the motor's output torque and other performance characteristics remain essentially unchanged, attracting increasing attention.

[0003] However, due to the unipolar excitation of permanent magnets and the participation of salient pole cores in the magnetic circuit construction, coupled with the periodic variation of air gap permeability caused by the doubly salient pole structure, the air gap magnetic field contains abundant spatiotemporal harmonic components. These harmonics easily cause significant torque ripple, reducing the smoothness of motor operation. Existing research mostly uses the one-dimensional Fourier method to decompose the air gap magnetic flux density into spatial harmonics, which can only analyze the contribution of each spatial harmonic to the average torque, making it difficult to reveal the influence of spatiotemporal harmonics formed by time-space coupling on torque ripple, and lacking quantitative analysis methods for torque ripple under alternating pole structures. Therefore, in order to analyze the torque quality of alternating pole permanent magnet vernier motors and provide a theoretical basis for motor structure optimization and performance improvement, it is necessary to start from the doubly salient pole structure of alternating pole permanent magnet vernier motors, systematically analyze the sources of spatiotemporal harmonics and their interaction relationships, and study the decomposition and quantification of average torque and torque ripple. Summary of the Invention

[0004] The purpose of this invention is to propose a torque quality analysis and design method for an alternating pole permanent magnet vernier motor. This method mainly includes quantifying the contribution of spatiotemporal harmonics to the motor torque and establishing the permanent magnet magnetomotive force and armature magnetomotive force under the modulation of a doubly salient pole stator and rotor. The derived magnetomotive force is used to accurately analyze the order and source of spatiotemporal harmonics, providing a theoretical basis for the low torque ripple design of the alternating pole permanent magnet vernier motor.

[0005] Specifically, the technical solution adopted in this invention is: a torque quality analysis and design method for an alternating pole permanent magnet vernier motor, comprising the following steps:

[0006] Step 1: Using two-dimensional Fourier analysis, the amplitude and phase of the radial and tangential air gap magnetic flux density of the motor are obtained. Based on Maxwell's stress tensor method, the conditional relationship that contributes to the average torque and torque ripple of the motor is clarified, and the contribution of spatiotemporal harmonics to the average torque and torque ripple of the motor is accurately calculated.

[0007] Step 2: Based on the unipolar excitation and double salient pole structure of the motor rotor, construct the stator and rotor magnetic permeability modulation function expression of the air gap magnetic field of the motor, so as to provide a unified theoretical basis for the subsequent mathematical derivation of the permanent magnet magnetic field and armature magnetic field modulation process.

[0008] Step 3: Considering the coexistence of odd and even harmonics in the permanent magnet magnetic field under unipolar excitation of the permanent magnet, the expression of the air gap magnetic flux density of the permanent magnet magnetic field is derived under the action of stator and rotor magnetic permeability modulation. The main spatiotemporal harmonic orders and distribution laws generated after modulation are clarified, and the spatiotemporal harmonic order characteristics generated by the magnetic field are analyzed.

[0009] Step 4: After the motor is supplied with a five-phase symmetrical sinusoidal current, a synchronous rotating armature magnetic field is generated. Under the action of stator and rotor magnetic permeability modulation, the expression of the air gap magnetic flux density of the armature magnetic field is derived, and the spatiotemporal harmonic order distribution law generated by the armature magnetic field is analyzed, thereby fully revealing the composition characteristics of the armature side harmonic order.

[0010] Step 5: By calculating the spatiotemporal harmonic orders that significantly contribute to the average torque and torque ripple, and using magnetomotive force and magnetic permeability as a bridge, the motor design parameters associated with the source of the spatiotemporal harmonic orders are clarified. Thus, the key motor design parameters as design variables can be determined, and the average torque and torque ripple of the motor can be used as optimization targets.

[0011] Step 6: After determining the key design variables, select several parameter combination sample points through experimental design methods and calculate and analyze the motor torque performance; establish a response surface model between design parameters and performance indicators based on the obtained data results, obtain the approximate functional relationship between each design variable and the objective function, and analyze the influence trend of each design parameter on the performance indicators to evaluate the model accuracy.

[0012] Step 7: Based on the established high-precision model, a multi-objective genetic algorithm is introduced to optimize and solve the key design parameters of the motor. By comprehensively analyzing the Pareto solution set obtained during the optimization process, the relationship between various optimization objectives is weighed, and the Pareto optimum is selected as the final optimized design parameter.

[0013] Step 8: Substitute the optimized design parameters into the motor model to establish the electromagnetic analysis model of the optimized motor. Calculate the electromagnetic torque of the optimized motor using the finite element simulation method, and compare the optimized torque performance with the torque performance before optimization to verify the effectiveness and accuracy of the motor design method guided by the spatiotemporal harmonic theory.

[0014] Furthermore, the alternating pole permanent magnet vernier motor has 20 slots and 23 pole pairs. The motor consists of three parts: a hybrid stator, an air gap, and a salient pole rotor. The hybrid stator is composed of alternating straight teeth and split teeth, wherein the straight teeth contain one modulation pole and the split teeth contain two modulation poles. These modulation poles can increase additional operating harmonics. The rotor permanent magnets are embedded in the grooves of the salient pole rotor, which adopts an alternating pole radial magnetization structure. The stator winding adopts a five-phase single-layer concentrated winding, with the coil wound on the split teeth and the straight teeth not wound. The rotor is on the outside and the stator is on the inside. The air gap is located between the stator and the rotor and has a thickness of 0.5 mm.

[0015] Furthermore, in step 1, for the air gap magnetic flux density of one cycle, a one-dimensional Fourier analysis is first performed on each spatial location to obtain the spatial harmonics, which can be expressed as:

[0016] ;

[0017] In the formula, B(t1,θ1) is the instantaneous value of the air gap magnetic flux density at the first sampling time and the first spatial sampling position, B(t1,θ2) is the instantaneous value of the air gap magnetic flux density at the first sampling time and the second spatial sampling position, ..., B(t... 2M ,θ 2N B represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time and the 2Nth spatial sampling position. x,M×2N and B y,M×2N The real and imaginary parts of the air gap magnetic flux density are represented by a one-dimensional Fourier decomposition of the real part over one period:

[0018] ;

[0019] In the formula, B x (t 2M ,θ N () represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time and the Nth spatial sampling position of the real part.

[0020] Perform a one-dimensional Fourier decomposition of the time within one period of the imaginary part:

[0021] ;

[0022] In the formula, B y (t 2M ,θ N() represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time of the imaginary part and the Nth spatial sampling position.

[0023] Taking the counterclockwise rotation of the air gap magnetic flux density as positive, the air gap magnetic flux density can be written as a function of time and space as follows:

[0024] ;

[0025] In the formula, m is the spatial order and n is the time order;

[0026] Suppose that a point on the air gap magnetic flux density rotates from π / 2 to the origin 0, the air gap magnetic flux density can be expressed as:

[0027] ;

[0028] Taking clockwise rotation of the air gap magnetic flux density as negative, the air gap magnetic flux density can be written as a function of time and space as follows:

[0029] ;

[0030] Suppose that when a point on the air gap magnetic flux density rotates from the origin 0 to π / 2, the air gap magnetic flux density can be expressed as:

[0031] ;

[0032] When the air gap magnetic flux density undergoes circular motion, its motion relationship can be expressed as:

[0033] ;

[0034] After obtaining amplitude and phase information through spatiotemporal harmonic analysis, the electromagnetic torque can be expressed as follows based on Maxwell's stress tensor method:

[0035] ;

[0036] In the formula, r g and l st Let represent the air gap radius and axial length, respectively, and μ0 be the free permeability. B r and B t It can be written in spacetime harmonic form as follows:

[0037] ;

[0038] In the formula, B r,mr,nr and B t,mt,nt It represents the spatiotemporal harmonic amplitudes of the radial and tangential air gap magnetic flux densities. m r and m t n represents the spatial order of the spatiotemporal harmonics of the radial and tangential air gap magnetic flux densities, respectively. r and n tThis represents the time order of the spatiotemporal harmonics of the radial and tangential air gap magnetic flux densities. φ r,mt,nt and φ t,mt,nt Let be the phase angle of the spacetime harmonics of the radial and tangential air gap magnetic flux densities. Then the electromagnetic torque can be further written as:

[0039] ;

[0040] This formula shows that when the time order and spatial order are the same, an average torque is generated; when the spatial order is the same but the time order is different, torque pulsation is generated.

[0041] ;

[0042] In the formula u=n t -n r Due to the properties of integrals, the condition for generating average torque is m. r =m t and u=0. When m r =m t When u≠0, a u-order torque pulsation T is generated. u Furthermore, the magnitude of the average torque and torque ripple depends on the phase angle and amplitude of the radial and tangential air gap flux densities.

[0043] Furthermore, in step 2, the rotor magnetic permeability M r (θ,t) in a magnetic field changes continuously with time and spatial position; it is a function of time and spatial position, and can be written as:

[0044] ;

[0045] In the formula θ m It is the radius of the permanent magnet, w s where g is the rotor slot pitch, and a is the air gap length. r ε1 is the permanent magnet pole arc coefficient, and θ and Ω are intermediate variables. r Let represent the mechanical angle and mechanical angular velocity, respectively, and t be the motor rotation time. r,0 and a r,i These are the zero-order and iP of the rotor magnetic permeability, respectively. r The amplitude of the second harmonic, P r This represents the number of pole pairs of the permanent magnet. From this, we can see the design parameter θ of the rotor teeth. m It has a significant impact on the distribution of magnetic permeability.

[0046] Stator magnetic permeability M s (θ) is stationary in the magnetic field and depends only on its position, which is determined by the split-tooth modulation factor M. sm (θ) and winding slot modulation factor M ss Together with (θ), the expression can be written as:

[0047] ;

[0048] In the formula w ks w is the width of the split teeth. m =18°−w st / 2+w ks / 2+w kp , where w is the width of the air gap on both sides of the split tooth. st w is the width of the straight tooth. kp a is the width of the split tooth tip. s To modulate the tooth polar arc coefficient, ε2 is an intermediate variable; a sm,0 and a sm,j These are the zeroth and jP of the split tooth groove modulation factor, respectively. s The amplitude of the second harmonic, a ss,0 and a ss,j These are the zeroth order and jP of the winding slot modulation factor, respectively. s The amplitude of the second harmonic, a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, P s This represents the number of 10 stator tooth units. j is a positive integer. From this, we can see the design parameters w of the stator teeth. ks w kp w st It also has a significant impact on the distribution of magnetic permeability.

[0049] Furthermore, in step 3, the permanent magnet magnetomotive force can be expressed as a Fourier expression:

[0050] ;

[0051] In the formula, F pm，k The amplitude of the permanent magnet magnetomotive force harmonic of the k-th order alternating pole permanent magnet vernier motor can be written as:

[0052] ;

[0053] In the formula, whether it is an odd or even harmonic, F pm,k The values ​​are not necessarily zero. This indicates that the magnetomotive force harmonics of the alternating pole permanent magnet vernier motor contain both odd and even harmonics, resulting in a rich harmonic content. m and F iron It is the maximum value of the magnetomotive force amplitude under the permanent magnet pole and the iron pole, which can be expressed as:

[0054] ;

[0055] In the formula h m For the thickness of the permanent magnet, l mμ is the length of the magnetic circuit of the iron core. r μ is the relative permeability of the permanent magnet. rp b is the relative permeability of silicon steel sheet r It is remanence, b p Here, μ is the magnetic flux density of the iron core, and μ0 is the permeability of free space. Therefore, the design parameter h of the permanent magnet... m It also has a significant impact on the magnetomotive force distribution of permanent magnets. The modulated air gap magnetic flux density B of the permanent magnet... pm (θ,t) can be written as:

[0056] ;

[0057] In the formula, μ0 is the free permeability, g is the air gap length, and a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, a r,0 and a r,i These are the zero order and iP of the stator magnetic permeability, respectively. r The amplitude of the second harmonic, P r P is the number of pole pairs of a permanent magnet. s Let i and j be the number of 10 stator tooth units, where i and j are both positive integers. The harmonics of the air gap magnetic flux density in the permanent magnet exhibit spatiotemporal variations. Its spatiotemporal harmonics can be expressed as (k, kP) r Spacetime harmonics can be divided into four categories, namely spacetime harmonics (k, kP). r ), (k, kP r ±jP s (k±i, kP) r ±iP r ) and (k±i, kP r ±iP r ±jP s Spacetime harmonics (k, kP) r The modulation (k, kP) is generated by the interaction between the zero-order stator and rotor magnetic permeability and the permanent magnet magnetomotive force. Therefore, they do not introduce stator and rotor magnetic permeability harmonics into the spacetime harmonics; this modulation behavior is called unit modulation. Similarly, the modulation of the permanent magnet magnetomotive force by the non-zero-order stator magnetic permeability and the zero-order rotor magnetic permeability will generate spacetime harmonics (k, kP). r ±jP s The non-zero-order rotor permeability and zero-order stator permeability modulate the permanent magnet magnetomotive force, generating spatiotemporal harmonics (k±i, kP). r ±iP r The modulation of the permanent magnet magnetomotive force by the non-zero order stator and rotor magnetic permeability generates spatiotemporal harmonics (k±i, kP). r ±iP r ±jP s ).

[0058] Furthermore, in step 4, the initial magnetomotive force of the armature can be expressed as:

[0059] ;

[0060] In the formula, P w It is the number of pole pairs in the five-phase armature winding. F v It is the vP of the armature winding magnetomotive force. w Spatial harmonic amplitude. Modulated armature air gap magnetic flux density B aw (θ,t) can be expressed as:

[0061] ;

[0062] In the formula, μ0 is the free permeability, g is the air gap length, and a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, a r,0 and a r,i These are the zero order and iP of the stator magnetic permeability, respectively. r The amplitude of the second harmonic, P r P is the number of pole pairs of a permanent magnet. s For 10 stator tooth units, P w Let v, i, and j be the pole pairs of the armature winding, and v, i, and j be positive integers. Similar to the permanent magnet magnetic field, the spatiotemporal harmonics of the armature air gap magnetic flux density also consist of four modulation behaviors, namely (1, vP). w (1,vP) w ±jP s (1±i,vP) w ±iP r ) and (1±i,vP w ±jP s ±iP r Due to the doubly salient pole structure of the alternating pole permanent magnet vernier motor, the stator and rotor magnetic permeability exhibit obvious periodic changes. Under the action of magnetic permeability modulation, multiple spatiotemporal harmonic components are generated in the armature magnetic field, which significantly increases both the time order and spatial order of the motor magnetic field and presents complex spatiotemporal harmonic distribution characteristics.

[0063] Furthermore, in step 5, after calculating the contribution of each order of spatiotemporal harmonic to the average torque and torque ripple of the motor, the beneficial spatiotemporal harmonics that significantly contribute to the average torque and the detrimental spatiotemporal harmonics that significantly affect the torque ripple are selected based on the calculation results. The sources of spatiotemporal harmonic generation are analyzed from the perspective of magnetomotive force and stator and rotor magnetic permeability. The motor parameters related to the order of spatiotemporal harmonics are identified, and the relationship between motor design parameters and average torque and torque ripple is established. The average torque T0 and torque ripple T0 of the motor are then compared. uThe objective function for motor optimization is F[Max(T0),Min(T0)]. u The key parameters of the motor selected based on the analysis of magnetomotive force and magnetic permeability angle are taken as design variables x, and their value ranges are determined. From the above analysis, it can be seen that the variable θ... m w st w ks w kp h m As key design parameters of the motor, they affect the average torque T0 and torque ripple T. u .

[0064] Furthermore, in step 6, the motor design parameters are sampled using the design of experiments method. The design of experiments method is a statistical approach that obtains good results with a small number of experiments. Among various design of experiments methods, the central composite design is widely used due to its high fitting accuracy and strong predictive ability. Using the five design parameters of the motor as variables, based on the analysis results of the design of experiments method, two objective second-order polynomial response surface models are established, which can be expressed as:

[0065] ;

[0066] Where f1 is the predicted value of the design target average torque T0, and f2 is the design target torque ripple T. u The predicted value is given by β, where β is the regression coefficient and ε is the random error. The accuracy of the model can be evaluated using the coefficient R0. 2 The accuracy of the model can be evaluated as follows:

[0067] ;

[0068] Where y i and f i These are the target values ​​obtained from the finite element method and the response surface model, respectively. It is y i The average value of R is given by n, where n is the number of design points in the experimental design method. 2 When the value is close to 1, it indicates that the response surface model has a more accurate approximation.

[0069] Furthermore, in step 7, based on the established high-precision model, a multi-objective genetic algorithm is introduced to optimize and solve the key design parameters of the motor. By comprehensively analyzing the Pareto solution set obtained during the optimization process and balancing the relationship between various optimization objectives, the Pareto optimum is selected as the final optimized design parameter, thereby effectively reducing torque ripple while improving the average torque and enhancing the torque quality of the motor.

[0070] Furthermore, in step 8, the optimized design parameters are substituted into the motor model to establish the electromagnetic analysis model of the optimized motor. The electromagnetic torque of the optimized motor is calculated using the finite element simulation method, and the torque performance effect after optimization is compared with the torque performance before optimization to verify the effectiveness and accuracy of the method for calculating the average torque and torque pulsation of the motor based on the spatiotemporal harmonic theory.

[0071] The present invention has the following beneficial effects:

[0072] 1. Based on spatiotemporal harmonic analysis, this invention can construct functional expressions for the contributions of each order of spatiotemporal harmonics to average torque and torque ripple in an alternating pole permanent magnet vernier motor. This enables quantitative calculation and separate analysis of the contributions of different orders of spatiotemporal harmonics to average torque and torque ripple. This method clarifies the magnitude of the contribution of different spatiotemporal harmonics to torque performance, achieves decoupled analysis of average torque and torque ripple, and effectively analyzes and evaluates the torque quality of the alternating pole permanent magnet vernier motor. The analysis method is concise in calculation, has clear physical meaning, and explicit analytical relationships. It can not only accurately analyze the output torque performance of the alternating pole permanent magnet vernier motor but also provide theoretical basis and design guidance for optimizing stator and rotor structural parameters, designing torque ripple suppression, and improving torque performance.

[0073] 2. This invention fully considers the magnetic circuit asymmetry and magnetic permeability periodic modulation characteristics caused by the unipolar excitation and doubly salient pole structure of the alternating pole permanent magnet vernier motor, and establishes complete and accurate analytical expressions for the air gap magnetic flux density of the permanent magnet and the armature air gap magnetic flux density. Based on this, it reveals the generation law and distribution characteristics of spatiotemporal harmonics generated by the modulation behavior of the two magnetic fields, thus enabling the clear identification of the sources of various major spatiotemporal harmonics in the motor. Using magnetomotive force and magnetic permeability as bridges, it clarifies the motor design parameters associated with spatiotemporal harmonics. Furthermore, using spatiotemporal harmonics as a bridge, it clarifies the relationship between spatiotemporal harmonics and average torque and torque ripple. Finally, it optimizes motor parameters as design variables and average torque and torque ripple as targets, thereby improving average torque, suppressing torque ripple, and achieving a design improvement in motor torque quality. Attached Figure Description

[0074] Figure 1 This is the topology of the alternating pole permanent magnet vernier motor according to an embodiment of the present invention;

[0075] Among them, 1-split tooth, 2-winding, 3-rotor, 4-permanent magnet, 5-stator;

[0076] Figure 2 The vector synthesis diagram of the motor's 10th torque ripple;

[0077] Figure 3 A diagram showing the structural parameters of the motor;

[0078] Figure 4A comparison chart showing the torque performance of the motor before and after optimization; Detailed Implementation

[0079] The following will be combined with the appendix Figures 1-4 The technical solutions in the embodiments of the present invention will be clearly and completely described.

[0080] To more concisely illustrate the beneficial effects of this invention, a detailed description is provided below using a specific alternating pole permanent magnet vernier motor: Figure 1 The topology of the motor is as follows: 1-split teeth, 2-winding, 3-rotor, 4-permanent magnet, 5-stator; in this embodiment, there are 20 slots and 23 pairs of permanent magnet poles. The motor includes a hybrid stator, an air gap, and a salient pole rotor. The hybrid stator is formed by alternating straight teeth and split teeth, where the straight teeth contain one modulation pole and the split teeth contain two modulation poles, which can increase additional operating harmonics. The rotor permanent magnet is embedded in the groove of the salient pole rotor and is radially magnetized. The stator winding adopts a five-phase single-layer concentrated winding structure, with the coil wound on the split teeth and no winding on the straight teeth, which serves as phase isolation. The air gap is located between the stator and rotor and has a thickness of 0.5 mm.

[0081] The present invention discloses a torque quality analysis and design method for an alternating pole permanent magnet vernier motor, the specific implementation object of which is as follows. Figure 1 As shown, it includes the following steps:

[0082] Step 1: For the air gap magnetic flux density of one cycle, first perform a one-dimensional Fourier analysis at each spatial location to obtain the spatial harmonics, which can be expressed as:

[0083] ;

[0084] In the formula, B(t1,θ1) is the instantaneous value of the air gap magnetic flux density at the first sampling time and the first spatial sampling position, B(t1,θ2) is the instantaneous value of the air gap magnetic flux density at the first sampling time and the second spatial sampling position, ..., B(t... 2M ,θ 2N B represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time and the 2Nth spatial sampling position. x,M×2N and B y,M×2N The real and imaginary parts of the air gap magnetic flux density are represented by a one-dimensional Fourier decomposition of the real part over one period:

[0085] ;

[0086] In the formula, B x (t 2M ,θ N () represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time and the Nth spatial sampling position of the real part.

[0087] Perform a one-dimensional Fourier decomposition of the time within one period of the imaginary part:

[0088] ;

[0089] In the formula, B y (t 2M ,θ N () represents the instantaneous value of the air gap magnetic flux density at the 2Mth sampling time of the imaginary part and the Nth spatial sampling position.

[0090] Taking the counterclockwise rotation of the air gap magnetic flux density as positive, the air gap magnetic flux density can be written as a function of time and space as follows:

[0091] ;

[0092] In the formula, m is the spatial order and n is the time order;

[0093] Suppose that a point on the air gap magnetic flux density rotates from π / 2 to the origin 0, the air gap magnetic flux density can be expressed as:

[0094] ;

[0095] Taking clockwise rotation of the air gap magnetic flux density as negative, the air gap magnetic flux density can be written as a function of time and space as follows:

[0096] ;

[0097] Suppose that when a point on the air gap magnetic flux density rotates from the origin 0 to π / 2, the air gap magnetic flux density can be expressed as:

[0098] ;

[0099] When the air gap magnetic flux density undergoes circular motion, it can be represented as:

[0100] ;

[0101] Therefore, when the air gap magnetic flux density undergoes circular motion, its motion relationship can be expressed as shown in Table 1.

[0102] Table 1 shows:

[0103] ;

[0104] Using spatiotemporal harmonic analysis based on Maxwell's stress tensor method, the electromagnetic torque can be expressed as:

[0105] ;

[0106] In the formula, r g and l st Let represent the air gap radius and axial length, respectively, and μ0 be the free permeability. B rand B t It can be written in spacetime harmonic form as follows:

[0107] ;

[0108] In the formula, B r,mr,nr and B t,mt,nt It represents the spatiotemporal harmonic amplitudes of the radial and tangential air gap magnetic flux densities. m r and m t n represents the spatial order of the spatiotemporal harmonics of the radial and tangential air gap magnetic flux densities, respectively. r and n t This represents the time order of the spatiotemporal harmonics of the radial and tangential air gap magnetic flux densities. φ r,mt,nt and φ t,mt,nt Let be the phase angle of the spacetime harmonics of the radial and tangential air gap magnetic flux densities. The electromagnetic torque can be further written as:

[0109] ;

[0110] This indicates that when the time order and spatial order are the same, an average torque is generated; when the spatial order is the same but the time order is different, torque pulsation is generated. Electromagnetic torque can be simplified as:

[0111] ;

[0112] In the formula u=n t -n r Due to the properties of integrals, the condition for generating average torque is m. r =m t and u=0. When m r =m t When u≠0, a u-order torque pulsation T is generated. u Furthermore, the magnitude of the average torque and torque ripple depends on the phase angle and amplitude of the radial and tangential air gap flux densities.

[0113] The amplitude and phase of each order of spacetime harmonics can be obtained through the two-dimensional Fourier analysis described above. According to Maxwell's stress tensor method, substituting the amplitude and phase into the formula yields the contribution of each order of spacetime harmonics to the average torque and torque ripple, as shown in Table 2. The table lists the amplitude, phase, and order of each order of radial and tangential air gap magnetic flux density obtained using two-dimensional Fourier analysis. Substituting these values ​​into the formula yields the average torque and torque ripple of the motor. The calculation results show that the 10th order torque ripple dominates the total torque ripple of the motor. Furthermore, the average torque T0 of the motor is generated by a pair of radial and tangential air gap magnetic flux density harmonics with the same time and spatial order. The 10th order torque ripple T... 10Composed of the same spatial order but with a time order difference of 10, Table 2 lists the five pairs of spatiotemporal harmonic orders that have a significant impact on the average torque and their contribution to the average torque. Similarly, the five pairs of spatiotemporal harmonics that have a significant impact on torque pulsation are also listed in Table 2.

[0114] Table 2 shows:

[0115] ;

[0116] Because of T 10 It is a phasor, therefore the 10th torque pulsation T 10 The principle of phasor superposition should be followed. Figure 2 T is generated by multiple spatiotemporal harmonics. 10 The synthesis process is illustrated, where each arrow represents a 10th torque ripple generated by a pair of spatiotemporal harmonics. The length and angle of the arrow represent the amplitude and phase of the 10th torque ripple. The longer the arrow, the greater the contribution of that spatiotemporal harmonic to the 10th torque ripple.

[0117] Step 2: Based on the unipolar excitation and doubly salient pole structure of the motor, establish the stator and rotor magnetic permeability modulation model of the motor, where the rotor magnetic permeability M... r (θ,t) in a magnetic field changes continuously with time and spatial position; it is a function of time and spatial position, and can be written as:

[0118] ;

[0119] In the formula θ m It is the radius of the permanent magnet, w s where g is the rotor slot pitch, and a is the air gap length. r ε1 is the permanent magnet pole arc coefficient, and θ and Ω are intermediate variables. r Let represent the mechanical angle and mechanical angular velocity, respectively, and t be the motor rotation time. r,0 and a r,i These are the zero-order and iP of the rotor magnetic permeability, respectively. r The amplitude of the second harmonic, P r This represents the number of pole pairs of the permanent magnet. From this, we can see the design parameter θ of the rotor teeth. m It has a significant impact on the distribution of magnetic permeability.

[0120] Stator magnetic permeability M s (θ) is stationary in the magnetic field and depends only on its position, which is determined by the split-tooth modulation factor M. sm (θ) and winding slot modulation factor M ss Together with (θ), the expression can be written as:

[0121] ;

[0122] In the formula wks w is the width of the split teeth. m =18°−w st / 2+w ks / 2+w kp , where w is the width of the air gap on both sides of the split tooth. st w is the width of the straight tooth. kp a is the width of the split tooth tip. s To modulate the tooth polar arc coefficient, ε2 is an intermediate variable; a sm,0 and a sm,j These are the zeroth and jP of the split tooth groove modulation factor, respectively. s The amplitude of the second harmonic, a ss,0 and a ss,j These are the zeroth order and jP of the winding slot modulation factor, respectively. s The amplitude of the second harmonic, a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, P s This represents the number of 10 stator tooth units. Both i and j are positive integers. From this, we can see the design parameters w of the stator teeth. ks w kp w st It also has a significant impact on the distribution of magnetic permeability.

[0123] Step 3, the permanent magnet magnetomotive force can be written as a Fourier expression as follows:

[0124] ;

[0125] In the formula, F pm，k The amplitude of the permanent magnet magnetomotive force harmonic of the k-th order alternating pole permanent magnet vernier motor can be written as:

[0126] ;

[0127] In the formula, whether it is an odd or even harmonic, F pm,k The values ​​are not necessarily zero. This indicates that the magnetomotive force harmonics of the alternating pole permanent magnet vernier motor contain both odd and even harmonics, resulting in a rich harmonic content. m and F iron It is the maximum value of the magnetomotive force amplitude under the permanent magnet pole and the iron pole, which can be expressed as:

[0128] ;

[0129] In the formula h m For the thickness of the permanent magnet, l m μ is the length of the magnetic circuit of the iron core. r μ is the relative permeability of the permanent magnet. rp b is the relative permeability of silicon steel sheet r It is remanence, bp Here, μ is the magnetic flux density of the iron core, and μ0 is the permeability of free space. Therefore, the design parameter h of the permanent magnet... m It also has a significant impact on the magnetomotive force distribution of permanent magnets. The modulated air gap magnetic flux density B of the permanent magnet... pm (θ,t) can be written as:

[0130] ;

[0131] In the formula, μ0 is the free permeability, g is the air gap length, and a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, a r,0 and a r,i These are the zero order and iP of the stator magnetic permeability, respectively. r The amplitude of the second harmonic, P r P is the number of pole pairs of a permanent magnet. s Let i and j be the number of 10 stator tooth units, where i and j are both positive integers. Clearly, the air gap flux density harmonics of the permanent magnet exhibit spatiotemporal variations. Its spatiotemporal harmonics can be expressed as (k, kP) r Spacetime harmonics can be divided into four categories, namely spacetime harmonics (k, kP). r ), (k, kP r ±jP s (k±i, kP) r ±iP r ) and (k±i, kP r ±iP r ±jP s Spacetime harmonics (k, kP) r The modulation (k, kP) is generated by the interaction between the zero-order stator and rotor magnetic permeability and the permanent magnet magnetomotive force. Therefore, they do not introduce stator and rotor magnetic permeability harmonics into the spacetime harmonics; this modulation behavior is called unit modulation. Similarly, the modulation of the permanent magnet magnetomotive force by the non-zero-order stator magnetic permeability and the zero-order rotor magnetic permeability will generate spacetime harmonics (k, kP). r ±jP s This modulation behavior is called stator modulation. The non-zero-order rotor permeability and the zero-order stator permeability modulate the permanent magnet magnetomotive force, generating spatiotemporal harmonics (k±i, kP). r ±iP r This refers to rotor modulation behavior. The modulation of the permanent magnet magnetomotive force by the non-zero order stator and rotor magnetic permeability generates spatiotemporal harmonics (k±i, kP). r ±iP r ±jP s This modulation behavior is called stator-rotor modulation.

[0132] For unit modulation (k, kP) r When k=1, kP rWhen =23, it indicates that the permanent magnet magnetomotive force generates a spatiotemporal harmonic (1,23) with a time order of 1 and a spatial order of 23; for the stator modulation behavior (k, kP) r ±jP s When k=1 and j=1, kP r -jP s When k = 13, a first-order stator magnetic permeability is introduced, which means that the permanent magnet magnetomotive force generates a spatiotemporal harmonic (1, 13) with a time order of 1 and a spatial order of 13; for rotor modulation (k ± i, kP) r ±iP r ), when k=2, i=1, kP r -iP r When the coefficient of magnetomotive force is 23, a first-order rotor permeability is introduced, which means that the permanent magnet magnetomotive force generates a spatiotemporal harmonic (1,23) with a time order of 1 and a spatial order of 23; similarly, for the stator-rotor modulation behavior (k±i, kP) r ±iP r ±jP s When k=1, i=1, j=2, kP r +iP r -jP s When the magnetic permeability is 26, the first-order rotor permeability and the second-order stator permeability are introduced, indicating that the magnetomotive force of the permanent magnet generates a spatiotemporal harmonic (2,26) with a time order of 2 and a spatial order of 26, as shown in Table 3 below. The spatiotemporal harmonic laws generated by the permanent magnet magnetic field under magnetic permeability modulation are systematically summarized. Since the alternating pole permanent magnet vernier motor exhibits four typical modulation behaviors simultaneously—stator magnetic permeability modulation, rotor magnetic permeability modulation, stator-rotor joint modulation, and unit modulation—the modulation process of the permanent magnet magnetic field in the air gap exhibits multiple modulation superposition characteristics. Under the above multiple modulation effects, the fundamental wave and its odd and even spatial harmonics in the original permanent magnet magnetic field undergo order transformation, forming a series of new spatiotemporal harmonic components, causing the air gap magnetic field to exhibit rich distribution characteristics in both spatial and temporal dimensions.

[0133] Table 3 shows:

[0134] ;

[0135] Step 4: After the motor is supplied with a five-phase sinusoidal symmetrical current, the motor rotates synchronously. The initial magnetomotive force of the armature can be expressed as:

[0136] ;

[0137] In the formula, P w It is the number of pole pairs in the five-phase armature winding. F v It is the vP of the armature winding magnetomotive force. w Spatial harmonic amplitude. Modulated armature air gap magnetic flux density Baw (θ,t) can be expressed as:

[0138] ;

[0139] In the formula, μ0 is the free permeability, g is the air gap length, and a s,0 and a s,j These are the zero order and jP of the stator magnetic permeability, respectively. s The amplitude of the second harmonic, a r,0 and a r,i These are the zero order and iP of the stator magnetic permeability, respectively. r The amplitude of the second harmonic, P r P is the number of pole pairs of a permanent magnet. s For 10 stator tooth units, P w Let v, i, and j be the pole pairs of the armature winding, and v, i, and j be positive integers. Similar to the permanent magnet magnetic field, the spatiotemporal harmonics of the armature air gap magnetic flux density also consist of four types of modulation behaviors. That is, (1, vP) w (1,vP) w ±jP s (1±i,vP) w ±iP r ) and (1±i,vP w ±jP s ±iP r As shown in Table 4 below, the spatiotemporal harmonics generated by the armature magnetic field under magnetic permeability modulation are systematically summarized. Unlike the odd and even harmonics of the permanent magnet magnetic field, the initial magnetomotive force of the armature only contains harmonics such as 3, 13, 23… and 7, 17, 27… The generation rules of its spatiotemporal harmonics are the same as those of the permanent magnet magnetomotive force. Due to the four typical modulation behaviors in the alternating pole permanent magnet vernier motor, the armature magnetic field also exhibits multiple modulation superposition characteristics in the air gap.

[0140] Table 4 shows:

[0141] ;

[0142] Step 5: In this case, key motor design parameters are selected as design variables, including θ. m h m w ks w kp w st ,like Figure 3 As shown. The average torque T0 is the maximum, and the torque pulsation T10 times is the maximum. 10 The minimum construction of the multi-objective optimization model allows us to define the objective function and design variables as follows:

[0143] ;

[0144] Step 6: Sample the motor design parameters using the design-of-experiments method. The design-of-experiments method is a statistical approach that obtains good results with a small number of experiments. Among various design-of-experiments methods, the central composite design is widely used due to its high fitting accuracy and strong predictive ability. Using the five motor design parameters as variables, and based on the analysis results of the design-of-experiments method, a second-order polynomial response surface model is established, which can be expressed as:

[0145] ;

[0146] Where f1 is the predicted value of the design target average torque T0, and f2 is the design target torque ripple T. 10 The predicted value is given by β, where β is the regression coefficient and ε is the random error. The accuracy of the model can be evaluated using the coefficient R0. 2 The accuracy of the model can be evaluated as follows:

[0147] ;

[0148] Where y i and f i These are the target values ​​obtained from the finite element method and the response surface model, respectively. It is y i The average value of R is given by n, where n is the number of design points in the experimental design method. 2 When the value is close to 1, it indicates that the response surface model has a more accurate approximation.

[0149] Step 7: Based on the established high-precision model, a multi-objective genetic algorithm is introduced to optimize and solve the key design parameters of the motor. By comprehensively analyzing the Pareto solution set obtained during the optimization process, the relationship between various optimization objectives is weighed, and the Pareto optimum is selected as the final optimized design parameter.

[0150] Step 8: Substitute the optimized design parameters into the motor model, as shown in Table 5, to establish the electromagnetic analysis model of the optimized motor and calculate its torque performance index. Based on the spatiotemporal harmonic decomposition of the air gap magnetic field, analyze the contribution of each order of spatiotemporal harmonics of the optimized motor to the average torque and torque ripple. At the same time, calculate the electromagnetic torque of the optimized motor using the finite element simulation method, and compare the torque performance effect after optimization with the torque performance before optimization to verify the effectiveness and accuracy of the motor design method based on spatiotemporal harmonics.

[0151] Table 5 shows:

[0152] ;

[0153] Figure 4A comparison of the torque performance of the motor before and after optimization shows that the average torque increased by 44.9% and the torque ripple decreased by 82.3%, proving the effectiveness of the invention.

[0154] In summary, this invention discloses a torque quality analysis and design method for an alternating pole permanent magnet vernier motor. The motor's permanent magnet is unipolar excitation, with a salient core acting as the permanent magnet. The stator employs a hybrid stator structure consisting of alternating straight and split teeth, resulting in a doubly salient pole structure that presents significant challenges in torque performance analysis. To clarify the intrinsic causes of torque generation, the amplitude and phase of the air gap magnetic flux density spatiotemporal harmonics are calculated using two-dimensional Fourier analysis. The contribution of spatiotemporal harmonics to the average torque and torque pulsation of the motor is calculated based on the Maxwell stress tensor method. From the perspective of magnetomotive force and permeability, the magnetic field is divided into two models: the armature magnetic field and the permanent magnet magnetic field. The expression for the air gap magnetic flux density is derived, clarifying the source of the spatiotemporal harmonics generated therein. The average torque T0 and the 10th order torque pulsation T are selected. 10 Using key motor design parameters as design variables, this invention optimizes the torque quality of alternating pole permanent magnet vernier motors to improve average torque and reduce torque ripple. The torque quality analysis and design method for alternating pole permanent magnet vernier motors is characterized by its simple calculation process, clear physical meaning, and explicit analytical relationships. It can not only accurately analyze the output torque performance of alternating pole permanent magnet vernier motors but also achieve optimization of stator and rotor structural parameters, torque ripple suppression design, and torque quality improvement design.

[0155] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "illustrative embodiment," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0156] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A method for torque quality analysis and design of an alternating pole permanent magnet vernier motor, characterized in that, Includes the following steps: Step S1: Obtain the air gap magnetic flux density data of the target alternating pole permanent magnet vernier motor, and perform spatiotemporal harmonic decomposition on the air gap magnetic flux density data to obtain the amplitude and phase information of each order of spatiotemporal harmonics of radial air gap magnetic flux density and tangential air gap magnetic flux density; Step S2: Based on Maxwell's stress tensor method, construct an analytical expression for electromagnetic torque according to the amplitude and phase information, determine the conditional relationship of the spatiotemporal harmonics that generate average torque and torque pulsation according to the analytical expression for electromagnetic torque, and quantitatively calculate and separate the contribution of different order spatiotemporal harmonics to average torque and torque pulsation. Step S3: Based on the stator and rotor structures of the target alternating pole permanent magnet vernier motor, construct stator magnetic permeability modulation function expressions and rotor magnetic permeability modulation function expressions. Based on the stator magnetic permeability modulation function expressions and rotor magnetic permeability modulation function expressions, construct permanent magnet magnetic field air gap magnetic flux density function expressions and armature magnetic field air gap magnetic flux density function expressions respectively, so as to clarify the key design parameters of the motor associated with each order of spatiotemporal harmonics. Step S4: Based on the analysis results of Step S2 and the key motor design parameters of Step S3, establish a multi-objective optimization model with motor torque quality as the optimization objective, optimize and solve the key motor design parameters, and obtain the optimized motor design parameters to improve the torque quality of the motor.

2. The method according to claim 1, characterized in that, In step S2, the analytical expression for the electromagnetic torque is: ； In the formula, r g and l st These represent the air gap radius and axial length, respectively, where μ0 is the free permeability, and B... r and B t These are the radial air gap magnetic flux density and the tangential air gap magnetic flux density, respectively. According to the analytical expression of electromagnetic torque, the condition for generating average torque is that the spatiotemporal harmonics of the radial air gap magnetic flux density and the tangential air gap magnetic flux density have the same spatial order and the same time order. The condition for generating torque pulsation is that the spatiotemporal harmonics of the radial air gap magnetic flux density and the tangential air gap magnetic flux density have the same spatial order but different time orders.

3. The method according to claim 1, characterized in that, In step S3, constructing the stator magnetic permeability modulation function expression and the rotor magnetic permeability modulation function expression further includes: The rotor magnetic permeability modulation function expression is constructed based on the rotor's salient pole structure and the permanent magnet curvature. The rotor magnetic permeability modulation function M r The expression for (θ,t) is: ； In the formula θ m It is the radius of the permanent magnet, w s where g is the rotor slot pitch, and a is the air gap length. r ε1 is the permanent magnet pole arc coefficient, θ and Ω are intermediate variables; r These represent the mechanical angle and mechanical angular velocity, respectively, where t is the motor rotation time; a r,0 and a r,i These are the zero-order and iP of the rotor magnetic permeability, respectively. r The amplitude of the second harmonic, P r The number of permanent magnet pole pairs is i; i is a positive integer; the design parameter θ of the rotor teeth is... m It has a significant impact on the magnetic permeability distribution; the stator magnetic permeability modulation function expression is constructed based on the straight tooth and split tooth structure of the stator.

4. The method according to claim 1, characterized in that, In step S3, constructing the permanent magnet magnetic field air gap magnetic flux density function expression further includes: considering the coexistence of odd and even harmonics in the permanent magnet magnetic field under unipolar excitation of the permanent magnet, constructing a permanent magnet magnetomotive force Fourier expression; modulating the permanent magnet magnetomotive force Fourier expression according to the stator magnetic permeability modulation function expression and the rotor magnetic permeability modulation function expression to obtain the permanent magnet magnetic field air gap magnetic flux density function expression; wherein, the spatiotemporal harmonics in the permanent magnet magnetic field air gap magnetic flux density function expression include at least the first type of spatiotemporal harmonics generated by unit modulation behavior, the second type of spatiotemporal harmonics generated by stator modulation behavior, the third type of spatiotemporal harmonics generated by rotor modulation behavior, and the fourth type of spatiotemporal harmonics generated by the joint modulation behavior of stator and rotor.

5. The method according to claim 1, characterized in that, In step S3, constructing the armature magnetic field air gap magnetic flux density function expression further includes: constructing the armature magnetomotive force Fourier expression after passing a five-phase symmetrical sinusoidal current; modulating the armature magnetomotive force Fourier expression according to the stator magnetic permeability modulation function expression and the rotor magnetic permeability modulation function expression to obtain the armature magnetic field air gap magnetic flux density function expression; wherein, the spatiotemporal harmonics in the armature magnetic field air gap magnetic flux density function expression include at least the fifth type of spatiotemporal harmonic generated by unit modulation behavior, the sixth type of spatiotemporal harmonic generated by stator modulation behavior, the seventh type of spatiotemporal harmonic generated by rotor modulation behavior, and the eighth type of spatiotemporal harmonic generated by the joint modulation behavior of stator and rotor.

6. The method according to claim 1, characterized in that, Step S4 further includes: based on the quantitative calculation and separation analysis results of step S2, selecting spatiotemporal harmonics that contribute positively to the average torque as favorable harmonics, and selecting spatiotemporal harmonics that contribute significantly to torque ripple as unfavorable harmonics; establishing the correlation between the favorable harmonics, the unfavorable harmonics, and the key design parameters of the motor based on the key design parameters of the motor in step S3; using the correlation as the constraint condition of the multi-objective optimization model, and optimizing the key design parameters of the motor with the optimization objectives of maximizing the average torque and minimizing the torque ripple.

7. The method according to claim 1, characterized in that, In step S4, establishing a multi-objective optimization model further includes: sampling the key design parameters of the motor using an experimental design method to construct a response surface model, which is used to characterize the approximate functional relationship between the key design parameters of the motor and the average torque and torque ripple; based on the response surface model, using a multi-objective genetic algorithm to optimize and solve the key design parameters of the motor to obtain a Pareto optimal solution set; and selecting the optimal combination of design parameters from the Pareto optimal solution set as the optimized motor design parameters.

8. The method according to claim 1, characterized in that, It also includes: Step S5: Substitute the optimized motor design parameters into the finite element simulation model of the target alternating pole permanent magnet vernier motor, calculate the average torque and torque ripple of the optimized motor, and compare and verify the torque performance of the motor before optimization.

9. The method according to claim 1, characterized in that, The alternating pole permanent magnet vernier motor includes a hybrid stator, an air gap, and a salient pole rotor. The hybrid stator is composed of alternating straight teeth and split teeth, wherein the straight teeth contain one modulation pole and the split teeth contain two modulation poles, and the modulation poles add additional operating harmonics. The rotor permanent magnets are embedded in the grooves of the salient pole rotor, which adopts an alternating pole radial magnetization structure. The stator winding adopts a five-phase single-layer concentrated winding, with the coils wound on the split teeth and the straight teeth not wound. The salient pole rotor is on the outside and the stator is on the inside, and the air gap is located between the hybrid stator and the salient pole rotor.

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