Analytical modeling method for surface-embedded trapezoidal-pole halbach motor

Abstract

The application discloses a surface-embedded trapezoidal magnetic pole Halbach motor analytical modeling method. Compared with a conventional tile-shaped Halbach structure, the trapezoidal magnetic pole Halbach structure uses a layered and divided domain method to ensure the same overall shape and magnetic consumption, change the shape of the segmented magnetic pole to improve the air gap magnetic flux density waveform and reduce the total harmonic distortion (THD), thereby inhibiting the torque pulsation to a certain extent. By making a three-dimensional slice graph of the influence of the inner and outer pole arc proportion of the middle trapezoidal magnetic pole and the magnetization angle of the two side magnetic poles on the fundamental amplitude and THD, according to the numerical value size distribution in the graph, a relatively optimal variable distribution range can be obtained. By selecting a larger fundamental amplitude and a lower THD, a relatively optimal combination is determined. Therefore, after parameter optimization, the performance of the motor is improved.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of Halbach permanent magnet motors, and particularly relates to a surface-embedded trapezoidal magnetic pole Halbach motor analytical modeling method. BACKGROUND

[0002] Permanent magnet machines with segmented Halbach arrays can make the air gap magnetic field closer to sinusoidal distribution, thus having many higher requirement applications. Surface-mounted Halbach machines have the advantages of simple manufacturing and high efficiency. Due to their relatively simple structure, they are very suitable for analytical modeling. Unlike surface-mounted Halbach, surface-embedded Halbach has a reluctance torque, so it is difficult to analytically model and analyze. A two-dimensional subdomain model method is used to solve the magnetic field of surface-embedded Halbach. The double-sided linear servo motor of the surface-mounted trapezoidal magnet can reduce the thrust ripple, and a related paper studies the trapezoidal double-section Halbach of the surface-mounted and slotless pipe type actuator. However, surface-embedded trapezoidal Halbach has not been mentioned. SUMMARY

[0003] The application aims to make up for the defects of the prior art, and provides a surface-embedded trapezoidal magnetic pole Halbach motor analytical modeling method, which provides a magnetic pole shape for conventional tile-shaped Halbach, improves the air gap magnetic flux density waveform, reduces the total harmonic distortion (THD), and thus suppresses the torque ripple.

[0004] The application is implemented by the following technical solutions:

[0005] The surface-embedded trapezoidal magnetic pole Halbach motor analytical modeling method changes the three-section conventional tile-shaped Halbach magnetic pole shape into a trapezoidal structure to improve the motor performance, and optimizes the inner and outer pole arc ratio of the middle trapezoidal magnetic pole and the magnetization angle of the two side magnetic poles, so that a relatively optimal parameter combination is obtained, and the motor performance is improved.

[0006] The analytical solution of the model is obtained by the layered and domain linear superposition method, and after the analytical solution is obtained, the inner and outer arc pole arc ratios of the three-section trapezoidal magnetic pole Halbach and the magnetization angles of the two side magnetic poles are parameter scanned and optimized, and a relatively optimal radial air gap magnetic flux density and electromagnetic torque are obtained; the trapezoidal magnetic pole Halbach shape is a trapezoid with different inner and outer pole arc ratios, the trapezoidal shape with different inner and outer pole arc ratios, the middle magnetic pole is in parallel magnetization mode, and the two side magnetic poles are magnetized with magnetization angles.

[0007] The radial air gap magnetic flux density of the surface-embedded trapezoidal magnetic pole Halbach is obtained by the analytical method, and the mathematical expression of the air gap magnetic flux density contains the inner and outer pole arc ratios of the middle trapezoidal magnetic pole C and αA magnetization angle θ of two side poles m These three parameters. Then form three-dimensional slice map with these three parameters as variables, study the influence of these three variables on the fundamental amplitude value of radial air gap magnetic flux density and THD size. In the three-dimensional slice map of fundamental amplitude value and THD, find the optimal variable range, so as to obtain the optimal trapezoidal magnetic pole structure parameters.

[0008] The specific calculation process is as follows:

[0009] The proportion of each section of trapezoidal Halbach magnetic pole is as follows:

[0010]

[0011] In the formula: τ r is the arc length of the rotor slot, τ p is the pole pitch and is equal to π / p, p is the number of pole pairs, τ1 is the pole arc length, τ A and τ C are the outer arc and inner arc length of the middle magnetic pole respectively; α r is the proportion of rotor slot pole arc, α1 is the permanent magnet pole arc proportion, α A and α C are the ratio of the outer arc / inner arc of the middle trapezoidal magnetic pole to the pole arc respectively.

[0012] For any jth layer, according to linear layering, the shape and position parameters of the middle magnetic pole can be obtained:

[0013]

[0014] In the formula: τ j is the arc length of the jth layer middle magnetic pole, α j is the ratio of the jth layer middle magnetic pole arc length to the pole arc, R r and R m are the inner and outer radii of Halbach permanent magnet, R j-1 and R j are the inner and outer radii of the jth layer permanent magnet.

[0015] The layering form of magnetization intensity of Halbach array composed of p-pole magnetic poles in one electrical cycle is as follows:

[0016]

[0017]

[0018]

[0019]

[0020] In the formula: M rjand M θj B r is the residual magnetization of the pole, μ0 is the vacuum permeability, θ m is the magnetization angle of the two-side pole, θ is the rotor position angle, is the offset angle of the two-side pole.

[0021] (1) Air-gap magnetic field generated by the outermost pole

[0022] The solution region is divided into air-gap region I, permanent magnet region II, and rotor slot air region III. According to Laplace equation, the scalar magnetic potential functions of the three regions can be obtained:

[0023]

[0024]

[0025]

[0026] In the two-dimensional polar coordinate system, the two components of the magnetic field intensity vector and the two components of the flux density vector can be represented by the magnetic potential function as follows:

[0027]

[0028] where, μ r is the relative permeability of the permanent magnet, M r and M θ are its two magnetization vector components, when in the air region, μ r = 1, M r = 0, M θ = 0.

[0029] According to the quasi-Poisson equation and the boundary conditions, we have:

[0030]

[0031]

[0032]

[0033] R s is the inner diameter of the stator, at r = R m , the following equation is satisfied:

[0034]

[0035]

[0036] By solving the matrix equation of the boundary conditions, the magnetic field generated by the outermost magnet in the slotless air gap region can be obtained, and thus the air gap magnetic flux density BrA-slotless The air-gap magnetic field generated by the innermost layer magnetic pole can be obtained.

[0037] (2) Air-gap magnetic field generated by the intermediate layer magnetic pole

[0038] The intermediate layer contains w-2 equal-thickness sub-layers, and the solution region is divided into air gap region I, air region above the permanent magnet region II, permanent magnet region III, and air region below the permanent magnet region IV, one more solution region than the outermost layer. Take any sub-layer of the intermediate layer for calculation and analysis, and the scalar magnetic potential functions of the four regions can be obtained according to Laplace equation:

[0039]

[0040]

[0041]

[0042]

[0043] The boundary conditions are as follows:

[0044]

[0045]

[0046]

[0047]

[0048] At r = R m , the following equation is satisfied:

[0049]

[0050]

[0051] The magnetic field generated by any intermediate layer magnet in the slotless air gap region can be obtained by solving the matrix equation of the boundary conditions, and thus the air-gap magnetic flux density B rB-slotless generated by the intermediate layer magnet can be obtained.

[0052] (3) Air-gap magnetic field generated by the innermost layer magnetic pole

[0053] The solution region is divided into air gap region I, air region above the permanent magnet region II, and permanent magnet region III. The scalar magnetic potential functions of the three regions can be obtained according to Laplace equation:

[0054]

[0055]

[0056]

[0057] The boundary conditions of the innermost layer are as follows:

[0058]

[0059]

[0060]

[0061] At r=R m , the following equation is satisfied:

[0062]

[0063]

[0064] The magnetic field generated by the innermost magnet in the slotless air gap region can be obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rC-slotless generated by the innermost magnet can be obtained.

[0065] The air gap magnetic field generated by the entire magnetic pole can be obtained by superimposing the outermost layer, the middle layer, and the innermost layer, and thus the air gap flux density generated by the w-th layer magnetic pole is:

[0066]

[0067] For the trapezoidal Halbach magnet, the inner and outer pole arc ratios and the magnetization angle of the middle magnetic pole affect the fundamental amplitude and the total harmonic distortion (THD) of the slotless air gap flux density, which can be written as:

[0068] B rf1 = B r-slotless (α A ,α C ,θ m , k = 1) (35)

[0069]

[0070] Due to the large number of variables, it is difficult to obtain the optimal solution using the analytical method, and therefore three-dimensional slice graphs of the effects of α A , α C , and θ m on the fundamental amplitude and THD are made. According to the distribution of the numerical values in the graphs, a relatively optimal variable distribution range can be obtained. By selecting a larger fundamental amplitude and a lower THD, a set of relatively optimal combinations can be determined.

[0071] The advantages of the present application are: compared with the conventional tile-shaped Halbach motor, the trapezoidal pole Halbach motor provided by the present application uses the method of layered and divided domains to improve the air gap flux density waveform and reduce the total harmonic distortion (THD) by changing the shape of the segmented pole under the condition of ensuring the same overall shape and magnetic consumption, thereby suppressing the torque pulsation to a certain extent. BRIEF DESCRIPTION OF DRAWINGS

[0072] Figure 1 is a structure schematic diagram of trapezoidal pole three-section Halbach.

[0073] Figure 2 is a structure schematic diagram of the surface embedded trapezoidal pole three-section Halbach motor of the present application.

[0074] Figure 3 is an equivalent layered schematic diagram of trapezoidal pole three-section Halbach in polar coordinates.

[0075] Figure 4 is a schematic diagram of three different solving domains of the layered trapezoidal pole. Among them, Figure 4(a) is the outermost solving domain, and Figures 4(b) and 4(c) are the solving domains of the middle layer and the innermost layer, respectively.

[0076] Figure 5 is the influence of the ratio of the inner / outer pole arc of the middle trapezoidal pole α C / α A and the magnetization angle θ m of the two side poles on the fundamental amplitude three-dimensional slice diagram.

[0077] Figure 6 is the influence of the ratio of the inner / outer pole arc of the middle trapezoidal pole α C / α A and the magnetization angle θ m of the two side poles on the THD three-dimensional slice diagram.

[0078] Figure 7 is the comparison of electromagnetic torque between the optimized conventional tile-shaped surface embedded Halbach motor and the surface embedded trapezoidal Halbach motor.

[0079] Figure 8 is the comparison of the analytical method and the finite element of the air gap flux density waveform of the optimized slotted motor.

[0080] Figure 9 is the comparison of the analytical method and the finite element of the electromagnetic torque waveform of the optimized slotted motor. DETAILED DESCRIPTION

[0081] The analytic modeling method of surface-embedded trapezoidal magnetic pole Halbach motor is used to improve the motor performance by changing the shape of three conventional tile-shaped Halbach magnetic poles into trapezoidal structure, and a set of relatively optimal parameter combinations are obtained by optimizing the inner and outer pole arc ratio of the middle trapezoidal magnetic pole and the magnetization angle of the two side magnetic poles, so that the motor performance is improved.

[0082] The analytic solution of the model is obtained by the method of linear superposition in layers and domains, and the inner and outer arc pole arc ratio of the three trapezoidal Halbach magnetic poles and the magnetization angle of the two side magnetic poles are optimized by parameter scanning, so that the relatively optimal radial air gap flux density and electromagnetic torque are obtained; the trapezoidal Halbach magnetic pole shape is different in inner and outer pole arc ratio for each section, the middle magnetic pole is parallel magnetization, and the two side magnetic poles are magnetized with an angle.

[0083] The radial air gap flux density of the surface-embedded trapezoidal magnetic pole Halbach motor is obtained by the analytic method, and the mathematical expression of the air gap flux density contains three parameters: the inner and outer pole arc ratio of the middle trapezoidal magnetic pole α C and α A , and the magnetization angle of the two side magnetic poles θ m . Then a three-dimensional slice graph is formed with these three parameters as variables to study the influence of these three variables on the fundamental amplitude and THD size of the radial air gap flux density. In the three-dimensional slice graph of the fundamental amplitude and THD, the optimal variable range is found to obtain the optimal trapezoidal magnetic pole structure parameters.

[0084] The pole ratio of each section of the trapezoidal Halbach is as follows:

[0085]

[0086] In the formula: τ r is the arc length of the rotor slot, τ p is the pole pitch and is equal to π / p, p is the number of pole pairs, τ1 is the pole arc length, τ A and τ C are the outer arc and inner arc lengths of the middle magnetic pole respectively; α r is the rotor slot pole arc ratio, α1 is the permanent magnet pole arc ratio, α A and α C are the outer arc / inner arc and pole arc ratios of the middle trapezoidal magnetic pole respectively.

[0087] For any jth layer, the shape and position parameters of the middle magnetic pole can be obtained according to the linear layering:

[0088]

[0089] In the formula: τ j is the arc length of the jth layer middle magnetic pole, α jRj is the ratio of the arc length of the middle pole to the pole arc in the jth layer r and R m are the inner and outer radius of the Halbach permanent magnet, R j-1 and R j are the inner and outer radius of the jth layer permanent magnet.

[0090] The layered form of the magnetization of the Halbach array composed of p pairs of poles in one electrical period is as follows:

[0091]

[0092]

[0093]

[0094]

[0095] where: M rj and M θj are the radial and tangential components of the magnetization, B r is the residual magnetization of the pole, μ0 is the vacuum permeability, θ m is the magnetization angle of the two side poles, θ is the rotor position angle, is the offset angle of the two side poles.

[0096] (1) The air gap magnetic field generated by the outermost layer of poles

[0097] The solution region is divided into air gap domain I, permanent magnet domain II, and rotor slot air domain III. According to the Laplace equation, the scalar magnetic potential functions of the three regions can be obtained:

[0098]

[0099]

[0100]

[0101] In the two-dimensional polar coordinate system, the two components of the magnetic field intensity vector and the two components of the flux density vector can be represented by the magnetic potential function as follows:

[0102]

[0103] μ r is the relative permeability of the permanent magnet, M r and M θ are its two magnetization vector components, when in the air region, μ r = 1, M r = 0, M θ = 0.

[0104] According to the quasi-Poisson equation and the boundary conditions, we have:

[0105]

[0106]

[0107]

[0108] R s At r = R m , the following equation is satisfied:

[0109]

[0110]

[0111] The magnetic field generated by the outermost magnet in the slotless air gap region can be obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rA-slotless generated by the outermost magnet can be obtained.

[0112] (2) Air gap magnetic field generated by the middle layer magnetic pole

[0113] The middle layer contains w-2 equal-thickness small layers, and the solution region is divided into air gap domain I, air above permanent magnet domain II, permanent magnet domain III, and air below permanent magnet domain IV, with one more solution region than the outermost layer. Take any small layer of the middle layer for calculation and analysis. According to the Laplace equation, the scalar magnetic potential functions of the four regions can be obtained:

[0114]

[0115]

[0116]

[0117]

[0118] The boundary conditions are as follows:

[0119]

[0120]

[0121]

[0122]

[0123] At r = R m , the following equation is satisfied:

[0124]

[0125]

[0126] The magnetic field generated by any intermediate layer magnet in the slotless air gap region can be obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rB-slotless can be solved.

[0127] (3) Air gap magnetic field generated by the innermost layer magnetic pole

[0128] The solving region is divided into air gap region I, air region above permanent magnet II, and permanent magnet region III. According to the Laplace equation, the scalar magnetic potential functions of the three regions can be obtained:

[0129]

[0130]

[0131]

[0132] The boundary conditions of the innermost layer are as follows:

[0133]

[0134]

[0135]

[0136] At r = R m , the following equation is satisfied:

[0137]

[0138]

[0139] The magnetic field generated by the innermost layer magnet in the slotless air gap region can be obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rC-slotless generated by the innermost layer magnet can be solved.

[0140] Superimposing the outermost layer, the intermediate layer, and the innermost layer can obtain the air gap magnetic field generated by the entire magnetic pole, and thus the air gap flux density of the w-th layer magnetic pole is:

[0141]

[0142] For trapezoidal Halbach magnets, the inner and outer pole arc ratios and magnetization angles of the intermediate magnetic pole affect the fundamental amplitude and total harmonic distortion (THD) of the slotless air gap flux density, which can be written as:

[0143] B rf1= B r-slotless (α A ,α C ,θ m ,k=1) (35)

[0144]

[0145] Due to the large number of variables, it is difficult to obtain the optimal solution by using the analytical method, so the alpha A , alpha C , theta m The influence of the three-dimensional slice diagram on the fundamental amplitude and THD, according to the value size distribution in the diagram, the optimal variable distribution range can be obtained. By selecting a larger fundamental amplitude and a lower THD, a group of optimal combination can be determined.

[0146] Figure 1 is the structure diagram of the trapezoidal magnetic pole three-section Halbach. Each pole is composed of three trapezoidal permanent magnets, and the symmetry axis is the geometric center of the middle magnetic pole. The middle trapezoidal magnetic pole is parallel magnetization, and the two sides of the magnetic pole are magnetized with an angle, and the magnetization angle theta m is defined as follows: the magnetization angle of the left trapezoidal permanent magnet 1.1 of the N pole is the included angle between the magnetization direction and the clockwise circumferential tangential direction; the middle trapezoidal permanent magnet 1.2 of the N pole adopts parallel magnetization, and the magnetization direction is the center line of the parallel magnetic pole and points outward; the magnetization angle of the right trapezoidal permanent magnet 1.3 of the N pole is the included angle between the magnetization direction and the counterclockwise circumferential tangential direction; the magnetization angle of the left trapezoidal permanent magnet 1.5 of the S pole is the included angle between the magnetization direction and the counterclockwise circumferential tangential direction; the middle trapezoidal permanent magnet 1.6 of the S pole adopts parallel magnetization, and the magnetization direction is the center line of the parallel magnetic pole and points inward; the magnetization angle of the right trapezoidal permanent magnet 1.7 of the S pole is the included angle between the magnetization direction and the clockwise circumferential tangential direction. There is a certain thickness of pole interval iron 1.4 between the N and S poles.

[0147] Figure 2 is the structure diagram of the surface embedded trapezoidal magnetic pole three-section Halbach motor of the application. The motor model adopts a 6-pole 9-slot parallel tooth structure, and the rotating speed of the motor is 3000r / min. The stator core and the rotor core both adopt 50W470 silicon steel sheets, and the permanent magnet adopts neodymium iron boron N35H. The main parameters of the motor are as follows: the outer diameter and the inner diameter of the stator are 50mm and 30.2mm, the outer diameter and the inner diameter of the permanent magnet are 29.2mm and 25.2mm, the axial length of the motor is 40mm, the coil turns are 40 turns, the angular velocity of the rotor is 100pi, the pole arc and pole pitch ratio is 0.8, the rotor slot arc and pole pitch ratio is 0.9, the relative magnetic permeability is 1.05, the tooth width is 7mm, the slot opening width is 1.5mm, the permanent magnet thickness is 4mm, and the residual magnetization is 1.2T. The parameter variables of the optimized trapezoidal magnetic pole are as follows: alpha A =0.49, alpha C= 0.30, θ m = 71.34°.

[0148] Figure 3 is the equivalent layered diagram of trapezoidal pole three-segment Halbach in polar coordinates. Each layer is an equidistant three-segment conventional tile-shaped Halbach. In the figure, τ j is the arc length of the middle pole in the jth layer, α j is the ratio of the arc length of the middle pole to the pole arc in the jth layer, R r and R m are the inner and outer radii of the Halbach permanent magnet, R j-1 and R j are the inner and outer radii of the jth layer of permanent magnet.

[0149] Figure 4 is a schematic diagram of three different solving domains of the trapezoidal pole after layering. Among them, Figure 4(a) is the outermost layer solving domain, and Figures 4(b) and 4(c) are the solving domains of the middle layer and the innermost layer, respectively. For different solving domains and different boundary conditions, the outermost layer, the middle layer and the innermost layer of the magnet need to be analyzed respectively. The outermost layer and the innermost layer contain three solving domains, and the middle layer contains four solving domains.

[0150] Figure 5 is the ratio of the inner / outer pole arc of the middle trapezoidal pole α C / α A and the magnetization angle θ m of the two side poles to the fundamental amplitude. Three-dimensional slice diagram. According to the depth of the gray in the figure, the amplitude of the fundamental amplitude in different regions can be judged, so as to divide the change range. From the figure, when 0.1 < α A < 0.4, 0.2 < α C < 0.6, 69° < θ m < 76°, the fundamental amplitude is larger.

[0151] Figure 6 is the ratio of the inner / outer pole arc of the middle trapezoidal pole α C / α A and the magnetization angle θ m of the two side poles to the THD. Three-dimensional slice diagram. As Figure 5 the same, according to the different gray in the region, the size distribution range of THD can be obtained. From the figure, when 0.1 < α A < 0.5, 0.1 < α C < 0.45, 70° < θ m < 72°, THD is smaller. From Figure 6 and Figure 5 , it can be seen that the fundamental amplitude of the air gap flux density and the THD cannot be simultaneously optimized, but the range of both being better can be selected for parameter optimization.

[0152] Figure 7 is the optimized electromagnetic torque comparison of the conventional tile surface embedded Halbach motor and the surface embedded trapezoidal Halbach motor. As can be seen from the figure, the average value of the electromagnetic torque of the optimized trapezoidal Halbach motor increases, and the torque ripple decreases.

[0153] Figure 8 is the comparison of the analytical method and the finite element method of the optimized air gap flux waveform of the slotted motor. As can be seen from the figure, due to the limited number of layer superposition and the approximate calculation of the Carter coefficient algorithm when calculating the stator slot, there is a negligible calculation error between the analytical method and the finite element method. Further verify the correctness of the proposed calculation method.

[0154] Figure 9 is the comparison of the analytical method and the finite element method of the optimized electromagnetic torque waveform of the slotted motor. As can be seen from the figure, the torque amplitude and phase of the analytical method and the finite element method are well matched, which shows the correctness of the analytical modeling.

Claims

1. A method for analytical modeling of surface-embedded trapezoidal-pole Halbach machines, characterized in that: An analytic solution of the model is obtained by a method of linear superposition by layer and area, and after obtaining the analytic solution, inner and outer arc pole arc ratios of the three-section trapezoidal Halbach and magnetization angles of the two side poles are scanned and optimized to obtain optimal radial air gap flux density and electromagnetic torque. The pole ratio of the trapezoidal Halbach is as follows: In the formula: τ r is the arc length of the rotor slot, τ p is the pole pitch and is equal to π / p, p is the pole pair number, τ1 is the pole arc length, τ A and τ C are the outer arc length and the inner arc length of the middle magnetic pole respectively; α r is the ratio of the rotor slot pole arc, α1 is the ratio of the permanent magnet pole arc, α A and α C are the ratios of the outer arc / inner arc of the middle trapezoidal magnetic pole to the pole arc respectively; The analytic solution of the model is obtained by a method of linear superposition by layer and area, and the specific content is as follows: the poles are equivalently approximated by layer, i.e., the entire pole is evenly divided into w layers along the radial direction, each layer is a three-section Halbach, in the w layers, the first and w layers are the innermost and outermost layers respectively, and the remaining w-2 layers are the middle layers, and the three different layers correspond to three different solving regions; the air gap magnetic field obtained by analyzing the three different layers is superposed to obtain the magnetic field generated by the entire pole in the air gap; for any j layer, the shape and position parameters of the middle pole are obtained according to linear layering: In the formula: τ j is the arc length of the jth layer of intermediate magnetic poles, α j is the ratio of the arc length of the jth layer of intermediate magnetic poles to the pole arc, R r and R m are the inner and outer radii of the Halbach permanent magnet, R j-1 and R j are the inner and outer radii of the jth layer of permanent magnets; The inner and outer arc pole arc ratios of the three-section trapezoidal Halbach and the magnetization angles of the two side poles are scanned and optimized to obtain optimal radial air gap flux density and electromagnetic torque, and the specific content is as follows: The layering form of the magnetization intensity of the Halbach array composed of p pole pairs in one electrical cycle is as follows: where M rj and M θj are the radial and tangential components of the magnetization, B r is the residual magnetization of the pole, μ0is the vacuum permeability, θ m is the magnetization angle of the pole, θ is the rotor position angle, is the offset angle of the pole. The radial component M rj and tangential component M θj of the magnetization of any layer is obtained After that, the solution of the air-gap field is carried out in layers and regions, as follows: (1) Air gap magnetic field generated by the outermost layer When the outermost layer is calculated, the middle layer and the innermost layer are treated as air; at this time, the solving region is divided into air gap region I, permanent magnet region II, and rotor slot air region III; according to Laplace equation, the scalar magnetic potential functions of the three regions are obtained: where X mAI , Y mAI , X nAII , Y nAII , X nAIII and Y nAIII are undetermined coefficients to be solved by boundary conditions; r is the radius distance from the measurement location to the center of the rotor, n is the harmonic number of the magnetic potential function of the permanent magnet region and the air region in the rotor slot, and m is the harmonic number of the magnetic potential function of the air gap region; in a two-dimensional polar coordinate system, the two components of the magnetic field intensity vector and the two components of the flux density vector are represented by the magnetic potential function as follows: μ r is the relative permeability of the permanent magnet, M r and M θ are its two magnetization vector components, μ r = 1, M r = 0, M θ = 0 when in air. According to the quasi-Poisson equation and the boundary conditions, we have: where: Hg is the tangential magnetic field strength at r = R s for the air gap region, s R is the stator inner diameter, Hg is the tangential magnetic field strength at r = R r for the rotor slot air region, Hg is the tangential magnetic field strength at r = R w-1 for the permanent magnet region and the rotor slot air region, respectively, Hg is the radial magnetization strength at r = R w-1 for the permanent magnet region and the rotor slot air region, respectively, At r = R m the following equation is satisfied: The magnetic field generated by the outermost magnets in the slotless air gap region is obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rA-slotless The air gap magnetic field is found; (2) Air gap magnetic field generated by the middle layer The middle layer contains w-2 small layers of equal thickness; when the middle layer is calculated, the outermost layer and the innermost layer are treated as air; at this time, the solving region is divided into air gap region I, permanent magnet upper air region II, permanent magnet region III, and permanent magnet lower air region IV; compared with the outermost layer, one more solving region is added; take any small layer of the middle layer for calculation and analysis, and according to Laplace equation, the scalar magnetic potential functions of the four regions are obtained: where X jmBI , Y jmBI , X jnBII , Y jnBII , X jnBIII , Y jnBIII , X jnBIV and Y jnBIV are unknown coefficients, solved by boundary conditions; The boundary conditions are as follows: where: Hg is the tangential magnetic field strength at r = R s of the air gap region, Hg is the tangential magnetic field strength at r = R r of the air region below the permanent magnet, Hg and Hg are the tangential magnetic field strength at r = R j and the radial magnetization strength of the air region above the permanent magnet and the permanent magnet region, respectively, Hg and Hg are the tangential magnetic field strength at r = R j-1 and the radial magnetization strength of the permanent magnet region and the air region below the permanent magnet, respectively. At r = R m the following equation is satisfied: The magnetic field generated by any of the intermediate layer magnets in the slotless air gap region is obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rB-slotless The air gap field is found; (3) Air gap magnetic field generated by the innermost layer When the innermost layer is calculated, the middle layer and the outermost layer are treated as air; at this time, the solving region is divided into air gap region I, permanent magnet upper air region II, and permanent magnet region III; according to Laplace equation, the scalar magnetic potential functions of the three regions are obtained: where X mCI , Y mCI , X nCII , Y nCII , X nCIII and Y nCIII are undetermined coefficients of the magnetic potential function; The boundary conditions of the innermost layer are as follows: In the formula: For the air gap region at r = R s Tangential magnetic field strength at the location, For the permanent magnet domain in r = R r Tangential magnetic field strength at the location, These represent the tangential magnetic field strengths at r = R1 in the air domain above the permanent magnet and the permanent magnet domain, respectively. These are the radial magnetization intensities of the air domain above the permanent magnet and the permanent magnet domain at r = R1, respectively; At r = R m the following equation is satisfied: The magnetic field generated by the innermost magnets in the slotless air gap region is obtained by solving the matrix equation of the boundary conditions, and thus the air gap flux density B rC-slotless The air gap magnetic field is found; The air gap magnetic field generated by the entire pole is obtained by superimposing the outermost layer, the middle layer, and the innermost layer, so the air gap flux density generated by the w-layer pole is: The air gap flux density when there is a slot is obtained by using the Carter coefficient: B r-slotted = K c (θ) x B r-slotless (35) According to Faraday's electromagnetic induction principle, the magnetic flux of the three-phase winding with N p turns of the coil is obtained: where: L a is the axial length of the motor, ω r is the rotor angular velocity, θ sp is the coil span angle, the three-phase currents obtained from the fluxes are I A , I B , I C the counter electromotive forces and electromagnetic torque: The variables of a surface-embedded trapezoidal magnetic pole Halbach motor are optimized to obtain better motor performance, the variables being the inner and outer pole arc proportioning alpha of the middle trapezoidal magnetic pole C , alpha A , and the magnetization angle theta of the two side magnetic poles m ​ For the trapezoidal Halbach, the inner and outer pole arc ratios of the middle pole and the magnetization angle affect the fundamental wave amplitude and total harmonic distortion THD of the air gap flux density, which is written as: B rf1 = B r-slotless (α A ,α C ,θ m , k = 1) (39) Make α A , α C , θ m The influence of fundamental amplitude and THD Three-dimensional slice map, according to the value distribution in the figure, obtain the optimal variable distribution range, by selecting larger fundamental amplitude and lower THD, determine a set of optimal combination.

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