CN114844267B - Double-stator permanent magnet synchronous motor torque pulsation weakening method based on single-side Halbach array - Google Patents
Double-stator permanent magnet synchronous motor torque pulsation weakening method based on single-side Halbach array Download PDFInfo
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- 230000010349 pulsation Effects 0.000 title claims abstract description 43
- 230000001360 synchronised effect Effects 0.000 title claims abstract description 26
- 238000000034 method Methods 0.000 title claims abstract description 21
- 230000003313 weakening effect Effects 0.000 title claims abstract description 8
- 238000004088 simulation Methods 0.000 claims abstract description 9
- 230000003247 decreasing effect Effects 0.000 claims description 8
- 230000008859 change Effects 0.000 claims description 6
- 238000010586 diagram Methods 0.000 claims description 5
- 230000005284 excitation Effects 0.000 claims description 2
- 230000005415 magnetization Effects 0.000 claims 1
- 238000004364 calculation method Methods 0.000 abstract description 2
- 238000005457 optimization Methods 0.000 description 5
- 238000004891 communication Methods 0.000 description 2
- 238000000354 decomposition reaction Methods 0.000 description 1
- 230000007423 decrease Effects 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 230000004907 flux Effects 0.000 description 1
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- 230000005389 magnetism Effects 0.000 description 1
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Classifications
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02K—DYNAMO-ELECTRIC MACHINES
- H02K1/00—Details of the magnetic circuit
- H02K1/06—Details of the magnetic circuit characterised by the shape, form or construction
- H02K1/22—Rotating parts of the magnetic circuit
- H02K1/27—Rotor cores with permanent magnets
- H02K1/2706—Inner rotors
- H02K1/272—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis
- H02K1/274—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis the rotor consisting of two or more circumferentially positioned magnets
- H02K1/2753—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis the rotor consisting of two or more circumferentially positioned magnets the rotor consisting of magnets or groups of magnets arranged with alternating polarity
- H02K1/278—Surface mounted magnets; Inset magnets
- H02K1/2783—Surface mounted magnets; Inset magnets with magnets arranged in Halbach arrays
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
-
- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02K—DYNAMO-ELECTRIC MACHINES
- H02K1/00—Details of the magnetic circuit
- H02K1/06—Details of the magnetic circuit characterised by the shape, form or construction
- H02K1/22—Rotating parts of the magnetic circuit
- H02K1/27—Rotor cores with permanent magnets
- H02K1/2786—Outer rotors
- H02K1/2787—Outer rotors the magnetisation axis of the magnets being perpendicular to the rotor axis
- H02K1/2789—Outer rotors the magnetisation axis of the magnets being perpendicular to the rotor axis the rotor consisting of two or more circumferentially positioned magnets
- H02K1/2791—Surface mounted magnets; Inset magnets
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02K—DYNAMO-ELECTRIC MACHINES
- H02K16/00—Machines with more than one rotor or stator
- H02K16/04—Machines with one rotor and two stators
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02K—DYNAMO-ELECTRIC MACHINES
- H02K21/00—Synchronous motors having permanent magnets; Synchronous generators having permanent magnets
- H02K21/12—Synchronous motors having permanent magnets; Synchronous generators having permanent magnets with stationary armatures and rotating magnets
-
- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02K—DYNAMO-ELECTRIC MACHINES
- H02K2213/00—Specific aspects, not otherwise provided for and not covered by codes H02K2201/00 - H02K2211/00
- H02K2213/03—Machines characterised by numerical values, ranges, mathematical expressions or similar information
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T10/00—Road transport of goods or passengers
- Y02T10/60—Other road transportation technologies with climate change mitigation effect
- Y02T10/64—Electric machine technologies in electromobility
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- Engineering & Computer Science (AREA)
- Power Engineering (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Permanent Field Magnets Of Synchronous Machinery (AREA)
Abstract
The invention discloses a torque pulsation weakening method of a double-stator permanent magnet synchronous motor based on a unilateral Halbach array, which is used for splitting the double-stator permanent magnet synchronous motor into an inner motor and an outer motor based on a superposition method; performing finite element simulation calculation on the split two motors respectively to obtain the amplitude values of main harmonic components in the torques of the internal motor and the external motor; the structure of the inner motor is unchanged, and the permanent magnet of the outer motor is changed into a 5-section magnetized Halbach array; the magnetizing angle of the magnetic pole at the middle position of the Halbach array is 90 degrees, and the magnetizing angles of the two magnetic poles at the same side of the middle magnetic pole are optimized step by utilizing finite elements, so that the optimal magnetizing angle is obtained. The invention can obviously reduce the torque pulsation of the motor.
Description
Technical Field
The invention relates to the technical field of motor design methods, in particular to a torque pulsation weakening method of a surface-mounted radial magnetizing double-stator permanent magnet synchronous motor with the same number of inner and outer stator slots and with an inner and outer stator position deviation of 90 degrees.
Background
Torque ripple is a key indicator that characterizes how well the motor output torque is stable, and lower torque ripple means that the motor output torque is more stable, less prone to change over time, and less fluctuating.
In the traditional single-stator permanent magnet synchronous motor, a Halbach permanent magnet array is adopted as a better means for reducing the torque pulsation effect of the motor, and similarly, a motor with double-stator topology can also adopt the means for reducing the torque pulsation, but on one hand, the cost of the Halbach array is relatively higher, and on the other hand, the torque pulsation of the double-stator motor is jointly determined by an inner motor and an outer motor, and the outer motor is usually larger in torque pulsation, so that a permanent magnet of the outer motor can be replaced by the Halbach array, and the magnetizing angle of the array is adjusted so as to achieve the aim of reducing the torque pulsation, but due to the structure of the double stators, the searching of the magnetizing angle is different from that of the single stator.
Disclosure of Invention
The invention provides a torque pulsation weakening method of a double-stator permanent magnet synchronous motor based on a single-side Halbach array aiming at torque pulsation of the double-stator surface-mounted permanent magnet synchronous motor, which is a torque pulsation reducing method of the double-stator permanent magnet synchronous motor based on the single-side Halbach array to find an optimal magnetizing angle.
The invention realizes the aim through the following technical scheme, and the torque pulsation weakening method of the double-stator permanent magnet synchronous motor based on the unilateral Halbach array comprises the following steps:
S1, splitting a surface-mounted double-stator permanent magnet synchronous motor into an inner motor and an outer motor, specifically, in the step S1, splitting the surface-mounted double-stator permanent magnet synchronous motor into the inner motor and the outer motor to obtain an outer motor, taking the outer stator, the rotor and a permanent magnet on one side of the rotor close to the outer stator as a whole to obtain the outer motor during splitting, and in actual simulation, only exciting the inner stator and keeping the residual magnetism of the permanent magnet between the inner stator and the rotor to be zero (namely, the split inner motor and the split inner motor are realized in simulation software);
s2, under the condition that given excitation is kept unchanged, finite element simulation is conducted on the torque of the inner motor and the torque of the outer motor respectively, and the amplitude of the torque harmonic wave of the inner motor and the torque harmonic wave of the outer motor are obtained;
s3, comparing the maximum amplitude of the torque harmonic waves of the inner motor and the outer motor, if the maximum amplitude of the torque harmonic wave of the outer motor is 1.5 times or more than that of the inner motor, continuing the step S4, otherwise, not applying the method;
S4, replacing the permanent magnet of each magnetic pole of the outer motor with 5 permanent magnets with different magnetizing angles, wherein the magnetizing angles are beta 1、β2 and beta 3 respectively, the magnetizing angle of the permanent magnet with the largest volume is beta 1, and the magnetizing angle of beta 1 is 90 degrees;
S5, initially selecting magnetizing angles beta 2 and beta 3 to be 80 degrees and 60 degrees respectively;
S6, adjusting the magnetizing angle in the increasing direction to obtain a torque pulsation statistical result diagram of the whole motor, wherein in the step, the magnetizing angles beta 2 and beta 3 are increased simultaneously to obtain the torque pulsation statistical result diagram of the whole motor, and specifically, the magnetizing angles beta 2 and beta 3 are increased in a mode that the step size is 1 degree;
s7, observing the change trend of torque pulsation along with the increase of the magnetizing angle, if a minimum value point appears in the observed magnetizing angle, namely, a concave area appears in the graph, no further optimization is needed, and if the minimum value point does not appear, S8 is carried out;
S8, fixing a magnetizing angle beta 2 according to the change trend of the torque pulsation, enabling the magnetizing angle beta 3 to be adjusted in the increasing or decreasing direction, obtaining the condition that the torque pulsation of the whole motor changes along with the magnetizing angle beta 3, and finding out the magnetizing angle beta 3' when the torque pulsation is the lowest;
And S9, on the premise that the found fixed magnetizing angle beta 3 'is unchanged, adjusting the magnetizing angle beta 2 in the increasing or decreasing direction to obtain the condition that the torque pulsation of the whole motor changes along with the magnetizing angle beta 2, and finding the magnetizing angle beta 2′,β2′,β3' with the lowest torque pulsation as the optimal magnetizing angle for torque pulsation optimization.
As a further optimized scheme of the invention, step S4 replaces the permanent magnet of each magnetic pole of the outer motor with 5 permanent magnets with different magnetizing angles, and the magnetizing angle of the Halbach permanent magnet array of S4 takes the center of each permanent magnet as a reference point;
In the step S4, the permanent magnet of each magnetic pole of the outer motor is replaced by 5 permanent magnets with different magnetizing angles, and the ratio of the pole arc coefficients of the Halbach permanent magnet array of the step S4 is 0.5:0.75:0.8;
In step S6, the direction of increasing the magnetizing angles is adjusted, the changing ranges of the magnetizing angles beta 2 and beta 3 are respectively [80 degrees, 85 degrees ] and [60 degrees, 65 degrees ];
As a further optimized scheme of the invention, in step S8, if the magnetizing angle β 3 is increased and the torque ripple is increased, the magnetizing angle β 3 should be adjusted in a decreasing direction, and the variation range is [55 °,60 ° ], the step size in the variation range is 1 °, and in this case, β 3 when the torque ripple of the whole motor has a minimum value is β 3';
As a further optimized scheme of the present invention, in step S8, if the magnetizing angle β 2 is increased and the torque ripple is increased, the magnetizing angle β 2 should be adjusted in a decreasing direction, and the variation range is [80 °,75 ° ], and the step size in the variation range is 1 °, where β 2 when the absolute value of the difference value of the harmonic amplitude of the internal and external motor torque appears as a minimum value is β 2'.
The invention is characterized in that the torque pulsation of the double-stator motor is jointly determined by an inner motor and an outer motor, and the double-stator permanent magnet synchronous motor is split into the inner motor and the outer motor based on a superposition method; performing finite element simulation calculation on the split two motors respectively to obtain curves of corresponding torque change along with time; performing fast Fourier decomposition on the obtained torque-time curve to obtain the amplitude of the main harmonic component in the torque of the internal and external motors; the structure of the inner motor is unchanged, and the permanent magnet of the outer motor is changed into a 5-section magnetized Halbach array; the magnetizing angle of the magnetic pole at the middle position of the Halbach array is 90 degrees, and the magnetizing angles of the two magnetic poles at the same side of the middle magnetic pole are optimized step by utilizing finite elements, so that the optimal magnetizing angle is obtained. The invention can obviously reduce the torque pulsation of the surface-mounted double-stator permanent magnet synchronous motor with the same number of inner and outer stator slots and the position deviation of the inner and outer stators by 90 degrees.
Compared with the prior art, the invention has the advantages that:
The invention aims to provide a searching method for an optimal magnetizing angle under the condition of adopting a unilateral Halbach permanent magnet array, so that torque pulsation of a double-stator permanent magnet synchronous motor is minimized.
The adoption of the unilateral Halbach permanent magnet array reduces engineering complexity, integrates the characteristics of the double-stator motor, provides a selection method of a magnetizing angle of the Halbach array when the torque ripple of the motor is low, and well reduces the torque ripple of the motor.
Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
Fig. 1 is a flow chart of a torque pulsation attenuation method of a double-stator permanent magnet synchronous motor based on a single-side Halbach array.
Fig. 2 is a model diagram of the motor before permanent magnet optimization.
Fig. 3 is a torque ripple result before motor permanent magnet optimization.
Fig. 4 is a model diagram of a unilateral Halbach array.
Fig. 5 is a variation of torque ripple of the entire motor as a function of magnetizing angle.
Fig. 6 is a plot of torque for the entire motor with β 2 unchanged and β 3 increased.
Fig. 7 is the torque profile of the entire motor as β 2 increases, with β 3 unchanged.
Detailed Description
Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein like or similar symbols indicate like or similar elements or elements having like or similar functions throughout. The embodiments described below by referring to the drawings are exemplary only for explaining the present invention and are not to be construed as limiting the present invention.
It is to be understood that the terms "center," "longitudinal," "transverse," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counter-clockwise," "axial," "radial," "circumferential," and the like are directional or positional relationships as indicated based on the drawings, merely to facilitate describing the invention and to simplify the description, and do not indicate or imply that the devices or elements referred to must have a particular orientation, be configured and operated in a particular orientation, and therefore should not be construed as limiting the invention.
Furthermore, the terms "first," "second," and the like, are used for descriptive purposes only and are not to be construed as indicating or implying a relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first" or "a second" may explicitly or implicitly include at least one such feature. In the description of the present invention, the meaning of "plurality" means at least two, for example, two, three, etc., unless specifically defined otherwise.
In the present invention, unless explicitly specified and limited otherwise, the terms "mounted," "connected," "secured," and the like are to be construed broadly, and may be, for example, fixedly connected, detachably connected, or integrally formed; may be mechanically connected, may be electrically connected or may be in communication with each other; either directly or indirectly, through intermediaries, or both, may be in communication with each other or in interaction with each other, unless expressly defined otherwise. The specific meaning of the above terms in the present invention can be understood by those of ordinary skill in the art according to the specific circumstances.
In the present invention, unless expressly stated or limited otherwise, a first feature "up" or "down" a second feature may be the first and second features in direct contact, or the first and second features in indirect contact via an intervening medium. Moreover, a first feature being "above," "over" and "on" a second feature may be a first feature being directly above or obliquely above the second feature, or simply indicating that the first feature is level higher than the second feature. The first feature being "under", "below" and "beneath" the second feature may be the first feature being directly under or obliquely below the second feature, or simply indicating that the first feature is less level than the second feature.
As shown in fig. 1, the invention discloses a torque pulsation attenuation method of a double-stator permanent magnet synchronous motor based on a single-side Halbach array, which comprises the following steps:
(1) The method comprises the steps of dividing a permanent magnet with a rated torque of 573Nm, a rated phase voltage of 220V, a power frequency of 10Hz, a pole number of 16 and an inner and outer motor power ratio of 1:3, wherein the inner and outer stators staggered by 90 degrees are all radial-type magnetized double-stator permanent magnet synchronous motors into an inner motor and an outer motor, dividing the surface-mounted double-stator permanent magnet synchronous motor into an inner motor and an outer motor in step S1, taking the outer motor as an example, dividing the outer motor into the outer motor, and taking the outer stator, the rotor and the permanent magnet on the side, close to the outer stator, of the rotor as a whole to obtain the outer motor, wherein in actual simulation, only the inner stator is excited and the residual magnetic flux of the permanent magnet between the inner stator and the rotor is zero; as shown in fig. 2, the inside of the dotted line frame is an outer motor, and the inside of the dotted line frame is an inner motor;
(2) When the whole motor is output 573Nm, simulation results in that the torque pulsation of the whole motor is 5.06%, as shown in fig. 3, finite element simulation is performed on the torque of the inner motor and the torque of the outer motor respectively, and the obtained amplitude of the torque harmonic of the inner motor and the torque harmonic of the outer motor are recorded in a table 1;
TABLE 1
(3) The data in the table 1 are observed, and the ratio of the maximum amplitude of the torque harmonic of the motor outside and inside the maximum amplitude of the torque harmonic is 14.416:6.851=2.103 & gt 1.5, so that the motor can meet the amplitude requirement;
(4) Adopting permanent magnet topology as shown in fig. 4, replacing the permanent magnet of each magnetic pole of the outer motor with permanent magnets with 5 different magnetizing angles, wherein the magnetizing angles are beta 1、β2 and beta 3 respectively, the magnetizing angle of the permanent magnet with the maximum volume is beta 1, and the magnetizing angle of beta 1 is 90 degrees;
(5) Optimizing motor torque pulsation by adopting a permanent magnet topology as shown in fig. 4, and selecting initial values of magnetizing angles beta 2 and beta 3 to be 80 degrees and 60 degrees respectively;
(6) The variation ranges of the magnetizing angles beta 2 and beta 3 are respectively [80 degrees, 85 degrees ] and [60 degrees, 65 degrees ], the step length selected in the variation ranges is 1 degree, and meanwhile, the magnetizing angles beta 2 and beta 3 are added for simulation to obtain the torque pulsation variation trend of the whole motor as shown in figure 5;
(7) From the results of fig. 5, the torque ripple variation tendency was observed. Torque ripple gradually decreases with increasing magnetizing angle, and no minimum point appears;
(8) When the torque pulsation is minimum, beta 2 and beta 3 are respectively 85 degrees and 65 degrees, beta 2 is kept to be 85 degrees, the magnetizing angle beta 3 is increased from 65 degrees, the condition that the amplitude of the external motor torque harmonic wave changes along with the magnetizing angle beta 3 is obtained as shown in fig. 6, and therefore the optimal magnetizing angle beta 3' when the torque pulsation of the whole motor is found to be 66 degrees;
(9) On the basis of the previous step, when the torque pulsation is minimum, beta 3 'is kept to be 66 degrees, the magnetizing angle is increased by beta 2 from 85 degrees, the condition that the harmonic amplitude of the external motor torque changes along with the magnetizing angle beta 2 is obtained, as shown in fig. 7, and therefore the optimal magnetizing angle beta 2' when the torque pulsation of the whole motor is found to be 85 degrees.
Therefore, the optimal magnetizing angles beta 2 'and beta 3' are 85 degrees and 66 degrees respectively, the torque pulsation is 1.26 percent, and the torque pulsation is obviously reduced compared with the prior 5.06 percent before the optimization.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (6)
1. The torque pulsation weakening method of the double-stator permanent magnet synchronous motor based on the unilateral Halbach array is characterized by comprising the following steps of:
S1, splitting a surface-mounted double-stator permanent magnet synchronous motor into an inner motor and an outer motor;
s2, under the condition that given excitation is kept unchanged, finite element simulation is conducted on the torque of the inner motor and the torque of the outer motor respectively, and the amplitude of the torque harmonic wave of the inner motor and the torque harmonic wave of the outer motor are obtained;
S3, comparing the maximum amplitude of the torque harmonic waves of the inner motor and the outer motor, and if the maximum amplitude of the torque harmonic wave of the outer motor is 1.5 times or more than that of the inner motor, continuing the step S4;
S4, replacing the permanent magnet of each magnetic pole of the outer motor with 5 permanent magnets with different magnetizing angles, wherein the magnetizing angles are beta 1、β2 and beta 3 respectively, the magnetizing angle of the permanent magnet with the largest volume is beta 1, and the magnetizing angle of beta 1 is 90 degrees;
S5, initially selecting magnetizing angles beta 2 and beta 3 to be 80 degrees and 60 degrees respectively;
S6, adjusting the magnetizing angle in the increasing direction to obtain a torque pulsation statistical result diagram of the whole motor;
S7, observing the change trend of torque pulsation along with the increase of the magnetizing angle, if a minimum value point appears in the observed magnetizing angle, namely, when a concave area appears in the drawing, the magnetizing angle does not need to be further selected, and if the minimum value point does not exist, S8 is carried out;
s8, according to the change trend of the torque pulsation, the fixed magnetizing angle beta 2 is unchanged, the magnetizing angle beta 3 is adjusted in the increasing or decreasing direction, the condition that the torque pulsation of the whole motor changes along with the magnetizing angle beta 3 is obtained, and the magnetizing angle beta 3' when the torque pulsation is the lowest is found;
S9, when the found fixed magnetizing angle beta 3 'is unchanged, the magnetizing angle beta 2 is adjusted to be increased or decreased, the condition that the torque ripple magnetizing angle beta 2 of the whole motor is changed is obtained, and the magnetizing angle beta 2′,β2′,β3' when the torque ripple is the lowest is found to be the optimal magnetizing angle optimized for the torque ripple.
2. The torque ripple reducing method of the double-stator permanent magnet synchronous motor based on the single-side Halbach array according to claim 1, wherein the magnetization angle of the Halbach permanent magnet array in the step S4 takes the center of each permanent magnet as a reference point.
3. The torque ripple reduction method of the double-stator permanent magnet synchronous motor based on the single-side Halbach array according to claim 1, wherein the ratio of pole arc coefficients of the Halbach permanent magnet array of S4 is 0.5:0.75:0.8.
4. The torque ripple weakening method of the double-stator permanent magnet synchronous motor based on the single-side Halbach array according to claim 1, wherein in the step S6, the variation ranges of magnetizing angles beta 2 and beta 3 are respectively [80 degrees, 85 degrees ] and [60 degrees, 65 degrees ], and the step length selected in the variation ranges is 1 degree.
5. The torque ripple reducing method of double-stator permanent magnet synchronous motor based on single-side Halbach array according to claim 1, wherein if the magnetizing angle β 3 is increased and the torque ripple is increased in step S8, the magnetizing angle β 3 is adjusted in a decreasing direction, and the variation range is [55 °,60 ° ], the step size in the variation range is 1 °, and in this case, β 3 when the torque ripple has a minimum value is β 3'.
6. The torque ripple reducing method of double-stator permanent magnet synchronous motor based on single-side Halbach array according to claim 1, wherein if the magnetizing angle β 2 is increased and the torque ripple is increased in step S9, the magnetizing angle β 2 is adjusted in a decreasing direction, and the variation range is [80 °,75 ° ], the step size in the variation range is 1 °, and in this case, β 2 when the torque ripple has a minimum value is β 2'.
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