An analytical method of cogging torque of permanent magnet motor considering core saturation

Abstract

The application discloses an analytical method of cogging torque of a permanent magnet motor considering core saturation. Different from the existing analytical method without considering core saturation, after considering core saturation, air gap permeance is changed, a saturation coefficient is introduced, and modulation units are changed from being modulated by stator slots alone to being modulated by a combination of stator slots and saturation coefficients, so that the harmonic components of air gap flux density are changed. The number of harmonic pole pairs and the rotating speed after the air gap flux density is changed are determined by Fourier decomposition, the traveling wave pairs generating the cogging torque are found according to the condition that the cogging torque is generated by the same number of pole pairs and different rotating speeds, and the existing method for analyzing the cogging torque is effectively supplemented.

Description

Technical Field

[0001] This invention belongs to the field of motor body design technology, and in particular relates to a method for analyzing the cogging torque of a permanent magnet motor. Background Technology

[0002] Permanent magnet motors are widely used in aerospace, electric vehicles, and home appliances due to their advantages of high power density, high power factor, and high efficiency. However, in permanent magnet motors, the cogging torque generated by the interaction between the permanent magnet and the stator teeth produces additional torque ripple, affecting the motor control accuracy. Therefore, how to reduce cogging torque has always been one of the research hotspots in the field of permanent magnet motors.

[0003] The main methods for analyzing cogging torque include the finite element method and analytical methods. The finite element method offers fast calculation speed and high accuracy, but it cannot reveal the generation mechanism of cogging torque. Analytical methods, through theoretical derivation, can explain the causes of cogging torque and provide guidance for reducing it. Currently, researchers typically use energy methods for analytical analysis of cogging torque.

[0004] In recent years, with the rapid rise of field-modulated motors, a corresponding new motor theory—field modulation theory—has attracted widespread attention from scholars. Initially, field modulation theory was only applied to a few special types of motors, such as magnetic gear motors, vernier motors, reverse flux motors, and field-modulated dual-rotor motors. Later, scholars gradually discovered that many magnetic problems in traditional motors could be explained by the principle of field modulation. This led to the development of a new motor research theory based on the principle of field modulation, providing new ideas for the design and analysis of traditional motors.

[0005] In 2020, Liu Jiaqi et al. from Harbin Institute of Technology published a paper entitled "Research on Cogging Torque Based on Magnetic Field Modulation Principle" in the Journal of Electrical Engineering Technology. They proposed a new method for analyzing the cogging torque of permanent magnet motors based on magnetic field modulation and explained the generation mechanism of cogging torque. However, the method proposed in the article was based on the premise of not considering core saturation. Maxwell simulation showed that the cogging torque of the permanent magnet motor changed significantly with and without core saturation. Therefore, the analysis method for cogging torque still has defects.

[0006] For example, in a single-pole permanent magnet motor, the permanent magnets are magnetized in parallel, and the air gap magnetic field of the permanent magnets only has a fundamental wave. According to the traditional conditions for generating positioning torque, even considering the modulation effect of the stator teeth, the motor will not generate two sets of traveling waves with equal pole pairs but different mechanical speeds, and therefore will not generate positioning torque. However, through simulation and experiments, this motor exhibits a certain value of positioning torque. Traditional positioning torque theory cannot explain this phenomenon, and therefore cannot analyze or interpret this positioning torque. Summary of the Invention

[0007] To address the problems mentioned in the background section, the present invention aims to provide an analytical method for calculating the cogging torque of a permanent magnet motor considering core saturation. By introducing a saturation coefficient and remodeling the variable air gap permeability considering core saturation, the effective traveling wave pairs of the air gap magnetic flux density of the permanent magnet motor can be completely determined, thereby accurately calculating the traveling wave pairs that generate cogging torque (based on the condition that different rotational speeds with the same number of pole pairs will generate cogging torque, the corresponding harmonic order is found, and the cause of cogging torque is obtained).

[0008] The objective of this invention can be achieved through the following technical solutions:

[0009] An analytical method for cogging torque of a permanent magnet motor considering core saturation consists of five main steps: parallel magnetization of the permanent magnet, introducing a saturation coefficient to obtain the variable air gap permeability, obtaining the modified air gap magnetic flux density, Fourier decomposition of the air gap magnetic flux density, and obtaining the harmonics that generate the cogging torque.

[0010] 1. The permanent magnet is magnetized in parallel, and the magnetomotive force is a perfect sine wave. There is no need to perform Fourier decomposition on the magnetomotive force, and the magnetomotive force only has the fundamental wave.

[0011] Mathematical model of magnetomotive force: F(θ,t)=Acosp r (ω r t-θ)

[0012] A is the magnetomotive force amplitude, p r ω is the number of rotor pole pairs. r θ is the rotor mechanical angular frequency, and θ is the magnetomotive force phase angle.

[0013] 2. Mathematical model of air gap permeability after Fourier decomposition without considering core saturation:

[0014]

[0015] P j N is the magnetic permeability amplitude. s θ represents the number of stator slots and θ represents the rotor angle.

[0016] Considering the variation in air gap permeability after core saturation, we introduce the saturation coefficient obtained from Fourier decomposition:

[0017]

[0018] a0 is the saturation coefficient constant term, P r ω is the number of rotor pole pairs. r Let a be the rotor mechanical angular frequency. n is the amplitude of each saturation coefficient harmonic, n is the order of the saturation coefficient harmonic, and t is time.

[0019] The final air gap permeability is the product of the original permeability and the saturation coefficient:

[0020]

[0021] 3. The air gap magnetic flux density B is equal to the product of the permanent magnet magnetomotive force F and the air gap magnetic permeability P:

[0022]

[0023] 4. Perform Fourier decomposition on the air gap magnetic flux density:

[0024]

[0025] 5. Based on the expressions derived in step 4, different harmonic pole pair numbers and rotational velocities can be obtained. These are the pole pair numbers |p r -jN s | Rotation speed pole number p r +jN s Rotation speed Extreme logarithm | (2n+1)p r -jN s | Rotation speed Extreme logarithm (2n+1)p r +jN s Rotation speed Extreme logarithm |(2n-1)p r -jN s | Rotation speed The number of extreme pairs (2n-1)p r +jN s Rotation speed Six different harmonics.

[0026] The obtained harmonics are shown in Tables 1, 2, and 3 below.

[0027] Table 1 Harmonics generated only by stator slot modulation

[0028]

[0029]

[0030] Table 2 shows harmonics generated only by saturation coefficient modulation.

[0031] Harmonic order rotational speed 1 (1 is adjusted to 1 by the saturation coefficient 2) <![CDATA[ω r t]]> 3 (1 is adjusted to 3 by the saturation coefficient 2) <![CDATA[ω r t]]> 5 (1 is adjusted to 5 by the saturation coefficient 4) <![CDATA[ω r t]]> 7 (1 is adjusted to 7 by the saturation coefficient of 6) <![CDATA[ω r t]]>

[0032] Table 3 Harmonics generated by simultaneous modulation of stator slot and saturation coefficient.

[0033]

[0034] Based on the condition that only harmonics with the same number of pole pairs but different speeds can generate cogging torque, if the saturation effect is not considered, this motor does not have cogging torque, as shown in Table 1. Similarly, the magnitude of the cogging torque cannot be accurately described, failing to provide a basis for compensation in the control system, and the motor speed fluctuates significantly.

[0035] Based on Tables 2 and 3 established according to this invention, it is found that harmonics 1, 3, 5, and 7 all have different rotational speeds, thus generating cogging torque. The cause of the cogging torque in this motor has been identified, and the harmonic orders generating the cogging torque have been determined. The amplitude and phase of these harmonics are obtained based on specific motor parameters. Substituting these amplitudes and phases into the cogging torque expression allows for accurate calculation of the cogging torque value. This provides a basis for compensation in the control system, reducing motor speed fluctuations.

[0036] It also revealed why, according to the traditional positioning torque theory, the motor should not generate cogging torque, but the actual prototype did.

[0037] The beneficial effects that can be achieved by adopting the above technical solutions conceived in this invention are as follows:

[0038] By considering core saturation and introducing a saturation coefficient, the effective traveling wave pair of the air gap magnetic flux density in a permanent magnet motor is fully obtained. Traditional theory, which does not consider core saturation, assumes parallel magnetization of the permanent magnets, resulting in no cogging torque in the permanent magnet motor. However, considering core saturation, the cogging torque significantly increases. Core saturation has a significant impact on the cogging torque of a permanent magnet motor. Therefore, this invention considers core saturation, introduces a saturation coefficient, and derives a modified air gap permeability formula to obtain the harmonics that generate cogging torque. This supplements and improves the existing magnetic field modulation theory's analysis of the causes of cogging torque in permanent magnet motors. For a series of motors where traditional cogging torque theory considers cogging torque to be zero, the cause of cogging torque in these motors is found, and the harmonic order of the cogging torque is clarified. The numerical value of the cogging torque is accurately calculated, providing a basis for compensation in the control system and reducing motor speed fluctuations. Attached Figure Description

[0039] Figure 1 A mathematical model of magnetomotive force established for parallel magnetization of the permanent magnet in this invention;

[0040] Figure 2 The mathematical model established for the air gap permeability of this invention does not consider core saturation;

[0041] Figure 3 The mathematical model for the saturation coefficient is established to account for core saturation in this invention.

[0042] Figure 4 The 6-slot, 1-pole permanent magnet synchronous motor topology used as an example in this invention;

[0043] Figure 5 The cogging torque waveform of a 6-slot, 1-pole permanent magnet motor without considering core saturation;

[0044] Figure 6 The radial air gap magnetic flux density FFT decomposition diagram of a 6-slot, 1-pole permanent magnet motor without considering core saturation.

[0045] Figure 7 The cogging torque waveform of a 6-slot, 1-pole permanent magnet motor considering core saturation;

[0046] Figure 8 FFT decomposition diagram of radial air gap magnetic flux density for a 6-slot, 1-pole permanent magnet motor considering core saturation;

[0047] Figure 9 The present invention provides a schematic diagram of the cogging torque of a permanent magnet motor considering core saturation. Detailed Implementation

[0048] like Figure 4 The image shows a fractional-slot permanent magnet motor with 6 stator teeth and 1 pair of poles.

[0049] A method for analyzing the cogging torque of a permanent magnet motor considering core saturation consists of five main steps: parallel magnetization of the permanent magnet, introducing a saturation coefficient to obtain the variable air gap permeability, obtaining the altered air gap magnetic flux density, Fourier decomposition of the air gap magnetic flux density, and obtaining the harmonics that generate the cogging torque.

[0050] 1. The permanent magnet is magnetized in parallel, and the magnetomotive force is a perfect sine wave, such as... Figure 1 As shown, Fourier decomposition of the magnetomotive force is not required; the magnetomotive force only has a fundamental wave, and the permanent magnet has one pole pair. The mathematical model of the magnetomotive force is: F(θ,t)=Acosp r (ω r t-θ).

[0051] 2. Air gap permeability without considering core saturation, such as Figure 2 As shown, N s The value is 6. The mathematical model after Fourier decomposition:

[0052]

[0053] Considering the variation in air gap permeability after core saturation, a saturation coefficient is introduced, such as... Figure 3 As shown. The saturation coefficient after Fourier decomposition is:

[0054]

[0055] The final air gap permeability is the product of the original permeability and the saturation coefficient:

[0056]

[0057] 3. The air gap magnetic flux density B is equal to the product of the permanent magnet magnetomotive force F and the air gap magnetic permeability P:

[0058]

[0059] 4. Perform Fourier decomposition on the air gap magnetic flux density:

[0060]

[0061] 5. Obtain different harmonic pole pair numbers and rotational speeds. (The last part, "N," appears to be a typo and can be omitted.) s =6, p r Substituting 1 into the equation, we get the number of pole pairs |1-j6| and the rotational speed, respectively. Number of pole pairs 1+j6, rotational speed Number of pole pairs |(2n+1)-j6|, rotational speed Number of pole pairs (2n+1)+j6, rotational speed Number of pole pairs |(2n-1)-j6|, rotational speed Number of pole pairs (2n-1)+j6, rotational speed Six different harmonics.

[0062] The obtained harmonics are shown in Tables 1, 2, and 3 below.

[0063] Table 1 Harmonics generated only by stator slot modulation

[0064]

[0065] Table 2 shows harmonics generated only by saturation coefficient modulation.

[0066] Harmonic order rotational speed 1 (1 is adjusted to 1 by the saturation coefficient 2) <![CDATA[ω r t]]> 3 (1 is adjusted to 3 by the saturation coefficient 2) <![CDATA[ω r t]]> 5 (1 is adjusted to 5 by the saturation coefficient 4) <![CDATA[ω r t]]> 7 (1 is adjusted to 7 by the saturation coefficient of 6) <![CDATA[ω r t]]>

[0067] Table 3 Harmonics generated by simultaneous modulation of stator slot and saturation coefficient.

[0068]

[0069] According to the condition that only two harmonics with the same number of pole pairs but different rotational speeds can generate cogging torque, there are no harmonics in Table 1 that meet this condition, and therefore no cogging torque will be generated. Figure 5 As shown, the cogging torque is very small and can be ignored.

[0070] However, considering the saturation effect proposed in this invention, and introducing the analysis of the saturation coefficient, in Table 3, taking the 5th harmonic as an example, two 5th harmonics appear, and their rotational speeds are not equal, generating cogging torque. All harmonics in Table 3 that meet the condition that "only two harmonics with the same number of pole pairs but different rotational speeds will generate cogging torque" will generate cogging torque.

[0071] contrast Figure 6 and Figure 8 The FFT analysis of the air gap magnetic flux density of the permanent magnet motor (obtained using finite element software) shows that, taking the 5th harmonic as an example, after considering the saturation effect, Figure 8 The increase in the amplitude of the fifth harmonic, resulting in fifth harmonics with different rotational speeds, verifies the correctness of the invention.

[0072] In summary, this invention identifies the cause of cogging torque in the motor and clarifies the harmonic order that generates it. By calculating the amplitude and phase of these harmonics based on specific motor parameters and substituting these values ​​into the cogging torque expression, the numerical value of the cogging torque can be accurately calculated. The analytical method of this invention yields a peak cogging torque of 0.8 Nm, which aligns with actual conditions. This provides a basis for compensation in the control system, reducing motor speed fluctuations.

Claims

1. An analytical method of cogging torque of a permanent magnet motor considering core saturation, characterized by, Comprising the following steps: Multiply the mathematical model of air gap permeance after Fourier decomposition without considering the core saturation with saturation coefficient to obtain air gap permeance P(θ, t) considering core saturation, and the saturation coefficient is: where a0 is a saturation coefficient constant term, P r is the number of rotor pole pairs, ω r is the rotor mechanical angular frequency, θ is the phase angle of the magnetomotive force, a n is the amplitude of each saturation coefficient harmonic, n is the number of saturation coefficient harmonics, and t is time. Multiply P(θ, t) with permanent magnet magnetomotive force F(θ, t) to obtain air gap flux density B(θ, t): Fourier decomposition is performed on air gap flux density B(θ, t) to obtain different harmonic pole pair numbers and rotating speeds; According to the condition that the same pole pair number and different rotating speeds will produce slot torque, find the harmonic order that produces slot torque, and according to the specific motor parameters, the amplitude and phase of these harmonics are obtained, and the amplitude and phase of these harmonics are brought into the expression of slot torque, so that the numerical value of slot torque can be calculated, and then the torque compensation of permanent magnet motor control system is provided, and the speed fluctuation of permanent magnet motor is reduced.

2. The analytical method for considering the cogging torque of a permanent magnet motor with core saturation according to claim 1, characterized in that, Permanent magnet parallel magnetization, perfect sine magnetomotive force, no Fourier decomposition is needed for magnetomotive force, and the magnetomotive force only has fundamental wave.

3. The analytical method for considering the cogging torque of a permanent magnet motor with core saturation according to claim 1, characterized in that, Fourier decomposition is performed on air gap flux density B(θ, t) to obtain: where A is the magnetic motive force amplitude, P j is the permeance amplitude, N s is the number of stator slots.

4. The analytical method for considering the cogging torque of a permanent magnet motor with core saturation according to claim 3, characterized in that, By Fourier decomposition of the air-gap flux B(θ, t), the different air-gap flux harmonic pole pair numbers and rotational speeds are obtained, which are pole pair number |p r -jN s |, rotational speed pole pair number p r +jN s , rotational speed pole pair number |(2n+1)p r -jN s |, rotational speed pole pair number (2n+1)p r +jN s , rotational speed pole pair number |(2n-1)p r -jN s |, rotational speed pole pair number (2n-1)p r +jN s , rotational speed six different harmonics.

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