Numerical simulation and optimization for electromagnetic noises of permanent magnet synchronous motors in vehicles

2.1. Numerical computation of electromagnetic forces of motors

In the paper, the Pro-engineer software is used to establish a motor model. Before it is input into ANSYS, the CAD model is simplified, and some secondary structures are neglected, so mesh division and computation will become more convenient. For example, parts such as gaskets, velometers, detailed chamfers and drawing dies are neglected. Fig. 1 shows the established geometric model of the motor. The motor contains 24 slots and 12 poles. The rated rotation speed is 3000 rpm. Other related parameters are shown in Table 1.

Fig. 1Geometric model of permanent magnet synchronous motors

During electromagnetic analysis on a permanent magnet synchronous motor, a finite element model of the motor should be established firstly. Electromagnetic analysis on the motor mainly studied characteristics of gap flux density acting between the motor stator and the rotor, so the finite element model of the stator and rotor should be established during modeling, and some secondary parts should be neglected. The electromagnetic simulation analysis software Ansoft is used to establish a two-dimensional finite element analysis model of the motor and then a three-dimensional model is obtained after stretching, as shown in Fig. 2. Under the condition of steady and symmetric loads, the fundamental wave magnetic potentials of the motor stator and rotor are rotated under a synchronous speed. They can be deemed to be relatively stationary. Therefore, the electromagnetic field at any instant state can be deemed as a constant magnetic field during analysis. ANSYS is used to conduct finite element analysis. The element type is a default three-dimensional solid element. Material properties such as elasticity modulus, Poisson’s ratio and density of each part are defined. The motor is an assembled body, so related contact types should be defined. Then, mesh division of the complete model is carried out. Mesh division has obvious impacts on computational results. In general, some meshes must had high quality. Nodes should be sufficient, and relative element distribution should be rational during division of the finite element model. Regarding the stator, density is 7700 kg/m³; elasticity modulus is 2.05e¹¹Pa; Poisson’s ratio is 0.27.

Fig. 2Finite element model of electromagnetic forces of motors

According to the given structural parameters of the motor, a finite element simulation model of the motor is established. Through simulation computation, contours of magnetic force distribution and flux density in the motor at the time $t=$ 0.04 s are obtained, as shown in Fig. 3. Obviously, the magnetic pole directly faces the stator tooth; nearly all the magnetic force lines enter the stator tooth along the axial direction; the tooth flux density is high; radial electromagnetic force density is high and nearly distributed on the complete tooth uniformly, which will cause radial telescopic vibration of the stator tooth. At the positions between poles, which directly faces the stator tooth, the tangential components of magnetic force lines are large; at the position, radial electromagnetic forces borne by the stator tooth have relatively small density, which will cause uneven radial force on the stator tooth as well as sway and vibration of the tooth. Therefore, radial force waves of the motor are the main reason for causing electromagnetic vibration and noise.

Fig. 3Contours of flux density and magnetic force lines of motors

a) Flux density of motors

b) Magnetic force lines of motors

The time-frequency spectrum of radial electromagnetic force density of the motor under 3000 rpm is shown in Fig. 4. It is shown in Fig. 4(a) that the time-domain electromagnetic force density presents an obvious periodicity, where the maximum electromagnetic force density is 6e⁵N/m². The electromagnetic force density under time-domain is processed by Fourier transform, so results of the frequency domain are obtained, as shown in Fig. 4(b). It is shown in the figure that the frequency-domain electromagnetic force density has many peak points, where peaks are especially obvious at 50 Hz, 900 Hz, 1200 Hz and 5800 Hz, and specific values are 112 dB, 110 dB, 100 dB and 65 dB, respectively. Peak frequencies of electromagnetic forces are caused by that the motor has obvious orders as a rotational body. In addition, the electromagnetic force spectrum of the motor is mainly concentrated within 4000 Hz.

Fig. 4Time-domain and frequency-domain electromagnetic force density

a) Time-domain electromagnetic force density

b) Frequency-domain electromagnetic force density

Electromagnetic vibration of the permanent magnet synchronous motor can be studied by ANSYS transient dynamics analysis. Radial electromagnetic forces of each tooth varying with time can be obtained through electromagnetic analysis. As an excitation source of dynamic simulation, it is loaded into a three-dimensional finite element model of the motor, as shown in Fig. 5. It is shown in Fig. 5 that the finite element model of the structure contains end cap and cover of the motor. In this way, electromagnetic vibration of the motor under radial electromagnetic forces can be computed, as shown in Fig. 6. It is shown in the figure that the motor displacement fluctuates seriously around 0, but fails to show an obvious periodicity like the electromagnetic force density. The motor vibration displacement has the maximum vibration amplitude around 0.05 s, where the maximum value is 6e^-5m. Large vibration responses do not appear in low-order harmonic waves, while they are at high orders, namely large time points. In vibration responses, the vibration amplitude is in direct proportion to excitation force wave amplitudes, while large amplitudes of low-order harmonic waves can easily cause vibration. However, when the frequency of excitation force waves under specific orders approach natural frequencies of corresponding modes, even very small force wave amplitudes can lead to large vibration responses.

Fig. 5Finite element model of the structural vibration of motors

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Fig. 6Electromagnetic vibration displacement of permanent magnet synchronous motors

2.2. Experimental verification of the electromagnetic vibration of motors

Numerical computation models of vibration characteristics of motors are very complicated and can be obtained indirectly through electromagnetic force density. Therefore, it is necessary to verify correctness of computational results by experimental test. Experimental equipment includes QDAC signal generation modules of LMS, power amplifiers and exciters, such as an acceleration sensor, a force sensor, an LMS signal amplification and intelligent collection system, and LMS vibration analysis software Test.lab. Vibration characteristics within 0-10000 Hz are mainly studied in experiments. The motor is powered and started working under a no-load state. Signal sampling frequency is 25000 Hz; the sensor is surrounded on the outer surface of the stator near the end part; motor vibration signals tested by the sensor are input into Test.lab for analysis and processing. In order to avoid interference of loads on electromagnetic vibration of the motor and ensure accurate test on the motor stator surface, the permanent magnet synchronous motor in the paper is hung on a rigid support by a soft rubber band, as shown in Fig. 7.

Fig. 7Experimental test on vibration characteristics of motors

Experimental results of the vibration displacement is compared with numerical simulation results, as shown in Fig. 8. It is shown in the figure that changing trends between experimental and numerical simulation results are consistent while peak points of the vibration displacement are completely consistent. Results of the experimental test are slightly more than those of numerical simulation because the vibration displacement of experimental test not only contains the electromagnetic vibration, mechanical vibration and aerodynamic vibration, but also contains tangential electromagnetic forces. Numerical simulation results only involve vibration results under the radial electromagnetic force. Despite of the slight difference in values, the numerical simulation can still replace experimental test.

Fig. 8Comparison of the electromagnetic vibration between experiment and simulation

3.1. Numerical computation of electromagnetic noises

Next, modal participation factors will be used to recognize modals which affect the motor electromagnetic noise obviously, in order to improve the structure and reduce the noise. Therefore, the finite element model is used to compute modals of the motor, as shown in Fig. 9. It is shown in the figure that modals of the motor at each order present an obvious symmetry while positions with serious vibration are located at the radial cover surface of the motor as electromagnetic forces of the motor are also distributed symmetrically in the radial direction, and positions with serious vibration are under different rotation angles of the motor. With the increased analysis order, multiple wave nodes are at the motor end cap such as 7th and 8th modal shapes.

Fig. 9Modal shapes of motor structures at top 8 orders

a) First order

b) Second order

c) Third order

d) Fourth order

e) Fifth order

f) Sixth order

g) Seventh order

h) Eighth order

The FEM/AML method is used to compute radiation noises of the motor. The motor AML model should be established at first. During establishment of the AML model, the mesh size should be more than that of the finite element meshes. In general, the characteristic size $L$ of the AML model should satisfy the following requirements:

where: $C_0$ is the velocity of sound waves, which is generally set to be 340 m/s; $F_{max}$ is the maximum frequency, equal to 10000 Hz; $N$ is the multiple of the sound wave length relative to the element mesh size, where larger $N$ leads to higher accuracy of computational results and generally $N\ge$ 6, and $N$ is 8 in the paper. Therefore, in the paper, the element size is 4 mm during division of the AML meshes. During mesh division, meshes should also be simplified in combination with the specific finite element model.

The vibration displacement obtained by harmonic response analysis is taken as the boundary condition of acoustic analysis. Acoustic meshes and structural meshes of the motor are inconsistent in element size, element density and node coordinates, so the input vibration displacement should be transferred, and thus corresponding relations between vibration data and meshes could be established. The AML model has 13250 nodes and 9816 elements, as shown in Fig. 10. In the computation, it is assumed that the motor is located in free space and free of reflection face structures, and the fluid medium is air. In the end, some basic acoustic analysis items are defined, so the motor sound field responses can be computed.

Fig. 10AML model of electromagnetic noises of permanent magnet synchronous motors

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When sound waves are propagated in the medium, the medium will absorb the sound waves, so sound waves will be attenuated with the increased propagation distance. Therefore, during studying a small-power motor, the motor can be processed as a point sound source within the distance of 1.5 m away from the motor. At this moment, radiation attenuation of the noise is very small and can be neglected. Based on these theories, a semi-spherical field point with the center radius of 1 m is selected. In this way, the situations will be consistent with subsequent experimental test, as shown in Fig. 11. In addition, in order to observe sound field distribution of the motor in $X$, $Y$ and $Z$ directions, field point models of three planes are established, as shown in Fig. 12.

Fig. 11Semi-spherical field point meshes of permanent magnet synchronous motors

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Fig. 12Panel field point meshes of permanent magnet synchronous motors

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Sound pressure levels of electromagnetic noises of the motor are extracted, as shown in Fig. 13. It is shown in the figure that when the analyzed frequency is lower than 1000 Hz, the sound pressure level gradually increased with the analyzed frequency and reached an obvious peak value. When the analyzed frequency is within 1000 Hz-3000 Hz, the sound pressure level gradually decreased with the analyzed frequency. Within 3000 Hz-10000 Hz, the sound pressure level reaches a stable value 35 dB, and obvious peak noises are only at frequency points of some orders. Regarding a rotational body, there are obvious rotation orders. According to Eq. (2), we can find that the first order of motors is 50 Hz, so peak noises which are not very obvious on the sound pressure level curve. As well, the motor also has obvious peak noises at 300 Hz, 400 Hz, 500 Hz, 600 HZ, 900 Hz, 1200 Hz, 1500 Hz and 1800 Hz, where they are corresponding to 6th, 8th, 10th, 12th, 18th, 24th, 30th and 36th orders of the motor. The motor has not order frequencies at some frequency points, but peak noises are still on sound pressure level curves because of resonance of the motor structure. According to Eq. (3), the total sound pressure level of electromagnetic noises can be computed, where the value is 58 dB.

Through FFT, frequency spectrum results under the constant rotation speed can be obtained. Based on order analysis theories, the relations between order and rotational speed can generally be expressed as follows:

where: $f$ is signal frequency (Hz), $O$ is the order, and $rpm$ is a referential axis rotation speed (r/min). Obviously, direct correspondence relations are between order and rotational speed. Essentially, order analysis is a signal processing method, where signals with equal time intervals are converted into signals with equal angle intervals, and then frequency spectrum analysis is conducted to the signals:

where: $L_i$ is the sound pressure level under the $i$th frequency; $n$ is the amount of frequency points. $L$ is the total sound pressure level.

Sound pressure contours of electromagnetic noises are extracted, as shown in Fig. 14. It is shown in Fig. 14 that the electromagnetic noise contour of the motor presents an obvious symmetry, where electromagnetic noises of the motor are axis-symmetric in plane $X$ and plane $Z$. In plane $Y$, the electromagnetic noise of the motor is skew-symmetric relative to a 45° angle. In plane $X$ and plane $Z$, the noise is generated from vibration of the end cap, while noises in plane $Y$ are caused by electromagnetic radial force. Obviously, the electromagnetic noise has an obvious directivity.

Fig. 13Sound pressure levels of electromagnetic noises of motors

Fig. 14Sound pressure contours of electromagnetic noises of motors

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3.2. Experimental verification of electromagnetic noises of motors

In order to verify the correctness of the method proposed in the paper, experimental test on noises is conducted on the motor. The experiment is carried out at 3000 rpm and the average background noise is 66 dB. The perpendicular projection center of the motor on the frame is taken as the spherical point. Observation points are located on the semi-spherical surface with radius of 1 m. Experimental results are revised, and a sound level meter is used to test A-weighting sound pressure levels of each observation point. Frequency of 20 kHz is adopted for each microphone. Fig. 15 shows chromatogram which is the motor noise with the rotational speed at 2 observation points. It is shown in Fig. 13 that main orders of the motor electromagnetic noise are 6th, 8th, 10th, 12th, 18th, 24th, 30th and 36th orders, etc. The result is consistent with the simulation conditions, verifying the correctness of computational results in the paper.

Sound pressures of observation points obtained in the experimental test are further extracted and compared with the numerical simulation results, as shown in Fig. 16. It is shown in the figure that each peak noise is completely consistent between experimental and numerical simulation results. However, peak noises in experimental test are more obvious because radiation noises in experimental test not only contains electromagnetic noises, mechanical noises and aerodynamic noises, but also contains electromagnetic noises caused by tangential electromagnetic forces. Numerical results only involve electromagnetic noises under radial electromagnetic forces. Despite of the slight differences in values, the numerical simulation can still replace experimental test.

Fig. 15Chromatogram of noise test on permanent magnet synchronous motors

a) Noise observation point 1

b) Noise observation point 2

Fig. 16Comparison of electromagnetic noises between experiment and simulation