Analysis of Vibration and Noise in a Permanent Magnet Synchronous Motor Based on Temperature-Dependent Characteristics of Permanent Magnet

Abstract

Interior permanent magnet synchronous motors (IPMSMs) are widely utilized due to their high power density. However, noise and vibration issues are often encountered in these motors. While researchers have extensively investigated individual aspects such as noise, vibration, and heat generation in PMSMs, there has been a lack of comprehensive studies examining the interrelationships among these factors. In this paper, a novel approach is proposed for predicting vibration by considering the radial force in the air gap as the exciting force, while also accounting for the changes in the permanent magnet (PM) characteristics caused by heat generation during motor operation. The method involves decomposing and identifying vibration components associated with each vibration mode and predicting noise based on the sound radiation efficiency of each mode. By constructing a vibration map based on current and temperature at a specific frequency, the components most affected by current variations and PM characteristics can be identified. This allows for the proposal of design improvements aimed at reducing vibration. Furthermore, by comparing the vibration map with the noise map, it is confirmed that vibration serves as a source of noise and influences its generation. However, it is found that vibration and noise are not strictly proportional. Overall, a comprehensive analysis of the correlations between vibration, noise, and other factors in IPMSMs is presented in this study. The proposed method and findings contribute to the understanding of the complex dynamics involved and provide valuable insights for the design of quieter and more efficient motor systems.

1. Introduction

Compressors are widely used in various industrial and commercial applications, including air conditioning systems and refrigeration equipment. They play a crucial role in compressing refrigerants, facilitating heat absorption and release for effective cooling and heat management. In precision-driven environments like factories and automotive engines, compressors generate air movement by compressing air or refrigerant. They also contribute to the comfortable driving experience in vehicles as an integral part of the overall cooling system. Additionally, compressors are utilized in everyday household appliances like air conditioners and refrigerators to cool or regulate the air. These operations rely on motors that convert electrical energy into mechanical energy to perform the necessary compression work.

Permanent magnet synchronous motors (PMSMs) are widely employed to drive compressors due to their high torque density and efficiency [1,2,3,4]. However, vibration and noise issues during operation remain significant concerns. These problems can lead to discomfort, performance degradation, and equipment damage. The generation of vibration and noise in compressors is attributed to the electromagnetic force resulting from the interaction between the motor’s magnetic field and the current flowing through the windings. Understanding these electromagnetic sources is vital for improving the performance, reliability, and sustainability of compressor motors. Temperature is also a critical factor as it significantly affects the motor’s performance, efficiency, reliability, and lifespan. Therefore, a comprehensive analysis of temperature’s impact on motor components and performance is essential for optimizing operations and ensuring safe and sustainable usage.

Numerous studies have been conducted on the vibration and noise generated by motors. Generally, vibration and noise are considered to have two main causes: electronic and mechanical factors. Studies have proposed pole/slot combinations that are advantageous for reducing vibration and noise by analyzing the characteristics of motors based on pole/slot combinations [5,6,7]. Another study focused on altering the stator shape in IPMSMs to reduce vibration and noise, providing an analysis process [8]. Additionally, a study utilized finite element method (FEM) to calculate displacement, vibration, and noise at the stator tooth of a PMSM [9]. There are also papers that use numerical methods to optimize the design of PMSMs in agreement with FEM results [10,11]. Optimizing stator parameters, such as slot length, yoke thickness, and tooth thickness, was explored in [12] to minimize vibration and noise. In [13], a vibration acceleration reduction model was proposed by optimizing stator slot parameters using an algorithm. A study in [14] utilized response surface methodology to optimize IPMSMs (Interior PMSMs), maintaining average torque while reducing vibration and torque ripple. Vibration mode, radial force, and vibration frequency in PMSM stators were compared using FEM in [15]. The mechanism, prediction methods, suppression techniques, and pros and cons of current technology for predicting electronic vibration and noise in PMSMs were extensively discussed in [16]. Furthermore, a multi-physics workflow considering electromagnetic, thermal, and noise, vibration, and harshness (NVH) analysis was proposed for PMSMs used in traction motors [17]. An electronic vibration analysis process considering time and spatial harmonics was proposed in another study in [18]. The impact of radial harmonics with low mode numbers on vibration was investigated in [19], while vibration reduction was achieved through modal analysis optimization of the stator, rotor, and housing shell of a PMSM [20]. The consideration of both mechanical bearing vibration and electronic vibration was addressed in [21]. The influence of current harmonics on vibration and noise was analyzed in [22], and an increase in switching frequency was claimed to suppress vibration [23]. Additionally, vibration characteristics were reduced by applying phase-shifted SVPWM to carrier harmonics during PMSM operation [24].

Previous studies have explored the significance of temperature on the electromagnetic vibration and noise produced by motors, specifically focusing on the temperature of permanent magnets (PMs). While these studies have presented experimental NVH analysis results, a definitive theoretical correlation between temperature and these factors remains elusive. This paper aims to address this gap by conducting an in-depth analysis of the vibration and noise of an electric motor in relation to temperature. Outside factors not relating to the motor will not be discussed in this paper. The base model for this study is a compressor motor commercially employed in laundry dryers.

2. Electromagnetic Characteristics of Study Motor

2.1. Configuration of Compressor Motor

The compressor in the dryer plays a crucial role in pressurizing and increasing the temperature of the refrigerant, leading to heat release in the condenser. The expanded refrigerant, at lower pressure and temperature, absorbs moisture in the evaporator. Efficient and high-power density IPMSMs are commonly employed in compressors for effective and rapid drying. A six-pole nine-slot IPMSM, as shown in Figure 1, is used in this paper.

Table 1. Specifications of compressor motor.

Parameters	Value	Unit
Number of poles	6	-
Number of slots	9	-
Number of phases	3	-
Stator outer diameter	90	mm
Rotor outer diameter	48.8	mm
Shaft diameter	13	mm
Stack length	30	mm
Air-gap length	0.6	mm
Magnet length	20.5	mm
Magnet height	1.8	mm
Number of turns/pole	130	turns
DC-link voltage	310	V
Phase resistance	6.7	ohm
Rated torque	1.09	Nm
Rated speed	2700	rpm
Rated power	308	W
Efficiency	95.8	%
Stator, rotor core material	27PNX1350F	-
Magnet material	G54SH	-

shows the detailed specifications of the compressor motor.

To comprehensively evaluate the electrical characteristics, vibration, and noise levels of the PM relative to temperature, various operating conditions are considered. This includes investigating the motor’s response to temperature variations under constant current to validate the significance of considering the temperature effect on the PMs. Additionally, the study also analyzes the motor’s performance and characteristics at the rated point while varying the temperature conditions. All analyses were performed under idealized sinusoidal current conditions.

2.2. Electromagnetic Characteristics of Permanent Magnets

The performance of the PMs in an IPMSM is greatly influenced by temperature, as evident from previous studies examining various motor components such as the rotor, stator, and PMs. Particularly, PMs exhibit significant changes and exert a notable impact on the motor’s performance [25]. Thus, this study aims to analyze the effects of PM temperature on the motor’s electrical characteristics, as well as its noise and vibration.

In Figure 2, variations in coercive force and residual flux density of the G54SH PM material, used in the reference model, are depicted as a function of temperature. The solid and dashed lines represent the hysteresis and magnet demagnetization curves, respectively. Experimental data from the manufacturer are utilized to determine the temperature coefficients ${\alpha }_B$ and ${\alpha }_H$ for residual flux density and coercive force, respectively, using Equations (1) and (2). These coefficients allow for the calculation of residual flux density and coercive force at a specific temperature based on their values at the reference temperature. Here, $B_r\left(T\right)$ and $B_r\left(T_0\right)$ represent the residual flux density at temperatures $T$ and $T_0$, respectively [21].

$$B_r\left(T\right)=B_r\left(T_0\right)\left[1+{\alpha }_B\left(T-T_0\right)\right]$$

(1)

$$H_c\left(T\right)=H_c\left(T_0\right)\left[1+{\alpha }_H\left(T-T_0\right)\right]$$

(2)

Table 2. Residual magnetic flux density and coercive force with respect to temperature.

	Temperature (°C)
	−40	−20	0	20	40	60	80	100	120	140	160	180
$B_r$  $\left(T\right)$	1.54	1.51	1.47	1.44	1.41	1.38	1.35	1.32	1.28	1.24	1.19	1.14
$-H_c$  $\left(kA/m\right)$	1203	1170	1139	1109	1080	1053	1028	1004	950	885	695	572

shows the calculated values of residual flux density and coercive force of the G54SH for temperature intervals from −40 °C to 180 °C, using temperature coefficients. As temperature increases, both coercive force and residual flux density decrease due to PM demagnetization. This suggests that the motor’s performance is expected to deteriorate with higher temperatures under the same current.

Figure 3 depicts the distribution of flux density at four different temperatures (−40 °C, 20 °C, 100 °C, and 180 °C). It shows that as the temperature increases, the residual flux density of the PM decreases. This results in a decrease in the overall flux density distribution within the motor. It emphasizes the significance of accounting for temperature effects during operation, especially in applications with significant temperature variations like compressors. Neglecting temperature considerations can lead to deviations in performance from the desired operating point.

Figure 4 displays torque analysis results at different temperatures (−40 °C, 20 °C, 100 °C, and 180 °C) for the motor under the same current. The torque decreases with increasing temperature, reflecting the trend observed in Figure 3. Notably, temperatures exceeding the maximum limit of 150 °C for the PM, such as 160 °C and 180 °C, result in a significant drop in average torque. Neglecting temperature variations can lead to performance deviations when comparing cases with the same output power.

2.3. Electromagnetic Characteristics under Constant Current Condition

The operational stability of an electric motor is determined by its temperature, which is influenced by heat generation and dissipation. Heat is generated through various losses, including copper, iron, mechanical, and stray load losses. Iron losses depend on the motor’s rotational speed and magnet strength, while copper losses are influenced by the current magnitude and winding resistance. Mechanical losses are associated with factors like bearing friction and windage. Copper losses increase with the square of the current and the winding resistance, and temperature impacts both PM characteristics and the copper winding resistance. The temperature coefficient of copper at 20 °C is 0.00393/°C, indicating that the resistance of copper windings increases by approximately 0.4% for a 1 °C temperature rise from 20 °C. The resistance of copper windings can be calculated as follows,

$$R_T=R_{20}\left[1+0.00393\left(T-T_{20}\right)\right]$$

(3)

where $R_T$ represents the resistance at temperature $T$. $R_{20}$ and $T_{20}$ are the resistance and temperature at 20 °C, respectively.

As temperature increases, the resistance of the winding rises, causing an increase in copper loss. Simultaneously, the characteristics of PMs lead to a decrease in core loss. Figure 5 shows the motor’s efficiency, copper loss, and core loss at the rated speed. With a higher temperature under the same speed and current, the torque and output power decrease. However, the increase in copper loss surpasses the decrease in core loss, resulting in higher total loss and reduced efficiency. The motor’s efficiency is calculated as follows,

$$\eta =\frac{P_{mech}}{P_{mech}+P_{loss}}=\frac{\tau \omega }{\tau \omega +P_{loss}}$$

(4)

where $P_{mech}$ is the mechanical output power of the motor, $P_{loss}$ is the total sum of losses, $\tau$ is the magnitude of average torque, and $\omega$ is the angular velocity of the rotor.

Assuming the absence of mechanical losses such as bearing friction and windage, the efficiency is calculated by considering the core loss, copper loss, and stray load loss. With the same speed and current conditions, the increase in copper losses exceeds the decrease in iron losses, resulting in higher total losses and a decrease in average torque. As shown in Figure 5, the motor efficiency gradually decreases as the temperature increases.

2.4. Electromagnetic Characteristics under Constant Torque Condition

Under constant torque conditions, the influence of temperature on the motor’s electromagnetic characteristics becomes evident. As discussed earlier, neglecting the demagnetization effect of the PMs due to temperature variations leads to performance discrepancies. As the temperature rises and the residual PM flux density decreases, a higher current is required to achieve the same rated torque. Figure 6 illustrates a comparison between torque ripple and cogging torque under equal output power conditions. As the temperature increases, the current needed to maintain the same torque level gradually rises due to the diminishing residual flux density. At temperatures exceeding the permissible limit for the PM, such as 160 °C and 180 °C, a significant increase in current is observed as rapid demagnetization occurs due to the sharp reduction in residual flux density. Additionally, the interaction between PMs and the motor results in a decrease in cogging torque as temperature rises. This reduction in cogging torque, which is one of the factors contributing to torque ripple, consequently leads to a decline in overall torque.

Figure 7 presents a comparison of copper losses, iron losses, and efficiency under identical output conditions. The armature resistance, influenced by the temperature coefficient of copper, exhibits non-linear growth as temperature increases, demanding a higher current to maintain the same torque output. In contrast, iron losses remain relatively stable across the range of current and output conditions. Consequently, the efficiency under identical output conditions gradually declines with rising temperature, in contrast to the efficiency observed under identical current conditions at lower temperatures.

3. Proposed Electromagnetic Vibration Analysis

We previously published a comparison between the numerical vibration calculation results and the experimental results, so these numerical vibration calculation results are used for the analysis in this paper [26]. The mechanism of vibration and noise generation in the electric motor is schematically illustrated in Figure 8. The electromagnetic forces distributed in space and time act on the stator teeth, resulting in the radial deformation of the stator, which can be expressed as follows [25],

$$A_{nm}=\frac{\pi D_{si}{L}_{stk}{P}_{nm}/M_t}{\sqrt{{\left({\omega }_m^2-{\omega }_n^2\right)}^2+4{\zeta }_m^2{\omega }_m^2{\omega }_n^2}}$$

(5)

where $P_{nm}$ represents the electromagnetic force, which means the harmonic component of the force density in the radial direction with respect to space and time. $M_t$ is mass of the stator, $D_{si}$ is diameter of stator, $L_{stk}$ is stack length of motor, ${\omega }_m$ is the angular natural frequency of the mode $m$, ${\omega }_n$ is the angular frequency of $P_{nm}$, and ${\zeta }_m$ is the modal damping ratio.

The vibration displacement ($A_{nm}$) produced in electric motors is converted into acoustic noise that propagates through the surrounding air. This vibration and noise are direct responses to the electromagnetic forces acting as the source of vibration and the mechanical structure that undergoes the vibration. Additionally, when the frequency of the electromagnetic forces aligns closely with the natural vibration frequency of the stator, resonance can occur, resulting in significant vibrations and noise. Therefore, it is crucial to analyze not only the frequency components of the radiated forces but also the structural characteristics of the stator to accurately predict the vibration behavior of electric motors.

Figure 9 depicts the proposed analysis process for vibration and noise. It consists of three categories: electrical factor (electromagnetic force analysis), mechanical factor (natural vibration frequency calculation), and vibration and noise analysis (integration of both factors). To be noted, the vibration and noise analysis performed in this paper took into consideration an ideal, lossless environment.

3.1. Electrical Factor Analysis

Vibration in the stator primarily results from electromagnetic forces acting on its inner surface. These forces can be categorized into radial and tangential components, both of which contribute to radial vibrations. This study focuses on the radial force and aims to explore its correlation with stator electromagnetic vibrations. The radial force density is determined by the difference between the squared radial and tangential flux densities, as follows:

$$P_r\left(t, \theta \right)=\frac{1}{2{\mu }_0}\left[B_r^2\left(t, \theta \right)-B_t^2\left(t, \theta \right)\right]$$

(6)

where the permeability of air (${\mu }_0$) is $4\pi \times {10}^{-7} \mathrm{H}/\mathrm{m}$. The electromagnetic forces in the air gap exhibit complex temporal and spatial magnetic flux distributions, influenced by factors such as the rotor and stator shape, materials, winding method, current, and others. To consider these complexities, FEM software (Ansys Electronics Desktop 2022) can be employed to calculate the magnetic field distribution in the air gap [25].

Figure 10 demonstrates the calculation process and 2D FFT results for the radial force density of the reference model. The calculation was performed using MATLAB. The x-axis represents the electric angle, and the y-axis represents the mechanical angle. The complex geometry of the stator and rotor leads to multiple harmonics in the radial force, necessitating the identification of temporal and spatial harmonics using 2D FFT. The temporal harmonic order (THO), denoted as n, signifies the periodicity in time.

The sixth harmonic component emerges dominantly in the temporal domain (sixth-order component of THO on the x-axis) due to the pole number. At 270 Hz (2700 rpm, n = 6th) in the spatial y-axis, two prominent components are observed: the third and sixth harmonics. The third harmonic is determined by the motor’s symmetry, while the dominant sixth harmonic is derived from the pole number, with a magnitude of 1.56 × 10⁶ N/m².

3.2. Mechanical Factor Analysis

The radial vibration in PMSMs is caused by the radial electromagnetic pressure on the stator, leading to vibration displacement. Therefore, accurate calculation of the natural frequency is essential for predicting vibration. In this study, the accuracy of numerically calculated natural frequencies is validated by comparing and verifying a numerical method for calculating natural frequencies and performing modal analysis using FEM. The natural frequency of the m^th-order circumferential vibration mode of the stator is determined as follows,

$$f_m=\frac{1}{2\pi }\sqrt{\frac{K_m}{M_m}}$$

(7)

where $K_m$ and $M_m$ represent the stiffness and mass properties, respectively.

The natural frequencies of the reference model were numerically calculated and compared with the FEA modal analysis results in

Table 3. Comparison of the results between compressor motor’s numerical natural frequency calculation and modal analysis through FEM.

Mode	Calculation				FEA
	K	M	f (Hz)	Shape	f (Hz)	Shape
2nd	$9.095\times {10}^7$	1.547	1221		1259
3rd	$6.468\times {10}^8$	1.579	3221		3145

. The geometric characteristics of the natural frequencies corresponding to the circumferential modes are provided. The numerical calculation results were validated with a maximum error of 3.0%. However, the limited consideration of complex mechanical geometries, such as windings and teeth, contributed to this error. These numerical analysis results are used as inputs for further vibration and noise analyses.

The magnification factor, which represents the amplification of vibration based on the frequency of the force, can be calculated assuming equal harmonic components of the force. It is expressed as follows,

$$h_{nm}=\frac{A_m}{F_m/M{\omega }_m^2}=\frac{1}{\sqrt{{\left[1-{\left(f_r/f_m\right)}^2\right]}^2+{\left[2{\zeta }_m\left(f_r/f_m\right)\right]}^2}}$$

(8)

where $M$ is mass of the stator, $f_m$ is the natural frequency of the mode $m$, $f_r$ is the frequency of $F_m$, and ${\zeta }_m$ is the modal damping ratio.

In Figure 11, the magnification factor of the reference model is shown, with THO on the x-axis and velocity on the y-axis. Resonance points are observed for the second, third, fourth, fifth, and sixth orders within the examined velocity range. Lower-order resonances, particularly the second and third orders, exhibit significant magnification. As the mode increases, the natural frequencies and magnification factors during resonance decrease. The reference motor, a six-pole nine-slot configuration, predicts vibrations due to the third order, as previously indicated in Figure 10.

3.3. Vibration Analysis under Constant Current Condition

In this section, the vibration is observed at the rated point under constant current conditions. The radial force obtained from electromagnetic analysis is subjected to 2D FFT to determine its harmonic components and gain insights into the electromagnetic factors involved. Modal analysis using FEM is utilized to calculate and validate the natural frequencies, providing insights into the mechanical factors. By considering both factors together, numerical predictions can be made regarding the vibrations generated within the motor. Specifically, the vibration displacement is observed to have significant components at frequencies of 270 Hz and 540 Hz, which correspond to the multiple components of the sixth order. Further analysis was conducted to investigate the impact of temperature on these specific frequency orders.

In Figure 12, the vibration displacement at 270 Hz and 540 Hz is shown as a function of temperature under the same current conditions. It can be observed that as temperature increases, the vibration displacement decreases for both frequencies. However, the decrease in vibration is more significant at 540 Hz compared to 270 Hz.

At 270 Hz, the third-order component of radial pressure decreases by −12.6% as temperature increases from −40 to 180 °C. On the other hand, the dominant sixth-order component experiences a larger decrease of −40.7% at −40 °C. From Figure 12, it can be seen that the sixth-order component has a lesser impact on vibration compared to the third, resulting in a vibration reduction of −14.7%, similar to the third-order component.

3.4. Vibration Analysis under Constant Torque Condition

In this section, the vibration performance of the motor is analyzed by fixing the torque output and examining its variation with temperature. As the temperature increases, the demagnetization of PMs occurs, resulting in the need for more current to generate the same output. This enables the simultaneous study of the influences of both the PMs and the current on the vibration characteristics of the motor.

Figure 13 compares vibration displacements at different temperatures under the same output condition, specifically at frequencies of 270 Hz and 540 Hz, which are known for significant vibrations. The results differ from those obtained under the same current condition. At 270 Hz, vibration gradually increases as temperature rises, contrary to the decrease observed under the same current condition. At 540 Hz, the vibration decreases with temperature, but the decrease is more pronounced under the same output condition compared to the same current condition.

At 270 Hz, the third-order component increases by 9.9% with rising temperature, while the dominant sixth-order component decreases by −36.8%, resulting in a 6.3% increase in vibration displacement. Under the same output condition at 180 °C, the third mode’s radial pressure experiences a significant 23.6% increase compared to the same current condition, influenced by the current and winding’s counter-electromotive force.

For 540 Hz, the third-order component is 48.6% smaller at 180 °C compared to the same current condition. Interestingly, at this frequency, the current reduces the third-order component. This indicates that the components are influenced differently by the current and the residual magnetic flux density of the PMs at each frequency. By analyzing the vibration map for a wider range of torque, it is possible to examine the individual influences of the current and the residual magnetic flux density of the PMs for each frequency.

3.5. Vibration Maps as a Function of Torque and Temperature

For vibration analysis, a temperature range of −40 to 180 °C and a torque range of 0 to 1.5 Nm were chosen, as mentioned earlier. Figure 14 displays the frequency-dependent distribution of vibration displacement, with torque on the x-axis and temperature on the y-axis. The graph also includes a dashed line indicating the magnitude of the current, which allows for the examination of the relationship between torque and current magnitude.

The analyzed frequencies range from 270 Hz to 2700 Hz, corresponding to multiples of the sixth-order harmonic component. Distinct vibration displacement distributions were observed for each frequency. At frequencies like 270 Hz, 1080 Hz, 1620 Hz, 1890 Hz, and 2160 Hz, increasing torque (current) led to higher vibration displacement. Conversely, at frequencies such as 540 Hz, 810 Hz, and 1350 Hz, higher temperatures resulted in greater vibration displacement. However, in most frequency ranges, vibration displacement was influenced by both torque and temperature, rather than being solely influenced by one factor. To further analyze the dominant components, two representative frequencies, 270 Hz and 810 Hz, will be examined in detail.

At 270 Hz, an increase in current, indicating an amplification of torque, corresponds to an increase in vibration displacement. Figure 15 provides a breakdown of the vibration components associated with each circumferential mode. It is clear that the predominant vibration mode at 270 Hz is the third-order mode. While there are components attributed to the sixth-order mode, they are relatively insignificant compared to the vibration displacement induced by the third-order mode.

At 810 Hz, the vibration displacement is mainly affected by the temperature of the PMs rather than the current, as shown in Figure 16. The decomposition of spatial mode components confirms this relationship. Additionally, the dominant vibration mode at 810 Hz is the ninth-order mode. In contrast to the third-order dominance observed at 270 Hz, the ninth-order mode plays a significant role in the overall vibration displacement at 810 Hz.

3.6. Experimental Verification of Vibration Analysis [26]

The electromagnetic vibrations in PMSMs were calculated numerically and validated experimentally by the authors, as published in a separate paper [26]. A sixteen-pole eighteen-slot (16p/18s) PMSM was investigated to verify the vibration results of both calculation and experimentation. The electromagnetic and vibration performance of the 16p/18s model were analyzed using the same methodology as outlined in this paper. However, the paper did not present the methodology clearly; it primarily focuses on the vibration characteristics of that specific motor. Figure 17 shows the cross-sectional view of 16p/18s PMSM and the motor components.

In order to validate the calculated results, vibration experiments were conducted under both no-load and loaded conditions for the 16p/18s PMSM. The experimental setup comprises a B&K (Brüel & Kjær, London, UK) FFT analyzer, a tachometer, and an accelerometer, as illustrated in Figure 18.

The experimental results are presented in Figure 19 and

Table 4. Comparison of vibration acceleration between calculation and measurement in 16p/18s at no-load and full-load points.

		$\mathbf{Vibration} \mathbf{Acceleration} \left(\mathsf{m}/{\mathsf{s}}^2\right)$
		Calculation		Measurement
Pole/slot	RPM	No load	Full load	No load	Full load
16p/18s	2000	0.65 (57.0%)	1.14	1.09 (60.9%)	1.79

. Figure 19 depicts the order-tracking waterfall chart of measured vibration acceleration under both no-load and full-load conditions. On the x-axis, the vibration acceleration of each order is plotted with respect to the speed, up to 2000 rpm, and on the y-axis, in a three-dimensional representation. The comparison between measurement and calculation is performed in terms of the root mean square (RMS) of vibration acceleration, as summarized in Table 4. While the deviation between measurement and calculation is not negligible, the no-load vibration acceleration constitutes 57.0% and 60.9% of the full-load value in the calculation and measurement, respectively. It is noteworthy that the measurement reasonably agrees with the calculation, with an error of only 3.9 percentage points. Given that experimental results inherently incorporate mechanical vibrations (such as those from bearings, shaft misalignment, and eccentricity), perfect alignment between numerical calculations and experimental results cannot be expected. Nevertheless, these findings are sufficiently reasonable to confirm the overall trend.

4. Noise Analysis

This section focuses on analyzing the noise generated by the vibrations in the motor structure. When a motor operates, the electromagnetic forces acting on the stator cause the structure to deform, resulting in the production of noise. However, determining the exact relationship between vibration and noise in a machine is not straightforward and relies on various factors such as dimensions, boundary conditions, and material properties. In general, larger vibrations in simple structures tend to generate higher sound pressure and noise. However, in complex structures like motors, the relationship between noise and vibration is not always directly proportional. For the purpose of this paper, the analysis will specifically consider radial vibration and noise, assuming no axial vibration.

The noise in the radial direction originating from the stator is amplified when the spatial and temporal harmonics of the electromagnetic forces align closely with the natural frequencies of the stator vibration modes. To analyze the sound waves, the sound power frequencies for each vibration mode of the electric motor are calculated and combined. The sound power of the vibration mode ‘m’ occurring on the stator surface can be calculated using the following equation [23],

$$\mathsf{\Pi }\approx {\mathsf{\Pi }}_m={\rho }_0{c}_0{\left(\frac{\omega A_m}{\sqrt{2}}\right)}^2{\sigma }_m{S}_f$$

(9)

where $S_f$ represents the external cross-sectional area of the stator, and $A_m$ denotes the vibration displacement in the radial direction. The density of air ${\rho }_0$ and the velocity of the air $c_0$, are determined under temperature condition $20 °\mathrm{C}$. ${\sigma }_m$ represents the acoustic radiation efficiency for each vibration mode.

The Sound Power Level (SWL) can be calculated using the obtained acoustic output as follows [23],

$$L_{\omega }=10{\log }_{10}\frac{\mathsf{\Pi }}{{\mathsf{\Pi }}_{ref}}=10{\log }_{10}\frac{\mathsf{\Pi }}{{10}^{-12}}\mathrm{dB}$$

(10)

Then, using the equation below, SWL can be converted into Sound Pressure Level (SPL) as a means of assessing the magnitude of the noise.

$$L_p=L_{\omega }-10\times \log \left(\frac{Q}{4\pi }\times r^2\right)\mathrm{dB}$$

(11)

where $Q$ represents the directivity factor, and $r$ denotes the distance from the microphone. In this study, a distance of $1 \mathrm{m}$, which is commonly assumed in experimental measurements, is used for the calculation.

The ${\sigma }_m$ in Equation (9), which is used to calculate the sound power, is determined by Equation (12). $J_m\left(k_{0}a\right)$ and $J_{m+1}\left(k_{0}a\right)$ are the Bessel functions of first kind for $m_{th}$ and ${\left(m+1\right)}_{th}$ orders, while $Y_m\left(k_{0}a\right)$ and $Y_{m+1}\left(k_{0}a\right)$ are the Bessel functions of second kind for $m_{th}$ and ${\left(m+1\right)}_{th}$ orders. Here, $\omega /c_0$ is calculated using $c_0=344 \mathrm{m}/\mathrm{s}$ to determine $k_0$, and $a$ represents the outer radius of the housing, including the electric motor.

$$\begin{gathered} \sigma \left(m, k_{0}a\right)=\frac{2}{\pi k_{0}a{\left|\frac{d{H}_m^{\left(2\right)}\left(k_{0}a\right)}{d\left(k_{0}a\right)}\right|}^2} \\ \sigma \left(m, k_{0}a\right)=\frac{{\left(k_{0}a\right)}^2\left[Y_m\left(k_{0}a\right)J_{m+1}\left(k_{0}a\right)-J_m\left(k_{0}a\right)Y_{m+1}\left(k_{0}a\right)\right]}{{[m{J}_m\left(k_{0}a\right)-\left(k_{0}a\right)J_{m+1}\left(k_{0}a\right)]}^2+{\left[m{Y}_m\left(k_{0}a\right)Y_{m+1}\left(k_{0}a\right)\right]}^2} \end{gathered}$$

(12)

In Figure 20, the acoustic radiation efficiency of the compressor motor is depicted based on mode ‘$m$’ and wavenumber ‘$k_{0}a$’. It can be observed that as the mode decreases, the acoustic radiation efficiency starts from lower frequencies and gradually converges toward a value close to one as the frequency increases. This suggests that noise generated by the third mode is expected to occur only below approximately 4000 Hz. Hence, a separate analysis of the noise impact for each mode is required.

In Figure 21, the SPL at the rated torque is compared for different temperatures. The highest noise level is observed at a frequency of 3145 Hz, which corresponds to the third vibration mode. This consistent noise occurrence is observed across all temperature ranges. Furthermore, the noise level varies with temperature, indicating the influence of temperature on noise generation. To gain a deeper understanding, the noise will be decomposed into components corresponding to each vibration mode, allowing for a detailed examination of the impact of each mode.

In Figure 22, SPL is analyzed for an ambient temperature of 20 °C. Figure 22a compares vibration acceleration and noise, with vibration acceleration peaking at 1350 Hz and noise at 2970 Hz. The presence of acoustic radiation efficiency from the third mode aligns overall noise and third mode noise up to around 6000 Hz (Figure 22b). The sixth mode does not contribute to noise below 4000 Hz, but as frequency increases, the noise caused by the sixth mode becomes similar to overall noise (Figure 22c).

The noise analysis is performed using a noise map, similar to the vibration analysis, taking into account the temperature and current. Noise is influenced by both vibration and acoustic radiation efficiency, which leads to the combination of noise from each vibration mode. Therefore, understanding the impact of each vibration mode is crucial for a comprehensive noise analysis.

Table 5. Comparison of vibration map and noise map by frequency.

Frequency	Vibration Map	Noise Map
270 Hz
810 Hz
2970 Hz

compares the noise maps for dominant modes at specific frequencies: 270 Hz (third mode dominant), 810 Hz (ninth mode dominant), and 2970 Hz (third mode natural frequency range). At 270 Hz, where the third mode dominates, the noise map shows a similar trend to the vibration map, with noise proportional to the square of the vibration displacement. However, at 810 Hz, the vibration is primarily influenced by the temperature of the PMs rather than the current variation. Interestingly, the noise increases as the current increases. This discrepancy occurs because, as shown in Figure 16, the vibration displacement at 810 Hz is minimally affected by the third mode, and the dominant component is the ninth mode. Since there is no sound radiation efficiency caused by the ninth mode at this frequency, the vibration map and noise map at 810 Hz do not align. At 2970 Hz, which corresponds to the third natural frequency, both the vibration map and noise map exhibit a similar trend influenced by the third mode.

The inclusion of sound radiation efficiency in noise prediction is crucial, as demonstrated by the findings. Simultaneous prediction of vibration and noise can be achieved by analyzing components for each vibration mode. Applying the methodology outlined in this paper to analyze specific frequencies allows for valuable insights into the dominant factors of vibration and noise concerning temperature and current. These insights can guide improvements in electric motor design.

5. Conclusions

This paper presents a process for analyzing the electromagnetic vibration and noise in IPMSMs. The analysis takes into account the effects of temperature, as temperature changes during motor operation can greatly influence the motor’s performance. Specifically, the paper focuses on the temperature variation of the PMs, which has a significant impact on the motor’s behavior.

The forces in the motor were computed using FEM based on the air gap flux density, enabling the numerical prediction of vibration and noise. The force density in the radiated direction was determined by calculating the air gap flux density, and harmonic orders were analyzed using a 2D FFT in both time and space domains. The natural vibration frequencies of the stator were calculated using the stator dimensions, and the numerical results were validated by comparing them with modal analysis. By utilizing the computed radiated force density and natural vibration frequencies, vibration and frequency maps were generated and analyzed numerically. This analysis aimed to identify the dominant component in vibration and noise considering the influence of temperature and current.

In this paper, the calculation of the acoustic radiation efficiency, which serves as the transfer function from electromagnetic vibration to noise, was conducted using Bessel functions. The principle that noise is generated by the overlapping sound pressures produced by each mode was identified, and individual vibrations corresponding to each vibration mode were derived, leading to the determination of the associated noise. It was confirmed that even with significant vibration, the manifestation of noise depends on the characteristics of the acoustic radiation efficiency. By comparing the vibration maps and noise maps, the importance and necessity of identifying the spatial vibration mode that generates vibration or noise were recognized. Furthermore, by creating vibration maps and noise maps based on the current and PM temperature, the factors that dominate at each frequency can be determined. Therefore, an easy way to propose design directions for reducing vibration or noise at specific frequencies in improvement studies is provided by this research. In the future, further experiments will be performed to provide support for the proposed methodology.