## 2 Acoustic Noise Sources of Electric Machines  
For several years, the search to guarantee the best operating performance for an IM powered by static converters has been the subject of numerous research and development projects. Electrical machines, including IMs, constitute major noise sources in industry. Noise exposure leads to psycho-acoustic and health problems for human beings. Figure 1 demonstrates the mechanism for generating noise in an electrical machine. Thus, this system is characterized, on the one hand, by the noise source (electric motor) as a principal cause of acoustic phenomena and, on the other hand, by the elements related to perception and sensation by the human ear. The overall noise level in an electrical machine originates from four main sources. Considerable efforts have been made to identify the acoustic noise sources of an electrical machines [1; 2; 3; 4; 22]. These can be further subdivided into four main sources: electronic, mechanical, magnetic and aerodynamic. These sources provide also the excitation forces that act on the internal structure of the stator core. The pressure of the ambient air varies periodically under the effect of vibrations, and this creates noise. Consequently, the sound wave is audible to the human ear. In this study, electronic noise, as well as its switching harmonics, are studied. A voltage inverter produces voltages containing harmonics, which can cause considerable noise, particularly when their frequencies are close to those proper to the machine. Figure 1: Noise generation in electrical machines and different noise sources. Furthermore, the harmonics absorbed by the stator windings produce additional stator flux density components, hence generating other magnetic force components. The frequency of voltage harmonics can be obtained as follows:  
\[\mathrm{F_{n}}=|\mathrm{n_{1}F_{c}+n_{2}F}| \tag{1}\]  
where F is the fundamental frequency, \(\mathrm{F_{c}}\) is the switching frequency, and \(\mathrm{n_{1}}\) and \(\mathrm{n_{2}}\) are integers. An example of voltage waveform at the output of a three-phase inverter and its Fast Fourier Transform (FFT) for an \(\mathrm{F_{c}}\) switching frequency of 2.5 kHz are presented in Figure 2. The harmonic components are distributed in groups of sidebands near the switching frequency, and its multiple is shown. In electric machines, the harmonics components in the air gap generate radial forces applied to the stator, tending to dynamically deform its structure. In this way, the stator is stressed by multiple deflections. Generally, the mechanical behavior of the machine stator can be characterized by its natural modes of vibration. Each mode, corresponding to a proper frequency, represents a particular deformation of the structure [8]. When designing electric rotating machinery, it is essential to take into account both the frequencies of the applied forces and the proper frequencies of the stator system. Various vibration modes (0, 1, 2, 3...) are linked to the natural frequencies, and they are the lower modes that significantly influence the increase in electromagnetic noise. Figure 3 illustrates the spatial distribution of forces generating vibration modes of different orders (m = 0, 1, 2, 3, 4) and the geometry of the stator ring and the teeth. Equation (2) gives the natural frequency \(\mathrm{F_{0}}\) of the order vibration mode m = 0 [8]:  
\[\mathrm{F_{0}}=\frac{1}{\pi\mathrm{D_{c}}}\sqrt{\frac{E_{c}}{\rho_{c}k_{i}k_{md }}} \tag{2}\]  
where \(\mathrm{D_{c}}\) is the average diameter of the stator core, \(\mathrm{E_{c}}\) is the elasticity modulus, \(\rho_{c}\) is the density, \(k_{i}\) is the piling factor, and \(k_{md}\) is the mass addition factor for the displacement. Figure 3: Principal radial deformation modes of stator and geometry of stator ring and teeth. Figure 2: Waveform of output voltage and its FFT.