# Faults and Diagnosis Methods of Permanent Magnet Synchronous Motors: A Review
## 3 Fault Diagnosis Methods
### Signal Processing Methods
#### 3.2.1 Frequency Domain Methods  
In general, signal processing methods typically include time domain methods, frequency domain methods, and time-frequency analysis methods. Time domain statistical indicators are used as fault characteristics in the initial period of fault diagnosis technology development, such as mean, peak, root mean square, kurtosis values, etc. [PERSON] _et al._ (2015); however, they are not accurate enough. The classical method of frequency domain analysis is fast Fourier transform (FFT). The Fourier transform represents a signal as a superposition of several sine or cosine functions. It can clearly show the frequency distribution of the signal, where the amplitude and frequency of harmonic components can be used as the features of different faults [54]. The harmonic features of different faults in current and vibration signals are as follows. (1) Current  
The inter-turn short circuit fault can lead to PMSM stator current imbalance and amplitude change [55]. When the winding is short-circuited, in addition to the amplitude of the fundamental frequency, the amplitude of the high frequency harmonics of the current increases. Let \(f_{\text{s}}\) be the frequency of the power supply, \(k\) be a positive integer. \(z\) is the number of stator slots, and \(p\) is the number of pole pairs. Thus, the harmonics at the following frequencies will increase [56]:  
\[f_{\text{itsc}}=f_{\text{s}}\bigg{(}1\pm k\frac{z}{p}\bigg{)} \tag{1}\]  
Among them, the most obvious feature is the increase of the third harmonic amplitude [57]. Furthermore, [PERSON] proposed a three-phase current balance indicator [58] which is:  
\[f_{i}=|I_{a}-I_{b}|+|I_{b}-I_{c}|+|I_{c}-I_{a}| \tag{2}\]  
where \(I_{a}\), \(I_{b}\) and \(I_{c}\) are the fundamental frequency amplitudes of the three-phase current, respectively. As for the mechanical faults, they will cause distortions in the flux distribution inside the motor, and will also lead to some current harmonics in the stator current in turn [59]. According to the research of [PERSON] et al. [21], the characteristic frequencies for eccentric faults of PMSM is as follows:  
\[f_{\text{eccentricity}}=f_{\text{s}}\bigg{(}1\pm\frac{2k-1}{p}\bigg{)} \tag{3}\]  
Furthermore, according to the research of [PERSON] et al. [60], the harmonic at \(f_{\text{s}}(1-3/p)\) is more related to static eccentricity faults, while the harmonic at \(f_{\text{s}}(1+1/p)\) is related to both static eccentricity and dynamic eccentricity. In addition, when a demagnetization fault occurs, the distributed magnetomotive force (MMF) will not be sinusoidal. Therefore, the normal and failed portions of the MMF will together produce current with multiple frequencies, which means, if there is demagnetization, low frequency components will appear near the fundamental wave in the current signal [61]. Thus, the harmonic frequencies that can be used as characteristic frequencies for demagnetization faults are as follows [62]:  
\[f_{\text{d}mg}=f_{\text{s}}(1\pm\frac{k}{p}) \tag{4}\]  
As can be seen, unlike asynchronous motors, the above fault pattern does not depend on slip, which means that for all types of faults, features could appear at the same point in the current spectrum. To solve this problem, according to the research of [PERSON] et al. [63], 0.25 th, 0.5 th and 0.75 th harmonics are the ideal indicators to separate a demagnetization fault from the static eccentricity fault. (2) Vibration  
When the motor is running normally, the stator vibration frequency should be twice the frequency of the power supply. But inter-turn short-circuit fault and three-phase asymmetry of the motor windings will cause magnetic field asymmetry, resulting in abnormal vibration, which will produce the harmonic components of \(4f_{\text{s}}\) and \(8f_{\text{s}}\) in addition to the fundamental frequency of \(2f_{\text{s}}\) in the vibration [64]. As for the mechanical faults, the method based on vibration signal is widely considered the most effective one.