# Fault Detection of Stator Inter-Turn Short-Circuit in PMSM on Stator Current and Vibration Signal
## 2 Materials and Methods  
Stator inter-turn short circuit fault is a common fault in PMSM, which is due to the problems in stator winding insulation. If not diagnosed in time, it will spread to more stator turns and even cause other motor faults. When short circuit occurs in the stator winding of a motor, the harmonic component in the stator current will increase. According to some related research [20], the harmonic components at the following frequencies will increase:  
\[f_{\textit{itsc}}=f_{\textit{s}}\bigg{(}v\frac{Z}{p}\pm 1\bigg{)} \tag{1}\]  
where \(f_{\textit{s}}\) is the frequency of the source and \(v\) is a positive integer number, \(Z\) is the number of stator slots and \(p\) is the number of pole pairs. In PMSM, the most obvious of these harmonic components is the third harmonic component [21; 22]. Furthermore, the stator inter-turn short circuit fault and the three-phase asymmetry of the motor windings can bring magnetic field asymmetry, which can in turn create abnormal vibrations. When the motor is running normally, the vibration frequency of the stator should be twice the frequency of the source. However, these abnormal vibrations could cause the vibration signal to have the harmonic components of 4\(f_{\textit{s}}\) and 8\(f_{\textit{s}}\) in addition to the fundamental frequency of 2\(f_{\textit{s}}\). As for the signal processing method, wavelet analysis can decompose signals into time and frequency domain, simultaneously [23]. Wavelet analysis offers a windowing technique with variable-sized regions, which is stable in FFT and STFT. Because of this, wavelet analysis has become a commonly used time-frequency analysis method of signal processing. The most basic wavelet transform is calculated as:  
\[W_{\psi}f(a,b)=\int f(t)\overline{|a|^{-\frac{1}{2}}\psi(\frac{t-b}{a})}dt \tag{2}\]where \(\psi(x)\) is a wavelet basis function, \(a\) is a scaling factor which represents the frequency component of the signal, and \(b\) is a translation factor which represents the time position. In wavelet analysis, wavelet transform can only analyze the low frequency part of the signal, ignoring the high frequency part. However, the wavelet packet transform can further analyze all frequency components. The flow of the fast algorithm for binary wavelet packet analysis is shown in Figure 1.  
The fast algorithm for binary wavelet packet decomposition is:  
\[\left\{ \begin{array}{l}p_{0}^{1}(t)=f(t)\\ p_{j}^{2i-1}=\sum\limits_{k}H(k-2t)p_{j-1}^{i}(t)\\ p_{j}^{2i}=\sum\limits_{k}G(k-2t)p_{j-1}^{i}(t)\end{array} \right. \tag{3}\]  
where \(t\) = 1, 2,..., 2\({}^{l-j}\), \(i\) = 1, 2,..., 2\({}^{i}\), and \(J\) = log\({}_{2}\)\(N\). \(f(t)\) is a time signal. \(P(t)\) is called the wavelet packet coefficient. \(H\) stands for wavelet decomposition low frequency filters and \(G\) for high frequency. The fast algorithm for binary wavelet packet reconstruction is:  
\[p_{j}^{i}=2\Bigg{[}\sum\limits_{k}h(t-2k)p_{j+1}^{2i-1}(t)+\sum\limits_{k}g(t -2k)p_{j+1}^{2i}(t)\Bigg{]} \tag{4}\]  
where \(j\) = \(J\) - 1, \(J\) - 2,..., 0, \(i\) = 2\({}^{j}\), 2\({}^{j-1}\),..., 1. \(h\) stands for wavelet reconstruction low frequency filters and \(g\) for high frequency. However, the above decomposition and reconstruction algorithms are also flawed because the wavelet packet fast algorithm includes three operations: wavelet filter convolution, interlaced sampling  
Figure 1: The flow chart of the fast algorithm for binary wavelet packet decomposition. and interleaved zero, which could cause frequency confusion. To solve this problem, the improved wavelet packet transform was invented. To eliminate the frequency aliasing, the improved wavelet packet decomposition and reconstruction algorithms have two operators: C and \(D\). The formula of operator C is:  
\[\left\{ \begin{array}{ll}X(k)=\sum\limits_{n=0}^{N_{j}-1}x(n)W^{kn}&,0\leq k \leq\frac{N_{j}}{4}\text{ or }\frac{3N_{j}}{4}\leq k\leq N_{j}\\ X(k)=0&,\text{else}\end{array} \right.