# A Diagnostic Method for the Saturable Reactor Core Looseness Degree of Thyristor Converter Valves
## 4 Diagnostic Method for Saturable Reactor Core Looseness Degrees  
### Synchrosqueezed Wavelet Transform  
Synchrosqueezed wavelet transform is the redistribution of the time and frequency domains based on wavelet transform [21]. It uses synchronous compression to transform the energy of the time scale plane into the energy of the time frequency plane, which makes the curve expressing frequency more concentrated.  
For a given signal \(s(t)\), its continuous wavelet transform \(W_{s}\) is defined as:  
\[W_{s}(a,b)=\int s(t)a^{-\frac{1}{2}\overline{\Psi}\left(\frac{t-b}{a}\right)}dt \tag{6}\]  
where \(\Psi\) is the selected mother wavelet, \(a\) and \(b\) are the scale parameter and translation parameter of the mother wavelet transform, respectively, and \(\overline{\Psi(t)}\) represents the complex conjugation of \(\Psi(t)\). For any \((a,b)\) satisfying \(W_{s}(a,b)\
eq 0\), the instantaneous phase and frequency of signal \(s(t)\) is:  
\[W_{s}(a,b)=-i(W_{s}(a,b))^{-1}\frac{\partial}{\partial b}W_{s}(a,b) \tag{7}\]  
According to the mapping relationship \(\left(b,a\right)\rightarrow\left(b,W_{s}(a,b)\right),\) the transformation information \(W_{s}(a,b)\) is transformed from the time-scale plane of the wavelet transform to the time-frequency plane; this process is called synchronous compression. \(a\) and \(\omega\) are divided into several small boxes, with \(a_{k}-a_{k-1}=\left(\Delta a\right)_{k},\ \omega_{l}-\omega_{l-1}=\Delta\omega\). The synchronous compression transformation \(T_{s}(\omega,b)\) is only determined by \(W_{s}(a,b)\) in a box with a continuous center \(\omega_{l}\) and a width \(\Delta\omega\):  
\[T_{s}(\ \omega_{l},b)=\Delta\omega^{-1}\sum\
olimits_{a_{k}:\left|\omega(a_{k},b )-\omega_{l}\right|\leq\Delta\omega/2}W_{s}(a_{k},b)a_{k}^{-\frac{3}{2}}( \Delta a)_{k} \tag{8}\]  
where \(a_{k}\) is the \(k\)-th discrete scale and \(\omega_{l}\) is the first discrete angular frequency. Through the above transformation, the wavelet factor spectrum can be squeezed along the scale axis and the energy distribution of the spectrogram can be more concentrated. Therefore, the spectrogram of the SST has a higher aggregation and resolution than the general wavelet factor spectrum.  
SST is applied to convert the vibration signal in Figure 4 into a 2D figure, and CWT was also used for comparison with SST; these data are shown in Figures 5 and 6.  
In terms of the comparison between the CWT and SST methods, the mother wavelet functions they use are all Morlet wavelet functions. The original spectrogram directly obtained through calculation can only highlight the main frequency components. In order to improve the ability to reflect the non-main frequency components, the original spectrogram is logarithmically treated as \(T_{st}=\log(T_{s}+\alpha)\). For the spectrogram of the wavelet transform, the value of parameter \(\alpha\) is taken as \(\alpha=0.05\). For the SST spectrogram, the value of parameter \(\alpha\) is \(\alpha=0.0005\). Figures 5 and 6 are the vibration spectrogram spectra of the two methods under three typical working conditions, which correspond to Figure 4. It can be seen from the two figures that the working conditions and the iron core looseness will affect the color and shape of the time spectrum, such as the brightness, holes, vortices, etc. The SST time-frequency figures are generally clearer than the wavelet transform time-frequency map, with more obvious boundaries. When viewed with the naked eye, the spectrogram spectrum of wavelet transform is fuzzy. Dark spots, holes and other features can be clearly seen in the SST spectrogram; this is more beneficial for identifying the iron core looseness.  
Figure 5: Spectrogram of CWT under Different Working Conditions.  
Figure 6: Spectrogram of SST under Different Working Conditions.