# Research on the High Reliable Electromagnetic Stirring Power Supply
## 2 Materials and Methods
### Fault Diagnosis and Reconstruction of the Back-Stage Two-Phase Quadrature Inverter
In this article, set \(\Delta u_{th}\) as \(1/10\)\(\Delta u_{x0 max}\)namely \(u_{dc}/30\). Only when \(\mid\!\Delta u_{x0}\!\mid\!>\!\Delta u_{th}\), the polarity of the neutral point voltage deviation value is considered to have changed. In Table 1, if two switching tubes of the same bridge arm fail at the same time, such as V\({}_{1}\) & V\({}_{2}\), V\({}_{3}\) & V\({}_{4}\), V\({}_{5}\) & V\({}_{6}\), and if the upper tube is open, the polarity of the midpoint voltage of the bridge arm is \(N\), and otherwise \(P\). When the fault is detected in V, immediately reconstruct the inverter topology: first cut off V\({}_{1}\), turn on TR\({}_{\rm w}\) and TR\({}_{13}\), stop triggering V\({}_{2}\), and then send the pulse signals of V\({}_{1}\) and V\({}_{2}\) to V\({}_{3}\) and V\({}_{4}\) respectively. Namely the \(\alpha\)-phase is reconstituted by V\({}_{3}\) and V\({}_{4}\), and the DC-side capacitor as the \(w\)-phase. The reconstruction process is shown in the Figure 2.  
The rest of the reconstruction status is shown in Table 2. It can be seen from Table 2 that no matter which IGBT is reconstructed after failure, it can be reconstructed to the structure shown in Figure 2b, and the \(w\)-phase is always replaced by the DC side capacitors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Primary Fault** & \(J_{\mathbf{\alpha}\mathbf{\omega}}\) & \(J_{\mathbf{two}}\) & \(J_{\mathbf{\beta}\mathbf{\omega}}\) \\ \hline
1 & / & Z & Z & Z \\  
& N/P \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Primary fault diagnosis. Figure 2: Inverter reconstruction process when the fault occurs in V1. (**a**) Switching process; (**b**) Reconstruction result. 2.2 The Fault Diagnosis Algorithm and Reconstruction Strategy after the Secondary Fault of the Back-Stage Two-Phase Quadrature Inverter  
The inverter is reconstructed after primary failure, so that the inverter can work in fault-tolerant conditions. It can be seen from Table 1 that some IGBTs can be used for reconstruction after some faults have occurred. Take case 2 in Table 1 as an example, when the fault of V\({}_{1}\) is removed, V\({}_{2}\) is still normal. Therefore, when a secondary failure occurs in the inverter, V\({}_{2}\) can be used for secondary reconstruction. Firstly, perform a secondary fault diagnosis on the inverter after the first reconstruction shown in Figure 2b. Take the IGBT V\({}_{4}\) failure as an example. V\({}_{3}\) and V\({}_{4}\) construct \(\alpha\)-phase in the first reconstruction, so the midpoint voltage of the arm also satisfies Equation (2). Similarly, when V\({}_{4}\) has an open-circuit fault and S\({}_{\alpha}\)= 1, both V\({}_{3}\) and V\({}_{4}\) are in a disconnected state, and the \(\alpha\) point is suspended. Besides, \(i_{\alpha}=0\) and \(u_{\alpha\alpha^{\prime}}=0\). The potential at point \(\alpha\) is equivalent to the potential at point \(O^{\prime}\), that is, \(u_{\alpha\alpha F}=u_{\alpha^{\prime}o}=-u_{\alpha\alpha^{\prime}}\). when S\({}_{\alpha}\)= \(-1\), the equivalent circuit is shown in Figure 3.  
From Figure 3a, we can obtain:  
\[u_{\alpha^{\prime}o}=\frac{u_{dc}}{4}\times\frac{r_{o}+L_{o}}{2(r_{o}+L_{o})+L_ {\beta}} \tag{12}\]  
The \(r_{o}\) is the inner resistance of the inductance \(L_{o}\). The midpoint voltage deviation of \(\alpha\)-phase is:  
\[\Delta u_{\alpha o}=\frac{u_{dc}}{4}\times\frac{r_{o}+L_{o}}{2(r_{o}+L_{o})+L_ {\beta}}+\frac{u_{dc}}{2}>0 \tag{13}\]  
The polarity of the midpoint voltage deviation is:  
\[J_{\alpha o}=J[\Delta u_{\alpha o}]=P \tag{14}\]