# 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  
This section mainly analyzes the fault characteristics of the voltage between the midpoint potential of the arm and the midpoint potential of the DC side after the open-circuit failure of the IGBT in the subsequent two-phase quadrature inverter, that is the three-phase arm midpoint voltage \(u_{\alpha o}\), \(u_{\alpha w}\), and \(u_{\beta o}\) of the inverter in Figure 1, and use these features to diagnose faults. This section proposes a fault diagnosis algorithm based on the voltage polarity at the midpoint of the arm can realize secondary fault diagnosis. 2.1 The Fault Diagnosis Algorithm and Reconstruction Strategy of the Back-Stage Two-Phase Quadrature Inverter after Primary Failure  
Suppose the bridge arm switch function is:  
\[S_{x}=\left\{ \begin{array}{ll}1&\text{upper bridge \ on}\\ -1&\text{lower bridge \ on}\end{array} \right. \tag{1}\]  
where \(x\) is one of the three phases (\(\alpha\), \(w\), \(\beta\)). The midpoint voltage of the bridge arm can be expressed as:  
\[u_{x0}=\frac{u_{dc}}{2}S_{x} \tag{2}\]  
Figure 1: The structure of the electromagnetic stirring power supply. Take the fault of IGBT V\({}_{1}\) as an example. When V\({}_{1}\) works under normal conditions, the midpoint voltage of the bridge arm is:  
\[u_{aoN}=\left\{ \begin{array}{ll}\frac{u_{dc}}{2}&S_{a}=1\\ -\frac{u_{dc}}{2}&S_{a}=-1\end{array} \right. \tag{3}\]  
When an open-circuit fault occurs in V\({}_{1}\) and \(S_{a}=1\), both V\({}_{1}\) and V\({}_{2}\) are in a disconnected state, and point \(\alpha\) is suspended. At this time, \(i_{a}\) = 0 and \(u_{ao^{\prime}}\) = 0. The potential at point \(\alpha\) is equivalent to the potential at point \(O^{\prime}\), that is, \(u_{aoF}=u_{o^{\prime}o}=-u_{ao^{\prime}}\). From the circuit of the inverter in Figure 1, \(u_{ao^{\prime}}\) can be obtained as:  
\[u_{oo^{\prime}}=-\frac{u_{ao^{\prime}}+u_{\beta o^{\prime}}}{3}=-\frac{u_{dc }}{3}\big{(}S_{\omega}+S_{\beta}\big{)}+u_{dc} \tag{4}\]  
Then:  
\[u_{aoF}=-u_{oo^{\prime}}=\frac{u_{dc}}{3}\big{(}S_{\omega}+S_{\beta}\big{)}-u _{dc} \tag{5}\]  
In Equation (5), \(u_{aoF}\) represents \(u_{ao}\) after failure. Substituting different values of \(S_{w}\) and \(S_{\beta}\), the midpoint voltage of the \(\alpha\)-phase arm after the fault can be obtained as:  
\[u_{aoF}<\frac{u_{dc}}{2} \tag{6}\]  
Define the voltage deviation at the midpoint of the bridge arm as:  
\[\Delta u_{x0}=u_{x0F}-u_{x0N} \tag{7}\]  
Then the voltage deviation at the midpoint of the \(\alpha\)-phase arm is:  
\[\Delta u_{ao}=u_{aoF}-u_{aoN}<0 \tag{8}\]  
Define the voltage deviation polarity at the midpoint of the bridge arm as:  
\[J_{x0}=\left\{ \begin{array}{ll}P&\Delta u_{x0}>0\\ Z&\Delta u_{x0}=0\\ N&\Delta u_{x0}<0\end{array} \right. \tag{9}\]  
where \(P\), \(Z\), \(N\) stand for positive, zero and negative, respectively. Thus there is:  
\[J_{ao}=N \tag{10}\]  
The voltage polarity at the midpoint of the \(\alpha\)-phase arm is negative \(N\). In other words, when V\({}_{1}\) has an open-circuit fault and \(S_{\alpha}\) = 1, the polarity of the voltage at the midpoint of the bridge arm is negative. In the same way, the polarity of the midpoint voltage of other bridge arms can be obtained when other IGBTs fail. In the actual engineering environment, due to various interferences on the DC side capacitors, the midpoint voltage deviation will not be exactly zero. In order to avoid misjudgment, the mid-point voltage deviation threshold \(\Delta u_{th}\) is set to eliminate the zero-point drift of the mid-point voltage deviation value, so that the polarity of the mid-point voltage deviation can be redefined as:  
\[J_{x0}=\left\{ \begin{array}{ll}P&\Delta u_{x0}>\Delta u_{th}\\ Z&|\Delta u_{x0}|\leq\Delta u_{th}\\ N&\Delta u_{x0}<-\Delta u_{th}\end{array} \right. \tag{11}\]  
In the above equation, \(P\), \(Z\), and \(N\) respectively represent the positive, zero, and negative polarity of the midpoint voltage deviation.