## 3 Detection Mechanism Based on Cri Markers  
For the correct operation, the drive system with PMSM requires at least two current sensors. In the analyzed example, the third sensor is used only for diagnostics. The basis of \begin{table}
\begin{tabular}{c c} \hline \hline
**Type of the Fault** & **Current Value** \\ \hline Variable gain & \(i_{s}^{m}=(1-\gamma)i_{a}\) \\ Signal limit & \(i_{s}^{m}=i_{sat}\) \\ Noise & \(i_{s}^{m}=i_{4}+n(t)\) \\ Lack of signal & \(i_{s}^{m}=0\) \\ Intermittent signal & \(i_{s}^{m}=[0,1]\) \\ \hline \end{tabular} where \(i_{s}^{m}\)—fault current, \(i_{s}\)—measured current, \(\gamma\)—constant value from the range \(\epsilon\)\(-\)1, 15.  
\end{table} 
Table 1: Equations to simulate individual failures of current sensors. Figure 1: Block diagram of the control structure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**P\({}_{\mathbf{N}}\)** & **P\({}_{\mathbf{P}}\)** & **\(\Omega_{\mathbf{N}}\)** & **T\({}_{\mathbf{N}}\)** & **I\({}_{\mathbf{N}}\)** & **J** & **R\({}_{\mathbf{s}}\)** \\
**[kW]** & **[-J]** & **[rpm]** & **[Nm]** & **[A]** & **[kg\(\cdot\)m\({}^{2}\)]** & **[\(\Omega\)]** \\ \hline
0.894 & 4 & 6200 & 1.4 & 1.9 & 0.000039 & 4.6615 \\ \hline \hline \end{tabular}  
\(P_{\mathbf{N}}\)—nominal power, Pp—pool pairs, \(\Omega_{\mathbf{N}}\)—nominal speed, \(T_{\mathbf{N}}\)—nominal torque, \(I_{\mathbf{N}}\)—nominal stator current, \(J\)—inertia, Rs—stator phase resistance.  
\end{table} 
Table 2: Parameters of the motor used in simulation and experimental results. the detection algorithm is the fact that the \(i_{\alpha}\) and \(i_{\beta}\) current components used in the control structure can be determined using different equations depending on the phases in which the measurement is performed [24]:  
\[i_{\alpha 1}=\frac{2}{3}(i_{A}-\frac{1}{2}(i_{B}+i_{C})),i_{\beta 1}=\frac{ \sqrt{3}}{3}(i_{B}-i_{C}), \tag{1}\]  
\[i_{\alpha 3}=-(i_{B}+i_{C}),i_{\beta 3}=-\frac{\sqrt{3}}{3}(i_{B}-i_{C}), \tag{2}\]  
\[i_{\alpha 2}=i_{A},i_{\beta 2}=\frac{\sqrt{3}}{3}(i_{A}+2i_{B}) \tag{3}\]  
On the basis of these equations, it is possible to determine current markers that are insensitive to the measurement of one of the phases:  
\[C_{ri1}=(i_{\alpha 3}^{2}+i_{\beta 1}^{2}),C_{ri2}=(i_{\alpha 2}^{2}+i_{\beta 3} ^{2}),C_{ri3}=(i_{\alpha 2}^{2}+i_{\beta 2}^{2}), \tag{4}\]  
After transformations, the following formulae are obtained:  
\[C_{ri1}=(-(i_{B}+i_{C}))^{2}+(\frac{\sqrt{3}}{3}(i_{B}-i_{C}))^{2}, \tag{5}\]  
\[C_{ri2}=(i_{A})^{2}+(-\frac{\sqrt{3}}{3}(i_{A}+2i_{C}))^{2}, \tag{6}\]  
\[C_{ri3}=(i_{A})^{2}+(\frac{\sqrt{3}}{3}(i_{A}+2i_{B}))^{2}. \tag{7}\]  
However, based on the values of the markers themselves, the location of the damaged phase would not always be clear and stable. Therefore, the algorithm uses the difference in the value of markers from the current and previous samples--the so-called marker errors. This ensures stable detection:  
\[\Delta C_{rij}=\big{|}C_{rij}(k)-C_{rij}(k-1)\big{|}. \tag{8}\]  
The last element of the algorithm is an additional condition--_Delta_ relating to the \(i_{\alpha}\) and \(i_{\beta}\) values determined with the use of various current sensors:  
\[(i_{\alpha 1}=i_{\alpha 2}=i_{\alpha 3})\wedge(i_{\beta 1}=i_{\beta 2}=i_{\beta 3 }). \tag{9}\]  
The relations between the marker error values depending on the damaged phase are repeatable. On this basis, the detector not only detects damage to the current sensor, but also locates the damaged phase. Table 3 summarizes the relationship between marker errors, which are confirmed by the following simulation and experimental results. Figures 2-4 show the marker error transients and detector responses at the time of different failures in phases A and B in simulation and experimental tests. The detector has two outputs. The first output, D1, refers to phase A, and the second, D2, refers to phase B. Value 0 at the detector output means no failure, and 1 at the output means failure in the corresponding phase.