## 3 Inductance Identification and Inductance Analytical Expressions
### The Proposed NAM  
#### 3.2.1 Inductances Identification of Fault Coils in the Slot Considering Spatial Position  
Since conductors in the same slot do not share the slot-leakage flux equally, the inductance differs as the position changes, as shown in Figure 2. Figure 3 shows the distribution of the turns in the stator slot. It can be seen that all turns in slot are numbered from 1 to 48 and divided into four layers and two turns per layer. Additionally, the rectangular coordinate is established to accurately define the location of the fault turns in the slot. X is the number of layers, and Y is the slot depth of the turns at each layer. To indicate the inter-layer short circuit fault, the symbol of W is introduced. When W = 1, it means that the short circuit fault is occurred between the first and second layers. W = 2 indicates that the turns of the second and third layers are short-circuit. W = 3 shows the short circuit fault is occurred between the first and second layers.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Items (Unit)** & **Value** & **Items (Unit)** & **Value** \\ \hline Wire diameter of winding (mm) & 1.3 & Number of coils & 36 \\ Area of slot (mm\({}^{2}\)) & 154 & Coil turns & 48 \\ Thickness of PM (mm) & 5.3 & Parallel-circuits per phase & 3 \\ Pole-arc coefficient & 0.83 & Slot-pole combination & 72–66 \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 2: The armature flux linkage of A34.  
Therefore, when the coordinate of inter-layer short circuit fault is (W,Y), the number of the first turn in the faulty part can be obtained by Equation (4). Similarly, the number of the last turn in the faulty part can be obtained according to Equation (5). For example, when the inter-layer short circuit occurs at (2,5), it can be calculated from Equations (4) and (5) that the fault turns are located from turn no. 20 to turn no. 29, as shown in Figure 4.  
\[b_{\rm s}=-13W^{2}+64W-51+(-1)^{W+1}Y \tag{4}\]  
\[b_{f}=-13W^{2}+64W-51+(-1)^{W+1}Y \tag{5}\]  
When the short circuit fault occurs in the same layer and the coordinate of the faulty part is from (X\({}_{1}\), Y\({}_{1}\)) to (X\({}_{1}\), Y\({}_{2}\)). The number of the first and last turn in the faulty part can be obtained by Equations (6) and (7), respectively. As a result, the fault turns consist of the turns numbered \(b_{\rm s}\)-\(b_{f}\). For example, when the inter-layer short circuit fault occurs from (2,5) to (2,7), the faulty part is composed of the turns at no. 18, no. 19, and no. 20, as shown in Figure 5.  
\[b_{\rm s}=12X_{1}-1+Y_{1} \tag{6}\]  
\[b_{f}=12X_{2}-1+Y_{1} \tag{7}\]  
Figure 4: Turn distribution when the inter-layer fault occurs at (W = 2, Y = 5).  
Figure 5: Turn distribution when the inter-layer fault occurs at (X\({}_{1}\) = 2, Y\({}_{1}\) = 5) to (X\({}_{2}\) = 2, Y\({}_{1}\) = 7).  
Figure 3: The distribution of the turns in the slot.  
After getting the turn numbers of the faulty part, the position matrix \(C_{2,48}\) of the faulty coil is established according to the position of the turns. In the first row of the position matrix, all the column elements whose serial number is the fault turn number are set to 1, and the other column elements are all set to 0. Additionally, all the column elements in the second row are opposite to elements in the first row. Then, it is necessary to establish the inductance matrix \(L\) (Equation (8)), in which \(L_{ij}\) means the mutual inductance between the turns numbered \(i\) and \(j\). When \(i=j\), \(L_{ij}\) means the self-inductance of the turn numbered \(i\). Afterwards, a second-order matrix \(L_{XQ}\) is calculated by Equation (9), and it consists of the inductances of healthy and fault parts:  
\[L=\left[\begin{array}{cccccc}L_{1}&M_{1,2}&&&M_{1,47}&M_{1,48}\\ M_{2,1}&L_{2}&&\ldots&M_{2,47}&M_{2,48}\\ \vdots&\vdots&\ldots&&\vdots&\vdots&\\ M_{47,1}&M_{47,2}&\ldots&L_{47}&M_{47,48}\\ M_{48,1}&M_{48,2}&&M_{48,47}&L_{48}\\ \end{array}\right] \tag{8}\]  
\[L_{XQ}=C_{2,48}\cdot L\cdot C_{2,48}^{T} \tag{9}\]