## 4 Model of DDPMSM with SWF  
Figure 7 shows the stator circuit schematic with ITSCF in the A phase. It can be seen that fault branch A1 is divided into the healthy part A11 and the faulty part A12. Additionally, resistor \(R_{f}\), connected in parallel on the faulty coil, means the insulation fails between turns, where its value depends on the fault severity. When \(R_{f}\) decreases toward zero, the insulation faults evolve toward an inter-turn full short-circuit. In this paper, the value of the value of \(R_{f}\) is set to 0.4 \(\Omega\).  
For establishing the analytical model of the DDPMSM with SWF, we assume that the magnetic permeability of iron is infinite and the magnetic saturation is not taken into account. Furthermore, the eddy current losses, hysteresis loss and higher harmonics are also negligible. Additionally, the ITSCF is represented using an additional circuit (subscript\(f\)), which increases the order of analytical model compared to the model in the healthy situation.  
With the above assumptions and using the circuit scheme presented in Figure 7, the electrical equation of such model can be written as [14]:  
\[v_{ABCf}=R_{ABCf}i_{ABCf}+L_{ABCf}.\frac{di_{ABCf}}{dt}+e_{ABCf} \tag{12}\]  
where:  
\[\left\{\begin{array}{l}i_{ABCf}=\left[i_{A1}\;\;i_{A2}\;\;i_{A3}\;\;i_{B1}\;\; i_{B2}\;\;i_{B3}\;\;i_{C1}\;\;i_{C2}\;\;i_{C3}\;\;i_{f}\right]^{T}\\ v_{ABCf}=\left[v_{A1}\;\;v_{A2}\;\;v_{A3}\;\;v_{B1}\;\;v_{B2}\;\;v_{B3}\;\;v_{ C1}\;\;v_{C2}\;\;v_{C3}\;\;v_{f}\right]^{T}\\ e_{ABCf}=\left[e_{A1}\;\;e_{A2}\;\;e_{A3}\;\;e_{B1}\;\;e_{B2}\;\;e_{B3}\;\;e_{ C1}\;\;e_{C2}\;\;e_{C3}\;\;e_{f}\right]^{T}\end{array}\right. \tag{13}\]  
\[R_{ABCf}=\left[\begin{array}{cccccc}R_{A1}&0&\cdots&0&0&-R_{A12}\\ 0&R_{A2}&\cdots&0&0&0\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0&R_{C3}&0\\ -R_{A12}&0&\cdots&0&0&R_{A12}+R_{f}\end{array}\right] \tag{14}\]  
Figure 7: The stator circuit schematic with ITSF in the A phase.  
\[L_{ABC}\!\!\left(\begin{array}{cccccccc}L_{A11}&M_{A11A2}&M_{A11A3}&M_{A11C2}&M_{A11 C3}&-(M_{A12A11}+L_{A12})\\ M_{A2A11}&L_{A2}&M_{A2A3}&\ldots&M_{A2C2}&M_{A2C3}&-M_{A2A12}\\ M_{A3A11}&M_{A3A2}&L_{A3}&\ldots&M_{A3C2}&M_{A3C3}&-M_{A3A12}\\ \vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\vdots&\vdots\\ M_{C3A11}&M_{C3A2}&M_{C3A3}&\cdots&M_{C3C2}&L_{C3}&-M_{C3A12}\\ -(M_{A12A11}+L_{A12})&-M_{A12A2}&-M_{A12A3}&-M_{A12C2}&-M_{A12C3}&L_{A12}\\ \end{array}\right) \tag{15}\]  
The stator windings in the healthy motor are supposed to be star connected; this implies that the zero sequence component of the stator current is zero [8]:  
\[i_{A1}+i_{A2}+i_{A3}+i_{B1}+i_{B2}+i_{B3}+i_{C1}+i_{C2}+i_{C3}=0 \tag{16}\]  
where \(i_{f}\) is the fault current, \(v_{f}\) is the fault voltage, and \(e_{f}\) denotes the fault back electromotive force. Generally, the value of \(v_{f}\) is equal to 0.  
Additionally, we have the following considerations: \(R_{j}\) (\(j\) = A11, A12, A2, A3, B1, B2, B3, C1, C2, C3) is the resistance of the winding \(j\). \(L_{j}\) and \(M_{ij}\) (\(i\) and \(j\) = A, B, C) are, respectively, the self and mutual inductances of the healthy motor. \(L_{j}\) (\(j\) = A11, A12), \(M_{A12k}\) (\(k\) = B, C, A2, A3), and \(M_{A11k}\) (\(k\) = B, C, A12) are the self and mutual induction involving the fault circuit. \(R_{f}\) and \(i_{f}\) are the fault resistance and current, as shown in the Figure 7. If the healthy motor is balanced, we have:  
\[R_{A11}+R_{A12}=\ R_{A1}=\ R_{A2}=\ R_{A3}=\ R_{B1}=\ R_{B2}=\ R_{B3}=\ R_{C1}= \ R_{C2}=\ R_{C3} \tag{17}\]  
\[L_{A11}+L_{A12}+2\!\cdot\!M_{A11A12}=L_{A1}=L_{A2}=L_{A3}=L_{B1}=L_{B2}=L_{B3} =L_{C1}=L_{C2}=L_{C3} \tag{18}\]  
where, according to the value of \(\mu\), it can be obtained that \(R_{A12}=\mu R_{A1}\) and \(R_{A11}=(1-\mu)R_{A1}\)[15].