## 3 ITSCF Model and Feature
### The Design of ITSCF Model  
This research presents a design methodology for an ITSCF model [8], aimed at facilitating the development and application of deep learning models for motor fault diagnosis and analysis. Derive the \(d\)-\(q\) axis mathematical model for motor ITSCF from the three-phase ITSC mathematical model and compare the voltage characteristic differences between ITSC and healthy motors through the mathematical model. Subsequently, perform signal analysis and feature extraction using DWT as preprocessing steps, and convert the fault features into a format suitable for input into deep learning networks. The ITSC structure of the motor system model is shown in Figure 2, where \(N_{1}\) represents the number of turns in the healthy coil, \(N_{2}\) represents the number of turns in the short-circuited coil, and \(\mu\) denotes the internal short-circuit ratio as described by Equation (1). \[\mu=\frac{N_{2}}{N_{1}+N_{2}}, \tag{1}\]  
The ITSC model of the motor is given by Equation (2). \[\mathbf{v}_{\textit{abcf}}=\mathbf{r}_{fs}\mathbf{i}_{\textit{abcf}}+\mathbf{L}_{fs}\frac{ \mathrm{d}}{\mathrm{d}t}\Big{(}\mathbf{i}_{\textit{abcf}}\Big{)}+\mathbf{e}_{\textit{ abcf}}+\mathbf{v}_{f}, \tag{2}\]  
The voltage and current with ITSC are represented by Equations (3) and (4), respectively Voltage with ITSC:  
\[\mathbf{v}_{\textit{abcf}}=\big{[}\mathbf{v}_{\textit{a}}\quad\mathbf{v}_{\textit{b}} \quad\mathbf{v}_{\textit{c}}\quad\mathbf{0}\big{]}^{\mathrm{T}}, \tag{3}\]  
where the fourth item represents the shorted coil voltage, with a value of 0 V.  
\[\mathbf{i}_{\textit{abcf}}=\big{[}i_{a}\quad i_{b}\quad i_{c}\quad i_{f}\big{]}^{ \mathrm{T}}, \tag{4}\]  
where \(i_{f}\) refers to the shorted coil fault current; the resistance and inductance under ITSC are given by Equations (5) and (6). The ITSC resistance is expressed as follows:  
\[\mathbf{r}_{fs}= \begin{bmatrix}r_{s}&0&0&-\mu r_{s}\\ 0&r_{s}&0&0\\ 0&0&r_{s}&0\\ \mu r_{s}&0&0&-\mu r_{s}-r_{f}\end{bmatrix} , \tag{5}\]  
where \(\mu r_{s}\) represents the shorted coil resistance, and \(r_{f}\) is the external parallel resistance of the shorted coil. According to reference [8], when an internal short circuit occurs in phase **a** coil, the phase **a** current \(i_{a}\) still flows through the entire phase a coil, so the corresponding phase resistance remains \(r_{s}\). However, the fault current \(i_{f}\) primarily constitutes a circulating current within the shorted coil, thus corresponding to a resistance value of \(\mu r_{s}\). Additionally, the shorted coil may form a short circuit through other mediums; thus, \(r_{f}\) is introduced as a model parameter. This parameter influences the response of the short-circuit fault current \(i_{f}\). The ITSC inductance is expressed as follows:  
\[\mathbf{L}_{fs}= \begin{bmatrix}L&M&M&-\mu L\\ M&L&M&-\mu M\\ M&M&L&-\mu M\\ \mu L&\mu M&\mu M&-\mu^{2}L\end{bmatrix} , \tag{6}\]  
where \(\mu L\) represents the self-inductance of the phase **a** shorted coil, \(\mu M\) denotes the mutual inductance between the phase **b** and phase **c** coils with respect to the shorted coil, and \(\mu^{2}L\) refers to the self-inductance of the shorted coil. Similarly, based on reference [8], Figure 2, and the aforementioned resistance model, the phase **a** current \(i_{a}\) still flows through the entire phase **a** coil; thus, the corresponding phase inductance remains \(L\). Considering the response of the shorted coil fault current \(i_{f}\), the parameters \(\mu L\), \(\mu M\), and \(\mu^{2}L\) are defined toestablish the inductance model for each phase coil in relation to the shorted coil. The back electromotive force with ITSC is represented by Equation (7). \[e_{abcf}=\left[ \begin{array}{c}\lambda^{\prime}m\omega_{r}\cos\left(\theta_{r }\right)\\ \lambda^{\prime}m\omega_{r}\cos\left(\theta_{r}-\frac{2}{3}\pi\right)\\ \lambda^{\prime}m\omega_{r}\cos\left(\theta_{r}+\frac{2}{3}\pi\right)\\ -\mu\lambda^{\prime}m\omega_{r}\cos\left(\theta_{r}\right)\end{array}