## 2 Turn Short and D-Q Transformation
### D-Q Transformation Analysis According to Turn Short  
When turn short circuits occurs in one phase as shown in Figure 4a, an independent closed circuit is created through \(R_{s}\), and as shown in Figure 4b, the independent closed loop is also created during D-Q transformation. If the number of turns with the short circuit is \(N_{S}\), the number of turns with no short circuit is \(N_{N}\), the total number of turns is \(N_{T}\), \(N_{T}=N_{N}+N_{S}\). Figure 3: Current change due to D-Q transformation (**a**) 3-phase input current, (**b**) D-Q stationary coordinate system, (**c**) D-Q synchronous coordinate system. Figure 2: Direct-quadrature (D-Q) transformation process: (**a**) D-Q stationary reference frame (**b**) D-Q synchronous reference frame. Shorted and unshorted windings have the same inductance, resulting in direct interference. This interference generates an induced electromotive force in the shorted windings when it is present in inductances with the same magnetic flux, and this induced electromotive force generates a short circuit current, \(I_{S}\). When a turn short occurs, \(N_{S}\) is large; therefore, the induced electromotive force becomes large. As the short-circuit resistance \(R_{S}\) becomes small, the short-circuit current \(I_{S}\) becomes large. As shown in Figure 5a, two inductances exist in the inductance on one phase due to the occurrence of \(N_{S}\), as shown in Figure 5b. Between two inductances, the magnetic flux is interlinked with each other. In this case, if the inductance on one phase according to the number of turns is \(L_{T}\) and the inductance by a short circuit is \(L_{S}\), the interlinked magnetic flux can be defined as Equations (3) and (4). \[\lambda_{1S}=N_{T}(\phi_{1S}-\phi_{2S})=N_{T}\Big{(}\frac{L_{T}I_{d}}{N_{T}}- \frac{L_{S}I_{S}}{N_{T}}\Big{)}=L_{T}I_{d}-L_{S}I_{S} \tag{3}\]  
\[\lambda_{2S}=N_{S}(\phi_{2S}-\phi_{1S})=N_{S}\Big{(}\frac{L_{S}I_{S}}{N_{T}}- \frac{L_{T}I_{d}}{N_{T}}\Big{)}=L_{T}\frac{N_{S}}{N_{T}}I_{S}-L_{S}I_{d} \tag{4}\]  
where \(\lambda_{1S}\) is the interlinked magnetic flux in one phase, \(\lambda_{2S}\) is the interlinked magnetic flux of a fault inductance generated at the turn short circuit, \(\Phi_{1S}\) is the flux in one phase, \(\Phi_{2S}\) is the magnetic flux of a fault inductance, and \(I_{d}\) is the steady current in one phase. When a turn short occurs, the two magnetic fluxes behave like a transformer, and the two interlinked magnetic fluxes generate leakage flux. Accordingly, the sum of the magnetic flux leakage by a short circuit, \(\lambda_{ST}=\lambda_{1S}+\lambda_{2S}\), is given by Equation (5). \[\lambda_{ST}=(L_{T}-L_{S})I_{d}-\Big{(}L_{S}-L_{S}\frac{N_{S}}{N_{T}}\Big{)}I _{S}=(L_{T}-L_{S})I_{d}-L_{S}\Big{(}1-\frac{N_{S}}{N_{T}}\Big{)}I_{S} \tag{5}\]  
Figure 4: (a) Turn short in phase A, (b) D-Q Transformation equivalent model by turn short circuit. Figure 5: (a) Definition of inductance and number of turns by turn short circuit and (b) two inductances due to turn short circuit. In Equation (5), the total flux decreases as a result of \((L_{T}-L_{S})I_{d}\). The generated short circuit current becomes \(L_{S}(1-N_{S}/N_{T})I_{S}\), and the total magnetic flux is further reduced. If \((L_{T}-L_{S})I_{d}\) is term A, \(L_{S}(1-N_{S}/N_{T})I_{S}\) is term B, \((1-N_{S}/N_{T})_{\_1}\) is the case when the first short circuit occurs, and \(I_{S}\) is negligible because it is small for a negligible turn short circuit, then the variation of current can be confirmed by the relational analysis of \(L_{S}\). In this case, it can be found that \(L_{S}\) in term B is reduced by \((1-N_{S}/N_{T})_{\_1}\) compared to \(L_{S}\) in term A. Furthermore, it can be found that the total magnetic flux becomes smaller when the first short circuit occurs. On the other hand, if the case where the short circuit becomes significant is \((1-N_{S}{{}^{\prime}}/N_{T})_{\_2}\). As \(N_{S}\) increases, the value of \((1-N_{S}{{}^{\prime}}/N_{T})_{\_2}\) becomes smaller than that of \((1-N_{S}/N_{T})_{\_1}\).