# Linear Method for Diagnosis of Inter-Turn Short Circuits in 3-Phase Induction Motors
## 4 Turn-to-Turn Short and Inter Turn Short Circuit  
### Magnetic Flux Linkage Quantity according to a Turn-to-Turn Short  
Industrial sites generally use 3-phase induction motors that are difficult to analyze when faults occur. Therefore, the initial 3-phase condition is usually reduced to 2-phase to simplify analysis using a D-Q transformation. Figure 3: Changes to the current by the D-Q transformation: (**a**) 3-phase input current; (**b**) D-Q stationary reference frame; (**c**) D-Q synchronous reference frame over time. As shown in Figure 4a, in the event of a turn-to-turn short in one phase, an independent closed circuit to \(R_{s}\) is generated, and another independent closed circuit occurs in the D-Q transformation, as shown in Figure 4b. If the turn number at which the short circuit occurred is \(N_{S}\), the winding turn number at which no short circuit occurred is \(N_{N}\), the total number of turns is \(N_{T}\), then \(N_{T}=N_{N}+N_{S}\). The shorted winding and a non-shorted winding have the same inductance and direct interference occurs. This interference causes an induced electromotive force in the shorted winding when magnetic flux exists with the same inductance, and this induced electromotive force generates the short circuit current \(I_{S}\). When a turn-to-turn short occurs, \(N_{S}\) is large, and the induced electromotive force is large as a result. Then the short circuit \(R_{S}\) becomes small and a large short circuit current \(I_{S}\) is generated. As shown in Figure 5a, due to the generation of \(N_{S}\), two inductances exist in one phase, as shown in Figure 5b, and the magnetic fluxes link the two inductances. If the inductance of one phase is \(L_{T}\) according to the winding number, the inductance \(L_{S}\) by short circuit can be expressed by the short circuit ratio as follows:  
\[L_{S}=L_{T}\frac{N_{S}}{N_{T}} \tag{10}\]  
Figure 5 gives fragmentary indication that the inductance of one phase will decrease and the phase current will increase. However, change of the current in the event of a fault can be known only by investigating various kinds of information including the inductance, winding number, and magnetic  
Figure 4: (**a**) Occurrence of turn-to-turn short of phase R and (**b**) D-Q transformation equivalent model of a turn-to-turn short. Figure 5: (**a**) Inductance by turn-to-turn short and definition of winding number and (**b**) two inductances by a turn-to-turn short. flux linkage quantity. In the event of a fault, using the conditions inductance, winding number, and flux, the magnetic flux linkage quantity can be defined as follows:  
\[\lambda_{1S}=N_{T}(\phi_{1S}-\phi_{2S})=N_{T}\Big{(}\frac{L_{T}I_{d}}{N_{T}}- \frac{L_{S}I_{S}}{N_{T}}\Big{)}=L_{T}I_{d}-L_{S}I_{S} \tag{11}\]  
\[\lambda_{2S}=N_{S}(\phi_{2S}-\phi_{1S})=N_{S}\Big{(}\frac{L_{S}I_{S}}{N_{T}}- \frac{L_{T}I_{d}}{N_{T}}\Big{)}=L_{T}\frac{N_{S}}{N_{T}}I_{S}-L_{S}I_{d} \tag{12}\]  
where \(\lambda_{1S}\) is the magnetic flux linkage quantity of one phase, \(\lambda_{2S}\) is the magnetic flux linkage quantity of the fault inductance generated at the turn-to-turn short, \(\phi_{1S}\) is the flux of one phase, \(\phi_{2S}\) is the flux of the fault inductance, and \(I_{d}\) is the normal current of one phase. When a turn-to-turn short occurs, two fluxes appear that resemble a two-winding transformer, and the quantities of the two magnetic flux linkages generate a leakage flux. The total leakage fluxes of the short circuit \(\lambda_{ST}=\lambda_{1S}+\lambda_{2S}\), can be expressed as follows:  
\[\lambda_{ST}=(L_{T}-L_{S})I_{d}-\Big{(}L_{S}-L_{S}\frac{N_{S}}{N_{T}}\Big{)}I_ {S}=(L_{T}I_{d}-L_{S}I_{S})\Big{(}1-\frac{N_{S}}{N_{T}}\Big{)} \tag{13}\]  
Based on Equation (13) only, the total flux decreases by the result of \((L_{T}I_{d}-L_{S}I_{S})\), and the total leakage flux diminishes further according to the short circuit ratio. It can also be seen that the more severe the short circuit is, the lower the current becomes.