# Efficient Zero-Sequence Impedance Measurement in Autotransformers Using Low-Voltage Excitation
## 2 YNa Autotransformer Zero-Sequence Impedance Field Test Method  
### One-Phase Disconnection Test on the High-Voltage Side  
The wiring for the one-phase disconnection test on the high-voltage side of an autotransformer with YNa connection is shown in Figure 1 (phase A as the special phase). The test is achieved by applying a three-phase power source on the low-voltage side, leaving the A-phase winding on the high-voltage side open, and the remaining two phases short circuited to earth. The corresponding complex sequence network is shown in Figure 2. Here, \(Z_{\text{H-1}}\), \(Z_{\text{H-2}}\) and \(Z_{\text{H-0}}\) are the positive-, negative- and zero-sequence impedances of the three phases on the high-voltage side of the autotransformer, respectively; \(Z_{\text{L-1}}\), \(Z_{\text{L-2}}\) and \(Z_{\text{L-0}}\) are the positive-, negative- and zero-sequence impedances of the three phases on the low-voltage side, respectively; \(Z_{\text{g-0}}\) is the zero-sequence excitation impedance. The core structure of automatic transformers usually adopts a three-column core design [x]. In this construction, the core of the transformer consists of three cylindrical sections, each of which is wrapped around a winding. This design is common in three-phase transformers and allows for efficient transfer of electrical energy. It is worth noting that the automatic transformer has only one winding, in which different parts of the same winding serve as both the main winding and the secondary winding of the transformer [18]. Therefore, the zero-sequence flux can only form a closed loop through the oil gap and the box, as shown in Figure 3. This makes the reluctance of the zero-sequence flux large, while the excitation reactance is small and cannot be ignored. Based on Figure 3, the zero-sequence impedance can be expressed as:  
\[Z_{\text{H(L|g)}-0}=Z_{\text{H-0}}+Z_{\text{(L|g)}-0}=\frac{V_{1}-I_{1}Z_{ \text{HL}-1}}{I_{0}} \tag{1}\]  
Figure 1: YNa autotransformer test diagram for the high-voltage side One-Open-Phase method. Figure 3: The magnetic circuit of the zero-sequence flux. (**a**) Core and zero-sequence magnetic flux path. (**b**) Induced current in the fuel tank wall. Figure 2: Sequence network of the YNa autotransformer high-voltage One-Open-Phase test. The leakage impedance of the autotransformer is inductive, and the proportion of resistance value is small, so it can be ignored to simplify the calculation. Figure 3 can be considered as a purely inductive circuit. Phases of each sequence current vector are the same. The vector relationship can be expressed directly in terms of numerical magnitude:  
\[\left\{ \begin{array}{l}\dot{I}_{ab}=\frac{3}{2}(I_{1}+I_{2})+j\frac{\sqrt{3}}{2 }(I_{1}-I_{2})\\ \dot{I}_{bc}=-j\sqrt{3}(I_{1}-I_{2})\\ \dot{I}_{ca}=-\frac{3}{2}(I_{1}+I_{2})+j\frac{\sqrt{3}}{2}(I_{1}-I_{2})\\ \end{array} \right. \tag{2}\]  
Here, \(\dot{I}_{ab}\), \(\dot{I}_{bc}\), \(\dot{I}_{ca}\) are the difference between two different phase line currents on the low-voltage side of the autotransformer, respectively, and are positive-sequence currents; \(I_{1}\), \(I_{2}\), \(I_{0}\) are negative- and zero-sequence currents. From Equation (3), it can be seen that the real part of \(\dot{I}_{b}\) and \(\dot{I}_{c}\) is the same, and the imaginary part is opposite. Assume that the angle between \(\dot{I}_{ab}\) and \(\dot{I}_{1}\) is \(\theta\), \(\left|\dot{I}_{ab}\right|=\left|\dot{I}_{ca}\right|=I\), and \(\dot{I}_{0}=-\left(\dot{I}_{1}+\dot{I}_{2}\right)\). It can be found that:  
\[\left\{ \begin{array}{l}I_{1}+I_{2}=\frac{2}{3}I\cos\theta=\left|\dot{I}_{0} \right|\\ I_{1}-I_{2}=\frac{2}{\sqrt{3}}I\sin\theta\\ \end{array} \right. \tag{3}\]  
Then it can be found that:  
\[\left\{ \begin{array}{l}\theta=\sin^{-1}\frac{\left|\dot{I}_{bc}\right|}{2I} \\ I_{1}=\frac{2}{3}I\cos(\theta-60^{\circ})\\ I_{2}=-\frac{2}{3}I\cos(\theta+60^{\circ})\\ \end{array} \right.