# Detection of Arc Faults in Transformer Windings via Transient Signal Analysis
## 2 Material and Method
### Arc Model  
Electrical arcs exhibit nonlinear characteristics, and for a realistic representation, a dynamic arc model should be employed for arc simulation [22]. Arc models can be categorized into three groups [23]: physical models, black box models, and models based on graphs or diagrams. Black box models solely describe the relationship between input and output signals. In these models, the arc is represented by one or more differential equations that characterize the arc conductivity and the energy balance of the arc column. Numerous accepted arc models exist in the literature. The Cassie, Mayr, Habedank, Schavemaker, and Kema models are based on differential equations and have been validated through various studies [24; 25]. In this study, the Habedank model was used to represent the arc. The Habedank arc model is a series connection of the [PERSON] and Mayr models and consists of two differential equations and one algebraic equation [26]:  
\[u_{s}=Ri+L\frac{di}{dt}+u \tag{1}\]  
\begin{table}
\begin{tabular}{l c l c} \hline \hline
**Label Information** & **Value** & **Label Information** & **Value** \\ \hline Rated Power & 15 MVA & Core Material & M125-027S \\ \hline HV/LV Voltages & 33,000/11,000 V & Filling Factors & 0.93 \\ \hline HV/LV Connection & \(\Delta\)/Y & Core Type & Step Lap \\ \hline HV/LV Turns & 665/119 & Core Material Mass & \\  & Density & 7650 kg/m3 \\ \hline Current Density (HV Winding) & 3.68 A/mm2 & Core K\({}_{\text{h}}\) Material & \\  & & Lamination & 43.59 W/m3 \\ \hline Current Density (LV Winding) & 3.19 A/mm2 & Core K\({}_{\text{c}}\) Material & \\  & & Lamination & 0.599 W/m3 \\ \hline HV Winding & 1.7 \(\Omega\) & Core Material & \\  & & Thickness & \\ \hline LV Winding & 40 m\(\Omega\) & Core Material & \\  & Resistance & & Conductance & 5,000,000 S/m \\ \hline Frequency & 50 Hz & Core Diameter & 430 mm \\ \hline Percentage Short & 10\% & Core Distyoke & 1416 mm \\ \hline Copper Loss & 97,000 W & Core DistLeg & 839 mm \\ \hline Core Loss & 12,000 W & Core Cross-section & 1270 cm2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: ANSYS 3-D modeled transformer values.  
Figure 1: Transformer Maxwell 3D model.  
where \(i\) is the arc current, \(L\) is the coil inductance, \(R\) is the coil resistance, \(u\) is the voltage across the arc, and \(u_{s}\) is the voltage between the supply terminals. When Cassie and Mayr differential equations are written, the following equations are obtained:  
\[\frac{1}{g_{c}}\frac{dg_{c}}{dt}=\frac{1}{\tau_{c}}\left(\left(\frac{ug}{U_{c}g _{c}}\right)^{2}-1\right) \tag{2}\]  
\[\frac{1}{g_{m}}\frac{dg_{m}}{dt}=\frac{1}{\tau_{m}}\left(\left(\frac{u^{2}g^{2 }}{C_{0}g_{m}}\right)-1\right) \tag{3}\]  
\[\frac{1}{g}=\frac{1}{g_{m}}+\frac{1}{g_{c}} \tag{4}\]  
where \(g\) is the arc conductance, \(g_{c}\) is the arc conductance in the Cassie equation, \(\tau_{c}\) is the Cassie time constant, \(g_{m}\) is the arc conductance in the Mayr equation, \(\tau_{m}\) is the Mayr time constant, and \(C_{0}\) is a constant. Using these equations, a complete arc model is constructed and solved differentially using block diagrams in the Matlab@Simulink environment.