# Six Sigma-Based Frequency Response Analysis for Power Transformer Winding Deformation
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Frequency Range (Hz)** & \
\begin{tabular}{c} **FRA6\(\sigma\) Results** \\ **(Proposed Approach)** \\ \end{tabular} &
\begin{tabular}{c} **Physical Inspection** \\ **Findings** \\ \end{tabular} & **Remarks on Fault Detection** \\ \hline \multirow{2}{*}{Case 3} & \multirow{2}{*}{10–1000 (Low frequency)} & Core movement minimal; & \multirow{2}{*}{No core issues found} & \multirow{2}{*}{No significant deviation detected} \\  & & response within & & \\  & & expected limits & & \\  & 1000–10,000 & No major clamping loss; & & \\  & (Mid–low frequency) & response remains stable & & \\  & 10,000–100,000 & Minor winding & & \\  & (Mid frequency) & displacement; Cpk remains & & \\  & above 1.33 & displacement observed & & \\  & 100,000–1,000,000 & No insulation issues found; & & \\  & (High frequency) & minor response variations & & \\  & 1,000,000–2,000,000 & No floating parts or & & \\  & (Upper high frequency) & PDs detected & No defects found & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._stable response, showcasing the effectiveness of the statistical approach in distinguishing between developing and non-developing faults. This analysis confirms that the proposed method offers a more granular view of transformer winding integrity, particularly in scenarios where mechanical stress accumulates over time. Traditional FRA analysis often fails to provide such early warnings, as it lacks the statistical rigor necessary to isolate these incipient shifts. In the high-frequency range, insulation breakdown between winding turns, tap changer defects, and grounding issues become the primary concerns. In Case 1, no major insulation breakdown was detected, as FRA6\(\sigma\) results remained stable, aligning with physical inspection findings. However, in Case 2, early signs of insulation degradation were observed, with \(C_{pk}\) values falling below 1.0, indicating that the insulation system was beginning to weaken. Notably, physical inspection did not identify any immediate signs of insulation failure, reinforcing the predictive capability of the FRA6\(\sigma\) framework. Case 3 showed minor insulation deviations but remained within acceptable limits, as confirmed by both FRA6\(\sigma\) and inspection. The ability of the proposed approach to quantify insulation degradation before physical failure occurs suggests its superiority over traditional FRA methods, which often fail to capture insulation weaknesses until they manifest as electrical failures. The upper high-frequency range primarily detects partial discharge (PD) activity, floating metal parts, and loose clamps or connections. In Case 1, no floating conductive parts or PD activity was found, with FRA6\(\sigma\) confirming a stable response, validated by physical inspection. However, in Case 2, FRA6\(\sigma\) detected incipient PD activity, indicated by deviations beyond the UCL. Despite this, physical inspection did not reveal any loose connections or floating metal elements, reinforcing the predictive strength of the FRA6\(\sigma\) model in identifying pre-failure insulation issues. Case 3 exhibited a normal response, with both FRA6\(\sigma\) and physical inspection agreeing on the absence of high-risk defects. This consistency highlights that the proposed approach not only detects emerging faults but also reliably confirms healthy operating conditions. The comparative analysis between FRA6\(\sigma\) results and physical inspection findings underscores the transformative potential of the proposed approach in transformer fault diagnostics. While traditional FRA methods depend heavily on expert interpretation, FRA6\(\sigma\) introduces a structured, statistically driven framework that objectively identifies incipient mechanical and electrical failures.