Detecting inter-turn short circuits in ultra-high voltage shunt reactors is challenging due to weak fault signatures. This paper establishes an accurate three-segment fault model and derives

for the first time

explicit analytical expressions for the short-circuit phase equivalent resistance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\boldsymbol{R}_{\mathbf{eq}})$ as a function of both short-circuit turn ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\boldsymbol{\alpha})$ and transition resistance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\boldsymbol{R}_{\mathbf{k}})$. In particular

this model reveals specific <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{R}_{\mathbf{eq}}$ variation laws: monotonic decrease with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{\alpha}$ for metal faults and non-monotonic behavior for transition resistance faults. Leveraging these insights

an innovative identification method is proposed. Its core innovation is a theory-driven

flexible threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\boldsymbol{R}_{\mathbf{set}})$ calculation based solely on the derived <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{R}_{\mathbf{eq}}=\boldsymbol{f}(\boldsymbol{\alpha}

\boldsymbol{R}_{\mathbf{k}})$ function and user-defined pa-rameters (minimum detectable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{\alpha}_{\mathbf{min}}$ and maximum tolerable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{R}_{\mathbf{k}\_\mathbf{max}}$)

thereby reducing reliance on empirical values. Simulation results validate the proposed model and method

demonstrating superior sensitivity and ac-curacy

especially for small-turn inter-turn short circuits with transition resistance

compared to conventional ze-ro-sequence methods. This provides a robust theoretical foundation for practical fault detection.