Analytical Expressions to Link SCNF and OCNF of Transformer Windings to Their Inductances and Capacitances for 1-$\Phi$, 3-$\Phi$ Y and $\Delta$ Configurations

Recently, the present authors' research group derived an analytical expression to link the harmonic sum of the squares of short circuit natural frequencies (SCNFs) of a single, isolated winding to its elementary inductances and capacitances. Also, it was demonstrated how this expression could be practically used for locating radial and axial displacements, along with assessing its severity on a single, isolated, actual transformer winding. The next logical task was to extend the derived expression to multi-phase windings, viz., star/delta connections and for any terminal condition of the neutral. Actually, it would be desirable to have a similar expression for open circuit natural frequency (OCNF) as well. A major highlight of this paper is that it presents a single, unified approach (using salient features of coefficients of the numerator and denominator polynomial of driving-point impedance) by which compact analytical expressions can be simultaneously derived for both SCNF and OCNF, for any multi-phase transformer configuration, and for any condition of the neutral terminal. Complete derivation details of the proposed method are presented, and then, the expressions for SCNF and OCNF are derived for 1-Φ isolated windings as well as 3-Φ (Y, Δ) configurations. Finally, simulation results for all the cases are reported.