Generalized Analytical Expression for Natural Frequencies of a Single Isolated Air-Core Inhomogeneous Transformer Winding

Given a completely inhomogeneous, fully-coupled, $N$ -section ladder network in which elements of each section are distinct from the others, there exists no closed-form solution which connects the ladder network elements to its natural frequencies. Instead of the present practice of comparing individual natural frequencies, finding such a generalized solution would not only permit quantification of deviations between two frequency responses (FRA) but also provides a generic platform for its interpretation. Presently, interpretation of FRA is mostly empirical and difficult to generalize. Although pioneering contributions by Bewley et al., Abetti and Maginniss, Heller and Veverka, and many others, were made towards developing analytical solutions, they were essentially suitable for a homogeneous winding. For any formulation to become suitable for FRA interpretation (corresponding to a pre and postdamage condition), it must obviously be applicable to an inhomogeneous winding structure. Pursuing this motivation, this paper presents complete details of derivation of analytical expressions that aims to correlate natural frequencies (and their deviations as well) of the ladder network to its basic inductances and capacitances. For this, both short-circuit and open-circuit natural frequencies are examined. Finally, the analytical solution is extended from the discrete-domain to the continuous-domain (transformer winding). Recently, authors have shown practical usefulness of this derived formula for localization and severity assessment of radial/axial displacements in an actual single-isolated continuous-disk transformer winding.