0885-8977 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRD.2018.2881771, IEEE Transactions on Power Delivery Analytical Expression for Resonances of an Inhomogeneous, Radial, Lossless LC Network Pritam Mukherjee, Member, IEEE and L. Satish Abstract—This paper is a sequel to the authors’ previous work wherein an analytical expression was derived for the harmonic sum of squares of natural frequencies of a lossless LC ladder network which is a widely accepted means to model a single, isolated winding for analyzing impulse behavior. The present paper extends the previous result so that it is applicable to multiple windings in actual transformers. The complexities of the previous work, viz., inhomogeneity and an exhaustive consideration of inductive coupling are retained, and additionally all possible capacitive couplings are now considered. Further, the topology considered in this paper is generic enough to accommodate inductive branches emerging from any node. To the best of authors’ knowledge, there exists no compact closed-form expression to relate the natural frequencies to the inductances and capacitances for a network having this extent of arbitrariness in topology. The success of this method can be ascribed to careful inspection of the inductance matrix to identify hidden pattern and/or structure, followed by their deft manipulation guided by the basic law of mutual coupling between inductors. Interestingly, even a physical meaning can be assigned to each term of the derived expression, and this is done with the intention of assisting its use to conjure new applications. Index Terms—electric network, eigenvalue, frequency response, natural frequencies, resonance. I. I NTRODUCTION The transmission line type ladder network representation of transformer winding was first used to capture the impulse behaviour of transformers [1]-[5]. Study of such networks have gained popularity in recent years with frequency response analysis (FRA) emerging to be an ultra-sensitive, off-line, on-site, non-destructive tool to detect damages in transform- ers [6]. The predisposition of FRA to reveal even minute alterations in geometry, including clearances, is driving the current research to also focus on its online implementation [7]-[10]. The frequency response of a transformer, as it often happens with hypersensitive measurements, is susceptible to a variety of factors, viz., temperature, terminal connection, etc.; influence of these factors are also being investigated [11]. This remarkable sensitivity of FRA is, quite regrettably, accompanied by an ad hoc diagnosis at present, i.e., most of the changes detected by FRA cannot be deterministically and reliably diagnosed from the observed deviation. Rigorous interpretation of FRA is largely based on the analysis of the ladder network equivalent circuit of transformer winding. The winding system has to be treated as a distributed parameter system at high frequencies and discretization of Pritam Mukherjee (mukherjee.hve@gmail.com) and L. Satish (satish@iisc.ac.in) are with the High Voltage Laboratory, Dept. of Electrical Engg., Indian Institute of Science, Bangalore - 560012, India. such a distributed system leads to a ladder network of finite sections. Although these ladder networks strongly resemble the transmission line representation, transformer’s winding geometry and structure introduces additional complexities, viz., mutual inductance and series capacitance. The complexity increases while dealing with damaged windings as it can no longer be modelled with a uniform ladder network. Number of recent articles attempt to improve the synthesis of such ladder networks [12]-[16]. Deviation in FRA is largely visualized as observed shifts in resonances (and also in the magnitudes) [17]-[19] but there is no analytical solution for resonances for these complicated ladder networks. This makes diagnosing damages from the shifts in resonant frequencies an unresolved task. The current approach of FRA interpretation is predom- inantly based on statistical and numerical indices [20]-[25], while image processing and data intelligence based techniques are also reported [26]-[28]. No interpretation framework devel- oped so far is universally accepted and reliable. This interpre- tational predicament of FRA seems to be rooted in the absence of a compact and meaningful analytical correlation between the parameters of the system and the natural frequencies. This was the motivation. II. PRIOR WORK AND DIFFICULTIES IN MULTIWINDING SCENARIO Authors’ previous work reported an analytical relation- ship for resonant frequencies of an unbranched, inhomoge- neous, fully inductively-coupled, lossless LC ladder network as shown in Fig. 1 (Note: By the term “unbranched”, the authors imply that the inductive path is unbranched). L 1,1 L 2,2 L N,N C g0 C g1 C g2 C gN−1 C gN C s1 C s2 C sN Fig. 1. An inhomogeneous, inductively coupled, lossless LC ladder network It was shown in [29] and [30] that the harmonic sum of squares of resonant frequencies can be expressed as N  i=1 1 ω 2 i = N  i=1 L i,i C si + N  i=1 M 0i C gi (1) where M 0i = ∑ i k=1 ∑ i k ′ =1 L k,k ′ Although this relationship does not yield the individual resonant frequencies which would have been incredibly useful