Improved Understanding of Standing Waves in Single Layer Coils and Elegant Methods to Estimate Transformer Winding Parameters
Analyzing the effect of impulse voltages (like lightning, switching) on transformer winding has occupied centerstage in core electrical engineering research for over a century. These investigations gather great significance and relevance as it eventually governs the design of insulation in the winding. Notwithstanding the colossal contribution this domain has witnessed from stalwarts in the past century, a closer scrutiny surprisingly reveals that there still exists tiny grey areas that demands attention. Pursuing this line of thought, the first part of this thesis aims to clearly describe what this grey area is and resolving it would provide a deeper insight about fundamental understanding of surge response in transformer windings – with special emphasis on its standing wave phenomenon. Following this, in the latter part, elegant procedures are stitched together to determine a few electrical parameters of the transformer winding equivalent circuit that have the potential to help in assessing mechanical status of windings. Objectives of the thesis are - 1. Formulate an analytical method to determine the exact shape of standing waves for all modes in a uniform single layer coil as a solution of its governing partial differential equation 2. Estimate series capacitance of a uniform transformer winding from its measured driving point impedance 3. Determine effective air-core inductance of an iron-core uniform winding as a function of its axial length from measured driving point impedance First part of the thesis revisits a century-old classical theory of standing waves on uniform single layer coils. Accurate information about natural frequencies and shapes of the corresponding standing waves are essential for gaining a deeper understanding of the response of coils to impulse excitations. Analytical studies on coils have largely been based on the assumption that standing waves are sinusoids in both space and time. However, this contradicts the results from numerical circuit analysis and practical measurements. So, this thesis attempts to bridge this discrepancy by revisiting the classical standing wave phenomena in coils. It not only assesses the reason for the aforementioned inconsistency, but also makes a contribution by analytically deriving the exact mode shape of standing waves for both neutral open/short conditions. For this, the coil is modelled as a distributed network of elemental inductances and capacitances, while an exponential function describes the spatial variation of mutual inductance between turns. Initially, an elegant derivation of the governing partial differential equation (in terms of voltage as the variable instead of flux) for surge distribution is presented and to the best of our knowledge, for the first time, an analytical solution for the same has been found by the variable-separable method to find the complete solution (sum of time and spatial terms). Hyperbolic terms in the spatial part of the solution have always been neglected but are included here, thus, yielding the exact mode shapes. For verification, both voltage and current standing waves computed from the analytical solution were plotted and compared with PSPICE simulation results on a 100-section ladder network representing a uniform single-layer coil. Then, practical measurements were made on a tailor-made large-sized single layer coil with a length of 2.02 m, diameter of ~1 m and having 640 turns. It turns out that even in such simple single layer coils, the shape of standing waves of all modes deviates considerably from being sinusoidal. It was further observed that this deviation depends on spatial variation of mutual inductance, capacitive coupling, and order of the standing waves. In the second part, an elegant method for determining the series capacitance (Cs) and air-core equivalent inductance of a uniform winding as a function of its axial length (termed as M0x in this thesis) of a uniform transformer winding, from its measured DPI magnitude, is discussed. Knowledge about the series capacitance of the winding is essential, which along with shunt capacitance, determines the initial impulse voltage distribution when a surge impinges on the winding. Unlike previously published approaches, the proposed method does not involve any cumbersome and time-consuming curve-fitting or running of optimization/search algorithms. Neither does it require winding geometry data. The proposed procedure for finding series capacitance relies on a property that is observable in the driving point impedance (DPI) function of a lossless winding with an open neutral condition, viz., the ratio of the product of squares of open circuit natural frequencies to the product of squares of short circuit natural frequencies bears a particular relation to the coefficients of the DPI function. A simple procedure involving a deft manipulation and combination of a few well-known properties that correlate the roots of a polynomial to its coefficients are then utilized for determining series capacitance. Knowledge about equivalent air-core inductance distribution as a function of its axial length (i.e., M0x) is useful for localizing a minor/incipient mechanical fault in the winding. A physically realizable empirical relationship to estimate M0x is initially proposed. The corresponding constants of the empirical relationship are then calculated from the measured DPI. The proposed method requires three DPI measurements: one with neutral-end open and the other with neutral-end shorted. The third DPI is measured with a known external lumped capacitance connected between the neutral and ground. This method requires only the first few dominant natural frequencies observable in the first two of the DPIs. Feasibility of proposed methods for estimating Cs and M0x was initially verified by simulation on an N-section ladder network and then by experiments on small-sized continuous-disk and interleaved-disk windings, and finally on a large-sized 33 kV, 3.5 MVA continuous-disk winding. Salient features of the proposed methods are – they are simple, elegant and involve minimum post-processing after measuring the DPI. Given its inherent simplicity and their relevance, the author is hopeful that industry will come forward to implement these procedures on an existing FRA measuring instruments – thus opening a new dimension to FRA measurements.