In this paper, self inductance for a conductor with rectangular cross section is investigated. Using the three-dimensional Fredholm's integral equation of the second kind with weakly singular kernel we obtain an equation for the complex voltage drop in the conductor. Self impedance appearing in the equation is expressed in the form of integral relation for any current density distribution. The imaginary part of this impedance divided by angular frequency is the self inductance of a conductor of any shape and finite length. In the case of direct current (DC), low frequency (LF) or thin strip conductor of rectangular cross section the formulae for the self inductances are given for any length and for length much greater than the other dimensions.

Abstract-In this paper, a new numerical method of calculating rectangular busbar impedance is proposed. This method is based on integral equation method and partial inductance theory. In particular, impedances of shielded and unshielded three-phase systems with rectangular phase and neutral busbars, conductive enclosure, and use of the method are described. Results for resistances and reactances for these systems of multiple rectangular conductors have been obtained, and skin and proximity effects have also been taken into consideration. The impact of the enclosure on impedances is also presented. Finally, two applications to three-phase shielded and unshielded systems busbars are described. The validation of the proposed method is carried out through FEM and laboratory measurements, and a reasonable level of accuracy is demonstrated.

The present paper describes the research on the mechanism of inrush current formation in a modern single-phase transformer, in which the insulation system must withstand the stresses arising during these transient states. A complete and measurement-verified method for determining the transformer inrush current waveforms and other signals (e.g., fluxes and voltages) is developed. This method makes it possible to determine a steady state solution. However, on account of the electromagnetic phenomena, the solving process is long. To analyze the transient dynamic response of the tested transformer, a nonlinear model was assumed, from which the stiff differential equations were derived. The simulation analyses were performed using dedicated software written in C# with the original implementation of the five-stage Radau IIA algorithm selected from the known variants of the Runge–Kutta implicit methods. The calculations were based on the measurement waveforms recorded during transient (switch-on) and steady-state states when the transformer was not loaded. The full magnetization curve of the core sheets of the tested transformer with an original implementation of the polynomial fitting mechanism was applied. As a representative example and for the purposes of experimental verification of numerical tests, the worst case scenario for switch-on of an unloaded transformer was applied (switch-on is performed when the supplied voltage is zero). Phenomena related to the obtained experimental results, such as saturation and hysteresis, are discussed as well.

The methodology and test results of a three-phase three-column transformer with a Dy connection group are presented in this paper. This study covers the dynamics of events that took place in the first period of the transient state caused by the energizing of the transformer under no-load conditions. The origin of inrush currents was analyzed. The influence of factors accompanying the switch-on and the impact of the model parameters on the distribution and maximum values of these currents was studied. In particular, the computational methods of taking into account the influence of residual magnetism in different columns of the transformer core, as well as the impact of the time instant determined in the voltage waveform at which the indicated voltage is supplied to a given transformer winding, were examined. The study was carried out using a nonlinear model constructed on the basis of classical modeling, in which hysteresis is not taken into account. Such a formulated model requires simplification, which is discussed in this paper. The model is described using a system of stiff nonlinear ordinary differential equations. In order to solve the stiff differential state equations set for the transient states of a three-phase transformer in a no-load condition, a Runge–Kutta method, namely the Radau IIA method, with ninth-order quadrature formulas was applied. All calculations were carried out using the authors’ own software, written in C#. A ready-made strategy for energizing a three-column three-phase transformer with a suitable pre-magnetization of its columns is given.

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