The Telegrapher’s equations, two linear partial differential equations, one for voltages and the other for currents, describe the behavior of voltages and currents in each moment and at any point along an electrical line. They can only be integrated in special cases e. g. if the voltage is sinusoidal function of time and the obtained solution is well-known as the General Line Equations. However, these equations are limited in that they are only convenient for analysis of the transmission lines operating in steady state regimes. Also, these equations are not sufficiently accurate in the case of a lumped parameter ladder circuit that is formed in practice by transmission line ground wires and the belonging tower footing electrodes, especially in the cases when the number of the considered spans is small.
Furthermore, the Telegrapher’s equations can be integrated in the transient regime only using operational calculus, i.e. the Laplas Transformation. However, on the basis of such solution the finally obtained analytical expressions are not convenient for the interpretation and analysis of resonant phenomena, as well as for determination of transient over-voltages in transmission lines.
Conversely, knowledge exists that lumped parameter ladder circuits represent universal physical and mathematical models of systems having distributed parameters and as such can also be used for analyses of electrical lines. However, these circuits were without the adequate general solutions, i.e. without the equations analogous to the General Line Equations.
Dr. Ljubivoje Popovic at J.P. Elekdrodistribucija-Beograd in Serbia has managed to develop an alternative analytical procedure for obtaining the well-known General Line Equations. Also, on the basis of the same procedure he has obtained the, so far unknown equations and named them “General Equations of Lumped Parameter Ladder Circuits”. These equations enable correct determination of ground fault current distribution for a fault at any point along an HV transmission line. Moreover, these equations enable the analysis of electrical quantities along any actual (with distributed parameters) line in steady state by applying relatively simple mathematical operations and with a desired degree of accuracy.
However, the most important research result is the developed analytical procedure itself, because it opens possibilities for a new approach in analyzing resonant phenomena and transient states in electric-power lines. The author began by representing an electrical line through its lumped parameter model as a base and then applied the principle of superposition and a summation of the especially formed finite and infinite geometric series. The researcher was able to make observations such as: the developed analytical procedure resulted in relatively simple analytical expressions for the relationship between currents and voltages at different points of transmission lines in transient state conditions. Secondly, the developed analytical procedure was performed to follow, one-by-one, all the phases of the actual physical process occurring during transient states in electrical lines. Eventually, the researcher applied his analytical procedure in a numerical example concerning determination of switching over-voltages in transmission lines and show that this problem can be solved without specially developed computer programs. His research work is now published in the peer-reviewed journal, Electrical Power and Energy Systems.
Ljubivoje M. Popovic. General equations of lumped parameter ladder circuits and a special approach to analyzing electrical line transient states. Electrical Power and Energy Systems, volume 95 (2018) pages 568–576.
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