# General Equations of Lumped Parameter Ladder Circuits and a Special Approach to Analyzing Electrical Line Transient States

### Significance Statement

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. Popović was born in Markovac (at Mladenovac), Serbia, in 1944, graduated (1969) and received Master (1983) and Doctor (1991) degrees, all at the School of Electrical Engineering, University of Belgrade.

In 1969 joined the Electric Power Distribution Company of Belgrade, where he stayed until retirement in 2007.  In 1999 he was elected an associated professor at the School of Electrical Engineering, University of Belgrade and in 2010 he was elected an IEEE R8 Industry Lecturer (Industry Continuing Education Program).

At the beginning of his professional carrier he worked on design of different power installations, including the first 110/10 kV substations in the power distribution network of Belgrade. At the end of the seventies, he moved to the Development and Research Department of the same company.

His research work has been mainly focused on the following  topics:

– Grounding systems of HV substations located in urban areas,
– Ground fault current distribution along the overhead and cable feeding lines,
– Fault locator algorithms,
– Resonant phenomena in the transmission lines and transformer windings,
– Influence of metal installations surrounding the feeding line on the ground fault current distribution,
– Influence of electric-power lines on surrounding metal installations,
– Influence of surrounding metal installations on the transfer characteristics of distribution lines.

In addition to realization of the numerous studies and projects, in the area of power delivery he published: over 50 research papers in international journals and proceedings of international conferences, 2 chapters in two international scientific books and one scientific book, Actual Parameters of Power Lines Passing through Urban Areas.

Some of his papers and research results have had an impact on the following IEC publications: Technical Report IEC 60909-2, Ed 1(1992-09) and IEC standard 60909-3, Ed 2 (2003-09)), and was specially highlighted by: ”Vertical News”, ”High-beam Research”, ”High-beam Business”, ”Goliath Business News” and ”News-edge”. Two of his papers published in 2014 have been selected by ”Renewably Energy Global Innovations” as the key scientific articles.

He was elected a member of the IEC Technical Committee – IEC/TC73- Short Circuit Currents from 2004. A member of IEEE since 1987, he became Senior Member in 1991 and was the Chair of IEEE PES Serbia and Montenegro Chapter from 2002 until 2009.

He received: national ”Nikola Tesla” Award in 2006, IEEE PES Chapter Outstanding Engineer Award, and Certificate of Appreciation for Notable Services and Contributions towards advancement if IEEE and Engineering Professions.

### Reference

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.