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Quadratic polynomial form of electric arc furnace equation

Dariusz Grabowski (Faculty of Electrical Engineering, Silesian University of Technology, Gliwice, Poland)



Electric arc furnaces are very often modeled using combined models which cover separately deterministic and stochastic phenomena taking place in the furnace. The deterministic part is expressed by nonlinear differential equations. A closed form of the solution to one of the most popular nonlinear differential equations used for the AC electric arc modeling does not exist for some values of the parameters. The purpose of this paper is to convert electric arc furnace equation for these parameters to the quadratic polynomial form which significantly simplifies solution.


The solution has been obtained in the time domain by a sequence of transformations of the original nonlinear equation which lead to a system of quadratic equations, for which a periodic solution can be found easily using harmonic balance method (HBM).


Quadratic polynomial form of electric arc furnace nonlinear equation in the case for which the solution to the nonlinear differential equation describing electric arc cannot be obtained in a closed form.

Research limitations/implications

The complete model of the arc requires extension of the deterministic solution obtained for the quadratic polynomial form using stochastic or chaotic component.

Practical implications

The obtained results simplify determination of the arc voltage or radius time waveforms if a closed form solution does not exist. The arc model can be used to evaluate the impact of arc furnaces on power quality during the planning stage of new plants. The proposed approach facilitates calculation of the arc characteristic.


In order to avoid problems occurring when a large number of harmonics is required or the system contains strong nonlinearities, a transformation of the original equation has been proposed. Nonlinearities present in the equation have been transformed into purely quadratic polynomial terms. It facilitates application of the classical HBM and allows to follow periodic solutions of the arc equation when its parameters are varied. It also enables better understanding of the phenomenon described by the equation and makes easier the extension of the arc model in order to cover the time-varying character of the arc waveforms.



Grabowski, D. (2014), "Quadratic polynomial form of electric arc furnace equation", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 33 No. 6, pp. 2030-2042.



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