Minimization of Gibb's oscillations in transients' simulations using damping resistance

Considering simulations of electromagnetic transients and searching for better results, it is investigated the limits of the change of the representation of the longitudinal parameters in the model of infinitesimal unit of transmission lines' model. It is introduced a damping resistance in parallel to the impedance that represents the longitudinal parameters of transmission lines. Several simulations are made, analyzing how the best relation between the damping resistance and the numeric of infinitesimal units' model for minimizing the peak of numeric oscillations or Gibb's oscillations generated by the trapezoidal rule. The trapezoidal rule is used for solving the linear system applied to the electromagnetic transient simulations in transmission lines. Numerical methods are used to solve ordinary differential equations. Ordinary differential equations (ODE) are equations with one or more differential relations of an unknown function, and all are considered to be derived in respect of a single independent variable. Modeling linear systems of first order or linearized at a point of interest, Heun's method is applied to simulations of electromagnetic transients in transmission lines. The analysis investigates the introduction of a damping resistance in a simple model applied to obtain the mentioned simulations. So, the damping resistance is introduced in the π circuit that represents the infinitesimal unit of transmission lines. Using a cascade with a great number of these infinitesimal units that is solving using Heun's method, Gibb's oscillations leads to undesirable voltage peaks of 25% overvalue without the application of the damping resistance. If this resistance is used, Gibb's oscillations become negligible. So, several simulations are carried out applying different values of the damping resistance and the π circuits' number for the mentioned model. Considering the damping resistance, the π circuits' number and overvalue of the first peak of voltage at the transmission line end, three dimension graphic is used for determining the best values of the damping resistance and the π circuits' number for decreasing the voltage peaks caused by Gibb's oscillations. In Fig. 1, the structures of π circuits without the damping resistance (Fig. 1(a)) and with the damping resistance (Fig. 1(b)) are shown.