A new approach to steady state analysis of nonlinear electrical circuits

Abstract

The paper presents a new method of determining the steady-state of electrical circuits with nonlinear elements whereas periodic solution can be predicted. This method allows for calculating steady-state wave-forms directly in time domain. A new discrete differential operator has been defined. It reduces nonlinear differential equations of a circuit to a set of algebraic equations for the values of the steady state solution at discrete time instants. Based on it an algorithm for solving nonlinear differential equations has been proposed. Numerical tests have been performed for elementary electrical circuits with a nonlinear coil and a power electronic switching element.

Direct steady-state solutions for circuit models of nonlinear electromagnetic devices

“…According to Sobczyk and Radzik (2017), the first-order DDOs are applied to determine the steady-state solution of equation (1) directly, rewritten in the form: if its solution can be forecast as a periodic function approximated by the Fourier series with a limited number of terms, (2 R + 1), where R is the number of highest harmonics. Thus, we obtain: …”

“…Steady-state solutions are the most interesting ones for engineering applications, and direct steady-state determination for electromagnetic devices remains as a research topic. The nonlinearities embedded in their mathematical models make the time-domain approach (Hara et al , 1985; Biro and Preis, 2006; Garcia, 2010); Lelarasmee et al , 1982; Caron et al , 2016; Sobczyk and Radzik, 2017) preferable over the frequency-domain approach (Yamada et al , 1989; Semlyen et al , 1991; Sobczyk, 1994, 2004; Gyselinck et al , 2006; Deblecker and Lobry, 2006).…”

“…The idea of an alternative approach for determining a periodic steady-state solution was presented by Sobczyk and Radzik (2017, 2019) using the so-called first-order discrete differential operator (DDO) developed by Sobczyk et al (2019). This operator relates the first-derivative values of a periodic function to values of that function in a set of time instants that are uniformly distributed over the period.…”

“…Both algorithms address nonlinear ordinary differential equations written in the normal form. The first one, published by Sobczyk and Radzik (2017), engages both DDO and the right-hand side nonlinear equations to create nonlinear FDEs, which must be solved numerically. The second algorithm, presented by Sobczyk and Radzik (2019), creates FDEs based on DDO only.…”

“…As a result, the algorithm applying DIO reduces the problem of finding the steady-state solution to the multiplication of matrices only, instead of solving the FDEs, thereby significantly reducing the computational complexity. The proposed algorithm was tested for steady-state analysis of an elementary circuit with a nonlinear coil, as presented by Sobczyk and Radzik (2017) and Sobczyk and Radzik( 2019). The comparison results show the advantages of the algorithm based on DIO.…”