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.

Purpose To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions. Design/methodology/approach The development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions. Findings Novel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic. Research limitations/implications Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist. Practical implications Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time. Originality/value Identify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

Purpose This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models. Design/methodology/approach Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only. Findings An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain. Research limitations/implications Reduction of software and hardware requirements for determining steady-states of electromagnetic. Practical implications A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form. Originality/value Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

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