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The purpose of this study is to identify a novel methodology for direct calculation of steady-state periodic solutions for electrical circuits described by nonlinear differential equations, in the time domain.

An iterative algorithm was created to determine periodic steady-state solutions for circuits with nonlinear elements in a chosen set of time instants.

This study found a novel differential operator for periodic functions and its application in the steady-state analysis.

This approach can be extended to the determination of two- or multi-periodic solutions of nonlinear dynamic systems.

The complexity of the steady-state analysis can be reduced in comparison with the frequency-domain approach.

This study identified novel difference equations for direct steady-state analysis of nonlinear electrical circuits.

The purpose of this paper is to present a new type of equivalent scheme of multi‐winding transformers.

An inventory representation of relations between currents and flux linkages has been interpreted as a multi‐port purely inductive circuit.

An equivalent scheme in the form of a multi‐port circuit, and a method of its parameters determination from field computations.

Core losses are not considered in the multi‐port equivalent scheme.

A new equivalent scheme could become a basic tool for modeling multi‐winding transformers.

The introduced multi‐port equivalent scheme eliminates disadvantages of classical T‐type equivalent scheme of transformers.

To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions.

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.

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.

Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist.

Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time.

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.

This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models.

Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only.

An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain.

Reduction of software and hardware requirements for determining steady-states of electromagnetic.

A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form.

Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.

This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.

An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.

The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.

This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.

The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.

The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.

An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.

The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.

Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.

New finite-difference expressions for the first- and second-order derivatives have been applied.

The method presented here concerns the analysis of the steady‐state performance of semiconductor switching devices (SSDs) on the assumption that they are represented by resistances whose values vary periodically in time. The method is reduced to the analysis of a system of linear differential equations with periodic coefficients and the method consists of two stages. The first is the calculation of characteristic exponents in order to affirm that the steady state does exist and the second stage is the calculation of the frequency spectrum of currents in the steady state. These calculations are reduced to one of solving the eigen‐problem of an infinite matrix and an infinite system of linear algebraic equations, respectively. Criteria are given which allow one to obtain results with the required accuracy, restricting oneself to finite dimensions. Finally, the method is illustrated by an example which confirms its effectiveness.

This paper presents an approach which transforms the problem of finding the general solutions of linear ordinary differential equation systems with periodic coefficients in eigenvalue and eigenvector problems of an infinite matrix. The problem of determining particular integrals for almost periodic input functions is also presented. This is equivalent to a problem of solving infinite linear algebraic equations. The paper includes an example application of the approach to the analysis of a simple electromechanical system. Results of numerical tests are also given.

The purpose of this paper is to reduce issues arising when computing steady‐state solutions for AC machine models using the harmonic balance method.

Generally, currents at steady‐states of AC machines are described by periodic or quasi‐periodic time functions, which Fourier spectra are determined by an infinite set of algebraic equations obtained from a harmonic balance method. To solve them, after reducing to finite dimensions, an iterative algorithm is developed in this paper. It bases on the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine. Since that decomposition is done separately, due to a band type form of this matrix, the equation set determining the Fourier spectra of currents is solved recurrently.

An algorithm for the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine and an iterative algorithm for determining AC machine steady‐state currents in a recursive manner.

The approach is limited to solving of so‐called “circuital” models of AC voltage supplied machines. The approach breaks the large dimension barrier when solving steady‐state equations for AC machines.

Reducing computer requirements in terms of computer memory, workload and computing time to determine a steady‐state solution for AC machines.

A separation of the LU decomposition of an infinite matrix representing the inductance matrix in AC machine steady‐state model from the solution method.

To study effects in the Fourier spectra of cage motor phase currents due to saturation of the main magnetic circuit by the fundamental MMF harmonic.

An idea of an equivalent magnetizing current is applied, which allows to consider an influence of all currents of stator and rotor windings on the main magnetic circuit permeability. The energy base approach is used to write machine equations and the harmonic balance method is used to determine the Fourier spectra of currents.

It has been shown that the saturation generate additional harmonics in phase currents, which are shifted by 100 Hz from the so called slot harmonics.

A model and a solving method allows to predict all slot harmonics quantitatively, but qualitative difference of the Fourier spectra to measurement still exist.

More precise prediction of the Fourier spectra of stator phase currents for on‐line diagnostic systems.

A circuit model of a cage motor accounting for saturation by slot harmonics and an algorithm for determination of additional components in the phase current Fourier spectra.