# Algorithm for determining two-periodic steady-states in AC machines directly in time domain

Open access

## Abstract

This paper describes an algorithm for finding steady states in AC machines for the cases of their two-periodic nature. The algorithm enables to specify the steady-state solution identified directly in time domain despite of the fact that two-periodic waveforms are not repeated in any finite time interval. The basis for such an algorithm is a discrete differential operator that specifies the temporary values of the derivative of the two-periodic function in the selected set of points on the basis of the values of that function in the same set of points. It allows to develop algebraic equations defining the steady state solution reached in a chosen point set for the nonlinear differential equations describing the AC machines when electrical and mechanical equations should be solved together. That set of those values allows determining the steady state solution at any time instant up to infinity. The algorithm described in this paper is competitive with respect to the one known in literature an approach based on the harmonic balance method operated in frequency domain.

[1] Sobczyk T., Infinitely-dimensional linear and quadratic forms of electric machines, Rozprawy Elektrotechniczne 29: 697-707 (1983).

[2] Sobczyk T., A reinterpretation of the Floquet solution of the ordinary differential equation system with periodic coefficients as a problem of infinite matrix, Compel 5: 1-22 (1986).

[3] Rusek J., Computer analysis of induction machines by using the harmonic balance method, Pub. University of Mining & Metallurgy (AGH), Cracow (2000) (in Polish).

[4] Sobczyk T., Methodological aspects of mathematical modelling of induction machines, WNT, Warsaw (2004) (in Polish).

[5] Sobczyk T., Direct determination of two-periodic solution for nonlinear dynamic systems, Compel 13: 509-529 (1994).

[6] Radzik M., Algorithm for direct determination of steady-states in synchronous machines accounting for the equation of motion, PhD Thesis, Cracow Univ. of Technology, Faculty of Electrical & Computer Eng. (2011) (in Polish).

[7] Radzik M., Sobczyk T.J., Steady-state analysis of synchronous machines loaded by an angle depended torque, Przegląd Elektrotechniczny 89(2b): 158-161 (2013).

[8] Sobczyk T., Direct determination of periodic solution in time domain for differential equations, Proc of Conf. “Selected Issues of Electrical Engineering & Electronics”, Rzeszów-Czarna (2013) (in Polish).

[9] Sobczyk T.J., Radzik M., Direct determination of periodic solution in the time domain for electromechanical converters, Technical Transactions - Electrical Eng., Pub. Cracow University of Technology 112(2-E): 73-82 (2015).

[10] Sobczyk T.J., An algorithm for determining steady-states for AC machines directly in time domain, Electrical Machines - Transaction Journal 3(107): 1-5, 2015 (in Polish).

# Archives of Electrical Engineering

## The Journal of Polish Academy of Sciences

### Journal Information

CiteScore 2016: 0.71

SCImago Journal Rank (SJR) 2016: 0.238
Source Normalized Impact per Paper (SNIP) 2016: 0.535

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 75 75 13