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Abstract

This paper presents novel discrete differential operators for periodic functions of one- and two-variables, which relate the values of the derivatives to the values of the function itself for a set of arbitrarily chosen points over the function’s area. It is very characteristic, that the values of the derivatives at each point depend on the function values at all points in that area. Such operators allow one to easily create finite-difference equations for boundaryvalue problems. The operators are addressed especially to nonlinear differential equations.
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Bibliography

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[9] Demenko A., Sykulski J., On the equivalence of finite difference and edge element formulations in magnetic field analysis using vector potential, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 33, no. 1/2, pp. 47–55 (2014).
[10] Huang J., LiaoW., Li Z., A multi-block finite difference method for seismic wave equation in auxiliary coordinate system with irregular fluid–solid interface, Engineering Computations, vol. 35, no. 1, pp. 334–362 (2018).
[11] Chapwanya M., Dozva R., Gift Muchatibaya G., A nonstandard finite difference technique for singular Lane-Emden type equations, Engineering Computations, vol. 36, no. 5, pp. 1566–1578 (2019).
[12] Mawlood M., Basri S., AsrarW., Omar A., Mokhtar A., Ahmad M., Solution of Navier-Stokes equations by fourth-order compact schemes and AUSM flux splitting, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 16, no. 1, pp. 107–120 (2006).
[13] Ivanovic M., Svicevic M., Savovic S., Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 27, no. 12, pp. 2682–2695 (2017).
[14] Sobczyk T.J., Algorithm for determining two-periodic steady-states in AC machines directly in time domain, Archives of Electrical Engineering, Polish Academy of Science, Electrical Engineering Committee, vol. 65, no. 3, pp. 575–583 (2016), DOI: 10.1515/aee-2016-0041.
[15] Sobczyk T.J., Radzik M., Radwan-Pragłowska N., Discrete differential operators for periodic and two-periodic time functions, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 38, no. 1, pp. 325–347 (2019).
[16] Sobczyk T.J., Radzik M., A new approach to steady state analysis of nonlinear electrical circuits, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 37, no. 3, pp. 716–728 (2017).
[17] Sobczyk T.J., Radzik M., Tulicki J., Direct steady-state solutions for circuit models of nonlinear electromagnetic devices, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 40, no. 3, pp. 660–675 (2021), DOI: 10.1108/COMPEL-10-2020-0324.
[18] Sobczyk T.J., Jaraczewski M., Application of discrete differential operators of periodic functions to solve 1D boundary-value problems, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 39, no. 4, pp. 885–897 (2020).
[19] Sobczyk T.J., 2D discrete operators for periodic functions, Proceedings IEEE Conference Selected Issues of Electrical Engineering and Electronics (WZZE), Zakopane, Poland, pp. 1–5 (2019), https://ieeexplore.ieee.org/document/8979992.
[20] Jaraczewski M., Sobczyk T., Leakage Inductances of Transformers at Arbitrarily Located Windings, Energies, vol. 13, no. 23, 6464 (2020), DOI: 10.3390/en13236464.

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Authors and Affiliations

Tadeusz Jan Sobczyk
1
ORCID: ORCID

  1. Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Cracow University of Technology, 24 Warszawska str., 31-155 Kraków, Poland
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Abstract

This study presents a method to directly calculate the stator current Fourier spectra in double-cage induction motors to diagnose faults in rotor cages. A circuit model is developed for this purpose, allowing the modelling of any asymmetry in the outer and inner rotor cages. The model extends the conventional model of a cage motor by considering the higher space harmonics generated by the stator windings. The asymmetry of the cages is modelled by growing the resistance of any of the rotor bars. This results in various model equations, to be solved by looking for diagnostic signals. Motor current signature analysis is typically used to diagnose cage motors based on the Fourier spectra of the stator currents during steady-state operation. This study determines these spectra for double cage motors using the harmonic balance method, omitting the transient calculations. The calculation results confirmed the sensitivity of the stator current Fourier spectra as a diagnostic signal to distinguish faults in the outer and inner cages.
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Authors and Affiliations

Jarosław Tulicki
1
ORCID: ORCID
Tadeusz Jan Sobczyk
1
ORCID: ORCID
Maciej Sułowicz
1
ORCID: ORCID

  1. Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Cracow University of Technology, 24 Warszawska str., 31-155 Kraków, Poland

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