Improvement of the finite element method equations conditioning for the magnetic field-circuital problems

• Autorzy: Gołębiowski M.

Identyfikatory

DOI

10.1515/aee-2017-0024

• Języki publikacji: EN

Abstrakty

EN

The presented systems with magnetically coupled windings are solved with the finite element method. If the issue of voltage supply is analyzed, a system of linear equations with a partially skew-symmetric sparse matrix is obtained. Iterative methods used to solve a system of equations are particularly effective for symmetric matrices. Resultant equations can be reduced to this symmetrical form by using the method known from the literature [1]. The ratio of the maximum to the minimum eigenvalue of the main matrix of this circuit, which is the condition number, is however very high. This means that the problem is ill-conditioned and leads to a very long iterative solution process. The method presented in the article allows for a direct solution of a system of equations on its part, corresponding to high eigenvalues of the system matrix. The remaining part of the system of equations is solved by iterative methods. This part has much better condition number, and therefore the computational process is fast. The proposed iterative process depends on multiplication of a sparse matrix by vectors. It is not necessary (and possible) to store the entire matrix. This is especially important for larger sizes of a matrix.

Słowa kluczowe

EN

magnetic field-circuital problem; condition number; iterative method

• Wydawca: Polish Academy of Sciences, Committee on Electrical Engineering
• Czasopismo: Archives of Electrical Engineering
• Rocznik: 2017
• Tom: Vol. 66, nr 2
• Strony: 325--337
• Opis fizyczny: Bibliogr. 14 poz., rys., tab., wz.

Twórcy

autor

Gołębiowski M.

• Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, Wincentego Pola 2, 35-959 Rzeszów, Poland

Bibliografia

• [1] Demenko A., Wojciechowski R., Sykulski J., 2-D versus 3-D electromagnetic field modeling in electromechanical energy converters, IEEE Transactions on Magnetics, vol. 50, no. 2, pp. 897-900 (2014).
• [2] Saad Y., Iterative Methods for Sparse Linear Systems, Second Edition, Society for Industrial and Applied Mathematics (2003).
• [3] Traub J.F., Woźniakowski H., On the Optimal Solution of Large Linear Systems, Department of Computer Science Columbia University (1982).
• [4] Higham N.J., Accuracy and Stability of Numerical Algorithms, Second Edition, Society for Industrial and Applied Mathematics (2002).
• [5] Saad Y., Numerical Methods for Large Eigenvalues Problems, Second Edition, Society for Industrial and Applied Mathematics (2011).
• [6] Gołębiowski M., The effect of the leakage inductance auto-transformers, which supply a multipulse rectifier systems on the network current distortion factor (dissertation in Polish), PhD Thesis, The Faculty of Electrical and Computer Engineering, Rzeszów University of Technology, Rzeszów (2010).
• [7] Gołębiowski M., Mazur D., Calculating the main and leakage inductance matrix of the 3-column 15-winding autotransformer, Archives of Electrical Engineering, vol. 60, no. 3, pp. 215-222 (2011).
• [8] Gołębiowski M., Mazur D., Measurement and calculation of 3-column 15-winding autotransformer, Archives of Electrical Engineering, vol. 60, no. 3, pp. 223-230 (2011).
• [9] Jin J.M., The Finite Element Method in Electromagnetics, 3rd Edition, Wiley-IEEE Press (2014).
• [10] Tomczuk B., Koteras D., Waindok A., 3D Field Calculations of the Modular Transformer Heating under High Frequency Operation, Proceedings of the XIX International Conference on the Computation of Electromagnetic Fields, Budapest, Hungary (2013).
• [11] Gołębiowski M., Gołębiowski L., Mazur D., Voltages in the Shaft of the Induction Motor in 3D FEM Formulation, IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, Valencia, Spain, pp. 142-145 (2007).
• [12] Choi S.C.T., Minimal Residual Methods for Complex Symmetric, Skew Symmetric and Skew Hermitian Systems, arXiv: 1304.6782v2 [cs.MS] (2014).
• [13] Nowak L., Field Models of Transients in Electromechanical Converters (in Polish), Poznan University of Technology Publishing House (1999).
• [14] Berhausen S., Paszek S., Use of the finite element method for parameter estimation of the circuit model of a high power synchronous generator, Bulletin of the Polish Academy of Science, vol. 63, no. 3, pp. 575-582 (2015).

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).