# Analysis of Symmetric and Periodic Open Boundary Problem by Coupling of FEM and Fourier Series

• Kim, Young Sun (Department of Electrical and Electronic Engineering, Joongbu University)
• Accepted : 2013.04.11
• Published : 2013.06.30
• 38 63

#### Abstract

Most electrical machines like motor, generator and transformer are symmetric in terms of magnetic field distribution and mechanical structure. In order to analyze these problems effectively, many coupling techniques have been introduced. This paper deals with a coupling scheme for open boundary problem of symmetric and periodic structure. It couples an analytical solution of Fourier series expansion with the standard finite element method. The analytical solution is derived for the magnetic field in the outside of the boundary, and the finite element method is for the magnetic field in the inside with source current and magnetic materials. The main advantage of the proposed method is that it retains sparsity and symmetry of system matrix like the standard FEM and it can also be easily applied to symmetric and periodic problems. Also, unknowns of finite elements at the boundary are coupled with Fourier series coefficients. The boundary conditions are used to derive a coupled system equation expressed in matrix form. The proposed algorithm is validated using a test model of a bush bar for the power supply. And the each result is compared with analytical solution respectively.

#### Keywords

analytical solution;finite element method;Fourier series expansion;open boundary;symmetric and periodic structure

#### References

1. Onuki, T., Im Jee-won, and Kuwahara, T., IEEE Trans. Magn. 81, 4076 (1997).
2. I. Kwak, S. Lee, D. Lee, and I. Park, IEEE Trans. Appl. Supercond. 14, 1870 (2004). https://doi.org/10.1109/TASC.2004.830886
3. J. X. Guo, and P. P. Biringer, IEEE Trans. Magn. 27, 4097 (1991). https://doi.org/10.1109/20.105002
4. T. Nakata, N. Takahashi, K. Fujiwara, and M. Sakaguchi, IEEE Trans. Magn. 26, 368 (1990). https://doi.org/10.1109/20.106330
5. M. Chari, and G. Bedrosian, IEEE Trans. Magn. 23, 3572 (1987). https://doi.org/10.1109/TMAG.1987.1065766
6. Z. Luo, and N. A. Demerdash, IEEE Trans. Magn. 28, 2241 (1992). https://doi.org/10.1109/20.179456
7. U. Kim, J. Yang, K. Lee, and D. Yoo, The Trans. KIEE 41, 362 (1992).
8. S. J. Salon, and J. D'Angelo, IEEE Trans. Magn. 24, 80 (1988). https://doi.org/10.1109/20.43861
9. Sheppard J. Salon, Finite element analysis of electrical machines, Kluwer Academic Publishers (1995).
10. Y. S. Kim, H. S. Choi, I. H. Park, and K. S. Lee, IEEE Trans. Magn. 44, 1290 (2008). https://doi.org/10.1109/TMAG.2007.915990

#### Cited by

1. A Type of Subsection Model for a Permanent Magnet Bar and its Leakage Permeance Calculation Method in an Open Magnetic Circuit vol.19, pp.1, 2014, https://doi.org/10.4283/JMAG.2014.19.1.037