# Plasma and Fusion Research

## Volume 16, 2405037 (2021)

# Regular Articles

- Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasugakoen, Kasuga 816-8580, Japan
- 1)
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasugakoen, Kasuga 816-8580, Japan
- 2)
- Institute for Advanced Fusion & Physics Education, 2-14-8 Tokuou, Kita-ku, Kumamoto 861-5525, Japan

### Abstract

In a three-phase matrix converter based on space-vector modulation (SVM), nine switches are controlled so that the instantaneous space vector of the line-to-line voltage rotates smoothly in two-dimensional space. The quaternion is a four-dimensional hypercomplex number that is good at describing three-dimensional rotation, such as that seen in three-dimensional game graphics programming theory. Utilizing the quaternion capability, we analyze a matrix converter by three-dimensional rotation instead of transforming to two-dimensional rotation in alpha-beta coordinates. It was clarified that the projection of the quaternion locus in three-dimensional space in the (1,1,1) direction is the same as an alpha-beta transformation locus in two-dimensional space. Concerning the direct matrix converter, we clarified that the (1,1,1)-directional superposition of three-fold higher harmonics cannot be eliminated. The quaternion can rotate and divide a three-dimensional vector. When the output voltage quaternion is divided by input one, the switching quaternion is obtained. The quaternion characteristics will be utilized to analyze a matrix converter based on direct SVM in more detail.

### Keywords

circulant matrix, quaternion, direct matrix converter, space vector modulation, three-phase to three-phase

### Full Text

### References

- [1] L. Huber and D. Borojevic, IEEE Trans. Ind. Electron. 31, No. 6, 1234 (1995).
- [2] D. Casadei, A. Tani and L. Zarri, IEEE Trans. Ind. Electron. 49, No. 2, 370 (2002).
- [3] J.H. Conway and D. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry (A.K. Perters, Ltd., 2003).
- [4] K. Nakamura, I. Jamil, X.L. Liu, O. Mitarai, M. Hasegawa, K. Tokunaga, K. Araki, H. Zushi, K. Hanada, A. Fujisawa, H. Idei, Y. Nagashima, S. Kawasaki, H. Nakashima and A. Higashijima, Quaternion Analysis of Three-Phase Power Electronic Circuit by Using Conjugation, International Conference on Electrical Engineering, ICEE 2015, 15A-476 (2015).
- [5] K. Nakamura, M. Hasegawa, K. Tokunaga, K. Araki, I. Jamil, X.L. Liu, O. Mitarai, H. Zushi, K. Hanada, A. Fujisawa, H. Idei, Y. Nagashima, S. Kawasaki, H. Nakashima, A. Higashijima and T. Nagata, Quaternion Analysis of Three-Phase Matrix Converter Switching Method, International Conference on Electrical Engineering, ICEE 2016, D2-4-90432 (2016).
- [6] P.W.Wheeler, J. Rodriguez, J. Clare, L. Empringham and A. Weinstein, IEEE Trans. Ind. Electron. 49, No.2, 274 (2002).