Modification of the Clarke Transformation Matrices for Controlling Electric Machines with Non-Equally Spaced Windings

The Clarke transformation is mathematically a Vandermonde matrix of a discrete Fourier transformation. Its purpose is to transform space-vector quantities in the space domain. One necessary condition is that the base function forms an n-th root of unity. This condition translates into requiring all sampling points to be equally spaced. However, manufacturing constraints and imperfections in electric machines may result in non-equally spaced magnetic axes, especially in the case of multiphase electrical machines, making the conventional inverse Clarke transformation used in classic vector control inaccurate. The mathematical inverse Clarke transformation solves this issue, and the existence of this matrix is always ensured. This avoids additional control action and for model predictive controllers, an accurate model is crucial. Analytical results and a simple experimental validation on a multiphase induction machine show the improvements introduced by changing the transformation matrix.