A closed-form approach for contiguous and non-contiguous harmonic elimination with application to a three-level switching waveform

In this paper, an analytical approach is proposed for solving a special set of transcendental equations, which are often encountered for eliminating harmonics in converters. Typically, numerical solutions are used for solving the transcendental equations. The disadvantage of numerical approach is overhead in computational effort as it requires a PC with expensive software and solution time can also be a problem in real-time operations. In the proposed approach, the transcendental equations are converted to power-sum non-linear polynomials using Chebyshev expansion. Their solutions through Girard-Waring equations for higher order power-sum polynomials are prohibitive to find closed-form solutions. Here, the non-linear polynomials have been reformulated using an extraction method that simplifies them and their solution leads to a polynomial in terms of elementary functions. Its solution provides the desired switching angles of the waveform. As an application of the proposed approach, the complete solution for unipolar switching pattern is given, eliminating up to ninth harmonic in single-phase and up to thirteenth harmonic in three-phase while controlling the fundamental independently. Contrary to two solutions in existing literature, three solutions have been found for the triplen system with four switching angles.