A Unified Analytical Solution for Bipolar, Unipolar, and Multistep SHE Converters

In this paper, a unified analytical approach is presented for solving a set of transcendental equations commonly encountered in Selective Harmonic Elimination (SHE). Traditionally numerical solutions are used that require an initial value and a valid value of modulation index m. Not only the initial value but also an invalid m will affect convergence. Such difficulties are not associated with analytical solution. The solution set will determine the range of m valid for that modulation. The analytical approach is also suitable for real-time applications where a microcontroller/microprocessor can be programmed for the application making it attractive for low cost application. In the proposed approach, the transcendental equations are converted to power-sum non-linear polynomials using Chebyshev expansion. These non-linear polynomials have been simplified using a successive polynomial extraction method and lead to a solution set for elementary functions. These have been formulated for the three types of modulation for a 4
^th
order system with contiguous harmonic elimination that eliminates up to ninth harmonic. The solution trajectories as a function of m have been presented and valid range of m values has been determined. Simulation results verify the harmonic cancellations.