Solution trajectories of the harmonic-elimination problem

The paper considers the problem of finding the notch angles which are required for implementing programmed-harmonic-elimination switching in pulse-width-modulated (PWM) invertors. For a desired fundamental output voltage, the proposed methodology employs the Krawczyk-Moore test within a linear-programming-based search to determine, with mathematical certainty, a set of N-dimensional boxes enclosing all solutions of the harmonic-elimination problem. Each of these boxes contains a distinct and unique solution, and is a safe starting region for a Newton iterative method applied to solve the transcendental harmonic-elimination equations. Once all solutions for a given fundamental amplitude are determined, the solution trajectories for all feasible amplitudes are traced using a simple continuation method which is essentially a numerical adaptation of the implicit-function theorem. The proposed technique is used for optimising line-to-neutral quarter-wave symmetric PWM waveforms employed in three-phase applications. Switching-angle spreads from 0° to 60° and 0° to 90° are considered. Solution trajectories are computed for up to eight switching angles.