Complete Fast Analytical Solution of the Optimal Odd Single-Phase Multilevel Problem

In this paper, we focus on the computation of optimal switching angles for general multilevel (ML) odd symmetry waveforms. We show that this problem is similar to (but more general than) the optimal pulsewidth modulation (PWM) problem, which is an established method of generating PWM waveforms with low baseband distortion. We introduce a new general modulation strategy for ML inverters, which takes an analytic form and is very fast, with a complexity of only ${\cal O}(n\log^{2}n)$ arithmetic operations, where $n$ is the number of controlled harmonics. This algorithm is based on a transformation of appropriate trigonometric equations for each controlled harmonics to a polynomial system of equations that is further transformed to a special system of composite sum of powers. The solution of this system is carried out by a modification of the Newton's identity via Padé approximation, formal orthogonal polynomials (FOPs) theory, and properties of symmetric polynomials. Finally, the optimal switching sequence is obtained by computing zeros of two FOP polynomials in one variable or, alternatively, by a special recurrence formula and eigenvalues computation.