Analytical modelling of single-phase stacked multicell multilevel converters exploiting Kapteyn (Fourier–Bessel) series

This study proposes a mathematical model for stacked multicell (SM) converters (SMCs), to be exploited in the analytic determination of the natural voltage balancing dynamics of the SMCs, that is, investigation of the start-up behaviour, dynamic response and natural voltage balancing phenomenon. The crux of the proposed strategy is based on the closed-form analytic solution derivation for the switching functions used in the switching of the SMCs operated under phase disposition (PD) and phase-shifted carrier (PSC) pulse-width modulation (PD-PSC-PWM) technique. Hence, the suggested approach develops an analytic solution for the Fourier series and associated Fourier coefficients pertinent to the switching functions of the SMCs by obtaining the switching instants of the PD-PSC-PWM modulator in terms of `Kapteyn series´ when the frequency of the triangular carrier waveform (f<sub>c</sub>) and that of the sinusoidal reference waveform (f<sub>r</sub>) have an integer ratio, that is, f<sub>c</sub>f<sub>r</sub><sup>-1</sup> = k,k∈N. This strategy results into a model (`first-order differential equation based model´) which can be readily developed for the SMCs with any number of levels expediting the investigation of their performance. Numeric computation results of the proposed analytic model for the SMCs and simulation results as well as measurements taken from an experimental set-up are presented in order to validate the suggested approach and derived model.