Analytical Formulas for Phase Voltage RMS Squared and THD in PWM Multiphase Systems

The analysis and assessment of the pulsewidth modulation (PWM) techniques is commonly based on the comparison of the total harmonic distortion (THD) results. THD is usually calculated by application of the Fourier transformation and by taking a limited number of harmonics into the consideration. In this paper, derivation of analytical formulas for the phase voltage THD is presented. The considered system is a symmetrical multiphase star-connected load, supplied from a multilevel pulsewidth-modulated voltage-source inverter (VSI, three-phase case is also covered). The solution is based on the Parseval´s theorem, which links frequency spectrum and time domain through the average power (i.e., root-mean-square (rms) squared value) of the signal. The assumption throughout the derivations is that the ratio of the switching to fundamental frequency is high. Derivations are based on the integration of the power of the PWM signal in a single switching period over the fundamental period of the signal. Only ideal sinusoidal reference voltages are analyzed, and no injection of any type is considered. Formulas for phase voltage THD for any number of phases are derived for two- and three-level cases, for the most commonly used carrier-based methods. Comparison of the analytically obtained curves with simulation and experimental results shows a high level of agreement and validates the analysis and derivations.