Theoretical Harmonic Spectra of PWM Waveforms Including DC Bus Voltage Ripple—Application to a Low-Capacitance Modular Multilevel Converter

This article develops a new closed-form analytical solution to the harmonic spectrum of the pulsewidth modulated (PWM) output voltage of the single-phase inverter connected to the dc bus with considerable voltage ripple. The solution is based on a double Fourier series expansion in two variables. As the single-phase inverter forms the basic building block of most converters, the developed expressions can be applied to various topologies. This article first identifies the interactions of a single-frequency sinusoidal modulation signal and the carrier signal with the dc bus voltage harmonics. The dc bus voltage harmonics are constrained in this article to multiples of the inverter output voltage fundamental frequency. The analysis is then extended to include a multifrequency modulation signal. Starting from the general solution, new expressions for the output voltage harmonics of the modular multilevel converter (MMC) are developed. The low-capacitance MMC is chosen in this analysis due to inherent low-frequency voltage oscillations in the converter internal capacitors. The MMC analytical harmonic spectrum can incorporate the effects of the circulating current control and third harmonic injection PWM, by including the second- and third-order harmonic components in the modulation signal, respectively. The MMC analytical harmonic spectrum is benchmarked against the experimental results.