Frequency spectra based approach to analytical formulation and minimization of voltage THD in staircase modulated multilevel inverters

Eli Barbie, Dmitry Baimel, Alon Kuperman

Research output: Contribution to journalArticlepeer-review


Staircase Modulation (SCM) is a popular switching strategy for multilevel inverters (MLI), which is often a preferable alternative to Pulse Width Modulation (PWM) due to its reduced switching losses, especially for MLIs with a relatively high number of voltage levels (N). In some applications, such as motor drives, or when output filters are applicable, SCM may also suit MLIs with moderate N values. While MLIs with Equal DC Sources (EDCS) are more common, MLIs with Unequal DC Sources (UDCS), such as asymmetric Cascaded H-Bridge (CHB) MLIs are known to further reduce the waveforms’ Total Harmonic Distortion (THD). The main objective of SCM is to adjust the waveform's fundamental component, while employing some harmonic mitigation strategy, such as THD minimization. Such an optimization approach relies on an accurate THD formulation, which eliminates underestimation errors associated with numerical THD approximations. This paper reveals novel generic analytical formulations of both Phase-voltage THD (PTHD) for single-phase MLIs and Line-voltage THD (LTHD) for three-phase MLIs. The revealed exact formulation approach is derived from Fourier series representation of THD, resulting in simple closed-form PTHD and LTHD expressions, applicable to any MLI topology, with either EDCS or UDCS configurations, and arbitrary value of N (odd and even). Given an arbitrary N value, the proposed formulations can be used to generate symbolic N-level PTHD or LTHD expressions, which are symbolic functions of the Phase Switching Angles (PSA) and (optionally) the DC source Ratios (DCR). The revealed THD formulations are verified and compared against recently introduced time-domain-based THD formulations. It is shown that when both time and frequency-based formulations are employed in THD minimizations to form a novel Hybrid Temporal-Spectral (HTS) Optimal Minimization of THD (OMTHD) approach, even better results can be achieved. The novel OMTHD is deeply explored and verified by both digital simulations and Controller + Hardware in Loop (C-HIL) based real-time experiments using 7- and 8-level three-phase MLI configurations. A downloadable supplemental file containing Maple and MATLAB functions of the proposed THD expressions, as well as pre-calculated sets of optimum variables for different values of N is provided for readers' convenience.

Original languageAmerican English
Pages (from-to)7781-7809
Number of pages29
JournalAlexandria Engineering Journal
Issue number10
StatePublished - 1 Oct 2022


  • Line-voltage
  • Multilevel inverters
  • Optimization
  • Phase-voltage
  • Spectral
  • Staircase modulation
  • THD
  • Temporal

All Science Journal Classification (ASJC) codes

  • General Engineering


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