*2.1. Proposed Topology*

The circuit diagram of the proposed three-phase six-level MLI topology is depicted in Figure 1. Six unidirectional switches formed by (S1~S6) and (Da1~Dc2) diodes are connected with a classical three-phase six-switch (Q1~Q6) to build the main bridge. On the other hand, a single DC voltage supply, a half-bridge cell consisting of two switches (Ta1, Ta2) and a single DC supply along with a full-bridge power cell comprising four switches (Tb1~Tb4) and a DC supply are connected together to form a multilevel DC-link voltage. The DC voltages for half-bridge and full-bridge power cells are 3Vdc and Vdc, respectively, while the DC voltage for a single supply is 2Vdc. The power cell switches are turned on/off to synthesize four consecutive voltage levels: 4Vdc, 3Vdc, 2Vdc and Vdc at the middle point (o) relating to the inverter's ground. Table 1 illustrates the switching operation of the proposed inverter.

**Figure 1.** Proposed structure of a three-phase six-level inverter.



It is worth mentioning that the switching states demonstrated in Table 1 are applicable for all three phases. For phase b and c, it should be Q3, Q4, S3, S4 and Q5, Q6, S5, S6, respectively instead of Q1, Q2, S1, S2. The switching configuration of the proposed MLI is given in the following equation:

$$
\begin{bmatrix} Vag\\Vbg\\Vcg \end{bmatrix} = \frac{5Vdc}{N-1} \times \begin{bmatrix} Sa\\Sb\\Sc \end{bmatrix} \tag{1}
$$

where *Sa*, *Sb* and *Sc* are switching positions. *Vag*, *Vbg* and *Vcg* are the MLI's line to ground voltages and *N* = 6 is the number of voltage levels. Since the topology employed for the proposed MLI is modeled to obtain the three-phase well-adjusted line to line output voltages producing the highest number of voltage levels, a suitable switching structure is needed for generating the MLI's gate pulses. Consecutively, these switching states generate the gate pulses and drive the studied MLI under all modulation indexes. Different modulation techniques have been suggested to control the multilevel inverters. These modulation methods are categorized depending on the operating switching frequency [25]. The low frequency modulation method known as selective harmonic elimination (SHE-PWM) is usually employed to predefine and pre-calculate the optimal switching angles. To obtain a solution for the angles, a set of equations is normally solved offline using numerical methods [26,27]. The proposed MLI can be easily controlled, and the desired outputs can be simply achieved with regard to the MLI's switching states *Sa*, *Sb* and *Sc*. Therefore, a new modulation method was recently suggested in [28] where each phase in the proposed inverter is controlled independently. For a given modulation index *Ma*, the inverter's switching states *Sa*, *Sb* and *Sc* are determined with respect to the MLI's line to ground reference voltages *Vag\_ref*, *Vbg\_ref* and *Vcg\_ref*. The correlation among the MLI's reference line to ground voltages and the respective switching sequences is:

$$
\begin{bmatrix} Sa \\ Sb \\ Sc \end{bmatrix} = \left( \frac{N - 1}{5Vdc} \times \begin{bmatrix} Vag\\_ref \\ Vbg\\_ref \\ Vcg\\_ref \end{bmatrix} \right) \tag{2}
$$

$$
\begin{bmatrix}
\text{Vag\\_ref} \\
\text{Vbg\\_ref} \\
\text{Vcg\\_ref}
\end{bmatrix} = \frac{Ma \times 5Vdc}{2} \times \begin{bmatrix}
\cos(\omega t) \\
\cos(\omega t - \frac{2\pi}{3}) \\
\cos(\omega t + \frac{2\pi}{3})
\end{bmatrix} + \frac{5Vdc}{2} \times \left[1 - \frac{Ma}{6}\cos(3\omega t)\right] \times \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.
\tag{3}
$$

 

According to (3), the 3rd harmonic element is added to MLI's line to ground reference voltage waveforms. Therefore, *Ma* can extend to 1.15 without instigating over modulation.
