**3. Simulation Results**

According to these space vector diagrams shown in Figure 4, the switching states of the proposed MLI can be classified into four categories.

(1) In case of *Ma* ≥ 1.2, the resultant switching state compromises of 30 various groupings of switching sequences (active vectors) are as follows: {055-054-053-052-051-050-150-250- 350-450-550-540-530-520-510-500-501-502-503-504-505-405-305-205-105-005-015-025-035-045}.

For instance: at *Ma* = 1.3, Figure 5a shows the instantaneous values of *Sa*, *Sb* and *Sc* and switching pulses arrangements within a full cycle of operation. These switching states allow the proposed MLI to produce a three-phase stable staircase 11 level line to a line voltage with mutual variance of *Vdc*, which is as follows: 5*Vdc*, +4*Vdc*, +3*Vdc*, +2*Vdc*, +*Vdc*,

0, −*Vdc*, −2*Vdc*, −3*Vdc*, −4*Vdc*, −5*Vdc*) where *Vab*, *Vbc* and *Vca* are related to *Sa*, *Sb* and *Sc* by:

$$
\begin{bmatrix} Vab \\ Vbc \\ Vca \end{bmatrix} = \frac{5Vdc}{N-1} \times \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} Sa \\ Sb \\ Sc \end{bmatrix} \tag{20}
$$

**Figure 5.** Simulation results: (**a**) switching states at *Ma* = 1.3, (**b**) simulated waveforms at *Ma* = 1.3 (**c**) switching states at *Ma* = 1.15, (**d**) simulated waveforms at *Ma* = 1.15, (**e**) switching states at *Ma* = 0.98, (**f**) simulated waveforms at *Ma* = 0.98, (**g**) switching states at *Ma* = 0.8 and (**h**) simulated waveforms *Ma* = 0.8.

The simulated outputs including the MLI's line to line (*Vab*), line to neutral (*VaN*), line to ground (*Vag*), line to center (*Vao*) and center to ground (*Vog*) voltage outputs are illustrated in Figure 5b.

(2) For 1.2 > *Ma* > 0.98, the maneuver of the proposed MLI is organized inside 30 other combinations of switching sequences (active vectors) as follows: {044 (155)-054-053-052-051- 151 (040)-150-250-350-450-440-(551)-540-530-520-510-511 (400)-501-502-503-504-404 (515)- 405-305-205-105-115 (004)-015-025-035-045}.

The switching sequences of the proposed MLI and its equivalent switching pulses at (for instance: *Ma* = 1.15) are shown in Figure 5c. Reducing the modulation index affects the peak value voltage. Thus, the amount of line to neutral (*VaN*) voltage levels of the MLI, in this case, is decreased to 14 voltage levels, as shown in Figure 5d. These voltage levels are as follows: (+9*Vdc/*3, +8*Vdc/*3, +7*Vdc/*3, +6*Vdc/*3, +4*Vdc/*3, +3*Vdc/*3, +*Vdc/*3, −*Vdc/*3, −3*Vdc/*3, −4*Vdc/*3, −6*Vdc/*3, −7*Vdc/*3, −8*Vdc/*3 and −9*Vdc/*3). Although this switching category has decreased the peak voltage and the line to neutral voltage, the number of line to line voltage (*Vab*) outputs is maintained at 5*Vdc* with 11 voltage levels (+5*Vdc*, +4*Vdc*, +3*Vdc*, +2*Vdc*, +*Vdc*, *0*, −*Vdc*, −2*Vdc*, −3*Vdc*, −4*Vdc*, −5*Vdc*).

From Figure 5d, it can be further observed that utilizing the synthesized switching states, six voltage steps can be generated using multiple combinations of switching sequences. For example, the voltage levels *Vab* = −4*Vdc*, *Vbc* = 0 and *Vca* = +4*Vdc* can be produced applying two valid redundant switching states 044 or (155). Since the switching sequence design should minimize the number of switching instances, the redundant switching states: {(155)-(040)-(551)-(400)-(515)-(004)} are not utilized.

(3) For *Ma* = 0.98, 18 different combinations of switching states (active vectors) are generated to form the new valid switching sequence which is as follows: {044 (155)-053-052- 151 (040)-250-350-440 (551)-530-520-511 (400)-502-503-404 (515)-305-205-115 (004)-025-035}.

Figure 5e depicts the switching sequences of the projected MLI and switching pulses under the condition of *Ma* = 0.98. It can be observed that the MLI's line to line voltage (*Vab*) waveforms reach their highest values of +5*Vdc* compromising nine voltage levels (+5*Vdc*, +4*Vdc*, +3*Vdc*, +2*Vdc*, *0*, −2*Vdc*, −3*Vdc*, −4*Vdc*, −5*Vdc*), while the inverter line to neutral voltages (*VaN*) reach their maximum value of +8*Vdc/*3 with eight voltage steps (+8*Vdc/*3, +7*Vdc/*3, +4*Vdc/*3, +*Vdc/*3, −*Vdc/*3, −4*Vdc/*3, +7*Vdc/*3, −8*Vdc/*3). It is worth mentioning that the operation under the condition of *Ma* = 0.98 makes the common voltage deference in staircase waveform of nine line to line voltage levels fluctuate between *Vdc* and 2*Vdc*. As a result, the MLI functions like a three-phase five-level asymmetrical MLI.

The simulated staircase waveforms of voltages at *Ma* = 0.98 are shown in Figure 5f. Similar to the previous case, the six redundant switching states in this switching sequence: {(155)-(040)-(551)-(400)-(515)-(004)} are also not utilized.

(4) For *Ma* < 0.98, the MLI functions like a traditional three-phase two-level MLI. The novel switching sequences are demonstrated in Table 2 (For instance: at *Ma* = 0.8). The switching sequences tabulated in Table 2 are also applicable for other phases.


**Table 2.** Switching states for the proposed six-level inverter (*Ma* < 0.98).

In this case, the MLI's switching pulses are generated utilizing the switching sequence shown in Figure 5g. It is identified that the final switching state comprises of six groupings of active vectors: {011-010-110-100-101-001}. Following this switching state leads the projected MLI to obtain the desired output voltage of *Vab*, *VaN* and *Vag* as depicted in Figure 5h.

The extended configuration is also verified by simulating a model for (*n* = 2). The simulation is done utilizing 10 V voltage supply and a three-phase resistive-inductive load

of 237 Ω–0.53 H as output. The nearest vector modulation technique is implemented with a nominal frequency of 50 Hz. The simulation results of fifteen-level line to line output voltage (*Vab*) built on the extended model and its equivalent switching pulses are illustrated in Figure 6a. In addition, the output voltages of *VaN*, *Vao*, *Vog* and *Vag* are also shown in Figure 6b.

**Figure 6.** For (*n* = 2, *N* = 15, *f* = 50 Hz) (**a**) Switching gate signals and *Vab*, *Vbc* and *Vca*, (**b**) Simulated waveforms of *VaN*, *Vao*, *Vog* and *Vag*.
