*2.2. Proposed Space Vector PWM Techniques*

The circuit theory states that a three-phase system can be represented on a *α*-*β* plane by means of the Clarke transformation, as shown in (3)–(5). Note that a simplification using cosine functions for the three-phase voltage components (*va*, *v<sup>b</sup>* , and *vc*) is considered.

$$
\begin{bmatrix} v\_{\alpha} \\ v\_{\beta} \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} v\_{a} \\ v\_{b} \\ v\_{c} \end{bmatrix} \tag{3}
$$

$$v\_{\mathfrak{A}} = V\_{\mathfrak{m}} \cos(\omega t), \tag{4}$$

$$
v\_{\beta} = V\_{m} \sin(\omega t).\tag{5}$$

Based on (4) and (5), the module and the angle of the reference vector, *Vre f* , can be calculated as

$$|V\_{ref}| = \sqrt{v\_a^2 + v\_{\beta}^2} \,\,\,\,\tag{6}$$

$$\tan(\theta) = \frac{v\_{\beta}}{v\_{a}}.\tag{7}$$

Then, substituting (4) and (5) in (7), the module of the reference vector can be redefined as

$$|V\_{ref}| = \sqrt{(V\_m \cos(\omega t))^2 + (V\_m \sin(\omega t))^2}$$

$$|V\_{ref}| = V\_{m\nu} \tag{8}$$

and finally, from (7), the angle *θ* can be calculated as follows:

$$\theta = \tan^{-1} \left( \frac{v\_{\beta}}{v\_{a}} \right). \tag{9}$$

From the above analysis, the eight states of the DCM-232 inverter can be represented on the *α*-*β* plane, as shown in Figure 3. Considering this representation, it should be noted that this is similar to that of the space vector representation for a 3P-FB inverter; however, in this particular case, the zero vectors imply the decoupling of the DC sources performed by switches *S*7*<sup>a</sup>* , *S*7*<sup>b</sup>* , *S*8*<sup>a</sup>* , and *S*8*<sup>b</sup>* .

**Figure 3.** Space vector representation of the DCM-232 states.

Considering the polar representation, the eight vectors in the complex plane can be written as

$$V\_n = \begin{cases} \frac{2}{3} V\_{\rm DC} e^{j\frac{\pi}{3}(n-1)} & n = 1, \dots, 6\\ 0 & n = 0, 7 \end{cases} \tag{10}$$

For a balanced three-phase system, *Vre f* can be expressed as

$$V\_{\nu\varepsilon f} = V e^{j\omega t}.\tag{11}$$

In order to synthesize *Vre f* , three successive space vectors can be applied along a switching period (*T<sup>s</sup>* = <sup>1</sup> *fs* ). Therefore, the addition of the applied vectors (active and/or null) must satisfy

$$V\_a t\_a + V\_b t\_b + V\_N t\_0 = V\_{ref} T\_{s\nu} \tag{12}$$

Notice that the switching period is the sum of the times of each applied vector:

$$t\_a + t\_b + t\_0 = T\_\text{s.} \tag{13}$$

To determine the duty cycles for each applied vector, the complex components (*V<sup>A</sup>* and *VB*) in the *α*-*β* plane for *Vre f* can be defined as

$$V\_{r\!\!f} = \begin{cases} \ V\_A = \frac{1}{T\_s} V\_a t\_a \\\ V\_B = \frac{1}{T\_s} V\_b t\_b. \end{cases} \tag{14}$$

The complex components *V<sup>A</sup>* and *V<sup>B</sup>* defined in (14) can be represented in Sector 1, as shown in Figure 4. As can be observed, *V<sup>a</sup>* = *V*<sup>1</sup> and *V<sup>b</sup>* = *V*2. Therefore, analyzing and performing the projections of *Vre f* over the *α* and *β* axis yields

$$|V\_A|\angle 0^\circ \, \_\prime$$

$$|V\_B|\angle 60^\circ \, \_\prime$$

$$|V\_{ref}| = V\_{m\prime}$$

$$|V\_{ref}|\cos(\theta) = |V\_A| + |V\_B|\sin(60^\circ) \, \_\prime \tag{15}$$

$$|V\_{ref}| = |V\_A| + |V\_B| \sin(60^\circ) \, \_\prime \tag{16}$$

$$|V\_{ref}|\sin(\theta) = |V\_B|\sin(60^\circ). \tag{16}$$

**Figure 4.** Analysis of *Vre f* at the Sector 1.

Solving for |*VB*| from (16),

$$|V\_B| = \frac{|V\_{ref}|\sin(\theta)}{\sin(60^\circ)};$$

solving for |*VA*| from (15); and substituting (17) yields

$$|V\_A| = |V\_{ref}|\cos(\theta) - |V\_B|\sin(60^\circ),$$

$$|V\_A| = |V\_{ref}| \left(\frac{\sin(60^\circ)\cos(\theta) - \cos(60^\circ)\sin(\theta)}{\sin(60^\circ)}\right).$$

Then, using the following trigonometric identity,

$$
\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b),
$$

yields

$$|V\_A| = \frac{|V\_{ref}|\sin(60^\circ - \theta)}{\sin(60^\circ)}.\tag{18}$$

Substituting the components (14) in (17) and (18),

$$\frac{1}{T\_s}V\_bt\_b = \frac{|V\_{ref}|\sin(\theta)}{\sin(60^\circ)},$$

$$\frac{1}{T\_s}V\_at\_a = \frac{|V\_{ref}|\sin(60^\circ - \theta)}{\sin(60^\circ)}.$$

Now, solving for *ta*,

$$t\_a = \frac{|V\_{ref}|T\_s \sin(60^\circ - \theta)}{V\_a \sin(60^\circ)}$$

and then solving for *t<sup>b</sup>* ,

$$t\_b = \frac{|V\_{ref}|T\_s \sin(\theta)}{V\_b \sin(60^\circ)}\text{.}$$

,

and since *V<sup>a</sup>* = *V<sup>b</sup>* = <sup>2</sup> <sup>3</sup>*VDC* and sin(60◦ ) = √ 3 2

$$t\_a = \frac{\sqrt{3}|V\_{ref}|T\_s \sin(60^\circ - \theta)}{V\_{DC}}\tag{19}$$

,

and

$$t\_b = \frac{\sqrt{3}|V\_{ref}|T\_s \sin(\theta)}{V\_{DC}}.\tag{20}$$

Finally, solving for *t*<sup>0</sup> from (13) yields

$$t\_0 = T\_s - t\_a - t\_b.\tag{21}$$

Equations (19)–(21) are a general solution for *ta*, *t<sup>b</sup>* , and *t*<sup>0</sup> since the times used for each active or null vector along the grid period are the same. The evolution of the calculated times and PWM switching signals along a grid period is depicted in Figure 5, where *A* is the waveform for the evolution of *ta*, *B* is the waveform for the evolution of *t<sup>b</sup>* , and *C* is the waveform for the evolution of *t*0.

By using these time calculations and by considering that the main objective of the DCM-232 topology is to reduce the CMC by decoupling the DC sources, a simple way to obtain a constant common mode voltage for different vector sequences is proposed. Considering the active vectors given in Table 1 and the operation states of the DCM-232 inverter as logic states, the following Boolean expressions are obtained:

$$S\_7 = S\_1 \overline{S\_3 S\_5} + \overline{S\_1} S\_3 \overline{S\_5} + \overline{S\_1} \overline{S\_3} S\_5 \tag{22}$$

$$S\_8 = S\_1 S\_3 \overline{S\_5} + \overline{S\_1} S\_3 S\_5 + S\_1 \overline{S\_3} S\_5. \tag{23}$$

According to (22) and (23), the switches *S*<sup>7</sup> and *S*<sup>8</sup> are in the active state when the corresponding logic states involved in each equation comply with the logic conditions. Therefore, only when the active vectors appear in the modulation sequences are *S*<sup>7</sup> and *S*<sup>8</sup> turned on.

Based on the above analysis, three different modulation strategies are proposed in this paper to control the DCM-232 topology using the proposed technique. Note that any vector sequence can be adopted to control the inverter. In this paper, the proposed SVM strategies are based in the conventional SVM for a three-phase full-bridge inverter, named Conventional Symmetric Space Vector Modulation (CSSVM), Conventional Asymmetric Space Vector Modulation (CASVM), and Discontinuous Space Vector Modulation Maximum (DSVMMAX). The switching patterns for these three proposed SVM strategies are depicted in Figure 6. Note that the switching pattern for CSSVM and CASVM strategies is the same, and the main difference is the way in which the times *ta*, *t<sup>b</sup>* , and *t*<sup>0</sup> are computed, as shown below.

**Figure 6.** Switching patterns for (**a**) CSSVM and CASVM, and (**b**) DSVMMAX.
