*2.3. PWM Duty Cycles Calculation for CMCP and Topology 5* × *5*

Referred to the triangle Δ[1,2,3] in Figure 6, the reference output voltage *v*<sup>P</sup> is synthesized using 3 switches: *h*11, *h*21, and *h*31. Taking into account previous considerations, the following formulas can be proposed for the calculation of PWM duty cycles,

$$d\_{\rm I} \mathbf{p} = \mathbf{\tilde{q}} \cdot \left| \det \begin{bmatrix} \upsilon\_{\rm I2x} - \upsilon\_{\rm Px} & \upsilon\_{\rm I2y} - \upsilon\_{\rm Py} \\ \upsilon\_{\rm I3x} - \upsilon\_{\rm Px} & \upsilon\_{\rm I3y} - \upsilon\_{\rm Py} \end{bmatrix} \right| = \frac{\Delta\_{\rm [2,P3]}}{\Delta\_{\rm [1,2,3]}} \tag{12}$$

$$d\_{\rm 3P} = \,\_{\rm \bf \bf \bf \bf} \cdot \left| \det \begin{bmatrix} \upsilon\_{\rm 11\chi} - \upsilon\_{\rm P\chi} & \upsilon\_{\rm 1\chi} - \upsilon\_{\rm P\chi} \\ \upsilon\_{\rm 2\chi} - \upsilon\_{\rm P\chi} & \upsilon\_{\rm 2\chi} - \upsilon\_{\rm P\chi} \end{bmatrix} \right| \right| = \frac{\Delta\_{\rm [2,P,1]}}{\Delta\_{\rm [1,2,3]}} \tag{13}$$

$$d\_{\rm 2P} = 1 - d\_{\rm 1P} - d\_{\rm 3P} = \frac{\Delta\_{[3, \rm 1P, 1]}}{\Delta\_{[1, 2, 3]}} \tag{14}$$

where det is the determinant of the second-order matrix, and

$$\mathbf{q}\_{\mathbf{P}}^{\mathbf{x}} = \left| \det \begin{bmatrix} \upsilon\_{\mathbf{l}2\mathbf{x}} - \upsilon\_{\mathbf{l}1\mathbf{x}} & \upsilon\_{\mathbf{l}2\mathbf{y}} - \upsilon\_{\mathbf{l}1\mathbf{y}} \\ \upsilon\_{\mathbf{l}3\mathbf{x}} - \upsilon\_{\mathbf{l}1\mathbf{x}} & \upsilon\_{\mathbf{l}3\mathbf{y}} - \upsilon\_{\mathbf{l}1\mathbf{y}} \end{bmatrix} \right|^{-1} \tag{15}$$

is the scaling factor, which is equal to the triangle Δ[1,2,3] surface. Thus, the average value of the CMCP output voltage can be expressed by the following formula.

$$
\overline{\boldsymbol{\upsilon}}\_{\rm P} = d\_{\rm 1P} \cdot \boldsymbol{\upsilon}\_{\rm i1} + d\_{\rm 2P} \cdot \boldsymbol{\upsilon}\_{\rm i2} + d\_{\rm 3P} \cdot \boldsymbol{\upsilon}\_{\rm i3} \tag{16}
$$
