## 4 Interleaved Inverter
### Modulation  
In each sampling time (\(t_{m+1}\), \(t_{m+2}\)) the voltages \(\mathbf{v_{r}^{\mathbf{p}^{\ast}}}\) and \(\mathbf{v_{r}^{\mathbf{n}^{\ast}}}\) can be obtained by a synchronous space vector modulation using a triangular carrier of a proper frequency value. The switching frequency can be fixed either equal to the sampling frequency or half of the sampling frequency. Denoting with \(T_{m}\) the modulation period and \(T_{s}\) the sampling time, it will be \(T_{m}=T_{s}\) or \(T_{m}=2\)\(T_{s}\). Furthermore, it is advantageous to fix a phase shift between the upper and lower converter triangular carrier. Indeed, by choosing this phase shift correspondent to the half of \(T_{m}\), the main voltage harmonic, due to the modulation, will be null on the total interleaved inverter voltage space vector.  
In each modulation period, the space vector modulation pattern applies four vectors, as follows:  
\[\left\{\begin{array}{l}\mathbf{v_{s_{1}}^{p}}\left(t_{m+1}\right)\to \mathbf{v_{\alpha}^{p}}\left(t_{m+1}\right)\to\mathbf{v_{\beta}^{p}}\left(t_{ m+1}\right)\to\mathbf{v_{\beta_{2}}^{p}}\left(t_{m+1}\right)\\ \mathbf{v_{\delta_{1}}^{p}}\left(t_{m+1}\right)\to\mathbf{v_{\alpha}^{n}}\left( t_{m+1}\right)\to\mathbf{v_{\beta_{2}}^{p}}\left(t_{m+1}\right)\end{array}\right., \tag{32}\]  
The vectors \(\mathbf{v_{s_{1}}}\left(t_{m+1}\right)\), \(\mathbf{v_{s_{2}}}\left(t_{mSetting \(c^{p}\left(t_{m+1}\right)=-c^{n}\left(t_{m+1}\right)=c\left(t_{m+1}\right)\), the homopolar components of vectors \(\mathbf{v_{\alpha}}\left(t_{m+1}\right)\) and \(\mathbf{v_{\beta}}\left(t_{m+1}\right)\) will be expressed as follows:  
\[\left\{\begin{array}{l}v_{0}^{p}\left(t_{m+1}\right)=v_{\alpha,0}^{p}\left(t_ {m+1}\right)\,\alpha^{p}\left(t_{m+1}\right)+v_{\beta,0}^{p}\left(t_{m+1} \right)\,\beta^{p}\left(t_{m+1}\right)+\left[1+c\left(t_{m+1}\right)\right] \frac{\delta^{p}\left(t_{m+1}\right)}{2}v_{c}\left(t_{m}\right)\\ v_{0}^{n}\left(t_{m+1}\right)=v_{\alpha,0}^{n}\left(t_{m+1}\right)\,\alpha^{n} \left(t_{m+1}\right)+v_{\beta,0}^{n}\left(t_{m+1}\right)\,\beta^{n}\left(t_{m+ 1}\right)+\left[1-c\left(t_{m+1}\right)\right]\frac{\delta^{n}\left(t_{m+1} \right)}{2}v_{c}\left(t_{m}\right)\end{array}\right., \tag{35}\]  
The semi-difference of the two equations in Equation (35) gives:  
\[\Delta v_{r,0}\left(t_{m+1}\right)=\Delta\overline{v}_{r,0}\left(t_{m+1} \right)+\frac{v_{c}\left(t_{m}\right)}{4}\left[\delta^{p}\left(t_{m+1}\right) +\delta^{n}\left(t_{m+1}\right)\right]c\left(t_{m+1}\right), \tag{36}\]  
where, \(\Delta\overline{v}_{r,0}\left(t_{m+1}\right)\) is the component of \(\Delta v_{r,0}\left(t_{m+1}\right)\), which is invariant with \(c\left(t_{m+1}\right)\). In particular, \(\Delta\overline{v}_{r,0}\left(t_{m+1}\right)=0\) when \(\mathbf{v_{r}^{p*}}\left(t_{m}\right)=\mathbf{v_{r}^{n*}}\left(t_{m}\right)\). By imposing \(\Delta v_{r,0}\left(t_{m+1}\right)=\Delta v_{r,0}^{*}\left(t_{m}\right)\), \(c\left(t_{m+1}\right)\) is then computed as follows:  
\[c\left(t_{m+1}\right)=\frac{4\Delta v_{r,0}^{*}\left(t_{m}\right)}{v_{c}\left( t_{m}\right)\left[\delta^{p}\left(t_{m+1}\right)+\delta^{n}\left(t_{m+1}\right) \right]} \tag{37}\]