## 3 Control of the Inverter
### Analysis of the Dynamic Behavior using Harmonic Balance  
From the analysis in Section 2, the reference required to enforce the desired shape in the capacitor voltages has the following form:  
\[v_{1e}=V_{dc}-V_{s1}\sin\omega t+V_{c2}\cos 2\omega t+V_{c4}\cos 4\omega t \tag{31}\]  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t+V_{c2}\cos 2\omega t+V_{c4}\cos 4\omega t \tag{32}\]  
The corresponding time derivatives are given by:  
\[\frac{dv_{1e}}{dt}=-\omega V_{s1}\cos\omega t-2\omega V_{c2}\sin 2\omega t-4 \omega V_{c4}\sin 4\omega t \tag{33}\]  
\[\frac{dv_{2e}}{dt}=\omega V_{s1}\cos\omega t-2\omega V_{c2}\sin 2\omega t-4 \omega V_{c4}\sin 4\omega t \tag{34}\]  
Now, suppose that the stationary periodic behavior of the inductor currents can be approximated, in a satisfactory way by means of the following waveforms and its derivatives:  
\[i_{1e}=I_{dc}-I_{s1}\sin\omega t-I_{c1}\cos\omega t+\sum_{n=2}^{4}(-1)^{n+1}[I _{sn}\sin n\omega t+I_{cn}\cos n\omega t] \tag{35}\]  
\[i_{2e}=I_{dc}+I_{s1}\sin\omega t+I_{c1}\cos\omega t-\sum_{n=2}^{4}\left[I_{sn }\sin n\omega t+I_{cn}\cos n\omega t\right] \tag{36}\]  
Figure 5: Block diagram of the outer voltage controllers. \[\frac{di_{e1}}{dt}=\omega l_{c1}\sin\omega t-\omega l_{s1}\cos\omega t+\omega \sum_{n=2}^{4}n\Big{[}(-1)^{n}I_{cn}\sin n\omega t+(-1)^{n+1}I_{sn}\cos n\omega t \Big{]} \tag{37}\]  
\[\frac{di_{2e}}{dt}=-\omega l_{c1}\sin\omega t+\omega l_{s1}\cos\omega t+\omega \sum_{n=2}^{4}n[I_{cn}\sin n\omega t-I_{sn}\cos n\omega t] \tag{38}\]  
On the other hand, expressions Equations (18) and (19) can be interpreted as power balance constraints for the boost inverter cells. Then, they can be rewritten as follows:  
\[Li_{1e}\frac{di_{1e}}{dt}+Cv_{1e}\frac{dv_{1}}{dt}-\left(v_{in}i_{1e}+\frac{v_{ 1}v_{0e}}{r_{0}}\right)=0 \tag{39}\]  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t+V_{c2}\cos 2\omega t+V_{c4}\cos 4\omega t \tag{32}\]0  
Summing Equations (39) and (40), the power balance equation for the complete inverter can be written as:  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t+V_{c2}\cos 2\omega t+V_{c4}\cos 4\omega t \tag{32}\]1  
By replacing Equations (31)-(38) into Equation (41) and applying the harmonic balance technique, it is obtained a set of nonlinear algebraic equations, whose unknowns are the amplitudes of the harmonic components of the currents. It can be compactly written as:  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t+V_{c2}\cos 2\omega t+V_{c4}\cos 4\omega t \tag{32}\]2  
where the components of \(F_{B}\) are expressed by:  
\[\begin{array}{l}\mathrm{F}_{dc}=2V_{in}I_{dc}-\frac{2V_{s1}{}^{2}}{r_{0}}\\ \mathrm{F}_{s2}=\omega L\left(I_{s1}{}^{2}-I_{c1}{}^{2}+4I_{dc}I_{c2}+2I_{s1}I_ {s3}+2I_{c1}I_{c3}-2I_{s2}I_{s4}-2I_{c2}I_{c4}\right)+\omega C\big{(}V_{s1}{}^ {2}-4V_{dc}V_{c2}-2V_{c2}V_{c4}\big{)}+2V_{in}I_{s2}\\ \mathrm{F}_{c2}=2\omega L(I_{s1}{}I_{c1}-2I_{dc}I_{c2}+2I_{s1}I_{c3}-2I_{s3}I_ {c1}+2I_{s4}I_{c2}-2I_{s2}I_{c4})-\left(-2V_{in}I_{c2}+\frac{2V_{s1}{}^{2}}{ r_{0}}\right)\\ \mathrm{F}_{s4}=2\omega L\left(I_{s2}{}^{2}-I_{c2}{}^{2}-2I_{s1}I_{s3}+2I_{c1 }I_{s3}+4I_{dc}I_{c4}\right)+2\omega C\big{(}-V_{c2}{}^{2}-4V_{dc}V_{c4}\big{)} +2V_{in}I_{s4}\\ \mathrm{F}_{c4}=4\omega L\left(I_{s2}{}^{2}I_{c2}-I_{s3}I_{c1}-I_{s1}I_{c3}-2I _{c4}I_{s4}\right)+2V_{in}I_{c4}\\ \mathrm{F}_{s6}=3\omega L\left(I_{s3}{}^{2}-I_{s3}{}^{2}+2I_{s2}I_{s4}-2I_{c2} I_{c4}\right)-6\omega CV_{c2}V_{c4}\\ \mathrm{F}_{c6}=6\omega L\left(I_{s3}{}^{2}I_{c3}+I_{s2}I_{c4}+I_{s4}I_{c2} \right)\\ \mathrm{F}_{s7}=7\omega L\left(-I_{s3}I_{s4}-I_{s4}I_{c3}\right)\\ \mathrm{F}_{c7}=7\omega L\left(-I_{s3}I_{c4}-I_{s4}I_{c3}\right)\\ \mathrm{F}_{s8}=4\omega L\left(I_{sBy evaluating the periodic terms of Equations (16) and (17), it is possible to observe that their values are always lower than any value of the input voltage in the range of operation of the converter. Figure 6 shows three cycles of these terms evaluated for 30 coordinates into the range of input voltage and output load including extreme values. A coincidence of the produced waveforms around zero is recognized.