# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 6 Numerical Examples
### Boost Inverter for Stand-Alone Applications  
The converter considered in this paragraph is the inverter proposed in [33], whose topology is represented in Figure 5. The parameters used for the simulations are given in Table 4.  
Figure 4: Simulated waveforms of the full-bridge inverter feeding a CFL-type nonlinear load. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency (Hz)** & **Amplitude (A)** & **Phase (Degrees)** \\ \hline
60 & 1.524 & \(-\)30 \\
180 & 0.8185 & 90 \\
300 & 0.2878 & \(-\)90 \\
420 & 0.2728 & 90 \\
540 & 0.1636 & \(-\)90 \\
660 & 0.1636 & 90 \\
780 & 0.1168 & \(-\)90 \\
900 & 0.1168 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the CFL-type nonlinear load. To model the converter circuit, consider that if \(u_{1}=1\) when \(S_{1}=\) OFF and \(S_{2}=\) ON, \(u_{1}=0\) when \(S_{1}=\) ON and \(S_{2}=\) OFF, and similarly for switches \(S_{3}\) and \(S_{4}\); the elementary electrical equations are given by  
\[\left\{ \begin{array}{rcl}L_{1}\dfrac{di_{1}(t)}{dt}&=&-r_{1}i_{1}(t)+(u_{1}(t )-1)v_{C_{1}}(t)+V_{in}\\ C_{1}\dfrac{dv_{C_{1}}(t)}{dt}&=&(1-u_{1}(t))i_{1}(t)-\dfrac{v_{C_{1}}(t)}{R}+ \dfrac{v_{C_{2}}(t)}{R}\\ L_{2}\dfrac{di_{2}(t)}{dt}&=&-r_{2}i_{2}(t)+(u_{2}(t)-1)v_{C_{2}}(t)+V_{in}\\ C_{2}\dfrac{dv_{C_{2}}(t)}{dt}&=&(1-u_{2}(t))i_{2}(t)-\dfrac{v_{C_{2}}(t)}{R}+ \dfrac{v_{C_{1}}(t)}{R}\\ \end{array} \right.\]  
Taking the state vector \(x=\left[ \begin{array}{cccc}i_{1}&v_{C_{1}}&i_{2}&v_{C_{2}}\end{array} \right]^ {T}\), model (1) is expressed as  
\[\dfrac{dx(t)}{dt} = \left[\left[ \begin{array}{cccc}-\frac{r_{1}}{L_{1}}&-\frac{1}{ L_{2}}&0&0\\ \frac{1}{C_{1}}&-\frac{1}{RC_{1}}&0&\frac{1}{RC_{1}}\\ 0&0&-\frac{r_{2}}{L_{2}}&-\frac{1}{RC_{2}}\\ 0&\frac{1}{RC_{2}}&\frac{1}{C_{2}}&-\frac{1}{RC_{2}}\\ \end{array} \right]+u_{1}(t)\underbrace{\left[ \begin{array}{cccc}0&\frac{1}{ L_{1}}&0&0\\ -\frac{1}{C_{1}}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array} \right]}_{A_{u_{1}}}+\underbrace{\left[ \begin{array}{cccc}0&0&0&0 \\ 0&0&0&0\\ 0&0&0&\frac{1}{L_{2}}\\ 0&0&-\frac{1}{C_{2}}&0\\ \end{array} \right]}_{A_{u_{2}}}\right]x(t)\] \[+\underbrace{\left[ \begin{array}{cccc}\frac{1}{L_{1}}\\ 0\\ \frac{1}{L_{2}}\\ 0\\ \end{array} \right]}_{B_{0}}V_{in}\] \[y(t) = \underbrace{\left[ \begin{array}{cccc}0&1&0&-1\\ \zeta_{0}&0\\ \end{array} \right]}_{C_{0}}x(t)\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline \(V_{in}\) & 150 & V \\ \(V_{\max}\) & 220 & V \\ \(V_{dc}\) & 300 & V \\ \(L_{1}=L_{2}\) & 100 & \(\mu\)H \\ \(C_{1}=C_{2}\) & 2 & \(\mu\)F \\ \(R\) & 100 & \(\Omega\) \\ \(r_{1}=r_{2}\) & 2 & \(\Omega\) \\ \(\omega_{0}\) & \(2\pi 50\) & rad/s \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of the boost inverter. Figure 5: Circuit diagram of the boost inverter. The idea is to control the inverter while ensuring that the output of each of the two internal boost converters tracks the following reference signals:  
\[v_{C_{1_{ref}}}(t)=V_{dc}+\frac{V_{\max}}{2}\sin(\omega_{0}t)\text{ and }v_{C_{2_{ref}}}(t)=V_{dc}-\frac{V_{\max}}{2}\sin(\omega_{0}t)\]  
Because the output voltage of a boost converter is greater than the DC voltage source, the following conditions must be satisfied:  
\[V_{dc}-\frac{V_{\max}}{2}>V_{in}\Rightarrow V_{dc}>V_{in}+\frac{V_{\max}}{2}\]  
Under these conditions, the reference of the output \(y(t)\) becomes  
\[y_{ref}(t)=v_{C_{1_{ref}}}(t)-v_{C_{2_{ref}}}(t)=V_{\max}\sin(\omega_{0}t)\]  
The power balance can be obtained from the state space model and is given by  
\[Li_{i}\frac{di_{i}}{dt}+Cv_{C_{i}}\frac{dv_{C_{i}}}{dt}=-r_{L}i_{i}^{2}+V_{in}i _{i}-\frac{v_{C_{i}}^{2}}{R}+\frac{v_{C_{1}}v_{C_{2}}}{R},\ \ i=1,2 \tag{21}\]  
As pointed out, determining the inductor current references is quite tedious because such references must satisfy the two previous nonlinear power balance equations, which are difficult to solve analytically. The idea is to approximate the current references using the harmonic balance method [45].