# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 3 Input Filter
### Input Filter Optimization  
In this subsection, an optimization problem corresponding to the input filter is proposed, minimizing the response in the filter amplitude in a frequency analysis to obtain the magnitude of the passive elements of the filter. The problem is solved by means of the DE algorithm. A second order filter is proposed, as shown in Figure 3, with the elements \(C_{i}\), \(L_{i}\), and \(R_{d_{i}}\) and considering the buffer \(\zeta_{i}\). Therefore, the design variables are as in Expression (12):  
\[\mathbf{x}=\{C_{i},\ L_{i},\ \zeta_{i}\}^{T} \tag{12}\]  
The design vector can be expressed in terms of the optimization problem, as stated in Equation (13):  
\[\mathbf{x}=\{x_{1},\ x_{2},\ x_{3}\}^{T} \tag{13}\]  
Considering that the filter is designed from a fixed resonance frequency, an expression for conditioning the value of \(C_{i}\) and \(L_{i}\) is obtained by means of Equation (6), as observed in Equation (14):  
\[L_{i}\geq\frac{1}{4\pi^{2}}f_{i0}^{2}C_{i} \tag{14}\]  
The goal is to minimize the objective function, which corresponds to the amplitude response of the filter in the frequency behavior of Equation (5), evaluated at three frequencies to obtain the numerical behavior of the function at three points, and the main value is the switching frequency, which in this design is proposed as 1200 Hz and two frequencies plus one with a higher value and the other with a lower value, as expressed in (15):  
minimize:  
\[\begin{split} f(x_{1},x_{2},x_{3})&=-20 log|H_{1}(j \omega_{1})|_{\omega_{1}=900 Hz}\\ &\quad-20 log|H_{2}(j\omega_{2})|_{\omega_{2}=1200 Hz}\\ &\quad-20 log|H_{3}(j\omega_{3})|_{\omega_{3}=3000 Hz}\end{split} \tag{15}\]  
subject to Expressions (16) to (19):  
\[g_{1}(x_{1},x_{2},x_{3})=x_{2}-\frac{1}{4\pi^{2}f_{i0}^{2}x_{1}}\geq 0 \tag{16}\]  
\[1\times 10^{-6}\leq x_{1}\leq 2.8\times 10^{-6} \tag{17}\]  
\[1\times 10^{-4}\leq x_{2}\leq 1\times 10^{-2} \tag{18}\]  
\[0.01\leq x_{3}\leq 0.9 \tag{19}\]  
The expression \(H_{i}(j\omega_{i})\) in Equation (15) is defined by Equations (20)-(22):  
\[H_{i}(j\omega_{i})=\sqrt{\left[G_{1}\right]^{2}\ \ +\left[G_{2}\right]^{2}} \ \ \ \ \ i=1,2,3 \tag{20}\]  
\[G_{1i}=\frac{B_{i}(B_{i}-\omega_{i}^{2})+A_{i}C_{i}\omega_{i}^{2}}{(B_{i}- \omega_{i})^{2}+C_{i}^{2}\omega_{i}^{2}} \ \ \ \ i=1,2,3 \tag{21}\]\[G_{2i}=\frac{A_{i}\omega_{i}(B_{i}-\omega_{i}^{2})-B_{i}C_{i}\omega_{i}}{(B_{i}- \omega_{i})^{2}+C_{i}^{2}\omega_{i}^{2}}\qquad i=1,2,3 \tag{22}\]  
where:  
\[A_{i}=\frac{1}{R_{d_{i}}x_{1}}\qquad i=1,2,3 \tag{23}\]  
\[B_{i}=\frac{1}{x_{2}x_{1}}+\frac{r_{i}}{R_{d_{i}}x_{1}x_{2}}\qquad i=1,2,3 \tag{24}\]  
\[C_{i}=\frac{r_{i}}{x_{2}}+\frac{1}{R_{d_{i}}x_{1}}\qquad i=1,2,3 \tag{25}\]  
\[R_{d_{i}}=\frac{1}{2x_{3}}\sqrt{\frac{x_{2}}{x_{1}}}\qquad i=1,2,3 \tag{26}\]  
The search space for the design variables is limited to a margin in which it is considered feasible to build these elements in a future CM; however, in the use of this methodology for the design of another filter, other search limitations can be established depending on the desired nominal power.  
In order to test the proposed methodology, it was applied to the design of a converter with a nominal power of \(P_{n}=746\) W, with a three-phase voltage source \(V_{i}=117\) V and a frequency \(f_{i0}=60\) Hz. The switching frequency of the bidirectional power switches was \(f_{s}=1200\) Hz, with a total power factor of the converter \(PF=0.8\).