# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 2 Matrix Converter  
The matrix converter consists of an array of electronic bidirectional power switches and passive elements that act as filters for the input and output signals, as seen in Figure 1. The power switches carry out the control by means of different modulation techniques. A matrix converter of dimension \((n\times m)\) consists of an array of \((n\cdot m)\) switches. This device is connected directly to the power lines, as input, and the load is connected directly to the converter output. In the case of a three-phase converter (Figure 1), there are three input signals and three output signals, so the matrix dimension is \((3\times 3)\) with nine switches [6].  
The advantages of an MC with respect to a conventional AC-AC are:* The waveform of the input current is sinusoidal, and this allows minimizing the harmonic and subharmonic content injected by the converter into the AC power supply.
* The energy flow can be bidirectional.
* The minimum requirement of passive components, which implies a reduction of their resignations by up to 50% compared to other converter topologies.
* Power-factor control. Disadvantages:
* Complex implementation.
* It requires bidirectional switches, so the number of electronic devices in the switching matrix increases, depending on the number of stands and outputs.
* It presents a problem of both synchronization and protection.
* In open loop operation, there is distortion of the input and output waveforms due to the high sensitivity of the converter.
* The maximum voltage ratio is 0.866.  
The behavior of the modulation function for an output phase is defined by the activation and deactivation of the switches that are connected to each input phase, as indicated in Expression (1):  
\[\text{switch}\;s_{jK}=\left\{\begin{array}{ll}1,&\text{close}\\ 0,&\text{open}\end{array}\right. \tag{1}\]  
where \(K\in\{a,b,c\}\) are the output phases and \(j\in\{A,B,C\}\) are the input phases. Two switches should not be activated at the same time to avoid short circuits in the input phases. In the same way, it should be avoided that at any moment of operation, all the switches are deactivated, since it would cause a high voltage differential at the output. This operation constraint is expressed in Equation (2):  
\[s_{ja}+s_{jb}+s_{jc}=1 \tag{2}\]  
The voltages and the input and output currents referred to the neutral line can be expressed with their vector representation, as seen in (3) and (4):  
\[\left[\begin{array}{c}u_{a}(t)\\ u_{b}(t)\\ u_{c}(t)\end{array}\right]=\left[\begin{array}{ccc}s_{aA}(t)&s_{aB}(t)&s_{aC} (t)\\ s_{bA}(t)&s_{bB}(t)&s_{bC}(t)\\ s_{CA}(t)&s_{eB}(t)&s_{C}(t)\end{array}\right]\left[\begin{array}{c}u_{A}(t) \\ u_{B}(t)\\ u_{C}(t)\end{array}\right] \tag{3}\]  
\[\left[\begin{array}{c}i_{A}(t)\\ i_{B}(t)\\ i_{C}(t)\end{array}\right]=\left[\begin{array}{ccc}s_{aA}(t)&s_{aB}(t)&s_{aC} (t)\\ s_{bA}(t)&s_{bB}(t)&s_{bC}(t)\\ s_{CA}(t)&s_{CB}(t)&s_{CC}(t)\end{array}\right]\left[\begin{array}{c}i_{a}(t) \\ i_{b}(t)\\ i_{c}(t)\end{array}\right] \tag{4}\]  
Figure 1: Three-phase matrix converter.  
The power filters \((L_{Li},L_{si},C_{si})\) located at the input and output of each of the lines of the converter mitigate the high-frequency components, generating current signals from input and sinusoidal output voltages, avoiding the generation of over-voltages. The over-voltage is caused by the fast switching in the input currents due to the presence of the short-circuit reactance of any real power supply. An adequate design of these filters is very important in the operation of matrix converters [23; 31].