# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 3 Input Filter  
As mentioned previously, the bidirectional switches in the design of an MC require the use of input filters. Disturbances in the input voltages can cause an erroneous measurement of the instantaneous values and affect the calculations made by the modulation strategy. To avoid these unwanted harmonic signals in the input, a low pass filter \(LC\) is implemented. There are different \(LC\) filter arrays that can be integrated into the MC topology [19; 20], as shown in Figure 2:  
For this work, a single-state \(LC\) filter was selected because it uses few components and has a proper response in the frequency domain [19]. The presence of reactive elements improves the behavior of the MC in the following circumstances:  
* \(C_{F}\): Limits the over-voltages of the input originated by the distortions or disturbances of the power supply network. The parasitic inductances through the switches of the MC can generate over-voltages on both sides of the converter due to the high values of \(\frac{di}{dt}\) in the switching periods. To reduce this effect, capacitor \(C_{F}\) should be placed as close as possible to the MC switches. * \(L_{F}\): The switching process in the MC causes the appearance of over-currents, which are generated from small discontinuities that occur when switching from a switch on an input phase to another switch on a different input phase. The inductor \(L_{F}\) is responsible for smoothing the slope of that over-current. Figure 3 shows the \(LC\) input filter with a damping resistance in parallel with the filter inductor. The internal resistance of the inductor is represented by \(r_{i}\). Figure 2: Basic settings of LC input filters used in MC design: (**a**) multi-state, (**b**) single-state, (**c**) filter in connection, and (**d**) filter with one bypass of a state. The transfer function of the input filter is expressed in Equation (5):  
\[\frac{V_{o}(s)}{V_{i}(s)}=\frac{\left[\frac{1}{R_{d}C_{i}}\right]s+\left[\frac{1} {L_{i}C_{i}}+\frac{r_{i}}{R_{d}L_{i}C_{i}}\right]}{s^{2}+\left[\frac{r_{i}}{L_{ i}}+\frac{1}{R_{d}C_{i}}\right]s+\left[\frac{1}{L_{i}C_{i}}+\frac{r_{i}}{R_{d}L_{i}C_{i}}\right]} \tag{5}\]  
The input frequency is the frequency of the electric network, 60 Hz. The input filter must be able to attenuate the multiples of higher order harmonics of the power frequency. The values of its components should be as small as possible to minimize the dimensions of the converter. The harmonic reduction effect of the filter depends on its resonance frequency, which is determined by Equation (6):  
\[f_{i0}=\frac{1}{2\pi\sqrt{L_{i}C_{i}}} \tag{6}\]  
The value of the capacitance must be selected according to the nominal power value of the MC. There is no criterion to establish the appropriate power factor at the input of the inverter in low power conditions, but a power factor higher than 0.8 when the input power is 10% of its nominal value is acceptable [32]. The capacitive reactive power \(Q_{C}\) is selected according to Equation (7). \[Q_{C}=3\omega_{i}C_{i}V_{i}^{2}\leqslant\frac{3}{4}10\%P_{n} \tag{7}\]  
where \(C_{i}\) is the value of the input capacitance, \(V_{i}\) is the input RMS voltage, \(P_{n}\) is the nominal power of the MC at the input expressed in watts, and \(\omega_{i}\) the angular frequency in the input. The value for the capacitor should be in the range according to Equation (8):  
\[C_{i}\leqslant\frac{P_{n}}{40\omega_{i}V_{i}^{2}} \tag{8}\]  
The design of the input filter is related to the switching frequency and the voltage drops in the \(LC\) filter inductor. The voltage drops must not be higher than 5% of the input voltage. The resonance frequency \((f_{i0})\) must be higher than the fundamental frequency of the voltage supply and lower than the switching frequency. Normally, \(f_{i0}\) must be higher than 20 for the power frequency \((f_{i})\) and lower than one third of the frequency in the load \((f_{s})\), as in Expression (9).