# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 4 Output Filter  
Passive \(LC\) filters are commonly used in the output terminals of power converters when the output voltages of the system are the main control objective. Am MC requires low pass \(LC\) filters in its outputs in order to reduce the harmonics generated in the output voltage waveform, due to the bidirectional switching. A power converter with a high resonance frequency will result in a small filter size [33]. The output impedance of the power converter must be close to zero to operate as an ideal voltage source, since there must be no distortion in the output voltage even under unbalanced conditions in the connected load.  
In motor control applications, it is advantageous to use \(LC\) filters in the MC output. The filter can compensate the reactive power consumption needed by the motor, which is supplied by the filter capacitor [34]. There are two variants shown in Figure 4. The resonance frequency \(f_{out0}\) of the filter for the inverter output attenuates the voltage laps caused by the switching frequency \(f_{s}\). The attenuation effect is increased when the resonance frequency is low compared to the switching frequency of the converter according to the relationship in Expression (27):  
\[-40 log\left(\frac{f_{s}}{f_{out0}}\right)dB \tag{27}\]  
There are diverse combinations of inductor and capacitor values for a given resonance frequency. The currents in the transient state of the converter can be reduced by increasing the inductance. However, the output voltages can be affected by small disturbances in the load due to the small values of the capacitance compared to the inductance. On the other hand, large inductors not only increase the cost and size of the output filter, they also increase the stress on the voltage of the inverter switches, because the voltage drops of the inductor result in losses in the output voltage. The resonance frequency determines the damping and stability of the output filter [35]. Its resonance frequency \(f_{out0}\) is determined by the product of the inductance and capacitance, as show in Equation (28):  
\[f_{out0}=\frac{1}{2\pi\sqrt{L_{o}C_{0}}} \tag{28}\]where \(L_{o}\) and \(C_{o}\) are the inductance and capacitance of the filter, respectively. The optimal design of the \(LC\) output filter has been addressed in various publications, and several investigations are based on a filter cost function. This cost function is defined by the typical reactive power in the filter components under steady state conditions [33; 36].  
From the point of view of cost and weight, the capacitor is the most economical design element. As for the output current of the converter, a large inductance has the advantage of limiting abrupt variations, but it is the most expensive design element. Figure 5 shows the scheme of the output filter in the frequency domain \((s)\), where \(r_{o}\) represents the internal resistance of the inductor. The transfer function of the output filter is determined with Equation (29):  
\[\frac{V_{o}(s)}{V_{i}(s)}=\left(\frac{\frac{1}{L_{o}C_{o}}}{s^{2}+\left(\frac{ r_{0}}{L_{o}}\right)s+\left(\frac{1}{L_{o}C_{o}}\right)}\right) \tag{29}\]  
Figure 4: Topologies of filters \(LC\) for output stages: (**a**) star connection and (**b**) delta connection.  
Figure 5: Scheme of the output filter \(LC\).  
The second order characteristic equation in the denominator of Equation (29) gives the relation between the damping factor \(\zeta\) and the values of \(L_{o}\) and \(C_{o}\), in Equation (30):  
\[L_{o}=\frac{r_{o}^{2}C_{o}}{4\zeta_{o}^{2}} \tag{30}\]  
Like in the input filter, the resonance frequency for the output filter must be higher than the fundamental frequency of the output voltage and lower than one third of the switching frequency to avoid resonance problems, as in expression (31):  
\[f_{o}\leq f_{out0}\leq\frac{1}{3}f_{s} \tag{31}\]