# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 1 Introduction  
An AC-AC Matrix Converter (MC) is an array of bidirectional switches that fit a three-phase signal by controlling its characteristics, as required for the process in the following stage. They are of great interest in industrial applications due mainly to their reduced dimension compared to other types of converters and because they have the ability to control the amplitude, frequency, and phase of the output voltage signal independently, converting both the input and output currents into sinusoidal signals. By their nature, they have the additional advantage of operating in all four quadrants, allowing the regeneration of energy since their switching elements are bidirectional. However, the design of this kind of converter is a difficult issue, due mainly to the technical limitations found in their construction, the number of semiconductor devices required, and the complexity in the control stage. Different MC topologies have been developed despite their complexity, to provide better performance and efficient energy consumption [1]. The study of the three-phase MC was presented for the first time by [PERSON] and [PERSON] [2] in 1980. They developed the basic configuration of a power circuit as a matrix of bidirectional switches that connects each of the output phases to the input phases, introducing the term of the matrix converter. Their work included a rigorous mathematical model, in which they demonstrated that it was possible to obtain an ideal electronic converter, transforming an alternating polyphase electric input signal withfixed characteristics (magnitude, frequency, and phase) into an alternating polyphase signal of variable parameters, independently controlling each of them [3]. They proposed a control strategy known as Venturini modulation [2], based on a matrix of low-frequency modulation for the voltage and current, generating a maximum output of 50% with respect to the input (\(q=0.5\)), without the ability to control the power factor. [PERSON] and [PERSON] [4; 5] also presented a method to increase the maximum output voltage of the converter at a ratio of 0.866 with respect to the input. This was achieved by including the third harmonic in the modulation equation for both the voltage and current signals, with a technique known as the Venturini optimo method. A transfer ratio between the input and output voltages lower than one is the main disadvantage of the matrix converters [1; 6]. In 1983, [PERSON] [7] proposed a new control technique based on a virtual direct current bus approach. In that modulation, the voltage inputs were first rectified in virtual form and then transformed to an alternating current, generating a variable-frequency output signal. In this method, each output line switches between positive and negative values of the input lines using pulse width modulation (PWM) techniques, similar to those used in back-to-back inverters. This control strategy was developed by [PERSON] with a method known as indirect transfer [8; 9], achieving a maximum input/output ratio of 1053 in an MC. However, the maximum voltage obtained presented low-frequency distortions. In 1987, [PERSON] and [PERSON] [10] proposed a new control strategy called the scalar method, where the signals for the switch activation were calculated by measuring the input voltage lines. Similar to the Venturini method, the transfer ratio was limited to a value of 0.5 with a maximum of 0.866, due to low-frequency disturbances in the power supply and to variable loads [11; 12]. Another limitation was that the method could not control the power factor. A similar approach to the scalar method was proposed by [PERSON] [13], generating the switching signals from the measurement of the line-to-line input voltage, different from the scalar method, where the signals were measured with reference to neutral. The maximum voltage ratio was 0.75. Spatial vector modulation (SVM) was first proposed by [PERSON] [14; 15] to be applied to matrix converters.