# Design of the Input and Output Filter for a Matrix Converter Using Evolutionary Techniques
## 1 Introduction
This is based on the instantaneous representation of spatial vectors for voltages and currents, for both the input and output. One of the most outstanding works was presented by [PERSON] [16], called the spatial vector approach of service cycle, applied to matrix converters. It uses the spatial vector notation and introduces the concept of the spatial vector of service cycle. These levels are obtained by a high switching speed, but implying a greater loss of energy. There are additional works in which other types of modulation and control schemes for matrix converters were planned, such as control by sliding modes [17] and PWM control based on carrier signals [18]. In the design of matrix converters, the use of electronic power devices such as bidirectional switches requires input filters. This reduces the high harmonic content in the input current, increasing the system protection against transitory effects coming from the power supply. Furthermore, the ripple in the waveform and excessive currents or voltages in the converter are reduced. The peaks generated in the waveform of the input voltages can cause an erroneous measurement of the instantaneous values and can affect the calculations made by any implemented modulation strategy. A low-pass filter can be employed to avoid these unwanted harmonic signals in the input. In [19; 20], different filter arrays suitable to be integrated into the MC topology were proposed, since the only inertial elements in these topologies were the filter components. However, the filter parameters significantly affected the system dynamics. The input filter design was discussed in [21; 22; 23], where due to the cost and weight considerations, it was found that the \(LC\) single-stage filter was the most suitable topology, although the set of filter elements \((L_{s},C_{s})\) for a particular resonance frequency was infinite. In [22], a very large \(C_{f}\) was proposed to ensure a minimum input displacement factor for low loads. A comprehensive treatment of the input/output filters focused on reducing electromagnetic interference and common mode voltages was done in [24; 25]. The works so far in the design and analysis of the filters were developed independently and were focused on satisfying the stability, power, or cost requirements. They were based on the arbitrary choice of some of the elements, and the values of the remaining parts were determined by means of the characteristic equations of the filter, without establishing if these elements generated the best dynamic response. Most of these works did not consider the integral design of the filters in the context of dynamic performance improvement or the reliability of the switching hardware. The only optimal approach so far was [26], which was proposed for the input filter and considered the objectives of both stable and dynamic states. It used a genetic algorithm to derive the values of the parameters and a test with three different topologies. A metaheuristic is a non-deterministic approximated method for solving complex optimization problems. Diverse metaheuristics have been developed [27]. The Differential Evolution (DE) algorithm was a metaheuristic presented by [PERSON] and [PERSON] [28] that has been applied to the solution of complex engineering problems. It is one of the most popular metaheuristics [29; 30] because of its fast convergence and easy implementation. In this paper, an alternative design technique for the input and output filters of an MC is presented. This approach is based on the modeling of an optimization problem, which to the best of the authors' knowledge has not been proposed in any previous work. This technique proposes the solution in the design of the filters directly, without the use of analytical substitution techniques, which allows guaranteeing the optimal solution to the optimization problem and obtaining an efficient response in the filters. This problem is solved by applying differential evolution.