## 1 Introduction  
Since brownmillerites have compositional flexibility and are applicable as oxygen ion conductors in solid oxide fuel cells (SOFC) [1], as a membrane for oxygen separation [2], as an electrocatalyst for oxygen or hydrogen evolution reactions, or a catalyst for hydrocarbon oxidation [3; 4], there have been intense studies conducted on them. In addition, brownmillerite-type compounds have also been studied for their significant magnetoresistance effects [5; 6]. Oxygen defects in the oxygen-deficient perovskites play a considerable role in their properties and can be distributed randomly or in an ordered fashion. Ca\({}_{2}\)FeO\({}_{5}\) is one of the most popular long-range defect-ordered perovskites. Ca\({}_{2}\)FeO\({}_{5}\) has a brownmillerite structure where tetrahedral structures are formed due to defects, and these tetrahedra form alternate layers with octahedra in the structure of a crystal. Due to the long-range defect order, this material has been in focus for oxide ion conductivity and other properties and applications. It is also used as a parent compound to study the doping effect on the structure and properties [7]. Ca\({}_{2}\)FeO\({}_{5}\) has been studied for a wide variety of purposes, such as the production of biodiesel using soybean and Jatropha oils by magnetic CaFe\({}_{2}\)O\({}_{4}\)-Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\)-based catalysts [8], ion conductivity in Li-ion batteries [9], and as an electrocatalyst for oxygen evolution reactions [3]. Therefore, Ca\({}_{2}\)FeO\({}_{5}\) is one of the most popular oxygen-deficient perovskites in defect-ordered materials research. Brownmillerites have also been studied for their electric and magnetic properties due to possible application of their ferroelectricity, piezoelectric, and pyroelectric properties. The electrical properties of brownmillerites were generally studied by measuring the electrical conductivity by the direct current (DC) method or electrochemical impedance method. The DC method is generally used to obtain the information of the total electrical conductivity at a given temperature, which is the sum of the electronic and ionic conductivities. The DC technique was also found to apply to measuring oxide ion conductivity in brownmillerite-type compounds [10; 11]. DC conductivity measurement was applied in Ca\({}_{2}\)FeO\({}_{5}\) to demonstrate the dependence of charge transport properties on the synthetic technique [12]. We measured the DC conductivity of this material to show the correlation of the electrocatalytic activity of the oxygen evolution reaction with electrical conductivity [3; 12]. However, the complete electrical properties of a material cannot be studied by DC measurement. For example, if the conductivity or resistivity contribution of the grains, grain boundaries, and electrode-material interface is to be studied, an electrical impedance spectroscopic (EIS) study is performed. EIS is a powerful tool to measure various electrochemical and electromagnetic properties as well as the charge transport mechanisms. It provides information on the grains, grain boundary, and electrode material interface's contributions to the impedance during charge transport [13; 14]. EIS studies can be found to apply during the study of the electrochemical properties of brownmillerites or other materials [15]. It is used to study the magnetoelectric properties as well [16]. It can be found that different spectroscopic techniques have been applied for the study of brownmillerite-type Ca\({}_{2}\)FeO\({}_{5}\). For example, Mossbauer spectroscopy was used to study the structural relation to the nuclear electric field at the iron site [17] and determination of the nuclear electric field gradient (EFG) tensor parameters at the Fe\({}^{3+}\) sites in Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\)[18]. Direct current and Seebeck coefficient (thermopower) measurements were performed to study the conductivity [19] and thermoelectric properties [20].