## 2 Impedance Spectroscopy
### Basics of Impedance Spectroscopy and Modelling  
In the study of the electrical properties of materials, it is common to apply the term electrical resistance for data analysis. This term can be interpreted as the ability of a material to oppose the passage of an electric current. The so-called Ohm's Law defines ohmic resistance (\(R_{\Omega}\)) in terms of the ratio between voltage (\(V\)) and current (\(I\)), \(R_{\Omega}=V/I\). Although this relationship is well known, its use is limited to direct current, DC, mode excitation. More generally, and for a deeper understanding of the electrical response of a system, a more comprehensive approach needs to be used, known as the generalized Ohm's Law, where an impedance is calculated instead. In it, an alternate current, AC, typically with a sinusoidal form (\(I=I_{m}cos(wt+\varnothing)\), where \(I_{m}\) is the current maximum amplitude, \(w\) is the angular velocity given \(w=2\pi f\), and \(\varnothing\) is the respective phase angle), is applied as a stimulus, and the voltage produced at the system terminals is measured (\(V=V_{m}cos(wt+\theta)\), where \(V_{m}\) is the current maximum amplitude, and \(\theta\) is the respective phase angle. The impedance is then determined by dividing the measured voltage by the used current; if performed in the frequency domain, the impedance \(Z\) is given by \(Z=\left(\frac{V_{m}}{I_{m}}\right)\times e^{-i\left(\theta-\varnothing\right)}\), or in the complex form, \(Z=R+jX\), where \(R\) is the resistance of the system (\(Z^{\prime}=R=Zcos(\theta-\varnothing)\)), and \(X\) is the reactance (\(Z^{\prime\prime}=X=Zsin(\theta-\varnothing)\)). \begin{table}
\begin{tabular}{c c c c} \hline
**Cell Configuration Approach** & **Frequency Range (Hz)** & **Use of Bias Signal** & **Reported in** \\ \hline EleIS & Starting from 0.01 Hz up to 10 MHz & No & [38, 51, 53, 54, 55, 56, 66, 75, 76, 77, 79, 82, 85, 87, 88, 91, 93, 96, 98] \\ EIS & Starting from 0.01 Hz up to 100 kHz & Yes & [41, 42, 43, 44, 45, 50, 89, 94, 95] \\ \hline \end{tabular}
\end{table} 
Table 1: Comparative summary of Impedance Spectroscopy data. If a purely resistive system is under evaluation (for instance, a single resistor), then the alternating current and respective voltage across the resistor will both be in phase. However, other circuit elements, such as inductors and capacitors, must be considered for a more detailed analysis of the electrical behavior of systems. Inductors and other known electrical components of systems represented (each displaying its own distinctive electrical behavior) are characterized by their inductance (L) and capacitance (C), respectively. Both are reactive in nature, with the main difference being that, while resistances dissipate energy in the form of heat, reactances store energy in the form of electric and magnetic fields. The impedance is commonly measured using a low amplitude excitation signal to obtain its value as close as possible to the linearity condition (similar to DC mode excitation). In addition, it is possible to evaluate a material sample by analyzing its electrical response over a wide range of frequencies or frequency sweeps since the voltage potentials comply with a sinusoidal wave [5; 99]. The main advantage of IS is that it allows for the identification of each electrical phenomenon present in a system once it reaches its maximum contribution at different excitation frequencies of the applied current. Besides, an equivalent electrical circuit that models the system response can be constructed. From those models, it is possible to analyze and interpret the transport of charges in solids and liquids, conductivity of films, capacitance of the electric double layer (resulting from the charge separation at the electrode/material interface), and the diffusion coefficients of charge carriers, among other phenomena [5; 13; 99]. If the under-evaluation system comprises a capacitor, the current and voltage observed will be out of phase.