## 3 Theoretical Background  
A lithium-ion battery cell has a negative electrode, commonly made of graphite, and a positive electrode based on Li\({}^{+}\). Both are separated by an electrolyte and a separator (see Figure 2). Electrons are transported from one electrode to the other during charge or discharge processes through the external circuit. Inside the battery cell, this current is compensated by ions movement in two steps: Dissolution of Li\({}^{+}\) in the electrolyte from the emitting electrode and rejoining a Li\({}^{+}\) from the electrolyte with the electron coming from the external circuit to form Li in the receiving electrode [1]. During the battery's first cycles, a solid electrolyte interphase (SEI) layer is formed by inactive lithium due to electrolyte decomposition over the negative electrode. The SEI continues to grow, affecting both capacity and power to fade. Moreover, several pores will be filled by covering the negative electrodes, causing more power to fade [6; 17]. Among other reasons for negative electrode aging are metallic lithium plating, decomposition of the binder, the loss of contact of active material particles due to a volume change, which forms capacity fade. Similar aging is perceivable in the positive electrode [17]. In summary, most of the aging is caused by lithium ions stalling. The mechanism within a battery cell and the construction of the battery itself could be described with the help of a lumped equivalent circuit model (ECM), as shown in Figure 2. A detailed ECM impedance model is elaborated in this figure based on [1; 18]. The positive electrode can be modeled with a Randles circuit (R\({}_{\text{PE}}\), C\({}_{\text{PE}}\), and W\({}_{\text{PE}}\)) [18]. R\({}_{\text{PE}}\) is the charge-transfer resistance within the positive electrode and can be described using a Nernst equation as the electrons (de)intercalcalating the positive electrode to/from the electrolyte. It is inversely proportional to the number of electrons n\({}_{\text{e,PE}}\) involved in the charge transfer in  
Figure 2: Decomposition of a battery impedance spectrum. the positive electrode, and the exchange-current density \(\mathrm{i_{0,ct}}\) as depicted in [19] and shown in Equation (1):  
\[\mathrm{R_{PE}}=\frac{\alpha_{ct}}{n_{ct,PE}} \tag{1}\]  
where \(\alpha_{ct}=\frac{RT}{F_{0,ct}}\) is a constant [19] which contains the universal gas constant (R), the temperature in Kelvin (T), and the Faraday constant (F). \(\mathrm{C_{PE}}\) is the double-layer capacitance to model the charge difference between the electrolyte and the positive electrode. \(\mathrm{W_{PE}}\) is the diffusion process to the electrode. Similarly, the negative electrodes share the same Randles-like structure (\(\mathrm{R_{NE}}\), \(\mathrm{Q_{NE}}\), and \(\mathrm{W_{NE}}\)). Due to the pores inside the negative electrode during the constructions, the capacitor is \"imperfect\" and is often substituted with a constant-phase element named \(\mathrm{Q_{NE}}\). In most lithium-ion cells, separating the charge transfer within positive and negative electrodes using impedance spectroscopy is not feasible without accessing internal battery cell components, and a composite electrode model can be instead considered [4]. In this case, the lumped ECM of the charge transfer is as described in Equation (2):  
\[\mathrm{R_{ct}}=\frac{\alpha_{ct}}{n_{ct}} \tag{2}\]  
here \(n_{ct}\) is the charge-transfer electrons involved in the charge-transfer phenomena and \(\mathrm{i_{0}}\) is the exchange current. Next is the separator, which is often viewed as a non-conductive RC circuit (\(\mathrm{R_{sep}}\), \(\mathrm{C_{sep}}\)). The electrolyte is modeled as a resistance (\(\mathrm{R_{E}}\)). Finally, the side reaction, which consists of lithium plating and SEI layer, and mainly represented as a thin film on top of the positive and negative electrodes, which are more impacted in the \"SEI layer\", can be described by an RQ circuit \(\mathrm{R_{SEI}}\) and \(\mathrm{Q_{SEI}}\) considering both of them are a reduction reaction.