The equivalent circuit model and Nyquist plot for a parallel R-C circuit are shown in Fig. 4.

The diameter of the semicircle shown along the real part in Fig. 4(b) indicates the resistance of R₁ in Fig. 4(a). Therefore, the resistance of R₁ is 100 Ω, which is the end of the semicircle along the real axis. The Nyquist plot was simulated using a capacitance of 100 mF. This capacitance value can be determined from the maximum point of the semicircle in the Nyquist plot. At the center of the semicircle, where Z_img is at a minimum, the values of ω_min, R₁ and C₁ satisfy the following equation.

Therefore, by identifying the R₁ and ω values when Z_img is at a minimum value, the capacitance value can be calculated by the above equation (6).

The simplified Randles model includes a double layer capacitor with bulk resistance (current collector, solution, etc.) and polarization resistance. The polarization resistor (R₁) is connected in series with the bulk resistor (R₂) and in parallel with the surface deposited double layer capacitor, as shown in Fig. 5(a). This model is fundamental for analyzing an electrochemical system. As real systems tend to be more complicated, additional elements are connected in this simplified Randles cell. The Nyquist plot of a simplified Randles cell is shown in Fig. 5(b).

In the Nyquist plot of a simplified Randles cell, the resistance of R₂ shifts the starting point of the semicircle to higher Z_real values. Considering that this spectrum was generated by R₂ = 20 Ω and R₁= 100 Ω, the 120 Ω at the end of this semicircle (low-frequency) intercept on the x-axis is the sum of the polarization resistance and the bulk resistance. Additionally, the diameter of the semicircle represents the polarization resistance (in this case, 100 Ω).

In special cases where semi-infinite linear diffusion affects an electrochemical system by kinetic and diffusion control, the Warburg impedance is used in the equivalent circuit model.

This model was presented by the Faraday Society in 1947, and the name of this model is taken from the name of the creator J.E.B. Randles[26]. The circuit model and the Nyquist plot for this cell are shown in Fig. 6. The Warburg impedance in Fig. 6(b) appears as a straight line with a 45° slope under semi-infinite conditions, and one-dimensional diffusion that is bound only by a large planar electrode on one side [25]. Along with the bulk and polarization resistance losses, this model represents the polarization in the lithium diffusion reaction as well. The Warburg coefficient, σ, in this case is 1 Ω·s^−1/2, and the other values are R₂ = 20 Ω, R₁ = 100 Ω, and C₁ = 100 mF, which are similar to the previous models.

An entire lithium-ion battery cell can be modeled using the equivalent circuit shown in Fig. 7. Although there are still challenges in interpreting the meaning of each semicircle in EIS spectra[27], this model follows the traditional interpretation of each arc. However, since this circuit model has many variables, it is necessary to minimize the circuit elements for practical impedance analysis.

Among these circuit elements, the capacitive effect of the separator is minimal compared to that of the other components. Therefore, it is not necessary to consider C_sep in setting up a circuit model[28]. Furthermore, the EIS spectra mainly come from the contribution of the working electrode in half-cell conditions, so the model does not need to consider the counter electrode. Additionally, the resistances R_{current collector}, R_electrolyte, and R_separator can be combined into the bulk resistance R_b. Moreover, since the capacitances of lithium-ion battery components are not ideal values, it is possible to obtain a more accurate model by replacing them with a constant phase element. The overall simplified equivalent circuit of a lithium-ion battery half-cell is shown in Fig. 8.

However, in the case of a half-cell system, the impedance value coming from the lithium counter electrode may affect the entire impedance spectrum. Therefore, in a certain case where the lithium counter electrode causes a relatively large impedance, there is an advanced measuring method assembling a symmetric cell [29] and a three-electrode cell [27,30] that only get the EIS spectra from the target electrode.

The above circuit model is based on the most fundamental lithium-ion battery half-cell system. It is useful to understand fundamental cell models; however, an equivalent circuit model representing a lithium-ion battery can be diverse depending on the electrode characteristics, cell type, and storage and cycling conditions [31]. Therefore, there is no single standard circuit model that may fit all battery systems. A customized circuit model can be devised by adding or subtracting electronic elements depending on the electrochemical cell characteristics [32–35].

After configuring the circuit model, it is necessary to ensure its accuracy. Zhai et al. reported the EIS determination algorithm using the digital signal processing (DSP) system and PC communication through ethernet/RS232 communication [36]. The EIS determination algorithm repeatedly matches the EIS fitting curve with the experimental data until the unknown value is obtained based on constructed circuit model. An open-source platform, the Impedance Analyzer, for easy-to-use physics-based analysis of experimental EIS spectra is available as well [37].

R_b represents the internal resistance value of the bulk materials in a battery, such as the current collector, electrolyte and separator. Since these values have a resistance only without capacitance or inductance, their effect can be observed in Z_real. This bulk resistance does not change significantly with the state of charge, but as cycling continues, the bulk resistance increases due to the depletion of the electrolyte and microcrack formation in the particles. Therefore, the bulk resistance is generally used as an indicator to determine the state of health (SOH) of aged cells.

Westerhoff et al. monitored the change in the internal resistance of a lithium-ion battery while the state of health decreased from 100 % to 86 %, as shown in Fig. 9(a) [28]. During the aging process, the impedance spectra moved gradually to higher Z_real values, and the shape of the semicircle changed as well. This change in the semicircle can be explained based on the frequency defined in Fig. 9(b). The starting point of impedance spectroscopy is referred to as f_ZIM,0 indicating purely ohmic resistance, and the maximum of the semicircle in the middle of the impedance spectrum correlates to the charge transfer process and is expressed as f_ZIM,max. The starting frequency of the diffusion process is defined as f_ZIM,min. In Fig. 9(c), the charge transfer associated with f_ZIM,max and lithium diffusion associated with f_ZIM,min moved to the lower f values as SOH decreased from 100 % to 86 %. However, the f_ZIM,0 shifts to higher f values, which demonstrates the internal resistance. Therefore, it can be noticed that the total resistance of the aged batteries are coming from the increase in the internal resistance and then the R_b can be used as an indicator of battery aging during cycling. An increase in the internal resistance was also observed during long-term storage. Eddahech et al. focused on performance degradation with storage time by comparing four different cathode materials (NMC, NCA, LFP, and LMO) with graphite counter electrodes and all of four aged cathodes show increased R_b value [38]. In addition, Schuster et al. investigated the aging process using a widespread aging matrix [39,40]. These reports suggested the use of R_b, which is sensitive to operating and storage conditions, as an indicator of the SOH rather than the use of R_SEI or R_ct.

The aging of lithium-ion batteries is caused by a combination of diverse phenomena. Formation of microcracks, gas evolution, binder decomposition, corrosion of the current collector and electrolyte depletion have all been reported as the cause of battery aging and performance degradation [41]. It is challenging to distinguish these various processes and identify the characteristics of aging. In many cases, measuring the R_b using EIS is a very convenient and critical method to evaluate the condition of the aged battery.

The first semicircle of the EIS spectrum in Fig. 8 originates from the impedance of a layer that forms on the interface between the electrode and electrolyte. This interfacial layer is called the solid electrolyte interphase (SEI) layer, and it is generated by the decomposition of the electrolyte. The SEI layer has a more profound influence on the electrochemical characteristics of the anode material than the cathode material. The characteristics of the layer, such as surface coating, gradient, surface area, etc., are changed. In addition, some nanostructured anode materials, such as Co₃O₄, Fe₂O₃, and MnO, shows a reversible film formation that contributes to additional capacity beyond the theoretical capacity [42–46]. Therefore, by monitoring the R_SEI, we can observe the reversible organic film formation as well as identify the characteristics of the SEI layer.

Zhang et al. studied the formation of the SEI layer on the surface of graphite using the EIS technique under various electrochemical cycling conditions [47]. First, the characteristics of the SEI layer were studied at different states of charge/discharge. A relatively resistive preliminary SEI is formed at approximately 0.15 V, and then a highly conductive SEI layer forms between 0.15–0.04 V. Accordingly, it can be confirmed that SEI layers with two different characteristics are formed according to the voltage region in the graphite anode. The resistance of the SEI layer also depends on the constituents of diverse electrolytes, such as solvents (EC, EMC, MB, etc.) and lithium salts (LiBF₄, LiSO₃CF₃, LiBOB, LiPF₆, etc.). It was found that R_SEI is very sensitive to the reactivity of the electrolyte during the first lithiation cycle. Furthermore, Steinhauer et al. monitored the SEI formation by calculating the distribution relaxation time (DRT) from the measured impedance spectra to gain more insight into the number of processes involved in the cell [48]. They compared the EIS spectra for 2- and 3- electrode cells to study the effect of SEI film formation on the lithium counter electrode. Abarbanel et al. investigated the cause of the impedance increase when Li[Ni_0.42Mn_0.42Co_0.16]O₂ (NMC442)/ graphite lithium-ion cells operate at high voltages beyond 4.4 V using the transmission line model (TLM) [35]. It was found that the impedance increases in the high-frequency region when operating at high voltage are mainly caused by the transfer resistance through the SEI layer along with the diverse influences of the contact resistance (between the carbon conductor and active material), the electrical path resistance through carbon black and the ionic path resistance through the electrolyte solution imbedded in the pore structure.

Moreover, in the impedance spectra of Li/C cells, an inductive loop can be observed in some cases because of the unstable nature of the SEI layer. Once a stable SEI has formed, the inductive loop does not appear in the impedance spectra. The formation and disappearance of the inductive loop in the impedance spectra can be used to assess the robustness of the SEI film on the anode [49].

In addition, there is a previous report that reversible SEI formation contributes to extra capacity in high-energy sodium conversion systems. Shubhra et al. observed the impedance spectra of a MoS₂ electrode at eight specified potentials, as shown in Fig. 10 [50]. In the initial stage of discharge, 0.8 V, two semicircles are observed, which represent SEI formation and Na ion intercalated in MoS₂ on the electrode surface, respectively. Then, at 0.4 V, the small semicircle in the intermediate frequency region disappears, and a semicircle with a high R_sl (migration resistance through SEI layer) value was observed due to the formation of a thick SEI layer on the surface and distorting of the host material. At the final stage of discharge, 0.01 V, a single semicircle in the entire frequency range, dominates the impedance value of the SEI layer. However, during charging, R_sl at 0.5–1.5 V starts to decrease dramatically and continues to decline until the end of the charging process. This phenomenon indicates that the thickly formed SEI layer was dissolved or destroyed during discharge due to the volume change of the electrode material. As the electrode is charged to 2.0 V, the semicircle correlating to the SEI layer is further reduced, and incomplete MoS₂ formation generates a middle-frequency semicircle. When charged to 2.6 V, the high-frequency region representing the SEI layer is stably maintained without any change. This partial decomposition of the SEI layer resulted in the extra capacity over the theoretical value.

The R_SEI indicates resistance from an interfacial layer that is generated by the decomposition of the electrolyte. Initially, the studies on this SEI layer focused on understanding the degradation of graphite electrodes by revealing the relationship between the resistance characteristics of the SEI layer and the cycling conditions. However, in recent studies on nanoscale electrode materials for high-performance lithium-ion batteries, the SEI layer began to be considered as another factor resulting in extra capacity through reversible reactions. The reversible SEI formation with extra capacity has not yet been clarified in detail. Therefore, analyzing the SEI layer using EIS is very useful for observing not only the resistance of the anode material but also extra capacity from reversible SEI layer formation, which is generally called a polymeric gel-like film.

The second semicircle in the EIS results in Fig. 8 is related to the R_ct. R_ct is linked to the kinetics of an electrochemical reaction and is changed by the surface coating, phase transition, bandgap structure and particle size. Moreover, R_ct shows a dramatic change in response to a temperature change. In particular, at low temperatures (≤ 20°C), the total resistance of the lithium-ion cell is determined by R_ct. Therefore, monitoring R_ct is useful for understanding electrode reaction processes that occur during the charge-discharge process and the characteristics of lithium-ion batteries depending on the temperature changes.

Lee et al. compared the R_ct of pristine LiNi_0.5+x Co_0.2Mn_0.3−xO₂ (x= 0, 0.1, and 0.2) and clearly identified the impedance difference according to the Ni contents of the electrode material[51]. Each pristine electrode had only one charge transfer-related semicircle in the impedance spectra. The fitting results showed the lower Ni-content NCM (523) has a higher R_ct value of 127.42 Ω, and the higher Ni-content NCM (721) has a lower R_ct value of 79.45 Ω. This report suggested that the increase in the Ni contents in the Co-fixed NCM system will have a beneficial effect on the electron conductivity, showing R_ct dependence on the transition metal composition. In addition, Zhang et al. applied the EIS analysis to confirm the R_ct dependence on voltage and temperature [52]. The Ni-based cathode/graphite cell was cycled in the voltage range of 3.1–4.2 V, and it was found that R_ct changed more than R_b or R_SEI. R_ct, which accounts for more than 60% of the total cell resistance, predominantly determines the cell resistance. Especially below 3.0 V, the ratio of R_ct to the total resistance exceeds 90% and reaches almost 100% in a completely discharged state. Ogihara et al. reported that the tendency of cell resistance changes at porous electrodes depends on the temperature by using four parameters: electric resistance (R_e), electrolyte bulk resistance (R_sol), ionic resistance in pores (R_ion), and charge transfer resistance for lithium intercalation (R_ct) in Fig. 11(a) [53]. The plots can be described as TLM for cylindrical pores for faradaic processes at all temperatures. Upon decreasing temperature, the real impedance component at 100 kHz, which determines the value of R_sol, slightly shifts toward more positive values, and the length of the 45° slope line, which reflects R_ion, increases, as shown in Fig. 11(b). This indicates that two resistances somewhat increase with decreasing temperature. However, the semicircle in the low-frequency region, which corresponds to R_ct, is significantly enlarged with decreasing temperature. It seems that the temperature dependence of R_ct is larger than that of R_sol or R_ion. Based on recorded EIS spectra, fitting was performed for a clear comparison of each resistance parameter. As shown in Fig. 11(c), R_e is approximately 2~4 orders smaller in magnitude than the others, and there is only a small change in temperature. R_ion and R_sol have relatively large values and moderate temperature dependencies. On the other hand, R_ct exhibits a strong temperature dependence and becomes the largest resistance value among the four resistance parameters in the electrode when the temperature is lower than 34.8°C.

Ogihara et al. investigated the relationship between the energy density and power density of LiNiO₂-based porous electrodes by monitoring the resistance change according to the coated electrode thickness on aluminum foil [54]. A symmetric cell was assembled to remove the effects of the Li counter electrode, and the TLM was used to separate R_ion and R_ct from the porous electrode resistance. The electrochemical results for various thicknesses (25~75 μm) show that as the thickness increases, the energy density increases while the power density decreases. The improved energy density is easily understood as an increase in the portion of the active material relative to the total cell volume. However, the cause of the power density change is divided into two regions according to decreasing tendency: region I (thin electrode), the power density slightly decreases; region II (thick electrode), the power density dramatically decreases. These two separate regions are determined by the correlation between R_ct and R_ion, which accounts for the largest portion of the total cell resistance. The EIS fitting results demonstrate that the R_ct value is the main influence on the thin electrode, while the R_ion value is the dominant factor in the thick electrode. Since the R_ct value is inversely proportional to the charge-transfer active area, it gradually decreases as the thickness increases. Conversely, because the R_ion value increases in proportion to the lithium diffusion length, the larger the thickness is, the larger the amount of R_ion. Therefore, the power density slightly decreases in region I because only R_ct affects the rate capability. However, region II shows a dramatic decrease in power density since an increase in R_ct as well as R_ion affects the power density.

R_ct is the largest value among the total cell resistance at room temperature. The power density of a lithium-ion battery is mainly determined by R_ct. To reduce the R_ct value, many studies have examined the effect of electrode properties such as the Ni content, temperature and electrode thickness on the R_ct value. In addition, since R_ct varies depending on the charge/ discharge state, it can contribute to understanding the reaction mechanism by comparing the resistance changes over voltage ranges. Therefore, an analysis of the R_ct value would be helpful not only in understanding the reaction mechanism of the lithium-ion battery but also in research for the optimization of the cycling performance of commercial cell systems.

The straight line immediately after the semicircles in Fig. 8 represents the Warburg impedance associated with the lithium-ion diffusion process of the electrode material. The D_Li+ calculation generally proceeds using the following equation proposed by Ho et al. [55].

Using equation (7), the diffusion coefficient can be easily calculated from the low-frequency area. However, since the above equation does not consider the phase transition of the electrode material, the calculation of the diffusion coefficient is only possible for solid-solution behavior (i.e., the single-phase region) [56]. Accordingly, the Warburg impedance analysis is mainly used to calculate the diffusion coefficient of a cathode material that shows a single-phase reaction, combined with galvanostatic intermittent titration (GITT), potentiostatic intermittent titration technique (PITT) and cyclic voltammetry (CV)[57–60]. Xia et al. measured the chemical diffusion coefficients of c-axis oriented LiCoO₂ thin films using EIS and PITT and clearly observed the order/disorder transition phase boundaries near the composition Li_0.5CoO₂ [57]. Liu et al. synthesized a LiFePO₄/C nanocomposite and confirmed the lithium intercalation and deintercalation diffusivity by CV and EIS [60]. Xie et al. prepared a LiMn₂O₄ thin film by pulse laser deposition on stainless-steel substrates and characterized the lithium diffusion using CV, PITT, and EIS [61]. Rui et al. measured the chemical diffusion coefficients of lithium ions (D_Li+) in Li₃V₂(PO₄)₃ by dividing the system into a two-phase region and single-phase region [62]. The chemical diffusion coefficient of the two-phase region was measured through GITT and CV. The single-phase region from 4.70 to 3.68 V(Li_2.5V₂(PO₄)₃→Li₃V₂(PO₄)₃) during the discharge process was calculated integrally using GITT, CV, and EIS techniques. In addition, the Warburg impedance has been used to calculate the chemical diffusion coefficient for negative electrode materials such as Si. Ding et al. measured the diffusion coefficient by EIS as mentioned above and compared it to the results from different analysis techniques, such as CV and GITT methods[63]. The diffusion coefficients of lithium ions in nano-Si at different charge-discharge states were determined by EIS and GITT. The D_Li+ values calculated from EIS range from 10⁻¹³ to 10⁻¹² cm² s⁻¹. The results from GITT are slightly higher than those from EIS due to its nonequilibrium states (30 minutes was given before measurement for stabilization in EIS, but GITT gave only 10 minutes). Although the actual values of the EIS and GITT analytical methods were different, it was found that the general trend in the graph shown by the variation in the lithium concentration in the silicon electrode was the same as a “W” type.

The Warburg impedance at low-frequency is directly related to the diffusion of lithium-ions. However, there are limitations to EIS measurements at two-phase transitions during lithium insertion/ extraction. Therefore, it is necessary to comprehensively use several techniques for accurate analysis.