Lithium ion cells are currently among the common products of a number of renowned companies (SAFT, VARTA, Sony, Duracell, etc.). The charged cell of the usual design has an open circuit voltage of 2.4–3.7 V and its energy density ranges from 80 to 260 Wh/kg. Self-discharge is about 5–10% of the capacity per month. The cell has a long cycle life and after 500 of cycles capacity of LI-ion cell decreases by 10–20%. The electrodes of these cells are very thin (around 200 µm) and are made of intercalating compounds (compounds that can accept an atom or molecule into their crystal lattice).

The active material of the positive electrode are metal compounds, the 6 most common types of Li-ion batteries include Lithium Cobalt Oxide (LiCoO2) - LCO (ICR), Lithium Manganese Oxide (LiMn2O4) - LMO (IMR), Lithium Nickel Manganese Cobalt Oxide (LiNiMnCoO2) - NMC (INR), Lithium Iron Phosphate (LiFePO4) - LFP, Lithium Nickel Cobalt Aluminum Oxide (LiNiCoAlO2) - NCA (NCR) and Lithium Titanate (Li2TiO3) - LTO. The negative electrode is carbon (graphite). These substances must be sufficiently porous. The matrix must very well receive (intercalate) lithium ions and again easily release them. The collectors of the negative electrodes are usually made of copper foil, the positive electrodes are aluminum foil. Active electrode materials are applied to the collectors. The separators are usually made of a very thin porous film made of polyethylene or polypropylene or a microporous polymer film. The electrolyte is a lithium salt (LiPF6, LiBF4, or LiClO4) and an organic solvent (ether, various mixtures of ethylene-, propylene-, dimethyl- or diethyl carbonate, etc.). The liquid electrolyte is conductive for lithium ions, which travel between the electrodes during discharging and charging [4].

The chemical process in the cell consists only in the transport of lithium ions. During charging, positively charged lithium ions travel to the negative electrode, where they are deposited in free spaces in the carbon structure. When discharging, the opposite process takes place - lithium ions travel to the positive electrode, where they are deposited in free spaces in the crystal lattice of the positive active material. During the discharge, the impedance of the Li-ion cell is changing, including ohmic resistance *Rs*, resistance and capacitance of SEI layer (*Rsei, Csei*), charge transfer resistance *Rct*, double layer capacity *Cdl* and Warburg impedance *Zw*. Electrochemical impedance spectroscopy (EIS) is a suitable method for determining these parameters. This method is suitable for monitoring the impedance of a Li-ion cell over a wide range of frequencies. Results are displayed by the impedance diagrams [2].

The electrochemical impedance of a battery *Z* is a complex number, frequency-dependent, described either by its real and imaginary parts Re (*Z*) and Im (*Z*), or by its modulus |*Z*| and phase angle *ϕ*. Different frequencies reflect different parameters of the Li-ion cell, from ohmic resistance through resistance and capacitance of the SEI layer, charge transfer resistance at the electrodes, diffusion double layer capacity to Warburg impedance related to ion diffusion in the electrolyte and electrode pores. The SEI layer is a passivation layer formed on the surface of the negative electrode materials of lithium-ion batteries.

The general shape of the Nyquist diagram of the complex electrochemical impedance of the battery is shown in Fig. 1. This diagram starts at the highest frequencies around 5 kHz below the x-axis, where the circuit inductance and skin effect predominate. At frequencies around 1 kHz, the ohmic resistance of the *Rs* cell is presented in the range of mΩ, the imaginary part of the impedance is close to zero. The ohmic resistance includes the resistance of the interconnection, the separator, the electrolyte and the both electrodes. The first arc appears at frequencies in the hundreds of Hz. Here, the phenomena in the SEI layer are applied, which represents the resistance and capacity of the SEI layer. The resistance of the SEI layer indicates the size of the first arc. The second arc appears at frequencies in tens of Hz. Here, the charge transfer resistance at the electrodes *Rct* and the capacitance of the double layer *Cdl* caused by the distribution of the space charge in the electrochemical double layers are presented. The charge transfer resistance indicates the size of the second arc. At the lowest frequencies, the Warburg impedance is presented due to the diffusion of ions in the electrolyte and in the electrodes. All of these parameters can be represented by an equivalent circuit [1–3].

The cell impedance is:

$$Z={R}_{S}+\frac{{R}_{sei}}{j\omega {C}_{sei}{R}_{sei}+1}+\frac{{R}_{ct}}{j\omega {C}_{dl}{R}_{ct}+1}+{Z}_{w}$$

,

where \({Z}_{w}=\frac{\sigma }{\sqrt{\omega }}-j\frac{\sigma }{\sqrt{\omega }}\), \({C}_{sei}{R}_{sei}={\tau }_{sei}\) and \({C}_{dl}{R}_{ct}={\tau }_{dl}\) (1)

*Zw* is the Warburg impedance, *σ* the Warburg coefficient [Ωs− 1/2] and *ω* the angular frequency [s− 1], *τsei* is time constant of SEI layer, *τdl* time constant of diffuse double layer [s].

From the measured values of the cell impedance at different frequencies, individual parameters of the measured impedance can be found by fitting. When fitting the Nyquist diagram around the middle frequencies corresponding to the SEI layer area, it was shown that the impedance values obtained from the equivalent circuit do not correspond to the measured values and it is necessary to use the constant phase element (CPE) *Q*, better corresponds to the measured data [5–8]. For a constant phase element, the phase angle of its impedance is independent of frequency and its value is (90°. *α*). It is caused by the dispersion of capacity caused by inhomogeneities, which result in uneven current and potential distribution. CPE is related to the two-dimensional distribution of frequencies or time constants (due to the surface heterogeneities, non-uniform distribution of charge on the electrode surface) and to the three-dimensional distribution of frequencies or time constants (due to the porosity and roughness of the electrode surface).

The impedance of the CPE element is equal to:

$${Z}_{CPE}=\frac{1}{{\left(j\omega \right)}^{\propto }{Q}_{sei}}$$

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,

where *Q**SEI* is a type of capacitance with a unit dependent on *α* [F.sα − 1].

Together with the *R**SEI* resistor of the SEI layer connected in parallel to the CPE, an *RQ* element, or ZARC element, is created. Its impedance is calculated as:

$${Z}_{ARC}=\frac{{R}_{sei}}{{\left(j\omega \right)}^{\propto }{Q}_{sei}{R}_{sei}+1}$$

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In the Nyquist diagram, the ZARC element creates an arc whose center is for *α* < 1 below the x-axis, see Fig. 2.

The cell impedance with using the ZARC element is:

$$Z={R}_{S}+\frac{{R}_{sei}}{{\left(j\omega \right)}^{\propto }{Q}_{sei}{R}_{sei}+1}+\frac{{R}_{ct}}{j\omega {C}_{dl}{R}_{ct}+1}+{Z}_{w}$$

4

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