Introduction

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 510% of the capacity per month. The cell has a long cycle life and after 500 cycles, the 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 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, and lithium nickel cobalt aluminum oxide (LiNiCoAlO2)—NCA (NCR). The negative electrode is carbon (graphite). The second possible material for the negative electrode is lithium titanate (Li2TiO3)—LTO. 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 [1].

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 intercalated in the carbon structure. When discharging, the opposite process takes place—lithium ions travel to the positive electrode, where they are intercalated in the positive active material structure. 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 [3].

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 m\(\Omega\), 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 [4].

Fig. 1
figure 1

Nyquist impedance diagram and equivalent Li-ion cell circuit

Full size image

The cell impedance is:

$$ Z = R_{{{S}}} + \frac{{R_{{{{sei}}}} }}{{j\omega C_{{{{sei}}}} R_{{{{sei}}}} + 1}} + \frac{{R_{{{{ct}}}} }}{{j\omega C_{{{{dl}}}} R_{{{{ct}}}} + 1}} + Z_{{\text{w}}} , $$
(1)

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

Zw is the Warburg impedance, σ the Warburg coefficient [\(\Omega\)s−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 [5,6,7,8]. 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:

$$ Z_{{{\text{CPE}}}} = \frac{1}{{\left( {j\omega } \right)^{ \propto } Q_{{{\text{sei}}}} }}, $$
(2)

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

Together with the RSEI resistor of the SEI layer connected in parallel to the CPE, an RQ element, or ZARC element, is created [9]. Its impedance is calculated as:

$$ Z_{{{\text{ARC}}}} = \frac{{R_{{{{sei}}}} }}{{\left( {j\omega } \right)^{ \propto } Q_{{{\text{sei}}}} R_{{{{sei}}}} + 1}}. $$
(3)

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

Fig. 2
figure 2

Geometry of a ZARC element with α = 0.8

Full size image

The cell impedance with using the ZARC element is:

$$Z = R_{S} + \frac{R_{\text{sei}}}{\left( {j\omega } \right)^{ \propto } Q_{{\text{sei}}} R_{{\text{sei}}} + 1} + \frac{R_{ct}}{{j\omega}C_{dl} R_{ct} + 1} + Z_{w}.$$
(4)

The aim of this study was to compare the two models that describe the SEI layer, i.e. using the parallel connection of the resistance and capacity of the SEI layer or using the ZARC element—the parallel connection of the resistance and constant phase element QSEI. From the results of the Nyquist diagram, it is clear that the use of the ZARC element more accurately describes the properties of the SEI layer and the Li-ion cell [9].

Results and discussion

Figure 3 is Nyquist diagram during intermittent discharge of a Li-ion cell. SoC corresponds to a state of charge from 100 to 0%. The measured impedance values were compared with the values obtained from impedance calculations from the cell equivalent circuit. These dependences for a charged cell with an equivalent circuit without CPE and with CPE are shown in Figs. 4 and 5 in the form of Nyquist diagram and in Figs. 6 and 7 in the form of frequency dependence of real and imaginary Z values.

Fig. 3
figure 3

Nyquist diagram of Li-ion cell during intermittent discharge (SoC = 100–0%)

Full size image
Fig. 4
figure 4

Nyquist diagram of a charged Li-ion cell (SoC = 100%) for measured and fitted impedance values without CPE element

Full size image
Fig. 5
figure 5

Nyquist diagram of a charged Li-ion cell (SoC = 100%) for measured and fitted impedance values with CPE element

Full size image
Fig. 6
figure 6

Frequency dependence of real and imaginary Z values for measured and fitted impedance values without CPE element

Full size image
Fig. 7
figure 7

Frequency dependence of real and imaginary Z values for measured and fitted impedance values with CPE element

Full size image

When compare Figs. 4 and 5, or 6 and 7, it is clear that the use of a equivalent circuit with a CPE element gives an impedance curve that corresponds much better to the measured data. Coefficient of determination without CPE element R2 = 0.987, with CPE element R2 = 0.997. The highest difference between the measured and fitted data are for equivalent circuit without a CPE element in the area of medium frequencies (hundreds of Hz), where the influence of the SEI layer is manifested.

The changes in the values of the cell equivalent circuit parameters during discharge without CPE and with CPE are shown in Figs. 8, 9, 10 and 11. DoD (depth of discharge) varies from 100 to 0%.

Fig. 8
figure 8

The dependence of Rs, Rct, and Rsei of the battery during intermittent discharge without CPE (DoD = 0–100%)

Full size image
Fig. 9
figure 9

The dependence of Rs, Rct, and Rsei of the battery during intermittent discharge with CPE (DoD = 0–100%)

Full size image
Fig. 10
figure 10

The dependence of Cdl and Csei of the battery during intermittent discharge without CPE (DoD = 0–100%)

Full size image
Fig. 11
figure 11

The dependence of Cdl and Qsei of the battery during intermittent discharge with CPE (DoD = 0–100%)

Full size image

It can be seen that the ohmic resistance Rs gradually increases during discharging. This increase is accelerated at the end of discharge. During discharge lithium intercalates into the structure of the positive electrode and increase its resistance. Measured results reveal a drastic change in the electronic conductivity occurring at an early stage of lithium deintercalation (or at the end of lithium intercalation), which may be caused by a transition from polaron to metal. In the case of LixNiCoAlO2 (NCR) or LixCoO2 (LCO) materials, such a transition has been proposed to be responsible for the existence of a two-phase region between the lithium concentrations of x = 0.95 and x = 0.75, namely, LixNiCoAlO2 or LixCoO2 is a semiconductor for x > 0.95 (end of discharge), and is of metallic property for x < 0.75 [10].

The charge transfer resistance Rct decreases up to about 50% of DoD during discharging. At the end of the discharge, on the other hand, Rct rises sharply. It is obvious that at the end of the discharge, lithium is added to the active mass of the positive electrode and fills the remaining free spaces in the structure of the positive electrode, and this is reflected in the increase of the charge transfer resistance.

The resistance of the SEI layer, Rsei, does not change much during discharge, its magnitude is about 0.05 \(\Omega\) for an equivalent circuit without CPE, comparable to the value of Rct between 20 and 80% of discharge. For a equivalent circuit with CPE, the size of Rsei is higher, around 0.09 \(\Omega\).

The course of capacities differs for both equivalent circuits. When using a equivalent circuit without CPE, the capacity of the double layer Cdl during discharging decreases up to about 50% DoD and at the end of discharge Cdl increases sharply.

The capacity of the SEI layer, Csei, has a similar course as the capacity of the Cdl double layer. Compared to Cdl, it is much lower by 1 to 2 orders of magnitude. When using equivalent circuit with CPE, the capacity of the double layer Cdl increases during discharge. Its size is comparable to the capacity of a double layer Cdl from equivalent circuit without CPE.

The course and size of the Qsei from equivalent circuit with CPE differs from the Csei from equivalent circuit without CPE. After an initial sharp decrease, Qsei increases slightly during discharge. The size of Qsei is four times higher than Csei. The value of the parameter α is around 0.7.

Conclusion

Nyquist diagrams of the Li-ion cell during discharge were measured. Important parameters of the battery equivalent circuit were found from the diagrams. These parameters indicate the changes of the impedance of the Li-ion cell in different states of discharge. The use of CPE element in the cell equivalent circuit increased the accuracy of the regression function parameter estimation.

Experimental

A Li-ion cell NCR18650B with a nominal capacity of 3350 mAh, a nominal voltage of 3.6 V, an energy density: 676 Wh/l, 243 Wh/kg was used to measure the impedance changes. The positive electrode is composed of LiNiCoAlO2, the negative electrode of graphite, electrolyte: ethylene carbonate, diethyl carbonate, salt: lithium hexafluorophosphate. The cell was placed in a temperature chamber with a temperature of 34.4 °C and subjected to intermittent discharging with a current of 0.4 A for 1 h, followed by current off period for 2 h. The measured discharged capacity was 3314 mAh. The voltage limitation at the end of discharging was 2.5 V. Impedance changes were measured by EIS during the experiment. The voltage amplitude at EIS was set at 5 mV, frequency from 6 kHz to 50 mHz, 6 measurements per decade.