# Detailed explanation of the application of electrode feature analysis

Figure 1 shows a single porous electrode for a lithium secondary battery, and its equivalent circuit is shown in Figure 2. Figure 3 shows the corresponding Nyquist diagram, consisting of a high-frequency region, a diffusion-controlled mid-frequency region (Warburg impedance), and a charge-saturated low-frequency region.

Figures 4 and 5 are the impedance spectra of the “carbon/electrolyte/lithium” battery and the “LiCo0z/electrolyte/lithium” battery, respectively. Since the impedance characteristics include the test (working) electrode and the lithium counter electrode, they differ from the theoretical map explained above. Given the presence of current between the working and counter electrodes, it is not possible to directly distinguish these results. When testing the AC impedance, even if the working electrode and the counter electrode are exchanged, the result is still the same.

1. Application analysis (1): AI/LiCoOz/electrolyte/carbon/Cu battery

The above battery is shown in FIG. 6 , and its equivalent circuit is shown in FIG. 7 . In this system, an electric double layer is formed at the electrode-electrolyte interface. Assuming there is a charge transfer reaction, impedance analysis can be performed as follows.

Impedance as defined by Equation (Figure 8) can be expressed in terms of resistance, capacitance, inductance and Warburg impedance.

Here, Zw is the Warburg impedance, resulting from diffusion within the cell. When the interfacial reaction is dominated by charge transfer and one-dimensional diffusion, the Zpw of the positive electrode is a frequency-dependent function. As shown in formula (Figure 9), formula (Figure 10) and formula (Figure 11), it is represented by the series connection of resistor Rf and capacitor Cf.

where Sp is a function related to the diffusion coefficient (Fig. 12).

In the formula, A represents the area; n is the number of electrons; F is the Faraday constant; D is the diffusion coefficient;
The impedance analysis of the negative electrode is similar to that of the positive electrode, and its impedance is defined by Eq. (Figure 13). Placing the formula (Fig. 13) in the complex plane yields the Cole-Cole diagram of Fig. 14.

1. High frequency region
Impedance in this frequency range is related to the movement of electrons at a certain frequency, the Warburg impedance and RC are connected in parallel, and the circuit is ignored. Ion motion is impossible, and the impedance can be represented by Eq. (Fig. 15).
2. Intermediate frequency area
At this frequency the Warburg impedance and inductance are negligible, and the impedance is represented by Eq. (Figure 16).

The above equation corresponds to the equivalent circuit of two RC parallel circuits with a resistor (RS) in series, when CPRP,
, the term containing ω can be omitted. To draw a Nyquist diagram, it can also be represented by Eq. (Fig. 17). A semicircle of radius (RP+Rn)/2 is shown on the Nyquist diagram.

Due to the different materials used in the negative and positive electrodes, the electrodes have different relaxation times. When r, >>T, the arc corresponding to the smaller r appears in the high frequency region and the two arcs are separated. The size of , has a great influence on the output of information at the electrode and electrolyte interface. Taking into account the incomplete semicircle and scattered relaxation times, it can be represented by the non-idealized components in Eq. (Fig. 18).

Figure 19, Table 1, and Figure 20 correspond to the two parallel RC equivalent circuits, the parameters for establishing the model, and the Nyquist diagram, respectively. If the resistance and capacitance are the same, the two RC circuits behave as a single RC circuit, forming curve a in Figure 20. If the capacitance on one side is small, the Nyquist plot is shown in curve d. In most cases, the battery shows an intermediate curve between curve a in Figure 20 and curve d in Figure 20.

1. Low frequency region dominated by Warburg impedance
Within this region, all other terms except ω-1/2 can be omitted (Figure 21).

After omitting other terms, the graph forms a straight line with a slope of 45° in the low-frequency region of the complex plane. This region is dominated by ion diffusion (Figure 22).

1. Ultra-low frequency region
The impedance curve in the complex plane has a rapid upward trend in the low frequency region and can be expressed as follows. As shown in the equation, B represents the electrode potential as a function of concentration (Figure 23).

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