Time Domain Analysis on Current Response of Equivalent Circuit Involving Constant Phase Element

Electrochemical impedance spectroscopy (EIS) is powerful method to investigate electrolyte/electrode interface of various energy devices and electrochemical phenomena. The electrolyte/electrode interface is simply represented by a parallel circuit of charge transfer resistance (R_ct) and double-layer capacitance (C_dl), and its impedance spectrum describes a true semicircle on the Nyquist plane theoretically. However, the depressed semicircle is often observed in the experimental result of the impedance spectrum. The depressed semicircle has been explained by distribution of various relaxations at electrolyte/electrode interface [1, 2]. Burg and co-workers [3] reported the relation between constant phase element (CPE) and distribution of time constants and proposed the calculation method of a capacitance from CPE parameters on the basis of ladder equivalent circuit. In the present paper, a time domain data of current response by parallel circuit of CPE and resistance were calculated by the following algorithm in order to attempt the separation of time constants in the current response in the time domain.

(1) The pulse voltage V (t) in the time domain including the data of the 2^N unit is converted into the voltage V (ω) in the frequency domain by a fast Fourier transform (FFT).

(2) The impedance Z (ω) is calculated as a complex number at each frequency from the equivalent circuit involving CPE.

(3) The current I (ω) in the frequency domain is calculated by dividing Z (ω) by V (ω) in the frequency domain.

(4) I (ω) is converted into the time domain current I (t) by an inverse fast Fourier transform (IFFT)

(5) The number and value of the time constant of I (t) are determined by curve fitting of the ladder equivalent circuit.

(6) The impedance spectrum calculated by (5) is compared with that by the equivalent circuit involving CPE.

The calculated results indicate that the current response by the equivalent circuit involving CPE is a sum of those with plural time constants. It was found that the distribution of time constants is concerned with the CPE index (p) remarkably. In addition, the relation between the CPE constant (T_CPE) and C_dl was discussed.

References

[1] Z. Kener, T. Pajkossy, J. Electroanal. Chem. 448 (1998) 139.

[2] C. A. Schiller, W. Strunz, Electrochim. Acta 46 (2001) 3619.

[3] G. J. Burg, V. D. Eeden, M. Sluyters-Rehbach, J. H. Sluyters, J. Electroanal Chem., 176 (1984) 275.