The Method of Kramers-Kronig Transform Effective to the Impedance Spectrum of Lithium Battery
Authors: Kobayashi Kiyoshi , Sakka Yoshio , Suzuki Tohru S. 
Published: February 2020

Introduction

Kramers-Kronig (KK) transform test is well known to check the linearity, causality, and stability of measured impedance spectrum.1) By contrast, satisfaction of the KK transform indicates that the measured impedance spectrum can be reproduced by adequate equivalent electrical circuit using lumped parameter elements. Although the conventional KK transform test is convenient as pre-analysis of the impedance spectrum, this algorithm is difficult to apply to the impedance of the lithium battery because impedance spectrum of primary and secondary batteries is not satisfied the necessity condition of the conventional KK transform algorithm. The condition is that value of the imaginary part for impedance approaches to zero at angular frequency is close to 0.
In 2000, Ehm et al. derived a new KK relationship (ZHIT) which is applicable to the impedance spectrum of constant phase element (CPE).2) This means that the ZHIT algorithm has a possibility to use an impedance spectrum of lithium battery. However, we could not find reports on the ZHIT application to the impedance of lithium battery. In this study, two calculation algorithms of the KK transformation were examined to the impedance spectrum of lithium battery and usefulness of the ZHIT algorithm will be discussed.

Experimental

Two KK transformation programs were coded using the IgorPro software.3). One is based on the traditional equations which correspond to the linear transform.1) The second is based on the ZHIT equation which corresponds to the logarithmic transform.2) On both of the programs, calculation procedures of smoothing and expansion of the angular frequency range were not employed.
The impedance spectrum was measured using the purchased coin-type lithium battery (type CR1620) by combination of frequency response analyzer (1255, Solartron) with potentio/galvanostat (1287, Solartron). The spectrum was obtained at open cell circuit condition convoluting an amplitude voltage at 20 mV at room temperature. The frequency range was from 1MHz to 0.1Hz.

Results and discussion

In the case of the traditional KK transform algorithm, the KK transformability was not maintained because the calculated spectrum at low frequency region was completely different from the measured spectrum as shown in the figure. The difference between the measured spectrum and calculated one becomes large at the frequency region where the Warburg impedance is predominant.
Meanwhile, ZHIT result shows good agreement with the measured spectrum as shown in the figure. Actually, the measured spectrum can fit by some equivalent electrical circuit model and therefore, this spectrum would maintain the KK transformability in the meaning of a basic concept on the KK transform.1)

Although the KK transform check has not been applied to primary and secondary battery system, the ZHIT algorithm is found to be useful to lithium battery system. This method is effective to prejudge the impedance spectrum which can be solved by equivalent electrical circuit model or not. When one measures an impedance spectrum using deteriorated battery system, the system is not maintain the causality and stability of the fitting parameters. In this case, deterioration can be detected by the ZHIT analysis.

References

1. M. E. Orazem, B. Tribollet, “Electrochemical impedance spectroscopy”, Wiley, New Jersey, 2008
2 W. Ehm, H. Göhr, R. Kaus, B. Röseler, C. A. Schiller, ACH Models Chem., 137, 145-157 (2000) .
3. WaveMetrics, http://www.wavemetrics.com

Figure 1