Impedance Evaluation By Distribution of Relaxation Times Applied to a Lead-Acid Storage Battery
Authors: Vanysek Petr , Baca Petr , Křivík Petr 
Published: July 2021

Introduction – electrical impedance and its evaluation.

Electrical Impedance Spectroscopy (EIS) is used to identify the physical characteristics of a system or, in the absence of adequate equivalent circuits; it is at least used to compare overall impedance response as a change of a particular parameter. A novel approach to acquired impedance data treatment is to process them mathematically and obtain what is known as distribution of relaxation times, abbreviated as DRT. Here the method is evaluated to explore its suitability for evaluation of state of charge of a lead-acid battery.
The typical impedance evaluation is based on fitting to equivalent circuits. However, oftentimes both finding the equivalent circuit and the physical component assignment are tedious, if not impossible. Recently a new approach to data reduction has been treatment via distribution of relaxation times (DRT) [1, 2]. In principle, this approach has been outlined already in 1936 [3]. Its present use was made possible by available computation power of current PCs [4, 5]. While this is not a substitute to equivalent circuits and subsequent physical data interpretation, it is possibly a method, when used appropriately and in automated way, that can provide more information that simple comparison of the Bode or Nyquist graphs.
Here we treated impedance data from lead-acid battery studies by the DRT approach. Of interest are charged and discharged states of the battery as well as aging of the battery. During the operation of a lead-acid battery, its operating properties gradually deteriorate.

Experimental part

A lead-acid commercial maintenance free cell Hawker Cyclon 2 V/2.5 Ah (EnerSys) of spiral construction with AGM separator was used. The discharge step was carried out at current 0.4 A until the voltage dropped to 1.6 V. After that charging commenced using current of 0.4 A, with upper voltage limit 2.4 V, which took about 7 hours. During the experiment impedance of the cell was measured every approximately 150 s. The frequency range was 6 kHz to 351 mHz, 6 steps per decade, amplitude 5 mV.

DRT – resolving model data

Application of the DRT method makes it possible, without any preliminary considerations, to estimate how many meaningful time constants can be expected in the evaluated system. It is possible to illustrate, for example, that an impedance response generated by five RC circuits separated by 1/2 order of magnitude in a time constant, which in a Nyquist plot looks like a depressed semicircle, can be resolved into five peaks in DRT.
State of charge of a lead-acid battery can be investigated by impedance measurements [6-8]. A large series of the impedance data obtained in our laboratory previously from gradual discharge of a lead-acid battery was treated by the DRT routine and the response is shown in the attached figure (Impedance data treated by the DRT method for a lead-acid battery. Front to back a fully charged to a discharged state.). Remarkable is the growing-in feature, with also changing its time-constant over four orders of magnitude can be associated with the state of charge.

Conclusion

The mathematical method of applying the distribution of relaxation times numerical method to evaluation of large sets of data appears to be a promising approach to data reduction and subsequent studies of devices and material properties with changing impedance properties.

Acknowledgement

This work was supported with the help of the grant for specific research at the Brno University of Technology (FEKT-S-20-6206).

References

[1] BioLogic, Distribution of Relaxation Times (DRT): an introduction, BioLogic, Editor. 2017.
[2] Weese, J., A reliable and fast method for the solution of Fredhol integral equations of the first kind based on Tikhonov regularization. Computer Physics Communications, 1992. 69(1): p. 99-111.
[3] Yager, W.A., The Distribution of Relaxation Times in Typical Dielectrics. Physics, 1936. 7(12): p. 434-450.
[4] Shoar Abouzari, M.R., et al., On the physical interpretation of constant phase elements. Solid State Ionics, 2009. 180(14): p. 922-927.
[5] Wan, T.H., et al., Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools. Electrochimica Acta, 2015. 184: p. 483-499.
[6] Křivík, P., et al., Determination of state of charge of lead-acid battery by EIS. Journal of Energy Storage, 2019. 21: p. 581-585.
[7] Rodrigues, S., N. Munichandraiah, and A.K. Shukla, A review of state-of-charge indication of batteries by means of a.c. impedance measurements. Journal of Power Sources, 2000. 87(1): p. 12-20.
[8] Murariu, T. and C. Morari, Time-dependent analysis of the state-of-health for lead-acid batteries: An EIS study. Journal of Energy Storage, 2019. 21: p. 87-93.

Figure 1