Equivalent circuit models (ECMs) of measured impedance typically have been the basis for organizing and estimating the impedance of lithium-ion batteries and other energy storage devices. Assuming a general impedance measurement may satisfy the Kramers-Kronig relations of linearity, it could be reasoned that an ECM of linear components should model a pseudo-linear impedance measurement with considerable confidence [1, 2]. Fundamentally, this is based on the optimization an objective function that represents the total impedance of the ECM, with the coefficients of the function being the circuit parameters of the model. However, there is always an inherent uncertainty in impedance measurements and model results due to either stochastic noise or bias from statistical or computational error in the measurement device and by the modeling device [3]. Despite this uncertainty in the impedance measurements to the manufacturer’s noise specifications for the potentiostat used and uncertainty in ECM parameters due to the computational error in fitting an ECM to measured impedance, ECMs demonstrate an ability to have a good fit. However, what determines that the estimated parameters are the truly accurate values for an impedance measurement when there are many sets of parameter values that can produce a model that fits an impedance pattern? For one, a knowledge or assumption of the distribution of the variation in the impedance measurements must be made. Then a Monte Carlo approach of generating further impedance samples similar to that distribution can be simulated, with the general ECM fitted to each of the simulations. When this is performed a large number of times (≈ 1000), the spread of the impedance parameters should provide a distribution and variance that is statistically significant and accurate to the type of distribution specified [4]. The most likely case is a Gaussian distribution, and this method is performed under this condition to due general noise behavior in electrical signals. Furthermore, resampling via bootstrapping and simulating the bootstrapped impedance values or even simply simulating the impedance of the used ECM with the mean values each of the parameter spreads should provide a demonstration of validity to the parameter values as an accurate representation of the impedance of the tested device.

This work is applied to the impedance spectra of an in-house fabricated Li-O₂battery to determine the impedance of parameters of the cell within a 95% confidence. The high level of confidence provides some accuracy in describing the overall impedance organization, and more interestingly, the type of diffusion that is occurring, as depicted in Figure 1. Typically, diffusion of these devices or any Li-ion battery may be generally described as simply Warburg diffusion when a deeper investigation shows that a more complex, anomalous diffusion can be the more accurate classification of the observed phenomena with a high level of confidence.

 

References:

 

1. P. Agarwal, M. E. Orazem, Application of Measurement Models to Impedance Spectroscopy: Evolution of Consistency with the Kramers-Kronig Relations, J. Electrochem. Soc., Vol. 142, 12, Dec. 1995.
2.  S. Erol, M. E. Orazem, Influence of Overcharge of Over-Discharge on the Impedance Response of LiCoO2|C Batteries, J. Power Sources, Vol. 270, 15, Dec 2014.
3. J. Randa, Noise-Parameter Uncertainties: A Monte-Carlo Simulation, J. Res. Natl. Inst. Stand. Technol., Vol. 107, 5, Oct 2002.
4. R. J. Smith, M. H. Weatherspoon, Noise Temperature Meausrement Uncertainty Analysis Using Monte Carlo Simulations, 69^thARFTG Conference, 2007.