(Invited) Joint Time-frequency Analysis of Lithium-ion Batteries Undergoing Dynamic EIS
Authors: Schwartz Daniel T 
Published: February 2020

Joint time-frequency analysis (JTFA) is an important domain of signal processing that is widely applied as a real time diagnostic tool in acoustics, optics, and electronics. The conversion of time series data into dynamic spectral components can be accomplished using a variety of mathematical transforms, such as short-time Fourier transforms (st-FTs), Gabor expansions, wavelets, and others.[1] Electrochemists have used JTFA methods for probing dynamic change in lithium ion batteries undergoing active charge or discharge when the applied current has a superimposed small amplitude oscillation of frequency w. These methods are sometimes called dynamic electrochemical impedance spectroscopy (DEIS) or in situ EIS, and typically involve st-FTs of the voltage and current to get an impedance-like response function.[2,3] Physics-based analysis of DEIS experiments, when performed at all, has involved computationally expensive brute force calculations of the full time series response, followed by st-FT analysis.[4]
In this invited talk, we explore the math and physics associated with efficient computations of the dynamic st-FT voltage spectrum generated by batteries undergoing active charge or discharge when the applied current has a superimposed small amplitude oscillation of frequency w. Computationally efficient physics-based predictions of battery st-FT voltage spectra have the potential to significantly improve the value of this diagnostic method for applications such as super fast charging, especially if the underlying physics revealed by the st-FT spectrum can be connected to degradation and safety.
Our approach is to apply st-FTs to the physics-based pseudo-two dimensional (P2D) model of a lithium ion battery prior to solving, transforming the time domain equations into a hierarchical set of mixed time and frequency domain equations that are separated by time-scales and sequentially solvable. Slow dynamics associated with the charge-discharge cycle are solved in the time domain and faster dynamics from current oscillations are solved in the frequency domain. Slow time domain terms are included in the frequency domain solutions, creating a mixed time-frequency set of equations for the st-FT voltage spectra that is efficient to solve. The governing equations show the impedance-like nature of the dynamic voltage signal as a function of frequency w, while also clearly revealing time-dependent differences between DEIS and stationary EIS that offers insights into the method as a real-time diagnostic of batteries.
Using realistic Li-ion battery model parameters, we show how the selected frequency, C-rate, and cell charge-discharge history affects the st-FT voltage signal amplitude and phase. We evaluate the dynamic results as the battery cycles through different states of charge, and compare DEIS impedance-like response to the stationary EIS signals at the same states. Finally, we show some of our initial experimental results looking at this method from a theory-based perspective.

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