Obtaining the Low Frequency Impedance of Li-Ion Batteries from Arbitrary Excitation Signals
Authors: Schoenleber Michael , Weber André , Ivers-Tiffée Ellen 
Published: February 2020

Electrochemical Impedance Spectroscopy (EIS) has proven to be a very powerful tool for investigating the properties of all sorts of electrochemical systems. By sequentially applying sinusoidal excitations of different frequencies and observing the system’s response, the impedance spectrum of the system is measured. In this representation, processes taking place on different time scales are well-separated and therefore can be studied easily.
However, in the case of Lithium-Ion batteries the problem arises, that investigating very slow processes like solid state diffusion requires the impedance spectrum to be measured down to the range of microhertz. Due to its sequential nature, application of regular EIS there leads to measurement times of several days which are not only inconvenient, but also run contrary to the requirement on the battery to be time-invariant during the measurement.
In order to minimize measurement time, one has to choose the excitation signal in a way that the frequencies of interest are simultaneously excited. Depending on how the excitation signal is chosen, literature distinguishes between multi-sine methods, where sinusoidals of all frequencies of interest are superposed [1], and time-domain methods, which allow arbitrary excitation signals like steps and pulses [2][3]. While multi-sine methods benefit from a very easy calculation of the impedance spectrum via Fast Fourier Transformation (FFT), excitation by arbitrary excitation signals is much easier implemented and offers endless possibilities of tailoring the excitation signals according to one’s specific needs. However, calculating impedance spectra from these signals is challenging: Without further measures the quality of the obtained spectra is usually quite poor due to measurement noise and spectral leakage.
Overcoming these drawbacks, a way of computing impedance spectra from arbitrary excitation signals will be presented, which is tailored for the needs of Lithium-Ion batteries. It comprises optimal leakage rejection by Gaussian windowing and, other than existing approaches [2][3], does not require merging of subspectra. Additionally, improved noise rejection, as depicted in Figure 1, could be achieved by segmenting the measurement data and averaging its power spectral densities [4]. This newly developed method gains access to an impedance spectrum of a Lithium-Ion battery containing the lowest possible frequency for a given measurement time with the best possible quality for a given arbitrary excitation signal.
REFERENCES
[1] B. Evgenij, J. R. Macdonald, “Impedance Spectroscopy: Theory, Experiment, and Applications”, Wiley & Sons Inc., 2005
[2] K. Takano, K. Nozaki, Y. Saito, K. Kato, A. Negishi, “Impedance Spectroscopy by Voltage‐Step Chronoamperometry Using the Laplace Transform Method in a Lithium‐Ion Battery,” J. Electrochem. Soc., 147 (2000) 922
[3] D. Klotz, M. Schoenleber, J. P. Schmidt, E. Ivers-Tiffée, „New Approach for the Calculation of Impedance Spectra out of Time Domain Data,” Electrochimica Acta, 56 (2011) 8763
[4] P.D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms," IEEE Transactions on Audio Electroacoustics, 15 (1967)

Figure Caption:
Figure 1:
Simulation Result: True impedance (blue) versus impedance computed from time-domain data (red).
(Upper part) Without segmenting and averaging
(Lower part) With segmenting and averaging