Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method
Tanmoy Paul, P. W., Phillip M. Wu, Maw‐Kuen Wu
Abstract
Abstract In this paper, the distribution of relaxation times (DRTs) functions are calculated numerically in Matlab for synthetic impedance data from single parallel $$RC$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RC</mml:mi> </mml:mrow> </mml:math> circuit and two parallel $$RC$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>RC</mml:mi> </mml:mrow> </mml:math> circuits connected in series, experimental impedance data from supercapacitors and α-LiFeO 2 anode based Li ion batteries. The quality of the impedance data is checked with the Kramers–Krönig (KK) relations. The DRTs are calculated within the KK compatible regime for all the systems using Tikhonov regularization (TR) method. Here we use a fast and simple L-curve method to estimate the TR parameter (λ) for regularization of the Fredholm integral equations of first kind in impedance. Estimation of the regularization parameters are performed effectively from the offset of the global corner of the L-curve rather than simply using the global corner. The physical significances of DRT peaks are also discussed by calculating the effective resistances and capacitances coupled with peak fitting program. For instance, two peaks in the DRTs justify the electrical double layer capacitance and ion diffusion phenomena for supercapacitors in low to intermediate frequencies respectively. Moreover, the surface film effect, Li/electrolyte and electrode/electrolyte charge transfer related processes are identified for α-LiFeO 2 anode based Li-ion batteries. This estimation of the offset of the global corner extends the L-curve approach coupled with the Tikhonov regularization in the field of electrochemistry and can also be applied in similar process detection methods.