2.2.1. General Approach

Figure 2a shows the simulation of the cell voltage of an LIB during a charge pulse with constant current and the subsequent relaxation period. The impedance of the battery was modelled by three RC-elements and internal resistance. White noise was added to the voltage signal to take measurement inaccuracies of real-world applications into account.

**Figure 2.** (**a**) Simulated cell voltage of a LIB during a pulse test. (**b**) DRT derived from the voltage relaxation.

For the proposed method of an DRT analysis of time domain data, the voltage relaxation is evaluated. Similar to the DRT analysis of frequency domain data, an attempt is made to reconstruct the voltage behaviour with a series of RC-elements. The voltage course of a single RC-element during a relaxation period can be described as follows:

$$
\mu\_{RC}(t) = \mathcal{U}\_0 \cdot \varepsilon^{-\frac{t}{\tau\_{RC}}} \tag{3}
$$

where *τRC* = *R* · *C*, the characteristic time constant of a RC-element. The voltage of the RC-element at the beginning of the relaxation period *U*<sup>0</sup> at *t* = 0 s can be calculated if the duration *tcp* and current *Icp* of the previous pulse are known:

$$
\hbar L\_0 = R \cdot I\_{cp} \left( 1 - e^{-\frac{\Gamma\_{cp}}{T\_{\rm RC}}} \right) \tag{4}
$$

Equation (4) is only valid if the cell has been sufficiently relaxed before the current pulse and any overvoltages can be neglected. Assuming a finite number *m* of RC-elements, the overall voltage response can be computed numerically:

$$\mathcal{U}\_{\text{RC,series}}(t) = \sum\_{k=1}^{m} \mathcal{R}\_k \cdot I\_{\text{cp}} \left( 1 - e^{-\frac{t\_{\text{cp}}}{\tau\_k}} \right) \cdot e^{-\frac{t}{\tau\_k}} \tag{5}$$

By defining fixed time constants *τk*, the optimization problem becomes linear. The values of the resistances *Rk* must be set in such a way that the error between the voltage prediction by the series of RC-elements and the measured voltage data (or in this case the simulated voltage data) is minimized. Using the sum of squared errors as a cost function leads to the following optimization problem *min*{*J*}:

$$J = \left\| A \cdot R\_{\text{vec}} - lI\_{\text{vec}} \right\|^2 \tag{6}$$

where the vector *Rvec* corresponds to the unknown polarization contributions of the RCelements

$$R\_{\text{vec}} = \begin{bmatrix} R\_1 \dots \ R\_k \dots \ R\_m \end{bmatrix}^T \tag{7}$$

and *Uvec* is the vector of the measured (or simulated) voltage course:

$$\mathcal{U}\_{\text{vec}} = [\mathcal{U}\_1 \dots \mathcal{U}\_n] \tag{8}$$

The number *n* corresponds to the number of measurement points. The measurement data should be preprocessed in advance. In Section 2.2.3, a detailed description of the process is given. The dimension of the matrix *A* is *n* · *m* and the matrix is calculated for the different predefined constants and the time after the end of the pulse *t* which corresponds to the respective voltage measurement in *Uvec*.

$$A = I\_{cp} \cdot \begin{bmatrix} \left(1 - e^{-\frac{t\_{cp}}{\tau\_1}}\right) \cdot e^{-\frac{t\_1}{\tau\_1}} & \cdots & \left(1 - e^{-\frac{t\_{cp}}{\tau\_m}}\right) \cdot e^{-\frac{t\_1}{\tau\_m}} \\ \vdots & \ddots & \vdots \\ \left(1 - e^{-\frac{t\_{cp}}{\tau\_1}}\right) \cdot e^{-\frac{t\_n}{\tau\_1}} & \cdots & \left(1 - e^{-\frac{t\_{cp}}{\tau\_m}}\right) \cdot e^{-\frac{t\_n}{\tau\_m}} \end{bmatrix} \tag{9}$$
