# Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra

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## Abstract

**:**

## 1. Introduction

#### 1.1. Generalized Impedance Spectroscopy

#### 1.2. Distribution of Relaxation Times Analysis

#### 1.3. Solving the Optimization Problem of DRT Using Regularization

#### 1.4. Analyzing Distribution Functions by Peak Analysis

#### 1.5. Discussion of Shortcomings of DRT

## 2. Generalized Distribution of Relaxation Times Analysis

## 3. Results of GDRT Analysis

#### 3.1. Lithium-Ion Battery

#### 3.2. Vanadium Redox Flow Battery

#### 3.3. Double Layer Capacitor

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamic, frequency dependent processes of an electrochemical system (in green) and battery test methods (in blue) with typical ranges of characteristic time constants.

**Figure 2.**Example spectrum of series connection of an ohmic resistance, an RC element, and a ZARC element. (

**a**) Nyquist plot of the spectrum with the sum of squared errors ($sse$) according to Equation (5); (

**b**) Distribution of relaxation times ${h}_{k}$ of the spectrum with fitted peaks and the sum of squared errors of their residual. In legends the engineering notation is used for $sse$ values, where, e.g., 1.62e-08 corresponds to $1.62\cdot {10}^{-8}$.

**Figure 3.**(

**a**) Work flow of state of the art DRT analysis with two model-based steps for preprocessing measured impedance data to extract the resistive-capacitive behavior of the measured spectrum; (

**b**) Reduced work flow of model-free generalized DRT analysis without the need of data preprocessing resulting in two distribution functions of relaxation times, one for the resistive-capacitive behavior and one for the resistive-inductive behavior.

**Figure 4.**Generalized DRT analysis of a lithium-ion battery (A123 Systems, LFP-graphite, 2.6 Ah, 26650) with the sum of squared errors ($sse$) of each plot. (

**a**) Measured and calculated impedance spectrum; (

**b**) Normalized residuals of real and imaginary part of the impedance spectra of subplot (

**a**); (

**c**) Identified resistive-capacitive distribution of relaxation times function; (

**d**) Identified resistive-inductive distribution of relaxation times function.

**Figure 5.**Generalized DRT analysis of a vanadium redox-flow battery (Micro Flow Cell, Separator Nafion, Electrodes SGL GFA6) with the sum of squared errors ($sse$) of each plot. (

**a**) Measured and calculated impedance spectrum; (

**b**) Normalized residuals of real and imaginary part of the impedance spectra of subplot (

**a**); (

**c**) Identified resistive-capacitive distribution of relaxation times function; (

**d**) Identified resistive-inductive distribution of relaxation times function.

**Figure 6.**Generalized DRT analysis of a double layer capacitor (Maxwell, 3400 F, 2.85 V) with the sum of squared errors ($sse$) of each plot. (

**a**) Measured and calculated impedance spectrum; (

**b**) Normalized residuals of real and imaginary part of the impedance spectra of subplot (

**a**); (

**c**) Identified resistive-capacitive distribution of relaxation times function; (

**d**) Identified resistive-inductive distribution of relaxation times function.

**Table 1.**Generalized impedances of different physical domains in analogy to the standard impedance definition of the electrical domain.

Physical Domain | Generalized Voltage (Intensive, Across Variable) | Generalized Current (Extensive, Through Variable) | Generalized Impedance (Transfer Function) |
---|---|---|---|

Electrical | Voltage $V$ | Current $I$ | Impedance $Z$ |

Electrostatic | Voltage $V$ | Dielectric flow ${\varphi}_{\mathrm{D}}$ | Dielectric impedance ${Z}_{\mathrm{D}}$ |

Magnetostatic | Magnetic voltage ${V}_{\mathrm{m}}$ | Magnetic flow ${\varphi}_{\mathrm{B}}$ | Magnetic resistance ${Z}_{\mathrm{m}}$ |

Mechanical (translatory) | Force $F$ | Velocity $v$ | Mass $m$ |

Mechanical (rotatory) | Torque $M$ | Angular velocity $\omega $ | Inertia $J$ |

Hydro-mechanical | Pressure $p$ | Volume flow $\dot{V}$, Mass flow $\dot{m}$ | Pneumatic impedance ${Z}_{\mathrm{p}}$ |

Thermal conductivity | Temperature difference $\Delta T$ | Heat flow $W$ | Thermal impedance ${Z}_{\mathrm{T}}$ |

**Table 2.**Model parameter values of the impedance simulation of series connection of an ohmic resistance, an RC element, and a ZARC element.

Impedance Element | Parameter | Value |
---|---|---|

Ohmic resistance | ${R}_{\mathsf{\Omega}}$ | $3\mathrm{m}\mathsf{\Omega}$ |

RC element | ${R}_{1}$ | $4\mathrm{m}\mathsf{\Omega}$ |

${\tau}_{1}$ | $0.5\mathrm{ms}$ | |

ZARC element | ${R}_{2}$ | $7\mathrm{m}\mathsf{\Omega}$ |

${\tau}_{2}$ | $5\mathrm{ms}$ | |

${\phi}_{2}$ | $0.8$ |

Degree of Freedom | Variable |
---|---|

Definition of error and optimization function | $e,\text{}J,\text{}\underset{{h}_{k}}{\mathrm{min}}\left\{J\right\}$ |

Modelling of impedance spectrum | ${Z}_{\mathrm{EIS},\mathrm{mod}}$ |

Exclusion of non-resistive-capacitive elements | ${Z}_{\mathrm{reduced}}={Z}_{\mathrm{meas}}-{Z}_{\mathrm{mod},\text{}\mathrm{excl}}$ |

Number of time constants | ${N}_{\tau}$ |

Min/max time constant | ${\tau}_{\mathrm{min}},{\tau}_{\mathrm{max}}$ |

Distribution of predefined time constants | $\left[{\tau}_{\mathrm{min}}\cdots {\tau}_{k}\cdots {\tau}_{\mathrm{max}}\right]$ |

Choice of regularization parameter | $\lambda $ |

**Table 4.**Identified polarization contributions of the lithium-ion battery comprising ohmic resistance, inductivity, capacitance, resistive-capacitive, and resistive-inductive elements.

Variable | Value | Variable | Value |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ | $7.97\text{}\mathrm{m}\mathsf{\Omega}$ | ||

$L$ | $29.3\mathrm{nH}$ | ||

$C$ | $854\mathrm{mF}$ | ||

${R}_{1}$ | $6.35\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{1}$ | $58.7\mathsf{\mu}\mathrm{s}$ |

${R}_{2}$ | $1.41\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{2}$ | $421\mathsf{\mu}\mathrm{s}$ |

${R}_{3}$ | $1.55\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{3}$ | $2.29\mathrm{ms}$ |

${R}_{4}$ | $2.43\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{4}$ | $13.5\mathrm{ms}$ |

${R}_{5}$ | $0.499\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{5}$ | $99.5\mathrm{ms}$ |

${R}_{6}$ | $1.03\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{6}$ | $455\mathrm{ms}$ |

${R}_{7}$ | $3.22\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{7}$ | $2.97\mathrm{s}$ |

${R}_{8}$ | $1.41\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{8}$ | $9.20\mathsf{\mu}\mathrm{s}$ |

**Table 5.**Identified polarization contributions of the vanadium redox-flow battery comprising ohmic resistance, inductivity, capacitance, resistive-capacitive, and resistive-inductive elements.

Variable | Value | Variable | Value |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ | $160\text{}\mathrm{m}\mathsf{\Omega}$ | ||

$L$ | $223\mathrm{nH}$ | ||

$C$ | $0\mathrm{mF}$ | ||

${R}_{1}$ | $42.9\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{1}$ | $28.8\mathsf{\mu}\mathrm{s}$ |

${R}_{2}$ | $30.5\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{2}$ | $272\mathsf{\mu}\mathrm{s}$ |

${R}_{3}$ | $252\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{3}$ | $2.46\mathrm{ms}$ |

${R}_{4}$ | $14.9\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{4}$ | $31.8\mathrm{ms}$ |

${R}_{5}$ | $16.7\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{5}$ | $1.39\mathrm{s}$ |

${R}_{6}$ | $1.10\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{6}$ | $4.99\mathsf{\mu}\mathrm{s}$ |

${R}_{7}$ | $35.1\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{7}$ | $5.45\mathrm{s}$ |

**Table 6.**Identified polarization contributions of the double-layer capacitor comprising ohmic resistance, inductivity, capacitance, resistive-capacitive, and resistive-inductive elements.

Variable | Value | Variable | Value |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ | $7.97\text{}\mathrm{m}\mathsf{\Omega}$ | ||

$L$ | $29.3\mathrm{nH}$ | ||

$C$ | $854\mathrm{mF}$ | ||

${R}_{1}$ | $6.35\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{1}$ | $58.7\mathsf{\mu}\mathrm{s}$ |

${R}_{2}$ | $1.41\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{2}$ | $421\mathsf{\mu}\mathrm{s}$ |

${R}_{3}$ | $1.55\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{3}$ | $2.29\mathrm{ms}$ |

${R}_{4}$ | $2.43\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{4}$ | $13.5\mathrm{ms}$ |

${R}_{5}$ | $0.499\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{5}$ | $99.5\mathrm{ms}$ |

${R}_{6}$ | $1.03\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{6}$ | $455\mathrm{ms}$ |

${R}_{7}$ | $3.22\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{7}$ | $2.97\mathrm{s}$ |

${R}_{8}$ | $1.41\text{}\mathrm{m}\mathsf{\Omega}$ | ${\tau}_{8}$ | $9.20\mathsf{\mu}\mathrm{s}$ |

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**MDPI and ACS Style**

Danzer, M.A.
Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra. *Batteries* **2019**, *5*, 53.
https://doi.org/10.3390/batteries5030053

**AMA Style**

Danzer MA.
Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra. *Batteries*. 2019; 5(3):53.
https://doi.org/10.3390/batteries5030053

**Chicago/Turabian Style**

Danzer, Michael A.
2019. "Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra" *Batteries* 5, no. 3: 53.
https://doi.org/10.3390/batteries5030053