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Chapter 3. Characterisation Methods The weighting factors in Equation 3.58 weigh the significance of each data point on the sum of squares. There are several types of weighting factors, including unit weighting (wi,Re = wi,Im = 1), modulus weighting (wi,Re = wi,Im = |Zi|2), and proportional weighting (wi,Re = 1/Z2 and wi,Im = 1/Z2 ) [67]. Unit weighting was used for fitting impedance i,Re i,Im data throughout this work, as it is an appropriate choice for impedance data with similar values of the real and imaginary parts that does not change by orders of magnitude from high to low frequency. The modelled impedance is compared qualitatively to the experimental impedance by graphical inspection of the Nyquist and Bode plots and quantitatively by calculating the relative fit residuals, which are the difference between experimental and model impedance divided by the modulus, and plotting them against the frequency. Residuals randomly distributed around and close to 0% over the applied frequency range means that the experimental data was modelled well. One must remember, however, that this does not say anything about the appropriateness of the used model. The impedance modelling presented in this work was performed by passing a set of fixed and free fit parameters to a CNLS algorithm in Python, using the impedance.py package developed for analysis of electrochemical impedance data [75]. Data handling and analysis in general was carried out using the numpy, pandas, and scipy libraries [76–78], and data visualisation with the matplotlib library [79]. 3.6.4 Data Validation The involvement of complex numbers in EIS allows one to validate the data quality. This is made possible by Cauchy’s integral theorem, which implies that there is a relation between the real and imaginary parts of a complex function [67]. This was formulated into mathematical relations by Kramers and Kronig, who stated how the imaginary component could be calculated from the real component, and vice versa [67, 80]. A complex function must satisfy four conditions to be compliant with the Kramers-Kronig relations [67]: linearity, causality, stability, and finiteness. Linearity means the system can be described by a set of linear differential equations and that the responses are independent of amplitude. Causality means the response to an excitation signal depends solely on the excitation signal. Stability means the system remains stable until excited by an external source and returns to its original state once the excitation signal is removed. The real part of the impedance must also be above 0 for all applied frequencies and the impedance must be independent of time. Finally, finiteness means the impedance must have a finite value over the frequency range 0 < ω < ∞, and that the real part tends towards a constant value as ω → 0 and ω → ∞. The Kramers-Kronig relations require evaluation of the impedance over a frequency range from zero to infinity, which is not possible in practice. The relations are therefore replaced by approximations, which must still comply with the four stated conditions. One such approximation was suggested by Boukamp [80], which involves fitting an equivalent circuit consisting of a finite number of Voigt elements (parallel connection of a resistor and capacitor) to the experimental data. The schematic of a Voigt circuit is shown in Figure 3.19. Since a Voigt circuit is linear and Kramers-Kronig compliant, any system that can be modelled well by the Voigt circuit must itself be Kramers-Kronig compliant. It should be 42PDF Image | Organic Redox Flow Batteries 2023
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