Z-HIT, also denoted as ZHIT, Z-HIT relationship, is a bidirectional mathematical transformation, connecting the two parts of a complex function—i.e., its modulus and its phase. Z-HIT relations are somewhat similar to the Kramers–Kronig relations, where the real part can be computed from the imaginary part (or vice versa). In contrast to the Kramers–Kronig relations, in the Z-HIT the impedance modulus is computed from the course of the phase angle (or vice versa). The main practical advantage of Z-HIT relationships over Kramers–Kronig relationships is, that the Z-HIT integration limits do not require any extrapolation: instead, an integration over the experimentally available frequency range provides accurate data. More specifically, the angular frequency (ω) boundaries for computing one component of the complex function from the other one using the Kramers-Kronig relations, are ω=0 and ω=∞; these boundaries require extrapolation procedures of the measured impedance spectra. Concerning the ZHIT however, the computing of the course of the impedance modulus from the course of the phase shift can be performed within the measured frequency range, without the need of extrapolation. This avoids complications which may arise from the fact that impedance spectra can only be measured in a limited frequency range. Therefore, the Z-HIT algorithm allows for verification of the stationarity of the measured test object as well as calculating the impedance values using the phase data. The latter property becomes important when drift effects are present in the impedance spectra which had to be detected or even removed when analysing and/or interpreting the spectra. Z-HIT relations find use in Dielectric spectroscopy and in Electrochemical Impedance Spectroscopy.
Motivation An important application of Z-HIT is the examination of experimental impedance spectra for artifacts. The examination of EIS series measurements is often difficult due to the tendency of examined objects to undergo changes during the measurement. This may occur in many standard EIS applications such as the evaluation of fuel cells or batteries during discharge. Further examples include the investigation of light-sensitive systems under illumination (e.g. Photoelectrochemistry) or the analysis of water uptake of lacquers on metal surfaces (e.g. corrosion-protection). A descriptive example for an unsteady system is a Lithium-ion battery. Under cyclization or discharging, the amount of charge in the battery changes over time. The change in charge is coupled with a chemical redox reaction, transferring to a change in concentrations of the involved substances. This violates the principles of stationarity and causality which are prerequisites for proper EIS measurements. In theory, this would exclude drift-affected samples from valid evaluation. Using the ZHIT-algorithm, these and similar artifacts can be recognized and spectra following causality can even be reconstructed, which are consistent with the Kramers–Kronig relations and thereby valid for analysis.
Mathematical Formulation Z-HIT is a special case of the Hilbert transform and through restriction by the Kramers–Kronig relations it can be derived for one-Port-systems. The frequency-dependent relationship between impedance and phase angle can be observed in the Bode plot of an impedance spectrum. Equation (1) is obtained as a general solution of the correlation between impedance modulus and phase shift.
( 1 ) ln [ Z ( ω o ) ] − ln [ Z ( 0 ) ] = 2 π ⋅ ∫ ω S ω O φ ( ω ) d l n ( ω ) + γ k ⋅ ∑ k = 1 ∞ d k φ ( ω 0 ) d ln ( ω ) k w i t h k = 1 , 3 , 5 , 7 , … ( k = odd ) {\displaystyle (1){\text{ }}\ln \left[Z\left(\omega _{o}\right)\right]-\ln \left[Z\left(0\right)\right]{\text{ }}={\text{ }}{\frac {2}{\pi }}\cdot \int \limits _{\omega _{S}}^{\omega _{O}}\varphi \left(\omega \right)dln\left(\omega \right){\text{ }}+{\text{ }}\gamma _{k}\cdot \sum _{k=1}^{\infty }{\frac {d^{k}\varphi \left(\omega _{0}\right)}{d\ln {\left(\omega \right)}^{k}}}{\text{ }}{\text{ }}{\text{ }}{\text{ }}with{\text{ }}k{\text{ }}={\text{ }}1,3,5,7,\ldots (k{\text{ }}={\text{odd}})}
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