User:Werner S58/blafusel

ZHIT, also denoted as ZHIT-Algorithmn or ZHIT-Approximation, is a tool in Elektrochemical Impedance-Spektroscopy (EIS – see also Impedance spectroscopy). In EIS, ZHIT establishes a relationship between the two measured values, modulus of the impedance and phase shift of one-Port-systems in the form of a integral equation. The ZHIT-Algorithm allows to verify the stationarity of the measured test object as well as calculating the impedance values using the phase data.

Motivation
An important appication of ZHIT is the examination of experimental impedance spectra for artifacts. The examination of impedance sepctroscopy measurement series is often made difficult by the tendency of examinded objects to change while being measured. This happens with many default applications of EIS like the evaluation of fuel cells and  batteries under discharge. Further examples include the investigation of light-sensitive systems under illumination (e.g. Photoelectrochemistry) or the analysis of water uptake of varnish on metal surfaces (e.g. corrosion-protection). A demonstrative example for a instationary system is a Lithium-ion battery. While cyclization or discharging the amount of charge in the battery and the whole system itself changes over time. The change in the level of charge is coupled with a chemical redox reaction which means the concentration of the invovled substances changes. This violates the stationarity and causality. Theoretically this would mean that impedance spectra of such systems could not be analyzed properly. Using the ZHIT-algorithm, these and similar artifacts can be recognized. It is also possible to reconstruct causal spectra, which are consistent with the Kramers–Kronig relations and thereby valid for analysis.

Mathematical Formulation
ZHIT 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 relationship between impedance and phase angle can be depicted from the Bode plot of a impedance spectrum. The equation (1) is obtained as a general solution of the correlation between modulus of the impedance and the phase shift. $$ (1)\text{ } \ln \left[Z \left(\omega_o \right) \right] - \ln \left[Z \left(0 \right) \right] \text{ }= \text{ }\frac{2}{\pi}\cdot \int\limits^{\omega_O}_{\omega_S}\varphi \left(\omega \right)dln \left(\omega \right) \text{ }+ \text{ }\gamma_k \cdot \sum^{\infty}_{k=1}\frac{d^{k}\varphi \left(\omega_0 \right)}{d \ln {\left(\omega \right)}^{k}} \text { } \text { } \text { } \text { } mit \text { } k \text { }= \text { }1, 3, 5, 7, \ldots (k\text { } = \text {odd})$$

Equation (1) indicates, that the logarithm of the impedance ($$ \ln \left[Z \left(\omega_o \right) \right] $$) in one point $$ \omega_O $$ can be calculated up to a constant value of ($$ \ln \left[Z \left(0 \right) \right] $$), if the phase shift $$ \varphi \left(\omega \right) $$ is integrated up to the point of intrest $$ \omega_O $$, while the starting value $$ \omega_S $$ of the integral can be choosen at will. As additional contribution to the calculation of $$ \ln \left[Z \left(\omega_o \right) \right] $$ are the odd-numbered derivatives of the phase shift at the point $$ \omega_O $$, weighted with the factors $$ \gamma_k $$ which have to be added. The factors $$ \gamma_k $$ can be calculated according to equation (2), whereat $$ \zeta \left(k + 1 \right) $$ represents the Riemann ζ-function.

$$ (2)\text{ } \gamma_k = {\left(- 1 \right)}^k \cdot \frac{2}{\pi}\cdot \frac{1}{2^k}\cdot \zeta \left(k + 1 \right) \text { } \text { } \text { } \text { } mit \text { } k \text { }= \text { } 1, 3, 5, 7, \ldots (k\text { } = \text {odd numbered}) $$

The ZHIT approximation applied in practice is obtained from equation (1) by restricting to the first derivative of the phase shift, neglecting higher derivatives (equation (3)), where C represents a constant.

$$(3)\text{ } \ln \left[Z \left(\omega_o \right) \right] \text{ }= \text{ }\frac{2}{\pi}\text{ }\int\limits^{\omega_O}_{\omega_S}\varphi \left(\omega \right)dln \left(\omega \right) \text{ }\text{ }+ \text{ }\text{ }\gamma_{1} \text{ }\frac{d\varphi \left(\omega_O \right)}{d \ln \left(\omega \right)}\text{ }+ \text{ }C$$

The free choice of the integration boundaries in the ZHIT algorithm is a fundamental difference concerning the Kramers-Kronig relations; in ZHIT the integration boundaries are $$ \omega = 0 \text{ } $$ und $$ \omega = \infty \text{ } $$. The advantage of the ZHIT results from the fact that both integration boundaries can be chosen within the measured spectrum and not as with the Kramers-Kronig relations against the (not real) frequencies 0 und $$ \infty $$ which requires an extrapolation procedure where the frequency behavior of the system is clearly unknown.

Practical implementation
The practical implementation of the ZHIT approximation is shown schematically in Figure 1. From the measured data points of impedance and phase, a continuous curve (spline) for each of the two independent measured quantities (impedance and phase) is created by smoothing (part 1 in Figure (1)). With the help of the spline for the phase shift, values for the impedance are now calculated. First, the integral of the phase shift is calculated up to the corresponding frequency $$ \omega_0 $$ aufintegriert, where suitably the highest measured frequency is selected as starting $$ \omega_S $$ - see part 2 in Figure (1). From the spline of the phase shift the slope of the phase shift can be calculated at $$ \omega_0 $$ too (part 3 in figure (1)). Thus, a reconstructed curve for the impedance is obtained which - in the ideal case - is (only) parallel shifted to the original measured curve of the impedance. To determine the constant C in the ZHIT equation (part 4 in Figure (1)) there are several possibilities. One possibility is to perform the parallel shift of the reconstructed impedance in a frequency range that is not affected by the occurrence of artifacts (see notes). This shift is performed by a linear regression procedure. Comparing the resulting reconstructed impedance curve with the original measured one (or the Splines of the impedance), artifacts can be detected. These are usually in the high frequency range (caused by induction or mutual induction, especially when low impedance systems are investigated) or in the low frequency range (caused by the change of the system during the measurement (=drift)).

Notes (Time requirements to measure an impedance spectrum)
The measurement time required for a single impedance measurement point depends very much on the frequency of interest. While frequencies above about 1 Hz can be measured practically in seconds, the measurement time increases exponentially in the lower frequency range.
 * Although the exact duration for measuring a complete impedance spectrum depends on the measurement system itself as well as on internal settings, the following measurement times can be assumed as rules of thumb when measuring the frequency measurement points sequentially, with the upper frequency assumed to be 100 kHz or 1 MHz.
 * Up to approx. 1 Hz the measuring time is approx. 1 minute, up to 0.1 Hz approx. 5 minutes, up to 0.05 Hz approx. 10 minutes, up to 0.02 Hz approx. 15 minutes and up to 0.01 Hz approx. 30 minutes.
 * Measurements down to or below 0.01 Hz can be associated with measurement times in the range of several hours.
 * As a consequence of this time requirements of the measurement downto the different lower frequencies, a spectrum can be roughly divided into three sub-ranges as far as the occurrence of artifacts is concerned: high-frequency (approx. > 100 to 1000 Hz) induction or mutual induction can dominate. Low frequency (at frequencies < 1 Hz) drift can occur due to noticeable change in the system.
 * The range between about 1 Hz and 1000 Hz is usually not affected by either high-frequency or low-frequency artifacts; however, one has to exclude the mains frequency (50/60 Hz) in this consideration.

Notes (Application procedure)
In addition to the reconstructing of the impedance from the phase shift, the reverse approach is also possible. However, the procedure outlined here possesses different advantages.


 * KTY.png When calculating the phase shift from the impedance, instead of the constant C in equation (3), a function of the angular frequency &omega; occurs which is more difficult to determine.
 * "The phase shift is more stable than the impedance". Behind this statement is the fact that for impedance elements (more precisely: constant phase elements, CPE )the property "phase shift" remains constant even if the value of the impedance changes. Such Constant Phase-elements are among others the typical electronic elements like electrical resistor, capacitor and coil. For illustration, Figure 2 shows the impedance spectrum of an NTC resistor heated during the measurement (starting between 1 kHz and 10 kHz downto to lower frequencies). It can be clearly seen that the value of the impedance (red curve) changes with temperature, while the phase shift (blue curve) remains constant, i.e. "a resistor remains a resistor".
 * The reconstruction of the impedance from the phase shift further restores the "inner (=complex)" relationship between these two quantities. This relationship is lost by the independent construction of the supporting point splines for impedance and phase (Figure 1). Depending on the system under investigation, this restored correlation - even in the absence of artifacts - can lead to an improved evaluation of the spectra. In such cases, the gain in accuracy due to the reconstruction of the complex impedance outweighs the approximation error according to equation (3), which results from the neglection of the higher derivatives.

Applications
Figure 3 shows an impedance spectrum of a series of measurements of a painted steel sample during water uptake (upper part in Figure 3). The symbols in the diagram represent the interpolation points (nodes) of the measurement, while the solid lines represent the theoretical values simulated according to an appropriate model. The interpolation points for the impedance were obtained according to the ZHIT reconstruction of the phase shift. In the plower part of Figure 3, the normalized error (ZZHIT − Zsmooth)/ZZHIT·100 of the impedance is depicted. For the calculation of the error, two different procedures are used to determine the impedance values of the "extrapolated impedance values": The simulation according to the appropriate model is performed on the base of the two different impedance courses and the corresponding residuals are calculated and depicted in the lower part of the diagram depicted in Figure (3). ''Remark: Error patterns like in the lower partial diagram (magenta) can often be the reason to extend an existing model for the simulation with additional elements to minimize the error. However, this is not possible in principle. The drift in the impedance spectrum manifests itself in the low-frequency part by the fact that the system changes during the measurement. The spectrum in Figure 3 is caused by water penetrating into the pores. This reduces the impedance (resistance) of the coating. De facto, during water uptake, the system behaves as if at each low-frequency measurement point the resistance of the coating has been replaced by another, smaller resistance. However, there is no impedance element that exhibits such behavior. Therefore, any extension of the model would only result in "smearing" the error over a wider frequency range without reducing the error itself. Only the removal of the drift by reconstructing the impedance using ZHIT leads to a significantly better agreement between measurement and model.''
 * once the "extrapolated impedance values" are calculated from the "splined (=Zsmooth)" data of the impedance (magenta)
 * secondly from the reconstruction of the impedance values (blue) according to the ZHIT (= ZZHIT) using the spline of the phase shift

In Figure 4, a Bode plot of a series impedance measurement is depicted, taken on a fuel cell where the hydrogen of the fuel gas was deliberately poisoned by the addition of carbon monoxide. Due to the carbon monoxide poisoning, active centers of the platinum catalyst are blocked, which severely impairs the performance of the fuel cell. The blocking of the catalyst is thereby potential-dependent, whereby an alternating sorption and desorption of the carbon monoxide on the catalyst surface occurs in the cell. This ("cyclic") change of the active catalyst surface manifests itself in pseudo-inductive behavior, which can be observed in the impedance spectrum of Figure 4 at low frequencies (< 3 Hz). In this figure, the impedance curve reconstructed by the ZHIT is represented by the purple line, while the supporting points from the original measured values are represented by the blue circles. One can very clearly see the deviation in the low frequency part of the measurement between these two curves. Evaluation of the spectra according to a chosen model shows that significantly better agreement between model and measurement can be obtained if the reconstructed ZHIT impedances are used to calculate the impedances instead of the original measured data.

Individual references
category:Electrochemistry