ZHIT

Zhit , even zhit algorithm or zhit approximation , is used in the e lektrochemische I mpedanz s spectroscopy (EIS) . There it establishes a relationship between the two measurands modulus of impedance and phase shift of two-pole systems in the form of an integral equation. The ZHIT algorithm enables the stationarity of the test object to be checked and the impedance data to be recalculated from the phase data. The abbreviation "zhit" represents Z weipol- Hi lbert- T ransformation.

motivation

An important application of the ZHIT is the checking of experimental impedance spectra for artifacts (see Artifact (Technology) ). The evaluation of impedance spectroscopic examinations is often made more difficult by the fact that the objects to be examined can change during the measurement. This applies to many standard applications of electrochemical impedance spectroscopy, such as the examination of fuel cells and accumulators while drawing electricity, the examination of light-sensitive systems under illumination (e.g. photo- electrochromy ) or the examination of the water absorption of paints on metal surfaces ( corrosion protection ). A vivid example of a transient system is a lithium-ion accumulator during cycling or discharge: the charge state of the accumulator and thus the system itself changes when the current is drawn, since the change in charge state is accompanied by a chemical redox reaction and the concentration changes of the substances involved changes. This leads to a violation of stationarity and causality, so that impedance spectra of such systems cannot be evaluated from a theoretical point of view. With the help of the ZHIT algorithm, such and similar artifacts can be recognized and, if necessary, causal spectra can be reconstructed that are consistent with the Kramers-Kronig relationships and can thus be evaluated.

Mathematical formulation

ZHIT is a special case of the Hilbert transformation and can be derived by restricting the Kramers-Kronig relationships to two-pole systems. The relationships between impedance and phase can be clearly derived from the Bode diagram of an impedance spectrum . Equation (1) is obtained as a general solution of the relationship between the modulus of the impedance and the phase shift. ${\displaystyle {(1)\quad \ln \left[Z\left(\omega _{o}\right)\right]-\ln \left[Z\left(0\right)\right]={\frac {2}{\pi }}\cdot \int \limits _{\omega _{S}}^{\omega _{O}}\varphi \left(\omega \right)dln\left(\omega \right)+\gamma _{k}\cdot \sum _{k=1}^{\infty }{\frac {d^{k}\varphi \left(\omega _{0}\right)}{d\ln {\left(\omega \right)}^{k}}}\qquad {\text{mit}}\quad k=1,3,5,7,\ldots (k={\text{ungerade}})}}$

Equation (1) says that one can calculate the logarithm of the impedance ( ) at one point up to a constant value ( ) by integrating the phase shift up to the point of interest , whereby the starting value of the integral can be chosen arbitrarily. As an additional contribution to the calculation of , the odd derivative of the phase shift at the point , weighted with the factors, must be added. The factors can be calculated according to equation (2), where the Riemann ζ function means. ${\displaystyle \ln \left[Z\left(\omega _{o}\right)\right]}$ ${\displaystyle \omega _{O}}$ ${\displaystyle \ln \left[Z\left(0\right)\right]}$ ${\displaystyle \varphi \left(\omega \right)}$ ${\displaystyle \omega _{O}}$ ${\displaystyle \omega _{S}}$ ${\displaystyle \ln \left[Z\left(\omega _{o}\right)\right]}$ ${\displaystyle \omega _{O}}$ ${\displaystyle \gamma _{k}}$ ${\displaystyle \gamma _{k}}$ ${\displaystyle \zeta \left(k+1\right)}$

${\displaystyle {(2)\quad \gamma _{k}=\left(-1\right)^{k}\cdot {\frac {2}{\pi }}\cdot {\frac {1}{2^{k}}}\cdot \zeta \left(k+1\right)\qquad {\text{mit}}\quad k=1,3,5,7,\ldots (k={\text{ungerade}})}}$

Table 1: Pre-factors for determining the slope of the phase shift (numerical values ​​for the zeta function).

${\displaystyle k+1}$	${\displaystyle \zeta (k+1)}$	${\displaystyle {\frac {2}{\pi }}\cdot {\frac {1}{2^{k}}}\cdot \zeta (k+1)}$
2	${\displaystyle {\frac {\pi ^{2}}{6}}}$	${\displaystyle {\frac {\pi }{6}}\approx 0{,}52}$
4th	${\displaystyle {\frac {\pi ^{4}}{90}}}$	${\displaystyle {\frac {\pi ^{3}}{360}}\approx 0{,}086}$
6th	${\displaystyle {\frac {\pi ^{6}}{945}}}$	${\displaystyle {\frac {\pi ^{5}}{15120}}\approx 0{,}0202}$
8th	${\displaystyle {\frac {\pi ^{8}}{9450}}}$	${\displaystyle {\frac {\pi ^{7}}{604800}}\approx 0{,}005}$

The ZHIT approximation as it is used in practice can be obtained from equation (1) by restricting the phase shift to the first derivative and neglecting the higher derivatives (equation (3)), where C represents a constant.

${\displaystyle {(3)\quad \ln \left[Z\left(\omega _{o}\right)\right]={\frac {2}{\pi }}\int \limits _{\omega _{S}}^{\omega _{O}}\varphi \left(\omega \right)dln\left(\omega \right)+\gamma _{1}{\frac {d\varphi \left(\omega _{O}\right)}{d\ln \left(\omega \right)}}+C}}$

The fact that the integration limits can be freely selected in the ZHIT algorithm is a fundamental difference to the Kramers-Kronig relationships; with her are the integration limits and . The advantage of the ZHIT results from the fact that both integration limits can be selected within the measured spectrum and do not have to be extrapolated against the (non-real) frequencies 0 and 0 as in the Kramers-Kronig relationships . ${\displaystyle \omega =0\,}$ ${\displaystyle \omega =\infty \,}$ ${\displaystyle \infty }$

Practical implementation

Figure 1: Smoothing of the measurement data and calculation of the components of the ZHIT equation

The practical implementation of the ZHIT approximation is shown schematically in Figure 1. From the measurement points of impedance and phase, a continuous curve ( spline ) is created for the two independent measured variables impedance and phase by smoothing (part (1)). Function values ​​for the impedance are now determined with the aid of the spline for the phase shift. First, the phase shift is integrated up to the corresponding frequency , in which case the highest frequency of interest is expediently selected as the starting frequency (part (1)). The slope of the phase shift an can also be determined from the spline of the phase shift (partial image (3)). A reconstructed curve for the impedance is obtained, which (in the ideal case) is (only) shifted parallel to the original measurement curve for the impedance. There are several options for determining the constant C in the ZHIT equation (part (4)). One possibility is to carry out the parallel shift of the reconstructed impedance in a frequency range that is not influenced by the occurrence of artifacts (see notes). This shift is done by linear regression . By comparing the resulting, reconstructed impedance curve with the originally measured one (or the spline of the impedance), artifacts can now be detected. These are usually in the high-frequency range (caused by induction or mutual induction , especially with low-resistance systems) or in the low-frequency range (caused by changes in the system during the measurement (= drift)). ${\displaystyle \omega _{0}}$ ${\displaystyle \omega _{S}}$ ${\displaystyle \omega _{0}}$

Comments (time required for impedance measurement)

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 in a matter of seconds, the measurement time increases exponentially in the lower-frequency range.

• Although the exact duration for measuring a complete impedance spectrum depends on the measuring system itself and internal settings, the following measuring times can be assumed as rules of thumb for sequential measurement of the frequency measuring points, whereby the upper frequency is assumed to be 100 kHz or 1 MHz.
• Up to approx. 1 Hz the measurement 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 below 0.01 Hz can be associated with measurement times in the range of several hours.
• As a consequence of this time-dependency of the measurement at the different frequencies, a spectrum can roughly be divided into three sub-areas with regard to the occurrence of artifacts: high-frequency (approx.> 100 to 1000 Hz) induction or mutual induction can dominate. At low frequencies (at frequencies <1 Hz), drift can occur due to noticeable changes in the system.
• The range between about 1 Hz and 1000 Hz is usually not influenced by high-frequency or low-frequency artifacts, whereby the mains frequency (50 Hz) must also be excluded.

Notes (procedure)

In addition to the reconstruction of the impedance from the phase shift, the reverse is also possible. However, the procedure described here offers several advantages.

• 

  Figure 2: Impedance measurement of a temperature sensor KTY (10 KΩ), whereby the sensor was heated during the measurement
  When calculating the phase shift from the impedance, instead of the constant C in equation (3), a function of the angular frequency ω occurs, which is more difficult to determine
• “The phase shift is more stable than the impedance.” This statement hides the fact that for impedance elements (more precisely: constant phase elements , CPE ) the phase shift property remains constant, even if the value of the impedance changes. Such CP elements include the typical electronic elements such as electrical resistance , capacitor and coil . To illustrate this, Figure 2 shows the impedance spectrum of an NTC resistor that was heated during the measurement (starting between 1 kHz and 10 kHz down to low frequencies). One can clearly see that the value of the impedance (red curve) changes with temperature, while the amount of phase shift (blue curve) remains constant (“a resistor remains a resistor”).
• The reconstruction of the impedance from the phase shift also restores the "internal (= complex)" relationship between these two measured variables. This relationship is lost due to the independent construction of the support point splines for impedance and phase (Figure 1). Depending on the system being examined, this restored connection - 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 arises from neglecting the higher derivatives.

Applications

Figure 3: Above, impedance spectrum (symbols) and model simulation (lines) of a painted steel during water absorption. Bottom: resulting error without (magenta) and with (blue) ZHIT reconstruction of the impedance

Figure 3 shows an impedance spectrum of a series of measurements on a painted steel sample during water absorption (upper partial diagram). The symbols in the diagram represent the support points of the measurement, while the solid lines represent the theoretical values ​​simulated according to a specific model. The support points for the impedance were obtained by the ZHIT reconstruction from the phase shift. The lower part of the diagram shows the normalized error (Z _ZHIT - Z _smooth ) / Z _ZHIT · 100 of the impedance that arises when the measurement is simulated with the model, with the reference points of the impedance from the "splined (= Z _smooth ) "Measured values ​​themselves (magenta) - and a second time with the impedance values ​​(blue) reconstructed after the ZHIT (= Z _ZHIT ). The improvement from using the reconstructed data is significant.

Note: Defect patterns like the one in the lower part diagram (magenta) can often be the reason to expand an existing model for the simulation with additional elements in order to minimize the error. In principle, however, this is not possible. The drift in the impedance spectrum is expressed in the low-frequency part by the fact that the system changes during the measurement. The spectrum in Figure 3 results from the fact that water penetrates the pores. This reduces the impedance (resistance) of the coating. De facto, the system behaves during water absorption as if the resistance of the coating had been replaced by another, smaller resistance at each low-frequency measuring point. But there is no impedance element that shows such behavior. Any expansion of the model would therefore only lead to the error being "smeared" over a larger frequency range without the error itself being able to be reduced. Only the removal of the drift through the reconstruction of the impedance using ZHIT leads to a significantly better correspondence between measurement and model.

Figure 4: Impedance spectrum of a fuel cell, the fuel gas being poisoned by carbon monoxide

Figure 4 shows the Bode diagram of a series impedance measurement that was measured on a fuel cell, where the hydrogen in the fuel gas was intentionally poisoned by the addition of carbon monoxide. Due to the poisoning with carbon monoxide, active centers of the platinum catalyst are blocked, which severely affects the performance of the fuel cell. The blocking of the catalyst is dependent on the potential, with an alternating sorption and desorption of the carbon monoxide on the catalyst surface in the cell. This ("cyclical") change in the active catalyst surface is expressed in pseudo-inductive behavior, which can be observed in the impedance spectrum in Figure 4 at low frequencies (<3 Hz). In this figure, the impedance curve reconstructed by the ZHIT is shown by the violet line, while the support points from the original measured values ​​are shown by the blue circles. One can see very clearly the deviation in the low-frequency part of the measurement between these two curves. The evaluation of the spectra according to a selected model shows that a significantly better correspondence between model and measurement can be obtained if the reconstructed ZHIT impedances are used to calculate the impedances instead of the original measurement data.

literature

Original works:

• CA Schiller, F. Richter, E. Gülzow, N. Wagner: Validation and evaluation of electrochemical impedance spectra of systems with states that change with time . In: Physical Chemistry Chemical Physics . tape 3 , no. 3 , January 1, 2001, p. 374-378 , doi : 10.1039 / B007678N .

Further reading:

• W. Ehm, R. Kaus, CA Schiller, W. Strunz: Z-HIT - A Simple Relation Between Impedance Modulus and Phase Angle. Providing a New Way to the Validation of Electrochemical Impedance Spectra . In: F. Mansfeld, F. Huet, OR Mattos (Ed.): New Trends in Electrochemical Impedance Spectroscopy and Electrochemical Noise Analysis . Electrochemical Society Inc., Pennington, NJ, 2001, vol. 2000-24, ISBN 1-56677-291-5 , pp. 1-10.
• Andrzej Lasia: Z-HIT Transform . In: Electrochemical Impedance Spectroscopy and its Application . Springer New York Heidelberg Dordrecht London, 2014, ISBN 978-1-4614-8932-0 , p. 299.

Individual evidence

1. ^ W. Ehm, H. Gohr, R. Kaus, B. Roseler, CA Schiller: The evaluation of electrochemical impedance spectra using a modified logarithmic Hilbert transform . In: ACH-Models in Chemistry . tape 137 , no. 2–3 , 2000, pp. 145-157 .
2. ↑ ^{a b} W. Ehml: Expansion for the Logarithmic Kramers-Kronig relations . 1998 ( PDF on zahner.de [accessed on November 29, 2014] unpublished work).
3. ↑ Numerical values ​​zeta function
4. ↑ CPE (mathematical)
5. ↑ CPE (physical)
6. ^ W. Strunz, CA Schiller, J. Vogelsang: The change of dielectric properties of barrier coatings during the initial state of immersion . In: Materials and Corrosion . tape 59 , no. 2 , February 1, 2008, p. 159–166 , doi : 10.1002 / maco.200804156 .
7. ↑ ^{a b} C. A. Schiller, F. Richter, E. Gülzow, N. Wagner: Relaxation impedance as a model for the deactivation mechanism of fuel cells due to carbon monoxide poisoning . In: Physical Chemistry Chemical Physics . tape 3 , no. 11 , January 1, 2001, p. 2113-2116 , doi : 10.1039 / B007674K .