It is well known that the real and imaginary parts of certain complex function
Kramers  and Kronig  developed the equations that bear their names. When working on linear optical spectroscopy, they can be applied to a large number of physical processes and are one of the most general relationships in electrodynamics. KK relationships are established between the real part of certain quantities that characterize the physical dispersion processes (dielectric permittivity, magnetic permeability, conductivity, etc.) and the imaginary part of these quantities that characterize the physical absorption processes.
If instead of
On the other hand, if we know Z′(ω), equation (4) allows us to obtain Z′′(ω).
If Z′′(ω) has the formula:
If now Z′(ω) is expressed in terms of ω as shown in equation (8)
The integral (4) gives as a result:
In many situations, especially when conducting experiments, a limited number of measured values of Z′ and Z′′ are available in a narrow range of frequencies. This limited frequency range avoids knowing ω →∞ or ω → 0, limiting the validity of the KK equations. The limitation may have different origins, for example, it may be purely technological, such as the Solartron 12508W or HP 4291 impedance analyzers, which have ranges from 10 μHz to 65 kHz and 1 MHz to 1.8 GHz respectively.
A frequency range is also limited when separating a phenomenon of interest from others, reducing the range of measured frequencies, as in the monitoring of α, β and δ dispersions of biological suspensions. For example, all the information needed to characterize the β dispersion of a yeast suspension from an industrial fermenter is in the range of 0.1 MHz to 10 MHz . Other dispersions outside this range can also be observed, but they are not of practical interest. The necessary information to monitor yeast growth is only in the mentioned range.
In the case of yeasts, real and imaginary components for ω → ∞ and ω → 0 can be obtained, simply by assuming that the values outside the working range remain constant. This leads us to assume that there are measurements
between 0 and ∞. This is not the case when impedance is used to monitor electrochemical sensors. Here, the ranges of frequency to observe the behavior of the electrode-electrolyte interfaces must include very low frequencies, difficult to measure experimentally.
For example, to complete the sigmoid curve of interface impedance modelled with the Randles circuit, it is necessary to measure frequencies lower than 0.1 Hz . This is impractical because it is very difficult to keep stable measurements for enough time to complete a cycle at very low frequencies, where a cycle can take several minutes.
All these limitations of information that Z′ and Z′′ give us, play an important role when using KK equations to verify that the system of measurement is working correctly.
Several works about extrapolating experimental results below the frequency at which the imaginary part of the impedance presents the maximum have already been published.
These methodologies are based on the symmetry of the imaginary impedance around the frequency of the maximum and are limited to systems having one time constant. Kendig and Mansfeld , for example, obtained the polarization resistance from the following equation:
Macdonald and Urquidi-Macdonald  presented a method of experimental data extrapolation consisting of polynomial fit. The authors evaluated the integral in equation (11) by segments, adjusting the experimental data of Z′′ versus frequency with a fifth-order polynomial, given by equation (12), using the least squares technique.
The segments on which the integral is evaluated are chosen to coincide with the sign of Z′′ changes or with changes in the gradient of Z′′, This methodology of experimental data extrapolation is also presented in ref .
Esteban and Orazem  proposed an approach that avoids the inconveniences derived from extrapolations with polynomials and also does not require making assumptions about asymptotical behaviors. Authors use an algorithm to determine the functions of the real and imaginary impedance components in a region of ω for which no experimental data are available. Then, for each frequency, the polynomial functions of K and M order given by
Due to the complicated form of the integrands, the integrals were numerically solved.
Another approach shown in the literature is based on the expected asymptotical behavior of a typical electrochemical system .
That is, to extrapolate the Z′′(ω) value to ω = 0, it is assumed that the function is proportional to ω, when ω → 0, This is consistent with the behaviour in frequency of Z′′(ω) in a Randles circuit.
For its part, the real part of the impedance tends to a constant value, which is the sum of the electrolytic solution resistance and the charge transfer resistance. The last resistance models the non-linearity of the electrode-electrolyte interface.
Finally, it should be mentioned that the lack of agreement between experimental data and the corresponding KK transformations can be attributed to two factors: on the one hand, the questions associated with the non-linearity or non-causality of the system and, on the other hand, the questions related to the low precision of the measurement systems at a very low frequency.
In order to overcome those limitations, we propose to extrapolate data outside of the available range of measurement, significantly improving the results of applying the KK transformation. The method, which is based on the symmetry of Z′′(ω), can be applied in impedance measurements of the electrode-electrolyte interface (EEI), when the expected behavior responds to the Randles model.
In order to illustrate the application of the KK transformation and validate the method, measurements were made using a tripolar cell in an electrolytic solution.
Several frequency sweeping experiments were carried out at constant overpotential and using electrodes polished with sandpaper #180, The three-electrode electrochemical cell used (Fig. 1) is composed of a working electrode (WE) and an Ag/AgCl reference electrode (Re1). An AISI 304 stainless steel concave counter-electrode (CE) 85 mm was also used. The WE is a solid cylinder (1.5 cm long) made also of AISI 304 stainless steel with only 1 cm2 of its transversal section exposed; the rest was insulated with Grilon. The CE area was larger than the working electrode to minimize its impedance. The sampling frequency was such that the measurement points were equally spaced in a logarithmic scale taking 5 points per decade. The integration time of the measurements was 1 cycle for frequencies lower than 6.28 rad/s and 16 cycles for the rest.
In every case, the potential was stabilized in an open circuit until the voltage shift was lower than 0.05 mV/seg. Then, the equivalent series resistance and reactance measurements were carried out. The overpotentials used were 10 mV. The electrolyte solution used was NaCl 0.9%.
Electrochemical measurements were performed with a Solartron 12508W Impedance Analyzer composed of a Solartron 1287 Electrochemical Analyzer and a Solartron 1250 Frequency Response Analyzer, commanded by the software provided by the manufacturer (ZPlot®, Scribner Associates Incorporated).
The method consists of expanding the amount of data available in a range of frequencies greater than that measured, extrapolating data from the available measurements.
The method requires two conditions for its application: (i) the curves of Z′(ω) and Z′′(ω) must be symmetric respect to the central frequency, a condition that is fulfilled by dispersions with a single time constant, and (ii) the frequency range of the measurements must cover at least the central frequency of the dispersion until the percentage change of Z′(ω) is below 0.1, Extrapolation allows us to reduce the difference between the curves of the measured and the calculated data using the KK transformation.
Extrapolation is performed at low frequency, from the lowest limit of frequency ωmin up to the central frequency of the dispersion ωp, taking into account that Z′(ω) − Z′(∞) is symmetric respect to ωp , This frequency ωp is also the frequency in which Z′′(ω) is maximum. ωp , often called the peak frequency.
In this work, we used
Figure 2 graphically shows how to extrapolate frequency and amplitude of
The conducted research is not related to either human or animal use.
Figure 3 shows the plots of
Equation (4) solved numerically is used to obtain
Figure 5 shows the expanded dataset of
It should be noted that the agreement between maximum of
Ideally, equations (2) through (4) of KK allow us to obtain
These restrictions are fulfilled in many physical and biophysical systems for certain frequency ranges.
However, unlike published works, the method presented does not use polynomial fit or assumptions about asymptotic behaviors. For frequencies lower than ωp, the method replicates the behaviour observed for
Figure 7 shows how the theoretical adjustment depends on the amplitude of the extrapolated frequency range, by representing the percentage change of
In conclusion, we present a simple method to extrapolate impedance data sets from linear, causal systems and with a single time constant. The new method allows us to apply the KK equations with a minimum estimation error of the real or imaginary parts of impedance.