Measurement errors Impedance analysis

In addition, the variance of impedance measurements depends strongly on frequency, and this variation needs to be addressed by the regression strategies employed. An assumed dependence of the variance of the impedance measurement on impedance values was employed in early stages of regression analysis, and this gave rise to some controversy over what assumed error structure was most appropriate. An experimental approach using measurement models, described in Chapter 21, was later developed, which eliminated the need for assumed error structures. 

Figure 23.1 Schematic flowchart showing the relationship between impedance measurements, error analysis, supporting observations, model development, and weighted regression analysis. (Taken from Orazem and Tribollet and reproduced with permission of Elsevier, Inc.)...

Impedance analysis is also suggested when properties of an attached film, a liquid, or interfaces are of interest. Due to the weak frequency dependence of the acoustic load within a typical measurement range of some 10 kHz at fundamental mode, one measurement point would be sufficient to calculate Zl (Eq. 2). An effective method to decrease statistical errors is to first fit a theoretical curve to the experimental curve or a specific segment, secondly to calculate Zl from the fit, and finally to extract (material) parameters of interest using separate models describing how the acoustic load is generated [37]. 

A number of other operational problems exist when using the FFT algorithm. The most important of these, as far as electrochemistry is concerned, is due to the inherently nonlinear nature of the system. When Eq. (56) is used to measure the impedance with an arbitrary time domain input function (i.e. not a single-frequency sinusoidal perturbation), then the Fourier analysis will incorrectly ascribe the harmonic responses due to system nonlinearity, to input signal components which may or may not be present at higher frequencies. As a consequence, the measured impedance spectrum may be seriously in error. 

A systematic error analysis demonstrates that, in spite of differences between sequential impedance scans and sometimes the appearance of inductive and incomplete capacitive loops, the individual data sets represented a pseu-dostationary system and could be interpreted in terms of a stationary model [37]. An example of such a system can be impedance analysis of corrosion, which usually includes both capacitive and inductive elements. The residual errors impedance measurement can be expressed in terms of the... 

It is particularly interesting to test whether an analysis of a conventional macroscopic impedance measurement based on a brick layer model leads to the same results as obtained with microcontacts. The relaxation frequency of the grain boundary arc measured in the conventional impedance experiment (1.2 Hz) is similar to the mean value (3.0 Hz) deduced from the microelectrode experiments, although not identical. One possible reason for the moderate discrepancy is the inaccuracy with respect to the temperature measurement, which is somewhat difficult in the case of the microelectrode set-up. A temperature error of about 20 K could already explain the difference. 

As with any analytical technique, it is important for US spectrometry users to have a thorough understanding of its underlying physical principles and of potential sources of errors adversely affecting measurements. The basis of ultrasonic analyses in a number of fields (particularly in food analysis) is the relationship between the measurable ultrasonic properties (velocity, attenuation and impedance, mainly) and the physicochemical properties of the sample (e.g. composition, structure, physical state). Such a relationship can be established empirically from a calibration curve that relates the property of interest to the measured ultrasonic property, or theoretically from equations describing the propagation of ultrasound through materials. 

Access to powerful computers and to commercial partial-differential-equation (PDE) solvers has facilitated modeling of the impedance response of electrodes exhibiting distributions of reactivity. Use of these tools, coupled with development of localized impedance measurements, has introduced a renewed emphasis on the study of heterogenous surfaces. This coupling provides a nice example for the integration of experiment, modeling, and error analysis described in Chapter 23. 

The magnitude of the stocheistic errors in impedance measurements depends on the selection of experimental parameters as detailed in Chapter 8. The simulation results described by Carson et a 00,25i,255 particular provide insight into differences between commonly used impedance instrumentation, including methods based on Fourier analysis and on phase-sensitive detection. ... 

Some general properties for stochastic errors have been established for impedance measurements through experimental observation and simulations. The results described here correspond to additive time-domain errors. The comparison between simulations and experimental results obtained via Fourier analysis supports the suggestion that the nature of experimental time-domain errors is likely to be additive rather than proportional ... 

In the absence of instmment-induced correlations, stochastic errors in the frequency-domain are normally distributed. The appearance of a normal distribution of frequency-domain stochastic errors can be regarded to be a consequence of the Central Limit Theorem applied to the methodology used to measure the complex impedance. ° This result validates an essential assumption routinely used during regression analysis of impedance (and other frequency-domain) data. 

It should be noted that the error analysis methods using measurement models are sensitive to data outliers. Occasionally, outliers can be attributed to external influences. Most often, outliers appear near the line frequency and at the beginning of an impedance measurement. Data collected within 5 Hz of the line frequency and its first harmonic (e.g., 50 and 100 Hz in Europe or 60 and 120 Hz in the United States) should be deleted. Startup transients cause some systems to exhibit a detectable artifact at the first frequency measured. This point, too, should be deleted. 

Impedance Measurements Integrated with Error Analysis... 

S. L. Carson, M. E. Orazem, O. D. Crisalle, and L. H. Garcia-Rubio, "On the Error Structure of Impedance Measurements Simulation of Frequency Response Analysis (FRA) Instrumentation," Journal of The Electrochemical Society, 150 (2003) E477-E490. 

A wide variety of ion-selective electrodes are now available (see Table 11.7) and the only instrumentation required is a high-impedance voltmeter to monitor the potential difference between the measuring electrode and the external reference electrode. The voltmeter must, however, be capable of accurate measurement since an error of 0.1 mV in the measurement of potential introduces an inaccuracy of almost 1% in the ion analysis (a 1 mV error in potential leads to a concentration error of 4% and 8% for singly and doubly charged ions respectively). 

The author would like to thank Thomas Springer (Los Alamos) for many contributions to the mathematical modelling aspects mentioned here, Ruth Sherman (Los Alamos) who gathered much experimental data on RuOj, Judith Rishpon (University of Tel Aviv) who made the quartz crystal microbalance measurements of Fig. 14, and Bernard Boukamp (University of Twente) who has allowed the author to use his non-linear least-squared error fitting routines for impedance data analysis. 

Linearity, causality, stability, consistency, and error analysis of impedance measurements... 

Such initial experimental and data-assessment procedures should be supported by a series of measurements at different potentials, temperatures, concentrations, and convections, with the data to be combined with the error analysis. After the data is acquired, it can be initially represented by an equivalent circuit, physical, or continuum level model that is consistent with physical and chemical information and is comparable to previously published EIS and other analytical results on identical or at least similar systems. The preliminary selection of the data representation, such as complex impedance, modulus, and phase- angle notations, is often helpful, as quite often some of these graphic notations are more informative than others. 

Another problem of data modeling is due to the fact that the same data may often be represented by very different equivalent circuits. The model is chosen to give the best possible match between the calculated and the measured impedance data. However, there is not a unique equivalent circuit that describes the spectrum, and one cannot assume that an equivalent circuit that produces a good fit to a data set represents an accurate physical model of the cell. The exclusive use of equivalent circuit models for the analysis of largely unknown materials and systems is fraught with potential sources of error. 

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