Error Structure of Impedance Measurements

While the nature of the error structure of the measurements is often ignored or understated in electrochemical impedance spectroscopy, recent developments have made possible experimental identification of error structure. Quantitative assessment of stochastic and experimental bias errors has been used to filter data, to design experiments, and to assess the validity of regression assumptions. 

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The contributions to the error structure of impedance measurements Eire described in Section 21.1. Impedance measurements entail a compromise between minimizing bias errors, minimizing stochastic errors, and maximizing the information content of the resulting spectrum. The parameter settings described in this section may not apply to all impedance instrumentation. 

S. L. Carson, M. E. Orazem, O. D. Crisalle, and L. H. Garcia-Rubio, "On the Error Structure of Impedance Measurements Series Expansions," Journal of The Electrochemical Society, 150 (2003) E501-E511. 

M. E. Orazem, T. E. Moustafid, C. Deslouis, and B. Tribollet, "The Error Structure of Impedance Spectroscopy Measurements for Systems with a Large Ohmic Resistance with Respect to the Polarization Impedance/ Journal of The Electrochemical Society, 143 (1996) 3880-3890. 

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. 

Carson et al. °° showed that phase-sensitive detection measurements with a single reference signal biases the error structure of the impedance data due to errors introduced when the square-wave reference signal is in pheise with the measured signal. Modem phase-sensitive detection instruments employ more than one reference signal and may thereby avoid this imdesired correlation. 

Exercise 14.1 Determine the error structure of the impedance in the files Z1. z, Z2.z, Z3.Z, and Z4.z using Orazem s measurement model approach and determine the impedance parameters for Zl.z using the circuit / sCCdiC ctZpLw)). where ZpLw is the finite-length transmissive mass transfer impedance. 

E>ue to the appearance of the variance in equation (19.24), the statistic provides a useful measure of the quality of a fit only if the variance of the measurement is known. The techniques described in Qiapter 21 may be used to assess the standard deviation of an impedance measurement as a function of frequency. In the absence of such an assessment, researchers have used assumed error structures, but, in this case, the numerical value of the statistic cannot be used to assess the... 

In a general sense, the frequency-domain error structure is determined by the nature of errors in the time-domain signals and by the method used to process the time-domain data into the frequency domain. The ceU impedance influences the frequency-dependence of the variance of the measurements, but the cell impedance does not influence whether the variances of real and imaginary components are equal or whether errors in the real and imaginary components are uncorrelated. 

The third approach is to use experimental methods to assess the error structure. Independent identification of error structure is the preferred approach, but even minor nonstationarity between repeated measurements introduces a significant bias error in the estimation of the stocheistic variance. Dygas emd Breiter report on the use of intermediate results from a frequency-response analyzer to estimate the variance of real and imaginary components of the impedance. Their approach allows assessment of the variance of the stochastic component without the need for replicate experiments. The drawback is that their approach cannot be used to assess bias errors and is specific to a particular commercial impedance instrumentation. Van Gheem et have proposed a structured multi-sine... 

Measurement models, developed for impedance spectroscopy by Agarwal et 56,86,262 generally applicable and can be used to estimate both stochastic and bias errors of a measxuement from imperfectly replicated impedance measurements. Orazem et al. used a measurement model approach to show that a general model for the error structure could take the form... 

The concept of the measurement model as a tool for assessment of the error structure was applied to impedance spectroscopy initially by Agarwal et The... 

Dielectric Constant and Dielectric Loss. Impedance measurement bridges are the most commonly used instruments for dielectric constant and dielectric loss measnrements for lie-quencies up to 10 MHz. A capacitor structure is built as a bottom electrode, top electrode, and two or more separation layers of dielectric material. Figure 8.44 shows a typical electrode pattern. The outside guard ring on the top electrode pattern prevents measurement errors due to fringe effects. Dielectric constant and dielectric loss measurements at frequencies... 

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. 

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