On the Error Structure of Impedance Measurements

Abstract

A framework is presented for characterizing the statistical properties of frequency-domain measurements. The correlation between the variance in the complex impedance and the variances in the complex current and voltage are examined by means of a Taylor series expansion for the impedance. Both sufficient-only and necessary and sufficient conditions for equality of the variances of real and imaginary impedance are identified and verified using stochastic simulations of techniques commonly employed in frequency-domain measurements.

A systematic approach toward error structure identification for impedance spectroscopy

“…Carson et al [27] used numerical simulations to show that, when a frequency-response analyzer algorithm is used to obtain the impedance response from time-domain signals containing normally distributed errors, the real and imaginary parts of the impedance were uncorrelated. In contrast, use of a phase-sensitivedetection algorithm yielded correlation between real and imaginary components of the impedance [28].…”

On the Error Structure of Impedance Measurements

“…In addition, the impedance values calculated using the PSD algorithm were not normally distributed for all frequencies; whereas, the impedance values were normally distributed when calculated using the FRA algorithm. Finally, as discussed in a companion paper, 22 the different statistical properties obtained with PSD simulations could be attributed in part to bias errors introduced when the square-wave reference signal was in phase with the measured signal. Thus, it can be argued that, of the two approaches, it is the FRA which provides statistical properties intrinsic to transferfunction measurements.…”

“…The present simulation was conducted under galvanostatic modulation, in which the current was the input signal, and potential was the output. The analysis of the sta-tistics of input and output signals, presented by Carson et al, 22 showed that the PSD algorithm used reduced the apparent variance of the real part of the input signal. The simulations were conducted under galvanostatic modulation, in which current was the input signal.…”

Finding the Optimal Regularization Parameter in Distribution of Relaxation Times Analysis

“…The error term in equation (11) consists of a scaling factor which is multiplied with the modulus of the respective frequency, ensuring the heteroscedastic structure of stochastic error in impedance measurements, and a normally distributed random number ( and ), which is different for each frequency and for the imaginary and the real part as well. This leads to a damped error structure which is randomly distributed around an EV of 0 . The error was added separately to the imaginary and the real part, as well as for individual frequencies.…”