Assessing Impedance Analyzer Data Quality by Fractional Order Calculus: A QCM Sensor Case Study

Abstract

The paper presents the theoretical, simulation, and experimental results on the QCM sensor based on the Butterworth van Dyke (BVD) model with lumped reactive motional circuit elements of fractional order. The equation of the fractional order BVD model of the QCM sensor has been derived based on Caputo definitions and its behavior around the resonant frequencies has been simulated. The simulations confirm the ability of fractional order calculus to cover a wide range of behaviors beyond those found in experimental practice. The fractional order BVD model of the QCM sensor is considered from the perspective of impedance spectroscopy to give an idea of the advantages that fractional order calculus brings to its modeling. For the true values of the electrical parameters of the QCM sensor based on the standard BVD model, the experimental investigations confirm the equivalence of the measurements after the standard compensation of the virtual impedance analyzer (VIA) and the measurements without compensation by fitting with the fractional order BVD model. From an experimental point of view, using fractional order calculus brings a new dimension to impedance analyzer compensation procedures, as well as a new method for validating the compensation.

Effect of Load on Quartz Crystal Microbalance Sensor Response Addressed Using Fractional Order Calculus

“…The experimental study in air involved the following steps: (i) BVD compensation of the VIA [ 28 ] in air followed by verification of the compensation using the fractional order BVD model [ 38 ]; (ii) determination of the BVD parameters in air based on the experimental resonance frequencies [ 28 ]; and (iii) using the Levenberg–Marquardt (LM) algorithm to fit the BVD model and fine-tune the raw electrical parameters. To evaluate the fractional order effect, the value of the electrical parameters of the previously determined BVD model was considered, and a rerun with the fractional order BVD model was performed.…”

“…Two distinct applications of FOC in QCM technology have been highlighted. The first application presented in detail in the literature [ 38 ] is essentially a highly sensitive method for verifying the optimal compensation of an impedance analyzer. By compensating the VIA using the FOC method any fractional order behavior for the reactive motional circuit elements of the BVD model is eliminated.…”

“…A large spectrum of QCM sensor behavior can be covered by the FOC according to extensive simulations of the fractional order BVD model. An advantage in experimental practice is the ability to evaluate the α and β parameters of the fractional order BVD model and determine whether the VIA compensation is accurate [ 38 ]. A comprehensive understanding of the fractional order BVD model, followed by simulation of commonly encountered situations in experimental practice, provides the necessary support.…”

“…The impedance analyzer uses a passive approach; therefore, under ideal conditions, the electrical parameters of the BVD model or the derived physical parameters of the QCM sensor are unaffected [ 35 ]. In reality, the experimental setup and topology for adapting the QCM sensor to a general-purpose impedance analyzer are sources of measurement uncertainty [ 38 ].…”

“…This section concludes with a simulation of the frequency response of a QCM sensor using a fractional order BVD model and an experimental setup involving a virtual impedance analyzer (VIA). In Section 3 , the benefits of the fractional order BVD model and the experimental outcomes of impedance spectroscopy augmented by VIA compensation using a FOC-based approach in air [ 38 ] are presented. Section 4 and Section 5 present the final discussion and the conclusions, respectively.…”