Computation of Equivalent Circuit Parameters of QCM and Evaluation of the Measurement Uncertainties

Abstract

This paper presents an approach of computing the parameters of quartz crystal microbalance (QCM) equivalent circuit model using the least squares algorithm in MATLAB. Modification of the measured data’s order in the least squares algorithm was implemented so that the precision was improved. Compared with nonlinear least squares algorithm, the computation method in this paper was faster and simpler with little lower accuracy. Moreover, Monte Carlo Method was utilized to evaluate the measurement uncertainties. An experiment on the temperature response of QCM’s equivalent circuit parameters was performed to show that the proposed approach was accurate and fast. The measurement errors are less than 0.6%. Introduction QCM has been used as a gravimetric sensor in many sensing applications for its high sensitivity. Since Kanazawa demonstrated that QCM could be applied in liquid phase , applications of QCM have extended to numerous fields like physics, electrochemistry and biology. In some applications resonant frequency is not the only parameter of interest, for properties of the added loading on the quartz sometimes are simply related to the parameters of the crystal’s equivalent circuit model. Each parameter of the QCM’s Butterworth-Van-Dyke (BVD) equivalent circuit model (shown in Fig. 1) represents a physical property of the quartz and its coating . On the other hand, oscillators are always a good choice for QCM sensors with the merits of low cost, continuous measurement and integration capability . The resonant frequency of oscillators is mainly determined by the equivalent circuit parameters. When loadings, especially in liquid occur on the surface of QCM, the oscillator’s resonant frequency may have a big difference with the fundamental resonant frequency of QCM. Hence, the exact measurement of the equivalent circuit parameters can make a big difference in improving the precision of oscillators. Impedance analysis or network analysis are the most common means to obtain QCM’s equivalent circuit parameters. Responses of QCM in a small range of spectrum near the resonant frequency are usually measured through a high precision instrument. With the conductance and admittance spectra, the parameters can be estimated. A highly accurate and reliable non-linear Levenberg-Marquadt least squares fitting algorithm has been used for estimating the parameters, however the algorithm suffers from the defects of requiring large space and time [2, . Fig. 1 Equivalent circuit model of QCM In this paper, an algorithm for computing the equivalent circuit parameters of QCM based on least squares in MATLAB is described. Aimed at a high precision, the order of the measured data in least squares algorithm is modified. And such a computing method’s uncertainty is evaluated through Monte Carlo mathematical simulation. An experiment about the temperature effects on the QCM C0 Cq Rq Lq 1919 2nd International Conference on Electronic & Mechanical Engineering and Information Technology (EMEIT-2012) Published by Atlantis Press, Paris, France. © the authors equivalent circuit parameters is performed and the results were compared with the algorithm of computing resonant frequency demonstrated in [5]. Experimental Method The sensing element consisted of an AT-cut quartz crystal vibrating at 4MHz. The quartz was stuck directly on a Peltier element by a PTFE gasket, which ensured heat transfer meanwhile maintained the fundamental resonant frequency of the quartz. The radiator underneath the Peltier element was employed to play a better cooling effect. A platinum resistance thermometer was placed on the quartz surface in order to follow the quartz temperature. The admittance spectra of QCM was measured by an impedance analyzer (Agilent 4294A) connected to a computer via GPIB interface. The measurement system is given in Fig. 2. Fig. 2 Schematic diagram of the measurement system The temperature of QCM was determined by the cooling power for the Peltier element. In this experiment, we set three different values of cooling power to get the QCM temperature response at 24°C, 26 °C and 29°C. When the temperature stabilized at the desired value, the measurement began. The spectrum was configured around the resonant frequency of 2 KHz here. The measuring points were set at 501 so that the resolution reached 4Hz. Result and Analyses Computing the equivalent circuit parameters of QCM in impedance analysis method. As shown in Fig. 1, the QCM equivalent circuit model consists of the motional branch q R , q L , q C and the static capacitor 0 C . The motional series resonant frequency s f is usually regarded as the response of QCM sensors in many applications. And s f can be obtained through q L and q C as follows: 1 2 s q q f L C π = . (1) The admittance spectra of the quartz generated from the BVD model can be expressed as: ( ) ( ) 0 j 1 j 1 j j Q q q q Y G B R L C C ω ω ω ω = + = + + + , (2) where G and B can be expressed respectively as: ( )2 2 1 q q q q G R R L C ω ω   = + −     , ( ) ( ) 2 2 0 1 1 q q q q q B C L C R L C ω ω ω ω ω   = − − + −     . (3) Firstly, the motional branch parameters were computed through the expression of G in Eq. 3 by least-square fitting and the expression were transferred to the general form as: ( ) ( ) 1 2 3 ; , , , Y f f β β β = = 1 2 3 X , X , X X β , (4) Agilent 4294A Impedance Analyzer QCM Peltier element Radiator PRT