Butterworth-van Dyke equivalent circuit

Fig. 8 a Experimental setup to perform impedance analysis of the shear oscillation. Quartz resonators are used as the bottom plate of a measuring chamber that is mounted in a temperature-controlled Faraday cage. Impedance data is recorded with a gain/phase analyzer in the vicinity of the fundamental resonance of 5 MHz (typically from 4.97 MHz to 5.04 MHz). b Butterworth-Van Dyke equivalent circuit to analyze the impedance raw data of the loaded resonator. All parameters except Zl are assigned to the unperturbed resonator whereas Zl denotes the impedance of the load material (cell layer) on the resonator surface... 

The electrical characteristics of a QCM are well represented by a simple RLC damped resonator equivalent circuit [8], termed the Butterworth-Van Dyke equivalent circuit (Figure 11). 

FIG. 3. The Butterworth-van Dyke equivalent circuit of the loaded quartz crystal resonator. The parameters R, C, and L describe the behavior of the unloaded quartz resonator Zout is the impedance of the contacting medium. 

This relation is only valid for acoustically thin films. Concretely, it means that the relation is not respected when thick layers are used because the influence of the viscoelastic properties of the layer may appear in the frequency change. This effect is, in general, amplified when polymer Aims are used. To determine the maximum useful thickness, electroacoustic measurements allowed a pertinent value to be evaluated. In a first step, a classical Butterworth-Van Dyke equivalent circuit of the loaded quartz (Fig. 13a) is extracted... 

Figure 13. Electroacoustic characterization of the quartz crystal loaded by a polypyrrole film, (a) Butterworth-Van Dyke equivalent circuit, (b) Change of the frequency, fs = l/(2w vTinOn), and motional resistance, Rm, with respect to the polypyrrole film thickness. From AI-Sana et al. ...

Passive oscillator mode Impedance analysis of the forced oscillation of the quartz plate provides valuable information about the coating even if the active mode is not applicable anymore. For impedance analysis, a frequency generator is used to excite the crystal to a constraint vibration near resonance while monitoring the complex electrical impedance and admittance, respectively, dependent on the applied frequency (Figure 2B). For low load situations near resonance, an equivalent circuit with lumped elements - the so-called Butterworth—van-Dyke (BVD) circuit — can be applied to model the impedance data. The BVD circuit combines a parallel and series (motional branch) resonance circuit. The motional branch consists of an inductance Lq, a capacitance Cq, and a resistance Rq. An additional parallel capacitance Co arises primarily from the presence of the dielectric quartz material between the two surface electrodes (parallel plate capacitor) also containing parasitic contributions of the wiring and the crystal holder (Figure 2B). 

In contrast to the case of uniform mass loading, dm/ (R) = 0, two values of the resonance frequency appear. This effect can be simulated by a simple equivalent circuit consisting of two Butterworth-van Dyke [BBSS] circuits in series with the inductances corresponding to the two different values of the surface mass densities. Am/ + Am/ and Am/ — Am/. Due to overlap of these two resonance states, splitting can manifest itself as a broadening of the resonance, which will have an effective width of the... 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

Fig. 13.8. Equivalent circuit models for crystal impedance responses (a) transmission line model (b) lumped clement (modified Butterworth van Dyke) model.

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

As shown in the previous section, the mechanical properties of a quartz crystal close to resonance frequency can be expressed by means of a motional impedance. To complete the equivalent circuit of a quartz crystal, the capacitance, Co, must be added in parallel to the motional impedance. It results in the Butterworth-Van Dyke (BVD) equivalent circuit of a quartz crystal, as shown again in Fig. 8 for an unloaded quartz crystal [32]. In this notation common in electronic Hterature, Is is the dynamic inductance and is imder-stood here as a representation of the oscillating mass of the quartz crystal. Cs is the dynamic capacitance and reflects the elasticity of the oscillating body. Rs is the dynamic resistance and returns friction of the quartz slice as well as all kinds of acoustic damping. 

In the following, we derive the Butterworth-van Dyke (BvD) equivalent circuit (Fig. 7) from the Mason circuit (Fig. 6c). The Mason circuit itself is derived in detail in [4]. The BvD circuit approximates the Mason circuit close to the resonances. The BvD circuit accounts for piezoelectric stiffening and can also be extended in a simple way to include an acoustic load on one side of the crystal. In the derivation of the BvD circuits, one assiunes small frequency shifts as well as small loads and apphes Taylor expansions in the frequency shift (or the load) whenever these variables occur. The condition of A/// load impedance of the sample, Zi, is much smaller than the impedance of crystalhne quartz, Zq (where the latter, as opposed to Zl, is a material constant). Zq sets the scale of the impedances contained in the Mason circuit. Generally speaking, the QCM only works properly if ZL Zq.ii... 

Fig. 13 Steps in the derivation of the Butterworth-van Dyke circuit, a Same as Fig. 6c with the circuit elements rearranged, b Norton equivalence Zx = aZ/, Zy = aZ, a = Z + Z )jZ. c Norton equivalence applied to b. c Same as c where the relation - 2/ sin(2x) + tan(x) =- cot(x) has been used. The two transformers have been merged...

The quartz disk is used as the bottom plate of a cell culture vessel and is moimted in a temperature controlled crystal holder (37 °C). The surface electrodes on either side of the quartz are connected to an impedance analyzer (Solatron Instruments, SI-1260) operating in continuous wave mode. The frequency-dependent complex impedance Z(J) returned by the impedance analyzer is expressed as magnitude of impedance Z (f) and phase shift between voltage and current (f). The raw data is analyzed by the well-known Butterworth-Van Dyke (BVD) equivalent circuit with the liunped impedance elements Co, Rq, iq, Cq and Zl. Rq, Lq and Cq represent the piezoelectric properties of the unperturbed resonator itself, whereas Co summarizes its dielectric properties and all parasitic contributions arising from contacts and wiring. The load material in contact with the resonator surface is represented by the complex impedance Zl. As long as the resonator is not loaded too... 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

Generally, electroacoustical resonators can be described by mechanical and electrical equivalent circuits. For the quartz, two electrical models are often used the transmission line model and the Butterworth-van Dyke circuit (BVD circuit). These models were made in order to describe the propagation of the acoustic wave in analogy with the electrical waves. More detailed descriptions of electrical equivalent circuits can be found, for example, in [4, 11, 26,48,49]. 

The conference persuaded Cady to turn his interest to piezoelectricity. In 1919 Cady initiated the study of resonators and the first report on piezoelectric resonator was presented to the American Physical Society in 1921. He proposed the piezoelectric quartz resonator as a frequency standard or a filter. Cady showed how to connect a resonating quartz crystal to an electrical oscillator and in this way to achieve frequency stability. Studies of properties of crystal resonator represented by its equivalent electrical circuit were undertaken by Butterworth, Dye, Van Dyke and Mason. They led to a better imderstanding of crystal resonators used in filters and... 

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