Equivalent circuit Mason

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

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... 

The Mason equivalent circuit may be derived directly from Eq. 19. It is sometimes called a transmission-line circuit model since the transcendental terms in the matrix appear in the same way when modeling power transmission lines. Most importantly, the circuit represents more than one resonance with these transcendental terms. Consider first an element that does not have piezoelectricity, implying the piezoelectric stress coefficient e = 0. The force-velocity relationships in the nonpiezoelectric element would then be... 

So far various analyses of PZT sensors have been performed. These can be classified into two groups. One employed equivalent electric circuit (Mason 1958) and the other applied solutions of the field equations (Auld 1973). These are based on one-dimensional analysis, and thus results can not be readily extended to three-dimensional (3-D) analysis. This is because the PZT element used in an AE sensor is neither an infinite bar nor an infinite plate. 

T. Nakamoto, and T. Moriizumi, A theory of a quartz crystal microbalance based upon a mason equivalent-circuit,ypn.y. Appl. Phys. 1,29, 963-969 C1990],... 

In this section, the Mason circuit does account for piezoelectric stiffening, whereas piezoelectric stiffening is neglected in Sects. 4 and 5. In order to find exact equivalence between the three models, the element Z), (deaUng with piezoelectric stiffening) must be deleted from the Mason circuit. 

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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