Piezoelectric equivalent circuit

Fig. 6.3 (a) Equivalent circuit for a piezoelectric specimen vibrating close to resonance (b)... 

Piezoelectric crystals, notably quartz, are used to control or limit the operating frequency of electrical circuits. A well-known example is their use in quartz clocks . The fact that a dielectric body vibrating at a resonant frequency can absorb considerably more energy than at other frequencies provides the basis for piezoelectric wave filters. The equivalent circuit for a piezoelectric body vibrating at frequencies close to a natural frequency is given in Fig. 6.3. At resonance the impedance due to L, and C falls to zero and, provided that Rx is small, the overall impedance is small. 

In the previous section we considered the conditions under which mechanical resonances would occur in a TSM resonator. In considering only the mechanical properties of the crystal, however, we neglected consideration of how these resonances would actually be excited or detected. The device uses a piezoelectric substrate material in which the electric field generated between electrodes couples to mechanical displacement. This allows electrical excitation and detection of mechanical resonances. In constructing a practical sensor, changes in resonant frequency of the device are measured electrically. The electrical characteristics of the resonator can be described in terms of an equivalent-circuit model that describes the impedance (ratio of applied voltage to current) or admittance (reciprocal of impedance) over a range of frequencies near resonance. 

The derivation above ignores piezoelectricity (Sect. 6). The theory of the piezoelectric plate has been worked out by Tiersten [56]. Kanazawa has applied this theory rigorously to the case of a crystal loaded with a liquid or a viscoelastic film [54]. These treatments are equivalent to the treatment with equivalent circuits (Sect. 6), and we therefore defer the discussion of piezoelectricity to that section. 

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.

Fig. 6 Equivalent circuits of a a viscoelastic layer of thickness 2h, b two viscoelastic layers of thickness 2hi and 2h2 (where 1 denotes the quartz crystal and 2 denotes the film), and c a piezoelectric plate loaded on one side with a load AZi. The parameter h is half of the thickness of the respective layer...

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. 1 Equivalent circuit representation of the quartz crystal including a load. Piezoelectric stiffening (described by the element 42 in Fig. 13, Chap. 2 in this volume) was neglected. The sample is represented by the load Zl...

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... 

It is important to be able to calculate the system parameters from the model of the sensor. Let us determine the cut-off frequency for a piezoelectric sensor which has C = 400 pF and leakage resistance of 10 Gfi. The amplifier input impedance is 10 MS2. If we use the modified piezoelectric sensor equivalent circuit (figure 2.1(b)) for this calculation, we find that the cut-off frequency for this circuit is... 

The high-frequency equivalent circuit for a piezoelectric sensor is complex because of its mechanical resonance. This can be modeled by adding a series RLC circuit in parallel with the sensor capacitance and leakage resistance. The high-frequency equivalent circuit and its frequency response are shown in figure 2.3. In some applications, the mechanical resonance is desirable for accurate frequency control, as in the case of crystal filters. 

Figure. I. Equivalent circuit for piezoelectric crystal resonator.

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... 

From the expression in Eq. 19, most forms of equivalent circuit models of piezoelectric elements may be found. The Van Dyke circuit [4] is the simplest, using discrete electrical components combined to approximate the piezoelectric... 

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... 

Leedom D, Krimholtz R, Matthaei G (1971) Equivalent circuits for transducers having arbitrary even-or odd-symmetry piezoelectric excitation. IEEE Trans SonUltrason 18(3) 128-141... 

Equivalent circuit models have been used to represent piezoelectric material behavior for many years, but are not without limitations. The most common representation for a piezoelectric material, as recommended by the IRE/IEEE standards on piezoelectricity, is the Van Dyke model as described in detail in piezoelectric materials in microfluidics, with a capacitor, resistor, and inductor in series representing a single resonance, or motional component of the piezoelectric element, all set in parallel with a second capacitor, representing the shunt capacitance. Additional resonances may be included by placing additional... 

Krimholtz R, Leedom DA and Matthai G (1970) New equivalent Circuit for Elementary Piezoelectric Transducers. Electronics Letters 6 398... 

Leedom DA, Krimholtz R, Matthai GL (1971) Equivalent Circuit for Transducers Having Arbitrary Even- or Odd-Symmetry Piezoelectric Excitation. IEEE Transactions on Sonics and ultrasonics, SU-18 128-141... 

The equivalent circuit for the piezoelectric acuator is represented by a combination of L,C and R. Figure 4.1.1 la shows an equivalent circuit for the resonance state, which has a very low impedance. Ca corresponds to the electrostatic capacitance, and the components La and Ca in a series resonance circuit are related to the piezoelectric motion. For example, in the case of the longitudinal vibration of the above rectangular plate through dji, these components are represented by... 

FIGURE 4.1.11 Equivalent circuit of a piezoelectric device for (a) the resonance and (b) the antiresonance states. 

Piezoelectric crystals can be used as the basis for force measurement. An equivalent circuit is shown in Figure 27.9, which captures the electrical characteristics of this material under an applied force. 

Most piezoelectric characterization methods were developed for crystalline ceramics and had to be adapted for piezoelectric polymers. Methods based on resonance analysis and equivalent circuits that can be used to characterize semicrystalline PVDF and its copolymers are outlined in IEEE standards (66). Details for applying resonance analysis to piezoelectric polymers have recently been explored (67). Because of the lossy nature of some polymers, the IEEE standards are not adequate, and other techniques are needed to describe piezoelectric properties more accurately. 

FIGURE 3.6 Passive JT network for measurement of equivalent circuit parameters. (From Zelenka, J. 1986. Piezoelectric Resonators and Their Applications, Chap. 5. Elsevier, Amsterdam.)... 

Fig. 6.16. Piezoelectric stack translator, a Structure, b electromechanical equivalent circuit and amplitude responses of the actuator and sensor transfer behaviour in small signal operation (derived from [5])...

The interpretation of (6.44), (6.45), (6.46) and (6.47) is illustrated in Fig. 6.130, showing an electromechanical equivalent circuit diagram. Accordingly, the input of a piezoelectric transducer can be considered as an electrical capacitor with the capacitance C and its output as a mechanical spring with the stiffness cp. As in reality C is always lossy and cp has always a mass and a structural damping behaviour, the amplitude response IV /Fgl of the piezoelectric transducer has a definite lower cut-off frequency /u and a mechanically determined natural frequency /o for an open electrical port (Jg = 0), and the amplitude response s/V has a mechanically determined natural frequency /o for an open mechanical port F = 0). 

SAW component design is based on the application of an equivalent circuit model [29] using the values of the piezoelectric coupling coefficient of the material, Fo. and the static capacitance [30], The frequency at which the AW device operates depends on ... 

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