Piezoelectricity piezoelectric charge

Fig. 4.4. The piezoelectric charge produced by elastic strain in x-cut quartz and z-cut lithium niobate is well represented by a quadratic relationship without a need for fourth-order contributions.

The piezoelectric effect entails a linear coupling between electrical and mechanical energies. Numerous piezoelectric coefficients are in use, depending on the electrical and mechanical boundary conditions imposed on the part under test. Each of the piezoelectric d, e, g, and h coefficients can be defined in terms of a direct and a converse effect the two sets of coefficients are related by thermodynamics. For example, the piezoelectric charge coefficient, dkjk, can be defined via [1] ... 

Because of the very large differences in mechanical compliance between the two phases, the hydrostatic pressures experienced by the polymer phase transfer forces to the rods magnifying the stress in them in the poled direction. This magnified stress increases the charge induced on to the electrode tending to compensate for the inactive polymer. As a result, in typical practical cases, the piezoelectric charge coefficient fi/13) is not very sensitive to volume fraction of PZT . 

The piezoelectric charge coefficient is improved, and the materials is more difficult to depole. A typical composition is Pbo,94 Sro.oeCZro.sa Tio.47)03.o. 

We can estimate the piezoelectric charge and piezoelectric constant in the Bond Orbital Approximation with no additional assumptions or parameters (Harrison, 1974). Return to the geometry of Fig. 8-5 there we found, as shown in Eq. (8-24), a change in every bond length of magnitude dd = — C) d/3. Proceeding just as... 

In this system of equations the piezoelectric charge constant d indicates the intensity of the piezo effect is the dielectric constant for constant T and is the elastic compliance for constant E eft is the transpose of matrix d. The mentioned parameters are tensors of the first to fourth order. A simplification is possible by using the symmetry properties of tensors. Usually, the Cartesian coordinate system in Fig. 6.12a is used, with axis 3 pointing in the direction of polarization of the piezo substance (see below) [5,6]. 

The parameters in these equations are the electrical small-signal capacitance C, the small-signal stiffness cp and the effective piezoelectric charge constant dp, compare Fig. 6.130. 

Here the coefficients 7e, 7s = 7a, and 7m correspond to the small-signal capacitance C, the effective piezoelectric charge constant dp and the inverse of the small-signal stiffness cp for a piezoelectric transducer and to the small-signal inductance L, the effective magnetostrictive constant du and the inverse of the small-signal stiffness cm for a magnetostrictive transducer, respectively. 

Figure 6.140 shows the measurement results obtained with the two selfsensing actuator principles for electrical large-signal operation. They display the characteristics of the three transfer paths of the bidirectional actuator, illustrated in the form of s-Sd, Sr S and Fr F trajectories. In this case, the values X, and ym in the (6.80), (6.77) and (6.78) correspond to the inverse control voltage, the measured control voltage and the measured piezoelectric charge. 

The piezo effect produced after the poling is quantified by the tensor coefEcients of the piezoelectric charge coefficients d.33, dis and d.15. For a clear indexing the Cartesian xs-coordinate (i. e. the -axis) is applied as a reference axis in parallel direction to the polarization vector in general [90-92]. 

Piezoelectric Charge and Voltage Coefflcient/-Constant. For a piezoelectric material interactions between the electrical field and mechanical quantities have to be considered. In a good approximation this can be described via the linear context... 

Here D is the vector of the dielectric displacement (size 3x1, unit C/m ), S is the strain (size 6x1, dimension 1), E is a vector of the electric field strength (size 3x1, unit V/m) and T is a vector of the mechanical tension (size 6x1, unit N/m ). As the piezoelectric constants depend on the direction in space they are described as tensors e- is the permittivity constant also called dielectric permittivity at constant mechanical tension T (size 3x3, unit F/m) and 5 , is the elastic compliance matrix (size 6x6, unit m /N). The piezoelectric charge coefficient df " (size 6x3, unit C/N) defines the dielectric displacement per mechanical tension at constant electrical field and (size 3x6, unit m/V) defines the strain per eiectric fieid at constant mechanical tension [84], The first equation describes the direct piezo effect (sensor equation) and the second the inverse piezo effect (actuator equation). 

The piezoelectric voltage constant, also known as the g factor, denotes the electric field generated by materials per unit of mechanical stress applied. Like the piezoelectric charge constant, these values can also be classed in terms of directions (i.e. g y)-... 

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