THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

462 16. Ultrasonicstransducer, it has mislabeled as being “piezoelectric.” Among the principal ferroelectrics,barium titanate, lead meta-niobate, and lead zirconate titanate are greatlyused for electrostrictive applications. To construct this type of transducer, manysmall crystallites of ferroelectric substances are bonded together to form a ceramicof the appropriate shape. Because such materials are polycrystalline, they may beconsidered as being isotropic and thus do not have to be cut along specific axes.This renders possible the construction of a concave transducer so that the ultrasonicradiation can be focused without the need for an auxiliary lens system.Fundamental Piezoelectric RelationshipsFor a given temperature, consider a piezoelectric element having cross-sectionalarea A, thickness t, with electrodes attached to the opposite faces. A voltage V isapplied across the electrodes to generate an electric field E = V/t, and a constanttensile stress σ is applied to the surfaces. Within the elastic limits, the resultantmechanical strain s relates to the stress as follows:s = aσ + bE (16.19)and we also haveD = cE + dσ (16.20)where D represents the electric displacement and a, b, c, d are coefficients definedbelow.Let us now short circuit the electrodes so that E = 0. Equation (16.20) nowreadsD = dσWe note here that the electric displacement D equals the dielectric polarization P,or the charge per unit area. HenceP = dσ (16.21)under short-circuit conditions. The coefficient d constitutes the piezoelectric strainconstant, which is defined as the charge-density output per unit applied stress underthe conditions of short-circuited electrodes. Now if the stress σ is reduced to zero,Equation (16.19) modifies toσ = bEThe principle of conservation of energy dictates that b = d, resulting inσ = dE (16.22)for the no-load condition. The coefficient d may also be described as the mechanicalstrain produced by a unit applied field under the conditions of no loading and it isexpressed either in units of coulomb/newton (C/N) or meter/volt (m/V). Equation(16.19) becomes altered tos = aσ + dE (16.23)