On the design of piezoelectric sensors and actuators Rogelio Luck *, Emmanuel I. Agba Mechanical Engineering Department, Mississippi State University, P.O. Drawer ME, MSU, MS 39762, U.S.A. Abstract Piezoelectric devices are becoming more common with the introduction of smart sensors and actuators, and new developments in piezoelectric polymers and ceramics. However, as the authors found, obtaining relevant information for a comprehensive design of piezoelectric sensors and actuators is quite an ordeal. The literature on piezoelectricity is mainly divided on oversimpli®ed explanations based on quartz crystals or on journals and books requiring an extensive background in solid state physics. This paper provides a non-trivial description of piezoelectricity from the perspective of sensor/actuator design. Contributions include: (a) using a simple matrix approach to (i) clearly describe the piezo- electric relationships, and (ii) clearly show how to manipulate the boundary conditions to experimentally obtain the constitutive parameters; (b) showing how the coecients in the transfer function of a pressure sensor vary as the mode of operation change from isothermal to adiabatic by de®ning eective capacitances, permitivity, piezoelectric, and compliance constants; (c) showing that the wave equation is a natural result of introducing a kinetic energy term into the energy balance. # 1998 Elsevier Science Ltd. All rights reserved. 1. Introduction The applications of piezoelectricity has recently expanded with the introduction of piezo-ceramics and piezo-polymers [1,2]. A notable enhancement in instrumentation involves the combination of piezoelectric sensors with microprocessors to pro- duce ``smart'' devices [3,4]. This paper provides the sensor/actuator/instrumentation designer with an overview and insights on how to design piezo- electric devices. It ®lls a gap in the existing litera- ture in piezoelectricity where the pragmatic information on sensor/actuator design is either missing or embedded in an exhaustive exposition of continuum mechanics, acoustics, and solid state physics [5,6], or where the characterization of the piezoelectric material is oversimpli®ed [7,8]. Transfer functions that directly relate the thermo- electromechanical responses to the physical prop- erties of piezoelectric materials such as stiness and dieletric permittivity are used in the deriva- tions. The eects of physical constraints that can be imposed on the piezoelectric material to experimentally determine the values of some phy- sical constants used in the constitutive equations are discussed as well. In order to illustrate the type of physical inter- actions involved in piezoelectricity consider a dif- ferential element of piezoelectric material. Assume that the dierential element is of rectangular shape with edges aligned with a Cartesian coordinate system. An electric ®eld E can be applied in three possible directions: x, y, and z. Six stresses are identi®ed with the dierential element: three nor- mal stresses:  xx ,  yy , and  zz , and three shear stresses:  xy ,  xz , and  yz . In continuum mechanics, ISA TRANSACTIONS 1 ISA Transactions 37 (1998) 65±72 0968-0896/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved PII: S0019-0578(98)00007-X * Corresponding author. Tel: (601) 325-7307; Fax: (601) 32507223; e-mail: rll@me.msstate.edu