The superscript E indicates a constant electric field (which corresponds for example to a short circuit condition, where E=0), as well as the superscript S stands for a condition of constant strain.For each couple of constitutive equations there is a different piezoelectric coefficient, defined as:eip?(��Di��Sp)E=?(��Tp��Ei)Sdip?(��Di��Tp)E=(��Sp��Ei)Tgip??(��Ei��Tp)D=(��Sp��Di)Thip??(��Ei��Sp)D=?(��Tp��Di)S(2)Related to each other as follows:dip=?ikTgkpeip=diqcqpEgip=dkp/?ikThip=giqcqpD(3)An important parameter is the electromechanical coupling factor, kiq, which describes the conversion between mechanical and electrical energy.

It can be written in terms of coefficients of the material:kiq2?Wi(electrical)Wq(mechanical)=eiq2?ikTcpqE=eiq2?ikScpqE+eiq2(4)The efficiency of energy conversion, ��, is described, at resonance, as follows:��=k22(1?k2)1Q+k22(1?k2)(5)where, k2 is the coupling factor as defined in Equation (4) and Q is the quality factor of the generator [21,27].To understand how the electrical quantities (V and I) are related to the mechanical ones (force F and displacement z), the particular case of a piezoelectric disk can be considered. In this case, from Equation (1) the following relationships can be obtained [28]:{FP=kPEz+��VIp=��z�B?CpdVdt(6)In which the featuring quantities are the restoring force Fp of the piezoelectric material, its stiffness when it is short-circuited kPE, the displacement z, the force factor ��, the voltage across the electrodes V and the outgoing current Ip, and the clamped capacitance Cp.

These equations are derived considering the following approximations:E=?VH;S=zH;Ip=AdDdt;FP=ATp(7)and the featured quantities can be written as:kPE=cpqEAH;Cp=?ikSAH;��=eiqAH(8)where, A and H are the section and thickness of the piezoelectric disk.In a more generic case of a mechanical stress in direction p and an induced electric field in direction i, the open-circuit voltage of a piezoelectric device can be written as follows:V=Tpgipl(9)Assuming that the voltage coefficient gip is constant with the stress, and where l is the gap between the electrodes.3.?MaterialsEach piezoelectric material can be characterized with a set of parameters. For example, considering a stress Tp as input, the strain coefficient dip gives the relationship between the applied stress and the electric induction Di (therefore, current density is Jp=dDdt), while the voltage coefficient gip gives the voltage Equation (9).

Thus, a high energy density piezoelectric material is characterized by a large product of the strain coefficient (dip) and the voltage constant (gip) [29]. The coupling factor kiq, combining the piezoelectric properties of the material with its mechanical and electrical properties, gives the converted Drug_discovery energy and efficiency of the harvester, as remarked by Equations (4) and (5).