## 3 Overview of Modelling Issues for Energy Recovery Systems  
By knowing the basics of the mathematical and physical description of the piezoelectric phenomenon, it is possible to describe and then create mathematical models of transducers, i.e., piezoelectric sensors or actuators. In general terms, sensors are used to convert measurands into corresponding measurands. Piezoelectric (and other electromechanical) sensors convert mechanical quantities (e.g., deformations, stresses, forces) into electrical quantities (e.g., current, voltage). A simple piezoelectric phenomenon is used for this. A transducer dealing with energy recovery from mechanical oscillations can be interpreted as a sensor whose measurable quantity (input) is the mechanical energy of the system's oscillations and whose output is electrical energy [11]. Electromechanical actuators operate on exactly the opposite principle: they convert electrical quantities into mechanical quantities in the case of piezoelectric actuators based on the reverse piezoelectric phenomenon. From a systems point of view, it can therefore be said that sensors are the inputs to the system and actuators are its outputs. It should also be noted that not all existing conversion laws work both ways [11]. As an example of a sensor, one can consider a block of quartz on the surface of an arbitrarily vibrating (and therefore deforming) body. To do so, use the relevant pair of constitutive equations for piezoelectric materials [10; 19; 20]:  
\[S_{p}=s_{pq}^{E}T_{q}+d_{kq}^{t}E_{k}, \tag{1}\]  
\[D_{i}=d_{iq}T_{q}+\epsilon_{ik}^{T}E_{k}. \tag{2}\]  
From the constitutive equations, the stresses within the piezoelectric material must be determined, assuming knowledge of the piezoelectric constants, as well as the dielectric and susceptibility constants. When analysing the sensor, it can also be assumed that the deformation of the material to which it is rigidly attached is known, so that it deforms along with it [21]. On this basis, Equation (1) can be derived:  
\[s_{pq}^{E}T_{q}=S_{p}-d_{kq}^{t}E_{k}. \tag{3}\]  
The result can be obtained by any calculation method, for example by inverting the matrix \(s_{pq}^{E}\) or by solving the equation by Gauss elimination. The \(T_{q}\) matrix thus obtained should then be substituted into Equation (2), resulting in the D\({}_{i}\) matrix. From this, the total accumulated charge on the transducer can be obtained using the following equation [21]:  
\[\mathrm{q}\ =\ \iint\left[ \begin{array}{cc}\mathrm{D}_{1}&\mathrm{D}_{2}& \mathrm{D}_{3}\end{array} \right] \begin{bmatrix}dA_{1}\\ dA_{2}\\ dA_{3}\end{bmatrix} , \tag{4}\]  
where \(dA_{1}\), \(dA_{2}\), and \(dA_{3}\) [m\({}^{2}\)] are the partial fields of the electrodes perpendicular to the axes 1, 2, and 3 of the coordinate system, respectively. It may be noted that with the equation written in this way, only the charge in those directions in which the electrodes are located will be taken into account, and in the others, their area will be zero. Equation (4) is presented in integral form because the shapes of the transducers and electrodes can be arbitrary. Note, however, that the most common transducer shapes are disks, toruses, cylindrical shells, and rectangular blocks. Analogous shapes are adopted by the cooperating electrodes, making it possible to determine their surface area from simple geometrical relationships [16]. Assuming the piezoelectric transducer to be a capacitive element and knowing the charge accumulated on the electrodes, the voltage generated by the transducer can be calculated from the relationship [21]:  
\[U=\frac{q}{C}, \tag{5}\]  
where \(U\) [V] is the electrical voltage, \(q\) [C] is the calculated charge, and \(C\) [F] is the capacitance of the piezoelectric sensor. The mathematical model used ignores the influence of a number of factors that can affect its representation of reality and, in some cases, can lead to situations where the model is inadequate.