# Optimizing Energy Conversion in a Piezo Disk Using a Controlled Supply of Electrical Load
## 3 Investigation of the Problem of Optimizing the EMCO Value  
To determine the measure of the efficiency of energy conversion--the dynamic coefficient of electromechanical coupling (EMCO)--there is an energy criterion that allows determination of this coefficient at the resonant frequency. According to this criterion, the EMCO is defined as the ratio of the electrical (mechanical) energy accumulated in the volume of the body capable of rotation to the total mechanical (electrical) energy supplied to the body [2]:  
\[k_{d}^{2}=\dfrac{U_{e}^{rot}}{U^{n}}=\dfrac{U_{p}-U_{k}}{(U_{p}-U_{k})+U_{k}}, \tag{19}\]  
where \(U_{k}\) is energy in the volume of the body with short-circuited electrodes under the condition \(E_{z}=0\), \(U_{p}\) is full energy in the body with open electrodes provided  
\[\iint_{S^{+}}D_{z}dS^{+}=0. \tag{20}\]  
In both cases, the total energy in the body is calculated on the same constant deformation field. Figure 2: Distribution of normal stresses and distribution of tangential stresses. The general view of the expression of the total electromechanical energy in the volume of the piezoceramic body is  
\[U=\frac{1}{2}\iiint_{V}\left(\sigma_{x}\varepsilon_{x}+\sigma_{y}\varepsilon_{y} \sigma_{z}\varepsilon_{z}+\sigma_{xy}\varepsilon_{xy}+\sigma_{xz}\varepsilon_{ xx}+\sigma_{yz}\varepsilon_{yz}+D_{x}E_{x}+D_{y}E_{y}+D_{z}E_{z}\right)dV. \tag{21}\]  
Having applied the transformations corresponding to the applied hypotheses of electroelasticity, analogous to the Kirchhoff-Leav hypotheses of the theory of plates and shells in Expression (20), we exclude zero factors, including non-zero deformation \(\varepsilon_{33}\). We have the following estimate:  
\[\varepsilon_{z}=-\frac{s_{13}^{E}}{s_{11}^{E}}\frac{1}{1-\upsilon}div\vec{u}+ \left(d_{33}+2d_{31}\frac{1}{1+\upsilon}\frac{s_{13}^{E}}{s_{11}^{E}}\right)E _{z}. \tag{22}\]  
The obtained estimate explains the effect of the electric load during the thickness polarization of the plate and the implicit consideration of this non-zero deformation in this problem. We write down the total electromechanical energy in the plate in a simplified form for the selected type of body deformation:  
\[U=\frac{1}{2}\iiint_{V}\left(\sigma_{r}\varepsilon_{r}+\sigma_{\varphi} \varepsilon_{\varphi}+\tau_{r\varphi}\varepsilon_{r\varphi}+D_{z}E_{z} \right)dV. \tag{23}\]  
Then, for the amplitude of the energy difference in Numerator (19) after passing to the repeated integral, taking into account the orthogonality of the system of functions, we obtain  
\[U_{p}-U_{k}=\frac{\pi h}{2}\int_{0}^{a}\left(2(\sigma_{0}^{1(p)}- \sigma_{0}^{1(k)})\varepsilon_{0}^{1}+2(\sigma_{0}^{2(p)}-\sigma_{0}^{2(k)}) \varepsilon_{0}^{2}\right)\rho d\rho\] \[+\frac{\pi h}{2}\int_{0}^{a}\sum_{k=1}^{a}\left((\sigma_{k}^{1(p )}-\sigma_{k}^{1(k)})\varepsilon_{k}^{1}+(\sigma_{k}^{2(p)}-\sigma_{k}^{2(k)} )\varepsilon_{k}^{2}+(t_{k}^{p}-t_{k}^{k})t_{k}\right)\rho d\rho.\]  
The amplitude of the total mechanical energy accumulated in a plate with short-circited electrodes is determined by the following integral:  
\[U_{k}=\frac{\pi h}{2}\int_{0}^{a}\left(2(\sigma_{0}^{1(k)}\varepsilon_{0}^{1} +\sigma_{0}^{2(k)}\varepsilon_{0}^{2})+\sum_{k=1}^{a}(\sigma_{k}^{1(k)} \varepsilon_{k}^{1}+\sigma_{k}^{2(k)}\varepsilon_{k}^{2})\right)\rho d\rho.\]  
To analyze the integral, we consider the expressions of radial, circumferential, and shear stresses due to deformations and the nonzero component of the electric field intensity. Obviously, it follows from (15) that the tangential stresses \(\tau_{r\varphi}\) do not depend on the intensity of the electric field; therefore, \(t_{k}^{p}-t_{k}^{k}=0\), \(k=1,2,3,\ldots\) Thus, the corresponding term of the integrand is equal to zero. Expressions for the corresponding normal breakdown voltages in the case of short-circited electrodes do not contain electrical quantities. For open electrodes, they contain an addition with a thickness component of electrical induction \(E_{z}^{(p)}\), different from that caused by the initial load.