## 3 Theory and Modeling
### Principal and Modeling of Piezoelectric Transducer Component
 \tag{8}\]  
where \(M(x)\) represents the bending moment, \(z\) is the coordinate in the thickness direction relative to the neutral plane, \(I(x)\) is the moment of inertia of the cross-section, \(Y\) denotes the elastic modulus, and \(l_{p}\) and \(l_{m}\) are the lengths of the piezoelectric section and the excitation section, respectively. \(\gamma\) is the system modulus, which is the equivalent elastic modulus of the entire cantilever beam calculated by combining the layer thicknesses and elastic moduli of the piezoelectric layer and the copper electrode layer, and is a constant when the material and thickness are determined. \(b(x)\) is the width of the beam at position \(x\). If the width of the fixed base is equal while the width of the movable base is relatively reduced, \(b(x)\) becomes smaller, which leads to an increase in strain at position \(x\). Conversely, increasing the width of the fixed base results in an increase in \(b(x)\) and a decrease in strain. The variation in \(b(x)\) not only affects the strain but also alters the area of the piezoelectric layer. The working principle of the piezoelectric cantilever beam is based on surface charge generated during deformation of the piezoelectric material. In the simplified model of piezoelectric component, the upper and lower surfaces could be regarded as capacitor plates, while the open-circuit voltage arises from the charge difference between the surfaces [12; 13; 24]. Deformation will alter the surface charge and also change the open-circuit  
Figure 9: Piezoelectric energy harvester structure diagram, optimized by non-uniform cross-section method. voltage; the piezoelectric output of the capacitance-like simplified model can be jointly solved according to Equation (9):  
\[\left\{ \begin{array}{l}D(x,z)=d_{31}\widetilde{Y}S(x,z)\\ C_{p}=\frac{\epsilon_{0}\epsilon_{p}J_{0}^{\dagger p}b(x)dx}{z_{p}}\\ Q_{p}=\int_{0}^{\dagger p}D(x,z)b(x)dx\\ U_{p}=\frac{Q_{p}}{C_{p}}\end{array} \right. \tag{9}\]  
where \(D(x,z)\) represents the distribution of the electric displacement vector on the surface of the piezoelectric layer, \(d_{31}\) is the piezoelectric constant of the cantilever beam structure, and \(\widetilde{Y}\) is the elastic modulus of the composite material, which is dependent only on the materials and layer thickness. \(C_{p}\), \(\epsilon_{p}\), and \(z_{p}\) denote the internal equivalent capacitance, dielectric constant, and thickness of the piezoelectric material, respectively. Due to the assumption of linear elastic small deformation, the strain on the upper and lower surfaces during the loading and deformation process has a negligible effect on the capacitance surface area and spacing, which could be considered constant. \(Q_{p}\) and \(U_{p}\) are the total surface charge and open-circuit voltage characterizing the output performance of the piezoelectric layer, respectively. Since the discussion is based on a linear elastic deformation model, it can be deduced from the principle of symmetry that the strains on the upper and lower surfaces of the same material section of the cantilever beam are equal in magnitude but opposite in direction, resulting in equal magnitude but opposite charges on the upper and lower surfaces of the piezoelectric layer. Organize Equations (8) and (9) to obtain the expression Equation (10) for \(Q_{p}\) and \(U_{p}\) of the model:  
\[\left\{ \begin{array}{l}Q_{p}=\frac{0\xi d_{31}\widetilde{Y}Fz_{p}(l_{p}+l_{m} )}{\gamma}\\ U_{p}=\frac{d_{31}\widetilde{Y}Fz_{p}(l_{p}+l_{m})}{\gamma\epsilon_{0}\epsilon_ {p}(b_{1}+b_{2})}\end{array} \right. \tag{10}\]  
where \(b_{1}\) and \(b_{2}\) refer to the fixed end and the active end widths shown in Figure 9, respectively. It can be concluded that, in this model, changing either \(b_{1}\) or \(b_{2}\) to gain the non-uniform cross-section piezoelectric cantilever beam will alter the strain distribution and the open-circuit voltage of the component while keeping the same applied force, but it will not affect the total surface charge.