## 6 Discussion about the Applied Load Resistance  
The maximum power is generated at an excitation frequency that equals the resonance frequency and at an applied load resistance that equals the magnitude of the internal impedance [10]. By replacing the excitation frequency with the resonance frequency in the equation of the power output (Equation (4)), the power output at resonance is obtained as follows:  
\[P|_{\Omega=\omega_{n}}=\frac{\alpha^{2}\omega_{n}^{2}u^{2}\left(\frac{0.375+ \beta}{0.2357+\beta}\right)^{2}}{\left(\frac{1}{R}+C_{p}^{2}\omega_{n}^{2}R \right)\left(\left(-\frac{\alpha^{2}C_{p}}{m_{eq}\left(\frac{1}{R^{2}}+C_{p}^ {2}\omega_{n}^{2}\right)}\right)^{2}+\left(2\zeta_{m}+\frac{\alpha^{2}}{m_{eq} \omega_{n}R\left(\frac{1}{R^{2}}+C_{p}^{2}\omega_{n}^{2}\right)}\right)^{2} \right)}. \tag{10}\]  
By replacing the applied load \(R\) in Equation (10) by the optimal load resistance \(R_{op\_res}\), the maximum power is obtained as expressed by Equation (11). In this equation, the maximum power (power output at \(\omega_{n}\) and the corresponding \(R_{op\_res}\)) is independent of the mechanical damping ratio (\(\zeta_{m}\)). This can be explained by the fact that \(\zeta_{m}\) is implicitly included in the optimal load resistance where \(R_{op\_res}=f(\zeta_{m})\). By multiplying the maximum power by \(R_{op\_res}\), the magnitude of \(P_{\max}R_{op\_res}\), which equals the square power of the output voltage, is independent of the mechanical damping ratio at the corresponding \(R_{op\_res}\).  
\[P_{\max}=\frac{m_{eq}^{2}\omega_{n}^{4}u^{2}}{4\alpha^{2}}\left(\frac{0.375+ \beta}{0.2357+\beta}\right)^{2}\frac{1}{R_{op\_res}}. \tag{11}\]  
Figure 10 displays the impact of the load resistance on the power output at resonance. Different mechanical damping ratios are considered for the investigation. An increase in the mechanical damping ratio results in an increase in the optimal load resistance and a decrease in the maximum power by the same magnitude so that the magnitude of \(P_{\max}R_{op\_res}\) is constant at different mechanical damping ratios.  
The magnitude of \(R_{op\_res}\) equals the magnitude of the internal impedance of the PEH and is therefore dependent on multiple parameters, such as the mechanical damping, the mass, the coupling factor, the material properties, the geometrical parameters, and the resonance frequency. At very low mechanical damping, the \(R_{op\_res}\) will accordingly be very low. During the use of the PEH in the real environment, the mechanical damping may differ, and the applied load will thus no longer be optimal. If the excitation frequency is deviated from the resonance frequency, the internal impedance will differ, and accordingly, the pervious optimal load resistance at resonance will no longer be the optimal load  
Figure 9: Power output at different quality factors at the corresponding \(R_{op\_res}\) (solid curves) and at \(R_{d}\) (dashed curves).  
for the new conditions. This means that applying a load resistance of \(R_{op\_res}\) limits the applicability of the proposed piezoelectric energy harvester.  
The \(R_{d}\) is obtained by multiplying \((1/C_{p})\) by \((1/\omega_{d})\). The change in the value \((1/\omega_{d})\) is negligible in comparison with the value \((1/C_{p})\). This means that the value of \(R_{d}\) is less sensitive to the excitation frequency than the value of \(R_{op\_res}\). An applied load resistance of \(R_{d}\) is of more robustness than \(R_{op\_res}\), especially for a wide band of excitation frequencies and broadband piezoelectric energy harvesters. The maximum harvested power when the applied load is \(R_{d}\) and the excitation frequency is \(\omega_{d}\) is obtained as expressed by Equation (12) by substituting \(R_{d}\) and \(\omega_{d}\) in Equation (4) of the power output.  
\[P|_{R_{d},\omega_{d}}=\frac{2\alpha^{2}m_{eq}^{2}\omega_{d}^{5}u^{2}C_{p}}{( \alpha^{2}+4\zeta_{m}\omega_{n}\omega_{d}m_{eq}C_{p})^{2}}\bigg{(}\frac{0.375+ \beta}{0.2357+\beta}\bigg{)}^{2}. \tag{12}\]