# A Novel Ropes-Driven Wideband Piezoelectric Vibration Energy Harvester
## 2 Working Principle
### Unlimited Bandwidth with Unchanged Output  
For the traditional AB-VEH, which is composed of multiple beams attached with piezoelectric material as generators, the equivalent circuit model connected in parallel and in series are shown in Figure 2a,b respectively. For convenience of discussion, when ambient vibration frequency is closed to the operation frequency of one generating beam, only this beam works effectively while the others hardly work, which can be regarded as a capacitor with the same value of \(C_{p}\). Therefore, the equivalent circuit can be simplified to a voltage source \(\dot{U}_{S}\), and an equivalent impedance \(1/j\mu\omega C_{p}\) in parallel or \(n/j\omega C_{p}\) in series (\(n=1\), \(2\), \(3\), \(\cdots\)... ), then the maximum output power \(P_{Lmax}\) delivered to load resistance \(R_{L}\) can be determined by  
\[P_{L\max}=\left|\frac{\dot{I}_{S}}{\sqrt{\left(\left|1/j\mu\omega C_{p}\right| \right)^{2}+{R_{L}}^{2}}}\right|^{2}\times R_{L}=\ \frac{{I_{S}}^{2}}{2\mu\omega C_{p}},\ R_{L}=R_{m}= \frac{1}{n\omega C_{p}} \tag{6}\]  
\[P_{L\max}=\left|\frac{\dot{U}_{S}}{\sqrt{\left(\left|n/j\omega C_{p}\right| \right)^{2}+{R_{L}}^{2}}}\right|^{2}\times R_{L}=\frac{{I_{S}}^{2}}{2\mu\omega C _{p}},\ R_{L}=R_{m}=\frac{n}{\omega C_{p}} \tag{7}\]  
where Equations (6) and (7) are the maximum output power in parallel and in series, respectively, \(\omega\) is vibration frequency, and \(R_{m}\) is optimal load resistance for maximum output power. As seen from Equations (6) and (7), the maximum output power is inversely proportional to the number of beams  
Figure 1: (**a**) Architecture of the novel wideband vibration energy harvester (VEH), (**b**) architecture, (**c**) schematic model, and (**d**) operation mechanism of the individual low-frequency driving beam (LFDB) driving a high-frequency generating beam (HFGB) VEH system.  
arrayed in either a parallel or series connection. Obviously, it is not practical to realize unlimited wideband energy-harvesting for real applications.  
The equivalent circuit model of the proposed ropes-driven wideband VEH is illustrated in Figure 3. Different from AB-VEH, the novel device only has one generating beam. Here, assuming each LFDB works individually for convenience of discussion, when one LFDB\(i\) (\(i\) = 1, 2, 3,..., \(n\)) drives HFGB to oscillate during its own operation frequency \(\Delta f_{i}\), the switch corresponding to LFDB\(i\) is closed and others are still opened. Thus, if the closed circuit is composed of an equivalent voltage source \(\dot{U}_{S}\), impedance \(1/j\omega C_{p}\), and load resistance \(R_{L}\), the maximum output power \(P_{Lmax}\) of \(R_{L}\) can be given by  
\[P_{Lmax}=\left|\frac{\dot{U}_{S}}{\sqrt{\left(\left|1/j\omega C_{p}\right| \right)^{2}+{R_{L}}^{2}}}\right|^{2}\times R_{L}=\ \frac{{U_{S}}^{2}}{2/\omega C_{p}}=\frac{{I_{S}}^{2}}{2\omega C_{p}},\ R_{L}= R_{m}=\frac{1}{\omega C_{p}} \tag{8}\]  
where \(\dot{U}_{S}\) and \(1/j\omega C_{p}\) are the equivalent voltage source and internal impedance of HFGB, and \(\omega\) is the resonant frequency of HFGB during the oscillation. Equation (8) illustrates that the maximum output power is irrelevant to the number of LFDBs. Hence, the novel device can achieve wider or even unlimited bandwidth without sacrificing output power.  
Figure 3: Equivalent circuit of the novel ropes-driven wideband VEH system.  
Figure 2: Equivalent circuit of an arrays of beams vibration energy harvester (AB-VEH) system (**a**) in parallel and (**b**) in series.