## 3 Results and Discussion
### Comparison with Experiment
The curves \"Simulation FB\" and \"Experiment FB\" compare simulated and experimental dependences of the output power on the excitation frequency for these two PECs with respective electrical rectification (full bridge, FB) as shown in Figure 11.  
It can be seen in Figure 10 that, in some frequency regions, the resulting electric power of serial and parallel connections is much lower than the optimum power. The power loss is caused by different amplitudes and phases of the output voltage of two PECs. Without additional electrical components the maximum optimum power cannot be achieved with this setup in the broad frequency range. Figure 11 shows as an example a simple approach to scavenge the energy from two cantilevers close to the optimum value. This approach contains two standard FB rectifiers in parallel connection with the load \(R_{\mathrm{L}}\). Figure 10 compares the optimum power of two harvesters (curve \"PEC 1 + PEC 2\") with the measurement of the parallel connection with two full wave bridge rectifiers and \(\Delta f\) of 1 Hz. It is seen that the measured power (curve \"Experiment FB\") is close to the optimum power. The observed  
Figure 11: Two piezoelectric cantilevers (denoted as PZ1, PZ2) with the individual full wave bridge rectifiers connected parallel to the resistance \(R_{\mathrm{L}}\). Figure 10: Power of parallel and series connection compared with the optimum power (designated as “PEC 1 + PEC 2”). FB: full bridge. difference is caused by the power consumption of the diodes. Nevertheless, it is approximately possible to achieve the theoretical maximum power with a simple circuit. However, the effectiveness of this circuit depends on the load resistance \(R_{\mathrm{L}}\). Figure 12 presents the dependence of the power dissipated the load resistance \(R_{\mathrm{L}}\) for two PECs with \(\Delta f\) of 1 Hz at the excitation frequency of 40 Hz. The curve \"PEC 1 + PEC 2\" is the experimentally obtained power limit of the two separate PEC's without the full bridge rectifier. The curve \"full bridge\" shows the parallel connection with two full wave rectifiers in dependence on the load resistance. To facilitate the comparison, the electrical power is normalized to the total electrical power of two PECs individually loaded by the optimum resistance \(R_{\mathrm{L}}\) = 275 k\(\Omega\). The interesting aspect is that the behavior of this circuit can be divided into three cases. At small load resistances (\(R_{\mathrm{L}}\) < 75 k\(\Omega\)) it is possible to scavenge more energy than the theoretical sum of two separate PEC's operating at the optimum load resistance of 275 k\(\Omega\). The reason for this behavior is the smaller internal resistance of the parallel connection. In the second case (75 k\(\Omega\) < \(R_{\mathrm{L}}\) < 220 k\(\Omega\)), both PECs generate less electrical energy than the theoretical optimum. For large resistance loads (\(R_{\mathrm{L}}\) > 220 k\(\Omega\)), only one PEC with a higher voltage amplitude delivers the electrical energy to the load resistance. The reason is the higher output voltage of PEC 1 as compared to PEC 2. The higher output voltage blocks the diodes of the bridge rectifier of PEC 2 and only the diodes of PEC 1 are conductive. To verify theoretical considerations, the dependence of achieved bandwidth on frequency detuning \(\Delta f\) was investigated experimentally. Figure 13 presents two cases, (a) and (b) with two cantilevers, and (c) and (d) with three cantilevers. There are three graphs compared in each figure. The first graph shows the simulated theoretical maximum bandwidth as described in Section 3.1, while the second graph represents the simulated parallel connection with rectifiers at its optimum resistance. The third curve demonstrates the measured results for the parallel connection with rectifiers at its optimum resistance. From the diagrams, it can be seen that the agreement between theoretical and experimental bandwidths at low frequency detuning (\(\Delta f\) < 1.7 Hz) is good.