## 6 Results and Discussion  
To measure the optimal load resistance matched to the coil, the piezoelectric layers are short-circuit. There are two peak output powers for the configuration of two magnetic coupled oscillators. The first peak is induced by the magnetic oscillator, while the second one is induced by the piezoelectric oscillator. Figure 9 shows the first peak output power delivered to the load resistance \(R_{2}\) for the MEMEH at resonant frequency. The maximum power is 1.2 mW at 22.8 Hz with the load resistance 370 \(\Omega\). The second peak output power delivered to the load resistance \(R_{1}\) for the MPEH at resonant frequency is shown in Figure 10. The induction coil around the magnetic oscillator is open-circuit. With the increasing of \(R_{1}\), the power reaches the maximum 2.57 mW at 57 k\(\Omega\). The corresponding resonant frequency is 25.8 Hz. By using the logarithmic decrement method, the measured mechanical damping ratios of piezoelectric beam and magnetic oscillator with magnetic interaction are 0.023 and 0.024, respectively. Figure 11a,b shows the experimental and numerical comparisons of the frequency-response for the HEH, MPEH and MEMEH, respectively. The initial distance \(d_{0}\) is still 52.5 mm. The loads connected to the HEH are the optimal values determined from the MPEH and MEMEH. It can be seen in Figure 11a that each harvester has two peak output powers. For HEH, the measured peak output powers are 1.40 mW at 22.6 Hz and 2.82 mW at 25.6 Hz. In view of the low-frequency applications, the first peak output power is the research focus. The first peak of the HEH increases by 16.7% and 833.3% compared with the first peaks of the MEMEH (1.2 mW) and the MPEH (0.15 mW), respectively. The frequency response of the HEH completely covers that of the MPEH and MEMEH. That is to say, the  
Figure 10: Output power with different load resistances for the MPEH. Figure 9: Output power with different load resistances for the MEMEH. hybrid transduction mechanism is superior to the multi-frequency devices with single conversion mechanism in broadening the bandwidth and improving the generating performance. We can find that the simulation results of the HEH are in good agreement with the experimental results. The inaccuracies between theoretical and simulation results for the MPEH and MEMEH are significant. We think that these errors may be due to the damping change between the magnetic oscillator and the aluminum rod with varying vibration frequencies. Besides, the effective stiffness of the spring varies with the change of effective length during the oscillation, which may be another factor. Figure 12a,b illustrates the numerical and experimental comparisons of the output power for the HEH, LEMEH, LPEH and LVEHs, respectively. The connected load resistances keep still. Due to the repulsive magnetic force, the measured first resonance of the HEH (22.6 Hz) is higher than that of the LVEHs (21.6 Hz) and the second one (25.6 Hz) is higher than that of the LVEHs (25 Hz). Moreover, the first peak power of the HEH (1.4 mW) is 2.36 times more than that of the LVEHs (the sum of 0.08 mW from LPEH and 0.49 mW from LEMEH) at the same excitation frequency. Meanwhile, it increases by 45.8% compared to that of the LVEHs, although the second peak power of the HEH decreases by 13.8%. Obviously, the repulsive magnetic force can suppress the oscillation of the piezoelectric oscillator, but enhance the oscillation of the magnetic oscillator. The half-power bandwidth of the first peak power for the HEH is about 2.8 Hz, which is wider than that of the LEMEH (1.9 Hz). It is clear that the generating performances of the LPEH and LEMEH are greatly improved by the magnetic coupling multi-frequency structure and hybrid transduction mechanism in the low-frequency range. Consequently, magnetic interaction is an effective approach to improve the performance of the micro energy harvester in the low-frequency range. The numerical results show good agreement with the experimental results.