## 3 Analysis of the Electromagnetic Energy Harvester
### Voltage Response Analysis  
In this section, we analyze and compare the induced voltage in a single coil under multiple parameters. The phase difference in the coil's induced voltage at different positions is analyzed. The periodic curve of induced voltage versus time in the coil of the REMEH with the Halbach array is shown in Figure 8. When the gap between the coil and the magnet array increases, the amplitude of the induced voltage decreases, while the frequency and period of the induced voltage remain unchanged. The decrease in magnetic induction as the gap increases is the reason for the decrease in the induced voltage in the coil. Figure 6: Magnetic flux density versus angle at four different gaps for the two-directional magnet array. Figure 7: The relationship between the \(\Delta\)B value and the angle when the two arrays are applied with different gap diameters. TD\(-\)1 mm, two directional array with a gap diameter of 1 mm; HB\(-\)1 mm, Halbach array with a gap diameter of 1 mm; TD\(-\)1 mm, two–directional array with a gap diameter of 3 mm; HB\(-\)1 mm, Halbach array with a gap diameter of 3 mm. The periodic curve of induced voltage versus time in the coil of the REMEH with the two-directional magnet array is shown in Figure 9. The magnetization of eight magnets in the two-directional magnet array is in the z-axis direction, and the magnetization of the other eight magnets is in the opposite direction. The period number of the induced voltage in the coil with the two-directional magnet array is twice that of the coil with the Halbach array. The amplitude of the induced voltage in the coil using the Halbach array exceeds the induced voltage in the coil using the two-directional magnet array when the gap diameter is 2 mm. Different rotation speeds result in different amplitudes and frequencies of the periodic induced voltage under the same gap diameter. The induced voltage's amplitude doubles and the frequency doubles when the speed is increased from 5 r/s to 10 r/s, as is shown in Figures 10 and 11. The induced voltage in the coil of the REMEH using the Halbach array exhibits four different phases under the same gap diameter and rotation speed, as shown in Figures 12 and 13. The different initial positions of the coil caused four different voltage phases. The amplitude, frequency, and RMS value of the voltage in each coil are the same. Figure 8: Induced voltage in the coil of the REMEH with the Halbach array versus time under different gaps at a rotation speed of 10 r/s. Figure 9: Induced voltage in the coil of the REMEH with the two-directional magnet array under different gaps at a rotation speed of 10 r/s. Figure 11: Induced voltage in the coil of the REMEH with the two-directional magnet array versus time under different rotation speeds when the gap diameter is 2 mm. Figure 12: Four phases of the induced voltage in the coil of the REMEH with the Halbach array and two phases of the induced voltage in the coil of the REMEH with the two-directional array. The number of turns of each coil in the calculation is 50. **(a)** REMEH with the Halbach array; **(b)** REMEH with the two-directional magnet array. Figure 10: Induced voltage in the coil of the REMEH with the Halbach array versus time under different rotation speeds when the gap is 2 mm. The induced voltage in the coil of the REMEH using the two-directional magnet array exhibits two different phases, as shown in Figure 14. The induced voltages of the two coils have the same value and opposite directions at the same time. Two induced voltages that are opposite in phase should be connected in series oppositely, thus enhancing the voltage by 2 times [30]. The RMS value of an induced voltage is an effective way to evaluate the average power of a REMEH. For the proposed REMEH with the Halbach array, as the gap between the magnet and the coil increases, the RMS value of the induced voltage decreases. At the same gap diameter, the RMS value of the induced voltage increases proportionally as the rotation speed increases.