## 3 Theoretical Modeling  
### Electromagnetic Coupling Coefficient and Magnetic Force  
The electromagnetic transduction mechanism is based on [PERSON]'s law of induction. In this paper, the magnetic oscillator is arranged to move inside the induction coil along the center axis, as depicted in Figure 2. The electromotive force \(U_{\mathrm{emf}}\) is derived from the rate of change of the magnetic flux \(\Phi_{B}\) as follows [34]  
\[U_{\mathrm{emf}}=-\frac{\mathrm{d}\Phi_{B}}{\mathrm{d}t}=-\frac{\mathrm{d} \Phi_{B}}{\mathrm{d}z}\cdot\frac{\mathrm{d}z}{\mathrm{d}t}=-\frac{A\mathrm{d} B}{\mathrm{d}z}\cdot\dot{z}=\theta_{\mathrm{em}}\dot{z} \tag{1}\]  
where \(B\) is the magnetic flux density. \(A\) is the area enclosed by the wire loop, which is considered as a constant value. \(\theta_{\mathrm{em}}\) is called the electromagnetic coupling coefficient. The magnet is considered as a dipole with magnetic moment vector \(\mathbf{m}_{2}\)[35], the magnetic field generated by the magnet can be expressed as  
\[\mathbf{B}=-\frac{\mu_{0}}{4\pi}\left[\frac{\mathbf{m}_{2}}{r_{0}^{3}}-\frac{3\tau_{0} (\mathbf{m}_{2}\cdot\mathbf{r}_{0})}{r_{0}^{5}}\right] \tag{2}\]  
where \(\mu_{0}=4\pi\times 10^{-7}\) Hm\({}^{-1}\) is the permeability of vacuum. \(\mathbf{r}_{0}\) is the distance vector from the center of the magnet to an arbitrary single wire. Assumed that the position coordinates of an arbitrary wire and magnet core are \((r,z_{1})\) and \((0,z_{2})\), respectively. The corresponding expression of \(\mathbf{B}\) is  
Figure 1: Schematic diagram of the proposed hybrid energy harvester. Figure 2: The relative position between magnet and coil. \[B=\frac{B_{r}V_{2}}{4\pi}\cdot\frac{3\left(z_{1}-z_{2}\right)re_{r}-\left[r^{2}-2 \left(z_{1}-z_{2}\right)^{2}\right]e_{z}}{\left[r^{2}+\left(z_{1}-z_{2}\right)^ {2}\right]^{5/2}} \tag{3}\]  
where \(B_{r}\) and \(V_{2}\) are the residual magnetic flux density and volume of the source magnet. \(e_{r}\) and \(e_{z}\) are the unit vectors along \(r\)- and \(z\)-axis, respectively. As can be seen from Equation (1), the magnetic flux gradient along the \(z\)-axis direction is related to the induced voltage. For this reason, the magnetic flux density along the \(z\)-axis is the focus of research. The magnetic flux through the area enclosed by the wire is  
\[\Phi=\oint_{s}B\mathbf{d}A=\frac{B_{r}V_{2}r^{2}}{2\left[r^{2}+\left(z_{1}-z_{ 2}\right)^{2}\right]^{3/2}} \tag{4}\]  
The average magnetic flux over the cross-section of the coil can be given as  
\[\overline{\Phi}=\frac{1}{A_{c}}\cdot\int_{0}^{h_{c}}\int_{R_{\mathrm{i}}}^{R_ {\mathrm{o}}}\Phi\mathrm{d}r\mathrm{d}z_{1} \tag{5}\]  
where \(A_{c}=(R_{\mathrm{o}}-R_{\mathrm{i}})h_{c}\) is the coil cross-sectional area [36]. \(h_{c}\), \(R_{\mathrm{o}}\), and \(R_{\mathrm{i}}\) are the height, outer radius, and inner radius of the coil, respectively. Correspondingly, the total magnetic flux through the coil can be derived from \(\Phi_{B}=f_{c}N\overline{\Phi}\). \(f_{c}\)=\(NA_{\mathrm{w}}/A_{c}\) is the coil fill factor [37]. \(N\) is the number of turns in the coil. \(A_{\mathrm{w}}\) is the cross-sectional area of single wire. Consequently, the electromotive force \(\mathrm{U_{emf}}\) can be expressed as  
\[U_{\mathrm{emf}}=-\frac{\mathrm{d}\left(f_{\mathrm{c}}N\overline{\Phi}\right) }{\mathrm{d}z}\dot{z}=\theta_{\mathrm{em}}\dot{z} \tag{6}\]  
where \(\theta_{\mathrm{em}}\) is  
\[ \begin{split}\theta_{\mathrm{em}}&=-\frac{B_{r}V_{ 2}f_{\mathrm{c}}N}{2A_{\mathrm{c}}}\left[\ln\frac{R_{\mathrm{i}}+\sqrt{R_{ \mathrm{i}}^{2}+\left(z_{2}-h_{\mathrm{c}}\right)^{2}}}{R_{\mathrm{o}}+\sqrt{ R_{\mathrm{o}}^{2}+\left(z_{2}-h_{\mathrm{c}}\right)^{2}}}+\ln\frac{R_{ \mathrm{o}}+\sqrt{R_{\mathrm{o}}^{2}+z_{2}^{2}}}{R_{\mathrm{i}}+\sqrt{R_{ \mathrm{i}}^{2}+z_{2}^{2}}}+\right.\\ &\left.\frac{R_{\mathrm{o}}}{\sqrt{R_{\mathrm{o}}^{2}+\left(z_{ 2}-h_{\mathrm{c}}\right)^{2}}}-\frac{R_{\mathrm{o}}}{\sqrt{R_{\mathrm{i}}^{2} +\left(z_{2}-h_{\mathrm{c}}\right)^{2}}}+\frac{R_{\mathrm{i}}}{\sqrt{R_{ \mathrm{i}}^{2}+z_{2}^{2}}}\right]\end{split}