# Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters
## 2 Materials and Methods
### Analytical Model  
The analytical model of the generator consists of a 1 DOF non-linear mass-spring-damper system [27]. The seismic mass m of the system is the moving magnet which is suspended thanks to the repulsive force of the fixed magnet at the bottom. The stiffness \(k\) is related to the magnetic suspension. The damping characteristic derives from two components: mechanical due to the viscous action of the air and friction (\(c_{vis}\)), electromagnetic due to the induction of parasitic currents in the coils (\(c_{em}\)) [28]. External inputs on the system generate out-of-phase oscillation motions of the moving magnet. The transducing mechanism consists of the time variation of the magnetic flux linkage of the coils due to \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{Moving magnet} & **EH1** & **EH2** & **EH3** \\ \hline \multirow{3}{*}{Fixed magnet} & \(H_{mov}=30\) mm & \(H_{mov}=30\) mm & \(H_{mov}=30\) mm \\  & \(D_{mov,ext}=20\) mm & \(D_{mov,ext}=30\) mm & \(D_{mov,ext}=50\) mm \\  & \(D_{mov,int}=10\) mm & \(D_{mov,int}=/\) & \(D_{mov,int}=5\) mm \\ \hline \multirow{3}{*}{Fixed magnet} & \(H_{fix}=6\) mm & \(H_{fix}=12\) mm & \(H_{fix}=18\) mm \\  & \(D_{fix,ext}=20\) mm & \(D_{fix,ext}=20\) mm & \(D_{fix,ext}=20\) mm \\  & \(D_{fix,int}=10\) mm & \(D_{fix,int}=10\) mm & \(D_{fix,int}=10\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main dimensions of the magnets. Figure 1: Schematic representation of the harvesters. the relative motion of the mass inside the tube, thus inducing an electromotive force. A representation of the system analytical model is shown in Figure 2a. The equation of motion of the system (1) is a second-order differential equation expressed in the variable \(z\), which is the relative motion coordinate of the moving magnet with origin in the equilibrium position [20]. \[\underline{z}=-\frac{c_{tot}(z)}{m}\underline{z}-\frac{k(z)}{m}z+Y_{0}\sin( \Omega t) \tag{1}\]  
where:  
* \(z\) is the relative motion coordinate of the moving magnet;
* \(m\) is the seismic mass of the moving magnet;
* \(c_{tot}\) is the sum of viscous damping \(c_{vis}\) and electromagnetic damping \(c_{em}\);
* \(k\) is the non-linear stiffness of the magnetic spring;
* \(Y_{0}\) is the external acceleration amplitude of sinusoidal input;
* \(\Omega\) is the pulsation of the external excitation. The device configuration is characterized by an electro-mechanical coupling. The induced electromotive force and the generated power depend on the moving magnet speed and displacement. Therefore, the electromagnetic subsystem needs to be studied separately and coupled with the mechanical subsystem to correctly analyze the system behavior. The electromagnetic subsystem is modeled as a purely resistive circuit where the harvester is the AC source, having an internal resistance \(R_{coil}\), connected to the external resistance \(R_{load}\) that needs to be properly optimized for maximum power generation in each configuration. The scheme of the electromagnetic circuit of the harvester is reported in Figure 2b. The electromagnetic coupling coefficient \(k_{em}(z)\) derives from the variation of the magnetic flux during the moving magnet oscillation (2). The coefficient directly links the voltage to the moving magnet speed (3) and defines the electromagnetic damping force acting on the moving mass (4)-(5). \[k_{em}(z)=-\frac{d\Phi}{dz} \tag{2}\]  
\[V_{eh}=-\frac{d\Phi}{dt}=-\frac{d\Phi}{dz}\frac{dz}{dt}=k_{em}\underline{z}(t) \tag{3}\]  
\[F_{em}=k_{em}(z)\times i(t)=k_{em}^{2}(z)\times\frac{z(t)}{R_{coil}+R_{load}}= c_{em}(z)\times z(t) \tag{4}\]  
\[c_{em}(z)=\frac{k_{em}^{2}(z)}{R_{coil}+R_{load}} \tag{5}\]  
Figure 2: (**a**) Scheme of harvester mechanical subsystem. (**b**) Scheme of harvester electric circuit. Coils dimensions and location along the tube need to be properly designed in order to extract the maximum amount of power from the system.