## 2 Double Coil Electromagnetic Harvester  
In order to predict the maximum power that can be extracted by a standard single coil harvester and by a double coil electromagnetic harvester, the single degree of freedom models of both harvesters are presented here. Firstly, the single coil electromagnetic harvester is studied and then the double coil harvester is analyzed. A standard single coil electromagnetic harvester is made up of a vibrating case which houses a permanent magnet linked to a spring system and a coil fixed to the housing and connected to the electrical load, as shown in Figure 1. When a vibration is applied to the case, the spring system makes the permanent magnet moving out of phase with respect to the coil and the relative displacement between the magnet and the coil leads to the conversion of mechanical energy into electrical energy. During its movement, the permanent magnet is subject to the viscous damping force of the medium, to the elastic force of the spring and to the electromagnetic (Lorentz) force due to the interaction between the current flowing into the coil and the induction field of the magnet. If a single degree of freedom model is employed for the sake of simplicity, [PERSON]'s second law can be written as [28]  
\[m\;\ddot{x}(t)+c\;\dot{x}(t)+k_{s}\;x(t)+\theta\;i_{LOAD}(t)=-m\;\ddot{y}(t) \tag{1}\]  
where \(\ddot{y}(t)\) denotes the acceleration of the case, \(x(t)\) the relative displacement between the magnet and the coil, \(m\) the magnet mass, \(c\) the viscous damping coefficient, \(k_{s}\) the equivalent stiffness of the spring system, and \(i_{LOAD}(t)\) the electrical current flowing into the coil. \(\theta\) is the electromechanical coupling coefficient of the coil that takes into account its geometrical properties, its number of turns and the magnetic field strength, as shown in detail in [28]. This current is due to the electromotive force (e.m.f.) induced in the coil:  
\[\varepsilon(t)=\theta\;\dot{x}(t) \tag{2}\]  
and flows through the coil resistance and inductance as well as through the electrical load connected to the coil. Equations (1) and (2) can be represented by the equivalent electric circuit shown in Figure 2, where \(k=m/\theta\), \(R_{m}=\theta^{2}/c\), \(L_{m}=\theta^{2}/k_{s}\), \(C_{m}=m/\theta^{2}\), being \(R_{C}\) and \(L_{C}\) the resistance and the inductance of the coil, respectively [17]. When the single coil electromagnetic harvester is driven by a sinusoidal vibration at frequency \(\omega_{VIB}=2\;\pi\;f_{VIB}\), i.e., \(\ddot{y}(t)=A_{VIB}\sin(2\;\pi\;f_{VIB}\;t)\), the average power that can be delivered to the load can be predicted by applying the phasor method to the equivalent circuit in Figure 2, i.e.,  
\[P_{LOAD}=\frac{\left|\overline{I}_{LOAD}\right|^{2}}{2}\cdot R_{LOAD} \tag{3}\]  
where \(\overline{I}_{LOAD}\) is the phasor of the load current \(i_{LOAD}(t)\) and \(R_{LOAD}\) is the resistive part of the load. According to the maximum power transfer theorem, the maximum average power is delivered to the load when its impedance is equal to the complex conjugate of the source impedance, i.e., \(\dot{Z}_{LOAD}=\dot{Z}_{OPT}=R_{OPT}+j\cdot X_{OPT}\) where,  
\[R_{OPT}(\omega_{VIB})=\frac{\theta^{2}\cdot\omega_{VIB}^{2}\cdot c}{\left(k_{ s}-m\cdot\omega_{VIB}^{2}\right)^{2}+\left(\omega_{VIB}\cdot c\right)^{2}}+R_{c} \tag{4}\]  
Figure 1: Schematic representation of a standard single coil resonant electromagnetic vibration energy harvester. Figure 2: Equivalent circuit of a standard single coil resonant electromagnetic vibration energy harvester. \[X_{OPT}(\omega_{VIB})=\frac{\theta^{2}\cdot\omega_{VIB}\cdot\left(m\cdot\omega_{VIB} ^{2}-k_{s}\right)}{\left(k_{s}-m\cdot\omega_{VIB}^{2}\right)^{2}+\left(\omega_{ VIB}\cdot c\right)^{2}}-X_{c}(\omega_{VIB}) \tag{5}\]  
being \(X_{c}(\omega_{VIB})=\omega_{VIB}\ L_{\mathrm{C}}\).