# A Robust Generator-Harvester for Independent Sensor Systems +
## 3 Mathematical-Physical Model  
The basic formulation for deriving the mathematical expression to characterize the electromagnetic part of the model is based on [PERSON]'s induction law (1); see, for example, the interpretation by [PERSON] [1]. The use of 3D modeling and simulation, analysis, and comparisons with experimental models of a linear generator is discussed in multiple sources, including [8; 18; 19; 20]. The simulation and experimental verification of the parameters of the equivalent circuits and the rotor/ stator core losses via the FEM are addressed in article [37]. An evaluation of the related impact of the electrical parameters, load, and drive dynamics is proposed in [7; 8]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reference** & **Permanent Magnet** & \begin{tabular}{c} **Generator Body** \\ **Size** \\ **x,y,z [m]** \\ \end{tabular} & \begin{tabular}{c} **Output Power** \\ **\(\mathbf{P_{\text{out}}}\) [W]** \\ \end{tabular} & \
\begin{tabular}{c} **Output Voltage** \\ **\(\mathbf{U_{\text{out}}}\) [V]** \\ \end{tabular} &
\begin{tabular}{c} **Effective Power** \\ **Density \(p_{\text{efd}}\)** \\ **[W/m\({}^{3}\)]** \\ \end{tabular} \\ \hline [PERSON] et al. [28], 2007 & – & 375 mm\({}^{3}\) & 2 \(\times\) 10\({}^{-6}\) & 0.428 RMS & \(\approx\)6 \\ [PERSON] et al. [29], 2010 & FeNdB & 2000 mm\({}^{3}\) & 61.6\(-\)156.6 \(\times\) 10\({}^{-6}\) & – & \(\approx\)30\(-\)80 \\ [PERSON] et al. [27], 2008 & FeNdB & 3375 mm\({}^{3}\) & 0.6 \(\times\) 10\({}^{-6}\) & 0.025 & \(\approx\)0.2 \\ [PERSON] et al. [31], 2007 & FeNdB & 256 mm\({}^{3}\) & - & 0.06 & - \\ [PERSON] et al. [33], 2012 & FeNdB & 1.4 \(\times\) 10\({}^{-4}\) m\({}^{3}\) & 1.52 \(\times\) 10\({}^{-3}\) & 4.8 & \(\approx\)10 \\ [PERSON] et al. [32], 2014. & – & 50,000 mm\({}^{3}\) & 13.4 \(\times\) 10\({}^{-3}\) & 0.7\(-\)2.0 & \(\approx\)270 \\ [PERSON] et al. [30], 2011 & – & 15,000 mm\({}^{3}\) & 4 \(\times\) 10\({}^{-6}\) & 0.007 & \(\approx\)0.26 \\ MG 1[18], 2006 & FeNdB & 90, 40, 30 mm & 70 \(\times\) 10\({}^{-3}\) & 4\(-\)60 (300) p-p & \(\approx\)650 \\ MG II [18], 2006 & FeNdB & 50, 27, 25 mm & 19.5 \(\times\) 10\({}^{-3}\) & 6\(-\)15 & \(\approx\)60 \\ MG III & FeNdB & 50, 25, 25 mm & 5.0 \(\times\) 10\({}^{-3}\) & 1.0\(-\)2.5 & \(\approx\)15 \\ MG IV & FeNdB & 50, 35, 25 mm & 8.0 \(\times\) 10\({}^{-3}\) & 1.0\(-\)2.5 & \(\approx\)18 \\ \hline \hline \end{tabular}  
* Lith. battery [34], 2018
* Supercap [35], 2010
* \end{table} 
Table 1: The parameters of selected mini-generators Reprinted/adapted with permission from Ref. [20]. Copyright 2024, copyright [PERSON] [20]. According to [8; 20], the change in the magnetic flux \(\Phi\) according to [PERSON]'s induction law is  
\[\underbrace{\oint\limits_{\ell}\mathbf{E}(t)\cdot d\ell}_{\Phi}=-\int\limits_{S} \frac{\partial\mathbf{B}(t)}{\partial t}\,dS+\oint\limits_{\ell}\left(\mathbf{v}(t) \times\mathbf{B}(x,y,z,t)\right)\cdot d\ell, \tag{1}\]  
where \(E(t)\) is the electric field intensity vector, \(\mathbf{B}(t)\) denotes the magnetic flux density vector (induction), \(\mathbf{v}(t)\) indicates the speed of the shift of the generator core position in time (instantaneous speed), \(S\) denotes the cross-section of the magnetic flux regions, and \(\lambda\) denotes the curve along the boundary of area \(S\). The simple concept, which was experimented with previously [2; 3; 4; 8; 20], can be applied to the electromagnetic part of the harvester.