# A Robust Generator-Harvester for Independent Sensor Systems +
## 3 Mathematical-Physical Model
\begin{array}{l}m\frac{d^{2}x}{dt^{2}}\ +l_{c}\ \frac{dx}{dt}+k\ x=m_{m} \frac{d^{2}x}{dt^{2}}-\int\limits_{V_{\rm f}}\left(\left(\frac{dx}{dt}\mathbf{u}_ {x}\times\mathbf{B}_{br}(t)\right)\times\mathbf{B}\right)\cdot\mathbf{n}\ dV-\int\limits_{V _{\rm f}}\left(\frac{i(t)}{S_{\rm v}}\mathbf{n}\times\mathbf{B}\right)\cdot\mathbf{n}\ dV,\\ m\frac{d^{2}x}{dt^{2}}\ +l_{c}\ \frac{dx}{dt}+k\ x=m_{m}\frac{d^{2}x}{dt^{2}}- \int\limits_{V_{\rm f}}\left(\left(\frac{dx}{dt}\mathbf{u}_{x}\times\mathbf{B}_{br}(t )\right)\times\mathbf{B}\right)\cdot\mathbf{n}\ dV-\int\limits_{V_{\rm f}}\left(i(t) \mathbf{n}\times\mathbf{B}\right)\cdot\mathbf{n}\ d\ell,\\ \end{array}  \tag{16}\]  
where \(\mathbf{B}_{\rm br}\) is the braking component of the magnetic induction vector, \(\mathbf{J}_{\rm v}\) denotes the current density vector of the electrically conductive components due to eddy currents, \(\mathbf{J}_{\rm circ}\) represents the current density vector in the coil winding, \(i\)(t) stands for the instantaneous value of the coil electric current, and \(\mathbf{u}_{\rm x}\) is the unit vector of the coordinate system. Interestingly, in this context, the approaches described in research articles [45,46] can be considered inspiring for further characterization of the modeling procedure. The model described above was compiled using the finite element method (FEM) in ANSYS [47]. Both the dynamics of the model in the Ansys Multiphysics module [47] and the magnetic field distribution subproblems for the static and the quasi-stationary arrangements were analyzed, as shown below  
To increase the yield (efficiency) of the dynamic energy transformation (Figure 1a) generated by the motion, the magnetic circuit layout, according to Figure 1b, is applicable. In order to achieve a high efficiency in the Faraday induction law-based conversion (1) of the kinetic energy of the generator into electricity, the arrangement of the magnetic circuit, shown schematically in Figure 1a, is utilized. Due to the arrangement of the magnetic circuit in Figure 1b, the change in the flux \(\mathrm{d}\Phi/\mathrm{d}x\) or \(\mathrm{d}\Phi/\mathrm{d}t\) in Figure 1b is multiplied. A significant increase in the magnitude of the induced electric voltage \(u\)(t) can therefore be expected in a comparable winding of the electric generator. This corresponds to a change in the proposed arrangement (Figures 2 and 3) for the resulting electric voltage \(u\)(t). The segment of the magnetic arrangement of the linear generator will be periodically repeated during the designing part, and therefore the sections and subchapters below will deal with the fundamental parts of the periodic arrangement of the generator elements. Figure 3: The segments I and II of the double-acting magnetic circuit of the periodic structure of the linear motion to achieve the maximum change in the magnetic flux of the moving and fixed parts: (**a**) principal geometrical arrangement, and (**b**) a symbolic representation of the magnetic circuit (a model with concentrated parameters). Figure 2: The segments I and II of a single-acting magnetic circuit arrangement of the periodic structure of the linear motion shown below. Such configurations allow us to achieve the maximum change in the magnetic flux of the moving and fixed parts: (**a**) a principal geometrical arrangement, and (**b**) a symbolic representation of the magnetic circuit (a model with concentrated parameters). The magnetic circuit arrangement and efficiency are related to the arrangement of the transition of the moving part of the magnetic circuit to the circuit of the fixed part of the generator. The detail is provided in Figure 1b--magnetic flux converter. Several pole mounting configurations for transmitting the magnetic flux through the air gap of the moving and the static parts of the generator are illustrated in Figure 4.