# A Robust Generator-Harvester for Independent Sensor Systems +
## 3 Mathematical-Physical Model
The equation of the motion relating to the arrangement of the electromagnetic part of the generator/harvester, the force field couplings, and the linear motion of the moving part are expressible as  
\[m\ \ddot{x}+l_{c}\ \ddot{x}\ +k\ x=f_{mag}\big{(}B,\dot{x},t\big{)}+f_{mech}(t), \tag{2}\]  
where \(m\) is the mass, \(m_{m}\) denotes the mass of the moving part of the generator system, \(l_{c}\) is the damping coefficient, \(k\) represents the stiffness coefficient, \(x\) stands for the position of the body, \(\dot{x}\) expresses the velocity of the moving part (\(\mathrm{d}x/\mathrm{d}t\)), \(\ddot{x}\) is the acceleration of the moving part (\(\mathrm{d}^{2}x/\mathrm{d}t^{2}\)),\(f_{mag}\) stands for the force acting on the moving part via interaction with the magnetic field, and \(f_{mech}\) denotes the force of the mechanical motion of the moving part of the linear actuator. In order to obtain a concept to deliver an efficient harvester yield with linear motion, we have to accept the dynamic parameters of the system to prevent the the conceptual design from markedly reducing the energy conversion efficiency of the generator system already at the beginning of the actual designing task. As already determined via a comparison with other motion/vibration harvester designs [20], the moving segment of the electromagnetic part of the generator for the zenith linear actuator concept must, in the moving part, exhibit a mass parameter \(m_{m}\) close to the minimum value (solution of Equation (2), Figure 1a). In the case of the moving part of the generator model, the mass must be close to the minimum. The instantaneous value of the electric current through the winding \(i(t)\) at an electric load on the terminals of the winding \(R_{z}\) can be derived and formulated according to the procedure in [2]. If the electromagnetic arrangement of a linear generator is considered, as shown in, for example in [36], it will be driven by the corresponding unit (an internal combustion engine) in the dynamic non-volatile state (2); to characterize such an application, we employ a general mathematical model. [PERSON]'s law of induction (1) is employed to formulate the principle and the design of the active part of the generator, also enabling the motion energy of the system to transform into electrical energy, through induction, to the electrical voltage in the inserted conductor of the electric winding; the relevant part is described as  
\[\oint\limits_{\ell}E(t)\cdot dl=-\frac{d\Phi(t)}{dt} \tag{3}\]  
If a thin conductor in the form of a closed loop is inserted into such a variable electric field, an electric current \(i\)(t) starts to flow therein. Now, let us denote the current that flows through the closed loop without the presence of an external source of electric voltage as the induced current. The magnetic flux \(\Phi\) is generated by an external magnetic field. We then have  
\[u(t)=-\frac{d\Phi(t)}{dt} \tag{4}\]  
\[u(t)=u_{1}(t)+u_{2}(t) \tag{5}\]  
\[u(t)=U_{mag}|g(t)| \tag{6}\]  
Figure 1: (**a**) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right) in a linear motion system with minimized dynamic losses (double-acting arrangement). (**b**) A modified magnetic circuit: a model with concentrated parameters (left) and its design scheme (right), representing an arrangement for the maximum yield efficiency of the linear motion system with minimized dynamic losses (a double-acting arrangement). where \(U_{\rm mag}\) is the maximum value of the induced coil winding voltage for a single-line arrangement (Figure 1a) under the given conditions, and \(g(t)\) denotes a function of the time dependence of the magnetic field corresponding to the system parameters and the dynamics of the moving part of the generator. The resulting induced voltage for the double-acting arrangement (Figure 1b) changes with respect to the single magnetic circuit arrangement into the form (6).