# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 2 The Applied Methodology  
This paper is concerned with the analysis of an overhead line energy harvester (OLMEH) illustrated in Figure 1. The OLMEH in Figure 1 is of a clamped-on CT type, the concept for which can be found in [22; 23; 24; 25; 26]. An analysis of the OLMEH's behavior, while modeling the nonlinear BH characteristic using the _atan(x)_ function, was described in [23]. The choice of such an approximation function results in analytical complications and necessitates the application of numerical solutions to attain the final results. Alternatively, ref. [26] opted to model the BH curve by a simple piecewise linear function for the sake of deriving crude analytical results. In the low flux density region, the linear approximation allows easy analytical description of the device; however, in the high flux density region, where the core approaches the saturation knee, the linear approach is inadequate.  
One can argue that, in our age, a researcher assisted with modern computational tools can crack any nonlinear problem. To attain the solution, it is sufficient to just apply simulation or to properly introduce basic laws and system descriptions to the computer and then obtain the desired numerical results and plots. Yet, there is a clear and undistputable advantage to the old school approach of deriving an analytical solution to the problem. The latter can show the explicit effect each variable has on the end result and so provide a deeper understanding of the physical process. The analytical solutions can also facilitate the derivation of straightforward designing rules. Hence, an extra effort was put in to revisit the earlier findings, particularly of [23]. This paper aims to develop a complete analytical model that faithfully describes OLMEH operation. The proposed modeling approach is based on the Froelich equation, which represents the nonlinearity of the BH curve by a first-order rational polynomial function. The choice of the Froelich equation as the approximation function is imperative. The advantage of the undertaken approach over the numerical [23] is that the Froelich approximation allows deriving closed-form analytical solutions that consider the complex behavior of the magnetic core, yet provides meaningful analytical results suited for engineering applications. In summary, the contribution of this paper over the earlier counterpart is the analytical solutions for the: (a) power transfer window; (b) charging output power, and (c) prediction of the operational modes boundary.  
The rest of this paper is organized as follows: Section 3 reviews the modeling of the BH curve by the Froelich equation. Section 4 presents basic assumptions, preliminary simulation waveforms, and OLMEH models. Section 5 presents the analysis of the discontinuous current mode. The continuous current mode of OLMEHs is described in Section 6. Section 7 discusses the boundary conditions for continuous to discontinuous current mode change. Analytical solutions for output current and power are derived in Section 8. Verification and experimental results are given in Section 9. Finally, conclusions are given.