# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 9 Conclusions  
This paper offers a model, analysis, and analytical solutions for a clamped-type overhead line energy harvester with a full-wave rectifier with natural commutation loaded by a constant voltage load. This paper also reviews the Froelich equation and adopts it as a main tool to investigate harvester operation. This paper further suggests a model and qualitative description of the energy conversion process, and presents the simulation results followed by a full nonlinear analysis supported by experimental results. The contribution of this paper is the complete analytical solutions for charging and idle time intervals, the solution for the average output current, and the solution for the average output power. The proposed theory was verified by comparison to simulation and experimental data collected from a laboratory prototype. An excellent match was observed. The prototype harvester was shown to provide a substantial charging power of up to 50 W in 25 V CVL at 100 A rms line current. Compared to the earlier models [23; 26], the choice of the Froelich equation as the approximation function to describe the core BH characteristic is critical, and proved to be well-suited to facilitate the derivation of complete analytical solutions to the key parameters of OLMEHs. The proposed approach offers both meaningful engineering insight and an accurate analytical description of the OLMEH's performance. The undertaken approach also allowed finding the CCM-DCM regime boundary that has not yet been discussed in the literature. Yet, the solutions are still an approximation. Firstly, because the Froelich equation does not consider the hysteresis feature and, secondly, since (16) is valid in a limited range of the power line angle the analysis result can be applied in cases where the width of the power transfer window exceeds 100\({}^{\circ}\). This experimental work revealed that handling the OLMEH prototype affects its magnetic properties. This is mainly attributed to the variations of the technological air gap remaining between the two C-core halves of a split magnetic core that depends on the mounting force, surface imperfections, etc. Therefore, in the authors' opinion, the derivation of a more precise and, thus, more complex theoretical model (that includes the hysteresis effect) is of questionable practical value due to the mentioned, and unavoidable in practice, core parameter variations. Conceptualization, A.K. ; methodology, A.A.; software, A.A.; validation, A.A. and M.S. ; formal analysis, A.A.; investigation, A.A.; resources, A.K. ; data curation, M.S. ; writing--original draft preparation, A.A.; writing--review and editing, A.A.; visualization, A.A. and  
Figure 12: Comparison of the calculated power, Pocalc, vs. the measured power, Pomeas, output power (DCM) as a function of the CVL voltage at a fixed line current (100 A rms). M.S. ; supervision, A.K. ; project administration, A.K. ; funding acquisition, A.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data is presented in this manuscript. The authors declare no conflicts of interest.