# Nonlinear Analysis and Closed-Form Solution for Overhead Line Magnetic Energy Harvester Behavior
## 1 Introduction  
Advanced ultra-low power VLSI and wireless communications techniques spurred the development of various sensor networks and wearable electronics [1; 2; 3]. Such devices can be used to great advantage in various domestic, commercial, vehicular, military, and health applications. A common concern of these technologies is their dependence on power sources. Batteries, biofuel cells, and supercapacitors have to be periodically replaced or recharged, posing certain application difficulties, whereas energy sources like solar cells, thermoelectric generators, piezoelectric and triboelectric generators, and various energy scavengers/harvesters can allow for a long service life while avoiding the need for maintenance and battery replacement. Energy harvesting is a particularly attractive approach to developing miniature self-powered systems designed to operate in remote locations or locations with restricted access [4]. The power distribution network crisscrossing the urban environment is a readily available, predictable, and reliable energy source. Therefore, recent studies have investigated the possibility of energy scavenging from power lines [5]. Thermal energy harvesting exploits the temperature difference developed by the current-carrying conductor and the ambient. DC electric current can be attained by applying thermo-electric generators clamped to the cable [6; 7]. Energy can also be harvested from the electric field of the power line [8; 9; 10; 11]. This suggests creating a capacitor-like structure by wrapping a conductive surface around the (isolated) line conductor and taking advantage of the displacement currents. Alternatively, the harvester can be designed to operate at a safe standoff distance from a high-voltage line. Since the line voltage is well stabilized, the electric field harvesters have the advantage of a nearly constant output current. Magnetic energy harvesters (MEHs) rely on the magnetic field generated by a current-carrying conductor. Several types of MEHs were developed [4] that exploit different physical principles of interaction with the magnetic field such as variable reluctance [12; 13], magnetostrictive [14; 15], and ferrofluid [16; 17; 18] generators. The magnetic field generated by the line current can also be converted into the mechanical vibration of a miniature resonant cantilever. The output power can then be obtained by using a piezoelectric [19; 20] or inductive [21] transducer mounted on the free end of the cantilever beam. Yet, the mentioned harvester types can develop only a tiny amount of power in the range of up to a few mW. This may be sufficient to operate miniature wireless sensors; however, in case a substantially higher power is needed, a different approach is needed. A current transformer (CT) type MEH is, perhaps, the most promising concept that can allow the harvesting of a significant amount of power from a power line. Two configurations are mainly considered in the current literature: the \"stick on\" or \"freestanding\" harvester, which is mounted in the close vicinity of a bus bar, and the \"clamp on\"-type harvester, which is configured to \"embrace\" the current-carrying conductor. Although the output power of MEHs varies with the line current amplitude, the average daily current profile can be measured and the system can be designed to its minimum expected value. Thus, MEHs can be considered as a quite dependable energy source. Traditionally, CTs are used as sensors to accurately measure high AC currents. The most desirable feature of a CT is its high linearity in a wide range of line currents. Thus, the preferable operating conditions of a CT's magnetic core are in the linear region of its BH characteristic. To help attain such a condition, the CT is usually operated under a nearly short-circuited secondary. The construction of an overhead line magnetic energy harvester (OLMEH) is quite similar to that of a CT; yet, since the task of an OLMEH is often charging a battery, the operating conditions of an OLMEH are quite different.