# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 1 Introduction  
The simplicity of the design of dynamic inductive wireless power transfer (DIWPT) with oblong primary coil configurations, which had long ago been invented [1, 2], was rediscovered [3, 4, 5, 6] in the late 20 th century, as a result of a growing interest in electric vehicles (EVs). After 2010, the quest for sustainable EV transportation motivated the use of dynamic inductive wireless power transfer (DIWPT) for automotive applications in the design, implementation, and testing of more advanced and practical prototypes, for instance, the OLEV Bus from the Korea Advanced Institute of Science & Technology (KAIST) [7, 8] and Section 9.4 of [9], which, by its innovative conception and extensive public demonstration, can be considered disruptive. While researchers at KAIST were searching for high-end, on-the-road recharging of electric buses, it was the smaller conceptual demonstration prototype developed at NISSAN, in Japan [10], that most attracted our attention, for its simplicity, and for handling lower power levels, around 1 kW, more adequate for lightweight electric vehicles. So,inspired by the work of all these researchers, we set the target of implementing an inductive lane for electrically assisted bicycles and other similar cycles [11]. Our prototype was then devised for the continuous operation of a PEDELEC-type bicycle, which, according to guidelines [12], could only receive up to 250 W of electric assistance to the powertrain. The lane was capable of transferring the full power limit of 250 W inside a 27 cm wide corridor, center-aligned with the bike lane, but more, 400 W, over a 20 cm wide corridor within it. A standard bike was specially adapted to dynamically harvest energy from this inductive lane. Depending on the lateral misalignment of the bike within the 27 cm wide central corridor of the lane, up to 150 W of excess power could be harvested over the 250 W limit. This was enough to store short-term energy that could allow the smooth activation of the powertrain during the transit of the bike between neighbor primary coils lying along the lane. The geometric configuration of the protype that made this possible was derived by installing an as large as possible secondary coil, wound around its rear wheel and fixed to the frame of the bike as close as possible to the lane. This was carried out in such a way that this secondary coil would not mechanically interfere with the normal maneuvering of the bike, nor with the pedaling action of the rider. Targeting a solution with a reduced electronic component count (per unit of length) on the lane side, the use of oblong primary coils seemed to be the most natural design paradigm to be adopted. It was then necessary to design the cross-section of the magnetic link, due to the intrinsic symmetry of the configuration. After gross estimations, we searched, by experimentation, for a set of combinations of primary current values and relative sizes and positions of the coils that would successfully yield the nominal 36 VDC required by the powertrain, at a chosen operation frequency. The onboard circuitry was set to rectify the induced voltage, filter it, and, finally, regulate it by the use of a step-down DC-DC converter. In the tests, as intuitively expected, it became apparent that for a given primary current and a fixed distance imposed between the planes containing the primary and secondary coils, there would be an optimum width for the oblong primary coil that would maximize the induced voltage on the secondary coil. This condition would lead to the maximum possible power to be extracted from the secondary coil, if impedance matching and other circuit parameters were carefully adjusted. Conversely, this same configuration would also minimize the primary current that was necessary to obtain some given desired induced voltage or power level on the powertrain, thus diminishing losses on the primary winding and favoring an improved electric efficiency of the system.