# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 1 Introduction
During the continuation of our investigation, almost immediately after our first publication was released, additional tests seemed to indicate that, when implemented by single-turn coils, the induced voltage would be maximized when the coils were under center alignment conditions and the centers of the primary coil wiring cross-sections formed the diameter of a semi-circle approximately containing the centers of the wiring cross-sections of the secondary coil as well. However, our interest at that time was redirected to the design of a technique for automatically sensing the presence of the bike over the primary coil to enable its inductive field [13], and this interesting observation was momentarily forgotten. By the end of 2017, the simple mathematics justifying our results (Theorem 2, Section 3.2) was finally established, but it was initially perceived as a mere curiosity, and kept undocumented until it was included as part of the thesis work of the first author [14], remaining otherwise unpublished. The recent accomplishment of several large-scale DIWPT prototypes and pilot installations, such as the FABRIC project [15; 16] and the Smartroad Gotland project, in Sweden [17], brought a whole new interest in the subject. But at the same time, they revealed the relatively much higher costs involved in the deployment of this technology with respect to any other conductive road electrification techniques [18]. However, it is our impression that, when lightweight vehicles are considered, instead of standard, heavier road vehicles, the decrease in the required power levels can make the system costs drop significantly. On one hand, this is because the components used in lower-power circuits are less expensive than those used in higher-power ones. On the other hand, when dealing with lightweight, low-power vehicles, lower electric efficiencies can be better tolerated, and then oblong primary coils (those that are much longer than wider), also much longer than the secondary coils, can be used, despite the lower magnetic coupling coefficients obtained. The consequence is that DIWPT for lightweight EVs could also require much fewer inverter modules per kilometer, helping the maintainability and further decreasing the overall costs. Considering these possible future DIWPT systems for lightweight EV applications, we reunite in this paper some theorems that will help the design of inductive lanes constituted by oblong primary coils. The theorems indicate the optimum dimensions of the other coil in a magnetic link, which can be either the primary (transmitter) coil or the secondary (receiver) coil. Optimality here refers to two distinct possible design goals: (i) obtaining the maximum open-circuit voltage induced on the secondary coil, or (ii) obtaining the maximally flat response for the open-circuit induced voltage, as the coils are laterally displaced from the center alignment position. Although all the theorems were derived under the hypothetical situation of filamentary (circular with zero diameter) wires and single-turn coils, we infer that the width estimation given by these theorems will reasonably approximate the actual value of the respective optimum width in a real system if (i) the wire diameters are much smaller than both the coil widths and the distance between coils, thus resulting in a relatively large distance between any two conductors in different coils, and (ii) we use the abstraction of replacing the multiple turns of a coil winding by the same number of turns coinciding, but electrically isolated, on the same support line in a suitably interpolated position. This approximation has been actually experimentally verified for the case of Theorem 1 (general expression for induced voltage) and Theorem 2 (optimum primary coil width that maximizes the induced voltage on given rectangular secondary coil). The experimental verification of Theorems 3 to 5 are left for future work.