# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 5 Conclusions and Discussion  
Five theorems were introduced, aiming to facilitate the design of DIWPT configurations for lightweight vehicular applications, where an oblong coil and a parallel rectangular coil form an air-cored magnetic link, which allows energy to be inductively harvested by wireless power transfer. The first theorem can be seen as a mere manipulation of the basic equations of the mutual inductance between parallel wires, long ago derived [25; 35] (pp. 45-47). Theorems 1 and 2 were developed to explain previously developed experimental work, involving the construction and testing of an inductive lane for electric bikes [11; 14]. The experimental evaluations of Theorems 3, 4 and 5, which are based on different constraints and desired optimized target behaviors for the induced voltage, are left for future work. Hopefully, all the theorems will serve in other WPT applications where either the primary or the secondary coils are of the oblong type, and the other coil, rectangular, is positioned over the first coil, on a plane that is parallel to that containing the firstly considered oblong coil. In particular, they might find applications in the design of inductive lanes for lower power, lightweight electric vehicles. Theorems 2 and 3, geometrically interpreted in Figure 10, establish that the width of a first given coil (which can be either the primary or secondary) and the double of the  
Figure 15: Geometric construction of a DIWPT configuration based on oblong primary coils, using Theorem 4.  
distance separation between the first and the second coil planes form the perpendicular sides of a right triangle where the hypotenuse determines the optimum width of the other coil (respectively, the secondary or primary first coil), maximizing the open secondary induced voltage. Noticeably, the other coil should always be of larger width than that of the first given coil, their size converging to the same value when the distance between them converges to zero, as expected. Hence, these two theorems become more relevant as the distance being imposed between the coils is particularly significant with respect to the dimensions of the first given coil, a typical situation found in DIWPT in electric mobility applications. Theorems 4 and 5 establish that if a right triangle is formed with the perpendicular sides numerically given by the geometric mean of the optimum width of the second coil for a maximum induced voltage (as calculated by either Theorem 2 or 3) and the double of the distance separation between the first and the second coil planes, then this triangle will have the length of its hypotenuse equal to the width of the primary coil that would yield a maximally flat induced voltage on the secondary coil. In all cases, the theorems only hold when an alternated current of constant amplitude is being imposed on the primary coil. While the design for maximum induced voltage can help increase both the peak power transferred and the resulting obtained electric efficiency, its flat response to the lateral displacement of the vehicle would simplify the design of the electronic circuitry by diminishing the dynamic input voltage range requirements for the DC-DC converter that, in the secondary coil, regulates the voltage applied to the powertrain. According to the designer's priorities and, ultimately, to the system requirements, a compromise width for the other coil (which can be either the primary or the secondary coil) can possibly be found between the value given Theorems 2 and 3, and that given by Theorems 4 and 5. The theorems are expected to be applicable to magnetic link configurations with air gaps constituted by air or other non-conductive materials as long as they exhibit constant and isotropic magnetic permeability. While inductive lanes for lightweight electric vehicles can have advantages for the overall conservation of energy, as well as for the reduction in both gas emissions and use of natural resources, their adoption is presumed to be a difficult and slow process, not even guaranteed to happen.