# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 2 Wireless Power Transfer
### Mutual Inductance of a Magnetic Link in DIWPT Configurations  
A calculation tool for the mutual inductance, \(M\), between two filamentary coils with windings modeled by curves \(\mathrm{k}_{1}\) and \(\mathrm{k}_{2}\) in \(\mathbb{R}^{3}\) was established by [PERSON] [19; 20], and it is now expressed in the International System of Units by  
\[M=\frac{\mu}{4\pi}\underset{\mathrm{k}_{1}}{\oint}\underset{\mathrm{k}_{2}}{ \oint}\underset{\mathrm{k}_{2}}{\overset{\rightarrow}{\oint}}\frac{d\ell_{1}.d\overset{\rightarrow}{\ell_{2}}}{r}, \tag{3}\]  
where \(r\) is the distance between the points of \(\mathrm{k}_{1}\) and \(\mathrm{k}_{2}\), for which the infinitesimal oriented elements \(d\overset{\rightarrow}{\ell_{1}}\) and \(d\overset{\rightarrow}{\ell_{2}}\) are, respectively, considered, and \(\mu\) is magnetic permeability of the medium in which the coils are immersed. Thus, \(M\) is purely dependent on the geometry of the magnetic link. For real, non-filamentary coils, the distribution of the current density inside the conductors along each cross-section has to be considered, and a dependence of \(M\) on the inducing frequency starts to show when the frequency itself is high enough to make the skin depth in the conductors become comparable to or smaller than the diameter of the wires used in the coil windings. But, in many practical cases, designers tend to avoid this condition to reduce winding losses, and a good estimate of the mutual inductance can still be provided by (3). But (3) is usually difficult to solve analytically. When some closed-form expression for the mutual inductance is found, often it is not a simple one, preventing intuition from building up in the designer. In fact, most of the design cases can be exclusively accomplished by numeric computation. This has been particularly observed when dealing with \"short\" coils, that is, coils of about the same size, where each of them also has its a longer dimension of the same magnitude as its shorter dimension. Notably, in this class of coils, formulation and precise computation methods for self- and mutual inductances have been determined for circular [21; 22; 23; 24] and rectangular shapes [25] Chapter 8, [26; 27; 28; 29; 30], and even for some hybrid variants [31]. Short coils of other shapes have also been studied, such as the double \"D\" coils [32], which, in addition to the good power transfer capabilities of other short coils, exhibit a diminished fluctuation of the induced voltage and the resulting transferred power (\"power pulsation\") as the vehicle moves along its path. The electric losses in oblong primary coils, for a given alternate current, tend to be higher than those in shorter length ones when using the same type of wiring for the windings of both types of coils. In this manner, the electrical efficiencies obtained in inductive lanes built with oblong coils with the same conductive technology tend to be lower than those obtained with short coils. This is especially important in high-power heavy road EVs, ones that can be a direct functional replacement for their fossil fuel engine counterparts. Possibly because of this, inductive lanes constituted by oblong primary coils have been somewhat forgotten in recent years, although constituting the basis for the first known DIWPT prototypes [2; 4; 5]. Except in dedicated industrial applications, where they are appreciated for allowing the dependable implementation of wireless power transfer with a low component count, resulting in more simplicity and robustness. However, when dealing with lightweight electric vehicles (LEVs), which intrinsically require less energy to move than heavier road vehicles, lower electric efficiencies in the WPT can be tolerated, and result in systems with a potentially much lower cost to build, to operate, and to maintain than short-coil-based systems. This possibility brings back interest to DIWPT with oblong coils, and to the search for analytical design tools for these systems.