# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 4 Application of Theorems 1 to 5 in DIWPT Design
The two blue curves are the maximally flat induced voltage profiles, obtained for \(D=w_{F}\) (continuous blue) and for \(D=w_{F}^{\prime}\) (dashed blue), given by Theorem 4. The green curve is the result of setting the width of the primary coil to a value that is less than \(w_{P}\), but still higher than \(w_{F}^{\prime}\), and the brown curve is an example of a profile generated when the width of the primary coil is larger then \(w_{F}\). The criterion for a DIWPT design will be strongly dependent on the application. However, in general, the transfer of maximum power for the same primary current is one of the main goals, not only for attaining the required power levels, but also for attaining better efficiency levels, which can be obtained by drawing more power from the same primary current. This is due to the primary losses being strongly related to the square of the rms value of the primary current, which is fixed by hypothesis. An alternative convenient design goal is to try to keep the amplitude of the induced voltage as constant as possible, while the electric vehicle (secondary coil) harvesting power from the lane (primary coil) drifts laterally around the center of the lane, constantly generating a misalignment input \(x\) to the function \(\xi\). This property is obtained when \(D\) is close to \(w_{F}\), and useful when a DC-DC converter is used to match the load impedance, and the acceptable ratio between the maximum and minimum input to the converter should be kept as low as possible. An in-depth analysis of the application, supported by the tools presented in this section, will lead to the best design choices. In general, unless there is some additional restriction imposed on the width \(D\) of the primary coil, there will be no good argument for picking a \(D<w_{P}\) or \(D\) much greater than \(w_{F}\). Figure 14: Curves of normalized voltage induced on a single-turn rectangular secondary coil by a single-turn oblong primary coil for different primary coil widths, under the conditions of Theorem 1. Concerning electric mobility, although Theorems 3 and 5 provide solutions for optimum geometries of the secondary pick-up coils to be installed on vehicles, the most usual application is to design the primary coil with an optimal width, given a secondary coil of fixed dimensions. This is because whilst the coils should be as large as possible for a good transfer of power, the maximum dimensions of the secondary coil are a priori constrained by the size of the vehicle and mechanical restrictions imposed by the installation of the coil on the vehicle. An example of a graphical construction of the optimum cross-section of an inductive coupling, by the determination of the width \(w_{F}\) of an oblong primary coil (in red) using Theorem 4, given a rectangular secondary coil of width \(d\) (in blue), is shown in Figure 15.  
Once the most favorable dimensions for the coils are fixed, their equivalent circuit parameters, such as inductances and mutual inductance, can be derived and serve as the basis for the circuit design of the primary and the secondary power electronics circuits. Under the hypothesis of concentrated filamentary coil turns, changing the number of turns will vary the self-inductances of the coils and their mutual inductances, providing a means for impedance matching, but the coupling coefficient should remain unaltered if the proximity effect between turns within each coil and the changes in the cross-section of the coils are neglected.