# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
While both of these values of \(D\) will yield a flat response under the center alignment condition, the normalized induced voltage on \(s\), under the central alignment condition, will be significant less at \(D=w^{\prime}_{F}\) than at \(D=w_{F}\), except when the ratio \(h/d\) is very low (less than 10%), in which case in which the magnitude of the ratio of the induced voltages will be closer to unity (higher than 90%), as shown in \[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]9 However, a too low value of \(h/d\) is usually not practical to implement in a real DIWPT application for electric mobility, due to mechanical restrictions on the minimum required air gap. In summary, although two optimum values were found for a maximum flatness in the induced voltage response, which is a desirable characteristic is some applications, one of them, \(w^{\prime}_{F}\), will only exist if \(h/d<\)28.8%. Even then, the induced voltage obtained at \(D=w^{\prime}_{F}\), would be much less than that provided at \(D=w_{F}\), unless \(h/d<\)10%, which is likely to be difficult or impractical to implement if the inductive configuration is used for transferring power to terrestrial vehicles. So, the option for the primary coil width of \(w^{\prime}_{F}\) is apparently not as useful for a DIWPT design aiming for lightweight electric mobility as the larger alternative, \(w_{F}\). Thus, in principle, the existence of \(w^{\prime}_{F}\) may be disregarded. Geometric Interpretation for Theorem 4: A geometrical interpretation is given for the main optimum value \(w_{F}\). Despite not being as straightforward as the geometric construction provided for Theorems 2 and 3, it provides a tool that enhances the visualization, and a  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]0 Condition for the existence of a second smaller width for maximum flatness in the induced voltage. more intuitive design of DIWPT configurations using oblong coils, when a maximally flat induced voltage in a secondary coil of given width, is sought. Figure 12 represents a cross-section of the inductive configuration formed by coils \(p\), whose width is a free design parameter, and \(s\), of fixed width \(d\), given by points \(S_{1}\) and \(S_{2}\), at distance \(h\) from the support plane of \(p\). For a maximally flat induced voltage, the positioning of the conductors of the primary coil, \(w_{F}\), far apart from each other, is found by first constructing points \(P_{1}\) and \(P_{2}\), such that \(\overline{P_{1}P_{2}}\) is a diameter of a circle also containing \(S_{1}\) and \(S_{2}\), centered at \(\mathbb{C}_{1}\), the intersection of the mediatrix of \(\overline{S_{1}S_{2}}\) with the support line at distance \(h\) from \(\overline{S_{1}S_{2}}\) (\(P_{1}\) and \(P_{2}\) are the loci for the conductors of \(p\) for a maximum induced voltage on \(s\), as seen in Theorem 2). Then, point \(\mathbb{H}_{2}\) is constructed by making \(\overline{P_{2}\mathbb{H}_{2}}=h\). The intersection of a circle with center at \(\mathbb{C}_{2}\) and radius \(\overline{\mathbb{C}_{2}\mathbb{H}_{2}}\) with the perpendicular to the support line of the conductors of \(p\), drawn from \(P_{2}\), is point \(\mathbb{z}_{2}\). Another circle centered at \(\mathbb{C}_{1}\) and drawn with radius \(\overline{\mathbb{C}_{1}\mathbb{Z}_{2}}\) will intercept the support line for coil \(p\) projections at points \(\mathbb{F}_{1}\) and \(\mathbb{F}_{2}\), which are the positioning solutions for the conductors of \(p\) for a maximally flat induced voltage on \(s\). Incidentally, \(\mathbb{F}_{1}\) and \(\mathbb{F}_{2}\) would be the loci of the conductors of \(p\) for maximum voltage on \(s\), if \(s\), with a distinct width, had its conductors placed at \(\mathbb{z}_{1}\) and \(\mathbb{z}_{2}\), \(\mathbb{z}_{1}\) being the point symmetric to \(\mathbb{Z}_{2}\) with respect to the center plane of \(s\) that is perpendicular to the support line of \(p\).