# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]6 Normalized voltage induced on a single-turn secondary coil \(s\) by a single-turn primary coil \(p\) as a function of the relative position of the center of \(s\) to \(p\), under the hypotheses of Theorem 1. The conductors of the primary coil \(p\), indicated in Figure 6 by two red dots, are placed centered at \(h=0\). The conductors of the secondary coil s are indicated by two dark blue dots, \(S_{1}\) and \(S_{2}\), and probe the induced voltage when translated over the cross-section, a typical situation of interest in the design of DIWPT configurations. Specifically, for this example, the ratio \(D/d\) was set to \(2/\sqrt{3}\). The red curve exemplifies a possible path for the displacement of \(s\), with the red cross denoting the center of the secondary coil \(s\). The color of the plot shows how the value of the normalized induced voltage, \(\xi\), changes when the center of the secondary coil is displaced on the cross-section plane. The blue regions indicate the positions of the cross-section, where \(\xi>0\), i.e., \(\dot{V}_{ind}\) is given by \(ja\dot{I}_{p}\), where \(\alpha>0\), and the phase of \(V_{ind}\) is \(\pi/2\) ahead of the phase of \(I_{p}\). Similarly, in the pink regions, the phase of \(V_{ind}\) is a \(\pi/2\) delayed form of the phase of \(I_{p}\). The black dashed curves are the loci of the zero induced voltage, defined by the following equation:  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]7  
where \(d\) and \(D\) are the fixed widths of, respectively, the secondary and primary coils, and \((x,h)\) is the coordinate pair of a point of the cross-section belonging to the dashed curve where \(V_{ind}=0\). Developing (56) results in an equation of \(x\) and \(h\):  
\[ \begin{array}{l}\left(\left(x-\frac{D+d}{2}\right)^{2}+h^{2}\right)\left( \left(x+\frac{D+d}{2}\right)^{2}+h^{2}\right)\\ =\left(\left(x+\frac{D-d}{2}\right)^{2}+h^{2}\right)\left(\left(x-\frac{D-d}{ 2}\right)^{2}+h^{2}\right)\end{array}  \tag{57}\]  
which simplifies to  
\[d^{2}+D^{2}+4h^{2}-4x^{2}=0\, \tag{58}\]  
which is the equation of a rectangular hyperbola on the plane \((x,h)\), with vertices \(\mathbb{H}_{1}\) and \(\mathbb{H}_{2}\) at  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]  
\[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]  
Remarkably, the fact that the loci of \(V_{ind}=0\) form a rectangular hyperbola (eccentricity \(\sqrt{2}\)) does not depend on the ratio \(d/D\), only its scale factor will depend on the indistinct sum of the squares of the coil widths. Since in most vehicular DIWPT applications, the secondary coil is installed at a fixed distance from the primary coil, another useful application for Theorem 1 is to predict the induced voltage on a rectangular secondary coil as a function of the lateral misalignment on the lane, as exemplified in the curves of Figure 7, plotted in different colors tones for different ratios of \(D/d\), increasing from 0.25 (red curve) to 2.5 (blue curve), and \(h/d=1/2\sqrt{3}\). One key consideration to obtain a good numerical approximation for this prediction, using Theorem 1, is to require that the middle cross-section under analysis is far enough from the extremities of the primary coil. However, this is not a problem: while no actual oblong primary coil \(p\) has infinite length, in practice, it is verified that the magnetic flux distribution generated by current \(I_{p}\) is fairly regular along all its length, except very close to the extremities of \(p\), when this coil is much longer than wide, and also much longer than the secondary coil. Width of an Oblong Primary Coil for Maximum Induced Voltage on a Given Secondary Rectangular Coil at a Given Distance from the Primary Coil, Under Cater Alignment Conditions  
**Theorem 2**.