# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
\(\Box\)  
Geometric Interpretation for Theorem 2: Despite the tedious algebraic manipulation involved in Theorem 2, it has a simple immediate geometric interpretation: The infinitely long filamentary single-turn primary coil carrying a fixed sinusoidal current \(I_{p}\) that will induce the maximum voltage on a parallel rectangular single-turn filamentary coil \(s\) at distance \(h\) from \(p\) is the one in which the projection on a cross-section of this inductive configuration of the support lines of the conductors of \(p\), \(P_{1}\) and \(P_{2}\), constitute the diameter \(\overline{P_{1}P_{2}}\) of a circle also containing the projection of the support lines of the conductors of \(s\), \(S_{1}\) and \(S_{2}\), as shown in \[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]0  
This relative positioning of \(S_{1}\), \(S_{2}\), \(P_{1}\), and \(P_{2}\) creates a right-angled rectangle where the vertical cathetus is \(h\), the horizontal cathetus is \(d/2\), and the hypotenuse, \(D/2\), is calculated using Pythagoras' theorem:  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]1  
Hence, the geometrical construction in Figure 8 assures that \(D=\sqrt{d^{2}+4h^{2}}\), which is the required width \(w_{P}\) of \(p\) for maximum induction in \(s\). Width of a Secondary Rectangular Coil at a Given Distance from a Given Oblong Primary Coil, for the Maximum Induced Voltage on the Secondary Coil Under Center Alignment Conditions  
**Theorem 3**. : _The maximum value for the normalized voltage \(\xi=\dot{V}_{ind}/-juf_{0}\dot{I}_{p}e^{\prime}\) induced by a stationary, infinitely long filamentary coil \(p\), of fixed width \(D\), that is constituted of a single turn of wire carrying a sinusoidal current \(I_{p}\) of constant amplitude and frequency \(f_{0}=2\pi\omega_{0}\), on a parallel rectangular coil \(s\), of width \(d\) and length \(e^{\prime}\), also constituted of a single turn of filamentary wire, located on a plane that is parallel to the plane that contains \(p\), at a fixed distance \(h\) to that plane, is obtained when the coils are symmetrically aligned, with central planes coinciding, and the width \(d\) of coil \(s\) is equal to \(w_{S}\):_  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]2  
_and can be calculated by_  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]3  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]4 Geometrical interpretation of Theorem 2. _Additionally, when the coils remain center-aligned and the width \(d\) of the primary coil \(s\) is varied, the normalized induced voltage observed with \(d=\lambda w_{S}\) is the same as that observed when \(d=w_{S}/\lambda\), for any \(\lambda>0\)._  
There is a duality relating Theorem 3 and Theorem 2; the expressions (77) and (78) are exactly the same as those in (61) and (62), except that \(D\) and \(w_{S}\) are used in place of \(d\) and \(w_{P}\). The physical situation in each case is different, though: In Theorem 2, the width of the secondary coil is fixed at \(d\), and an optimal diameter of the primary coil, \(D\), that maximizes the induced voltage on \(s\) is to be found. Meanwhile, in Theorem 3, the search is reversed: the width \(D\) of the primary coil itself is fixed, and an optimal width \(d\) for the secondary coil that maximizes the induced on voltage \(s\) is to be determined.