# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
Proof of Theorem 3.: The reason for the above-mentioned duality, which is the proof for Theorem 3, is that, by inspection of the algebraic expression (49) of the function \(\xi\), both the induced voltage from \(p\) on \(s\), \(\tilde{V}_{ind}\), as well as the associated normalized induced voltage, \(\xi\), given by Theorem 1, are symmetric on \(d\) and \(D\):  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]5  
and  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]6  
Hence, the points of maxima in \(\xi\), with respect to \(d\), when \(x=0\), are expressed in terms of fixed \(h\) and \(D\) by the same function as the points of maxima in \(\xi\), when \(x=0\), with respect to \(D\), when expressed in terms of fixed \(h\) and \(d\), causing the width of the maximum induced voltage on \(s\), at \(x=0\), to also be found at an optimum value of \(d\) and \(w_{S}\), given as the function of \(h\) and \(D\):  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]7  
In the same manner, the value of the maxima will be given by the same function by which it was calculated in the case of the optimization in \(D\), but evaluated at the pair \((h,D)\), rather than at the pair \((h,d)\):  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]8  
To prove the last statement of Theorem 3, it can be realized that, for all \(\lambda>0\),  
\[\mathbb{H}_{1}=\left(-\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{59}\]9 \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]0 \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]1  
with the first and last equalities being due to the existing indistinguishability in \(d\) and \(D\), as given by (78), and the equality in the middle, due to Theorem 2. Hence,  
\[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]2  
which constitutes the last part of Theorem 3. \(\square\)Geometric Interpretation for Theorem 3: Similarly to Theorem 2, the dimensions and location of the optimum filamentary secondary coil \(s\) for a maximum induced voltage from a filamentary primary coil \(p\), at a fixed distance \(h\), also has a simple immediate geometric interpretation: if the projections of the support lines of the conductors of \(p\), in a common cross-section with s, are \(P_{1}\) and \(P_{2}\), and the projections of the support lines of the conductors of s parallel to \(p\) are \(S_{1}\) and \(S_{2}\), then \(S_{1}\) and \(S_{2}\) constitute the diameter \(\overline{S_{1}S_{2}}\) of a circle also containing \(P_{1}\) and \(P_{2}\), as shown in \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]3  
Attention should be paid when comparing, using Theorem 2 and Theorem 3, which geometries can be visualized, respectively, in Figures 8 and 9: the optimum width for the other coil, in order to attain maximum induced voltage on the secondary, is always larger than that of a first coil whose width has been fixed, no matter whether the first coil to have its dimensions fixed beforehand is the secondary or the primary coil, as synthetically represented in \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]4  
Width of an Oblong Primary Coil for a Maximally Flat Induced Voltage on a Given Secondary Rectangular Coil at a Given Distance from the Primary Coil, Under Center Alignment Conditions  
**Theorem 4**. : _Let \(p\) be a stationary, infinitely long filamentary coil \(p\), of width \(D\), that is constituted of a single turn of wire carrying a sinusoidal current \(I_{p}\) of constant amplitude and frequency \(f_{0}~{}=~{}2\pi\omega_{0}\); let \(s\) be a rectangular coil parallel to \(p\) of fixed width \(d\) and length \(\epsilon\), also constituted of a single turn of filamentary wire and located on a plane that is parallel to the plane that contains \(p\), at a fixed distance \(h\) to that plane.