# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
From this geometric construction, it can be shown that the length of \(\overline{\mathbb{F}_{1}\mathbb{F}_{2}}\) is exactly given by the expression derived for \(w_{F}\) (95). For quick estimations of the position of \(\mathbb{F}_{1}\) and \(\mathbb{F}_{2}\), it can also be verified that  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]1  
Width of a Secondary Rectangular Coil for a Maximally Flat Induced Voltage by a Given Oblong Primary Coil at a Given Distance from the Secondary Coil, Under Center Alignment Conditions  
**Theorem 5**. : _Let \(p\) be 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 \(f_{0}=2\pi\omega_{0}\); let \(s\) be a rectangular coil parallel to \(p\) of width \(d\) and fixed length \(\ell\), 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 hto that plane. Then, the profile of the normalized voltage \(\xi=\dot{V}_{ind}/\ j  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]2  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]3  
_where \(w_{S}\) is the width of \(p\) for the maximum induced voltage on \(s\), as in Theorem 3:_  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]4  
_and \(\gamma_{h}\) is the optimum \(h/D\) ratio for maximum linearity of the induced voltage on \(s\) due to a single conductor of \(p\), as given by_  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]5  
Notably, Theorem 5 can be seen as Theorem 4 with \(d\) and \(D\) reversed. That is, in Theorem 4 the secondary width is fixed at \(d\), and an optimum is sought for the width \(D\) of the primary coil, while in Theorem 5 the primary coil has a fixed width \(D\), and the problem to be solved is the determination of an optimum secondary width \(d\) that will maximize the flatness of the response of the induced voltage as a function of the lateral misalignment of the primary and secondary coils. Again, likewise to the case of Theorems 2 and 3, there is a duality between Theorems 4 and 5, and the proof of Theorem 5 is based on the same relations and expressions derived for Theorem 4. Proof of Theorem 5.: Since the inductive configuration and its parameters are equivalent and named exactly as in Theorem 4, the expression for the normalized induced voltage for the present situation is the same as (92), and so are valid expressions (93) and (94). In order to find the optimum, \(d\) is then given by the real positive roots of (94), having this time \(d\) as the free variable, not \(D\) as in the proof of Theorem 4. Since (94) is symmetric with respect to \(d\) and \(D\), the optimum values for \(d\), \(w_{I}\) and \(w_{J}^{\prime}\), are given by  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]6  
\[\xi_{max_{h,d}}=\ln\frac{w_{P}+d}{w_{P}-d} \tag{62}\]7  
It can be recognized from Theorem 3 (77) that \(D^{2}+4h^{2}=w_{S}^{2}\), and from the development of Lemma 1 (32) that \(\gamma_{h}=1/2\sqrt{3}\). So, expressions (102) and (103) can be rewritten, respectively, as (99) and (100). \(\square\)  
By analogy to Theorem 4, the geometric evaluation of optimum widths \(w_{I}\) and \(w_{I}^{\prime}\) for a secondary coil \(s\) at fixed distance \(h\) to a given primary coil with width \(D\) will be entirely similar to that shown in Figure 12, but with the drawing up-side down and the location of points \(P_{1}\) and \(P_{2}\), respectively, reversed with \(S_{1}\) and \(S_{2}\). A general visualization tool for optimum widths of a second coil for the maximum induced voltage (red line segment) and for a maximally flat induced voltage (green line segment) is seen in Figure 13, where the width of the first given coil is represented by the black horizontal line segment and the fixed distance between the two coils.