# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
Then, the profile of the normalized voltage \(\xi=\dot{V}_{ind}/~{}j\mu f_{0}I_{p}\epsilon\), induced by \(p\) on \(s\), as a function of the lateral misalignment of \(p\) and \(s\), is maximally flat in the vicinity of the center alignment condition when the width \(D\) of coil \(p\) is either equal to \(w_{F}\) or to \(w_{F}^{\prime}\), as given by the following equations and conditions:_  
\[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]5  
\[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]6 Geometric interpretation of the optimum width of the other coil in an oblong coil configuration for maximum induced voltage. \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]7 Geometrical interpretation of Theorem 3. \[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]8  
_where_ \(w_{P}\) _is the width of_ \(p\) _for the maximum induced voltage on_ \(s\)_, as in Theorem 2:_  
\[\mathbb{H}_{2}=\left(+\frac{\sqrt{d^{2}+D^{2}}}{2},0\right) \tag{60}\]9  
_and_ \(\gamma_{h}\) _is the optimum_ \(h/d\) _ratio for the maximum linearity of the induced voltage on_ \(s\) _due to a single conductor of_ \(p\) _(32), as given by_  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]0  
**Proof of Theorem 4. ** By analogous argumentation to that used in Lemma 1, when considering expressions (22)-(24), \(\xi\), as given by (49), is also continuous and continuously n-differentiable in \(\mathbb{R}\), and, by inspection of (49), clearly symmetric in the variable \(u\):  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]1  
This implies that  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]2  
and thus  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]3  
The requirement for a maximally flat response to lateral misalignment between coils \(p\) and \(s\) can be traduced by a zero in the first- and second-order derivatives of the normalized induced voltage with respect to the lateral displacement, at the point where central alignment condition is met. The first order derivative \(\partial\xi/\partial u\) at \(u=0\) is already guaranteed to be zero by (91). The same holds for the third order derivative. By developing \(\partial^{2}\xi/\partial u^{2}\), it is found that:  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]4  
Considering the Taylor expansion of \(\xi\) to the third order, at the vicinity of \(u=0\), the center of alignment between coils \(s\) and \(p\), with \(h\) and \(d\) fixed, it is then possible to approximate \(\xi(x,h,d,D)\) by  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]5  
So, given that the first and the third derivatives of \(\xi\) with respect to \(u\) are already zero at \(u=0\), the variation in the induced voltage \(\xi\) in response to the lateral displacement \(x\), in the vicinity of the center of alignment of coils \(s\) and \(p\), will be minimized when the second derivative of \(\xi\) at \(u=0\), given by (92), is also zero, under which condition \(\xi\) will be approximated by the constant \(\xi(0,h,d,D)\). This happens when  
\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]6  
As \(D>0\) is the free variable geometrical parameter considered in Theorem 4, the only two possibly real positive roots of (94) in D are given by \(w_{F}\) and \(w^{\prime}_{F}\):\[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]7 \[w_{P}=\sqrt{d^{2}+4h^{2}} \tag{61}\]8  
By recognizing that \(d^{2}+4h^{2}=w_{P}^{2}\), from Theorem 2 (61), and that \(\gamma_{h}=1/2\sqrt{3}\), from the development of Lemma 1 (32), expressions (95) and (96) can be rewritten, respectively, as (85) and (86). \(\square\)  
The two possible solutions found, \(w_{F}\) and \(w^{\prime}_{F}\), have different properties concerning the magnitude of voltage that can be induced in each configuration derived. Also, \(w_{F}\) is visibly a larger width than \(w_{P}\), and it is always available as a design option, while \(w^{\prime}_{F}\), when it is defined, is smaller, and associated with lower induced voltages.