# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems  
Throughout this section, the following general symbology convention is used, unless otherwise explicitly defined:  
* \(j\) is the complex number \(\sqrt{-1}\). * Other Latin letters denote unidimensional variables, e.g., \(d\), \(h\), and \(D\). * An arrow over an italic letter denotes a vector in \(\mathbb{R}^{3}\) associated with the corresponding physical unit. For instance, \(\overset{\rightarrow}{E}\) denotes an electric field. * Non-italic uppercase letters denote points drawn on a cross-section of the magnetic link, representing the position of parallel filamentary wires. * A dot over a letter indicates a phasor, a complex number representing the rms value and phase of a sinusoidal time signal. \(V\) indicates voltages; \(I\) indicates currents. **Lemma 1**. : _The voltage \(V_{ind}\) induced on a rectangular coil \(s\), constituted by a single turn of filamentary wire, due to an infinitely long, straight, parallel filamentary wire \(p\), carrying a sinusoidal current \(I_{p}\) of constant amplitude and frequency \(f_{0}=2\pi\omega_{0}\), when \(s\) is stationary relative to \(p\) and located at distance \(h\) from the plane that contains \(p\), is given by_  
\[\dot{V}_{ind}=-j\;\mu\;f_{0}\;\dot{I}_{p}\;\epsilon\frac{1}{2}\text{ln}\frac{ \left(u-\frac{d}{2}\right)^{2}+h^{2}}{\left(u+\frac{d}{2}\right)^{2}+h^{2}}, \tag{4}\]  
_where_  
* \(\dot{V}_{ind}\) _is the phasor representation of the voltage_ \(V_{ind}\)_;_
* \(\mu\) _is the magnetic permeability of the medium where the conductors are immersed;_
* \(\dot{I}_{p}\) _is the phasor representing current_ \(I_{p}\)_;_
* \(\ell\) _is the length of the side of the rectangular coil_ \(s\) _that is parallel to the wire_ \(p\)_;_
* \(d\) _is the length of the side of the rectangular coil_ \(s\) _that is perpendicular to_ \(p\)_;_
* \(u\) _is the oriented displacement of_ \(p\) _relative to the center plane of_ \(s\)_, which is perpendicular to the plane of_ \(s\) _and parallel to_ \(p\)_._
* \(j\in\mathbb{C}\)_,_ \(j^{2}=-1\)__  
With \(p\) being a filamentary wire, it is assumed that, regarding its effect on other conductors, the current traversing wire \(p\) can be considered concentrated on a single line support of \(p\), analogous to what is happening to the filamentary wire \(s\), which is assumed to only contain a single path where the electric field can be integrated. This assumption will approximate real problems where the distances between conductors are much greater than their diameters. Remarkably, when this assumption does not hold, (4) will visibly fail when \(p\) excessively approaches one of the conductors that are the parallel to \(p\) sides of the rectangular coil \(s\). That is, expression (4) clearly diverges to infinity when, simultaneously, \(h\to 0\) and either \(u\;\to\;d/2\) or \(u\;\to\;-d/2\), while it is known from experimentation that, in this case, the induced voltage \(V_{ind}\) approaches the voltage across the segment of \(p\) with length \(\ell\) that is closest to \(s\). Proof of Lemma 1.: For the visualization of the geometry involved in the problem, let us start by considering the cross-section of this configuration, where \(p\) is inducing a voltage in coil \(s\), as shown in Figure 2, where the above linear dimensions are indicated, except \(\ell\), which is taken as perpendicular to the cross-section. Figure 2: Reference cross-section for Lemma 1. Let \(S_{1}\), \(S_{2}\), and P be, respectively, the points where coil \(s\) and wire \(p\) intercept this cross-section. Let \(\stackrel{{\rightarrow}}{{B}}\) be the magnetic field generated by \(\dot{I}_{p}\) flowing on p, and \(\dot{B}_{z}\) its complex scalar component on axis \(z\), which is perpendicular to the plane containing \(s\).