# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 3 Theorems
: _The voltage \(V_{ind}\) induced by a stationary, infinitely long coil \(p\), constituted of a single turn of a filamentary wire carrying a sinusoidal current \(I_{p}\) of constant amplitude and frequency \(f_{0}=2\pi\omega_{0}\), on a parallel rectangular coil \(s\), stationary relative to \(p\) and also constituted of a single turn of a filamentary wire, which is located on a plane that is parallel to the plane that contains \(p\), at a distance \(h\) to that plane, is given by_  
\[\dot{V}_{ind}=-\int\limits_{-\frac{d}{2}}^{\frac{d}{2}}\frac{d}{dt}\dot{B}_{z }\;\ell\;dv=-j\omega\epsilon\int\limits_{-\frac{d}{2}}^{\frac{d}{2}}\dot{B}_{z }\;dv \tag{7}\]8  
_with the real function \(\xi\) given by_  
\[\dot{V}_{ind}=-\int\limits_{-\frac{d}{2}}^{\frac{d}{2}}\frac{d}{dt}\dot{B}_{z }\;\ell\;dv=-j\omega\epsilon\int\limits_{-\frac{d}{2}}^{\frac{d}{2}}\dot{B}_{z }\;dv \tag{7}\]9  
_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 width of the infinitely long coil_ \(p\)_, i.e., the distance between its two conductors;_
* \(d\) _is the length of the side of the rectangular coil_ \(s\) _that is perpendicular to_ \(p\)_;_
* \(x\) _is the oriented displacement of the projection of the longitudinal axis of_ \(s\) _on the plane of_ \(p\) _to the longitudinal axis of_ \(p\) _(if_ \(p\) _and_ \(s\) _are contained in horizontal planes,_ \(x\) _is the lateral misalignment between these coils)._
* \(j\in\mathbb{C}\)_,_ \(j^{2}=-1\)__  
Again, the term filamentary wire is used here in the same sense and with the same assumptions as used for Lemmas 1 and 2. Proof of Theorem 1.: For the visualization of the geometry involved in the problem, let us start by considering a cross-section of this configuration, where coil \(p\) is inducing a voltage in coil \(s\), as shown in Figure 5, with equivalently defined parameters as in Figure 2, where \(S_{1}\). Applying the superposition principle to the magnetic fields generated by the currents traversing the two wires of coil p, it can be derived that the total induced voltage will be the sum of the two voltages separately induced by currents \(I_{p}\) flowing out of the paper at \(P_{1}\) and inwards at \(P_{2}\), which is equivalent to a current \(-I_{p}\) flowing out of the paper. Then, from Lemma 1, it is possible to use (18) twice, adding up the voltages induced on \(s\) by the conductor of \(p\) at \(P_{1}\), \(\dot{V}_{1}\), and by the conductor of \(p\) at \(P_{2}\), \(\dot{V}_{1}\):  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]0  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]1Applying (19) to expand the function g, (52) becomes  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]2  
Reorganizing the terms of (53) results in  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]3  
which can be decomposed in expressions (48) and (50). \(\square\)  
The expression for \(\dot{V}_{ind}\) (48) can be compared to the model in Figure 1b, permitting the mutual inductance \(M\) between \(s\) and the long coil \(p\) to be equivalently expressed as follows:  
\[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]4  
Illustrating a possible application of Theorem 1, Figure 6 shows the scalar field of the normalized induced voltage, \(\xi\), computed using (49), on the cross-section of an inductive configuration with an infinitely long primary filamentary coil \(p\) and a parallel rectangular filamentary coil \(s\), each consisting of a single turn. \[\dot{B}=\frac{\mu\dot{I}_{p}}{2\pi r} \tag{8}\]5 Cross-section of the inductive configuration formed by coils \(p\) and \(s\).