# Optimum Geometries for Wireless Power Transfer Between an Oblong Coil and a Rectangular Coil
## 4 Application of Theorems 1 to 5 in DIWPT Design  
Considering the possibility of approximating real coils by filamentary windings, for both primary and secondary coils, and provided that their relative velocity is such that the quasi-stationary hypothesis can be applied [34], Theorems 1 to 5 can help the design of DIWPT configurations where the primary coil is an oblong coil and the secondary coil is a rectangular parallel coil at a fixed distance \(h\) to the primary coil. This happens because the magnetic field in a cross-section of the primary coil is approximately the same as if the primary coil had infinite length, if the considered cross-section is away from the extremities of the primary coil (by at least one primary coil width of distance or one secondary coil length along the main axis of the primary coil, whichever is larger). When the secondary coil is compensated, the induced voltage ultimately determines the available power in the secondary coil for a given connected resistive load. In the simplified model considered, the power delivered to the load will be proportional to the squared normalized induced voltage. This is a good reason for the concern in maximizing the induced voltage in practical applications, as established in Theorems 2 and 4. Alternatively, depending on the circuit design of the secondary coil, the design target might be to have the induced voltage with a flat as possible characteristic, in response to small misalignment variations of the primary and secondary coils, and that is the utility of Theorem 3 and Theorem 5. Theorem 1, on the other hand, provides an analytic method for calculating the induced voltage or the power delivered to \(R_{L}\) as a function of the circuit parameters and the geometry of the coils, allowing the DIWPT design to be refined. However, in order to fully benefit from the predictions provided by these theorems, the hypothesis of a constant current in the primary coil must hold for a good approximation to be obtained. This will naturally happen if the back-induced voltage from the current secondary coil current is small compared to the voltage established on the primary coil. Noticeably, in DIWPT configurations, where the magnetic coupling coefficient is low, the secondary will likely have more turns than the primary coil, as resulted in our DIWPT application that was developed for e-bikes. It will also happen if the primary circuit is specifically designed to keep the current amplitude constant. As an illustration of the main mathematical results achieved, the plots of the normalized induced voltage, calculated on basis of the analytical expressions derived for designs with different optimal properties, are exemplified in Figure 14. The configuration corre  
Figure 13: Geometric visualization for the calculation of optimum width coils. sponds to a given secondary coil with fixed width \(d\), at a fixed distance \(h\) from a primary coil in which the width \(D\) is a parameter of free election by design. The fixed location of the secondary coil is indicated by points \(S_{1}\) and \(S_{2}\), plotted with the vertical axis scale equal to the normalized horizontal scale, differently for the vertical scale for the curves (normalized voltage), for improved perception of the proportions of the geometry in this example, where \(h/d\) was set to \(1/4\). In the chart of Figure 14, the induced voltage \(\xi\) is normalized by \(\xi_{max}\) before being plotted, so that the maximum value observed on the vertical axis is one, facilitating the visual comparison of the induced voltage profile of the various configurations. The colors of the curves are in correspondence with the color of the small circles, which represent the location of the conductors of the primary coil that generate each curve. The red curve corresponds to the maximum induced voltage condition, obtained for \(D=w_{P}\), as given by Theorem 2.