# Retractable Wireless Charging Windings for Inspection Robots
## 3 Design and Simulation of Retractable Wireless Power Transfer Coils
### Parametric Analysis of Coil Geometries  
Three subcategories of geometric shapes of WPT coils were selected as shown in Figure 5. Group 1 represents the triangle group (equilateral and right-angled). Group 2 includes quadrilaterals, squares, and rectangles with a 2:1 aspect ratio. Group 3 consists of polygons such as pentagons, hexagons, octagons, and circles with 16 and 32 edges. Table 3 presents the simulation results for the groups of coil shapes under a fixed input power of 200 W (where \(N\) represents the coil's number of edges). Observations were made on the WPT system's coil resistance (\(r_{coil}\)), mutual inductance (\(M\)), coupling coefficient (\(k\)), transmitted power (\(P_{out}\)), and energy efficiency (\(\eta\)).  
The \(f_{s}\) across different WPT coil shapes is varied. While the coil cross-sectional area (\(W_{csr}\)) remains consistent across all shapes and the coil diameter (\(C_{d}\)) or aperture size is maintained at 1 m, the coil length is adjusted. This variation affects self-inductance, which in turn influences both the transmitted power and energy efficiency. Based on the simulation  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Conceptual Designs** & **Potential** & **Retractable 2** & **Coil** & **Mechanical** & **Electrical** & **Overall Score** \\  & **Shapes 1** & & **Management 3** & **Complexity 4** & **Complexity 4** & **Complexity 4** & **Overall Score** \\ \hline Sunshade Folding & C, S & 5 & 5 & 5 & 5 & 20 \\ Rotation Folding & C & 4 & 5 & 4 & 4 & 17 \\ Two-piece Folding & C, O, S & 3 & 4 & 3 & 3 & 13 \\ Telescopic & O, H & 4 & 4 & 4 & 4 & 16 \\ Umbrella & C, O, H & 4 & 5 & 4 & 4 & 17 \\ Scissor-linkages & H, P, S, T & 3 & 3 & 3 & 3 & 12 \\ \hline \hline \end{tabular}  
* C = circular, O = octagon, H = hexagon, P = pentagon, S = square, T = triangle. 2 1 = easy to form, 2 = basic, 3 = moderate, 4 = complex, 5 = high mechanical complexity. 3 1 = easy to manage, 2 = basic, 3 = moderate, 4 = complex, 5 = high entanglement. 4 1 = simple, 2 = basic, 3 = moderate, 4 = complex, 5 = sophisticated.  
\end{table}
Table 2: Conceptual design matrix.  
results, square-shaped coils achieve the highest energy efficiency (90.67%), followed closely by pentagon, hexagon, octagon, and circular coils, while triangle and rectangle shapes perform significantly worse. This is mainly due to better area-to-perimeter ratios and more uniform magnetic flux distribution in compact, symmetric shapes. In summary, other shapes could perform better; however, based on basic physics (area-to-perimeter ratio and field uniformity), the simple compact shapes (squares and circles) are already very close to optimal. Fine-tuning shapes, such as rounding edges, adjusting polygon sides, or using hybrid designs, could bring small further improvements, but dramatic changes beyond approximately 91-92% efficiency are unlikely in practice.  
This parametric analysis directly informed the final design decision. The square coil geometry was selected due to its optimal balance between electromagnetic performance. It demonstrated a high coupling coefficient and energy efficiency and mechanical compatibility with the retractable scissor-linkage mechanism. The simulation results guided both the physical implementation and the subsequent experimental validation.