# Design of a Compact Planar Magnetic Levitation System with Wrench-Current Decoupling Enhancement
## 2 System Design Procedure
### Electromagnet Design  
A comparison between square and cylindrical coils of similar size in [30] indicated that an average magnetic flux density generated by an array of square coils was stronger than that of cylindrical coils. Thus, a square coil is preferable. Figure 1 presents the top and side views of the round-corner square coil with inner radius \(R_{i}\) and outer radius \(R_{o}\). There were three key dimensions to be designed in this work, the inner width \(D_{i}\), the outer width \(D_{o}\), and the height \(H\). The coil was designed with the objective of this research to propose a compact and portable design of the planar maglev system. First, the inner width was obtained from the width of the coil bobbin spool. The bobbin spool was made of a 31.75 mm (1-1/4-inch) aluminum square tube welded on a 76.2 mm (3-inch) aluminum square base flange. The advantages of using the hollow aluminum bobbin are a good machinability, an efficient heat dissipation from the coil, and a slight influence on the magnetic field generated by the coil. Next, the outer width was determined by the thickness of the winding. The winding was assumed to be evenly distributed with the orthocyclic winding pattern as shown in Figure 1b, which ideally provides an optimal fill factor of 90.7% [31]. In this work, an insulated magnet wire gauge of 16 AWG with a diameter \(d\) of 1.31 mm was used to make the coils with a maximum current of 7 A. The winding thickness can be calculated from the magnet wire diameter and the number of radial layers but must not exceed the width of the bobbin base flange. By considering a 10% tolerance from the material dimension and winding process, the number of radial layers of 16 was selected. As a result, the total outer width of the coil was approximately 75 mm. Since the inner and outer widths of the coil were decided, the height of the coil was designed to achieve an optimal energy efficiency and levitation effort. In terms of energy use, the Fabry factor was used in this work. This factor considers the coil geometry to evaluate the efficiency in generating a magnetic field. Even though the factor was derived from the cylindrical coil, it was adopted in this initial design since the square coil in this work had a large corner radius which resembled that of a cylindrical coil. The Fabry factor \(G\) of the uniform-current-density coil was expressed in [32] as  
\[G(\alpha,\beta)=\frac{1}{5}\sqrt{\frac{2\pi\beta}{\alpha^{2}-1}}\,\ln\left( \frac{\alpha+\sqrt{\alpha^{2}+\beta^{2}}}{1+\sqrt{1+\beta^{2}}}\right), \tag{1}\]  
where \(\alpha=D_{o}/D_{i}\) and \(\beta=H/D_{i}\). From the designed outer and inner widths, the ratio of the coil's outer-inner width was \(\alpha=2.36\). The remaining variable \(\beta\) depended on the height of the coil. With the designed \(\alpha\) value, the best \(G\) value this design could achieve was 0.175, where \(\beta=1.59\). Therefore, the optimal coil height from the Fabry factor was 50.48 mm, which was equivalent to 35 vertical winding turns or 50.435 mm. Simulations of a square and a cylindrical coil with parameters in Table 1 were conducted using finite element analysis software ANSYS Maxwell. Both coils were comparable in size with the designed inner-outer widths and optimal height from the Fabry factor. The comparison of the z-component of the magnetic flux density \(B_{z}\) between the two coils is shown in Figure 2. The magnetic flux density is observed along the diagonal distance from center to corner at different levels from 18 mm to 26 mm above the coil top surface. The square coil could generate a stronger magnetic flux density than the cylindrical coil, especially at the corner, with higher percentage differences as shown in Figure 3. With an average difference of 18%, the height of the square coil was determined to attain at least 82% of the best achievable \(G\) value (\(G\geq 0.144\)). Accordingly, the range of \(\beta\) was \(0.60\leq\beta\leq 4.22\), and the range of the coil height was 19.05 mm \(\leq H\leq\) 133.98 mm.