## 4 Prototype Experiment and Results  
### Resistance and Inductance Test  
The model and prototype of the square coil used in the proposed planar motor is shown in Figure 10. The actual sizes of coil and the conductor gauge is given in Table 1.  
Figure 10: Model and prototype of the square coil: (**a**) model and sizes; (**b**) actual square coil.  
Figure 9: Acceleration versus parameters \(\gamma\) and \(\lambda\): (**a**) with different \(\lambda\); (**b**) with different \(\gamma\).  
The resistance calculation formula for the square coil is  
\[R=\frac{\rho N(4\times d+2\pi\frac{D-d}{4})}{\frac{1}{4}\pi d_{c1}^{2}} \tag{31}\]  
where \(\rho\) is the resistivity.  
Based on Equation (31), the analytical value of the resistance is 2.427 \(\Omega\). The actual resistance test results for the four square coils are shown in Table 2.  
The estimated value of resistance agrees well with the actual values. The error between them is less than 3.5%. The main reason for the error is that the actual shape of the square coil is slightly different from the one calculated in the model.  
The inductance calculation model for the square coil is shown in Figure 11. First, according to the Biot-Savart Law, the magnetic field from one square conductor can be obtained. Then, we can get the magnetic field generated by one square coil with current. Second, the magnetic flux for a single-turn square coil can be calculated with double integral of the magnetic field within the _xoy_ coordinate system. Finally, the total magnetic flux for the square coil with \(N\) turns can be calculated using the superposition principle. The inductance calculation formula for the square coil is  
\[L=\frac{\Psi}{I}=\frac{\mu_{0}N}{2\pi(D-d)h_{c}}\sum_{z=-nd_{c}}^{nd_{c}}\sum_{ p=0}^{m}\int_{-(\frac{d}{2}+pd_{c})}^{(\frac{d}{2}+pd_{c})}\int_{-(\frac{d}{2}+pd_{c}) }^{(\frac{d}{2}+pd_{c})}\int_{z-\delta_{c}/2}^{z+\delta_{c}/2}\int_{d/2}^{D/2} B_{z}dtdzdxdy \tag{32}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Symbol** & **Value** & **Unit** \\ \hline inner width & \(d\) & 35.76 & mm \\ outer width & \(D\) & 71.28 & mm \\ thickness & \(h_{c}\) & 9.60 & mm \\ wire diameter & \(d_{c}\) & 0.76 & mm \\ conductor diameter & \(d_{c1}\) & 0.71 & mm \\ number of turns & \(N\) & 276 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: The actual sizes of coil and the conductor gauge.  
Figure 11: Inductance calculation model for the square coil.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Coil** & **Value** & **Unit** \\ \hline A & 2.341 & \(\Omega\) \\ B & 2.348 & \(\Omega\) \\ C & 2.350 & \(\Omega\) \\ D & 2.349 & \(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The resistance test results for the four square coils.  
Based on Equation (32), the analytical value of the inductance is 4.216 mH. The actual inductance test results for the four square coils are shown in Table 3.  
The inductance error between the actual values and the estimated values is about 8%. Although the inductance error is larger compared to the resistance error, it can provide the theoretical basis for the motor design and be beneficial to the parametric modeling due to the analytical expressions.