## 4 3D FEM Simulation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotation Angle** & **Total Torque Output** & **Lorentz Torque Output** & **Attractive Torque Output** & **Ratio of Lorentz and** \\
**(Deg)** & **(mN\(\cdot\)m)** & **(mN\(\cdot\)m)** & **(mN\(\cdot\)m)** & **Attractive Torque Output** \\ \hline \hline \multicolumn{5}{c}{Rotate from \(-25^{\circ}\) to \(+25^{\circ}\) in a counter-clockwise direction} \\ \hline \(-25\) & 6.64 & 5.72 & 0.93 & 6.18 \\ \(-20\) & 7.26 & 5.49 & 1.77 & 3.11 \\ \(-15\) & 7.03 & 5.03 & 2.00 & 2.52 \\ \(-10\) & 6.22 & 4.39 & 1.84 & 2.38 \\ \(-5\) & 5.13 & 3.69 & 1.45 & 2.55 \\ \(0\) & 4.09 & 3.23 & 0.86 & 3.77 \\ \(5\) & 4.58 & 4.27 & 0.31 & 13.66 \\ \(10\) & 4.87 & 4.95 & \(-\)0.08 & \(-\)61.36 \\ \(15\) & 5.24 & 5.52 & \(-\)0.29 & \(-\)19.33 \\ \(20\) & 5.66 & 5.90 & \(-\)0.24 & \(-\)24.94 \\ \(25\) & 6.07 & 6.02 & 0.05 & 123.17 \\ \hline \hline \multicolumn{5}{c}{Rotate from \(+25^{\circ}\) to \(-25^{\circ}\) in a clockwise direction} \\ \hline \(25\) & \(-6.67\) & \(-5.71\) & \(-0.96\) & 5.94 \\ \(20\) & \(-7.20\) & \(-5.51\) & \(-1.69\) & 3.25 \\ \(15\) & \(-7.04\) & \(-5.02\) & \(-\)2.02 & 2.49 \\ \(10\) & \(-6.27\) & \(-4.38\) & \(-\)1.89 & 2.32 \\ \(5\) & \(-5.16\) & \(-3.68\) & \(-\)1.48 & 2.49 \\ \(0\) & \(-4.11\) & \(-3.23\) & \(-\)0.87 & 3.71 \\ \(-5\) & \(-4.62\) & \(-4.28\) & \(-\)0.35 & 12.39 \\ \(-10\) & \(-4.92\) & \Finally, from the ANSYS dynamic simulation, the response time of the rotor moving from \(-25^{\circ}\) to \(+25^{\circ}\) in a counter-clockwise direction are 0.0763 s and 0.0699 s for the original and proposed 3-DOF spherical motor, respectively. There is a better performance in response time for the proposed 3-DOF spherical motor. Detailly, the calculation process of the dynamic simulation is to compute the time step to reach the next angle by angular acceleration and angular velocity, which are calculated from the torque generated from the magnetostatic analysis. Therefore, the actual parameter setting for simulation and step size of the rotation angle is \(1^{\circ}\) in this paper. The torque and time step are the output, in which the minimum time step is 0.0007 s by calculating. In addition, the proposed 3-DOF spherical motor has a greater torque output in the large rotation angle and has an average improvement percentage of 5.22% when compared to the original 3-DOF spherical motor. In other words, the proposed 3-DOF spherical motor has a nice improvement in the large rotation angle by using the novel electromagnetic structure with a particular controlling strategy. In this paper, we mainly focus on the improvement of the torque output for the rotation around the \(X\)- and \(Y\)-axes. Avoiding the text repeatability, we ignored the detail information. Note that for a more comprehensive description of the original system, the reader is referred to [25; 26]. The main contribution of this study is to propose this design and verify its performance by using a 3D FEM. A prototype of the proposed 3-DOF spherical motor will be built to demonstrate its dynamic performance in a future work, and the driving circuit will also be designed by referring to the previous study [25; 26].