# A Linear Brushless Direct Current Motor Design Approach for Seismic Shake Tables
## 3 Electromagnetic Analysis and Results  
A linear BLDC motor design involves a number of assumptions and constraints, where the final design parameters are generally obtained by using the FEM package in the Cartesian coordinates [32]. The magnetic material properties are input to the software including their nonlinearities. By means of automatic mesh generation, the vector potential values are calculated for the different motor parts. The current sheet (\(J\)) in the windings located in the slots is,  
\[\overrightarrow{J}(t)=J(t)\overrightarrow{k} \tag{16}\]  
For 2D magnetic field analysis, the flux density, \(\overrightarrow{B}\), and field intensity of \(\overrightarrow{H}\) are related as follows,  
\[\overrightarrow{H}=\frac{1}{\mu}\overrightarrow{B}-\frac{1}{\mu} \overrightarrow{M_{0}} \tag{17}\]  
where \(\mu\) and \(M_{0}\) are the magnetic permeability of the medium and the magnetization vector of the PMs, respectively. The magnetic properties of the PMs and the back iron, which is used for transferring the flux between PMs, are single values. Thus, the [PERSON]'s nonlinear equation is satisfied as,  
\[\frac{\partial}{\partial x}\frac{1}{\mu}\frac{\partial A}{\partial x}+\frac{ \partial}{\partial y}\frac{1}{\mu}\frac{\partial A}{\partial y}=-(J+J_{m}) \tag{18}\]  
where \(A\) and \(J_{m}\) are the vector potential and the current density, corresponding to the magnets. \(J_{m}\) is given as follows,  
\[J_{m}=rot_{z}(\mu^{-1}\overrightarrow{M_{0}})=\frac{\partial(\mu^{-1} \overrightarrow{M_{0y}})}{\partial x}\frac{\partial(\mu^{-1}\overrightarrow {M_{0x}})}{\partial y} \tag{19}\]  
The calculated longitudinal force can be found by using the volumetric integral of the vector product of the current sheet, \(J\), and the flux density, \(B\), as given in Equation (20). \[\overrightarrow{F}=\int_{v}\overrightarrow{J}\times\overrightarrow{B}\cdot dV \tag{20}\]  
By using FEM, the effect of each parameter on the dynamic performance of the motor is investigated. If the output dataset is considered in Section 2, all calculations are well suited for the requirements of the seismic shake table driving. After completing the analysis of the designed linear BLDC motor, a detailed electromagnetic FEA was made to provide proper data to manufacturing process. A two-dimensional magnetic analysis using Ansys Maxwell was implemented by means of a 2D model of the designed motor, which is shown in Figure 3. 2D FEA shows that the optimized motor has admissible magnetic flux airgap density (approximately 0.8 T) that allows the proper force production with the back-iron fully covered. Afterwards, as the 2D model was built to validate the overall design, the sketch of 3D linear BLDC motor, which is shown in Figure 4, was created by using a physical modeling environment. Moreover, a 3D magnetic analysis using ANSYS Maxwell was performed with an adaptive mesh option, which is presented in Figure 5. To have realistic solutions, all parts used for assembly were taken into account according to their magnetic properties. Finally, the coils were sketched and the model was prepared for FEA. In total, 148,332 elements were used for FEA calculations. In the mo \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Linear BLDC Motor Part** & **Length** & **Thickness** & **Height** \\ \hline Primary & 182.5 mm & 80 mm & 25 mm \\ Back Iron & 1590 mm & 80 mm & 5 mm \\ Magnets & 20 mm \(\times\) 65 mm & 80 mm & 5 mm \\ Slot & 7.7 mm & 80 mm & 15 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fundamental parameters of designed linear BLDC motor. motion performance of the motor, a sinusoidal function for speed is given for having the motor moves until the position of 338 mm and goes back to the initial state to define the motion mechanism of the motor. In the analyses, the motor speed was set as 0.6 m/s. In Figure 6, the armature teeth have a little more than 2.5 T flux density, which implies the presence of saturation. As can be seen from the magnetic flux density, a heavy saturation occurs in the back iron.