# A Linear Brushless Direct Current Motor Design Approach for Seismic Shake Tables
## 2 Design Approach
Thus, by using the linear motion equation (Equation (1)), the maximum force is calculated as 2.5 kN for 460 kg of total moving mass. \[F_{e}=m_{L}a+\beta v \tag{1}\]  
where \(F_{e}\), \(m_{L}\), \(a\), \(\beta\) and \(v\) are the stroke force, payload, acceleration, damping coefficient of the system and motion speed, respectively. Although there are rubber and spring stoppers in the seismic shake table, the spring component of the motion equation is assumed nearly zero, because the spring action is not included in the shaker motion. The initial design parameters are shown in Table 1. The crucial point taken into consideration in the sizing of the magnets is to prevent the increase of short-circuit flux lines between the two adjoining inverse polarized magnets by leaving a gap or a non-magnetic spacing between the magnets. Thus, the flux lines of PMs can be directed to the airgap. For this reason, the magnet embrace factor is determined as 85% according to the number of poles selected for no-load speed determined for the seismic shake table. The calculation of the flux concentration factor, \(C_{\phi}\), for the linear BLDC motor is shown in Equation (2). \[C_{\phi}=\frac{A_{m}}{A_{g}}=\frac{\tau_{m}L_{a}}{\tau_{p}L_{a}} \tag{2}\]  
\[H_{m}=-[\frac{1}{\mu_{0}}\frac{g}{d_{m}}\frac{A_{m}}{A_{g}}]B_{m} \tag{3}\]  
where \(\tau_{m}\), \(\tau_{p}\), \(A_{m}\), \(A_{g}\), \(\mu_{0}\), \(d_{m}\), \(L_{a}\) and \(B_{m}\) are magnet width, pole pitch, magnet area, gap area, permeability, magnet depth, magnet length (perpendicular to the plane) and magnet flux density, respectively. The flux concentration factor is calculated as 0.8772. The calculation of the \(H_{m}\)-\(B_{m}\) curve is shown in Equation (3) [30]. The tooth, core and yoke ampere-turn losses are neglected during calculations. The intersection of the curve with the PM operation line yields approximately 1.08 T by taking structural dimensions and material features into account. \(A_{g}\) is assumed slightly larger than \(A_{m}\), by the factor of 10% due to practical issues. The calculation of flux-per-pole, \(\phi_{p}\) is shown in Equation (4) [31]. \[\phi_{p}=B_{m}\tau_{m}L_{a} \tag{4}\]  
As it is seen, the flux per pole value is determined depending on the flux density. The demagnetization characteristic is obtained out of \(H_{m}\)-\(B_{m}\) curve according to the operating \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Supply Voltage (\(V_{d}\)) & 36 V \\ Maximum Targeted Force (\(F_{max}\)) & 2.5 kN \\ Motion Speed (\(v\)) & 0.6 m/s \\ Number of Phase (\(N_{ph}\)) & 3 \\ Number of Magnet Poles Per Three Phase Windings (\(N_{m}\)) & 6 \\ Number of Slots-Per-Pole-Per-Phase (\(N_{spp}\)) & 0.5 \\ Number of Slots-Per-Phase (\(N_{sph}\)) & 4 \\ Slot/Pole Ratio (\(N_{sp}\)) & 1.5 \\ Air Gap Length (\(g\)) & 1 mm \\ Magnet Length (\(L_{a}\)) & 80 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial design parameters. point (\(B_{m}=-10.027\times 10^{-6}H_{m}\)) and is chosen as 1.1 T. The tooth flux density, \(B_{t}\), is obtained using Equation (5). \[B_{t}=\frac{\phi_{p}}{N_{sp}L_{a}w_{sb}} \tag{5}\]  
where \(w_{sb}\) is the tooth width. The value of \(w_{sb}\) is targeted as 7.7 \(\times\) 10\({}^{-3}\) m and the value of \(B_{t}\) is calculated as 1.904 T which can be provided by high quality steels. Flux density at back-iron, \(B_{bi}\), the tooth flux density in the core, \(B_{yoke}\), and air gap flux density, \(B_{g}\), are calculated, respectively, in Equations (6)-(8). \[B_{bi}=\frac{\phi_{p}/2}{L_{bi}d_{bi}} \tag{6}\]  
\[B_{yoke}=\frac{\phi_{p}/2}{L_{a}d_{yoke}} \tag{7}\]  
\[B_{g}=B_{m}C_{\phi} \tag{8}\]  
where \(L_{bi}\) is the back-iron length. In Equation (6), the flux density at back-iron, \(B_{bi}\), is calculated depending on the depth of back-iron, \(d_{bi}\) value.