## 3 Influence of Stator Configuration on Thrust Force  
### Goal and Method of Comparative Design  
The preliminary design goals of the proposed actuator are to maximize its thrust force density and to minimize the ripple of thrust force. Copper loss and core loss decreased correspondingly and the efficiency improved. The thrust force per area \(f_{\rho}\), and the thrust force ripple \(f_{\sigma}\), which greatly influence the operating capability of the LOA [10; 11], are defined as  
\[f_{\sigma}=\frac{F_{\rm max}-F_{\rm min}}{F_{\rm max}+F_{\rm min}}\times 100\%, \tag{1}\]  
Figure 2: Topology of short stroke single-phase quasi-Halbach PM linear oscillating actuators (LOA). (**a**) C-core with one slot in stator; (**b**) E-core with two slots in stator. \[f_{\sigma}=\frac{F_{\rm max}-F_{\rm min}}{F_{\rm max}+F_{\rm min}}\times 100\%, \tag{2}\]  
Here, \(F_{\rm avg}\) is the average thrust force over the stroke, \(\tau_{p}\) is the width of the unit actuator, \(b_{e}\) is the added width of each side of the stator teeth, and \(L_{fe}\) is the length of stator core. \(F_{\rm max}\) is the maximum force over the stroke, and \(F_{\rm min}\) is the minimum thrust over the stroke. The aim of the comparative design is to find a good balance between the maximum of the thrust force density and the minimum of the thrust force ripple, and is also to control the core loss. According to the theory of electrical machine design, the air-gap shear stress is  
\[\sigma\approx A\times B, \tag{3}\]  
Here, \(A\) is the electrical loading of a machine, and \(B\) is the magnetic loading. The permanent magnet can be designed to improve the magnetic loading [8]. Increasing the electrical loading \(A\) can improve the force density of a machine. At the same time, in order to restrict the temperature rise, the force density of a machine is affected by the heat loading \(q\)  
\[q\approx A\times J, \tag{4}\]  
Here, \(J\) is current density of coil. In air-cooled machines, the product \(AJ\) ranges from 100 A\({}^{2}\)/mm\({}^{3}\) to 400 A\({}^{2}\)/mm\({}^{3}\), in the case of direct water cooling, \(AJ\) can reach 2000 A\({}^{2}\)/mm\({}^{3}\). Because electrical loading \(A\) appears in shear stress and heat loading at the same time, choosing a small current density and a large electrical loading can improve the thrust force density of LOA. The current density was kept at \(J\) = 4.5 A/mm\({}^{2}\) and electrical loading at \(A\) = 90 A/mm, so the product \(AJ\) was 405 A\({}^{2}\)/mm\({}^{3}\), which is a common value in electrical machines, and the LOA can operate in air-cooled mode. The cross-section of the designed LOA is shown in Figure 3, and the design parameters are listed in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameter** & **Value** \\ \hline \hline Stator core width \(2\tau_{p}\) (mm) & 64 \\ Stator core length \(L_{fe}\) (mm) & 50 \\ Air-gap length \(g\) (mm) & 2 \\ Magnet thickness \(h_{m}\) (mm) & 10 \\ Magnet (N38 SH) remanence \(B_{r}\) (T) & 1.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the E-cored LOA  
Figure 3: Cross-section of the E-cored LOA. The total current is distributed in the slot of LOA, and the needed slot area is as follows  
\[S=\frac{A\times\tau_{p}}{J\times k_{f}}, \tag{5}\]  
Here, \(k_{f}\) is the slot-fill factor and was kept to be 0.67, so the slot area was 960 mm\({}^{2}\). As shown in Figure 3, the slot area was defined as  
\[S=\frac{1}{2}[(b_{0}+b_{1})h_{1}+(b_{1}+b_{2})h_{2}], \tag{6}\]  
The area of the slot was kept constant, and the design parameters of the stator core were kept within a reasonable range to produce an optimal combination of parameters that could maximize the thrust force density and keep the thrust force ripple at a low level, the copper loss will keep low at the same time. Slot height \(h_{s}\) is firstly considered during the optimization since \(h_{s}\) has the most considerable influence on thrust force. \(b_{0}\) is then considered in the process to avoid tooth tips to be saturated.