## 3 Modeling and Design Optimization
### Optimization Design
#### 3.2.2 Force Density  
As a type of short-stroke planar motor, it needs to generate forces as high as possible with a smallest volume (i.e., to achieve an optimal force density). The definition of the force density is the ratio between motor force and volume, which can be written as  
Figure 5: Side view of one coil and its relating permanent magnets.  
Figure 6: Force characteristic versus parameter \(\gamma\): (**a**) with different \(H\) when \(\tau\) = 50 mm; (**b**) with different \(\tau\) when \(H\) = 20 mm.  
\[G=\frac{F}{V} \tag{20}\]  
where \(V\) is the volume of the motor.  
To obtain the analytical force density expression, the volume expression is required. The top and side view of the planar motor is shown in Figure 7, and the motor volume, ignoring the back irons, can be obtained as  
\[V=9\tau^{2}(2h_{M}+h) \tag{21}\]  
Substituting Equations (18) and (21) into Equation (20), the force density expression of the planar motor can be derived as  
\[G=\frac{12.8\mu_{0}M_{\tau}\alpha_{i}J_{c}C_{fw}}{9\pi}\frac{X(\lambda,\gamma) }{2\gamma+1}=K_{G}\frac{X(\lambda,\gamma)}{2\gamma+1} \tag{22}\]  
where \(K_{G}\) is the force density coefficient.  
As shown in Equation (22), the force density of the planar motor mainly depends on the main parameters \(\gamma\) and \(\lambda\) due to the small variation range of the coefficient \(K_{G}\). Figure 8 shows the force density characteristics versus parameters \(\gamma\) and \(\lambda\), in which the force density variation trend can be found.  
Figure 8: Force characteristic versus parameters \(\gamma\) and \(\lambda\): (**a**) with different \(\lambda\); (**b**) with different \(\gamma\).  
Figure 7: Top and side view of the planar motor.  
In Figure 8a, when the parameter \(\lambda\) is invariant, the force density increases first and then decreases with the increase of parameter \(\gamma\). There is an optimal value of \(\gamma\) between 0.2 and 0.5. The reason is that, increasing the magnet thickness can increase the air-gap magnetic field and the force, but it also inevitably increases the motor volume. When the value of \(\gamma\) is small, the magnetic field is more sensitive to the magnet thickness rather than the volume, so the force density increases with the increase of \(\gamma\). However, the force increment will be less than the volume increment when the magnet thickness increases to a certain value, because the force tends to saturate and the volume still increases linearly. Additionally, it can be found that the force density increases with the increase of parameter \(\lambda\).  
In Figure 8b, the force density characteristic with the parameter \(\lambda\) is shown. The force density increases with the increase of parameter \(\lambda\) when the parameter \(\gamma\) is invariant, but the increasing trend gradually becomes saturated. In other words, the force density depends on the shape of the planar motor. The force density is higher in a flat shape than in a high-vertical shape. To obtain the optimal parameters for a maximum force density, the follow equation must be satisfied  
\[\frac{\partial G}{\partial\gamma}=0 \tag{23}\]