# Rapid and Precise Zoom Lens Design Based on Voice Coil Motors with Tunnel Magnetoresistance Sensors
## 3 Rapid and Precise Zoom System Design Example
### VCM Structural Design
#### 3.3.1 Magnetic Circuit Design  
According to the principle of the Lorentz force, the motor driving force \(F_{d}\) that the VCM can output is given in Equation (5). In this equation, \(B\) represents the magnetic field strength, \(I\) represents the current through the energized coil, and \(L\) represents the total length of the coil in the magnetic field. \[F_{d}=BIL \tag{5}\]  
According to Equation (5), under the constraint of the motor power, the peak current of the motor is also restricted. To meet the driving force requirements under the maximum acceleration conditions, it is necessary to optimize the dimensions of the magnets and the yoke within the limited spatial constraints to achieve the maximum magnetic field strength, thereby maximizing the electromagnetic driving force. However, for systems with enough space, the magnetic flux density \(B\) is greater with larger magnets and yokes, and a greater driving force can be provided. In this study, Zoom Group 1 is the heaviest lens group and has the longest travel distance of the two zoom groups discussed. Therefore, the design method of the VCM for Zoom Group 1 is used as a case study. The primary function of the permanent magnet is to generate a stable magnetic field, which interacts with the energized coil to produce an electromagnetic force, thereby driving the movement of the coil and the zoom group. The residual magnetic flux density \(B_{r}\), maximum energy product \(BH_{MAX}\), and coercive force \(H_{cb}\) of the magnetic material directly affect the efficiency and power output of the VCM. Higher values of these parameters indicate that the magnetic material can provide a stronger magnetic field and better magnetic stability for the same volume. Considering these characteristics, along with the operational temperature tolerance and cost, neodymium iron boron (NdFeB) 45 SH was selected as the magnetic material for this study. The yoke primarily serves to guide and concentrate the magnetic field. By utilizing high-permeability materials, the yoke effectively enhances the magnetic effect, reduces magnetic field leakage, and provides structural support, thereby improving the efficiency and performance of the VCM while protecting the external environment from magnetic interference. In selecting the yoke material, both the magnetic permeability and hardness must be considered. Taking the cost into account, low-carbon steel was chosen as the yoke material. The design of the dimensions of the magnetic circuit includes the width of the magnets and yoke (\(W_{\rm m}\) and \(W_{\rm y}\)), the thickness of the magnets and yoke (\(T_{\rm m}\) and \(T_{\rm y}\)), and the length  
Figure 6: VCM components including permanent magnet, magnetic yoke, and energized coil. of the magnets and yoke (\(L_{\rm m}\) and \(L_{\rm y}\)), as shown in Figure 7. Theoretically, the larger the width (\(W_{m}\) and \(W_{y}\)) and thickness (\(T_{m}\) and \(T_{y}\)) of the magnets and yoke, the stronger the magnetic field that they generate. However, in practical optical systems, the sizes of these components are often constrained by the available space within the lens and cannot be increased indefinitely. To avoid magnetic leakage, the widths of the magnets and the yoke should be consistent. Theoretically, a larger magnet width can provide a stronger magnetic field. However, due to the radial constraints of the lens, the maximum allowable width (\(W_{\rm m}\) and \(W_{\rm y}\)) for both the magnets and the yoke should be less than 21 mm. The primary goal in designing the thickness of the magnets and yoke is to achieve high magnetic field strength and low magnetic leakage while considering the space constraints and mechanical strength requirements. Increasing the thickness of the magnets and yoke can indeed enhance the magnetic field strength, but the marginal gain in the magnetic field strength diminishes when the thickness is beyond a certain value. Additionally, when the yoke thickness equals the magnet thickness, magnetic leakage is minimized.