# Magnetic Field Analysis and Optimization of the Gauge of Hybrid Maglev Needles
## 3 Performance Analysis of Hybrid Maglev Needle Drive  
### Mathematical Model of the Magnetic Force for Knitting Needle Gauge  
In the structure of the needle plate, the close arrangement of the permanent magnet needles leads to magnetic disturbance forces between the permanent magnets. One needle is driven by an electromagnet, and the adjacent needles follow the moving needle. It is very  
Figure 1: Driving structure diagram of hybrid maglev needle array.  
difficult to control every needle independently. In order to study the magnetic disturbance between permanent magnet knitting needles, the disturbance force between permanent magnet knitting needles is simplified to the magnetic disturbance force between permanent magnets [16]. On the basis of the magnetic dipole method, the simplified magnetic force model of a permanent magnet needle can be obtained as shown in Figure 2.  
The magnetic flux density generated by B at A is:  
\[\textbf{B}_{BA}=-\frac{\mu_{0}}{4\pi}\
abla\frac{\textbf{m}_{B}\cdot\textbf{r}_{ BA}}{\left|\textbf{r}_{BA}\right|^{3}} \tag{1}\]  
where \(\mu_{0}\) is the vacuum permeability, \(\
abla\) is the vector gradient, \(\textbf{m}_{A}\) and \(\textbf{m}_{B}\) are the magnetic moments of the permanent magnet, and \(\textbf{r}_{BA}\) is the direction vector from the permanent magnet B to permanent magnet A. \(\textbf{m}_{A}\), \(\textbf{m}_{B}\) and \(\textbf{r}_{BA}\) are, respectively, expressed as:  
\[\textbf{m}_{A}=M_{A}V_{A}\cos\theta\cdot\textbf{i}+M_{A}V_{A}\sin\theta\cdot \textbf{j} \tag{2}\]  
\[\textbf{m}_{B}=M_{B}V_{B}\cdot\textbf{j} \tag{3}\]  
\[\textbf{r}_{BA}=-x\cdot\textbf{i}+Y\cdot\textbf{j} \tag{4}\]  
where \(M_{A}\) and \(M_{B}\) are the magnetization of the permanent magnet, \(V_{A}\) and \(V_{B}\) are the volume of the permanent magnet, \(\theta\) is the deflection angle of the permanent magnet A, \(x\) is the center distance between the permanent magnet A and permanent magnet B, \(Y\) is the vertical distance between the permanent magnet A and permanent magnet B, and \(\textbf{i}\) and \(\textbf{j}\) are vector coordinates. The potential energy between two permanent magnets is:  
\[\textbf{U}_{m}(x,Y,\theta)=-\textbf{m}_{A}\cdot\textbf{B}_{BA} \tag{5}\]  
Substituting Equations (1)-(4) into Equation (5) results in:  
\[\textbf{U}_{m}(x,Y,\theta)=(M_{A}V_{A}\cos\theta\cdot\textbf{i}+M_{A}V_{A}\sin \theta\cdot\textbf{j})\cdot\left[\frac{\mu_{0}}{4\pi}\
abla\frac{(M_{B}V_{B} \cdot\textbf{j})\cdot(-x\cdot\textbf{i}+Y\cdot\textbf{j})}{(\sqrt{x^{2}+Y^{2} })^{3}}\right] \tag{6}\]  
According to Equation (6), the smaller the center distance \(x\) and vertical distance \(Y\) between the two permanent magnets, the greater the magnetic potential between the two permanent magnets. The deflection angle caused by the movement of the permanent magnet needle in the needle slot is \(90^{\circ}\). When \(\theta=90^{\circ},\ Y=0\). Therefore, Equation (6) can be simplified as:  
\[U_{m}(x)=-\frac{\mu_{0}}{4\pi}\frac{M_{A}V_{A}M_{B}V_{B}}{x^{3}} \tag{7}\]  
Figure 2: Simplified model of interaction between two permanent magnet needles.  
Based on Equation (7), the magnetic force of mutual interference between permanent magnetic needles is:  
\[F_{m}=U_{m}^{\prime}(x)=\frac{3\mu_{0}}{4\pi}\frac{M_{A}V_{A}M_{B}V_{B}}{x^{4}} \tag{8}\]  
The relationship between residual magnetic induction and residual magnetization meets the following requirements:  
\[B_{r}=\mu_{0}M_{r} \tag{9}\]  
Equation (8) is transformed into:  
\[\Gamma_{m}=\frac{3}{4\pi\mu_{0}}\frac{B_{rA}B_{rB}V_{A}V_{B}}{x^{4}} \tag{10}\]  
According to Equation (10), the magnetic force between permanent magnets is inversely proportional to the Fourth Square of the center distance of permanent magnet needles. A dense needle pitch will produce a large interference magnetic force.