## 4 Analysis of Electromagnetic Weft Insertion Characteristics
### Electromagnetic Force Analysis  
From the point of view of micro-elements, the weft gripper is regarded as a small cylindrical element segment with a radius of \(r\) and a height of \(\Delta h\). Because the net magnetic  
Figure 6: Schematic diagram of acceleration coil. Figure 7: Magnetic induction intensity inside the coil. flux of the magnetic field emitted by the object is zero, each small element segment has radial and axial flux; the radial flux is recorded as \(2\pi\pi\Delta hB_{j}\), and the axial flux is recorded as \(\pi\pi^{2}[-B(x)+B(x+\Delta h)]\). Take the first-level approximation to the height of the micro-element section \(\Delta h\) as:  
\[\pi r^{2}\frac{\partial B}{\partial x}\Delta h \tag{5}\]  
If the total magnetic flux is zero, we can get:  
\[\pi r^{2}\frac{\partial B}{\partial x}\Delta h+2\pi r\Delta hB_{j}=0 \tag{6}\]  
Simplified:  
\[B_{j}=-\frac{r}{2}\frac{\partial B}{\partial x} \tag{7}\]  
Use the gradient of the magnetic field B to express the force acting on the dipole:  
\[F=\frac{2\pi rI}{c}\cdot\frac{r}{2}\frac{\partial B}{\partial x}=\frac{\pi r ^{2}I}{c}\cdot\frac{\partial B}{\partial x} \tag{8}\]  
The circuit with the product of the same current and area produces the same magnetic field at a distance. The product \(la/c\) is called the magnetic dipole moment of the current circuit, and it is represented by \(m\). The magnetic dipole moment is a vector, and the direction is the normal direction of the circuit. In other words, it is the direction of vector \(a\), which is the area of the direction surrounded by the loop. \[m=\frac{I}{c}a \tag{9}\]  
where \(I=Mcdx\). The unit of \(I\): electrostatic unit/second; the unit of \(a\): square centimeter (cm\({}^{2}\)); the unit of \(c\): centimeter/second (cm/s); and the unit of \(m\): erg (\(10^{-7}\) J)/Gauss (\(10^{-4}\) T). If a piece of material with a thickness of \(dx\) is cut perpendicular to the magnetization direction and divided into small micro-elements, the cross-sectional area of each small micro-element is \(ds\). Since \(M\) is the dipole moment per unit volume, each small micro-element contains the total dipole moment \(Mdsdx\). The magnetic dipole moment can be further simplified as:  
\[m=\frac{I}{c}a=\frac{Mcdx}{c}ds=Mdsdx=\chi_{m}Hdsdx \tag{10}\]  
The factor \(\pi r^{2}I/c\) is the value \(m\) of the magnetic dipole moment of the current loop, so Equation (8) is further simplified as:  
\[F=m\frac{\partial B}{\partial x} \tag{11}\]  
\[F_{x}=m\cdot gradB_{x} \tag{12}\]  
Combining Equations (10) and (12) together, we get:  
Figure 7:0  
The unit of \(F\) in the above two formulas: dyne (\(10^{-5}\) N); the unit of magnetic field gradient: Gauss/cm (G/cm). InFigure 7:1 Figure 7:2 Figure 7:3  
The electromagnetic force curve on the center axis of the coil is shown in Figure 7:4 After the gripper enters the accelerating coil, the electromagnetic force gradually increases. When the center of the gripper and the center of the coil are about to coincide, that is, when the top of the weft gripper is about to fly out of the accelerating coil, the electromagnetic force decreases sharply to zero and increases in the opposite direction. As the weft gripper flies forward, the end of the weft gripper flies away from the coil, and the electromagnetic force on the weft gripper continues to decrease until it reaches zero.