# Insertion Performance Study of an Inductive Weft Insertion System for Wide Weaving Machines
## 3 Calculation of Weft Insertion Performance Parameters  
### Equivalent Circuit of the Weft Insertion System  
The driving part of the weft insertion system consists of the guideway and the conductive plate, with the permanent magnet array installed on both sides of the guideway. The impact of the permanent magnet array on the weft insertion system's performance can be neglected. Therefore, when studying the weft insertion system's performance, it is sufficient to focus on the guideway and the conductive plate and establish an equivalent circuit. The electromagnetic traction coils are wound around the core of the weft insertion guideway in a \"half-filled slot\" manner, and there is an end effect at both ends of the core. In situations where the thickness of the conductive plate and the air gap are relatively large, the impact of the leakage reactance and skin effect of the conductive plate cannot be ignored. Let \(K^{\prime}_{r}(s)\) and \(K^{\prime}_{x}(s)\) be correction factors for the resistance and magnetizing reactance of the conductive plate due to the longitudinal dynamic end effect, and let \(C_{r}(s)\) and \(C_{x}(s)\) be correction factors for the resistance and magnetizing reactance of the conductive plate due to the transverse end effect, respectively. \(K_{f}\) is the correction factor for the resistance of the conductive plate due to the skin effect. The equivalent circuit of the weft insertion guideway and the conductive plate can be represented as shown in Figure 2.  
In this figure, \(r_{1}\) is the resistance of the electromagnetic traction coil, \(x_{1}\) is the leakage reactance of the electromagnetic traction coil, \(x_{m0}\) is the magnetizing reactance, \(r^{\prime}_{2}\) is the resistance of the conductive plate referred to the electromagnetic traction coil, and \(x_{2}\) is the leakage reactance of the conductive plate. In one-dimensional field analysis [13], we can separate the longitudinal and transverse end effects and then superimpose the separately processed results to obtain the analytical result that includes the total end effects. At the same time, according to the relationship of equal complex power in the field circuit, we can obtain the equations for \(K^{\prime}_{r}(s)\), \(K^{\prime}_{x}(s)\), \(C_{r}(s)\), and \(C_{x}(s)\). These are all equations about the slip ratio, and when the slip ratio is very small, \(C_{r}(s)\) and \(C_{x}(s)\) can be written separately as  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]  
\[C_{x}(s)=\frac{\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Im}[T]} \tag{2}\]  
In the above equation,  
\[T=j\bigg{[}r^{2}+(1-r)\frac{\lambda}{\alpha a}\text{tanh}\alpha a\bigg{]} \tag{3}\]  
\[r^{2}=\frac{1}{1+j\text{s}G} \tag{4}\]  
Figure 2: Equivalent circuit of weft insertion guideway and conductive plate. \[\lambda=\frac{1}{1+\frac{1}{r}\text{tanh}ha\cdot\text{tanh}k(c-a)} \tag{5}\]  
where \(G\) is the quality factor, \(2a\) is the width of the weft insertion guideway core, \(2c\) is the transverse length of the conductive plate, \(\alpha=\frac{c}{r},k=\frac{\pi}{\tau}\), and \(\tau\) is the polar distance of the weft insertion guideway. In the process of warp insertion, the angle between the upper warp and the lower warp is small, and the space available for installing the warp insertion guideway is limited. Therefore, the gap thickness of the designed guideway and the thickness of the conductive plate in this paper are relatively small. Considering the uniformity of the conductive plate [14], the influence of secondary leakage inductance and the skin effect of the secondary conductor on the electromagnetic warp-insertion guideway characteristics can be neglected. Moreover, during the warp-insertion motion, the shuttle primarily operates in the central portion of the warp-insertion guideway. At this point, the influence of end effects can also be neglected, i.e., setting \(K_{r}=K_{x}=C_{x}=K_{f}=1\) and \(x_{2}=0\).