# 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 equivalent circuit depicted in Figure 2 can be simplified to the approximate equivalent circuit shown in Figure 3.  
The reactive power equality relationship gives the magnetizing reactance as in Equation (6):  
\[x_{m0}=\frac{4m_{1}\mu_{0}(W_{1}k_{w1})^{2}aV_{s}}{\pi k_{\delta}k_{\mu}\wp_{e}} \tag{6}\]  
where \(m_{1}\) is the number of phases in the power source, \(\mu_{0}\) is the vacuum permeability, \(W_{1}\) is the number of turns per phase connected in series in the weft insertion guideway winding, \(k_{w1}\) is the winding factor, \(k_{\delta}\) and \(k_{\mu}\) are the gap factor and saturation factor of the weft insertion guideway, \(\delta\) is the gap thickness between the weft insertion guideway and the conducting plate, and \(p_{e}\) is the equivalent number of pole pairs. Based on the principle of equal complex power in the field path and [PERSON]'s law, the equivalent resistance of the conductor plate assigned to the weft insertion guideway is obtained as follows:  
\[r^{\prime}{}_{2}=4m_{1}\rho_{2}\frac{(W_{1}k_{w1})^{2}}{2p_{e}}\frac{2a}{\tau d} \tag{7}\]  
where \(\rho_{2}\) is the volume conductivity of the conductor plate and \(d\) is the thickness of the conductor plate. The starting internal power factor refers to the internal power factor of the weft insertion system during the startup process, reflecting the power factor condition of the internal components of the weft insertion system. Its expression is as follows:  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]0  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]1 Weft insertion guideway and conductive plate approximate equivalent circuits. The primary per-phase resistance is given by Equation (9):  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]2  
where \(\rho_{1}\) is the resistivity of the guideway coil winding, \(L_{cp}\) is the average half-turn length of the winding, and \(A\) is the cross-sectional area of the winding. The per-phase leakage inductance of the guideway coil winding is given by Equation (10):  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]3  
where \(f\) is the power supply frequency, \(q_{1}\) is the number of slots per phase per pole of the guideway, \(p\) is the pole pair number, and \(\lambda_{s},\lambda_{t},\lambda_{e},\) and \(\lambda_{d}\) are the leakage magnetic conductances for the slot, tooth end, winding end, and harmonic, respectively. Letting \(k_{\mu}=1\), by combining Equations (6)-(8), we can derive the formula for calculating the thickness of the conductive plate as given in Equation (11):  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]4  
Also, since the synchronous speed of the warp insertion guideway is \(V_{S}=2f\tau\), the above equation can be expressed as a linear equation in terms of \(\delta\) as  
\[C_{r}(s)=\frac{sG\Big{\{}\text{Re}^{2}[T]+\text{Im}^{2}[T]\Big{\}}}{\text{Re} [T]} \tag{1}\]5  
The above equation indicates that, in the case where other parameters of the weft insertion system have been determined, the thickness can be expressed as having a proportional relationship with the air gap. In order to obtain the maximum traction force, the thickness of the conductive plate is determined only by the gap thickness between the warp insertion guideway and the shuttle. Moreover, the selection of the conductive plate thickness should monotonically increase with the increase in gap thickness.