## 2 Configuration and Design Scheme of the Feeding Structure
### Equivalent Circuit Model  
Figure 2b shows the equivalent transmission line model of the proposed SR feed network. The set of dimensions \(l\), \(w\), \(z\) in Figure 2a represents the length, width, and impedance of the branches of the feeding network. The impedances \(Z_{p1}\) to \(Z_{p5}\) are the port impedances which is equal to 50 \(\Omega\). The impedance \(Z_{0}\) to \(Z_{7}\) are the characteristic impedances of the transmission lines which are used as quarter-wave transformers. \(Z_{j1}\) to \(Z_{j4}\) are the impedances at the junctions and \(Z_{A}\) to \(Z_{C}\) are the impedances following the junction points on the circular section of the SR phase feed. The impedances \(Z_{A}^{\prime}\) to \(Z_{C}^{\prime}\) are the impedances following the junction points to the output ports and \(Z_{S}\) denotes the stub impedance. In Figure 2b, \(P_{in}\) is the total input power flow to the circuit. Each part of the SR phase feed is shown with some fraction of the input power that is based on the equal  
power flow of \(P_{in}/4\) to each of the output ports. From the basic transmission line theory, the following equations can be obtained:  
\[Z_{C}=\frac{Z_{6}^{2}}{Z_{j4}},Z_{B}=\frac{Z_{4}^{2}}{Z_{j3}},Z_{A}=\frac{Z_{2}^{ 2}}{Z_{j2}} \tag{1}\]  
To allow equal power to the output ports, the following equation can be obtained between the impedances just following the junction points:  
\[Z_{C}^{\prime}=Z_{C},Z_{B}^{\prime}=2.Z_{B},Z_{A}^{\prime}=3.Z_{A} \tag{2}\]  
The impedance at the junction points, \(Z_{j1}\) to \(Z_{j3}\) can be calculated as:  
\[Z_{j3}=\frac{Z_{C}\cdot Z_{S}}{2\cdot Z_{S}+Z_{C}},Z_{j2}=\frac{2\cdot Z_{B}}{ 3},Z_{j1}=\frac{3\cdot Z_{A}}{4} \tag{3}\]  
To provide a sequential phase at the output branches, the lengths of the branches with the impedances \(Z_{0}\), \(Z_{2}\), \(Z_{4}\), and \(Z_{6}\) play a vital role. By optimizing the lengths and widths of all the branches, equal power is delivered to all ports. All of the above equations are used for the initial values of the impedances. After that, the lengths and widths are optimized in CST for better performance. Thus, the final values of the impedances may not be the same as the initial calculated values. It should be noted that the impedances \(Z_{A}\) to \(Z_{C}\), \(Z_{A}^{\prime}\) to \(Z_{C}^{\prime}\), and \(Z_{j1}\) to \(Z_{j4}\) are not shown in the transmission line model. These are omitted by using the tapering near the junction points to reduce losses. Owing to the phase arrangement and equal power distribution capabilities, an SR phase feeding network can significantly improve the performance of the CP array antenna. Figure 3a,b depict the developed SR phase feed's simulated performances in terms of S-parameters and relative phase differences between consecutive ports with open stub. The 10 dB bandwidth covers the desired X-band, as can be seen. In terms of the transmission performances, the input power is evenly distributed through the SP-feed ports, \(P_{2}\) to \(P_{5}\). The optimal design's S_i1, where \(i\) = 2, 3, 4, 5, are close to \(-6\) dB for all ports, ensuring that equal power is transmitted to each port. The simulated transmission phase balance of the SP feed is shown in Figure 3b. The progressive phase difference between the adjacent ports is plotted against frequencies in the targeted band that are around \(90^{\circ}\). The offset phase \(5^{\circ}\) or \(10^{\circ}\) with regard to the nominal \(90^{\circ}\) can be used to specify the bandwidth of phase balancing. It is seen that the bandwidth of the \(10^{\circ}\) phase balance is ranging from 7.8 GHz to 8.425 GHz. This type of feed ensures that the array antenna achieves a good CP performance in the targeted frequency range. The stub in Figure 2a has arbitrary position. The radial length and angular width of the stub help in adjusting the phase as well as the impedance on the line. For reference, the performance of the SR phase feed without stub is shown in Figure 3c,d.