## 6 Design of a Feed Network  
To cover the worldwide GNSS frequencies, a feed network for the crossed-dipole antenna should provide two balanced outputs with a \(90^{\circ}\) phase difference over a broad bandwidth. In this work, we employ a three-branch line microstrip quadrature hybrid coupler (QHC), which is printed on the backside of the circular ground plane. The use of the QHC enables the antenna to be used for the LHCP \(L\)-band satellite phone (1.525-1.661 GHz) as well.  
Figure 12: Axial ratio patterns of the crossed-dipole antenna at (**a**) 1.16 GHz, (**b**) 1.30 GHz, and (**c**) 1.61 GHz.  
The two-branch line coupler has a bandwidth of 15-20%, which is not sufficient to cover 1.16-1.61 GHz (32.5% bandwidth) [38], so we employ a three-branch line coupler. Starting with an unreduced-size QHC shown in Figure 12(a), a reduced-size QHC shown in Figure 12(b) is designed. The QHC of Figure 12(a) is designed at the center frequency of 1.39 GHz for Butterworth response with the following theoretical parameter values [39]; \(Z_{1}=120.5\)\(\Omega\), \(Z_{2}=37.2\)\(\Omega\), \(Z_{3}=36.3\)\(\Omega\), \(L_{1}=0.249\)\(\lambda\), \(L_{2}=0.264\)\(\lambda\), and \(L_{3}=0.270\)\(\lambda\), where \(\lambda\) is the wavelength of the corresponding microstrip lines. This design gives 2.4 dB and 3.3 dB differences in transmission coefficients at 1.16 GHz and 1.61 GHz, respectively. To improve the magnitude balance, line widths and lengths are optimized, resulting in the following values: \(Z_{1}=99.7\)\(\Omega\), \(Z_{2}=31.2\)\(\Omega\), \(Z_{3}=62.0\)\(\Omega\), \(L_{1}=0.253\)\(\lambda\), \(L_{2}=0.265\)\(\lambda\), and \(L_{3}=0.261\)\(\lambda\). The improved design shows 0.33-0.38 dB in amplitude balance and 88.9\({}^{\circ}\)-90.5\({}^{\circ}\) in phase difference at 1.16-1.61 GHz.  
The reduced QHC shown in Figure 12(b) is obtained from the optimized design of the QHC shown in Figure 12(a) by meandering the main lines so that the coupler's size can be fitted within the boundary of the crossed-dipole antenna's ground plane. The line widths and lengths are readjusted after meandering. In Figure 12(b), Port 1 and Port 4 are the RHCP and LHCP inputs, respectively, while Port 2 is the coupled output port (connected to the Dipole \(B\) feed line) with a 0\({}^{\circ}\) phase reference, and Port 3 is the coupled output (connected to the Dipole \(A\) feed line) with a \(-90^{\circ}\) phase with respect to the 0\({}^{\circ}\) phase reference. Note that the QHC shown in Figure 1 is the one seen from the top side of the ground plane, while the QHC of Figure 12(b) is the one seen from the back side of the ground plane. We can call Port 2 and Port 3 the in-phase port and the quadrature-phase port, respectively.  
Figure 12: Structure of a three-branch line coupler (**a**) before size reduction and (**b**) after size reduction.  
Table 4 shows the dimensions of the designed QHC. The multi-branch line QHC is notorious for very narrow branch lines resulting in difficulties in accurate fabrication. In our design of QHC, the smallest line width (\(W_{1}\)) is 0.36 mm, which can be realized using standard PCB fabrication techniques.  
Figure 14a shows the reflection and transmission coefficients of the designed QHC. The reflection coefficient (S1,1) at the input Port 1 is less than \(-17\) dB at 1.07-1.75 GHz and is \(-28.9\) dB and \(-26.2\) dB at 1.16 GHz and 1.61 GHz, respectively. The transmission from the input port to the isolated port (S4,1) closely tracks the reflection coefficient. The transmission from the input port to the in-phase port (S2,1) and to the quadrature-phase port (S3,1) is \(-3.80\) dB and \(-3.44\) dB at 1.16 GHz (0.36 dB imbalance) and \(-3.54\) dB and \(-3.97\) dB at 1.61 GHz (0.43 dB imbalance). Figure 14b shows the phase difference between the in-phase output and the quadrature phase output (Arg (S2,1)-Arg (3,1)). The transmission phase difference is \(89.3^{\circ}\) and \(90.8^{\circ}\) at 1.16 GHz and 1.61 GHz, respectively (\(\pm 0.8^{\circ}\) imbalance).