# C Band 360\({}^{\circ}\) Triangular Phase Shift Detector for Precise Vertical Landing RF System
## 3 Circuit Design
### Experimental Characterization  
Once the generators are calibrated in the operating frequency (5.3 GHz) [42; 46], the detector curves are obtained for each of the combinations in Table 2. The generators are connected to two inputs and the third one is loaded with 50 \(\Omega\). This operation is performed with the three possible combinations: 1-2, 2-3, and 3-1. In each connection, the phase of one of the reference generators is modified and the detector output data is taken. This operation is repeated to obtain the curves Q \(\times\) Q (in phase) and I \(\times\) Q (shifted phase). Figure 11 shows the six curves corresponding to the IQ-\(\beta\) products.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Switches**} & \multicolumn{3}{c}{**Output Voltage of Phase Detector**} \\ \hline
**S4** & **S5** & **S6** & \(\Delta\theta_{12}\) & \(\Delta\theta_{23}\) & \(\Delta\theta_{31}\) \\ \hline
0 & 0 & 0 & \(V_{12\text{QQ}}\) & \(V_{23\text{QQ}}\) & \(  
The curves have a sinusoidal aspect, so a piecewise characterization is not considered [42]. Equation (4) is the function selected to model the detector behavior, and by means of a simple least squares algorithm, the values shown in Table 3 are obtained. Figure 12 shows the best and worst case error curves (difference between modeled equations and data), which do not exceed 0.15 V error.  
\[DC+A\cdot cos(wt+d_{phasedeg}\cdot\pi/180) \tag{4}\]  
However, what is important is to know how these amplitude errors are transformed into phase errors, which will be the difference between the measured and calculated values for each pair of \(V_{dI}\) and \(V_{dQ}\) values. The calculation of the phase errors is performed from the IQ-\(\beta\) voltages (six voltages corresponding to \(V_{dI}=V_{ijQQ}\) and \(V_{dQ}=V_{ijIJQ}\) for \(i=1,2,3\) and \(j=2,3,1\)) and the modeled sinusoidal curves of each detector (Table 3), as indicated in Section 2.3 (Figure 6b). Figure 13 shows the phase errors for each detector, which do not  
Figure 11: IQ-\(\beta\) data curves for each of the three detectors at 5.3 GHz, according to the different switch positions detailed in Table 2.  
Figure 10: Detail photograph of one of the circuit branches, including the location of the most significant RF components.  
exceed \(\pm 4^{\circ}\) in any case. In fact, the phase shifts provided by curves 1-2 and 2-3, which do not exceed \(\pm 3^{\circ}\), can be used to obtain the third one, taking into account that the sum of the phase shifts is zero [25]. This error translates into 1.7% losses (3/180%) over the detector theoretical range in the worst case. That is, the unambiguous range of phase detection is reduced to \(\pm 177^{\circ}\).  
The bandwidth of the circuit was obtained from the minimum input power that stabilizes the reference point power to 0 dBm (Figure 7), i.e., when the AGC is limiting the  
Figure 12: Amplitude errors in the modeled detector curves at 5.3 GHz. (**a**) Best case corresponding to the 1 \(\times\) Q curve and inputs 2-3. (**b**) Worst case corresponding to the 1 \(\times\) Q curve and inputs 3-1.  
Figure 13: Phase errors calculated from the detector curve models and the original data used for modeling at 5.3 GHz.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{DC}\) & \(\mathbf{A}\) & \(\mathbf{d_{phase_{\mu x}}}\) & **Curve** & **Inputs** \\ \hline
2.3169 & 2.0173 & 243.5288 & \(Q\times Q\) & 1-2 \\ \hline
2.3044 & 1.8025 & 127.1475 & \(I\times Q\) & 1-2 \\ \hline
2.3161 & 2.0857 & 84.8720 & \(Q\times Q\) & 2-3 \\ \hline
2.3444 & 1.8893 & 320.1708 & \(I\times Q\) & 2-3 \\ \hline
2.3636 & 1.9995 & 343.6281 & \(Q\times Q\) & 3-1 \\ \hline
2.3558 & 1.7404 & 214.9538 & \(I\times Q\) & 3-1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of the function modeling the phase detector curves.  
signal. Table 4 represents the measured powers for each frequency, showing that the 3 dB bandwidth extends from 4.8 GHz to 5.6 GHz.