# A Reflection-Type Dual-Band Phase Shifter with an Independently Tunable Phase
## 2 Design Method of Proposed Dual-Band Phase Shifter
### Analysis of Phase Shifting Range
#### 2.2.1 Phase Shifting Range at Low-Band Operation  
During \(f_{\rm L}\) operation, the \(L_{\rm c}\) is used to compensate for the unwanted effect of a \(\lambda/4\) shunt open stub, as shown Figure 3a. The reflection load of the circuit at \(f_{\rm L}\) (\(I_{\rm L}^{f_{\rm L}}\)) can be determined using Equation (7):  
\[\Gamma_{\rm L}^{f_{\rm L}}=\frac{-X_{\rm A}Z_{0}+jX_{\rm B}Z_{\rm Ho}}{X_{\rm A }Z_{0}+jX_{\rm B}Z_{\rm Ho}}, \tag{7}\]  
in which:  
\[X_{\rm A}=Z_{\rm Ho}Z_{\rm Hs}\cot\theta_{\rm Hs}-\omega L_{\rm c}Z_{\rm Ho}- \omega L_{\rm c}Z_{\rm Hs}-\omega^{2}C_{\rm v}L_{\rm c}Z_{\rm Ho}Z_{\rm Hs} \cot\theta_{\rm Hs} \tag{8a}\] \[X_{\rm B}=Z_{\rm Ho}Z_{\rm Hs}+\omega L_{\rm c}Z_{\rm Hs}\cot\theta_{\rm Hs }-\omega L_{\rm c}Z_{\rm Ho}\tan\theta_{\rm Ho}-\omega^{2}C_{\rm v}L_{\rm c}Z_{ \rm Ho}Z_{\rm Hs}\] (8b) \[\theta_{\rm Hs}=\frac{\pi f}{2f_{\rm H}},\theta_{\rm Ho}=\frac{\pi f}{2f_{\rm H}} \tag{8c}\]  
and \(f\) is an operating frequency. From (7), the magnitude and phase of \(I_{\rm L}^{f_{\rm L}}\) are determined using Equation (9a,b):  
\[\left|I_{\rm L}^{f_{\rm L}}\right|=1 \tag{9a}\]  
Figure 3: Each reflection load according to operating band: (**a**) Low-band operation; (**b**) High-band operation. \[\angle I_{\rm L}^{f_{\rm L}}=\phi_{f_{\rm L}}=\pi-2\tan^{-1}\left(\frac{X_{\rm B}Z_{ \rm Ho}}{X_{\rm A}Z_{0}}\right) \tag{9b}\]  
Using (9b), the PSR of the proposed structure at \(f_{\rm L}\) can be expressed as the result of Equation (10), in which subscript \(V\) represents the bias voltage that changes the capacitance of \(C_{\rm v}\) by varying \(V_{\rm min}\) to \(V_{\rm max}\):  
\[\Delta\phi_{f_{\rm L}}\Big{|}_{V}=\left\{ \begin{array}{l|l}\phi_{f_{\rm L}} \Big{|}_{V}-\phi_{f_{\rm L}}\Big{|}_{V_{\rm min}}\ {\rm if}\phi_{f_{\rm L}}\Big{|}_{V_{\rm min}}<\phi_{f_{\rm L}} \Big{|}_{V_{\rm max}}\\ \phi_{f_{\rm L}}\Big{|}_{V}-\phi_{f_{\rm L}}\Big{|}_{V_{\rm max}}\ {\rm if}\phi_{f_{\rm L}} \Big{|}_{V_{\rm min}}>\phi_{f_{\rm L}}\Big{|}_{V_{\rm max}}\end{array} \right. \tag{10}\]  
Table 1 shows the calculated values of PSR and PD according to \(Z_{\rm Ho}\) and \(Z_{\rm Hs}\). For demonstrations, \(f_{\rm L}=1.88\) GHz, \(f_{\rm H}=2.44\) GHz, and BW was 100 MHz. The value of \(L_{\rm c}\) was calculated using (3), while \(C_{\rm v}\) was implemented using a SMV-1231 from SKYWORKS, which provided variable capacitance by varying the bias voltage from 0 V to 16 V. PSR and PD at the \(f_{\rm L}\) band increased along with \(Z_{\rm Ho}\). The PSR of the \(f_{\rm L}\) band was remained constant, even though \(Z_{\rm Hs}\) increased. However, the PD of the \(f_{\rm L}\) band decreased and the PD of the \(f_{\rm H}\) band increased as \(Z_{\rm Hs}\) increased. These results highlight that an appropriate \(Z_{\rm Ho}\) and \(Z_{\rm Hs}\) must be selected to achieve the desired PSR and PD depending on the requirements of a specific application. The design goals were specified as PSR \(>100^{\circ}\) at the \(f_{\rm L}\) band, PD \(<15^{\circ}\) at the \(f_{\rm L}\) band, and PD \(<1.5^{\circ}\) at the \(f_{\rm H}\) band. For these specified design goals, the circuit parameters selected were \(Z_{\rm Ho}=50\)\(\Omega\) and \(Z_{\rm Hs}=120\)\(\Omega\). Figure 4 shows the calculated PSR according to \(C_{\rm v}\) variation. \begin{table}
\begin{tabular}{c c c c} \hline \(Z_{\rm Ho}\) [\(\Omega\)] & \(Z_{\rm Hs}\) [\(\Omega\)] & **PSR** [\({}^{\circ}\)] **(f\({}_{\rm L}\)/f\({}_{\rm H}\))** & **PD** [\({}^{\circ}\)] **(f\({}_{\rm L}\)/f\({}_{\rm H}\))** \\ \hline
30 & & 56.3/0 & \(\pm 1.19\)/\(\pm 3.04\) \\
40 & & 84.21/0 & \(\pm 4.69\)/\(\pm 2.15\) \\
50 & 120 & 107.74/0 & \(\pm 10.69\)/\(\pm 1.4\) \\
60 & & 124.9/0 & \(\pm 18.45\)/\(\pm 0.98\) \\
70 & & 136.31/0 & \(\pm 27.22\)/\(\pm 0.72\) \\ \hline  & 100 & 107.74/0 & \(\pm 13.07\)/\(\pm 0.92\) \\  & 110 & 107.74/0 & \(\pm 11.77\)/\(\pm 1.14\) \\
50 & 120 & 107.74/0 & \(\pm 10.69\)/\(\pm 1.4\) \\  & 130 & 107.74/0 & \(\pm 9.76\)/\(\pm 1.69\) \\  & 140 & 107.74/0 & \(\pm 8.97\)/\(\pm 2.01\) \\ \hline \end{tabular}
\end{table}