# 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
#### 2.2.2 Phase Shifting Range at High-Band Operation  
During \(f_{\rm H}\) operation, the unwanted effect of a \(\lambda/4\) shunt open stub was compensated by \(\rm C_{c}\), as shown in Figure 3b. The reflection load at \(f_{\rm H}\) (\(I_{\rm L}^{\rm f_{H}}\)) is determined using Equation (11):  
\[I_{\rm L}^{\rm f_{H}}=\frac{-X_{\rm C}Z_{0}+jX_{\rm D}Z_{Lo}}{X_{\rm C}Z_{0}+jX _{\rm D}Z_{Lo}}, \tag{11}\]  
in which:  
\[X_{\rm C}=Z_{\rm Lo}+Z_{\rm Ls}\cot\theta_{\rm Ls}+2\pi f(C_{\rm v}+C_{\rm c} )Z_{\rm Lo}Z_{\rm Ls}\cot\theta_{\rm Ls} \tag{12a}\] \[X_{\rm D}=Z_{\rm Lo}\tan\theta_{\rm Lo}-Z_{\rm Ls}\cot\theta_{\rm Ls}+2\pi f (C_{\rm v}+C_{\rm c})Z_{\rm Lo}Z_{\rm Ls}\] (12b) \[\theta_{\rm Ls}=\frac{\pi f}{2f_{\rm L}},\theta_{\rm Lo}=\frac{\pi f}{2f_{ \rm L}} \tag{12c}\]  
From (12), the magnitude and phase of \(I_{\rm L}^{\rm f_{H}}\) can be determined using (13):  
\[\left|I_{\rm L}^{\rm f_{H}}\right|=1 \tag{13a}\] \[\angle I_{\rm L}^{\rm f_{H}}=\phi_{\rm f_{H}}=\pi-2\tan^{-1}\left(\frac{X_{ \rm D}Z_{\rm Lo}}{X_{\rm C}Z_{0}}\right) \tag{13b}\]  
Using (13b), the PSR of the proposed structure can be expressed using Equation (14), in which subscript \(V\) represents a bias voltage of varactor:  
\[\Delta\phi_{\rm f_{H}}\Big{|}_{V}=\left\{\begin{array}{c|c}\phi_{\rm f_{H}} \Big{|}_{V}-\phi_{\rm f_{H}}\Big{|}_{V\rm min}\ {\rm if}\phi_{\rm f_{H}}\Big{|}_{V\rm min}<\phi_{\rm f_{H}}\Big{|}_{V\rm max }\\ \phi_{\rm f_{H}}\Big{|}_{V}-\phi_{\rm f_{H}}\Big{|}_{V\rm max}\ {\rm if}\phi_{\rm f_{H}}\Big{|}_{V\rm min }>\phi_{\rm f_{H}}\Big{|}_{V\rm max}\end{array}\right. \tag{14}\]  
Table 2 shows that the calculated PSR and PD values as \(Z_{\rm Lo}\) and \(Z_{\rm Ls}\) varied. Consistent with \(f_{\rm L}\) band operation, \(f_{\rm L}=1.88\) GHz, \(f_{\rm H}=2.44\) GHz, and BW was 100 MHz. The value of \(C_{\rm c}\) that compensated for the effect of a \(\lambda/4\) shunt open stub was calculated using (6), while \(C_{\rm v}\) was implemented using a varactor SMV-1231 from SKYWORKS, which provided a variable capacitance of 4.7 pF to 0.3 pF by varying the bias voltage from 0 V to 16 V. The PSR at the \(f_{\rm H}\) band decreased as \(Z_{\rm Lo}\) increased, however, PD at the \(f_{\rm H}\) band increased. Similarly, PSR remained constant as \(Z_{\rm Ls}\) increased. PD, however, increased at the \(f_{\rm L}\) band  
Figure 5: PSR with and without compensating for the effects of a of \(\lambda/4\) shunt open stub at the low\(-\)band.  
while PD decreased at the \(f_{\rm H}\) band. Therefore, appropriate \(Z_{\rm Lo}\) and \(Z_{\rm Ls}\) values should be selected to achieve the desired PSR and PD, keeping in mind the specific application.  
Like the \(f_{\rm L}\) operation, the design goals were specified as PSR \(>100^{\circ}\) at the \(f_{\rm H}\) band and PD \(<2^{\circ}\) and \(10^{\circ}\) at the \(f_{\rm L}\) and \(f_{\rm H}\) band, respectively. To achieve these design goals, the circuit parameters were selected as \(Z_{\rm Lo}\) = 50 \(\Omega\) and \(Z_{\rm Ls}\) = 120 \(\Omega\). Figure 6 shows the calculated PSR according to \(C_{\rm v}\) variations.  
Figure 7 depicts the PSR with and without compensation for the \(\lambda/4\) shunt open stub. A PSR of \(116^{\circ}\) at the \(f_{\rm H}\) band was achieved with compensation. However, the PSR increased to \(203^{\circ}\) in the absence of compensation. In the event that a wider PSR was desired, a circuit lacking compensation would be appropriate. We, however, assessed a circuit with compensation in order to obtain the same PSR as the varactor diode.