## 2 Design Method
### Arbitrary Termination Impedance Reflection-Type Phase Shifter  
Figure 5a shows the proposed structure of the arbitrary termination impedance reflection-type phase shifter that consists of the arbitrary termination impedance 3 dB hybrid, where the through and coupled ports are terminated with two identical reflection loads. In this proposed reflection-type phase shifter, the input, through, and coupled ports are terminated with \(Z_{0}\) (\(Z_{1}=Z_{2}=Z_{3}=Z_{0}\)), while the output (or isolation) port is terminated with arbitrary termination impedance \(Z_{4}=R_{4}\pm jX_{4}\). Figure 5b depicts the structure of reflection load, which is composed of a short-circuited TL with characteristic impedance \(Z_{r}\) and electrical length \(\theta_{r}\), in a series with a varactor diode with capacitance \(C_{v}\). By considering ideal TLs (i.e., no loss and no dispersion) and an ideal varactor diode with no junction resistance (i.e., \(R_{j}\) = 0 \(\Omega\)), the input impedance of the reflection load \(Z_{\textit{ref}}\) can be defined as follows:  
\[Z_{\textit{ref}}=j[X_{1}(f)+X_{2}(f)] \tag{7}\]  
where  
\[X_{1}(f)=Z_{r}\tan\theta_{r} \tag{8}\]  
\[X_{2}(f)=-\frac{1}{\omega C_{v}} \tag{9}\]  
Assuming ideal perfect characteristics of the 3 dB arbitrary termination impedance hybrid, the transmission phase of the reflection-type phase shifter is equal to the phase of the reflection coefficient (\(\Gamma\)) of the load. Therefore, the \(\Gamma\) of the reflection load can be expressed as follows:  
\[\Gamma=\frac{j[X_{1}(f)+X_{2}(f)]-Z_{0}}{j[X_{1}(f)+X_{2}(f)]+Z_{0}} \tag{10}\]  
As shown in Figure 5b, the reflection load consists of two reactive elements, which are \(jX_{1}(f)\) and \(jX_{2}(f)\). By properly selecting \(\theta_{r}\) of the short-circuited TL between 0\({}^{\circ}\) to 90\({}^{\circ}\)  
Figure 5: Proposed arbitrary termination impedance reflection-type phase shifter: (**a**) overall structure, (**b**) equivalent circuit of reflection load, and (**c**) phase shift range concept (ATI: arbitrary termination impedance). \(jX_{1}(f)\) can cover part of the upper side of the Smith chart, shown as \(jX_{1}(f)\) in Figure 5c. Simultaneously, \(jX_{2}(f)\) consists of a varactor diode that can cover the lower part of Smith chart, transitioning from \(C_{\text{p, max}}\) to \(C_{\text{p, min}}\) as the bias voltage varies from \(V_{min}\) to \(V_{max}\), as shown in \(jX_{2}(f)\) in Figure 5c. Once \(jX_{1}(f)\) is series connected with \(jX_{2}(f)\), their combined effect enables a wider reactance range, thereby broadening the achievable phase shift range, as demonstrated by \(jX_{1}(f)+jX_{2}(f)\) in Figure 5c. Therefore, the transmission phase (\(\phi\)) of the reflection-type phase shifter can be calculated as (11) using the reflection coefficient of the load:  
\[\phi=\angle\Gamma=-2\tan^{-1}\bigg{[}\frac{X_{1}(f)+X_{2}(f)}{Z_{0}}\bigg{]} \tag{11}\]  
As a result, the phase shift range (\(\Delta\phi\)) of the proposed arbitrary termination impedance reflection-type phase shifter can be expressed in the following equation, where \(V\) varies from \(V_{min}\) to \(V_{max}\)[9]:  
\[\Delta\phi|_{V}=\left\{ \begin{array}{c}\phi|_{V}-\phi|_{V_{\min}}\text{if } \phi|_{V_{\min}}<\phi|_{V_{\max}}\\ \phi|_{V}-\phi|_{V_{\max}}\text{if }\phi|_{V_{\max}}<\phi|_{V_{\min}}\end{array} \right. \tag{12}\]  
Lastly, the in-band phase derivation error (\(\phi_{err}\)) within the operating bandwidth (BW) can be defined as the difference between the maximum and minimum values of \(\Delta\phi\mid\
u\):  
\[\phi_{err}=\pm\frac{\max(\Delta\phi)|_{V}-\min(\Delta\phi)|_{V}}{2} \tag{13}\]  
To achieve high phase shift range with low phase deviation error within the operating BW, values of \(Z_{r}\), \(\theta_{r}\), and \(C_{\text{p}}\) should be chosen carefully. In this work, the SMV1231 varactor diode from Skyworks Inc. is used as variable capacitor \(C_{\text{p}}\), which provides a tuning capacitance range of 0.33 pF to 4.78 pF, as shown in Figure 5a.