## 2 Design Method
### Arbitrary Termination Impedance 3 dB Branch-Line Hybrid  
Figure 3a illustrates the proposed structure of the arbitrary termination impedance 3 dB branch-line hybrid, in which the input, through, coupled, and isolation ports are terminated with \(Z_{i}=R_{i}\pm jX_{i}\), where \(i\) = 1, 2, 3, and 4, respectively. For convenience, the intersection points of each branch are denoted by nodes 1, 2, 3, and 4. To match the complex impedance to real impedance (\(R_{i}\)), the imaginary part (\(X_{i}\)) of \(Z_{i}\) must be canceled using  
Figure 2: Structure of the proposed co-design of the microwave amplifier–phase shifter (RTPS: reflection-type phase shifter; IMN: input matching network; ATI: arbitrary termination impedance; MT: microwave transistor; COMN: conventional output matching network). either the capacitor (\(C_{i}\)) or inductor (\(L_{i}\)). Depending on the sign of the imaginary part of \(Z_{i}\), the values of \(C_{i}\) and \(L_{i}\) can be determined as follows:  
\[C_{i}=\frac{1}{\omega_{0}|X_{i}|}\ \ \ \text{for}\ +X_{i} \tag{1}\]  
\[L_{i}=\frac{|X_{i}|}{\omega_{0}}\ \ \ \text{for}\ -X_{i} \tag{2}\]  
where \(\omega_{0}\) is the angular center frequency. Once \(X_{i}\) is cancelled, pure real impedance \(R_{i}\) remains at each node, which will transform from one node impedance to another node impedance by the proposed 3 dB branch-line hybrid. Since the 3 dB branch-line hybrid consists of quarter-wavelength (\(\lambda/4\)) transmission lines (TLs), the characteristic impedances can be found by using equivalent structure with isolation characteristics shown in Figure 3b,c. If the isolation port is perfectly isolated (i.e., \(\mid S_{41}\mid\) = 0), there is no signal coupled to node 4 when the signal is excited at node 1. Under this condition, the input power is equally divided between nodes 2 and 3, denoted as \(d_{1}\) and \(d_{2}\), respectively, as shown in Figure 3b. The resulting power ratio between \(d_{1}\) and \(d_{2}\) is 1:1. So, the voltage across \(R_{2}\) at node 2 is equal to that across TL \(Z_{B}\) terminated with \(R_{3}\) at node 3. Therefore, the impedance (\(Z_{\text{in1}}\)) looking into the TL \(Z_{B}\) terminated with \(R_{3}\) must be \(R_{2}\)[24]. Denoting \(M=R_{2}/2\) as the equivalent impedance of parallel \(Z_{\text{in1}}\) with \(R_{2}\) at node 2, the characteristic impedances \(Z_{A}\) and \(Z_{B}\) of TL can be defined as follows:  
\[Z_{A}=\sqrt{\frac{R_{1}R_{2}}{2}} \tag{3}\]  
\[Z_{B}=\sqrt{R_{2}R_{3}} \tag{4}\]  
Similarly, due to reciprocal characteristics, the circuit in Figure 3c is the simplified equivalent circuit in condition input signal exited at node 3. The voltage across \(R_{4}\) at node 4 is equal to that across TL \(Z_{D}\) terminated with \(R_{1}\) at node 1. Thus, the impedance (\(Z_{\text{in2}}\)  
Figure 3: Arbitrary termination impedance 3 dB branch–line coupler: (**a**) full structure, (**b**) simplified structure under the assumption of \(\mid S_{41}\mid\) = 0, and (**c**) reciprocal structure of (**b**) (ATI: arbitrary termination impedance). of TL \(Z_{D}\) terminated with \(R_{1}\) must be \(R_{4}\). Assuming \(N=R_{4}/2\) as equivalent impedances of parallel \(Z_{\text{in2}}\) with \(R_{4}\) at node 4, the characteristic impedances \(Z_{C}\) and \(Z_{D}\) of TL can be expressed as follows:  
\[Z_{C} =\sqrt{\frac{R_{3}R_{4}}{2}} \tag{5}\] \[Z_{D} =\sqrt{R_{1}R_{4}}. \tag{6}\]  
For equation validation, the calculated parameters for arbitrary terminated 3 dB branch-line hybrids with different termination impedances are shown in Table 1. The simulation was performed using advanced design system (ADS) software. Based on (1)-(6), the values of \(C_{m}\), \(L_{m}\), \(Z_{A}\), \(Z_{B}\), \(Z_{C}\), and \(Z_{D}\) are calculated at \(f_{0}\) of 2.45 GHz. The simulated electrical performances of the designed 3 dB branch-line hybrids are shown in Figure 4. Although termination impedances are different from 50 \(\Omega\), the optimum RL characteristic can be achieved at \(f_{0}\), as can be seen in Figure 4a,b.