# A CMOS-Based Terahertz Reconfigurable Reflectarray with Amplitude Control: Design and Validation
## 3 Element Design
### Amplitude-Control RRA Element
#### 3.2.2 Validation of the Design Concept for the Passive Part in the RRA Element  
To validate the proposed design concept, the passive part of the element is simulated, and the impedance parameters of the two-port network are extracted at 290 GHz as presented in Table 2. Here, \(Z_{11}\), \(Z_{21}\), \(Z_{12}\), and \(Z_{22}\) represent the impedance parameters of the two-port network. \(\eta_{0}\) denotes the free-space impedance at Port 1.  
For a lossless network, it is sufficient to directly calculate the input impedance at Port 2 and conjugate-match it to the switch impedance, as derived in [38]. The formulas are shown below:  
\[Z_{\text{in2}}=Z_{22}-\frac{Z_{12}Z_{21}}{Z_{11}+\eta_{0}}, \tag{3}\]  
\[Z_{\text{in2}}=\overline{Z}_{\text{ON/OFF}}. \tag{4}\]  
Here, \(Z_{\text{in1}}\) and \(Z_{\text{in2}}\) denote the input impedances at Ports 1 and 2, respectively. \(Z_{\text{ON/OFF}}\) corresponds to the switch impedance in ON or OFF state.  
Assume that the switch parameter \(Z_{\text{ON/OFF}}\) equals the conjugate of \(Z_{\text{in2}}\), as shown in Equation (4). This implies conjugate matching at Port 2, while the spatial wave Port 1 is also matched. At this point, the design of the lossless element can be considered complete. However, at terahertz frequencies, losses are significant, and element designs often involve  
\begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline \(Z_{11}\) & \(91.57-290.08\times i\,\Omega\) \\ \hline \(Z_{12}\) & \(20.86-67.60\times i\,\Omega\) \\ \hline \(Z_{21}\) & \(20.86-67.60\times i\,\Omega\) \\ \hline \(Z_{22}\) & \(8.26+48.86\times i\,\Omega\) \\ \hline \(\eta_{0}\) & \(377\ \Omega\) \\ \hline \end{tabular}
\end{table}
Table 2: Impedance parameters of the two-port network.  
Figure 6: Element of the amplitude-control RRA in 3D view.  
highly lossy networks. If this method is blindly applied to lossy networks, severe errors will occur. To verify this, the two-port network impedance parameters listed in Table 2 are substituted for validation.  
After calculations using Equations (3) and (4), we obtain \(Z_{\text{ON/OFF}}=11.95-57.16\times i\,\Omega\). Assuming this lossy network is loaded with this impedance, the input impedance at Port 1 can be calculated as follows:  
\[Z_{\text{in1}}=Z_{11}-\frac{Z_{12}Z_{21}}{Z_{22}+Z_{\text{ON/OFF}}}. \tag{5}\]  
The calculated input impedance (\(Z_{\text{in1}}\)) at Port 1 is valued as \(217.59\,-\,98.73\times i\,\Omega\), which mismatches the free-space impedance (\(\eta_{0}\)), leading to design failure.  
To ensure Port 1 remains matched in the lossy network, set \(Z_{02}\) as the characteristic impedance of Port 2 and equate the characteristic impedance (\(Z_{\text{in1}}\)) of Port 1 to the free-space impedance (\(\eta_{0}\)). This yields the following equation:  
\[Z_{\text{in1}}=Z_{11}-\frac{Z_{12}Z_{21}}{Z_{22}+Z_{02}}, \tag{6}\]  
\[Z_{\text{in1}}=\eta_{0}. \tag{7}\]  
The calculation method for the characteristic impedance (\(Z_{02}\)) of Port 2 can now be derived and set equal to the switch impedance \(Z_{\text{ON/OFF}}\), as follows:  
\[Z_{02}=\frac{Z_{12}Z_{21}}{Z_{11}-\eta_{0}}-Z_{22}, \tag{8}\]  
\[Z_{02}=Z_{\text{ON/OFF}} \tag{9}\]  
The calculated characteristic impedance (\(Z_{02}\)) of Port 2 is valued as \(3.80\,-\,51.24\times i\,\Omega\).  
It is evident that this proposed design concept differs from the previous method in calculating switch parameters. However, it maintains Port 1 matching even in lossy networks. Matching at Port 1 ensures a low reflection coefficient for the final element. This methodology effectively enables the design of amplitude-control elements, even in THz bands.