## 3 Proposed Design Methodology
### Two-Stage LNA Design
To meet the SSNM condition, the source termination \(\Gamma_{S}\) is selected to meet the optimum termination for noise measurement, \(\Gamma_{optM}\)[4], the correct equations for calculating the optimum termination can be found in [20]. The input and output reflection coefficients for each stage can be calculated using Equations (1) and (2) [15]. \[\Gamma_{in}=\frac{S_{11}-\Gamma_{L}\cdot\Delta}{1-\Gamma_{L}\cdot S_{22}}, \tag{1}\]  
\[\Gamma_{out}=\frac{S_{22}-\Gamma_{optM}\cdot\Delta}{1-\Gamma_{optM}\cdot S_{11}}, \tag{2}\]  
where \(\Delta\) is the determinant of the matrix of the S parameters of the active device, \([S]\). The mismatch values at the input and output of each stage can be calculated using Equations (3) and (4), respectively [15]. In all equations, we assume that \(\Gamma_{S}=\Gamma_{opt,\ M}\). Figure 5.0  
Figure 5.1  
The minimum output mismatch level when the input is matched to \(Z_{0}\) (\(IM_{1}=0\)), is known as \(OM0_{1}\), while the minimum input mismatch level when the output is matched to \(Z_{0}\) (\(OM_{2}=0\)), is defined as \(IM0_{2}\)[15]. This special scenario can only be met if the input reflection coefficients of the two stages (\(\Gamma_{S,1}\) and \(\Gamma_{S,2}\)) are fixed to satisfy the optimum termination of \(\Gamma_{opt,M,1}\) and \(\Gamma_{opt,M,2}\), respectively. The third step of the design is to find the exact values of the feedback inductor. To do so, we first determine the minimum level of mismatch in the interstage network. Using the equations available in [11], the \(OM0_{1}\) vs. \(L_{s,1}\) and \(IM0_{2}\) vs. \(L_{s,2}\) graph is drawn and it is clear from Figure 7 that the minimum value of interstage mismatch is equal to 0.2. The reader is free to choose the level of interstage mismatch within the range of \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(S_{11}\) & \(0.875/-115.5^{\circ}\) \\ \(S_{21}\) & \(4.184/104.2^{\circ}\) \\ \(S_{12}\) & \(0.144/22.1^{\circ}\) \\ \(S_{22}\) & \(0.524/-68.9^{\circ}\) \\ \(\Gamma_{opt}\) & \(0.600/59.9^{\circ}\) \\ \(NF_{min}\) (dB) & \(0.46\) \\ \(R_{n}\) (\(\Omega\)) & \(10.25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: FET small signal and noise parameters with \(2\times 50\) μm gate width at 15 GHz. The Source terminal is ideally Grounded. \(0.2\leq(OM0_{1}=IM0_{2})<1\). A higher mismatch level will result in a smaller inductance but less stability, Figure 6. In this design, the level of interstage mismatch is set to 0.26, that is, \(OM0_{1}=IM0_{2}=0.26\). The corresponding feedback inductor values of these mismatch levels are 0.4 nH and 0.311 nH, to be connected to the first and second stages' transistors, respectively, as shown in Figure 7.  
The fourth step is to design the IMN, ISMN1, and OMN of the 2-stage LNA, Figure 4. As mentioned earlier, the input matching network is designed to meet the optimum noise measure termination, \(\Gamma_{opt,\ M}\). The input load is synthesized to satisfy \(\Gamma_{L,1}=\Gamma_{LCin}\), the latter is the load termination that makes the LNA input perfectly matched to \(Z_{0}\), and its value can be calculated from Equation (5) [15]. \[\Gamma_{LCin}=\frac{S_{11_{1}}-\Gamma_{optM,1}^{*}}{\Delta_{1}-S_{22_{1}}\cdot \Gamma_{optM,1}^{*}}, \tag{5}\]  
where \(\Delta\) is the determinant of the matrix of the S-parameters of the active device, \([S]\). The ISMN1 can be designed simply by matching \(\Gamma_{S,2}\) to \(\Gamma_{LC,in}\). On the other hand, the second stage output termination that results in \(OM_{2}=0\), is simply \(\Gamma_{L,2}=\Gamma_{out}^{*}\). Table 2 summarizes the reflection coefficient values that are required to design the input, interstage, and output matching networks of the two-stage LNA. The steps for designing the 2-stage LNA can be summarized as follows:  
1. Select the appropriate device for each stage by comparing the performance of available options. 2. Perform stability analyses across the entire operating bandwidth. 3. Plot \(OM0_{1}\) vs. \(L_{s,1}\) and \(IM0_{2}\) vs. \(L_{s,2}\). 4.