## 2 LNA Design Methodology
### Two-Stage Gain Compensation  
An important design criterion for broadband LNAs is flat gain, and one way is to synthesize a load with a high-order matching network such that the load impedance does not vary much with frequency. However, the tradeoff is that it may take up more die space, which incurs higher costs. Thus, in this work, we create a two-stage broadband LNA, where each stage has complementary gain responses across frequencies to each other. Figure 3 shows a simplified schematic of the two-stage LNA, where all passives and active components are integrated on-chip. The first stage's load \(Z_{load1}\), shown in Figure 3b, is calculated as  
\[ \begin{split} Z_{load1}=\left[j\omega L_{4}\right]\left\|\left(Z_{ in5}+\frac{1}{j\omega C_{3}}+j\omega L_{5}\right)\right]=\\ \omega^{2}Z_{in5}L_{4}\left[L_{5}(\omega^{2}-1)+L_{4}\omega^{2} \right]+j\left\{\omega^{3}Z_{in5}^{2}+\\ \frac{\omega\frac{L_{4}}{C_{3}}(\frac{1}{C_{3}}+C_{3}L_{4}L_{5} \omega^{2}(L_{4}+L_{5})-L_{5}(\omega^{2}+1)+L_{4}]}{[\frac{1}{C_{3}}-\omega^{2 }(L_{4}+L_{5})]^{2}+\omega^{2}Z_{in5}^{2}}\end{split}  \tag{1}\]  
\(Z_{load1}\) is a resonant circuit that was designed to provide two resonant frequencies, resulting in high voltage gains with large impedances at these frequencies, approximately around 10 and 41 GHz. Components \(C_{3}\), \(L_{4}\) and \(L_{5}\) are tuned such that \(Z_{load1}\) provides larger power transfer at these frequencies. The values of \(L_{4}\) and \(C_{3}\) are responsible for controlling the first (lower) resonant frequency, while \(C_{3}\), \(L_{5}\), and the gate-source capacitance of \(M_{3}\) are used to adjust the second (higher) resonant frequency. The separation of the gain peaks should be carefully chosen. If the peak separation is small, we cannot achieve wide \(S_{21}\) bandwidth; however, if the separation is too wide, a simple L section output matching network in the second stage would be unable to compensate for the gain droop sufficiently because a one-stage L matching network provides narrowband frequency response. Since a one-stage cascade alone cannot provide wide \(S_{21}\) bandwidth, the second stage of the LNA's gain frequency response is purposely designed to align its gain peak with the gain trough of the first stage, as shown in Figure 4. This design arrangement ensures that the overall gain remains relatively consistent across a wider range of frequencies, achieving a broadband LNA. To synthesize the second stage's load to achieve complementary gain responses in frequency between the first and second stages, we may observe the power gain circles for the full LNA minus the second stage's load at various frequencies such as 24, 30 and 40 GHz in Figure 5. Using these gain circles, we would know how the load impedance should vary on the Smith Chart such that the second stage's load can mitigate the gain droops in the first stage. Instead of aiming to achieve relatively smaller changes in load impedance versus frequency, we chose to design the output matching network that achieves a similar distance from the gain circles at their given frequencies. In Figure 5, even though the load impedance locus changes from 24 GHz to 40 GHz, the impedances reside in the 24, 26 and 25 dB power gain circles for 24, 30 and 40 GHz, respectively, meaning a frequency-dependent load can provide the LNA a wide gain BW. We would like to point out that the gain from the gain circles assumes matching using lossless components, so it is naturally higher than matching the load using real on-chip components. \begin{table}
\begin{tabular}{c c c c c} \hline W/L & 8 \(\mu\)m/18 nm & 16 \(\mu\)m/18 nm & 32 \(\mu\)m/18 nm & 64 \(\mu\)m/18 nm \\ \hline CPP & 1x/2x & 1x/2x & 1x/2x & 1x/2x \\ \hline \(N\)\(F_{min}\) @ 28 GHz (dB) & 0.691/0.636 & 0.839/0.709 & 1.175/0.866 & 1.926/1.168 \\ \hline \(f_{MAX}\) (GHz) & 303/309 & 253/273 & 182/222 & 139/188 \\ \hline \(f_{T}\) (GHz) & 349/344 & 371/372 & 381/386 & 386/391 \\ \hline \end{tabular}
\end{table}