# Frequency-Bounded Matching Strategy for Wideband LNA Design Utilising a Relaxed SSNM Approach
## 2 Relaxed SSNM Conditions: Background and Choice of Source Degenerative Inductors  
In traditional multistage LNA design, SSNM is achieved by enforcing strict mismatch conditions at each stage [9; 10; 11; 16]. At the design frequency \(f_{0}\), and assuming lossless matching networks, the following relationships are imposed:  
1. Mismatch at external ports: IM\({}_{1}\) = OM\({}_{n}\) = 0
2. Mismatch at inter-stages: OM\({}_{n}\) = IM\({}_{n+1}\)  
This strict SSNM condition is enforced throughout a narrowband design procedure. This enforcement is used even when selecting source degeneration (L\({}_{S}\)); an appropriate L\({}_{S}\) is crucial in the case of LNA, as it must satisfy not only the gain-noise trade-off but also ensure signal matching. Figure 2 shows the IM-OM pairs obtained by sweeping L\({}_{S}\) from 0 pH to 300 pH in 10 pH steps. Both IM and OM decrease as L\({}_{S}\) increases. For this process, the grounded FET (0 pH) yields IM = 0.891 and OM = 1.358. The geometrical stability factor (\(\upmu\)) remains above unity at \(f_{0}\) = 31 GHz for L\({}_{S}\) = 45 pH to 200 pH, allowing designers to choose L\({}_{S}\) safely within the unshaded regions (for Tr2: \(\upmu>1\)). The complex chart in Figure 2 is further disintegrated into smaller, more interpretable zones. Region A is unusable, as the IM-OM trace at 0 pH lies below it. The pale blue (B) and green areas (including C) correspond to conditional stable regions (\(\upmu<1\)). [PERSON] et al. [14] developed a chart-based method to select L\({}_{S}\) values according to mismatch levels that forces the strict SSNM conditions in multistage, narrowband amplifiers. To understand the method, consider a three-stage LNA with identical biasing and transistor sizes, and assume chart shown in Figure 2 represents the second stage of the cascade network and the Y-axis equates to OM\({}_{1}\) = IM\({}_{2}\), while the X-axis represents OM\({}_{2}\) = IM\({}_{3}\). For this, we obtain an unconditional stable region for Tr1 (between D1-D2 boundaries) and Tr2 (between C1-C2). Their sole intersection region (C), where both transistors are stable, lies in the region that corresponds to \(\upmu<1\) for Tr2. Any inductor value chosen in this region will  
Figure 2: IM-OM vs. L\({}_{S}\) variation chart with 10 pH steps (unshaded zones denote unconditional stable regions, allowing a visual one-shot selection of safe inductance values, shaded zones including B, C, A represent conditional stability). come with certain limitations of source and load terminations for Tr2. This illustrates how some methods may offer limited flexibility for selecting inductor values. That limitation, however, depends on factors like transistor sizing, biasing, or technology; under different conditions, [PERSON]'s system works effectively. A possible solution to this problem is to relax the SSNM constraint by deliberately introducing a small mismatch at each external port. Even a mismatch of 0.1 would be a great help to reduce the limitations, and a 0.1 mismatch will equate to \(-\)20 dB reflection, which is not ideal, but still an excellent match in a practical RF system. Easing this constraint allows more flexibility in choosing appropriate source degeneration values, enabling better gain and increased stability. Figure 3 demonstrates this perfectly. Here, introducing a mismatch of 0.1 will move the IM-OM axis to a new x\({}^{\prime}\)-y\({}^{\prime}\) axis, and a mismatch of 0.2 moves it to an x\({}^{\prime\prime}\)-y\({}^{\prime\prime}\) axis. This gives us an extended region to select L\({}_{\text{S}}\) value where all transistors are within the stability margin. Also, introducing deliberate mismatches eventually leads to low inductance values. For example, in Figure 3, Point \"p\" in the stable region gives a low OM value compared to Figure 2, and low L\({}_{\text{S}}\) and high OM both result in a high gain. The chart shown in Figure 3 can be modified for various pairs of IM-OM values by simply shifting the axes according to the desired mismatches.