# Frequency-Bounded Matching Strategy for Wideband LNA Design Utilising a Relaxed SSNM Approach
## 3 IM-OM Circles and Frequency-Bounded Mismatch Envelope
### Frequency-Bounded Envelope of Various Mismatch Regions  
The IM-OM circles introduced in Section 3.1 are extended to construct a frequency-bounded mismatch envelope on the Smith chart. This approach offers a structured graphical tool to select \(\Gamma_{\text{L}}\), based on relaxed SSNM conditions, for wideband LNA design. The first step involves decomposing the operating frequency band into three discrete points (illustrated in Figure 6):  
* \(f_{1}\): Starting frequency of the band (29 GHz)
* \(f_{2}\) (or \(f_{0}\)): Centre frequency (31 GHz)
* \(f_{3}\): Ending frequency (33 GHz)  
The second step is the selection of appropriate source reflection coefficients \(\Gamma_{\text{S}}\) at the three chosen frequencies:  
* \(\Gamma_{\text{S1}}=\Gamma_{\text{OPT}}\) @ 29 GHz
* \(\Gamma_{\text{S2}}=\Gamma_{\text{OPT}}\) @ 31 GHz
* \(\Gamma_{\text{S3}}=\Gamma_{\text{OPT}}\) @ 33 GHz  
In the third step, the independent sets of IM and OM circles at each frequency are generated using the equations defined in Section 3.1 and using \(\Gamma_{\text{S}}\) values defined in step 2. The fourth step involves tracing the loci of tangency points between the IM and OM circles for all three frequencies. These points define the practical IM-OM pairs across the band. The lower tangency points from \(f_{1}\) to \(f_{3}\) are connected to form the lower boundary, and the upper tangency points define the upper boundary of the frequency-bounded mismatch region. As shown in Figure 7a, the left and right edges of the envelope correspond to tangency points at \(f_{1}\) and \(f_{3}\), respectively. Figure 6: Frequency decomposition for wideband design (28–34 GHz). The interior of the envelope shown in Figure 7a is subdivided into several mismatch zones (Figure 7b), which guide the selection of appropriate reflection coefficients for each stage of the LNA. 1. Low mismatch zones * A1: 0 \(<\) \(|\)OM\(|\) \(<\) 0.1 (red shaded area) This region is suitable for designing the last stage, or in between stages, where strong output matching is critical and a higher gain is desired. It typically offers output return loss better than \(-\)20 dB. However, the input mismatch in this region may be higher. * A1\({}^{\prime}\): 0 \(<\) \(|\)OM\(|\) \(<\) 0.1 (light green shaded area) This region lies opposite side of A1, with \(\Gamma_{OUT}\)* being the centre. In this region, OM is the same, but IM is comparatively larger than A1. It may be useful for inter stages focused on good S\({}_{22}\) performance. * B1: 0 \(<\) \(|\)IM\(|\) \(<\) 0.1 (pink shaded area) Similar to A1, this zone supports input matching with excellent return loss. In this region, OM is higher. This region can be used to design the initial stages. * B1\({}^{\prime}\): 0 \(<\) \(|\)IM\(|\) \(<\) 0.1 (black shaded area) This represents the mirrored region of B1, with excellent IM but much higher OM. It is suitable for stages where input reflection dominates performance considerations. 2. Acceptable mismatch zones * A2: 0.1 \(<\) \(|\)OM\(|\) \(<\) 0.25 (blue shaded area) * Represents a more relaxed output match. It can be used where \(-\)12 dB to \(-\)20 dB return loss is sufficient. In this region, IM is higher than OM but lower than the A1 region. * A2\({}^{\prime}\): 0.1 \(<\) \(|\)OM\(|\) \(<\) 0.25 (green shaded area) * This region has the same output mismatch as A2, but IM is comparatively higher than A2 and A1. Figure 7: (a) Frequency-bounded mismatch region, formed using tangency loci at \(f_{1},f_{2},\) and \(f_{3}\); (b) categorised mismatch zones based on IM and OM levels. B2: \(0.1<|\text{IM}|<0.25\) (yellow shaded area) It is similar to A2; in this region, IM remains in an acceptable range. These regions are suitable for intermediate stages where a balance between input and output matching is acceptable, and design constraints are less strict. 3. High mismatch zone * C1: Corresponds to areas where mismatch exceeds its largest possible limits. These regions may be avoided in critical input-output paths but are occasionally useful in buffering stages or gain shaping.