# Frequency-Bounded Matching Strategy for Wideband LNA Design Utilising a Relaxed SSNM Approach
## 3 IM-OM Circles and Frequency-Bounded Mismatch Envelope
### IM-OM Circles for a Single Frequency Point  
To illustrate the behaviour of mismatch at a single frequency point, we analyse the IM-OM circles using the equations introduced in [15]. For simplification, the source reflection coefficient is set to a predefined value \(\Gamma_{S}\) (\(\Gamma_{OPT}\)), Based on this assumption, the centre and radius of the OM circle can be calculated using the following equations:  
\[Center=\frac{1-\ OM^{2}}{1-\ OM^{2}\left|\Gamma_{OUT}\right|^{2}}\Gamma_{OUT}* \tag{7}\]  
\[Radius=\frac{1-\left|\Gamma_{OUT}\right|^{2}}{1-\ OM^{2}\left|\Gamma_{OUT} \right|^{2}}\ OM \tag{8}\]Here OM = 0 represents a perfect match at output, which means \(\Gamma_{L}\) = \(\Gamma_{OUT}*\); this is also equal to the centre of \"OM = 0\" circle. Similarly, IM = 0 circle can be obtained at \(\Gamma_{L}\) = \(\Gamma_{LCin}\) which means perfect input match and can be evaluated from following equation:  
\[\Gamma_{LCin}=\frac{(S_{11}-\;\Gamma_{S}*)}{(\Delta-\;\Gamma_{S}*S_{22})} \tag{9}\]  
This equation is simply obtained by putting \(\Gamma_{S}*\) = \(\Gamma_{in}\) in Equation (3). This defines the centre of the IM = 0 circle on the Smith chart. To plot corresponding IM circles for various OM values, the following equations can be used:  
\[Center=\frac{\Gamma_{LCin}\;-\;q^{2}\Gamma_{OUT}*}{1-\;q^{2}\;|\Gamma_{OUT}|^{ 2}} \tag{10}\]  
\[Radius=\frac{1-|\Gamma_{OUT}|^{2}}{1-q^{2}\;|\Gamma_{OUT}|^{2}}q \tag{11}\]  
where the scaling factor (\(q\)) is defined as:  
\[q=IM\left|\frac{1-S_{11}\Gamma_{S}}{\Delta-S_{22}\Gamma_{s}^{*}}\right| \tag{12}\]  
Using the equations described above and the IM-OM values listed in Table 1, the corresponding mismatch circles are plotted in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline OM (Linear) & 0 & 0.1 & 0.25 & 0.3 & 0.6 & 0.65 & 0.7 & 0.8 & 0.895 & 0.952 & 1 \\ \hline OM [dB] & \(-\infty\) & \(-20\) & \(-12\) & \(-10.46\) & \(-4.44\) & \(-3.74\) & \(-3.10\) & \(-1.94\) & \(-0.96\) & \(-0.43\) & 0 \\ \hline IM (Linear) & 0.47 & 0.36 & 0.3 & 0.25 & 0.1 & 0.08 & 0.05 & 0 & 0.05 & 0.08 & 0.1 \\ \hline IM [dB] & \(-6.56\) & \(-8.87\) & \(-10.46\) & \(-12\) & \(-20\) & \(-17.20\) & \(-20.90\) & \(-\infty\) & \(-26\) & \(-21.9\) & \(-20\) \\ \hline \end{tabular}
\end{table} 
Table 1: OM values and corresponding IM Values. Figure 4: (**a**) IM-OM circles (@31 GHz) for Ls = 70 pH; (**b**) IM-OM circles along with stability circles (zoomed out view). As illustrated in Figure 4a, each IM or OM value corresponds to both a minimum and maximum counterpart; that is, every OM circle (blue) is associated with two potential IM values (red), and vice versa. The tangency points between these circles do not lie along a straight path but instead trace a curved trajectory, which becomes more noticeable as they move away from the centre of the Smith chart. Between the points \(\Gamma_{\text{OUT}}\)* and \(\Gamma_{\text{Lin}}\) no IM circle fully encloses an OM circle, and nor does any OM circle fully enclose an IM circle. Instead, these circles intersect at discrete points, forming a curved locus that defines Region B, a zone where the mismatch levels are balanced. By increasing the IM level further, IM circles begin to fully enclose OM circles, defining Region A. A similar reversal occurs in Region C, where elevated OM levels cause OM circles to enclose IM circles. This dual behaviour demonstrates that, for any fixed OM, there exists both a minimum and maximum IM value, and vice versa, outlining a bounded range of feasible matching conditions. Notably, the OM = 1 circle aligns exactly with the boundary of the Smith chart (dotted black line), indicating \(\Gamma_{L}=1\), or a completely reactive load. The corresponding IM values for this OM = 1 condition are 0.1 and 0.9. When OM exceeds 1, the resulting circles completely encompass the Smith chart, reflecting load conditions with a negative real part. Such terminations imply instability and are therefore excluded from practical amplifier design.