Question 6.6: Show an example of a two-port having K > 1 but being pote......

Consider a two-port with the scattering matrix:

S =\left(\begin{array}{rr} 0.5292- \text{j} 0.6643 & 0.1375- \text{j} 0.1346 \\ -5.3756+ \text{j} 2.9848 & 0.5918- \text{j} 0.5800 \end{array}\right) .

The Linville coefficient is K = 1.2787 > 1; however, the two-port is not unconditionally stable. For example, close port 2 with the passive load \Gamma_L=0.3762+ \text{j} 0.5264; we have \Gamma_{in}=-1.0464- \text{j} 0.4481 with magnitude larger than one. This is confirmed by the fact that \left|\Delta_S\right|=1.8528>1, contrarily to what requested by condition (6.53c).

\left|\Delta_S\right|<1 \hspace{30 pt} \text{(6.53c)}