Question 6.2: Show that the constant operational gain curves are circles i......

Reworking (6.22)

we obtain:

and, rearranging:

\left|\Gamma_L\right|^2-2 \text{Re}\left[\Gamma_L \frac{G_{ \text{op} }\left(S_{22}-S_{11}^* \Delta_S\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2}\right]=\frac{\left|S_{21}\right|^2-G_{ \text{op} }\left(1-\left|S_{11}\right|^2\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2} . \hspace{30 pt} \text{(6.23)}

Adding to both sides of (6.23) the parameter α, defined as:

\alpha=\frac{\left|G_{ \text{op} }\left(S_{22}-S_{11}^* \Delta_S\right)\right|^2}{\left[G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2\right]^2}

and applying the equality |a-b|^2=|a|^2+|b|^2-2 \text{Re}\left[a b^*\right], Eq. (6.23) can be rewritten as:

\left|\Gamma_L-C\right|^2=\frac{\left|S_{21}\right|^2-G_{ \text{op} }\left(1-\left|S_{11}\right|^2\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2}+\alpha=R^2 ,

that describes a circle with center:

C=\left[\frac{G_{ \text{op} }\left(S_{22}-S_{11}^* \Delta_S\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2}\right]^*=\frac{G_{ \text{op} }\left(S_{22}^*-S_{11} \Delta_S^*\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2}. \hspace{30 pt} \text{(6.24)}

and radius R, defined by its square:

R^2=\frac{\left|S_{21}\right|^2-G_{ \text{op} }\left(1-\left|S_{11}\right|^2\right)}{G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\left|S_{21}\right|^2}+\alpha \geq 0 .

Thus, the constant operational gain curves are circles with center C and radius R. Introducing the factor C_2=S_{22}-S_{11}^* \Delta_S, and expanding the expression for \left|C_2\right|^2 as follows:

we obtain for the radius the explicit expression:

R=\left|S_{21}\right| \frac{\sqrt{\left|S_{21}\right|^2-2 K\left|S_{21}\right|\left|S_{12}\right| G_{ \text{op} }+\left|S_{12}\right|^2 G_{ \text{op} }^2}}{\left.\left|G_{ \text{op} }\left(\left|S_{22}\right|^2-\left|\Delta_S\right|^2\right)+\right| S_{21}\right|^2 \mid}, \hspace{30 pt} \text{(6.25)}

where we have introduced the real parameter K defined in (6.52),

K=\frac{1-\left|S_{22}\right|^2-\left|S_{11}\right|^2+\left|\Delta_S\right|^2}{2\left|S_{21} S_{12}\right|}>1 \hspace{30 pt} \text{(6.52)}

called the Linville or Rollet coefficient, which plays a fundamental role in assessing the two-port stability. From (6.24) we also find that, varying G_{\text {op }}, the centers of the circles in the \Gamma_L plane lie on a straight line, crossing the origin of the \Gamma_L plane (for G_{ \text{op} }=0 ), with slope:

\arg \left[C_2^*\right]=-\arg \left[C_2\right]=-\tan ^{-1} \frac{\operatorname{Im}\; \left[C_2\right]}{\operatorname{Re}\; \left[C_2\right]} .

In conclusion, if the radius R defined in (6.25) is real, the constant gain curves in plane \Gamma_L are non-concentric circles, whose centers lie on a straight line. The circle G_{ \text{op} }=0 correspond to the unit circle of the Smith chart, with center in \Gamma_L=0. A qualitative plot of the constant level circles is provided in Fig. 6.6.