Question 6.5: Demonstrate Eqs. (6.46) and (6.47). ΓLC = S11 Δ^∗ S − S^∗ 22......
Demonstrate Eqs. (6.46) and (6.47).
\begin{aligned} \Gamma_{L C} & =\frac{S_{11} \Delta_S^*-S_{22}^*}{\left|\Delta_S\right|^2-\left|S_{22}\right|^2}\hspace{30 pt} \text{(6.46a)} \\ R_{L C} & =\frac{\left|S_{12} S_{21}\right|}{\left.|| \Delta_S\right|^2-\left|S_{22}\right|^2 \mid} \hspace{30 pt} \text{(6.46b)} \end{aligned}
\begin{aligned} \Gamma_{G C} & =\frac{S_{22} \Delta_S^*-S_{11}^*}{\left|\Delta_S\right|^2-\left|S_{11}\right|^2}\hspace{30 pt} \text{(6.47a)} \\ R_{G C} & =\frac{\left|S_{12} S_{21}\right|}{\left.|| \Delta_S\right|^2-\left|S_{11}\right|^2 \mid}.\hspace{30 pt} \text{(6.47b)} \end{aligned}
The relation (6.11)
\Gamma_{i n}=\frac{b_1}{a_1}=S_{11}+\frac{S_{12} S_{21} \Gamma_L}{1-S_{22} \Gamma_L}=\frac{S_{11}-\Delta_S \Gamma_L}{1-S_{22} \Gamma_L} . \hspace{30 pt} \text{(6.11)}
yielding \Gamma_{in} as a function of \Gamma_L is a linear fractional transformation between complex variables of the kind:
w=\frac{a z+b}{c z+d}, \hspace{30 pt} \text{(6.48)}
that transforms the circles of z plane into circles of w plane. The unit circle in w plane will therefore correspond to the condition:
\left|\frac{a z+b}{c z+d}\right|^2=\frac{(a z+b)\left(a^* z^*+b^*\right)}{(c z+d)\left(c^* z^*+d^*\right)}=1
and thus:
|z|^2+z \frac{\left(a b^*-c d^*\right)}{|a|^2-|c|^2}+z^* \frac{\left(a^* b-c^* d\right)}{|a|^2-|c|^2}=\frac{|d|^2-|b|^2}{|a|^2-|c|^2}. \hspace{30 pt} \text{(6.49)}
Eq. (6.49) is the equation of a circle in the z plane, as it is clear if we sum and subtract the factor
\frac{\left|c^* d-a^* b\right|^2}{\left(|a|^2-|c|^2\right)^2},
to the left-hand side of (6.49), that becomes:
\left|z-\frac{c^* d-a^* b}{|a|^2-|c|^2}\right|^2=\frac{|d|^2-|b|^2}{|a|^2-|c|^2}+\frac{\left|c^* d-a^* b\right|^2}{\left(|a|^2-|c|^2\right)^2}=\frac{|a d-c b|^2}{\left(|a|^2-|c|^2\right)^2} . \hspace{30 pt} \text{(6.50)}
From (6.50) we immediately obtain that the center C and radius R of the afore mentioned circle are given by:
\left\{\begin{array}{l} C=\frac{c^* d-a^* b}{|a|^2-|c|^2}\\ \\ R=\left|\frac{a d-c b}{|a|^2-|c|^2}\right| . \end{array}\right. \hspace{30 pt} \text{(6.51)}
From (6.11), comparing with (6.48), we obtain:
\left\{\begin{array}{l} a=\Delta_S \\ b=-S_{11} \\ c=S_{22} \\ d=-1, \end{array}\right.
that, after substitution into (6.51), yield the first two equations (6.46). If we exchange \Gamma_L with \Gamma_G and \Gamma_{in} with \Gamma_{out} we similarly obtain Eq. (6.47).