一般化散乱パラメータによる利得

 2端子対回路の伝送利得(transducer gain)\(G\)は,一般化散乱パラメータ\(\hat{\mathcal{S}}_{21}\)より, \begin{eqnarray} G &=& \frac{P_{L,ava}}{P_{ava}} = |\hat{\mathcal{S}}_{21}|^2 \nonumber \\ &=& \left| \frac{S_{21}}{W} \frac{(1-\gamma_2) (1-\gamma_1)}{|1-\gamma_2| |1-\gamma_1|} \sqrt{1-|\gamma_2|^2} \sqrt{1-|\gamma_1|^2} \right|^2 \nonumber \\ &=& \frac{|S_{21}|^2}{|W|^2} (1-|\gamma_1|^2) (1-|\gamma_2|^2) \nonumber \\ &=& \frac{(1-|\gamma_1|^2) (1-|\gamma_2|^2) |S_{21}|^2}{|(1-\gamma_1 S_{11}) (1-\gamma_2 S_{22}) - \gamma_1 \gamma_2 S_{12} S_{21} |^2} \end{eqnarray} ただし,\(P_{ava}\)は信号源からの有効電力(available power from source), \(P_{L,ava}\)は終端負荷への供給電力(power delivered to load)を示す.

 また,電力利得(power gain)\(G_p\)は, \begin{eqnarray} G_p &=& \frac{P_{L, ava}}{P_{\mathrm{IN}}} = \frac{P_{L, ava}}{(1-|\hat{\mathcal{S}}_{11}|^2) P_{ava}} \nonumber \\ &=& \frac{G}{1-|\hat{\mathcal{S}}_{11}|^2} \nonumber \\ &=& \frac{|\hat{\mathcal{S}}_{21}|^2}{1-|\hat{\mathcal{S}}_{11}|^2} \end{eqnarray} ただし,\(P_{\mathrm{IN}}\)は入力電力を示す.

\begin{gather} P_{\mathrm{IN}} = (1-|\hat{\mathcal{S}}_{11}|^2) P_{ava} \end{gather}