一般化散乱行列と散乱行列の関係
一般化散乱行列を散乱行列を用いて表す式[3],[4]を導出する．
- [3] R. Mavaddat, “Network Scattering Parameters,” World Scientific, 1996.
- [4] Robert E. Collin, “Foundations for Microwave Engineering,” 2nd ed., Wiley-IEEE Press, 2001.
一般化散乱行列\([\hat{\boldsymbol{S}}]\)をインピーダンス行列\([\boldsymbol{Z}]\)から求める式
\begin{gather}
[\hat{\boldsymbol{S}}]
= \left[ \sqrt{\hat{R}} \right]^{-1} \big( [\boldsymbol{Z}] - [\hat{Z}]^* \big)
\big( [\boldsymbol{Z}] + [\hat{Z}] \big) ^{-1} \left[ \sqrt{\hat{R}} \right]
\end{gather}
に，インピーダンス行列\([\boldsymbol{Z}]\)を通常の散乱行列\([\boldsymbol{S}]\)から求める式
\begin{gather}
[\boldsymbol{Z}]
= \left[ \sqrt{R_0} \right] \big( [U] - [\boldsymbol{S}] \big) ^{-1}
\big( [U] + [\boldsymbol{S}] \big) \left[ \sqrt{R_0} \right]
\tag{1} \label{eq:ZRUSRSR}
\end{gather}
を代入して，インピーダンス行列\([\boldsymbol{Z}]\)を消去して，式を整理していく．まず，
\begin{eqnarray}
[ \boldsymbol{A}_1] &\equiv& \left[ \sqrt{\hat{R}} \right]^{-1} \big( [\boldsymbol{Z}] - [\hat{Z}]^* \big)
\tag{2} \label{eq:A1} \\
[ \boldsymbol{A}_2] &\equiv& \big( [\boldsymbol{Z}] + [\hat{Z}] \big) ^{-1} \left[ \sqrt{\hat{R}} \right]
\tag{3} \label{eq:A2}
\end{eqnarray}
とおくと，
\begin{gather}
[\boldsymbol{S}]
= \left[ \sqrt{\hat{R}} \right]^{-1} \big( [\boldsymbol{Z}] - [\hat{Z}]^* \big)
\big( [\boldsymbol{Z}] + [\hat{Z}] \big) ^{-1} \left[ \sqrt{\hat{R}} \right]
= [ \boldsymbol{A}_1] [ \boldsymbol{A}_2]
\nonumber
\end{gather}
先に求めた式を再記して，
\begin{eqnarray}
\left[ \sqrt{\hat{R}} \right]^{-1}
&=& \left[ \sqrt{R_0} \right]^{-1} [\Lambda]^{-1*} \big( [U] - [\gamma] \big)
\tag{4} \label{eq:hatRi}
\\
[\hat{Z}]^*
&=& \left[ \sqrt{R_0} \right] \big( [U] -[\gamma]^* \big) ^{-1} \big( [U] +[\gamma]^* \big)
\left[ \sqrt{R_0} \right]
\tag{5} \label{eq:hatZcR0}
\end{eqnarray}
式\eqref{eq:ZRUSRSR}，式\eqref{eq:hatRi}，式\eqref{eq:hatZcR0}を式\eqref{eq:A1}に代入して，
\begin{eqnarray}
[ \boldsymbol{A}_1]
&=& \left[ \sqrt{\hat{R}} \right]^{-1} \big( [\boldsymbol{Z}] - [\hat{Z}]^* \big)
\nonumber \\
&=& \left[ \sqrt{R_0} \right]^{-1} [\Lambda]^{-1} \big( [U] - [\gamma]^* \big)
\nonumber \\
&& \cdot \left[ \sqrt{R_0} \right]
\left\{ \big( [U] - [\boldsymbol{S}] \big) ^{-1} \big( [U] + [\boldsymbol{S}] \big)
- \big( [U] - [\gamma]^* \big) ^{-1} \big( [U] +[\gamma]^* \big) \right\}
\left[ \sqrt{R_0} \right]
\nonumber \\
&=& [\Lambda]^{-1} \left\{
\big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} \big( [U] + [\boldsymbol{S}] \big)
- \big( [U] +[\gamma]^* \big) \right\} \left[ \sqrt{R_0} \right]
\end{eqnarray}
ここで，
\begin{eqnarray}
\big( [U] - [\boldsymbol{S}] \big) ^{-1} \big( [U] + [\boldsymbol{S}] \big)
&=& \big( [U] - [\boldsymbol{S}] \big) ^{-1} \left\{ \big( [U] - [\boldsymbol{S}] \big) + 2[\boldsymbol{S}] \right\}
\nonumber \\
&=& [U] + \big( [U] - [\boldsymbol{S}] \big) ^{-1} 2[\boldsymbol{S}]
\nonumber
\end{eqnarray}
より，
\begin{eqnarray}
[ \boldsymbol{A}_1] &=& [\Lambda]^{-1} \left\{
\big( [U] - [\gamma]^* \big) \big\{
[U] + \big( [U] - [\boldsymbol{S}] \big) ^{-1} 2[\boldsymbol{S}] \big\}
- \big( [U] +[\gamma]^* \big) \right\} \left[ \sqrt{R_0} \right]
\nonumber \\ \hspace{7.3mm}
&=& 2[\Lambda]^{-1} \left\{
-[\gamma]^* + \big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\right\} \left[ \sqrt{R_0} \right]
\nonumber
\end{eqnarray}
また，先に求めた式を再記して，
\begin{eqnarray}
[\hat{Z}]
&=& \left[ \sqrt{R_0} \right] \big( [U] -[\gamma] \big) ^{-1} \big( [U] +[\gamma] \big)
\left[ \sqrt{R_0} \right]
\tag{6} \label{eq:hatZR0}
\\
\left[ \sqrt{\hat{R}} \right]
&=& \left[ \sqrt{R_0} \right] \big( [U] - [\gamma] \big) ^{-1} [\Lambda]^*
\tag{7} \label{eq:hatR}
\end{eqnarray}
式\eqref{eq:ZRUSRSR}，式\eqref{eq:hatZR0}，式\eqref{eq:hatR}を式\eqref{eq:A2}に代入して
\begin{eqnarray}
[ \boldsymbol{A}_2] &=& \big( [\boldsymbol{Z}] + [\hat{Z}] \big) ^{-1} \left[ \sqrt{\hat{R}} \right]
\nonumber \\
&=& \left( \left[ \sqrt{R_0} \right]
\left\{ \big( [U] - [\boldsymbol{S}] \big) ^{-1} \big( [U] + [\boldsymbol{S}] \big)
+ \big( [U] - [\gamma] \big) ^{-1} \big( [U] +[\gamma] \big) \right\}
\left[ \sqrt{R_0} \right] \right)^{-1}
\nonumber \\
&& \cdot \left[ \sqrt{R_0} \right] \big( [U] - [\gamma] \big) ^{-1} [\Lambda]^*
\end{eqnarray}
変形して，
\begin{eqnarray}
[U] &=& [\Lambda]^{-1*} \big( [U] - [\gamma] \big) \left[ \sqrt{R_0} \right]^{-1}
\nonumber \\
&& \cdot \left[ \sqrt{R_0} \right]
\left\{ \big( [U] - [\boldsymbol{S}] \big) ^{-1} \big( [U] + [\boldsymbol{S}] \big)
+ \big( [U] - [\gamma] \big) ^{-1} \big( [U] +[\gamma] \big) \right\}
\left[ \sqrt{R_0} \right] [\boldsymbol{A}_2]
\nonumber \\
&=& [\Lambda]^{-1*} \big( [U] - [\gamma] \big)
\left\{ [U] + \big( [U] - [\boldsymbol{S}] \big) ^{-1} 2[\boldsymbol{S}]
+ \big( [U] - [\gamma] \big) ^{-1} \big( [U] +[\gamma] \big) \right\}
\left[ \sqrt{R_0} \right]
[\boldsymbol{A}_2]
\nonumber \\
&=& [\Lambda]^{-1*}
\left\{ \big( [U] - [\gamma] \big) + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} 2[\boldsymbol{S}]
+ \big( [U] +[\gamma] \big) \right\}
\left[ \sqrt{R_0} \right] [\boldsymbol{A}_2]
\nonumber \\
&=& 2[\Lambda]^{-1*}
\left\{ [U] + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big)^{-1} [\boldsymbol{S}] \right\}
\left[ \sqrt{R_0} \right]
[\boldsymbol{A}_2]
\end{eqnarray}
したがって，
\begin{gather}
[\boldsymbol{A}_2] = \frac{1}{2} \left[ \sqrt{R_0} \right]^{-1}
\left\{ [U] + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big)^{-1} [\boldsymbol{S}] \right\}^{-1}
[\Lambda]^{*}
\end{gather}
これより，
\begin{eqnarray}
[\hat{\boldsymbol{S}}]
&=& \left\{ \left[ \sqrt{\hat{R}} \right]^{-1} \big( [\boldsymbol{Z}] - [\hat{Z}]^* \big) \right\}
\left\{ \big( [\boldsymbol{Z}] + [\hat{Z}] \big)^{-1}\left[ \sqrt{\hat{R}} \right] \right\}
\nonumber \\
&=& [\boldsymbol{A}_1] [\boldsymbol{A}_2]
\nonumber \\
&=& 2[\Lambda]^{-1} \left\{
-[\gamma]^* + \big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\right\} \left[ \sqrt{R_0} \right]
\nonumber \\
&& \cdot \frac{1}{2} \left[ \sqrt{R_0} \right]^{-1}
\left\{ [U] + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big)^{-1} [\boldsymbol{S}] \right\}^{-1}
[\Lambda]^{*}
\nonumber \\
&=& [\Lambda]^{-1} \left\{
-[\gamma]^* + \big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}] \right\}
\nonumber \\
&& \cdot \left\{ [U] + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big)^{-1} [\boldsymbol{S}] \right\}^{-1}
[\Lambda]^{*}
\end{eqnarray}
次に，
\begin{eqnarray}
&&-[\gamma]^* + \big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\nonumber \\
&\equiv& \Big\{ -[\gamma]^* [\boldsymbol{A}_3] + \big( [U] - [\gamma]^* \big) \Big\} \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\end{eqnarray}
とおいて，\([\boldsymbol{A}_3]\)を求めると，
\begin{eqnarray}
[U] &=& [\boldsymbol{A}_3] \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\nonumber \\
[\boldsymbol{A}_3] &=& [\boldsymbol{S}]^{-1} \big( [U] - [\boldsymbol{S}] \big)
\nonumber \\
&=& [\boldsymbol{S}]^{-1} - [U]
\end{eqnarray}
これより，
\begin{eqnarray}
&&-[\gamma]^* + \big( [U] - [\gamma]^* \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\nonumber \\
&=& \Big\{ -[\gamma]^* ([\boldsymbol{S}]^{-1} - [U]) + \big( [U] - [\gamma]^* \big) \Big\} \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\nonumber \\
&=& \big( -[\gamma]^* [\boldsymbol{S}]^{-1} + [U] \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\end{eqnarray}
同様にして，
\begin{eqnarray}
\left\{ [U] + ( [U] - [\gamma] ) ( [U] - [\boldsymbol{S}] )^{-1} [\boldsymbol{S}] \right\}^{-1}
&\equiv& [\boldsymbol{S}]^{-1} \big( [U] - [\boldsymbol{S}] \big) [\boldsymbol{A}_4]
\nonumber \\
&=& \big( [\boldsymbol{S}]^{-1} - [U] \big) [\boldsymbol{A}_4]
\nonumber
\end{eqnarray}
とおいて，\([\boldsymbol{A}_4]\)を求めよう．
\begin{gather}
[\boldsymbol{A}_4] \left\{ [U] + \big( [U] - [\gamma] \big) \big( [U] - [\boldsymbol{S}] \big)^{-1} [\boldsymbol{S}] \right\}
[\boldsymbol{S}]^{-1} \big( [U] - [\boldsymbol{S}] \big) = [U]
\nonumber
\end{gather}
変形して，
\begin{gather}
[\boldsymbol{A}_4] \left\{ \big( [\boldsymbol{S}]^{-1} - [U] \big) + \big( [U] - [\gamma] \big) \right\}
= [\boldsymbol{A}_4] \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)
= [U]
\nonumber \\
[\boldsymbol{A}_4] = \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)^{-1}
\end{gather}
よって，
\begin{gather}
\left\{ [U] + ( [U] - [\gamma] ) ( [U] - [\boldsymbol{S}] )^{-1} [\boldsymbol{S}] \right\}^{-1}
= [\boldsymbol{S}]^{-1} \big( [U] - [\boldsymbol{S}] \big) \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)^{-1}
\nonumber
\end{gather}
したがって，
\begin{eqnarray}
[\hat{\boldsymbol{S}}]
&=& [\Lambda]^{-1}
\big( -[\gamma]^* [\boldsymbol{S}]^{-1} + [U] \big) \big( [U] - [\boldsymbol{S}] \big) ^{-1} [\boldsymbol{S}]
\cdot [\boldsymbol{S}]^{-1} \big( [U] - [\boldsymbol{S}] \big) \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)^{-1}
[\Lambda]^{*}
\nonumber \\
&=& [\Lambda]^{-1} \big( [U]-[\gamma]^* [\boldsymbol{S}]^{-1} \big)
\big( [\boldsymbol{S}]^{-1} - [\gamma] \big)^{-1} [\Lambda]^{*}
\end{eqnarray}
ここで，
\begin{gather}
[\boldsymbol{A}_5]
\equiv \big( [U]-[\gamma]^* [\boldsymbol{S}]^{-1} \big) \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)^{-1}
\end{gather}
とおいて変形すると，
\begin{gather}
[U]-[\gamma]^* [\boldsymbol{S}]^{-1}
= [\boldsymbol{A}_5] \big( [\boldsymbol{S}]^{-1} - [\gamma] \big)
= [\boldsymbol{A}_5] \big( [U] - [\gamma] [\boldsymbol{S}] \big) [\boldsymbol{S}]^{-1}
\nonumber \\
[\boldsymbol{S}] - [\gamma]^*= [\boldsymbol{A}_5] \big( [U] - [\gamma] [\boldsymbol{S}] \big)
\nonumber \\
[\boldsymbol{A}_5] = \big( [\boldsymbol{S}] - [\gamma]^* \big) \big( [U] - [\gamma] [\boldsymbol{S}] \big)^{-1}
\end{gather}
よって，
\begin{eqnarray}
[\hat{\boldsymbol{S}}]
&=& [\Lambda]^{-1} [\boldsymbol{A}_5] [\Lambda]^{*}
\nonumber \\
&=& [\Lambda]^{-1} \big( [\boldsymbol{S}] - [\gamma]^* \big) \big( [U] - [\gamma] [\boldsymbol{S}] \big)^{-1} [\Lambda]^{*}
\end{eqnarray}
ここで，
\begin{gather}
[\boldsymbol{S}] =
\begin{pmatrix}
S_{11} & S_{12} \\ S_{21} & S_{22}
\end{pmatrix}, \ \ \ \ \
[\gamma] =
\begin{pmatrix}
\gamma_1 & 0 \\ 0 & \gamma_2
\end{pmatrix}, \ \ \ \ \
[\Lambda] =
\begin{pmatrix}
\Lambda_1 & 0 \\ 0 & \Lambda_2
\end{pmatrix}
\end{gather}
ただし，\(\gamma_1\)，\(\gamma_2\)は基準反射係数を示し，通常の散乱行列の基準インピーダンス\(R_{01}\)，\(R_{02}\)と一般化散乱行列の複素基準インピーダンス\(Z_1\)，\(Z_2\)より，
\begin{gather}
\gamma_1 = \frac{Z_1 - R_{01}}{Z_1 + R_{01}}, \ \ \ \ \
\gamma_2 = \frac{Z_2 - R_{02}}{Z_2 + R_{02}}
\end{gather}
通常の散乱行列の基準インピーダンスを\(R_{01}=R_{02}=Z_c\)とすると，
\begin{gather}
\gamma_1 = \frac{Z_1 - Z_c}{Z_1 + Z_c}, \ \ \ \ \
\gamma_2 = \frac{Z_2 - Z_c}{Z_2 + Z_c}
\end{gather}
また，
\begin{eqnarray}
\Lambda_i &\equiv& (1-\gamma_i^*) \sqrt{\frac{1-\gamma_i \gamma_i^*}{(1-\gamma_i)(1-\gamma_i^*)}}
\nonumber \\
&=& \frac{1-\gamma_i^*}{|1-\gamma_i^*|} \sqrt{1-|\gamma_i|^2}
\nonumber \\
&=& \sqrt{\frac{ (1-\gamma_i^*)^2}{(1-\gamma_i)(1-\gamma_i^*)}} \sqrt{1-|\gamma_i|^2}
\nonumber \\
&=& \sqrt{\frac{1-\gamma_i^*}{1-\gamma_i}} \sqrt{1-|\gamma_i|^2} \ \ \ (i=1,2)
\end{eqnarray}
一般化散乱行列\([\hat{\boldsymbol{S}}]\)の要素を求めると，
\begin{eqnarray}
[\hat{\boldsymbol{S}}] &=&
\begin{pmatrix}
\hat{\mathcal{S}}_{11} & \hat{\mathcal{S}}_{12} \\
\hat{\mathcal{S}}_{21} & \hat{\mathcal{S}}_{22}
\end{pmatrix}
=
\begin{pmatrix}
\Lambda_1^{-1} & 0 \\
0 & \Lambda_2^{-1}
\end{pmatrix}
\begin{pmatrix}
S_{11}-\gamma_1^* & S_{12} \\
S_{21} & S_{22} - \gamma_2^*
\end{pmatrix}
\nonumber \\
&& \cdot \frac{1}{W}
\begin{pmatrix}
1-\gamma_2 S_{22} & \gamma_1 S_{12} \\
\gamma_2 S_{21} & 1-\gamma_1 S_{11}
\end{pmatrix}
\begin{pmatrix}
\Lambda_1^* & 0 \\
0 & \Lambda_2^*
\end{pmatrix}
\nonumber \\
&=& \frac{1}{W}
\begin{pmatrix}
\Lambda_1^{-1} (S_{11}-\gamma_1^*) & \Lambda_1^{-1 }S_{12} \\
\Lambda_2^{-1} S_{21} & \Lambda_2^{-1} (S_{22} - \gamma_2^*)
\end{pmatrix}
\begin{pmatrix}
(1-\gamma_2 S_{22}) \Lambda_1^* & \gamma_1 S_{12}\Lambda_2^* \\
\gamma_2 S_{21} \Lambda_1^* & (1-\gamma_1 S_{11}) \Lambda_2^*
\end{pmatrix}
\end{eqnarray}
ここで，
\begin{gather}
W = (1-\gamma_1 S_{11}) (1-\gamma_2 S_{22}) - \gamma_1 S_{12} \gamma_2 S_{21}
\nonumber
\end{gather}
これより，\(\hat{\mathcal{S}}_{11}\)は，
\begin{eqnarray}
\hat{\mathcal{S}}_{11} &=& \frac{1}{W}
\Big\{ \Lambda_1^{-1} (S_{11}-\gamma_1^*) (1-\gamma_2 S_{22}) \Lambda_1^*
+ \Lambda_1^{-1 }S_{12} \gamma_2 S_{21} \Lambda_1^* \Big\}
\nonumber \\
&=& \frac{1}{W} \cdot \frac{1-\gamma_1}{1-\gamma_1^*}
\Big\{ (S_{11}-\gamma_1^*) (1-\gamma_2 S_{22}) + S_{12} \gamma_2 S_{21} \Big\}
\end{eqnarray}
ここで，
\begin{gather}
\Lambda_1^{-1} \Lambda_1^* = \frac{1-\gamma_1}{1-\gamma_1^*}
\nonumber
\end{gather}
同様にして，\(\hat{\mathcal{S}}_{22}\)は，
\begin{gather}
\hat{\mathcal{S}}_{22} = \frac{1}{W} \cdot \frac{1-\gamma_2}{1-\gamma_2^*}
\Big\{ (S_{22}-\gamma_2^*) (1-\gamma_1 S_{11}) + S_{21} \gamma_1 S_{12} \Big\}
\end{gather}
また，\(\hat{\mathcal{S}}_{12}\)は，
\begin{eqnarray}
\hat{\mathcal{S}}_{12} &=& \frac{1}{W}
\Big\{ \Lambda_1^{-1} (S_{11}-\gamma_1^*) \gamma_1 S_{12}\Lambda_2^*
+ \Lambda_1^{-1 }S_{12} (1-\gamma_1 S_{11}) \Lambda_2^* \Big\}
\nonumber \\
&=& \frac{S_{12}}{W} \Lambda_1^{-1} \Lambda_2^*
\Big\{ (S_{11}-\gamma_1^*) \gamma_1+ (1-\gamma_1 S_{11}) \Big\}
\nonumber \\
&=& \frac{S_{12}}{W} \Lambda_1^{-1} \Lambda_2^* ( 1-\gamma_1 \gamma_1^*)
\nonumber \\
&=& \frac{S_{12}}{W}
\sqrt{\frac{1-\gamma_1}{1-\gamma_1^*}} \frac{1}{\sqrt{1-|\gamma_1|^2}}
\left( \sqrt{\frac{1-\gamma_2^*}{1-\gamma_2}} \sqrt{1-|\gamma_2|^2} \right)^*
(1-|\gamma_1|^2)
\nonumber \\
&=& \frac{S_{12}}{W}
\sqrt{\frac{1-\gamma_1}{1-\gamma_1^*}} \cdot
\sqrt{\frac{1-\gamma_2}{1-\gamma_2^*}} \cdot
\sqrt{1-|\gamma_1|^2} \sqrt{1-|\gamma_2|^2}
\nonumber \\
&=& \frac{S_{12}}{W}
\frac{(1-\gamma_1) (1-\gamma_2)}{|1-\gamma_1| |1-\gamma_2|}
\sqrt{1-|\gamma_1|^2} \sqrt{1-|\gamma_2|^2}
\end{eqnarray}
同様にして，\(\hat{\mathcal{S}}_{21}\)は，
\begin{eqnarray}
\hat{\mathcal{S}}_{21}
&=& \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}
\nonumber \\
&=& \frac{S_{21}}{S_{12}} \hat{\mathcal{S}}_{12}
\end{eqnarray}