3.ＹマトリクスからFマトリクスへの変換
\(\displaystyle\rm A=-\frac{Y_{22}}{Y_{21}}\)
\(\displaystyle\rm B=-\frac{1}{Y_{21}}\)
\(\displaystyle\rm C=\frac{-Y_{11}Y_{22}+Y_{12}Y_{21}}{Y_{21}}\)
\(\displaystyle\rm D=-\frac{Y_{11}}{Y_{21}}\)
YマトリクスからFマトリクスへの変換の詳細を下記に示す。
まず、Yマトリクスは以下のとおりである(電流I2はFマトリクスとYマトリクスで向きが逆なので負となる)。
\(\left[\begin{array}{c}I_{1} \\-I_{2}\end{array}\right]=\left[\begin{array}{cc}Y_{11} &Y_{12}\\Y_{21} &Y_{22}\end{array}\right]\left[\begin{array}{c}V_{1} \\V_{2}\end{array}\right]\tag{3.1}\)
(3.1)式を Fマトリクスの定義式の形に整理すると、下記のとおり変換式が導かれる。
\(\displaystyle\rm \left[\begin{array}{cc} Y_{11} & -１ \\ Y_{21} & 0 \end{array}\right]\left[\begin{array}{c}V_{1} \\I_{1}\end{array}\right]=\left[\begin{array}{cc} -Y_{12} & 0 \\ -Y_{22} & -1 \end{array}\right]\left[\begin{array}{c}V_{2} \\I_{2}\end{array}\right]\)
\(\displaystyle\rm \left[\begin{array}{c}V_{1} \\I_{1}\end{array}\right]=\frac{1}{Y_{21}} \left[\begin{array}{cc} 0 & １ \\ -Y_{21} & Y_{11} \end{array}\right] \left[\begin{array}{cc} -Y_{12} & 0 \\ -Y_{22} & -1 \end{array}\right]\left[\begin{array}{c}V_{2} \\I_{2}\end{array}\right]\)
\(\displaystyle\rm= \left[\begin{array}{cc} -\frac{Y_{22}}{Y_{21}} & -\frac{1}{Y_{21}} \\ \frac{-Y_{11}Y_{22}+Y_{12}Y_{21}}{Y_{21}} & -\frac{Y_{11}}{Y_{21}} \end{array}\right]\left[\begin{array}{c}V_{2} \\I_{2}\end{array}\right]\)
4.FマトリクスからYマトリクスへの変換
\(\displaystyle\rm Y_{11}=\frac{D}{B}\)
\(\displaystyle\rm Y_{12}=\frac{BC-AD}{B}\)
\(\displaystyle\rm Y_{21}=-\frac{1}{B}\)
\(\displaystyle\rm Y_{22}=\frac{A}{B}\)
FマトリクスからYマトリクスへの変換の詳細を下記に示す。
まず、Fマトリクスは以下のとおりである (電流I2はFマトリクスとYマトリクスで向きが逆なので負となる)。
\(\left[\begin{array}{c}V_{1} \\I_{1}\end{array}\right]=\left[\begin{array}{cc}A &B \\ C & D \end{array}\right]\left[\begin{array}{c}V_{2} \\-I_{2}\end{array}\right]\tag{4.1}\)
(4.1)式を Yマトリクスの定義式の形に整理すると、下記のとおり変換式が導かれる。
\(\displaystyle\rm \left[\begin{array}{cc} 0 & B \\ 1 & D \end{array}\right]\left[\begin{array}{c}I_{1} \\I_{2}\end{array}\right]=\left[\begin{array}{cc} -1 & A \\ 0 & C \end{array}\right]\left[\begin{array}{c}V_{1} \\V_{2}\end{array}\right]\)
\(\displaystyle\rm \left[\begin{array}{c}I_{1} \\I_{2}\end{array}\right]=-\frac{1}{B} \left[\begin{array}{cc} D & -B \\ -1 & 0 \end{array}\right] \left[\begin{array}{cc} -1 & A \\ 0 & C \end{array}\right]\left[\begin{array}{c}V_{1} \\V_{2}\end{array}\right]\)
\(\displaystyle\rm= \left[\begin{array}{cc} \frac{D}{B} & \frac{BC-AD}{B} \\ -\frac{1}{B} & \frac{A}{B} \end{array}\right]\left[\begin{array}{c}V_{1} \\V_{2}\end{array}\right]\)