38. 임피던스 정수

38. 임피던스 정수

독립전원을 포함하지 않는 선형 2포트

\(\mathbf{V}_{1}=\mathbf{z}_{11}\mathbf{I}_{1}+\mathbf{z}_{12}\mathbf{I}_{2}, \mathbf{V}_{2}=\mathbf{z}_{21}\mathbf{I}_{1}+\mathbf{z}_{22}\mathbf{I}_{2}\)

\(\mathbf{V}=\begin{pmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{pmatrix},\,\mathbf{z}=\begin{pmatrix}\mathbf{z}_{11}&\mathbf{z}_{12}\\ \mathbf{z}_{21}&\mathbf{z}_{22}\end{pmatrix},\,\mathbf{I}=\begin{pmatrix}\mathbf{I}_{1}\\ \mathbf{I}_{2}\end{pmatrix}\)라고 하면 \(\mathbf{V}=\mathbf{z}\mathbf{I}\), \(\begin{pmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{pmatrix}=\begin{pmatrix}\mathbf{z}_{11}&\mathbf{z}_{12}\\ \mathbf{z}_{21}&\mathbf{z}_{22}\end{pmatrix}\begin{pmatrix}\mathbf{I}_{1}\\ \mathbf{I}_{2}\end{pmatrix}\)

\(\displaystyle\mathbf{z}_{11}=\frac{\mathbf{V}_{1}}{\mathbf{I}_{1}}|_{\mathbf{I}_{2}=0}\)(출력단자의 개방): 개방회로 입력 임피던스.

\(\displaystyle\mathbf{z}_{21}=\frac{\mathbf{V}_{2}}{\mathbf{I}_{2}}|_{\mathbf{I}_{2}=0}\)(출력단자의 개방): 개방회로 전달 임피던스.

\(\displaystyle\mathbf{z}_{22}=\frac{\mathbf{V}_{2}}{\mathbf{I}_{1}}|_{\mathbf{I}_{1}=0}\)(입력단자의 개방): 개방회로 출력 임피던스.

\(\displaystyle\mathbf{z}_{12}=\frac{\mathbf{V}_{1}}{\mathbf{I}_{2}}|_{\mathbf{I}_{1}=0}\)(입력단자의 개방): 개방회로 전달 임피던스.

\(\displaystyle\mathbf{I}_{1}=\frac{\left|\begin{matrix}\mathbf{V}_{1}&\mathbf{z}_{12}\\ \mathbf{V}_{2}&\mathbf{z}_{22}\end{matrix}\right|}{\left|\begin{matrix}\mathbf{z}_{11}&\mathbf{z}_{12}\\ \mathbf{z}_{21}&\mathbf{z}_{22}\end{matrix}\right|}=\left(\frac{\mathbf{z}_{22}}{\mathbf{z}_{11}\mathbf{z}_{22}-\mathbf{z}_{12}\mathbf{z}_{21}}\right)\mathbf{V}_{1}-\left(\frac{\mathbf{z}_{12}}{\mathbf{z}_{11}\mathbf{z}_{22}-\mathbf{z}_{12}\mathbf{z}_{21}}\right)\mathbf{V}_{2}\)

\(\displaystyle\mathbf{y}_{11}=\frac{\mathbf{z}_{22}}{\Delta_{\mathbf{z}}},\,\mathbf{y}_{12}=-\frac{\mathbf{z}_{12}}{\Delta_{\mathbf{z}}},\,\mathbf{y}_{21}=-\frac{\mathbf{z}_{21}}{\Delta_{\mathbf{z}}},\,\mathbf{y}_{22}=\frac{\mathbf{z}_{11}}{\Delta_{\mathbf{z}}}\,\left(\Delta_{\mathbf{z}}=\det\begin{pmatrix}\mathbf{z}_{11}&\mathbf{z}_{12}\\ \mathbf{z}_{21}&\mathbf{z}_{22}\end{pmatrix}\right)\)

\(\displaystyle\mathbf{z}_{11}=\frac{\mathbf{z}_{22}}{\Delta_{\mathbf{Y}}},\,\mathbf{z}_{12}=-\frac{\mathbf{y}_{12}}{\Delta_{\mathbf{Y}}},\,\mathbf{z}_{21}=-\frac{\mathbf{y}_{21}}{\Delta_{\mathbf{Y}}},\,\mathbf{z}_{22}=\frac{\mathbf{y}_{11}}{\Delta_{\mathbf{Y}}}\,\left(\Delta_{\mathbf{Y}}=\det\begin{pmatrix}\mathbf{y}_{11}&\mathbf{y}_{12}\\ \mathbf{y}_{21}&\mathbf{y}_{22}\end{pmatrix}\right)\)

입력과 출력에 공통인 기준마디를 가진 2포트

\(\mathbf{I}=\mathbf{I}_{A}=\mathbf{I}_{B}\), \(\mathbf{V}=\mathbf{V}_{A}+\mathbf{V}_{B}=\mathbf{z}_{A}\mathbf{I}_{A}+\mathbf{z}_{B}\mathbf{I}_{B}=(\mathbf{z}_{A}+\mathbf{z}_{B})\mathbf{I}=\mathbf{z}\mathbf{I}\)

\(\Rightarrow\mathbf{V}=\mathbf{zI},\,\mathbf{z}=\mathbf{z}_{A}+\mathbf{z}_{B}\).

\(\Rightarrow\mathbf{z}_{11}=\mathbf{z}_{11A}+\mathbf{z}_{11B}\).

\(\mathbf{V}_{1}=45\mathbf{I}_{1}+25\mathbf{I}_{2}+0.5\mathbf{V}_{2}\), \(\mathbf{V}_{2}=25\mathbf{I}_{1}+75\mathbf{I}_{2}+0.5\mathbf{V}_{2}\rightarrow0.5\mathbf{V}_{2}=25\mathbf{I}_{1}+75\mathbf{I}_{2}\)

\(\Rightarrow\,\mathbf{V}_{1}=70\mathbf{I}_{1}+100\mathbf{I}_{2},\,\mathbf{V}_{2}=50\mathbf{I}_{1}+150\mathbf{I}_{2}\)

\(\mathbf{Z}=\begin{pmatrix}70&100\\50&150\end{pmatrix}(\Omega)\)

참고자료:

Engineering Circuit Analysis 8th edition, Hayt, Kemmerly, Durbin, McGraw-Hill