4단자망(T형회로,파이형회로),Z파라미터(T형 회로),Y형 파라미터(파이형 회로)/회로이론

 

 

●회로이론 32강: 4단자망(T형회로,$\pi$형회로) ,Z파라미터(T형 회로),Y형 파라미터($\pi$형 회로)

$V_1=AV_2+BI_2$

$I_1=CV_2+DI_2$

행렬식으로 나타내면...

$\begin{pmatrix}V_1\\I_1\end{pmatrix}=\begin{pmatrix}A&B\\C&D\\ \end{pmatrix}\begin{pmatrix}V_2\\I_2\end{pmatrix}$

$A=\frac{V_1}{V_2}\mid_{I_2=0}\,\,\,\to$개방   전압이득.

$B=\frac{V_1}{I_2}\mid_{V_2=0}\,\,\, \to$ 단락 임피던스.

$C=\frac{I_1}{V_2}\mid_{I_2=0}\,\,\, \to$  개방 어드미턴스.

$D+\frac{I_1}{I_2}\mid_{V_2=0}\,\,\, \to$ 단락 전류이득.

 

예시)직렬에서..

$V_1=1V_2+ZI_2$

$I_1=0V_2+1I_2$

$\binom{V_1}{I_1}=\begin{pmatrix}1&Z\\0&1\end{pmatrix}\binom{V_2}{I_2}$

 

예시)병렬에서

$V_1=1V_2+0I_2$

$I_1=YV_2+1I_2$

$\binom{V_1}{I_1}=\begin{pmatrix}1&0\\Y&1\end{pmatrix}\binom{V_2}{I_2}$

 

*4단자망 성질

①AD-BC=1

②A=D

 

●4단자망 T형 회로

$\begin{pmatrix}1&Z_1\\0&1\end{pmatrix}\begin{pmatrix}1&0\\ \frac{1}{Z_3}&1\end{pmatrix}\begin{pmatrix}1&Z_2\\0&1\end{pmatrix}$

$=\begin{pmatrix}1+\frac{Z_1}{Z_3}\,\,\,&\frac{Z_1Z_2+Z_2Z_3+Z_3Z_1}{Z_3}\\ \frac{1}{Z_3}\,\,\,&1+\frac{Z_2}{Z_3}\end{pmatrix}$

$=\begin{pmatrix}A&B\\C&D\end{pmatrix}$

 

●4단자망 $\pi$형 회로

$\begin{pmatrix}A&B\\C&D\end{pmatrix}=\begin{pmatrix}1+\frac{Z_2}{Z_3}&Z_2\\ \frac{Z_1+Z_2+Z_3}{Z_1Z_3}&1+\frac{Z_2}{Z_1}\end{pmatrix}$

 

●Z파라미터(전압을 전류로)

$V_1=Z_{11}I_1+Z_{12}I_2$

$V_2=Z_{21}I_1+Z_{22}I_2$

$\binom{V_1}{V_2}=\begin{pmatrix}Z_{11}\,\,\,&Z_{12}\\Z_{21}\,\,\,&Z_{22}\end{pmatrix}\binom{I_1}{I_2}$

※위에서 $Z_{11}\,\,\, Z_{12}\,\,\, Z_{21}\,\,\, Z_{22}$를 임피던스(Z) 파라미터라고 한다.

$Z_{11}=\frac{V_1}{I_1}\,\,\,|_{I_{2}=0}$

$Z_{12}=\frac{V_1}{I_2}\,\,\,|_{I_{1}=0}$

$Z_{21}=\frac{V_2}{I_1}\,\,\,|_{I_{2}=0}$

$Z_{22}=\frac{V_2}{I_2}\,\,\,|_{I_{1}=0}$

 

●T형회로(Z파라미터)

$Z_{11}=Z_1+Z_2$

$Z_{12}=Z_3$

$Z_{21}=Z_3$

$Z_{22}=Z_2+Z_3$

 

●Y파라미터(전류을 전압으로)

$I_1=Y_{11}V_1+Y_{12}V_2$

$I_2=Y_{21}V_1+Y_{22}V_2$

$\binom{I_1}{I_2}=\begin{pmatrix}Y_{11}\,\,\,&Y_{12}\\Y_{21}\,\,\,&Y_{22} \end{pmatrix} \binom{V_1}{V_2}$

※위에서 $Y_{11}\,\,\, Y_{12}\,\,\, Y_{21}\,\,\, Y_{22}$를 어드미턴스(Y) 파라미터라고 한다.

$Y_{11}=\frac{I_1}{V_1}|_{V_{2}=0}$

$Y_{12}=\frac{I_1}{V_2}|_{V_{1}=0}$

$Y_{21}=\frac{I_2}{V_1}|_{V_{2}=0}$

$Y_{22}=\frac{I_2}{V_2}|_{V_{1}=0}$

 

●$\pi$형 회로(Y파라미터)

$Y_{11}=Y_1+Y_2$

$Y_{12}=-Y_2$

$Y_{21}=-Y_2$

$Y_{22}=Y_2+Y_3$

 

●이상변압기 4단자 정수

$\underset{(권수비)}{a}=\frac{V_1}{V_2}=\frac{I_2}{I_1}$

$V_1=aV_2+0I_2$

$I_1=0V_2+\frac{1}{a}I_2$

$\begin{pmatrix}a\,\,\,&0\\0\,\,\,&\frac{1}{a}\end{pmatrix}$