[Circuit] Two-Port Networks 分析

電路知識：Two-Port Networks 分析
工具：Qucs

功能：分析電路用

需知

• Port 指的是電流的進出路徑，$R,L,C$ 則是 one-port 元件
  而電路分析往往是 two-port，因為不知頭也不知尾
• 中間的黑盒子必須為 linear system
• 黑盒子不含任何 independent source，但可有 dependent source
• 可用 Qucs 輔助，利用輸入或輸出 1V or 1A 得到參數

參數類型

• $\mathbf{Z}$
• $\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ {\mathbf{V}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{z}}_{\mathbf{11}}& {\mathbf{z}}_{\mathbf{12}}\\ {\mathbf{z}}_{\mathbf{21}}& {\mathbf{z}}_{\mathbf{22}}\end{array}\right]\left[\begin{array}{c}{\mathbf{I}}_{\mathbf{1}}\\ {\mathbf{I}}_{\mathbf{2}}\end{array}\right]$

  ${\begin{array}{c}{\mathbf{z}}_{\mathbf{11}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{z}}_{\mathbf{12}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}{\mathbf{z}}_{\mathbf{21}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{z}}_{\mathbf{22}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$

• 等效電路
  
• $\mathbf{Y}$
• $\left[\begin{array}{c}{\mathbf{I}}_{\mathbf{1}}\\ {\mathbf{I}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{y}}_{\mathbf{11}}& {\mathbf{y}}_{\mathbf{12}}\\ {\mathbf{y}}_{\mathbf{21}}& {\mathbf{y}}_{\mathbf{22}}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ {\mathbf{V}}_{\mathbf{2}}\end{array}\right]$

  ${\begin{array}{c}{\mathbf{y}}_{\mathbf{11}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{V}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{y}}_{\mathbf{12}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{V}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}{\mathbf{y}}_{\mathbf{21}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{2}}}{{\mathbf{V}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{y}}_{\mathbf{22}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{2}}}{{\mathbf{V}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$

• 等效電路
  
• $\mathbf{H}$
• $\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ {\mathbf{I}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{h}}_{\mathbf{11}}& {\mathbf{h}}_{\mathbf{12}}\\ {\mathbf{h}}_{\mathbf{21}}& {\mathbf{h}}_{\mathbf{22}}\end{array}\right]\left[\begin{array}{c}{\mathbf{I}}_{\mathbf{1}}\\ {\mathbf{V}}_{\mathbf{2}}\end{array}\right]$

  ${\begin{array}{c}{\mathbf{h}}_{\mathbf{11}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{h}}_{\mathbf{12}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{1}}}{{\mathbf{V}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}{\mathbf{h}}_{\mathbf{21}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{h}}_{\mathbf{22}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{2}}}{{\mathbf{V}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$

• 等效電路
  
• $\mathbf{G}$
• $\left[\begin{array}{c}{\mathbf{I}}_{\mathbf{1}}\\ {\mathbf{V}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{g}}_{\mathbf{11}}& {\mathbf{g}}_{\mathbf{12}}\\ {\mathbf{g}}_{\mathbf{21}}& {\mathbf{g}}_{\mathbf{22}}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ {\mathbf{I}}_{\mathbf{2}}\end{array}\right]$

  ${\begin{array}{c}{\mathbf{g}}_{\mathbf{11}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{V}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{g}}_{\mathbf{12}}\mathbf{=}\frac{{\mathbf{I}}_{\mathbf{1}}}{{\mathbf{I}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}{\mathbf{g}}_{\mathbf{21}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{2}}}{{\mathbf{V}}_{\mathbf{1}}}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}{\mathbf{g}}_{\mathbf{22}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{2}}}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$

• 等效電路
  
• $\mathbf{T}$
• $\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ {\mathbf{I}}_{\mathbf{1}}\end{array}\right]=\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\\ \mathbf{C}& \mathbf{D}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{2}}\\ \mathbf{-}{\mathbf{I}}_{\mathbf{2}}\end{array}\right]$

  ${\begin{array}{c}\mathbf{A}=\frac{V_1}{V_2}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}\mathbf{B}=-\frac{V_1}{I_2}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}\mathbf{C}=\frac{I_1}{V_2}\end{array}|}_{{\mathbf{I}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}\mathbf{D}=-\frac{I_1}{I_2}\end{array}|}_{{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}}$

• $\mathbf{t}$
• $\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{2}}\\ {\mathbf{I}}_{\mathbf{2}}\end{array}\right]=\left[\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{c}& \mathbf{d}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{\mathbf{1}}\\ \mathbf{-}{\mathbf{I}}_{\mathbf{1}}\end{array}\right]$

  ${\begin{array}{c}\mathbf{a}=\frac{V_2}{V_1}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}\mathbf{b}=-\frac{V_2}{I_1}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$
  ${\begin{array}{c}\mathbf{c}=\frac{I_2}{V_1}\end{array}|}_{{\mathbf{I}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$	${\begin{array}{c}\mathbf{d}=-\frac{I_2}{I_1}\end{array}|}_{{\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{0}}$

• $S$
• $\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$  

  ${\begin{array}{c}{S}_{11}=\frac{b_1}{a_1}\end{array}|}_{a_2=0}$	${\begin{array}{c}{S}_{12}=\frac{b_1}{a_2}\end{array}|}_{a_1=0}$
  ${\begin{array}{c}{S}_{21}=\frac{b_2}{a_1}\end{array}|}_{a_2=0}$	${\begin{array}{c}{S}_{22}=\frac{b_2}{a_2}\end{array}|}_{a_1=0}$

• [Circuit] S-Parameters (Scattering parameters) 

合併方法

• 利用轉換表，轉換成對應參數
• Series 連接 $\mathbf{Z}={\mathbf{Z}}_a+{\mathbf{Z}}_b$
  
  $$\begin{array}{rl}{\mathbf{V}}_{1a}& ={\mathbf{z}}_{11a}{\mathbf{I}}_{1a}+{\mathbf{z}}_{12a}{\mathbf{I}}_{2a}\\ {\mathbf{V}}_{2a}& ={\mathbf{z}}_{21a}{\mathbf{I}}_{1a}+{\mathbf{z}}_{22a}{\mathbf{I}}_{2a}\\ \\ {\mathbf{V}}_{1b}& ={\mathbf{z}}_{11b}{\mathbf{I}}_{1b}+{\mathbf{z}}_{12a}{\mathbf{I}}_{2b}\\ {\mathbf{V}}_{2b}& ={\mathbf{z}}_{21b}{\mathbf{I}}_{1b}+{\mathbf{z}}_{22a}{\mathbf{I}}_{2b}\\ \\ {\mathbf{I}}_1& ={\mathbf{I}}_{1a}={\mathbf{I}}_{1b}\\ {\mathbf{I}}_2& ={\mathbf{I}}_{2a}={\mathbf{I}}_{2b}\\ \\ {\mathbf{V}}_1& ={\mathbf{V}}_{1a}+{\mathbf{V}}_{1b}=({\mathbf{z}}_{11a}+{\mathbf{z}}_{11b}){\mathbf{I}}_1+({\mathbf{z}}_{12a}+{\mathbf{z}}_{12b}){\mathbf{I}}_2\\ {\mathbf{V}}_2& ={\mathbf{V}}_{2a}+{\mathbf{V}}_{2b}=({\mathbf{z}}_{21a}+{\mathbf{z}}_{21b}){\mathbf{I}}_1+({\mathbf{z}}_{22a}+{\mathbf{z}}_{22b}){\mathbf{I}}_2\end{array}$$ $$\left[\begin{array}{c}{\mathbf{V}}_1\\ {\mathbf{V}}_2\end{array}\right]=\left[\begin{array}{cc}{\mathbf{z}}_{11a}+{\mathbf{z}}_{11b}& {\mathbf{z}}_{12a}+{\mathbf{z}}_{12b}\\ {\mathbf{z}}_{21a}+{\mathbf{z}}_{21b}& {\mathbf{z}}_{22a}+{\mathbf{z}}_{22b}\end{array}\right]\left[\begin{array}{c}{\mathbf{I}}_1\\ {\mathbf{I}}_2\end{array}\right]=({\mathbf{Z}}_a+{\mathbf{Z}}_b)\left[\begin{array}{c}{\mathbf{I}}_1\\ {\mathbf{I}}_2\end{array}\right]=\mathbf{Z}\left[\begin{array}{c}{\mathbf{I}}_1\\ {\mathbf{I}}_2\end{array}\right]$$
• Parallel 連接 $\mathbf{Y}={\mathbf{Y}}_a+{\mathbf{Y}}_b$
  
  $$\begin{array}{rl}{\mathbf{I}}_{1a}& ={\mathbf{y}}_{11a}{\mathbf{V}}_{1a}+{\mathbf{y}}_{12a}{\mathbf{V}}_{2a}\\ {\mathbf{I}}_{2a}& ={\mathbf{y}}_{21a}{\mathbf{V}}_{1a}+{\mathbf{y}}_{22a}{\mathbf{V}}_{2a}\\ \\ {\mathbf{I}}_{1b}& ={\mathbf{y}}_{11b}{\mathbf{V}}_{1b}+{\mathbf{y}}_{12a}{\mathbf{V}}_{2b}\\ {\mathbf{I}}_{2b}& ={\mathbf{y}}_{21b}{\mathbf{V}}_{1b}+{\mathbf{y}}_{22a}{\mathbf{V}}_{2b}\\ \\ {\mathbf{V}}_1& ={\mathbf{V}}_{1a}={\mathbf{V}}_{1b}\\ {\mathbf{V}}_2& ={\mathbf{V}}_{2a}={\mathbf{V}}_{2b}\\ \\ {\mathbf{I}}_1& ={\mathbf{I}}_{1a}+{\mathbf{I}}_{1b}=({\mathbf{y}}_{11a}+{\mathbf{y}}_{11b}){\mathbf{V}}_1+({\mathbf{y}}_{12a}+{\mathbf{y}}_{12b}){\mathbf{V}}_2\\ {\mathbf{I}}_2& ={\mathbf{I}}_{2a}+{\mathbf{I}}_{2b}=({\mathbf{y}}_{21a}+{\mathbf{y}}_{21b}){\mathbf{V}}_1+({\mathbf{y}}_{22a}+{\mathbf{y}}_{22b}){\mathbf{V}}_2\end{array}$$ $$\left[\begin{array}{c}{\mathbf{I}}_1\\ {\mathbf{I}}_2\end{array}\right]=\left[\begin{array}{cc}{\mathbf{y}}_{11a}+{\mathbf{y}}_{11b}& {\mathbf{y}}_{12a}+{\mathbf{y}}_{12b}\\ {\mathbf{y}}_{21a}+{\mathbf{y}}_{21b}& {\mathbf{y}}_{22a}+{\mathbf{y}}_{22b}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_1\\ {\mathbf{V}}_2\end{array}\right]=({\mathbf{Y}}_a+{\mathbf{Y}}_b)\left[\begin{array}{c}{\mathbf{V}}_1\\ {\mathbf{V}}_2\end{array}\right]=\mathbf{Y}\left[\begin{array}{c}{\mathbf{V}}_1\\ {\mathbf{V}}_2\end{array}\right]$$
• Cascade 連接 $\mathbf{T}={\mathbf{T}}_a\times {\mathbf{T}}_b$
  
  $$\begin{array}{rl}\left[\begin{array}{c}{\mathbf{V}}_{1a}\\ {\mathbf{I}}_{1a}\end{array}\right]& =\left[\begin{array}{cc}{\mathbf{A}}_a& {\mathbf{B}}_a\\ {\mathbf{C}}_a& {\mathbf{D}}_a\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{2a}\\ -{\mathbf{I}}_{2a}\end{array}\right]\\ \left[\begin{array}{c}{\mathbf{V}}_{1b}\\ {\mathbf{I}}_{1b}\end{array}\right]& =\left[\begin{array}{cc}{\mathbf{A}}_b& {\mathbf{B}}_b\\ {\mathbf{C}}_b& {\mathbf{D}}_b\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{2b}\\ -{\mathbf{I}}_{2b}\end{array}\right]\\ \\ \left[\begin{array}{c}{\mathbf{V}}_1\\ {\mathbf{I}}_1\end{array}\right]& =\left[\begin{array}{c}{\mathbf{V}}_{1a}\\ {\mathbf{I}}_{1a}\end{array}\right]\\ & =\left[\begin{array}{cc}{\mathbf{A}}_a& {\mathbf{B}}_a\\ {\mathbf{C}}_a& {\mathbf{D}}_a\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{2a}\\ -{\mathbf{I}}_{2a}\end{array}\right]\\ & =\left[\begin{array}{cc}{\mathbf{A}}_a& {\mathbf{B}}_a\\ {\mathbf{C}}_a& {\mathbf{D}}_a\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{1b}\\ {\mathbf{I}}_{1b}\end{array}\right]\\ & =\left[\begin{array}{cc}{\mathbf{A}}_a& {\mathbf{B}}_a\\ {\mathbf{C}}_a& {\mathbf{D}}_a\end{array}\right]\left[\begin{array}{cc}{\mathbf{A}}_b& {\mathbf{B}}_b\\ {\mathbf{C}}_b& {\mathbf{D}}_b\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{2b}\\ -{\mathbf{I}}_{2b}\end{array}\right]\\ & =\left[\begin{array}{cc}{\mathbf{A}}_a& {\mathbf{B}}_a\\ {\mathbf{C}}_a& {\mathbf{D}}_a\end{array}\right]\left[\begin{array}{cc}{\mathbf{A}}_b& {\mathbf{B}}_b\\ {\mathbf{C}}_b& {\mathbf{D}}_b\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_2\\ -{\mathbf{I}}_2\end{array}\right]\\ & ={\mathbf{T}}_a{\mathbf{T}}_b\left[\begin{array}{c}{\mathbf{V}}_2\\ -{\mathbf{I}}_2\end{array}\right]\\ & =\mathbf{T}\left[\begin{array}{c}{\mathbf{V}}_2\\ -{\mathbf{I}}_2\end{array}\right]\end{array}$$

參考

Alexander & Sadiku, “Fundamentals of Electric Circuits, Second Edition”, McGraw-Hill, New York, NY, 2004.
Wiki Two-port network
Two-Port Networks