[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

${\bf Z}$

$ \begin{bmatrix} \mathbf{V_1}\\ \mathbf{V_2} \end{bmatrix} = \begin{bmatrix} \mathbf{z_{11}} & \mathbf{z_{12}}\\ \mathbf{z_{21}} & \mathbf{z_{22}} \end{bmatrix} \begin{bmatrix} \mathbf{I_1}\\ \mathbf{I_2} \end{bmatrix} $

等效電路

${\bf Y}$

$ \begin{bmatrix} \mathbf{I_1}\\ \mathbf{I_2} \end{bmatrix} = \begin{bmatrix} \mathbf{y_{11}} & \mathbf{y_{12}}\\ \mathbf{y_{21}} & \mathbf{y_{22}} \end{bmatrix} \begin{bmatrix} \mathbf{V_1}\\ \mathbf{V_2} \end{bmatrix} $

等效電路

${\bf H}$

$ \begin{bmatrix} \mathbf{V_1}\\ \mathbf{I_2} \end{bmatrix} = \begin{bmatrix} \mathbf{h_{11}} & \mathbf{h_{12}}\\ \mathbf{h_{21}} & \mathbf{h_{22}} \end{bmatrix} \begin{bmatrix} \mathbf{I_1}\\ \mathbf{V_2} \end{bmatrix} $

等效電路

${\bf G}$

$ \begin{bmatrix} \mathbf{I_1}\\ \mathbf{V_2} \end{bmatrix} = \begin{bmatrix} \mathbf{g_{11}} & \mathbf{g_{12}}\\ \mathbf{g_{21}} & \mathbf{g_{22}} \end{bmatrix} \begin{bmatrix} \mathbf{V_1}\\ \mathbf{I_2} \end{bmatrix} $

等效電路

${\bf T}$

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

${\bf t}$

$ \begin{bmatrix} \mathbf{V_2}\\ \mathbf{I_2} \end{bmatrix} = \begin{bmatrix} \mathbf{a} & \mathbf{b}\\ \mathbf{c} & \mathbf{d} \end{bmatrix} \begin{bmatrix} \mathbf{V_1}\\ \mathbf{-I_1} \end{bmatrix} $

${S}$

$ \begin{bmatrix} b_1\\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_{1}\\ a_{2} \end{bmatrix} $

[Circuit] S-Parameters (Scattering parameters)

合併方法

利用轉換表，轉換成對應參數
Series 連接 ${\bf Z}={\bf Z}_a+{\bf Z}_b$

$$ \begin{align*} \mathbf{V}_{1a} &= \mathbf{z}_{11a}\mathbf{I}_{1a}+\mathbf{z}_{12a}\mathbf{I}_{2a}\\ \mathbf{V}_{2a} &= \mathbf{z}_{21a}\mathbf{I}_{1a}+\mathbf{z}_{22a}\mathbf{I}_{2a}\\ \\ \mathbf{V}_{1b} &= \mathbf{z}_{11b}\mathbf{I}_{1b}+\mathbf{z}_{12a}\mathbf{I}_{2b}\\ \mathbf{V}_{2b} &= \mathbf{z}_{21b}\mathbf{I}_{1b}+\mathbf{z}_{22a}\mathbf{I}_{2b}\\ \\ \mathbf{I}_{1} &= \mathbf{I}_{1a}=\mathbf{I}_{1b} \\ \mathbf{I}_{2} &= \mathbf{I}_{2a}=\mathbf{I}_{2b} \\ \\ \mathbf{V}_{1} &= \mathbf{V}_{1a}+\mathbf{V}_{1b} =(\mathbf{z}_{11a}+\mathbf{z}_{11b})\mathbf{I}_{1}+(\mathbf{z}_{12a}+\mathbf{z}_{12b})\mathbf{I}_{2}\\ \mathbf{V}_{2} &= \mathbf{V}_{2a}+\mathbf{V}_{2b} =(\mathbf{z}_{21a}+\mathbf{z}_{21b})\mathbf{I}_{1}+(\mathbf{z}_{22a}+\mathbf{z}_{22b})\mathbf{I}_{2}\\ \end{align*} $$ $$ \begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{V}_{2} \end{bmatrix} = \begin{bmatrix} \mathbf{z}_{11a}+\mathbf{z}_{11b} & \mathbf{z}_{12a}+\mathbf{z}_{12b}\\ \mathbf{z}_{21a}+\mathbf{z}_{21b} & \mathbf{z}_{22a}+\mathbf{z}_{22b} \end{bmatrix} \begin{bmatrix} \mathbf{I}_{1}\\ \mathbf{I}_{2} \end{bmatrix} = (\mathbf{Z}_{a}+\mathbf{Z}_{b}) \begin{bmatrix} \mathbf{I}_{1}\\ \mathbf{I}_{2} \end{bmatrix} = \mathbf{Z} \begin{bmatrix} \mathbf{I}_{1}\\ \mathbf{I}_{2} \end{bmatrix} $$
Parallel 連接 ${\bf Y}={\bf Y}_a+{\bf Y}_b$

$$ \begin{align*} \mathbf{I}_{1a} &= \mathbf{y}_{11a}\mathbf{V}_{1a}+\mathbf{y}_{12a}\mathbf{V}_{2a}\\ \mathbf{I}_{2a} &= \mathbf{y}_{21a}\mathbf{V}_{1a}+\mathbf{y}_{22a}\mathbf{V}_{2a}\\ \\ \mathbf{I}_{1b} &= \mathbf{y}_{11b}\mathbf{V}_{1b}+\mathbf{y}_{12a}\mathbf{V}_{2b}\\ \mathbf{I}_{2b} &= \mathbf{y}_{21b}\mathbf{V}_{1b}+\mathbf{y}_{22a}\mathbf{V}_{2b}\\ \\ \mathbf{V}_{1} &= \mathbf{V}_{1a}=\mathbf{V}_{1b} \\ \mathbf{V}_{2} &= \mathbf{V}_{2a}=\mathbf{V}_{2b} \\ \\ \mathbf{I}_{1} &= \mathbf{I}_{1a}+\mathbf{I}_{1b} =(\mathbf{y}_{11a}+\mathbf{y}_{11b})\mathbf{V}_{1}+(\mathbf{y}_{12a}+\mathbf{y}_{12b})\mathbf{V}_{2}\\ \mathbf{I}_{2} &= \mathbf{I}_{2a}+\mathbf{I}_{2b} =(\mathbf{y}_{21a}+\mathbf{y}_{21b})\mathbf{V}_{1}+(\mathbf{y}_{22a}+\mathbf{y}_{22b})\mathbf{V}_{2}\\ \end{align*} $$ $$ \begin{bmatrix} \mathbf{I}_{1}\\ \mathbf{I}_{2} \end{bmatrix} = \begin{bmatrix} \mathbf{y}_{11a}+\mathbf{y}_{11b} & \mathbf{y}_{12a}+\mathbf{y}_{12b}\\ \mathbf{y}_{21a}+\mathbf{y}_{21b} & \mathbf{y}_{22a}+\mathbf{y}_{22b} \end{bmatrix} \begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{V}_{2} \end{bmatrix} = (\mathbf{Y}_{a}+\mathbf{Y}_{b}) \begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{V}_{2} \end{bmatrix} = \mathbf{Y} \begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{V}_{2} \end{bmatrix} $$
Cascade 連接 ${\bf T}={\bf T}_a \times {\bf T}_b$

$$ \begin{align*} \begin{bmatrix} \mathbf{V}_{1a}\\ \mathbf{I}_{1a} \end{bmatrix} &= \begin{bmatrix} \mathbf{A}_a & \mathbf{B}_a\\ \mathbf{C}_a & \mathbf{D}_a \end{bmatrix} \begin{bmatrix} \mathbf{V}_{2a}\\ -\mathbf{I}_{2a} \end{bmatrix} \\ \begin{bmatrix} \mathbf{V}_{1b}\\ \mathbf{I}_{1b} \end{bmatrix} &= \begin{bmatrix} \mathbf{A}_b & \mathbf{B}_b\\ \mathbf{C}_b & \mathbf{D}_b \end{bmatrix} \begin{bmatrix} \mathbf{V}_{2b}\\ -\mathbf{I}_{2b} \end{bmatrix}\\ \\ \begin{bmatrix} \mathbf{V}_{1}\\ \mathbf{I}_{1} \end{bmatrix} &= \begin{bmatrix} \mathbf{V}_{1a}\\ \mathbf{I}_{1a} \end{bmatrix}\\ &= \begin{bmatrix} \mathbf{A}_a & \mathbf{B}_a\\ \mathbf{C}_a & \mathbf{D}_a \end{bmatrix} \begin{bmatrix} \mathbf{V}_{2a}\\ -\mathbf{I}_{2a} \end{bmatrix} \\ &= \begin{bmatrix} \mathbf{A}_a & \mathbf{B}_a\\ \mathbf{C}_a & \mathbf{D}_a \end{bmatrix} \begin{bmatrix} \mathbf{V}_{1b}\\ \mathbf{I}_{1b} \end{bmatrix}\\ &= \begin{bmatrix} \mathbf{A}_a & \mathbf{B}_a\\ \mathbf{C}_a & \mathbf{D}_a \end{bmatrix} \begin{bmatrix} \mathbf{A}_b & \mathbf{B}_b\\ \mathbf{C}_b & \mathbf{D}_b \end{bmatrix} \begin{bmatrix} \mathbf{V}_{2b}\\ -\mathbf{I}_{2b} \end{bmatrix}\\ &= \begin{bmatrix} \mathbf{A}_a & \mathbf{B}_a\\ \mathbf{C}_a & \mathbf{D}_a \end{bmatrix} \begin{bmatrix} \mathbf{A}_b & \mathbf{B}_b\\ \mathbf{C}_b & \mathbf{D}_b \end{bmatrix} \begin{bmatrix} \mathbf{V}_{2}\\ -\mathbf{I}_{2} \end{bmatrix}\\ &= \mathbf{T}_a\mathbf{T}_b \begin{bmatrix} \mathbf{V}_{2}\\ -\mathbf{I}_{2} \end{bmatrix}\\ &= \mathbf{T} \begin{bmatrix} \mathbf{V}_{2}\\ -\mathbf{I}_{2} \end{bmatrix} \end{align*} $$

參考

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

\( \left.\begin{matrix} \mathbf{z_{11}=\frac{V_1}{I_1}} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{z_{12}=\frac{V_1}{I_2}} \end{matrix}\right\|_\mathbf{{I_1=0}} \)
\( \left.\begin{matrix} \mathbf{z_{21}=\frac{V_2}{I_1}} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{z_{22}=\frac{V_2}{I_2}} \end{matrix}\right\|_\mathbf{{I_1=0}} \)

\( \left.\begin{matrix} \mathbf{y_{11}=\frac{I_1}{V_1}} \end{matrix}\right\|_\mathbf{{V_2=0}} \)	\( \left.\begin{matrix} \mathbf{y_{12}=\frac{I_1}{V_2}} \end{matrix}\right\|_\mathbf{{V_1=0}} \)
\( \left.\begin{matrix} \mathbf{y_{21}=\frac{I_2}{V_1}} \end{matrix}\right\|_\mathbf{{V_2=0}} \)	\( \left.\begin{matrix} \mathbf{y_{22}=\frac{I_2}{V_2}} \end{matrix}\right\|_\mathbf{{V_1=0}} \)

\( \left.\begin{matrix} \mathbf{h_{11}=\frac{V_1}{I_1}} \end{matrix}\right\|_\mathbf{{V_2=0}} \)	\( \left.\begin{matrix} \mathbf{h_{12}=\frac{V_1}{V_2}} \end{matrix}\right\|_\mathbf{{I_1=0}} \)
\( \left.\begin{matrix} \mathbf{h_{21}=\frac{I_2}{I_1}} \end{matrix}\right\|_\mathbf{{V_2=0}} \)	\( \left.\begin{matrix} \mathbf{h_{22}=\frac{I_2}{V_2}} \end{matrix}\right\|_\mathbf{{I_1=0}} \)

\( \left.\begin{matrix} \mathbf{g_{11}=\frac{I_1}{V_1}} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{g_{12}=\frac{I_1}{I_2}} \end{matrix}\right\|_\mathbf{{V_1=0}} \)
\( \left.\begin{matrix} \mathbf{g_{21}=\frac{V_2}{V_1}} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{g_{22}=\frac{V_2}{I_2}} \end{matrix}\right\|_\mathbf{{V_1=0}} \)

\( \left.\begin{matrix} \mathbf{A}=\frac{V_1}{V_2} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{B}=-\frac{V_1}{I_2} \end{matrix}\right\|_\mathbf{{V_2=0}} \)
\( \left.\begin{matrix} \mathbf{C}=\frac{I_1}{V_2} \end{matrix}\right\|_\mathbf{{I_2=0}} \)	\( \left.\begin{matrix} \mathbf{D}=-\frac{I_1}{I_2} \end{matrix}\right\|_\mathbf{{V_2=0}} \)

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