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## 概要

${\displaystyle {V_{1}}}$ 入力電圧
${\displaystyle {V_{2}}}$ 出力電圧
${\displaystyle {I_{1}}}$ 入力電流
${\displaystyle {I_{2}}}$ 出力電流

## Zパラメータ

インピーダンス行列、Z行列とも。

${\displaystyle {V_{1} \choose V_{2}}={\begin{pmatrix}Z_{11}&Z_{12}\\Z_{21}&Z_{22}\end{pmatrix}}{I_{1} \choose I_{2}}}$

${\displaystyle Z_{11}}$${\displaystyle Z_{12}}$${\displaystyle Z_{21}}$${\displaystyle Z_{22}}$の各インピーダンスパラメータは以下のとおり。

${\displaystyle Z_{11}={V_{1} \over I_{1}}{\bigg |}_{I_{2}=0}\qquad Z_{12}={V_{1} \over I_{2}}{\bigg |}_{I_{1}=0}}$
${\displaystyle Z_{21}={V_{2} \over I_{1}}{\bigg |}_{I_{2}=0}\qquad Z_{22}={V_{2} \over I_{2}}{\bigg |}_{I_{1}=0}}$

## Yパラメータ

アドミタンス行列、Y行列とも。

${\displaystyle {I_{1} \choose I_{2}}={\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}{V_{1} \choose V_{2}}}$

${\displaystyle Y_{11}}$${\displaystyle Y_{12}}$${\displaystyle Y_{21}}$${\displaystyle Y_{22}}$の各アドミタンスパラメータは以下のとおり。

${\displaystyle Y_{11}={I_{1} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{12}={I_{1} \over V_{2}}{\bigg |}_{V_{1}=0}}$
${\displaystyle Y_{21}={I_{2} \over V_{1}}{\bigg |}_{V_{2}=0}\qquad Y_{22}={I_{2} \over V_{2}}{\bigg |}_{V_{1}=0}}$

## hパラメータ

ハイブリッド行列、h行列とも。

${\displaystyle {V_{1} \choose I_{2}}={\begin{pmatrix}h_{11}&h_{12}\\h_{21}&h_{22}\end{pmatrix}}{I_{1} \choose V_{2}}}$

${\displaystyle h_{11}}$${\displaystyle h_{12}}$${\displaystyle h_{21}}$${\displaystyle h_{22}}$の各ハイブリッドパラメータは以下のとおり。

${\displaystyle h_{11}={V_{1} \over I_{1}}{\bigg |}_{V_{2}=0}\qquad h_{12}={V_{1} \over V_{2}}{\bigg |}_{I_{1}=0}}$
${\displaystyle h_{21}={I_{2} \over I_{1}}{\bigg |}_{V_{2}=0}\qquad h_{22}={I_{2} \over V_{2}}{\bigg |}_{I_{1}=0}}$

## gパラメータ

hパラメータの逆行列で定義される。

${\displaystyle {I_{1} \choose V_{2}}={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}{V_{1} \choose I_{2}}}$

${\displaystyle g_{11}}$${\displaystyle g_{12}}$${\displaystyle g_{21}}$${\displaystyle g_{22}}$各パラメータは以下のとおり。

${\displaystyle g_{11}={I_{1} \over V_{1}}{\bigg |}_{I_{2}=0}\qquad g_{12}={I_{1} \over I_{2}}{\bigg |}_{V_{1}=0}}$
${\displaystyle g_{21}={V_{2} \over V_{1}}{\bigg |}_{I_{2}=0}\qquad g_{22}={V_{2} \over I_{2}}{\bigg |}_{V_{1}=0}}$

## Fパラメータ (ABCDパラメータ)

${\displaystyle {V_{1} \choose I_{1}}={\begin{pmatrix}A&B\\C&D\end{pmatrix}}{V_{2} \choose I_{2}}}$

ここで、${\displaystyle I_{2}}$は上の図とは逆の向きを正にとる。

A・B・C・Dの各四端子定数は以下のとおり。

${\displaystyle A={V_{1} \over V_{2}}{\bigg |}_{I_{2}=0}\qquad B={V_{1} \over I_{2}}{\bigg |}_{V_{2}=0}}$
${\displaystyle C={I_{1} \over V_{2}}{\bigg |}_{I_{2}=0}\qquad D={I_{1} \over I_{2}}{\bigg |}_{V_{2}=0}}$

## 縦続接続

${\displaystyle {V_{1} \choose I_{1}}={\begin{pmatrix}A_{1}&B_{1}\\C_{1}&D_{1}\end{pmatrix}}{V_{2} \choose I_{2}}}$
${\displaystyle {V_{2} \choose I_{2}}={\begin{pmatrix}A_{2}&B_{2}\\C_{2}&D_{2}\end{pmatrix}}{V_{3} \choose I_{3}}}$

このとき、${\displaystyle V_{1}}$${\displaystyle I_{1}}$${\displaystyle V_{3}}$${\displaystyle I_{3}}$には以下の関係が成り立つ。

${\displaystyle {V_{1} \choose I_{1}}={\begin{pmatrix}A_{1}&B_{1}\\C_{1}&D_{1}\end{pmatrix}}{\begin{pmatrix}A_{2}&B_{2}\\C_{2}&D_{2}\end{pmatrix}}{V_{3} \choose I_{3}}}$

よって縦続接続したときの回路全体のFパラメータは以下となる。

${\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}A_{1}&B_{1}\\C_{1}&D_{1}\end{pmatrix}}{\begin{pmatrix}A_{2}&B_{2}\\C_{2}&D_{2}\end{pmatrix}}={\begin{pmatrix}A_{1}A_{2}+B_{1}C_{2}&A_{1}B_{2}+B_{1}D_{2}\\C_{1}A_{2}+D_{1}C_{2}&C_{1}B_{2}+D_{1}D_{2}\end{pmatrix}}}$

## 直列接続

2つの異なる二端子対回路を直列に接続することを「直列接続」と呼ぶ。Zパラメータを用いると都合がよい。 ここで、2つの異なる二端子対回路を以下のZパラメータとする。

${\displaystyle {V_{1}' \choose V_{2}'}={\begin{pmatrix}Z_{11}'&Z_{12}'\\Z_{21}'&Z_{22}'\end{pmatrix}}{I_{1}' \choose I_{2}'}}$
${\displaystyle {V_{1}'' \choose V_{2}''}={\begin{pmatrix}Z_{11}''&Z_{12}''\\Z_{21}''&Z_{22}''\end{pmatrix}}{I_{1}'' \choose I_{2}''}}$

このとき、${\displaystyle V_{1}}$${\displaystyle V_{2}}$${\displaystyle I_{1}}$${\displaystyle I_{2}}$は、${\displaystyle V_{1}=V_{1}'+V_{1}''}$${\displaystyle V_{2}=V_{2}'+V_{2}''}$${\displaystyle I_{1}=I_{1}'+I_{1}''}$${\displaystyle I_{2}=I_{2}'+I_{2}''}$の関係があるので、以下の関係が成り立つ。

${\displaystyle {V_{1} \choose V_{2}}={V_{1}' \choose V_{2}'}+{V_{1}'' \choose V_{2}''}={\begin{pmatrix}Z_{11}'+Z_{11}''&Z_{12}'+Z_{12}''\\Z_{21}'+Z_{21}''&Z_{22}'+Z_{22}''\end{pmatrix}}{I_{1} \choose I_{2}}}$

よって直列接続したときの回路全体のZパラメータは以下となる。

${\displaystyle {\begin{pmatrix}Z_{11}&Z_{12}\\Z_{21}&Z_{22}\end{pmatrix}}={\begin{pmatrix}Z_{11}'+Z_{11}''&Z_{12}'+Z_{12}''\\Z_{21}'+Z_{21}''&Z_{22}'+Z_{22}''\end{pmatrix}}}$

## 並列接続

2つの異なる二端子対回路を並列に接続することを「並列接続」と呼ぶ。Yパラメータを用いると都合がよい。 ここで、2つの異なる二端子対回路を以下のYパラメータとする

${\displaystyle {I_{1}' \choose I_{2}'}={\begin{pmatrix}Y_{11}'&Y_{12}'\\Y_{21}'&Y_{22}'\end{pmatrix}}{V_{1}' \choose V_{2}'}}$
${\displaystyle {I_{1}'' \choose I_{2}''}={\begin{pmatrix}Y_{11}''&Y_{12}''\\Y_{21}''&Y_{22}''\end{pmatrix}}{V_{1}'' \choose V_{2}''}}$

このとき、${\displaystyle I_{1}}$${\displaystyle I_{2}}$${\displaystyle V_{1}}$${\displaystyle V_{2}}$は、${\displaystyle I_{1}=I_{1}'+I_{1}''}$${\displaystyle I_{2}=I_{2}'+I_{2}''}$${\displaystyle V_{1}=V_{1}'+V_{1}''}$${\displaystyle V_{2}=V_{2}'+V_{2}''}$の関係があるので、以下の関係が成り立つ。

${\displaystyle {I_{1} \choose I_{2}}={I_{1}' \choose I_{2}'}+{I_{1}'' \choose I_{2}''}={\begin{pmatrix}Y_{11}'+Y_{11}''&Y_{12}'+Y_{12}''\\Y_{21}'+Y_{21}''&Y_{22}'+Y_{22}''\end{pmatrix}}{V_{1} \choose V_{2}}}$

よって並列接続したときの回路全体のYパラメータは以下となる。

${\displaystyle {\begin{pmatrix}Y_{11}&Y_{12}\\Y_{21}&Y_{22}\end{pmatrix}}={\begin{pmatrix}Y_{11}'+Y_{11}''&Y_{12}'+Y_{12}''\\Y_{21}'+Y_{21}''&Y_{22}'+Y_{22}''\end{pmatrix}}}$

## 関連項目

This article uses material from the Wikipedia article "二端子対回路", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia

### Electrical Engineering

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