If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 叠加

### 我们要做的是什么

$f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)$

• 若要抑制电压源，请用短路替换它。
• 若要抑制电流源，请用开路替换它。

## 把电路描述成一个函数

$i=f\left({v}_{i}\right)=\frac{1}{\text{R}}\phantom{\rule{0.167em}{0ex}}{v}_{i}$

(请告诉我 线性的含义。)

## 使用叠加来帮助解决电路问题

(这是一个“玩具”示例，目的是让您了解如何使用叠加。)

### 传统解决方案

$\text{Vs1}+\text{Vs2}-i\phantom{\rule{0.167em}{0ex}}\text{R}=0$

$i=f\left(\text{Vs1}+\text{Vs2}\right)=\frac{\text{Vs1}+\text{Vs2}}{\text{R}}$
(传统的解决方案)

### 解决方案使用叠加原理

$f\left({x}_{1}+{x}_{2}\right)=f\left({x}_{1}\right)+f\left({x}_{2}\right)$

$i=f\left(\text{Vs1}+\text{Vs2}\right)$

$i=f\left(\text{Vs1}\right)+f\left(\text{Vs2}\right)$

### 使用叠加

$f\left(\text{Vs1}+0\right)\to f\left(\text{Vs1}\right)$ and $f\left(\text{0 + Vs2}\right)\to f\left(\text{Vs2}\right)$

${i}_{1}=\frac{\text{Vs1}}{\text{R}}\phantom{\rule{2em}{0ex}}{i}_{2}=\frac{\text{Vs2}}{\text{R}}$

$i={i}_{1}+{i}_{2}$
$i=\frac{\text{Vs1}}{\text{R}}+\frac{\text{Vs2}}{\text{R}}$
$i=\frac{\text{Vs1}+\text{Vs2}}{\text{R}}$
(叠加的解决方案)

### 例题1

${i}_{1}={f}_{1}\left(\text{Is},\text{Vs)}\phantom{\rule{0.167em}{0ex}}$$\phantom{\rule{0.167em}{0ex}}{i}_{2}={f}_{2}\left(\text{Is},\text{Vs)}$

${i}_{1V}=0\phantom{\rule{2em}{0ex}}{i}_{2V}=\frac{\text{Vs}}{\text{R}2}$

${i}_{1I}=\text{Is}\phantom{\rule{2em}{0ex}}{i}_{2I}=0$

${i}_{1}={i}_{1V}+{i}_{1I}=0+\text{Is}=\text{Is}$
${i}_{2}={i}_{2V}+{i}_{2I}=\frac{\text{Vs}}{\text{R}2}+0=\frac{\text{Vs}}{\text{R}2}$

### 例题 2

#### 传统解决方案

$\begin{array}{cccc}+{i}_{\text{R}1}& -{i}_{\text{R}2}& +\text{Is}& =0\\ \\ +\frac{\text{Vs}-v}{\text{R1}}& -\frac{v}{\text{R2}}& +\text{Is}& =0\end{array}$

$v=\frac{\text{R2}}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Vs}+\frac{\text{R}1\phantom{\rule{0.167em}{0ex}}\text{R}2}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Is}$
(传统的解决方案)

#### 使用叠加的解决方案

${v}_{Vs}=\text{Vs}\phantom{\rule{0.167em}{0ex}}\frac{\text{R}2}{\text{R}1+\text{R}2}$

${v}_{Is}=\text{Is}\phantom{\rule{0.167em}{0ex}}\frac{\text{R}1\cdot \text{R}2}{\text{R}1+\text{R}2}$

$v={v}_{Vs}+{v}_{Is}$
$v=\frac{\text{R2}}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Vs}+\frac{\text{R}1\cdot \text{R}2}{\text{R}1+\text{R}2}\phantom{\rule{0.167em}{0ex}}\text{Is}$
(叠加的解决方案)