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# 电容i-v方程

$i=\text{C}\phantom{\rule{0.167em}{0ex}}\frac{dv}{dt}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}$ $v=\frac{1}{\text{C}}\phantom{\rule{0.167em}{0ex}}{\int }_{\phantom{\rule{0.167em}{0ex}}0}^{\phantom{\rule{0.167em}{0ex}}T}i\phantom{\rule{0.167em}{0ex}}\text{d}t+{v}_{0}$
$\text{C}$电容, 电容的物理属性。
$\text{C}$ $i$$dv/dt$之间关系的比例因子。
$\text{C}$ 确定给定数量的$dv/dt$生成多少$i$
${v}_{0}$是在$t=0$时跨电容的初始电压。

## 电流脉冲前、中、后的电压

$\text{C}=1\phantom{\rule{0.167em}{0ex}}\mu \text{F}$
${v}_{0}=0$

### 脉冲期间

$v\left(T\right)=\frac{1}{\text{C}}\phantom{\rule{0.167em}{0ex}}{\int }_{\phantom{\rule{0.167em}{0ex}}0}^{\phantom{\rule{0.167em}{0ex}}T}i\phantom{\rule{0.167em}{0ex}}\text{d}t+{v}_{0}$

$v\left(T\right)=\frac{i}{\text{C}}\phantom{\rule{0.167em}{0ex}}{\int }_{\phantom{\rule{0.167em}{0ex}}0}^{\phantom{\rule{0.167em}{0ex}}T}\text{d}t$
$v\left(T\right)=\frac{i}{\text{C}}\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}t{|}_{\phantom{\rule{0.167em}{0ex}}0}^{\phantom{\rule{0.167em}{0ex}}T}$
$v\left(T\right)=\frac{i}{\text{C}}\phantom{\rule{0.167em}{0ex}}T\phantom{\rule{2em}{0ex}}\text{伏}$

$\frac{i}{\text{C}}=\frac{2×{10}^{-3}\phantom{\rule{0.167em}{0ex}}\text{A}}{1×{10}^{-6}\phantom{\rule{0.167em}{0ex}}\text{F}}=2000\phantom{\rule{0.167em}{0ex}}\text{伏/秒}$

${v}_{\left(T=3\phantom{\rule{0.167em}{0ex}}\text{ms}\right)}=2000\phantom{\rule{0.167em}{0ex}}\text{伏/秒}\phantom{\rule{0.167em}{0ex}}\cdot \phantom{\rule{0.167em}{0ex}}0.003\phantom{\rule{0.167em}{0ex}}\text{s}=6\phantom{\rule{0.167em}{0ex}}\text{伏}$

### 脉冲之后

$v=\frac{1}{\text{C}}\phantom{\rule{0.167em}{0ex}}{\int }_{\phantom{\rule{0.167em}{0ex}}3\phantom{\rule{0.167em}{0ex}}\text{ms}}^{\phantom{\rule{0.167em}{0ex}}T}0\phantom{\rule{0.167em}{0ex}}\text{d}t+6=6\phantom{\rule{0.167em}{0ex}}\text{伏特}$

• 有多少种方法可以实现$4\phantom{\rule{0.167em}{0ex}}\text{V}$的结束电压?
• 如果当前脉冲变为负值，$v\left(t\right)$会发生什么