RC电路瞬时电压计算
RC电路（电阻-电容电路）的充放电过程是电子学的基础。电容两端的电压随时间按指数规律变化。
一、时间常数
$\tau = RC$文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
$\tau$（tau，单位：秒）是RC电路最重要的参数。经过 $1\tau$ 后电容电压变化约63.2%，经过 $5\tau$ 后基本达到稳态（99.3%）。文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
二、充电过程
电容初始电压为零，通过电阻 $R$ 接入直流电源 $U_s$：文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
$u_c(t) = U_s\left(1 - e^{-t/RC}\right)$文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
充电电流：$i(t) = \dfrac{U_s}{R}e^{-t/RC}$文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
三、放电过程
电容初始电压为 $U_0$，通过电阻 $R$ 放电：文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
$u_c(t) = U_0 \cdot e^{-t/RC}$文章源自公式库网-https://www.gongshiku.com/html/201910/dianzidianlurcdianlushunshidianyazaixianjisuanqi.html
四、关键时间点
| 时间 | 充电电压 | 放电电压 |
|---|---|---|
| $t = \tau$ | $0.632U_s$ | $0.368U_0$ |
| $t = 2\tau$ | $0.865U_s$ | $0.135U_0$ |
| $t = 3\tau$ | $0.950U_s$ | $0.050U_0$ |
| $t = 5\tau$ | $0.993U_s$ | $0.007U_0$ |
