RC电路简介

Consider a battery with voltage $V$, a resistor with resistance $R$ and a capacitor with capacitance $C$ all connected in series.

At any time $t$, let $q(t)$ be the charge on one plate of the capacitor and let $I(t)$ be the current through the circuit.

In a small time interval $dt$, a current of $I(t)$ will flow into one of the capacitor plates. Since no charge can leave the same capacitor plate on the other side, the capacitor plate's charge increases by $I(t)dt$. So, $dq(t) = I(t)dt$, or $\frac{dq(t)}{dt}=I(t)$.

The voltage across the battery is $V$.

The voltage across the resistor is $-I(t)R$.

The voltage across the capacitor is $-q(t)/C$.

We can rearrange the equation as $\frac{CV-q}{C} = \frac{dq}{dt}R$, or $\frac{dt}{RC} = \frac{dq}{CV-q}$. Integrating both sides we get $\frac{t}{RC} = -\ln{(CV-q)} + K$, where $K$ is some constant.

Therefore, we find $q(t) = CV - Ae^{\frac{-t}{RC}}$, where $A$ is also some constant that is easy to find. Of course, if the capacitor is uncharged at $t=0$, $A=CV$.

Thus, qualitatively the capacitor's charge starts at zero and exponentially approaches the maximum charge $CV$. A graphic representation of charge vs. time is shown below:

The quantity $RC$ is the system's "time constant."

Looking back at the function $q(t)$, we see that $RC$ is actually the time it takes for the difference between capacitor charge and maximum charge in a charging capacitor to decrease by a factor of $\frac{1}{e}$.

While there is probably nothing very interesting about this number itself, the time constant is still important because it is constant for any Resistor-Capacitor series system regardless of the applied voltage. It also lets you accurately compare the charging rates of different Resistor-Capacitor systems (i.e. if one system has twice the time constant, it charges half as fast).