Natural Response
The simplest RC circuit is a capacitor and a resistor in series. When a circuit consists of only a charged capacitor and a resistor, the capacitor will discharge its stored energy through the resistor. The voltage across the capacitor, which is time dependent, can be found by using Kirchhoff's current law, where the current through the capacitor must equal the current through the resistor. This results in the linear differential equation
.
Solving this equation for V yields the formula for exponential decay:
where V0 is the capacitor voltage at time t = 0.
The time required for the voltage to fall to is called the RC time constant and is given by
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