# Capacitator - Theory of Operation - DC Circuits

DC Circuits

A series circuit containing only a resistor, a capacitor, a switch and a constant DC source of voltage V0 is known as a charging circuit. If the capacitor is initially uncharged while the switch is open, and the switch is closed at t = 0, it follows from Kirchhoff's voltage law that

Taking the derivative and multiplying by C, gives a first-order differential equation,

At t = 0, the voltage across the capacitor is zero and the voltage across the resistor is V0. The initial current is then i (0) =V0 /R. With this assumption, the differential equation yields

where is the time constant of the system.

As the capacitor reaches equilibrium with the source voltage, the voltages across the resistor and the current through the entire circuit decay exponentially. The case of discharging a charged capacitor likewise demonstrates exponential decay, but with the initial capacitor voltage replacing V0 and the final voltage being zero.