RC, RL, and LC Circuits Cheat Sheet

Key Facts

• The RC time constant is $\tau = RC$, and after one time constant a charging capacitor reaches about $63\%$ of its final voltage.
• For a charging RC circuit connected to a battery $V_0$, the capacitor voltage is $V_C(t) = V_0\left(1 - e^{-t/(RC)}\right)$.
• For a discharging RC circuit, the capacitor voltage is $V_C(t) = V_i e^{-t/(RC)}$ and the current magnitude is $I(t) = \frac{V_i}{R}e^{-t/(RC)}$.
• The RL time constant is $\tau = \frac{L}{R}$, and after one time constant the current rises to about $63\%$ of its final value.
• For an RL circuit connected to a battery $V_0$, the current is $I(t) = \frac{V_0}{R}\left(1 - e^{-Rt/L}\right)$.
• For an RL circuit when the source is removed, the current decays as $I(t) = I_i e^{-Rt/L}$.
• In an ideal LC circuit, the angular frequency is $\omega = \frac{1}{\sqrt{LC}}$ and the period is $T = 2\pi\sqrt{LC}$.
• The stored energies are $U_C = \frac{1}{2}CV^2$ for a capacitor and $U_L = \frac{1}{2}LI^2$ for an inductor.

Vocabulary

Time constant

The characteristic time $\tau$ that describes how quickly an exponential circuit response rises or decays.

RC circuit

A circuit containing a resistor and capacitor where charge, voltage, and current change according to $\tau = RC$.

RL circuit

A circuit containing a resistor and inductor where current changes according to $\tau = \frac{L}{R}$.

LC circuit

A circuit containing an inductor and capacitor that can oscillate by transferring energy between electric and magnetic fields.

Inductance

The property of an inductor that opposes changes in current and stores magnetic energy according to $U_L = \frac{1}{2}LI^2$.

Capacitance

The property of a capacitor that stores charge and electric energy according to $U_C = \frac{1}{2}CV^2$.