Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

RC, RL, and LC Circuits cheat sheet - grade 11-12

Click image to open full size

Physics Grade 11-12

RC, RL, and LC Circuits Cheat Sheet

A printable reference covering RC time constants, RL transients, LC oscillations, exponential charging, current decay, and resonance for grades 11-12.

TopicsCircuitsElectricityElectromagnetismTimePhysicsStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

RC, RL, and LC circuits describe how capacitors and inductors store and exchange energy in electric circuits. Students need this cheat sheet because these circuits introduce time-dependent behavior instead of simple steady current. The key skill is recognizing which quantity changes exponentially, which quantity stays continuous, and which time constant controls the rate of change.

These ideas connect directly to filters, oscillators, power supplies, radios, and many electronic devices.

In an RC circuit, the capacitor voltage changes with time constant τ=RC\tau = RC. In an RL circuit, the inductor current changes with time constant τ=LR\tau = \frac{L}{R}. In an ideal LC circuit, energy oscillates between the capacitor electric field and the inductor magnetic field with angular frequency ω=1LC\omega = \frac{1}{\sqrt{LC}}.

For all three circuit types, Kirchhoff's laws, initial conditions, and energy storage formulas are the main tools.

Key Facts

  • The RC time constant is τ=RC\tau = RC, and after one time constant a charging capacitor reaches about 63%63\% of its final voltage.
  • For a charging RC circuit connected to a battery V0V_0, the capacitor voltage is VC(t)=V0(1−e−t/(RC))V_C(t) = V_0\left(1 - e^{-t/(RC)}\right).
  • For a discharging RC circuit, the capacitor voltage is VC(t)=Vie−t/(RC)V_C(t) = V_i e^{-t/(RC)} and the current magnitude is I(t)=ViRe−t/(RC)I(t) = \frac{V_i}{R}e^{-t/(RC)}.
  • The RL time constant is τ=LR\tau = \frac{L}{R}, and after one time constant the current rises to about 63%63\% of its final value.
  • For an RL circuit connected to a battery V0V_0, the current is I(t)=V0R(1−e−Rt/L)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)=Iie−Rt/LI(t) = I_i e^{-Rt/L}.
  • In an ideal LC circuit, the angular frequency is ω=1LC\omega = \frac{1}{\sqrt{LC}} and the period is T=2πLCT = 2\pi\sqrt{LC}.
  • The stored energies are UC=12CV2U_C = \frac{1}{2}CV^2 for a capacitor and UL=12LI2U_L = \frac{1}{2}LI^2 for an inductor.

Vocabulary

Time constant
The characteristic time τ\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 τ=RC\tau = RC.
RL circuit
A circuit containing a resistor and inductor where current changes according to τ=LR\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 UL=12LI2U_L = \frac{1}{2}LI^2.
Capacitance
The property of a capacitor that stores charge and electric energy according to UC=12CV2U_C = \frac{1}{2}CV^2.

Common Mistakes to Avoid

  • Using τ=RL\tau = \frac{R}{L} for an RL circuit is wrong because the correct time constant is τ=LR\tau = \frac{L}{R}.
  • Assuming capacitor voltage changes instantly is wrong because a capacitor voltage must change continuously over time in a real circuit.
  • Assuming inductor current changes instantly is wrong because an inductor opposes sudden changes in current.
  • Mixing up charging and discharging equations is wrong because charging approaches a final value such as V0V_0, while discharging decays toward 00.
  • Forgetting unit conversions is wrong because using μF\mu\text{F}, mH\text{mH}, or kΩ\text{k}\Omega without converting can make τ\tau, ω\omega, or TT off by powers of ten.

Practice Questions

  1. 1 An RC circuit has R=2.0 kΩR = 2.0\,\text{k}\Omega and C=470 μFC = 470\,\mu\text{F}. Find the time constant τ=RC\tau = RC.
  2. 2 A discharging capacitor starts at Vi=12 VV_i = 12\,\text{V} in a circuit with R=5.0 kΩR = 5.0\,\text{k}\Omega and C=100 μFC = 100\,\mu\text{F}. Find VC(t)V_C(t) at t=0.50 st = 0.50\,\text{s} using VC(t)=Vie−t/(RC)V_C(t) = V_i e^{-t/(RC)}.
  3. 3 An LC circuit has L=25 mHL = 25\,\text{mH} and C=4.0 μFC = 4.0\,\mu\text{F}. Find the oscillation period using T=2πLCT = 2\pi\sqrt{LC}.
  4. 4 Explain why the current in an inductor cannot jump instantly when a switch is closed or opened.

Understanding RC, RL, and LC Circuits

The most important idea at the switching moment is continuity. The voltage across an ideal capacitor cannot jump suddenly. A sudden voltage jump would require an infinite current.

The current through an ideal inductor cannot jump suddenly. A sudden current jump would require an infinite voltage. These rules let students set the starting values before doing any algebra.

After a switch has been closed for a long time with a direct current battery, a capacitor in series behaves like an open gap because its current has fallen to zero. An inductor behaves like an ordinary wire in the ideal model because its voltage has fallen to zero.

Kirchhoff's loop rule explains why the curves are exponential rather than straight lines. In a charging resistor capacitor circuit, the battery voltage is shared between the resistor and capacitor. At first, the capacitor voltage is low, so the resistor gets most of the battery voltage and the current is large.

As charge builds up, capacitor voltage rises. Less voltage remains across the resistor, so current decreases. The rate of charging slows because the circuit is getting closer to its final state.

In a resistor inductor circuit, the inductor initially opposes the growth of current by producing a voltage in the opposite direction. This opposing voltage becomes smaller as the current settles. The same feedback creates the curved rise or decay.

An LC circuit has no resistor in the ideal case, so it has no final steady state. A charged capacitor pushes current through the inductor. As the capacitor loses charge, its electric field energy falls while the inductor magnetic field energy grows.

When the capacitor is briefly uncharged, the current is greatest. The magnetic field then keeps the current moving and charges the capacitor with reversed polarity. Voltage and current are out of step by one quarter of a cycle.

Real circuits always have some resistance. It removes energy as thermal energy, so the oscillations shrink over time. A driving signal can replace this lost energy.

When its frequency matches the circuit's natural frequency, the response becomes especially large. This is resonance, which is used when radio receivers select a station.

Graphs are often the clearest way to check an answer. At the start of a capacitor charging graph, voltage rises steeply, while the current is largest. At the start of an inductor current growth graph, the slope depends on the applied voltage and inductance.

A larger resistance makes charging or decay finish sooner in both resistor capacitor and resistor inductor circuits, though it changes the final current in the resistor inductor case. Students should keep track of units before calculating. Capacitance is measured in farads, inductance in henrys, resistance in ohms, and time in seconds.

In real devices, these ideas appear in camera flashes, timing circuits, motor coils, speaker crossovers, power supplies, and radio tuning circuits. A careful circuit diagram, a stated switch position before and after the change, and correct signs for current direction prevent most mistakes.