Source: University Physics: Elec & Mag, UIUC
Tags: LC circuit, RLC circuit, oscillation, natural frequency, damping factor, energy conservation, simple harmonic motion, charge oscillation, angular frequency
Difficulty: Intermediate Prerequisites: Inductor behaviour and RL circuits, capacitor basics (C = Q/V), Kirchhoff's voltage law.
Big picture: LC circuits are where inductors and capacitors meet, and the result is oscillation. Energy sloshes back and forth between the inductor's magnetic field and the capacitor's electric field, much like a mass on a spring trades kinetic and potential energy. Adding a resistor (RLC) introduces damping, and the oscillation dies out over time. This is the physics behind radio tuning, signal filtering, and every resonant electronic system. If you are comfortable with RL circuits and basic differential equations, this section extends those ideas into oscillatory territory.
An LC circuit oscillates: charge on the capacitor varies as Q(t) = Q_max cos(ωt + φ), with natural frequency ω₀ = 1/√(LC). Total energy is conserved, bouncing between inductor and capacitor. Adding resistance (RLC) damps the oscillation exponentially, with a damping factor β = R/(2L).
Natural frequency (ω₀)
The angular frequency at which a lossless LC circuit oscillates: ω₀ = 1/√(LC). In simple terms, this is how fast the charge and current swing back and forth when nothing is draining energy from the system.
Damping factor (β)
In an RLC circuit, β = R/(2L). It measures how quickly the resistance saps energy from the oscillation. A larger β means the oscillation dies out faster.
Oscillation frequency (ω')
The actual angular frequency of a damped RLC circuit: ω'² = ω₀² − β². When R is small, ω' is close to ω₀. When β ≥ ω₀, the system is overdamped and no longer oscillates at all.
Q factor (quality factor)
Defined as Q² = L/(R²C). A higher Q factor means the resonance peak is sharper and the oscillation rings for longer before dying out. Think of it as a measure of how "pure" the oscillation is.
The circuit has an inductor (L) and capacitor (C) in a loop, with no resistance.
Recall: V_L = L(dI/dt) and V_C = Q/C.
Apply KVL around the loop: L(dI/dt) + Q/C = 0.
Since I = dQ/dt, we get dI/dt = d²Q/dt².
Substituting: L(d²Q/dt²) + Q/C = 0, which rearranges to:
d²Q/dt² = −Q/(LC) = −ω₀² Q
This is the equation for simple harmonic motion.
General solution: Q(t) = Q_max cos(ωt + φ)
ω₀ = 1/√(LC)
Current: I(t) = dQ/dt = −ω Q_max sin(ωt + φ)
The current leads (or lags) the charge by 90°, just as velocity and displacement are 90° out of phase in a mass-spring system.
Maximum current amplitude: I_max = ω Q_max.
Energy stored in the inductor: U_L = ½ L I²
Energy stored in the capacitor: U_C = ½ Q²/C = ½ C V_C²
Maximum values:
U_L(max) = ½ L I_max²
U_C(max) = ½ Q_max² / C
Total energy is conserved: E = U_L + U_C = constant at all times.
When U_L is at its maximum, U_C = 0, and vice versa. The energy graphs are complementary: one peaks when the other is at zero.
Adding a resistor R in series introduces energy loss.
The differential equation becomes: L(d²Q/dt²) + R(dQ/dt) + Q/C = 0.
The solution is a damped oscillation:
Q(t) = Q₀ e^(−βt) cos(ω't + φ)
β = R/(2L) (damping factor)
ω'² = ω₀² − β² (damped frequency)
The exponential envelope e^(−βt) shrinks the amplitude over time. The cosine inside still oscillates, but within a decaying window.
These two rules underpin every inductor and capacitor problem:
Current through an inductor cannot change abruptly.
Voltage across a capacitor cannot change abruptly.
Keep these in mind whenever a switch is thrown in any circuit containing L or C. They set the initial conditions for every transient problem.
Quantity | Formula |
|---|---|
Natural frequency | ω₀ = 1/√(LC) |
LC charge oscillation | Q(t) = Q_max cos(ωt + φ) |
LC current | I(t) = −ω Q_max sin(ωt + φ) |
Inductor energy | U_L = ½ L I² |
Capacitor energy | U_C = ½ Q²/C |
Damping factor | β = R/(2L) |
Damped frequency | ω'² = ω₀² − β² |
RLC damped oscillation | Q(t) = Q₀ e^(−βt) cos(ω't + φ) |
Q factor | Q² = L/(R²C) |
LC oscillations are the basis of radio tuning: adjusting C (or L) changes ω₀, selecting different broadcast frequencies. RLC damping governs how quickly a speaker cone settles after a bass hit, and it determines the bandwidth of band-pass filters in audio equipment and communications hardware.
Students sometimes think energy is lost in an ideal LC circuit. It is not. There is no resistance, so total energy is perfectly conserved; it just moves between the inductor and capacitor.
Confusing ω₀ with frequency f. Remember ω₀ = 2πf₀, so f₀ = 1/(2π√(LC)). Exam questions may ask for either one.
Forgetting that the RLC damped frequency ω' is less than ω₀. The resistance slows the oscillation slightly, not just the amplitude.
Thinking the Q factor and charge Q are the same symbol. Context usually makes this clear, but double-check on exams.
⚠️ You will be asked to derive the LC differential equation from KVL. Practise the steps: V_L + V_C = 0, substitute I = dQ/dt, arrive at d²Q/dt² = −ω₀²Q.
⚠️ Energy conservation questions are common: if given Q_max and C, find I_max using ½ Q_max²/C = ½ L I_max².
⚠️ For RLC problems, know how to identify whether a circuit is underdamped (ω₀ > β), critically damped (ω₀ = β), or overdamped (ω₀ < β), and what the charge looks like in each case.
⚠️ The "golden principles" (current through L and voltage across C are continuous) appear in initial-condition problems. Expect a switch-flip scenario where you need to state I_L and V_C at t = 0⁺.
Fill in the blank: The natural frequency of an LC circuit is ω₀ = ______.
True or false: In an ideal LC circuit, energy is gradually lost to the magnetic field.
Fill in the blank: The damping factor in an RLC circuit is β = ______.
True or false: Voltage across a capacitor can jump instantaneously when a switch is thrown.
True or false: A higher Q factor means the oscillation dies out more quickly.
Answers: 1. 1/√(LC). 2. False (energy is conserved). 3. R/(2L). 4. False. 5. False (higher Q means the oscillation persists longer).
Q: An LC circuit has L = 10 mH and C = 100 μF. What is the natural frequency of oscillation?
A: ω₀ = 1/√(LC) = 1/√(0.01 × 0.0001) = 1/√(10⁻⁶) = 1000 rad/s.
Q: At the instant when the capacitor is fully charged (Q = Q_max), what is the current in the circuit and the energy stored in the inductor?
A: I = 0 and U_L = 0. All the energy is in the capacitor: U_C = ½ Q_max²/C.
Q: In an RLC circuit with L = 0.5 H, R = 10 Ω, and C = 200 μF, find the damping factor and determine whether the circuit oscillates.
A: β = R/(2L) = 10/(2 × 0.5) = 10 s⁻¹. ω₀ = 1/√(0.5 × 0.0002) = 1/√(10⁻⁴) = 100 rad/s. Since ω₀ > β, the circuit is underdamped and oscillates.
Q: State the two "golden principles" for circuits containing inductors and capacitors.
A: (1) Current through an inductor cannot change abruptly. (2) Voltage across a capacitor cannot change abruptly.
LC oscillations are the undamped foundation for the AC circuits topic that follows: a driven RLC circuit at resonance is simply an LC oscillation sustained by an external source. The maths here (SHM differential equation, natural frequency) connects to mechanical oscillation problems in earlier physics courses, so if you understood mass-on-a-spring, this is the same structure with electrical quantities. The Q factor introduced here reappears when studying resonance bandwidth in AC circuits.
Related Terms / Search Tags: LC circuit, RLC circuit, natural frequency, damping, damped oscillation, quality factor, Q factor, simple harmonic motion, energy conservation, inductor energy, capacitor energy, angular frequency, underdamped, overdamped, critically damped, PHY 212, UIUC, midterm 3