Source: University Physics, Electromagnetism Handout (UIUC)
Tags: RLC circuit, damped oscillations, underdamped, overdamped, critically damped, natural frequency, angular frequency, exponential decay, LC circuit, electromagnetism, PHY 142
Difficulty: Intermediate | Prerequisites: LC oscillations (Ch. 30), basic differential equations, complex exponentials
This section extends the ideal LC circuit (where energy sloshes back and forth forever between the capacitor and inductor) by adding a resistor. The resistor dissipates energy as heat, so the oscillations gradually die out. This is the electrical analogue of a damped mass-on-a-spring system from mechanics. If you are comfortable with LC circuits and the idea of exponential decay, you have everything you need here.
An RLC circuit oscillates like an LC circuit, but the resistor bleeds energy on every cycle, so the amplitude decays exponentially. How fast it decays depends on R relative to L and C. Small R gives underdamped (ringing) behaviour; large R kills the oscillation entirely.
RLC circuit
A closed loop containing a resistor (R), inductor (L), and capacitor (C) in series. The governing equation is a second-order linear ODE identical in form to the damped harmonic oscillator.
Think of it as: a swing that loses a little height on every pass because of air resistance.
Damped oscillation
An oscillation whose amplitude decreases over time due to energy dissipation. In the RLC circuit, the resistor converts electromagnetic energy into thermal energy.
In simple terms, the signal rings but gets quieter and quieter until it stops.
Natural angular frequency (ω₀)
The frequency at which the circuit would oscillate with no resistance. Defined as ω₀ = 1/√(LC).
Think of it as: the "pure" ringing frequency of the LC part of the circuit, before the resistor slows things down.
Damped angular frequency (ω')
The actual oscillation frequency of the underdamped RLC circuit, always slightly lower than ω₀. Given by ω' = √(ω₀² − (R/2L)²).
In simple terms, adding resistance makes the circuit oscillate a bit more slowly than it would without resistance.
Underdamped (R small)
The regime where R/2L < ω₀. The circuit still oscillates, but the peaks shrink exponentially. This is the most common case tested in PHY 142.
Overdamped (R large)
The regime where R/2L > ω₀. No oscillation occurs; the charge simply decays back to zero without crossing it.
Critically damped
The boundary case where R/2L = ω₀ exactly. The charge returns to zero as fast as possible without oscillating. Used in instrument design where you want quick settling with no ringing.
Applying Kirchhoff's voltage rule around the RLC loop gives:
L (dI/dt) + IR + Q/C = 0
Since I = dQ/dt, this rewrites as:
L (d²Q/dt²) + R (dQ/dt) + Q/C = 0
This is mathematically identical to the damped harmonic oscillator equation from mechanics: m·x'' + b·x' + k·x = 0, with the mapping L ↔ m, R ↔ b, 1/C ↔ k.
When R is small enough that ω₀² > (R/2L)², the solution is:
Q(t) = Q_max · e^(−Rt/2L) · cos(ω't + φ)
The exponential envelope e^(−Rt/2L) controls how fast the amplitude dies.
The cosine factor gives the oscillation at the shifted frequency ω'.
φ is the phase constant, set by initial conditions (how much charge and current the circuit starts with).
Natural angular frequency: ω₀ = 1/√(LC)
Damped angular frequency: ω' = √(ω₀² − (R/2L)²)
Decay time constant: τ = 2L/R (the time for the envelope to fall to 1/e of its initial value)
Total energy in the circuit decays as roughly E(t) ∝ e^(−Rt/L).
Energy oscillates between the capacitor (electric field) and inductor (magnetic field), but some is lost to the resistor on every half-cycle.
When all the energy has been dissipated, the current and charge are both zero.
Quantity | Formula |
|---|---|
Circuit equation | L (d²Q/dt²) + R (dQ/dt) + Q/C = 0 |
Charge (underdamped) | Q(t) = Q_max · e^(−Rt/2L) · cos(ω't + φ) |
Natural frequency | ω₀ = 1/√(LC) |
Damped frequency | ω' = √(ω₀² − (R/2L)²) |
Circuit diagram: a series loop with R (resistor, zigzag symbol), L (inductor, coil symbol), and C (capacitor, parallel plates symbol), all connected in a single closed loop.
Damped RLC circuits appear wherever you need a signal to ring briefly and then settle. The tuning circuit in an AM radio is an RLC circuit; its damping determines how sharply it selects one station over its neighbours. Critically damped circuits are used in measuring instruments (like galvanometers) so the needle settles quickly without overshooting.
Students often assume ω' = ω₀. It does not. The damped frequency is always lower than the natural frequency when R > 0. They are only equal when R = 0 (the ideal LC case).
Students sometimes think "overdamped" means the circuit oscillates more. The opposite is true: overdamped means no oscillation at all, just a slow exponential return to zero.
Confusing the decay of charge amplitude (time constant 2L/R) with the decay of energy (time constant L/R). Energy decays twice as fast because it goes as the square of the amplitude.
Forgetting that the phase constant φ depends on initial conditions. It is not always zero.
⚠️ You will almost certainly be asked to identify whether a circuit is underdamped, critically damped, or overdamped given values of R, L, and C. Compare R/2L to ω₀ = 1/√(LC).
⚠️ Be ready to sketch Q(t) vs. t for the underdamped case: a cosine wave inside a decaying exponential envelope.
⚠️ Know how to extract the decay time constant τ = 2L/R and the oscillation period T = 2π/ω' from the solution.
⚠️ The mechanical analogy (L ↔ mass, R ↔ damping, 1/C ↔ spring constant) is a common conceptual question.
True or false: In an underdamped RLC circuit, the oscillation frequency equals 1/√(LC).
Fill in the blank: The amplitude of charge oscillations decays with a time constant of ______.
True or false: If you double R while keeping L and C fixed, the damped frequency ω' increases.
Fill in the blank: The three damping regimes are underdamped, ______, and overdamped.
True or false: Energy in the RLC circuit decays at the same rate as the charge amplitude.
Answers: 1. False (it equals √(ω₀² − (R/2L)²), which is less than ω₀). 2. τ = 2L/R. 3. False (increasing R lowers ω'). 4. Critically damped. 5. False (energy decays twice as fast).
Q: An RLC circuit has L = 0.5 H, C = 2 μF, and R = 100 Ω. Is the circuit underdamped, critically damped, or overdamped?
A: ω₀ = 1/√(LC) = 1/√(0.5 × 2×10⁻⁶) = 1000 rad/s. R/2L = 100/(2 × 0.5) = 100 s⁻¹. Since R/2L (100) < ω₀ (1000), the circuit is underdamped.
Q: For the circuit above, what is the damped angular frequency ω'?
A: ω' = √(ω₀² − (R/2L)²) = √(1000² − 100²) = √(990000) ≈ 995 rad/s.
Q: Write the expression for Q(t) in a general underdamped RLC circuit, and identify what each term represents physically.
A: Q(t) = Q_max · e^(−Rt/2L) · cos(ω't + φ). Q_max is the initial maximum charge, the exponential is the decaying envelope (energy loss to R), the cosine is the oscillation, and φ is set by initial conditions.
Q: What happens to the oscillation if R is increased until R/2L = ω₀?
A: The circuit becomes critically damped. ω' drops to zero, and the charge returns to zero as quickly as possible without oscillating.
This connects directly to LC oscillations (Ch. 30), which are just the R = 0 special case. The mathematics is identical to the damped harmonic oscillator from mechanics (Ch. 15), so practising one reinforces the other. In the next sections (31.3–31.4), you will see what happens when you drive this circuit with an external AC source, which leads to forced oscillations and resonance.
RLC circuit, damped oscillations, underdamped oscillations, overdamped, critically damped, exponential decay, natural frequency, damped frequency, LC circuit, electromagnetic oscillations, Kirchhoff's voltage law, second-order ODE, damped harmonic oscillator, quality factor, PHY 142, UIUC physics, electromagnetism