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UY1: L-R-C Series Circuit

Analyze the free transient of a series RLC circuit and classify underdamped, critically damped, and overdamped behavior.

  • University Physics Year 1
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Learning objectives

  • Use vector and differential or integral calculus to express physical change and accumulation.
  • Use complex numbers, linear algebra, and differential equations to solve coupled physical models.
  • Choose an efficient mathematical method and validate the result physically.
Why this matters + quick links

This page gives the UY1 working model/result for L-R-C Series Circuit. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

  • A series RLC circuit without AC drive behaves like damped SHM.
  • Governing equation:
d²q/dt² + (R/L)dq/dt + (1/LC)q = 0.
  • Damping classification uses α = R/(2L) and ω₀ = 1/square root of LC.
  • This page is the free transient (no sinusoidal driving source). For steady-state AC phasor analysis, see L-R-C Series Circuit With A.C..
  • Quick check: R = 0 gives ideal LC oscillations; increasing R moves the system toward critical damping then overdamping.

Prerequisites: L-C Circuit, Second Order Differential Equation
Next uses: R-L Circuit, L-R-C Series Circuit With A.C.

2) Setup

We study a charged capacitor released into a series path containing R, L, and C (no source).

Define:

i = dq/dt.

Loop rule with passive signs gives:

Ldi/dt + Ri + q/C = 0.

Modelling assumptions (UY1 level):

  • Lumped-element circuit (circuit size small compared with the wavelength associated with the relevant time variation).
  • Constant R, L, C (linear elements; no saturation).
  • No external drive during the transient.

3) Core derivation/explanation

Substitute i = dq/dt:

d²q/dt² + (R/L)dq/dt + (1/LC)q = 0.

Define:

α = R/2L, ω₀ = 1/(square root of LC).

Why this step matters

The pair (α,ω₀) turns the differential equation into a simple regime test. Comparing them gives behavior (oscillatory vs non-oscillatory) before solving full constants from initial conditions.

Regimes:

  • Underdamped: α < ω₀ (oscillatory decay),
  • Critically damped: α = ω₀ (fastest non-oscillatory return),
  • Overdamped: α > ω₀ (non-oscillatory slow return).

Underdamped solution:

q(t) = Q₀e^(-α t) cos(ω_d t + φ), ω_d = square root of (ω₀²-α²) .

Resistance is the energy-loss channel (Joule heating).

Common traps (damping + units)
  • Using ω_d = ω₀ in underdamped cases. The actual oscillation is slower: ω_d = square root of (ω₀²-α²).
  • Mixing units between α (s⁻¹) and ω (rad s⁻¹) without dimensional checks.
  • Classifying damping from intuition instead of testing α against ω₀.

Checks (sanity)

  • Regime test: underdamped α < ω₀, critically damped α = ω₀, overdamped α > ω₀.
  • Energy: with R > 0 the total stored energy decays (Joule heating in R); with R = 0 energy is constant.

4) Worked example(s)

Given L = 0.50 H, C = 20 μF, R = 100 Ω.

ω₀ = 1/(square root of LC) = 316 rad s⁻¹, α = R/2L = 100 s⁻¹.

Since α < ω₀, the response is underdamped.

Damped angular frequency:

ω_d = square root of (316²-100²) ≈ 300 rad s⁻¹.

Decay envelope time constant:

τ = 1/α = 0.010 s.

5) Practice set (with hints + answers)

  1. For fixed L,C, what effect does increasing R have on damping? Hint: inspect α = R/(2L). Answer: damping increases.

  2. Condition for critical damping in series RLC? Hint: set α = ω₀. Answer: R = 2 square root of (L/C).

  3. If R = 0, what equation remains? Hint: drop damping term. Answer: q'' + ω₀²q = 0 (ideal LC oscillation).

6) Summary + next steps

  • Free RLC transients are damped oscillators in electrical form.
  • R controls decay; L and C set natural timescale.
  • Driven AC RLC circuits use phasors/impedance in steady state.

Next: R-L Circuit Previous: Magnetic-Field Energy In Inductor Back To Electromagnetism (UY1)