RLC Circuits Analysis

An RLC series circuit with no e.m.f. source is a second-order transient system. Depending on R, it can:

• oscillate with decaying amplitude (underdamped), or
• return to equilibrium without oscillating (critical/overdamped).

In H3, you are not expected to solve the most general second-order differential equation from scratch, but you should be able to set up the equation, identify the regime, and use/verify standard solution forms.

1. Definitions (Must Know)

• State variable (common choice): capacitor charge q(t) (C).
• Current: I(t) = dq/dt
• KVL for source-free series RLC: V_L + V_R + V_C = 0
• Using V_L = LdI/dt = Ld²q/dt², V_R = IR = Rdq/dt, V_C = q/C gives: Ld²q/dt² + Rdq/dt + q/C = 0
• Natural angular frequency (LC): ω₀ = 1/(square root of LC)
• Damping factor: γ = R/2L
• Damped angular frequency (underdamped): ω_d = square root of (ω₀²-γ²)
• Critical resistance: R_c = 2 square root of (L/C)
• Symbols used in this lesson: L (H), R (Ω), R_c critical resistance (Ω), C (F), q (C), I (A), t (s), ω₀,ω_d (rad s⁻¹), γ (s⁻¹).

2. Key Ideas (What Earns Marks)

• The governing equation is: q double dot + R/Lq dot + (1/LC)q = 0 where dots denote time derivatives.
• Determine the behaviour from R relative to R_c:
  • Underdamped: R < R_c (oscillatory, decaying)
  • Critical: R = R_c (fastest return without oscillation)
  • Overdamped: R > R_c (no oscillation, slower return)
• In underdamped case, amplitude decays as e^(-γ t): q(t) = Qe^(-γ t) cos(ω_dt + φ) and I(t) = dq/dt is also sinusoidal with the same envelope.
• The resistor dissipates energy. For weak damping, the energy envelope and cycle-averaged stored energy decay approximately as e^(-2γ t); the instantaneous total stored energy is not a perfect exponential throughout each cycle.

Quick classification table:

3. Detailed Explanations

A. Setting up the ODE (the main H3 skill)

Use KVL and write each element voltage in terms of q:

• V_L = Ld²q/dt²
• V_R = Rdq/dt
• V_C = q/C

So: Lq double dot + Rq dot + q/C = 0

B. How to classify the response quickly

Define: ω₀ = 1/(square root of LC), γ = R/2L

• If γ < ω₀, ω_d is real and you get oscillations (underdamped).
• If γ = ω₀, you are critically damped.
• If γ > ω₀, you are overdamped.

Equivalent resistance test: R_c = 2 square root of (L/C)

C. “Use/verify” a given solution form (what the syllabus allows)

If you are given a proposed form for q(t), you can verify it by:

1. differentiating to get q dot and q double dot, and
2. substituting into Lq double dot + Rq dot + q/C = 0 to check it holds.

This is often faster than solving from scratch and is explicitly within H3 expectations.

4. Common Mistakes

• Using the wrong variable: mixing q(t) and I(t) without using I = dq/dt consistently.
• Forgetting the capacitor relation V_C = q/C.
• Using ω = 1/square root of LC even when R is not negligible (use ω_d for underdamped).
• Confusing γ = R/2L with τ = L/R (RL time constant).

5. Exam Tips

• Start from the canonical ODE: Lq double dot + Rq dot + q/C = 0

• Immediately compute R_c = 2 square root of (L/C) to classify the regime.

• If the question gives/assumes “underdamped oscillations”, use: γ = R/2L

  ω_d = square root of (1/LC-(R/2L)²)

• Useful decay facts (underdamped):

  • amplitude time constant: τₐₘₚ = 1/γ = 2L/R
  • energy decays with half the time constant: τ_U = 1/2γ = L/R

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Full Solutions (Not Required)

Full derivations use the characteristic equation of the second-order ODE and produce different functional forms for under/critical/over damping. H3 typically focuses on using the standard forms and verifying them when needed.

Next step

Return to the RLC Circuits hub for mixed revision, then continue to Special Relativity.