ELECTENG291 · Fundamentals of Electrical Engineering
Second-Order Transients: RLC Circuits
This Module 2 chapter of University of Auckland ELECTENG 291 steps up to circuits with two energy-storage elements — the RLC circuit — governed by a second-order ODE. From its characteristic equation the response is classified as overdamped, critically damped or underdamped (oscillatory, with ringing), by comparing the damping coefficient with the undamped natural frequency ω0 = 1/√(LC). Recognising the damping case and writing the correct response form is the examinable core here in the tests and the exam.
What this chapter covers
- 01Second-order circuit = two energy-storage elements → second-order ODE and a characteristic (auxiliary) equation
- 02Undamped natural frequency ω0 = 1/√(LC) — the ideal loss-less LC oscillation rate [rad/s]
- 03Characteristic equation s² + 2αs + ω0² = 0 with roots s = −α ± √(α² − ω0²)
- 04Damping coefficient α (its form depends on the topology: series RLC α = R/2L, parallel RLC α = 1/2RC — confirm which against the Module 2.2 notes)
- 05Overdamped (α > ω0): two real distinct roots → sum of two decaying exponentials, no oscillation
- 06Critically damped (α = ω0): a repeated real root → the fastest non-oscillatory settling
- 07Underdamped (α < ω0): complex roots → a decaying sinusoid (ringing) at the damped frequency ω_d = √(ω0² − α²)
Classify a second-order response from its characteristic equation
- +1Match to the standard form s² + 2αs + ω0² = 0: the s-coefficient gives 2α = 6 → α = 3 s⁻¹, and the constant gives ω0² = 8 → ω0 = √8 ≈ 2.83 rad/s.
- +1Compare α with ω0: α = 3 > ω0 = 2.83, so α > ω0 → the response is overdamped (no oscillation).
- +1Find the roots: s = −α ± √(α² − ω0²) = −3 ± √(9 − 8) = −3 ± 1, giving s1 = −2 s⁻¹ and s2 = −4 s⁻¹ — two real, distinct, negative roots, confirming overdamped. (Equivalently s² + 6s + 8 = (s + 2)(s + 4).)
- +1Write the natural response as a sum of the two decaying exponentials: x(t) = A1 e^(−2t) + A2 e^(−4t), with A1, A2 fixed by the initial conditions. Add the steady-state value for a full (forced) response.
Key terms
- Second-order circuit
- A circuit with two (irreducible) energy-storage elements, such as an RLC circuit, whose behaviour obeys a second-order differential equation and a corresponding characteristic (auxiliary) equation.
- Characteristic equation
- The auxiliary polynomial s² + 2αs + ω0² = 0 whose roots s = −α ± √(α² − ω0²) determine the form of the natural response. The nature of the roots (real, repeated or complex) fixes the damping case.
- Natural frequency (ω0)
- The undamped natural (resonant) frequency ω0 = 1/√(LC), the rate at which an ideal loss-less LC circuit would oscillate. Comparing α with ω0 decides the damping regime.
- Damping coefficient (α)
- The neper frequency that sets how fast the response decays. Its form depends on topology — for a series RLC α = R/2L and for a parallel RLC α = 1/2RC — so confirm which applies from the Module 2.2 notes before using it.
- Overdamped / critically damped
- Overdamped (α > ω0): two real distinct roots, a non-oscillatory sum of decaying exponentials. Critically damped (α = ω0): a repeated real root, the fastest possible settling without overshoot.
- Underdamped response
- The case α < ω0, giving complex-conjugate roots and a decaying sinusoid (ringing) at the damped frequency ω_d = √(ω0² − α²). The smaller α is relative to ω0, the more the circuit oscillates before settling.
Second-Order Transients: RLC Circuits FAQ
How do I classify a second-order (RLC) response?
Compare the damping coefficient α with the natural frequency ω0 = 1/√(LC), or look at the roots of the characteristic equation s² + 2αs + ω0² = 0. Two real distinct roots (α > ω0) → overdamped; a repeated root (α = ω0) → critically damped; complex roots (α < ω0) → underdamped, an oscillatory 'ringing' response at ω_d = √(ω0² − α²). Naming the case and writing the matching response form is the examinable skill.
What is the difference between the series and parallel RLC damping formula?
The natural frequency is the same, ω0 = 1/√(LC), but the damping coefficient α differs with topology: for a series RLC it is α = R/2L, and for a parallel RLC it is α = 1/2RC. Using the wrong one flips your damping classification, so first decide whether the R, L and C are in series or in parallel and check the exact form against the Module 2.2 course notes.
What does underdamped mean physically?
Underdamped means there is too little damping to stop the energy sloshing back and forth between the capacitor and inductor, so the response overshoots and rings — a decaying sinusoid at the damped frequency ω_d = √(ω0² − α²). As α is reduced toward zero the oscillation lasts longer; at α = 0 an ideal LC would oscillate forever at ω0 = 1/√(LC).
Can Sia help me with RLC and second-order circuits in ELECTENG 291?
Yes, as a study aid. Sia can form the characteristic equation, identify α and ω0, classify the damping, find the roots and write the response form, and check your working. It explains and drills you on fresh numbers; it does not do graded assessment for you, and University of Auckland academic-integrity rules apply — confirm what is permitted on Canvas.
Exam move
Lead with the classification skill, because it is where the marks are: form the characteristic equation, read off α and ω0 = 1/√(LC), and compare them (or inspect the roots) to name overdamped, critically damped or underdamped, then write the matching response form. Be deliberate about the α formula — series RLC uses R/2L, parallel uses 1/2RC — and confirm which topology you have against the Module 2.2 notes, since this is the easiest thing to get backwards. Because the second-order transcription (α, ω0, ω_d) is detailed, work through the course's own second-order supplementary problems and tutorials rather than relying on memory. Keep your initial-condition skills from the previous chapter sharp, since they fix the constants A1, A2. Confirm assessment details on Canvas.
Working through Second-Order Transients: RLC Circuits in ELECTENG 291? Sia is AskSia’s AI Electrical Engineering tutor — ask any ELECTENG 291 Second-Order Transients: RLC Circuits question and get a clear, step-by-step explanation grounded in how ELECTENG 291 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
