Second Order RLC Series

A second order RLC series circuit contains a resistor, inductor, and capacitor connected in series. When a step voltage is applied, the interaction between the inductor and capacitor creates a response that can oscillate, die out gradually, or decay without oscillation depending on the amount of damping. This is the simplest example of a second-order system and appears throughout control systems, communications, and power electronics.

Core Concept Explanation

When the switch is closed in a series RLC circuit, the differential equation governing the capacitor voltage becomes second order. Unlike first-order RC and RL circuits, this system can exhibit oscillatory behavior. The key parameter that determines the type of response is the damping ratio ζ (zeta) or equivalently the damping coefficient α compared to the natural frequency ω₀.

The characteristic equation of the circuit is: s² + (R/L)s + 1/(LC) = 0. The roots of this equation are s = -α ± √(α² - ω₀²), where α = R/(2L) and ω₀ = 1/√(LC). The nature of these roots determines the type of transient response. If the roots are real and distinct, the system is overdamped. If they are complex conjugates, the system is underdamped. If they are equal, the system is critically damped.

In the overdamped case (α > ω₀ or ζ > 1), the response decays exponentially without oscillation, similar to a first-order circuit but with two time constants. In the underdamped case (α < ω₀ or ζ < 1), the response oscillates with a decaying amplitude — the circuit rings. In the critically damped case (α = ω₀ or ζ = 1), the response decays as fast as possible without oscillating.

Mathematical Expression

The second-order differential equation for the series RLC circuit is: L·d²i/dt² + R·di/dt + (1/C)·i = dVs/dt. For capacitor voltage: d²Vc/dt² + (R/L)·dVc/dt + (1/LC)·Vc = Vs/LC. The damped natural frequency for the underdamped case is ωd = √(ω₀² - α²). The complete solution for the underdamped step response is: Vc(t) = Vs + e^(-αt)·[A·cos(ωd·t) + B·sin(ωd·t)], where A and B are determined by initial conditions.

For the overdamped case: Vc(t) = Vs + A₁·e^(s₁t) + A₂·e^(s₂t), where s₁ and s₂ are the two distinct real negative roots. For critically damped: Vc(t) = Vs + (A + Bt)·e^(-αt). The natural frequency ω₀ = 1/√(LC) is a property of the LC combination alone, independent of R.

Practical Understanding

In communications, LC tank circuits are used as bandpass filters and oscillators. An underdamped RLC circuit can sustain oscillations if the resistance is very small. In practice, a small series resistance always damps the oscillation eventually. The Q factor (quality factor) Q = ω₀L/R = (1/R)·√(L/C) describes how sharply the circuit resonates and how long oscillations persist.

In power systems, RLC transients appear when a capacitor bank is switched onto a transmission line. The resulting oscillatory current can cause problems if the circuit is underdamped. Engineers choose R values to ensure the transient damps out quickly, aiming for critical or slight overdamping.

Given:
R = 4 Ω, L = 1 H, C = 0.25 F, Vs = 10 V (step)
Initial conditions: Vc(0) = 0, i(0) = 0

Why this formula applies:
Compare α and ω₀ to determine damping type.

Formula:
α = R/(2L) = 4/(2×1) = 2 rad/s
ω₀ = 1/√(LC) = 1/√(1×0.25) = 1/0.5 = 2 rad/s

Substitution:
α = ω₀ = 2 rad/s  →  Critically damped

Calculation:
Vc(t) = Vs + (A + Bt)·e^(-αt)
At t=0: Vc(0) = 0 → Vs + A = 0 → A = -Vs = -10
dVc/dt|₀ = i(0)/C = 0 → -αA + B = 0 → B = αA = 2×(-10) = -20
Vc(t) = 10 + (-10 - 20t)·e^(-2t)

Final Answer: Vc(t) = 10 - (10 + 20t)·e^(-2t). Circuit is critically damped. Vc approaches 10 V without oscillation, reaching ~8.6 V at t = 1s.

Exam Tip: GATE frequently tests the condition for each damping type. Memorize: ζ > 1 (overdamped), ζ = 1 (critically damped), ζ < 1 (underdamped). Also, ω₀ = 1/√(LC) is independent of R. The underdamped case shows overshoot; overdamped does not.

Quick Revision

• α = R/(2L) is the damping coefficient. ω₀ = 1/√(LC) is the undamped natural frequency. ζ = α/ω₀.
• Overdamped (ζ > 1): two real distinct negative roots. Response is sum of two decaying exponentials — no oscillation.
• Critically damped (ζ = 1): two equal real roots. Fastest response without oscillation. Vc(t) = Vs + (A + Bt)·e^(-αt).
• Underdamped (ζ < 1): complex conjugate roots. Response oscillates with damped frequency ωd = √(ω₀² - α²).
• Q factor = ω₀L/R = (1/R)√(L/C). Higher Q means sharper resonance and longer ringing.
• Initial conditions: Vc(0+) = Vc(0−) and iL(0+) = iL(0−). Both energy storage elements maintain continuity.
• Exam trap: In series RLC, resonance occurs at ω₀ regardless of R. R only affects damping, not the natural frequency.