Transient Analysis of RC, RL and RLC Circuits

When a switch opens or closes, currents and voltages move from one steady state to another. This transient follows simple exponential or oscillating forms.

Initial and final conditions

• A capacitor voltage cannot change instantly: v_C(0⁺) = v_C(0⁻). In DC steady state a capacitor is an open circuit.
• An inductor current cannot change instantly: i_L(0⁺) = i_L(0⁻). In DC steady state an inductor is a short circuit.
• At t = 0⁺, an uncharged capacitor acts like a short; an unenergised inductor acts like an open.

First-order circuits

Any variable x in an RC or RL circuit with a DC source follows:

x(t) = x(∞) + [x(0⁺) − x(∞)] e^(−t/τ)

• τ = RC or τ = L/R, with R the Thevenin resistance seen by C or L.
• After 1τ the change is 63.2% complete; after 5τ it is over 99%.
• Energy stored: ½CV² and ½LI².

Second-order RLC circuits

Series RLC: s² + (R/L)s + 1/(LC) = 0.

• Natural frequency ω₀ = 1/√(LC).
• Damping ratio ζ = (R/2)√(C/L) for series RLC.
• ζ > 1 overdamped (two real roots), ζ = 1 critically damped, ζ < 1 underdamped (oscillates at ω_d = ω₀√(1 − ζ²)).
• Critical resistance for series RLC: R = 2√(L/C).

Laplace method

Transform elements with their initial conditions:

• R → R; L → sL with a source L i(0⁻); C → 1/(sC) with a source v(0⁻)/s.
• Solve algebraically in s, then take the inverse transform. Initial value theorem: f(0⁺) = lim s F(s) as s → ∞; final value: f(∞) = lim s F(s) as s → 0 (if stable).

Worked example

A 10 V source charges a 100 µF capacitor through 10 kΩ, starting from 0 V.

• τ = 10⁴ × 10⁻⁴ = 1 s. v_C(t) = 10(1 − e^(−t)).
• At t = 2 s: v_C = 10(1 − e⁻²) = 8.65 V.

Practice