How it works
For an RC circuit with step input V_s applied at t = 0 with zero initial condition: v_C(t) = V_s(1 − e^(−t/RC))u(t). Time constant τ = RC in seconds. For an RL circuit: i_L(t) = (V_s/R)(1 − e^(−Rt/L))u(t), τ = L/R. The complete response = natural response + forced response. Natural response is determined by initial conditions and decays as e^(−t/τ); forced response is the steady-state value. General formula: x(t) = x(∞) + [x(0⁺) − x(∞)]e^(−t/τ), where x(0⁺) is the initial value and x(∞) is the final (steady-state) value.
Key points to remember
Key initial and final conditions: capacitor voltage cannot change instantaneously (v_C(0⁺) = v_C(0⁻)); inductor current cannot change instantaneously (i_L(0⁺) = i_L(0⁻)). At t = 0⁺, a capacitor with zero initial charge acts as a short circuit; a charged capacitor acts as a voltage source. At steady state (t → ∞), a capacitor is open circuit and an inductor is short circuit for DC. For a source-free RC discharge: v_C(t) = V₀e^(−t/RC). Energy stored in capacitor at steady state: W_C = ½CV². Only half the source energy is stored; the other half is dissipated in R regardless of R value.
Exam tip
The examiner always asks you to find the complete response of an RL or RC circuit by identifying x(0⁺) and x(∞) and substituting into the single-formula solution — draw the equivalent circuits at t=0⁺ and t=∞ separately before writing any equation.